LMFDB - Embedded newform 400.6.n.g.207.4

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			207.4
Root			$5.50401 - 11.9953i$ of defining polynomial
Character	$\chi$	$=$	400.207
Dual form			400.6.n.g.143.4

$q$-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 $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

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

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	$-0.360238\pi$
$3$	−7.48311	−	7.48311i	−0.480042	−	0.480042i	−0.905145	+	0.425103i	$0.860238\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−19.2260	+	19.2260i	−0.148301	+	0.148301i	−0.777359	−	0.629058i	$-0.783441\pi$
$7$	−19.2260	+	19.2260i	−0.148301	+	0.148301i	0.629058	+	0.777359i	$0.283441\pi$
$8$		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	−0.742449	−	0.669903i	$-0.766336\pi$
$13$	−44.2050	+	44.2050i	−0.0725459	+	0.0725459i	0.669903	+	0.742449i	$0.266336\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	621.037	+	621.037i	0.521189	+	0.521189i	0.917930	−	0.396742i	$-0.129859\pi$
$17$	621.037	+	621.037i	0.521189	+	0.521189i	−0.396742	+	0.917930i	$0.629859\pi$
$18$		0			0
$19$	2674.30			1.69952			0.849759	−	0.527172i	$-0.176747\pi$
$19$	2674.30			1.69952			0.849759	+	0.527172i	$0.176747\pi$
$20$		0			0
$21$	287.741			0.142381
$22$		0			0
$23$	−2233.28	−	2233.28i	−0.880285	−	0.880285i	0.113278	−	0.993563i	$-0.463865\pi$
$23$	−2233.28	−	2233.28i	−0.880285	−	0.880285i	−0.993563	+	0.113278i	$0.963865\pi$
$24$		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	$-0.975171\pi$
$29$		−	705.810i		−	0.155845i	0.996959	−	0.0779225i	$-0.0248287\pi$
$30$		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	0.887061	−	0.461652i	$-0.152743\pi$
$37$	3542.51	+	3542.51i	0.425409	+	0.425409i	−0.461652	+	0.887061i	$0.652743\pi$
$38$		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	$-0.350993\pi$
$43$	−5349.56	−	5349.56i	−0.441212	−	0.441212i	−0.892419	+	0.451207i	$0.850993\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	13279.7	−	13279.7i	0.876884	−	0.876884i	−0.116327	−	0.993211i	$-0.537112\pi$
$47$	13279.7	−	13279.7i	0.876884	−	0.876884i	0.993211	+	0.116327i	$0.0371121\pi$
$48$		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	−0.210763	−	0.977537i	$-0.567595\pi$
$53$	15680.4	−	15680.4i	0.766774	−	0.766774i	0.977537	+	0.210763i	$0.0675948\pi$
$54$		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	$-0.828860\pi$
$59$	−45931.3			−1.71783			−0.858913	+	0.512122i	$0.828860\pi$
$60$		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	$-0.545851\pi$
$67$	31089.0	−	31089.0i	0.846097	−	0.846097i	0.989644	+	0.143547i	$0.0458508\pi$
$68$		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	−0.127447	+	0.991845i	$0.540678\pi$
$73$	−50962.5	+	50962.5i	−1.11929	+	1.11929i	−0.991845	+	0.127447i	$0.959322\pi$
$74$		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	$-0.571674\pi$
$79$	−24770.6			−0.446548			−0.223274	+	0.974756i	$0.571674\pi$
$80$		0			0
$81$	10051.9			0.170231
$82$		0			0
$83$	−48930.7	−	48930.7i	−0.779626	−	0.779626i	0.200141	−	0.979767i	$-0.435860\pi$
$83$	−48930.7	−	48930.7i	−0.779626	−	0.779626i	−0.979767	+	0.200141i	$0.935860\pi$
$84$		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	$-0.944107\pi$
$89$		−	26108.1i		−	0.349382i	0.984623	−	0.174691i	$-0.0558926\pi$
$90$		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	0.457770	−	0.889071i	$-0.348648\pi$
$97$	−39967.8	−	39967.8i	−0.431301	−	0.431301i	−0.889071	+	0.457770i	$0.848648\pi$
$98$		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	−0.853282	−	0.521450i	$-0.825391\pi$
$103$	−148017.	−	148017.i	−1.37473	−	1.37473i	−0.521450	−	0.853282i	$-0.674609\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−40600.5	+	40600.5i	−0.342824	+	0.342824i	−0.857428	−	0.514604i	$-0.827939\pi$
$107$	−40600.5	+	40600.5i	−0.342824	+	0.342824i	0.514604	+	0.857428i	$0.327939\pi$
$108$		0			0
$109$		−	50412.1i		−	0.406414i	−0.979136	−	0.203207i	$-0.934864\pi$
$109$		−	50412.1i		−	0.406414i	0.979136	−	0.203207i	$-0.0651364\pi$
$110$		0			0
$111$		−	53018.0i		−	0.408428i
$112$		0			0
$113$	−152938.	+	152938.i	−1.12673	+	1.12673i	−0.136024	+	0.990706i	$0.543432\pi$
$113$	−152938.	+	152938.i	−1.12673	+	1.12673i	−0.990706	+	0.136024i	$0.956568\pi$
$114$		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	0.0470414	−	0.998893i	$-0.485021\pi$
$127$	190114.	−	190114.i	1.04593	−	1.04593i	0.998893	−	0.0470414i	$-0.0149793\pi$
$128$		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	0.378477	−	0.925611i	$-0.376448\pi$
$137$	−120197.	−	120197.i	−0.547133	−	0.547133i	−0.925611	+	0.378477i	$0.876448\pi$
$138$		0			0
$139$	−336335.			−1.47651			−0.738253	−	0.674524i	$-0.764349\pi$
$139$	−336335.			−1.47651			−0.738253	+	0.674524i	$0.764349\pi$
$140$		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	$0.215828\pi$
$149$			339978.i			1.25454i	−0.778802	+	0.627270i	$0.784172\pi$
$150$		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	0.378789	−	0.925483i	$-0.376340\pi$
$157$	−168847.	−	168847.i	−0.546694	−	0.546694i	−0.925483	+	0.378789i	$0.876340\pi$
$158$		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	−0.999912	−	0.0133036i	$-0.995765\pi$
$163$	−343693.	−	343693.i	−1.01322	−	1.01322i	−0.0133036	−	0.999912i	$-0.504235\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	220038.	−	220038.i	0.610529	−	0.610529i	−0.332555	−	0.943084i	$-0.607911\pi$
$167$	220038.	−	220038.i	0.610529	−	0.610529i	0.943084	+	0.332555i	$0.107911\pi$
$168$		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	−0.563389	−	0.826192i	$-0.690503\pi$
$173$	103453.	−	103453.i	0.262802	−	0.262802i	0.826192	+	0.563389i	$0.190503\pi$
$174$		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	$-0.645295\pi$
$179$	−377900.			−0.881544			−0.440772	+	0.897619i	$0.645295\pi$
$180$		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	−0.851617	−	0.524164i	$-0.824378\pi$
$193$	−169450.	+	169450.i	−0.327453	+	0.327453i	0.524164	+	0.851617i	$0.324378\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−219866.	−	219866.i	−0.403638	−	0.403638i	0.475875	−	0.879513i	$-0.342132\pi$
$197$	−219866.	−	219866.i	−0.403638	−	0.403638i	−0.879513	+	0.475875i	$0.842132\pi$
$198$		0			0
$199$	672141.			1.20317			0.601586	−	0.798808i	$-0.294536\pi$
$199$	672141.			1.20317			0.601586	+	0.798808i	$0.294536\pi$
$200$		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	$-0.381809\pi$
$223$	−422559.	−	422559.i	−0.569017	−	0.569017i	−0.931853	+	0.362836i	$0.881809\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−747586.	+	747586.i	−0.962934	+	0.962934i	−0.999337	−	0.0364031i	$-0.988410\pi$
$227$	−747586.	+	747586.i	−0.962934	+	0.962934i	0.0364031	+	0.999337i	$0.488410\pi$
$228$		0			0
$229$		−	740343.i		−	0.932920i	−0.884542	−	0.466460i	$-0.845529\pi$
$229$		−	740343.i		−	0.932920i	0.884542	−	0.466460i	$-0.154471\pi$
$230$		0			0
$231$		−	51978.1i		−	0.0640901i
$232$		0			0
$233$	427262.	−	427262.i	0.515590	−	0.515590i	−0.400644	−	0.916234i	$-0.631214\pi$
$233$	427262.	−	427262.i	0.515590	−	0.515590i	0.916234	+	0.400644i	$0.131214\pi$
$234$		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	$-0.406962\pi$
$239$	508899.			0.576284			0.288142	+	0.957588i	$0.406962\pi$
$240$		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	0.999907	+	0.0136415i	$0.00434237\pi$
$257$	1.07319e6	+	1.07319e6i	1.01355	+	1.01355i	0.0136415	+	0.999907i	$0.495658\pi$
$258$		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	0.999965	+	0.00839852i	$0.00267336\pi$
$263$	1.13111e6	+	1.13111e6i	1.00836	+	1.00836i	0.00839852	+	0.999965i	$0.497327\pi$
$264$		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	$-0.765343\pi$
$269$		−	1.59558e6i		−	1.34443i	0.740356	−	0.672216i	$-0.234657\pi$
$270$		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	−0.979425	−	0.201809i	$-0.935318\pi$
$277$	−1.50847e6	−	1.50847e6i	−1.18123	−	1.18123i	−0.201809	−	0.979425i	$-0.564682\pi$
$278$		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	$-0.131339\pi$
$283$	693960.	+	693960.i	0.515072	+	0.515072i	−0.401004	+	0.916076i	$0.631339\pi$
$284$		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	−0.0182181	−	0.999834i	$-0.505799\pi$
$293$	1.44248e6	−	1.44248e6i	0.981616	−	0.981616i	0.999834	+	0.0182181i	$0.00579933\pi$
$294$		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	$-0.693027\pi$
$307$	415774.	−	415774.i	0.251774	−	0.251774i	0.821698	+	0.569924i	$0.193027\pi$
$308$		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	−0.449655	−	0.893202i	$-0.648453\pi$
$313$	768778.	−	768778.i	0.443547	−	0.443547i	0.893202	+	0.449655i	$0.148453\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−342207.	−	342207.i	−0.191267	−	0.191267i	0.604976	−	0.796244i	$-0.293183\pi$
$317$	−342207.	−	342207.i	−0.191267	−	0.191267i	−0.796244	+	0.604976i	$0.793183\pi$
$318$		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	0.972423	−	0.233224i	$-0.0749276\pi$
$337$	1.54112e6	+	1.54112e6i	0.739199	+	0.739199i	−0.233224	+	0.972423i	$0.574928\pi$
$338$		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	$-0.783612\pi$
$347$	−334328.	+	334328.i	−0.149056	+	0.149056i	0.628640	+	0.777696i	$0.283612\pi$
$348$		0			0
$349$		−	3.55159e6i		−	1.56084i	−0.625254	−	0.780422i	$-0.715005\pi$
$349$		−	3.55159e6i		−	1.56084i	0.625254	−	0.780422i	$-0.284995\pi$
$350$		0			0
$351$		−	247436.i		−	0.107200i
$352$		0			0
$353$	−942189.	+	942189.i	−0.402440	+	0.402440i	−0.879092	−	0.476652i	$-0.841850\pi$
$353$	−942189.	+	942189.i	−0.402440	+	0.402440i	0.476652	+	0.879092i	$0.341850\pi$
$354$		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	$-0.382849\pi$
$359$	1.75716e6			0.719575			0.359787	+	0.933034i	$0.382849\pi$
$360$		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	$-0.971001\pi$
$367$	−2.33482e6	+	2.33482e6i	−0.904876	+	0.904876i	0.0909771	+	0.995853i	$0.471001\pi$
$368$		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	−0.0857724	−	0.996315i	$-0.527336\pi$
$373$	2.44665e6	−	2.44665e6i	0.910542	−	0.910542i	0.996315	+	0.0857724i	$0.0273358\pi$
$374$		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	$-0.370624\pi$
$379$	2.21109e6			0.790694			0.395347	+	0.918532i	$0.370624\pi$
$380$		0			0
$381$	−2.84529e6			−1.00418
$382$		0			0
$383$	−1.12909e6	−	1.12909e6i	−0.393307	−	0.393307i	0.482557	−	0.875865i	$-0.339708\pi$
$383$	−1.12909e6	−	1.12909e6i	−0.393307	−	0.393307i	−0.875865	+	0.482557i	$0.839708\pi$
$384$		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	$0.0267792\pi$
$389$			501579.i			0.168060i	−0.996463	+	0.0840302i	$0.973221\pi$
$390$		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	0.952945	−	0.303143i	$-0.0980361\pi$
$397$	2.04060e6	+	2.04060e6i	0.649802	+	0.649802i	−0.303143	+	0.952945i	$0.598036\pi$
$398$		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	$0.211189\pi$
$409$			4.16692e6i			1.23171i	−0.787861	+	0.615853i	$0.788811\pi$
$410$		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	$-0.488644\pi$
$419$	256351.			0.0713346			0.0356673	+	0.999364i	$0.488644\pi$
$420$		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	−0.999087	−	0.0427150i	$-0.986399\pi$
$433$	−3.73118e6	+	3.73118e6i	−0.956372	+	0.956372i	0.0427150	+	0.999087i	$0.486399\pi$
$434$		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	$-0.682385\pi$
$439$	−4.37825e6			−1.08427			−0.542137	+	0.840290i	$0.682385\pi$
$440$		0			0
$441$	2.10497e6			0.515406
$442$		0			0
$443$	−880315.	−	880315.i	−0.213122	−	0.213122i	0.592470	−	0.805592i	$-0.298153\pi$
$443$	−880315.	−	880315.i	−0.213122	−	0.213122i	−0.805592	+	0.592470i	$0.798153\pi$
$444$		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	$-0.957396\pi$
$449$		−	1.14012e6i		−	0.266892i	0.991056	−	0.133446i	$-0.0426042\pi$
$450$		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	0.464025	−	0.885822i	$-0.346405\pi$
$457$	−1.88319e6	−	1.88319e6i	−0.421796	−	0.421796i	−0.885822	+	0.464025i	$0.846405\pi$
$458$		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	$-0.0762534\pi$
$463$	3.38649e6	+	3.38649e6i	0.734171	+	0.734171i	−0.237273	+	0.971443i	$0.576253\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.16438e6	−	5.16438e6i	1.09579	−	1.09579i	0.100888	−	0.994898i	$-0.467832\pi$
$467$	5.16438e6	−	5.16438e6i	1.09579	−	1.09579i	0.994898	−	0.100888i	$-0.0321685\pi$
$468$		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	$-0.779560\pi$
$479$	−7.72951e6			−1.53926			−0.769632	+	0.638488i	$0.779560\pi$
$480$		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	$-0.760213\pi$
$487$	−237457.	+	237457.i	−0.0453694	+	0.0453694i	0.684058	+	0.729428i	$0.260213\pi$
$488$		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	$-0.701495\pi$
$499$	−6.58102e6			−1.18316			−0.591578	+	0.806248i	$0.701495\pi$
$500$		0			0
$501$	−3.29314e6			−0.586159
$502$		0			0
$503$	−4.79249e6	−	4.79249e6i	−0.844582	−	0.844582i	0.144869	−	0.989451i	$-0.453724\pi$
$503$	−4.79249e6	−	4.79249e6i	−0.844582	−	0.844582i	−0.989451	+	0.144869i	$0.953724\pi$
$504$		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	$-0.882690\pi$
$509$		−	4.21148e6i		−	0.720511i	0.932854	−	0.360255i	$-0.117310\pi$
$510$		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	$-0.393733\pi$
$523$	−3.86026e6	−	3.86026e6i	−0.617109	−	0.617109i	−0.944789	+	0.327680i	$0.893733\pi$
$524$		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	$-0.962811\pi$
$547$	−6.13449e6	+	6.13449e6i	−0.876617	+	0.876617i	0.116566	+	0.993183i	$0.462811\pi$
$548$		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	0.469523	−	0.882920i	$-0.344426\pi$
$557$	−3.02695e6	−	3.02695e6i	−0.413397	−	0.413397i	−0.882920	+	0.469523i	$0.844426\pi$
$558$		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	$-0.307567\pi$
$563$	−1.91311e6	−	1.91311e6i	−0.254372	−	0.254372i	−0.822760	+	0.568388i	$0.807567\pi$
$564$		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	$0.00823418\pi$
$569$			399515.i			0.0517311i	−0.999665	+	0.0258656i	$0.991766\pi$
$570$		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	0.907057	−	0.421007i	$-0.138323\pi$
$577$	3.88705e6	+	3.88705e6i	0.486050	+	0.486050i	−0.421007	+	0.907057i	$0.638323\pi$
$578$		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	$-0.614564\pi$
$587$	4.87317e6	−	4.87317e6i	0.583735	−	0.583735i	0.935928	+	0.352192i	$0.114564\pi$
$588$		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	−0.827813	−	0.561004i	$-0.810415\pi$
$593$	−2.28474e6	+	2.28474e6i	−0.266808	+	0.266808i	0.561004	+	0.827813i	$0.310415\pi$
$594$		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	$-0.303444\pi$
$599$	1.01689e7			1.15799			0.578997	+	0.815330i	$0.303444\pi$
$600$		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	$-0.778354\pi$
$607$	−1.14202e6	+	1.14202e6i	−0.125806	+	0.125806i	0.641400	+	0.767206i	$0.278354\pi$
$608$		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	−0.867444	−	0.497536i	$-0.834238\pi$
$613$	−3.44148e6	+	3.44148e6i	−0.369908	+	0.369908i	0.497536	+	0.867444i	$0.334238\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.50497e6	+	7.50497e6i	0.793663	+	0.793663i	0.982088	−	0.188425i	$-0.0603381\pi$
$617$	7.50497e6	+	7.50497e6i	0.793663	+	0.793663i	−0.188425	+	0.982088i	$0.560338\pi$
$618$		0			0
$619$	−2.89736e6			−0.303931			−0.151966	−	0.988386i	$-0.548560\pi$
$619$	−2.89736e6			−0.303931			−0.151966	+	0.988386i	$0.548560\pi$
$620$		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	$-0.252723\pi$
$643$	−126819.	−	126819.i	−0.0120964	−	0.0120964i	−0.713129	+	0.701033i	$0.752723\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.07986e7	−	1.07986e7i	1.01416	−	1.01416i	0.0142641	−	0.999898i	$-0.495459\pi$
$647$	1.07986e7	−	1.07986e7i	1.01416	−	1.01416i	0.999898	−	0.0142641i	$-0.00454055\pi$
$648$		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	−0.403390	−	0.915028i	$-0.632168\pi$
$653$	5.57502e6	−	5.57502e6i	0.511639	−	0.511639i	0.915028	+	0.403390i	$0.132168\pi$
$654$		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	$-0.638150\pi$
$659$	−9.37611e6			−0.841026			−0.420513	+	0.907287i	$0.638150\pi$
$660$		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	0.323834	−	0.946114i	$-0.395028\pi$
$673$	1.49219e7	−	1.49219e7i	1.26995	−	1.26995i	0.946114	−	0.323834i	$-0.104972\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.54101e7	−	1.54101e7i	−1.29221	−	1.29221i	−0.933418	−	0.358791i	$-0.883189\pi$
$677$	−1.54101e7	−	1.54101e7i	−1.29221	−	1.29221i	−0.358791	−	0.933418i	$-0.616811\pi$
$678$		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	$-0.0600584\pi$
$683$	9.68836e6	+	9.68836e6i	0.794691	+	0.794691i	−0.187562	+	0.982253i	$0.560058\pi$
$684$		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	$-0.877746\pi$
$709$		−	1.00306e7i		−	0.749395i	0.927147	−	0.374698i	$-0.122254\pi$
$710$		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	$-0.444186\pi$
$719$	4.83635e6			0.348895			0.174448	+	0.984666i	$0.444186\pi$
$720$		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	$-0.786516\pi$
$727$	−2.30692e6	+	2.30692e6i	−0.161881	+	0.161881i	0.621519	+	0.783399i	$0.286516\pi$
$728$		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	−0.881261	−	0.472629i	$-0.843305\pi$
$733$	−5.94419e6	+	5.94419e6i	−0.408632	+	0.408632i	0.472629	+	0.881261i	$0.343305\pi$
$734$		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	$-0.512781\pi$
$739$	−1.19188e6			−0.0802829			−0.0401415	+	0.999194i	$0.512781\pi$
$740$		0			0
$741$	1.76927e6			0.118372
$742$		0			0
$743$	1.87097e7	+	1.87097e7i	1.24336	+	1.24336i	0.958600	+	0.284758i	$0.0919132\pi$
$743$	1.87097e7	+	1.87097e7i	1.24336	+	1.24336i	0.284758	+	0.958600i	$0.408087\pi$
$744$		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	0.911447	+	0.411417i	$0.134966\pi$
$757$	2.08571e7	+	2.08571e7i	1.32286	+	1.32286i	0.411417	+	0.911447i	$0.365034\pi$
$758$		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	$0.163611\pi$
$769$			1.61255e7i			0.983325i	−0.870786	+	0.491663i	$0.836389\pi$
$770$		0			0
$771$		−	1.60616e7i		−	0.973092i
$772$		0			0
$773$	1.32344e7	−	1.32344e7i	0.796630	−	0.796630i	−0.185933	−	0.982563i	$-0.559531\pi$
$773$	1.32344e7	−	1.32344e7i	0.796630	−	0.796630i	0.982563	+	0.185933i	$0.0595306\pi$
$774$		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	$-0.978741\pi$
$787$	−1.61771e7	+	1.61771e7i	−0.931032	+	0.931032i	0.0667386	+	0.997770i	$0.478741\pi$
$788$		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	0.785609	−	0.618723i	$-0.212350\pi$
$797$	2.99273e6	+	2.99273e6i	0.166887	+	0.166887i	−0.618723	+	0.785609i	$0.712350\pi$
$798$		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	$-0.604585\pi$
$809$		−	3.52391e7i		−	1.89301i	0.322683	−	0.946507i	$-0.395415\pi$
$810$		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	$-0.387580\pi$
$823$	−1.15110e7	−	1.15110e7i	−0.592399	−	0.592399i	−0.938279	+	0.345880i	$0.887580\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.12263e7	+	1.12263e7i	−0.570787	+	0.570787i	−0.932348	−	0.361561i	$-0.882244\pi$
$827$	−1.12263e7	+	1.12263e7i	−0.570787	+	0.570787i	0.361561	+	0.932348i	$0.382244\pi$
$828$		0			0
$829$		−	8.81335e6i		−	0.445405i	−0.974887	−	0.222702i	$-0.928512\pi$
$829$		−	8.81335e6i		−	0.445405i	0.974887	−	0.222702i	$-0.0714878\pi$
$830$		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	$-0.306910\pi$
$839$	2.32474e7			1.14017			0.570086	+	0.821585i	$0.306910\pi$
$840$		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	−0.997177	−	0.0750917i	$-0.976075\pi$
$853$	−1.95949e7	+	1.95949e7i	−0.922085	+	0.922085i	0.0750917	+	0.997177i	$0.476075\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.52052e7	+	1.52052e7i	0.707197	+	0.707197i	0.965945	−	0.258748i	$-0.0833099\pi$
$857$	1.52052e7	+	1.52052e7i	0.707197	+	0.707197i	−0.258748	+	0.965945i	$0.583310\pi$
$858$		0			0
$859$	1.55233e7			0.717795			0.358897	−	0.933377i	$-0.383153\pi$
$859$	1.55233e7			0.717795			0.358897	+	0.933377i	$0.383153\pi$
$860$		0			0
$861$	3.13851e6			0.144283
$862$		0			0
$863$	1.63784e7	+	1.63784e7i	0.748590	+	0.748590i	0.974214	−	0.225624i	$-0.0724421\pi$
$863$	1.63784e7	+	1.63784e7i	0.748590	+	0.748590i	−0.225624	+	0.974214i	$0.572442\pi$
$864$		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	0.803886	−	0.594783i	$-0.202762\pi$
$877$	4.76278e6	+	4.76278e6i	0.209103	+	0.209103i	−0.594783	+	0.803886i	$0.702762\pi$
$878$		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	$-0.111955\pi$
$883$	1.37685e7	+	1.37685e7i	0.594272	+	0.594272i	−0.344511	+	0.938782i	$0.611955\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.78096e6	−	1.78096e6i	0.0760056	−	0.0760056i	−0.668082	−	0.744088i	$-0.732884\pi$
$887$	1.78096e6	−	1.78096e6i	0.0760056	−	0.0760056i	0.744088	+	0.668082i	$0.232884\pi$
$888$		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	$-0.608009\pi$
$907$	1.51162e7	−	1.51162e7i	0.610134	−	0.610134i	0.942981	+	0.332847i	$0.108009\pi$
$908$		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	$-0.654975\pi$
$919$	−2.39571e7			−0.935720			−0.467860	+	0.883803i	$0.654975\pi$
$920$		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	$0.164379\pi$
$929$			2.59769e7i			0.987526i	−0.869597	+	0.493763i	$0.835621\pi$
$930$		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	0.00797527	−	0.999968i	$-0.497461\pi$
$937$	−2.66598e7	−	2.66598e7i	−0.991993	−	0.991993i	−0.999968	+	0.00797527i	$0.997461\pi$
$938$		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	0.414470	−	0.910063i	$-0.363967\pi$
$947$	3.65542e7	−	3.65542e7i	1.32453	−	1.32453i	0.910063	−	0.414470i	$-0.136033\pi$
$948$		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	0.292008	−	0.956416i	$-0.405677\pi$
$953$	3.50021e7	−	3.50021e7i	1.24842	−	1.24842i	0.956416	−	0.292008i	$-0.0943234\pi$
$954$		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	$-0.698280\pi$
$967$	6.65238e6	−	6.65238e6i	0.228776	−	0.228776i	0.812181	+	0.583405i	$0.198280\pi$
$968$		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	−0.982888	−	0.184204i	$-0.941029\pi$
$977$	−3.48210e7	−	3.48210e7i	−1.16709	−	1.16709i	−0.184204	−	0.982888i	$-0.558971\pi$
$978$		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	$-0.0723140\pi$
$983$	2.26938e7	+	2.26938e7i	0.749073	+	0.749073i	−0.225232	+	0.974305i	$0.572314\pi$
$984$		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	0.979484	+	0.201521i	$0.0645885\pi$
$997$	3.70672e7	+	3.70672e7i	1.18101	+	1.18101i	0.201521	+	0.979484i	$0.435411\pi$
$998$		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.207.4		20
4.3	odd	2	inner	400.6.n.g.207.7		20
5.2	odd	4		80.6.n.d.63.4	yes	20
5.3	odd	4	inner	400.6.n.g.143.7		20
5.4	even	2		80.6.n.d.47.7	yes	20
20.3	even	4	inner	400.6.n.g.143.4		20
20.7	even	4		80.6.n.d.63.7	yes	20
20.19	odd	2		80.6.n.d.47.4	✓	20

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