LMFDB - Embedded newform 7616.2.a.bo.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:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.546295$ 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)

$f(q)$	$=$	$q+0.706585 q^{3} -4.40815 q^{5} -1.00000 q^{7} -2.50074 q^{9} +1.09259 q^{11} -2.00000 q^{13} -3.11473 q^{15} +1.00000 q^{17} -0.706585 q^{21} -1.41317 q^{23} +14.4318 q^{25} -3.88674 q^{27} +5.32206 q^{29} +10.1319 q^{31} +0.772008 q^{33} +4.40815 q^{35} -3.09259 q^{37} -1.41317 q^{39} +11.2023 q^{41} -3.21235 q^{43} +11.0236 q^{45} +11.0458 q^{47} +1.00000 q^{49} +0.706585 q^{51} +6.31703 q^{53} -4.81630 q^{55} -2.18518 q^{59} -1.29341 q^{61} +2.50074 q^{63} +8.81630 q^{65} +2.08757 q^{67} -0.998526 q^{69} -11.8178 q^{71} -7.84346 q^{73} +10.1973 q^{75} -1.09259 q^{77} -11.0458 q^{79} +4.75590 q^{81} -13.8721 q^{83} -4.40815 q^{85} +3.76049 q^{87} -7.82782 q^{89} +2.00000 q^{91} +7.15902 q^{93} +12.9039 q^{97} -2.73228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} - 8 q^{13} + 4 q^{17} - 3 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 8 q^{29} + 7 q^{31} - 6 q^{33} + 5 q^{35} - 8 q^{37} - 6 q^{39} + 15 q^{41} - 9 q^{43} + 2 q^{45}+ \cdots - 50 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 - 8 * q^13 + 4 * q^17 - 3 * q^21 - 6 * q^23 + 3 * q^25 + 6 * q^27 - 8 * q^29 + 7 * q^31 - 6 * q^33 + 5 * q^35 - 8 * q^37 - 6 * q^39 + 15 * q^41 - 9 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 3 * q^51 - 17 * q^53 + 6 * q^55 - 5 * q^61 - 7 * q^63 + 10 * q^65 - 9 * q^67 - 38 * q^69 + 12 * q^71 - 11 * q^73 - 2 * q^75 + 6 * q^79 + 16 * q^81 - 6 * q^83 - 5 * q^85 + 14 * q^87 + 2 * q^89 + 8 * q^91 + 9 * q^97 - 50 * 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
$3$	0.706585	0.407947	0.203974	−	0.978976i	$-0.434614\pi$
$3$	0.706585	0.407947	0.203974	+	0.978976i	$0.434614\pi$
$4$	0	0
$5$	−4.40815	−1.97138	−0.985692	−	0.168558i	$-0.946089\pi$
$5$	−4.40815	−1.97138	−0.985692	+	0.168558i	$0.946089\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.50074	−0.833579
$10$	0	0
$11$	1.09259	0.329428	0.164714	−	0.986341i	$-0.447330\pi$
$11$	1.09259	0.329428	0.164714	+	0.986341i	$0.447330\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−3.11473	−0.804221
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−0.706585	−0.154190
$22$	0	0
$23$	−1.41317	−0.294667	−0.147333	−	0.989087i	$-0.547069\pi$
$23$	−1.41317	−0.294667	−0.147333	+	0.989087i	$0.547069\pi$
$24$	0	0
$25$	14.4318	2.88635
$26$	0	0
$27$	−3.88674	−0.748004
$28$	0	0
$29$	5.32206	0.988281	0.494140	−	0.869382i	$-0.335483\pi$
$29$	5.32206	0.988281	0.494140	+	0.869382i	$0.335483\pi$
$30$	0	0
$31$	10.1319	1.81973	0.909867	−	0.414899i	$-0.136183\pi$
$31$	10.1319	1.81973	0.909867	+	0.414899i	$0.136183\pi$
$32$	0	0
$33$	0.772008	0.134389
$34$	0	0
$35$	4.40815	0.745113
$36$	0	0
$37$	−3.09259	−0.508419	−0.254209	−	0.967149i	$-0.581815\pi$
$37$	−3.09259	−0.508419	−0.254209	+	0.967149i	$0.581815\pi$
$38$	0	0
$39$	−1.41317	−0.226288
$40$	0	0
$41$	11.2023	1.74951	0.874753	−	0.484570i	$-0.161024\pi$
$41$	11.2023	1.74951	0.874753	+	0.484570i	$0.161024\pi$
$42$	0	0
$43$	−3.21235	−0.489878	−0.244939	−	0.969538i	$-0.578768\pi$
$43$	−3.21235	−0.489878	−0.244939	+	0.969538i	$0.578768\pi$
$44$	0	0
$45$	11.0236	1.64330
$46$	0	0
$47$	11.0458	1.61119	0.805595	−	0.592467i	$-0.201846\pi$
$47$	11.0458	1.61119	0.805595	+	0.592467i	$0.201846\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.706585	0.0989418
$52$	0	0
$53$	6.31703	0.867711	0.433856	−	0.900982i	$-0.357153\pi$
$53$	6.31703	0.867711	0.433856	+	0.900982i	$0.357153\pi$
$54$	0	0
$55$	−4.81630	−0.649429
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.18518	−0.284486	−0.142243	−	0.989832i	$-0.545431\pi$
$59$	−2.18518	−0.284486	−0.142243	+	0.989832i	$0.545431\pi$
$60$	0	0
$61$	−1.29341	−0.165605	−0.0828024	−	0.996566i	$-0.526387\pi$
$61$	−1.29341	−0.165605	−0.0828024	+	0.996566i	$0.526387\pi$
$62$	0	0
$63$	2.50074	0.315063
$64$	0	0
$65$	8.81630	1.09353
$66$	0	0
$67$	2.08757	0.255037	0.127518	−	0.991836i	$-0.459299\pi$
$67$	2.08757	0.255037	0.127518	+	0.991836i	$0.459299\pi$
$68$	0	0
$69$	−0.998526	−0.120208
$70$	0	0
$71$	−11.8178	−1.40251	−0.701256	−	0.712910i	$-0.747377\pi$
$71$	−11.8178	−1.40251	−0.701256	+	0.712910i	$0.747377\pi$
$72$	0	0
$73$	−7.84346	−0.918008	−0.459004	−	0.888434i	$-0.651794\pi$
$73$	−7.84346	−0.918008	−0.459004	+	0.888434i	$0.651794\pi$
$74$	0	0
$75$	10.1973	1.17748
$76$	0	0
$77$	−1.09259	−0.124512
$78$	0	0
$79$	−11.0458	−1.24275	−0.621373	−	0.783515i	$-0.713425\pi$
$79$	−11.0458	−1.24275	−0.621373	+	0.783515i	$0.713425\pi$
$80$	0	0
$81$	4.75590	0.528433
$82$	0	0
$83$	−13.8721	−1.52266	−0.761331	−	0.648364i	$-0.775454\pi$
$83$	−13.8721	−1.52266	−0.761331	+	0.648364i	$0.775454\pi$
$84$	0	0
$85$	−4.40815	−0.478131
$86$	0	0
$87$	3.76049	0.403167
$88$	0	0
$89$	−7.82782	−0.829747	−0.414873	−	0.909879i	$-0.636174\pi$
$89$	−7.82782	−0.829747	−0.414873	+	0.909879i	$0.636174\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	7.15902	0.742356
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.9039	1.31019	0.655094	−	0.755547i	$-0.272629\pi$
$97$	12.9039	1.31019	0.655094	+	0.755547i	$0.272629\pi$
$98$	0	0
$99$	−2.73228	−0.274604
$100$	0	0
$101$	−8.22947	−0.818862	−0.409431	−	0.912341i	$-0.634273\pi$
$101$	−8.22947	−0.818862	−0.409431	+	0.912341i	$0.634273\pi$
$102$	0	0
$103$	−14.0342	−1.38283	−0.691417	−	0.722456i	$-0.743013\pi$
$103$	−14.0342	−1.38283	−0.691417	+	0.722456i	$0.743013\pi$
$104$	0	0
$105$	3.11473	0.303967
$106$	0	0
$107$	−10.7252	−1.03684	−0.518421	−	0.855125i	$-0.673480\pi$
$107$	−10.7252	−1.03684	−0.518421	+	0.855125i	$0.673480\pi$
$108$	0	0
$109$	−7.13688	−0.683589	−0.341794	−	0.939775i	$-0.611035\pi$
$109$	−7.13688	−0.683589	−0.341794	+	0.939775i	$0.611035\pi$
$110$	0	0
$111$	−2.18518	−0.207408
$112$	0	0
$113$	−0.542542	−0.0510380	−0.0255190	−	0.999674i	$-0.508124\pi$
$113$	−0.542542	−0.0510380	−0.0255190	+	0.999674i	$0.508124\pi$
$114$	0	0
$115$	6.22947	0.580901
$116$	0	0
$117$	5.00147	0.462386
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−9.80625	−0.891477
$122$	0	0
$123$	7.91538	0.713706
$124$	0	0
$125$	−41.5766	−3.71873
$126$	0	0
$127$	−0.718682	−0.0637728	−0.0318864	−	0.999492i	$-0.510151\pi$
$127$	−0.718682	−0.0637728	−0.0318864	+	0.999492i	$0.510151\pi$
$128$	0	0
$129$	−2.26980	−0.199844
$130$	0	0
$131$	8.49571	0.742274	0.371137	−	0.928578i	$-0.378968\pi$
$131$	8.49571	0.742274	0.371137	+	0.928578i	$0.378968\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	17.1333	1.47460
$136$	0	0
$137$	−9.83342	−0.840125	−0.420063	−	0.907495i	$-0.637992\pi$
$137$	−9.83342	−0.840125	−0.420063	+	0.907495i	$0.637992\pi$
$138$	0	0
$139$	−19.2687	−1.63435	−0.817176	−	0.576388i	$-0.804462\pi$
$139$	−19.2687	−1.63435	−0.817176	+	0.576388i	$0.804462\pi$
$140$	0	0
$141$	7.80477	0.657281
$142$	0	0
$143$	−2.18518	−0.182734
$144$	0	0
$145$	−23.4604	−1.94828
$146$	0	0
$147$	0.706585	0.0582782
$148$	0	0
$149$	5.83342	0.477892	0.238946	−	0.971033i	$-0.423198\pi$
$149$	5.83342	0.477892	0.238946	+	0.971033i	$0.423198\pi$
$150$	0	0
$151$	21.2153	1.72648	0.863238	−	0.504797i	$-0.168433\pi$
$151$	21.2153	1.72648	0.863238	+	0.504797i	$0.168433\pi$
$152$	0	0
$153$	−2.50074	−0.202173
$154$	0	0
$155$	−44.6627	−3.58739
$156$	0	0
$157$	14.4146	1.15041	0.575207	−	0.818008i	$-0.304922\pi$
$157$	14.4146	1.15041	0.575207	+	0.818008i	$0.304922\pi$
$158$	0	0
$159$	4.46352	0.353980
$160$	0	0
$161$	1.41317	0.111373
$162$	0	0
$163$	14.7695	1.15683	0.578417	−	0.815741i	$-0.303671\pi$
$163$	14.7695	1.15683	0.578417	+	0.815741i	$0.303671\pi$
$164$	0	0
$165$	−3.40312	−0.264933
$166$	0	0
$167$	12.8006	0.990544	0.495272	−	0.868738i	$-0.335068\pi$
$167$	12.8006	0.990544	0.495272	+	0.868738i	$0.335068\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.1112	−0.996825	−0.498412	−	0.866940i	$-0.666083\pi$
$173$	−13.1112	−0.996825	−0.498412	+	0.866940i	$0.666083\pi$
$174$	0	0
$175$	−14.4318	−1.09094
$176$	0	0
$177$	−1.54402	−0.116055
$178$	0	0
$179$	−1.79917	−0.134477	−0.0672383	−	0.997737i	$-0.521419\pi$
$179$	−1.79917	−0.134477	−0.0672383	+	0.997737i	$0.521419\pi$
$180$	0	0
$181$	1.48419	0.110319	0.0551596	−	0.998478i	$-0.482433\pi$
$181$	1.48419	0.110319	0.0551596	+	0.998478i	$0.482433\pi$
$182$	0	0
$183$	−0.913908	−0.0675580
$184$	0	0
$185$	13.6326	1.00229
$186$	0	0
$187$	1.09259	0.0798980
$188$	0	0
$189$	3.88674	0.282719
$190$	0	0
$191$	−3.09909	−0.224242	−0.112121	−	0.993695i	$-0.535764\pi$
$191$	−3.09909	−0.224242	−0.112121	+	0.993695i	$0.535764\pi$
$192$	0	0
$193$	−14.9914	−1.07911	−0.539553	−	0.841951i	$-0.681407\pi$
$193$	−14.9914	−1.07911	−0.539553	+	0.841951i	$0.681407\pi$
$194$	0	0
$195$	6.22947	0.446101
$196$	0	0
$197$	2.08107	0.148270	0.0741350	−	0.997248i	$-0.476380\pi$
$197$	2.08107	0.148270	0.0741350	+	0.997248i	$0.476380\pi$
$198$	0	0
$199$	15.7891	1.11926	0.559631	−	0.828742i	$-0.310943\pi$
$199$	15.7891	1.11926	0.559631	+	0.828742i	$0.310943\pi$
$200$	0	0
$201$	1.47504	0.104042
$202$	0	0
$203$	−5.32206	−0.373535
$204$	0	0
$205$	−49.3814	−3.44895
$206$	0	0
$207$	3.53397	0.245628
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.7352	−1.15210	−0.576050	−	0.817414i	$-0.695407\pi$
$211$	−16.7352	−1.15210	−0.576050	+	0.817414i	$0.695407\pi$
$212$	0	0
$213$	−8.35026	−0.572151
$214$	0	0
$215$	14.1605	0.965738
$216$	0	0
$217$	−10.1319	−0.687795
$218$	0	0
$219$	−5.54208	−0.374499
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	23.4474	1.57015	0.785077	−	0.619398i	$-0.212623\pi$
$223$	23.4474	1.57015	0.785077	+	0.619398i	$0.212623\pi$
$224$	0	0
$225$	−36.0900	−2.40600
$226$	0	0
$227$	21.6491	1.43690	0.718452	−	0.695577i	$-0.244851\pi$
$227$	21.6491	1.43690	0.718452	+	0.695577i	$0.244851\pi$
$228$	0	0
$229$	−2.64116	−0.174533	−0.0872665	−	0.996185i	$-0.527813\pi$
$229$	−2.64116	−0.174533	−0.0872665	+	0.996185i	$0.527813\pi$
$230$	0	0
$231$	−0.772008	−0.0507944
$232$	0	0
$233$	13.9799	0.915854	0.457927	−	0.888990i	$-0.348592\pi$
$233$	13.9799	0.915854	0.457927	+	0.888990i	$0.348592\pi$
$234$	0	0
$235$	−48.6913	−3.17627
$236$	0	0
$237$	−7.80477	−0.506975
$238$	0	0
$239$	−0.146491	−0.00947572	−0.00473786	−	0.999989i	$-0.501508\pi$
$239$	−0.146491	−0.00947572	−0.00473786	+	0.999989i	$0.501508\pi$
$240$	0	0
$241$	−14.2727	−0.919388	−0.459694	−	0.888077i	$-0.652041\pi$
$241$	−14.2727	−0.919388	−0.459694	+	0.888077i	$0.652041\pi$
$242$	0	0
$243$	15.0207	0.963576
$244$	0	0
$245$	−4.40815	−0.281626
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−9.80183	−0.621166
$250$	0	0
$251$	27.9063	1.76143	0.880716	−	0.473644i	$-0.157062\pi$
$251$	27.9063	1.76143	0.880716	+	0.473644i	$0.157062\pi$
$252$	0	0
$253$	−1.54402	−0.0970714
$254$	0	0
$255$	−3.11473	−0.195052
$256$	0	0
$257$	27.6899	1.72725	0.863623	−	0.504138i	$-0.168190\pi$
$257$	27.6899	1.72725	0.863623	+	0.504138i	$0.168190\pi$
$258$	0	0
$259$	3.09259	0.192164
$260$	0	0
$261$	−13.3091	−0.823810
$262$	0	0
$263$	−4.53249	−0.279486	−0.139743	−	0.990188i	$-0.544628\pi$
$263$	−4.53249	−0.279486	−0.139743	+	0.990188i	$0.544628\pi$
$264$	0	0
$265$	−27.8464	−1.71059
$266$	0	0
$267$	−5.53102	−0.338493
$268$	0	0
$269$	26.1513	1.59448	0.797238	−	0.603665i	$-0.206294\pi$
$269$	26.1513	1.59448	0.797238	+	0.603665i	$0.206294\pi$
$270$	0	0
$271$	−20.5898	−1.25074	−0.625370	−	0.780328i	$-0.715052\pi$
$271$	−20.5898	−1.25074	−0.625370	+	0.780328i	$0.715052\pi$
$272$	0	0
$273$	1.41317	0.0855290
$274$	0	0
$275$	15.7680	0.950846
$276$	0	0
$277$	6.32058	0.379767	0.189883	−	0.981807i	$-0.439189\pi$
$277$	6.32058	0.379767	0.189883	+	0.981807i	$0.439189\pi$
$278$	0	0
$279$	−25.3371	−1.51689
$280$	0	0
$281$	−12.6974	−0.757466	−0.378733	−	0.925506i	$-0.623640\pi$
$281$	−12.6974	−0.757466	−0.378733	+	0.925506i	$0.623640\pi$
$282$	0	0
$283$	3.18016	0.189041	0.0945203	−	0.995523i	$-0.469868\pi$
$283$	3.18016	0.189041	0.0945203	+	0.995523i	$0.469868\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.2023	−0.661251
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	9.11768	0.534488
$292$	0	0
$293$	13.4161	0.783778	0.391889	−	0.920012i	$-0.371822\pi$
$293$	13.4161	0.783778	0.391889	+	0.920012i	$0.371822\pi$
$294$	0	0
$295$	9.63259	0.560831
$296$	0	0
$297$	−4.24661	−0.246413
$298$	0	0
$299$	2.82634	0.163452
$300$	0	0
$301$	3.21235	0.185157
$302$	0	0
$303$	−5.81482	−0.334053
$304$	0	0
$305$	5.70156	0.326470
$306$	0	0
$307$	2.54254	0.145111	0.0725553	−	0.997364i	$-0.476885\pi$
$307$	2.54254	0.145111	0.0725553	+	0.997364i	$0.476885\pi$
$308$	0	0
$309$	−9.91639	−0.564124
$310$	0	0
$311$	−16.4993	−0.935587	−0.467794	−	0.883838i	$-0.654951\pi$
$311$	−16.4993	−0.935587	−0.467794	+	0.883838i	$0.654951\pi$
$312$	0	0
$313$	−9.09909	−0.514311	−0.257155	−	0.966370i	$-0.582785\pi$
$313$	−9.09909	−0.514311	−0.257155	+	0.966370i	$0.582785\pi$
$314$	0	0
$315$	−11.0236	−0.621110
$316$	0	0
$317$	24.3678	1.36863	0.684316	−	0.729185i	$-0.260101\pi$
$317$	24.3678	1.36863	0.684316	+	0.729185i	$0.260101\pi$
$318$	0	0
$319$	5.81482	0.325567
$320$	0	0
$321$	−7.57826	−0.422977
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−28.8635	−1.60106
$326$	0	0
$327$	−5.04281	−0.278868
$328$	0	0
$329$	−11.0458	−0.608973
$330$	0	0
$331$	−10.0876	−0.554463	−0.277231	−	0.960803i	$-0.589417\pi$
$331$	−10.0876	−0.554463	−0.277231	+	0.960803i	$0.589417\pi$
$332$	0	0
$333$	7.73375	0.423807
$334$	0	0
$335$	−9.20230	−0.502775
$336$	0	0
$337$	−10.5969	−0.577249	−0.288624	−	0.957442i	$-0.593198\pi$
$337$	−10.5969	−0.577249	−0.288624	+	0.957442i	$0.593198\pi$
$338$	0	0
$339$	−0.383352	−0.0208208
$340$	0	0
$341$	11.0700	0.599472
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.40165	0.236977
$346$	0	0
$347$	6.90741	0.370809	0.185405	−	0.982662i	$-0.440640\pi$
$347$	6.90741	0.370809	0.185405	+	0.982662i	$0.440640\pi$
$348$	0	0
$349$	1.68692	0.0902990	0.0451495	−	0.998980i	$-0.485624\pi$
$349$	1.68692	0.0902990	0.0451495	+	0.998980i	$0.485624\pi$
$350$	0	0
$351$	7.77348	0.414918
$352$	0	0
$353$	−22.4388	−1.19430	−0.597149	−	0.802130i	$-0.703700\pi$
$353$	−22.4388	−1.19430	−0.597149	+	0.802130i	$0.703700\pi$
$354$	0	0
$355$	52.0945	2.76489
$356$	0	0
$357$	−0.706585	−0.0373965
$358$	0	0
$359$	3.81381	0.201285	0.100643	−	0.994923i	$-0.467910\pi$
$359$	3.81381	0.201285	0.100643	+	0.994923i	$0.467910\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−6.92895	−0.363676
$364$	0	0
$365$	34.5751	1.80975
$366$	0	0
$367$	1.78618	0.0932379	0.0466189	−	0.998913i	$-0.485155\pi$
$367$	1.78618	0.0932379	0.0466189	+	0.998913i	$0.485155\pi$
$368$	0	0
$369$	−28.0140	−1.45835
$370$	0	0
$371$	−6.31703	−0.327964
$372$	0	0
$373$	−11.1893	−0.579360	−0.289680	−	0.957124i	$-0.593549\pi$
$373$	−11.1893	−0.579360	−0.289680	+	0.957124i	$0.593549\pi$
$374$	0	0
$375$	−29.3774	−1.51704
$376$	0	0
$377$	−10.6441	−0.548200
$378$	0	0
$379$	−13.6251	−0.699874	−0.349937	−	0.936773i	$-0.613797\pi$
$379$	−13.6251	−0.699874	−0.349937	+	0.936773i	$0.613797\pi$
$380$	0	0
$381$	−0.507811	−0.0260159
$382$	0	0
$383$	−30.2295	−1.54465	−0.772327	−	0.635225i	$-0.780907\pi$
$383$	−30.2295	−1.54465	−0.772327	+	0.635225i	$0.780907\pi$
$384$	0	0
$385$	4.81630	0.245461
$386$	0	0
$387$	8.03323	0.408352
$388$	0	0
$389$	−29.1363	−1.47727	−0.738634	−	0.674107i	$-0.764529\pi$
$389$	−29.1363	−1.47727	−0.738634	+	0.674107i	$0.764529\pi$
$390$	0	0
$391$	−1.41317	−0.0714671
$392$	0	0
$393$	6.00295	0.302809
$394$	0	0
$395$	48.6913	2.44993
$396$	0	0
$397$	−29.8786	−1.49956	−0.749782	−	0.661685i	$-0.769842\pi$
$397$	−29.8786	−1.49956	−0.749782	+	0.661685i	$0.769842\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.38188	0.168883	0.0844415	−	0.996428i	$-0.473089\pi$
$401$	3.38188	0.168883	0.0844415	+	0.996428i	$0.473089\pi$
$402$	0	0
$403$	−20.2637	−1.00941
$404$	0	0
$405$	−20.9647	−1.04174
$406$	0	0
$407$	−3.37893	−0.167487
$408$	0	0
$409$	−13.8048	−0.682602	−0.341301	−	0.939954i	$-0.610868\pi$
$409$	−13.8048	−0.682602	−0.341301	+	0.939954i	$0.610868\pi$
$410$	0	0
$411$	−6.94815	−0.342727
$412$	0	0
$413$	2.18518	0.107526
$414$	0	0
$415$	61.1503	3.00175
$416$	0	0
$417$	−13.6150	−0.666730
$418$	0	0
$419$	−0.637614	−0.0311495	−0.0155747	−	0.999879i	$-0.504958\pi$
$419$	−0.637614	−0.0311495	−0.0155747	+	0.999879i	$0.504958\pi$
$420$	0	0
$421$	−10.8650	−0.529529	−0.264765	−	0.964313i	$-0.585294\pi$
$421$	−10.8650	−0.529529	−0.264765	+	0.964313i	$0.585294\pi$
$422$	0	0
$423$	−27.6225	−1.34305
$424$	0	0
$425$	14.4318	0.700043
$426$	0	0
$427$	1.29341	0.0625927
$428$	0	0
$429$	−1.54402	−0.0745458
$430$	0	0
$431$	−38.4489	−1.85202	−0.926009	−	0.377502i	$-0.876783\pi$
$431$	−38.4489	−1.85202	−0.926009	+	0.377502i	$0.876783\pi$
$432$	0	0
$433$	−9.11932	−0.438247	−0.219123	−	0.975697i	$-0.570320\pi$
$433$	−9.11932	−0.438247	−0.219123	+	0.975697i	$0.570320\pi$
$434$	0	0
$435$	−16.5768	−0.794796
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−4.63851	−0.221384	−0.110692	−	0.993855i	$-0.535307\pi$
$439$	−4.63851	−0.221384	−0.110692	+	0.993855i	$0.535307\pi$
$440$	0	0
$441$	−2.50074	−0.119083
$442$	0	0
$443$	23.8178	1.13162	0.565808	−	0.824537i	$-0.308564\pi$
$443$	23.8178	1.13162	0.565808	+	0.824537i	$0.308564\pi$
$444$	0	0
$445$	34.5062	1.63575
$446$	0	0
$447$	4.12181	0.194955
$448$	0	0
$449$	9.61134	0.453587	0.226794	−	0.973943i	$-0.427176\pi$
$449$	9.61134	0.453587	0.226794	+	0.973943i	$0.427176\pi$
$450$	0	0
$451$	12.2395	0.576336
$452$	0	0
$453$	14.9904	0.704311
$454$	0	0
$455$	−8.81630	−0.413314
$456$	0	0
$457$	−29.2249	−1.36708	−0.683540	−	0.729913i	$-0.739561\pi$
$457$	−29.2249	−1.36708	−0.683540	+	0.729913i	$0.739561\pi$
$458$	0	0
$459$	−3.88674	−0.181418
$460$	0	0
$461$	−16.4247	−0.764974	−0.382487	−	0.923961i	$-0.624932\pi$
$461$	−16.4247	−0.764974	−0.382487	+	0.923961i	$0.624932\pi$
$462$	0	0
$463$	8.04164	0.373726	0.186863	−	0.982386i	$-0.440168\pi$
$463$	8.04164	0.373726	0.186863	+	0.982386i	$0.440168\pi$
$464$	0	0
$465$	−31.5580	−1.46347
$466$	0	0
$467$	11.0216	0.510017	0.255009	−	0.966939i	$-0.417922\pi$
$467$	11.0216	0.510017	0.255009	+	0.966939i	$0.417922\pi$
$468$	0	0
$469$	−2.08757	−0.0963948
$470$	0	0
$471$	10.1852	0.469308
$472$	0	0
$473$	−3.50977	−0.161380
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−15.7972	−0.723306
$478$	0	0
$479$	22.6527	1.03503	0.517513	−	0.855675i	$-0.326858\pi$
$479$	22.6527	1.03503	0.517513	+	0.855675i	$0.326858\pi$
$480$	0	0
$481$	6.18518	0.282020
$482$	0	0
$483$	0.998526	0.0454345
$484$	0	0
$485$	−56.8821	−2.58288
$486$	0	0
$487$	−21.2551	−0.963162	−0.481581	−	0.876402i	$-0.659937\pi$
$487$	−21.2551	−0.963162	−0.481581	+	0.876402i	$0.659937\pi$
$488$	0	0
$489$	10.4359	0.471927
$490$	0	0
$491$	−36.9068	−1.66558	−0.832790	−	0.553588i	$-0.813258\pi$
$491$	−36.9068	−1.66558	−0.832790	+	0.553588i	$0.813258\pi$
$492$	0	0
$493$	5.32206	0.239693
$494$	0	0
$495$	12.0443	0.541350
$496$	0	0
$497$	11.8178	0.530099
$498$	0	0
$499$	34.7825	1.55708	0.778538	−	0.627597i	$-0.215961\pi$
$499$	34.7825	1.55708	0.778538	+	0.627597i	$0.215961\pi$
$500$	0	0
$501$	9.04475	0.404090
$502$	0	0
$503$	−21.7544	−0.969981	−0.484990	−	0.874520i	$-0.661177\pi$
$503$	−21.7544	−0.969981	−0.484990	+	0.874520i	$0.661177\pi$
$504$	0	0
$505$	36.2767	1.61429
$506$	0	0
$507$	−6.35927	−0.282425
$508$	0	0
$509$	−26.5032	−1.17473	−0.587367	−	0.809321i	$-0.699836\pi$
$509$	−26.5032	−1.17473	−0.587367	+	0.809321i	$0.699836\pi$
$510$	0	0
$511$	7.84346	0.346974
$512$	0	0
$513$	0	0
$514$	0	0
$515$	61.8650	2.72610
$516$	0	0
$517$	12.0685	0.530771
$518$	0	0
$519$	−9.26417	−0.406652
$520$	0	0
$521$	10.9093	0.477946	0.238973	−	0.971026i	$-0.423189\pi$
$521$	10.9093	0.477946	0.238973	+	0.971026i	$0.423189\pi$
$522$	0	0
$523$	−12.6784	−0.554386	−0.277193	−	0.960814i	$-0.589404\pi$
$523$	−12.6784	−0.554386	−0.277193	+	0.960814i	$0.589404\pi$
$524$	0	0
$525$	−10.1973	−0.445046
$526$	0	0
$527$	10.1319	0.441350
$528$	0	0
$529$	−21.0029	−0.913172
$530$	0	0
$531$	5.46456	0.237142
$532$	0	0
$533$	−22.4046	−0.970451
$534$	0	0
$535$	47.2782	2.04401
$536$	0	0
$537$	−1.27127	−0.0548594
$538$	0	0
$539$	1.09259	0.0470612
$540$	0	0
$541$	−42.7594	−1.83837	−0.919186	−	0.393825i	$-0.871152\pi$
$541$	−42.7594	−1.83837	−0.919186	+	0.393825i	$0.871152\pi$
$542$	0	0
$543$	1.04871	0.0450044
$544$	0	0
$545$	31.4604	1.34762
$546$	0	0
$547$	2.70099	0.115486	0.0577429	−	0.998331i	$-0.481610\pi$
$547$	2.70099	0.115486	0.0577429	+	0.998331i	$0.481610\pi$
$548$	0	0
$549$	3.23449	0.138045
$550$	0	0
$551$	0	0
$552$	0	0
$553$	11.0458	0.469714
$554$	0	0
$555$	9.63259	0.408881
$556$	0	0
$557$	−13.4474	−0.569785	−0.284893	−	0.958559i	$-0.591958\pi$
$557$	−13.4474	−0.569785	−0.284893	+	0.958559i	$0.591958\pi$
$558$	0	0
$559$	6.42469	0.271736
$560$	0	0
$561$	0.772008	0.0325942
$562$	0	0
$563$	1.93267	0.0814524	0.0407262	−	0.999170i	$-0.487033\pi$
$563$	1.93267	0.0814524	0.0407262	+	0.999170i	$0.487033\pi$
$564$	0	0
$565$	2.39160	0.100616
$566$	0	0
$567$	−4.75590	−0.199729
$568$	0	0
$569$	−21.3745	−0.896065	−0.448032	−	0.894017i	$-0.647875\pi$
$569$	−21.3745	−0.896065	−0.448032	+	0.894017i	$0.647875\pi$
$570$	0	0
$571$	−25.6694	−1.07423	−0.537115	−	0.843509i	$-0.680486\pi$
$571$	−25.6694	−1.07423	−0.537115	+	0.843509i	$0.680486\pi$
$572$	0	0
$573$	−2.18977	−0.0914789
$574$	0	0
$575$	−20.3946	−0.850512
$576$	0	0
$577$	10.2525	0.426817	0.213409	−	0.976963i	$-0.431543\pi$
$577$	10.2525	0.426817	0.213409	+	0.976963i	$0.431543\pi$
$578$	0	0
$579$	−10.5927	−0.440219
$580$	0	0
$581$	13.8721	0.575512
$582$	0	0
$583$	6.90192	0.285848
$584$	0	0
$585$	−22.0472	−0.911541
$586$	0	0
$587$	39.9193	1.64765	0.823824	−	0.566846i	$-0.191837\pi$
$587$	39.9193	1.64765	0.823824	+	0.566846i	$0.191837\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	1.47045	0.0604864
$592$	0	0
$593$	4.57383	0.187825	0.0939124	−	0.995580i	$-0.470063\pi$
$593$	4.57383	0.187825	0.0939124	+	0.995580i	$0.470063\pi$
$594$	0	0
$595$	4.40815	0.180716
$596$	0	0
$597$	11.1564	0.456600
$598$	0	0
$599$	7.26963	0.297029	0.148514	−	0.988910i	$-0.452551\pi$
$599$	7.26963	0.297029	0.148514	+	0.988910i	$0.452551\pi$
$600$	0	0
$601$	5.37893	0.219411	0.109706	−	0.993964i	$-0.465009\pi$
$601$	5.37893	0.219411	0.109706	+	0.993964i	$0.465009\pi$
$602$	0	0
$603$	−5.22045	−0.212593
$604$	0	0
$605$	43.2274	1.75744
$606$	0	0
$607$	11.0418	0.448173	0.224087	−	0.974569i	$-0.428060\pi$
$607$	11.0418	0.448173	0.224087	+	0.974569i	$0.428060\pi$
$608$	0	0
$609$	−3.76049	−0.152383
$610$	0	0
$611$	−22.0915	−0.893727
$612$	0	0
$613$	−18.6744	−0.754252	−0.377126	−	0.926162i	$-0.623088\pi$
$613$	−18.6744	−0.754252	−0.377126	+	0.926162i	$0.623088\pi$
$614$	0	0
$615$	−34.8922	−1.40699
$616$	0	0
$617$	35.2421	1.41880	0.709398	−	0.704809i	$-0.248967\pi$
$617$	35.2421	1.41880	0.709398	+	0.704809i	$0.248967\pi$
$618$	0	0
$619$	−14.2220	−0.571629	−0.285814	−	0.958285i	$-0.592264\pi$
$619$	−14.2220	−0.571629	−0.285814	+	0.958285i	$0.592264\pi$
$620$	0	0
$621$	5.49263	0.220412
$622$	0	0
$623$	7.82782	0.313615
$624$	0	0
$625$	111.117	4.44468
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.09259	−0.123310
$630$	0	0
$631$	−23.9969	−0.955300	−0.477650	−	0.878550i	$-0.658511\pi$
$631$	−23.9969	−0.955300	−0.477650	+	0.878550i	$0.658511\pi$
$632$	0	0
$633$	−11.8249	−0.469996
$634$	0	0
$635$	3.16806	0.125721
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	29.5531	1.16910
$640$	0	0
$641$	6.23951	0.246446	0.123223	−	0.992379i	$-0.460677\pi$
$641$	6.23951	0.246446	0.123223	+	0.992379i	$0.460677\pi$
$642$	0	0
$643$	48.0090	1.89329	0.946644	−	0.322281i	$-0.104449\pi$
$643$	48.0090	1.89329	0.946644	+	0.322281i	$0.104449\pi$
$644$	0	0
$645$	10.0056	0.393970
$646$	0	0
$647$	9.69697	0.381227	0.190614	−	0.981665i	$-0.438952\pi$
$647$	9.69697	0.381227	0.190614	+	0.981665i	$0.438952\pi$
$648$	0	0
$649$	−2.38750	−0.0937177
$650$	0	0
$651$	−7.15902	−0.280584
$652$	0	0
$653$	46.5642	1.82220	0.911099	−	0.412188i	$-0.135235\pi$
$653$	46.5642	1.82220	0.911099	+	0.412188i	$0.135235\pi$
$654$	0	0
$655$	−37.4504	−1.46331
$656$	0	0
$657$	19.6144	0.765232
$658$	0	0
$659$	−15.7072	−0.611864	−0.305932	−	0.952053i	$-0.598968\pi$
$659$	−15.7072	−0.611864	−0.305932	+	0.952053i	$0.598968\pi$
$660$	0	0
$661$	−23.6527	−0.919983	−0.459991	−	0.887923i	$-0.652148\pi$
$661$	−23.6527	−0.919983	−0.459991	+	0.887923i	$0.652148\pi$
$662$	0	0
$663$	−1.41317	−0.0548830
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.52097	−0.291213
$668$	0	0
$669$	16.5676	0.640540
$670$	0	0
$671$	−1.41317	−0.0545549
$672$	0	0
$673$	−25.8751	−0.997410	−0.498705	−	0.866772i	$-0.666191\pi$
$673$	−25.8751	−0.997410	−0.498705	+	0.866772i	$0.666191\pi$
$674$	0	0
$675$	−56.0925	−2.15900
$676$	0	0
$677$	5.91714	0.227414	0.113707	−	0.993514i	$-0.463727\pi$
$677$	5.91714	0.227414	0.113707	+	0.993514i	$0.463727\pi$
$678$	0	0
$679$	−12.9039	−0.495205
$680$	0	0
$681$	15.2970	0.586181
$682$	0	0
$683$	39.6723	1.51802	0.759010	−	0.651079i	$-0.225684\pi$
$683$	39.6723	1.51802	0.759010	+	0.651079i	$0.225684\pi$
$684$	0	0
$685$	43.3471	1.65621
$686$	0	0
$687$	−1.86621	−0.0712003
$688$	0	0
$689$	−12.6341	−0.481320
$690$	0	0
$691$	−2.86047	−0.108817	−0.0544087	−	0.998519i	$-0.517327\pi$
$691$	−2.86047	−0.108817	−0.0544087	+	0.998519i	$0.517327\pi$
$692$	0	0
$693$	2.73228	0.103791
$694$	0	0
$695$	84.9394	3.22194
$696$	0	0
$697$	11.2023	0.424317
$698$	0	0
$699$	9.87800	0.373620
$700$	0	0
$701$	−25.4273	−0.960377	−0.480188	−	0.877165i	$-0.659432\pi$
$701$	−25.4273	−0.960377	−0.480188	+	0.877165i	$0.659432\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−34.4046	−1.29575
$706$	0	0
$707$	8.22947	0.309501
$708$	0	0
$709$	14.1513	0.531465	0.265732	−	0.964047i	$-0.414386\pi$
$709$	14.1513	0.531465	0.265732	+	0.964047i	$0.414386\pi$
$710$	0	0
$711$	27.6225	1.03593
$712$	0	0
$713$	−14.3180	−0.536215
$714$	0	0
$715$	9.63259	0.360238
$716$	0	0
$717$	−0.103508	−0.00386559
$718$	0	0
$719$	−36.2794	−1.35299	−0.676496	−	0.736447i	$-0.736502\pi$
$719$	−36.2794	−1.35299	−0.676496	+	0.736447i	$0.736502\pi$
$720$	0	0
$721$	14.0342	0.522662
$722$	0	0
$723$	−10.0849	−0.375062
$724$	0	0
$725$	76.8067	2.85253
$726$	0	0
$727$	14.0360	0.520568	0.260284	−	0.965532i	$-0.416184\pi$
$727$	14.0360	0.520568	0.260284	+	0.965532i	$0.416184\pi$
$728$	0	0
$729$	−3.65430	−0.135345
$730$	0	0
$731$	−3.21235	−0.118813
$732$	0	0
$733$	−31.2309	−1.15354	−0.576771	−	0.816906i	$-0.695687\pi$
$733$	−31.2309	−1.15354	−0.576771	+	0.816906i	$0.695687\pi$
$734$	0	0
$735$	−3.11473	−0.114889
$736$	0	0
$737$	2.28085	0.0840163
$738$	0	0
$739$	−35.4318	−1.30338	−0.651689	−	0.758486i	$-0.725939\pi$
$739$	−35.4318	−1.30338	−0.651689	+	0.758486i	$0.725939\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.7658	−1.42218	−0.711089	−	0.703102i	$-0.751798\pi$
$743$	−38.7658	−1.42218	−0.711089	+	0.703102i	$0.751798\pi$
$744$	0	0
$745$	−25.7146	−0.942108
$746$	0	0
$747$	34.6905	1.26926
$748$	0	0
$749$	10.7252	0.391890
$750$	0	0
$751$	21.8408	0.796982	0.398491	−	0.917172i	$-0.369534\pi$
$751$	21.8408	0.796982	0.398491	+	0.917172i	$0.369534\pi$
$752$	0	0
$753$	19.7182	0.718572
$754$	0	0
$755$	−93.5201	−3.40355
$756$	0	0
$757$	−3.74189	−0.136001	−0.0680007	−	0.997685i	$-0.521662\pi$
$757$	−3.74189	−0.136001	−0.0680007	+	0.997685i	$0.521662\pi$
$758$	0	0
$759$	−1.09098	−0.0396000
$760$	0	0
$761$	35.2108	1.27639	0.638196	−	0.769874i	$-0.279681\pi$
$761$	35.2108	1.27639	0.638196	+	0.769874i	$0.279681\pi$
$762$	0	0
$763$	7.13688	0.258372
$764$	0	0
$765$	11.0236	0.398560
$766$	0	0
$767$	4.37036	0.157804
$768$	0	0
$769$	−31.8591	−1.14887	−0.574434	−	0.818551i	$-0.694778\pi$
$769$	−31.8591	−1.14887	−0.574434	+	0.818551i	$0.694778\pi$
$770$	0	0
$771$	19.5653	0.704626
$772$	0	0
$773$	27.6196	0.993408	0.496704	−	0.867920i	$-0.334543\pi$
$773$	27.6196	0.993408	0.496704	+	0.867920i	$0.334543\pi$
$774$	0	0
$775$	146.221	5.25240
$776$	0	0
$777$	2.18518	0.0783929
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−12.9120	−0.462027
$782$	0	0
$783$	−20.6855	−0.739238
$784$	0	0
$785$	−63.5419	−2.26791
$786$	0	0
$787$	49.4136	1.76140	0.880702	−	0.473671i	$-0.157071\pi$
$787$	49.4136	1.76140	0.880702	+	0.473671i	$0.157071\pi$
$788$	0	0
$789$	−3.20260	−0.114015
$790$	0	0
$791$	0.542542	0.0192906
$792$	0	0
$793$	2.58683	0.0918610
$794$	0	0
$795$	−19.6759	−0.697831
$796$	0	0
$797$	0.542542	0.0192178	0.00960891	−	0.999954i	$-0.496941\pi$
$797$	0.542542	0.0192178	0.00960891	+	0.999954i	$0.496941\pi$
$798$	0	0
$799$	11.0458	0.390771
$800$	0	0
$801$	19.5753	0.691660
$802$	0	0
$803$	−8.56968	−0.302418
$804$	0	0
$805$	−6.22947	−0.219560
$806$	0	0
$807$	18.4782	0.650462
$808$	0	0
$809$	−10.7850	−0.379180	−0.189590	−	0.981863i	$-0.560716\pi$
$809$	−10.7850	−0.379180	−0.189590	+	0.981863i	$0.560716\pi$
$810$	0	0
$811$	−43.3306	−1.52154	−0.760772	−	0.649020i	$-0.775179\pi$
$811$	−43.3306	−1.52154	−0.760772	+	0.649020i	$0.775179\pi$
$812$	0	0
$813$	−14.5484	−0.510236
$814$	0	0
$815$	−65.1060	−2.28056
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−5.00147	−0.174766
$820$	0	0
$821$	49.8052	1.73821	0.869106	−	0.494625i	$-0.164695\pi$
$821$	49.8052	1.73821	0.869106	+	0.494625i	$0.164695\pi$
$822$	0	0
$823$	−46.4749	−1.62001	−0.810006	−	0.586421i	$-0.800536\pi$
$823$	−46.4749	−1.62001	−0.810006	+	0.586421i	$0.800536\pi$
$824$	0	0
$825$	11.1414	0.387895
$826$	0	0
$827$	−41.3120	−1.43656	−0.718280	−	0.695755i	$-0.755070\pi$
$827$	−41.3120	−1.43656	−0.718280	+	0.695755i	$0.755070\pi$
$828$	0	0
$829$	28.2537	0.981290	0.490645	−	0.871360i	$-0.336761\pi$
$829$	28.2537	0.981290	0.490645	+	0.871360i	$0.336761\pi$
$830$	0	0
$831$	4.46603	0.154925
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−56.4271	−1.95274
$836$	0	0
$837$	−39.3799	−1.36117
$838$	0	0
$839$	1.99705	0.0689459	0.0344729	−	0.999406i	$-0.489025\pi$
$839$	1.99705	0.0689459	0.0344729	+	0.999406i	$0.489025\pi$
$840$	0	0
$841$	−0.675725	−0.0233009
$842$	0	0
$843$	−8.97182	−0.309006
$844$	0	0
$845$	39.6733	1.36480
$846$	0	0
$847$	9.80625	0.336947
$848$	0	0
$849$	2.24705	0.0771186
$850$	0	0
$851$	4.37036	0.149814
$852$	0	0
$853$	−49.6672	−1.70057	−0.850287	−	0.526319i	$-0.823572\pi$
$853$	−49.6672	−1.70057	−0.850287	+	0.526319i	$0.823572\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37.7131	−1.28825	−0.644127	−	0.764919i	$-0.722779\pi$
$857$	−37.7131	−1.28825	−0.644127	+	0.764919i	$0.722779\pi$
$858$	0	0
$859$	23.8951	0.815292	0.407646	−	0.913140i	$-0.366350\pi$
$859$	23.8951	0.815292	0.407646	+	0.913140i	$0.366350\pi$
$860$	0	0
$861$	−7.91538	−0.269755
$862$	0	0
$863$	−18.3513	−0.624685	−0.312342	−	0.949970i	$-0.601114\pi$
$863$	−18.3513	−0.624685	−0.312342	+	0.949970i	$0.601114\pi$
$864$	0	0
$865$	57.7960	1.96512
$866$	0	0
$867$	0.706585	0.0239969
$868$	0	0
$869$	−12.0685	−0.409395
$870$	0	0
$871$	−4.17513	−0.141469
$872$	0	0
$873$	−32.2692	−1.09215
$874$	0	0
$875$	41.5766	1.40555
$876$	0	0
$877$	−40.1525	−1.35585	−0.677927	−	0.735129i	$-0.737121\pi$
$877$	−40.1525	−1.35585	−0.677927	+	0.735129i	$0.737121\pi$
$878$	0	0
$879$	9.47963	0.319740
$880$	0	0
$881$	7.28427	0.245413	0.122707	−	0.992443i	$-0.460843\pi$
$881$	7.28427	0.245413	0.122707	+	0.992443i	$0.460843\pi$
$882$	0	0
$883$	52.0299	1.75095	0.875474	−	0.483266i	$-0.160550\pi$
$883$	52.0299	1.75095	0.875474	+	0.483266i	$0.160550\pi$
$884$	0	0
$885$	6.80625	0.228790
$886$	0	0
$887$	−15.6889	−0.526780	−0.263390	−	0.964689i	$-0.584841\pi$
$887$	−15.6889	−0.526780	−0.263390	+	0.964689i	$0.584841\pi$
$888$	0	0
$889$	0.718682	0.0241038
$890$	0	0
$891$	5.19624	0.174081
$892$	0	0
$893$	0	0
$894$	0	0
$895$	7.93103	0.265105
$896$	0	0
$897$	1.99705	0.0666796
$898$	0	0
$899$	53.9223	1.79841
$900$	0	0
$901$	6.31703	0.210451
$902$	0	0
$903$	2.26980	0.0755341
$904$	0	0
$905$	−6.54254	−0.217481
$906$	0	0
$907$	−38.6315	−1.28274	−0.641369	−	0.767232i	$-0.721633\pi$
$907$	−38.6315	−1.28274	−0.641369	+	0.767232i	$0.721633\pi$
$908$	0	0
$909$	20.5797	0.682587
$910$	0	0
$911$	−28.2324	−0.935382	−0.467691	−	0.883892i	$-0.654914\pi$
$911$	−28.2324	−0.935382	−0.467691	+	0.883892i	$0.654914\pi$
$912$	0	0
$913$	−15.1565	−0.501607
$914$	0	0
$915$	4.02864	0.133183
$916$	0	0
$917$	−8.49571	−0.280553
$918$	0	0
$919$	30.1494	0.994538	0.497269	−	0.867596i	$-0.334336\pi$
$919$	30.1494	0.994538	0.497269	+	0.867596i	$0.334336\pi$
$920$	0	0
$921$	1.79652	0.0591974
$922$	0	0
$923$	23.6355	0.777973
$924$	0	0
$925$	−44.6315	−1.46748
$926$	0	0
$927$	35.0959	1.15270
$928$	0	0
$929$	−44.5595	−1.46195	−0.730975	−	0.682404i	$-0.760934\pi$
$929$	−44.5595	−1.46195	−0.730975	+	0.682404i	$0.760934\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−11.6581	−0.381670
$934$	0	0
$935$	−4.81630	−0.157510
$936$	0	0
$937$	−7.05581	−0.230503	−0.115252	−	0.993336i	$-0.536767\pi$
$937$	−7.05581	−0.230503	−0.115252	+	0.993336i	$0.536767\pi$
$938$	0	0
$939$	−6.42928	−0.209812
$940$	0	0
$941$	37.6274	1.22662	0.613309	−	0.789843i	$-0.289838\pi$
$941$	37.6274	1.22662	0.613309	+	0.789843i	$0.289838\pi$
$942$	0	0
$943$	−15.8308	−0.515521
$944$	0	0
$945$	−17.1333	−0.557347
$946$	0	0
$947$	41.5464	1.35008	0.675039	−	0.737782i	$-0.264127\pi$
$947$	41.5464	1.35008	0.675039	+	0.737782i	$0.264127\pi$
$948$	0	0
$949$	15.6869	0.509219
$950$	0	0
$951$	17.2179	0.558330
$952$	0	0
$953$	19.7595	0.640072	0.320036	−	0.947405i	$-0.396305\pi$
$953$	19.7595	0.640072	0.320036	+	0.947405i	$0.396305\pi$
$954$	0	0
$955$	13.6612	0.442067
$956$	0	0
$957$	4.10867	0.132814
$958$	0	0
$959$	9.83342	0.317538
$960$	0	0
$961$	71.6545	2.31143
$962$	0	0
$963$	26.8209	0.864290
$964$	0	0
$965$	66.0844	2.12733
$966$	0	0
$967$	−26.2581	−0.844404	−0.422202	−	0.906502i	$-0.638743\pi$
$967$	−26.2581	−0.844404	−0.422202	+	0.906502i	$0.638743\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.48018	0.207959	0.103979	−	0.994579i	$-0.466842\pi$
$971$	6.48018	0.207959	0.103979	+	0.994579i	$0.466842\pi$
$972$	0	0
$973$	19.2687	0.617727
$974$	0	0
$975$	−20.3946	−0.653148
$976$	0	0
$977$	−13.5893	−0.434761	−0.217380	−	0.976087i	$-0.569751\pi$
$977$	−13.5893	−0.434761	−0.217380	+	0.976087i	$0.569751\pi$
$978$	0	0
$979$	−8.55259	−0.273342
$980$	0	0
$981$	17.8475	0.569825
$982$	0	0
$983$	0.296939	0.00947087	0.00473544	−	0.999989i	$-0.498493\pi$
$983$	0.296939	0.00947087	0.00473544	+	0.999989i	$0.498493\pi$
$984$	0	0
$985$	−9.17366	−0.292297
$986$	0	0
$987$	−7.80477	−0.248429
$988$	0	0
$989$	4.53959	0.144351
$990$	0	0
$991$	−24.0502	−0.763979	−0.381990	−	0.924167i	$-0.624761\pi$
$991$	−24.0502	−0.763979	−0.381990	+	0.924167i	$0.624761\pi$
$992$	0	0
$993$	−7.12773	−0.226192
$994$	0	0
$995$	−69.6008	−2.20649
$996$	0	0
$997$	−9.29341	−0.294325	−0.147163	−	0.989112i	$-0.547014\pi$
$997$	−9.29341	−0.294325	−0.147163	+	0.989112i	$0.547014\pi$
$998$	0	0
$999$	12.0201	0.380299

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bo.1.2		4
4.3	odd	2		7616.2.a.bi.1.3		4
8.3	odd	2		952.2.a.h.1.2	✓	4
8.5	even	2		1904.2.a.r.1.3		4
24.11	even	2		8568.2.a.bg.1.1		4
56.27	even	2		6664.2.a.n.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.h.1.2	✓	4	8.3	odd	2
1904.2.a.r.1.3		4	8.5	even	2
6664.2.a.n.1.3		4	56.27	even	2
7616.2.a.bi.1.3		4	4.3	odd	2
7616.2.a.bo.1.2		4	1.1	even	1	trivial
8568.2.a.bg.1.1		4	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.706585 q^{3} -4.40815 q^{5} -1.00000 q^{7} -2.50074 q^{9} +1.09259 q^{11} -2.00000 q^{13} -3.11473 q^{15} +1.00000 q^{17} -0.706585 q^{21} -1.41317 q^{23} +14.4318 q^{25} -3.88674 q^{27} +5.32206 q^{29} +10.1319 q^{31} +0.772008 q^{33} +4.40815 q^{35} -3.09259 q^{37} -1.41317 q^{39} +11.2023 q^{41} -3.21235 q^{43} +11.0236 q^{45} +11.0458 q^{47} +1.00000 q^{49} +0.706585 q^{51} +6.31703 q^{53} -4.81630 q^{55} -2.18518 q^{59} -1.29341 q^{61} +2.50074 q^{63} +8.81630 q^{65} +2.08757 q^{67} -0.998526 q^{69} -11.8178 q^{71} -7.84346 q^{73} +10.1973 q^{75} -1.09259 q^{77} -11.0458 q^{79} +4.75590 q^{81} -13.8721 q^{83} -4.40815 q^{85} +3.76049 q^{87} -7.82782 q^{89} +2.00000 q^{91} +7.15902 q^{93} +12.9039 q^{97} -2.73228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} - 8 q^{13} + 4 q^{17} - 3 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 8 q^{29} + 7 q^{31} - 6 q^{33} + 5 q^{35} - 8 q^{37} - 6 q^{39} + 15 q^{41} - 9 q^{43} + 2 q^{45}+ \cdots - 50 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 - 8 * q^13 + 4 * q^17 - 3 * q^21 - 6 * q^23 + 3 * q^25 + 6 * q^27 - 8 * q^29 + 7 * q^31 - 6 * q^33 + 5 * q^35 - 8 * q^37 - 6 * q^39 + 15 * q^41 - 9 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 3 * q^51 - 17 * q^53 + 6 * q^55 - 5 * q^61 - 7 * q^63 + 10 * q^65 - 9 * q^67 - 38 * q^69 + 12 * q^71 - 11 * q^73 - 2 * q^75 + 6 * q^79 + 16 * q^81 - 6 * q^83 - 5 * q^85 + 14 * q^87 + 2 * q^89 + 8 * q^91 + 9 * q^97 - 50 * q^99

Introduction

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