LMFDB - Embedded newform 162.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,4,Mod(1,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

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:	yes
Analytic conductor:	$9.55830942093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{105}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26 $ x^2 - x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.62348$ of defining polynomial
Character	$\chi$	$=$	162.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +19.8704 q^{5} -5.87043 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +19.8704 q^{5} -5.87043 q^{7} +8.00000 q^{8} +39.7409 q^{10} -18.7409 q^{11} +45.8704 q^{13} -11.7409 q^{14} +16.0000 q^{16} -16.8704 q^{17} -10.3521 q^{19} +79.4817 q^{20} -37.4817 q^{22} -49.8704 q^{23} +269.834 q^{25} +91.7409 q^{26} -23.4817 q^{28} -10.9070 q^{29} +151.611 q^{31} +32.0000 q^{32} -33.7409 q^{34} -116.648 q^{35} +346.186 q^{37} -20.7043 q^{38} +158.963 q^{40} -264.741 q^{41} -411.890 q^{43} -74.9634 q^{44} -99.7409 q^{46} -472.056 q^{47} -308.538 q^{49} +539.668 q^{50} +183.482 q^{52} -290.186 q^{53} -372.389 q^{55} -46.9634 q^{56} -21.8140 q^{58} -53.2591 q^{59} -293.944 q^{61} +303.223 q^{62} +64.0000 q^{64} +911.465 q^{65} -398.555 q^{67} -67.4817 q^{68} -233.296 q^{70} +647.854 q^{71} -478.279 q^{73} +692.372 q^{74} -41.4085 q^{76} +110.017 q^{77} +374.316 q^{79} +317.927 q^{80} -529.482 q^{82} -933.279 q^{83} -335.223 q^{85} -823.780 q^{86} -149.927 q^{88} -368.817 q^{89} -269.279 q^{91} -199.482 q^{92} -944.113 q^{94} -205.701 q^{95} +274.149 q^{97} -617.076 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 9 * q^5 + 19 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8} + 18 q^{10} + 24 q^{11} + 61 q^{13} + 38 q^{14} + 32 q^{16} - 3 q^{17} + 133 q^{19} + 36 q^{20} + 48 q^{22} - 69 q^{23} + 263 q^{25} + 122 q^{26} + 76 q^{28} - 237 q^{29} + 211 q^{31} + 64 q^{32} - 6 q^{34} - 387 q^{35} + 262 q^{37} + 266 q^{38} + 72 q^{40} - 468 q^{41} - 86 q^{43} + 96 q^{44} - 138 q^{46} - 483 q^{47} - 33 q^{49} + 526 q^{50} + 244 q^{52} - 150 q^{53} - 837 q^{55} + 152 q^{56} - 474 q^{58} - 168 q^{59} - 1049 q^{61} + 422 q^{62} + 128 q^{64} + 747 q^{65} - 1166 q^{67} - 12 q^{68} - 774 q^{70} + 312 q^{71} - 311 q^{73} + 524 q^{74} + 532 q^{76} + 1173 q^{77} + 349 q^{79} + 144 q^{80} - 936 q^{82} - 1221 q^{83} - 486 q^{85} - 172 q^{86} + 192 q^{88} + 492 q^{89} + 107 q^{91} - 276 q^{92} - 966 q^{94} - 1764 q^{95} - 128 q^{97} - 66 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 9 * q^5 + 19 * q^7 + 16 * q^8 + 18 * q^10 + 24 * q^11 + 61 * q^13 + 38 * q^14 + 32 * q^16 - 3 * q^17 + 133 * q^19 + 36 * q^20 + 48 * q^22 - 69 * q^23 + 263 * q^25 + 122 * q^26 + 76 * q^28 - 237 * q^29 + 211 * q^31 + 64 * q^32 - 6 * q^34 - 387 * q^35 + 262 * q^37 + 266 * q^38 + 72 * q^40 - 468 * q^41 - 86 * q^43 + 96 * q^44 - 138 * q^46 - 483 * q^47 - 33 * q^49 + 526 * q^50 + 244 * q^52 - 150 * q^53 - 837 * q^55 + 152 * q^56 - 474 * q^58 - 168 * q^59 - 1049 * q^61 + 422 * q^62 + 128 * q^64 + 747 * q^65 - 1166 * q^67 - 12 * q^68 - 774 * q^70 + 312 * q^71 - 311 * q^73 + 524 * q^74 + 532 * q^76 + 1173 * q^77 + 349 * q^79 + 144 * q^80 - 936 * q^82 - 1221 * q^83 - 486 * q^85 - 172 * q^86 + 192 * q^88 + 492 * q^89 + 107 * q^91 - 276 * q^92 - 966 * q^94 - 1764 * q^95 - 128 * q^97 - 66 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	19.8704	1.77726	0.888632	−	0.458620i	$-0.151656\pi$
$5$	19.8704	1.77726	0.888632	+	0.458620i	$0.151656\pi$
$6$	0	0
$7$	−5.87043	−0.316973	−0.158487	−	0.987361i	$-0.550661\pi$
$7$	−5.87043	−0.316973	−0.158487	+	0.987361i	$0.550661\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	39.7409	1.25672
$11$	−18.7409	−0.513689	−0.256845	−	0.966453i	$-0.582683\pi$
$11$	−18.7409	−0.513689	−0.256845	+	0.966453i	$0.582683\pi$
$12$	0	0
$13$	45.8704	0.978628	0.489314	−	0.872108i	$-0.337247\pi$
$13$	45.8704	0.978628	0.489314	+	0.872108i	$0.337247\pi$
$14$	−11.7409	−0.224134
$15$	0	0
$16$	16.0000	0.250000
$17$	−16.8704	−0.240687	−0.120344	−	0.992732i	$-0.538400\pi$
$17$	−16.8704	−0.240687	−0.120344	+	0.992732i	$0.538400\pi$
$18$	0	0
$19$	−10.3521	−0.124997	−0.0624985	−	0.998045i	$-0.519907\pi$
$19$	−10.3521	−0.124997	−0.0624985	+	0.998045i	$0.519907\pi$
$20$	79.4817	0.888632
$21$	0	0
$22$	−37.4817	−0.363233
$23$	−49.8704	−0.452118	−0.226059	−	0.974114i	$-0.572584\pi$
$23$	−49.8704	−0.452118	−0.226059	+	0.974114i	$0.572584\pi$
$24$	0	0
$25$	269.834	2.15867
$26$	91.7409	0.691995
$27$	0	0
$28$	−23.4817	−0.158487
$29$	−10.9070	−0.0698408	−0.0349204	−	0.999390i	$-0.511118\pi$
$29$	−10.9070	−0.0698408	−0.0349204	+	0.999390i	$0.511118\pi$
$30$	0	0
$31$	151.611	0.878393	0.439197	−	0.898391i	$-0.355263\pi$
$31$	151.611	0.878393	0.439197	+	0.898391i	$0.355263\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−33.7409	−0.170191
$35$	−116.648	−0.563345
$36$	0	0
$37$	346.186	1.53818	0.769089	−	0.639141i	$-0.220710\pi$
$37$	346.186	1.53818	0.769089	+	0.639141i	$0.220710\pi$
$38$	−20.7043	−0.0883862
$39$	0	0
$40$	158.963	0.628358
$41$	−264.741	−1.00843	−0.504214	−	0.863579i	$-0.668218\pi$
$41$	−264.741	−1.00843	−0.504214	+	0.863579i	$0.668218\pi$
$42$	0	0
$43$	−411.890	−1.46076	−0.730380	−	0.683041i	$-0.760657\pi$
$43$	−411.890	−1.46076	−0.730380	+	0.683041i	$0.760657\pi$
$44$	−74.9634	−0.256845
$45$	0	0
$46$	−99.7409	−0.319695
$47$	−472.056	−1.46503	−0.732516	−	0.680750i	$-0.761654\pi$
$47$	−472.056	−1.46503	−0.732516	+	0.680750i	$0.761654\pi$
$48$	0	0
$49$	−308.538	−0.899528
$50$	539.668	1.52641
$51$	0	0
$52$	183.482	0.489314
$53$	−290.186	−0.752078	−0.376039	−	0.926604i	$-0.622714\pi$
$53$	−290.186	−0.752078	−0.376039	+	0.926604i	$0.622714\pi$
$54$	0	0
$55$	−372.389	−0.912962
$56$	−46.9634	−0.112067
$57$	0	0
$58$	−21.8140	−0.0493849
$59$	−53.2591	−0.117521	−0.0587606	−	0.998272i	$-0.518715\pi$
$59$	−53.2591	−0.117521	−0.0587606	+	0.998272i	$0.518715\pi$
$60$	0	0
$61$	−293.944	−0.616977	−0.308489	−	0.951228i	$-0.599823\pi$
$61$	−293.944	−0.616977	−0.308489	+	0.951228i	$0.599823\pi$
$62$	303.223	0.621118
$63$	0	0
$64$	64.0000	0.125000
$65$	911.465	1.73928
$66$	0	0
$67$	−398.555	−0.726735	−0.363367	−	0.931646i	$-0.618373\pi$
$67$	−398.555	−0.726735	−0.363367	+	0.931646i	$0.618373\pi$
$68$	−67.4817	−0.120344
$69$	0	0
$70$	−233.296	−0.398345
$71$	647.854	1.08290	0.541451	−	0.840732i	$-0.317875\pi$
$71$	647.854	1.08290	0.541451	+	0.840732i	$0.317875\pi$
$72$	0	0
$73$	−478.279	−0.766826	−0.383413	−	0.923577i	$-0.625251\pi$
$73$	−478.279	−0.766826	−0.383413	+	0.923577i	$0.625251\pi$
$74$	692.372	1.08766
$75$	0	0
$76$	−41.4085	−0.0624985
$77$	110.017	0.162826
$78$	0	0
$79$	374.316	0.533086	0.266543	−	0.963823i	$-0.414119\pi$
$79$	374.316	0.533086	0.266543	+	0.963823i	$0.414119\pi$
$80$	317.927	0.444316
$81$	0	0
$82$	−529.482	−0.713067
$83$	−933.279	−1.23422	−0.617112	−	0.786875i	$-0.711698\pi$
$83$	−933.279	−1.23422	−0.617112	+	0.786875i	$0.711698\pi$
$84$	0	0
$85$	−335.223	−0.427765
$86$	−823.780	−1.03291
$87$	0	0
$88$	−149.927	−0.181617
$89$	−368.817	−0.439264	−0.219632	−	0.975583i	$-0.570486\pi$
$89$	−368.817	−0.439264	−0.219632	+	0.975583i	$0.570486\pi$
$90$	0	0
$91$	−269.279	−0.310199
$92$	−199.482	−0.226059
$93$	0	0
$94$	−944.113	−1.03593
$95$	−205.701	−0.222153
$96$	0	0
$97$	274.149	0.286965	0.143483	−	0.989653i	$-0.454170\pi$
$97$	274.149	0.286965	0.143483	+	0.989653i	$0.454170\pi$
$98$	−617.076	−0.636062
$99$	0	0
$100$	1079.34	1.07934
$101$	9.57168	0.00942987	0.00471494	−	0.999989i	$-0.498499\pi$
$101$	9.57168	0.00942987	0.00471494	+	0.999989i	$0.498499\pi$
$102$	0	0
$103$	1971.50	1.88600	0.943001	−	0.332791i	$-0.107990\pi$
$103$	1971.50	1.88600	0.943001	+	0.332791i	$0.107990\pi$
$104$	366.963	0.345997
$105$	0	0
$106$	−580.372	−0.531799
$107$	−1441.13	−1.30205	−0.651025	−	0.759057i	$-0.725661\pi$
$107$	−1441.13	−1.30205	−0.651025	+	0.759057i	$0.725661\pi$
$108$	0	0
$109$	−90.3323	−0.0793786	−0.0396893	−	0.999212i	$-0.512637\pi$
$109$	−90.3323	−0.0793786	−0.0396893	+	0.999212i	$0.512637\pi$
$110$	−744.777	−0.645561
$111$	0	0
$112$	−93.9268	−0.0792433
$113$	1650.73	1.37422	0.687112	−	0.726551i	$-0.258878\pi$
$113$	1650.73	1.37422	0.687112	+	0.726551i	$0.258878\pi$
$114$	0	0
$115$	−990.947	−0.803533
$116$	−43.6281	−0.0349204
$117$	0	0
$118$	−106.518	−0.0831000
$119$	99.0366	0.0762913
$120$	0	0
$121$	−979.780	−0.736124
$122$	−587.887	−0.436269
$123$	0	0
$124$	606.445	0.439197
$125$	2877.91	2.05926
$126$	0	0
$127$	1997.45	1.39563	0.697814	−	0.716279i	$-0.254156\pi$
$127$	1997.45	1.39563	0.697814	+	0.716279i	$0.254156\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	1822.93	1.22986
$131$	123.645	0.0824649	0.0412324	−	0.999150i	$-0.486872\pi$
$131$	123.645	0.0824649	0.0412324	+	0.999150i	$0.486872\pi$
$132$	0	0
$133$	60.7714	0.0396207
$134$	−797.110	−0.513879
$135$	0	0
$136$	−134.963	−0.0850957
$137$	−113.406	−0.0707218	−0.0353609	−	0.999375i	$-0.511258\pi$
$137$	−113.406	−0.0707218	−0.0353609	+	0.999375i	$0.511258\pi$
$138$	0	0
$139$	−393.598	−0.240176	−0.120088	−	0.992763i	$-0.538318\pi$
$139$	−393.598	−0.240176	−0.120088	+	0.992763i	$0.538318\pi$
$140$	−466.591	−0.281673
$141$	0	0
$142$	1295.71	0.765728
$143$	−859.651	−0.502711
$144$	0	0
$145$	−216.727	−0.124126
$146$	−956.558	−0.542228
$147$	0	0
$148$	1384.74	0.769089
$149$	−1507.21	−0.828695	−0.414348	−	0.910119i	$-0.635990\pi$
$149$	−1507.21	−0.828695	−0.414348	+	0.910119i	$0.635990\pi$
$150$	0	0
$151$	3161.40	1.70378	0.851890	−	0.523720i	$-0.175456\pi$
$151$	3161.40	1.70378	0.851890	+	0.523720i	$0.175456\pi$
$152$	−82.8170	−0.0441931
$153$	0	0
$154$	220.034	0.115135
$155$	3012.58	1.56114
$156$	0	0
$157$	1375.61	0.699272	0.349636	−	0.936886i	$-0.386305\pi$
$157$	1375.61	0.699272	0.349636	+	0.936886i	$0.386305\pi$
$158$	748.631	0.376949
$159$	0	0
$160$	635.854	0.314179
$161$	292.761	0.143309
$162$	0	0
$163$	542.073	0.260481	0.130241	−	0.991482i	$-0.458425\pi$
$163$	542.073	0.260481	0.130241	+	0.991482i	$0.458425\pi$
$164$	−1058.96	−0.504214
$165$	0	0
$166$	−1866.56	−0.872729
$167$	−3129.62	−1.45016	−0.725081	−	0.688664i	$-0.758198\pi$
$167$	−3129.62	−1.45016	−0.725081	+	0.688664i	$0.758198\pi$
$168$	0	0
$169$	−92.9040	−0.0422868
$170$	−670.445	−0.302475
$171$	0	0
$172$	−1647.56	−0.730380
$173$	2518.28	1.10671	0.553355	−	0.832945i	$-0.313347\pi$
$173$	2518.28	1.10671	0.553355	+	0.832945i	$0.313347\pi$
$174$	0	0
$175$	−1584.04	−0.684241
$176$	−299.854	−0.128422
$177$	0	0
$178$	−737.634	−0.310607
$179$	2331.26	0.973443	0.486721	−	0.873557i	$-0.338193\pi$
$179$	2331.26	0.973443	0.486721	+	0.873557i	$0.338193\pi$
$180$	0	0
$181$	734.969	0.301822	0.150911	−	0.988547i	$-0.451779\pi$
$181$	734.969	0.301822	0.150911	+	0.988547i	$0.451779\pi$
$182$	−538.558	−0.219344
$183$	0	0
$184$	−398.963	−0.159848
$185$	6878.86	2.73375
$186$	0	0
$187$	316.166	0.123638
$188$	−1888.23	−0.732516
$189$	0	0
$190$	−411.402	−0.157086
$191$	−3617.76	−1.37053	−0.685266	−	0.728293i	$-0.740314\pi$
$191$	−3617.76	−1.37053	−0.685266	+	0.728293i	$0.740314\pi$
$192$	0	0
$193$	4341.12	1.61907	0.809536	−	0.587070i	$-0.199719\pi$
$193$	4341.12	1.61907	0.809536	+	0.587070i	$0.199719\pi$
$194$	548.299	0.202915
$195$	0	0
$196$	−1234.15	−0.449764
$197$	3286.20	1.18849	0.594243	−	0.804285i	$-0.297452\pi$
$197$	3286.20	1.18849	0.594243	+	0.804285i	$0.297452\pi$
$198$	0	0
$199$	332.265	0.118360	0.0591800	−	0.998247i	$-0.481151\pi$
$199$	332.265	0.118360	0.0591800	+	0.998247i	$0.481151\pi$
$200$	2158.67	0.763205
$201$	0	0
$202$	19.1434	0.00666793
$203$	64.0288	0.0221377
$204$	0	0
$205$	−5260.51	−1.79224
$206$	3943.01	1.33360
$207$	0	0
$208$	733.927	0.244657
$209$	194.008	0.0642096
$210$	0	0
$211$	−5742.47	−1.87359	−0.936797	−	0.349874i	$-0.886224\pi$
$211$	−5742.47	−1.87359	−0.936797	+	0.349874i	$0.886224\pi$
$212$	−1160.74	−0.376039
$213$	0	0
$214$	−2882.26	−0.920688
$215$	−8184.43	−2.59616
$216$	0	0
$217$	−890.023	−0.278427
$218$	−180.665	−0.0561291
$219$	0	0
$220$	−1489.55	−0.456481
$221$	−773.854	−0.235543
$222$	0	0
$223$	−2462.61	−0.739502	−0.369751	−	0.929131i	$-0.620557\pi$
$223$	−2462.61	−0.739502	−0.369751	+	0.929131i	$0.620557\pi$
$224$	−187.854	−0.0560335
$225$	0	0
$226$	3301.45	0.971723
$227$	−2799.34	−0.818496	−0.409248	−	0.912423i	$-0.634209\pi$
$227$	−2799.34	−0.818496	−0.409248	+	0.912423i	$0.634209\pi$
$228$	0	0
$229$	1412.85	0.407701	0.203850	−	0.979002i	$-0.434654\pi$
$229$	1412.85	0.407701	0.203850	+	0.979002i	$0.434654\pi$
$230$	−1981.89	−0.568183
$231$	0	0
$232$	−87.2561	−0.0246924
$233$	−2033.81	−0.571843	−0.285922	−	0.958253i	$-0.592300\pi$
$233$	−2033.81	−0.571843	−0.285922	+	0.958253i	$0.592300\pi$
$234$	0	0
$235$	−9379.96	−2.60375
$236$	−213.037	−0.0587606
$237$	0	0
$238$	198.073	0.0539461
$239$	521.611	0.141173	0.0705863	−	0.997506i	$-0.477513\pi$
$239$	521.611	0.141173	0.0705863	+	0.997506i	$0.477513\pi$
$240$	0	0
$241$	5915.65	1.58116	0.790582	−	0.612356i	$-0.209778\pi$
$241$	5915.65	1.58116	0.790582	+	0.612356i	$0.209778\pi$
$242$	−1959.56	−0.520518
$243$	0	0
$244$	−1175.77	−0.308489
$245$	−6130.78	−1.59870
$246$	0	0
$247$	−474.857	−0.122326
$248$	1212.89	0.310559
$249$	0	0
$250$	5755.82	1.45612
$251$	−710.127	−0.178577	−0.0892884	−	0.996006i	$-0.528459\pi$
$251$	−710.127	−0.178577	−0.0892884	+	0.996006i	$0.528459\pi$
$252$	0	0
$253$	934.614	0.232248
$254$	3994.90	0.986859
$255$	0	0
$256$	256.000	0.0625000
$257$	−6301.01	−1.52936	−0.764681	−	0.644409i	$-0.777103\pi$
$257$	−6301.01	−1.52936	−0.764681	+	0.644409i	$0.777103\pi$
$258$	0	0
$259$	−2032.26	−0.487562
$260$	3645.86	0.869641
$261$	0	0
$262$	247.290	0.0583115
$263$	4921.82	1.15396	0.576981	−	0.816757i	$-0.304231\pi$
$263$	4921.82	1.15396	0.576981	+	0.816757i	$0.304231\pi$
$264$	0	0
$265$	−5766.12	−1.33664
$266$	121.543	0.0280161
$267$	0	0
$268$	−1594.22	−0.363367
$269$	5454.50	1.23631	0.618154	−	0.786057i	$-0.287881\pi$
$269$	5454.50	1.23631	0.618154	+	0.786057i	$0.287881\pi$
$270$	0	0
$271$	2797.10	0.626981	0.313491	−	0.949591i	$-0.398502\pi$
$271$	2797.10	0.626981	0.313491	+	0.949591i	$0.398502\pi$
$272$	−269.927	−0.0601718
$273$	0	0
$274$	−226.811	−0.0500079
$275$	−5056.92	−1.10889
$276$	0	0
$277$	−2145.17	−0.465310	−0.232655	−	0.972559i	$-0.574741\pi$
$277$	−2145.17	−0.465310	−0.232655	+	0.972559i	$0.574741\pi$
$278$	−787.195	−0.169830
$279$	0	0
$280$	−933.183	−0.199173
$281$	5509.77	1.16970	0.584849	−	0.811142i	$-0.301154\pi$
$281$	5509.77	1.16970	0.584849	+	0.811142i	$0.301154\pi$
$282$	0	0
$283$	4007.84	0.841842	0.420921	−	0.907097i	$-0.361707\pi$
$283$	4007.84	0.841842	0.420921	+	0.907097i	$0.361707\pi$
$284$	2591.41	0.541451
$285$	0	0
$286$	−1719.30	−0.355470
$287$	1554.14	0.319645
$288$	0	0
$289$	−4628.39	−0.942070
$290$	−433.454	−0.0877700
$291$	0	0
$292$	−1913.12	−0.383413
$293$	5809.24	1.15829	0.579146	−	0.815224i	$-0.303386\pi$
$293$	5809.24	1.15829	0.579146	+	0.815224i	$0.303386\pi$
$294$	0	0
$295$	−1058.28	−0.208866
$296$	2769.49	0.543828
$297$	0	0
$298$	−3014.42	−0.585976
$299$	−2287.58	−0.442455
$300$	0	0
$301$	2417.97	0.463022
$302$	6322.80	1.20475
$303$	0	0
$304$	−165.634	−0.0312492
$305$	−5840.78	−1.09653
$306$	0	0
$307$	8688.30	1.61520	0.807602	−	0.589728i	$-0.200765\pi$
$307$	8688.30	1.61520	0.807602	+	0.589728i	$0.200765\pi$
$308$	440.067	0.0814128
$309$	0	0
$310$	6025.16	1.10389
$311$	−1432.01	−0.261099	−0.130550	−	0.991442i	$-0.541674\pi$
$311$	−1432.01	−0.261099	−0.130550	+	0.991442i	$0.541674\pi$
$312$	0	0
$313$	6614.52	1.19449	0.597244	−	0.802059i	$-0.296262\pi$
$313$	6614.52	1.19449	0.597244	+	0.802059i	$0.296262\pi$
$314$	2751.22	0.494460
$315$	0	0
$316$	1497.26	0.266543
$317$	−1424.09	−0.252318	−0.126159	−	0.992010i	$-0.540265\pi$
$317$	−1424.09	−0.252318	−0.126159	+	0.992010i	$0.540265\pi$
$318$	0	0
$319$	204.407	0.0358764
$320$	1271.71	0.222158
$321$	0	0
$322$	585.521	0.101335
$323$	174.645	0.0300851
$324$	0	0
$325$	12377.4	2.11254
$326$	1084.15	0.184188
$327$	0	0
$328$	−2117.93	−0.356533
$329$	2771.17	0.464376
$330$	0	0
$331$	3537.65	0.587453	0.293726	−	0.955890i	$-0.405105\pi$
$331$	3537.65	0.587453	0.293726	+	0.955890i	$0.405105\pi$
$332$	−3733.12	−0.617112
$333$	0	0
$334$	−6259.23	−1.02542
$335$	−7919.46	−1.29160
$336$	0	0
$337$	−1760.28	−0.284536	−0.142268	−	0.989828i	$-0.545439\pi$
$337$	−1760.28	−0.284536	−0.142268	+	0.989828i	$0.545439\pi$
$338$	−185.808	−0.0299013
$339$	0	0
$340$	−1340.89	−0.213882
$341$	−2841.32	−0.451221
$342$	0	0
$343$	3824.81	0.602099
$344$	−3295.12	−0.516457
$345$	0	0
$346$	5036.55	0.782563
$347$	1605.44	0.248370	0.124185	−	0.992259i	$-0.460368\pi$
$347$	1605.44	0.248370	0.124185	+	0.992259i	$0.460368\pi$
$348$	0	0
$349$	−6641.89	−1.01872	−0.509358	−	0.860555i	$-0.670117\pi$
$349$	−6641.89	−1.01872	−0.509358	+	0.860555i	$0.670117\pi$
$350$	−3168.08	−0.483831
$351$	0	0
$352$	−599.707	−0.0908083
$353$	3056.27	0.460818	0.230409	−	0.973094i	$-0.425994\pi$
$353$	3056.27	0.460818	0.230409	+	0.973094i	$0.425994\pi$
$354$	0	0
$355$	12873.1	1.92460
$356$	−1475.27	−0.219632
$357$	0	0
$358$	4662.51	0.688328
$359$	2489.46	0.365985	0.182993	−	0.983114i	$-0.441422\pi$
$359$	2489.46	0.365985	0.182993	+	0.983114i	$0.441422\pi$
$360$	0	0
$361$	−6751.83	−0.984376
$362$	1469.94	0.213421
$363$	0	0
$364$	−1077.12	−0.155099
$365$	−9503.61	−1.36285
$366$	0	0
$367$	2355.05	0.334966	0.167483	−	0.985875i	$-0.446436\pi$
$367$	2355.05	0.334966	0.167483	+	0.985875i	$0.446436\pi$
$368$	−797.927	−0.113029
$369$	0	0
$370$	13757.7	1.93305
$371$	1703.52	0.238388
$372$	0	0
$373$	−2757.03	−0.382717	−0.191358	−	0.981520i	$-0.561289\pi$
$373$	−2757.03	−0.382717	−0.191358	+	0.981520i	$0.561289\pi$
$374$	632.332	0.0874255
$375$	0	0
$376$	−3776.45	−0.517967
$377$	−500.310	−0.0683481
$378$	0	0
$379$	246.459	0.0334030	0.0167015	−	0.999861i	$-0.494683\pi$
$379$	246.459	0.0334030	0.0167015	+	0.999861i	$0.494683\pi$
$380$	−822.805	−0.111076
$381$	0	0
$382$	−7235.52	−0.969113
$383$	−650.575	−0.0867959	−0.0433979	−	0.999058i	$-0.513818\pi$
$383$	−650.575	−0.0867959	−0.0433979	+	0.999058i	$0.513818\pi$
$384$	0	0
$385$	2186.08	0.289384
$386$	8682.25	1.14486
$387$	0	0
$388$	1096.60	0.143483
$389$	10246.2	1.33548	0.667739	−	0.744395i	$-0.267262\pi$
$389$	10246.2	1.33548	0.667739	+	0.744395i	$0.267262\pi$
$390$	0	0
$391$	841.335	0.108819
$392$	−2468.30	−0.318031
$393$	0	0
$394$	6572.40	0.840387
$395$	7437.81	0.947435
$396$	0	0
$397$	−9453.68	−1.19513	−0.597565	−	0.801820i	$-0.703865\pi$
$397$	−9453.68	−1.19513	−0.597565	+	0.801820i	$0.703865\pi$
$398$	664.530	0.0836932
$399$	0	0
$400$	4317.34	0.539668
$401$	−270.723	−0.0337138	−0.0168569	−	0.999858i	$-0.505366\pi$
$401$	−270.723	−0.0337138	−0.0168569	+	0.999858i	$0.505366\pi$
$402$	0	0
$403$	6954.47	0.859620
$404$	38.2867	0.00471494
$405$	0	0
$406$	128.058	0.0156537
$407$	−6487.82	−0.790146
$408$	0	0
$409$	−11586.2	−1.40073	−0.700366	−	0.713784i	$-0.746980\pi$
$409$	−11586.2	−1.40073	−0.700366	+	0.713784i	$0.746980\pi$
$410$	−10521.0	−1.26731
$411$	0	0
$412$	7886.02	0.943001
$413$	312.654	0.0372511
$414$	0	0
$415$	−18544.7	−2.19354
$416$	1467.85	0.172999
$417$	0	0
$418$	388.016	0.0454030
$419$	16133.7	1.88111	0.940554	−	0.339644i	$-0.110307\pi$
$419$	16133.7	1.88111	0.940554	+	0.339644i	$0.110307\pi$
$420$	0	0
$421$	4991.93	0.577890	0.288945	−	0.957346i	$-0.406696\pi$
$421$	4991.93	0.577890	0.288945	+	0.957346i	$0.406696\pi$
$422$	−11484.9	−1.32483
$423$	0	0
$424$	−2321.49	−0.265900
$425$	−4552.21	−0.519564
$426$	0	0
$427$	1725.57	0.195565
$428$	−5764.52	−0.651025
$429$	0	0
$430$	−16368.9	−1.83576
$431$	−8184.74	−0.914722	−0.457361	−	0.889281i	$-0.651205\pi$
$431$	−8184.74	−0.914722	−0.457361	+	0.889281i	$0.651205\pi$
$432$	0	0
$433$	8663.17	0.961490	0.480745	−	0.876860i	$-0.340366\pi$
$433$	8663.17	0.961490	0.480745	+	0.876860i	$0.340366\pi$
$434$	−1780.05	−0.196878
$435$	0	0
$436$	−361.329	−0.0396893
$437$	516.265	0.0565133
$438$	0	0
$439$	−15864.3	−1.72474	−0.862371	−	0.506277i	$-0.831021\pi$
$439$	−15864.3	−1.72474	−0.862371	+	0.506277i	$0.831021\pi$
$440$	−2979.11	−0.322781
$441$	0	0
$442$	−1547.71	−0.166554
$443$	399.641	0.0428612	0.0214306	−	0.999770i	$-0.493178\pi$
$443$	399.641	0.0428612	0.0214306	+	0.999770i	$0.493178\pi$
$444$	0	0
$445$	−7328.55	−0.780689
$446$	−4925.23	−0.522907
$447$	0	0
$448$	−375.707	−0.0396217
$449$	−5924.51	−0.622706	−0.311353	−	0.950294i	$-0.600782\pi$
$449$	−5924.51	−0.622706	−0.311353	+	0.950294i	$0.600782\pi$
$450$	0	0
$451$	4961.47	0.518019
$452$	6602.91	0.687112
$453$	0	0
$454$	−5598.68	−0.578764
$455$	−5350.69	−0.551306
$456$	0	0
$457$	−7891.14	−0.807728	−0.403864	−	0.914819i	$-0.632333\pi$
$457$	−7891.14	−0.807728	−0.403864	+	0.914819i	$0.632333\pi$
$458$	2825.69	0.288288
$459$	0	0
$460$	−3963.79	−0.401766
$461$	3262.18	0.329577	0.164789	−	0.986329i	$-0.447306\pi$
$461$	3262.18	0.329577	0.164789	+	0.986329i	$0.447306\pi$
$462$	0	0
$463$	3690.40	0.370427	0.185213	−	0.982698i	$-0.440702\pi$
$463$	3690.40	0.370427	0.185213	+	0.982698i	$0.440702\pi$
$464$	−174.512	−0.0174602
$465$	0	0
$466$	−4067.62	−0.404354
$467$	10193.2	1.01003	0.505017	−	0.863110i	$-0.331486\pi$
$467$	10193.2	1.01003	0.505017	+	0.863110i	$0.331486\pi$
$468$	0	0
$469$	2339.69	0.230355
$470$	−18759.9	−1.84113
$471$	0	0
$472$	−426.073	−0.0415500
$473$	7719.17	0.750376
$474$	0	0
$475$	−2793.36	−0.269827
$476$	396.146	0.0381457
$477$	0	0
$478$	1043.22	0.0998240
$479$	−840.068	−0.0801330	−0.0400665	−	0.999197i	$-0.512757\pi$
$479$	−840.068	−0.0801330	−0.0400665	+	0.999197i	$0.512757\pi$
$480$	0	0
$481$	15879.7	1.50531
$482$	11831.3	1.11805
$483$	0	0
$484$	−3919.12	−0.368062
$485$	5447.46	0.510014
$486$	0	0
$487$	−3367.28	−0.313319	−0.156659	−	0.987653i	$-0.550072\pi$
$487$	−3367.28	−0.313319	−0.156659	+	0.987653i	$0.550072\pi$
$488$	−2351.55	−0.218134
$489$	0	0
$490$	−12261.6	−1.13045
$491$	18869.1	1.73432	0.867160	−	0.498030i	$-0.165943\pi$
$491$	18869.1	1.73432	0.867160	+	0.498030i	$0.165943\pi$
$492$	0	0
$493$	184.006	0.0168098
$494$	−949.713	−0.0864972
$495$	0	0
$496$	2425.78	0.219598
$497$	−3803.18	−0.343251
$498$	0	0
$499$	−14566.0	−1.30674	−0.653371	−	0.757038i	$-0.726646\pi$
$499$	−14566.0	−1.30674	−0.653371	+	0.757038i	$0.726646\pi$
$500$	11511.6	1.02963
$501$	0	0
$502$	−1420.25	−0.126273
$503$	17361.6	1.53900	0.769499	−	0.638648i	$-0.220506\pi$
$503$	17361.6	1.53900	0.769499	+	0.638648i	$0.220506\pi$
$504$	0	0
$505$	190.193	0.0167594
$506$	1869.23	0.164224
$507$	0	0
$508$	7989.79	0.697814
$509$	−13933.3	−1.21332	−0.606661	−	0.794961i	$-0.707491\pi$
$509$	−13933.3	−1.21332	−0.606661	+	0.794961i	$0.707491\pi$
$510$	0	0
$511$	2807.70	0.243063
$512$	512.000	0.0441942
$513$	0	0
$514$	−12602.0	−1.08142
$515$	39174.6	3.35192
$516$	0	0
$517$	8846.74	0.752571
$518$	−4064.52	−0.344758
$519$	0	0
$520$	7291.72	0.614929
$521$	−5024.22	−0.422486	−0.211243	−	0.977434i	$-0.567751\pi$
$521$	−5024.22	−0.422486	−0.211243	+	0.977434i	$0.567751\pi$
$522$	0	0
$523$	16008.7	1.33845	0.669226	−	0.743059i	$-0.266626\pi$
$523$	16008.7	1.33845	0.669226	+	0.743059i	$0.266626\pi$
$524$	494.579	0.0412324
$525$	0	0
$526$	9843.63	0.815975
$527$	−2557.75	−0.211418
$528$	0	0
$529$	−9679.94	−0.795590
$530$	−11532.2	−0.945148
$531$	0	0
$532$	243.086	0.0198103
$533$	−12143.8	−0.986877
$534$	0	0
$535$	−28635.9	−2.31409
$536$	−3188.44	−0.256940
$537$	0	0
$538$	10909.0	0.874202
$539$	5782.27	0.462078
$540$	0	0
$541$	−10094.1	−0.802179	−0.401089	−	0.916039i	$-0.631368\pi$
$541$	−10094.1	−0.802179	−0.401089	+	0.916039i	$0.631368\pi$
$542$	5594.21	0.443343
$543$	0	0
$544$	−539.854	−0.0425479
$545$	−1794.94	−0.141077
$546$	0	0
$547$	−6060.54	−0.473730	−0.236865	−	0.971543i	$-0.576120\pi$
$547$	−6060.54	−0.473730	−0.236865	+	0.971543i	$0.576120\pi$
$548$	−453.622	−0.0353609
$549$	0	0
$550$	−10113.8	−0.784100
$551$	112.911	0.00872988
$552$	0	0
$553$	−2197.39	−0.168974
$554$	−4290.34	−0.329024
$555$	0	0
$556$	−1574.39	−0.120088
$557$	−13688.4	−1.04128	−0.520642	−	0.853775i	$-0.674307\pi$
$557$	−13688.4	−1.04128	−0.520642	+	0.853775i	$0.674307\pi$
$558$	0	0
$559$	−18893.6	−1.42954
$560$	−1866.37	−0.140836
$561$	0	0
$562$	11019.5	0.827102
$563$	−22346.6	−1.67282	−0.836411	−	0.548103i	$-0.815350\pi$
$563$	−22346.6	−1.67282	−0.836411	+	0.548103i	$0.815350\pi$
$564$	0	0
$565$	32800.7	2.44236
$566$	8015.67	0.595272
$567$	0	0
$568$	5182.83	0.382864
$569$	−8117.94	−0.598105	−0.299052	−	0.954237i	$-0.596671\pi$
$569$	−8117.94	−0.598105	−0.299052	+	0.954237i	$0.596671\pi$
$570$	0	0
$571$	6982.97	0.511783	0.255892	−	0.966705i	$-0.417631\pi$
$571$	6982.97	0.511783	0.255892	+	0.966705i	$0.417631\pi$
$572$	−3438.60	−0.251355
$573$	0	0
$574$	3108.28	0.226023
$575$	−13456.7	−0.975973
$576$	0	0
$577$	13972.4	1.00811	0.504055	−	0.863671i	$-0.331841\pi$
$577$	13972.4	1.00811	0.504055	+	0.863671i	$0.331841\pi$
$578$	−9256.78	−0.666144
$579$	0	0
$580$	−866.908	−0.0620628
$581$	5478.75	0.391216
$582$	0	0
$583$	5438.33	0.386334
$584$	−3826.23	−0.271114
$585$	0	0
$586$	11618.5	0.819036
$587$	1509.51	0.106140	0.0530701	−	0.998591i	$-0.483099\pi$
$587$	1509.51	0.106140	0.0530701	+	0.998591i	$0.483099\pi$
$588$	0	0
$589$	−1569.50	−0.109796
$590$	−2116.56	−0.147691
$591$	0	0
$592$	5538.98	0.384545
$593$	21495.3	1.48854	0.744270	−	0.667879i	$-0.232798\pi$
$593$	21495.3	1.48854	0.744270	+	0.667879i	$0.232798\pi$
$594$	0	0
$595$	1967.90	0.135590
$596$	−6028.85	−0.414348
$597$	0	0
$598$	−4575.16	−0.312863
$599$	−10104.9	−0.689272	−0.344636	−	0.938736i	$-0.611998\pi$
$599$	−10104.9	−0.689272	−0.344636	+	0.938736i	$0.611998\pi$
$600$	0	0
$601$	−11008.2	−0.747143	−0.373572	−	0.927601i	$-0.621867\pi$
$601$	−11008.2	−0.747143	−0.373572	+	0.927601i	$0.621867\pi$
$602$	4835.94	0.327406
$603$	0	0
$604$	12645.6	0.851890
$605$	−19468.7	−1.30829
$606$	0	0
$607$	14680.2	0.981629	0.490815	−	0.871264i	$-0.336699\pi$
$607$	14680.2	0.981629	0.490815	+	0.871264i	$0.336699\pi$
$608$	−331.268	−0.0220965
$609$	0	0
$610$	−11681.6	−0.775365
$611$	−21653.4	−1.43372
$612$	0	0
$613$	235.863	0.0155407	0.00777033	−	0.999970i	$-0.497527\pi$
$613$	235.863	0.0155407	0.00777033	+	0.999970i	$0.497527\pi$
$614$	17376.6	1.14212
$615$	0	0
$616$	880.134	0.0575676
$617$	−18313.1	−1.19490	−0.597452	−	0.801904i	$-0.703820\pi$
$617$	−18313.1	−1.19490	−0.597452	+	0.801904i	$0.703820\pi$
$618$	0	0
$619$	6849.46	0.444755	0.222377	−	0.974961i	$-0.428618\pi$
$619$	6849.46	0.444755	0.222377	+	0.974961i	$0.428618\pi$
$620$	12050.3	0.780569
$621$	0	0
$622$	−2864.02	−0.184625
$623$	2165.11	0.139235
$624$	0	0
$625$	23456.1	1.50119
$626$	13229.0	0.844631
$627$	0	0
$628$	5502.45	0.349636
$629$	−5840.30	−0.370220
$630$	0	0
$631$	22464.3	1.41726	0.708629	−	0.705581i	$-0.249314\pi$
$631$	22464.3	1.41726	0.708629	+	0.705581i	$0.249314\pi$
$632$	2994.52	0.188474
$633$	0	0
$634$	−2848.18	−0.178416
$635$	39690.1	2.48040
$636$	0	0
$637$	−14152.8	−0.880303
$638$	408.814	0.0253685
$639$	0	0
$640$	2543.41	0.157090
$641$	−18888.8	−1.16390	−0.581952	−	0.813223i	$-0.697711\pi$
$641$	−18888.8	−1.16390	−0.581952	+	0.813223i	$0.697711\pi$
$642$	0	0
$643$	6793.95	0.416683	0.208342	−	0.978056i	$-0.433193\pi$
$643$	6793.95	0.416683	0.208342	+	0.978056i	$0.433193\pi$
$644$	1171.04	0.0716546
$645$	0	0
$646$	349.290	0.0212734
$647$	−5277.92	−0.320706	−0.160353	−	0.987060i	$-0.551263\pi$
$647$	−5277.92	−0.320706	−0.160353	+	0.987060i	$0.551263\pi$
$648$	0	0
$649$	998.122	0.0603694
$650$	24754.8	1.49379
$651$	0	0
$652$	2168.29	0.130241
$653$	16564.2	0.992659	0.496330	−	0.868134i	$-0.334681\pi$
$653$	16564.2	0.992659	0.496330	+	0.868134i	$0.334681\pi$
$654$	0	0
$655$	2456.88	0.146562
$656$	−4235.85	−0.252107
$657$	0	0
$658$	5542.34	0.328363
$659$	3301.02	0.195128	0.0975641	−	0.995229i	$-0.468895\pi$
$659$	3301.02	0.195128	0.0975641	+	0.995229i	$0.468895\pi$
$660$	0	0
$661$	12990.3	0.764395	0.382198	−	0.924081i	$-0.375167\pi$
$661$	12990.3	0.764395	0.382198	+	0.924081i	$0.375167\pi$
$662$	7075.30	0.415392
$663$	0	0
$664$	−7466.23	−0.436364
$665$	1207.55	0.0704164
$666$	0	0
$667$	543.938	0.0315762
$668$	−12518.5	−0.725081
$669$	0	0
$670$	−15838.9	−0.913299
$671$	5508.75	0.316935
$672$	0	0
$673$	18514.3	1.06044	0.530219	−	0.847861i	$-0.322110\pi$
$673$	18514.3	1.06044	0.530219	+	0.847861i	$0.322110\pi$
$674$	−3520.56	−0.201197
$675$	0	0
$676$	−371.616	−0.0211434
$677$	6099.99	0.346295	0.173148	−	0.984896i	$-0.444606\pi$
$677$	6099.99	0.346295	0.173148	+	0.984896i	$0.444606\pi$
$678$	0	0
$679$	−1609.37	−0.0909604
$680$	−2681.78	−0.151238
$681$	0	0
$682$	−5682.65	−0.319061
$683$	22630.0	1.26781	0.633905	−	0.773411i	$-0.281451\pi$
$683$	22630.0	1.26781	0.633905	+	0.773411i	$0.281451\pi$
$684$	0	0
$685$	−2253.42	−0.125691
$686$	7649.61	0.425749
$687$	0	0
$688$	−6590.24	−0.365190
$689$	−13311.0	−0.736004
$690$	0	0
$691$	22986.8	1.26550	0.632750	−	0.774356i	$-0.281926\pi$
$691$	22986.8	1.26550	0.632750	+	0.774356i	$0.281926\pi$
$692$	10073.1	0.553355
$693$	0	0
$694$	3210.88	0.175624
$695$	−7820.95	−0.426857
$696$	0	0
$697$	4466.29	0.242716
$698$	−13283.8	−0.720342
$699$	0	0
$700$	−6336.16	−0.342120
$701$	−27015.5	−1.45558	−0.727790	−	0.685800i	$-0.759452\pi$
$701$	−27015.5	−1.45558	−0.727790	+	0.685800i	$0.759452\pi$
$702$	0	0
$703$	−3583.76	−0.192268
$704$	−1199.41	−0.0642111
$705$	0	0
$706$	6112.54	0.325848
$707$	−56.1898	−0.00298902
$708$	0	0
$709$	−19177.3	−1.01582	−0.507912	−	0.861409i	$-0.669583\pi$
$709$	−19177.3	−1.01582	−0.507912	+	0.861409i	$0.669583\pi$
$710$	25746.3	1.36090
$711$	0	0
$712$	−2950.54	−0.155303
$713$	−7560.92	−0.397137
$714$	0	0
$715$	−17081.6	−0.893450
$716$	9325.02	0.486721
$717$	0	0
$718$	4978.92	0.258791
$719$	−25931.8	−1.34505	−0.672526	−	0.740073i	$-0.734791\pi$
$719$	−25931.8	−1.34505	−0.672526	+	0.740073i	$0.734791\pi$
$720$	0	0
$721$	−11573.6	−0.597812
$722$	−13503.7	−0.696059
$723$	0	0
$724$	2939.88	0.150911
$725$	−2943.08	−0.150763
$726$	0	0
$727$	5830.50	0.297443	0.148722	−	0.988879i	$-0.452484\pi$
$727$	5830.50	0.297443	0.148722	+	0.988879i	$0.452484\pi$
$728$	−2154.23	−0.109672
$729$	0	0
$730$	−19007.2	−0.963683
$731$	6948.76	0.351586
$732$	0	0
$733$	−23154.6	−1.16676	−0.583379	−	0.812200i	$-0.698270\pi$
$733$	−23154.6	−1.16676	−0.583379	+	0.812200i	$0.698270\pi$
$734$	4710.09	0.236857
$735$	0	0
$736$	−1595.85	−0.0799238
$737$	7469.26	0.373316
$738$	0	0
$739$	−27085.8	−1.34827	−0.674133	−	0.738610i	$-0.735483\pi$
$739$	−27085.8	−1.34827	−0.674133	+	0.738610i	$0.735483\pi$
$740$	27515.5	1.36688
$741$	0	0
$742$	3407.03	0.168566
$743$	−32187.6	−1.58930	−0.794650	−	0.607068i	$-0.792345\pi$
$743$	−32187.6	−1.58930	−0.794650	+	0.607068i	$0.792345\pi$
$744$	0	0
$745$	−29948.9	−1.47281
$746$	−5514.05	−0.270622
$747$	0	0
$748$	1264.66	0.0618191
$749$	8460.04	0.412715
$750$	0	0
$751$	−2732.93	−0.132791	−0.0663954	−	0.997793i	$-0.521150\pi$
$751$	−2732.93	−0.132791	−0.0663954	+	0.997793i	$0.521150\pi$
$752$	−7552.90	−0.366258
$753$	0	0
$754$	−1000.62	−0.0483294
$755$	62818.3	3.02807
$756$	0	0
$757$	−6315.62	−0.303230	−0.151615	−	0.988440i	$-0.548447\pi$
$757$	−6315.62	−0.303230	−0.151615	+	0.988440i	$0.548447\pi$
$758$	492.918	0.0236195
$759$	0	0
$760$	−1645.61	−0.0785428
$761$	−31481.3	−1.49960	−0.749801	−	0.661663i	$-0.769851\pi$
$761$	−31481.3	−1.49960	−0.749801	+	0.661663i	$0.769851\pi$
$762$	0	0
$763$	530.289	0.0251609
$764$	−14471.0	−0.685266
$765$	0	0
$766$	−1301.15	−0.0613739
$767$	−2443.02	−0.115010
$768$	0	0
$769$	−27296.6	−1.28003	−0.640014	−	0.768363i	$-0.721071\pi$
$769$	−27296.6	−1.28003	−0.640014	+	0.768363i	$0.721071\pi$
$770$	4372.16	0.204626
$771$	0	0
$772$	17364.5	0.809536
$773$	−24025.2	−1.11789	−0.558943	−	0.829206i	$-0.688793\pi$
$773$	−24025.2	−1.11789	−0.558943	+	0.829206i	$0.688793\pi$
$774$	0	0
$775$	40909.9	1.89616
$776$	2193.20	0.101458
$777$	0	0
$778$	20492.3	0.944326
$779$	2740.63	0.126050
$780$	0	0
$781$	−12141.3	−0.556275
$782$	1682.67	0.0769465
$783$	0	0
$784$	−4936.61	−0.224882
$785$	27334.0	1.24279
$786$	0	0
$787$	−10558.9	−0.478251	−0.239125	−	0.970989i	$-0.576861\pi$
$787$	−10558.9	−0.478251	−0.239125	+	0.970989i	$0.576861\pi$
$788$	13144.8	0.594243
$789$	0	0
$790$	14875.6	0.669938
$791$	−9690.47	−0.435592
$792$	0	0
$793$	−13483.3	−0.603792
$794$	−18907.4	−0.845084
$795$	0	0
$796$	1329.06	0.0591800
$797$	26889.1	1.19506	0.597529	−	0.801847i	$-0.296149\pi$
$797$	26889.1	1.19506	0.597529	+	0.801847i	$0.296149\pi$
$798$	0	0
$799$	7963.79	0.352614
$800$	8634.68	0.381603
$801$	0	0
$802$	−541.446	−0.0238393
$803$	8963.36	0.393910
$804$	0	0
$805$	5817.28	0.254698
$806$	13908.9	0.607843
$807$	0	0
$808$	76.5734	0.00333396
$809$	18643.8	0.810236	0.405118	−	0.914264i	$-0.367230\pi$
$809$	18643.8	0.810236	0.405118	+	0.914264i	$0.367230\pi$
$810$	0	0
$811$	−23870.8	−1.03356	−0.516781	−	0.856118i	$-0.672870\pi$
$811$	−23870.8	−1.03356	−0.516781	+	0.856118i	$0.672870\pi$
$812$	256.115	0.0110688
$813$	0	0
$814$	−12975.6	−0.558717
$815$	10771.2	0.462944
$816$	0	0
$817$	4263.94	0.182590
$818$	−23172.3	−0.990467
$819$	0	0
$820$	−21042.1	−0.896122
$821$	−14533.0	−0.617791	−0.308895	−	0.951096i	$-0.599959\pi$
$821$	−14533.0	−0.617791	−0.308895	+	0.951096i	$0.599959\pi$
$822$	0	0
$823$	8868.42	0.375618	0.187809	−	0.982206i	$-0.439861\pi$
$823$	8868.42	0.375618	0.187809	+	0.982206i	$0.439861\pi$
$824$	15772.0	0.666802
$825$	0	0
$826$	625.308	0.0263405
$827$	−25059.3	−1.05369	−0.526843	−	0.849963i	$-0.676624\pi$
$827$	−25059.3	−1.05369	−0.526843	+	0.849963i	$0.676624\pi$
$828$	0	0
$829$	−22556.3	−0.945009	−0.472505	−	0.881328i	$-0.656650\pi$
$829$	−22556.3	−0.945009	−0.472505	+	0.881328i	$0.656650\pi$
$830$	−37089.3	−1.55107
$831$	0	0
$832$	2935.71	0.122329
$833$	5205.17	0.216505
$834$	0	0
$835$	−62186.8	−2.57732
$836$	776.031	0.0321048
$837$	0	0
$838$	32267.5	1.33014
$839$	−21209.6	−0.872748	−0.436374	−	0.899765i	$-0.643738\pi$
$839$	−21209.6	−0.872748	−0.436374	+	0.899765i	$0.643738\pi$
$840$	0	0
$841$	−24270.0	−0.995122
$842$	9983.86	0.408630
$843$	0	0
$844$	−22969.9	−0.936797
$845$	−1846.04	−0.0751548
$846$	0	0
$847$	5751.73	0.233331
$848$	−4642.98	−0.188019
$849$	0	0
$850$	−9104.42	−0.367387
$851$	−17264.4	−0.695438
$852$	0	0
$853$	28116.2	1.12858	0.564292	−	0.825576i	$-0.309149\pi$
$853$	28116.2	1.12858	0.564292	+	0.825576i	$0.309149\pi$
$854$	3451.15	0.138286
$855$	0	0
$856$	−11529.0	−0.460344
$857$	9477.64	0.377771	0.188886	−	0.981999i	$-0.439512\pi$
$857$	9477.64	0.377771	0.188886	+	0.981999i	$0.439512\pi$
$858$	0	0
$859$	29545.8	1.17356	0.586780	−	0.809746i	$-0.300395\pi$
$859$	29545.8	1.17356	0.586780	+	0.809746i	$0.300395\pi$
$860$	−32737.7	−1.29808
$861$	0	0
$862$	−16369.5	−0.646806
$863$	−9932.53	−0.391781	−0.195891	−	0.980626i	$-0.562760\pi$
$863$	−9932.53	−0.391781	−0.195891	+	0.980626i	$0.562760\pi$
$864$	0	0
$865$	50039.2	1.96692
$866$	17326.3	0.679876
$867$	0	0
$868$	−3560.09	−0.139214
$869$	−7014.99	−0.273840
$870$	0	0
$871$	−18281.9	−0.711203
$872$	−722.659	−0.0280646
$873$	0	0
$874$	1032.53	0.0399609
$875$	−16894.6	−0.652732
$876$	0	0
$877$	−27025.5	−1.04058	−0.520288	−	0.853991i	$-0.674175\pi$
$877$	−27025.5	−1.04058	−0.520288	+	0.853991i	$0.674175\pi$
$878$	−31728.6	−1.21958
$879$	0	0
$880$	−5958.22	−0.228240
$881$	12798.1	0.489422	0.244711	−	0.969596i	$-0.421307\pi$
$881$	12798.1	0.489422	0.244711	+	0.969596i	$0.421307\pi$
$882$	0	0
$883$	27016.7	1.02965	0.514826	−	0.857294i	$-0.327856\pi$
$883$	27016.7	1.02965	0.514826	+	0.857294i	$0.327856\pi$
$884$	−3095.41	−0.117772
$885$	0	0
$886$	799.281	0.0303074
$887$	20150.1	0.762768	0.381384	−	0.924417i	$-0.375448\pi$
$887$	20150.1	0.762768	0.381384	+	0.924417i	$0.375448\pi$
$888$	0	0
$889$	−11725.9	−0.442377
$890$	−14657.1	−0.552031
$891$	0	0
$892$	−9850.46	−0.369751
$893$	4886.79	0.183124
$894$	0	0
$895$	46323.1	1.73007
$896$	−751.415	−0.0280167
$897$	0	0
$898$	−11849.0	−0.440320
$899$	−1653.63	−0.0613477
$900$	0	0
$901$	4895.56	0.181015
$902$	9922.94	0.366295
$903$	0	0
$904$	13205.8	0.485862
$905$	14604.2	0.536418
$906$	0	0
$907$	9279.91	0.339729	0.169865	−	0.985467i	$-0.445667\pi$
$907$	9279.91	0.339729	0.169865	+	0.985467i	$0.445667\pi$
$908$	−11197.4	−0.409248
$909$	0	0
$910$	−10701.4	−0.389832
$911$	−16400.6	−0.596461	−0.298231	−	0.954494i	$-0.596396\pi$
$911$	−16400.6	−0.596461	−0.298231	+	0.954494i	$0.596396\pi$
$912$	0	0
$913$	17490.4	0.634008
$914$	−15782.3	−0.571150
$915$	0	0
$916$	5651.38	0.203850
$917$	−725.848	−0.0261392
$918$	0	0
$919$	−11704.6	−0.420128	−0.210064	−	0.977688i	$-0.567367\pi$
$919$	−11704.6	−0.420128	−0.210064	+	0.977688i	$0.567367\pi$
$920$	−7927.57	−0.284092
$921$	0	0
$922$	6524.37	0.233046
$923$	29717.3	1.05976
$924$	0	0
$925$	93412.7	3.32042
$926$	7380.81	0.261931
$927$	0	0
$928$	−349.025	−0.0123462
$929$	−16324.4	−0.576520	−0.288260	−	0.957552i	$-0.593077\pi$
$929$	−16324.4	−0.576520	−0.288260	+	0.957552i	$0.593077\pi$
$930$	0	0
$931$	3194.03	0.112438
$932$	−8135.25	−0.285922
$933$	0	0
$934$	20386.4	0.714201
$935$	6282.36	0.219738
$936$	0	0
$937$	−7397.79	−0.257925	−0.128962	−	0.991649i	$-0.541165\pi$
$937$	−7397.79	−0.257925	−0.128962	+	0.991649i	$0.541165\pi$
$938$	4679.37	0.162886
$939$	0	0
$940$	−37519.8	−1.30187
$941$	−7426.88	−0.257289	−0.128645	−	0.991691i	$-0.541063\pi$
$941$	−7426.88	−0.257289	−0.128645	+	0.991691i	$0.541063\pi$
$942$	0	0
$943$	13202.7	0.455928
$944$	−852.146	−0.0293803
$945$	0	0
$946$	15438.3	0.530596
$947$	23051.9	0.791011	0.395505	−	0.918464i	$-0.370570\pi$
$947$	23051.9	0.791011	0.395505	+	0.918464i	$0.370570\pi$
$948$	0	0
$949$	−21938.9	−0.750438
$950$	−5586.71	−0.190797
$951$	0	0
$952$	792.293	0.0269731
$953$	23445.7	0.796936	0.398468	−	0.917182i	$-0.369542\pi$
$953$	23445.7	0.796936	0.398468	+	0.917182i	$0.369542\pi$
$954$	0	0
$955$	−71886.4	−2.43580
$956$	2086.45	0.0705863
$957$	0	0
$958$	−1680.14	−0.0566626
$959$	665.739	0.0224169
$960$	0	0
$961$	−6805.02	−0.228425
$962$	31759.4	1.06441
$963$	0	0
$964$	23662.6	0.790582
$965$	86260.0	2.87752
$966$	0	0
$967$	−16741.9	−0.556756	−0.278378	−	0.960472i	$-0.589797\pi$
$967$	−16741.9	−0.556756	−0.278378	+	0.960472i	$0.589797\pi$
$968$	−7838.24	−0.260259
$969$	0	0
$970$	10894.9	0.360634
$971$	43037.7	1.42240	0.711198	−	0.702992i	$-0.248153\pi$
$971$	43037.7	1.42240	0.711198	+	0.702992i	$0.248153\pi$
$972$	0	0
$973$	2310.59	0.0761295
$974$	−6734.57	−0.221550
$975$	0	0
$976$	−4703.10	−0.154244
$977$	17899.8	0.586147	0.293074	−	0.956090i	$-0.405322\pi$
$977$	17899.8	0.586147	0.293074	+	0.956090i	$0.405322\pi$
$978$	0	0
$979$	6911.95	0.225645
$980$	−24523.1	−0.799350
$981$	0	0
$982$	37738.2	1.22635
$983$	46300.3	1.50229	0.751144	−	0.660138i	$-0.229502\pi$
$983$	46300.3	1.50229	0.751144	+	0.660138i	$0.229502\pi$
$984$	0	0
$985$	65298.2	2.11226
$986$	368.012	0.0118863
$987$	0	0
$988$	−1899.43	−0.0611628
$989$	20541.1	0.660435
$990$	0	0
$991$	4621.74	0.148148	0.0740739	−	0.997253i	$-0.476400\pi$
$991$	4621.74	0.148148	0.0740739	+	0.997253i	$0.476400\pi$
$992$	4851.56	0.155279
$993$	0	0
$994$	−7606.35	−0.242715
$995$	6602.25	0.210357
$996$	0	0
$997$	19944.9	0.633561	0.316780	−	0.948499i	$-0.397398\pi$
$997$	19944.9	0.633561	0.316780	+	0.948499i	$0.397398\pi$
$998$	−29132.0	−0.924006
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.a.g.1.2		2
3.2	odd	2		162.4.a.f.1.1		2
4.3	odd	2		1296.4.a.r.1.2		2
9.2	odd	6		18.4.c.b.13.1	yes	4
9.4	even	3		54.4.c.b.19.1		4
9.5	odd	6		18.4.c.b.7.1	✓	4
9.7	even	3		54.4.c.b.37.1		4
12.11	even	2		1296.4.a.l.1.1		2
36.7	odd	6		432.4.i.b.145.1		4
36.11	even	6		144.4.i.b.49.2		4
36.23	even	6		144.4.i.b.97.2		4
36.31	odd	6		432.4.i.b.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.b.7.1	✓	4	9.5	odd	6
18.4.c.b.13.1	yes	4	9.2	odd	6
54.4.c.b.19.1		4	9.4	even	3
54.4.c.b.37.1		4	9.7	even	3
144.4.i.b.49.2		4	36.11	even	6
144.4.i.b.97.2		4	36.23	even	6
162.4.a.f.1.1		2	3.2	odd	2
162.4.a.g.1.2		2	1.1	even	1	trivial
432.4.i.b.145.1		4	36.7	odd	6
432.4.i.b.289.1		4	36.31	odd	6
1296.4.a.l.1.1		2	12.11	even	2
1296.4.a.r.1.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +19.8704 q^{5} -5.87043 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +19.8704 q^{5} -5.87043 q^{7} +8.00000 q^{8} +39.7409 q^{10} -18.7409 q^{11} +45.8704 q^{13} -11.7409 q^{14} +16.0000 q^{16} -16.8704 q^{17} -10.3521 q^{19} +79.4817 q^{20} -37.4817 q^{22} -49.8704 q^{23} +269.834 q^{25} +91.7409 q^{26} -23.4817 q^{28} -10.9070 q^{29} +151.611 q^{31} +32.0000 q^{32} -33.7409 q^{34} -116.648 q^{35} +346.186 q^{37} -20.7043 q^{38} +158.963 q^{40} -264.741 q^{41} -411.890 q^{43} -74.9634 q^{44} -99.7409 q^{46} -472.056 q^{47} -308.538 q^{49} +539.668 q^{50} +183.482 q^{52} -290.186 q^{53} -372.389 q^{55} -46.9634 q^{56} -21.8140 q^{58} -53.2591 q^{59} -293.944 q^{61} +303.223 q^{62} +64.0000 q^{64} +911.465 q^{65} -398.555 q^{67} -67.4817 q^{68} -233.296 q^{70} +647.854 q^{71} -478.279 q^{73} +692.372 q^{74} -41.4085 q^{76} +110.017 q^{77} +374.316 q^{79} +317.927 q^{80} -529.482 q^{82} -933.279 q^{83} -335.223 q^{85} -823.780 q^{86} -149.927 q^{88} -368.817 q^{89} -269.279 q^{91} -199.482 q^{92} -944.113 q^{94} -205.701 q^{95} +274.149 q^{97} -617.076 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 9 * q^5 + 19 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8} + 18 q^{10} + 24 q^{11} + 61 q^{13} + 38 q^{14} + 32 q^{16} - 3 q^{17} + 133 q^{19} + 36 q^{20} + 48 q^{22} - 69 q^{23} + 263 q^{25} + 122 q^{26} + 76 q^{28} - 237 q^{29} + 211 q^{31} + 64 q^{32} - 6 q^{34} - 387 q^{35} + 262 q^{37} + 266 q^{38} + 72 q^{40} - 468 q^{41} - 86 q^{43} + 96 q^{44} - 138 q^{46} - 483 q^{47} - 33 q^{49} + 526 q^{50} + 244 q^{52} - 150 q^{53} - 837 q^{55} + 152 q^{56} - 474 q^{58} - 168 q^{59} - 1049 q^{61} + 422 q^{62} + 128 q^{64} + 747 q^{65} - 1166 q^{67} - 12 q^{68} - 774 q^{70} + 312 q^{71} - 311 q^{73} + 524 q^{74} + 532 q^{76} + 1173 q^{77} + 349 q^{79} + 144 q^{80} - 936 q^{82} - 1221 q^{83} - 486 q^{85} - 172 q^{86} + 192 q^{88} + 492 q^{89} + 107 q^{91} - 276 q^{92} - 966 q^{94} - 1764 q^{95} - 128 q^{97} - 66 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 9 * q^5 + 19 * q^7 + 16 * q^8 + 18 * q^10 + 24 * q^11 + 61 * q^13 + 38 * q^14 + 32 * q^16 - 3 * q^17 + 133 * q^19 + 36 * q^20 + 48 * q^22 - 69 * q^23 + 263 * q^25 + 122 * q^26 + 76 * q^28 - 237 * q^29 + 211 * q^31 + 64 * q^32 - 6 * q^34 - 387 * q^35 + 262 * q^37 + 266 * q^38 + 72 * q^40 - 468 * q^41 - 86 * q^43 + 96 * q^44 - 138 * q^46 - 483 * q^47 - 33 * q^49 + 526 * q^50 + 244 * q^52 - 150 * q^53 - 837 * q^55 + 152 * q^56 - 474 * q^58 - 168 * q^59 - 1049 * q^61 + 422 * q^62 + 128 * q^64 + 747 * q^65 - 1166 * q^67 - 12 * q^68 - 774 * q^70 + 312 * q^71 - 311 * q^73 + 524 * q^74 + 532 * q^76 + 1173 * q^77 + 349 * q^79 + 144 * q^80 - 936 * q^82 - 1221 * q^83 - 486 * q^85 - 172 * q^86 + 192 * q^88 + 492 * q^89 + 107 * q^91 - 276 * q^92 - 966 * q^94 - 1764 * q^95 - 128 * q^97 - 66 * q^98

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