LMFDB - Embedded newform 64.24.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,24,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, 24, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 24 $
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:	$214.530583901$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 389656213x^{4} + 47522643058672215x^{2} - 1756479932541937262634975 $ x^6 - 389656213x^4 + 47522643058672215x^2 - 1756479932541937262634975
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{71}\cdot 3^{6}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$11884.9$ 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)

$f(q)$	$=$	$q+216966. q^{3} -1.90927e8 q^{5} -9.74240e9 q^{7} -4.70690e10 q^{9} +O(q^{10})$	$q+216966. q^{3} -1.90927e8 q^{5} -9.74240e9 q^{7} -4.70690e10 q^{9} +9.58624e11 q^{11} +2.14517e12 q^{13} -4.14246e13 q^{15} +2.65620e13 q^{17} +7.63669e14 q^{19} -2.11377e15 q^{21} -3.75905e15 q^{23} +2.45321e16 q^{25} -3.06382e16 q^{27} -5.01247e16 q^{29} +1.90008e17 q^{31} +2.07989e17 q^{33} +1.86008e18 q^{35} +1.56774e18 q^{37} +4.65429e17 q^{39} -3.05851e18 q^{41} +3.07525e18 q^{43} +8.98674e18 q^{45} -2.72401e18 q^{47} +6.75455e19 q^{49} +5.76305e18 q^{51} +2.52631e19 q^{53} -1.83027e20 q^{55} +1.65690e20 q^{57} -2.29901e20 q^{59} +6.13479e20 q^{61} +4.58565e20 q^{63} -4.09571e20 q^{65} -1.20452e21 q^{67} -8.15585e20 q^{69} +2.43377e20 q^{71} -3.45063e20 q^{73} +5.32263e21 q^{75} -9.33929e21 q^{77} -3.08236e21 q^{79} -2.21622e21 q^{81} -1.36342e22 q^{83} -5.07140e21 q^{85} -1.08753e22 q^{87} -1.67114e22 q^{89} -2.08991e22 q^{91} +4.12252e22 q^{93} -1.45805e23 q^{95} +7.71584e22 q^{97} -4.51215e22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 163121700 q^{5} + 14120212542 q^{9}+O(q^{10}) $ 6 * q - 163121700 * q^5 + 14120212542 * q^9	$ 6 q - 163121700 q^{5} + 14120212542 q^{9} - 1101295489524 q^{13} + 56435517243468 q^{17} - 14\!\cdots\!84 q^{21}+ \cdots + 25\!\cdots\!16 q^{97}+O(q^{100}) $ 6 * q - 163121700 * q^5 + 14120212542 * q^9 - 1101295489524 * q^13 + 56435517243468 * q^17 - 1447661324841984 * q^21 + 17512304669291850 * q^25 - 87161858953770708 * q^29 + 48462097564970496 * q^33 + 1542766250573484636 * q^37 - 865941451897399716 * q^41 + 9286254082406210700 * q^45 + 52551978665245784886 * q^49 + 10735070383373044668 * q^53 + 155104978099636154880 * q^57 + 432169207960503656364 * q^61 - 316526040451899107400 * q^65 - 695880182828346153984 * q^69 + 1758226166115151758204 * q^73 - 11926085657398731979776 * q^77 + 4010608843245537630678 * q^81 - 20251531460297137542600 * q^85 + 20087131519604301542940 * q^89 - 67310470168488243892224 * q^93 + 254665147475934678043116 * 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
$3$	216966.	0.707126	0.353563	−	0.935411i	$-0.384970\pi$
$3$	216966.	0.707126	0.353563	+	0.935411i	$0.384970\pi$
$4$	0	0
$5$	−1.90927e8	−1.74869	−0.874343	−	0.485308i	$-0.838707\pi$
$5$	−1.90927e8	−1.74869	−0.874343	+	0.485308i	$0.838707\pi$
$6$	0	0
$7$	−9.74240e9	−1.86225	−0.931126	−	0.364698i	$-0.881172\pi$
$7$	−9.74240e9	−1.86225	−0.931126	+	0.364698i	$0.881172\pi$
$8$	0	0
$9$	−4.70690e10	−0.499973
$10$	0	0
$11$	9.58624e11	1.01305	0.506527	−	0.862224i	$-0.330929\pi$
$11$	9.58624e11	1.01305	0.506527	+	0.862224i	$0.330929\pi$
$12$	0	0
$13$	2.14517e12	0.331981	0.165991	−	0.986127i	$-0.446918\pi$
$13$	2.14517e12	0.331981	0.165991	+	0.986127i	$0.446918\pi$
$14$	0	0
$15$	−4.14246e13	−1.23654
$16$	0	0
$17$	2.65620e13	0.187974	0.0939872	−	0.995573i	$-0.470039\pi$
$17$	2.65620e13	0.187974	0.0939872	+	0.995573i	$0.470039\pi$
$18$	0	0
$19$	7.63669e14	1.50397	0.751985	−	0.659181i	$-0.229097\pi$
$19$	7.63669e14	1.50397	0.751985	+	0.659181i	$0.229097\pi$
$20$	0	0
$21$	−2.11377e15	−1.31685
$22$	0	0
$23$	−3.75905e15	−0.822636	−0.411318	−	0.911492i	$-0.634931\pi$
$23$	−3.75905e15	−0.822636	−0.411318	+	0.911492i	$0.634931\pi$
$24$	0	0
$25$	2.45321e16	2.05790
$26$	0	0
$27$	−3.06382e16	−1.06067
$28$	0	0
$29$	−5.01247e16	−0.762912	−0.381456	−	0.924387i	$-0.624577\pi$
$29$	−5.01247e16	−0.762912	−0.381456	+	0.924387i	$0.624577\pi$
$30$	0	0
$31$	1.90008e17	1.34311	0.671556	−	0.740954i	$-0.265626\pi$
$31$	1.90008e17	1.34311	0.671556	+	0.740954i	$0.265626\pi$
$32$	0	0
$33$	2.07989e17	0.716357
$34$	0	0
$35$	1.86008e18	3.25649
$36$	0	0
$37$	1.56774e18	1.44862	0.724311	−	0.689474i	$-0.242158\pi$
$37$	1.56774e18	1.44862	0.724311	+	0.689474i	$0.242158\pi$
$38$	0	0
$39$	4.65429e17	0.234753
$40$	0	0
$41$	−3.05851e18	−0.867951	−0.433975	−	0.900925i	$-0.642890\pi$
$41$	−3.05851e18	−0.867951	−0.433975	+	0.900925i	$0.642890\pi$
$42$	0	0
$43$	3.07525e18	0.504653	0.252327	−	0.967642i	$-0.418804\pi$
$43$	3.07525e18	0.504653	0.252327	+	0.967642i	$0.418804\pi$
$44$	0	0
$45$	8.98674e18	0.874295
$46$	0	0
$47$	−2.72401e18	−0.160725	−0.0803626	−	0.996766i	$-0.525608\pi$
$47$	−2.72401e18	−0.160725	−0.0803626	+	0.996766i	$0.525608\pi$
$48$	0	0
$49$	6.75455e19	2.46798
$50$	0	0
$51$	5.76305e18	0.132922
$52$	0	0
$53$	2.52631e19	0.374381	0.187191	−	0.982324i	$-0.440062\pi$
$53$	2.52631e19	0.374381	0.187191	+	0.982324i	$0.440062\pi$
$54$	0	0
$55$	−1.83027e20	−1.77151
$56$	0	0
$57$	1.65690e20	1.06350
$58$	0	0
$59$	−2.29901e20	−0.992527	−0.496263	−	0.868172i	$-0.665295\pi$
$59$	−2.29901e20	−0.992527	−0.496263	+	0.868172i	$0.665295\pi$
$60$	0	0
$61$	6.13479e20	1.80512	0.902561	−	0.430562i	$-0.141685\pi$
$61$	6.13479e20	1.80512	0.902561	+	0.430562i	$0.141685\pi$
$62$	0	0
$63$	4.58565e20	0.931075
$64$	0	0
$65$	−4.09571e20	−0.580531
$66$	0	0
$67$	−1.20452e21	−1.20491	−0.602456	−	0.798152i	$-0.705811\pi$
$67$	−1.20452e21	−1.20491	−0.602456	+	0.798152i	$0.705811\pi$
$68$	0	0
$69$	−8.15585e20	−0.581707
$70$	0	0
$71$	2.43377e20	0.124971	0.0624854	−	0.998046i	$-0.480097\pi$
$71$	2.43377e20	0.124971	0.0624854	+	0.998046i	$0.480097\pi$
$72$	0	0
$73$	−3.45063e20	−0.128732	−0.0643659	−	0.997926i	$-0.520502\pi$
$73$	−3.45063e20	−0.128732	−0.0643659	+	0.997926i	$0.520502\pi$
$74$	0	0
$75$	5.32263e21	1.45520
$76$	0	0
$77$	−9.33929e21	−1.88656
$78$	0	0
$79$	−3.08236e21	−0.463630	−0.231815	−	0.972760i	$-0.574466\pi$
$79$	−3.08236e21	−0.463630	−0.231815	+	0.972760i	$0.574466\pi$
$80$	0	0
$81$	−2.21622e21	−0.250055
$82$	0	0
$83$	−1.36342e22	−1.16207	−0.581036	−	0.813878i	$-0.697352\pi$
$83$	−1.36342e22	−1.16207	−0.581036	+	0.813878i	$0.697352\pi$
$84$	0	0
$85$	−5.07140e21	−0.328708
$86$	0	0
$87$	−1.08753e22	−0.539475
$88$	0	0
$89$	−1.67114e22	−0.638303	−0.319152	−	0.947704i	$-0.603398\pi$
$89$	−1.67114e22	−0.638303	−0.319152	+	0.947704i	$0.603398\pi$
$90$	0	0
$91$	−2.08991e22	−0.618232
$92$	0	0
$93$	4.12252e22	0.949749
$94$	0	0
$95$	−1.45805e23	−2.62997
$96$	0	0
$97$	7.71584e22	1.09524	0.547619	−	0.836728i	$-0.315534\pi$
$97$	7.71584e22	1.09524	0.547619	+	0.836728i	$0.315534\pi$
$98$	0	0
$99$	−4.51215e22	−0.506499
$100$	0	0
$101$	−1.31460e23	−1.17246	−0.586231	−	0.810144i	$-0.699389\pi$
$101$	−1.31460e23	−1.17246	−0.586231	+	0.810144i	$0.699389\pi$
$102$	0	0
$103$	2.43701e23	1.73472	0.867361	−	0.497680i	$-0.165815\pi$
$103$	2.43701e23	1.73472	0.867361	+	0.497680i	$0.165815\pi$
$104$	0	0
$105$	4.03575e23	2.30275
$106$	0	0
$107$	−1.04639e23	−0.480598	−0.240299	−	0.970699i	$-0.577246\pi$
$107$	−1.04639e23	−0.480598	−0.240299	+	0.970699i	$0.577246\pi$
$108$	0	0
$109$	−5.91477e22	−0.219550	−0.109775	−	0.993956i	$-0.535013\pi$
$109$	−5.91477e22	−0.219550	−0.109775	+	0.993956i	$0.535013\pi$
$110$	0	0
$111$	3.40147e23	1.02436
$112$	0	0
$113$	1.55264e23	0.380774	0.190387	−	0.981709i	$-0.439026\pi$
$113$	1.55264e23	0.380774	0.190387	+	0.981709i	$0.439026\pi$
$114$	0	0
$115$	7.17703e23	1.43853
$116$	0	0
$117$	−1.00971e23	−0.165981
$118$	0	0
$119$	−2.58778e23	−0.350056
$120$	0	0
$121$	2.35294e22	0.0262772
$122$	0	0
$123$	−6.63591e23	−0.613751
$124$	0	0
$125$	−2.40781e24	−1.84994
$126$	0	0
$127$	−2.80213e24	−1.79368	−0.896841	−	0.442353i	$-0.854144\pi$
$127$	−2.80213e24	−1.79368	−0.896841	+	0.442353i	$0.854144\pi$
$128$	0	0
$129$	6.67224e23	0.356854
$130$	0	0
$131$	4.42927e23	0.198478	0.0992390	−	0.995064i	$-0.468359\pi$
$131$	4.42927e23	0.198478	0.0992390	+	0.995064i	$0.468359\pi$
$132$	0	0
$133$	−7.43997e24	−2.80077
$134$	0	0
$135$	5.84966e24	1.85478
$136$	0	0
$137$	5.28208e24	1.41423	0.707113	−	0.707101i	$-0.249997\pi$
$137$	5.28208e24	1.41423	0.707113	+	0.707101i	$0.249997\pi$
$138$	0	0
$139$	1.03441e24	0.234435	0.117218	−	0.993106i	$-0.462603\pi$
$139$	1.03441e24	0.234435	0.117218	+	0.993106i	$0.462603\pi$
$140$	0	0
$141$	−5.91018e23	−0.113653
$142$	0	0
$143$	2.05641e24	0.336315
$144$	0	0
$145$	9.57014e24	1.33409
$146$	0	0
$147$	1.46551e25	1.74517
$148$	0	0
$149$	9.38525e23	0.0956762	0.0478381	−	0.998855i	$-0.484767\pi$
$149$	9.38525e23	0.0956762	0.0478381	+	0.998855i	$0.484767\pi$
$150$	0	0
$151$	−6.67720e23	−0.0583928	−0.0291964	−	0.999574i	$-0.509295\pi$
$151$	−6.67720e23	−0.0583928	−0.0291964	+	0.999574i	$0.509295\pi$
$152$	0	0
$153$	−1.25025e24	−0.0939820
$154$	0	0
$155$	−3.62776e25	−2.34868
$156$	0	0
$157$	1.43848e25	0.803635	0.401818	−	0.915720i	$-0.368379\pi$
$157$	1.43848e25	0.803635	0.401818	+	0.915720i	$0.368379\pi$
$158$	0	0
$159$	5.48123e24	0.264735
$160$	0	0
$161$	3.66221e25	1.53195
$162$	0	0
$163$	2.39784e25	0.870289	0.435145	−	0.900361i	$-0.356697\pi$
$163$	2.39784e25	0.870289	0.435145	+	0.900361i	$0.356697\pi$
$164$	0	0
$165$	−3.97106e25	−1.25268
$166$	0	0
$167$	1.51560e25	0.416242	0.208121	−	0.978103i	$-0.433265\pi$
$167$	1.51560e25	0.416242	0.208121	+	0.978103i	$0.433265\pi$
$168$	0	0
$169$	−3.71521e25	−0.889788
$170$	0	0
$171$	−3.59452e25	−0.751943
$172$	0	0
$173$	−5.92919e25	−1.08509	−0.542545	−	0.840027i	$-0.682539\pi$
$173$	−5.92919e25	−1.08509	−0.542545	+	0.840027i	$0.682539\pi$
$174$	0	0
$175$	−2.39002e26	−3.83233
$176$	0	0
$177$	−4.98806e25	−0.701842
$178$	0	0
$179$	−6.32487e24	−0.0782063	−0.0391031	−	0.999235i	$-0.512450\pi$
$179$	−6.32487e24	−0.0782063	−0.0391031	+	0.999235i	$0.512450\pi$
$180$	0	0
$181$	2.91917e25	0.317655	0.158828	−	0.987306i	$-0.449229\pi$
$181$	2.91917e25	0.317655	0.158828	+	0.987306i	$0.449229\pi$
$182$	0	0
$183$	1.33104e26	1.27645
$184$	0	0
$185$	−2.99324e26	−2.53318
$186$	0	0
$187$	2.54630e25	0.190428
$188$	0	0
$189$	2.98490e26	1.97523
$190$	0	0
$191$	3.38328e26	1.98360	0.991800	−	0.127796i	$-0.0407904\pi$
$191$	3.38328e26	1.98360	0.991800	+	0.127796i	$0.0407904\pi$
$192$	0	0
$193$	−1.27890e26	−0.665163	−0.332581	−	0.943075i	$-0.607920\pi$
$193$	−1.27890e26	−0.665163	−0.332581	+	0.943075i	$0.607920\pi$
$194$	0	0
$195$	−8.88629e25	−0.410509
$196$	0	0
$197$	−3.48404e26	−1.43127	−0.715635	−	0.698474i	$-0.753863\pi$
$197$	−3.48404e26	−1.43127	−0.715635	+	0.698474i	$0.753863\pi$
$198$	0	0
$199$	−1.41137e26	−0.516214	−0.258107	−	0.966116i	$-0.583099\pi$
$199$	−1.41137e26	−0.516214	−0.258107	+	0.966116i	$0.583099\pi$
$200$	0	0
$201$	−2.61341e26	−0.852025
$202$	0	0
$203$	4.88334e26	1.42073
$204$	0	0
$205$	5.83951e26	1.51777
$206$	0	0
$207$	1.76935e26	0.411295
$208$	0	0
$209$	7.32072e26	1.52360
$210$	0	0
$211$	−4.52456e26	−0.843972	−0.421986	−	0.906602i	$-0.638667\pi$
$211$	−4.52456e26	−0.843972	−0.421986	+	0.906602i	$0.638667\pi$
$212$	0	0
$213$	5.28045e25	0.0883701
$214$	0	0
$215$	−5.87148e26	−0.882480
$216$	0	0
$217$	−1.85113e27	−2.50121
$218$	0	0
$219$	−7.48669e25	−0.0910296
$220$	0	0
$221$	5.69801e25	0.0624040
$222$	0	0
$223$	9.10896e26	0.899421	0.449711	−	0.893174i	$-0.351527\pi$
$223$	9.10896e26	0.899421	0.449711	+	0.893174i	$0.351527\pi$
$224$	0	0
$225$	−1.15470e27	−1.02890
$226$	0	0
$227$	4.31344e26	0.347158	0.173579	−	0.984820i	$-0.444467\pi$
$227$	4.31344e26	0.347158	0.173579	+	0.984820i	$0.444467\pi$
$228$	0	0
$229$	−1.66792e27	−1.21357	−0.606787	−	0.794864i	$-0.707542\pi$
$229$	−1.66792e27	−1.21357	−0.606787	+	0.794864i	$0.707542\pi$
$230$	0	0
$231$	−2.02631e27	−1.33404
$232$	0	0
$233$	−5.01267e26	−0.298866	−0.149433	−	0.988772i	$-0.547745\pi$
$233$	−5.01267e26	−0.298866	−0.149433	+	0.988772i	$0.547745\pi$
$234$	0	0
$235$	5.20088e26	0.281058
$236$	0	0
$237$	−6.68766e26	−0.327845
$238$	0	0
$239$	2.25295e27	1.00271	0.501355	−	0.865241i	$-0.332835\pi$
$239$	2.25295e27	1.00271	0.501355	+	0.865241i	$0.332835\pi$
$240$	0	0
$241$	−6.48722e26	−0.262339	−0.131169	−	0.991360i	$-0.541873\pi$
$241$	−6.48722e26	−0.262339	−0.131169	+	0.991360i	$0.541873\pi$
$242$	0	0
$243$	2.40354e27	0.883850
$244$	0	0
$245$	−1.28963e28	−4.31572
$246$	0	0
$247$	1.63820e27	0.499290
$248$	0	0
$249$	−2.95817e27	−0.821732
$250$	0	0
$251$	−7.85626e26	−0.199053	−0.0995264	−	0.995035i	$-0.531733\pi$
$251$	−7.85626e26	−0.199053	−0.0995264	+	0.995035i	$0.531733\pi$
$252$	0	0
$253$	−3.60351e27	−0.833374
$254$	0	0
$255$	−1.10032e27	−0.232438
$256$	0	0
$257$	6.31330e27	1.21906	0.609531	−	0.792762i	$-0.291358\pi$
$257$	6.31330e27	1.21906	0.609531	+	0.792762i	$0.291358\pi$
$258$	0	0
$259$	−1.52736e28	−2.69770
$260$	0	0
$261$	2.35932e27	0.381435
$262$	0	0
$263$	−2.63348e27	−0.389977	−0.194989	−	0.980806i	$-0.562467\pi$
$263$	−2.63348e27	−0.389977	−0.194989	+	0.980806i	$0.562467\pi$
$264$	0	0
$265$	−4.82340e27	−0.654675
$266$	0	0
$267$	−3.62580e27	−0.451361
$268$	0	0
$269$	−8.67658e27	−0.991282	−0.495641	−	0.868527i	$-0.665067\pi$
$269$	−8.67658e27	−0.991282	−0.495641	+	0.868527i	$0.665067\pi$
$270$	0	0
$271$	−1.35097e28	−1.41742	−0.708711	−	0.705499i	$-0.750723\pi$
$271$	−1.35097e28	−1.41742	−0.708711	+	0.705499i	$0.750723\pi$
$272$	0	0
$273$	−4.53439e27	−0.437168
$274$	0	0
$275$	2.35171e28	2.08477
$276$	0	0
$277$	2.14474e27	0.174927	0.0874635	−	0.996168i	$-0.472124\pi$
$277$	2.14474e27	0.174927	0.0874635	+	0.996168i	$0.472124\pi$
$278$	0	0
$279$	−8.94348e27	−0.671519
$280$	0	0
$281$	1.26495e28	0.874883	0.437441	−	0.899247i	$-0.355885\pi$
$281$	1.26495e28	0.874883	0.437441	+	0.899247i	$0.355885\pi$
$282$	0	0
$283$	−3.33583e27	−0.212647	−0.106324	−	0.994332i	$-0.533908\pi$
$283$	−3.33583e27	−0.212647	−0.106324	+	0.994332i	$0.533908\pi$
$284$	0	0
$285$	−3.16347e28	−1.85972
$286$	0	0
$287$	2.97972e28	1.61634
$288$	0	0
$289$	−1.92620e28	−0.964666
$290$	0	0
$291$	1.67407e28	0.774472
$292$	0	0
$293$	−2.39590e28	−1.02445	−0.512225	−	0.858851i	$-0.671179\pi$
$293$	−2.39590e28	−1.02445	−0.512225	+	0.858851i	$0.671179\pi$
$294$	0	0
$295$	4.38942e28	1.73562
$296$	0	0
$297$	−2.93705e28	−1.07452
$298$	0	0
$299$	−8.06380e27	−0.273100
$300$	0	0
$301$	−2.99603e28	−0.939792
$302$	0	0
$303$	−2.85224e28	−0.829078
$304$	0	0
$305$	−1.17130e29	−3.15659
$306$	0	0
$307$	4.06329e28	1.01575	0.507874	−	0.861431i	$-0.330431\pi$
$307$	4.06329e28	1.01575	0.507874	+	0.861431i	$0.330431\pi$
$308$	0	0
$309$	5.28749e28	1.22667
$310$	0	0
$311$	2.40957e27	0.0519033	0.0259517	−	0.999663i	$-0.491738\pi$
$311$	2.40957e27	0.0519033	0.0259517	+	0.999663i	$0.491738\pi$
$312$	0	0
$313$	−3.08964e28	−0.618226	−0.309113	−	0.951025i	$-0.600032\pi$
$313$	−3.08964e28	−0.618226	−0.309113	+	0.951025i	$0.600032\pi$
$314$	0	0
$315$	−8.75523e28	−1.62816
$316$	0	0
$317$	4.47714e28	0.774140	0.387070	−	0.922050i	$-0.373487\pi$
$317$	4.47714e28	0.774140	0.387070	+	0.922050i	$0.373487\pi$
$318$	0	0
$319$	−4.80507e28	−0.772871
$320$	0	0
$321$	−2.27032e28	−0.339843
$322$	0	0
$323$	2.02846e28	0.282708
$324$	0	0
$325$	5.26256e28	0.683185
$326$	0	0
$327$	−1.28330e28	−0.155249
$328$	0	0
$329$	2.65384e28	0.299311
$330$	0	0
$331$	−5.57485e28	−0.586423	−0.293212	−	0.956048i	$-0.594724\pi$
$331$	−5.57485e28	−0.586423	−0.293212	+	0.956048i	$0.594724\pi$
$332$	0	0
$333$	−7.37921e28	−0.724271
$334$	0	0
$335$	2.29976e29	2.10701
$336$	0	0
$337$	6.21767e28	0.531966	0.265983	−	0.963978i	$-0.414303\pi$
$337$	6.21767e28	0.531966	0.265983	+	0.963978i	$0.414303\pi$
$338$	0	0
$339$	3.36869e28	0.269255
$340$	0	0
$341$	1.82146e29	1.36064
$342$	0	0
$343$	−3.91418e29	−2.73375
$344$	0	0
$345$	1.55717e29	1.01722
$346$	0	0
$347$	1.40504e29	0.858817	0.429409	−	0.903110i	$-0.358722\pi$
$347$	1.40504e29	0.858817	0.429409	+	0.903110i	$0.358722\pi$
$348$	0	0
$349$	9.62640e28	0.550772	0.275386	−	0.961334i	$-0.411194\pi$
$349$	9.62640e28	0.550772	0.275386	+	0.961334i	$0.411194\pi$
$350$	0	0
$351$	−6.57242e28	−0.352122
$352$	0	0
$353$	−2.84480e29	−1.42772	−0.713860	−	0.700288i	$-0.753055\pi$
$353$	−2.84480e29	−1.42772	−0.713860	+	0.700288i	$0.753055\pi$
$354$	0	0
$355$	−4.64672e28	−0.218535
$356$	0	0
$357$	−5.61459e28	−0.247533
$358$	0	0
$359$	1.00468e29	0.415376	0.207688	−	0.978195i	$-0.433406\pi$
$359$	1.00468e29	0.415376	0.207688	+	0.978195i	$0.433406\pi$
$360$	0	0
$361$	3.25361e29	1.26192
$362$	0	0
$363$	5.10508e27	0.0185813
$364$	0	0
$365$	6.58818e28	0.225112
$366$	0	0
$367$	5.45272e29	1.74966	0.874829	−	0.484431i	$-0.160973\pi$
$367$	5.45272e29	1.74966	0.874829	+	0.484431i	$0.160973\pi$
$368$	0	0
$369$	1.43961e29	0.433952
$370$	0	0
$371$	−2.46123e29	−0.697192
$372$	0	0
$373$	−1.93840e29	−0.516170	−0.258085	−	0.966122i	$-0.583091\pi$
$373$	−1.93840e29	−0.516170	−0.258085	+	0.966122i	$0.583091\pi$
$374$	0	0
$375$	−5.22413e29	−1.30814
$376$	0	0
$377$	−1.07526e29	−0.253272
$378$	0	0
$379$	−2.74346e29	−0.608063	−0.304031	−	0.952662i	$-0.598333\pi$
$379$	−2.74346e29	−0.608063	−0.304031	+	0.952662i	$0.598333\pi$
$380$	0	0
$381$	−6.07967e29	−1.26836
$382$	0	0
$383$	4.18842e29	0.822744	0.411372	−	0.911468i	$-0.365050\pi$
$383$	4.18842e29	0.822744	0.411372	+	0.911468i	$0.365050\pi$
$384$	0	0
$385$	1.78312e30	3.29900
$386$	0	0
$387$	−1.44749e29	−0.252313
$388$	0	0
$389$	−4.88162e29	−0.801944	−0.400972	−	0.916090i	$-0.631328\pi$
$389$	−4.88162e29	−0.801944	−0.400972	+	0.916090i	$0.631328\pi$
$390$	0	0
$391$	−9.98479e28	−0.154634
$392$	0	0
$393$	9.61000e28	0.140349
$394$	0	0
$395$	5.88505e29	0.810743
$396$	0	0
$397$	1.29652e30	1.68535	0.842673	−	0.538425i	$-0.180981\pi$
$397$	1.29652e30	1.68535	0.842673	+	0.538425i	$0.180981\pi$
$398$	0	0
$399$	−1.61422e30	−1.98050
$400$	0	0
$401$	1.12391e30	1.30188	0.650939	−	0.759130i	$-0.274375\pi$
$401$	1.12391e30	1.30188	0.650939	+	0.759130i	$0.274375\pi$
$402$	0	0
$403$	4.07599e29	0.445888
$404$	0	0
$405$	4.23136e29	0.437267
$406$	0	0
$407$	1.50288e30	1.46753
$408$	0	0
$409$	2.82215e29	0.260472	0.130236	−	0.991483i	$-0.458426\pi$
$409$	2.82215e29	0.260472	0.130236	+	0.991483i	$0.458426\pi$
$410$	0	0
$411$	1.14603e30	1.00004
$412$	0	0
$413$	2.23978e30	1.84833
$414$	0	0
$415$	2.60314e30	2.03210
$416$	0	0
$417$	2.24432e29	0.165775
$418$	0	0
$419$	2.72724e30	1.90661	0.953305	−	0.302009i	$-0.0976572\pi$
$419$	2.72724e30	1.90661	0.953305	+	0.302009i	$0.0976572\pi$
$420$	0	0
$421$	1.64535e30	1.08897	0.544484	−	0.838771i	$-0.316726\pi$
$421$	1.64535e30	1.08897	0.544484	+	0.838771i	$0.316726\pi$
$422$	0	0
$423$	1.28217e29	0.0803582
$424$	0	0
$425$	6.51623e29	0.386833
$426$	0	0
$427$	−5.97675e30	−3.36159
$428$	0	0
$429$	4.46171e29	0.237817
$430$	0	0
$431$	−2.50630e30	−1.26632	−0.633162	−	0.774020i	$-0.718243\pi$
$431$	−2.50630e30	−1.26632	−0.633162	+	0.774020i	$0.718243\pi$
$432$	0	0
$433$	2.04414e30	0.979262	0.489631	−	0.871930i	$-0.337131\pi$
$433$	2.04414e30	0.979262	0.489631	+	0.871930i	$0.337131\pi$
$434$	0	0
$435$	2.07639e30	0.943373
$436$	0	0
$437$	−2.87067e30	−1.23722
$438$	0	0
$439$	−4.46273e30	−1.82498	−0.912492	−	0.409095i	$-0.865844\pi$
$439$	−4.46273e30	−1.82498	−0.912492	+	0.409095i	$0.865844\pi$
$440$	0	0
$441$	−3.17930e30	−1.23392
$442$	0	0
$443$	−1.66470e30	−0.613330	−0.306665	−	0.951817i	$-0.599213\pi$
$443$	−1.66470e30	−0.613330	−0.306665	+	0.951817i	$0.599213\pi$
$444$	0	0
$445$	3.19065e30	1.11619
$446$	0	0
$447$	2.03628e29	0.0676551
$448$	0	0
$449$	−3.50605e30	−1.10659	−0.553293	−	0.832986i	$-0.686629\pi$
$449$	−3.50605e30	−1.10659	−0.553293	+	0.832986i	$0.686629\pi$
$450$	0	0
$451$	−2.93196e30	−0.879281
$452$	0	0
$453$	−1.44872e29	−0.0412911
$454$	0	0
$455$	3.99020e30	1.08109
$456$	0	0
$457$	−5.37923e30	−1.38575	−0.692873	−	0.721059i	$-0.743655\pi$
$457$	−5.37923e30	−1.38575	−0.692873	+	0.721059i	$0.743655\pi$
$458$	0	0
$459$	−8.13813e29	−0.199379
$460$	0	0
$461$	−2.22741e30	−0.519086	−0.259543	−	0.965731i	$-0.583572\pi$
$461$	−2.22741e30	−0.519086	−0.259543	+	0.965731i	$0.583572\pi$
$462$	0	0
$463$	−2.60053e29	−0.0576608	−0.0288304	−	0.999584i	$-0.509178\pi$
$463$	−2.60053e29	−0.0576608	−0.0288304	+	0.999584i	$0.509178\pi$
$464$	0	0
$465$	−7.87100e30	−1.66081
$466$	0	0
$467$	−8.02144e30	−1.61105	−0.805524	−	0.592563i	$-0.798116\pi$
$467$	−8.02144e30	−1.61105	−0.805524	+	0.592563i	$0.798116\pi$
$468$	0	0
$469$	1.17350e31	2.24385
$470$	0	0
$471$	3.12102e30	0.568272
$472$	0	0
$473$	2.94801e30	0.511241
$474$	0	0
$475$	1.87344e31	3.09502
$476$	0	0
$477$	−1.18911e30	−0.187180
$478$	0	0
$479$	−9.99745e30	−1.49979	−0.749896	−	0.661556i	$-0.769896\pi$
$479$	−9.99745e30	−1.49979	−0.749896	+	0.661556i	$0.769896\pi$
$480$	0	0
$481$	3.36308e30	0.480915
$482$	0	0
$483$	7.94575e30	1.08329
$484$	0	0
$485$	−1.47316e31	−1.91523
$486$	0	0
$487$	5.15003e30	0.638597	0.319298	−	0.947654i	$-0.396553\pi$
$487$	5.15003e30	0.638597	0.319298	+	0.947654i	$0.396553\pi$
$488$	0	0
$489$	5.20250e30	0.615404
$490$	0	0
$491$	−6.37446e30	−0.719460	−0.359730	−	0.933056i	$-0.617131\pi$
$491$	−6.37446e30	−0.719460	−0.359730	+	0.933056i	$0.617131\pi$
$492$	0	0
$493$	−1.33141e30	−0.143408
$494$	0	0
$495$	8.61490e30	0.885708
$496$	0	0
$497$	−2.37108e30	−0.232727
$498$	0	0
$499$	9.09248e30	0.852170	0.426085	−	0.904683i	$-0.359892\pi$
$499$	9.09248e30	0.852170	0.426085	+	0.904683i	$0.359892\pi$
$500$	0	0
$501$	3.28834e30	0.294335
$502$	0	0
$503$	−2.41376e30	−0.206377	−0.103189	−	0.994662i	$-0.532905\pi$
$503$	−2.41376e30	−0.206377	−0.103189	+	0.994662i	$0.532905\pi$
$504$	0	0
$505$	2.50993e31	2.05027
$506$	0	0
$507$	−8.06075e30	−0.629193
$508$	0	0
$509$	−2.40043e31	−1.79075	−0.895376	−	0.445312i	$-0.853093\pi$
$509$	−2.40043e31	−1.79075	−0.895376	+	0.445312i	$0.853093\pi$
$510$	0	0
$511$	3.36174e30	0.239731
$512$	0	0
$513$	−2.33975e31	−1.59522
$514$	0	0
$515$	−4.65291e31	−3.03348
$516$	0	0
$517$	−2.61131e30	−0.162823
$518$	0	0
$519$	−1.28643e31	−0.767295
$520$	0	0
$521$	−3.16451e31	−1.80581	−0.902907	−	0.429835i	$-0.858572\pi$
$521$	−3.16451e31	−1.80581	−0.902907	+	0.429835i	$0.858572\pi$
$522$	0	0
$523$	2.09227e31	1.14248	0.571239	−	0.820784i	$-0.306463\pi$
$523$	2.09227e31	1.14248	0.571239	+	0.820784i	$0.306463\pi$
$524$	0	0
$525$	−5.18552e31	−2.70994
$526$	0	0
$527$	5.04699e30	0.252471
$528$	0	0
$529$	−6.75004e30	−0.323270
$530$	0	0
$531$	1.08212e31	0.496236
$532$	0	0
$533$	−6.56102e30	−0.288143
$534$	0	0
$535$	1.99785e31	0.840415
$536$	0	0
$537$	−1.37228e30	−0.0553017
$538$	0	0
$539$	6.47508e31	2.50020
$540$	0	0
$541$	−1.40632e31	−0.520375	−0.260187	−	0.965558i	$-0.583784\pi$
$541$	−1.40632e31	−0.520375	−0.260187	+	0.965558i	$0.583784\pi$
$542$	0	0
$543$	6.33360e30	0.224622
$544$	0	0
$545$	1.12929e31	0.383924
$546$	0	0
$547$	−2.95880e31	−0.964410	−0.482205	−	0.876058i	$-0.660164\pi$
$547$	−2.95880e31	−0.964410	−0.482205	+	0.876058i	$0.660164\pi$
$548$	0	0
$549$	−2.88758e31	−0.902511
$550$	0	0
$551$	−3.82787e31	−1.14740
$552$	0	0
$553$	3.00295e31	0.863395
$554$	0	0
$555$	−6.49431e31	−1.79128
$556$	0	0
$557$	7.87490e30	0.208406	0.104203	−	0.994556i	$-0.466771\pi$
$557$	7.87490e30	0.208406	0.104203	+	0.994556i	$0.466771\pi$
$558$	0	0
$559$	6.59694e30	0.167535
$560$	0	0
$561$	5.52460e30	0.134657
$562$	0	0
$563$	4.45515e31	1.04235	0.521177	−	0.853448i	$-0.325493\pi$
$563$	4.45515e31	1.04235	0.521177	+	0.853448i	$0.325493\pi$
$564$	0	0
$565$	−2.96440e31	−0.665854
$566$	0	0
$567$	2.15913e31	0.465665
$568$	0	0
$569$	−5.59955e31	−1.15974	−0.579872	−	0.814708i	$-0.696897\pi$
$569$	−5.59955e31	−1.15974	−0.579872	+	0.814708i	$0.696897\pi$
$570$	0	0
$571$	−9.51176e29	−0.0189211	−0.00946053	−	0.999955i	$-0.503011\pi$
$571$	−9.51176e29	−0.0189211	−0.00946053	+	0.999955i	$0.503011\pi$
$572$	0	0
$573$	7.34056e31	1.40266
$574$	0	0
$575$	−9.22174e31	−1.69290
$576$	0	0
$577$	−1.28399e31	−0.226485	−0.113243	−	0.993567i	$-0.536124\pi$
$577$	−1.28399e31	−0.226485	−0.113243	+	0.993567i	$0.536124\pi$
$578$	0	0
$579$	−2.77478e31	−0.470354
$580$	0	0
$581$	1.32830e32	2.16407
$582$	0	0
$583$	2.42178e31	0.379268
$584$	0	0
$585$	1.92781e31	0.290250
$586$	0	0
$587$	−7.40632e31	−1.07217	−0.536086	−	0.844163i	$-0.680098\pi$
$587$	−7.40632e31	−1.07217	−0.536086	+	0.844163i	$0.680098\pi$
$588$	0	0
$589$	1.45103e32	2.02000
$590$	0	0
$591$	−7.55918e31	−1.01209
$592$	0	0
$593$	−8.21493e31	−1.05797	−0.528987	−	0.848630i	$-0.677428\pi$
$593$	−8.21493e31	−1.05797	−0.528987	+	0.848630i	$0.677428\pi$
$594$	0	0
$595$	4.94076e31	0.612137
$596$	0	0
$597$	−3.06218e31	−0.365028
$598$	0	0
$599$	1.31000e32	1.50267	0.751333	−	0.659924i	$-0.229411\pi$
$599$	1.31000e32	1.50267	0.751333	+	0.659924i	$0.229411\pi$
$600$	0	0
$601$	1.05699e31	0.116685	0.0583424	−	0.998297i	$-0.481418\pi$
$601$	1.05699e31	0.116685	0.0583424	+	0.998297i	$0.481418\pi$
$602$	0	0
$603$	5.66958e31	0.602423
$604$	0	0
$605$	−4.49239e30	−0.0459506
$606$	0	0
$607$	−1.29237e31	−0.127267	−0.0636337	−	0.997973i	$-0.520269\pi$
$607$	−1.29237e31	−0.127267	−0.0636337	+	0.997973i	$0.520269\pi$
$608$	0	0
$609$	1.05952e32	1.00464
$610$	0	0
$611$	−5.84348e30	−0.0533578
$612$	0	0
$613$	−8.92595e31	−0.784981	−0.392490	−	0.919756i	$-0.628386\pi$
$613$	−8.92595e31	−0.784981	−0.392490	+	0.919756i	$0.628386\pi$
$614$	0	0
$615$	1.26697e32	1.07326
$616$	0	0
$617$	−1.18347e32	−0.965777	−0.482889	−	0.875682i	$-0.660412\pi$
$617$	−1.18347e32	−0.965777	−0.482889	+	0.875682i	$0.660412\pi$
$618$	0	0
$619$	−2.35060e32	−1.84815	−0.924073	−	0.382217i	$-0.875161\pi$
$619$	−2.35060e32	−1.84815	−0.924073	+	0.382217i	$0.875161\pi$
$620$	0	0
$621$	1.15170e32	0.872545
$622$	0	0
$623$	1.62809e32	1.18868
$624$	0	0
$625$	1.67271e32	1.17706
$626$	0	0
$627$	1.58835e32	1.07738
$628$	0	0
$629$	4.16424e31	0.272304
$630$	0	0
$631$	−1.77460e32	−1.11883	−0.559415	−	0.828888i	$-0.688974\pi$
$631$	−1.77460e32	−1.11883	−0.559415	+	0.828888i	$0.688974\pi$
$632$	0	0
$633$	−9.81674e31	−0.596795
$634$	0	0
$635$	5.35002e32	3.13659
$636$	0	0
$637$	1.44897e32	0.819323
$638$	0	0
$639$	−1.14555e31	−0.0624820
$640$	0	0
$641$	1.82879e32	0.962270	0.481135	−	0.876646i	$-0.340225\pi$
$641$	1.82879e32	0.962270	0.481135	+	0.876646i	$0.340225\pi$
$642$	0	0
$643$	1.51276e32	0.767971	0.383986	−	0.923339i	$-0.374551\pi$
$643$	1.51276e32	0.767971	0.383986	+	0.923339i	$0.374551\pi$
$644$	0	0
$645$	−1.27391e32	−0.624025
$646$	0	0
$647$	−1.45858e32	−0.689495	−0.344747	−	0.938695i	$-0.612035\pi$
$647$	−1.45858e32	−0.689495	−0.344747	+	0.938695i	$0.612035\pi$
$648$	0	0
$649$	−2.20388e32	−1.00548
$650$	0	0
$651$	−4.01632e32	−1.76867
$652$	0	0
$653$	2.02823e32	0.862216	0.431108	−	0.902300i	$-0.358123\pi$
$653$	2.02823e32	0.862216	0.431108	+	0.902300i	$0.358123\pi$
$654$	0	0
$655$	−8.45666e31	−0.347076
$656$	0	0
$657$	1.62418e31	0.0643624
$658$	0	0
$659$	−3.26709e32	−1.25020	−0.625099	−	0.780545i	$-0.714942\pi$
$659$	−3.26709e32	−1.25020	−0.625099	+	0.780545i	$0.714942\pi$
$660$	0	0
$661$	5.26703e31	0.194648	0.0973238	−	0.995253i	$-0.468972\pi$
$661$	5.26703e31	0.194648	0.0973238	+	0.995253i	$0.468972\pi$
$662$	0	0
$663$	1.23627e31	0.0441275
$664$	0	0
$665$	1.42049e33	4.89767
$666$	0	0
$667$	1.88421e32	0.627599
$668$	0	0
$669$	1.97633e32	0.636004
$670$	0	0
$671$	5.88095e32	1.82868
$672$	0	0
$673$	2.50043e32	0.751349	0.375675	−	0.926752i	$-0.377411\pi$
$673$	2.50043e32	0.751349	0.375675	+	0.926752i	$0.377411\pi$
$674$	0	0
$675$	−7.51620e32	−2.18276
$676$	0	0
$677$	1.88330e31	0.0528627	0.0264313	−	0.999651i	$-0.491586\pi$
$677$	1.88330e31	0.0528627	0.0264313	+	0.999651i	$0.491586\pi$
$678$	0	0
$679$	−7.51708e32	−2.03961
$680$	0	0
$681$	9.35870e31	0.245484
$682$	0	0
$683$	−1.97248e32	−0.500236	−0.250118	−	0.968215i	$-0.580469\pi$
$683$	−1.97248e32	−0.500236	−0.250118	+	0.968215i	$0.580469\pi$
$684$	0	0
$685$	−1.00849e33	−2.47304
$686$	0	0
$687$	−3.61881e32	−0.858150
$688$	0	0
$689$	5.41937e31	0.124288
$690$	0	0
$691$	5.10523e32	1.13245	0.566223	−	0.824252i	$-0.308404\pi$
$691$	5.10523e32	1.13245	0.566223	+	0.824252i	$0.308404\pi$
$692$	0	0
$693$	4.39591e32	0.943228
$694$	0	0
$695$	−1.97497e32	−0.409954
$696$	0	0
$697$	−8.12402e31	−0.163153
$698$	0	0
$699$	−1.08758e32	−0.211336
$700$	0	0
$701$	−5.10061e32	−0.959103	−0.479551	−	0.877514i	$-0.659201\pi$
$701$	−5.10061e32	−0.959103	−0.479551	+	0.877514i	$0.659201\pi$
$702$	0	0
$703$	1.19724e33	2.17868
$704$	0	0
$705$	1.12841e32	0.198743
$706$	0	0
$707$	1.28074e33	2.18342
$708$	0	0
$709$	2.56920e32	0.424000	0.212000	−	0.977270i	$-0.432002\pi$
$709$	2.56920e32	0.424000	0.212000	+	0.977270i	$0.432002\pi$
$710$	0	0
$711$	1.45083e32	0.231802
$712$	0	0
$713$	−7.14248e32	−1.10489
$714$	0	0
$715$	−3.92624e32	−0.588109
$716$	0	0
$717$	4.88813e32	0.709043
$718$	0	0
$719$	6.23870e32	0.876419	0.438209	−	0.898873i	$-0.355613\pi$
$719$	6.23870e32	0.876419	0.438209	+	0.898873i	$0.355613\pi$
$720$	0	0
$721$	−2.37423e33	−3.23049
$722$	0	0
$723$	−1.40750e32	−0.185507
$724$	0	0
$725$	−1.22966e33	−1.57000
$726$	0	0
$727$	−1.55385e32	−0.192205	−0.0961024	−	0.995371i	$-0.530638\pi$
$727$	−1.55385e32	−0.192205	−0.0961024	+	0.995371i	$0.530638\pi$
$728$	0	0
$729$	7.30127e32	0.875048
$730$	0	0
$731$	8.16849e31	0.0948619
$732$	0	0
$733$	−6.32529e32	−0.711844	−0.355922	−	0.934516i	$-0.615833\pi$
$733$	−6.32529e32	−0.711844	−0.355922	+	0.934516i	$0.615833\pi$
$734$	0	0
$735$	−2.79805e33	−3.05176
$736$	0	0
$737$	−1.15469e33	−1.22064
$738$	0	0
$739$	−1.11960e33	−1.14723	−0.573615	−	0.819125i	$-0.694459\pi$
$739$	−1.11960e33	−1.14723	−0.573615	+	0.819125i	$0.694459\pi$
$740$	0	0
$741$	3.55434e32	0.353061
$742$	0	0
$743$	9.97216e32	0.960325	0.480162	−	0.877180i	$-0.340578\pi$
$743$	9.97216e32	0.960325	0.480162	+	0.877180i	$0.340578\pi$
$744$	0	0
$745$	−1.79190e32	−0.167308
$746$	0	0
$747$	6.41750e32	0.581004
$748$	0	0
$749$	1.01944e33	0.894994
$750$	0	0
$751$	4.68302e32	0.398719	0.199359	−	0.979926i	$-0.436114\pi$
$751$	4.68302e32	0.398719	0.199359	+	0.979926i	$0.436114\pi$
$752$	0	0
$753$	−1.70454e32	−0.140755
$754$	0	0
$755$	1.27486e32	0.102111
$756$	0	0
$757$	−1.06644e33	−0.828576	−0.414288	−	0.910146i	$-0.635969\pi$
$757$	−1.06644e33	−0.828576	−0.414288	+	0.910146i	$0.635969\pi$
$758$	0	0
$759$	−7.81839e32	−0.589301
$760$	0	0
$761$	1.10705e33	0.809550	0.404775	−	0.914416i	$-0.367350\pi$
$761$	1.10705e33	0.809550	0.404775	+	0.914416i	$0.367350\pi$
$762$	0	0
$763$	5.76240e32	0.408857
$764$	0	0
$765$	2.38706e32	0.164345
$766$	0	0
$767$	−4.93177e32	−0.329500
$768$	0	0
$769$	−9.85970e32	−0.639309	−0.319655	−	0.947534i	$-0.603567\pi$
$769$	−9.85970e32	−0.639309	−0.319655	+	0.947534i	$0.603567\pi$
$770$	0	0
$771$	1.36977e33	0.862030
$772$	0	0
$773$	2.87456e33	1.75593	0.877965	−	0.478725i	$-0.158901\pi$
$773$	2.87456e33	1.75593	0.877965	+	0.478725i	$0.158901\pi$
$774$	0	0
$775$	4.66129e33	2.76399
$776$	0	0
$777$	−3.31384e33	−1.90761
$778$	0	0
$779$	−2.33569e33	−1.30537
$780$	0	0
$781$	2.33307e32	0.126602
$782$	0	0
$783$	1.53573e33	0.809198
$784$	0	0
$785$	−2.74645e33	−1.40531
$786$	0	0
$787$	7.82813e32	0.388999	0.194500	−	0.980903i	$-0.437692\pi$
$787$	7.82813e32	0.388999	0.194500	+	0.980903i	$0.437692\pi$
$788$	0	0
$789$	−5.71375e32	−0.275763
$790$	0	0
$791$	−1.51264e33	−0.709097
$792$	0	0
$793$	1.31602e33	0.599266
$794$	0	0
$795$	−1.04651e33	−0.462938
$796$	0	0
$797$	1.46880e33	0.631235	0.315617	−	0.948887i	$-0.397788\pi$
$797$	1.46880e33	0.631235	0.315617	+	0.948887i	$0.397788\pi$
$798$	0	0
$799$	−7.23554e31	−0.0302122
$800$	0	0
$801$	7.86588e32	0.319134
$802$	0	0
$803$	−3.30786e32	−0.130412
$804$	0	0
$805$	−6.99214e33	−2.67891
$806$	0	0
$807$	−1.88252e33	−0.700962
$808$	0	0
$809$	−3.87756e33	−1.40330	−0.701650	−	0.712522i	$-0.747553\pi$
$809$	−3.87756e33	−1.40330	−0.701650	+	0.712522i	$0.747553\pi$
$810$	0	0
$811$	3.25547e33	1.14518	0.572591	−	0.819841i	$-0.305938\pi$
$811$	3.25547e33	1.14518	0.572591	+	0.819841i	$0.305938\pi$
$812$	0	0
$813$	−2.93115e33	−1.00230
$814$	0	0
$815$	−4.57813e33	−1.52186
$816$	0	0
$817$	2.34847e33	0.758983
$818$	0	0
$819$	9.83700e32	0.309099
$820$	0	0
$821$	−3.90085e33	−1.19183	−0.595914	−	0.803049i	$-0.703210\pi$
$821$	−3.90085e33	−1.19183	−0.595914	+	0.803049i	$0.703210\pi$
$822$	0	0
$823$	−1.74515e33	−0.518482	−0.259241	−	0.965813i	$-0.583472\pi$
$823$	−1.74515e33	−0.518482	−0.259241	+	0.965813i	$0.583472\pi$
$824$	0	0
$825$	5.10240e33	1.47419
$826$	0	0
$827$	−4.60301e33	−1.29339	−0.646694	−	0.762749i	$-0.723849\pi$
$827$	−4.60301e33	−1.29339	−0.646694	+	0.762749i	$0.723849\pi$
$828$	0	0
$829$	−7.40688e31	−0.0202422	−0.0101211	−	0.999949i	$-0.503222\pi$
$829$	−7.40688e31	−0.0202422	−0.0101211	+	0.999949i	$0.503222\pi$
$830$	0	0
$831$	4.65335e32	0.123695
$832$	0	0
$833$	1.79415e33	0.463917
$834$	0	0
$835$	−2.89369e33	−0.727876
$836$	0	0
$837$	−5.82150e33	−1.42460
$838$	0	0
$839$	−2.97493e33	−0.708294	−0.354147	−	0.935190i	$-0.615229\pi$
$839$	−2.97493e33	−0.708294	−0.354147	+	0.935190i	$0.615229\pi$
$840$	0	0
$841$	−1.80424e33	−0.417965
$842$	0	0
$843$	2.74450e33	0.618652
$844$	0	0
$845$	7.09334e33	1.55596
$846$	0	0
$847$	−2.29233e32	−0.0489348
$848$	0	0
$849$	−7.23761e32	−0.150368
$850$	0	0
$851$	−5.89322e33	−1.19169
$852$	0	0
$853$	5.32152e32	0.104742	0.0523711	−	0.998628i	$-0.483322\pi$
$853$	5.32152e32	0.104742	0.0523711	+	0.998628i	$0.483322\pi$
$854$	0	0
$855$	6.86290e33	1.31491
$856$	0	0
$857$	6.39595e33	1.19296	0.596480	−	0.802628i	$-0.296566\pi$
$857$	6.39595e33	1.19296	0.596480	+	0.802628i	$0.296566\pi$
$858$	0	0
$859$	−5.22319e32	−0.0948450	−0.0474225	−	0.998875i	$-0.515101\pi$
$859$	−5.22319e32	−0.0948450	−0.0474225	+	0.998875i	$0.515101\pi$
$860$	0	0
$861$	6.46497e33	1.14296
$862$	0	0
$863$	7.61664e33	1.31111	0.655555	−	0.755147i	$-0.272435\pi$
$863$	7.61664e33	1.31111	0.655555	+	0.755147i	$0.272435\pi$
$864$	0	0
$865$	1.13204e34	1.89748
$866$	0	0
$867$	−4.17920e33	−0.682140
$868$	0	0
$869$	−2.95482e33	−0.469682
$870$	0	0
$871$	−2.58391e33	−0.400008
$872$	0	0
$873$	−3.63177e33	−0.547589
$874$	0	0
$875$	2.34579e34	3.44506
$876$	0	0
$877$	5.20154e33	0.744109	0.372055	−	0.928211i	$-0.378653\pi$
$877$	5.20154e33	0.744109	0.372055	+	0.928211i	$0.378653\pi$
$878$	0	0
$879$	−5.19828e33	−0.724415
$880$	0	0
$881$	−2.52401e33	−0.342664	−0.171332	−	0.985213i	$-0.554807\pi$
$881$	−2.52401e33	−0.342664	−0.171332	+	0.985213i	$0.554807\pi$
$882$	0	0
$883$	5.17998e33	0.685141	0.342570	−	0.939492i	$-0.388702\pi$
$883$	5.17998e33	0.685141	0.342570	+	0.939492i	$0.388702\pi$
$884$	0	0
$885$	9.52355e33	1.22730
$886$	0	0
$887$	1.35171e34	1.69731	0.848654	−	0.528948i	$-0.177413\pi$
$887$	1.35171e34	1.69731	0.848654	+	0.528948i	$0.177413\pi$
$888$	0	0
$889$	2.72995e34	3.34029
$890$	0	0
$891$	−2.12452e33	−0.253319
$892$	0	0
$893$	−2.08025e33	−0.241726
$894$	0	0
$895$	1.20759e33	0.136758
$896$	0	0
$897$	−1.74957e33	−0.193116
$898$	0	0
$899$	−9.52408e33	−1.02468
$900$	0	0
$901$	6.71039e32	0.0703741
$902$	0	0
$903$	−6.50036e33	−0.664551
$904$	0	0
$905$	−5.57348e33	−0.555479
$906$	0	0
$907$	−4.97097e32	−0.0483011	−0.0241505	−	0.999708i	$-0.507688\pi$
$907$	−4.97097e32	−0.0483011	−0.0241505	+	0.999708i	$0.507688\pi$
$908$	0	0
$909$	6.18770e33	0.586199
$910$	0	0
$911$	8.11942e33	0.750004	0.375002	−	0.927024i	$-0.377642\pi$
$911$	8.11942e33	0.750004	0.375002	+	0.927024i	$0.377642\pi$
$912$	0	0
$913$	−1.30701e34	−1.17724
$914$	0	0
$915$	−2.54131e34	−2.23211
$916$	0	0
$917$	−4.31517e33	−0.369616
$918$	0	0
$919$	−1.62024e34	−1.35348	−0.676741	−	0.736221i	$-0.736608\pi$
$919$	−1.62024e34	−1.35348	−0.676741	+	0.736221i	$0.736608\pi$
$920$	0	0
$921$	8.81595e33	0.718262
$922$	0	0
$923$	5.22086e32	0.0414880
$924$	0	0
$925$	3.84600e34	2.98112
$926$	0	0
$927$	−1.14708e34	−0.867313
$928$	0	0
$929$	−1.55887e34	−1.14982	−0.574910	−	0.818216i	$-0.694963\pi$
$929$	−1.55887e34	−1.14982	−0.574910	+	0.818216i	$0.694963\pi$
$930$	0	0
$931$	5.15825e34	3.71177
$932$	0	0
$933$	5.22795e32	0.0367022
$934$	0	0
$935$	−4.86157e33	−0.332999
$936$	0	0
$937$	−4.49867e33	−0.300662	−0.150331	−	0.988636i	$-0.548034\pi$
$937$	−4.49867e33	−0.300662	−0.150331	+	0.988636i	$0.548034\pi$
$938$	0	0
$939$	−6.70345e33	−0.437164
$940$	0	0
$941$	−5.17928e32	−0.0329601	−0.0164801	−	0.999864i	$-0.505246\pi$
$941$	−5.17928e32	−0.0329601	−0.0164801	+	0.999864i	$0.505246\pi$
$942$	0	0
$943$	1.14971e34	0.714007
$944$	0	0
$945$	−5.69897e34	−3.45406
$946$	0	0
$947$	4.30688e33	0.254764	0.127382	−	0.991854i	$-0.459343\pi$
$947$	4.30688e33	0.254764	0.127382	+	0.991854i	$0.459343\pi$
$948$	0	0
$949$	−7.40220e32	−0.0427365
$950$	0	0
$951$	9.71387e33	0.547414
$952$	0	0
$953$	2.71314e34	1.49246	0.746231	−	0.665687i	$-0.231862\pi$
$953$	2.71314e34	1.49246	0.746231	+	0.665687i	$0.231862\pi$
$954$	0	0
$955$	−6.45959e34	−3.46870
$956$	0	0
$957$	−1.04254e34	−0.546517
$958$	0	0
$959$	−5.14601e34	−2.63364
$960$	0	0
$961$	1.60897e34	0.803948
$962$	0	0
$963$	4.92528e33	0.240286
$964$	0	0
$965$	2.44176e34	1.16316
$966$	0	0
$967$	4.19669e34	1.95210	0.976050	−	0.217547i	$-0.0698056\pi$
$967$	4.19669e34	1.95210	0.976050	+	0.217547i	$0.0698056\pi$
$968$	0	0
$969$	4.40107e33	0.199910
$970$	0	0
$971$	2.05790e34	0.912857	0.456428	−	0.889760i	$-0.349129\pi$
$971$	2.05790e34	0.912857	0.456428	+	0.889760i	$0.349129\pi$
$972$	0	0
$973$	−1.00776e34	−0.436577
$974$	0	0
$975$	1.14180e34	0.483098
$976$	0	0
$977$	1.04532e34	0.431979	0.215989	−	0.976396i	$-0.430702\pi$
$977$	1.04532e34	0.431979	0.215989	+	0.976396i	$0.430702\pi$
$978$	0	0
$979$	−1.60199e34	−0.646635
$980$	0	0
$981$	2.78402e33	0.109769
$982$	0	0
$983$	3.14322e34	1.21063	0.605313	−	0.795988i	$-0.293048\pi$
$983$	3.14322e34	1.21063	0.605313	+	0.795988i	$0.293048\pi$
$984$	0	0
$985$	6.65197e34	2.50284
$986$	0	0
$987$	5.75793e33	0.211651
$988$	0	0
$989$	−1.15600e34	−0.415146
$990$	0	0
$991$	1.36431e34	0.478704	0.239352	−	0.970933i	$-0.423065\pi$
$991$	1.36431e34	0.478704	0.239352	+	0.970933i	$0.423065\pi$
$992$	0	0
$993$	−1.20955e34	−0.414675
$994$	0	0
$995$	2.69468e34	0.902696
$996$	0	0
$997$	−2.61404e34	−0.855693	−0.427847	−	0.903851i	$-0.640728\pi$
$997$	−2.61404e34	−0.855693	−0.427847	+	0.903851i	$0.640728\pi$
$998$	0	0
$999$	−4.80328e34	−1.53651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.n.1.5		6
4.3	odd	2	inner	64.24.a.n.1.2		6
8.3	odd	2		32.24.a.d.1.5	yes	6
8.5	even	2		32.24.a.d.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.24.a.d.1.2	✓	6	8.5	even	2
32.24.a.d.1.5	yes	6	8.3	odd	2
64.24.a.n.1.2		6	4.3	odd	2	inner
64.24.a.n.1.5		6	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q+216966. q^{3} -1.90927e8 q^{5} -9.74240e9 q^{7} -4.70690e10 q^{9} +O(q^{10})\)	\(q+216966. q^{3} -1.90927e8 q^{5} -9.74240e9 q^{7} -4.70690e10 q^{9} +9.58624e11 q^{11} +2.14517e12 q^{13} -4.14246e13 q^{15} +2.65620e13 q^{17} +7.63669e14 q^{19} -2.11377e15 q^{21} -3.75905e15 q^{23} +2.45321e16 q^{25} -3.06382e16 q^{27} -5.01247e16 q^{29} +1.90008e17 q^{31} +2.07989e17 q^{33} +1.86008e18 q^{35} +1.56774e18 q^{37} +4.65429e17 q^{39} -3.05851e18 q^{41} +3.07525e18 q^{43} +8.98674e18 q^{45} -2.72401e18 q^{47} +6.75455e19 q^{49} +5.76305e18 q^{51} +2.52631e19 q^{53} -1.83027e20 q^{55} +1.65690e20 q^{57} -2.29901e20 q^{59} +6.13479e20 q^{61} +4.58565e20 q^{63} -4.09571e20 q^{65} -1.20452e21 q^{67} -8.15585e20 q^{69} +2.43377e20 q^{71} -3.45063e20 q^{73} +5.32263e21 q^{75} -9.33929e21 q^{77} -3.08236e21 q^{79} -2.21622e21 q^{81} -1.36342e22 q^{83} -5.07140e21 q^{85} -1.08753e22 q^{87} -1.67114e22 q^{89} -2.08991e22 q^{91} +4.12252e22 q^{93} -1.45805e23 q^{95} +7.71584e22 q^{97} -4.51215e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 163121700 q^{5} + 14120212542 q^{9}+O(q^{10}) \) 6 * q - 163121700 * q^5 + 14120212542 * q^9	\( 6 q - 163121700 q^{5} + 14120212542 q^{9} - 1101295489524 q^{13} + 56435517243468 q^{17} - 14\!\cdots\!84 q^{21}+ \cdots + 25\!\cdots\!16 q^{97}+O(q^{100}) \) 6 * q - 163121700 * q^5 + 14120212542 * q^9 - 1101295489524 * q^13 + 56435517243468 * q^17 - 1447661324841984 * q^21 + 17512304669291850 * q^25 - 87161858953770708 * q^29 + 48462097564970496 * q^33 + 1542766250573484636 * q^37 - 865941451897399716 * q^41 + 9286254082406210700 * q^45 + 52551978665245784886 * q^49 + 10735070383373044668 * q^53 + 155104978099636154880 * q^57 + 432169207960503656364 * q^61 - 316526040451899107400 * q^65 - 695880182828346153984 * q^69 + 1758226166115151758204 * q^73 - 11926085657398731979776 * q^77 + 4010608843245537630678 * q^81 - 20251531460297137542600 * q^85 + 20087131519604301542940 * q^89 - 67310470168488243892224 * q^93 + 254665147475934678043116 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more