LMFDB - Embedded newform 225.9.d.c.224.8

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			224.8
Character	$\chi$	$=$	225.224
Dual form			225.9.d.c.224.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-19.1733 q^{2} +111.617 q^{4} +367.611i q^{7} +2768.31 q^{8} +9369.90i q^{11} +28955.0i q^{13} -7048.33i q^{14} -81651.6 q^{16} -51768.6 q^{17} +66954.5 q^{19} -179652. i q^{22} -118225. q^{23} -555165. i q^{26} +41031.5i q^{28} +1.18592e6i q^{29} -231474. q^{31} +856845. q^{32} +992576. q^{34} -1.63980e6i q^{37} -1.28374e6 q^{38} +2.74156e6i q^{41} -1.44556e6i q^{43} +1.04584e6i q^{44} +2.26677e6 q^{46} -6.90683e6 q^{47} +5.62966e6 q^{49} +3.23186e6i q^{52} +1.26381e7 q^{53} +1.01766e6i q^{56} -2.27381e7i q^{58} +4.74840e6i q^{59} +7.53776e6 q^{61} +4.43813e6 q^{62} +4.47424e6 q^{64} +2.64406e7i q^{67} -5.77823e6 q^{68} +3.33257e7i q^{71} -4.47705e7i q^{73} +3.14404e7i q^{74} +7.47323e6 q^{76} -3.44448e6 q^{77} +7.76913e7 q^{79} -5.25649e7i q^{82} +2.37418e7 q^{83} +2.77161e7i q^{86} +2.59388e7i q^{88} -522787. i q^{89} -1.06442e7 q^{91} -1.31959e7 q^{92} +1.32427e8 q^{94} -3.64707e7i q^{97} -1.07939e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−19.1733	−1.19833	−0.599167	−	0.800624i	$-0.704501\pi$
$2$	−19.1733	−1.19833	−0.599167	+	0.800624i	$0.704501\pi$
$3$	0	0
$3$	0	0
$4$	111.617	0.436002
$5$	0	0
$5$	0	0
$6$	0	0
$7$	367.611i	0.153108i	0.997065	+	0.0765538i	$0.0243917\pi$
$7$	367.611i	0.153108i	−0.997065	+	0.0765538i	$0.975608\pi$
$8$	2768.31	0.675857
$9$	0	0
$10$	0	0
$11$	9369.90i	0.639977i	0.947421	+	0.319988i	$0.103679\pi$
$11$	9369.90i	0.639977i	−0.947421	+	0.319988i	$0.896321\pi$
$12$	0	0
$13$	28955.0i	1.01380i	0.862006	+	0.506898i	$0.169208\pi$
$13$	28955.0i	1.01380i	−0.862006	+	0.506898i	$0.830792\pi$
$14$	− 7048.33i	− 0.183474i
$15$	0	0
$16$	−81651.6	−1.24590
$17$	−51768.6	−0.619827	−0.309914	−	0.950765i	$-0.600300\pi$
$17$	−51768.6	−0.619827	−0.309914	+	0.950765i	$0.600300\pi$
$18$	0	0
$19$	66954.5	0.513766	0.256883	−	0.966442i	$-0.417304\pi$
$19$	66954.5	0.513766	0.256883	+	0.966442i	$0.417304\pi$
$20$	0	0
$21$	0	0
$22$	− 179652.i	− 0.766905i
$23$	−118225.	−0.422472	−0.211236	−	0.977435i	$-0.567749\pi$
$23$	−118225.	−0.422472	−0.211236	+	0.977435i	$0.567749\pi$
$24$	0	0
$25$	0	0
$26$	− 555165.i	− 1.21487i
$27$	0	0
$28$	41031.5i	0.0667552i
$29$	1.18592e6i	1.67674i	0.545105	+	0.838368i	$0.316490\pi$
$29$	1.18592e6i	1.67674i	−0.545105	+	0.838368i	$0.683510\pi$
$30$	0	0
$31$	−231474.	−0.250643	−0.125322	−	0.992116i	$-0.539996\pi$
$31$	−231474.	−0.250643	−0.125322	+	0.992116i	$0.539996\pi$
$32$	856845.	0.817151
$33$	0	0
$34$	992576.	0.742759
$35$	0	0
$36$	0	0
$37$	− 1.63980e6i	− 0.874951i	−0.899230	−	0.437476i	$-0.855873\pi$
$37$	− 1.63980e6i	− 0.874951i	0.899230	−	0.437476i	$-0.144127\pi$
$38$	−1.28374e6	−0.615663
$39$	0	0
$40$	0	0
$41$	2.74156e6i	0.970203i	0.874458	+	0.485102i	$0.161217\pi$
$41$	2.74156e6i	0.970203i	−0.874458	+	0.485102i	$0.838783\pi$
$42$	0	0
$43$	− 1.44556e6i	− 0.422826i	−0.977397	−	0.211413i	$-0.932194\pi$
$43$	− 1.44556e6i	− 0.422826i	0.977397	−	0.211413i	$-0.0678065\pi$
$44$	1.04584e6i	0.279031i
$45$	0	0
$46$	2.26677e6	0.506263
$47$	−6.90683e6	−1.41543	−0.707714	−	0.706499i	$-0.750273\pi$
$47$	−6.90683e6	−1.41543	−0.707714	+	0.706499i	$0.750273\pi$
$48$	0	0
$49$	5.62966e6	0.976558
$50$	0	0
$51$	0	0
$52$	3.23186e6i	0.442018i
$53$	1.26381e7	1.60168	0.800842	−	0.598876i	$-0.204386\pi$
$53$	1.26381e7	1.60168	0.800842	+	0.598876i	$0.204386\pi$
$54$	0	0
$55$	0	0
$56$	1.01766e6i	0.103479i
$57$	0	0
$58$	− 2.27381e7i	− 2.00929i
$59$	4.74840e6i	0.391867i	0.980617	+	0.195934i	$0.0627737\pi$
$59$	4.74840e6i	0.391867i	−0.980617	+	0.195934i	$0.937226\pi$
$60$	0	0
$61$	7.53776e6	0.544406	0.272203	−	0.962240i	$-0.412248\pi$
$61$	7.53776e6	0.544406	0.272203	+	0.962240i	$0.412248\pi$
$62$	4.43813e6	0.300354
$63$	0	0
$64$	4.47424e6	0.266685
$65$	0	0
$66$	0	0
$67$	2.64406e7i	1.31212i	0.754711	+	0.656058i	$0.227777\pi$
$67$	2.64406e7i	1.31212i	−0.754711	+	0.656058i	$0.772223\pi$
$68$	−5.77823e6	−0.270246
$69$	0	0
$70$	0	0
$71$	3.33257e7i	1.31143i	0.755007	+	0.655716i	$0.227633\pi$
$71$	3.33257e7i	1.31143i	−0.755007	+	0.655716i	$0.772367\pi$
$72$	0	0
$73$	− 4.47705e7i	− 1.57653i	−0.615339	−	0.788263i	$-0.710981\pi$
$73$	− 4.47705e7i	− 1.57653i	0.615339	−	0.788263i	$-0.289019\pi$
$74$	3.14404e7i	1.04848i
$75$	0	0
$76$	7.47323e6	0.224003
$77$	−3.44448e6	−0.0979852
$78$	0	0
$79$	7.76913e7	1.99464	0.997318	−	0.0731864i	$-0.0233168\pi$
$79$	7.76913e7	1.99464	0.997318	+	0.0731864i	$0.0233168\pi$
$80$	0	0
$81$	0	0
$82$	− 5.25649e7i	− 1.16263i
$83$	2.37418e7	0.500266	0.250133	−	0.968211i	$-0.419526\pi$
$83$	2.37418e7	0.500266	0.250133	+	0.968211i	$0.419526\pi$
$84$	0	0
$85$	0	0
$86$	2.77161e7i	0.506686i
$87$	0	0
$88$	2.59388e7i	0.432533i
$89$	− 522787.i	− 0.00833230i	−0.999991	−	0.00416615i	$-0.998674\pi$
$89$	− 522787.i	− 0.00833230i	0.999991	−	0.00416615i	$-0.00132613\pi$
$90$	0	0
$91$	−1.06442e7	−0.155220
$92$	−1.31959e7	−0.184199
$93$	0	0
$94$	1.32427e8	1.69615
$95$	0	0
$96$	0	0
$97$	− 3.64707e7i	− 0.411962i	−0.978556	−	0.205981i	$-0.933961\pi$
$97$	− 3.64707e7i	− 0.411962i	0.978556	−	0.205981i	$-0.0660385\pi$
$98$	−1.07939e8	−1.17024
$99$	0	0
$100$	0	0
$101$	4.14714e7i	0.398532i	0.979945	+	0.199266i	$0.0638557\pi$
$101$	4.14714e7i	0.398532i	−0.979945	+	0.199266i	$0.936144\pi$
$102$	0	0
$103$	− 1.94830e8i	− 1.73104i	−0.500872	−	0.865521i	$-0.666987\pi$
$103$	− 1.94830e8i	− 1.73104i	0.500872	−	0.865521i	$-0.333013\pi$
$104$	8.01566e7i	0.685182i
$105$	0	0
$106$	−2.42314e8	−1.91935
$107$	−3.97292e7	−0.303092	−0.151546	−	0.988450i	$-0.548425\pi$
$107$	−3.97292e7	−0.303092	−0.151546	+	0.988450i	$0.548425\pi$
$108$	0	0
$109$	−2.05517e8	−1.45593	−0.727967	−	0.685613i	$-0.759534\pi$
$109$	−2.05517e8	−1.45593	−0.727967	+	0.685613i	$0.759534\pi$
$110$	0	0
$111$	0	0
$112$	− 3.00160e7i	− 0.190757i
$113$	−2.40531e8	−1.47522	−0.737610	−	0.675227i	$-0.764046\pi$
$113$	−2.40531e8	−1.47522	−0.737610	+	0.675227i	$0.764046\pi$
$114$	0	0
$115$	0	0
$116$	1.32369e8i	0.731060i
$117$	0	0
$118$	− 9.10426e7i	− 0.469588i
$119$	− 1.90307e7i	− 0.0949002i
$120$	0	0
$121$	1.26564e8	0.590430
$122$	−1.44524e8	−0.652379
$123$	0	0
$124$	−2.58364e7	−0.109281
$125$	0	0
$126$	0	0
$127$	1.42272e8i	0.546895i	0.961887	+	0.273448i	$0.0881640\pi$
$127$	1.42272e8i	0.546895i	−0.961887	+	0.273448i	$0.911836\pi$
$128$	−3.05138e8	−1.13673
$129$	0	0
$130$	0	0
$131$	1.82412e8i	0.619397i	0.950835	+	0.309698i	$0.100228\pi$
$131$	1.82412e8i	0.619397i	−0.950835	+	0.309698i	$0.899772\pi$
$132$	0	0
$133$	2.46132e7i	0.0786615i
$134$	− 5.06954e8i	− 1.57235i
$135$	0	0
$136$	−1.43312e8	−0.418915
$137$	−4.63606e8	−1.31603	−0.658016	−	0.753004i	$-0.728604\pi$
$137$	−4.63606e8	−1.31603	−0.658016	+	0.753004i	$0.728604\pi$
$138$	0	0
$139$	−6.61605e8	−1.77231	−0.886154	−	0.463390i	$-0.846633\pi$
$139$	−6.61605e8	−1.77231	−0.886154	+	0.463390i	$0.846633\pi$
$140$	0	0
$141$	0	0
$142$	− 6.38965e8i	− 1.57153i
$143$	−2.71306e8	−0.648806
$144$	0	0
$145$	0	0
$146$	8.58400e8i	1.88920i
$147$	0	0
$148$	− 1.83029e8i	− 0.381481i
$149$	− 5.85816e8i	− 1.18855i	−0.804263	−	0.594273i	$-0.797440\pi$
$149$	− 5.85816e8i	− 1.18855i	0.804263	−	0.594273i	$-0.202560\pi$
$150$	0	0
$151$	−8.85940e8	−1.70411	−0.852053	−	0.523455i	$-0.824643\pi$
$151$	−8.85940e8	−1.70411	−0.852053	+	0.523455i	$0.824643\pi$
$152$	1.85351e8	0.347233
$153$	0	0
$154$	6.60421e7	0.117419
$155$	0	0
$156$	0	0
$157$	2.79032e8i	0.459257i	0.973278	+	0.229629i	$0.0737512\pi$
$157$	2.79032e8i	0.459257i	−0.973278	+	0.229629i	$0.926249\pi$
$158$	−1.48960e9	−2.39024
$159$	0	0
$160$	0	0
$161$	− 4.34609e7i	− 0.0646837i
$162$	0	0
$163$	2.91436e7i	0.0412851i	0.999787	+	0.0206425i	$0.00657119\pi$
$163$	2.91436e7i	0.0412851i	−0.999787	+	0.0206425i	$0.993429\pi$
$164$	3.06004e8i	0.423011i
$165$	0	0
$166$	−4.55209e8	−0.599486
$167$	4.73192e8	0.608376	0.304188	−	0.952612i	$-0.401615\pi$
$167$	4.73192e8	0.608376	0.304188	+	0.952612i	$0.401615\pi$
$168$	0	0
$169$	−2.26641e7	−0.0277838
$170$	0	0
$171$	0	0
$172$	− 1.61348e8i	− 0.184353i
$173$	−6.61443e8	−0.738428	−0.369214	−	0.929344i	$-0.620373\pi$
$173$	−6.61443e8	−0.738428	−0.369214	+	0.929344i	$0.620373\pi$
$174$	0	0
$175$	0	0
$176$	− 7.65067e8i	− 0.797350i
$177$	0	0
$178$	1.00236e7i	0.00998487i
$179$	− 6.35798e8i	− 0.619308i	−0.950849	−	0.309654i	$-0.899787\pi$
$179$	− 6.35798e8i	− 0.619308i	0.950849	−	0.309654i	$-0.100213\pi$
$180$	0	0
$181$	−1.03463e9	−0.963987	−0.481993	−	0.876175i	$-0.660087\pi$
$181$	−1.03463e9	−0.963987	−0.481993	+	0.876175i	$0.660087\pi$
$182$	2.04085e8	0.186005
$183$	0	0
$184$	−3.27284e8	−0.285531
$185$	0	0
$186$	0	0
$187$	− 4.85066e8i	− 0.396675i
$188$	−7.70917e8	−0.617129
$189$	0	0
$190$	0	0
$191$	5.10686e8i	0.383725i	0.981422	+	0.191863i	$0.0614528\pi$
$191$	5.10686e8i	0.383725i	−0.981422	+	0.191863i	$0.938547\pi$
$192$	0	0
$193$	4.69155e8i	0.338133i	0.985605	+	0.169066i	$0.0540753\pi$
$193$	4.69155e8i	0.338133i	−0.985605	+	0.169066i	$0.945925\pi$
$194$	6.99265e8i	0.493668i
$195$	0	0
$196$	6.28363e8	0.425781
$197$	−2.47608e9	−1.64399	−0.821997	−	0.569492i	$-0.807140\pi$
$197$	−2.47608e9	−1.64399	−0.821997	+	0.569492i	$0.807140\pi$
$198$	0	0
$199$	1.69051e9	1.07797	0.538983	−	0.842317i	$-0.318809\pi$
$199$	1.69051e9	1.07797	0.538983	+	0.842317i	$0.318809\pi$
$200$	0	0
$201$	0	0
$202$	− 7.95144e8i	− 0.477574i
$203$	−4.35959e8	−0.256721
$204$	0	0
$205$	0	0
$206$	3.73555e9i	2.07437i
$207$	0	0
$208$	− 2.36423e9i	− 1.26309i
$209$	6.27357e8i	0.328798i
$210$	0	0
$211$	−9.62123e8	−0.485401	−0.242701	−	0.970101i	$-0.578033\pi$
$211$	−9.62123e8	−0.485401	−0.242701	+	0.970101i	$0.578033\pi$
$212$	1.41062e9	0.698337
$213$	0	0
$214$	7.61741e8	0.363205
$215$	0	0
$216$	0	0
$217$	− 8.50925e7i	− 0.0383754i
$218$	3.94044e9	1.74469
$219$	0	0
$220$	0	0
$221$	− 1.49896e9i	− 0.628379i
$222$	0	0
$223$	− 3.57620e9i	− 1.44611i	−0.690789	−	0.723056i	$-0.742737\pi$
$223$	− 3.57620e9i	− 1.44611i	0.690789	−	0.723056i	$-0.257263\pi$
$224$	3.14986e8i	0.125112i
$225$	0	0
$226$	4.61178e9	1.76781
$227$	−6.58041e8	−0.247828	−0.123914	−	0.992293i	$-0.539545\pi$
$227$	−6.58041e8	−0.247828	−0.123914	+	0.992293i	$0.539545\pi$
$228$	0	0
$229$	2.03547e8	0.0740154	0.0370077	−	0.999315i	$-0.488217\pi$
$229$	2.03547e8	0.0740154	0.0370077	+	0.999315i	$0.488217\pi$
$230$	0	0
$231$	0	0
$232$	3.28300e9i	1.13323i
$233$	2.56183e9	0.869214	0.434607	−	0.900620i	$-0.356887\pi$
$233$	2.56183e9	0.869214	0.434607	+	0.900620i	$0.356887\pi$
$234$	0	0
$235$	0	0
$236$	5.30000e8i	0.170855i
$237$	0	0
$238$	3.64882e8i	0.113722i
$239$	5.62808e9i	1.72492i	0.506127	+	0.862459i	$0.331077\pi$
$239$	5.62808e9i	1.72492i	−0.506127	+	0.862459i	$0.668923\pi$
$240$	0	0
$241$	1.91504e9	0.567687	0.283843	−	0.958871i	$-0.408390\pi$
$241$	1.91504e9	0.567687	0.283843	+	0.958871i	$0.408390\pi$
$242$	−2.42665e9	−0.707532
$243$	0	0
$244$	8.41338e8	0.237362
$245$	0	0
$246$	0	0
$247$	1.93867e9i	0.520855i
$248$	−6.40793e8	−0.169399
$249$	0	0
$250$	0	0
$251$	− 5.39207e9i	− 1.35850i	−0.733906	−	0.679252i	$-0.762305\pi$
$251$	− 5.39207e9i	− 1.35850i	0.733906	−	0.679252i	$-0.237695\pi$
$252$	0	0
$253$	− 1.10776e9i	− 0.270372i
$254$	− 2.72782e9i	− 0.655362i
$255$	0	0
$256$	4.70511e9	1.09549
$257$	3.32416e9	0.761990	0.380995	−	0.924577i	$-0.375581\pi$
$257$	3.32416e9	0.761990	0.380995	+	0.924577i	$0.375581\pi$
$258$	0	0
$259$	6.02809e8	0.133962
$260$	0	0
$261$	0	0
$262$	− 3.49745e9i	− 0.742244i
$263$	1.80944e9	0.378199	0.189099	−	0.981958i	$-0.439443\pi$
$263$	1.80944e9	0.378199	0.189099	+	0.981958i	$0.439443\pi$
$264$	0	0
$265$	0	0
$266$	− 4.71918e8i	− 0.0942627i
$267$	0	0
$268$	2.95121e9i	0.572085i
$269$	− 1.19480e8i	− 0.0228184i	−0.999935	−	0.0114092i	$-0.996368\pi$
$269$	− 1.19480e8i	− 0.0228184i	0.999935	−	0.0114092i	$-0.00363175\pi$
$270$	0	0
$271$	−9.26363e9	−1.71753	−0.858764	−	0.512371i	$-0.828767\pi$
$271$	−9.26363e9	−1.71753	−0.858764	+	0.512371i	$0.828767\pi$
$272$	4.22699e9	0.772245
$273$	0	0
$274$	8.88886e9	1.57704
$275$	0	0
$276$	0	0
$277$	5.45978e9i	0.927376i	0.885998	+	0.463688i	$0.153474\pi$
$277$	5.45978e9i	0.927376i	−0.885998	+	0.463688i	$0.846526\pi$
$278$	1.26852e10	2.12382
$279$	0	0
$280$	0	0
$281$	− 2.23569e9i	− 0.358581i	−0.983796	−	0.179290i	$-0.942620\pi$
$281$	− 2.23569e9i	− 0.358581i	0.983796	−	0.179290i	$-0.0573801\pi$
$282$	0	0
$283$	1.19652e10i	1.86540i	0.360648	+	0.932702i	$0.382556\pi$
$283$	1.19652e10i	1.86540i	−0.360648	+	0.932702i	$0.617444\pi$
$284$	3.71970e9i	0.571787i
$285$	0	0
$286$	5.20184e9	0.777486
$287$	−1.00783e9	−0.148545
$288$	0	0
$289$	−4.29577e9	−0.615814
$290$	0	0
$291$	0	0
$292$	− 4.99713e9i	− 0.687368i
$293$	1.64130e9	0.222699	0.111349	−	0.993781i	$-0.464483\pi$
$293$	1.64130e9	0.222699	0.111349	+	0.993781i	$0.464483\pi$
$294$	0	0
$295$	0	0
$296$	− 4.53948e9i	− 0.591342i
$297$	0	0
$298$	1.12320e10i	1.42428i
$299$	− 3.42321e9i	− 0.428301i
$300$	0	0
$301$	5.31403e8	0.0647378
$302$	1.69864e10	2.04209
$303$	0	0
$304$	−5.46694e9	−0.640104
$305$	0	0
$306$	0	0
$307$	− 7.60508e8i	− 0.0856150i	−0.999083	−	0.0428075i	$-0.986370\pi$
$307$	− 7.60508e8i	− 0.0856150i	0.999083	−	0.0428075i	$-0.0136302\pi$
$308$	−3.84461e8	−0.0427218
$309$	0	0
$310$	0	0
$311$	− 1.70174e10i	− 1.81908i	−0.415621	−	0.909538i	$-0.636436\pi$
$311$	− 1.70174e10i	− 1.81908i	0.415621	−	0.909538i	$-0.363564\pi$
$312$	0	0
$313$	− 3.58579e7i	− 0.00373601i	−0.999998	−	0.00186800i	$-0.999405\pi$
$313$	− 3.58579e7i	− 0.00373601i	0.999998	−	0.00186800i	$-0.000594605\pi$
$314$	− 5.34998e9i	− 0.550343i
$315$	0	0
$316$	8.67163e9	0.869666
$317$	−7.55878e9	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$317$	−7.55878e9	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$318$	0	0
$319$	−1.11120e10	−1.07307
$320$	0	0
$321$	0	0
$322$	8.33290e8i	0.0775126i
$323$	−3.46614e9	−0.318446
$324$	0	0
$325$	0	0
$326$	− 5.58780e8i	− 0.0494733i
$327$	0	0
$328$	7.58950e9i	0.655719i
$329$	− 2.53903e9i	− 0.216713i
$330$	0	0
$331$	−5.31316e8	−0.0442630	−0.0221315	−	0.999755i	$-0.507045\pi$
$331$	−5.31316e8	−0.0442630	−0.0221315	+	0.999755i	$0.507045\pi$
$332$	2.64998e9	0.218117
$333$	0	0
$334$	−9.07267e9	−0.729037
$335$	0	0
$336$	0	0
$337$	1.49671e10i	1.16042i	0.814466	+	0.580212i	$0.197030\pi$
$337$	1.49671e10i	1.16042i	−0.814466	+	0.580212i	$0.802970\pi$
$338$	4.34547e8	0.0332943
$339$	0	0
$340$	0	0
$341$	− 2.16889e9i	− 0.160406i
$342$	0	0
$343$	4.18873e9i	0.302626i
$344$	− 4.00175e9i	− 0.285770i
$345$	0	0
$346$	1.26821e10	0.884882
$347$	−1.64170e10	−1.13234	−0.566170	−	0.824288i	$-0.691576\pi$
$347$	−1.64170e10	−1.13234	−0.566170	+	0.824288i	$0.691576\pi$
$348$	0	0
$349$	−2.55575e10	−1.72273	−0.861363	−	0.507990i	$-0.830389\pi$
$349$	−2.55575e10	−1.72273	−0.861363	+	0.507990i	$0.830389\pi$
$350$	0	0
$351$	0	0
$352$	8.02855e9i	0.522957i
$353$	9.98085e9	0.642789	0.321395	−	0.946945i	$-0.395848\pi$
$353$	9.98085e9	0.642789	0.321395	+	0.946945i	$0.395848\pi$
$354$	0	0
$355$	0	0
$356$	− 5.83517e7i	− 0.00363290i
$357$	0	0
$358$	1.21904e10i	0.742138i
$359$	− 1.89263e10i	− 1.13943i	−0.821843	−	0.569715i	$-0.807054\pi$
$359$	− 1.89263e10i	− 1.13943i	0.821843	−	0.569715i	$-0.192946\pi$
$360$	0	0
$361$	−1.25007e10	−0.736044
$362$	1.98373e10	1.15518
$363$	0	0
$364$	−1.18807e9	−0.0676762
$365$	0	0
$366$	0	0
$367$	− 1.65661e10i	− 0.913180i	−0.889677	−	0.456590i	$-0.849071\pi$
$367$	− 1.65661e10i	− 0.913180i	0.889677	−	0.456590i	$-0.150929\pi$
$368$	9.65327e9	0.526360
$369$	0	0
$370$	0	0
$371$	4.64589e9i	0.245230i
$372$	0	0
$373$	− 8.78219e9i	− 0.453698i	−0.973930	−	0.226849i	$-0.927158\pi$
$373$	− 8.78219e9i	− 0.453698i	0.973930	−	0.226849i	$-0.0728425\pi$
$374$	9.30033e9i	0.475349i
$375$	0	0
$376$	−1.91203e10	−0.956627
$377$	−3.43385e10	−1.69987
$378$	0	0
$379$	4.27282e9	0.207090	0.103545	−	0.994625i	$-0.466981\pi$
$379$	4.27282e9	0.207090	0.103545	+	0.994625i	$0.466981\pi$
$380$	0	0
$381$	0	0
$382$	− 9.79155e9i	− 0.459831i
$383$	−7.68672e9	−0.357229	−0.178614	−	0.983919i	$-0.557161\pi$
$383$	−7.68672e9	−0.357229	−0.178614	+	0.983919i	$0.557161\pi$
$384$	0	0
$385$	0	0
$386$	− 8.99527e9i	− 0.405196i
$387$	0	0
$388$	− 4.07074e9i	− 0.179616i
$389$	6.14689e9i	0.268446i	0.990951	+	0.134223i	$0.0428539\pi$
$389$	6.14689e9i	0.268446i	−0.990951	+	0.134223i	$0.957146\pi$
$390$	0	0
$391$	6.12035e9	0.261860
$392$	1.55847e10	0.660014
$393$	0	0
$394$	4.74747e10	1.97005
$395$	0	0
$396$	0	0
$397$	2.20936e9i	0.0889413i	0.999011	+	0.0444707i	$0.0141601\pi$
$397$	2.20936e9i	0.0889413i	−0.999011	+	0.0444707i	$0.985840\pi$
$398$	−3.24127e10	−1.29176
$399$	0	0
$400$	0	0
$401$	1.68791e10i	0.652786i	0.945234	+	0.326393i	$0.105833\pi$
$401$	1.68791e10i	0.652786i	−0.945234	+	0.326393i	$0.894167\pi$
$402$	0	0
$403$	− 6.70235e9i	− 0.254101i
$404$	4.62889e9i	0.173761i
$405$	0	0
$406$	8.35878e9	0.307637
$407$	1.53648e10	0.559948
$408$	0	0
$409$	−1.62285e10	−0.579944	−0.289972	−	0.957035i	$-0.593646\pi$
$409$	−1.62285e10	−0.579944	−0.289972	+	0.957035i	$0.593646\pi$
$410$	0	0
$411$	0	0
$412$	− 2.17463e10i	− 0.754738i
$413$	−1.74556e9	−0.0599979
$414$	0	0
$415$	0	0
$416$	2.48100e10i	0.828425i
$417$	0	0
$418$	− 1.20285e10i	− 0.394010i
$419$	4.80952e10i	1.56043i	0.625509	+	0.780217i	$0.284891\pi$
$419$	4.80952e10i	1.56043i	−0.625509	+	0.780217i	$0.715109\pi$
$420$	0	0
$421$	8.87073e9	0.282378	0.141189	−	0.989983i	$-0.454907\pi$
$421$	8.87073e9	0.282378	0.141189	+	0.989983i	$0.454907\pi$
$422$	1.84471e10	0.581672
$423$	0	0
$424$	3.49861e10	1.08251
$425$	0	0
$426$	0	0
$427$	2.77096e9i	0.0833526i
$428$	−4.43444e9	−0.132149
$429$	0	0
$430$	0	0
$431$	− 3.08343e10i	− 0.893564i	−0.894643	−	0.446782i	$-0.852570\pi$
$431$	− 3.08343e10i	− 0.893564i	0.894643	−	0.446782i	$-0.147430\pi$
$432$	0	0
$433$	− 5.61968e10i	− 1.59867i	−0.600883	−	0.799337i	$-0.705184\pi$
$433$	− 5.61968e10i	− 1.59867i	0.600883	−	0.799337i	$-0.294816\pi$
$434$	1.63151e9i	0.0459865i
$435$	0	0
$436$	−2.29391e10	−0.634790
$437$	−7.91571e9	−0.217052
$438$	0	0
$439$	5.11998e10	1.37851	0.689256	−	0.724518i	$-0.257938\pi$
$439$	5.11998e10	1.37851	0.689256	+	0.724518i	$0.257938\pi$
$440$	0	0
$441$	0	0
$442$	2.87401e10i	0.753007i
$443$	−2.08745e10	−0.542001	−0.271001	−	0.962579i	$-0.587355\pi$
$443$	−2.08745e10	−0.542001	−0.271001	+	0.962579i	$0.587355\pi$
$444$	0	0
$445$	0	0
$446$	6.85676e10i	1.73292i
$447$	0	0
$448$	1.64478e9i	0.0408315i
$449$	1.21070e10i	0.297888i	0.988846	+	0.148944i	$0.0475874\pi$
$449$	1.21070e10i	0.297888i	−0.988846	+	0.148944i	$0.952413\pi$
$450$	0	0
$451$	−2.56882e10	−0.620907
$452$	−2.68472e10	−0.643199
$453$	0	0
$454$	1.26168e10	0.296980
$455$	0	0
$456$	0	0
$457$	− 2.67225e9i	− 0.0612650i	−0.999531	−	0.0306325i	$-0.990248\pi$
$457$	− 2.67225e9i	− 0.0612650i	0.999531	−	0.0306325i	$-0.00975215\pi$
$458$	−3.90266e9	−0.0886950
$459$	0	0
$460$	0	0
$461$	6.53060e10i	1.44594i	0.690881	+	0.722968i	$0.257223\pi$
$461$	6.53060e10i	1.44594i	−0.690881	+	0.722968i	$0.742777\pi$
$462$	0	0
$463$	− 3.77041e10i	− 0.820474i	−0.911979	−	0.410237i	$-0.865446\pi$
$463$	− 3.77041e10i	− 0.820474i	0.911979	−	0.410237i	$-0.134554\pi$
$464$	− 9.68325e10i	− 2.08905i
$465$	0	0
$466$	−4.91188e10	−1.04161
$467$	−5.49774e10	−1.15589	−0.577946	−	0.816075i	$-0.696145\pi$
$467$	−5.49774e10	−1.15589	−0.577946	+	0.816075i	$0.696145\pi$
$468$	0	0
$469$	−9.71986e9	−0.200895
$470$	0	0
$471$	0	0
$472$	1.31450e10i	0.264846i
$473$	1.35447e10	0.270599
$474$	0	0
$475$	0	0
$476$	− 2.12414e9i	− 0.0413767i
$477$	0	0
$478$	− 1.07909e11i	− 2.06703i
$479$	8.38732e10i	1.59324i	0.604481	+	0.796620i	$0.293381\pi$
$479$	8.38732e10i	1.59324i	−0.604481	+	0.796620i	$0.706619\pi$
$480$	0	0
$481$	4.74805e10	0.887023
$482$	−3.67176e10	−0.680278
$483$	0	0
$484$	1.41266e10	0.257429
$485$	0	0
$486$	0	0
$487$	− 8.98198e10i	− 1.59682i	−0.602113	−	0.798411i	$-0.705674\pi$
$487$	− 8.98198e10i	− 1.59682i	0.602113	−	0.798411i	$-0.294326\pi$
$488$	2.08669e10	0.367941
$489$	0	0
$490$	0	0
$491$	5.33288e10i	0.917563i	0.888549	+	0.458782i	$0.151714\pi$
$491$	5.33288e10i	0.917563i	−0.888549	+	0.458782i	$0.848286\pi$
$492$	0	0
$493$	− 6.13935e10i	− 1.03929i
$494$	− 3.71708e10i	− 0.624157i
$495$	0	0
$496$	1.89002e10	0.312277
$497$	−1.22509e10	−0.200790
$498$	0	0
$499$	−7.91014e9	−0.127580	−0.0637899	−	0.997963i	$-0.520319\pi$
$499$	−7.91014e9	−0.127580	−0.0637899	+	0.997963i	$0.520319\pi$
$500$	0	0
$501$	0	0
$502$	1.03384e11i	1.62794i
$503$	−1.24445e11	−1.94404	−0.972019	−	0.234902i	$-0.924523\pi$
$503$	−1.24445e11	−1.94404	−0.972019	+	0.234902i	$0.924523\pi$
$504$	0	0
$505$	0	0
$506$	2.12394e10i	0.323996i
$507$	0	0
$508$	1.58799e10i	0.238447i
$509$	1.71577e9i	0.0255616i	0.999918	+	0.0127808i	$0.00406837\pi$
$509$	1.71577e9i	0.0255616i	−0.999918	+	0.0127808i	$0.995932\pi$
$510$	0	0
$511$	1.64582e10	0.241378
$512$	−1.20973e10	−0.176038
$513$	0	0
$514$	−6.37352e10	−0.913118
$515$	0	0
$516$	0	0
$517$	− 6.47163e10i	− 0.905840i
$518$	−1.15579e10	−0.160531
$519$	0	0
$520$	0	0
$521$	− 9.22868e10i	− 1.25253i	−0.779610	−	0.626266i	$-0.784582\pi$
$521$	− 9.22868e10i	− 1.25253i	0.779610	−	0.626266i	$-0.215418\pi$
$522$	0	0
$523$	1.19364e11i	1.59539i	0.603060	+	0.797696i	$0.293948\pi$
$523$	1.19364e11i	1.59539i	−0.603060	+	0.797696i	$0.706052\pi$
$524$	2.03602e10i	0.270058i
$525$	0	0
$526$	−3.46929e10	−0.453208
$527$	1.19831e10	0.155355
$528$	0	0
$529$	−6.43338e10	−0.821517
$530$	0	0
$531$	0	0
$532$	2.74724e9i	0.0342966i
$533$	−7.93821e10	−0.983589
$534$	0	0
$535$	0	0
$536$	7.31958e10i	0.886803i
$537$	0	0
$538$	2.29083e9i	0.0273441i
$539$	5.27494e10i	0.624974i
$540$	0	0
$541$	−2.52941e10	−0.295277	−0.147639	−	0.989041i	$-0.547167\pi$
$541$	−2.52941e10	−0.295277	−0.147639	+	0.989041i	$0.547167\pi$
$542$	1.77615e11	2.05817
$543$	0	0
$544$	−4.43576e10	−0.506492
$545$	0	0
$546$	0	0
$547$	− 7.76113e10i	− 0.866913i	−0.901174	−	0.433457i	$-0.857294\pi$
$547$	− 7.76113e10i	− 0.866913i	0.901174	−	0.433457i	$-0.142706\pi$
$548$	−5.17461e10	−0.573793
$549$	0	0
$550$	0	0
$551$	7.94029e10i	0.861450i
$552$	0	0
$553$	2.85602e10i	0.305394i
$554$	− 1.04682e11i	− 1.11131i
$555$	0	0
$556$	−7.38460e10	−0.772730
$557$	−1.57686e11	−1.63822	−0.819109	−	0.573638i	$-0.805532\pi$
$557$	−1.57686e11	−1.63822	−0.819109	+	0.573638i	$0.805532\pi$
$558$	0	0
$559$	4.18562e10	0.428659
$560$	0	0
$561$	0	0
$562$	4.28657e10i	0.429699i
$563$	−1.71164e10	−0.170364	−0.0851822	−	0.996365i	$-0.527147\pi$
$563$	−1.71164e10	−0.170364	−0.0851822	+	0.996365i	$0.527147\pi$
$564$	0	0
$565$	0	0
$566$	− 2.29412e11i	− 2.23538i
$567$	0	0
$568$	9.22559e10i	0.886341i
$569$	− 3.67008e10i	− 0.350128i	−0.984557	−	0.175064i	$-0.943987\pi$
$569$	− 3.67008e10i	− 0.350128i	0.984557	−	0.175064i	$-0.0560132\pi$
$570$	0	0
$571$	−1.67307e11	−1.57387	−0.786936	−	0.617034i	$-0.788334\pi$
$571$	−1.67307e11	−1.57387	−0.786936	+	0.617034i	$0.788334\pi$
$572$	−3.02822e10	−0.282881
$573$	0	0
$574$	1.93234e10	0.178007
$575$	0	0
$576$	0	0
$577$	− 1.32539e11i	− 1.19575i	−0.801590	−	0.597874i	$-0.796012\pi$
$577$	− 1.32539e11i	− 1.19575i	0.801590	−	0.597874i	$-0.203988\pi$
$578$	8.23643e10	0.737951
$579$	0	0
$580$	0	0
$581$	8.72775e9i	0.0765946i
$582$	0	0
$583$	1.18417e11i	1.02504i
$584$	− 1.23939e11i	− 1.06551i
$585$	0	0
$586$	−3.14692e10	−0.266867
$587$	1.04094e11	0.876745	0.438372	−	0.898793i	$-0.355555\pi$
$587$	1.04094e11	0.876745	0.438372	+	0.898793i	$0.355555\pi$
$588$	0	0
$589$	−1.54983e10	−0.128772
$590$	0	0
$591$	0	0
$592$	1.33892e11i	1.09011i
$593$	3.74947e10	0.303215	0.151608	−	0.988441i	$-0.451555\pi$
$593$	3.74947e10	0.303215	0.151608	+	0.988441i	$0.451555\pi$
$594$	0	0
$595$	0	0
$596$	− 6.53868e10i	− 0.518209i
$597$	0	0
$598$	6.56344e10i	0.513247i
$599$	− 5.15028e10i	− 0.400058i	−0.979790	−	0.200029i	$-0.935896\pi$
$599$	− 5.15028e10i	− 0.400058i	0.979790	−	0.200029i	$-0.0641037\pi$
$600$	0	0
$601$	2.14744e11	1.64597	0.822987	−	0.568061i	$-0.192306\pi$
$601$	2.14744e11	1.64597	0.822987	+	0.568061i	$0.192306\pi$
$602$	−1.01888e10	−0.0775774
$603$	0	0
$604$	−9.88856e10	−0.742994
$605$	0	0
$606$	0	0
$607$	− 1.52816e11i	− 1.12568i	−0.826566	−	0.562839i	$-0.809709\pi$
$607$	− 1.52816e11i	− 1.12568i	0.826566	−	0.562839i	$-0.190291\pi$
$608$	5.73696e10	0.419825
$609$	0	0
$610$	0	0
$611$	− 1.99988e11i	− 1.43496i
$612$	0	0
$613$	1.27601e11i	0.903672i	0.892101	+	0.451836i	$0.149231\pi$
$613$	1.27601e11i	0.903672i	−0.892101	+	0.451836i	$0.850769\pi$
$614$	1.45815e10i	0.102595i
$615$	0	0
$616$	−9.53539e9	−0.0662240
$617$	9.12840e10	0.629874	0.314937	−	0.949113i	$-0.398017\pi$
$617$	9.12840e10	0.629874	0.314937	+	0.949113i	$0.398017\pi$
$618$	0	0
$619$	−2.09670e11	−1.42815	−0.714076	−	0.700068i	$-0.753153\pi$
$619$	−2.09670e11	−1.42815	−0.714076	+	0.700068i	$0.753153\pi$
$620$	0	0
$621$	0	0
$622$	3.26280e11i	2.17986i
$623$	1.92182e8	0.00127574
$624$	0	0
$625$	0	0
$626$	6.87516e8i	0.00447698i
$627$	0	0
$628$	3.11446e10i	0.200237i
$629$	8.48901e10i	0.542319i
$630$	0	0
$631$	1.84086e11	1.16119	0.580595	−	0.814192i	$-0.302820\pi$
$631$	1.84086e11	1.16119	0.580595	+	0.814192i	$0.302820\pi$
$632$	2.15074e11	1.34809
$633$	0	0
$634$	1.44927e11	0.896999
$635$	0	0
$636$	0	0
$637$	1.63007e11i	0.990031i
$638$	2.13054e11	1.28590
$639$	0	0
$640$	0	0
$641$	2.67533e11i	1.58469i	0.610073	+	0.792345i	$0.291140\pi$
$641$	2.67533e11i	1.58469i	−0.610073	+	0.792345i	$0.708860\pi$
$642$	0	0
$643$	− 2.16257e11i	− 1.26510i	−0.774519	−	0.632551i	$-0.782008\pi$
$643$	− 2.16257e11i	− 1.26510i	0.774519	−	0.632551i	$-0.217992\pi$
$644$	− 4.85095e9i	− 0.0282022i
$645$	0	0
$646$	6.64575e10	0.381605
$647$	−9.78698e10	−0.558511	−0.279255	−	0.960217i	$-0.590088\pi$
$647$	−9.78698e10	−0.558511	−0.279255	+	0.960217i	$0.590088\pi$
$648$	0	0
$649$	−4.44920e10	−0.250786
$650$	0	0
$651$	0	0
$652$	3.25291e9i	0.0180004i
$653$	1.06692e11	0.586787	0.293394	−	0.955992i	$-0.405215\pi$
$653$	1.06692e11	0.586787	0.293394	+	0.955992i	$0.405215\pi$
$654$	0	0
$655$	0	0
$656$	− 2.23853e11i	− 1.20878i
$657$	0	0
$658$	4.86816e10i	0.259694i
$659$	2.41804e11i	1.28210i	0.767498	+	0.641051i	$0.221501\pi$
$659$	2.41804e11i	1.28210i	−0.767498	+	0.641051i	$0.778499\pi$
$660$	0	0
$661$	3.16280e11	1.65678	0.828392	−	0.560149i	$-0.189256\pi$
$661$	3.16280e11	1.65678	0.828392	+	0.560149i	$0.189256\pi$
$662$	1.01871e10	0.0530418
$663$	0	0
$664$	6.57247e10	0.338109
$665$	0	0
$666$	0	0
$667$	− 1.40206e11i	− 0.708374i
$668$	5.28161e10	0.265253
$669$	0	0
$670$	0	0
$671$	7.06280e10i	0.348407i
$672$	0	0
$673$	− 8.47380e10i	− 0.413065i	−0.978440	−	0.206532i	$-0.933782\pi$
$673$	− 8.47380e10i	− 0.413065i	0.978440	−	0.206532i	$-0.0662179\pi$
$674$	− 2.86968e11i	− 1.39057i
$675$	0	0
$676$	−2.52969e9	−0.0121138
$677$	−1.37689e11	−0.655460	−0.327730	−	0.944771i	$-0.606284\pi$
$677$	−1.37689e11	−0.655460	−0.327730	+	0.944771i	$0.606284\pi$
$678$	0	0
$679$	1.34070e10	0.0630745
$680$	0	0
$681$	0	0
$682$	4.15848e10i	0.192220i
$683$	3.20959e11	1.47491	0.737457	−	0.675394i	$-0.236026\pi$
$683$	3.20959e11	1.47491	0.737457	+	0.675394i	$0.236026\pi$
$684$	0	0
$685$	0	0
$686$	− 8.03119e10i	− 0.362647i
$687$	0	0
$688$	1.18032e11i	0.526800i
$689$	3.65935e11i	1.62378i
$690$	0	0
$691$	9.13185e10	0.400540	0.200270	−	0.979741i	$-0.435818\pi$
$691$	9.13185e10	0.400540	0.200270	+	0.979741i	$0.435818\pi$
$692$	−7.38280e10	−0.321956
$693$	0	0
$694$	3.14769e11	1.35692
$695$	0	0
$696$	0	0
$697$	− 1.41927e11i	− 0.601358i
$698$	4.90022e11	2.06440
$699$	0	0
$700$	0	0
$701$	− 3.75287e11i	− 1.55415i	−0.629411	−	0.777073i	$-0.716704\pi$
$701$	− 3.75287e11i	− 1.55415i	0.629411	−	0.777073i	$-0.283296\pi$
$702$	0	0
$703$	− 1.09792e11i	− 0.449521i
$704$	4.19231e10i	0.170672i
$705$	0	0
$706$	−1.91366e11	−0.770276
$707$	−1.52453e10	−0.0610182
$708$	0	0
$709$	2.45402e11	0.971167	0.485583	−	0.874190i	$-0.338607\pi$
$709$	2.45402e11	0.971167	0.485583	+	0.874190i	$0.338607\pi$
$710$	0	0
$711$	0	0
$712$	− 1.44724e9i	− 0.00563145i
$713$	2.73661e10	0.105890
$714$	0	0
$715$	0	0
$716$	− 7.09656e10i	− 0.270020i
$717$	0	0
$718$	3.62880e11i	1.36542i
$719$	9.18110e10i	0.343541i	0.985137	+	0.171771i	$0.0549488\pi$
$719$	9.18110e10i	0.343541i	−0.985137	+	0.171771i	$0.945051\pi$
$720$	0	0
$721$	7.16218e10	0.265036
$722$	2.39679e11	0.882026
$723$	0	0
$724$	−1.15482e11	−0.420300
$725$	0	0
$726$	0	0
$727$	2.00140e11i	0.716466i	0.933632	+	0.358233i	$0.116621\pi$
$727$	2.00140e11i	0.716466i	−0.933632	+	0.358233i	$0.883379\pi$
$728$	−2.94665e10	−0.104907
$729$	0	0
$730$	0	0
$731$	7.48344e10i	0.262079i
$732$	0	0
$733$	− 2.83723e11i	− 0.982831i	−0.870925	−	0.491415i	$-0.836480\pi$
$733$	− 2.83723e11i	− 0.982831i	0.870925	−	0.491415i	$-0.163520\pi$
$734$	3.17628e11i	1.09429i
$735$	0	0
$736$	−1.01301e11	−0.345224
$737$	−2.47746e11	−0.839723
$738$	0	0
$739$	2.87048e10	0.0962448	0.0481224	−	0.998841i	$-0.484676\pi$
$739$	2.87048e10	0.0962448	0.0481224	+	0.998841i	$0.484676\pi$
$740$	0	0
$741$	0	0
$742$	− 8.90772e10i	− 0.293867i
$743$	−3.52977e11	−1.15822	−0.579110	−	0.815249i	$-0.696600\pi$
$743$	−3.52977e11	−1.15822	−0.579110	+	0.815249i	$0.696600\pi$
$744$	0	0
$745$	0	0
$746$	1.68384e11i	0.543682i
$747$	0	0
$748$	− 5.41414e10i	− 0.172951i
$749$	− 1.46049e10i	− 0.0464057i
$750$	0	0
$751$	1.47680e11	0.464260	0.232130	−	0.972685i	$-0.425431\pi$
$751$	1.47680e11	0.464260	0.232130	+	0.972685i	$0.425431\pi$
$752$	5.63954e11	1.76349
$753$	0	0
$754$	6.58383e11	2.03701
$755$	0	0
$756$	0	0
$757$	− 1.11874e10i	− 0.0340679i	−0.999855	−	0.0170339i	$-0.994578\pi$
$757$	− 1.11874e10i	− 0.0340679i	0.999855	−	0.0170339i	$-0.00542233\pi$
$758$	−8.19243e10	−0.248162
$759$	0	0
$760$	0	0
$761$	− 1.96694e9i	− 0.00586480i	−0.999996	−	0.00293240i	$-0.999067\pi$
$761$	− 1.96694e9i	− 0.00586480i	0.999996	−	0.00293240i	$-0.000933414\pi$
$762$	0	0
$763$	− 7.55503e10i	− 0.222914i
$764$	5.70010e10i	0.167305i
$765$	0	0
$766$	1.47380e11	0.428079
$767$	−1.37490e11	−0.397274
$768$	0	0
$769$	−2.76788e11	−0.791484	−0.395742	−	0.918362i	$-0.629513\pi$
$769$	−2.76788e11	−0.791484	−0.395742	+	0.918362i	$0.629513\pi$
$770$	0	0
$771$	0	0
$772$	5.23655e10i	0.147427i
$773$	−1.07076e11	−0.299897	−0.149949	−	0.988694i	$-0.547911\pi$
$773$	−1.07076e11	−0.299897	−0.149949	+	0.988694i	$0.547911\pi$
$774$	0	0
$775$	0	0
$776$	− 1.00962e11i	− 0.278428i
$777$	0	0
$778$	− 1.17856e11i	− 0.321688i
$779$	1.83560e11i	0.498458i
$780$	0	0
$781$	−3.12258e11	−0.839286
$782$	−1.17347e11	−0.313795
$783$	0	0
$784$	−4.59671e11	−1.21670
$785$	0	0
$786$	0	0
$787$	9.64061e10i	0.251308i	0.992074	+	0.125654i	$0.0401028\pi$
$787$	9.64061e10i	0.251308i	−0.992074	+	0.125654i	$0.959897\pi$
$788$	−2.76372e11	−0.716784
$789$	0	0
$790$	0	0
$791$	− 8.84218e10i	− 0.225867i
$792$	0	0
$793$	2.18256e11i	0.551917i
$794$	− 4.23607e10i	− 0.106581i
$795$	0	0
$796$	1.88689e11	0.469995
$797$	2.37223e11	0.587928	0.293964	−	0.955817i	$-0.405025\pi$
$797$	2.37223e11	0.587928	0.293964	+	0.955817i	$0.405025\pi$
$798$	0	0
$799$	3.57557e11	0.877320
$800$	0	0
$801$	0	0
$802$	− 3.23628e11i	− 0.782255i
$803$	4.19495e11	1.00894
$804$	0	0
$805$	0	0
$806$	1.28506e11i	0.304498i
$807$	0	0
$808$	1.14806e11i	0.269351i
$809$	− 7.32794e11i	− 1.71075i	−0.518006	−	0.855377i	$-0.673325\pi$
$809$	− 7.32794e11i	− 1.71075i	0.518006	−	0.855377i	$-0.326675\pi$
$810$	0	0
$811$	7.41211e10	0.171340	0.0856699	−	0.996324i	$-0.472697\pi$
$811$	7.41211e10	0.171340	0.0856699	+	0.996324i	$0.472697\pi$
$812$	−4.86602e10	−0.111931
$813$	0	0
$814$	−2.94594e11	−0.671005
$815$	0	0
$816$	0	0
$817$	− 9.67866e10i	− 0.217234i
$818$	3.11155e11	0.694966
$819$	0	0
$820$	0	0
$821$	6.64630e11i	1.46288i	0.681908	+	0.731438i	$0.261150\pi$
$821$	6.64630e11i	1.46288i	−0.681908	+	0.731438i	$0.738850\pi$
$822$	0	0
$823$	8.12755e10i	0.177158i	0.996069	+	0.0885789i	$0.0282325\pi$
$823$	8.12755e10i	0.177158i	−0.996069	+	0.0885789i	$0.971767\pi$
$824$	− 5.39351e11i	− 1.16994i
$825$	0	0
$826$	3.34683e10	0.0718974
$827$	−6.94609e11	−1.48497	−0.742487	−	0.669861i	$-0.766354\pi$
$827$	−6.94609e11	−1.48497	−0.742487	+	0.669861i	$0.766354\pi$
$828$	0	0
$829$	−4.71468e11	−0.998239	−0.499119	−	0.866533i	$-0.666343\pi$
$829$	−4.71468e11	−0.998239	−0.499119	+	0.866533i	$0.666343\pi$
$830$	0	0
$831$	0	0
$832$	1.29552e11i	0.270365i
$833$	−2.91440e11	−0.605297
$834$	0	0
$835$	0	0
$836$	7.00234e10i	0.143357i
$837$	0	0
$838$	− 9.22145e11i	− 1.86992i
$839$	− 7.29115e11i	− 1.47146i	−0.677275	−	0.735730i	$-0.736839\pi$
$839$	− 7.29115e11i	− 1.47146i	0.677275	−	0.735730i	$-0.263161\pi$
$840$	0	0
$841$	−9.06167e11	−1.81144
$842$	−1.70081e11	−0.338383
$843$	0	0
$844$	−1.07389e11	−0.211636
$845$	0	0
$846$	0	0
$847$	4.65263e10i	0.0903993i
$848$	−1.03192e12	−1.99554
$849$	0	0
$850$	0	0
$851$	1.93866e11i	0.369643i
$852$	0	0
$853$	6.28399e11i	1.18697i	0.804845	+	0.593484i	$0.202248\pi$
$853$	6.28399e11i	1.18697i	−0.804845	+	0.593484i	$0.797752\pi$
$854$	− 5.31286e10i	− 0.0998842i
$855$	0	0
$856$	−1.09983e11	−0.204847
$857$	3.34649e11	0.620391	0.310196	−	0.950673i	$-0.399605\pi$
$857$	3.34649e11	0.620391	0.310196	+	0.950673i	$0.399605\pi$
$858$	0	0
$859$	−7.69062e11	−1.41250	−0.706250	−	0.707962i	$-0.749615\pi$
$859$	−7.69062e11	−1.41250	−0.706250	+	0.707962i	$0.749615\pi$
$860$	0	0
$861$	0	0
$862$	5.91197e11i	1.07079i
$863$	−4.07704e11	−0.735024	−0.367512	−	0.930019i	$-0.619790\pi$
$863$	−4.07704e11	−0.735024	−0.367512	+	0.930019i	$0.619790\pi$
$864$	0	0
$865$	0	0
$866$	1.07748e12i	1.91574i
$867$	0	0
$868$	− 9.49773e9i	− 0.0167317i
$869$	7.27959e11i	1.27652i
$870$	0	0
$871$	−7.65589e11	−1.33022
$872$	−5.68935e11	−0.984003
$873$	0	0
$874$	1.51770e11	0.260101
$875$	0	0
$876$	0	0
$877$	− 3.77721e11i	− 0.638517i	−0.947668	−	0.319258i	$-0.896566\pi$
$877$	− 3.77721e11i	− 0.638517i	0.947668	−	0.319258i	$-0.103434\pi$
$878$	−9.81671e11	−1.65192
$879$	0	0
$880$	0	0
$881$	− 8.54455e11i	− 1.41836i	−0.705029	−	0.709179i	$-0.749066\pi$
$881$	− 8.54455e11i	− 1.41836i	0.705029	−	0.709179i	$-0.250934\pi$
$882$	0	0
$883$	− 3.81956e11i	− 0.628305i	−0.949373	−	0.314153i	$-0.898280\pi$
$883$	− 3.81956e11i	− 0.628305i	0.949373	−	0.314153i	$-0.101720\pi$
$884$	− 1.67309e11i	− 0.273974i
$885$	0	0
$886$	4.00233e11	0.649498
$887$	−1.16568e11	−0.188314	−0.0941571	−	0.995557i	$-0.530016\pi$
$887$	−1.16568e11	−0.188314	−0.0941571	+	0.995557i	$0.530016\pi$
$888$	0	0
$889$	−5.23007e10	−0.0837338
$890$	0	0
$891$	0	0
$892$	− 3.99163e11i	− 0.630508i
$893$	−4.62444e11	−0.727199
$894$	0	0
$895$	0	0
$896$	− 1.12172e11i	− 0.174042i
$897$	0	0
$898$	− 2.32132e11i	− 0.356969i
$899$	− 2.74511e11i	− 0.420262i
$900$	0	0
$901$	−6.54254e11	−0.992767
$902$	4.92527e11	0.744054
$903$	0	0
$904$	−6.65864e11	−0.997039
$905$	0	0
$906$	0	0
$907$	1.36281e11i	0.201375i	0.994918	+	0.100688i	$0.0321043\pi$
$907$	1.36281e11i	0.201375i	−0.994918	+	0.100688i	$0.967896\pi$
$908$	−7.34483e10	−0.108053
$909$	0	0
$910$	0	0
$911$	− 2.64274e11i	− 0.383691i	−0.981425	−	0.191845i	$-0.938553\pi$
$911$	− 2.64274e11i	− 0.383691i	0.981425	−	0.191845i	$-0.0614472\pi$
$912$	0	0
$913$	2.22458e11i	0.320159i
$914$	5.12359e10i	0.0734159i
$915$	0	0
$916$	2.27192e10	0.0322708
$917$	−6.70568e10	−0.0948343
$918$	0	0
$919$	−4.88253e11	−0.684515	−0.342257	−	0.939606i	$-0.611191\pi$
$919$	−4.88253e11	−0.684515	−0.342257	+	0.939606i	$0.611191\pi$
$920$	0	0
$921$	0	0
$922$	− 1.25213e12i	− 1.73271i
$923$	−9.64947e11	−1.32953
$924$	0	0
$925$	0	0
$926$	7.22914e11i	0.983202i
$927$	0	0
$928$	1.01615e12i	1.37015i
$929$	3.78797e11i	0.508562i	0.967130	+	0.254281i	$0.0818388\pi$
$929$	3.78797e11i	0.508562i	−0.967130	+	0.254281i	$0.918161\pi$
$930$	0	0
$931$	3.76931e11	0.501723
$932$	2.85943e11	0.378979
$933$	0	0
$934$	1.05410e12	1.38514
$935$	0	0
$936$	0	0
$937$	1.32228e10i	0.0171539i	0.999963	+	0.00857697i	$0.00273017\pi$
$937$	1.32228e10i	0.0171539i	−0.999963	+	0.00857697i	$0.997270\pi$
$938$	1.86362e11	0.240739
$939$	0	0
$940$	0	0
$941$	− 8.58191e11i	− 1.09452i	−0.836961	−	0.547262i	$-0.815670\pi$
$941$	− 8.58191e11i	− 1.09452i	0.836961	−	0.547262i	$-0.184330\pi$
$942$	0	0
$943$	− 3.24122e11i	− 0.409884i
$944$	− 3.87714e11i	− 0.488229i
$945$	0	0
$946$	−2.59697e11	−0.324267
$947$	−5.90260e11	−0.733911	−0.366955	−	0.930239i	$-0.619600\pi$
$947$	−5.90260e11	−0.733911	−0.366955	+	0.930239i	$0.619600\pi$
$948$	0	0
$949$	1.29633e12	1.59828
$950$	0	0
$951$	0	0
$952$	− 5.26829e10i	− 0.0641390i
$953$	−4.29217e11	−0.520362	−0.260181	−	0.965560i	$-0.583782\pi$
$953$	−4.29217e11	−0.520362	−0.260181	+	0.965560i	$0.583782\pi$
$954$	0	0
$955$	0	0
$956$	6.28186e11i	0.752068i
$957$	0	0
$958$	− 1.60813e12i	− 1.90923i
$959$	− 1.70427e11i	− 0.201494i
$960$	0	0
$961$	−7.99311e11	−0.937178
$962$	−9.10359e11	−1.06295
$963$	0	0
$964$	2.13750e11	0.247513
$965$	0	0
$966$	0	0
$967$	− 5.19725e11i	− 0.594385i	−0.954818	−	0.297193i	$-0.903950\pi$
$967$	− 5.19725e11i	− 0.594385i	0.954818	−	0.297193i	$-0.0960503\pi$
$968$	3.50368e11	0.399046
$969$	0	0
$970$	0	0
$971$	8.29188e11i	0.932773i	0.884581	+	0.466386i	$0.154444\pi$
$971$	8.29188e11i	0.932773i	−0.884581	+	0.466386i	$0.845556\pi$
$972$	0	0
$973$	− 2.43213e11i	− 0.271354i
$974$	1.72215e12i	1.91352i
$975$	0	0
$976$	−6.15470e11	−0.678278
$977$	5.88558e11	0.645968	0.322984	−	0.946404i	$-0.395314\pi$
$977$	5.88558e11	0.645968	0.322984	+	0.946404i	$0.395314\pi$
$978$	0	0
$979$	4.89846e9	0.00533248
$980$	0	0
$981$	0	0
$982$	− 1.02249e12i	− 1.09955i
$983$	8.98175e11	0.961938	0.480969	−	0.876738i	$-0.340285\pi$
$983$	8.98175e11	0.961938	0.480969	+	0.876738i	$0.340285\pi$
$984$	0	0
$985$	0	0
$986$	1.17712e12i	1.24541i
$987$	0	0
$988$	2.16388e11i	0.227094i
$989$	1.70901e11i	0.178632i
$990$	0	0
$991$	−3.95110e11	−0.409660	−0.204830	−	0.978798i	$-0.565664\pi$
$991$	−3.95110e11	−0.409660	−0.204830	+	0.978798i	$0.565664\pi$
$992$	−1.98338e11	−0.204813
$993$	0	0
$994$	2.34891e11	0.240614
$995$	0	0
$996$	0	0
$997$	1.72914e12i	1.75004i	0.484083	+	0.875022i	$0.339153\pi$
$997$	1.72914e12i	1.75004i	−0.484083	+	0.875022i	$0.660847\pi$
$998$	1.51664e11	0.152883
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.9.d.c.224.8		24
3.2	odd	2	inner	225.9.d.c.224.18		24
5.2	odd	4		225.9.c.d.26.4		12
5.3	odd	4		45.9.c.a.26.9	yes	12
5.4	even	2	inner	225.9.d.c.224.17		24
15.2	even	4		225.9.c.d.26.9		12
15.8	even	4		45.9.c.a.26.4	✓	12
15.14	odd	2	inner	225.9.d.c.224.7		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.9.c.a.26.4	✓	12	15.8	even	4
45.9.c.a.26.9	yes	12	5.3	odd	4
225.9.c.d.26.4		12	5.2	odd	4
225.9.c.d.26.9		12	15.2	even	4
225.9.d.c.224.7		24	15.14	odd	2	inner
225.9.d.c.224.8		24	1.1	even	1	trivial
225.9.d.c.224.17		24	5.4	even	2	inner
225.9.d.c.224.18		24	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-19.1733 q^{2} +111.617 q^{4} +367.611i q^{7} +2768.31 q^{8} +9369.90i q^{11} +28955.0i q^{13} -7048.33i q^{14} -81651.6 q^{16} -51768.6 q^{17} +66954.5 q^{19} -179652. i q^{22} -118225. q^{23} -555165. i q^{26} +41031.5i q^{28} +1.18592e6i q^{29} -231474. q^{31} +856845. q^{32} +992576. q^{34} -1.63980e6i q^{37} -1.28374e6 q^{38} +2.74156e6i q^{41} -1.44556e6i q^{43} +1.04584e6i q^{44} +2.26677e6 q^{46} -6.90683e6 q^{47} +5.62966e6 q^{49} +3.23186e6i q^{52} +1.26381e7 q^{53} +1.01766e6i q^{56} -2.27381e7i q^{58} +4.74840e6i q^{59} +7.53776e6 q^{61} +4.43813e6 q^{62} +4.47424e6 q^{64} +2.64406e7i q^{67} -5.77823e6 q^{68} +3.33257e7i q^{71} -4.47705e7i q^{73} +3.14404e7i q^{74} +7.47323e6 q^{76} -3.44448e6 q^{77} +7.76913e7 q^{79} -5.25649e7i q^{82} +2.37418e7 q^{83} +2.77161e7i q^{86} +2.59388e7i q^{88} -522787. i q^{89} -1.06442e7 q^{91} -1.31959e7 q^{92} +1.32427e8 q^{94} -3.64707e7i q^{97} -1.07939e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 5048 q^{4} + 595144 q^{16} - 249920 q^{19} - 1241936 q^{31} - 14972992 q^{34} - 14323536 q^{46} - 42020312 q^{49} + 56906192 q^{61} + 338779896 q^{64} - 511347840 q^{76} + 127619024 q^{79} - 593698576 q^{91}+ \cdots - 51203776 q^{94}+O(q^{100}) \) 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

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