LMFDB - Embedded newform 3700.2.a.p.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3700,2,Mod(1,3700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3700.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3700 = 2^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3700.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.5446487479$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 13x^{7} + 57x^{5} - 2x^{4} - 96x^{3} + 12x^{2} + 48x - 12 $ x^9 - 13x^7 + 57x^5 - 2x^4 - 96x^3 + 12x^2 + 48x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 740)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.807220$ of defining polynomial
Character	$\chi$	$=$	3700.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.05716 q^{3} -0.448446 q^{7} +6.34624 q^{9} +1.79699 q^{11} +3.55175 q^{13} +1.08306 q^{17} +0.502334 q^{19} -1.37097 q^{21} +1.38561 q^{23} +10.2300 q^{27} -3.33072 q^{29} +2.17657 q^{31} +5.49370 q^{33} +1.00000 q^{37} +10.8583 q^{39} -0.833432 q^{41} -11.0621 q^{43} +5.06914 q^{47} -6.79890 q^{49} +3.31109 q^{51} +2.12975 q^{53} +1.53572 q^{57} +8.02989 q^{59} +10.9546 q^{61} -2.84595 q^{63} -7.15244 q^{67} +4.23603 q^{69} -8.82409 q^{71} +14.3995 q^{73} -0.805855 q^{77} +8.21536 q^{79} +12.2360 q^{81} -13.0316 q^{83} -10.1826 q^{87} -4.73198 q^{89} -1.59277 q^{91} +6.65412 q^{93} -15.4101 q^{97} +11.4041 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 4 q^{7} + 9 q^{9} + 10 q^{13} + 18 q^{17} - 2 q^{19} - 8 q^{21} + 14 q^{23} + 2 q^{29} + 4 q^{31} + 26 q^{33} + 9 q^{37} - 4 q^{39} - 2 q^{41} + 8 q^{43} + 14 q^{47} - 3 q^{49} - 20 q^{51} + 36 q^{53}+ \cdots - 8 q^{99}+O(q^{100}) $ 9 * q + 4 * q^7 + 9 * q^9 + 10 * q^13 + 18 * q^17 - 2 * q^19 - 8 * q^21 + 14 * q^23 + 2 * q^29 + 4 * q^31 + 26 * q^33 + 9 * q^37 - 4 * q^39 - 2 * q^41 + 8 * q^43 + 14 * q^47 - 3 * q^49 - 20 * q^51 + 36 * q^53 + 12 * q^57 - 4 * q^59 + 42 * q^63 - 8 * q^67 - 14 * q^69 - 12 * q^71 + 40 * q^73 + 26 * q^77 + 12 * q^79 + 17 * q^81 + 6 * q^83 - 28 * q^87 + 18 * q^91 + 38 * q^93 + 22 * q^97 - 8 * q^99

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
$3$	3.05716	1.76505	0.882527	−	0.470262i	$-0.155841\pi$
$3$	3.05716	1.76505	0.882527	+	0.470262i	$0.155841\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.448446	−0.169497	−0.0847484	−	0.996402i	$-0.527009\pi$
$7$	−0.448446	−0.169497	−0.0847484	+	0.996402i	$0.527009\pi$
$8$	0	0
$9$	6.34624	2.11541
$10$	0	0
$11$	1.79699	0.541814	0.270907	−	0.962606i	$-0.412676\pi$
$11$	1.79699	0.541814	0.270907	+	0.962606i	$0.412676\pi$
$12$	0	0
$13$	3.55175	0.985078	0.492539	−	0.870290i	$-0.336069\pi$
$13$	3.55175	0.985078	0.492539	+	0.870290i	$0.336069\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.08306	0.262681	0.131340	−	0.991337i	$-0.458072\pi$
$17$	1.08306	0.262681	0.131340	+	0.991337i	$0.458072\pi$
$18$	0	0
$19$	0.502334	0.115243	0.0576217	−	0.998338i	$-0.481648\pi$
$19$	0.502334	0.115243	0.0576217	+	0.998338i	$0.481648\pi$
$20$	0	0
$21$	−1.37097	−0.299171
$22$	0	0
$23$	1.38561	0.288919	0.144460	−	0.989511i	$-0.453856\pi$
$23$	1.38561	0.288919	0.144460	+	0.989511i	$0.453856\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	10.2300	1.96876
$28$	0	0
$29$	−3.33072	−0.618499	−0.309250	−	0.950981i	$-0.600078\pi$
$29$	−3.33072	−0.618499	−0.309250	+	0.950981i	$0.600078\pi$
$30$	0	0
$31$	2.17657	0.390923	0.195462	−	0.980711i	$-0.437379\pi$
$31$	2.17657	0.390923	0.195462	+	0.980711i	$0.437379\pi$
$32$	0	0
$33$	5.49370	0.956330
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	10.8583	1.73872
$40$	0	0
$41$	−0.833432	−0.130160	−0.0650801	−	0.997880i	$-0.520730\pi$
$41$	−0.833432	−0.130160	−0.0650801	+	0.997880i	$0.520730\pi$
$42$	0	0
$43$	−11.0621	−1.68696	−0.843480	−	0.537160i	$-0.819497\pi$
$43$	−11.0621	−1.68696	−0.843480	+	0.537160i	$0.819497\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.06914	0.739409	0.369705	−	0.929149i	$-0.379459\pi$
$47$	5.06914	0.739409	0.369705	+	0.929149i	$0.379459\pi$
$48$	0	0
$49$	−6.79890	−0.971271
$50$	0	0
$51$	3.31109	0.463646
$52$	0	0
$53$	2.12975	0.292544	0.146272	−	0.989244i	$-0.453273\pi$
$53$	2.12975	0.292544	0.146272	+	0.989244i	$0.453273\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.53572	0.203411
$58$	0	0
$59$	8.02989	1.04540	0.522701	−	0.852516i	$-0.324924\pi$
$59$	8.02989	1.04540	0.522701	+	0.852516i	$0.324924\pi$
$60$	0	0
$61$	10.9546	1.40259	0.701294	−	0.712872i	$-0.252606\pi$
$61$	10.9546	1.40259	0.701294	+	0.712872i	$0.252606\pi$
$62$	0	0
$63$	−2.84595	−0.358556
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.15244	−0.873810	−0.436905	−	0.899508i	$-0.643925\pi$
$67$	−7.15244	−0.873810	−0.436905	+	0.899508i	$0.643925\pi$
$68$	0	0
$69$	4.23603	0.509958
$70$	0	0
$71$	−8.82409	−1.04723	−0.523613	−	0.851956i	$-0.675416\pi$
$71$	−8.82409	−1.04723	−0.523613	+	0.851956i	$0.675416\pi$
$72$	0	0
$73$	14.3995	1.68534	0.842669	−	0.538431i	$-0.180983\pi$
$73$	14.3995	1.68534	0.842669	+	0.538431i	$0.180983\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.805855	−0.0918357
$78$	0	0
$79$	8.21536	0.924300	0.462150	−	0.886802i	$-0.347078\pi$
$79$	8.21536	0.924300	0.462150	+	0.886802i	$0.347078\pi$
$80$	0	0
$81$	12.2360	1.35956
$82$	0	0
$83$	−13.0316	−1.43041	−0.715204	−	0.698915i	$-0.753666\pi$
$83$	−13.0316	−1.43041	−0.715204	+	0.698915i	$0.753666\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−10.1826	−1.09168
$88$	0	0
$89$	−4.73198	−0.501589	−0.250794	−	0.968040i	$-0.580692\pi$
$89$	−4.73198	−0.501589	−0.250794	+	0.968040i	$0.580692\pi$
$90$	0	0
$91$	−1.59277	−0.166968
$92$	0	0
$93$	6.65412	0.690000
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.4101	−1.56466	−0.782332	−	0.622862i	$-0.785970\pi$
$97$	−15.4101	−1.56466	−0.782332	+	0.622862i	$0.785970\pi$
$98$	0	0
$99$	11.4041	1.14616
$100$	0	0
$101$	11.6226	1.15649	0.578247	−	0.815862i	$-0.303737\pi$
$101$	11.6226	1.15649	0.578247	+	0.815862i	$0.303737\pi$
$102$	0	0
$103$	10.9452	1.07846	0.539231	−	0.842158i	$-0.318715\pi$
$103$	10.9452	1.07846	0.539231	+	0.842158i	$0.318715\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.3841	−1.29389	−0.646945	−	0.762537i	$-0.723954\pi$
$107$	−13.3841	−1.29389	−0.646945	+	0.762537i	$0.723954\pi$
$108$	0	0
$109$	−10.2736	−0.984029	−0.492014	−	0.870587i	$-0.663739\pi$
$109$	−10.2736	−0.984029	−0.492014	+	0.870587i	$0.663739\pi$
$110$	0	0
$111$	3.05716	0.290173
$112$	0	0
$113$	18.5406	1.74416	0.872078	−	0.489366i	$-0.162772\pi$
$113$	18.5406	1.74416	0.872078	+	0.489366i	$0.162772\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	22.5403	2.08385
$118$	0	0
$119$	−0.485695	−0.0445236
$120$	0	0
$121$	−7.77082	−0.706438
$122$	0	0
$123$	−2.54794	−0.229740
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.07107	0.804927	0.402464	−	0.915436i	$-0.368154\pi$
$127$	9.07107	0.804927	0.402464	+	0.915436i	$0.368154\pi$
$128$	0	0
$129$	−33.8188	−2.97758
$130$	0	0
$131$	−4.71173	−0.411665	−0.205833	−	0.978587i	$-0.565990\pi$
$131$	−4.71173	−0.411665	−0.205833	+	0.978587i	$0.565990\pi$
$132$	0	0
$133$	−0.225270	−0.0195334
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.64911	−0.226329	−0.113164	−	0.993576i	$-0.536099\pi$
$137$	−2.64911	−0.226329	−0.113164	+	0.993576i	$0.536099\pi$
$138$	0	0
$139$	−17.1883	−1.45789	−0.728946	−	0.684571i	$-0.759989\pi$
$139$	−17.1883	−1.45789	−0.728946	+	0.684571i	$0.759989\pi$
$140$	0	0
$141$	15.4972	1.30510
$142$	0	0
$143$	6.38247	0.533729
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−20.7853	−1.71434
$148$	0	0
$149$	15.6090	1.27874	0.639369	−	0.768900i	$-0.279196\pi$
$149$	15.6090	1.27874	0.639369	+	0.768900i	$0.279196\pi$
$150$	0	0
$151$	20.0474	1.63143	0.815715	−	0.578454i	$-0.196344\pi$
$151$	20.0474	1.63143	0.815715	+	0.578454i	$0.196344\pi$
$152$	0	0
$153$	6.87337	0.555679
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.3913	0.829315	0.414657	−	0.909978i	$-0.363901\pi$
$157$	10.3913	0.829315	0.414657	+	0.909978i	$0.363901\pi$
$158$	0	0
$159$	6.51099	0.516355
$160$	0	0
$161$	−0.621371	−0.0489709
$162$	0	0
$163$	18.1408	1.42090	0.710450	−	0.703748i	$-0.248491\pi$
$163$	18.1408	1.42090	0.710450	+	0.703748i	$0.248491\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.5406	1.58948	0.794741	−	0.606949i	$-0.207607\pi$
$167$	20.5406	1.58948	0.794741	+	0.606949i	$0.207607\pi$
$168$	0	0
$169$	−0.385066	−0.0296205
$170$	0	0
$171$	3.18793	0.243787
$172$	0	0
$173$	10.1606	0.772499	0.386249	−	0.922394i	$-0.373770\pi$
$173$	10.1606	0.772499	0.386249	+	0.922394i	$0.373770\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	24.5487	1.84519
$178$	0	0
$179$	8.33088	0.622679	0.311340	−	0.950299i	$-0.399222\pi$
$179$	8.33088	0.622679	0.311340	+	0.950299i	$0.399222\pi$
$180$	0	0
$181$	−18.7286	−1.39208	−0.696042	−	0.718001i	$-0.745057\pi$
$181$	−18.7286	−1.39208	−0.696042	+	0.718001i	$0.745057\pi$
$182$	0	0
$183$	33.4899	2.47564
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.94625	0.142324
$188$	0	0
$189$	−4.58760	−0.333699
$190$	0	0
$191$	−11.9574	−0.865208	−0.432604	−	0.901584i	$-0.642405\pi$
$191$	−11.9574	−0.865208	−0.432604	+	0.901584i	$0.642405\pi$
$192$	0	0
$193$	−12.9063	−0.929012	−0.464506	−	0.885570i	$-0.653768\pi$
$193$	−12.9063	−0.929012	−0.464506	+	0.885570i	$0.653768\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.94271	−0.494647	−0.247324	−	0.968933i	$-0.579551\pi$
$197$	−6.94271	−0.494647	−0.247324	+	0.968933i	$0.579551\pi$
$198$	0	0
$199$	−24.6083	−1.74444	−0.872220	−	0.489114i	$-0.837320\pi$
$199$	−24.6083	−1.74444	−0.872220	+	0.489114i	$0.837320\pi$
$200$	0	0
$201$	−21.8662	−1.54232
$202$	0	0
$203$	1.49365	0.104834
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.79341	0.611184
$208$	0	0
$209$	0.902691	0.0624404
$210$	0	0
$211$	13.2009	0.908788	0.454394	−	0.890801i	$-0.349856\pi$
$211$	13.2009	0.908788	0.454394	+	0.890801i	$0.349856\pi$
$212$	0	0
$213$	−26.9767	−1.84841
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.976074	−0.0662602
$218$	0	0
$219$	44.0217	2.97471
$220$	0	0
$221$	3.84676	0.258761
$222$	0	0
$223$	−4.62772	−0.309895	−0.154947	−	0.987923i	$-0.549521\pi$
$223$	−4.62772	−0.309895	−0.154947	+	0.987923i	$0.549521\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.20796	0.412037	0.206018	−	0.978548i	$-0.433949\pi$
$227$	6.20796	0.412037	0.206018	+	0.978548i	$0.433949\pi$
$228$	0	0
$229$	−26.6510	−1.76115	−0.880573	−	0.473911i	$-0.842842\pi$
$229$	−26.6510	−1.76115	−0.880573	+	0.473911i	$0.842842\pi$
$230$	0	0
$231$	−2.46363	−0.162095
$232$	0	0
$233$	20.2645	1.32757	0.663787	−	0.747922i	$-0.268948\pi$
$233$	20.2645	1.32757	0.663787	+	0.747922i	$0.268948\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	25.1157	1.63144
$238$	0	0
$239$	−8.60759	−0.556779	−0.278389	−	0.960468i	$-0.589801\pi$
$239$	−8.60759	−0.556779	−0.278389	+	0.960468i	$0.589801\pi$
$240$	0	0
$241$	−0.683555	−0.0440316	−0.0220158	−	0.999758i	$-0.507008\pi$
$241$	−0.683555	−0.0440316	−0.0220158	+	0.999758i	$0.507008\pi$
$242$	0	0
$243$	6.71757	0.430932
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.78417	0.113524
$248$	0	0
$249$	−39.8398	−2.52475
$250$	0	0
$251$	−18.5993	−1.17398	−0.586989	−	0.809595i	$-0.699687\pi$
$251$	−18.5993	−1.17398	−0.586989	+	0.809595i	$0.699687\pi$
$252$	0	0
$253$	2.48993	0.156540
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.0027	0.623948	0.311974	−	0.950091i	$-0.399010\pi$
$257$	10.0027	0.623948	0.311974	+	0.950091i	$0.399010\pi$
$258$	0	0
$259$	−0.448446	−0.0278651
$260$	0	0
$261$	−21.1375	−1.30838
$262$	0	0
$263$	−8.05851	−0.496909	−0.248455	−	0.968644i	$-0.579923\pi$
$263$	−8.05851	−0.496909	−0.248455	+	0.968644i	$0.579923\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.4664	−0.885331
$268$	0	0
$269$	−19.6458	−1.19782	−0.598912	−	0.800815i	$-0.704400\pi$
$269$	−19.6458	−1.19782	−0.598912	+	0.800815i	$0.704400\pi$
$270$	0	0
$271$	0.816358	0.0495902	0.0247951	−	0.999693i	$-0.492107\pi$
$271$	0.816358	0.0495902	0.0247951	+	0.999693i	$0.492107\pi$
$272$	0	0
$273$	−4.86935	−0.294707
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.7950	1.42970	0.714852	−	0.699276i	$-0.246494\pi$
$277$	23.7950	1.42970	0.714852	+	0.699276i	$0.246494\pi$
$278$	0	0
$279$	13.8130	0.826964
$280$	0	0
$281$	−7.04816	−0.420458	−0.210229	−	0.977652i	$-0.567421\pi$
$281$	−7.04816	−0.420458	−0.210229	+	0.977652i	$0.567421\pi$
$282$	0	0
$283$	−4.00339	−0.237977	−0.118988	−	0.992896i	$-0.537965\pi$
$283$	−4.00339	−0.237977	−0.118988	+	0.992896i	$0.537965\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.373750	0.0220617
$288$	0	0
$289$	−15.8270	−0.930999
$290$	0	0
$291$	−47.1113	−2.76171
$292$	0	0
$293$	−15.1899	−0.887404	−0.443702	−	0.896174i	$-0.646335\pi$
$293$	−15.1899	−0.887404	−0.443702	+	0.896174i	$0.646335\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18.3832	1.06670
$298$	0	0
$299$	4.92134	0.284608
$300$	0	0
$301$	4.96078	0.285934
$302$	0	0
$303$	35.5322	2.04127
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−13.4367	−0.766875	−0.383438	−	0.923567i	$-0.625260\pi$
$307$	−13.4367	−0.766875	−0.383438	+	0.923567i	$0.625260\pi$
$308$	0	0
$309$	33.4612	1.90354
$310$	0	0
$311$	−27.1028	−1.53686	−0.768430	−	0.639934i	$-0.778962\pi$
$311$	−27.1028	−1.53686	−0.768430	+	0.639934i	$0.778962\pi$
$312$	0	0
$313$	−13.1432	−0.742899	−0.371450	−	0.928453i	$-0.621139\pi$
$313$	−13.1432	−0.742899	−0.371450	+	0.928453i	$0.621139\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.69453	0.488333	0.244167	−	0.969733i	$-0.421486\pi$
$317$	8.69453	0.488333	0.244167	+	0.969733i	$0.421486\pi$
$318$	0	0
$319$	−5.98528	−0.335111
$320$	0	0
$321$	−40.9174	−2.28379
$322$	0	0
$323$	0.544059	0.0302722
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−31.4080	−1.73686
$328$	0	0
$329$	−2.27324	−0.125328
$330$	0	0
$331$	6.66210	0.366182	0.183091	−	0.983096i	$-0.441390\pi$
$331$	6.66210	0.366182	0.183091	+	0.983096i	$0.441390\pi$
$332$	0	0
$333$	6.34624	0.347772
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−30.6358	−1.66884	−0.834419	−	0.551131i	$-0.814196\pi$
$337$	−30.6358	−1.66884	−0.834419	+	0.551131i	$0.814196\pi$
$338$	0	0
$339$	56.6817	3.07853
$340$	0	0
$341$	3.91128	0.211808
$342$	0	0
$343$	6.18806	0.334124
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.70306	−0.520888	−0.260444	−	0.965489i	$-0.583869\pi$
$347$	−9.70306	−0.520888	−0.260444	+	0.965489i	$0.583869\pi$
$348$	0	0
$349$	−13.2195	−0.707623	−0.353812	−	0.935317i	$-0.615115\pi$
$349$	−13.2195	−0.707623	−0.353812	+	0.935317i	$0.615115\pi$
$350$	0	0
$351$	36.3344	1.93939
$352$	0	0
$353$	12.1558	0.646986	0.323493	−	0.946231i	$-0.395143\pi$
$353$	12.1558	0.646986	0.323493	+	0.946231i	$0.395143\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.48485	−0.0785865
$358$	0	0
$359$	−12.3936	−0.654109	−0.327055	−	0.945005i	$-0.606056\pi$
$359$	−12.3936	−0.654109	−0.327055	+	0.945005i	$0.606056\pi$
$360$	0	0
$361$	−18.7477	−0.986719
$362$	0	0
$363$	−23.7567	−1.24690
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.5473	0.550567	0.275283	−	0.961363i	$-0.411228\pi$
$367$	10.5473	0.550567	0.275283	+	0.961363i	$0.411228\pi$
$368$	0	0
$369$	−5.28916	−0.275343
$370$	0	0
$371$	−0.955078	−0.0495852
$372$	0	0
$373$	3.16321	0.163785	0.0818923	−	0.996641i	$-0.473904\pi$
$373$	3.16321	0.163785	0.0818923	+	0.996641i	$0.473904\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11.8299	−0.609270
$378$	0	0
$379$	26.2277	1.34723	0.673614	−	0.739084i	$-0.264741\pi$
$379$	26.2277	1.34723	0.673614	+	0.739084i	$0.264741\pi$
$380$	0	0
$381$	27.7317	1.42074
$382$	0	0
$383$	−6.10204	−0.311799	−0.155900	−	0.987773i	$-0.549828\pi$
$383$	−6.10204	−0.311799	−0.155900	+	0.987773i	$0.549828\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−70.2030	−3.56862
$388$	0	0
$389$	−11.0799	−0.561771	−0.280886	−	0.959741i	$-0.590628\pi$
$389$	−11.0799	−0.561771	−0.280886	+	0.959741i	$0.590628\pi$
$390$	0	0
$391$	1.50070	0.0758936
$392$	0	0
$393$	−14.4045	−0.726611
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.3837	−0.571329	−0.285665	−	0.958330i	$-0.592214\pi$
$397$	−11.3837	−0.571329	−0.285665	+	0.958330i	$0.592214\pi$
$398$	0	0
$399$	−0.688687	−0.0344775
$400$	0	0
$401$	−7.96999	−0.398002	−0.199001	−	0.979999i	$-0.563770\pi$
$401$	−7.96999	−0.398002	−0.199001	+	0.979999i	$0.563770\pi$
$402$	0	0
$403$	7.73063	0.385090
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.79699	0.0890736
$408$	0	0
$409$	35.7633	1.76838	0.884191	−	0.467126i	$-0.154711\pi$
$409$	35.7633	1.76838	0.884191	+	0.467126i	$0.154711\pi$
$410$	0	0
$411$	−8.09876	−0.399482
$412$	0	0
$413$	−3.60098	−0.177192
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−52.5474	−2.57326
$418$	0	0
$419$	−30.4526	−1.48771	−0.743853	−	0.668343i	$-0.767004\pi$
$419$	−30.4526	−1.48771	−0.743853	+	0.668343i	$0.767004\pi$
$420$	0	0
$421$	6.37088	0.310498	0.155249	−	0.987875i	$-0.450382\pi$
$421$	6.37088	0.310498	0.155249	+	0.987875i	$0.450382\pi$
$422$	0	0
$423$	32.1700	1.56416
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.91253	−0.237734
$428$	0	0
$429$	19.5122	0.942060
$430$	0	0
$431$	25.5381	1.23013	0.615064	−	0.788477i	$-0.289130\pi$
$431$	25.5381	1.23013	0.615064	+	0.788477i	$0.289130\pi$
$432$	0	0
$433$	−19.2955	−0.927285	−0.463642	−	0.886022i	$-0.653458\pi$
$433$	−19.2955	−0.927285	−0.463642	+	0.886022i	$0.653458\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.696039	0.0332961
$438$	0	0
$439$	−12.9250	−0.616876	−0.308438	−	0.951244i	$-0.599806\pi$
$439$	−12.9250	−0.616876	−0.308438	+	0.951244i	$0.599806\pi$
$440$	0	0
$441$	−43.1474	−2.05464
$442$	0	0
$443$	−19.8707	−0.944085	−0.472042	−	0.881576i	$-0.656483\pi$
$443$	−19.8707	−0.944085	−0.472042	+	0.881576i	$0.656483\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	47.7192	2.25704
$448$	0	0
$449$	23.1727	1.09359	0.546793	−	0.837268i	$-0.315848\pi$
$449$	23.1727	1.09359	0.546793	+	0.837268i	$0.315848\pi$
$450$	0	0
$451$	−1.49767	−0.0705226
$452$	0	0
$453$	61.2880	2.87956
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.5062	0.585013	0.292507	−	0.956264i	$-0.405511\pi$
$457$	12.5062	0.585013	0.292507	+	0.956264i	$0.405511\pi$
$458$	0	0
$459$	11.0797	0.517157
$460$	0	0
$461$	−41.4167	−1.92897	−0.964484	−	0.264143i	$-0.914911\pi$
$461$	−41.4167	−1.92897	−0.964484	+	0.264143i	$0.914911\pi$
$462$	0	0
$463$	−14.2735	−0.663344	−0.331672	−	0.943395i	$-0.607613\pi$
$463$	−14.2735	−0.663344	−0.331672	+	0.943395i	$0.607613\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	38.3655	1.77534	0.887672	−	0.460475i	$-0.152321\pi$
$467$	38.3655	1.77534	0.887672	+	0.460475i	$0.152321\pi$
$468$	0	0
$469$	3.20749	0.148108
$470$	0	0
$471$	31.7678	1.46378
$472$	0	0
$473$	−19.8786	−0.914018
$474$	0	0
$475$	0	0
$476$	0	0
$477$	13.5159	0.618851
$478$	0	0
$479$	−6.76138	−0.308935	−0.154468	−	0.987998i	$-0.549366\pi$
$479$	−6.76138	−0.308935	−0.154468	+	0.987998i	$0.549366\pi$
$480$	0	0
$481$	3.55175	0.161946
$482$	0	0
$483$	−1.89963	−0.0864363
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.55868	−0.251888	−0.125944	−	0.992037i	$-0.540196\pi$
$487$	−5.55868	−0.251888	−0.125944	+	0.992037i	$0.540196\pi$
$488$	0	0
$489$	55.4595	2.50796
$490$	0	0
$491$	−15.4135	−0.695600	−0.347800	−	0.937569i	$-0.613071\pi$
$491$	−15.4135	−0.695600	−0.347800	+	0.937569i	$0.613071\pi$
$492$	0	0
$493$	−3.60737	−0.162468
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.95713	0.177502
$498$	0	0
$499$	−31.7897	−1.42310	−0.711550	−	0.702635i	$-0.752007\pi$
$499$	−31.7897	−1.42310	−0.711550	+	0.702635i	$0.752007\pi$
$500$	0	0
$501$	62.7960	2.80552
$502$	0	0
$503$	−30.1261	−1.34326	−0.671628	−	0.740889i	$-0.734404\pi$
$503$	−30.1261	−1.34326	−0.671628	+	0.740889i	$0.734404\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.17721	−0.0522818
$508$	0	0
$509$	−37.0537	−1.64238	−0.821189	−	0.570657i	$-0.806689\pi$
$509$	−37.0537	−1.64238	−0.821189	+	0.570657i	$0.806689\pi$
$510$	0	0
$511$	−6.45742	−0.285659
$512$	0	0
$513$	5.13888	0.226887
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.10920	0.400622
$518$	0	0
$519$	31.0627	1.36350
$520$	0	0
$521$	5.20283	0.227940	0.113970	−	0.993484i	$-0.463643\pi$
$521$	5.20283	0.227940	0.113970	+	0.993484i	$0.463643\pi$
$522$	0	0
$523$	−31.0528	−1.35784	−0.678922	−	0.734210i	$-0.737553\pi$
$523$	−31.0528	−1.35784	−0.678922	+	0.734210i	$0.737553\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.35736	0.102688
$528$	0	0
$529$	−21.0801	−0.916526
$530$	0	0
$531$	50.9596	2.21146
$532$	0	0
$533$	−2.96014	−0.128218
$534$	0	0
$535$	0	0
$536$	0	0
$537$	25.4689	1.09906
$538$	0	0
$539$	−12.2176	−0.526248
$540$	0	0
$541$	21.7498	0.935097	0.467548	−	0.883967i	$-0.345137\pi$
$541$	21.7498	0.935097	0.467548	+	0.883967i	$0.345137\pi$
$542$	0	0
$543$	−57.2563	−2.45710
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.97017	0.298023	0.149012	−	0.988835i	$-0.452391\pi$
$547$	6.97017	0.298023	0.149012	+	0.988835i	$0.452391\pi$
$548$	0	0
$549$	69.5203	2.96705
$550$	0	0
$551$	−1.67313	−0.0712779
$552$	0	0
$553$	−3.68415	−0.156666
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.7192	1.13213	0.566065	−	0.824361i	$-0.308465\pi$
$557$	26.7192	1.13213	0.566065	+	0.824361i	$0.308465\pi$
$558$	0	0
$559$	−39.2900	−1.66179
$560$	0	0
$561$	5.95001	0.251210
$562$	0	0
$563$	38.2345	1.61139	0.805695	−	0.592331i	$-0.201792\pi$
$563$	38.2345	1.61139	0.805695	+	0.592331i	$0.201792\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5.48721	−0.230441
$568$	0	0
$569$	−2.63777	−0.110581	−0.0552904	−	0.998470i	$-0.517608\pi$
$569$	−2.63777	−0.110581	−0.0552904	+	0.998470i	$0.517608\pi$
$570$	0	0
$571$	11.0844	0.463867	0.231934	−	0.972732i	$-0.425495\pi$
$571$	11.0844	0.463867	0.231934	+	0.972732i	$0.425495\pi$
$572$	0	0
$573$	−36.5557	−1.52714
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.87140	−0.244430	−0.122215	−	0.992504i	$-0.539000\pi$
$577$	−5.87140	−0.244430	−0.122215	+	0.992504i	$0.539000\pi$
$578$	0	0
$579$	−39.4565	−1.63976
$580$	0	0
$581$	5.84399	0.242450
$582$	0	0
$583$	3.82714	0.158504
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.9503	1.48383	0.741915	−	0.670494i	$-0.233918\pi$
$587$	35.9503	1.48383	0.741915	+	0.670494i	$0.233918\pi$
$588$	0	0
$589$	1.09336	0.0450513
$590$	0	0
$591$	−21.2250	−0.873079
$592$	0	0
$593$	11.3178	0.464768	0.232384	−	0.972624i	$-0.425347\pi$
$593$	11.3178	0.464768	0.232384	+	0.972624i	$0.425347\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−75.2317	−3.07903
$598$	0	0
$599$	28.4210	1.16125	0.580626	−	0.814171i	$-0.302808\pi$
$599$	28.4210	1.16125	0.580626	+	0.814171i	$0.302808\pi$
$600$	0	0
$601$	23.0837	0.941602	0.470801	−	0.882239i	$-0.343965\pi$
$601$	23.0837	0.941602	0.470801	+	0.882239i	$0.343965\pi$
$602$	0	0
$603$	−45.3911	−1.84847
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.55759	−0.347342	−0.173671	−	0.984804i	$-0.555563\pi$
$607$	−8.55759	−0.347342	−0.173671	+	0.984804i	$0.555563\pi$
$608$	0	0
$609$	4.56633	0.185037
$610$	0	0
$611$	18.0043	0.728376
$612$	0	0
$613$	−6.23495	−0.251827	−0.125914	−	0.992041i	$-0.540186\pi$
$613$	−6.23495	−0.251827	−0.125914	+	0.992041i	$0.540186\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.1159	1.61500	0.807502	−	0.589865i	$-0.200819\pi$
$617$	40.1159	1.61500	0.807502	+	0.589865i	$0.200819\pi$
$618$	0	0
$619$	−41.2861	−1.65943	−0.829714	−	0.558188i	$-0.811497\pi$
$619$	−41.2861	−1.65943	−0.829714	+	0.558188i	$0.811497\pi$
$620$	0	0
$621$	14.1748	0.568814
$622$	0	0
$623$	2.12204	0.0850177
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.75967	0.110211
$628$	0	0
$629$	1.08306	0.0431845
$630$	0	0
$631$	−12.6926	−0.505284	−0.252642	−	0.967560i	$-0.581299\pi$
$631$	−12.6926	−0.505284	−0.252642	+	0.967560i	$0.581299\pi$
$632$	0	0
$633$	40.3573	1.60406
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−24.1480	−0.956778
$638$	0	0
$639$	−55.9998	−2.21532
$640$	0	0
$641$	25.3440	1.00103	0.500514	−	0.865729i	$-0.333145\pi$
$641$	25.3440	1.00103	0.500514	+	0.865729i	$0.333145\pi$
$642$	0	0
$643$	−23.5901	−0.930304	−0.465152	−	0.885231i	$-0.654000\pi$
$643$	−23.5901	−0.930304	−0.465152	+	0.885231i	$0.654000\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.3132	0.916535	0.458267	−	0.888814i	$-0.348470\pi$
$647$	23.3132	0.916535	0.458267	+	0.888814i	$0.348470\pi$
$648$	0	0
$649$	14.4297	0.566414
$650$	0	0
$651$	−2.98402	−0.116953
$652$	0	0
$653$	40.8357	1.59802	0.799012	−	0.601315i	$-0.205356\pi$
$653$	40.8357	1.59802	0.799012	+	0.601315i	$0.205356\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	91.3829	3.56519
$658$	0	0
$659$	22.6517	0.882384	0.441192	−	0.897413i	$-0.354556\pi$
$659$	22.6517	0.882384	0.441192	+	0.897413i	$0.354556\pi$
$660$	0	0
$661$	0.230985	0.00898429	0.00449214	−	0.999990i	$-0.498570\pi$
$661$	0.230985	0.00898429	0.00449214	+	0.999990i	$0.498570\pi$
$662$	0	0
$663$	11.7602	0.456728
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.61508	−0.178696
$668$	0	0
$669$	−14.1477	−0.546981
$670$	0	0
$671$	19.6853	0.759941
$672$	0	0
$673$	−23.5437	−0.907543	−0.453771	−	0.891118i	$-0.649922\pi$
$673$	−23.5437	−0.907543	−0.453771	+	0.891118i	$0.649922\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.61986	0.215989	0.107994	−	0.994151i	$-0.465557\pi$
$677$	5.61986	0.215989	0.107994	+	0.994151i	$0.465557\pi$
$678$	0	0
$679$	6.91062	0.265205
$680$	0	0
$681$	18.9787	0.727267
$682$	0	0
$683$	9.37947	0.358896	0.179448	−	0.983767i	$-0.442569\pi$
$683$	9.37947	0.358896	0.179448	+	0.983767i	$0.442569\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−81.4763	−3.10852
$688$	0	0
$689$	7.56434	0.288178
$690$	0	0
$691$	−2.58554	−0.0983584	−0.0491792	−	0.998790i	$-0.515661\pi$
$691$	−2.58554	−0.0983584	−0.0491792	+	0.998790i	$0.515661\pi$
$692$	0	0
$693$	−5.11415	−0.194270
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.902658	−0.0341906
$698$	0	0
$699$	61.9520	2.34324
$700$	0	0
$701$	25.1954	0.951618	0.475809	−	0.879549i	$-0.342155\pi$
$701$	25.1954	0.951618	0.475809	+	0.879549i	$0.342155\pi$
$702$	0	0
$703$	0.502334	0.0189459
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.21212	−0.196022
$708$	0	0
$709$	42.7884	1.60695	0.803476	−	0.595337i	$-0.202981\pi$
$709$	42.7884	1.60695	0.803476	+	0.595337i	$0.202981\pi$
$710$	0	0
$711$	52.1366	1.95528
$712$	0	0
$713$	3.01587	0.112945
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−26.3148	−0.982744
$718$	0	0
$719$	−21.4988	−0.801771	−0.400886	−	0.916128i	$-0.631298\pi$
$719$	−21.4988	−0.801771	−0.400886	+	0.916128i	$0.631298\pi$
$720$	0	0
$721$	−4.90833	−0.182796
$722$	0	0
$723$	−2.08974	−0.0777182
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.85916	0.143128	0.0715641	−	0.997436i	$-0.477201\pi$
$727$	3.85916	0.143128	0.0715641	+	0.997436i	$0.477201\pi$
$728$	0	0
$729$	−16.1714	−0.598942
$730$	0	0
$731$	−11.9810	−0.443133
$732$	0	0
$733$	18.9340	0.699345	0.349672	−	0.936872i	$-0.386293\pi$
$733$	18.9340	0.699345	0.349672	+	0.936872i	$0.386293\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.8529	−0.473442
$738$	0	0
$739$	−28.4508	−1.04658	−0.523290	−	0.852155i	$-0.675296\pi$
$739$	−28.4508	−1.04658	−0.523290	+	0.852155i	$0.675296\pi$
$740$	0	0
$741$	5.45448	0.200375
$742$	0	0
$743$	−45.2282	−1.65926	−0.829630	−	0.558313i	$-0.811449\pi$
$743$	−45.2282	−1.65926	−0.829630	+	0.558313i	$0.811449\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−82.7019	−3.02591
$748$	0	0
$749$	6.00205	0.219310
$750$	0	0
$751$	28.6127	1.04409	0.522046	−	0.852917i	$-0.325169\pi$
$751$	28.6127	1.04409	0.522046	+	0.852917i	$0.325169\pi$
$752$	0	0
$753$	−56.8611	−2.07213
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−37.0006	−1.34481	−0.672406	−	0.740183i	$-0.734739\pi$
$757$	−37.0006	−1.34481	−0.672406	+	0.740183i	$0.734739\pi$
$758$	0	0
$759$	7.61212	0.276302
$760$	0	0
$761$	−23.8512	−0.864605	−0.432303	−	0.901729i	$-0.642299\pi$
$761$	−23.8512	−0.864605	−0.432303	+	0.901729i	$0.642299\pi$
$762$	0	0
$763$	4.60714	0.166790
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.5202	1.02980
$768$	0	0
$769$	18.8200	0.678665	0.339333	−	0.940666i	$-0.389799\pi$
$769$	18.8200	0.678665	0.339333	+	0.940666i	$0.389799\pi$
$770$	0	0
$771$	30.5797	1.10130
$772$	0	0
$773$	8.85107	0.318351	0.159175	−	0.987250i	$-0.449116\pi$
$773$	8.85107	0.318351	0.159175	+	0.987250i	$0.449116\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.37097	−0.0491834
$778$	0	0
$779$	−0.418662	−0.0150001
$780$	0	0
$781$	−15.8568	−0.567402
$782$	0	0
$783$	−34.0733	−1.21768
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−50.4985	−1.80008	−0.900039	−	0.435810i	$-0.856462\pi$
$787$	−50.4985	−1.80008	−0.900039	+	0.435810i	$0.856462\pi$
$788$	0	0
$789$	−24.6362	−0.877071
$790$	0	0
$791$	−8.31448	−0.295629
$792$	0	0
$793$	38.9079	1.38166
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.6495	1.01482	0.507408	−	0.861706i	$-0.330604\pi$
$797$	28.6495	1.01482	0.507408	+	0.861706i	$0.330604\pi$
$798$	0	0
$799$	5.49018	0.194229
$800$	0	0
$801$	−30.0303	−1.06107
$802$	0	0
$803$	25.8759	0.913139
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−60.0603	−2.11422
$808$	0	0
$809$	−42.7386	−1.50261	−0.751305	−	0.659955i	$-0.770575\pi$
$809$	−42.7386	−1.50261	−0.751305	+	0.659955i	$0.770575\pi$
$810$	0	0
$811$	−2.86550	−0.100621	−0.0503107	−	0.998734i	$-0.516021\pi$
$811$	−2.86550	−0.100621	−0.0503107	+	0.998734i	$0.516021\pi$
$812$	0	0
$813$	2.49574	0.0875294
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−5.55689	−0.194411
$818$	0	0
$819$	−10.1081	−0.353206
$820$	0	0
$821$	36.5616	1.27601	0.638004	−	0.770033i	$-0.279760\pi$
$821$	36.5616	1.27601	0.638004	+	0.770033i	$0.279760\pi$
$822$	0	0
$823$	31.5554	1.09995	0.549976	−	0.835180i	$-0.314637\pi$
$823$	31.5554	1.09995	0.549976	+	0.835180i	$0.314637\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.47860	0.225283	0.112642	−	0.993636i	$-0.464069\pi$
$827$	6.47860	0.225283	0.112642	+	0.993636i	$0.464069\pi$
$828$	0	0
$829$	−0.642832	−0.0223265	−0.0111632	−	0.999938i	$-0.503553\pi$
$829$	−0.642832	−0.0223265	−0.0111632	+	0.999938i	$0.503553\pi$
$830$	0	0
$831$	72.7452	2.52350
$832$	0	0
$833$	−7.36362	−0.255134
$834$	0	0
$835$	0	0
$836$	0	0
$837$	22.2663	0.769636
$838$	0	0
$839$	−29.4919	−1.01817	−0.509087	−	0.860715i	$-0.670017\pi$
$839$	−29.4919	−1.01817	−0.509087	+	0.860715i	$0.670017\pi$
$840$	0	0
$841$	−17.9063	−0.617459
$842$	0	0
$843$	−21.5474	−0.742130
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.48479	0.119739
$848$	0	0
$849$	−12.2390	−0.420042
$850$	0	0
$851$	1.38561	0.0474981
$852$	0	0
$853$	−4.06198	−0.139080	−0.0695398	−	0.997579i	$-0.522153\pi$
$853$	−4.06198	−0.139080	−0.0695398	+	0.997579i	$0.522153\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.9330	0.544260	0.272130	−	0.962260i	$-0.412272\pi$
$857$	15.9330	0.544260	0.272130	+	0.962260i	$0.412272\pi$
$858$	0	0
$859$	−3.60380	−0.122960	−0.0614801	−	0.998108i	$-0.519582\pi$
$859$	−3.60380	−0.122960	−0.0614801	+	0.998108i	$0.519582\pi$
$860$	0	0
$861$	1.14261	0.0389401
$862$	0	0
$863$	−24.7833	−0.843634	−0.421817	−	0.906681i	$-0.638607\pi$
$863$	−24.7833	−0.843634	−0.421817	+	0.906681i	$0.638607\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−48.3856	−1.64326
$868$	0	0
$869$	14.7629	0.500798
$870$	0	0
$871$	−25.4037	−0.860771
$872$	0	0
$873$	−97.7965	−3.30991
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21.3618	0.721337	0.360669	−	0.932694i	$-0.382549\pi$
$877$	21.3618	0.721337	0.360669	+	0.932694i	$0.382549\pi$
$878$	0	0
$879$	−46.4380	−1.56632
$880$	0	0
$881$	47.1125	1.58726	0.793630	−	0.608401i	$-0.208189\pi$
$881$	47.1125	1.58726	0.793630	+	0.608401i	$0.208189\pi$
$882$	0	0
$883$	−43.0287	−1.44803	−0.724015	−	0.689784i	$-0.757705\pi$
$883$	−43.0287	−1.44803	−0.724015	+	0.689784i	$0.757705\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.2127	−1.31663	−0.658317	−	0.752741i	$-0.728731\pi$
$887$	−39.2127	−1.31663	−0.658317	+	0.752741i	$0.728731\pi$
$888$	0	0
$889$	−4.06789	−0.136433
$890$	0	0
$891$	21.9881	0.736628
$892$	0	0
$893$	2.54640	0.0852120
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15.0453	0.502349
$898$	0	0
$899$	−7.24954	−0.241786
$900$	0	0
$901$	2.30665	0.0768456
$902$	0	0
$903$	15.1659	0.504690
$904$	0	0
$905$	0	0
$906$	0	0
$907$	37.0368	1.22979	0.614893	−	0.788610i	$-0.289199\pi$
$907$	37.0368	1.22979	0.614893	+	0.788610i	$0.289199\pi$
$908$	0	0
$909$	73.7599	2.44646
$910$	0	0
$911$	−36.5966	−1.21250	−0.606250	−	0.795274i	$-0.707327\pi$
$911$	−36.5966	−1.21250	−0.606250	+	0.795274i	$0.707327\pi$
$912$	0	0
$913$	−23.4178	−0.775015
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.11296	0.0697760
$918$	0	0
$919$	15.2311	0.502426	0.251213	−	0.967932i	$-0.419171\pi$
$919$	15.2311	0.502426	0.251213	+	0.967932i	$0.419171\pi$
$920$	0	0
$921$	−41.0783	−1.35358
$922$	0	0
$923$	−31.3410	−1.03160
$924$	0	0
$925$	0	0
$926$	0	0
$927$	69.4609	2.28139
$928$	0	0
$929$	−12.2944	−0.403365	−0.201683	−	0.979451i	$-0.564641\pi$
$929$	−12.2944	−0.403365	−0.201683	+	0.979451i	$0.564641\pi$
$930$	0	0
$931$	−3.41532	−0.111933
$932$	0	0
$933$	−82.8577	−2.71264
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−32.9738	−1.07721	−0.538603	−	0.842560i	$-0.681048\pi$
$937$	−32.9738	−1.07721	−0.538603	+	0.842560i	$0.681048\pi$
$938$	0	0
$939$	−40.1810	−1.31126
$940$	0	0
$941$	−35.0646	−1.14307	−0.571537	−	0.820576i	$-0.693653\pi$
$941$	−35.0646	−1.14307	−0.571537	+	0.820576i	$0.693653\pi$
$942$	0	0
$943$	−1.15481	−0.0376058
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31.6836	−1.02958	−0.514789	−	0.857317i	$-0.672130\pi$
$947$	−31.6836	−1.02958	−0.514789	+	0.857317i	$0.672130\pi$
$948$	0	0
$949$	51.1436	1.66019
$950$	0	0
$951$	26.5806	0.861934
$952$	0	0
$953$	16.1636	0.523589	0.261795	−	0.965124i	$-0.415686\pi$
$953$	16.1636	0.523589	0.261795	+	0.965124i	$0.415686\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−18.2980	−0.591489
$958$	0	0
$959$	1.18798	0.0383620
$960$	0	0
$961$	−26.2626	−0.847179
$962$	0	0
$963$	−84.9387	−2.73711
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−28.4425	−0.914649	−0.457325	−	0.889300i	$-0.651192\pi$
$967$	−28.4425	−0.914649	−0.457325	+	0.889300i	$0.651192\pi$
$968$	0	0
$969$	1.66328	0.0534321
$970$	0	0
$971$	−48.2297	−1.54776	−0.773882	−	0.633329i	$-0.781688\pi$
$971$	−48.2297	−1.54776	−0.773882	+	0.633329i	$0.781688\pi$
$972$	0	0
$973$	7.70802	0.247108
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.29722	0.201466	0.100733	−	0.994913i	$-0.467881\pi$
$977$	6.29722	0.201466	0.100733	+	0.994913i	$0.467881\pi$
$978$	0	0
$979$	−8.50333	−0.271768
$980$	0	0
$981$	−65.1985	−2.08163
$982$	0	0
$983$	44.7044	1.42585	0.712924	−	0.701241i	$-0.247370\pi$
$983$	44.7044	1.42585	0.712924	+	0.701241i	$0.247370\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.94965	−0.221210
$988$	0	0
$989$	−15.3278	−0.487396
$990$	0	0
$991$	−48.1874	−1.53072	−0.765362	−	0.643600i	$-0.777440\pi$
$991$	−48.1874	−1.53072	−0.765362	+	0.643600i	$0.777440\pi$
$992$	0	0
$993$	20.3671	0.646331
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.1005	0.383225	0.191613	−	0.981471i	$-0.438628\pi$
$997$	12.1005	0.383225	0.191613	+	0.981471i	$0.438628\pi$
$998$	0	0
$999$	10.2300	0.323663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3700.2.a.p.1.9		9
5.2	odd	4		740.2.d.a.149.2	✓	18
5.3	odd	4		740.2.d.a.149.17	yes	18
5.4	even	2		3700.2.a.o.1.1		9
15.2	even	4		6660.2.f.c.5329.10		18
15.8	even	4		6660.2.f.c.5329.9		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.d.a.149.2	✓	18	5.2	odd	4
740.2.d.a.149.17	yes	18	5.3	odd	4
3700.2.a.o.1.1		9	5.4	even	2
3700.2.a.p.1.9		9	1.1	even	1	trivial
6660.2.f.c.5329.9		18	15.8	even	4
6660.2.f.c.5329.10		18	15.2	even	4

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3700.a (trivial)

\(f(q)\)	\(=\)	\(q+3.05716 q^{3} -0.448446 q^{7} +6.34624 q^{9} +1.79699 q^{11} +3.55175 q^{13} +1.08306 q^{17} +0.502334 q^{19} -1.37097 q^{21} +1.38561 q^{23} +10.2300 q^{27} -3.33072 q^{29} +2.17657 q^{31} +5.49370 q^{33} +1.00000 q^{37} +10.8583 q^{39} -0.833432 q^{41} -11.0621 q^{43} +5.06914 q^{47} -6.79890 q^{49} +3.31109 q^{51} +2.12975 q^{53} +1.53572 q^{57} +8.02989 q^{59} +10.9546 q^{61} -2.84595 q^{63} -7.15244 q^{67} +4.23603 q^{69} -8.82409 q^{71} +14.3995 q^{73} -0.805855 q^{77} +8.21536 q^{79} +12.2360 q^{81} -13.0316 q^{83} -10.1826 q^{87} -4.73198 q^{89} -1.59277 q^{91} +6.65412 q^{93} -15.4101 q^{97} +11.4041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 4 q^{7} + 9 q^{9} + 10 q^{13} + 18 q^{17} - 2 q^{19} - 8 q^{21} + 14 q^{23} + 2 q^{29} + 4 q^{31} + 26 q^{33} + 9 q^{37} - 4 q^{39} - 2 q^{41} + 8 q^{43} + 14 q^{47} - 3 q^{49} - 20 q^{51} + 36 q^{53}+ \cdots - 8 q^{99}+O(q^{100}) \) 9 * q + 4 * q^7 + 9 * q^9 + 10 * q^13 + 18 * q^17 - 2 * q^19 - 8 * q^21 + 14 * q^23 + 2 * q^29 + 4 * q^31 + 26 * q^33 + 9 * q^37 - 4 * q^39 - 2 * q^41 + 8 * q^43 + 14 * q^47 - 3 * q^49 - 20 * q^51 + 36 * q^53 + 12 * q^57 - 4 * q^59 + 42 * q^63 - 8 * q^67 - 14 * q^69 - 12 * q^71 + 40 * q^73 + 26 * q^77 + 12 * q^79 + 17 * q^81 + 6 * q^83 - 28 * q^87 + 18 * q^91 + 38 * q^93 + 22 * q^97 - 8 * q^99

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