LMFDB - Embedded newform 4304.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 $ x^7 - x^6 - 10x^5 + 7x^4 + 27x^3 - 15x^2 - 20*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.49844$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.370477 q^{3} +2.98038 q^{5} +2.56529 q^{7} -2.86275 q^{9} +5.77573 q^{11} -0.953732 q^{13} +1.10416 q^{15} +2.51668 q^{17} +7.19922 q^{19} +0.950379 q^{21} +5.17399 q^{23} +3.88265 q^{25} -2.17201 q^{27} +4.63236 q^{29} -8.16921 q^{31} +2.13977 q^{33} +7.64552 q^{35} -9.43775 q^{37} -0.353336 q^{39} +2.46810 q^{41} -5.86496 q^{43} -8.53207 q^{45} +9.50969 q^{47} -0.419304 q^{49} +0.932373 q^{51} -7.51226 q^{53} +17.2139 q^{55} +2.66714 q^{57} +1.01485 q^{59} -5.86716 q^{61} -7.34377 q^{63} -2.84248 q^{65} -12.1288 q^{67} +1.91685 q^{69} +12.3537 q^{71} +0.650878 q^{73} +1.43843 q^{75} +14.8164 q^{77} -15.5158 q^{79} +7.78356 q^{81} +8.27171 q^{83} +7.50067 q^{85} +1.71618 q^{87} +2.89852 q^{89} -2.44660 q^{91} -3.02650 q^{93} +21.4564 q^{95} +0.956247 q^{97} -16.5345 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{3} - 6 q^{5} + 3 q^{7} + 7 q^{9} + 12 q^{11} + 3 q^{13} + 6 q^{15} - 8 q^{17} + 7 q^{19} - 5 q^{21} + 22 q^{23} + 9 q^{25} + 22 q^{27} - 7 q^{29} + 3 q^{31} - 19 q^{33} + 25 q^{35} - 13 q^{37}+ \cdots + 9 q^{99}+O(q^{100}) $ 7 * q + 4 * q^3 - 6 * q^5 + 3 * q^7 + 7 * q^9 + 12 * q^11 + 3 * q^13 + 6 * q^15 - 8 * q^17 + 7 * q^19 - 5 * q^21 + 22 * q^23 + 9 * q^25 + 22 * q^27 - 7 * q^29 + 3 * q^31 - 19 * q^33 + 25 * q^35 - 13 * q^37 - q^39 - 11 * q^41 + 14 * q^43 - 17 * q^45 + 23 * q^47 + 6 * q^49 + 21 * q^51 - 21 * q^53 - 15 * q^55 - 14 * q^57 + 20 * q^59 + 13 * q^61 + 12 * q^63 - 29 * q^65 + 30 * q^67 + 13 * q^69 + 23 * q^71 + 9 * q^73 - 3 * q^75 + 8 * q^77 - 11 * q^79 + 43 * q^81 + 3 * q^83 + 21 * q^85 - 28 * q^87 + 16 * q^89 + 10 * q^91 - 13 * q^93 + 21 * q^97 + 9 * 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
$3$	0.370477	0.213895	0.106947	−	0.994265i	$-0.465892\pi$
$3$	0.370477	0.213895	0.106947	+	0.994265i	$0.465892\pi$
$4$	0	0
$5$	2.98038	1.33287	0.666433	−	0.745565i	$-0.267820\pi$
$5$	2.98038	1.33287	0.666433	+	0.745565i	$0.267820\pi$
$6$	0	0
$7$	2.56529	0.969587	0.484794	−	0.874629i	$-0.338895\pi$
$7$	2.56529	0.969587	0.484794	+	0.874629i	$0.338895\pi$
$8$	0	0
$9$	−2.86275	−0.954249
$10$	0	0
$11$	5.77573	1.74145	0.870724	−	0.491772i	$-0.163651\pi$
$11$	5.77573	1.74145	0.870724	+	0.491772i	$0.163651\pi$
$12$	0	0
$13$	−0.953732	−0.264518	−0.132259	−	0.991215i	$-0.542223\pi$
$13$	−0.953732	−0.264518	−0.132259	+	0.991215i	$0.542223\pi$
$14$	0	0
$15$	1.10416	0.285093
$16$	0	0
$17$	2.51668	0.610385	0.305193	−	0.952291i	$-0.401279\pi$
$17$	2.51668	0.610385	0.305193	+	0.952291i	$0.401279\pi$
$18$	0	0
$19$	7.19922	1.65161	0.825807	−	0.563953i	$-0.190720\pi$
$19$	7.19922	1.65161	0.825807	+	0.563953i	$0.190720\pi$
$20$	0	0
$21$	0.950379	0.207390
$22$	0	0
$23$	5.17399	1.07885	0.539426	−	0.842033i	$-0.318641\pi$
$23$	5.17399	1.07885	0.539426	+	0.842033i	$0.318641\pi$
$24$	0	0
$25$	3.88265	0.776530
$26$	0	0
$27$	−2.17201	−0.418004
$28$	0	0
$29$	4.63236	0.860208	0.430104	−	0.902779i	$-0.358477\pi$
$29$	4.63236	0.860208	0.430104	+	0.902779i	$0.358477\pi$
$30$	0	0
$31$	−8.16921	−1.46723	−0.733617	−	0.679563i	$-0.762169\pi$
$31$	−8.16921	−1.46723	−0.733617	+	0.679563i	$0.762169\pi$
$32$	0	0
$33$	2.13977	0.372487
$34$	0	0
$35$	7.64552	1.29233
$36$	0	0
$37$	−9.43775	−1.55156	−0.775778	−	0.631006i	$-0.782642\pi$
$37$	−9.43775	−1.55156	−0.775778	+	0.631006i	$0.782642\pi$
$38$	0	0
$39$	−0.353336	−0.0565790
$40$	0	0
$41$	2.46810	0.385453	0.192726	−	0.981253i	$-0.438267\pi$
$41$	2.46810	0.385453	0.192726	+	0.981253i	$0.438267\pi$
$42$	0	0
$43$	−5.86496	−0.894398	−0.447199	−	0.894435i	$-0.647578\pi$
$43$	−5.86496	−0.894398	−0.447199	+	0.894435i	$0.647578\pi$
$44$	0	0
$45$	−8.53207	−1.27189
$46$	0	0
$47$	9.50969	1.38713	0.693565	−	0.720394i	$-0.256039\pi$
$47$	9.50969	1.38713	0.693565	+	0.720394i	$0.256039\pi$
$48$	0	0
$49$	−0.419304	−0.0599006
$50$	0	0
$51$	0.932373	0.130558
$52$	0	0
$53$	−7.51226	−1.03189	−0.515944	−	0.856622i	$-0.672559\pi$
$53$	−7.51226	−1.03189	−0.515944	+	0.856622i	$0.672559\pi$
$54$	0	0
$55$	17.2139	2.32112
$56$	0	0
$57$	2.66714	0.353272
$58$	0	0
$59$	1.01485	0.132122	0.0660610	−	0.997816i	$-0.478957\pi$
$59$	1.01485	0.132122	0.0660610	+	0.997816i	$0.478957\pi$
$60$	0	0
$61$	−5.86716	−0.751212	−0.375606	−	0.926779i	$-0.622565\pi$
$61$	−5.86716	−0.751212	−0.375606	+	0.926779i	$0.622565\pi$
$62$	0	0
$63$	−7.34377	−0.925228
$64$	0	0
$65$	−2.84248	−0.352566
$66$	0	0
$67$	−12.1288	−1.48177	−0.740883	−	0.671634i	$-0.765593\pi$
$67$	−12.1288	−1.48177	−0.740883	+	0.671634i	$0.765593\pi$
$68$	0	0
$69$	1.91685	0.230761
$70$	0	0
$71$	12.3537	1.46611	0.733057	−	0.680167i	$-0.238093\pi$
$71$	12.3537	1.46611	0.733057	+	0.680167i	$0.238093\pi$
$72$	0	0
$73$	0.650878	0.0761795	0.0380897	−	0.999274i	$-0.487873\pi$
$73$	0.650878	0.0761795	0.0380897	+	0.999274i	$0.487873\pi$
$74$	0	0
$75$	1.43843	0.166096
$76$	0	0
$77$	14.8164	1.68849
$78$	0	0
$79$	−15.5158	−1.74567	−0.872834	−	0.488018i	$-0.837720\pi$
$79$	−15.5158	−1.74567	−0.872834	+	0.488018i	$0.837720\pi$
$80$	0	0
$81$	7.78356	0.864840
$82$	0	0
$83$	8.27171	0.907939	0.453969	−	0.891017i	$-0.350008\pi$
$83$	8.27171	0.907939	0.453969	+	0.891017i	$0.350008\pi$
$84$	0	0
$85$	7.50067	0.813562
$86$	0	0
$87$	1.71618	0.183994
$88$	0	0
$89$	2.89852	0.307242	0.153621	−	0.988130i	$-0.450907\pi$
$89$	2.89852	0.307242	0.153621	+	0.988130i	$0.450907\pi$
$90$	0	0
$91$	−2.44660	−0.256473
$92$	0	0
$93$	−3.02650	−0.313834
$94$	0	0
$95$	21.4564	2.20138
$96$	0	0
$97$	0.956247	0.0970922	0.0485461	−	0.998821i	$-0.484541\pi$
$97$	0.956247	0.0970922	0.0485461	+	0.998821i	$0.484541\pi$
$98$	0	0
$99$	−16.5345	−1.66178
$100$	0	0
$101$	5.92549	0.589608	0.294804	−	0.955558i	$-0.404746\pi$
$101$	5.92549	0.589608	0.294804	+	0.955558i	$0.404746\pi$
$102$	0	0
$103$	7.57015	0.745909	0.372955	−	0.927850i	$-0.378345\pi$
$103$	7.57015	0.745909	0.372955	+	0.927850i	$0.378345\pi$
$104$	0	0
$105$	2.83249	0.276423
$106$	0	0
$107$	−16.0455	−1.55118	−0.775589	−	0.631238i	$-0.782547\pi$
$107$	−16.0455	−1.55118	−0.775589	+	0.631238i	$0.782547\pi$
$108$	0	0
$109$	2.50942	0.240359	0.120179	−	0.992752i	$-0.461653\pi$
$109$	2.50942	0.240359	0.120179	+	0.992752i	$0.461653\pi$
$110$	0	0
$111$	−3.49647	−0.331870
$112$	0	0
$113$	−13.4589	−1.26611	−0.633055	−	0.774107i	$-0.718199\pi$
$113$	−13.4589	−1.26611	−0.633055	+	0.774107i	$0.718199\pi$
$114$	0	0
$115$	15.4205	1.43796
$116$	0	0
$117$	2.73029	0.252416
$118$	0	0
$119$	6.45602	0.591822
$120$	0	0
$121$	22.3591	2.03264
$122$	0	0
$123$	0.914375	0.0824464
$124$	0	0
$125$	−3.33013	−0.297856
$126$	0	0
$127$	−2.35435	−0.208915	−0.104458	−	0.994529i	$-0.533311\pi$
$127$	−2.35435	−0.208915	−0.104458	+	0.994529i	$0.533311\pi$
$128$	0	0
$129$	−2.17283	−0.191307
$130$	0	0
$131$	−8.14252	−0.711415	−0.355708	−	0.934597i	$-0.615760\pi$
$131$	−8.14252	−0.711415	−0.355708	+	0.934597i	$0.615760\pi$
$132$	0	0
$133$	18.4681	1.60138
$134$	0	0
$135$	−6.47342	−0.557143
$136$	0	0
$137$	−13.5630	−1.15876	−0.579382	−	0.815056i	$-0.696706\pi$
$137$	−13.5630	−1.15876	−0.579382	+	0.815056i	$0.696706\pi$
$138$	0	0
$139$	17.6145	1.49405	0.747023	−	0.664798i	$-0.231482\pi$
$139$	17.6145	1.49405	0.747023	+	0.664798i	$0.231482\pi$
$140$	0	0
$141$	3.52312	0.296700
$142$	0	0
$143$	−5.50850	−0.460644
$144$	0	0
$145$	13.8062	1.14654
$146$	0	0
$147$	−0.155342	−0.0128124
$148$	0	0
$149$	4.73354	0.387787	0.193893	−	0.981023i	$-0.437888\pi$
$149$	4.73354	0.387787	0.193893	+	0.981023i	$0.437888\pi$
$150$	0	0
$151$	−18.5426	−1.50897	−0.754486	−	0.656316i	$-0.772114\pi$
$151$	−18.5426	−1.50897	−0.754486	+	0.656316i	$0.772114\pi$
$152$	0	0
$153$	−7.20463	−0.582460
$154$	0	0
$155$	−24.3473	−1.95562
$156$	0	0
$157$	6.62583	0.528799	0.264399	−	0.964413i	$-0.414826\pi$
$157$	6.62583	0.528799	0.264399	+	0.964413i	$0.414826\pi$
$158$	0	0
$159$	−2.78312	−0.220716
$160$	0	0
$161$	13.2728	1.04604
$162$	0	0
$163$	7.35217	0.575867	0.287933	−	0.957650i	$-0.407032\pi$
$163$	7.35217	0.575867	0.287933	+	0.957650i	$0.407032\pi$
$164$	0	0
$165$	6.37734	0.496475
$166$	0	0
$167$	18.4854	1.43045	0.715223	−	0.698896i	$-0.246325\pi$
$167$	18.4854	1.43045	0.715223	+	0.698896i	$0.246325\pi$
$168$	0	0
$169$	−12.0904	−0.930030
$170$	0	0
$171$	−20.6095	−1.57605
$172$	0	0
$173$	−15.2035	−1.15590	−0.577951	−	0.816071i	$-0.696148\pi$
$173$	−15.2035	−1.15590	−0.577951	+	0.816071i	$0.696148\pi$
$174$	0	0
$175$	9.96011	0.752913
$176$	0	0
$177$	0.375978	0.0282602
$178$	0	0
$179$	8.92937	0.667412	0.333706	−	0.942677i	$-0.391701\pi$
$179$	8.92937	0.667412	0.333706	+	0.942677i	$0.391701\pi$
$180$	0	0
$181$	8.96265	0.666189	0.333094	−	0.942894i	$-0.391907\pi$
$181$	8.96265	0.666189	0.333094	+	0.942894i	$0.391907\pi$
$182$	0	0
$183$	−2.17365	−0.160680
$184$	0	0
$185$	−28.1280	−2.06802
$186$	0	0
$187$	14.5357	1.06295
$188$	0	0
$189$	−5.57183	−0.405291
$190$	0	0
$191$	2.71179	0.196218	0.0981091	−	0.995176i	$-0.468721\pi$
$191$	2.71179	0.196218	0.0981091	+	0.995176i	$0.468721\pi$
$192$	0	0
$193$	−10.4616	−0.753042	−0.376521	−	0.926408i	$-0.622880\pi$
$193$	−10.4616	−0.753042	−0.376521	+	0.926408i	$0.622880\pi$
$194$	0	0
$195$	−1.05307	−0.0754122
$196$	0	0
$197$	−8.17270	−0.582280	−0.291140	−	0.956680i	$-0.594035\pi$
$197$	−8.17270	−0.582280	−0.291140	+	0.956680i	$0.594035\pi$
$198$	0	0
$199$	−23.4576	−1.66287	−0.831433	−	0.555625i	$-0.812479\pi$
$199$	−23.4576	−1.66287	−0.831433	+	0.555625i	$0.812479\pi$
$200$	0	0
$201$	−4.49343	−0.316942
$202$	0	0
$203$	11.8833	0.834047
$204$	0	0
$205$	7.35587	0.513757
$206$	0	0
$207$	−14.8118	−1.02949
$208$	0	0
$209$	41.5807	2.87620
$210$	0	0
$211$	−23.7236	−1.63320	−0.816600	−	0.577203i	$-0.804144\pi$
$211$	−23.7236	−1.63320	−0.816600	+	0.577203i	$0.804144\pi$
$212$	0	0
$213$	4.57676	0.313594
$214$	0	0
$215$	−17.4798	−1.19211
$216$	0	0
$217$	−20.9564	−1.42261
$218$	0	0
$219$	0.241135	0.0162944
$220$	0	0
$221$	−2.40024	−0.161458
$222$	0	0
$223$	4.63802	0.310585	0.155292	−	0.987869i	$-0.450368\pi$
$223$	4.63802	0.310585	0.155292	+	0.987869i	$0.450368\pi$
$224$	0	0
$225$	−11.1150	−0.741003
$226$	0	0
$227$	−2.30295	−0.152852	−0.0764261	−	0.997075i	$-0.524351\pi$
$227$	−2.30295	−0.152852	−0.0764261	+	0.997075i	$0.524351\pi$
$228$	0	0
$229$	1.98890	0.131430	0.0657150	−	0.997838i	$-0.479067\pi$
$229$	1.98890	0.131430	0.0657150	+	0.997838i	$0.479067\pi$
$230$	0	0
$231$	5.48913	0.361159
$232$	0	0
$233$	5.58928	0.366166	0.183083	−	0.983097i	$-0.441392\pi$
$233$	5.58928	0.366166	0.183083	+	0.983097i	$0.441392\pi$
$234$	0	0
$235$	28.3425	1.84886
$236$	0	0
$237$	−5.74826	−0.373389
$238$	0	0
$239$	3.34553	0.216404	0.108202	−	0.994129i	$-0.465491\pi$
$239$	3.34553	0.216404	0.108202	+	0.994129i	$0.465491\pi$
$240$	0	0
$241$	26.7139	1.72079	0.860395	−	0.509627i	$-0.170217\pi$
$241$	26.7139	1.72079	0.860395	+	0.509627i	$0.170217\pi$
$242$	0	0
$243$	9.39967	0.602989
$244$	0	0
$245$	−1.24968	−0.0798394
$246$	0	0
$247$	−6.86613	−0.436881
$248$	0	0
$249$	3.06448	0.194203
$250$	0	0
$251$	29.3076	1.84988	0.924940	−	0.380112i	$-0.124115\pi$
$251$	29.3076	1.84988	0.924940	+	0.380112i	$0.124115\pi$
$252$	0	0
$253$	29.8836	1.87877
$254$	0	0
$255$	2.77882	0.174017
$256$	0	0
$257$	10.3803	0.647504	0.323752	−	0.946142i	$-0.395056\pi$
$257$	10.3803	0.647504	0.323752	+	0.946142i	$0.395056\pi$
$258$	0	0
$259$	−24.2105	−1.50437
$260$	0	0
$261$	−13.2613	−0.820853
$262$	0	0
$263$	8.83916	0.545046	0.272523	−	0.962149i	$-0.412142\pi$
$263$	8.83916	0.545046	0.272523	+	0.962149i	$0.412142\pi$
$264$	0	0
$265$	−22.3894	−1.37537
$266$	0	0
$267$	1.07383	0.0657175
$268$	0	0
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	−7.39604	−0.449277	−0.224639	−	0.974442i	$-0.572120\pi$
$271$	−7.39604	−0.449277	−0.224639	+	0.974442i	$0.572120\pi$
$272$	0	0
$273$	−0.906407	−0.0548583
$274$	0	0
$275$	22.4251	1.35229
$276$	0	0
$277$	13.5773	0.815779	0.407890	−	0.913031i	$-0.366265\pi$
$277$	13.5773	0.815779	0.407890	+	0.913031i	$0.366265\pi$
$278$	0	0
$279$	23.3864	1.40011
$280$	0	0
$281$	13.5723	0.809653	0.404826	−	0.914394i	$-0.367332\pi$
$281$	13.5723	0.809653	0.404826	+	0.914394i	$0.367332\pi$
$282$	0	0
$283$	−20.5026	−1.21875	−0.609377	−	0.792880i	$-0.708581\pi$
$283$	−20.5026	−1.21875	−0.609377	+	0.792880i	$0.708581\pi$
$284$	0	0
$285$	7.94909	0.470864
$286$	0	0
$287$	6.33139	0.373730
$288$	0	0
$289$	−10.6663	−0.627430
$290$	0	0
$291$	0.354267	0.0207675
$292$	0	0
$293$	−26.0133	−1.51971	−0.759857	−	0.650090i	$-0.774731\pi$
$293$	−26.0133	−1.51971	−0.759857	+	0.650090i	$0.774731\pi$
$294$	0	0
$295$	3.02463	0.176101
$296$	0	0
$297$	−12.5450	−0.727932
$298$	0	0
$299$	−4.93460	−0.285376
$300$	0	0
$301$	−15.0453	−0.867197
$302$	0	0
$303$	2.19526	0.126114
$304$	0	0
$305$	−17.4863	−1.00126
$306$	0	0
$307$	17.3993	0.993031	0.496516	−	0.868028i	$-0.334613\pi$
$307$	17.3993	0.993031	0.496516	+	0.868028i	$0.334613\pi$
$308$	0	0
$309$	2.80457	0.159546
$310$	0	0
$311$	13.4740	0.764038	0.382019	−	0.924155i	$-0.375229\pi$
$311$	13.4740	0.764038	0.382019	+	0.924155i	$0.375229\pi$
$312$	0	0
$313$	−9.89195	−0.559126	−0.279563	−	0.960127i	$-0.590190\pi$
$313$	−9.89195	−0.559126	−0.279563	+	0.960127i	$0.590190\pi$
$314$	0	0
$315$	−21.8872	−1.23320
$316$	0	0
$317$	19.2200	1.07950	0.539751	−	0.841825i	$-0.318518\pi$
$317$	19.2200	1.07950	0.539751	+	0.841825i	$0.318518\pi$
$318$	0	0
$319$	26.7553	1.49801
$320$	0	0
$321$	−5.94449	−0.331789
$322$	0	0
$323$	18.1182	1.00812
$324$	0	0
$325$	−3.70301	−0.205406
$326$	0	0
$327$	0.929682	0.0514115
$328$	0	0
$329$	24.3951	1.34494
$330$	0	0
$331$	10.6168	0.583550	0.291775	−	0.956487i	$-0.405754\pi$
$331$	10.6168	0.583550	0.291775	+	0.956487i	$0.405754\pi$
$332$	0	0
$333$	27.0179	1.48057
$334$	0	0
$335$	−36.1483	−1.97499
$336$	0	0
$337$	−4.73621	−0.257998	−0.128999	−	0.991645i	$-0.541176\pi$
$337$	−4.73621	−0.257998	−0.128999	+	0.991645i	$0.541176\pi$
$338$	0	0
$339$	−4.98622	−0.270814
$340$	0	0
$341$	−47.1832	−2.55511
$342$	0	0
$343$	−19.0326	−1.02767
$344$	0	0
$345$	5.71292	0.307573
$346$	0	0
$347$	9.70333	0.520902	0.260451	−	0.965487i	$-0.416129\pi$
$347$	9.70333	0.520902	0.260451	+	0.965487i	$0.416129\pi$
$348$	0	0
$349$	−34.9874	−1.87283	−0.936416	−	0.350893i	$-0.885878\pi$
$349$	−34.9874	−1.87283	−0.936416	+	0.350893i	$0.885878\pi$
$350$	0	0
$351$	2.07152	0.110569
$352$	0	0
$353$	17.9950	0.957777	0.478888	−	0.877876i	$-0.341040\pi$
$353$	17.9950	0.957777	0.478888	+	0.877876i	$0.341040\pi$
$354$	0	0
$355$	36.8187	1.95413
$356$	0	0
$357$	2.39180	0.126588
$358$	0	0
$359$	8.64349	0.456186	0.228093	−	0.973639i	$-0.426751\pi$
$359$	8.64349	0.456186	0.228093	+	0.973639i	$0.426751\pi$
$360$	0	0
$361$	32.8287	1.72783
$362$	0	0
$363$	8.28351	0.434772
$364$	0	0
$365$	1.93986	0.101537
$366$	0	0
$367$	25.4599	1.32899	0.664497	−	0.747291i	$-0.268646\pi$
$367$	25.4599	1.32899	0.664497	+	0.747291i	$0.268646\pi$
$368$	0	0
$369$	−7.06555	−0.367818
$370$	0	0
$371$	−19.2711	−1.00051
$372$	0	0
$373$	−5.63673	−0.291859	−0.145929	−	0.989295i	$-0.546617\pi$
$373$	−5.63673	−0.291859	−0.145929	+	0.989295i	$0.546617\pi$
$374$	0	0
$375$	−1.23374	−0.0637098
$376$	0	0
$377$	−4.41803	−0.227540
$378$	0	0
$379$	16.2741	0.835944	0.417972	−	0.908460i	$-0.362741\pi$
$379$	16.2741	0.835944	0.417972	+	0.908460i	$0.362741\pi$
$380$	0	0
$381$	−0.872233	−0.0446859
$382$	0	0
$383$	−17.3292	−0.885483	−0.442742	−	0.896649i	$-0.645994\pi$
$383$	−17.3292	−0.885483	−0.442742	+	0.896649i	$0.645994\pi$
$384$	0	0
$385$	44.1585	2.25052
$386$	0	0
$387$	16.7899	0.853478
$388$	0	0
$389$	32.7406	1.66002	0.830008	−	0.557752i	$-0.188336\pi$
$389$	32.7406	1.66002	0.830008	+	0.557752i	$0.188336\pi$
$390$	0	0
$391$	13.0213	0.658516
$392$	0	0
$393$	−3.01661	−0.152168
$394$	0	0
$395$	−46.2430	−2.32674
$396$	0	0
$397$	−10.1894	−0.511392	−0.255696	−	0.966757i	$-0.582305\pi$
$397$	−10.1894	−0.511392	−0.255696	+	0.966757i	$0.582305\pi$
$398$	0	0
$399$	6.84199	0.342528
$400$	0	0
$401$	27.6214	1.37935	0.689673	−	0.724121i	$-0.257754\pi$
$401$	27.6214	1.37935	0.689673	+	0.724121i	$0.257754\pi$
$402$	0	0
$403$	7.79124	0.388109
$404$	0	0
$405$	23.1979	1.15272
$406$	0	0
$407$	−54.5099	−2.70195
$408$	0	0
$409$	−32.3617	−1.60018	−0.800092	−	0.599877i	$-0.795216\pi$
$409$	−32.3617	−1.60018	−0.800092	+	0.599877i	$0.795216\pi$
$410$	0	0
$411$	−5.02477	−0.247854
$412$	0	0
$413$	2.60338	0.128104
$414$	0	0
$415$	24.6528	1.21016
$416$	0	0
$417$	6.52578	0.319569
$418$	0	0
$419$	20.7391	1.01317	0.506586	−	0.862189i	$-0.330907\pi$
$419$	20.7391	1.01317	0.506586	+	0.862189i	$0.330907\pi$
$420$	0	0
$421$	−2.74188	−0.133631	−0.0668155	−	0.997765i	$-0.521284\pi$
$421$	−2.74188	−0.133631	−0.0668155	+	0.997765i	$0.521284\pi$
$422$	0	0
$423$	−27.2238	−1.32367
$424$	0	0
$425$	9.77140	0.473983
$426$	0	0
$427$	−15.0509	−0.728366
$428$	0	0
$429$	−2.04077	−0.0985294
$430$	0	0
$431$	1.93294	0.0931065	0.0465533	−	0.998916i	$-0.485176\pi$
$431$	1.93294	0.0931065	0.0465533	+	0.998916i	$0.485176\pi$
$432$	0	0
$433$	4.35794	0.209429	0.104715	−	0.994502i	$-0.466607\pi$
$433$	4.35794	0.209429	0.104715	+	0.994502i	$0.466607\pi$
$434$	0	0
$435$	5.11487	0.245239
$436$	0	0
$437$	37.2487	1.78185
$438$	0	0
$439$	−8.19465	−0.391109	−0.195555	−	0.980693i	$-0.562651\pi$
$439$	−8.19465	−0.391109	−0.195555	+	0.980693i	$0.562651\pi$
$440$	0	0
$441$	1.20036	0.0571601
$442$	0	0
$443$	5.51425	0.261990	0.130995	−	0.991383i	$-0.458183\pi$
$443$	5.51425	0.261990	0.130995	+	0.991383i	$0.458183\pi$
$444$	0	0
$445$	8.63867	0.409512
$446$	0	0
$447$	1.75367	0.0829456
$448$	0	0
$449$	−19.1545	−0.903956	−0.451978	−	0.892029i	$-0.649281\pi$
$449$	−19.1545	−0.903956	−0.451978	+	0.892029i	$0.649281\pi$
$450$	0	0
$451$	14.2551	0.671246
$452$	0	0
$453$	−6.86959	−0.322762
$454$	0	0
$455$	−7.29178	−0.341844
$456$	0	0
$457$	−41.6610	−1.94882	−0.974409	−	0.224783i	$-0.927833\pi$
$457$	−41.6610	−1.94882	−0.974409	+	0.224783i	$0.927833\pi$
$458$	0	0
$459$	−5.46627	−0.255144
$460$	0	0
$461$	28.7942	1.34108	0.670539	−	0.741874i	$-0.266063\pi$
$461$	28.7942	1.34108	0.670539	+	0.741874i	$0.266063\pi$
$462$	0	0
$463$	−1.89638	−0.0881321	−0.0440660	−	0.999029i	$-0.514031\pi$
$463$	−1.89638	−0.0881321	−0.0440660	+	0.999029i	$0.514031\pi$
$464$	0	0
$465$	−9.02012	−0.418298
$466$	0	0
$467$	9.53499	0.441227	0.220613	−	0.975361i	$-0.429194\pi$
$467$	9.53499	0.441227	0.220613	+	0.975361i	$0.429194\pi$
$468$	0	0
$469$	−31.1138	−1.43670
$470$	0	0
$471$	2.45472	0.113107
$472$	0	0
$473$	−33.8744	−1.55755
$474$	0	0
$475$	27.9520	1.28253
$476$	0	0
$477$	21.5057	0.984678
$478$	0	0
$479$	−11.2845	−0.515602	−0.257801	−	0.966198i	$-0.582998\pi$
$479$	−11.2845	−0.515602	−0.257801	+	0.966198i	$0.582998\pi$
$480$	0	0
$481$	9.00108	0.410414
$482$	0	0
$483$	4.91726	0.223743
$484$	0	0
$485$	2.84998	0.129411
$486$	0	0
$487$	−13.3515	−0.605014	−0.302507	−	0.953147i	$-0.597824\pi$
$487$	−13.3515	−0.605014	−0.302507	+	0.953147i	$0.597824\pi$
$488$	0	0
$489$	2.72381	0.123175
$490$	0	0
$491$	−37.9516	−1.71273	−0.856365	−	0.516370i	$-0.827283\pi$
$491$	−37.9516	−1.71273	−0.856365	+	0.516370i	$0.827283\pi$
$492$	0	0
$493$	11.6582	0.525058
$494$	0	0
$495$	−49.2789	−2.21492
$496$	0	0
$497$	31.6908	1.42153
$498$	0	0
$499$	27.6365	1.23718	0.618590	−	0.785714i	$-0.287704\pi$
$499$	27.6365	1.23718	0.618590	+	0.785714i	$0.287704\pi$
$500$	0	0
$501$	6.84842	0.305965
$502$	0	0
$503$	−20.9442	−0.933855	−0.466927	−	0.884296i	$-0.654639\pi$
$503$	−20.9442	−0.933855	−0.466927	+	0.884296i	$0.654639\pi$
$504$	0	0
$505$	17.6602	0.785868
$506$	0	0
$507$	−4.47921	−0.198929
$508$	0	0
$509$	−25.5377	−1.13194	−0.565969	−	0.824426i	$-0.691498\pi$
$509$	−25.5377	−1.13194	−0.565969	+	0.824426i	$0.691498\pi$
$510$	0	0
$511$	1.66969	0.0738627
$512$	0	0
$513$	−15.6368	−0.690381
$514$	0	0
$515$	22.5619	0.994197
$516$	0	0
$517$	54.9254	2.41562
$518$	0	0
$519$	−5.63255	−0.247242
$520$	0	0
$521$	−38.8849	−1.70358	−0.851788	−	0.523886i	$-0.824482\pi$
$521$	−38.8849	−1.70358	−0.851788	+	0.523886i	$0.824482\pi$
$522$	0	0
$523$	25.0299	1.09448	0.547240	−	0.836976i	$-0.315679\pi$
$523$	25.0299	1.09448	0.547240	+	0.836976i	$0.315679\pi$
$524$	0	0
$525$	3.68999	0.161044
$526$	0	0
$527$	−20.5593	−0.895578
$528$	0	0
$529$	3.77022	0.163922
$530$	0	0
$531$	−2.90525	−0.126077
$532$	0	0
$533$	−2.35391	−0.101959
$534$	0	0
$535$	−47.8217	−2.06751
$536$	0	0
$537$	3.30812	0.142756
$538$	0	0
$539$	−2.42179	−0.104314
$540$	0	0
$541$	0.997803	0.0428989	0.0214494	−	0.999770i	$-0.493172\pi$
$541$	0.997803	0.0428989	0.0214494	+	0.999770i	$0.493172\pi$
$542$	0	0
$543$	3.32046	0.142494
$544$	0	0
$545$	7.47902	0.320366
$546$	0	0
$547$	15.8011	0.675608	0.337804	−	0.941217i	$-0.390316\pi$
$547$	15.8011	0.675608	0.337804	+	0.941217i	$0.390316\pi$
$548$	0	0
$549$	16.7962	0.716843
$550$	0	0
$551$	33.3494	1.42073
$552$	0	0
$553$	−39.8026	−1.69258
$554$	0	0
$555$	−10.4208	−0.442338
$556$	0	0
$557$	−36.3525	−1.54031	−0.770153	−	0.637860i	$-0.779820\pi$
$557$	−36.3525	−1.54031	−0.770153	+	0.637860i	$0.779820\pi$
$558$	0	0
$559$	5.59360	0.236584
$560$	0	0
$561$	5.38514	0.227361
$562$	0	0
$563$	20.1622	0.849736	0.424868	−	0.905255i	$-0.360320\pi$
$563$	20.1622	0.849736	0.424868	+	0.905255i	$0.360320\pi$
$564$	0	0
$565$	−40.1127	−1.68755
$566$	0	0
$567$	19.9671	0.838538
$568$	0	0
$569$	−16.0424	−0.672530	−0.336265	−	0.941767i	$-0.609164\pi$
$569$	−16.0424	−0.672530	−0.336265	+	0.941767i	$0.609164\pi$
$570$	0	0
$571$	32.2187	1.34831	0.674156	−	0.738589i	$-0.264507\pi$
$571$	32.2187	1.34831	0.674156	+	0.738589i	$0.264507\pi$
$572$	0	0
$573$	1.00466	0.0419701
$574$	0	0
$575$	20.0888	0.837761
$576$	0	0
$577$	42.6582	1.77589	0.887943	−	0.459954i	$-0.152134\pi$
$577$	42.6582	1.77589	0.887943	+	0.459954i	$0.152134\pi$
$578$	0	0
$579$	−3.87578	−0.161072
$580$	0	0
$581$	21.2193	0.880326
$582$	0	0
$583$	−43.3888	−1.79698
$584$	0	0
$585$	8.13731	0.336436
$586$	0	0
$587$	13.8741	0.572646	0.286323	−	0.958133i	$-0.407567\pi$
$587$	13.8741	0.572646	0.286323	+	0.958133i	$0.407567\pi$
$588$	0	0
$589$	−58.8119	−2.42330
$590$	0	0
$591$	−3.02779	−0.124547
$592$	0	0
$593$	33.6435	1.38157	0.690786	−	0.723059i	$-0.257265\pi$
$593$	33.6435	1.38157	0.690786	+	0.723059i	$0.257265\pi$
$594$	0	0
$595$	19.2414	0.788819
$596$	0	0
$597$	−8.69051	−0.355679
$598$	0	0
$599$	−12.8677	−0.525761	−0.262881	−	0.964828i	$-0.584673\pi$
$599$	−12.8677	−0.525761	−0.262881	+	0.964828i	$0.584673\pi$
$600$	0	0
$601$	17.2785	0.704804	0.352402	−	0.935849i	$-0.385365\pi$
$601$	17.2785	0.704804	0.352402	+	0.935849i	$0.385365\pi$
$602$	0	0
$603$	34.7216	1.41397
$604$	0	0
$605$	66.6384	2.70924
$606$	0	0
$607$	36.7366	1.49109	0.745547	−	0.666453i	$-0.232188\pi$
$607$	36.7366	1.49109	0.745547	+	0.666453i	$0.232188\pi$
$608$	0	0
$609$	4.40250	0.178398
$610$	0	0
$611$	−9.06970	−0.366921
$612$	0	0
$613$	−35.5029	−1.43395	−0.716975	−	0.697099i	$-0.754474\pi$
$613$	−35.5029	−1.43395	−0.716975	+	0.697099i	$0.754474\pi$
$614$	0	0
$615$	2.72518	0.109890
$616$	0	0
$617$	5.97403	0.240505	0.120253	−	0.992743i	$-0.461630\pi$
$617$	5.97403	0.240505	0.120253	+	0.992743i	$0.461630\pi$
$618$	0	0
$619$	−1.54631	−0.0621513	−0.0310757	−	0.999517i	$-0.509893\pi$
$619$	−1.54631	−0.0621513	−0.0310757	+	0.999517i	$0.509893\pi$
$620$	0	0
$621$	−11.2380	−0.450965
$622$	0	0
$623$	7.43552	0.297898
$624$	0	0
$625$	−29.3383	−1.17353
$626$	0	0
$627$	15.4047	0.615205
$628$	0	0
$629$	−23.7518	−0.947047
$630$	0	0
$631$	−33.5953	−1.33741	−0.668705	−	0.743528i	$-0.733151\pi$
$631$	−33.5953	−1.33741	−0.668705	+	0.743528i	$0.733151\pi$
$632$	0	0
$633$	−8.78905	−0.349333
$634$	0	0
$635$	−7.01686	−0.278456
$636$	0	0
$637$	0.399904	0.0158448
$638$	0	0
$639$	−35.3655	−1.39904
$640$	0	0
$641$	22.6820	0.895884	0.447942	−	0.894063i	$-0.352157\pi$
$641$	22.6820	0.895884	0.447942	+	0.894063i	$0.352157\pi$
$642$	0	0
$643$	−35.9742	−1.41868	−0.709341	−	0.704865i	$-0.751007\pi$
$643$	−35.9742	−1.41868	−0.709341	+	0.704865i	$0.751007\pi$
$644$	0	0
$645$	−6.47586	−0.254987
$646$	0	0
$647$	29.8868	1.17497	0.587485	−	0.809235i	$-0.300118\pi$
$647$	29.8868	1.17497	0.587485	+	0.809235i	$0.300118\pi$
$648$	0	0
$649$	5.86149	0.230084
$650$	0	0
$651$	−7.76385	−0.304289
$652$	0	0
$653$	−48.1187	−1.88303	−0.941516	−	0.336968i	$-0.890598\pi$
$653$	−48.1187	−1.88303	−0.941516	+	0.336968i	$0.890598\pi$
$654$	0	0
$655$	−24.2678	−0.948220
$656$	0	0
$657$	−1.86330	−0.0726942
$658$	0	0
$659$	27.3372	1.06491	0.532454	−	0.846459i	$-0.321270\pi$
$659$	27.3372	1.06491	0.532454	+	0.846459i	$0.321270\pi$
$660$	0	0
$661$	−20.6868	−0.804623	−0.402311	−	0.915503i	$-0.631793\pi$
$661$	−20.6868	−0.804623	−0.402311	+	0.915503i	$0.631793\pi$
$662$	0	0
$663$	−0.889234	−0.0345350
$664$	0	0
$665$	55.0418	2.13443
$666$	0	0
$667$	23.9678	0.928037
$668$	0	0
$669$	1.71828	0.0664325
$670$	0	0
$671$	−33.8871	−1.30820
$672$	0	0
$673$	−3.68048	−0.141872	−0.0709360	−	0.997481i	$-0.522599\pi$
$673$	−3.68048	−0.141872	−0.0709360	+	0.997481i	$0.522599\pi$
$674$	0	0
$675$	−8.43316	−0.324593
$676$	0	0
$677$	−17.3358	−0.666268	−0.333134	−	0.942880i	$-0.608106\pi$
$677$	−17.3358	−0.666268	−0.333134	+	0.942880i	$0.608106\pi$
$678$	0	0
$679$	2.45305	0.0941393
$680$	0	0
$681$	−0.853190	−0.0326943
$682$	0	0
$683$	−2.26489	−0.0866636	−0.0433318	−	0.999061i	$-0.513797\pi$
$683$	−2.26489	−0.0866636	−0.0433318	+	0.999061i	$0.513797\pi$
$684$	0	0
$685$	−40.4228	−1.54448
$686$	0	0
$687$	0.736840	0.0281122
$688$	0	0
$689$	7.16468	0.272953
$690$	0	0
$691$	34.9262	1.32865	0.664327	−	0.747442i	$-0.268718\pi$
$691$	34.9262	1.32865	0.664327	+	0.747442i	$0.268718\pi$
$692$	0	0
$693$	−42.4156	−1.61124
$694$	0	0
$695$	52.4980	1.99136
$696$	0	0
$697$	6.21143	0.235275
$698$	0	0
$699$	2.07070	0.0783210
$700$	0	0
$701$	−8.67601	−0.327688	−0.163844	−	0.986486i	$-0.552389\pi$
$701$	−8.67601	−0.327688	−0.163844	+	0.986486i	$0.552389\pi$
$702$	0	0
$703$	−67.9444	−2.56257
$704$	0	0
$705$	10.5002	0.395461
$706$	0	0
$707$	15.2006	0.571676
$708$	0	0
$709$	−52.4850	−1.97111	−0.985557	−	0.169342i	$-0.945836\pi$
$709$	−52.4850	−1.97111	−0.985557	+	0.169342i	$0.945836\pi$
$710$	0	0
$711$	44.4179	1.66580
$712$	0	0
$713$	−42.2675	−1.58293
$714$	0	0
$715$	−16.4174	−0.613976
$716$	0	0
$717$	1.23944	0.0462878
$718$	0	0
$719$	−12.2225	−0.455822	−0.227911	−	0.973682i	$-0.573190\pi$
$719$	−12.2225	−0.455822	−0.227911	+	0.973682i	$0.573190\pi$
$720$	0	0
$721$	19.4196	0.723224
$722$	0	0
$723$	9.89686	0.368068
$724$	0	0
$725$	17.9858	0.667977
$726$	0	0
$727$	−32.8117	−1.21692	−0.608460	−	0.793585i	$-0.708212\pi$
$727$	−32.8117	−1.21692	−0.608460	+	0.793585i	$0.708212\pi$
$728$	0	0
$729$	−19.8683	−0.735864
$730$	0	0
$731$	−14.7602	−0.545927
$732$	0	0
$733$	−11.8567	−0.437937	−0.218968	−	0.975732i	$-0.570269\pi$
$733$	−11.8567	−0.437937	−0.218968	+	0.975732i	$0.570269\pi$
$734$	0	0
$735$	−0.462979	−0.0170772
$736$	0	0
$737$	−70.0525	−2.58042
$738$	0	0
$739$	−23.7096	−0.872173	−0.436086	−	0.899905i	$-0.643636\pi$
$739$	−23.7096	−0.872173	−0.436086	+	0.899905i	$0.643636\pi$
$740$	0	0
$741$	−2.54374	−0.0934466
$742$	0	0
$743$	−16.3166	−0.598600	−0.299300	−	0.954159i	$-0.596753\pi$
$743$	−16.3166	−0.598600	−0.299300	+	0.954159i	$0.596753\pi$
$744$	0	0
$745$	14.1077	0.516867
$746$	0	0
$747$	−23.6798	−0.866399
$748$	0	0
$749$	−41.1613	−1.50400
$750$	0	0
$751$	−30.5188	−1.11365	−0.556824	−	0.830630i	$-0.687980\pi$
$751$	−30.5188	−1.11365	−0.556824	+	0.830630i	$0.687980\pi$
$752$	0	0
$753$	10.8578	0.395680
$754$	0	0
$755$	−55.2639	−2.01126
$756$	0	0
$757$	21.0756	0.766007	0.383003	−	0.923747i	$-0.374890\pi$
$757$	21.0756	0.766007	0.383003	+	0.923747i	$0.374890\pi$
$758$	0	0
$759$	11.0712	0.401858
$760$	0	0
$761$	−9.37156	−0.339719	−0.169859	−	0.985468i	$-0.554331\pi$
$761$	−9.37156	−0.339719	−0.169859	+	0.985468i	$0.554331\pi$
$762$	0	0
$763$	6.43738	0.233049
$764$	0	0
$765$	−21.4725	−0.776340
$766$	0	0
$767$	−0.967894	−0.0349486
$768$	0	0
$769$	26.9367	0.971363	0.485681	−	0.874136i	$-0.338572\pi$
$769$	26.9367	0.971363	0.485681	+	0.874136i	$0.338572\pi$
$770$	0	0
$771$	3.84565	0.138498
$772$	0	0
$773$	−14.7111	−0.529121	−0.264560	−	0.964369i	$-0.585227\pi$
$773$	−14.7111	−0.529121	−0.264560	+	0.964369i	$0.585227\pi$
$774$	0	0
$775$	−31.7182	−1.13935
$776$	0	0
$777$	−8.96944	−0.321777
$778$	0	0
$779$	17.7684	0.636619
$780$	0	0
$781$	71.3516	2.55316
$782$	0	0
$783$	−10.0615	−0.359570
$784$	0	0
$785$	19.7475	0.704818
$786$	0	0
$787$	−3.73905	−0.133283	−0.0666414	−	0.997777i	$-0.521228\pi$
$787$	−3.73905	−0.133283	−0.0666414	+	0.997777i	$0.521228\pi$
$788$	0	0
$789$	3.27471	0.116583
$790$	0	0
$791$	−34.5260	−1.22760
$792$	0	0
$793$	5.59569	0.198709
$794$	0	0
$795$	−8.29474	−0.294184
$796$	0	0
$797$	7.11196	0.251919	0.125959	−	0.992035i	$-0.459799\pi$
$797$	7.11196	0.251919	0.125959	+	0.992035i	$0.459799\pi$
$798$	0	0
$799$	23.9329	0.846685
$800$	0	0
$801$	−8.29772	−0.293185
$802$	0	0
$803$	3.75929	0.132663
$804$	0	0
$805$	39.5579	1.39423
$806$	0	0
$807$	−0.370477	−0.0130414
$808$	0	0
$809$	−10.2508	−0.360398	−0.180199	−	0.983630i	$-0.557674\pi$
$809$	−10.2508	−0.360398	−0.180199	+	0.983630i	$0.557674\pi$
$810$	0	0
$811$	17.6750	0.620655	0.310327	−	0.950630i	$-0.399561\pi$
$811$	17.6750	0.620655	0.310327	+	0.950630i	$0.399561\pi$
$812$	0	0
$813$	−2.74006	−0.0960981
$814$	0	0
$815$	21.9123	0.767553
$816$	0	0
$817$	−42.2231	−1.47720
$818$	0	0
$819$	7.00399	0.244739
$820$	0	0
$821$	14.4456	0.504153	0.252077	−	0.967707i	$-0.418886\pi$
$821$	14.4456	0.504153	0.252077	+	0.967707i	$0.418886\pi$
$822$	0	0
$823$	5.39437	0.188036	0.0940179	−	0.995571i	$-0.470029\pi$
$823$	5.39437	0.188036	0.0940179	+	0.995571i	$0.470029\pi$
$824$	0	0
$825$	8.30799	0.289247
$826$	0	0
$827$	40.7336	1.41645	0.708223	−	0.705989i	$-0.249497\pi$
$827$	40.7336	1.41645	0.708223	+	0.705989i	$0.249497\pi$
$828$	0	0
$829$	−15.9379	−0.553547	−0.276773	−	0.960935i	$-0.589265\pi$
$829$	−15.9379	−0.553547	−0.276773	+	0.960935i	$0.589265\pi$
$830$	0	0
$831$	5.03007	0.174491
$832$	0	0
$833$	−1.05526	−0.0365625
$834$	0	0
$835$	55.0936	1.90659
$836$	0	0
$837$	17.7436	0.613309
$838$	0	0
$839$	−47.0290	−1.62362	−0.811811	−	0.583921i	$-0.801518\pi$
$839$	−47.0290	−1.62362	−0.811811	+	0.583921i	$0.801518\pi$
$840$	0	0
$841$	−7.54122	−0.260042
$842$	0	0
$843$	5.02821	0.173181
$844$	0	0
$845$	−36.0339	−1.23961
$846$	0	0
$847$	57.3574	1.97082
$848$	0	0
$849$	−7.59575	−0.260685
$850$	0	0
$851$	−48.8308	−1.67390
$852$	0	0
$853$	5.50121	0.188358	0.0941790	−	0.995555i	$-0.469977\pi$
$853$	5.50121	0.188358	0.0941790	+	0.995555i	$0.469977\pi$
$854$	0	0
$855$	−61.4242	−2.10066
$856$	0	0
$857$	−6.74343	−0.230351	−0.115176	−	0.993345i	$-0.536743\pi$
$857$	−6.74343	−0.230351	−0.115176	+	0.993345i	$0.536743\pi$
$858$	0	0
$859$	−38.3733	−1.30928	−0.654641	−	0.755940i	$-0.727180\pi$
$859$	−38.3733	−1.30928	−0.654641	+	0.755940i	$0.727180\pi$
$860$	0	0
$861$	2.34563	0.0799390
$862$	0	0
$863$	−27.5099	−0.936449	−0.468224	−	0.883610i	$-0.655106\pi$
$863$	−27.5099	−0.936449	−0.468224	+	0.883610i	$0.655106\pi$
$864$	0	0
$865$	−45.3122	−1.54066
$866$	0	0
$867$	−3.95162	−0.134204
$868$	0	0
$869$	−89.6153	−3.03999
$870$	0	0
$871$	11.5676	0.391953
$872$	0	0
$873$	−2.73749	−0.0926501
$874$	0	0
$875$	−8.54273	−0.288797
$876$	0	0
$877$	40.4307	1.36525	0.682625	−	0.730769i	$-0.260838\pi$
$877$	40.4307	1.36525	0.682625	+	0.730769i	$0.260838\pi$
$878$	0	0
$879$	−9.63733	−0.325059
$880$	0	0
$881$	14.9852	0.504866	0.252433	−	0.967614i	$-0.418769\pi$
$881$	14.9852	0.504866	0.252433	+	0.967614i	$0.418769\pi$
$882$	0	0
$883$	−45.8998	−1.54465	−0.772325	−	0.635227i	$-0.780906\pi$
$883$	−45.8998	−1.54465	−0.772325	+	0.635227i	$0.780906\pi$
$884$	0	0
$885$	1.12056	0.0376671
$886$	0	0
$887$	−38.4897	−1.29236	−0.646179	−	0.763186i	$-0.723634\pi$
$887$	−38.4897	−1.29236	−0.646179	+	0.763186i	$0.723634\pi$
$888$	0	0
$889$	−6.03959	−0.202561
$890$	0	0
$891$	44.9557	1.50607
$892$	0	0
$893$	68.4623	2.29100
$894$	0	0
$895$	26.6129	0.889570
$896$	0	0
$897$	−1.82816	−0.0610404
$898$	0	0
$899$	−37.8427	−1.26213
$900$	0	0
$901$	−18.9060	−0.629849
$902$	0	0
$903$	−5.57393	−0.185489
$904$	0	0
$905$	26.7121	0.887940
$906$	0	0
$907$	3.30242	0.109655	0.0548275	−	0.998496i	$-0.482539\pi$
$907$	3.30242	0.109655	0.0548275	+	0.998496i	$0.482539\pi$
$908$	0	0
$909$	−16.9632	−0.562633
$910$	0	0
$911$	−0.593599	−0.0196668	−0.00983341	−	0.999952i	$-0.503130\pi$
$911$	−0.593599	−0.0196668	−0.00983341	+	0.999952i	$0.503130\pi$
$912$	0	0
$913$	47.7752	1.58113
$914$	0	0
$915$	−6.47828	−0.214165
$916$	0	0
$917$	−20.8879	−0.689779
$918$	0	0
$919$	−7.37035	−0.243125	−0.121563	−	0.992584i	$-0.538791\pi$
$919$	−7.37035	−0.243125	−0.121563	+	0.992584i	$0.538791\pi$
$920$	0	0
$921$	6.44604	0.212404
$922$	0	0
$923$	−11.7821	−0.387813
$924$	0	0
$925$	−36.6435	−1.20483
$926$	0	0
$927$	−21.6714	−0.711783
$928$	0	0
$929$	0.386422	0.0126781	0.00633905	−	0.999980i	$-0.497982\pi$
$929$	0.386422	0.0126781	0.00633905	+	0.999980i	$0.497982\pi$
$930$	0	0
$931$	−3.01866	−0.0989327
$932$	0	0
$933$	4.99179	0.163424
$934$	0	0
$935$	43.3218	1.41678
$936$	0	0
$937$	−29.0617	−0.949405	−0.474702	−	0.880146i	$-0.657444\pi$
$937$	−29.0617	−0.949405	−0.474702	+	0.880146i	$0.657444\pi$
$938$	0	0
$939$	−3.66474	−0.119594
$940$	0	0
$941$	−22.0276	−0.718081	−0.359040	−	0.933322i	$-0.616896\pi$
$941$	−22.0276	−0.718081	−0.359040	+	0.933322i	$0.616896\pi$
$942$	0	0
$943$	12.7699	0.415847
$944$	0	0
$945$	−16.6062	−0.540199
$946$	0	0
$947$	6.64851	0.216047	0.108024	−	0.994148i	$-0.465548\pi$
$947$	6.64851	0.216047	0.108024	+	0.994148i	$0.465548\pi$
$948$	0	0
$949$	−0.620763	−0.0201508
$950$	0	0
$951$	7.12056	0.230900
$952$	0	0
$953$	−57.4933	−1.86239	−0.931195	−	0.364521i	$-0.881233\pi$
$953$	−57.4933	−1.86239	−0.931195	+	0.364521i	$0.881233\pi$
$954$	0	0
$955$	8.08216	0.261532
$956$	0	0
$957$	9.91221	0.320416
$958$	0	0
$959$	−34.7929	−1.12352
$960$	0	0
$961$	35.7360	1.15277
$962$	0	0
$963$	45.9342	1.48021
$964$	0	0
$965$	−31.1795	−1.00370
$966$	0	0
$967$	−44.0592	−1.41685	−0.708424	−	0.705788i	$-0.750593\pi$
$967$	−44.0592	−1.41685	−0.708424	+	0.705788i	$0.750593\pi$
$968$	0	0
$969$	6.71236	0.215632
$970$	0	0
$971$	−25.8900	−0.830850	−0.415425	−	0.909627i	$-0.636367\pi$
$971$	−25.8900	−0.830850	−0.415425	+	0.909627i	$0.636367\pi$
$972$	0	0
$973$	45.1864	1.44861
$974$	0	0
$975$	−1.37188	−0.0439353
$976$	0	0
$977$	16.1799	0.517642	0.258821	−	0.965925i	$-0.416666\pi$
$977$	16.1799	0.517642	0.258821	+	0.965925i	$0.416666\pi$
$978$	0	0
$979$	16.7410	0.535046
$980$	0	0
$981$	−7.18383	−0.229362
$982$	0	0
$983$	27.7004	0.883504	0.441752	−	0.897137i	$-0.354357\pi$
$983$	27.7004	0.883504	0.441752	+	0.897137i	$0.354357\pi$
$984$	0	0
$985$	−24.3577	−0.776101
$986$	0	0
$987$	9.03781	0.287677
$988$	0	0
$989$	−30.3453	−0.964923
$990$	0	0
$991$	15.6306	0.496523	0.248262	−	0.968693i	$-0.420141\pi$
$991$	15.6306	0.496523	0.248262	+	0.968693i	$0.420141\pi$
$992$	0	0
$993$	3.93326	0.124818
$994$	0	0
$995$	−69.9126	−2.21638
$996$	0	0
$997$	59.0197	1.86917	0.934586	−	0.355737i	$-0.115770\pi$
$997$	59.0197	1.86917	0.934586	+	0.355737i	$0.115770\pi$
$998$	0	0
$999$	20.4989	0.648557

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.i.1.4		7
4.3	odd	2		538.2.a.d.1.4	✓	7
12.11	even	2		4842.2.a.o.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.d.1.4	✓	7	4.3	odd	2
4304.2.a.i.1.4		7	1.1	even	1	trivial
4842.2.a.o.1.1		7	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q+0.370477 q^{3} +2.98038 q^{5} +2.56529 q^{7} -2.86275 q^{9} +5.77573 q^{11} -0.953732 q^{13} +1.10416 q^{15} +2.51668 q^{17} +7.19922 q^{19} +0.950379 q^{21} +5.17399 q^{23} +3.88265 q^{25} -2.17201 q^{27} +4.63236 q^{29} -8.16921 q^{31} +2.13977 q^{33} +7.64552 q^{35} -9.43775 q^{37} -0.353336 q^{39} +2.46810 q^{41} -5.86496 q^{43} -8.53207 q^{45} +9.50969 q^{47} -0.419304 q^{49} +0.932373 q^{51} -7.51226 q^{53} +17.2139 q^{55} +2.66714 q^{57} +1.01485 q^{59} -5.86716 q^{61} -7.34377 q^{63} -2.84248 q^{65} -12.1288 q^{67} +1.91685 q^{69} +12.3537 q^{71} +0.650878 q^{73} +1.43843 q^{75} +14.8164 q^{77} -15.5158 q^{79} +7.78356 q^{81} +8.27171 q^{83} +7.50067 q^{85} +1.71618 q^{87} +2.89852 q^{89} -2.44660 q^{91} -3.02650 q^{93} +21.4564 q^{95} +0.956247 q^{97} -16.5345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{3} - 6 q^{5} + 3 q^{7} + 7 q^{9} + 12 q^{11} + 3 q^{13} + 6 q^{15} - 8 q^{17} + 7 q^{19} - 5 q^{21} + 22 q^{23} + 9 q^{25} + 22 q^{27} - 7 q^{29} + 3 q^{31} - 19 q^{33} + 25 q^{35} - 13 q^{37}+ \cdots + 9 q^{99}+O(q^{100}) \) 7 * q + 4 * q^3 - 6 * q^5 + 3 * q^7 + 7 * q^9 + 12 * q^11 + 3 * q^13 + 6 * q^15 - 8 * q^17 + 7 * q^19 - 5 * q^21 + 22 * q^23 + 9 * q^25 + 22 * q^27 - 7 * q^29 + 3 * q^31 - 19 * q^33 + 25 * q^35 - 13 * q^37 - q^39 - 11 * q^41 + 14 * q^43 - 17 * q^45 + 23 * q^47 + 6 * q^49 + 21 * q^51 - 21 * q^53 - 15 * q^55 - 14 * q^57 + 20 * q^59 + 13 * q^61 + 12 * q^63 - 29 * q^65 + 30 * q^67 + 13 * q^69 + 23 * q^71 + 9 * q^73 - 3 * q^75 + 8 * q^77 - 11 * q^79 + 43 * q^81 + 3 * q^83 + 21 * q^85 - 28 * q^87 + 16 * q^89 + 10 * q^91 - 13 * q^93 + 21 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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