LMFDB - Embedded newform 225.9.d.c.224.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,9,Mod(224,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.224"); S:= CuspForms(chi, 9); N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	225.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$91.6601872638$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			224.5
Character	$\chi$	$=$	225.224
Dual form			225.9.d.c.224.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-23.2704 q^{2} +285.513 q^{4} -4399.16i q^{7} -686.784 q^{8} -16184.6i q^{11} -2771.50i q^{13} +102370. i q^{14} -57109.6 q^{16} -26479.8 q^{17} -186497. q^{19} +376622. i q^{22} -52174.8 q^{23} +64494.0i q^{26} -1.25602e6i q^{28} +928476. i q^{29} -960982. q^{31} +1.50478e6 q^{32} +616197. q^{34} +2.30066e6i q^{37} +4.33986e6 q^{38} -3.41833e6i q^{41} +4.56761e6i q^{43} -4.62091e6i q^{44} +1.21413e6 q^{46} -3.39999e6 q^{47} -1.35878e7 q^{49} -791299. i q^{52} +6.12740e6 q^{53} +3.02127e6i q^{56} -2.16060e7i q^{58} -1.07944e7i q^{59} -7.04198e6 q^{61} +2.23625e7 q^{62} -2.03969e7 q^{64} -2.46307e7i q^{67} -7.56034e6 q^{68} +3.67232e7i q^{71} -3.11078e7i q^{73} -5.35374e7i q^{74} -5.32472e7 q^{76} -7.11986e7 q^{77} -6.46442e7 q^{79} +7.95461e7i q^{82} +7.65633e7 q^{83} -1.06290e8i q^{86} +1.11153e7i q^{88} -5.07750e7i q^{89} -1.21923e7 q^{91} -1.48966e7 q^{92} +7.91192e7 q^{94} -8.64077e7i q^{97} +3.16194e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 5048 q^{4} + 595144 q^{16} - 249920 q^{19} - 1241936 q^{31} - 14972992 q^{34} - 14323536 q^{46} - 42020312 q^{49} + 56906192 q^{61} + 338779896 q^{64} - 511347840 q^{76} + 127619024 q^{79} - 593698576 q^{91}+ \cdots - 51203776 q^{94}+O(q^{100}) $ 24 * q + 5048 * q^4 + 595144 * q^16 - 249920 * q^19 - 1241936 * q^31 - 14972992 * q^34 - 14323536 * q^46 - 42020312 * q^49 + 56906192 * q^61 + 338779896 * q^64 - 511347840 * q^76 + 127619024 * q^79 - 593698576 * q^91 - 51203776 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−23.2704	−1.45440	−0.727201	−	0.686424i	$-0.759179\pi$
$2$	−23.2704	−1.45440	−0.727201	+	0.686424i	$0.759179\pi$
$3$	0	0
$3$	0	0
$4$	285.513	1.11529
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4399.16i	− 1.83222i	−0.400927	−	0.916110i	$-0.631312\pi$
$7$	− 4399.16i	− 1.83222i	0.400927	−	0.916110i	$-0.368688\pi$
$8$	−686.784	−0.167672
$9$	0	0
$10$	0	0
$11$	− 16184.6i	− 1.10543i	−0.833371	−	0.552714i	$-0.813592\pi$
$11$	− 16184.6i	− 1.10543i	0.833371	−	0.552714i	$-0.186408\pi$
$12$	0	0
$13$	− 2771.50i	− 0.0970379i	−0.998822	−	0.0485189i	$-0.984550\pi$
$13$	− 2771.50i	− 0.0970379i	0.998822	−	0.0485189i	$-0.0154501\pi$
$14$	102370.i	2.66478i
$15$	0	0
$16$	−57109.6	−0.871424
$17$	−26479.8	−0.317044	−0.158522	−	0.987355i	$-0.550673\pi$
$17$	−26479.8	−0.317044	−0.158522	+	0.987355i	$0.550673\pi$
$18$	0	0
$19$	−186497.	−1.43106	−0.715528	−	0.698584i	$-0.753814\pi$
$19$	−186497.	−1.43106	−0.715528	+	0.698584i	$0.753814\pi$
$20$	0	0
$21$	0	0
$22$	376622.i	1.60774i
$23$	−52174.8	−0.186445	−0.0932223	−	0.995645i	$-0.529717\pi$
$23$	−52174.8	−0.186445	−0.0932223	+	0.995645i	$0.529717\pi$
$24$	0	0
$25$	0	0
$26$	64494.0i	0.141132i
$27$	0	0
$28$	− 1.25602e6i	− 2.04345i
$29$	928476.i	1.31274i	0.754439	+	0.656370i	$0.227909\pi$
$29$	928476.i	1.31274i	−0.754439	+	0.656370i	$0.772091\pi$
$30$	0	0
$31$	−960982.	−1.04056	−0.520282	−	0.853995i	$-0.674173\pi$
$31$	−960982.	−1.04056	−0.520282	+	0.853995i	$0.674173\pi$
$32$	1.50478e6	1.43507
$33$	0	0
$34$	616197.	0.461109
$35$	0	0
$36$	0	0
$37$	2.30066e6i	1.22757i	0.789474	+	0.613784i	$0.210354\pi$
$37$	2.30066e6i	1.22757i	−0.789474	+	0.613784i	$0.789646\pi$
$38$	4.33986e6	2.08133
$39$	0	0
$40$	0	0
$41$	− 3.41833e6i	− 1.20970i	−0.796338	−	0.604852i	$-0.793232\pi$
$41$	− 3.41833e6i	− 1.20970i	0.796338	−	0.604852i	$-0.206768\pi$
$42$	0	0
$43$	4.56761e6i	1.33603i	0.744149	+	0.668013i	$0.232855\pi$
$43$	4.56761e6i	1.33603i	−0.744149	+	0.668013i	$0.767145\pi$
$44$	− 4.62091e6i	− 1.23287i
$45$	0	0
$46$	1.21413e6	0.271165
$47$	−3.39999e6	−0.696764	−0.348382	−	0.937353i	$-0.613269\pi$
$47$	−3.39999e6	−0.696764	−0.348382	+	0.937353i	$0.613269\pi$
$48$	0	0
$49$	−1.35878e7	−2.35703
$50$	0	0
$51$	0	0
$52$	− 791299.i	− 0.108225i
$53$	6.12740e6	0.776556	0.388278	−	0.921542i	$-0.373070\pi$
$53$	6.12740e6	0.776556	0.388278	+	0.921542i	$0.373070\pi$
$54$	0	0
$55$	0	0
$56$	3.02127e6i	0.307211i
$57$	0	0
$58$	− 2.16060e7i	− 1.90925i
$59$	− 1.07944e7i	− 0.890824i	−0.895326	−	0.445412i	$-0.853057\pi$
$59$	− 1.07944e7i	− 0.890824i	0.895326	−	0.445412i	$-0.146943\pi$
$60$	0	0
$61$	−7.04198e6	−0.508599	−0.254300	−	0.967125i	$-0.581845\pi$
$61$	−7.04198e6	−0.508599	−0.254300	+	0.967125i	$0.581845\pi$
$62$	2.23625e7	1.51340
$63$	0	0
$64$	−2.03969e7	−1.21575
$65$	0	0
$66$	0	0
$67$	− 2.46307e7i	− 1.22230i	−0.791514	−	0.611151i	$-0.790707\pi$
$67$	− 2.46307e7i	− 1.22230i	0.791514	−	0.611151i	$-0.209293\pi$
$68$	−7.56034e6	−0.353594
$69$	0	0
$70$	0	0
$71$	3.67232e7i	1.44513i	0.691303	+	0.722565i	$0.257037\pi$
$71$	3.67232e7i	1.44513i	−0.691303	+	0.722565i	$0.742963\pi$
$72$	0	0
$73$	− 3.11078e7i	− 1.09541i	−0.836671	−	0.547706i	$-0.815501\pi$
$73$	− 3.11078e7i	− 1.09541i	0.836671	−	0.547706i	$-0.184499\pi$
$74$	− 5.35374e7i	− 1.78538i
$75$	0	0
$76$	−5.32472e7	−1.59604
$77$	−7.11986e7	−2.02539
$78$	0	0
$79$	−6.46442e7	−1.65967	−0.829834	−	0.558010i	$-0.811565\pi$
$79$	−6.46442e7	−1.65967	−0.829834	+	0.558010i	$0.811565\pi$
$80$	0	0
$81$	0	0
$82$	7.95461e7i	1.75940i
$83$	7.65633e7	1.61328	0.806638	−	0.591046i	$-0.201285\pi$
$83$	7.65633e7	1.61328	0.806638	+	0.591046i	$0.201285\pi$
$84$	0	0
$85$	0	0
$86$	− 1.06290e8i	− 1.94312i
$87$	0	0
$88$	1.11153e7i	0.185349i
$89$	− 5.07750e7i	− 0.809264i	−0.914480	−	0.404632i	$-0.867400\pi$
$89$	− 5.07750e7i	− 0.809264i	0.914480	−	0.404632i	$-0.132600\pi$
$90$	0	0
$91$	−1.21923e7	−0.177795
$92$	−1.48966e7	−0.207939
$93$	0	0
$94$	7.91192e7	1.01338
$95$	0	0
$96$	0	0
$97$	− 8.64077e7i	− 0.976035i	−0.872834	−	0.488017i	$-0.837720\pi$
$97$	− 8.64077e7i	− 0.976035i	0.872834	−	0.488017i	$-0.162280\pi$
$98$	3.16194e8	3.42807
$99$	0	0
$100$	0	0
$101$	1.25688e8i	1.20783i	0.797047	+	0.603917i	$0.206394\pi$
$101$	1.25688e8i	1.20783i	−0.797047	+	0.603917i	$0.793606\pi$
$102$	0	0
$103$	− 2.63087e7i	− 0.233750i	−0.993147	−	0.116875i	$-0.962712\pi$
$103$	− 2.63087e7i	− 0.233750i	0.993147	−	0.116875i	$-0.0372876\pi$
$104$	1.90342e6i	0.0162705i
$105$	0	0
$106$	−1.42587e8	−1.12942
$107$	2.49819e8	1.90586	0.952929	−	0.303193i	$-0.0980527\pi$
$107$	2.49819e8	1.90586	0.952929	+	0.303193i	$0.0980527\pi$
$108$	0	0
$109$	5.80899e6	0.0411523	0.0205762	−	0.999788i	$-0.493450\pi$
$109$	5.80899e6	0.0411523	0.0205762	+	0.999788i	$0.493450\pi$
$110$	0	0
$111$	0	0
$112$	2.51234e8i	1.59664i
$113$	−1.50367e8	−0.922228	−0.461114	−	0.887341i	$-0.652550\pi$
$113$	−1.50367e8	−0.922228	−0.461114	+	0.887341i	$0.652550\pi$
$114$	0	0
$115$	0	0
$116$	2.65092e8i	1.46408i
$117$	0	0
$118$	2.51191e8i	1.29562i
$119$	1.16489e8i	0.580894i
$120$	0	0
$121$	−4.75819e7	−0.221973
$122$	1.63870e8	0.739708
$123$	0	0
$124$	−2.74373e8	−1.16053
$125$	0	0
$126$	0	0
$127$	2.68993e8i	1.03401i	0.855982	+	0.517006i	$0.172954\pi$
$127$	2.68993e8i	1.03401i	−0.855982	+	0.517006i	$0.827046\pi$
$128$	8.94199e7	0.333115
$129$	0	0
$130$	0	0
$131$	2.59837e8i	0.882298i	0.897434	+	0.441149i	$0.145429\pi$
$131$	2.59837e8i	0.882298i	−0.897434	+	0.441149i	$0.854571\pi$
$132$	0	0
$133$	8.20428e8i	2.62201i
$134$	5.73168e8i	1.77772i
$135$	0	0
$136$	1.81859e7	0.0531593
$137$	2.23476e8	0.634378	0.317189	−	0.948362i	$-0.397261\pi$
$137$	2.23476e8	0.634378	0.317189	+	0.948362i	$0.397261\pi$
$138$	0	0
$139$	3.10970e8	0.833028	0.416514	−	0.909129i	$-0.363252\pi$
$139$	3.10970e8	0.833028	0.416514	+	0.909129i	$0.363252\pi$
$140$	0	0
$141$	0	0
$142$	− 8.54564e8i	− 2.10180i
$143$	−4.48556e7	−0.107269
$144$	0	0
$145$	0	0
$146$	7.23891e8i	1.59317i
$147$	0	0
$148$	6.56869e8i	1.36909i
$149$	191472.i	0 0.000388473i	1.00000		0.000194236i	$6.18274e-5\pi$
$149$	191472.i	0 0.000388473i	−1.00000		0.000194236i	$0.999938\pi$
$150$	0	0
$151$	5.84355e8	1.12401	0.562004	−	0.827135i	$-0.310031\pi$
$151$	5.84355e8	1.12401	0.562004	+	0.827135i	$0.310031\pi$
$152$	1.28083e8	0.239948
$153$	0	0
$154$	1.65682e9	2.94573
$155$	0	0
$156$	0	0
$157$	7.91625e7i	0.130293i	0.997876	+	0.0651464i	$0.0207515\pi$
$157$	7.91625e7i	0.130293i	−0.997876	+	0.0651464i	$0.979249\pi$
$158$	1.50430e9	2.41383
$159$	0	0
$160$	0	0
$161$	2.29525e8i	0.341607i
$162$	0	0
$163$	3.52707e8i	0.499648i	0.968291	+	0.249824i	$0.0803727\pi$
$163$	3.52707e8i	0.499648i	−0.968291	+	0.249824i	$0.919627\pi$
$164$	− 9.75979e8i	− 1.34917i
$165$	0	0
$166$	−1.78166e9	−2.34635
$167$	−7.67131e8	−0.986288	−0.493144	−	0.869948i	$-0.664152\pi$
$167$	−7.67131e8	−0.986288	−0.493144	+	0.869948i	$0.664152\pi$
$168$	0	0
$169$	8.08050e8	0.990584
$170$	0	0
$171$	0	0
$172$	1.30411e9i	1.49005i
$173$	−3.74444e8	−0.418025	−0.209012	−	0.977913i	$-0.567025\pi$
$173$	−3.74444e8	−0.418025	−0.209012	+	0.977913i	$0.567025\pi$
$174$	0	0
$175$	0	0
$176$	9.24295e8i	0.963297i
$177$	0	0
$178$	1.18156e9i	1.17699i
$179$	− 5.04945e8i	− 0.491849i	−0.969289	−	0.245924i	$-0.920909\pi$
$179$	− 5.04945e8i	− 0.491849i	0.969289	−	0.245924i	$-0.0790915\pi$
$180$	0	0
$181$	1.13823e9	1.06051	0.530255	−	0.847838i	$-0.322096\pi$
$181$	1.13823e9	1.06051	0.530255	+	0.847838i	$0.322096\pi$
$182$	2.83719e8	0.258585
$183$	0	0
$184$	3.58328e7	0.0312615
$185$	0	0
$186$	0	0
$187$	4.28565e8i	0.350469i
$188$	−9.70741e8	−0.777091
$189$	0	0
$190$	0	0
$191$	− 5.34154e8i	− 0.401359i	−0.979657	−	0.200680i	$-0.935685\pi$
$191$	− 5.34154e8i	− 0.401359i	0.979657	−	0.200680i	$-0.0643150\pi$
$192$	0	0
$193$	− 1.65692e9i	− 1.19418i	−0.802173	−	0.597092i	$-0.796323\pi$
$193$	− 1.65692e9i	− 1.19418i	0.802173	−	0.597092i	$-0.203677\pi$
$194$	2.01074e9i	1.41955i
$195$	0	0
$196$	−3.87950e9	−2.62876
$197$	1.87487e8	0.124482	0.0622408	−	0.998061i	$-0.480175\pi$
$197$	1.87487e8	0.124482	0.0622408	+	0.998061i	$0.480175\pi$
$198$	0	0
$199$	−4.54531e8	−0.289835	−0.144918	−	0.989444i	$-0.546292\pi$
$199$	−4.54531e8	−0.289835	−0.144918	+	0.989444i	$0.546292\pi$
$200$	0	0
$201$	0	0
$202$	− 2.92481e9i	− 1.75668i
$203$	4.08451e9	2.40523
$204$	0	0
$205$	0	0
$206$	6.12215e8i	0.339966i
$207$	0	0
$208$	1.58279e8i	0.0845611i
$209$	3.01837e9i	1.58193i
$210$	0	0
$211$	2.53860e9	1.28075	0.640376	−	0.768062i	$-0.278779\pi$
$211$	2.53860e9	1.28075	0.640376	+	0.768062i	$0.278779\pi$
$212$	1.74945e9	0.866082
$213$	0	0
$214$	−5.81340e9	−2.77188
$215$	0	0
$216$	0	0
$217$	4.22751e9i	1.90654i
$218$	−1.35178e8	−0.0598521
$219$	0	0
$220$	0	0
$221$	7.33888e7i	0.0307653i
$222$	0	0
$223$	− 1.35915e9i	− 0.549602i	−0.961501	−	0.274801i	$-0.911388\pi$
$223$	− 1.35915e9i	− 0.549602i	0.961501	−	0.274801i	$-0.0886120\pi$
$224$	− 6.61978e9i	− 2.62937i
$225$	0	0
$226$	3.49910e9	1.34129
$227$	9.20041e8	0.346500	0.173250	−	0.984878i	$-0.444573\pi$
$227$	9.20041e8	0.346500	0.173250	+	0.984878i	$0.444573\pi$
$228$	0	0
$229$	−4.11927e9	−1.49789	−0.748943	−	0.662634i	$-0.769438\pi$
$229$	−4.11927e9	−1.49789	−0.748943	+	0.662634i	$0.769438\pi$
$230$	0	0
$231$	0	0
$232$	− 6.37662e8i	− 0.220109i
$233$	−1.65614e9	−0.561918	−0.280959	−	0.959720i	$-0.590653\pi$
$233$	−1.65614e9	−0.561918	−0.280959	+	0.959720i	$0.590653\pi$
$234$	0	0
$235$	0	0
$236$	− 3.08195e9i	− 0.993523i
$237$	0	0
$238$	− 2.71075e9i	− 0.844853i
$239$	− 2.82808e9i	− 0.866761i	−0.901211	−	0.433380i	$-0.857321\pi$
$239$	− 2.82808e9i	− 0.866761i	0.901211	−	0.433380i	$-0.142679\pi$
$240$	0	0
$241$	−5.61470e9	−1.66440	−0.832201	−	0.554474i	$-0.812920\pi$
$241$	−5.61470e9	−1.66440	−0.832201	+	0.554474i	$0.812920\pi$
$242$	1.10725e9	0.322838
$243$	0	0
$244$	−2.01058e9	−0.567233
$245$	0	0
$246$	0	0
$247$	5.16875e8i	0.138867i
$248$	6.59987e8	0.174473
$249$	0	0
$250$	0	0
$251$	3.82921e9i	0.964749i	0.875965	+	0.482375i	$0.160226\pi$
$251$	3.82921e9i	0.964749i	−0.875965	+	0.482375i	$0.839774\pi$
$252$	0	0
$253$	8.44428e8i	0.206101i
$254$	− 6.25958e9i	− 1.50387i
$255$	0	0
$256$	3.14076e9	0.731265
$257$	−5.39550e9	−1.23680	−0.618399	−	0.785864i	$-0.712219\pi$
$257$	−5.39550e9	−1.23680	−0.618399	+	0.785864i	$0.712219\pi$
$258$	0	0
$259$	1.01210e10	2.24918
$260$	0	0
$261$	0	0
$262$	− 6.04651e9i	− 1.28322i
$263$	2.65139e9	0.554180	0.277090	−	0.960844i	$-0.410630\pi$
$263$	2.65139e9	0.554180	0.277090	+	0.960844i	$0.410630\pi$
$264$	0	0
$265$	0	0
$266$	− 1.90917e10i	− 3.81346i
$267$	0	0
$268$	− 7.03240e9i	− 1.36322i
$269$	5.01721e9i	0.958193i	0.877762	+	0.479097i	$0.159036\pi$
$269$	5.01721e9i	0.958193i	−0.877762	+	0.479097i	$0.840964\pi$
$270$	0	0
$271$	−1.66334e9	−0.308393	−0.154196	−	0.988040i	$-0.549279\pi$
$271$	−1.66334e9	−0.308393	−0.154196	+	0.988040i	$0.549279\pi$
$272$	1.51225e9	0.276279
$273$	0	0
$274$	−5.20038e9	−0.922641
$275$	0	0
$276$	0	0
$277$	3.31124e9i	0.562434i	0.959644	+	0.281217i	$0.0907381\pi$
$277$	3.31124e9i	0.562434i	−0.959644	+	0.281217i	$0.909262\pi$
$278$	−7.23641e9	−1.21156
$279$	0	0
$280$	0	0
$281$	6.52601e9i	1.04670i	0.852118	+	0.523350i	$0.175318\pi$
$281$	6.52601e9i	1.04670i	−0.852118	+	0.523350i	$0.824682\pi$
$282$	0	0
$283$	− 1.10332e10i	− 1.72010i	−0.510207	−	0.860052i	$-0.670431\pi$
$283$	− 1.10332e10i	− 1.72010i	0.510207	−	0.860052i	$-0.329569\pi$
$284$	1.04849e10i	1.61173i
$285$	0	0
$286$	1.04381e9	0.156012
$287$	−1.50378e10	−2.21644
$288$	0	0
$289$	−6.27458e9	−0.899483
$290$	0	0
$291$	0	0
$292$	− 8.88167e9i	− 1.22170i
$293$	−1.28273e10	−1.74046	−0.870230	−	0.492646i	$-0.836030\pi$
$293$	−1.28273e10	−1.74046	−0.870230	+	0.492646i	$0.836030\pi$
$294$	0	0
$295$	0	0
$296$	− 1.58006e9i	− 0.205829i
$297$	0	0
$298$	− 4.45564e6i	0 0.000564996i
$299$	1.44603e8i	0.0180922i
$300$	0	0
$301$	2.00936e10	2.44789
$302$	−1.35982e10	−1.63476
$303$	0	0
$304$	1.06508e10	1.24706
$305$	0	0
$306$	0	0
$307$	8.62442e9i	0.970904i	0.874263	+	0.485452i	$0.161345\pi$
$307$	8.62442e9i	0.970904i	−0.874263	+	0.485452i	$0.838655\pi$
$308$	−2.03281e10	−2.25889
$309$	0	0
$310$	0	0
$311$	− 2.03281e9i	− 0.217298i	−0.994080	−	0.108649i	$-0.965348\pi$
$311$	− 2.03281e9i	− 0.217298i	0.994080	−	0.108649i	$-0.0346524\pi$
$312$	0	0
$313$	7.06926e8i	0.0736541i	0.999322	+	0.0368270i	$0.0117251\pi$
$313$	7.06926e8i	0.0736541i	−0.999322	+	0.0368270i	$0.988275\pi$
$314$	− 1.84215e9i	− 0.189498i
$315$	0	0
$316$	−1.84568e10	−1.85100
$317$	1.51681e10	1.50208	0.751041	−	0.660255i	$-0.229552\pi$
$317$	1.51681e10	1.50208	0.751041	+	0.660255i	$0.229552\pi$
$318$	0	0
$319$	1.50270e10	1.45114
$320$	0	0
$321$	0	0
$322$	− 5.34116e9i	− 0.496835i
$323$	4.93840e9	0.453707
$324$	0	0
$325$	0	0
$326$	− 8.20765e9i	− 0.726689i
$327$	0	0
$328$	2.34766e9i	0.202833i
$329$	1.49571e10i	1.27663i
$330$	0	0
$331$	2.23026e10	1.85799	0.928994	−	0.370095i	$-0.120675\pi$
$331$	2.23026e10	1.85799	0.928994	+	0.370095i	$0.120675\pi$
$332$	2.18598e10	1.79926
$333$	0	0
$334$	1.78515e10	1.43446
$335$	0	0
$336$	0	0
$337$	− 4.40375e9i	− 0.341431i	−0.985320	−	0.170716i	$-0.945392\pi$
$337$	− 4.40375e9i	− 0.341431i	0.985320	−	0.170716i	$-0.0546079\pi$
$338$	−1.88037e10	−1.44071
$339$	0	0
$340$	0	0
$341$	1.55531e10i	1.15027i
$342$	0	0
$343$	3.44146e10i	2.48637i
$344$	− 3.13696e9i	− 0.224014i
$345$	0	0
$346$	8.71347e9	0.607976
$347$	5.58698e9	0.385353	0.192677	−	0.981262i	$-0.438283\pi$
$347$	5.58698e9	0.385353	0.192677	+	0.981262i	$0.438283\pi$
$348$	0	0
$349$	−1.47015e10	−0.990969	−0.495485	−	0.868617i	$-0.665010\pi$
$349$	−1.47015e10	−0.990969	−0.495485	+	0.868617i	$0.665010\pi$
$350$	0	0
$351$	0	0
$352$	− 2.43543e10i	− 1.58637i
$353$	−5.68423e7	−0.00366078	−0.00183039	−	0.999998i	$-0.500583\pi$
$353$	−5.68423e7	−0.00366078	−0.00183039	+	0.999998i	$0.500583\pi$
$354$	0	0
$355$	0	0
$356$	− 1.44969e10i	− 0.902560i
$357$	0	0
$358$	1.17503e10i	0.715346i
$359$	9.56361e9i	0.575763i	0.957666	+	0.287882i	$0.0929510\pi$
$359$	9.56361e9i	0.575763i	−0.957666	+	0.287882i	$0.907049\pi$
$360$	0	0
$361$	1.77974e10	1.04792
$362$	−2.64870e10	−1.54241
$363$	0	0
$364$	−3.48105e9	−0.198292
$365$	0	0
$366$	0	0
$367$	2.94912e10i	1.62565i	0.582505	+	0.812827i	$0.302073\pi$
$367$	2.94912e10i	1.62565i	−0.582505	+	0.812827i	$0.697927\pi$
$368$	2.97968e9	0.162472
$369$	0	0
$370$	0	0
$371$	− 2.69554e10i	− 1.42282i
$372$	0	0
$373$	− 6.87435e9i	− 0.355137i	−0.984108	−	0.177569i	$-0.943177\pi$
$373$	− 6.87435e9i	− 0.355137i	0.984108	−	0.177569i	$-0.0568232\pi$
$374$	− 9.97289e9i	− 0.509723i
$375$	0	0
$376$	2.33506e9	0.116828
$377$	2.57327e9	0.127385
$378$	0	0
$379$	−6.23700e9	−0.302287	−0.151143	−	0.988512i	$-0.548296\pi$
$379$	−6.23700e9	−0.302287	−0.151143	+	0.988512i	$0.548296\pi$
$380$	0	0
$381$	0	0
$382$	1.24300e10i	0.583738i
$383$	2.51985e10	1.17106	0.585530	−	0.810651i	$-0.300886\pi$
$383$	2.51985e10	1.17106	0.585530	+	0.810651i	$0.300886\pi$
$384$	0	0
$385$	0	0
$386$	3.85572e10i	1.73682i
$387$	0	0
$388$	− 2.46705e10i	− 1.08856i
$389$	1.81758e10i	0.793771i	0.917868	+	0.396886i	$0.129909\pi$
$389$	1.81758e10i	0.793771i	−0.917868	+	0.396886i	$0.870091\pi$
$390$	0	0
$391$	1.38158e9	0.0591111
$392$	9.33188e9	0.395207
$393$	0	0
$394$	−4.36290e9	−0.181046
$395$	0	0
$396$	0	0
$397$	− 7.27352e9i	− 0.292808i	−0.989225	−	0.146404i	$-0.953230\pi$
$397$	− 7.27352e9i	− 0.292808i	0.989225	−	0.146404i	$-0.0467699\pi$
$398$	1.05771e10	0.421537
$399$	0	0
$400$	0	0
$401$	2.20001e10i	0.850839i	0.904996	+	0.425420i	$0.139874\pi$
$401$	2.20001e10i	0.850839i	−0.904996	+	0.425420i	$0.860126\pi$
$402$	0	0
$403$	2.66336e9i	0.100974i
$404$	3.58855e10i	1.34708i
$405$	0	0
$406$	−9.50484e10	−3.49817
$407$	3.72352e10	1.35699
$408$	0	0
$409$	1.79326e8	0.00640840	0.00320420	−	0.999995i	$-0.498980\pi$
$409$	1.79326e8	0.00640840	0.00320420	+	0.999995i	$0.498980\pi$
$410$	0	0
$411$	0	0
$412$	− 7.51148e9i	− 0.260698i
$413$	−4.74864e10	−1.63218
$414$	0	0
$415$	0	0
$416$	− 4.17050e9i	− 0.139256i
$417$	0	0
$418$	− 7.02388e10i	− 2.30076i
$419$	− 3.72051e10i	− 1.20711i	−0.797322	−	0.603554i	$-0.793751\pi$
$419$	− 3.72051e10i	− 1.20711i	0.797322	−	0.603554i	$-0.206249\pi$
$420$	0	0
$421$	−4.38448e9	−0.139569	−0.0697847	−	0.997562i	$-0.522231\pi$
$421$	−4.38448e9	−0.139569	−0.0697847	+	0.997562i	$0.522231\pi$
$422$	−5.90744e10	−1.86273
$423$	0	0
$424$	−4.20820e9	−0.130206
$425$	0	0
$426$	0	0
$427$	3.09788e10i	0.931866i
$428$	7.13266e10	2.12558
$429$	0	0
$430$	0	0
$431$	− 2.16425e10i	− 0.627190i	−0.949557	−	0.313595i	$-0.898467\pi$
$431$	− 2.16425e10i	− 0.627190i	0.949557	−	0.313595i	$-0.101533\pi$
$432$	0	0
$433$	− 6.74698e9i	− 0.191937i	−0.995384	−	0.0959683i	$-0.969405\pi$
$433$	− 6.74698e9i	− 0.191937i	0.995384	−	0.0959683i	$-0.0305948\pi$
$434$	− 9.83761e10i	− 2.77288i
$435$	0	0
$436$	1.65854e9	0.0458966
$437$	9.73043e9	0.266813
$438$	0	0
$439$	9.51260e9	0.256119	0.128059	−	0.991767i	$-0.459125\pi$
$439$	9.51260e9	0.256119	0.128059	+	0.991767i	$0.459125\pi$
$440$	0	0
$441$	0	0
$442$	− 1.70779e9i	− 0.0447451i
$443$	3.20737e10	0.832788	0.416394	−	0.909184i	$-0.363294\pi$
$443$	3.20737e10	0.832788	0.416394	+	0.909184i	$0.363294\pi$
$444$	0	0
$445$	0	0
$446$	3.16281e10i	0.799343i
$447$	0	0
$448$	8.97291e10i	2.22752i
$449$	− 2.94112e10i	− 0.723648i	−0.932246	−	0.361824i	$-0.882154\pi$
$449$	− 2.94112e10i	− 0.723648i	0.932246	−	0.361824i	$-0.117846\pi$
$450$	0	0
$451$	−5.53243e10	−1.33724
$452$	−4.29317e10	−1.02855
$453$	0	0
$454$	−2.14098e10	−0.503951
$455$	0	0
$456$	0	0
$457$	− 9.62260e9i	− 0.220611i	−0.993898	−	0.110306i	$-0.964817\pi$
$457$	− 9.62260e9i	− 0.220611i	0.993898	−	0.110306i	$-0.0351830\pi$
$458$	9.58573e10	2.17853
$459$	0	0
$460$	0	0
$461$	− 3.66842e9i	− 0.0812222i	−0.999175	−	0.0406111i	$-0.987070\pi$
$461$	− 3.66842e9i	− 0.0812222i	0.999175	−	0.0406111i	$-0.0129305\pi$
$462$	0	0
$463$	− 2.55583e10i	− 0.556170i	−0.960556	−	0.278085i	$-0.910300\pi$
$463$	− 2.55583e10i	− 0.556170i	0.960556	−	0.278085i	$-0.0896997\pi$
$464$	− 5.30249e10i	− 1.14395i
$465$	0	0
$466$	3.85391e10	0.817255
$467$	−3.61615e10	−0.760290	−0.380145	−	0.924927i	$-0.624126\pi$
$467$	−3.61615e10	−0.760290	−0.380145	+	0.924927i	$0.624126\pi$
$468$	0	0
$469$	−1.08355e11	−2.23952
$470$	0	0
$471$	0	0
$472$	7.41344e9i	0.149366i
$473$	7.39249e10	1.47688
$474$	0	0
$475$	0	0
$476$	3.32591e10i	0.647863i
$477$	0	0
$478$	6.58105e10i	1.26062i
$479$	4.16338e10i	0.790868i	0.918494	+	0.395434i	$0.129406\pi$
$479$	4.16338e10i	0.790868i	−0.918494	+	0.395434i	$0.870594\pi$
$480$	0	0
$481$	6.37628e9	0.119121
$482$	1.30657e11	2.42071
$483$	0	0
$484$	−1.35853e10	−0.247564
$485$	0	0
$486$	0	0
$487$	− 9.11488e8i	− 0.0162045i	−0.999967	−	0.00810224i	$-0.997421\pi$
$487$	− 9.11488e8i	− 0.0162045i	0.999967	−	0.00810224i	$-0.00257905\pi$
$488$	4.83632e9	0.0852777
$489$	0	0
$490$	0	0
$491$	5.05964e10i	0.870551i	0.900297	+	0.435275i	$0.143349\pi$
$491$	5.05964e10i	0.870551i	−0.900297	+	0.435275i	$0.856651\pi$
$492$	0	0
$493$	− 2.45859e10i	− 0.416196i
$494$	− 1.20279e10i	− 0.201968i
$495$	0	0
$496$	5.48813e10	0.906772
$497$	1.61551e11	2.64779
$498$	0	0
$499$	2.13194e10	0.343853	0.171926	−	0.985110i	$-0.445001\pi$
$499$	2.13194e10	0.343853	0.171926	+	0.985110i	$0.445001\pi$
$500$	0	0
$501$	0	0
$502$	− 8.91074e10i	− 1.40313i
$503$	8.15896e10	1.27457	0.637284	−	0.770629i	$-0.280058\pi$
$503$	8.15896e10	1.27457	0.637284	+	0.770629i	$0.280058\pi$
$504$	0	0
$505$	0	0
$506$	− 1.96502e10i	− 0.299754i
$507$	0	0
$508$	7.68010e10i	1.15322i
$509$	2.44976e10i	0.364966i	0.983209	+	0.182483i	$0.0584135\pi$
$509$	2.44976e10i	0.364966i	−0.983209	+	0.182483i	$0.941587\pi$
$510$	0	0
$511$	−1.36848e11	−2.00703
$512$	−9.59783e10	−1.39667
$513$	0	0
$514$	1.25556e11	1.79880
$515$	0	0
$516$	0	0
$517$	5.50274e10i	0.770223i
$518$	−2.35519e11	−3.27121
$519$	0	0
$520$	0	0
$521$	1.37580e11i	1.86726i	0.358235	+	0.933631i	$0.383379\pi$
$521$	1.37580e11i	1.86726i	−0.358235	+	0.933631i	$0.616621\pi$
$522$	0	0
$523$	1.09839e10i	0.146808i	0.997302	+	0.0734039i	$0.0233862\pi$
$523$	1.09839e10i	0.146808i	−0.997302	+	0.0734039i	$0.976614\pi$
$524$	7.41868e10i	0.984015i
$525$	0	0
$526$	−6.16990e10	−0.806001
$527$	2.54466e10	0.329904
$528$	0	0
$529$	−7.55888e10	−0.965238
$530$	0	0
$531$	0	0
$532$	2.34243e11i	2.92429i
$533$	−9.47391e9	−0.117387
$534$	0	0
$535$	0	0
$536$	1.69160e10i	0.204945i
$537$	0	0
$538$	− 1.16753e11i	− 1.39360i
$539$	2.19913e11i	2.60553i
$540$	0	0
$541$	6.88987e10	0.804307	0.402154	−	0.915572i	$-0.368262\pi$
$541$	6.88987e10	0.804307	0.402154	+	0.915572i	$0.368262\pi$
$542$	3.87066e10	0.448527
$543$	0	0
$544$	−3.98464e10	−0.454981
$545$	0	0
$546$	0	0
$547$	7.36294e10i	0.822436i	0.911537	+	0.411218i	$0.134897\pi$
$547$	7.36294e10i	0.822436i	−0.911537	+	0.411218i	$0.865103\pi$
$548$	6.38053e10	0.707513
$549$	0	0
$550$	0	0
$551$	− 1.73158e11i	− 1.87860i
$552$	0	0
$553$	2.84380e11i	3.04088i
$554$	− 7.70540e10i	− 0.818005i
$555$	0	0
$556$	8.87861e10	0.929064
$557$	−7.86391e10	−0.816992	−0.408496	−	0.912760i	$-0.633947\pi$
$557$	−7.86391e10	−0.816992	−0.408496	+	0.912760i	$0.633947\pi$
$558$	0	0
$559$	1.26591e10	0.129645
$560$	0	0
$561$	0	0
$562$	− 1.51863e11i	− 1.52232i
$563$	−6.63370e10	−0.660271	−0.330135	−	0.943934i	$-0.607094\pi$
$563$	−6.63370e10	−0.660271	−0.330135	+	0.943934i	$0.607094\pi$
$564$	0	0
$565$	0	0
$566$	2.56747e11i	2.50172i
$567$	0	0
$568$	− 2.52209e10i	− 0.242307i
$569$	5.16708e10i	0.492943i	0.969150	+	0.246471i	$0.0792711\pi$
$569$	5.16708e10i	0.492943i	−0.969150	+	0.246471i	$0.920729\pi$
$570$	0	0
$571$	−1.04745e11	−0.985343	−0.492672	−	0.870215i	$-0.663980\pi$
$571$	−1.04745e11	−0.985343	−0.492672	+	0.870215i	$0.663980\pi$
$572$	−1.28069e10	−0.119635
$573$	0	0
$574$	3.49936e11	3.22360
$575$	0	0
$576$	0	0
$577$	− 1.65645e11i	− 1.49443i	−0.664582	−	0.747216i	$-0.731390\pi$
$577$	− 1.65645e11i	− 1.49443i	0.664582	−	0.747216i	$-0.268610\pi$
$578$	1.46012e11	1.30821
$579$	0	0
$580$	0	0
$581$	− 3.36814e11i	− 2.95587i
$582$	0	0
$583$	− 9.91694e10i	− 0.858427i
$584$	2.13643e10i	0.183670i
$585$	0	0
$586$	2.98496e11	2.53133
$587$	2.42404e10	0.204168	0.102084	−	0.994776i	$-0.467449\pi$
$587$	2.42404e10	0.204168	0.102084	+	0.994776i	$0.467449\pi$
$588$	0	0
$589$	1.79220e11	1.48910
$590$	0	0
$591$	0	0
$592$	− 1.31390e11i	− 1.06973i
$593$	3.38219e10	0.273514	0.136757	−	0.990605i	$-0.456332\pi$
$593$	3.38219e10	0.273514	0.136757	+	0.990605i	$0.456332\pi$
$594$	0	0
$595$	0	0
$596$	5.46678e7i	0 0.000433258i
$597$	0	0
$598$	− 3.36496e9i	− 0.0263133i
$599$	− 1.38008e10i	− 0.107200i	−0.998562	−	0.0536002i	$-0.982930\pi$
$599$	− 1.38008e10i	− 0.107200i	0.998562	−	0.0536002i	$-0.0170697\pi$
$600$	0	0
$601$	−1.27071e11	−0.973978	−0.486989	−	0.873408i	$-0.661905\pi$
$601$	−1.27071e11	−0.973978	−0.486989	+	0.873408i	$0.661905\pi$
$602$	−4.67588e11	−3.56022
$603$	0	0
$604$	1.66841e11	1.25359
$605$	0	0
$606$	0	0
$607$	− 2.61072e10i	− 0.192312i	−0.995366	−	0.0961558i	$-0.969345\pi$
$607$	− 2.61072e10i	− 0.192312i	0.995366	−	0.0961558i	$-0.0306547\pi$
$608$	−2.80637e11	−2.05367
$609$	0	0
$610$	0	0
$611$	9.42306e9i	0.0676125i
$612$	0	0
$613$	1.31329e10i	0.0930075i	0.998918	+	0.0465037i	$0.0148079\pi$
$613$	1.31329e10i	0.0930075i	−0.998918	+	0.0465037i	$0.985192\pi$
$614$	− 2.00694e11i	− 1.41208i
$615$	0	0
$616$	4.88980e10	0.339600
$617$	2.62461e11	1.81102	0.905512	−	0.424321i	$-0.139487\pi$
$617$	2.62461e11	1.81102	0.905512	+	0.424321i	$0.139487\pi$
$618$	0	0
$619$	−4.91912e10	−0.335062	−0.167531	−	0.985867i	$-0.553579\pi$
$619$	−4.91912e10	−0.335062	−0.167531	+	0.985867i	$0.553579\pi$
$620$	0	0
$621$	0	0
$622$	4.73044e10i	0.316039i
$623$	−2.23367e11	−1.48275
$624$	0	0
$625$	0	0
$626$	− 1.64505e10i	− 0.107123i
$627$	0	0
$628$	2.26019e10i	0.145314i
$629$	− 6.09211e10i	− 0.389193i
$630$	0	0
$631$	2.00621e11	1.26549	0.632745	−	0.774361i	$-0.281928\pi$
$631$	2.00621e11	1.26549	0.632745	+	0.774361i	$0.281928\pi$
$632$	4.43966e10	0.278280
$633$	0	0
$634$	−3.52968e11	−2.18463
$635$	0	0
$636$	0	0
$637$	3.76586e10i	0.228721i
$638$	−3.49685e11	−2.11054
$639$	0	0
$640$	0	0
$641$	− 8.38334e10i	− 0.496575i	−0.968686	−	0.248288i	$-0.920132\pi$
$641$	− 8.38334e10i	− 0.496575i	0.968686	−	0.248288i	$-0.0798678\pi$
$642$	0	0
$643$	5.19580e10i	0.303954i	0.988384	+	0.151977i	$0.0485640\pi$
$643$	5.19580e10i	0.303954i	−0.988384	+	0.151977i	$0.951436\pi$
$644$	6.55325e10i	0.380990i
$645$	0	0
$646$	−1.14919e11	−0.659873
$647$	1.60226e11	0.914358	0.457179	−	0.889375i	$-0.348860\pi$
$647$	1.60226e11	0.914358	0.457179	+	0.889375i	$0.348860\pi$
$648$	0	0
$649$	−1.74703e11	−0.984742
$650$	0	0
$651$	0	0
$652$	1.00703e11i	0.557250i
$653$	6.01694e10	0.330920	0.165460	−	0.986217i	$-0.447089\pi$
$653$	6.01694e10	0.330920	0.165460	+	0.986217i	$0.447089\pi$
$654$	0	0
$655$	0	0
$656$	1.95220e11i	1.05416i
$657$	0	0
$658$	− 3.48058e11i	− 1.85673i
$659$	− 5.94975e10i	− 0.315469i	−0.987482	−	0.157735i	$-0.949581\pi$
$659$	− 5.94975e10i	− 0.315469i	0.987482	−	0.157735i	$-0.0504190\pi$
$660$	0	0
$661$	−2.59920e10	−0.136155	−0.0680775	−	0.997680i	$-0.521686\pi$
$661$	−2.59920e10	−0.136155	−0.0680775	+	0.997680i	$0.521686\pi$
$662$	−5.18990e11	−2.70226
$663$	0	0
$664$	−5.25824e10	−0.270501
$665$	0	0
$666$	0	0
$667$	− 4.84431e10i	− 0.244753i
$668$	−2.19026e11	−1.09999
$669$	0	0
$670$	0	0
$671$	1.13972e11i	0.562220i
$672$	0	0
$673$	− 1.53578e11i	− 0.748634i	−0.927301	−	0.374317i	$-0.877877\pi$
$673$	− 1.53578e11i	− 0.748634i	0.927301	−	0.374317i	$-0.122123\pi$
$674$	1.02477e11i	0.496578i
$675$	0	0
$676$	2.30709e11	1.10478
$677$	3.15811e11	1.50339	0.751697	−	0.659509i	$-0.229236\pi$
$677$	3.15811e11	1.50339	0.751697	+	0.659509i	$0.229236\pi$
$678$	0	0
$679$	−3.80121e11	−1.78831
$680$	0	0
$681$	0	0
$682$	− 3.61927e11i	− 1.67295i
$683$	1.63003e11	0.749052	0.374526	−	0.927216i	$-0.377805\pi$
$683$	1.63003e11	0.749052	0.374526	+	0.927216i	$0.377805\pi$
$684$	0	0
$685$	0	0
$686$	− 8.00843e11i	− 3.61619i
$687$	0	0
$688$	− 2.60854e11i	− 1.16425i
$689$	− 1.69821e10i	− 0.0753553i
$690$	0	0
$691$	3.15836e10	0.138532	0.0692659	−	0.997598i	$-0.477934\pi$
$691$	3.15836e10	0.138532	0.0692659	+	0.997598i	$0.477934\pi$
$692$	−1.06909e11	−0.466217
$693$	0	0
$694$	−1.30011e11	−0.560459
$695$	0	0
$696$	0	0
$697$	9.05169e10i	0.383529i
$698$	3.42111e11	1.44127
$699$	0	0
$700$	0	0
$701$	− 8.43107e10i	− 0.349149i	−0.984644	−	0.174574i	$-0.944145\pi$
$701$	− 8.43107e10i	− 0.349149i	0.984644	−	0.174574i	$-0.0558550\pi$
$702$	0	0
$703$	− 4.29066e11i	− 1.75672i
$704$	3.30115e11i	1.34392i
$705$	0	0
$706$	1.32275e9	0.00532424
$707$	5.52920e11	2.21302
$708$	0	0
$709$	−1.36293e11	−0.539374	−0.269687	−	0.962948i	$-0.586920\pi$
$709$	−1.36293e11	−0.539374	−0.269687	+	0.962948i	$0.586920\pi$
$710$	0	0
$711$	0	0
$712$	3.48714e10i	0.135691i
$713$	5.01391e10	0.194007
$714$	0	0
$715$	0	0
$716$	− 1.44168e11i	− 0.548552i
$717$	0	0
$718$	− 2.22549e11i	− 0.837392i
$719$	3.82144e11i	1.42992i	0.699165	+	0.714960i	$0.253555\pi$
$719$	3.82144e11i	1.42992i	−0.699165	+	0.714960i	$0.746445\pi$
$720$	0	0
$721$	−1.15736e11	−0.428281
$722$	−4.14154e11	−1.52410
$723$	0	0
$724$	3.24979e11	1.18277
$725$	0	0
$726$	0	0
$727$	1.37646e11i	0.492751i	0.969174	+	0.246375i	$0.0792396\pi$
$727$	1.37646e11i	0.492751i	−0.969174	+	0.246375i	$0.920760\pi$
$728$	8.37345e9	0.0298112
$729$	0	0
$730$	0	0
$731$	− 1.20949e11i	− 0.423579i
$732$	0	0
$733$	3.87192e11i	1.34125i	0.741796	+	0.670626i	$0.233974\pi$
$733$	3.87192e11i	1.34125i	−0.741796	+	0.670626i	$0.766026\pi$
$734$	− 6.86273e11i	− 2.36435i
$735$	0	0
$736$	−7.85117e10	−0.267561
$737$	−3.98638e11	−1.35117
$738$	0	0
$739$	−5.79542e11	−1.94315	−0.971577	−	0.236726i	$-0.923926\pi$
$739$	−5.79542e11	−1.94315	−0.971577	+	0.236726i	$0.923926\pi$
$740$	0	0
$741$	0	0
$742$	6.27264e11i	2.06935i
$743$	1.93713e11	0.635630	0.317815	−	0.948153i	$-0.397051\pi$
$743$	1.93713e11	0.635630	0.317815	+	0.948153i	$0.397051\pi$
$744$	0	0
$745$	0	0
$746$	1.59969e11i	0.516512i
$747$	0	0
$748$	1.22361e11i	0.390874i
$749$	− 1.09899e12i	− 3.49195i
$750$	0	0
$751$	−1.80812e11	−0.568418	−0.284209	−	0.958762i	$-0.591731\pi$
$751$	−1.80812e11	−0.568418	−0.284209	+	0.958762i	$0.591731\pi$
$752$	1.94172e11	0.607177
$753$	0	0
$754$	−5.98811e10	−0.185270
$755$	0	0
$756$	0	0
$757$	3.05384e11i	0.929958i	0.885322	+	0.464979i	$0.153938\pi$
$757$	3.05384e11i	0.929958i	−0.885322	+	0.464979i	$0.846062\pi$
$758$	1.45138e11	0.439646
$759$	0	0
$760$	0	0
$761$	− 2.84245e11i	− 0.847529i	−0.905772	−	0.423765i	$-0.860708\pi$
$761$	− 2.84245e11i	− 0.847529i	0.905772	−	0.423765i	$-0.139292\pi$
$762$	0	0
$763$	− 2.55547e10i	− 0.0754001i
$764$	− 1.52508e11i	− 0.447630i
$765$	0	0
$766$	−5.86380e11	−1.70319
$767$	−2.99168e10	−0.0864437
$768$	0	0
$769$	5.48712e11	1.56906	0.784529	−	0.620092i	$-0.212905\pi$
$769$	5.48712e11	1.56906	0.784529	+	0.620092i	$0.212905\pi$
$770$	0	0
$771$	0	0
$772$	− 4.73071e11i	− 1.33186i
$773$	−2.22203e11	−0.622346	−0.311173	−	0.950353i	$-0.600722\pi$
$773$	−2.22203e11	−0.622346	−0.311173	+	0.950353i	$0.600722\pi$
$774$	0	0
$775$	0	0
$776$	5.93434e10i	0.163653i
$777$	0	0
$778$	− 4.22959e11i	− 1.15446i
$779$	6.37508e11i	1.73115i
$780$	0	0
$781$	5.94349e11	1.59749
$782$	−3.21500e10	−0.0859713
$783$	0	0
$784$	7.75994e11	2.05397
$785$	0	0
$786$	0	0
$787$	− 5.76769e10i	− 0.150350i	−0.997170	−	0.0751749i	$-0.976048\pi$
$787$	− 5.76769e10i	− 0.150350i	0.997170	−	0.0751749i	$-0.0239515\pi$
$788$	5.35299e10	0.138833
$789$	0	0
$790$	0	0
$791$	6.61488e11i	1.68972i
$792$	0	0
$793$	1.95169e10i	0.0493534i
$794$	1.69258e11i	0.425860i
$795$	0	0
$796$	−1.29775e11	−0.323249
$797$	−5.38839e11	−1.33544	−0.667722	−	0.744411i	$-0.732730\pi$
$797$	−5.38839e11	−1.33544	−0.667722	+	0.744411i	$0.732730\pi$
$798$	0	0
$799$	9.00310e10	0.220905
$800$	0	0
$801$	0	0
$802$	− 5.11952e11i	− 1.23746i
$803$	−5.03466e11	−1.21090
$804$	0	0
$805$	0	0
$806$	− 6.19776e10i	− 0.146857i
$807$	0	0
$808$	− 8.63203e10i	− 0.202520i
$809$	7.75938e11i	1.81148i	0.423837	+	0.905738i	$0.360683\pi$
$809$	7.75938e11i	1.81148i	−0.423837	+	0.905738i	$0.639317\pi$
$810$	0	0
$811$	−1.26913e11	−0.293375	−0.146688	−	0.989183i	$-0.546861\pi$
$811$	−1.26913e11	−0.293375	−0.146688	+	0.989183i	$0.546861\pi$
$812$	1.16618e12	2.68251
$813$	0	0
$814$	−8.66480e11	−1.97361
$815$	0	0
$816$	0	0
$817$	− 8.51844e11i	− 1.91193i
$818$	−4.17299e9	−0.00932039
$819$	0	0
$820$	0	0
$821$	3.93640e11i	0.866417i	0.901294	+	0.433208i	$0.142619\pi$
$821$	3.93640e11i	0.866417i	−0.901294	+	0.433208i	$0.857381\pi$
$822$	0	0
$823$	8.13687e11i	1.77361i	0.462144	+	0.886805i	$0.347080\pi$
$823$	8.13687e11i	1.77361i	−0.462144	+	0.886805i	$0.652920\pi$
$824$	1.80684e10i	0.0391932i
$825$	0	0
$826$	1.10503e12	2.37385
$827$	−4.48558e11	−0.958950	−0.479475	−	0.877555i	$-0.659173\pi$
$827$	−4.48558e11	−0.958950	−0.479475	+	0.877555i	$0.659173\pi$
$828$	0	0
$829$	−3.50125e11	−0.741319	−0.370660	−	0.928769i	$-0.620868\pi$
$829$	−3.50125e11	−0.741319	−0.370660	+	0.928769i	$0.620868\pi$
$830$	0	0
$831$	0	0
$832$	5.65299e10i	0.117974i
$833$	3.59803e11	0.747281
$834$	0	0
$835$	0	0
$836$	8.61785e11i	1.76430i
$837$	0	0
$838$	8.65779e11i	1.75562i
$839$	− 4.80271e10i	− 0.0969256i	−0.998825	−	0.0484628i	$-0.984568\pi$
$839$	− 4.80271e10i	− 0.0969256i	0.998825	−	0.0484628i	$-0.0154322\pi$
$840$	0	0
$841$	−3.61820e11	−0.723284
$842$	1.02029e11	0.202990
$843$	0	0
$844$	7.24804e11	1.42840
$845$	0	0
$846$	0	0
$847$	2.09320e11i	0.406704i
$848$	−3.49933e11	−0.676709
$849$	0	0
$850$	0	0
$851$	− 1.20037e11i	− 0.228873i
$852$	0	0
$853$	2.42085e11i	0.457270i	0.973512	+	0.228635i	$0.0734262\pi$
$853$	2.42085e11i	0.457270i	−0.973512	+	0.228635i	$0.926574\pi$
$854$	− 7.20890e11i	− 1.35531i
$855$	0	0
$856$	−1.71572e11	−0.319559
$857$	−2.23435e11	−0.414218	−0.207109	−	0.978318i	$-0.566405\pi$
$857$	−2.23435e11	−0.414218	−0.207109	+	0.978318i	$0.566405\pi$
$858$	0	0
$859$	6.68565e11	1.22792	0.613961	−	0.789336i	$-0.289575\pi$
$859$	6.68565e11	1.22792	0.613961	+	0.789336i	$0.289575\pi$
$860$	0	0
$861$	0	0
$862$	5.03631e11i	0.912186i
$863$	3.96860e11	0.715475	0.357738	−	0.933822i	$-0.383548\pi$
$863$	3.96860e11	0.715475	0.357738	+	0.933822i	$0.383548\pi$
$864$	0	0
$865$	0	0
$866$	1.57005e11i	0.279153i
$867$	0	0
$868$	1.20701e12i	2.12634i
$869$	1.04624e12i	1.83465i
$870$	0	0
$871$	−6.82641e10	−0.118610
$872$	−3.98952e9	−0.00690009
$873$	0	0
$874$	−2.26431e11	−0.388053
$875$	0	0
$876$	0	0
$877$	1.09419e12i	1.84967i	0.380363	+	0.924837i	$0.375799\pi$
$877$	1.09419e12i	1.84967i	−0.380363	+	0.924837i	$0.624201\pi$
$878$	−2.21362e11	−0.372500
$879$	0	0
$880$	0	0
$881$	7.89783e11i	1.31100i	0.755193	+	0.655502i	$0.227543\pi$
$881$	7.89783e11i	1.31100i	−0.755193	+	0.655502i	$0.772457\pi$
$882$	0	0
$883$	2.41466e11i	0.397203i	0.980080	+	0.198601i	$0.0636399\pi$
$883$	2.41466e11i	0.397203i	−0.980080	+	0.198601i	$0.936360\pi$
$884$	2.09535e10i	0.0343121i
$885$	0	0
$886$	−7.46369e11	−1.21121
$887$	1.16400e11	0.188043	0.0940216	−	0.995570i	$-0.470028\pi$
$887$	1.16400e11	0.188043	0.0940216	+	0.995570i	$0.470028\pi$
$888$	0	0
$889$	1.18334e12	1.89454
$890$	0	0
$891$	0	0
$892$	− 3.88056e11i	− 0.612964i
$893$	6.34086e11	0.997109
$894$	0	0
$895$	0	0
$896$	− 3.93372e11i	− 0.610340i
$897$	0	0
$898$	6.84412e11i	1.05248i
$899$	− 8.92249e11i	− 1.36599i
$900$	0	0
$901$	−1.62252e11	−0.246202
$902$	1.28742e12	1.94489
$903$	0	0
$904$	1.03269e11	0.154632
$905$	0	0
$906$	0	0
$907$	− 1.88146e11i	− 0.278013i	−0.990291	−	0.139007i	$-0.955609\pi$
$907$	− 1.88146e11i	− 0.278013i	0.990291	−	0.139007i	$-0.0443909\pi$
$908$	2.62684e11	0.386447
$909$	0	0
$910$	0	0
$911$	3.22353e11i	0.468014i	0.972235	+	0.234007i	$0.0751838\pi$
$911$	3.22353e11i	0.468014i	−0.972235	+	0.234007i	$0.924816\pi$
$912$	0	0
$913$	− 1.23915e12i	− 1.78336i
$914$	2.23922e11i	0.320858i
$915$	0	0
$916$	−1.17611e12	−1.67057
$917$	1.14306e12	1.61656
$918$	0	0
$919$	5.59401e11	0.784262	0.392131	−	0.919909i	$-0.371738\pi$
$919$	5.59401e11	0.784262	0.392131	+	0.919909i	$0.371738\pi$
$920$	0	0
$921$	0	0
$922$	8.53656e10i	0.118130i
$923$	1.01778e11	0.140232
$924$	0	0
$925$	0	0
$926$	5.94752e11i	0.808895i
$927$	0	0
$928$	1.39715e12i	1.88388i
$929$	1.26527e12i	1.69872i	0.527817	+	0.849358i	$0.323011\pi$
$929$	1.26527e12i	1.69872i	−0.527817	+	0.849358i	$0.676989\pi$
$930$	0	0
$931$	2.53408e12	3.37304
$932$	−4.72849e11	−0.626699
$933$	0	0
$934$	8.41495e11	1.10577
$935$	0	0
$936$	0	0
$937$	1.14305e12i	1.48289i	0.671015	+	0.741444i	$0.265859\pi$
$937$	1.14305e12i	1.48289i	−0.671015	+	0.741444i	$0.734141\pi$
$938$	2.52146e12	3.25717
$939$	0	0
$940$	0	0
$941$	− 7.23423e11i	− 0.922643i	−0.887233	−	0.461322i	$-0.847375\pi$
$941$	− 7.23423e11i	− 0.922643i	0.887233	−	0.461322i	$-0.152625\pi$
$942$	0	0
$943$	1.78351e11i	0.225543i
$944$	6.16466e11i	0.776285i
$945$	0	0
$946$	−1.72026e12	−2.14798
$947$	3.09470e11	0.384786	0.192393	−	0.981318i	$-0.438375\pi$
$947$	3.09470e11	0.384786	0.192393	+	0.981318i	$0.438375\pi$
$948$	0	0
$949$	−8.62151e10	−0.106296
$950$	0	0
$951$	0	0
$952$	− 8.00027e10i	− 0.0973995i
$953$	3.15952e11	0.383045	0.191522	−	0.981488i	$-0.438658\pi$
$953$	3.15952e11	0.383045	0.191522	+	0.981488i	$0.438658\pi$
$954$	0	0
$955$	0	0
$956$	− 8.07453e11i	− 0.966686i
$957$	0	0
$958$	− 9.68837e11i	− 1.15024i
$959$	− 9.83106e11i	− 1.16232i
$960$	0	0
$961$	7.05960e10	0.0827726
$962$	−1.48379e11	−0.173249
$963$	0	0
$964$	−1.60307e12	−1.85628
$965$	0	0
$966$	0	0
$967$	2.47951e11i	0.283570i	0.989897	+	0.141785i	$0.0452841\pi$
$967$	2.47951e11i	0.283570i	−0.989897	+	0.141785i	$0.954716\pi$
$968$	3.26785e10	0.0372186
$969$	0	0
$970$	0	0
$971$	7.68280e11i	0.864256i	0.901812	+	0.432128i	$0.142237\pi$
$971$	7.68280e11i	0.864256i	−0.901812	+	0.432128i	$0.857763\pi$
$972$	0	0
$973$	− 1.36801e12i	− 1.52629i
$974$	2.12107e10i	0.0235678i
$975$	0	0
$976$	4.02165e11	0.443205
$977$	−1.31954e12	−1.44825	−0.724127	−	0.689666i	$-0.757757\pi$
$977$	−1.31954e12	−1.44825	−0.724127	+	0.689666i	$0.757757\pi$
$978$	0	0
$979$	−8.21773e11	−0.894584
$980$	0	0
$981$	0	0
$982$	− 1.17740e12i	− 1.26613i
$983$	−6.17012e11	−0.660815	−0.330407	−	0.943838i	$-0.607186\pi$
$983$	−6.17012e11	−0.660815	−0.330407	+	0.943838i	$0.607186\pi$
$984$	0	0
$985$	0	0
$986$	5.72124e11i	0.605316i
$987$	0	0
$988$	1.47575e11i	0.154876i
$989$	− 2.38314e11i	− 0.249095i
$990$	0	0
$991$	−8.44484e11	−0.875582	−0.437791	−	0.899077i	$-0.644239\pi$
$991$	−8.44484e11	−0.875582	−0.437791	+	0.899077i	$0.644239\pi$
$992$	−1.44607e12	−1.49328
$993$	0	0
$994$	−3.75936e12	−3.85096
$995$	0	0
$996$	0	0
$997$	1.27342e12i	1.28881i	0.764683	+	0.644406i	$0.222895\pi$
$997$	1.27342e12i	1.28881i	−0.764683	+	0.644406i	$0.777105\pi$
$998$	−4.96112e11	−0.500100
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.9.d.c.224.5		24
3.2	odd	2	inner	225.9.d.c.224.19		24
5.2	odd	4		45.9.c.a.26.3	✓	12
5.3	odd	4		225.9.c.d.26.10		12
5.4	even	2	inner	225.9.d.c.224.20		24
15.2	even	4		45.9.c.a.26.10	yes	12
15.8	even	4		225.9.c.d.26.3		12
15.14	odd	2	inner	225.9.d.c.224.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.9.c.a.26.3	✓	12	5.2	odd	4
45.9.c.a.26.10	yes	12	15.2	even	4
225.9.c.d.26.3		12	15.8	even	4
225.9.c.d.26.10		12	5.3	odd	4
225.9.d.c.224.5		24	1.1	even	1	trivial
225.9.d.c.224.6		24	15.14	odd	2	inner
225.9.d.c.224.19		24	3.2	odd	2	inner
225.9.d.c.224.20		24	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	225.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-23.2704 q^{2} +285.513 q^{4} -4399.16i q^{7} -686.784 q^{8} -16184.6i q^{11} -2771.50i q^{13} +102370. i q^{14} -57109.6 q^{16} -26479.8 q^{17} -186497. q^{19} +376622. i q^{22} -52174.8 q^{23} +64494.0i q^{26} -1.25602e6i q^{28} +928476. i q^{29} -960982. q^{31} +1.50478e6 q^{32} +616197. q^{34} +2.30066e6i q^{37} +4.33986e6 q^{38} -3.41833e6i q^{41} +4.56761e6i q^{43} -4.62091e6i q^{44} +1.21413e6 q^{46} -3.39999e6 q^{47} -1.35878e7 q^{49} -791299. i q^{52} +6.12740e6 q^{53} +3.02127e6i q^{56} -2.16060e7i q^{58} -1.07944e7i q^{59} -7.04198e6 q^{61} +2.23625e7 q^{62} -2.03969e7 q^{64} -2.46307e7i q^{67} -7.56034e6 q^{68} +3.67232e7i q^{71} -3.11078e7i q^{73} -5.35374e7i q^{74} -5.32472e7 q^{76} -7.11986e7 q^{77} -6.46442e7 q^{79} +7.95461e7i q^{82} +7.65633e7 q^{83} -1.06290e8i q^{86} +1.11153e7i q^{88} -5.07750e7i q^{89} -1.21923e7 q^{91} -1.48966e7 q^{92} +7.91192e7 q^{94} -8.64077e7i q^{97} +3.16194e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 5048 q^{4} + 595144 q^{16} - 249920 q^{19} - 1241936 q^{31} - 14972992 q^{34} - 14323536 q^{46} - 42020312 q^{49} + 56906192 q^{61} + 338779896 q^{64} - 511347840 q^{76} + 127619024 q^{79} - 593698576 q^{91}+ \cdots - 51203776 q^{94}+O(q^{100}) \) 24 * q + 5048 * q^4 + 595144 * q^16 - 249920 * q^19 - 1241936 * q^31 - 14972992 * q^34 - 14323536 * q^46 - 42020312 * q^49 + 56906192 * q^61 + 338779896 * q^64 - 511347840 * q^76 + 127619024 * q^79 - 593698576 * q^91 - 51203776 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more