LMFDB - Embedded newform 64.12.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,12,Mod(1,64)]

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, 0]))

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

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	64.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:	$49.1739635558$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-252.000 q^{3} -4830.00 q^{5} -16744.0 q^{7} -113643. q^{9} +O(q^{10})$

$q-252.000 q^{3} -4830.00 q^{5} -16744.0 q^{7} -113643. q^{9} -534612. q^{11} +577738. q^{13} +1.21716e6 q^{15} -6.90593e6 q^{17} -1.06614e7 q^{19} +4.21949e6 q^{21} +1.86433e7 q^{23} -2.54992e7 q^{25} +7.32791e7 q^{27} -1.28407e8 q^{29} -5.28432e7 q^{31} +1.34722e8 q^{33} +8.08735e7 q^{35} +1.82213e8 q^{37} -1.45590e8 q^{39} +3.08120e8 q^{41} +1.71257e7 q^{43} +5.48896e8 q^{45} +2.68735e9 q^{47} -1.69697e9 q^{49} +1.74030e9 q^{51} +1.59606e9 q^{53} +2.58218e9 q^{55} +2.68668e9 q^{57} +5.18920e9 q^{59} -6.95648e9 q^{61} +1.90284e9 q^{63} -2.79047e9 q^{65} +1.54818e10 q^{67} -4.69810e9 q^{69} +9.79149e9 q^{71} +1.46379e9 q^{73} +6.42580e9 q^{75} +8.95154e9 q^{77} +3.81168e10 q^{79} +1.66519e9 q^{81} +2.93351e10 q^{83} +3.33557e10 q^{85} +3.23585e10 q^{87} -2.49929e10 q^{89} -9.67365e9 q^{91} +1.33165e10 q^{93} +5.14947e10 q^{95} +7.50136e10 q^{97} +6.07549e10 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	−252.000	−0.598734	−0.299367	−	0.954138i	$-0.596775\pi$
$3$	−252.000	−0.598734	−0.299367	+	0.954138i	$0.596775\pi$
$4$	0	0
$5$	−4830.00	−0.691213	−0.345607	−	0.938379i	$-0.612327\pi$
$5$	−4830.00	−0.691213	−0.345607	+	0.938379i	$0.612327\pi$
$6$	0	0
$7$	−16744.0	−0.376548	−0.188274	−	0.982117i	$-0.560289\pi$
$7$	−16744.0	−0.376548	−0.188274	+	0.982117i	$0.560289\pi$
$8$	0	0
$9$	−113643.	−0.641518
$10$	0	0
$11$	−534612.	−1.00087	−0.500436	−	0.865773i	$-0.666827\pi$
$11$	−534612.	−1.00087	−0.500436	+	0.865773i	$0.666827\pi$
$12$	0	0
$13$	577738.	0.431561	0.215781	−	0.976442i	$-0.430770\pi$
$13$	577738.	0.431561	0.215781	+	0.976442i	$0.430770\pi$
$14$	0	0
$15$	1.21716e6	0.413853
$16$	0	0
$17$	−6.90593e6	−1.17965	−0.589825	−	0.807531i	$-0.700803\pi$
$17$	−6.90593e6	−1.17965	−0.589825	+	0.807531i	$0.700803\pi$
$18$	0	0
$19$	−1.06614e7	−0.987803	−0.493901	−	0.869518i	$-0.664430\pi$
$19$	−1.06614e7	−0.987803	−0.493901	+	0.869518i	$0.664430\pi$
$20$	0	0
$21$	4.21949e6	0.225452
$22$	0	0
$23$	1.86433e7	0.603975	0.301988	−	0.953312i	$-0.402350\pi$
$23$	1.86433e7	0.603975	0.301988	+	0.953312i	$0.402350\pi$
$24$	0	0
$25$	−2.54992e7	−0.522224
$26$	0	0
$27$	7.32791e7	0.982832
$28$	0	0
$29$	−1.28407e8	−1.16251	−0.581257	−	0.813720i	$-0.697439\pi$
$29$	−1.28407e8	−1.16251	−0.581257	+	0.813720i	$0.697439\pi$
$30$	0	0
$31$	−5.28432e7	−0.331512	−0.165756	−	0.986167i	$-0.553006\pi$
$31$	−5.28432e7	−0.331512	−0.165756	+	0.986167i	$0.553006\pi$
$32$	0	0
$33$	1.34722e8	0.599256
$34$	0	0
$35$	8.08735e7	0.260275
$36$	0	0
$37$	1.82213e8	0.431987	0.215993	−	0.976395i	$-0.430701\pi$
$37$	1.82213e8	0.431987	0.215993	+	0.976395i	$0.430701\pi$
$38$	0	0
$39$	−1.45590e8	−0.258390
$40$	0	0
$41$	3.08120e8	0.415345	0.207673	−	0.978198i	$-0.433411\pi$
$41$	3.08120e8	0.415345	0.207673	+	0.978198i	$0.433411\pi$
$42$	0	0
$43$	1.71257e7	0.0177653	0.00888264	−	0.999961i	$-0.497173\pi$
$43$	1.71257e7	0.0177653	0.00888264	+	0.999961i	$0.497173\pi$
$44$	0	0
$45$	5.48896e8	0.443426
$46$	0	0
$47$	2.68735e9	1.70917	0.854586	−	0.519310i	$-0.173811\pi$
$47$	2.68735e9	1.70917	0.854586	+	0.519310i	$0.173811\pi$
$48$	0	0
$49$	−1.69697e9	−0.858212
$50$	0	0
$51$	1.74030e9	0.706296
$52$	0	0
$53$	1.59606e9	0.524241	0.262120	−	0.965035i	$-0.415578\pi$
$53$	1.59606e9	0.524241	0.262120	+	0.965035i	$0.415578\pi$
$54$	0	0
$55$	2.58218e9	0.691817
$56$	0	0
$57$	2.68668e9	0.591431
$58$	0	0
$59$	5.18920e9	0.944963	0.472481	−	0.881341i	$-0.343358\pi$
$59$	5.18920e9	0.944963	0.472481	+	0.881341i	$0.343358\pi$
$60$	0	0
$61$	−6.95648e9	−1.05457	−0.527285	−	0.849689i	$-0.676790\pi$
$61$	−6.95648e9	−1.05457	−0.527285	+	0.849689i	$0.676790\pi$
$62$	0	0
$63$	1.90284e9	0.241562
$64$	0	0
$65$	−2.79047e9	−0.298301
$66$	0	0
$67$	1.54818e10	1.40091	0.700456	−	0.713696i	$-0.252980\pi$
$67$	1.54818e10	1.40091	0.700456	+	0.713696i	$0.252980\pi$
$68$	0	0
$69$	−4.69810e9	−0.361620
$70$	0	0
$71$	9.79149e9	0.644062	0.322031	−	0.946729i	$-0.395634\pi$
$71$	9.79149e9	0.644062	0.322031	+	0.946729i	$0.395634\pi$
$72$	0	0
$73$	1.46379e9	0.0826425	0.0413212	−	0.999146i	$-0.486843\pi$
$73$	1.46379e9	0.0826425	0.0413212	+	0.999146i	$0.486843\pi$
$74$	0	0
$75$	6.42580e9	0.312673
$76$	0	0
$77$	8.95154e9	0.376876
$78$	0	0
$79$	3.81168e10	1.39370	0.696848	−	0.717219i	$-0.254585\pi$
$79$	3.81168e10	1.39370	0.696848	+	0.717219i	$0.254585\pi$
$80$	0	0
$81$	1.66519e9	0.0530635
$82$	0	0
$83$	2.93351e10	0.817444	0.408722	−	0.912659i	$-0.365975\pi$
$83$	2.93351e10	0.817444	0.408722	+	0.912659i	$0.365975\pi$
$84$	0	0
$85$	3.33557e10	0.815390
$86$	0	0
$87$	3.23585e10	0.696037
$88$	0	0
$89$	−2.49929e10	−0.474430	−0.237215	−	0.971457i	$-0.576235\pi$
$89$	−2.49929e10	−0.474430	−0.237215	+	0.971457i	$0.576235\pi$
$90$	0	0
$91$	−9.67365e9	−0.162503
$92$	0	0
$93$	1.33165e10	0.198488
$94$	0	0
$95$	5.14947e10	0.682782
$96$	0	0
$97$	7.50136e10	0.886942	0.443471	−	0.896289i	$-0.353747\pi$
$97$	7.50136e10	0.886942	0.443471	+	0.896289i	$0.353747\pi$
$98$	0	0
$99$	6.07549e10	0.642078
$100$	0	0
$101$	−8.17430e10	−0.773896	−0.386948	−	0.922101i	$-0.626471\pi$
$101$	−8.17430e10	−0.773896	−0.386948	+	0.922101i	$0.626471\pi$
$102$	0	0
$103$	−2.25755e11	−1.91881	−0.959407	−	0.282025i	$-0.908994\pi$
$103$	−2.25755e11	−1.91881	−0.959407	+	0.282025i	$0.908994\pi$
$104$	0	0
$105$	−2.03801e10	−0.155835
$106$	0	0
$107$	−9.02413e10	−0.622006	−0.311003	−	0.950409i	$-0.600665\pi$
$107$	−9.02413e10	−0.622006	−0.311003	+	0.950409i	$0.600665\pi$
$108$	0	0
$109$	−7.34827e10	−0.457445	−0.228723	−	0.973492i	$-0.573455\pi$
$109$	−7.34827e10	−0.457445	−0.228723	+	0.973492i	$0.573455\pi$
$110$	0	0
$111$	−4.59178e10	−0.258645
$112$	0	0
$113$	−8.51469e10	−0.434748	−0.217374	−	0.976088i	$-0.569749\pi$
$113$	−8.51469e10	−0.434748	−0.217374	+	0.976088i	$0.569749\pi$
$114$	0	0
$115$	−9.00470e10	−0.417476
$116$	0	0
$117$	−6.56559e10	−0.276854
$118$	0	0
$119$	1.15633e11	0.444195
$120$	0	0
$121$	4.98320e8	0.00174658
$122$	0	0
$123$	−7.76464e10	−0.248681
$124$	0	0
$125$	3.59001e11	1.05218
$126$	0	0
$127$	−2.62717e11	−0.705615	−0.352808	−	0.935696i	$-0.614773\pi$
$127$	−2.62717e11	−0.705615	−0.352808	+	0.935696i	$0.614773\pi$
$128$	0	0
$129$	−4.31568e9	−0.0106367
$130$	0	0
$131$	−6.31529e11	−1.43021	−0.715107	−	0.699015i	$-0.753622\pi$
$131$	−6.31529e11	−1.43021	−0.715107	+	0.699015i	$0.753622\pi$
$132$	0	0
$133$	1.78515e11	0.371955
$134$	0	0
$135$	−3.53938e11	−0.679347
$136$	0	0
$137$	−2.97199e11	−0.526119	−0.263059	−	0.964780i	$-0.584732\pi$
$137$	−2.97199e11	−0.526119	−0.263059	+	0.964780i	$0.584732\pi$
$138$	0	0
$139$	−5.96794e11	−0.975535	−0.487767	−	0.872974i	$-0.662189\pi$
$139$	−5.96794e11	−0.975535	−0.487767	+	0.872974i	$0.662189\pi$
$140$	0	0
$141$	−6.77212e11	−1.02334
$142$	0	0
$143$	−3.08866e11	−0.431938
$144$	0	0
$145$	6.20204e11	0.803546
$146$	0	0
$147$	4.27635e11	0.513840
$148$	0	0
$149$	1.11543e12	1.24428	0.622142	−	0.782905i	$-0.286263\pi$
$149$	1.11543e12	1.24428	0.622142	+	0.782905i	$0.286263\pi$
$150$	0	0
$151$	−8.24447e11	−0.854653	−0.427326	−	0.904097i	$-0.640544\pi$
$151$	−8.24447e11	−0.854653	−0.427326	+	0.904097i	$0.640544\pi$
$152$	0	0
$153$	7.84811e11	0.756767
$154$	0	0
$155$	2.55233e11	0.229146
$156$	0	0
$157$	−1.31512e12	−1.10031	−0.550156	−	0.835062i	$-0.685432\pi$
$157$	−1.31512e12	−1.10031	−0.550156	+	0.835062i	$0.685432\pi$
$158$	0	0
$159$	−4.02206e11	−0.313881
$160$	0	0
$161$	−3.12163e11	−0.227425
$162$	0	0
$163$	3.57833e11	0.243584	0.121792	−	0.992556i	$-0.461136\pi$
$163$	3.57833e11	0.243584	0.121792	+	0.992556i	$0.461136\pi$
$164$	0	0
$165$	−6.50708e11	−0.414214
$166$	0	0
$167$	2.75483e12	1.64117	0.820587	−	0.571521i	$-0.193646\pi$
$167$	2.75483e12	1.64117	0.820587	+	0.571521i	$0.193646\pi$
$168$	0	0
$169$	−1.45838e12	−0.813755
$170$	0	0
$171$	1.21160e12	0.633693
$172$	0	0
$173$	9.50387e11	0.466280	0.233140	−	0.972443i	$-0.425100\pi$
$173$	9.50387e11	0.466280	0.233140	+	0.972443i	$0.425100\pi$
$174$	0	0
$175$	4.26959e11	0.196642
$176$	0	0
$177$	−1.30768e12	−0.565781
$178$	0	0
$179$	−1.68138e12	−0.683873	−0.341936	−	0.939723i	$-0.611083\pi$
$179$	−1.68138e12	−0.683873	−0.341936	+	0.939723i	$0.611083\pi$
$180$	0	0
$181$	9.96774e11	0.381386	0.190693	−	0.981650i	$-0.438927\pi$
$181$	9.96774e11	0.381386	0.190693	+	0.981650i	$0.438927\pi$
$182$	0	0
$183$	1.75303e12	0.631406
$184$	0	0
$185$	−8.80090e11	−0.298595
$186$	0	0
$187$	3.69200e12	1.18068
$188$	0	0
$189$	−1.22698e12	−0.370083
$190$	0	0
$191$	2.76240e12	0.786328	0.393164	−	0.919468i	$-0.371381\pi$
$191$	2.76240e12	0.786328	0.393164	+	0.919468i	$0.371381\pi$
$192$	0	0
$193$	5.44239e12	1.46293	0.731466	−	0.681878i	$-0.238836\pi$
$193$	5.44239e12	1.46293	0.731466	+	0.681878i	$0.238836\pi$
$194$	0	0
$195$	7.03200e11	0.178603
$196$	0	0
$197$	2.87609e12	0.690619	0.345309	−	0.938489i	$-0.387774\pi$
$197$	2.87609e12	0.690619	0.345309	+	0.938489i	$0.387774\pi$
$198$	0	0
$199$	7.28391e11	0.165452	0.0827262	−	0.996572i	$-0.473637\pi$
$199$	7.28391e11	0.165452	0.0827262	+	0.996572i	$0.473637\pi$
$200$	0	0
$201$	−3.90142e12	−0.838773
$202$	0	0
$203$	2.15004e12	0.437742
$204$	0	0
$205$	−1.48822e12	−0.287092
$206$	0	0
$207$	−2.11868e12	−0.387461
$208$	0	0
$209$	5.69972e12	0.988665
$210$	0	0
$211$	6.79317e12	1.11820	0.559099	−	0.829101i	$-0.311147\pi$
$211$	6.79317e12	1.11820	0.559099	+	0.829101i	$0.311147\pi$
$212$	0	0
$213$	−2.46745e12	−0.385622
$214$	0	0
$215$	−8.27172e10	−0.0122796
$216$	0	0
$217$	8.84806e11	0.124830
$218$	0	0
$219$	−3.68875e11	−0.0494808
$220$	0	0
$221$	−3.98982e12	−0.509092
$222$	0	0
$223$	7.33486e12	0.890667	0.445333	−	0.895365i	$-0.353085\pi$
$223$	7.33486e12	0.890667	0.445333	+	0.895365i	$0.353085\pi$
$224$	0	0
$225$	2.89781e12	0.335016
$226$	0	0
$227$	1.35984e12	0.149743	0.0748713	−	0.997193i	$-0.476145\pi$
$227$	1.35984e12	0.149743	0.0748713	+	0.997193i	$0.476145\pi$
$228$	0	0
$229$	1.18244e13	1.24075	0.620375	−	0.784305i	$-0.286980\pi$
$229$	1.18244e13	1.24075	0.620375	+	0.784305i	$0.286980\pi$
$230$	0	0
$231$	−2.25579e12	−0.225649
$232$	0	0
$233$	−1.75634e13	−1.67552	−0.837761	−	0.546038i	$-0.816135\pi$
$233$	−1.75634e13	−1.67552	−0.837761	+	0.546038i	$0.816135\pi$
$234$	0	0
$235$	−1.29799e13	−1.18140
$236$	0	0
$237$	−9.60545e12	−0.834452
$238$	0	0
$239$	−7.13958e12	−0.592221	−0.296111	−	0.955154i	$-0.595690\pi$
$239$	−7.13958e12	−0.592221	−0.296111	+	0.955154i	$0.595690\pi$
$240$	0	0
$241$	−2.31307e11	−0.0183271	−0.00916357	−	0.999958i	$-0.502917\pi$
$241$	−2.31307e11	−0.0183271	−0.00916357	+	0.999958i	$0.502917\pi$
$242$	0	0
$243$	−1.34008e13	−1.01460
$244$	0	0
$245$	8.19634e12	0.593207
$246$	0	0
$247$	−6.15951e12	−0.426297
$248$	0	0
$249$	−7.39245e12	−0.489431
$250$	0	0
$251$	−1.29831e13	−0.822567	−0.411284	−	0.911507i	$-0.634919\pi$
$251$	−1.29831e13	−0.822567	−0.411284	+	0.911507i	$0.634919\pi$
$252$	0	0
$253$	−9.96692e12	−0.604502
$254$	0	0
$255$	−8.40563e12	−0.488201
$256$	0	0
$257$	2.39612e13	1.33314	0.666571	−	0.745442i	$-0.267761\pi$
$257$	2.39612e13	1.33314	0.666571	+	0.745442i	$0.267761\pi$
$258$	0	0
$259$	−3.05098e12	−0.162664
$260$	0	0
$261$	1.45925e13	0.745774
$262$	0	0
$263$	−2.42737e13	−1.18954	−0.594771	−	0.803895i	$-0.702757\pi$
$263$	−2.42737e13	−1.18954	−0.594771	+	0.803895i	$0.702757\pi$
$264$	0	0
$265$	−7.70895e12	−0.362362
$266$	0	0
$267$	6.29822e12	0.284057
$268$	0	0
$269$	−2.58377e13	−1.11845	−0.559225	−	0.829016i	$-0.688901\pi$
$269$	−2.58377e13	−1.11845	−0.559225	+	0.829016i	$0.688901\pi$
$270$	0	0
$271$	−3.76793e12	−0.156593	−0.0782964	−	0.996930i	$-0.524948\pi$
$271$	−3.76793e12	−0.156593	−0.0782964	+	0.996930i	$0.524948\pi$
$272$	0	0
$273$	2.43776e12	0.0972963
$274$	0	0
$275$	1.36322e13	0.522680
$276$	0	0
$277$	1.64189e13	0.604931	0.302466	−	0.953160i	$-0.402190\pi$
$277$	1.64189e13	0.604931	0.302466	+	0.953160i	$0.402190\pi$
$278$	0	0
$279$	6.00526e12	0.212671
$280$	0	0
$281$	2.10357e13	0.716263	0.358132	−	0.933671i	$-0.383414\pi$
$281$	2.10357e13	0.716263	0.358132	+	0.933671i	$0.383414\pi$
$282$	0	0
$283$	−1.67132e13	−0.547310	−0.273655	−	0.961828i	$-0.588233\pi$
$283$	−1.67132e13	−0.547310	−0.273655	+	0.961828i	$0.588233\pi$
$284$	0	0
$285$	−1.29767e13	−0.408805
$286$	0	0
$287$	−5.15917e12	−0.156397
$288$	0	0
$289$	1.34200e13	0.391575
$290$	0	0
$291$	−1.89034e13	−0.531042
$292$	0	0
$293$	2.39269e13	0.647312	0.323656	−	0.946175i	$-0.395088\pi$
$293$	2.39269e13	0.647312	0.323656	+	0.946175i	$0.395088\pi$
$294$	0	0
$295$	−2.50639e13	−0.653171
$296$	0	0
$297$	−3.91759e13	−0.983690
$298$	0	0
$299$	1.07709e13	0.260652
$300$	0	0
$301$	−2.86753e11	−0.00668947
$302$	0	0
$303$	2.05992e13	0.463358
$304$	0	0
$305$	3.35998e13	0.728933
$306$	0	0
$307$	−1.53111e13	−0.320439	−0.160219	−	0.987081i	$-0.551220\pi$
$307$	−1.53111e13	−0.320439	−0.160219	+	0.987081i	$0.551220\pi$
$308$	0	0
$309$	5.68903e13	1.14886
$310$	0	0
$311$	4.98752e13	0.972080	0.486040	−	0.873936i	$-0.338441\pi$
$311$	4.98752e13	0.972080	0.486040	+	0.873936i	$0.338441\pi$
$312$	0	0
$313$	−9.94808e13	−1.87174	−0.935870	−	0.352345i	$-0.885384\pi$
$313$	−9.94808e13	−1.87174	−0.935870	+	0.352345i	$0.885384\pi$
$314$	0	0
$315$	−9.19071e12	−0.166971
$316$	0	0
$317$	−8.33692e13	−1.46278	−0.731392	−	0.681958i	$-0.761129\pi$
$317$	−8.33692e13	−1.46278	−0.731392	+	0.681958i	$0.761129\pi$
$318$	0	0
$319$	6.86477e13	1.16353
$320$	0	0
$321$	2.27408e13	0.372416
$322$	0	0
$323$	7.36271e13	1.16526
$324$	0	0
$325$	−1.47319e13	−0.225372
$326$	0	0
$327$	1.85176e13	0.273888
$328$	0	0
$329$	−4.49970e13	−0.643585
$330$	0	0
$331$	6.35840e13	0.879618	0.439809	−	0.898091i	$-0.355046\pi$
$331$	6.35840e13	0.879618	0.439809	+	0.898091i	$0.355046\pi$
$332$	0	0
$333$	−2.07073e13	−0.277127
$334$	0	0
$335$	−7.47772e13	−0.968329
$336$	0	0
$337$	1.21001e14	1.51644	0.758221	−	0.651997i	$-0.226069\pi$
$337$	1.21001e14	1.51644	0.758221	+	0.651997i	$0.226069\pi$
$338$	0	0
$339$	2.14570e13	0.260298
$340$	0	0
$341$	2.82506e13	0.331802
$342$	0	0
$343$	6.15223e13	0.699705
$344$	0	0
$345$	2.26918e13	0.249957
$346$	0	0
$347$	1.55662e14	1.66100	0.830499	−	0.557020i	$-0.188055\pi$
$347$	1.55662e14	1.66100	0.830499	+	0.557020i	$0.188055\pi$
$348$	0	0
$349$	2.56430e13	0.265112	0.132556	−	0.991176i	$-0.457682\pi$
$349$	2.56430e13	0.265112	0.132556	+	0.991176i	$0.457682\pi$
$350$	0	0
$351$	4.23361e13	0.424152
$352$	0	0
$353$	2.49098e13	0.241885	0.120943	−	0.992659i	$-0.461408\pi$
$353$	2.49098e13	0.241885	0.120943	+	0.992659i	$0.461408\pi$
$354$	0	0
$355$	−4.72929e13	−0.445184
$356$	0	0
$357$	−2.91395e13	−0.265954
$358$	0	0
$359$	1.57584e14	1.39474	0.697370	−	0.716712i	$-0.254354\pi$
$359$	1.57584e14	1.39474	0.697370	+	0.716712i	$0.254354\pi$
$360$	0	0
$361$	−2.82438e12	−0.0242457
$362$	0	0
$363$	−1.25577e11	−0.00104574
$364$	0	0
$365$	−7.07011e12	−0.0571236
$366$	0	0
$367$	−1.77901e14	−1.39481	−0.697406	−	0.716676i	$-0.745662\pi$
$367$	−1.77901e14	−1.39481	−0.697406	+	0.716676i	$0.745662\pi$
$368$	0	0
$369$	−3.50157e13	−0.266452
$370$	0	0
$371$	−2.67244e13	−0.197402
$372$	0	0
$373$	5.51617e13	0.395585	0.197792	−	0.980244i	$-0.436623\pi$
$373$	5.51617e13	0.395585	0.197792	+	0.980244i	$0.436623\pi$
$374$	0	0
$375$	−9.04683e13	−0.629976
$376$	0	0
$377$	−7.41854e13	−0.501696
$378$	0	0
$379$	−1.46463e14	−0.962083	−0.481042	−	0.876698i	$-0.659741\pi$
$379$	−1.46463e14	−0.962083	−0.481042	+	0.876698i	$0.659741\pi$
$380$	0	0
$381$	6.62047e13	0.422476
$382$	0	0
$383$	2.31450e14	1.43504	0.717519	−	0.696539i	$-0.245278\pi$
$383$	2.31450e14	1.43504	0.717519	+	0.696539i	$0.245278\pi$
$384$	0	0
$385$	−4.32360e13	−0.260502
$386$	0	0
$387$	−1.94622e12	−0.0113967
$388$	0	0
$389$	1.49872e14	0.853093	0.426547	−	0.904466i	$-0.359730\pi$
$389$	1.49872e14	0.853093	0.426547	+	0.904466i	$0.359730\pi$
$390$	0	0
$391$	−1.28749e14	−0.712480
$392$	0	0
$393$	1.59145e14	0.856317
$394$	0	0
$395$	−1.84104e14	−0.963341
$396$	0	0
$397$	−2.08111e14	−1.05912	−0.529562	−	0.848271i	$-0.677644\pi$
$397$	−2.08111e14	−1.05912	−0.529562	+	0.848271i	$0.677644\pi$
$398$	0	0
$399$	−4.49857e13	−0.222702
$400$	0	0
$401$	−1.33408e14	−0.642521	−0.321261	−	0.946991i	$-0.604107\pi$
$401$	−1.33408e14	−0.642521	−0.321261	+	0.946991i	$0.604107\pi$
$402$	0	0
$403$	−3.05295e13	−0.143068
$404$	0	0
$405$	−8.04286e12	−0.0366782
$406$	0	0
$407$	−9.74134e13	−0.432364
$408$	0	0
$409$	−2.06168e14	−0.890722	−0.445361	−	0.895351i	$-0.646925\pi$
$409$	−2.06168e14	−0.890722	−0.445361	+	0.895351i	$0.646925\pi$
$410$	0	0
$411$	7.48941e13	0.315005
$412$	0	0
$413$	−8.68880e13	−0.355824
$414$	0	0
$415$	−1.41689e14	−0.565028
$416$	0	0
$417$	1.50392e14	0.584085
$418$	0	0
$419$	−7.34035e13	−0.277677	−0.138838	−	0.990315i	$-0.544337\pi$
$419$	−7.34035e13	−0.277677	−0.138838	+	0.990315i	$0.544337\pi$
$420$	0	0
$421$	−1.71112e14	−0.630563	−0.315282	−	0.948998i	$-0.602099\pi$
$421$	−1.71112e14	−0.630563	−0.315282	+	0.948998i	$0.602099\pi$
$422$	0	0
$423$	−3.05398e14	−1.09646
$424$	0	0
$425$	1.76096e14	0.616042
$426$	0	0
$427$	1.16479e14	0.397096
$428$	0	0
$429$	7.78341e13	0.258616
$430$	0	0
$431$	−7.17758e13	−0.232463	−0.116231	−	0.993222i	$-0.537081\pi$
$431$	−7.17758e13	−0.232463	−0.116231	+	0.993222i	$0.537081\pi$
$432$	0	0
$433$	9.98812e13	0.315356	0.157678	−	0.987491i	$-0.449599\pi$
$433$	9.98812e13	0.315356	0.157678	+	0.987491i	$0.449599\pi$
$434$	0	0
$435$	−1.56291e14	−0.481110
$436$	0	0
$437$	−1.98764e14	−0.596608
$438$	0	0
$439$	−2.90312e13	−0.0849788	−0.0424894	−	0.999097i	$-0.513529\pi$
$439$	−2.90312e13	−0.0849788	−0.0424894	+	0.999097i	$0.513529\pi$
$440$	0	0
$441$	1.92848e14	0.550558
$442$	0	0
$443$	−3.28370e14	−0.914414	−0.457207	−	0.889360i	$-0.651150\pi$
$443$	−3.28370e14	−0.914414	−0.457207	+	0.889360i	$0.651150\pi$
$444$	0	0
$445$	1.20716e14	0.327932
$446$	0	0
$447$	−2.81089e14	−0.744994
$448$	0	0
$449$	−6.12368e14	−1.58364	−0.791822	−	0.610752i	$-0.790867\pi$
$449$	−6.12368e14	−1.58364	−0.791822	+	0.610752i	$0.790867\pi$
$450$	0	0
$451$	−1.64725e14	−0.415708
$452$	0	0
$453$	2.07761e14	0.511709
$454$	0	0
$455$	4.67237e13	0.112325
$456$	0	0
$457$	3.03483e14	0.712189	0.356095	−	0.934450i	$-0.384108\pi$
$457$	3.03483e14	0.712189	0.356095	+	0.934450i	$0.384108\pi$
$458$	0	0
$459$	−5.06060e14	−1.15940
$460$	0	0
$461$	7.29308e14	1.63138	0.815691	−	0.578487i	$-0.196357\pi$
$461$	7.29308e14	1.63138	0.815691	+	0.578487i	$0.196357\pi$
$462$	0	0
$463$	1.22188e14	0.266891	0.133445	−	0.991056i	$-0.457396\pi$
$463$	1.22188e14	0.266891	0.133445	+	0.991056i	$0.457396\pi$
$464$	0	0
$465$	−6.43186e13	−0.137197
$466$	0	0
$467$	6.17381e14	1.28621	0.643103	−	0.765780i	$-0.277647\pi$
$467$	6.17381e14	1.28621	0.643103	+	0.765780i	$0.277647\pi$
$468$	0	0
$469$	−2.59228e14	−0.527510
$470$	0	0
$471$	3.31409e14	0.658794
$472$	0	0
$473$	−9.15561e12	−0.0177808
$474$	0	0
$475$	2.71858e14	0.515854
$476$	0	0
$477$	−1.81381e14	−0.336310
$478$	0	0
$479$	1.05084e15	1.90410	0.952052	−	0.305938i	$-0.0989700\pi$
$479$	1.05084e15	1.90410	0.952052	+	0.305938i	$0.0989700\pi$
$480$	0	0
$481$	1.05272e14	0.186429
$482$	0	0
$483$	7.86651e13	0.136167
$484$	0	0
$485$	−3.62316e14	−0.613066
$486$	0	0
$487$	−2.19910e14	−0.363777	−0.181889	−	0.983319i	$-0.558221\pi$
$487$	−2.19910e14	−0.363777	−0.181889	+	0.983319i	$0.558221\pi$
$488$	0	0
$489$	−9.01739e13	−0.145842
$490$	0	0
$491$	4.83863e14	0.765199	0.382599	−	0.923914i	$-0.375029\pi$
$491$	4.83863e14	0.765199	0.382599	+	0.923914i	$0.375029\pi$
$492$	0	0
$493$	8.86768e14	1.37136
$494$	0	0
$495$	−2.93446e14	−0.443813
$496$	0	0
$497$	−1.63949e14	−0.242520
$498$	0	0
$499$	1.08878e14	0.157538	0.0787691	−	0.996893i	$-0.474901\pi$
$499$	1.08878e14	0.157538	0.0787691	+	0.996893i	$0.474901\pi$
$500$	0	0
$501$	−6.94218e14	−0.982626
$502$	0	0
$503$	5.06588e14	0.701506	0.350753	−	0.936468i	$-0.385926\pi$
$503$	5.06588e14	0.701506	0.350753	+	0.936468i	$0.385926\pi$
$504$	0	0
$505$	3.94818e14	0.534927
$506$	0	0
$507$	3.67512e14	0.487222
$508$	0	0
$509$	−8.57534e13	−0.111251	−0.0556254	−	0.998452i	$-0.517715\pi$
$509$	−8.57534e13	−0.111251	−0.0556254	+	0.998452i	$0.517715\pi$
$510$	0	0
$511$	−2.45097e13	−0.0311188
$512$	0	0
$513$	−7.81259e14	−0.970844
$514$	0	0
$515$	1.09040e15	1.32631
$516$	0	0
$517$	−1.43669e15	−1.71066
$518$	0	0
$519$	−2.39498e14	−0.279178
$520$	0	0
$521$	9.27575e14	1.05862	0.529312	−	0.848428i	$-0.322450\pi$
$521$	9.27575e14	1.05862	0.529312	+	0.848428i	$0.322450\pi$
$522$	0	0
$523$	2.18187e13	0.0243820	0.0121910	−	0.999926i	$-0.496119\pi$
$523$	2.18187e13	0.0243820	0.0121910	+	0.999926i	$0.496119\pi$
$524$	0	0
$525$	−1.07594e14	−0.117736
$526$	0	0
$527$	3.64931e14	0.391069
$528$	0	0
$529$	−6.05238e14	−0.635214
$530$	0	0
$531$	−5.89717e14	−0.606211
$532$	0	0
$533$	1.78013e14	0.179247
$534$	0	0
$535$	4.35865e14	0.429939
$536$	0	0
$537$	4.23709e14	0.409458
$538$	0	0
$539$	9.07218e14	0.858961
$540$	0	0
$541$	1.69527e15	1.57273	0.786363	−	0.617765i	$-0.211962\pi$
$541$	1.69527e15	1.57273	0.786363	+	0.617765i	$0.211962\pi$
$542$	0	0
$543$	−2.51187e14	−0.228349
$544$	0	0
$545$	3.54921e14	0.316192
$546$	0	0
$547$	−7.52145e14	−0.656706	−0.328353	−	0.944555i	$-0.606494\pi$
$547$	−7.52145e14	−0.656706	−0.328353	+	0.944555i	$0.606494\pi$
$548$	0	0
$549$	7.90555e14	0.676526
$550$	0	0
$551$	1.36900e15	1.14834
$552$	0	0
$553$	−6.38228e14	−0.524793
$554$	0	0
$555$	2.21783e14	0.178779
$556$	0	0
$557$	−1.87489e14	−0.148174	−0.0740870	−	0.997252i	$-0.523604\pi$
$557$	−1.87489e14	−0.148174	−0.0740870	+	0.997252i	$0.523604\pi$
$558$	0	0
$559$	9.89417e12	0.00766681
$560$	0	0
$561$	−9.30383e14	−0.706913
$562$	0	0
$563$	−2.44971e14	−0.182524	−0.0912618	−	0.995827i	$-0.529090\pi$
$563$	−2.44971e14	−0.182524	−0.0912618	+	0.995827i	$0.529090\pi$
$564$	0	0
$565$	4.11259e14	0.300503
$566$	0	0
$567$	−2.78819e13	−0.0199809
$568$	0	0
$569$	1.35243e15	0.950596	0.475298	−	0.879825i	$-0.342340\pi$
$569$	1.35243e15	0.950596	0.475298	+	0.879825i	$0.342340\pi$
$570$	0	0
$571$	−1.43223e15	−0.987447	−0.493723	−	0.869619i	$-0.664364\pi$
$571$	−1.43223e15	−0.987447	−0.493723	+	0.869619i	$0.664364\pi$
$572$	0	0
$573$	−6.96126e14	−0.470801
$574$	0	0
$575$	−4.75389e14	−0.315410
$576$	0	0
$577$	−8.77659e14	−0.571293	−0.285647	−	0.958335i	$-0.592208\pi$
$577$	−8.77659e14	−0.571293	−0.285647	+	0.958335i	$0.592208\pi$
$578$	0	0
$579$	−1.37148e15	−0.875907
$580$	0	0
$581$	−4.91187e14	−0.307807
$582$	0	0
$583$	−8.53271e14	−0.524698
$584$	0	0
$585$	3.17118e14	0.191365
$586$	0	0
$587$	2.43425e15	1.44164	0.720818	−	0.693124i	$-0.243766\pi$
$587$	2.43425e15	1.44164	0.720818	+	0.693124i	$0.243766\pi$
$588$	0	0
$589$	5.63383e14	0.327469
$590$	0	0
$591$	−7.24775e14	−0.413497
$592$	0	0
$593$	−3.03318e14	−0.169863	−0.0849313	−	0.996387i	$-0.527067\pi$
$593$	−3.03318e14	−0.169863	−0.0849313	+	0.996387i	$0.527067\pi$
$594$	0	0
$595$	−5.58507e14	−0.307033
$596$	0	0
$597$	−1.83555e14	−0.0990619
$598$	0	0
$599$	−1.70198e15	−0.901795	−0.450898	−	0.892576i	$-0.648896\pi$
$599$	−1.70198e15	−0.901795	−0.450898	+	0.892576i	$0.648896\pi$
$600$	0	0
$601$	2.33922e15	1.21692	0.608458	−	0.793586i	$-0.291788\pi$
$601$	2.33922e15	1.21692	0.608458	+	0.793586i	$0.291788\pi$
$602$	0	0
$603$	−1.75940e15	−0.898710
$604$	0	0
$605$	−2.40689e12	−0.00120726
$606$	0	0
$607$	−2.49607e15	−1.22947	−0.614737	−	0.788732i	$-0.710738\pi$
$607$	−2.49607e15	−1.22947	−0.614737	+	0.788732i	$0.710738\pi$
$608$	0	0
$609$	−5.41810e14	−0.262091
$610$	0	0
$611$	1.55258e15	0.737612
$612$	0	0
$613$	−2.47301e15	−1.15397	−0.576983	−	0.816756i	$-0.695770\pi$
$613$	−2.47301e15	−1.15397	−0.576983	+	0.816756i	$0.695770\pi$
$614$	0	0
$615$	3.75032e14	0.171892
$616$	0	0
$617$	2.43368e13	0.0109571	0.00547854	−	0.999985i	$-0.498256\pi$
$617$	2.43368e13	0.0109571	0.00547854	+	0.999985i	$0.498256\pi$
$618$	0	0
$619$	−4.22545e15	−1.86885	−0.934425	−	0.356160i	$-0.884086\pi$
$619$	−4.22545e15	−1.86885	−0.934425	+	0.356160i	$0.884086\pi$
$620$	0	0
$621$	1.36616e15	0.593606
$622$	0	0
$623$	4.18481e14	0.178645
$624$	0	0
$625$	−4.88896e14	−0.205058
$626$	0	0
$627$	−1.43633e15	−0.591947
$628$	0	0
$629$	−1.25835e15	−0.509594
$630$	0	0
$631$	−4.26326e15	−1.69660	−0.848302	−	0.529513i	$-0.822375\pi$
$631$	−4.26326e15	−1.69660	−0.848302	+	0.529513i	$0.822375\pi$
$632$	0	0
$633$	−1.71188e15	−0.669503
$634$	0	0
$635$	1.26892e15	0.487731
$636$	0	0
$637$	−9.80401e14	−0.370371
$638$	0	0
$639$	−1.11273e15	−0.413177
$640$	0	0
$641$	1.00830e15	0.368018	0.184009	−	0.982925i	$-0.441092\pi$
$641$	1.00830e15	0.368018	0.184009	+	0.982925i	$0.441092\pi$
$642$	0	0
$643$	−3.03982e14	−0.109066	−0.0545328	−	0.998512i	$-0.517367\pi$
$643$	−3.03982e14	−0.109066	−0.0545328	+	0.998512i	$0.517367\pi$
$644$	0	0
$645$	2.08447e13	0.00735221
$646$	0	0
$647$	3.43583e15	1.19140	0.595700	−	0.803207i	$-0.296875\pi$
$647$	3.43583e15	1.19140	0.595700	+	0.803207i	$0.296875\pi$
$648$	0	0
$649$	−2.77421e15	−0.945788
$650$	0	0
$651$	−2.22971e14	−0.0747400
$652$	0	0
$653$	1.18539e15	0.390695	0.195347	−	0.980734i	$-0.437417\pi$
$653$	1.18539e15	0.390695	0.195347	+	0.980734i	$0.437417\pi$
$654$	0	0
$655$	3.05028e15	0.988583
$656$	0	0
$657$	−1.66350e14	−0.0530167
$658$	0	0
$659$	2.26510e15	0.709934	0.354967	−	0.934879i	$-0.384492\pi$
$659$	2.26510e15	0.709934	0.354967	+	0.934879i	$0.384492\pi$
$660$	0	0
$661$	5.33012e15	1.64297	0.821484	−	0.570232i	$-0.193147\pi$
$661$	5.33012e15	1.64297	0.821484	+	0.570232i	$0.193147\pi$
$662$	0	0
$663$	1.00543e15	0.304810
$664$	0	0
$665$	−8.62227e14	−0.257100
$666$	0	0
$667$	−2.39392e15	−0.702130
$668$	0	0
$669$	−1.84839e15	−0.533272
$670$	0	0
$671$	3.71902e15	1.05549
$672$	0	0
$673$	4.74120e15	1.32375	0.661874	−	0.749615i	$-0.269761\pi$
$673$	4.74120e15	1.32375	0.661874	+	0.749615i	$0.269761\pi$
$674$	0	0
$675$	−1.86856e15	−0.513259
$676$	0	0
$677$	1.41307e15	0.381880	0.190940	−	0.981602i	$-0.438846\pi$
$677$	1.41307e15	0.381880	0.190940	+	0.981602i	$0.438846\pi$
$678$	0	0
$679$	−1.25603e15	−0.333976
$680$	0	0
$681$	−3.42680e14	−0.0896559
$682$	0	0
$683$	3.03116e15	0.780359	0.390180	−	0.920739i	$-0.372413\pi$
$683$	3.03116e15	0.780359	0.390180	+	0.920739i	$0.372413\pi$
$684$	0	0
$685$	1.43547e15	0.363660
$686$	0	0
$687$	−2.97975e15	−0.742879
$688$	0	0
$689$	9.22102e14	0.226242
$690$	0	0
$691$	2.74731e15	0.663405	0.331703	−	0.943384i	$-0.392377\pi$
$691$	2.74731e15	0.663405	0.331703	+	0.943384i	$0.392377\pi$
$692$	0	0
$693$	−1.01728e15	−0.241773
$694$	0	0
$695$	2.88251e15	0.674303
$696$	0	0
$697$	−2.12786e15	−0.489962
$698$	0	0
$699$	4.42597e15	1.00319
$700$	0	0
$701$	−5.72747e15	−1.27795	−0.638974	−	0.769228i	$-0.720641\pi$
$701$	−5.72747e15	−1.27795	−0.638974	+	0.769228i	$0.720641\pi$
$702$	0	0
$703$	−1.94265e15	−0.426718
$704$	0	0
$705$	3.27093e15	0.707345
$706$	0	0
$707$	1.36870e15	0.291409
$708$	0	0
$709$	−6.98326e14	−0.146388	−0.0731938	−	0.997318i	$-0.523319\pi$
$709$	−6.98326e14	−0.146388	−0.0731938	+	0.997318i	$0.523319\pi$
$710$	0	0
$711$	−4.33171e15	−0.894081
$712$	0	0
$713$	−9.85170e14	−0.200225
$714$	0	0
$715$	1.49182e15	0.298561
$716$	0	0
$717$	1.79917e15	0.354583
$718$	0	0
$719$	9.70979e15	1.88452	0.942260	−	0.334882i	$-0.108696\pi$
$719$	9.70979e15	1.88452	0.942260	+	0.334882i	$0.108696\pi$
$720$	0	0
$721$	3.78004e15	0.722525
$722$	0	0
$723$	5.82893e13	0.0109731
$724$	0	0
$725$	3.27427e15	0.607093
$726$	0	0
$727$	2.46469e15	0.450114	0.225057	−	0.974346i	$-0.427743\pi$
$727$	2.46469e15	0.450114	0.225057	+	0.974346i	$0.427743\pi$
$728$	0	0
$729$	3.08202e15	0.554413
$730$	0	0
$731$	−1.18269e14	−0.0209568
$732$	0	0
$733$	−7.91285e15	−1.38121	−0.690607	−	0.723230i	$-0.742657\pi$
$733$	−7.91285e15	−1.38121	−0.690607	+	0.723230i	$0.742657\pi$
$734$	0	0
$735$	−2.06548e15	−0.355173
$736$	0	0
$737$	−8.27677e15	−1.40213
$738$	0	0
$739$	8.40694e15	1.40312	0.701558	−	0.712613i	$-0.252488\pi$
$739$	8.40694e15	1.40312	0.701558	+	0.712613i	$0.252488\pi$
$740$	0	0
$741$	1.55220e15	0.255239
$742$	0	0
$743$	1.36287e15	0.220809	0.110404	−	0.993887i	$-0.464785\pi$
$743$	1.36287e15	0.220809	0.110404	+	0.993887i	$0.464785\pi$
$744$	0	0
$745$	−5.38754e15	−0.860065
$746$	0	0
$747$	−3.33373e15	−0.524405
$748$	0	0
$749$	1.51100e15	0.234215
$750$	0	0
$751$	6.81722e15	1.04133	0.520664	−	0.853762i	$-0.325684\pi$
$751$	6.81722e15	1.04133	0.520664	+	0.853762i	$0.325684\pi$
$752$	0	0
$753$	3.27173e15	0.492499
$754$	0	0
$755$	3.98208e15	0.590747
$756$	0	0
$757$	6.67049e14	0.0975282	0.0487641	−	0.998810i	$-0.484472\pi$
$757$	6.67049e14	0.0975282	0.0487641	+	0.998810i	$0.484472\pi$
$758$	0	0
$759$	2.51166e15	0.361936
$760$	0	0
$761$	−7.74408e15	−1.09990	−0.549951	−	0.835197i	$-0.685354\pi$
$761$	−7.74408e15	−1.09990	−0.549951	+	0.835197i	$0.685354\pi$
$762$	0	0
$763$	1.23039e15	0.172250
$764$	0	0
$765$	−3.79064e15	−0.523088
$766$	0	0
$767$	2.99800e15	0.407809
$768$	0	0
$769$	2.52411e15	0.338465	0.169232	−	0.985576i	$-0.445871\pi$
$769$	2.52411e15	0.338465	0.169232	+	0.985576i	$0.445871\pi$
$770$	0	0
$771$	−6.03822e15	−0.798197
$772$	0	0
$773$	1.11453e16	1.45246	0.726229	−	0.687453i	$-0.241271\pi$
$773$	1.11453e16	1.45246	0.726229	+	0.687453i	$0.241271\pi$
$774$	0	0
$775$	1.34746e15	0.173124
$776$	0	0
$777$	7.68847e14	0.0973922
$778$	0	0
$779$	−3.28500e15	−0.410279
$780$	0	0
$781$	−5.23465e15	−0.644624
$782$	0	0
$783$	−9.40952e15	−1.14256
$784$	0	0
$785$	6.35201e15	0.760551
$786$	0	0
$787$	−1.32271e16	−1.56172	−0.780861	−	0.624705i	$-0.785219\pi$
$787$	−1.32271e16	−1.56172	−0.780861	+	0.624705i	$0.785219\pi$
$788$	0	0
$789$	6.11698e15	0.712219
$790$	0	0
$791$	1.42570e15	0.163703
$792$	0	0
$793$	−4.01902e15	−0.455112
$794$	0	0
$795$	1.94266e15	0.216958
$796$	0	0
$797$	−2.30248e15	−0.253615	−0.126807	−	0.991927i	$-0.540473\pi$
$797$	−2.30248e15	−0.253615	−0.126807	+	0.991927i	$0.540473\pi$
$798$	0	0
$799$	−1.85587e16	−2.01623
$800$	0	0
$801$	2.84027e15	0.304355
$802$	0	0
$803$	−7.82560e14	−0.0827146
$804$	0	0
$805$	1.50775e15	0.157199
$806$	0	0
$807$	6.51110e15	0.669653
$808$	0	0
$809$	5.60472e15	0.568639	0.284320	−	0.958730i	$-0.408232\pi$
$809$	5.60472e15	0.568639	0.284320	+	0.958730i	$0.408232\pi$
$810$	0	0
$811$	5.08516e15	0.508968	0.254484	−	0.967077i	$-0.418094\pi$
$811$	5.08516e15	0.508968	0.254484	+	0.967077i	$0.418094\pi$
$812$	0	0
$813$	9.49519e14	0.0937574
$814$	0	0
$815$	−1.72833e15	−0.168368
$816$	0	0
$817$	−1.82584e14	−0.0175486
$818$	0	0
$819$	1.09934e15	0.104249
$820$	0	0
$821$	−2.79111e14	−0.0261150	−0.0130575	−	0.999915i	$-0.504156\pi$
$821$	−2.79111e14	−0.0261150	−0.0130575	+	0.999915i	$0.504156\pi$
$822$	0	0
$823$	−1.35265e16	−1.24878	−0.624391	−	0.781112i	$-0.714653\pi$
$823$	−1.35265e16	−1.24878	−0.624391	+	0.781112i	$0.714653\pi$
$824$	0	0
$825$	−3.43531e15	−0.312946
$826$	0	0
$827$	−2.72544e14	−0.0244994	−0.0122497	−	0.999925i	$-0.503899\pi$
$827$	−2.72544e14	−0.0244994	−0.0122497	+	0.999925i	$0.503899\pi$
$828$	0	0
$829$	−1.80459e16	−1.60077	−0.800385	−	0.599486i	$-0.795372\pi$
$829$	−1.80459e16	−1.60077	−0.800385	+	0.599486i	$0.795372\pi$
$830$	0	0
$831$	−4.13757e15	−0.362193
$832$	0	0
$833$	1.17191e16	1.01239
$834$	0	0
$835$	−1.33058e16	−1.13440
$836$	0	0
$837$	−3.87230e15	−0.325821
$838$	0	0
$839$	−7.96183e15	−0.661184	−0.330592	−	0.943774i	$-0.607248\pi$
$839$	−7.96183e15	−0.661184	−0.330592	+	0.943774i	$0.607248\pi$
$840$	0	0
$841$	4.28775e15	0.351440
$842$	0	0
$843$	−5.30100e15	−0.428851
$844$	0	0
$845$	7.04397e15	0.562478
$846$	0	0
$847$	−8.34387e12	−0.000657671 0
$848$	0	0
$849$	4.21172e15	0.327693
$850$	0	0
$851$	3.39705e15	0.260909
$852$	0	0
$853$	1.49826e16	1.13598	0.567988	−	0.823037i	$-0.307722\pi$
$853$	1.49826e16	1.13598	0.567988	+	0.823037i	$0.307722\pi$
$854$	0	0
$855$	−5.85201e15	−0.438017
$856$	0	0
$857$	−2.22561e16	−1.64458	−0.822290	−	0.569068i	$-0.807304\pi$
$857$	−2.22561e16	−1.64458	−0.822290	+	0.569068i	$0.807304\pi$
$858$	0	0
$859$	−5.44237e15	−0.397032	−0.198516	−	0.980098i	$-0.563612\pi$
$859$	−5.44237e15	−0.397032	−0.198516	+	0.980098i	$0.563612\pi$
$860$	0	0
$861$	1.30011e15	0.0936403
$862$	0	0
$863$	1.08110e16	0.768787	0.384393	−	0.923169i	$-0.374411\pi$
$863$	1.08110e16	0.768787	0.384393	+	0.923169i	$0.374411\pi$
$864$	0	0
$865$	−4.59037e15	−0.322299
$866$	0	0
$867$	−3.38185e15	−0.234449
$868$	0	0
$869$	−2.03777e16	−1.39491
$870$	0	0
$871$	8.94444e15	0.604579
$872$	0	0
$873$	−8.52477e15	−0.568989
$874$	0	0
$875$	−6.01111e15	−0.396197
$876$	0	0
$877$	2.81024e16	1.82914	0.914568	−	0.404431i	$-0.132530\pi$
$877$	2.81024e16	1.82914	0.914568	+	0.404431i	$0.132530\pi$
$878$	0	0
$879$	−6.02957e15	−0.387568
$880$	0	0
$881$	4.22209e15	0.268016	0.134008	−	0.990980i	$-0.457215\pi$
$881$	4.22209e15	0.268016	0.134008	+	0.990980i	$0.457215\pi$
$882$	0	0
$883$	−5.16092e14	−0.0323551	−0.0161776	−	0.999869i	$-0.505150\pi$
$883$	−5.16092e14	−0.0323551	−0.0161776	+	0.999869i	$0.505150\pi$
$884$	0	0
$885$	6.31609e15	0.391075
$886$	0	0
$887$	5.71906e15	0.349740	0.174870	−	0.984592i	$-0.444050\pi$
$887$	5.71906e15	0.349740	0.174870	+	0.984592i	$0.444050\pi$
$888$	0	0
$889$	4.39894e15	0.265698
$890$	0	0
$891$	−8.90230e14	−0.0531098
$892$	0	0
$893$	−2.86510e16	−1.68832
$894$	0	0
$895$	8.12109e15	0.472702
$896$	0	0
$897$	−2.71427e15	−0.156061
$898$	0	0
$899$	6.78541e15	0.385388
$900$	0	0
$901$	−1.10223e16	−0.618421
$902$	0	0
$903$	7.22617e13	0.00400521
$904$	0	0
$905$	−4.81442e15	−0.263619
$906$	0	0
$907$	8.43778e13	0.00456445	0.00228222	−	0.999997i	$-0.499274\pi$
$907$	8.43778e13	0.00456445	0.00228222	+	0.999997i	$0.499274\pi$
$908$	0	0
$909$	9.28952e15	0.496468
$910$	0	0
$911$	−1.10091e16	−0.581298	−0.290649	−	0.956830i	$-0.593871\pi$
$911$	−1.10091e16	−0.581298	−0.290649	+	0.956830i	$0.593871\pi$
$912$	0	0
$913$	−1.56829e16	−0.818158
$914$	0	0
$915$	−8.46715e15	−0.436437
$916$	0	0
$917$	1.05743e16	0.538544
$918$	0	0
$919$	−4.86351e15	−0.244746	−0.122373	−	0.992484i	$-0.539050\pi$
$919$	−4.86351e15	−0.244746	−0.122373	+	0.992484i	$0.539050\pi$
$920$	0	0
$921$	3.85840e15	0.191857
$922$	0	0
$923$	5.65691e15	0.277952
$924$	0	0
$925$	−4.64630e15	−0.225594
$926$	0	0
$927$	2.56555e16	1.23095
$928$	0	0
$929$	3.57534e15	0.169524	0.0847620	−	0.996401i	$-0.472987\pi$
$929$	3.57534e15	0.169524	0.0847620	+	0.996401i	$0.472987\pi$
$930$	0	0
$931$	1.80921e16	0.847744
$932$	0	0
$933$	−1.25685e16	−0.582017
$934$	0	0
$935$	−1.78323e16	−0.816102
$936$	0	0
$937$	3.86373e16	1.74759	0.873795	−	0.486295i	$-0.161652\pi$
$937$	3.86373e16	1.74759	0.873795	+	0.486295i	$0.161652\pi$
$938$	0	0
$939$	2.50692e16	1.12067
$940$	0	0
$941$	3.48997e16	1.54198	0.770991	−	0.636846i	$-0.219761\pi$
$941$	3.48997e16	1.54198	0.770991	+	0.636846i	$0.219761\pi$
$942$	0	0
$943$	5.74437e15	0.250858
$944$	0	0
$945$	5.92634e15	0.255806
$946$	0	0
$947$	2.85123e16	1.21649	0.608243	−	0.793751i	$-0.291875\pi$
$947$	2.85123e16	1.21649	0.608243	+	0.793751i	$0.291875\pi$
$948$	0	0
$949$	8.45688e14	0.0356653
$950$	0	0
$951$	2.10091e16	0.875817
$952$	0	0
$953$	4.00334e16	1.64973	0.824863	−	0.565332i	$-0.191252\pi$
$953$	4.00334e16	1.64973	0.824863	+	0.565332i	$0.191252\pi$
$954$	0	0
$955$	−1.33424e16	−0.543520
$956$	0	0
$957$	−1.72992e16	−0.696644
$958$	0	0
$959$	4.97630e15	0.198109
$960$	0	0
$961$	−2.26161e16	−0.890100
$962$	0	0
$963$	1.02553e16	0.399028
$964$	0	0
$965$	−2.62867e16	−1.01120
$966$	0	0
$967$	1.84953e16	0.703422	0.351711	−	0.936109i	$-0.385600\pi$
$967$	1.84953e16	0.703422	0.351711	+	0.936109i	$0.385600\pi$
$968$	0	0
$969$	−1.85540e16	−0.697682
$970$	0	0
$971$	2.14877e16	0.798884	0.399442	−	0.916759i	$-0.369204\pi$
$971$	2.14877e16	0.798884	0.399442	+	0.916759i	$0.369204\pi$
$972$	0	0
$973$	9.99271e15	0.367335
$974$	0	0
$975$	3.71243e15	0.134938
$976$	0	0
$977$	−8.73880e15	−0.314074	−0.157037	−	0.987593i	$-0.550194\pi$
$977$	−8.73880e15	−0.314074	−0.157037	+	0.987593i	$0.550194\pi$
$978$	0	0
$979$	1.33615e16	0.474844
$980$	0	0
$981$	8.35079e15	0.293459
$982$	0	0
$983$	1.18924e16	0.413263	0.206631	−	0.978419i	$-0.433750\pi$
$983$	1.18924e16	0.413263	0.206631	+	0.978419i	$0.433750\pi$
$984$	0	0
$985$	−1.38915e16	−0.477365
$986$	0	0
$987$	1.13392e16	0.385336
$988$	0	0
$989$	3.19279e14	0.0107298
$990$	0	0
$991$	2.34409e16	0.779056	0.389528	−	0.921015i	$-0.372638\pi$
$991$	2.34409e16	0.779056	0.389528	+	0.921015i	$0.372638\pi$
$992$	0	0
$993$	−1.60232e16	−0.526657
$994$	0	0
$995$	−3.51813e15	−0.114363
$996$	0	0
$997$	2.14004e16	0.688016	0.344008	−	0.938967i	$-0.388215\pi$
$997$	2.14004e16	0.688016	0.344008	+	0.938967i	$0.388215\pi$
$998$	0	0
$999$	1.33524e16	0.424571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.12.a.b.1.1		1
4.3	odd	2		64.12.a.f.1.1		1
8.3	odd	2		16.12.a.a.1.1		1
8.5	even	2		1.12.a.a.1.1	✓	1
16.3	odd	4		256.12.b.c.129.2		2
16.5	even	4		256.12.b.e.129.2		2
16.11	odd	4		256.12.b.c.129.1		2
16.13	even	4		256.12.b.e.129.1		2
24.5	odd	2		9.12.a.b.1.1		1
24.11	even	2		144.12.a.d.1.1		1
40.13	odd	4		25.12.b.b.24.2		2
40.29	even	2		25.12.a.b.1.1		1
40.37	odd	4		25.12.b.b.24.1		2
56.5	odd	6		49.12.c.c.18.1		2
56.13	odd	2		49.12.a.a.1.1		1
56.37	even	6		49.12.c.b.18.1		2
56.45	odd	6		49.12.c.c.30.1		2
56.53	even	6		49.12.c.b.30.1		2
72.5	odd	6		81.12.c.b.55.1		2
72.13	even	6		81.12.c.d.55.1		2
72.29	odd	6		81.12.c.b.28.1		2
72.61	even	6		81.12.c.d.28.1		2
88.21	odd	2		121.12.a.b.1.1		1
104.77	even	2		169.12.a.a.1.1		1
120.29	odd	2		225.12.a.b.1.1		1
120.53	even	4		225.12.b.d.199.1		2
120.77	even	4		225.12.b.d.199.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.12.a.a.1.1	✓	1	8.5	even	2
9.12.a.b.1.1		1	24.5	odd	2
16.12.a.a.1.1		1	8.3	odd	2
25.12.a.b.1.1		1	40.29	even	2
25.12.b.b.24.1		2	40.37	odd	4
25.12.b.b.24.2		2	40.13	odd	4
49.12.a.a.1.1		1	56.13	odd	2
49.12.c.b.18.1		2	56.37	even	6
49.12.c.b.30.1		2	56.53	even	6
49.12.c.c.18.1		2	56.5	odd	6
49.12.c.c.30.1		2	56.45	odd	6
64.12.a.b.1.1		1	1.1	even	1	trivial
64.12.a.f.1.1		1	4.3	odd	2
81.12.c.b.28.1		2	72.29	odd	6
81.12.c.b.55.1		2	72.5	odd	6
81.12.c.d.28.1		2	72.61	even	6
81.12.c.d.55.1		2	72.13	even	6
121.12.a.b.1.1		1	88.21	odd	2
144.12.a.d.1.1		1	24.11	even	2
169.12.a.a.1.1		1	104.77	even	2
225.12.a.b.1.1		1	120.29	odd	2
225.12.b.d.199.1		2	120.53	even	4
225.12.b.d.199.2		2	120.77	even	4
256.12.b.c.129.1		2	16.11	odd	4
256.12.b.c.129.2		2	16.3	odd	4
256.12.b.e.129.1		2	16.13	even	4
256.12.b.e.129.2		2	16.5	even	4

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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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