LMFDB - Embedded newform 216.7.m.a.17.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,7,Mod(17,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("216.17");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	216.m (of order $6$, degree $2$, not 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:	$49.6916820619$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$18$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.5
Character	$\chi$	$=$	216.17
Dual form			216.7.m.a.89.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-86.3751 - 49.8687i) q^{5} +(-207.661 - 359.679i) q^{7} +O(q^{10})$	$q+(-86.3751 - 49.8687i) q^{5} +(-207.661 - 359.679i) q^{7} +(-1641.05 + 947.463i) q^{11} +(-335.248 + 580.666i) q^{13} +594.749i q^{17} -9514.73 q^{19} +(13766.9 + 7948.35i) q^{23} +(-2838.72 - 4916.81i) q^{25} +(14337.7 - 8277.87i) q^{29} +(22971.5 - 39787.8i) q^{31} +41423.1i q^{35} -20749.4 q^{37} +(45300.1 + 26154.0i) q^{41} +(51531.5 + 89255.1i) q^{43} +(42344.5 - 24447.6i) q^{47} +(-27421.6 + 47495.5i) q^{49} +152594. i q^{53} +188995. q^{55} +(-297704. - 171879. i) q^{59} +(33078.1 + 57293.0i) q^{61} +(57914.1 - 33436.7i) q^{65} +(173996. - 301369. i) q^{67} +564987. i q^{71} +675795. q^{73} +(681565. + 393502. i) q^{77} +(239627. + 415047. i) q^{79} +(-520059. + 300256. i) q^{83} +(29659.4 - 51371.6i) q^{85} -333280. i q^{89} +278471. q^{91} +(821836. + 474487. i) q^{95} +(272344. + 471714. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q+O(q^{10}) $ 36 * q	$ 36 q - 1350 q^{11} + 9540 q^{19} - 30888 q^{23} + 56250 q^{25} - 38556 q^{29} + 27720 q^{31} - 179226 q^{41} + 15930 q^{43} - 187596 q^{47} - 198774 q^{49} - 197064 q^{55} + 408618 q^{59} + 17136 q^{61} + 125712 q^{65} + 27090 q^{67} - 534060 q^{73} - 48168 q^{77} + 172620 q^{79} - 1801980 q^{83} - 791568 q^{85} + 538560 q^{91} - 1832652 q^{95} + 770706 q^{97}+O(q^{100}) $ 36 * q - 1350 * q^11 + 9540 * q^19 - 30888 * q^23 + 56250 * q^25 - 38556 * q^29 + 27720 * q^31 - 179226 * q^41 + 15930 * q^43 - 187596 * q^47 - 198774 * q^49 - 197064 * q^55 + 408618 * q^59 + 17136 * q^61 + 125712 * q^65 + 27090 * q^67 - 534060 * q^73 - 48168 * q^77 + 172620 * q^79 - 1801980 * q^83 - 791568 * q^85 + 538560 * q^91 - 1832652 * q^95 + 770706 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−86.3751	−	49.8687i	−0.691001	−	0.398950i	0.112986	−	0.993597i	$-0.463959\pi$
$5$	−86.3751	−	49.8687i	−0.691001	−	0.398950i	−0.803987	+	0.594647i	$0.797292\pi$
$6$		0			0
$7$	−207.661	−	359.679i	−0.605425	−	1.04863i	−0.991984	−	0.126363i	$-0.959670\pi$
$7$	−207.661	−	359.679i	−0.605425	−	1.04863i	0.386559	−	0.922265i	$-0.373664\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1641.05	+	947.463i	−1.23295	+	0.711843i	−0.967643	−	0.252321i	$-0.918806\pi$
$11$	−1641.05	+	947.463i	−1.23295	+	0.711843i	−0.265305	+	0.964165i	$0.585473\pi$
$12$		0			0
$13$	−335.248	+	580.666i	−0.152593	+	0.264300i	−0.932180	−	0.361995i	$-0.882096\pi$
$13$	−335.248	+	580.666i	−0.152593	+	0.264300i	0.779587	+	0.626294i	$0.215429\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			594.749i			0.121056i	0.998166	+	0.0605281i	$0.0192785\pi$
$17$			594.749i			0.121056i	−0.998166	+	0.0605281i	$0.980722\pi$
$18$		0			0
$19$	−9514.73			−1.38719			−0.693594	−	0.720366i	$-0.743974\pi$
$19$	−9514.73			−1.38719			−0.693594	+	0.720366i	$0.743974\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	13766.9	+	7948.35i	1.13150	+	0.653271i	0.944311	−	0.329053i	$-0.106729\pi$
$23$	13766.9	+	7948.35i	1.13150	+	0.653271i	0.187187	+	0.982324i	$0.440063\pi$
$24$		0			0
$25$	−2838.72	−	4916.81i	−0.181678	−	0.314676i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	14337.7	−	8277.87i	0.587876	−	0.339410i	−0.176381	−	0.984322i	$-0.556439\pi$
$29$	14337.7	−	8277.87i	0.587876	−	0.339410i	0.764257	+	0.644912i	$0.223106\pi$
$30$		0			0
$31$	22971.5	−	39787.8i	0.771088	−	1.33556i	−0.165879	−	0.986146i	$-0.553046\pi$
$31$	22971.5	−	39787.8i	0.771088	−	1.33556i	0.936967	−	0.349417i	$-0.113620\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			41423.1i			0.966137i
$36$		0			0
$37$	−20749.4			−0.409637			−0.204819	−	0.978800i	$-0.565660\pi$
$37$	−20749.4			−0.409637			−0.204819	+	0.978800i	$0.565660\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	45300.1	+	26154.0i	0.657275	+	0.379478i	0.791238	−	0.611508i	$-0.209437\pi$
$41$	45300.1	+	26154.0i	0.657275	+	0.379478i	−0.133963	+	0.990986i	$0.542770\pi$
$42$		0			0
$43$	51531.5	+	89255.1i	0.648138	+	1.12261i	0.983567	+	0.180542i	$0.0577851\pi$
$43$	51531.5	+	89255.1i	0.648138	+	1.12261i	−0.335430	+	0.942065i	$0.608882\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	42344.5	−	24447.6i	0.407853	−	0.235474i	−0.282014	−	0.959410i	$-0.591002\pi$
$47$	42344.5	−	24447.6i	0.407853	−	0.235474i	0.689867	+	0.723936i	$0.257669\pi$
$48$		0			0
$49$	−27421.6	+	47495.5i	−0.233079	+	0.403705i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			152594.i			1.02496i	0.858698	+	0.512482i	$0.171274\pi$
$53$			152594.i			1.02496i	−0.858698	+	0.512482i	$0.828726\pi$
$54$		0			0
$55$	188995.			1.13596
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−297704.	−	171879.i	−1.44953	−	0.836888i	−0.451079	−	0.892484i	$-0.648961\pi$
$59$	−297704.	−	171879.i	−1.44953	−	0.836888i	−0.998453	+	0.0555960i	$0.982294\pi$
$60$		0			0
$61$	33078.1	+	57293.0i	0.145731	+	0.252413i	0.929645	−	0.368456i	$-0.120113\pi$
$61$	33078.1	+	57293.0i	0.145731	+	0.252413i	−0.783915	+	0.620869i	$0.786780\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	57914.1	−	33436.7i	0.210884	−	0.121754i
$66$		0			0
$67$	173996.	−	301369.i	0.578514	−	1.00202i	−0.417136	−	0.908844i	$-0.636966\pi$
$67$	173996.	−	301369.i	0.578514	−	1.00202i	0.995650	−	0.0931713i	$-0.0297004\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			564987.i			1.57857i	0.614029	+	0.789284i	$0.289548\pi$
$71$			564987.i			1.57857i	−0.614029	+	0.789284i	$0.710452\pi$
$72$		0			0
$73$	675795.			1.73719			0.868594	−	0.495525i	$-0.165024\pi$
$73$	675795.			1.73719			0.868594	+	0.495525i	$0.165024\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	681565.	+	393502.i	1.49292	+	0.861936i
$78$		0			0
$79$	239627.	+	415047.i	0.486021	+	0.841814i	0.999871	−	0.0160667i	$-0.00511442\pi$
$79$	239627.	+	415047.i	0.486021	+	0.841814i	−0.513850	+	0.857880i	$0.671781\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−520059.	+	300256.i	−0.909534	+	0.525119i	−0.880281	−	0.474453i	$-0.842646\pi$
$83$	−520059.	+	300256.i	−0.909534	+	0.525119i	−0.0292525	+	0.999572i	$0.509313\pi$
$84$		0			0
$85$	29659.4	−	51371.6i	0.0482954	−	0.0836500i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	333280.i		−	0.472759i	−0.971661	−	0.236379i	$-0.924039\pi$
$89$		−	333280.i		−	0.472759i	0.971661	−	0.236379i	$-0.0759608\pi$
$90$		0			0
$91$	278471.			0.369536
$92$		0			0
$93$		0			0
$94$		0			0
$95$	821836.	+	474487.i	0.958549	+	0.553418i
$96$		0			0
$97$	272344.	+	471714.i	0.298403	+	0.516848i	0.975771	−	0.218796i	$-0.0702129\pi$
$97$	272344.	+	471714.i	0.298403	+	0.516848i	−0.677368	+	0.735644i	$0.736880\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	188900.	−	109062.i	0.183345	−	0.105854i	−0.405519	−	0.914087i	$-0.632909\pi$
$101$	188900.	−	109062.i	0.183345	−	0.105854i	0.588863	+	0.808233i	$0.299576\pi$
$102$		0			0
$103$	−275093.	+	476475.i	−0.251749	+	0.436042i	−0.964007	−	0.265875i	$-0.914339\pi$
$103$	−275093.	+	476475.i	−0.251749	+	0.436042i	0.712259	+	0.701917i	$0.247672\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1.00684e6i			0.821879i	0.911663	+	0.410939i	$0.134799\pi$
$107$			1.00684e6i			0.821879i	−0.911663	+	0.410939i	$0.865201\pi$
$108$		0			0
$109$	1.53682e6			1.18670			0.593352	−	0.804943i	$-0.297804\pi$
$109$	1.53682e6			1.18670			0.593352	+	0.804943i	$0.297804\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.47341e6	+	850672.i	1.02114	+	0.589558i	0.914435	−	0.404732i	$-0.132635\pi$
$113$	1.47341e6	+	850672.i	1.02114	+	0.589558i	0.106709	+	0.994290i	$0.465969\pi$
$114$		0			0
$115$	−792748.	−	1.37308e6i	−0.521245	−	0.902822i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	213919.	−	123506.i	0.126943	−	0.0732905i
$120$		0			0
$121$	909592.	−	1.57546e6i	0.513441	−	0.889306i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			2.12465e6i			1.08782i
$126$		0			0
$127$	−3.04921e6			−1.48859			−0.744296	−	0.667850i	$-0.767215\pi$
$127$	−3.04921e6			−1.48859			−0.744296	+	0.667850i	$0.767215\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.02242e6	+	590292.i	0.454793	+	0.262575i	0.709852	−	0.704351i	$-0.248762\pi$
$131$	1.02242e6	+	590292.i	0.454793	+	0.262575i	−0.255059	+	0.966925i	$0.582095\pi$
$132$		0			0
$133$	1.97584e6	+	3.42225e6i	0.839839	+	1.45464i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	252872.	−	145995.i	0.0983418	−	0.0567777i	−0.450022	−	0.893017i	$-0.648584\pi$
$137$	252872.	−	145995.i	0.0983418	−	0.0567777i	0.548364	+	0.836240i	$0.315251\pi$
$138$		0			0
$139$	2.48488e6	−	4.30393e6i	0.925253	−	1.60259i	0.134099	−	0.990968i	$-0.457186\pi$
$139$	2.48488e6	−	4.30393e6i	0.925253	−	1.60259i	0.791154	−	0.611617i	$-0.209481\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	1.27054e6i		−	0.434490i
$144$		0			0
$145$	−1.65123e6			−0.541630
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.22477e6	−	2.43917e6i	−1.27716	−	0.737367i	−0.300833	−	0.953677i	$-0.597265\pi$
$149$	−4.22477e6	−	2.43917e6i	−1.27716	−	0.737367i	−0.976325	+	0.216310i	$0.930598\pi$
$150$		0			0
$151$	626266.	+	1.08472e6i	0.181898	+	0.315056i	0.942527	−	0.334131i	$-0.108443\pi$
$151$	626266.	+	1.08472e6i	0.181898	+	0.315056i	−0.760629	+	0.649187i	$0.775109\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.96833e6	+	2.29112e6i	−1.06565	+	0.615251i
$156$		0			0
$157$	−1.63418e6	+	2.83048e6i	−0.422280	+	0.731410i	−0.996162	−	0.0875279i	$-0.972103\pi$
$157$	−1.63418e6	+	2.83048e6i	−0.422280	+	0.731410i	0.573882	+	0.818938i	$0.305437\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	6.60225e6i		−	1.58203i
$162$		0			0
$163$	−5.96607e6			−1.37761			−0.688803	−	0.724948i	$-0.741864\pi$
$163$	−5.96607e6			−1.37761			−0.688803	+	0.724948i	$0.741864\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.47983e6	−	2.58643e6i	−0.961860	−	0.555330i	−0.0651149	−	0.997878i	$-0.520741\pi$
$167$	−4.47983e6	−	2.58643e6i	−0.961860	−	0.555330i	−0.896745	+	0.442548i	$0.854075\pi$
$168$		0			0
$169$	2.18862e6	+	3.79081e6i	0.453430	+	0.785365i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.54989e6	−	3.20423e6i	1.07188	−	0.618850i	0.143185	−	0.989696i	$-0.454266\pi$
$173$	5.54989e6	−	3.20423e6i	1.07188	−	0.618850i	0.928694	+	0.370846i	$0.120932\pi$
$174$		0			0
$175$	−1.17898e6	+	2.04206e6i	−0.219985	+	0.381026i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1.83866e6i		−	0.320585i	−0.987070	−	0.160292i	$-0.948756\pi$
$179$		−	1.83866e6i		−	0.320585i	0.987070	−	0.160292i	$-0.0512437\pi$
$180$		0			0
$181$	5.78967e6			0.976378			0.488189	−	0.872738i	$-0.337658\pi$
$181$	5.78967e6			0.976378			0.488189	+	0.872738i	$0.337658\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.79223e6	+	1.03474e6i	0.283060	+	0.163425i
$186$		0			0
$187$	−563503.	−	976016.i	−0.0861731	−	0.149256i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.06864e6	+	2.92638e6i	−0.727430	+	0.419982i	−0.817481	−	0.575955i	$-0.804630\pi$
$191$	−5.06864e6	+	2.92638e6i	−0.727430	+	0.419982i	0.0900514	+	0.995937i	$0.471297\pi$
$192$		0			0
$193$	3.91069e6	−	6.77351e6i	0.543977	−	0.942197i	−0.454693	−	0.890648i	$-0.650251\pi$
$193$	3.91069e6	−	6.77351e6i	0.543977	−	0.942197i	0.998670	−	0.0515484i	$-0.0164156\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1.13162e6i			0.148014i	0.997258	+	0.0740069i	$0.0235787\pi$
$197$			1.13162e6i			0.148014i	−0.997258	+	0.0740069i	$0.976421\pi$
$198$		0			0
$199$	−1.44240e7			−1.83032			−0.915160	−	0.403090i	$-0.867936\pi$
$199$	−1.44240e7			−1.83032			−0.915160	+	0.403090i	$0.867936\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.95476e6	−	3.43798e6i	−0.711829	−	0.410975i
$204$		0			0
$205$	−2.60853e6	−	4.51811e6i	−0.302785	−	0.524439i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.56142e7	−	9.01485e6i	1.71033	−	0.987461i
$210$		0			0
$211$	142920.	−	247545.i	0.0152141	−	0.0263516i	−0.858318	−	0.513118i	$-0.828490\pi$
$211$	142920.	−	247545.i	0.0152141	−	0.0263516i	0.873532	+	0.486766i	$0.161824\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.02792e7i		−	1.03430i
$216$		0			0
$217$	−1.90811e7			−1.86734
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−345351.	−	199388.i	−0.0319951	−	0.0184724i
$222$		0			0
$223$	−6.86168e6	−	1.18848e7i	−0.618751	−	1.07171i	−0.989714	−	0.143060i	$-0.954306\pi$
$223$	−6.86168e6	−	1.18848e7i	−0.618751	−	1.07171i	0.370963	−	0.928648i	$-0.379028\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.39089e7	+	8.03031e6i	−1.18909	+	0.686523i	−0.958100	−	0.286433i	$-0.907530\pi$
$227$	−1.39089e7	+	8.03031e6i	−1.18909	+	0.686523i	−0.230992	+	0.972956i	$0.574197\pi$
$228$		0			0
$229$	−2.67395e6	+	4.63142e6i	−0.222662	+	0.385663i	−0.955616	−	0.294616i	$-0.904808\pi$
$229$	−2.67395e6	+	4.63142e6i	−0.222662	+	0.385663i	0.732953	+	0.680279i	$0.238141\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1.52125e6i			0.120263i	0.998190	+	0.0601315i	$0.0191520\pi$
$233$			1.52125e6i			0.120263i	−0.998190	+	0.0601315i	$0.980848\pi$
$234$		0			0
$235$	−4.87668e6			−0.375769
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.31852e6	−	4.22535e6i	−0.536080	−	0.309506i	0.207409	−	0.978254i	$-0.433497\pi$
$239$	−7.31852e6	−	4.22535e6i	−0.536080	−	0.309506i	−0.743489	+	0.668749i	$0.766830\pi$
$240$		0			0
$241$	8.23037e6	+	1.42554e7i	0.587988	+	1.01842i	0.994496	+	0.104778i	$0.0334131\pi$
$241$	8.23037e6	+	1.42554e7i	0.587988	+	1.01842i	−0.406508	+	0.913647i	$0.633254\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.73708e6	−	2.73496e6i	0.322116	−	0.185974i
$246$		0			0
$247$	3.18979e6	−	5.52488e6i	0.211676	−	0.366633i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			2.09175e7i			1.32278i	0.750040	+	0.661392i	$0.230034\pi$
$251$			2.09175e7i			1.32278i	−0.750040	+	0.661392i	$0.769966\pi$
$252$		0			0
$253$	−3.01231e7			−1.86011
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.28454e6	−	1.89633e6i	−0.193497	−	0.111716i	0.400122	−	0.916462i	$-0.368968\pi$
$257$	−3.28454e6	−	1.89633e6i	−0.193497	−	0.111716i	−0.593619	+	0.804746i	$0.702301\pi$
$258$		0			0
$259$	4.30883e6	+	7.46311e6i	0.248005	+	0.429557i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.09725e7	−	1.21085e7i	1.15288	−	0.665614i	0.203290	−	0.979118i	$-0.434836\pi$
$263$	2.09725e7	−	1.21085e7i	1.15288	−	0.665614i	0.949587	+	0.313505i	$0.101503\pi$
$264$		0			0
$265$	7.60964e6	−	1.31803e7i	0.408909	−	0.708251i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	6.42268e6i		−	0.329959i	−0.986297	−	0.164979i	$-0.947244\pi$
$269$		−	6.42268e6i		−	0.329959i	0.986297	−	0.164979i	$-0.0527557\pi$
$270$		0			0
$271$	1.79911e7			0.903960			0.451980	−	0.892028i	$-0.350718\pi$
$271$	1.79911e7			0.903960			0.451980	+	0.892028i	$0.350718\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.31700e6	+	5.37917e6i	0.448000	+	0.258653i
$276$		0			0
$277$	1.40065e7	+	2.42599e7i	0.659006	+	1.14143i	0.980873	+	0.194646i	$0.0623560\pi$
$277$	1.40065e7	+	2.42599e7i	0.659006	+	1.14143i	−0.321868	+	0.946785i	$0.604311\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.79637e7	+	1.61449e7i	−1.26031	+	0.727638i	−0.973134	−	0.230241i	$-0.926048\pi$
$281$	−2.79637e7	+	1.61449e7i	−1.26031	+	0.727638i	−0.287172	+	0.957879i	$0.592715\pi$
$282$		0			0
$283$	1.66288e7	−	2.88019e7i	0.733670	−	1.27075i	−0.221635	−	0.975130i	$-0.571139\pi$
$283$	1.66288e7	−	2.88019e7i	0.733670	−	1.27075i	0.955305	−	0.295623i	$-0.0955273\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	2.17246e7i		−	0.918982i
$288$		0			0
$289$	2.37838e7			0.985345
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.77415e7	+	1.02431e7i	0.705323	+	0.407218i	0.809327	−	0.587359i	$-0.199832\pi$
$293$	1.77415e7	+	1.02431e7i	0.705323	+	0.407218i	−0.104004	+	0.994577i	$0.533165\pi$
$294$		0			0
$295$	1.71428e7	+	2.96922e7i	0.667752	+	1.15658i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.23068e6	+	5.32933e6i	−0.345319	+	0.199370i
$300$		0			0
$301$	2.14021e7	−	3.70696e7i	0.784798	−	1.35931i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	6.59825e6i		−	0.232557i
$306$		0			0
$307$	−1.62714e7			−0.562353			−0.281176	−	0.959656i	$-0.590725\pi$
$307$	−1.62714e7			−0.562353			−0.281176	+	0.959656i	$0.590725\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.16807e7	+	1.82909e7i	1.05321	+	0.608069i	0.923545	−	0.383489i	$-0.125278\pi$
$311$	3.16807e7	+	1.82909e7i	1.05321	+	0.608069i	0.129661	+	0.991558i	$0.458611\pi$
$312$		0			0
$313$	2.24297e7	+	3.88493e7i	0.731459	+	1.26692i	0.956260	+	0.292519i	$0.0944935\pi$
$313$	2.24297e7	+	3.88493e7i	0.731459	+	1.26692i	−0.224801	+	0.974405i	$0.572173\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.39103e6	+	2.53516e6i	−0.137844	+	0.0795844i	−0.567336	−	0.823486i	$-0.692026\pi$
$317$	−4.39103e6	+	2.53516e6i	−0.137844	+	0.0795844i	0.429492	+	0.903071i	$0.358693\pi$
$318$		0			0
$319$	−1.56860e7	+	2.71689e7i	−0.483213	+	0.836950i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5.65888e6i		−	0.167928i
$324$		0			0
$325$	3.80670e6			0.110892
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.75866e7	−	1.01536e7i	−0.493848	−	0.285124i
$330$		0			0
$331$	1.82785e7	+	3.16593e7i	0.504031	+	0.873006i	0.999989	+	0.00466031i	$0.00148343\pi$
$331$	1.82785e7	+	3.16593e7i	0.504031	+	0.873006i	−0.495959	+	0.868346i	$0.665183\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.00578e7	+	1.73539e7i	−0.799507	+	0.461596i
$336$		0			0
$337$	1.06337e7	−	1.84181e7i	0.277839	−	0.481232i	−0.693008	−	0.720930i	$-0.743715\pi$
$337$	1.06337e7	−	1.84181e7i	0.277839	−	0.481232i	0.970848	+	0.239698i	$0.0770484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			8.70585e7i			2.19557i
$342$		0			0
$343$	−2.60846e7			−0.646402
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.12339e7	+	2.38064e7i	0.986883	+	0.569777i	0.904341	−	0.426811i	$-0.140363\pi$
$347$	4.12339e7	+	2.38064e7i	0.986883	+	0.569777i	0.0825418	+	0.996588i	$0.473696\pi$
$348$		0			0
$349$	−2.81464e7	−	4.87510e7i	−0.662135	−	1.14685i	−0.980054	−	0.198733i	$-0.936317\pi$
$349$	−2.81464e7	−	4.87510e7i	−0.662135	−	1.14685i	0.317919	−	0.948118i	$-0.397016\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.59643e7	+	9.21697e6i	−0.362932	+	0.209539i	−0.670366	−	0.742031i	$-0.733863\pi$
$353$	−1.59643e7	+	9.21697e6i	−0.362932	+	0.209539i	0.307434	+	0.951569i	$0.400530\pi$
$354$		0			0
$355$	2.81752e7	−	4.88008e7i	0.629769	−	1.09079i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.17453e7i			0.902245i	0.892462	+	0.451122i	$0.148976\pi$
$359$			4.17453e7i			0.902245i	−0.892462	+	0.451122i	$0.851024\pi$
$360$		0			0
$361$	4.34841e7			0.924292
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.83719e7	−	3.37010e7i	−1.20040	−	0.693050i
$366$		0			0
$367$	1.25909e7	+	2.18082e7i	0.254718	+	0.441185i	0.964819	−	0.262915i	$-0.0846839\pi$
$367$	1.25909e7	+	2.18082e7i	0.254718	+	0.441185i	−0.710101	+	0.704100i	$0.751351\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.48847e7	−	3.16877e7i	1.07481	−	0.620539i
$372$		0			0
$373$	−1.29804e7	+	2.24827e7i	−0.250128	+	0.433234i	−0.963561	−	0.267489i	$-0.913806\pi$
$373$	−1.29804e7	+	2.24827e7i	−0.250128	+	0.433234i	0.713433	+	0.700723i	$0.247139\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1.11006e7i			0.207167i
$378$		0			0
$379$	5.80301e7			1.06595			0.532974	−	0.846132i	$-0.321074\pi$
$379$	5.80301e7			1.06595			0.532974	+	0.846132i	$0.321074\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.29275e7	+	4.21047e7i	1.29806	+	0.749435i	0.980069	−	0.198658i	$-0.0636584\pi$
$383$	7.29275e7	+	4.21047e7i	1.29806	+	0.749435i	0.317991	+	0.948094i	$0.396992\pi$
$384$		0			0
$385$	−3.92469e7	−	6.79776e7i	−0.687738	−	1.19120i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.25057e7	−	4.18612e7i	1.23175	−	0.711153i	0.264358	−	0.964425i	$-0.414840\pi$
$389$	7.25057e7	−	4.18612e7i	1.23175	−	0.711153i	0.967395	+	0.253272i	$0.0815067\pi$
$390$		0			0
$391$	−4.72728e6	+	8.18788e6i	−0.0790826	+	0.136975i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	4.77996e7i		−	0.775592i
$396$		0			0
$397$	−8.51334e7			−1.36059			−0.680297	−	0.732937i	$-0.738149\pi$
$397$	−8.51334e7			−1.36059			−0.680297	+	0.732937i	$0.738149\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.13758e7	+	2.96618e7i	0.796756	+	0.460007i	0.842336	−	0.538953i	$-0.181180\pi$
$401$	5.13758e7	+	2.96618e7i	0.796756	+	0.460007i	−0.0455795	+	0.998961i	$0.514513\pi$
$402$		0			0
$403$	1.54023e7	+	2.66775e7i	0.235326	+	0.407596i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.40508e7	−	1.96593e7i	0.505062	−	0.291597i
$408$		0			0
$409$	−2.05055e7	+	3.55165e7i	−0.299709	+	0.519111i	−0.976069	−	0.217460i	$-0.930223\pi$
$409$	−2.05055e7	+	3.55165e7i	−0.299709	+	0.519111i	0.676360	+	0.736571i	$0.263556\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1.42770e8i			2.02669i
$414$		0			0
$415$	5.98936e7			0.837985
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.77563e7	−	2.17986e7i	−0.513272	−	0.296338i	0.220906	−	0.975295i	$-0.429099\pi$
$419$	−3.77563e7	−	2.17986e7i	−0.513272	−	0.296338i	−0.734178	+	0.678958i	$0.762432\pi$
$420$		0			0
$421$	2.33594e7	+	4.04597e7i	0.313051	+	0.542221i	0.979021	−	0.203758i	$-0.0653155\pi$
$421$	2.33594e7	+	4.04597e7i	0.313051	+	0.542221i	−0.665970	+	0.745978i	$0.731982\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.92427e6	−	1.68833e6i	0.0380935	−	0.0219933i
$426$		0			0
$427$	1.37381e7	−	2.37950e7i	0.176458	−	0.305635i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1.19205e8i		−	1.48889i	−0.667685	−	0.744443i	$-0.732715\pi$
$431$		−	1.19205e8i		−	1.48889i	0.667685	−	0.744443i	$-0.267285\pi$
$432$		0			0
$433$	1.04949e6			0.0129275			0.00646377	−	0.999979i	$-0.497943\pi$
$433$	1.04949e6			0.0129275			0.00646377	+	0.999979i	$0.497943\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.30989e8	−	7.56264e7i	−1.56960	−	0.906210i
$438$		0			0
$439$	2.53760e7	+	4.39526e7i	0.299937	+	0.519506i	0.976121	−	0.217226i	$-0.0697010\pi$
$439$	2.53760e7	+	4.39526e7i	0.299937	+	0.519506i	−0.676184	+	0.736733i	$0.736368\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.07260e7	+	4.66072e7i	−0.928544	+	0.536095i	−0.886351	−	0.463014i	$-0.846768\pi$
$443$	−8.07260e7	+	4.66072e7i	−0.928544	+	0.536095i	−0.0421931	+	0.999109i	$0.513434\pi$
$444$		0			0
$445$	−1.66203e7	+	2.87871e7i	−0.188607	+	0.326677i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5.96004e7i		−	0.658431i	−0.944255	−	0.329216i	$-0.893216\pi$
$449$		−	5.96004e7i		−	0.658431i	0.944255	−	0.329216i	$-0.106784\pi$
$450$		0			0
$451$	−9.91198e7			−1.08051
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.40530e7	−	1.38870e7i	−0.255350	−	0.147426i
$456$		0			0
$457$	−8.79814e7	−	1.52388e8i	−0.921812	−	1.59663i	−0.796610	−	0.604494i	$-0.793375\pi$
$457$	−8.79814e7	−	1.52388e8i	−0.921812	−	1.59663i	−0.125203	−	0.992131i	$-0.539958\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.06549e8	+	6.15164e7i	−1.08755	+	0.627896i	−0.932923	−	0.360077i	$-0.882750\pi$
$461$	−1.06549e8	+	6.15164e7i	−1.08755	+	0.627896i	−0.154626	+	0.987973i	$0.549417\pi$
$462$		0			0
$463$	8.76488e6	−	1.51812e7i	0.0883086	−	0.152955i	−0.818488	−	0.574524i	$-0.805187\pi$
$463$	8.76488e6	−	1.51812e7i	0.0883086	−	0.152955i	0.906796	+	0.421569i	$0.138521\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.37634e8i		−	1.35137i	−0.737189	−	0.675687i	$-0.763847\pi$
$467$		−	1.37634e8i		−	1.35137i	0.737189	−	0.675687i	$-0.236153\pi$
$468$		0			0
$469$	−1.44528e8			−1.40099
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.69132e8	−	9.76483e7i	−1.59824	−	0.922744i
$474$		0			0
$475$	2.70097e7	+	4.67821e7i	0.252022	+	0.436515i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.55663e7	+	5.51752e7i	−0.869558	+	0.502039i	−0.867201	−	0.497958i	$-0.834083\pi$
$479$	−9.55663e7	+	5.51752e7i	−0.869558	+	0.502039i	−0.00235639	+	0.999997i	$0.500750\pi$
$480$		0			0
$481$	6.95618e6	−	1.20484e7i	0.0625080	−	0.108267i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	5.43258e7i		−	0.476190i
$486$		0			0
$487$	1.64810e8			1.42691			0.713454	−	0.700702i	$-0.247130\pi$
$487$	1.64810e8			1.42691			0.713454	+	0.700702i	$0.247130\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.49352e7	+	4.32639e7i	0.633055	+	0.365494i	0.781934	−	0.623361i	$-0.214233\pi$
$491$	7.49352e7	+	4.32639e7i	0.633055	+	0.365494i	−0.148879	+	0.988855i	$0.547567\pi$
$492$		0			0
$493$	4.92326e6	+	8.52734e6i	0.0410877	+	0.0711660i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.03214e8	−	1.17326e8i	1.65533	−	0.955705i
$498$		0			0
$499$	−1.21159e8	+	2.09854e8i	−0.975113	+	1.68894i	−0.295550	+	0.955327i	$0.595503\pi$
$499$	−1.21159e8	+	2.09854e8i	−0.975113	+	1.68894i	−0.679563	+	0.733617i	$0.737831\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			8.54544e6i			0.0671476i	0.999436	+	0.0335738i	$0.0106889\pi$
$503$			8.54544e6i			0.0671476i	−0.999436	+	0.0335738i	$0.989311\pi$
$504$		0			0
$505$	−2.17550e7			−0.168922
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.07522e7	+	4.66223e7i	0.612352	+	0.353541i	0.773885	−	0.633326i	$-0.218311\pi$
$509$	8.07522e7	+	4.66223e7i	0.612352	+	0.353541i	−0.161534	+	0.986867i	$0.551644\pi$
$510$		0			0
$511$	−1.40336e8	−	2.43070e8i	−1.05174	−	1.82166i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.75224e7	−	2.74370e7i	0.347917	−	0.200870i
$516$		0			0
$517$	−4.63264e7	+	8.02397e7i	−0.335241	+	0.580654i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1.51554e8i		−	1.07165i	−0.844328	−	0.535827i	$-0.820000\pi$
$521$		−	1.51554e8i		−	1.07165i	0.844328	−	0.535827i	$-0.180000\pi$
$522$		0			0
$523$	−6.92213e7			−0.483877			−0.241938	−	0.970292i	$-0.577783\pi$
$523$	−6.92213e7			−0.483877			−0.241938	+	0.970292i	$0.577783\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.36638e7	+	1.36623e7i	0.161678	+	0.0933450i
$528$		0			0
$529$	5.23346e7	+	9.06462e7i	0.353527	+	0.612326i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.03735e7	+	1.75361e7i	−0.200592	+	0.115812i
$534$		0			0
$535$	5.02096e7	−	8.69656e7i	0.327888	−	0.567919i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1.03924e8i		−	0.663664i
$540$		0			0
$541$	−4.28051e7			−0.270336			−0.135168	−	0.990823i	$-0.543157\pi$
$541$	−4.28051e7			−0.270336			−0.135168	+	0.990823i	$0.543157\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.32743e8	−	7.66390e7i	−0.820014	−	0.473435i
$546$		0			0
$547$	−2.73793e7	−	4.74223e7i	−0.167286	−	0.289748i	0.770179	−	0.637828i	$-0.220167\pi$
$547$	−2.73793e7	−	4.74223e7i	−0.167286	−	0.289748i	−0.937465	+	0.348080i	$0.886834\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.36419e8	+	7.87617e7i	−0.815494	+	0.470826i
$552$		0			0
$553$	9.95225e7	−	1.72378e8i	0.588499	−	1.01931i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			2.76165e8i			1.59809i	0.601268	+	0.799047i	$0.294662\pi$
$557$			2.76165e8i			1.59809i	−0.601268	+	0.799047i	$0.705338\pi$
$558$		0			0
$559$	−6.91032e7			−0.395606
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.09201e7	−	6.30471e6i	−0.0611929	−	0.0353297i	0.469091	−	0.883150i	$-0.344581\pi$
$563$	−1.09201e7	−	6.30471e6i	−0.0611929	−	0.0353297i	−0.530284	+	0.847820i	$0.677915\pi$
$564$		0			0
$565$	−8.48438e7	−	1.46954e8i	−0.470408	−	0.814771i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.91111e8	−	1.10338e8i	1.03741	−	0.598948i	0.118310	−	0.992977i	$-0.462252\pi$
$569$	1.91111e8	−	1.10338e8i	1.03741	−	0.598948i	0.919098	+	0.394029i	$0.128919\pi$
$570$		0			0
$571$	−1.14680e8	+	1.98631e8i	−0.615996	+	1.06694i	0.374213	+	0.927343i	$0.377913\pi$
$571$	−1.14680e8	+	1.98631e8i	−0.615996	+	1.06694i	−0.990209	+	0.139593i	$0.955421\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	9.02527e7i		−	0.474741i
$576$		0			0
$577$	2.80584e8			1.46061			0.730307	−	0.683119i	$-0.239377\pi$
$577$	2.80584e8			1.46061			0.730307	+	0.683119i	$0.239377\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.15992e8	+	1.24703e8i	1.10131	+	0.635841i
$582$		0			0
$583$	−1.44577e8	−	2.50414e8i	−0.729614	−	1.26373i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.27003e7	−	4.77470e7i	0.408877	−	0.236065i	−0.281430	−	0.959582i	$-0.590809\pi$
$587$	8.27003e7	−	4.77470e7i	0.408877	−	0.236065i	0.690307	+	0.723517i	$0.257475\pi$
$588$		0			0
$589$	−2.18567e8	+	3.78570e8i	−1.06964	+	1.85268i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1.21878e8i			0.584471i	0.956346	+	0.292235i	$0.0943991\pi$
$593$			1.21878e8i			0.584471i	−0.956346	+	0.292235i	$0.905601\pi$
$594$		0			0
$595$	−2.46364e7			−0.116957
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.69933e6	−	2.13581e6i	−0.0172124	−	0.00993760i	0.491369	−	0.870951i	$-0.336497\pi$
$599$	−3.69933e6	−	2.13581e6i	−0.0172124	−	0.00993760i	−0.508582	+	0.861014i	$0.669830\pi$
$600$		0			0
$601$	−1.51180e7	−	2.61851e7i	−0.0696418	−	0.120623i	0.829102	−	0.559098i	$-0.188852\pi$
$601$	−1.51180e7	−	2.61851e7i	−0.0696418	−	0.120623i	−0.898744	+	0.438475i	$0.855519\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.57132e8	+	9.07204e7i	−0.709577	+	0.409674i
$606$		0			0
$607$	−1.30421e8	+	2.25896e8i	−0.583153	+	1.01005i	0.411950	+	0.911206i	$0.364848\pi$
$607$	−1.30421e8	+	2.25896e8i	−0.583153	+	1.01005i	−0.995103	+	0.0988436i	$0.968486\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			3.27840e7i			0.143727i
$612$		0			0
$613$	2.88624e8			1.25300			0.626500	−	0.779422i	$-0.284487\pi$
$613$	2.88624e8			1.25300			0.626500	+	0.779422i	$0.284487\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.35310e8	−	7.81215e7i	−0.576070	−	0.332594i	0.183500	−	0.983020i	$-0.441257\pi$
$617$	−1.35310e8	−	7.81215e7i	−0.576070	−	0.332594i	−0.759570	+	0.650425i	$0.774591\pi$
$618$		0			0
$619$	−4.48054e6	−	7.76052e6i	−0.0188911	−	0.0327204i	0.856425	−	0.516271i	$-0.172680\pi$
$619$	−4.48054e6	−	7.76052e6i	−0.0188911	−	0.0327204i	−0.875316	+	0.483551i	$0.839347\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.19874e8	+	6.92093e7i	−0.495748	+	0.286220i
$624$		0			0
$625$	6.15985e7	−	1.06692e8i	0.252308	−	0.437010i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.23407e7i		−	0.0495892i
$630$		0			0
$631$	1.20098e8			0.478021			0.239010	−	0.971017i	$-0.423177\pi$
$631$	1.20098e8			0.478021			0.239010	+	0.971017i	$0.423177\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.63376e8	+	1.52060e8i	1.02862	+	0.593873i
$636$		0			0
$637$	−1.83860e7	−	3.18455e7i	−0.0711328	−	0.123206i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.12042e8	−	6.46875e7i	0.425409	−	0.245610i	−0.271980	−	0.962303i	$-0.587678\pi$
$641$	1.12042e8	−	6.46875e7i	0.425409	−	0.245610i	0.697389	+	0.716693i	$0.254345\pi$
$642$		0			0
$643$	1.17521e8	−	2.03552e8i	0.442060	−	0.765670i	−0.555782	−	0.831328i	$-0.687581\pi$
$643$	1.17521e8	−	2.03552e8i	0.442060	−	0.765670i	0.997842	+	0.0656577i	$0.0209145\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			3.71853e8i			1.37296i	0.727148	+	0.686481i	$0.240846\pi$
$647$			3.71853e8i			1.37296i	−0.727148	+	0.686481i	$0.759154\pi$
$648$		0			0
$649$	6.51397e8			2.38293
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.70500e8	+	9.84380e7i	0.612328	+	0.353528i	0.773876	−	0.633337i	$-0.218315\pi$
$653$	1.70500e8	+	9.84380e7i	0.612328	+	0.353528i	−0.161548	+	0.986865i	$0.551649\pi$
$654$		0			0
$655$	−5.88742e7	−	1.01973e8i	−0.209508	−	0.362879i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.36750e8	+	7.89529e7i	−0.477829	+	0.275875i	−0.719511	−	0.694481i	$-0.755634\pi$
$659$	−1.36750e8	+	7.89529e7i	−0.477829	+	0.275875i	0.241682	+	0.970355i	$0.422301\pi$
$660$		0			0
$661$	−4.50661e7	+	7.80568e7i	−0.156044	+	0.270275i	−0.933439	−	0.358737i	$-0.883207\pi$
$661$	−4.50661e7	+	7.80568e7i	−0.156044	+	0.270275i	0.777395	+	0.629013i	$0.216541\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	3.94130e8i		−	1.34021i
$666$		0			0
$667$	2.63182e8			0.886907
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.08566e8	−	6.26806e7i	−0.359357	−	0.207475i
$672$		0			0
$673$	1.51945e8	+	2.63177e8i	0.498473	+	0.863381i	0.999998	−	0.00176204i	$-0.000560877\pi$
$673$	1.51945e8	+	2.63177e8i	0.498473	+	0.863381i	−0.501525	+	0.865143i	$0.667228\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.56583e7	−	9.04033e6i	0.0504637	−	0.0291352i	−0.474556	−	0.880225i	$-0.657391\pi$
$677$	1.56583e7	−	9.04033e6i	0.0504637	−	0.0291352i	0.525020	+	0.851090i	$0.324058\pi$
$678$		0			0
$679$	1.13110e8	−	1.95913e8i	0.361321	−	0.625826i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			2.51133e8i			0.788209i	0.919066	+	0.394105i	$0.128945\pi$
$683$			2.51133e8i			0.788209i	−0.919066	+	0.394105i	$0.871055\pi$
$684$		0			0
$685$	−2.91224e7			−0.0906058
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−8.86059e7	−	5.11567e7i	−0.270898	−	0.156403i
$690$		0			0
$691$	−8.37607e7	−	1.45078e8i	−0.253867	−	0.439710i	0.710720	−	0.703475i	$-0.248369\pi$
$691$	−8.37607e7	−	1.45078e8i	−0.253867	−	0.439710i	−0.964587	+	0.263764i	$0.915036\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.29263e8	+	2.47835e8i	−1.27870	+	0.738259i
$696$		0			0
$697$	−1.55551e7	+	2.69422e7i	−0.0459382	+	0.0795673i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			4.19959e8i			1.21914i	0.792733	+	0.609569i	$0.208657\pi$
$701$			4.19959e8i			1.21914i	−0.792733	+	0.609569i	$0.791343\pi$
$702$		0			0
$703$	1.97424e8			0.568244
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.84543e7	−	4.52956e7i	−0.222003	−	0.128173i
$708$		0			0
$709$	2.92143e8	+	5.06006e8i	0.819702	+	1.41977i	0.905902	+	0.423488i	$0.139194\pi$
$709$	2.92143e8	+	5.06006e8i	0.819702	+	1.41977i	−0.0861995	+	0.996278i	$0.527472\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.32494e8	−	3.65171e8i	1.74497	−	1.00746i
$714$		0			0
$715$	−6.33602e7	+	1.09743e8i	−0.173340	+	0.300233i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1.23867e8i		−	0.333249i	−0.986020	−	0.166625i	$-0.946713\pi$
$719$		−	1.23867e8i		−	0.333249i	0.986020	−	0.166625i	$-0.0532868\pi$
$720$		0			0
$721$	2.28504e8			0.609660
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.14015e7	−	4.69972e7i	−0.213608	−	0.123327i
$726$		0			0
$727$	2.59463e8	+	4.49403e8i	0.675262	+	1.16959i	0.976392	+	0.216005i	$0.0693028\pi$
$727$	2.59463e8	+	4.49403e8i	0.675262	+	1.16959i	−0.301130	+	0.953583i	$0.597364\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.30844e7	+	3.06483e7i	−0.135899	+	0.0784611i
$732$		0			0
$733$	1.52477e7	−	2.64097e7i	0.0387161	−	0.0670582i	−0.846018	−	0.533154i	$-0.821007\pi$
$733$	1.52477e7	−	2.64097e7i	0.0387161	−	0.0670582i	0.884734	+	0.466096i	$0.154340\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			6.59417e8i			1.64724i
$738$		0			0
$739$	−4.95085e7			−0.122672			−0.0613361	−	0.998117i	$-0.519536\pi$
$739$	−4.95085e7			−0.122672			−0.0613361	+	0.998117i	$0.519536\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.72630e8	+	2.15138e8i	0.908473	+	0.524507i	0.879939	−	0.475086i	$-0.157583\pi$
$743$	3.72630e8	+	2.15138e8i	0.908473	+	0.524507i	0.0285333	+	0.999593i	$0.490916\pi$
$744$		0			0
$745$	2.43277e8	+	4.21368e8i	0.588345	+	1.01904i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.62138e8	−	2.09081e8i	0.861844	−	0.497586i
$750$		0			0
$751$	−1.53847e8	+	2.66471e8i	−0.363219	+	0.629114i	−0.988489	−	0.151295i	$-0.951656\pi$
$751$	−1.53847e8	+	2.66471e8i	−0.363219	+	0.629114i	0.625269	+	0.780409i	$0.284989\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1.24924e8i		−	0.290272i
$756$		0			0
$757$	−6.63191e8			−1.52880			−0.764400	−	0.644742i	$-0.776965\pi$
$757$	−6.63191e8			−1.52880			−0.764400	+	0.644742i	$0.776965\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.60784e7	−	2.08299e7i	−0.0818641	−	0.0472642i	0.458509	−	0.888690i	$-0.348384\pi$
$761$	−3.60784e7	−	2.08299e7i	−0.0818641	−	0.0472642i	−0.540373	+	0.841425i	$0.681717\pi$
$762$		0			0
$763$	−3.19137e8	−	5.52761e8i	−0.718461	−	1.24441i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.99609e8	−	1.15244e8i	0.442378	−	0.255407i
$768$		0			0
$769$	2.66593e8	−	4.61753e8i	0.586232	−	1.01538i	−0.408488	−	0.912764i	$-0.633944\pi$
$769$	2.66593e8	−	4.61753e8i	0.586232	−	1.01538i	0.994721	−	0.102621i	$-0.0327228\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			3.84433e8i			0.832304i	0.909295	+	0.416152i	$0.136622\pi$
$773$			3.84433e8i			0.832304i	−0.909295	+	0.416152i	$0.863378\pi$
$774$		0			0
$775$	−2.60839e8			−0.560360
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.31018e8	−	2.48848e8i	−0.911764	−	0.526407i
$780$		0			0
$781$	−5.35304e8	−	9.27174e8i	−1.12369	−	1.94629i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.82305e8	−	1.62989e8i	0.583591	−	0.336937i
$786$		0			0
$787$	3.48013e8	−	6.02776e8i	0.713956	−	1.23661i	−0.249405	−	0.968399i	$-0.580235\pi$
$787$	3.48013e8	−	6.02776e8i	0.713956	−	1.23661i	0.963361	−	0.268208i	$-0.0864315\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	7.06605e8i		−	1.42773i
$792$		0			0
$793$	−4.43575e7			−0.0889503
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.61308e8	+	3.81806e8i	1.30626	+	0.754168i	0.981469	−	0.191619i	$-0.0613737\pi$
$797$	6.61308e8	+	3.81806e8i	1.30626	+	0.754168i	0.324788	+	0.945787i	$0.394707\pi$
$798$		0			0
$799$	1.45402e7	+	2.51844e7i	0.0285056	+	0.0493731i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.10902e9	+	6.40291e8i	−2.14186	+	1.23660i
$804$		0			0
$805$	−3.29245e8	+	5.70270e8i	−0.631149	+	1.09318i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	4.08074e8i		−	0.770714i	−0.922768	−	0.385357i	$-0.874078\pi$
$809$		−	4.08074e8i		−	0.770714i	0.922768	−	0.385357i	$-0.125922\pi$
$810$		0			0
$811$	3.33812e8			0.625806			0.312903	−	0.949785i	$-0.398699\pi$
$811$	3.33812e8			0.625806			0.312903	+	0.949785i	$0.398699\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5.15320e8	+	2.97520e8i	0.951928	+	0.549596i
$816$		0			0
$817$	−4.90308e8	−	8.49238e8i	−0.899089	−	1.55727i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.65460e8	+	2.68733e8i	−0.841110	+	0.485615i	−0.857641	−	0.514249i	$-0.828071\pi$
$821$	−4.65460e8	+	2.68733e8i	−0.841110	+	0.485615i	0.0165317	+	0.999863i	$0.494738\pi$
$822$		0			0
$823$	2.46925e7	−	4.27687e7i	0.0442962	−	0.0767233i	−0.843027	−	0.537871i	$-0.819229\pi$
$823$	2.46925e7	−	4.27687e7i	0.0442962	−	0.0767233i	0.887323	+	0.461148i	$0.152562\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2.72468e8i			0.481726i	0.970559	+	0.240863i	$0.0774304\pi$
$827$			2.72468e8i			0.481726i	−0.970559	+	0.240863i	$0.922570\pi$
$828$		0			0
$829$	1.23939e8			0.217542			0.108771	−	0.994067i	$-0.465308\pi$
$829$	1.23939e8			0.217542			0.108771	+	0.994067i	$0.465308\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.82479e7	−	1.63090e7i	−0.0488711	−	0.0282157i
$834$		0			0
$835$	2.57964e8	+	4.46806e8i	0.443097	+	0.767467i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.50756e8	−	2.60244e8i	0.763231	−	0.440651i	−0.0672238	−	0.997738i	$-0.521414\pi$
$839$	4.50756e8	−	2.60244e8i	0.763231	−	0.440651i	0.830454	+	0.557087i	$0.188081\pi$
$840$		0			0
$841$	−1.60365e8	+	2.77761e8i	−0.269602	+	0.466964i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	4.36575e8i		−	0.723584i
$846$		0			0
$847$	−7.55547e8			−1.24340
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.85655e8	−	1.64923e8i	−0.463504	−	0.267604i
$852$		0			0
$853$	1.67400e6	+	2.89945e6i	0.00269716	+	0.00467163i	0.867371	−	0.497662i	$-0.165808\pi$
$853$	1.67400e6	+	2.89945e6i	0.00269716	+	0.00467163i	−0.864674	+	0.502334i	$0.832475\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.23377e8	+	2.44437e8i	−0.672643	+	0.388351i	−0.797077	−	0.603877i	$-0.793622\pi$
$857$	−4.23377e8	+	2.44437e8i	−0.672643	+	0.388351i	0.124434	+	0.992228i	$0.460288\pi$
$858$		0			0
$859$	−3.06897e8	+	5.31560e8i	−0.484186	+	0.838635i	−0.999835	−	0.0181649i	$-0.994218\pi$
$859$	−3.06897e8	+	5.31560e8i	−0.484186	+	0.838635i	0.515649	+	0.856800i	$0.327551\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.53387e8i		−	0.238647i	−0.992855	−	0.119324i	$-0.961927\pi$
$863$		−	1.53387e8i		−	0.238647i	0.992855	−	0.119324i	$-0.0380726\pi$
$864$		0			0
$865$	−6.39163e8			−0.987559
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.86483e8	−	4.54076e8i	−1.19848	−	0.691942i
$870$		0			0
$871$	1.16663e8	+	2.02067e8i	0.176555	+	0.305802i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.64193e8	−	4.41207e8i	1.14072	−	0.658595i
$876$		0			0
$877$	−5.58483e7	+	9.67321e7i	−0.0827963	+	0.143407i	−0.904450	−	0.426580i	$-0.859718\pi$
$877$	−5.58483e7	+	9.67321e7i	−0.0827963	+	0.143407i	0.821654	+	0.569987i	$0.193052\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1.19379e9i		−	1.74583i	−0.487873	−	0.872915i	$-0.662227\pi$
$881$		−	1.19379e9i		−	1.74583i	0.487873	−	0.872915i	$-0.337773\pi$
$882$		0			0
$883$	−6.64444e8			−0.965109			−0.482554	−	0.875866i	$-0.660291\pi$
$883$	−6.64444e8			−0.965109			−0.482554	+	0.875866i	$0.660291\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2.83018e8	−	1.63401e8i	−0.405550	−	0.234144i	0.283326	−	0.959024i	$-0.408562\pi$
$887$	−2.83018e8	−	1.63401e8i	−0.405550	−	0.234144i	−0.688876	+	0.724879i	$0.741895\pi$
$888$		0			0
$889$	6.33201e8	+	1.09674e9i	0.901231	+	1.56098i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.02896e8	+	2.32612e8i	−0.565768	+	0.326647i
$894$		0			0
$895$	−9.16917e7	+	1.58815e8i	−0.127897	+	0.221524i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	7.60620e8i		−	1.04686i
$900$		0			0
$901$	−9.07549e7			−0.124078
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.00084e8	−	2.88723e8i	−0.674679	−	0.389526i
$906$		0			0
$907$	−1.78077e8	−	3.08439e8i	−0.238664	−	0.413377i	0.721667	−	0.692240i	$-0.243376\pi$
$907$	−1.78077e8	−	3.08439e8i	−0.238664	−	0.413377i	−0.960331	+	0.278862i	$0.910043\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.50415e7	−	5.48722e7i	0.125707	−	0.0725768i	−0.435828	−	0.900030i	$-0.643544\pi$
$911$	9.50415e7	−	5.48722e7i	0.125707	−	0.0725768i	0.561535	+	0.827453i	$0.310211\pi$
$912$		0			0
$913$	5.68964e8	−	9.85474e8i	0.747605	−	1.29489i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	4.90322e8i		−	0.635878i
$918$		0			0
$919$	−6.99498e8			−0.901239			−0.450620	−	0.892716i	$-0.648797\pi$
$919$	−6.99498e8			−0.901239			−0.450620	+	0.892716i	$0.648797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.28069e8	−	1.89411e8i	−0.417215	−	0.240879i
$924$		0			0
$925$	5.89017e7	+	1.02021e8i	0.0744222	+	0.128903i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.93754e8	+	3.42804e8i	−0.740558	+	0.427561i	−0.822272	−	0.569094i	$-0.807294\pi$
$929$	−5.93754e8	+	3.42804e8i	−0.740558	+	0.427561i	0.0817140	+	0.996656i	$0.473961\pi$
$930$		0			0
$931$	2.60909e8	−	4.51907e8i	0.323325	−	0.560016i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			1.12405e8i			0.137515i
$936$		0			0
$937$	1.04563e9			1.27104			0.635520	−	0.772084i	$-0.280786\pi$
$937$	1.04563e9			1.27104			0.635520	+	0.772084i	$0.280786\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.81017e7	−	5.08655e7i	−0.105734	−	0.0610456i	0.446200	−	0.894933i	$-0.352777\pi$
$941$	−8.81017e7	−	5.08655e7i	−0.105734	−	0.0610456i	−0.551935	+	0.833887i	$0.686110\pi$
$942$		0			0
$943$	4.15762e8	+	7.20121e8i	0.495804	+	0.858758i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.09024e9	−	6.29449e8i	1.28372	−	0.741158i	0.306196	−	0.951968i	$-0.400944\pi$
$947$	1.09024e9	−	6.29449e8i	1.28372	−	0.741158i	0.977527	+	0.210811i	$0.0676103\pi$
$948$		0			0
$949$	−2.26559e8	+	3.92412e8i	−0.265083	+	0.459138i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	2.21916e8i		−	0.256396i	−0.991749	−	0.128198i	$-0.959081\pi$
$953$		−	2.21916e8i		−	0.256396i	0.991749	−	0.128198i	$-0.0409192\pi$
$954$		0			0
$955$	5.83739e8			0.670206
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.05023e8	−	6.06351e7i	−0.119077	−	0.0687493i
$960$		0			0
$961$	−6.11626e8	−	1.05937e9i	−0.689153	−	1.19365i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.75572e8	+	3.90042e8i	−0.751778	+	0.434039i
$966$		0			0
$967$	3.89189e8	−	6.74095e8i	0.430409	−	0.745490i	−0.566500	−	0.824062i	$-0.691703\pi$
$967$	3.89189e8	−	6.74095e8i	0.430409	−	0.745490i	0.996908	+	0.0785723i	$0.0250361\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.29298e9i			1.41232i	0.708053	+	0.706159i	$0.249574\pi$
$971$			1.29298e9i			1.41232i	−0.708053	+	0.706159i	$0.750426\pi$
$972$		0			0
$973$	−2.06405e9			−2.24069
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.37032e8	−	3.67791e8i	−0.683090	−	0.394382i	0.117928	−	0.993022i	$-0.462375\pi$
$977$	−6.37032e8	−	3.67791e8i	−0.683090	−	0.394382i	−0.801018	+	0.598640i	$0.795708\pi$
$978$		0			0
$979$	3.15771e8	+	5.46931e8i	0.336530	+	0.582887i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.38767e8	−	5.41997e8i	0.988319	−	0.570606i	0.0835479	−	0.996504i	$-0.473375\pi$
$983$	9.38767e8	−	5.41997e8i	0.988319	−	0.570606i	0.904772	+	0.425897i	$0.140042\pi$
$984$		0			0
$985$	5.64325e7	−	9.77439e7i	0.0590501	−	0.102278i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.63836e9i			1.69364i
$990$		0			0
$991$	−1.15250e7			−0.0118419			−0.00592095	−	0.999982i	$-0.501885\pi$
$991$	−1.15250e7			−0.0118419			−0.00592095	+	0.999982i	$0.501885\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.24588e9	+	7.19307e8i	1.26475	+	0.730206i
$996$		0			0
$997$	−6.25279e7	−	1.08302e8i	−0.0630941	−	0.109282i	0.832753	−	0.553645i	$-0.186763\pi$
$997$	−6.25279e7	−	1.08302e8i	−0.0630941	−	0.109282i	−0.895847	+	0.444363i	$0.853430\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.7.m.a.17.5		36
3.2	odd	2		72.7.m.a.41.10	✓	36
4.3	odd	2		432.7.q.d.17.5		36
9.2	odd	6	inner	216.7.m.a.89.5		36
9.4	even	3		648.7.e.c.161.25		36
9.5	odd	6		648.7.e.c.161.12		36
9.7	even	3		72.7.m.a.65.10	yes	36
12.11	even	2		144.7.q.d.113.9		36
36.7	odd	6		144.7.q.d.65.9		36
36.11	even	6		432.7.q.d.305.5		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.m.a.41.10	✓	36	3.2	odd	2
72.7.m.a.65.10	yes	36	9.7	even	3
144.7.q.d.65.9		36	36.7	odd	6
144.7.q.d.113.9		36	12.11	even	2
216.7.m.a.17.5		36	1.1	even	1	trivial
216.7.m.a.89.5		36	9.2	odd	6	inner
432.7.q.d.17.5		36	4.3	odd	2
432.7.q.d.305.5		36	36.11	even	6
648.7.e.c.161.12		36	9.5	odd	6
648.7.e.c.161.25		36	9.4	even	3

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	216.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-86.3751 - 49.8687i) q^{5} +(-207.661 - 359.679i) q^{7} +O(q^{10})\)	\(q+(-86.3751 - 49.8687i) q^{5} +(-207.661 - 359.679i) q^{7} +(-1641.05 + 947.463i) q^{11} +(-335.248 + 580.666i) q^{13} +594.749i q^{17} -9514.73 q^{19} +(13766.9 + 7948.35i) q^{23} +(-2838.72 - 4916.81i) q^{25} +(14337.7 - 8277.87i) q^{29} +(22971.5 - 39787.8i) q^{31} +41423.1i q^{35} -20749.4 q^{37} +(45300.1 + 26154.0i) q^{41} +(51531.5 + 89255.1i) q^{43} +(42344.5 - 24447.6i) q^{47} +(-27421.6 + 47495.5i) q^{49} +152594. i q^{53} +188995. q^{55} +(-297704. - 171879. i) q^{59} +(33078.1 + 57293.0i) q^{61} +(57914.1 - 33436.7i) q^{65} +(173996. - 301369. i) q^{67} +564987. i q^{71} +675795. q^{73} +(681565. + 393502. i) q^{77} +(239627. + 415047. i) q^{79} +(-520059. + 300256. i) q^{83} +(29659.4 - 51371.6i) q^{85} -333280. i q^{89} +278471. q^{91} +(821836. + 474487. i) q^{95} +(272344. + 471714. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q+O(q^{10}) \) 36 * q	\( 36 q - 1350 q^{11} + 9540 q^{19} - 30888 q^{23} + 56250 q^{25} - 38556 q^{29} + 27720 q^{31} - 179226 q^{41} + 15930 q^{43} - 187596 q^{47} - 198774 q^{49} - 197064 q^{55} + 408618 q^{59} + 17136 q^{61} + 125712 q^{65} + 27090 q^{67} - 534060 q^{73} - 48168 q^{77} + 172620 q^{79} - 1801980 q^{83} - 791568 q^{85} + 538560 q^{91} - 1832652 q^{95} + 770706 q^{97}+O(q^{100}) \) 36 * q - 1350 * q^11 + 9540 * q^19 - 30888 * q^23 + 56250 * q^25 - 38556 * q^29 + 27720 * q^31 - 179226 * q^41 + 15930 * q^43 - 187596 * q^47 - 198774 * q^49 - 197064 * q^55 + 408618 * q^59 + 17136 * q^61 + 125712 * q^65 + 27090 * q^67 - 534060 * q^73 - 48168 * q^77 + 172620 * q^79 - 1801980 * q^83 - 791568 * q^85 + 538560 * q^91 - 1832652 * q^95 + 770706 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more