LMFDB - Embedded newform 324.5.k.a.89.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,5,Mod(17,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 11]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("324.17");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.k (of order $18$, degree $6$, 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:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			89.1
Character	$\chi$	$=$	324.89
Dual form			324.5.k.a.233.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-29.8374 - 35.5588i) q^{5} +(-70.9709 + 25.8313i) q^{7} +O(q^{10})$	$q+(-29.8374 - 35.5588i) q^{5} +(-70.9709 + 25.8313i) q^{7} +(37.6687 - 44.8918i) q^{11} +(-56.3644 - 319.658i) q^{13} +(-143.731 + 82.9833i) q^{17} +(-17.1368 + 29.6819i) q^{19} +(43.9406 - 120.726i) q^{23} +(-265.630 + 1506.46i) q^{25} +(-1026.95 - 181.080i) q^{29} +(713.964 + 259.862i) q^{31} +(3036.12 + 1752.90i) q^{35} +(508.520 + 880.783i) q^{37} +(-23.6179 + 4.16447i) q^{41} +(1834.16 + 1539.05i) q^{43} +(-651.646 - 1790.38i) q^{47} +(2530.34 - 2123.21i) q^{49} +1365.27i q^{53} -2720.24 q^{55} +(1297.94 + 1546.83i) q^{59} +(1446.06 - 526.322i) q^{61} +(-9684.92 + 11542.0i) q^{65} +(30.0066 + 170.176i) q^{67} +(-6082.30 + 3511.62i) q^{71} +(2717.42 - 4706.72i) q^{73} +(-1513.77 + 4159.04i) q^{77} +(-220.907 + 1252.83i) q^{79} +(11696.3 + 2062.37i) q^{83} +(7239.36 + 2634.91i) q^{85} +(-5912.93 - 3413.83i) q^{89} +(12257.4 + 21230.5i) q^{91} +(1566.77 - 276.264i) q^{95} +(-5288.41 - 4437.50i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - 9 q^{5}+O(q^{10}) $ 72 * q - 9 * q^5	$ 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) $ 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{18}\right)$

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
$5$	−29.8374	−	35.5588i	−1.19350	−	1.42235i	−0.881444	−	0.472289i	$-0.843428\pi$
$5$	−29.8374	−	35.5588i	−1.19350	−	1.42235i	−0.312053	−	0.950065i	$-0.601017\pi$
$6$		0			0
$7$	−70.9709	+	25.8313i	−1.44839	+	0.527169i	−0.942140	−	0.335220i	$-0.891189\pi$
$7$	−70.9709	+	25.8313i	−1.44839	+	0.527169i	−0.506246	+	0.862389i	$0.668967\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	37.6687	−	44.8918i	0.311312	−	0.371007i	−0.587589	−	0.809160i	$-0.699923\pi$
$11$	37.6687	−	44.8918i	0.311312	−	0.371007i	0.898900	+	0.438153i	$0.144367\pi$
$12$		0			0
$13$	−56.3644	−	319.658i	−0.333517	−	1.89147i	−0.441406	−	0.897307i	$-0.645520\pi$
$13$	−56.3644	−	319.658i	−0.333517	−	1.89147i	0.107889	−	0.994163i	$-0.465591\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−143.731	+	82.9833i	−0.497340	+	0.287140i	−0.727614	−	0.685986i	$-0.759371\pi$
$17$	−143.731	+	82.9833i	−0.497340	+	0.287140i	0.230274	+	0.973126i	$0.426038\pi$
$18$		0			0
$19$	−17.1368	+	29.6819i	−0.0474705	+	0.0822212i	−0.888784	−	0.458326i	$-0.848449\pi$
$19$	−17.1368	+	29.6819i	−0.0474705	+	0.0822212i	0.841314	+	0.540547i	$0.181783\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	43.9406	−	120.726i	0.0830635	−	0.228215i	−0.891207	−	0.453597i	$-0.850141\pi$
$23$	43.9406	−	120.726i	0.0830635	−	0.228215i	0.974271	+	0.225382i	$0.0723629\pi$
$24$		0			0
$25$	−265.630	+	1506.46i	−0.425008	+	2.41034i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1026.95	−	181.080i	−1.22111	−	0.215315i	−0.474306	−	0.880360i	$-0.657301\pi$
$29$	−1026.95	−	181.080i	−1.22111	−	0.215315i	−0.746803	+	0.665045i	$0.768412\pi$
$30$		0			0
$31$	713.964	+	259.862i	0.742939	+	0.270408i	0.685631	−	0.727949i	$-0.259526\pi$
$31$	713.964	+	259.862i	0.742939	+	0.270408i	0.0573074	+	0.998357i	$0.481748\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3036.12	+	1752.90i	2.47846	+	1.43094i
$36$		0			0
$37$	508.520	+	880.783i	0.371454	+	0.643377i	0.989789	−	0.142537i	$-0.0455261\pi$
$37$	508.520	+	880.783i	0.371454	+	0.643377i	−0.618336	+	0.785914i	$0.712193\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−23.6179	+	4.16447i	−0.0140499	+	0.00247738i	−0.180669	−	0.983544i	$-0.557826\pi$
$41$	−23.6179	+	4.16447i	−0.0140499	+	0.00247738i	0.166619	+	0.986021i	$0.446715\pi$
$42$		0			0
$43$	1834.16	+	1539.05i	0.991976	+	0.832367i	0.985853	−	0.167614i	$-0.0536064\pi$
$43$	1834.16	+	1539.05i	0.991976	+	0.832367i	0.00612354	+	0.999981i	$0.498051\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−651.646	−	1790.38i	−0.294996	−	0.810494i	−0.995317	−	0.0966663i	$-0.969182\pi$
$47$	−651.646	−	1790.38i	−0.294996	−	0.810494i	0.700321	−	0.713828i	$-0.253040\pi$
$48$		0			0
$49$	2530.34	−	2123.21i	1.05387	−	0.884301i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1365.27i			0.486034i	0.970022	+	0.243017i	$0.0781371\pi$
$53$			1365.27i			0.486034i	−0.970022	+	0.243017i	$0.921863\pi$
$54$		0			0
$55$	−2720.24			−0.899252
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1297.94	+	1546.83i	0.372865	+	0.444364i	0.919549	−	0.392976i	$-0.128554\pi$
$59$	1297.94	+	1546.83i	0.372865	+	0.444364i	−0.546683	+	0.837339i	$0.684110\pi$
$60$		0			0
$61$	1446.06	−	526.322i	0.388621	−	0.141447i	−0.140318	−	0.990107i	$-0.544812\pi$
$61$	1446.06	−	526.322i	0.388621	−	0.141447i	0.528939	+	0.848660i	$0.322590\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9684.92	+	11542.0i	−2.29229	+	2.73184i
$66$		0			0
$67$	30.0066	+	170.176i	0.00668447	+	0.0379095i	0.987968	−	0.154661i	$-0.0494286\pi$
$67$	30.0066	+	170.176i	0.00668447	+	0.0379095i	−0.981283	+	0.192571i	$0.938318\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6082.30	+	3511.62i	−1.20657	+	0.696611i	−0.962007	−	0.273024i	$-0.911976\pi$
$71$	−6082.30	+	3511.62i	−1.20657	+	0.696611i	−0.244558	+	0.969635i	$0.578643\pi$
$72$		0			0
$73$	2717.42	−	4706.72i	0.509931	−	0.883227i	−0.490002	−	0.871721i	$-0.663004\pi$
$73$	2717.42	−	4706.72i	0.509931	−	0.883227i	0.999934	−	0.0115061i	$-0.00366259\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1513.77	+	4159.04i	−0.255316	+	0.701475i
$78$		0			0
$79$	−220.907	+	1252.83i	−0.0353961	+	0.200741i	−0.997378	−	0.0723733i	$-0.976943\pi$
$79$	−220.907	+	1252.83i	−0.0353961	+	0.200741i	0.961982	+	0.273115i	$0.0880538\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11696.3	+	2062.37i	1.69782	+	0.299371i	0.936931	−	0.349513i	$-0.113653\pi$
$83$	11696.3	+	2062.37i	1.69782	+	0.299371i	0.760887	+	0.648884i	$0.224764\pi$
$84$		0			0
$85$	7239.36	+	2634.91i	1.00199	+	0.364694i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5912.93	−	3413.83i	−0.746488	−	0.430985i	0.0779356	−	0.996958i	$-0.475167\pi$
$89$	−5912.93	−	3413.83i	−0.746488	−	0.430985i	−0.824423	+	0.565973i	$0.808500\pi$
$90$		0			0
$91$	12257.4	+	21230.5i	1.48019	+	2.56376i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1566.77	−	276.264i	0.173604	−	0.0306110i
$96$		0			0
$97$	−5288.41	−	4437.50i	−0.562058	−	0.471623i	0.316942	−	0.948445i	$-0.397344\pi$
$97$	−5288.41	−	4437.50i	−0.562058	−	0.471623i	−0.879000	+	0.476822i	$0.841789\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−986.584	−	2710.62i	−0.0967144	−	0.265721i	0.881896	−	0.471445i	$-0.156267\pi$
$101$	−986.584	−	2710.62i	−0.0967144	−	0.265721i	−0.978610	+	0.205724i	$0.934045\pi$
$102$		0			0
$103$	−4959.73	+	4161.71i	−0.467502	+	0.392281i	−0.845883	−	0.533369i	$-0.820926\pi$
$103$	−4959.73	+	4161.71i	−0.467502	+	0.392281i	0.378380	+	0.925650i	$0.376481\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	12145.6i		−	1.06084i	−0.847735	−	0.530420i	$-0.822034\pi$
$107$		−	12145.6i		−	1.06084i	0.847735	−	0.530420i	$-0.177966\pi$
$108$		0			0
$109$	−1439.21			−0.121135			−0.0605676	−	0.998164i	$-0.519291\pi$
$109$	−1439.21			−0.121135			−0.0605676	+	0.998164i	$0.519291\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13564.9	+	16166.1i	1.06233	+	1.26604i	0.962571	+	0.271029i	$0.0873638\pi$
$113$	13564.9	+	16166.1i	1.06233	+	1.26604i	0.0997625	+	0.995011i	$0.468192\pi$
$114$		0			0
$115$	−5603.95	+	2039.67i	−0.423739	+	0.154228i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8057.17	−	9602.16i	0.568969	−	0.678071i
$120$		0			0
$121$	1946.04	+	11036.5i	0.132917	+	0.753810i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	36368.9	−	20997.6i	2.32761	−	1.34385i
$126$		0			0
$127$	10172.9	−	17620.0i	0.630722	−	1.09244i	−0.356682	−	0.934226i	$-0.616092\pi$
$127$	10172.9	−	17620.0i	0.630722	−	1.09244i	0.987404	−	0.158217i	$-0.0505746\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8755.93	+	24056.7i	−0.510223	+	1.40183i	0.370783	+	0.928719i	$0.379089\pi$
$131$	−8755.93	+	24056.7i	−0.510223	+	1.40183i	−0.881006	+	0.473106i	$0.843133\pi$
$132$		0			0
$133$	449.495	−	2549.21i	0.0254110	−	0.144113i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−19915.7	−	3511.68i	−1.06110	−	0.187100i	−0.384253	−	0.923228i	$-0.625541\pi$
$137$	−19915.7	−	3511.68i	−1.06110	−	0.187100i	−0.676843	+	0.736128i	$0.736652\pi$
$138$		0			0
$139$	−34308.1	−	12487.1i	−1.77569	−	0.646298i	−0.999883	−	0.0153267i	$-0.995121\pi$
$139$	−34308.1	−	12487.1i	−1.77569	−	0.646298i	−0.775806	−	0.630971i	$-0.782657\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−16473.2	−	9510.82i	−0.805576	−	0.465099i
$144$		0			0
$145$	24202.6	+	41920.2i	1.15114	+	1.99383i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−25352.5	+	4470.34i	−1.14196	+	0.201358i	−0.712461	−	0.701712i	$-0.752419\pi$
$149$	−25352.5	+	4470.34i	−1.14196	+	0.201358i	−0.429494	+	0.903069i	$0.641308\pi$
$150$		0			0
$151$	−14793.5	−	12413.2i	−0.648809	−	0.544416i	0.257900	−	0.966172i	$-0.416969\pi$
$151$	−14793.5	−	12413.2i	−0.648809	−	0.544416i	−0.906709	+	0.421756i	$0.861414\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12062.5	−	33141.3i	−0.502080	−	1.37945i
$156$		0			0
$157$	25816.0	−	21662.2i	1.04734	−	0.878825i	0.0545315	−	0.998512i	$-0.482633\pi$
$157$	25816.0	−	21662.2i	1.04734	−	0.878825i	0.992812	+	0.119687i	$0.0381890\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			9703.06i			0.374332i
$162$		0			0
$163$	−4497.89			−0.169291			−0.0846455	−	0.996411i	$-0.526976\pi$
$163$	−4497.89			−0.169291			−0.0846455	+	0.996411i	$0.526976\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4828.62	+	5754.52i	0.173137	+	0.206337i	0.845634	−	0.533763i	$-0.179223\pi$
$167$	4828.62	+	5754.52i	0.173137	+	0.206337i	−0.672497	+	0.740100i	$0.734778\pi$
$168$		0			0
$169$	−72166.0	+	26266.3i	−2.52673	+	0.919656i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−21750.5	+	25921.2i	−0.726737	+	0.866091i	−0.995267	−	0.0971801i	$-0.969018\pi$
$173$	−21750.5	+	25921.2i	−0.726737	+	0.866091i	0.268530	+	0.963271i	$0.413462\pi$
$174$		0			0
$175$	−20061.9	−	113777.i	−0.655082	−	3.71515i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15549.3	−	8977.38i	0.485293	−	0.280184i	−0.237327	−	0.971430i	$-0.576271\pi$
$179$	15549.3	−	8977.38i	0.485293	−	0.280184i	0.722620	+	0.691246i	$0.242938\pi$
$180$		0			0
$181$	16596.9	−	28746.6i	0.506605	−	0.877465i	−0.493366	−	0.869822i	$-0.664234\pi$
$181$	16596.9	−	28746.6i	0.506605	−	0.877465i	0.999971	−	0.00764330i	$-0.00243296\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16146.7	−	44362.7i	0.471781	−	1.29621i
$186$		0			0
$187$	−1688.90	+	9578.23i	−0.0482971	+	0.273906i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−42708.8	−	7530.72i	−1.17071	−	0.206428i	−0.445713	−	0.895176i	$-0.647050\pi$
$191$	−42708.8	−	7530.72i	−1.17071	−	0.206428i	−0.725001	+	0.688748i	$0.758161\pi$
$192$		0			0
$193$	32342.6	+	11771.8i	0.868282	+	0.316029i	0.737471	−	0.675378i	$-0.236020\pi$
$193$	32342.6	+	11771.8i	0.868282	+	0.316029i	0.130811	+	0.991407i	$0.458242\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	43545.9	+	25141.3i	1.12206	+	0.647820i	0.941925	−	0.335822i	$-0.109014\pi$
$197$	43545.9	+	25141.3i	1.12206	+	0.647820i	0.180132	+	0.983642i	$0.442347\pi$
$198$		0			0
$199$	−7325.12	−	12687.5i	−0.184973	−	0.320383i	0.758594	−	0.651563i	$-0.225886\pi$
$199$	−7325.12	−	12687.5i	−0.184973	−	0.320383i	−0.943567	+	0.331180i	$0.892553\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	77561.3	−	13676.1i	1.88214	−	0.331873i
$204$		0			0
$205$	852.780	+	715.567i	0.0202922	+	0.0170272i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	686.951	+	1887.38i	0.0157265	+	0.0432083i
$210$		0			0
$211$	25955.0	−	21778.8i	0.582983	−	0.489180i	−0.302943	−	0.953009i	$-0.597969\pi$
$211$	25955.0	−	21778.8i	0.582983	−	0.489180i	0.885925	+	0.463828i	$0.153525\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	111142.i		−	2.40437i
$216$		0			0
$217$	−57383.2			−1.21861
$218$		0			0
$219$		0			0
$220$		0			0
$221$	34627.7	+	41267.6i	0.708987	+	0.844938i
$222$		0			0
$223$	27809.3	−	10121.8i	0.559218	−	0.203539i	−0.0469196	−	0.998899i	$-0.514940\pi$
$223$	27809.3	−	10121.8i	0.559218	−	0.203539i	0.606137	+	0.795360i	$0.292718\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8954.28	−	10671.3i	0.173772	−	0.207093i	−0.672128	−	0.740435i	$-0.734620\pi$
$227$	8954.28	−	10671.3i	0.173772	−	0.207093i	0.845900	+	0.533342i	$0.179064\pi$
$228$		0			0
$229$	4011.14	+	22748.3i	0.0764886	+	0.433788i	0.998871	+	0.0475097i	$0.0151285\pi$
$229$	4011.14	+	22748.3i	0.0764886	+	0.433788i	−0.922382	+	0.386278i	$0.873760\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12190.7	−	7038.30i	0.224552	−	0.129645i	−0.383504	−	0.923539i	$-0.625283\pi$
$233$	12190.7	−	7038.30i	0.224552	−	0.129645i	0.608056	+	0.793894i	$0.291950\pi$
$234$		0			0
$235$	−44220.5	+	76592.1i	−0.800733	+	1.38691i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12450.9	−	34208.6i	0.217974	−	0.598880i	−0.781719	−	0.623630i	$-0.785657\pi$
$239$	12450.9	−	34208.6i	0.217974	−	0.598880i	0.999694	+	0.0247508i	$0.00787923\pi$
$240$		0			0
$241$	−12755.6	+	72340.4i	−0.219617	+	1.24551i	0.653096	+	0.757275i	$0.273470\pi$
$241$	−12755.6	+	72340.4i	−0.219617	+	1.24551i	−0.872713	+	0.488234i	$0.837641\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−150997.	−	26624.9i	−2.51558	−	0.443564i
$246$		0			0
$247$	10454.0	+	3804.93i	0.171351	+	0.0623668i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	29488.8	+	17025.4i	0.468068	+	0.270239i	0.715431	−	0.698684i	$-0.246230\pi$
$251$	29488.8	+	17025.4i	0.468068	+	0.270239i	−0.247362	+	0.968923i	$0.579564\pi$
$252$		0			0
$253$	−3764.42	−	6520.16i	−0.0588107	−	0.101863i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−64456.2	+	11365.4i	−0.975884	+	0.172075i	−0.638777	−	0.769392i	$-0.720560\pi$
$257$	−64456.2	+	11365.4i	−0.975884	+	0.172075i	−0.337107	+	0.941466i	$0.609448\pi$
$258$		0			0
$259$	−58841.9	−	49374.2i	−0.877177	−	0.736039i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	21591.9	+	59323.2i	0.312161	+	0.857656i	0.992220	+	0.124498i	$0.0397321\pi$
$263$	21591.9	+	59323.2i	0.312161	+	0.857656i	−0.680059	+	0.733158i	$0.738046\pi$
$264$		0			0
$265$	48547.4	−	40736.1i	0.691312	−	0.580080i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	42398.0i		−	0.585924i	−0.956124	−	0.292962i	$-0.905359\pi$
$269$		−	42398.0i		−	0.585924i	0.956124	−	0.292962i	$-0.0946409\pi$
$270$		0			0
$271$	72484.3			0.986973			0.493487	−	0.869753i	$-0.335722\pi$
$271$	72484.3			0.986973			0.493487	+	0.869753i	$0.335722\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	57621.9	+	68671.1i	0.761943	+	0.908048i
$276$		0			0
$277$	−58035.1	+	21123.0i	−0.756365	+	0.275294i	−0.691281	−	0.722586i	$-0.742953\pi$
$277$	−58035.1	+	21123.0i	−0.756365	+	0.275294i	−0.0650831	+	0.997880i	$0.520731\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	40685.2	−	48486.8i	0.515257	−	0.614060i	−0.444195	−	0.895930i	$-0.646510\pi$
$281$	40685.2	−	48486.8i	0.515257	−	0.614060i	0.959453	+	0.281870i	$0.0909547\pi$
$282$		0			0
$283$	2623.64	+	14879.4i	0.0327591	+	0.185786i	0.996796	−	0.0799809i	$-0.0254859\pi$
$283$	2623.64	+	14879.4i	0.0327591	+	0.185786i	−0.964037	+	0.265767i	$0.914375\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1568.61	−	905.636i	0.0190437	−	0.0109949i
$288$		0			0
$289$	−27988.0	+	48476.7i	−0.335102	+	0.580413i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−18253.5	+	50151.1i	−0.212623	+	0.584178i	−0.999456	−	0.0329889i	$-0.989497\pi$
$293$	−18253.5	+	50151.1i	−0.212623	+	0.584178i	0.786832	+	0.617167i	$0.211720\pi$
$294$		0			0
$295$	16276.2	−	92306.8i	0.187029	−	1.06069i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−41067.7	−	7241.35i	−0.459365	−	0.0809985i
$300$		0			0
$301$	−169928.	−	61848.7i	−1.87556	−	0.682649i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−61862.1	−	35716.1i	−0.665005	−	0.383941i
$306$		0			0
$307$	16543.4	+	28653.9i	0.175528	+	0.304024i	0.940344	−	0.340225i	$-0.110503\pi$
$307$	16543.4	+	28653.9i	0.175528	+	0.304024i	−0.764816	+	0.644249i	$0.777170\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−76370.1	+	13466.1i	−0.789592	+	0.139226i	−0.553880	−	0.832597i	$-0.686853\pi$
$311$	−76370.1	+	13466.1i	−0.789592	+	0.139226i	−0.235712	+	0.971823i	$0.575742\pi$
$312$		0			0
$313$	−2179.02	−	1828.42i	−0.0222420	−	0.0186632i	0.631599	−	0.775295i	$-0.282399\pi$
$313$	−2179.02	−	1828.42i	−0.0222420	−	0.0186632i	−0.653841	+	0.756632i	$0.726843\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	25997.9	+	71428.7i	0.258714	+	0.710811i	0.999247	+	0.0387913i	$0.0123507\pi$
$317$	25997.9	+	71428.7i	0.258714	+	0.710811i	−0.740533	+	0.672020i	$0.765427\pi$
$318$		0			0
$319$	−46813.0	+	39280.7i	−0.460029	+	0.386010i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5688.29i		−	0.0545226i
$324$		0			0
$325$	496526.			4.70084
$326$		0			0
$327$		0			0
$328$		0			0
$329$	92495.7	+	110232.i	0.854535	+	1.01840i
$330$		0			0
$331$	−70097.5	+	25513.4i	−0.639803	+	0.232869i	−0.641493	−	0.767129i	$-0.721684\pi$
$331$	−70097.5	+	25513.4i	−0.639803	+	0.232869i	0.00168942	+	0.999999i	$0.499462\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5155.93	−	6144.60i	0.0459428	−	0.0547525i
$336$		0			0
$337$	−9149.89	−	51891.6i	−0.0805668	−	0.456917i	−0.998225	−	0.0595474i	$-0.981034\pi$
$337$	−9149.89	−	51891.6i	−0.0805668	−	0.456917i	0.917659	−	0.397370i	$-0.130077\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	38559.8	−	22262.5i	0.331608	−	0.191454i
$342$		0			0
$343$	−34066.7	+	59005.2i	−0.289562	+	0.501536i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−42269.4	+	116134.i	−0.351048	+	0.964498i	0.630986	+	0.775794i	$0.282651\pi$
$347$	−42269.4	+	116134.i	−0.351048	+	0.964498i	−0.982034	+	0.188703i	$0.939572\pi$
$348$		0			0
$349$	−33080.1	+	187607.i	−0.271592	+	1.54027i	0.477992	+	0.878364i	$0.341365\pi$
$349$	−33080.1	+	187607.i	−0.271592	+	1.54027i	−0.749584	+	0.661909i	$0.769746\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	226109.	+	39869.0i	1.81454	+	0.319953i	0.974809	−	0.223043i	$-0.0715989\pi$
$353$	226109.	+	39869.0i	1.81454	+	0.319953i	0.839735	+	0.542996i	$0.182710\pi$
$354$		0			0
$355$	306349.	+	111502.i	2.43086	+	0.884760i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10865.6	−	6273.29i	−0.0843076	−	0.0486750i	0.457254	−	0.889336i	$-0.348833\pi$
$359$	−10865.6	−	6273.29i	−0.0843076	−	0.0486750i	−0.541561	+	0.840661i	$0.682167\pi$
$360$		0			0
$361$	64573.2	+	111844.i	0.495493	+	0.858219i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−248446.	+	43807.8i	−1.86486	+	0.328826i
$366$		0			0
$367$	29799.7	+	25004.9i	0.221248	+	0.185649i	0.746674	−	0.665190i	$-0.231650\pi$
$367$	29799.7	+	25004.9i	0.221248	+	0.185649i	−0.525426	+	0.850839i	$0.676094\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−35266.7	−	96894.4i	−0.256222	−	0.703964i
$372$		0			0
$373$	−8604.07	+	7219.67i	−0.0618424	+	0.0518919i	−0.673185	−	0.739474i	$-0.735074\pi$
$373$	−8604.07	+	7219.67i	−0.0618424	+	0.0518919i	0.611343	+	0.791366i	$0.290630\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			338481.i			2.38150i
$378$		0			0
$379$	−15774.6			−0.109819			−0.0549097	−	0.998491i	$-0.517487\pi$
$379$	−15774.6			−0.109819			−0.0549097	+	0.998491i	$0.517487\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	119843.	+	142824.i	0.816989	+	0.973650i	0.999955	−	0.00946947i	$-0.00301427\pi$
$383$	119843.	+	142824.i	0.816989	+	0.973650i	−0.182966	+	0.983119i	$0.558570\pi$
$384$		0			0
$385$	193058.	−	70267.2i	1.30246	−	0.474058i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−130987.	+	156104.i	−0.865624	+	1.03161i	0.133553	+	0.991042i	$0.457361\pi$
$389$	−130987.	+	156104.i	−0.865624	+	1.03161i	−0.999177	+	0.0405684i	$0.987083\pi$
$390$		0			0
$391$	3702.59	+	20998.4i	0.0242188	+	0.137351i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	51140.4	−	29525.9i	0.327770	−	0.189238i
$396$		0			0
$397$	−63046.4	+	109200.i	−0.400018	+	0.692851i	−0.993728	−	0.111828i	$-0.964329\pi$
$397$	−63046.4	+	109200.i	−0.400018	+	0.692851i	0.593710	+	0.804679i	$0.297663\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11690.9	+	32120.6i	−0.0727044	+	0.199754i	−0.970722	−	0.240206i	$-0.922785\pi$
$401$	−11690.9	+	32120.6i	−0.0727044	+	0.199754i	0.898018	+	0.439960i	$0.145007\pi$
$402$		0			0
$403$	42824.8	−	242872.i	0.263685	−	1.49543i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	58695.2	+	10349.6i	0.354335	+	0.0624788i
$408$		0			0
$409$	−142388.	−	51825.1i	−0.851193	−	0.309809i	−0.120666	−	0.992693i	$-0.538503\pi$
$409$	−142388.	−	51825.1i	−0.851193	−	0.309809i	−0.730527	+	0.682884i	$0.760725\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−132073.	−	76252.3i	−0.774308	−	0.447047i
$414$		0			0
$415$	−275651.	−	477442.i	−1.60053	−	2.77220i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−91436.8	+	16122.8i	−0.520827	+	0.0918358i	−0.427880	−	0.903835i	$-0.640740\pi$
$419$	−91436.8	+	16122.8i	−0.520827	+	0.0918358i	−0.0929462	+	0.995671i	$0.529628\pi$
$420$		0			0
$421$	140455.	+	117856.i	0.792452	+	0.664946i	0.946351	−	0.323140i	$-0.104739\pi$
$421$	140455.	+	117856.i	0.792452	+	0.664946i	−0.153899	+	0.988087i	$0.549183\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−86832.0	−	238569.i	−0.480731	−	1.32080i
$426$		0			0
$427$	−89032.5	+	74707.1i	−0.488307	+	0.409738i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	192840.i		−	1.03811i	−0.854741	−	0.519055i	$-0.826284\pi$
$431$		−	192840.i		−	1.03811i	0.854741	−	0.519055i	$-0.173716\pi$
$432$		0			0
$433$	−316690.			−1.68911			−0.844555	−	0.535469i	$-0.820135\pi$
$433$	−316690.			−1.68911			−0.844555	+	0.535469i	$0.820135\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2830.37	+	3373.10i	0.0148211	+	0.0176631i
$438$		0			0
$439$	−46022.1	+	16750.7i	−0.238802	+	0.0869167i	−0.458649	−	0.888618i	$-0.651666\pi$
$439$	−46022.1	+	16750.7i	−0.238802	+	0.0869167i	0.219847	+	0.975534i	$0.429444\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−244104.	+	290912.i	−1.24385	+	1.48236i	−0.428342	+	0.903616i	$0.640902\pi$
$443$	−244104.	+	290912.i	−1.24385	+	1.48236i	−0.815508	+	0.578746i	$0.803542\pi$
$444$		0			0
$445$	55034.6	+	312117.i	0.277918	+	1.57615i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−80703.0	+	46593.9i	−0.400311	+	0.231119i	−0.686618	−	0.727018i	$-0.740905\pi$
$449$	−80703.0	+	46593.9i	−0.400311	+	0.231119i	0.286307	+	0.958138i	$0.407572\pi$
$450$		0			0
$451$	−702.704	+	1217.12i	−0.00345477	+	0.00598384i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	389202.	−	1.06932e6i	1.87997	−	5.16519i
$456$		0			0
$457$	−4588.04	+	26020.1i	−0.0219682	+	0.124588i	−0.993820	−	0.111006i	$-0.964593\pi$
$457$	−4588.04	+	26020.1i	−0.0219682	+	0.124588i	0.971851	+	0.235594i	$0.0757037\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	35162.5	+	6200.09i	0.165454	+	0.0291740i	0.255761	−	0.966740i	$-0.417674\pi$
$461$	35162.5	+	6200.09i	0.165454	+	0.0291740i	−0.0903074	+	0.995914i	$0.528785\pi$
$462$		0			0
$463$	102480.	+	37299.7i	0.478055	+	0.173998i	0.569798	−	0.821785i	$-0.307022\pi$
$463$	102480.	+	37299.7i	0.478055	+	0.173998i	−0.0917430	+	0.995783i	$0.529244\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	190248.	+	109840.i	0.872341	+	0.503646i	0.868125	−	0.496345i	$-0.165325\pi$
$467$	190248.	+	109840.i	0.872341	+	0.503646i	0.00421546	+	0.999991i	$0.498658\pi$
$468$		0			0
$469$	−6525.45	−	11302.4i	−0.0296664	−	0.0513837i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	138181.	−	24365.1i	0.617627	−	0.108904i
$474$		0			0
$475$	−40162.6	−	33700.4i	−0.178006	−	0.149365i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	83633.5	+	229781.i	0.364510	+	1.00148i	0.977416	+	0.211326i	$0.0677782\pi$
$479$	83633.5	+	229781.i	0.364510	+	1.00148i	−0.612906	+	0.790156i	$0.710000\pi$
$480$		0			0
$481$	252887.	−	212198.i	1.09304	−	0.917171i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			320453.i			1.36233i
$486$		0			0
$487$	−325279.			−1.37151			−0.685753	−	0.727835i	$-0.740527\pi$
$487$	−325279.			−1.37151			−0.685753	+	0.727835i	$0.740527\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−175636.	−	209315.i	−0.728537	−	0.868237i	0.266893	−	0.963726i	$-0.414003\pi$
$491$	−175636.	−	209315.i	−0.728537	−	0.868237i	−0.995430	+	0.0954893i	$0.969558\pi$
$492$		0			0
$493$	162632.	−	59193.2i	0.669132	−	0.243544i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	340956.	−	406336.i	1.38034	−	1.64503i
$498$		0			0
$499$	−28154.6	−	159673.i	−0.113070	−	0.641253i	−0.987688	−	0.156439i	$-0.949998\pi$
$499$	−28154.6	−	159673.i	−0.113070	−	0.641253i	0.874617	−	0.484814i	$-0.161113\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−399222.	+	230491.i	−1.57790	+	0.910999i	−0.582744	+	0.812656i	$0.698021\pi$
$503$	−399222.	+	230491.i	−1.57790	+	0.910999i	−0.995153	+	0.0983431i	$0.968646\pi$
$504$		0			0
$505$	−66949.3	+	115960.i	−0.262521	+	0.454699i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	23099.1	−	63464.4i	0.0891580	−	0.244960i	−0.887097	−	0.461582i	$-0.847282\pi$
$509$	23099.1	−	63464.4i	0.0891580	−	0.244960i	0.976255	+	0.216623i	$0.0695041\pi$
$510$		0			0
$511$	−71277.5	+	404235.i	−0.272967	+	1.54807i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	295971.	+	52187.7i	1.11593	+	0.196768i
$516$		0			0
$517$	−104920.	−	38187.8i	−0.392534	−	0.142871i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	69074.8	+	39880.3i	0.254474	+	0.146921i	0.621811	−	0.783167i	$-0.286397\pi$
$521$	69074.8	+	39880.3i	0.254474	+	0.146921i	−0.367337	+	0.930088i	$0.619730\pi$
$522$		0			0
$523$	40324.6	+	69844.2i	0.147423	+	0.255345i	0.930274	−	0.366864i	$-0.119569\pi$
$523$	40324.6	+	69844.2i	0.147423	+	0.255345i	−0.782851	+	0.622209i	$0.786235\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−124183.	+	21896.8i	−0.447138	+	0.0788425i
$528$		0			0
$529$	201727.	+	169269.i	0.720862	+	0.604875i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2662.42	+	7314.93i	0.00937176	+	0.0257487i
$534$		0			0
$535$	−431882.	+	362392.i	−1.50889	+	1.26611i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	193570.i		−	0.666285i
$540$		0			0
$541$	−189478.			−0.647388			−0.323694	−	0.946162i	$-0.604925\pi$
$541$	−189478.			−0.647388			−0.323694	+	0.946162i	$0.604925\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	42942.2	+	51176.5i	0.144574	+	0.172297i
$546$		0			0
$547$	71091.7	−	25875.3i	0.237599	−	0.0864789i	−0.220476	−	0.975392i	$-0.570761\pi$
$547$	71091.7	−	25875.3i	0.237599	−	0.0864789i	0.458075	+	0.888913i	$0.348539\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22973.5	−	27378.8i	0.0756700	−	0.0901800i
$552$		0			0
$553$	−16684.1	−	94620.5i	−0.0545574	−	0.309410i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	241397.	−	139371.i	0.778076	−	0.449222i	−0.0576722	−	0.998336i	$-0.518368\pi$
$557$	241397.	−	139371.i	0.778076	−	0.449222i	0.835748	+	0.549113i	$0.185035\pi$
$558$		0			0
$559$	388588.	−	673054.i	1.24356	−	2.15390i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	144626.	−	397355.i	0.456277	−	1.25361i	−0.471960	−	0.881620i	$-0.656453\pi$
$563$	144626.	−	397355.i	0.456277	−	1.25361i	0.928237	−	0.371990i	$-0.121324\pi$
$564$		0			0
$565$	170104.	−	964707.i	0.532865	−	3.02203i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−73347.1	−	12933.1i	−0.226547	−	0.0399463i	0.0592222	−	0.998245i	$-0.481138\pi$
$569$	−73347.1	−	12933.1i	−0.226547	−	0.0399463i	−0.285769	+	0.958298i	$0.592249\pi$
$570$		0			0
$571$	158268.	+	57604.7i	0.485423	+	0.176679i	0.573126	−	0.819467i	$-0.305731\pi$
$571$	158268.	+	57604.7i	0.485423	+	0.176679i	−0.0877033	+	0.996147i	$0.527953\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	170197.	+	98263.3i	0.514774	+	0.297205i
$576$		0			0
$577$	−237397.	−	411184.i	−0.713057	−	1.23505i	−0.963704	−	0.266972i	$-0.913977\pi$
$577$	−237397.	−	411184.i	−0.713057	−	1.23505i	0.250647	−	0.968078i	$-0.419356\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−883368.	+	155762.i	−2.61691	+	0.461433i
$582$		0			0
$583$	61289.4	+	51427.9i	0.180322	+	0.151308i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−88461.3	−	243045.i	−0.256730	−	0.705361i	−0.999364	−	0.0356627i	$-0.988646\pi$
$587$	−88461.3	−	243045.i	−0.256730	−	0.705361i	0.742634	−	0.669698i	$-0.233576\pi$
$588$		0			0
$589$	−19948.3	+	16738.6i	−0.0575009	+	0.0482490i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	394190.i		−	1.12098i	−0.828162	−	0.560488i	$-0.810613\pi$
$593$		−	394190.i		−	1.12098i	0.828162	−	0.560488i	$-0.189387\pi$
$594$		0			0
$595$	−581847.			−1.64352
$596$		0			0
$597$		0			0
$598$		0			0
$599$	335953.	+	400373.i	0.936321	+	1.11586i	0.993076	+	0.117475i	$0.0374801\pi$
$599$	335953.	+	400373.i	0.936321	+	1.11586i	−0.0567553	+	0.998388i	$0.518075\pi$
$600$		0			0
$601$	−196127.	+	71384.4i	−0.542986	+	0.197631i	−0.598927	−	0.800803i	$-0.704406\pi$
$601$	−196127.	+	71384.4i	−0.542986	+	0.197631i	0.0559415	+	0.998434i	$0.482184\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	334382.	−	398501.i	0.913549	−	1.08873i
$606$		0			0
$607$	44185.9	+	250591.i	0.119924	+	0.680123i	0.984194	+	0.177095i	$0.0566699\pi$
$607$	44185.9	+	250591.i	0.119924	+	0.680123i	−0.864270	+	0.503029i	$0.832219\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−535581.	+	309218.i	−1.43464	+	0.828290i
$612$		0			0
$613$	328178.	−	568422.i	0.873351	−	1.51269i	0.0148429	−	0.999890i	$-0.495275\pi$
$613$	328178.	−	568422.i	0.873351	−	1.51269i	0.858509	−	0.512799i	$-0.171391\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−32385.1	+	88977.4i	−0.0850697	+	0.233727i	−0.974932	−	0.222501i	$-0.928578\pi$
$617$	−32385.1	+	88977.4i	−0.0850697	+	0.233727i	0.889863	+	0.456229i	$0.150800\pi$
$618$		0			0
$619$	15188.9	−	86140.4i	0.0396410	−	0.224815i	−0.958551	−	0.284921i	$-0.908033\pi$
$619$	15188.9	−	86140.4i	0.0396410	−	0.224815i	0.998192	+	0.0601057i	$0.0191438\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	507830.	+	89544.1i	1.30840	+	0.230707i
$624$		0			0
$625$	−933399.	−	339729.i	−2.38950	−	0.869707i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−146181.	−	84397.4i	−0.369478	−	0.213318i
$630$		0			0
$631$	249361.	+	431906.i	0.626281	+	1.08475i	0.988292	+	0.152577i	$0.0487572\pi$
$631$	249361.	+	431906.i	0.626281	+	1.08475i	−0.362010	+	0.932174i	$0.617909\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−930081.	+	163998.i	−2.30661	+	0.406717i
$636$		0			0
$637$	−821322.	−	689171.i	−2.02411	−	1.69843i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−158883.	−	436528.i	−0.386689	−	1.06242i	−0.968482	−	0.249083i	$-0.919871\pi$
$641$	−158883.	−	436528.i	−0.386689	−	1.06242i	0.581793	−	0.813337i	$-0.302352\pi$
$642$		0			0
$643$	−326693.	+	274128.i	−0.790166	+	0.663028i	−0.945787	−	0.324789i	$-0.894707\pi$
$643$	−326693.	+	274128.i	−0.790166	+	0.663028i	0.155620	+	0.987817i	$0.450262\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			560204.i			1.33825i	0.743149	+	0.669125i	$0.233331\pi$
$647$			560204.i			1.33825i	−0.743149	+	0.669125i	$0.766669\pi$
$648$		0			0
$649$	118332.			0.280939
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−68768.6	−	81955.3i	−0.161274	−	0.192199i	0.679356	−	0.733809i	$-0.262259\pi$
$653$	−68768.6	−	81955.3i	−0.161274	−	0.192199i	−0.840630	+	0.541610i	$0.817815\pi$
$654$		0			0
$655$	1.11668e6	−	406440.i	2.60284	−	0.947356i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	154521.	−	184151.i	0.355809	−	0.424036i	−0.558215	−	0.829696i	$-0.688514\pi$
$659$	154521.	−	184151.i	0.355809	−	0.424036i	0.914024	+	0.405660i	$0.132958\pi$
$660$		0			0
$661$	70851.2	+	401817.i	0.162160	+	0.919656i	0.951944	+	0.306271i	$0.0990814\pi$
$661$	70851.2	+	401817.i	0.162160	+	0.919656i	−0.789784	+	0.613385i	$0.789808\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−104059.	+	60078.5i	−0.235308	+	0.135855i
$666$		0			0
$667$	−66985.9	+	116023.i	−0.150568	+	0.260791i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	30843.6	−	84742.1i	0.0685046	−	0.188215i
$672$		0			0
$673$	−148100.	+	839918.i	−0.326983	+	1.85441i	0.168371	+	0.985724i	$0.446149\pi$
$673$	−148100.	+	839918.i	−0.326983	+	1.85441i	−0.495355	+	0.868691i	$0.664962\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	425195.	+	74973.3i	0.927706	+	0.163580i	0.617033	−	0.786937i	$-0.288334\pi$
$677$	425195.	+	74973.3i	0.927706	+	0.163580i	0.310673	+	0.950517i	$0.399446\pi$
$678$		0			0
$679$	489949.	+	178327.i	1.06270	+	0.386792i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−527711.	−	304674.i	−1.13124	−	0.653122i	−0.186993	−	0.982361i	$-0.559874\pi$
$683$	−527711.	−	304674.i	−1.13124	−	0.653122i	−0.944246	+	0.329240i	$0.893208\pi$
$684$		0			0
$685$	469362.	+	812959.i	1.00029	+	1.73256i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	436420.	−	76952.6i	0.919319	−	0.162101i
$690$		0			0
$691$	249223.	+	209123.i	0.521954	+	0.437971i	0.865312	−	0.501233i	$-0.167120\pi$
$691$	249223.	+	209123.i	0.521954	+	0.437971i	−0.343359	+	0.939204i	$0.611565\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	579637.	+	1.59254e6i	1.20001	+	3.29701i
$696$		0			0
$697$	3049.05	−	2558.45i	0.00627623	−	0.00526638i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	110954.i		−	0.225791i	−0.993607	−	0.112896i	$-0.963987\pi$
$701$		−	110954.i		−	0.225791i	0.993607	−	0.112896i	$-0.0360126\pi$
$702$		0			0
$703$	−34857.7			−0.0705323
$704$		0			0
$705$		0			0
$706$		0			0
$707$	140037.	+	166890.i	0.280159	+	0.333881i
$708$		0			0
$709$	−369839.	+	134610.i	−0.735733	+	0.267785i	−0.682589	−	0.730802i	$-0.739146\pi$
$709$	−369839.	+	134610.i	−0.735733	+	0.267785i	−0.0531432	+	0.998587i	$0.516924\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	62744.0	−	74775.4i	0.123422	−	0.147089i
$714$		0			0
$715$	153325.	+	869547.i	0.299916	+	1.70091i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	355152.	−	205047.i	0.687000	−	0.396640i	−0.115487	−	0.993309i	$-0.536843\pi$
$719$	355152.	−	205047.i	0.687000	−	0.396640i	0.802487	+	0.596669i	$0.203510\pi$
$720$		0			0
$721$	244494.	−	423477.i	0.470325	−	0.814627i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	545579.	−	1.49897e6i	1.03796	−	2.85178i
$726$		0			0
$727$	136857.	−	776156.i	0.258940	−	1.46852i	−0.526814	−	0.849981i	$-0.676613\pi$
$727$	136857.	−	776156.i	0.258940	−	1.46852i	0.785753	−	0.618540i	$-0.212275\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−391342.	−	69004.2i	−0.732355	−	0.129134i
$732$		0			0
$733$	322242.	+	117287.i	0.599756	+	0.218293i	0.624015	−	0.781412i	$-0.285500\pi$
$733$	322242.	+	117287.i	0.599756	+	0.218293i	−0.0242588	+	0.999706i	$0.507723\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8769.81	+	5063.25i	0.0161456	+	0.00932168i
$738$		0			0
$739$	−24921.7	−	43165.7i	−0.0456341	−	0.0790406i	0.842306	−	0.538999i	$-0.181197\pi$
$739$	−24921.7	−	43165.7i	−0.0456341	−	0.0790406i	−0.887940	+	0.459959i	$0.847864\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	114120.	−	20122.5i	0.206721	−	0.0364505i	−0.0693285	−	0.997594i	$-0.522086\pi$
$743$	114120.	−	20122.5i	0.206721	−	0.0364505i	0.276050	+	0.961143i	$0.410975\pi$
$744$		0			0
$745$	915415.	+	768124.i	1.64932	+	1.38394i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	313735.	+	861981.i	0.559242	+	1.53650i
$750$		0			0
$751$	−389838.	+	327113.i	−0.691201	+	0.579986i	−0.919255	−	0.393662i	$-0.871208\pi$
$751$	−389838.	+	327113.i	−0.691201	+	0.579986i	0.228054	+	0.973648i	$0.426764\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			896418.i			1.57259i
$756$		0			0
$757$	−746088.			−1.30196			−0.650981	−	0.759094i	$-0.725642\pi$
$757$	−746088.			−1.30196			−0.650981	+	0.759094i	$0.725642\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	552859.	+	658871.i	0.954652	+	1.13771i	0.990384	+	0.138348i	$0.0441793\pi$
$761$	552859.	+	658871.i	0.954652	+	1.13771i	−0.0357320	+	0.999361i	$0.511376\pi$
$762$		0			0
$763$	102142.	−	37176.6i	0.175450	−	0.0638587i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	421299.	−	502085.i	0.716144	−	0.853467i
$768$		0			0
$769$	−98528.6	−	558783.i	−0.166613	−	0.944911i	−0.947385	−	0.320095i	$-0.896285\pi$
$769$	−98528.6	−	558783.i	−0.166613	−	0.944911i	0.780772	−	0.624816i	$-0.214826\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	464118.	−	267958.i	0.776728	−	0.448444i	−0.0585413	−	0.998285i	$-0.518645\pi$
$773$	464118.	−	267958.i	0.776728	−	0.448444i	0.835270	+	0.549841i	$0.185312\pi$
$774$		0			0
$775$	−581122.	+	1.00653e6i	−0.967530	+	1.67581i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	281.126	−	772.388i	0.000463262	−	0.00127280i
$780$		0			0
$781$	−71469.4	+	405323.i	−0.117171	+	0.664507i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.54056e6	−	271643.i	−2.50000	−	0.440818i
$786$		0			0
$787$	963736.	+	350771.i	1.55600	+	0.566336i	0.969815	−	0.243843i	$-0.0784082\pi$
$787$	963736.	+	350771.i	1.55600	+	0.566336i	0.586182	+	0.810180i	$0.300630\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.38031e6	−	796920.i	−2.20609	−	1.27368i
$792$		0			0
$793$	−249750.	−	432579.i	−0.397154	−	0.687890i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−942058.	+	166110.i	−1.48307	+	0.261505i	−0.855803	−	0.517302i	$-0.826936\pi$
$797$	−942058.	+	166110.i	−1.48307	+	0.261505i	−0.627264	+	0.778807i	$0.715825\pi$
$798$		0			0
$799$	242234.	+	203258.i	0.379438	+	0.318386i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−108931.	−	299286.i	−0.168936	−	0.464147i
$804$		0			0
$805$	345030.	−	289514.i	0.532433	−	0.446764i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			747059.i			1.14145i	0.821141	+	0.570726i	$0.193338\pi$
$809$			747059.i			1.14145i	−0.821141	+	0.570726i	$0.806662\pi$
$810$		0			0
$811$	−320839.			−0.487805			−0.243902	−	0.969800i	$-0.578428\pi$
$811$	−320839.			−0.487805			−0.243902	+	0.969800i	$0.578428\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	134205.	+	159940.i	0.202048	+	0.240792i
$816$		0			0
$817$	−77113.5	+	28067.0i	−0.115528	+	0.0420487i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−255352.	+	304317.i	−0.378838	+	0.451482i	−0.921447	−	0.388504i	$-0.872992\pi$
$821$	−255352.	+	304317.i	−0.378838	+	0.451482i	0.542609	+	0.839985i	$0.317437\pi$
$822$		0			0
$823$	−154568.	−	876600.i	−0.228203	−	1.29420i	−0.856467	−	0.516202i	$-0.827345\pi$
$823$	−154568.	−	876600.i	−0.228203	−	1.29420i	0.628264	−	0.778000i	$-0.283766\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−694642.	+	401052.i	−1.01566	+	0.586394i	−0.912845	−	0.408306i	$-0.866120\pi$
$827$	−694642.	+	401052.i	−1.01566	+	0.586394i	−0.102819	+	0.994700i	$0.532786\pi$
$828$		0			0
$829$	255138.	−	441913.i	0.371250	−	0.643024i	−0.618508	−	0.785779i	$-0.712263\pi$
$829$	255138.	−	441913.i	0.371250	−	0.643024i	0.989758	+	0.142754i	$0.0455959\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−187498.	+	515147.i	−0.270213	+	0.742405i
$834$		0			0
$835$	60550.7	−	343400.i	0.0868453	−	0.492524i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−419091.	−	73897.0i	−0.595366	−	0.104979i	−0.132158	−	0.991229i	$-0.542191\pi$
$839$	−419091.	−	73897.0i	−0.595366	−	0.104979i	−0.463208	+	0.886250i	$0.653302\pi$
$840$		0			0
$841$	357216.	+	130016.i	0.505055	+	0.183825i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.08725e6	+	1.78242e6i	4.32372	+	2.49630i
$846$		0			0
$847$	−423200.	−	733004.i	−0.589901	−	1.02174i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	128678.	−	22689.4i	0.177683	−	0.0313302i
$852$		0			0
$853$	−436922.	−	366622.i	−0.600491	−	0.503872i	0.291113	−	0.956689i	$-0.405975\pi$
$853$	−436922.	−	366622.i	−0.600491	−	0.503872i	−0.891603	+	0.452817i	$0.850419\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−100509.	−	276146.i	−0.136849	−	0.375990i	0.852271	−	0.523101i	$-0.175225\pi$
$857$	−100509.	−	276146.i	−0.136849	−	0.375990i	−0.989120	+	0.147111i	$0.953003\pi$
$858$		0			0
$859$	970270.	−	814153.i	1.31494	−	1.10337i	0.327590	−	0.944820i	$-0.393764\pi$
$859$	970270.	−	814153.i	1.31494	−	1.10337i	0.987351	−	0.158547i	$-0.0506808\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			247377.i			0.332153i	0.986113	+	0.166076i	$0.0531098\pi$
$863$			247377.i			0.332153i	−0.986113	+	0.166076i	$0.946890\pi$
$864$		0			0
$865$	1.57071e6			2.09925
$866$		0			0
$867$		0			0
$868$		0			0
$869$	47920.4	+	57109.3i	0.0634571	+	0.0756253i
$870$		0			0
$871$	52706.8	−	19183.7i	0.0694753	−	0.0252869i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.03874e6	+	2.42967e6i	−2.66284	+	3.17345i
$876$		0			0
$877$	59522.7	+	337570.i	0.0773898	+	0.438899i	0.998741	+	0.0501677i	$0.0159756\pi$
$877$	59522.7	+	337570.i	0.0773898	+	0.438899i	−0.921351	+	0.388732i	$0.872913\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−65862.4	+	38025.7i	−0.0848566	+	0.0489920i	−0.541828	−	0.840489i	$-0.682268\pi$
$881$	−65862.4	+	38025.7i	−0.0848566	+	0.0489920i	0.456971	+	0.889481i	$0.348934\pi$
$882$		0			0
$883$	−219060.	+	379423.i	−0.280958	+	0.486634i	−0.971621	−	0.236543i	$-0.923985\pi$
$883$	−219060.	+	379423.i	−0.280958	+	0.486634i	0.690663	+	0.723177i	$0.257319\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−91782.8	+	252171.i	−0.116658	+	0.320515i	−0.984255	−	0.176752i	$-0.943441\pi$
$887$	−91782.8	+	252171.i	−0.116658	+	0.320515i	0.867597	+	0.497267i	$0.165663\pi$
$888$		0			0
$889$	−266833.	+	1.51329e6i	−0.337627	+	1.91478i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	64309.0	+	11339.4i	0.0806434	+	0.0142196i
$894$		0			0
$895$	−783176.	−	285053.i	−0.977717	−	0.355860i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−686152.	−	396150.i	−0.848987	−	0.490163i
$900$		0			0
$901$	−113295.	−	196232.i	−0.139560	−	0.241724i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.51741e6	+	267559.i	−1.85270	+	0.326680i
$906$		0			0
$907$	805157.	+	675607.i	0.978737	+	0.821258i	0.983898	−	0.178729i	$-0.0571986\pi$
$907$	805157.	+	675607.i	0.978737	+	0.821258i	−0.00516164	+	0.999987i	$0.501643\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−533448.	−	1.46564e6i	−0.642770	−	1.76600i	−0.642846	−	0.765995i	$-0.722246\pi$
$911$	−533448.	−	1.46564e6i	−0.642770	−	1.76600i	7.62429e−5	−	1.00000i	$-0.499976\pi$
$912$		0			0
$913$	533167.	−	447380.i	0.639619	−	0.536704i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1.93350e6i		−	2.29936i
$918$		0			0
$919$	508174.			0.601702			0.300851	−	0.953671i	$-0.402729\pi$
$919$	508174.			0.601702			0.300851	+	0.953671i	$0.402729\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.46534e6	+	1.74633e6i	1.72003	+	2.04985i
$924$		0			0
$925$	−1.46195e6	+	532105.i	−1.70863	+	0.621890i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	752601.	−	896915.i	0.872034	−	1.03925i	−0.126845	−	0.991923i	$-0.540485\pi$
$929$	752601.	−	896915.i	0.872034	−	1.03925i	0.998879	−	0.0473274i	$-0.0150704\pi$
$930$		0			0
$931$	19658.7	+	111490.i	0.0226807	+	0.128629i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	390983.	−	225734.i	0.447234	−	0.258211i
$936$		0			0
$937$	−676172.	+	1.17116e6i	−0.770154	+	1.33395i	0.167324	+	0.985902i	$0.446488\pi$
$937$	−676172.	+	1.17116e6i	−0.770154	+	1.33395i	−0.937478	+	0.348044i	$0.886846\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	428606.	−	1.17759e6i	0.484038	−	1.32988i	−0.421966	−	0.906612i	$-0.638660\pi$
$941$	428606.	−	1.17759e6i	0.484038	−	1.32988i	0.906003	−	0.423271i	$-0.139118\pi$
$942$		0			0
$943$	−535.025	+	3034.28i	−0.000601659	+	0.00341218i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.33018e6	+	234546.i	1.48323	+	0.261534i	0.855869	−	0.517193i	$-0.173023\pi$
$947$	1.33018e6	+	234546.i	1.48323	+	0.261534i	0.627363	+	0.778727i	$0.284134\pi$
$948$		0			0
$949$	−1.65771e6	−	603357.i	−1.84067	−	0.669949i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	614026.	+	354508.i	0.676085	+	0.390338i	0.798378	−	0.602156i	$-0.205692\pi$
$953$	614026.	+	354508.i	0.676085	+	0.390338i	−0.122294	+	0.992494i	$0.539025\pi$
$954$		0			0
$955$	1.00654e6	+	1.74337e6i	1.10363	+	1.91154i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.50415e6	−	265222.i	1.63551	−	0.288384i
$960$		0			0
$961$	−265242.	−	222564.i	−0.287207	−	0.240995i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−546431.	−	1.50131e6i	−0.586787	−	1.61218i
$966$		0			0
$967$	543985.	−	456457.i	0.581746	−	0.488143i	−0.303774	−	0.952744i	$-0.598247\pi$
$967$	543985.	−	456457.i	0.581746	−	0.488143i	0.885520	+	0.464601i	$0.153802\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.18416e6i		−	1.25595i	−0.778234	−	0.627975i	$-0.783884\pi$
$971$		−	1.18416e6i		−	1.25595i	0.778234	−	0.627975i	$-0.216116\pi$
$972$		0			0
$973$	2.75743e6			2.91259
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−619350.	−	738113.i	−0.648855	−	0.773275i	0.336886	−	0.941545i	$-0.390626\pi$
$977$	−619350.	−	738113.i	−0.648855	−	0.773275i	−0.985741	+	0.168271i	$0.946182\pi$
$978$		0			0
$979$	−375986.	+	136848.i	−0.392289	+	0.142781i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	325530.	−	387951.i	0.336886	−	0.401486i	−0.570831	−	0.821068i	$-0.693379\pi$
$983$	325530.	−	387951.i	0.336886	−	0.401486i	0.907717	+	0.419582i	$0.137823\pi$
$984$		0			0
$985$	−405304.	−	2.29859e6i	−0.417742	−	2.36913i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	266397.	−	153804.i	0.272356	−	0.157245i
$990$		0			0
$991$	−212890.	+	368736.i	−0.216774	+	0.375464i	−0.953820	−	0.300379i	$-0.902887\pi$
$991$	−212890.	+	368736.i	−0.216774	+	0.375464i	0.737046	+	0.675843i	$0.236220\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−232589.	+	639034.i	−0.234933	+	0.645473i
$996$		0			0
$997$	13863.8	−	78625.3i	0.0139473	−	0.0790992i	−0.977040	−	0.213058i	$-0.931658\pi$
$997$	13863.8	−	78625.3i	0.0139473	−	0.0790992i	0.990987	+	0.133959i	$0.0427689\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.k.a.89.1		72
3.2	odd	2		108.5.k.a.29.9	✓	72
27.13	even	9		108.5.k.a.41.9	yes	72
27.14	odd	18	inner	324.5.k.a.233.1		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.9	✓	72	3.2	odd	2
108.5.k.a.41.9	yes	72	27.13	even	9
324.5.k.a.89.1		72	1.1	even	1	trivial
324.5.k.a.233.1		72	27.14	odd	18	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.k (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-29.8374 - 35.5588i) q^{5} +(-70.9709 + 25.8313i) q^{7} +O(q^{10})\)	\(q+(-29.8374 - 35.5588i) q^{5} +(-70.9709 + 25.8313i) q^{7} +(37.6687 - 44.8918i) q^{11} +(-56.3644 - 319.658i) q^{13} +(-143.731 + 82.9833i) q^{17} +(-17.1368 + 29.6819i) q^{19} +(43.9406 - 120.726i) q^{23} +(-265.630 + 1506.46i) q^{25} +(-1026.95 - 181.080i) q^{29} +(713.964 + 259.862i) q^{31} +(3036.12 + 1752.90i) q^{35} +(508.520 + 880.783i) q^{37} +(-23.6179 + 4.16447i) q^{41} +(1834.16 + 1539.05i) q^{43} +(-651.646 - 1790.38i) q^{47} +(2530.34 - 2123.21i) q^{49} +1365.27i q^{53} -2720.24 q^{55} +(1297.94 + 1546.83i) q^{59} +(1446.06 - 526.322i) q^{61} +(-9684.92 + 11542.0i) q^{65} +(30.0066 + 170.176i) q^{67} +(-6082.30 + 3511.62i) q^{71} +(2717.42 - 4706.72i) q^{73} +(-1513.77 + 4159.04i) q^{77} +(-220.907 + 1252.83i) q^{79} +(11696.3 + 2062.37i) q^{83} +(7239.36 + 2634.91i) q^{85} +(-5912.93 - 3413.83i) q^{89} +(12257.4 + 21230.5i) q^{91} +(1566.77 - 276.264i) q^{95} +(-5288.41 - 4437.50i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{5}+O(q^{10}) \) 72 * q - 9 * q^5	\( 72 q - 9 q^{5} - 18 q^{11} + 1278 q^{23} + 441 q^{25} - 1854 q^{29} - 1665 q^{31} + 2673 q^{35} + 5472 q^{41} + 1260 q^{43} - 5103 q^{47} - 5904 q^{49} + 10944 q^{59} + 8352 q^{61} - 8757 q^{65} + 378 q^{67} + 19764 q^{71} + 6111 q^{73} + 5679 q^{77} - 5652 q^{79} + 20061 q^{83} + 26100 q^{85} - 15633 q^{89} - 6039 q^{91} - 48024 q^{95} - 37530 q^{97}+O(q^{100}) \) 72 * q - 9 * q^5 - 18 * q^11 + 1278 * q^23 + 441 * q^25 - 1854 * q^29 - 1665 * q^31 + 2673 * q^35 + 5472 * q^41 + 1260 * q^43 - 5103 * q^47 - 5904 * q^49 + 10944 * q^59 + 8352 * q^61 - 8757 * q^65 + 378 * q^67 + 19764 * q^71 + 6111 * q^73 + 5679 * q^77 - 5652 * q^79 + 20061 * q^83 + 26100 * q^85 - 15633 * q^89 - 6039 * q^91 - 48024 * q^95 - 37530 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more