LMFDB - Embedded newform 90.12.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,12,Mod(1,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("90.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

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:	yes
Analytic conductor:	$69.1508862504$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	90.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+32.0000 q^{2} +1024.00 q^{4} +3125.00 q^{5} -22876.0 q^{7} +32768.0 q^{8} +100000. q^{10} +259128. q^{11} -2.32346e6 q^{13} -732032. q^{14} +1.04858e6 q^{16} +1.91827e6 q^{17} -1.61374e6 q^{19} +3.20000e6 q^{20} +8.29210e6 q^{22} -3.28492e7 q^{23} +9.76562e6 q^{25} -7.43508e7 q^{26} -2.34250e7 q^{28} -1.52590e8 q^{29} +5.68101e7 q^{31} +3.35544e7 q^{32} +6.13845e7 q^{34} -7.14875e7 q^{35} +5.71517e8 q^{37} -5.16397e7 q^{38} +1.02400e8 q^{40} -7.34063e8 q^{41} -5.80697e8 q^{43} +2.65347e8 q^{44} -1.05117e9 q^{46} -4.78847e8 q^{47} -1.45402e9 q^{49} +3.12500e8 q^{50} -2.37923e9 q^{52} +2.55379e9 q^{53} +8.09775e8 q^{55} -7.49601e8 q^{56} -4.88289e9 q^{58} -6.31792e9 q^{59} -7.86138e8 q^{61} +1.81792e9 q^{62} +1.07374e9 q^{64} -7.26082e9 q^{65} -2.12877e10 q^{67} +1.96430e9 q^{68} -2.28760e9 q^{70} -1.17195e10 q^{71} +2.31756e10 q^{73} +1.82886e10 q^{74} -1.65247e9 q^{76} -5.92781e9 q^{77} +2.69897e9 q^{79} +3.27680e9 q^{80} -2.34900e10 q^{82} -3.26587e10 q^{83} +5.99458e9 q^{85} -1.85823e10 q^{86} +8.49111e9 q^{88} -7.45563e10 q^{89} +5.31515e10 q^{91} -3.36376e10 q^{92} -1.53231e10 q^{94} -5.04294e9 q^{95} +4.14349e10 q^{97} -4.65285e10 q^{98} +O(q^{100})\)

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 $

$2$	32.0000	0.707107
$2$	32.0000	0.707107
$3$	0	0
$3$	0	0
$4$	1024.00	0.500000
$5$	3125.00	0.447214
$5$	3125.00	0.447214
$6$	0	0
$7$	−22876.0	−0.514447	−0.257224	−	0.966352i	$-0.582808\pi$
$7$	−22876.0	−0.514447	−0.257224	+	0.966352i	$0.582808\pi$
$8$	32768.0	0.353553
$9$	0	0
$10$	100000.	0.316228
$11$	259128.	0.485126	0.242563	−	0.970136i	$-0.422012\pi$
$11$	259128.	0.485126	0.242563	+	0.970136i	$0.422012\pi$
$12$	0	0
$13$	−2.32346e6	−1.73559	−0.867795	−	0.496922i	$-0.834463\pi$
$13$	−2.32346e6	−1.73559	−0.867795	+	0.496922i	$0.834463\pi$
$14$	−732032.	−0.363769
$15$	0	0
$16$	1.04858e6	0.250000
$17$	1.91827e6	0.327672	0.163836	−	0.986488i	$-0.447613\pi$
$17$	1.91827e6	0.327672	0.163836	+	0.986488i	$0.447613\pi$
$18$	0	0
$19$	−1.61374e6	−0.149516	−0.0747582	−	0.997202i	$-0.523818\pi$
$19$	−1.61374e6	−0.149516	−0.0747582	+	0.997202i	$0.523818\pi$
$20$	3.20000e6	0.223607
$21$	0	0
$22$	8.29210e6	0.343036
$23$	−3.28492e7	−1.06420	−0.532098	−	0.846683i	$-0.678596\pi$
$23$	−3.28492e7	−1.06420	−0.532098	+	0.846683i	$0.678596\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	−7.43508e7	−1.22725
$27$	0	0
$28$	−2.34250e7	−0.257224
$29$	−1.52590e8	−1.38146	−0.690729	−	0.723113i	$-0.742710\pi$
$29$	−1.52590e8	−1.38146	−0.690729	+	0.723113i	$0.742710\pi$
$30$	0	0
$31$	5.68101e7	0.356399	0.178199	−	0.983994i	$-0.442973\pi$
$31$	5.68101e7	0.356399	0.178199	+	0.983994i	$0.442973\pi$
$32$	3.35544e7	0.176777
$33$	0	0
$34$	6.13845e7	0.231699
$35$	−7.14875e7	−0.230068
$36$	0	0
$37$	5.71517e8	1.35494	0.677470	−	0.735551i	$-0.263076\pi$
$37$	5.71517e8	1.35494	0.677470	+	0.735551i	$0.263076\pi$
$38$	−5.16397e7	−0.105724
$39$	0	0
$40$	1.02400e8	0.158114
$41$	−7.34063e8	−0.989515	−0.494757	−	0.869031i	$-0.664743\pi$
$41$	−7.34063e8	−0.989515	−0.494757	+	0.869031i	$0.664743\pi$
$42$	0	0
$43$	−5.80697e8	−0.602383	−0.301192	−	0.953564i	$-0.597384\pi$
$43$	−5.80697e8	−0.602383	−0.301192	+	0.953564i	$0.597384\pi$
$44$	2.65347e8	0.242563
$45$	0	0
$46$	−1.05117e9	−0.752501
$47$	−4.78847e8	−0.304550	−0.152275	−	0.988338i	$-0.548660\pi$
$47$	−4.78847e8	−0.304550	−0.152275	+	0.988338i	$0.548660\pi$
$48$	0	0
$49$	−1.45402e9	−0.735344
$50$	3.12500e8	0.141421
$51$	0	0
$52$	−2.37923e9	−0.867795
$53$	2.55379e9	0.838818	0.419409	−	0.907797i	$-0.362237\pi$
$53$	2.55379e9	0.838818	0.419409	+	0.907797i	$0.362237\pi$
$54$	0	0
$55$	8.09775e8	0.216955
$56$	−7.49601e8	−0.181885
$57$	0	0
$58$	−4.88289e9	−0.976839
$59$	−6.31792e9	−1.15050	−0.575252	−	0.817976i	$-0.695096\pi$
$59$	−6.31792e9	−1.15050	−0.575252	+	0.817976i	$0.695096\pi$
$60$	0	0
$61$	−7.86138e8	−0.119175	−0.0595874	−	0.998223i	$-0.518979\pi$
$61$	−7.86138e8	−0.119175	−0.0595874	+	0.998223i	$0.518979\pi$
$62$	1.81792e9	0.252012
$63$	0	0
$64$	1.07374e9	0.125000
$65$	−7.26082e9	−0.776179
$66$	0	0
$67$	−2.12877e10	−1.92627	−0.963135	−	0.269020i	$-0.913300\pi$
$67$	−2.12877e10	−1.92627	−0.963135	+	0.269020i	$0.913300\pi$
$68$	1.96430e9	0.163836
$69$	0	0
$70$	−2.28760e9	−0.162683
$71$	−1.17195e10	−0.770881	−0.385441	−	0.922733i	$-0.625951\pi$
$71$	−1.17195e10	−0.770881	−0.385441	+	0.922733i	$0.625951\pi$
$72$	0	0
$73$	2.31756e10	1.30845	0.654223	−	0.756302i	$-0.272996\pi$
$73$	2.31756e10	1.30845	0.654223	+	0.756302i	$0.272996\pi$
$74$	1.82886e10	0.958087
$75$	0	0
$76$	−1.65247e9	−0.0747582
$77$	−5.92781e9	−0.249572
$78$	0	0
$79$	2.69897e9	0.0986845	0.0493423	−	0.998782i	$-0.484287\pi$
$79$	2.69897e9	0.0986845	0.0493423	+	0.998782i	$0.484287\pi$
$80$	3.27680e9	0.111803
$81$	0	0
$82$	−2.34900e10	−0.699692
$83$	−3.26587e10	−0.910059	−0.455029	−	0.890476i	$-0.650371\pi$
$83$	−3.26587e10	−0.910059	−0.455029	+	0.890476i	$0.650371\pi$
$84$	0	0
$85$	5.99458e9	0.146540
$86$	−1.85823e10	−0.425949
$87$	0	0
$88$	8.49111e9	0.171518
$89$	−7.45563e10	−1.41527	−0.707635	−	0.706578i	$-0.750238\pi$
$89$	−7.45563e10	−1.41527	−0.707635	+	0.706578i	$0.750238\pi$
$90$	0	0
$91$	5.31515e10	0.892870
$92$	−3.36376e10	−0.532098
$93$	0	0
$94$	−1.53231e10	−0.215349
$95$	−5.04294e9	−0.0668658
$96$	0	0
$97$	4.14349e10	0.489916	0.244958	−	0.969534i	$-0.421226\pi$
$97$	4.14349e10	0.489916	0.244958	+	0.969534i	$0.421226\pi$
$98$	−4.65285e10	−0.519967
$99$	0	0
$100$	1.00000e10	0.100000
$101$	−1.16600e11	−1.10391	−0.551954	−	0.833875i	$-0.686118\pi$
$101$	−1.16600e11	−1.10391	−0.551954	+	0.833875i	$0.686118\pi$
$102$	0	0
$103$	4.30231e10	0.365676	0.182838	−	0.983143i	$-0.441472\pi$
$103$	4.30231e10	0.365676	0.182838	+	0.983143i	$0.441472\pi$
$104$	−7.61352e10	−0.613624
$105$	0	0
$106$	8.17212e10	0.593134
$107$	2.15323e11	1.48416	0.742078	−	0.670313i	$-0.233840\pi$
$107$	2.15323e11	1.48416	0.742078	+	0.670313i	$0.233840\pi$
$108$	0	0
$109$	−2.56069e11	−1.59409	−0.797044	−	0.603922i	$-0.793604\pi$
$109$	−2.56069e11	−1.59409	−0.797044	+	0.603922i	$0.793604\pi$
$110$	2.59128e10	0.153410
$111$	0	0
$112$	−2.39872e10	−0.128612
$113$	−1.32713e11	−0.677613	−0.338807	−	0.940856i	$-0.610023\pi$
$113$	−1.32713e11	−0.677613	−0.338807	+	0.940856i	$0.610023\pi$
$114$	0	0
$115$	−1.02654e11	−0.475923
$116$	−1.56252e11	−0.690729
$117$	0	0
$118$	−2.02173e11	−0.813529
$119$	−4.38823e10	−0.168570
$120$	0	0
$121$	−2.18164e11	−0.764653
$122$	−2.51564e10	−0.0842694
$123$	0	0
$124$	5.81735e10	0.178199
$125$	3.05176e10	0.0894427
$126$	0	0
$127$	−5.99930e11	−1.61132	−0.805658	−	0.592382i	$-0.798188\pi$
$127$	−5.99930e11	−1.61132	−0.805658	+	0.592382i	$0.798188\pi$
$128$	3.43597e10	0.0883883
$129$	0	0
$130$	−2.32346e11	−0.548842
$131$	2.86450e11	0.648719	0.324360	−	0.945934i	$-0.394851\pi$
$131$	2.86450e11	0.648719	0.324360	+	0.945934i	$0.394851\pi$
$132$	0	0
$133$	3.69159e10	0.0769183
$134$	−6.81206e11	−1.36208
$135$	0	0
$136$	6.28577e10	0.115850
$137$	2.68219e11	0.474817	0.237408	−	0.971410i	$-0.423702\pi$
$137$	2.68219e11	0.474817	0.237408	+	0.971410i	$0.423702\pi$
$138$	0	0
$139$	1.00413e12	1.64138	0.820692	−	0.571371i	$-0.193588\pi$
$139$	1.00413e12	1.64138	0.820692	+	0.571371i	$0.193588\pi$
$140$	−7.32032e10	−0.115034
$141$	0	0
$142$	−3.75023e11	−0.545095
$143$	−6.02074e11	−0.841980
$144$	0	0
$145$	−4.76845e11	−0.617807
$146$	7.41620e11	0.925211
$147$	0	0
$148$	5.85234e11	0.677470
$149$	−6.60408e11	−0.736695	−0.368348	−	0.929688i	$-0.620076\pi$
$149$	−6.60408e11	−0.736695	−0.368348	+	0.929688i	$0.620076\pi$
$150$	0	0
$151$	1.86615e11	0.193453	0.0967263	−	0.995311i	$-0.469163\pi$
$151$	1.86615e11	0.193453	0.0967263	+	0.995311i	$0.469163\pi$
$152$	−5.28790e10	−0.0528620
$153$	0	0
$154$	−1.89690e11	−0.176474
$155$	1.77531e11	0.159386
$156$	0	0
$157$	4.52456e11	0.378554	0.189277	−	0.981924i	$-0.439386\pi$
$157$	4.52456e11	0.378554	0.189277	+	0.981924i	$0.439386\pi$
$158$	8.63670e10	0.0697805
$159$	0	0
$160$	1.04858e11	0.0790569
$161$	7.51458e11	0.547473
$162$	0	0
$163$	1.51667e12	1.03243	0.516213	−	0.856460i	$-0.327341\pi$
$163$	1.51667e12	1.03243	0.516213	+	0.856460i	$0.327341\pi$
$164$	−7.51681e11	−0.494757
$165$	0	0
$166$	−1.04508e12	−0.643509
$167$	2.35061e12	1.40036	0.700181	−	0.713965i	$-0.253103\pi$
$167$	2.35061e12	1.40036	0.700181	+	0.713965i	$0.253103\pi$
$168$	0	0
$169$	3.60632e12	2.01227
$170$	1.91827e11	0.103619
$171$	0	0
$172$	−5.94633e11	−0.301192
$173$	−9.01111e11	−0.442104	−0.221052	−	0.975262i	$-0.570949\pi$
$173$	−9.01111e11	−0.442104	−0.221052	+	0.975262i	$0.570949\pi$
$174$	0	0
$175$	−2.23398e11	−0.102889
$176$	2.71715e11	0.121282
$177$	0	0
$178$	−2.38580e12	−1.00075
$179$	9.77006e11	0.397379	0.198690	−	0.980062i	$-0.436331\pi$
$179$	9.77006e11	0.397379	0.198690	+	0.980062i	$0.436331\pi$
$180$	0	0
$181$	3.83020e12	1.46551	0.732756	−	0.680491i	$-0.238234\pi$
$181$	3.83020e12	1.46551	0.732756	+	0.680491i	$0.238234\pi$
$182$	1.70085e12	0.631354
$183$	0	0
$184$	−1.07640e12	−0.376250
$185$	1.78599e12	0.605947
$186$	0	0
$187$	4.97076e11	0.158962
$188$	−4.90339e11	−0.152275
$189$	0	0
$190$	−1.61374e11	−0.0472812
$191$	2.49694e12	0.710764	0.355382	−	0.934721i	$-0.384351\pi$
$191$	2.49694e12	0.710764	0.355382	+	0.934721i	$0.384351\pi$
$192$	0	0
$193$	5.19794e12	1.39722	0.698612	−	0.715500i	$-0.253801\pi$
$193$	5.19794e12	1.39722	0.698612	+	0.715500i	$0.253801\pi$
$194$	1.32592e12	0.346423
$195$	0	0
$196$	−1.48891e12	−0.367672
$197$	5.70399e12	1.36967	0.684833	−	0.728700i	$-0.259875\pi$
$197$	5.70399e12	1.36967	0.684833	+	0.728700i	$0.259875\pi$
$198$	0	0
$199$	−6.39817e11	−0.145333	−0.0726664	−	0.997356i	$-0.523151\pi$
$199$	−6.39817e11	−0.145333	−0.0726664	+	0.997356i	$0.523151\pi$
$200$	3.20000e11	0.0707107
$201$	0	0
$202$	−3.73122e12	−0.780581
$203$	3.49066e12	0.710688
$204$	0	0
$205$	−2.29395e12	−0.442524
$206$	1.37674e12	0.258572
$207$	0	0
$208$	−2.43633e12	−0.433897
$209$	−4.18165e11	−0.0725343
$210$	0	0
$211$	2.47025e12	0.406618	0.203309	−	0.979115i	$-0.434830\pi$
$211$	2.47025e12	0.406618	0.203309	+	0.979115i	$0.434830\pi$
$212$	2.61508e12	0.419409
$213$	0	0
$214$	6.89034e12	1.04946
$215$	−1.81468e12	−0.269394
$216$	0	0
$217$	−1.29959e12	−0.183348
$218$	−8.19422e12	−1.12719
$219$	0	0
$220$	8.29210e11	0.108477
$221$	−4.45702e12	−0.568705
$222$	0	0
$223$	−5.26711e12	−0.639581	−0.319790	−	0.947488i	$-0.603613\pi$
$223$	−5.26711e12	−0.639581	−0.319790	+	0.947488i	$0.603613\pi$
$224$	−7.67591e11	−0.0909423
$225$	0	0
$226$	−4.24682e12	−0.479145
$227$	1.14447e13	1.26027	0.630133	−	0.776488i	$-0.283001\pi$
$227$	1.14447e13	1.26027	0.630133	+	0.776488i	$0.283001\pi$
$228$	0	0
$229$	3.62497e12	0.380372	0.190186	−	0.981748i	$-0.439091\pi$
$229$	3.62497e12	0.380372	0.190186	+	0.981748i	$0.439091\pi$
$230$	−3.28492e12	−0.336528
$231$	0	0
$232$	−5.00008e12	−0.488419
$233$	−1.44062e13	−1.37433	−0.687165	−	0.726501i	$-0.741145\pi$
$233$	−1.44062e13	−1.37433	−0.687165	+	0.726501i	$0.741145\pi$
$234$	0	0
$235$	−1.49640e12	−0.136199
$236$	−6.46955e12	−0.575252
$237$	0	0
$238$	−1.40423e12	−0.119197
$239$	1.33198e13	1.10487	0.552433	−	0.833557i	$-0.313699\pi$
$239$	1.33198e13	1.10487	0.552433	+	0.833557i	$0.313699\pi$
$240$	0	0
$241$	1.13769e13	0.901428	0.450714	−	0.892668i	$-0.351169\pi$
$241$	1.13769e13	0.901428	0.450714	+	0.892668i	$0.351169\pi$
$242$	−6.98126e12	−0.540691
$243$	0	0
$244$	−8.05005e11	−0.0595874
$245$	−4.54380e12	−0.328856
$246$	0	0
$247$	3.74946e12	0.259499
$248$	1.86155e12	0.126006
$249$	0	0
$250$	9.76562e11	0.0632456
$251$	2.83662e12	0.179719	0.0898597	−	0.995954i	$-0.471358\pi$
$251$	2.83662e12	0.179719	0.0898597	+	0.995954i	$0.471358\pi$
$252$	0	0
$253$	−8.51215e12	−0.516269
$254$	−1.91978e13	−1.13937
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	−3.01443e13	−1.67715	−0.838576	−	0.544784i	$-0.816612\pi$
$257$	−3.01443e13	−1.67715	−0.838576	+	0.544784i	$0.816612\pi$
$258$	0	0
$259$	−1.30740e13	−0.697045
$260$	−7.43508e12	−0.388090
$261$	0	0
$262$	9.16641e12	0.458714
$263$	−3.80917e13	−1.86670	−0.933349	−	0.358971i	$-0.883128\pi$
$263$	−3.80917e13	−1.86670	−0.933349	+	0.358971i	$0.883128\pi$
$264$	0	0
$265$	7.98059e12	0.375131
$266$	1.18131e12	0.0543894
$267$	0	0
$268$	−2.17986e13	−0.963135
$269$	−2.00988e12	−0.0870026	−0.0435013	−	0.999053i	$-0.513851\pi$
$269$	−2.00988e12	−0.0870026	−0.0435013	+	0.999053i	$0.513851\pi$
$270$	0	0
$271$	−3.06081e13	−1.27205	−0.636027	−	0.771667i	$-0.719423\pi$
$271$	−3.06081e13	−1.27205	−0.636027	+	0.771667i	$0.719423\pi$
$272$	2.01145e12	0.0819181
$273$	0	0
$274$	8.58300e12	0.335746
$275$	2.53055e12	0.0970252
$276$	0	0
$277$	2.77790e13	1.02348	0.511738	−	0.859142i	$-0.329002\pi$
$277$	2.77790e13	1.02348	0.511738	+	0.859142i	$0.329002\pi$
$278$	3.21323e13	1.16063
$279$	0	0
$280$	−2.34250e12	−0.0813413
$281$	3.95175e13	1.34557	0.672783	−	0.739840i	$-0.265099\pi$
$281$	3.95175e13	1.34557	0.672783	+	0.739840i	$0.265099\pi$
$282$	0	0
$283$	−3.57563e13	−1.17092	−0.585460	−	0.810701i	$-0.699086\pi$
$283$	−3.57563e13	−1.17092	−0.585460	+	0.810701i	$0.699086\pi$
$284$	−1.20007e13	−0.385441
$285$	0	0
$286$	−1.92664e13	−0.595370
$287$	1.67924e13	0.509053
$288$	0	0
$289$	−3.05922e13	−0.892631
$290$	−1.52590e13	−0.436856
$291$	0	0
$292$	2.37318e13	0.654223
$293$	−1.16731e13	−0.315801	−0.157901	−	0.987455i	$-0.550473\pi$
$293$	−1.16731e13	−0.315801	−0.157901	+	0.987455i	$0.550473\pi$
$294$	0	0
$295$	−1.97435e13	−0.514521
$296$	1.87275e13	0.479043
$297$	0	0
$298$	−2.11331e13	−0.520922
$299$	7.63239e13	1.84701
$300$	0	0
$301$	1.32840e13	0.309894
$302$	5.97169e12	0.136792
$303$	0	0
$304$	−1.69213e12	−0.0373791
$305$	−2.45668e12	−0.0532966
$306$	0	0
$307$	8.98114e13	1.87962	0.939811	−	0.341695i	$-0.111001\pi$
$307$	8.98114e13	1.87962	0.939811	+	0.341695i	$0.111001\pi$
$308$	−6.07008e12	−0.124786
$309$	0	0
$310$	5.68101e12	0.112703
$311$	−6.75879e13	−1.31731	−0.658653	−	0.752446i	$-0.728874\pi$
$311$	−6.75879e13	−1.31731	−0.658653	+	0.752446i	$0.728874\pi$
$312$	0	0
$313$	9.87171e13	1.85737	0.928685	−	0.370869i	$-0.120940\pi$
$313$	9.87171e13	1.85737	0.928685	+	0.370869i	$0.120940\pi$
$314$	1.44786e13	0.267678
$315$	0	0
$316$	2.76374e12	0.0493423
$317$	−1.10523e14	−1.93921	−0.969605	−	0.244674i	$-0.921319\pi$
$317$	−1.10523e14	−1.93921	−0.969605	+	0.244674i	$0.921319\pi$
$318$	0	0
$319$	−3.95404e13	−0.670182
$320$	3.35544e12	0.0559017
$321$	0	0
$322$	2.40467e13	0.387122
$323$	−3.09558e12	−0.0489924
$324$	0	0
$325$	−2.26901e13	−0.347118
$326$	4.85334e13	0.730036
$327$	0	0
$328$	−2.40538e13	−0.349846
$329$	1.09541e13	0.156675
$330$	0	0
$331$	−5.00155e13	−0.691912	−0.345956	−	0.938251i	$-0.612445\pi$
$331$	−5.00155e13	−0.691912	−0.345956	+	0.938251i	$0.612445\pi$
$332$	−3.34425e13	−0.455029
$333$	0	0
$334$	7.52196e13	0.990205
$335$	−6.65240e13	−0.861454
$336$	0	0
$337$	−3.62478e12	−0.0454274	−0.0227137	−	0.999742i	$-0.507231\pi$
$337$	−3.62478e12	−0.0454274	−0.0227137	+	0.999742i	$0.507231\pi$
$338$	1.15402e14	1.42289
$339$	0	0
$340$	6.13845e12	0.0732698
$341$	1.47211e13	0.172898
$342$	0	0
$343$	7.84954e13	0.892743
$344$	−1.90283e13	−0.212975
$345$	0	0
$346$	−2.88355e13	−0.312615
$347$	−6.56058e12	−0.0700051	−0.0350026	−	0.999387i	$-0.511144\pi$
$347$	−6.56058e12	−0.0700051	−0.0350026	+	0.999387i	$0.511144\pi$
$348$	0	0
$349$	−8.14353e13	−0.841924	−0.420962	−	0.907078i	$-0.638307\pi$
$349$	−8.14353e13	−0.841924	−0.420962	+	0.907078i	$0.638307\pi$
$350$	−7.14875e12	−0.0727538
$351$	0	0
$352$	8.69489e12	0.0857590
$353$	−1.03792e14	−1.00787	−0.503934	−	0.863742i	$-0.668115\pi$
$353$	−1.03792e14	−1.00787	−0.503934	+	0.863742i	$0.668115\pi$
$354$	0	0
$355$	−3.66234e13	−0.344749
$356$	−7.63457e13	−0.707635
$357$	0	0
$358$	3.12642e13	0.280990
$359$	1.38258e14	1.22369	0.611845	−	0.790978i	$-0.290428\pi$
$359$	1.38258e14	1.22369	0.611845	+	0.790978i	$0.290428\pi$
$360$	0	0
$361$	−1.13886e14	−0.977645
$362$	1.22566e14	1.03627
$363$	0	0
$364$	5.44272e13	0.446435
$365$	7.24239e13	0.585155
$366$	0	0
$367$	1.16357e14	0.912282	0.456141	−	0.889908i	$-0.349231\pi$
$367$	1.16357e14	0.912282	0.456141	+	0.889908i	$0.349231\pi$
$368$	−3.44449e13	−0.266049
$369$	0	0
$370$	5.71517e13	0.428470
$371$	−5.84205e13	−0.431527
$372$	0	0
$373$	−6.72186e13	−0.482049	−0.241024	−	0.970519i	$-0.577483\pi$
$373$	−6.72186e13	−0.482049	−0.241024	+	0.970519i	$0.577483\pi$
$374$	1.59064e13	0.112403
$375$	0	0
$376$	−1.56908e13	−0.107675
$377$	3.54538e14	2.39765
$378$	0	0
$379$	−1.32840e14	−0.872598	−0.436299	−	0.899802i	$-0.643711\pi$
$379$	−1.32840e14	−0.872598	−0.436299	+	0.899802i	$0.643711\pi$
$380$	−5.16397e12	−0.0334329
$381$	0	0
$382$	7.99022e13	0.502586
$383$	2.69810e14	1.67288	0.836439	−	0.548060i	$-0.184633\pi$
$383$	2.69810e14	1.67288	0.836439	+	0.548060i	$0.184633\pi$
$384$	0	0
$385$	−1.85244e13	−0.111612
$386$	1.66334e14	0.987987
$387$	0	0
$388$	4.24293e13	0.244958
$389$	−9.81084e13	−0.558449	−0.279224	−	0.960226i	$-0.590077\pi$
$389$	−9.81084e13	−0.558449	−0.279224	+	0.960226i	$0.590077\pi$
$390$	0	0
$391$	−6.30135e13	−0.348708
$392$	−4.76452e13	−0.259983
$393$	0	0
$394$	1.82528e14	0.968500
$395$	8.43428e12	0.0441331
$396$	0	0
$397$	−6.44136e13	−0.327816	−0.163908	−	0.986476i	$-0.552410\pi$
$397$	−6.44136e13	−0.327816	−0.163908	+	0.986476i	$0.552410\pi$
$398$	−2.04741e13	−0.102766
$399$	0	0
$400$	1.02400e13	0.0500000
$401$	−2.05799e14	−0.991172	−0.495586	−	0.868559i	$-0.665047\pi$
$401$	−2.05799e14	−0.991172	−0.495586	+	0.868559i	$0.665047\pi$
$402$	0	0
$403$	−1.31996e14	−0.618562
$404$	−1.19399e14	−0.551954
$405$	0	0
$406$	1.11701e14	0.502532
$407$	1.48096e14	0.657316
$408$	0	0
$409$	5.95615e13	0.257328	0.128664	−	0.991688i	$-0.458931\pi$
$409$	5.95615e13	0.257328	0.128664	+	0.991688i	$0.458931\pi$
$410$	−7.34063e13	−0.312912
$411$	0	0
$412$	4.40556e13	0.182838
$413$	1.44529e14	0.591874
$414$	0	0
$415$	−1.02058e14	−0.406991
$416$	−7.79624e13	−0.306812
$417$	0	0
$418$	−1.33813e13	−0.0512895
$419$	8.26291e13	0.312576	0.156288	−	0.987712i	$-0.450047\pi$
$419$	8.26291e13	0.312576	0.156288	+	0.987712i	$0.450047\pi$
$420$	0	0
$421$	−1.55708e14	−0.573800	−0.286900	−	0.957961i	$-0.592625\pi$
$421$	−1.55708e14	−0.573800	−0.286900	+	0.957961i	$0.592625\pi$
$422$	7.90479e13	0.287523
$423$	0	0
$424$	8.36825e13	0.296567
$425$	1.87331e13	0.0655345
$426$	0	0
$427$	1.79837e13	0.0613092
$428$	2.20491e14	0.742078
$429$	0	0
$430$	−5.80697e13	−0.190490
$431$	−4.40261e14	−1.42589	−0.712944	−	0.701221i	$-0.752639\pi$
$431$	−4.40261e14	−1.42589	−0.712944	+	0.701221i	$0.752639\pi$
$432$	0	0
$433$	3.21850e14	1.01618	0.508089	−	0.861304i	$-0.330352\pi$
$433$	3.21850e14	1.01618	0.508089	+	0.861304i	$0.330352\pi$
$434$	−4.15868e13	−0.129647
$435$	0	0
$436$	−2.62215e14	−0.797044
$437$	5.30101e13	0.159115
$438$	0	0
$439$	6.53476e14	1.91282	0.956411	−	0.292025i	$-0.0943289\pi$
$439$	6.53476e14	1.91282	0.956411	+	0.292025i	$0.0943289\pi$
$440$	2.65347e13	0.0767052
$441$	0	0
$442$	−1.42625e14	−0.402135
$443$	−5.50129e14	−1.53195	−0.765974	−	0.642872i	$-0.777743\pi$
$443$	−5.50129e14	−1.53195	−0.765974	+	0.642872i	$0.777743\pi$
$444$	0	0
$445$	−2.32989e14	−0.632928
$446$	−1.68547e14	−0.452252
$447$	0	0
$448$	−2.45629e13	−0.0643059
$449$	−2.56147e13	−0.0662421	−0.0331210	−	0.999451i	$-0.510545\pi$
$449$	−2.56147e13	−0.0662421	−0.0331210	+	0.999451i	$0.510545\pi$
$450$	0	0
$451$	−1.90216e14	−0.480039
$452$	−1.35898e14	−0.338807
$453$	0	0
$454$	3.66230e14	0.891142
$455$	1.66098e14	0.399303
$456$	0	0
$457$	1.73882e13	0.0408052	0.0204026	−	0.999792i	$-0.493505\pi$
$457$	1.73882e13	0.0408052	0.0204026	+	0.999792i	$0.493505\pi$
$458$	1.15999e14	0.268964
$459$	0	0
$460$	−1.05117e14	−0.237962
$461$	−7.61820e14	−1.70411	−0.852055	−	0.523453i	$-0.824644\pi$
$461$	−7.61820e14	−1.70411	−0.852055	+	0.523453i	$0.824644\pi$
$462$	0	0
$463$	3.88545e14	0.848684	0.424342	−	0.905502i	$-0.360505\pi$
$463$	3.88545e14	0.848684	0.424342	+	0.905502i	$0.360505\pi$
$464$	−1.60003e14	−0.345365
$465$	0	0
$466$	−4.60998e14	−0.971799
$467$	−1.22217e14	−0.254618	−0.127309	−	0.991863i	$-0.540634\pi$
$467$	−1.22217e14	−0.254618	−0.127309	+	0.991863i	$0.540634\pi$
$468$	0	0
$469$	4.86977e14	0.990964
$470$	−4.78847e13	−0.0963071
$471$	0	0
$472$	−2.07026e14	−0.406765
$473$	−1.50475e14	−0.292232
$474$	0	0
$475$	−1.57592e13	−0.0299033
$476$	−4.49354e13	−0.0842851
$477$	0	0
$478$	4.26234e14	0.781259
$479$	−4.35912e14	−0.789866	−0.394933	−	0.918710i	$-0.629232\pi$
$479$	−4.35912e14	−0.789866	−0.394933	+	0.918710i	$0.629232\pi$
$480$	0	0
$481$	−1.32790e15	−2.35162
$482$	3.64061e14	0.637406
$483$	0	0
$484$	−2.23400e14	−0.382326
$485$	1.29484e14	0.219097
$486$	0	0
$487$	6.66191e14	1.10202	0.551010	−	0.834499i	$-0.314243\pi$
$487$	6.66191e14	1.10202	0.551010	+	0.834499i	$0.314243\pi$
$488$	−2.57602e13	−0.0421347
$489$	0	0
$490$	−1.45402e14	−0.232536
$491$	−2.23817e13	−0.0353953	−0.0176976	−	0.999843i	$-0.505634\pi$
$491$	−2.23817e13	−0.0353953	−0.0176976	+	0.999843i	$0.505634\pi$
$492$	0	0
$493$	−2.92709e14	−0.452666
$494$	1.19983e14	0.183494
$495$	0	0
$496$	5.95697e13	0.0890997
$497$	2.68095e14	0.396578
$498$	0	0
$499$	−4.87506e13	−0.0705386	−0.0352693	−	0.999378i	$-0.511229\pi$
$499$	−4.87506e13	−0.0705386	−0.0352693	+	0.999378i	$0.511229\pi$
$500$	3.12500e13	0.0447214
$501$	0	0
$502$	9.07717e13	0.127081
$503$	−2.64333e14	−0.366039	−0.183019	−	0.983109i	$-0.558587\pi$
$503$	−2.64333e14	−0.366039	−0.183019	+	0.983109i	$0.558587\pi$
$504$	0	0
$505$	−3.64377e14	−0.493683
$506$	−2.72389e14	−0.365058
$507$	0	0
$508$	−6.14329e14	−0.805658
$509$	−8.09106e13	−0.104968	−0.0524841	−	0.998622i	$-0.516714\pi$
$509$	−8.09106e13	−0.104968	−0.0524841	+	0.998622i	$0.516714\pi$
$510$	0	0
$511$	−5.30166e14	−0.673127
$512$	3.51844e13	0.0441942
$513$	0	0
$514$	−9.64617e14	−1.18593
$515$	1.34447e14	0.163535
$516$	0	0
$517$	−1.24083e14	−0.147745
$518$	−4.18369e14	−0.492885
$519$	0	0
$520$	−2.37923e14	−0.274421
$521$	1.10833e14	0.126491	0.0632456	−	0.997998i	$-0.479855\pi$
$521$	1.10833e14	0.126491	0.0632456	+	0.997998i	$0.479855\pi$
$522$	0	0
$523$	6.91114e14	0.772308	0.386154	−	0.922434i	$-0.373803\pi$
$523$	6.91114e14	0.772308	0.386154	+	0.922434i	$0.373803\pi$
$524$	2.93325e14	0.324360
$525$	0	0
$526$	−1.21894e15	−1.31995
$527$	1.08977e14	0.116782
$528$	0	0
$529$	1.26261e14	0.132514
$530$	2.55379e14	0.265257
$531$	0	0
$532$	3.78019e13	0.0384591
$533$	1.70557e15	1.71739
$534$	0	0
$535$	6.72884e14	0.663735
$536$	−6.97555e14	−0.681039
$537$	0	0
$538$	−6.43161e13	−0.0615201
$539$	−3.76776e14	−0.356735
$540$	0	0
$541$	−2.02808e15	−1.88148	−0.940741	−	0.339125i	$-0.889869\pi$
$541$	−2.02808e15	−1.88148	−0.940741	+	0.339125i	$0.889869\pi$
$542$	−9.79460e14	−0.899478
$543$	0	0
$544$	6.43663e13	0.0579248
$545$	−8.00217e14	−0.712897
$546$	0	0
$547$	2.06348e15	1.80165	0.900826	−	0.434180i	$-0.142962\pi$
$547$	2.06348e15	1.80165	0.900826	+	0.434180i	$0.142962\pi$
$548$	2.74656e14	0.237408
$549$	0	0
$550$	8.09775e13	0.0686072
$551$	2.46241e14	0.206551
$552$	0	0
$553$	−6.17416e13	−0.0507680
$554$	8.88927e14	0.723707
$555$	0	0
$556$	1.02823e15	0.820692
$557$	2.11828e15	1.67409	0.837047	−	0.547130i	$-0.184280\pi$
$557$	2.11828e15	1.67409	0.837047	+	0.547130i	$0.184280\pi$
$558$	0	0
$559$	1.34923e15	1.04549
$560$	−7.49601e13	−0.0575170
$561$	0	0
$562$	1.26456e15	0.951458
$563$	1.16211e15	0.865870	0.432935	−	0.901425i	$-0.357478\pi$
$563$	1.16211e15	0.865870	0.432935	+	0.901425i	$0.357478\pi$
$564$	0	0
$565$	−4.14728e14	−0.303038
$566$	−1.14420e15	−0.827965
$567$	0	0
$568$	−3.84024e14	−0.272548
$569$	−1.19782e15	−0.841927	−0.420964	−	0.907077i	$-0.638308\pi$
$569$	−1.19782e15	−0.841927	−0.420964	+	0.907077i	$0.638308\pi$
$570$	0	0
$571$	1.46355e15	1.00904	0.504522	−	0.863399i	$-0.331669\pi$
$571$	1.46355e15	1.00904	0.504522	+	0.863399i	$0.331669\pi$
$572$	−6.16524e14	−0.420990
$573$	0	0
$574$	5.37358e14	0.359955
$575$	−3.20793e14	−0.212839
$576$	0	0
$577$	1.97111e15	1.28305	0.641526	−	0.767101i	$-0.278302\pi$
$577$	1.97111e15	1.28305	0.641526	+	0.767101i	$0.278302\pi$
$578$	−9.78949e14	−0.631185
$579$	0	0
$580$	−4.88289e14	−0.308904
$581$	7.47101e14	0.468177
$582$	0	0
$583$	6.61758e14	0.406932
$584$	7.59419e14	0.462606
$585$	0	0
$586$	−3.73539e14	−0.223305
$587$	−3.58593e14	−0.212369	−0.106185	−	0.994346i	$-0.533863\pi$
$587$	−3.58593e14	−0.212369	−0.106185	+	0.994346i	$0.533863\pi$
$588$	0	0
$589$	−9.16767e13	−0.0532874
$590$	−6.31792e14	−0.363821
$591$	0	0
$592$	5.99279e14	0.338735
$593$	1.43925e15	0.806003	0.403001	−	0.915199i	$-0.367967\pi$
$593$	1.43925e15	0.806003	0.403001	+	0.915199i	$0.367967\pi$
$594$	0	0
$595$	−1.37132e14	−0.0753869
$596$	−6.76258e14	−0.368348
$597$	0	0
$598$	2.44236e15	1.30603
$599$	3.41497e15	1.80942	0.904710	−	0.426028i	$-0.140088\pi$
$599$	3.41497e15	1.80942	0.904710	+	0.426028i	$0.140088\pi$
$600$	0	0
$601$	2.49987e15	1.30049	0.650245	−	0.759725i	$-0.274666\pi$
$601$	2.49987e15	1.30049	0.650245	+	0.759725i	$0.274666\pi$
$602$	4.25089e14	0.219128
$603$	0	0
$604$	1.91094e14	0.0967263
$605$	−6.81764e14	−0.341963
$606$	0	0
$607$	−2.34097e13	−0.0115308	−0.00576538	−	0.999983i	$-0.501835\pi$
$607$	−2.34097e13	−0.0115308	−0.00576538	+	0.999983i	$0.501835\pi$
$608$	−5.41481e13	−0.0264310
$609$	0	0
$610$	−7.86138e13	−0.0376864
$611$	1.11258e15	0.528573
$612$	0	0
$613$	−1.82595e15	−0.852032	−0.426016	−	0.904716i	$-0.640083\pi$
$613$	−1.82595e15	−0.852032	−0.426016	+	0.904716i	$0.640083\pi$
$614$	2.87397e15	1.32909
$615$	0	0
$616$	−1.94243e14	−0.0882369
$617$	9.58777e14	0.431667	0.215834	−	0.976430i	$-0.430753\pi$
$617$	9.58777e14	0.431667	0.215834	+	0.976430i	$0.430753\pi$
$618$	0	0
$619$	−6.16431e13	−0.0272638	−0.0136319	−	0.999907i	$-0.504339\pi$
$619$	−6.16431e13	−0.0272638	−0.0136319	+	0.999907i	$0.504339\pi$
$620$	1.81792e14	0.0796932
$621$	0	0
$622$	−2.16281e15	−0.931477
$623$	1.70555e15	0.728082
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	3.15895e15	1.31336
$627$	0	0
$628$	4.63315e14	0.189277
$629$	1.09632e15	0.443976
$630$	0	0
$631$	−3.38937e15	−1.34883	−0.674416	−	0.738352i	$-0.735604\pi$
$631$	−3.38937e15	−1.34883	−0.674416	+	0.738352i	$0.735604\pi$
$632$	8.84398e13	0.0348902
$633$	0	0
$634$	−3.53672e15	−1.37123
$635$	−1.87478e15	−0.720602
$636$	0	0
$637$	3.37835e15	1.27626
$638$	−1.26529e15	−0.473890
$639$	0	0
$640$	1.07374e14	0.0395285
$641$	−2.53363e15	−0.924749	−0.462375	−	0.886685i	$-0.653002\pi$
$641$	−2.53363e15	−0.924749	−0.462375	+	0.886685i	$0.653002\pi$
$642$	0	0
$643$	−4.33651e15	−1.55589	−0.777946	−	0.628331i	$-0.783739\pi$
$643$	−4.33651e15	−1.55589	−0.777946	+	0.628331i	$0.783739\pi$
$644$	7.69493e14	0.273736
$645$	0	0
$646$	−9.90586e13	−0.0346428
$647$	1.86323e15	0.646090	0.323045	−	0.946384i	$-0.395294\pi$
$647$	1.86323e15	0.646090	0.323045	+	0.946384i	$0.395294\pi$
$648$	0	0
$649$	−1.63715e15	−0.558139
$650$	−7.26082e14	−0.245449
$651$	0	0
$652$	1.55307e15	0.516213
$653$	−5.87157e15	−1.93523	−0.967613	−	0.252440i	$-0.918767\pi$
$653$	−5.87157e15	−1.93523	−0.967613	+	0.252440i	$0.918767\pi$
$654$	0	0
$655$	8.95157e14	0.290116
$656$	−7.69721e14	−0.247379
$657$	0	0
$658$	3.50531e14	0.110786
$659$	1.57270e14	0.0492920	0.0246460	−	0.999696i	$-0.492154\pi$
$659$	1.57270e14	0.0492920	0.0246460	+	0.999696i	$0.492154\pi$
$660$	0	0
$661$	−4.86756e15	−1.50039	−0.750193	−	0.661219i	$-0.770039\pi$
$661$	−4.86756e15	−1.50039	−0.750193	+	0.661219i	$0.770039\pi$
$662$	−1.60050e15	−0.489256
$663$	0	0
$664$	−1.07016e15	−0.321754
$665$	1.15362e14	0.0343989
$666$	0	0
$667$	5.01247e15	1.47014
$668$	2.40703e15	0.700181
$669$	0	0
$670$	−2.12877e15	−0.609140
$671$	−2.03710e14	−0.0578148
$672$	0	0
$673$	−5.94031e14	−0.165854	−0.0829270	−	0.996556i	$-0.526427\pi$
$673$	−5.94031e14	−0.165854	−0.0829270	+	0.996556i	$0.526427\pi$
$674$	−1.15993e14	−0.0321220
$675$	0	0
$676$	3.69287e15	1.00614
$677$	−3.03615e15	−0.820513	−0.410257	−	0.911970i	$-0.634561\pi$
$677$	−3.03615e15	−0.820513	−0.410257	+	0.911970i	$0.634561\pi$
$678$	0	0
$679$	−9.47864e14	−0.252036
$680$	1.96430e14	0.0518095
$681$	0	0
$682$	4.71075e14	0.122258
$683$	−2.38176e15	−0.613174	−0.306587	−	0.951843i	$-0.599187\pi$
$683$	−2.38176e15	−0.613174	−0.306587	+	0.951843i	$0.599187\pi$
$684$	0	0
$685$	8.38184e14	0.212345
$686$	2.51185e15	0.631265
$687$	0	0
$688$	−6.08905e14	−0.150596
$689$	−5.93363e15	−1.45584
$690$	0	0
$691$	1.02015e15	0.246339	0.123170	−	0.992386i	$-0.460694\pi$
$691$	1.02015e15	0.246339	0.123170	+	0.992386i	$0.460694\pi$
$692$	−9.22737e14	−0.221052
$693$	0	0
$694$	−2.09938e14	−0.0495011
$695$	3.13792e15	0.734049
$696$	0	0
$697$	−1.40813e15	−0.324237
$698$	−2.60593e15	−0.595330
$699$	0	0
$700$	−2.28760e14	−0.0514447
$701$	9.47724e14	0.211462	0.105731	−	0.994395i	$-0.466282\pi$
$701$	9.47724e14	0.211462	0.105731	+	0.994395i	$0.466282\pi$
$702$	0	0
$703$	−9.22280e14	−0.202586
$704$	2.78237e14	0.0606408
$705$	0	0
$706$	−3.32135e15	−0.712671
$707$	2.66735e15	0.567902
$708$	0	0
$709$	1.41385e15	0.296380	0.148190	−	0.988959i	$-0.452655\pi$
$709$	1.41385e15	0.296380	0.148190	+	0.988959i	$0.452655\pi$
$710$	−1.17195e15	−0.243774
$711$	0	0
$712$	−2.44306e15	−0.500374
$713$	−1.86617e15	−0.379278
$714$	0	0
$715$	−1.88148e15	−0.376545
$716$	1.00045e15	0.198690
$717$	0	0
$718$	4.42426e15	0.865280
$719$	−7.85082e15	−1.52372	−0.761862	−	0.647739i	$-0.775715\pi$
$719$	−7.85082e15	−1.52372	−0.761862	+	0.647739i	$0.775715\pi$
$720$	0	0
$721$	−9.84196e14	−0.188121
$722$	−3.64436e15	−0.691299
$723$	0	0
$724$	3.92213e15	0.732756
$725$	−1.49014e15	−0.276292
$726$	0	0
$727$	−6.17476e15	−1.12767	−0.563833	−	0.825889i	$-0.690674\pi$
$727$	−6.17476e15	−1.12767	−0.563833	+	0.825889i	$0.690674\pi$
$728$	1.74167e15	0.315677
$729$	0	0
$730$	2.31756e15	0.413767
$731$	−1.11393e15	−0.197384
$732$	0	0
$733$	−7.25819e15	−1.26694	−0.633471	−	0.773767i	$-0.718370\pi$
$733$	−7.25819e15	−1.26694	−0.633471	+	0.773767i	$0.718370\pi$
$734$	3.72342e15	0.645081
$735$	0	0
$736$	−1.10224e15	−0.188125
$737$	−5.51623e15	−0.934483
$738$	0	0
$739$	1.04737e16	1.74806	0.874030	−	0.485872i	$-0.161498\pi$
$739$	1.04737e16	1.74806	0.874030	+	0.485872i	$0.161498\pi$
$740$	1.82886e15	0.302974
$741$	0	0
$742$	−1.86945e15	−0.305136
$743$	−2.62290e15	−0.424956	−0.212478	−	0.977166i	$-0.568153\pi$
$743$	−2.62290e15	−0.424956	−0.212478	+	0.977166i	$0.568153\pi$
$744$	0	0
$745$	−2.06377e15	−0.329460
$746$	−2.15099e15	−0.340860
$747$	0	0
$748$	5.09006e14	0.0794812
$749$	−4.92573e15	−0.763520
$750$	0	0
$751$	3.09275e15	0.472417	0.236208	−	0.971702i	$-0.424095\pi$
$751$	3.09275e15	0.472417	0.236208	+	0.971702i	$0.424095\pi$
$752$	−5.02107e14	−0.0761374
$753$	0	0
$754$	1.13452e16	1.69539
$755$	5.83173e14	0.0865146
$756$	0	0
$757$	−5.54759e15	−0.811105	−0.405553	−	0.914072i	$-0.632921\pi$
$757$	−5.54759e15	−0.811105	−0.405553	+	0.914072i	$0.632921\pi$
$758$	−4.25089e15	−0.617020
$759$	0	0
$760$	−1.65247e14	−0.0236406
$761$	−1.79365e13	−0.00254754	−0.00127377	−	0.999999i	$-0.500405\pi$
$761$	−1.79365e13	−0.00254754	−0.00127377	+	0.999999i	$0.500405\pi$
$762$	0	0
$763$	5.85784e15	0.820074
$764$	2.55687e15	0.355382
$765$	0	0
$766$	8.63391e15	1.18290
$767$	1.46794e16	1.99680
$768$	0	0
$769$	−1.07234e16	−1.43794	−0.718968	−	0.695043i	$-0.755385\pi$
$769$	−1.07234e16	−1.43794	−0.718968	+	0.695043i	$0.755385\pi$
$770$	−5.92781e14	−0.0789215
$771$	0	0
$772$	5.32269e15	0.698612
$773$	−6.56995e15	−0.856199	−0.428100	−	0.903732i	$-0.640817\pi$
$773$	−6.56995e15	−0.856199	−0.428100	+	0.903732i	$0.640817\pi$
$774$	0	0
$775$	5.54786e14	0.0712797
$776$	1.35774e15	0.173211
$777$	0	0
$778$	−3.13947e15	−0.394883
$779$	1.18459e15	0.147949
$780$	0	0
$781$	−3.03685e15	−0.373975
$782$	−2.01643e15	−0.246574
$783$	0	0
$784$	−1.52465e15	−0.183836
$785$	1.41392e15	0.169295
$786$	0	0
$787$	−1.38290e16	−1.63279	−0.816394	−	0.577495i	$-0.804030\pi$
$787$	−1.38290e16	−1.63279	−0.816394	+	0.577495i	$0.804030\pi$
$788$	5.84088e15	0.684833
$789$	0	0
$790$	2.69897e14	0.0312068
$791$	3.03594e15	0.348596
$792$	0	0
$793$	1.82656e15	0.206839
$794$	−2.06124e15	−0.231801
$795$	0	0
$796$	−6.55172e14	−0.0726664
$797$	5.66909e15	0.624442	0.312221	−	0.950009i	$-0.398927\pi$
$797$	5.66909e15	0.624442	0.312221	+	0.950009i	$0.398927\pi$
$798$	0	0
$799$	−9.18555e14	−0.0997925
$800$	3.27680e14	0.0353553
$801$	0	0
$802$	−6.58557e15	−0.700865
$803$	6.00546e15	0.634761
$804$	0	0
$805$	2.34831e15	0.244837
$806$	−4.22387e15	−0.437389
$807$	0	0
$808$	−3.82077e15	−0.390290
$809$	−1.22726e16	−1.24514	−0.622572	−	0.782563i	$-0.713912\pi$
$809$	−1.22726e16	−1.24514	−0.622572	+	0.782563i	$0.713912\pi$
$810$	0	0
$811$	2.48630e15	0.248851	0.124426	−	0.992229i	$-0.460291\pi$
$811$	2.48630e15	0.248851	0.124426	+	0.992229i	$0.460291\pi$
$812$	3.57443e15	0.355344
$813$	0	0
$814$	4.73908e15	0.464793
$815$	4.73959e15	0.461715
$816$	0	0
$817$	9.37093e14	0.0900661
$818$	1.90597e15	0.181959
$819$	0	0
$820$	−2.34900e15	−0.221262
$821$	1.90122e16	1.77887	0.889435	−	0.457062i	$-0.151098\pi$
$821$	1.90122e16	1.77887	0.889435	+	0.457062i	$0.151098\pi$
$822$	0	0
$823$	2.42079e15	0.223490	0.111745	−	0.993737i	$-0.464356\pi$
$823$	2.42079e15	0.223490	0.111745	+	0.993737i	$0.464356\pi$
$824$	1.40978e15	0.129286
$825$	0	0
$826$	4.62492e15	0.418518
$827$	1.83310e16	1.64781	0.823903	−	0.566731i	$-0.191792\pi$
$827$	1.83310e16	1.64781	0.823903	+	0.566731i	$0.191792\pi$
$828$	0	0
$829$	−7.71702e15	−0.684541	−0.342271	−	0.939601i	$-0.611196\pi$
$829$	−7.71702e15	−0.684541	−0.342271	+	0.939601i	$0.611196\pi$
$830$	−3.26587e15	−0.287786
$831$	0	0
$832$	−2.49480e15	−0.216949
$833$	−2.78919e15	−0.240952
$834$	0	0
$835$	7.34566e15	0.626261
$836$	−4.28201e14	−0.0362671
$837$	0	0
$838$	2.64413e15	0.221025
$839$	1.33884e16	1.11183	0.555914	−	0.831240i	$-0.312368\pi$
$839$	1.33884e16	1.11183	0.555914	+	0.831240i	$0.312368\pi$
$840$	0	0
$841$	1.10833e16	0.908428
$842$	−4.98267e15	−0.405738
$843$	0	0
$844$	2.52953e15	0.203309
$845$	1.12697e16	0.899916
$846$	0	0
$847$	4.99073e15	0.393374
$848$	2.67784e15	0.209704
$849$	0	0
$850$	5.99458e14	0.0463399
$851$	−1.87739e16	−1.44192
$852$	0	0
$853$	3.99518e15	0.302912	0.151456	−	0.988464i	$-0.451604\pi$
$853$	3.99518e15	0.302912	0.151456	+	0.988464i	$0.451604\pi$
$854$	5.75478e14	0.0433522
$855$	0	0
$856$	7.05570e15	0.524728
$857$	−1.50656e16	−1.11324	−0.556622	−	0.830766i	$-0.687903\pi$
$857$	−1.50656e16	−1.11324	−0.556622	+	0.830766i	$0.687903\pi$
$858$	0	0
$859$	−2.66635e16	−1.94516	−0.972579	−	0.232572i	$-0.925286\pi$
$859$	−2.66635e16	−1.94516	−0.972579	+	0.232572i	$0.925286\pi$
$860$	−1.85823e15	−0.134697
$861$	0	0
$862$	−1.40884e16	−1.00826
$863$	−1.36614e16	−0.971486	−0.485743	−	0.874102i	$-0.661451\pi$
$863$	−1.36614e16	−0.971486	−0.485743	+	0.874102i	$0.661451\pi$
$864$	0	0
$865$	−2.81597e15	−0.197715
$866$	1.02992e16	0.718547
$867$	0	0
$868$	−1.33078e15	−0.0916742
$869$	6.99379e14	0.0478744
$870$	0	0
$871$	4.94611e16	3.34321
$872$	−8.39088e15	−0.563595
$873$	0	0
$874$	1.69632e15	0.112511
$875$	−6.98120e14	−0.0460136
$876$	0	0
$877$	−2.44903e16	−1.59403	−0.797016	−	0.603958i	$-0.793590\pi$
$877$	−2.44903e16	−1.59403	−0.797016	+	0.603958i	$0.793590\pi$
$878$	2.09112e16	1.35257
$879$	0	0
$880$	8.49111e14	0.0542387
$881$	−9.28368e15	−0.589322	−0.294661	−	0.955602i	$-0.595207\pi$
$881$	−9.28368e15	−0.589322	−0.294661	+	0.955602i	$0.595207\pi$
$882$	0	0
$883$	1.88185e16	1.17978	0.589890	−	0.807484i	$-0.299171\pi$
$883$	1.88185e16	1.17978	0.589890	+	0.807484i	$0.299171\pi$
$884$	−4.56399e15	−0.284352
$885$	0	0
$886$	−1.76041e16	−1.08325
$887$	6.50887e14	0.0398039	0.0199020	−	0.999802i	$-0.493665\pi$
$887$	6.50887e14	0.0398039	0.0199020	+	0.999802i	$0.493665\pi$
$888$	0	0
$889$	1.37240e16	0.828937
$890$	−7.45563e15	−0.447548
$891$	0	0
$892$	−5.39352e15	−0.319790
$893$	7.72734e14	0.0455352
$894$	0	0
$895$	3.05314e15	0.177713
$896$	−7.86013e14	−0.0454711
$897$	0	0
$898$	−8.19670e14	−0.0468402
$899$	−8.66867e15	−0.492350
$900$	0	0
$901$	4.89884e15	0.274857
$902$	−6.08692e15	−0.339439
$903$	0	0
$904$	−4.34874e15	−0.239573
$905$	1.19694e16	0.655397
$906$	0	0
$907$	2.01411e16	1.08954	0.544770	−	0.838585i	$-0.316617\pi$
$907$	2.01411e16	1.08954	0.544770	+	0.838585i	$0.316617\pi$
$908$	1.17194e16	0.630133
$909$	0	0
$910$	5.31515e15	0.282350
$911$	4.85179e15	0.256184	0.128092	−	0.991762i	$-0.459115\pi$
$911$	4.85179e15	0.256184	0.128092	+	0.991762i	$0.459115\pi$
$912$	0	0
$913$	−8.46279e15	−0.441493
$914$	5.56422e14	0.0288536
$915$	0	0
$916$	3.71197e15	0.190186
$917$	−6.55283e15	−0.333732
$918$	0	0
$919$	7.50130e15	0.377486	0.188743	−	0.982026i	$-0.439559\pi$
$919$	7.50130e15	0.377486	0.188743	+	0.982026i	$0.439559\pi$
$920$	−3.36376e15	−0.168264
$921$	0	0
$922$	−2.43782e16	−1.20499
$923$	2.72298e16	1.33793
$924$	0	0
$925$	5.58122e15	0.270988
$926$	1.24335e16	0.600111
$927$	0	0
$928$	−5.12008e15	−0.244210
$929$	6.04589e15	0.286664	0.143332	−	0.989675i	$-0.454218\pi$
$929$	6.04589e15	0.286664	0.143332	+	0.989675i	$0.454218\pi$
$930$	0	0
$931$	2.34640e15	0.109946
$932$	−1.47519e16	−0.687165
$933$	0	0
$934$	−3.91094e15	−0.180042
$935$	1.55336e15	0.0710901
$936$	0	0
$937$	1.17484e16	0.531389	0.265694	−	0.964057i	$-0.414399\pi$
$937$	1.17484e16	0.531389	0.265694	+	0.964057i	$0.414399\pi$
$938$	1.55833e16	0.700717
$939$	0	0
$940$	−1.53231e15	−0.0680994
$941$	5.89103e15	0.260284	0.130142	−	0.991495i	$-0.458457\pi$
$941$	5.89103e15	0.260284	0.130142	+	0.991495i	$0.458457\pi$
$942$	0	0
$943$	2.41134e16	1.05304
$944$	−6.62482e15	−0.287626
$945$	0	0
$946$	−4.81519e15	−0.206639
$947$	−2.96672e15	−0.126576	−0.0632881	−	0.997995i	$-0.520159\pi$
$947$	−2.96672e15	−0.126576	−0.0632881	+	0.997995i	$0.520159\pi$
$948$	0	0
$949$	−5.38477e16	−2.27093
$950$	−5.04294e14	−0.0211448
$951$	0	0
$952$	−1.43793e15	−0.0595985
$953$	−1.10371e15	−0.0454826	−0.0227413	−	0.999741i	$-0.507239\pi$
$953$	−1.10371e15	−0.0454826	−0.0227413	+	0.999741i	$0.507239\pi$
$954$	0	0
$955$	7.80295e15	0.317863
$956$	1.36395e16	0.552433
$957$	0	0
$958$	−1.39492e16	−0.558520
$959$	−6.13577e15	−0.244268
$960$	0	0
$961$	−2.21811e16	−0.872980
$962$	−4.24928e16	−1.66285
$963$	0	0
$964$	1.16500e16	0.450714
$965$	1.62436e16	0.624858
$966$	0	0
$967$	−1.84533e15	−0.0701826	−0.0350913	−	0.999384i	$-0.511172\pi$
$967$	−1.84533e15	−0.0701826	−0.0350913	+	0.999384i	$0.511172\pi$
$968$	−7.14881e15	−0.270346
$969$	0	0
$970$	4.14349e15	0.154925
$971$	1.46125e16	0.543273	0.271636	−	0.962400i	$-0.412435\pi$
$971$	1.46125e16	0.543273	0.271636	+	0.962400i	$0.412435\pi$
$972$	0	0
$973$	−2.29706e16	−0.844405
$974$	2.13181e16	0.779245
$975$	0	0
$976$	−8.24326e14	−0.0297937
$977$	−1.87225e16	−0.672891	−0.336445	−	0.941703i	$-0.609225\pi$
$977$	−1.87225e16	−0.672891	−0.336445	+	0.941703i	$0.609225\pi$
$978$	0	0
$979$	−1.93196e16	−0.686585
$980$	−4.65285e15	−0.164428
$981$	0	0
$982$	−7.16215e14	−0.0250282
$983$	−2.79209e16	−0.970253	−0.485126	−	0.874444i	$-0.661226\pi$
$983$	−2.79209e16	−0.970253	−0.485126	+	0.874444i	$0.661226\pi$
$984$	0	0
$985$	1.78250e16	0.612533
$986$	−9.36668e15	−0.320083
$987$	0	0
$988$	3.83945e15	0.129750
$989$	1.90754e16	0.641054
$990$	0	0
$991$	2.46391e16	0.818878	0.409439	−	0.912337i	$-0.365724\pi$
$991$	2.46391e16	0.818878	0.409439	+	0.912337i	$0.365724\pi$
$992$	1.90623e15	0.0630030
$993$	0	0
$994$	8.57903e15	0.280423
$995$	−1.99943e15	−0.0649948
$996$	0	0
$997$	4.80148e16	1.54366	0.771831	−	0.635828i	$-0.219341\pi$
$997$	4.80148e16	1.54366	0.771831	+	0.635828i	$0.219341\pi$
$998$	−1.56002e15	−0.0498783
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.12.a.j.1.1		1
3.2	odd	2		30.12.a.a.1.1	✓	1
12.11	even	2		240.12.a.d.1.1		1
15.2	even	4		150.12.c.g.49.1		2
15.8	even	4		150.12.c.g.49.2		2
15.14	odd	2		150.12.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.12.a.a.1.1	✓	1	3.2	odd	2
90.12.a.j.1.1		1	1.1	even	1	trivial
150.12.a.h.1.1		1	15.14	odd	2
150.12.c.g.49.1		2	15.2	even	4
150.12.c.g.49.2		2	15.8	even	4
240.12.a.d.1.1		1	12.11	even	2

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:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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