LMFDB - Embedded newform 736.4.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [736,4,Mod(1,736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("736.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 736 = 2^{5} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	736.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:	$43.4254057642$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 $ x^8 - 4x^7 - 137x^6 + 344x^5 + 6175x^4 - 7924x^3 - 89643x^2 + 45072*x + 51084
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.03676$ of defining polynomial
Character	$\chi$	$=$	736.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.963239 q^{3} +5.79749 q^{5} -15.5937 q^{7} -26.0722 q^{9} +25.2920 q^{11} +27.0039 q^{13} -5.58437 q^{15} +75.9614 q^{17} -27.9509 q^{19} +15.0205 q^{21} +23.0000 q^{23} -91.3891 q^{25} +51.1212 q^{27} +141.807 q^{29} -318.869 q^{31} -24.3622 q^{33} -90.4043 q^{35} +35.1933 q^{37} -26.0112 q^{39} -20.6356 q^{41} -181.307 q^{43} -151.153 q^{45} -423.443 q^{47} -99.8367 q^{49} -73.1690 q^{51} +167.525 q^{53} +146.630 q^{55} +26.9234 q^{57} +72.6487 q^{59} -744.947 q^{61} +406.561 q^{63} +156.555 q^{65} -538.772 q^{67} -22.1545 q^{69} +737.505 q^{71} +4.93206 q^{73} +88.0295 q^{75} -394.395 q^{77} -216.545 q^{79} +654.707 q^{81} -1478.42 q^{83} +440.386 q^{85} -136.594 q^{87} +1097.58 q^{89} -421.090 q^{91} +307.147 q^{93} -162.045 q^{95} -633.529 q^{97} -659.416 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} - 12 q^{5} + 14 q^{7} + 90 q^{9} - 88 q^{11} - 30 q^{13} - 30 q^{15} + 58 q^{17} - 190 q^{19} - 66 q^{21} + 184 q^{23} + 28 q^{25} - 432 q^{27} + 190 q^{29} + 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) $ 8 * q - 12 * q^3 - 12 * q^5 + 14 * q^7 + 90 * q^9 - 88 * q^11 - 30 * q^13 - 30 * q^15 + 58 * q^17 - 190 * q^19 - 66 * q^21 + 184 * q^23 + 28 * q^25 - 432 * q^27 + 190 * q^29 + 60 * q^31 + 346 * q^33 - 192 * q^35 - 156 * q^37 - 156 * q^39 + 282 * q^41 - 810 * q^43 + 358 * q^45 - 564 * q^47 + 752 * q^49 - 1614 * q^51 - 230 * q^53 - 1292 * q^55 + 796 * q^57 - 1916 * q^59 + 22 * q^61 - 1036 * q^63 + 222 * q^65 - 2292 * q^67 - 276 * q^69 - 2376 * q^71 - 630 * q^73 - 3952 * q^75 - 948 * q^77 - 892 * q^79 - 1044 * q^81 - 2800 * q^83 + 288 * q^85 - 2756 * q^87 - 308 * q^89 - 4610 * q^91 - 982 * q^93 - 1300 * q^95 + 854 * q^97 - 5986 * 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
$3$	−0.963239	−0.185375	−0.0926877	−	0.995695i	$-0.529546\pi$
$3$	−0.963239	−0.185375	−0.0926877	+	0.995695i	$0.529546\pi$
$4$	0	0
$5$	5.79749	0.518544	0.259272	−	0.965804i	$-0.416517\pi$
$5$	5.79749	0.518544	0.259272	+	0.965804i	$0.416517\pi$
$6$	0	0
$7$	−15.5937	−0.841980	−0.420990	−	0.907065i	$-0.638317\pi$
$7$	−15.5937	−0.841980	−0.420990	+	0.907065i	$0.638317\pi$
$8$	0	0
$9$	−26.0722	−0.965636
$10$	0	0
$11$	25.2920	0.693256	0.346628	−	0.938003i	$-0.387327\pi$
$11$	25.2920	0.693256	0.346628	+	0.938003i	$0.387327\pi$
$12$	0	0
$13$	27.0039	0.576117	0.288059	−	0.957613i	$-0.406990\pi$
$13$	27.0039	0.576117	0.288059	+	0.957613i	$0.406990\pi$
$14$	0	0
$15$	−5.58437	−0.0961252
$16$	0	0
$17$	75.9614	1.08373	0.541863	−	0.840467i	$-0.317719\pi$
$17$	75.9614	1.08373	0.541863	+	0.840467i	$0.317719\pi$
$18$	0	0
$19$	−27.9509	−0.337494	−0.168747	−	0.985659i	$-0.553972\pi$
$19$	−27.9509	−0.337494	−0.168747	+	0.985659i	$0.553972\pi$
$20$	0	0
$21$	15.0205	0.156082
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−91.3891	−0.731113
$26$	0	0
$27$	51.1212	0.364381
$28$	0	0
$29$	141.807	0.908031	0.454015	−	0.890994i	$-0.349991\pi$
$29$	141.807	0.908031	0.454015	+	0.890994i	$0.349991\pi$
$30$	0	0
$31$	−318.869	−1.84744	−0.923718	−	0.383073i	$-0.874866\pi$
$31$	−318.869	−1.84744	−0.923718	+	0.383073i	$0.874866\pi$
$32$	0	0
$33$	−24.3622	−0.128513
$34$	0	0
$35$	−90.4043	−0.436603
$36$	0	0
$37$	35.1933	0.156371	0.0781857	−	0.996939i	$-0.475087\pi$
$37$	35.1933	0.156371	0.0781857	+	0.996939i	$0.475087\pi$
$38$	0	0
$39$	−26.0112	−0.106798
$40$	0	0
$41$	−20.6356	−0.0786033	−0.0393016	−	0.999227i	$-0.512513\pi$
$41$	−20.6356	−0.0786033	−0.0393016	+	0.999227i	$0.512513\pi$
$42$	0	0
$43$	−181.307	−0.643000	−0.321500	−	0.946910i	$-0.604187\pi$
$43$	−181.307	−0.643000	−0.321500	+	0.946910i	$0.604187\pi$
$44$	0	0
$45$	−151.153	−0.500724
$46$	0	0
$47$	−423.443	−1.31416	−0.657080	−	0.753821i	$-0.728209\pi$
$47$	−423.443	−1.31416	−0.657080	+	0.753821i	$0.728209\pi$
$48$	0	0
$49$	−99.8367	−0.291069
$50$	0	0
$51$	−73.1690	−0.200896
$52$	0	0
$53$	167.525	0.434176	0.217088	−	0.976152i	$-0.430344\pi$
$53$	167.525	0.434176	0.217088	+	0.976152i	$0.430344\pi$
$54$	0	0
$55$	146.630	0.359483
$56$	0	0
$57$	26.9234	0.0625630
$58$	0	0
$59$	72.6487	0.160306	0.0801530	−	0.996783i	$-0.474459\pi$
$59$	72.6487	0.160306	0.0801530	+	0.996783i	$0.474459\pi$
$60$	0	0
$61$	−744.947	−1.56362	−0.781809	−	0.623518i	$-0.785703\pi$
$61$	−744.947	−1.56362	−0.781809	+	0.623518i	$0.785703\pi$
$62$	0	0
$63$	406.561	0.813046
$64$	0	0
$65$	156.555	0.298742
$66$	0	0
$67$	−538.772	−0.982410	−0.491205	−	0.871044i	$-0.663443\pi$
$67$	−538.772	−0.982410	−0.491205	+	0.871044i	$0.663443\pi$
$68$	0	0
$69$	−22.1545	−0.0386534
$70$	0	0
$71$	737.505	1.23276	0.616378	−	0.787450i	$-0.288599\pi$
$71$	737.505	1.23276	0.616378	+	0.787450i	$0.288599\pi$
$72$	0	0
$73$	4.93206	0.00790759	0.00395380	−	0.999992i	$-0.498741\pi$
$73$	4.93206	0.00790759	0.00395380	+	0.999992i	$0.498741\pi$
$74$	0	0
$75$	88.0295	0.135530
$76$	0	0
$77$	−394.395	−0.583708
$78$	0	0
$79$	−216.545	−0.308395	−0.154197	−	0.988040i	$-0.549279\pi$
$79$	−216.545	−0.308395	−0.154197	+	0.988040i	$0.549279\pi$
$80$	0	0
$81$	654.707	0.898089
$82$	0	0
$83$	−1478.42	−1.95515	−0.977576	−	0.210581i	$-0.932465\pi$
$83$	−1478.42	−1.95515	−0.977576	+	0.210581i	$0.932465\pi$
$84$	0	0
$85$	440.386	0.561959
$86$	0	0
$87$	−136.594	−0.168327
$88$	0	0
$89$	1097.58	1.30723	0.653613	−	0.756829i	$-0.273252\pi$
$89$	1097.58	1.30723	0.653613	+	0.756829i	$0.273252\pi$
$90$	0	0
$91$	−421.090	−0.485079
$92$	0	0
$93$	307.147	0.342469
$94$	0	0
$95$	−162.045	−0.175005
$96$	0	0
$97$	−633.529	−0.663146	−0.331573	−	0.943430i	$-0.607579\pi$
$97$	−633.529	−0.663146	−0.331573	+	0.943430i	$0.607579\pi$
$98$	0	0
$99$	−659.416	−0.669433
$100$	0	0
$101$	688.551	0.678351	0.339175	−	0.940723i	$-0.389852\pi$
$101$	688.551	0.678351	0.339175	+	0.940723i	$0.389852\pi$
$102$	0	0
$103$	−1161.45	−1.11108	−0.555541	−	0.831489i	$-0.687489\pi$
$103$	−1161.45	−1.11108	−0.555541	+	0.831489i	$0.687489\pi$
$104$	0	0
$105$	87.0810	0.0809355
$106$	0	0
$107$	−1132.42	−1.02313	−0.511565	−	0.859244i	$-0.670934\pi$
$107$	−1132.42	−1.02313	−0.511565	+	0.859244i	$0.670934\pi$
$108$	0	0
$109$	233.810	0.205458	0.102729	−	0.994709i	$-0.467243\pi$
$109$	233.810	0.205458	0.102729	+	0.994709i	$0.467243\pi$
$110$	0	0
$111$	−33.8995	−0.0289874
$112$	0	0
$113$	−1374.29	−1.14409	−0.572046	−	0.820222i	$-0.693850\pi$
$113$	−1374.29	−1.14409	−0.572046	+	0.820222i	$0.693850\pi$
$114$	0	0
$115$	133.342	0.108124
$116$	0	0
$117$	−704.049	−0.556320
$118$	0	0
$119$	−1184.52	−0.912476
$120$	0	0
$121$	−691.316	−0.519396
$122$	0	0
$123$	19.8770	0.0145711
$124$	0	0
$125$	−1254.51	−0.897657
$126$	0	0
$127$	−924.040	−0.645632	−0.322816	−	0.946462i	$-0.604630\pi$
$127$	−924.040	−0.645632	−0.322816	+	0.946462i	$0.604630\pi$
$128$	0	0
$129$	174.642	0.119196
$130$	0	0
$131$	−2029.38	−1.35349	−0.676747	−	0.736215i	$-0.736611\pi$
$131$	−2029.38	−1.35349	−0.676747	+	0.736215i	$0.736611\pi$
$132$	0	0
$133$	435.858	0.284163
$134$	0	0
$135$	296.375	0.188947
$136$	0	0
$137$	1521.76	0.948996	0.474498	−	0.880257i	$-0.342630\pi$
$137$	1521.76	0.948996	0.474498	+	0.880257i	$0.342630\pi$
$138$	0	0
$139$	−1506.73	−0.919417	−0.459708	−	0.888070i	$-0.652046\pi$
$139$	−1506.73	−0.919417	−0.459708	+	0.888070i	$0.652046\pi$
$140$	0	0
$141$	407.877	0.243613
$142$	0	0
$143$	682.981	0.399397
$144$	0	0
$145$	822.125	0.470853
$146$	0	0
$147$	96.1666	0.0539571
$148$	0	0
$149$	−896.765	−0.493059	−0.246530	−	0.969135i	$-0.579290\pi$
$149$	−896.765	−0.493059	−0.246530	+	0.969135i	$0.579290\pi$
$150$	0	0
$151$	1166.44	0.628633	0.314316	−	0.949318i	$-0.398225\pi$
$151$	1166.44	0.628633	0.314316	+	0.949318i	$0.398225\pi$
$152$	0	0
$153$	−1980.48	−1.04648
$154$	0	0
$155$	−1848.64	−0.957976
$156$	0	0
$157$	−1924.53	−0.978306	−0.489153	−	0.872198i	$-0.662694\pi$
$157$	−1924.53	−0.978306	−0.489153	+	0.872198i	$0.662694\pi$
$158$	0	0
$159$	−161.367	−0.0804856
$160$	0	0
$161$	−358.655	−0.175565
$162$	0	0
$163$	−2375.34	−1.14141	−0.570707	−	0.821153i	$-0.693331\pi$
$163$	−2375.34	−1.14141	−0.570707	+	0.821153i	$0.693331\pi$
$164$	0	0
$165$	−141.240	−0.0666394
$166$	0	0
$167$	387.710	0.179652	0.0898262	−	0.995957i	$-0.471369\pi$
$167$	387.710	0.179652	0.0898262	+	0.995957i	$0.471369\pi$
$168$	0	0
$169$	−1467.79	−0.668089
$170$	0	0
$171$	728.741	0.325896
$172$	0	0
$173$	2879.18	1.26532	0.632660	−	0.774430i	$-0.281963\pi$
$173$	2879.18	1.26532	0.632660	+	0.774430i	$0.281963\pi$
$174$	0	0
$175$	1425.09	0.615582
$176$	0	0
$177$	−69.9781	−0.0297168
$178$	0	0
$179$	−635.069	−0.265181	−0.132590	−	0.991171i	$-0.542329\pi$
$179$	−635.069	−0.265181	−0.132590	+	0.991171i	$0.542329\pi$
$180$	0	0
$181$	532.117	0.218519	0.109260	−	0.994013i	$-0.465152\pi$
$181$	532.117	0.218519	0.109260	+	0.994013i	$0.465152\pi$
$182$	0	0
$183$	717.562	0.289856
$184$	0	0
$185$	204.033	0.0810853
$186$	0	0
$187$	1921.21	0.751299
$188$	0	0
$189$	−797.168	−0.306801
$190$	0	0
$191$	3890.10	1.47370	0.736852	−	0.676054i	$-0.236311\pi$
$191$	3890.10	1.47370	0.736852	+	0.676054i	$0.236311\pi$
$192$	0	0
$193$	−224.643	−0.0837832	−0.0418916	−	0.999122i	$-0.513338\pi$
$193$	−224.643	−0.0837832	−0.0418916	+	0.999122i	$0.513338\pi$
$194$	0	0
$195$	−150.800	−0.0553794
$196$	0	0
$197$	4903.68	1.77346	0.886732	−	0.462284i	$-0.152970\pi$
$197$	4903.68	1.77346	0.886732	+	0.462284i	$0.152970\pi$
$198$	0	0
$199$	3130.19	1.11504	0.557521	−	0.830163i	$-0.311753\pi$
$199$	3130.19	1.11504	0.557521	+	0.830163i	$0.311753\pi$
$200$	0	0
$201$	518.966	0.182115
$202$	0	0
$203$	−2211.29	−0.764544
$204$	0	0
$205$	−119.635	−0.0407592
$206$	0	0
$207$	−599.660	−0.201349
$208$	0	0
$209$	−706.934	−0.233969
$210$	0	0
$211$	−4087.22	−1.33354	−0.666768	−	0.745266i	$-0.732323\pi$
$211$	−4087.22	−1.33354	−0.666768	+	0.745266i	$0.732323\pi$
$212$	0	0
$213$	−710.393	−0.228523
$214$	0	0
$215$	−1051.12	−0.333424
$216$	0	0
$217$	4972.34	1.55551
$218$	0	0
$219$	−4.75075	−0.00146587
$220$	0	0
$221$	2051.25	0.624353
$222$	0	0
$223$	−454.686	−0.136538	−0.0682692	−	0.997667i	$-0.521748\pi$
$223$	−454.686	−0.136538	−0.0682692	+	0.997667i	$0.521748\pi$
$224$	0	0
$225$	2382.71	0.705989
$226$	0	0
$227$	2912.76	0.851661	0.425830	−	0.904803i	$-0.359982\pi$
$227$	2912.76	0.851661	0.425830	+	0.904803i	$0.359982\pi$
$228$	0	0
$229$	−2140.49	−0.617676	−0.308838	−	0.951115i	$-0.599940\pi$
$229$	−2140.49	−0.617676	−0.308838	+	0.951115i	$0.599940\pi$
$230$	0	0
$231$	379.897	0.108205
$232$	0	0
$233$	−2820.37	−0.792998	−0.396499	−	0.918035i	$-0.629775\pi$
$233$	−2820.37	−0.792998	−0.396499	+	0.918035i	$0.629775\pi$
$234$	0	0
$235$	−2454.91	−0.681449
$236$	0	0
$237$	208.584	0.0571688
$238$	0	0
$239$	3656.67	0.989667	0.494834	−	0.868988i	$-0.335229\pi$
$239$	3656.67	0.989667	0.494834	+	0.868988i	$0.335229\pi$
$240$	0	0
$241$	−5033.21	−1.34530	−0.672651	−	0.739960i	$-0.734844\pi$
$241$	−5033.21	−1.34530	−0.672651	+	0.739960i	$0.734844\pi$
$242$	0	0
$243$	−2010.91	−0.530864
$244$	0	0
$245$	−578.803	−0.150932
$246$	0	0
$247$	−754.783	−0.194436
$248$	0	0
$249$	1424.07	0.362437
$250$	0	0
$251$	7900.45	1.98674	0.993370	−	0.114961i	$-0.0366745\pi$
$251$	7900.45	1.98674	0.993370	+	0.114961i	$0.0366745\pi$
$252$	0	0
$253$	581.715	0.144554
$254$	0	0
$255$	−424.196	−0.104173
$256$	0	0
$257$	3227.01	0.783250	0.391625	−	0.920125i	$-0.371913\pi$
$257$	3227.01	0.783250	0.391625	+	0.920125i	$0.371913\pi$
$258$	0	0
$259$	−548.793	−0.131662
$260$	0	0
$261$	−3697.22	−0.876827
$262$	0	0
$263$	−2252.34	−0.528081	−0.264041	−	0.964512i	$-0.585055\pi$
$263$	−2252.34	−0.528081	−0.264041	+	0.964512i	$0.585055\pi$
$264$	0	0
$265$	971.225	0.225139
$266$	0	0
$267$	−1057.23	−0.242328
$268$	0	0
$269$	−2834.50	−0.642462	−0.321231	−	0.947001i	$-0.604097\pi$
$269$	−2834.50	−0.642462	−0.321231	+	0.947001i	$0.604097\pi$
$270$	0	0
$271$	5134.97	1.15102	0.575511	−	0.817794i	$-0.304803\pi$
$271$	5134.97	1.15102	0.575511	+	0.817794i	$0.304803\pi$
$272$	0	0
$273$	405.610	0.0899218
$274$	0	0
$275$	−2311.41	−0.506848
$276$	0	0
$277$	6802.11	1.47545	0.737724	−	0.675103i	$-0.235901\pi$
$277$	6802.11	1.47545	0.737724	+	0.675103i	$0.235901\pi$
$278$	0	0
$279$	8313.60	1.78395
$280$	0	0
$281$	−6923.59	−1.46985	−0.734923	−	0.678151i	$-0.762782\pi$
$281$	−6923.59	−1.46985	−0.734923	+	0.678151i	$0.762782\pi$
$282$	0	0
$283$	−253.507	−0.0532488	−0.0266244	−	0.999646i	$-0.508476\pi$
$283$	−253.507	−0.0532488	−0.0266244	+	0.999646i	$0.508476\pi$
$284$	0	0
$285$	156.088	0.0324417
$286$	0	0
$287$	321.785	0.0661824
$288$	0	0
$289$	857.131	0.174462
$290$	0	0
$291$	610.240	0.122931
$292$	0	0
$293$	−4711.13	−0.939342	−0.469671	−	0.882842i	$-0.655627\pi$
$293$	−4711.13	−0.939342	−0.469671	+	0.882842i	$0.655627\pi$
$294$	0	0
$295$	421.180	0.0831257
$296$	0	0
$297$	1292.96	0.252609
$298$	0	0
$299$	621.089	0.120129
$300$	0	0
$301$	2827.24	0.541394
$302$	0	0
$303$	−663.239	−0.125750
$304$	0	0
$305$	−4318.83	−0.810804
$306$	0	0
$307$	3136.97	0.583180	0.291590	−	0.956543i	$-0.405816\pi$
$307$	3136.97	0.583180	0.291590	+	0.956543i	$0.405816\pi$
$308$	0	0
$309$	1118.76	0.205967
$310$	0	0
$311$	6117.79	1.11546	0.557730	−	0.830023i	$-0.311673\pi$
$311$	6117.79	1.11546	0.557730	+	0.830023i	$0.311673\pi$
$312$	0	0
$313$	−1124.42	−0.203054	−0.101527	−	0.994833i	$-0.532373\pi$
$313$	−1124.42	−0.203054	−0.101527	+	0.994833i	$0.532373\pi$
$314$	0	0
$315$	2357.04	0.421600
$316$	0	0
$317$	−1970.60	−0.349149	−0.174574	−	0.984644i	$-0.555855\pi$
$317$	−1970.60	−0.349149	−0.174574	+	0.984644i	$0.555855\pi$
$318$	0	0
$319$	3586.58	0.629498
$320$	0	0
$321$	1090.79	0.189663
$322$	0	0
$323$	−2123.19	−0.365751
$324$	0	0
$325$	−2467.86	−0.421207
$326$	0	0
$327$	−225.214	−0.0380868
$328$	0	0
$329$	6603.04	1.10650
$330$	0	0
$331$	3482.81	0.578345	0.289173	−	0.957277i	$-0.406620\pi$
$331$	3482.81	0.578345	0.289173	+	0.957277i	$0.406620\pi$
$332$	0	0
$333$	−917.565	−0.150998
$334$	0	0
$335$	−3123.53	−0.509422
$336$	0	0
$337$	2280.37	0.368604	0.184302	−	0.982870i	$-0.440998\pi$
$337$	2280.37	0.368604	0.184302	+	0.982870i	$0.440998\pi$
$338$	0	0
$339$	1323.77	0.212086
$340$	0	0
$341$	−8064.82	−1.28075
$342$	0	0
$343$	6905.46	1.08705
$344$	0	0
$345$	−128.441	−0.0200435
$346$	0	0
$347$	1575.82	0.243788	0.121894	−	0.992543i	$-0.461103\pi$
$347$	1575.82	0.243788	0.121894	+	0.992543i	$0.461103\pi$
$348$	0	0
$349$	−826.377	−0.126748	−0.0633738	−	0.997990i	$-0.520186\pi$
$349$	−826.377	−0.126748	−0.0633738	+	0.997990i	$0.520186\pi$
$350$	0	0
$351$	1380.47	0.209926
$352$	0	0
$353$	4271.66	0.644073	0.322036	−	0.946727i	$-0.395633\pi$
$353$	4271.66	0.644073	0.322036	+	0.946727i	$0.395633\pi$
$354$	0	0
$355$	4275.68	0.639238
$356$	0	0
$357$	1140.97	0.169151
$358$	0	0
$359$	−2312.24	−0.339931	−0.169965	−	0.985450i	$-0.554366\pi$
$359$	−2312.24	−0.339931	−0.169965	+	0.985450i	$0.554366\pi$
$360$	0	0
$361$	−6077.75	−0.886098
$362$	0	0
$363$	665.903	0.0962833
$364$	0	0
$365$	28.5936	0.00410043
$366$	0	0
$367$	9873.11	1.40428	0.702142	−	0.712037i	$-0.252227\pi$
$367$	9873.11	1.40428	0.702142	+	0.712037i	$0.252227\pi$
$368$	0	0
$369$	538.014	0.0759021
$370$	0	0
$371$	−2612.33	−0.365568
$372$	0	0
$373$	10604.4	1.47206	0.736029	−	0.676950i	$-0.236699\pi$
$373$	10604.4	1.47206	0.736029	+	0.676950i	$0.236699\pi$
$374$	0	0
$375$	1208.40	0.166404
$376$	0	0
$377$	3829.34	0.523132
$378$	0	0
$379$	−6553.06	−0.888148	−0.444074	−	0.895990i	$-0.646467\pi$
$379$	−6553.06	−0.888148	−0.444074	+	0.895990i	$0.646467\pi$
$380$	0	0
$381$	890.071	0.119684
$382$	0	0
$383$	−7584.99	−1.01195	−0.505973	−	0.862550i	$-0.668866\pi$
$383$	−7584.99	−1.01195	−0.505973	+	0.862550i	$0.668866\pi$
$384$	0	0
$385$	−2286.50	−0.302678
$386$	0	0
$387$	4727.06	0.620904
$388$	0	0
$389$	−4659.70	−0.607342	−0.303671	−	0.952777i	$-0.598212\pi$
$389$	−4659.70	−0.607342	−0.303671	+	0.952777i	$0.598212\pi$
$390$	0	0
$391$	1747.11	0.225972
$392$	0	0
$393$	1954.78	0.250905
$394$	0	0
$395$	−1255.42	−0.159916
$396$	0	0
$397$	861.297	0.108885	0.0544424	−	0.998517i	$-0.482662\pi$
$397$	861.297	0.108885	0.0544424	+	0.998517i	$0.482662\pi$
$398$	0	0
$399$	−419.835	−0.0526768
$400$	0	0
$401$	7647.62	0.952379	0.476189	−	0.879343i	$-0.342018\pi$
$401$	7647.62	0.952379	0.476189	+	0.879343i	$0.342018\pi$
$402$	0	0
$403$	−8610.69	−1.06434
$404$	0	0
$405$	3795.66	0.465698
$406$	0	0
$407$	890.107	0.108405
$408$	0	0
$409$	−15987.3	−1.93282	−0.966410	−	0.257005i	$-0.917264\pi$
$409$	−15987.3	−1.93282	−0.966410	+	0.257005i	$0.917264\pi$
$410$	0	0
$411$	−1465.81	−0.175920
$412$	0	0
$413$	−1132.86	−0.134975
$414$	0	0
$415$	−8571.13	−1.01383
$416$	0	0
$417$	1451.34	0.170437
$418$	0	0
$419$	14798.1	1.72538	0.862688	−	0.505736i	$-0.168779\pi$
$419$	14798.1	1.72538	0.862688	+	0.505736i	$0.168779\pi$
$420$	0	0
$421$	−7180.62	−0.831263	−0.415632	−	0.909533i	$-0.636439\pi$
$421$	−7180.62	−0.831263	−0.415632	+	0.909533i	$0.636439\pi$
$422$	0	0
$423$	11040.1	1.26900
$424$	0	0
$425$	−6942.04	−0.792326
$426$	0	0
$427$	11616.5	1.31654
$428$	0	0
$429$	−657.874	−0.0740383
$430$	0	0
$431$	−3904.27	−0.436339	−0.218169	−	0.975911i	$-0.570008\pi$
$431$	−3904.27	−0.436339	−0.218169	+	0.975911i	$0.570008\pi$
$432$	0	0
$433$	5988.08	0.664594	0.332297	−	0.943175i	$-0.392176\pi$
$433$	5988.08	0.664594	0.332297	+	0.943175i	$0.392176\pi$
$434$	0	0
$435$	−791.903	−0.0872846
$436$	0	0
$437$	−642.871	−0.0703723
$438$	0	0
$439$	6197.16	0.673746	0.336873	−	0.941550i	$-0.390631\pi$
$439$	6197.16	0.673746	0.336873	+	0.941550i	$0.390631\pi$
$440$	0	0
$441$	2602.96	0.281067
$442$	0	0
$443$	2804.51	0.300782	0.150391	−	0.988627i	$-0.451947\pi$
$443$	2804.51	0.300782	0.150391	+	0.988627i	$0.451947\pi$
$444$	0	0
$445$	6363.20	0.677854
$446$	0	0
$447$	863.799	0.0914011
$448$	0	0
$449$	2534.47	0.266389	0.133195	−	0.991090i	$-0.457476\pi$
$449$	2534.47	0.266389	0.133195	+	0.991090i	$0.457476\pi$
$450$	0	0
$451$	−521.914	−0.0544922
$452$	0	0
$453$	−1123.56	−0.116533
$454$	0	0
$455$	−2441.27	−0.251535
$456$	0	0
$457$	1846.14	0.188969	0.0944843	−	0.995526i	$-0.469880\pi$
$457$	1846.14	0.188969	0.0944843	+	0.995526i	$0.469880\pi$
$458$	0	0
$459$	3883.24	0.394889
$460$	0	0
$461$	8386.06	0.847241	0.423620	−	0.905840i	$-0.360759\pi$
$461$	8386.06	0.847241	0.423620	+	0.905840i	$0.360759\pi$
$462$	0	0
$463$	8337.69	0.836901	0.418451	−	0.908240i	$-0.362573\pi$
$463$	8337.69	0.836901	0.418451	+	0.908240i	$0.362573\pi$
$464$	0	0
$465$	1780.68	0.177585
$466$	0	0
$467$	4008.06	0.397154	0.198577	−	0.980085i	$-0.436368\pi$
$467$	4008.06	0.397154	0.198577	+	0.980085i	$0.436368\pi$
$468$	0	0
$469$	8401.45	0.827170
$470$	0	0
$471$	1853.78	0.181354
$472$	0	0
$473$	−4585.60	−0.445764
$474$	0	0
$475$	2554.41	0.246746
$476$	0	0
$477$	−4367.74	−0.419256
$478$	0	0
$479$	−15688.6	−1.49652	−0.748258	−	0.663408i	$-0.769109\pi$
$479$	−15688.6	−1.49652	−0.748258	+	0.663408i	$0.769109\pi$
$480$	0	0
$481$	950.354	0.0900882
$482$	0	0
$483$	345.470	0.0325454
$484$	0	0
$485$	−3672.88	−0.343870
$486$	0	0
$487$	16389.1	1.52497	0.762487	−	0.647003i	$-0.223978\pi$
$487$	16389.1	1.52497	0.762487	+	0.647003i	$0.223978\pi$
$488$	0	0
$489$	2288.02	0.211590
$490$	0	0
$491$	−12191.8	−1.12059	−0.560293	−	0.828295i	$-0.689311\pi$
$491$	−12191.8	−1.12059	−0.560293	+	0.828295i	$0.689311\pi$
$492$	0	0
$493$	10771.9	0.984056
$494$	0	0
$495$	−3822.96	−0.347130
$496$	0	0
$497$	−11500.4	−1.03796
$498$	0	0
$499$	−16889.6	−1.51520	−0.757598	−	0.652722i	$-0.773627\pi$
$499$	−16889.6	−1.51520	−0.757598	+	0.652722i	$0.773627\pi$
$500$	0	0
$501$	−373.458	−0.0333031
$502$	0	0
$503$	−1648.20	−0.146103	−0.0730513	−	0.997328i	$-0.523274\pi$
$503$	−1648.20	−0.146103	−0.0730513	+	0.997328i	$0.523274\pi$
$504$	0	0
$505$	3991.87	0.351754
$506$	0	0
$507$	1413.83	0.123847
$508$	0	0
$509$	−4038.00	−0.351633	−0.175816	−	0.984423i	$-0.556256\pi$
$509$	−4038.00	−0.351633	−0.175816	+	0.984423i	$0.556256\pi$
$510$	0	0
$511$	−76.9091	−0.00665804
$512$	0	0
$513$	−1428.88	−0.122976
$514$	0	0
$515$	−6733.52	−0.576145
$516$	0	0
$517$	−10709.7	−0.911049
$518$	0	0
$519$	−2773.34	−0.234559
$520$	0	0
$521$	−2300.49	−0.193448	−0.0967240	−	0.995311i	$-0.530836\pi$
$521$	−2300.49	−0.193448	−0.0967240	+	0.995311i	$0.530836\pi$
$522$	0	0
$523$	5859.00	0.489859	0.244929	−	0.969541i	$-0.421235\pi$
$523$	5859.00	0.489859	0.244929	+	0.969541i	$0.421235\pi$
$524$	0	0
$525$	−1372.71	−0.114114
$526$	0	0
$527$	−24221.7	−2.00211
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−1894.11	−0.154797
$532$	0	0
$533$	−557.240	−0.0452847
$534$	0	0
$535$	−6565.18	−0.530538
$536$	0	0
$537$	611.724	0.0491580
$538$	0	0
$539$	−2525.07	−0.201785
$540$	0	0
$541$	−22814.9	−1.81311	−0.906553	−	0.422092i	$-0.861296\pi$
$541$	−22814.9	−1.81311	−0.906553	+	0.422092i	$0.861296\pi$
$542$	0	0
$543$	−512.556	−0.0405081
$544$	0	0
$545$	1355.51	0.106539
$546$	0	0
$547$	7081.15	0.553506	0.276753	−	0.960941i	$-0.410742\pi$
$547$	7081.15	0.553506	0.276753	+	0.960941i	$0.410742\pi$
$548$	0	0
$549$	19422.4	1.50989
$550$	0	0
$551$	−3963.63	−0.306455
$552$	0	0
$553$	3376.73	0.259662
$554$	0	0
$555$	−196.532	−0.0150312
$556$	0	0
$557$	−1526.87	−0.116150	−0.0580750	−	0.998312i	$-0.518496\pi$
$557$	−1526.87	−0.116150	−0.0580750	+	0.998312i	$0.518496\pi$
$558$	0	0
$559$	−4895.98	−0.370444
$560$	0	0
$561$	−1850.59	−0.139272
$562$	0	0
$563$	−17867.4	−1.33752	−0.668759	−	0.743479i	$-0.733174\pi$
$563$	−17867.4	−1.33752	−0.668759	+	0.743479i	$0.733174\pi$
$564$	0	0
$565$	−7967.44	−0.593261
$566$	0	0
$567$	−10209.3	−0.756173
$568$	0	0
$569$	1888.06	0.139107	0.0695534	−	0.997578i	$-0.477843\pi$
$569$	1888.06	0.139107	0.0695534	+	0.997578i	$0.477843\pi$
$570$	0	0
$571$	13391.6	0.981475	0.490737	−	0.871308i	$-0.336727\pi$
$571$	13391.6	0.981475	0.490737	+	0.871308i	$0.336727\pi$
$572$	0	0
$573$	−3747.09	−0.273189
$574$	0	0
$575$	−2101.95	−0.152448
$576$	0	0
$577$	−6168.96	−0.445090	−0.222545	−	0.974922i	$-0.571436\pi$
$577$	−6168.96	−0.445090	−0.222545	+	0.974922i	$0.571436\pi$
$578$	0	0
$579$	216.385	0.0155313
$580$	0	0
$581$	23054.0	1.64620
$582$	0	0
$583$	4237.04	0.300995
$584$	0	0
$585$	−4081.72	−0.288476
$586$	0	0
$587$	6940.95	0.488047	0.244024	−	0.969769i	$-0.421533\pi$
$587$	6940.95	0.488047	0.244024	+	0.969769i	$0.421533\pi$
$588$	0	0
$589$	8912.68	0.623498
$590$	0	0
$591$	−4723.41	−0.328757
$592$	0	0
$593$	−6994.17	−0.484344	−0.242172	−	0.970233i	$-0.577860\pi$
$593$	−6994.17	−0.484344	−0.242172	+	0.970233i	$0.577860\pi$
$594$	0	0
$595$	−6867.24	−0.473158
$596$	0	0
$597$	−3015.12	−0.206701
$598$	0	0
$599$	−11522.0	−0.785934	−0.392967	−	0.919553i	$-0.628551\pi$
$599$	−11522.0	−0.785934	−0.392967	+	0.919553i	$0.628551\pi$
$600$	0	0
$601$	5272.41	0.357847	0.178923	−	0.983863i	$-0.442739\pi$
$601$	5272.41	0.357847	0.178923	+	0.983863i	$0.442739\pi$
$602$	0	0
$603$	14047.0	0.948651
$604$	0	0
$605$	−4007.90	−0.269330
$606$	0	0
$607$	−27509.3	−1.83949	−0.919743	−	0.392521i	$-0.871603\pi$
$607$	−27509.3	−1.83949	−0.919743	+	0.392521i	$0.871603\pi$
$608$	0	0
$609$	2130.00	0.141728
$610$	0	0
$611$	−11434.6	−0.757110
$612$	0	0
$613$	−750.042	−0.0494191	−0.0247095	−	0.999695i	$-0.507866\pi$
$613$	−750.042	−0.0494191	−0.0247095	+	0.999695i	$0.507866\pi$
$614$	0	0
$615$	115.237	0.00755576
$616$	0	0
$617$	23823.7	1.55447	0.777234	−	0.629211i	$-0.216622\pi$
$617$	23823.7	1.55447	0.777234	+	0.629211i	$0.216622\pi$
$618$	0	0
$619$	24366.0	1.58215	0.791075	−	0.611719i	$-0.209522\pi$
$619$	24366.0	1.58215	0.791075	+	0.611719i	$0.209522\pi$
$620$	0	0
$621$	1175.79	0.0759786
$622$	0	0
$623$	−17115.3	−1.10066
$624$	0	0
$625$	4150.60	0.265638
$626$	0	0
$627$	680.946	0.0433722
$628$	0	0
$629$	2673.33	0.169464
$630$	0	0
$631$	−24469.9	−1.54379	−0.771895	−	0.635751i	$-0.780691\pi$
$631$	−24469.9	−1.54379	−0.771895	+	0.635751i	$0.780691\pi$
$632$	0	0
$633$	3936.97	0.247205
$634$	0	0
$635$	−5357.12	−0.334789
$636$	0	0
$637$	−2695.98	−0.167690
$638$	0	0
$639$	−19228.3	−1.19039
$640$	0	0
$641$	−9957.55	−0.613572	−0.306786	−	0.951779i	$-0.599254\pi$
$641$	−9957.55	−0.613572	−0.306786	+	0.951779i	$0.599254\pi$
$642$	0	0
$643$	5726.97	0.351243	0.175622	−	0.984458i	$-0.443806\pi$
$643$	5726.97	0.351243	0.175622	+	0.984458i	$0.443806\pi$
$644$	0	0
$645$	1012.48	0.0618086
$646$	0	0
$647$	−3278.53	−0.199216	−0.0996078	−	0.995027i	$-0.531759\pi$
$647$	−3278.53	−0.199216	−0.0996078	+	0.995027i	$0.531759\pi$
$648$	0	0
$649$	1837.43	0.111133
$650$	0	0
$651$	−4789.55	−0.288352
$652$	0	0
$653$	−21747.0	−1.30326	−0.651628	−	0.758539i	$-0.725914\pi$
$653$	−21747.0	−1.30326	−0.651628	+	0.758539i	$0.725914\pi$
$654$	0	0
$655$	−11765.3	−0.701846
$656$	0	0
$657$	−128.590	−0.00763586
$658$	0	0
$659$	−6140.69	−0.362986	−0.181493	−	0.983392i	$-0.558093\pi$
$659$	−6140.69	−0.362986	−0.181493	+	0.983392i	$0.558093\pi$
$660$	0	0
$661$	−22278.8	−1.31096	−0.655479	−	0.755213i	$-0.727533\pi$
$661$	−22278.8	−1.31096	−0.655479	+	0.755213i	$0.727533\pi$
$662$	0	0
$663$	−1975.84	−0.115740
$664$	0	0
$665$	2526.88	0.147351
$666$	0	0
$667$	3261.56	0.189337
$668$	0	0
$669$	437.972	0.0253109
$670$	0	0
$671$	−18841.2	−1.08399
$672$	0	0
$673$	18127.3	1.03827	0.519134	−	0.854693i	$-0.326254\pi$
$673$	18127.3	1.03827	0.519134	+	0.854693i	$0.326254\pi$
$674$	0	0
$675$	−4671.92	−0.266403
$676$	0	0
$677$	−1644.29	−0.0933460	−0.0466730	−	0.998910i	$-0.514862\pi$
$677$	−1644.29	−0.0933460	−0.0466730	+	0.998910i	$0.514862\pi$
$678$	0	0
$679$	9879.06	0.558356
$680$	0	0
$681$	−2805.69	−0.157877
$682$	0	0
$683$	16616.0	0.930883	0.465441	−	0.885079i	$-0.345896\pi$
$683$	16616.0	0.930883	0.465441	+	0.885079i	$0.345896\pi$
$684$	0	0
$685$	8822.37	0.492096
$686$	0	0
$687$	2061.81	0.114502
$688$	0	0
$689$	4523.82	0.250136
$690$	0	0
$691$	12878.7	0.709014	0.354507	−	0.935053i	$-0.384649\pi$
$691$	12878.7	0.709014	0.354507	+	0.935053i	$0.384649\pi$
$692$	0	0
$693$	10282.7	0.563649
$694$	0	0
$695$	−8735.24	−0.476758
$696$	0	0
$697$	−1567.51	−0.0851844
$698$	0	0
$699$	2716.69	0.147002
$700$	0	0
$701$	13974.9	0.752959	0.376480	−	0.926425i	$-0.377134\pi$
$701$	13974.9	0.752959	0.376480	+	0.926425i	$0.377134\pi$
$702$	0	0
$703$	−983.684	−0.0527743
$704$	0	0
$705$	2364.66	0.126324
$706$	0	0
$707$	−10737.1	−0.571158
$708$	0	0
$709$	4977.21	0.263643	0.131822	−	0.991273i	$-0.457917\pi$
$709$	4977.21	0.263643	0.131822	+	0.991273i	$0.457917\pi$
$710$	0	0
$711$	5645.79	0.297797
$712$	0	0
$713$	−7333.98	−0.385217
$714$	0	0
$715$	3959.58	0.207105
$716$	0	0
$717$	−3522.25	−0.183460
$718$	0	0
$719$	−26016.5	−1.34944	−0.674722	−	0.738072i	$-0.735737\pi$
$719$	−26016.5	−1.34944	−0.674722	+	0.738072i	$0.735737\pi$
$720$	0	0
$721$	18111.4	0.935509
$722$	0	0
$723$	4848.19	0.249386
$724$	0	0
$725$	−12959.6	−0.663873
$726$	0	0
$727$	22984.9	1.17258	0.586288	−	0.810103i	$-0.300589\pi$
$727$	22984.9	1.17258	0.586288	+	0.810103i	$0.300589\pi$
$728$	0	0
$729$	−15740.1	−0.799680
$730$	0	0
$731$	−13772.3	−0.696836
$732$	0	0
$733$	12741.6	0.642047	0.321024	−	0.947071i	$-0.395973\pi$
$733$	12741.6	0.642047	0.321024	+	0.947071i	$0.395973\pi$
$734$	0	0
$735$	557.525	0.0279791
$736$	0	0
$737$	−13626.6	−0.681062
$738$	0	0
$739$	−12371.8	−0.615839	−0.307920	−	0.951412i	$-0.599633\pi$
$739$	−12371.8	−0.615839	−0.307920	+	0.951412i	$0.599633\pi$
$740$	0	0
$741$	727.036	0.0360436
$742$	0	0
$743$	−15596.6	−0.770098	−0.385049	−	0.922896i	$-0.625815\pi$
$743$	−15596.6	−0.770098	−0.385049	+	0.922896i	$0.625815\pi$
$744$	0	0
$745$	−5198.99	−0.255673
$746$	0	0
$747$	38545.6	1.88797
$748$	0	0
$749$	17658.6	0.861456
$750$	0	0
$751$	15907.2	0.772917	0.386458	−	0.922307i	$-0.373698\pi$
$751$	15907.2	0.772917	0.386458	+	0.922307i	$0.373698\pi$
$752$	0	0
$753$	−7610.02	−0.368293
$754$	0	0
$755$	6762.43	0.325974
$756$	0	0
$757$	−24564.2	−1.17940	−0.589698	−	0.807624i	$-0.700753\pi$
$757$	−24564.2	−1.17940	−0.589698	+	0.807624i	$0.700753\pi$
$758$	0	0
$759$	−560.331	−0.0267967
$760$	0	0
$761$	−3196.90	−0.152283	−0.0761417	−	0.997097i	$-0.524260\pi$
$761$	−3196.90	−0.152283	−0.0761417	+	0.997097i	$0.524260\pi$
$762$	0	0
$763$	−3645.95	−0.172991
$764$	0	0
$765$	−11481.8	−0.542648
$766$	0	0
$767$	1961.80	0.0923551
$768$	0	0
$769$	−35102.3	−1.64606	−0.823031	−	0.567997i	$-0.807718\pi$
$769$	−35102.3	−1.64606	−0.823031	+	0.567997i	$0.807718\pi$
$770$	0	0
$771$	−3108.38	−0.145195
$772$	0	0
$773$	16140.2	0.750998	0.375499	−	0.926823i	$-0.377471\pi$
$773$	16140.2	0.750998	0.375499	+	0.926823i	$0.377471\pi$
$774$	0	0
$775$	29141.1	1.35068
$776$	0	0
$777$	528.619	0.0244068
$778$	0	0
$779$	576.783	0.0265281
$780$	0	0
$781$	18652.9	0.854616
$782$	0	0
$783$	7249.34	0.330869
$784$	0	0
$785$	−11157.4	−0.507294
$786$	0	0
$787$	−29569.8	−1.33933	−0.669663	−	0.742665i	$-0.733561\pi$
$787$	−29569.8	−1.33933	−0.669663	+	0.742665i	$0.733561\pi$
$788$	0	0
$789$	2169.54	0.0978933
$790$	0	0
$791$	21430.3	0.963303
$792$	0	0
$793$	−20116.5	−0.900828
$794$	0	0
$795$	−935.522	−0.0417353
$796$	0	0
$797$	10435.7	0.463803	0.231901	−	0.972739i	$-0.425505\pi$
$797$	10435.7	0.463803	0.231901	+	0.972739i	$0.425505\pi$
$798$	0	0
$799$	−32165.3	−1.42419
$800$	0	0
$801$	−28616.3	−1.26230
$802$	0	0
$803$	124.742	0.00548199
$804$	0	0
$805$	−2079.30	−0.0910381
$806$	0	0
$807$	2730.30	0.119097
$808$	0	0
$809$	29073.7	1.26350	0.631752	−	0.775170i	$-0.282336\pi$
$809$	29073.7	1.26350	0.631752	+	0.775170i	$0.282336\pi$
$810$	0	0
$811$	−4533.07	−0.196273	−0.0981367	−	0.995173i	$-0.531288\pi$
$811$	−4533.07	−0.196273	−0.0981367	+	0.995173i	$0.531288\pi$
$812$	0	0
$813$	−4946.20	−0.213371
$814$	0	0
$815$	−13771.0	−0.591873
$816$	0	0
$817$	5067.69	0.217009
$818$	0	0
$819$	10978.7	0.468410
$820$	0	0
$821$	8267.86	0.351462	0.175731	−	0.984438i	$-0.443771\pi$
$821$	8267.86	0.351462	0.175731	+	0.984438i	$0.443771\pi$
$822$	0	0
$823$	−1310.08	−0.0554879	−0.0277440	−	0.999615i	$-0.508832\pi$
$823$	−1310.08	−0.0554879	−0.0277440	+	0.999615i	$0.508832\pi$
$824$	0	0
$825$	2226.44	0.0939572
$826$	0	0
$827$	−40271.1	−1.69330	−0.846652	−	0.532147i	$-0.821385\pi$
$827$	−40271.1	−1.69330	−0.846652	+	0.532147i	$0.821385\pi$
$828$	0	0
$829$	−38395.1	−1.60858	−0.804292	−	0.594235i	$-0.797455\pi$
$829$	−38395.1	−1.60858	−0.804292	+	0.594235i	$0.797455\pi$
$830$	0	0
$831$	−6552.05	−0.273512
$832$	0	0
$833$	−7583.74	−0.315439
$834$	0	0
$835$	2247.75	0.0931575
$836$	0	0
$837$	−16300.9	−0.673170
$838$	0	0
$839$	790.966	0.0325473	0.0162736	−	0.999868i	$-0.494820\pi$
$839$	790.966	0.0325473	0.0162736	+	0.999868i	$0.494820\pi$
$840$	0	0
$841$	−4279.79	−0.175480
$842$	0	0
$843$	6669.07	0.272473
$844$	0	0
$845$	−8509.51	−0.346433
$846$	0	0
$847$	10780.2	0.437321
$848$	0	0
$849$	244.188	0.00987102
$850$	0	0
$851$	809.445	0.0326057
$852$	0	0
$853$	29265.1	1.17470	0.587350	−	0.809333i	$-0.300171\pi$
$853$	29265.1	1.17470	0.587350	+	0.809333i	$0.300171\pi$
$854$	0	0
$855$	4224.87	0.168991
$856$	0	0
$857$	24737.1	0.985999	0.493000	−	0.870029i	$-0.335900\pi$
$857$	24737.1	0.985999	0.493000	+	0.870029i	$0.335900\pi$
$858$	0	0
$859$	5563.86	0.220997	0.110498	−	0.993876i	$-0.464755\pi$
$859$	5563.86	0.220997	0.110498	+	0.993876i	$0.464755\pi$
$860$	0	0
$861$	−309.956	−0.0122686
$862$	0	0
$863$	−3383.53	−0.133461	−0.0667305	−	0.997771i	$-0.521257\pi$
$863$	−3383.53	−0.133461	−0.0667305	+	0.997771i	$0.521257\pi$
$864$	0	0
$865$	16692.0	0.656123
$866$	0	0
$867$	−825.622	−0.0323409
$868$	0	0
$869$	−5476.84	−0.213796
$870$	0	0
$871$	−14548.9	−0.565983
$872$	0	0
$873$	16517.5	0.640357
$874$	0	0
$875$	19562.5	0.755810
$876$	0	0
$877$	29297.8	1.12807	0.564034	−	0.825752i	$-0.309249\pi$
$877$	29297.8	1.12807	0.564034	+	0.825752i	$0.309249\pi$
$878$	0	0
$879$	4537.94	0.174131
$880$	0	0
$881$	15974.0	0.610871	0.305436	−	0.952213i	$-0.401198\pi$
$881$	15974.0	0.610871	0.305436	+	0.952213i	$0.401198\pi$
$882$	0	0
$883$	−43736.3	−1.66687	−0.833433	−	0.552621i	$-0.813628\pi$
$883$	−43736.3	−1.66687	−0.833433	+	0.552621i	$0.813628\pi$
$884$	0	0
$885$	−405.697	−0.0154095
$886$	0	0
$887$	−14583.1	−0.552032	−0.276016	−	0.961153i	$-0.589014\pi$
$887$	−14583.1	−0.552032	−0.276016	+	0.961153i	$0.589014\pi$
$888$	0	0
$889$	14409.2	0.543610
$890$	0	0
$891$	16558.8	0.622605
$892$	0	0
$893$	11835.6	0.443521
$894$	0	0
$895$	−3681.81	−0.137508
$896$	0	0
$897$	−598.257	−0.0222689
$898$	0	0
$899$	−45217.8	−1.67753
$900$	0	0
$901$	12725.4	0.470528
$902$	0	0
$903$	−2723.31	−0.100361
$904$	0	0
$905$	3084.95	0.113312
$906$	0	0
$907$	34233.8	1.25327	0.626635	−	0.779313i	$-0.284432\pi$
$907$	34233.8	1.25327	0.626635	+	0.779313i	$0.284432\pi$
$908$	0	0
$909$	−17952.0	−0.655040
$910$	0	0
$911$	−13961.5	−0.507756	−0.253878	−	0.967236i	$-0.581706\pi$
$911$	−13961.5	−0.507756	−0.253878	+	0.967236i	$0.581706\pi$
$912$	0	0
$913$	−37392.2	−1.35542
$914$	0	0
$915$	4160.06	0.150303
$916$	0	0
$917$	31645.5	1.13962
$918$	0	0
$919$	12100.2	0.434330	0.217165	−	0.976135i	$-0.430319\pi$
$919$	12100.2	0.434330	0.217165	+	0.976135i	$0.430319\pi$
$920$	0	0
$921$	−3021.65	−0.108107
$922$	0	0
$923$	19915.5	0.710212
$924$	0	0
$925$	−3216.28	−0.114325
$926$	0	0
$927$	30281.6	1.07290
$928$	0	0
$929$	−39336.8	−1.38924	−0.694618	−	0.719379i	$-0.744426\pi$
$929$	−39336.8	−1.38924	−0.694618	+	0.719379i	$0.744426\pi$
$930$	0	0
$931$	2790.53	0.0982340
$932$	0	0
$933$	−5892.89	−0.206779
$934$	0	0
$935$	11138.2	0.389581
$936$	0	0
$937$	−8756.73	−0.305304	−0.152652	−	0.988280i	$-0.548781\pi$
$937$	−8756.73	−0.305304	−0.152652	+	0.988280i	$0.548781\pi$
$938$	0	0
$939$	1083.08	0.0376412
$940$	0	0
$941$	1141.38	0.0395408	0.0197704	−	0.999805i	$-0.493706\pi$
$941$	1141.38	0.0395408	0.0197704	+	0.999805i	$0.493706\pi$
$942$	0	0
$943$	−474.618	−0.0163899
$944$	0	0
$945$	−4621.58	−0.159090
$946$	0	0
$947$	49023.2	1.68220	0.841098	−	0.540883i	$-0.181910\pi$
$947$	49023.2	1.68220	0.841098	+	0.540883i	$0.181910\pi$
$948$	0	0
$949$	133.185	0.00455570
$950$	0	0
$951$	1898.16	0.0647236
$952$	0	0
$953$	6650.09	0.226042	0.113021	−	0.993593i	$-0.463947\pi$
$953$	6650.09	0.226042	0.113021	+	0.993593i	$0.463947\pi$
$954$	0	0
$955$	22552.8	0.764180
$956$	0	0
$957$	−3454.73	−0.116693
$958$	0	0
$959$	−23729.8	−0.799036
$960$	0	0
$961$	71886.3	2.41302
$962$	0	0
$963$	29524.6	0.987971
$964$	0	0
$965$	−1302.37	−0.0434452
$966$	0	0
$967$	24268.0	0.807039	0.403519	−	0.914971i	$-0.367787\pi$
$967$	24268.0	0.807039	0.403519	+	0.914971i	$0.367787\pi$
$968$	0	0
$969$	2045.14	0.0678012
$970$	0	0
$971$	−4819.80	−0.159294	−0.0796471	−	0.996823i	$-0.525379\pi$
$971$	−4819.80	−0.159294	−0.0796471	+	0.996823i	$0.525379\pi$
$972$	0	0
$973$	23495.4	0.774131
$974$	0	0
$975$	2377.14	0.0780813
$976$	0	0
$977$	34624.7	1.13382	0.566909	−	0.823780i	$-0.308139\pi$
$977$	34624.7	1.13382	0.566909	+	0.823780i	$0.308139\pi$
$978$	0	0
$979$	27759.9	0.906242
$980$	0	0
$981$	−6095.92	−0.198397
$982$	0	0
$983$	25428.2	0.825061	0.412531	−	0.910944i	$-0.364645\pi$
$983$	25428.2	0.825061	0.412531	+	0.910944i	$0.364645\pi$
$984$	0	0
$985$	28429.0	0.919618
$986$	0	0
$987$	−6360.31	−0.205117
$988$	0	0
$989$	−4170.05	−0.134075
$990$	0	0
$991$	59011.0	1.89157	0.945786	−	0.324792i	$-0.105294\pi$
$991$	59011.0	1.89157	0.945786	+	0.324792i	$0.105294\pi$
$992$	0	0
$993$	−3354.77	−0.107211
$994$	0	0
$995$	18147.3	0.578198
$996$	0	0
$997$	−36222.9	−1.15064	−0.575321	−	0.817928i	$-0.695123\pi$
$997$	−36222.9	−1.15064	−0.575321	+	0.817928i	$0.695123\pi$
$998$	0	0
$999$	1799.12	0.0569787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	736.4.a.e.1.5	✓	8
4.3	odd	2		736.4.a.f.1.4	yes	8
8.3	odd	2		1472.4.a.bg.1.5		8
8.5	even	2		1472.4.a.bh.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.e.1.5	✓	8	1.1	even	1	trivial
736.4.a.f.1.4	yes	8	4.3	odd	2
1472.4.a.bg.1.5		8	8.3	odd	2
1472.4.a.bh.1.4		8	8.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	736.a (trivial)

\(f(q)\)	\(=\)	\(q-0.963239 q^{3} +5.79749 q^{5} -15.5937 q^{7} -26.0722 q^{9} +25.2920 q^{11} +27.0039 q^{13} -5.58437 q^{15} +75.9614 q^{17} -27.9509 q^{19} +15.0205 q^{21} +23.0000 q^{23} -91.3891 q^{25} +51.1212 q^{27} +141.807 q^{29} -318.869 q^{31} -24.3622 q^{33} -90.4043 q^{35} +35.1933 q^{37} -26.0112 q^{39} -20.6356 q^{41} -181.307 q^{43} -151.153 q^{45} -423.443 q^{47} -99.8367 q^{49} -73.1690 q^{51} +167.525 q^{53} +146.630 q^{55} +26.9234 q^{57} +72.6487 q^{59} -744.947 q^{61} +406.561 q^{63} +156.555 q^{65} -538.772 q^{67} -22.1545 q^{69} +737.505 q^{71} +4.93206 q^{73} +88.0295 q^{75} -394.395 q^{77} -216.545 q^{79} +654.707 q^{81} -1478.42 q^{83} +440.386 q^{85} -136.594 q^{87} +1097.58 q^{89} -421.090 q^{91} +307.147 q^{93} -162.045 q^{95} -633.529 q^{97} -659.416 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} - 12 q^{5} + 14 q^{7} + 90 q^{9} - 88 q^{11} - 30 q^{13} - 30 q^{15} + 58 q^{17} - 190 q^{19} - 66 q^{21} + 184 q^{23} + 28 q^{25} - 432 q^{27} + 190 q^{29} + 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 8 * q - 12 * q^3 - 12 * q^5 + 14 * q^7 + 90 * q^9 - 88 * q^11 - 30 * q^13 - 30 * q^15 + 58 * q^17 - 190 * q^19 - 66 * q^21 + 184 * q^23 + 28 * q^25 - 432 * q^27 + 190 * q^29 + 60 * q^31 + 346 * q^33 - 192 * q^35 - 156 * q^37 - 156 * q^39 + 282 * q^41 - 810 * q^43 + 358 * q^45 - 564 * q^47 + 752 * q^49 - 1614 * q^51 - 230 * q^53 - 1292 * q^55 + 796 * q^57 - 1916 * q^59 + 22 * q^61 - 1036 * q^63 + 222 * q^65 - 2292 * q^67 - 276 * q^69 - 2376 * q^71 - 630 * q^73 - 3952 * q^75 - 948 * q^77 - 892 * q^79 - 1044 * q^81 - 2800 * q^83 + 288 * q^85 - 2756 * q^87 - 308 * q^89 - 4610 * q^91 - 982 * q^93 - 1300 * q^95 + 854 * q^97 - 5986 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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