LMFDB - Embedded newform 4761.2.a.bm.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.284630$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.236479 q^{2} -1.94408 q^{4} +1.00714 q^{5} +2.05529 q^{7} +0.932691 q^{8} -0.238168 q^{10} +4.39011 q^{11} -2.08816 q^{13} -0.486033 q^{14} +3.66759 q^{16} -2.16917 q^{17} -2.66759 q^{19} -1.95796 q^{20} -1.03817 q^{22} -3.98567 q^{25} +0.493805 q^{26} -3.99565 q^{28} -7.97428 q^{29} -7.82464 q^{31} -2.73269 q^{32} +0.512963 q^{34} +2.06997 q^{35} +3.92676 q^{37} +0.630830 q^{38} +0.939352 q^{40} -6.30788 q^{41} +2.16627 q^{43} -8.53471 q^{44} +7.22297 q^{47} -2.77577 q^{49} +0.942526 q^{50} +4.05954 q^{52} -11.2938 q^{53} +4.42146 q^{55} +1.91695 q^{56} +1.88575 q^{58} -12.1922 q^{59} +14.1129 q^{61} +1.85036 q^{62} -6.68896 q^{64} -2.10307 q^{65} -0.712023 q^{67} +4.21704 q^{68} -0.489504 q^{70} +4.11767 q^{71} -0.0586401 q^{73} -0.928595 q^{74} +5.18601 q^{76} +9.02295 q^{77} +16.2978 q^{79} +3.69379 q^{80} +1.49168 q^{82} -14.5437 q^{83} -2.18466 q^{85} -0.512278 q^{86} +4.09461 q^{88} +2.75372 q^{89} -4.29177 q^{91} -1.70808 q^{94} -2.68665 q^{95} -5.90566 q^{97} +0.656411 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{4} - 7 q^{5} - 3 q^{7} + 9 q^{8} - 6 q^{10} - 2 q^{11} - 7 q^{13} + 10 q^{14} + 6 q^{16} - 16 q^{17} - q^{19} - 21 q^{20} + 14 q^{22} + 20 q^{25} + 5 q^{26} + 2 q^{28} - 18 q^{29}+ \cdots - 14 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^4 - 7 * q^5 - 3 * q^7 + 9 * q^8 - 6 * q^10 - 2 * q^11 - 7 * q^13 + 10 * q^14 + 6 * q^16 - 16 * q^17 - q^19 - 21 * q^20 + 14 * q^22 + 20 * q^25 + 5 * q^26 + 2 * q^28 - 18 * q^29 + 5 * q^31 + 6 * q^32 + 2 * q^34 + 24 * q^35 + 26 * q^37 - 26 * q^38 - 17 * q^40 - 9 * q^41 + 22 * q^43 - 28 * q^44 - 2 * q^47 + 2 * q^49 + 14 * q^50 + q^52 - 35 * q^53 + 16 * q^55 + 10 * q^56 + 5 * q^58 - 6 * q^59 - 7 * q^61 - 13 * q^62 - 21 * q^64 - 10 * q^65 + 5 * q^67 - 15 * q^68 - 47 * q^70 + 18 * q^71 + 4 * q^73 - 28 * q^74 - 3 * q^76 + 21 * q^77 + 13 * q^79 - 37 * q^80 - 47 * q^82 - 24 * q^83 + 18 * q^85 - 11 * q^86 + 14 * q^88 + 7 * q^89 - 9 * q^91 - 8 * q^94 + 30 * q^95 + 6 * q^97 - 14 * q^98

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 $

$2$	−0.236479	−0.167216	−0.0836079	−	0.996499i	$-0.526644\pi$
$2$	−0.236479	−0.167216	−0.0836079	+	0.996499i	$0.526644\pi$
$3$	0	0
$3$	0	0
$4$	−1.94408	−0.972039
$5$	1.00714	0.450408	0.225204	−	0.974312i	$-0.427695\pi$
$5$	1.00714	0.450408	0.225204	+	0.974312i	$0.427695\pi$
$6$	0	0
$7$	2.05529	0.776828	0.388414	−	0.921485i	$-0.373023\pi$
$7$	2.05529	0.776828	0.388414	+	0.921485i	$0.373023\pi$
$8$	0.932691	0.329756
$9$	0	0
$10$	−0.238168	−0.0753153
$11$	4.39011	1.32367	0.661833	−	0.749651i	$-0.269779\pi$
$11$	4.39011	1.32367	0.661833	+	0.749651i	$0.269779\pi$
$12$	0	0
$13$	−2.08816	−0.579150	−0.289575	−	0.957155i	$-0.593514\pi$
$13$	−2.08816	−0.579150	−0.289575	+	0.957155i	$0.593514\pi$
$14$	−0.486033	−0.129898
$15$	0	0
$16$	3.66759	0.916898
$17$	−2.16917	−0.526101	−0.263050	−	0.964782i	$-0.584729\pi$
$17$	−2.16917	−0.526101	−0.263050	+	0.964782i	$0.584729\pi$
$18$	0	0
$19$	−2.66759	−0.611988	−0.305994	−	0.952033i	$-0.598989\pi$
$19$	−2.66759	−0.611988	−0.305994	+	0.952033i	$0.598989\pi$
$20$	−1.95796	−0.437814
$21$	0	0
$22$	−1.03817	−0.221338
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.98567	−0.797133
$26$	0.493805	0.0968431
$27$	0	0
$28$	−3.99565	−0.755107
$29$	−7.97428	−1.48079	−0.740393	−	0.672174i	$-0.765361\pi$
$29$	−7.97428	−1.48079	−0.740393	+	0.672174i	$0.765361\pi$
$30$	0	0
$31$	−7.82464	−1.40535	−0.702674	−	0.711512i	$-0.748011\pi$
$31$	−7.82464	−1.40535	−0.702674	+	0.711512i	$0.748011\pi$
$32$	−2.73269	−0.483076
$33$	0	0
$34$	0.512963	0.0879724
$35$	2.06997	0.349889
$36$	0	0
$37$	3.92676	0.645555	0.322777	−	0.946475i	$-0.395383\pi$
$37$	3.92676	0.645555	0.322777	+	0.946475i	$0.395383\pi$
$38$	0.630830	0.102334
$39$	0	0
$40$	0.939352	0.148525
$41$	−6.30788	−0.985126	−0.492563	−	0.870277i	$-0.663940\pi$
$41$	−6.30788	−0.985126	−0.492563	+	0.870277i	$0.663940\pi$
$42$	0	0
$43$	2.16627	0.330354	0.165177	−	0.986264i	$-0.447181\pi$
$43$	2.16627	0.330354	0.165177	+	0.986264i	$0.447181\pi$
$44$	−8.53471	−1.28666
$45$	0	0
$46$	0	0
$47$	7.22297	1.05358	0.526789	−	0.849996i	$-0.323396\pi$
$47$	7.22297	1.05358	0.526789	+	0.849996i	$0.323396\pi$
$48$	0	0
$49$	−2.77577	−0.396539
$50$	0.942526	0.133293
$51$	0	0
$52$	4.05954	0.562957
$53$	−11.2938	−1.55132	−0.775662	−	0.631149i	$-0.782584\pi$
$53$	−11.2938	−1.55132	−0.775662	+	0.631149i	$0.782584\pi$
$54$	0	0
$55$	4.42146	0.596189
$56$	1.91695	0.256164
$57$	0	0
$58$	1.88575	0.247611
$59$	−12.1922	−1.58729	−0.793643	−	0.608384i	$-0.791818\pi$
$59$	−12.1922	−1.58729	−0.793643	+	0.608384i	$0.791818\pi$
$60$	0	0
$61$	14.1129	1.80697	0.903485	−	0.428620i	$-0.141000\pi$
$61$	14.1129	1.80697	0.903485	+	0.428620i	$0.141000\pi$
$62$	1.85036	0.234996
$63$	0	0
$64$	−6.68896	−0.836120
$65$	−2.10307	−0.260854
$66$	0	0
$67$	−0.712023	−0.0869874	−0.0434937	−	0.999054i	$-0.513849\pi$
$67$	−0.712023	−0.0869874	−0.0434937	+	0.999054i	$0.513849\pi$
$68$	4.21704	0.511391
$69$	0	0
$70$	−0.489504	−0.0585070
$71$	4.11767	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$71$	4.11767	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$72$	0	0
$73$	−0.0586401	−0.00686331	−0.00343166	−	0.999994i	$-0.501092\pi$
$73$	−0.0586401	−0.00686331	−0.00343166	+	0.999994i	$0.501092\pi$
$74$	−0.928595	−0.107947
$75$	0	0
$76$	5.18601	0.594876
$77$	9.02295	1.02826
$78$	0	0
$79$	16.2978	1.83365	0.916823	−	0.399295i	$-0.130745\pi$
$79$	16.2978	1.83365	0.916823	+	0.399295i	$0.130745\pi$
$80$	3.69379	0.412978
$81$	0	0
$82$	1.49168	0.164729
$83$	−14.5437	−1.59638	−0.798191	−	0.602404i	$-0.794209\pi$
$83$	−14.5437	−1.59638	−0.798191	+	0.602404i	$0.794209\pi$
$84$	0	0
$85$	−2.18466	−0.236960
$86$	−0.512278	−0.0552403
$87$	0	0
$88$	4.09461	0.436487
$89$	2.75372	0.291893	0.145947	−	0.989292i	$-0.453377\pi$
$89$	2.75372	0.291893	0.145947	+	0.989292i	$0.453377\pi$
$90$	0	0
$91$	−4.29177	−0.449900
$92$	0	0
$93$	0	0
$94$	−1.70808	−0.176175
$95$	−2.68665	−0.275644
$96$	0	0
$97$	−5.90566	−0.599628	−0.299814	−	0.953998i	$-0.596925\pi$
$97$	−5.90566	−0.599628	−0.299814	+	0.953998i	$0.596925\pi$
$98$	0.656411	0.0663076
$99$	0	0
$100$	7.74844	0.774844
$101$	5.33760	0.531111	0.265556	−	0.964096i	$-0.414445\pi$
$101$	5.33760	0.531111	0.265556	+	0.964096i	$0.414445\pi$
$102$	0	0
$103$	4.95919	0.488643	0.244322	−	0.969694i	$-0.421435\pi$
$103$	4.95919	0.488643	0.244322	+	0.969694i	$0.421435\pi$
$104$	−1.94760	−0.190978
$105$	0	0
$106$	2.67075	0.259406
$107$	3.53964	0.342190	0.171095	−	0.985255i	$-0.445270\pi$
$107$	3.53964	0.342190	0.171095	+	0.985255i	$0.445270\pi$
$108$	0	0
$109$	−3.81953	−0.365845	−0.182922	−	0.983127i	$-0.558556\pi$
$109$	−3.81953	−0.365845	−0.182922	+	0.983127i	$0.558556\pi$
$110$	−1.04558	−0.0996923
$111$	0	0
$112$	7.53798	0.712272
$113$	−18.3439	−1.72565	−0.862825	−	0.505503i	$-0.831307\pi$
$113$	−18.3439	−1.72565	−0.862825	+	0.505503i	$0.831307\pi$
$114$	0	0
$115$	0	0
$116$	15.5026	1.43938
$117$	0	0
$118$	2.88319	0.265419
$119$	−4.45828	−0.408690
$120$	0	0
$121$	8.27303	0.752094
$122$	−3.33740	−0.302154
$123$	0	0
$124$	15.2117	1.36605
$125$	−9.04984	−0.809442
$126$	0	0
$127$	1.04594	0.0928121	0.0464060	−	0.998923i	$-0.485223\pi$
$127$	1.04594	0.0928121	0.0464060	+	0.998923i	$0.485223\pi$
$128$	7.04718	0.622889
$129$	0	0
$130$	0.497331	0.0436189
$131$	−9.18162	−0.802202	−0.401101	−	0.916034i	$-0.631372\pi$
$131$	−9.18162	−0.802202	−0.401101	+	0.916034i	$0.631372\pi$
$132$	0	0
$133$	−5.48269	−0.475409
$134$	0.168378	0.0145457
$135$	0	0
$136$	−2.02317	−0.173485
$137$	−14.8173	−1.26592	−0.632962	−	0.774183i	$-0.718161\pi$
$137$	−14.8173	−1.26592	−0.632962	+	0.774183i	$0.718161\pi$
$138$	0	0
$139$	16.3675	1.38827	0.694137	−	0.719843i	$-0.255786\pi$
$139$	16.3675	1.38827	0.694137	+	0.719843i	$0.255786\pi$
$140$	−4.02419	−0.340106
$141$	0	0
$142$	−0.973742	−0.0817146
$143$	−9.16723	−0.766602
$144$	0	0
$145$	−8.03123	−0.666957
$146$	0.0138672	0.00114765
$147$	0	0
$148$	−7.63392	−0.627504
$149$	0.714546	0.0585379	0.0292689	−	0.999572i	$-0.490682\pi$
$149$	0.714546	0.0585379	0.0292689	+	0.999572i	$0.490682\pi$
$150$	0	0
$151$	7.21853	0.587436	0.293718	−	0.955892i	$-0.405107\pi$
$151$	7.21853	0.587436	0.293718	+	0.955892i	$0.405107\pi$
$152$	−2.48804	−0.201807
$153$	0	0
$154$	−2.13374	−0.171941
$155$	−7.88052	−0.632979
$156$	0	0
$157$	−17.9878	−1.43558	−0.717792	−	0.696257i	$-0.754847\pi$
$157$	−17.9878	−1.43558	−0.717792	+	0.696257i	$0.754847\pi$
$158$	−3.85408	−0.306614
$159$	0	0
$160$	−2.75221	−0.217581
$161$	0	0
$162$	0	0
$163$	4.58162	0.358860	0.179430	−	0.983771i	$-0.442575\pi$
$163$	4.58162	0.358860	0.179430	+	0.983771i	$0.442575\pi$
$164$	12.2630	0.957581
$165$	0	0
$166$	3.43929	0.266940
$167$	16.8037	1.30031	0.650155	−	0.759801i	$-0.274704\pi$
$167$	16.8037	1.30031	0.650155	+	0.759801i	$0.274704\pi$
$168$	0	0
$169$	−8.63960	−0.664585
$170$	0.516626	0.0396234
$171$	0	0
$172$	−4.21140	−0.321116
$173$	5.79309	0.440441	0.220220	−	0.975450i	$-0.429322\pi$
$173$	5.79309	0.440441	0.220220	+	0.975450i	$0.429322\pi$
$174$	0	0
$175$	−8.19171	−0.619235
$176$	16.1011	1.21367
$177$	0	0
$178$	−0.651196	−0.0488092
$179$	−17.1995	−1.28555	−0.642777	−	0.766053i	$-0.722218\pi$
$179$	−17.1995	−1.28555	−0.642777	+	0.766053i	$0.722218\pi$
$180$	0	0
$181$	2.18141	0.162143	0.0810715	−	0.996708i	$-0.474166\pi$
$181$	2.18141	0.162143	0.0810715	+	0.996708i	$0.474166\pi$
$182$	1.01491	0.0752304
$183$	0	0
$184$	0	0
$185$	3.95480	0.290763
$186$	0	0
$187$	−9.52289	−0.696382
$188$	−14.0420	−1.02412
$189$	0	0
$190$	0.635335	0.0460920
$191$	−17.3153	−1.25289	−0.626445	−	0.779465i	$-0.715491\pi$
$191$	−17.3153	−1.25289	−0.626445	+	0.779465i	$0.715491\pi$
$192$	0	0
$193$	−17.9909	−1.29501	−0.647506	−	0.762060i	$-0.724188\pi$
$193$	−17.9909	−1.29501	−0.647506	+	0.762060i	$0.724188\pi$
$194$	1.39656	0.100267
$195$	0	0
$196$	5.39632	0.385451
$197$	4.90355	0.349363	0.174682	−	0.984625i	$-0.444110\pi$
$197$	4.90355	0.349363	0.174682	+	0.984625i	$0.444110\pi$
$198$	0	0
$199$	−12.4893	−0.885341	−0.442671	−	0.896684i	$-0.645969\pi$
$199$	−12.4893	−0.885341	−0.442671	+	0.896684i	$0.645969\pi$
$200$	−3.71739	−0.262859
$201$	0	0
$202$	−1.26223	−0.0888102
$203$	−16.3895	−1.15032
$204$	0	0
$205$	−6.35293	−0.443708
$206$	−1.17274	−0.0817089
$207$	0	0
$208$	−7.65851	−0.531022
$209$	−11.7110	−0.810068
$210$	0	0
$211$	−18.0552	−1.24297	−0.621485	−	0.783426i	$-0.713470\pi$
$211$	−18.0552	−1.24297	−0.621485	+	0.783426i	$0.713470\pi$
$212$	21.9560	1.50795
$213$	0	0
$214$	−0.837049	−0.0572195
$215$	2.18174	0.148794
$216$	0	0
$217$	−16.0819	−1.09171
$218$	0.903239	0.0611750
$219$	0	0
$220$	−8.59566	−0.579519
$221$	4.52957	0.304692
$222$	0	0
$223$	−8.73096	−0.584668	−0.292334	−	0.956316i	$-0.594432\pi$
$223$	−8.73096	−0.584668	−0.292334	+	0.956316i	$0.594432\pi$
$224$	−5.61648	−0.375267
$225$	0	0
$226$	4.33795	0.288556
$227$	23.7174	1.57418	0.787090	−	0.616838i	$-0.211587\pi$
$227$	23.7174	1.57418	0.787090	+	0.616838i	$0.211587\pi$
$228$	0	0
$229$	6.81277	0.450201	0.225100	−	0.974336i	$-0.427729\pi$
$229$	6.81277	0.450201	0.225100	+	0.974336i	$0.427729\pi$
$230$	0	0
$231$	0	0
$232$	−7.43754	−0.488298
$233$	1.36738	0.0895797	0.0447899	−	0.998996i	$-0.485738\pi$
$233$	1.36738	0.0895797	0.0447899	+	0.998996i	$0.485738\pi$
$234$	0	0
$235$	7.27455	0.474540
$236$	23.7025	1.54290
$237$	0	0
$238$	1.05429	0.0683394
$239$	18.0099	1.16497	0.582483	−	0.812843i	$-0.302081\pi$
$239$	18.0099	1.16497	0.582483	+	0.812843i	$0.302081\pi$
$240$	0	0
$241$	0.295761	0.0190516	0.00952582	−	0.999955i	$-0.496968\pi$
$241$	0.295761	0.0190516	0.00952582	+	0.999955i	$0.496968\pi$
$242$	−1.95640	−0.125762
$243$	0	0
$244$	−27.4365	−1.75644
$245$	−2.79560	−0.178604
$246$	0	0
$247$	5.57035	0.354433
$248$	−7.29797	−0.463422
$249$	0	0
$250$	2.14010	0.135352
$251$	15.0335	0.948906	0.474453	−	0.880281i	$-0.342646\pi$
$251$	15.0335	0.948906	0.474453	+	0.880281i	$0.342646\pi$
$252$	0	0
$253$	0	0
$254$	−0.247342	−0.0155196
$255$	0	0
$256$	11.7114	0.731964
$257$	−14.5046	−0.904774	−0.452387	−	0.891822i	$-0.649427\pi$
$257$	−14.5046	−0.904774	−0.452387	+	0.891822i	$0.649427\pi$
$258$	0	0
$259$	8.07064	0.501485
$260$	4.08853	0.253560
$261$	0	0
$262$	2.17126	0.134141
$263$	15.2707	0.941634	0.470817	−	0.882231i	$-0.343959\pi$
$263$	15.2707	0.941634	0.470817	+	0.882231i	$0.343959\pi$
$264$	0	0
$265$	−11.3745	−0.698728
$266$	1.29654	0.0794959
$267$	0	0
$268$	1.38423	0.0845552
$269$	−13.6130	−0.830000	−0.415000	−	0.909821i	$-0.636218\pi$
$269$	−13.6130	−0.830000	−0.415000	+	0.909821i	$0.636218\pi$
$270$	0	0
$271$	4.41606	0.268257	0.134128	−	0.990964i	$-0.457177\pi$
$271$	4.41606	0.268257	0.134128	+	0.990964i	$0.457177\pi$
$272$	−7.95563	−0.482381
$273$	0	0
$274$	3.50397	0.211682
$275$	−17.4975	−1.05514
$276$	0	0
$277$	16.7884	1.00872	0.504358	−	0.863495i	$-0.331729\pi$
$277$	16.7884	1.00872	0.504358	+	0.863495i	$0.331729\pi$
$278$	−3.87057	−0.232141
$279$	0	0
$280$	1.93064	0.115378
$281$	−28.0522	−1.67345	−0.836726	−	0.547622i	$-0.815533\pi$
$281$	−28.0522	−1.67345	−0.836726	+	0.547622i	$0.815533\pi$
$282$	0	0
$283$	−28.4068	−1.68861	−0.844304	−	0.535865i	$-0.819986\pi$
$283$	−28.4068	−1.68861	−0.844304	+	0.535865i	$0.819986\pi$
$284$	−8.00507	−0.475014
$285$	0	0
$286$	2.16786	0.128188
$287$	−12.9645	−0.765273
$288$	0	0
$289$	−12.2947	−0.723218
$290$	1.89922	0.111526
$291$	0	0
$292$	0.114001	0.00667140
$293$	−15.4269	−0.901251	−0.450625	−	0.892713i	$-0.648799\pi$
$293$	−15.4269	−0.901251	−0.450625	+	0.892713i	$0.648799\pi$
$294$	0	0
$295$	−12.2793	−0.714926
$296$	3.66245	0.212876
$297$	0	0
$298$	−0.168975	−0.00978846
$299$	0	0
$300$	0	0
$301$	4.45232	0.256628
$302$	−1.70703	−0.0982285
$303$	0	0
$304$	−9.78365	−0.561131
$305$	14.2137	0.813873
$306$	0	0
$307$	32.8470	1.87468	0.937339	−	0.348419i	$-0.113281\pi$
$307$	32.8470	1.87468	0.937339	+	0.348419i	$0.113281\pi$
$308$	−17.5413	−0.999510
$309$	0	0
$310$	1.86358	0.105844
$311$	2.02085	0.114592	0.0572959	−	0.998357i	$-0.481752\pi$
$311$	2.02085	0.114592	0.0572959	+	0.998357i	$0.481752\pi$
$312$	0	0
$313$	22.3502	1.26331	0.631655	−	0.775250i	$-0.282376\pi$
$313$	22.3502	1.26331	0.631655	+	0.775250i	$0.282376\pi$
$314$	4.25374	0.240052
$315$	0	0
$316$	−31.6842	−1.78237
$317$	−14.0996	−0.791915	−0.395957	−	0.918269i	$-0.629587\pi$
$317$	−14.0996	−0.791915	−0.395957	+	0.918269i	$0.629587\pi$
$318$	0	0
$319$	−35.0079	−1.96007
$320$	−6.73674	−0.376595
$321$	0	0
$322$	0	0
$323$	5.78646	0.321967
$324$	0	0
$325$	8.32269	0.461660
$326$	−1.08346	−0.0600071
$327$	0	0
$328$	−5.88331	−0.324851
$329$	14.8453	0.818449
$330$	0	0
$331$	0.786431	0.0432262	0.0216131	−	0.999766i	$-0.493120\pi$
$331$	0.786431	0.0432262	0.0216131	+	0.999766i	$0.493120\pi$
$332$	28.2741	1.55175
$333$	0	0
$334$	−3.97372	−0.217433
$335$	−0.717108	−0.0391798
$336$	0	0
$337$	−5.20000	−0.283262	−0.141631	−	0.989919i	$-0.545235\pi$
$337$	−5.20000	−0.283262	−0.141631	+	0.989919i	$0.545235\pi$
$338$	2.04308	0.111129
$339$	0	0
$340$	4.24715	0.230334
$341$	−34.3510	−1.86021
$342$	0	0
$343$	−20.0921	−1.08487
$344$	2.02046	0.108936
$345$	0	0
$346$	−1.36994	−0.0736486
$347$	−10.4512	−0.561050	−0.280525	−	0.959847i	$-0.590509\pi$
$347$	−10.4512	−0.561050	−0.280525	+	0.959847i	$0.590509\pi$
$348$	0	0
$349$	15.7564	0.843421	0.421710	−	0.906731i	$-0.361430\pi$
$349$	15.7564	0.843421	0.421710	+	0.906731i	$0.361430\pi$
$350$	1.93717	0.103546
$351$	0	0
$352$	−11.9968	−0.639432
$353$	−20.1148	−1.07060	−0.535302	−	0.844661i	$-0.679802\pi$
$353$	−20.1148	−1.07060	−0.535302	+	0.844661i	$0.679802\pi$
$354$	0	0
$355$	4.14708	0.220104
$356$	−5.35344	−0.283732
$357$	0	0
$358$	4.06733	0.214965
$359$	11.0102	0.581097	0.290548	−	0.956860i	$-0.406162\pi$
$359$	11.0102	0.581097	0.290548	+	0.956860i	$0.406162\pi$
$360$	0	0
$361$	−11.8839	−0.625471
$362$	−0.515858	−0.0271129
$363$	0	0
$364$	8.34354	0.437320
$365$	−0.0590590	−0.00309129
$366$	0	0
$367$	−20.5922	−1.07490	−0.537452	−	0.843294i	$-0.680613\pi$
$367$	−20.5922	−1.07490	−0.537452	+	0.843294i	$0.680613\pi$
$368$	0	0
$369$	0	0
$370$	−0.935227	−0.0486201
$371$	−23.2121	−1.20511
$372$	0	0
$373$	23.0849	1.19529	0.597644	−	0.801761i	$-0.296103\pi$
$373$	23.0849	1.19529	0.597644	+	0.801761i	$0.296103\pi$
$374$	2.25196	0.116446
$375$	0	0
$376$	6.73680	0.347424
$377$	16.6515	0.857598
$378$	0	0
$379$	1.22394	0.0628696	0.0314348	−	0.999506i	$-0.489992\pi$
$379$	1.22394	0.0628696	0.0314348	+	0.999506i	$0.489992\pi$
$380$	5.22305	0.267937
$381$	0	0
$382$	4.09470	0.209503
$383$	−7.51089	−0.383789	−0.191894	−	0.981416i	$-0.561463\pi$
$383$	−7.51089	−0.383789	−0.191894	+	0.981416i	$0.561463\pi$
$384$	0	0
$385$	9.08739	0.463136
$386$	4.25447	0.216547
$387$	0	0
$388$	11.4811	0.582862
$389$	−17.9573	−0.910473	−0.455237	−	0.890370i	$-0.650445\pi$
$389$	−17.9573	−0.910473	−0.455237	+	0.890370i	$0.650445\pi$
$390$	0	0
$391$	0	0
$392$	−2.58894	−0.130761
$393$	0	0
$394$	−1.15959	−0.0584191
$395$	16.4142	0.825888
$396$	0	0
$397$	−33.3520	−1.67389	−0.836944	−	0.547289i	$-0.815660\pi$
$397$	−33.3520	−1.67389	−0.836944	+	0.547289i	$0.815660\pi$
$398$	2.95345	0.148043
$399$	0	0
$400$	−14.6178	−0.730890
$401$	−8.12602	−0.405794	−0.202897	−	0.979200i	$-0.565036\pi$
$401$	−8.12602	−0.405794	−0.202897	+	0.979200i	$0.565036\pi$
$402$	0	0
$403$	16.3391	0.813907
$404$	−10.3767	−0.516261
$405$	0	0
$406$	3.87577	0.192351
$407$	17.2389	0.854500
$408$	0	0
$409$	−15.4452	−0.763718	−0.381859	−	0.924221i	$-0.624716\pi$
$409$	−15.4452	−0.763718	−0.381859	+	0.924221i	$0.624716\pi$
$410$	1.50233	0.0741950
$411$	0	0
$412$	−9.64104	−0.474980
$413$	−25.0585	−1.23305
$414$	0	0
$415$	−14.6476	−0.719023
$416$	5.70628	0.279774
$417$	0	0
$418$	2.76941	0.135456
$419$	−21.6131	−1.05587	−0.527935	−	0.849285i	$-0.677034\pi$
$419$	−21.6131	−1.05587	−0.527935	+	0.849285i	$0.677034\pi$
$420$	0	0
$421$	7.89428	0.384744	0.192372	−	0.981322i	$-0.438382\pi$
$421$	7.89428	0.384744	0.192372	+	0.981322i	$0.438382\pi$
$422$	4.26967	0.207844
$423$	0	0
$424$	−10.5336	−0.511558
$425$	8.64559	0.419372
$426$	0	0
$427$	29.0061	1.40370
$428$	−6.88133	−0.332622
$429$	0	0
$430$	−0.515936	−0.0248807
$431$	3.16611	0.152506	0.0762531	−	0.997088i	$-0.475704\pi$
$431$	3.16611	0.152506	0.0762531	+	0.997088i	$0.475704\pi$
$432$	0	0
$433$	16.9520	0.814662	0.407331	−	0.913280i	$-0.366459\pi$
$433$	16.9520	0.814662	0.407331	+	0.913280i	$0.366459\pi$
$434$	3.80304	0.182552
$435$	0	0
$436$	7.42547	0.355615
$437$	0	0
$438$	0	0
$439$	13.7946	0.658379	0.329189	−	0.944264i	$-0.393225\pi$
$439$	13.7946	0.658379	0.329189	+	0.944264i	$0.393225\pi$
$440$	4.12386	0.196597
$441$	0	0
$442$	−1.07115	−0.0509492
$443$	7.94775	0.377609	0.188805	−	0.982015i	$-0.439539\pi$
$443$	7.94775	0.377609	0.188805	+	0.982015i	$0.439539\pi$
$444$	0	0
$445$	2.77338	0.131471
$446$	2.06469	0.0977658
$447$	0	0
$448$	−13.7478	−0.649521
$449$	31.4846	1.48585	0.742925	−	0.669374i	$-0.233438\pi$
$449$	31.4846	1.48585	0.742925	+	0.669374i	$0.233438\pi$
$450$	0	0
$451$	−27.6923	−1.30398
$452$	35.6620	1.67740
$453$	0	0
$454$	−5.60867	−0.263228
$455$	−4.32242	−0.202638
$456$	0	0
$457$	21.3571	0.999041	0.499521	−	0.866302i	$-0.333509\pi$
$457$	21.3571	0.999041	0.499521	+	0.866302i	$0.333509\pi$
$458$	−1.61108	−0.0752807
$459$	0	0
$460$	0	0
$461$	−1.51050	−0.0703510	−0.0351755	−	0.999381i	$-0.511199\pi$
$461$	−1.51050	−0.0703510	−0.0351755	+	0.999381i	$0.511199\pi$
$462$	0	0
$463$	−3.05242	−0.141858	−0.0709289	−	0.997481i	$-0.522596\pi$
$463$	−3.05242	−0.141858	−0.0709289	+	0.997481i	$0.522596\pi$
$464$	−29.2464	−1.35773
$465$	0	0
$466$	−0.323355	−0.0149791
$467$	−5.32804	−0.246552	−0.123276	−	0.992372i	$-0.539340\pi$
$467$	−5.32804	−0.246552	−0.123276	+	0.992372i	$0.539340\pi$
$468$	0	0
$469$	−1.46342	−0.0675742
$470$	−1.72028	−0.0793505
$471$	0	0
$472$	−11.3715	−0.523417
$473$	9.51017	0.437278
$474$	0	0
$475$	10.6321	0.487836
$476$	8.66724	0.397262
$477$	0	0
$478$	−4.25897	−0.194801
$479$	9.46457	0.432447	0.216224	−	0.976344i	$-0.430626\pi$
$479$	9.46457	0.432447	0.216224	+	0.976344i	$0.430626\pi$
$480$	0	0
$481$	−8.19968	−0.373873
$482$	−0.0699412	−0.00318573
$483$	0	0
$484$	−16.0834	−0.731064
$485$	−5.94783	−0.270077
$486$	0	0
$487$	13.6397	0.618075	0.309037	−	0.951050i	$-0.399993\pi$
$487$	13.6397	0.618075	0.309037	+	0.951050i	$0.399993\pi$
$488$	13.1630	0.595859
$489$	0	0
$490$	0.661099	0.0298654
$491$	−2.93931	−0.132649	−0.0663247	−	0.997798i	$-0.521127\pi$
$491$	−2.93931	−0.132649	−0.0663247	+	0.997798i	$0.521127\pi$
$492$	0	0
$493$	17.2976	0.779043
$494$	−1.31727	−0.0592668
$495$	0	0
$496$	−28.6976	−1.28856
$497$	8.46302	0.379618
$498$	0	0
$499$	11.4719	0.513553	0.256776	−	0.966471i	$-0.417340\pi$
$499$	11.4719	0.513553	0.256776	+	0.966471i	$0.417340\pi$
$500$	17.5936	0.786809
$501$	0	0
$502$	−3.55511	−0.158672
$503$	−23.2357	−1.03603	−0.518014	−	0.855372i	$-0.673329\pi$
$503$	−23.2357	−1.03603	−0.518014	+	0.855372i	$0.673329\pi$
$504$	0	0
$505$	5.37572	0.239217
$506$	0	0
$507$	0	0
$508$	−2.03339	−0.0902169
$509$	3.96686	0.175828	0.0879139	−	0.996128i	$-0.471980\pi$
$509$	3.96686	0.175828	0.0879139	+	0.996128i	$0.471980\pi$
$510$	0	0
$511$	−0.120523	−0.00533161
$512$	−16.8639	−0.745284
$513$	0	0
$514$	3.43004	0.151292
$515$	4.99460	0.220089
$516$	0	0
$517$	31.7096	1.39459
$518$	−1.90853	−0.0838562
$519$	0	0
$520$	−1.96151	−0.0860181
$521$	−3.28183	−0.143779	−0.0718897	−	0.997413i	$-0.522903\pi$
$521$	−3.28183	−0.143779	−0.0718897	+	0.997413i	$0.522903\pi$
$522$	0	0
$523$	−0.172624	−0.00754832	−0.00377416	−	0.999993i	$-0.501201\pi$
$523$	−0.172624	−0.00754832	−0.00377416	+	0.999993i	$0.501201\pi$
$524$	17.8498	0.779771
$525$	0	0
$526$	−3.61121	−0.157456
$527$	16.9730	0.739354
$528$	0	0
$529$	0	0
$530$	2.68982	0.116838
$531$	0	0
$532$	10.6588	0.462116
$533$	13.1718	0.570536
$534$	0	0
$535$	3.56492	0.154125
$536$	−0.664097	−0.0286846
$537$	0	0
$538$	3.21919	0.138789
$539$	−12.1859	−0.524885
$540$	0	0
$541$	16.4406	0.706838	0.353419	−	0.935465i	$-0.385019\pi$
$541$	16.4406	0.706838	0.353419	+	0.935465i	$0.385019\pi$
$542$	−1.04431	−0.0448568
$543$	0	0
$544$	5.92767	0.254147
$545$	−3.84681	−0.164779
$546$	0	0
$547$	0.317709	0.0135842	0.00679212	−	0.999977i	$-0.497838\pi$
$547$	0.317709	0.0135842	0.00679212	+	0.999977i	$0.497838\pi$
$548$	28.8059	1.23053
$549$	0	0
$550$	4.13779	0.176436
$551$	21.2721	0.906223
$552$	0	0
$553$	33.4967	1.42443
$554$	−3.97009	−0.168673
$555$	0	0
$556$	−31.8197	−1.34946
$557$	−1.82207	−0.0772036	−0.0386018	−	0.999255i	$-0.512290\pi$
$557$	−1.82207	−0.0772036	−0.0386018	+	0.999255i	$0.512290\pi$
$558$	0	0
$559$	−4.52352	−0.191324
$560$	7.59181	0.320813
$561$	0	0
$562$	6.63374	0.279828
$563$	2.56170	0.107963	0.0539814	−	0.998542i	$-0.482809\pi$
$563$	2.56170	0.107963	0.0539814	+	0.998542i	$0.482809\pi$
$564$	0	0
$565$	−18.4749	−0.777246
$566$	6.71760	0.282362
$567$	0	0
$568$	3.84052	0.161144
$569$	11.7193	0.491300	0.245650	−	0.969359i	$-0.420999\pi$
$569$	11.7193	0.491300	0.245650	+	0.969359i	$0.420999\pi$
$570$	0	0
$571$	−0.594181	−0.0248657	−0.0124329	−	0.999923i	$-0.503958\pi$
$571$	−0.594181	−0.0248657	−0.0124329	+	0.999923i	$0.503958\pi$
$572$	17.8218	0.745167
$573$	0	0
$574$	3.06584	0.127966
$575$	0	0
$576$	0	0
$577$	4.11715	0.171399	0.0856995	−	0.996321i	$-0.472687\pi$
$577$	4.11715	0.171399	0.0856995	+	0.996321i	$0.472687\pi$
$578$	2.90744	0.120933
$579$	0	0
$580$	15.6133	0.648308
$581$	−29.8916	−1.24011
$582$	0	0
$583$	−49.5810	−2.05344
$584$	−0.0546931	−0.00226322
$585$	0	0
$586$	3.64814	0.150703
$587$	−41.6561	−1.71933	−0.859666	−	0.510857i	$-0.829328\pi$
$587$	−41.6561	−1.71933	−0.859666	+	0.510857i	$0.829328\pi$
$588$	0	0
$589$	20.8730	0.860055
$590$	2.90378	0.119547
$591$	0	0
$592$	14.4018	0.591908
$593$	−1.37079	−0.0562914	−0.0281457	−	0.999604i	$-0.508960\pi$
$593$	−1.37079	−0.0562914	−0.0281457	+	0.999604i	$0.508960\pi$
$594$	0	0
$595$	−4.49012	−0.184077
$596$	−1.38913	−0.0569011
$597$	0	0
$598$	0	0
$599$	26.0672	1.06508	0.532538	−	0.846406i	$-0.321238\pi$
$599$	26.0672	1.06508	0.532538	+	0.846406i	$0.321238\pi$
$600$	0	0
$601$	−0.0655862	−0.00267532	−0.00133766	−	0.999999i	$-0.500426\pi$
$601$	−0.0655862	−0.00267532	−0.00133766	+	0.999999i	$0.500426\pi$
$602$	−1.05288	−0.0429122
$603$	0	0
$604$	−14.0334	−0.571010
$605$	8.33211	0.338749
$606$	0	0
$607$	−15.0668	−0.611544	−0.305772	−	0.952105i	$-0.598915\pi$
$607$	−15.0668	−0.611544	−0.305772	+	0.952105i	$0.598915\pi$
$608$	7.28971	0.295637
$609$	0	0
$610$	−3.36123	−0.136092
$611$	−15.0827	−0.610180
$612$	0	0
$613$	−45.3834	−1.83302	−0.916509	−	0.400014i	$-0.869005\pi$
$613$	−45.3834	−1.83302	−0.916509	+	0.400014i	$0.869005\pi$
$614$	−7.76762	−0.313476
$615$	0	0
$616$	8.41563	0.339075
$617$	44.0445	1.77316	0.886582	−	0.462571i	$-0.153073\pi$
$617$	44.0445	1.77316	0.886582	+	0.462571i	$0.153073\pi$
$618$	0	0
$619$	19.9977	0.803776	0.401888	−	0.915689i	$-0.368354\pi$
$619$	19.9977	0.803776	0.401888	+	0.915689i	$0.368354\pi$
$620$	15.3204	0.615280
$621$	0	0
$622$	−0.477888	−0.0191615
$623$	5.65969	0.226751
$624$	0	0
$625$	10.8139	0.432554
$626$	−5.28536	−0.211245
$627$	0	0
$628$	34.9697	1.39544
$629$	−8.51780	−0.339627
$630$	0	0
$631$	−40.2554	−1.60254	−0.801271	−	0.598301i	$-0.795843\pi$
$631$	−40.2554	−1.60254	−0.801271	+	0.598301i	$0.795843\pi$
$632$	15.2008	0.604656
$633$	0	0
$634$	3.33427	0.132421
$635$	1.05341	0.0418033
$636$	0	0
$637$	5.79624	0.229656
$638$	8.27864	0.327754
$639$	0	0
$640$	7.09751	0.280554
$641$	−23.0020	−0.908526	−0.454263	−	0.890868i	$-0.650097\pi$
$641$	−23.0020	−0.908526	−0.454263	+	0.890868i	$0.650097\pi$
$642$	0	0
$643$	−6.29862	−0.248393	−0.124197	−	0.992258i	$-0.539635\pi$
$643$	−6.29862	−0.248393	−0.124197	+	0.992258i	$0.539635\pi$
$644$	0	0
$645$	0	0
$646$	−1.36838	−0.0538381
$647$	0.235819	0.00927101	0.00463550	−	0.999989i	$-0.498524\pi$
$647$	0.235819	0.00927101	0.00463550	+	0.999989i	$0.498524\pi$
$648$	0	0
$649$	−53.5250	−2.10104
$650$	−1.96814	−0.0771968
$651$	0	0
$652$	−8.90702	−0.348826
$653$	−19.7596	−0.773253	−0.386626	−	0.922236i	$-0.626360\pi$
$653$	−19.7596	−0.773253	−0.386626	+	0.922236i	$0.626360\pi$
$654$	0	0
$655$	−9.24719	−0.361318
$656$	−23.1348	−0.903260
$657$	0	0
$658$	−3.51060	−0.136858
$659$	−39.6658	−1.54516	−0.772579	−	0.634918i	$-0.781034\pi$
$659$	−39.6658	−1.54516	−0.772579	+	0.634918i	$0.781034\pi$
$660$	0	0
$661$	36.8354	1.43273	0.716366	−	0.697725i	$-0.245804\pi$
$661$	36.8354	1.43273	0.716366	+	0.697725i	$0.245804\pi$
$662$	−0.185974	−0.00722810
$663$	0	0
$664$	−13.5648	−0.526417
$665$	−5.52184	−0.214128
$666$	0	0
$667$	0	0
$668$	−32.6677	−1.26395
$669$	0	0
$670$	0.169581	0.00655148
$671$	61.9571	2.39183
$672$	0	0
$673$	50.6016	1.95055	0.975274	−	0.220999i	$-0.0709317\pi$
$673$	50.6016	1.95055	0.975274	+	0.220999i	$0.0709317\pi$
$674$	1.22969	0.0473659
$675$	0	0
$676$	16.7961	0.646002
$677$	−27.0137	−1.03822	−0.519111	−	0.854707i	$-0.673737\pi$
$677$	−27.0137	−1.03822	−0.519111	+	0.854707i	$0.673737\pi$
$678$	0	0
$679$	−12.1378	−0.465808
$680$	−2.03761	−0.0781390
$681$	0	0
$682$	8.12329	0.311057
$683$	27.5008	1.05229	0.526144	−	0.850396i	$-0.323637\pi$
$683$	27.5008	1.05229	0.526144	+	0.850396i	$0.323637\pi$
$684$	0	0
$685$	−14.9231	−0.570181
$686$	4.75135	0.181407
$687$	0	0
$688$	7.94501	0.302901
$689$	23.5832	0.898449
$690$	0	0
$691$	−3.58673	−0.136446	−0.0682229	−	0.997670i	$-0.521733\pi$
$691$	−3.58673	−0.136446	−0.0682229	+	0.997670i	$0.521733\pi$
$692$	−11.2622	−0.428125
$693$	0	0
$694$	2.47149	0.0938165
$695$	16.4844	0.625289
$696$	0	0
$697$	13.6829	0.518276
$698$	−3.72606	−0.141033
$699$	0	0
$700$	15.9253	0.601920
$701$	29.0797	1.09833	0.549163	−	0.835716i	$-0.314947\pi$
$701$	29.0797	1.09833	0.549163	+	0.835716i	$0.314947\pi$
$702$	0	0
$703$	−10.4750	−0.395072
$704$	−29.3653	−1.10674
$705$	0	0
$706$	4.75673	0.179022
$707$	10.9703	0.412582
$708$	0	0
$709$	−2.59976	−0.0976360	−0.0488180	−	0.998808i	$-0.515545\pi$
$709$	−2.59976	−0.0976360	−0.0488180	+	0.998808i	$0.515545\pi$
$710$	−0.980697	−0.0368049
$711$	0	0
$712$	2.56837	0.0962536
$713$	0	0
$714$	0	0
$715$	−9.23270	−0.345283
$716$	33.4373	1.24961
$717$	0	0
$718$	−2.60368	−0.0971686
$719$	−18.9338	−0.706110	−0.353055	−	0.935603i	$-0.614857\pi$
$719$	−18.9338	−0.706110	−0.353055	+	0.935603i	$0.614857\pi$
$720$	0	0
$721$	10.1926	0.379591
$722$	2.81030	0.104589
$723$	0	0
$724$	−4.24083	−0.157609
$725$	31.7828	1.18038
$726$	0	0
$727$	8.08294	0.299780	0.149890	−	0.988703i	$-0.452108\pi$
$727$	8.08294	0.299780	0.149890	+	0.988703i	$0.452108\pi$
$728$	−4.00290	−0.148357
$729$	0	0
$730$	0.0139662	0.000516912 0
$731$	−4.69901	−0.173799
$732$	0	0
$733$	6.31984	0.233429	0.116714	−	0.993166i	$-0.462764\pi$
$733$	6.31984	0.233429	0.116714	+	0.993166i	$0.462764\pi$
$734$	4.86962	0.179741
$735$	0	0
$736$	0	0
$737$	−3.12586	−0.115142
$738$	0	0
$739$	−23.0985	−0.849691	−0.424845	−	0.905266i	$-0.639672\pi$
$739$	−23.0985	−0.849691	−0.424845	+	0.905266i	$0.639672\pi$
$740$	−7.68844	−0.282633
$741$	0	0
$742$	5.48917	0.201514
$743$	−26.6717	−0.978491	−0.489246	−	0.872146i	$-0.662728\pi$
$743$	−26.6717	−0.978491	−0.489246	+	0.872146i	$0.662728\pi$
$744$	0	0
$745$	0.719649	0.0263659
$746$	−5.45908	−0.199871
$747$	0	0
$748$	18.5132	0.676911
$749$	7.27499	0.265822
$750$	0	0
$751$	31.6095	1.15345	0.576724	−	0.816939i	$-0.304331\pi$
$751$	31.6095	1.15345	0.576724	+	0.816939i	$0.304331\pi$
$752$	26.4909	0.966024
$753$	0	0
$754$	−3.93774	−0.143404
$755$	7.27008	0.264585
$756$	0	0
$757$	28.9705	1.05295	0.526475	−	0.850190i	$-0.323513\pi$
$757$	28.9705	1.05295	0.526475	+	0.850190i	$0.323513\pi$
$758$	−0.289436	−0.0105128
$759$	0	0
$760$	−2.50581	−0.0908953
$761$	45.0660	1.63364	0.816821	−	0.576891i	$-0.195734\pi$
$761$	45.0660	1.63364	0.816821	+	0.576891i	$0.195734\pi$
$762$	0	0
$763$	−7.85026	−0.284198
$764$	33.6623	1.21786
$765$	0	0
$766$	1.77617	0.0641755
$767$	25.4592	0.919277
$768$	0	0
$769$	20.5509	0.741084	0.370542	−	0.928816i	$-0.379172\pi$
$769$	20.5509	0.741084	0.370542	+	0.928816i	$0.379172\pi$
$770$	−2.14898	−0.0774437
$771$	0	0
$772$	34.9757	1.25880
$773$	−18.4221	−0.662598	−0.331299	−	0.943526i	$-0.607487\pi$
$773$	−18.4221	−0.662598	−0.331299	+	0.943526i	$0.607487\pi$
$774$	0	0
$775$	31.1864	1.12025
$776$	−5.50815	−0.197731
$777$	0	0
$778$	4.24653	0.152246
$779$	16.8269	0.602885
$780$	0	0
$781$	18.0770	0.646846
$782$	0	0
$783$	0	0
$784$	−10.1804	−0.363586
$785$	−18.1163	−0.646598
$786$	0	0
$787$	19.8522	0.707654	0.353827	−	0.935311i	$-0.384880\pi$
$787$	19.8522	0.707654	0.353827	+	0.935311i	$0.384880\pi$
$788$	−9.53288	−0.339595
$789$	0	0
$790$	−3.88161	−0.138101
$791$	−37.7021	−1.34053
$792$	0	0
$793$	−29.4699	−1.04651
$794$	7.88704	0.279900
$795$	0	0
$796$	24.2801	0.860586
$797$	−25.6946	−0.910148	−0.455074	−	0.890454i	$-0.650387\pi$
$797$	−25.6946	−0.910148	−0.455074	+	0.890454i	$0.650387\pi$
$798$	0	0
$799$	−15.6678	−0.554289
$800$	10.8916	0.385076
$801$	0	0
$802$	1.92163	0.0678552
$803$	−0.257436	−0.00908474
$804$	0	0
$805$	0	0
$806$	−3.86385	−0.136098
$807$	0	0
$808$	4.97834	0.175137
$809$	−15.4647	−0.543710	−0.271855	−	0.962338i	$-0.587637\pi$
$809$	−15.4647	−0.543710	−0.271855	+	0.962338i	$0.587637\pi$
$810$	0	0
$811$	−20.7503	−0.728642	−0.364321	−	0.931273i	$-0.618699\pi$
$811$	−20.7503	−0.728642	−0.364321	+	0.931273i	$0.618699\pi$
$812$	31.8624	1.11815
$813$	0	0
$814$	−4.07663	−0.142886
$815$	4.61434	0.161633
$816$	0	0
$817$	−5.77874	−0.202172
$818$	3.65247	0.127706
$819$	0	0
$820$	12.3506	0.431302
$821$	−35.6399	−1.24384	−0.621921	−	0.783080i	$-0.713648\pi$
$821$	−35.6399	−1.24384	−0.621921	+	0.783080i	$0.713648\pi$
$822$	0	0
$823$	46.5608	1.62301	0.811503	−	0.584348i	$-0.198650\pi$
$823$	46.5608	1.62301	0.811503	+	0.584348i	$0.198650\pi$
$824$	4.62539	0.161133
$825$	0	0
$826$	5.92580	0.206185
$827$	46.6143	1.62094	0.810468	−	0.585782i	$-0.199213\pi$
$827$	46.6143	1.62094	0.810468	+	0.585782i	$0.199213\pi$
$828$	0	0
$829$	−41.8838	−1.45468	−0.727342	−	0.686276i	$-0.759244\pi$
$829$	−41.8838	−1.45468	−0.727342	+	0.686276i	$0.759244\pi$
$830$	3.46385	0.120232
$831$	0	0
$832$	13.9676	0.484239
$833$	6.02112	0.208619
$834$	0	0
$835$	16.9237	0.585670
$836$	22.7671	0.787418
$837$	0	0
$838$	5.11105	0.176558
$839$	−27.7532	−0.958146	−0.479073	−	0.877775i	$-0.659027\pi$
$839$	−27.7532	−0.958146	−0.479073	+	0.877775i	$0.659027\pi$
$840$	0	0
$841$	34.5891	1.19273
$842$	−1.86683	−0.0643352
$843$	0	0
$844$	35.1007	1.20822
$845$	−8.70131	−0.299334
$846$	0	0
$847$	17.0035	0.584247
$848$	−41.4211	−1.42241
$849$	0	0
$850$	−2.04450	−0.0701257
$851$	0	0
$852$	0	0
$853$	−43.3281	−1.48353	−0.741763	−	0.670662i	$-0.766010\pi$
$853$	−43.3281	−1.48353	−0.741763	+	0.670662i	$0.766010\pi$
$854$	−6.85933	−0.234722
$855$	0	0
$856$	3.30139	0.112839
$857$	−50.2087	−1.71510	−0.857549	−	0.514402i	$-0.828014\pi$
$857$	−50.2087	−1.71510	−0.857549	+	0.514402i	$0.828014\pi$
$858$	0	0
$859$	−38.0827	−1.29936	−0.649682	−	0.760206i	$-0.725098\pi$
$859$	−38.0827	−1.29936	−0.649682	+	0.760206i	$0.725098\pi$
$860$	−4.24148	−0.144633
$861$	0	0
$862$	−0.748719	−0.0255015
$863$	−19.4101	−0.660728	−0.330364	−	0.943854i	$-0.607171\pi$
$863$	−19.4101	−0.660728	−0.330364	+	0.943854i	$0.607171\pi$
$864$	0	0
$865$	5.83447	0.198378
$866$	−4.00880	−0.136224
$867$	0	0
$868$	31.2645	1.06119
$869$	71.5490	2.42713
$870$	0	0
$871$	1.48681	0.0503788
$872$	−3.56244	−0.120640
$873$	0	0
$874$	0	0
$875$	−18.6001	−0.628797
$876$	0	0
$877$	−21.3238	−0.720054	−0.360027	−	0.932942i	$-0.617233\pi$
$877$	−21.3238	−0.720054	−0.360027	+	0.932942i	$0.617233\pi$
$878$	−3.26212	−0.110091
$879$	0	0
$880$	16.2161	0.546645
$881$	14.0783	0.474311	0.237155	−	0.971472i	$-0.423785\pi$
$881$	14.0783	0.474311	0.237155	+	0.971472i	$0.423785\pi$
$882$	0	0
$883$	56.8259	1.91234	0.956172	−	0.292804i	$-0.0945885\pi$
$883$	56.8259	1.91234	0.956172	+	0.292804i	$0.0945885\pi$
$884$	−8.80583	−0.296172
$885$	0	0
$886$	−1.87948	−0.0631422
$887$	20.5838	0.691137	0.345568	−	0.938394i	$-0.387686\pi$
$887$	20.5838	0.691137	0.345568	+	0.938394i	$0.387686\pi$
$888$	0	0
$889$	2.14971	0.0720990
$890$	−0.655847	−0.0219840
$891$	0	0
$892$	16.9737	0.568320
$893$	−19.2679	−0.644777
$894$	0	0
$895$	−17.3224	−0.579023
$896$	14.4840	0.483877
$897$	0	0
$898$	−7.44545	−0.248458
$899$	62.3959	2.08102
$900$	0	0
$901$	24.4982	0.816153
$902$	6.54864	0.218046
$903$	0	0
$904$	−17.1092	−0.569044
$905$	2.19699	0.0730304
$906$	0	0
$907$	−0.380471	−0.0126333	−0.00631667	−	0.999980i	$-0.502011\pi$
$907$	−0.380471	−0.0126333	−0.00631667	+	0.999980i	$0.502011\pi$
$908$	−46.1085	−1.53016
$909$	0	0
$910$	1.02216	0.0338843
$911$	5.82616	0.193029	0.0965146	−	0.995332i	$-0.469231\pi$
$911$	5.82616	0.193029	0.0965146	+	0.995332i	$0.469231\pi$
$912$	0	0
$913$	−63.8485	−2.11308
$914$	−5.05049	−0.167056
$915$	0	0
$916$	−13.2446	−0.437613
$917$	−18.8709	−0.623172
$918$	0	0
$919$	−8.02685	−0.264781	−0.132391	−	0.991198i	$-0.542265\pi$
$919$	−8.02685	−0.264781	−0.132391	+	0.991198i	$0.542265\pi$
$920$	0	0
$921$	0	0
$922$	0.357201	0.0117638
$923$	−8.59834	−0.283018
$924$	0	0
$925$	−15.6507	−0.514593
$926$	0.721832	0.0237209
$927$	0	0
$928$	21.7912	0.715332
$929$	4.06182	0.133264	0.0666319	−	0.997778i	$-0.478775\pi$
$929$	4.06182	0.133264	0.0666319	+	0.997778i	$0.478775\pi$
$930$	0	0
$931$	7.40463	0.242677
$932$	−2.65828	−0.0870750
$933$	0	0
$934$	1.25997	0.0412274
$935$	−9.59090	−0.313656
$936$	0	0
$937$	−30.2035	−0.986706	−0.493353	−	0.869829i	$-0.664229\pi$
$937$	−30.2035	−0.986706	−0.493353	+	0.869829i	$0.664229\pi$
$938$	0.346067	0.0112995
$939$	0	0
$940$	−14.1423	−0.461271
$941$	45.1432	1.47163	0.735813	−	0.677185i	$-0.236800\pi$
$941$	45.1432	1.47163	0.735813	+	0.677185i	$0.236800\pi$
$942$	0	0
$943$	0	0
$944$	−44.7160	−1.45538
$945$	0	0
$946$	−2.24895	−0.0731198
$947$	52.7617	1.71452	0.857262	−	0.514881i	$-0.172164\pi$
$947$	52.7617	1.71452	0.857262	+	0.514881i	$0.172164\pi$
$948$	0	0
$949$	0.122450	0.00397489
$950$	−2.51428	−0.0815739
$951$	0	0
$952$	−4.15820	−0.134768
$953$	−24.2187	−0.784521	−0.392261	−	0.919854i	$-0.628307\pi$
$953$	−24.2187	−0.784521	−0.392261	+	0.919854i	$0.628307\pi$
$954$	0	0
$955$	−17.4390	−0.564311
$956$	−35.0127	−1.13239
$957$	0	0
$958$	−2.23817	−0.0723120
$959$	−30.4538	−0.983404
$960$	0	0
$961$	30.2250	0.975000
$962$	1.93905	0.0625175
$963$	0	0
$964$	−0.574982	−0.0185189
$965$	−18.1194	−0.583284
$966$	0	0
$967$	5.47721	0.176135	0.0880675	−	0.996115i	$-0.471931\pi$
$967$	5.47721	0.176135	0.0880675	+	0.996115i	$0.471931\pi$
$968$	7.71618	0.248007
$969$	0	0
$970$	1.40654	0.0451612
$971$	−10.5100	−0.337283	−0.168641	−	0.985677i	$-0.553938\pi$
$971$	−10.5100	−0.337283	−0.168641	+	0.985677i	$0.553938\pi$
$972$	0	0
$973$	33.6400	1.07845
$974$	−3.22551	−0.103352
$975$	0	0
$976$	51.7603	1.65681
$977$	23.6399	0.756306	0.378153	−	0.925743i	$-0.376559\pi$
$977$	23.6399	0.756306	0.378153	+	0.925743i	$0.376559\pi$
$978$	0	0
$979$	12.0891	0.386370
$980$	5.43486	0.173610
$981$	0	0
$982$	0.695085	0.0221811
$983$	18.5143	0.590514	0.295257	−	0.955418i	$-0.404595\pi$
$983$	18.5143	0.590514	0.295257	+	0.955418i	$0.404595\pi$
$984$	0	0
$985$	4.93857	0.157356
$986$	−4.09051	−0.130268
$987$	0	0
$988$	−10.8292	−0.344523
$989$	0	0
$990$	0	0
$991$	17.9995	0.571772	0.285886	−	0.958264i	$-0.407712\pi$
$991$	17.9995	0.571772	0.285886	+	0.958264i	$0.407712\pi$
$992$	21.3823	0.678889
$993$	0	0
$994$	−2.00133	−0.0634782
$995$	−12.5785	−0.398764
$996$	0	0
$997$	−1.92620	−0.0610034	−0.0305017	−	0.999535i	$-0.509710\pi$
$997$	−1.92620	−0.0610034	−0.0305017	+	0.999535i	$0.509710\pi$
$998$	−2.71286	−0.0858742
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bm.1.4		5
3.2	odd	2		1587.2.a.r.1.2		5
23.17	odd	22		207.2.i.a.82.1		10
23.19	odd	22		207.2.i.a.154.1		10
23.22	odd	2		4761.2.a.bp.1.4		5
69.17	even	22		69.2.e.b.13.1	✓	10
69.65	even	22		69.2.e.b.16.1	yes	10
69.68	even	2		1587.2.a.q.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.b.13.1	✓	10	69.17	even	22
69.2.e.b.16.1	yes	10	69.65	even	22
207.2.i.a.82.1		10	23.17	odd	22
207.2.i.a.154.1		10	23.19	odd	22
1587.2.a.q.1.2		5	69.68	even	2
1587.2.a.r.1.2		5	3.2	odd	2
4761.2.a.bm.1.4		5	1.1	even	1	trivial
4761.2.a.bp.1.4		5	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q-0.236479 q^{2} -1.94408 q^{4} +1.00714 q^{5} +2.05529 q^{7} +0.932691 q^{8} -0.238168 q^{10} +4.39011 q^{11} -2.08816 q^{13} -0.486033 q^{14} +3.66759 q^{16} -2.16917 q^{17} -2.66759 q^{19} -1.95796 q^{20} -1.03817 q^{22} -3.98567 q^{25} +0.493805 q^{26} -3.99565 q^{28} -7.97428 q^{29} -7.82464 q^{31} -2.73269 q^{32} +0.512963 q^{34} +2.06997 q^{35} +3.92676 q^{37} +0.630830 q^{38} +0.939352 q^{40} -6.30788 q^{41} +2.16627 q^{43} -8.53471 q^{44} +7.22297 q^{47} -2.77577 q^{49} +0.942526 q^{50} +4.05954 q^{52} -11.2938 q^{53} +4.42146 q^{55} +1.91695 q^{56} +1.88575 q^{58} -12.1922 q^{59} +14.1129 q^{61} +1.85036 q^{62} -6.68896 q^{64} -2.10307 q^{65} -0.712023 q^{67} +4.21704 q^{68} -0.489504 q^{70} +4.11767 q^{71} -0.0586401 q^{73} -0.928595 q^{74} +5.18601 q^{76} +9.02295 q^{77} +16.2978 q^{79} +3.69379 q^{80} +1.49168 q^{82} -14.5437 q^{83} -2.18466 q^{85} -0.512278 q^{86} +4.09461 q^{88} +2.75372 q^{89} -4.29177 q^{91} -1.70808 q^{94} -2.68665 q^{95} -5.90566 q^{97} +0.656411 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} - 7 q^{5} - 3 q^{7} + 9 q^{8} - 6 q^{10} - 2 q^{11} - 7 q^{13} + 10 q^{14} + 6 q^{16} - 16 q^{17} - q^{19} - 21 q^{20} + 14 q^{22} + 20 q^{25} + 5 q^{26} + 2 q^{28} - 18 q^{29}+ \cdots - 14 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^4 - 7 * q^5 - 3 * q^7 + 9 * q^8 - 6 * q^10 - 2 * q^11 - 7 * q^13 + 10 * q^14 + 6 * q^16 - 16 * q^17 - q^19 - 21 * q^20 + 14 * q^22 + 20 * q^25 + 5 * q^26 + 2 * q^28 - 18 * q^29 + 5 * q^31 + 6 * q^32 + 2 * q^34 + 24 * q^35 + 26 * q^37 - 26 * q^38 - 17 * q^40 - 9 * q^41 + 22 * q^43 - 28 * q^44 - 2 * q^47 + 2 * q^49 + 14 * q^50 + q^52 - 35 * q^53 + 16 * q^55 + 10 * q^56 + 5 * q^58 - 6 * q^59 - 7 * q^61 - 13 * q^62 - 21 * q^64 - 10 * q^65 + 5 * q^67 - 15 * q^68 - 47 * q^70 + 18 * q^71 + 4 * q^73 - 28 * q^74 - 3 * q^76 + 21 * q^77 + 13 * q^79 - 37 * q^80 - 47 * q^82 - 24 * q^83 + 18 * q^85 - 11 * q^86 + 14 * q^88 + 7 * q^89 - 9 * q^91 - 8 * q^94 + 30 * q^95 + 6 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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