LMFDB - Embedded newform 225.12.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,12,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	225.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:	$172.877215626$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} - 8438 x^{6} - 12900 x^{5} + 24607297 x^{4} + 71220406 x^{3} - 29215500956 x^{2} + \cdots + 11837228681952 $ x^8 - 2x^7 - 8438x^6 - 12900x^5 + 24607297x^4 + 71220406x^3 - 29215500956x^2 - 69590816160*x + 11837228681952
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{12}\cdot 5^{10} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$26.7797$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-22.0579 q^{2} -1561.45 q^{4} -55546.4 q^{7} +79616.9 q^{8} -390374. q^{11} +294278. q^{13} +1.22524e6 q^{14} +1.44167e6 q^{16} -1.34966e6 q^{17} -4.87782e6 q^{19} +8.61083e6 q^{22} -2.15035e7 q^{23} -6.49116e6 q^{26} +8.67329e7 q^{28} +1.06704e8 q^{29} +8.85723e7 q^{31} -1.94855e8 q^{32} +2.97707e7 q^{34} +7.15174e8 q^{37} +1.07594e8 q^{38} -1.01025e9 q^{41} +6.84508e7 q^{43} +6.09549e8 q^{44} +4.74321e8 q^{46} -1.18367e9 q^{47} +1.10807e9 q^{49} -4.59501e8 q^{52} +5.58230e9 q^{53} -4.42243e9 q^{56} -2.35366e9 q^{58} +8.83822e9 q^{59} +9.40678e9 q^{61} -1.95372e9 q^{62} +1.34557e9 q^{64} -1.67537e10 q^{67} +2.10743e9 q^{68} +9.80178e9 q^{71} +7.18622e9 q^{73} -1.57752e10 q^{74} +7.61646e9 q^{76} +2.16839e10 q^{77} -2.66926e9 q^{79} +2.22841e10 q^{82} -4.33938e10 q^{83} -1.50988e9 q^{86} -3.10803e10 q^{88} -5.89906e10 q^{89} -1.63461e10 q^{91} +3.35766e10 q^{92} +2.61093e10 q^{94} +9.59614e10 q^{97} -2.44418e10 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8976 q^{4} + 15740992 q^{16} - 61220032 q^{19} + 279698464 q^{31} - 309623120 q^{34} - 13964841760 q^{46} + 1596098056 q^{49} - 4923969584 q^{61} - 40933220096 q^{64} - 128254720704 q^{76} - 87780965728 q^{79}+ \cdots + 460752353440 q^{94}+O(q^{100}) $ 8 * q + 8976 * q^4 + 15740992 * q^16 - 61220032 * q^19 + 279698464 * q^31 - 309623120 * q^34 - 13964841760 * q^46 + 1596098056 * q^49 - 4923969584 * q^61 - 40933220096 * q^64 - 128254720704 * q^76 - 87780965728 * q^79 + 71755992000 * q^91 + 460752353440 * q^94

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 $

$2$	−22.0579	−0.487415	−0.243708	−	0.969849i	$-0.578364\pi$
$2$	−22.0579	−0.487415	−0.243708	+	0.969849i	$0.578364\pi$
$3$	0	0
$3$	0	0
$4$	−1561.45	−0.762426
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−55546.4	−1.24916	−0.624578	−	0.780963i	$-0.714729\pi$
$7$	−55546.4	−1.24916	−0.624578	+	0.780963i	$0.714729\pi$
$8$	79616.9	0.859033
$9$	0	0
$10$	0	0
$11$	−390374.	−0.730838	−0.365419	−	0.930843i	$-0.619074\pi$
$11$	−390374.	−0.730838	−0.365419	+	0.930843i	$0.619074\pi$
$12$	0	0
$13$	294278.	0.219821	0.109911	−	0.993941i	$-0.464944\pi$
$13$	294278.	0.219821	0.109911	+	0.993941i	$0.464944\pi$
$14$	1.22524e6	0.608858
$15$	0	0
$16$	1.44167e6	0.343720
$17$	−1.34966e6	−0.230545	−0.115272	−	0.993334i	$-0.536774\pi$
$17$	−1.34966e6	−0.230545	−0.115272	+	0.993334i	$0.536774\pi$
$18$	0	0
$19$	−4.87782e6	−0.451940	−0.225970	−	0.974134i	$-0.572555\pi$
$19$	−4.87782e6	−0.451940	−0.225970	+	0.974134i	$0.572555\pi$
$20$	0	0
$21$	0	0
$22$	8.61083e6	0.356221
$23$	−2.15035e7	−0.696635	−0.348318	−	0.937377i	$-0.613247\pi$
$23$	−2.15035e7	−0.696635	−0.348318	+	0.937377i	$0.613247\pi$
$24$	0	0
$25$	0	0
$26$	−6.49116e6	−0.107144
$27$	0	0
$28$	8.67329e7	0.952389
$29$	1.06704e8	0.966030	0.483015	−	0.875612i	$-0.339542\pi$
$29$	1.06704e8	0.966030	0.483015	+	0.875612i	$0.339542\pi$
$30$	0	0
$31$	8.85723e7	0.555659	0.277830	−	0.960630i	$-0.410385\pi$
$31$	8.85723e7	0.555659	0.277830	+	0.960630i	$0.410385\pi$
$32$	−1.94855e8	−1.02657
$33$	0	0
$34$	2.97707e7	0.112371
$35$	0	0
$36$	0	0
$37$	7.15174e8	1.69552	0.847759	−	0.530382i	$-0.177951\pi$
$37$	7.15174e8	1.69552	0.847759	+	0.530382i	$0.177951\pi$
$38$	1.07594e8	0.220282
$39$	0	0
$40$	0	0
$41$	−1.01025e9	−1.36182	−0.680909	−	0.732368i	$-0.738415\pi$
$41$	−1.01025e9	−1.36182	−0.680909	+	0.732368i	$0.738415\pi$
$42$	0	0
$43$	6.84508e7	0.0710072	0.0355036	−	0.999370i	$-0.488696\pi$
$43$	6.84508e7	0.0710072	0.0355036	+	0.999370i	$0.488696\pi$
$44$	6.09549e8	0.557210
$45$	0	0
$46$	4.74321e8	0.339551
$47$	−1.18367e9	−0.752822	−0.376411	−	0.926453i	$-0.622842\pi$
$47$	−1.18367e9	−0.752822	−0.376411	+	0.926453i	$0.622842\pi$
$48$	0	0
$49$	1.10807e9	0.560390
$50$	0	0
$51$	0	0
$52$	−4.59501e8	−0.167598
$53$	5.58230e9	1.83356	0.916781	−	0.399389i	$-0.130778\pi$
$53$	5.58230e9	1.83356	0.916781	+	0.399389i	$0.130778\pi$
$54$	0	0
$55$	0	0
$56$	−4.42243e9	−1.07307
$57$	0	0
$58$	−2.35366e9	−0.470858
$59$	8.83822e9	1.60945	0.804727	−	0.593645i	$-0.202312\pi$
$59$	8.83822e9	1.60945	0.804727	+	0.593645i	$0.202312\pi$
$60$	0	0
$61$	9.40678e9	1.42602	0.713012	−	0.701152i	$-0.247330\pi$
$61$	9.40678e9	1.42602	0.713012	+	0.701152i	$0.247330\pi$
$62$	−1.95372e9	−0.270837
$63$	0	0
$64$	1.34557e9	0.156644
$65$	0	0
$66$	0	0
$67$	−1.67537e10	−1.51600	−0.758002	−	0.652253i	$-0.773824\pi$
$67$	−1.67537e10	−1.51600	−0.758002	+	0.652253i	$0.773824\pi$
$68$	2.10743e9	0.175774
$69$	0	0
$70$	0	0
$71$	9.80178e9	0.644739	0.322370	−	0.946614i	$-0.395521\pi$
$71$	9.80178e9	0.644739	0.322370	+	0.946614i	$0.395521\pi$
$72$	0	0
$73$	7.18622e9	0.405718	0.202859	−	0.979208i	$-0.434977\pi$
$73$	7.18622e9	0.405718	0.202859	+	0.979208i	$0.434977\pi$
$74$	−1.57752e10	−0.826421
$75$	0	0
$76$	7.61646e9	0.344571
$77$	2.16839e10	0.912930
$78$	0	0
$79$	−2.66926e9	−0.0975983	−0.0487991	−	0.998809i	$-0.515539\pi$
$79$	−2.66926e9	−0.0975983	−0.0487991	+	0.998809i	$0.515539\pi$
$80$	0	0
$81$	0	0
$82$	2.22841e10	0.663771
$83$	−4.33938e10	−1.20920	−0.604600	−	0.796529i	$-0.706667\pi$
$83$	−4.33938e10	−1.20920	−0.604600	+	0.796529i	$0.706667\pi$
$84$	0	0
$85$	0	0
$86$	−1.50988e9	−0.0346100
$87$	0	0
$88$	−3.10803e10	−0.627814
$89$	−5.89906e10	−1.11979	−0.559896	−	0.828563i	$-0.689159\pi$
$89$	−5.89906e10	−1.11979	−0.559896	+	0.828563i	$0.689159\pi$
$90$	0	0
$91$	−1.63461e10	−0.274591
$92$	3.35766e10	0.531133
$93$	0	0
$94$	2.61093e10	0.366937
$95$	0	0
$96$	0	0
$97$	9.59614e10	1.13462	0.567312	−	0.823503i	$-0.307983\pi$
$97$	9.59614e10	1.13462	0.567312	+	0.823503i	$0.307983\pi$
$98$	−2.44418e10	−0.273143
$99$	0	0
$100$	0	0
$101$	1.58971e11	1.50505	0.752524	−	0.658565i	$-0.228836\pi$
$101$	1.58971e11	1.50505	0.752524	+	0.658565i	$0.228836\pi$
$102$	0	0
$103$	1.21843e11	1.03561	0.517805	−	0.855499i	$-0.326749\pi$
$103$	1.21843e11	1.03561	0.517805	+	0.855499i	$0.326749\pi$
$104$	2.34295e10	0.188834
$105$	0	0
$106$	−1.23134e11	−0.893707
$107$	2.64305e11	1.82178	0.910888	−	0.412653i	$-0.135398\pi$
$107$	2.64305e11	1.82178	0.910888	+	0.412653i	$0.135398\pi$
$108$	0	0
$109$	−1.84618e11	−1.14929	−0.574644	−	0.818403i	$-0.694860\pi$
$109$	−1.84618e11	−1.14929	−0.574644	+	0.818403i	$0.694860\pi$
$110$	0	0
$111$	0	0
$112$	−8.00794e10	−0.429360
$113$	2.06690e11	1.05533	0.527664	−	0.849453i	$-0.323068\pi$
$113$	2.06690e11	1.05533	0.527664	+	0.849453i	$0.323068\pi$
$114$	0	0
$115$	0	0
$116$	−1.66613e11	−0.736527
$117$	0	0
$118$	−1.94952e11	−0.784472
$119$	7.49688e10	0.287987
$120$	0	0
$121$	−1.32920e11	−0.465876
$122$	−2.07494e11	−0.695066
$123$	0	0
$124$	−1.38301e11	−0.423649
$125$	0	0
$126$	0	0
$127$	−3.53722e11	−0.950039	−0.475020	−	0.879975i	$-0.657559\pi$
$127$	−3.53722e11	−0.950039	−0.475020	+	0.879975i	$0.657559\pi$
$128$	3.69384e11	0.950217
$129$	0	0
$130$	0	0
$131$	−4.97397e11	−1.12645	−0.563223	−	0.826305i	$-0.690439\pi$
$131$	−4.97397e11	−1.12645	−0.563223	+	0.826305i	$0.690439\pi$
$132$	0	0
$133$	2.70945e11	0.564543
$134$	3.69552e11	0.738923
$135$	0	0
$136$	−1.07456e11	−0.198046
$137$	3.10911e11	0.550393	0.275196	−	0.961388i	$-0.411257\pi$
$137$	3.10911e11	0.550393	0.275196	+	0.961388i	$0.411257\pi$
$138$	0	0
$139$	−1.80237e11	−0.294620	−0.147310	−	0.989090i	$-0.547062\pi$
$139$	−1.80237e11	−0.294620	−0.147310	+	0.989090i	$0.547062\pi$
$140$	0	0
$141$	0	0
$142$	−2.16207e11	−0.314256
$143$	−1.14879e11	−0.160654
$144$	0	0
$145$	0	0
$146$	−1.58513e11	−0.197753
$147$	0	0
$148$	−1.11671e12	−1.29271
$149$	−2.91760e11	−0.325463	−0.162731	−	0.986670i	$-0.552030\pi$
$149$	−2.91760e11	−0.325463	−0.162731	+	0.986670i	$0.552030\pi$
$150$	0	0
$151$	−2.33328e11	−0.241876	−0.120938	−	0.992660i	$-0.538590\pi$
$151$	−2.33328e11	−0.241876	−0.120938	+	0.992660i	$0.538590\pi$
$152$	−3.88356e11	−0.388232
$153$	0	0
$154$	−4.78300e11	−0.444976
$155$	0	0
$156$	0	0
$157$	1.54856e12	1.29563	0.647813	−	0.761800i	$-0.275684\pi$
$157$	1.54856e12	1.29563	0.647813	+	0.761800i	$0.275684\pi$
$158$	5.88783e10	0.0475709
$159$	0	0
$160$	0	0
$161$	1.19444e12	0.870206
$162$	0	0
$163$	1.69870e12	1.15634	0.578168	−	0.815917i	$-0.303768\pi$
$163$	1.69870e12	1.15634	0.578168	+	0.815917i	$0.303768\pi$
$164$	1.57746e12	1.03829
$165$	0	0
$166$	9.57176e11	0.589383
$167$	1.59377e12	0.949481	0.474740	−	0.880126i	$-0.342542\pi$
$167$	1.59377e12	0.949481	0.474740	+	0.880126i	$0.342542\pi$
$168$	0	0
$169$	−1.70556e12	−0.951679
$170$	0	0
$171$	0	0
$172$	−1.06883e11	−0.0541378
$173$	−1.16978e12	−0.573917	−0.286958	−	0.957943i	$-0.592644\pi$
$173$	−1.16978e12	−0.573917	−0.286958	+	0.957943i	$0.592644\pi$
$174$	0	0
$175$	0	0
$176$	−5.62789e11	−0.251204
$177$	0	0
$178$	1.30121e12	0.545804
$179$	−4.37250e12	−1.77844	−0.889218	−	0.457484i	$-0.848751\pi$
$179$	−4.37250e12	−1.77844	−0.889218	+	0.457484i	$0.848751\pi$
$180$	0	0
$181$	5.31067e11	0.203197	0.101599	−	0.994825i	$-0.467604\pi$
$181$	5.31067e11	0.203197	0.101599	+	0.994825i	$0.467604\pi$
$182$	3.60561e11	0.133840
$183$	0	0
$184$	−1.71204e12	−0.598433
$185$	0	0
$186$	0	0
$187$	5.26872e11	0.168491
$188$	1.84824e12	0.573971
$189$	0	0
$190$	0	0
$191$	4.26612e12	1.21437	0.607183	−	0.794562i	$-0.292300\pi$
$191$	4.26612e12	1.21437	0.607183	+	0.794562i	$0.292300\pi$
$192$	0	0
$193$	−3.58148e12	−0.962713	−0.481356	−	0.876525i	$-0.659856\pi$
$193$	−3.58148e12	−0.962713	−0.481356	+	0.876525i	$0.659856\pi$
$194$	−2.11671e12	−0.553033
$195$	0	0
$196$	−1.73020e12	−0.427256
$197$	3.14971e12	0.756322	0.378161	−	0.925740i	$-0.376557\pi$
$197$	3.14971e12	0.756322	0.378161	+	0.925740i	$0.376557\pi$
$198$	0	0
$199$	−7.71437e12	−1.75230	−0.876150	−	0.482039i	$-0.839896\pi$
$199$	−7.71437e12	−1.75230	−0.876150	+	0.482039i	$0.839896\pi$
$200$	0	0
$201$	0	0
$202$	−3.50656e12	−0.733583
$203$	−5.92701e12	−1.20672
$204$	0	0
$205$	0	0
$206$	−2.68760e12	−0.504772
$207$	0	0
$208$	4.24252e11	0.0755571
$209$	1.90417e12	0.330295
$210$	0	0
$211$	−1.06063e13	−1.74586	−0.872931	−	0.487843i	$-0.837784\pi$
$211$	−1.06063e13	−1.74586	−0.872931	+	0.487843i	$0.837784\pi$
$212$	−8.71648e12	−1.39796
$213$	0	0
$214$	−5.83002e12	−0.887962
$215$	0	0
$216$	0	0
$217$	−4.91987e12	−0.694105
$218$	4.07229e12	0.560181
$219$	0	0
$220$	0	0
$221$	−3.97176e11	−0.0506787
$222$	0	0
$223$	−1.19680e12	−0.145326	−0.0726630	−	0.997357i	$-0.523150\pi$
$223$	−1.19680e12	−0.145326	−0.0726630	+	0.997357i	$0.523150\pi$
$224$	1.08235e13	1.28234
$225$	0	0
$226$	−4.55914e12	−0.514383
$227$	−1.43614e13	−1.58145	−0.790724	−	0.612172i	$-0.790296\pi$
$227$	−1.43614e13	−1.58145	−0.790724	+	0.612172i	$0.790296\pi$
$228$	0	0
$229$	−1.73779e13	−1.82349	−0.911744	−	0.410758i	$-0.865264\pi$
$229$	−1.73779e13	−1.82349	−0.911744	+	0.410758i	$0.865264\pi$
$230$	0	0
$231$	0	0
$232$	8.49542e12	0.829852
$233$	−4.48575e12	−0.427935	−0.213968	−	0.976841i	$-0.568639\pi$
$233$	−4.48575e12	−0.427935	−0.213968	+	0.976841i	$0.568639\pi$
$234$	0	0
$235$	0	0
$236$	−1.38004e13	−1.22709
$237$	0	0
$238$	−1.65365e12	−0.140369
$239$	7.64513e10	0.00634157	0.00317078	−	0.999995i	$-0.498991\pi$
$239$	7.64513e10	0.00634157	0.00317078	+	0.999995i	$0.498991\pi$
$240$	0	0
$241$	−1.06281e13	−0.842097	−0.421049	−	0.907038i	$-0.638338\pi$
$241$	−1.06281e13	−0.842097	−0.421049	+	0.907038i	$0.638338\pi$
$242$	2.93193e12	0.227075
$243$	0	0
$244$	−1.46882e13	−1.08724
$245$	0	0
$246$	0	0
$247$	−1.43544e12	−0.0993461
$248$	7.05184e12	0.477330
$249$	0	0
$250$	0	0
$251$	4.24494e12	0.268946	0.134473	−	0.990917i	$-0.457066\pi$
$251$	4.24494e12	0.268946	0.134473	+	0.990917i	$0.457066\pi$
$252$	0	0
$253$	8.39439e12	0.509127
$254$	7.80236e12	0.463064
$255$	0	0
$256$	−1.09035e13	−0.619795
$257$	−1.26005e13	−0.701059	−0.350529	−	0.936552i	$-0.613998\pi$
$257$	−1.26005e13	−0.701059	−0.350529	+	0.936552i	$0.613998\pi$
$258$	0	0
$259$	−3.97253e13	−2.11797
$260$	0	0
$261$	0	0
$262$	1.09715e13	0.549047
$263$	7.46619e12	0.365883	0.182942	−	0.983124i	$-0.441438\pi$
$263$	7.46619e12	0.365883	0.182942	+	0.983124i	$0.441438\pi$
$264$	0	0
$265$	0	0
$266$	−5.97648e12	−0.275167
$267$	0	0
$268$	2.61601e13	1.15584
$269$	−1.69203e13	−0.732435	−0.366218	−	0.930529i	$-0.619347\pi$
$269$	−1.69203e13	−0.732435	−0.366218	+	0.930529i	$0.619347\pi$
$270$	0	0
$271$	2.59160e12	0.107705	0.0538526	−	0.998549i	$-0.482850\pi$
$271$	2.59160e12	0.107705	0.0538526	+	0.998549i	$0.482850\pi$
$272$	−1.94576e12	−0.0792430
$273$	0	0
$274$	−6.85804e12	−0.268270
$275$	0	0
$276$	0	0
$277$	8.12272e12	0.299270	0.149635	−	0.988741i	$-0.452190\pi$
$277$	8.12272e12	0.299270	0.149635	+	0.988741i	$0.452190\pi$
$278$	3.97565e12	0.143602
$279$	0	0
$280$	0	0
$281$	1.40507e13	0.478425	0.239213	−	0.970967i	$-0.423111\pi$
$281$	1.40507e13	0.478425	0.239213	+	0.970967i	$0.423111\pi$
$282$	0	0
$283$	−5.56339e13	−1.82186	−0.910929	−	0.412563i	$-0.864634\pi$
$283$	−5.56339e13	−1.82186	−0.910929	+	0.412563i	$0.864634\pi$
$284$	−1.53050e13	−0.491566
$285$	0	0
$286$	2.53398e12	0.0783051
$287$	5.61159e13	1.70112
$288$	0	0
$289$	−3.24503e13	−0.946849
$290$	0	0
$291$	0	0
$292$	−1.12209e13	−0.309330
$293$	−3.50276e13	−0.947631	−0.473815	−	0.880624i	$-0.657124\pi$
$293$	−3.50276e13	−0.947631	−0.473815	+	0.880624i	$0.657124\pi$
$294$	0	0
$295$	0	0
$296$	5.69399e13	1.45651
$297$	0	0
$298$	6.43561e12	0.158635
$299$	−6.32801e12	−0.153135
$300$	0	0
$301$	−3.80220e12	−0.0886990
$302$	5.14672e12	0.117894
$303$	0	0
$304$	−7.03219e12	−0.155341
$305$	0	0
$306$	0	0
$307$	−4.17912e13	−0.874628	−0.437314	−	0.899309i	$-0.644070\pi$
$307$	−4.17912e13	−0.874628	−0.437314	+	0.899309i	$0.644070\pi$
$308$	−3.38582e13	−0.696042
$309$	0	0
$310$	0	0
$311$	−1.46541e13	−0.285612	−0.142806	−	0.989751i	$-0.545612\pi$
$311$	−1.46541e13	−0.285612	−0.142806	+	0.989751i	$0.545612\pi$
$312$	0	0
$313$	4.36887e13	0.822007	0.411004	−	0.911634i	$-0.365178\pi$
$313$	4.36887e13	0.822007	0.411004	+	0.911634i	$0.365178\pi$
$314$	−3.41579e13	−0.631507
$315$	0	0
$316$	4.16792e12	0.0744115
$317$	−8.62787e12	−0.151383	−0.0756916	−	0.997131i	$-0.524116\pi$
$317$	−8.62787e12	−0.151383	−0.0756916	+	0.997131i	$0.524116\pi$
$318$	0	0
$319$	−4.16544e13	−0.706011
$320$	0	0
$321$	0	0
$322$	−2.63468e13	−0.424151
$323$	6.58340e12	0.104192
$324$	0	0
$325$	0	0
$326$	−3.74697e13	−0.563616
$327$	0	0
$328$	−8.04332e13	−1.16985
$329$	6.57486e13	0.940392
$330$	0	0
$331$	−2.29938e13	−0.318095	−0.159048	−	0.987271i	$-0.550842\pi$
$331$	−2.29938e13	−0.318095	−0.159048	+	0.987271i	$0.550842\pi$
$332$	6.77572e13	0.921926
$333$	0	0
$334$	−3.51553e13	−0.462791
$335$	0	0
$336$	0	0
$337$	2.07560e13	0.260123	0.130062	−	0.991506i	$-0.458483\pi$
$337$	2.07560e13	0.260123	0.130062	+	0.991506i	$0.458483\pi$
$338$	3.76211e13	0.463863
$339$	0	0
$340$	0	0
$341$	−3.45763e13	−0.406097
$342$	0	0
$343$	4.82838e13	0.549141
$344$	5.44984e12	0.0609975
$345$	0	0
$346$	2.58028e13	0.279736
$347$	−4.77409e13	−0.509423	−0.254711	−	0.967017i	$-0.581980\pi$
$347$	−4.77409e13	−0.509423	−0.254711	+	0.967017i	$0.581980\pi$
$348$	0	0
$349$	−3.89716e13	−0.402910	−0.201455	−	0.979498i	$-0.564567\pi$
$349$	−3.89716e13	−0.402910	−0.201455	+	0.979498i	$0.564567\pi$
$350$	0	0
$351$	0	0
$352$	7.60665e13	0.750255
$353$	1.70145e14	1.65218	0.826091	−	0.563536i	$-0.190559\pi$
$353$	1.70145e14	1.65218	0.826091	+	0.563536i	$0.190559\pi$
$354$	0	0
$355$	0	0
$356$	9.21108e13	0.853760
$357$	0	0
$358$	9.64481e13	0.866837
$359$	2.00471e13	0.177432	0.0887160	−	0.996057i	$-0.471724\pi$
$359$	2.00471e13	0.177432	0.0887160	+	0.996057i	$0.471724\pi$
$360$	0	0
$361$	−9.26972e13	−0.795750
$362$	−1.17142e13	−0.0990413
$363$	0	0
$364$	2.55236e13	0.209356
$365$	0	0
$366$	0	0
$367$	1.03112e14	0.808433	0.404217	−	0.914663i	$-0.367544\pi$
$367$	1.03112e14	0.808433	0.404217	+	0.914663i	$0.367544\pi$
$368$	−3.10009e13	−0.239448
$369$	0	0
$370$	0	0
$371$	−3.10077e14	−2.29041
$372$	0	0
$373$	−1.04079e13	−0.0746386	−0.0373193	−	0.999303i	$-0.511882\pi$
$373$	−1.04079e13	−0.0746386	−0.0373193	+	0.999303i	$0.511882\pi$
$374$	−1.16217e13	−0.0821251
$375$	0	0
$376$	−9.42400e13	−0.646699
$377$	3.14006e13	0.212354
$378$	0	0
$379$	2.25424e14	1.48076	0.740380	−	0.672188i	$-0.234645\pi$
$379$	2.25424e14	1.48076	0.740380	+	0.672188i	$0.234645\pi$
$380$	0	0
$381$	0	0
$382$	−9.41016e13	−0.591900
$383$	5.14914e13	0.319258	0.159629	−	0.987177i	$-0.448970\pi$
$383$	5.14914e13	0.319258	0.159629	+	0.987177i	$0.448970\pi$
$384$	0	0
$385$	0	0
$386$	7.89998e13	0.469241
$387$	0	0
$388$	−1.49839e14	−0.865068
$389$	3.01596e14	1.71673	0.858366	−	0.513037i	$-0.171480\pi$
$389$	3.01596e14	1.71673	0.858366	+	0.513037i	$0.171480\pi$
$390$	0	0
$391$	2.90224e13	0.160606
$392$	8.82214e13	0.481394
$393$	0	0
$394$	−6.94760e13	−0.368643
$395$	0	0
$396$	0	0
$397$	1.38344e14	0.704065	0.352033	−	0.935988i	$-0.385491\pi$
$397$	1.38344e14	0.704065	0.352033	+	0.935988i	$0.385491\pi$
$398$	1.70163e14	0.854097
$399$	0	0
$400$	0	0
$401$	2.15849e14	1.03958	0.519788	−	0.854295i	$-0.326011\pi$
$401$	2.15849e14	1.03958	0.519788	+	0.854295i	$0.326011\pi$
$402$	0	0
$403$	2.60649e13	0.122146
$404$	−2.48225e14	−1.14749
$405$	0	0
$406$	1.30737e14	0.588175
$407$	−2.79185e14	−1.23915
$408$	0	0
$409$	2.13616e14	0.922901	0.461450	−	0.887166i	$-0.347329\pi$
$409$	2.13616e14	0.922901	0.461450	+	0.887166i	$0.347329\pi$
$410$	0	0
$411$	0	0
$412$	−1.90252e14	−0.789576
$413$	−4.90931e14	−2.01046
$414$	0	0
$415$	0	0
$416$	−5.73418e13	−0.225662
$417$	0	0
$418$	−4.20020e13	−0.160991
$419$	−3.09233e14	−1.16979	−0.584896	−	0.811108i	$-0.698865\pi$
$419$	−3.09233e14	−1.16979	−0.584896	+	0.811108i	$0.698865\pi$
$420$	0	0
$421$	2.18306e14	0.804476	0.402238	−	0.915535i	$-0.368232\pi$
$421$	2.18306e14	0.804476	0.402238	+	0.915535i	$0.368232\pi$
$422$	2.33953e14	0.850960
$423$	0	0
$424$	4.44445e14	1.57509
$425$	0	0
$426$	0	0
$427$	−5.22513e14	−1.78133
$428$	−4.12699e14	−1.38897
$429$	0	0
$430$	0	0
$431$	4.05282e14	1.31260	0.656300	−	0.754500i	$-0.272121\pi$
$431$	4.05282e14	1.31260	0.656300	+	0.754500i	$0.272121\pi$
$432$	0	0
$433$	1.06068e14	0.334888	0.167444	−	0.985882i	$-0.446449\pi$
$433$	1.06068e14	0.334888	0.167444	+	0.985882i	$0.446449\pi$
$434$	1.08522e14	0.338317
$435$	0	0
$436$	2.88272e14	0.876248
$437$	1.04890e14	0.314837
$438$	0	0
$439$	2.34311e14	0.685864	0.342932	−	0.939360i	$-0.388580\pi$
$439$	2.34311e14	0.685864	0.342932	+	0.939360i	$0.388580\pi$
$440$	0	0
$441$	0	0
$442$	8.76087e12	0.0247016
$443$	−1.59155e13	−0.0443199	−0.0221600	−	0.999754i	$-0.507054\pi$
$443$	−1.59155e13	−0.0443199	−0.0221600	+	0.999754i	$0.507054\pi$
$444$	0	0
$445$	0	0
$446$	2.63988e13	0.0708341
$447$	0	0
$448$	−7.47413e13	−0.195673
$449$	−5.35002e14	−1.38357	−0.691784	−	0.722104i	$-0.743175\pi$
$449$	−5.35002e14	−1.38357	−0.691784	+	0.722104i	$0.743175\pi$
$450$	0	0
$451$	3.94377e14	0.995268
$452$	−3.22735e14	−0.804610
$453$	0	0
$454$	3.16783e14	0.770822
$455$	0	0
$456$	0	0
$457$	5.44568e14	1.27795	0.638973	−	0.769229i	$-0.279359\pi$
$457$	5.44568e14	1.27795	0.638973	+	0.769229i	$0.279359\pi$
$458$	3.83321e14	0.888796
$459$	0	0
$460$	0	0
$461$	−3.08085e14	−0.689154	−0.344577	−	0.938758i	$-0.611978\pi$
$461$	−3.08085e14	−0.689154	−0.344577	+	0.938758i	$0.611978\pi$
$462$	0	0
$463$	1.40262e14	0.306370	0.153185	−	0.988198i	$-0.451047\pi$
$463$	1.40262e14	0.306370	0.153185	+	0.988198i	$0.451047\pi$
$464$	1.53831e14	0.332044
$465$	0	0
$466$	9.89463e13	0.208582
$467$	−3.04077e14	−0.633491	−0.316746	−	0.948511i	$-0.602590\pi$
$467$	−3.04077e14	−0.633491	−0.316746	+	0.948511i	$0.602590\pi$
$468$	0	0
$469$	9.30609e14	1.89372
$470$	0	0
$471$	0	0
$472$	7.03671e14	1.38257
$473$	−2.67214e13	−0.0518947
$474$	0	0
$475$	0	0
$476$	−1.17060e14	−0.219569
$477$	0	0
$478$	−1.68635e12	−0.00309098
$479$	8.80901e14	1.59618	0.798090	−	0.602538i	$-0.205844\pi$
$479$	8.80901e14	1.59618	0.798090	+	0.602538i	$0.205844\pi$
$480$	0	0
$481$	2.10460e14	0.372711
$482$	2.34434e14	0.410451
$483$	0	0
$484$	2.07548e14	0.355196
$485$	0	0
$486$	0	0
$487$	−3.64694e14	−0.603280	−0.301640	−	0.953422i	$-0.597534\pi$
$487$	−3.64694e14	−0.603280	−0.301640	+	0.953422i	$0.597534\pi$
$488$	7.48938e14	1.22500
$489$	0	0
$490$	0	0
$491$	−2.81758e14	−0.445582	−0.222791	−	0.974866i	$-0.571517\pi$
$491$	−2.81758e14	−0.445582	−0.222791	+	0.974866i	$0.571517\pi$
$492$	0	0
$493$	−1.44014e14	−0.222713
$494$	3.16627e13	0.0484228
$495$	0	0
$496$	1.27692e14	0.190991
$497$	−5.44454e14	−0.805380
$498$	0	0
$499$	5.04088e14	0.729379	0.364690	−	0.931129i	$-0.381175\pi$
$499$	5.04088e14	0.729379	0.364690	+	0.931129i	$0.381175\pi$
$500$	0	0
$501$	0	0
$502$	−9.36344e13	−0.131089
$503$	−7.09290e14	−0.982200	−0.491100	−	0.871103i	$-0.663405\pi$
$503$	−7.09290e14	−0.982200	−0.491100	+	0.871103i	$0.663405\pi$
$504$	0	0
$505$	0	0
$506$	−1.85163e14	−0.248156
$507$	0	0
$508$	5.52319e14	0.724335
$509$	3.89311e12	0.00505067	0.00252534	−	0.999997i	$-0.499196\pi$
$509$	3.89311e12	0.00505067	0.00252534	+	0.999997i	$0.499196\pi$
$510$	0	0
$511$	−3.99169e14	−0.506806
$512$	−5.15989e14	−0.648120
$513$	0	0
$514$	2.77940e14	0.341707
$515$	0	0
$516$	0	0
$517$	4.62074e14	0.550191
$518$	8.76257e14	1.03233
$519$	0	0
$520$	0	0
$521$	6.67006e14	0.761241	0.380621	−	0.924731i	$-0.375710\pi$
$521$	6.67006e14	0.761241	0.380621	+	0.924731i	$0.375710\pi$
$522$	0	0
$523$	−9.54669e14	−1.06683	−0.533413	−	0.845855i	$-0.679091\pi$
$523$	−9.54669e14	−1.06683	−0.533413	+	0.845855i	$0.679091\pi$
$524$	7.76660e14	0.858833
$525$	0	0
$526$	−1.64689e14	−0.178337
$527$	−1.19543e14	−0.128104
$528$	0	0
$529$	−4.90411e14	−0.514700
$530$	0	0
$531$	0	0
$532$	−4.23067e14	−0.430423
$533$	−2.97296e14	−0.299357
$534$	0	0
$535$	0	0
$536$	−1.33388e15	−1.30230
$537$	0	0
$538$	3.73225e14	0.357000
$539$	−4.32563e14	−0.409554
$540$	0	0
$541$	−2.01083e14	−0.186548	−0.0932739	−	0.995640i	$-0.529733\pi$
$541$	−2.01083e14	−0.186548	−0.0932739	+	0.995640i	$0.529733\pi$
$542$	−5.71653e13	−0.0524972
$543$	0	0
$544$	2.62989e14	0.236670
$545$	0	0
$546$	0	0
$547$	−1.97509e15	−1.72447	−0.862236	−	0.506507i	$-0.830936\pi$
$547$	−1.97509e15	−1.72447	−0.862236	+	0.506507i	$0.830936\pi$
$548$	−4.85472e14	−0.419634
$549$	0	0
$550$	0	0
$551$	−5.20481e14	−0.436588
$552$	0	0
$553$	1.48268e14	0.121915
$554$	−1.79170e14	−0.145869
$555$	0	0
$556$	2.81431e14	0.224626
$557$	1.73969e15	1.37489	0.687444	−	0.726238i	$-0.258733\pi$
$557$	1.73969e15	1.37489	0.687444	+	0.726238i	$0.258733\pi$
$558$	0	0
$559$	2.01436e13	0.0156089
$560$	0	0
$561$	0	0
$562$	−3.09929e14	−0.233192
$563$	−7.00557e14	−0.521972	−0.260986	−	0.965343i	$-0.584048\pi$
$563$	−7.00557e14	−0.521972	−0.260986	+	0.965343i	$0.584048\pi$
$564$	0	0
$565$	0	0
$566$	1.22717e15	0.888001
$567$	0	0
$568$	7.80387e14	0.553853
$569$	−2.47841e15	−1.74203	−0.871014	−	0.491257i	$-0.836537\pi$
$569$	−2.47841e15	−1.74203	−0.871014	+	0.491257i	$0.836537\pi$
$570$	0	0
$571$	−1.29540e15	−0.893111	−0.446556	−	0.894756i	$-0.647350\pi$
$571$	−1.29540e15	−0.893111	−0.446556	+	0.894756i	$0.647350\pi$
$572$	1.79377e14	0.122487
$573$	0	0
$574$	−1.23780e15	−0.829153
$575$	0	0
$576$	0	0
$577$	8.93465e14	0.581582	0.290791	−	0.956787i	$-0.406082\pi$
$577$	8.93465e14	0.581582	0.290791	+	0.956787i	$0.406082\pi$
$578$	7.15786e14	0.461509
$579$	0	0
$580$	0	0
$581$	2.41037e15	1.51048
$582$	0	0
$583$	−2.17918e15	−1.34004
$584$	5.72144e14	0.348526
$585$	0	0
$586$	7.72636e14	0.461890
$587$	2.15415e15	1.27575	0.637877	−	0.770138i	$-0.279813\pi$
$587$	2.15415e15	1.27575	0.637877	+	0.770138i	$0.279813\pi$
$588$	0	0
$589$	−4.32039e14	−0.251125
$590$	0	0
$591$	0	0
$592$	1.03104e15	0.582784
$593$	−1.51707e15	−0.849582	−0.424791	−	0.905291i	$-0.639652\pi$
$593$	−1.51707e15	−0.849582	−0.424791	+	0.905291i	$0.639652\pi$
$594$	0	0
$595$	0	0
$596$	4.55568e14	0.248141
$597$	0	0
$598$	1.39583e14	0.0746405
$599$	4.59566e14	0.243501	0.121750	−	0.992561i	$-0.461149\pi$
$599$	4.59566e14	0.243501	0.121750	+	0.992561i	$0.461149\pi$
$600$	0	0
$601$	−1.61074e15	−0.837944	−0.418972	−	0.907999i	$-0.637610\pi$
$601$	−1.61074e15	−0.837944	−0.418972	+	0.907999i	$0.637610\pi$
$602$	8.38685e13	0.0432333
$603$	0	0
$604$	3.64330e14	0.184413
$605$	0	0
$606$	0	0
$607$	2.27195e15	1.11908	0.559541	−	0.828803i	$-0.310978\pi$
$607$	2.27195e15	1.11908	0.559541	+	0.828803i	$0.310978\pi$
$608$	9.50469e14	0.463947
$609$	0	0
$610$	0	0
$611$	−3.48328e14	−0.165486
$612$	0	0
$613$	2.51698e15	1.17448	0.587242	−	0.809411i	$-0.300214\pi$
$613$	2.51698e15	1.17448	0.587242	+	0.809411i	$0.300214\pi$
$614$	9.21825e14	0.426307
$615$	0	0
$616$	1.72640e15	0.784238
$617$	2.36230e15	1.06357	0.531785	−	0.846879i	$-0.321521\pi$
$617$	2.36230e15	1.06357	0.531785	+	0.846879i	$0.321521\pi$
$618$	0	0
$619$	−3.01613e15	−1.33399	−0.666993	−	0.745064i	$-0.732419\pi$
$619$	−3.01613e15	−1.33399	−0.666993	+	0.745064i	$0.732419\pi$
$620$	0	0
$621$	0	0
$622$	3.23238e14	0.139211
$623$	3.27671e15	1.39880
$624$	0	0
$625$	0	0
$626$	−9.63681e14	−0.400659
$627$	0	0
$628$	−2.41799e15	−0.987819
$629$	−9.65243e14	−0.390893
$630$	0	0
$631$	4.31590e14	0.171755	0.0858776	−	0.996306i	$-0.472631\pi$
$631$	4.31590e14	0.171755	0.0858776	+	0.996306i	$0.472631\pi$
$632$	−2.12518e14	−0.0838402
$633$	0	0
$634$	1.90313e14	0.0737865
$635$	0	0
$636$	0	0
$637$	3.26083e14	0.123186
$638$	9.18807e14	0.344121
$639$	0	0
$640$	0	0
$641$	6.67595e13	0.0243666	0.0121833	−	0.999926i	$-0.496122\pi$
$641$	6.67595e13	0.0243666	0.0121833	+	0.999926i	$0.496122\pi$
$642$	0	0
$643$	3.74464e15	1.34354	0.671769	−	0.740760i	$-0.265535\pi$
$643$	3.74464e15	1.34354	0.671769	+	0.740760i	$0.265535\pi$
$644$	−1.86506e15	−0.663468
$645$	0	0
$646$	−1.45216e14	−0.0507850
$647$	−3.70369e15	−1.28428	−0.642142	−	0.766585i	$-0.721954\pi$
$647$	−3.70369e15	−1.28428	−0.642142	+	0.766585i	$0.721954\pi$
$648$	0	0
$649$	−3.45021e15	−1.17625
$650$	0	0
$651$	0	0
$652$	−2.65243e15	−0.881622
$653$	−4.04155e15	−1.33207	−0.666033	−	0.745922i	$-0.732009\pi$
$653$	−4.04155e15	−1.33207	−0.666033	+	0.745922i	$0.732009\pi$
$654$	0	0
$655$	0	0
$656$	−1.45645e15	−0.468085
$657$	0	0
$658$	−1.45027e15	−0.458361
$659$	4.16521e15	1.30547	0.652735	−	0.757586i	$-0.273622\pi$
$659$	4.16521e15	1.30547	0.652735	+	0.757586i	$0.273622\pi$
$660$	0	0
$661$	−1.23316e15	−0.380111	−0.190056	−	0.981773i	$-0.560867\pi$
$661$	−1.23316e15	−0.380111	−0.190056	+	0.981773i	$0.560867\pi$
$662$	5.07195e14	0.155045
$663$	0	0
$664$	−3.45488e15	−1.03874
$665$	0	0
$666$	0	0
$667$	−2.29450e15	−0.672971
$668$	−2.48860e15	−0.723909
$669$	0	0
$670$	0	0
$671$	−3.67216e15	−1.04219
$672$	0	0
$673$	−7.13397e14	−0.199181	−0.0995905	−	0.995029i	$-0.531753\pi$
$673$	−7.13397e14	−0.199181	−0.0995905	+	0.995029i	$0.531753\pi$
$674$	−4.57833e14	−0.126788
$675$	0	0
$676$	2.66315e15	0.725585
$677$	−3.93889e15	−1.06448	−0.532239	−	0.846594i	$-0.678649\pi$
$677$	−3.93889e15	−1.06448	−0.532239	+	0.846594i	$0.678649\pi$
$678$	0	0
$679$	−5.33031e15	−1.41732
$680$	0	0
$681$	0	0
$682$	7.62680e14	0.197938
$683$	5.84741e15	1.50539	0.752696	−	0.658369i	$-0.228753\pi$
$683$	5.84741e15	1.50539	0.752696	+	0.658369i	$0.228753\pi$
$684$	0	0
$685$	0	0
$686$	−1.06504e15	−0.267660
$687$	0	0
$688$	9.86834e13	0.0244066
$689$	1.64275e15	0.403056
$690$	0	0
$691$	−6.16517e15	−1.48873	−0.744364	−	0.667774i	$-0.767247\pi$
$691$	−6.16517e15	−1.48873	−0.744364	+	0.667774i	$0.767247\pi$
$692$	1.82654e15	0.437569
$693$	0	0
$694$	1.05306e15	0.248301
$695$	0	0
$696$	0	0
$697$	1.36350e15	0.313960
$698$	8.59632e14	0.196385
$699$	0	0
$700$	0	0
$701$	−1.34268e15	−0.299587	−0.149793	−	0.988717i	$-0.547861\pi$
$701$	−1.34268e15	−0.299587	−0.149793	+	0.988717i	$0.547861\pi$
$702$	0	0
$703$	−3.48849e15	−0.766272
$704$	−5.25274e14	−0.114482
$705$	0	0
$706$	−3.75304e15	−0.805299
$707$	−8.83026e15	−1.88004
$708$	0	0
$709$	−6.27775e15	−1.31598	−0.657991	−	0.753026i	$-0.728594\pi$
$709$	−6.27775e15	−1.31598	−0.657991	+	0.753026i	$0.728594\pi$
$710$	0	0
$711$	0	0
$712$	−4.69665e15	−0.961940
$713$	−1.90461e15	−0.387092
$714$	0	0
$715$	0	0
$716$	6.82744e15	1.35593
$717$	0	0
$718$	−4.42197e14	−0.0864830
$719$	8.75562e15	1.69933	0.849665	−	0.527322i	$-0.176804\pi$
$719$	8.75562e15	1.69933	0.849665	+	0.527322i	$0.176804\pi$
$720$	0	0
$721$	−6.76794e15	−1.29364
$722$	2.04470e15	0.387861
$723$	0	0
$724$	−8.29235e14	−0.154923
$725$	0	0
$726$	0	0
$727$	−4.60975e15	−0.841857	−0.420929	−	0.907094i	$-0.638296\pi$
$727$	−4.60975e15	−0.841857	−0.420929	+	0.907094i	$0.638296\pi$
$728$	−1.30143e15	−0.235883
$729$	0	0
$730$	0	0
$731$	−9.23854e13	−0.0163704
$732$	0	0
$733$	7.71163e14	0.134609	0.0673046	−	0.997732i	$-0.478560\pi$
$733$	7.71163e14	0.134609	0.0673046	+	0.997732i	$0.478560\pi$
$734$	−2.27443e15	−0.394043
$735$	0	0
$736$	4.19007e15	0.715143
$737$	6.54022e15	1.10795
$738$	0	0
$739$	−2.51328e15	−0.419466	−0.209733	−	0.977759i	$-0.567259\pi$
$739$	−2.51328e15	−0.419466	−0.209733	+	0.977759i	$0.567259\pi$
$740$	0	0
$741$	0	0
$742$	6.83964e15	1.11638
$743$	−1.08955e15	−0.176526	−0.0882631	−	0.996097i	$-0.528132\pi$
$743$	−1.08955e15	−0.176526	−0.0882631	+	0.996097i	$0.528132\pi$
$744$	0	0
$745$	0	0
$746$	2.29576e14	0.0363800
$747$	0	0
$748$	−8.22685e14	−0.128462
$749$	−1.46812e16	−2.27568
$750$	0	0
$751$	−3.61242e15	−0.551795	−0.275898	−	0.961187i	$-0.588975\pi$
$751$	−3.61242e15	−0.551795	−0.275898	+	0.961187i	$0.588975\pi$
$752$	−1.70646e15	−0.258760
$753$	0	0
$754$	−6.92632e14	−0.103505
$755$	0	0
$756$	0	0
$757$	1.16985e16	1.71041	0.855207	−	0.518287i	$-0.173430\pi$
$757$	1.16985e16	1.71041	0.855207	+	0.518287i	$0.173430\pi$
$758$	−4.97238e15	−0.721745
$759$	0	0
$760$	0	0
$761$	3.29015e15	0.467304	0.233652	−	0.972320i	$-0.424932\pi$
$761$	3.29015e15	0.467304	0.233652	+	0.972320i	$0.424932\pi$
$762$	0	0
$763$	1.02549e16	1.43564
$764$	−6.66133e15	−0.925864
$765$	0	0
$766$	−1.13579e15	−0.155611
$767$	2.60090e15	0.353793
$768$	0	0
$769$	−6.00040e15	−0.804610	−0.402305	−	0.915506i	$-0.631791\pi$
$769$	−6.00040e15	−0.804610	−0.402305	+	0.915506i	$0.631791\pi$
$770$	0	0
$771$	0	0
$772$	5.59229e15	0.733998
$773$	3.98905e15	0.519856	0.259928	−	0.965628i	$-0.416301\pi$
$773$	3.98905e15	0.519856	0.259928	+	0.965628i	$0.416301\pi$
$774$	0	0
$775$	0	0
$776$	7.64015e15	0.974681
$777$	0	0
$778$	−6.65257e15	−0.836762
$779$	4.92783e15	0.615460
$780$	0	0
$781$	−3.82636e15	−0.471200
$782$	−6.40173e14	−0.0782817
$783$	0	0
$784$	1.59748e15	0.192618
$785$	0	0
$786$	0	0
$787$	−1.17707e16	−1.38976	−0.694882	−	0.719123i	$-0.744544\pi$
$787$	−1.17707e16	−1.38976	−0.694882	+	0.719123i	$0.744544\pi$
$788$	−4.91812e15	−0.576640
$789$	0	0
$790$	0	0
$791$	−1.14809e16	−1.31827
$792$	0	0
$793$	2.76821e15	0.313471
$794$	−3.05158e15	−0.343172
$795$	0	0
$796$	1.20456e16	1.33600
$797$	−6.56353e15	−0.722965	−0.361482	−	0.932379i	$-0.617729\pi$
$797$	−6.56353e15	−0.722965	−0.361482	+	0.932379i	$0.617729\pi$
$798$	0	0
$799$	1.59755e15	0.173559
$800$	0	0
$801$	0	0
$802$	−4.76118e15	−0.506705
$803$	−2.80531e15	−0.296514
$804$	0	0
$805$	0	0
$806$	−5.74937e14	−0.0595357
$807$	0	0
$808$	1.26568e16	1.29289
$809$	1.26091e15	0.127928	0.0639641	−	0.997952i	$-0.479626\pi$
$809$	1.26091e15	0.127928	0.0639641	+	0.997952i	$0.479626\pi$
$810$	0	0
$811$	−1.13759e15	−0.113860	−0.0569302	−	0.998378i	$-0.518131\pi$
$811$	−1.13759e15	−0.113860	−0.0569302	+	0.998378i	$0.518131\pi$
$812$	9.25472e15	0.920037
$813$	0	0
$814$	6.15824e15	0.603980
$815$	0	0
$816$	0	0
$817$	−3.33891e14	−0.0320910
$818$	−4.71191e15	−0.449836
$819$	0	0
$820$	0	0
$821$	1.31809e16	1.23326	0.616632	−	0.787251i	$-0.288497\pi$
$821$	1.31809e16	1.23326	0.616632	+	0.787251i	$0.288497\pi$
$822$	0	0
$823$	−1.98574e16	−1.83326	−0.916628	−	0.399742i	$-0.869100\pi$
$823$	−1.98574e16	−1.83326	−0.916628	+	0.399742i	$0.869100\pi$
$824$	9.70076e15	0.889623
$825$	0	0
$826$	1.08289e16	0.979928
$827$	−1.25548e16	−1.12857	−0.564285	−	0.825580i	$-0.690848\pi$
$827$	−1.25548e16	−1.12857	−0.564285	+	0.825580i	$0.690848\pi$
$828$	0	0
$829$	1.26916e16	1.12581	0.562905	−	0.826521i	$-0.309684\pi$
$829$	1.26916e16	1.12581	0.562905	+	0.826521i	$0.309684\pi$
$830$	0	0
$831$	0	0
$832$	3.95971e14	0.0344338
$833$	−1.49553e15	−0.129195
$834$	0	0
$835$	0	0
$836$	−2.97327e15	−0.251825
$837$	0	0
$838$	6.82103e15	0.570174
$839$	1.32436e15	0.109981	0.0549903	−	0.998487i	$-0.482487\pi$
$839$	1.32436e15	0.109981	0.0549903	+	0.998487i	$0.482487\pi$
$840$	0	0
$841$	−8.14816e14	−0.0667854
$842$	−4.81536e15	−0.392114
$843$	0	0
$844$	1.65612e16	1.33109
$845$	0	0
$846$	0	0
$847$	7.38322e15	0.581952
$848$	8.04782e15	0.630233
$849$	0	0
$850$	0	0
$851$	−1.53787e16	−1.18116
$852$	0	0
$853$	−1.99538e16	−1.51289	−0.756443	−	0.654060i	$-0.773064\pi$
$853$	−1.99538e16	−1.51289	−0.756443	+	0.654060i	$0.773064\pi$
$854$	1.15255e16	0.868245
$855$	0	0
$856$	2.10432e16	1.56497
$857$	2.21687e16	1.63812	0.819060	−	0.573709i	$-0.194496\pi$
$857$	2.21687e16	1.63812	0.819060	+	0.573709i	$0.194496\pi$
$858$	0	0
$859$	9.18375e15	0.669973	0.334987	−	0.942223i	$-0.391268\pi$
$859$	9.18375e15	0.669973	0.334987	+	0.942223i	$0.391268\pi$
$860$	0	0
$861$	0	0
$862$	−8.93967e15	−0.639781
$863$	2.31329e16	1.64502	0.822511	−	0.568750i	$-0.192573\pi$
$863$	2.31329e16	1.64502	0.822511	+	0.568750i	$0.192573\pi$
$864$	0	0
$865$	0	0
$866$	−2.33963e15	−0.163229
$867$	0	0
$868$	7.68213e15	0.529204
$869$	1.04201e15	0.0713285
$870$	0	0
$871$	−4.93026e15	−0.333250
$872$	−1.46987e16	−0.987277
$873$	0	0
$874$	−2.31365e15	−0.153456
$875$	0	0
$876$	0	0
$877$	−2.28779e16	−1.48908	−0.744541	−	0.667577i	$-0.767331\pi$
$877$	−2.28779e16	−1.48908	−0.744541	+	0.667577i	$0.767331\pi$
$878$	−5.16841e15	−0.334301
$879$	0	0
$880$	0	0
$881$	−3.00048e15	−0.190469	−0.0952344	−	0.995455i	$-0.530360\pi$
$881$	−3.00048e15	−0.190469	−0.0952344	+	0.995455i	$0.530360\pi$
$882$	0	0
$883$	−1.43409e15	−0.0899066	−0.0449533	−	0.998989i	$-0.514314\pi$
$883$	−1.43409e15	−0.0899066	−0.0449533	+	0.998989i	$0.514314\pi$
$884$	6.20171e14	0.0386388
$885$	0	0
$886$	3.51062e14	0.0216022
$887$	4.51694e15	0.276226	0.138113	−	0.990416i	$-0.455896\pi$
$887$	4.51694e15	0.276226	0.138113	+	0.990416i	$0.455896\pi$
$888$	0	0
$889$	1.96480e16	1.18675
$890$	0	0
$891$	0	0
$892$	1.86873e15	0.110800
$893$	5.77372e15	0.340230
$894$	0	0
$895$	0	0
$896$	−2.05179e16	−1.18697
$897$	0	0
$898$	1.18010e16	0.674373
$899$	9.45099e15	0.536783
$900$	0	0
$901$	−7.53421e15	−0.422719
$902$	−8.69912e15	−0.485109
$903$	0	0
$904$	1.64560e16	0.906562
$905$	0	0
$906$	0	0
$907$	3.33072e16	1.80176	0.900882	−	0.434064i	$-0.142921\pi$
$907$	3.33072e16	1.80176	0.900882	+	0.434064i	$0.142921\pi$
$908$	2.24246e16	1.20574
$909$	0	0
$910$	0	0
$911$	9.75985e15	0.515338	0.257669	−	0.966233i	$-0.417046\pi$
$911$	9.75985e15	0.515338	0.257669	+	0.966233i	$0.417046\pi$
$912$	0	0
$913$	1.69398e16	0.883729
$914$	−1.20120e16	−0.622891
$915$	0	0
$916$	2.71348e16	1.39028
$917$	2.76286e16	1.40711
$918$	0	0
$919$	8.87018e15	0.446372	0.223186	−	0.974776i	$-0.428354\pi$
$919$	8.87018e15	0.446372	0.223186	+	0.974776i	$0.428354\pi$
$920$	0	0
$921$	0	0
$922$	6.79572e15	0.335904
$923$	2.88445e15	0.141728
$924$	0	0
$925$	0	0
$926$	−3.09389e15	−0.149329
$927$	0	0
$928$	−2.07918e16	−0.991696
$929$	−2.79034e16	−1.32303	−0.661517	−	0.749930i	$-0.730087\pi$
$929$	−2.79034e16	−1.32303	−0.661517	+	0.749930i	$0.730087\pi$
$930$	0	0
$931$	−5.40499e15	−0.253263
$932$	7.00428e15	0.326269
$933$	0	0
$934$	6.70730e15	0.308773
$935$	0	0
$936$	0	0
$937$	−1.42598e16	−0.644978	−0.322489	−	0.946573i	$-0.604520\pi$
$937$	−1.42598e16	−0.644978	−0.322489	+	0.946573i	$0.604520\pi$
$938$	−2.05273e16	−0.923030
$939$	0	0
$940$	0	0
$941$	4.11724e16	1.81913	0.909563	−	0.415566i	$-0.136416\pi$
$941$	4.11724e16	1.81913	0.909563	+	0.415566i	$0.136416\pi$
$942$	0	0
$943$	2.17239e16	0.948690
$944$	1.27418e16	0.553202
$945$	0	0
$946$	5.89418e14	0.0252943
$947$	2.03073e16	0.866419	0.433209	−	0.901293i	$-0.357381\pi$
$947$	2.03073e16	0.866419	0.433209	+	0.901293i	$0.357381\pi$
$948$	0	0
$949$	2.11475e15	0.0891856
$950$	0	0
$951$	0	0
$952$	5.96878e15	0.247390
$953$	−3.65143e16	−1.50471	−0.752353	−	0.658760i	$-0.771081\pi$
$953$	−3.65143e16	−1.50471	−0.752353	+	0.658760i	$0.771081\pi$
$954$	0	0
$955$	0	0
$956$	−1.19375e14	−0.00483498
$957$	0	0
$958$	−1.94308e16	−0.778003
$959$	−1.72700e16	−0.687527
$960$	0	0
$961$	−1.75634e16	−0.691243
$962$	−4.64231e15	−0.181665
$963$	0	0
$964$	1.65953e16	0.642037
$965$	0	0
$966$	0	0
$967$	2.33229e16	0.887027	0.443513	−	0.896268i	$-0.353732\pi$
$967$	2.33229e16	0.887027	0.443513	+	0.896268i	$0.353732\pi$
$968$	−1.05827e16	−0.400203
$969$	0	0
$970$	0	0
$971$	1.97670e16	0.734910	0.367455	−	0.930041i	$-0.380229\pi$
$971$	1.97670e16	0.734910	0.367455	+	0.930041i	$0.380229\pi$
$972$	0	0
$973$	1.00115e16	0.368027
$974$	8.04437e15	0.294048
$975$	0	0
$976$	1.35615e16	0.490154
$977$	1.51393e16	0.544109	0.272055	−	0.962282i	$-0.412297\pi$
$977$	1.51393e16	0.544109	0.272055	+	0.962282i	$0.412297\pi$
$978$	0	0
$979$	2.30284e16	0.818387
$980$	0	0
$981$	0	0
$982$	6.21498e15	0.217183
$983$	1.59131e16	0.552982	0.276491	−	0.961017i	$-0.410828\pi$
$983$	1.59131e16	0.552982	0.276491	+	0.961017i	$0.410828\pi$
$984$	0	0
$985$	0	0
$986$	3.17664e15	0.108554
$987$	0	0
$988$	2.24136e15	0.0757441
$989$	−1.47193e15	−0.0494661
$990$	0	0
$991$	−3.31820e16	−1.10280	−0.551402	−	0.834240i	$-0.685907\pi$
$991$	−3.31820e16	−1.10280	−0.551402	+	0.834240i	$0.685907\pi$
$992$	−1.72588e16	−0.570422
$993$	0	0
$994$	1.20095e16	0.392554
$995$	0	0
$996$	0	0
$997$	−3.89617e16	−1.25261	−0.626303	−	0.779579i	$-0.715433\pi$
$997$	−3.89617e16	−1.25261	−0.626303	+	0.779579i	$0.715433\pi$
$998$	−1.11191e16	−0.355511
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.12.a.z.1.3		8
3.2	odd	2	inner	225.12.a.z.1.5		8
5.2	odd	4		45.12.b.c.19.4	yes	8
5.3	odd	4		45.12.b.c.19.6	yes	8
5.4	even	2	inner	225.12.a.z.1.6		8
15.2	even	4		45.12.b.c.19.5	yes	8
15.8	even	4		45.12.b.c.19.3	✓	8
15.14	odd	2	inner	225.12.a.z.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.12.b.c.19.3	✓	8	15.8	even	4
45.12.b.c.19.4	yes	8	5.2	odd	4
45.12.b.c.19.5	yes	8	15.2	even	4
45.12.b.c.19.6	yes	8	5.3	odd	4
225.12.a.z.1.3		8	1.1	even	1	trivial
225.12.a.z.1.4		8	15.14	odd	2	inner
225.12.a.z.1.5		8	3.2	odd	2	inner
225.12.a.z.1.6		8	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-22.0579 q^{2} -1561.45 q^{4} -55546.4 q^{7} +79616.9 q^{8} -390374. q^{11} +294278. q^{13} +1.22524e6 q^{14} +1.44167e6 q^{16} -1.34966e6 q^{17} -4.87782e6 q^{19} +8.61083e6 q^{22} -2.15035e7 q^{23} -6.49116e6 q^{26} +8.67329e7 q^{28} +1.06704e8 q^{29} +8.85723e7 q^{31} -1.94855e8 q^{32} +2.97707e7 q^{34} +7.15174e8 q^{37} +1.07594e8 q^{38} -1.01025e9 q^{41} +6.84508e7 q^{43} +6.09549e8 q^{44} +4.74321e8 q^{46} -1.18367e9 q^{47} +1.10807e9 q^{49} -4.59501e8 q^{52} +5.58230e9 q^{53} -4.42243e9 q^{56} -2.35366e9 q^{58} +8.83822e9 q^{59} +9.40678e9 q^{61} -1.95372e9 q^{62} +1.34557e9 q^{64} -1.67537e10 q^{67} +2.10743e9 q^{68} +9.80178e9 q^{71} +7.18622e9 q^{73} -1.57752e10 q^{74} +7.61646e9 q^{76} +2.16839e10 q^{77} -2.66926e9 q^{79} +2.22841e10 q^{82} -4.33938e10 q^{83} -1.50988e9 q^{86} -3.10803e10 q^{88} -5.89906e10 q^{89} -1.63461e10 q^{91} +3.35766e10 q^{92} +2.61093e10 q^{94} +9.59614e10 q^{97} -2.44418e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8976 q^{4} + 15740992 q^{16} - 61220032 q^{19} + 279698464 q^{31} - 309623120 q^{34} - 13964841760 q^{46} + 1596098056 q^{49} - 4923969584 q^{61} - 40933220096 q^{64} - 128254720704 q^{76} - 87780965728 q^{79}+ \cdots + 460752353440 q^{94}+O(q^{100}) \) 8 * q + 8976 * q^4 + 15740992 * q^16 - 61220032 * q^19 + 279698464 * q^31 - 309623120 * q^34 - 13964841760 * q^46 + 1596098056 * q^49 - 4923969584 * q^61 - 40933220096 * q^64 - 128254720704 * q^76 - 87780965728 * q^79 + 71755992000 * q^91 + 460752353440 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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