LMFDB - Embedded newform 96.8.f.c.47.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,8,Mod(47,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("96.47");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 96 = 2^{5} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	96.f (of order $2$, degree $1$, 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:	$29.9889624465$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 274 x^{18} + 45864 x^{16} - 5031360 x^{14} + 776389632 x^{12} - 102828146688 x^{10} + \cdots + 11\!\cdots\!24 $ x^20 - 274x^18 + 45864x^16 - 5031360x^14 + 776389632x^12 - 102828146688x^10 + 12720367730688x^8 - 1350595415900160x^6 + 201712005185273856x^4 - 19743780766392254464*x^2 + 1180591620717411303424
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{100}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.15
Root			$-10.4702 - 4.28654i$ of defining polynomial
Character	$\chi$	$=$	96.47
Dual form			96.8.f.c.47.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(24.1914 + 40.0222i) q^{3} -477.520 q^{5} -632.635i q^{7} +(-1016.56 + 1936.38i) q^{9} +O(q^{10})$	\(q+(24.1914 + 40.0222i) q^{3} -477.520 q^{5} -632.635i q^{7} +(-1016.56 + 1936.38i) q^{9} +4849.09i q^{11} -3229.57i q^{13} +(-11551.9 - 19111.4i) q^{15} -32288.3i q^{17} +21527.2 q^{19} +(25319.5 - 15304.3i) q^{21} -5755.11 q^{23} +149900. q^{25} +(-102090. + 6159.01i) q^{27} +131879. q^{29} -147516. i q^{31} +(-194071. + 117306. i) q^{33} +302096. i q^{35} +19019.9i q^{37} +(129254. - 78127.7i) q^{39} -42800.6i q^{41} +325765. q^{43} +(485425. - 924662. i) q^{45} +976107. q^{47} +423316. q^{49} +(1.29225e6 - 781099. i) q^{51} -554605. q^{53} -2.31554e6i q^{55} +(520772. + 861566. i) q^{57} -535369. i q^{59} -3.23726e6i q^{61} +(1.22502e6 + 643109. i) q^{63} +1.54218e6i q^{65} -3.19490e6 q^{67} +(-139224. - 230332. i) q^{69} -2.17151e6 q^{71} +569345. q^{73} +(3.62629e6 + 5.99934e6i) q^{75} +3.06771e6 q^{77} -1.27485e6i q^{79} +(-2.71620e6 - 3.93688e6i) q^{81} -3.57654e6i q^{83} +1.54183e7i q^{85} +(3.19035e6 + 5.27811e6i) q^{87} -6.34057e6i q^{89} -2.04314e6 q^{91} +(5.90394e6 - 3.56863e6i) q^{93} -1.02797e7 q^{95} +7.13949e6 q^{97} +(-9.38970e6 - 4.92937e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 120 q^{3} + 5076 q^{9}+O(q^{10}) $ 20 * q - 120 * q^3 + 5076 * q^9	$ 20 q - 120 q^{3} + 5076 q^{9} - 133744 q^{19} + 711596 q^{25} - 85320 q^{27} + 263640 q^{33} + 2292080 q^{43} + 2495228 q^{49} + 7417536 q^{51} - 5067120 q^{57} - 3654640 q^{67} + 19892680 q^{73} + 8612088 q^{75} - 6817932 q^{81} - 67826976 q^{91} + 5561800 q^{97} - 10152864 q^{99}+O(q^{100}) $ 20 * q - 120 * q^3 + 5076 * q^9 - 133744 * q^19 + 711596 * q^25 - 85320 * q^27 + 263640 * q^33 + 2292080 * q^43 + 2495228 * q^49 + 7417536 * q^51 - 5067120 * q^57 - 3654640 * q^67 + 19892680 * q^73 + 8612088 * q^75 - 6817932 * q^81 - 67826976 * q^91 + 5561800 * q^97 - 10152864 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$65$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	24.1914	+	40.0222i	0.517292	+	0.855809i
$3$	24.1914	+	40.0222i	0.517292	+	0.855809i
$4$		0			0
$5$	−477.520			−1.70843			−0.854214	−	0.519922i	$-0.825961\pi$
$5$	−477.520			−1.70843			−0.854214	+	0.519922i	$0.825961\pi$
$6$		0			0
$7$		−	632.635i		−	0.697124i	−0.937286	−	0.348562i	$-0.886670\pi$
$7$		−	632.635i		−	0.697124i	0.937286	−	0.348562i	$-0.113330\pi$
$8$		0			0
$9$	−1016.56	+	1936.38i	−0.464817	+	0.885407i
$10$		0			0
$11$			4849.09i			1.09846i	0.835670	+	0.549232i	$0.185080\pi$
$11$			4849.09i			1.09846i	−0.835670	+	0.549232i	$0.814920\pi$
$12$		0			0
$13$		−	3229.57i		−	0.407702i	−0.979002	−	0.203851i	$-0.934654\pi$
$13$		−	3229.57i		−	0.407702i	0.979002	−	0.203851i	$-0.0653458\pi$
$14$		0			0
$15$	−11551.9	−	19111.4i	−0.883756	−	1.46209i
$16$		0			0
$17$		−	32288.3i		−	1.59395i	−0.604013	−	0.796974i	$-0.706432\pi$
$17$		−	32288.3i		−	1.59395i	0.604013	−	0.796974i	$-0.293568\pi$
$18$		0			0
$19$	21527.2			0.720029			0.360014	−	0.932947i	$-0.382772\pi$
$19$	21527.2			0.720029			0.360014	+	0.932947i	$0.382772\pi$
$20$		0			0
$21$	25319.5	−	15304.3i	0.596605	−	0.360617i
$22$		0			0
$23$	−5755.11			−0.0986294			−0.0493147	−	0.998783i	$-0.515704\pi$
$23$	−5755.11			−0.0986294			−0.0493147	+	0.998783i	$0.515704\pi$
$24$		0			0
$25$	149900.			1.91872
$26$		0			0
$27$	−102090.	+	6159.01i	−0.998185	+	0.0602195i
$28$		0			0
$29$	131879.			1.00412			0.502058	−	0.864834i	$-0.332576\pi$
$29$	131879.			1.00412			0.502058	+	0.864834i	$0.332576\pi$
$30$		0			0
$31$		−	147516.i		−	0.889354i	−0.895691	−	0.444677i	$-0.853318\pi$
$31$		−	147516.i		−	0.889354i	0.895691	−	0.444677i	$-0.146682\pi$
$32$		0			0
$33$	−194071.	+	117306.i	−0.940075	+	0.568227i
$34$		0			0
$35$			302096.i			1.19099i
$36$		0			0
$37$			19019.9i			0.0617309i	0.999524	+	0.0308655i	$0.00982634\pi$
$37$			19019.9i			0.0617309i	−0.999524	+	0.0308655i	$0.990174\pi$
$38$		0			0
$39$	129254.	−	78127.7i	0.348915	−	0.210901i
$40$		0			0
$41$		−	42800.6i		−	0.0969855i	−0.998824	−	0.0484927i	$-0.984558\pi$
$41$		−	42800.6i		−	0.0969855i	0.998824	−	0.0484927i	$-0.0154418\pi$
$42$		0			0
$43$	325765.			0.624834			0.312417	−	0.949945i	$-0.398861\pi$
$43$	325765.			0.624834			0.312417	+	0.949945i	$0.398861\pi$
$44$		0			0
$45$	485425.	−	924662.i	0.794106	−	1.51265i
$46$		0			0
$47$	976107.			1.37137			0.685686	−	0.727898i	$-0.259502\pi$
$47$	976107.			1.37137			0.685686	+	0.727898i	$0.259502\pi$
$48$		0			0
$49$	423316.			0.514018
$50$		0			0
$51$	1.29225e6	−	781099.i	1.36412	−	0.824538i
$52$		0			0
$53$	−554605.			−0.511703			−0.255852	−	0.966716i	$-0.582356\pi$
$53$	−554605.			−0.511703			−0.255852	+	0.966716i	$0.582356\pi$
$54$		0			0
$55$		−	2.31554e6i		−	1.87665i
$56$		0			0
$57$	520772.	+	861566.i	0.372465	+	0.616207i
$58$		0			0
$59$		−	535369.i		−	0.339368i	−0.985498	−	0.169684i	$-0.945725\pi$
$59$		−	535369.i		−	0.339368i	0.985498	−	0.169684i	$-0.0542748\pi$
$60$		0			0
$61$		−	3.23726e6i		−	1.82609i	−0.407856	−	0.913046i	$-0.633724\pi$
$61$		−	3.23726e6i		−	1.82609i	0.407856	−	0.913046i	$-0.366276\pi$
$62$		0			0
$63$	1.22502e6	+	643109.i	0.617238	+	0.324035i
$64$		0			0
$65$			1.54218e6i			0.696529i
$66$		0			0
$67$	−3.19490e6			−1.29776			−0.648882	−	0.760889i	$-0.724763\pi$
$67$	−3.19490e6			−1.29776			−0.648882	+	0.760889i	$0.724763\pi$
$68$		0			0
$69$	−139224.	−	230332.i	−0.0510203	−	0.0844079i
$70$		0			0
$71$	−2.17151e6			−0.720042			−0.360021	−	0.932944i	$-0.617230\pi$
$71$	−2.17151e6			−0.720042			−0.360021	+	0.932944i	$0.617230\pi$
$72$		0			0
$73$	569345.			0.171295			0.0856477	−	0.996325i	$-0.472704\pi$
$73$	569345.			0.171295			0.0856477	+	0.996325i	$0.472704\pi$
$74$		0			0
$75$	3.62629e6	+	5.99934e6i	0.992541	+	1.64206i
$76$		0			0
$77$	3.06771e6			0.765766
$78$		0			0
$79$		−	1.27485e6i		−	0.290914i	−0.989365	−	0.145457i	$-0.953535\pi$
$79$		−	1.27485e6i		−	0.290914i	0.989365	−	0.145457i	$-0.0464652\pi$
$80$		0			0
$81$	−2.71620e6	−	3.93688e6i	−0.567890	−	0.823104i
$82$		0			0
$83$		−	3.57654e6i		−	0.686578i	−0.939230	−	0.343289i	$-0.888459\pi$
$83$		−	3.57654e6i		−	0.686578i	0.939230	−	0.343289i	$-0.111541\pi$
$84$		0			0
$85$			1.54183e7i			2.72315i
$86$		0			0
$87$	3.19035e6	+	5.27811e6i	0.519422	+	0.859332i
$88$		0			0
$89$		−	6.34057e6i		−	0.953374i	−0.879073	−	0.476687i	$-0.841837\pi$
$89$		−	6.34057e6i		−	0.953374i	0.879073	−	0.476687i	$-0.158163\pi$
$90$		0			0
$91$	−2.04314e6			−0.284219
$92$		0			0
$93$	5.90394e6	−	3.56863e6i	0.761117	−	0.460056i
$94$		0			0
$95$	−1.02797e7			−1.23012
$96$		0			0
$97$	7.13949e6			0.794266			0.397133	−	0.917761i	$-0.370005\pi$
$97$	7.13949e6			0.794266			0.397133	+	0.917761i	$0.370005\pi$
$98$		0			0
$99$	−9.38970e6	−	4.92937e6i	−0.972588	−	0.510585i
$100$		0			0
$101$	−1.83155e7			−1.76886			−0.884429	−	0.466675i	$-0.845452\pi$
$101$	−1.83155e7			−1.76886			−0.884429	+	0.466675i	$0.845452\pi$
$102$		0			0
$103$			7.88089e6i			0.710633i	0.934746	+	0.355316i	$0.115627\pi$
$103$			7.88089e6i			0.710633i	−0.934746	+	0.355316i	$0.884373\pi$
$104$		0			0
$105$	−1.20905e7	+	7.30811e6i	−1.01926	+	0.616088i
$106$		0			0
$107$			5.65137e6i			0.445975i	0.974821	+	0.222988i	$0.0715809\pi$
$107$			5.65137e6i			0.445975i	−0.974821	+	0.222988i	$0.928419\pi$
$108$		0			0
$109$			525457.i			0.0388637i	0.999811	+	0.0194319i	$0.00618574\pi$
$109$			525457.i			0.0388637i	−0.999811	+	0.0194319i	$0.993814\pi$
$110$		0			0
$111$	−761219.	+	460118.i	−0.0528299	+	0.0319329i
$112$		0			0
$113$			2.69478e6i			0.175691i	0.996134	+	0.0878454i	$0.0279982\pi$
$113$			2.69478e6i			0.175691i	−0.996134	+	0.0878454i	$0.972002\pi$
$114$		0			0
$115$	2.74818e6			0.168501
$116$		0			0
$117$	6.25369e6	+	3.28303e6i	0.360982	+	0.189507i
$118$		0			0
$119$	−2.04267e7			−1.11118
$120$		0			0
$121$	−4.02651e6			−0.206624
$122$		0			0
$123$	1.71298e6	−	1.03541e6i	0.0830010	−	0.0501699i
$124$		0			0
$125$	−3.42741e7			−1.56957
$126$		0			0
$127$			2.13324e7i			0.924117i	0.886849	+	0.462059i	$0.152889\pi$
$127$			2.13324e7i			0.924117i	−0.886849	+	0.462059i	$0.847111\pi$
$128$		0			0
$129$	7.88069e6	+	1.30378e7i	0.323222	+	0.534738i
$130$		0			0
$131$		−	4.33587e7i		−	1.68510i	−0.538615	−	0.842552i	$-0.681052\pi$
$131$		−	4.33587e7i		−	1.68510i	0.538615	−	0.842552i	$-0.318948\pi$
$132$		0			0
$133$		−	1.36189e7i		−	0.501950i
$134$		0			0
$135$	4.87501e7	−	2.94105e6i	1.70533	−	0.102881i
$136$		0			0
$137$		−	3.25183e7i		−	1.08045i	−0.841520	−	0.540227i	$-0.818338\pi$
$137$		−	3.25183e7i		−	1.08045i	0.841520	−	0.540227i	$-0.181662\pi$
$138$		0			0
$139$	1.58632e7			0.501000			0.250500	−	0.968117i	$-0.419405\pi$
$139$	1.58632e7			0.501000			0.250500	+	0.968117i	$0.419405\pi$
$140$		0			0
$141$	2.36134e7	+	3.90660e7i	0.709400	+	1.17363i
$142$		0			0
$143$	1.56605e7			0.447846
$144$		0			0
$145$	−6.29751e7			−1.71546
$146$		0			0
$147$	1.02406e7	+	1.69420e7i	0.265898	+	0.439901i
$148$		0			0
$149$	9.55034e6			0.236519			0.118260	−	0.992983i	$-0.462268\pi$
$149$	9.55034e6			0.236519			0.118260	+	0.992983i	$0.462268\pi$
$150$		0			0
$151$		−	1.00995e7i		−	0.238715i	−0.992851	−	0.119358i	$-0.961917\pi$
$151$		−	1.00995e7i		−	0.238715i	0.992851	−	0.119358i	$-0.0380835\pi$
$152$		0			0
$153$	6.25226e7	+	3.28229e7i	1.41129	+	0.740895i
$154$		0			0
$155$			7.04420e7i			1.51940i
$156$		0			0
$157$		−	1.83230e7i		−	0.377875i	−0.981989	−	0.188938i	$-0.939496\pi$
$157$		−	1.83230e7i		−	0.377875i	0.981989	−	0.188938i	$-0.0605044\pi$
$158$		0			0
$159$	−1.34167e7	−	2.21965e7i	−0.264700	−	0.437920i
$160$		0			0
$161$			3.64089e6i			0.0687570i
$162$		0			0
$163$	−4.97051e7			−0.898968			−0.449484	−	0.893288i	$-0.648392\pi$
$163$	−4.97051e7			−0.898968			−0.449484	+	0.893288i	$0.648392\pi$
$164$		0			0
$165$	9.26729e7	−	5.60160e7i	1.60605	−	0.970775i
$166$		0			0
$167$	−9.61596e7			−1.59766			−0.798832	−	0.601555i	$-0.794548\pi$
$167$	−9.61596e7			−1.59766			−0.798832	+	0.601555i	$0.794548\pi$
$168$		0			0
$169$	5.23184e7			0.833779
$170$		0			0
$171$	−2.18836e7	+	4.16849e7i	−0.334682	+	0.637518i
$172$		0			0
$173$	−2.49679e7			−0.366624			−0.183312	−	0.983055i	$-0.558682\pi$
$173$	−2.49679e7			−0.366624			−0.183312	+	0.983055i	$0.558682\pi$
$174$		0			0
$175$		−	9.48322e7i		−	1.33759i
$176$		0			0
$177$	2.14267e7	−	1.29513e7i	0.290434	−	0.175553i
$178$		0			0
$179$		−	7.35170e7i		−	0.958081i	−0.877793	−	0.479040i	$-0.840985\pi$
$179$		−	7.35170e7i		−	0.958081i	0.877793	−	0.479040i	$-0.159015\pi$
$180$		0			0
$181$		−	1.26712e8i		−	1.58833i	−0.607699	−	0.794167i	$-0.707907\pi$
$181$		−	1.26712e8i		−	1.58833i	0.607699	−	0.794167i	$-0.292093\pi$
$182$		0			0
$183$	1.29562e8	−	7.83137e7i	1.56279	−	0.944624i
$184$		0			0
$185$		−	9.08239e6i		−	0.105463i
$186$		0			0
$187$	1.56569e8			1.75090
$188$		0			0
$189$	3.89640e6	+	6.45859e7i	0.0419805	+	0.695859i
$190$		0			0
$191$	4.33405e7			0.450067			0.225034	−	0.974351i	$-0.427751\pi$
$191$	4.33405e7			0.450067			0.225034	+	0.974351i	$0.427751\pi$
$192$		0			0
$193$	7.11547e7			0.712448			0.356224	−	0.934401i	$-0.384064\pi$
$193$	7.11547e7			0.712448			0.356224	+	0.934401i	$0.384064\pi$
$194$		0			0
$195$	−6.17216e7	+	3.73075e7i	−0.596096	+	0.360309i
$196$		0			0
$197$	2.11807e7			0.197383			0.0986913	−	0.995118i	$-0.468534\pi$
$197$	2.11807e7			0.197383			0.0986913	+	0.995118i	$0.468534\pi$
$198$		0			0
$199$			2.72858e7i			0.245443i	0.992441	+	0.122722i	$0.0391622\pi$
$199$			2.72858e7i			0.245443i	−0.992441	+	0.122722i	$0.960838\pi$
$200$		0			0
$201$	−7.72891e7	−	1.27867e8i	−0.671323	−	1.11064i
$202$		0			0
$203$		−	8.34316e7i		−	0.699994i
$204$		0			0
$205$			2.04382e7i			0.165693i
$206$		0			0
$207$	5.85039e6	−	1.11441e7i	0.0458447	−	0.0873272i
$208$		0			0
$209$			1.04387e8i			0.790926i
$210$		0			0
$211$	1.10815e8			0.812102			0.406051	−	0.913850i	$-0.366905\pi$
$211$	1.10815e8			0.812102			0.406051	+	0.913850i	$0.366905\pi$
$212$		0			0
$213$	−5.25318e7	−	8.69087e7i	−0.372472	−	0.616218i
$214$		0			0
$215$	−1.55559e8			−1.06748
$216$		0			0
$217$	−9.33241e7			−0.619990
$218$		0			0
$219$	1.37732e7	+	2.27865e7i	0.0886098	+	0.146596i
$220$		0			0
$221$	−1.04277e8			−0.649856
$222$		0			0
$223$			2.73113e8i			1.64921i	0.565710	+	0.824604i	$0.308602\pi$
$223$			2.73113e8i			1.64921i	−0.565710	+	0.824604i	$0.691398\pi$
$224$		0			0
$225$	−1.52382e8	+	2.90264e8i	−0.891855	+	1.69885i
$226$		0			0
$227$			1.32485e8i			0.751753i	0.926670	+	0.375876i	$0.122658\pi$
$227$			1.32485e8i			0.751753i	−0.926670	+	0.375876i	$0.877342\pi$
$228$		0			0
$229$			3.24000e8i			1.78287i	0.453145	+	0.891437i	$0.350302\pi$
$229$			3.24000e8i			1.78287i	−0.453145	+	0.891437i	$0.649698\pi$
$230$		0			0
$231$	7.42120e7	+	1.22776e8i	0.396125	+	0.655349i
$232$		0			0
$233$		−	5.56285e7i		−	0.288106i	−0.989570	−	0.144053i	$-0.953986\pi$
$233$		−	5.56285e7i		−	0.288106i	0.989570	−	0.144053i	$-0.0460135\pi$
$234$		0			0
$235$	−4.66111e8			−2.34289
$236$		0			0
$237$	5.10223e7	−	3.08404e7i	0.248966	−	0.150487i
$238$		0			0
$239$	3.45579e8			1.63740			0.818699	−	0.574222i	$-0.194696\pi$
$239$	3.45579e8			1.63740			0.818699	+	0.574222i	$0.194696\pi$
$240$		0			0
$241$	−2.73635e7			−0.125925			−0.0629625	−	0.998016i	$-0.520055\pi$
$241$	−2.73635e7			−0.125925			−0.0629625	+	0.998016i	$0.520055\pi$
$242$		0			0
$243$	9.18542e7	−	2.03947e8i	0.410655	−	0.911791i
$244$		0			0
$245$	−2.02142e8			−0.878162
$246$		0			0
$247$		−	6.95236e7i		−	0.293557i
$248$		0			0
$249$	1.43141e8	−	8.65215e7i	0.587580	−	0.355162i
$250$		0			0
$251$		−	2.95932e8i		−	1.18123i	−0.806955	−	0.590614i	$-0.798886\pi$
$251$		−	2.95932e8i		−	1.18123i	0.806955	−	0.590614i	$-0.201114\pi$
$252$		0			0
$253$		−	2.79071e7i		−	0.108341i
$254$		0			0
$255$	−6.17075e8	+	3.72990e8i	−2.33049	+	1.40866i
$256$		0			0
$257$		−	2.71775e8i		−	0.998720i	−0.866395	−	0.499360i	$-0.833568\pi$
$257$		−	2.71775e8i		−	0.998720i	0.866395	−	0.499360i	$-0.166432\pi$
$258$		0			0
$259$	1.20327e7			0.0430341
$260$		0			0
$261$	−1.34063e8	+	2.55369e8i	−0.466731	+	0.889052i
$262$		0			0
$263$	−3.32668e8			−1.12763			−0.563814	−	0.825901i	$-0.690667\pi$
$263$	−3.32668e8			−1.12763			−0.563814	+	0.825901i	$0.690667\pi$
$264$		0			0
$265$	2.64835e8			0.874208
$266$		0			0
$267$	2.53764e8	−	1.53387e8i	0.815906	−	0.493173i
$268$		0			0
$269$	3.63197e8			1.13765			0.568825	−	0.822458i	$-0.307398\pi$
$269$	3.63197e8			1.13765			0.568825	+	0.822458i	$0.307398\pi$
$270$		0			0
$271$		−	5.75287e8i		−	1.75587i	−0.478782	−	0.877934i	$-0.658922\pi$
$271$		−	5.75287e8i		−	1.75587i	0.478782	−	0.877934i	$-0.341078\pi$
$272$		0			0
$273$	−4.94263e7	−	8.17709e7i	−0.147024	−	0.243237i
$274$		0			0
$275$			7.26880e8i			2.10765i
$276$		0			0
$277$			3.54054e8i			1.00090i	0.865765	+	0.500450i	$0.166832\pi$
$277$			3.54054e8i			1.00090i	−0.865765	+	0.500450i	$0.833168\pi$
$278$		0			0
$279$	2.85649e8	+	1.49959e8i	0.787440	+	0.413387i
$280$		0			0
$281$		−	3.30323e8i		−	0.888110i	−0.896000	−	0.444055i	$-0.853539\pi$
$281$		−	3.30323e8i		−	0.888110i	0.896000	−	0.444055i	$-0.146461\pi$
$282$		0			0
$283$	−2.36463e8			−0.620169			−0.310085	−	0.950709i	$-0.600357\pi$
$283$	−2.36463e8			−0.620169			−0.310085	+	0.950709i	$0.600357\pi$
$284$		0			0
$285$	−2.48679e8	−	4.11415e8i	−0.636330	−	1.05274i
$286$		0			0
$287$	−2.70772e7			−0.0676109
$288$		0			0
$289$	−6.32198e8			−1.54067
$290$		0			0
$291$	1.72714e8	+	2.85738e8i	0.410868	+	0.679740i
$292$		0			0
$293$	1.62422e8			0.377232			0.188616	−	0.982051i	$-0.439600\pi$
$293$	1.62422e8			0.377232			0.188616	+	0.982051i	$0.439600\pi$
$294$		0			0
$295$			2.55649e8i			0.579786i
$296$		0			0
$297$	−2.98656e7	−	4.95045e8i	−0.0661490	−	1.09647i
$298$		0			0
$299$			1.85865e7i			0.0402114i
$300$		0			0
$301$		−	2.06090e8i		−	0.435587i
$302$		0			0
$303$	−4.43076e8	−	7.33025e8i	−0.915016	−	1.51380i
$304$		0			0
$305$			1.54585e9i			3.11975i
$306$		0			0
$307$	−1.99590e8			−0.393690			−0.196845	−	0.980435i	$-0.563070\pi$
$307$	−1.99590e8			−0.393690			−0.196845	+	0.980435i	$0.563070\pi$
$308$		0			0
$309$	−3.15411e8	+	1.90650e8i	−0.608166	+	0.367605i
$310$		0			0
$311$	5.88253e8			1.10893			0.554464	−	0.832208i	$-0.312923\pi$
$311$	5.88253e8			1.10893			0.554464	+	0.832208i	$0.312923\pi$
$312$		0			0
$313$	9.59109e8			1.76792			0.883961	−	0.467561i	$-0.154867\pi$
$313$	9.59109e8			1.76792			0.883961	+	0.467561i	$0.154867\pi$
$314$		0			0
$315$	−5.84974e8	−	3.07097e8i	−1.05451	−	0.553591i
$316$		0			0
$317$	5.95992e8			1.05083			0.525416	−	0.850846i	$-0.323910\pi$
$317$	5.95992e8			1.05083			0.525416	+	0.850846i	$0.323910\pi$
$318$		0			0
$319$			6.39496e8i			1.10299i
$320$		0			0
$321$	−2.26180e8	+	1.36714e8i	−0.381669	+	0.230700i
$322$		0			0
$323$		−	6.95077e8i		−	1.14769i
$324$		0			0
$325$		−	4.84113e8i		−	0.782267i
$326$		0			0
$327$	−2.10300e7	+	1.27115e7i	−0.0332599	+	0.0201039i
$328$		0			0
$329$		−	6.17520e8i		−	0.956016i
$330$		0			0
$331$	1.23121e9			1.86609			0.933047	−	0.359755i	$-0.117140\pi$
$331$	1.23121e9			1.86609			0.933047	+	0.359755i	$0.117140\pi$
$332$		0			0
$333$	−3.68299e7	−	1.93348e7i	−0.0546570	−	0.0286936i
$334$		0			0
$335$	1.52563e9			2.21714
$336$		0			0
$337$	1.59873e8			0.227546			0.113773	−	0.993507i	$-0.463706\pi$
$337$	1.59873e8			0.227546			0.113773	+	0.993507i	$0.463706\pi$
$338$		0			0
$339$	−1.07851e8	+	6.51905e7i	−0.150358	+	0.0908835i
$340$		0			0
$341$	7.15321e8			0.976923
$342$		0			0
$343$		−	7.88807e8i		−	1.05546i
$344$		0			0
$345$	6.64823e7	+	1.09988e8i	0.0871644	+	0.144205i
$346$		0			0
$347$			2.81294e8i			0.361416i	0.983537	+	0.180708i	$0.0578390\pi$
$347$			2.81294e8i			0.361416i	−0.983537	+	0.180708i	$0.942161\pi$
$348$		0			0
$349$		−	3.00712e8i		−	0.378671i	−0.981912	−	0.189336i	$-0.939367\pi$
$349$		−	3.00712e8i		−	0.378671i	0.981912	−	0.189336i	$-0.0606334\pi$
$350$		0			0
$351$	1.98909e7	+	3.29707e8i	0.0245516	+	0.406962i
$352$		0			0
$353$			6.30246e8i			0.762603i	0.924451	+	0.381302i	$0.124524\pi$
$353$			6.30246e8i			0.762603i	−0.924451	+	0.381302i	$0.875476\pi$
$354$		0			0
$355$	1.03694e9			1.23014
$356$		0			0
$357$	−4.94151e8	−	8.17523e8i	−0.574805	−	0.950958i
$358$		0			0
$359$	−1.05605e7			−0.0120463			−0.00602316	−	0.999982i	$-0.501917\pi$
$359$	−1.05605e7			−0.0120463			−0.00602316	+	0.999982i	$0.501917\pi$
$360$		0			0
$361$	−4.30452e8			−0.481559
$362$		0			0
$363$	−9.74068e7	−	1.61150e8i	−0.106885	−	0.176830i
$364$		0			0
$365$	−2.71874e8			−0.292646
$366$		0			0
$367$		−	5.74355e8i		−	0.606526i	−0.952907	−	0.303263i	$-0.901924\pi$
$367$		−	5.74355e8i		−	0.606526i	0.952907	−	0.303263i	$-0.0980761\pi$
$368$		0			0
$369$	8.28785e7	+	4.35092e7i	0.0858716	+	0.0450805i
$370$		0			0
$371$			3.50863e8i			0.356721i
$372$		0			0
$373$		−	1.54165e9i		−	1.53817i	−0.639148	−	0.769084i	$-0.720713\pi$
$373$		−	1.54165e9i		−	1.53817i	0.639148	−	0.769084i	$-0.279287\pi$
$374$		0			0
$375$	−8.29137e8	−	1.37173e9i	−0.811927	−	1.34325i
$376$		0			0
$377$		−	4.25914e8i		−	0.409380i
$378$		0			0
$379$	−8.07688e8			−0.762090			−0.381045	−	0.924556i	$-0.624436\pi$
$379$	−8.07688e8			−0.762090			−0.381045	+	0.924556i	$0.624436\pi$
$380$		0			0
$381$	−8.53771e8	+	5.16060e8i	−0.790868	+	0.478039i
$382$		0			0
$383$	−7.50741e8			−0.682801			−0.341401	−	0.939918i	$-0.610901\pi$
$383$	−7.50741e8			−0.682801			−0.341401	+	0.939918i	$0.610901\pi$
$384$		0			0
$385$	−1.46489e9			−1.30826
$386$		0			0
$387$	−3.31158e8	+	6.30806e8i	−0.290433	+	0.553232i
$388$		0			0
$389$	5.35813e8			0.461519			0.230760	−	0.973011i	$-0.425879\pi$
$389$	5.35813e8			0.461519			0.230760	+	0.973011i	$0.425879\pi$
$390$		0			0
$391$			1.85823e8i			0.157210i
$392$		0			0
$393$	1.73531e9	−	1.04891e9i	1.44213	−	0.871692i
$394$		0			0
$395$			6.08766e8i			0.497005i
$396$		0			0
$397$			1.67867e9i			1.34648i	0.739426	+	0.673238i	$0.235097\pi$
$397$			1.67867e9i			1.34648i	−0.739426	+	0.673238i	$0.764903\pi$
$398$		0			0
$399$	5.45057e8	−	3.29459e8i	0.429573	−	0.259655i
$400$		0			0
$401$			1.55050e9i			1.20079i	0.799705	+	0.600393i	$0.204989\pi$
$401$			1.55050e9i			1.20079i	−0.799705	+	0.600393i	$0.795011\pi$
$402$		0			0
$403$	−4.76415e8			−0.362591
$404$		0			0
$405$	1.29704e9	+	1.87994e9i	0.970199	+	1.40621i
$406$		0			0
$407$	−9.22293e7			−0.0678092
$408$		0			0
$409$	6.81961e8			0.492865			0.246432	−	0.969160i	$-0.420742\pi$
$409$	6.81961e8			0.492865			0.246432	+	0.969160i	$0.420742\pi$
$410$		0			0
$411$	1.30145e9	−	7.86663e8i	0.924662	−	0.558910i
$412$		0			0
$413$	−3.38693e8			−0.236582
$414$		0			0
$415$			1.70787e9i			1.17297i
$416$		0			0
$417$	3.83751e8	+	6.34878e8i	0.259163	+	0.428760i
$418$		0			0
$419$		−	1.11095e8i		−	0.0737811i	−0.999319	−	0.0368906i	$-0.988255\pi$
$419$		−	1.11095e8i		−	0.0737811i	0.999319	−	0.0368906i	$-0.0117453\pi$
$420$		0			0
$421$		−	1.07896e9i		−	0.704722i	−0.935864	−	0.352361i	$-0.885379\pi$
$421$		−	1.07896e9i		−	0.704722i	0.935864	−	0.352361i	$-0.114621\pi$
$422$		0			0
$423$	−9.92267e8	+	1.89012e9i	−0.637437	+	1.21422i
$424$		0			0
$425$		−	4.84003e9i		−	3.05835i
$426$		0			0
$427$	−2.04800e9			−1.27301
$428$		0			0
$429$	3.78848e8	+	6.26767e8i	0.231667	+	0.383271i
$430$		0			0
$431$	−1.33369e9			−0.802388			−0.401194	−	0.915993i	$-0.631405\pi$
$431$	−1.33369e9			−0.802388			−0.401194	+	0.915993i	$0.631405\pi$
$432$		0			0
$433$	−5.75068e8			−0.340417			−0.170209	−	0.985408i	$-0.554444\pi$
$433$	−5.75068e8			−0.340417			−0.170209	+	0.985408i	$0.554444\pi$
$434$		0			0
$435$	−1.52345e9	−	2.52040e9i	−0.887395	−	1.46811i
$436$		0			0
$437$	−1.23891e8			−0.0710160
$438$		0			0
$439$			2.40435e9i			1.35635i	0.734899	+	0.678176i	$0.237229\pi$
$439$			2.40435e9i			1.35635i	−0.734899	+	0.678176i	$0.762771\pi$
$440$		0			0
$441$	−4.30324e8	+	8.19702e8i	−0.238924	+	0.455115i
$442$		0			0
$443$			1.19487e9i			0.652990i	0.945199	+	0.326495i	$0.105868\pi$
$443$			1.19487e9i			0.652990i	−0.945199	+	0.326495i	$0.894132\pi$
$444$		0			0
$445$			3.02775e9i			1.62877i
$446$		0			0
$447$	2.31036e8	+	3.82226e8i	0.122350	+	0.202415i
$448$		0			0
$449$			1.27909e9i			0.666867i	0.942774	+	0.333433i	$0.108207\pi$
$449$			1.27909e9i			0.666867i	−0.942774	+	0.333433i	$0.891793\pi$
$450$		0			0
$451$	2.07544e8			0.106535
$452$		0			0
$453$	4.04204e8	−	2.44320e8i	0.204294	−	0.123485i
$454$		0			0
$455$	9.75639e8			0.485567
$456$		0			0
$457$	−1.30300e9			−0.638612			−0.319306	−	0.947652i	$-0.603450\pi$
$457$	−1.30300e9			−0.638612			−0.319306	+	0.947652i	$0.603450\pi$
$458$		0			0
$459$	1.98864e8	+	3.29632e9i	0.0959869	+	1.59106i
$460$		0			0
$461$	−3.73192e9			−1.77410			−0.887052	−	0.461669i	$-0.847251\pi$
$461$	−3.73192e9			−1.77410			−0.887052	+	0.461669i	$0.847251\pi$
$462$		0			0
$463$		−	1.78625e9i		−	0.836391i	−0.908357	−	0.418196i	$-0.862663\pi$
$463$		−	1.78625e9i		−	0.836391i	0.908357	−	0.418196i	$-0.137337\pi$
$464$		0			0
$465$	−2.81925e9	+	1.70409e9i	−1.30031	+	0.785972i
$466$		0			0
$467$		−	1.96394e9i		−	0.892319i	−0.894953	−	0.446159i	$-0.852791\pi$
$467$		−	1.96394e9i		−	0.892319i	0.894953	−	0.446159i	$-0.147209\pi$
$468$		0			0
$469$			2.02121e9i			0.904703i
$470$		0			0
$471$	7.33328e8	−	4.43259e8i	0.323389	−	0.195472i
$472$		0			0
$473$			1.57966e9i			0.686357i
$474$		0			0
$475$	3.22693e9			1.38154
$476$		0			0
$477$	5.63787e8	−	1.07393e9i	0.237849	−	0.453066i
$478$		0			0
$479$	−2.97496e9			−1.23682			−0.618410	−	0.785856i	$-0.712223\pi$
$479$	−2.97496e9			−1.23682			−0.618410	+	0.785856i	$0.712223\pi$
$480$		0			0
$481$	6.14261e7			0.0251678
$482$		0			0
$483$	−1.45716e8	+	8.80781e7i	−0.0588428	+	0.0355675i
$484$		0			0
$485$	−3.40925e9			−1.35695
$486$		0			0
$487$		−	1.25599e9i		−	0.492759i	−0.969173	−	0.246379i	$-0.920759\pi$
$487$		−	1.25599e9i		−	0.492759i	0.969173	−	0.246379i	$-0.0792409\pi$
$488$		0			0
$489$	−1.20243e9	−	1.98931e9i	−0.465029	−	0.769344i
$490$		0			0
$491$		−	2.96100e9i		−	1.12890i	−0.825469	−	0.564448i	$-0.809089\pi$
$491$		−	2.96100e9i		−	1.12890i	0.825469	−	0.564448i	$-0.190911\pi$
$492$		0			0
$493$		−	4.25817e9i		−	1.60051i
$494$		0			0
$495$	4.48377e9	+	2.35387e9i	1.66159	+	0.872297i
$496$		0			0
$497$			1.37377e9i			0.501959i
$498$		0			0
$499$	−1.51883e9			−0.547214			−0.273607	−	0.961842i	$-0.588217\pi$
$499$	−1.51883e9			−0.547214			−0.273607	+	0.961842i	$0.588217\pi$
$500$		0			0
$501$	−2.32623e9	−	3.84852e9i	−0.826459	−	1.36729i
$502$		0			0
$503$	5.67657e8			0.198883			0.0994417	−	0.995043i	$-0.468294\pi$
$503$	5.67657e8			0.198883			0.0994417	+	0.995043i	$0.468294\pi$
$504$		0			0
$505$	8.74599e9			3.02196
$506$		0			0
$507$	1.26565e9	+	2.09390e9i	0.431308	+	0.713555i
$508$		0			0
$509$	4.49543e9			1.51098			0.755490	−	0.655160i	$-0.227399\pi$
$509$	4.49543e9			1.51098			0.755490	+	0.655160i	$0.227399\pi$
$510$		0			0
$511$		−	3.60188e8i		−	0.119414i
$512$		0			0
$513$	−2.19772e9	+	1.32586e8i	−0.718722	+	0.0433598i
$514$		0			0
$515$		−	3.76328e9i		−	1.21406i
$516$		0			0
$517$			4.73323e9i			1.50640i
$518$		0			0
$519$	−6.04007e8	−	9.99270e8i	−0.189652	−	0.313760i
$520$		0			0
$521$		−	2.67557e9i		−	0.828865i	−0.910080	−	0.414432i	$-0.863980\pi$
$521$		−	2.67557e9i		−	0.828865i	0.910080	−	0.414432i	$-0.136020\pi$
$522$		0			0
$523$	1.14640e9			0.350414			0.175207	−	0.984532i	$-0.443940\pi$
$523$	1.14640e9			0.350414			0.175207	+	0.984532i	$0.443940\pi$
$524$		0			0
$525$	3.79539e9	−	2.29412e9i	1.14472	−	0.691924i
$526$		0			0
$527$	−4.76306e9			−1.41758
$528$		0			0
$529$	−3.37170e9			−0.990272
$530$		0			0
$531$	1.03668e9	+	5.44232e8i	0.300479	+	0.157744i
$532$		0			0
$533$	−1.38228e8			−0.0395412
$534$		0			0
$535$		−	2.69864e9i		−	0.761916i
$536$		0			0
$537$	2.94231e9	−	1.77848e9i	0.819934	−	0.495608i
$538$		0			0
$539$			2.05270e9i			0.564630i
$540$		0			0
$541$		−	3.82086e9i		−	1.03746i	−0.854938	−	0.518730i	$-0.826405\pi$
$541$		−	3.82086e9i		−	1.03746i	0.854938	−	0.518730i	$-0.173595\pi$
$542$		0			0
$543$	5.07129e9	−	3.06533e9i	1.35931	−	0.821633i
$544$		0			0
$545$		−	2.50916e8i		−	0.0663959i
$546$		0			0
$547$	−2.86256e9			−0.747823			−0.373911	−	0.927464i	$-0.621984\pi$
$547$	−2.86256e9			−0.747823			−0.373911	+	0.927464i	$0.621984\pi$
$548$		0			0
$549$	6.26857e9	+	3.29085e9i	1.61683	+	0.848799i
$550$		0			0
$551$	2.83900e9			0.722993
$552$		0			0
$553$	−8.06515e8			−0.202803
$554$		0			0
$555$	3.63497e8	−	2.19716e8i	0.0902560	−	0.0545551i
$556$		0			0
$557$	−1.15130e9			−0.282289			−0.141144	−	0.989989i	$-0.545078\pi$
$557$	−1.15130e9			−0.282289			−0.141144	+	0.989989i	$0.545078\pi$
$558$		0			0
$559$		−	1.05208e9i		−	0.254746i
$560$		0			0
$561$	3.78762e9	+	6.26624e9i	0.905725	+	1.49843i
$562$		0			0
$563$			7.38735e9i			1.74465i	0.488923	+	0.872327i	$0.337390\pi$
$563$			7.38735e9i			1.74465i	−0.488923	+	0.872327i	$0.662610\pi$
$564$		0			0
$565$		−	1.28681e9i		−	0.300155i
$566$		0			0
$567$	−2.49061e9	+	1.71836e9i	−0.573806	+	0.395890i
$568$		0			0
$569$			3.67854e9i			0.837109i	0.908192	+	0.418555i	$0.137463\pi$
$569$			3.67854e9i			0.837109i	−0.908192	+	0.418555i	$0.862537\pi$
$570$		0			0
$571$	−6.87971e9			−1.54648			−0.773239	−	0.634114i	$-0.781365\pi$
$571$	−6.87971e9			−1.54648			−0.773239	+	0.634114i	$0.781365\pi$
$572$		0			0
$573$	1.04847e9	+	1.73458e9i	0.232816	+	0.385171i
$574$		0			0
$575$	−8.62693e8			−0.189243
$576$		0			0
$577$	2.23949e9			0.485326			0.242663	−	0.970111i	$-0.421979\pi$
$577$	2.23949e9			0.485326			0.242663	+	0.970111i	$0.421979\pi$
$578$		0			0
$579$	1.72133e9	+	2.84777e9i	0.368544	+	0.609719i
$580$		0			0
$581$	−2.26265e9			−0.478630
$582$		0			0
$583$		−	2.68933e9i		−	0.562088i
$584$		0			0
$585$	−2.98626e9	−	1.56771e9i	−0.616711	−	0.323759i
$586$		0			0
$587$			2.74509e9i			0.560174i	0.959975	+	0.280087i	$0.0903633\pi$
$587$			2.74509e9i			0.560174i	−0.959975	+	0.280087i	$0.909637\pi$
$588$		0			0
$589$		−	3.17562e9i		−	0.640360i
$590$		0			0
$591$	5.12390e8	+	8.47699e8i	0.102104	+	0.168922i
$592$		0			0
$593$		−	6.07047e9i		−	1.19545i	−0.801702	−	0.597724i	$-0.796072\pi$
$593$		−	6.07047e9i		−	1.19545i	0.801702	−	0.597724i	$-0.203928\pi$
$594$		0			0
$595$	9.75417e9			1.89837
$596$		0			0
$597$	−1.09204e9	+	6.60081e8i	−0.210052	+	0.126966i
$598$		0			0
$599$	2.94693e9			0.560242			0.280121	−	0.959965i	$-0.409625\pi$
$599$	2.94693e9			0.560242			0.280121	+	0.959965i	$0.409625\pi$
$600$		0			0
$601$	−9.07995e9			−1.70617			−0.853086	−	0.521771i	$-0.825272\pi$
$601$	−9.07995e9			−1.70617			−0.853086	+	0.521771i	$0.825272\pi$
$602$		0			0
$603$	3.24779e9	−	6.18656e9i	0.603223	−	1.14905i
$604$		0			0
$605$	1.92274e9			0.353002
$606$		0			0
$607$		−	6.49347e9i		−	1.17846i	−0.807964	−	0.589232i	$-0.799430\pi$
$607$		−	6.49347e9i		−	1.17846i	0.807964	−	0.589232i	$-0.200570\pi$
$608$		0			0
$609$	3.33912e9	−	2.01832e9i	0.599061	−	0.362102i
$610$		0			0
$611$		−	3.15241e9i		−	0.559111i
$612$		0			0
$613$			6.06636e8i			0.106369i	0.998585	+	0.0531847i	$0.0169372\pi$
$613$			6.06636e8i			0.106369i	−0.998585	+	0.0531847i	$0.983063\pi$
$614$		0			0
$615$	−8.17980e8	+	4.94427e8i	−0.141801	+	0.0857115i
$616$		0			0
$617$		−	6.41175e9i		−	1.09895i	−0.835510	−	0.549476i	$-0.814827\pi$
$617$		−	6.41175e9i		−	1.09895i	0.835510	−	0.549476i	$-0.185173\pi$
$618$		0			0
$619$	−2.94457e9			−0.499005			−0.249503	−	0.968374i	$-0.580267\pi$
$619$	−2.94457e9			−0.499005			−0.249503	+	0.968374i	$0.580267\pi$
$620$		0			0
$621$	5.87541e8	−	3.54458e7i	0.0984504	−	0.00593942i
$622$		0			0
$623$	−4.01127e9			−0.664620
$624$		0			0
$625$	4.65561e9			0.762775
$626$		0			0
$627$	−4.17781e9	+	2.52527e9i	−0.676881	+	0.409140i
$628$		0			0
$629$	6.14122e8			0.0983959
$630$		0			0
$631$			8.02707e9i			1.27190i	0.771729	+	0.635952i	$0.219392\pi$
$631$			8.02707e9i			1.27190i	−0.771729	+	0.635952i	$0.780608\pi$
$632$		0			0
$633$	2.68077e9	+	4.43507e9i	0.420094	+	0.695004i
$634$		0			0
$635$		−	1.01867e10i		−	1.57879i
$636$		0			0
$637$		−	1.36713e9i		−	0.209566i
$638$		0			0
$639$	2.20746e9	−	4.20488e9i	0.334688	−	0.637530i
$640$		0			0
$641$			5.90994e9i			0.886299i	0.896448	+	0.443149i	$0.146139\pi$
$641$			5.90994e9i			0.886299i	−0.896448	+	0.443149i	$0.853861\pi$
$642$		0			0
$643$	4.88948e9			0.725311			0.362656	−	0.931923i	$-0.381870\pi$
$643$	4.88948e9			0.725311			0.362656	+	0.931923i	$0.381870\pi$
$644$		0			0
$645$	−3.76319e9	−	6.22582e9i	−0.552201	−	0.913561i
$646$		0			0
$647$	−4.04673e9			−0.587408			−0.293704	−	0.955896i	$-0.594888\pi$
$647$	−4.04673e9			−0.587408			−0.293704	+	0.955896i	$0.594888\pi$
$648$		0			0
$649$	2.59605e9			0.372784
$650$		0			0
$651$	−2.25764e9	−	3.73504e9i	−0.320716	−	0.530593i
$652$		0			0
$653$	−7.55767e9			−1.06216			−0.531082	−	0.847320i	$-0.678214\pi$
$653$	−7.55767e9			−1.06216			−0.531082	+	0.847320i	$0.678214\pi$
$654$		0			0
$655$			2.07046e10i			2.87888i
$656$		0			0
$657$	−5.78771e8	+	1.10247e9i	−0.0796210	+	0.151666i
$658$		0			0
$659$		−	1.79440e9i		−	0.244242i	−0.992515	−	0.122121i	$-0.961030\pi$
$659$		−	1.79440e9i		−	0.244242i	0.992515	−	0.122121i	$-0.0389697\pi$
$660$		0			0
$661$			1.49795e9i			0.201741i	0.994900	+	0.100870i	$0.0321627\pi$
$661$			1.49795e9i			0.201741i	−0.994900	+	0.100870i	$0.967837\pi$
$662$		0			0
$663$	−2.52261e9	−	4.17341e9i	−0.336166	−	0.556152i
$664$		0			0
$665$			6.50328e9i			0.857544i
$666$		0			0
$667$	−7.58982e8			−0.0990355
$668$		0			0
$669$	−1.09306e10	+	6.60698e9i	−1.41141	+	0.853123i
$670$		0			0
$671$	1.56978e10			2.00590
$672$		0			0
$673$	−1.02064e10			−1.29069			−0.645343	−	0.763893i	$-0.723286\pi$
$673$	−1.02064e10			−1.29069			−0.645343	+	0.763893i	$0.723286\pi$
$674$		0			0
$675$	−1.53034e10	+	9.23236e8i	−1.91524	+	0.115545i
$676$		0			0
$677$	2.46012e8			0.0304716			0.0152358	−	0.999884i	$-0.495150\pi$
$677$	2.46012e8			0.0304716			0.0152358	+	0.999884i	$0.495150\pi$
$678$		0			0
$679$		−	4.51669e9i		−	0.553702i
$680$		0			0
$681$	−5.30233e9	+	3.20498e9i	−0.643356	+	0.388876i
$682$		0			0
$683$			7.16878e9i			0.860940i	0.902605	+	0.430470i	$0.141652\pi$
$683$			7.16878e9i			0.860940i	−0.902605	+	0.430470i	$0.858348\pi$
$684$		0			0
$685$			1.55281e10i			1.84588i
$686$		0			0
$687$	−1.29672e10	+	7.83800e9i	−1.52580	+	0.922267i
$688$		0			0
$689$			1.79113e9i			0.208622i
$690$		0			0
$691$	1.06347e10			1.22617			0.613087	−	0.790016i	$-0.289928\pi$
$691$	1.06347e10			1.22617			0.613087	+	0.790016i	$0.289928\pi$
$692$		0			0
$693$	−3.11849e9	+	5.94026e9i	−0.355941	+	0.678014i
$694$		0			0
$695$	−7.57497e9			−0.855922
$696$		0			0
$697$	−1.38196e9			−0.154590
$698$		0			0
$699$	2.22638e9	−	1.34573e9i	0.246563	−	0.149035i
$700$		0			0
$701$	4.61502e8			0.0506012			0.0253006	−	0.999680i	$-0.491946\pi$
$701$	4.61502e8			0.0506012			0.0253006	+	0.999680i	$0.491946\pi$
$702$		0			0
$703$			4.09446e8i			0.0444480i
$704$		0			0
$705$	−1.12759e10	−	1.86548e10i	−1.21196	−	2.00506i
$706$		0			0
$707$			1.15870e10i			1.23311i
$708$		0			0
$709$		−	1.37159e10i		−	1.44532i	−0.691206	−	0.722658i	$-0.742920\pi$
$709$		−	1.37159e10i		−	1.44532i	0.691206	−	0.722658i	$-0.257080\pi$
$710$		0			0
$711$	2.46860e9	+	1.29595e9i	0.257577	+	0.135222i
$712$		0			0
$713$			8.48974e8i			0.0877165i
$714$		0			0
$715$	−7.47819e9			−0.765112
$716$		0			0
$717$	8.36002e9	+	1.38308e10i	0.847014	+	1.40130i
$718$		0			0
$719$	−6.27355e8			−0.0629452			−0.0314726	−	0.999505i	$-0.510020\pi$
$719$	−6.27355e8			−0.0629452			−0.0314726	+	0.999505i	$0.510020\pi$
$720$		0			0
$721$	4.98573e9			0.495399
$722$		0			0
$723$	−6.61960e8	−	1.09515e9i	−0.0651400	−	0.107768i
$724$		0			0
$725$	1.97688e10			1.92662
$726$		0			0
$727$			1.42234e10i			1.37288i	0.727184	+	0.686442i	$0.240828\pi$
$727$			1.42234e10i			1.37288i	−0.727184	+	0.686442i	$0.759172\pi$
$728$		0			0
$729$	1.03845e10	−	1.25755e9i	0.992747	−	0.120221i
$730$		0			0
$731$		−	1.05184e10i		−	0.995953i
$732$		0			0
$733$			7.60365e9i			0.713113i	0.934274	+	0.356556i	$0.116049\pi$
$733$			7.60365e9i			0.713113i	−0.934274	+	0.356556i	$0.883951\pi$
$734$		0			0
$735$	−4.89008e9	−	8.09016e9i	−0.454266	−	0.751539i
$736$		0			0
$737$		−	1.54924e10i		−	1.42555i
$738$		0			0
$739$	−3.49238e9			−0.318321			−0.159161	−	0.987253i	$-0.550879\pi$
$739$	−3.49238e9			−0.318321			−0.159161	+	0.987253i	$0.550879\pi$
$740$		0			0
$741$	2.78249e9	−	1.68187e9i	0.251229	−	0.151855i
$742$		0			0
$743$	7.32860e9			0.655481			0.327741	−	0.944768i	$-0.393713\pi$
$743$	7.32860e9			0.655481			0.327741	+	0.944768i	$0.393713\pi$
$744$		0			0
$745$	−4.56048e9			−0.404076
$746$		0			0
$747$	6.92556e9	+	3.63575e9i	0.607901	+	0.319133i
$748$		0			0
$749$	3.57526e9			0.310900
$750$		0			0
$751$		−	7.77694e9i		−	0.669991i	−0.942220	−	0.334996i	$-0.891265\pi$
$751$		−	7.77694e9i		−	0.669991i	0.942220	−	0.334996i	$-0.108735\pi$
$752$		0			0
$753$	1.18438e10	−	7.15899e9i	1.01090	−	0.611040i
$754$		0			0
$755$			4.82270e9i			0.407827i
$756$		0			0
$757$			1.65306e9i			0.138501i	0.997599	+	0.0692504i	$0.0220607\pi$
$757$			1.65306e9i			0.138501i	−0.997599	+	0.0692504i	$0.977939\pi$
$758$		0			0
$759$	1.11690e9	−	6.75110e8i	0.0927191	−	0.0560439i
$760$		0			0
$761$			1.81550e10i			1.49331i	0.665211	+	0.746655i	$0.268341\pi$
$761$			1.81550e10i			1.49331i	−0.665211	+	0.746655i	$0.731659\pi$
$762$		0			0
$763$	3.32423e8			0.0270929
$764$		0			0
$765$	−2.98558e10	−	1.56736e10i	−2.41109	−	1.26576i
$766$		0			0
$767$	−1.72901e9			−0.138361
$768$		0			0
$769$	1.35979e10			1.07827			0.539137	−	0.842218i	$-0.318750\pi$
$769$	1.35979e10			1.07827			0.539137	+	0.842218i	$0.318750\pi$
$770$		0			0
$771$	1.08770e10	−	6.57461e9i	0.854714	−	0.516630i
$772$		0			0
$773$	−9.94999e7			−0.00774808			−0.00387404	−	0.999992i	$-0.501233\pi$
$773$	−9.94999e7			−0.00774808			−0.00387404	+	0.999992i	$0.501233\pi$
$774$		0			0
$775$		−	2.21128e10i		−	1.70642i
$776$		0			0
$777$	2.91087e8	+	4.81574e8i	0.0222612	+	0.0368290i
$778$		0			0
$779$		−	9.21378e8i		−	0.0698324i
$780$		0			0
$781$		−	1.05299e10i		−	0.790940i
$782$		0			0
$783$	−1.34636e10	+	8.12247e8i	−1.00229	+	0.0604675i
$784$		0			0
$785$			8.74961e9i			0.645573i
$786$		0			0
$787$	−1.60892e10			−1.17658			−0.588291	−	0.808650i	$-0.700199\pi$
$787$	−1.60892e10			−1.17658			−0.588291	+	0.808650i	$0.700199\pi$
$788$		0			0
$789$	−8.04770e9	−	1.33141e10i	−0.583314	−	0.965035i
$790$		0			0
$791$	1.70481e9			0.122478
$792$		0			0
$793$	−1.04549e10			−0.744501
$794$		0			0
$795$	6.40672e9	+	1.05993e10i	0.452221	+	0.748155i
$796$		0			0
$797$	2.62634e10			1.83758			0.918791	−	0.394745i	$-0.129167\pi$
$797$	2.62634e10			1.83758			0.918791	+	0.394745i	$0.129167\pi$
$798$		0			0
$799$		−	3.15169e10i		−	2.18590i
$800$		0			0
$801$	1.22778e10	+	6.44554e9i	0.844124	+	0.443145i
$802$		0			0
$803$			2.76081e9i			0.188162i
$804$		0			0
$805$		−	1.73860e9i		−	0.117466i
$806$		0			0
$807$	8.78622e9	+	1.45359e10i	0.588498	+	0.973611i
$808$		0			0
$809$			6.29708e9i			0.418138i	0.977901	+	0.209069i	$0.0670433\pi$
$809$			6.29708e9i			0.418138i	−0.977901	+	0.209069i	$0.932957\pi$
$810$		0			0
$811$	−2.22720e10			−1.46617			−0.733087	−	0.680134i	$-0.761921\pi$
$811$	−2.22720e10			−1.46617			−0.733087	+	0.680134i	$0.761921\pi$
$812$		0			0
$813$	2.30242e10	−	1.39170e10i	1.50269	−	0.908297i
$814$		0			0
$815$	2.37352e10			1.53582
$816$		0			0
$817$	7.01280e9			0.449898
$818$		0			0
$819$	2.07696e9	−	3.95630e9i	0.132110	−	0.251649i
$820$		0			0
$821$	−6.22873e9			−0.392825			−0.196412	−	0.980521i	$-0.562929\pi$
$821$	−6.22873e9			−0.392825			−0.196412	+	0.980521i	$0.562929\pi$
$822$		0			0
$823$			5.68307e9i			0.355372i	0.984087	+	0.177686i	$0.0568612\pi$
$823$			5.68307e9i			0.355372i	−0.984087	+	0.177686i	$0.943139\pi$
$824$		0			0
$825$	−2.90913e10	+	1.75842e10i	−1.80374	+	1.09027i
$826$		0			0
$827$			5.74742e9i			0.353349i	0.984269	+	0.176674i	$0.0565340\pi$
$827$			5.74742e9i			0.353349i	−0.984269	+	0.176674i	$0.943466\pi$
$828$		0			0
$829$		−	1.72035e10i		−	1.04876i	−0.851484	−	0.524381i	$-0.824297\pi$
$829$		−	1.72035e10i		−	1.04876i	0.851484	−	0.524381i	$-0.175703\pi$
$830$		0			0
$831$	−1.41700e10	+	8.56506e9i	−0.856579	+	0.517758i
$832$		0			0
$833$		−	1.36682e10i		−	0.819318i
$834$		0			0
$835$	4.59181e10			2.72949
$836$		0			0
$837$	9.08555e8	+	1.50600e10i	0.0535565	+	0.887740i
$838$		0			0
$839$	1.00089e10			0.585085			0.292543	−	0.956252i	$-0.405499\pi$
$839$	1.00089e10			0.585085			0.292543	+	0.956252i	$0.405499\pi$
$840$		0			0
$841$	1.42324e8			0.00825073
$842$		0			0
$843$	1.32203e10	−	7.99097e9i	0.760052	−	0.459413i
$844$		0			0
$845$	−2.49831e10			−1.42445
$846$		0			0
$847$			2.54731e9i			0.144042i
$848$		0			0
$849$	−5.72036e9	−	9.46376e9i	−0.320809	−	0.530746i
$850$		0			0
$851$		−	1.09462e8i		−	0.00608849i
$852$		0			0
$853$			4.78706e9i			0.264087i	0.991244	+	0.132044i	$0.0421539\pi$
$853$			4.78706e9i			0.264087i	−0.991244	+	0.132044i	$0.957846\pi$
$854$		0			0
$855$	1.04498e10	−	1.99054e10i	0.571779	−	1.08915i
$856$		0			0
$857$		−	6.87322e9i		−	0.373016i	−0.982453	−	0.186508i	$-0.940283\pi$
$857$		−	6.87322e9i		−	0.373016i	0.982453	−	0.186508i	$-0.0597171\pi$
$858$		0			0
$859$	1.29437e10			0.696760			0.348380	−	0.937353i	$-0.386732\pi$
$859$	1.29437e10			0.696760			0.348380	+	0.937353i	$0.386732\pi$
$860$		0			0
$861$	−6.55034e8	−	1.08369e9i	−0.0349746	−	0.0578620i
$862$		0			0
$863$	3.10825e10			1.64618			0.823091	−	0.567909i	$-0.192248\pi$
$863$	3.10825e10			1.64618			0.823091	+	0.567909i	$0.192248\pi$
$864$		0			0
$865$	1.19227e10			0.626350
$866$		0			0
$867$	−1.52937e10	−	2.53020e10i	−0.796979	−	1.31852i
$868$		0			0
$869$	6.18186e9			0.319558
$870$		0			0
$871$			1.03182e10i			0.529101i
$872$		0			0
$873$	−7.25768e9	+	1.38248e10i	−0.369189	+	0.703249i
$874$		0			0
$875$			2.16830e10i			1.09419i
$876$		0			0
$877$			1.59764e10i			0.799799i	0.916559	+	0.399900i	$0.130955\pi$
$877$			1.59764e10i			0.799799i	−0.916559	+	0.399900i	$0.869045\pi$
$878$		0			0
$879$	3.92922e9	+	6.50050e9i	0.195139	+	0.322839i
$880$		0			0
$881$		−	2.70049e10i		−	1.33054i	−0.746604	−	0.665269i	$-0.768317\pi$
$881$		−	2.70049e10i		−	1.33054i	0.746604	−	0.665269i	$-0.231683\pi$
$882$		0			0
$883$	−1.73646e10			−0.848794			−0.424397	−	0.905476i	$-0.639514\pi$
$883$	−1.73646e10			−0.848794			−0.424397	+	0.905476i	$0.639514\pi$
$884$		0			0
$885$	−1.02317e10	+	6.18451e9i	−0.496186	+	0.299919i
$886$		0			0
$887$	−2.27435e10			−1.09427			−0.547136	−	0.837043i	$-0.684282\pi$
$887$	−2.27435e10			−1.09427			−0.547136	+	0.837043i	$0.684282\pi$
$888$		0			0
$889$	1.34956e10			0.644224
$890$		0			0
$891$	1.90903e10	−	1.31711e10i	0.904151	−	0.623807i
$892$		0			0
$893$	2.10129e10			0.987427
$894$		0			0
$895$			3.51058e10i			1.63681i
$896$		0			0
$897$	−7.43874e8	+	4.49634e8i	−0.0344133	+	0.0208011i
$898$		0			0
$899$		−	1.94544e10i		−	0.893015i
$900$		0			0
$901$			1.79073e10i			0.815629i
$902$		0			0
$903$	8.24819e9	−	4.98560e9i	0.372779	−	0.225326i
$904$		0			0
$905$			6.05074e10i			2.71355i
$906$		0			0
$907$	−1.70413e10			−0.758362			−0.379181	−	0.925322i	$-0.623794\pi$
$907$	−1.70413e10			−0.758362			−0.379181	+	0.925322i	$0.623794\pi$
$908$		0			0
$909$	1.86187e10	−	3.54658e10i	0.822195	−	1.56616i
$910$		0			0
$911$	−1.00724e10			−0.441385			−0.220692	−	0.975343i	$-0.570832\pi$
$911$	−1.00724e10			−0.441385			−0.220692	+	0.975343i	$0.570832\pi$
$912$		0			0
$913$	1.73430e10			0.754182
$914$		0			0
$915$	−6.18685e10	+	3.73963e10i	−2.66991	+	1.61382i
$916$		0			0
$917$	−2.74302e10			−1.17473
$918$		0			0
$919$		−	3.16312e10i		−	1.34435i	−0.740393	−	0.672174i	$-0.765361\pi$
$919$		−	3.16312e10i		−	1.34435i	0.740393	−	0.672174i	$-0.234639\pi$
$920$		0			0
$921$	−4.82835e9	−	7.98802e9i	−0.203653	−	0.336923i
$922$		0			0
$923$			7.01304e9i			0.293562i
$924$		0			0
$925$			2.85109e9i			0.118445i
$926$		0			0
$927$	−1.52604e10	−	8.01136e9i	−0.629199	−	0.330314i
$928$		0			0
$929$		−	1.74385e10i		−	0.713597i	−0.934181	−	0.356799i	$-0.883868\pi$
$929$		−	1.74385e10i		−	0.713597i	0.934181	−	0.356799i	$-0.116132\pi$
$930$		0			0
$931$	9.11280e9			0.370108
$932$		0			0
$933$	1.42307e10	+	2.35432e10i	0.573640	+	0.949029i
$934$		0			0
$935$	−7.47648e10			−2.99128
$936$		0			0
$937$	−3.67462e10			−1.45923			−0.729615	−	0.683858i	$-0.760301\pi$
$937$	−3.67462e10			−1.45923			−0.729615	+	0.683858i	$0.760301\pi$
$938$		0			0
$939$	2.32022e10	+	3.83857e10i	0.914532	+	1.51300i
$940$		0			0
$941$	9.56897e8			0.0374371			0.0187185	−	0.999825i	$-0.494041\pi$
$941$	9.56897e8			0.0374371			0.0187185	+	0.999825i	$0.494041\pi$
$942$		0			0
$943$			2.46323e8i			0.00956563i
$944$		0			0
$945$	−1.86061e9	−	3.08410e10i	−0.0717206	−	1.18882i
$946$		0			0
$947$		−	4.40484e10i		−	1.68541i	−0.538377	−	0.842704i	$-0.680962\pi$
$947$		−	4.40484e10i		−	1.68541i	0.538377	−	0.842704i	$-0.319038\pi$
$948$		0			0
$949$		−	1.83874e9i		−	0.0698375i
$950$		0			0
$951$	1.44179e10	+	2.38529e10i	0.543587	+	0.899311i
$952$		0			0
$953$		−	1.16259e10i		−	0.435113i	−0.976048	−	0.217557i	$-0.930191\pi$
$953$		−	1.16259e10i		−	0.435113i	0.976048	−	0.217557i	$-0.0698087\pi$
$954$		0			0
$955$	−2.06960e10			−0.768907
$956$		0			0
$957$	−2.55940e10	+	1.54703e10i	−0.943945	+	0.570567i
$958$		0			0
$959$	−2.05722e10			−0.753210
$960$		0			0
$961$	5.75150e9			0.209050
$962$		0			0
$963$	−1.09432e10	−	5.74493e9i	−0.394869	−	0.207297i
$964$		0			0
$965$	−3.39778e10			−1.21717
$966$		0			0
$967$			1.71012e10i			0.608183i	0.952643	+	0.304092i	$0.0983529\pi$
$967$			1.71012e10i			0.608183i	−0.952643	+	0.304092i	$0.901647\pi$
$968$		0			0
$969$	2.78185e10	−	1.68149e10i	0.982202	−	0.593691i
$970$		0			0
$971$			2.77609e10i			0.973121i	0.873647	+	0.486560i	$0.161749\pi$
$971$			2.77609e10i			0.973121i	−0.873647	+	0.486560i	$0.838251\pi$
$972$		0			0
$973$		−	1.00356e10i		−	0.349259i
$974$		0			0
$975$	1.93753e10	−	1.17114e10i	0.669471	−	0.404661i
$976$		0			0
$977$		−	3.13262e10i		−	1.07467i	−0.843368	−	0.537337i	$-0.819430\pi$
$977$		−	3.13262e10i		−	1.07467i	0.843368	−	0.537337i	$-0.180570\pi$
$978$		0			0
$979$	3.07460e10			1.04725
$980$		0			0
$981$	−1.01749e9	−	5.34156e8i	−0.0344102	−	0.0180645i
$982$		0			0
$983$	4.78593e9			0.160705			0.0803524	−	0.996767i	$-0.474395\pi$
$983$	4.78593e9			0.160705			0.0803524	+	0.996767i	$0.474395\pi$
$984$		0			0
$985$	−1.01142e10			−0.337214
$986$		0			0
$987$	2.47145e10	−	1.49387e10i	0.818167	−	0.494540i
$988$		0			0
$989$	−1.87481e9			−0.0616270
$990$		0			0
$991$			1.38921e10i			0.453431i	0.973961	+	0.226715i	$0.0727987\pi$
$991$			1.38921e10i			0.453431i	−0.973961	+	0.226715i	$0.927201\pi$
$992$		0			0
$993$	2.97846e10	+	4.92757e10i	0.965316	+	1.59702i
$994$		0			0
$995$		−	1.30295e10i		−	0.419321i
$996$		0			0
$997$			2.60033e10i			0.830989i	0.909596	+	0.415494i	$0.136391\pi$
$997$			2.60033e10i			0.830989i	−0.909596	+	0.415494i	$0.863609\pi$
$998$		0			0
$999$	−1.17144e8	−	1.94175e9i	−0.00371741	−	0.0616189i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.8.f.c.47.15		20
3.2	odd	2	inner	96.8.f.c.47.14		20
4.3	odd	2		24.8.f.c.11.17	yes	20
8.3	odd	2	inner	96.8.f.c.47.16		20
8.5	even	2		24.8.f.c.11.3	✓	20
12.11	even	2		24.8.f.c.11.4	yes	20
24.5	odd	2		24.8.f.c.11.18	yes	20
24.11	even	2	inner	96.8.f.c.47.13		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.f.c.11.3	✓	20	8.5	even	2
24.8.f.c.11.4	yes	20	12.11	even	2
24.8.f.c.11.17	yes	20	4.3	odd	2
24.8.f.c.11.18	yes	20	24.5	odd	2
96.8.f.c.47.13		20	24.11	even	2	inner
96.8.f.c.47.14		20	3.2	odd	2	inner
96.8.f.c.47.15		20	1.1	even	1	trivial
96.8.f.c.47.16		20	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	96.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(24.1914 + 40.0222i) q^{3} -477.520 q^{5} -632.635i q^{7} +(-1016.56 + 1936.38i) q^{9} +O(q^{10})\)	\(q+(24.1914 + 40.0222i) q^{3} -477.520 q^{5} -632.635i q^{7} +(-1016.56 + 1936.38i) q^{9} +4849.09i q^{11} -3229.57i q^{13} +(-11551.9 - 19111.4i) q^{15} -32288.3i q^{17} +21527.2 q^{19} +(25319.5 - 15304.3i) q^{21} -5755.11 q^{23} +149900. q^{25} +(-102090. + 6159.01i) q^{27} +131879. q^{29} -147516. i q^{31} +(-194071. + 117306. i) q^{33} +302096. i q^{35} +19019.9i q^{37} +(129254. - 78127.7i) q^{39} -42800.6i q^{41} +325765. q^{43} +(485425. - 924662. i) q^{45} +976107. q^{47} +423316. q^{49} +(1.29225e6 - 781099. i) q^{51} -554605. q^{53} -2.31554e6i q^{55} +(520772. + 861566. i) q^{57} -535369. i q^{59} -3.23726e6i q^{61} +(1.22502e6 + 643109. i) q^{63} +1.54218e6i q^{65} -3.19490e6 q^{67} +(-139224. - 230332. i) q^{69} -2.17151e6 q^{71} +569345. q^{73} +(3.62629e6 + 5.99934e6i) q^{75} +3.06771e6 q^{77} -1.27485e6i q^{79} +(-2.71620e6 - 3.93688e6i) q^{81} -3.57654e6i q^{83} +1.54183e7i q^{85} +(3.19035e6 + 5.27811e6i) q^{87} -6.34057e6i q^{89} -2.04314e6 q^{91} +(5.90394e6 - 3.56863e6i) q^{93} -1.02797e7 q^{95} +7.13949e6 q^{97} +(-9.38970e6 - 4.92937e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 120 q^{3} + 5076 q^{9}+O(q^{10}) \) 20 * q - 120 * q^3 + 5076 * q^9	\( 20 q - 120 q^{3} + 5076 q^{9} - 133744 q^{19} + 711596 q^{25} - 85320 q^{27} + 263640 q^{33} + 2292080 q^{43} + 2495228 q^{49} + 7417536 q^{51} - 5067120 q^{57} - 3654640 q^{67} + 19892680 q^{73} + 8612088 q^{75} - 6817932 q^{81} - 67826976 q^{91} + 5561800 q^{97} - 10152864 q^{99}+O(q^{100}) \) 20 * q - 120 * q^3 + 5076 * q^9 - 133744 * q^19 + 711596 * q^25 - 85320 * q^27 + 263640 * q^33 + 2292080 * q^43 + 2495228 * q^49 + 7417536 * q^51 - 5067120 * q^57 - 3654640 * q^67 + 19892680 * q^73 + 8612088 * q^75 - 6817932 * q^81 - 67826976 * q^91 + 5561800 * q^97 - 10152864 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more