LMFDB - Embedded newform 378.4.g.b.163.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(109,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.g (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.11184604443.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 43x^{4} - 210x^{3} + 1849x^{2} - 4515x + 11025 $ x^6 + 43x^4 - 210x^3 + 1849x^2 - 4515x + 11025
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.3
Root			$2.16527 - 3.75036i$ of defining polynomial
Character	$\chi$	$=$	378.163
Dual form			378.4.g.b.109.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(6.70319 - 11.6103i) q^{5} +(5.32531 + 17.7381i) q^{7} -8.00000 q^{8} +(-13.4064 - 23.2205i) q^{10} +(-21.8169 - 37.7881i) q^{11} -8.40639 q^{13} +(36.0486 + 8.51443i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-36.3086 - 62.8883i) q^{17} +(29.8496 - 51.7011i) q^{19} -53.6255 q^{20} -87.2678 q^{22} +(38.9919 - 67.5359i) q^{23} +(-27.3656 - 47.3986i) q^{25} +(-8.40639 + 14.5603i) q^{26} +(50.7961 - 53.9237i) q^{28} -59.0953 q^{29} +(-41.9845 - 72.7193i) q^{31} +(16.0000 + 27.7128i) q^{32} -145.234 q^{34} +(241.641 + 57.0738i) q^{35} +(1.78183 - 3.08622i) q^{37} +(-59.6993 - 103.402i) q^{38} +(-53.6255 + 92.8822i) q^{40} +50.0978 q^{41} -556.622 q^{43} +(-87.2678 + 151.152i) q^{44} +(-77.9838 - 135.072i) q^{46} +(1.82554 - 3.16193i) q^{47} +(-286.282 + 188.922i) q^{49} -109.462 q^{50} +(16.8128 + 29.1206i) q^{52} +(160.893 + 278.675i) q^{53} -584.973 q^{55} +(-42.6024 - 141.905i) q^{56} +(-59.0953 + 102.356i) q^{58} +(-427.740 - 740.867i) q^{59} +(-3.53933 + 6.13030i) q^{61} -167.938 q^{62} +64.0000 q^{64} +(-56.3496 + 97.6004i) q^{65} +(-243.423 - 421.621i) q^{67} +(-145.234 + 251.553i) q^{68} +(340.496 - 361.461i) q^{70} +564.619 q^{71} +(-91.9378 - 159.241i) q^{73} +(-3.56366 - 6.17243i) q^{74} -238.797 q^{76} +(554.108 - 588.225i) q^{77} +(656.542 - 1137.16i) q^{79} +(107.251 + 185.764i) q^{80} +(50.0978 - 86.7719i) q^{82} +941.530 q^{83} -973.534 q^{85} +(-556.622 + 964.097i) q^{86} +(174.536 + 302.305i) q^{88} +(-399.453 + 691.873i) q^{89} +(-44.7666 - 149.114i) q^{91} -311.935 q^{92} +(-3.65109 - 6.32387i) q^{94} +(-400.176 - 693.124i) q^{95} +524.765 q^{97} +(40.9401 + 684.777i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - 12 q^{4} + 5 q^{5} + 8 q^{7} - 48 q^{8} - 10 q^{10} - 29 q^{11} + 20 q^{13} + 20 q^{14} - 48 q^{16} - 38 q^{17} + 57 q^{19} - 40 q^{20} - 116 q^{22} - 14 q^{23} + 134 q^{25} + 20 q^{26}+ \cdots + 720 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - 12 * q^4 + 5 * q^5 + 8 * q^7 - 48 * q^8 - 10 * q^10 - 29 * q^11 + 20 * q^13 + 20 * q^14 - 48 * q^16 - 38 * q^17 + 57 * q^19 - 40 * q^20 - 116 * q^22 - 14 * q^23 + 134 * q^25 + 20 * q^26 + 8 * q^28 + 362 * q^29 - 88 * q^31 + 96 * q^32 - 152 * q^34 + 490 * q^35 + 384 * q^37 - 114 * q^38 - 40 * q^40 - 432 * q^41 - 726 * q^43 - 116 * q^44 + 28 * q^46 - 183 * q^47 + 372 * q^49 + 536 * q^50 - 40 * q^52 - 396 * q^53 - 1268 * q^55 - 64 * q^56 + 362 * q^58 - 427 * q^59 - 427 * q^61 - 352 * q^62 + 384 * q^64 - 216 * q^65 - 32 * q^67 - 152 * q^68 + 1162 * q^70 + 790 * q^71 + 373 * q^73 - 768 * q^74 - 456 * q^76 + 1211 * q^77 + 1364 * q^79 + 80 * q^80 - 432 * q^82 + 154 * q^83 - 2678 * q^85 - 726 * q^86 + 232 * q^88 - 1233 * q^89 + 131 * q^91 + 112 * q^92 + 366 * q^94 - 464 * q^95 + 1180 * q^97 + 720 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	6.70319	−	11.6103i	0.599552	−	1.03845i	−0.393335	−	0.919395i	$-0.628679\pi$
$5$	6.70319	−	11.6103i	0.599552	−	1.03845i	0.992887	−	0.119059i	$-0.0379878\pi$
$6$		0			0
$7$	5.32531	+	17.7381i	0.287539	+	0.957769i
$7$	5.32531	+	17.7381i	0.287539	+	0.957769i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−13.4064	−	23.2205i	−0.423947	−	0.734298i
$11$	−21.8169	−	37.7881i	−0.598005	−	1.03578i	−0.993115	−	0.117142i	$-0.962627\pi$
$11$	−21.8169	−	37.7881i	−0.598005	−	1.03578i	0.395110	−	0.918634i	$-0.370706\pi$
$12$		0			0
$13$	−8.40639			−0.179347			−0.0896735	−	0.995971i	$-0.528582\pi$
$13$	−8.40639			−0.179347			−0.0896735	+	0.995971i	$0.528582\pi$
$14$	36.0486	+	8.51443i	0.688172	+	0.162541i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	−36.3086	−	62.8883i	−0.518007	−	0.897215i	−0.999781	−	0.0209196i	$-0.993341\pi$
$17$	−36.3086	−	62.8883i	−0.518007	−	0.897215i	0.481774	−	0.876296i	$-0.339993\pi$
$18$		0			0
$19$	29.8496	−	51.7011i	0.360420	−	0.624265i	−0.627610	−	0.778528i	$-0.715967\pi$
$19$	29.8496	−	51.7011i	0.360420	−	0.624265i	0.988030	+	0.154263i	$0.0493002\pi$
$20$	−53.6255			−0.599552
$21$		0			0
$22$	−87.2678			−0.845707
$23$	38.9919	−	67.5359i	0.353494	−	0.612270i	−0.633365	−	0.773853i	$-0.718327\pi$
$23$	38.9919	−	67.5359i	0.353494	−	0.612270i	0.986859	+	0.161583i	$0.0516600\pi$
$24$		0			0
$25$	−27.3656	−	47.3986i	−0.218925	−	0.379189i
$26$	−8.40639	+	14.5603i	−0.0634088	+	0.109827i
$27$		0			0
$28$	50.7961	−	53.9237i	0.342841	−	0.363950i
$29$	−59.0953			−0.378404			−0.189202	−	0.981938i	$-0.560590\pi$
$29$	−59.0953			−0.378404			−0.189202	+	0.981938i	$0.560590\pi$
$30$		0			0
$31$	−41.9845	−	72.7193i	−0.243246	−	0.421315i	0.718391	−	0.695640i	$-0.244879\pi$
$31$	−41.9845	−	72.7193i	−0.243246	−	0.421315i	−0.961637	+	0.274325i	$0.911546\pi$
$32$	16.0000	+	27.7128i	0.0883883	+	0.153093i
$33$		0			0
$34$	−145.234			−0.732573
$35$	241.641	+	57.0738i	1.16699	+	0.275635i
$36$		0			0
$37$	1.78183	−	3.08622i	0.00791705	−	0.0137127i	−0.862040	−	0.506841i	$-0.830813\pi$
$37$	1.78183	−	3.08622i	0.00791705	−	0.0137127i	0.869957	+	0.493128i	$0.164147\pi$
$38$	−59.6993	−	103.402i	−0.254855	−	0.441422i
$39$		0			0
$40$	−53.6255	+	92.8822i	−0.211974	+	0.367149i
$41$	50.0978			0.190828			0.0954141	−	0.995438i	$-0.469582\pi$
$41$	50.0978			0.190828			0.0954141	+	0.995438i	$0.469582\pi$
$42$		0			0
$43$	−556.622			−1.97405			−0.987023	−	0.160578i	$-0.948664\pi$
$43$	−556.622			−1.97405			−0.987023	+	0.160578i	$0.948664\pi$
$44$	−87.2678	+	151.152i	−0.299003	+	0.517888i
$45$		0			0
$46$	−77.9838	−	135.072i	−0.249958	−	0.432940i
$47$	1.82554	−	3.16193i	0.00566559	−	0.00981310i	−0.863179	−	0.504899i	$-0.831530\pi$
$47$	1.82554	−	3.16193i	0.00566559	−	0.00981310i	0.868844	+	0.495085i	$0.164863\pi$
$48$		0			0
$49$	−286.282	+	188.922i	−0.834642	+	0.550793i
$50$	−109.462			−0.309606
$51$		0			0
$52$	16.8128	+	29.1206i	0.0448368	+	0.0776596i
$53$	160.893	+	278.675i	0.416988	+	0.722244i	0.995635	−	0.0933338i	$-0.0297524\pi$
$53$	160.893	+	278.675i	0.416988	+	0.722244i	−0.578647	+	0.815578i	$0.696419\pi$
$54$		0			0
$55$	−584.973			−1.43414
$56$	−42.6024	−	141.905i	−0.101661	−	0.338622i
$57$		0			0
$58$	−59.0953	+	102.356i	−0.133786	+	0.231724i
$59$	−427.740	−	740.867i	−0.943847	−	1.63479i	−0.758042	−	0.652205i	$-0.773844\pi$
$59$	−427.740	−	740.867i	−0.943847	−	1.63479i	−0.185805	−	0.982587i	$-0.559489\pi$
$60$		0			0
$61$	−3.53933	+	6.13030i	−0.00742894	+	0.0128673i	−0.869716	−	0.493553i	$-0.835698\pi$
$61$	−3.53933	+	6.13030i	−0.00742894	+	0.0128673i	0.862287	+	0.506420i	$0.169031\pi$
$62$	−167.938			−0.344002
$63$		0			0
$64$	64.0000			0.125000
$65$	−56.3496	+	97.6004i	−0.107528	+	0.186244i
$66$		0			0
$67$	−243.423	−	421.621i	−0.443863	−	0.768793i	0.554109	−	0.832444i	$-0.313059\pi$
$67$	−243.423	−	421.621i	−0.443863	−	0.768793i	−0.997972	+	0.0636506i	$0.979726\pi$
$68$	−145.234	+	251.553i	−0.259004	+	0.448608i
$69$		0			0
$70$	340.496	−	361.461i	0.581386	−	0.617183i
$71$	564.619			0.943774			0.471887	−	0.881659i	$-0.343573\pi$
$71$	564.619			0.943774			0.471887	+	0.881659i	$0.343573\pi$
$72$		0			0
$73$	−91.9378	−	159.241i	−0.147404	−	0.255312i	0.782863	−	0.622194i	$-0.213758\pi$
$73$	−91.9378	−	159.241i	−0.147404	−	0.255312i	−0.930267	+	0.366882i	$0.880425\pi$
$74$	−3.56366	−	6.17243i	−0.00559820	−	0.00969636i
$75$		0			0
$76$	−238.797			−0.360420
$77$	554.108	−	588.225i	0.820083	−	0.870577i
$78$		0			0
$79$	656.542	−	1137.16i	0.935022	−	1.61950i	0.160426	−	0.987048i	$-0.448713\pi$
$79$	656.542	−	1137.16i	0.935022	−	1.61950i	0.774595	−	0.632457i	$-0.217954\pi$
$80$	107.251	+	185.764i	0.149888	+	0.259614i
$81$		0			0
$82$	50.0978	−	86.7719i	0.0674680	−	0.116858i
$83$	941.530			1.24514			0.622568	−	0.782565i	$-0.286089\pi$
$83$	941.530			1.24514			0.622568	+	0.782565i	$0.286089\pi$
$84$		0			0
$85$	−973.534			−1.24229
$86$	−556.622	+	964.097i	−0.697931	+	1.20885i
$87$		0			0
$88$	174.536	+	302.305i	0.211427	+	0.366202i
$89$	−399.453	+	691.873i	−0.475752	+	0.824027i	−0.999614	−	0.0277760i	$-0.991157\pi$
$89$	−399.453	+	691.873i	−0.475752	+	0.824027i	0.523862	+	0.851803i	$0.324491\pi$
$90$		0			0
$91$	−44.7666	−	149.114i	−0.0515694	−	0.171773i
$92$	−311.935			−0.353494
$93$		0			0
$94$	−3.65109	−	6.32387i	−0.00400618	−	0.00693891i
$95$	−400.176	−	693.124i	−0.432181	−	0.748559i
$96$		0			0
$97$	524.765			0.549297			0.274648	−	0.961545i	$-0.411439\pi$
$97$	524.765			0.549297			0.274648	+	0.961545i	$0.411439\pi$
$98$	40.9401	+	684.777i	0.0421997	+	0.705846i
$99$		0			0
$100$	−109.462	+	189.594i	−0.109462	+	0.189594i
$101$	337.266	+	584.162i	0.332270	+	0.575508i	0.982957	−	0.183838i	$-0.0588522\pi$
$101$	337.266	+	584.162i	0.332270	+	0.575508i	−0.650687	+	0.759346i	$0.725519\pi$
$102$		0			0
$103$	−624.773	+	1082.14i	−0.597676	+	1.03521i	0.395487	+	0.918472i	$0.370576\pi$
$103$	−624.773	+	1082.14i	−0.597676	+	1.03521i	−0.993163	+	0.116734i	$0.962757\pi$
$104$	67.2511			0.0634088
$105$		0			0
$106$	643.572			0.589710
$107$	−886.082	+	1534.74i	−0.800568	+	1.38662i	0.118675	+	0.992933i	$0.462135\pi$
$107$	−886.082	+	1534.74i	−0.800568	+	1.38662i	−0.919243	+	0.393691i	$0.871198\pi$
$108$		0			0
$109$	682.211	+	1181.62i	0.599486	+	1.03834i	0.992897	+	0.118978i	$0.0379617\pi$
$109$	682.211	+	1181.62i	0.599486	+	1.03834i	−0.393411	+	0.919363i	$0.628705\pi$
$110$	−584.973	+	1013.20i	−0.507045	+	0.878228i
$111$		0			0
$112$	−288.389	−	68.1154i	−0.243305	−	0.0574670i
$113$	780.182			0.649499			0.324749	−	0.945800i	$-0.394720\pi$
$113$	780.182			0.649499			0.324749	+	0.945800i	$0.394720\pi$
$114$		0			0
$115$	−522.740	−	905.413i	−0.423876	−	0.734175i
$116$	118.191	+	204.712i	0.0946011	+	0.163854i
$117$		0			0
$118$	−1710.96			−1.33480
$119$	922.167	−	978.946i	0.710377	−	0.754116i
$120$		0			0
$121$	−286.459	+	496.161i	−0.215221	+	0.372773i
$122$	7.07867	+	12.2606i	0.00525305	+	0.00909855i
$123$		0			0
$124$	−167.938	+	290.877i	−0.121623	+	0.210658i
$125$	942.051			0.674077
$126$		0			0
$127$	851.395			0.594875			0.297437	−	0.954741i	$-0.403868\pi$
$127$	851.395			0.594875			0.297437	+	0.954741i	$0.403868\pi$
$128$	64.0000	−	110.851i	0.0441942	−	0.0765466i
$129$		0			0
$130$	112.699	+	195.201i	0.0760337	+	0.131694i
$131$	751.502	−	1301.64i	0.501214	−	0.868129i	−0.498785	−	0.866726i	$-0.666220\pi$
$131$	751.502	−	1301.64i	0.501214	−	0.868129i	0.999999	−	0.00140275i	$-0.000446511\pi$
$132$		0			0
$133$	1076.04	+	254.152i	0.701537	+	0.165698i
$134$	−973.691			−0.627717
$135$		0			0
$136$	290.469	+	503.107i	0.183143	+	0.317214i
$137$	−868.400	−	1504.11i	−0.541551	−	0.937993i	−0.998815	−	0.0486627i	$-0.984504\pi$
$137$	−868.400	−	1504.11i	−0.541551	−	0.937993i	0.457264	−	0.889331i	$-0.348829\pi$
$138$		0			0
$139$	3116.41			1.90166			0.950828	−	0.309720i	$-0.100235\pi$
$139$	3116.41			1.90166			0.950828	+	0.309720i	$0.100235\pi$
$140$	−285.572	−	951.217i	−0.172395	−	0.574232i
$141$		0			0
$142$	564.619	−	977.949i	0.333675	−	0.577941i
$143$	183.402	+	317.661i	0.107250	+	0.185763i
$144$		0			0
$145$	−396.127	+	686.113i	−0.226873	+	0.392956i
$146$	−367.751			−0.208461
$147$		0			0
$148$	−14.2546			−0.00791705
$149$	1475.41	−	2555.49i	0.811211	−	1.40506i	−0.100805	−	0.994906i	$-0.532142\pi$
$149$	1475.41	−	2555.49i	0.811211	−	1.40506i	0.912017	−	0.410153i	$-0.134525\pi$
$150$		0			0
$151$	644.154	+	1115.71i	0.347156	+	0.601292i	0.985743	−	0.168258i	$-0.0538142\pi$
$151$	644.154	+	1115.71i	0.347156	+	0.601292i	−0.638587	+	0.769549i	$0.720481\pi$
$152$	−238.797	+	413.609i	−0.127428	+	0.220711i
$153$		0			0
$154$	−464.728	−	1547.97i	−0.243174	−	0.809992i
$155$	−1125.72			−0.583355
$156$		0			0
$157$	1666.45	+	2886.38i	0.847116	+	1.46725i	0.883771	+	0.467920i	$0.154996\pi$
$157$	1666.45	+	2886.38i	0.847116	+	1.46725i	−0.0366552	+	0.999328i	$0.511670\pi$
$158$	−1313.08	−	2274.33i	−0.661160	−	1.14516i
$159$		0			0
$160$	429.004			0.211974
$161$	1405.60	+	331.993i	0.688057	+	0.162514i
$162$		0			0
$163$	1825.57	−	3161.97i	0.877235	−	1.51942i	0.0228725	−	0.999738i	$-0.492719\pi$
$163$	1825.57	−	3161.97i	0.877235	−	1.51942i	0.854363	−	0.519677i	$-0.173948\pi$
$164$	−100.196	−	173.544i	−0.0477070	−	0.0826310i
$165$		0			0
$166$	941.530	−	1630.78i	0.440222	−	0.762487i
$167$	−3389.98			−1.57081			−0.785403	−	0.618985i	$-0.787544\pi$
$167$	−3389.98			−1.57081			−0.785403	+	0.618985i	$0.787544\pi$
$168$		0			0
$169$	−2126.33			−0.967835
$170$	−973.534	+	1686.21i	−0.439216	+	0.760744i
$171$		0			0
$172$	1113.24	+	1928.19i	0.493512	+	0.854787i
$173$	314.750	−	545.164i	0.138324	−	0.239584i	−0.788538	−	0.614986i	$-0.789162\pi$
$173$	314.750	−	545.164i	0.138324	−	0.239584i	0.926862	+	0.375402i	$0.122495\pi$
$174$		0			0
$175$	695.032	−	737.826i	0.300226	−	0.318711i
$176$	698.142			0.299003
$177$		0			0
$178$	798.907	+	1383.75i	0.336408	+	0.582675i
$179$	−121.747	−	210.872i	−0.0508369	−	0.0880520i	0.839487	−	0.543380i	$-0.182856\pi$
$179$	−121.747	−	210.872i	−0.0508369	−	0.0880520i	−0.890324	+	0.455327i	$0.849522\pi$
$180$		0			0
$181$	4216.08			1.73137			0.865687	−	0.500585i	$-0.166882\pi$
$181$	4216.08			1.73137			0.865687	+	0.500585i	$0.166882\pi$
$182$	−303.039	−	71.5755i	−0.123422	−	0.0291513i
$183$		0			0
$184$	−311.935	+	540.287i	−0.124979	+	0.216470i
$185$	−23.8879	−	41.3750i	−0.00949336	−	0.0164430i
$186$		0			0
$187$	−1584.29	+	2744.06i	−0.619542	+	1.07308i
$188$	−14.6044			−0.00566559
$189$		0			0
$190$	−1600.70			−0.611196
$191$	1900.44	−	3291.66i	0.719953	−	1.24700i	−0.241065	−	0.970509i	$-0.577497\pi$
$191$	1900.44	−	3291.66i	0.719953	−	1.24700i	0.961018	−	0.276486i	$-0.0891700\pi$
$192$		0			0
$193$	−1923.01	−	3330.75i	−0.717209	−	1.24224i	−0.962101	−	0.272692i	$-0.912086\pi$
$193$	−1923.01	−	3330.75i	−0.717209	−	1.24224i	0.244893	−	0.969550i	$-0.421247\pi$
$194$	524.765	−	908.919i	0.194206	−	0.336374i
$195$		0			0
$196$	1227.01	+	613.867i	0.447161	+	0.223712i
$197$	4739.52			1.71410			0.857048	−	0.515237i	$-0.172296\pi$
$197$	4739.52			1.71410			0.857048	+	0.515237i	$0.172296\pi$
$198$		0			0
$199$	2073.53	+	3591.46i	0.738637	+	1.27936i	0.953109	+	0.302627i	$0.0978637\pi$
$199$	2073.53	+	3591.46i	0.738637	+	1.27936i	−0.214472	+	0.976730i	$0.568803\pi$
$200$	218.925	+	379.189i	0.0774016	+	0.134063i
$201$		0			0
$202$	1349.07			0.469900
$203$	−314.701	−	1048.24i	−0.108806	−	0.362424i
$204$		0			0
$205$	335.815	−	581.649i	0.114411	−	0.198166i
$206$	1249.55	+	2164.28i	0.422621	+	0.732001i
$207$		0			0
$208$	67.2511	−	116.482i	0.0224184	−	0.0388298i
$209$	−2604.91			−0.862131
$210$		0			0
$211$	226.994			0.0740612			0.0370306	−	0.999314i	$-0.488210\pi$
$211$	226.994			0.0740612			0.0370306	+	0.999314i	$0.488210\pi$
$212$	643.572	−	1114.70i	0.208494	−	0.361122i
$213$		0			0
$214$	1772.16	+	3069.48i	0.566087	+	0.980491i
$215$	−3731.14	+	6462.53i	−1.18354	+	2.04996i
$216$		0			0
$217$	1066.32	−	1131.98i	0.333580	−	0.354119i
$218$	2728.84			0.847801
$219$		0			0
$220$	1169.95	+	2026.41i	0.358535	+	0.621001i
$221$	305.224	+	528.664i	0.0929031	+	0.160913i
$222$		0			0
$223$	−4919.61			−1.47732			−0.738658	−	0.674080i	$-0.764540\pi$
$223$	−4919.61			−1.47732			−0.738658	+	0.674080i	$0.764540\pi$
$224$	−406.368	+	431.389i	−0.121213	+	0.128676i
$225$		0			0
$226$	780.182	−	1351.32i	0.229633	−	0.397735i
$227$	751.565	+	1301.75i	0.219749	+	0.380617i	0.954731	−	0.297470i	$-0.0961427\pi$
$227$	751.565	+	1301.75i	0.219749	+	0.380617i	−0.734982	+	0.678087i	$0.762809\pi$
$228$		0			0
$229$	1870.02	−	3238.98i	0.539628	−	0.934662i	−0.459296	−	0.888283i	$-0.651898\pi$
$229$	1870.02	−	3238.98i	0.539628	−	0.934662i	0.998924	−	0.0463793i	$-0.0147683\pi$
$230$	−2090.96			−0.599452
$231$		0			0
$232$	472.762			0.133786
$233$	617.169	−	1068.97i	0.173528	−	0.300560i	−0.766123	−	0.642694i	$-0.777817\pi$
$233$	617.169	−	1068.97i	0.173528	−	0.300560i	0.939651	+	0.342135i	$0.111150\pi$
$234$		0			0
$235$	−24.4739	−	42.3901i	−0.00679364	−	0.0117669i
$236$	−1710.96	+	2963.47i	−0.471924	+	0.817396i
$237$		0			0
$238$	−773.417	−	2576.19i	−0.210644	−	0.701636i
$239$	−407.673			−0.110335			−0.0551677	−	0.998477i	$-0.517569\pi$
$239$	−407.673			−0.110335			−0.0551677	+	0.998477i	$0.517569\pi$
$240$		0			0
$241$	−62.5169	−	108.282i	−0.0167098	−	0.0289422i	0.857550	−	0.514401i	$-0.171986\pi$
$241$	−62.5169	−	108.282i	−0.0167098	−	0.0289422i	−0.874259	+	0.485459i	$0.838652\pi$
$242$	572.917	+	992.322i	0.152184	+	0.263590i
$243$		0			0
$244$	28.3147			0.00742894
$245$	274.429	+	4590.19i	0.0715618	+	1.19697i
$246$		0			0
$247$	−250.927	+	434.619i	−0.0646402	+	0.111960i
$248$	335.876	+	581.754i	0.0860006	+	0.148957i
$249$		0			0
$250$	942.051	−	1631.68i	0.238322	−	0.412786i
$251$	−4144.15			−1.04214			−0.521068	−	0.853515i	$-0.674466\pi$
$251$	−4144.15			−1.04214			−0.521068	+	0.853515i	$0.674466\pi$
$252$		0			0
$253$	−3402.74			−0.845566
$254$	851.395	−	1474.66i	0.210320	−	0.364285i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	−274.649	+	475.706i	−0.0666619	+	0.115462i	−0.897430	−	0.441157i	$-0.854568\pi$
$257$	−274.649	+	475.706i	−0.0666619	+	0.115462i	0.830768	+	0.556619i	$0.187902\pi$
$258$		0			0
$259$	64.2325	+	15.1712i	0.0154101	+	0.00363975i
$260$	450.797			0.107528
$261$		0			0
$262$	−1503.00	−	2603.28i	−0.354412	−	0.613860i
$263$	−3079.75	−	5334.28i	−0.722074	−	1.25067i	−0.960167	−	0.279427i	$-0.909856\pi$
$263$	−3079.75	−	5334.28i	−0.722074	−	1.25067i	0.238093	−	0.971242i	$-0.423478\pi$
$264$		0			0
$265$	4313.99			1.00002
$266$	1516.24	−	1609.60i	0.349499	−	0.371019i
$267$		0			0
$268$	−973.691	+	1686.48i	−0.221932	+	0.384397i
$269$	3198.75	+	5540.40i	0.725024	+	1.25578i	0.958964	+	0.283528i	$0.0915047\pi$
$269$	3198.75	+	5540.40i	0.725024	+	1.25578i	−0.233940	+	0.972251i	$0.575162\pi$
$270$		0			0
$271$	−3215.20	+	5568.89i	−0.720700	+	1.24829i	0.240020	+	0.970768i	$0.422846\pi$
$271$	−3215.20	+	5568.89i	−0.720700	+	1.24829i	−0.960720	+	0.277521i	$0.910487\pi$
$272$	1161.88			0.259004
$273$		0			0
$274$	−3473.60			−0.765868
$275$	−1194.07	+	2068.19i	−0.261836	+	0.453514i
$276$		0			0
$277$	−2033.41	−	3521.97i	−0.441068	−	0.763953i	0.556701	−	0.830713i	$-0.312067\pi$
$277$	−2033.41	−	3521.97i	−0.441068	−	0.763953i	−0.997769	+	0.0667605i	$0.978734\pi$
$278$	3116.41	−	5397.77i	0.672337	−	1.16452i
$279$		0			0
$280$	−1933.13	−	456.591i	−0.412595	−	0.0974518i
$281$	2270.90			0.482101			0.241050	−	0.970513i	$-0.422508\pi$
$281$	2270.90			0.482101			0.241050	+	0.970513i	$0.422508\pi$
$282$		0			0
$283$	3554.58	+	6156.71i	0.746635	+	1.29321i	0.949427	+	0.313988i	$0.101665\pi$
$283$	3554.58	+	6156.71i	0.746635	+	1.29321i	−0.202792	+	0.979222i	$0.565001\pi$
$284$	−1129.24	−	1955.90i	−0.235944	−	0.408666i
$285$		0			0
$286$	733.607			0.151675
$287$	266.786	+	888.640i	0.0548706	+	0.182769i
$288$		0			0
$289$	−180.128	+	311.991i	−0.0366635	+	0.0635031i
$290$	792.255	+	1372.23i	0.160423	+	0.277862i
$291$		0			0
$292$	−367.751	+	636.964i	−0.0737021	+	0.127656i
$293$	1088.93			0.217119			0.108560	−	0.994090i	$-0.465376\pi$
$293$	1088.93			0.217119			0.108560	+	0.994090i	$0.465376\pi$
$294$		0			0
$295$	−11468.9			−2.26354
$296$	−14.2546	+	24.6897i	−0.00279910	+	0.00484818i
$297$		0			0
$298$	−2950.82	−	5110.98i	−0.573613	−	0.993527i
$299$	−327.781	+	567.733i	−0.0633982	+	0.109809i
$300$		0			0
$301$	−2964.18	−	9873.42i	−0.567616	−	1.89068i
$302$	2576.62			0.490952
$303$		0			0
$304$	477.594	+	827.217i	0.0901049	+	0.156066i
$305$	47.4497	+	82.1852i	0.00890806	+	0.0154292i
$306$		0			0
$307$	1736.56			0.322836			0.161418	−	0.986886i	$-0.448393\pi$
$307$	1736.56			0.322836			0.161418	+	0.986886i	$0.448393\pi$
$308$	−3145.89	−	743.035i	−0.581992	−	0.137462i
$309$		0			0
$310$	−1125.72	+	1949.81i	−0.206247	+	0.357231i
$311$	−1164.74	−	2017.38i	−0.212367	−	0.367830i	0.740088	−	0.672510i	$-0.234784\pi$
$311$	−1164.74	−	2017.38i	−0.212367	−	0.367830i	−0.952455	+	0.304680i	$0.901451\pi$
$312$		0			0
$313$	−1744.00	+	3020.69i	−0.314941	+	0.545494i	−0.979425	−	0.201808i	$-0.935318\pi$
$313$	−1744.00	+	3020.69i	−0.314941	+	0.545494i	0.664484	+	0.747303i	$0.268652\pi$
$314$	6665.80			1.19800
$315$		0			0
$316$	−5252.33			−0.935022
$317$	−1348.77	+	2336.14i	−0.238974	+	0.413914i	−0.960420	−	0.278556i	$-0.910144\pi$
$317$	−1348.77	+	2336.14i	−0.238974	+	0.413914i	0.721446	+	0.692470i	$0.243478\pi$
$318$		0			0
$319$	1289.28	+	2233.10i	0.226288	+	0.391942i
$320$	429.004	−	743.057i	0.0749440	−	0.129807i
$321$		0			0
$322$	1980.63	−	2102.58i	0.342784	−	0.363890i
$323$	−4335.19			−0.746800
$324$		0			0
$325$	230.046	+	398.451i	0.0392635	+	0.0680064i
$326$	−3651.13	−	6323.94i	−0.620299	−	1.07439i
$327$		0			0
$328$	−400.782			−0.0674680
$329$	65.8084	+	15.5435i	0.0110278	+	0.00260468i
$330$		0			0
$331$	4985.88	−	8635.80i	0.827942	−	1.43404i	−0.0717073	−	0.997426i	$-0.522845\pi$
$331$	4985.88	−	8635.80i	0.827942	−	1.43404i	0.899650	−	0.436612i	$-0.143822\pi$
$332$	−1883.06	−	3261.56i	−0.311284	−	0.539160i
$333$		0			0
$334$	−3389.98	+	5871.62i	−0.555364	+	0.961918i
$335$	−6526.84			−1.06448
$336$		0			0
$337$	−8607.45			−1.39133			−0.695664	−	0.718367i	$-0.744890\pi$
$337$	−8607.45			−1.39133			−0.695664	+	0.718367i	$0.744890\pi$
$338$	−2126.33	+	3682.92i	−0.342181	+	0.592675i
$339$		0			0
$340$	1947.07	+	3372.42i	0.310572	+	0.537927i
$341$	−1831.95	+	3173.03i	−0.290925	+	0.503897i
$342$		0			0
$343$	−4875.66	−	4072.04i	−0.767525	−	0.641019i
$344$	4452.97			0.697931
$345$		0			0
$346$	−629.501	−	1090.33i	−0.0978097	−	0.169411i
$347$	2770.44	+	4798.54i	0.428602	+	0.742361i	0.996749	−	0.0805663i	$-0.0256729\pi$
$347$	2770.44	+	4798.54i	0.428602	+	0.742361i	−0.568147	+	0.822927i	$0.692340\pi$
$348$		0			0
$349$	−1450.89			−0.222535			−0.111267	−	0.993791i	$-0.535491\pi$
$349$	−1450.89			−0.222535			−0.111267	+	0.993791i	$0.535491\pi$
$350$	−582.921	−	1941.66i	−0.0890240	−	0.296531i
$351$		0			0
$352$	698.142	−	1209.22i	0.105713	−	0.183101i
$353$	2464.39	+	4268.45i	0.371576	+	0.643588i	0.989808	−	0.142407i	$-0.0454843\pi$
$353$	2464.39	+	4268.45i	0.371576	+	0.643588i	−0.618232	+	0.785995i	$0.712151\pi$
$354$		0			0
$355$	3784.75	−	6555.38i	0.565841	−	0.980066i
$356$	3195.63			0.475752
$357$		0			0
$358$	−486.988			−0.0718942
$359$	−624.930	+	1082.41i	−0.0918733	+	0.159129i	−0.908299	−	0.418321i	$-0.862619\pi$
$359$	−624.930	+	1082.41i	−0.0918733	+	0.159129i	0.816426	+	0.577450i	$0.195952\pi$
$360$		0			0
$361$	1647.50	+	2853.55i	0.240195	+	0.416030i
$362$	4216.08	−	7302.47i	0.612133	−	1.06025i
$363$		0			0
$364$	−427.011	+	453.303i	−0.0614876	+	0.0652734i
$365$	−2465.11			−0.353506
$366$		0			0
$367$	1807.17	+	3130.11i	0.257040	+	0.445206i	0.965448	−	0.260597i	$-0.0839195\pi$
$367$	1807.17	+	3130.11i	0.257040	+	0.445206i	−0.708408	+	0.705803i	$0.750586\pi$
$368$	623.870	+	1080.57i	0.0883736	+	0.153068i
$369$		0			0
$370$	−95.5515			−0.0134256
$371$	−4086.37	+	4337.97i	−0.571843	+	0.607052i
$372$		0			0
$373$	−2785.71	+	4824.98i	−0.386698	+	0.669781i	−0.992003	−	0.126213i	$-0.959718\pi$
$373$	−2785.71	+	4824.98i	−0.386698	+	0.669781i	0.605305	+	0.795994i	$0.293051\pi$
$374$	3168.57	+	5488.13i	0.438083	+	0.758781i
$375$		0			0
$376$	−14.6044	+	25.2955i	−0.00200309	+	0.00346945i
$377$	496.778			0.0678657
$378$		0			0
$379$	2674.89			0.362533			0.181266	−	0.983434i	$-0.441980\pi$
$379$	2674.89			0.362533			0.181266	+	0.983434i	$0.441980\pi$
$380$	−1600.70	+	2772.50i	−0.216090	+	0.374279i
$381$		0			0
$382$	−3800.88	−	6583.32i	−0.509084	−	0.881759i
$383$	1675.22	−	2901.57i	0.223498	−	0.387111i	−0.732369	−	0.680907i	$-0.761586\pi$
$383$	1675.22	−	2901.57i	0.223498	−	0.387111i	0.955868	+	0.293797i	$0.0949189\pi$
$384$		0			0
$385$	−3115.16	−	10376.3i	−0.412372	−	1.37358i
$386$	−7692.04			−1.01429
$387$		0			0
$388$	−1049.53	−	1817.84i	−0.137324	−	0.237852i
$389$	3529.71	+	6113.63i	0.460060	+	0.796847i	0.998963	−	0.0455204i	$-0.0144946\pi$
$389$	3529.71	+	6113.63i	0.460060	+	0.796847i	−0.538904	+	0.842367i	$0.681161\pi$
$390$		0			0
$391$	−5662.96			−0.732451
$392$	2290.26	−	1511.38i	0.295091	−	0.194735i
$393$		0			0
$394$	4739.52	−	8209.09i	0.606024	−	1.04966i
$395$	−8801.85	−	15245.3i	−1.12119	−	1.94195i
$396$		0			0
$397$	−5510.26	+	9544.05i	−0.696604	+	1.20655i	0.273032	+	0.962005i	$0.411973\pi$
$397$	−5510.26	+	9544.05i	−0.696604	+	1.20655i	−0.969637	+	0.244549i	$0.921360\pi$
$398$	8294.13			1.04459
$399$		0			0
$400$	875.699			0.109462
$401$	−1458.98	+	2527.02i	−0.181690	+	0.314697i	−0.942456	−	0.334330i	$-0.891490\pi$
$401$	−1458.98	+	2527.02i	−0.181690	+	0.314697i	0.760766	+	0.649026i	$0.224823\pi$
$402$		0			0
$403$	352.938	+	611.307i	0.0436255	+	0.0755617i
$404$	1349.07	−	2336.65i	0.166135	−	0.287754i
$405$		0			0
$406$	−2130.31	−	503.163i	−0.260407	−	0.0615063i
$407$	−155.496			−0.0189377
$408$		0			0
$409$	1400.02	+	2424.91i	0.169259	+	0.293165i	0.938159	−	0.346204i	$-0.112529\pi$
$409$	1400.02	+	2424.91i	0.169259	+	0.293165i	−0.768901	+	0.639368i	$0.779196\pi$
$410$	−671.630	−	1163.30i	−0.0809011	−	0.140125i
$411$		0			0
$412$	4998.18			0.597676
$413$	10863.8	−	11532.7i	1.29436	−	1.37405i
$414$		0			0
$415$	6311.26	−	10931.4i	0.746524	−	1.29302i
$416$	−134.502	−	232.965i	−0.0158522	−	0.0274568i
$417$		0			0
$418$	−2604.91	+	4511.84i	−0.304810	+	0.527946i
$419$	−7229.43			−0.842913			−0.421456	−	0.906849i	$-0.638481\pi$
$419$	−7229.43			−0.842913			−0.421456	+	0.906849i	$0.638481\pi$
$420$		0			0
$421$	−10949.7			−1.26759			−0.633795	−	0.773501i	$-0.718504\pi$
$421$	−10949.7			−1.26759			−0.633795	+	0.773501i	$0.718504\pi$
$422$	226.994	−	393.165i	0.0261846	−	0.0453530i
$423$		0			0
$424$	−1287.14	−	2229.40i	−0.147428	−	0.255352i
$425$	−1987.21	+	3441.95i	−0.226809	+	0.392845i
$426$		0			0
$427$	−127.588	−	30.1354i	−0.0144600	−	0.00341535i
$428$	7088.66			0.800568
$429$		0			0
$430$	7462.28	+	12925.1i	0.836891	+	1.44954i
$431$	−5046.99	−	8741.64i	−0.564048	−	0.976960i	−0.997138	−	0.0756091i	$-0.975910\pi$
$431$	−5046.99	−	8741.64i	−0.564048	−	0.976960i	0.433089	−	0.901351i	$-0.357423\pi$
$432$		0			0
$433$	11040.9			1.22539			0.612694	−	0.790320i	$-0.290086\pi$
$433$	11040.9			1.22539			0.612694	+	0.790320i	$0.290086\pi$
$434$	−894.321	−	2978.91i	−0.0989143	−	0.329475i
$435$		0			0
$436$	2728.84	−	4726.50i	0.299743	−	0.519170i
$437$	−2327.79	−	4031.84i	−0.254813	−	0.441348i
$438$		0			0
$439$	−566.206	+	980.697i	−0.0615570	+	0.106620i	−0.895162	−	0.445742i	$-0.852940\pi$
$439$	−566.206	+	980.697i	−0.0615570	+	0.106620i	0.833605	+	0.552362i	$0.186273\pi$
$440$	4679.78			0.507045
$441$		0			0
$442$	1220.90			0.131385
$443$	4719.63	−	8174.65i	0.506177	−	0.876725i	−0.493797	−	0.869577i	$-0.664391\pi$
$443$	4719.63	−	8174.65i	0.506177	−	0.876725i	0.999974	−	0.00714780i	$-0.00227523\pi$
$444$		0			0
$445$	5355.22	+	9275.52i	0.570476	+	0.988094i
$446$	−4919.61	+	8521.01i	−0.522310	+	0.904668i
$447$		0			0
$448$	340.820	+	1135.24i	0.0359424	+	0.119721i
$449$	−8855.77			−0.930801			−0.465401	−	0.885100i	$-0.654090\pi$
$449$	−8855.77			−0.930801			−0.465401	+	0.885100i	$0.654090\pi$
$450$		0			0
$451$	−1092.98	−	1893.10i	−0.114116	−	0.197655i
$452$	−1560.36	−	2702.63i	−0.162375	−	0.281241i
$453$		0			0
$454$	3006.26			0.310772
$455$	−2031.33	−	479.785i	−0.209297	−	0.0494344i
$456$		0			0
$457$	−2650.62	+	4591.01i	−0.271315	+	0.469931i	−0.969199	−	0.246280i	$-0.920792\pi$
$457$	−2650.62	+	4591.01i	−0.271315	+	0.469931i	0.697884	+	0.716211i	$0.254125\pi$
$458$	−3740.05	−	6477.96i	−0.381574	−	0.660906i
$459$		0			0
$460$	−2090.96	+	3621.65i	−0.211938	+	0.367088i
$461$	8217.88			0.830249			0.415124	−	0.909765i	$-0.363738\pi$
$461$	8217.88			0.830249			0.415124	+	0.909765i	$0.363738\pi$
$462$		0			0
$463$	−14957.4			−1.50136			−0.750682	−	0.660664i	$-0.770275\pi$
$463$	−14957.4			−1.50136			−0.750682	+	0.660664i	$0.770275\pi$
$464$	472.762	−	818.849i	0.0473005	−	0.0819269i
$465$		0			0
$466$	−1234.34	−	2137.94i	−0.122703	−	0.212528i
$467$	−121.131	+	209.805i	−0.0120027	+	0.0207893i	−0.871964	−	0.489569i	$-0.837154\pi$
$467$	−121.131	+	209.805i	−0.0120027	+	0.0207893i	0.859962	+	0.510359i	$0.170487\pi$
$468$		0			0
$469$	6182.46	−	6563.12i	0.608698	−	0.646177i
$470$	−97.8958			−0.00960765
$471$		0			0
$472$	3421.92	+	5926.94i	0.333700	+	0.577986i
$473$	12143.8	+	21033.7i	1.18049	+	2.04467i
$474$		0			0
$475$	−3267.41			−0.315619
$476$	−5235.50	−	1236.59i	−0.504136	−	0.119073i
$477$		0			0
$478$	−407.673	+	706.110i	−0.0390095	+	0.0675664i
$479$	−4879.07	−	8450.79i	−0.465408	−	0.806110i	0.533812	−	0.845603i	$-0.320759\pi$
$479$	−4879.07	−	8450.79i	−0.465408	−	0.806110i	−0.999220	+	0.0394932i	$0.987426\pi$
$480$		0			0
$481$	−14.9787	+	25.9439i	−0.00141990	+	0.00245934i
$482$	−250.067			−0.0236312
$483$		0			0
$484$	2291.67			0.215221
$485$	3517.60	−	6092.66i	0.329332	−	0.570419i
$486$		0			0
$487$	6608.98	+	11447.1i	0.614952	+	1.06513i	0.990393	+	0.138282i	$0.0441580\pi$
$487$	6608.98	+	11447.1i	0.614952	+	1.06513i	−0.375441	+	0.926846i	$0.622509\pi$
$488$	28.3147	−	49.0424i	0.00262653	−	0.00454928i
$489$		0			0
$490$	8224.88	+	4114.87i	0.758290	+	0.379369i
$491$	−13789.5			−1.26744			−0.633720	−	0.773562i	$-0.718473\pi$
$491$	−13789.5			−1.26744			−0.633720	+	0.773562i	$0.718473\pi$
$492$		0			0
$493$	2145.67	+	3716.41i	0.196016	+	0.339510i
$494$	501.855	+	869.238i	0.0457075	+	0.0791678i
$495$		0			0
$496$	1343.50			0.121623
$497$	3006.77	+	10015.3i	0.271372	+	0.903917i
$498$		0			0
$499$	995.188	−	1723.72i	0.0892800	−	0.154638i	−0.817927	−	0.575322i	$-0.804877\pi$
$499$	995.188	−	1723.72i	0.0892800	−	0.154638i	0.907207	+	0.420684i	$0.138210\pi$
$500$	−1884.10	−	3263.36i	−0.168519	−	0.291884i
$501$		0			0
$502$	−4144.15	+	7177.87i	−0.368451	+	0.638176i
$503$	221.800			0.0196612			0.00983059	−	0.999952i	$-0.496871\pi$
$503$	221.800			0.0196612			0.00983059	+	0.999952i	$0.496871\pi$
$504$		0			0
$505$	9043.04			0.796852
$506$	−3402.74	+	5893.71i	−0.298953	+	0.517801i
$507$		0			0
$508$	−1702.79	−	2949.32i	−0.148719	−	0.257588i
$509$	−2603.26	+	4508.99i	−0.226695	+	0.392647i	−0.956827	−	0.290659i	$-0.906125\pi$
$509$	−2603.26	+	4508.99i	−0.226695	+	0.392647i	0.730132	+	0.683306i	$0.239459\pi$
$510$		0			0
$511$	2335.04	−	2478.81i	0.202145	−	0.214591i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	549.298	+	951.411i	0.0471371	+	0.0816439i
$515$	8375.94	+	14507.6i	0.716676	+	1.24132i
$516$		0			0
$517$	−159.311			−0.0135522
$518$	90.5098	−	96.0827i	0.00767717	−	0.00814986i
$519$		0			0
$520$	450.797	−	780.803i	0.0380168	−	0.0658471i
$521$	−6123.38	−	10606.0i	−0.514914	−	0.891858i	−0.999850	−	0.0173079i	$-0.994490\pi$
$521$	−6123.38	−	10606.0i	−0.514914	−	0.891858i	0.484936	−	0.874550i	$-0.338843\pi$
$522$		0			0
$523$	6370.75	−	11034.5i	0.532646	−	0.922569i	−0.466628	−	0.884454i	$-0.654531\pi$
$523$	6370.75	−	11034.5i	0.532646	−	0.922569i	0.999273	−	0.0381155i	$-0.0121355\pi$
$524$	−6012.02			−0.501214
$525$		0			0
$526$	−12319.0			−1.02117
$527$	−3048.80	+	5280.67i	−0.252007	+	0.436489i
$528$		0			0
$529$	3042.77	+	5270.23i	0.250084	+	0.433157i
$530$	4313.99	−	7472.05i	0.353562	−	0.612387i
$531$		0			0
$532$	−1271.67	−	4235.81i	−0.103635	−	0.345199i
$533$	−421.141			−0.0342245
$534$		0			0
$535$	11879.2	+	20575.3i	0.959964	+	1.66271i
$536$	1947.38	+	3372.97i	0.156929	+	0.271810i
$537$		0			0
$538$	12795.0			1.02534
$539$	13384.8	+	6696.35i	1.06962	+	0.535125i
$540$		0			0
$541$	1757.33	−	3043.78i	0.139655	−	0.241890i	−0.787711	−	0.616045i	$-0.788734\pi$
$541$	1757.33	−	3043.78i	0.139655	−	0.241890i	0.927366	+	0.374155i	$0.122067\pi$
$542$	6430.40	+	11137.8i	0.509612	+	0.882673i
$543$		0			0
$544$	1161.88	−	2012.43i	0.0915717	−	0.158607i
$545$	18292.0			1.43769
$546$		0			0
$547$	8362.90			0.653696			0.326848	−	0.945077i	$-0.394013\pi$
$547$	8362.90			0.653696			0.326848	+	0.945077i	$0.394013\pi$
$548$	−3473.60	+	6016.45i	−0.270775	+	0.468997i
$549$		0			0
$550$	2388.13	+	4136.37i	0.185146	+	0.320683i
$551$	−1763.97	+	3055.29i	−0.136384	+	0.236225i
$552$		0			0
$553$	23667.4	+	5590.08i	1.81997	+	0.429863i
$554$	−8133.65			−0.623765
$555$		0			0
$556$	−6232.81	−	10795.5i	−0.475414	−	0.823441i
$557$	6241.84	+	10811.2i	0.474821	+	0.822414i	0.999584	−	0.0288343i	$-0.00917952\pi$
$557$	6241.84	+	10811.2i	0.474821	+	0.822414i	−0.524763	+	0.851248i	$0.675846\pi$
$558$		0			0
$559$	4679.18			0.354039
$560$	−2723.97	+	2891.69i	−0.205551	+	0.218207i
$561$		0			0
$562$	2270.90	−	3933.31i	0.170448	−	0.295225i
$563$	5609.22	+	9715.46i	0.419894	+	0.727278i	0.995928	−	0.0901473i	$-0.0287338\pi$
$563$	5609.22	+	9715.46i	0.419894	+	0.727278i	−0.576034	+	0.817426i	$0.695400\pi$
$564$		0			0
$565$	5229.71	−	9058.13i	0.389408	−	0.674475i
$566$	14218.3			1.05590
$567$		0			0
$568$	−4516.95			−0.333675
$569$	10878.7	−	18842.5i	0.801511	−	1.38826i	−0.117110	−	0.993119i	$-0.537363\pi$
$569$	10878.7	−	18842.5i	0.801511	−	1.38826i	0.918621	−	0.395139i	$-0.129304\pi$
$570$		0			0
$571$	−1353.09	−	2343.62i	−0.0991683	−	0.171765i	0.812172	−	0.583418i	$-0.198285\pi$
$571$	−1353.09	−	2343.62i	−0.0991683	−	0.171765i	−0.911341	+	0.411653i	$0.864952\pi$
$572$	733.607	−	1270.64i	0.0536252	−	0.0928817i
$573$		0			0
$574$	1805.96	+	426.554i	0.131323	+	0.0310174i
$575$	−4268.14			−0.309555
$576$		0			0
$577$	3842.52	+	6655.44i	0.277238	+	0.480190i	0.970697	−	0.240306i	$-0.0772477\pi$
$577$	3842.52	+	6655.44i	0.277238	+	0.480190i	−0.693460	+	0.720496i	$0.743914\pi$
$578$	360.256	+	623.982i	0.0259250	+	0.0449035i
$579$		0			0
$580$	3169.02			0.226873
$581$	5013.94	+	16701.0i	0.358026	+	1.19255i
$582$		0			0
$583$	7020.39	−	12159.7i	0.498722	−	0.863812i
$584$	735.502	+	1273.93i	0.0521153	+	0.0902663i
$585$		0			0
$586$	1088.93	−	1886.08i	0.0767632	−	0.132958i
$587$	−24730.3			−1.73889			−0.869445	−	0.494030i	$-0.835523\pi$
$587$	−24730.3			−1.73889			−0.869445	+	0.494030i	$0.835523\pi$
$588$		0			0
$589$	−5012.89			−0.350683
$590$	−11468.9	+	19864.7i	−0.800283	+	1.38613i
$591$		0			0
$592$	28.5092	+	49.3795i	0.00197926	+	0.00342818i
$593$	8612.12	−	14916.6i	0.596387	−	1.03297i	−0.396963	−	0.917835i	$-0.629936\pi$
$593$	8612.12	−	14916.6i	0.596387	−	1.03297i	0.993350	−	0.115138i	$-0.0367309\pi$
$594$		0			0
$595$	−5184.37	−	17268.7i	−0.357207	−	1.18983i
$596$	−11803.3			−0.811211
$597$		0			0
$598$	655.562	+	1135.47i	0.0448293	+	0.0776466i
$599$	4841.41	+	8385.56i	0.330241	+	0.571995i	0.982559	−	0.185951i	$-0.0595366\pi$
$599$	4841.41	+	8385.56i	0.330241	+	0.571995i	−0.652318	+	0.757946i	$0.726203\pi$
$600$		0			0
$601$	16084.7			1.09169			0.545847	−	0.837885i	$-0.316208\pi$
$601$	16084.7			1.09169			0.545847	+	0.837885i	$0.316208\pi$
$602$	−20065.5	−	4739.31i	−1.35848	−	0.320864i
$603$		0			0
$604$	2576.62	−	4462.83i	0.173578	−	0.300646i
$605$	3840.37	+	6651.72i	0.258072	+	0.446993i
$606$		0			0
$607$	−3883.60	+	6726.59i	−0.259688	+	0.449792i	−0.966158	−	0.257950i	$-0.916953\pi$
$607$	−3883.60	+	6726.59i	−0.259688	+	0.449792i	0.706470	+	0.707743i	$0.250286\pi$
$608$	1910.38			0.127428
$609$		0			0
$610$	189.799			0.0125979
$611$	−15.3462	+	26.5804i	−0.00101611	+	0.00175995i
$612$		0			0
$613$	14001.4	+	24251.2i	0.922531	+	1.59787i	0.795484	+	0.605974i	$0.207217\pi$
$613$	14001.4	+	24251.2i	0.922531	+	1.59787i	0.127047	+	0.991897i	$0.459450\pi$
$614$	1736.56	−	3007.81i	0.114140	−	0.197696i
$615$		0			0
$616$	−4432.86	+	4705.80i	−0.289943	+	0.307795i
$617$	10440.8			0.681247			0.340624	−	0.940200i	$-0.389362\pi$
$617$	10440.8			0.681247			0.340624	+	0.940200i	$0.389362\pi$
$618$		0			0
$619$	14166.3	+	24536.7i	0.919857	+	1.59324i	0.799630	+	0.600493i	$0.205029\pi$
$619$	14166.3	+	24536.7i	0.919857	+	1.59324i	0.120227	+	0.992746i	$0.461638\pi$
$620$	2251.44	+	3899.61i	0.145839	+	0.252600i
$621$		0			0
$622$	−4658.95			−0.300332
$623$	−14399.7	−	3401.12i	−0.926025	−	0.218720i
$624$		0			0
$625$	9735.45	−	16862.3i	0.623069	−	1.07919i
$626$	3488.00	+	6041.39i	0.222697	+	0.385723i
$627$		0			0
$628$	6665.80	−	11545.5i	0.423558	−	0.733624i
$629$	−258.783			−0.0164044
$630$		0			0
$631$	−6567.27			−0.414325			−0.207162	−	0.978307i	$-0.566423\pi$
$631$	−6567.27			−0.414325			−0.207162	+	0.978307i	$0.566423\pi$
$632$	−5252.33	+	9097.31i	−0.330580	+	0.572581i
$633$		0			0
$634$	2697.55	+	4672.29i	0.168980	+	0.292682i
$635$	5707.07	−	9884.93i	0.356658	−	0.617750i
$636$		0			0
$637$	2406.60	−	1588.15i	0.149691	−	0.0987831i
$638$	5157.12			0.320019
$639$		0			0
$640$	−858.009	−	1486.11i	−0.0529934	−	0.0917872i
$641$	−3966.30	−	6869.84i	−0.244399	−	0.423311i	0.717564	−	0.696493i	$-0.245257\pi$
$641$	−3966.30	−	6869.84i	−0.244399	−	0.423311i	−0.961962	+	0.273182i	$0.911924\pi$
$642$		0			0
$643$	−21042.2			−1.29055			−0.645274	−	0.763951i	$-0.723257\pi$
$643$	−21042.2			−1.29055			−0.645274	+	0.763951i	$0.723257\pi$
$644$	−1661.15	−	5533.14i	−0.101644	−	0.338566i
$645$		0			0
$646$	−4335.19	+	7508.77i	−0.264034	+	0.457320i
$647$	−14683.0	−	25431.7i	−0.892193	−	1.54532i	−0.837241	−	0.546834i	$-0.815833\pi$
$647$	−14683.0	−	25431.7i	−0.892193	−	1.54532i	−0.0549522	−	0.998489i	$-0.517501\pi$
$648$		0			0
$649$	−18664.0	+	32326.9i	−1.12885	+	1.95523i
$650$	920.183			0.0555270
$651$		0			0
$652$	−14604.5			−0.877235
$653$	12269.9	−	21252.2i	0.735314	−	1.27360i	−0.219271	−	0.975664i	$-0.570368\pi$
$653$	12269.9	−	21252.2i	0.735314	−	1.27360i	0.954585	−	0.297937i	$-0.0962987\pi$
$654$		0			0
$655$	−10074.9	−	17450.3i	−0.601008	−	1.04098i
$656$	−400.782	+	694.175i	−0.0238535	+	0.0413155i
$657$		0			0
$658$	92.7304	−	98.4400i	0.00549393	−	0.00583220i
$659$	−4246.88			−0.251039			−0.125520	−	0.992091i	$-0.540060\pi$
$659$	−4246.88			−0.251039			−0.125520	+	0.992091i	$0.540060\pi$
$660$		0			0
$661$	−10274.9	−	17796.7i	−0.604611	−	1.04722i	−0.992113	−	0.125348i	$-0.959995\pi$
$661$	−10274.9	−	17796.7i	−0.604611	−	1.04722i	0.387502	−	0.921869i	$-0.373338\pi$
$662$	−9971.77	−	17271.6i	−0.585444	−	1.01402i
$663$		0			0
$664$	−7532.24			−0.440222
$665$	10163.7	−	10789.5i	0.592677	−	0.629169i
$666$		0			0
$667$	−2304.24	+	3991.06i	−0.133764	+	0.231686i
$668$	6779.96	+	11743.2i	0.392701	+	0.680179i
$669$		0			0
$670$	−6526.84	+	11304.8i	−0.376349	+	0.651856i
$671$	308.870			0.0177702
$672$		0			0
$673$	−6712.91			−0.384493			−0.192246	−	0.981347i	$-0.561577\pi$
$673$	−6712.91			−0.384493			−0.192246	+	0.981347i	$0.561577\pi$
$674$	−8607.45	+	14908.5i	−0.491909	+	0.852011i
$675$		0			0
$676$	4252.67	+	7365.83i	0.241959	+	0.419085i
$677$	12236.0	−	21193.3i	0.694632	−	1.20314i	−0.275673	−	0.961252i	$-0.588901\pi$
$677$	12236.0	−	21193.3i	0.694632	−	1.20314i	0.970305	−	0.241886i	$-0.0777661\pi$
$678$		0			0
$679$	2794.53	+	9308.34i	0.157944	+	0.526099i
$680$	7788.27			0.439216
$681$		0			0
$682$	3663.90	+	6346.05i	0.205715	+	0.356309i
$683$	−10284.1	−	17812.5i	−0.576148	−	0.997918i	−0.995916	−	0.0902863i	$-0.971222\pi$
$683$	−10284.1	−	17812.5i	−0.576148	−	0.997918i	0.419768	−	0.907632i	$-0.362112\pi$
$684$		0			0
$685$	−23284.2			−1.29875
$686$	−11928.6	+	4372.85i	−0.663904	+	0.243376i
$687$		0			0
$688$	4452.97	−	7712.77i	0.246756	−	0.427394i
$689$	−1352.53	−	2342.65i	−0.0747856	−	0.129532i
$690$		0			0
$691$	3635.32	−	6296.55i	0.200136	−	0.346646i	−0.748436	−	0.663207i	$-0.769195\pi$
$691$	3635.32	−	6296.55i	0.200136	−	0.346646i	0.948572	+	0.316561i	$0.102528\pi$
$692$	−2518.00			−0.138324
$693$		0			0
$694$	11081.8			0.606135
$695$	20889.9	−	36182.3i	1.14014	−	1.97478i
$696$		0			0
$697$	−1818.98	−	3150.56i	−0.0988504	−	0.171214i
$698$	−1450.89	+	2513.02i	−0.0786779	+	0.136274i
$699$		0			0
$700$	−3945.97	−	932.009i	−0.213062	−	0.0503238i
$701$	3876.20			0.208848			0.104424	−	0.994533i	$-0.466700\pi$
$701$	3876.20			0.208848			0.104424	+	0.994533i	$0.466700\pi$
$702$		0			0
$703$	−106.374	−	184.245i	−0.00570692	−	0.00988467i
$704$	−1396.28	−	2418.44i	−0.0747507	−	0.129472i
$705$		0			0
$706$	9857.56			0.525487
$707$	−8565.90	+	9093.32i	−0.455663	+	0.483719i
$708$		0			0
$709$	−10514.6	+	18211.8i	−0.556960	+	0.964682i	0.440789	+	0.897611i	$0.354699\pi$
$709$	−10514.6	+	18211.8i	−0.556960	+	0.964682i	−0.997748	+	0.0670713i	$0.978634\pi$
$710$	−7569.50	−	13110.8i	−0.400110	−	0.693011i
$711$		0			0
$712$	3195.63	−	5534.99i	0.168204	−	0.291338i
$713$	−6548.22			−0.343945
$714$		0			0
$715$	4917.51			0.257209
$716$	−486.988	+	843.488i	−0.0254184	+	0.0440260i
$717$		0			0
$718$	1249.86	+	2164.82i	0.0649642	+	0.112521i
$719$	5169.58	−	8953.98i	0.268140	−	0.464433i	−0.700241	−	0.713906i	$-0.746924\pi$
$719$	5169.58	−	8953.98i	0.268140	−	0.464433i	0.968382	+	0.249474i	$0.0802576\pi$
$720$		0			0
$721$	−22522.2	−	5319.58i	−1.16334	−	0.274773i
$722$	6590.00			0.339687
$723$		0			0
$724$	−8432.16	−	14604.9i	−0.432844	−	0.749707i
$725$	1617.18	+	2801.03i	0.0828421	+	0.143487i
$726$		0			0
$727$	9537.43			0.486552			0.243276	−	0.969957i	$-0.421778\pi$
$727$	9537.43			0.486552			0.243276	+	0.969957i	$0.421778\pi$
$728$	358.133	+	1192.91i	0.0182325	+	0.0607309i
$729$		0			0
$730$	−2465.11	+	4269.69i	−0.124983	+	0.216477i
$731$	20210.1	+	35005.0i	1.02257	+	1.77114i
$732$		0			0
$733$	6617.25	−	11461.4i	0.333443	−	0.577540i	−0.649742	−	0.760155i	$-0.725123\pi$
$733$	6617.25	−	11461.4i	0.333443	−	0.577540i	0.983184	+	0.182615i	$0.0584562\pi$
$734$	7228.69			0.363509
$735$		0			0
$736$	2495.48			0.124979
$737$	−10621.5	+	18397.0i	−0.530865	+	0.919485i
$738$		0			0
$739$	−18675.7	−	32347.2i	−0.929628	−	1.61016i	−0.783943	−	0.620833i	$-0.786795\pi$
$739$	−18675.7	−	32347.2i	−0.929628	−	1.61016i	−0.145685	−	0.989331i	$-0.546539\pi$
$740$	−95.5515	+	165.500i	−0.00474668	+	0.00822149i
$741$		0			0
$742$	3427.22	+	11415.8i	0.169565	+	0.564806i
$743$	−23.0684			−0.00113903			−0.000569515	−	1.00000i	$-0.500181\pi$
$743$	−23.0684			−0.00113903			−0.000569515		1.00000i	$0.500181\pi$
$744$		0			0
$745$	−19779.9	−	34259.9i	−0.972727	−	1.68481i
$746$	5571.41	+	9649.97i	0.273437	+	0.473607i
$747$		0			0
$748$	12674.3			0.619542
$749$	−31942.0	−	7544.48i	−1.55826	−	0.368050i
$750$		0			0
$751$	9842.85	−	17048.3i	0.478257	−	0.828365i	−0.521433	−	0.853292i	$-0.674602\pi$
$751$	9842.85	−	17048.3i	0.478257	−	0.828365i	0.999689	+	0.0249278i	$0.00793559\pi$
$752$	29.2087	+	50.5910i	0.00141640	+	0.00245327i
$753$		0			0
$754$	496.778	−	860.445i	0.0239941	−	0.0415591i
$755$	17271.6			0.832552
$756$		0			0
$757$	5562.94			0.267092			0.133546	−	0.991043i	$-0.457364\pi$
$757$	5562.94			0.267092			0.133546	+	0.991043i	$0.457364\pi$
$758$	2674.89	−	4633.05i	0.128175	−	0.222005i
$759$		0			0
$760$	3201.40	+	5545.00i	0.152799	+	0.264655i
$761$	−4584.56	+	7940.68i	−0.218384	+	0.378252i	−0.954314	−	0.298806i	$-0.903412\pi$
$761$	−4584.56	+	7940.68i	−0.218384	+	0.378252i	0.735930	+	0.677057i	$0.236745\pi$
$762$		0			0
$763$	−17326.8	+	18393.7i	−0.822114	+	0.872733i
$764$	−15203.5			−0.719953
$765$		0			0
$766$	−3350.45	−	5803.14i	−0.158037	−	0.273729i
$767$	3595.75	+	6228.02i	0.169276	+	0.293195i
$768$		0			0
$769$	−12864.9			−0.603277			−0.301638	−	0.953422i	$-0.597534\pi$
$769$	−12864.9			−0.603277			−0.301638	+	0.953422i	$0.597534\pi$
$770$	−21087.5	−	4980.71i	−0.986935	−	0.233107i
$771$		0			0
$772$	−7692.04	+	13323.0i	−0.358604	+	0.621121i
$773$	−460.408	−	797.450i	−0.0214227	−	0.0371052i	0.855115	−	0.518438i	$-0.173486\pi$
$773$	−460.408	−	797.450i	−0.0214227	−	0.0371052i	−0.876538	+	0.481333i	$0.840153\pi$
$774$		0			0
$775$	−2297.86	+	3980.01i	−0.106505	+	0.184473i
$776$	−4198.12			−0.194206
$777$		0			0
$778$	14118.8			0.650623
$779$	1495.40	−	2590.11i	0.0687782	−	0.119127i
$780$		0			0
$781$	−12318.3	−	21335.9i	−0.564382	−	0.977538i
$782$	−5662.96	+	9808.54i	−0.258960	+	0.448533i
$783$		0			0
$784$	−327.521	−	5478.22i	−0.0149199	−	0.249554i
$785$	44682.1			2.03156
$786$		0			0
$787$	2159.63	+	3740.60i	0.0978179	+	0.169426i	0.910781	−	0.412889i	$-0.135480\pi$
$787$	2159.63	+	3740.60i	0.0978179	+	0.169426i	−0.812963	+	0.582315i	$0.802147\pi$
$788$	−9479.04	−	16418.2i	−0.428524	−	0.742225i
$789$		0			0
$790$	−35207.4			−1.58560
$791$	4154.71	+	13839.0i	0.186757	+	0.622070i
$792$		0			0
$793$	29.7530	−	51.5337i	0.00133236	−	0.00230771i
$794$	11020.5	+	19088.1i	0.492574	+	0.853163i
$795$		0			0
$796$	8294.13	−	14365.9i	0.369319	−	0.639679i
$797$	−20369.7			−0.905308			−0.452654	−	0.891686i	$-0.649523\pi$
$797$	−20369.7			−0.905308			−0.452654	+	0.891686i	$0.649523\pi$
$798$		0			0
$799$	−265.132			−0.0117393
$800$	875.699	−	1516.76i	0.0387008	−	0.0670317i
$801$		0			0
$802$	2917.95	+	5054.04i	0.128474	+	0.222524i
$803$	−4011.60	+	6948.30i	−0.176297	+	0.305355i
$804$		0			0
$805$	13276.6	−	14094.0i	0.581289	−	0.617080i
$806$	1411.75			0.0616958
$807$		0			0
$808$	−2698.13	−	4673.30i	−0.117475	−	0.203473i
$809$	18244.3	+	31600.1i	0.792876	+	1.37330i	0.924179	+	0.381959i	$0.124750\pi$
$809$	18244.3	+	31600.1i	0.792876	+	1.37330i	−0.131304	+	0.991342i	$0.541916\pi$
$810$		0			0
$811$	−5722.53			−0.247775			−0.123887	−	0.992296i	$-0.539536\pi$
$811$	−5722.53			−0.247775			−0.123887	+	0.992296i	$0.539536\pi$
$812$	−3001.81	+	3186.64i	−0.129733	+	0.137720i
$813$		0			0
$814$	−155.496	+	269.327i	−0.00669550	+	0.0115969i
$815$	−24474.2	−	42390.6i	−1.05190	−	1.82194i
$816$		0			0
$817$	−16614.9	+	28777.9i	−0.711485	+	1.23233i
$818$	5600.10			0.239368
$819$		0			0
$820$	−2686.52			−0.114411
$821$	15351.2	−	26589.1i	0.652572	−	1.13029i	−0.329925	−	0.944007i	$-0.607023\pi$
$821$	15351.2	−	26589.1i	0.652572	−	1.13029i	0.982497	−	0.186280i	$-0.0596432\pi$
$822$		0			0
$823$	2340.54	+	4053.94i	0.0991326	+	0.171703i	0.911326	−	0.411686i	$-0.135060\pi$
$823$	2340.54	+	4053.94i	0.0991326	+	0.171703i	−0.812193	+	0.583388i	$0.801727\pi$
$824$	4998.18	−	8657.10i	0.211311	−	0.366001i
$825$		0			0
$826$	−9111.38	−	30349.2i	−0.383808	−	1.27843i
$827$	10970.9			0.461300			0.230650	−	0.973037i	$-0.425915\pi$
$827$	10970.9			0.461300			0.230650	+	0.973037i	$0.425915\pi$
$828$		0			0
$829$	−9850.98	−	17062.4i	−0.412713	−	0.714839i	0.582473	−	0.812850i	$-0.302085\pi$
$829$	−9850.98	−	17062.4i	−0.412713	−	0.714839i	−0.995185	+	0.0980110i	$0.968752\pi$
$830$	−12622.5	−	21862.8i	−0.527872	−	0.914301i
$831$		0			0
$832$	−538.009			−0.0224184
$833$	22275.5	+	11144.3i	0.926530	+	0.463539i
$834$		0			0
$835$	−22723.7	+	39358.6i	−0.941779	+	1.63121i
$836$	5209.82	+	9023.68i	0.215533	+	0.373314i
$837$		0			0
$838$	−7229.43	+	12521.7i	−0.298015	+	0.516177i
$839$	−22274.4			−0.916564			−0.458282	−	0.888807i	$-0.651535\pi$
$839$	−22274.4			−0.916564			−0.458282	+	0.888807i	$0.651535\pi$
$840$		0			0
$841$	−20896.7			−0.856810
$842$	−10949.7	+	18965.4i	−0.448160	+	0.776237i
$843$		0			0
$844$	−453.988	−	786.330i	−0.0185153	−	0.0320694i
$845$	−14253.2	+	24687.3i	−0.580267	+	1.00505i
$846$		0			0
$847$	−10326.4	−	2439.03i	−0.418915	−	0.0989446i
$848$	−5148.58			−0.208494
$849$		0			0
$850$	3974.42	+	6883.91i	0.160378	+	0.277784i
$851$	−138.954	−	240.675i	−0.00559726	−	0.00969474i
$852$		0			0
$853$	28462.2			1.14247			0.571234	−	0.820787i	$-0.306465\pi$
$853$	28462.2			1.14247			0.571234	+	0.820787i	$0.306465\pi$
$854$	−179.784	+	190.854i	−0.00720385	+	0.00764740i
$855$		0			0
$856$	7088.66	−	12277.9i	0.283044	−	0.490246i
$857$	15320.4	+	26535.7i	0.610658	+	1.05769i	0.991130	+	0.132898i	$0.0424284\pi$
$857$	15320.4	+	26535.7i	0.610658	+	1.05769i	−0.380472	+	0.924793i	$0.624238\pi$
$858$		0			0
$859$	−2623.56	+	4544.14i	−0.104208	+	0.180494i	−0.913414	−	0.407031i	$-0.866564\pi$
$859$	−2623.56	+	4544.14i	−0.104208	+	0.180494i	0.809206	+	0.587525i	$0.199897\pi$
$860$	29849.1			1.18354
$861$		0			0
$862$	−20187.9			−0.797685
$863$	8166.76	−	14145.2i	0.322132	−	0.557949i	−0.658796	−	0.752322i	$-0.728934\pi$
$863$	8166.76	−	14145.2i	0.322132	−	0.557949i	0.980928	+	0.194373i	$0.0622672\pi$
$864$		0			0
$865$	−4219.67	−	7308.67i	−0.165865	−	0.287286i
$866$	11040.9	−	19123.5i	0.433240	−	0.750394i
$867$		0			0
$868$	−6053.94	−	1429.90i	−0.236733	−	0.0559146i
$869$	−57294.9			−2.23659
$870$		0			0
$871$	2046.31	+	3544.31i	0.0796055	+	0.137881i
$872$	−5457.69	−	9453.00i	−0.211950	−	0.367109i
$873$		0			0
$874$	−9311.14			−0.360359
$875$	5016.71	+	16710.2i	0.193824	+	0.645610i
$876$		0			0
$877$	−2357.53	+	4083.35i	−0.0907731	+	0.157224i	−0.907837	−	0.419324i	$-0.862267\pi$
$877$	−2357.53	+	4083.35i	−0.0907731	+	0.157224i	0.817064	+	0.576548i	$0.195600\pi$
$878$	1132.41	+	1961.39i	0.0435274	+	0.0753917i
$879$		0			0
$880$	4679.78	−	8105.62i	0.179268	−	0.310501i
$881$	−31965.3			−1.22240			−0.611202	−	0.791475i	$-0.709314\pi$
$881$	−31965.3			−1.22240			−0.611202	+	0.791475i	$0.709314\pi$
$882$		0			0
$883$	20209.4			0.770217			0.385109	−	0.922871i	$-0.374164\pi$
$883$	20209.4			0.770217			0.385109	+	0.922871i	$0.374164\pi$
$884$	1220.90	−	2114.65i	0.0464516	−	0.0804565i
$885$		0			0
$886$	−9439.27	−	16349.3i	−0.357921	−	0.619938i
$887$	8313.28	−	14399.0i	0.314693	−	0.545064i	−0.664679	−	0.747129i	$-0.731432\pi$
$887$	8313.28	−	14399.0i	0.314693	−	0.545064i	0.979372	+	0.202065i	$0.0647651\pi$
$888$		0			0
$889$	4533.94	+	15102.2i	0.171050	+	0.569753i
$890$	21420.9			0.806775
$891$		0			0
$892$	9839.22	+	17042.0i	0.369329	+	0.639697i
$893$	−108.984	−	188.765i	−0.00408398	−	0.00707367i
$894$		0			0
$895$	−3264.37			−0.121917
$896$	2307.11	+	544.923i	0.0860215	+	0.0203176i
$897$		0			0
$898$	−8855.77	+	15338.6i	−0.329088	+	0.569997i
$899$	2481.09	+	4297.37i	0.0920455	+	0.159427i
$900$		0			0
$901$	11683.6	−	20236.6i	0.432006	−	0.748256i
$902$	−4371.92			−0.161385
$903$		0			0
$904$	−6241.46			−0.229633
$905$	28261.2	−	48949.8i	1.03805	−	1.79795i
$906$		0			0
$907$	9941.67	+	17219.5i	0.363956	+	0.630390i	0.988608	−	0.150513i	$-0.0480927\pi$
$907$	9941.67	+	17219.5i	0.363956	+	0.630390i	−0.624652	+	0.780903i	$0.714759\pi$
$908$	3006.26	−	5206.99i	0.109875	−	0.190308i
$909$		0			0
$910$	−2862.34	+	3038.58i	−0.104270	+	0.110690i
$911$	−4520.76			−0.164412			−0.0822060	−	0.996615i	$-0.526197\pi$
$911$	−4520.76			−0.164412			−0.0822060	+	0.996615i	$0.526197\pi$
$912$		0			0
$913$	−20541.3	−	35578.6i	−0.744598	−	1.28968i
$914$	5301.24	+	9182.02i	0.191849	+	0.332291i
$915$		0			0
$916$	−14960.2			−0.539628
$917$	27090.6	+	6398.61i	0.975585	+	0.230426i
$918$		0			0
$919$	−262.053	+	453.889i	−0.00940623	+	0.0162921i	−0.870690	−	0.491832i	$-0.836327\pi$
$919$	−262.053	+	453.889i	−0.00940623	+	0.0162921i	0.861284	+	0.508124i	$0.169661\pi$
$920$	4181.92	+	7243.30i	0.149863	+	0.259570i
$921$		0			0
$922$	8217.88	−	14233.8i	0.293537	−	0.508421i
$923$	−4746.41			−0.169263
$924$		0			0
$925$	−195.043			−0.00693295
$926$	−14957.4	+	25907.1i	−0.530812	+	0.919394i
$927$		0			0
$928$	−945.525	−	1637.70i	−0.0334465	−	0.0579311i
$929$	−2348.23	+	4067.25i	−0.0829310	+	0.143641i	−0.904508	−	0.426457i	$-0.859761\pi$
$929$	−2348.23	+	4067.25i	−0.0829310	+	0.143641i	0.821577	+	0.570098i	$0.193095\pi$
$930$		0			0
$931$	1222.05	+	20440.3i	0.0430193	+	0.719555i
$932$	−4937.35			−0.173528
$933$		0			0
$934$	242.262	+	419.610i	0.00848720	+	0.0147003i
$935$	21239.5	+	36788.0i	0.742896	+	1.28673i
$936$		0			0
$937$	28172.9			0.982250			0.491125	−	0.871089i	$-0.336586\pi$
$937$	28172.9			0.982250			0.491125	+	0.871089i	$0.336586\pi$
$938$	−5185.20	−	17271.5i	−0.180493	−	0.601208i
$939$		0			0
$940$	−97.8958	+	169.560i	−0.00339682	+	0.00588346i
$941$	−2698.64	−	4674.19i	−0.0934891	−	0.161928i	0.815488	−	0.578774i	$-0.196469\pi$
$941$	−2698.64	−	4674.19i	−0.0934891	−	0.161928i	−0.908977	+	0.416846i	$0.863135\pi$
$942$		0			0
$943$	1953.41	−	3383.40i	0.0674567	−	0.116838i
$944$	13687.7			0.471924
$945$		0			0
$946$	48575.1			1.66947
$947$	−8835.13	+	15302.9i	−0.303171	+	0.525108i	−0.976852	−	0.213914i	$-0.931379\pi$
$947$	−8835.13	+	15302.9i	−0.303171	+	0.525108i	0.673681	+	0.739022i	$0.264712\pi$
$948$		0			0
$949$	772.865	+	1338.64i	0.0264365	+	0.0457894i
$950$	−3267.41	+	5659.32i	−0.111588	+	0.193276i
$951$		0			0
$952$	−7377.33	+	7831.57i	−0.251156	+	0.266620i
$953$	−54328.8			−1.84668			−0.923338	−	0.383989i	$-0.874550\pi$
$953$	−54328.8			−1.84668			−0.923338	+	0.383989i	$0.874550\pi$
$954$		0			0
$955$	−25478.0	−	44129.3i	−0.863298	−	1.49528i
$956$	815.346	+	1412.22i	0.0275838	+	0.0477766i
$957$		0			0
$958$	−19516.3			−0.658186
$959$	22055.7	−	23413.7i	0.742664	−	0.788391i
$960$		0			0
$961$	11370.1	−	19693.6i	0.381662	−	0.661058i
$962$	29.9575	+	51.8879i	0.00100402	+	0.00173901i
$963$		0			0
$964$	−250.067	+	433.130i	−0.00835491	+	0.0144711i
$965$	−51561.2			−1.72002
$966$		0			0
$967$	37517.1			1.24764			0.623821	−	0.781567i	$-0.285579\pi$
$967$	37517.1			1.24764			0.623821	+	0.781567i	$0.285579\pi$
$968$	2291.67	−	3969.29i	0.0760920	−	0.131795i
$969$		0			0
$970$	−7035.20	−	12185.3i	−0.232873	−	0.403347i
$971$	−26650.0	+	46159.1i	−0.880781	+	1.52556i	−0.0303077	+	0.999541i	$0.509649\pi$
$971$	−26650.0	+	46159.1i	−0.880781	+	1.52556i	−0.850474	+	0.526018i	$0.823685\pi$
$972$		0			0
$973$	16595.8	+	55279.2i	0.546801	+	1.82135i
$974$	26435.9			0.869674
$975$		0			0
$976$	−56.6293	−	98.0849i	−0.00185723	−	0.00321682i
$977$	−12188.1	−	21110.4i	−0.399111	−	0.691281i	0.594505	−	0.804092i	$-0.297348\pi$
$977$	−12188.1	−	21110.4i	−0.399111	−	0.691281i	−0.993616	+	0.112811i	$0.964015\pi$
$978$		0			0
$979$	34859.4			1.13801
$980$	15352.0	−	10131.0i	0.500411	−	0.330229i
$981$		0			0
$982$	−13789.5	+	23884.2i	−0.448108	+	0.776146i
$983$	220.918	+	382.641i	0.00716804	+	0.0124154i	0.869587	−	0.493779i	$-0.164385\pi$
$983$	220.918	+	382.641i	0.00716804	+	0.0124154i	−0.862419	+	0.506195i	$0.831052\pi$
$984$		0			0
$985$	31769.9	−	55027.1i	1.02769	−	1.78001i
$986$	8582.67			0.277209
$987$		0			0
$988$	2007.42			0.0646402
$989$	−21703.7	+	37591.9i	−0.697814	+	1.20865i
$990$		0			0
$991$	−22694.0	−	39307.2i	−0.727447	−	1.25998i	−0.957959	−	0.286906i	$-0.907373\pi$
$991$	−22694.0	−	39307.2i	−0.727447	−	1.25998i	0.230512	−	0.973070i	$-0.425960\pi$
$992$	1343.50	−	2327.02i	0.0430003	−	0.0744787i
$993$		0			0
$994$	20353.8	+	4807.41i	0.649479	+	0.153402i
$995$	55597.2			1.77141
$996$		0			0
$997$	19853.0	+	34386.3i	0.630641	+	1.09230i	0.987421	+	0.158114i	$0.0505414\pi$
$997$	19853.0	+	34386.3i	0.630641	+	1.09230i	−0.356780	+	0.934189i	$0.616125\pi$
$998$	−1990.38	−	3447.43i	−0.0631305	−	0.109345i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.g.b.163.3	yes	6
3.2	odd	2		378.4.g.a.163.1	yes	6
7.4	even	3	inner	378.4.g.b.109.3	yes	6
21.11	odd	6		378.4.g.a.109.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.a.109.1	✓	6	21.11	odd	6
378.4.g.a.163.1	yes	6	3.2	odd	2
378.4.g.b.109.3	yes	6	7.4	even	3	inner
378.4.g.b.163.3	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(6.70319 - 11.6103i) q^{5} +(5.32531 + 17.7381i) q^{7} -8.00000 q^{8} +(-13.4064 - 23.2205i) q^{10} +(-21.8169 - 37.7881i) q^{11} -8.40639 q^{13} +(36.0486 + 8.51443i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(-36.3086 - 62.8883i) q^{17} +(29.8496 - 51.7011i) q^{19} -53.6255 q^{20} -87.2678 q^{22} +(38.9919 - 67.5359i) q^{23} +(-27.3656 - 47.3986i) q^{25} +(-8.40639 + 14.5603i) q^{26} +(50.7961 - 53.9237i) q^{28} -59.0953 q^{29} +(-41.9845 - 72.7193i) q^{31} +(16.0000 + 27.7128i) q^{32} -145.234 q^{34} +(241.641 + 57.0738i) q^{35} +(1.78183 - 3.08622i) q^{37} +(-59.6993 - 103.402i) q^{38} +(-53.6255 + 92.8822i) q^{40} +50.0978 q^{41} -556.622 q^{43} +(-87.2678 + 151.152i) q^{44} +(-77.9838 - 135.072i) q^{46} +(1.82554 - 3.16193i) q^{47} +(-286.282 + 188.922i) q^{49} -109.462 q^{50} +(16.8128 + 29.1206i) q^{52} +(160.893 + 278.675i) q^{53} -584.973 q^{55} +(-42.6024 - 141.905i) q^{56} +(-59.0953 + 102.356i) q^{58} +(-427.740 - 740.867i) q^{59} +(-3.53933 + 6.13030i) q^{61} -167.938 q^{62} +64.0000 q^{64} +(-56.3496 + 97.6004i) q^{65} +(-243.423 - 421.621i) q^{67} +(-145.234 + 251.553i) q^{68} +(340.496 - 361.461i) q^{70} +564.619 q^{71} +(-91.9378 - 159.241i) q^{73} +(-3.56366 - 6.17243i) q^{74} -238.797 q^{76} +(554.108 - 588.225i) q^{77} +(656.542 - 1137.16i) q^{79} +(107.251 + 185.764i) q^{80} +(50.0978 - 86.7719i) q^{82} +941.530 q^{83} -973.534 q^{85} +(-556.622 + 964.097i) q^{86} +(174.536 + 302.305i) q^{88} +(-399.453 + 691.873i) q^{89} +(-44.7666 - 149.114i) q^{91} -311.935 q^{92} +(-3.65109 - 6.32387i) q^{94} +(-400.176 - 693.124i) q^{95} +524.765 q^{97} +(40.9401 + 684.777i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - 12 q^{4} + 5 q^{5} + 8 q^{7} - 48 q^{8} - 10 q^{10} - 29 q^{11} + 20 q^{13} + 20 q^{14} - 48 q^{16} - 38 q^{17} + 57 q^{19} - 40 q^{20} - 116 q^{22} - 14 q^{23} + 134 q^{25} + 20 q^{26}+ \cdots + 720 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - 12 * q^4 + 5 * q^5 + 8 * q^7 - 48 * q^8 - 10 * q^10 - 29 * q^11 + 20 * q^13 + 20 * q^14 - 48 * q^16 - 38 * q^17 + 57 * q^19 - 40 * q^20 - 116 * q^22 - 14 * q^23 + 134 * q^25 + 20 * q^26 + 8 * q^28 + 362 * q^29 - 88 * q^31 + 96 * q^32 - 152 * q^34 + 490 * q^35 + 384 * q^37 - 114 * q^38 - 40 * q^40 - 432 * q^41 - 726 * q^43 - 116 * q^44 + 28 * q^46 - 183 * q^47 + 372 * q^49 + 536 * q^50 - 40 * q^52 - 396 * q^53 - 1268 * q^55 - 64 * q^56 + 362 * q^58 - 427 * q^59 - 427 * q^61 - 352 * q^62 + 384 * q^64 - 216 * q^65 - 32 * q^67 - 152 * q^68 + 1162 * q^70 + 790 * q^71 + 373 * q^73 - 768 * q^74 - 456 * q^76 + 1211 * q^77 + 1364 * q^79 + 80 * q^80 - 432 * q^82 + 154 * q^83 - 2678 * q^85 - 726 * q^86 + 232 * q^88 - 1233 * q^89 + 131 * q^91 + 112 * q^92 + 366 * q^94 - 464 * q^95 + 1180 * q^97 + 720 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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