LMFDB - Embedded newform 378.4.g.b.109.3

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			109.3
Root			$2.16527 + 3.75036i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.b.163.3

$q$-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 $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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	0.992887	+	0.119059i	$0.0379878\pi$
$5$	6.70319	+	11.6103i	0.599552	+	1.03845i	−0.393335	+	0.919395i	$0.628679\pi$
$6$		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	0.395110	+	0.918634i	$0.370706\pi$
$11$	−21.8169	+	37.7881i	−0.598005	+	1.03578i	−0.993115	+	0.117142i	$0.962627\pi$
$12$		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	0.481774	+	0.876296i	$0.339993\pi$
$17$	−36.3086	+	62.8883i	−0.518007	+	0.897215i	−0.999781	+	0.0209196i	$0.993341\pi$
$18$		0			0
$19$	29.8496	+	51.7011i	0.360420	+	0.624265i	0.988030	−	0.154263i	$-0.0493002\pi$
$19$	29.8496	+	51.7011i	0.360420	+	0.624265i	−0.627610	+	0.778528i	$0.715967\pi$
$20$	−53.6255			−0.599552
$21$		0			0
$22$	−87.2678			−0.845707
$23$	38.9919	+	67.5359i	0.353494	+	0.612270i	0.986859	−	0.161583i	$-0.0516600\pi$
$23$	38.9919	+	67.5359i	0.353494	+	0.612270i	−0.633365	+	0.773853i	$0.718327\pi$
$24$		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	−0.961637	−	0.274325i	$-0.911546\pi$
$31$	−41.9845	+	72.7193i	−0.243246	+	0.421315i	0.718391	+	0.695640i	$0.244879\pi$
$32$	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	0.869957	−	0.493128i	$-0.164147\pi$
$37$	1.78183	+	3.08622i	0.00791705	+	0.0137127i	−0.862040	+	0.506841i	$0.830813\pi$
$38$	−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	0.868844	−	0.495085i	$-0.164863\pi$
$47$	1.82554	+	3.16193i	0.00566559	+	0.00981310i	−0.863179	+	0.504899i	$0.831530\pi$
$48$		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	−0.578647	−	0.815578i	$-0.696419\pi$
$53$	160.893	−	278.675i	0.416988	−	0.722244i	0.995635	+	0.0933338i	$0.0297524\pi$
$54$		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	−0.185805	+	0.982587i	$0.559489\pi$
$59$	−427.740	+	740.867i	−0.943847	+	1.63479i	−0.758042	+	0.652205i	$0.773844\pi$
$60$		0			0
$61$	−3.53933	−	6.13030i	−0.00742894	−	0.0128673i	0.862287	−	0.506420i	$-0.169031\pi$
$61$	−3.53933	−	6.13030i	−0.00742894	−	0.0128673i	−0.869716	+	0.493553i	$0.835698\pi$
$62$	−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	−0.997972	−	0.0636506i	$-0.979726\pi$
$67$	−243.423	+	421.621i	−0.443863	+	0.768793i	0.554109	+	0.832444i	$0.313059\pi$
$68$	−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	−0.930267	−	0.366882i	$-0.880425\pi$
$73$	−91.9378	+	159.241i	−0.147404	+	0.255312i	0.782863	+	0.622194i	$0.213758\pi$
$74$	−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	0.774595	+	0.632457i	$0.217954\pi$
$79$	656.542	+	1137.16i	0.935022	+	1.61950i	0.160426	+	0.987048i	$0.448713\pi$
$80$	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	0.523862	−	0.851803i	$-0.324491\pi$
$89$	−399.453	−	691.873i	−0.475752	−	0.824027i	−0.999614	+	0.0277760i	$0.991157\pi$
$90$		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	−0.650687	−	0.759346i	$-0.725519\pi$
$101$	337.266	−	584.162i	0.332270	−	0.575508i	0.982957	+	0.183838i	$0.0588522\pi$
$102$		0			0
$103$	−624.773	−	1082.14i	−0.597676	−	1.03521i	−0.993163	−	0.116734i	$-0.962757\pi$
$103$	−624.773	−	1082.14i	−0.597676	−	1.03521i	0.395487	−	0.918472i	$-0.370576\pi$
$104$	67.2511			0.0634088
$105$		0			0
$106$	643.572			0.589710
$107$	−886.082	−	1534.74i	−0.800568	−	1.38662i	−0.919243	−	0.393691i	$-0.871198\pi$
$107$	−886.082	−	1534.74i	−0.800568	−	1.38662i	0.118675	−	0.992933i	$-0.462135\pi$
$108$		0			0
$109$	682.211	−	1181.62i	0.599486	−	1.03834i	−0.393411	−	0.919363i	$-0.628705\pi$
$109$	682.211	−	1181.62i	0.599486	−	1.03834i	0.992897	−	0.118978i	$-0.0379617\pi$
$110$	−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	0.999999	+	0.00140275i	$0.000446511\pi$
$131$	751.502	+	1301.64i	0.501214	+	0.868129i	−0.498785	+	0.866726i	$0.666220\pi$
$132$		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	0.457264	+	0.889331i	$0.348829\pi$
$137$	−868.400	+	1504.11i	−0.541551	+	0.937993i	−0.998815	+	0.0486627i	$0.984504\pi$
$138$		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	0.912017	+	0.410153i	$0.134525\pi$
$149$	1475.41	+	2555.49i	0.811211	+	1.40506i	−0.100805	+	0.994906i	$0.532142\pi$
$150$		0			0
$151$	644.154	−	1115.71i	0.347156	−	0.601292i	−0.638587	−	0.769549i	$-0.720481\pi$
$151$	644.154	−	1115.71i	0.347156	−	0.601292i	0.985743	+	0.168258i	$0.0538142\pi$
$152$	−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	−0.0366552	−	0.999328i	$-0.511670\pi$
$157$	1666.45	−	2886.38i	0.847116	−	1.46725i	0.883771	−	0.467920i	$-0.154996\pi$
$158$	−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	0.854363	+	0.519677i	$0.173948\pi$
$163$	1825.57	+	3161.97i	0.877235	+	1.51942i	0.0228725	+	0.999738i	$0.492719\pi$
$164$	−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	0.926862	−	0.375402i	$-0.122495\pi$
$173$	314.750	+	545.164i	0.138324	+	0.239584i	−0.788538	+	0.614986i	$0.789162\pi$
$174$		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	−0.890324	−	0.455327i	$-0.849522\pi$
$179$	−121.747	+	210.872i	−0.0508369	+	0.0880520i	0.839487	+	0.543380i	$0.182856\pi$
$180$		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	0.961018	+	0.276486i	$0.0891700\pi$
$191$	1900.44	+	3291.66i	0.719953	+	1.24700i	−0.241065	+	0.970509i	$0.577497\pi$
$192$		0			0
$193$	−1923.01	+	3330.75i	−0.717209	+	1.24224i	0.244893	+	0.969550i	$0.421247\pi$
$193$	−1923.01	+	3330.75i	−0.717209	+	1.24224i	−0.962101	+	0.272692i	$0.912086\pi$
$194$	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	−0.214472	−	0.976730i	$-0.568803\pi$
$199$	2073.53	−	3591.46i	0.738637	−	1.27936i	0.953109	−	0.302627i	$-0.0978637\pi$
$200$	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	−0.734982	−	0.678087i	$-0.762809\pi$
$227$	751.565	−	1301.75i	0.219749	−	0.380617i	0.954731	+	0.297470i	$0.0961427\pi$
$228$		0			0
$229$	1870.02	+	3238.98i	0.539628	+	0.934662i	0.998924	+	0.0463793i	$0.0147683\pi$
$229$	1870.02	+	3238.98i	0.539628	+	0.934662i	−0.459296	+	0.888283i	$0.651898\pi$
$230$	−2090.96			−0.599452
$231$		0			0
$232$	472.762			0.133786
$233$	617.169	+	1068.97i	0.173528	+	0.300560i	0.939651	−	0.342135i	$-0.111150\pi$
$233$	617.169	+	1068.97i	0.173528	+	0.300560i	−0.766123	+	0.642694i	$0.777817\pi$
$234$		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	−0.874259	−	0.485459i	$-0.838652\pi$
$241$	−62.5169	+	108.282i	−0.0167098	+	0.0289422i	0.857550	+	0.514401i	$0.171986\pi$
$242$	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	0.830768	−	0.556619i	$-0.187902\pi$
$257$	−274.649	−	475.706i	−0.0666619	−	0.115462i	−0.897430	+	0.441157i	$0.854568\pi$
$258$		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	0.238093	+	0.971242i	$0.423478\pi$
$263$	−3079.75	+	5334.28i	−0.722074	+	1.25067i	−0.960167	+	0.279427i	$0.909856\pi$
$264$		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	−0.233940	−	0.972251i	$-0.575162\pi$
$269$	3198.75	−	5540.40i	0.725024	−	1.25578i	0.958964	−	0.283528i	$-0.0915047\pi$
$270$		0			0
$271$	−3215.20	−	5568.89i	−0.720700	−	1.24829i	−0.960720	−	0.277521i	$-0.910487\pi$
$271$	−3215.20	−	5568.89i	−0.720700	−	1.24829i	0.240020	−	0.970768i	$-0.422846\pi$
$272$	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	−0.997769	−	0.0667605i	$-0.978734\pi$
$277$	−2033.41	+	3521.97i	−0.441068	+	0.763953i	0.556701	+	0.830713i	$0.312067\pi$
$278$	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	−0.202792	−	0.979222i	$-0.565001\pi$
$283$	3554.58	−	6156.71i	0.746635	−	1.29321i	0.949427	−	0.313988i	$-0.101665\pi$
$284$	−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	−0.952455	−	0.304680i	$-0.901451\pi$
$311$	−1164.74	+	2017.38i	−0.212367	+	0.367830i	0.740088	+	0.672510i	$0.234784\pi$
$312$		0			0
$313$	−1744.00	−	3020.69i	−0.314941	−	0.545494i	0.664484	−	0.747303i	$-0.268652\pi$
$313$	−1744.00	−	3020.69i	−0.314941	−	0.545494i	−0.979425	+	0.201808i	$0.935318\pi$
$314$	6665.80			1.19800
$315$		0			0
$316$	−5252.33			−0.935022
$317$	−1348.77	−	2336.14i	−0.238974	−	0.413914i	0.721446	−	0.692470i	$-0.243478\pi$
$317$	−1348.77	−	2336.14i	−0.238974	−	0.413914i	−0.960420	+	0.278556i	$0.910144\pi$
$318$		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	0.899650	+	0.436612i	$0.143822\pi$
$331$	4985.88	+	8635.80i	0.827942	+	1.43404i	−0.0717073	+	0.997426i	$0.522845\pi$
$332$	−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	−0.568147	−	0.822927i	$-0.692340\pi$
$347$	2770.44	−	4798.54i	0.428602	−	0.742361i	0.996749	+	0.0805663i	$0.0256729\pi$
$348$		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	−0.618232	−	0.785995i	$-0.712151\pi$
$353$	2464.39	−	4268.45i	0.371576	−	0.643588i	0.989808	+	0.142407i	$0.0454843\pi$
$354$		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	0.816426	−	0.577450i	$-0.195952\pi$
$359$	−624.930	−	1082.41i	−0.0918733	−	0.159129i	−0.908299	+	0.418321i	$0.862619\pi$
$360$		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	−0.708408	−	0.705803i	$-0.750586\pi$
$367$	1807.17	−	3130.11i	0.257040	−	0.445206i	0.965448	+	0.260597i	$0.0839195\pi$
$368$	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	0.605305	−	0.795994i	$-0.293051\pi$
$373$	−2785.71	−	4824.98i	−0.386698	−	0.669781i	−0.992003	+	0.126213i	$0.959718\pi$
$374$	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	0.955868	−	0.293797i	$-0.0949189\pi$
$383$	1675.22	+	2901.57i	0.223498	+	0.387111i	−0.732369	+	0.680907i	$0.761586\pi$
$384$		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	−0.538904	−	0.842367i	$-0.681161\pi$
$389$	3529.71	−	6113.63i	0.460060	−	0.796847i	0.998963	+	0.0455204i	$0.0144946\pi$
$390$		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	−0.969637	−	0.244549i	$-0.921360\pi$
$397$	−5510.26	−	9544.05i	−0.696604	−	1.20655i	0.273032	−	0.962005i	$-0.411973\pi$
$398$	8294.13			1.04459
$399$		0			0
$400$	875.699			0.109462
$401$	−1458.98	−	2527.02i	−0.181690	−	0.314697i	0.760766	−	0.649026i	$-0.224823\pi$
$401$	−1458.98	−	2527.02i	−0.181690	−	0.314697i	−0.942456	+	0.334330i	$0.891490\pi$
$402$		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	−0.768901	−	0.639368i	$-0.779196\pi$
$409$	1400.02	−	2424.91i	0.169259	−	0.293165i	0.938159	+	0.346204i	$0.112529\pi$
$410$	−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	0.433089	+	0.901351i	$0.357423\pi$
$431$	−5046.99	+	8741.64i	−0.564048	+	0.976960i	−0.997138	+	0.0756091i	$0.975910\pi$
$432$		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	0.833605	−	0.552362i	$-0.186273\pi$
$439$	−566.206	−	980.697i	−0.0615570	−	0.106620i	−0.895162	+	0.445742i	$0.852940\pi$
$440$	4679.78			0.507045
$441$		0			0
$442$	1220.90			0.131385
$443$	4719.63	+	8174.65i	0.506177	+	0.876725i	0.999974	+	0.00714780i	$0.00227523\pi$
$443$	4719.63	+	8174.65i	0.506177	+	0.876725i	−0.493797	+	0.869577i	$0.664391\pi$
$444$		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	0.697884	−	0.716211i	$-0.254125\pi$
$457$	−2650.62	−	4591.01i	−0.271315	−	0.469931i	−0.969199	+	0.246280i	$0.920792\pi$
$458$	−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	0.859962	−	0.510359i	$-0.170487\pi$
$467$	−121.131	−	209.805i	−0.0120027	−	0.0207893i	−0.871964	+	0.489569i	$0.837154\pi$
$468$		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	−0.999220	−	0.0394932i	$-0.987426\pi$
$479$	−4879.07	+	8450.79i	−0.465408	+	0.806110i	0.533812	+	0.845603i	$0.320759\pi$
$480$		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	−0.375441	−	0.926846i	$-0.622509\pi$
$487$	6608.98	−	11447.1i	0.614952	−	1.06513i	0.990393	−	0.138282i	$-0.0441580\pi$
$488$	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	0.907207	−	0.420684i	$-0.138210\pi$
$499$	995.188	+	1723.72i	0.0892800	+	0.154638i	−0.817927	+	0.575322i	$0.804877\pi$
$500$	−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	0.730132	−	0.683306i	$-0.239459\pi$
$509$	−2603.26	−	4508.99i	−0.226695	−	0.392647i	−0.956827	+	0.290659i	$0.906125\pi$
$510$		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	0.484936	+	0.874550i	$0.338843\pi$
$521$	−6123.38	+	10606.0i	−0.514914	+	0.891858i	−0.999850	+	0.0173079i	$0.994490\pi$
$522$		0			0
$523$	6370.75	+	11034.5i	0.532646	+	0.922569i	0.999273	+	0.0381155i	$0.0121355\pi$
$523$	6370.75	+	11034.5i	0.532646	+	0.922569i	−0.466628	+	0.884454i	$0.654531\pi$
$524$	−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	0.927366	−	0.374155i	$-0.122067\pi$
$541$	1757.33	+	3043.78i	0.139655	+	0.241890i	−0.787711	+	0.616045i	$0.788734\pi$
$542$	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	−0.524763	−	0.851248i	$-0.675846\pi$
$557$	6241.84	−	10811.2i	0.474821	−	0.822414i	0.999584	+	0.0288343i	$0.00917952\pi$
$558$		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	−0.576034	−	0.817426i	$-0.695400\pi$
$563$	5609.22	−	9715.46i	0.419894	−	0.727278i	0.995928	+	0.0901473i	$0.0287338\pi$
$564$		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	0.918621	+	0.395139i	$0.129304\pi$
$569$	10878.7	+	18842.5i	0.801511	+	1.38826i	−0.117110	+	0.993119i	$0.537363\pi$
$570$		0			0
$571$	−1353.09	+	2343.62i	−0.0991683	+	0.171765i	−0.911341	−	0.411653i	$-0.864952\pi$
$571$	−1353.09	+	2343.62i	−0.0991683	+	0.171765i	0.812172	+	0.583418i	$0.198285\pi$
$572$	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	−0.693460	−	0.720496i	$-0.743914\pi$
$577$	3842.52	−	6655.44i	0.277238	−	0.480190i	0.970697	+	0.240306i	$0.0772477\pi$
$578$	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	0.993350	+	0.115138i	$0.0367309\pi$
$593$	8612.12	+	14916.6i	0.596387	+	1.03297i	−0.396963	+	0.917835i	$0.629936\pi$
$594$		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	−0.652318	−	0.757946i	$-0.726203\pi$
$599$	4841.41	−	8385.56i	0.330241	−	0.571995i	0.982559	+	0.185951i	$0.0595366\pi$
$600$		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	0.706470	−	0.707743i	$-0.250286\pi$
$607$	−3883.60	−	6726.59i	−0.259688	−	0.449792i	−0.966158	+	0.257950i	$0.916953\pi$
$608$	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	0.127047	−	0.991897i	$-0.459450\pi$
$613$	14001.4	−	24251.2i	0.922531	−	1.59787i	0.795484	−	0.605974i	$-0.207217\pi$
$614$	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	0.120227	−	0.992746i	$-0.461638\pi$
$619$	14166.3	−	24536.7i	0.919857	−	1.59324i	0.799630	−	0.600493i	$-0.205029\pi$
$620$	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	−0.961962	−	0.273182i	$-0.911924\pi$
$641$	−3966.30	+	6869.84i	−0.244399	+	0.423311i	0.717564	+	0.696493i	$0.245257\pi$
$642$		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	−0.0549522	+	0.998489i	$0.517501\pi$
$647$	−14683.0	+	25431.7i	−0.892193	+	1.54532i	−0.837241	+	0.546834i	$0.815833\pi$
$648$		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	0.954585	+	0.297937i	$0.0962987\pi$
$653$	12269.9	+	21252.2i	0.735314	+	1.27360i	−0.219271	+	0.975664i	$0.570368\pi$
$654$		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	0.387502	+	0.921869i	$0.373338\pi$
$661$	−10274.9	+	17796.7i	−0.604611	+	1.04722i	−0.992113	+	0.125348i	$0.959995\pi$
$662$	−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	0.970305	+	0.241886i	$0.0777661\pi$
$677$	12236.0	+	21193.3i	0.694632	+	1.20314i	−0.275673	+	0.961252i	$0.588901\pi$
$678$		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	0.419768	+	0.907632i	$0.362112\pi$
$683$	−10284.1	+	17812.5i	−0.576148	+	0.997918i	−0.995916	+	0.0902863i	$0.971222\pi$
$684$		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	0.948572	−	0.316561i	$-0.102528\pi$
$691$	3635.32	+	6296.55i	0.200136	+	0.346646i	−0.748436	+	0.663207i	$0.769195\pi$
$692$	−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	−0.997748	−	0.0670713i	$-0.978634\pi$
$709$	−10514.6	−	18211.8i	−0.556960	−	0.964682i	0.440789	−	0.897611i	$-0.354699\pi$
$710$	−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	0.968382	−	0.249474i	$-0.0802576\pi$
$719$	5169.58	+	8953.98i	0.268140	+	0.464433i	−0.700241	+	0.713906i	$0.746924\pi$
$720$		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	0.983184	−	0.182615i	$-0.0584562\pi$
$733$	6617.25	+	11461.4i	0.333443	+	0.577540i	−0.649742	+	0.760155i	$0.725123\pi$
$734$	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	−0.145685	+	0.989331i	$0.546539\pi$
$739$	−18675.7	+	32347.2i	−0.929628	+	1.61016i	−0.783943	+	0.620833i	$0.786795\pi$
$740$	−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	0.999689	−	0.0249278i	$-0.00793559\pi$
$751$	9842.85	+	17048.3i	0.478257	+	0.828365i	−0.521433	+	0.853292i	$0.674602\pi$
$752$	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	0.735930	−	0.677057i	$-0.236745\pi$
$761$	−4584.56	−	7940.68i	−0.218384	−	0.378252i	−0.954314	+	0.298806i	$0.903412\pi$
$762$		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	−0.876538	−	0.481333i	$-0.840153\pi$
$773$	−460.408	+	797.450i	−0.0214227	+	0.0371052i	0.855115	+	0.518438i	$0.173486\pi$
$774$		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	−0.812963	−	0.582315i	$-0.802147\pi$
$787$	2159.63	−	3740.60i	0.0978179	−	0.169426i	0.910781	+	0.412889i	$0.135480\pi$
$788$	−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	−0.131304	−	0.991342i	$-0.541916\pi$
$809$	18244.3	−	31600.1i	0.792876	−	1.37330i	0.924179	−	0.381959i	$-0.124750\pi$
$810$		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	0.982497	+	0.186280i	$0.0596432\pi$
$821$	15351.2	+	26589.1i	0.652572	+	1.13029i	−0.329925	+	0.944007i	$0.607023\pi$
$822$		0			0
$823$	2340.54	−	4053.94i	0.0991326	−	0.171703i	−0.812193	−	0.583388i	$-0.801727\pi$
$823$	2340.54	−	4053.94i	0.0991326	−	0.171703i	0.911326	+	0.411686i	$0.135060\pi$
$824$	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	−0.995185	−	0.0980110i	$-0.968752\pi$
$829$	−9850.98	+	17062.4i	−0.412713	+	0.714839i	0.582473	+	0.812850i	$0.302085\pi$
$830$	−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	−0.380472	−	0.924793i	$-0.624238\pi$
$857$	15320.4	−	26535.7i	0.610658	−	1.05769i	0.991130	−	0.132898i	$-0.0424284\pi$
$858$		0			0
$859$	−2623.56	−	4544.14i	−0.104208	−	0.180494i	0.809206	−	0.587525i	$-0.199897\pi$
$859$	−2623.56	−	4544.14i	−0.104208	−	0.180494i	−0.913414	+	0.407031i	$0.866564\pi$
$860$	29849.1			1.18354
$861$		0			0
$862$	−20187.9			−0.797685
$863$	8166.76	+	14145.2i	0.322132	+	0.557949i	0.980928	−	0.194373i	$-0.0622672\pi$
$863$	8166.76	+	14145.2i	0.322132	+	0.557949i	−0.658796	+	0.752322i	$0.728934\pi$
$864$		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	0.817064	−	0.576548i	$-0.195600\pi$
$877$	−2357.53	−	4083.35i	−0.0907731	−	0.157224i	−0.907837	+	0.419324i	$0.862267\pi$
$878$	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	0.979372	−	0.202065i	$-0.0647651\pi$
$887$	8313.28	+	14399.0i	0.314693	+	0.545064i	−0.664679	+	0.747129i	$0.731432\pi$
$888$		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	−0.624652	−	0.780903i	$-0.714759\pi$
$907$	9941.67	−	17219.5i	0.363956	−	0.630390i	0.988608	+	0.150513i	$0.0480927\pi$
$908$	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	0.861284	−	0.508124i	$-0.169661\pi$
$919$	−262.053	−	453.889i	−0.00940623	−	0.0162921i	−0.870690	+	0.491832i	$0.836327\pi$
$920$	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	0.821577	−	0.570098i	$-0.193095\pi$
$929$	−2348.23	−	4067.25i	−0.0829310	−	0.143641i	−0.904508	+	0.426457i	$0.859761\pi$
$930$		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	−0.908977	−	0.416846i	$-0.863135\pi$
$941$	−2698.64	+	4674.19i	−0.0934891	+	0.161928i	0.815488	+	0.578774i	$0.196469\pi$
$942$		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	0.673681	−	0.739022i	$-0.264712\pi$
$947$	−8835.13	−	15302.9i	−0.303171	−	0.525108i	−0.976852	+	0.213914i	$0.931379\pi$
$948$		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	−0.850474	−	0.526018i	$-0.823685\pi$
$971$	−26650.0	−	46159.1i	−0.880781	−	1.52556i	−0.0303077	−	0.999541i	$-0.509649\pi$
$972$		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	−0.993616	−	0.112811i	$-0.964015\pi$
$977$	−12188.1	+	21110.4i	−0.399111	+	0.691281i	0.594505	+	0.804092i	$0.297348\pi$
$978$		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	−0.862419	−	0.506195i	$-0.831052\pi$
$983$	220.918	−	382.641i	0.00716804	−	0.0124154i	0.869587	+	0.493779i	$0.164385\pi$
$984$		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	0.230512	+	0.973070i	$0.425960\pi$
$991$	−22694.0	+	39307.2i	−0.727447	+	1.25998i	−0.957959	+	0.286906i	$0.907373\pi$
$992$	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	−0.356780	−	0.934189i	$-0.616125\pi$
$997$	19853.0	−	34386.3i	0.630641	−	1.09230i	0.987421	−	0.158114i	$-0.0505414\pi$
$998$	−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.109.3	yes	6
3.2	odd	2		378.4.g.a.109.1	✓	6
7.2	even	3	inner	378.4.g.b.163.3	yes	6
21.2	odd	6		378.4.g.a.163.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.g.a.109.1	✓	6	3.2	odd	2
378.4.g.a.163.1	yes	6	21.2	odd	6
378.4.g.b.109.3	yes	6	1.1	even	1	trivial
378.4.g.b.163.3	yes	6	7.2	even	3	inner

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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