LMFDB - Embedded newform 225.10.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$115.883063137$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 652x + 4000 $ x^3 - x^2 - 652*x + 4000
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-27.7229$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+31.7828 q^{2} +498.143 q^{4} +637.237 q^{7} -440.406 q^{8} +O(q^{10})$	$q+31.7828 q^{2} +498.143 q^{4} +637.237 q^{7} -440.406 q^{8} +49042.6 q^{11} +72726.2 q^{13} +20253.2 q^{14} -269047. q^{16} -67319.3 q^{17} +341136. q^{19} +1.55871e6 q^{22} +134355. q^{23} +2.31144e6 q^{26} +317435. q^{28} -4.45784e6 q^{29} +456520. q^{31} -8.32555e6 q^{32} -2.13959e6 q^{34} +1.30516e7 q^{37} +1.08422e7 q^{38} +2.56667e7 q^{41} +3.42710e6 q^{43} +2.44302e7 q^{44} +4.27016e6 q^{46} +3.39814e7 q^{47} -3.99475e7 q^{49} +3.62281e7 q^{52} +8.42456e7 q^{53} -280643. q^{56} -1.41682e8 q^{58} +7.46358e7 q^{59} +1.78017e8 q^{61} +1.45094e7 q^{62} -1.26857e8 q^{64} +6.94299e7 q^{67} -3.35347e7 q^{68} +2.07860e8 q^{71} -3.02516e8 q^{73} +4.14815e8 q^{74} +1.69935e8 q^{76} +3.12518e7 q^{77} +3.72244e8 q^{79} +8.15758e8 q^{82} -4.50079e8 q^{83} +1.08923e8 q^{86} -2.15987e7 q^{88} -5.82741e7 q^{89} +4.63438e7 q^{91} +6.69279e7 q^{92} +1.08002e9 q^{94} +7.85850e8 q^{97} -1.26964e9 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 33 q^{2} + 341 q^{4} - 5258 q^{7} - 105 q^{8}+O(q^{10}) $ 3 * q + 33 * q^2 + 341 * q^4 - 5258 * q^7 - 105 * q^8	$ 3 q + 33 q^{2} + 341 q^{4} - 5258 q^{7} - 105 q^{8} + 54699 q^{11} - 215884 q^{13} - 272922 q^{14} - 699247 q^{16} + 334983 q^{17} + 818845 q^{19} - 72761 q^{22} + 3526854 q^{23} + 1280004 q^{26} + 428134 q^{28} - 2175480 q^{29} + 4274066 q^{31} - 9464577 q^{32} - 6838963 q^{34} + 10305042 q^{37} - 24180495 q^{38} - 5926311 q^{41} + 24429956 q^{43} + 21995703 q^{44} + 14223246 q^{46} + 66858708 q^{47} - 6453929 q^{49} + 57862932 q^{52} + 132620514 q^{53} + 169538130 q^{56} - 201908320 q^{58} - 5670960 q^{59} + 125306926 q^{61} - 39831174 q^{62} + 167542401 q^{64} - 88829483 q^{67} - 71162559 q^{68} - 297550596 q^{71} - 181321729 q^{73} + 251507358 q^{74} + 89414865 q^{76} + 561214086 q^{77} - 310025170 q^{79} + 1368322979 q^{82} - 731088801 q^{83} - 33947196 q^{86} + 943671285 q^{88} + 1103860035 q^{89} + 1183187656 q^{91} - 190024242 q^{92} + 1727891132 q^{94} + 332236842 q^{97} + 457927581 q^{98}+O(q^{100}) $ 3 * q + 33 * q^2 + 341 * q^4 - 5258 * q^7 - 105 * q^8 + 54699 * q^11 - 215884 * q^13 - 272922 * q^14 - 699247 * q^16 + 334983 * q^17 + 818845 * q^19 - 72761 * q^22 + 3526854 * q^23 + 1280004 * q^26 + 428134 * q^28 - 2175480 * q^29 + 4274066 * q^31 - 9464577 * q^32 - 6838963 * q^34 + 10305042 * q^37 - 24180495 * q^38 - 5926311 * q^41 + 24429956 * q^43 + 21995703 * q^44 + 14223246 * q^46 + 66858708 * q^47 - 6453929 * q^49 + 57862932 * q^52 + 132620514 * q^53 + 169538130 * q^56 - 201908320 * q^58 - 5670960 * q^59 + 125306926 * q^61 - 39831174 * q^62 + 167542401 * q^64 - 88829483 * q^67 - 71162559 * q^68 - 297550596 * q^71 - 181321729 * q^73 + 251507358 * q^74 + 89414865 * q^76 + 561214086 * q^77 - 310025170 * q^79 + 1368322979 * q^82 - 731088801 * q^83 - 33947196 * q^86 + 943671285 * q^88 + 1103860035 * q^89 + 1183187656 * q^91 - 190024242 * q^92 + 1727891132 * q^94 + 332236842 * q^97 + 457927581 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	31.7828	1.40461	0.702306	−	0.711875i	$-0.252154\pi$
$2$	31.7828	1.40461	0.702306	+	0.711875i	$0.252154\pi$
$3$	0	0
$3$	0	0
$4$	498.143	0.972936
$5$	0	0
$5$	0	0
$6$	0	0
$7$	637.237	0.100314	0.0501568	−	0.998741i	$-0.484028\pi$
$7$	637.237	0.100314	0.0501568	+	0.998741i	$0.484028\pi$
$8$	−440.406	−0.0380144
$9$	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
$13$	72726.2	0.706229	0.353115	−	0.935580i	$-0.385123\pi$
$13$	72726.2	0.706229	0.353115	+	0.935580i	$0.385123\pi$
$14$	20253.2	0.140902
$15$	0	0
$16$	−269047.	−1.02633
$17$	−67319.3	−0.195488	−0.0977439	−	0.995212i	$-0.531163\pi$
$17$	−67319.3	−0.195488	−0.0977439	+	0.995212i	$0.531163\pi$
$18$	0	0
$19$	341136.	0.600532	0.300266	−	0.953855i	$-0.402925\pi$
$19$	341136.	0.600532	0.300266	+	0.953855i	$0.402925\pi$
$20$	0	0
$21$	0	0
$22$	1.55871e6	1.41861
$23$	134355.	0.100110	0.0500550	−	0.998746i	$-0.484060\pi$
$23$	134355.	0.100110	0.0500550	+	0.998746i	$0.484060\pi$
$24$	0	0
$25$	0	0
$26$	2.31144e6	0.991978
$27$	0	0
$28$	317435.	0.0975988
$29$	−4.45784e6	−1.17040	−0.585199	−	0.810890i	$-0.698984\pi$
$29$	−4.45784e6	−1.17040	−0.585199	+	0.810890i	$0.698984\pi$
$30$	0	0
$31$	456520.	0.0887834	0.0443917	−	0.999014i	$-0.485865\pi$
$31$	456520.	0.0887834	0.0443917	+	0.999014i	$0.485865\pi$
$32$	−8.32555e6	−1.40358
$33$	0	0
$34$	−2.13959e6	−0.274585
$35$	0	0
$36$	0	0
$37$	1.30516e7	1.14487	0.572433	−	0.819951i	$-0.305999\pi$
$37$	1.30516e7	1.14487	0.572433	+	0.819951i	$0.305999\pi$
$38$	1.08422e7	0.843515
$39$	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
$43$	3.42710e6	0.152869	0.0764344	−	0.997075i	$-0.475646\pi$
$43$	3.42710e6	0.152869	0.0764344	+	0.997075i	$0.475646\pi$
$44$	2.44302e7	0.982631
$45$	0	0
$46$	4.27016e6	0.140616
$47$	3.39814e7	1.01578	0.507891	−	0.861421i	$-0.330425\pi$
$47$	3.39814e7	1.01578	0.507891	+	0.861421i	$0.330425\pi$
$48$	0	0
$49$	−3.99475e7	−0.989937
$50$	0	0
$51$	0	0
$52$	3.62281e7	0.687116
$53$	8.42456e7	1.46658	0.733290	−	0.679916i	$-0.237984\pi$
$53$	8.42456e7	1.46658	0.733290	+	0.679916i	$0.237984\pi$
$54$	0	0
$55$	0	0
$56$	−280643.	−0.00381337
$57$	0	0
$58$	−1.41682e8	−1.64396
$59$	7.46358e7	0.801887	0.400944	−	0.916103i	$-0.368682\pi$
$59$	7.46358e7	0.801887	0.400944	+	0.916103i	$0.368682\pi$
$60$	0	0
$61$	1.78017e8	1.64618	0.823088	−	0.567913i	$-0.192249\pi$
$61$	1.78017e8	1.64618	0.823088	+	0.567913i	$0.192249\pi$
$62$	1.45094e7	0.124706
$63$	0	0
$64$	−1.26857e8	−0.945159
$65$	0	0
$66$	0	0
$67$	6.94299e7	0.420930	0.210465	−	0.977601i	$-0.432502\pi$
$67$	6.94299e7	0.420930	0.210465	+	0.977601i	$0.432502\pi$
$68$	−3.35347e7	−0.190197
$69$	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
$73$	−3.02516e8	−1.24680	−0.623398	−	0.781905i	$-0.714248\pi$
$73$	−3.02516e8	−1.24680	−0.623398	+	0.781905i	$0.714248\pi$
$74$	4.14815e8	1.60809
$75$	0	0
$76$	1.69935e8	0.584279
$77$	3.12518e7	0.101313
$78$	0	0
$79$	3.72244e8	1.07524	0.537621	−	0.843187i	$-0.319323\pi$
$79$	3.72244e8	1.07524	0.537621	+	0.843187i	$0.319323\pi$
$80$	0	0
$81$	0	0
$82$	8.15758e8	1.99250
$83$	−4.50079e8	−1.04097	−0.520484	−	0.853872i	$-0.674248\pi$
$83$	−4.50079e8	−1.04097	−0.520484	+	0.853872i	$0.674248\pi$
$84$	0	0
$85$	0	0
$86$	1.08923e8	0.214721
$87$	0	0
$88$	−2.15987e7	−0.0383932
$89$	−5.82741e7	−0.0984511	−0.0492255	−	0.998788i	$-0.515675\pi$
$89$	−5.82741e7	−0.0984511	−0.0492255	+	0.998788i	$0.515675\pi$
$90$	0	0
$91$	4.63438e7	0.0708444
$92$	6.69279e7	0.0974006
$93$	0	0
$94$	1.08002e9	1.42678
$95$	0	0
$96$	0	0
$97$	7.85850e8	0.901295	0.450647	−	0.892702i	$-0.351193\pi$
$97$	7.85850e8	0.901295	0.450647	+	0.892702i	$0.351193\pi$
$98$	−1.26964e9	−1.39048
$99$	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
$103$	−1.39454e9	−1.22085	−0.610425	−	0.792074i	$-0.709001\pi$
$103$	−1.39454e9	−1.22085	−0.610425	+	0.792074i	$0.709001\pi$
$104$	−3.20291e7	−0.0268469
$105$	0	0
$106$	2.67756e9	2.05998
$107$	1.56463e9	1.15394	0.576972	−	0.816764i	$-0.304234\pi$
$107$	1.56463e9	1.15394	0.576972	+	0.816764i	$0.304234\pi$
$108$	0	0
$109$	1.24703e9	0.846172	0.423086	−	0.906089i	$-0.360947\pi$
$109$	1.24703e9	0.846172	0.423086	+	0.906089i	$0.360947\pi$
$110$	0	0
$111$	0	0
$112$	−1.71447e8	−0.102955
$113$	1.81056e9	1.04463	0.522313	−	0.852754i	$-0.325069\pi$
$113$	1.81056e9	1.04463	0.522313	+	0.852754i	$0.325069\pi$
$114$	0	0
$115$	0	0
$116$	−2.22064e9	−1.13872
$117$	0	0
$118$	2.37213e9	1.12634
$119$	−4.28984e7	−0.0196101
$120$	0	0
$121$	4.72275e7	0.0200291
$122$	5.65786e9	2.31224
$123$	0	0
$124$	2.27412e8	0.0863806
$125$	0	0
$126$	0	0
$127$	−3.06491e9	−1.04545	−0.522723	−	0.852503i	$-0.675084\pi$
$127$	−3.06491e9	−1.04545	−0.522723	+	0.852503i	$0.675084\pi$
$128$	2.30815e8	0.0760010
$129$	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
$133$	2.17385e8	0.0602416
$134$	2.20667e9	0.591243
$135$	0	0
$136$	2.96478e7	0.00743136
$137$	4.62426e9	1.12150	0.560751	−	0.827984i	$-0.310512\pi$
$137$	4.62426e9	1.12150	0.560751	+	0.827984i	$0.310512\pi$
$138$	0	0
$139$	2.57140e8	0.0584255	0.0292128	−	0.999573i	$-0.490700\pi$
$139$	2.57140e8	0.0584255	0.0292128	+	0.999573i	$0.490700\pi$
$140$	0	0
$141$	0	0
$142$	6.60637e9	1.36353
$143$	3.56668e9	0.713267
$144$	0	0
$145$	0	0
$146$	−9.61479e9	−1.75127
$147$	0	0
$148$	6.50155e9	1.11388
$149$	−3.14809e9	−0.523250	−0.261625	−	0.965170i	$-0.584258\pi$
$149$	−3.14809e9	−0.523250	−0.261625	+	0.965170i	$0.584258\pi$
$150$	0	0
$151$	−1.02772e10	−1.60871	−0.804356	−	0.594147i	$-0.797490\pi$
$151$	−1.02772e10	−1.60871	−0.804356	+	0.594147i	$0.797490\pi$
$152$	−1.50238e8	−0.0228289
$153$	0	0
$154$	9.93267e8	0.142306
$155$	0	0
$156$	0	0
$157$	3.25077e9	0.427009	0.213505	−	0.976942i	$-0.431512\pi$
$157$	3.25077e9	0.427009	0.213505	+	0.976942i	$0.431512\pi$
$158$	1.18309e10	1.51030
$159$	0	0
$160$	0	0
$161$	8.56158e7	0.0100424
$162$	0	0
$163$	3.40049e9	0.377309	0.188655	−	0.982043i	$-0.439587\pi$
$163$	3.40049e9	0.377309	0.188655	+	0.982043i	$0.439587\pi$
$164$	1.27857e10	1.38015
$165$	0	0
$166$	−1.43047e10	−1.46216
$167$	−5.27785e9	−0.525089	−0.262544	−	0.964920i	$-0.584562\pi$
$167$	−5.27785e9	−0.525089	−0.262544	+	0.964920i	$0.584562\pi$
$168$	0	0
$169$	−5.31540e9	−0.501240
$170$	0	0
$171$	0	0
$172$	1.70719e9	0.148732
$173$	−6.58861e9	−0.559224	−0.279612	−	0.960113i	$-0.590206\pi$
$173$	−6.58861e9	−0.559224	−0.279612	+	0.960113i	$0.590206\pi$
$174$	0	0
$175$	0	0
$176$	−1.31947e10	−1.03656
$177$	0	0
$178$	−1.85211e9	−0.138286
$179$	1.23663e10	0.900326	0.450163	−	0.892946i	$-0.351366\pi$
$179$	1.23663e10	0.900326	0.450163	+	0.892946i	$0.351366\pi$
$180$	0	0
$181$	−2.59914e10	−1.80002	−0.900009	−	0.435872i	$-0.856440\pi$
$181$	−2.59914e10	−1.80002	−0.900009	+	0.435872i	$0.856440\pi$
$182$	1.47293e9	0.0995090
$183$	0	0
$184$	−5.91706e7	−0.00380562
$185$	0	0
$186$	0	0
$187$	−3.30151e9	−0.197436
$188$	1.69276e10	0.988292
$189$	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
$193$	1.40329e10	0.728014	0.364007	−	0.931396i	$-0.381408\pi$
$193$	1.40329e10	0.728014	0.364007	+	0.931396i	$0.381408\pi$
$194$	2.49765e10	1.26597
$195$	0	0
$196$	−1.98996e10	−0.963146
$197$	2.57285e10	1.21707	0.608536	−	0.793526i	$-0.291757\pi$
$197$	2.57285e10	1.21707	0.608536	+	0.793526i	$0.291757\pi$
$198$	0	0
$199$	−2.94367e10	−1.33061	−0.665303	−	0.746573i	$-0.731698\pi$
$199$	−2.94367e10	−1.33061	−0.665303	+	0.746573i	$0.731698\pi$
$200$	0	0
$201$	0	0
$202$	5.09985e9	0.215515
$203$	−2.84070e9	−0.117407
$204$	0	0
$205$	0	0
$206$	−4.43222e10	−1.71482
$207$	0	0
$208$	−1.95667e10	−0.724825
$209$	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	$0.434716\pi$
$212$	4.19664e10	1.42689
$213$	0	0
$214$	4.97283e10	1.62084
$215$	0	0
$216$	0	0
$217$	2.90911e8	0.00890619
$218$	3.96341e10	1.18854
$219$	0	0
$220$	0	0
$221$	−4.89588e9	−0.138059
$222$	0	0
$223$	2.58340e10	0.699550	0.349775	−	0.936834i	$-0.386258\pi$
$223$	2.58340e10	0.699550	0.349775	+	0.936834i	$0.386258\pi$
$224$	−5.30535e9	−0.140799
$225$	0	0
$226$	5.75447e10	1.46729
$227$	2.50896e10	0.627159	0.313580	−	0.949562i	$-0.398472\pi$
$227$	2.50896e10	0.627159	0.313580	+	0.949562i	$0.398472\pi$
$228$	0	0
$229$	−2.30463e10	−0.553785	−0.276892	−	0.960901i	$-0.589305\pi$
$229$	−2.30463e10	−0.553785	−0.276892	+	0.960901i	$0.589305\pi$
$230$	0	0
$231$	0	0
$232$	1.96326e9	0.0444920
$233$	−3.17197e10	−0.705062	−0.352531	−	0.935800i	$-0.614679\pi$
$233$	−3.17197e10	−0.705062	−0.352531	+	0.935800i	$0.614679\pi$
$234$	0	0
$235$	0	0
$236$	3.71793e10	0.780185
$237$	0	0
$238$	−1.36343e9	−0.0275446
$239$	−2.23794e10	−0.443668	−0.221834	−	0.975084i	$-0.571204\pi$
$239$	−2.23794e10	−0.443668	−0.221834	+	0.975084i	$0.571204\pi$
$240$	0	0
$241$	−2.10823e10	−0.402569	−0.201285	−	0.979533i	$-0.564512\pi$
$241$	−2.10823e10	−0.402569	−0.201285	+	0.979533i	$0.564512\pi$
$242$	1.50102e9	0.0281331
$243$	0	0
$244$	8.86778e10	1.60162
$245$	0	0
$246$	0	0
$247$	2.48095e10	0.424113
$248$	−2.01054e8	−0.00337505
$249$	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
$253$	6.58910e9	0.101108
$254$	−9.74114e10	−1.46845
$255$	0	0
$256$	7.22868e10	1.05191
$257$	3.30815e10	0.473027	0.236514	−	0.971628i	$-0.423995\pi$
$257$	3.30815e10	0.473027	0.236514	+	0.971628i	$0.423995\pi$
$258$	0	0
$259$	8.31695e9	0.114846
$260$	0	0
$261$	0	0
$262$	5.83240e10	0.764701
$263$	−6.05028e10	−0.779784	−0.389892	−	0.920861i	$-0.627488\pi$
$263$	−6.05028e10	−0.779784	−0.389892	+	0.920861i	$0.627488\pi$
$264$	0	0
$265$	0	0
$266$	6.90908e9	0.0846161
$267$	0	0
$268$	3.45860e10	0.409538
$269$	−2.61933e10	−0.305004	−0.152502	−	0.988303i	$-0.548733\pi$
$269$	−2.61933e10	−0.305004	−0.152502	+	0.988303i	$0.548733\pi$
$270$	0	0
$271$	5.42857e10	0.611398	0.305699	−	0.952128i	$-0.401110\pi$
$271$	5.42857e10	0.611398	0.305699	+	0.952128i	$0.401110\pi$
$272$	1.81120e10	0.200635
$273$	0	0
$274$	1.46972e11	1.57528
$275$	0	0
$276$	0	0
$277$	1.76561e11	1.80192	0.900961	−	0.433900i	$-0.142863\pi$
$277$	1.76561e11	1.80192	0.900961	+	0.433900i	$0.142863\pi$
$278$	8.17261e9	0.0820652
$279$	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
$283$	−1.34806e11	−1.24931	−0.624657	−	0.780899i	$-0.714761\pi$
$283$	−1.34806e11	−1.24931	−0.624657	+	0.780899i	$0.714761\pi$
$284$	1.03544e11	0.944480
$285$	0	0
$286$	1.13359e11	1.00186
$287$	1.63558e10	0.142299
$288$	0	0
$289$	−1.14056e11	−0.961785
$290$	0	0
$291$	0	0
$292$	−1.50696e11	−1.21305
$293$	−1.02784e11	−0.814741	−0.407370	−	0.913263i	$-0.633554\pi$
$293$	−1.02784e11	−0.814741	−0.407370	+	0.913263i	$0.633554\pi$
$294$	0	0
$295$	0	0
$296$	−5.74799e9	−0.0435215
$297$	0	0
$298$	−1.00055e11	−0.734964
$299$	9.77110e9	0.0707006
$300$	0	0
$301$	2.18388e9	0.0153348
$302$	−3.26638e11	−2.25962
$303$	0	0
$304$	−9.17815e10	−0.616345
$305$	0	0
$306$	0	0
$307$	−2.61472e11	−1.67998	−0.839988	−	0.542606i	$-0.817438\pi$
$307$	−2.61472e11	−1.67998	−0.839988	+	0.542606i	$0.817438\pi$
$308$	1.55679e10	0.0985713
$309$	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
$313$	−1.35766e11	−0.799540	−0.399770	−	0.916615i	$-0.630910\pi$
$313$	−1.35766e11	−0.799540	−0.399770	+	0.916615i	$0.630910\pi$
$314$	1.03318e11	0.599783
$315$	0	0
$316$	1.85431e11	1.04614
$317$	9.06853e10	0.504394	0.252197	−	0.967676i	$-0.418847\pi$
$317$	9.06853e10	0.504394	0.252197	+	0.967676i	$0.418847\pi$
$318$	0	0
$319$	−2.18624e11	−1.18206
$320$	0	0
$321$	0	0
$322$	2.72111e9	0.0141057
$323$	−2.29650e10	−0.117397
$324$	0	0
$325$	0	0
$326$	1.08077e11	0.529973
$327$	0	0
$328$	−1.13038e10	−0.0539251
$329$	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	$0.790060\pi$
$332$	−2.24204e11	−1.01279
$333$	0	0
$334$	−1.67745e11	−0.737546
$335$	0	0
$336$	0	0
$337$	8.39418e10	0.354523	0.177261	−	0.984164i	$-0.443276\pi$
$337$	8.39418e10	0.354523	0.177261	+	0.984164i	$0.443276\pi$
$338$	−1.68938e11	−0.704048
$339$	0	0
$340$	0	0
$341$	2.23889e10	0.0896681
$342$	0	0
$343$	−5.11709e10	−0.199618
$344$	−1.50932e9	−0.00581122
$345$	0	0
$346$	−2.09404e11	−0.785493
$347$	−1.56153e11	−0.578188	−0.289094	−	0.957301i	$-0.593354\pi$
$347$	−1.56153e11	−0.578188	−0.289094	+	0.957301i	$0.593354\pi$
$348$	0	0
$349$	6.31862e9	0.0227986	0.0113993	−	0.999935i	$-0.496371\pi$
$349$	6.31862e9	0.0227986	0.0113993	+	0.999935i	$0.496371\pi$
$350$	0	0
$351$	0	0
$352$	−4.08307e11	−1.41757
$353$	2.83875e11	0.973064	0.486532	−	0.873663i	$-0.338262\pi$
$353$	2.83875e11	0.973064	0.486532	+	0.873663i	$0.338262\pi$
$354$	0	0
$355$	0	0
$356$	−2.90288e10	−0.0957866
$357$	0	0
$358$	3.93034e11	1.26461
$359$	−6.00733e11	−1.90878	−0.954391	−	0.298558i	$-0.903494\pi$
$359$	−6.00733e11	−1.90878	−0.954391	+	0.298558i	$0.903494\pi$
$360$	0	0
$361$	−2.06314e11	−0.639361
$362$	−8.26079e11	−2.52833
$363$	0	0
$364$	2.30859e10	0.0689271
$365$	0	0
$366$	0	0
$367$	4.97260e11	1.43082	0.715412	−	0.698702i	$-0.246239\pi$
$367$	4.97260e11	1.43082	0.715412	+	0.698702i	$0.246239\pi$
$368$	−3.61477e10	−0.102746
$369$	0	0
$370$	0	0
$371$	5.36845e10	0.147118
$372$	0	0
$373$	1.39183e11	0.372303	0.186151	−	0.982521i	$-0.440398\pi$
$373$	1.39183e11	0.372303	0.186151	+	0.982521i	$0.440398\pi$
$374$	−1.04931e11	−0.277321
$375$	0	0
$376$	−1.49656e10	−0.0386144
$377$	−3.24202e11	−0.826569
$378$	0	0
$379$	−5.65247e11	−1.40722	−0.703610	−	0.710587i	$-0.748430\pi$
$379$	−5.65247e11	−1.40722	−0.703610	+	0.710587i	$0.748430\pi$
$380$	0	0
$381$	0	0
$382$	−4.78350e11	−1.14937
$383$	5.38864e11	1.27963	0.639816	−	0.768528i	$-0.279011\pi$
$383$	5.38864e11	1.27963	0.639816	+	0.768528i	$0.279011\pi$
$384$	0	0
$385$	0	0
$386$	4.46004e11	1.02258
$387$	0	0
$388$	3.91466e11	0.876902
$389$	−7.67552e11	−1.69955	−0.849776	−	0.527144i	$-0.823263\pi$
$389$	−7.67552e11	−1.69955	−0.849776	+	0.527144i	$0.823263\pi$
$390$	0	0
$391$	−9.04466e9	−0.0195703
$392$	1.75931e10	0.0376319
$393$	0	0
$394$	8.17722e11	1.70951
$395$	0	0
$396$	0	0
$397$	−6.25258e11	−1.26329	−0.631643	−	0.775259i	$-0.717619\pi$
$397$	−6.25258e11	−1.26329	−0.631643	+	0.775259i	$0.717619\pi$
$398$	−9.35578e11	−1.86899
$399$	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
$403$	3.32009e10	0.0627014
$404$	7.99320e10	0.149281
$405$	0	0
$406$	−9.02853e10	−0.164911
$407$	6.40083e11	1.15628
$408$	0	0
$409$	−7.24004e11	−1.27934	−0.639670	−	0.768649i	$-0.720929\pi$
$409$	−7.24004e11	−1.27934	−0.639670	+	0.768649i	$0.720929\pi$
$410$	0	0
$411$	0	0
$412$	−6.94679e11	−1.18781
$413$	4.75607e10	0.0804403
$414$	0	0
$415$	0	0
$416$	−6.05486e11	−0.991252
$417$	0	0
$418$	5.31731e11	0.851920
$419$	−2.80040e11	−0.443871	−0.221936	−	0.975061i	$-0.571237\pi$
$419$	−2.80040e11	−0.443871	−0.221936	+	0.975061i	$0.571237\pi$
$420$	0	0
$421$	−6.55915e10	−0.101760	−0.0508801	−	0.998705i	$-0.516203\pi$
$421$	−6.55915e10	−0.101760	−0.0508801	+	0.998705i	$0.516203\pi$
$422$	3.72732e11	0.572125
$423$	0	0
$424$	−3.71023e10	−0.0557512
$425$	0	0
$426$	0	0
$427$	1.13439e11	0.165134
$428$	7.79410e11	1.12271
$429$	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
$433$	−7.98482e11	−1.09162	−0.545808	−	0.837910i	$-0.683777\pi$
$433$	−7.98482e11	−1.09162	−0.545808	+	0.837910i	$0.683777\pi$
$434$	9.24596e9	0.0125097
$435$	0	0
$436$	6.21201e11	0.823271
$437$	4.58332e10	0.0601193
$438$	0	0
$439$	7.98379e11	1.02593	0.512966	−	0.858409i	$-0.328547\pi$
$439$	7.98379e11	1.02593	0.512966	+	0.858409i	$0.328547\pi$
$440$	0	0
$441$	0	0
$442$	−1.55604e11	−0.193920
$443$	−7.56642e11	−0.933413	−0.466707	−	0.884412i	$-0.654560\pi$
$443$	−7.56642e11	−0.933413	−0.466707	+	0.884412i	$0.654560\pi$
$444$	0	0
$445$	0	0
$446$	8.21074e11	0.982597
$447$	0	0
$448$	−8.08381e10	−0.0948124
$449$	6.39419e11	0.742467	0.371234	−	0.928540i	$-0.378935\pi$
$449$	6.39419e11	0.742467	0.371234	+	0.928540i	$0.378935\pi$
$450$	0	0
$451$	1.25876e12	1.43268
$452$	9.01919e11	1.01635
$453$	0	0
$454$	7.97418e11	0.880916
$455$	0	0
$456$	0	0
$457$	1.34276e12	1.44005	0.720024	−	0.693949i	$-0.244131\pi$
$457$	1.34276e12	1.44005	0.720024	+	0.693949i	$0.244131\pi$
$458$	−7.32474e11	−0.777853
$459$	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
$463$	−7.20214e11	−0.728362	−0.364181	−	0.931328i	$-0.618651\pi$
$463$	−7.20214e11	−0.728362	−0.364181	+	0.931328i	$0.618651\pi$
$464$	1.19937e12	1.20122
$465$	0	0
$466$	−1.00814e12	−0.990339
$467$	−7.31553e10	−0.0711737	−0.0355869	−	0.999367i	$-0.511330\pi$
$467$	−7.31553e10	−0.0711737	−0.0355869	+	0.999367i	$0.511330\pi$
$468$	0	0
$469$	4.42433e10	0.0422250
$470$	0	0
$471$	0	0
$472$	−3.28701e10	−0.0304833
$473$	1.68074e11	0.154392
$474$	0	0
$475$	0	0
$476$	−2.13695e10	−0.0190794
$477$	0	0
$478$	−7.11279e11	−0.623181
$479$	4.80307e10	0.0416878	0.0208439	−	0.999783i	$-0.493365\pi$
$479$	4.80307e10	0.0416878	0.0208439	+	0.999783i	$0.493365\pi$
$480$	0	0
$481$	9.49191e11	0.808539
$482$	−6.70052e11	−0.565454
$483$	0	0
$484$	2.35261e10	0.0194870
$485$	0	0
$486$	0	0
$487$	1.24384e12	1.00204	0.501021	−	0.865435i	$-0.332958\pi$
$487$	1.24384e12	1.00204	0.501021	+	0.865435i	$0.332958\pi$
$488$	−7.83997e10	−0.0625785
$489$	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
$493$	3.00099e11	0.228798
$494$	7.88515e11	0.595715
$495$	0	0
$496$	−1.22825e11	−0.0911212
$497$	1.32456e11	0.0973798
$498$	0	0
$499$	3.75452e10	0.0271083	0.0135542	−	0.999908i	$-0.495685\pi$
$499$	3.75452e10	0.0271083	0.0135542	+	0.999908i	$0.495685\pi$
$500$	0	0
$501$	0	0
$502$	2.53153e12	1.77916
$503$	−2.86189e11	−0.199341	−0.0996707	−	0.995020i	$-0.531779\pi$
$503$	−2.86189e11	−0.199341	−0.0996707	+	0.995020i	$0.531779\pi$
$504$	0	0
$505$	0	0
$506$	2.09420e11	0.142017
$507$	0	0
$508$	−1.52677e12	−1.01715
$509$	1.48505e12	0.980646	0.490323	−	0.871541i	$-0.336879\pi$
$509$	1.48505e12	0.980646	0.490323	+	0.871541i	$0.336879\pi$
$510$	0	0
$511$	−1.92774e11	−0.125071
$512$	2.17930e12	1.40153
$513$	0	0
$514$	1.05142e12	0.664420
$515$	0	0
$516$	0	0
$517$	1.66654e12	1.02590
$518$	2.64336e11	0.161314
$519$	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
$523$	1.20614e12	0.704922	0.352461	−	0.935826i	$-0.385345\pi$
$523$	1.20614e12	0.704922	0.352461	+	0.935826i	$0.385345\pi$
$524$	9.14135e11	0.529687
$525$	0	0
$526$	−1.92294e12	−1.09529
$527$	−3.07326e10	−0.0173561
$528$	0	0
$529$	−1.78310e12	−0.989978
$530$	0	0
$531$	0	0
$532$	1.08289e11	0.0586112
$533$	1.86664e12	1.00182
$534$	0	0
$535$	0	0
$536$	−3.05773e10	−0.0160014
$537$	0	0
$538$	−8.32495e11	−0.428412
$539$	−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	$0.197528\pi$
$542$	1.72535e12	0.858777
$543$	0	0
$544$	5.60471e11	0.274383
$545$	0	0
$546$	0	0
$547$	9.33463e11	0.445814	0.222907	−	0.974840i	$-0.428445\pi$
$547$	9.33463e11	0.445814	0.222907	+	0.974840i	$0.428445\pi$
$548$	2.30355e12	1.09115
$549$	0	0
$550$	0	0
$551$	−1.52073e12	−0.702861
$552$	0	0
$553$	2.37208e11	0.107861
$554$	5.61160e12	2.53100
$555$	0	0
$556$	1.28092e11	0.0568443
$557$	1.31760e12	0.580012	0.290006	−	0.957025i	$-0.406343\pi$
$557$	1.31760e12	0.580012	0.290006	+	0.957025i	$0.406343\pi$
$558$	0	0
$559$	2.49240e11	0.107960
$560$	0	0
$561$	0	0
$562$	2.51442e11	0.106322
$563$	4.33363e12	1.81788	0.908938	−	0.416931i	$-0.136894\pi$
$563$	4.33363e12	1.81788	0.908938	+	0.416931i	$0.136894\pi$
$564$	0	0
$565$	0	0
$566$	−4.28452e12	−1.75480
$567$	0	0
$568$	−9.15429e10	−0.0369026
$569$	−3.08940e12	−1.23558	−0.617788	−	0.786345i	$-0.711971\pi$
$569$	−3.08940e12	−1.23558	−0.617788	+	0.786345i	$0.711971\pi$
$570$	0	0
$571$	−7.43095e11	−0.292538	−0.146269	−	0.989245i	$-0.546726\pi$
$571$	−7.43095e11	−0.292538	−0.146269	+	0.989245i	$0.546726\pi$
$572$	1.77672e12	0.693963
$573$	0	0
$574$	5.19832e11	0.199875
$575$	0	0
$576$	0	0
$577$	−2.25852e12	−0.848269	−0.424134	−	0.905599i	$-0.639422\pi$
$577$	−2.25852e12	−0.848269	−0.424134	+	0.905599i	$0.639422\pi$
$578$	−3.62501e12	−1.35093
$579$	0	0
$580$	0	0
$581$	−2.86807e11	−0.104423
$582$	0	0
$583$	4.13162e12	1.48119
$584$	1.33230e11	0.0473962
$585$	0	0
$586$	−3.26674e12	−1.14440
$587$	4.75794e12	1.65405	0.827023	−	0.562169i	$-0.190033\pi$
$587$	4.75794e12	1.65405	0.827023	+	0.562169i	$0.190033\pi$
$588$	0	0
$589$	1.55735e11	0.0533173
$590$	0	0
$591$	0	0
$592$	−3.51148e12	−1.17501
$593$	2.01077e12	0.667755	0.333878	−	0.942616i	$-0.391643\pi$
$593$	2.01077e12	0.667755	0.333878	+	0.942616i	$0.391643\pi$
$594$	0	0
$595$	0	0
$596$	−1.56820e12	−0.509089
$597$	0	0
$598$	3.10552e11	0.0993070
$599$	3.29427e12	1.04554	0.522768	−	0.852475i	$-0.324899\pi$
$599$	3.29427e12	1.04554	0.522768	+	0.852475i	$0.324899\pi$
$600$	0	0
$601$	−1.98887e12	−0.621830	−0.310915	−	0.950438i	$-0.600635\pi$
$601$	−1.98887e12	−0.621830	−0.310915	+	0.950438i	$0.600635\pi$
$602$	6.94096e10	0.0215395
$603$	0	0
$604$	−5.11952e12	−1.56517
$605$	0	0
$606$	0	0
$607$	−4.11631e12	−1.23072	−0.615360	−	0.788246i	$-0.710989\pi$
$607$	−4.11631e12	−1.23072	−0.615360	+	0.788246i	$0.710989\pi$
$608$	−2.84015e12	−0.842897
$609$	0	0
$610$	0	0
$611$	2.47134e12	0.717376
$612$	0	0
$613$	2.06864e12	0.591715	0.295858	−	0.955232i	$-0.404395\pi$
$613$	2.06864e12	0.591715	0.295858	+	0.955232i	$0.404395\pi$
$614$	−8.31031e12	−2.35971
$615$	0	0
$616$	−1.37635e10	−0.00385137
$617$	−5.42058e12	−1.50578	−0.752891	−	0.658145i	$-0.771341\pi$
$617$	−5.42058e12	−1.50578	−0.752891	+	0.658145i	$0.771341\pi$
$618$	0	0
$619$	3.87240e12	1.06016	0.530081	−	0.847947i	$-0.322162\pi$
$619$	3.87240e12	1.06016	0.530081	+	0.847947i	$0.322162\pi$
$620$	0	0
$621$	0	0
$622$	2.62153e12	0.702260
$623$	−3.71344e10	−0.00987599
$624$	0	0
$625$	0	0
$626$	−4.31500e12	−1.12304
$627$	0	0
$628$	1.61935e12	0.415453
$629$	−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	$0.632891\pi$
$632$	−1.63939e11	−0.0408747
$633$	0	0
$634$	2.88223e12	0.708478
$635$	0	0
$636$	0	0
$637$	−2.90523e12	−0.699123
$638$	−6.94847e12	−1.66034
$639$	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
$643$	2.83993e12	0.655177	0.327588	−	0.944821i	$-0.393764\pi$
$643$	2.83993e12	0.655177	0.327588	+	0.944821i	$0.393764\pi$
$644$	4.26489e10	0.00977061
$645$	0	0
$646$	−7.29892e11	−0.164897
$647$	−2.66962e12	−0.598935	−0.299467	−	0.954107i	$-0.596809\pi$
$647$	−2.66962e12	−0.598935	−0.299467	+	0.954107i	$0.596809\pi$
$648$	0	0
$649$	3.66033e12	0.809878
$650$	0	0
$651$	0	0
$652$	1.69393e12	0.367098
$653$	4.72156e12	1.01619	0.508097	−	0.861300i	$-0.330349\pi$
$653$	4.72156e12	1.01619	0.508097	+	0.861300i	$0.330349\pi$
$654$	0	0
$655$	0	0
$656$	−6.90554e12	−1.45590
$657$	0	0
$658$	6.88231e11	0.143126
$659$	−4.01448e12	−0.829173	−0.414586	−	0.910010i	$-0.636074\pi$
$659$	−4.01448e12	−0.829173	−0.414586	+	0.910010i	$0.636074\pi$
$660$	0	0
$661$	−8.01591e12	−1.63323	−0.816613	−	0.577186i	$-0.804151\pi$
$661$	−8.01591e12	−1.63323	−0.816613	+	0.577186i	$0.804151\pi$
$662$	−1.09704e13	−2.22005
$663$	0	0
$664$	1.98218e11	0.0395718
$665$	0	0
$666$	0	0
$667$	−5.98931e11	−0.117168
$668$	−2.62912e12	−0.510878
$669$	0	0
$670$	0	0
$671$	8.73040e12	1.66258
$672$	0	0
$673$	−3.55147e12	−0.667330	−0.333665	−	0.942692i	$-0.608286\pi$
$673$	−3.55147e12	−0.667330	−0.333665	+	0.942692i	$0.608286\pi$
$674$	2.66790e12	0.497967
$675$	0	0
$676$	−2.64783e12	−0.487675
$677$	6.06872e12	1.11032	0.555160	−	0.831744i	$-0.312657\pi$
$677$	6.06872e12	1.11032	0.555160	+	0.831744i	$0.312657\pi$
$678$	0	0
$679$	5.00773e11	0.0904122
$680$	0	0
$681$	0	0
$682$	7.11581e11	0.125949
$683$	−9.96514e11	−0.175223	−0.0876113	−	0.996155i	$-0.527923\pi$
$683$	−9.96514e11	−0.175223	−0.0876113	+	0.996155i	$0.527923\pi$
$684$	0	0
$685$	0	0
$686$	−1.62635e12	−0.280386
$687$	0	0
$688$	−9.22050e11	−0.156894
$689$	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	$0.172428\pi$
$692$	−3.28207e12	−0.544090
$693$	0	0
$694$	−4.96299e12	−0.812130
$695$	0	0
$696$	0	0
$697$	−1.72786e12	−0.277308
$698$	2.00823e11	0.0320232
$699$	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
$703$	4.45236e12	0.687529
$704$	−6.22140e12	−0.954578
$705$	0	0
$706$	9.02234e12	1.36678
$707$	1.02251e11	0.0153915
$708$	0	0
$709$	−8.11428e12	−1.20598	−0.602992	−	0.797747i	$-0.706025\pi$
$709$	−8.11428e12	−1.20598	−0.602992	+	0.797747i	$0.706025\pi$
$710$	0	0
$711$	0	0
$712$	2.56643e10	0.00374256
$713$	6.13355e10	0.00888810
$714$	0	0
$715$	0	0
$716$	6.16017e12	0.875960
$717$	0	0
$718$	−1.90929e13	−2.68110
$719$	−1.06399e13	−1.48476	−0.742380	−	0.669979i	$-0.766303\pi$
$719$	−1.06399e13	−1.48476	−0.742380	+	0.669979i	$0.766303\pi$
$720$	0	0
$721$	−8.88651e11	−0.122468
$722$	−6.55723e12	−0.898055
$723$	0	0
$724$	−1.29475e13	−1.75130
$725$	0	0
$726$	0	0
$727$	−1.35040e13	−1.79290	−0.896450	−	0.443145i	$-0.853863\pi$
$727$	−1.35040e13	−1.79290	−0.896450	+	0.443145i	$0.853863\pi$
$728$	−2.04101e10	−0.00269311
$729$	0	0
$730$	0	0
$731$	−2.30710e11	−0.0298840
$732$	0	0
$733$	3.03764e12	0.388659	0.194329	−	0.980936i	$-0.437747\pi$
$733$	3.03764e12	0.388659	0.194329	+	0.980936i	$0.437747\pi$
$734$	1.58043e13	2.00975
$735$	0	0
$736$	−1.11858e12	−0.140513
$737$	3.40502e12	0.425124
$738$	0	0
$739$	7.29483e12	0.899736	0.449868	−	0.893095i	$-0.351471\pi$
$739$	7.29483e12	0.899736	0.449868	+	0.893095i	$0.351471\pi$
$740$	0	0
$741$	0	0
$742$	1.70624e12	0.206644
$743$	1.18034e13	1.42088	0.710442	−	0.703756i	$-0.248495\pi$
$743$	1.18034e13	1.42088	0.710442	+	0.703756i	$0.248495\pi$
$744$	0	0
$745$	0	0
$746$	4.42362e12	0.522941
$747$	0	0
$748$	−1.64463e12	−0.192092
$749$	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	$0.256202\pi$
$752$	−9.14258e12	−1.04253
$753$	0	0
$754$	−1.03040e13	−1.16101
$755$	0	0
$756$	0	0
$757$	1.48077e13	1.63891	0.819455	−	0.573144i	$-0.194276\pi$
$757$	1.48077e13	1.63891	0.819455	+	0.573144i	$0.194276\pi$
$758$	−1.79651e13	−1.97660
$759$	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
$763$	7.94656e11	0.0848826
$764$	−7.49736e12	−0.796138
$765$	0	0
$766$	1.71266e13	1.79739
$767$	5.42798e12	0.566316
$768$	0	0
$769$	4.16729e12	0.429720	0.214860	−	0.976645i	$-0.431070\pi$
$769$	4.16729e12	0.429720	0.214860	+	0.976645i	$0.431070\pi$
$770$	0	0
$771$	0	0
$772$	6.99039e12	0.708311
$773$	1.05896e13	1.06677	0.533386	−	0.845872i	$-0.320919\pi$
$773$	1.05896e13	1.06677	0.533386	+	0.845872i	$0.320919\pi$
$774$	0	0
$775$	0	0
$776$	−3.46093e11	−0.0342622
$777$	0	0
$778$	−2.43949e13	−2.38721
$779$	8.75583e12	0.851881
$780$	0	0
$781$	1.01940e13	0.980426
$782$	−2.87464e11	−0.0274887
$783$	0	0
$784$	1.07478e13	1.01600
$785$	0	0
$786$	0	0
$787$	2.50861e12	0.233102	0.116551	−	0.993185i	$-0.462816\pi$
$787$	2.50861e12	0.233102	0.116551	+	0.993185i	$0.462816\pi$
$788$	1.28165e13	1.18413
$789$	0	0
$790$	0	0
$791$	1.15376e12	0.104790
$792$	0	0
$793$	1.29465e13	1.16258
$794$	−1.98724e13	−1.77443
$795$	0	0
$796$	−1.46637e13	−1.29460
$797$	−1.85822e13	−1.63131	−0.815653	−	0.578541i	$-0.803622\pi$
$797$	−1.85822e13	−1.63131	−0.815653	+	0.578541i	$0.803622\pi$
$798$	0	0
$799$	−2.28760e12	−0.198573
$800$	0	0
$801$	0	0
$802$	5.12746e12	0.437641
$803$	−1.48362e13	−1.25922
$804$	0	0
$805$	0	0
$806$	1.05522e12	0.0880712
$807$	0	0
$808$	−7.06675e10	−0.00583268
$809$	4.47333e12	0.367166	0.183583	−	0.983004i	$-0.441230\pi$
$809$	4.47333e12	0.367166	0.183583	+	0.983004i	$0.441230\pi$
$810$	0	0
$811$	−3.49582e12	−0.283763	−0.141881	−	0.989884i	$-0.545315\pi$
$811$	−3.49582e12	−0.283763	−0.141881	+	0.989884i	$0.545315\pi$
$812$	−1.41508e12	−0.114229
$813$	0	0
$814$	2.03436e13	1.62412
$815$	0	0
$816$	0	0
$817$	1.16911e12	0.0918026
$818$	−2.30108e13	−1.79698
$819$	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
$823$	8.11058e12	0.616244	0.308122	−	0.951347i	$-0.400300\pi$
$823$	8.11058e12	0.616244	0.308122	+	0.951347i	$0.400300\pi$
$824$	6.14163e11	0.0464099
$825$	0	0
$826$	1.51161e12	0.112987
$827$	5.06349e10	0.00376422	0.00188211	−	0.999998i	$-0.499401\pi$
$827$	5.06349e10	0.00376422	0.00188211	+	0.999998i	$0.499401\pi$
$828$	0	0
$829$	−1.50474e13	−1.10654	−0.553269	−	0.833002i	$-0.686620\pi$
$829$	−1.50474e13	−1.10654	−0.553269	+	0.833002i	$0.686620\pi$
$830$	0	0
$831$	0	0
$832$	−9.22584e12	−0.667499
$833$	2.68924e12	0.193521
$834$	0	0
$835$	0	0
$836$	8.33403e12	0.590102
$837$	0	0
$838$	−8.90044e12	−0.623467
$839$	−2.68352e13	−1.86972	−0.934859	−	0.355020i	$-0.884474\pi$
$839$	−2.68352e13	−1.86972	−0.934859	+	0.355020i	$0.884474\pi$
$840$	0	0
$841$	5.36519e12	0.369831
$842$	−2.08468e12	−0.142934
$843$	0	0
$844$	5.84197e12	0.396295
$845$	0	0
$846$	0	0
$847$	3.00951e10	0.00200919
$848$	−2.26660e13	−1.50520
$849$	0	0
$850$	0	0
$851$	1.75354e12	0.114613
$852$	0	0
$853$	4.22107e12	0.272993	0.136497	−	0.990641i	$-0.456416\pi$
$853$	4.22107e12	0.272993	0.136497	+	0.990641i	$0.456416\pi$
$854$	3.60540e12	0.231949
$855$	0	0
$856$	−6.89073e11	−0.0438665
$857$	7.57740e12	0.479851	0.239926	−	0.970791i	$-0.422877\pi$
$857$	7.57740e12	0.479851	0.239926	+	0.970791i	$0.422877\pi$
$858$	0	0
$859$	1.41463e13	0.886486	0.443243	−	0.896401i	$-0.353828\pi$
$859$	1.41463e13	0.886486	0.443243	+	0.896401i	$0.353828\pi$
$860$	0	0
$861$	0	0
$862$	−3.18757e13	−1.96642
$863$	−1.52498e13	−0.935868	−0.467934	−	0.883763i	$-0.655002\pi$
$863$	−1.52498e13	−0.935868	−0.467934	+	0.883763i	$0.655002\pi$
$864$	0	0
$865$	0	0
$866$	−2.53780e13	−1.53330
$867$	0	0
$868$	1.44915e11	0.00866515
$869$	1.82558e13	1.08596
$870$	0	0
$871$	5.04937e12	0.297273
$872$	−5.49201e11	−0.0321668
$873$	0	0
$874$	1.45671e12	0.0844443
$875$	0	0
$876$	0	0
$877$	3.30350e12	0.188572	0.0942858	−	0.995545i	$-0.469943\pi$
$877$	3.30350e12	0.188572	0.0942858	+	0.995545i	$0.469943\pi$
$878$	2.53747e13	1.44104
$879$	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
$883$	1.10669e13	0.612637	0.306318	−	0.951929i	$-0.400903\pi$
$883$	1.10669e13	0.612637	0.306318	+	0.951929i	$0.400903\pi$
$884$	−2.43885e12	−0.134323
$885$	0	0
$886$	−2.40482e13	−1.31108
$887$	6.92035e12	0.375381	0.187690	−	0.982228i	$-0.439900\pi$
$887$	6.92035e12	0.375381	0.187690	+	0.982228i	$0.439900\pi$
$888$	0	0
$889$	−1.95308e12	−0.104872
$890$	0	0
$891$	0	0
$892$	1.28690e13	0.680618
$893$	1.15923e13	0.610010
$894$	0	0
$895$	0	0
$896$	1.47084e11	0.00762394
$897$	0	0
$898$	2.03225e13	1.04288
$899$	−2.03509e12	−0.103912
$900$	0	0
$901$	−5.67136e12	−0.286699
$902$	4.00069e13	2.01236
$903$	0	0
$904$	−7.97383e11	−0.0397108
$905$	0	0
$906$	0	0
$907$	−2.00486e13	−0.983676	−0.491838	−	0.870687i	$-0.663675\pi$
$907$	−2.00486e13	−0.983676	−0.491838	+	0.870687i	$0.663675\pi$
$908$	1.24982e13	0.610186
$909$	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
$913$	−2.20730e13	−1.05134
$914$	4.26768e13	2.02271
$915$	0	0
$916$	−1.14803e13	−0.538797
$917$	1.16938e12	0.0546129
$918$	0	0
$919$	−6.15926e12	−0.284845	−0.142423	−	0.989806i	$-0.545489\pi$
$919$	−6.15926e12	−0.284845	−0.142423	+	0.989806i	$0.545489\pi$
$920$	0	0
$921$	0	0
$922$	−1.10322e13	−0.502774
$923$	1.51169e13	0.685574
$924$	0	0
$925$	0	0
$926$	−2.28904e13	−1.02307
$927$	0	0
$928$	3.71140e13	1.64275
$929$	−2.20000e13	−0.969061	−0.484531	−	0.874774i	$-0.661010\pi$
$929$	−2.20000e13	−0.969061	−0.484531	+	0.874774i	$0.661010\pi$
$930$	0	0
$931$	−1.36275e13	−0.594489
$932$	−1.58010e13	−0.685980
$933$	0	0
$934$	−2.32508e12	−0.0999715
$935$	0	0
$936$	0	0
$937$	2.49274e12	0.105645	0.0528225	−	0.998604i	$-0.483178\pi$
$937$	2.49274e12	0.105645	0.0528225	+	0.998604i	$0.483178\pi$
$938$	1.40617e12	0.0593098
$939$	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
$943$	3.44844e12	0.142010
$944$	−2.00805e13	−0.823002
$945$	0	0
$946$	5.34185e12	0.216861
$947$	2.05809e13	0.831551	0.415775	−	0.909467i	$-0.363510\pi$
$947$	2.05809e13	0.831551	0.415775	+	0.909467i	$0.363510\pi$
$948$	0	0
$949$	−2.20008e13	−0.880524
$950$	0	0
$951$	0	0
$952$	1.88927e10	0.000745466 0
$953$	2.97887e13	1.16986	0.584929	−	0.811084i	$-0.301122\pi$
$953$	2.97887e13	1.16986	0.584929	+	0.811084i	$0.301122\pi$
$954$	0	0
$955$	0	0
$956$	−1.11481e13	−0.431660
$957$	0	0
$958$	1.52655e12	0.0585553
$959$	2.94675e12	0.112502
$960$	0	0
$961$	−2.62312e13	−0.992118
$962$	3.01679e13	1.13568
$963$	0	0
$964$	−1.05020e13	−0.391674
$965$	0	0
$966$	0	0
$967$	1.74473e13	0.641664	0.320832	−	0.947136i	$-0.396037\pi$
$967$	1.74473e13	0.641664	0.320832	+	0.947136i	$0.396037\pi$
$968$	−2.07993e10	−0.000761394 0
$969$	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
$973$	1.63859e11	0.00586088
$974$	3.95328e13	1.40748
$975$	0	0
$976$	−4.78948e13	−1.68952
$977$	−3.23029e13	−1.13427	−0.567135	−	0.823625i	$-0.691948\pi$
$977$	−3.23029e13	−1.13427	−0.567135	+	0.823625i	$0.691948\pi$
$978$	0	0
$979$	−2.85791e12	−0.0994321
$980$	0	0
$981$	0	0
$982$	8.36348e12	0.287002
$983$	1.78014e13	0.608083	0.304041	−	0.952659i	$-0.401664\pi$
$983$	1.78014e13	0.608083	0.304041	+	0.952659i	$0.401664\pi$
$984$	0	0
$985$	0	0
$986$	9.53796e12	0.321373
$987$	0	0
$988$	1.23587e13	0.412635
$989$	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	$0.517760\pi$
$992$	−3.80078e12	−0.124615
$993$	0	0
$994$	4.20982e12	0.136781
$995$	0	0
$996$	0	0
$997$	4.17766e13	1.33908	0.669538	−	0.742778i	$-0.266492\pi$
$997$	4.17766e13	1.33908	0.669538	+	0.742778i	$0.266492\pi$
$998$	1.19329e12	0.0380767
$999$	0	0

Twists

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

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

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.a (trivial)

\(f(q)\)	\(=\)	\(q+31.7828 q^{2} +498.143 q^{4} +637.237 q^{7} -440.406 q^{8} +O(q^{10})\)	\(q+31.7828 q^{2} +498.143 q^{4} +637.237 q^{7} -440.406 q^{8} +49042.6 q^{11} +72726.2 q^{13} +20253.2 q^{14} -269047. q^{16} -67319.3 q^{17} +341136. q^{19} +1.55871e6 q^{22} +134355. q^{23} +2.31144e6 q^{26} +317435. q^{28} -4.45784e6 q^{29} +456520. q^{31} -8.32555e6 q^{32} -2.13959e6 q^{34} +1.30516e7 q^{37} +1.08422e7 q^{38} +2.56667e7 q^{41} +3.42710e6 q^{43} +2.44302e7 q^{44} +4.27016e6 q^{46} +3.39814e7 q^{47} -3.99475e7 q^{49} +3.62281e7 q^{52} +8.42456e7 q^{53} -280643. q^{56} -1.41682e8 q^{58} +7.46358e7 q^{59} +1.78017e8 q^{61} +1.45094e7 q^{62} -1.26857e8 q^{64} +6.94299e7 q^{67} -3.35347e7 q^{68} +2.07860e8 q^{71} -3.02516e8 q^{73} +4.14815e8 q^{74} +1.69935e8 q^{76} +3.12518e7 q^{77} +3.72244e8 q^{79} +8.15758e8 q^{82} -4.50079e8 q^{83} +1.08923e8 q^{86} -2.15987e7 q^{88} -5.82741e7 q^{89} +4.63438e7 q^{91} +6.69279e7 q^{92} +1.08002e9 q^{94} +7.85850e8 q^{97} -1.26964e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 33 q^{2} + 341 q^{4} - 5258 q^{7} - 105 q^{8}+O(q^{10}) \) 3 * q + 33 * q^2 + 341 * q^4 - 5258 * q^7 - 105 * q^8	\( 3 q + 33 q^{2} + 341 q^{4} - 5258 q^{7} - 105 q^{8} + 54699 q^{11} - 215884 q^{13} - 272922 q^{14} - 699247 q^{16} + 334983 q^{17} + 818845 q^{19} - 72761 q^{22} + 3526854 q^{23} + 1280004 q^{26} + 428134 q^{28} - 2175480 q^{29} + 4274066 q^{31} - 9464577 q^{32} - 6838963 q^{34} + 10305042 q^{37} - 24180495 q^{38} - 5926311 q^{41} + 24429956 q^{43} + 21995703 q^{44} + 14223246 q^{46} + 66858708 q^{47} - 6453929 q^{49} + 57862932 q^{52} + 132620514 q^{53} + 169538130 q^{56} - 201908320 q^{58} - 5670960 q^{59} + 125306926 q^{61} - 39831174 q^{62} + 167542401 q^{64} - 88829483 q^{67} - 71162559 q^{68} - 297550596 q^{71} - 181321729 q^{73} + 251507358 q^{74} + 89414865 q^{76} + 561214086 q^{77} - 310025170 q^{79} + 1368322979 q^{82} - 731088801 q^{83} - 33947196 q^{86} + 943671285 q^{88} + 1103860035 q^{89} + 1183187656 q^{91} - 190024242 q^{92} + 1727891132 q^{94} + 332236842 q^{97} + 457927581 q^{98}+O(q^{100}) \) 3 * q + 33 * q^2 + 341 * q^4 - 5258 * q^7 - 105 * q^8 + 54699 * q^11 - 215884 * q^13 - 272922 * q^14 - 699247 * q^16 + 334983 * q^17 + 818845 * q^19 - 72761 * q^22 + 3526854 * q^23 + 1280004 * q^26 + 428134 * q^28 - 2175480 * q^29 + 4274066 * q^31 - 9464577 * q^32 - 6838963 * q^34 + 10305042 * q^37 - 24180495 * q^38 - 5926311 * q^41 + 24429956 * q^43 + 21995703 * q^44 + 14223246 * q^46 + 66858708 * q^47 - 6453929 * q^49 + 57862932 * q^52 + 132620514 * q^53 + 169538130 * q^56 - 201908320 * q^58 - 5670960 * q^59 + 125306926 * q^61 - 39831174 * q^62 + 167542401 * q^64 - 88829483 * q^67 - 71162559 * q^68 - 297550596 * q^71 - 181321729 * q^73 + 251507358 * q^74 + 89414865 * q^76 + 561214086 * q^77 - 310025170 * q^79 + 1368322979 * q^82 - 731088801 * q^83 - 33947196 * q^86 + 943671285 * q^88 + 1103860035 * q^89 + 1183187656 * q^91 - 190024242 * q^92 + 1727891132 * q^94 + 332236842 * q^97 + 457927581 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more