LMFDB - Embedded newform 4176.2.a.bx.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4176,2,Mod(1,4176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4176, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4176 = 2^{4} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4176.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:	$33.3455278841$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	4176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.508203 q^{5} -3.68133 q^{7} +O(q^{10})$	$q-0.508203 q^{5} -3.68133 q^{7} -0.318669 q^{11} +4.18953 q^{13} -3.17313 q^{17} +5.87086 q^{19} +2.50820 q^{23} -4.74173 q^{25} -1.00000 q^{29} -2.50820 q^{31} +1.87086 q^{35} +7.87086 q^{37} -8.72532 q^{41} +10.7253 q^{43} -11.0440 q^{47} +6.55220 q^{49} +8.24993 q^{53} +0.161949 q^{55} -11.3627 q^{59} -3.87086 q^{61} -2.12914 q^{65} -7.04399 q^{67} +6.24993 q^{71} -7.87086 q^{73} +1.17313 q^{77} +4.85446 q^{79} -8.37907 q^{83} +1.61259 q^{85} +15.9313 q^{89} -15.4231 q^{91} -2.98359 q^{95} +11.2335 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{7}+O(q^{10}) $ 3 * q - 4 * q^7	$ 3 q - 4 q^{7} - 8 q^{11} + 4 q^{13} - 4 q^{17} + 2 q^{19} + 6 q^{23} + 17 q^{25} - 3 q^{29} - 6 q^{31} - 10 q^{35} + 8 q^{37} + 2 q^{41} + 4 q^{43} - 12 q^{47} - 3 q^{49} - 8 q^{53} + 10 q^{55} - 20 q^{59} + 4 q^{61} - 22 q^{65} - 14 q^{71} - 8 q^{73} - 2 q^{77} + 2 q^{79} - 8 q^{83} - 42 q^{85} + 8 q^{89} - 8 q^{91} - 12 q^{95} + 4 q^{97}+O(q^{100}) $ 3 * q - 4 * q^7 - 8 * q^11 + 4 * q^13 - 4 * q^17 + 2 * q^19 + 6 * q^23 + 17 * q^25 - 3 * q^29 - 6 * q^31 - 10 * q^35 + 8 * q^37 + 2 * q^41 + 4 * q^43 - 12 * q^47 - 3 * q^49 - 8 * q^53 + 10 * q^55 - 20 * q^59 + 4 * q^61 - 22 * q^65 - 14 * q^71 - 8 * q^73 - 2 * q^77 + 2 * q^79 - 8 * q^83 - 42 * q^85 + 8 * q^89 - 8 * q^91 - 12 * q^95 + 4 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.508203	−0.227275	−0.113638	−	0.993522i	$-0.536250\pi$
$5$	−0.508203	−0.227275	−0.113638	+	0.993522i	$0.536250\pi$
$6$	0	0
$7$	−3.68133	−1.39141	−0.695706	−	0.718327i	$-0.744908\pi$
$7$	−3.68133	−1.39141	−0.695706	+	0.718327i	$0.744908\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.318669	−0.0960824	−0.0480412	−	0.998845i	$-0.515298\pi$
$11$	−0.318669	−0.0960824	−0.0480412	+	0.998845i	$0.515298\pi$
$12$	0	0
$13$	4.18953	1.16197	0.580984	−	0.813915i	$-0.302668\pi$
$13$	4.18953	1.16197	0.580984	+	0.813915i	$0.302668\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.17313	−0.769596	−0.384798	−	0.923001i	$-0.625729\pi$
$17$	−3.17313	−0.769596	−0.384798	+	0.923001i	$0.625729\pi$
$18$	0	0
$19$	5.87086	1.34687	0.673434	−	0.739247i	$-0.264818\pi$
$19$	5.87086	1.34687	0.673434	+	0.739247i	$0.264818\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.50820	0.522997	0.261498	−	0.965204i	$-0.415783\pi$
$23$	2.50820	0.522997	0.261498	+	0.965204i	$0.415783\pi$
$24$	0	0
$25$	−4.74173	−0.948346
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.50820	−0.450487	−0.225243	−	0.974303i	$-0.572318\pi$
$31$	−2.50820	−0.450487	−0.225243	+	0.974303i	$0.572318\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.87086	0.316234
$36$	0	0
$37$	7.87086	1.29396	0.646981	−	0.762506i	$-0.276031\pi$
$37$	7.87086	1.29396	0.646981	+	0.762506i	$0.276031\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.72532	−1.36267	−0.681333	−	0.731973i	$-0.738600\pi$
$41$	−8.72532	−1.36267	−0.681333	+	0.731973i	$0.738600\pi$
$42$	0	0
$43$	10.7253	1.63560	0.817798	−	0.575505i	$-0.195195\pi$
$43$	10.7253	1.63560	0.817798	+	0.575505i	$0.195195\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.0440	−1.61093	−0.805466	−	0.592642i	$-0.798085\pi$
$47$	−11.0440	−1.61093	−0.805466	+	0.592642i	$0.798085\pi$
$48$	0	0
$49$	6.55220	0.936028
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.24993	1.13322	0.566608	−	0.823988i	$-0.308256\pi$
$53$	8.24993	1.13322	0.566608	+	0.823988i	$0.308256\pi$
$54$	0	0
$55$	0.161949	0.0218372
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.3627	−1.47929	−0.739646	−	0.672996i	$-0.765007\pi$
$59$	−11.3627	−1.47929	−0.739646	+	0.672996i	$0.765007\pi$
$60$	0	0
$61$	−3.87086	−0.495613	−0.247807	−	0.968809i	$-0.579710\pi$
$61$	−3.87086	−0.495613	−0.247807	+	0.968809i	$0.579710\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.12914	−0.264087
$66$	0	0
$67$	−7.04399	−0.860561	−0.430280	−	0.902695i	$-0.641585\pi$
$67$	−7.04399	−0.860561	−0.430280	+	0.902695i	$0.641585\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.24993	0.741731	0.370865	−	0.928687i	$-0.379061\pi$
$71$	6.24993	0.741731	0.370865	+	0.928687i	$0.379061\pi$
$72$	0	0
$73$	−7.87086	−0.921215	−0.460608	−	0.887604i	$-0.652368\pi$
$73$	−7.87086	−0.921215	−0.460608	+	0.887604i	$0.652368\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.17313	0.133690
$78$	0	0
$79$	4.85446	0.546169	0.273085	−	0.961990i	$-0.411956\pi$
$79$	4.85446	0.546169	0.273085	+	0.961990i	$0.411956\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.37907	−0.919722	−0.459861	−	0.887991i	$-0.652101\pi$
$83$	−8.37907	−0.919722	−0.459861	+	0.887991i	$0.652101\pi$
$84$	0	0
$85$	1.61259	0.174910
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.9313	1.68871	0.844355	−	0.535784i	$-0.179984\pi$
$89$	15.9313	1.68871	0.844355	+	0.535784i	$0.179984\pi$
$90$	0	0
$91$	−15.4231	−1.61678
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.98359	−0.306110
$96$	0	0
$97$	11.2335	1.14059	0.570296	−	0.821439i	$-0.306829\pi$
$97$	11.2335	1.14059	0.570296	+	0.821439i	$0.306829\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.9149	−1.08607	−0.543034	−	0.839710i	$-0.682725\pi$
$101$	−10.9149	−1.08607	−0.543034	+	0.839710i	$0.682725\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.83805	0.371038	0.185519	−	0.982641i	$-0.440603\pi$
$107$	3.83805	0.371038	0.185519	+	0.982641i	$0.440603\pi$
$108$	0	0
$109$	−8.18953	−0.784415	−0.392208	−	0.919877i	$-0.628288\pi$
$109$	−8.18953	−0.784415	−0.392208	+	0.919877i	$0.628288\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.2059	−1.24231	−0.621155	−	0.783688i	$-0.713336\pi$
$113$	−13.2059	−1.24231	−0.621155	+	0.783688i	$0.713336\pi$
$114$	0	0
$115$	−1.27468	−0.118864
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.6813	1.07083
$120$	0	0
$121$	−10.8984	−0.990768
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.95078	0.442811
$126$	0	0
$127$	−9.39547	−0.833714	−0.416857	−	0.908972i	$-0.636868\pi$
$127$	−9.39547	−0.833714	−0.416857	+	0.908972i	$0.636868\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.0768	−1.49201	−0.746004	−	0.665942i	$-0.768030\pi$
$131$	−17.0768	−1.49201	−0.746004	+	0.665942i	$0.768030\pi$
$132$	0	0
$133$	−21.6126	−1.87405
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.0164	−1.28294	−0.641469	−	0.767149i	$-0.721675\pi$
$137$	−15.0164	−1.28294	−0.641469	+	0.767149i	$0.721675\pi$
$138$	0	0
$139$	−7.68133	−0.651522	−0.325761	−	0.945452i	$-0.605620\pi$
$139$	−7.68133	−0.651522	−0.325761	+	0.945452i	$0.605620\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.33508	−0.111645
$144$	0	0
$145$	0.508203	0.0422040
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.3955	0.933554	0.466777	−	0.884375i	$-0.345415\pi$
$149$	11.3955	0.933554	0.466777	+	0.884375i	$0.345415\pi$
$150$	0	0
$151$	2.03281	0.165428	0.0827140	−	0.996573i	$-0.473641\pi$
$151$	2.03281	0.165428	0.0827140	+	0.996573i	$0.473641\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.27468	0.102385
$156$	0	0
$157$	−10.7581	−0.858593	−0.429296	−	0.903164i	$-0.641238\pi$
$157$	−10.7581	−0.858593	−0.429296	+	0.903164i	$0.641238\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.23353	−0.727704
$162$	0	0
$163$	−21.8297	−1.70984	−0.854918	−	0.518764i	$-0.826392\pi$
$163$	−21.8297	−1.70984	−0.854918	+	0.518764i	$0.826392\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.4999	−1.58633	−0.793164	−	0.609009i	$-0.791567\pi$
$167$	−20.4999	−1.58633	−0.793164	+	0.609009i	$0.791567\pi$
$168$	0	0
$169$	4.55220	0.350169
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.62093	−0.123237	−0.0616186	−	0.998100i	$-0.519626\pi$
$173$	−1.62093	−0.123237	−0.0616186	+	0.998100i	$0.519626\pi$
$174$	0	0
$175$	17.4559	1.31954
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.47539	0.334506	0.167253	−	0.985914i	$-0.446510\pi$
$179$	4.47539	0.334506	0.167253	+	0.985914i	$0.446510\pi$
$180$	0	0
$181$	1.20594	0.0896369	0.0448184	−	0.998995i	$-0.485729\pi$
$181$	1.20594	0.0896369	0.0448184	+	0.998995i	$0.485729\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	1.01118	0.0739447
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.758136	0.0548568	0.0274284	−	0.999624i	$-0.491268\pi$
$191$	0.758136	0.0548568	0.0274284	+	0.999624i	$0.491268\pi$
$192$	0	0
$193$	−16.4671	−1.18532	−0.592662	−	0.805451i	$-0.701923\pi$
$193$	−16.4671	−1.18532	−0.592662	+	0.805451i	$0.701923\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.9588	1.27951	0.639757	−	0.768577i	$-0.279035\pi$
$197$	17.9588	1.27951	0.639757	+	0.768577i	$0.279035\pi$
$198$	0	0
$199$	−1.33508	−0.0946410	−0.0473205	−	0.998880i	$-0.515068\pi$
$199$	−1.33508	−0.0946410	−0.0473205	+	0.998880i	$0.515068\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.68133	0.258379
$204$	0	0
$205$	4.43424	0.309701
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.87086	−0.129410
$210$	0	0
$211$	−8.37907	−0.576839	−0.288419	−	0.957504i	$-0.593130\pi$
$211$	−8.37907	−0.576839	−0.288419	+	0.957504i	$0.593130\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.45065	−0.371731
$216$	0	0
$217$	9.23353	0.626813
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−13.2939	−0.894246
$222$	0	0
$223$	6.66492	0.446316	0.223158	−	0.974782i	$-0.428363\pi$
$223$	6.66492	0.446316	0.223158	+	0.974782i	$0.428363\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.379068	0.0251596	0.0125798	−	0.999921i	$-0.495996\pi$
$227$	0.379068	0.0251596	0.0125798	+	0.999921i	$0.495996\pi$
$228$	0	0
$229$	9.26634	0.612337	0.306168	−	0.951977i	$-0.400953\pi$
$229$	9.26634	0.612337	0.306168	+	0.951977i	$0.400953\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.7417	−0.900251	−0.450125	−	0.892965i	$-0.648621\pi$
$233$	−13.7417	−0.900251	−0.450125	+	0.892965i	$0.648621\pi$
$234$	0	0
$235$	5.61259	0.366125
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.8709	1.15597	0.577985	−	0.816047i	$-0.303839\pi$
$239$	17.8709	1.15597	0.577985	+	0.816047i	$0.303839\pi$
$240$	0	0
$241$	−14.2223	−0.916142	−0.458071	−	0.888916i	$-0.651459\pi$
$241$	−14.2223	−0.916142	−0.458071	+	0.888916i	$0.651459\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.32985	−0.212736
$246$	0	0
$247$	24.5962	1.56502
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.43947	0.532694	0.266347	−	0.963877i	$-0.414183\pi$
$251$	8.43947	0.532694	0.266347	+	0.963877i	$0.414183\pi$
$252$	0	0
$253$	−0.799288	−0.0502508
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.3627	−1.58208	−0.791040	−	0.611765i	$-0.790460\pi$
$257$	−25.3627	−1.58208	−0.791040	+	0.611765i	$0.790460\pi$
$258$	0	0
$259$	−28.9753	−1.80043
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.4835	1.44805	0.724026	−	0.689773i	$-0.242290\pi$
$263$	23.4835	1.44805	0.724026	+	0.689773i	$0.242290\pi$
$264$	0	0
$265$	−4.19264	−0.257552
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.10155	0.128134	0.0640669	−	0.997946i	$-0.479593\pi$
$269$	2.10155	0.128134	0.0640669	+	0.997946i	$0.479593\pi$
$270$	0	0
$271$	−3.10439	−0.188578	−0.0942892	−	0.995545i	$-0.530058\pi$
$271$	−3.10439	−0.188578	−0.0942892	+	0.995545i	$0.530058\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.51104	0.0911194
$276$	0	0
$277$	−13.9641	−0.839020	−0.419510	−	0.907751i	$-0.637798\pi$
$277$	−13.9641	−0.839020	−0.419510	+	0.907751i	$0.637798\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.4342	0.861074	0.430537	−	0.902573i	$-0.358324\pi$
$281$	14.4342	0.861074	0.430537	+	0.902573i	$0.358324\pi$
$282$	0	0
$283$	−8.25827	−0.490903	−0.245452	−	0.969409i	$-0.578936\pi$
$283$	−8.25827	−0.490903	−0.245452	+	0.969409i	$0.578936\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	32.1208	1.89603
$288$	0	0
$289$	−6.93126	−0.407721
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.4478	0.727209	0.363604	−	0.931554i	$-0.381546\pi$
$293$	12.4478	0.727209	0.363604	+	0.931554i	$0.381546\pi$
$294$	0	0
$295$	5.77454	0.336207
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.5082	0.607705
$300$	0	0
$301$	−39.4835	−2.27579
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.96719	0.112641
$306$	0	0
$307$	21.9917	1.25513	0.627565	−	0.778564i	$-0.284052\pi$
$307$	21.9917	1.25513	0.627565	+	0.778564i	$0.284052\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.0440	−1.07989	−0.539943	−	0.841702i	$-0.681554\pi$
$311$	−19.0440	−1.07989	−0.539943	+	0.841702i	$0.681554\pi$
$312$	0	0
$313$	9.89845	0.559493	0.279747	−	0.960074i	$-0.409750\pi$
$313$	9.89845	0.559493	0.279747	+	0.960074i	$0.409750\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.568602	−0.0319359	−0.0159679	−	0.999873i	$-0.505083\pi$
$317$	−0.568602	−0.0319359	−0.0159679	+	0.999873i	$0.505083\pi$
$318$	0	0
$319$	0.318669	0.0178421
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−18.6290	−1.03655
$324$	0	0
$325$	−19.8656	−1.10195
$326$	0	0
$327$	0	0
$328$	0	0
$329$	40.6566	2.24147
$330$	0	0
$331$	−1.17836	−0.0647683	−0.0323841	−	0.999475i	$-0.510310\pi$
$331$	−1.17836	−0.0647683	−0.0323841	+	0.999475i	$0.510310\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.57978	0.195584
$336$	0	0
$337$	17.2007	0.936983	0.468491	−	0.883468i	$-0.344798\pi$
$337$	17.2007	0.936983	0.468491	+	0.883468i	$0.344798\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.799288	0.0432838
$342$	0	0
$343$	1.64852	0.0890116
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.2663	0.819540	0.409770	−	0.912189i	$-0.365609\pi$
$347$	15.2663	0.819540	0.409770	+	0.912189i	$0.365609\pi$
$348$	0	0
$349$	−27.7089	−1.48322	−0.741612	−	0.670829i	$-0.765938\pi$
$349$	−27.7089	−1.48322	−0.741612	+	0.670829i	$0.765938\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.79929	−0.148991	−0.0744955	−	0.997221i	$-0.523735\pi$
$353$	−2.79929	−0.148991	−0.0744955	+	0.997221i	$0.523735\pi$
$354$	0	0
$355$	−3.17624	−0.168577
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.2583	0.646967	0.323483	−	0.946234i	$-0.395146\pi$
$359$	12.2583	0.646967	0.323483	+	0.946234i	$0.395146\pi$
$360$	0	0
$361$	15.4671	0.814055
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	4.25827	0.222280	0.111140	−	0.993805i	$-0.464550\pi$
$367$	4.25827	0.222280	0.111140	+	0.993805i	$0.464550\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−30.3707	−1.57677
$372$	0	0
$373$	25.7417	1.33286	0.666428	−	0.745569i	$-0.267822\pi$
$373$	25.7417	1.33286	0.666428	+	0.745569i	$0.267822\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.18953	−0.215772
$378$	0	0
$379$	−19.6454	−1.00912	−0.504558	−	0.863378i	$-0.668345\pi$
$379$	−19.6454	−1.00912	−0.504558	+	0.863378i	$0.668345\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.5634	−1.15293	−0.576467	−	0.817120i	$-0.695569\pi$
$383$	−22.5634	−1.15293	−0.576467	+	0.817120i	$0.695569\pi$
$384$	0	0
$385$	−0.596187	−0.0303845
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−15.8105	−0.801622	−0.400811	−	0.916161i	$-0.631272\pi$
$389$	−15.8105	−0.801622	−0.400811	+	0.916161i	$0.631272\pi$
$390$	0	0
$391$	−7.95885	−0.402496
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.46705	−0.124131
$396$	0	0
$397$	−3.27468	−0.164351	−0.0821757	−	0.996618i	$-0.526187\pi$
$397$	−3.27468	−0.164351	−0.0821757	+	0.996618i	$0.526187\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.47539	−0.123615	−0.0618075	−	0.998088i	$-0.519686\pi$
$401$	−2.47539	−0.123615	−0.0618075	+	0.998088i	$0.519686\pi$
$402$	0	0
$403$	−10.5082	−0.523451
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.50820	−0.124327
$408$	0	0
$409$	15.0164	0.742514	0.371257	−	0.928530i	$-0.378927\pi$
$409$	15.0164	0.742514	0.371257	+	0.928530i	$0.378927\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	41.8297	2.05831
$414$	0	0
$415$	4.25827	0.209030
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13.8709	−0.677636	−0.338818	−	0.940852i	$-0.610027\pi$
$419$	−13.8709	−0.677636	−0.338818	+	0.940852i	$0.610027\pi$
$420$	0	0
$421$	−7.45065	−0.363122	−0.181561	−	0.983380i	$-0.558115\pi$
$421$	−7.45065	−0.363122	−0.181561	+	0.983380i	$0.558115\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.0461	0.729844
$426$	0	0
$427$	14.2499	0.689603
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.7253	0.709294	0.354647	−	0.935000i	$-0.384601\pi$
$431$	14.7253	0.709294	0.354647	+	0.935000i	$0.384601\pi$
$432$	0	0
$433$	1.52461	0.0732681	0.0366340	−	0.999329i	$-0.488336\pi$
$433$	1.52461	0.0732681	0.0366340	+	0.999329i	$0.488336\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.7253	0.704408
$438$	0	0
$439$	−23.3023	−1.11216	−0.556078	−	0.831130i	$-0.687694\pi$
$439$	−23.3023	−1.11216	−0.556078	+	0.831130i	$0.687694\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.0440	1.09485	0.547427	−	0.836854i	$-0.315608\pi$
$443$	23.0440	1.09485	0.547427	+	0.836854i	$0.315608\pi$
$444$	0	0
$445$	−8.09632	−0.383802
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.53579	0.308443	0.154221	−	0.988036i	$-0.450713\pi$
$449$	6.53579	0.308443	0.154221	+	0.988036i	$0.450713\pi$
$450$	0	0
$451$	2.78049	0.130928
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.83805	0.367454
$456$	0	0
$457$	10.2223	0.478181	0.239091	−	0.970997i	$-0.423151\pi$
$457$	10.2223	0.478181	0.239091	+	0.970997i	$0.423151\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	24.1812	1.12380	0.561898	−	0.827207i	$-0.310071\pi$
$463$	24.1812	1.12380	0.561898	+	0.827207i	$0.310071\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.96719	−0.461226	−0.230613	−	0.973046i	$-0.574073\pi$
$467$	−9.96719	−0.461226	−0.230613	+	0.973046i	$0.574073\pi$
$468$	0	0
$469$	25.9313	1.19739
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.41783	−0.157152
$474$	0	0
$475$	−27.8381	−1.27730
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−21.7745	−0.994904	−0.497452	−	0.867491i	$-0.665731\pi$
$479$	−21.7745	−0.994904	−0.497452	+	0.867491i	$0.665731\pi$
$480$	0	0
$481$	32.9753	1.50354
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.70892	−0.259229
$486$	0	0
$487$	−37.4506	−1.69705	−0.848525	−	0.529155i	$-0.822509\pi$
$487$	−37.4506	−1.69705	−0.848525	+	0.529155i	$0.822509\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.6925	−0.753322	−0.376661	−	0.926351i	$-0.622928\pi$
$491$	−16.6925	−0.753322	−0.376661	+	0.926351i	$0.622928\pi$
$492$	0	0
$493$	3.17313	0.142910
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−23.0081	−1.03205
$498$	0	0
$499$	−43.8573	−1.96332	−0.981661	−	0.190634i	$-0.938946\pi$
$499$	−43.8573	−1.96332	−0.981661	+	0.190634i	$0.938946\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.4067	1.17741	0.588707	−	0.808346i	$-0.299637\pi$
$503$	26.4067	1.17741	0.588707	+	0.808346i	$0.299637\pi$
$504$	0	0
$505$	5.54697	0.246837
$506$	0	0
$507$	0	0
$508$	0	0
$509$	36.1432	1.60202	0.801009	−	0.598653i	$-0.204297\pi$
$509$	36.1432	1.60202	0.801009	+	0.598653i	$0.204297\pi$
$510$	0	0
$511$	28.9753	1.28179
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.06563	0.179153
$516$	0	0
$517$	3.51938	0.154782
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.1208	−1.14437	−0.572186	−	0.820124i	$-0.693905\pi$
$521$	−26.1208	−1.14437	−0.572186	+	0.820124i	$0.693905\pi$
$522$	0	0
$523$	2.35148	0.102823	0.0514116	−	0.998678i	$-0.483628\pi$
$523$	2.35148	0.102823	0.0514116	+	0.998678i	$0.483628\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.95885	0.346693
$528$	0	0
$529$	−16.7089	−0.726475
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−36.5550	−1.58337
$534$	0	0
$535$	−1.95051	−0.0843279
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.08798	−0.0899358
$540$	0	0
$541$	12.2499	0.526666	0.263333	−	0.964705i	$-0.415178\pi$
$541$	12.2499	0.526666	0.263333	+	0.964705i	$0.415178\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.16195	0.178278
$546$	0	0
$547$	11.3023	0.483250	0.241625	−	0.970370i	$-0.422320\pi$
$547$	11.3023	0.483250	0.241625	+	0.970370i	$0.422320\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.87086	−0.250107
$552$	0	0
$553$	−17.8709	−0.759946
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.4671	−1.37567	−0.687837	−	0.725866i	$-0.741439\pi$
$557$	−32.4671	−1.37567	−0.687837	+	0.725866i	$0.741439\pi$
$558$	0	0
$559$	44.9341	1.90051
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.5439	0.486516	0.243258	−	0.969962i	$-0.421784\pi$
$563$	11.5439	0.486516	0.243258	+	0.969962i	$0.421784\pi$
$564$	0	0
$565$	6.71130	0.282347
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−34.5358	−1.44782	−0.723908	−	0.689897i	$-0.757656\pi$
$569$	−34.5358	−1.44782	−0.723908	+	0.689897i	$0.757656\pi$
$570$	0	0
$571$	−21.7089	−0.908490	−0.454245	−	0.890877i	$-0.650091\pi$
$571$	−21.7089	−0.908490	−0.454245	+	0.890877i	$0.650091\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.8932	−0.495982
$576$	0	0
$577$	−3.13720	−0.130604	−0.0653018	−	0.997866i	$-0.520801\pi$
$577$	−3.13720	−0.130604	−0.0653018	+	0.997866i	$0.520801\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	30.8461	1.27971
$582$	0	0
$583$	−2.62900	−0.108882
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.5386	1.30174	0.650869	−	0.759190i	$-0.274405\pi$
$587$	31.5386	1.30174	0.650869	+	0.759190i	$0.274405\pi$
$588$	0	0
$589$	−14.7253	−0.606746
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26.9753	1.10774	0.553870	−	0.832603i	$-0.313150\pi$
$593$	26.9753	1.10774	0.553870	+	0.832603i	$0.313150\pi$
$594$	0	0
$595$	−5.93649	−0.243372
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.2364	−0.826836	−0.413418	−	0.910541i	$-0.635665\pi$
$599$	−20.2364	−0.826836	−0.413418	+	0.910541i	$0.635665\pi$
$600$	0	0
$601$	−42.7826	−1.74514	−0.872570	−	0.488490i	$-0.837548\pi$
$601$	−42.7826	−1.74514	−0.872570	+	0.488490i	$0.837548\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.53863	0.225177
$606$	0	0
$607$	−3.42829	−0.139150	−0.0695750	−	0.997577i	$-0.522164\pi$
$607$	−3.42829	−0.139150	−0.0695750	+	0.997577i	$0.522164\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−46.2692	−1.87185
$612$	0	0
$613$	−21.3267	−0.861379	−0.430689	−	0.902500i	$-0.641730\pi$
$613$	−21.3267	−0.861379	−0.430689	+	0.902500i	$0.641730\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.8074	1.20000	0.599999	−	0.800000i	$-0.295167\pi$
$617$	29.8074	1.20000	0.599999	+	0.800000i	$0.295167\pi$
$618$	0	0
$619$	−5.87086	−0.235970	−0.117985	−	0.993015i	$-0.537643\pi$
$619$	−5.87086	−0.235970	−0.117985	+	0.993015i	$0.537643\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−58.6482	−2.34969
$624$	0	0
$625$	21.1926	0.847706
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24.9753	−0.995829
$630$	0	0
$631$	−20.7529	−0.826160	−0.413080	−	0.910695i	$-0.635547\pi$
$631$	−20.7529	−0.826160	−0.413080	+	0.910695i	$0.635547\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.77481	0.189483
$636$	0	0
$637$	27.4506	1.08763
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.9149	0.747092	0.373546	−	0.927612i	$-0.378142\pi$
$641$	18.9149	0.747092	0.373546	+	0.927612i	$0.378142\pi$
$642$	0	0
$643$	47.1648	1.86000	0.929999	−	0.367562i	$-0.119808\pi$
$643$	47.1648	1.86000	0.929999	+	0.367562i	$0.119808\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.20071	0.283089	0.141545	−	0.989932i	$-0.454793\pi$
$647$	7.20071	0.283089	0.141545	+	0.989932i	$0.454793\pi$
$648$	0	0
$649$	3.62093	0.142134
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−42.5191	−1.66390	−0.831951	−	0.554850i	$-0.812776\pi$
$653$	−42.5191	−1.66390	−0.831951	+	0.554850i	$0.812776\pi$
$654$	0	0
$655$	8.67849	0.339097
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.90679	−0.0742779	−0.0371390	−	0.999310i	$-0.511824\pi$
$659$	−1.90679	−0.0742779	−0.0371390	+	0.999310i	$0.511824\pi$
$660$	0	0
$661$	35.9969	1.40012	0.700058	−	0.714086i	$-0.253157\pi$
$661$	35.9969	1.40012	0.700058	+	0.714086i	$0.253157\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.9836	0.425925
$666$	0	0
$667$	−2.50820	−0.0971180
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.23353	0.0476197
$672$	0	0
$673$	9.46421	0.364819	0.182409	−	0.983223i	$-0.441610\pi$
$673$	9.46421	0.364819	0.182409	+	0.983223i	$0.441610\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.81047	0.146448	0.0732241	−	0.997316i	$-0.476671\pi$
$677$	3.81047	0.146448	0.0732241	+	0.997316i	$0.476671\pi$
$678$	0	0
$679$	−41.3543	−1.58703
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.9341	1.26019	0.630094	−	0.776519i	$-0.283016\pi$
$683$	32.9341	1.26019	0.630094	+	0.776519i	$0.283016\pi$
$684$	0	0
$685$	7.63139	0.291580
$686$	0	0
$687$	0	0
$688$	0	0
$689$	34.5634	1.31676
$690$	0	0
$691$	−0.752908	−0.0286420	−0.0143210	−	0.999897i	$-0.504559\pi$
$691$	−0.752908	−0.0286420	−0.0143210	+	0.999897i	$0.504559\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.90368	0.148075
$696$	0	0
$697$	27.6866	1.04870
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.3543	1.03316	0.516579	−	0.856239i	$-0.327205\pi$
$701$	27.3543	1.03316	0.516579	+	0.856239i	$0.327205\pi$
$702$	0	0
$703$	46.2088	1.74280
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40.1812	1.51117
$708$	0	0
$709$	50.1760	1.88440	0.942199	−	0.335054i	$-0.108754\pi$
$709$	50.1760	1.88440	0.942199	+	0.335054i	$0.108754\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.29108	−0.235603
$714$	0	0
$715$	0.678490	0.0253741
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.4014	1.13378	0.566891	−	0.823793i	$-0.308146\pi$
$719$	30.4014	1.13378	0.566891	+	0.823793i	$0.308146\pi$
$720$	0	0
$721$	29.4506	1.09680
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.74173	0.176103
$726$	0	0
$727$	46.1676	1.71226	0.856131	−	0.516758i	$-0.172861\pi$
$727$	46.1676	1.71226	0.856131	+	0.516758i	$0.172861\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−34.0328	−1.25875
$732$	0	0
$733$	−34.1208	−1.26028	−0.630140	−	0.776481i	$-0.717003\pi$
$733$	−34.1208	−1.26028	−0.630140	+	0.776481i	$0.717003\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.24470	0.0826847
$738$	0	0
$739$	40.7826	1.50021	0.750106	−	0.661317i	$-0.230002\pi$
$739$	40.7826	1.50021	0.750106	+	0.661317i	$0.230002\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.3515	−0.526505	−0.263252	−	0.964727i	$-0.584795\pi$
$743$	−14.3515	−0.526505	−0.263252	+	0.964727i	$0.584795\pi$
$744$	0	0
$745$	−5.79122	−0.212174
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−14.1291	−0.516267
$750$	0	0
$751$	21.4506	0.782745	0.391373	−	0.920232i	$-0.372000\pi$
$751$	21.4506	0.782745	0.391373	+	0.920232i	$0.372000\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.03308	−0.0375977
$756$	0	0
$757$	0.862796	0.0313588	0.0156794	−	0.999877i	$-0.495009\pi$
$757$	0.862796	0.0313588	0.0156794	+	0.999877i	$0.495009\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25.4423	−0.922283	−0.461141	−	0.887327i	$-0.652560\pi$
$761$	−25.4423	−0.922283	−0.461141	+	0.887327i	$0.652560\pi$
$762$	0	0
$763$	30.1484	1.09144
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−47.6043	−1.71889
$768$	0	0
$769$	−0.194762	−0.00702331	−0.00351165	−	0.999994i	$-0.501118\pi$
$769$	−0.194762	−0.00702331	−0.00351165	+	0.999994i	$0.501118\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−44.5327	−1.60173	−0.800865	−	0.598846i	$-0.795626\pi$
$773$	−44.5327	−1.60173	−0.800865	+	0.598846i	$0.795626\pi$
$774$	0	0
$775$	11.8932	0.427217
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−51.2252	−1.83533
$780$	0	0
$781$	−1.99166	−0.0712673
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.46732	0.195137
$786$	0	0
$787$	24.4999	0.873326	0.436663	−	0.899625i	$-0.356160\pi$
$787$	24.4999	0.873326	0.436663	+	0.899625i	$0.356160\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	48.6154	1.72857
$792$	0	0
$793$	−16.2171	−0.575887
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.7253	−0.450754	−0.225377	−	0.974272i	$-0.572361\pi$
$797$	−12.7253	−0.450754	−0.225377	+	0.974272i	$0.572361\pi$
$798$	0	0
$799$	35.0440	1.23977
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.50820	0.0885126
$804$	0	0
$805$	4.69251	0.165389
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−13.6402	−0.479563	−0.239782	−	0.970827i	$-0.577076\pi$
$809$	−13.6402	−0.479563	−0.239782	+	0.970827i	$0.577076\pi$
$810$	0	0
$811$	−15.9948	−0.561652	−0.280826	−	0.959759i	$-0.590608\pi$
$811$	−15.9948	−0.561652	−0.280826	+	0.959759i	$0.590608\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.0939	0.388604
$816$	0	0
$817$	62.9669	2.20293
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.70892	0.129442	0.0647210	−	0.997903i	$-0.479384\pi$
$821$	3.70892	0.129442	0.0647210	+	0.997903i	$0.479384\pi$
$822$	0	0
$823$	48.8685	1.70345	0.851724	−	0.523991i	$-0.175557\pi$
$823$	48.8685	1.70345	0.851724	+	0.523991i	$0.175557\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.20905	0.0420428	0.0210214	−	0.999779i	$-0.493308\pi$
$827$	1.20905	0.0420428	0.0210214	+	0.999779i	$0.493308\pi$
$828$	0	0
$829$	−4.67015	−0.162201	−0.0811005	−	0.996706i	$-0.525843\pi$
$829$	−4.67015	−0.162201	−0.0811005	+	0.996706i	$0.525843\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20.7909	−0.720364
$834$	0	0
$835$	10.4181	0.360533
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.8513	−0.374630	−0.187315	−	0.982300i	$-0.559979\pi$
$839$	−10.8513	−0.374630	−0.187315	+	0.982300i	$0.559979\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.31344	−0.0795848
$846$	0	0
$847$	40.1208	1.37857
$848$	0	0
$849$	0	0
$850$	0	0
$851$	19.7417	0.676738
$852$	0	0
$853$	44.3296	1.51782	0.758908	−	0.651198i	$-0.225733\pi$
$853$	44.3296	1.51782	0.758908	+	0.651198i	$0.225733\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.23353	−0.110455	−0.0552276	−	0.998474i	$-0.517588\pi$
$857$	−3.23353	−0.110455	−0.0552276	+	0.998474i	$0.517588\pi$
$858$	0	0
$859$	−38.3463	−1.30836	−0.654179	−	0.756340i	$-0.726986\pi$
$859$	−38.3463	−1.30836	−0.654179	+	0.756340i	$0.726986\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.4751	1.41183	0.705915	−	0.708297i	$-0.250536\pi$
$863$	41.4751	1.41183	0.705915	+	0.708297i	$0.250536\pi$
$864$	0	0
$865$	0.823763	0.0280088
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.54697	−0.0524773
$870$	0	0
$871$	−29.5110	−0.999944
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−18.2255	−0.616133
$876$	0	0
$877$	13.0492	0.440641	0.220320	−	0.975428i	$-0.429290\pi$
$877$	13.0492	0.440641	0.220320	+	0.975428i	$0.429290\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39.6074	1.33441	0.667203	−	0.744876i	$-0.267491\pi$
$881$	39.6074	1.33441	0.667203	+	0.744876i	$0.267491\pi$
$882$	0	0
$883$	−29.9505	−1.00791	−0.503957	−	0.863728i	$-0.668123\pi$
$883$	−29.9505	−1.00791	−0.503957	+	0.863728i	$0.668123\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.5439	1.59637	0.798183	−	0.602415i	$-0.205795\pi$
$887$	47.5439	1.59637	0.798183	+	0.602415i	$0.205795\pi$
$888$	0	0
$889$	34.5878	1.16004
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−64.8378	−2.16971
$894$	0	0
$895$	−2.27441	−0.0760251
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.50820	0.0836533
$900$	0	0
$901$	−26.1781	−0.872119
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.612863	−0.0203723
$906$	0	0
$907$	−14.8628	−0.493511	−0.246756	−	0.969078i	$-0.579364\pi$
$907$	−14.8628	−0.493511	−0.246756	+	0.969078i	$0.579364\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−28.5603	−0.946244	−0.473122	−	0.880997i	$-0.656873\pi$
$911$	−28.5603	−0.946244	−0.473122	+	0.880997i	$0.656873\pi$
$912$	0	0
$913$	2.67015	0.0883691
$914$	0	0
$915$	0	0
$916$	0	0
$917$	62.8654	2.07600
$918$	0	0
$919$	9.32462	0.307591	0.153795	−	0.988103i	$-0.450850\pi$
$919$	9.32462	0.307591	0.153795	+	0.988103i	$0.450850\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	26.1843	0.861867
$924$	0	0
$925$	−37.3215	−1.22712
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.29108	0.140786	0.0703930	−	0.997519i	$-0.477575\pi$
$929$	4.29108	0.140786	0.0703930	+	0.997519i	$0.477575\pi$
$930$	0	0
$931$	38.4671	1.26071
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.513884	−0.0168058
$936$	0	0
$937$	−4.94767	−0.161633	−0.0808167	−	0.996729i	$-0.525753\pi$
$937$	−4.94767	−0.161633	−0.0808167	+	0.996729i	$0.525753\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17.7969	0.580162	0.290081	−	0.957002i	$-0.406318\pi$
$941$	17.7969	0.580162	0.290081	+	0.957002i	$0.406318\pi$
$942$	0	0
$943$	−21.8849	−0.712670
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−53.7693	−1.74727	−0.873634	−	0.486584i	$-0.838243\pi$
$947$	−53.7693	−1.74727	−0.873634	+	0.486584i	$0.838243\pi$
$948$	0	0
$949$	−32.9753	−1.07042
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−50.5550	−1.63764	−0.818819	−	0.574052i	$-0.805371\pi$
$953$	−50.5550	−1.63764	−0.818819	+	0.574052i	$0.805371\pi$
$954$	0	0
$955$	−0.385287	−0.0124676
$956$	0	0
$957$	0	0
$958$	0	0
$959$	55.2804	1.78510
$960$	0	0
$961$	−24.7089	−0.797062
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.36861	0.269395
$966$	0	0
$967$	13.1784	0.423787	0.211894	−	0.977293i	$-0.432037\pi$
$967$	13.1784	0.423787	0.211894	+	0.977293i	$0.432037\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16.3686	−0.525294	−0.262647	−	0.964892i	$-0.584595\pi$
$971$	−16.3686	−0.525294	−0.262647	+	0.964892i	$0.584595\pi$
$972$	0	0
$973$	28.2775	0.906536
$974$	0	0
$975$	0	0
$976$	0	0
$977$	59.7229	1.91071	0.955353	−	0.295467i	$-0.0954752\pi$
$977$	59.7229	1.91071	0.955353	+	0.295467i	$0.0954752\pi$
$978$	0	0
$979$	−5.07681	−0.162255
$980$	0	0
$981$	0	0
$982$	0	0
$983$	55.4835	1.76965	0.884824	−	0.465926i	$-0.154279\pi$
$983$	55.4835	1.76965	0.884824	+	0.465926i	$0.154279\pi$
$984$	0	0
$985$	−9.12675	−0.290802
$986$	0	0
$987$	0	0
$988$	0	0
$989$	26.9013	0.855411
$990$	0	0
$991$	−12.8737	−0.408947	−0.204473	−	0.978872i	$-0.565548\pi$
$991$	−12.8737	−0.408947	−0.204473	+	0.978872i	$0.565548\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.678490	0.0215096
$996$	0	0
$997$	−16.0880	−0.509512	−0.254756	−	0.967005i	$-0.581995\pi$
$997$	−16.0880	−0.509512	−0.254756	+	0.967005i	$0.581995\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4176.2.a.bx.1.2		3
3.2	odd	2		1392.2.a.u.1.2		3
4.3	odd	2		261.2.a.e.1.3		3
12.11	even	2		87.2.a.b.1.1	✓	3
20.19	odd	2		6525.2.a.bg.1.1		3
24.5	odd	2		5568.2.a.bx.1.2		3
24.11	even	2		5568.2.a.cb.1.2		3
60.23	odd	4		2175.2.c.l.349.5		6
60.47	odd	4		2175.2.c.l.349.2		6
60.59	even	2		2175.2.a.t.1.3		3
84.83	odd	2		4263.2.a.m.1.1		3
116.115	odd	2		7569.2.a.t.1.1		3
348.347	even	2		2523.2.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.a.b.1.1	✓	3	12.11	even	2
261.2.a.e.1.3		3	4.3	odd	2
1392.2.a.u.1.2		3	3.2	odd	2
2175.2.a.t.1.3		3	60.59	even	2
2175.2.c.l.349.2		6	60.47	odd	4
2175.2.c.l.349.5		6	60.23	odd	4
2523.2.a.h.1.3		3	348.347	even	2
4176.2.a.bx.1.2		3	1.1	even	1	trivial
4263.2.a.m.1.1		3	84.83	odd	2
5568.2.a.bx.1.2		3	24.5	odd	2
5568.2.a.cb.1.2		3	24.11	even	2
6525.2.a.bg.1.1		3	20.19	odd	2
7569.2.a.t.1.1		3	116.115	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:	\( N \)	\(=\)	\( 4176 = 2^{4} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4176.a (trivial)

\(f(q)\)	\(=\)	\(q-0.508203 q^{5} -3.68133 q^{7} +O(q^{10})\)	\(q-0.508203 q^{5} -3.68133 q^{7} -0.318669 q^{11} +4.18953 q^{13} -3.17313 q^{17} +5.87086 q^{19} +2.50820 q^{23} -4.74173 q^{25} -1.00000 q^{29} -2.50820 q^{31} +1.87086 q^{35} +7.87086 q^{37} -8.72532 q^{41} +10.7253 q^{43} -11.0440 q^{47} +6.55220 q^{49} +8.24993 q^{53} +0.161949 q^{55} -11.3627 q^{59} -3.87086 q^{61} -2.12914 q^{65} -7.04399 q^{67} +6.24993 q^{71} -7.87086 q^{73} +1.17313 q^{77} +4.85446 q^{79} -8.37907 q^{83} +1.61259 q^{85} +15.9313 q^{89} -15.4231 q^{91} -2.98359 q^{95} +11.2335 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{7}+O(q^{10}) \) 3 * q - 4 * q^7	\( 3 q - 4 q^{7} - 8 q^{11} + 4 q^{13} - 4 q^{17} + 2 q^{19} + 6 q^{23} + 17 q^{25} - 3 q^{29} - 6 q^{31} - 10 q^{35} + 8 q^{37} + 2 q^{41} + 4 q^{43} - 12 q^{47} - 3 q^{49} - 8 q^{53} + 10 q^{55} - 20 q^{59} + 4 q^{61} - 22 q^{65} - 14 q^{71} - 8 q^{73} - 2 q^{77} + 2 q^{79} - 8 q^{83} - 42 q^{85} + 8 q^{89} - 8 q^{91} - 12 q^{95} + 4 q^{97}+O(q^{100}) \) 3 * q - 4 * q^7 - 8 * q^11 + 4 * q^13 - 4 * q^17 + 2 * q^19 + 6 * q^23 + 17 * q^25 - 3 * q^29 - 6 * q^31 - 10 * q^35 + 8 * q^37 + 2 * q^41 + 4 * q^43 - 12 * q^47 - 3 * q^49 - 8 * q^53 + 10 * q^55 - 20 * q^59 + 4 * q^61 - 22 * q^65 - 14 * q^71 - 8 * q^73 - 2 * q^77 + 2 * q^79 - 8 * q^83 - 42 * q^85 + 8 * q^89 - 8 * q^91 - 12 * q^95 + 4 * q^97

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