LMFDB - Embedded newform 4012.2.a.h.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 $ x^15 - 5x^14 - 16x^13 + 123x^12 - 76x^11 - 630x^10 + 937x^9 + 1083x^8 - 2602x^7 - 249x^6 + 2736x^5 - 801x^4 - 900x^3 + 429x^2 - 36x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.64466$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48779 q^{3} +1.28557 q^{5} +0.0101628 q^{7} -0.786472 q^{9} +O(q^{10})$	$q+1.48779 q^{3} +1.28557 q^{5} +0.0101628 q^{7} -0.786472 q^{9} -2.81865 q^{11} -4.93841 q^{13} +1.91266 q^{15} +1.00000 q^{17} +0.137777 q^{19} +0.0151201 q^{21} +2.88816 q^{23} -3.34731 q^{25} -5.63349 q^{27} +0.119759 q^{29} +1.47393 q^{31} -4.19356 q^{33} +0.0130650 q^{35} -7.73302 q^{37} -7.34733 q^{39} +0.257127 q^{41} +3.65719 q^{43} -1.01107 q^{45} +4.25148 q^{47} -6.99990 q^{49} +1.48779 q^{51} +7.93129 q^{53} -3.62357 q^{55} +0.204983 q^{57} -1.00000 q^{59} -15.4016 q^{61} -0.00799274 q^{63} -6.34867 q^{65} -9.20159 q^{67} +4.29699 q^{69} -13.5926 q^{71} +7.46098 q^{73} -4.98010 q^{75} -0.0286453 q^{77} -8.24751 q^{79} -6.02205 q^{81} +8.73422 q^{83} +1.28557 q^{85} +0.178177 q^{87} -7.42934 q^{89} -0.0501879 q^{91} +2.19290 q^{93} +0.177122 q^{95} +16.9457 q^{97} +2.21679 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.48779	0.858978	0.429489	−	0.903072i	$-0.358694\pi$
$3$	1.48779	0.858978	0.429489	+	0.903072i	$0.358694\pi$
$4$	0	0
$5$	1.28557	0.574925	0.287462	−	0.957792i	$-0.407188\pi$
$5$	1.28557	0.574925	0.287462	+	0.957792i	$0.407188\pi$
$6$	0	0
$7$	0.0101628	0.00384117	0.00192058	−	0.999998i	$-0.499389\pi$
$7$	0.0101628	0.00384117	0.00192058	+	0.999998i	$0.499389\pi$
$8$	0	0
$9$	−0.786472	−0.262157
$10$	0	0
$11$	−2.81865	−0.849854	−0.424927	−	0.905228i	$-0.639700\pi$
$11$	−2.81865	−0.849854	−0.424927	+	0.905228i	$0.639700\pi$
$12$	0	0
$13$	−4.93841	−1.36967	−0.684834	−	0.728699i	$-0.740125\pi$
$13$	−4.93841	−1.36967	−0.684834	+	0.728699i	$0.740125\pi$
$14$	0	0
$15$	1.91266	0.493847
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.137777	0.0316081	0.0158041	−	0.999875i	$-0.494969\pi$
$19$	0.137777	0.0316081	0.0158041	+	0.999875i	$0.494969\pi$
$20$	0	0
$21$	0.0151201	0.00329948
$22$	0	0
$23$	2.88816	0.602223	0.301112	−	0.953589i	$-0.402642\pi$
$23$	2.88816	0.602223	0.301112	+	0.953589i	$0.402642\pi$
$24$	0	0
$25$	−3.34731	−0.669462
$26$	0	0
$27$	−5.63349	−1.08416
$28$	0	0
$29$	0.119759	0.0222388	0.0111194	−	0.999938i	$-0.496461\pi$
$29$	0.119759	0.0222388	0.0111194	+	0.999938i	$0.496461\pi$
$30$	0	0
$31$	1.47393	0.264725	0.132363	−	0.991201i	$-0.457744\pi$
$31$	1.47393	0.264725	0.132363	+	0.991201i	$0.457744\pi$
$32$	0	0
$33$	−4.19356	−0.730006
$34$	0	0
$35$	0.0130650	0.00220838
$36$	0	0
$37$	−7.73302	−1.27130	−0.635650	−	0.771977i	$-0.719268\pi$
$37$	−7.73302	−1.27130	−0.635650	+	0.771977i	$0.719268\pi$
$38$	0	0
$39$	−7.34733	−1.17651
$40$	0	0
$41$	0.257127	0.0401565	0.0200783	−	0.999798i	$-0.493608\pi$
$41$	0.257127	0.0401565	0.0200783	+	0.999798i	$0.493608\pi$
$42$	0	0
$43$	3.65719	0.557717	0.278858	−	0.960332i	$-0.410044\pi$
$43$	3.65719	0.557717	0.278858	+	0.960332i	$0.410044\pi$
$44$	0	0
$45$	−1.01107	−0.150721
$46$	0	0
$47$	4.25148	0.620142	0.310071	−	0.950713i	$-0.399647\pi$
$47$	4.25148	0.620142	0.310071	+	0.950713i	$0.399647\pi$
$48$	0	0
$49$	−6.99990	−0.999985
$50$	0	0
$51$	1.48779	0.208333
$52$	0	0
$53$	7.93129	1.08945	0.544723	−	0.838616i	$-0.316635\pi$
$53$	7.93129	1.08945	0.544723	+	0.838616i	$0.316635\pi$
$54$	0	0
$55$	−3.62357	−0.488602
$56$	0	0
$57$	0.204983	0.0271507
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−15.4016	−1.97198	−0.985988	−	0.166815i	$-0.946652\pi$
$61$	−15.4016	−1.97198	−0.985988	+	0.166815i	$0.946652\pi$
$62$	0	0
$63$	−0.00799274	−0.00100699
$64$	0	0
$65$	−6.34867	−0.787456
$66$	0	0
$67$	−9.20159	−1.12415	−0.562077	−	0.827085i	$-0.689997\pi$
$67$	−9.20159	−1.12415	−0.562077	+	0.827085i	$0.689997\pi$
$68$	0	0
$69$	4.29699	0.517296
$70$	0	0
$71$	−13.5926	−1.61315	−0.806573	−	0.591134i	$-0.798680\pi$
$71$	−13.5926	−1.61315	−0.806573	+	0.591134i	$0.798680\pi$
$72$	0	0
$73$	7.46098	0.873242	0.436621	−	0.899646i	$-0.356175\pi$
$73$	7.46098	0.873242	0.436621	+	0.899646i	$0.356175\pi$
$74$	0	0
$75$	−4.98010	−0.575053
$76$	0	0
$77$	−0.0286453	−0.00326443
$78$	0	0
$79$	−8.24751	−0.927917	−0.463958	−	0.885857i	$-0.653571\pi$
$79$	−8.24751	−0.927917	−0.463958	+	0.885857i	$0.653571\pi$
$80$	0	0
$81$	−6.02205	−0.669116
$82$	0	0
$83$	8.73422	0.958705	0.479352	−	0.877623i	$-0.340872\pi$
$83$	8.73422	0.958705	0.479352	+	0.877623i	$0.340872\pi$
$84$	0	0
$85$	1.28557	0.139440
$86$	0	0
$87$	0.178177	0.0191026
$88$	0	0
$89$	−7.42934	−0.787509	−0.393754	−	0.919216i	$-0.628824\pi$
$89$	−7.42934	−0.787509	−0.393754	+	0.919216i	$0.628824\pi$
$90$	0	0
$91$	−0.0501879	−0.00526112
$92$	0	0
$93$	2.19290	0.227393
$94$	0	0
$95$	0.177122	0.0181723
$96$	0	0
$97$	16.9457	1.72058	0.860289	−	0.509806i	$-0.170283\pi$
$97$	16.9457	1.72058	0.860289	+	0.509806i	$0.170283\pi$
$98$	0	0
$99$	2.21679	0.222795
$100$	0	0
$101$	4.18850	0.416771	0.208386	−	0.978047i	$-0.433179\pi$
$101$	4.18850	0.416771	0.208386	+	0.978047i	$0.433179\pi$
$102$	0	0
$103$	−9.83881	−0.969446	−0.484723	−	0.874668i	$-0.661080\pi$
$103$	−9.83881	−0.969446	−0.484723	+	0.874668i	$0.661080\pi$
$104$	0	0
$105$	0.0194380	0.00189695
$106$	0	0
$107$	−15.4692	−1.49547	−0.747733	−	0.664000i	$-0.768858\pi$
$107$	−15.4692	−1.49547	−0.747733	+	0.664000i	$0.768858\pi$
$108$	0	0
$109$	−12.1063	−1.15957	−0.579786	−	0.814769i	$-0.696864\pi$
$109$	−12.1063	−1.15957	−0.579786	+	0.814769i	$0.696864\pi$
$110$	0	0
$111$	−11.5051	−1.09202
$112$	0	0
$113$	−14.6496	−1.37811	−0.689057	−	0.724707i	$-0.741975\pi$
$113$	−14.6496	−1.37811	−0.689057	+	0.724707i	$0.741975\pi$
$114$	0	0
$115$	3.71293	0.346233
$116$	0	0
$117$	3.88392	0.359069
$118$	0	0
$119$	0.0101628	0.000931620 0
$120$	0	0
$121$	−3.05523	−0.277748
$122$	0	0
$123$	0.382552	0.0344936
$124$	0	0
$125$	−10.7311	−0.959815
$126$	0	0
$127$	−6.17962	−0.548352	−0.274176	−	0.961680i	$-0.588405\pi$
$127$	−6.17962	−0.548352	−0.274176	+	0.961680i	$0.588405\pi$
$128$	0	0
$129$	5.44115	0.479066
$130$	0	0
$131$	0.110665	0.00966883	0.00483441	−	0.999988i	$-0.498461\pi$
$131$	0.110665	0.00966883	0.00483441	+	0.999988i	$0.498461\pi$
$132$	0	0
$133$	0.00140019	0.000121412 0
$134$	0	0
$135$	−7.24224	−0.623313
$136$	0	0
$137$	−14.7277	−1.25827	−0.629136	−	0.777295i	$-0.716591\pi$
$137$	−14.7277	−1.25827	−0.629136	+	0.777295i	$0.716591\pi$
$138$	0	0
$139$	1.32494	0.112380	0.0561898	−	0.998420i	$-0.482105\pi$
$139$	1.32494	0.112380	0.0561898	+	0.998420i	$0.482105\pi$
$140$	0	0
$141$	6.32532	0.532688
$142$	0	0
$143$	13.9196	1.16402
$144$	0	0
$145$	0.153959	0.0127856
$146$	0	0
$147$	−10.4144	−0.858965
$148$	0	0
$149$	7.54774	0.618335	0.309168	−	0.951008i	$-0.399950\pi$
$149$	7.54774	0.618335	0.309168	+	0.951008i	$0.399950\pi$
$150$	0	0
$151$	−18.0754	−1.47096	−0.735478	−	0.677549i	$-0.763042\pi$
$151$	−18.0754	−1.47096	−0.735478	+	0.677549i	$0.763042\pi$
$152$	0	0
$153$	−0.786472	−0.0635825
$154$	0	0
$155$	1.89484	0.152197
$156$	0	0
$157$	18.9299	1.51077	0.755386	−	0.655280i	$-0.227449\pi$
$157$	18.9299	1.51077	0.755386	+	0.655280i	$0.227449\pi$
$158$	0	0
$159$	11.8001	0.935810
$160$	0	0
$161$	0.0293517	0.00231324
$162$	0	0
$163$	18.9178	1.48176	0.740878	−	0.671640i	$-0.234410\pi$
$163$	18.9178	1.48176	0.740878	+	0.671640i	$0.234410\pi$
$164$	0	0
$165$	−5.39112	−0.419698
$166$	0	0
$167$	21.2006	1.64055	0.820275	−	0.571969i	$-0.193820\pi$
$167$	21.2006	1.64055	0.820275	+	0.571969i	$0.193820\pi$
$168$	0	0
$169$	11.3879	0.875992
$170$	0	0
$171$	−0.108358	−0.00828631
$172$	0	0
$173$	−8.10335	−0.616086	−0.308043	−	0.951372i	$-0.599674\pi$
$173$	−8.10335	−0.616086	−0.308043	+	0.951372i	$0.599674\pi$
$174$	0	0
$175$	−0.0340179	−0.00257151
$176$	0	0
$177$	−1.48779	−0.111829
$178$	0	0
$179$	17.3901	1.29979	0.649897	−	0.760022i	$-0.274812\pi$
$179$	17.3901	1.29979	0.649897	+	0.760022i	$0.274812\pi$
$180$	0	0
$181$	23.3137	1.73289	0.866445	−	0.499272i	$-0.166399\pi$
$181$	23.3137	1.73289	0.866445	+	0.499272i	$0.166399\pi$
$182$	0	0
$183$	−22.9144	−1.69388
$184$	0	0
$185$	−9.94134	−0.730902
$186$	0	0
$187$	−2.81865	−0.206120
$188$	0	0
$189$	−0.0572518	−0.00416446
$190$	0	0
$191$	12.5308	0.906698	0.453349	−	0.891333i	$-0.350229\pi$
$191$	12.5308	0.906698	0.453349	+	0.891333i	$0.350229\pi$
$192$	0	0
$193$	20.6319	1.48512	0.742559	−	0.669781i	$-0.233612\pi$
$193$	20.6319	1.48512	0.742559	+	0.669781i	$0.233612\pi$
$194$	0	0
$195$	−9.44551	−0.676407
$196$	0	0
$197$	12.0643	0.859548	0.429774	−	0.902936i	$-0.358593\pi$
$197$	12.0643	0.859548	0.429774	+	0.902936i	$0.358593\pi$
$198$	0	0
$199$	8.30489	0.588718	0.294359	−	0.955695i	$-0.404894\pi$
$199$	8.30489	0.588718	0.294359	+	0.955695i	$0.404894\pi$
$200$	0	0
$201$	−13.6901	−0.965623
$202$	0	0
$203$	0.00121709	8.54228e−5 0
$204$	0	0
$205$	0.330555	0.0230870
$206$	0	0
$207$	−2.27146	−0.157877
$208$	0	0
$209$	−0.388344	−0.0268623
$210$	0	0
$211$	−0.876507	−0.0603412	−0.0301706	−	0.999545i	$-0.509605\pi$
$211$	−0.876507	−0.0603412	−0.0301706	+	0.999545i	$0.509605\pi$
$212$	0	0
$213$	−20.2230	−1.38566
$214$	0	0
$215$	4.70158	0.320645
$216$	0	0
$217$	0.0149792	0.00101685
$218$	0	0
$219$	11.1004	0.750095
$220$	0	0
$221$	−4.93841	−0.332193
$222$	0	0
$223$	−16.1819	−1.08362	−0.541810	−	0.840501i	$-0.682261\pi$
$223$	−16.1819	−1.08362	−0.541810	+	0.840501i	$0.682261\pi$
$224$	0	0
$225$	2.63257	0.175504
$226$	0	0
$227$	1.58684	0.105322	0.0526612	−	0.998612i	$-0.483230\pi$
$227$	1.58684	0.105322	0.0526612	+	0.998612i	$0.483230\pi$
$228$	0	0
$229$	9.77465	0.645927	0.322964	−	0.946411i	$-0.395321\pi$
$229$	9.77465	0.645927	0.322964	+	0.946411i	$0.395321\pi$
$230$	0	0
$231$	−0.0426182	−0.00280407
$232$	0	0
$233$	−11.6499	−0.763208	−0.381604	−	0.924326i	$-0.624628\pi$
$233$	−11.6499	−0.763208	−0.381604	+	0.924326i	$0.624628\pi$
$234$	0	0
$235$	5.46558	0.356535
$236$	0	0
$237$	−12.2706	−0.797060
$238$	0	0
$239$	−26.1228	−1.68974	−0.844870	−	0.534971i	$-0.820322\pi$
$239$	−26.1228	−1.68974	−0.844870	+	0.534971i	$0.820322\pi$
$240$	0	0
$241$	16.0197	1.03192	0.515960	−	0.856613i	$-0.327435\pi$
$241$	16.0197	1.03192	0.515960	+	0.856613i	$0.327435\pi$
$242$	0	0
$243$	7.94090	0.509409
$244$	0	0
$245$	−8.99886	−0.574916
$246$	0	0
$247$	−0.680398	−0.0432927
$248$	0	0
$249$	12.9947	0.823506
$250$	0	0
$251$	−6.41936	−0.405187	−0.202593	−	0.979263i	$-0.564937\pi$
$251$	−6.41936	−0.405187	−0.202593	+	0.979263i	$0.564937\pi$
$252$	0	0
$253$	−8.14071	−0.511802
$254$	0	0
$255$	1.91266	0.119776
$256$	0	0
$257$	−11.4862	−0.716490	−0.358245	−	0.933628i	$-0.616625\pi$
$257$	−11.4862	−0.716490	−0.358245	+	0.933628i	$0.616625\pi$
$258$	0	0
$259$	−0.0785889	−0.00488328
$260$	0	0
$261$	−0.0941874	−0.00583005
$262$	0	0
$263$	−3.13659	−0.193411	−0.0967053	−	0.995313i	$-0.530830\pi$
$263$	−3.13659	−0.193411	−0.0967053	+	0.995313i	$0.530830\pi$
$264$	0	0
$265$	10.1962	0.626349
$266$	0	0
$267$	−11.0533	−0.676452
$268$	0	0
$269$	−4.24523	−0.258836	−0.129418	−	0.991590i	$-0.541311\pi$
$269$	−4.24523	−0.258836	−0.129418	+	0.991590i	$0.541311\pi$
$270$	0	0
$271$	−4.98709	−0.302944	−0.151472	−	0.988462i	$-0.548401\pi$
$271$	−4.98709	−0.302944	−0.151472	+	0.988462i	$0.548401\pi$
$272$	0	0
$273$	−0.0746692	−0.00451919
$274$	0	0
$275$	9.43488	0.568945
$276$	0	0
$277$	−1.81269	−0.108914	−0.0544569	−	0.998516i	$-0.517343\pi$
$277$	−1.81269	−0.108914	−0.0544569	+	0.998516i	$0.517343\pi$
$278$	0	0
$279$	−1.15920	−0.0693997
$280$	0	0
$281$	29.0233	1.73139	0.865693	−	0.500575i	$-0.166878\pi$
$281$	29.0233	1.73139	0.865693	+	0.500575i	$0.166878\pi$
$282$	0	0
$283$	−27.2647	−1.62072	−0.810359	−	0.585934i	$-0.800728\pi$
$283$	−27.2647	−1.62072	−0.810359	+	0.585934i	$0.800728\pi$
$284$	0	0
$285$	0.263520	0.0156096
$286$	0	0
$287$	0.00261313	0.000154248 0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	25.2117	1.47794
$292$	0	0
$293$	23.7291	1.38627	0.693135	−	0.720807i	$-0.256229\pi$
$293$	23.7291	1.38627	0.693135	+	0.720807i	$0.256229\pi$
$294$	0	0
$295$	−1.28557	−0.0748488
$296$	0	0
$297$	15.8788	0.921382
$298$	0	0
$299$	−14.2629	−0.824846
$300$	0	0
$301$	0.0371672	0.00214228
$302$	0	0
$303$	6.23162	0.357997
$304$	0	0
$305$	−19.7999	−1.13374
$306$	0	0
$307$	−28.0628	−1.60163	−0.800815	−	0.598912i	$-0.795600\pi$
$307$	−28.0628	−1.60163	−0.800815	+	0.598912i	$0.795600\pi$
$308$	0	0
$309$	−14.6381	−0.832733
$310$	0	0
$311$	2.83575	0.160801	0.0804003	−	0.996763i	$-0.474380\pi$
$311$	2.83575	0.160801	0.0804003	+	0.996763i	$0.474380\pi$
$312$	0	0
$313$	−24.4516	−1.38209	−0.691044	−	0.722813i	$-0.742849\pi$
$313$	−24.4516	−1.38209	−0.691044	+	0.722813i	$0.742849\pi$
$314$	0	0
$315$	−0.0102752	−0.000578943 0
$316$	0	0
$317$	10.2204	0.574034	0.287017	−	0.957925i	$-0.407336\pi$
$317$	10.2204	0.574034	0.287017	+	0.957925i	$0.407336\pi$
$318$	0	0
$319$	−0.337559	−0.0188997
$320$	0	0
$321$	−23.0150	−1.28457
$322$	0	0
$323$	0.137777	0.00766610
$324$	0	0
$325$	16.5304	0.916941
$326$	0	0
$327$	−18.0116	−0.996046
$328$	0	0
$329$	0.0432068	0.00238207
$330$	0	0
$331$	26.4120	1.45174	0.725868	−	0.687834i	$-0.241438\pi$
$331$	26.4120	1.45174	0.725868	+	0.687834i	$0.241438\pi$
$332$	0	0
$333$	6.08180	0.333281
$334$	0	0
$335$	−11.8293	−0.646303
$336$	0	0
$337$	7.91774	0.431307	0.215653	−	0.976470i	$-0.430812\pi$
$337$	7.91774	0.431307	0.215653	+	0.976470i	$0.430812\pi$
$338$	0	0
$339$	−21.7955	−1.18377
$340$	0	0
$341$	−4.15448	−0.224978
$342$	0	0
$343$	−0.142278	−0.00768228
$344$	0	0
$345$	5.52408	0.297406
$346$	0	0
$347$	0.433667	0.0232805	0.0116402	−	0.999932i	$-0.496295\pi$
$347$	0.433667	0.0232805	0.0116402	+	0.999932i	$0.496295\pi$
$348$	0	0
$349$	−2.39255	−0.128070	−0.0640350	−	0.997948i	$-0.520397\pi$
$349$	−2.39255	−0.128070	−0.0640350	+	0.997948i	$0.520397\pi$
$350$	0	0
$351$	27.8205	1.48495
$352$	0	0
$353$	−18.7968	−1.00045	−0.500227	−	0.865894i	$-0.666750\pi$
$353$	−18.7968	−1.00045	−0.500227	+	0.865894i	$0.666750\pi$
$354$	0	0
$355$	−17.4743	−0.927437
$356$	0	0
$357$	0.0151201	0.000800241 0
$358$	0	0
$359$	14.1592	0.747296	0.373648	−	0.927571i	$-0.378107\pi$
$359$	14.1592	0.747296	0.373648	+	0.927571i	$0.378107\pi$
$360$	0	0
$361$	−18.9810	−0.999001
$362$	0	0
$363$	−4.54555	−0.238580
$364$	0	0
$365$	9.59162	0.502048
$366$	0	0
$367$	−5.46543	−0.285293	−0.142647	−	0.989774i	$-0.545561\pi$
$367$	−5.46543	−0.285293	−0.142647	+	0.989774i	$0.545561\pi$
$368$	0	0
$369$	−0.202223	−0.0105273
$370$	0	0
$371$	0.0806039	0.00418474
$372$	0	0
$373$	14.4329	0.747306	0.373653	−	0.927568i	$-0.378105\pi$
$373$	14.4329	0.747306	0.373653	+	0.927568i	$0.378105\pi$
$374$	0	0
$375$	−15.9656	−0.824459
$376$	0	0
$377$	−0.591421	−0.0304597
$378$	0	0
$379$	3.72208	0.191190	0.0955951	−	0.995420i	$-0.469525\pi$
$379$	3.72208	0.191190	0.0955951	+	0.995420i	$0.469525\pi$
$380$	0	0
$381$	−9.19399	−0.471022
$382$	0	0
$383$	−6.81409	−0.348184	−0.174092	−	0.984729i	$-0.555699\pi$
$383$	−6.81409	−0.348184	−0.174092	+	0.984729i	$0.555699\pi$
$384$	0	0
$385$	−0.0368255	−0.00187680
$386$	0	0
$387$	−2.87628	−0.146210
$388$	0	0
$389$	21.4562	1.08787	0.543937	−	0.839126i	$-0.316933\pi$
$389$	21.4562	1.08787	0.543937	+	0.839126i	$0.316933\pi$
$390$	0	0
$391$	2.88816	0.146061
$392$	0	0
$393$	0.164646	0.00830531
$394$	0	0
$395$	−10.6028	−0.533482
$396$	0	0
$397$	17.4129	0.873931	0.436965	−	0.899478i	$-0.356053\pi$
$397$	17.4129	0.873931	0.436965	+	0.899478i	$0.356053\pi$
$398$	0	0
$399$	0.00208320	0.000104290 0
$400$	0	0
$401$	0.380633	0.0190079	0.00950394	−	0.999955i	$-0.496975\pi$
$401$	0.380633	0.0190079	0.00950394	+	0.999955i	$0.496975\pi$
$402$	0	0
$403$	−7.27886	−0.362586
$404$	0	0
$405$	−7.74176	−0.384691
$406$	0	0
$407$	21.7966	1.08042
$408$	0	0
$409$	−27.4681	−1.35821	−0.679105	−	0.734042i	$-0.737632\pi$
$409$	−27.4681	−1.35821	−0.679105	+	0.734042i	$0.737632\pi$
$410$	0	0
$411$	−21.9118	−1.08083
$412$	0	0
$413$	−0.0101628	−0.000500077 0
$414$	0	0
$415$	11.2285	0.551183
$416$	0	0
$417$	1.97123	0.0965316
$418$	0	0
$419$	21.5070	1.05069	0.525343	−	0.850891i	$-0.323937\pi$
$419$	21.5070	1.05069	0.525343	+	0.850891i	$0.323937\pi$
$420$	0	0
$421$	−18.4991	−0.901590	−0.450795	−	0.892628i	$-0.648859\pi$
$421$	−18.4991	−0.901590	−0.450795	+	0.892628i	$0.648859\pi$
$422$	0	0
$423$	−3.34367	−0.162575
$424$	0	0
$425$	−3.34731	−0.162368
$426$	0	0
$427$	−0.156523	−0.00757469
$428$	0	0
$429$	20.7095	0.999866
$430$	0	0
$431$	−22.6567	−1.09133	−0.545666	−	0.838003i	$-0.683723\pi$
$431$	−22.6567	−1.09133	−0.545666	+	0.838003i	$0.683723\pi$
$432$	0	0
$433$	22.1525	1.06458	0.532292	−	0.846561i	$-0.321331\pi$
$433$	22.1525	1.06458	0.532292	+	0.846561i	$0.321331\pi$
$434$	0	0
$435$	0.229059	0.0109826
$436$	0	0
$437$	0.397921	0.0190352
$438$	0	0
$439$	1.37871	0.0658021	0.0329010	−	0.999459i	$-0.489525\pi$
$439$	1.37871	0.0658021	0.0329010	+	0.999459i	$0.489525\pi$
$440$	0	0
$441$	5.50522	0.262154
$442$	0	0
$443$	−11.1019	−0.527465	−0.263733	−	0.964596i	$-0.584954\pi$
$443$	−11.1019	−0.527465	−0.263733	+	0.964596i	$0.584954\pi$
$444$	0	0
$445$	−9.55094	−0.452758
$446$	0	0
$447$	11.2295	0.531136
$448$	0	0
$449$	−31.8084	−1.50113	−0.750565	−	0.660797i	$-0.770218\pi$
$449$	−31.8084	−1.50113	−0.750565	+	0.660797i	$0.770218\pi$
$450$	0	0
$451$	−0.724751	−0.0341272
$452$	0	0
$453$	−26.8925	−1.26352
$454$	0	0
$455$	−0.0645201	−0.00302475
$456$	0	0
$457$	6.68605	0.312760	0.156380	−	0.987697i	$-0.450018\pi$
$457$	6.68605	0.312760	0.156380	+	0.987697i	$0.450018\pi$
$458$	0	0
$459$	−5.63349	−0.262949
$460$	0	0
$461$	30.8485	1.43676	0.718380	−	0.695651i	$-0.244884\pi$
$461$	30.8485	1.43676	0.718380	+	0.695651i	$0.244884\pi$
$462$	0	0
$463$	7.51644	0.349319	0.174659	−	0.984629i	$-0.444118\pi$
$463$	7.51644	0.349319	0.174659	+	0.984629i	$0.444118\pi$
$464$	0	0
$465$	2.81913	0.130734
$466$	0	0
$467$	−11.1214	−0.514635	−0.257318	−	0.966327i	$-0.582839\pi$
$467$	−11.1214	−0.514635	−0.257318	+	0.966327i	$0.582839\pi$
$468$	0	0
$469$	−0.0935137	−0.00431806
$470$	0	0
$471$	28.1638	1.29772
$472$	0	0
$473$	−10.3083	−0.473978
$474$	0	0
$475$	−0.461181	−0.0211604
$476$	0	0
$477$	−6.23774	−0.285606
$478$	0	0
$479$	−12.7433	−0.582256	−0.291128	−	0.956684i	$-0.594031\pi$
$479$	−12.7433	−0.582256	−0.291128	+	0.956684i	$0.594031\pi$
$480$	0	0
$481$	38.1888	1.74126
$482$	0	0
$483$	0.0436693	0.00198702
$484$	0	0
$485$	21.7849	0.989203
$486$	0	0
$487$	21.5048	0.974476	0.487238	−	0.873269i	$-0.338004\pi$
$487$	21.5048	0.974476	0.487238	+	0.873269i	$0.338004\pi$
$488$	0	0
$489$	28.1458	1.27280
$490$	0	0
$491$	−19.9468	−0.900184	−0.450092	−	0.892982i	$-0.648609\pi$
$491$	−19.9468	−0.900184	−0.450092	+	0.892982i	$0.648609\pi$
$492$	0	0
$493$	0.119759	0.00539369
$494$	0	0
$495$	2.84984	0.128091
$496$	0	0
$497$	−0.138139	−0.00619636
$498$	0	0
$499$	40.8526	1.82881	0.914406	−	0.404799i	$-0.132659\pi$
$499$	40.8526	1.82881	0.914406	+	0.404799i	$0.132659\pi$
$500$	0	0
$501$	31.5421	1.40920
$502$	0	0
$503$	32.2663	1.43868	0.719341	−	0.694657i	$-0.244444\pi$
$503$	32.2663	1.43868	0.719341	+	0.694657i	$0.244444\pi$
$504$	0	0
$505$	5.38461	0.239612
$506$	0	0
$507$	16.9428	0.752457
$508$	0	0
$509$	−23.4913	−1.04123	−0.520617	−	0.853790i	$-0.674298\pi$
$509$	−23.4913	−1.04123	−0.520617	+	0.853790i	$0.674298\pi$
$510$	0	0
$511$	0.0758243	0.00335427
$512$	0	0
$513$	−0.776163	−0.0342684
$514$	0	0
$515$	−12.6485	−0.557359
$516$	0	0
$517$	−11.9834	−0.527030
$518$	0	0
$519$	−12.0561	−0.529204
$520$	0	0
$521$	−4.30265	−0.188502	−0.0942512	−	0.995548i	$-0.530046\pi$
$521$	−4.30265	−0.188502	−0.0942512	+	0.995548i	$0.530046\pi$
$522$	0	0
$523$	36.7826	1.60839	0.804196	−	0.594364i	$-0.202596\pi$
$523$	36.7826	1.60839	0.804196	+	0.594364i	$0.202596\pi$
$524$	0	0
$525$	−0.0506116	−0.00220887
$526$	0	0
$527$	1.47393	0.0642053
$528$	0	0
$529$	−14.6585	−0.637327
$530$	0	0
$531$	0.786472	0.0341300
$532$	0	0
$533$	−1.26980	−0.0550011
$534$	0	0
$535$	−19.8868	−0.859780
$536$	0	0
$537$	25.8728	1.11649
$538$	0	0
$539$	19.7302	0.849841
$540$	0	0
$541$	−29.3239	−1.26073	−0.630367	−	0.776297i	$-0.717096\pi$
$541$	−29.3239	−1.26073	−0.630367	+	0.776297i	$0.717096\pi$
$542$	0	0
$543$	34.6859	1.48851
$544$	0	0
$545$	−15.5635	−0.666666
$546$	0	0
$547$	−6.33188	−0.270732	−0.135366	−	0.990796i	$-0.543221\pi$
$547$	−6.33188	−0.270732	−0.135366	+	0.990796i	$0.543221\pi$
$548$	0	0
$549$	12.1130	0.516968
$550$	0	0
$551$	0.0165000	0.000702926 0
$552$	0	0
$553$	−0.0838175	−0.00356428
$554$	0	0
$555$	−14.7907	−0.627828
$556$	0	0
$557$	−29.1078	−1.23334	−0.616668	−	0.787224i	$-0.711518\pi$
$557$	−29.1078	−1.23334	−0.616668	+	0.787224i	$0.711518\pi$
$558$	0	0
$559$	−18.0607	−0.763887
$560$	0	0
$561$	−4.19356	−0.177052
$562$	0	0
$563$	−28.7455	−1.21148	−0.605739	−	0.795663i	$-0.707122\pi$
$563$	−28.7455	−1.21148	−0.605739	+	0.795663i	$0.707122\pi$
$564$	0	0
$565$	−18.8330	−0.792312
$566$	0	0
$567$	−0.0612007	−0.00257019
$568$	0	0
$569$	14.1022	0.591196	0.295598	−	0.955312i	$-0.404481\pi$
$569$	14.1022	0.591196	0.295598	+	0.955312i	$0.404481\pi$
$570$	0	0
$571$	−36.1094	−1.51113	−0.755566	−	0.655072i	$-0.772638\pi$
$571$	−36.1094	−1.51113	−0.755566	+	0.655072i	$0.772638\pi$
$572$	0	0
$573$	18.6433	0.778833
$574$	0	0
$575$	−9.66757	−0.403165
$576$	0	0
$577$	27.8565	1.15968	0.579840	−	0.814730i	$-0.303115\pi$
$577$	27.8565	1.15968	0.579840	+	0.814730i	$0.303115\pi$
$578$	0	0
$579$	30.6960	1.27568
$580$	0	0
$581$	0.0887638	0.00368254
$582$	0	0
$583$	−22.3555	−0.925870
$584$	0	0
$585$	4.99306	0.206437
$586$	0	0
$587$	−28.8068	−1.18898	−0.594491	−	0.804102i	$-0.702647\pi$
$587$	−28.8068	−1.18898	−0.594491	+	0.804102i	$0.702647\pi$
$588$	0	0
$589$	0.203073	0.00836747
$590$	0	0
$591$	17.9492	0.738333
$592$	0	0
$593$	13.9225	0.571731	0.285865	−	0.958270i	$-0.407719\pi$
$593$	13.9225	0.571731	0.285865	+	0.958270i	$0.407719\pi$
$594$	0	0
$595$	0.0130650	0.000535611 0
$596$	0	0
$597$	12.3560	0.505696
$598$	0	0
$599$	−10.2615	−0.419273	−0.209636	−	0.977779i	$-0.567228\pi$
$599$	−10.2615	−0.419273	−0.209636	+	0.977779i	$0.567228\pi$
$600$	0	0
$601$	24.4515	0.997399	0.498700	−	0.866775i	$-0.333811\pi$
$601$	24.4515	0.997399	0.498700	+	0.866775i	$0.333811\pi$
$602$	0	0
$603$	7.23680	0.294705
$604$	0	0
$605$	−3.92771	−0.159684
$606$	0	0
$607$	31.1414	1.26399	0.631996	−	0.774972i	$-0.282236\pi$
$607$	31.1414	1.26399	0.631996	+	0.774972i	$0.282236\pi$
$608$	0	0
$609$	0.00181077	7.33763e−5 0
$610$	0	0
$611$	−20.9955	−0.849389
$612$	0	0
$613$	−23.5065	−0.949420	−0.474710	−	0.880142i	$-0.657447\pi$
$613$	−23.5065	−0.949420	−0.474710	+	0.880142i	$0.657447\pi$
$614$	0	0
$615$	0.491798	0.0198312
$616$	0	0
$617$	−42.2334	−1.70025	−0.850125	−	0.526580i	$-0.823474\pi$
$617$	−42.2334	−1.70025	−0.850125	+	0.526580i	$0.823474\pi$
$618$	0	0
$619$	37.2874	1.49871	0.749353	−	0.662171i	$-0.230365\pi$
$619$	37.2874	1.49871	0.749353	+	0.662171i	$0.230365\pi$
$620$	0	0
$621$	−16.2704	−0.652909
$622$	0	0
$623$	−0.0755027	−0.00302495
$624$	0	0
$625$	2.94102	0.117641
$626$	0	0
$627$	−0.577775	−0.0230741
$628$	0	0
$629$	−7.73302	−0.308336
$630$	0	0
$631$	−33.7939	−1.34532	−0.672658	−	0.739954i	$-0.734847\pi$
$631$	−33.7939	−1.34532	−0.672658	+	0.739954i	$0.734847\pi$
$632$	0	0
$633$	−1.30406	−0.0518318
$634$	0	0
$635$	−7.94433	−0.315261
$636$	0	0
$637$	34.5684	1.36965
$638$	0	0
$639$	10.6902	0.422898
$640$	0	0
$641$	−13.0726	−0.516337	−0.258168	−	0.966100i	$-0.583119\pi$
$641$	−13.0726	−0.516337	−0.258168	+	0.966100i	$0.583119\pi$
$642$	0	0
$643$	−21.6224	−0.852706	−0.426353	−	0.904557i	$-0.640202\pi$
$643$	−21.6224	−0.852706	−0.426353	+	0.904557i	$0.640202\pi$
$644$	0	0
$645$	6.99498	0.275427
$646$	0	0
$647$	9.75399	0.383469	0.191734	−	0.981447i	$-0.438589\pi$
$647$	9.75399	0.383469	0.191734	+	0.981447i	$0.438589\pi$
$648$	0	0
$649$	2.81865	0.110642
$650$	0	0
$651$	0.0222859	0.000873455 0
$652$	0	0
$653$	34.8042	1.36199	0.680996	−	0.732287i	$-0.261547\pi$
$653$	34.8042	1.36199	0.680996	+	0.732287i	$0.261547\pi$
$654$	0	0
$655$	0.142267	0.00555885
$656$	0	0
$657$	−5.86786	−0.228927
$658$	0	0
$659$	−13.5091	−0.526240	−0.263120	−	0.964763i	$-0.584752\pi$
$659$	−13.5091	−0.526240	−0.263120	+	0.964763i	$0.584752\pi$
$660$	0	0
$661$	30.3000	1.17853	0.589267	−	0.807938i	$-0.299417\pi$
$661$	30.3000	1.17853	0.589267	+	0.807938i	$0.299417\pi$
$662$	0	0
$663$	−7.34733	−0.285347
$664$	0	0
$665$	0.00180005	6.98028e−5 0
$666$	0	0
$667$	0.345884	0.0133927
$668$	0	0
$669$	−24.0753	−0.930806
$670$	0	0
$671$	43.4117	1.67589
$672$	0	0
$673$	7.75150	0.298799	0.149399	−	0.988777i	$-0.452266\pi$
$673$	7.75150	0.298799	0.149399	+	0.988777i	$0.452266\pi$
$674$	0	0
$675$	18.8570	0.725807
$676$	0	0
$677$	49.9314	1.91902	0.959510	−	0.281676i	$-0.0908902\pi$
$677$	49.9314	1.91902	0.959510	+	0.281676i	$0.0908902\pi$
$678$	0	0
$679$	0.172216	0.00660903
$680$	0	0
$681$	2.36089	0.0904696
$682$	0	0
$683$	−4.35358	−0.166585	−0.0832925	−	0.996525i	$-0.526544\pi$
$683$	−4.35358	−0.166585	−0.0832925	+	0.996525i	$0.526544\pi$
$684$	0	0
$685$	−18.9335	−0.723412
$686$	0	0
$687$	14.5427	0.554837
$688$	0	0
$689$	−39.1679	−1.49218
$690$	0	0
$691$	−45.6328	−1.73595	−0.867977	−	0.496604i	$-0.834580\pi$
$691$	−45.6328	−1.73595	−0.867977	+	0.496604i	$0.834580\pi$
$692$	0	0
$693$	0.0225287	0.000855795 0
$694$	0	0
$695$	1.70330	0.0646098
$696$	0	0
$697$	0.257127	0.00973939
$698$	0	0
$699$	−17.3326	−0.655579
$700$	0	0
$701$	18.2497	0.689283	0.344641	−	0.938734i	$-0.388001\pi$
$701$	18.2497	0.689283	0.344641	+	0.938734i	$0.388001\pi$
$702$	0	0
$703$	−1.06543	−0.0401834
$704$	0	0
$705$	8.13165	0.306256
$706$	0	0
$707$	0.0425668	0.00160089
$708$	0	0
$709$	−40.5368	−1.52239	−0.761196	−	0.648521i	$-0.775388\pi$
$709$	−40.5368	−1.52239	−0.761196	+	0.648521i	$0.775388\pi$
$710$	0	0
$711$	6.48643	0.243260
$712$	0	0
$713$	4.25694	0.159424
$714$	0	0
$715$	17.8947	0.669223
$716$	0	0
$717$	−38.8653	−1.45145
$718$	0	0
$719$	−41.4536	−1.54596	−0.772979	−	0.634431i	$-0.781234\pi$
$719$	−41.4536	−1.54596	−0.772979	+	0.634431i	$0.781234\pi$
$720$	0	0
$721$	−0.0999895	−0.00372381
$722$	0	0
$723$	23.8340	0.886396
$724$	0	0
$725$	−0.400872	−0.0148880
$726$	0	0
$727$	22.2947	0.826863	0.413432	−	0.910535i	$-0.364330\pi$
$727$	22.2947	0.826863	0.413432	+	0.910535i	$0.364330\pi$
$728$	0	0
$729$	29.8806	1.10669
$730$	0	0
$731$	3.65719	0.135266
$732$	0	0
$733$	11.0544	0.408304	0.204152	−	0.978939i	$-0.434556\pi$
$733$	11.0544	0.408304	0.204152	+	0.978939i	$0.434556\pi$
$734$	0	0
$735$	−13.3884	−0.493840
$736$	0	0
$737$	25.9360	0.955366
$738$	0	0
$739$	8.19489	0.301454	0.150727	−	0.988575i	$-0.451839\pi$
$739$	8.19489	0.301454	0.150727	+	0.988575i	$0.451839\pi$
$740$	0	0
$741$	−1.01229	−0.0371874
$742$	0	0
$743$	−38.6352	−1.41739	−0.708694	−	0.705516i	$-0.750716\pi$
$743$	−38.6352	−1.41739	−0.708694	+	0.705516i	$0.750716\pi$
$744$	0	0
$745$	9.70315	0.355496
$746$	0	0
$747$	−6.86922	−0.251332
$748$	0	0
$749$	−0.157210	−0.00574433
$750$	0	0
$751$	41.3266	1.50803	0.754014	−	0.656858i	$-0.228115\pi$
$751$	41.3266	1.50803	0.754014	+	0.656858i	$0.228115\pi$
$752$	0	0
$753$	−9.55068	−0.348046
$754$	0	0
$755$	−23.2372	−0.845688
$756$	0	0
$757$	27.8880	1.01361	0.506804	−	0.862062i	$-0.330827\pi$
$757$	27.8880	1.01361	0.506804	+	0.862062i	$0.330827\pi$
$758$	0	0
$759$	−12.1117	−0.439626
$760$	0	0
$761$	−20.0352	−0.726276	−0.363138	−	0.931735i	$-0.618295\pi$
$761$	−20.0352	−0.726276	−0.363138	+	0.931735i	$0.618295\pi$
$762$	0	0
$763$	−0.123033	−0.00445411
$764$	0	0
$765$	−1.01107	−0.0365551
$766$	0	0
$767$	4.93841	0.178316
$768$	0	0
$769$	−20.5370	−0.740582	−0.370291	−	0.928916i	$-0.620742\pi$
$769$	−20.5370	−0.740582	−0.370291	+	0.928916i	$0.620742\pi$
$770$	0	0
$771$	−17.0891	−0.615449
$772$	0	0
$773$	36.5245	1.31369	0.656847	−	0.754024i	$-0.271890\pi$
$773$	36.5245	1.31369	0.656847	+	0.754024i	$0.271890\pi$
$774$	0	0
$775$	−4.93369	−0.177223
$776$	0	0
$777$	−0.116924	−0.00419462
$778$	0	0
$779$	0.0354261	0.00126927
$780$	0	0
$781$	38.3128	1.37094
$782$	0	0
$783$	−0.674663	−0.0241105
$784$	0	0
$785$	24.3357	0.868580
$786$	0	0
$787$	−45.4715	−1.62089	−0.810443	−	0.585818i	$-0.800773\pi$
$787$	−45.4715	−1.62089	−0.810443	+	0.585818i	$0.800773\pi$
$788$	0	0
$789$	−4.66660	−0.166135
$790$	0	0
$791$	−0.148880	−0.00529357
$792$	0	0
$793$	76.0595	2.70095
$794$	0	0
$795$	15.1699	0.538020
$796$	0	0
$797$	−45.6786	−1.61802	−0.809009	−	0.587796i	$-0.799996\pi$
$797$	−45.6786	−1.61802	−0.809009	+	0.587796i	$0.799996\pi$
$798$	0	0
$799$	4.25148	0.150407
$800$	0	0
$801$	5.84297	0.206451
$802$	0	0
$803$	−21.0299	−0.742128
$804$	0	0
$805$	0.0377337	0.00132994
$806$	0	0
$807$	−6.31602	−0.222334
$808$	0	0
$809$	−10.4027	−0.365740	−0.182870	−	0.983137i	$-0.558539\pi$
$809$	−10.4027	−0.365740	−0.182870	+	0.983137i	$0.558539\pi$
$810$	0	0
$811$	−13.1144	−0.460508	−0.230254	−	0.973131i	$-0.573956\pi$
$811$	−13.1144	−0.460508	−0.230254	+	0.973131i	$0.573956\pi$
$812$	0	0
$813$	−7.41976	−0.260222
$814$	0	0
$815$	24.3202	0.851898
$816$	0	0
$817$	0.503876	0.0176284
$818$	0	0
$819$	0.0394714	0.00137924
$820$	0	0
$821$	−4.95032	−0.172767	−0.0863837	−	0.996262i	$-0.527531\pi$
$821$	−4.95032	−0.172767	−0.0863837	+	0.996262i	$0.527531\pi$
$822$	0	0
$823$	20.5661	0.716890	0.358445	−	0.933551i	$-0.383307\pi$
$823$	20.5661	0.716890	0.358445	+	0.933551i	$0.383307\pi$
$824$	0	0
$825$	14.0371	0.488711
$826$	0	0
$827$	34.7900	1.20977	0.604883	−	0.796314i	$-0.293220\pi$
$827$	34.7900	1.20977	0.604883	+	0.796314i	$0.293220\pi$
$828$	0	0
$829$	43.7897	1.52088	0.760439	−	0.649410i	$-0.224984\pi$
$829$	43.7897	1.52088	0.760439	+	0.649410i	$0.224984\pi$
$830$	0	0
$831$	−2.69690	−0.0935546
$832$	0	0
$833$	−6.99990	−0.242532
$834$	0	0
$835$	27.2548	0.943193
$836$	0	0
$837$	−8.30336	−0.287006
$838$	0	0
$839$	0.702445	0.0242511	0.0121256	−	0.999926i	$-0.496140\pi$
$839$	0.702445	0.0242511	0.0121256	+	0.999926i	$0.496140\pi$
$840$	0	0
$841$	−28.9857	−0.999505
$842$	0	0
$843$	43.1807	1.48722
$844$	0	0
$845$	14.6399	0.503629
$846$	0	0
$847$	−0.0310496	−0.00106688
$848$	0	0
$849$	−40.5642	−1.39216
$850$	0	0
$851$	−22.3342	−0.765606
$852$	0	0
$853$	21.2522	0.727662	0.363831	−	0.931465i	$-0.381469\pi$
$853$	21.2522	0.727662	0.363831	+	0.931465i	$0.381469\pi$
$854$	0	0
$855$	−0.139301	−0.00476400
$856$	0	0
$857$	39.2898	1.34212	0.671058	−	0.741405i	$-0.265840\pi$
$857$	39.2898	1.34212	0.671058	+	0.741405i	$0.265840\pi$
$858$	0	0
$859$	−42.8377	−1.46160	−0.730801	−	0.682591i	$-0.760853\pi$
$859$	−42.8377	−1.46160	−0.730801	+	0.682591i	$0.760853\pi$
$860$	0	0
$861$	0.00388779	0.000132496 0
$862$	0	0
$863$	34.2577	1.16614	0.583072	−	0.812420i	$-0.301850\pi$
$863$	34.2577	1.16614	0.583072	+	0.812420i	$0.301850\pi$
$864$	0	0
$865$	−10.4174	−0.354203
$866$	0	0
$867$	1.48779	0.0505281
$868$	0	0
$869$	23.2468	0.788594
$870$	0	0
$871$	45.4412	1.53972
$872$	0	0
$873$	−13.3273	−0.451062
$874$	0	0
$875$	−0.109057	−0.00368681
$876$	0	0
$877$	45.3016	1.52973	0.764863	−	0.644193i	$-0.222807\pi$
$877$	45.3016	1.52973	0.764863	+	0.644193i	$0.222807\pi$
$878$	0	0
$879$	35.3040	1.19078
$880$	0	0
$881$	28.8779	0.972922	0.486461	−	0.873702i	$-0.338287\pi$
$881$	28.8779	0.972922	0.486461	+	0.873702i	$0.338287\pi$
$882$	0	0
$883$	−56.0214	−1.88527	−0.942635	−	0.333824i	$-0.891661\pi$
$883$	−56.0214	−1.88527	−0.942635	+	0.333824i	$0.891661\pi$
$884$	0	0
$885$	−1.91266	−0.0642935
$886$	0	0
$887$	−12.1486	−0.407908	−0.203954	−	0.978980i	$-0.565379\pi$
$887$	−12.1486	−0.407908	−0.203954	+	0.978980i	$0.565379\pi$
$888$	0	0
$889$	−0.0628020	−0.00210631
$890$	0	0
$891$	16.9740	0.568651
$892$	0	0
$893$	0.585755	0.0196015
$894$	0	0
$895$	22.3562	0.747284
$896$	0	0
$897$	−21.2203	−0.708524
$898$	0	0
$899$	0.176517	0.00588716
$900$	0	0
$901$	7.93129	0.264230
$902$	0	0
$903$	0.0552971	0.00184017
$904$	0	0
$905$	29.9713	0.996281
$906$	0	0
$907$	−56.9893	−1.89230	−0.946149	−	0.323732i	$-0.895062\pi$
$907$	−56.9893	−1.89230	−0.946149	+	0.323732i	$0.895062\pi$
$908$	0	0
$909$	−3.29414	−0.109260
$910$	0	0
$911$	−46.0046	−1.52420	−0.762100	−	0.647459i	$-0.775832\pi$
$911$	−46.0046	−1.52420	−0.762100	+	0.647459i	$0.775832\pi$
$912$	0	0
$913$	−24.6187	−0.814759
$914$	0	0
$915$	−29.4581	−0.973855
$916$	0	0
$917$	0.00112466	3.71396e−5 0
$918$	0	0
$919$	−19.8158	−0.653663	−0.326831	−	0.945083i	$-0.605981\pi$
$919$	−19.8158	−0.653663	−0.326831	+	0.945083i	$0.605981\pi$
$920$	0	0
$921$	−41.7517	−1.37576
$922$	0	0
$923$	67.1259	2.20948
$924$	0	0
$925$	25.8848	0.851087
$926$	0	0
$927$	7.73795	0.254148
$928$	0	0
$929$	32.2198	1.05710	0.528549	−	0.848903i	$-0.322736\pi$
$929$	32.2198	1.05710	0.528549	+	0.848903i	$0.322736\pi$
$930$	0	0
$931$	−0.964422	−0.0316077
$932$	0	0
$933$	4.21901	0.138124
$934$	0	0
$935$	−3.62357	−0.118503
$936$	0	0
$937$	−8.04436	−0.262798	−0.131399	−	0.991330i	$-0.541947\pi$
$937$	−8.04436	−0.262798	−0.131399	+	0.991330i	$0.541947\pi$
$938$	0	0
$939$	−36.3790	−1.18718
$940$	0	0
$941$	−53.1023	−1.73109	−0.865543	−	0.500834i	$-0.833027\pi$
$941$	−53.1023	−1.73109	−0.865543	+	0.500834i	$0.833027\pi$
$942$	0	0
$943$	0.742625	0.0241832
$944$	0	0
$945$	−0.0736013	−0.00239425
$946$	0	0
$947$	−39.0994	−1.27056	−0.635281	−	0.772281i	$-0.719116\pi$
$947$	−39.0994	−1.27056	−0.635281	+	0.772281i	$0.719116\pi$
$948$	0	0
$949$	−36.8454	−1.19605
$950$	0	0
$951$	15.2058	0.493083
$952$	0	0
$953$	35.6931	1.15621	0.578107	−	0.815961i	$-0.303792\pi$
$953$	35.6931	1.15621	0.578107	+	0.815961i	$0.303792\pi$
$954$	0	0
$955$	16.1092	0.521283
$956$	0	0
$957$	−0.502218	−0.0162344
$958$	0	0
$959$	−0.149674	−0.00483323
$960$	0	0
$961$	−28.8275	−0.929920
$962$	0	0
$963$	12.1661	0.392047
$964$	0	0
$965$	26.5238	0.853830
$966$	0	0
$967$	−18.3975	−0.591623	−0.295812	−	0.955246i	$-0.595590\pi$
$967$	−18.3975	−0.591623	−0.295812	+	0.955246i	$0.595590\pi$
$968$	0	0
$969$	0.204983	0.00658501
$970$	0	0
$971$	10.8643	0.348652	0.174326	−	0.984688i	$-0.444225\pi$
$971$	10.8643	0.348652	0.174326	+	0.984688i	$0.444225\pi$
$972$	0	0
$973$	0.0134650	0.000431669 0
$974$	0	0
$975$	24.5938	0.787631
$976$	0	0
$977$	15.1930	0.486066	0.243033	−	0.970018i	$-0.421858\pi$
$977$	15.1930	0.486066	0.243033	+	0.970018i	$0.421858\pi$
$978$	0	0
$979$	20.9407	0.669267
$980$	0	0
$981$	9.52125	0.303990
$982$	0	0
$983$	−14.6448	−0.467097	−0.233548	−	0.972345i	$-0.575034\pi$
$983$	−14.6448	−0.467097	−0.233548	+	0.972345i	$0.575034\pi$
$984$	0	0
$985$	15.5096	0.494176
$986$	0	0
$987$	0.0642828	0.00204614
$988$	0	0
$989$	10.5626	0.335870
$990$	0	0
$991$	−3.49079	−0.110889	−0.0554443	−	0.998462i	$-0.517658\pi$
$991$	−3.49079	−0.110889	−0.0554443	+	0.998462i	$0.517658\pi$
$992$	0	0
$993$	39.2956	1.24701
$994$	0	0
$995$	10.6765	0.338469
$996$	0	0
$997$	18.6090	0.589354	0.294677	−	0.955597i	$-0.404788\pi$
$997$	18.6090	0.589354	0.294677	+	0.955597i	$0.404788\pi$
$998$	0	0
$999$	43.5638	1.37830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.10	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.10	✓	15	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48779 q^{3} +1.28557 q^{5} +0.0101628 q^{7} -0.786472 q^{9} +O(q^{10})\)	\(q+1.48779 q^{3} +1.28557 q^{5} +0.0101628 q^{7} -0.786472 q^{9} -2.81865 q^{11} -4.93841 q^{13} +1.91266 q^{15} +1.00000 q^{17} +0.137777 q^{19} +0.0151201 q^{21} +2.88816 q^{23} -3.34731 q^{25} -5.63349 q^{27} +0.119759 q^{29} +1.47393 q^{31} -4.19356 q^{33} +0.0130650 q^{35} -7.73302 q^{37} -7.34733 q^{39} +0.257127 q^{41} +3.65719 q^{43} -1.01107 q^{45} +4.25148 q^{47} -6.99990 q^{49} +1.48779 q^{51} +7.93129 q^{53} -3.62357 q^{55} +0.204983 q^{57} -1.00000 q^{59} -15.4016 q^{61} -0.00799274 q^{63} -6.34867 q^{65} -9.20159 q^{67} +4.29699 q^{69} -13.5926 q^{71} +7.46098 q^{73} -4.98010 q^{75} -0.0286453 q^{77} -8.24751 q^{79} -6.02205 q^{81} +8.73422 q^{83} +1.28557 q^{85} +0.178177 q^{87} -7.42934 q^{89} -0.0501879 q^{91} +2.19290 q^{93} +0.177122 q^{95} +16.9457 q^{97} +2.21679 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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