LMFDB - Embedded newform 720.6.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(1,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("720.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.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:	$115.476350265$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2521041.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 344x + 384 $ x^3 - x^2 - 344*x + 384
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.11670$ of defining polynomial
Character	$\chi$	$=$	720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.0000 q^{5} -15.4004 q^{7} +O(q^{10})$	$q+25.0000 q^{5} -15.4004 q^{7} +492.209 q^{11} +678.410 q^{13} +910.808 q^{17} +1714.42 q^{19} -2420.43 q^{23} +625.000 q^{25} +1757.21 q^{29} -682.804 q^{31} -385.011 q^{35} +7971.25 q^{37} -15330.6 q^{41} -14129.8 q^{43} +5678.01 q^{47} -16569.8 q^{49} +16411.9 q^{53} +12305.2 q^{55} +19398.0 q^{59} +49359.2 q^{61} +16960.2 q^{65} -24137.3 q^{67} -40417.9 q^{71} +46654.2 q^{73} -7580.22 q^{77} -81597.6 q^{79} +80160.8 q^{83} +22770.2 q^{85} +41686.3 q^{89} -10447.8 q^{91} +42860.6 q^{95} +6327.48 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 75 q^{5} - 18 q^{7}+O(q^{10}) $ 3 * q + 75 * q^5 - 18 * q^7	$ 3 q + 75 q^{5} - 18 q^{7} - 558 q^{11} - 84 q^{13} + 726 q^{17} - 876 q^{19} - 960 q^{23} + 1875 q^{25} + 1710 q^{29} - 324 q^{31} - 450 q^{35} + 10320 q^{37} + 4080 q^{41} + 1752 q^{43} + 8976 q^{47} + 48855 q^{49} - 16098 q^{53} - 13950 q^{55} + 23106 q^{59} + 51270 q^{61} - 2100 q^{65} - 69084 q^{67} + 55932 q^{71} + 100914 q^{73} + 65172 q^{77} - 159492 q^{79} - 16428 q^{83} + 18150 q^{85} + 120852 q^{89} - 235176 q^{91} - 21900 q^{95} + 223734 q^{97}+O(q^{100}) $ 3 * q + 75 * q^5 - 18 * q^7 - 558 * q^11 - 84 * q^13 + 726 * q^17 - 876 * q^19 - 960 * q^23 + 1875 * q^25 + 1710 * q^29 - 324 * q^31 - 450 * q^35 + 10320 * q^37 + 4080 * q^41 + 1752 * q^43 + 8976 * q^47 + 48855 * q^49 - 16098 * q^53 - 13950 * q^55 + 23106 * q^59 + 51270 * q^61 - 2100 * q^65 - 69084 * q^67 + 55932 * q^71 + 100914 * q^73 + 65172 * q^77 - 159492 * q^79 - 16428 * q^83 + 18150 * q^85 + 120852 * q^89 - 235176 * q^91 - 21900 * q^95 + 223734 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−15.4004	−0.118792	−0.0593961	−	0.998234i	$-0.518917\pi$
$7$	−15.4004	−0.118792	−0.0593961	+	0.998234i	$0.518917\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	492.209	1.22650	0.613250	−	0.789889i	$-0.289862\pi$
$11$	492.209	1.22650	0.613250	+	0.789889i	$0.289862\pi$
$12$	0	0
$13$	678.410	1.11336	0.556678	−	0.830729i	$-0.312076\pi$
$13$	678.410	1.11336	0.556678	+	0.830729i	$0.312076\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	910.808	0.764372	0.382186	−	0.924085i	$-0.375171\pi$
$17$	910.808	0.764372	0.382186	+	0.924085i	$0.375171\pi$
$18$	0	0
$19$	1714.42	1.08952	0.544759	−	0.838593i	$-0.316621\pi$
$19$	1714.42	1.08952	0.544759	+	0.838593i	$0.316621\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2420.43	−0.954053	−0.477027	−	0.878889i	$-0.658286\pi$
$23$	−2420.43	−0.954053	−0.477027	+	0.878889i	$0.658286\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1757.21	0.387997	0.193999	−	0.981002i	$-0.437854\pi$
$29$	1757.21	0.387997	0.193999	+	0.981002i	$0.437854\pi$
$30$	0	0
$31$	−682.804	−0.127612	−0.0638060	−	0.997962i	$-0.520324\pi$
$31$	−682.804	−0.127612	−0.0638060	+	0.997962i	$0.520324\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−385.011	−0.0531254
$36$	0	0
$37$	7971.25	0.957243	0.478621	−	0.878021i	$-0.341137\pi$
$37$	7971.25	0.957243	0.478621	+	0.878021i	$0.341137\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−15330.6	−1.42430	−0.712148	−	0.702029i	$-0.752277\pi$
$41$	−15330.6	−1.42430	−0.712148	+	0.702029i	$0.752277\pi$
$42$	0	0
$43$	−14129.8	−1.16537	−0.582686	−	0.812698i	$-0.697998\pi$
$43$	−14129.8	−1.16537	−0.582686	+	0.812698i	$0.697998\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5678.01	0.374931	0.187466	−	0.982271i	$-0.439973\pi$
$47$	5678.01	0.374931	0.187466	+	0.982271i	$0.439973\pi$
$48$	0	0
$49$	−16569.8	−0.985888
$50$	0	0
$51$	0	0
$52$	0	0
$53$	16411.9	0.802544	0.401272	−	0.915959i	$-0.368568\pi$
$53$	16411.9	0.802544	0.401272	+	0.915959i	$0.368568\pi$
$54$	0	0
$55$	12305.2	0.548507
$56$	0	0
$57$	0	0
$58$	0	0
$59$	19398.0	0.725482	0.362741	−	0.931890i	$-0.381841\pi$
$59$	19398.0	0.725482	0.362741	+	0.931890i	$0.381841\pi$
$60$	0	0
$61$	49359.2	1.69842	0.849208	−	0.528059i	$-0.177080\pi$
$61$	49359.2	1.69842	0.849208	+	0.528059i	$0.177080\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16960.2	0.497908
$66$	0	0
$67$	−24137.3	−0.656902	−0.328451	−	0.944521i	$-0.606527\pi$
$67$	−24137.3	−0.656902	−0.328451	+	0.944521i	$0.606527\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−40417.9	−0.951543	−0.475771	−	0.879569i	$-0.657831\pi$
$71$	−40417.9	−0.951543	−0.475771	+	0.879569i	$0.657831\pi$
$72$	0	0
$73$	46654.2	1.02467	0.512334	−	0.858786i	$-0.328781\pi$
$73$	46654.2	1.02467	0.512334	+	0.858786i	$0.328781\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7580.22	−0.145699
$78$	0	0
$79$	−81597.6	−1.47099	−0.735495	−	0.677530i	$-0.763050\pi$
$79$	−81597.6	−1.47099	−0.735495	+	0.677530i	$0.763050\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	80160.8	1.27722	0.638611	−	0.769529i	$-0.279509\pi$
$83$	80160.8	1.27722	0.638611	+	0.769529i	$0.279509\pi$
$84$	0	0
$85$	22770.2	0.341837
$86$	0	0
$87$	0	0
$88$	0	0
$89$	41686.3	0.557851	0.278925	−	0.960313i	$-0.410022\pi$
$89$	41686.3	0.557851	0.278925	+	0.960313i	$0.410022\pi$
$90$	0	0
$91$	−10447.8	−0.132258
$92$	0	0
$93$	0	0
$94$	0	0
$95$	42860.6	0.487247
$96$	0	0
$97$	6327.48	0.0682812	0.0341406	−	0.999417i	$-0.489131\pi$
$97$	6327.48	0.0682812	0.0341406	+	0.999417i	$0.489131\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	33256.1	0.324390	0.162195	−	0.986759i	$-0.448143\pi$
$101$	33256.1	0.324390	0.162195	+	0.986759i	$0.448143\pi$
$102$	0	0
$103$	−78465.9	−0.728766	−0.364383	−	0.931249i	$-0.618720\pi$
$103$	−78465.9	−0.728766	−0.364383	+	0.931249i	$0.618720\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−78863.0	−0.665908	−0.332954	−	0.942943i	$-0.608045\pi$
$107$	−78863.0	−0.665908	−0.332954	+	0.942943i	$0.608045\pi$
$108$	0	0
$109$	168744.	1.36039	0.680193	−	0.733033i	$-0.261896\pi$
$109$	168744.	1.36039	0.680193	+	0.733033i	$0.261896\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	52704.0	0.388282	0.194141	−	0.980974i	$-0.437808\pi$
$113$	52704.0	0.388282	0.194141	+	0.980974i	$0.437808\pi$
$114$	0	0
$115$	−60510.7	−0.426666
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−14026.8	−0.0908013
$120$	0	0
$121$	81218.3	0.504302
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−293003.	−1.61199	−0.805995	−	0.591923i	$-0.798369\pi$
$127$	−293003.	−1.61199	−0.805995	+	0.591923i	$0.798369\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	158580.	0.807364	0.403682	−	0.914899i	$-0.367730\pi$
$131$	158580.	0.807364	0.403682	+	0.914899i	$0.367730\pi$
$132$	0	0
$133$	−26402.9	−0.129426
$134$	0	0
$135$	0	0
$136$	0	0
$137$	109251.	0.497305	0.248652	−	0.968593i	$-0.420012\pi$
$137$	109251.	0.497305	0.248652	+	0.968593i	$0.420012\pi$
$138$	0	0
$139$	−167560.	−0.735587	−0.367793	−	0.929908i	$-0.619887\pi$
$139$	−167560.	−0.735587	−0.367793	+	0.929908i	$0.619887\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	333919.	1.36553
$144$	0	0
$145$	43930.2	0.173518
$146$	0	0
$147$	0	0
$148$	0	0
$149$	73315.5	0.270539	0.135270	−	0.990809i	$-0.456810\pi$
$149$	73315.5	0.270539	0.135270	+	0.990809i	$0.456810\pi$
$150$	0	0
$151$	−141222.	−0.504036	−0.252018	−	0.967723i	$-0.581094\pi$
$151$	−141222.	−0.504036	−0.252018	+	0.967723i	$0.581094\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−17070.1	−0.0570699
$156$	0	0
$157$	118324.	0.383111	0.191555	−	0.981482i	$-0.438647\pi$
$157$	118324.	0.383111	0.191555	+	0.981482i	$0.438647\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	37275.6	0.113334
$162$	0	0
$163$	218124.	0.643034	0.321517	−	0.946904i	$-0.395807\pi$
$163$	218124.	0.643034	0.321517	+	0.946904i	$0.395807\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	537207.	1.49056	0.745282	−	0.666749i	$-0.232315\pi$
$167$	537207.	1.49056	0.745282	+	0.666749i	$0.232315\pi$
$168$	0	0
$169$	88947.0	0.239560
$170$	0	0
$171$	0	0
$172$	0	0
$173$	562692.	1.42941	0.714703	−	0.699428i	$-0.246562\pi$
$173$	562692.	1.42941	0.714703	+	0.699428i	$0.246562\pi$
$174$	0	0
$175$	−9625.27	−0.0237584
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23559.8	−0.0549590	−0.0274795	−	0.999622i	$-0.508748\pi$
$179$	−23559.8	−0.0549590	−0.0274795	+	0.999622i	$0.508748\pi$
$180$	0	0
$181$	49433.0	0.112155	0.0560777	−	0.998426i	$-0.482141\pi$
$181$	49433.0	0.112155	0.0560777	+	0.998426i	$0.482141\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	199281.	0.428092
$186$	0	0
$187$	448308.	0.937502
$188$	0	0
$189$	0	0
$190$	0	0
$191$	324435.	0.643493	0.321746	−	0.946826i	$-0.395730\pi$
$191$	324435.	0.643493	0.321746	+	0.946826i	$0.395730\pi$
$192$	0	0
$193$	631048.	1.21946	0.609732	−	0.792608i	$-0.291277\pi$
$193$	631048.	1.21946	0.609732	+	0.792608i	$0.291277\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	168287.	0.308948	0.154474	−	0.987997i	$-0.450632\pi$
$197$	168287.	0.308948	0.154474	+	0.987997i	$0.450632\pi$
$198$	0	0
$199$	649346.	1.16237	0.581184	−	0.813772i	$-0.302590\pi$
$199$	649346.	1.16237	0.581184	+	0.813772i	$0.302590\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−27061.8	−0.0460910
$204$	0	0
$205$	−383266.	−0.636964
$206$	0	0
$207$	0	0
$208$	0	0
$209$	843855.	1.33629
$210$	0	0
$211$	−75931.0	−0.117412	−0.0587061	−	0.998275i	$-0.518697\pi$
$211$	−75931.0	−0.117412	−0.0587061	+	0.998275i	$0.518697\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−353245.	−0.521170
$216$	0	0
$217$	10515.5	0.0151593
$218$	0	0
$219$	0	0
$220$	0	0
$221$	617901.	0.851017
$222$	0	0
$223$	−459251.	−0.618426	−0.309213	−	0.950993i	$-0.600066\pi$
$223$	−459251.	−0.618426	−0.309213	+	0.950993i	$0.600066\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	583371.	0.751415	0.375708	−	0.926738i	$-0.377400\pi$
$227$	583371.	0.751415	0.375708	+	0.926738i	$0.377400\pi$
$228$	0	0
$229$	−976861.	−1.23096	−0.615480	−	0.788152i	$-0.711038\pi$
$229$	−976861.	−1.23096	−0.615480	+	0.788152i	$0.711038\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.22206e6	−1.47470	−0.737349	−	0.675512i	$-0.763923\pi$
$233$	−1.22206e6	−1.47470	−0.737349	+	0.675512i	$0.763923\pi$
$234$	0	0
$235$	141950.	0.167674
$236$	0	0
$237$	0	0
$238$	0	0
$239$	653604.	0.740151	0.370075	−	0.929002i	$-0.379332\pi$
$239$	653604.	0.740151	0.370075	+	0.929002i	$0.379332\pi$
$240$	0	0
$241$	−1.52747e6	−1.69406	−0.847031	−	0.531544i	$-0.821612\pi$
$241$	−1.52747e6	−1.69406	−0.847031	+	0.531544i	$0.821612\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−414246.	−0.440903
$246$	0	0
$247$	1.16308e6	1.21302
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.49551e6	1.49832	0.749160	−	0.662389i	$-0.230457\pi$
$251$	1.49551e6	1.49832	0.749160	+	0.662389i	$0.230457\pi$
$252$	0	0
$253$	−1.19136e6	−1.17015
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.61118e6	1.52164	0.760818	−	0.648965i	$-0.224798\pi$
$257$	1.61118e6	1.52164	0.760818	+	0.648965i	$0.224798\pi$
$258$	0	0
$259$	−122761.	−0.113713
$260$	0	0
$261$	0	0
$262$	0	0
$263$	971046.	0.865666	0.432833	−	0.901474i	$-0.357514\pi$
$263$	971046.	0.865666	0.432833	+	0.901474i	$0.357514\pi$
$264$	0	0
$265$	410297.	0.358908
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.25433e6	1.89949	0.949744	−	0.313029i	$-0.101344\pi$
$269$	2.25433e6	1.89949	0.949744	+	0.313029i	$0.101344\pi$
$270$	0	0
$271$	−290005.	−0.239873	−0.119937	−	0.992782i	$-0.538269\pi$
$271$	−290005.	−0.239873	−0.119937	+	0.992782i	$0.538269\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	307630.	0.245300
$276$	0	0
$277$	−98437.6	−0.0770835	−0.0385418	−	0.999257i	$-0.512271\pi$
$277$	−98437.6	−0.0770835	−0.0385418	+	0.999257i	$0.512271\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	711495.	0.537534	0.268767	−	0.963205i	$-0.413384\pi$
$281$	711495.	0.537534	0.268767	+	0.963205i	$0.413384\pi$
$282$	0	0
$283$	2.10014e6	1.55877	0.779385	−	0.626545i	$-0.215532\pi$
$283$	2.10014e6	1.55877	0.779385	+	0.626545i	$0.215532\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	236098.	0.169195
$288$	0	0
$289$	−590285.	−0.415736
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.21760e6	−0.828580	−0.414290	−	0.910145i	$-0.635970\pi$
$293$	−1.21760e6	−0.828580	−0.414290	+	0.910145i	$0.635970\pi$
$294$	0	0
$295$	484950.	0.324446
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.64204e6	−1.06220
$300$	0	0
$301$	217605.	0.138437
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.23398e6	0.759554
$306$	0	0
$307$	−971884.	−0.588530	−0.294265	−	0.955724i	$-0.595075\pi$
$307$	−971884.	−0.588530	−0.294265	+	0.955724i	$0.595075\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	515383.	0.302154	0.151077	−	0.988522i	$-0.451726\pi$
$311$	515383.	0.302154	0.151077	+	0.988522i	$0.451726\pi$
$312$	0	0
$313$	−1.24081e6	−0.715887	−0.357943	−	0.933743i	$-0.616522\pi$
$313$	−1.24081e6	−0.715887	−0.357943	+	0.933743i	$0.616522\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.96757e6	−1.09972	−0.549859	−	0.835257i	$-0.685319\pi$
$317$	−1.96757e6	−1.09972	−0.549859	+	0.835257i	$0.685319\pi$
$318$	0	0
$319$	864914.	0.475878
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.56151e6	0.832797
$324$	0	0
$325$	424006.	0.222671
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−87443.8	−0.0445389
$330$	0	0
$331$	3.19929e6	1.60503	0.802515	−	0.596632i	$-0.203495\pi$
$331$	3.19929e6	1.60503	0.802515	+	0.596632i	$0.203495\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−603431.	−0.293776
$336$	0	0
$337$	−2.96894e6	−1.42406	−0.712028	−	0.702151i	$-0.752223\pi$
$337$	−2.96894e6	−1.42406	−0.712028	+	0.702151i	$0.752223\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−336082.	−0.156516
$342$	0	0
$343$	514017.	0.235908
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−780880.	−0.348146	−0.174073	−	0.984733i	$-0.555693\pi$
$347$	−780880.	−0.348146	−0.174073	+	0.984733i	$0.555693\pi$
$348$	0	0
$349$	1.80457e6	0.793069	0.396535	−	0.918020i	$-0.370213\pi$
$349$	1.80457e6	0.793069	0.396535	+	0.918020i	$0.370213\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1882.62	0.000804130 0	0.000402065	−	1.00000i	$-0.499872\pi$
$353$	1882.62	0.000804130 0	0.000402065		1.00000i	$0.499872\pi$
$354$	0	0
$355$	−1.01045e6	−0.425543
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.19425e6	−0.489057	−0.244528	−	0.969642i	$-0.578633\pi$
$359$	−1.19425e6	−0.489057	−0.244528	+	0.969642i	$0.578633\pi$
$360$	0	0
$361$	463153.	0.187049
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.16635e6	0.458246
$366$	0	0
$367$	−1.77262e6	−0.686989	−0.343495	−	0.939155i	$-0.611611\pi$
$367$	−1.77262e6	−0.686989	−0.343495	+	0.939155i	$0.611611\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−252750.	−0.0953359
$372$	0	0
$373$	2.63660e6	0.981233	0.490617	−	0.871375i	$-0.336772\pi$
$373$	2.63660e6	0.981233	0.490617	+	0.871375i	$0.336772\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.19211e6	0.431979
$378$	0	0
$379$	−3.58048e6	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$379$	−3.58048e6	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.63240e6	0.916969	0.458485	−	0.888702i	$-0.348392\pi$
$383$	2.63240e6	0.916969	0.458485	+	0.888702i	$0.348392\pi$
$384$	0	0
$385$	−189506.	−0.0651584
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.54651e6	1.52336	0.761682	−	0.647951i	$-0.224374\pi$
$389$	4.54651e6	1.52336	0.761682	+	0.647951i	$0.224374\pi$
$390$	0	0
$391$	−2.20455e6	−0.729252
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.03994e6	−0.657847
$396$	0	0
$397$	−474849.	−0.151210	−0.0756048	−	0.997138i	$-0.524089\pi$
$397$	−474849.	−0.151210	−0.0756048	+	0.997138i	$0.524089\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.18207e6	−1.60932	−0.804659	−	0.593737i	$-0.797652\pi$
$401$	−5.18207e6	−1.60932	−0.804659	+	0.593737i	$0.797652\pi$
$402$	0	0
$403$	−463221.	−0.142078
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.92352e6	1.17406
$408$	0	0
$409$	236791.	0.0699933	0.0349966	−	0.999387i	$-0.488858\pi$
$409$	236791.	0.0699933	0.0349966	+	0.999387i	$0.488858\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−298737.	−0.0861816
$414$	0	0
$415$	2.00402e6	0.571191
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.75329e6	−1.87923	−0.939616	−	0.342232i	$-0.888817\pi$
$419$	−6.75329e6	−1.87923	−0.939616	+	0.342232i	$0.888817\pi$
$420$	0	0
$421$	−219653.	−0.0603994	−0.0301997	−	0.999544i	$-0.509614\pi$
$421$	−219653.	−0.0603994	−0.0301997	+	0.999544i	$0.509614\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	569255.	0.152874
$426$	0	0
$427$	−760153.	−0.201758
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.54792e6	0.660683	0.330341	−	0.943862i	$-0.392836\pi$
$431$	2.54792e6	0.660683	0.330341	+	0.943862i	$0.392836\pi$
$432$	0	0
$433$	5.45373e6	1.39789	0.698946	−	0.715174i	$-0.253653\pi$
$433$	5.45373e6	1.39789	0.698946	+	0.715174i	$0.253653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.14964e6	−1.03946
$438$	0	0
$439$	5.05122e6	1.25094	0.625468	−	0.780250i	$-0.284908\pi$
$439$	5.05122e6	1.25094	0.625468	+	0.780250i	$0.284908\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.54264e6	0.857665	0.428833	−	0.903384i	$-0.358925\pi$
$443$	3.54264e6	0.857665	0.428833	+	0.903384i	$0.358925\pi$
$444$	0	0
$445$	1.04216e6	0.249479
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.43350e6	−0.335569	−0.167784	−	0.985824i	$-0.553661\pi$
$449$	−1.43350e6	−0.335569	−0.167784	+	0.985824i	$0.553661\pi$
$450$	0	0
$451$	−7.54587e6	−1.74690
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−261195.	−0.0591475
$456$	0	0
$457$	5.06705e6	1.13492	0.567460	−	0.823401i	$-0.307926\pi$
$457$	5.06705e6	1.13492	0.567460	+	0.823401i	$0.307926\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.62406e6	−1.45168	−0.725842	−	0.687861i	$-0.758550\pi$
$461$	−6.62406e6	−1.45168	−0.725842	+	0.687861i	$0.758550\pi$
$462$	0	0
$463$	−2.51099e6	−0.544367	−0.272184	−	0.962245i	$-0.587746\pi$
$463$	−2.51099e6	−0.544367	−0.272184	+	0.962245i	$0.587746\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.43548e6	−1.78985	−0.894927	−	0.446212i	$-0.852773\pi$
$467$	−8.43548e6	−1.78985	−0.894927	+	0.446212i	$0.852773\pi$
$468$	0	0
$469$	371724.	0.0780348
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.95480e6	−1.42933
$474$	0	0
$475$	1.07152e6	0.217904
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.53497e6	−1.69966	−0.849832	−	0.527054i	$-0.823297\pi$
$479$	−8.53497e6	−1.69966	−0.849832	+	0.527054i	$0.823297\pi$
$480$	0	0
$481$	5.40778e6	1.06575
$482$	0	0
$483$	0	0
$484$	0	0
$485$	158187.	0.0305363
$486$	0	0
$487$	−5.12561e6	−0.979316	−0.489658	−	0.871915i	$-0.662878\pi$
$487$	−5.12561e6	−0.979316	−0.489658	+	0.871915i	$0.662878\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.04216e6	−1.50546	−0.752730	−	0.658329i	$-0.771263\pi$
$491$	−8.04216e6	−1.50546	−0.752730	+	0.658329i	$0.771263\pi$
$492$	0	0
$493$	1.60048e6	0.296574
$494$	0	0
$495$	0	0
$496$	0	0
$497$	622453.	0.113036
$498$	0	0
$499$	−3.68195e6	−0.661953	−0.330976	−	0.943639i	$-0.607378\pi$
$499$	−3.68195e6	−0.661953	−0.330976	+	0.943639i	$0.607378\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.05832e6	1.59635	0.798174	−	0.602427i	$-0.205799\pi$
$503$	9.05832e6	1.59635	0.798174	+	0.602427i	$0.205799\pi$
$504$	0	0
$505$	831403.	0.145072
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.80987e6	1.67830	0.839148	−	0.543903i	$-0.183054\pi$
$509$	9.80987e6	1.67830	0.839148	+	0.543903i	$0.183054\pi$
$510$	0	0
$511$	−718494.	−0.121723
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.96165e6	−0.325914
$516$	0	0
$517$	2.79477e6	0.459853
$518$	0	0
$519$	0	0
$520$	0	0
$521$	551174.	0.0889599	0.0444799	−	0.999010i	$-0.485837\pi$
$521$	551174.	0.0889599	0.0444799	+	0.999010i	$0.485837\pi$
$522$	0	0
$523$	314830.	0.0503294	0.0251647	−	0.999683i	$-0.491989\pi$
$523$	314830.	0.0503294	0.0251647	+	0.999683i	$0.491989\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−621903.	−0.0975431
$528$	0	0
$529$	−577867.	−0.0897819
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.04004e7	−1.58575
$534$	0	0
$535$	−1.97158e6	−0.297803
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−8.15581e6	−1.20919
$540$	0	0
$541$	−1.19713e7	−1.75853	−0.879264	−	0.476336i	$-0.841965\pi$
$541$	−1.19713e7	−1.75853	−0.879264	+	0.476336i	$0.841965\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.21860e6	0.608383
$546$	0	0
$547$	2.33368e6	0.333483	0.166741	−	0.986001i	$-0.446676\pi$
$547$	2.33368e6	0.333483	0.166741	+	0.986001i	$0.446676\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.01260e6	0.422730
$552$	0	0
$553$	1.25664e6	0.174742
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.10982e7	−1.51571	−0.757855	−	0.652423i	$-0.773753\pi$
$557$	−1.10982e7	−1.51571	−0.757855	+	0.652423i	$0.773753\pi$
$558$	0	0
$559$	−9.58578e6	−1.29747
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.68074e6	−0.755325	−0.377663	−	0.925943i	$-0.623272\pi$
$563$	−5.68074e6	−0.755325	−0.377663	+	0.925943i	$0.623272\pi$
$564$	0	0
$565$	1.31760e6	0.173645
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.85613e6	0.240341	0.120171	−	0.992753i	$-0.461656\pi$
$569$	1.85613e6	0.240341	0.120171	+	0.992753i	$0.461656\pi$
$570$	0	0
$571$	−9.05836e6	−1.16268	−0.581339	−	0.813662i	$-0.697471\pi$
$571$	−9.05836e6	−1.16268	−0.581339	+	0.813662i	$0.697471\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.51277e6	−0.190811
$576$	0	0
$577$	1.00802e6	0.126046	0.0630232	−	0.998012i	$-0.479926\pi$
$577$	1.00802e6	0.126046	0.0630232	+	0.998012i	$0.479926\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.23451e6	−0.151724
$582$	0	0
$583$	8.07807e6	0.984320
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.31524e6	0.636690	0.318345	−	0.947975i	$-0.396873\pi$
$587$	5.31524e6	0.636690	0.318345	+	0.947975i	$0.396873\pi$
$588$	0	0
$589$	−1.17062e6	−0.139036
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.38796e7	−1.62084	−0.810422	−	0.585846i	$-0.800762\pi$
$593$	−1.38796e7	−1.62084	−0.810422	+	0.585846i	$0.800762\pi$
$594$	0	0
$595$	−350671.	−0.0406076
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.17883e6	−0.703622	−0.351811	−	0.936071i	$-0.614434\pi$
$599$	−6.17883e6	−0.703622	−0.351811	+	0.936071i	$0.614434\pi$
$600$	0	0
$601$	−1.18727e7	−1.34079	−0.670397	−	0.742002i	$-0.733876\pi$
$601$	−1.18727e7	−1.34079	−0.670397	+	0.742002i	$0.733876\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.03046e6	0.225531
$606$	0	0
$607$	8.75134e6	0.964057	0.482029	−	0.876155i	$-0.339900\pi$
$607$	8.75134e6	0.964057	0.482029	+	0.876155i	$0.339900\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.85202e6	0.417432
$612$	0	0
$613$	1.31559e7	1.41407	0.707033	−	0.707180i	$-0.250033\pi$
$613$	1.31559e7	1.41407	0.707033	+	0.707180i	$0.250033\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−152712.	−0.0161495	−0.00807475	−	0.999967i	$-0.502570\pi$
$617$	−152712.	−0.0161495	−0.00807475	+	0.999967i	$0.502570\pi$
$618$	0	0
$619$	5.44904e6	0.571601	0.285801	−	0.958289i	$-0.407741\pi$
$619$	5.44904e6	0.571601	0.285801	+	0.958289i	$0.407741\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−641987.	−0.0662683
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.26028e6	0.731689
$630$	0	0
$631$	−3.80451e6	−0.380387	−0.190193	−	0.981747i	$-0.560912\pi$
$631$	−3.80451e6	−0.380387	−0.190193	+	0.981747i	$0.560912\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7.32506e6	−0.720904
$636$	0	0
$637$	−1.12411e7	−1.09764
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.11903e6	−0.780475	−0.390237	−	0.920714i	$-0.627607\pi$
$641$	−8.11903e6	−0.780475	−0.390237	+	0.920714i	$0.627607\pi$
$642$	0	0
$643$	1.93317e7	1.84392	0.921959	−	0.387288i	$-0.126588\pi$
$643$	1.93317e7	1.84392	0.921959	+	0.387288i	$0.126588\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.22083e6	−0.396403	−0.198202	−	0.980161i	$-0.563510\pi$
$647$	−4.22083e6	−0.396403	−0.198202	+	0.980161i	$0.563510\pi$
$648$	0	0
$649$	9.54786e6	0.889804
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.59385e7	−1.46274	−0.731368	−	0.681983i	$-0.761118\pi$
$653$	−1.59385e7	−1.46274	−0.731368	+	0.681983i	$0.761118\pi$
$654$	0	0
$655$	3.96449e6	0.361064
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.72241e7	−1.54498	−0.772492	−	0.635024i	$-0.780990\pi$
$659$	−1.72241e7	−1.54498	−0.772492	+	0.635024i	$0.780990\pi$
$660$	0	0
$661$	−848047.	−0.0754947	−0.0377473	−	0.999287i	$-0.512018\pi$
$661$	−848047.	−0.0754947	−0.0377473	+	0.999287i	$0.512018\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−660072.	−0.0578811
$666$	0	0
$667$	−4.25320e6	−0.370170
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.42950e7	2.08311
$672$	0	0
$673$	−1.67528e7	−1.42577	−0.712886	−	0.701280i	$-0.752612\pi$
$673$	−1.67528e7	−1.42577	−0.712886	+	0.701280i	$0.752612\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.04111e7	0.873022	0.436511	−	0.899699i	$-0.356214\pi$
$677$	1.04111e7	0.873022	0.436511	+	0.899699i	$0.356214\pi$
$678$	0	0
$679$	−97445.8	−0.00811127
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.10857e7	−0.909311	−0.454655	−	0.890667i	$-0.650238\pi$
$683$	−1.10857e7	−0.909311	−0.454655	+	0.890667i	$0.650238\pi$
$684$	0	0
$685$	2.73127e6	0.222401
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.11340e7	0.893516
$690$	0	0
$691$	1.50089e7	1.19579	0.597894	−	0.801575i	$-0.296004\pi$
$691$	1.50089e7	1.19579	0.597894	+	0.801575i	$0.296004\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.18901e6	−0.328964
$696$	0	0
$697$	−1.39633e7	−1.08869
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4.15952e6	−0.319704	−0.159852	−	0.987141i	$-0.551102\pi$
$701$	−4.15952e6	−0.319704	−0.159852	+	0.987141i	$0.551102\pi$
$702$	0	0
$703$	1.36661e7	1.04293
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−512158.	−0.0385350
$708$	0	0
$709$	−9.90459e6	−0.739982	−0.369991	−	0.929035i	$-0.620639\pi$
$709$	−9.90459e6	−0.739982	−0.369991	+	0.929035i	$0.620639\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.65268e6	0.121749
$714$	0	0
$715$	8.34798e6	0.610684
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.45179e6	0.609714	0.304857	−	0.952398i	$-0.401391\pi$
$719$	8.45179e6	0.609714	0.304857	+	0.952398i	$0.401391\pi$
$720$	0	0
$721$	1.20841e6	0.0865716
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.09826e6	0.0775994
$726$	0	0
$727$	−1.09776e7	−0.770317	−0.385159	−	0.922850i	$-0.625853\pi$
$727$	−1.09776e7	−0.770317	−0.385159	+	0.922850i	$0.625853\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.28695e7	−0.890777
$732$	0	0
$733$	1.83926e7	1.26439	0.632197	−	0.774808i	$-0.282153\pi$
$733$	1.83926e7	1.26439	0.632197	+	0.774808i	$0.282153\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.18806e7	−0.805691
$738$	0	0
$739$	2.32355e7	1.56510	0.782549	−	0.622589i	$-0.213919\pi$
$739$	2.32355e7	1.56510	0.782549	+	0.622589i	$0.213919\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.79553e7	1.85777	0.928887	−	0.370364i	$-0.120767\pi$
$743$	2.79553e7	1.85777	0.928887	+	0.370364i	$0.120767\pi$
$744$	0	0
$745$	1.83289e6	0.120989
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.21452e6	0.0791046
$750$	0	0
$751$	−3.86213e6	−0.249877	−0.124939	−	0.992164i	$-0.539873\pi$
$751$	−3.86213e6	−0.249877	−0.124939	+	0.992164i	$0.539873\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.53056e6	−0.225412
$756$	0	0
$757$	−8.96551e6	−0.568637	−0.284319	−	0.958730i	$-0.591767\pi$
$757$	−8.96551e6	−0.568637	−0.284319	+	0.958730i	$0.591767\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.69920e6	−0.106361	−0.0531806	−	0.998585i	$-0.516936\pi$
$761$	−1.69920e6	−0.106361	−0.0531806	+	0.998585i	$0.516936\pi$
$762$	0	0
$763$	−2.59873e6	−0.161603
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.31598e7	0.807720
$768$	0	0
$769$	−3.10686e7	−1.89455	−0.947275	−	0.320422i	$-0.896175\pi$
$769$	−3.10686e7	−1.89455	−0.947275	+	0.320422i	$0.896175\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.20292e7	0.724084	0.362042	−	0.932162i	$-0.382080\pi$
$773$	1.20292e7	0.724084	0.362042	+	0.932162i	$0.382080\pi$
$774$	0	0
$775$	−426752.	−0.0255224
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.62832e7	−1.55180
$780$	0	0
$781$	−1.98941e7	−1.16707
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.95811e6	0.171332
$786$	0	0
$787$	−1.58306e7	−0.911087	−0.455543	−	0.890214i	$-0.650555\pi$
$787$	−1.58306e7	−0.911087	−0.455543	+	0.890214i	$0.650555\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−811664.	−0.0461248
$792$	0	0
$793$	3.34858e7	1.89094
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.57037e7	0.875699	0.437850	−	0.899048i	$-0.355740\pi$
$797$	1.57037e7	0.875699	0.437850	+	0.899048i	$0.355740\pi$
$798$	0	0
$799$	5.17158e6	0.286587
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.29636e7	1.25676
$804$	0	0
$805$	931891.	0.0506845
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.01239e7	1.61823	0.809114	−	0.587651i	$-0.199947\pi$
$809$	3.01239e7	1.61823	0.809114	+	0.587651i	$0.199947\pi$
$810$	0	0
$811$	−1.54587e7	−0.825315	−0.412658	−	0.910886i	$-0.635399\pi$
$811$	−1.54587e7	−0.825315	−0.412658	+	0.910886i	$0.635399\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.45309e6	0.287573
$816$	0	0
$817$	−2.42244e7	−1.26969
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.18634e7	0.614260	0.307130	−	0.951667i	$-0.400631\pi$
$821$	1.18634e7	0.614260	0.307130	+	0.951667i	$0.400631\pi$
$822$	0	0
$823$	2.77574e7	1.42850	0.714249	−	0.699892i	$-0.246769\pi$
$823$	2.77574e7	1.42850	0.714249	+	0.699892i	$0.246769\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.75820e6	−0.445298	−0.222649	−	0.974899i	$-0.571470\pi$
$827$	−8.75820e6	−0.445298	−0.222649	+	0.974899i	$0.571470\pi$
$828$	0	0
$829$	−2.73617e6	−0.138279	−0.0691395	−	0.997607i	$-0.522025\pi$
$829$	−2.73617e6	−0.138279	−0.0691395	+	0.997607i	$0.522025\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.50919e7	−0.753585
$834$	0	0
$835$	1.34302e7	0.666601
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.34019e7	1.14775	0.573873	−	0.818945i	$-0.305440\pi$
$839$	2.34019e7	1.14775	0.573873	+	0.818945i	$0.305440\pi$
$840$	0	0
$841$	−1.74234e7	−0.849458
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.22367e6	0.107135
$846$	0	0
$847$	−1.25080e6	−0.0599071
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.92938e7	−0.913261
$852$	0	0
$853$	−1.41447e7	−0.665611	−0.332806	−	0.942995i	$-0.607995\pi$
$853$	−1.41447e7	−0.665611	−0.332806	+	0.942995i	$0.607995\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.83888e6	0.178547	0.0892734	−	0.996007i	$-0.471546\pi$
$857$	3.83888e6	0.178547	0.0892734	+	0.996007i	$0.471546\pi$
$858$	0	0
$859$	−2.90812e7	−1.34471	−0.672357	−	0.740227i	$-0.734718\pi$
$859$	−2.90812e7	−1.34471	−0.672357	+	0.740227i	$0.734718\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.80704e7	1.28298	0.641491	−	0.767130i	$-0.278316\pi$
$863$	2.80704e7	1.28298	0.641491	+	0.767130i	$0.278316\pi$
$864$	0	0
$865$	1.40673e7	0.639250
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.01631e7	−1.80417
$870$	0	0
$871$	−1.63749e7	−0.731366
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−240632.	−0.0106251
$876$	0	0
$877$	2.46494e7	1.08220	0.541101	−	0.840958i	$-0.318008\pi$
$877$	2.46494e7	1.08220	0.541101	+	0.840958i	$0.318008\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.27656e7	0.554119	0.277060	−	0.960853i	$-0.410640\pi$
$881$	1.27656e7	0.554119	0.277060	+	0.960853i	$0.410640\pi$
$882$	0	0
$883$	−1.06326e7	−0.458920	−0.229460	−	0.973318i	$-0.573696\pi$
$883$	−1.06326e7	−0.458920	−0.229460	+	0.973318i	$0.573696\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.62218e7	1.54583	0.772913	−	0.634512i	$-0.218799\pi$
$887$	3.62218e7	1.54583	0.772913	+	0.634512i	$0.218799\pi$
$888$	0	0
$889$	4.51236e6	0.191492
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.73452e6	0.408494
$894$	0	0
$895$	−588995.	−0.0245784
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.19983e6	−0.0495131
$900$	0	0
$901$	1.49481e7	0.613442
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.23582e6	0.0501574
$906$	0	0
$907$	3.57116e7	1.44142	0.720711	−	0.693236i	$-0.243816\pi$
$907$	3.57116e7	1.44142	0.720711	+	0.693236i	$0.243816\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.50186e7	1.79720	0.898599	−	0.438771i	$-0.144586\pi$
$911$	4.50186e7	1.79720	0.898599	+	0.438771i	$0.144586\pi$
$912$	0	0
$913$	3.94558e7	1.56651
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.44219e6	−0.0959084
$918$	0	0
$919$	−4.04547e7	−1.58008	−0.790042	−	0.613053i	$-0.789941\pi$
$919$	−4.04547e7	−1.58008	−0.790042	+	0.613053i	$0.789941\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.74199e7	−1.05940
$924$	0	0
$925$	4.98203e6	0.191449
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.32731e7	1.26489	0.632447	−	0.774604i	$-0.282051\pi$
$929$	3.32731e7	1.26489	0.632447	+	0.774604i	$0.282051\pi$
$930$	0	0
$931$	−2.84077e7	−1.07414
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.12077e7	0.419264
$936$	0	0
$937$	2.84377e7	1.05815	0.529073	−	0.848576i	$-0.322540\pi$
$937$	2.84377e7	1.05815	0.529073	+	0.848576i	$0.322540\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.82692e6	−0.0672582	−0.0336291	−	0.999434i	$-0.510706\pi$
$941$	−1.82692e6	−0.0672582	−0.0336291	+	0.999434i	$0.510706\pi$
$942$	0	0
$943$	3.71067e7	1.35885
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.37392e7	−1.58488	−0.792439	−	0.609952i	$-0.791189\pi$
$947$	−4.37392e7	−1.58488	−0.792439	+	0.609952i	$0.791189\pi$
$948$	0	0
$949$	3.16507e7	1.14082
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.48447e7	1.95615	0.978076	−	0.208246i	$-0.0667755\pi$
$953$	5.48447e7	1.95615	0.978076	+	0.208246i	$0.0667755\pi$
$954$	0	0
$955$	8.11087e6	0.287779
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.68251e6	−0.0590759
$960$	0	0
$961$	−2.81629e7	−0.983715
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.57762e7	0.545361
$966$	0	0
$967$	−4.23376e7	−1.45599	−0.727997	−	0.685580i	$-0.759549\pi$
$967$	−4.23376e7	−1.45599	−0.727997	+	0.685580i	$0.759549\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.93560e7	0.658821	0.329411	−	0.944187i	$-0.393150\pi$
$971$	1.93560e7	0.658821	0.329411	+	0.944187i	$0.393150\pi$
$972$	0	0
$973$	2.58050e6	0.0873819
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.00733e7	−0.337626	−0.168813	−	0.985648i	$-0.553993\pi$
$977$	−1.00733e7	−0.337626	−0.168813	+	0.985648i	$0.553993\pi$
$978$	0	0
$979$	2.05183e7	0.684204
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.09761e7	0.362296	0.181148	−	0.983456i	$-0.442019\pi$
$983$	1.09761e7	0.362296	0.181148	+	0.983456i	$0.442019\pi$
$984$	0	0
$985$	4.20718e6	0.138166
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.42001e7	1.11183
$990$	0	0
$991$	−3.39971e7	−1.09966	−0.549830	−	0.835277i	$-0.685307\pi$
$991$	−3.39971e7	−1.09966	−0.549830	+	0.835277i	$0.685307\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.62337e7	0.519827
$996$	0	0
$997$	−3.79808e7	−1.21011	−0.605056	−	0.796183i	$-0.706849\pi$
$997$	−3.79808e7	−1.21011	−0.605056	+	0.796183i	$0.706849\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.a.bj.1.2		3
3.2	odd	2		720.6.a.bi.1.2		3
4.3	odd	2		360.6.a.p.1.2	yes	3
12.11	even	2		360.6.a.o.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.6.a.o.1.2	✓	3	12.11	even	2
360.6.a.p.1.2	yes	3	4.3	odd	2
720.6.a.bi.1.2		3	3.2	odd	2
720.6.a.bj.1.2		3	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

\(f(q)\)	\(=\)	\(q+25.0000 q^{5} -15.4004 q^{7} +O(q^{10})\)	\(q+25.0000 q^{5} -15.4004 q^{7} +492.209 q^{11} +678.410 q^{13} +910.808 q^{17} +1714.42 q^{19} -2420.43 q^{23} +625.000 q^{25} +1757.21 q^{29} -682.804 q^{31} -385.011 q^{35} +7971.25 q^{37} -15330.6 q^{41} -14129.8 q^{43} +5678.01 q^{47} -16569.8 q^{49} +16411.9 q^{53} +12305.2 q^{55} +19398.0 q^{59} +49359.2 q^{61} +16960.2 q^{65} -24137.3 q^{67} -40417.9 q^{71} +46654.2 q^{73} -7580.22 q^{77} -81597.6 q^{79} +80160.8 q^{83} +22770.2 q^{85} +41686.3 q^{89} -10447.8 q^{91} +42860.6 q^{95} +6327.48 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 75 q^{5} - 18 q^{7}+O(q^{10}) \) 3 * q + 75 * q^5 - 18 * q^7	\( 3 q + 75 q^{5} - 18 q^{7} - 558 q^{11} - 84 q^{13} + 726 q^{17} - 876 q^{19} - 960 q^{23} + 1875 q^{25} + 1710 q^{29} - 324 q^{31} - 450 q^{35} + 10320 q^{37} + 4080 q^{41} + 1752 q^{43} + 8976 q^{47} + 48855 q^{49} - 16098 q^{53} - 13950 q^{55} + 23106 q^{59} + 51270 q^{61} - 2100 q^{65} - 69084 q^{67} + 55932 q^{71} + 100914 q^{73} + 65172 q^{77} - 159492 q^{79} - 16428 q^{83} + 18150 q^{85} + 120852 q^{89} - 235176 q^{91} - 21900 q^{95} + 223734 q^{97}+O(q^{100}) \) 3 * q + 75 * q^5 - 18 * q^7 - 558 * q^11 - 84 * q^13 + 726 * q^17 - 876 * q^19 - 960 * q^23 + 1875 * q^25 + 1710 * q^29 - 324 * q^31 - 450 * q^35 + 10320 * q^37 + 4080 * q^41 + 1752 * q^43 + 8976 * q^47 + 48855 * q^49 - 16098 * q^53 - 13950 * q^55 + 23106 * q^59 + 51270 * q^61 - 2100 * q^65 - 69084 * q^67 + 55932 * q^71 + 100914 * q^73 + 65172 * q^77 - 159492 * q^79 - 16428 * q^83 + 18150 * q^85 + 120852 * q^89 - 235176 * q^91 - 21900 * q^95 + 223734 * q^97

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