LMFDB - Embedded newform 64.12.a.j.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
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)

$f(q)$	$=$	$q+415.692 q^{3} +10.0000 q^{5} +25772.9 q^{7} -4347.00 q^{9} +O(q^{10})$	$q+415.692 q^{3} +10.0000 q^{5} +25772.9 q^{7} -4347.00 q^{9} -813510. q^{11} +148722. q^{13} +4156.92 q^{15} -1.90461e6 q^{17} -636425. q^{19} +1.07136e7 q^{21} -5.24928e7 q^{23} -4.88280e7 q^{25} -7.54456e7 q^{27} +1.18020e8 q^{29} +9.85789e7 q^{31} -3.38170e8 q^{33} +257729. q^{35} +3.90851e8 q^{37} +6.18226e7 q^{39} -9.92147e8 q^{41} +1.29209e9 q^{43} -43470.0 q^{45} +4.87040e8 q^{47} -1.31308e9 q^{49} -7.91730e8 q^{51} -3.53821e9 q^{53} -8.13510e6 q^{55} -2.64557e8 q^{57} -5.94555e9 q^{59} -1.19579e10 q^{61} -1.12035e8 q^{63} +1.48722e6 q^{65} +1.00118e10 q^{67} -2.18208e10 q^{69} -2.29930e10 q^{71} -7.51976e9 q^{73} -2.02974e10 q^{75} -2.09665e10 q^{77} +2.94658e10 q^{79} -3.05921e10 q^{81} +3.55008e10 q^{83} -1.90461e7 q^{85} +4.90599e10 q^{87} -5.90580e10 q^{89} +3.83300e9 q^{91} +4.09785e10 q^{93} -6.36425e6 q^{95} -7.67665e10 q^{97} +3.53633e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{5} - 8694 q^{9}+O(q^{10}) $ 2 * q + 20 * q^5 - 8694 * q^9	$ 2 q + 20 q^{5} - 8694 q^{9} + 297444 q^{13} - 3809212 q^{17} + 21427200 q^{21} - 97656050 q^{25} + 236039620 q^{29} - 676339200 q^{33} + 781701268 q^{37} - 1984294220 q^{41} - 86940 q^{45} - 2626167086 q^{49} - 7076426444 q^{53} - 529113600 q^{57} - 23915839420 q^{61} + 2974440 q^{65} - 43641676800 q^{69} - 15039524076 q^{73} - 41933030400 q^{77} - 61184210382 q^{81} - 38092120 q^{85} - 118115907980 q^{89} + 81956966400 q^{93} - 153533015388 q^{97}+O(q^{100}) $ 2 * q + 20 * q^5 - 8694 * q^9 + 297444 * q^13 - 3809212 * q^17 + 21427200 * q^21 - 97656050 * q^25 + 236039620 * q^29 - 676339200 * q^33 + 781701268 * q^37 - 1984294220 * q^41 - 86940 * q^45 - 2626167086 * q^49 - 7076426444 * q^53 - 529113600 * q^57 - 23915839420 * q^61 + 2974440 * q^65 - 43641676800 * q^69 - 15039524076 * q^73 - 41933030400 * q^77 - 61184210382 * q^81 - 38092120 * q^85 - 118115907980 * q^89 + 81956966400 * q^93 - 153533015388 * q^97

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
$3$	415.692	0.987654	0.493827	−	0.869560i	$-0.335598\pi$
$3$	415.692	0.987654	0.493827	+	0.869560i	$0.335598\pi$
$4$	0	0
$5$	10.0000	0.00143108	0.000715542	−	1.00000i	$-0.499772\pi$
$5$	10.0000	0.00143108	0.000715542		1.00000i	$0.499772\pi$
$6$	0	0
$7$	25772.9	0.579595	0.289797	−	0.957088i	$-0.406412\pi$
$7$	25772.9	0.579595	0.289797	+	0.957088i	$0.406412\pi$
$8$	0	0
$9$	−4347.00	−0.0245389
$10$	0	0
$11$	−813510.	−1.52301	−0.761505	−	0.648159i	$-0.775539\pi$
$11$	−813510.	−1.52301	−0.761505	+	0.648159i	$0.775539\pi$
$12$	0	0
$13$	148722.	0.111093	0.0555465	−	0.998456i	$-0.482310\pi$
$13$	148722.	0.111093	0.0555465	+	0.998456i	$0.482310\pi$
$14$	0	0
$15$	4156.92	0.00141342
$16$	0	0
$17$	−1.90461e6	−0.325339	−0.162669	−	0.986681i	$-0.552010\pi$
$17$	−1.90461e6	−0.325339	−0.162669	+	0.986681i	$0.552010\pi$
$18$	0	0
$19$	−636425.	−0.0589661	−0.0294830	−	0.999565i	$-0.509386\pi$
$19$	−636425.	−0.0589661	−0.0294830	+	0.999565i	$0.509386\pi$
$20$	0	0
$21$	1.07136e7	0.572439
$22$	0	0
$23$	−5.24928e7	−1.70058	−0.850289	−	0.526316i	$-0.823573\pi$
$23$	−5.24928e7	−1.70058	−0.850289	+	0.526316i	$0.823573\pi$
$24$	0	0
$25$	−4.88280e7	−0.999998
$26$	0	0
$27$	−7.54456e7	−1.01189
$28$	0	0
$29$	1.18020e8	1.06848	0.534239	−	0.845333i	$-0.320598\pi$
$29$	1.18020e8	1.06848	0.534239	+	0.845333i	$0.320598\pi$
$30$	0	0
$31$	9.85789e7	0.618436	0.309218	−	0.950991i	$-0.399933\pi$
$31$	9.85789e7	0.618436	0.309218	+	0.950991i	$0.399933\pi$
$32$	0	0
$33$	−3.38170e8	−1.50421
$34$	0	0
$35$	257729.	0.000829448 0
$36$	0	0
$37$	3.90851e8	0.926619	0.463310	−	0.886196i	$-0.346662\pi$
$37$	3.90851e8	0.926619	0.463310	+	0.886196i	$0.346662\pi$
$38$	0	0
$39$	6.18226e7	0.109722
$40$	0	0
$41$	−9.92147e8	−1.33741	−0.668705	−	0.743527i	$-0.733151\pi$
$41$	−9.92147e8	−1.33741	−0.668705	+	0.743527i	$0.733151\pi$
$42$	0	0
$43$	1.29209e9	1.34034	0.670170	−	0.742208i	$-0.266221\pi$
$43$	1.29209e9	1.34034	0.670170	+	0.742208i	$0.266221\pi$
$44$	0	0
$45$	−43470.0	−3.51173e−5 0
$46$	0	0
$47$	4.87040e8	0.309761	0.154880	−	0.987933i	$-0.450501\pi$
$47$	4.87040e8	0.309761	0.154880	+	0.987933i	$0.450501\pi$
$48$	0	0
$49$	−1.31308e9	−0.664070
$50$	0	0
$51$	−7.91730e8	−0.321322
$52$	0	0
$53$	−3.53821e9	−1.16216	−0.581081	−	0.813846i	$-0.697370\pi$
$53$	−3.53821e9	−1.16216	−0.581081	+	0.813846i	$0.697370\pi$
$54$	0	0
$55$	−8.13510e6	−0.00217956
$56$	0	0
$57$	−2.64557e8	−0.0582381
$58$	0	0
$59$	−5.94555e9	−1.08270	−0.541348	−	0.840799i	$-0.682086\pi$
$59$	−5.94555e9	−1.08270	−0.541348	+	0.840799i	$0.682086\pi$
$60$	0	0
$61$	−1.19579e10	−1.81277	−0.906383	−	0.422458i	$-0.861167\pi$
$61$	−1.19579e10	−1.81277	−0.906383	+	0.422458i	$0.861167\pi$
$62$	0	0
$63$	−1.12035e8	−0.0142226
$64$	0	0
$65$	1.48722e6	0.000158983 0
$66$	0	0
$67$	1.00118e10	0.905942	0.452971	−	0.891525i	$-0.350364\pi$
$67$	1.00118e10	0.905942	0.452971	+	0.891525i	$0.350364\pi$
$68$	0	0
$69$	−2.18208e10	−1.67958
$70$	0	0
$71$	−2.29930e10	−1.51243	−0.756213	−	0.654325i	$-0.772953\pi$
$71$	−2.29930e10	−1.51243	−0.756213	+	0.654325i	$0.772953\pi$
$72$	0	0
$73$	−7.51976e9	−0.424550	−0.212275	−	0.977210i	$-0.568087\pi$
$73$	−7.51976e9	−0.424550	−0.212275	+	0.977210i	$0.568087\pi$
$74$	0	0
$75$	−2.02974e10	−0.987652
$76$	0	0
$77$	−2.09665e10	−0.882729
$78$	0	0
$79$	2.94658e10	1.07738	0.538691	−	0.842504i	$-0.318919\pi$
$79$	2.94658e10	1.07738	0.538691	+	0.842504i	$0.318919\pi$
$80$	0	0
$81$	−3.05921e10	−0.974859
$82$	0	0
$83$	3.55008e10	0.989255	0.494628	−	0.869105i	$-0.335304\pi$
$83$	3.55008e10	0.989255	0.494628	+	0.869105i	$0.335304\pi$
$84$	0	0
$85$	−1.90461e7	−0.000465587 0
$86$	0	0
$87$	4.90599e10	1.05529
$88$	0	0
$89$	−5.90580e10	−1.12107	−0.560536	−	0.828130i	$-0.689405\pi$
$89$	−5.90580e10	−1.12107	−0.560536	+	0.828130i	$0.689405\pi$
$90$	0	0
$91$	3.83300e9	0.0643889
$92$	0	0
$93$	4.09785e10	0.610801
$94$	0	0
$95$	−6.36425e6	−8.43854e−5 0
$96$	0	0
$97$	−7.67665e10	−0.907668	−0.453834	−	0.891086i	$-0.649944\pi$
$97$	−7.67665e10	−0.907668	−0.453834	+	0.891086i	$0.649944\pi$
$98$	0	0
$99$	3.53633e9	0.0373731
$100$	0	0
$101$	6.61729e10	0.626488	0.313244	−	0.949673i	$-0.398584\pi$
$101$	6.61729e10	0.626488	0.313244	+	0.949673i	$0.398584\pi$
$102$	0	0
$103$	−2.16018e11	−1.83605	−0.918027	−	0.396517i	$-0.870219\pi$
$103$	−2.16018e11	−1.83605	−0.918027	+	0.396517i	$0.870219\pi$
$104$	0	0
$105$	1.07136e8	0.000819208 0
$106$	0	0
$107$	−4.83022e10	−0.332932	−0.166466	−	0.986047i	$-0.553236\pi$
$107$	−4.83022e10	−0.332932	−0.166466	+	0.986047i	$0.553236\pi$
$108$	0	0
$109$	9.39196e10	0.584669	0.292335	−	0.956316i	$-0.405568\pi$
$109$	9.39196e10	0.584669	0.292335	+	0.956316i	$0.405568\pi$
$110$	0	0
$111$	1.62474e11	0.915180
$112$	0	0
$113$	1.35852e11	0.693641	0.346820	−	0.937932i	$-0.387261\pi$
$113$	1.35852e11	0.693641	0.346820	+	0.937932i	$0.387261\pi$
$114$	0	0
$115$	−5.24928e8	−0.00243367
$116$	0	0
$117$	−6.46495e8	−0.00272611
$118$	0	0
$119$	−4.90873e10	−0.188565
$120$	0	0
$121$	3.76486e11	1.31956
$122$	0	0
$123$	−4.12428e11	−1.32090
$124$	0	0
$125$	−9.76562e8	−0.00286216
$126$	0	0
$127$	6.80045e10	0.182649	0.0913245	−	0.995821i	$-0.470890\pi$
$127$	6.80045e10	0.182649	0.0913245	+	0.995821i	$0.470890\pi$
$128$	0	0
$129$	5.37110e11	1.32379
$130$	0	0
$131$	4.32075e11	0.978515	0.489258	−	0.872139i	$-0.337268\pi$
$131$	4.32075e11	0.978515	0.489258	+	0.872139i	$0.337268\pi$
$132$	0	0
$133$	−1.64025e10	−0.0341764
$134$	0	0
$135$	−7.54456e8	−0.00144810
$136$	0	0
$137$	7.04409e11	1.24699	0.623493	−	0.781829i	$-0.285713\pi$
$137$	7.04409e11	1.24699	0.623493	+	0.781829i	$0.285713\pi$
$138$	0	0
$139$	8.57162e11	1.40114	0.700570	−	0.713584i	$-0.252929\pi$
$139$	8.57162e11	1.40114	0.700570	+	0.713584i	$0.252929\pi$
$140$	0	0
$141$	2.02459e11	0.305937
$142$	0	0
$143$	−1.20987e11	−0.169196
$144$	0	0
$145$	1.18020e9	0.00152908
$146$	0	0
$147$	−5.45839e11	−0.655872
$148$	0	0
$149$	1.15383e11	0.128711	0.0643557	−	0.997927i	$-0.479501\pi$
$149$	1.15383e11	0.128711	0.0643557	+	0.997927i	$0.479501\pi$
$150$	0	0
$151$	1.12673e12	1.16801	0.584005	−	0.811750i	$-0.301485\pi$
$151$	1.12673e12	1.16801	0.584005	+	0.811750i	$0.301485\pi$
$152$	0	0
$153$	8.27932e9	0.00798347
$154$	0	0
$155$	9.85789e8	0.000885033 0
$156$	0	0
$157$	1.22523e12	1.02511	0.512555	−	0.858654i	$-0.328699\pi$
$157$	1.22523e12	1.02511	0.512555	+	0.858654i	$0.328699\pi$
$158$	0	0
$159$	−1.47081e12	−1.14781
$160$	0	0
$161$	−1.35289e12	−0.985646
$162$	0	0
$163$	−1.23126e12	−0.838141	−0.419070	−	0.907954i	$-0.637644\pi$
$163$	−1.23126e12	−0.838141	−0.419070	+	0.907954i	$0.637644\pi$
$164$	0	0
$165$	−3.38170e9	−0.00215265
$166$	0	0
$167$	1.32035e12	0.786588	0.393294	−	0.919413i	$-0.371335\pi$
$167$	1.32035e12	0.786588	0.393294	+	0.919413i	$0.371335\pi$
$168$	0	0
$169$	−1.77004e12	−0.987658
$170$	0	0
$171$	2.76654e9	0.00144697
$172$	0	0
$173$	4.75014e11	0.233052	0.116526	−	0.993188i	$-0.462824\pi$
$173$	4.75014e11	0.233052	0.116526	+	0.993188i	$0.462824\pi$
$174$	0	0
$175$	−1.25844e12	−0.579593
$176$	0	0
$177$	−2.47152e12	−1.06933
$178$	0	0
$179$	−3.73857e11	−0.152060	−0.0760299	−	0.997106i	$-0.524224\pi$
$179$	−3.73857e11	−0.152060	−0.0760299	+	0.997106i	$0.524224\pi$
$180$	0	0
$181$	−1.00500e12	−0.384535	−0.192267	−	0.981343i	$-0.561584\pi$
$181$	−1.00500e12	−0.384535	−0.192267	+	0.981343i	$0.561584\pi$
$182$	0	0
$183$	−4.97081e12	−1.79039
$184$	0	0
$185$	3.90851e9	0.00132607
$186$	0	0
$187$	1.54942e12	0.495495
$188$	0	0
$189$	−1.94445e12	−0.586486
$190$	0	0
$191$	3.26123e12	0.928322	0.464161	−	0.885751i	$-0.346356\pi$
$191$	3.26123e12	0.928322	0.464161	+	0.885751i	$0.346356\pi$
$192$	0	0
$193$	−3.76328e12	−1.01158	−0.505792	−	0.862656i	$-0.668800\pi$
$193$	−3.76328e12	−1.01158	−0.505792	+	0.862656i	$0.668800\pi$
$194$	0	0
$195$	6.18226e8	0.000157021 0
$196$	0	0
$197$	−7.22764e12	−1.73553	−0.867765	−	0.496974i	$-0.834444\pi$
$197$	−7.22764e12	−1.73553	−0.867765	+	0.496974i	$0.834444\pi$
$198$	0	0
$199$	7.80075e12	1.77192	0.885960	−	0.463761i	$-0.153500\pi$
$199$	7.80075e12	1.77192	0.885960	+	0.463761i	$0.153500\pi$
$200$	0	0
$201$	4.16183e12	0.894758
$202$	0	0
$203$	3.04171e12	0.619285
$204$	0	0
$205$	−9.92147e9	−0.00191395
$206$	0	0
$207$	2.28186e11	0.0417304
$208$	0	0
$209$	5.17738e11	0.0898060
$210$	0	0
$211$	−1.72695e12	−0.284267	−0.142133	−	0.989848i	$-0.545396\pi$
$211$	−1.72695e12	−0.284267	−0.142133	+	0.989848i	$0.545396\pi$
$212$	0	0
$213$	−9.55800e12	−1.49375
$214$	0	0
$215$	1.29209e10	0.00191814
$216$	0	0
$217$	2.54067e12	0.358442
$218$	0	0
$219$	−3.12591e12	−0.419308
$220$	0	0
$221$	−2.83257e11	−0.0361429
$222$	0	0
$223$	−8.58280e12	−1.04220	−0.521102	−	0.853495i	$-0.674479\pi$
$223$	−8.58280e12	−1.04220	−0.521102	+	0.853495i	$0.674479\pi$
$224$	0	0
$225$	2.12255e11	0.0245389
$226$	0	0
$227$	3.70966e12	0.408500	0.204250	−	0.978919i	$-0.434524\pi$
$227$	3.70966e12	0.408500	0.204250	+	0.978919i	$0.434524\pi$
$228$	0	0
$229$	3.30481e12	0.346778	0.173389	−	0.984853i	$-0.444528\pi$
$229$	3.30481e12	0.346778	0.173389	+	0.984853i	$0.444528\pi$
$230$	0	0
$231$	−8.71562e12	−0.871831
$232$	0	0
$233$	1.00569e13	0.959412	0.479706	−	0.877429i	$-0.340743\pi$
$233$	1.00569e13	0.959412	0.479706	+	0.877429i	$0.340743\pi$
$234$	0	0
$235$	4.87040e9	0.000443293 0
$236$	0	0
$237$	1.22487e13	1.06408
$238$	0	0
$239$	9.62317e12	0.798233	0.399116	−	0.916900i	$-0.369317\pi$
$239$	9.62317e12	0.798233	0.399116	+	0.916900i	$0.369317\pi$
$240$	0	0
$241$	−1.70105e13	−1.34780	−0.673898	−	0.738825i	$-0.735381\pi$
$241$	−1.70105e13	−1.34780	−0.673898	+	0.738825i	$0.735381\pi$
$242$	0	0
$243$	6.48069e11	0.0490667
$244$	0	0
$245$	−1.31308e10	−0.000950340 0
$246$	0	0
$247$	−9.46504e10	−0.00655072
$248$	0	0
$249$	1.47574e13	0.977042
$250$	0	0
$251$	−1.95983e13	−1.24169	−0.620844	−	0.783934i	$-0.713210\pi$
$251$	−1.95983e13	−1.24169	−0.620844	+	0.783934i	$0.713210\pi$
$252$	0	0
$253$	4.27034e13	2.59000
$254$	0	0
$255$	−7.91730e9	−0.000459839 0
$256$	0	0
$257$	−6.76937e12	−0.376631	−0.188315	−	0.982109i	$-0.560303\pi$
$257$	−6.76937e12	−0.376631	−0.188315	+	0.982109i	$0.560303\pi$
$258$	0	0
$259$	1.00734e13	0.537064
$260$	0	0
$261$	−5.13032e11	−0.0262193
$262$	0	0
$263$	−1.54011e13	−0.754738	−0.377369	−	0.926063i	$-0.623171\pi$
$263$	−1.54011e13	−0.754738	−0.377369	+	0.926063i	$0.623171\pi$
$264$	0	0
$265$	−3.53821e10	−0.00166315
$266$	0	0
$267$	−2.45499e13	−1.10723
$268$	0	0
$269$	1.76171e13	0.762598	0.381299	−	0.924452i	$-0.375477\pi$
$269$	1.76171e13	0.762598	0.381299	+	0.924452i	$0.375477\pi$
$270$	0	0
$271$	3.81101e13	1.58383	0.791916	−	0.610630i	$-0.209084\pi$
$271$	3.81101e13	1.58383	0.791916	+	0.610630i	$0.209084\pi$
$272$	0	0
$273$	1.59335e12	0.0635940
$274$	0	0
$275$	3.97221e13	1.52301
$276$	0	0
$277$	1.24363e13	0.458198	0.229099	−	0.973403i	$-0.426422\pi$
$277$	1.24363e13	0.458198	0.229099	+	0.973403i	$0.426422\pi$
$278$	0	0
$279$	−4.28523e11	−0.0151758
$280$	0	0
$281$	4.01925e13	1.36855	0.684275	−	0.729224i	$-0.260119\pi$
$281$	4.01925e13	1.36855	0.684275	+	0.729224i	$0.260119\pi$
$282$	0	0
$283$	5.59771e13	1.83309	0.916547	−	0.399928i	$-0.130965\pi$
$283$	5.59771e13	1.83309	0.916547	+	0.399928i	$0.130965\pi$
$284$	0	0
$285$	−2.64557e9	−8.33436e−5 0
$286$	0	0
$287$	−2.55705e13	−0.775156
$288$	0	0
$289$	−3.06444e13	−0.894155
$290$	0	0
$291$	−3.19112e13	−0.896463
$292$	0	0
$293$	−6.38889e13	−1.72844	−0.864218	−	0.503117i	$-0.832187\pi$
$293$	−6.38889e13	−1.72844	−0.864218	+	0.503117i	$0.832187\pi$
$294$	0	0
$295$	−5.94555e10	−0.00154943
$296$	0	0
$297$	6.13758e13	1.54112
$298$	0	0
$299$	−7.80683e12	−0.188922
$300$	0	0
$301$	3.33008e13	0.776854
$302$	0	0
$303$	2.75075e13	0.618753
$304$	0	0
$305$	−1.19579e11	−0.00259422
$306$	0	0
$307$	−6.63349e13	−1.38829	−0.694147	−	0.719834i	$-0.744218\pi$
$307$	−6.63349e13	−1.38829	−0.694147	+	0.719834i	$0.744218\pi$
$308$	0	0
$309$	−8.97971e13	−1.81339
$310$	0	0
$311$	1.92176e13	0.374556	0.187278	−	0.982307i	$-0.440034\pi$
$311$	1.92176e13	0.374556	0.187278	+	0.982307i	$0.440034\pi$
$312$	0	0
$313$	−1.47273e13	−0.277095	−0.138547	−	0.990356i	$-0.544243\pi$
$313$	−1.47273e13	−0.277095	−0.138547	+	0.990356i	$0.544243\pi$
$314$	0	0
$315$	−1.12035e9	−2.03538e−5 0
$316$	0	0
$317$	−4.67009e13	−0.819407	−0.409703	−	0.912219i	$-0.634368\pi$
$317$	−4.67009e13	−0.819407	−0.409703	+	0.912219i	$0.634368\pi$
$318$	0	0
$319$	−9.60103e13	−1.62730
$320$	0	0
$321$	−2.00788e13	−0.328822
$322$	0	0
$323$	1.21214e12	0.0191840
$324$	0	0
$325$	−7.26180e12	−0.111093
$326$	0	0
$327$	3.90416e13	0.577451
$328$	0	0
$329$	1.25524e13	0.179536
$330$	0	0
$331$	−1.88279e13	−0.260464	−0.130232	−	0.991484i	$-0.541572\pi$
$331$	−1.88279e13	−0.260464	−0.130232	+	0.991484i	$0.541572\pi$
$332$	0	0
$333$	−1.69903e12	−0.0227383
$334$	0	0
$335$	1.00118e11	0.00129648
$336$	0	0
$337$	−1.53250e14	−1.92060	−0.960299	−	0.278971i	$-0.910007\pi$
$337$	−1.53250e14	−1.92060	−0.960299	+	0.278971i	$0.910007\pi$
$338$	0	0
$339$	5.64726e13	0.685077
$340$	0	0
$341$	−8.01949e13	−0.941884
$342$	0	0
$343$	−8.48035e13	−0.964486
$344$	0	0
$345$	−2.18208e11	−0.00240362
$346$	0	0
$347$	1.54844e13	0.165228	0.0826138	−	0.996582i	$-0.473673\pi$
$347$	1.54844e13	0.165228	0.0826138	+	0.996582i	$0.473673\pi$
$348$	0	0
$349$	1.23055e14	1.27221	0.636104	−	0.771604i	$-0.280545\pi$
$349$	1.23055e14	1.27221	0.636104	+	0.771604i	$0.280545\pi$
$350$	0	0
$351$	−1.12204e13	−0.112414
$352$	0	0
$353$	−1.12950e14	−1.09680	−0.548399	−	0.836217i	$-0.684762\pi$
$353$	−1.12950e14	−1.09680	−0.548399	+	0.836217i	$0.684762\pi$
$354$	0	0
$355$	−2.29930e11	−0.00216441
$356$	0	0
$357$	−2.04052e13	−0.186237
$358$	0	0
$359$	1.81299e14	1.60463	0.802316	−	0.596900i	$-0.203601\pi$
$359$	1.81299e14	1.60463	0.802316	+	0.596900i	$0.203601\pi$
$360$	0	0
$361$	−1.16085e14	−0.996523
$362$	0	0
$363$	1.56502e14	1.30327
$364$	0	0
$365$	−7.51976e10	−0.000607566 0
$366$	0	0
$367$	1.29565e14	1.01584	0.507918	−	0.861405i	$-0.330415\pi$
$367$	1.29565e14	1.01584	0.507918	+	0.861405i	$0.330415\pi$
$368$	0	0
$369$	4.31286e12	0.0328186
$370$	0	0
$371$	−9.11901e13	−0.673583
$372$	0	0
$373$	1.40379e14	1.00671	0.503355	−	0.864079i	$-0.332099\pi$
$373$	1.40379e14	1.00671	0.503355	+	0.864079i	$0.332099\pi$
$374$	0	0
$375$	−4.05949e11	−0.00282683
$376$	0	0
$377$	1.75521e13	0.118701
$378$	0	0
$379$	−1.32268e14	−0.868836	−0.434418	−	0.900711i	$-0.643046\pi$
$379$	−1.32268e14	−0.868836	−0.434418	+	0.900711i	$0.643046\pi$
$380$	0	0
$381$	2.82690e13	0.180394
$382$	0	0
$383$	−2.35418e14	−1.45964	−0.729821	−	0.683639i	$-0.760396\pi$
$383$	−2.35418e14	−1.45964	−0.729821	+	0.683639i	$0.760396\pi$
$384$	0	0
$385$	−2.09665e11	−0.00126326
$386$	0	0
$387$	−5.61670e12	−0.0328905
$388$	0	0
$389$	−1.16649e14	−0.663985	−0.331993	−	0.943282i	$-0.607721\pi$
$389$	−1.16649e14	−0.663985	−0.331993	+	0.943282i	$0.607721\pi$
$390$	0	0
$391$	9.99781e13	0.553264
$392$	0	0
$393$	1.79610e14	0.966435
$394$	0	0
$395$	2.94658e11	0.00154182
$396$	0	0
$397$	1.21031e14	0.615954	0.307977	−	0.951394i	$-0.400348\pi$
$397$	1.21031e14	0.615954	0.307977	+	0.951394i	$0.400348\pi$
$398$	0	0
$399$	−6.81840e12	−0.0337545
$400$	0	0
$401$	1.55669e14	0.749736	0.374868	−	0.927078i	$-0.377688\pi$
$401$	1.55669e14	0.749736	0.374868	+	0.927078i	$0.377688\pi$
$402$	0	0
$403$	1.46609e13	0.0687039
$404$	0	0
$405$	−3.05921e11	−0.00139510
$406$	0	0
$407$	−3.17961e14	−1.41125
$408$	0	0
$409$	2.14249e14	0.925636	0.462818	−	0.886453i	$-0.346838\pi$
$409$	2.14249e14	0.925636	0.462818	+	0.886453i	$0.346838\pi$
$410$	0	0
$411$	2.92817e14	1.23159
$412$	0	0
$413$	−1.53234e14	−0.627524
$414$	0	0
$415$	3.55008e11	0.00141571
$416$	0	0
$417$	3.56316e14	1.38384
$418$	0	0
$419$	−1.62799e14	−0.615851	−0.307926	−	0.951410i	$-0.599635\pi$
$419$	−1.62799e14	−0.615851	−0.307926	+	0.951410i	$0.599635\pi$
$420$	0	0
$421$	1.59992e14	0.589586	0.294793	−	0.955561i	$-0.404749\pi$
$421$	1.59992e14	0.589586	0.294793	+	0.955561i	$0.404749\pi$
$422$	0	0
$423$	−2.11716e12	−0.00760120
$424$	0	0
$425$	9.29981e13	0.325338
$426$	0	0
$427$	−3.08190e14	−1.05067
$428$	0	0
$429$	−5.02933e13	−0.167107
$430$	0	0
$431$	4.45504e14	1.44287	0.721434	−	0.692483i	$-0.243483\pi$
$431$	4.45504e14	1.44287	0.721434	+	0.692483i	$0.243483\pi$
$432$	0	0
$433$	2.24720e14	0.709509	0.354754	−	0.934960i	$-0.384564\pi$
$433$	2.24720e14	0.709509	0.354754	+	0.934960i	$0.384564\pi$
$434$	0	0
$435$	4.90599e11	0.00151020
$436$	0	0
$437$	3.34077e13	0.100276
$438$	0	0
$439$	−2.79027e14	−0.816755	−0.408377	−	0.912813i	$-0.633905\pi$
$439$	−2.79027e14	−0.816755	−0.408377	+	0.912813i	$0.633905\pi$
$440$	0	0
$441$	5.70797e12	0.0162956
$442$	0	0
$443$	−5.42400e14	−1.51042	−0.755212	−	0.655480i	$-0.772466\pi$
$443$	−5.42400e14	−1.51042	−0.755212	+	0.655480i	$0.772466\pi$
$444$	0	0
$445$	−5.90580e11	−0.00160435
$446$	0	0
$447$	4.79638e13	0.127122
$448$	0	0
$449$	4.38213e12	0.0113326	0.00566631	−	0.999984i	$-0.498196\pi$
$449$	4.38213e12	0.0113326	0.00566631	+	0.999984i	$0.498196\pi$
$450$	0	0
$451$	8.07121e14	2.03689
$452$	0	0
$453$	4.68373e14	1.15359
$454$	0	0
$455$	3.83300e10	9.21459e−5 0
$456$	0	0
$457$	−2.03550e14	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$457$	−2.03550e14	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$458$	0	0
$459$	1.43694e14	0.329207
$460$	0	0
$461$	−8.72976e13	−0.195275	−0.0976376	−	0.995222i	$-0.531129\pi$
$461$	−8.72976e13	−0.195275	−0.0976376	+	0.995222i	$0.531129\pi$
$462$	0	0
$463$	5.09678e14	1.11327	0.556635	−	0.830757i	$-0.312092\pi$
$463$	5.09678e14	1.11327	0.556635	+	0.830757i	$0.312092\pi$
$464$	0	0
$465$	4.09785e11	0.000874107 0
$466$	0	0
$467$	5.65357e14	1.17782	0.588911	−	0.808198i	$-0.299557\pi$
$467$	5.65357e14	1.17782	0.588911	+	0.808198i	$0.299557\pi$
$468$	0	0
$469$	2.58033e14	0.525079
$470$	0	0
$471$	5.09320e14	1.01245
$472$	0	0
$473$	−1.05112e15	−2.04135
$474$	0	0
$475$	3.10754e13	0.0589660
$476$	0	0
$477$	1.53806e13	0.0285182
$478$	0	0
$479$	−7.34756e14	−1.33137	−0.665683	−	0.746234i	$-0.731860\pi$
$479$	−7.34756e14	−1.33137	−0.665683	+	0.746234i	$0.731860\pi$
$480$	0	0
$481$	5.81281e13	0.102941
$482$	0	0
$483$	−5.62387e14	−0.973477
$484$	0	0
$485$	−7.67665e11	−0.00129895
$486$	0	0
$487$	−8.21258e14	−1.35853	−0.679267	−	0.733892i	$-0.737702\pi$
$487$	−8.21258e14	−1.35853	−0.679267	+	0.733892i	$0.737702\pi$
$488$	0	0
$489$	−5.11824e14	−0.827793
$490$	0	0
$491$	4.63353e14	0.732763	0.366381	−	0.930465i	$-0.380597\pi$
$491$	4.63353e14	0.732763	0.366381	+	0.930465i	$0.380597\pi$
$492$	0	0
$493$	−2.24781e14	−0.347618
$494$	0	0
$495$	3.53633e10	5.34840e−5 0
$496$	0	0
$497$	−5.92596e14	−0.876594
$498$	0	0
$499$	−9.03216e14	−1.30689	−0.653444	−	0.756975i	$-0.726677\pi$
$499$	−9.03216e14	−1.30689	−0.653444	+	0.756975i	$0.726677\pi$
$500$	0	0
$501$	5.48858e14	0.776877
$502$	0	0
$503$	6.41196e14	0.887905	0.443953	−	0.896050i	$-0.353576\pi$
$503$	6.41196e14	0.887905	0.443953	+	0.896050i	$0.353576\pi$
$504$	0	0
$505$	6.61729e11	0.000896556 0
$506$	0	0
$507$	−7.35793e14	−0.975465
$508$	0	0
$509$	−1.44008e15	−1.86826	−0.934131	−	0.356931i	$-0.883823\pi$
$509$	−1.44008e15	−1.86826	−0.934131	+	0.356931i	$0.883823\pi$
$510$	0	0
$511$	−1.93806e14	−0.246067
$512$	0	0
$513$	4.80155e13	0.0596672
$514$	0	0
$515$	−2.16018e12	−0.00262755
$516$	0	0
$517$	−3.96212e14	−0.471769
$518$	0	0
$519$	1.97460e14	0.230175
$520$	0	0
$521$	−1.25869e15	−1.43652	−0.718261	−	0.695774i	$-0.755061\pi$
$521$	−1.25869e15	−1.43652	−0.718261	+	0.695774i	$0.755061\pi$
$522$	0	0
$523$	4.26785e13	0.0476925	0.0238463	−	0.999716i	$-0.492409\pi$
$523$	4.26785e13	0.0476925	0.0238463	+	0.999716i	$0.492409\pi$
$524$	0	0
$525$	−5.23124e14	−0.572438
$526$	0	0
$527$	−1.87754e14	−0.201201
$528$	0	0
$529$	1.80268e15	1.89196
$530$	0	0
$531$	2.58453e13	0.0265682
$532$	0	0
$533$	−1.47554e14	−0.148577
$534$	0	0
$535$	−4.83022e11	−0.000476454 0
$536$	0	0
$537$	−1.55410e14	−0.150182
$538$	0	0
$539$	1.06821e15	1.01139
$540$	0	0
$541$	−1.52186e15	−1.41185	−0.705927	−	0.708285i	$-0.749469\pi$
$541$	−1.52186e15	−1.41185	−0.705927	+	0.708285i	$0.749469\pi$
$542$	0	0
$543$	−4.17773e14	−0.379788
$544$	0	0
$545$	9.39196e11	0.000836711 0
$546$	0	0
$547$	1.54822e14	0.135176	0.0675882	−	0.997713i	$-0.478470\pi$
$547$	1.54822e14	0.135176	0.0675882	+	0.997713i	$0.478470\pi$
$548$	0	0
$549$	5.19811e13	0.0444833
$550$	0	0
$551$	−7.51107e13	−0.0630040
$552$	0	0
$553$	7.59420e14	0.624444
$554$	0	0
$555$	1.62474e12	0.00130970
$556$	0	0
$557$	−6.84793e14	−0.541198	−0.270599	−	0.962692i	$-0.587222\pi$
$557$	−6.84793e14	−0.541198	−0.270599	+	0.962692i	$0.587222\pi$
$558$	0	0
$559$	1.92162e14	0.148902
$560$	0	0
$561$	6.44080e14	0.489377
$562$	0	0
$563$	2.97022e14	0.221306	0.110653	−	0.993859i	$-0.464706\pi$
$563$	2.97022e14	0.221306	0.110653	+	0.993859i	$0.464706\pi$
$564$	0	0
$565$	1.35852e12	0.000992658 0
$566$	0	0
$567$	−7.88448e14	−0.565023
$568$	0	0
$569$	−6.33864e14	−0.445532	−0.222766	−	0.974872i	$-0.571509\pi$
$569$	−6.33864e14	−0.445532	−0.222766	+	0.974872i	$0.571509\pi$
$570$	0	0
$571$	2.05989e15	1.42018	0.710092	−	0.704109i	$-0.248653\pi$
$571$	2.05989e15	1.42018	0.710092	+	0.704109i	$0.248653\pi$
$572$	0	0
$573$	1.35567e15	0.916861
$574$	0	0
$575$	2.56312e15	1.70057
$576$	0	0
$577$	1.96534e15	1.27930	0.639648	−	0.768668i	$-0.279080\pi$
$577$	1.96534e15	1.27930	0.639648	+	0.768668i	$0.279080\pi$
$578$	0	0
$579$	−1.56437e15	−0.999095
$580$	0	0
$581$	9.14959e14	0.573367
$582$	0	0
$583$	2.87837e15	1.76999
$584$	0	0
$585$	−6.46495e9	−3.90128e−6 0
$586$	0	0
$587$	−1.74515e15	−1.03353	−0.516766	−	0.856126i	$-0.672864\pi$
$587$	−1.74515e15	−1.03353	−0.516766	+	0.856126i	$0.672864\pi$
$588$	0	0
$589$	−6.27381e13	−0.0364667
$590$	0	0
$591$	−3.00447e15	−1.71410
$592$	0	0
$593$	−2.01638e15	−1.12920	−0.564601	−	0.825364i	$-0.690970\pi$
$593$	−2.01638e15	−1.12920	−0.564601	+	0.825364i	$0.690970\pi$
$594$	0	0
$595$	−4.90873e11	−0.000269852 0
$596$	0	0
$597$	3.24271e15	1.75005
$598$	0	0
$599$	−6.85882e14	−0.363414	−0.181707	−	0.983353i	$-0.558162\pi$
$599$	−6.85882e14	−0.363414	−0.181707	+	0.983353i	$0.558162\pi$
$600$	0	0
$601$	7.52985e14	0.391721	0.195860	−	0.980632i	$-0.437250\pi$
$601$	7.52985e14	0.391721	0.195860	+	0.980632i	$0.437250\pi$
$602$	0	0
$603$	−4.35213e13	−0.0222309
$604$	0	0
$605$	3.76486e12	0.00188840
$606$	0	0
$607$	−9.97425e14	−0.491295	−0.245648	−	0.969359i	$-0.579001\pi$
$607$	−9.97425e14	−0.491295	−0.245648	+	0.969359i	$0.579001\pi$
$608$	0	0
$609$	1.26442e15	0.611639
$610$	0	0
$611$	7.24336e13	0.0344123
$612$	0	0
$613$	2.31569e15	1.08056	0.540279	−	0.841486i	$-0.318319\pi$
$613$	2.31569e15	1.08056	0.540279	+	0.841486i	$0.318319\pi$
$614$	0	0
$615$	−4.12428e12	−0.00189032
$616$	0	0
$617$	−2.30058e15	−1.03578	−0.517891	−	0.855447i	$-0.673283\pi$
$617$	−2.30058e15	−1.03578	−0.517891	+	0.855447i	$0.673283\pi$
$618$	0	0
$619$	2.25568e13	0.00997652	0.00498826	−	0.999988i	$-0.498412\pi$
$619$	2.25568e13	0.00997652	0.00498826	+	0.999988i	$0.498412\pi$
$620$	0	0
$621$	3.96035e15	1.72080
$622$	0	0
$623$	−1.52210e15	−0.649767
$624$	0	0
$625$	2.38417e15	0.999994
$626$	0	0
$627$	2.15220e14	0.0886972
$628$	0	0
$629$	−7.44416e14	−0.301465
$630$	0	0
$631$	1.16010e15	0.461673	0.230836	−	0.972993i	$-0.425854\pi$
$631$	1.16010e15	0.461673	0.230836	+	0.972993i	$0.425854\pi$
$632$	0	0
$633$	−7.17879e14	−0.280757
$634$	0	0
$635$	6.80045e11	0.000261386 0
$636$	0	0
$637$	−1.95284e14	−0.0737736
$638$	0	0
$639$	9.99505e13	0.0371133
$640$	0	0
$641$	−2.92958e15	−1.06927	−0.534633	−	0.845084i	$-0.679550\pi$
$641$	−2.92958e15	−1.06927	−0.534633	+	0.845084i	$0.679550\pi$
$642$	0	0
$643$	4.29003e13	0.0153922	0.00769608	−	0.999970i	$-0.497550\pi$
$643$	4.29003e13	0.0153922	0.00769608	+	0.999970i	$0.497550\pi$
$644$	0	0
$645$	5.37110e12	0.00189446
$646$	0	0
$647$	−3.38184e15	−1.17268	−0.586340	−	0.810065i	$-0.699432\pi$
$647$	−3.38184e15	−1.17268	−0.586340	+	0.810065i	$0.699432\pi$
$648$	0	0
$649$	4.83676e15	1.64896
$650$	0	0
$651$	1.05614e15	0.354017
$652$	0	0
$653$	1.18601e15	0.390899	0.195449	−	0.980714i	$-0.437383\pi$
$653$	1.18601e15	0.390899	0.195449	+	0.980714i	$0.437383\pi$
$654$	0	0
$655$	4.32075e12	0.00140034
$656$	0	0
$657$	3.26884e13	0.0104180
$658$	0	0
$659$	−1.02501e14	−0.0321260	−0.0160630	−	0.999871i	$-0.505113\pi$
$659$	−1.02501e14	−0.0321260	−0.0160630	+	0.999871i	$0.505113\pi$
$660$	0	0
$661$	−8.25683e14	−0.254510	−0.127255	−	0.991870i	$-0.540617\pi$
$661$	−8.25683e14	−0.254510	−0.127255	+	0.991870i	$0.540617\pi$
$662$	0	0
$663$	−1.17748e14	−0.0356967
$664$	0	0
$665$	−1.64025e11	−4.89093e−5 0
$666$	0	0
$667$	−6.19519e15	−1.81703
$668$	0	0
$669$	−3.56780e15	−1.02934
$670$	0	0
$671$	9.72788e15	2.76086
$672$	0	0
$673$	−2.01981e13	−0.00563932	−0.00281966	−	0.999996i	$-0.500898\pi$
$673$	−2.01981e13	−0.00563932	−0.00281966	+	0.999996i	$0.500898\pi$
$674$	0	0
$675$	3.68386e15	1.01189
$676$	0	0
$677$	−6.02569e15	−1.62843	−0.814216	−	0.580563i	$-0.802833\pi$
$677$	−6.02569e15	−1.62843	−0.814216	+	0.580563i	$0.802833\pi$
$678$	0	0
$679$	−1.97850e15	−0.526080
$680$	0	0
$681$	1.54208e15	0.403457
$682$	0	0
$683$	−6.29107e14	−0.161961	−0.0809805	−	0.996716i	$-0.525805\pi$
$683$	−6.29107e14	−0.161961	−0.0809805	+	0.996716i	$0.525805\pi$
$684$	0	0
$685$	7.04409e12	0.00178454
$686$	0	0
$687$	1.37378e15	0.342497
$688$	0	0
$689$	−5.26210e14	−0.129108
$690$	0	0
$691$	4.75485e15	1.14817	0.574087	−	0.818794i	$-0.305357\pi$
$691$	4.75485e15	1.14817	0.574087	+	0.818794i	$0.305357\pi$
$692$	0	0
$693$	9.11414e13	0.0216612
$694$	0	0
$695$	8.57162e12	0.00200515
$696$	0	0
$697$	1.88965e15	0.435112
$698$	0	0
$699$	4.18056e15	0.947567
$700$	0	0
$701$	−4.97599e15	−1.11027	−0.555137	−	0.831759i	$-0.687334\pi$
$701$	−4.97599e15	−1.11027	−0.555137	+	0.831759i	$0.687334\pi$
$702$	0	0
$703$	−2.48747e14	−0.0546391
$704$	0	0
$705$	2.02459e12	0.000437821 0
$706$	0	0
$707$	1.70547e15	0.363109
$708$	0	0
$709$	−7.74296e15	−1.62313	−0.811564	−	0.584263i	$-0.801384\pi$
$709$	−7.74296e15	−1.62313	−0.811564	+	0.584263i	$0.801384\pi$
$710$	0	0
$711$	−1.28088e14	−0.0264378
$712$	0	0
$713$	−5.17468e15	−1.05170
$714$	0	0
$715$	−1.20987e12	−0.000242133 0
$716$	0	0
$717$	4.00028e15	0.788378
$718$	0	0
$719$	−6.74480e15	−1.30906	−0.654531	−	0.756035i	$-0.727134\pi$
$719$	−6.74480e15	−1.30906	−0.654531	+	0.756035i	$0.727134\pi$
$720$	0	0
$721$	−5.56742e15	−1.06417
$722$	0	0
$723$	−7.07114e15	−1.33116
$724$	0	0
$725$	−5.76267e15	−1.06848
$726$	0	0
$727$	−2.28261e15	−0.416862	−0.208431	−	0.978037i	$-0.566836\pi$
$727$	−2.28261e15	−0.416862	−0.208431	+	0.978037i	$0.566836\pi$
$728$	0	0
$729$	5.68870e15	1.02332
$730$	0	0
$731$	−2.46091e15	−0.436065
$732$	0	0
$733$	5.82179e15	1.01621	0.508106	−	0.861294i	$-0.330346\pi$
$733$	5.82179e15	1.01621	0.508106	+	0.861294i	$0.330346\pi$
$734$	0	0
$735$	−5.45839e12	−0.000938607 0
$736$	0	0
$737$	−8.14469e15	−1.37976
$738$	0	0
$739$	−4.69328e15	−0.783307	−0.391654	−	0.920113i	$-0.628097\pi$
$739$	−4.69328e15	−0.783307	−0.391654	+	0.920113i	$0.628097\pi$
$740$	0	0
$741$	−3.93454e13	−0.00646985
$742$	0	0
$743$	−7.35195e15	−1.19114	−0.595572	−	0.803302i	$-0.703074\pi$
$743$	−7.35195e15	−1.19114	−0.595572	+	0.803302i	$0.703074\pi$
$744$	0	0
$745$	1.15383e12	0.000184197 0
$746$	0	0
$747$	−1.54322e14	−0.0242753
$748$	0	0
$749$	−1.24489e15	−0.192966
$750$	0	0
$751$	2.55261e15	0.389911	0.194955	−	0.980812i	$-0.437544\pi$
$751$	2.55261e15	0.389911	0.194955	+	0.980812i	$0.437544\pi$
$752$	0	0
$753$	−8.14686e15	−1.22636
$754$	0	0
$755$	1.12673e13	0.00167152
$756$	0	0
$757$	4.96461e15	0.725868	0.362934	−	0.931815i	$-0.381775\pi$
$757$	4.96461e15	0.725868	0.362934	+	0.931815i	$0.381775\pi$
$758$	0	0
$759$	1.77515e16	2.55802
$760$	0	0
$761$	−4.37753e15	−0.621747	−0.310873	−	0.950451i	$-0.600622\pi$
$761$	−4.37753e15	−0.621747	−0.310873	+	0.950451i	$0.600622\pi$
$762$	0	0
$763$	2.42058e15	0.338871
$764$	0	0
$765$	8.27932e10	1.14250e−5 0
$766$	0	0
$767$	−8.84234e14	−0.120280
$768$	0	0
$769$	−2.08611e15	−0.279732	−0.139866	−	0.990170i	$-0.544667\pi$
$769$	−2.08611e15	−0.279732	−0.139866	+	0.990170i	$0.544667\pi$
$770$	0	0
$771$	−2.81397e15	−0.371981
$772$	0	0
$773$	9.36216e14	0.122008	0.0610041	−	0.998138i	$-0.480570\pi$
$773$	9.36216e14	0.122008	0.0610041	+	0.998138i	$0.480570\pi$
$774$	0	0
$775$	−4.81341e15	−0.618435
$776$	0	0
$777$	4.18742e15	0.530433
$778$	0	0
$779$	6.31427e14	0.0788619
$780$	0	0
$781$	1.87050e16	2.30344
$782$	0	0
$783$	−8.90408e15	−1.08118
$784$	0	0
$785$	1.22523e13	0.00146702
$786$	0	0
$787$	5.38189e15	0.635438	0.317719	−	0.948185i	$-0.397083\pi$
$787$	5.38189e15	0.635438	0.317719	+	0.948185i	$0.397083\pi$
$788$	0	0
$789$	−6.40213e15	−0.745420
$790$	0	0
$791$	3.50130e15	0.402030
$792$	0	0
$793$	−1.77841e15	−0.201386
$794$	0	0
$795$	−1.47081e13	−0.00164262
$796$	0	0
$797$	−2.82315e15	−0.310966	−0.155483	−	0.987839i	$-0.549693\pi$
$797$	−2.82315e15	−0.310966	−0.155483	+	0.987839i	$0.549693\pi$
$798$	0	0
$799$	−9.27619e14	−0.100777
$800$	0	0
$801$	2.56725e14	0.0275099
$802$	0	0
$803$	6.11740e15	0.646593
$804$	0	0
$805$	−1.35289e13	−0.00141054
$806$	0	0
$807$	7.32327e15	0.753183
$808$	0	0
$809$	1.58216e16	1.60521	0.802607	−	0.596508i	$-0.203445\pi$
$809$	1.58216e16	1.60521	0.802607	+	0.596508i	$0.203445\pi$
$810$	0	0
$811$	2.59483e15	0.259714	0.129857	−	0.991533i	$-0.458548\pi$
$811$	2.59483e15	0.259714	0.129857	+	0.991533i	$0.458548\pi$
$812$	0	0
$813$	1.58421e16	1.56428
$814$	0	0
$815$	−1.23126e13	−0.00119945
$816$	0	0
$817$	−8.22315e14	−0.0790346
$818$	0	0
$819$	−1.66620e13	−0.00158004
$820$	0	0
$821$	−3.78317e15	−0.353972	−0.176986	−	0.984213i	$-0.556635\pi$
$821$	−3.78317e15	−0.353972	−0.176986	+	0.984213i	$0.556635\pi$
$822$	0	0
$823$	−1.01417e16	−0.936292	−0.468146	−	0.883651i	$-0.655078\pi$
$823$	−1.01417e16	−0.936292	−0.468146	+	0.883651i	$0.655078\pi$
$824$	0	0
$825$	1.65122e16	1.50420
$826$	0	0
$827$	1.02758e15	0.0923709	0.0461854	−	0.998933i	$-0.485293\pi$
$827$	1.02758e15	0.0923709	0.0461854	+	0.998933i	$0.485293\pi$
$828$	0	0
$829$	−7.33308e15	−0.650484	−0.325242	−	0.945631i	$-0.605446\pi$
$829$	−7.33308e15	−0.650484	−0.325242	+	0.945631i	$0.605446\pi$
$830$	0	0
$831$	5.16969e15	0.452542
$832$	0	0
$833$	2.50091e15	0.216048
$834$	0	0
$835$	1.32035e13	0.00112567
$836$	0	0
$837$	−7.43735e15	−0.625789
$838$	0	0
$839$	8.80240e15	0.730988	0.365494	−	0.930814i	$-0.380900\pi$
$839$	8.80240e15	0.730988	0.365494	+	0.930814i	$0.380900\pi$
$840$	0	0
$841$	1.72817e15	0.141647
$842$	0	0
$843$	1.67077e16	1.35165
$844$	0	0
$845$	−1.77004e13	−0.00141342
$846$	0	0
$847$	9.70315e15	0.764811
$848$	0	0
$849$	2.32692e16	1.81046
$850$	0	0
$851$	−2.05168e16	−1.57579
$852$	0	0
$853$	2.21269e16	1.67765	0.838825	−	0.544401i	$-0.183243\pi$
$853$	2.21269e16	1.67765	0.838825	+	0.544401i	$0.183243\pi$
$854$	0	0
$855$	2.76654e10	2.07073e−6 0
$856$	0	0
$857$	8.13932e15	0.601442	0.300721	−	0.953712i	$-0.402773\pi$
$857$	8.13932e15	0.601442	0.300721	+	0.953712i	$0.402773\pi$
$858$	0	0
$859$	−1.77641e16	−1.29592	−0.647962	−	0.761672i	$-0.724379\pi$
$859$	−1.77641e16	−1.29592	−0.647962	+	0.761672i	$0.724379\pi$
$860$	0	0
$861$	−1.06295e16	−0.765586
$862$	0	0
$863$	−7.10195e15	−0.505031	−0.252516	−	0.967593i	$-0.581258\pi$
$863$	−7.10195e15	−0.505031	−0.252516	+	0.967593i	$0.581258\pi$
$864$	0	0
$865$	4.75014e12	0.000333517 0
$866$	0	0
$867$	−1.27386e16	−0.883116
$868$	0	0
$869$	−2.39707e16	−1.64086
$870$	0	0
$871$	1.48897e15	0.100644
$872$	0	0
$873$	3.33704e14	0.0222732
$874$	0	0
$875$	−2.51688e13	−0.00165889
$876$	0	0
$877$	1.79549e16	1.16865	0.584326	−	0.811519i	$-0.301359\pi$
$877$	1.79549e16	1.16865	0.584326	+	0.811519i	$0.301359\pi$
$878$	0	0
$879$	−2.65581e16	−1.70710
$880$	0	0
$881$	−7.81664e15	−0.496196	−0.248098	−	0.968735i	$-0.579805\pi$
$881$	−7.81664e15	−0.496196	−0.248098	+	0.968735i	$0.579805\pi$
$882$	0	0
$883$	5.72282e15	0.358778	0.179389	−	0.983778i	$-0.442588\pi$
$883$	5.72282e15	0.358778	0.179389	+	0.983778i	$0.442588\pi$
$884$	0	0
$885$	−2.47152e13	−0.00153030
$886$	0	0
$887$	−1.15649e16	−0.707232	−0.353616	−	0.935391i	$-0.615048\pi$
$887$	−1.15649e16	−0.707232	−0.353616	+	0.935391i	$0.615048\pi$
$888$	0	0
$889$	1.75268e15	0.105862
$890$	0	0
$891$	2.48870e16	1.48472
$892$	0	0
$893$	−3.09964e14	−0.0182654
$894$	0	0
$895$	−3.73857e12	−0.000217610 0
$896$	0	0
$897$	−3.24524e15	−0.186590
$898$	0	0
$899$	1.16343e16	0.660786
$900$	0	0
$901$	6.73890e15	0.378097
$902$	0	0
$903$	1.38429e16	0.767263
$904$	0	0
$905$	−1.00500e13	−0.000550302 0
$906$	0	0
$907$	−2.00209e16	−1.08304	−0.541520	−	0.840688i	$-0.682151\pi$
$907$	−2.00209e16	−1.08304	−0.541520	+	0.840688i	$0.682151\pi$
$908$	0	0
$909$	−2.87653e14	−0.0153733
$910$	0	0
$911$	1.08866e16	0.574833	0.287417	−	0.957806i	$-0.407204\pi$
$911$	1.08866e16	0.574833	0.287417	+	0.957806i	$0.407204\pi$
$912$	0	0
$913$	−2.88802e16	−1.50665
$914$	0	0
$915$	−4.97081e13	−0.00256219
$916$	0	0
$917$	1.11358e16	0.567142
$918$	0	0
$919$	−1.86543e16	−0.938735	−0.469367	−	0.883003i	$-0.655518\pi$
$919$	−1.86543e16	−0.938735	−0.469367	+	0.883003i	$0.655518\pi$
$920$	0	0
$921$	−2.75749e16	−1.37115
$922$	0	0
$923$	−3.41956e15	−0.168020
$924$	0	0
$925$	−1.90845e16	−0.926617
$926$	0	0
$927$	9.39031e14	0.0450548
$928$	0	0
$929$	1.40081e16	0.664191	0.332096	−	0.943246i	$-0.392244\pi$
$929$	1.40081e16	0.664191	0.332096	+	0.943246i	$0.392244\pi$
$930$	0	0
$931$	8.35679e14	0.0391576
$932$	0	0
$933$	7.98859e15	0.369931
$934$	0	0
$935$	1.54942e13	0.000709094 0
$936$	0	0
$937$	−3.91751e16	−1.77191	−0.885956	−	0.463769i	$-0.846497\pi$
$937$	−3.91751e16	−1.77191	−0.885956	+	0.463769i	$0.846497\pi$
$938$	0	0
$939$	−6.12201e15	−0.273674
$940$	0	0
$941$	1.62834e16	0.719454	0.359727	−	0.933058i	$-0.382870\pi$
$941$	1.62834e16	0.719454	0.359727	+	0.933058i	$0.382870\pi$
$942$	0	0
$943$	5.20806e16	2.27437
$944$	0	0
$945$	−1.94445e13	−0.000839311 0
$946$	0	0
$947$	2.90606e16	1.23988	0.619940	−	0.784649i	$-0.287157\pi$
$947$	2.90606e16	1.23988	0.619940	+	0.784649i	$0.287157\pi$
$948$	0	0
$949$	−1.11835e15	−0.0471645
$950$	0	0
$951$	−1.94132e16	−0.809290
$952$	0	0
$953$	−2.06101e16	−0.849316	−0.424658	−	0.905354i	$-0.639606\pi$
$953$	−2.06101e16	−0.849316	−0.424658	+	0.905354i	$0.639606\pi$
$954$	0	0
$955$	3.26123e13	0.00132851
$956$	0	0
$957$	−3.99107e16	−1.60721
$958$	0	0
$959$	1.81547e16	0.722747
$960$	0	0
$961$	−1.56907e16	−0.617537
$962$	0	0
$963$	2.09969e14	0.00816980
$964$	0	0
$965$	−3.76328e13	−0.00144766
$966$	0	0
$967$	1.34802e16	0.512683	0.256342	−	0.966586i	$-0.417483\pi$
$967$	1.34802e16	0.512683	0.256342	+	0.966586i	$0.417483\pi$
$968$	0	0
$969$	5.03876e14	0.0189471
$970$	0	0
$971$	−4.23511e16	−1.57456	−0.787279	−	0.616597i	$-0.788511\pi$
$971$	−4.23511e16	−1.57456	−0.787279	+	0.616597i	$0.788511\pi$
$972$	0	0
$973$	2.20916e16	0.812093
$974$	0	0
$975$	−3.01867e15	−0.109721
$976$	0	0
$977$	7.56455e15	0.271871	0.135936	−	0.990718i	$-0.456596\pi$
$977$	7.56455e15	0.271871	0.135936	+	0.990718i	$0.456596\pi$
$978$	0	0
$979$	4.80442e16	1.70740
$980$	0	0
$981$	−4.08268e14	−0.0143472
$982$	0	0
$983$	1.58800e16	0.551830	0.275915	−	0.961182i	$-0.411019\pi$
$983$	1.58800e16	0.551830	0.275915	+	0.961182i	$0.411019\pi$
$984$	0	0
$985$	−7.22764e13	−0.00248369
$986$	0	0
$987$	5.21795e15	0.177319
$988$	0	0
$989$	−6.78252e16	−2.27935
$990$	0	0
$991$	−5.65504e16	−1.87945	−0.939724	−	0.341933i	$-0.888918\pi$
$991$	−5.65504e16	−1.87945	−0.939724	+	0.341933i	$0.888918\pi$
$992$	0	0
$993$	−7.82661e15	−0.257248
$994$	0	0
$995$	7.80075e13	0.00253577
$996$	0	0
$997$	−1.84756e16	−0.593986	−0.296993	−	0.954880i	$-0.595984\pi$
$997$	−1.84756e16	−0.593986	−0.296993	+	0.954880i	$0.595984\pi$
$998$	0	0
$999$	−2.94880e16	−0.937637

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.12.a.j.1.2		2
4.3	odd	2	inner	64.12.a.j.1.1		2
8.3	odd	2		32.12.a.b.1.2	yes	2
8.5	even	2		32.12.a.b.1.1	✓	2
16.3	odd	4		256.12.b.j.129.1		4
16.5	even	4		256.12.b.j.129.2		4
16.11	odd	4		256.12.b.j.129.4		4
16.13	even	4		256.12.b.j.129.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.12.a.b.1.1	✓	2	8.5	even	2
32.12.a.b.1.2	yes	2	8.3	odd	2
64.12.a.j.1.1		2	4.3	odd	2	inner
64.12.a.j.1.2		2	1.1	even	1	trivial
256.12.b.j.129.1		4	16.3	odd	4
256.12.b.j.129.2		4	16.5	even	4
256.12.b.j.129.3		4	16.13	even	4
256.12.b.j.129.4		4	16.11	odd	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)

\(f(q)\)	\(=\)	\(q+415.692 q^{3} +10.0000 q^{5} +25772.9 q^{7} -4347.00 q^{9} +O(q^{10})\)	\(q+415.692 q^{3} +10.0000 q^{5} +25772.9 q^{7} -4347.00 q^{9} -813510. q^{11} +148722. q^{13} +4156.92 q^{15} -1.90461e6 q^{17} -636425. q^{19} +1.07136e7 q^{21} -5.24928e7 q^{23} -4.88280e7 q^{25} -7.54456e7 q^{27} +1.18020e8 q^{29} +9.85789e7 q^{31} -3.38170e8 q^{33} +257729. q^{35} +3.90851e8 q^{37} +6.18226e7 q^{39} -9.92147e8 q^{41} +1.29209e9 q^{43} -43470.0 q^{45} +4.87040e8 q^{47} -1.31308e9 q^{49} -7.91730e8 q^{51} -3.53821e9 q^{53} -8.13510e6 q^{55} -2.64557e8 q^{57} -5.94555e9 q^{59} -1.19579e10 q^{61} -1.12035e8 q^{63} +1.48722e6 q^{65} +1.00118e10 q^{67} -2.18208e10 q^{69} -2.29930e10 q^{71} -7.51976e9 q^{73} -2.02974e10 q^{75} -2.09665e10 q^{77} +2.94658e10 q^{79} -3.05921e10 q^{81} +3.55008e10 q^{83} -1.90461e7 q^{85} +4.90599e10 q^{87} -5.90580e10 q^{89} +3.83300e9 q^{91} +4.09785e10 q^{93} -6.36425e6 q^{95} -7.67665e10 q^{97} +3.53633e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{5} - 8694 q^{9}+O(q^{10}) \) 2 * q + 20 * q^5 - 8694 * q^9	\( 2 q + 20 q^{5} - 8694 q^{9} + 297444 q^{13} - 3809212 q^{17} + 21427200 q^{21} - 97656050 q^{25} + 236039620 q^{29} - 676339200 q^{33} + 781701268 q^{37} - 1984294220 q^{41} - 86940 q^{45} - 2626167086 q^{49} - 7076426444 q^{53} - 529113600 q^{57} - 23915839420 q^{61} + 2974440 q^{65} - 43641676800 q^{69} - 15039524076 q^{73} - 41933030400 q^{77} - 61184210382 q^{81} - 38092120 q^{85} - 118115907980 q^{89} + 81956966400 q^{93} - 153533015388 q^{97}+O(q^{100}) \) 2 * q + 20 * q^5 - 8694 * q^9 + 297444 * q^13 - 3809212 * q^17 + 21427200 * q^21 - 97656050 * q^25 + 236039620 * q^29 - 676339200 * q^33 + 781701268 * q^37 - 1984294220 * q^41 - 86940 * q^45 - 2626167086 * q^49 - 7076426444 * q^53 - 529113600 * q^57 - 23915839420 * q^61 + 2974440 * q^65 - 43641676800 * q^69 - 15039524076 * q^73 - 41933030400 * q^77 - 61184210382 * q^81 - 38092120 * q^85 - 118115907980 * q^89 + 81956966400 * q^93 - 153533015388 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more