LMFDB - Embedded newform 864.5.t.b.559.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,5,Mod(559,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4]))

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

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

chi := DirichletCharacter("864.559");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	864.t (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:	$89.3116481044$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$44$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			559.20
Character	$\chi$	$=$	864.559
Dual form			864.5.t.b.847.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.53773 + 1.46516i) q^{5} +(-39.8426 - 23.0032i) q^{7} +O(q^{10})$	$q+(-2.53773 + 1.46516i) q^{5} +(-39.8426 - 23.0032i) q^{7} +(27.1224 - 46.9774i) q^{11} +(174.611 - 100.811i) q^{13} +88.7146 q^{17} +357.083 q^{19} +(-214.383 + 123.774i) q^{23} +(-308.207 + 533.830i) q^{25} +(18.7746 + 10.8395i) q^{29} +(349.016 - 201.504i) q^{31} +134.813 q^{35} +1263.37i q^{37} +(1106.42 + 1916.37i) q^{41} +(-678.384 + 1175.00i) q^{43} +(-3120.48 - 1801.61i) q^{47} +(-142.209 - 246.314i) q^{49} -4122.82i q^{53} +158.955i q^{55} +(-2643.76 - 4579.12i) q^{59} +(4778.25 + 2758.72i) q^{61} +(-295.410 + 511.664i) q^{65} +(-3090.68 - 5353.21i) q^{67} -4832.30i q^{71} +9435.38 q^{73} +(-2161.26 + 1247.80i) q^{77} +(-1099.86 - 635.004i) q^{79} +(1266.75 - 2194.08i) q^{83} +(-225.134 + 129.981i) q^{85} -5907.27 q^{89} -9275.93 q^{91} +(-906.180 + 523.183i) q^{95} +(-928.999 + 1609.07i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q+O(q^{10}) $ 88 * q	$ 88 q + 44 q^{11} + 1156 q^{17} - 860 q^{19} + 5998 q^{25} - 2508 q^{35} - 2348 q^{41} - 3500 q^{43} + 16462 q^{49} - 3508 q^{59} + 2502 q^{65} - 5132 q^{67} + 19004 q^{73} + 17090 q^{83} + 8272 q^{89} + 9612 q^{91} + 9980 q^{97}+O(q^{100}) $ 88 * q + 44 * q^11 + 1156 * q^17 - 860 * q^19 + 5998 * q^25 - 2508 * q^35 - 2348 * q^41 - 3500 * q^43 + 16462 * q^49 - 3508 * q^59 + 2502 * q^65 - 5132 * q^67 + 19004 * q^73 + 17090 * q^83 + 8272 * q^89 + 9612 * q^91 + 9980 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.53773	+	1.46516i	−0.101509	+	0.0586064i	−0.549895	−	0.835234i	$-0.685332\pi$
$5$	−2.53773	+	1.46516i	−0.101509	+	0.0586064i	0.448386	+	0.893840i	$0.351999\pi$
$6$		0			0
$7$	−39.8426	−	23.0032i	−0.813115	−	0.469452i	0.0349214	−	0.999390i	$-0.488882\pi$
$7$	−39.8426	−	23.0032i	−0.813115	−	0.469452i	−0.848036	+	0.529938i	$0.822215\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	27.1224	−	46.9774i	0.224152	−	0.388243i	−0.731913	−	0.681398i	$-0.761372\pi$
$11$	27.1224	−	46.9774i	0.224152	−	0.388243i	0.956065	+	0.293156i	$0.0947054\pi$
$12$		0			0
$13$	174.611	−	100.811i	1.03320	−	0.596517i	0.115299	−	0.993331i	$-0.463217\pi$
$13$	174.611	−	100.811i	1.03320	−	0.596517i	0.917899	+	0.396813i	$0.129884\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	88.7146			0.306971			0.153485	−	0.988151i	$-0.450950\pi$
$17$	88.7146			0.306971			0.153485	+	0.988151i	$0.450950\pi$
$18$		0			0
$19$	357.083			0.989149			0.494575	−	0.869135i	$-0.335324\pi$
$19$	357.083			0.989149			0.494575	+	0.869135i	$0.335324\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−214.383	+	123.774i	−0.405261	+	0.233977i	−0.688751	−	0.724998i	$-0.741841\pi$
$23$	−214.383	+	123.774i	−0.405261	+	0.233977i	0.283491	+	0.958975i	$0.408508\pi$
$24$		0			0
$25$	−308.207	+	533.830i	−0.493131	+	0.854127i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	18.7746	+	10.8395i	0.0223242	+	0.0128889i	0.511121	−	0.859509i	$-0.329231\pi$
$29$	18.7746	+	10.8395i	0.0223242	+	0.0128889i	−0.488796	+	0.872398i	$0.662564\pi$
$30$		0			0
$31$	349.016	−	201.504i	0.363180	−	0.209682i	−0.307295	−	0.951614i	$-0.599424\pi$
$31$	349.016	−	201.504i	0.363180	−	0.209682i	0.670475	+	0.741932i	$0.266090\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	134.813			0.110052
$36$		0			0
$37$			1263.37i			0.922839i	0.887182	+	0.461419i	$0.152660\pi$
$37$			1263.37i			0.922839i	−0.887182	+	0.461419i	$0.847340\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1106.42	+	1916.37i	0.658189	+	1.14002i	0.981084	+	0.193582i	$0.0620106\pi$
$41$	1106.42	+	1916.37i	0.658189	+	1.14002i	−0.322895	+	0.946435i	$0.604656\pi$
$42$		0			0
$43$	−678.384	+	1175.00i	−0.366892	+	0.635476i	−0.989078	−	0.147393i	$-0.952912\pi$
$43$	−678.384	+	1175.00i	−0.366892	+	0.635476i	0.622185	+	0.782870i	$0.286245\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3120.48	−	1801.61i	−1.41262	−	0.815577i	−0.416985	−	0.908913i	$-0.636913\pi$
$47$	−3120.48	−	1801.61i	−1.41262	−	0.815577i	−0.995635	+	0.0933366i	$0.970247\pi$
$48$		0			0
$49$	−142.209	−	246.314i	−0.0592292	−	0.102588i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4122.82i		−	1.46772i	−0.679303	−	0.733858i	$-0.737718\pi$
$53$		−	4122.82i		−	1.46772i	0.679303	−	0.733858i	$-0.262282\pi$
$54$		0			0
$55$			158.955i			0.0525469i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2643.76	−	4579.12i	−0.759482	−	1.31546i	−0.943115	−	0.332466i	$-0.892119\pi$
$59$	−2643.76	−	4579.12i	−0.759482	−	1.31546i	0.183634	−	0.982995i	$-0.441214\pi$
$60$		0			0
$61$	4778.25	+	2758.72i	1.28413	+	0.741393i	0.977601	−	0.210468i	$-0.0674987\pi$
$61$	4778.25	+	2758.72i	1.28413	+	0.741393i	0.306530	+	0.951861i	$0.400832\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−295.410	+	511.664i	−0.0699194	+	0.121104i
$66$		0			0
$67$	−3090.68	−	5353.21i	−0.688500	−	1.19252i	−0.972323	−	0.233641i	$-0.924936\pi$
$67$	−3090.68	−	5353.21i	−0.688500	−	1.19252i	0.283822	−	0.958877i	$-0.408397\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4832.30i		−	0.958600i	−0.877651	−	0.479300i	$-0.840891\pi$
$71$		−	4832.30i		−	0.958600i	0.877651	−	0.479300i	$-0.159109\pi$
$72$		0			0
$73$	9435.38			1.77057			0.885286	−	0.465047i	$-0.153963\pi$
$73$	9435.38			1.77057			0.885286	+	0.465047i	$0.153963\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2161.26	+	1247.80i	−0.364523	+	0.210457i
$78$		0			0
$79$	−1099.86	−	635.004i	−0.176231	−	0.101747i	0.409290	−	0.912405i	$-0.365777\pi$
$79$	−1099.86	−	635.004i	−0.176231	−	0.101747i	−0.585521	+	0.810657i	$0.699110\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1266.75	−	2194.08i	0.183881	−	0.318490i	−0.759318	−	0.650720i	$-0.774467\pi$
$83$	1266.75	−	2194.08i	0.183881	−	0.318490i	0.943199	+	0.332229i	$0.107801\pi$
$84$		0			0
$85$	−225.134	+	129.981i	−0.0311604	+	0.0179904i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5907.27			−0.745773			−0.372887	−	0.927877i	$-0.621632\pi$
$89$	−5907.27			−0.745773			−0.372887	+	0.927877i	$0.621632\pi$
$90$		0			0
$91$	−9275.93			−1.12015
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−906.180	+	523.183i	−0.100408	+	0.0579704i
$96$		0			0
$97$	−928.999	+	1609.07i	−0.0987351	+	0.171014i	−0.911161	−	0.412050i	$-0.864813\pi$
$97$	−928.999	+	1609.07i	−0.0987351	+	0.171014i	0.812426	+	0.583064i	$0.198146\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13841.8	−	7991.56i	−1.35690	−	0.783409i	−0.367699	−	0.929945i	$-0.619854\pi$
$101$	−13841.8	−	7991.56i	−1.35690	−	0.783409i	−0.989205	+	0.146536i	$0.953188\pi$
$102$		0			0
$103$	17094.2	−	9869.37i	1.61130	−	0.930283i	0.622227	−	0.782837i	$-0.286228\pi$
$103$	17094.2	−	9869.37i	1.61130	−	0.930283i	0.989070	−	0.147445i	$-0.0471051\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8611.81			0.752189			0.376095	−	0.926581i	$-0.377267\pi$
$107$	8611.81			0.752189			0.376095	+	0.926581i	$0.377267\pi$
$108$		0			0
$109$		−	19759.0i		−	1.66307i	−0.555470	−	0.831537i	$-0.687461\pi$
$109$		−	19759.0i		−	1.66307i	0.555470	−	0.831537i	$-0.312539\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8203.89	+	14209.6i	0.642485	+	1.11282i	0.984876	+	0.173259i	$0.0554297\pi$
$113$	8203.89	+	14209.6i	0.642485	+	1.11282i	−0.342392	+	0.939557i	$0.611237\pi$
$114$		0			0
$115$	362.697	−	628.210i	0.0274251	−	0.0475017i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3534.62	−	2040.72i	−0.249603	−	0.144108i
$120$		0			0
$121$	5849.25	+	10131.2i	0.399512	+	0.691975i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	3637.74i		−	0.232815i
$126$		0			0
$127$			190.402i			0.0118049i	0.999983	+	0.00590247i	$0.00187883\pi$
$127$			190.402i			0.0118049i	−0.999983	+	0.00590247i	$0.998121\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5339.62	−	9248.49i	−0.311148	−	0.538925i	0.667463	−	0.744643i	$-0.267380\pi$
$131$	−5339.62	−	9248.49i	−0.311148	−	0.538925i	−0.978611	+	0.205718i	$0.934047\pi$
$132$		0			0
$133$	−14227.1	−	8214.03i	−0.804292	−	0.464358i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2673.96	−	4631.43i	0.142467	−	0.246759i	−0.785958	−	0.618280i	$-0.787830\pi$
$137$	2673.96	−	4631.43i	0.142467	−	0.246759i	0.928425	+	0.371520i	$0.121163\pi$
$138$		0			0
$139$	4160.16	+	7205.61i	0.215318	+	0.372942i	0.953371	−	0.301801i	$-0.0975878\pi$
$139$	4160.16	+	7205.61i	0.215318	+	0.372942i	−0.738053	+	0.674743i	$0.764254\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	10937.0i		−	0.534842i
$144$		0			0
$145$	−63.5266			−0.00302148
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8232.13	+	4752.82i	−0.370800	+	0.214082i	−0.673808	−	0.738907i	$-0.735342\pi$
$149$	−8232.13	+	4752.82i	−0.370800	+	0.214082i	0.303008	+	0.952988i	$0.402009\pi$
$150$		0			0
$151$	−23033.8	−	13298.5i	−1.01021	−	0.583244i	−0.0989556	−	0.995092i	$-0.531550\pi$
$151$	−23033.8	−	13298.5i	−1.01021	−	0.583244i	−0.911253	+	0.411848i	$0.864884\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−590.472	+	1022.73i	−0.0245774	+	0.0425693i
$156$		0			0
$157$	14257.2	−	8231.41i	0.578410	−	0.333945i	−0.182091	−	0.983282i	$-0.558287\pi$
$157$	14257.2	−	8231.41i	0.578410	−	0.333945i	0.760501	+	0.649336i	$0.224953\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	11388.8			0.439365
$162$		0			0
$163$	6664.66			0.250843			0.125422	−	0.992104i	$-0.459972\pi$
$163$	6664.66			0.250843			0.125422	+	0.992104i	$0.459972\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	19196.1	−	11082.8i	0.688302	−	0.397391i	−0.114674	−	0.993403i	$-0.536582\pi$
$167$	19196.1	−	11082.8i	0.688302	−	0.397391i	0.802976	+	0.596012i	$0.203249\pi$
$168$		0			0
$169$	6045.39	−	10470.9i	0.211666	−	0.366616i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9709.71	−	5605.90i	−0.324425	−	0.187307i	0.328938	−	0.944351i	$-0.393309\pi$
$173$	−9709.71	−	5605.90i	−0.324425	−	0.187307i	−0.653363	+	0.757045i	$0.726643\pi$
$174$		0			0
$175$	24559.5	−	14179.5i	0.801944	−	0.463002i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16987.5			−0.530179			−0.265090	−	0.964224i	$-0.585402\pi$
$179$	−16987.5			−0.530179			−0.265090	+	0.964224i	$0.585402\pi$
$180$		0			0
$181$		−	48494.7i		−	1.48026i	−0.672465	−	0.740129i	$-0.734764\pi$
$181$		−	48494.7i		−	1.48026i	0.672465	−	0.740129i	$-0.265236\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1851.03	−	3206.08i	−0.0540842	−	0.0936766i
$186$		0			0
$187$	2406.15	−	4167.58i	0.0688082	−	0.119179i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−31625.9	−	18259.2i	−0.866915	−	0.500514i	−0.000593302	−	1.00000i	$-0.500189\pi$
$191$	−31625.9	−	18259.2i	−0.866915	−	0.500514i	−0.866322	+	0.499486i	$0.833522\pi$
$192$		0			0
$193$	−12557.9	−	21750.9i	−0.337133	−	0.583932i	0.646759	−	0.762694i	$-0.276124\pi$
$193$	−12557.9	−	21750.9i	−0.337133	−	0.583932i	−0.983892	+	0.178762i	$0.942791\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	28539.7i		−	0.735388i	−0.929947	−	0.367694i	$-0.880147\pi$
$197$		−	28539.7i		−	0.735388i	0.929947	−	0.367694i	$-0.119853\pi$
$198$		0			0
$199$		−	63914.2i		−	1.61395i	−0.590583	−	0.806977i	$-0.701102\pi$
$199$		−	63914.2i		−	1.61395i	0.590583	−	0.806977i	$-0.298898\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−498.687	−	863.752i	−0.0121014	−	0.0209603i
$204$		0			0
$205$	−5615.57	−	3242.15i	−0.133624	−	0.0771481i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9684.94	−	16774.8i	0.221720	−	0.384030i
$210$		0			0
$211$	7822.71	+	13549.3i	0.175708	+	0.304336i	0.940406	−	0.340053i	$-0.110445\pi$
$211$	7822.71	+	13549.3i	0.175708	+	0.304336i	−0.764698	+	0.644389i	$0.777112\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3975.76i		−	0.0860089i
$216$		0			0
$217$	−18541.0			−0.393743
$218$		0			0
$219$		0			0
$220$		0			0
$221$	15490.5	−	8943.44i	0.317162	−	0.183113i
$222$		0			0
$223$	−44835.1	−	25885.6i	−0.901589	−	0.520532i	−0.0238733	−	0.999715i	$-0.507600\pi$
$223$	−44835.1	−	25885.6i	−0.901589	−	0.520532i	−0.877715	+	0.479183i	$0.840933\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−28229.2	+	48894.4i	−0.547831	+	0.948871i	0.450592	+	0.892730i	$0.351213\pi$
$227$	−28229.2	+	48894.4i	−0.547831	+	0.948871i	−0.998423	+	0.0561410i	$0.982120\pi$
$228$		0			0
$229$	48956.6	−	28265.1i	0.933556	−	0.538989i	0.0456218	−	0.998959i	$-0.485473\pi$
$229$	48956.6	−	28265.1i	0.933556	−	0.538989i	0.887935	+	0.459970i	$0.152140\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−72322.4			−1.33217			−0.666087	−	0.745874i	$-0.732032\pi$
$233$	−72322.4			−1.33217			−0.666087	+	0.745874i	$0.732032\pi$
$234$		0			0
$235$	10558.6			0.191192
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−86456.7	+	49915.8i	−1.51357	+	0.873861i	−0.513698	+	0.857971i	$0.671725\pi$
$239$	−86456.7	+	49915.8i	−1.51357	+	0.873861i	−0.999874	+	0.0158894i	$0.994942\pi$
$240$		0			0
$241$	−13108.7	+	22704.9i	−0.225697	+	0.390918i	−0.956528	−	0.291640i	$-0.905799\pi$
$241$	−13108.7	+	22704.9i	−0.225697	+	0.390918i	0.730832	+	0.682558i	$0.239132\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	721.778	+	416.719i	0.0120246	+	0.00694242i
$246$		0			0
$247$	62350.4	−	35998.0i	1.02199	−	0.590045i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	110168.			1.74866			0.874332	−	0.485328i	$-0.161300\pi$
$251$	110168.			1.74866			0.874332	+	0.485328i	$0.161300\pi$
$252$		0			0
$253$			13428.2i			0.209786i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−21395.2	−	37057.7i	−0.323930	−	0.561063i	0.657365	−	0.753572i	$-0.271671\pi$
$257$	−21395.2	−	37057.7i	−0.323930	−	0.561063i	−0.981295	+	0.192509i	$0.938338\pi$
$258$		0			0
$259$	29061.4	−	50335.8i	0.433229	−	0.750374i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−70761.8	−	40854.3i	−1.02303	−	0.590645i	−0.108048	−	0.994146i	$-0.534460\pi$
$263$	−70761.8	−	40854.3i	−1.02303	−	0.590645i	−0.914979	+	0.403501i	$0.867793\pi$
$264$		0			0
$265$	6040.58	+	10462.6i	0.0860175	+	0.148987i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	71867.3i		−	0.993177i	−0.867986	−	0.496589i	$-0.834586\pi$
$269$		−	71867.3i		−	0.993177i	0.867986	−	0.496589i	$-0.165414\pi$
$270$		0			0
$271$			57719.4i			0.785929i	0.919554	+	0.392964i	$0.128550\pi$
$271$			57719.4i			0.785929i	−0.919554	+	0.392964i	$0.871450\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16718.6	+	28957.5i	0.221072	+	0.382909i
$276$		0			0
$277$	17169.2	+	9912.65i	0.223764	+	0.129190i	0.607692	−	0.794173i	$-0.292095\pi$
$277$	17169.2	+	9912.65i	0.223764	+	0.129190i	−0.383928	+	0.923363i	$0.625429\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	62271.1	−	107857.i	0.788631	−	1.36595i	−0.138175	−	0.990408i	$-0.544124\pi$
$281$	62271.1	−	107857.i	0.788631	−	1.36595i	0.926806	−	0.375541i	$-0.122543\pi$
$282$		0			0
$283$	28575.7	+	49494.6i	0.356799	+	0.617995i	0.987424	−	0.158093i	$-0.0505347\pi$
$283$	28575.7	+	49494.6i	0.356799	+	0.617995i	−0.630625	+	0.776088i	$0.717201\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	101804.i		−	1.23595i
$288$		0			0
$289$	−75650.7			−0.905769
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−25459.0	+	14698.8i	−0.296555	+	0.171216i	−0.640894	−	0.767629i	$-0.721436\pi$
$293$	−25459.0	+	14698.8i	−0.296555	+	0.171216i	0.344339	+	0.938845i	$0.388103\pi$
$294$		0			0
$295$	13418.3	+	7747.04i	0.154189	+	0.0890209i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−24955.7	+	43224.5i	−0.279143	+	0.483490i
$300$		0			0
$301$	54057.2	−	31210.0i	0.596652	−	0.344477i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−16167.9			−0.173801
$306$		0			0
$307$	77280.8			0.819964			0.409982	−	0.912094i	$-0.365535\pi$
$307$	77280.8			0.819964			0.409982	+	0.912094i	$0.365535\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	88843.4	−	51293.7i	0.918553	−	0.530327i	0.0353799	−	0.999374i	$-0.488736\pi$
$311$	88843.4	−	51293.7i	0.918553	−	0.530327i	0.883173	+	0.469047i	$0.155403\pi$
$312$		0			0
$313$	20566.6	−	35622.4i	0.209930	−	0.363609i	−0.741762	−	0.670663i	$-0.766010\pi$
$313$	20566.6	−	35622.4i	0.209930	−	0.363609i	0.951692	+	0.307054i	$0.0993431\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−109026.	−	62945.9i	−1.08495	−	0.626396i	−0.152723	−	0.988269i	$-0.548804\pi$
$317$	−109026.	−	62945.9i	−1.08495	−	0.626396i	−0.932227	+	0.361873i	$0.882137\pi$
$318$		0			0
$319$	1018.43	−	587.989i	0.0100080	−	0.00577813i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	31678.5			0.303640
$324$		0			0
$325$			124283.i			1.17664i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	82885.4	+	143562.i	0.765748	+	1.32632i
$330$		0			0
$331$	−54232.0	+	93932.5i	−0.494993	+	0.857354i	−0.999983	−	0.00577154i	$-0.998163\pi$
$331$	−54232.0	+	93932.5i	−0.494993	+	0.857354i	0.504990	+	0.863125i	$0.331496\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	15686.6	+	9056.67i	0.139778	+	0.0807010i
$336$		0			0
$337$	−1823.71	−	3158.77i	−0.0160582	−	0.0278136i	0.857885	−	0.513842i	$-0.171778\pi$
$337$	−1823.71	−	3158.77i	−0.0160582	−	0.0278136i	−0.873943	+	0.486029i	$0.838445\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	21861.1i		−	0.188003i
$342$		0			0
$343$			123546.i			1.05013i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−86264.4	−	149414.i	−0.716428	−	1.24089i	−0.962406	−	0.271615i	$-0.912442\pi$
$347$	−86264.4	−	149414.i	−0.716428	−	1.24089i	0.245978	−	0.969275i	$-0.420891\pi$
$348$		0			0
$349$	−138147.	−	79759.0i	−1.13420	−	0.654831i	−0.189212	−	0.981936i	$-0.560593\pi$
$349$	−138147.	−	79759.0i	−1.13420	−	0.654831i	−0.944988	+	0.327106i	$0.893927\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	20555.4	−	35603.0i	0.164959	−	0.285718i	−0.771681	−	0.636009i	$-0.780584\pi$
$353$	20555.4	−	35603.0i	0.164959	−	0.285718i	0.936641	+	0.350291i	$0.113917\pi$
$354$		0			0
$355$	7080.09	+	12263.1i	0.0561800	+	0.0973067i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			76360.8i			0.592491i	0.955112	+	0.296245i	$0.0957346\pi$
$359$			76360.8i			0.592491i	−0.955112	+	0.296245i	$0.904265\pi$
$360$		0			0
$361$	−2812.82			−0.0215838
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−23944.4	+	13824.3i	−0.179729	+	0.103767i
$366$		0			0
$367$	102925.	+	59423.6i	0.764165	+	0.441191i	0.830789	−	0.556587i	$-0.187889\pi$
$367$	102925.	+	59423.6i	0.764165	+	0.441191i	−0.0666239	+	0.997778i	$0.521223\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−94837.8	+	164264.i	−0.689023	+	1.19342i
$372$		0			0
$373$	115434.	−	66645.9i	0.829691	−	0.479023i	−0.0240556	−	0.999711i	$-0.507658\pi$
$373$	115434.	−	66645.9i	0.829691	−	0.479023i	0.853747	+	0.520688i	$0.174325\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4371.00			0.0307537
$378$		0			0
$379$	−141694.			−0.986447			−0.493224	−	0.869902i	$-0.664182\pi$
$379$	−141694.			−0.986447			−0.493224	+	0.869902i	$0.664182\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	43983.8	−	25394.1i	0.299844	−	0.173115i	−0.342529	−	0.939507i	$-0.611283\pi$
$383$	43983.8	−	25394.1i	0.299844	−	0.173115i	0.642373	+	0.766392i	$0.277950\pi$
$384$		0			0
$385$	3656.46	−	6333.17i	0.0246683	−	0.0427267i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	103180.	+	59570.8i	0.681860	+	0.393672i	0.800555	−	0.599259i	$-0.204538\pi$
$389$	103180.	+	59570.8i	0.681860	+	0.393672i	−0.118696	+	0.992931i	$0.537871\pi$
$390$		0			0
$391$	−19018.9	+	10980.6i	−0.124403	+	0.0718242i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3721.53			0.0238521
$396$		0			0
$397$			200049.i			1.26927i	0.772812	+	0.634635i	$0.218850\pi$
$397$			200049.i			1.26927i	−0.772812	+	0.634635i	$0.781150\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13977.6	+	24209.9i	0.0869248	+	0.150558i	0.906210	−	0.422828i	$-0.138963\pi$
$401$	13977.6	+	24209.9i	0.0869248	+	0.150558i	−0.819285	+	0.573387i	$0.805629\pi$
$402$		0			0
$403$	40627.9	−	70369.6i	0.250158	−	0.433286i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	59349.6	+	34265.5i	0.358285	+	0.206856i
$408$		0			0
$409$	−27650.2	−	47891.6i	−0.165292	−	0.286294i	0.771467	−	0.636270i	$-0.219523\pi$
$409$	−27650.2	−	47891.6i	−0.165292	−	0.286294i	−0.936759	+	0.349975i	$0.886190\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			243259.i			1.42616i
$414$		0			0
$415$			7423.98i			0.0431063i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−76753.4	−	132941.i	−0.437189	−	0.757234i	0.560282	−	0.828302i	$-0.310693\pi$
$419$	−76753.4	−	132941.i	−0.437189	−	0.757234i	−0.997472	+	0.0710675i	$0.977359\pi$
$420$		0			0
$421$	−230275.	−	132949.i	−1.29922	−	0.750105i	−0.318950	−	0.947771i	$-0.603330\pi$
$421$	−230275.	−	132949.i	−1.29922	−	0.750105i	−0.980269	+	0.197667i	$0.936664\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−27342.4	+	47358.5i	−0.151377	+	0.262192i
$426$		0			0
$427$	−126919.	−	219830.i	−0.696097	−	1.20568i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			187074.i			1.00707i	0.863976	+	0.503533i	$0.167967\pi$
$431$			187074.i			1.00707i	−0.863976	+	0.503533i	$0.832033\pi$
$432$		0			0
$433$	252526.			1.34688			0.673441	−	0.739241i	$-0.264815\pi$
$433$	252526.			1.34688			0.673441	+	0.739241i	$0.264815\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−76552.5	+	44197.6i	−0.400863	+	0.231439i
$438$		0			0
$439$	−138415.	−	79914.0i	−0.718215	−	0.414662i	0.0958801	−	0.995393i	$-0.469433\pi$
$439$	−138415.	−	79914.0i	−0.718215	−	0.414662i	−0.814095	+	0.580731i	$0.802767\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	152813.	−	264680.i	0.778670	−	1.34870i	−0.154038	−	0.988065i	$-0.549228\pi$
$443$	152813.	−	264680.i	0.778670	−	1.34870i	0.932708	−	0.360632i	$-0.117439\pi$
$444$		0			0
$445$	14991.1	−	8655.09i	0.0757028	−	0.0437070i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−338152.			−1.67733			−0.838667	−	0.544645i	$-0.816664\pi$
$449$	−338152.			−1.67733			−0.838667	+	0.544645i	$0.816664\pi$
$450$		0			0
$451$	120035.			0.590138
$452$		0			0
$453$		0			0
$454$		0			0
$455$	23539.8	−	13590.7i	0.113705	−	0.0656477i
$456$		0			0
$457$	−1641.07	+	2842.42i	−0.00785770	+	0.0136099i	−0.869928	−	0.493180i	$-0.835835\pi$
$457$	−1641.07	+	2842.42i	−0.00785770	+	0.0136099i	0.862070	+	0.506790i	$0.169168\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	215070.	+	124171.i	1.01199	+	0.584275i	0.911775	−	0.410690i	$-0.134712\pi$
$461$	215070.	+	124171.i	1.01199	+	0.584275i	0.100220	+	0.994965i	$0.468045\pi$
$462$		0			0
$463$	−42958.9	+	24802.3i	−0.200397	+	0.115699i	−0.596841	−	0.802360i	$-0.703578\pi$
$463$	−42958.9	+	24802.3i	−0.200397	+	0.115699i	0.396444	+	0.918059i	$0.370244\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18727.6			0.0858712			0.0429356	−	0.999078i	$-0.486329\pi$
$467$	18727.6			0.0858712			0.0429356	+	0.999078i	$0.486329\pi$
$468$		0			0
$469$			284381.i			1.29287i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	36798.8	+	63737.4i	0.164479	+	0.284887i
$474$		0			0
$475$	−110055.	+	190621.i	−0.487780	+	0.844859i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	343234.	+	198166.i	1.49596	+	0.863691i	0.999989	−	0.00465021i	$-0.00148021\pi$
$479$	343234.	+	198166.i	1.49596	+	0.863691i	0.495967	+	0.868341i	$0.334814\pi$
$480$		0			0
$481$	127362.	+	220597.i	0.550489	+	0.953475i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	5444.52i		−	0.0231460i
$486$		0			0
$487$			240262.i			1.01304i	0.862228	+	0.506520i	$0.169068\pi$
$487$			240262.i			1.01304i	−0.862228	+	0.506520i	$0.830932\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	98040.3	+	169811.i	0.406670	+	0.704372i	0.994514	−	0.104601i	$-0.0333567\pi$
$491$	98040.3	+	169811.i	0.406670	+	0.704372i	−0.587845	+	0.808974i	$0.700023\pi$
$492$		0			0
$493$	1665.58	+	961.625i	0.00685287	+	0.00395651i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−111158.	+	192532.i	−0.450017	+	0.779452i
$498$		0			0
$499$	−130948.	−	226808.i	−0.525892	−	0.910871i	−0.999545	−	0.0301596i	$-0.990398\pi$
$499$	−130948.	−	226808.i	−0.525892	−	0.910871i	0.473654	−	0.880711i	$-0.342935\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			311251.i			1.23020i	0.788450	+	0.615099i	$0.210884\pi$
$503$			311251.i			1.23020i	−0.788450	+	0.615099i	$0.789116\pi$
$504$		0			0
$505$	46835.6			0.183651
$506$		0			0
$507$		0			0
$508$		0			0
$509$	317446.	−	183278.i	1.22528	−	0.707415i	0.259240	−	0.965813i	$-0.416528\pi$
$509$	317446.	−	183278.i	1.22528	−	0.707415i	0.966039	+	0.258398i	$0.0831945\pi$
$510$		0			0
$511$	−375930.	−	217043.i	−1.43968	−	0.831199i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−28920.4	+	50091.6i	−0.109041	+	0.188864i
$516$		0			0
$517$	−169270.	+	97727.9i	−0.633283	+	0.365626i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	526988.			1.94145			0.970723	−	0.240203i	$-0.0772142\pi$
$521$	526988.			1.94145			0.970723	+	0.240203i	$0.0772142\pi$
$522$		0			0
$523$	−311012.			−1.13703			−0.568517	−	0.822671i	$-0.692483\pi$
$523$	−311012.			−1.13703			−0.568517	+	0.822671i	$0.692483\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	30962.8	−	17876.4i	0.111486	−	0.0643663i
$528$		0			0
$529$	−109280.	+	189279.i	−0.390509	+	0.676382i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	386384.	+	223079.i	1.36008	+	0.785242i
$534$		0			0
$535$	−21854.5	+	12617.7i	−0.0763541	+	0.0440831i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15428.2			−0.0531054
$540$		0			0
$541$			377631.i			1.29025i	0.764078	+	0.645124i	$0.223194\pi$
$541$			377631.i			1.29025i	−0.764078	+	0.645124i	$0.776806\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	28950.0	+	50143.0i	0.0974667	+	0.168817i
$546$		0			0
$547$	51766.7	−	89662.5i	0.173012	−	0.299665i	−0.766460	−	0.642292i	$-0.777983\pi$
$547$	51766.7	−	89662.5i	0.173012	−	0.299665i	0.939471	+	0.342627i	$0.111317\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6704.10	+	3870.61i	0.0220819	+	0.0127490i
$552$		0			0
$553$	29214.2	+	50600.4i	0.0955308	+	0.165464i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			321643.i			1.03673i	0.855161	+	0.518363i	$0.173458\pi$
$557$			321643.i			1.03673i	−0.855161	+	0.518363i	$0.826542\pi$
$558$		0			0
$559$			273556.i			0.875431i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15717.8	−	27224.0i	−0.0495878	−	0.0858885i	0.840166	−	0.542329i	$-0.182457\pi$
$563$	−15717.8	−	27224.0i	−0.0495878	−	0.0858885i	−0.889754	+	0.456441i	$0.849124\pi$
$564$		0			0
$565$	−41638.5	−	24040.0i	−0.130436	−	0.0753074i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−118855.	+	205862.i	−0.367106	+	0.635846i	−0.989112	−	0.147166i	$-0.952985\pi$
$569$	−118855.	+	205862.i	−0.367106	+	0.635846i	0.622006	+	0.783013i	$0.286318\pi$
$570$		0			0
$571$	259870.	+	450108.i	0.797047	+	1.38053i	0.921531	+	0.388304i	$0.126939\pi$
$571$	259870.	+	450108.i	0.797047	+	1.38053i	−0.124484	+	0.992222i	$0.539728\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	152592.i		−	0.461526i
$576$		0			0
$577$	−150655.			−0.452513			−0.226257	−	0.974068i	$-0.572649\pi$
$577$	−150655.			−0.452513			−0.226257	+	0.974068i	$0.572649\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−100942.	+	58278.7i	−0.299032	+	0.172646i
$582$		0			0
$583$	−193679.	−	111821.i	−0.569830	−	0.328992i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	222883.	−	386045.i	0.646846	−	1.12037i	−0.337025	−	0.941496i	$-0.609421\pi$
$587$	222883.	−	386045.i	0.646846	−	1.12037i	0.983872	−	0.178875i	$-0.0572458\pi$
$588$		0			0
$589$	124628.	−	71953.8i	0.359239	−	0.207407i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	59064.6			0.167965			0.0839823	−	0.996467i	$-0.473236\pi$
$593$	59064.6			0.167965			0.0839823	+	0.996467i	$0.473236\pi$
$594$		0			0
$595$	11959.9			0.0337826
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−354748.	+	204814.i	−0.988703	+	0.570828i	−0.904886	−	0.425653i	$-0.860044\pi$
$599$	−354748.	+	204814.i	−0.988703	+	0.570828i	−0.0838164	+	0.996481i	$0.526711\pi$
$600$		0			0
$601$	268772.	−	465527.i	0.744107	−	1.28883i	−0.206505	−	0.978446i	$-0.566209\pi$
$601$	268772.	−	465527.i	0.744107	−	1.28883i	0.950611	−	0.310385i	$-0.100458\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−29687.6	−	17140.2i	−0.0811082	−	0.0468279i
$606$		0			0
$607$	377551.	−	217979.i	1.02470	−	0.591614i	0.109242	−	0.994015i	$-0.465158\pi$
$607$	377551.	−	217979.i	1.02470	−	0.591614i	0.915463	+	0.402402i	$0.131824\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−726491.			−1.94602
$612$		0			0
$613$			184751.i			0.491662i	0.969313	+	0.245831i	$0.0790608\pi$
$613$			184751.i			0.491662i	−0.969313	+	0.245831i	$0.920939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−70628.6	−	122332.i	−0.185528	−	0.321344i	0.758226	−	0.651992i	$-0.226066\pi$
$617$	−70628.6	−	122332.i	−0.185528	−	0.321344i	−0.943754	+	0.330647i	$0.892733\pi$
$618$		0			0
$619$	132429.	−	229374.i	0.345623	−	0.598637i	−0.639844	−	0.768505i	$-0.721001\pi$
$619$	132429.	−	229374.i	0.345623	−	0.598637i	0.985467	+	0.169868i	$0.0543342\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	235361.	+	135886.i	0.606399	+	0.350105i
$624$		0			0
$625$	−187299.	−	324412.i	−0.479486	−	0.830494i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			112079.i			0.283285i
$630$		0			0
$631$			699879.i			1.75778i	0.477026	+	0.878889i	$0.341715\pi$
$631$			699879.i			1.75778i	−0.477026	+	0.878889i	$0.658285\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−278.969	−	483.188i	−0.000691844	−	0.00119831i
$636$		0			0
$637$	−49662.5	−	28672.7i	−0.122391	−	0.0706625i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	159686.	−	276584.i	0.388643	−	0.673149i	−0.603625	−	0.797269i	$-0.706277\pi$
$641$	159686.	−	276584.i	0.388643	−	0.673149i	0.992267	+	0.124120i	$0.0396107\pi$
$642$		0			0
$643$	120033.	+	207903.i	0.290321	+	0.502850i	0.973886	−	0.227040i	$-0.0729047\pi$
$643$	120033.	+	207903.i	0.290321	+	0.502850i	−0.683565	+	0.729890i	$0.739571\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	160526.i		−	0.383475i	−0.981446	−	0.191737i	$-0.938588\pi$
$647$		−	160526.i		−	0.383475i	0.981446	−	0.191737i	$-0.0614122\pi$
$648$		0			0
$649$	−286820.			−0.680957
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−23755.5	+	13715.3i	−0.0557107	+	0.0321646i	−0.527597	−	0.849495i	$-0.676907\pi$
$653$	−23755.5	+	13715.3i	−0.0557107	+	0.0321646i	0.471886	+	0.881660i	$0.343573\pi$
$654$		0			0
$655$	27101.0	+	15646.8i	0.0631689	+	0.0364706i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−215117.	+	372594.i	−0.495341	+	0.857957i	−0.999986	−	0.00537093i	$-0.998290\pi$
$659$	−215117.	+	372594.i	−0.495341	+	0.857957i	0.504644	+	0.863327i	$0.331624\pi$
$660$		0			0
$661$	122972.	−	70998.0i	0.281451	−	0.162496i	−0.352629	−	0.935763i	$-0.614712\pi$
$661$	122972.	−	70998.0i	0.281451	−	0.162496i	0.634080	+	0.773267i	$0.281379\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	48139.5			0.108857
$666$		0			0
$667$	−5366.61			−0.0120628
$668$		0			0
$669$		0			0
$670$		0			0
$671$	259195.	−	149646.i	0.575681	−	0.332370i
$672$		0			0
$673$	−78759.2	+	136415.i	−0.173889	+	0.301184i	−0.939776	−	0.341791i	$-0.888967\pi$
$673$	−78759.2	+	136415.i	−0.173889	+	0.301184i	0.765887	+	0.642975i	$0.222300\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−606737.	−	350300.i	−1.32380	−	0.764298i	−0.339470	−	0.940617i	$-0.610248\pi$
$677$	−606737.	−	350300.i	−1.32380	−	0.764298i	−0.984333	+	0.176319i	$0.943581\pi$
$678$		0			0
$679$	74027.5	−	42739.8i	0.160566	−	0.0927028i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	408773.			0.876275			0.438137	−	0.898908i	$-0.355638\pi$
$683$	408773.			0.876275			0.438137	+	0.898908i	$0.355638\pi$
$684$		0			0
$685$			15671.1i			0.0333978i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−415627.	−	719887.i	−0.875518	−	1.51644i
$690$		0			0
$691$	309805.	−	536598.i	0.648832	−	1.12381i	−0.334570	−	0.942371i	$-0.608591\pi$
$691$	309805.	−	536598.i	0.648832	−	1.12381i	0.983402	−	0.181439i	$-0.0580756\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−21114.7	−	12190.6i	−0.0437135	−	0.0252380i
$696$		0			0
$697$	98155.2	+	170010.i	0.202045	+	0.349952i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	98886.5i		−	0.201234i	−0.994925	−	0.100617i	$-0.967918\pi$
$701$		−	98886.5i		−	0.201234i	0.994925	−	0.100617i	$-0.0320816\pi$
$702$		0			0
$703$			451126.i			0.912825i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	367662.	+	636809.i	0.735546	+	1.27400i
$708$		0			0
$709$	−16701.3	−	9642.51i	−0.0332245	−	0.0191822i	0.483296	−	0.875457i	$-0.339440\pi$
$709$	−16701.3	−	9642.51i	−0.0332245	−	0.0191822i	−0.516520	+	0.856275i	$0.672773\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−49882.0	+	86398.2i	−0.0981217	+	0.169952i
$714$		0			0
$715$	16024.4	+	27755.1i	0.0313452	+	0.0542914i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			124940.i			0.241681i	0.992672	+	0.120841i	$0.0385590\pi$
$719$			124940.i			0.241681i	−0.992672	+	0.120841i	$0.961441\pi$
$720$		0			0
$721$	−908107.			−1.74689
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−11572.9	+	6681.64i	−0.0220175	+	0.0127118i
$726$		0			0
$727$	237638.	+	137200.i	0.449621	+	0.259589i	0.707670	−	0.706543i	$-0.249746\pi$
$727$	237638.	+	137200.i	0.449621	+	0.259589i	−0.258049	+	0.966132i	$0.583080\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−60182.6	+	104239.i	−0.112625	+	0.195073i
$732$		0			0
$733$	196395.	−	113389.i	0.365530	−	0.211039i	−0.305974	−	0.952040i	$-0.598982\pi$
$733$	196395.	−	113389.i	0.365530	−	0.211039i	0.671504	+	0.741001i	$0.265649\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−335306.			−0.617315
$738$		0			0
$739$	173966.			0.318549			0.159274	−	0.987234i	$-0.449085\pi$
$739$	173966.			0.318549			0.159274	+	0.987234i	$0.449085\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	223552.	−	129068.i	0.404949	−	0.233797i	−0.283668	−	0.958923i	$-0.591551\pi$
$743$	223552.	−	129068.i	0.404949	−	0.233797i	0.688617	+	0.725125i	$0.258218\pi$
$744$		0			0
$745$	13927.3	−	24122.8i	0.0250931	−	0.0434625i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−343117.	−	198099.i	−0.611616	−	0.353117i
$750$		0			0
$751$	439854.	−	253950.i	0.779881	−	0.450264i	−0.0565073	−	0.998402i	$-0.517996\pi$
$751$	439854.	−	253950.i	0.779881	−	0.450264i	0.836388	+	0.548138i	$0.184663\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	77937.9			0.136727
$756$		0			0
$757$		−	146756.i		−	0.256097i	−0.991768	−	0.128048i	$-0.959129\pi$
$757$		−	146756.i		−	0.256097i	0.991768	−	0.128048i	$-0.0408713\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−106335.	−	184178.i	−0.183615	−	0.318030i	0.759494	−	0.650514i	$-0.225447\pi$
$761$	−106335.	−	184178.i	−0.183615	−	0.318030i	−0.943109	+	0.332484i	$0.892113\pi$
$762$		0			0
$763$	−454519.	+	787250.i	−0.780734	+	1.35227i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−923255.	−	533042.i	−1.56939	−	0.906088i
$768$		0			0
$769$	44021.9	+	76248.2i	0.0744417	+	0.128937i	0.900843	−	0.434144i	$-0.142949\pi$
$769$	44021.9	+	76248.2i	0.0744417	+	0.128937i	−0.826402	+	0.563081i	$0.809616\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	956337.i		−	1.60049i	−0.599675	−	0.800243i	$-0.704704\pi$
$773$		−	956337.i		−	1.60049i	0.599675	−	0.800243i	$-0.295296\pi$
$774$		0			0
$775$			248420.i			0.413602i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	395082.	+	684302.i	0.651047	+	1.12765i
$780$		0			0
$781$	−227009.	−	131064.i	−0.372169	−	0.214872i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24120.7	+	41778.2i	−0.0391426	+	0.0677970i
$786$		0			0
$787$	150134.	+	260040.i	0.242398	+	0.419846i	0.961397	−	0.275165i	$-0.0887326\pi$
$787$	150134.	+	260040.i	0.242398	+	0.419846i	−0.718999	+	0.695012i	$0.755399\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	754861.i		−	1.20646i
$792$		0			0
$793$	1.11244e6			1.76902
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−189436.	+	109371.i	−0.298226	+	0.172181i	−0.641646	−	0.767001i	$-0.721748\pi$
$797$	−189436.	+	109371.i	−0.298226	+	0.172181i	0.343420	+	0.939182i	$0.388415\pi$
$798$		0			0
$799$	−276832.	−	159829.i	−0.433633	−	0.250358i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	255910.	−	443249.i	0.396877	−	0.687412i
$804$		0			0
$805$	−28901.6	+	16686.4i	−0.0445996	+	0.0257496i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	373854.			0.571223			0.285611	−	0.958346i	$-0.407803\pi$
$809$	373854.			0.571223			0.285611	+	0.958346i	$0.407803\pi$
$810$		0			0
$811$	389516.			0.592220			0.296110	−	0.955154i	$-0.404310\pi$
$811$	389516.			0.592220			0.296110	+	0.955154i	$0.404310\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−16913.1	+	9764.78i	−0.0254629	+	0.0147010i
$816$		0			0
$817$	−242239.	+	419571.i	−0.362911	+	0.628581i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	766580.	+	442585.i	1.13729	+	0.656614i	0.945758	−	0.324872i	$-0.105321\pi$
$821$	766580.	+	442585.i	1.13729	+	0.656614i	0.191531	+	0.981486i	$0.438655\pi$
$822$		0			0
$823$	556310.	−	321186.i	0.821329	−	0.474195i	−0.0295455	−	0.999563i	$-0.509406\pi$
$823$	556310.	−	321186.i	0.821329	−	0.474195i	0.850875	+	0.525369i	$0.176073\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−144563.			−0.211372			−0.105686	−	0.994400i	$-0.533704\pi$
$827$	−144563.			−0.211372			−0.105686	+	0.994400i	$0.533704\pi$
$828$		0			0
$829$		−	390583.i		−	0.568335i	−0.958775	−	0.284167i	$-0.908283\pi$
$829$		−	390583.i		−	0.568335i	0.958775	−	0.284167i	$-0.0917171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12616.0	−	21851.6i	−0.0181816	−	0.0314915i
$834$		0			0
$835$	−32476.3	+	56250.6i	−0.0465793	+	0.0806778i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21529.2	+	12429.9i	0.0305846	+	0.0176580i	0.515214	−	0.857061i	$-0.327712\pi$
$839$	21529.2	+	12429.9i	0.0305846	+	0.0176580i	−0.484630	+	0.874719i	$0.661046\pi$
$840$		0			0
$841$	−353406.	−	612116.i	−0.499668	−	0.865450i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			35429.8i			0.0496199i
$846$		0			0
$847$		−	538205.i		−	0.750207i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−156372.	−	270844.i	−0.215923	−	0.373990i
$852$		0			0
$853$	264635.	+	152787.i	0.363705	+	0.209985i	0.670705	−	0.741724i	$-0.265992\pi$
$853$	264635.	+	152787.i	0.363705	+	0.209985i	−0.307000	+	0.951710i	$0.599325\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	221670.	−	383944.i	0.301818	−	0.522764i	−0.674730	−	0.738065i	$-0.735740\pi$
$857$	221670.	−	383944.i	0.301818	−	0.522764i	0.976548	+	0.215301i	$0.0690732\pi$
$858$		0			0
$859$	−228055.	−	395003.i	−0.309067	−	0.535320i	0.669091	−	0.743180i	$-0.266683\pi$
$859$	−228055.	−	395003.i	−0.309067	−	0.535320i	−0.978159	+	0.207860i	$0.933350\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			432830.i			0.581160i	0.956851	+	0.290580i	$0.0938483\pi$
$863$			432830.i			0.581160i	−0.956851	+	0.290580i	$0.906152\pi$
$864$		0			0
$865$	32854.2			0.0439095
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−59661.6	+	34445.6i	−0.0790052	+	0.0456136i
$870$		0			0
$871$	−1.07933e6	−	623151.i	−1.42272	−	0.821405i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−83679.4	+	144937.i	−0.109296	+	0.189305i
$876$		0			0
$877$	439842.	−	253943.i	0.571870	−	0.330169i	−0.186026	−	0.982545i	$-0.559561\pi$
$877$	439842.	−	253943.i	0.571870	−	0.330169i	0.757896	+	0.652376i	$0.226227\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	630033.			0.811730			0.405865	−	0.913933i	$-0.366970\pi$
$881$	630033.			0.811730			0.405865	+	0.913933i	$0.366970\pi$
$882$		0			0
$883$	111488.			0.142990			0.0714950	−	0.997441i	$-0.477223\pi$
$883$	111488.			0.142990			0.0714950	+	0.997441i	$0.477223\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−642963.	+	371215.i	−0.817220	+	0.471822i	−0.849457	−	0.527658i	$-0.823070\pi$
$887$	−642963.	+	371215.i	−0.817220	+	0.471822i	0.0322369	+	0.999480i	$0.489737\pi$
$888$		0			0
$889$	4379.84	−	7586.11i	0.00554185	−	0.00959877i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.11427e6	−	643324.i	−1.39729	−	0.806727i
$894$		0			0
$895$	43109.6	−	24889.4i	0.0538181	−	0.0310719i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8736.86			0.0108103
$900$		0			0
$901$		−	365754.i		−	0.450546i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	71052.5	+	123067.i	0.0867525	+	0.150260i
$906$		0			0
$907$	120922.	−	209443.i	0.146991	−	0.254596i	−0.783123	−	0.621867i	$-0.786374\pi$
$907$	120922.	−	209443.i	0.146991	−	0.254596i	0.930114	+	0.367271i	$0.119708\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−276542.	−	159661.i	−0.333214	−	0.192381i	0.324053	−	0.946039i	$-0.394954\pi$
$911$	−276542.	−	159661.i	−0.333214	−	0.192381i	−0.657267	+	0.753658i	$0.728288\pi$
$912$		0			0
$913$	−68714.8	−	119017.i	−0.0824344	−	0.142781i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			491312.i			0.584277i
$918$		0			0
$919$		−	1.48790e6i		−	1.76175i	−0.473350	−	0.880875i	$-0.656955\pi$
$919$		−	1.48790e6i		−	1.76175i	0.473350	−	0.880875i	$-0.343045\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−487151.	−	843771.i	−0.571821	−	0.990424i
$924$		0			0
$925$	−674422.	−	389378.i	−0.788222	−	0.455080i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−588618.	+	1.01952e6i	−0.682028	+	1.18131i	0.292333	+	0.956317i	$0.405569\pi$
$929$	−588618.	+	1.01952e6i	−0.682028	+	1.18131i	−0.974361	+	0.224991i	$0.927765\pi$
$930$		0			0
$931$	−50780.5	−	87954.5i	−0.0585865	−	0.101475i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			14101.6i			0.0161304i
$936$		0			0
$937$	904036.			1.02969			0.514845	−	0.857283i	$-0.327849\pi$
$937$	904036.			1.02969			0.514845	+	0.857283i	$0.327849\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	569128.	−	328586.i	0.642733	−	0.371082i	−0.142934	−	0.989732i	$-0.545654\pi$
$941$	569128.	−	328586.i	0.642733	−	0.371082i	0.785666	+	0.618650i	$0.212320\pi$
$942$		0			0
$943$	−474393.	−	273891.i	−0.533476	−	0.308003i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	100820.	−	174625.i	0.112420	−	0.194718i	−0.804325	−	0.594189i	$-0.797473\pi$
$947$	100820.	−	174625.i	0.112420	−	0.194718i	0.916746	+	0.399471i	$0.130806\pi$
$948$		0			0
$949$	1.64752e6	−	951194.i	1.82935	−	1.05618i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	638389.			0.702910			0.351455	−	0.936205i	$-0.385687\pi$
$953$	638389.			0.702910			0.351455	+	0.936205i	$0.385687\pi$
$954$		0			0
$955$	107011.			0.117333
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−213075.	+	123019.i	−0.231684	+	0.133763i
$960$		0			0
$961$	−380552.	+	659136.i	−0.412067	+	0.713721i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	63737.0	+	36798.6i	0.0684443	+	0.0395163i
$966$		0			0
$967$	961591.	−	555175.i	1.02834	−	0.593714i	0.111833	−	0.993727i	$-0.464328\pi$
$967$	961591.	−	555175.i	1.02834	−	0.593714i	0.916509	+	0.400013i	$0.130995\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.25193e6			−1.32783			−0.663913	−	0.747810i	$-0.731105\pi$
$971$	−1.25193e6			−1.32783			−0.663913	+	0.747810i	$0.731105\pi$
$972$		0			0
$973$		−	382787.i		−	0.404326i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−33214.5	−	57529.3i	−0.0347968	−	0.0602698i	0.848103	−	0.529832i	$-0.177745\pi$
$977$	−33214.5	−	57529.3i	−0.0347968	−	0.0602698i	−0.882899	+	0.469562i	$0.844412\pi$
$978$		0			0
$979$	−160219.	+	277508.i	−0.167167	+	0.289541i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	130314.	+	75236.9i	0.134860	+	0.0778617i	0.565912	−	0.824465i	$-0.308524\pi$
$983$	130314.	+	75236.9i	0.134860	+	0.0778617i	−0.431052	+	0.902327i	$0.641857\pi$
$984$		0			0
$985$	41815.2	+	72426.0i	0.0430984	+	0.0746487i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	335865.i		−	0.343378i
$990$		0			0
$991$		−	795870.i		−	0.810391i	−0.914230	−	0.405196i	$-0.867203\pi$
$991$		−	795870.i		−	0.810391i	0.914230	−	0.405196i	$-0.132797\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	93644.4	+	162197.i	0.0945879	+	0.163831i
$996$		0			0
$997$	−674456.	−	389398.i	−0.678521	−	0.391745i	0.120776	−	0.992680i	$-0.461462\pi$
$997$	−674456.	−	389398.i	−0.678521	−	0.391745i	−0.799298	+	0.600935i	$0.794795\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.5.t.b.559.20		88
3.2	odd	2		288.5.t.b.79.16		88
4.3	odd	2		216.5.p.b.19.6		88
8.3	odd	2	inner	864.5.t.b.559.25		88
8.5	even	2		216.5.p.b.19.35		88
9.4	even	3	inner	864.5.t.b.847.25		88
9.5	odd	6		288.5.t.b.175.15		88
12.11	even	2		72.5.p.b.43.39	yes	88
24.5	odd	2		72.5.p.b.43.10	✓	88
24.11	even	2		288.5.t.b.79.15		88
36.23	even	6		72.5.p.b.67.10	yes	88
36.31	odd	6		216.5.p.b.91.35		88
72.5	odd	6		72.5.p.b.67.39	yes	88
72.13	even	6		216.5.p.b.91.6		88
72.59	even	6		288.5.t.b.175.16		88
72.67	odd	6	inner	864.5.t.b.847.20		88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.5.p.b.43.10	✓	88	24.5	odd	2
72.5.p.b.43.39	yes	88	12.11	even	2
72.5.p.b.67.10	yes	88	36.23	even	6
72.5.p.b.67.39	yes	88	72.5	odd	6
216.5.p.b.19.6		88	4.3	odd	2
216.5.p.b.19.35		88	8.5	even	2
216.5.p.b.91.6		88	72.13	even	6
216.5.p.b.91.35		88	36.31	odd	6
288.5.t.b.79.15		88	24.11	even	2
288.5.t.b.79.16		88	3.2	odd	2
288.5.t.b.175.15		88	9.5	odd	6
288.5.t.b.175.16		88	72.59	even	6
864.5.t.b.559.20		88	1.1	even	1	trivial
864.5.t.b.559.25		88	8.3	odd	2	inner
864.5.t.b.847.20		88	72.67	odd	6	inner
864.5.t.b.847.25		88	9.4	even	3	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	864.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.53773 + 1.46516i) q^{5} +(-39.8426 - 23.0032i) q^{7} +O(q^{10})\)	\(q+(-2.53773 + 1.46516i) q^{5} +(-39.8426 - 23.0032i) q^{7} +(27.1224 - 46.9774i) q^{11} +(174.611 - 100.811i) q^{13} +88.7146 q^{17} +357.083 q^{19} +(-214.383 + 123.774i) q^{23} +(-308.207 + 533.830i) q^{25} +(18.7746 + 10.8395i) q^{29} +(349.016 - 201.504i) q^{31} +134.813 q^{35} +1263.37i q^{37} +(1106.42 + 1916.37i) q^{41} +(-678.384 + 1175.00i) q^{43} +(-3120.48 - 1801.61i) q^{47} +(-142.209 - 246.314i) q^{49} -4122.82i q^{53} +158.955i q^{55} +(-2643.76 - 4579.12i) q^{59} +(4778.25 + 2758.72i) q^{61} +(-295.410 + 511.664i) q^{65} +(-3090.68 - 5353.21i) q^{67} -4832.30i q^{71} +9435.38 q^{73} +(-2161.26 + 1247.80i) q^{77} +(-1099.86 - 635.004i) q^{79} +(1266.75 - 2194.08i) q^{83} +(-225.134 + 129.981i) q^{85} -5907.27 q^{89} -9275.93 q^{91} +(-906.180 + 523.183i) q^{95} +(-928.999 + 1609.07i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q+O(q^{10}) \) 88 * q	\( 88 q + 44 q^{11} + 1156 q^{17} - 860 q^{19} + 5998 q^{25} - 2508 q^{35} - 2348 q^{41} - 3500 q^{43} + 16462 q^{49} - 3508 q^{59} + 2502 q^{65} - 5132 q^{67} + 19004 q^{73} + 17090 q^{83} + 8272 q^{89} + 9612 q^{91} + 9980 q^{97}+O(q^{100}) \) 88 * q + 44 * q^11 + 1156 * q^17 - 860 * q^19 + 5998 * q^25 - 2508 * q^35 - 2348 * q^41 - 3500 * q^43 + 16462 * q^49 - 3508 * q^59 + 2502 * q^65 - 5132 * q^67 + 19004 * q^73 + 17090 * q^83 + 8272 * q^89 + 9612 * q^91 + 9980 * q^97

Introduction

L-functions

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