LMFDB - Embedded newform 225.9.c.d.26.3

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			26.3
Root			$10.5572i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.9.c.d.26.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-23.2704i q^{2} -285.513 q^{4} -4399.16 q^{7} +686.784i q^{8} +16184.6i q^{11} +2771.50 q^{13} +102370. i q^{14} -57109.6 q^{16} -26479.8i q^{17} +186497. q^{19} +376622. q^{22} +52174.8i q^{23} -64494.0i q^{26} +1.25602e6 q^{28} +928476. i q^{29} -960982. q^{31} +1.50478e6i q^{32} -616197. q^{34} +2.30066e6 q^{37} -4.33986e6i q^{38} +3.41833e6i q^{41} -4.56761e6 q^{43} -4.62091e6i q^{44} +1.21413e6 q^{46} -3.39999e6i q^{47} +1.35878e7 q^{49} -791299. q^{52} -6.12740e6i q^{53} -3.02127e6i q^{56} +2.16060e7 q^{58} -1.07944e7i q^{59} -7.04198e6 q^{61} +2.23625e7i q^{62} +2.03969e7 q^{64} -2.46307e7 q^{67} +7.56034e6i q^{68} -3.67232e7i q^{71} +3.11078e7 q^{73} -5.35374e7i q^{74} -5.32472e7 q^{76} -7.11986e7i q^{77} +6.46442e7 q^{79} +7.95461e7 q^{82} -7.65633e7i q^{83} +1.06290e8i q^{86} -1.11153e7 q^{88} -5.07750e7i q^{89} -1.21923e7 q^{91} -1.48966e7i q^{92} -7.91192e7 q^{94} -8.64077e7 q^{97} -3.16194e8i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 23.2704i	− 1.45440i	−0.686424	−	0.727201i	$-0.740821\pi$
$2$	− 23.2704i	− 1.45440i	0.686424	−	0.727201i	$-0.259179\pi$
$3$	0	0
$3$	0	0
$4$	−285.513	−1.11529
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4399.16	−1.83222	−0.916110	−	0.400927i	$-0.868688\pi$
$7$	−4399.16	−1.83222	−0.916110	+	0.400927i	$0.868688\pi$
$8$	686.784i	0.167672i
$9$	0	0
$10$	0	0
$11$	16184.6i	1.10543i	0.833371	+	0.552714i	$0.186408\pi$
$11$	16184.6i	1.10543i	−0.833371	+	0.552714i	$0.813592\pi$
$12$	0	0
$13$	2771.50	0.0970379	0.0485189	−	0.998822i	$-0.484550\pi$
$13$	2771.50	0.0970379	0.0485189	+	0.998822i	$0.484550\pi$
$14$	102370.i	2.66478i
$15$	0	0
$16$	−57109.6	−0.871424
$17$	− 26479.8i	− 0.317044i	−0.987355	−	0.158522i	$-0.949327\pi$
$17$	− 26479.8i	− 0.317044i	0.987355	−	0.158522i	$-0.0506728\pi$
$18$	0	0
$19$	186497.	1.43106	0.715528	−	0.698584i	$-0.246186\pi$
$19$	186497.	1.43106	0.715528	+	0.698584i	$0.246186\pi$
$20$	0	0
$21$	0	0
$22$	376622.	1.60774
$23$	52174.8i	0.186445i	0.995645	+	0.0932223i	$0.0297167\pi$
$23$	52174.8i	0.186445i	−0.995645	+	0.0932223i	$0.970283\pi$
$24$	0	0
$25$	0	0
$26$	− 64494.0i	− 0.141132i
$27$	0	0
$28$	1.25602e6	2.04345
$29$	928476.i	1.31274i	0.754439	+	0.656370i	$0.227909\pi$
$29$	928476.i	1.31274i	−0.754439	+	0.656370i	$0.772091\pi$
$30$	0	0
$31$	−960982.	−1.04056	−0.520282	−	0.853995i	$-0.674173\pi$
$31$	−960982.	−1.04056	−0.520282	+	0.853995i	$0.674173\pi$
$32$	1.50478e6i	1.43507i
$33$	0	0
$34$	−616197.	−0.461109
$35$	0	0
$36$	0	0
$37$	2.30066e6	1.22757	0.613784	−	0.789474i	$-0.289646\pi$
$37$	2.30066e6	1.22757	0.613784	+	0.789474i	$0.289646\pi$
$38$	− 4.33986e6i	− 2.08133i
$39$	0	0
$40$	0	0
$41$	3.41833e6i	1.20970i	0.796338	+	0.604852i	$0.206768\pi$
$41$	3.41833e6i	1.20970i	−0.796338	+	0.604852i	$0.793232\pi$
$42$	0	0
$43$	−4.56761e6	−1.33603	−0.668013	−	0.744149i	$-0.732855\pi$
$43$	−4.56761e6	−1.33603	−0.668013	+	0.744149i	$0.732855\pi$
$44$	− 4.62091e6i	− 1.23287i
$45$	0	0
$46$	1.21413e6	0.271165
$47$	− 3.39999e6i	− 0.696764i	−0.937353	−	0.348382i	$-0.886731\pi$
$47$	− 3.39999e6i	− 0.696764i	0.937353	−	0.348382i	$-0.113269\pi$
$48$	0	0
$49$	1.35878e7	2.35703
$50$	0	0
$51$	0	0
$52$	−791299.	−0.108225
$53$	− 6.12740e6i	− 0.776556i	−0.921542	−	0.388278i	$-0.873070\pi$
$53$	− 6.12740e6i	− 0.776556i	0.921542	−	0.388278i	$-0.126930\pi$
$54$	0	0
$55$	0	0
$56$	− 3.02127e6i	− 0.307211i
$57$	0	0
$58$	2.16060e7	1.90925
$59$	− 1.07944e7i	− 0.890824i	−0.895326	−	0.445412i	$-0.853057\pi$
$59$	− 1.07944e7i	− 0.890824i	0.895326	−	0.445412i	$-0.146943\pi$
$60$	0	0
$61$	−7.04198e6	−0.508599	−0.254300	−	0.967125i	$-0.581845\pi$
$61$	−7.04198e6	−0.508599	−0.254300	+	0.967125i	$0.581845\pi$
$62$	2.23625e7i	1.51340i
$63$	0	0
$64$	2.03969e7	1.21575
$65$	0	0
$66$	0	0
$67$	−2.46307e7	−1.22230	−0.611151	−	0.791514i	$-0.709293\pi$
$67$	−2.46307e7	−1.22230	−0.611151	+	0.791514i	$0.709293\pi$
$68$	7.56034e6i	0.353594i
$69$	0	0
$70$	0	0
$71$	− 3.67232e7i	− 1.44513i	−0.691303	−	0.722565i	$-0.742963\pi$
$71$	− 3.67232e7i	− 1.44513i	0.691303	−	0.722565i	$-0.257037\pi$
$72$	0	0
$73$	3.11078e7	1.09541	0.547706	−	0.836671i	$-0.315501\pi$
$73$	3.11078e7	1.09541	0.547706	+	0.836671i	$0.315501\pi$
$74$	− 5.35374e7i	− 1.78538i
$75$	0	0
$76$	−5.32472e7	−1.59604
$77$	− 7.11986e7i	− 2.02539i
$78$	0	0
$79$	6.46442e7	1.65967	0.829834	−	0.558010i	$-0.188435\pi$
$79$	6.46442e7	1.65967	0.829834	+	0.558010i	$0.188435\pi$
$80$	0	0
$81$	0	0
$82$	7.95461e7	1.75940
$83$	− 7.65633e7i	− 1.61328i	−0.591046	−	0.806638i	$-0.701285\pi$
$83$	− 7.65633e7i	− 1.61328i	0.591046	−	0.806638i	$-0.298715\pi$
$84$	0	0
$85$	0	0
$86$	1.06290e8i	1.94312i
$87$	0	0
$88$	−1.11153e7	−0.185349
$89$	− 5.07750e7i	− 0.809264i	−0.914480	−	0.404632i	$-0.867400\pi$
$89$	− 5.07750e7i	− 0.809264i	0.914480	−	0.404632i	$-0.132600\pi$
$90$	0	0
$91$	−1.21923e7	−0.177795
$92$	− 1.48966e7i	− 0.207939i
$93$	0	0
$94$	−7.91192e7	−1.01338
$95$	0	0
$96$	0	0
$97$	−8.64077e7	−0.976035	−0.488017	−	0.872834i	$-0.662280\pi$
$97$	−8.64077e7	−0.976035	−0.488017	+	0.872834i	$0.662280\pi$
$98$	− 3.16194e8i	− 3.42807i
$99$	0	0
$100$	0	0
$101$	− 1.25688e8i	− 1.20783i	−0.797047	−	0.603917i	$-0.793606\pi$
$101$	− 1.25688e8i	− 1.20783i	0.797047	−	0.603917i	$-0.206394\pi$
$102$	0	0
$103$	2.63087e7	0.233750	0.116875	−	0.993147i	$-0.462712\pi$
$103$	2.63087e7	0.233750	0.116875	+	0.993147i	$0.462712\pi$
$104$	1.90342e6i	0.0162705i
$105$	0	0
$106$	−1.42587e8	−1.12942
$107$	2.49819e8i	1.90586i	0.303193	+	0.952929i	$0.401947\pi$
$107$	2.49819e8i	1.90586i	−0.303193	+	0.952929i	$0.598053\pi$
$108$	0	0
$109$	−5.80899e6	−0.0411523	−0.0205762	−	0.999788i	$-0.506550\pi$
$109$	−5.80899e6	−0.0411523	−0.0205762	+	0.999788i	$0.506550\pi$
$110$	0	0
$111$	0	0
$112$	2.51234e8	1.59664
$113$	1.50367e8i	0.922228i	0.887341	+	0.461114i	$0.152550\pi$
$113$	1.50367e8i	0.922228i	−0.887341	+	0.461114i	$0.847450\pi$
$114$	0	0
$115$	0	0
$116$	− 2.65092e8i	− 1.46408i
$117$	0	0
$118$	−2.51191e8	−1.29562
$119$	1.16489e8i	0.580894i
$120$	0	0
$121$	−4.75819e7	−0.221973
$122$	1.63870e8i	0.739708i
$123$	0	0
$124$	2.74373e8	1.16053
$125$	0	0
$126$	0	0
$127$	2.68993e8	1.03401	0.517006	−	0.855982i	$-0.327046\pi$
$127$	2.68993e8	1.03401	0.517006	+	0.855982i	$0.327046\pi$
$128$	− 8.94199e7i	− 0.333115i
$129$	0	0
$130$	0	0
$131$	− 2.59837e8i	− 0.882298i	−0.897434	−	0.441149i	$-0.854571\pi$
$131$	− 2.59837e8i	− 0.882298i	0.897434	−	0.441149i	$-0.145429\pi$
$132$	0	0
$133$	−8.20428e8	−2.62201
$134$	5.73168e8i	1.77772i
$135$	0	0
$136$	1.81859e7	0.0531593
$137$	2.23476e8i	0.634378i	0.948362	+	0.317189i	$0.102739\pi$
$137$	2.23476e8i	0.634378i	−0.948362	+	0.317189i	$0.897261\pi$
$138$	0	0
$139$	−3.10970e8	−0.833028	−0.416514	−	0.909129i	$-0.636748\pi$
$139$	−3.10970e8	−0.833028	−0.416514	+	0.909129i	$0.636748\pi$
$140$	0	0
$141$	0	0
$142$	−8.54564e8	−2.10180
$143$	4.48556e7i	0.107269i
$144$	0	0
$145$	0	0
$146$	− 7.23891e8i	− 1.59317i
$147$	0	0
$148$	−6.56869e8	−1.36909
$149$	191472.i	0 0.000388473i	1.00000		0.000194236i	$6.18274e-5\pi$
$149$	191472.i	0 0.000388473i	−1.00000		0.000194236i	$0.999938\pi$
$150$	0	0
$151$	5.84355e8	1.12401	0.562004	−	0.827135i	$-0.310031\pi$
$151$	5.84355e8	1.12401	0.562004	+	0.827135i	$0.310031\pi$
$152$	1.28083e8i	0.239948i
$153$	0	0
$154$	−1.65682e9	−2.94573
$155$	0	0
$156$	0	0
$157$	7.91625e7	0.130293	0.0651464	−	0.997876i	$-0.479249\pi$
$157$	7.91625e7	0.130293	0.0651464	+	0.997876i	$0.479249\pi$
$158$	− 1.50430e9i	− 2.41383i
$159$	0	0
$160$	0	0
$161$	− 2.29525e8i	− 0.341607i
$162$	0	0
$163$	−3.52707e8	−0.499648	−0.249824	−	0.968291i	$-0.580373\pi$
$163$	−3.52707e8	−0.499648	−0.249824	+	0.968291i	$0.580373\pi$
$164$	− 9.75979e8i	− 1.34917i
$165$	0	0
$166$	−1.78166e9	−2.34635
$167$	− 7.67131e8i	− 0.986288i	−0.869948	−	0.493144i	$-0.835848\pi$
$167$	− 7.67131e8i	− 0.986288i	0.869948	−	0.493144i	$-0.164152\pi$
$168$	0	0
$169$	−8.08050e8	−0.990584
$170$	0	0
$171$	0	0
$172$	1.30411e9	1.49005
$173$	3.74444e8i	0.418025i	0.977913	+	0.209012i	$0.0670249\pi$
$173$	3.74444e8i	0.418025i	−0.977913	+	0.209012i	$0.932975\pi$
$174$	0	0
$175$	0	0
$176$	− 9.24295e8i	− 0.963297i
$177$	0	0
$178$	−1.18156e9	−1.17699
$179$	− 5.04945e8i	− 0.491849i	−0.969289	−	0.245924i	$-0.920909\pi$
$179$	− 5.04945e8i	− 0.491849i	0.969289	−	0.245924i	$-0.0790915\pi$
$180$	0	0
$181$	1.13823e9	1.06051	0.530255	−	0.847838i	$-0.322096\pi$
$181$	1.13823e9	1.06051	0.530255	+	0.847838i	$0.322096\pi$
$182$	2.83719e8i	0.258585i
$183$	0	0
$184$	−3.58328e7	−0.0312615
$185$	0	0
$186$	0	0
$187$	4.28565e8	0.350469
$188$	9.70741e8i	0.777091i
$189$	0	0
$190$	0	0
$191$	5.34154e8i	0.401359i	0.979657	+	0.200680i	$0.0643150\pi$
$191$	5.34154e8i	0.401359i	−0.979657	+	0.200680i	$0.935685\pi$
$192$	0	0
$193$	1.65692e9	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$193$	1.65692e9	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$194$	2.01074e9i	1.41955i
$195$	0	0
$196$	−3.87950e9	−2.62876
$197$	1.87487e8i	0.124482i	0.998061	+	0.0622408i	$0.0198247\pi$
$197$	1.87487e8i	0.124482i	−0.998061	+	0.0622408i	$0.980175\pi$
$198$	0	0
$199$	4.54531e8	0.289835	0.144918	−	0.989444i	$-0.453708\pi$
$199$	4.54531e8	0.289835	0.144918	+	0.989444i	$0.453708\pi$
$200$	0	0
$201$	0	0
$202$	−2.92481e9	−1.75668
$203$	− 4.08451e9i	− 2.40523i
$204$	0	0
$205$	0	0
$206$	− 6.12215e8i	− 0.339966i
$207$	0	0
$208$	−1.58279e8	−0.0845611
$209$	3.01837e9i	1.58193i
$210$	0	0
$211$	2.53860e9	1.28075	0.640376	−	0.768062i	$-0.278779\pi$
$211$	2.53860e9	1.28075	0.640376	+	0.768062i	$0.278779\pi$
$212$	1.74945e9i	0.866082i
$213$	0	0
$214$	5.81340e9	2.77188
$215$	0	0
$216$	0	0
$217$	4.22751e9	1.90654
$218$	1.35178e8i	0.0598521i
$219$	0	0
$220$	0	0
$221$	− 7.33888e7i	− 0.0307653i
$222$	0	0
$223$	1.35915e9	0.549602	0.274801	−	0.961501i	$-0.411388\pi$
$223$	1.35915e9	0.549602	0.274801	+	0.961501i	$0.411388\pi$
$224$	− 6.61978e9i	− 2.62937i
$225$	0	0
$226$	3.49910e9	1.34129
$227$	9.20041e8i	0.346500i	0.984878	+	0.173250i	$0.0554269\pi$
$227$	9.20041e8i	0.346500i	−0.984878	+	0.173250i	$0.944573\pi$
$228$	0	0
$229$	4.11927e9	1.49789	0.748943	−	0.662634i	$-0.230562\pi$
$229$	4.11927e9	1.49789	0.748943	+	0.662634i	$0.230562\pi$
$230$	0	0
$231$	0	0
$232$	−6.37662e8	−0.220109
$233$	1.65614e9i	0.561918i	0.959720	+	0.280959i	$0.0906525\pi$
$233$	1.65614e9i	0.561918i	−0.959720	+	0.280959i	$0.909347\pi$
$234$	0	0
$235$	0	0
$236$	3.08195e9i	0.993523i
$237$	0	0
$238$	2.71075e9	0.844853
$239$	− 2.82808e9i	− 0.866761i	−0.901211	−	0.433380i	$-0.857321\pi$
$239$	− 2.82808e9i	− 0.866761i	0.901211	−	0.433380i	$-0.142679\pi$
$240$	0	0
$241$	−5.61470e9	−1.66440	−0.832201	−	0.554474i	$-0.812920\pi$
$241$	−5.61470e9	−1.66440	−0.832201	+	0.554474i	$0.812920\pi$
$242$	1.10725e9i	0.322838i
$243$	0	0
$244$	2.01058e9	0.567233
$245$	0	0
$246$	0	0
$247$	5.16875e8	0.138867
$248$	− 6.59987e8i	− 0.174473i
$249$	0	0
$250$	0	0
$251$	− 3.82921e9i	− 0.964749i	−0.875965	−	0.482375i	$-0.839774\pi$
$251$	− 3.82921e9i	− 0.964749i	0.875965	−	0.482375i	$-0.160226\pi$
$252$	0	0
$253$	−8.44428e8	−0.206101
$254$	− 6.25958e9i	− 1.50387i
$255$	0	0
$256$	3.14076e9	0.731265
$257$	− 5.39550e9i	− 1.23680i	−0.785864	−	0.618399i	$-0.787781\pi$
$257$	− 5.39550e9i	− 1.23680i	0.785864	−	0.618399i	$-0.212219\pi$
$258$	0	0
$259$	−1.01210e10	−2.24918
$260$	0	0
$261$	0	0
$262$	−6.04651e9	−1.28322
$263$	− 2.65139e9i	− 0.554180i	−0.960844	−	0.277090i	$-0.910630\pi$
$263$	− 2.65139e9i	− 0.554180i	0.960844	−	0.277090i	$-0.0893700\pi$
$264$	0	0
$265$	0	0
$266$	1.90917e10i	3.81346i
$267$	0	0
$268$	7.03240e9	1.36322
$269$	5.01721e9i	0.958193i	0.877762	+	0.479097i	$0.159036\pi$
$269$	5.01721e9i	0.958193i	−0.877762	+	0.479097i	$0.840964\pi$
$270$	0	0
$271$	−1.66334e9	−0.308393	−0.154196	−	0.988040i	$-0.549279\pi$
$271$	−1.66334e9	−0.308393	−0.154196	+	0.988040i	$0.549279\pi$
$272$	1.51225e9i	0.276279i
$273$	0	0
$274$	5.20038e9	0.922641
$275$	0	0
$276$	0	0
$277$	3.31124e9	0.562434	0.281217	−	0.959644i	$-0.409262\pi$
$277$	3.31124e9	0.562434	0.281217	+	0.959644i	$0.409262\pi$
$278$	7.23641e9i	1.21156i
$279$	0	0
$280$	0	0
$281$	− 6.52601e9i	− 1.04670i	−0.852118	−	0.523350i	$-0.824682\pi$
$281$	− 6.52601e9i	− 1.04670i	0.852118	−	0.523350i	$-0.175318\pi$
$282$	0	0
$283$	1.10332e10	1.72010	0.860052	−	0.510207i	$-0.170431\pi$
$283$	1.10332e10	1.72010	0.860052	+	0.510207i	$0.170431\pi$
$284$	1.04849e10i	1.61173i
$285$	0	0
$286$	1.04381e9	0.156012
$287$	− 1.50378e10i	− 2.21644i
$288$	0	0
$289$	6.27458e9	0.899483
$290$	0	0
$291$	0	0
$292$	−8.88167e9	−1.22170
$293$	1.28273e10i	1.74046i	0.492646	+	0.870230i	$0.336030\pi$
$293$	1.28273e10i	1.74046i	−0.492646	+	0.870230i	$0.663970\pi$
$294$	0	0
$295$	0	0
$296$	1.58006e9i	0.205829i
$297$	0	0
$298$	4.45564e6	0.000564996 0
$299$	1.44603e8i	0.0180922i
$300$	0	0
$301$	2.00936e10	2.44789
$302$	− 1.35982e10i	− 1.63476i
$303$	0	0
$304$	−1.06508e10	−1.24706
$305$	0	0
$306$	0	0
$307$	8.62442e9	0.970904	0.485452	−	0.874263i	$-0.338655\pi$
$307$	8.62442e9	0.970904	0.485452	+	0.874263i	$0.338655\pi$
$308$	2.03281e10i	2.25889i
$309$	0	0
$310$	0	0
$311$	2.03281e9i	0.217298i	0.994080	+	0.108649i	$0.0346524\pi$
$311$	2.03281e9i	0.217298i	−0.994080	+	0.108649i	$0.965348\pi$
$312$	0	0
$313$	−7.06926e8	−0.0736541	−0.0368270	−	0.999322i	$-0.511725\pi$
$313$	−7.06926e8	−0.0736541	−0.0368270	+	0.999322i	$0.511725\pi$
$314$	− 1.84215e9i	− 0.189498i
$315$	0	0
$316$	−1.84568e10	−1.85100
$317$	1.51681e10i	1.50208i	0.660255	+	0.751041i	$0.270448\pi$
$317$	1.51681e10i	1.50208i	−0.660255	+	0.751041i	$0.729552\pi$
$318$	0	0
$319$	−1.50270e10	−1.45114
$320$	0	0
$321$	0	0
$322$	−5.34116e9	−0.496835
$323$	− 4.93840e9i	− 0.453707i
$324$	0	0
$325$	0	0
$326$	8.20765e9i	0.726689i
$327$	0	0
$328$	−2.34766e9	−0.202833
$329$	1.49571e10i	1.27663i
$330$	0	0
$331$	2.23026e10	1.85799	0.928994	−	0.370095i	$-0.120675\pi$
$331$	2.23026e10	1.85799	0.928994	+	0.370095i	$0.120675\pi$
$332$	2.18598e10i	1.79926i
$333$	0	0
$334$	−1.78515e10	−1.43446
$335$	0	0
$336$	0	0
$337$	−4.40375e9	−0.341431	−0.170716	−	0.985320i	$-0.554608\pi$
$337$	−4.40375e9	−0.341431	−0.170716	+	0.985320i	$0.554608\pi$
$338$	1.88037e10i	1.44071i
$339$	0	0
$340$	0	0
$341$	− 1.55531e10i	− 1.15027i
$342$	0	0
$343$	−3.44146e10	−2.48637
$344$	− 3.13696e9i	− 0.224014i
$345$	0	0
$346$	8.71347e9	0.607976
$347$	5.58698e9i	0.385353i	0.981262	+	0.192677i	$0.0617169\pi$
$347$	5.58698e9i	0.385353i	−0.981262	+	0.192677i	$0.938283\pi$
$348$	0	0
$349$	1.47015e10	0.990969	0.495485	−	0.868617i	$-0.334990\pi$
$349$	1.47015e10	0.990969	0.495485	+	0.868617i	$0.334990\pi$
$350$	0	0
$351$	0	0
$352$	−2.43543e10	−1.58637
$353$	5.68423e7i	0.00366078i	0.999998	+	0.00183039i	$0.000582631\pi$
$353$	5.68423e7i	0.00366078i	−0.999998	+	0.00183039i	$0.999417\pi$
$354$	0	0
$355$	0	0
$356$	1.44969e10i	0.902560i
$357$	0	0
$358$	−1.17503e10	−0.715346
$359$	9.56361e9i	0.575763i	0.957666	+	0.287882i	$0.0929510\pi$
$359$	9.56361e9i	0.575763i	−0.957666	+	0.287882i	$0.907049\pi$
$360$	0	0
$361$	1.77974e10	1.04792
$362$	− 2.64870e10i	− 1.54241i
$363$	0	0
$364$	3.48105e9	0.198292
$365$	0	0
$366$	0	0
$367$	2.94912e10	1.62565	0.812827	−	0.582505i	$-0.197927\pi$
$367$	2.94912e10	1.62565	0.812827	+	0.582505i	$0.197927\pi$
$368$	− 2.97968e9i	− 0.162472i
$369$	0	0
$370$	0	0
$371$	2.69554e10i	1.42282i
$372$	0	0
$373$	6.87435e9	0.355137	0.177569	−	0.984108i	$-0.443177\pi$
$373$	6.87435e9	0.355137	0.177569	+	0.984108i	$0.443177\pi$
$374$	− 9.97289e9i	− 0.509723i
$375$	0	0
$376$	2.33506e9	0.116828
$377$	2.57327e9i	0.127385i
$378$	0	0
$379$	6.23700e9	0.302287	0.151143	−	0.988512i	$-0.451704\pi$
$379$	6.23700e9	0.302287	0.151143	+	0.988512i	$0.451704\pi$
$380$	0	0
$381$	0	0
$382$	1.24300e10	0.583738
$383$	− 2.51985e10i	− 1.17106i	−0.810651	−	0.585530i	$-0.800886\pi$
$383$	− 2.51985e10i	− 1.17106i	0.810651	−	0.585530i	$-0.199114\pi$
$384$	0	0
$385$	0	0
$386$	− 3.85572e10i	− 1.73682i
$387$	0	0
$388$	2.46705e10	1.08856
$389$	1.81758e10i	0.793771i	0.917868	+	0.396886i	$0.129909\pi$
$389$	1.81758e10i	0.793771i	−0.917868	+	0.396886i	$0.870091\pi$
$390$	0	0
$391$	1.38158e9	0.0591111
$392$	9.33188e9i	0.395207i
$393$	0	0
$394$	4.36290e9	0.181046
$395$	0	0
$396$	0	0
$397$	−7.27352e9	−0.292808	−0.146404	−	0.989225i	$-0.546770\pi$
$397$	−7.27352e9	−0.292808	−0.146404	+	0.989225i	$0.546770\pi$
$398$	− 1.05771e10i	− 0.421537i
$399$	0	0
$400$	0	0
$401$	− 2.20001e10i	− 0.850839i	−0.904996	−	0.425420i	$-0.860126\pi$
$401$	− 2.20001e10i	− 0.850839i	0.904996	−	0.425420i	$-0.139874\pi$
$402$	0	0
$403$	−2.66336e9	−0.100974
$404$	3.58855e10i	1.34708i
$405$	0	0
$406$	−9.50484e10	−3.49817
$407$	3.72352e10i	1.35699i
$408$	0	0
$409$	−1.79326e8	−0.00640840	−0.00320420	−	0.999995i	$-0.501020\pi$
$409$	−1.79326e8	−0.00640840	−0.00320420	+	0.999995i	$0.501020\pi$
$410$	0	0
$411$	0	0
$412$	−7.51148e9	−0.260698
$413$	4.74864e10i	1.63218i
$414$	0	0
$415$	0	0
$416$	4.17050e9i	0.139256i
$417$	0	0
$418$	7.02388e10	2.30076
$419$	− 3.72051e10i	− 1.20711i	−0.797322	−	0.603554i	$-0.793751\pi$
$419$	− 3.72051e10i	− 1.20711i	0.797322	−	0.603554i	$-0.206249\pi$
$420$	0	0
$421$	−4.38448e9	−0.139569	−0.0697847	−	0.997562i	$-0.522231\pi$
$421$	−4.38448e9	−0.139569	−0.0697847	+	0.997562i	$0.522231\pi$
$422$	− 5.90744e10i	− 1.86273i
$423$	0	0
$424$	4.20820e9	0.130206
$425$	0	0
$426$	0	0
$427$	3.09788e10	0.931866
$428$	− 7.13266e10i	− 2.12558i
$429$	0	0
$430$	0	0
$431$	2.16425e10i	0.627190i	0.949557	+	0.313595i	$0.101533\pi$
$431$	2.16425e10i	0.627190i	−0.949557	+	0.313595i	$0.898467\pi$
$432$	0	0
$433$	6.74698e9	0.191937	0.0959683	−	0.995384i	$-0.469405\pi$
$433$	6.74698e9	0.191937	0.0959683	+	0.995384i	$0.469405\pi$
$434$	− 9.83761e10i	− 2.77288i
$435$	0	0
$436$	1.65854e9	0.0458966
$437$	9.73043e9i	0.266813i
$438$	0	0
$439$	−9.51260e9	−0.256119	−0.128059	−	0.991767i	$-0.540875\pi$
$439$	−9.51260e9	−0.256119	−0.128059	+	0.991767i	$0.540875\pi$
$440$	0	0
$441$	0	0
$442$	−1.70779e9	−0.0447451
$443$	− 3.20737e10i	− 0.832788i	−0.909184	−	0.416394i	$-0.863294\pi$
$443$	− 3.20737e10i	− 0.832788i	0.909184	−	0.416394i	$-0.136706\pi$
$444$	0	0
$445$	0	0
$446$	− 3.16281e10i	− 0.799343i
$447$	0	0
$448$	−8.97291e10	−2.22752
$449$	− 2.94112e10i	− 0.723648i	−0.932246	−	0.361824i	$-0.882154\pi$
$449$	− 2.94112e10i	− 0.723648i	0.932246	−	0.361824i	$-0.117846\pi$
$450$	0	0
$451$	−5.53243e10	−1.33724
$452$	− 4.29317e10i	− 1.02855i
$453$	0	0
$454$	2.14098e10	0.503951
$455$	0	0
$456$	0	0
$457$	−9.62260e9	−0.220611	−0.110306	−	0.993898i	$-0.535183\pi$
$457$	−9.62260e9	−0.220611	−0.110306	+	0.993898i	$0.535183\pi$
$458$	− 9.58573e10i	− 2.17853i
$459$	0	0
$460$	0	0
$461$	3.66842e9i	0.0812222i	0.999175	+	0.0406111i	$0.0129305\pi$
$461$	3.66842e9i	0.0812222i	−0.999175	+	0.0406111i	$0.987070\pi$
$462$	0	0
$463$	2.55583e10	0.556170	0.278085	−	0.960556i	$-0.410300\pi$
$463$	2.55583e10	0.556170	0.278085	+	0.960556i	$0.410300\pi$
$464$	− 5.30249e10i	− 1.14395i
$465$	0	0
$466$	3.85391e10	0.817255
$467$	− 3.61615e10i	− 0.760290i	−0.924927	−	0.380145i	$-0.875874\pi$
$467$	− 3.61615e10i	− 0.760290i	0.924927	−	0.380145i	$-0.124126\pi$
$468$	0	0
$469$	1.08355e11	2.23952
$470$	0	0
$471$	0	0
$472$	7.41344e9	0.149366
$473$	− 7.39249e10i	− 1.47688i
$474$	0	0
$475$	0	0
$476$	− 3.32591e10i	− 0.647863i
$477$	0	0
$478$	−6.58105e10	−1.26062
$479$	4.16338e10i	0.790868i	0.918494	+	0.395434i	$0.129406\pi$
$479$	4.16338e10i	0.790868i	−0.918494	+	0.395434i	$0.870594\pi$
$480$	0	0
$481$	6.37628e9	0.119121
$482$	1.30657e11i	2.42071i
$483$	0	0
$484$	1.35853e10	0.247564
$485$	0	0
$486$	0	0
$487$	−9.11488e8	−0.0162045	−0.00810224	−	0.999967i	$-0.502579\pi$
$487$	−9.11488e8	−0.0162045	−0.00810224	+	0.999967i	$0.502579\pi$
$488$	− 4.83632e9i	− 0.0852777i
$489$	0	0
$490$	0	0
$491$	− 5.05964e10i	− 0.870551i	−0.900297	−	0.435275i	$-0.856651\pi$
$491$	− 5.05964e10i	− 0.870551i	0.900297	−	0.435275i	$-0.143349\pi$
$492$	0	0
$493$	2.45859e10	0.416196
$494$	− 1.20279e10i	− 0.201968i
$495$	0	0
$496$	5.48813e10	0.906772
$497$	1.61551e11i	2.64779i
$498$	0	0
$499$	−2.13194e10	−0.343853	−0.171926	−	0.985110i	$-0.554999\pi$
$499$	−2.13194e10	−0.343853	−0.171926	+	0.985110i	$0.554999\pi$
$500$	0	0
$501$	0	0
$502$	−8.91074e10	−1.40313
$503$	− 8.15896e10i	− 1.27457i	−0.770629	−	0.637284i	$-0.780058\pi$
$503$	− 8.15896e10i	− 1.27457i	0.770629	−	0.637284i	$-0.219942\pi$
$504$	0	0
$505$	0	0
$506$	1.96502e10i	0.299754i
$507$	0	0
$508$	−7.68010e10	−1.15322
$509$	2.44976e10i	0.364966i	0.983209	+	0.182483i	$0.0584135\pi$
$509$	2.44976e10i	0.364966i	−0.983209	+	0.182483i	$0.941587\pi$
$510$	0	0
$511$	−1.36848e11	−2.00703
$512$	− 9.59783e10i	− 1.39667i
$513$	0	0
$514$	−1.25556e11	−1.79880
$515$	0	0
$516$	0	0
$517$	5.50274e10	0.770223
$518$	2.35519e11i	3.27121i
$519$	0	0
$520$	0	0
$521$	− 1.37580e11i	− 1.86726i	−0.358235	−	0.933631i	$-0.616621\pi$
$521$	− 1.37580e11i	− 1.86726i	0.358235	−	0.933631i	$-0.383379\pi$
$522$	0	0
$523$	−1.09839e10	−0.146808	−0.0734039	−	0.997302i	$-0.523386\pi$
$523$	−1.09839e10	−0.146808	−0.0734039	+	0.997302i	$0.523386\pi$
$524$	7.41868e10i	0.984015i
$525$	0	0
$526$	−6.16990e10	−0.806001
$527$	2.54466e10i	0.329904i
$528$	0	0
$529$	7.55888e10	0.965238
$530$	0	0
$531$	0	0
$532$	2.34243e11	2.92429
$533$	9.47391e9i	0.117387i
$534$	0	0
$535$	0	0
$536$	− 1.69160e10i	− 0.204945i
$537$	0	0
$538$	1.16753e11	1.39360
$539$	2.19913e11i	2.60553i
$540$	0	0
$541$	6.88987e10	0.804307	0.402154	−	0.915572i	$-0.368262\pi$
$541$	6.88987e10	0.804307	0.402154	+	0.915572i	$0.368262\pi$
$542$	3.87066e10i	0.448527i
$543$	0	0
$544$	3.98464e10	0.454981
$545$	0	0
$546$	0	0
$547$	7.36294e10	0.822436	0.411218	−	0.911537i	$-0.365103\pi$
$547$	7.36294e10	0.822436	0.411218	+	0.911537i	$0.365103\pi$
$548$	− 6.38053e10i	− 0.707513i
$549$	0	0
$550$	0	0
$551$	1.73158e11i	1.87860i
$552$	0	0
$553$	−2.84380e11	−3.04088
$554$	− 7.70540e10i	− 0.818005i
$555$	0	0
$556$	8.87861e10	0.929064
$557$	− 7.86391e10i	− 0.816992i	−0.912760	−	0.408496i	$-0.866053\pi$
$557$	− 7.86391e10i	− 0.816992i	0.912760	−	0.408496i	$-0.133947\pi$
$558$	0	0
$559$	−1.26591e10	−0.129645
$560$	0	0
$561$	0	0
$562$	−1.51863e11	−1.52232
$563$	6.63370e10i	0.660271i	0.943934	+	0.330135i	$0.107094\pi$
$563$	6.63370e10i	0.660271i	−0.943934	+	0.330135i	$0.892906\pi$
$564$	0	0
$565$	0	0
$566$	− 2.56747e11i	− 2.50172i
$567$	0	0
$568$	2.52209e10	0.242307
$569$	5.16708e10i	0.492943i	0.969150	+	0.246471i	$0.0792711\pi$
$569$	5.16708e10i	0.492943i	−0.969150	+	0.246471i	$0.920729\pi$
$570$	0	0
$571$	−1.04745e11	−0.985343	−0.492672	−	0.870215i	$-0.663980\pi$
$571$	−1.04745e11	−0.985343	−0.492672	+	0.870215i	$0.663980\pi$
$572$	− 1.28069e10i	− 0.119635i
$573$	0	0
$574$	−3.49936e11	−3.22360
$575$	0	0
$576$	0	0
$577$	−1.65645e11	−1.49443	−0.747216	−	0.664582i	$-0.768610\pi$
$577$	−1.65645e11	−1.49443	−0.747216	+	0.664582i	$0.768610\pi$
$578$	− 1.46012e11i	− 1.30821i
$579$	0	0
$580$	0	0
$581$	3.36814e11i	2.95587i
$582$	0	0
$583$	9.91694e10	0.858427
$584$	2.13643e10i	0.183670i
$585$	0	0
$586$	2.98496e11	2.53133
$587$	2.42404e10i	0.204168i	0.994776	+	0.102084i	$0.0325510\pi$
$587$	2.42404e10i	0.204168i	−0.994776	+	0.102084i	$0.967449\pi$
$588$	0	0
$589$	−1.79220e11	−1.48910
$590$	0	0
$591$	0	0
$592$	−1.31390e11	−1.06973
$593$	− 3.38219e10i	− 0.273514i	−0.990605	−	0.136757i	$-0.956332\pi$
$593$	− 3.38219e10i	− 0.273514i	0.990605	−	0.136757i	$-0.0436680\pi$
$594$	0	0
$595$	0	0
$596$	− 5.46678e7i	0 0.000433258i
$597$	0	0
$598$	3.36496e9	0.0263133
$599$	− 1.38008e10i	− 0.107200i	−0.998562	−	0.0536002i	$-0.982930\pi$
$599$	− 1.38008e10i	− 0.107200i	0.998562	−	0.0536002i	$-0.0170697\pi$
$600$	0	0
$601$	−1.27071e11	−0.973978	−0.486989	−	0.873408i	$-0.661905\pi$
$601$	−1.27071e11	−0.973978	−0.486989	+	0.873408i	$0.661905\pi$
$602$	− 4.67588e11i	− 3.56022i
$603$	0	0
$604$	−1.66841e11	−1.25359
$605$	0	0
$606$	0	0
$607$	−2.61072e10	−0.192312	−0.0961558	−	0.995366i	$-0.530655\pi$
$607$	−2.61072e10	−0.192312	−0.0961558	+	0.995366i	$0.530655\pi$
$608$	2.80637e11i	2.05367i
$609$	0	0
$610$	0	0
$611$	− 9.42306e9i	− 0.0676125i
$612$	0	0
$613$	−1.31329e10	−0.0930075	−0.0465037	−	0.998918i	$-0.514808\pi$
$613$	−1.31329e10	−0.0930075	−0.0465037	+	0.998918i	$0.514808\pi$
$614$	− 2.00694e11i	− 1.41208i
$615$	0	0
$616$	4.88980e10	0.339600
$617$	2.62461e11i	1.81102i	0.424321	+	0.905512i	$0.360513\pi$
$617$	2.62461e11i	1.81102i	−0.424321	+	0.905512i	$0.639487\pi$
$618$	0	0
$619$	4.91912e10	0.335062	0.167531	−	0.985867i	$-0.446421\pi$
$619$	4.91912e10	0.335062	0.167531	+	0.985867i	$0.446421\pi$
$620$	0	0
$621$	0	0
$622$	4.73044e10	0.316039
$623$	2.23367e11i	1.48275i
$624$	0	0
$625$	0	0
$626$	1.64505e10i	0.107123i
$627$	0	0
$628$	−2.26019e10	−0.145314
$629$	− 6.09211e10i	− 0.389193i
$630$	0	0
$631$	2.00621e11	1.26549	0.632745	−	0.774361i	$-0.281928\pi$
$631$	2.00621e11	1.26549	0.632745	+	0.774361i	$0.281928\pi$
$632$	4.43966e10i	0.278280i
$633$	0	0
$634$	3.52968e11	2.18463
$635$	0	0
$636$	0	0
$637$	3.76586e10	0.228721
$638$	3.49685e11i	2.11054i
$639$	0	0
$640$	0	0
$641$	8.38334e10i	0.496575i	0.968686	+	0.248288i	$0.0798678\pi$
$641$	8.38334e10i	0.496575i	−0.968686	+	0.248288i	$0.920132\pi$
$642$	0	0
$643$	−5.19580e10	−0.303954	−0.151977	−	0.988384i	$-0.548564\pi$
$643$	−5.19580e10	−0.303954	−0.151977	+	0.988384i	$0.548564\pi$
$644$	6.55325e10i	0.380990i
$645$	0	0
$646$	−1.14919e11	−0.659873
$647$	1.60226e11i	0.914358i	0.889375	+	0.457179i	$0.151140\pi$
$647$	1.60226e11i	0.914358i	−0.889375	+	0.457179i	$0.848860\pi$
$648$	0	0
$649$	1.74703e11	0.984742
$650$	0	0
$651$	0	0
$652$	1.00703e11	0.557250
$653$	− 6.01694e10i	− 0.330920i	−0.986217	−	0.165460i	$-0.947089\pi$
$653$	− 6.01694e10i	− 0.330920i	0.986217	−	0.165460i	$-0.0529109\pi$
$654$	0	0
$655$	0	0
$656$	− 1.95220e11i	− 1.05416i
$657$	0	0
$658$	3.48058e11	1.85673
$659$	− 5.94975e10i	− 0.315469i	−0.987482	−	0.157735i	$-0.949581\pi$
$659$	− 5.94975e10i	− 0.315469i	0.987482	−	0.157735i	$-0.0504190\pi$
$660$	0	0
$661$	−2.59920e10	−0.136155	−0.0680775	−	0.997680i	$-0.521686\pi$
$661$	−2.59920e10	−0.136155	−0.0680775	+	0.997680i	$0.521686\pi$
$662$	− 5.18990e11i	− 2.70226i
$663$	0	0
$664$	5.25824e10	0.270501
$665$	0	0
$666$	0	0
$667$	−4.84431e10	−0.244753
$668$	2.19026e11i	1.09999i
$669$	0	0
$670$	0	0
$671$	− 1.13972e11i	− 0.562220i
$672$	0	0
$673$	1.53578e11	0.748634	0.374317	−	0.927301i	$-0.377877\pi$
$673$	1.53578e11	0.748634	0.374317	+	0.927301i	$0.377877\pi$
$674$	1.02477e11i	0.496578i
$675$	0	0
$676$	2.30709e11	1.10478
$677$	3.15811e11i	1.50339i	0.659509	+	0.751697i	$0.270764\pi$
$677$	3.15811e11i	1.50339i	−0.659509	+	0.751697i	$0.729236\pi$
$678$	0	0
$679$	3.80121e11	1.78831
$680$	0	0
$681$	0	0
$682$	−3.61927e11	−1.67295
$683$	− 1.63003e11i	− 0.749052i	−0.927216	−	0.374526i	$-0.877805\pi$
$683$	− 1.63003e11i	− 0.749052i	0.927216	−	0.374526i	$-0.122195\pi$
$684$	0	0
$685$	0	0
$686$	8.00843e11i	3.61619i
$687$	0	0
$688$	2.60854e11	1.16425
$689$	− 1.69821e10i	− 0.0753553i
$690$	0	0
$691$	3.15836e10	0.138532	0.0692659	−	0.997598i	$-0.477934\pi$
$691$	3.15836e10	0.138532	0.0692659	+	0.997598i	$0.477934\pi$
$692$	− 1.06909e11i	− 0.466217i
$693$	0	0
$694$	1.30011e11	0.560459
$695$	0	0
$696$	0	0
$697$	9.05169e10	0.383529
$698$	− 3.42111e11i	− 1.44127i
$699$	0	0
$700$	0	0
$701$	8.43107e10i	0.349149i	0.984644	+	0.174574i	$0.0558550\pi$
$701$	8.43107e10i	0.349149i	−0.984644	+	0.174574i	$0.944145\pi$
$702$	0	0
$703$	4.29066e11	1.75672
$704$	3.30115e11i	1.34392i
$705$	0	0
$706$	1.32275e9	0.00532424
$707$	5.52920e11i	2.21302i
$708$	0	0
$709$	1.36293e11	0.539374	0.269687	−	0.962948i	$-0.413080\pi$
$709$	1.36293e11	0.539374	0.269687	+	0.962948i	$0.413080\pi$
$710$	0	0
$711$	0	0
$712$	3.48714e10	0.135691
$713$	− 5.01391e10i	− 0.194007i
$714$	0	0
$715$	0	0
$716$	1.44168e11i	0.548552i
$717$	0	0
$718$	2.22549e11	0.837392
$719$	3.82144e11i	1.42992i	0.699165	+	0.714960i	$0.253555\pi$
$719$	3.82144e11i	1.42992i	−0.699165	+	0.714960i	$0.746445\pi$
$720$	0	0
$721$	−1.15736e11	−0.428281
$722$	− 4.14154e11i	− 1.52410i
$723$	0	0
$724$	−3.24979e11	−1.18277
$725$	0	0
$726$	0	0
$727$	1.37646e11	0.492751	0.246375	−	0.969174i	$-0.420760\pi$
$727$	1.37646e11	0.492751	0.246375	+	0.969174i	$0.420760\pi$
$728$	− 8.37345e9i	− 0.0298112i
$729$	0	0
$730$	0	0
$731$	1.20949e11i	0.423579i
$732$	0	0
$733$	−3.87192e11	−1.34125	−0.670626	−	0.741796i	$-0.733974\pi$
$733$	−3.87192e11	−1.34125	−0.670626	+	0.741796i	$0.733974\pi$
$734$	− 6.86273e11i	− 2.36435i
$735$	0	0
$736$	−7.85117e10	−0.267561
$737$	− 3.98638e11i	− 1.35117i
$738$	0	0
$739$	5.79542e11	1.94315	0.971577	−	0.236726i	$-0.0760742\pi$
$739$	5.79542e11	1.94315	0.971577	+	0.236726i	$0.0760742\pi$
$740$	0	0
$741$	0	0
$742$	6.27264e11	2.06935
$743$	− 1.93713e11i	− 0.635630i	−0.948153	−	0.317815i	$-0.897051\pi$
$743$	− 1.93713e11i	− 0.635630i	0.948153	−	0.317815i	$-0.102949\pi$
$744$	0	0
$745$	0	0
$746$	− 1.59969e11i	− 0.516512i
$747$	0	0
$748$	−1.22361e11	−0.390874
$749$	− 1.09899e12i	− 3.49195i
$750$	0	0
$751$	−1.80812e11	−0.568418	−0.284209	−	0.958762i	$-0.591731\pi$
$751$	−1.80812e11	−0.568418	−0.284209	+	0.958762i	$0.591731\pi$
$752$	1.94172e11i	0.607177i
$753$	0	0
$754$	5.98811e10	0.185270
$755$	0	0
$756$	0	0
$757$	3.05384e11	0.929958	0.464979	−	0.885322i	$-0.346062\pi$
$757$	3.05384e11	0.929958	0.464979	+	0.885322i	$0.346062\pi$
$758$	− 1.45138e11i	− 0.439646i
$759$	0	0
$760$	0	0
$761$	2.84245e11i	0.847529i	0.905772	+	0.423765i	$0.139292\pi$
$761$	2.84245e11i	0.847529i	−0.905772	+	0.423765i	$0.860708\pi$
$762$	0	0
$763$	2.55547e10	0.0754001
$764$	− 1.52508e11i	− 0.447630i
$765$	0	0
$766$	−5.86380e11	−1.70319
$767$	− 2.99168e10i	− 0.0864437i
$768$	0	0
$769$	−5.48712e11	−1.56906	−0.784529	−	0.620092i	$-0.787095\pi$
$769$	−5.48712e11	−1.56906	−0.784529	+	0.620092i	$0.787095\pi$
$770$	0	0
$771$	0	0
$772$	−4.73071e11	−1.33186
$773$	2.22203e11i	0.622346i	0.950353	+	0.311173i	$0.100722\pi$
$773$	2.22203e11i	0.622346i	−0.950353	+	0.311173i	$0.899278\pi$
$774$	0	0
$775$	0	0
$776$	− 5.93434e10i	− 0.163653i
$777$	0	0
$778$	4.22959e11	1.15446
$779$	6.37508e11i	1.73115i
$780$	0	0
$781$	5.94349e11	1.59749
$782$	− 3.21500e10i	− 0.0859713i
$783$	0	0
$784$	−7.75994e11	−2.05397
$785$	0	0
$786$	0	0
$787$	−5.76769e10	−0.150350	−0.0751749	−	0.997170i	$-0.523952\pi$
$787$	−5.76769e10	−0.150350	−0.0751749	+	0.997170i	$0.523952\pi$
$788$	− 5.35299e10i	− 0.138833i
$789$	0	0
$790$	0	0
$791$	− 6.61488e11i	− 1.68972i
$792$	0	0
$793$	−1.95169e10	−0.0493534
$794$	1.69258e11i	0.425860i
$795$	0	0
$796$	−1.29775e11	−0.323249
$797$	− 5.38839e11i	− 1.33544i	−0.744411	−	0.667722i	$-0.767270\pi$
$797$	− 5.38839e11i	− 1.33544i	0.744411	−	0.667722i	$-0.232730\pi$
$798$	0	0
$799$	−9.00310e10	−0.220905
$800$	0	0
$801$	0	0
$802$	−5.11952e11	−1.23746
$803$	5.03466e11i	1.21090i
$804$	0	0
$805$	0	0
$806$	6.19776e10i	0.146857i
$807$	0	0
$808$	8.63203e10	0.202520
$809$	7.75938e11i	1.81148i	0.423837	+	0.905738i	$0.360683\pi$
$809$	7.75938e11i	1.81148i	−0.423837	+	0.905738i	$0.639317\pi$
$810$	0	0
$811$	−1.26913e11	−0.293375	−0.146688	−	0.989183i	$-0.546861\pi$
$811$	−1.26913e11	−0.293375	−0.146688	+	0.989183i	$0.546861\pi$
$812$	1.16618e12i	2.68251i
$813$	0	0
$814$	8.66480e11	1.97361
$815$	0	0
$816$	0	0
$817$	−8.51844e11	−1.91193
$818$	4.17299e9i	0.00932039i
$819$	0	0
$820$	0	0
$821$	− 3.93640e11i	− 0.866417i	−0.901294	−	0.433208i	$-0.857381\pi$
$821$	− 3.93640e11i	− 0.866417i	0.901294	−	0.433208i	$-0.142619\pi$
$822$	0	0
$823$	−8.13687e11	−1.77361	−0.886805	−	0.462144i	$-0.847080\pi$
$823$	−8.13687e11	−1.77361	−0.886805	+	0.462144i	$0.847080\pi$
$824$	1.80684e10i	0.0391932i
$825$	0	0
$826$	1.10503e12	2.37385
$827$	− 4.48558e11i	− 0.958950i	−0.877555	−	0.479475i	$-0.840827\pi$
$827$	− 4.48558e11i	− 0.958950i	0.877555	−	0.479475i	$-0.159173\pi$
$828$	0	0
$829$	3.50125e11	0.741319	0.370660	−	0.928769i	$-0.379132\pi$
$829$	3.50125e11	0.741319	0.370660	+	0.928769i	$0.379132\pi$
$830$	0	0
$831$	0	0
$832$	5.65299e10	0.117974
$833$	− 3.59803e11i	− 0.747281i
$834$	0	0
$835$	0	0
$836$	− 8.61785e11i	− 1.76430i
$837$	0	0
$838$	−8.65779e11	−1.75562
$839$	− 4.80271e10i	− 0.0969256i	−0.998825	−	0.0484628i	$-0.984568\pi$
$839$	− 4.80271e10i	− 0.0969256i	0.998825	−	0.0484628i	$-0.0154322\pi$
$840$	0	0
$841$	−3.61820e11	−0.723284
$842$	1.02029e11i	0.202990i
$843$	0	0
$844$	−7.24804e11	−1.42840
$845$	0	0
$846$	0	0
$847$	2.09320e11	0.406704
$848$	3.49933e11i	0.676709i
$849$	0	0
$850$	0	0
$851$	1.20037e11i	0.228873i
$852$	0	0
$853$	−2.42085e11	−0.457270	−0.228635	−	0.973512i	$-0.573426\pi$
$853$	−2.42085e11	−0.457270	−0.228635	+	0.973512i	$0.573426\pi$
$854$	− 7.20890e11i	− 1.35531i
$855$	0	0
$856$	−1.71572e11	−0.319559
$857$	− 2.23435e11i	− 0.414218i	−0.978318	−	0.207109i	$-0.933595\pi$
$857$	− 2.23435e11i	− 0.414218i	0.978318	−	0.207109i	$-0.0664054\pi$
$858$	0	0
$859$	−6.68565e11	−1.22792	−0.613961	−	0.789336i	$-0.710425\pi$
$859$	−6.68565e11	−1.22792	−0.613961	+	0.789336i	$0.710425\pi$
$860$	0	0
$861$	0	0
$862$	5.03631e11	0.912186
$863$	− 3.96860e11i	− 0.715475i	−0.933822	−	0.357738i	$-0.883548\pi$
$863$	− 3.96860e11i	− 0.715475i	0.933822	−	0.357738i	$-0.116452\pi$
$864$	0	0
$865$	0	0
$866$	− 1.57005e11i	− 0.279153i
$867$	0	0
$868$	−1.20701e12	−2.12634
$869$	1.04624e12i	1.83465i
$870$	0	0
$871$	−6.82641e10	−0.118610
$872$	− 3.98952e9i	− 0.00690009i
$873$	0	0
$874$	2.26431e11	0.388053
$875$	0	0
$876$	0	0
$877$	1.09419e12	1.84967	0.924837	−	0.380363i	$-0.124201\pi$
$877$	1.09419e12	1.84967	0.924837	+	0.380363i	$0.124201\pi$
$878$	2.21362e11i	0.372500i
$879$	0	0
$880$	0	0
$881$	− 7.89783e11i	− 1.31100i	−0.755193	−	0.655502i	$-0.772457\pi$
$881$	− 7.89783e11i	− 1.31100i	0.755193	−	0.655502i	$-0.227543\pi$
$882$	0	0
$883$	−2.41466e11	−0.397203	−0.198601	−	0.980080i	$-0.563640\pi$
$883$	−2.41466e11	−0.397203	−0.198601	+	0.980080i	$0.563640\pi$
$884$	2.09535e10i	0.0343121i
$885$	0	0
$886$	−7.46369e11	−1.21121
$887$	1.16400e11i	0.188043i	0.995570	+	0.0940216i	$0.0299723\pi$
$887$	1.16400e11i	0.188043i	−0.995570	+	0.0940216i	$0.970028\pi$
$888$	0	0
$889$	−1.18334e12	−1.89454
$890$	0	0
$891$	0	0
$892$	−3.88056e11	−0.612964
$893$	− 6.34086e11i	− 0.997109i
$894$	0	0
$895$	0	0
$896$	3.93372e11i	0.610340i
$897$	0	0
$898$	−6.84412e11	−1.05248
$899$	− 8.92249e11i	− 1.36599i
$900$	0	0
$901$	−1.62252e11	−0.246202
$902$	1.28742e12i	1.94489i
$903$	0	0
$904$	−1.03269e11	−0.154632
$905$	0	0
$906$	0	0
$907$	−1.88146e11	−0.278013	−0.139007	−	0.990291i	$-0.544391\pi$
$907$	−1.88146e11	−0.278013	−0.139007	+	0.990291i	$0.544391\pi$
$908$	− 2.62684e11i	− 0.386447i
$909$	0	0
$910$	0	0
$911$	− 3.22353e11i	− 0.468014i	−0.972235	−	0.234007i	$-0.924816\pi$
$911$	− 3.22353e11i	− 0.468014i	0.972235	−	0.234007i	$-0.0751838\pi$
$912$	0	0
$913$	1.23915e12	1.78336
$914$	2.23922e11i	0.320858i
$915$	0	0
$916$	−1.17611e12	−1.67057
$917$	1.14306e12i	1.61656i
$918$	0	0
$919$	−5.59401e11	−0.784262	−0.392131	−	0.919909i	$-0.628262\pi$
$919$	−5.59401e11	−0.784262	−0.392131	+	0.919909i	$0.628262\pi$
$920$	0	0
$921$	0	0
$922$	8.53656e10	0.118130
$923$	− 1.01778e11i	− 0.140232i
$924$	0	0
$925$	0	0
$926$	− 5.94752e11i	− 0.808895i
$927$	0	0
$928$	−1.39715e12	−1.88388
$929$	1.26527e12i	1.69872i	0.527817	+	0.849358i	$0.323011\pi$
$929$	1.26527e12i	1.69872i	−0.527817	+	0.849358i	$0.676989\pi$
$930$	0	0
$931$	2.53408e12	3.37304
$932$	− 4.72849e11i	− 0.626699i
$933$	0	0
$934$	−8.41495e11	−1.10577
$935$	0	0
$936$	0	0
$937$	1.14305e12	1.48289	0.741444	−	0.671015i	$-0.234141\pi$
$937$	1.14305e12	1.48289	0.741444	+	0.671015i	$0.234141\pi$
$938$	− 2.52146e12i	− 3.25717i
$939$	0	0
$940$	0	0
$941$	7.23423e11i	0.922643i	0.887233	+	0.461322i	$0.152625\pi$
$941$	7.23423e11i	0.922643i	−0.887233	+	0.461322i	$0.847375\pi$
$942$	0	0
$943$	−1.78351e11	−0.225543
$944$	6.16466e11i	0.776285i
$945$	0	0
$946$	−1.72026e12	−2.14798
$947$	3.09470e11i	0.384786i	0.981318	+	0.192393i	$0.0616248\pi$
$947$	3.09470e11i	0.384786i	−0.981318	+	0.192393i	$0.938375\pi$
$948$	0	0
$949$	8.62151e10	0.106296
$950$	0	0
$951$	0	0
$952$	−8.00027e10	−0.0973995
$953$	− 3.15952e11i	− 0.383045i	−0.981488	−	0.191522i	$-0.938658\pi$
$953$	− 3.15952e11i	− 0.383045i	0.981488	−	0.191522i	$-0.0613425\pi$
$954$	0	0
$955$	0	0
$956$	8.07453e11i	0.966686i
$957$	0	0
$958$	9.68837e11	1.15024
$959$	− 9.83106e11i	− 1.16232i
$960$	0	0
$961$	7.05960e10	0.0827726
$962$	− 1.48379e11i	− 0.173249i
$963$	0	0
$964$	1.60307e12	1.85628
$965$	0	0
$966$	0	0
$967$	2.47951e11	0.283570	0.141785	−	0.989897i	$-0.454716\pi$
$967$	2.47951e11	0.283570	0.141785	+	0.989897i	$0.454716\pi$
$968$	− 3.26785e10i	− 0.0372186i
$969$	0	0
$970$	0	0
$971$	− 7.68280e11i	− 0.864256i	−0.901812	−	0.432128i	$-0.857763\pi$
$971$	− 7.68280e11i	− 0.864256i	0.901812	−	0.432128i	$-0.142237\pi$
$972$	0	0
$973$	1.36801e12	1.52629
$974$	2.12107e10i	0.0235678i
$975$	0	0
$976$	4.02165e11	0.443205
$977$	− 1.31954e12i	− 1.44825i	−0.689666	−	0.724127i	$-0.742243\pi$
$977$	− 1.31954e12i	− 1.44825i	0.689666	−	0.724127i	$-0.257757\pi$
$978$	0	0
$979$	8.21773e11	0.894584
$980$	0	0
$981$	0	0
$982$	−1.17740e12	−1.26613
$983$	6.17012e11i	0.660815i	0.943838	+	0.330407i	$0.107186\pi$
$983$	6.17012e11i	0.660815i	−0.943838	+	0.330407i	$0.892814\pi$
$984$	0	0
$985$	0	0
$986$	− 5.72124e11i	− 0.605316i
$987$	0	0
$988$	−1.47575e11	−0.154876
$989$	− 2.38314e11i	− 0.249095i
$990$	0	0
$991$	−8.44484e11	−0.875582	−0.437791	−	0.899077i	$-0.644239\pi$
$991$	−8.44484e11	−0.875582	−0.437791	+	0.899077i	$0.644239\pi$
$992$	− 1.44607e12i	− 1.49328i
$993$	0	0
$994$	3.75936e12	3.85096
$995$	0	0
$996$	0	0
$997$	1.27342e12	1.28881	0.644406	−	0.764683i	$-0.277105\pi$
$997$	1.27342e12	1.28881	0.644406	+	0.764683i	$0.277105\pi$
$998$	4.96112e11i	0.500100i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.9.c.d.26.3		12
3.2	odd	2	inner	225.9.c.d.26.10		12
5.2	odd	4		225.9.d.c.224.19		24
5.3	odd	4		225.9.d.c.224.6		24
5.4	even	2		45.9.c.a.26.10	yes	12
15.2	even	4		225.9.d.c.224.5		24
15.8	even	4		225.9.d.c.224.20		24
15.14	odd	2		45.9.c.a.26.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.9.c.a.26.3	✓	12	15.14	odd	2
45.9.c.a.26.10	yes	12	5.4	even	2
225.9.c.d.26.3		12	1.1	even	1	trivial
225.9.c.d.26.10		12	3.2	odd	2	inner
225.9.d.c.224.5		24	15.2	even	4
225.9.d.c.224.6		24	5.3	odd	4
225.9.d.c.224.19		24	5.2	odd	4
225.9.d.c.224.20		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)

\(f(q)\)	\(=\)	\(q-23.2704i q^{2} -285.513 q^{4} -4399.16 q^{7} +686.784i q^{8} +16184.6i q^{11} +2771.50 q^{13} +102370. i q^{14} -57109.6 q^{16} -26479.8i q^{17} +186497. q^{19} +376622. q^{22} +52174.8i q^{23} -64494.0i q^{26} +1.25602e6 q^{28} +928476. i q^{29} -960982. q^{31} +1.50478e6i q^{32} -616197. q^{34} +2.30066e6 q^{37} -4.33986e6i q^{38} +3.41833e6i q^{41} -4.56761e6 q^{43} -4.62091e6i q^{44} +1.21413e6 q^{46} -3.39999e6i q^{47} +1.35878e7 q^{49} -791299. q^{52} -6.12740e6i q^{53} -3.02127e6i q^{56} +2.16060e7 q^{58} -1.07944e7i q^{59} -7.04198e6 q^{61} +2.23625e7i q^{62} +2.03969e7 q^{64} -2.46307e7 q^{67} +7.56034e6i q^{68} -3.67232e7i q^{71} +3.11078e7 q^{73} -5.35374e7i q^{74} -5.32472e7 q^{76} -7.11986e7i q^{77} +6.46442e7 q^{79} +7.95461e7 q^{82} -7.65633e7i q^{83} +1.06290e8i q^{86} -1.11153e7 q^{88} -5.07750e7i q^{89} -1.21923e7 q^{91} -1.48966e7i q^{92} -7.91192e7 q^{94} -8.64077e7 q^{97} -3.16194e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2524 q^{4} + 6928 q^{7} - 124408 q^{13} + 297572 q^{16} + 124960 q^{19} + 1370512 q^{22} - 1114496 q^{28} - 620968 q^{31} + 7486496 q^{34} + 11533176 q^{37} - 14405296 q^{43} - 7161768 q^{46} + 21010156 q^{49}+ \cdots + 302816184 q^{97}+O(q^{100}) \) 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

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Database

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