LMFDB - Embedded newform 225.8.a.ba.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,8,Mod(1,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,528,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$70.2866307339$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{65}, \sqrt{85})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 46x^{2} - 115x - 35 $ x^4 - x^3 - 46x^2 - 115x - 35
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.17720$ 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)

$f(q)$	$=$	$q+16.1245 q^{2} +132.000 q^{4} +891.964 q^{7} +64.4981 q^{8} -3872.21 q^{11} -9811.61 q^{13} +14382.5 q^{14} -15856.0 q^{16} -10916.3 q^{17} -14924.0 q^{19} -62437.5 q^{22} +225.743 q^{23} -158207. q^{26} +117739. q^{28} +136634. q^{29} -244192. q^{31} -263926. q^{32} -176020. q^{34} +511095. q^{37} -240642. q^{38} +616234. q^{41} -656486. q^{43} -511132. q^{44} +3640.00 q^{46} -1.32102e6 q^{47} -27943.0 q^{49} -1.29513e6 q^{52} -1.10593e6 q^{53} +57530.0 q^{56} +2.20315e6 q^{58} -1.96210e6 q^{59} +1.76244e6 q^{61} -3.93748e6 q^{62} -2.22611e6 q^{64} -993648. q^{67} -1.44095e6 q^{68} +454708. q^{71} -4.05665e6 q^{73} +8.24117e6 q^{74} -1.96997e6 q^{76} -3.45387e6 q^{77} -2.16822e6 q^{79} +9.93648e6 q^{82} +748371. q^{83} -1.05855e7 q^{86} -249750. q^{88} -5.31046e6 q^{89} -8.75160e6 q^{91} +29798.1 q^{92} -2.13008e7 q^{94} -1.13048e7 q^{97} -450567. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 528 q^{4} - 63424 q^{16} - 59696 q^{19} - 976768 q^{31} - 704080 q^{34} + 14560 q^{46} - 111772 q^{49} + 7049768 q^{61} - 8904448 q^{64} - 7879872 q^{76} - 8672864 q^{79} - 35006400 q^{91} - 85203040 q^{94}+O(q^{100}) $ 4 * q + 528 * q^4 - 63424 * q^16 - 59696 * q^19 - 976768 * q^31 - 704080 * q^34 + 14560 * q^46 - 111772 * q^49 + 7049768 * q^61 - 8904448 * q^64 - 7879872 * q^76 - 8672864 * q^79 - 35006400 * q^91 - 85203040 * q^94

Coefficient data

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

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

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

$2$	16.1245	1.42522	0.712610	−	0.701561i	$-0.247513\pi$
$2$	16.1245	1.42522	0.712610	+	0.701561i	$0.247513\pi$
$3$	0	0
$3$	0	0
$4$	132.000	1.03125
$5$	0	0
$5$	0	0
$6$	0	0
$7$	891.964	0.982888	0.491444	−	0.870909i	$-0.336469\pi$
$7$	891.964	0.982888	0.491444	+	0.870909i	$0.336469\pi$
$8$	64.4981	0.0445381
$9$	0	0
$10$	0	0
$11$	−3872.21	−0.877171	−0.438586	−	0.898689i	$-0.644520\pi$
$11$	−3872.21	−0.877171	−0.438586	+	0.898689i	$0.644520\pi$
$12$	0	0
$13$	−9811.61	−1.23862	−0.619310	−	0.785146i	$-0.712588\pi$
$13$	−9811.61	−1.23862	−0.619310	+	0.785146i	$0.712588\pi$
$14$	14382.5	1.40083
$15$	0	0
$16$	−15856.0	−0.967773
$17$	−10916.3	−0.538895	−0.269447	−	0.963015i	$-0.586841\pi$
$17$	−10916.3	−0.538895	−0.269447	+	0.963015i	$0.586841\pi$
$18$	0	0
$19$	−14924.0	−0.499169	−0.249585	−	0.968353i	$-0.580294\pi$
$19$	−14924.0	−0.499169	−0.249585	+	0.968353i	$0.580294\pi$
$20$	0	0
$21$	0	0
$22$	−62437.5	−1.25016
$23$	225.743	0.00386872	0.00193436	−	0.999998i	$-0.499384\pi$
$23$	225.743	0.00386872	0.00193436	+	0.999998i	$0.499384\pi$
$24$	0	0
$25$	0	0
$26$	−158207.	−1.76531
$27$	0	0
$28$	117739.	1.01360
$29$	136634.	1.04031	0.520157	−	0.854070i	$-0.325873\pi$
$29$	136634.	1.04031	0.520157	+	0.854070i	$0.325873\pi$
$30$	0	0
$31$	−244192.	−1.47220	−0.736098	−	0.676875i	$-0.763334\pi$
$31$	−244192.	−1.47220	−0.736098	+	0.676875i	$0.763334\pi$
$32$	−263926.	−1.42383
$33$	0	0
$34$	−176020.	−0.768043
$35$	0	0
$36$	0	0
$37$	511095.	1.65881	0.829404	−	0.558650i	$-0.188680\pi$
$37$	511095.	1.65881	0.829404	+	0.558650i	$0.188680\pi$
$38$	−240642.	−0.711425
$39$	0	0
$40$	0	0
$41$	616234.	1.39638	0.698188	−	0.715914i	$-0.253990\pi$
$41$	616234.	1.39638	0.698188	+	0.715914i	$0.253990\pi$
$42$	0	0
$43$	−656486.	−1.25917	−0.629587	−	0.776930i	$-0.716776\pi$
$43$	−656486.	−1.25917	−0.629587	+	0.776930i	$0.716776\pi$
$44$	−511132.	−0.904583
$45$	0	0
$46$	3640.00	0.00551377
$47$	−1.32102e6	−1.85595	−0.927974	−	0.372644i	$-0.878451\pi$
$47$	−1.32102e6	−1.85595	−0.927974	+	0.372644i	$0.878451\pi$
$48$	0	0
$49$	−27943.0	−0.0339302
$50$	0	0
$51$	0	0
$52$	−1.29513e6	−1.27733
$53$	−1.10593e6	−1.02038	−0.510191	−	0.860061i	$-0.670425\pi$
$53$	−1.10593e6	−1.02038	−0.510191	+	0.860061i	$0.670425\pi$
$54$	0	0
$55$	0	0
$56$	57530.0	0.0437760
$57$	0	0
$58$	2.20315e6	1.48268
$59$	−1.96210e6	−1.24377	−0.621885	−	0.783109i	$-0.713633\pi$
$59$	−1.96210e6	−1.24377	−0.621885	+	0.783109i	$0.713633\pi$
$60$	0	0
$61$	1.76244e6	0.994169	0.497085	−	0.867702i	$-0.334404\pi$
$61$	1.76244e6	0.994169	0.497085	+	0.867702i	$0.334404\pi$
$62$	−3.93748e6	−2.09820
$63$	0	0
$64$	−2.22611e6	−1.06149
$65$	0	0
$66$	0	0
$67$	−993648.	−0.403618	−0.201809	−	0.979425i	$-0.564682\pi$
$67$	−993648.	−0.403618	−0.201809	+	0.979425i	$0.564682\pi$
$68$	−1.44095e6	−0.555735
$69$	0	0
$70$	0	0
$71$	454708.	0.150775	0.0753873	−	0.997154i	$-0.475981\pi$
$71$	454708.	0.150775	0.0753873	+	0.997154i	$0.475981\pi$
$72$	0	0
$73$	−4.05665e6	−1.22050	−0.610250	−	0.792209i	$-0.708931\pi$
$73$	−4.05665e6	−1.22050	−0.610250	+	0.792209i	$0.708931\pi$
$74$	8.24117e6	2.36416
$75$	0	0
$76$	−1.96997e6	−0.514768
$77$	−3.45387e6	−0.862161
$78$	0	0
$79$	−2.16822e6	−0.494775	−0.247387	−	0.968917i	$-0.579572\pi$
$79$	−2.16822e6	−0.494775	−0.247387	+	0.968917i	$0.579572\pi$
$80$	0	0
$81$	0	0
$82$	9.93648e6	1.99014
$83$	748371.	0.143663	0.0718313	−	0.997417i	$-0.477116\pi$
$83$	748371.	0.143663	0.0718313	+	0.997417i	$0.477116\pi$
$84$	0	0
$85$	0	0
$86$	−1.05855e7	−1.79460
$87$	0	0
$88$	−249750.	−0.0390675
$89$	−5.31046e6	−0.798485	−0.399243	−	0.916845i	$-0.630727\pi$
$89$	−5.31046e6	−0.798485	−0.399243	+	0.916845i	$0.630727\pi$
$90$	0	0
$91$	−8.75160e6	−1.21743
$92$	29798.1	0.00398962
$93$	0	0
$94$	−2.13008e7	−2.64513
$95$	0	0
$96$	0	0
$97$	−1.13048e7	−1.25765	−0.628825	−	0.777547i	$-0.716464\pi$
$97$	−1.13048e7	−1.25765	−0.628825	+	0.777547i	$0.716464\pi$
$98$	−450567.	−0.0483580
$99$	0	0
$100$	0	0
$101$	9.78728e6	0.945230	0.472615	−	0.881269i	$-0.343310\pi$
$101$	9.78728e6	0.945230	0.472615	+	0.881269i	$0.343310\pi$
$102$	0	0
$103$	1.92673e7	1.73736	0.868682	−	0.495370i	$-0.164967\pi$
$103$	1.92673e7	1.73736	0.868682	+	0.495370i	$0.164967\pi$
$104$	−632830.	−0.0551658
$105$	0	0
$106$	−1.78326e7	−1.45427
$107$	2.24713e7	1.77331	0.886656	−	0.462429i	$-0.153022\pi$
$107$	2.24713e7	1.77331	0.886656	+	0.462429i	$0.153022\pi$
$108$	0	0
$109$	−1.70298e7	−1.25955	−0.629776	−	0.776777i	$-0.716853\pi$
$109$	−1.70298e7	−1.25955	−0.629776	+	0.776777i	$0.716853\pi$
$110$	0	0
$111$	0	0
$112$	−1.41430e7	−0.951213
$113$	1.37331e7	0.895355	0.447677	−	0.894195i	$-0.352251\pi$
$113$	1.37331e7	0.895355	0.447677	+	0.894195i	$0.352251\pi$
$114$	0	0
$115$	0	0
$116$	1.80356e7	1.07282
$117$	0	0
$118$	−3.16380e7	−1.77264
$119$	−9.73695e6	−0.529674
$120$	0	0
$121$	−4.49317e6	−0.230571
$122$	2.84185e7	1.41691
$123$	0	0
$124$	−3.22333e7	−1.51820
$125$	0	0
$126$	0	0
$127$	2.85330e7	1.23605	0.618024	−	0.786160i	$-0.287934\pi$
$127$	2.85330e7	1.23605	0.618024	+	0.786160i	$0.287934\pi$
$128$	−2.11244e6	−0.0890327
$129$	0	0
$130$	0	0
$131$	9.19760e6	0.357458	0.178729	−	0.983898i	$-0.442801\pi$
$131$	9.19760e6	0.357458	0.178729	+	0.983898i	$0.442801\pi$
$132$	0	0
$133$	−1.33117e7	−0.490627
$134$	−1.60221e7	−0.575244
$135$	0	0
$136$	−704080.	−0.0240014
$137$	3.64799e7	1.21208	0.606040	−	0.795434i	$-0.292757\pi$
$137$	3.64799e7	1.21208	0.606040	+	0.795434i	$0.292757\pi$
$138$	0	0
$139$	−1.05728e7	−0.333916	−0.166958	−	0.985964i	$-0.553394\pi$
$139$	−1.05728e7	−0.333916	−0.166958	+	0.985964i	$0.553394\pi$
$140$	0	0
$141$	0	0
$142$	7.33195e6	0.214887
$143$	3.79926e7	1.08648
$144$	0	0
$145$	0	0
$146$	−6.54116e7	−1.73948
$147$	0	0
$148$	6.74646e7	1.71065
$149$	2.63327e7	0.652143	0.326072	−	0.945345i	$-0.394275\pi$
$149$	2.63327e7	0.652143	0.326072	+	0.945345i	$0.394275\pi$
$150$	0	0
$151$	2.87125e7	0.678660	0.339330	−	0.940667i	$-0.389800\pi$
$151$	2.87125e7	0.678660	0.339330	+	0.940667i	$0.389800\pi$
$152$	−962569.	−0.0222320
$153$	0	0
$154$	−5.56920e7	−1.22877
$155$	0	0
$156$	0	0
$157$	994540.	0.0205104	0.0102552	−	0.999947i	$-0.496736\pi$
$157$	994540.	0.0205104	0.0102552	+	0.999947i	$0.496736\pi$
$158$	−3.49614e7	−0.705163
$159$	0	0
$160$	0	0
$161$	201355.	0.00380252
$162$	0	0
$163$	8.91072e6	0.161160	0.0805798	−	0.996748i	$-0.474323\pi$
$163$	8.91072e6	0.161160	0.0805798	+	0.996748i	$0.474323\pi$
$164$	8.13429e7	1.44001
$165$	0	0
$166$	1.20671e7	0.204751
$167$	−1.85304e7	−0.307877	−0.153938	−	0.988080i	$-0.549196\pi$
$167$	−1.85304e7	−0.307877	−0.153938	+	0.988080i	$0.549196\pi$
$168$	0	0
$169$	3.35191e7	0.534181
$170$	0	0
$171$	0	0
$172$	−8.66561e7	−1.29852
$173$	4.68600e7	0.688083	0.344042	−	0.938954i	$-0.388204\pi$
$173$	4.68600e7	0.688083	0.344042	+	0.938954i	$0.388204\pi$
$174$	0	0
$175$	0	0
$176$	6.13977e7	0.848903
$177$	0	0
$178$	−8.56286e7	−1.13802
$179$	−3.64596e6	−0.0475145	−0.0237573	−	0.999718i	$-0.507563\pi$
$179$	−3.64596e6	−0.0475145	−0.0237573	+	0.999718i	$0.507563\pi$
$180$	0	0
$181$	−8.98324e7	−1.12605	−0.563025	−	0.826440i	$-0.690363\pi$
$181$	−8.98324e7	−1.12605	−0.563025	+	0.826440i	$0.690363\pi$
$182$	−1.41115e8	−1.73510
$183$	0	0
$184$	14560.0	0.000172305 0
$185$	0	0
$186$	0	0
$187$	4.22702e7	0.472703
$188$	−1.74374e8	−1.91395
$189$	0	0
$190$	0	0
$191$	−1.69641e8	−1.76163	−0.880816	−	0.473459i	$-0.843005\pi$
$191$	−1.69641e8	−1.76163	−0.880816	+	0.473459i	$0.843005\pi$
$192$	0	0
$193$	−8.59407e7	−0.860495	−0.430248	−	0.902711i	$-0.641574\pi$
$193$	−8.59407e7	−0.860495	−0.430248	+	0.902711i	$0.641574\pi$
$194$	−1.82284e8	−1.79243
$195$	0	0
$196$	−3.68848e6	−0.0349905
$197$	−7.48293e7	−0.697333	−0.348666	−	0.937247i	$-0.613365\pi$
$197$	−7.48293e7	−0.697333	−0.348666	+	0.937247i	$0.613365\pi$
$198$	0	0
$199$	−8.39132e7	−0.754822	−0.377411	−	0.926046i	$-0.623186\pi$
$199$	−8.39132e7	−0.754822	−0.377411	+	0.926046i	$0.623186\pi$
$200$	0	0
$201$	0	0
$202$	1.57815e8	1.34716
$203$	1.21872e8	1.02251
$204$	0	0
$205$	0	0
$206$	3.10676e8	2.47613
$207$	0	0
$208$	1.55573e8	1.19870
$209$	5.77888e7	0.437857
$210$	0	0
$211$	−9.68059e7	−0.709436	−0.354718	−	0.934973i	$-0.615423\pi$
$211$	−9.68059e7	−0.709436	−0.354718	+	0.934973i	$0.615423\pi$
$212$	−1.45983e8	−1.05227
$213$	0	0
$214$	3.62339e8	2.52736
$215$	0	0
$216$	0	0
$217$	−2.17811e8	−1.44700
$218$	−2.74597e8	−1.79514
$219$	0	0
$220$	0	0
$221$	1.07106e8	0.667486
$222$	0	0
$223$	1.32857e8	0.802265	0.401133	−	0.916020i	$-0.368617\pi$
$223$	1.32857e8	0.802265	0.401133	+	0.916020i	$0.368617\pi$
$224$	−2.35413e8	−1.39946
$225$	0	0
$226$	2.21440e8	1.27608
$227$	−9.72571e7	−0.551862	−0.275931	−	0.961177i	$-0.588986\pi$
$227$	−9.72571e7	−0.551862	−0.275931	+	0.961177i	$0.588986\pi$
$228$	0	0
$229$	2.84016e7	0.156286	0.0781429	−	0.996942i	$-0.475101\pi$
$229$	2.84016e7	0.156286	0.0781429	+	0.996942i	$0.475101\pi$
$230$	0	0
$231$	0	0
$232$	8.81261e6	0.0463336
$233$	1.36738e8	0.708180	0.354090	−	0.935211i	$-0.384791\pi$
$233$	1.36738e8	0.708180	0.354090	+	0.935211i	$0.384791\pi$
$234$	0	0
$235$	0	0
$236$	−2.58998e8	−1.28264
$237$	0	0
$238$	−1.57004e8	−0.754901
$239$	2.93583e8	1.39104	0.695518	−	0.718509i	$-0.255175\pi$
$239$	2.93583e8	1.39104	0.695518	+	0.718509i	$0.255175\pi$
$240$	0	0
$241$	2.01709e8	0.928251	0.464126	−	0.885769i	$-0.346369\pi$
$241$	2.01709e8	0.928251	0.464126	+	0.885769i	$0.346369\pi$
$242$	−7.24502e7	−0.328614
$243$	0	0
$244$	2.32642e8	1.02524
$245$	0	0
$246$	0	0
$247$	1.46428e8	0.618281
$248$	−1.57499e7	−0.0655688
$249$	0	0
$250$	0	0
$251$	−2.23310e8	−0.891353	−0.445676	−	0.895194i	$-0.647037\pi$
$251$	−2.23310e8	−0.891353	−0.445676	+	0.895194i	$0.647037\pi$
$252$	0	0
$253$	−874125.	−0.00339353
$254$	4.60081e8	1.76164
$255$	0	0
$256$	2.50880e8	0.934602
$257$	3.81922e8	1.40349	0.701744	−	0.712430i	$-0.252405\pi$
$257$	3.81922e8	1.40349	0.701744	+	0.712430i	$0.252405\pi$
$258$	0	0
$259$	4.55879e8	1.63042
$260$	0	0
$261$	0	0
$262$	1.48307e8	0.509456
$263$	2.76045e8	0.935695	0.467848	−	0.883809i	$-0.345030\pi$
$263$	2.76045e8	0.935695	0.467848	+	0.883809i	$0.345030\pi$
$264$	0	0
$265$	0	0
$266$	−2.14644e8	−0.699252
$267$	0	0
$268$	−1.31162e8	−0.416231
$269$	−4.22435e8	−1.32321	−0.661603	−	0.749855i	$-0.730123\pi$
$269$	−4.22435e8	−1.32321	−0.661603	+	0.749855i	$0.730123\pi$
$270$	0	0
$271$	−1.71153e6	−0.00522386	−0.00261193	−	0.999997i	$-0.500831\pi$
$271$	−1.71153e6	−0.00522386	−0.00261193	+	0.999997i	$0.500831\pi$
$272$	1.73089e8	0.521528
$273$	0	0
$274$	5.88220e8	1.72748
$275$	0	0
$276$	0	0
$277$	−1.61917e8	−0.457735	−0.228868	−	0.973458i	$-0.573502\pi$
$277$	−1.61917e8	−0.457735	−0.228868	+	0.973458i	$0.573502\pi$
$278$	−1.70481e8	−0.475903
$279$	0	0
$280$	0	0
$281$	3.59164e8	0.965652	0.482826	−	0.875716i	$-0.339610\pi$
$281$	3.59164e8	0.965652	0.482826	+	0.875716i	$0.339610\pi$
$282$	0	0
$283$	−2.05767e8	−0.539664	−0.269832	−	0.962907i	$-0.586968\pi$
$283$	−2.05767e8	−0.539664	−0.269832	+	0.962907i	$0.586968\pi$
$284$	6.00214e7	0.155486
$285$	0	0
$286$	6.12612e8	1.54848
$287$	5.49659e8	1.37248
$288$	0	0
$289$	−2.91173e8	−0.709592
$290$	0	0
$291$	0	0
$292$	−5.35478e8	−1.25864
$293$	7.21304e8	1.67526	0.837629	−	0.546240i	$-0.183941\pi$
$293$	7.21304e8	1.67526	0.837629	+	0.546240i	$0.183941\pi$
$294$	0	0
$295$	0	0
$296$	3.29647e7	0.0738801
$297$	0	0
$298$	4.24602e8	0.929447
$299$	−2.21490e6	−0.00479188
$300$	0	0
$301$	−5.85562e8	−1.23763
$302$	4.62976e8	0.967239
$303$	0	0
$304$	2.36635e8	0.483083
$305$	0	0
$306$	0	0
$307$	2.99711e8	0.591177	0.295589	−	0.955315i	$-0.404484\pi$
$307$	2.99711e8	0.591177	0.295589	+	0.955315i	$0.404484\pi$
$308$	−4.55911e8	−0.889104
$309$	0	0
$310$	0	0
$311$	1.72058e8	0.324349	0.162175	−	0.986762i	$-0.448149\pi$
$311$	1.72058e8	0.324349	0.162175	+	0.986762i	$0.448149\pi$
$312$	0	0
$313$	3.05567e8	0.563251	0.281625	−	0.959524i	$-0.409126\pi$
$313$	3.05567e8	0.563251	0.281625	+	0.959524i	$0.409126\pi$
$314$	1.60365e7	0.0292318
$315$	0	0
$316$	−2.86205e8	−0.510237
$317$	4.39480e8	0.774875	0.387437	−	0.921896i	$-0.373360\pi$
$317$	4.39480e8	0.774875	0.387437	+	0.921896i	$0.373360\pi$
$318$	0	0
$319$	−5.29074e8	−0.912534
$320$	0	0
$321$	0	0
$322$	3.24675e6	0.00541943
$323$	1.62915e8	0.269000
$324$	0	0
$325$	0	0
$326$	1.43681e8	0.229688
$327$	0	0
$328$	3.97459e7	0.0621919
$329$	−1.17830e9	−1.82419
$330$	0	0
$331$	−9.61785e8	−1.45774	−0.728870	−	0.684652i	$-0.759954\pi$
$331$	−9.61785e8	−1.45774	−0.728870	+	0.684652i	$0.759954\pi$
$332$	9.87850e7	0.148152
$333$	0	0
$334$	−2.98794e8	−0.438792
$335$	0	0
$336$	0	0
$337$	5.23454e8	0.745031	0.372515	−	0.928026i	$-0.378495\pi$
$337$	5.23454e8	0.745031	0.372515	+	0.928026i	$0.378495\pi$
$338$	5.40479e8	0.761325
$339$	0	0
$340$	0	0
$341$	9.45562e8	1.29137
$342$	0	0
$343$	−7.59495e8	−1.01624
$344$	−4.23420e7	−0.0560812
$345$	0	0
$346$	7.55595e8	0.980670
$347$	−1.36908e8	−0.175905	−0.0879523	−	0.996125i	$-0.528032\pi$
$347$	−1.36908e8	−0.175905	−0.0879523	+	0.996125i	$0.528032\pi$
$348$	0	0
$349$	1.82702e8	0.230067	0.115034	−	0.993362i	$-0.463302\pi$
$349$	1.82702e8	0.230067	0.115034	+	0.993362i	$0.463302\pi$
$350$	0	0
$351$	0	0
$352$	1.02198e9	1.24894
$353$	−1.14182e9	−1.38161	−0.690804	−	0.723042i	$-0.742743\pi$
$353$	−1.14182e9	−1.38161	−0.690804	+	0.723042i	$0.742743\pi$
$354$	0	0
$355$	0	0
$356$	−7.00980e8	−0.823438
$357$	0	0
$358$	−5.87894e7	−0.0677186
$359$	1.46808e9	1.67463	0.837317	−	0.546718i	$-0.184123\pi$
$359$	1.46808e9	1.67463	0.837317	+	0.546718i	$0.184123\pi$
$360$	0	0
$361$	−6.71146e8	−0.750830
$362$	−1.44850e9	−1.60487
$363$	0	0
$364$	−1.15521e9	−1.25547
$365$	0	0
$366$	0	0
$367$	−3.71113e8	−0.391900	−0.195950	−	0.980614i	$-0.562779\pi$
$367$	−3.71113e8	−0.391900	−0.195950	+	0.980614i	$0.562779\pi$
$368$	−3.57938e6	−0.00374404
$369$	0	0
$370$	0	0
$371$	−9.86452e8	−1.00292
$372$	0	0
$373$	4.24232e8	0.423274	0.211637	−	0.977348i	$-0.432120\pi$
$373$	4.24232e8	0.423274	0.211637	+	0.977348i	$0.432120\pi$
$374$	6.81586e8	0.673706
$375$	0	0
$376$	−8.52030e7	−0.0826604
$377$	−1.34060e9	−1.28856
$378$	0	0
$379$	−1.03619e9	−0.977688	−0.488844	−	0.872371i	$-0.662581\pi$
$379$	−1.03619e9	−0.977688	−0.488844	+	0.872371i	$0.662581\pi$
$380$	0	0
$381$	0	0
$382$	−2.73539e9	−2.51071
$383$	7.40719e8	0.673687	0.336843	−	0.941561i	$-0.390641\pi$
$383$	7.40719e8	0.673687	0.336843	+	0.941561i	$0.390641\pi$
$384$	0	0
$385$	0	0
$386$	−1.38575e9	−1.22639
$387$	0	0
$388$	−1.49223e9	−1.29695
$389$	−1.22689e9	−1.05677	−0.528387	−	0.849003i	$-0.677203\pi$
$389$	−1.22689e9	−1.05677	−0.528387	+	0.849003i	$0.677203\pi$
$390$	0	0
$391$	−2.46428e6	−0.00208483
$392$	−1.80227e6	−0.00151119
$393$	0	0
$394$	−1.20659e9	−0.993852
$395$	0	0
$396$	0	0
$397$	−1.16332e9	−0.933109	−0.466555	−	0.884492i	$-0.654505\pi$
$397$	−1.16332e9	−0.933109	−0.466555	+	0.884492i	$0.654505\pi$
$398$	−1.35306e9	−1.07579
$399$	0	0
$400$	0	0
$401$	−7.05638e7	−0.0546483	−0.0273242	−	0.999627i	$-0.508699\pi$
$401$	−7.05638e7	−0.0546483	−0.0273242	+	0.999627i	$0.508699\pi$
$402$	0	0
$403$	2.39592e9	1.82349
$404$	1.29192e9	0.974768
$405$	0	0
$406$	1.96513e9	1.45731
$407$	−1.97907e9	−1.45506
$408$	0	0
$409$	−4.84419e8	−0.350098	−0.175049	−	0.984560i	$-0.556008\pi$
$409$	−4.84419e8	−0.350098	−0.175049	+	0.984560i	$0.556008\pi$
$410$	0	0
$411$	0	0
$412$	2.54329e9	1.79166
$413$	−1.75013e9	−1.22249
$414$	0	0
$415$	0	0
$416$	2.58954e9	1.76358
$417$	0	0
$418$	9.31817e8	0.624042
$419$	2.05312e9	1.36353	0.681767	−	0.731570i	$-0.261212\pi$
$419$	2.05312e9	1.36353	0.681767	+	0.731570i	$0.261212\pi$
$420$	0	0
$421$	−4.64439e8	−0.303348	−0.151674	−	0.988431i	$-0.548466\pi$
$421$	−4.64439e8	−0.303348	−0.151674	+	0.988431i	$0.548466\pi$
$422$	−1.56095e9	−1.01110
$423$	0	0
$424$	−7.13305e7	−0.0454459
$425$	0	0
$426$	0	0
$427$	1.57204e9	0.977158
$428$	2.96621e9	1.82873
$429$	0	0
$430$	0	0
$431$	−1.66845e9	−1.00379	−0.501896	−	0.864928i	$-0.667364\pi$
$431$	−1.66845e9	−1.00379	−0.501896	+	0.864928i	$0.667364\pi$
$432$	0	0
$433$	−1.08260e9	−0.640854	−0.320427	−	0.947273i	$-0.603826\pi$
$433$	−1.08260e9	−0.640854	−0.320427	+	0.947273i	$0.603826\pi$
$434$	−3.51209e9	−2.06230
$435$	0	0
$436$	−2.24793e9	−1.29891
$437$	−3.36899e6	−0.00193115
$438$	0	0
$439$	−1.94290e9	−1.09603	−0.548016	−	0.836468i	$-0.684617\pi$
$439$	−1.94290e9	−1.09603	−0.548016	+	0.836468i	$0.684617\pi$
$440$	0	0
$441$	0	0
$442$	1.72704e9	0.951315
$443$	−3.25401e9	−1.77830	−0.889150	−	0.457615i	$-0.848704\pi$
$443$	−3.25401e9	−1.77830	−0.889150	+	0.457615i	$0.848704\pi$
$444$	0	0
$445$	0	0
$446$	2.14226e9	1.14340
$447$	0	0
$448$	−1.98561e9	−1.04333
$449$	−1.59381e9	−0.830950	−0.415475	−	0.909605i	$-0.636385\pi$
$449$	−1.59381e9	−0.830950	−0.415475	+	0.909605i	$0.636385\pi$
$450$	0	0
$451$	−2.38619e9	−1.22486
$452$	1.81277e9	0.923335
$453$	0	0
$454$	−1.56822e9	−0.786525
$455$	0	0
$456$	0	0
$457$	−1.03584e9	−0.507675	−0.253838	−	0.967247i	$-0.581693\pi$
$457$	−1.03584e9	−0.507675	−0.253838	+	0.967247i	$0.581693\pi$
$458$	4.57963e8	0.222741
$459$	0	0
$460$	0	0
$461$	2.37166e9	1.12745	0.563727	−	0.825961i	$-0.309367\pi$
$461$	2.37166e9	1.12745	0.563727	+	0.825961i	$0.309367\pi$
$462$	0	0
$463$	−2.47025e9	−1.15666	−0.578332	−	0.815801i	$-0.696296\pi$
$463$	−2.47025e9	−1.15666	−0.578332	+	0.815801i	$0.696296\pi$
$464$	−2.16646e9	−1.00679
$465$	0	0
$466$	2.20484e9	1.00931
$467$	6.83727e8	0.310652	0.155326	−	0.987863i	$-0.450357\pi$
$467$	6.83727e8	0.310652	0.155326	+	0.987863i	$0.450357\pi$
$468$	0	0
$469$	−8.86298e8	−0.396712
$470$	0	0
$471$	0	0
$472$	−1.26552e8	−0.0553951
$473$	2.54205e9	1.10451
$474$	0	0
$475$	0	0
$476$	−1.28528e9	−0.546226
$477$	0	0
$478$	4.73389e9	1.98253
$479$	7.36812e8	0.306325	0.153162	−	0.988201i	$-0.451054\pi$
$479$	7.36812e8	0.306325	0.153162	+	0.988201i	$0.451054\pi$
$480$	0	0
$481$	−5.01467e9	−2.05463
$482$	3.25246e9	1.32296
$483$	0	0
$484$	−5.93099e8	−0.237776
$485$	0	0
$486$	0	0
$487$	−4.17299e9	−1.63718	−0.818589	−	0.574380i	$-0.805243\pi$
$487$	−4.17299e9	−1.63718	−0.818589	+	0.574380i	$0.805243\pi$
$488$	1.13674e8	0.0442784
$489$	0	0
$490$	0	0
$491$	−2.52822e9	−0.963894	−0.481947	−	0.876200i	$-0.660070\pi$
$491$	−2.52822e9	−0.963894	−0.481947	+	0.876200i	$0.660070\pi$
$492$	0	0
$493$	−1.49153e9	−0.560620
$494$	2.36109e9	0.881186
$495$	0	0
$496$	3.87191e9	1.42475
$497$	4.05583e8	0.148195
$498$	0	0
$499$	1.08872e9	0.392252	0.196126	−	0.980579i	$-0.437164\pi$
$499$	1.08872e9	0.392252	0.196126	+	0.980579i	$0.437164\pi$
$500$	0	0
$501$	0	0
$502$	−3.60076e9	−1.27037
$503$	−1.07775e9	−0.377600	−0.188800	−	0.982016i	$-0.560460\pi$
$503$	−1.07775e9	−0.377600	−0.188800	+	0.982016i	$0.560460\pi$
$504$	0	0
$505$	0	0
$506$	−1.40948e7	−0.00483652
$507$	0	0
$508$	3.76636e9	1.27467
$509$	3.85873e9	1.29698	0.648488	−	0.761225i	$-0.275401\pi$
$509$	3.85873e9	1.29698	0.648488	+	0.761225i	$0.275401\pi$
$510$	0	0
$511$	−3.61839e9	−1.19962
$512$	4.31571e9	1.42105
$513$	0	0
$514$	6.15830e9	2.00028
$515$	0	0
$516$	0	0
$517$	5.11525e9	1.62798
$518$	7.35082e9	2.32371
$519$	0	0
$520$	0	0
$521$	−6.39661e8	−0.198161	−0.0990805	−	0.995079i	$-0.531590\pi$
$521$	−6.39661e8	−0.198161	−0.0990805	+	0.995079i	$0.531590\pi$
$522$	0	0
$523$	9.53972e8	0.291595	0.145797	−	0.989314i	$-0.453425\pi$
$523$	9.53972e8	0.291595	0.145797	+	0.989314i	$0.453425\pi$
$524$	1.21408e9	0.368629
$525$	0	0
$526$	4.45109e9	1.33357
$527$	2.66567e9	0.793359
$528$	0	0
$529$	−3.40477e9	−0.999985
$530$	0	0
$531$	0	0
$532$	−1.75714e9	−0.505960
$533$	−6.04625e9	−1.72958
$534$	0	0
$535$	0	0
$536$	−6.40884e7	−0.0179764
$537$	0	0
$538$	−6.81156e9	−1.88586
$539$	1.08201e8	0.0297626
$540$	0	0
$541$	4.12373e9	1.11969	0.559847	−	0.828596i	$-0.310860\pi$
$541$	4.12373e9	1.11969	0.559847	+	0.828596i	$0.310860\pi$
$542$	−2.75976e7	−0.00744515
$543$	0	0
$544$	2.88110e9	0.767293
$545$	0	0
$546$	0	0
$547$	3.87126e9	1.01134	0.505669	−	0.862727i	$-0.331246\pi$
$547$	3.87126e9	1.01134	0.505669	+	0.862727i	$0.331246\pi$
$548$	4.81534e9	1.24996
$549$	0	0
$550$	0	0
$551$	−2.03912e9	−0.519293
$552$	0	0
$553$	−1.93397e9	−0.486309
$554$	−2.61084e9	−0.652373
$555$	0	0
$556$	−1.39560e9	−0.344350
$557$	1.38658e9	0.339978	0.169989	−	0.985446i	$-0.445627\pi$
$557$	1.38658e9	0.339978	0.169989	+	0.985446i	$0.445627\pi$
$558$	0	0
$559$	6.44118e9	1.55964
$560$	0	0
$561$	0	0
$562$	5.79134e9	1.37627
$563$	2.02834e9	0.479028	0.239514	−	0.970893i	$-0.423012\pi$
$563$	2.02834e9	0.479028	0.239514	+	0.970893i	$0.423012\pi$
$564$	0	0
$565$	0	0
$566$	−3.31790e9	−0.769140
$567$	0	0
$568$	2.93278e7	0.00671522
$569$	−7.16780e8	−0.163115	−0.0815573	−	0.996669i	$-0.525989\pi$
$569$	−7.16780e8	−0.163115	−0.0815573	+	0.996669i	$0.525989\pi$
$570$	0	0
$571$	3.19832e9	0.718944	0.359472	−	0.933156i	$-0.382957\pi$
$571$	3.19832e9	0.718944	0.359472	+	0.933156i	$0.382957\pi$
$572$	5.01502e9	1.12043
$573$	0	0
$574$	8.86298e9	1.95609
$575$	0	0
$576$	0	0
$577$	2.14993e9	0.465917	0.232959	−	0.972487i	$-0.425159\pi$
$577$	2.14993e9	0.465917	0.232959	+	0.972487i	$0.425159\pi$
$578$	−4.69503e9	−1.01132
$579$	0	0
$580$	0	0
$581$	6.67520e8	0.141204
$582$	0	0
$583$	4.28240e9	0.895050
$584$	−2.61646e8	−0.0543588
$585$	0	0
$586$	1.16307e10	2.38761
$587$	−6.42578e9	−1.31127	−0.655635	−	0.755078i	$-0.727599\pi$
$587$	−6.42578e9	−1.31127	−0.655635	+	0.755078i	$0.727599\pi$
$588$	0	0
$589$	3.64432e9	0.734874
$590$	0	0
$591$	0	0
$592$	−8.10393e9	−1.60535
$593$	−3.04761e9	−0.600162	−0.300081	−	0.953914i	$-0.597014\pi$
$593$	−3.04761e9	−0.600162	−0.300081	+	0.953914i	$0.597014\pi$
$594$	0	0
$595$	0	0
$596$	3.47591e9	0.672523
$597$	0	0
$598$	−3.57142e7	−0.00682947
$599$	−4.90779e9	−0.933023	−0.466512	−	0.884515i	$-0.654489\pi$
$599$	−4.90779e9	−0.933023	−0.466512	+	0.884515i	$0.654489\pi$
$600$	0	0
$601$	−3.13473e9	−0.589033	−0.294516	−	0.955646i	$-0.595159\pi$
$601$	−3.13473e9	−0.589033	−0.294516	+	0.955646i	$0.595159\pi$
$602$	−9.44190e9	−1.76389
$603$	0	0
$604$	3.79005e9	0.699868
$605$	0	0
$606$	0	0
$607$	−8.96199e9	−1.62646	−0.813231	−	0.581940i	$-0.802294\pi$
$607$	−8.96199e9	−1.62646	−0.813231	+	0.581940i	$0.802294\pi$
$608$	3.93883e9	0.710731
$609$	0	0
$610$	0	0
$611$	1.29613e10	2.29882
$612$	0	0
$613$	9.33008e9	1.63596	0.817982	−	0.575243i	$-0.195093\pi$
$613$	9.33008e9	1.63596	0.817982	+	0.575243i	$0.195093\pi$
$614$	4.83269e9	0.842558
$615$	0	0
$616$	−2.22768e8	−0.0383990
$617$	6.31024e7	0.0108155	0.00540777	−	0.999985i	$-0.498279\pi$
$617$	6.31024e7	0.0108155	0.00540777	+	0.999985i	$0.498279\pi$
$618$	0	0
$619$	−9.58488e9	−1.62431	−0.812156	−	0.583440i	$-0.801706\pi$
$619$	−9.58488e9	−1.62431	−0.812156	+	0.583440i	$0.801706\pi$
$620$	0	0
$621$	0	0
$622$	2.77435e9	0.462269
$623$	−4.73674e9	−0.784822
$624$	0	0
$625$	0	0
$626$	4.92712e9	0.802756
$627$	0	0
$628$	1.31279e8	0.0211513
$629$	−5.57927e9	−0.893923
$630$	0	0
$631$	−6.88427e9	−1.09083	−0.545413	−	0.838168i	$-0.683627\pi$
$631$	−6.88427e9	−1.09083	−0.545413	+	0.838168i	$0.683627\pi$
$632$	−1.39846e8	−0.0220363
$633$	0	0
$634$	7.08640e9	1.10437
$635$	0	0
$636$	0	0
$637$	2.74166e8	0.0420267
$638$	−8.53106e9	−1.30056
$639$	0	0
$640$	0	0
$641$	−4.62211e9	−0.693167	−0.346583	−	0.938019i	$-0.612658\pi$
$641$	−4.62211e9	−0.693167	−0.346583	+	0.938019i	$0.612658\pi$
$642$	0	0
$643$	−4.44867e9	−0.659921	−0.329961	−	0.943995i	$-0.607035\pi$
$643$	−4.44867e9	−0.659921	−0.329961	+	0.943995i	$0.607035\pi$
$644$	2.65788e7	0.00392135
$645$	0	0
$646$	2.62692e9	0.383384
$647$	−5.26642e9	−0.764453	−0.382226	−	0.924069i	$-0.624843\pi$
$647$	−5.26642e9	−0.764453	−0.382226	+	0.924069i	$0.624843\pi$
$648$	0	0
$649$	7.59767e9	1.09100
$650$	0	0
$651$	0	0
$652$	1.17622e9	0.166196
$653$	6.40447e9	0.900092	0.450046	−	0.893005i	$-0.351408\pi$
$653$	6.40447e9	0.900092	0.450046	+	0.893005i	$0.351408\pi$
$654$	0	0
$655$	0	0
$656$	−9.77101e9	−1.35138
$657$	0	0
$658$	−1.89995e10	−2.59987
$659$	−7.30551e9	−0.994378	−0.497189	−	0.867642i	$-0.665634\pi$
$659$	−7.30551e9	−0.994378	−0.497189	+	0.867642i	$0.665634\pi$
$660$	0	0
$661$	−1.00223e10	−1.34978	−0.674889	−	0.737919i	$-0.735809\pi$
$661$	−1.00223e10	−1.34978	−0.674889	+	0.737919i	$0.735809\pi$
$662$	−1.55083e10	−2.07760
$663$	0	0
$664$	4.82685e7	0.00639846
$665$	0	0
$666$	0	0
$667$	3.08441e7	0.00402469
$668$	−2.44601e9	−0.317498
$669$	0	0
$670$	0	0
$671$	−6.82454e9	−0.872057
$672$	0	0
$673$	3.65625e8	0.0462363	0.0231182	−	0.999733i	$-0.492641\pi$
$673$	3.65625e8	0.0462363	0.0231182	+	0.999733i	$0.492641\pi$
$674$	8.44045e9	1.06183
$675$	0	0
$676$	4.42452e9	0.550874
$677$	−9.06259e9	−1.12251	−0.561257	−	0.827641i	$-0.689682\pi$
$677$	−9.06259e9	−1.12251	−0.561257	+	0.827641i	$0.689682\pi$
$678$	0	0
$679$	−1.00834e10	−1.23613
$680$	0	0
$681$	0	0
$682$	1.52467e10	1.84048
$683$	−5.51289e9	−0.662075	−0.331038	−	0.943618i	$-0.607399\pi$
$683$	−5.51289e9	−0.662075	−0.331038	+	0.943618i	$0.607399\pi$
$684$	0	0
$685$	0	0
$686$	−1.22465e10	−1.44836
$687$	0	0
$688$	1.04092e10	1.21859
$689$	1.08510e10	1.26387
$690$	0	0
$691$	1.71515e9	0.197755	0.0988777	−	0.995100i	$-0.468475\pi$
$691$	1.71515e9	0.197755	0.0988777	+	0.995100i	$0.468475\pi$
$692$	6.18552e9	0.709586
$693$	0	0
$694$	−2.20758e9	−0.250703
$695$	0	0
$696$	0	0
$697$	−6.72700e9	−0.752500
$698$	2.94598e9	0.327896
$699$	0	0
$700$	0	0
$701$	1.10662e10	1.21335	0.606675	−	0.794950i	$-0.292503\pi$
$701$	1.10662e10	1.21335	0.606675	+	0.794950i	$0.292503\pi$
$702$	0	0
$703$	−7.62759e9	−0.828025
$704$	8.61997e9	0.931111
$705$	0	0
$706$	−1.84112e10	−1.96909
$707$	8.72991e9	0.929055
$708$	0	0
$709$	3.69647e9	0.389516	0.194758	−	0.980851i	$-0.437608\pi$
$709$	3.69647e9	0.389516	0.194758	+	0.980851i	$0.437608\pi$
$710$	0	0
$711$	0	0
$712$	−3.42514e8	−0.0355630
$713$	−5.51247e7	−0.00569551
$714$	0	0
$715$	0	0
$716$	−4.81267e8	−0.0489994
$717$	0	0
$718$	2.36721e10	2.38672
$719$	−4.99915e9	−0.501586	−0.250793	−	0.968041i	$-0.580691\pi$
$719$	−4.99915e9	−0.501586	−0.250793	+	0.968041i	$0.580691\pi$
$720$	0	0
$721$	1.71858e10	1.70764
$722$	−1.08219e10	−1.07010
$723$	0	0
$724$	−1.18579e10	−1.16124
$725$	0	0
$726$	0	0
$727$	2.91969e9	0.281817	0.140908	−	0.990023i	$-0.454998\pi$
$727$	2.91969e9	0.281817	0.140908	+	0.990023i	$0.454998\pi$
$728$	−5.64461e8	−0.0542218
$729$	0	0
$730$	0	0
$731$	7.16639e9	0.678562
$732$	0	0
$733$	5.21237e9	0.488845	0.244422	−	0.969669i	$-0.421402\pi$
$733$	5.21237e9	0.488845	0.244422	+	0.969669i	$0.421402\pi$
$734$	−5.98402e9	−0.558543
$735$	0	0
$736$	−5.95795e7	−0.00550839
$737$	3.84761e9	0.354042
$738$	0	0
$739$	5.26869e8	0.0480227	0.0240114	−	0.999712i	$-0.492356\pi$
$739$	5.26869e8	0.0480227	0.0240114	+	0.999712i	$0.492356\pi$
$740$	0	0
$741$	0	0
$742$	−1.59061e10	−1.42938
$743$	1.67008e10	1.49375	0.746875	−	0.664965i	$-0.231553\pi$
$743$	1.67008e10	1.49375	0.746875	+	0.664965i	$0.231553\pi$
$744$	0	0
$745$	0	0
$746$	6.84053e9	0.603259
$747$	0	0
$748$	5.57966e9	0.487475
$749$	2.00436e10	1.74297
$750$	0	0
$751$	−9.03609e8	−0.0778468	−0.0389234	−	0.999242i	$-0.512393\pi$
$751$	−9.03609e8	−0.0778468	−0.0389234	+	0.999242i	$0.512393\pi$
$752$	2.09460e10	1.79614
$753$	0	0
$754$	−2.16165e10	−1.83647
$755$	0	0
$756$	0	0
$757$	1.02084e10	0.855307	0.427654	−	0.903943i	$-0.359340\pi$
$757$	1.02084e10	0.855307	0.427654	+	0.903943i	$0.359340\pi$
$758$	−1.67080e10	−1.39342
$759$	0	0
$760$	0	0
$761$	−1.81052e10	−1.48922	−0.744608	−	0.667502i	$-0.767363\pi$
$761$	−1.81052e10	−1.48922	−0.744608	+	0.667502i	$0.767363\pi$
$762$	0	0
$763$	−1.51899e10	−1.23800
$764$	−2.23927e10	−1.81668
$765$	0	0
$766$	1.19437e10	0.960151
$767$	1.92514e10	1.54056
$768$	0	0
$769$	−4.21234e9	−0.334026	−0.167013	−	0.985955i	$-0.553412\pi$
$769$	−4.21234e9	−0.334026	−0.167013	+	0.985955i	$0.553412\pi$
$770$	0	0
$771$	0	0
$772$	−1.13442e10	−0.887386
$773$	1.99796e10	1.55582	0.777908	−	0.628379i	$-0.216281\pi$
$773$	1.99796e10	1.55582	0.777908	+	0.628379i	$0.216281\pi$
$774$	0	0
$775$	0	0
$776$	−7.29135e8	−0.0560134
$777$	0	0
$778$	−1.97830e10	−1.50614
$779$	−9.19668e9	−0.697028
$780$	0	0
$781$	−1.76072e9	−0.132255
$782$	−3.97353e7	−0.00297135
$783$	0	0
$784$	4.43064e8	0.0328368
$785$	0	0
$786$	0	0
$787$	1.58792e9	0.116123	0.0580613	−	0.998313i	$-0.481508\pi$
$787$	1.58792e9	0.116123	0.0580613	+	0.998313i	$0.481508\pi$
$788$	−9.87747e9	−0.719124
$789$	0	0
$790$	0	0
$791$	1.22495e10	0.880034
$792$	0	0
$793$	−1.72924e10	−1.23140
$794$	−1.87580e10	−1.32989
$795$	0	0
$796$	−1.10765e10	−0.778410
$797$	6.84247e9	0.478750	0.239375	−	0.970927i	$-0.423058\pi$
$797$	6.84247e9	0.478750	0.239375	+	0.970927i	$0.423058\pi$
$798$	0	0
$799$	1.44206e10	1.00016
$800$	0	0
$801$	0	0
$802$	−1.13781e9	−0.0778858
$803$	1.57082e10	1.07059
$804$	0	0
$805$	0	0
$806$	3.86330e10	2.59888
$807$	0	0
$808$	6.31261e8	0.0420987
$809$	−9.65265e9	−0.640954	−0.320477	−	0.947256i	$-0.603843\pi$
$809$	−9.65265e9	−0.640954	−0.320477	+	0.947256i	$0.603843\pi$
$810$	0	0
$811$	−1.52539e9	−0.100417	−0.0502086	−	0.998739i	$-0.515989\pi$
$811$	−1.52539e9	−0.100417	−0.0502086	+	0.998739i	$0.515989\pi$
$812$	1.60871e10	1.05447
$813$	0	0
$814$	−3.19115e10	−2.07378
$815$	0	0
$816$	0	0
$817$	9.79739e9	0.628540
$818$	−7.81102e9	−0.498966
$819$	0	0
$820$	0	0
$821$	−2.60390e10	−1.64219	−0.821096	−	0.570790i	$-0.806637\pi$
$821$	−2.60390e10	−1.64219	−0.821096	+	0.570790i	$0.806637\pi$
$822$	0	0
$823$	1.13082e10	0.707120	0.353560	−	0.935412i	$-0.384971\pi$
$823$	1.13082e10	0.707120	0.353560	+	0.935412i	$0.384971\pi$
$824$	1.24270e9	0.0773789
$825$	0	0
$826$	−2.82199e10	−1.74231
$827$	1.80755e10	1.11127	0.555637	−	0.831425i	$-0.312474\pi$
$827$	1.80755e10	1.11127	0.555637	+	0.831425i	$0.312474\pi$
$828$	0	0
$829$	−2.51806e10	−1.53506	−0.767531	−	0.641012i	$-0.778515\pi$
$829$	−2.51806e10	−1.53506	−0.767531	+	0.641012i	$0.778515\pi$
$830$	0	0
$831$	0	0
$832$	2.18417e10	1.31479
$833$	3.05034e8	0.0182848
$834$	0	0
$835$	0	0
$836$	7.62813e9	0.451540
$837$	0	0
$838$	3.31056e10	1.94333
$839$	2.72950e10	1.59557	0.797787	−	0.602939i	$-0.206004\pi$
$839$	2.72950e10	1.59557	0.797787	+	0.602939i	$0.206004\pi$
$840$	0	0
$841$	1.41888e9	0.0822544
$842$	−7.48885e9	−0.432338
$843$	0	0
$844$	−1.27784e10	−0.731606
$845$	0	0
$846$	0	0
$847$	−4.00775e9	−0.226625
$848$	1.75357e10	0.987499
$849$	0	0
$850$	0	0
$851$	1.15376e8	0.00641746
$852$	0	0
$853$	1.29870e10	0.716453	0.358226	−	0.933635i	$-0.383382\pi$
$853$	1.29870e10	0.716453	0.358226	+	0.933635i	$0.383382\pi$
$854$	2.53483e10	1.39266
$855$	0	0
$856$	1.44936e9	0.0789800
$857$	8.31784e9	0.451417	0.225708	−	0.974195i	$-0.427530\pi$
$857$	8.31784e9	0.451417	0.225708	+	0.974195i	$0.427530\pi$
$858$	0	0
$859$	−8.94316e9	−0.481410	−0.240705	−	0.970598i	$-0.577379\pi$
$859$	−8.94316e9	−0.481410	−0.240705	+	0.970598i	$0.577379\pi$
$860$	0	0
$861$	0	0
$862$	−2.69030e10	−1.43062
$863$	4.59600e9	0.243412	0.121706	−	0.992566i	$-0.461163\pi$
$863$	4.59600e9	0.243412	0.121706	+	0.992566i	$0.461163\pi$
$864$	0	0
$865$	0	0
$866$	−1.74563e10	−0.913357
$867$	0	0
$868$	−2.87510e10	−1.49222
$869$	8.39578e9	0.434002
$870$	0	0
$871$	9.74928e9	0.499930
$872$	−1.09839e9	−0.0560980
$873$	0	0
$874$	−5.43234e7	−0.00275231
$875$	0	0
$876$	0	0
$877$	−2.17651e10	−1.08959	−0.544795	−	0.838569i	$-0.683392\pi$
$877$	−2.17651e10	−1.08959	−0.544795	+	0.838569i	$0.683392\pi$
$878$	−3.13282e10	−1.56209
$879$	0	0
$880$	0	0
$881$	−3.32060e10	−1.63607	−0.818034	−	0.575170i	$-0.804936\pi$
$881$	−3.32060e10	−1.63607	−0.818034	+	0.575170i	$0.804936\pi$
$882$	0	0
$883$	3.47892e10	1.70052	0.850261	−	0.526361i	$-0.176444\pi$
$883$	3.47892e10	1.70052	0.850261	+	0.526361i	$0.176444\pi$
$884$	1.41380e10	0.688345
$885$	0	0
$886$	−5.24693e10	−2.53447
$887$	−3.48616e10	−1.67731	−0.838657	−	0.544660i	$-0.816659\pi$
$887$	−3.48616e10	−1.67731	−0.838657	+	0.544660i	$0.816659\pi$
$888$	0	0
$889$	2.54504e10	1.21490
$890$	0	0
$891$	0	0
$892$	1.75371e10	0.827336
$893$	1.97149e10	0.926432
$894$	0	0
$895$	0	0
$896$	−1.88422e9	−0.0875092
$897$	0	0
$898$	−2.56994e10	−1.18429
$899$	−3.33648e10	−1.53155
$900$	0	0
$901$	1.20727e10	0.549879
$902$	−3.84761e10	−1.74570
$903$	0	0
$904$	8.85761e8	0.0398774
$905$	0	0
$906$	0	0
$907$	1.82643e10	0.812788	0.406394	−	0.913698i	$-0.366786\pi$
$907$	1.82643e10	0.812788	0.406394	+	0.913698i	$0.366786\pi$
$908$	−1.28379e10	−0.569108
$909$	0	0
$910$	0	0
$911$	1.62009e10	0.709945	0.354973	−	0.934877i	$-0.384490\pi$
$911$	1.62009e10	0.709945	0.354973	+	0.934877i	$0.384490\pi$
$912$	0	0
$913$	−2.89785e9	−0.126017
$914$	−1.67024e10	−0.723548
$915$	0	0
$916$	3.74902e9	0.161170
$917$	8.20393e9	0.351341
$918$	0	0
$919$	−2.47808e10	−1.05320	−0.526599	−	0.850114i	$-0.676533\pi$
$919$	−2.47808e10	−1.05320	−0.526599	+	0.850114i	$0.676533\pi$
$920$	0	0
$921$	0	0
$922$	3.82418e10	1.60687
$923$	−4.46141e9	−0.186753
$924$	0	0
$925$	0	0
$926$	−3.98316e10	−1.64850
$927$	0	0
$928$	−3.60612e10	−1.48123
$929$	1.75952e9	0.0720011	0.0360005	−	0.999352i	$-0.488538\pi$
$929$	1.75952e9	0.0720011	0.0360005	+	0.999352i	$0.488538\pi$
$930$	0	0
$931$	4.17021e8	0.0169369
$932$	1.80494e10	0.730311
$933$	0	0
$934$	1.10248e10	0.442747
$935$	0	0
$936$	0	0
$937$	−4.17587e10	−1.65828	−0.829141	−	0.559040i	$-0.811170\pi$
$937$	−4.17587e10	−1.65828	−0.829141	+	0.559040i	$0.811170\pi$
$938$	−1.42911e10	−0.565401
$939$	0	0
$940$	0	0
$941$	2.01970e10	0.790173	0.395087	−	0.918644i	$-0.370715\pi$
$941$	2.01970e10	0.790173	0.395087	+	0.918644i	$0.370715\pi$
$942$	0	0
$943$	1.39111e8	0.00540219
$944$	3.11111e10	1.20369
$945$	0	0
$946$	4.09893e10	1.57417
$947$	−2.65395e10	−1.01547	−0.507736	−	0.861513i	$-0.669517\pi$
$947$	−2.65395e10	−1.01547	−0.507736	+	0.861513i	$0.669517\pi$
$948$	0	0
$949$	3.98023e10	1.51174
$950$	0	0
$951$	0	0
$952$	−6.28014e8	−0.0235907
$953$	−3.15558e9	−0.118101	−0.0590505	−	0.998255i	$-0.518807\pi$
$953$	−3.15558e9	−0.118101	−0.0590505	+	0.998255i	$0.518807\pi$
$954$	0	0
$955$	0	0
$956$	3.87530e10	1.43451
$957$	0	0
$958$	1.18807e10	0.436580
$959$	3.25387e10	1.19134
$960$	0	0
$961$	3.21171e10	1.16736
$962$	−8.08591e10	−2.92830
$963$	0	0
$964$	2.66256e10	0.957259
$965$	0	0
$966$	0	0
$967$	1.84578e10	0.656428	0.328214	−	0.944603i	$-0.393553\pi$
$967$	1.84578e10	0.656428	0.328214	+	0.944603i	$0.393553\pi$
$968$	−2.89801e8	−0.0102692
$969$	0	0
$970$	0	0
$971$	3.98868e10	1.39818	0.699089	−	0.715035i	$-0.253589\pi$
$971$	3.98868e10	1.39818	0.699089	+	0.715035i	$0.253589\pi$
$972$	0	0
$973$	−9.43053e9	−0.328202
$974$	−6.72874e10	−2.33334
$975$	0	0
$976$	−2.79453e10	−0.962131
$977$	−3.05548e10	−1.04821	−0.524106	−	0.851653i	$-0.675600\pi$
$977$	−3.05548e10	−1.04821	−0.524106	+	0.851653i	$0.675600\pi$
$978$	0	0
$979$	2.05632e10	0.700408
$980$	0	0
$981$	0	0
$982$	−4.07663e10	−1.37376
$983$	−3.06865e10	−1.03041	−0.515206	−	0.857067i	$-0.672284\pi$
$983$	−3.06865e10	−1.03041	−0.515206	+	0.857067i	$0.672284\pi$
$984$	0	0
$985$	0	0
$986$	−2.40503e10	−0.799007
$987$	0	0
$988$	1.93285e10	0.637602
$989$	−1.48197e8	−0.00487139
$990$	0	0
$991$	5.44706e10	1.77789	0.888944	−	0.458016i	$-0.151440\pi$
$991$	5.44706e10	1.77789	0.888944	+	0.458016i	$0.151440\pi$
$992$	6.44486e10	2.09615
$993$	0	0
$994$	6.53983e9	0.211210
$995$	0	0
$996$	0	0
$997$	−1.17569e10	−0.375717	−0.187858	−	0.982196i	$-0.560155\pi$
$997$	−1.17569e10	−0.375717	−0.187858	+	0.982196i	$0.560155\pi$
$998$	1.75551e10	0.559045
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.8.a.ba.1.4		4
3.2	odd	2	inner	225.8.a.ba.1.2		4
5.2	odd	4		45.8.b.c.19.4	yes	4
5.3	odd	4		45.8.b.c.19.2	yes	4
5.4	even	2	inner	225.8.a.ba.1.1		4
15.2	even	4		45.8.b.c.19.1	✓	4
15.8	even	4		45.8.b.c.19.3	yes	4
15.14	odd	2	inner	225.8.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.b.c.19.1	✓	4	15.2	even	4
45.8.b.c.19.2	yes	4	5.3	odd	4
45.8.b.c.19.3	yes	4	15.8	even	4
45.8.b.c.19.4	yes	4	5.2	odd	4
225.8.a.ba.1.1		4	5.4	even	2	inner
225.8.a.ba.1.2		4	3.2	odd	2	inner
225.8.a.ba.1.3		4	15.14	odd	2	inner
225.8.a.ba.1.4		4	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+16.1245 q^{2} +132.000 q^{4} +891.964 q^{7} +64.4981 q^{8} -3872.21 q^{11} -9811.61 q^{13} +14382.5 q^{14} -15856.0 q^{16} -10916.3 q^{17} -14924.0 q^{19} -62437.5 q^{22} +225.743 q^{23} -158207. q^{26} +117739. q^{28} +136634. q^{29} -244192. q^{31} -263926. q^{32} -176020. q^{34} +511095. q^{37} -240642. q^{38} +616234. q^{41} -656486. q^{43} -511132. q^{44} +3640.00 q^{46} -1.32102e6 q^{47} -27943.0 q^{49} -1.29513e6 q^{52} -1.10593e6 q^{53} +57530.0 q^{56} +2.20315e6 q^{58} -1.96210e6 q^{59} +1.76244e6 q^{61} -3.93748e6 q^{62} -2.22611e6 q^{64} -993648. q^{67} -1.44095e6 q^{68} +454708. q^{71} -4.05665e6 q^{73} +8.24117e6 q^{74} -1.96997e6 q^{76} -3.45387e6 q^{77} -2.16822e6 q^{79} +9.93648e6 q^{82} +748371. q^{83} -1.05855e7 q^{86} -249750. q^{88} -5.31046e6 q^{89} -8.75160e6 q^{91} +29798.1 q^{92} -2.13008e7 q^{94} -1.13048e7 q^{97} -450567. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 528 q^{4} - 63424 q^{16} - 59696 q^{19} - 976768 q^{31} - 704080 q^{34} + 14560 q^{46} - 111772 q^{49} + 7049768 q^{61} - 8904448 q^{64} - 7879872 q^{76} - 8672864 q^{79} - 35006400 q^{91} - 85203040 q^{94}+O(q^{100}) \) 4 * q + 528 * q^4 - 63424 * q^16 - 59696 * q^19 - 976768 * q^31 - 704080 * q^34 + 14560 * q^46 - 111772 * q^49 + 7049768 * q^61 - 8904448 * q^64 - 7879872 * q^76 - 8672864 * q^79 - 35006400 * q^91 - 85203040 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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