LMFDB - Embedded newform 64.22.b.c.33.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,22,Mod(33,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("64.33");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	64.b (of order $2$, degree $1$, 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:	$178.865500344$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.6
Character	$\chi$	$=$	64.33
Dual form			64.22.b.c.33.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-114658. i q^{3} +1.31189e7i q^{5} -8.14201e8 q^{7} -2.68616e9 q^{9} +O(q^{10})$	$q-114658. i q^{3} +1.31189e7i q^{5} -8.14201e8 q^{7} -2.68616e9 q^{9} +1.24608e11i q^{11} +1.76398e11i q^{13} +1.50419e12 q^{15} -3.32868e12 q^{17} +1.75236e13i q^{19} +9.33549e13i q^{21} +2.65572e14 q^{23} +3.04732e14 q^{25} -8.91375e14i q^{27} -7.38411e14i q^{29} -2.12499e15 q^{31} +1.42874e16 q^{33} -1.06814e16i q^{35} +3.13429e16i q^{37} +2.02255e16 q^{39} +3.67346e15 q^{41} +1.60426e17i q^{43} -3.52394e16i q^{45} -4.23709e17 q^{47} +1.04377e17 q^{49} +3.81660e17i q^{51} -1.51790e18i q^{53} -1.63472e18 q^{55} +2.00923e18 q^{57} -2.75219e18i q^{59} +1.67390e18i q^{61} +2.18708e18 q^{63} -2.31415e18 q^{65} -2.15114e19i q^{67} -3.04501e19i q^{69} -3.67913e19 q^{71} +5.73277e19 q^{73} -3.49401e19i q^{75} -1.01456e20i q^{77} -9.95602e19 q^{79} -1.30302e20 q^{81} +1.00024e20i q^{83} -4.36685e19i q^{85} -8.46649e19 q^{87} +3.88634e20 q^{89} -1.43624e20i q^{91} +2.43648e20i q^{93} -2.29890e20 q^{95} +4.38790e20 q^{97} -3.34719e20i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 77960422492 q^{9}+O(q^{10}) $ 28 * q - 77960422492 * q^9	$ 28 q - 77960422492 q^{9} + 16832040195288 q^{17} + 202504130118092 q^{25} - 55\!\cdots\!92 q^{33}+ \cdots - 19\!\cdots\!16 q^{97}+O(q^{100}) $ 28 * q - 77960422492 * q^9 + 16832040195288 * q^17 + 202504130118092 * q^25 - 55633095046450992 * q^33 + 265230847130533848 * q^41 + 5944576136704810940 * q^49 - 4544453360963256304 * q^57 - 63340883112821103360 * q^65 + 2760069698648092472 * q^73 + 934709246686586470348 * q^81 - 375882191672218968072 * q^89 - 1953791530192678912616 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$.

$n$	$5$	$63$
$\chi(n)$	$-1$	$1$

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
$3$	− 114658.i	− 1.12107i	−0.828131	−	0.560534i	$-0.810596\pi$
$3$	− 114658.i	− 1.12107i	0.828131	−	0.560534i	$-0.189404\pi$
$4$	0	0
$5$	1.31189e7i	0.600775i	0.953817	+	0.300387i	$0.0971159\pi$
$5$	1.31189e7i	0.600775i	−0.953817	+	0.300387i	$0.902884\pi$
$6$	0	0
$7$	−8.14201e8	−1.08944	−0.544719	−	0.838619i	$-0.683364\pi$
$7$	−8.14201e8	−1.08944	−0.544719	+	0.838619i	$0.683364\pi$
$8$	0	0
$9$	−2.68616e9	−0.256795
$10$	0	0
$11$	1.24608e11i	1.44852i	0.689527	+	0.724260i	$0.257818\pi$
$11$	1.24608e11i	1.44852i	−0.689527	+	0.724260i	$0.742182\pi$
$12$	0	0
$13$	1.76398e11i	0.354886i	0.984131	+	0.177443i	$0.0567825\pi$
$13$	1.76398e11i	0.354886i	−0.984131	+	0.177443i	$0.943217\pi$
$14$	0	0
$15$	1.50419e12	0.673509
$16$	0	0
$17$	−3.32868e12	−0.400459	−0.200229	−	0.979749i	$-0.564169\pi$
$17$	−3.32868e12	−0.400459	−0.200229	+	0.979749i	$0.564169\pi$
$18$	0	0
$19$	1.75236e13i	0.655708i	0.944728	+	0.327854i	$0.106325\pi$
$19$	1.75236e13i	0.655708i	−0.944728	+	0.327854i	$0.893675\pi$
$20$	0	0
$21$	9.33549e13i	1.22133i
$22$	0	0
$23$	2.65572e14	1.33672	0.668360	−	0.743838i	$-0.266997\pi$
$23$	2.65572e14	1.33672	0.668360	+	0.743838i	$0.266997\pi$
$24$	0	0
$25$	3.04732e14	0.639070
$26$	0	0
$27$	− 8.91375e14i	− 0.833184i
$28$	0	0
$29$	− 7.38411e14i	− 0.325926i	−0.986632	−	0.162963i	$-0.947895\pi$
$29$	− 7.38411e14i	− 0.325926i	0.986632	−	0.162963i	$-0.0521051\pi$
$30$	0	0
$31$	−2.12499e15	−0.465650	−0.232825	−	0.972519i	$-0.574797\pi$
$31$	−2.12499e15	−0.465650	−0.232825	+	0.972519i	$0.574797\pi$
$32$	0	0
$33$	1.42874e16	1.62389
$34$	0	0
$35$	− 1.06814e16i	− 0.654506i
$36$	0	0
$37$	3.13429e16i	1.07157i	0.844354	+	0.535785i	$0.179984\pi$
$37$	3.13429e16i	1.07157i	−0.844354	+	0.535785i	$0.820016\pi$
$38$	0	0
$39$	2.02255e16	0.397852
$40$	0	0
$41$	3.67346e15	0.0427411	0.0213705	−	0.999772i	$-0.493197\pi$
$41$	3.67346e15	0.0427411	0.0213705	+	0.999772i	$0.493197\pi$
$42$	0	0
$43$	1.60426e17i	1.13202i	0.824397	+	0.566012i	$0.191515\pi$
$43$	1.60426e17i	1.13202i	−0.824397	+	0.566012i	$0.808485\pi$
$44$	0	0
$45$	− 3.52394e16i	− 0.154276i
$46$	0	0
$47$	−4.23709e17	−1.17501	−0.587504	−	0.809222i	$-0.699889\pi$
$47$	−4.23709e17	−1.17501	−0.587504	+	0.809222i	$0.699889\pi$
$48$	0	0
$49$	1.04377e17	0.186874
$50$	0	0
$51$	3.81660e17i	0.448942i
$52$	0	0
$53$	− 1.51790e18i	− 1.19219i	−0.802912	−	0.596097i	$-0.796717\pi$
$53$	− 1.51790e18i	− 1.19219i	0.802912	−	0.596097i	$-0.203283\pi$
$54$	0	0
$55$	−1.63472e18	−0.870234
$56$	0	0
$57$	2.00923e18	0.735094
$58$	0	0
$59$	− 2.75219e18i	− 0.701022i	−0.936559	−	0.350511i	$-0.886008\pi$
$59$	− 2.75219e18i	− 0.701022i	0.936559	−	0.350511i	$-0.113992\pi$
$60$	0	0
$61$	1.67390e18i	0.300446i	0.988652	+	0.150223i	$0.0479991\pi$
$61$	1.67390e18i	0.300446i	−0.988652	+	0.150223i	$0.952001\pi$
$62$	0	0
$63$	2.18708e18	0.279762
$64$	0	0
$65$	−2.31415e18	−0.213207
$66$	0	0
$67$	− 2.15114e19i	− 1.44173i	−0.693075	−	0.720865i	$-0.743745\pi$
$67$	− 2.15114e19i	− 1.44173i	0.693075	−	0.720865i	$-0.256255\pi$
$68$	0	0
$69$	− 3.04501e19i	− 1.49855i
$70$	0	0
$71$	−3.67913e19	−1.34132	−0.670660	−	0.741765i	$-0.733989\pi$
$71$	−3.67913e19	−1.34132	−0.670660	+	0.741765i	$0.733989\pi$
$72$	0	0
$73$	5.73277e19	1.56126	0.780629	−	0.624994i	$-0.214899\pi$
$73$	5.73277e19	1.56126	0.780629	+	0.624994i	$0.214899\pi$
$74$	0	0
$75$	− 3.49401e19i	− 0.716441i
$76$	0	0
$77$	− 1.01456e20i	− 1.57807i
$78$	0	0
$79$	−9.95602e19	−1.18304	−0.591522	−	0.806289i	$-0.701473\pi$
$79$	−9.95602e19	−1.18304	−0.591522	+	0.806289i	$0.701473\pi$
$80$	0	0
$81$	−1.30302e20	−1.19085
$82$	0	0
$83$	1.00024e20i	0.707596i	0.935322	+	0.353798i	$0.115110\pi$
$83$	1.00024e20i	0.707596i	−0.935322	+	0.353798i	$0.884890\pi$
$84$	0	0
$85$	− 4.36685e19i	− 0.240585i
$86$	0	0
$87$	−8.46649e19	−0.365385
$88$	0	0
$89$	3.88634e20	1.32113	0.660566	−	0.750768i	$-0.270317\pi$
$89$	3.88634e20	1.32113	0.660566	+	0.750768i	$0.270317\pi$
$90$	0	0
$91$	− 1.43624e20i	− 0.386626i
$92$	0	0
$93$	2.43648e20i	0.522025i
$94$	0	0
$95$	−2.29890e20	−0.393933
$96$	0	0
$97$	4.38790e20	0.604162	0.302081	−	0.953282i	$-0.402319\pi$
$97$	4.38790e20	0.604162	0.302081	+	0.953282i	$0.402319\pi$
$98$	0	0
$99$	− 3.34719e20i	− 0.371972i
$100$	0	0
$101$	− 3.97827e20i	− 0.358360i	−0.983816	−	0.179180i	$-0.942656\pi$
$101$	− 3.97827e20i	− 0.358360i	0.983816	−	0.179180i	$-0.0573445\pi$
$102$	0	0
$103$	9.02232e20	0.661496	0.330748	−	0.943719i	$-0.392699\pi$
$103$	9.02232e20	0.661496	0.330748	+	0.943719i	$0.392699\pi$
$104$	0	0
$105$	−1.22471e21	−0.733746
$106$	0	0
$107$	8.14459e20i	0.400258i	0.979770	+	0.200129i	$0.0641361\pi$
$107$	8.14459e20i	0.400258i	−0.979770	+	0.200129i	$0.935864\pi$
$108$	0	0
$109$	3.54780e21i	1.43542i	0.696340	+	0.717712i	$0.254811\pi$
$109$	3.54780e21i	1.43542i	−0.696340	+	0.717712i	$0.745189\pi$
$110$	0	0
$111$	3.59372e21	1.20130
$112$	0	0
$113$	−4.17475e21	−1.15693	−0.578464	−	0.815708i	$-0.696348\pi$
$113$	−4.17475e21	−1.15693	−0.578464	+	0.815708i	$0.696348\pi$
$114$	0	0
$115$	3.48401e21i	0.803068i
$116$	0	0
$117$	− 4.73834e20i	− 0.0911328i
$118$	0	0
$119$	2.71021e21	0.436275
$120$	0	0
$121$	−8.12702e21	−1.09821
$122$	0	0
$123$	− 4.21193e20i	− 0.0479156i
$124$	0	0
$125$	1.02533e22i	0.984712i
$126$	0	0
$127$	−1.00609e22	−0.817896	−0.408948	−	0.912558i	$-0.634104\pi$
$127$	−1.00609e22	−0.817896	−0.408948	+	0.912558i	$0.634104\pi$
$128$	0	0
$129$	1.83942e22	1.26908
$130$	0	0
$131$	− 5.16162e21i	− 0.302996i	−0.988458	−	0.151498i	$-0.951590\pi$
$131$	− 5.16162e21i	− 0.302996i	0.988458	−	0.151498i	$-0.0484098\pi$
$132$	0	0
$133$	− 1.42677e22i	− 0.714353i
$134$	0	0
$135$	1.16938e22	0.500556
$136$	0	0
$137$	4.18447e21	0.153488	0.0767441	−	0.997051i	$-0.475548\pi$
$137$	4.18447e21	0.153488	0.0767441	+	0.997051i	$0.475548\pi$
$138$	0	0
$139$	− 6.91283e21i	− 0.217771i	−0.994054	−	0.108886i	$-0.965272\pi$
$139$	− 6.91283e21i	− 0.217771i	0.994054	−	0.108886i	$-0.0347282\pi$
$140$	0	0
$141$	4.85818e22i	1.31726i
$142$	0	0
$143$	−2.19807e22	−0.514059
$144$	0	0
$145$	9.68712e21	0.195808
$146$	0	0
$147$	− 1.19677e22i	− 0.209498i
$148$	0	0
$149$	− 6.29440e22i	− 0.956090i	−0.878335	−	0.478045i	$-0.841346\pi$
$149$	− 6.29440e22i	− 0.956090i	0.878335	−	0.478045i	$-0.158654\pi$
$150$	0	0
$151$	−1.40998e23	−1.86189	−0.930947	−	0.365155i	$-0.881016\pi$
$151$	−1.40998e23	−1.86189	−0.930947	+	0.365155i	$0.881016\pi$
$152$	0	0
$153$	8.94136e21	0.102836
$154$	0	0
$155$	− 2.78775e22i	− 0.279751i
$156$	0	0
$157$	− 1.68616e23i	− 1.47894i	−0.673187	−	0.739472i	$-0.735075\pi$
$157$	− 1.68616e23i	− 1.47894i	0.673187	−	0.739472i	$-0.264925\pi$
$158$	0	0
$159$	−1.74040e23	−1.33653
$160$	0	0
$161$	−2.16229e23	−1.45627
$162$	0	0
$163$	− 2.06801e23i	− 1.22344i	−0.791074	−	0.611720i	$-0.790478\pi$
$163$	− 2.06801e23i	− 1.22344i	0.791074	−	0.611720i	$-0.209522\pi$
$164$	0	0
$165$	1.87435e23i	0.975592i
$166$	0	0
$167$	2.28331e23	1.04723	0.523615	−	0.851955i	$-0.324583\pi$
$167$	2.28331e23	1.04723	0.523615	+	0.851955i	$0.324583\pi$
$168$	0	0
$169$	2.15948e23	0.874056
$170$	0	0
$171$	− 4.70712e22i	− 0.168382i
$172$	0	0
$173$	4.81019e23i	1.52292i	0.648210	+	0.761462i	$0.275518\pi$
$173$	4.81019e23i	1.52292i	−0.648210	+	0.761462i	$0.724482\pi$
$174$	0	0
$175$	−2.48113e23	−0.696227
$176$	0	0
$177$	−3.15561e23	−0.785894
$178$	0	0
$179$	1.75630e23i	0.388724i	0.980930	+	0.194362i	$0.0622636\pi$
$179$	1.75630e23i	0.388724i	−0.980930	+	0.194362i	$0.937736\pi$
$180$	0	0
$181$	5.48571e23i	1.08046i	0.841518	+	0.540229i	$0.181662\pi$
$181$	5.48571e23i	1.08046i	−0.841518	+	0.540229i	$0.818338\pi$
$182$	0	0
$183$	1.91926e23	0.336820
$184$	0	0
$185$	−4.11183e23	−0.643773
$186$	0	0
$187$	− 4.14781e23i	− 0.580072i
$188$	0	0
$189$	7.25759e23i	0.907702i
$190$	0	0
$191$	−1.02164e24	−1.14405	−0.572027	−	0.820235i	$-0.693843\pi$
$191$	−1.02164e24	−1.14405	−0.572027	+	0.820235i	$0.693843\pi$
$192$	0	0
$193$	1.01129e22	0.0101514	0.00507569	−	0.999987i	$-0.498384\pi$
$193$	1.01129e22	0.0101514	0.00507569	+	0.999987i	$0.498384\pi$
$194$	0	0
$195$	2.65336e23i	0.239019i
$196$	0	0
$197$	2.16736e24i	1.75402i	0.480468	+	0.877012i	$0.340467\pi$
$197$	2.16736e24i	1.75402i	−0.480468	+	0.877012i	$0.659533\pi$
$198$	0	0
$199$	−1.20832e24	−0.879477	−0.439738	−	0.898126i	$-0.644929\pi$
$199$	−1.20832e24	−0.879477	−0.439738	+	0.898126i	$0.644929\pi$
$200$	0	0
$201$	−2.46646e24	−1.61628
$202$	0	0
$203$	6.01215e23i	0.355076i
$204$	0	0
$205$	4.81917e22i	0.0256777i
$206$	0	0
$207$	−7.13370e23	−0.343262
$208$	0	0
$209$	−2.18359e24	−0.949806
$210$	0	0
$211$	− 2.51453e24i	− 0.989672i	−0.868987	−	0.494836i	$-0.835228\pi$
$211$	− 2.51453e24i	− 0.989672i	0.868987	−	0.494836i	$-0.164772\pi$
$212$	0	0
$213$	4.21843e24i	1.50371i
$214$	0	0
$215$	−2.10461e24	−0.680092
$216$	0	0
$217$	1.73017e24	0.507296
$218$	0	0
$219$	− 6.57310e24i	− 1.75028i
$220$	0	0
$221$	− 5.87172e23i	− 0.142117i
$222$	0	0
$223$	−7.94567e24	−1.74956	−0.874781	−	0.484519i	$-0.838995\pi$
$223$	−7.94567e24	−1.74956	−0.874781	+	0.484519i	$0.838995\pi$
$224$	0	0
$225$	−8.18560e23	−0.164110
$226$	0	0
$227$	− 7.16629e24i	− 1.30925i	−0.755954	−	0.654625i	$-0.772826\pi$
$227$	− 7.16629e24i	− 1.30925i	0.755954	−	0.654625i	$-0.227174\pi$
$228$	0	0
$229$	− 4.42950e23i	− 0.0738044i	−0.999319	−	0.0369022i	$-0.988251\pi$
$229$	− 4.42950e23i	− 0.0738044i	0.999319	−	0.0369022i	$-0.0117490\pi$
$230$	0	0
$231$	−1.16328e25	−1.76913
$232$	0	0
$233$	2.26907e24	0.315218	0.157609	−	0.987502i	$-0.449621\pi$
$233$	2.26907e24	0.315218	0.157609	+	0.987502i	$0.449621\pi$
$234$	0	0
$235$	− 5.55859e24i	− 0.705914i
$236$	0	0
$237$	1.14154e25i	1.32627i
$238$	0	0
$239$	−1.22558e25	−1.30366	−0.651828	−	0.758367i	$-0.725998\pi$
$239$	−1.22558e25	−1.30366	−0.651828	+	0.758367i	$0.725998\pi$
$240$	0	0
$241$	−1.81847e25	−1.77226	−0.886128	−	0.463440i	$-0.846615\pi$
$241$	−1.81847e25	−1.77226	−0.886128	+	0.463440i	$0.846615\pi$
$242$	0	0
$243$	5.61607e24i	0.501841i
$244$	0	0
$245$	1.36932e24i	0.112269i
$246$	0	0
$247$	−3.09113e24	−0.232702
$248$	0	0
$249$	1.14686e25	0.793263
$250$	0	0
$251$	1.93006e25i	1.22743i	0.789527	+	0.613716i	$0.210326\pi$
$251$	1.93006e25i	1.22743i	−0.789527	+	0.613716i	$0.789674\pi$
$252$	0	0
$253$	3.30926e25i	1.93627i
$254$	0	0
$255$	−5.00695e24	−0.269713
$256$	0	0
$257$	−3.60236e25	−1.78768	−0.893839	−	0.448387i	$-0.851999\pi$
$257$	−3.60236e25	−1.78768	−0.893839	+	0.448387i	$0.851999\pi$
$258$	0	0
$259$	− 2.55194e25i	− 1.16741i
$260$	0	0
$261$	1.98349e24i	0.0836960i
$262$	0	0
$263$	−2.63224e25	−1.02516	−0.512578	−	0.858641i	$-0.671309\pi$
$263$	−2.63224e25	−1.02516	−0.512578	+	0.858641i	$0.671309\pi$
$264$	0	0
$265$	1.99132e25	0.716240
$266$	0	0
$267$	− 4.45601e25i	− 1.48108i
$268$	0	0
$269$	1.54840e25i	0.475865i	0.971282	+	0.237933i	$0.0764698\pi$
$269$	1.54840e25i	0.475865i	−0.971282	+	0.237933i	$0.923530\pi$
$270$	0	0
$271$	5.42811e25	1.54337	0.771687	−	0.636002i	$-0.219413\pi$
$271$	5.42811e25	1.54337	0.771687	+	0.636002i	$0.219413\pi$
$272$	0	0
$273$	−1.64676e25	−0.433434
$274$	0	0
$275$	3.79722e25i	0.925705i
$276$	0	0
$277$	6.04508e25i	1.36573i	0.730545	+	0.682864i	$0.239266\pi$
$277$	6.04508e25i	1.36573i	−0.730545	+	0.682864i	$0.760734\pi$
$278$	0	0
$279$	5.70807e24	0.119576
$280$	0	0
$281$	−3.29873e25	−0.641106	−0.320553	−	0.947231i	$-0.603869\pi$
$281$	−3.29873e25	−0.641106	−0.320553	+	0.947231i	$0.603869\pi$
$282$	0	0
$283$	2.20492e25i	0.397773i	0.980022	+	0.198887i	$0.0637326\pi$
$283$	2.20492e25i	0.397773i	−0.980022	+	0.198887i	$0.936267\pi$
$284$	0	0
$285$	2.63588e25i	0.441626i
$286$	0	0
$287$	−2.99094e24	−0.0465637
$288$	0	0
$289$	−5.80119e25	−0.839633
$290$	0	0
$291$	− 5.03109e25i	− 0.677307i
$292$	0	0
$293$	− 6.77166e25i	− 0.848369i	−0.905576	−	0.424185i	$-0.860561\pi$
$293$	− 6.77166e25i	− 0.848369i	0.905576	−	0.424185i	$-0.139439\pi$
$294$	0	0
$295$	3.61056e25	0.421156
$296$	0	0
$297$	1.11073e26	1.20688
$298$	0	0
$299$	4.68465e25i	0.474383i
$300$	0	0
$301$	− 1.30619e26i	− 1.23327i
$302$	0	0
$303$	−4.56141e25	−0.401746
$304$	0	0
$305$	−2.19597e25	−0.180500
$306$	0	0
$307$	9.44941e25i	0.725190i	0.931947	+	0.362595i	$0.118109\pi$
$307$	9.44941e25i	0.725190i	−0.931947	+	0.362595i	$0.881891\pi$
$308$	0	0
$309$	− 1.03448e26i	− 0.741582i
$310$	0	0
$311$	−1.92541e26	−1.28985	−0.644925	−	0.764246i	$-0.723111\pi$
$311$	−1.92541e26	−1.28985	−0.644925	+	0.764246i	$0.723111\pi$
$312$	0	0
$313$	2.47024e26	1.54712	0.773558	−	0.633726i	$-0.218475\pi$
$313$	2.47024e26	1.54712	0.773558	+	0.633726i	$0.218475\pi$
$314$	0	0
$315$	2.86920e25i	0.168074i
$316$	0	0
$317$	− 2.73540e26i	− 1.49933i	−0.661816	−	0.749667i	$-0.730214\pi$
$317$	− 2.73540e26i	− 1.49933i	0.661816	−	0.749667i	$-0.269786\pi$
$318$	0	0
$319$	9.20123e25	0.472110
$320$	0	0
$321$	9.33845e25	0.448716
$322$	0	0
$323$	− 5.83304e25i	− 0.262584i
$324$	0	0
$325$	5.37542e25i	0.226797i
$326$	0	0
$327$	4.06784e26	1.60921
$328$	0	0
$329$	3.44985e26	1.28010
$330$	0	0
$331$	− 2.81293e26i	− 0.979410i	−0.871888	−	0.489705i	$-0.837104\pi$
$331$	− 2.81293e26i	− 0.979410i	0.871888	−	0.489705i	$-0.162896\pi$
$332$	0	0
$333$	− 8.41921e25i	− 0.275174i
$334$	0	0
$335$	2.82206e26	0.866155
$336$	0	0
$337$	−3.38854e26	−0.977010	−0.488505	−	0.872561i	$-0.662458\pi$
$337$	−3.38854e26	−0.977010	−0.488505	+	0.872561i	$0.662458\pi$
$338$	0	0
$339$	4.78669e26i	1.29700i
$340$	0	0
$341$	− 2.64792e26i	− 0.674503i
$342$	0	0
$343$	3.69784e26	0.885850
$344$	0	0
$345$	3.99470e26	0.900294
$346$	0	0
$347$	6.52881e26i	1.38476i	0.721533	+	0.692380i	$0.243438\pi$
$347$	6.52881e26i	1.38476i	−0.721533	+	0.692380i	$0.756562\pi$
$348$	0	0
$349$	− 8.30091e26i	− 1.65752i	−0.559605	−	0.828759i	$-0.689047\pi$
$349$	− 8.30091e26i	− 1.65752i	0.559605	−	0.828759i	$-0.310953\pi$
$350$	0	0
$351$	1.57237e26	0.295685
$352$	0	0
$353$	−6.48358e26	−1.14863	−0.574315	−	0.818634i	$-0.694732\pi$
$353$	−6.48358e26	−1.14863	−0.574315	+	0.818634i	$0.694732\pi$
$354$	0	0
$355$	− 4.82660e26i	− 0.805831i
$356$	0	0
$357$	− 3.10748e26i	− 0.489094i
$358$	0	0
$359$	8.89871e26	1.32079	0.660397	−	0.750917i	$-0.270388\pi$
$359$	8.89871e26	1.32079	0.660397	+	0.750917i	$0.270388\pi$
$360$	0	0
$361$	4.07133e26	0.570047
$362$	0	0
$363$	9.31830e26i	1.23117i
$364$	0	0
$365$	7.52075e26i	0.937964i
$366$	0	0
$367$	2.55739e26	0.301164	0.150582	−	0.988598i	$-0.451885\pi$
$367$	2.55739e26	0.301164	0.150582	+	0.988598i	$0.451885\pi$
$368$	0	0
$369$	−9.86752e24	−0.0109757
$370$	0	0
$371$	1.23588e27i	1.29882i
$372$	0	0
$373$	− 1.08913e26i	− 0.108178i	−0.998536	−	0.0540888i	$-0.982775\pi$
$373$	− 1.08913e26i	− 0.108178i	0.998536	−	0.0540888i	$-0.0172254\pi$
$374$	0	0
$375$	1.17563e27	1.10393
$376$	0	0
$377$	1.30254e26	0.115667
$378$	0	0
$379$	− 3.03293e26i	− 0.254771i	−0.991853	−	0.127386i	$-0.959341\pi$
$379$	− 3.03293e26i	− 0.254771i	0.991853	−	0.127386i	$-0.0406585\pi$
$380$	0	0
$381$	1.15357e27i	0.916917i
$382$	0	0
$383$	7.88044e26	0.592875	0.296438	−	0.955052i	$-0.404201\pi$
$383$	7.88044e26	0.592875	0.296438	+	0.955052i	$0.404201\pi$
$384$	0	0
$385$	1.33099e27	0.948065
$386$	0	0
$387$	− 4.30930e26i	− 0.290698i
$388$	0	0
$389$	− 5.00122e26i	− 0.319599i	−0.987150	−	0.159799i	$-0.948915\pi$
$389$	− 5.00122e26i	− 0.319599i	0.987150	−	0.159799i	$-0.0510847\pi$
$390$	0	0
$391$	−8.84004e26	−0.535301
$392$	0	0
$393$	−5.91822e26	−0.339680
$394$	0	0
$395$	− 1.30612e27i	− 0.710743i
$396$	0	0
$397$	− 2.48390e27i	− 1.28184i	−0.767607	−	0.640920i	$-0.778553\pi$
$397$	− 2.48390e27i	− 1.28184i	0.767607	−	0.640920i	$-0.221447\pi$
$398$	0	0
$399$	−1.63591e27	−0.800839
$400$	0	0
$401$	3.69255e27	1.71518	0.857590	−	0.514333i	$-0.171961\pi$
$401$	3.69255e27	1.71518	0.857590	+	0.514333i	$0.171961\pi$
$402$	0	0
$403$	− 3.74845e26i	− 0.165253i
$404$	0	0
$405$	− 1.70941e27i	− 0.715433i
$406$	0	0
$407$	−3.90559e27	−1.55219
$408$	0	0
$409$	−4.04322e27	−1.52627	−0.763137	−	0.646236i	$-0.776342\pi$
$409$	−4.04322e27	−1.52627	−0.763137	+	0.646236i	$0.776342\pi$
$410$	0	0
$411$	− 4.79784e26i	− 0.172071i
$412$	0	0
$413$	2.24083e27i	0.763720i
$414$	0	0
$415$	−1.31221e27	−0.425106
$416$	0	0
$417$	−7.92613e26	−0.244136
$418$	0	0
$419$	− 1.53519e27i	− 0.449693i	−0.974394	−	0.224846i	$-0.927812\pi$
$419$	− 1.53519e27i	− 0.449693i	0.974394	−	0.224846i	$-0.0721880\pi$
$420$	0	0
$421$	− 1.47068e27i	− 0.409786i	−0.978784	−	0.204893i	$-0.934315\pi$
$421$	− 1.47068e27i	− 0.409786i	0.978784	−	0.204893i	$-0.0656846\pi$
$422$	0	0
$423$	1.13815e27	0.301735
$424$	0	0
$425$	−1.01435e27	−0.255921
$426$	0	0
$427$	− 1.36289e27i	− 0.327317i
$428$	0	0
$429$	2.52027e27i	0.576296i
$430$	0	0
$431$	−2.28715e27	−0.498063	−0.249031	−	0.968495i	$-0.580112\pi$
$431$	−2.28715e27	−0.498063	−0.249031	+	0.968495i	$0.580112\pi$
$432$	0	0
$433$	−7.26337e27	−1.50666	−0.753330	−	0.657643i	$-0.771554\pi$
$433$	−7.26337e27	−1.50666	−0.753330	+	0.657643i	$0.771554\pi$
$434$	0	0
$435$	− 1.11071e27i	− 0.219514i
$436$	0	0
$437$	4.65378e27i	0.876498i
$438$	0	0
$439$	−5.82811e27	−1.04629	−0.523143	−	0.852245i	$-0.675241\pi$
$439$	−5.82811e27	−1.04629	−0.523143	+	0.852245i	$0.675241\pi$
$440$	0	0
$441$	−2.80375e26	−0.0479881
$442$	0	0
$443$	− 5.06036e27i	− 0.825928i	−0.910747	−	0.412964i	$-0.864494\pi$
$443$	− 5.06036e27i	− 0.825928i	0.910747	−	0.412964i	$-0.135506\pi$
$444$	0	0
$445$	5.09844e27i	0.793702i
$446$	0	0
$447$	−7.21705e27	−1.07184
$448$	0	0
$449$	1.03689e28	1.46943	0.734713	−	0.678378i	$-0.237317\pi$
$449$	1.03689e28	1.46943	0.734713	+	0.678378i	$0.237317\pi$
$450$	0	0
$451$	4.57745e26i	0.0619113i
$452$	0	0
$453$	1.61666e28i	2.08731i
$454$	0	0
$455$	1.88418e27	0.232275
$456$	0	0
$457$	−4.56627e27	−0.537578	−0.268789	−	0.963199i	$-0.586623\pi$
$457$	−4.56627e27	−0.537578	−0.268789	+	0.963199i	$0.586623\pi$
$458$	0	0
$459$	2.96710e27i	0.333656i
$460$	0	0
$461$	− 2.22037e27i	− 0.238542i	−0.992862	−	0.119271i	$-0.961944\pi$
$461$	− 2.22037e27i	− 0.238542i	0.992862	−	0.119271i	$-0.0380558\pi$
$462$	0	0
$463$	1.54922e28	1.59043	0.795213	−	0.606330i	$-0.207359\pi$
$463$	1.54922e28	1.59043	0.795213	+	0.606330i	$0.207359\pi$
$464$	0	0
$465$	−3.19639e27	−0.313620
$466$	0	0
$467$	− 9.66026e27i	− 0.906069i	−0.891493	−	0.453035i	$-0.850341\pi$
$467$	− 9.66026e27i	− 0.906069i	0.891493	−	0.453035i	$-0.149659\pi$
$468$	0	0
$469$	1.75146e28i	1.57067i
$470$	0	0
$471$	−1.93332e28	−1.65800
$472$	0	0
$473$	−1.99904e28	−1.63976
$474$	0	0
$475$	5.34001e27i	0.419043i
$476$	0	0
$477$	4.07733e27i	0.306149i
$478$	0	0
$479$	−6.55119e27	−0.470758	−0.235379	−	0.971904i	$-0.575633\pi$
$479$	−6.55119e27	−0.470758	−0.235379	+	0.971904i	$0.575633\pi$
$480$	0	0
$481$	−5.52883e27	−0.380286
$482$	0	0
$483$	2.47925e28i	1.63258i
$484$	0	0
$485$	5.75643e27i	0.362965i
$486$	0	0
$487$	3.88451e27	0.234575	0.117288	−	0.993098i	$-0.462580\pi$
$487$	3.88451e27	0.234575	0.117288	+	0.993098i	$0.462580\pi$
$488$	0	0
$489$	−2.37114e28	−1.37156
$490$	0	0
$491$	− 1.06166e28i	− 0.588342i	−0.955753	−	0.294171i	$-0.904956\pi$
$491$	− 1.06166e28i	− 0.588342i	0.955753	−	0.294171i	$-0.0950436\pi$
$492$	0	0
$493$	2.45793e27i	0.130520i
$494$	0	0
$495$	4.39113e27	0.223471
$496$	0	0
$497$	2.99555e28	1.46128
$498$	0	0
$499$	− 3.58203e28i	− 1.67522i	−0.546265	−	0.837612i	$-0.683951\pi$
$499$	− 3.58203e28i	− 1.67522i	0.546265	−	0.837612i	$-0.316049\pi$
$500$	0	0
$501$	− 2.61801e28i	− 1.17402i
$502$	0	0
$503$	−1.68066e28	−0.722798	−0.361399	−	0.932411i	$-0.617701\pi$
$503$	−1.68066e28	−0.722798	−0.361399	+	0.932411i	$0.617701\pi$
$504$	0	0
$505$	5.21904e27	0.215294
$506$	0	0
$507$	− 2.47602e28i	− 0.979877i
$508$	0	0
$509$	− 1.08041e28i	− 0.410255i	−0.978735	−	0.205127i	$-0.934239\pi$
$509$	− 1.08041e28i	− 0.410255i	0.978735	−	0.205127i	$-0.0657608\pi$
$510$	0	0
$511$	−4.66763e28	−1.70089
$512$	0	0
$513$	1.56201e28	0.546326
$514$	0	0
$515$	1.18363e28i	0.397410i
$516$	0	0
$517$	− 5.27978e28i	− 1.70202i
$518$	0	0
$519$	5.51528e28	1.70730
$520$	0	0
$521$	1.39234e28	0.413951	0.206976	−	0.978346i	$-0.433638\pi$
$521$	1.39234e28	0.413951	0.206976	+	0.978346i	$0.433638\pi$
$522$	0	0
$523$	1.30668e28i	0.373166i	0.982439	+	0.186583i	$0.0597413\pi$
$523$	1.30668e28i	0.373166i	−0.982439	+	0.186583i	$0.940259\pi$
$524$	0	0
$525$	2.84482e28i	0.780518i
$526$	0	0
$527$	7.07341e27	0.186474
$528$	0	0
$529$	3.10571e28	0.786821
$530$	0	0
$531$	7.39282e27i	0.180019i
$532$	0	0
$533$	6.47992e26i	0.0151682i
$534$	0	0
$535$	−1.06848e28	−0.240465
$536$	0	0
$537$	2.01374e28	0.435786
$538$	0	0
$539$	1.30063e28i	0.270690i
$540$	0	0
$541$	− 3.99689e28i	− 0.800111i	−0.916491	−	0.400056i	$-0.868991\pi$
$541$	− 3.99689e28i	− 0.800111i	0.916491	−	0.400056i	$-0.131009\pi$
$542$	0	0
$543$	6.28982e28	1.21127
$544$	0	0
$545$	−4.65431e28	−0.862367
$546$	0	0
$547$	4.09401e28i	0.729930i	0.931021	+	0.364965i	$0.118919\pi$
$547$	4.09401e28i	0.729930i	−0.931021	+	0.364965i	$0.881081\pi$
$548$	0	0
$549$	− 4.49636e27i	− 0.0771528i
$550$	0	0
$551$	1.29396e28	0.213712
$552$	0	0
$553$	8.10620e28	1.28885
$554$	0	0
$555$	4.71456e28i	0.721713i
$556$	0	0
$557$	− 9.63372e28i	− 1.42009i	−0.704158	−	0.710043i	$-0.748675\pi$
$557$	− 9.63372e28i	− 1.42009i	0.704158	−	0.710043i	$-0.251325\pi$
$558$	0	0
$559$	−2.82989e28	−0.401740
$560$	0	0
$561$	−4.75581e28	−0.650301
$562$	0	0
$563$	1.10402e29i	1.45425i	0.686503	+	0.727127i	$0.259145\pi$
$563$	1.10402e29i	1.45425i	−0.686503	+	0.727127i	$0.740855\pi$
$564$	0	0
$565$	− 5.47680e28i	− 0.695053i
$566$	0	0
$567$	1.06092e29	1.29736
$568$	0	0
$569$	4.27031e28	0.503247	0.251623	−	0.967825i	$-0.419036\pi$
$569$	4.27031e28	0.503247	0.251623	+	0.967825i	$0.419036\pi$
$570$	0	0
$571$	− 4.84895e28i	− 0.550767i	−0.961334	−	0.275384i	$-0.911195\pi$
$571$	− 4.84895e28i	− 0.550767i	0.961334	−	0.275384i	$-0.0888049\pi$
$572$	0	0
$573$	1.17139e29i	1.28256i
$574$	0	0
$575$	8.09284e28	0.854258
$576$	0	0
$577$	−3.65903e28	−0.372409	−0.186205	−	0.982511i	$-0.559619\pi$
$577$	−3.65903e28	−0.372409	−0.186205	+	0.982511i	$0.559619\pi$
$578$	0	0
$579$	− 1.15953e27i	− 0.0113804i
$580$	0	0
$581$	− 8.14398e28i	− 0.770881i
$582$	0	0
$583$	1.89144e29	1.72692
$584$	0	0
$585$	6.21617e27	0.0547503
$586$	0	0
$587$	8.80312e27i	0.0748060i	0.999300	+	0.0374030i	$0.0119085\pi$
$587$	8.80312e27i	0.0748060i	−0.999300	+	0.0374030i	$0.988091\pi$
$588$	0	0
$589$	− 3.72375e28i	− 0.305330i
$590$	0	0
$591$	2.48506e29	1.96638
$592$	0	0
$593$	−1.35815e29	−1.03723	−0.518613	−	0.855009i	$-0.673551\pi$
$593$	−1.35815e29	−1.03723	−0.518613	+	0.855009i	$0.673551\pi$
$594$	0	0
$595$	3.55549e28i	0.262103i
$596$	0	0
$597$	1.38544e29i	0.985954i
$598$	0	0
$599$	−7.49105e28	−0.514708	−0.257354	−	0.966317i	$-0.582851\pi$
$599$	−7.49105e28	−0.514708	−0.257354	+	0.966317i	$0.582851\pi$
$600$	0	0
$601$	2.82014e29	1.87106	0.935531	−	0.353244i	$-0.114922\pi$
$601$	2.82014e29	1.87106	0.935531	+	0.353244i	$0.114922\pi$
$602$	0	0
$603$	5.77832e28i	0.370229i
$604$	0	0
$605$	− 1.06617e29i	− 0.659776i
$606$	0	0
$607$	−1.17396e29	−0.701736	−0.350868	−	0.936425i	$-0.614113\pi$
$607$	−1.17396e29	−0.701736	−0.350868	+	0.936425i	$0.614113\pi$
$608$	0	0
$609$	6.89342e28	0.398065
$610$	0	0
$611$	− 7.47416e28i	− 0.416994i
$612$	0	0
$613$	− 3.83941e28i	− 0.206980i	−0.994630	−	0.103490i	$-0.966999\pi$
$613$	− 3.83941e28i	− 0.206980i	0.994630	−	0.103490i	$-0.0330011\pi$
$614$	0	0
$615$	5.52558e27	0.0287865
$616$	0	0
$617$	−6.53635e27	−0.0329110	−0.0164555	−	0.999865i	$-0.505238\pi$
$617$	−6.53635e27	−0.0329110	−0.0164555	+	0.999865i	$0.505238\pi$
$618$	0	0
$619$	− 1.35329e29i	− 0.658625i	−0.944221	−	0.329312i	$-0.893183\pi$
$619$	− 1.35329e29i	− 0.658625i	0.944221	−	0.329312i	$-0.106817\pi$
$620$	0	0
$621$	− 2.36725e29i	− 1.11373i
$622$	0	0
$623$	−3.16426e29	−1.43929
$624$	0	0
$625$	1.07957e28	0.0474801
$626$	0	0
$627$	2.50367e29i	1.06480i
$628$	0	0
$629$	− 1.04330e29i	− 0.429120i
$630$	0	0
$631$	−4.42416e29	−1.76004	−0.880019	−	0.474938i	$-0.842470\pi$
$631$	−4.42416e29	−1.76004	−0.880019	+	0.474938i	$0.842470\pi$
$632$	0	0
$633$	−2.88312e29	−1.10949
$634$	0	0
$635$	− 1.31988e29i	− 0.491371i
$636$	0	0
$637$	1.84120e28i	0.0663188i
$638$	0	0
$639$	9.88274e28	0.344444
$640$	0	0
$641$	5.08677e28	0.171566	0.0857832	−	0.996314i	$-0.472661\pi$
$641$	5.08677e28	0.171566	0.0857832	+	0.996314i	$0.472661\pi$
$642$	0	0
$643$	5.61728e28i	0.183363i	0.995788	+	0.0916813i	$0.0292241\pi$
$643$	5.61728e28i	0.183363i	−0.995788	+	0.0916813i	$0.970776\pi$
$644$	0	0
$645$	2.41311e29i	0.762429i
$646$	0	0
$647$	−1.68341e29	−0.514868	−0.257434	−	0.966296i	$-0.582877\pi$
$647$	−1.68341e29	−0.514868	−0.257434	+	0.966296i	$0.582877\pi$
$648$	0	0
$649$	3.42946e29	1.01544
$650$	0	0
$651$	− 1.98378e29i	− 0.568714i
$652$	0	0
$653$	− 1.42886e29i	− 0.396645i	−0.980137	−	0.198322i	$-0.936451\pi$
$653$	− 1.42886e29i	− 0.396645i	0.980137	−	0.198322i	$-0.0635493\pi$
$654$	0	0
$655$	6.77146e28	0.182033
$656$	0	0
$657$	−1.53992e29	−0.400923
$658$	0	0
$659$	− 2.69706e29i	− 0.680133i	−0.940401	−	0.340066i	$-0.889550\pi$
$659$	− 2.69706e29i	− 0.680133i	0.940401	−	0.340066i	$-0.110450\pi$
$660$	0	0
$661$	4.69427e29i	1.14671i	0.819307	+	0.573354i	$0.194358\pi$
$661$	4.69427e29i	1.14671i	−0.819307	+	0.573354i	$0.805642\pi$
$662$	0	0
$663$	−6.73242e28	−0.159323
$664$	0	0
$665$	1.87177e29	0.429165
$666$	0	0
$667$	− 1.96101e29i	− 0.435672i
$668$	0	0
$669$	9.11036e29i	1.96138i
$670$	0	0
$671$	−2.08582e29	−0.435202
$672$	0	0
$673$	−7.03975e29	−1.42364	−0.711818	−	0.702364i	$-0.752128\pi$
$673$	−7.03975e29	−1.42364	−0.711818	+	0.702364i	$0.752128\pi$
$674$	0	0
$675$	− 2.71631e29i	− 0.532463i
$676$	0	0
$677$	− 2.19335e29i	− 0.416798i	−0.978044	−	0.208399i	$-0.933175\pi$
$677$	− 2.19335e29i	− 0.416798i	0.978044	−	0.208399i	$-0.0668253\pi$
$678$	0	0
$679$	−3.57263e29	−0.658197
$680$	0	0
$681$	−8.21674e29	−1.46776
$682$	0	0
$683$	− 8.89835e29i	− 1.54132i	−0.637248	−	0.770659i	$-0.719927\pi$
$683$	− 8.89835e29i	− 1.54132i	0.637248	−	0.770659i	$-0.280073\pi$
$684$	0	0
$685$	5.48956e28i	0.0922118i
$686$	0	0
$687$	−5.07879e28	−0.0827398
$688$	0	0
$689$	2.67755e29	0.423093
$690$	0	0
$691$	9.42228e29i	1.44423i	0.691773	+	0.722115i	$0.256830\pi$
$691$	9.42228e29i	1.44423i	−0.691773	+	0.722115i	$0.743170\pi$
$692$	0	0
$693$	2.72528e29i	0.405240i
$694$	0	0
$695$	9.06886e28	0.130831
$696$	0	0
$697$	−1.22278e28	−0.0171160
$698$	0	0
$699$	− 2.60168e29i	− 0.353381i
$700$	0	0
$701$	1.18589e30i	1.56316i	0.623804	+	0.781581i	$0.285586\pi$
$701$	1.18589e30i	1.56316i	−0.623804	+	0.781581i	$0.714414\pi$
$702$	0	0
$703$	−5.49240e29	−0.702638
$704$	0	0
$705$	−6.37338e29	−0.791378
$706$	0	0
$707$	3.23911e29i	0.390411i
$708$	0	0
$709$	4.56974e29i	0.534695i	0.963600	+	0.267348i	$0.0861472\pi$
$709$	4.56974e29i	0.534695i	−0.963600	+	0.267348i	$0.913853\pi$
$710$	0	0
$711$	2.67435e29	0.303799
$712$	0	0
$713$	−5.64339e29	−0.622443
$714$	0	0
$715$	− 2.88362e29i	− 0.308834i
$716$	0	0
$717$	1.40523e30i	1.46149i
$718$	0	0
$719$	9.01293e28	0.0910359	0.0455179	−	0.998964i	$-0.485506\pi$
$719$	9.01293e28	0.0910359	0.0455179	+	0.998964i	$0.485506\pi$
$720$	0	0
$721$	−7.34598e29	−0.720658
$722$	0	0
$723$	2.08502e30i	1.98682i
$724$	0	0
$725$	− 2.25018e29i	− 0.208290i
$726$	0	0
$727$	3.62357e28	0.0325856	0.0162928	−	0.999867i	$-0.494814\pi$
$727$	3.62357e28	0.0325856	0.0162928	+	0.999867i	$0.494814\pi$
$728$	0	0
$729$	−7.19073e29	−0.628253
$730$	0	0
$731$	− 5.34006e29i	− 0.453329i
$732$	0	0
$733$	− 1.40449e30i	− 1.15858i	−0.815120	−	0.579292i	$-0.803329\pi$
$733$	− 1.40449e30i	− 1.15858i	0.815120	−	0.579292i	$-0.196671\pi$
$734$	0	0
$735$	1.57003e29	0.125861
$736$	0	0
$737$	2.68051e30	2.08838
$738$	0	0
$739$	− 1.28167e30i	− 0.970528i	−0.874368	−	0.485264i	$-0.838723\pi$
$739$	− 1.28167e30i	− 0.970528i	0.874368	−	0.485264i	$-0.161277\pi$
$740$	0	0
$741$	3.54424e29i	0.260875i
$742$	0	0
$743$	−2.67917e30	−1.91698	−0.958490	−	0.285128i	$-0.907964\pi$
$743$	−2.67917e30	−1.91698	−0.958490	+	0.285128i	$0.907964\pi$
$744$	0	0
$745$	8.25755e29	0.574395
$746$	0	0
$747$	− 2.68681e29i	− 0.181707i
$748$	0	0
$749$	− 6.63133e29i	− 0.436056i
$750$	0	0
$751$	−1.59720e30	−1.02127	−0.510635	−	0.859798i	$-0.670590\pi$
$751$	−1.59720e30	−1.02127	−0.510635	+	0.859798i	$0.670590\pi$
$752$	0	0
$753$	2.21298e30	1.37603
$754$	0	0
$755$	− 1.84973e30i	− 1.11858i
$756$	0	0
$757$	− 1.95679e30i	− 1.15090i	−0.817836	−	0.575451i	$-0.804826\pi$
$757$	− 1.95679e30i	− 1.15090i	0.817836	−	0.575451i	$-0.195174\pi$
$758$	0	0
$759$	3.79434e30	2.17069
$760$	0	0
$761$	−2.12382e30	−1.18189	−0.590947	−	0.806710i	$-0.701246\pi$
$761$	−2.12382e30	−1.18189	−0.590947	+	0.806710i	$0.701246\pi$
$762$	0	0
$763$	− 2.88862e30i	− 1.56381i
$764$	0	0
$765$	1.17301e29i	0.0617810i
$766$	0	0
$767$	4.85481e29	0.248783
$768$	0	0
$769$	1.94984e30	0.972236	0.486118	−	0.873893i	$-0.338413\pi$
$769$	1.94984e30	0.972236	0.486118	+	0.873893i	$0.338413\pi$
$770$	0	0
$771$	4.13040e30i	2.00411i
$772$	0	0
$773$	− 1.54317e30i	− 0.728667i	−0.931268	−	0.364334i	$-0.881297\pi$
$773$	− 1.54317e30i	− 0.728667i	0.931268	−	0.364334i	$-0.118703\pi$
$774$	0	0
$775$	−6.47553e29	−0.297583
$776$	0	0
$777$	−2.92601e30	−1.30875
$778$	0	0
$779$	6.43723e28i	0.0280257i
$780$	0	0
$781$	− 4.58451e30i	− 1.94293i
$782$	0	0
$783$	−6.58201e29	−0.271556
$784$	0	0
$785$	2.21205e30	0.888512
$786$	0	0
$787$	2.29401e30i	0.897140i	0.893748	+	0.448570i	$0.148067\pi$
$787$	2.29401e30i	0.897140i	−0.893748	+	0.448570i	$0.851933\pi$
$788$	0	0
$789$	3.01808e30i	1.14927i
$790$	0	0
$791$	3.39908e30	1.26040
$792$	0	0
$793$	−2.95273e29	−0.106624
$794$	0	0
$795$	− 2.28321e30i	− 0.802954i
$796$	0	0
$797$	− 1.11630e30i	− 0.382356i	−0.981555	−	0.191178i	$-0.938769\pi$
$797$	− 1.11630e30i	− 0.382356i	0.981555	−	0.191178i	$-0.0612308\pi$
$798$	0	0
$799$	1.41039e30	0.470542
$800$	0	0
$801$	−1.04393e30	−0.339259
$802$	0	0
$803$	7.14352e30i	2.26151i
$804$	0	0
$805$	− 2.83668e30i	− 0.874892i
$806$	0	0
$807$	1.77537e30	0.533478
$808$	0	0
$809$	4.22000e30	1.23553	0.617764	−	0.786363i	$-0.288039\pi$
$809$	4.22000e30	1.23553	0.617764	+	0.786363i	$0.288039\pi$
$810$	0	0
$811$	2.17983e29i	0.0621877i	0.999516	+	0.0310938i	$0.00989907\pi$
$811$	2.17983e29i	0.0621877i	−0.999516	+	0.0310938i	$0.990101\pi$
$812$	0	0
$813$	− 6.22378e30i	− 1.73023i
$814$	0	0
$815$	2.71300e30	0.735012
$816$	0	0
$817$	−2.81124e30	−0.742278
$818$	0	0
$819$	3.85796e29i	0.0992835i
$820$	0	0
$821$	1.48715e30i	0.373037i	0.982451	+	0.186518i	$0.0597204\pi$
$821$	1.48715e30i	0.373037i	−0.982451	+	0.186518i	$0.940280\pi$
$822$	0	0
$823$	−1.35473e29	−0.0331248	−0.0165624	−	0.999863i	$-0.505272\pi$
$823$	−1.35473e29	−0.0331248	−0.0165624	+	0.999863i	$0.505272\pi$
$824$	0	0
$825$	4.35383e30	1.03778
$826$	0	0
$827$	− 3.26860e30i	− 0.759546i	−0.925080	−	0.379773i	$-0.876002\pi$
$827$	− 3.26860e30i	− 0.759546i	0.925080	−	0.379773i	$-0.123998\pi$
$828$	0	0
$829$	6.32938e30i	1.43396i	0.697092	+	0.716982i	$0.254477\pi$
$829$	6.32938e30i	1.43396i	−0.697092	+	0.716982i	$0.745523\pi$
$830$	0	0
$831$	6.93118e30	1.53107
$832$	0	0
$833$	−3.47439e29	−0.0748352
$834$	0	0
$835$	2.99545e30i	0.629149i
$836$	0	0
$837$	1.89416e30i	0.387972i
$838$	0	0
$839$	3.73795e29	0.0746677	0.0373338	−	0.999303i	$-0.488114\pi$
$839$	3.73795e29	0.0746677	0.0373338	+	0.999303i	$0.488114\pi$
$840$	0	0
$841$	4.58759e30	0.893772
$842$	0	0
$843$	3.78226e30i	0.718724i
$844$	0	0
$845$	2.83300e30i	0.525111i
$846$	0	0
$847$	6.61703e30	1.19643
$848$	0	0
$849$	2.52813e30	0.445931
$850$	0	0
$851$	8.32380e30i	1.43239i
$852$	0	0
$853$	− 1.46122e30i	− 0.245329i	−0.992448	−	0.122665i	$-0.960856\pi$
$853$	− 1.46122e30i	− 0.245329i	0.992448	−	0.122665i	$-0.0391440\pi$
$854$	0	0
$855$	6.17522e29	0.101160
$856$	0	0
$857$	−4.94598e30	−0.790595	−0.395297	−	0.918553i	$-0.629358\pi$
$857$	−4.94598e30	−0.790595	−0.395297	+	0.918553i	$0.629358\pi$
$858$	0	0
$859$	− 9.79146e30i	− 1.52728i	−0.645641	−	0.763641i	$-0.723410\pi$
$859$	− 9.79146e30i	− 1.52728i	0.645641	−	0.763641i	$-0.276590\pi$
$860$	0	0
$861$	3.42936e29i	0.0522011i
$862$	0	0
$863$	−1.46950e30	−0.218301	−0.109151	−	0.994025i	$-0.534813\pi$
$863$	−1.46950e30	−0.218301	−0.109151	+	0.994025i	$0.534813\pi$
$864$	0	0
$865$	−6.31043e30	−0.914934
$866$	0	0
$867$	6.65154e30i	0.941286i
$868$	0	0
$869$	− 1.24060e31i	− 1.71366i
$870$	0	0
$871$	3.79458e30	0.511650
$872$	0	0
$873$	−1.17866e30	−0.155146
$874$	0	0
$875$	− 8.34826e30i	− 1.07278i
$876$	0	0
$877$	2.85727e30i	0.358472i	0.983806	+	0.179236i	$0.0573625\pi$
$877$	2.85727e30i	0.358472i	−0.983806	+	0.179236i	$0.942637\pi$
$878$	0	0
$879$	−7.76427e30	−0.951080
$880$	0	0
$881$	−3.34495e30	−0.400076	−0.200038	−	0.979788i	$-0.564107\pi$
$881$	−3.34495e30	−0.400076	−0.200038	+	0.979788i	$0.564107\pi$
$882$	0	0
$883$	3.96693e30i	0.463305i	0.972799	+	0.231652i	$0.0744132\pi$
$883$	3.96693e30i	0.463305i	−0.972799	+	0.231652i	$0.925587\pi$
$884$	0	0
$885$	− 4.13981e30i	− 0.472145i
$886$	0	0
$887$	−1.17546e31	−1.30921	−0.654607	−	0.755969i	$-0.727166\pi$
$887$	−1.17546e31	−1.30921	−0.654607	+	0.755969i	$0.727166\pi$
$888$	0	0
$889$	8.19160e30	0.891046
$890$	0	0
$891$	− 1.62367e31i	− 1.72497i
$892$	0	0
$893$	− 7.42491e30i	− 0.770462i
$894$	0	0
$895$	−2.30406e30	−0.233535
$896$	0	0
$897$	5.37133e30	0.531816
$898$	0	0
$899$	1.56912e30i	0.151767i
$900$	0	0
$901$	5.05260e30i	0.477425i
$902$	0	0
$903$	−1.49765e31	−1.38258
$904$	0	0
$905$	−7.19664e30	−0.649112
$906$	0	0
$907$	− 4.54097e30i	− 0.400195i	−0.979776	−	0.200098i	$-0.935874\pi$
$907$	− 4.54097e30i	− 0.400195i	0.979776	−	0.200098i	$-0.0641259\pi$
$908$	0	0
$909$	1.06863e30i	0.0920249i
$910$	0	0
$911$	−1.58505e31	−1.33383	−0.666916	−	0.745133i	$-0.732386\pi$
$911$	−1.58505e31	−1.33383	−0.666916	+	0.745133i	$0.732386\pi$
$912$	0	0
$913$	−1.24639e31	−1.02497
$914$	0	0
$915$	2.51786e30i	0.202353i
$916$	0	0
$917$	4.20260e30i	0.330096i
$918$	0	0
$919$	7.50687e30	0.576297	0.288148	−	0.957586i	$-0.406960\pi$
$919$	7.50687e30	0.576297	0.288148	+	0.957586i	$0.406960\pi$
$920$	0	0
$921$	1.08345e31	0.812988
$922$	0	0
$923$	− 6.48992e30i	− 0.476016i
$924$	0	0
$925$	9.55119e30i	0.684809i
$926$	0	0
$927$	−2.42354e30	−0.169869
$928$	0	0
$929$	−1.48338e30	−0.101645	−0.0508225	−	0.998708i	$-0.516184\pi$
$929$	−1.48338e30	−0.101645	−0.0508225	+	0.998708i	$0.516184\pi$
$930$	0	0
$931$	1.82907e30i	0.122535i
$932$	0	0
$933$	2.20764e31i	1.44601i
$934$	0	0
$935$	5.44146e30	0.348493
$936$	0	0
$937$	8.05541e30	0.504455	0.252227	−	0.967668i	$-0.418837\pi$
$937$	8.05541e30	0.504455	0.252227	+	0.967668i	$0.418837\pi$
$938$	0	0
$939$	− 2.83233e31i	− 1.73442i
$940$	0	0
$941$	2.58565e31i	1.54839i	0.632949	+	0.774193i	$0.281844\pi$
$941$	2.58565e31i	1.54839i	−0.632949	+	0.774193i	$0.718156\pi$
$942$	0	0
$943$	9.75570e29	0.0571328
$944$	0	0
$945$	−9.52114e30	−0.545324
$946$	0	0
$947$	2.22907e31i	1.24868i	0.781154	+	0.624338i	$0.214631\pi$
$947$	2.22907e31i	1.24868i	−0.781154	+	0.624338i	$0.785369\pi$
$948$	0	0
$949$	1.01125e31i	0.554069i
$950$	0	0
$951$	−3.13636e31	−1.68086
$952$	0	0
$953$	9.08594e29	0.0476315	0.0238157	−	0.999716i	$-0.492418\pi$
$953$	9.08594e29	0.0476315	0.0238157	+	0.999716i	$0.492418\pi$
$954$	0	0
$955$	− 1.34027e31i	− 0.687318i
$956$	0	0
$957$	− 1.05500e31i	− 0.529268i
$958$	0	0
$959$	−3.40700e30	−0.167216
$960$	0	0
$961$	−1.63099e31	−0.783170
$962$	0	0
$963$	− 2.18777e30i	− 0.102784i
$964$	0	0
$965$	1.32670e29i	0.00609870i
$966$	0	0
$967$	−1.53441e31	−0.690183	−0.345091	−	0.938569i	$-0.612152\pi$
$967$	−1.53441e31	−0.690183	−0.345091	+	0.938569i	$0.612152\pi$
$968$	0	0
$969$	−6.68806e30	−0.294375
$970$	0	0
$971$	8.48542e30i	0.365486i	0.983161	+	0.182743i	$0.0584977\pi$
$971$	8.48542e30i	0.365486i	−0.983161	+	0.182743i	$0.941502\pi$
$972$	0	0
$973$	5.62843e30i	0.237248i
$974$	0	0
$975$	6.16336e30	0.254255
$976$	0	0
$977$	−2.57089e31	−1.03798	−0.518992	−	0.854779i	$-0.673692\pi$
$977$	−2.57089e31	−1.03798	−0.518992	+	0.854779i	$0.673692\pi$
$978$	0	0
$979$	4.84271e31i	1.91368i
$980$	0	0
$981$	− 9.52995e30i	− 0.368609i
$982$	0	0
$983$	4.49539e31	1.70199	0.850993	−	0.525178i	$-0.176001\pi$
$983$	4.49539e31	1.70199	0.850993	+	0.525178i	$0.176001\pi$
$984$	0	0
$985$	−2.84333e31	−1.05377
$986$	0	0
$987$	− 3.95553e31i	− 1.43508i
$988$	0	0
$989$	4.26047e31i	1.51320i
$990$	0	0
$991$	−3.88864e31	−1.35215	−0.676075	−	0.736833i	$-0.736320\pi$
$991$	−3.88864e31	−1.35215	−0.676075	+	0.736833i	$0.736320\pi$
$992$	0	0
$993$	−3.22525e31	−1.09799
$994$	0	0
$995$	− 1.58518e31i	− 0.528367i
$996$	0	0
$997$	5.29787e31i	1.72903i	0.502608	+	0.864514i	$0.332374\pi$
$997$	5.29787e31i	1.72903i	−0.502608	+	0.864514i	$0.667626\pi$
$998$	0	0
$999$	2.79383e31	0.892816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.22.b.c.33.6	yes	28
4.3	odd	2	inner	64.22.b.c.33.24	yes	28
8.3	odd	2	inner	64.22.b.c.33.5	✓	28
8.5	even	2	inner	64.22.b.c.33.23	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.22.b.c.33.5	✓	28	8.3	odd	2	inner
64.22.b.c.33.6	yes	28	1.1	even	1	trivial
64.22.b.c.33.23	yes	28	8.5	even	2	inner
64.22.b.c.33.24	yes	28	4.3	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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	64.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-114658. i q^{3} +1.31189e7i q^{5} -8.14201e8 q^{7} -2.68616e9 q^{9} +O(q^{10})\)	\(q-114658. i q^{3} +1.31189e7i q^{5} -8.14201e8 q^{7} -2.68616e9 q^{9} +1.24608e11i q^{11} +1.76398e11i q^{13} +1.50419e12 q^{15} -3.32868e12 q^{17} +1.75236e13i q^{19} +9.33549e13i q^{21} +2.65572e14 q^{23} +3.04732e14 q^{25} -8.91375e14i q^{27} -7.38411e14i q^{29} -2.12499e15 q^{31} +1.42874e16 q^{33} -1.06814e16i q^{35} +3.13429e16i q^{37} +2.02255e16 q^{39} +3.67346e15 q^{41} +1.60426e17i q^{43} -3.52394e16i q^{45} -4.23709e17 q^{47} +1.04377e17 q^{49} +3.81660e17i q^{51} -1.51790e18i q^{53} -1.63472e18 q^{55} +2.00923e18 q^{57} -2.75219e18i q^{59} +1.67390e18i q^{61} +2.18708e18 q^{63} -2.31415e18 q^{65} -2.15114e19i q^{67} -3.04501e19i q^{69} -3.67913e19 q^{71} +5.73277e19 q^{73} -3.49401e19i q^{75} -1.01456e20i q^{77} -9.95602e19 q^{79} -1.30302e20 q^{81} +1.00024e20i q^{83} -4.36685e19i q^{85} -8.46649e19 q^{87} +3.88634e20 q^{89} -1.43624e20i q^{91} +2.43648e20i q^{93} -2.29890e20 q^{95} +4.38790e20 q^{97} -3.34719e20i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 77960422492 q^{9}+O(q^{10}) \) 28 * q - 77960422492 * q^9	\( 28 q - 77960422492 q^{9} + 16832040195288 q^{17} + 202504130118092 q^{25} - 55\!\cdots\!92 q^{33}+ \cdots - 19\!\cdots\!16 q^{97}+O(q^{100}) \) 28 * q - 77960422492 * q^9 + 16832040195288 * q^17 + 202504130118092 * q^25 - 55633095046450992 * q^33 + 265230847130533848 * q^41 + 5944576136704810940 * q^49 - 4544453360963256304 * q^57 - 63340883112821103360 * q^65 + 2760069698648092472 * q^73 + 934709246686586470348 * q^81 - 375882191672218968072 * q^89 - 1953791530192678912616 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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