LMFDB - Embedded newform 7600.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3800)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53209 q^{3} -2.22668 q^{7} -0.652704 q^{9} +3.22668 q^{11} -1.57398 q^{13} -3.53209 q^{17} -1.00000 q^{19} +3.41147 q^{21} +4.47565 q^{23} +5.59627 q^{27} +1.92127 q^{29} +3.81521 q^{31} -4.94356 q^{33} -11.3550 q^{37} +2.41147 q^{39} +3.47565 q^{41} -1.69459 q^{43} +1.57398 q^{47} -2.04189 q^{49} +5.41147 q^{51} +7.12836 q^{53} +1.53209 q^{57} +7.88713 q^{59} -2.79561 q^{61} +1.45336 q^{63} +3.22668 q^{67} -6.85710 q^{69} +4.38919 q^{71} +6.41147 q^{73} -7.18479 q^{77} +8.59627 q^{79} -6.61587 q^{81} -14.6236 q^{83} -2.94356 q^{87} +6.10607 q^{89} +3.50475 q^{91} -5.84524 q^{93} -15.7023 q^{97} -2.10607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{17} - 3 q^{19} - 6 q^{23} + 3 q^{27} - 3 q^{29} + 15 q^{31} - 9 q^{37} - 3 q^{39} - 9 q^{41} - 3 q^{43} - 3 q^{47} - 3 q^{49} + 6 q^{51} + 3 q^{53} - 6 q^{59}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^17 - 3 * q^19 - 6 * q^23 + 3 * q^27 - 3 * q^29 + 15 * q^31 - 9 * q^37 - 3 * q^39 - 9 * q^41 - 3 * q^43 - 3 * q^47 - 3 * q^49 + 6 * q^51 + 3 * q^53 - 6 * q^59 - 9 * q^61 - 9 * q^63 + 3 * q^67 - 21 * q^69 + 9 * q^71 + 9 * q^73 - 18 * q^77 + 12 * q^79 - 9 * q^81 - 9 * q^83 + 6 * q^87 + 6 * q^89 + 27 * q^91 + 9 * q^93 - 21 * q^97 + 6 * 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
$3$	−1.53209	−0.884552	−0.442276	−	0.896879i	$-0.645829\pi$
$3$	−1.53209	−0.884552	−0.442276	+	0.896879i	$0.645829\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.22668	−0.841607	−0.420803	−	0.907152i	$-0.638252\pi$
$7$	−2.22668	−0.841607	−0.420803	+	0.907152i	$0.638252\pi$
$8$	0	0
$9$	−0.652704	−0.217568
$10$	0	0
$11$	3.22668	0.972881	0.486441	−	0.873714i	$-0.338295\pi$
$11$	3.22668	0.972881	0.486441	+	0.873714i	$0.338295\pi$
$12$	0	0
$13$	−1.57398	−0.436543	−0.218271	−	0.975888i	$-0.570042\pi$
$13$	−1.57398	−0.436543	−0.218271	+	0.975888i	$0.570042\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.53209	−0.856657	−0.428329	−	0.903623i	$-0.640897\pi$
$17$	−3.53209	−0.856657	−0.428329	+	0.903623i	$0.640897\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.41147	0.744445
$22$	0	0
$23$	4.47565	0.933238	0.466619	−	0.884458i	$-0.345472\pi$
$23$	4.47565	0.933238	0.466619	+	0.884458i	$0.345472\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.59627	1.07700
$28$	0	0
$29$	1.92127	0.356772	0.178386	−	0.983961i	$-0.442912\pi$
$29$	1.92127	0.356772	0.178386	+	0.983961i	$0.442912\pi$
$30$	0	0
$31$	3.81521	0.685231	0.342616	−	0.939476i	$-0.388687\pi$
$31$	3.81521	0.685231	0.342616	+	0.939476i	$0.388687\pi$
$32$	0	0
$33$	−4.94356	−0.860564
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.3550	−1.86676	−0.933378	−	0.358894i	$-0.883154\pi$
$37$	−11.3550	−1.86676	−0.933378	+	0.358894i	$0.883154\pi$
$38$	0	0
$39$	2.41147	0.386145
$40$	0	0
$41$	3.47565	0.542806	0.271403	−	0.962466i	$-0.412512\pi$
$41$	3.47565	0.542806	0.271403	+	0.962466i	$0.412512\pi$
$42$	0	0
$43$	−1.69459	−0.258423	−0.129211	−	0.991617i	$-0.541245\pi$
$43$	−1.69459	−0.258423	−0.129211	+	0.991617i	$0.541245\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.57398	0.229588	0.114794	−	0.993389i	$-0.463379\pi$
$47$	1.57398	0.229588	0.114794	+	0.993389i	$0.463379\pi$
$48$	0	0
$49$	−2.04189	−0.291698
$50$	0	0
$51$	5.41147	0.757758
$52$	0	0
$53$	7.12836	0.979155	0.489577	−	0.871960i	$-0.337151\pi$
$53$	7.12836	0.979155	0.489577	+	0.871960i	$0.337151\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.53209	0.202930
$58$	0	0
$59$	7.88713	1.02682	0.513408	−	0.858145i	$-0.328383\pi$
$59$	7.88713	1.02682	0.513408	+	0.858145i	$0.328383\pi$
$60$	0	0
$61$	−2.79561	−0.357941	−0.178970	−	0.983854i	$-0.557277\pi$
$61$	−2.79561	−0.357941	−0.178970	+	0.983854i	$0.557277\pi$
$62$	0	0
$63$	1.45336	0.183107
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.22668	0.394202	0.197101	−	0.980383i	$-0.436847\pi$
$67$	3.22668	0.394202	0.197101	+	0.980383i	$0.436847\pi$
$68$	0	0
$69$	−6.85710	−0.825497
$70$	0	0
$71$	4.38919	0.520900	0.260450	−	0.965487i	$-0.416129\pi$
$71$	4.38919	0.520900	0.260450	+	0.965487i	$0.416129\pi$
$72$	0	0
$73$	6.41147	0.750406	0.375203	−	0.926943i	$-0.377573\pi$
$73$	6.41147	0.750406	0.375203	+	0.926943i	$0.377573\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.18479	−0.818783
$78$	0	0
$79$	8.59627	0.967156	0.483578	−	0.875301i	$-0.339337\pi$
$79$	8.59627	0.967156	0.483578	+	0.875301i	$0.339337\pi$
$80$	0	0
$81$	−6.61587	−0.735096
$82$	0	0
$83$	−14.6236	−1.60515	−0.802575	−	0.596552i	$-0.796537\pi$
$83$	−14.6236	−1.60515	−0.802575	+	0.596552i	$0.796537\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.94356	−0.315583
$88$	0	0
$89$	6.10607	0.647242	0.323621	−	0.946187i	$-0.395100\pi$
$89$	6.10607	0.647242	0.323621	+	0.946187i	$0.395100\pi$
$90$	0	0
$91$	3.50475	0.367397
$92$	0	0
$93$	−5.84524	−0.606123
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.7023	−1.59433	−0.797165	−	0.603761i	$-0.793668\pi$
$97$	−15.7023	−1.59433	−0.797165	+	0.603761i	$0.793668\pi$
$98$	0	0
$99$	−2.10607	−0.211668
$100$	0	0
$101$	4.35504	0.433342	0.216671	−	0.976245i	$-0.430480\pi$
$101$	4.35504	0.433342	0.216671	+	0.976245i	$0.430480\pi$
$102$	0	0
$103$	4.27126	0.420860	0.210430	−	0.977609i	$-0.432514\pi$
$103$	4.27126	0.420860	0.210430	+	0.977609i	$0.432514\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.106067	0.0102539	0.00512693	−	0.999987i	$-0.498368\pi$
$107$	0.106067	0.0102539	0.00512693	+	0.999987i	$0.498368\pi$
$108$	0	0
$109$	−14.4115	−1.38037	−0.690184	−	0.723634i	$-0.742471\pi$
$109$	−14.4115	−1.38037	−0.690184	+	0.723634i	$0.742471\pi$
$110$	0	0
$111$	17.3969	1.65124
$112$	0	0
$113$	16.5895	1.56061	0.780303	−	0.625402i	$-0.215065\pi$
$113$	16.5895	1.56061	0.780303	+	0.625402i	$0.215065\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.02734	0.0949777
$118$	0	0
$119$	7.86484	0.720968
$120$	0	0
$121$	−0.588526	−0.0535024
$122$	0	0
$123$	−5.32501	−0.480140
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.02229	−0.356920	−0.178460	−	0.983947i	$-0.557112\pi$
$127$	−4.02229	−0.356920	−0.178460	+	0.983947i	$0.557112\pi$
$128$	0	0
$129$	2.59627	0.228589
$130$	0	0
$131$	4.32770	0.378113	0.189056	−	0.981966i	$-0.439457\pi$
$131$	4.32770	0.378113	0.189056	+	0.981966i	$0.439457\pi$
$132$	0	0
$133$	2.22668	0.193078
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.0223	−0.941698	−0.470849	−	0.882214i	$-0.656052\pi$
$137$	−11.0223	−0.941698	−0.470849	+	0.882214i	$0.656052\pi$
$138$	0	0
$139$	17.2567	1.46370	0.731848	−	0.681468i	$-0.238658\pi$
$139$	17.2567	1.46370	0.731848	+	0.681468i	$0.238658\pi$
$140$	0	0
$141$	−2.41147	−0.203083
$142$	0	0
$143$	−5.07873	−0.424704
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.12836	0.258022
$148$	0	0
$149$	12.3824	1.01440	0.507202	−	0.861827i	$-0.330680\pi$
$149$	12.3824	1.01440	0.507202	+	0.861827i	$0.330680\pi$
$150$	0	0
$151$	−20.8016	−1.69281	−0.846405	−	0.532540i	$-0.821238\pi$
$151$	−20.8016	−1.69281	−0.846405	+	0.532540i	$0.821238\pi$
$152$	0	0
$153$	2.30541	0.186381
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.25402	0.419317	0.209658	−	0.977775i	$-0.432765\pi$
$157$	5.25402	0.419317	0.209658	+	0.977775i	$0.432765\pi$
$158$	0	0
$159$	−10.9213	−0.866113
$160$	0	0
$161$	−9.96585	−0.785419
$162$	0	0
$163$	−6.63816	−0.519940	−0.259970	−	0.965617i	$-0.583713\pi$
$163$	−6.63816	−0.519940	−0.259970	+	0.965617i	$0.583713\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.3824	−1.26771	−0.633853	−	0.773453i	$-0.718528\pi$
$167$	−16.3824	−1.26771	−0.633853	+	0.773453i	$0.718528\pi$
$168$	0	0
$169$	−10.5226	−0.809430
$170$	0	0
$171$	0.652704	0.0499135
$172$	0	0
$173$	4.01455	0.305220	0.152610	−	0.988286i	$-0.451232\pi$
$173$	4.01455	0.305220	0.152610	+	0.988286i	$0.451232\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0838	−0.908272
$178$	0	0
$179$	−22.2472	−1.66283	−0.831417	−	0.555648i	$-0.812470\pi$
$179$	−22.2472	−1.66283	−0.831417	+	0.555648i	$0.812470\pi$
$180$	0	0
$181$	−7.29591	−0.542301	−0.271150	−	0.962537i	$-0.587404\pi$
$181$	−7.29591	−0.542301	−0.271150	+	0.962537i	$0.587404\pi$
$182$	0	0
$183$	4.28312	0.316617
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−11.3969	−0.833426
$188$	0	0
$189$	−12.4611	−0.906412
$190$	0	0
$191$	9.55169	0.691136	0.345568	−	0.938394i	$-0.387686\pi$
$191$	9.55169	0.691136	0.345568	+	0.938394i	$0.387686\pi$
$192$	0	0
$193$	1.23442	0.0888557	0.0444278	−	0.999013i	$-0.485854\pi$
$193$	1.23442	0.0888557	0.0444278	+	0.999013i	$0.485854\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.5672	0.824127	0.412063	−	0.911155i	$-0.364808\pi$
$197$	11.5672	0.824127	0.412063	+	0.911155i	$0.364808\pi$
$198$	0	0
$199$	−12.0104	−0.851397	−0.425698	−	0.904865i	$-0.639972\pi$
$199$	−12.0104	−0.851397	−0.425698	+	0.904865i	$0.639972\pi$
$200$	0	0
$201$	−4.94356	−0.348692
$202$	0	0
$203$	−4.27807	−0.300261
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.92127	−0.203043
$208$	0	0
$209$	−3.22668	−0.223194
$210$	0	0
$211$	11.3696	0.782715	0.391357	−	0.920239i	$-0.372006\pi$
$211$	11.3696	0.782715	0.391357	+	0.920239i	$0.372006\pi$
$212$	0	0
$213$	−6.72462	−0.460764
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.49525	−0.576695
$218$	0	0
$219$	−9.82295	−0.663773
$220$	0	0
$221$	5.55943	0.373968
$222$	0	0
$223$	13.7493	0.920720	0.460360	−	0.887732i	$-0.347720\pi$
$223$	13.7493	0.920720	0.460360	+	0.887732i	$0.347720\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.06149	0.468688	0.234344	−	0.972154i	$-0.424706\pi$
$227$	7.06149	0.468688	0.234344	+	0.972154i	$0.424706\pi$
$228$	0	0
$229$	−7.05644	−0.466302	−0.233151	−	0.972440i	$-0.574904\pi$
$229$	−7.05644	−0.466302	−0.233151	+	0.972440i	$0.574904\pi$
$230$	0	0
$231$	11.0077	0.724256
$232$	0	0
$233$	−11.9659	−0.783909	−0.391955	−	0.919985i	$-0.628201\pi$
$233$	−11.9659	−0.783909	−0.391955	+	0.919985i	$0.628201\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.1702	−0.855499
$238$	0	0
$239$	−2.75103	−0.177949	−0.0889747	−	0.996034i	$-0.528359\pi$
$239$	−2.75103	−0.177949	−0.0889747	+	0.996034i	$0.528359\pi$
$240$	0	0
$241$	−7.95811	−0.512627	−0.256313	−	0.966594i	$-0.582508\pi$
$241$	−7.95811	−0.512627	−0.256313	+	0.966594i	$0.582508\pi$
$242$	0	0
$243$	−6.65270	−0.426771
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.57398	0.100150
$248$	0	0
$249$	22.4047	1.41984
$250$	0	0
$251$	13.6827	0.863646	0.431823	−	0.901958i	$-0.357870\pi$
$251$	13.6827	0.863646	0.431823	+	0.901958i	$0.357870\pi$
$252$	0	0
$253$	14.4415	0.907930
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.5517	−0.658196	−0.329098	−	0.944296i	$-0.606745\pi$
$257$	−10.5517	−0.658196	−0.329098	+	0.944296i	$0.606745\pi$
$258$	0	0
$259$	25.2841	1.57107
$260$	0	0
$261$	−1.25402	−0.0776221
$262$	0	0
$263$	−9.40642	−0.580025	−0.290012	−	0.957023i	$-0.593659\pi$
$263$	−9.40642	−0.580025	−0.290012	+	0.957023i	$0.593659\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.35504	−0.572519
$268$	0	0
$269$	9.02734	0.550407	0.275203	−	0.961386i	$-0.411255\pi$
$269$	9.02734	0.550407	0.275203	+	0.961386i	$0.411255\pi$
$270$	0	0
$271$	22.3354	1.35678	0.678391	−	0.734701i	$-0.262678\pi$
$271$	22.3354	1.35678	0.678391	+	0.734701i	$0.262678\pi$
$272$	0	0
$273$	−5.36959	−0.324982
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−17.2959	−1.03921	−0.519605	−	0.854406i	$-0.673921\pi$
$277$	−17.2959	−1.03921	−0.519605	+	0.854406i	$0.673921\pi$
$278$	0	0
$279$	−2.49020	−0.149084
$280$	0	0
$281$	−10.4534	−0.623595	−0.311798	−	0.950149i	$-0.600931\pi$
$281$	−10.4534	−0.623595	−0.311798	+	0.950149i	$0.600931\pi$
$282$	0	0
$283$	−23.3628	−1.38877	−0.694386	−	0.719602i	$-0.744324\pi$
$283$	−23.3628	−1.38877	−0.694386	+	0.719602i	$0.744324\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.73917	−0.456829
$288$	0	0
$289$	−4.52435	−0.266138
$290$	0	0
$291$	24.0574	1.41027
$292$	0	0
$293$	27.9641	1.63368	0.816840	−	0.576864i	$-0.195724\pi$
$293$	27.9641	1.63368	0.816840	+	0.576864i	$0.195724\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18.0574	1.04779
$298$	0	0
$299$	−7.04458	−0.407398
$300$	0	0
$301$	3.77332	0.217490
$302$	0	0
$303$	−6.67230	−0.383314
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−22.8452	−1.30385	−0.651923	−	0.758285i	$-0.726038\pi$
$307$	−22.8452	−1.30385	−0.651923	+	0.758285i	$0.726038\pi$
$308$	0	0
$309$	−6.54395	−0.372272
$310$	0	0
$311$	11.8699	0.673080	0.336540	−	0.941669i	$-0.390743\pi$
$311$	11.8699	0.673080	0.336540	+	0.941669i	$0.390743\pi$
$312$	0	0
$313$	−25.5672	−1.44514	−0.722571	−	0.691297i	$-0.757040\pi$
$313$	−25.5672	−1.44514	−0.722571	+	0.691297i	$0.757040\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.4020	−0.977392	−0.488696	−	0.872454i	$-0.662527\pi$
$317$	−17.4020	−0.977392	−0.488696	+	0.872454i	$0.662527\pi$
$318$	0	0
$319$	6.19934	0.347096
$320$	0	0
$321$	−0.162504	−0.00907008
$322$	0	0
$323$	3.53209	0.196531
$324$	0	0
$325$	0	0
$326$	0	0
$327$	22.0797	1.22101
$328$	0	0
$329$	−3.50475	−0.193223
$330$	0	0
$331$	6.77063	0.372147	0.186074	−	0.982536i	$-0.440424\pi$
$331$	6.77063	0.372147	0.186074	+	0.982536i	$0.440424\pi$
$332$	0	0
$333$	7.41147	0.406146
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.11112	−0.223947	−0.111973	−	0.993711i	$-0.535717\pi$
$337$	−4.11112	−0.223947	−0.111973	+	0.993711i	$0.535717\pi$
$338$	0	0
$339$	−25.4165	−1.38044
$340$	0	0
$341$	12.3105	0.666649
$342$	0	0
$343$	20.1334	1.08710
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.0247	−0.591834	−0.295917	−	0.955214i	$-0.595625\pi$
$347$	−11.0247	−0.591834	−0.295917	+	0.955214i	$0.595625\pi$
$348$	0	0
$349$	−22.8307	−1.22210	−0.611049	−	0.791592i	$-0.709252\pi$
$349$	−22.8307	−1.22210	−0.611049	+	0.791592i	$0.709252\pi$
$350$	0	0
$351$	−8.80840	−0.470158
$352$	0	0
$353$	−14.6732	−0.780978	−0.390489	−	0.920608i	$-0.627694\pi$
$353$	−14.6732	−0.780978	−0.390489	+	0.920608i	$0.627694\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−12.0496	−0.637734
$358$	0	0
$359$	−3.60307	−0.190163	−0.0950815	−	0.995469i	$-0.530311\pi$
$359$	−3.60307	−0.190163	−0.0950815	+	0.995469i	$0.530311\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.901674	0.0473256
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−19.8007	−1.03359	−0.516793	−	0.856110i	$-0.672874\pi$
$367$	−19.8007	−1.03359	−0.516793	+	0.856110i	$0.672874\pi$
$368$	0	0
$369$	−2.26857	−0.118097
$370$	0	0
$371$	−15.8726	−0.824063
$372$	0	0
$373$	17.6631	0.914562	0.457281	−	0.889322i	$-0.348823\pi$
$373$	17.6631	0.914562	0.457281	+	0.889322i	$0.348823\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.02404	−0.155746
$378$	0	0
$379$	−6.05644	−0.311098	−0.155549	−	0.987828i	$-0.549715\pi$
$379$	−6.05644	−0.311098	−0.155549	+	0.987828i	$0.549715\pi$
$380$	0	0
$381$	6.16250	0.315715
$382$	0	0
$383$	−20.8895	−1.06740	−0.533702	−	0.845673i	$-0.679199\pi$
$383$	−20.8895	−1.06740	−0.533702	+	0.845673i	$0.679199\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.10607	0.0562245
$388$	0	0
$389$	−2.30541	−0.116889	−0.0584444	−	0.998291i	$-0.518614\pi$
$389$	−2.30541	−0.116889	−0.0584444	+	0.998291i	$0.518614\pi$
$390$	0	0
$391$	−15.8084	−0.799465
$392$	0	0
$393$	−6.63041	−0.334460
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.2354	1.11596	0.557980	−	0.829854i	$-0.311576\pi$
$397$	22.2354	1.11596	0.557980	+	0.829854i	$0.311576\pi$
$398$	0	0
$399$	−3.41147	−0.170787
$400$	0	0
$401$	−14.1584	−0.707036	−0.353518	−	0.935428i	$-0.615015\pi$
$401$	−14.1584	−0.707036	−0.353518	+	0.935428i	$0.615015\pi$
$402$	0	0
$403$	−6.00505	−0.299133
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−36.6391	−1.81613
$408$	0	0
$409$	−4.22432	−0.208879	−0.104440	−	0.994531i	$-0.533305\pi$
$409$	−4.22432	−0.208879	−0.104440	+	0.994531i	$0.533305\pi$
$410$	0	0
$411$	16.8871	0.832980
$412$	0	0
$413$	−17.5621	−0.864175
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−26.4388	−1.29471
$418$	0	0
$419$	11.9632	0.584439	0.292219	−	0.956351i	$-0.405606\pi$
$419$	11.9632	0.584439	0.292219	+	0.956351i	$0.405606\pi$
$420$	0	0
$421$	−10.2395	−0.499041	−0.249521	−	0.968369i	$-0.580273\pi$
$421$	−10.2395	−0.499041	−0.249521	+	0.968369i	$0.580273\pi$
$422$	0	0
$423$	−1.02734	−0.0499510
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.22493	0.301245
$428$	0	0
$429$	7.78106	0.375673
$430$	0	0
$431$	−22.2517	−1.07182	−0.535912	−	0.844274i	$-0.680032\pi$
$431$	−22.2517	−1.07182	−0.535912	+	0.844274i	$0.680032\pi$
$432$	0	0
$433$	−24.9581	−1.19941	−0.599705	−	0.800221i	$-0.704715\pi$
$433$	−24.9581	−1.19941	−0.599705	+	0.800221i	$0.704715\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.47565	−0.214099
$438$	0	0
$439$	−16.0942	−0.768135	−0.384067	−	0.923305i	$-0.625477\pi$
$439$	−16.0942	−0.768135	−0.384067	+	0.923305i	$0.625477\pi$
$440$	0	0
$441$	1.33275	0.0634642
$442$	0	0
$443$	17.7270	0.842235	0.421117	−	0.907006i	$-0.361638\pi$
$443$	17.7270	0.842235	0.421117	+	0.907006i	$0.361638\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−18.9709	−0.897293
$448$	0	0
$449$	−37.6049	−1.77469	−0.887343	−	0.461109i	$-0.847452\pi$
$449$	−37.6049	−1.77469	−0.887343	+	0.461109i	$0.847452\pi$
$450$	0	0
$451$	11.2148	0.528085
$452$	0	0
$453$	31.8699	1.49738
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.1949	0.944677	0.472339	−	0.881417i	$-0.343410\pi$
$457$	20.1949	0.944677	0.472339	+	0.881417i	$0.343410\pi$
$458$	0	0
$459$	−19.7665	−0.922622
$460$	0	0
$461$	6.81521	0.317416	0.158708	−	0.987326i	$-0.449267\pi$
$461$	6.81521	0.317416	0.158708	+	0.987326i	$0.449267\pi$
$462$	0	0
$463$	−33.7083	−1.56656	−0.783279	−	0.621670i	$-0.786454\pi$
$463$	−33.7083	−1.56656	−0.783279	+	0.621670i	$0.786454\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.5648	1.27555	0.637774	−	0.770224i	$-0.279856\pi$
$467$	27.5648	1.27555	0.637774	+	0.770224i	$0.279856\pi$
$468$	0	0
$469$	−7.18479	−0.331763
$470$	0	0
$471$	−8.04963	−0.370907
$472$	0	0
$473$	−5.46791	−0.251415
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.65270	−0.213033
$478$	0	0
$479$	−26.1438	−1.19454	−0.597271	−	0.802039i	$-0.703748\pi$
$479$	−26.1438	−1.19454	−0.597271	+	0.802039i	$0.703748\pi$
$480$	0	0
$481$	17.8726	0.814919
$482$	0	0
$483$	15.2686	0.694744
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.63310	−0.255260	−0.127630	−	0.991822i	$-0.540737\pi$
$487$	−5.63310	−0.255260	−0.127630	+	0.991822i	$0.540737\pi$
$488$	0	0
$489$	10.1702	0.459914
$490$	0	0
$491$	−14.6682	−0.661966	−0.330983	−	0.943637i	$-0.607380\pi$
$491$	−14.6682	−0.661966	−0.330983	+	0.943637i	$0.607380\pi$
$492$	0	0
$493$	−6.78611	−0.305631
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.77332	−0.438393
$498$	0	0
$499$	−16.7793	−0.751145	−0.375572	−	0.926793i	$-0.622554\pi$
$499$	−16.7793	−0.751145	−0.375572	+	0.926793i	$0.622554\pi$
$500$	0	0
$501$	25.0993	1.12135
$502$	0	0
$503$	3.12330	0.139261	0.0696306	−	0.997573i	$-0.477818\pi$
$503$	3.12330	0.139261	0.0696306	+	0.997573i	$0.477818\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	16.1215	0.715983
$508$	0	0
$509$	−40.1566	−1.77991	−0.889956	−	0.456047i	$-0.849265\pi$
$509$	−40.1566	−1.77991	−0.889956	+	0.456047i	$0.849265\pi$
$510$	0	0
$511$	−14.2763	−0.631547
$512$	0	0
$513$	−5.59627	−0.247081
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.07873	0.223362
$518$	0	0
$519$	−6.15064	−0.269983
$520$	0	0
$521$	−8.75465	−0.383548	−0.191774	−	0.981439i	$-0.561424\pi$
$521$	−8.75465	−0.383548	−0.191774	+	0.981439i	$0.561424\pi$
$522$	0	0
$523$	−26.9522	−1.17854	−0.589270	−	0.807937i	$-0.700584\pi$
$523$	−26.9522	−1.17854	−0.589270	+	0.807937i	$0.700584\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13.4757	−0.587009
$528$	0	0
$529$	−2.96854	−0.129067
$530$	0	0
$531$	−5.14796	−0.223402
$532$	0	0
$533$	−5.47060	−0.236958
$534$	0	0
$535$	0	0
$536$	0	0
$537$	34.0847	1.47086
$538$	0	0
$539$	−6.58853	−0.283788
$540$	0	0
$541$	8.48751	0.364907	0.182453	−	0.983215i	$-0.441596\pi$
$541$	8.48751	0.364907	0.182453	+	0.983215i	$0.441596\pi$
$542$	0	0
$543$	11.1780	0.479693
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.62267	−0.154894	−0.0774472	−	0.996996i	$-0.524677\pi$
$547$	−3.62267	−0.154894	−0.0774472	+	0.996996i	$0.524677\pi$
$548$	0	0
$549$	1.82470	0.0778764
$550$	0	0
$551$	−1.92127	−0.0818490
$552$	0	0
$553$	−19.1411	−0.813964
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.01455	0.0853591	0.0426796	−	0.999089i	$-0.486411\pi$
$557$	2.01455	0.0853591	0.0426796	+	0.999089i	$0.486411\pi$
$558$	0	0
$559$	2.66725	0.112813
$560$	0	0
$561$	17.4611	0.737208
$562$	0	0
$563$	−16.5185	−0.696171	−0.348085	−	0.937463i	$-0.613168\pi$
$563$	−16.5185	−0.696171	−0.348085	+	0.937463i	$0.613168\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	14.7314	0.618662
$568$	0	0
$569$	−8.18304	−0.343051	−0.171525	−	0.985180i	$-0.554870\pi$
$569$	−8.18304	−0.343051	−0.171525	+	0.985180i	$0.554870\pi$
$570$	0	0
$571$	36.9796	1.54755	0.773774	−	0.633462i	$-0.218367\pi$
$571$	36.9796	1.54755	0.773774	+	0.633462i	$0.218367\pi$
$572$	0	0
$573$	−14.6340	−0.611346
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.7347	0.779937	0.389968	−	0.920828i	$-0.372486\pi$
$577$	18.7347	0.779937	0.389968	+	0.920828i	$0.372486\pi$
$578$	0	0
$579$	−1.89124	−0.0785975
$580$	0	0
$581$	32.5621	1.35090
$582$	0	0
$583$	23.0009	0.952601
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.2968	1.20921	0.604605	−	0.796525i	$-0.293331\pi$
$587$	29.2968	1.20921	0.604605	+	0.796525i	$0.293331\pi$
$588$	0	0
$589$	−3.81521	−0.157203
$590$	0	0
$591$	−17.7219	−0.728983
$592$	0	0
$593$	−22.4260	−0.920926	−0.460463	−	0.887679i	$-0.652317\pi$
$593$	−22.4260	−0.920926	−0.460463	+	0.887679i	$0.652317\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.4010	0.753105
$598$	0	0
$599$	33.1002	1.35244	0.676219	−	0.736701i	$-0.263617\pi$
$599$	33.1002	1.35244	0.676219	+	0.736701i	$0.263617\pi$
$600$	0	0
$601$	43.5613	1.77690	0.888451	−	0.458971i	$-0.151782\pi$
$601$	43.5613	1.77690	0.888451	+	0.458971i	$0.151782\pi$
$602$	0	0
$603$	−2.10607	−0.0857657
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2.03178	−0.0824676	−0.0412338	−	0.999150i	$-0.513129\pi$
$607$	−2.03178	−0.0824676	−0.0412338	+	0.999150i	$0.513129\pi$
$608$	0	0
$609$	6.55438	0.265597
$610$	0	0
$611$	−2.47741	−0.100225
$612$	0	0
$613$	−25.1266	−1.01485	−0.507427	−	0.861695i	$-0.669403\pi$
$613$	−25.1266	−1.01485	−0.507427	+	0.861695i	$0.669403\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.4979	−0.986250	−0.493125	−	0.869958i	$-0.664145\pi$
$617$	−24.4979	−0.986250	−0.493125	+	0.869958i	$0.664145\pi$
$618$	0	0
$619$	−9.05138	−0.363806	−0.181903	−	0.983316i	$-0.558226\pi$
$619$	−9.05138	−0.363806	−0.181903	+	0.983316i	$0.558226\pi$
$620$	0	0
$621$	25.0469	1.00510
$622$	0	0
$623$	−13.5963	−0.544723
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.94356	0.197427
$628$	0	0
$629$	40.1070	1.59917
$630$	0	0
$631$	−23.7324	−0.944770	−0.472385	−	0.881392i	$-0.656607\pi$
$631$	−23.7324	−0.944770	−0.472385	+	0.881392i	$0.656607\pi$
$632$	0	0
$633$	−17.4192	−0.692352
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.21389	0.127339
$638$	0	0
$639$	−2.86484	−0.113331
$640$	0	0
$641$	10.6067	0.418939	0.209470	−	0.977815i	$-0.432826\pi$
$641$	10.6067	0.418939	0.209470	+	0.977815i	$0.432826\pi$
$642$	0	0
$643$	31.1634	1.22897	0.614483	−	0.788930i	$-0.289365\pi$
$643$	31.1634	1.22897	0.614483	+	0.788930i	$0.289365\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.9154	0.507757	0.253878	−	0.967236i	$-0.418294\pi$
$647$	12.9154	0.507757	0.253878	+	0.967236i	$0.418294\pi$
$648$	0	0
$649$	25.4492	0.998970
$650$	0	0
$651$	13.0155	0.510117
$652$	0	0
$653$	10.8060	0.422873	0.211436	−	0.977392i	$-0.432186\pi$
$653$	10.8060	0.422873	0.211436	+	0.977392i	$0.432186\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.18479	−0.163264
$658$	0	0
$659$	28.3583	1.10468	0.552342	−	0.833618i	$-0.313734\pi$
$659$	28.3583	1.10468	0.552342	+	0.833618i	$0.313734\pi$
$660$	0	0
$661$	−28.9691	−1.12677	−0.563385	−	0.826195i	$-0.690501\pi$
$661$	−28.9691	−1.12677	−0.563385	+	0.826195i	$0.690501\pi$
$662$	0	0
$663$	−8.51754	−0.330794
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.59896	0.332953
$668$	0	0
$669$	−21.0651	−0.814424
$670$	0	0
$671$	−9.02053	−0.348234
$672$	0	0
$673$	−15.6973	−0.605086	−0.302543	−	0.953136i	$-0.597836\pi$
$673$	−15.6973	−0.605086	−0.302543	+	0.953136i	$0.597836\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.25578	0.125130	0.0625648	−	0.998041i	$-0.480072\pi$
$677$	3.25578	0.125130	0.0625648	+	0.998041i	$0.480072\pi$
$678$	0	0
$679$	34.9641	1.34180
$680$	0	0
$681$	−10.8188	−0.414578
$682$	0	0
$683$	−33.1834	−1.26973	−0.634863	−	0.772625i	$-0.718943\pi$
$683$	−33.1834	−1.26973	−0.634863	+	0.772625i	$0.718943\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.8111	0.412469
$688$	0	0
$689$	−11.2199	−0.427443
$690$	0	0
$691$	−41.1242	−1.56444	−0.782220	−	0.623002i	$-0.785913\pi$
$691$	−41.1242	−1.56444	−0.782220	+	0.623002i	$0.785913\pi$
$692$	0	0
$693$	4.68954	0.178141
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12.2763	−0.464998
$698$	0	0
$699$	18.3327	0.693408
$700$	0	0
$701$	44.9198	1.69660	0.848300	−	0.529517i	$-0.177627\pi$
$701$	44.9198	1.69660	0.848300	+	0.529517i	$0.177627\pi$
$702$	0	0
$703$	11.3550	0.428263
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.69728	−0.364704
$708$	0	0
$709$	−34.9240	−1.31160	−0.655798	−	0.754936i	$-0.727668\pi$
$709$	−34.9240	−1.31160	−0.655798	+	0.754936i	$0.727668\pi$
$710$	0	0
$711$	−5.61081	−0.210422
$712$	0	0
$713$	17.0755	0.639484
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.21482	0.157405
$718$	0	0
$719$	−29.0077	−1.08181	−0.540903	−	0.841085i	$-0.681917\pi$
$719$	−29.0077	−1.08181	−0.540903	+	0.841085i	$0.681917\pi$
$720$	0	0
$721$	−9.51073	−0.354198
$722$	0	0
$723$	12.1925	0.453445
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.27126	0.306764	0.153382	−	0.988167i	$-0.450983\pi$
$727$	8.27126	0.306764	0.153382	+	0.988167i	$0.450983\pi$
$728$	0	0
$729$	30.0401	1.11260
$730$	0	0
$731$	5.98545	0.221380
$732$	0	0
$733$	12.6696	0.467963	0.233981	−	0.972241i	$-0.424824\pi$
$733$	12.6696	0.467963	0.233981	+	0.972241i	$0.424824\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.4115	0.383512
$738$	0	0
$739$	−17.9733	−0.661157	−0.330579	−	0.943778i	$-0.607244\pi$
$739$	−17.9733	−0.661157	−0.330579	+	0.943778i	$0.607244\pi$
$740$	0	0
$741$	−2.41147	−0.0885877
$742$	0	0
$743$	−5.47977	−0.201033	−0.100517	−	0.994935i	$-0.532050\pi$
$743$	−5.47977	−0.201033	−0.100517	+	0.994935i	$0.532050\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.54488	0.349229
$748$	0	0
$749$	−0.236177	−0.00862972
$750$	0	0
$751$	−32.5803	−1.18887	−0.594436	−	0.804143i	$-0.702625\pi$
$751$	−32.5803	−1.18887	−0.594436	+	0.804143i	$0.702625\pi$
$752$	0	0
$753$	−20.9632	−0.763940
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4.97359	−0.180768	−0.0903841	−	0.995907i	$-0.528809\pi$
$757$	−4.97359	−0.180768	−0.0903841	+	0.995907i	$0.528809\pi$
$758$	0	0
$759$	−22.1257	−0.803111
$760$	0	0
$761$	24.5262	0.889075	0.444537	−	0.895760i	$-0.353368\pi$
$761$	24.5262	0.889075	0.444537	+	0.895760i	$0.353368\pi$
$762$	0	0
$763$	32.0898	1.16173
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.4142	−0.448249
$768$	0	0
$769$	−24.4688	−0.882369	−0.441185	−	0.897416i	$-0.645442\pi$
$769$	−24.4688	−0.882369	−0.441185	+	0.897416i	$0.645442\pi$
$770$	0	0
$771$	16.1661	0.582209
$772$	0	0
$773$	−4.43645	−0.159568	−0.0797840	−	0.996812i	$-0.525423\pi$
$773$	−4.43645	−0.159568	−0.0797840	+	0.996812i	$0.525423\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−38.7374	−1.38970
$778$	0	0
$779$	−3.47565	−0.124528
$780$	0	0
$781$	14.1625	0.506774
$782$	0	0
$783$	10.7520	0.384244
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.73143	0.0617188	0.0308594	−	0.999524i	$-0.490176\pi$
$787$	1.73143	0.0617188	0.0308594	+	0.999524i	$0.490176\pi$
$788$	0	0
$789$	14.4115	0.513062
$790$	0	0
$791$	−36.9394	−1.31342
$792$	0	0
$793$	4.40022	0.156257
$794$	0	0
$795$	0	0
$796$	0	0
$797$	48.3979	1.71434	0.857170	−	0.515033i	$-0.172221\pi$
$797$	48.3979	1.71434	0.857170	+	0.515033i	$0.172221\pi$
$798$	0	0
$799$	−5.55943	−0.196678
$800$	0	0
$801$	−3.98545	−0.140819
$802$	0	0
$803$	20.6878	0.730056
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−13.8307	−0.486863
$808$	0	0
$809$	45.7861	1.60975	0.804877	−	0.593442i	$-0.202231\pi$
$809$	45.7861	1.60975	0.804877	+	0.593442i	$0.202231\pi$
$810$	0	0
$811$	40.0779	1.40733	0.703663	−	0.710534i	$-0.251547\pi$
$811$	40.0779	1.40733	0.703663	+	0.710534i	$0.251547\pi$
$812$	0	0
$813$	−34.2199	−1.20014
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.69459	0.0592863
$818$	0	0
$819$	−2.28756	−0.0799339
$820$	0	0
$821$	−51.9208	−1.81205	−0.906024	−	0.423227i	$-0.860897\pi$
$821$	−51.9208	−1.81205	−0.906024	+	0.423227i	$0.860897\pi$
$822$	0	0
$823$	−18.0779	−0.630156	−0.315078	−	0.949066i	$-0.602031\pi$
$823$	−18.0779	−0.630156	−0.315078	+	0.949066i	$0.602031\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.5022	−1.61704	−0.808519	−	0.588469i	$-0.799731\pi$
$827$	−46.5022	−1.61704	−0.808519	+	0.588469i	$0.799731\pi$
$828$	0	0
$829$	−44.6946	−1.55231	−0.776154	−	0.630544i	$-0.782832\pi$
$829$	−44.6946	−1.55231	−0.776154	+	0.630544i	$0.782832\pi$
$830$	0	0
$831$	26.4989	0.919236
$832$	0	0
$833$	7.21213	0.249886
$834$	0	0
$835$	0	0
$836$	0	0
$837$	21.3509	0.737996
$838$	0	0
$839$	6.37134	0.219963	0.109982	−	0.993934i	$-0.464921\pi$
$839$	6.37134	0.219963	0.109982	+	0.993934i	$0.464921\pi$
$840$	0	0
$841$	−25.3087	−0.872714
$842$	0	0
$843$	16.0155	0.551602
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.31046	0.0450279
$848$	0	0
$849$	35.7939	1.22844
$850$	0	0
$851$	−50.8212	−1.74213
$852$	0	0
$853$	37.9813	1.30046	0.650228	−	0.759739i	$-0.274673\pi$
$853$	37.9813	1.30046	0.650228	+	0.759739i	$0.274673\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.2104	−1.61268	−0.806338	−	0.591455i	$-0.798554\pi$
$857$	−47.2104	−1.61268	−0.806338	+	0.591455i	$0.798554\pi$
$858$	0	0
$859$	−15.6587	−0.534268	−0.267134	−	0.963659i	$-0.586077\pi$
$859$	−15.6587	−0.534268	−0.267134	+	0.963659i	$0.586077\pi$
$860$	0	0
$861$	11.8571	0.404089
$862$	0	0
$863$	54.9026	1.86891	0.934453	−	0.356086i	$-0.115889\pi$
$863$	54.9026	1.86891	0.934453	+	0.356086i	$0.115889\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.93170	0.235413
$868$	0	0
$869$	27.7374	0.940927
$870$	0	0
$871$	−5.07873	−0.172086
$872$	0	0
$873$	10.2490	0.346875
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.74329	0.261472	0.130736	−	0.991417i	$-0.458266\pi$
$877$	7.74329	0.261472	0.130736	+	0.991417i	$0.458266\pi$
$878$	0	0
$879$	−42.8435	−1.44507
$880$	0	0
$881$	−48.6005	−1.63739	−0.818696	−	0.574227i	$-0.805303\pi$
$881$	−48.6005	−1.63739	−0.818696	+	0.574227i	$0.805303\pi$
$882$	0	0
$883$	31.1625	1.04870	0.524351	−	0.851502i	$-0.324308\pi$
$883$	31.1625	1.04870	0.524351	+	0.851502i	$0.324308\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.52259	−0.151854	−0.0759269	−	0.997113i	$-0.524192\pi$
$887$	−4.52259	−0.151854	−0.0759269	+	0.997113i	$0.524192\pi$
$888$	0	0
$889$	8.95636	0.300387
$890$	0	0
$891$	−21.3473	−0.715161
$892$	0	0
$893$	−1.57398	−0.0526712
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.7929	0.360365
$898$	0	0
$899$	7.33006	0.244471
$900$	0	0
$901$	−25.1780	−0.838800
$902$	0	0
$903$	−5.78106	−0.192382
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27.9777	−0.928985	−0.464492	−	0.885577i	$-0.653763\pi$
$907$	−27.9777	−0.928985	−0.464492	+	0.885577i	$0.653763\pi$
$908$	0	0
$909$	−2.84255	−0.0942814
$910$	0	0
$911$	−35.5895	−1.17913	−0.589566	−	0.807720i	$-0.700701\pi$
$911$	−35.5895	−1.17913	−0.589566	+	0.807720i	$0.700701\pi$
$912$	0	0
$913$	−47.1857	−1.56162
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.63640	−0.318222
$918$	0	0
$919$	−20.8854	−0.688945	−0.344472	−	0.938796i	$-0.611942\pi$
$919$	−20.8854	−0.688945	−0.344472	+	0.938796i	$0.611942\pi$
$920$	0	0
$921$	35.0009	1.15332
$922$	0	0
$923$	−6.90848	−0.227395
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.78787	−0.0915655
$928$	0	0
$929$	45.5885	1.49571	0.747856	−	0.663862i	$-0.231084\pi$
$929$	45.5885	1.49571	0.747856	+	0.663862i	$0.231084\pi$
$930$	0	0
$931$	2.04189	0.0669202
$932$	0	0
$933$	−18.1857	−0.595374
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34.7638	1.13568	0.567842	−	0.823137i	$-0.307778\pi$
$937$	34.7638	1.13568	0.567842	+	0.823137i	$0.307778\pi$
$938$	0	0
$939$	39.1712	1.27830
$940$	0	0
$941$	5.80066	0.189096	0.0945480	−	0.995520i	$-0.469859\pi$
$941$	5.80066	0.189096	0.0945480	+	0.995520i	$0.469859\pi$
$942$	0	0
$943$	15.5558	0.506567
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.4466	−0.599433	−0.299716	−	0.954028i	$-0.596892\pi$
$947$	−18.4466	−0.599433	−0.299716	+	0.954028i	$0.596892\pi$
$948$	0	0
$949$	−10.0915	−0.327585
$950$	0	0
$951$	26.6614	0.864554
$952$	0	0
$953$	−15.6568	−0.507174	−0.253587	−	0.967313i	$-0.581610\pi$
$953$	−15.6568	−0.507174	−0.253587	+	0.967313i	$0.581610\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9.49794	−0.307025
$958$	0	0
$959$	24.5431	0.792539
$960$	0	0
$961$	−16.4442	−0.530458
$962$	0	0
$963$	−0.0692302	−0.00223091
$964$	0	0
$965$	0	0
$966$	0	0
$967$	13.9923	0.449961	0.224980	−	0.974363i	$-0.427768\pi$
$967$	13.9923	0.449961	0.224980	+	0.974363i	$0.427768\pi$
$968$	0	0
$969$	−5.41147	−0.173842
$970$	0	0
$971$	−13.0487	−0.418753	−0.209376	−	0.977835i	$-0.567143\pi$
$971$	−13.0487	−0.418753	−0.209376	+	0.977835i	$0.567143\pi$
$972$	0	0
$973$	−38.4252	−1.23186
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.3327	−0.554524	−0.277262	−	0.960794i	$-0.589427\pi$
$977$	−17.3327	−0.554524	−0.277262	+	0.960794i	$0.589427\pi$
$978$	0	0
$979$	19.7023	0.629689
$980$	0	0
$981$	9.40642	0.300324
$982$	0	0
$983$	25.3114	0.807308	0.403654	−	0.914912i	$-0.367740\pi$
$983$	25.3114	0.807308	0.403654	+	0.914912i	$0.367740\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.36959	0.170916
$988$	0	0
$989$	−7.58441	−0.241170
$990$	0	0
$991$	−37.1762	−1.18094	−0.590471	−	0.807059i	$-0.701058\pi$
$991$	−37.1762	−1.18094	−0.590471	+	0.807059i	$0.701058\pi$
$992$	0	0
$993$	−10.3732	−0.329184
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.3865	0.835669	0.417834	−	0.908523i	$-0.362789\pi$
$997$	26.3865	0.835669	0.417834	+	0.908523i	$0.362789\pi$
$998$	0	0
$999$	−63.5458	−2.01050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bs.1.1		3
4.3	odd	2		3800.2.a.t.1.3	yes	3
5.4	even	2		7600.2.a.br.1.3		3
20.3	even	4		3800.2.d.o.3649.5		6
20.7	even	4		3800.2.d.o.3649.2		6
20.19	odd	2		3800.2.a.s.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.s.1.1	✓	3	20.19	odd	2
3800.2.a.t.1.3	yes	3	4.3	odd	2
3800.2.d.o.3649.2		6	20.7	even	4
3800.2.d.o.3649.5		6	20.3	even	4
7600.2.a.br.1.3		3	5.4	even	2
7600.2.a.bs.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53209 q^{3} -2.22668 q^{7} -0.652704 q^{9} +3.22668 q^{11} -1.57398 q^{13} -3.53209 q^{17} -1.00000 q^{19} +3.41147 q^{21} +4.47565 q^{23} +5.59627 q^{27} +1.92127 q^{29} +3.81521 q^{31} -4.94356 q^{33} -11.3550 q^{37} +2.41147 q^{39} +3.47565 q^{41} -1.69459 q^{43} +1.57398 q^{47} -2.04189 q^{49} +5.41147 q^{51} +7.12836 q^{53} +1.53209 q^{57} +7.88713 q^{59} -2.79561 q^{61} +1.45336 q^{63} +3.22668 q^{67} -6.85710 q^{69} +4.38919 q^{71} +6.41147 q^{73} -7.18479 q^{77} +8.59627 q^{79} -6.61587 q^{81} -14.6236 q^{83} -2.94356 q^{87} +6.10607 q^{89} +3.50475 q^{91} -5.84524 q^{93} -15.7023 q^{97} -2.10607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{17} - 3 q^{19} - 6 q^{23} + 3 q^{27} - 3 q^{29} + 15 q^{31} - 9 q^{37} - 3 q^{39} - 9 q^{41} - 3 q^{43} - 3 q^{47} - 3 q^{49} + 6 q^{51} + 3 q^{53} - 6 q^{59}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^17 - 3 * q^19 - 6 * q^23 + 3 * q^27 - 3 * q^29 + 15 * q^31 - 9 * q^37 - 3 * q^39 - 9 * q^41 - 3 * q^43 - 3 * q^47 - 3 * q^49 + 6 * q^51 + 3 * q^53 - 6 * q^59 - 9 * q^61 - 9 * q^63 + 3 * q^67 - 21 * q^69 + 9 * q^71 + 9 * q^73 - 18 * q^77 + 12 * q^79 - 9 * q^81 - 9 * q^83 + 6 * q^87 + 6 * q^89 + 27 * q^91 + 9 * q^93 - 21 * q^97 + 6 * q^99

Introduction

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