LMFDB - Embedded newform 225.10.b.m.199.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(199,225)]

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, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$115.883063137$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 1305x^{4} + 433104x^{2} + 16000000 $ x^6 + 1305x^4 + 433104x^2 + 16000000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.6
Root			$27.7229i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.10.b.m.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+31.7828i q^{2} -498.143 q^{4} +637.237i q^{7} +440.406i q^{8} +O(q^{10})$	$q+31.7828i q^{2} -498.143 q^{4} +637.237i q^{7} +440.406i q^{8} +49042.6 q^{11} -72726.2i q^{13} -20253.2 q^{14} -269047. q^{16} -67319.3i q^{17} -341136. q^{19} +1.55871e6i q^{22} -134355. i q^{23} +2.31144e6 q^{26} -317435. i q^{28} +4.45784e6 q^{29} +456520. q^{31} -8.32555e6i q^{32} +2.13959e6 q^{34} +1.30516e7i q^{37} -1.08422e7i q^{38} +2.56667e7 q^{41} -3.42710e6i q^{43} -2.44302e7 q^{44} +4.27016e6 q^{46} +3.39814e7i q^{47} +3.99475e7 q^{49} +3.62281e7i q^{52} -8.42456e7i q^{53} -280643. q^{56} +1.41682e8i q^{58} -7.46358e7 q^{59} +1.78017e8 q^{61} +1.45094e7i q^{62} +1.26857e8 q^{64} +6.94299e7i q^{67} +3.35347e7i q^{68} +2.07860e8 q^{71} +3.02516e8i q^{73} -4.14815e8 q^{74} +1.69935e8 q^{76} +3.12518e7i q^{77} -3.72244e8 q^{79} +8.15758e8i q^{82} +4.50079e8i q^{83} +1.08923e8 q^{86} +2.15987e7i q^{88} +5.82741e7 q^{89} +4.63438e7 q^{91} +6.69279e7i q^{92} -1.08002e9 q^{94} +7.85850e8i q^{97} +1.26964e9i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 682 q^{4}+O(q^{10}) $ 6 * q - 682 * q^4	$ 6 q - 682 q^{4} + 109398 q^{11} + 545844 q^{14} - 1398494 q^{16} - 1637690 q^{19} + 2560008 q^{26} + 4350960 q^{29} + 8548132 q^{31} + 13677926 q^{34} - 11852622 q^{41} - 43991406 q^{44} + 28446492 q^{46} + 12907858 q^{49} + 339076260 q^{56} + 11341920 q^{59} + 250613852 q^{61} - 335084802 q^{64} - 595101192 q^{71} - 503014716 q^{74} + 178829730 q^{76} + 620050340 q^{79} - 67894392 q^{86} - 2207720070 q^{89} + 2366375312 q^{91} - 3455782264 q^{94}+O(q^{100}) $ 6 * q - 682 * q^4 + 109398 * q^11 + 545844 * q^14 - 1398494 * q^16 - 1637690 * q^19 + 2560008 * q^26 + 4350960 * q^29 + 8548132 * q^31 + 13677926 * q^34 - 11852622 * q^41 - 43991406 * q^44 + 28446492 * q^46 + 12907858 * q^49 + 339076260 * q^56 + 11341920 * q^59 + 250613852 * q^61 - 335084802 * q^64 - 595101192 * q^71 - 503014716 * q^74 + 178829730 * q^76 + 620050340 * q^79 - 67894392 * q^86 - 2207720070 * q^89 + 2366375312 * q^91 - 3455782264 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	31.7828i	1.40461i	0.711875	+	0.702306i	$0.247846\pi$
$2$	31.7828i	1.40461i	−0.711875	+	0.702306i	$0.752154\pi$
$3$	0	0
$3$	0	0
$4$	−498.143	−0.972936
$5$	0	0
$5$	0	0
$6$	0	0
$7$	637.237i	0.100314i	0.998741	+	0.0501568i	$0.0159721\pi$
$7$	637.237i	0.100314i	−0.998741	+	0.0501568i	$0.984028\pi$
$8$	440.406i	0.0380144i
$9$	0	0
$10$	0	0
$11$	49042.6	1.00996	0.504982	−	0.863130i	$-0.331499\pi$
$11$	49042.6	1.00996	0.504982	+	0.863130i	$0.331499\pi$
$12$	0	0
$13$	− 72726.2i	− 0.706229i	−0.935580	−	0.353115i	$-0.885123\pi$
$13$	− 72726.2i	− 0.706229i	0.935580	−	0.353115i	$-0.114877\pi$
$14$	−20253.2	−0.140902
$15$	0	0
$16$	−269047.	−1.02633
$17$	− 67319.3i	− 0.195488i	−0.995212	−	0.0977439i	$-0.968837\pi$
$17$	− 67319.3i	− 0.195488i	0.995212	−	0.0977439i	$-0.0311626\pi$
$18$	0	0
$19$	−341136.	−0.600532	−0.300266	−	0.953855i	$-0.597075\pi$
$19$	−341136.	−0.600532	−0.300266	+	0.953855i	$0.597075\pi$
$20$	0	0
$21$	0	0
$22$	1.55871e6i	1.41861i
$23$	− 134355.i	− 0.100110i	−0.998746	−	0.0500550i	$-0.984060\pi$
$23$	− 134355.i	− 0.100110i	0.998746	−	0.0500550i	$-0.0159397\pi$
$24$	0	0
$25$	0	0
$26$	2.31144e6	0.991978
$27$	0	0
$28$	− 317435.i	− 0.0975988i
$29$	4.45784e6	1.17040	0.585199	−	0.810890i	$-0.301016\pi$
$29$	4.45784e6	1.17040	0.585199	+	0.810890i	$0.301016\pi$
$30$	0	0
$31$	456520.	0.0887834	0.0443917	−	0.999014i	$-0.485865\pi$
$31$	456520.	0.0887834	0.0443917	+	0.999014i	$0.485865\pi$
$32$	− 8.32555e6i	− 1.40358i
$33$	0	0
$34$	2.13959e6	0.274585
$35$	0	0
$36$	0	0
$37$	1.30516e7i	1.14487i	0.819951	+	0.572433i	$0.194001\pi$
$37$	1.30516e7i	1.14487i	−0.819951	+	0.572433i	$0.805999\pi$
$38$	− 1.08422e7i	− 0.843515i
$39$	0	0
$40$	0	0
$41$	2.56667e7	1.41854	0.709272	−	0.704935i	$-0.249024\pi$
$41$	2.56667e7	1.41854	0.709272	+	0.704935i	$0.249024\pi$
$42$	0	0
$43$	− 3.42710e6i	− 0.152869i	−0.997075	−	0.0764344i	$-0.975646\pi$
$43$	− 3.42710e6i	− 0.152869i	0.997075	−	0.0764344i	$-0.0243536\pi$
$44$	−2.44302e7	−0.982631
$45$	0	0
$46$	4.27016e6	0.140616
$47$	3.39814e7i	1.01578i	0.861421	+	0.507891i	$0.169575\pi$
$47$	3.39814e7i	1.01578i	−0.861421	+	0.507891i	$0.830425\pi$
$48$	0	0
$49$	3.99475e7	0.989937
$50$	0	0
$51$	0	0
$52$	3.62281e7i	0.687116i
$53$	− 8.42456e7i	− 1.46658i	−0.679916	−	0.733290i	$-0.737984\pi$
$53$	− 8.42456e7i	− 1.46658i	0.679916	−	0.733290i	$-0.262016\pi$
$54$	0	0
$55$	0	0
$56$	−280643.	−0.00381337
$57$	0	0
$58$	1.41682e8i	1.64396i
$59$	−7.46358e7	−0.801887	−0.400944	−	0.916103i	$-0.631318\pi$
$59$	−7.46358e7	−0.801887	−0.400944	+	0.916103i	$0.631318\pi$
$60$	0	0
$61$	1.78017e8	1.64618	0.823088	−	0.567913i	$-0.192249\pi$
$61$	1.78017e8	1.64618	0.823088	+	0.567913i	$0.192249\pi$
$62$	1.45094e7i	0.124706i
$63$	0	0
$64$	1.26857e8	0.945159
$65$	0	0
$66$	0	0
$67$	6.94299e7i	0.420930i	0.977601	+	0.210465i	$0.0674978\pi$
$67$	6.94299e7i	0.420930i	−0.977601	+	0.210465i	$0.932502\pi$
$68$	3.35347e7i	0.190197i
$69$	0	0
$70$	0	0
$71$	2.07860e8	0.970753	0.485376	−	0.874305i	$-0.338683\pi$
$71$	2.07860e8	0.970753	0.485376	+	0.874305i	$0.338683\pi$
$72$	0	0
$73$	3.02516e8i	1.24680i	0.781905	+	0.623398i	$0.214248\pi$
$73$	3.02516e8i	1.24680i	−0.781905	+	0.623398i	$0.785752\pi$
$74$	−4.14815e8	−1.60809
$75$	0	0
$76$	1.69935e8	0.584279
$77$	3.12518e7i	0.101313i
$78$	0	0
$79$	−3.72244e8	−1.07524	−0.537621	−	0.843187i	$-0.680677\pi$
$79$	−3.72244e8	−1.07524	−0.537621	+	0.843187i	$0.680677\pi$
$80$	0	0
$81$	0	0
$82$	8.15758e8i	1.99250i
$83$	4.50079e8i	1.04097i	0.853872	+	0.520484i	$0.174248\pi$
$83$	4.50079e8i	1.04097i	−0.853872	+	0.520484i	$0.825752\pi$
$84$	0	0
$85$	0	0
$86$	1.08923e8	0.214721
$87$	0	0
$88$	2.15987e7i	0.0383932i
$89$	5.82741e7	0.0984511	0.0492255	−	0.998788i	$-0.484325\pi$
$89$	5.82741e7	0.0984511	0.0492255	+	0.998788i	$0.484325\pi$
$90$	0	0
$91$	4.63438e7	0.0708444
$92$	6.69279e7i	0.0974006i
$93$	0	0
$94$	−1.08002e9	−1.42678
$95$	0	0
$96$	0	0
$97$	7.85850e8i	0.901295i	0.892702	+	0.450647i	$0.148807\pi$
$97$	7.85850e8i	0.901295i	−0.892702	+	0.450647i	$0.851193\pi$
$98$	1.26964e9i	1.39048i
$99$	0	0
$100$	0	0
$101$	1.60460e8	0.153433	0.0767167	−	0.997053i	$-0.475556\pi$
$101$	1.60460e8	0.153433	0.0767167	+	0.997053i	$0.475556\pi$
$102$	0	0
$103$	1.39454e9i	1.22085i	0.792074	+	0.610425i	$0.209001\pi$
$103$	1.39454e9i	1.22085i	−0.792074	+	0.610425i	$0.790999\pi$
$104$	3.20291e7	0.0268469
$105$	0	0
$106$	2.67756e9	2.05998
$107$	1.56463e9i	1.15394i	0.816764	+	0.576972i	$0.195766\pi$
$107$	1.56463e9i	1.15394i	−0.816764	+	0.576972i	$0.804234\pi$
$108$	0	0
$109$	−1.24703e9	−0.846172	−0.423086	−	0.906089i	$-0.639053\pi$
$109$	−1.24703e9	−0.846172	−0.423086	+	0.906089i	$0.639053\pi$
$110$	0	0
$111$	0	0
$112$	− 1.71447e8i	− 0.102955i
$113$	− 1.81056e9i	− 1.04463i	−0.852754	−	0.522313i	$-0.825069\pi$
$113$	− 1.81056e9i	− 1.04463i	0.852754	−	0.522313i	$-0.174931\pi$
$114$	0	0
$115$	0	0
$116$	−2.22064e9	−1.13872
$117$	0	0
$118$	− 2.37213e9i	− 1.12634i
$119$	4.28984e7	0.0196101
$120$	0	0
$121$	4.72275e7	0.0200291
$122$	5.65786e9i	2.31224i
$123$	0	0
$124$	−2.27412e8	−0.0863806
$125$	0	0
$126$	0	0
$127$	− 3.06491e9i	− 1.04545i	−0.852503	−	0.522723i	$-0.824916\pi$
$127$	− 3.06491e9i	− 1.04545i	0.852503	−	0.522723i	$-0.175084\pi$
$128$	− 2.30815e8i	− 0.0760010i
$129$	0	0
$130$	0	0
$131$	1.83508e9	0.544421	0.272211	−	0.962238i	$-0.412245\pi$
$131$	1.83508e9	0.544421	0.272211	+	0.962238i	$0.412245\pi$
$132$	0	0
$133$	− 2.17385e8i	− 0.0602416i
$134$	−2.20667e9	−0.591243
$135$	0	0
$136$	2.96478e7	0.00743136
$137$	4.62426e9i	1.12150i	0.827984	+	0.560751i	$0.189488\pi$
$137$	4.62426e9i	1.12150i	−0.827984	+	0.560751i	$0.810512\pi$
$138$	0	0
$139$	−2.57140e8	−0.0584255	−0.0292128	−	0.999573i	$-0.509300\pi$
$139$	−2.57140e8	−0.0584255	−0.0292128	+	0.999573i	$0.509300\pi$
$140$	0	0
$141$	0	0
$142$	6.60637e9i	1.36353i
$143$	− 3.56668e9i	− 0.713267i
$144$	0	0
$145$	0	0
$146$	−9.61479e9	−1.75127
$147$	0	0
$148$	− 6.50155e9i	− 1.11388i
$149$	3.14809e9	0.523250	0.261625	−	0.965170i	$-0.415742\pi$
$149$	3.14809e9	0.523250	0.261625	+	0.965170i	$0.415742\pi$
$150$	0	0
$151$	−1.02772e10	−1.60871	−0.804356	−	0.594147i	$-0.797490\pi$
$151$	−1.02772e10	−1.60871	−0.804356	+	0.594147i	$0.797490\pi$
$152$	− 1.50238e8i	− 0.0228289i
$153$	0	0
$154$	−9.93267e8	−0.142306
$155$	0	0
$156$	0	0
$157$	3.25077e9i	0.427009i	0.976942	+	0.213505i	$0.0684878\pi$
$157$	3.25077e9i	0.427009i	−0.976942	+	0.213505i	$0.931512\pi$
$158$	− 1.18309e10i	− 1.51030i
$159$	0	0
$160$	0	0
$161$	8.56158e7	0.0100424
$162$	0	0
$163$	− 3.40049e9i	− 0.377309i	−0.982043	−	0.188655i	$-0.939587\pi$
$163$	− 3.40049e9i	− 0.377309i	0.982043	−	0.188655i	$-0.0604127\pi$
$164$	−1.27857e10	−1.38015
$165$	0	0
$166$	−1.43047e10	−1.46216
$167$	− 5.27785e9i	− 0.525089i	−0.964920	−	0.262544i	$-0.915438\pi$
$167$	− 5.27785e9i	− 0.525089i	0.964920	−	0.262544i	$-0.0845616\pi$
$168$	0	0
$169$	5.31540e9	0.501240
$170$	0	0
$171$	0	0
$172$	1.70719e9i	0.148732i
$173$	6.58861e9i	0.559224i	0.960113	+	0.279612i	$0.0902059\pi$
$173$	6.58861e9i	0.559224i	−0.960113	+	0.279612i	$0.909794\pi$
$174$	0	0
$175$	0	0
$176$	−1.31947e10	−1.03656
$177$	0	0
$178$	1.85211e9i	0.138286i
$179$	−1.23663e10	−0.900326	−0.450163	−	0.892946i	$-0.648634\pi$
$179$	−1.23663e10	−0.900326	−0.450163	+	0.892946i	$0.648634\pi$
$180$	0	0
$181$	−2.59914e10	−1.80002	−0.900009	−	0.435872i	$-0.856440\pi$
$181$	−2.59914e10	−1.80002	−0.900009	+	0.435872i	$0.856440\pi$
$182$	1.47293e9i	0.0995090i
$183$	0	0
$184$	5.91706e7	0.00380562
$185$	0	0
$186$	0	0
$187$	− 3.30151e9i	− 0.197436i
$188$	− 1.69276e10i	− 0.988292i
$189$	0	0
$190$	0	0
$191$	−1.50506e10	−0.818284	−0.409142	−	0.912471i	$-0.634172\pi$
$191$	−1.50506e10	−0.818284	−0.409142	+	0.912471i	$0.634172\pi$
$192$	0	0
$193$	− 1.40329e10i	− 0.728014i	−0.931396	−	0.364007i	$-0.881408\pi$
$193$	− 1.40329e10i	− 0.728014i	0.931396	−	0.364007i	$-0.118592\pi$
$194$	−2.49765e10	−1.26597
$195$	0	0
$196$	−1.98996e10	−0.963146
$197$	2.57285e10i	1.21707i	0.793526	+	0.608536i	$0.208243\pi$
$197$	2.57285e10i	1.21707i	−0.793526	+	0.608536i	$0.791757\pi$
$198$	0	0
$199$	2.94367e10	1.33061	0.665303	−	0.746573i	$-0.268302\pi$
$199$	2.94367e10	1.33061	0.665303	+	0.746573i	$0.268302\pi$
$200$	0	0
$201$	0	0
$202$	5.09985e9i	0.215515i
$203$	2.84070e9i	0.117407i
$204$	0	0
$205$	0	0
$206$	−4.43222e10	−1.71482
$207$	0	0
$208$	1.95667e10i	0.724825i
$209$	−1.67302e10	−0.606516
$210$	0	0
$211$	1.17275e10	0.407319	0.203659	−	0.979042i	$-0.434716\pi$
$211$	1.17275e10	0.407319	0.203659	+	0.979042i	$0.434716\pi$
$212$	4.19664e10i	1.42689i
$213$	0	0
$214$	−4.97283e10	−1.62084
$215$	0	0
$216$	0	0
$217$	2.90911e8i	0.00890619i
$218$	− 3.96341e10i	− 1.18854i
$219$	0	0
$220$	0	0
$221$	−4.89588e9	−0.138059
$222$	0	0
$223$	− 2.58340e10i	− 0.699550i	−0.936834	−	0.349775i	$-0.886258\pi$
$223$	− 2.58340e10i	− 0.699550i	0.936834	−	0.349775i	$-0.113742\pi$
$224$	5.30535e9	0.140799
$225$	0	0
$226$	5.75447e10	1.46729
$227$	2.50896e10i	0.627159i	0.949562	+	0.313580i	$0.101528\pi$
$227$	2.50896e10i	0.627159i	−0.949562	+	0.313580i	$0.898472\pi$
$228$	0	0
$229$	2.30463e10	0.553785	0.276892	−	0.960901i	$-0.410695\pi$
$229$	2.30463e10	0.553785	0.276892	+	0.960901i	$0.410695\pi$
$230$	0	0
$231$	0	0
$232$	1.96326e9i	0.0444920i
$233$	3.17197e10i	0.705062i	0.935800	+	0.352531i	$0.114679\pi$
$233$	3.17197e10i	0.705062i	−0.935800	+	0.352531i	$0.885321\pi$
$234$	0	0
$235$	0	0
$236$	3.71793e10	0.780185
$237$	0	0
$238$	1.36343e9i	0.0275446i
$239$	2.23794e10	0.443668	0.221834	−	0.975084i	$-0.428796\pi$
$239$	2.23794e10	0.443668	0.221834	+	0.975084i	$0.428796\pi$
$240$	0	0
$241$	−2.10823e10	−0.402569	−0.201285	−	0.979533i	$-0.564512\pi$
$241$	−2.10823e10	−0.402569	−0.201285	+	0.979533i	$0.564512\pi$
$242$	1.50102e9i	0.0281331i
$243$	0	0
$244$	−8.86778e10	−1.60162
$245$	0	0
$246$	0	0
$247$	2.48095e10i	0.424113i
$248$	2.01054e8i	0.00337505i
$249$	0	0
$250$	0	0
$251$	7.96509e10	1.26666	0.633329	−	0.773883i	$-0.281688\pi$
$251$	7.96509e10	1.26666	0.633329	+	0.773883i	$0.281688\pi$
$252$	0	0
$253$	− 6.58910e9i	− 0.101108i
$254$	9.74114e10	1.46845
$255$	0	0
$256$	7.22868e10	1.05191
$257$	3.30815e10i	0.473027i	0.971628	+	0.236514i	$0.0760048\pi$
$257$	3.30815e10i	0.473027i	−0.971628	+	0.236514i	$0.923995\pi$
$258$	0	0
$259$	−8.31695e9	−0.114846
$260$	0	0
$261$	0	0
$262$	5.83240e10i	0.764701i
$263$	6.05028e10i	0.779784i	0.920861	+	0.389892i	$0.127488\pi$
$263$	6.05028e10i	0.779784i	−0.920861	+	0.389892i	$0.872512\pi$
$264$	0	0
$265$	0	0
$266$	6.90908e9	0.0846161
$267$	0	0
$268$	− 3.45860e10i	− 0.409538i
$269$	2.61933e10	0.305004	0.152502	−	0.988303i	$-0.451267\pi$
$269$	2.61933e10	0.305004	0.152502	+	0.988303i	$0.451267\pi$
$270$	0	0
$271$	5.42857e10	0.611398	0.305699	−	0.952128i	$-0.401110\pi$
$271$	5.42857e10	0.611398	0.305699	+	0.952128i	$0.401110\pi$
$272$	1.81120e10i	0.200635i
$273$	0	0
$274$	−1.46972e11	−1.57528
$275$	0	0
$276$	0	0
$277$	1.76561e11i	1.80192i	0.433900	+	0.900961i	$0.357137\pi$
$277$	1.76561e11i	1.80192i	−0.433900	+	0.900961i	$0.642863\pi$
$278$	− 8.17261e9i	− 0.0820652i
$279$	0	0
$280$	0	0
$281$	7.91126e9	0.0756950	0.0378475	−	0.999284i	$-0.487950\pi$
$281$	7.91126e9	0.0756950	0.0378475	+	0.999284i	$0.487950\pi$
$282$	0	0
$283$	1.34806e11i	1.24931i	0.780899	+	0.624657i	$0.214761\pi$
$283$	1.34806e11i	1.24931i	−0.780899	+	0.624657i	$0.785239\pi$
$284$	−1.03544e11	−0.944480
$285$	0	0
$286$	1.13359e11	1.00186
$287$	1.63558e10i	0.142299i
$288$	0	0
$289$	1.14056e11	0.961785
$290$	0	0
$291$	0	0
$292$	− 1.50696e11i	− 1.21305i
$293$	1.02784e11i	0.814741i	0.913263	+	0.407370i	$0.133554\pi$
$293$	1.02784e11i	0.814741i	−0.913263	+	0.407370i	$0.866446\pi$
$294$	0	0
$295$	0	0
$296$	−5.74799e9	−0.0435215
$297$	0	0
$298$	1.00055e11i	0.734964i
$299$	−9.77110e9	−0.0707006
$300$	0	0
$301$	2.18388e9	0.0153348
$302$	− 3.26638e11i	− 2.25962i
$303$	0	0
$304$	9.17815e10	0.616345
$305$	0	0
$306$	0	0
$307$	− 2.61472e11i	− 1.67998i	−0.542606	−	0.839988i	$-0.682562\pi$
$307$	− 2.61472e11i	− 1.67998i	0.542606	−	0.839988i	$-0.317438\pi$
$308$	− 1.55679e10i	− 0.0985713i
$309$	0	0
$310$	0	0
$311$	8.24828e10	0.499967	0.249984	−	0.968250i	$-0.419575\pi$
$311$	8.24828e10	0.499967	0.249984	+	0.968250i	$0.419575\pi$
$312$	0	0
$313$	1.35766e11i	0.799540i	0.916615	+	0.399770i	$0.130910\pi$
$313$	1.35766e11i	0.799540i	−0.916615	+	0.399770i	$0.869090\pi$
$314$	−1.03318e11	−0.599783
$315$	0	0
$316$	1.85431e11	1.04614
$317$	9.06853e10i	0.504394i	0.967676	+	0.252197i	$0.0811531\pi$
$317$	9.06853e10i	0.504394i	−0.967676	+	0.252197i	$0.918847\pi$
$318$	0	0
$319$	2.18624e11	1.18206
$320$	0	0
$321$	0	0
$322$	2.72111e9i	0.0141057i
$323$	2.29650e10i	0.117397i
$324$	0	0
$325$	0	0
$326$	1.08077e11	0.529973
$327$	0	0
$328$	1.13038e10i	0.0539251i
$329$	−2.16542e10	−0.101897
$330$	0	0
$331$	−3.45169e11	−1.58054	−0.790271	−	0.612758i	$-0.790060\pi$
$331$	−3.45169e11	−1.58054	−0.790271	+	0.612758i	$0.790060\pi$
$332$	− 2.24204e11i	− 1.01279i
$333$	0	0
$334$	1.67745e11	0.737546
$335$	0	0
$336$	0	0
$337$	8.39418e10i	0.354523i	0.984164	+	0.177261i	$0.0567238\pi$
$337$	8.39418e10i	0.354523i	−0.984164	+	0.177261i	$0.943276\pi$
$338$	1.68938e11i	0.704048i
$339$	0	0
$340$	0	0
$341$	2.23889e10	0.0896681
$342$	0	0
$343$	5.11709e10i	0.199618i
$344$	1.50932e9	0.00581122
$345$	0	0
$346$	−2.09404e11	−0.785493
$347$	− 1.56153e11i	− 0.578188i	−0.957301	−	0.289094i	$-0.906646\pi$
$347$	− 1.56153e11i	− 0.578188i	0.957301	−	0.289094i	$-0.0933540\pi$
$348$	0	0
$349$	−6.31862e9	−0.0227986	−0.0113993	−	0.999935i	$-0.503629\pi$
$349$	−6.31862e9	−0.0227986	−0.0113993	+	0.999935i	$0.503629\pi$
$350$	0	0
$351$	0	0
$352$	− 4.08307e11i	− 1.41757i
$353$	− 2.83875e11i	− 0.973064i	−0.873663	−	0.486532i	$-0.838262\pi$
$353$	− 2.83875e11i	− 0.973064i	0.873663	−	0.486532i	$-0.161738\pi$
$354$	0	0
$355$	0	0
$356$	−2.90288e10	−0.0957866
$357$	0	0
$358$	− 3.93034e11i	− 1.26461i
$359$	6.00733e11	1.90878	0.954391	−	0.298558i	$-0.0965058\pi$
$359$	6.00733e11	1.90878	0.954391	+	0.298558i	$0.0965058\pi$
$360$	0	0
$361$	−2.06314e11	−0.639361
$362$	− 8.26079e11i	− 2.52833i
$363$	0	0
$364$	−2.30859e10	−0.0689271
$365$	0	0
$366$	0	0
$367$	4.97260e11i	1.43082i	0.698702	+	0.715412i	$0.253761\pi$
$367$	4.97260e11i	1.43082i	−0.698702	+	0.715412i	$0.746239\pi$
$368$	3.61477e10i	0.102746i
$369$	0	0
$370$	0	0
$371$	5.36845e10	0.147118
$372$	0	0
$373$	− 1.39183e11i	− 0.372303i	−0.982521	−	0.186151i	$-0.940398\pi$
$373$	− 1.39183e11i	− 0.372303i	0.982521	−	0.186151i	$-0.0596015\pi$
$374$	1.04931e11	0.277321
$375$	0	0
$376$	−1.49656e10	−0.0386144
$377$	− 3.24202e11i	− 0.826569i
$378$	0	0
$379$	5.65247e11	1.40722	0.703610	−	0.710587i	$-0.251570\pi$
$379$	5.65247e11	1.40722	0.703610	+	0.710587i	$0.251570\pi$
$380$	0	0
$381$	0	0
$382$	− 4.78350e11i	− 1.14937i
$383$	− 5.38864e11i	− 1.27963i	−0.768528	−	0.639816i	$-0.779011\pi$
$383$	− 5.38864e11i	− 1.27963i	0.768528	−	0.639816i	$-0.220989\pi$
$384$	0	0
$385$	0	0
$386$	4.46004e11	1.02258
$387$	0	0
$388$	− 3.91466e11i	− 0.876902i
$389$	7.67552e11	1.69955	0.849776	−	0.527144i	$-0.176737\pi$
$389$	7.67552e11	1.69955	0.849776	+	0.527144i	$0.176737\pi$
$390$	0	0
$391$	−9.04466e9	−0.0195703
$392$	1.75931e10i	0.0376319i
$393$	0	0
$394$	−8.17722e11	−1.70951
$395$	0	0
$396$	0	0
$397$	− 6.25258e11i	− 1.26329i	−0.775259	−	0.631643i	$-0.782381\pi$
$397$	− 6.25258e11i	− 1.26329i	0.775259	−	0.631643i	$-0.217619\pi$
$398$	9.35578e11i	1.86899i
$399$	0	0
$400$	0	0
$401$	1.61329e11	0.311574	0.155787	−	0.987791i	$-0.450209\pi$
$401$	1.61329e11	0.311574	0.155787	+	0.987791i	$0.450209\pi$
$402$	0	0
$403$	− 3.32009e10i	− 0.0627014i
$404$	−7.99320e10	−0.149281
$405$	0	0
$406$	−9.02853e10	−0.164911
$407$	6.40083e11i	1.15628i
$408$	0	0
$409$	7.24004e11	1.27934	0.639670	−	0.768649i	$-0.279071\pi$
$409$	7.24004e11	1.27934	0.639670	+	0.768649i	$0.279071\pi$
$410$	0	0
$411$	0	0
$412$	− 6.94679e11i	− 1.18781i
$413$	− 4.75607e10i	− 0.0804403i
$414$	0	0
$415$	0	0
$416$	−6.05486e11	−0.991252
$417$	0	0
$418$	− 5.31731e11i	− 0.851920i
$419$	2.80040e11	0.443871	0.221936	−	0.975061i	$-0.428763\pi$
$419$	2.80040e11	0.443871	0.221936	+	0.975061i	$0.428763\pi$
$420$	0	0
$421$	−6.55915e10	−0.101760	−0.0508801	−	0.998705i	$-0.516203\pi$
$421$	−6.55915e10	−0.101760	−0.0508801	+	0.998705i	$0.516203\pi$
$422$	3.72732e11i	0.572125i
$423$	0	0
$424$	3.71023e10	0.0557512
$425$	0	0
$426$	0	0
$427$	1.13439e11i	0.165134i
$428$	− 7.79410e11i	− 1.12271i
$429$	0	0
$430$	0	0
$431$	−1.00292e12	−1.39997	−0.699987	−	0.714156i	$-0.746811\pi$
$431$	−1.00292e12	−1.39997	−0.699987	+	0.714156i	$0.746811\pi$
$432$	0	0
$433$	7.98482e11i	1.09162i	0.837910	+	0.545808i	$0.183777\pi$
$433$	7.98482e11i	1.09162i	−0.837910	+	0.545808i	$0.816223\pi$
$434$	−9.24596e9	−0.0125097
$435$	0	0
$436$	6.21201e11	0.823271
$437$	4.58332e10i	0.0601193i
$438$	0	0
$439$	−7.98379e11	−1.02593	−0.512966	−	0.858409i	$-0.671453\pi$
$439$	−7.98379e11	−1.02593	−0.512966	+	0.858409i	$0.671453\pi$
$440$	0	0
$441$	0	0
$442$	− 1.55604e11i	− 0.193920i
$443$	7.56642e11i	0.933413i	0.884412	+	0.466707i	$0.154560\pi$
$443$	7.56642e11i	0.933413i	−0.884412	+	0.466707i	$0.845440\pi$
$444$	0	0
$445$	0	0
$446$	8.21074e11	0.982597
$447$	0	0
$448$	8.08381e10i	0.0948124i
$449$	−6.39419e11	−0.742467	−0.371234	−	0.928540i	$-0.621065\pi$
$449$	−6.39419e11	−0.742467	−0.371234	+	0.928540i	$0.621065\pi$
$450$	0	0
$451$	1.25876e12	1.43268
$452$	9.01919e11i	1.01635i
$453$	0	0
$454$	−7.97418e11	−0.880916
$455$	0	0
$456$	0	0
$457$	1.34276e12i	1.44005i	0.693949	+	0.720024i	$0.255869\pi$
$457$	1.34276e12i	1.44005i	−0.693949	+	0.720024i	$0.744131\pi$
$458$	7.32474e11i	0.777853i
$459$	0	0
$460$	0	0
$461$	−3.47112e11	−0.357945	−0.178972	−	0.983854i	$-0.557277\pi$
$461$	−3.47112e11	−0.357945	−0.178972	+	0.983854i	$0.557277\pi$
$462$	0	0
$463$	7.20214e11i	0.728362i	0.931328	+	0.364181i	$0.118651\pi$
$463$	7.20214e11i	0.728362i	−0.931328	+	0.364181i	$0.881349\pi$
$464$	−1.19937e12	−1.20122
$465$	0	0
$466$	−1.00814e12	−0.990339
$467$	− 7.31553e10i	− 0.0711737i	−0.999367	−	0.0355869i	$-0.988670\pi$
$467$	− 7.31553e10i	− 0.0711737i	0.999367	−	0.0355869i	$-0.0113300\pi$
$468$	0	0
$469$	−4.42433e10	−0.0422250
$470$	0	0
$471$	0	0
$472$	− 3.28701e10i	− 0.0304833i
$473$	− 1.68074e11i	− 0.154392i
$474$	0	0
$475$	0	0
$476$	−2.13695e10	−0.0190794
$477$	0	0
$478$	7.11279e11i	0.623181i
$479$	−4.80307e10	−0.0416878	−0.0208439	−	0.999783i	$-0.506635\pi$
$479$	−4.80307e10	−0.0416878	−0.0208439	+	0.999783i	$0.506635\pi$
$480$	0	0
$481$	9.49191e11	0.808539
$482$	− 6.70052e11i	− 0.565454i
$483$	0	0
$484$	−2.35261e10	−0.0194870
$485$	0	0
$486$	0	0
$487$	1.24384e12i	1.00204i	0.865435	+	0.501021i	$0.167042\pi$
$487$	1.24384e12i	1.00204i	−0.865435	+	0.501021i	$0.832958\pi$
$488$	7.83997e10i	0.0625785i
$489$	0	0
$490$	0	0
$491$	2.63145e11	0.204328	0.102164	−	0.994768i	$-0.467423\pi$
$491$	2.63145e11	0.204328	0.102164	+	0.994768i	$0.467423\pi$
$492$	0	0
$493$	− 3.00099e11i	− 0.228798i
$494$	−7.88515e11	−0.595715
$495$	0	0
$496$	−1.22825e11	−0.0911212
$497$	1.32456e11i	0.0973798i
$498$	0	0
$499$	−3.75452e10	−0.0271083	−0.0135542	−	0.999908i	$-0.504315\pi$
$499$	−3.75452e10	−0.0271083	−0.0135542	+	0.999908i	$0.504315\pi$
$500$	0	0
$501$	0	0
$502$	2.53153e12i	1.77916i
$503$	2.86189e11i	0.199341i	0.995020	+	0.0996707i	$0.0317789\pi$
$503$	2.86189e11i	0.199341i	−0.995020	+	0.0996707i	$0.968221\pi$
$504$	0	0
$505$	0	0
$506$	2.09420e11	0.142017
$507$	0	0
$508$	1.52677e12i	1.01715i
$509$	−1.48505e12	−0.980646	−0.490323	−	0.871541i	$-0.663121\pi$
$509$	−1.48505e12	−0.980646	−0.490323	+	0.871541i	$0.663121\pi$
$510$	0	0
$511$	−1.92774e11	−0.125071
$512$	2.17930e12i	1.40153i
$513$	0	0
$514$	−1.05142e12	−0.664420
$515$	0	0
$516$	0	0
$517$	1.66654e12i	1.02590i
$518$	− 2.64336e11i	− 0.161314i
$519$	0	0
$520$	0	0
$521$	1.77521e12	1.05556	0.527778	−	0.849383i	$-0.323025\pi$
$521$	1.77521e12	1.05556	0.527778	+	0.849383i	$0.323025\pi$
$522$	0	0
$523$	− 1.20614e12i	− 0.704922i	−0.935826	−	0.352461i	$-0.885345\pi$
$523$	− 1.20614e12i	− 0.704922i	0.935826	−	0.352461i	$-0.114655\pi$
$524$	−9.14135e11	−0.529687
$525$	0	0
$526$	−1.92294e12	−1.09529
$527$	− 3.07326e10i	− 0.0173561i
$528$	0	0
$529$	1.78310e12	0.989978
$530$	0	0
$531$	0	0
$532$	1.08289e11i	0.0586112i
$533$	− 1.86664e12i	− 1.00182i
$534$	0	0
$535$	0	0
$536$	−3.05773e10	−0.0160014
$537$	0	0
$538$	8.32495e11i	0.428412i
$539$	1.95913e12	0.999802
$540$	0	0
$541$	3.24195e12	1.62711	0.813557	−	0.581485i	$-0.197528\pi$
$541$	3.24195e12	1.62711	0.813557	+	0.581485i	$0.197528\pi$
$542$	1.72535e12i	0.858777i
$543$	0	0
$544$	−5.60471e11	−0.274383
$545$	0	0
$546$	0	0
$547$	9.33463e11i	0.445814i	0.974840	+	0.222907i	$0.0715547\pi$
$547$	9.33463e11i	0.445814i	−0.974840	+	0.222907i	$0.928445\pi$
$548$	− 2.30355e12i	− 1.09115i
$549$	0	0
$550$	0	0
$551$	−1.52073e12	−0.702861
$552$	0	0
$553$	− 2.37208e11i	− 0.107861i
$554$	−5.61160e12	−2.53100
$555$	0	0
$556$	1.28092e11	0.0568443
$557$	1.31760e12i	0.580012i	0.957025	+	0.290006i	$0.0936572\pi$
$557$	1.31760e12i	0.580012i	−0.957025	+	0.290006i	$0.906343\pi$
$558$	0	0
$559$	−2.49240e11	−0.107960
$560$	0	0
$561$	0	0
$562$	2.51442e11i	0.106322i
$563$	− 4.33363e12i	− 1.81788i	−0.416931	−	0.908938i	$-0.636894\pi$
$563$	− 4.33363e12i	− 1.81788i	0.416931	−	0.908938i	$-0.363106\pi$
$564$	0	0
$565$	0	0
$566$	−4.28452e12	−1.75480
$567$	0	0
$568$	9.15429e10i	0.0369026i
$569$	3.08940e12	1.23558	0.617788	−	0.786345i	$-0.288029\pi$
$569$	3.08940e12	1.23558	0.617788	+	0.786345i	$0.288029\pi$
$570$	0	0
$571$	−7.43095e11	−0.292538	−0.146269	−	0.989245i	$-0.546726\pi$
$571$	−7.43095e11	−0.292538	−0.146269	+	0.989245i	$0.546726\pi$
$572$	1.77672e12i	0.693963i
$573$	0	0
$574$	−5.19832e11	−0.199875
$575$	0	0
$576$	0	0
$577$	− 2.25852e12i	− 0.848269i	−0.905599	−	0.424134i	$-0.860578\pi$
$577$	− 2.25852e12i	− 0.848269i	0.905599	−	0.424134i	$-0.139422\pi$
$578$	3.62501e12i	1.35093i
$579$	0	0
$580$	0	0
$581$	−2.86807e11	−0.104423
$582$	0	0
$583$	− 4.13162e12i	− 1.48119i
$584$	−1.33230e11	−0.0473962
$585$	0	0
$586$	−3.26674e12	−1.14440
$587$	4.75794e12i	1.65405i	0.562169	+	0.827023i	$0.309967\pi$
$587$	4.75794e12i	1.65405i	−0.562169	+	0.827023i	$0.690033\pi$
$588$	0	0
$589$	−1.55735e11	−0.0533173
$590$	0	0
$591$	0	0
$592$	− 3.51148e12i	− 1.17501i
$593$	− 2.01077e12i	− 0.667755i	−0.942616	−	0.333878i	$-0.891643\pi$
$593$	− 2.01077e12i	− 0.667755i	0.942616	−	0.333878i	$-0.108357\pi$
$594$	0	0
$595$	0	0
$596$	−1.56820e12	−0.509089
$597$	0	0
$598$	− 3.10552e11i	− 0.0993070i
$599$	−3.29427e12	−1.04554	−0.522768	−	0.852475i	$-0.675101\pi$
$599$	−3.29427e12	−1.04554	−0.522768	+	0.852475i	$0.675101\pi$
$600$	0	0
$601$	−1.98887e12	−0.621830	−0.310915	−	0.950438i	$-0.600635\pi$
$601$	−1.98887e12	−0.621830	−0.310915	+	0.950438i	$0.600635\pi$
$602$	6.94096e10i	0.0215395i
$603$	0	0
$604$	5.11952e12	1.56517
$605$	0	0
$606$	0	0
$607$	− 4.11631e12i	− 1.23072i	−0.788246	−	0.615360i	$-0.789011\pi$
$607$	− 4.11631e12i	− 1.23072i	0.788246	−	0.615360i	$-0.210989\pi$
$608$	2.84015e12i	0.842897i
$609$	0	0
$610$	0	0
$611$	2.47134e12	0.717376
$612$	0	0
$613$	− 2.06864e12i	− 0.591715i	−0.955232	−	0.295858i	$-0.904395\pi$
$613$	− 2.06864e12i	− 0.591715i	0.955232	−	0.295858i	$-0.0956054\pi$
$614$	8.31031e12	2.35971
$615$	0	0
$616$	−1.37635e10	−0.00385137
$617$	− 5.42058e12i	− 1.50578i	−0.658145	−	0.752891i	$-0.728659\pi$
$617$	− 5.42058e12i	− 1.50578i	0.658145	−	0.752891i	$-0.271341\pi$
$618$	0	0
$619$	−3.87240e12	−1.06016	−0.530081	−	0.847947i	$-0.677838\pi$
$619$	−3.87240e12	−1.06016	−0.530081	+	0.847947i	$0.677838\pi$
$620$	0	0
$621$	0	0
$622$	2.62153e12i	0.702260i
$623$	3.71344e10i	0.00987599i
$624$	0	0
$625$	0	0
$626$	−4.31500e12	−1.12304
$627$	0	0
$628$	− 1.61935e12i	− 0.415453i
$629$	8.78623e11	0.223807
$630$	0	0
$631$	−3.22937e12	−0.810933	−0.405467	−	0.914110i	$-0.632891\pi$
$631$	−3.22937e12	−0.810933	−0.405467	+	0.914110i	$0.632891\pi$
$632$	− 1.63939e11i	− 0.0408747i
$633$	0	0
$634$	−2.88223e12	−0.708478
$635$	0	0
$636$	0	0
$637$	− 2.90523e12i	− 0.699123i
$638$	6.94847e12i	1.66034i
$639$	0	0
$640$	0	0
$641$	−1.10935e12	−0.259543	−0.129771	−	0.991544i	$-0.541424\pi$
$641$	−1.10935e12	−0.259543	−0.129771	+	0.991544i	$0.541424\pi$
$642$	0	0
$643$	− 2.83993e12i	− 0.655177i	−0.944821	−	0.327588i	$-0.893764\pi$
$643$	− 2.83993e12i	− 0.655177i	0.944821	−	0.327588i	$-0.106236\pi$
$644$	−4.26489e10	−0.00977061
$645$	0	0
$646$	−7.29892e11	−0.164897
$647$	− 2.66962e12i	− 0.598935i	−0.954107	−	0.299467i	$-0.903191\pi$
$647$	− 2.66962e12i	− 0.598935i	0.954107	−	0.299467i	$-0.0968090\pi$
$648$	0	0
$649$	−3.66033e12	−0.809878
$650$	0	0
$651$	0	0
$652$	1.69393e12i	0.367098i
$653$	− 4.72156e12i	− 1.01619i	−0.861300	−	0.508097i	$-0.830349\pi$
$653$	− 4.72156e12i	− 1.01619i	0.861300	−	0.508097i	$-0.169651\pi$
$654$	0	0
$655$	0	0
$656$	−6.90554e12	−1.45590
$657$	0	0
$658$	− 6.88231e11i	− 0.143126i
$659$	4.01448e12	0.829173	0.414586	−	0.910010i	$-0.363926\pi$
$659$	4.01448e12	0.829173	0.414586	+	0.910010i	$0.363926\pi$
$660$	0	0
$661$	−8.01591e12	−1.63323	−0.816613	−	0.577186i	$-0.804151\pi$
$661$	−8.01591e12	−1.63323	−0.816613	+	0.577186i	$0.804151\pi$
$662$	− 1.09704e13i	− 2.22005i
$663$	0	0
$664$	−1.98218e11	−0.0395718
$665$	0	0
$666$	0	0
$667$	− 5.98931e11i	− 0.117168i
$668$	2.62912e12i	0.510878i
$669$	0	0
$670$	0	0
$671$	8.73040e12	1.66258
$672$	0	0
$673$	3.55147e12i	0.667330i	0.942692	+	0.333665i	$0.108286\pi$
$673$	3.55147e12i	0.667330i	−0.942692	+	0.333665i	$0.891714\pi$
$674$	−2.66790e12	−0.497967
$675$	0	0
$676$	−2.64783e12	−0.487675
$677$	6.06872e12i	1.11032i	0.831744	+	0.555160i	$0.187343\pi$
$677$	6.06872e12i	1.11032i	−0.831744	+	0.555160i	$0.812657\pi$
$678$	0	0
$679$	−5.00773e11	−0.0904122
$680$	0	0
$681$	0	0
$682$	7.11581e11i	0.125949i
$683$	9.96514e11i	0.175223i	0.996155	+	0.0876113i	$0.0279233\pi$
$683$	9.96514e11i	0.175223i	−0.996155	+	0.0876113i	$0.972077\pi$
$684$	0	0
$685$	0	0
$686$	−1.62635e12	−0.280386
$687$	0	0
$688$	9.22050e11i	0.156894i
$689$	−6.12686e12	−1.03574
$690$	0	0
$691$	1.02702e13	1.71367	0.856835	−	0.515591i	$-0.172428\pi$
$691$	1.02702e13	1.71367	0.856835	+	0.515591i	$0.172428\pi$
$692$	− 3.28207e12i	− 0.544090i
$693$	0	0
$694$	4.96299e12	0.812130
$695$	0	0
$696$	0	0
$697$	− 1.72786e12i	− 0.277308i
$698$	− 2.00823e11i	− 0.0320232i
$699$	0	0
$700$	0	0
$701$	8.26127e12	1.29216	0.646079	−	0.763270i	$-0.276408\pi$
$701$	8.26127e12	1.29216	0.646079	+	0.763270i	$0.276408\pi$
$702$	0	0
$703$	− 4.45236e12i	− 0.687529i
$704$	6.22140e12	0.954578
$705$	0	0
$706$	9.02234e12	1.36678
$707$	1.02251e11i	0.0153915i
$708$	0	0
$709$	8.11428e12	1.20598	0.602992	−	0.797747i	$-0.293975\pi$
$709$	8.11428e12	1.20598	0.602992	+	0.797747i	$0.293975\pi$
$710$	0	0
$711$	0	0
$712$	2.56643e10i	0.00374256i
$713$	− 6.13355e10i	− 0.00888810i
$714$	0	0
$715$	0	0
$716$	6.16017e12	0.875960
$717$	0	0
$718$	1.90929e13i	2.68110i
$719$	1.06399e13	1.48476	0.742380	−	0.669979i	$-0.233697\pi$
$719$	1.06399e13	1.48476	0.742380	+	0.669979i	$0.233697\pi$
$720$	0	0
$721$	−8.88651e11	−0.122468
$722$	− 6.55723e12i	− 0.898055i
$723$	0	0
$724$	1.29475e13	1.75130
$725$	0	0
$726$	0	0
$727$	− 1.35040e13i	− 1.79290i	−0.443145	−	0.896450i	$-0.646137\pi$
$727$	− 1.35040e13i	− 1.79290i	0.443145	−	0.896450i	$-0.353863\pi$
$728$	2.04101e10i	0.00269311i
$729$	0	0
$730$	0	0
$731$	−2.30710e11	−0.0298840
$732$	0	0
$733$	− 3.03764e12i	− 0.388659i	−0.980936	−	0.194329i	$-0.937747\pi$
$733$	− 3.03764e12i	− 0.388659i	0.980936	−	0.194329i	$-0.0622530\pi$
$734$	−1.58043e13	−2.00975
$735$	0	0
$736$	−1.11858e12	−0.140513
$737$	3.40502e12i	0.425124i
$738$	0	0
$739$	−7.29483e12	−0.899736	−0.449868	−	0.893095i	$-0.648529\pi$
$739$	−7.29483e12	−0.899736	−0.449868	+	0.893095i	$0.648529\pi$
$740$	0	0
$741$	0	0
$742$	1.70624e12i	0.206644i
$743$	− 1.18034e13i	− 1.42088i	−0.703756	−	0.710442i	$-0.748495\pi$
$743$	− 1.18034e13i	− 1.42088i	0.703756	−	0.710442i	$-0.251505\pi$
$744$	0	0
$745$	0	0
$746$	4.42362e12	0.522941
$747$	0	0
$748$	1.64463e12i	0.192092i
$749$	−9.97041e11	−0.115756
$750$	0	0
$751$	1.20855e13	1.38639	0.693197	−	0.720748i	$-0.256202\pi$
$751$	1.20855e13	1.38639	0.693197	+	0.720748i	$0.256202\pi$
$752$	− 9.14258e12i	− 1.04253i
$753$	0	0
$754$	1.03040e13	1.16101
$755$	0	0
$756$	0	0
$757$	1.48077e13i	1.63891i	0.573144	+	0.819455i	$0.305724\pi$
$757$	1.48077e13i	1.63891i	−0.573144	+	0.819455i	$0.694276\pi$
$758$	1.79651e13i	1.97660i
$759$	0	0
$760$	0	0
$761$	1.75185e13	1.89350	0.946749	−	0.321972i	$-0.104346\pi$
$761$	1.75185e13	1.89350	0.946749	+	0.321972i	$0.104346\pi$
$762$	0	0
$763$	− 7.94656e11i	− 0.0848826i
$764$	7.49736e12	0.796138
$765$	0	0
$766$	1.71266e13	1.79739
$767$	5.42798e12i	0.566316i
$768$	0	0
$769$	−4.16729e12	−0.429720	−0.214860	−	0.976645i	$-0.568930\pi$
$769$	−4.16729e12	−0.429720	−0.214860	+	0.976645i	$0.568930\pi$
$770$	0	0
$771$	0	0
$772$	6.99039e12i	0.708311i
$773$	− 1.05896e13i	− 1.06677i	−0.845872	−	0.533386i	$-0.820919\pi$
$773$	− 1.05896e13i	− 1.06677i	0.845872	−	0.533386i	$-0.179081\pi$
$774$	0	0
$775$	0	0
$776$	−3.46093e11	−0.0342622
$777$	0	0
$778$	2.43949e13i	2.38721i
$779$	−8.75583e12	−0.851881
$780$	0	0
$781$	1.01940e13	0.980426
$782$	− 2.87464e11i	− 0.0274887i
$783$	0	0
$784$	−1.07478e13	−1.01600
$785$	0	0
$786$	0	0
$787$	2.50861e12i	0.233102i	0.993185	+	0.116551i	$0.0371839\pi$
$787$	2.50861e12i	0.233102i	−0.993185	+	0.116551i	$0.962816\pi$
$788$	− 1.28165e13i	− 1.18413i
$789$	0	0
$790$	0	0
$791$	1.15376e12	0.104790
$792$	0	0
$793$	− 1.29465e13i	− 1.16258i
$794$	1.98724e13	1.77443
$795$	0	0
$796$	−1.46637e13	−1.29460
$797$	− 1.85822e13i	− 1.63131i	−0.578541	−	0.815653i	$-0.696378\pi$
$797$	− 1.85822e13i	− 1.63131i	0.578541	−	0.815653i	$-0.303622\pi$
$798$	0	0
$799$	2.28760e12	0.198573
$800$	0	0
$801$	0	0
$802$	5.12746e12i	0.437641i
$803$	1.48362e13i	1.25922i
$804$	0	0
$805$	0	0
$806$	1.05522e12	0.0880712
$807$	0	0
$808$	7.06675e10i	0.00583268i
$809$	−4.47333e12	−0.367166	−0.183583	−	0.983004i	$-0.558770\pi$
$809$	−4.47333e12	−0.367166	−0.183583	+	0.983004i	$0.558770\pi$
$810$	0	0
$811$	−3.49582e12	−0.283763	−0.141881	−	0.989884i	$-0.545315\pi$
$811$	−3.49582e12	−0.283763	−0.141881	+	0.989884i	$0.545315\pi$
$812$	− 1.41508e12i	− 0.114229i
$813$	0	0
$814$	−2.03436e13	−1.62412
$815$	0	0
$816$	0	0
$817$	1.16911e12i	0.0918026i
$818$	2.30108e13i	1.79698i
$819$	0	0
$820$	0	0
$821$	6.84197e12	0.525578	0.262789	−	0.964853i	$-0.415358\pi$
$821$	6.84197e12	0.525578	0.262789	+	0.964853i	$0.415358\pi$
$822$	0	0
$823$	− 8.11058e12i	− 0.616244i	−0.951347	−	0.308122i	$-0.900300\pi$
$823$	− 8.11058e12i	− 0.616244i	0.951347	−	0.308122i	$-0.0997005\pi$
$824$	−6.14163e11	−0.0464099
$825$	0	0
$826$	1.51161e12	0.112987
$827$	5.06349e10i	0.00376422i	0.999998	+	0.00188211i	$0.000599095\pi$
$827$	5.06349e10i	0.00376422i	−0.999998	+	0.00188211i	$0.999401\pi$
$828$	0	0
$829$	1.50474e13	1.10654	0.553269	−	0.833002i	$-0.313380\pi$
$829$	1.50474e13	1.10654	0.553269	+	0.833002i	$0.313380\pi$
$830$	0	0
$831$	0	0
$832$	− 9.22584e12i	− 0.667499i
$833$	− 2.68924e12i	− 0.193521i
$834$	0	0
$835$	0	0
$836$	8.33403e12	0.590102
$837$	0	0
$838$	8.90044e12i	0.623467i
$839$	2.68352e13	1.86972	0.934859	−	0.355020i	$-0.115526\pi$
$839$	2.68352e13	1.86972	0.934859	+	0.355020i	$0.115526\pi$
$840$	0	0
$841$	5.36519e12	0.369831
$842$	− 2.08468e12i	− 0.142934i
$843$	0	0
$844$	−5.84197e12	−0.396295
$845$	0	0
$846$	0	0
$847$	3.00951e10i	0.00200919i
$848$	2.26660e13i	1.50520i
$849$	0	0
$850$	0	0
$851$	1.75354e12	0.114613
$852$	0	0
$853$	− 4.22107e12i	− 0.272993i	−0.990641	−	0.136497i	$-0.956416\pi$
$853$	− 4.22107e12i	− 0.272993i	0.990641	−	0.136497i	$-0.0435843\pi$
$854$	−3.60540e12	−0.231949
$855$	0	0
$856$	−6.89073e11	−0.0438665
$857$	7.57740e12i	0.479851i	0.970791	+	0.239926i	$0.0771230\pi$
$857$	7.57740e12i	0.479851i	−0.970791	+	0.239926i	$0.922877\pi$
$858$	0	0
$859$	−1.41463e13	−0.886486	−0.443243	−	0.896401i	$-0.646172\pi$
$859$	−1.41463e13	−0.886486	−0.443243	+	0.896401i	$0.646172\pi$
$860$	0	0
$861$	0	0
$862$	− 3.18757e13i	− 1.96642i
$863$	1.52498e13i	0.935868i	0.883763	+	0.467934i	$0.155002\pi$
$863$	1.52498e13i	0.935868i	−0.883763	+	0.467934i	$0.844998\pi$
$864$	0	0
$865$	0	0
$866$	−2.53780e13	−1.53330
$867$	0	0
$868$	− 1.44915e11i	− 0.00866515i
$869$	−1.82558e13	−1.08596
$870$	0	0
$871$	5.04937e12	0.297273
$872$	− 5.49201e11i	− 0.0321668i
$873$	0	0
$874$	−1.45671e12	−0.0844443
$875$	0	0
$876$	0	0
$877$	3.30350e12i	0.188572i	0.995545	+	0.0942858i	$0.0300568\pi$
$877$	3.30350e12i	0.188572i	−0.995545	+	0.0942858i	$0.969943\pi$
$878$	− 2.53747e13i	− 1.44104i
$879$	0	0
$880$	0	0
$881$	−3.14728e13	−1.76013	−0.880063	−	0.474856i	$-0.842500\pi$
$881$	−3.14728e13	−1.76013	−0.880063	+	0.474856i	$0.842500\pi$
$882$	0	0
$883$	− 1.10669e13i	− 0.612637i	−0.951929	−	0.306318i	$-0.900903\pi$
$883$	− 1.10669e13i	− 0.612637i	0.951929	−	0.306318i	$-0.0990973\pi$
$884$	2.43885e12	0.134323
$885$	0	0
$886$	−2.40482e13	−1.31108
$887$	6.92035e12i	0.375381i	0.982228	+	0.187690i	$0.0601001\pi$
$887$	6.92035e12i	0.375381i	−0.982228	+	0.187690i	$0.939900\pi$
$888$	0	0
$889$	1.95308e12	0.104872
$890$	0	0
$891$	0	0
$892$	1.28690e13i	0.680618i
$893$	− 1.15923e13i	− 0.610010i
$894$	0	0
$895$	0	0
$896$	1.47084e11	0.00762394
$897$	0	0
$898$	− 2.03225e13i	− 1.04288i
$899$	2.03509e12	0.103912
$900$	0	0
$901$	−5.67136e12	−0.286699
$902$	4.00069e13i	2.01236i
$903$	0	0
$904$	7.97383e11	0.0397108
$905$	0	0
$906$	0	0
$907$	− 2.00486e13i	− 0.983676i	−0.870687	−	0.491838i	$-0.836325\pi$
$907$	− 2.00486e13i	− 0.983676i	0.870687	−	0.491838i	$-0.163675\pi$
$908$	− 1.24982e13i	− 0.610186i
$909$	0	0
$910$	0	0
$911$	−2.53798e13	−1.22083	−0.610416	−	0.792081i	$-0.708998\pi$
$911$	−2.53798e13	−1.22083	−0.610416	+	0.792081i	$0.708998\pi$
$912$	0	0
$913$	2.20730e13i	1.05134i
$914$	−4.26768e13	−2.02271
$915$	0	0
$916$	−1.14803e13	−0.538797
$917$	1.16938e12i	0.0546129i
$918$	0	0
$919$	6.15926e12	0.284845	0.142423	−	0.989806i	$-0.454511\pi$
$919$	6.15926e12	0.284845	0.142423	+	0.989806i	$0.454511\pi$
$920$	0	0
$921$	0	0
$922$	− 1.10322e13i	− 0.502774i
$923$	− 1.51169e13i	− 0.685574i
$924$	0	0
$925$	0	0
$926$	−2.28904e13	−1.02307
$927$	0	0
$928$	− 3.71140e13i	− 1.64275i
$929$	2.20000e13	0.969061	0.484531	−	0.874774i	$-0.338990\pi$
$929$	2.20000e13	0.969061	0.484531	+	0.874774i	$0.338990\pi$
$930$	0	0
$931$	−1.36275e13	−0.594489
$932$	− 1.58010e13i	− 0.685980i
$933$	0	0
$934$	2.32508e12	0.0999715
$935$	0	0
$936$	0	0
$937$	2.49274e12i	0.105645i	0.998604	+	0.0528225i	$0.0168217\pi$
$937$	2.49274e12i	0.105645i	−0.998604	+	0.0528225i	$0.983178\pi$
$938$	− 1.40617e12i	− 0.0593098i
$939$	0	0
$940$	0	0
$941$	1.26366e13	0.525382	0.262691	−	0.964880i	$-0.415390\pi$
$941$	1.26366e13	0.525382	0.262691	+	0.964880i	$0.415390\pi$
$942$	0	0
$943$	− 3.44844e12i	− 0.142010i
$944$	2.00805e13	0.823002
$945$	0	0
$946$	5.34185e12	0.216861
$947$	2.05809e13i	0.831551i	0.909467	+	0.415775i	$0.136490\pi$
$947$	2.05809e13i	0.831551i	−0.909467	+	0.415775i	$0.863510\pi$
$948$	0	0
$949$	2.20008e13	0.880524
$950$	0	0
$951$	0	0
$952$	1.88927e10i	0 0.000745466i
$953$	− 2.97887e13i	− 1.16986i	−0.811084	−	0.584929i	$-0.801122\pi$
$953$	− 2.97887e13i	− 1.16986i	0.811084	−	0.584929i	$-0.198878\pi$
$954$	0	0
$955$	0	0
$956$	−1.11481e13	−0.431660
$957$	0	0
$958$	− 1.52655e12i	− 0.0585553i
$959$	−2.94675e12	−0.112502
$960$	0	0
$961$	−2.62312e13	−0.992118
$962$	3.01679e13i	1.13568i
$963$	0	0
$964$	1.05020e13	0.391674
$965$	0	0
$966$	0	0
$967$	1.74473e13i	0.641664i	0.947136	+	0.320832i	$0.103963\pi$
$967$	1.74473e13i	0.641664i	−0.947136	+	0.320832i	$0.896037\pi$
$968$	2.07993e10i	0 0.000761394i
$969$	0	0
$970$	0	0
$971$	2.53844e13	0.916390	0.458195	−	0.888852i	$-0.348496\pi$
$971$	2.53844e13	0.916390	0.458195	+	0.888852i	$0.348496\pi$
$972$	0	0
$973$	− 1.63859e11i	− 0.00586088i
$974$	−3.95328e13	−1.40748
$975$	0	0
$976$	−4.78948e13	−1.68952
$977$	− 3.23029e13i	− 1.13427i	−0.823625	−	0.567135i	$-0.808052\pi$
$977$	− 3.23029e13i	− 1.13427i	0.823625	−	0.567135i	$-0.191948\pi$
$978$	0	0
$979$	2.85791e12	0.0994321
$980$	0	0
$981$	0	0
$982$	8.36348e12i	0.287002i
$983$	− 1.78014e13i	− 0.608083i	−0.952659	−	0.304041i	$-0.901664\pi$
$983$	− 1.78014e13i	− 0.608083i	0.952659	−	0.304041i	$-0.0983361\pi$
$984$	0	0
$985$	0	0
$986$	9.53796e12	0.321373
$987$	0	0
$988$	− 1.23587e13i	− 0.412635i
$989$	−4.60447e11	−0.0153037
$990$	0	0
$991$	−3.38638e12	−0.111533	−0.0557666	−	0.998444i	$-0.517760\pi$
$991$	−3.38638e12	−0.111533	−0.0557666	+	0.998444i	$0.517760\pi$
$992$	− 3.80078e12i	− 0.124615i
$993$	0	0
$994$	−4.20982e12	−0.136781
$995$	0	0
$996$	0	0
$997$	4.17766e13i	1.33908i	0.742778	+	0.669538i	$0.233508\pi$
$997$	4.17766e13i	1.33908i	−0.742778	+	0.669538i	$0.766492\pi$
$998$	− 1.19329e12i	− 0.0380767i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.b.m.199.6		6
3.2	odd	2		25.10.b.c.24.1		6
5.2	odd	4		225.10.a.m.1.1		3
5.3	odd	4		225.10.a.p.1.3		3
5.4	even	2	inner	225.10.b.m.199.1		6
12.11	even	2		400.10.c.q.49.1		6
15.2	even	4		25.10.a.d.1.3	yes	3
15.8	even	4		25.10.a.c.1.1	✓	3
15.14	odd	2		25.10.b.c.24.6		6
60.23	odd	4		400.10.a.y.1.3		3
60.47	odd	4		400.10.a.u.1.1		3
60.59	even	2		400.10.c.q.49.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.10.a.c.1.1	✓	3	15.8	even	4
25.10.a.d.1.3	yes	3	15.2	even	4
25.10.b.c.24.1		6	3.2	odd	2
25.10.b.c.24.6		6	15.14	odd	2
225.10.a.m.1.1		3	5.2	odd	4
225.10.a.p.1.3		3	5.3	odd	4
225.10.b.m.199.1		6	5.4	even	2	inner
225.10.b.m.199.6		6	1.1	even	1	trivial
400.10.a.u.1.1		3	60.47	odd	4
400.10.a.y.1.3		3	60.23	odd	4
400.10.c.q.49.1		6	12.11	even	2
400.10.c.q.49.6		6	60.59	even	2

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+31.7828i q^{2} -498.143 q^{4} +637.237i q^{7} +440.406i q^{8} +O(q^{10})\)	\(q+31.7828i q^{2} -498.143 q^{4} +637.237i q^{7} +440.406i q^{8} +49042.6 q^{11} -72726.2i q^{13} -20253.2 q^{14} -269047. q^{16} -67319.3i q^{17} -341136. q^{19} +1.55871e6i q^{22} -134355. i q^{23} +2.31144e6 q^{26} -317435. i q^{28} +4.45784e6 q^{29} +456520. q^{31} -8.32555e6i q^{32} +2.13959e6 q^{34} +1.30516e7i q^{37} -1.08422e7i q^{38} +2.56667e7 q^{41} -3.42710e6i q^{43} -2.44302e7 q^{44} +4.27016e6 q^{46} +3.39814e7i q^{47} +3.99475e7 q^{49} +3.62281e7i q^{52} -8.42456e7i q^{53} -280643. q^{56} +1.41682e8i q^{58} -7.46358e7 q^{59} +1.78017e8 q^{61} +1.45094e7i q^{62} +1.26857e8 q^{64} +6.94299e7i q^{67} +3.35347e7i q^{68} +2.07860e8 q^{71} +3.02516e8i q^{73} -4.14815e8 q^{74} +1.69935e8 q^{76} +3.12518e7i q^{77} -3.72244e8 q^{79} +8.15758e8i q^{82} +4.50079e8i q^{83} +1.08923e8 q^{86} +2.15987e7i q^{88} +5.82741e7 q^{89} +4.63438e7 q^{91} +6.69279e7i q^{92} -1.08002e9 q^{94} +7.85850e8i q^{97} +1.26964e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 682 q^{4}+O(q^{10}) \) 6 * q - 682 * q^4	\( 6 q - 682 q^{4} + 109398 q^{11} + 545844 q^{14} - 1398494 q^{16} - 1637690 q^{19} + 2560008 q^{26} + 4350960 q^{29} + 8548132 q^{31} + 13677926 q^{34} - 11852622 q^{41} - 43991406 q^{44} + 28446492 q^{46} + 12907858 q^{49} + 339076260 q^{56} + 11341920 q^{59} + 250613852 q^{61} - 335084802 q^{64} - 595101192 q^{71} - 503014716 q^{74} + 178829730 q^{76} + 620050340 q^{79} - 67894392 q^{86} - 2207720070 q^{89} + 2366375312 q^{91} - 3455782264 q^{94}+O(q^{100}) \) 6 * q - 682 * q^4 + 109398 * q^11 + 545844 * q^14 - 1398494 * q^16 - 1637690 * q^19 + 2560008 * q^26 + 4350960 * q^29 + 8548132 * q^31 + 13677926 * q^34 - 11852622 * q^41 - 43991406 * q^44 + 28446492 * q^46 + 12907858 * q^49 + 339076260 * q^56 + 11341920 * q^59 + 250613852 * q^61 - 335084802 * q^64 - 595101192 * q^71 - 503014716 * q^74 + 178829730 * q^76 + 620050340 * q^79 - 67894392 * q^86 - 2207720070 * q^89 + 2366375312 * q^91 - 3455782264 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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