LMFDB - Embedded newform 225.9.c.d.26.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,9,Mod(26,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.26"); S:= CuspForms(chi, 9); N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	225.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-2524,0,0,6928] 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:	no
Analytic conductor:	$91.6601872638$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 698x^{10} + 179931x^{8} + 20356724x^{6} + 872357011x^{4} + 2973132090x^{2} + 1458246969 $ x^12 + 698x^10 + 179931x^8 + 20356724x^6 + 872357011x^4 + 2973132090*x^2 + 1458246969
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{25}\cdot 5^{12} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.6
Root			$-14.9768i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.9.c.d.26.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-7.31273i q^{2} +202.524 q^{4} +3662.21 q^{7} -3353.06i q^{8} +13264.5i q^{11} -21608.3 q^{13} -26780.7i q^{14} +27326.1 q^{16} -117206. i q^{17} -255517. q^{19} +96999.9 q^{22} -421328. i q^{23} +158015. i q^{26} +741685. q^{28} -610903. i q^{29} -980464. q^{31} -1.05821e6i q^{32} -857097. q^{34} -300222. q^{37} +1.86853e6i q^{38} +1.06451e6i q^{41} +1.60840e6 q^{43} +2.68638e6i q^{44} -3.08106e6 q^{46} -757324. i q^{47} +7.64696e6 q^{49} -4.37619e6 q^{52} -3.30314e6i q^{53} -1.22796e7i q^{56} -4.46737e6 q^{58} -2.09416e7i q^{59} +2.16083e7 q^{61} +7.16987e6i q^{62} -742936. q^{64} +1.46747e7 q^{67} -2.37371e7i q^{68} +1.44676e7i q^{71} -2.16541e7 q^{73} +2.19544e6i q^{74} -5.17483e7 q^{76} +4.85774e7i q^{77} +4.05718e7 q^{79} +7.78449e6 q^{82} -4.84469e7i q^{83} -1.17618e7i q^{86} +4.44768e7 q^{88} +3.57780e7i q^{89} -7.91339e7 q^{91} -8.53291e7i q^{92} -5.53811e6 q^{94} +6.01429e7 q^{97} -5.59202e7i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2524 q^{4} + 6928 q^{7} - 124408 q^{13} + 297572 q^{16} + 124960 q^{19} + 1370512 q^{22} - 1114496 q^{28} - 620968 q^{31} + 7486496 q^{34} + 11533176 q^{37} - 14405296 q^{43} - 7161768 q^{46} + 21010156 q^{49}+ \cdots + 302816184 q^{97}+O(q^{100}) $ 12 * q - 2524 * q^4 + 6928 * q^7 - 124408 * q^13 + 297572 * q^16 + 124960 * q^19 + 1370512 * q^22 - 1114496 * q^28 - 620968 * q^31 + 7486496 * q^34 + 11533176 * q^37 - 14405296 * q^43 - 7161768 * q^46 + 21010156 * q^49 + 10459176 * q^52 - 22630504 * q^58 + 28453096 * q^61 - 169389948 * q^64 - 47768816 * q^67 - 38412808 * q^73 - 255673920 * q^76 - 63809512 * q^79 - 165335176 * q^82 - 679824192 * q^88 - 296849288 * q^91 + 25601888 * q^94 + 302816184 * q^97

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 $

$2$	− 7.31273i	− 0.457046i	−0.973539	−	0.228523i	$-0.926610\pi$
$2$	− 7.31273i	− 0.457046i	0.973539	−	0.228523i	$-0.0733895\pi$
$3$	0	0
$3$	0	0
$4$	202.524	0.791109
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3662.21	1.52528	0.762642	−	0.646821i	$-0.223902\pi$
$7$	3662.21	1.52528	0.762642	+	0.646821i	$0.223902\pi$
$8$	− 3353.06i	− 0.818619i
$9$	0	0
$10$	0	0
$11$	13264.5i	0.905985i	0.891514	+	0.452992i	$0.149644\pi$
$11$	13264.5i	0.905985i	−0.891514	+	0.452992i	$0.850356\pi$
$12$	0	0
$13$	−21608.3	−0.756565	−0.378283	−	0.925690i	$-0.623485\pi$
$13$	−21608.3	−0.756565	−0.378283	+	0.925690i	$0.623485\pi$
$14$	− 26780.7i	− 0.697124i
$15$	0	0
$16$	27326.1	0.416963
$17$	− 117206.i	− 1.40331i	−0.712515	−	0.701657i	$-0.752444\pi$
$17$	− 117206.i	− 1.40331i	0.712515	−	0.701657i	$-0.247556\pi$
$18$	0	0
$19$	−255517.	−1.96067	−0.980337	−	0.197330i	$-0.936773\pi$
$19$	−255517.	−1.96067	−0.980337	+	0.197330i	$0.936773\pi$
$20$	0	0
$21$	0	0
$22$	96999.9	0.414076
$23$	− 421328.i	− 1.50560i	−0.658250	−	0.752799i	$-0.728703\pi$
$23$	− 421328.i	− 1.50560i	0.658250	−	0.752799i	$-0.271297\pi$
$24$	0	0
$25$	0	0
$26$	158015.i	0.345785i
$27$	0	0
$28$	741685.	1.20667
$29$	− 610903.i	− 0.863735i	−0.901937	−	0.431867i	$-0.857855\pi$
$29$	− 610903.i	− 0.863735i	0.901937	−	0.431867i	$-0.142145\pi$
$30$	0	0
$31$	−980464.	−1.06166	−0.530829	−	0.847479i	$-0.678119\pi$
$31$	−980464.	−1.06166	−0.530829	+	0.847479i	$0.678119\pi$
$32$	− 1.05821e6i	− 1.00919i
$33$	0	0
$34$	−857097.	−0.641378
$35$	0	0
$36$	0	0
$37$	−300222.	−0.160190	−0.0800950	−	0.996787i	$-0.525522\pi$
$37$	−300222.	−0.160190	−0.0800950	+	0.996787i	$0.525522\pi$
$38$	1.86853e6i	0.896118i
$39$	0	0
$40$	0	0
$41$	1.06451e6i	0.376717i	0.982100	+	0.188359i	$0.0603167\pi$
$41$	1.06451e6i	0.376717i	−0.982100	+	0.188359i	$0.939683\pi$
$42$	0	0
$43$	1.60840e6	0.470458	0.235229	−	0.971940i	$-0.424416\pi$
$43$	1.60840e6	0.470458	0.235229	+	0.971940i	$0.424416\pi$
$44$	2.68638e6i	0.716733i
$45$	0	0
$46$	−3.08106e6	−0.688127
$47$	− 757324.i	− 0.155200i	−0.996985	−	0.0775998i	$-0.975274\pi$
$47$	− 757324.i	− 0.155200i	0.996985	−	0.0775998i	$-0.0247256\pi$
$48$	0	0
$49$	7.64696e6	1.32649
$50$	0	0
$51$	0	0
$52$	−4.37619e6	−0.598526
$53$	− 3.30314e6i	− 0.418623i	−0.977849	−	0.209311i	$-0.932878\pi$
$53$	− 3.30314e6i	− 0.418623i	0.977849	−	0.209311i	$-0.0671222\pi$
$54$	0	0
$55$	0	0
$56$	− 1.22796e7i	− 1.24863i
$57$	0	0
$58$	−4.46737e6	−0.394766
$59$	− 2.09416e7i	− 1.72823i	−0.503293	−	0.864116i	$-0.667878\pi$
$59$	− 2.09416e7i	− 1.72823i	0.503293	−	0.864116i	$-0.332122\pi$
$60$	0	0
$61$	2.16083e7	1.56063	0.780316	−	0.625385i	$-0.215058\pi$
$61$	2.16083e7	1.56063	0.780316	+	0.625385i	$0.215058\pi$
$62$	7.16987e6i	0.485226i
$63$	0	0
$64$	−742936.	−0.0442824
$65$	0	0
$66$	0	0
$67$	1.46747e7	0.728231	0.364115	−	0.931354i	$-0.381371\pi$
$67$	1.46747e7	0.728231	0.364115	+	0.931354i	$0.381371\pi$
$68$	− 2.37371e7i	− 1.11017i
$69$	0	0
$70$	0	0
$71$	1.44676e7i	0.569329i	0.958627	+	0.284665i	$0.0918823\pi$
$71$	1.44676e7i	0.569329i	−0.958627	+	0.284665i	$0.908118\pi$
$72$	0	0
$73$	−2.16541e7	−0.762516	−0.381258	−	0.924469i	$-0.624509\pi$
$73$	−2.16541e7	−0.762516	−0.381258	+	0.924469i	$0.624509\pi$
$74$	2.19544e6i	0.0732142i
$75$	0	0
$76$	−5.17483e7	−1.55111
$77$	4.85774e7i	1.38188i
$78$	0	0
$79$	4.05718e7	1.04164	0.520818	−	0.853667i	$-0.325627\pi$
$79$	4.05718e7	1.04164	0.520818	+	0.853667i	$0.325627\pi$
$80$	0	0
$81$	0	0
$82$	7.78449e6	0.172177
$83$	− 4.84469e7i	− 1.02083i	−0.859928	−	0.510416i	$-0.829492\pi$
$83$	− 4.84469e7i	− 1.02083i	0.859928	−	0.510416i	$-0.170508\pi$
$84$	0	0
$85$	0	0
$86$	− 1.17618e7i	− 0.215021i
$87$	0	0
$88$	4.44768e7	0.741656
$89$	3.57780e7i	0.570239i	0.958492	+	0.285119i	$0.0920332\pi$
$89$	3.57780e7i	0.570239i	−0.958492	+	0.285119i	$0.907967\pi$
$90$	0	0
$91$	−7.91339e7	−1.15398
$92$	− 8.53291e7i	− 1.19109i
$93$	0	0
$94$	−5.53811e6	−0.0709333
$95$	0	0
$96$	0	0
$97$	6.01429e7	0.679356	0.339678	−	0.940542i	$-0.389682\pi$
$97$	6.01429e7	0.679356	0.339678	+	0.940542i	$0.389682\pi$
$98$	− 5.59202e7i	− 0.606267i
$99$	0	0
$100$	0	0
$101$	− 1.09720e8i	− 1.05438i	−0.849746	−	0.527192i	$-0.823245\pi$
$101$	− 1.09720e8i	− 1.05438i	0.849746	−	0.527192i	$-0.176755\pi$
$102$	0	0
$103$	−7.14368e6	−0.0634707	−0.0317353	−	0.999496i	$-0.510103\pi$
$103$	−7.14368e6	−0.0634707	−0.0317353	+	0.999496i	$0.510103\pi$
$104$	7.24538e7i	0.619338i
$105$	0	0
$106$	−2.41549e7	−0.191330
$107$	1.37428e8i	1.04843i	0.851587	+	0.524214i	$0.175641\pi$
$107$	1.37428e8i	1.04843i	−0.851587	+	0.524214i	$0.824359\pi$
$108$	0	0
$109$	−6.92546e7	−0.490617	−0.245309	−	0.969445i	$-0.578889\pi$
$109$	−6.92546e7	−0.490617	−0.245309	+	0.969445i	$0.578889\pi$
$110$	0	0
$111$	0	0
$112$	1.00074e8	0.635988
$113$	− 1.50284e8i	− 0.921721i	−0.887473	−	0.460861i	$-0.847541\pi$
$113$	− 1.50284e8i	− 0.921721i	0.887473	−	0.460861i	$-0.152459\pi$
$114$	0	0
$115$	0	0
$116$	− 1.23723e8i	− 0.683309i
$117$	0	0
$118$	−1.53140e8	−0.789881
$119$	− 4.29233e8i	− 2.14045i
$120$	0	0
$121$	3.84112e7	0.179191
$122$	− 1.58015e8i	− 0.713280i
$123$	0	0
$124$	−1.98567e8	−0.839888
$125$	0	0
$126$	0	0
$127$	−3.74962e8	−1.44136	−0.720679	−	0.693269i	$-0.756170\pi$
$127$	−3.74962e8	−1.44136	−0.720679	+	0.693269i	$0.756170\pi$
$128$	− 2.65469e8i	− 0.988951i
$129$	0	0
$130$	0	0
$131$	− 4.86380e8i	− 1.65155i	−0.564003	−	0.825773i	$-0.690739\pi$
$131$	− 4.86380e8i	− 1.65155i	0.564003	−	0.825773i	$-0.309261\pi$
$132$	0	0
$133$	−9.35756e8	−2.99059
$134$	− 1.07312e8i	− 0.332835i
$135$	0	0
$136$	−3.93000e8	−1.14878
$137$	2.90334e8i	0.824168i	0.911146	+	0.412084i	$0.135199\pi$
$137$	2.90334e8i	0.824168i	−0.911146	+	0.412084i	$0.864801\pi$
$138$	0	0
$139$	1.58457e6	0.00424476	0.00212238	−	0.999998i	$-0.499324\pi$
$139$	1.58457e6	0.00424476	0.00212238	+	0.999998i	$0.499324\pi$
$140$	0	0
$141$	0	0
$142$	1.05798e8	0.260210
$143$	− 2.86623e8i	− 0.685437i
$144$	0	0
$145$	0	0
$146$	1.58351e8i	0.348505i
$147$	0	0
$148$	−6.08021e7	−0.126728
$149$	6.74094e8i	1.36765i	0.729645	+	0.683826i	$0.239685\pi$
$149$	6.74094e8i	1.36765i	−0.729645	+	0.683826i	$0.760315\pi$
$150$	0	0
$151$	1.30910e8	0.251805	0.125902	−	0.992043i	$-0.459817\pi$
$151$	1.30910e8	0.251805	0.125902	+	0.992043i	$0.459817\pi$
$152$	8.56764e8i	1.60504i
$153$	0	0
$154$	3.55234e8	0.631584
$155$	0	0
$156$	0	0
$157$	6.84598e8	1.12677	0.563387	−	0.826193i	$-0.309498\pi$
$157$	6.84598e8	1.12677	0.563387	+	0.826193i	$0.309498\pi$
$158$	− 2.96691e8i	− 0.476075i
$159$	0	0
$160$	0	0
$161$	− 1.54299e9i	− 2.29647i
$162$	0	0
$163$	−1.45371e8	−0.205934	−0.102967	−	0.994685i	$-0.532834\pi$
$163$	−1.45371e8	−0.205934	−0.102967	+	0.994685i	$0.532834\pi$
$164$	2.15589e8i	0.298024i
$165$	0	0
$166$	−3.54279e8	−0.466566
$167$	− 2.37839e8i	− 0.305786i	−0.988243	−	0.152893i	$-0.951141\pi$
$167$	− 2.37839e8i	− 0.305786i	0.988243	−	0.152893i	$-0.0488589\pi$
$168$	0	0
$169$	−3.48814e8	−0.427609
$170$	0	0
$171$	0	0
$172$	3.25740e8	0.372184
$173$	− 1.18265e9i	− 1.32029i	−0.751137	−	0.660146i	$-0.770494\pi$
$173$	− 1.18265e9i	− 1.32029i	0.751137	−	0.660146i	$-0.229506\pi$
$174$	0	0
$175$	0	0
$176$	3.62468e8i	0.377763i
$177$	0	0
$178$	2.61635e8	0.260625
$179$	9.82068e8i	0.956598i	0.878197	+	0.478299i	$0.158747\pi$
$179$	9.82068e8i	0.956598i	−0.878197	+	0.478299i	$0.841253\pi$
$180$	0	0
$181$	−2.06622e9	−1.92514	−0.962569	−	0.271036i	$-0.912634\pi$
$181$	−2.06622e9	−1.92514	−0.962569	+	0.271036i	$0.912634\pi$
$182$	5.78685e8i	0.527420i
$183$	0	0
$184$	−1.41274e9	−1.23251
$185$	0	0
$186$	0	0
$187$	1.55468e9	1.27138
$188$	− 1.53376e8i	− 0.122780i
$189$	0	0
$190$	0	0
$191$	− 5.26889e8i	− 0.395900i	−0.980212	−	0.197950i	$-0.936572\pi$
$191$	− 5.26889e8i	− 0.395900i	0.980212	−	0.197950i	$-0.0634284\pi$
$192$	0	0
$193$	−2.06017e9	−1.48482	−0.742412	−	0.669944i	$-0.766318\pi$
$193$	−2.06017e9	−1.48482	−0.742412	+	0.669944i	$0.766318\pi$
$194$	− 4.39809e8i	− 0.310497i
$195$	0	0
$196$	1.54869e9	1.04940
$197$	− 7.95356e8i	− 0.528077i	−0.964512	−	0.264038i	$-0.914945\pi$
$197$	− 7.95356e8i	− 0.528077i	0.964512	−	0.264038i	$-0.0850545\pi$
$198$	0	0
$199$	6.52669e8	0.416180	0.208090	−	0.978110i	$-0.433275\pi$
$199$	6.52669e8	0.416180	0.208090	+	0.978110i	$0.433275\pi$
$200$	0	0
$201$	0	0
$202$	−8.02349e8	−0.481901
$203$	− 2.23725e9i	− 1.31744i
$204$	0	0
$205$	0	0
$206$	5.22398e7i	0.0290090i
$207$	0	0
$208$	−5.90470e8	−0.315460
$209$	− 3.38931e9i	− 1.77634i
$210$	0	0
$211$	1.73985e7	0.00877774	0.00438887	−	0.999990i	$-0.498603\pi$
$211$	1.73985e7	0.00877774	0.00438887	+	0.999990i	$0.498603\pi$
$212$	− 6.68964e8i	− 0.331176i
$213$	0	0
$214$	1.00497e9	0.479179
$215$	0	0
$216$	0	0
$217$	−3.59066e9	−1.61933
$218$	5.06440e8i	0.224234i
$219$	0	0
$220$	0	0
$221$	2.53262e9i	1.06170i
$222$	0	0
$223$	4.65947e9	1.88416	0.942079	−	0.335392i	$-0.108869\pi$
$223$	4.65947e9	1.88416	0.942079	+	0.335392i	$0.108869\pi$
$224$	− 3.87539e9i	− 1.53930i
$225$	0	0
$226$	−1.09899e9	−0.421269
$227$	1.44662e9i	0.544819i	0.962181	+	0.272410i	$0.0878206\pi$
$227$	1.44662e9i	0.544819i	−0.962181	+	0.272410i	$0.912179\pi$
$228$	0	0
$229$	2.56679e8	0.0933358	0.0466679	−	0.998910i	$-0.485140\pi$
$229$	2.56679e8	0.0933358	0.0466679	+	0.998910i	$0.485140\pi$
$230$	0	0
$231$	0	0
$232$	−2.04840e9	−0.707069
$233$	3.63538e9i	1.23346i	0.787173	+	0.616732i	$0.211544\pi$
$233$	3.63538e9i	1.23346i	−0.787173	+	0.616732i	$0.788456\pi$
$234$	0	0
$235$	0	0
$236$	− 4.24118e9i	− 1.36722i
$237$	0	0
$238$	−3.13887e9	−0.978285
$239$	5.83715e9i	1.78900i	0.447072	+	0.894498i	$0.352467\pi$
$239$	5.83715e9i	1.78900i	−0.447072	+	0.894498i	$0.647533\pi$
$240$	0	0
$241$	3.30480e9	0.979663	0.489831	−	0.871817i	$-0.337058\pi$
$241$	3.30480e9	0.979663	0.489831	+	0.871817i	$0.337058\pi$
$242$	− 2.80891e8i	− 0.0818986i
$243$	0	0
$244$	4.37619e9	1.23463
$245$	0	0
$246$	0	0
$247$	5.52128e9	1.48338
$248$	3.28756e9i	0.869093i
$249$	0	0
$250$	0	0
$251$	4.21438e8i	0.106179i	0.998590	+	0.0530895i	$0.0169069\pi$
$251$	4.21438e8i	0.106179i	−0.998590	+	0.0530895i	$0.983093\pi$
$252$	0	0
$253$	5.58872e9	1.36405
$254$	2.74199e9i	0.658767i
$255$	0	0
$256$	−2.13150e9	−0.496278
$257$	− 3.67809e9i	− 0.843121i	−0.906800	−	0.421561i	$-0.861482\pi$
$257$	− 3.67809e9i	− 0.843121i	0.906800	−	0.421561i	$-0.138518\pi$
$258$	0	0
$259$	−1.09948e9	−0.244335
$260$	0	0
$261$	0	0
$262$	−3.55676e9	−0.754831
$263$	3.78322e9i	0.790750i	0.918520	+	0.395375i	$0.129385\pi$
$263$	3.78322e9i	0.790750i	−0.918520	+	0.395375i	$0.870615\pi$
$264$	0	0
$265$	0	0
$266$	6.84293e9i	1.36683i
$267$	0	0
$268$	2.97197e9	0.576110
$269$	− 4.61132e9i	− 0.880676i	−0.897832	−	0.440338i	$-0.854859\pi$
$269$	− 4.61132e9i	− 0.880676i	0.897832	−	0.440338i	$-0.145141\pi$
$270$	0	0
$271$	5.25945e9	0.975132	0.487566	−	0.873086i	$-0.337885\pi$
$271$	5.25945e9	0.975132	0.487566	+	0.873086i	$0.337885\pi$
$272$	− 3.20279e9i	− 0.585131i
$273$	0	0
$274$	2.12313e9	0.376682
$275$	0	0
$276$	0	0
$277$	6.49623e9	1.10342	0.551712	−	0.834035i	$-0.313975\pi$
$277$	6.49623e9	1.10342	0.551712	+	0.834035i	$0.313975\pi$
$278$	− 1.15875e7i	− 0.00194005i
$279$	0	0
$280$	0	0
$281$	1.12110e10i	1.79812i	0.437826	+	0.899060i	$0.355749\pi$
$281$	1.12110e10i	1.79812i	−0.437826	+	0.899060i	$0.644251\pi$
$282$	0	0
$283$	4.03832e9	0.629585	0.314793	−	0.949160i	$-0.398065\pi$
$283$	4.03832e9	0.629585	0.314793	+	0.949160i	$0.398065\pi$
$284$	2.93004e9i	0.450402i
$285$	0	0
$286$	−2.09600e9	−0.313276
$287$	3.89847e9i	0.574601i
$288$	0	0
$289$	−6.76153e9	−0.969290
$290$	0	0
$291$	0	0
$292$	−4.38548e9	−0.603234
$293$	− 9.12329e9i	− 1.23789i	−0.785435	−	0.618944i	$-0.787561\pi$
$293$	− 9.12329e9i	− 1.23789i	0.785435	−	0.618944i	$-0.212439\pi$
$294$	0	0
$295$	0	0
$296$	1.00666e9i	0.131135i
$297$	0	0
$298$	4.92947e9	0.625079
$299$	9.10417e9i	1.13908i
$300$	0	0
$301$	5.89030e9	0.717582
$302$	− 9.57307e8i	− 0.115086i
$303$	0	0
$304$	−6.98229e9	−0.817529
$305$	0	0
$306$	0	0
$307$	1.19827e9	0.134896	0.0674481	−	0.997723i	$-0.478514\pi$
$307$	1.19827e9	0.134896	0.0674481	+	0.997723i	$0.478514\pi$
$308$	9.83810e9i	1.09322i
$309$	0	0
$310$	0	0
$311$	7.25355e9i	0.775370i	0.921792	+	0.387685i	$0.126725\pi$
$311$	7.25355e9i	0.775370i	−0.921792	+	0.387685i	$0.873275\pi$
$312$	0	0
$313$	−7.48886e9	−0.780258	−0.390129	−	0.920760i	$-0.627570\pi$
$313$	−7.48886e9	−0.780258	−0.390129	+	0.920760i	$0.627570\pi$
$314$	− 5.00628e9i	− 0.514987i
$315$	0	0
$316$	8.21677e9	0.824049
$317$	2.06905e9i	0.204897i	0.994738	+	0.102448i	$0.0326676\pi$
$317$	2.06905e9i	0.204897i	−0.994738	+	0.102448i	$0.967332\pi$
$318$	0	0
$319$	8.10334e9	0.782531
$320$	0	0
$321$	0	0
$322$	−1.12835e10	−1.04959
$323$	2.99482e10i	2.75144i
$324$	0	0
$325$	0	0
$326$	1.06306e9i	0.0941211i
$327$	0	0
$328$	3.56938e9	0.308388
$329$	− 2.77348e9i	− 0.236723i
$330$	0	0
$331$	−1.57907e9	−0.131549	−0.0657747	−	0.997834i	$-0.520952\pi$
$331$	−1.57907e9	−0.131549	−0.0657747	+	0.997834i	$0.520952\pi$
$332$	− 9.81166e9i	− 0.807589i
$333$	0	0
$334$	−1.73925e9	−0.139758
$335$	0	0
$336$	0	0
$337$	−7.38336e9	−0.572446	−0.286223	−	0.958163i	$-0.592400\pi$
$337$	−7.38336e9	−0.572446	−0.286223	+	0.958163i	$0.592400\pi$
$338$	2.55078e9i	0.195437i
$339$	0	0
$340$	0	0
$341$	− 1.30054e10i	− 0.961847i
$342$	0	0
$343$	6.89287e9	0.497993
$344$	− 5.39307e9i	− 0.385126i
$345$	0	0
$346$	−8.64836e9	−0.603434
$347$	− 1.61084e10i	− 1.11105i	−0.831500	−	0.555525i	$-0.812517\pi$
$347$	− 1.61084e10i	− 1.11105i	0.831500	−	0.555525i	$-0.187483\pi$
$348$	0	0
$349$	1.35263e10	0.911755	0.455877	−	0.890043i	$-0.349326\pi$
$349$	1.35263e10	0.911755	0.455877	+	0.890043i	$0.349326\pi$
$350$	0	0
$351$	0	0
$352$	1.40367e10	0.914311
$353$	2.27293e10i	1.46382i	0.681402	+	0.731910i	$0.261371\pi$
$353$	2.27293e10i	1.46382i	−0.681402	+	0.731910i	$0.738629\pi$
$354$	0	0
$355$	0	0
$356$	7.24591e9i	0.451121i
$357$	0	0
$358$	7.18160e9	0.437209
$359$	3.00848e10i	1.81121i	0.424118	+	0.905607i	$0.360584\pi$
$359$	3.00848e10i	1.81121i	−0.424118	+	0.905607i	$0.639416\pi$
$360$	0	0
$361$	4.83054e10	2.84424
$362$	1.51097e10i	0.879876i
$363$	0	0
$364$	−1.60265e10	−0.912922
$365$	0	0
$366$	0	0
$367$	−2.14496e10	−1.18237	−0.591187	−	0.806535i	$-0.701340\pi$
$367$	−2.14496e10	−1.18237	−0.591187	+	0.806535i	$0.701340\pi$
$368$	− 1.15133e10i	− 0.627780i
$369$	0	0
$370$	0	0
$371$	− 1.20968e10i	− 0.638519i
$372$	0	0
$373$	−2.84075e10	−1.46757	−0.733783	−	0.679384i	$-0.762247\pi$
$373$	−2.84075e10	−1.46757	−0.733783	+	0.679384i	$0.762247\pi$
$374$	− 1.13690e10i	− 0.581079i
$375$	0	0
$376$	−2.53936e9	−0.127049
$377$	1.32006e10i	0.653472i
$378$	0	0
$379$	−6.45475e9	−0.312840	−0.156420	−	0.987691i	$-0.549995\pi$
$379$	−6.45475e9	−0.312840	−0.156420	+	0.987691i	$0.549995\pi$
$380$	0	0
$381$	0	0
$382$	−3.85299e9	−0.180944
$383$	− 1.84124e10i	− 0.855686i	−0.903853	−	0.427843i	$-0.859274\pi$
$383$	− 1.84124e10i	− 0.855686i	0.903853	−	0.427843i	$-0.140726\pi$
$384$	0	0
$385$	0	0
$386$	1.50655e10i	0.678632i
$387$	0	0
$388$	1.21804e10	0.537445
$389$	− 2.20288e10i	− 0.962037i	−0.876710	−	0.481019i	$-0.840267\pi$
$389$	− 2.20288e10i	− 0.962037i	0.876710	−	0.481019i	$-0.159733\pi$
$390$	0	0
$391$	−4.93823e10	−2.11283
$392$	− 2.56407e10i	− 1.08589i
$393$	0	0
$394$	−5.81623e9	−0.241355
$395$	0	0
$396$	0	0
$397$	1.19815e10	0.482336	0.241168	−	0.970483i	$-0.422470\pi$
$397$	1.19815e10	0.482336	0.241168	+	0.970483i	$0.422470\pi$
$398$	− 4.77279e9i	− 0.190213i
$399$	0	0
$400$	0	0
$401$	− 9.80577e9i	− 0.379231i	−0.981858	−	0.189616i	$-0.939276\pi$
$401$	− 9.80577e9i	− 0.379231i	0.981858	−	0.189616i	$-0.0607242\pi$
$402$	0	0
$403$	2.11861e10	0.803214
$404$	− 2.22208e10i	− 0.834133i
$405$	0	0
$406$	−1.63604e10	−0.602131
$407$	− 3.98230e9i	− 0.145130i
$408$	0	0
$409$	6.12299e9	0.218812	0.109406	−	0.993997i	$-0.465105\pi$
$409$	6.12299e9	0.218812	0.109406	+	0.993997i	$0.465105\pi$
$410$	0	0
$411$	0	0
$412$	−1.44677e9	−0.0502122
$413$	− 7.66925e10i	− 2.63604i
$414$	0	0
$415$	0	0
$416$	2.28661e10i	0.763518i
$417$	0	0
$418$	−2.47851e10	−0.811869
$419$	1.08044e10i	0.350546i	0.984520	+	0.175273i	$0.0560808\pi$
$419$	1.08044e10i	0.350546i	−0.984520	+	0.175273i	$0.943919\pi$
$420$	0	0
$421$	−1.55252e10	−0.494208	−0.247104	−	0.968989i	$-0.579479\pi$
$421$	−1.55252e10	−0.494208	−0.247104	+	0.968989i	$0.579479\pi$
$422$	− 1.27231e8i	− 0.00401183i
$423$	0	0
$424$	−1.10756e10	−0.342692
$425$	0	0
$426$	0	0
$427$	7.91340e10	2.38041
$428$	2.78324e10i	0.829421i
$429$	0	0
$430$	0	0
$431$	− 1.83165e10i	− 0.530802i	−0.964138	−	0.265401i	$-0.914496\pi$
$431$	− 1.83165e10i	− 0.530802i	0.964138	−	0.265401i	$-0.0855044\pi$
$432$	0	0
$433$	−3.10188e10	−0.882415	−0.441208	−	0.897405i	$-0.645450\pi$
$433$	−3.10188e10	−0.882415	−0.441208	+	0.897405i	$0.645450\pi$
$434$	2.62575e10i	0.740108i
$435$	0	0
$436$	−1.40257e10	−0.388132
$437$	1.07657e11i	2.95199i
$438$	0	0
$439$	2.60289e10	0.700805	0.350403	−	0.936599i	$-0.386045\pi$
$439$	2.60289e10	0.700805	0.350403	+	0.936599i	$0.386045\pi$
$440$	0	0
$441$	0	0
$442$	1.85204e10	0.485245
$443$	− 6.47619e10i	− 1.68153i	−0.541401	−	0.840765i	$-0.682106\pi$
$443$	− 6.47619e10i	− 1.68153i	0.541401	−	0.840765i	$-0.317894\pi$
$444$	0	0
$445$	0	0
$446$	− 3.40735e10i	− 0.861146i
$447$	0	0
$448$	−2.72078e9	−0.0675433
$449$	2.72508e10i	0.670493i	0.942130	+	0.335246i	$0.108820\pi$
$449$	2.72508e10i	0.670493i	−0.942130	+	0.335246i	$0.891180\pi$
$450$	0	0
$451$	−1.41203e10	−0.341300
$452$	− 3.04362e10i	− 0.729182i
$453$	0	0
$454$	1.05788e10	0.249007
$455$	0	0
$456$	0	0
$457$	−3.41731e10	−0.783465	−0.391733	−	0.920079i	$-0.628124\pi$
$457$	−3.41731e10	−0.783465	−0.391733	+	0.920079i	$0.628124\pi$
$458$	− 1.87702e9i	− 0.0426587i
$459$	0	0
$460$	0	0
$461$	− 3.70973e10i	− 0.821370i	−0.911777	−	0.410685i	$-0.865290\pi$
$461$	− 3.70973e10i	− 0.821370i	0.911777	−	0.410685i	$-0.134710\pi$
$462$	0	0
$463$	−5.13571e9	−0.111757	−0.0558787	−	0.998438i	$-0.517796\pi$
$463$	−5.13571e9	−0.111757	−0.0558787	+	0.998438i	$0.517796\pi$
$464$	− 1.66936e10i	− 0.360146i
$465$	0	0
$466$	2.65846e10	0.563749
$467$	− 5.10012e9i	− 0.107229i	−0.998562	−	0.0536146i	$-0.982926\pi$
$467$	− 5.10012e9i	− 0.107229i	0.998562	−	0.0536146i	$-0.0170742\pi$
$468$	0	0
$469$	5.37417e10	1.11076
$470$	0	0
$471$	0	0
$472$	−7.02185e10	−1.41476
$473$	2.13347e10i	0.426228i
$474$	0	0
$475$	0	0
$476$	− 8.69301e10i	− 1.69333i
$477$	0	0
$478$	4.26855e10	0.817653
$479$	− 9.29660e9i	− 0.176597i	−0.996094	−	0.0882983i	$-0.971857\pi$
$479$	− 9.29660e9i	− 0.176597i	0.996094	−	0.0882983i	$-0.0281429\pi$
$480$	0	0
$481$	6.48727e9	0.121194
$482$	− 2.41671e10i	− 0.447751i
$483$	0	0
$484$	7.77920e9	0.141760
$485$	0	0
$486$	0	0
$487$	6.22466e10	1.10662	0.553312	−	0.832974i	$-0.313364\pi$
$487$	6.22466e10	1.10662	0.553312	+	0.832974i	$0.313364\pi$
$488$	− 7.24539e10i	− 1.27756i
$489$	0	0
$490$	0	0
$491$	1.18310e10i	0.203561i	0.994807	+	0.101781i	$0.0324540\pi$
$491$	1.18310e10i	0.203561i	−0.994807	+	0.101781i	$0.967546\pi$
$492$	0	0
$493$	−7.16016e10	−1.21209
$494$	− 4.03756e10i	− 0.677971i
$495$	0	0
$496$	−2.67923e10	−0.442673
$497$	5.29834e10i	0.868389i
$498$	0	0
$499$	4.35370e7	0.000702193 0	0.000351096	−	1.00000i	$-0.499888\pi$
$499$	4.35370e7	0.000702193 0	0.000351096		1.00000i	$0.499888\pi$
$500$	0	0
$501$	0	0
$502$	3.08186e9	0.0485286
$503$	− 2.42920e10i	− 0.379483i	−0.981834	−	0.189741i	$-0.939235\pi$
$503$	− 2.42920e10i	− 0.379483i	0.981834	−	0.189741i	$-0.0607649\pi$
$504$	0	0
$505$	0	0
$506$	− 4.08688e10i	− 0.623433i
$507$	0	0
$508$	−7.59387e10	−1.14027
$509$	− 5.82458e10i	− 0.867748i	−0.900974	−	0.433874i	$-0.857146\pi$
$509$	− 5.82458e10i	− 0.867748i	0.900974	−	0.433874i	$-0.142854\pi$
$510$	0	0
$511$	−7.93019e10	−1.16305
$512$	− 5.23731e10i	− 0.762129i
$513$	0	0
$514$	−2.68969e10	−0.385345
$515$	0	0
$516$	0	0
$517$	1.00455e10	0.140608
$518$	8.04016e9i	0.111672i
$519$	0	0
$520$	0	0
$521$	4.94448e10i	0.671073i	0.942027	+	0.335536i	$0.108918\pi$
$521$	4.94448e10i	0.671073i	−0.942027	+	0.335536i	$0.891082\pi$
$522$	0	0
$523$	−7.26221e10	−0.970648	−0.485324	−	0.874334i	$-0.661298\pi$
$523$	−7.26221e10	−0.970648	−0.485324	+	0.874334i	$0.661298\pi$
$524$	− 9.85036e10i	− 1.30655i
$525$	0	0
$526$	2.76657e10	0.361409
$527$	1.14916e11i	1.48984i
$528$	0	0
$529$	−9.92066e10	−1.26683
$530$	0	0
$531$	0	0
$532$	−1.89513e11	−2.36588
$533$	− 2.30023e10i	− 0.285011i
$534$	0	0
$535$	0	0
$536$	− 4.92050e10i	− 0.596143i
$537$	0	0
$538$	−3.37213e10	−0.402509
$539$	1.01433e11i	1.20178i
$540$	0	0
$541$	−3.33242e10	−0.389019	−0.194509	−	0.980901i	$-0.562311\pi$
$541$	−3.33242e10	−0.389019	−0.194509	+	0.980901i	$0.562311\pi$
$542$	− 3.84610e10i	− 0.445680i
$543$	0	0
$544$	−1.24029e11	−1.41621
$545$	0	0
$546$	0	0
$547$	1.41404e11	1.57947	0.789736	−	0.613447i	$-0.210217\pi$
$547$	1.41404e11	1.57947	0.789736	+	0.613447i	$0.210217\pi$
$548$	5.87996e10i	0.652007i
$549$	0	0
$550$	0	0
$551$	1.56096e11i	1.69350i
$552$	0	0
$553$	1.48582e11	1.58879
$554$	− 4.75052e10i	− 0.504315i
$555$	0	0
$556$	3.20914e8	0.00335807
$557$	− 1.67388e11i	− 1.73901i	−0.493921	−	0.869507i	$-0.664437\pi$
$557$	− 1.67388e11i	− 1.73901i	0.493921	−	0.869507i	$-0.335563\pi$
$558$	0	0
$559$	−3.47548e10	−0.355932
$560$	0	0
$561$	0	0
$562$	8.19829e10	0.821822
$563$	− 1.23714e11i	− 1.23136i	−0.787997	−	0.615679i	$-0.788882\pi$
$563$	− 1.23714e11i	− 1.23136i	0.787997	−	0.615679i	$-0.211118\pi$
$564$	0	0
$565$	0	0
$566$	− 2.95311e10i	− 0.287749i
$567$	0	0
$568$	4.85108e10	0.466064
$569$	1.69396e11i	1.61604i	0.589152	+	0.808022i	$0.299462\pi$
$569$	1.69396e11i	1.61604i	−0.589152	+	0.808022i	$0.700538\pi$
$570$	0	0
$571$	1.05597e11	0.993359	0.496679	−	0.867934i	$-0.334553\pi$
$571$	1.05597e11	0.993359	0.496679	+	0.867934i	$0.334553\pi$
$572$	− 5.80481e10i	− 0.542255i
$573$	0	0
$574$	2.85084e10	0.262619
$575$	0	0
$576$	0	0
$577$	3.71561e10	0.335218	0.167609	−	0.985854i	$-0.446395\pi$
$577$	3.71561e10	0.335218	0.167609	+	0.985854i	$0.446395\pi$
$578$	4.94453e10i	0.443010i
$579$	0	0
$580$	0	0
$581$	− 1.77423e11i	− 1.55706i
$582$	0	0
$583$	4.38145e10	0.379266
$584$	7.26076e10i	0.624210i
$585$	0	0
$586$	−6.67162e10	−0.565771
$587$	1.85072e11i	1.55879i	0.626530	+	0.779397i	$0.284474\pi$
$587$	1.85072e11i	1.55879i	−0.626530	+	0.779397i	$0.715526\pi$
$588$	0	0
$589$	2.50525e11	2.08157
$590$	0	0
$591$	0	0
$592$	−8.20390e9	−0.0667934
$593$	5.13565e10i	0.415314i	0.978202	+	0.207657i	$0.0665838\pi$
$593$	5.13565e10i	0.415314i	−0.978202	+	0.207657i	$0.933416\pi$
$594$	0	0
$595$	0	0
$596$	1.36520e11i	1.08196i
$597$	0	0
$598$	6.65763e10	0.520613
$599$	1.99249e11i	1.54771i	0.633362	+	0.773855i	$0.281674\pi$
$599$	1.99249e11i	1.54771i	−0.633362	+	0.773855i	$0.718326\pi$
$600$	0	0
$601$	5.62386e10	0.431059	0.215530	−	0.976497i	$-0.430852\pi$
$601$	5.62386e10	0.431059	0.215530	+	0.976497i	$0.430852\pi$
$602$	− 4.30742e10i	− 0.327968i
$603$	0	0
$604$	2.65123e10	0.199205
$605$	0	0
$606$	0	0
$607$	−3.56822e10	−0.262843	−0.131422	−	0.991327i	$-0.541954\pi$
$607$	−3.56822e10	−0.262843	−0.131422	+	0.991327i	$0.541954\pi$
$608$	2.70391e11i	1.97869i
$609$	0	0
$610$	0	0
$611$	1.63645e10i	0.117419i
$612$	0	0
$613$	−2.17689e11	−1.54168	−0.770839	−	0.637030i	$-0.780163\pi$
$613$	−2.17689e11	−1.54168	−0.770839	+	0.637030i	$0.780163\pi$
$614$	− 8.76259e9i	− 0.0616537i
$615$	0	0
$616$	1.62883e11	1.13124
$617$	1.60556e11i	1.10786i	0.832563	+	0.553931i	$0.186873\pi$
$617$	1.60556e11i	1.10786i	−0.832563	+	0.553931i	$0.813127\pi$
$618$	0	0
$619$	−2.22627e11	−1.51640	−0.758202	−	0.652019i	$-0.773922\pi$
$619$	−2.22627e11	−1.51640	−0.758202	+	0.652019i	$0.773922\pi$
$620$	0	0
$621$	0	0
$622$	5.30432e10	0.354379
$623$	1.31027e11i	0.869776i
$624$	0	0
$625$	0	0
$626$	5.47640e10i	0.356613i
$627$	0	0
$628$	1.38647e11	0.891402
$629$	3.51879e10i	0.224797i
$630$	0	0
$631$	−9.64259e10	−0.608242	−0.304121	−	0.952633i	$-0.598363\pi$
$631$	−9.64259e10	−0.608242	−0.304121	+	0.952633i	$0.598363\pi$
$632$	− 1.36040e11i	− 0.852703i
$633$	0	0
$634$	1.51304e10	0.0936471
$635$	0	0
$636$	0	0
$637$	−1.65238e11	−1.00358
$638$	− 5.92575e10i	− 0.357652i
$639$	0	0
$640$	0	0
$641$	2.06616e11i	1.22386i	0.790912	+	0.611930i	$0.209607\pi$
$641$	2.06616e11i	1.22386i	−0.790912	+	0.611930i	$0.790393\pi$
$642$	0	0
$643$	5.65637e10	0.330898	0.165449	−	0.986218i	$-0.447093\pi$
$643$	5.65637e10	0.330898	0.165449	+	0.986218i	$0.447093\pi$
$644$	− 3.12493e11i	− 1.81676i
$645$	0	0
$646$	2.19003e11	1.25753
$647$	− 2.46730e10i	− 0.140801i	−0.997519	−	0.0704003i	$-0.977572\pi$
$647$	− 2.46730e10i	− 0.140801i	0.997519	−	0.0704003i	$-0.0224277\pi$
$648$	0	0
$649$	2.77780e11	1.56575
$650$	0	0
$651$	0	0
$652$	−2.94411e10	−0.162916
$653$	2.77278e11i	1.52498i	0.647002	+	0.762488i	$0.276023\pi$
$653$	2.77278e11i	1.52498i	−0.647002	+	0.762488i	$0.723977\pi$
$654$	0	0
$655$	0	0
$656$	2.90890e10i	0.157077i
$657$	0	0
$658$	−2.02817e10	−0.108193
$659$	− 2.79317e11i	− 1.48100i	−0.672054	−	0.740502i	$-0.734588\pi$
$659$	− 2.79317e11i	− 1.48100i	0.672054	−	0.740502i	$-0.265412\pi$
$660$	0	0
$661$	3.58762e9	0.0187932	0.00939661	−	0.999956i	$-0.497009\pi$
$661$	3.58762e9	0.0187932	0.00939661	+	0.999956i	$0.497009\pi$
$662$	1.15473e10i	0.0601241i
$663$	0	0
$664$	−1.62446e11	−0.835671
$665$	0	0
$666$	0	0
$667$	−2.57391e11	−1.30044
$668$	− 4.81681e10i	− 0.241910i
$669$	0	0
$670$	0	0
$671$	2.86623e11i	1.41391i
$672$	0	0
$673$	1.74585e11	0.851034	0.425517	−	0.904950i	$-0.360092\pi$
$673$	1.74585e11	0.851034	0.425517	+	0.904950i	$0.360092\pi$
$674$	5.39925e10i	0.261634i
$675$	0	0
$676$	−7.06432e10	−0.338286
$677$	9.83100e10i	0.467997i	0.972237	+	0.233999i	$0.0751811\pi$
$677$	9.83100e10i	0.467997i	−0.972237	+	0.233999i	$0.924819\pi$
$678$	0	0
$679$	2.20256e11	1.03621
$680$	0	0
$681$	0	0
$682$	−9.51049e10	−0.439608
$683$	− 1.22852e11i	− 0.564548i	−0.959334	−	0.282274i	$-0.908911\pi$
$683$	− 1.22852e11i	− 0.564548i	0.959334	−	0.282274i	$-0.0910888\pi$
$684$	0	0
$685$	0	0
$686$	− 5.04057e10i	− 0.227606i
$687$	0	0
$688$	4.39514e10	0.196164
$689$	7.13750e10i	0.316715i
$690$	0	0
$691$	2.59741e11	1.13927	0.569637	−	0.821896i	$-0.307084\pi$
$691$	2.59741e11	1.13927	0.569637	+	0.821896i	$0.307084\pi$
$692$	− 2.39514e11i	− 1.04450i
$693$	0	0
$694$	−1.17796e11	−0.507800
$695$	0	0
$696$	0	0
$697$	1.24767e11	0.528652
$698$	− 9.89143e10i	− 0.416714i
$699$	0	0
$700$	0	0
$701$	− 1.03431e11i	− 0.428332i	−0.976797	−	0.214166i	$-0.931297\pi$
$701$	− 1.03431e11i	− 0.428332i	0.976797	−	0.214166i	$-0.0687033\pi$
$702$	0	0
$703$	7.67118e10	0.314081
$704$	− 9.85469e9i	− 0.0401192i
$705$	0	0
$706$	1.66213e11	0.669032
$707$	− 4.01816e11i	− 1.60823i
$708$	0	0
$709$	1.57054e11	0.621533	0.310767	−	0.950486i	$-0.399414\pi$
$709$	1.57054e11	0.621533	0.310767	+	0.950486i	$0.399414\pi$
$710$	0	0
$711$	0	0
$712$	1.19966e11	0.466808
$713$	4.13097e11i	1.59843i
$714$	0	0
$715$	0	0
$716$	1.98892e11i	0.756774i
$717$	0	0
$718$	2.20002e11	0.827807
$719$	− 1.04101e11i	− 0.389527i	−0.980850	−	0.194764i	$-0.937606\pi$
$719$	− 1.04101e11i	− 0.389527i	0.980850	−	0.194764i	$-0.0623940\pi$
$720$	0	0
$721$	−2.61616e10	−0.0968108
$722$	− 3.53244e11i	− 1.29995i
$723$	0	0
$724$	−4.18459e11	−1.52300
$725$	0	0
$726$	0	0
$727$	−1.97631e10	−0.0707484	−0.0353742	−	0.999374i	$-0.511262\pi$
$727$	−1.97631e10	−0.0707484	−0.0353742	+	0.999374i	$0.511262\pi$
$728$	2.65341e11i	0.944667i
$729$	0	0
$730$	0	0
$731$	− 1.88515e11i	− 0.660201i
$732$	0	0
$733$	4.69549e11	1.62654	0.813270	−	0.581887i	$-0.197685\pi$
$733$	4.69549e11	1.62654	0.813270	+	0.581887i	$0.197685\pi$
$734$	1.56855e11i	0.540399i
$735$	0	0
$736$	−4.45855e11	−1.51944
$737$	1.94652e11i	0.659766i
$738$	0	0
$739$	2.48981e10	0.0834813	0.0417406	−	0.999128i	$-0.486710\pi$
$739$	2.48981e10	0.0834813	0.0417406	+	0.999128i	$0.486710\pi$
$740$	0	0
$741$	0	0
$742$	−8.84604e10	−0.291832
$743$	− 1.84452e11i	− 0.605241i	−0.953111	−	0.302621i	$-0.902139\pi$
$743$	− 1.84452e11i	− 0.605241i	0.953111	−	0.302621i	$-0.0978615\pi$
$744$	0	0
$745$	0	0
$746$	2.07736e11i	0.670744i
$747$	0	0
$748$	3.14861e11	1.00580
$749$	5.03288e11i	1.59915i
$750$	0	0
$751$	−4.71095e10	−0.148098	−0.0740489	−	0.997255i	$-0.523592\pi$
$751$	−4.71095e10	−0.148098	−0.0740489	+	0.997255i	$0.523592\pi$
$752$	− 2.06947e10i	− 0.0647125i
$753$	0	0
$754$	9.65321e10	0.298666
$755$	0	0
$756$	0	0
$757$	6.14374e11	1.87089	0.935447	−	0.353467i	$-0.114997\pi$
$757$	6.14374e11	1.87089	0.935447	+	0.353467i	$0.114997\pi$
$758$	4.72018e10i	0.142982i
$759$	0	0
$760$	0	0
$761$	5.20082e11i	1.55072i	0.631520	+	0.775359i	$0.282431\pi$
$761$	5.20082e11i	1.55072i	−0.631520	+	0.775359i	$0.717569\pi$
$762$	0	0
$763$	−2.53625e11	−0.748331
$764$	− 1.06708e11i	− 0.313200i
$765$	0	0
$766$	−1.34645e11	−0.391088
$767$	4.52512e11i	1.30752i
$768$	0	0
$769$	3.46901e11	0.991974	0.495987	−	0.868330i	$-0.334806\pi$
$769$	3.46901e11	0.991974	0.495987	+	0.868330i	$0.334806\pi$
$770$	0	0
$771$	0	0
$772$	−4.17235e11	−1.17466
$773$	− 7.16096e10i	− 0.200564i	−0.994959	−	0.100282i	$-0.968025\pi$
$773$	− 7.16096e10i	− 0.200564i	0.994959	−	0.100282i	$-0.0319745\pi$
$774$	0	0
$775$	0	0
$776$	− 2.01663e11i	− 0.556134i
$777$	0	0
$778$	−1.61090e11	−0.439695
$779$	− 2.72001e11i	− 0.738619i
$780$	0	0
$781$	−1.91906e11	−0.515804
$782$	3.61119e11i	0.965659i
$783$	0	0
$784$	2.08962e11	0.553099
$785$	0	0
$786$	0	0
$787$	2.58944e11	0.675004	0.337502	−	0.941325i	$-0.390418\pi$
$787$	2.58944e11	0.675004	0.337502	+	0.941325i	$0.390418\pi$
$788$	− 1.61079e11i	− 0.417766i
$789$	0	0
$790$	0	0
$791$	− 5.50372e11i	− 1.40589i
$792$	0	0
$793$	−4.66917e11	−1.18072
$794$	− 8.76175e10i	− 0.220449i
$795$	0	0
$796$	1.32181e11	0.329244
$797$	9.11965e10i	0.226019i	0.993594	+	0.113009i	$0.0360490\pi$
$797$	9.11965e10i	0.226019i	−0.993594	+	0.113009i	$0.963951\pi$
$798$	0	0
$799$	−8.87631e10	−0.217794
$800$	0	0
$801$	0	0
$802$	−7.17069e10	−0.173326
$803$	− 2.87232e11i	− 0.690828i
$804$	0	0
$805$	0	0
$806$	− 1.54928e11i	− 0.367105i
$807$	0	0
$808$	−3.67896e11	−0.863138
$809$	1.88801e11i	0.440767i	0.975413	+	0.220384i	$0.0707309\pi$
$809$	1.88801e11i	0.440767i	−0.975413	+	0.220384i	$0.929269\pi$
$810$	0	0
$811$	7.11503e11	1.64472	0.822362	−	0.568964i	$-0.192656\pi$
$811$	7.11503e11	1.64472	0.822362	+	0.568964i	$0.192656\pi$
$812$	− 4.53098e11i	− 1.04224i
$813$	0	0
$814$	−2.91215e10	−0.0663309
$815$	0	0
$816$	0	0
$817$	−4.10974e11	−0.922415
$818$	− 4.47758e10i	− 0.100007i
$819$	0	0
$820$	0	0
$821$	2.66425e10i	0.0586411i	0.999570	+	0.0293205i	$0.00933435\pi$
$821$	2.66425e10i	0.0586411i	−0.999570	+	0.0293205i	$0.990666\pi$
$822$	0	0
$823$	−5.69829e11	−1.24207	−0.621033	−	0.783784i	$-0.713287\pi$
$823$	−5.69829e11	−1.24207	−0.621033	+	0.783784i	$0.713287\pi$
$824$	2.39532e10i	0.0519583i
$825$	0	0
$826$	−5.60832e11	−1.20479
$827$	2.83844e11i	0.606816i	0.952861	+	0.303408i	$0.0981245\pi$
$827$	2.83844e11i	0.606816i	−0.952861	+	0.303408i	$0.901875\pi$
$828$	0	0
$829$	−8.79841e11	−1.86288	−0.931442	−	0.363889i	$-0.881449\pi$
$829$	−8.79841e11	−1.86288	−0.931442	+	0.363889i	$0.881449\pi$
$830$	0	0
$831$	0	0
$832$	1.60535e10	0.0335025
$833$	− 8.96271e11i	− 1.86149i
$834$	0	0
$835$	0	0
$836$	− 6.86417e11i	− 1.40528i
$837$	0	0
$838$	7.90098e10	0.160216
$839$	− 3.79708e11i	− 0.766305i	−0.923685	−	0.383153i	$-0.874838\pi$
$839$	− 3.79708e11i	− 0.766305i	0.923685	−	0.383153i	$-0.125162\pi$
$840$	0	0
$841$	1.27044e11	0.253962
$842$	1.13532e11i	0.225876i
$843$	0	0
$844$	3.52362e9	0.00694415
$845$	0	0
$846$	0	0
$847$	1.40670e11	0.273318
$848$	− 9.02618e10i	− 0.174550i
$849$	0	0
$850$	0	0
$851$	1.26492e11i	0.241182i
$852$	0	0
$853$	−4.18933e11	−0.791314	−0.395657	−	0.918398i	$-0.629483\pi$
$853$	−4.18933e11	−0.791314	−0.395657	+	0.918398i	$0.629483\pi$
$854$	− 5.78685e11i	− 1.08796i
$855$	0	0
$856$	4.60803e11	0.858263
$857$	9.86863e11i	1.82950i	0.404015	+	0.914752i	$0.367614\pi$
$857$	9.86863e11i	1.82950i	−0.404015	+	0.914752i	$0.632386\pi$
$858$	0	0
$859$	6.76444e11	1.24239	0.621197	−	0.783655i	$-0.286647\pi$
$859$	6.76444e11	1.24239	0.621197	+	0.783655i	$0.286647\pi$
$860$	0	0
$861$	0	0
$862$	−1.33943e11	−0.242601
$863$	1.65158e11i	0.297753i	0.988856	+	0.148876i	$0.0475657\pi$
$863$	1.65158e11i	0.297753i	−0.988856	+	0.148876i	$0.952434\pi$
$864$	0	0
$865$	0	0
$866$	2.26832e11i	0.403304i
$867$	0	0
$868$	−7.27195e11	−1.28107
$869$	5.38166e11i	0.943707i
$870$	0	0
$871$	−3.17094e11	−0.550954
$872$	2.32215e11i	0.401628i
$873$	0	0
$874$	7.87263e11	1.34919
$875$	0	0
$876$	0	0
$877$	5.00676e11	0.846366	0.423183	−	0.906044i	$-0.360913\pi$
$877$	5.00676e11	0.846366	0.423183	+	0.906044i	$0.360913\pi$
$878$	− 1.90342e11i	− 0.320300i
$879$	0	0
$880$	0	0
$881$	− 5.49244e11i	− 0.911720i	−0.890051	−	0.455860i	$-0.849332\pi$
$881$	− 5.49244e11i	− 0.911720i	0.890051	−	0.455860i	$-0.150668\pi$
$882$	0	0
$883$	−2.56467e10	−0.0421880	−0.0210940	−	0.999777i	$-0.506715\pi$
$883$	−2.56467e10	−0.0421880	−0.0210940	+	0.999777i	$0.506715\pi$
$884$	5.12917e11i	0.839920i
$885$	0	0
$886$	−4.73586e11	−0.768536
$887$	− 6.24010e11i	− 1.00809i	−0.863679	−	0.504043i	$-0.831845\pi$
$887$	− 6.24010e11i	− 1.00809i	0.863679	−	0.504043i	$-0.168155\pi$
$888$	0	0
$889$	−1.37319e12	−2.19848
$890$	0	0
$891$	0	0
$892$	9.43655e11	1.49057
$893$	1.93509e11i	0.304296i
$894$	0	0
$895$	0	0
$896$	− 9.72204e11i	− 1.50843i
$897$	0	0
$898$	1.99278e11	0.306446
$899$	5.98969e11i	0.916991i
$900$	0	0
$901$	−3.87148e11	−0.587459
$902$	1.03258e11i	0.155990i
$903$	0	0
$904$	−5.03912e11	−0.754538
$905$	0	0
$906$	0	0
$907$	−5.76568e11	−0.851964	−0.425982	−	0.904732i	$-0.640071\pi$
$907$	−5.76568e11	−0.851964	−0.425982	+	0.904732i	$0.640071\pi$
$908$	2.92976e11i	0.431012i
$909$	0	0
$910$	0	0
$911$	− 7.32970e11i	− 1.06417i	−0.846690	−	0.532087i	$-0.821408\pi$
$911$	− 7.32970e11i	− 1.06417i	0.846690	−	0.532087i	$-0.178592\pi$
$912$	0	0
$913$	6.42625e11	0.924858
$914$	2.49899e11i	0.358079i
$915$	0	0
$916$	5.19836e10	0.0738388
$917$	− 1.78122e12i	− 2.51908i
$918$	0	0
$919$	−4.30705e11	−0.603834	−0.301917	−	0.953334i	$-0.597627\pi$
$919$	−4.30705e11	−0.603834	−0.301917	+	0.953334i	$0.597627\pi$
$920$	0	0
$921$	0	0
$922$	−2.71283e11	−0.375403
$923$	− 3.12620e11i	− 0.430735i
$924$	0	0
$925$	0	0
$926$	3.75561e10i	0.0510783i
$927$	0	0
$928$	−6.46465e11	−0.871672
$929$	− 1.10110e12i	− 1.47831i	−0.673538	−	0.739153i	$-0.735226\pi$
$929$	− 1.10110e12i	− 1.47831i	0.673538	−	0.739153i	$-0.264774\pi$
$930$	0	0
$931$	−1.95393e12	−2.60082
$932$	7.36252e11i	0.975805i
$933$	0	0
$934$	−3.72958e10	−0.0490086
$935$	0	0
$936$	0	0
$937$	5.55540e11	0.720704	0.360352	−	0.932816i	$-0.382657\pi$
$937$	5.55540e11	0.720704	0.360352	+	0.932816i	$0.382657\pi$
$938$	− 3.92998e11i	− 0.507667i
$939$	0	0
$940$	0	0
$941$	1.21149e11i	0.154512i	0.997011	+	0.0772558i	$0.0246158\pi$
$941$	1.21149e11i	0.154512i	−0.997011	+	0.0772558i	$0.975384\pi$
$942$	0	0
$943$	4.48509e11	0.567185
$944$	− 5.72253e11i	− 0.720609i
$945$	0	0
$946$	1.56015e11	0.194806
$947$	6.98500e11i	0.868494i	0.900794	+	0.434247i	$0.142985\pi$
$947$	6.98500e11i	0.868494i	−0.900794	+	0.434247i	$0.857015\pi$
$948$	0	0
$949$	4.67908e11	0.576893
$950$	0	0
$951$	0	0
$952$	−1.43925e12	−1.75221
$953$	− 1.47296e12i	− 1.78575i	−0.450304	−	0.892875i	$-0.648685\pi$
$953$	− 1.47296e12i	− 1.78575i	0.450304	−	0.892875i	$-0.351315\pi$
$954$	0	0
$955$	0	0
$956$	1.18216e12i	1.41529i
$957$	0	0
$958$	−6.79835e10	−0.0807127
$959$	1.06326e12i	1.25709i
$960$	0	0
$961$	1.08419e11	0.127119
$962$	− 4.74397e10i	− 0.0553913i
$963$	0	0
$964$	6.69301e11	0.775020
$965$	0	0
$966$	0	0
$967$	1.18280e12	1.35271	0.676357	−	0.736574i	$-0.263557\pi$
$967$	1.18280e12	1.35271	0.676357	+	0.736574i	$0.263557\pi$
$968$	− 1.28795e11i	− 0.146689i
$969$	0	0
$970$	0	0
$971$	− 1.90694e11i	− 0.214517i	−0.994231	−	0.107258i	$-0.965793\pi$
$971$	− 1.90694e11i	− 0.214517i	0.994231	−	0.107258i	$-0.0342072\pi$
$972$	0	0
$973$	5.80303e9	0.00647446
$974$	− 4.55193e11i	− 0.505777i
$975$	0	0
$976$	5.90470e11	0.650727
$977$	− 8.63910e11i	− 0.948179i	−0.880477	−	0.474090i	$-0.842777\pi$
$977$	− 8.63910e11i	− 0.948179i	0.880477	−	0.474090i	$-0.157223\pi$
$978$	0	0
$979$	−4.74579e11	−0.516628
$980$	0	0
$981$	0	0
$982$	8.65169e10	0.0930368
$983$	− 5.38400e11i	− 0.576621i	−0.957537	−	0.288311i	$-0.906906\pi$
$983$	− 5.38400e11i	− 0.576621i	0.957537	−	0.288311i	$-0.0930936\pi$
$984$	0	0
$985$	0	0
$986$	5.23603e11i	0.553981i
$987$	0	0
$988$	1.11819e12	1.17351
$989$	− 6.77666e11i	− 0.708321i
$990$	0	0
$991$	5.65493e11	0.586317	0.293159	−	0.956064i	$-0.405294\pi$
$991$	5.65493e11	0.586317	0.293159	+	0.956064i	$0.405294\pi$
$992$	1.03754e12i	1.07142i
$993$	0	0
$994$	3.87453e11	0.396893
$995$	0	0
$996$	0	0
$997$	−7.32546e11	−0.741403	−0.370702	−	0.928752i	$-0.620883\pi$
$997$	−7.32546e11	−0.741403	−0.370702	+	0.928752i	$0.620883\pi$
$998$	− 3.18374e8i	0 0.000320934i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.9.c.d.26.6		12
3.2	odd	2	inner	225.9.c.d.26.7		12
5.2	odd	4		225.9.d.c.224.13		24
5.3	odd	4		225.9.d.c.224.12		24
5.4	even	2		45.9.c.a.26.7	yes	12
15.2	even	4		225.9.d.c.224.11		24
15.8	even	4		225.9.d.c.224.14		24
15.14	odd	2		45.9.c.a.26.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.9.c.a.26.6	✓	12	15.14	odd	2
45.9.c.a.26.7	yes	12	5.4	even	2
225.9.c.d.26.6		12	1.1	even	1	trivial
225.9.c.d.26.7		12	3.2	odd	2	inner
225.9.d.c.224.11		24	15.2	even	4
225.9.d.c.224.12		24	5.3	odd	4
225.9.d.c.224.13		24	5.2	odd	4
225.9.d.c.224.14		24	15.8	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.31273i q^{2} +202.524 q^{4} +3662.21 q^{7} -3353.06i q^{8} +13264.5i q^{11} -21608.3 q^{13} -26780.7i q^{14} +27326.1 q^{16} -117206. i q^{17} -255517. q^{19} +96999.9 q^{22} -421328. i q^{23} +158015. i q^{26} +741685. q^{28} -610903. i q^{29} -980464. q^{31} -1.05821e6i q^{32} -857097. q^{34} -300222. q^{37} +1.86853e6i q^{38} +1.06451e6i q^{41} +1.60840e6 q^{43} +2.68638e6i q^{44} -3.08106e6 q^{46} -757324. i q^{47} +7.64696e6 q^{49} -4.37619e6 q^{52} -3.30314e6i q^{53} -1.22796e7i q^{56} -4.46737e6 q^{58} -2.09416e7i q^{59} +2.16083e7 q^{61} +7.16987e6i q^{62} -742936. q^{64} +1.46747e7 q^{67} -2.37371e7i q^{68} +1.44676e7i q^{71} -2.16541e7 q^{73} +2.19544e6i q^{74} -5.17483e7 q^{76} +4.85774e7i q^{77} +4.05718e7 q^{79} +7.78449e6 q^{82} -4.84469e7i q^{83} -1.17618e7i q^{86} +4.44768e7 q^{88} +3.57780e7i q^{89} -7.91339e7 q^{91} -8.53291e7i q^{92} -5.53811e6 q^{94} +6.01429e7 q^{97} -5.59202e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2524 q^{4} + 6928 q^{7} - 124408 q^{13} + 297572 q^{16} + 124960 q^{19} + 1370512 q^{22} - 1114496 q^{28} - 620968 q^{31} + 7486496 q^{34} + 11533176 q^{37} - 14405296 q^{43} - 7161768 q^{46} + 21010156 q^{49}+ \cdots + 302816184 q^{97}+O(q^{100}) \) 12 * q - 2524 * q^4 + 6928 * q^7 - 124408 * q^13 + 297572 * q^16 + 124960 * q^19 + 1370512 * q^22 - 1114496 * q^28 - 620968 * q^31 + 7486496 * q^34 + 11533176 * q^37 - 14405296 * q^43 - 7161768 * q^46 + 21010156 * q^49 + 10459176 * q^52 - 22630504 * q^58 + 28453096 * q^61 - 169389948 * q^64 - 47768816 * q^67 - 38412808 * q^73 - 255673920 * q^76 - 63809512 * q^79 - 165335176 * q^82 - 679824192 * q^88 - 296849288 * q^91 + 25601888 * q^94 + 302816184 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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