LMFDB - Embedded newform 400.6.n.g.143.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.n (of order $4$, degree $2$, minimal)

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:	no
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + \cdots + 57\!\cdots\!24 $ x^20 + 271x^18 + 109637x^16 + 25993614x^14 + 5522961902x^12 + 881545050522x^10 + 133816049059481x^8 + 14779507781220031x^6 + 824105698447750789x^4 + 12044868290803250652*x^2 + 579398322543528055824
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{67}\cdot 3^{4}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.7
Root			$-5.50401 + 11.9953i$ of defining polynomial
Character	$\chi$	$=$	400.143
Dual form			400.6.n.g.207.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.48311 - 7.48311i) q^{3} +(19.2260 + 19.2260i) q^{7} +131.006i q^{9} +O(q^{10})$	$q+(7.48311 - 7.48311i) q^{3} +(19.2260 + 19.2260i) q^{7} +131.006i q^{9} -180.642i q^{11} +(-44.2050 - 44.2050i) q^{13} +(621.037 - 621.037i) q^{17} -2674.30 q^{19} +287.741 q^{21} +(2233.28 - 2233.28i) q^{23} +(2798.73 + 2798.73i) q^{27} +705.810i q^{29} -2761.15i q^{31} +(-1351.77 - 1351.77i) q^{33} +(3542.51 - 3542.51i) q^{37} -661.582 q^{39} +10907.4 q^{41} +(5349.56 - 5349.56i) q^{43} +(-13279.7 - 13279.7i) q^{47} -16067.7i q^{49} -9294.57i q^{51} +(15680.4 + 15680.4i) q^{53} +(-20012.1 + 20012.1i) q^{57} +45931.3 q^{59} -17547.9 q^{61} +(-2518.72 + 2518.72i) q^{63} +(-31089.0 - 31089.0i) q^{67} -33423.8i q^{69} -10610.1i q^{71} +(-50962.5 - 50962.5i) q^{73} +(3473.03 - 3473.03i) q^{77} +24770.6 q^{79} +10051.9 q^{81} +(48930.7 - 48930.7i) q^{83} +(5281.66 + 5281.66i) q^{87} +26108.1i q^{89} -1699.77i q^{91} +(-20662.0 - 20662.0i) q^{93} +(-39967.8 + 39967.8i) q^{97} +23665.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) $ 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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
$3$	7.48311	−	7.48311i	0.480042	−	0.480042i	−0.425103	−	0.905145i	$-0.639762\pi$
$3$	7.48311	−	7.48311i	0.480042	−	0.480042i	0.905145	+	0.425103i	$0.139762\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	19.2260	+	19.2260i	0.148301	+	0.148301i	0.777359	−	0.629058i	$-0.216559\pi$
$7$	19.2260	+	19.2260i	0.148301	+	0.148301i	−0.629058	+	0.777359i	$0.716559\pi$
$8$		0			0
$9$			131.006i			0.539120i
$10$		0			0
$11$		−	180.642i		−	0.450130i	−0.974344	−	0.225065i	$-0.927741\pi$
$11$		−	180.642i		−	0.450130i	0.974344	−	0.225065i	$-0.0722594\pi$
$12$		0			0
$13$	−44.2050	−	44.2050i	−0.0725459	−	0.0725459i	0.669903	−	0.742449i	$-0.266336\pi$
$13$	−44.2050	−	44.2050i	−0.0725459	−	0.0725459i	−0.742449	+	0.669903i	$0.766336\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	621.037	−	621.037i	0.521189	−	0.521189i	−0.396742	−	0.917930i	$-0.629859\pi$
$17$	621.037	−	621.037i	0.521189	−	0.521189i	0.917930	+	0.396742i	$0.129859\pi$
$18$		0			0
$19$	−2674.30			−1.69952			−0.849759	−	0.527172i	$-0.823253\pi$
$19$	−2674.30			−1.69952			−0.849759	+	0.527172i	$0.823253\pi$
$20$		0			0
$21$	287.741			0.142381
$22$		0			0
$23$	2233.28	−	2233.28i	0.880285	−	0.880285i	−0.113278	−	0.993563i	$-0.536135\pi$
$23$	2233.28	−	2233.28i	0.880285	−	0.880285i	0.993563	+	0.113278i	$0.0361351\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2798.73	+	2798.73i	0.738842	+	0.738842i
$28$		0			0
$29$			705.810i			0.155845i	0.996959	+	0.0779225i	$0.0248287\pi$
$29$			705.810i			0.155845i	−0.996959	+	0.0779225i	$0.975171\pi$
$30$		0			0
$31$		−	2761.15i		−	0.516043i	−0.966139	−	0.258021i	$-0.916930\pi$
$31$		−	2761.15i		−	0.516043i	0.966139	−	0.258021i	$-0.0830705\pi$
$32$		0			0
$33$	−1351.77	−	1351.77i	−0.216081	−	0.216081i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3542.51	−	3542.51i	0.425409	−	0.425409i	−0.461652	−	0.887061i	$-0.652743\pi$
$37$	3542.51	−	3542.51i	0.425409	−	0.425409i	0.887061	+	0.461652i	$0.152743\pi$
$38$		0			0
$39$	−661.582			−0.0696502
$40$		0			0
$41$	10907.4			1.01336			0.506679	−	0.862135i	$-0.330873\pi$
$41$	10907.4			1.01336			0.506679	+	0.862135i	$0.330873\pi$
$42$		0			0
$43$	5349.56	−	5349.56i	0.441212	−	0.441212i	−0.451207	−	0.892419i	$-0.649007\pi$
$43$	5349.56	−	5349.56i	0.441212	−	0.441212i	0.892419	+	0.451207i	$0.149007\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−13279.7	−	13279.7i	−0.876884	−	0.876884i	0.116327	−	0.993211i	$-0.462888\pi$
$47$	−13279.7	−	13279.7i	−0.876884	−	0.876884i	−0.993211	+	0.116327i	$0.962888\pi$
$48$		0			0
$49$		−	16067.7i		−	0.956014i
$50$		0			0
$51$		−	9294.57i		−	0.500385i
$52$		0			0
$53$	15680.4	+	15680.4i	0.766774	+	0.766774i	0.977537	−	0.210763i	$-0.0675948\pi$
$53$	15680.4	+	15680.4i	0.766774	+	0.766774i	−0.210763	+	0.977537i	$0.567595\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−20012.1	+	20012.1i	−0.815840	+	0.815840i
$58$		0			0
$59$	45931.3			1.71783			0.858913	−	0.512122i	$-0.171140\pi$
$59$	45931.3			1.71783			0.858913	+	0.512122i	$0.171140\pi$
$60$		0			0
$61$	−17547.9			−0.603809			−0.301904	−	0.953338i	$-0.597622\pi$
$61$	−17547.9			−0.603809			−0.301904	+	0.953338i	$0.597622\pi$
$62$		0			0
$63$	−2518.72	+	2518.72i	−0.0799519	+	0.0799519i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−31089.0	−	31089.0i	−0.846097	−	0.846097i	0.143547	−	0.989644i	$-0.454149\pi$
$67$	−31089.0	−	31089.0i	−0.846097	−	0.846097i	−0.989644	+	0.143547i	$0.954149\pi$
$68$		0			0
$69$		−	33423.8i		−	0.845148i
$70$		0			0
$71$		−	10610.1i		−	0.249788i	−0.992170	−	0.124894i	$-0.960141\pi$
$71$		−	10610.1i		−	0.249788i	0.992170	−	0.124894i	$-0.0398592\pi$
$72$		0			0
$73$	−50962.5	−	50962.5i	−1.11929	−	1.11929i	−0.991845	−	0.127447i	$-0.959322\pi$
$73$	−50962.5	−	50962.5i	−1.11929	−	1.11929i	−0.127447	−	0.991845i	$-0.540678\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3473.03	−	3473.03i	0.0667546	−	0.0667546i
$78$		0			0
$79$	24770.6			0.446548			0.223274	−	0.974756i	$-0.428326\pi$
$79$	24770.6			0.446548			0.223274	+	0.974756i	$0.428326\pi$
$80$		0			0
$81$	10051.9			0.170231
$82$		0			0
$83$	48930.7	−	48930.7i	0.779626	−	0.779626i	−0.200141	−	0.979767i	$-0.564140\pi$
$83$	48930.7	−	48930.7i	0.779626	−	0.779626i	0.979767	+	0.200141i	$0.0641401\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	5281.66	+	5281.66i	0.0748122	+	0.0748122i
$88$		0			0
$89$			26108.1i			0.349382i	0.984623	+	0.174691i	$0.0558926\pi$
$89$			26108.1i			0.349382i	−0.984623	+	0.174691i	$0.944107\pi$
$90$		0			0
$91$		−	1699.77i		−	0.0215173i
$92$		0			0
$93$	−20662.0	−	20662.0i	−0.247722	−	0.247722i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−39967.8	+	39967.8i	−0.431301	+	0.431301i	−0.889071	−	0.457770i	$-0.848648\pi$
$97$	−39967.8	+	39967.8i	−0.431301	+	0.431301i	0.457770	+	0.889071i	$0.348648\pi$
$98$		0			0
$99$	23665.2			0.242674
$100$		0			0
$101$	−196040.			−1.91223			−0.956115	−	0.292991i	$-0.905350\pi$
$101$	−196040.			−1.91223			−0.956115	+	0.292991i	$0.905350\pi$
$102$		0			0
$103$	148017.	−	148017.i	1.37473	−	1.37473i	0.521450	−	0.853282i	$-0.325391\pi$
$103$	148017.	−	148017.i	1.37473	−	1.37473i	0.853282	−	0.521450i	$-0.174609\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	40600.5	+	40600.5i	0.342824	+	0.342824i	0.857428	−	0.514604i	$-0.172061\pi$
$107$	40600.5	+	40600.5i	0.342824	+	0.342824i	−0.514604	+	0.857428i	$0.672061\pi$
$108$		0			0
$109$			50412.1i			0.406414i	0.979136	+	0.203207i	$0.0651364\pi$
$109$			50412.1i			0.406414i	−0.979136	+	0.203207i	$0.934864\pi$
$110$		0			0
$111$		−	53018.0i		−	0.408428i
$112$		0			0
$113$	−152938.	−	152938.i	−1.12673	−	1.12673i	−0.990706	−	0.136024i	$-0.956568\pi$
$113$	−152938.	−	152938.i	−1.12673	−	1.12673i	−0.136024	−	0.990706i	$-0.543432\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	5791.12	−	5791.12i	0.0391109	−	0.0391109i
$118$		0			0
$119$	23880.1			0.154586
$120$		0			0
$121$	128419.			0.797383
$122$		0			0
$123$	81621.6	−	81621.6i	0.486455	−	0.486455i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−190114.	−	190114.i	−1.04593	−	1.04593i	−0.998893	−	0.0470414i	$-0.985021\pi$
$127$	−190114.	−	190114.i	−1.04593	−	1.04593i	−0.0470414	−	0.998893i	$-0.514979\pi$
$128$		0			0
$129$		−	80062.7i		−	0.423600i
$130$		0			0
$131$		−	225175.i		−	1.14641i	−0.819411	−	0.573207i	$-0.805699\pi$
$131$		−	225175.i		−	1.14641i	0.819411	−	0.573207i	$-0.194301\pi$
$132$		0			0
$133$	−51416.1	−	51416.1i	−0.252040	−	0.252040i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−120197.	+	120197.i	−0.547133	+	0.547133i	−0.925611	−	0.378477i	$-0.876448\pi$
$137$	−120197.	+	120197.i	−0.547133	+	0.547133i	0.378477	+	0.925611i	$0.376448\pi$
$138$		0			0
$139$	336335.			1.47651			0.738253	−	0.674524i	$-0.235651\pi$
$139$	336335.			1.47651			0.738253	+	0.674524i	$0.235651\pi$
$140$		0			0
$141$	−198746.			−0.841882
$142$		0			0
$143$	−7985.29	+	7985.29i	−0.0326551	+	0.0326551i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−120237.	−	120237.i	−0.458927	−	0.458927i
$148$		0			0
$149$		−	339978.i		−	1.25454i	−0.778802	−	0.627270i	$-0.784172\pi$
$149$		−	339978.i		−	1.25454i	0.778802	−	0.627270i	$-0.215828\pi$
$150$		0			0
$151$			224030.i			0.799582i	0.916606	+	0.399791i	$0.130917\pi$
$151$			224030.i			0.799582i	−0.916606	+	0.399791i	$0.869083\pi$
$152$		0			0
$153$	81359.5	+	81359.5i	0.280983	+	0.280983i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−168847.	+	168847.i	−0.546694	+	0.546694i	−0.925483	−	0.378789i	$-0.876340\pi$
$157$	−168847.	+	168847.i	−0.546694	+	0.546694i	0.378789	+	0.925483i	$0.376340\pi$
$158$		0			0
$159$	234676.			0.736168
$160$		0			0
$161$	85874.1			0.261094
$162$		0			0
$163$	343693.	−	343693.i	1.01322	−	1.01322i	0.0133036	−	0.999912i	$-0.495765\pi$
$163$	343693.	−	343693.i	1.01322	−	1.01322i	0.999912	−	0.0133036i	$-0.00423478\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−220038.	−	220038.i	−0.610529	−	0.610529i	0.332555	−	0.943084i	$-0.392089\pi$
$167$	−220038.	−	220038.i	−0.610529	−	0.610529i	−0.943084	+	0.332555i	$0.892089\pi$
$168$		0			0
$169$		−	367385.i		−	0.989474i
$170$		0			0
$171$		−	350349.i		−	0.916243i
$172$		0			0
$173$	103453.	+	103453.i	0.262802	+	0.262802i	0.826192	−	0.563389i	$-0.190503\pi$
$173$	103453.	+	103453.i	0.262802	+	0.262802i	−0.563389	+	0.826192i	$0.690503\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	343709.	−	343709.i	0.824628	−	0.824628i
$178$		0			0
$179$	377900.			0.881544			0.440772	−	0.897619i	$-0.354705\pi$
$179$	377900.			0.881544			0.440772	+	0.897619i	$0.354705\pi$
$180$		0			0
$181$	583576.			1.32404			0.662020	−	0.749486i	$-0.269699\pi$
$181$	583576.			1.32404			0.662020	+	0.749486i	$0.269699\pi$
$182$		0			0
$183$	−131313.	+	131313.i	−0.289854	+	0.289854i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−112185.	−	112185.i	−0.234602	−	0.234602i
$188$		0			0
$189$			107617.i			0.219142i
$190$		0			0
$191$		−	368495.i		−	0.730884i	−0.930834	−	0.365442i	$-0.880918\pi$
$191$		−	368495.i		−	0.730884i	0.930834	−	0.365442i	$-0.119082\pi$
$192$		0			0
$193$	−169450.	−	169450.i	−0.327453	−	0.327453i	0.524164	−	0.851617i	$-0.324378\pi$
$193$	−169450.	−	169450.i	−0.327453	−	0.327453i	−0.851617	+	0.524164i	$0.824378\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−219866.	+	219866.i	−0.403638	+	0.403638i	−0.879513	−	0.475875i	$-0.842132\pi$
$197$	−219866.	+	219866.i	−0.403638	+	0.403638i	0.475875	+	0.879513i	$0.342132\pi$
$198$		0			0
$199$	−672141.			−1.20317			−0.601586	−	0.798808i	$-0.705464\pi$
$199$	−672141.			−1.20317			−0.601586	+	0.798808i	$0.705464\pi$
$200$		0			0
$201$	−465285.			−0.812324
$202$		0			0
$203$	−13569.9	+	13569.9i	−0.0231120	+	0.0231120i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	292573.	+	292573.i	0.474579	+	0.474579i
$208$		0			0
$209$			483091.i			0.765003i
$210$		0			0
$211$			594402.i			0.919124i	0.888146	+	0.459562i	$0.151994\pi$
$211$			594402.i			0.919124i	−0.888146	+	0.459562i	$0.848006\pi$
$212$		0			0
$213$	−79396.4	−	79396.4i	−0.119909	−	0.119909i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	53085.9	−	53085.9i	0.0765296	−	0.0765296i
$218$		0			0
$219$	−762716.			−1.07461
$220$		0			0
$221$	−54905.8			−0.0756202
$222$		0			0
$223$	422559.	−	422559.i	0.569017	−	0.569017i	−0.362836	−	0.931853i	$-0.618191\pi$
$223$	422559.	−	422559.i	0.569017	−	0.569017i	0.931853	+	0.362836i	$0.118191\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	747586.	+	747586.i	0.962934	+	0.962934i	0.999337	−	0.0364031i	$-0.0115900\pi$
$227$	747586.	+	747586.i	0.962934	+	0.962934i	−0.0364031	+	0.999337i	$0.511590\pi$
$228$		0			0
$229$			740343.i			0.932920i	0.884542	+	0.466460i	$0.154471\pi$
$229$			740343.i			0.932920i	−0.884542	+	0.466460i	$0.845529\pi$
$230$		0			0
$231$		−	51978.1i		−	0.0640901i
$232$		0			0
$233$	427262.	+	427262.i	0.515590	+	0.515590i	0.916234	−	0.400644i	$-0.131214\pi$
$233$	427262.	+	427262.i	0.515590	+	0.515590i	−0.400644	+	0.916234i	$0.631214\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	185361.	−	185361.i	0.214362	−	0.214362i
$238$		0			0
$239$	−508899.			−0.576284			−0.288142	−	0.957588i	$-0.593038\pi$
$239$	−508899.			−0.576284			−0.288142	+	0.957588i	$0.593038\pi$
$240$		0			0
$241$	5494.45			0.00609370			0.00304685	−	0.999995i	$-0.499030\pi$
$241$	5494.45			0.00609370			0.00304685	+	0.999995i	$0.499030\pi$
$242$		0			0
$243$	−604871.	+	604871.i	−0.657124	+	0.657124i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	118217.	+	118217.i	0.123293	+	0.123293i
$248$		0			0
$249$		−	732308.i		−	0.748506i
$250$		0			0
$251$		−	150381.i		−	0.150664i	−0.997159	−	0.0753321i	$-0.975998\pi$
$251$		−	150381.i		−	0.150664i	0.997159	−	0.0753321i	$-0.0240017\pi$
$252$		0			0
$253$	−403424.	−	403424.i	−0.396242	−	0.396242i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.07319e6	−	1.07319e6i	1.01355	−	1.01355i	0.0136415	−	0.999907i	$-0.495658\pi$
$257$	1.07319e6	−	1.07319e6i	1.01355	−	1.01355i	0.999907	−	0.0136415i	$-0.00434237\pi$
$258$		0			0
$259$	136217.			0.126177
$260$		0			0
$261$	−92465.4			−0.0840191
$262$		0			0
$263$	−1.13111e6	+	1.13111e6i	−1.00836	+	1.00836i	−0.00839852	+	0.999965i	$0.502673\pi$
$263$	−1.13111e6	+	1.13111e6i	−1.00836	+	1.00836i	−0.999965	+	0.00839852i	$0.997327\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	195370.	+	195370.i	0.167718	+	0.167718i
$268$		0			0
$269$			1.59558e6i			1.34443i	0.740356	+	0.672216i	$0.234657\pi$
$269$			1.59558e6i			1.34443i	−0.740356	+	0.672216i	$0.765343\pi$
$270$		0			0
$271$			2.05748e6i			1.70182i	0.525312	+	0.850910i	$0.323949\pi$
$271$			2.05748e6i			1.70182i	−0.525312	+	0.850910i	$0.676051\pi$
$272$		0			0
$273$	−12719.6	−	12719.6i	−0.0103292	−	0.0103292i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.50847e6	+	1.50847e6i	−1.18123	+	1.18123i	−0.201809	+	0.979425i	$0.564682\pi$
$277$	−1.50847e6	+	1.50847e6i	−1.18123	+	1.18123i	−0.979425	+	0.201809i	$0.935318\pi$
$278$		0			0
$279$	361727.			0.278209
$280$		0			0
$281$	−113566.			−0.0857993			−0.0428997	−	0.999079i	$-0.513660\pi$
$281$	−113566.			−0.0857993			−0.0428997	+	0.999079i	$0.513660\pi$
$282$		0			0
$283$	−693960.	+	693960.i	−0.515072	+	0.515072i	−0.916076	−	0.401004i	$-0.868661\pi$
$283$	−693960.	+	693960.i	−0.515072	+	0.515072i	0.401004	+	0.916076i	$0.368661\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	209706.	+	209706.i	0.150282	+	0.150282i
$288$		0			0
$289$			648484.i			0.456725i
$290$		0			0
$291$			598167.i			0.414085i
$292$		0			0
$293$	1.44248e6	+	1.44248e6i	0.981616	+	0.981616i	0.999834	−	0.0182181i	$-0.00579933\pi$
$293$	1.44248e6	+	1.44248e6i	0.981616	+	0.981616i	−0.0182181	+	0.999834i	$0.505799\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	505569.	−	505569.i	0.332575	−	0.332575i
$298$		0			0
$299$	−197444.			−0.127722
$300$		0			0
$301$	205701.			0.130864
$302$		0			0
$303$	−1.46699e6	+	1.46699e6i	−0.917951	+	0.917951i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−415774.	−	415774.i	−0.251774	−	0.251774i	0.569924	−	0.821698i	$-0.306973\pi$
$307$	−415774.	−	415774.i	−0.251774	−	0.251774i	−0.821698	+	0.569924i	$0.806973\pi$
$308$		0			0
$309$		−	2.21525e6i		−	1.31986i
$310$		0			0
$311$		−	722906.i		−	0.423819i	−0.977289	−	0.211910i	$-0.932032\pi$
$311$		−	722906.i		−	0.423819i	0.977289	−	0.211910i	$-0.0679683\pi$
$312$		0			0
$313$	768778.	+	768778.i	0.443547	+	0.443547i	0.893202	−	0.449655i	$-0.148453\pi$
$313$	768778.	+	768778.i	0.443547	+	0.443547i	−0.449655	+	0.893202i	$0.648453\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−342207.	+	342207.i	−0.191267	+	0.191267i	−0.796244	−	0.604976i	$-0.793183\pi$
$317$	−342207.	+	342207.i	−0.191267	+	0.191267i	0.604976	+	0.796244i	$0.293183\pi$
$318$		0			0
$319$	127499.			0.0701505
$320$		0			0
$321$	607636.			0.329140
$322$		0			0
$323$	−1.66084e6	+	1.66084e6i	−0.885769	+	0.885769i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	377239.	+	377239.i	0.195096	+	0.195096i
$328$		0			0
$329$		−	510629.i		−	0.260085i
$330$		0			0
$331$			595925.i			0.298966i	0.988764	+	0.149483i	$0.0477609\pi$
$331$			595925.i			0.298966i	−0.988764	+	0.149483i	$0.952239\pi$
$332$		0			0
$333$	464090.	+	464090.i	0.229346	+	0.229346i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.54112e6	−	1.54112e6i	0.739199	−	0.739199i	−0.233224	−	0.972423i	$-0.574928\pi$
$337$	1.54112e6	−	1.54112e6i	0.739199	−	0.739199i	0.972423	+	0.233224i	$0.0749276\pi$
$338$		0			0
$339$	−2.28891e6			−1.08175
$340$		0			0
$341$	−498780.			−0.232286
$342$		0			0
$343$	632050.	−	632050.i	0.290079	−	0.290079i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	334328.	+	334328.i	0.149056	+	0.149056i	0.777696	−	0.628640i	$-0.216388\pi$
$347$	334328.	+	334328.i	0.149056	+	0.149056i	−0.628640	+	0.777696i	$0.716388\pi$
$348$		0			0
$349$			3.55159e6i			1.56084i	0.625254	+	0.780422i	$0.284995\pi$
$349$			3.55159e6i			1.56084i	−0.625254	+	0.780422i	$0.715005\pi$
$350$		0			0
$351$		−	247436.i		−	0.107200i
$352$		0			0
$353$	−942189.	−	942189.i	−0.402440	−	0.402440i	0.476652	−	0.879092i	$-0.341850\pi$
$353$	−942189.	−	942189.i	−0.402440	−	0.402440i	−0.879092	+	0.476652i	$0.841850\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	178698.	−	178698.i	0.0742075	−	0.0742075i
$358$		0			0
$359$	−1.75716e6			−0.719575			−0.359787	−	0.933034i	$-0.617151\pi$
$359$	−1.75716e6			−0.719575			−0.359787	+	0.933034i	$0.617151\pi$
$360$		0			0
$361$	4.67577e6			1.88836
$362$		0			0
$363$	960977.	−	960977.i	0.382777	−	0.382777i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	2.33482e6	+	2.33482e6i	0.904876	+	0.904876i	0.995853	−	0.0909771i	$-0.0289990\pi$
$367$	2.33482e6	+	2.33482e6i	0.904876	+	0.904876i	−0.0909771	+	0.995853i	$0.528999\pi$
$368$		0			0
$369$			1.42894e6i			0.546321i
$370$		0			0
$371$			602943.i			0.227427i
$372$		0			0
$373$	2.44665e6	+	2.44665e6i	0.910542	+	0.910542i	0.996315	−	0.0857724i	$-0.0273358\pi$
$373$	2.44665e6	+	2.44665e6i	0.910542	+	0.910542i	−0.0857724	+	0.996315i	$0.527336\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	31200.4	−	31200.4i	0.0113059	−	0.0113059i
$378$		0			0
$379$	−2.21109e6			−0.790694			−0.395347	−	0.918532i	$-0.629376\pi$
$379$	−2.21109e6			−0.790694			−0.395347	+	0.918532i	$0.629376\pi$
$380$		0			0
$381$	−2.84529e6			−1.00418
$382$		0			0
$383$	1.12909e6	−	1.12909e6i	0.393307	−	0.393307i	−0.482557	−	0.875865i	$-0.660292\pi$
$383$	1.12909e6	−	1.12909e6i	0.393307	−	0.393307i	0.875865	+	0.482557i	$0.160292\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	700825.	+	700825.i	0.237866	+	0.237866i
$388$		0			0
$389$		−	501579.i		−	0.168060i	−0.996463	−	0.0840302i	$-0.973221\pi$
$389$		−	501579.i		−	0.168060i	0.996463	−	0.0840302i	$-0.0267792\pi$
$390$		0			0
$391$		−	2.77390e6i		−	0.917589i
$392$		0			0
$393$	−1.68501e6	−	1.68501e6i	−0.550327	−	0.550327i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.04060e6	−	2.04060e6i	0.649802	−	0.649802i	−0.303143	−	0.952945i	$-0.598036\pi$
$397$	2.04060e6	−	2.04060e6i	0.649802	−	0.649802i	0.952945	+	0.303143i	$0.0980361\pi$
$398$		0			0
$399$	−769504.			−0.241980
$400$		0			0
$401$	3.62084e6			1.12447			0.562236	−	0.826977i	$-0.309941\pi$
$401$	3.62084e6			1.12447			0.562236	+	0.826977i	$0.309941\pi$
$402$		0			0
$403$	−122057.	+	122057.i	−0.0374368	+	0.0374368i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−639926.	−	639926.i	−0.191489	−	0.191489i
$408$		0			0
$409$		−	4.16692e6i		−	1.23171i	−0.787861	−	0.615853i	$-0.788811\pi$
$409$		−	4.16692e6i		−	1.23171i	0.787861	−	0.615853i	$-0.211189\pi$
$410$		0			0
$411$			1.79890e6i			0.525294i
$412$		0			0
$413$	883076.	+	883076.i	0.254755	+	0.254755i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.51683e6	−	2.51683e6i	0.708784	−	0.708784i
$418$		0			0
$419$	−256351.			−0.0713346			−0.0356673	−	0.999364i	$-0.511356\pi$
$419$	−256351.			−0.0713346			−0.0356673	+	0.999364i	$0.511356\pi$
$420$		0			0
$421$	−5.13509e6			−1.41203			−0.706013	−	0.708199i	$-0.749508\pi$
$421$	−5.13509e6			−1.41203			−0.706013	+	0.708199i	$0.749508\pi$
$422$		0			0
$423$	1.73971e6	−	1.73971e6i	0.472745	−	0.472745i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−337375.	−	337375.i	−0.0895455	−	0.0895455i
$428$		0			0
$429$			119510.i			0.0313516i
$430$		0			0
$431$		−	541018.i		−	0.140287i	−0.997537	−	0.0701437i	$-0.977654\pi$
$431$		−	541018.i		−	0.140287i	0.997537	−	0.0701437i	$-0.0223458\pi$
$432$		0			0
$433$	−3.73118e6	−	3.73118e6i	−0.956372	−	0.956372i	0.0427150	−	0.999087i	$-0.486399\pi$
$433$	−3.73118e6	−	3.73118e6i	−0.956372	−	0.956372i	−0.999087	+	0.0427150i	$0.986399\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.97245e6	+	5.97245e6i	−1.49606	+	1.49606i
$438$		0			0
$439$	4.37825e6			1.08427			0.542137	−	0.840290i	$-0.317615\pi$
$439$	4.37825e6			1.08427			0.542137	+	0.840290i	$0.317615\pi$
$440$		0			0
$441$	2.10497e6			0.515406
$442$		0			0
$443$	880315.	−	880315.i	0.213122	−	0.213122i	−0.592470	−	0.805592i	$-0.701847\pi$
$443$	880315.	−	880315.i	0.213122	−	0.213122i	0.805592	+	0.592470i	$0.201847\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−2.54409e6	−	2.54409e6i	−0.602232	−	0.602232i
$448$		0			0
$449$			1.14012e6i			0.266892i	0.991056	+	0.133446i	$0.0426042\pi$
$449$			1.14012e6i			0.266892i	−0.991056	+	0.133446i	$0.957396\pi$
$450$		0			0
$451$		−	1.97034e6i		−	0.456143i
$452$		0			0
$453$	1.67644e6	+	1.67644e6i	0.383833	+	0.383833i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.88319e6	+	1.88319e6i	−0.421796	+	0.421796i	−0.885822	−	0.464025i	$-0.846405\pi$
$457$	−1.88319e6	+	1.88319e6i	−0.421796	+	0.421796i	0.464025	+	0.885822i	$0.346405\pi$
$458$		0			0
$459$	3.47623e6			0.770152
$460$		0			0
$461$	−3.75143e6			−0.822138			−0.411069	−	0.911604i	$-0.634845\pi$
$461$	−3.75143e6			−0.822138			−0.411069	+	0.911604i	$0.634845\pi$
$462$		0			0
$463$	−3.38649e6	+	3.38649e6i	−0.734171	+	0.734171i	−0.971443	−	0.237273i	$-0.923747\pi$
$463$	−3.38649e6	+	3.38649e6i	−0.734171	+	0.734171i	0.237273	+	0.971443i	$0.423747\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.16438e6	−	5.16438e6i	−1.09579	−	1.09579i	−0.994898	−	0.100888i	$-0.967832\pi$
$467$	−5.16438e6	−	5.16438e6i	−1.09579	−	1.09579i	−0.100888	−	0.994898i	$-0.532168\pi$
$468$		0			0
$469$		−	1.19544e6i		−	0.250954i
$470$		0			0
$471$			2.52700e6i			0.524872i
$472$		0			0
$473$	−966356.	−	966356.i	−0.198602	−	0.198602i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−2.05423e6	+	2.05423e6i	−0.413383	+	0.413383i
$478$		0			0
$479$	7.72951e6			1.53926			0.769632	−	0.638488i	$-0.220440\pi$
$479$	7.72951e6			1.53926			0.769632	+	0.638488i	$0.220440\pi$
$480$		0			0
$481$	−313193.			−0.0617234
$482$		0			0
$483$	642605.	−	642605.i	0.125336	−	0.125336i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	237457.	+	237457.i	0.0453694	+	0.0453694i	0.729428	−	0.684058i	$-0.239787\pi$
$487$	237457.	+	237457.i	0.0453694	+	0.0453694i	−0.684058	+	0.729428i	$0.739787\pi$
$488$		0			0
$489$		−	5.14379e6i		−	0.972771i
$490$		0			0
$491$		−	3.99956e6i		−	0.748701i	−0.927287	−	0.374350i	$-0.877866\pi$
$491$		−	3.99956e6i		−	0.748701i	0.927287	−	0.374350i	$-0.122134\pi$
$492$		0			0
$493$	438334.	+	438334.i	0.0812247	+	0.0812247i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	203989.	−	203989.i	0.0370439	−	0.0370439i
$498$		0			0
$499$	6.58102e6			1.18316			0.591578	−	0.806248i	$-0.298505\pi$
$499$	6.58102e6			1.18316			0.591578	+	0.806248i	$0.298505\pi$
$500$		0			0
$501$	−3.29314e6			−0.586159
$502$		0			0
$503$	4.79249e6	−	4.79249e6i	0.844582	−	0.844582i	−0.144869	−	0.989451i	$-0.546276\pi$
$503$	4.79249e6	−	4.79249e6i	0.844582	−	0.844582i	0.989451	+	0.144869i	$0.0462761\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.74918e6	−	2.74918e6i	−0.474989	−	0.474989i
$508$		0			0
$509$			4.21148e6i			0.720511i	0.932854	+	0.360255i	$0.117310\pi$
$509$			4.21148e6i			0.720511i	−0.932854	+	0.360255i	$0.882690\pi$
$510$		0			0
$511$		−	1.95961e6i		−	0.331984i
$512$		0			0
$513$	−7.48464e6	−	7.48464e6i	−1.25567	−	1.25567i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−2.39887e6	+	2.39887e6i	−0.394711	+	0.394711i
$518$		0			0
$519$	1.54831e6			0.252312
$520$		0			0
$521$	884172.			0.142706			0.0713531	−	0.997451i	$-0.477268\pi$
$521$	884172.			0.142706			0.0713531	+	0.997451i	$0.477268\pi$
$522$		0			0
$523$	3.86026e6	−	3.86026e6i	0.617109	−	0.617109i	−0.327680	−	0.944789i	$-0.606267\pi$
$523$	3.86026e6	−	3.86026e6i	0.617109	−	0.617109i	0.944789	+	0.327680i	$0.106267\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.71477e6	−	1.71477e6i	−0.268955	−	0.268955i
$528$		0			0
$529$		−	3.53873e6i		−	0.549804i
$530$		0			0
$531$			6.01728e6i			0.926114i
$532$		0			0
$533$	−482163.	−	482163.i	−0.0735150	−	0.0735150i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	2.82787e6	−	2.82787e6i	0.423178	−	0.423178i
$538$		0			0
$539$	−2.90251e6			−0.430330
$540$		0			0
$541$	−4.20713e6			−0.618007			−0.309003	−	0.951061i	$-0.599995\pi$
$541$	−4.20713e6			−0.618007			−0.309003	+	0.951061i	$0.599995\pi$
$542$		0			0
$543$	4.36697e6	−	4.36697e6i	0.635595	−	0.635595i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.13449e6	+	6.13449e6i	0.876617	+	0.876617i	0.993183	−	0.116566i	$-0.0371885\pi$
$547$	6.13449e6	+	6.13449e6i	0.876617	+	0.876617i	−0.116566	+	0.993183i	$0.537189\pi$
$548$		0			0
$549$		−	2.29888e6i		−	0.325525i
$550$		0			0
$551$		−	1.88755e6i		−	0.264861i
$552$		0			0
$553$	476239.	+	476239.i	0.0662235	+	0.0662235i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.02695e6	+	3.02695e6i	−0.413397	+	0.413397i	−0.882920	−	0.469523i	$-0.844426\pi$
$557$	−3.02695e6	+	3.02695e6i	−0.413397	+	0.413397i	0.469523	+	0.882920i	$0.344426\pi$
$558$		0			0
$559$	−472955.			−0.0640162
$560$		0			0
$561$	−1.67899e6			−0.225238
$562$		0			0
$563$	1.91311e6	−	1.91311e6i	0.254372	−	0.254372i	−0.568388	−	0.822760i	$-0.692433\pi$
$563$	1.91311e6	−	1.91311e6i	0.254372	−	0.254372i	0.822760	+	0.568388i	$0.192433\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	193259.	+	193259.i	0.0252454	+	0.0252454i
$568$		0			0
$569$		−	399515.i		−	0.0517311i	−0.999665	−	0.0258656i	$-0.991766\pi$
$569$		−	399515.i		−	0.0517311i	0.999665	−	0.0258656i	$-0.00823418\pi$
$570$		0			0
$571$		−	1.17755e7i		−	1.51143i	−0.654900	−	0.755716i	$-0.727289\pi$
$571$		−	1.17755e7i		−	1.51143i	0.654900	−	0.755716i	$-0.272711\pi$
$572$		0			0
$573$	−2.75749e6	−	2.75749e6i	−0.350855	−	0.350855i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	3.88705e6	−	3.88705e6i	0.486050	−	0.486050i	−0.421007	−	0.907057i	$-0.638323\pi$
$577$	3.88705e6	−	3.88705e6i	0.486050	−	0.486050i	0.907057	+	0.421007i	$0.138323\pi$
$578$		0			0
$579$	−2.53603e6			−0.314383
$580$		0			0
$581$	1.88148e6			0.231239
$582$		0			0
$583$	2.83254e6	−	2.83254e6i	0.345148	−	0.345148i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.87317e6	−	4.87317e6i	−0.583735	−	0.583735i	0.352192	−	0.935928i	$-0.385436\pi$
$587$	−4.87317e6	−	4.87317e6i	−0.583735	−	0.583735i	−0.935928	+	0.352192i	$0.885436\pi$
$588$		0			0
$589$			7.38413e6i			0.877023i
$590$		0			0
$591$			3.29056e6i			0.387527i
$592$		0			0
$593$	−2.28474e6	−	2.28474e6i	−0.266808	−	0.266808i	0.561004	−	0.827813i	$-0.310415\pi$
$593$	−2.28474e6	−	2.28474e6i	−0.266808	−	0.266808i	−0.827813	+	0.561004i	$0.810415\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5.02971e6	+	5.02971e6i	−0.577573	+	0.577573i
$598$		0			0
$599$	−1.01689e7			−1.15799			−0.578997	−	0.815330i	$-0.696556\pi$
$599$	−1.01689e7			−1.15799			−0.578997	+	0.815330i	$0.696556\pi$
$600$		0			0
$601$	−1.61358e6			−0.182223			−0.0911117	−	0.995841i	$-0.529042\pi$
$601$	−1.61358e6			−0.182223			−0.0911117	+	0.995841i	$0.529042\pi$
$602$		0			0
$603$	4.07285e6	−	4.07285e6i	0.456147	−	0.456147i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.14202e6	+	1.14202e6i	0.125806	+	0.125806i	0.767206	−	0.641400i	$-0.221646\pi$
$607$	1.14202e6	+	1.14202e6i	0.125806	+	0.125806i	−0.641400	+	0.767206i	$0.721646\pi$
$608$		0			0
$609$			203090.i			0.0221894i
$610$		0			0
$611$			1.17405e6i			0.127229i
$612$		0			0
$613$	−3.44148e6	−	3.44148e6i	−0.369908	−	0.369908i	0.497536	−	0.867444i	$-0.334238\pi$
$613$	−3.44148e6	−	3.44148e6i	−0.369908	−	0.369908i	−0.867444	+	0.497536i	$0.834238\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.50497e6	−	7.50497e6i	0.793663	−	0.793663i	−0.188425	−	0.982088i	$-0.560338\pi$
$617$	7.50497e6	−	7.50497e6i	0.793663	−	0.793663i	0.982088	+	0.188425i	$0.0603381\pi$
$618$		0			0
$619$	2.89736e6			0.303931			0.151966	−	0.988386i	$-0.451440\pi$
$619$	2.89736e6			0.303931			0.151966	+	0.988386i	$0.451440\pi$
$620$		0			0
$621$	1.25007e7			1.30078
$622$		0			0
$623$	−501955.	+	501955.i	−0.0518137	+	0.0518137i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	3.61502e6	+	3.61502e6i	0.367234	+	0.367234i
$628$		0			0
$629$		−	4.40005e6i		−	0.443436i
$630$		0			0
$631$		−	1.03301e7i		−	1.03284i	−0.856336	−	0.516420i	$-0.827264\pi$
$631$		−	1.03301e7i		−	1.03284i	0.856336	−	0.516420i	$-0.172736\pi$
$632$		0			0
$633$	4.44798e6	+	4.44798e6i	0.441218	+	0.441218i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−710274.	+	710274.i	−0.0693549	+	0.0693549i
$638$		0			0
$639$	1.38998e6			0.134666
$640$		0			0
$641$	−3.35436e6			−0.322452			−0.161226	−	0.986918i	$-0.551545\pi$
$641$	−3.35436e6			−0.322452			−0.161226	+	0.986918i	$0.551545\pi$
$642$		0			0
$643$	126819.	−	126819.i	0.0120964	−	0.0120964i	−0.701033	−	0.713129i	$-0.747277\pi$
$643$	126819.	−	126819.i	0.0120964	−	0.0120964i	0.713129	+	0.701033i	$0.247277\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.07986e7	−	1.07986e7i	−1.01416	−	1.01416i	−0.999898	−	0.0142641i	$-0.995459\pi$
$647$	−1.07986e7	−	1.07986e7i	−1.01416	−	1.01416i	−0.0142641	−	0.999898i	$-0.504541\pi$
$648$		0			0
$649$		−	8.29714e6i		−	0.773244i
$650$		0			0
$651$		−	794495.i		−	0.0734748i
$652$		0			0
$653$	5.57502e6	+	5.57502e6i	0.511639	+	0.511639i	0.915028	−	0.403390i	$-0.132168\pi$
$653$	5.57502e6	+	5.57502e6i	0.511639	+	0.511639i	−0.403390	+	0.915028i	$0.632168\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	6.67640e6	−	6.67640e6i	0.603432	−	0.603432i
$658$		0			0
$659$	9.37611e6			0.841026			0.420513	−	0.907287i	$-0.361850\pi$
$659$	9.37611e6			0.841026			0.420513	+	0.907287i	$0.361850\pi$
$660$		0			0
$661$	−1.43499e6			−0.127746			−0.0638729	−	0.997958i	$-0.520345\pi$
$661$	−1.43499e6			−0.127746			−0.0638729	+	0.997958i	$0.520345\pi$
$662$		0			0
$663$	−410867.	+	410867.i	−0.0363009	+	0.0363009i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.57627e6	+	1.57627e6i	0.137188	+	0.137188i
$668$		0			0
$669$		−	6.32411e6i		−	0.546304i
$670$		0			0
$671$			3.16988e6i			0.271792i
$672$		0			0
$673$	1.49219e7	+	1.49219e7i	1.26995	+	1.26995i	0.946114	+	0.323834i	$0.104972\pi$
$673$	1.49219e7	+	1.49219e7i	1.26995	+	1.26995i	0.323834	+	0.946114i	$0.395028\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.54101e7	+	1.54101e7i	−1.29221	+	1.29221i	−0.358791	+	0.933418i	$0.616811\pi$
$677$	−1.54101e7	+	1.54101e7i	−1.29221	+	1.29221i	−0.933418	+	0.358791i	$0.883189\pi$
$678$		0			0
$679$	−1.53684e6			−0.127925
$680$		0			0
$681$	1.11885e7			0.924497
$682$		0			0
$683$	−9.68836e6	+	9.68836e6i	−0.794691	+	0.794691i	−0.982253	−	0.187562i	$-0.939942\pi$
$683$	−9.68836e6	+	9.68836e6i	−0.794691	+	0.794691i	0.187562	+	0.982253i	$0.439942\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	5.54007e6	+	5.54007e6i	0.447841	+	0.447841i
$688$		0			0
$689$		−	1.38630e6i		−	0.111253i
$690$		0			0
$691$			1.49923e7i			1.19447i	0.802068	+	0.597233i	$0.203733\pi$
$691$			1.49923e7i			1.19447i	−0.802068	+	0.597233i	$0.796267\pi$
$692$		0			0
$693$	454988.	+	454988.i	0.0359887	+	0.0359887i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	6.77392e6	−	6.77392e6i	0.528151	−	0.528151i
$698$		0			0
$699$	6.39450e6			0.495010
$700$		0			0
$701$	−4.06612e6			−0.312525			−0.156263	−	0.987716i	$-0.549945\pi$
$701$	−4.06612e6			−0.312525			−0.156263	+	0.987716i	$0.549945\pi$
$702$		0			0
$703$	−9.47372e6	+	9.47372e6i	−0.722990	+	0.722990i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.76906e6	−	3.76906e6i	−0.283586	−	0.283586i
$708$		0			0
$709$			1.00306e7i			0.749395i	0.927147	+	0.374698i	$0.122254\pi$
$709$			1.00306e7i			0.749395i	−0.927147	+	0.374698i	$0.877746\pi$
$710$		0			0
$711$			3.24509e6i			0.240743i
$712$		0			0
$713$	−6.16641e6	−	6.16641e6i	−0.454265	−	0.454265i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−3.80815e6	+	3.80815e6i	−0.276640	+	0.276640i
$718$		0			0
$719$	−4.83635e6			−0.348895			−0.174448	−	0.984666i	$-0.555814\pi$
$719$	−4.83635e6			−0.348895			−0.174448	+	0.984666i	$0.555814\pi$
$720$		0			0
$721$	5.69154e6			0.407748
$722$		0			0
$723$	41115.6	−	41115.6i	0.00292523	−	0.00292523i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	2.30692e6	+	2.30692e6i	0.161881	+	0.161881i	0.783399	−	0.621519i	$-0.213484\pi$
$727$	2.30692e6	+	2.30692e6i	0.161881	+	0.161881i	−0.621519	+	0.783399i	$0.713484\pi$
$728$		0			0
$729$			1.14953e7i			0.801125i
$730$		0			0
$731$		−	6.64454e6i		−	0.459909i
$732$		0			0
$733$	−5.94419e6	−	5.94419e6i	−0.408632	−	0.408632i	0.472629	−	0.881261i	$-0.343305\pi$
$733$	−5.94419e6	−	5.94419e6i	−0.408632	−	0.408632i	−0.881261	+	0.472629i	$0.843305\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.61599e6	+	5.61599e6i	−0.380853	+	0.380853i
$738$		0			0
$739$	1.19188e6			0.0802829			0.0401415	−	0.999194i	$-0.487219\pi$
$739$	1.19188e6			0.0802829			0.0401415	+	0.999194i	$0.487219\pi$
$740$		0			0
$741$	1.76927e6			0.118372
$742$		0			0
$743$	−1.87097e7	+	1.87097e7i	−1.24336	+	1.24336i	−0.284758	+	0.958600i	$0.591913\pi$
$743$	−1.87097e7	+	1.87097e7i	−1.24336	+	1.24336i	−0.958600	+	0.284758i	$0.908087\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	6.41022e6	+	6.41022e6i	0.420312	+	0.420312i
$748$		0			0
$749$			1.56117e6i			0.101682i
$750$		0			0
$751$			2.35884e7i			1.52615i	0.646308	+	0.763076i	$0.276312\pi$
$751$			2.35884e7i			1.52615i	−0.646308	+	0.763076i	$0.723688\pi$
$752$		0			0
$753$	−1.12532e6	−	1.12532e6i	−0.0723251	−	0.0723251i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.08571e7	−	2.08571e7i	1.32286	−	1.32286i	0.411417	−	0.911447i	$-0.365034\pi$
$757$	2.08571e7	−	2.08571e7i	1.32286	−	1.32286i	0.911447	−	0.411417i	$-0.134966\pi$
$758$		0			0
$759$	−6.03774e6			−0.380426
$760$		0			0
$761$	−1.14245e7			−0.715114			−0.357557	−	0.933891i	$-0.616390\pi$
$761$	−1.14245e7			−0.715114			−0.357557	+	0.933891i	$0.616390\pi$
$762$		0			0
$763$	−969223.	+	969223.i	−0.0602716	+	0.0602716i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.03040e6	−	2.03040e6i	−0.124621	−	0.124621i
$768$		0			0
$769$		−	1.61255e7i		−	0.983325i	−0.870786	−	0.491663i	$-0.836389\pi$
$769$		−	1.61255e7i		−	0.983325i	0.870786	−	0.491663i	$-0.163611\pi$
$770$		0			0
$771$		−	1.60616e7i		−	0.973092i
$772$		0			0
$773$	1.32344e7	+	1.32344e7i	0.796630	+	0.796630i	0.982563	−	0.185933i	$-0.0595306\pi$
$773$	1.32344e7	+	1.32344e7i	0.796630	+	0.796630i	−0.185933	+	0.982563i	$0.559531\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1.01932e6	−	1.01932e6i	0.0605703	−	0.0605703i
$778$		0			0
$779$	−2.91697e7			−1.72222
$780$		0			0
$781$	−1.91663e6			−0.112437
$782$		0			0
$783$	−1.97537e6	+	1.97537e6i	−0.115145	+	0.115145i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	1.61771e7	+	1.61771e7i	0.931032	+	0.931032i	0.997770	−	0.0667386i	$-0.0212594\pi$
$787$	1.61771e7	+	1.61771e7i	0.931032	+	0.931032i	−0.0667386	+	0.997770i	$0.521259\pi$
$788$		0			0
$789$			1.69285e7i			0.968113i
$790$		0			0
$791$		−	5.88078e6i		−	0.334190i
$792$		0			0
$793$	775703.	+	775703.i	0.0438039	+	0.0438039i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.99273e6	−	2.99273e6i	0.166887	−	0.166887i	−0.618723	−	0.785609i	$-0.712350\pi$
$797$	2.99273e6	−	2.99273e6i	0.166887	−	0.166887i	0.785609	+	0.618723i	$0.212350\pi$
$798$		0			0
$799$	−1.64943e7			−0.914044
$800$		0			0
$801$	−3.42032e6			−0.188359
$802$		0			0
$803$	−9.20598e6	+	9.20598e6i	−0.503827	+	0.503827i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.19399e7	+	1.19399e7i	0.645383	+	0.645383i
$808$		0			0
$809$			3.52391e7i			1.89301i	0.322683	+	0.946507i	$0.395415\pi$
$809$			3.52391e7i			1.89301i	−0.322683	+	0.946507i	$0.604585\pi$
$810$		0			0
$811$			2.75053e7i			1.46847i	0.678897	+	0.734234i	$0.262458\pi$
$811$			2.75053e7i			1.46847i	−0.678897	+	0.734234i	$0.737542\pi$
$812$		0			0
$813$	1.53964e7	+	1.53964e7i	0.816945	+	0.816945i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.43063e7	+	1.43063e7i	−0.749847	+	0.749847i
$818$		0			0
$819$	222680.			0.0116004
$820$		0			0
$821$	−1.95665e7			−1.01311			−0.506554	−	0.862208i	$-0.669081\pi$
$821$	−1.95665e7			−1.01311			−0.506554	+	0.862208i	$0.669081\pi$
$822$		0			0
$823$	1.15110e7	−	1.15110e7i	0.592399	−	0.592399i	−0.345880	−	0.938279i	$-0.612420\pi$
$823$	1.15110e7	−	1.15110e7i	0.592399	−	0.592399i	0.938279	+	0.345880i	$0.112420\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.12263e7	+	1.12263e7i	0.570787	+	0.570787i	0.932348	−	0.361561i	$-0.117756\pi$
$827$	1.12263e7	+	1.12263e7i	0.570787	+	0.570787i	−0.361561	+	0.932348i	$0.617756\pi$
$828$		0			0
$829$			8.81335e6i			0.445405i	0.974887	+	0.222702i	$0.0714878\pi$
$829$			8.81335e6i			0.445405i	−0.974887	+	0.222702i	$0.928512\pi$
$830$		0			0
$831$			2.25760e7i			1.13408i
$832$		0			0
$833$	−9.97864e6	−	9.97864e6i	−0.498263	−	0.498263i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	7.72771e6	−	7.72771e6i	0.381274	−	0.381274i
$838$		0			0
$839$	−2.32474e7			−1.14017			−0.570086	−	0.821585i	$-0.693090\pi$
$839$	−2.32474e7			−1.14017			−0.570086	+	0.821585i	$0.693090\pi$
$840$		0			0
$841$	2.00130e7			0.975712
$842$		0			0
$843$	−849830.	+	849830.i	−0.0411873	+	0.0411873i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.46899e6	+	2.46899e6i	0.118253	+	0.118253i
$848$		0			0
$849$			1.03860e7i			0.494513i
$850$		0			0
$851$		−	1.58228e7i		−	0.748962i
$852$		0			0
$853$	−1.95949e7	−	1.95949e7i	−0.922085	−	0.922085i	0.0750917	−	0.997177i	$-0.476075\pi$
$853$	−1.95949e7	−	1.95949e7i	−0.922085	−	0.922085i	−0.997177	+	0.0750917i	$0.976075\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.52052e7	−	1.52052e7i	0.707197	−	0.707197i	−0.258748	−	0.965945i	$-0.583310\pi$
$857$	1.52052e7	−	1.52052e7i	0.707197	−	0.707197i	0.965945	+	0.258748i	$0.0833099\pi$
$858$		0			0
$859$	−1.55233e7			−0.717795			−0.358897	−	0.933377i	$-0.616847\pi$
$859$	−1.55233e7			−0.717795			−0.358897	+	0.933377i	$0.616847\pi$
$860$		0			0
$861$	3.13851e6			0.144283
$862$		0			0
$863$	−1.63784e7	+	1.63784e7i	−0.748590	+	0.748590i	−0.974214	−	0.225624i	$-0.927558\pi$
$863$	−1.63784e7	+	1.63784e7i	−0.748590	+	0.748590i	0.225624	+	0.974214i	$0.427558\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	4.85268e6	+	4.85268e6i	0.219247	+	0.219247i
$868$		0			0
$869$		−	4.47461e6i		−	0.201004i
$870$		0			0
$871$			2.74858e6i			0.122762i
$872$		0			0
$873$	−5.23602e6	−	5.23602e6i	−0.232523	−	0.232523i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	4.76278e6	−	4.76278e6i	0.209103	−	0.209103i	−0.594783	−	0.803886i	$-0.702762\pi$
$877$	4.76278e6	−	4.76278e6i	0.209103	−	0.209103i	0.803886	+	0.594783i	$0.202762\pi$
$878$		0			0
$879$	2.15885e7			0.942434
$880$		0			0
$881$	9.98884e6			0.433586			0.216793	−	0.976218i	$-0.430440\pi$
$881$	9.98884e6			0.433586			0.216793	+	0.976218i	$0.430440\pi$
$882$		0			0
$883$	−1.37685e7	+	1.37685e7i	−0.594272	+	0.594272i	−0.938782	−	0.344511i	$-0.888045\pi$
$883$	−1.37685e7	+	1.37685e7i	−0.594272	+	0.594272i	0.344511	+	0.938782i	$0.388045\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.78096e6	−	1.78096e6i	−0.0760056	−	0.0760056i	0.668082	−	0.744088i	$-0.267116\pi$
$887$	−1.78096e6	−	1.78096e6i	−0.0760056	−	0.0760056i	−0.744088	+	0.668082i	$0.767116\pi$
$888$		0			0
$889$		−	7.31026e6i		−	0.310226i
$890$		0			0
$891$		−	1.81581e6i		−	0.0766258i
$892$		0			0
$893$	3.55137e7	+	3.55137e7i	1.49028	+	1.49028i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.47750e6	+	1.47750e6i	−0.0613120	+	0.0613120i
$898$		0			0
$899$	1.94885e6			0.0804227
$900$		0			0
$901$	1.94762e7			0.799268
$902$		0			0
$903$	1.53929e6	−	1.53929e6i	0.0628203	−	0.0628203i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.51162e7	−	1.51162e7i	−0.610134	−	0.610134i	0.332847	−	0.942981i	$-0.391991\pi$
$907$	−1.51162e7	−	1.51162e7i	−0.610134	−	0.610134i	−0.942981	+	0.332847i	$0.891991\pi$
$908$		0			0
$909$		−	2.56824e7i		−	1.03092i
$910$		0			0
$911$		−	3.91576e7i		−	1.56322i	−0.623767	−	0.781611i	$-0.714398\pi$
$911$		−	3.91576e7i		−	1.56322i	0.623767	−	0.781611i	$-0.285602\pi$
$912$		0			0
$913$	−8.83895e6	−	8.83895e6i	−0.350933	−	0.350933i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.32921e6	−	4.32921e6i	0.170014	−	0.170014i
$918$		0			0
$919$	2.39571e7			0.935720			0.467860	−	0.883803i	$-0.345025\pi$
$919$	2.39571e7			0.935720			0.467860	+	0.883803i	$0.345025\pi$
$920$		0			0
$921$	−6.22256e6			−0.241724
$922$		0			0
$923$	−469018.	+	469018.i	−0.0181211	+	0.0181211i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.93911e7	+	1.93911e7i	0.741145	+	0.741145i
$928$		0			0
$929$		−	2.59769e7i		−	0.987526i	−0.869597	−	0.493763i	$-0.835621\pi$
$929$		−	2.59769e7i		−	0.987526i	0.869597	−	0.493763i	$-0.164379\pi$
$930$		0			0
$931$			4.29699e7i			1.62476i
$932$		0			0
$933$	−5.40959e6	−	5.40959e6i	−0.203451	−	0.203451i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.66598e7	+	2.66598e7i	−0.991993	+	0.991993i	−0.999968	−	0.00797527i	$-0.997461\pi$
$937$	−2.66598e7	+	2.66598e7i	−0.991993	+	0.991993i	0.00797527	+	0.999968i	$0.497461\pi$
$938$		0			0
$939$	1.15057e7			0.425843
$940$		0			0
$941$	−1.69843e7			−0.625280			−0.312640	−	0.949872i	$-0.601213\pi$
$941$	−1.69843e7			−0.625280			−0.312640	+	0.949872i	$0.601213\pi$
$942$		0			0
$943$	2.43594e7	−	2.43594e7i	0.892045	−	0.892045i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.65542e7	−	3.65542e7i	−1.32453	−	1.32453i	−0.910063	−	0.414470i	$-0.863967\pi$
$947$	−3.65542e7	−	3.65542e7i	−1.32453	−	1.32453i	−0.414470	−	0.910063i	$-0.636033\pi$
$948$		0			0
$949$			4.50560e6i			0.162400i
$950$		0			0
$951$			5.12155e6i			0.183633i
$952$		0			0
$953$	3.50021e7	+	3.50021e7i	1.24842	+	1.24842i	0.956416	+	0.292008i	$0.0943234\pi$
$953$	3.50021e7	+	3.50021e7i	1.24842	+	1.24842i	0.292008	+	0.956416i	$0.405677\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	954091.	−	954091.i	0.0336752	−	0.0336752i
$958$		0			0
$959$	−4.62183e6			−0.162281
$960$		0			0
$961$	2.10052e7			0.733700
$962$		0			0
$963$	−5.31891e6	+	5.31891e6i	−0.184823	+	0.184823i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−6.65238e6	−	6.65238e6i	−0.228776	−	0.228776i	0.583405	−	0.812181i	$-0.301720\pi$
$967$	−6.65238e6	−	6.65238e6i	−0.228776	−	0.228776i	−0.812181	+	0.583405i	$0.801720\pi$
$968$		0			0
$969$			2.48565e7i			0.850413i
$970$		0			0
$971$		−	2.00433e7i		−	0.682215i	−0.940024	−	0.341108i	$-0.889198\pi$
$971$		−	2.00433e7i		−	0.682215i	0.940024	−	0.341108i	$-0.110802\pi$
$972$		0			0
$973$	6.46638e6	+	6.46638e6i	0.218967	+	0.218967i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.48210e7	+	3.48210e7i	−1.16709	+	1.16709i	−0.184204	+	0.982888i	$0.558971\pi$
$977$	−3.48210e7	+	3.48210e7i	−1.16709	+	1.16709i	−0.982888	+	0.184204i	$0.941029\pi$
$978$		0			0
$979$	4.71623e6			0.157267
$980$		0			0
$981$	−6.60429e6			−0.219106
$982$		0			0
$983$	−2.26938e7	+	2.26938e7i	−0.749073	+	0.749073i	−0.974305	−	0.225232i	$-0.927686\pi$
$983$	−2.26938e7	+	2.26938e7i	−0.749073	+	0.749073i	0.225232	+	0.974305i	$0.427686\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−3.82110e6	−	3.82110e6i	−0.124852	−	0.124852i
$988$		0			0
$989$		−	2.38941e7i		−	0.776784i
$990$		0			0
$991$		−	7.33026e6i		−	0.237102i	−0.992948	−	0.118551i	$-0.962175\pi$
$991$		−	7.33026e6i		−	0.237102i	0.992948	−	0.118551i	$-0.0378249\pi$
$992$		0			0
$993$	4.45938e6	+	4.45938e6i	0.143516	+	0.143516i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.70672e7	−	3.70672e7i	1.18101	−	1.18101i	0.201521	−	0.979484i	$-0.435411\pi$
$997$	3.70672e7	−	3.70672e7i	1.18101	−	1.18101i	0.979484	−	0.201521i	$-0.0645885\pi$
$998$		0			0
$999$	1.98290e7			0.628620

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.n.g.143.7		20
4.3	odd	2	inner	400.6.n.g.143.4		20
5.2	odd	4	inner	400.6.n.g.207.4		20
5.3	odd	4		80.6.n.d.47.7	yes	20
5.4	even	2		80.6.n.d.63.4	yes	20
20.3	even	4		80.6.n.d.47.4	✓	20
20.7	even	4	inner	400.6.n.g.207.7		20
20.19	odd	2		80.6.n.d.63.7	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.6.n.d.47.4	✓	20	20.3	even	4
80.6.n.d.47.7	yes	20	5.3	odd	4
80.6.n.d.63.4	yes	20	5.4	even	2
80.6.n.d.63.7	yes	20	20.19	odd	2
400.6.n.g.143.4		20	4.3	odd	2	inner
400.6.n.g.143.7		20	1.1	even	1	trivial
400.6.n.g.207.4		20	5.2	odd	4	inner
400.6.n.g.207.7		20	20.7	even	4	inner

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(7.48311 - 7.48311i) q^{3} +(19.2260 + 19.2260i) q^{7} +131.006i q^{9} +O(q^{10})\)	\(q+(7.48311 - 7.48311i) q^{3} +(19.2260 + 19.2260i) q^{7} +131.006i q^{9} -180.642i q^{11} +(-44.2050 - 44.2050i) q^{13} +(621.037 - 621.037i) q^{17} -2674.30 q^{19} +287.741 q^{21} +(2233.28 - 2233.28i) q^{23} +(2798.73 + 2798.73i) q^{27} +705.810i q^{29} -2761.15i q^{31} +(-1351.77 - 1351.77i) q^{33} +(3542.51 - 3542.51i) q^{37} -661.582 q^{39} +10907.4 q^{41} +(5349.56 - 5349.56i) q^{43} +(-13279.7 - 13279.7i) q^{47} -16067.7i q^{49} -9294.57i q^{51} +(15680.4 + 15680.4i) q^{53} +(-20012.1 + 20012.1i) q^{57} +45931.3 q^{59} -17547.9 q^{61} +(-2518.72 + 2518.72i) q^{63} +(-31089.0 - 31089.0i) q^{67} -33423.8i q^{69} -10610.1i q^{71} +(-50962.5 - 50962.5i) q^{73} +(3473.03 - 3473.03i) q^{77} +24770.6 q^{79} +10051.9 q^{81} +(48930.7 - 48930.7i) q^{83} +(5281.66 + 5281.66i) q^{87} +26108.1i q^{89} -1699.77i q^{91} +(-20662.0 - 20662.0i) q^{93} +(-39967.8 + 39967.8i) q^{97} +23665.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) \) 20 * q - 804 * q^13 + 2236 * q^17 - 4520 * q^21 + 11096 * q^33 - 44260 * q^37 - 6760 * q^41 - 182452 * q^53 + 34288 * q^57 - 41080 * q^61 - 264372 * q^73 - 399304 * q^77 - 520220 * q^81 - 713496 * q^93 - 374772 * q^97

Introduction

L-functions

Modular forms

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Database

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