LMFDB - Embedded newform 1936.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,2,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1936.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1936.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:	$15.4590378313$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{3} +3.23607 q^{5} +2.00000 q^{7} -2.85410 q^{9} +1.23607 q^{13} -1.23607 q^{15} -0.618034 q^{17} +5.85410 q^{19} -0.763932 q^{21} -1.23607 q^{23} +5.47214 q^{25} +2.23607 q^{27} +4.47214 q^{29} -2.00000 q^{31} +6.47214 q^{35} -3.70820 q^{37} -0.472136 q^{39} +5.61803 q^{41} +8.56231 q^{43} -9.23607 q^{45} +6.47214 q^{47} -3.00000 q^{49} +0.236068 q^{51} -1.52786 q^{53} -2.23607 q^{57} +8.61803 q^{59} -2.47214 q^{61} -5.70820 q^{63} +4.00000 q^{65} -11.0902 q^{67} +0.472136 q^{69} +5.23607 q^{71} +10.3820 q^{73} -2.09017 q^{75} -13.4164 q^{79} +7.70820 q^{81} -9.32624 q^{83} -2.00000 q^{85} -1.70820 q^{87} -8.09017 q^{89} +2.47214 q^{91} +0.763932 q^{93} +18.9443 q^{95} +7.14590 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + 2 q^{5} + 4 q^{7} + q^{9} - 2 q^{13} + 2 q^{15} + q^{17} + 5 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{25} - 4 q^{31} + 4 q^{35} + 6 q^{37} + 8 q^{39} + 9 q^{41} - 3 q^{43} - 14 q^{45} + 4 q^{47}+ \cdots + 21 q^{97}+O(q^{100}) $ 2 * q - 3 * q^3 + 2 * q^5 + 4 * q^7 + q^9 - 2 * q^13 + 2 * q^15 + q^17 + 5 * q^19 - 6 * q^21 + 2 * q^23 + 2 * q^25 - 4 * q^31 + 4 * q^35 + 6 * q^37 + 8 * q^39 + 9 * q^41 - 3 * q^43 - 14 * q^45 + 4 * q^47 - 6 * q^49 - 4 * q^51 - 12 * q^53 + 15 * q^59 + 4 * q^61 + 2 * q^63 + 8 * q^65 - 11 * q^67 - 8 * q^69 + 6 * q^71 + 23 * q^73 + 7 * q^75 + 2 * q^81 - 3 * q^83 - 4 * q^85 + 10 * q^87 - 5 * q^89 - 4 * q^91 + 6 * q^93 + 20 * q^95 + 21 * q^97

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
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	0	0
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.85410	−0.951367
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	−0.618034	−0.149895	−0.0749476	−	0.997187i	$-0.523879\pi$
$17$	−0.618034	−0.149895	−0.0749476	+	0.997187i	$0.523879\pi$
$18$	0	0
$19$	5.85410	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$19$	5.85410	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$20$	0	0
$21$	−0.763932	−0.166704
$22$	0	0
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.47214	1.09399
$36$	0	0
$37$	−3.70820	−0.609625	−0.304812	−	0.952412i	$-0.598594\pi$
$37$	−3.70820	−0.609625	−0.304812	+	0.952412i	$0.598594\pi$
$38$	0	0
$39$	−0.472136	−0.0756023
$40$	0	0
$41$	5.61803	0.877390	0.438695	−	0.898636i	$-0.355441\pi$
$41$	5.61803	0.877390	0.438695	+	0.898636i	$0.355441\pi$
$42$	0	0
$43$	8.56231	1.30574	0.652870	−	0.757470i	$-0.273565\pi$
$43$	8.56231	1.30574	0.652870	+	0.757470i	$0.273565\pi$
$44$	0	0
$45$	−9.23607	−1.37683
$46$	0	0
$47$	6.47214	0.944058	0.472029	−	0.881583i	$-0.343522\pi$
$47$	6.47214	0.944058	0.472029	+	0.881583i	$0.343522\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0.236068	0.0330561
$52$	0	0
$53$	−1.52786	−0.209868	−0.104934	−	0.994479i	$-0.533463\pi$
$53$	−1.52786	−0.209868	−0.104934	+	0.994479i	$0.533463\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.23607	−0.296174
$58$	0	0
$59$	8.61803	1.12197	0.560986	−	0.827825i	$-0.310422\pi$
$59$	8.61803	1.12197	0.560986	+	0.827825i	$0.310422\pi$
$60$	0	0
$61$	−2.47214	−0.316525	−0.158262	−	0.987397i	$-0.550589\pi$
$61$	−2.47214	−0.316525	−0.158262	+	0.987397i	$0.550589\pi$
$62$	0	0
$63$	−5.70820	−0.719166
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−11.0902	−1.35488	−0.677440	−	0.735578i	$-0.736911\pi$
$67$	−11.0902	−1.35488	−0.677440	+	0.735578i	$0.736911\pi$
$68$	0	0
$69$	0.472136	0.0568385
$70$	0	0
$71$	5.23607	0.621407	0.310703	−	0.950507i	$-0.399435\pi$
$71$	5.23607	0.621407	0.310703	+	0.950507i	$0.399435\pi$
$72$	0	0
$73$	10.3820	1.21512	0.607559	−	0.794275i	$-0.292149\pi$
$73$	10.3820	1.21512	0.607559	+	0.794275i	$0.292149\pi$
$74$	0	0
$75$	−2.09017	−0.241352
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.4164	−1.50946	−0.754732	−	0.656033i	$-0.772233\pi$
$79$	−13.4164	−1.50946	−0.754732	+	0.656033i	$0.772233\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	0	0
$83$	−9.32624	−1.02369	−0.511844	−	0.859079i	$-0.671037\pi$
$83$	−9.32624	−1.02369	−0.511844	+	0.859079i	$0.671037\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−1.70820	−0.183139
$88$	0	0
$89$	−8.09017	−0.857556	−0.428778	−	0.903410i	$-0.641056\pi$
$89$	−8.09017	−0.857556	−0.428778	+	0.903410i	$0.641056\pi$
$90$	0	0
$91$	2.47214	0.259150
$92$	0	0
$93$	0.763932	0.0792161
$94$	0	0
$95$	18.9443	1.94364
$96$	0	0
$97$	7.14590	0.725556	0.362778	−	0.931876i	$-0.381828\pi$
$97$	7.14590	0.725556	0.362778	+	0.931876i	$0.381828\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.18034	−0.415959	−0.207980	−	0.978133i	$-0.566689\pi$
$101$	−4.18034	−0.415959	−0.207980	+	0.978133i	$0.566689\pi$
$102$	0	0
$103$	−15.7082	−1.54778	−0.773888	−	0.633323i	$-0.781691\pi$
$103$	−15.7082	−1.54778	−0.773888	+	0.633323i	$0.781691\pi$
$104$	0	0
$105$	−2.47214	−0.241256
$106$	0	0
$107$	1.14590	0.110778	0.0553891	−	0.998465i	$-0.482360\pi$
$107$	1.14590	0.110778	0.0553891	+	0.998465i	$0.482360\pi$
$108$	0	0
$109$	18.9443	1.81453	0.907266	−	0.420557i	$-0.138165\pi$
$109$	18.9443	1.81453	0.907266	+	0.420557i	$0.138165\pi$
$110$	0	0
$111$	1.41641	0.134439
$112$	0	0
$113$	−1.85410	−0.174419	−0.0872096	−	0.996190i	$-0.527795\pi$
$113$	−1.85410	−0.174419	−0.0872096	+	0.996190i	$0.527795\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	−3.52786	−0.326151
$118$	0	0
$119$	−1.23607	−0.113310
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−2.14590	−0.193489
$124$	0	0
$125$	1.52786	0.136656
$126$	0	0
$127$	10.9443	0.971147	0.485574	−	0.874196i	$-0.338611\pi$
$127$	10.9443	0.971147	0.485574	+	0.874196i	$0.338611\pi$
$128$	0	0
$129$	−3.27051	−0.287952
$130$	0	0
$131$	−6.79837	−0.593977	−0.296988	−	0.954881i	$-0.595982\pi$
$131$	−6.79837	−0.593977	−0.296988	+	0.954881i	$0.595982\pi$
$132$	0	0
$133$	11.7082	1.01523
$134$	0	0
$135$	7.23607	0.622782
$136$	0	0
$137$	16.0902	1.37468	0.687338	−	0.726338i	$-0.258779\pi$
$137$	16.0902	1.37468	0.687338	+	0.726338i	$0.258779\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−2.47214	−0.208191
$142$	0	0
$143$	0	0
$144$	0	0
$145$	14.4721	1.20185
$146$	0	0
$147$	1.14590	0.0945121
$148$	0	0
$149$	−6.18034	−0.506313	−0.253157	−	0.967425i	$-0.581469\pi$
$149$	−6.18034	−0.506313	−0.253157	+	0.967425i	$0.581469\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	1.76393	0.142605
$154$	0	0
$155$	−6.47214	−0.519854
$156$	0	0
$157$	9.70820	0.774799	0.387400	−	0.921912i	$-0.373373\pi$
$157$	9.70820	0.774799	0.387400	+	0.921912i	$0.373373\pi$
$158$	0	0
$159$	0.583592	0.0462819
$160$	0	0
$161$	−2.47214	−0.194832
$162$	0	0
$163$	−0.909830	−0.0712634	−0.0356317	−	0.999365i	$-0.511344\pi$
$163$	−0.909830	−0.0712634	−0.0356317	+	0.999365i	$0.511344\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.7639	1.14247	0.571234	−	0.820787i	$-0.306465\pi$
$167$	14.7639	1.14247	0.571234	+	0.820787i	$0.306465\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	−16.7082	−1.27771
$172$	0	0
$173$	21.8885	1.66416	0.832078	−	0.554659i	$-0.187151\pi$
$173$	21.8885	1.66416	0.832078	+	0.554659i	$0.187151\pi$
$174$	0	0
$175$	10.9443	0.827309
$176$	0	0
$177$	−3.29180	−0.247427
$178$	0	0
$179$	8.61803	0.644142	0.322071	−	0.946715i	$-0.395621\pi$
$179$	8.61803	0.644142	0.322071	+	0.946715i	$0.395621\pi$
$180$	0	0
$181$	−4.18034	−0.310722	−0.155361	−	0.987858i	$-0.549654\pi$
$181$	−4.18034	−0.310722	−0.155361	+	0.987858i	$0.549654\pi$
$182$	0	0
$183$	0.944272	0.0698026
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4.47214	0.325300
$190$	0	0
$191$	−4.76393	−0.344706	−0.172353	−	0.985035i	$-0.555137\pi$
$191$	−4.76393	−0.344706	−0.172353	+	0.985035i	$0.555137\pi$
$192$	0	0
$193$	17.4164	1.25366	0.626830	−	0.779156i	$-0.284352\pi$
$193$	17.4164	1.25366	0.626830	+	0.779156i	$0.284352\pi$
$194$	0	0
$195$	−1.52786	−0.109413
$196$	0	0
$197$	−20.9443	−1.49222	−0.746109	−	0.665824i	$-0.768080\pi$
$197$	−20.9443	−1.49222	−0.746109	+	0.665824i	$0.768080\pi$
$198$	0	0
$199$	18.9443	1.34292	0.671462	−	0.741039i	$-0.265667\pi$
$199$	18.9443	1.34292	0.671462	+	0.741039i	$0.265667\pi$
$200$	0	0
$201$	4.23607	0.298789
$202$	0	0
$203$	8.94427	0.627765
$204$	0	0
$205$	18.1803	1.26977
$206$	0	0
$207$	3.52786	0.245204
$208$	0	0
$209$	0	0
$210$	0	0
$211$	6.09017	0.419265	0.209632	−	0.977780i	$-0.432773\pi$
$211$	6.09017	0.419265	0.209632	+	0.977780i	$0.432773\pi$
$212$	0	0
$213$	−2.00000	−0.137038
$214$	0	0
$215$	27.7082	1.88968
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−3.96556	−0.267968
$220$	0	0
$221$	−0.763932	−0.0513876
$222$	0	0
$223$	−25.7082	−1.72155	−0.860774	−	0.508987i	$-0.830020\pi$
$223$	−25.7082	−1.72155	−0.860774	+	0.508987i	$0.830020\pi$
$224$	0	0
$225$	−15.6180	−1.04120
$226$	0	0
$227$	−14.5066	−0.962835	−0.481418	−	0.876491i	$-0.659878\pi$
$227$	−14.5066	−0.962835	−0.481418	+	0.876491i	$0.659878\pi$
$228$	0	0
$229$	1.70820	0.112881	0.0564406	−	0.998406i	$-0.482025\pi$
$229$	1.70820	0.112881	0.0564406	+	0.998406i	$0.482025\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.61803	−0.630098	−0.315049	−	0.949075i	$-0.602021\pi$
$233$	−9.61803	−0.630098	−0.315049	+	0.949075i	$0.602021\pi$
$234$	0	0
$235$	20.9443	1.36625
$236$	0	0
$237$	5.12461	0.332879
$238$	0	0
$239$	21.7082	1.40419	0.702093	−	0.712085i	$-0.252249\pi$
$239$	21.7082	1.40419	0.702093	+	0.712085i	$0.252249\pi$
$240$	0	0
$241$	−11.0902	−0.714381	−0.357190	−	0.934032i	$-0.616265\pi$
$241$	−11.0902	−0.714381	−0.357190	+	0.934032i	$0.616265\pi$
$242$	0	0
$243$	−9.65248	−0.619207
$244$	0	0
$245$	−9.70820	−0.620234
$246$	0	0
$247$	7.23607	0.460420
$248$	0	0
$249$	3.56231	0.225752
$250$	0	0
$251$	−20.9443	−1.32199	−0.660995	−	0.750390i	$-0.729866\pi$
$251$	−20.9443	−1.32199	−0.660995	+	0.750390i	$0.729866\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.763932	0.0478393
$256$	0	0
$257$	−5.61803	−0.350443	−0.175222	−	0.984529i	$-0.556064\pi$
$257$	−5.61803	−0.350443	−0.175222	+	0.984529i	$0.556064\pi$
$258$	0	0
$259$	−7.41641	−0.460833
$260$	0	0
$261$	−12.7639	−0.790068
$262$	0	0
$263$	23.2361	1.43280	0.716399	−	0.697691i	$-0.245789\pi$
$263$	23.2361	1.43280	0.716399	+	0.697691i	$0.245789\pi$
$264$	0	0
$265$	−4.94427	−0.303724
$266$	0	0
$267$	3.09017	0.189115
$268$	0	0
$269$	6.18034	0.376822	0.188411	−	0.982090i	$-0.439666\pi$
$269$	6.18034	0.376822	0.188411	+	0.982090i	$0.439666\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	0	0
$273$	−0.944272	−0.0571499
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.7082	−1.42449	−0.712244	−	0.701932i	$-0.752321\pi$
$277$	−23.7082	−1.42449	−0.712244	+	0.701932i	$0.752321\pi$
$278$	0	0
$279$	5.70820	0.341741
$280$	0	0
$281$	−16.0902	−0.959859	−0.479930	−	0.877307i	$-0.659338\pi$
$281$	−16.0902	−0.959859	−0.479930	+	0.877307i	$0.659338\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	0	0
$285$	−7.23607	−0.428628
$286$	0	0
$287$	11.2361	0.663244
$288$	0	0
$289$	−16.6180	−0.977531
$290$	0	0
$291$	−2.72949	−0.160006
$292$	0	0
$293$	−28.3607	−1.65685	−0.828424	−	0.560101i	$-0.810762\pi$
$293$	−28.3607	−1.65685	−0.828424	+	0.560101i	$0.810762\pi$
$294$	0	0
$295$	27.8885	1.62373
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.52786	−0.0883587
$300$	0	0
$301$	17.1246	0.987046
$302$	0	0
$303$	1.59675	0.0917308
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−27.7984	−1.58654	−0.793268	−	0.608872i	$-0.791622\pi$
$307$	−27.7984	−1.58654	−0.793268	+	0.608872i	$0.791622\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	0	0
$311$	−26.0689	−1.47823	−0.739115	−	0.673579i	$-0.764756\pi$
$311$	−26.0689	−1.47823	−0.739115	+	0.673579i	$0.764756\pi$
$312$	0	0
$313$	4.32624	0.244533	0.122267	−	0.992497i	$-0.460984\pi$
$313$	4.32624	0.244533	0.122267	+	0.992497i	$0.460984\pi$
$314$	0	0
$315$	−18.4721	−1.04079
$316$	0	0
$317$	−3.70820	−0.208273	−0.104137	−	0.994563i	$-0.533208\pi$
$317$	−3.70820	−0.208273	−0.104137	+	0.994563i	$0.533208\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−0.437694	−0.0244297
$322$	0	0
$323$	−3.61803	−0.201313
$324$	0	0
$325$	6.76393	0.375195
$326$	0	0
$327$	−7.23607	−0.400155
$328$	0	0
$329$	12.9443	0.713641
$330$	0	0
$331$	−6.27051	−0.344658	−0.172329	−	0.985039i	$-0.555129\pi$
$331$	−6.27051	−0.344658	−0.172329	+	0.985039i	$0.555129\pi$
$332$	0	0
$333$	10.5836	0.579977
$334$	0	0
$335$	−35.8885	−1.96080
$336$	0	0
$337$	−26.7984	−1.45980	−0.729900	−	0.683554i	$-0.760433\pi$
$337$	−26.7984	−1.45980	−0.729900	+	0.683554i	$0.760433\pi$
$338$	0	0
$339$	0.708204	0.0384644
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	1.52786	0.0822574
$346$	0	0
$347$	−16.6180	−0.892103	−0.446051	−	0.895007i	$-0.647170\pi$
$347$	−16.6180	−0.892103	−0.446051	+	0.895007i	$0.647170\pi$
$348$	0	0
$349$	−19.5967	−1.04899	−0.524495	−	0.851414i	$-0.675746\pi$
$349$	−19.5967	−1.04899	−0.524495	+	0.851414i	$0.675746\pi$
$350$	0	0
$351$	2.76393	0.147528
$352$	0	0
$353$	32.6180	1.73608	0.868041	−	0.496492i	$-0.165379\pi$
$353$	32.6180	1.73608	0.868041	+	0.496492i	$0.165379\pi$
$354$	0	0
$355$	16.9443	0.899309
$356$	0	0
$357$	0.472136	0.0249881
$358$	0	0
$359$	−5.12461	−0.270467	−0.135233	−	0.990814i	$-0.543178\pi$
$359$	−5.12461	−0.270467	−0.135233	+	0.990814i	$0.543178\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	0	0
$363$	0	0
$364$	0	0
$365$	33.5967	1.75853
$366$	0	0
$367$	−19.7082	−1.02876	−0.514380	−	0.857562i	$-0.671978\pi$
$367$	−19.7082	−1.02876	−0.514380	+	0.857562i	$0.671978\pi$
$368$	0	0
$369$	−16.0344	−0.834720
$370$	0	0
$371$	−3.05573	−0.158645
$372$	0	0
$373$	−4.29180	−0.222221	−0.111110	−	0.993808i	$-0.535441\pi$
$373$	−4.29180	−0.222221	−0.111110	+	0.993808i	$0.535441\pi$
$374$	0	0
$375$	−0.583592	−0.0301366
$376$	0	0
$377$	5.52786	0.284699
$378$	0	0
$379$	−14.2705	−0.733027	−0.366513	−	0.930413i	$-0.619449\pi$
$379$	−14.2705	−0.733027	−0.366513	+	0.930413i	$0.619449\pi$
$380$	0	0
$381$	−4.18034	−0.214165
$382$	0	0
$383$	28.3607	1.44916	0.724582	−	0.689189i	$-0.242033\pi$
$383$	28.3607	1.44916	0.724582	+	0.689189i	$0.242033\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−24.4377	−1.24224
$388$	0	0
$389$	−10.6525	−0.540102	−0.270051	−	0.962846i	$-0.587041\pi$
$389$	−10.6525	−0.540102	−0.270051	+	0.962846i	$0.587041\pi$
$390$	0	0
$391$	0.763932	0.0386337
$392$	0	0
$393$	2.59675	0.130989
$394$	0	0
$395$	−43.4164	−2.18452
$396$	0	0
$397$	−17.1246	−0.859460	−0.429730	−	0.902958i	$-0.641391\pi$
$397$	−17.1246	−0.859460	−0.429730	+	0.902958i	$0.641391\pi$
$398$	0	0
$399$	−4.47214	−0.223887
$400$	0	0
$401$	−17.7984	−0.888808	−0.444404	−	0.895826i	$-0.646585\pi$
$401$	−17.7984	−0.888808	−0.444404	+	0.895826i	$0.646585\pi$
$402$	0	0
$403$	−2.47214	−0.123146
$404$	0	0
$405$	24.9443	1.23949
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−13.4164	−0.663399	−0.331699	−	0.943385i	$-0.607622\pi$
$409$	−13.4164	−0.663399	−0.331699	+	0.943385i	$0.607622\pi$
$410$	0	0
$411$	−6.14590	−0.303155
$412$	0	0
$413$	17.2361	0.848131
$414$	0	0
$415$	−30.1803	−1.48149
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.8541	0.530258	0.265129	−	0.964213i	$-0.414586\pi$
$419$	10.8541	0.530258	0.265129	+	0.964213i	$0.414586\pi$
$420$	0	0
$421$	−31.4164	−1.53114	−0.765571	−	0.643351i	$-0.777543\pi$
$421$	−31.4164	−1.53114	−0.765571	+	0.643351i	$0.777543\pi$
$422$	0	0
$423$	−18.4721	−0.898146
$424$	0	0
$425$	−3.38197	−0.164049
$426$	0	0
$427$	−4.94427	−0.239270
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.9443	0.816177	0.408088	−	0.912942i	$-0.366196\pi$
$431$	16.9443	0.816177	0.408088	+	0.912942i	$0.366196\pi$
$432$	0	0
$433$	27.7426	1.33323	0.666613	−	0.745404i	$-0.267743\pi$
$433$	27.7426	1.33323	0.666613	+	0.745404i	$0.267743\pi$
$434$	0	0
$435$	−5.52786	−0.265041
$436$	0	0
$437$	−7.23607	−0.346148
$438$	0	0
$439$	6.58359	0.314218	0.157109	−	0.987581i	$-0.449783\pi$
$439$	6.58359	0.314218	0.157109	+	0.987581i	$0.449783\pi$
$440$	0	0
$441$	8.56231	0.407729
$442$	0	0
$443$	8.56231	0.406807	0.203404	−	0.979095i	$-0.434800\pi$
$443$	8.56231	0.406807	0.203404	+	0.979095i	$0.434800\pi$
$444$	0	0
$445$	−26.1803	−1.24107
$446$	0	0
$447$	2.36068	0.111656
$448$	0	0
$449$	9.27051	0.437502	0.218751	−	0.975781i	$-0.429802\pi$
$449$	9.27051	0.437502	0.218751	+	0.975781i	$0.429802\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−3.05573	−0.143571
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	−9.43769	−0.441477	−0.220738	−	0.975333i	$-0.570847\pi$
$457$	−9.43769	−0.441477	−0.220738	+	0.975333i	$0.570847\pi$
$458$	0	0
$459$	−1.38197	−0.0645046
$460$	0	0
$461$	1.34752	0.0627605	0.0313802	−	0.999508i	$-0.490010\pi$
$461$	1.34752	0.0627605	0.0313802	+	0.999508i	$0.490010\pi$
$462$	0	0
$463$	19.4164	0.902357	0.451178	−	0.892434i	$-0.351004\pi$
$463$	19.4164	0.902357	0.451178	+	0.892434i	$0.351004\pi$
$464$	0	0
$465$	2.47214	0.114643
$466$	0	0
$467$	33.3050	1.54117	0.770585	−	0.637338i	$-0.219964\pi$
$467$	33.3050	1.54117	0.770585	+	0.637338i	$0.219964\pi$
$468$	0	0
$469$	−22.1803	−1.02419
$470$	0	0
$471$	−3.70820	−0.170865
$472$	0	0
$473$	0	0
$474$	0	0
$475$	32.0344	1.46984
$476$	0	0
$477$	4.36068	0.199662
$478$	0	0
$479$	34.4721	1.57507	0.787536	−	0.616269i	$-0.211356\pi$
$479$	34.4721	1.57507	0.787536	+	0.616269i	$0.211356\pi$
$480$	0	0
$481$	−4.58359	−0.208994
$482$	0	0
$483$	0.944272	0.0429659
$484$	0	0
$485$	23.1246	1.05003
$486$	0	0
$487$	1.34752	0.0610621	0.0305311	−	0.999534i	$-0.490280\pi$
$487$	1.34752	0.0610621	0.0305311	+	0.999534i	$0.490280\pi$
$488$	0	0
$489$	0.347524	0.0157156
$490$	0	0
$491$	−22.8541	−1.03139	−0.515696	−	0.856772i	$-0.672466\pi$
$491$	−22.8541	−1.03139	−0.515696	+	0.856772i	$0.672466\pi$
$492$	0	0
$493$	−2.76393	−0.124481
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.4721	0.469739
$498$	0	0
$499$	30.8541	1.38122	0.690610	−	0.723228i	$-0.257342\pi$
$499$	30.8541	1.38122	0.690610	+	0.723228i	$0.257342\pi$
$500$	0	0
$501$	−5.63932	−0.251946
$502$	0	0
$503$	3.23607	0.144289	0.0721446	−	0.997394i	$-0.477016\pi$
$503$	3.23607	0.144289	0.0721446	+	0.997394i	$0.477016\pi$
$504$	0	0
$505$	−13.5279	−0.601982
$506$	0	0
$507$	4.38197	0.194610
$508$	0	0
$509$	−8.29180	−0.367527	−0.183764	−	0.982970i	$-0.558828\pi$
$509$	−8.29180	−0.367527	−0.183764	+	0.982970i	$0.558828\pi$
$510$	0	0
$511$	20.7639	0.918542
$512$	0	0
$513$	13.0902	0.577945
$514$	0	0
$515$	−50.8328	−2.23996
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−8.36068	−0.366993
$520$	0	0
$521$	−20.0344	−0.877725	−0.438862	−	0.898554i	$-0.644618\pi$
$521$	−20.0344	−0.877725	−0.438862	+	0.898554i	$0.644618\pi$
$522$	0	0
$523$	−9.20163	−0.402359	−0.201180	−	0.979554i	$-0.564477\pi$
$523$	−9.20163	−0.402359	−0.201180	+	0.979554i	$0.564477\pi$
$524$	0	0
$525$	−4.18034	−0.182445
$526$	0	0
$527$	1.23607	0.0538440
$528$	0	0
$529$	−21.4721	−0.933571
$530$	0	0
$531$	−24.5967	−1.06741
$532$	0	0
$533$	6.94427	0.300790
$534$	0	0
$535$	3.70820	0.160320
$536$	0	0
$537$	−3.29180	−0.142051
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.12461	−0.134338	−0.0671688	−	0.997742i	$-0.521397\pi$
$541$	−3.12461	−0.134338	−0.0671688	+	0.997742i	$0.521397\pi$
$542$	0	0
$543$	1.59675	0.0685230
$544$	0	0
$545$	61.3050	2.62602
$546$	0	0
$547$	−8.32624	−0.356004	−0.178002	−	0.984030i	$-0.556963\pi$
$547$	−8.32624	−0.356004	−0.178002	+	0.984030i	$0.556963\pi$
$548$	0	0
$549$	7.05573	0.301131
$550$	0	0
$551$	26.1803	1.11532
$552$	0	0
$553$	−26.8328	−1.14105
$554$	0	0
$555$	4.58359	0.194563
$556$	0	0
$557$	35.2361	1.49300	0.746500	−	0.665385i	$-0.231733\pi$
$557$	35.2361	1.49300	0.746500	+	0.665385i	$0.231733\pi$
$558$	0	0
$559$	10.5836	0.447638
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−0.381966	−0.0160979	−0.00804897	−	0.999968i	$-0.502562\pi$
$563$	−0.381966	−0.0160979	−0.00804897	+	0.999968i	$0.502562\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	15.4164	0.647428
$568$	0	0
$569$	38.2148	1.60205	0.801023	−	0.598633i	$-0.204289\pi$
$569$	38.2148	1.60205	0.801023	+	0.598633i	$0.204289\pi$
$570$	0	0
$571$	2.47214	0.103456	0.0517278	−	0.998661i	$-0.483527\pi$
$571$	2.47214	0.103456	0.0517278	+	0.998661i	$0.483527\pi$
$572$	0	0
$573$	1.81966	0.0760174
$574$	0	0
$575$	−6.76393	−0.282075
$576$	0	0
$577$	−6.79837	−0.283020	−0.141510	−	0.989937i	$-0.545196\pi$
$577$	−6.79837	−0.283020	−0.141510	+	0.989937i	$0.545196\pi$
$578$	0	0
$579$	−6.65248	−0.276467
$580$	0	0
$581$	−18.6525	−0.773835
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−11.4164	−0.472010
$586$	0	0
$587$	−22.1459	−0.914059	−0.457030	−	0.889452i	$-0.651087\pi$
$587$	−22.1459	−0.914059	−0.457030	+	0.889452i	$0.651087\pi$
$588$	0	0
$589$	−11.7082	−0.482428
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	−28.6869	−1.17803	−0.589015	−	0.808122i	$-0.700484\pi$
$593$	−28.6869	−1.17803	−0.589015	+	0.808122i	$0.700484\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	−7.23607	−0.296153
$598$	0	0
$599$	13.4164	0.548180	0.274090	−	0.961704i	$-0.411623\pi$
$599$	13.4164	0.548180	0.274090	+	0.961704i	$0.411623\pi$
$600$	0	0
$601$	−27.1459	−1.10730	−0.553652	−	0.832748i	$-0.686766\pi$
$601$	−27.1459	−1.10730	−0.553652	+	0.832748i	$0.686766\pi$
$602$	0	0
$603$	31.6525	1.28899
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.1246	−0.938599	−0.469300	−	0.883039i	$-0.655494\pi$
$607$	−23.1246	−0.938599	−0.469300	+	0.883039i	$0.655494\pi$
$608$	0	0
$609$	−3.41641	−0.138440
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	35.7082	1.44224	0.721120	−	0.692810i	$-0.243627\pi$
$613$	35.7082	1.44224	0.721120	+	0.692810i	$0.243627\pi$
$614$	0	0
$615$	−6.94427	−0.280020
$616$	0	0
$617$	23.4508	0.944096	0.472048	−	0.881573i	$-0.343515\pi$
$617$	23.4508	0.944096	0.472048	+	0.881573i	$0.343515\pi$
$618$	0	0
$619$	−5.20163	−0.209071	−0.104536	−	0.994521i	$-0.533336\pi$
$619$	−5.20163	−0.209071	−0.104536	+	0.994521i	$0.533336\pi$
$620$	0	0
$621$	−2.76393	−0.110913
$622$	0	0
$623$	−16.1803	−0.648252
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.29180	0.0913799
$630$	0	0
$631$	−23.7082	−0.943809	−0.471904	−	0.881650i	$-0.656433\pi$
$631$	−23.7082	−0.943809	−0.471904	+	0.881650i	$0.656433\pi$
$632$	0	0
$633$	−2.32624	−0.0924597
$634$	0	0
$635$	35.4164	1.40546
$636$	0	0
$637$	−3.70820	−0.146924
$638$	0	0
$639$	−14.9443	−0.591186
$640$	0	0
$641$	16.6738	0.658574	0.329287	−	0.944230i	$-0.393192\pi$
$641$	16.6738	0.658574	0.329287	+	0.944230i	$0.393192\pi$
$642$	0	0
$643$	−39.9787	−1.57661	−0.788303	−	0.615287i	$-0.789040\pi$
$643$	−39.9787	−1.57661	−0.788303	+	0.615287i	$0.789040\pi$
$644$	0	0
$645$	−10.5836	−0.416729
$646$	0	0
$647$	−29.0557	−1.14230	−0.571149	−	0.820846i	$-0.693502\pi$
$647$	−29.0557	−1.14230	−0.571149	+	0.820846i	$0.693502\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1.52786	0.0598817
$652$	0	0
$653$	−15.3475	−0.600595	−0.300298	−	0.953846i	$-0.597086\pi$
$653$	−15.3475	−0.600595	−0.300298	+	0.953846i	$0.597086\pi$
$654$	0	0
$655$	−22.0000	−0.859611
$656$	0	0
$657$	−29.6312	−1.15602
$658$	0	0
$659$	16.9098	0.658713	0.329357	−	0.944206i	$-0.393168\pi$
$659$	16.9098	0.658713	0.329357	+	0.944206i	$0.393168\pi$
$660$	0	0
$661$	−3.52786	−0.137218	−0.0686090	−	0.997644i	$-0.521856\pi$
$661$	−3.52786	−0.137218	−0.0686090	+	0.997644i	$0.521856\pi$
$662$	0	0
$663$	0.291796	0.0113324
$664$	0	0
$665$	37.8885	1.46925
$666$	0	0
$667$	−5.52786	−0.214040
$668$	0	0
$669$	9.81966	0.379650
$670$	0	0
$671$	0	0
$672$	0	0
$673$	9.85410	0.379848	0.189924	−	0.981799i	$-0.439176\pi$
$673$	9.85410	0.379848	0.189924	+	0.981799i	$0.439176\pi$
$674$	0	0
$675$	12.2361	0.470966
$676$	0	0
$677$	−14.3607	−0.551926	−0.275963	−	0.961168i	$-0.588997\pi$
$677$	−14.3607	−0.551926	−0.275963	+	0.961168i	$0.588997\pi$
$678$	0	0
$679$	14.2918	0.548469
$680$	0	0
$681$	5.54102	0.212332
$682$	0	0
$683$	−18.4721	−0.706817	−0.353408	−	0.935469i	$-0.614977\pi$
$683$	−18.4721	−0.706817	−0.353408	+	0.935469i	$0.614977\pi$
$684$	0	0
$685$	52.0689	1.98945
$686$	0	0
$687$	−0.652476	−0.0248935
$688$	0	0
$689$	−1.88854	−0.0719478
$690$	0	0
$691$	26.2148	0.997257	0.498629	−	0.866816i	$-0.333837\pi$
$691$	26.2148	0.997257	0.498629	+	0.866816i	$0.333837\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.47214	−0.131517
$698$	0	0
$699$	3.67376	0.138954
$700$	0	0
$701$	−14.8328	−0.560228	−0.280114	−	0.959967i	$-0.590372\pi$
$701$	−14.8328	−0.560228	−0.280114	+	0.959967i	$0.590372\pi$
$702$	0	0
$703$	−21.7082	−0.818740
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	−8.36068	−0.314436
$708$	0	0
$709$	24.4721	0.919070	0.459535	−	0.888160i	$-0.348016\pi$
$709$	24.4721	0.919070	0.459535	+	0.888160i	$0.348016\pi$
$710$	0	0
$711$	38.2918	1.43605
$712$	0	0
$713$	2.47214	0.0925822
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.29180	−0.309663
$718$	0	0
$719$	−26.8328	−1.00070	−0.500348	−	0.865825i	$-0.666794\pi$
$719$	−26.8328	−1.00070	−0.500348	+	0.865825i	$0.666794\pi$
$720$	0	0
$721$	−31.4164	−1.17001
$722$	0	0
$723$	4.23607	0.157541
$724$	0	0
$725$	24.4721	0.908872
$726$	0	0
$727$	−2.87539	−0.106642	−0.0533211	−	0.998577i	$-0.516981\pi$
$727$	−2.87539	−0.106642	−0.0533211	+	0.998577i	$0.516981\pi$
$728$	0	0
$729$	−19.4377	−0.719915
$730$	0	0
$731$	−5.29180	−0.195724
$732$	0	0
$733$	−9.41641	−0.347803	−0.173901	−	0.984763i	$-0.555637\pi$
$733$	−9.41641	−0.347803	−0.173901	+	0.984763i	$0.555637\pi$
$734$	0	0
$735$	3.70820	0.136779
$736$	0	0
$737$	0	0
$738$	0	0
$739$	37.5623	1.38175	0.690876	−	0.722973i	$-0.257225\pi$
$739$	37.5623	1.38175	0.690876	+	0.722973i	$0.257225\pi$
$740$	0	0
$741$	−2.76393	−0.101536
$742$	0	0
$743$	8.76393	0.321517	0.160759	−	0.986994i	$-0.448606\pi$
$743$	8.76393	0.321517	0.160759	+	0.986994i	$0.448606\pi$
$744$	0	0
$745$	−20.0000	−0.732743
$746$	0	0
$747$	26.6180	0.973903
$748$	0	0
$749$	2.29180	0.0837404
$750$	0	0
$751$	33.7771	1.23254	0.616272	−	0.787534i	$-0.288642\pi$
$751$	33.7771	1.23254	0.616272	+	0.787534i	$0.288642\pi$
$752$	0	0
$753$	8.00000	0.291536
$754$	0	0
$755$	25.8885	0.942181
$756$	0	0
$757$	−2.65248	−0.0964059	−0.0482029	−	0.998838i	$-0.515349\pi$
$757$	−2.65248	−0.0964059	−0.0482029	+	0.998838i	$0.515349\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.25735	−0.154329	−0.0771645	−	0.997018i	$-0.524587\pi$
$761$	−4.25735	−0.154329	−0.0771645	+	0.997018i	$0.524587\pi$
$762$	0	0
$763$	37.8885	1.37166
$764$	0	0
$765$	5.70820	0.206381
$766$	0	0
$767$	10.6525	0.384639
$768$	0	0
$769$	13.4164	0.483808	0.241904	−	0.970300i	$-0.422228\pi$
$769$	13.4164	0.483808	0.241904	+	0.970300i	$0.422228\pi$
$770$	0	0
$771$	2.14590	0.0772826
$772$	0	0
$773$	−3.88854	−0.139861	−0.0699306	−	0.997552i	$-0.522278\pi$
$773$	−3.88854	−0.139861	−0.0699306	+	0.997552i	$0.522278\pi$
$774$	0	0
$775$	−10.9443	−0.393130
$776$	0	0
$777$	2.83282	0.101627
$778$	0	0
$779$	32.8885	1.17835
$780$	0	0
$781$	0	0
$782$	0	0
$783$	10.0000	0.357371
$784$	0	0
$785$	31.4164	1.12130
$786$	0	0
$787$	−26.2148	−0.934456	−0.467228	−	0.884137i	$-0.654747\pi$
$787$	−26.2148	−0.934456	−0.467228	+	0.884137i	$0.654747\pi$
$788$	0	0
$789$	−8.87539	−0.315972
$790$	0	0
$791$	−3.70820	−0.131849
$792$	0	0
$793$	−3.05573	−0.108512
$794$	0	0
$795$	1.88854	0.0669797
$796$	0	0
$797$	−24.7639	−0.877183	−0.438592	−	0.898686i	$-0.644523\pi$
$797$	−24.7639	−0.877183	−0.438592	+	0.898686i	$0.644523\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	23.0902	0.815851
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	−2.36068	−0.0830999
$808$	0	0
$809$	16.9098	0.594518	0.297259	−	0.954797i	$-0.403928\pi$
$809$	16.9098	0.594518	0.297259	+	0.954797i	$0.403928\pi$
$810$	0	0
$811$	−8.38197	−0.294331	−0.147165	−	0.989112i	$-0.547015\pi$
$811$	−8.38197	−0.294331	−0.147165	+	0.989112i	$0.547015\pi$
$812$	0	0
$813$	0.763932	0.0267923
$814$	0	0
$815$	−2.94427	−0.103133
$816$	0	0
$817$	50.1246	1.75364
$818$	0	0
$819$	−7.05573	−0.246547
$820$	0	0
$821$	30.5410	1.06589	0.532944	−	0.846150i	$-0.321085\pi$
$821$	30.5410	1.06589	0.532944	+	0.846150i	$0.321085\pi$
$822$	0	0
$823$	−23.5967	−0.822531	−0.411265	−	0.911516i	$-0.634913\pi$
$823$	−23.5967	−0.822531	−0.411265	+	0.911516i	$0.634913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.3820	−1.36945	−0.684723	−	0.728804i	$-0.740077\pi$
$827$	−39.3820	−1.36945	−0.684723	+	0.728804i	$0.740077\pi$
$828$	0	0
$829$	31.0557	1.07861	0.539305	−	0.842111i	$-0.318687\pi$
$829$	31.0557	1.07861	0.539305	+	0.842111i	$0.318687\pi$
$830$	0	0
$831$	9.05573	0.314140
$832$	0	0
$833$	1.85410	0.0642408
$834$	0	0
$835$	47.7771	1.65339
$836$	0	0
$837$	−4.47214	−0.154580
$838$	0	0
$839$	−12.1115	−0.418134	−0.209067	−	0.977901i	$-0.567043\pi$
$839$	−12.1115	−0.418134	−0.209067	+	0.977901i	$0.567043\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	6.14590	0.211676
$844$	0	0
$845$	−37.1246	−1.27713
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.16718	0.314617
$850$	0	0
$851$	4.58359	0.157124
$852$	0	0
$853$	−30.4721	−1.04335	−0.521673	−	0.853146i	$-0.674692\pi$
$853$	−30.4721	−1.04335	−0.521673	+	0.853146i	$0.674692\pi$
$854$	0	0
$855$	−54.0689	−1.84912
$856$	0	0
$857$	−10.7426	−0.366962	−0.183481	−	0.983023i	$-0.558737\pi$
$857$	−10.7426	−0.366962	−0.183481	+	0.983023i	$0.558737\pi$
$858$	0	0
$859$	−46.5066	−1.58678	−0.793392	−	0.608711i	$-0.791687\pi$
$859$	−46.5066	−1.58678	−0.793392	+	0.608711i	$0.791687\pi$
$860$	0	0
$861$	−4.29180	−0.146264
$862$	0	0
$863$	33.2361	1.13137	0.565684	−	0.824622i	$-0.308612\pi$
$863$	33.2361	1.13137	0.565684	+	0.824622i	$0.308612\pi$
$864$	0	0
$865$	70.8328	2.40839
$866$	0	0
$867$	6.34752	0.215573
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−13.7082	−0.464485
$872$	0	0
$873$	−20.3951	−0.690270
$874$	0	0
$875$	3.05573	0.103302
$876$	0	0
$877$	−28.1803	−0.951582	−0.475791	−	0.879558i	$-0.657838\pi$
$877$	−28.1803	−0.951582	−0.475791	+	0.879558i	$0.657838\pi$
$878$	0	0
$879$	10.8328	0.365382
$880$	0	0
$881$	−46.3394	−1.56121	−0.780607	−	0.625022i	$-0.785090\pi$
$881$	−46.3394	−1.56121	−0.780607	+	0.625022i	$0.785090\pi$
$882$	0	0
$883$	3.43769	0.115688	0.0578438	−	0.998326i	$-0.481577\pi$
$883$	3.43769	0.115688	0.0578438	+	0.998326i	$0.481577\pi$
$884$	0	0
$885$	−10.6525	−0.358079
$886$	0	0
$887$	37.7771	1.26843	0.634215	−	0.773157i	$-0.281323\pi$
$887$	37.7771	1.26843	0.634215	+	0.773157i	$0.281323\pi$
$888$	0	0
$889$	21.8885	0.734118
$890$	0	0
$891$	0	0
$892$	0	0
$893$	37.8885	1.26789
$894$	0	0
$895$	27.8885	0.932211
$896$	0	0
$897$	0.583592	0.0194856
$898$	0	0
$899$	−8.94427	−0.298308
$900$	0	0
$901$	0.944272	0.0314583
$902$	0	0
$903$	−6.54102	−0.217672
$904$	0	0
$905$	−13.5279	−0.449681
$906$	0	0
$907$	−10.4377	−0.346578	−0.173289	−	0.984871i	$-0.555439\pi$
$907$	−10.4377	−0.346578	−0.173289	+	0.984871i	$0.555439\pi$
$908$	0	0
$909$	11.9311	0.395730
$910$	0	0
$911$	−18.1803	−0.602342	−0.301171	−	0.953570i	$-0.597377\pi$
$911$	−18.1803	−0.602342	−0.301171	+	0.953570i	$0.597377\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.05573	0.101019
$916$	0	0
$917$	−13.5967	−0.449004
$918$	0	0
$919$	−43.4164	−1.43218	−0.716088	−	0.698010i	$-0.754069\pi$
$919$	−43.4164	−1.43218	−0.716088	+	0.698010i	$0.754069\pi$
$920$	0	0
$921$	10.6180	0.349876
$922$	0	0
$923$	6.47214	0.213033
$924$	0	0
$925$	−20.2918	−0.667190
$926$	0	0
$927$	44.8328	1.47250
$928$	0	0
$929$	−16.2574	−0.533386	−0.266693	−	0.963781i	$-0.585931\pi$
$929$	−16.2574	−0.533386	−0.266693	+	0.963781i	$0.585931\pi$
$930$	0	0
$931$	−17.5623	−0.575581
$932$	0	0
$933$	9.95743	0.325992
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37.8541	−1.23664	−0.618320	−	0.785927i	$-0.712186\pi$
$937$	−37.8541	−1.23664	−0.618320	+	0.785927i	$0.712186\pi$
$938$	0	0
$939$	−1.65248	−0.0539265
$940$	0	0
$941$	48.1803	1.57063	0.785317	−	0.619094i	$-0.212500\pi$
$941$	48.1803	1.57063	0.785317	+	0.619094i	$0.212500\pi$
$942$	0	0
$943$	−6.94427	−0.226137
$944$	0	0
$945$	14.4721	0.470779
$946$	0	0
$947$	30.2148	0.981848	0.490924	−	0.871202i	$-0.336659\pi$
$947$	30.2148	0.981848	0.490924	+	0.871202i	$0.336659\pi$
$948$	0	0
$949$	12.8328	0.416571
$950$	0	0
$951$	1.41641	0.0459302
$952$	0	0
$953$	−11.3262	−0.366893	−0.183446	−	0.983030i	$-0.558725\pi$
$953$	−11.3262	−0.366893	−0.183446	+	0.983030i	$0.558725\pi$
$954$	0	0
$955$	−15.4164	−0.498863
$956$	0	0
$957$	0	0
$958$	0	0
$959$	32.1803	1.03916
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−3.27051	−0.105391
$964$	0	0
$965$	56.3607	1.81431
$966$	0	0
$967$	−38.0000	−1.22200	−0.610999	−	0.791632i	$-0.709232\pi$
$967$	−38.0000	−1.22200	−0.610999	+	0.791632i	$0.709232\pi$
$968$	0	0
$969$	1.38197	0.0443951
$970$	0	0
$971$	10.1115	0.324492	0.162246	−	0.986750i	$-0.448126\pi$
$971$	10.1115	0.324492	0.162246	+	0.986750i	$0.448126\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.58359	−0.0827412
$976$	0	0
$977$	−43.3050	−1.38545	−0.692724	−	0.721203i	$-0.743590\pi$
$977$	−43.3050	−1.38545	−0.692724	+	0.721203i	$0.743590\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−54.0689	−1.72629
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	−67.7771	−2.15956
$986$	0	0
$987$	−4.94427	−0.157378
$988$	0	0
$989$	−10.5836	−0.336539
$990$	0	0
$991$	−30.5410	−0.970167	−0.485084	−	0.874468i	$-0.661211\pi$
$991$	−30.5410	−0.970167	−0.485084	+	0.874468i	$0.661211\pi$
$992$	0	0
$993$	2.39512	0.0760069
$994$	0	0
$995$	61.3050	1.94350
$996$	0	0
$997$	8.65248	0.274027	0.137013	−	0.990569i	$-0.456250\pi$
$997$	8.65248	0.274027	0.137013	+	0.990569i	$0.456250\pi$
$998$	0	0
$999$	−8.29180	−0.262341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.2.a.o.1.2		2
4.3	odd	2		242.2.a.f.1.1		2
8.3	odd	2		7744.2.a.bm.1.2		2
8.5	even	2		7744.2.a.cz.1.1		2
11.5	even	5		176.2.m.c.113.1		4
11.9	even	5		176.2.m.c.81.1		4
11.10	odd	2		1936.2.a.n.1.2		2
12.11	even	2		2178.2.a.p.1.1		2
20.19	odd	2		6050.2.a.bs.1.2		2
44.3	odd	10		242.2.c.a.9.1		4
44.7	even	10		242.2.c.d.27.1		4
44.15	odd	10		242.2.c.a.27.1		4
44.19	even	10		242.2.c.d.9.1		4
44.27	odd	10		22.2.c.a.3.1	✓	4
44.31	odd	10		22.2.c.a.15.1	yes	4
44.35	even	10		242.2.c.c.81.1		4
44.39	even	10		242.2.c.c.3.1		4
44.43	even	2		242.2.a.d.1.1		2
88.5	even	10		704.2.m.a.641.1		4
88.21	odd	2		7744.2.a.cy.1.1		2
88.27	odd	10		704.2.m.h.641.1		4
88.43	even	2		7744.2.a.bn.1.2		2
88.53	even	10		704.2.m.a.257.1		4
88.75	odd	10		704.2.m.h.257.1		4
132.71	even	10		198.2.f.e.91.1		4
132.119	even	10		198.2.f.e.37.1		4
132.131	odd	2		2178.2.a.x.1.1		2
220.27	even	20		550.2.ba.c.399.1		8
220.119	odd	10		550.2.h.h.301.1		4
220.159	odd	10		550.2.h.h.201.1		4
220.163	even	20		550.2.ba.c.499.1		8
220.203	even	20		550.2.ba.c.399.2		8
220.207	even	20		550.2.ba.c.499.2		8
220.219	even	2		6050.2.a.ci.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.2.c.a.3.1	✓	4	44.27	odd	10
22.2.c.a.15.1	yes	4	44.31	odd	10
176.2.m.c.81.1		4	11.9	even	5
176.2.m.c.113.1		4	11.5	even	5
198.2.f.e.37.1		4	132.119	even	10
198.2.f.e.91.1		4	132.71	even	10
242.2.a.d.1.1		2	44.43	even	2
242.2.a.f.1.1		2	4.3	odd	2
242.2.c.a.9.1		4	44.3	odd	10
242.2.c.a.27.1		4	44.15	odd	10
242.2.c.c.3.1		4	44.39	even	10
242.2.c.c.81.1		4	44.35	even	10
242.2.c.d.9.1		4	44.19	even	10
242.2.c.d.27.1		4	44.7	even	10
550.2.h.h.201.1		4	220.159	odd	10
550.2.h.h.301.1		4	220.119	odd	10
550.2.ba.c.399.1		8	220.27	even	20
550.2.ba.c.399.2		8	220.203	even	20
550.2.ba.c.499.1		8	220.163	even	20
550.2.ba.c.499.2		8	220.207	even	20
704.2.m.a.257.1		4	88.53	even	10
704.2.m.a.641.1		4	88.5	even	10
704.2.m.h.257.1		4	88.75	odd	10
704.2.m.h.641.1		4	88.27	odd	10
1936.2.a.n.1.2		2	11.10	odd	2
1936.2.a.o.1.2		2	1.1	even	1	trivial
2178.2.a.p.1.1		2	12.11	even	2
2178.2.a.x.1.1		2	132.131	odd	2
6050.2.a.bs.1.2		2	20.19	odd	2
6050.2.a.ci.1.2		2	220.219	even	2
7744.2.a.bm.1.2		2	8.3	odd	2
7744.2.a.bn.1.2		2	88.43	even	2
7744.2.a.cy.1.1		2	88.21	odd	2
7744.2.a.cz.1.1		2	8.5	even	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:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{3} +3.23607 q^{5} +2.00000 q^{7} -2.85410 q^{9} +1.23607 q^{13} -1.23607 q^{15} -0.618034 q^{17} +5.85410 q^{19} -0.763932 q^{21} -1.23607 q^{23} +5.47214 q^{25} +2.23607 q^{27} +4.47214 q^{29} -2.00000 q^{31} +6.47214 q^{35} -3.70820 q^{37} -0.472136 q^{39} +5.61803 q^{41} +8.56231 q^{43} -9.23607 q^{45} +6.47214 q^{47} -3.00000 q^{49} +0.236068 q^{51} -1.52786 q^{53} -2.23607 q^{57} +8.61803 q^{59} -2.47214 q^{61} -5.70820 q^{63} +4.00000 q^{65} -11.0902 q^{67} +0.472136 q^{69} +5.23607 q^{71} +10.3820 q^{73} -2.09017 q^{75} -13.4164 q^{79} +7.70820 q^{81} -9.32624 q^{83} -2.00000 q^{85} -1.70820 q^{87} -8.09017 q^{89} +2.47214 q^{91} +0.763932 q^{93} +18.9443 q^{95} +7.14590 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + 2 q^{5} + 4 q^{7} + q^{9} - 2 q^{13} + 2 q^{15} + q^{17} + 5 q^{19} - 6 q^{21} + 2 q^{23} + 2 q^{25} - 4 q^{31} + 4 q^{35} + 6 q^{37} + 8 q^{39} + 9 q^{41} - 3 q^{43} - 14 q^{45} + 4 q^{47}+ \cdots + 21 q^{97}+O(q^{100}) \) 2 * q - 3 * q^3 + 2 * q^5 + 4 * q^7 + q^9 - 2 * q^13 + 2 * q^15 + q^17 + 5 * q^19 - 6 * q^21 + 2 * q^23 + 2 * q^25 - 4 * q^31 + 4 * q^35 + 6 * q^37 + 8 * q^39 + 9 * q^41 - 3 * q^43 - 14 * q^45 + 4 * q^47 - 6 * q^49 - 4 * q^51 - 12 * q^53 + 15 * q^59 + 4 * q^61 + 2 * q^63 + 8 * q^65 - 11 * q^67 - 8 * q^69 + 6 * q^71 + 23 * q^73 + 7 * q^75 + 2 * q^81 - 3 * q^83 - 4 * q^85 + 10 * q^87 - 5 * q^89 - 4 * q^91 + 6 * q^93 + 20 * q^95 + 21 * q^97

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