LMFDB - Embedded newform 7616.2.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, 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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.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:	$60.8140661794$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.147697840.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 10x^{3} + 18x^{2} - 16x + 3 $ x^6 - 2x^5 - 10x^4 + 10x^3 + 18x^2 - 16*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3808)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.75839$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75839 q^{3} -1.29021 q^{5} -1.00000 q^{7} +4.60870 q^{9} -3.92967 q^{11} -3.02708 q^{13} +3.55891 q^{15} +1.00000 q^{17} -1.45247 q^{19} +2.75839 q^{21} -2.74235 q^{23} -3.33535 q^{25} -4.43741 q^{27} +4.12449 q^{29} +5.90440 q^{31} +10.8396 q^{33} +1.29021 q^{35} +2.16107 q^{37} +8.34987 q^{39} +10.3997 q^{41} -0.144202 q^{43} -5.94621 q^{45} +0.778926 q^{47} +1.00000 q^{49} -2.75839 q^{51} +3.76742 q^{53} +5.07012 q^{55} +4.00646 q^{57} -10.9905 q^{59} +2.81623 q^{61} -4.60870 q^{63} +3.90559 q^{65} +2.35105 q^{67} +7.56447 q^{69} -9.21823 q^{71} -1.02354 q^{73} +9.20018 q^{75} +3.92967 q^{77} -6.93267 q^{79} -1.58599 q^{81} +2.72771 q^{83} -1.29021 q^{85} -11.3769 q^{87} -3.41589 q^{89} +3.02708 q^{91} -16.2866 q^{93} +1.87399 q^{95} +13.0575 q^{97} -18.1107 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 4 q^{13} + 6 q^{17} - 18 q^{19} + 4 q^{21} + 6 q^{23} + 12 q^{25} - 10 q^{27} - 4 q^{29} - 8 q^{31} - 6 q^{33} - 4 q^{35} + 8 q^{37} - 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 4 * q^13 + 6 * q^17 - 18 * q^19 + 4 * q^21 + 6 * q^23 + 12 * q^25 - 10 * q^27 - 4 * q^29 - 8 * q^31 - 6 * q^33 - 4 * q^35 + 8 * q^37 - 18 * q^39 - 4 * q^41 - 20 * q^43 + 24 * q^45 + 6 * q^47 + 6 * q^49 - 4 * q^51 - 14 * q^53 + 4 * q^55 - 4 * q^57 - 20 * q^59 - 2 * q^61 - 8 * q^63 + 6 * q^65 - 28 * q^67 + 4 * q^69 + 18 * q^71 + 4 * q^73 - 20 * q^75 + 8 * q^77 - 2 * q^79 + 34 * q^81 - 20 * q^83 + 4 * q^85 + 22 * q^87 - 6 * q^89 - 4 * q^91 - 40 * q^93 - 6 * q^95 + 30 * q^97 - 32 * 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
$3$	−2.75839	−1.59256	−0.796278	−	0.604931i	$-0.793201\pi$
$3$	−2.75839	−1.59256	−0.796278	+	0.604931i	$0.793201\pi$
$4$	0	0
$5$	−1.29021	−0.577001	−0.288501	−	0.957480i	$-0.593157\pi$
$5$	−1.29021	−0.577001	−0.288501	+	0.957480i	$0.593157\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.60870	1.53623
$10$	0	0
$11$	−3.92967	−1.18484	−0.592420	−	0.805629i	$-0.701828\pi$
$11$	−3.92967	−1.18484	−0.592420	+	0.805629i	$0.701828\pi$
$12$	0	0
$13$	−3.02708	−0.839562	−0.419781	−	0.907625i	$-0.637893\pi$
$13$	−3.02708	−0.839562	−0.419781	+	0.907625i	$0.637893\pi$
$14$	0	0
$15$	3.55891	0.918907
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−1.45247	−0.333219	−0.166609	−	0.986023i	$-0.553282\pi$
$19$	−1.45247	−0.333219	−0.166609	+	0.986023i	$0.553282\pi$
$20$	0	0
$21$	2.75839	0.601929
$22$	0	0
$23$	−2.74235	−0.571820	−0.285910	−	0.958256i	$-0.592296\pi$
$23$	−2.74235	−0.571820	−0.285910	+	0.958256i	$0.592296\pi$
$24$	0	0
$25$	−3.33535	−0.667069
$26$	0	0
$27$	−4.43741	−0.853981
$28$	0	0
$29$	4.12449	0.765899	0.382950	−	0.923769i	$-0.374908\pi$
$29$	4.12449	0.765899	0.382950	+	0.923769i	$0.374908\pi$
$30$	0	0
$31$	5.90440	1.06046	0.530231	−	0.847853i	$-0.322105\pi$
$31$	5.90440	1.06046	0.530231	+	0.847853i	$0.322105\pi$
$32$	0	0
$33$	10.8396	1.88692
$34$	0	0
$35$	1.29021	0.218086
$36$	0	0
$37$	2.16107	0.355277	0.177638	−	0.984096i	$-0.443154\pi$
$37$	2.16107	0.355277	0.177638	+	0.984096i	$0.443154\pi$
$38$	0	0
$39$	8.34987	1.33705
$40$	0	0
$41$	10.3997	1.62415	0.812077	−	0.583551i	$-0.198337\pi$
$41$	10.3997	1.62415	0.812077	+	0.583551i	$0.198337\pi$
$42$	0	0
$43$	−0.144202	−0.0219907	−0.0109953	−	0.999940i	$-0.503500\pi$
$43$	−0.144202	−0.0219907	−0.0109953	+	0.999940i	$0.503500\pi$
$44$	0	0
$45$	−5.94621	−0.886408
$46$	0	0
$47$	0.778926	0.113618	0.0568090	−	0.998385i	$-0.481907\pi$
$47$	0.778926	0.113618	0.0568090	+	0.998385i	$0.481907\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.75839	−0.386251
$52$	0	0
$53$	3.76742	0.517495	0.258747	−	0.965945i	$-0.416690\pi$
$53$	3.76742	0.517495	0.258747	+	0.965945i	$0.416690\pi$
$54$	0	0
$55$	5.07012	0.683655
$56$	0	0
$57$	4.00646	0.530669
$58$	0	0
$59$	−10.9905	−1.43084	−0.715421	−	0.698693i	$-0.753765\pi$
$59$	−10.9905	−1.43084	−0.715421	+	0.698693i	$0.753765\pi$
$60$	0	0
$61$	2.81623	0.360581	0.180291	−	0.983613i	$-0.442296\pi$
$61$	2.81623	0.360581	0.180291	+	0.983613i	$0.442296\pi$
$62$	0	0
$63$	−4.60870	−0.580641
$64$	0	0
$65$	3.90559	0.484428
$66$	0	0
$67$	2.35105	0.287227	0.143613	−	0.989634i	$-0.454128\pi$
$67$	2.35105	0.287227	0.143613	+	0.989634i	$0.454128\pi$
$68$	0	0
$69$	7.56447	0.910656
$70$	0	0
$71$	−9.21823	−1.09400	−0.547001	−	0.837132i	$-0.684231\pi$
$71$	−9.21823	−1.09400	−0.547001	+	0.837132i	$0.684231\pi$
$72$	0	0
$73$	−1.02354	−0.119796	−0.0598979	−	0.998205i	$-0.519078\pi$
$73$	−1.02354	−0.119796	−0.0598979	+	0.998205i	$0.519078\pi$
$74$	0	0
$75$	9.20018	1.06235
$76$	0	0
$77$	3.92967	0.447828
$78$	0	0
$79$	−6.93267	−0.779986	−0.389993	−	0.920818i	$-0.627523\pi$
$79$	−6.93267	−0.779986	−0.389993	+	0.920818i	$0.627523\pi$
$80$	0	0
$81$	−1.58599	−0.176221
$82$	0	0
$83$	2.72771	0.299405	0.149702	−	0.988731i	$-0.452168\pi$
$83$	2.72771	0.299405	0.149702	+	0.988731i	$0.452168\pi$
$84$	0	0
$85$	−1.29021	−0.139943
$86$	0	0
$87$	−11.3769	−1.21974
$88$	0	0
$89$	−3.41589	−0.362084	−0.181042	−	0.983475i	$-0.557947\pi$
$89$	−3.41589	−0.362084	−0.181042	+	0.983475i	$0.557947\pi$
$90$	0	0
$91$	3.02708	0.317325
$92$	0	0
$93$	−16.2866	−1.68884
$94$	0	0
$95$	1.87399	0.192268
$96$	0	0
$97$	13.0575	1.32579	0.662894	−	0.748714i	$-0.269328\pi$
$97$	13.0575	1.32579	0.662894	+	0.748714i	$0.269328\pi$
$98$	0	0
$99$	−18.1107	−1.82019
$100$	0	0
$101$	10.9873	1.09328	0.546639	−	0.837368i	$-0.315907\pi$
$101$	10.9873	1.09328	0.546639	+	0.837368i	$0.315907\pi$
$102$	0	0
$103$	10.1810	1.00316	0.501582	−	0.865110i	$-0.332752\pi$
$103$	10.1810	1.00316	0.501582	+	0.865110i	$0.332752\pi$
$104$	0	0
$105$	−3.55891	−0.347314
$106$	0	0
$107$	11.1548	1.07838	0.539189	−	0.842185i	$-0.318731\pi$
$107$	11.1548	1.07838	0.539189	+	0.842185i	$0.318731\pi$
$108$	0	0
$109$	−3.45012	−0.330462	−0.165231	−	0.986255i	$-0.552837\pi$
$109$	−3.45012	−0.330462	−0.165231	+	0.986255i	$0.552837\pi$
$110$	0	0
$111$	−5.96105	−0.565798
$112$	0	0
$113$	8.15505	0.767162	0.383581	−	0.923507i	$-0.374691\pi$
$113$	8.15505	0.767162	0.383581	+	0.923507i	$0.374691\pi$
$114$	0	0
$115$	3.53823	0.329941
$116$	0	0
$117$	−13.9509	−1.28976
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	4.44233	0.403848
$122$	0	0
$123$	−28.6863	−2.58655
$124$	0	0
$125$	10.7544	0.961901
$126$	0	0
$127$	−0.619018	−0.0549289	−0.0274645	−	0.999623i	$-0.508743\pi$
$127$	−0.619018	−0.0549289	−0.0274645	+	0.999623i	$0.508743\pi$
$128$	0	0
$129$	0.397766	0.0350214
$130$	0	0
$131$	−8.37716	−0.731916	−0.365958	−	0.930631i	$-0.619259\pi$
$131$	−8.37716	−0.731916	−0.365958	+	0.930631i	$0.619259\pi$
$132$	0	0
$133$	1.45247	0.125945
$134$	0	0
$135$	5.72521	0.492748
$136$	0	0
$137$	−2.17259	−0.185617	−0.0928085	−	0.995684i	$-0.529584\pi$
$137$	−2.17259	−0.185617	−0.0928085	+	0.995684i	$0.529584\pi$
$138$	0	0
$139$	−1.75152	−0.148562	−0.0742809	−	0.997237i	$-0.523666\pi$
$139$	−1.75152	−0.148562	−0.0742809	+	0.997237i	$0.523666\pi$
$140$	0	0
$141$	−2.14858	−0.180943
$142$	0	0
$143$	11.8954	0.994747
$144$	0	0
$145$	−5.32148	−0.441925
$146$	0	0
$147$	−2.75839	−0.227508
$148$	0	0
$149$	0.861610	0.0705858	0.0352929	−	0.999377i	$-0.488764\pi$
$149$	0.861610	0.0705858	0.0352929	+	0.999377i	$0.488764\pi$
$150$	0	0
$151$	3.59946	0.292920	0.146460	−	0.989217i	$-0.453212\pi$
$151$	3.59946	0.292920	0.146460	+	0.989217i	$0.453212\pi$
$152$	0	0
$153$	4.60870	0.372591
$154$	0	0
$155$	−7.61794	−0.611888
$156$	0	0
$157$	10.0755	0.804109	0.402055	−	0.915616i	$-0.368296\pi$
$157$	10.0755	0.804109	0.402055	+	0.915616i	$0.368296\pi$
$158$	0	0
$159$	−10.3920	−0.824139
$160$	0	0
$161$	2.74235	0.216128
$162$	0	0
$163$	2.85588	0.223689	0.111845	−	0.993726i	$-0.464324\pi$
$163$	2.85588	0.223689	0.111845	+	0.993726i	$0.464324\pi$
$164$	0	0
$165$	−13.9854	−1.08876
$166$	0	0
$167$	25.2347	1.95272	0.976360	−	0.216151i	$-0.0693502\pi$
$167$	25.2347	1.95272	0.976360	+	0.216151i	$0.0693502\pi$
$168$	0	0
$169$	−3.83677	−0.295136
$170$	0	0
$171$	−6.69398	−0.511901
$172$	0	0
$173$	17.1083	1.30072	0.650358	−	0.759628i	$-0.274619\pi$
$173$	17.1083	1.30072	0.650358	+	0.759628i	$0.274619\pi$
$174$	0	0
$175$	3.33535	0.252129
$176$	0	0
$177$	30.3161	2.27870
$178$	0	0
$179$	−5.37992	−0.402114	−0.201057	−	0.979579i	$-0.564438\pi$
$179$	−5.37992	−0.402114	−0.201057	+	0.979579i	$0.564438\pi$
$180$	0	0
$181$	12.6040	0.936851	0.468426	−	0.883503i	$-0.344821\pi$
$181$	12.6040	0.936851	0.468426	+	0.883503i	$0.344821\pi$
$182$	0	0
$183$	−7.76825	−0.574246
$184$	0	0
$185$	−2.78824	−0.204995
$186$	0	0
$187$	−3.92967	−0.287366
$188$	0	0
$189$	4.43741	0.322774
$190$	0	0
$191$	13.7934	0.998054	0.499027	−	0.866586i	$-0.333691\pi$
$191$	13.7934	0.998054	0.499027	+	0.866586i	$0.333691\pi$
$192$	0	0
$193$	6.65245	0.478854	0.239427	−	0.970914i	$-0.423040\pi$
$193$	6.65245	0.478854	0.239427	+	0.970914i	$0.423040\pi$
$194$	0	0
$195$	−10.7731	−0.771479
$196$	0	0
$197$	−9.86898	−0.703136	−0.351568	−	0.936162i	$-0.614351\pi$
$197$	−9.86898	−0.703136	−0.351568	+	0.936162i	$0.614351\pi$
$198$	0	0
$199$	−10.1040	−0.716250	−0.358125	−	0.933674i	$-0.616584\pi$
$199$	−10.1040	−0.716250	−0.358125	+	0.933674i	$0.616584\pi$
$200$	0	0
$201$	−6.48511	−0.457425
$202$	0	0
$203$	−4.12449	−0.289483
$204$	0	0
$205$	−13.4178	−0.937139
$206$	0	0
$207$	−12.6387	−0.878449
$208$	0	0
$209$	5.70772	0.394811
$210$	0	0
$211$	11.8529	0.815984	0.407992	−	0.912986i	$-0.366229\pi$
$211$	11.8529	0.815984	0.407992	+	0.912986i	$0.366229\pi$
$212$	0	0
$213$	25.4274	1.74226
$214$	0	0
$215$	0.186052	0.0126886
$216$	0	0
$217$	−5.90440	−0.400817
$218$	0	0
$219$	2.82331	0.190781
$220$	0	0
$221$	−3.02708	−0.203624
$222$	0	0
$223$	25.6464	1.71741	0.858704	−	0.512472i	$-0.171270\pi$
$223$	25.6464	1.71741	0.858704	+	0.512472i	$0.171270\pi$
$224$	0	0
$225$	−15.3716	−1.02477
$226$	0	0
$227$	−15.6657	−1.03977	−0.519884	−	0.854237i	$-0.674025\pi$
$227$	−15.6657	−1.03977	−0.519884	+	0.854237i	$0.674025\pi$
$228$	0	0
$229$	−8.93961	−0.590746	−0.295373	−	0.955382i	$-0.595444\pi$
$229$	−8.93961	−0.590746	−0.295373	+	0.955382i	$0.595444\pi$
$230$	0	0
$231$	−10.8396	−0.713190
$232$	0	0
$233$	−27.7483	−1.81785	−0.908925	−	0.416960i	$-0.863095\pi$
$233$	−27.7483	−1.81785	−0.908925	+	0.416960i	$0.863095\pi$
$234$	0	0
$235$	−1.00498	−0.0655578
$236$	0	0
$237$	19.1230	1.24217
$238$	0	0
$239$	−0.951517	−0.0615485	−0.0307743	−	0.999526i	$-0.509797\pi$
$239$	−0.951517	−0.0615485	−0.0307743	+	0.999526i	$0.509797\pi$
$240$	0	0
$241$	−4.53858	−0.292356	−0.146178	−	0.989258i	$-0.546697\pi$
$241$	−4.53858	−0.292356	−0.146178	+	0.989258i	$0.546697\pi$
$242$	0	0
$243$	17.6870	1.13462
$244$	0	0
$245$	−1.29021	−0.0824288
$246$	0	0
$247$	4.39674	0.279758
$248$	0	0
$249$	−7.52407	−0.476819
$250$	0	0
$251$	−9.51998	−0.600896	−0.300448	−	0.953798i	$-0.597136\pi$
$251$	−9.51998	−0.600896	−0.300448	+	0.953798i	$0.597136\pi$
$252$	0	0
$253$	10.7766	0.677516
$254$	0	0
$255$	3.55891	0.222868
$256$	0	0
$257$	−27.9156	−1.74133	−0.870665	−	0.491876i	$-0.836311\pi$
$257$	−27.9156	−1.74133	−0.870665	+	0.491876i	$0.836311\pi$
$258$	0	0
$259$	−2.16107	−0.134282
$260$	0	0
$261$	19.0085	1.17660
$262$	0	0
$263$	6.33759	0.390793	0.195396	−	0.980724i	$-0.437401\pi$
$263$	6.33759	0.390793	0.195396	+	0.980724i	$0.437401\pi$
$264$	0	0
$265$	−4.86078	−0.298595
$266$	0	0
$267$	9.42236	0.576639
$268$	0	0
$269$	20.9665	1.27835	0.639174	−	0.769062i	$-0.279276\pi$
$269$	20.9665	1.27835	0.639174	+	0.769062i	$0.279276\pi$
$270$	0	0
$271$	−24.7898	−1.50587	−0.752937	−	0.658092i	$-0.771364\pi$
$271$	−24.7898	−1.50587	−0.752937	+	0.658092i	$0.771364\pi$
$272$	0	0
$273$	−8.34987	−0.505357
$274$	0	0
$275$	13.1068	0.790371
$276$	0	0
$277$	−16.8201	−1.01062	−0.505312	−	0.862937i	$-0.668623\pi$
$277$	−16.8201	−1.01062	−0.505312	+	0.862937i	$0.668623\pi$
$278$	0	0
$279$	27.2116	1.62912
$280$	0	0
$281$	−25.5535	−1.52439	−0.762196	−	0.647346i	$-0.775879\pi$
$281$	−25.5535	−1.52439	−0.762196	+	0.647346i	$0.775879\pi$
$282$	0	0
$283$	−26.5441	−1.57788	−0.788942	−	0.614467i	$-0.789371\pi$
$283$	−26.5441	−1.57788	−0.788942	+	0.614467i	$0.789371\pi$
$284$	0	0
$285$	−5.16920	−0.306197
$286$	0	0
$287$	−10.3997	−0.613872
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−36.0176	−2.11139
$292$	0	0
$293$	−3.90321	−0.228028	−0.114014	−	0.993479i	$-0.536371\pi$
$293$	−3.90321	−0.228028	−0.114014	+	0.993479i	$0.536371\pi$
$294$	0	0
$295$	14.1801	0.825598
$296$	0	0
$297$	17.4376	1.01183
$298$	0	0
$299$	8.30133	0.480079
$300$	0	0
$301$	0.144202	0.00831169
$302$	0	0
$303$	−30.3072	−1.74111
$304$	0	0
$305$	−3.63354	−0.208056
$306$	0	0
$307$	−22.1879	−1.26633	−0.633166	−	0.774016i	$-0.718245\pi$
$307$	−22.1879	−1.26633	−0.633166	+	0.774016i	$0.718245\pi$
$308$	0	0
$309$	−28.0831	−1.59759
$310$	0	0
$311$	−12.2149	−0.692644	−0.346322	−	0.938116i	$-0.612569\pi$
$311$	−12.2149	−0.692644	−0.346322	+	0.938116i	$0.612569\pi$
$312$	0	0
$313$	22.0216	1.24473	0.622366	−	0.782726i	$-0.286171\pi$
$313$	22.0216	1.24473	0.622366	+	0.782726i	$0.286171\pi$
$314$	0	0
$315$	5.94621	0.335031
$316$	0	0
$317$	25.1191	1.41083	0.705414	−	0.708795i	$-0.250761\pi$
$317$	25.1191	1.41083	0.705414	+	0.708795i	$0.250761\pi$
$318$	0	0
$319$	−16.2079	−0.907469
$320$	0	0
$321$	−30.7694	−1.71738
$322$	0	0
$323$	−1.45247	−0.0808174
$324$	0	0
$325$	10.0964	0.560046
$326$	0	0
$327$	9.51677	0.526279
$328$	0	0
$329$	−0.778926	−0.0429436
$330$	0	0
$331$	−26.2391	−1.44223	−0.721117	−	0.692813i	$-0.756371\pi$
$331$	−26.2391	−1.44223	−0.721117	+	0.692813i	$0.756371\pi$
$332$	0	0
$333$	9.95970	0.545788
$334$	0	0
$335$	−3.03336	−0.165730
$336$	0	0
$337$	−29.8425	−1.62563	−0.812813	−	0.582525i	$-0.802065\pi$
$337$	−29.8425	−1.62563	−0.812813	+	0.582525i	$0.802065\pi$
$338$	0	0
$339$	−22.4948	−1.22175
$340$	0	0
$341$	−23.2024	−1.25648
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−9.75979	−0.525450
$346$	0	0
$347$	−11.2458	−0.603708	−0.301854	−	0.953354i	$-0.597606\pi$
$347$	−11.2458	−0.603708	−0.301854	+	0.953354i	$0.597606\pi$
$348$	0	0
$349$	−6.03325	−0.322953	−0.161476	−	0.986877i	$-0.551626\pi$
$349$	−6.03325	−0.322953	−0.161476	+	0.986877i	$0.551626\pi$
$350$	0	0
$351$	13.4324	0.716969
$352$	0	0
$353$	−1.29356	−0.0688495	−0.0344247	−	0.999407i	$-0.510960\pi$
$353$	−1.29356	−0.0688495	−0.0344247	+	0.999407i	$0.510960\pi$
$354$	0	0
$355$	11.8935	0.631241
$356$	0	0
$357$	2.75839	0.145989
$358$	0	0
$359$	−30.2891	−1.59860	−0.799299	−	0.600933i	$-0.794796\pi$
$359$	−30.2891	−1.59860	−0.799299	+	0.600933i	$0.794796\pi$
$360$	0	0
$361$	−16.8903	−0.888965
$362$	0	0
$363$	−12.2537	−0.643150
$364$	0	0
$365$	1.32058	0.0691223
$366$	0	0
$367$	−35.1168	−1.83308	−0.916541	−	0.399941i	$-0.869031\pi$
$367$	−35.1168	−1.83308	−0.916541	+	0.399941i	$0.869031\pi$
$368$	0	0
$369$	47.9289	2.49508
$370$	0	0
$371$	−3.76742	−0.195595
$372$	0	0
$373$	6.79540	0.351853	0.175926	−	0.984403i	$-0.443708\pi$
$373$	6.79540	0.351853	0.175926	+	0.984403i	$0.443708\pi$
$374$	0	0
$375$	−29.6648	−1.53188
$376$	0	0
$377$	−12.4852	−0.643020
$378$	0	0
$379$	23.6660	1.21564	0.607820	−	0.794075i	$-0.292044\pi$
$379$	23.6660	1.21564	0.607820	+	0.794075i	$0.292044\pi$
$380$	0	0
$381$	1.70749	0.0874774
$382$	0	0
$383$	35.6595	1.82211	0.911057	−	0.412280i	$-0.135267\pi$
$383$	35.6595	1.82211	0.911057	+	0.412280i	$0.135267\pi$
$384$	0	0
$385$	−5.07012	−0.258397
$386$	0	0
$387$	−0.664586	−0.0337828
$388$	0	0
$389$	−11.4453	−0.580299	−0.290150	−	0.956981i	$-0.593705\pi$
$389$	−11.4453	−0.580299	−0.290150	+	0.956981i	$0.593705\pi$
$390$	0	0
$391$	−2.74235	−0.138687
$392$	0	0
$393$	23.1074	1.16562
$394$	0	0
$395$	8.94463	0.450053
$396$	0	0
$397$	8.32376	0.417758	0.208879	−	0.977942i	$-0.433019\pi$
$397$	8.32376	0.417758	0.208879	+	0.977942i	$0.433019\pi$
$398$	0	0
$399$	−4.00646	−0.200574
$400$	0	0
$401$	−0.238688	−0.0119195	−0.00595976	−	0.999982i	$-0.501897\pi$
$401$	−0.238688	−0.0119195	−0.00595976	+	0.999982i	$0.501897\pi$
$402$	0	0
$403$	−17.8731	−0.890323
$404$	0	0
$405$	2.04627	0.101680
$406$	0	0
$407$	−8.49228	−0.420947
$408$	0	0
$409$	−13.9318	−0.688885	−0.344442	−	0.938807i	$-0.611932\pi$
$409$	−13.9318	−0.688885	−0.344442	+	0.938807i	$0.611932\pi$
$410$	0	0
$411$	5.99285	0.295605
$412$	0	0
$413$	10.9905	0.540808
$414$	0	0
$415$	−3.51933	−0.172757
$416$	0	0
$417$	4.83136	0.236593
$418$	0	0
$419$	−19.4728	−0.951309	−0.475655	−	0.879632i	$-0.657789\pi$
$419$	−19.4728	−0.951309	−0.475655	+	0.879632i	$0.657789\pi$
$420$	0	0
$421$	−2.48294	−0.121011	−0.0605055	−	0.998168i	$-0.519271\pi$
$421$	−2.48294	−0.121011	−0.0605055	+	0.998168i	$0.519271\pi$
$422$	0	0
$423$	3.58984	0.174544
$424$	0	0
$425$	−3.33535	−0.161788
$426$	0	0
$427$	−2.81623	−0.136287
$428$	0	0
$429$	−32.8122	−1.58419
$430$	0	0
$431$	7.27607	0.350476	0.175238	−	0.984526i	$-0.443931\pi$
$431$	7.27607	0.350476	0.175238	+	0.984526i	$0.443931\pi$
$432$	0	0
$433$	36.5277	1.75541	0.877704	−	0.479203i	$-0.159074\pi$
$433$	36.5277	1.75541	0.877704	+	0.479203i	$0.159074\pi$
$434$	0	0
$435$	14.6787	0.703790
$436$	0	0
$437$	3.98318	0.190541
$438$	0	0
$439$	−7.62099	−0.363730	−0.181865	−	0.983324i	$-0.558213\pi$
$439$	−7.62099	−0.363730	−0.181865	+	0.983324i	$0.558213\pi$
$440$	0	0
$441$	4.60870	0.219462
$442$	0	0
$443$	1.40984	0.0669837	0.0334919	−	0.999439i	$-0.489337\pi$
$443$	1.40984	0.0669837	0.0334919	+	0.999439i	$0.489337\pi$
$444$	0	0
$445$	4.40724	0.208923
$446$	0	0
$447$	−2.37665	−0.112412
$448$	0	0
$449$	17.9454	0.846896	0.423448	−	0.905921i	$-0.360820\pi$
$449$	17.9454	0.846896	0.423448	+	0.905921i	$0.360820\pi$
$450$	0	0
$451$	−40.8672	−1.92436
$452$	0	0
$453$	−9.92870	−0.466491
$454$	0	0
$455$	−3.90559	−0.183097
$456$	0	0
$457$	−26.3854	−1.23426	−0.617129	−	0.786862i	$-0.711704\pi$
$457$	−26.3854	−1.23426	−0.617129	+	0.786862i	$0.711704\pi$
$458$	0	0
$459$	−4.43741	−0.207121
$460$	0	0
$461$	39.4155	1.83576	0.917881	−	0.396856i	$-0.129899\pi$
$461$	39.4155	1.83576	0.917881	+	0.396856i	$0.129899\pi$
$462$	0	0
$463$	−6.01921	−0.279736	−0.139868	−	0.990170i	$-0.544668\pi$
$463$	−6.01921	−0.279736	−0.139868	+	0.990170i	$0.544668\pi$
$464$	0	0
$465$	21.0132	0.974465
$466$	0	0
$467$	22.1884	1.02675	0.513377	−	0.858163i	$-0.328394\pi$
$467$	22.1884	1.02675	0.513377	+	0.858163i	$0.328394\pi$
$468$	0	0
$469$	−2.35105	−0.108562
$470$	0	0
$471$	−27.7920	−1.28059
$472$	0	0
$473$	0.566669	0.0260554
$474$	0	0
$475$	4.84448	0.222280
$476$	0	0
$477$	17.3629	0.794993
$478$	0	0
$479$	8.93363	0.408188	0.204094	−	0.978951i	$-0.434575\pi$
$479$	8.93363	0.408188	0.204094	+	0.978951i	$0.434575\pi$
$480$	0	0
$481$	−6.54172	−0.298277
$482$	0	0
$483$	−7.56447	−0.344196
$484$	0	0
$485$	−16.8470	−0.764981
$486$	0	0
$487$	−8.56471	−0.388104	−0.194052	−	0.980991i	$-0.562163\pi$
$487$	−8.56471	−0.388104	−0.194052	+	0.980991i	$0.562163\pi$
$488$	0	0
$489$	−7.87761	−0.356238
$490$	0	0
$491$	43.9981	1.98561	0.992803	−	0.119757i	$-0.0382114\pi$
$491$	43.9981	1.98561	0.992803	+	0.119757i	$0.0382114\pi$
$492$	0	0
$493$	4.12449	0.185758
$494$	0	0
$495$	23.3667	1.05025
$496$	0	0
$497$	9.21823	0.413494
$498$	0	0
$499$	−7.03332	−0.314855	−0.157427	−	0.987531i	$-0.550320\pi$
$499$	−7.03332	−0.314855	−0.157427	+	0.987531i	$0.550320\pi$
$500$	0	0
$501$	−69.6071	−3.10981
$502$	0	0
$503$	11.8203	0.527040	0.263520	−	0.964654i	$-0.415116\pi$
$503$	11.8203	0.527040	0.263520	+	0.964654i	$0.415116\pi$
$504$	0	0
$505$	−14.1760	−0.630823
$506$	0	0
$507$	10.5833	0.470021
$508$	0	0
$509$	−20.0352	−0.888045	−0.444023	−	0.896016i	$-0.646449\pi$
$509$	−20.0352	−0.888045	−0.444023	+	0.896016i	$0.646449\pi$
$510$	0	0
$511$	1.02354	0.0452786
$512$	0	0
$513$	6.44519	0.284562
$514$	0	0
$515$	−13.1357	−0.578827
$516$	0	0
$517$	−3.06093	−0.134619
$518$	0	0
$519$	−47.1912	−2.07146
$520$	0	0
$521$	8.85121	0.387779	0.193889	−	0.981023i	$-0.437890\pi$
$521$	8.85121	0.387779	0.193889	+	0.981023i	$0.437890\pi$
$522$	0	0
$523$	32.0496	1.40143	0.700716	−	0.713441i	$-0.252864\pi$
$523$	32.0496	1.40143	0.700716	+	0.713441i	$0.252864\pi$
$524$	0	0
$525$	−9.20018	−0.401529
$526$	0	0
$527$	5.90440	0.257200
$528$	0	0
$529$	−15.4795	−0.673021
$530$	0	0
$531$	−50.6520	−2.19811
$532$	0	0
$533$	−31.4806	−1.36358
$534$	0	0
$535$	−14.3921	−0.622226
$536$	0	0
$537$	14.8399	0.640390
$538$	0	0
$539$	−3.92967	−0.169263
$540$	0	0
$541$	34.3819	1.47819	0.739097	−	0.673599i	$-0.235253\pi$
$541$	34.3819	1.47819	0.739097	+	0.673599i	$0.235253\pi$
$542$	0	0
$543$	−34.7668	−1.49199
$544$	0	0
$545$	4.45140	0.190677
$546$	0	0
$547$	2.22436	0.0951068	0.0475534	−	0.998869i	$-0.484858\pi$
$547$	2.22436	0.0951068	0.0475534	+	0.998869i	$0.484858\pi$
$548$	0	0
$549$	12.9792	0.553937
$550$	0	0
$551$	−5.99069	−0.255212
$552$	0	0
$553$	6.93267	0.294807
$554$	0	0
$555$	7.69104	0.326466
$556$	0	0
$557$	−19.3197	−0.818604	−0.409302	−	0.912399i	$-0.634228\pi$
$557$	−19.3197	−0.818604	−0.409302	+	0.912399i	$0.634228\pi$
$558$	0	0
$559$	0.436513	0.0184625
$560$	0	0
$561$	10.8396	0.457646
$562$	0	0
$563$	−15.9783	−0.673403	−0.336702	−	0.941611i	$-0.609311\pi$
$563$	−15.9783	−0.673403	−0.336702	+	0.941611i	$0.609311\pi$
$564$	0	0
$565$	−10.5218	−0.442654
$566$	0	0
$567$	1.58599	0.0666055
$568$	0	0
$569$	−28.0056	−1.17406	−0.587028	−	0.809567i	$-0.699702\pi$
$569$	−28.0056	−1.17406	−0.587028	+	0.809567i	$0.699702\pi$
$570$	0	0
$571$	−5.11050	−0.213868	−0.106934	−	0.994266i	$-0.534103\pi$
$571$	−5.11050	−0.213868	−0.106934	+	0.994266i	$0.534103\pi$
$572$	0	0
$573$	−38.0475	−1.58946
$574$	0	0
$575$	9.14670	0.381444
$576$	0	0
$577$	−32.9436	−1.37146	−0.685729	−	0.727857i	$-0.740517\pi$
$577$	−32.9436	−1.37146	−0.685729	+	0.727857i	$0.740517\pi$
$578$	0	0
$579$	−18.3500	−0.762601
$580$	0	0
$581$	−2.72771	−0.113164
$582$	0	0
$583$	−14.8047	−0.613149
$584$	0	0
$585$	17.9997	0.744195
$586$	0	0
$587$	−14.3178	−0.590959	−0.295479	−	0.955349i	$-0.595479\pi$
$587$	−14.3178	−0.590959	−0.295479	+	0.955349i	$0.595479\pi$
$588$	0	0
$589$	−8.57594	−0.353366
$590$	0	0
$591$	27.2225	1.11978
$592$	0	0
$593$	35.9635	1.47685	0.738423	−	0.674338i	$-0.235571\pi$
$593$	35.9635	1.47685	0.738423	+	0.674338i	$0.235571\pi$
$594$	0	0
$595$	1.29021	0.0528936
$596$	0	0
$597$	27.8706	1.14067
$598$	0	0
$599$	−9.72582	−0.397386	−0.198693	−	0.980062i	$-0.563670\pi$
$599$	−9.72582	−0.397386	−0.198693	+	0.980062i	$0.563670\pi$
$600$	0	0
$601$	−33.1659	−1.35287	−0.676433	−	0.736504i	$-0.736475\pi$
$601$	−33.1659	−1.35287	−0.676433	+	0.736504i	$0.736475\pi$
$602$	0	0
$603$	10.8353	0.441247
$604$	0	0
$605$	−5.73155	−0.233021
$606$	0	0
$607$	−14.7413	−0.598332	−0.299166	−	0.954201i	$-0.596708\pi$
$607$	−14.7413	−0.598332	−0.299166	+	0.954201i	$0.596708\pi$
$608$	0	0
$609$	11.3769	0.461017
$610$	0	0
$611$	−2.35787	−0.0953894
$612$	0	0
$613$	−34.7578	−1.40385	−0.701927	−	0.712249i	$-0.747677\pi$
$613$	−34.7578	−1.40385	−0.701927	+	0.712249i	$0.747677\pi$
$614$	0	0
$615$	37.0114	1.49245
$616$	0	0
$617$	−40.2723	−1.62130	−0.810650	−	0.585531i	$-0.800886\pi$
$617$	−40.2723	−1.62130	−0.810650	+	0.585531i	$0.800886\pi$
$618$	0	0
$619$	23.3453	0.938327	0.469164	−	0.883111i	$-0.344555\pi$
$619$	23.3453	0.938327	0.469164	+	0.883111i	$0.344555\pi$
$620$	0	0
$621$	12.1690	0.488324
$622$	0	0
$623$	3.41589	0.136855
$624$	0	0
$625$	2.80127	0.112051
$626$	0	0
$627$	−15.7441	−0.628759
$628$	0	0
$629$	2.16107	0.0861673
$630$	0	0
$631$	−41.2545	−1.64232	−0.821158	−	0.570701i	$-0.806672\pi$
$631$	−41.2545	−1.64232	−0.821158	+	0.570701i	$0.806672\pi$
$632$	0	0
$633$	−32.6948	−1.29950
$634$	0	0
$635$	0.798666	0.0316941
$636$	0	0
$637$	−3.02708	−0.119937
$638$	0	0
$639$	−42.4840	−1.68064
$640$	0	0
$641$	33.3902	1.31884	0.659418	−	0.751777i	$-0.270803\pi$
$641$	33.3902	1.31884	0.659418	+	0.751777i	$0.270803\pi$
$642$	0	0
$643$	−30.8226	−1.21553	−0.607763	−	0.794119i	$-0.707933\pi$
$643$	−30.8226	−1.21553	−0.607763	+	0.794119i	$0.707933\pi$
$644$	0	0
$645$	−0.513204	−0.0202074
$646$	0	0
$647$	−5.04316	−0.198267	−0.0991335	−	0.995074i	$-0.531607\pi$
$647$	−5.04316	−0.198267	−0.0991335	+	0.995074i	$0.531607\pi$
$648$	0	0
$649$	43.1891	1.69532
$650$	0	0
$651$	16.2866	0.638323
$652$	0	0
$653$	−24.8776	−0.973535	−0.486768	−	0.873532i	$-0.661824\pi$
$653$	−24.8776	−0.973535	−0.486768	+	0.873532i	$0.661824\pi$
$654$	0	0
$655$	10.8083	0.422316
$656$	0	0
$657$	−4.71717	−0.184034
$658$	0	0
$659$	20.1111	0.783416	0.391708	−	0.920090i	$-0.371884\pi$
$659$	20.1111	0.783416	0.391708	+	0.920090i	$0.371884\pi$
$660$	0	0
$661$	−29.3305	−1.14082	−0.570412	−	0.821359i	$-0.693216\pi$
$661$	−29.3305	−1.14082	−0.570412	+	0.821359i	$0.693216\pi$
$662$	0	0
$663$	8.34987	0.324282
$664$	0	0
$665$	−1.87399	−0.0726703
$666$	0	0
$667$	−11.3108	−0.437957
$668$	0	0
$669$	−70.7426	−2.73507
$670$	0	0
$671$	−11.0669	−0.427231
$672$	0	0
$673$	−2.92023	−0.112567	−0.0562834	−	0.998415i	$-0.517925\pi$
$673$	−2.92023	−0.112567	−0.0562834	+	0.998415i	$0.517925\pi$
$674$	0	0
$675$	14.8003	0.569664
$676$	0	0
$677$	2.20861	0.0848838	0.0424419	−	0.999099i	$-0.486486\pi$
$677$	2.20861	0.0848838	0.0424419	+	0.999099i	$0.486486\pi$
$678$	0	0
$679$	−13.0575	−0.501100
$680$	0	0
$681$	43.2120	1.65589
$682$	0	0
$683$	29.7005	1.13646	0.568230	−	0.822870i	$-0.307629\pi$
$683$	29.7005	1.13646	0.568230	+	0.822870i	$0.307629\pi$
$684$	0	0
$685$	2.80311	0.107101
$686$	0	0
$687$	24.6589	0.940795
$688$	0	0
$689$	−11.4043	−0.434469
$690$	0	0
$691$	−6.99178	−0.265980	−0.132990	−	0.991117i	$-0.542458\pi$
$691$	−6.99178	−0.265980	−0.132990	+	0.991117i	$0.542458\pi$
$692$	0	0
$693$	18.1107	0.687968
$694$	0	0
$695$	2.25983	0.0857203
$696$	0	0
$697$	10.3997	0.393915
$698$	0	0
$699$	76.5405	2.89503
$700$	0	0
$701$	−42.6627	−1.61135	−0.805674	−	0.592360i	$-0.798197\pi$
$701$	−42.6627	−1.61135	−0.805674	+	0.592360i	$0.798197\pi$
$702$	0	0
$703$	−3.13888	−0.118385
$704$	0	0
$705$	2.77213	0.104404
$706$	0	0
$707$	−10.9873	−0.413220
$708$	0	0
$709$	−19.4110	−0.728996	−0.364498	−	0.931204i	$-0.618759\pi$
$709$	−19.4110	−0.728996	−0.364498	+	0.931204i	$0.618759\pi$
$710$	0	0
$711$	−31.9506	−1.19824
$712$	0	0
$713$	−16.1920	−0.606393
$714$	0	0
$715$	−15.3477	−0.573970
$716$	0	0
$717$	2.62465	0.0980194
$718$	0	0
$719$	31.5643	1.17715	0.588576	−	0.808442i	$-0.299689\pi$
$719$	31.5643	1.17715	0.588576	+	0.808442i	$0.299689\pi$
$720$	0	0
$721$	−10.1810	−0.379160
$722$	0	0
$723$	12.5192	0.465592
$724$	0	0
$725$	−13.7566	−0.510908
$726$	0	0
$727$	25.9764	0.963410	0.481705	−	0.876333i	$-0.340018\pi$
$727$	25.9764	0.963410	0.481705	+	0.876333i	$0.340018\pi$
$728$	0	0
$729$	−44.0297	−1.63073
$730$	0	0
$731$	−0.144202	−0.00533352
$732$	0	0
$733$	29.2473	1.08027	0.540137	−	0.841577i	$-0.318373\pi$
$733$	29.2473	1.08027	0.540137	+	0.841577i	$0.318373\pi$
$734$	0	0
$735$	3.55891	0.131272
$736$	0	0
$737$	−9.23887	−0.340318
$738$	0	0
$739$	−1.50508	−0.0553652	−0.0276826	−	0.999617i	$-0.508813\pi$
$739$	−1.50508	−0.0553652	−0.0276826	+	0.999617i	$0.508813\pi$
$740$	0	0
$741$	−12.1279	−0.445530
$742$	0	0
$743$	0.915905	0.0336013	0.0168006	−	0.999859i	$-0.494652\pi$
$743$	0.915905	0.0336013	0.0168006	+	0.999859i	$0.494652\pi$
$744$	0	0
$745$	−1.11166	−0.0407281
$746$	0	0
$747$	12.5712	0.459955
$748$	0	0
$749$	−11.1548	−0.407589
$750$	0	0
$751$	−12.7202	−0.464166	−0.232083	−	0.972696i	$-0.574554\pi$
$751$	−12.7202	−0.464166	−0.232083	+	0.972696i	$0.574554\pi$
$752$	0	0
$753$	26.2598	0.956960
$754$	0	0
$755$	−4.64407	−0.169015
$756$	0	0
$757$	6.48632	0.235749	0.117875	−	0.993028i	$-0.462392\pi$
$757$	6.48632	0.235749	0.117875	+	0.993028i	$0.462392\pi$
$758$	0	0
$759$	−29.7259	−1.07898
$760$	0	0
$761$	−46.2819	−1.67772	−0.838858	−	0.544350i	$-0.816776\pi$
$761$	−46.2819	−1.67772	−0.838858	+	0.544350i	$0.816776\pi$
$762$	0	0
$763$	3.45012	0.124903
$764$	0	0
$765$	−5.94621	−0.214986
$766$	0	0
$767$	33.2692	1.20128
$768$	0	0
$769$	−43.9980	−1.58661	−0.793305	−	0.608825i	$-0.791641\pi$
$769$	−43.9980	−1.58661	−0.793305	+	0.608825i	$0.791641\pi$
$770$	0	0
$771$	77.0022	2.77316
$772$	0	0
$773$	−46.9173	−1.68750	−0.843749	−	0.536738i	$-0.819656\pi$
$773$	−46.9173	−1.68750	−0.843749	+	0.536738i	$0.819656\pi$
$774$	0	0
$775$	−19.6932	−0.707401
$776$	0	0
$777$	5.96105	0.213852
$778$	0	0
$779$	−15.1052	−0.541198
$780$	0	0
$781$	36.2246	1.29622
$782$	0	0
$783$	−18.3021	−0.654063
$784$	0	0
$785$	−12.9995	−0.463972
$786$	0	0
$787$	13.2391	0.471924	0.235962	−	0.971762i	$-0.424176\pi$
$787$	13.2391	0.471924	0.235962	+	0.971762i	$0.424176\pi$
$788$	0	0
$789$	−17.4815	−0.622359
$790$	0	0
$791$	−8.15505	−0.289960
$792$	0	0
$793$	−8.52496	−0.302730
$794$	0	0
$795$	13.4079	0.475530
$796$	0	0
$797$	8.53741	0.302411	0.151205	−	0.988502i	$-0.451685\pi$
$797$	8.53741	0.302411	0.151205	+	0.988502i	$0.451685\pi$
$798$	0	0
$799$	0.778926	0.0275564
$800$	0	0
$801$	−15.7428	−0.556246
$802$	0	0
$803$	4.02216	0.141939
$804$	0	0
$805$	−3.53823	−0.124706
$806$	0	0
$807$	−57.8337	−2.03584
$808$	0	0
$809$	37.8270	1.32993	0.664964	−	0.746875i	$-0.268447\pi$
$809$	37.8270	1.32993	0.664964	+	0.746875i	$0.268447\pi$
$810$	0	0
$811$	39.9844	1.40404	0.702021	−	0.712156i	$-0.252281\pi$
$811$	39.9844	1.40404	0.702021	+	0.712156i	$0.252281\pi$
$812$	0	0
$813$	68.3799	2.39819
$814$	0	0
$815$	−3.68469	−0.129069
$816$	0	0
$817$	0.209449	0.00732770
$818$	0	0
$819$	13.9509	0.487484
$820$	0	0
$821$	45.0117	1.57092	0.785459	−	0.618913i	$-0.212427\pi$
$821$	45.0117	1.57092	0.785459	+	0.618913i	$0.212427\pi$
$822$	0	0
$823$	43.6830	1.52270	0.761348	−	0.648344i	$-0.224538\pi$
$823$	43.6830	1.52270	0.761348	+	0.648344i	$0.224538\pi$
$824$	0	0
$825$	−36.1537	−1.25871
$826$	0	0
$827$	7.70896	0.268067	0.134033	−	0.990977i	$-0.457207\pi$
$827$	7.70896	0.268067	0.134033	+	0.990977i	$0.457207\pi$
$828$	0	0
$829$	−17.2803	−0.600171	−0.300086	−	0.953912i	$-0.597015\pi$
$829$	−17.2803	−0.600171	−0.300086	+	0.953912i	$0.597015\pi$
$830$	0	0
$831$	46.3964	1.60947
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−32.5582	−1.12672
$836$	0	0
$837$	−26.2003	−0.905613
$838$	0	0
$839$	−19.9727	−0.689533	−0.344766	−	0.938689i	$-0.612042\pi$
$839$	−19.9727	−0.689533	−0.344766	+	0.938689i	$0.612042\pi$
$840$	0	0
$841$	−11.9886	−0.413398
$842$	0	0
$843$	70.4863	2.42768
$844$	0	0
$845$	4.95025	0.170294
$846$	0	0
$847$	−4.44233	−0.152640
$848$	0	0
$849$	73.2190	2.51287
$850$	0	0
$851$	−5.92641	−0.203155
$852$	0	0
$853$	29.5400	1.01143	0.505715	−	0.862701i	$-0.331229\pi$
$853$	29.5400	1.01143	0.505715	+	0.862701i	$0.331229\pi$
$854$	0	0
$855$	8.63667	0.295368
$856$	0	0
$857$	21.9361	0.749323	0.374661	−	0.927162i	$-0.377759\pi$
$857$	21.9361	0.749323	0.374661	+	0.927162i	$0.377759\pi$
$858$	0	0
$859$	36.7294	1.25319	0.626596	−	0.779344i	$-0.284448\pi$
$859$	36.7294	1.25319	0.626596	+	0.779344i	$0.284448\pi$
$860$	0	0
$861$	28.6863	0.977625
$862$	0	0
$863$	54.6780	1.86126	0.930631	−	0.365958i	$-0.119259\pi$
$863$	54.6780	1.86126	0.930631	+	0.365958i	$0.119259\pi$
$864$	0	0
$865$	−22.0733	−0.750515
$866$	0	0
$867$	−2.75839	−0.0936797
$868$	0	0
$869$	27.2431	0.924159
$870$	0	0
$871$	−7.11683	−0.241145
$872$	0	0
$873$	60.1780	2.03672
$874$	0	0
$875$	−10.7544	−0.363565
$876$	0	0
$877$	−35.5141	−1.19923	−0.599613	−	0.800290i	$-0.704679\pi$
$877$	−35.5141	−1.19923	−0.599613	+	0.800290i	$0.704679\pi$
$878$	0	0
$879$	10.7666	0.363147
$880$	0	0
$881$	34.6323	1.16679	0.583397	−	0.812187i	$-0.301723\pi$
$881$	34.6323	1.16679	0.583397	+	0.812187i	$0.301723\pi$
$882$	0	0
$883$	20.7500	0.698294	0.349147	−	0.937068i	$-0.386471\pi$
$883$	20.7500	0.698294	0.349147	+	0.937068i	$0.386471\pi$
$884$	0	0
$885$	−39.1142	−1.31481
$886$	0	0
$887$	40.2243	1.35060	0.675300	−	0.737543i	$-0.264014\pi$
$887$	40.2243	1.35060	0.675300	+	0.737543i	$0.264014\pi$
$888$	0	0
$889$	0.619018	0.0207612
$890$	0	0
$891$	6.23243	0.208794
$892$	0	0
$893$	−1.13136	−0.0378597
$894$	0	0
$895$	6.94126	0.232021
$896$	0	0
$897$	−22.8983	−0.764552
$898$	0	0
$899$	24.3527	0.812206
$900$	0	0
$901$	3.76742	0.125511
$902$	0	0
$903$	−0.397766	−0.0132368
$904$	0	0
$905$	−16.2619	−0.540564
$906$	0	0
$907$	19.0850	0.633706	0.316853	−	0.948475i	$-0.397374\pi$
$907$	19.0850	0.633706	0.316853	+	0.948475i	$0.397374\pi$
$908$	0	0
$909$	50.6372	1.67953
$910$	0	0
$911$	41.0238	1.35918	0.679590	−	0.733593i	$-0.262158\pi$
$911$	41.0238	1.35918	0.679590	+	0.733593i	$0.262158\pi$
$912$	0	0
$913$	−10.7190	−0.354747
$914$	0	0
$915$	10.0227	0.331341
$916$	0	0
$917$	8.37716	0.276638
$918$	0	0
$919$	−21.7218	−0.716536	−0.358268	−	0.933619i	$-0.616633\pi$
$919$	−21.7218	−0.716536	−0.358268	+	0.933619i	$0.616633\pi$
$920$	0	0
$921$	61.2029	2.01670
$922$	0	0
$923$	27.9043	0.918483
$924$	0	0
$925$	−7.20790	−0.236994
$926$	0	0
$927$	46.9212	1.54109
$928$	0	0
$929$	−21.1377	−0.693505	−0.346752	−	0.937957i	$-0.612716\pi$
$929$	−21.1377	−0.693505	−0.346752	+	0.937957i	$0.612716\pi$
$930$	0	0
$931$	−1.45247	−0.0476027
$932$	0	0
$933$	33.6934	1.10307
$934$	0	0
$935$	5.07012	0.165811
$936$	0	0
$937$	−58.4445	−1.90930	−0.954649	−	0.297733i	$-0.903770\pi$
$937$	−58.4445	−1.90930	−0.954649	+	0.297733i	$0.903770\pi$
$938$	0	0
$939$	−60.7440	−1.98231
$940$	0	0
$941$	−7.34959	−0.239590	−0.119795	−	0.992799i	$-0.538224\pi$
$941$	−7.34959	−0.239590	−0.119795	+	0.992799i	$0.538224\pi$
$942$	0	0
$943$	−28.5195	−0.928724
$944$	0	0
$945$	−5.72521	−0.186241
$946$	0	0
$947$	−4.88408	−0.158711	−0.0793557	−	0.996846i	$-0.525286\pi$
$947$	−4.88408	−0.158711	−0.0793557	+	0.996846i	$0.525286\pi$
$948$	0	0
$949$	3.09833	0.100576
$950$	0	0
$951$	−69.2882	−2.24682
$952$	0	0
$953$	−35.5542	−1.15171	−0.575857	−	0.817550i	$-0.695331\pi$
$953$	−35.5542	−1.15171	−0.575857	+	0.817550i	$0.695331\pi$
$954$	0	0
$955$	−17.7964	−0.575878
$956$	0	0
$957$	44.7077	1.44519
$958$	0	0
$959$	2.17259	0.0701566
$960$	0	0
$961$	3.86192	0.124578
$962$	0	0
$963$	51.4093	1.65664
$964$	0	0
$965$	−8.58308	−0.276299
$966$	0	0
$967$	−4.27193	−0.137376	−0.0686880	−	0.997638i	$-0.521881\pi$
$967$	−4.27193	−0.137376	−0.0686880	+	0.997638i	$0.521881\pi$
$968$	0	0
$969$	4.00646	0.128706
$970$	0	0
$971$	33.9390	1.08915	0.544577	−	0.838711i	$-0.316690\pi$
$971$	33.9390	1.08915	0.544577	+	0.838711i	$0.316690\pi$
$972$	0	0
$973$	1.75152	0.0561511
$974$	0	0
$975$	−27.8497	−0.891904
$976$	0	0
$977$	−0.163504	−0.00523096	−0.00261548	−	0.999997i	$-0.500833\pi$
$977$	−0.163504	−0.00523096	−0.00261548	+	0.999997i	$0.500833\pi$
$978$	0	0
$979$	13.4233	0.429012
$980$	0	0
$981$	−15.9006	−0.507666
$982$	0	0
$983$	1.06979	0.0341209	0.0170604	−	0.999854i	$-0.494569\pi$
$983$	1.06979	0.0341209	0.0170604	+	0.999854i	$0.494569\pi$
$984$	0	0
$985$	12.7331	0.405710
$986$	0	0
$987$	2.14858	0.0683901
$988$	0	0
$989$	0.395454	0.0125747
$990$	0	0
$991$	−46.4429	−1.47531	−0.737654	−	0.675179i	$-0.764066\pi$
$991$	−46.4429	−1.47531	−0.737654	+	0.675179i	$0.764066\pi$
$992$	0	0
$993$	72.3777	2.29684
$994$	0	0
$995$	13.0363	0.413277
$996$	0	0
$997$	−20.7666	−0.657686	−0.328843	−	0.944385i	$-0.606659\pi$
$997$	−20.7666	−0.657686	−0.328843	+	0.944385i	$0.606659\pi$
$998$	0	0
$999$	−9.58954	−0.303400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bv.1.2		6
4.3	odd	2		7616.2.a.cd.1.5		6
8.3	odd	2		3808.2.a.g.1.2	✓	6
8.5	even	2		3808.2.a.o.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.g.1.2	✓	6	8.3	odd	2
3808.2.a.o.1.5	yes	6	8.5	even	2
7616.2.a.bv.1.2		6	1.1	even	1	trivial
7616.2.a.cd.1.5		6	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75839 q^{3} -1.29021 q^{5} -1.00000 q^{7} +4.60870 q^{9} -3.92967 q^{11} -3.02708 q^{13} +3.55891 q^{15} +1.00000 q^{17} -1.45247 q^{19} +2.75839 q^{21} -2.74235 q^{23} -3.33535 q^{25} -4.43741 q^{27} +4.12449 q^{29} +5.90440 q^{31} +10.8396 q^{33} +1.29021 q^{35} +2.16107 q^{37} +8.34987 q^{39} +10.3997 q^{41} -0.144202 q^{43} -5.94621 q^{45} +0.778926 q^{47} +1.00000 q^{49} -2.75839 q^{51} +3.76742 q^{53} +5.07012 q^{55} +4.00646 q^{57} -10.9905 q^{59} +2.81623 q^{61} -4.60870 q^{63} +3.90559 q^{65} +2.35105 q^{67} +7.56447 q^{69} -9.21823 q^{71} -1.02354 q^{73} +9.20018 q^{75} +3.92967 q^{77} -6.93267 q^{79} -1.58599 q^{81} +2.72771 q^{83} -1.29021 q^{85} -11.3769 q^{87} -3.41589 q^{89} +3.02708 q^{91} -16.2866 q^{93} +1.87399 q^{95} +13.0575 q^{97} -18.1107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 4 q^{13} + 6 q^{17} - 18 q^{19} + 4 q^{21} + 6 q^{23} + 12 q^{25} - 10 q^{27} - 4 q^{29} - 8 q^{31} - 6 q^{33} - 4 q^{35} + 8 q^{37} - 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 4 * q^13 + 6 * q^17 - 18 * q^19 + 4 * q^21 + 6 * q^23 + 12 * q^25 - 10 * q^27 - 4 * q^29 - 8 * q^31 - 6 * q^33 - 4 * q^35 + 8 * q^37 - 18 * q^39 - 4 * q^41 - 20 * q^43 + 24 * q^45 + 6 * q^47 + 6 * q^49 - 4 * q^51 - 14 * q^53 + 4 * q^55 - 4 * q^57 - 20 * q^59 - 2 * q^61 - 8 * q^63 + 6 * q^65 - 28 * q^67 + 4 * q^69 + 18 * q^71 + 4 * q^73 - 20 * q^75 + 8 * q^77 - 2 * q^79 + 34 * q^81 - 20 * q^83 + 4 * q^85 + 22 * q^87 - 6 * q^89 - 4 * q^91 - 40 * q^93 - 6 * q^95 + 30 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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