LMFDB - Embedded newform 1225.2.b.a.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1225.99
Dual form			1225.2.b.a.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +3.00000i q^{3} -2.00000 q^{4} -6.00000 q^{6} -6.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{2} +3.00000i q^{3} -2.00000 q^{4} -6.00000 q^{6} -6.00000 q^{9} +1.00000 q^{11} -6.00000i q^{12} +3.00000i q^{13} -4.00000 q^{16} +3.00000i q^{17} -12.0000i q^{18} -6.00000 q^{19} +2.00000i q^{22} -4.00000i q^{23} -6.00000 q^{26} -9.00000i q^{27} +1.00000 q^{29} +6.00000 q^{31} -8.00000i q^{32} +3.00000i q^{33} -6.00000 q^{34} +12.0000 q^{36} -12.0000i q^{38} -9.00000 q^{39} +6.00000 q^{41} -6.00000i q^{43} -2.00000 q^{44} +8.00000 q^{46} +9.00000i q^{47} -12.0000i q^{48} -9.00000 q^{51} -6.00000i q^{52} -10.0000i q^{53} +18.0000 q^{54} -18.0000i q^{57} +2.00000i q^{58} +6.00000 q^{59} +12.0000i q^{62} +8.00000 q^{64} -6.00000 q^{66} +14.0000i q^{67} -6.00000i q^{68} +12.0000 q^{69} -8.00000 q^{71} +6.00000i q^{73} +12.0000 q^{76} -18.0000i q^{78} +1.00000 q^{79} +9.00000 q^{81} +12.0000i q^{82} +12.0000i q^{83} +12.0000 q^{86} +3.00000i q^{87} -12.0000 q^{89} +8.00000i q^{92} +18.0000i q^{93} -18.0000 q^{94} +24.0000 q^{96} +15.0000i q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 12 q^{6} - 12 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 - 12 * q^6 - 12 * q^9	$ 2 q - 4 q^{4} - 12 q^{6} - 12 q^{9} + 2 q^{11} - 8 q^{16} - 12 q^{19} - 12 q^{26} + 2 q^{29} + 12 q^{31} - 12 q^{34} + 24 q^{36} - 18 q^{39} + 12 q^{41} - 4 q^{44} + 16 q^{46} - 18 q^{51} + 36 q^{54} + 12 q^{59} + 16 q^{64} - 12 q^{66} + 24 q^{69} - 16 q^{71} + 24 q^{76} + 2 q^{79} + 18 q^{81} + 24 q^{86} - 24 q^{89} - 36 q^{94} + 48 q^{96} - 12 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 - 12 * q^6 - 12 * q^9 + 2 * q^11 - 8 * q^16 - 12 * q^19 - 12 * q^26 + 2 * q^29 + 12 * q^31 - 12 * q^34 + 24 * q^36 - 18 * q^39 + 12 * q^41 - 4 * q^44 + 16 * q^46 - 18 * q^51 + 36 * q^54 + 12 * q^59 + 16 * q^64 - 12 * q^66 + 24 * q^69 - 16 * q^71 + 24 * q^76 + 2 * q^79 + 18 * q^81 + 24 * q^86 - 24 * q^89 - 36 * q^94 + 48 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$	2.00000i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	3.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	3.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	−6.00000	−2.44949
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	− 6.00000i	− 1.73205i
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	− 12.0000i	− 2.82843i
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	2.00000i	0.426401i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	−6.00000	−1.17670
$27$	− 9.00000i	− 1.73205i
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	− 8.00000i	− 1.41421i
$33$	3.00000i	0.522233i
$34$	−6.00000	−1.02899
$35$	0	0
$36$	12.0000	2.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	− 12.0000i	− 1.94666i
$39$	−9.00000	−1.44115
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	8.00000	1.17954
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	− 12.0000i	− 1.73205i
$49$	0	0
$50$	0	0
$51$	−9.00000	−1.26025
$52$	− 6.00000i	− 0.832050i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	18.0000	2.44949
$55$	0	0
$56$	0	0
$57$	− 18.0000i	− 2.38416i
$58$	2.00000i	0.262613i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	12.0000i	1.52400i
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	14.0000i	1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$	14.0000i	1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$	− 6.00000i	− 0.727607i
$69$	12.0000	1.44463
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	12.0000	1.37649
$77$	0	0
$78$	− 18.0000i	− 2.03810i
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	12.0000i	1.32518i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	12.0000	1.29399
$87$	3.00000i	0.321634i
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0	0
$92$	8.00000i	0.834058i
$93$	18.0000i	1.86651i
$94$	−18.0000	−1.85656
$95$	0	0
$96$	24.0000	2.44949
$97$	15.0000i	1.52302i	0.648154	+	0.761510i	$0.275541\pi$
$97$	15.0000i	1.52302i	−0.648154	+	0.761510i	$0.724459\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	− 18.0000i	− 1.78227i
$103$	− 9.00000i	− 0.886796i	−0.896325	−	0.443398i	$-0.853773\pi$
$103$	− 9.00000i	− 0.886796i	0.896325	−	0.443398i	$-0.146227\pi$
$104$	0	0
$105$	0	0
$106$	20.0000	1.94257
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	18.0000i	1.73205i
$109$	15.0000	1.43674	0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000	1.43674	0.718370	+	0.695662i	$0.244889\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.00000i	0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$	8.00000i	0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$	36.0000	3.37171
$115$	0	0
$116$	−2.00000	−0.185695
$117$	− 18.0000i	− 1.66410i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	18.0000i	1.62301i
$124$	−12.0000	−1.07763
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	18.0000	1.58481
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	− 6.00000i	− 0.522233i
$133$	0	0
$134$	−28.0000	−2.41883
$135$	0	0
$136$	0	0
$137$	− 8.00000i	− 0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	− 8.00000i	− 0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$	24.0000i	2.04302i
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−27.0000	−2.27381
$142$	− 16.0000i	− 1.34269i
$143$	3.00000i	0.250873i
$144$	24.0000	2.00000
$145$	0	0
$146$	−12.0000	−0.993127
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	0	0
$153$	− 18.0000i	− 1.45521i
$154$	0	0
$155$	0	0
$156$	18.0000	1.44115
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	2.00000i	0.159111i
$159$	30.0000	2.37915
$160$	0	0
$161$	0	0
$162$	18.0000i	1.41421i
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	−24.0000	−1.86276
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	36.0000	2.75299
$172$	12.0000i	0.914991i
$173$	− 3.00000i	− 0.228086i	−0.993476	−	0.114043i	$-0.963620\pi$
$173$	− 3.00000i	− 0.228086i	0.993476	−	0.114043i	$-0.0363801\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	18.0000i	1.35296i
$178$	− 24.0000i	− 1.79888i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	−36.0000	−2.63965
$187$	3.00000i	0.219382i
$188$	− 18.0000i	− 1.31278i
$189$	0	0
$190$	0	0
$191$	17.0000	1.23008	0.615038	−	0.788497i	$-0.289140\pi$
$191$	17.0000	1.23008	0.615038	+	0.788497i	$0.289140\pi$
$192$	24.0000i	1.73205i
$193$	12.0000i	0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$	12.0000i	0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$	−30.0000	−2.15387
$195$	0	0
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	− 12.0000i	− 0.852803i
$199$	−6.00000	−0.425329	−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000	−0.425329	−0.212664	+	0.977125i	$0.568214\pi$
$200$	0	0
$201$	−42.0000	−2.96245
$202$	− 36.0000i	− 2.53295i
$203$	0	0
$204$	18.0000	1.26025
$205$	0	0
$206$	18.0000	1.25412
$207$	24.0000i	1.66812i
$208$	− 12.0000i	− 0.832050i
$209$	−6.00000	−0.415029
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	20.0000i	1.37361i
$213$	− 24.0000i	− 1.64445i
$214$	−4.00000	−0.273434
$215$	0	0
$216$	0	0
$217$	0	0
$218$	30.0000i	2.03186i
$219$	−18.0000	−1.21633
$220$	0	0
$221$	−9.00000	−0.605406
$222$	0	0
$223$	3.00000i	0.200895i	0.994942	+	0.100447i	$0.0320274\pi$
$223$	3.00000i	0.200895i	−0.994942	+	0.100447i	$0.967973\pi$
$224$	0	0
$225$	0	0
$226$	−16.0000	−1.06430
$227$	− 3.00000i	− 0.199117i	−0.995032	−	0.0995585i	$-0.968257\pi$
$227$	− 3.00000i	− 0.199117i	0.995032	−	0.0995585i	$-0.0317430\pi$
$228$	36.0000i	2.38416i
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 8.00000i	− 0.524097i	−0.965055	−	0.262049i	$-0.915602\pi$
$233$	− 8.00000i	− 0.524097i	0.965055	−	0.262049i	$-0.0843981\pi$
$234$	36.0000	2.35339
$235$	0	0
$236$	−12.0000	−0.781133
$237$	3.00000i	0.194871i
$238$	0	0
$239$	−7.00000	−0.452792	−0.226396	−	0.974035i	$-0.572694\pi$
$239$	−7.00000	−0.452792	−0.226396	+	0.974035i	$0.572694\pi$
$240$	0	0
$241$	−24.0000	−1.54598	−0.772988	−	0.634421i	$-0.781239\pi$
$241$	−24.0000	−1.54598	−0.772988	+	0.634421i	$0.781239\pi$
$242$	− 20.0000i	− 1.28565i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	−36.0000	−2.29528
$247$	− 18.0000i	− 1.14531i
$248$	0	0
$249$	−36.0000	−2.28141
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	16.0000	1.00000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	36.0000i	2.24126i
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	24.0000i	1.48272i
$263$	− 10.0000i	− 0.616626i	−0.951285	−	0.308313i	$-0.900236\pi$
$263$	− 10.0000i	− 0.616626i	0.951285	−	0.308313i	$-0.0997645\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 36.0000i	− 2.20316i
$268$	− 28.0000i	− 1.71037i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	− 12.0000i	− 0.727607i
$273$	0	0
$274$	16.0000	0.966595
$275$	0	0
$276$	−24.0000	−1.44463
$277$	28.0000i	1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$	28.0000i	1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	−36.0000	−2.15526
$280$	0	0
$281$	−5.00000	−0.298275	−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000	−0.298275	−0.149137	+	0.988816i	$0.547650\pi$
$282$	− 54.0000i	− 3.21565i
$283$	21.0000i	1.24832i	0.781296	+	0.624160i	$0.214559\pi$
$283$	21.0000i	1.24832i	−0.781296	+	0.624160i	$0.785441\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	48.0000i	2.82843i
$289$	8.00000	0.470588
$290$	0	0
$291$	−45.0000	−2.63795
$292$	− 12.0000i	− 0.702247i
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 9.00000i	− 0.522233i
$298$	− 20.0000i	− 1.15857i
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	30.0000i	1.72631i
$303$	− 54.0000i	− 3.10222i
$304$	24.0000	1.37649
$305$	0	0
$306$	36.0000	2.05798
$307$	3.00000i	0.171219i	0.996329	+	0.0856095i	$0.0272838\pi$
$307$	3.00000i	0.171219i	−0.996329	+	0.0856095i	$0.972716\pi$
$308$	0	0
$309$	27.0000	1.53598
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	3.00000i	0.169570i	0.996399	+	0.0847850i	$0.0270203\pi$
$313$	3.00000i	0.169570i	−0.996399	+	0.0847850i	$0.972980\pi$
$314$	36.0000	2.03160
$315$	0	0
$316$	−2.00000	−0.112509
$317$	22.0000i	1.23564i	0.786318	+	0.617822i	$0.211985\pi$
$317$	22.0000i	1.23564i	−0.786318	+	0.617822i	$0.788015\pi$
$318$	60.0000i	3.36463i
$319$	1.00000	0.0559893
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	− 18.0000i	− 1.00155i
$324$	−18.0000	−1.00000
$325$	0	0
$326$	−32.0000	−1.77232
$327$	45.0000i	2.48851i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	− 24.0000i	− 1.31717i
$333$	0	0
$334$	6.00000	0.328305
$335$	0	0
$336$	0	0
$337$	24.0000i	1.30736i	0.756770	+	0.653682i	$0.226776\pi$
$337$	24.0000i	1.30736i	−0.756770	+	0.653682i	$0.773224\pi$
$338$	8.00000i	0.435143i
$339$	−24.0000	−1.30350
$340$	0	0
$341$	6.00000	0.324918
$342$	72.0000i	3.89331i
$343$	0	0
$344$	0	0
$345$	0	0
$346$	6.00000	0.322562
$347$	− 16.0000i	− 0.858925i	−0.903085	−	0.429463i	$-0.858703\pi$
$347$	− 16.0000i	− 0.858925i	0.903085	−	0.429463i	$-0.141297\pi$
$348$	− 6.00000i	− 0.321634i
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	27.0000	1.44115
$352$	− 8.00000i	− 0.426401i
$353$	− 3.00000i	− 0.159674i	−0.996808	−	0.0798369i	$-0.974560\pi$
$353$	− 3.00000i	− 0.159674i	0.996808	−	0.0798369i	$-0.0254400\pi$
$354$	−36.0000	−1.91338
$355$	0	0
$356$	24.0000	1.27200
$357$	0	0
$358$	− 8.00000i	− 0.422813i
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 12.0000i	− 0.630706i
$363$	− 30.0000i	− 1.57459i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	33.0000i	1.72259i	0.508109	+	0.861293i	$0.330345\pi$
$367$	33.0000i	1.72259i	−0.508109	+	0.861293i	$0.669655\pi$
$368$	16.0000i	0.834058i
$369$	−36.0000	−1.87409
$370$	0	0
$371$	0	0
$372$	− 36.0000i	− 1.86651i
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	−6.00000	−0.310253
$375$	0	0
$376$	0	0
$377$	3.00000i	0.154508i
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	34.0000i	1.73959i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−24.0000	−1.22157
$387$	36.0000i	1.82998i
$388$	− 30.0000i	− 1.52302i
$389$	−13.0000	−0.659126	−0.329563	−	0.944134i	$-0.606901\pi$
$389$	−13.0000	−0.659126	−0.329563	+	0.944134i	$0.606901\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	36.0000i	1.81596i
$394$	−4.00000	−0.201517
$395$	0	0
$396$	12.0000	0.603023
$397$	− 9.00000i	− 0.451697i	−0.974162	−	0.225849i	$-0.927485\pi$
$397$	− 9.00000i	− 0.451697i	0.974162	−	0.225849i	$-0.0725154\pi$
$398$	− 12.0000i	− 0.601506i
$399$	0	0
$400$	0	0
$401$	−19.0000	−0.948815	−0.474407	−	0.880305i	$-0.657338\pi$
$401$	−19.0000	−0.948815	−0.474407	+	0.880305i	$0.657338\pi$
$402$	− 84.0000i	− 4.18954i
$403$	18.0000i	0.896644i
$404$	36.0000	1.79107
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	24.0000	1.18383
$412$	18.0000i	0.886796i
$413$	0	0
$414$	−48.0000	−2.35907
$415$	0	0
$416$	24.0000	1.17670
$417$	0	0
$418$	− 12.0000i	− 0.586939i
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	30.0000i	1.46038i
$423$	− 54.0000i	− 2.62557i
$424$	0	0
$425$	0	0
$426$	48.0000	2.32561
$427$	0	0
$428$	− 4.00000i	− 0.193347i
$429$	−9.00000	−0.434524
$430$	0	0
$431$	5.00000	0.240842	0.120421	−	0.992723i	$-0.461576\pi$
$431$	5.00000	0.240842	0.120421	+	0.992723i	$0.461576\pi$
$432$	36.0000i	1.73205i
$433$	− 30.0000i	− 1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$	− 30.0000i	− 1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$	0	0
$435$	0	0
$436$	−30.0000	−1.43674
$437$	24.0000i	1.14808i
$438$	− 36.0000i	− 1.72015i
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	0	0
$442$	− 18.0000i	− 0.856173i
$443$	34.0000i	1.61539i	0.589601	+	0.807694i	$0.299285\pi$
$443$	34.0000i	1.61539i	−0.589601	+	0.807694i	$0.700715\pi$
$444$	0	0
$445$	0	0
$446$	−6.00000	−0.284108
$447$	− 30.0000i	− 1.41895i
$448$	0	0
$449$	−23.0000	−1.08544	−0.542719	−	0.839915i	$-0.682605\pi$
$449$	−23.0000	−1.08544	−0.542719	+	0.839915i	$0.682605\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	− 16.0000i	− 0.752577i
$453$	45.0000i	2.11428i
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	12.0000i	0.560723i
$459$	27.0000	1.26025
$460$	0	0
$461$	36.0000	1.67669	0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000	1.67669	0.838344	+	0.545142i	$0.183524\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	16.0000	0.741186
$467$	15.0000i	0.694117i	0.937843	+	0.347059i	$0.112820\pi$
$467$	15.0000i	0.694117i	−0.937843	+	0.347059i	$0.887180\pi$
$468$	36.0000i	1.66410i
$469$	0	0
$470$	0	0
$471$	54.0000	2.48819
$472$	0	0
$473$	− 6.00000i	− 0.275880i
$474$	−6.00000	−0.275589
$475$	0	0
$476$	0	0
$477$	60.0000i	2.74721i
$478$	− 14.0000i	− 0.640345i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	0	0
$482$	− 48.0000i	− 2.18634i
$483$	0	0
$484$	20.0000	0.909091
$485$	0	0
$486$	0	0
$487$	− 36.0000i	− 1.63132i	−0.578535	−	0.815658i	$-0.696375\pi$
$487$	− 36.0000i	− 1.63132i	0.578535	−	0.815658i	$-0.303625\pi$
$488$	0	0
$489$	−48.0000	−2.17064
$490$	0	0
$491$	23.0000	1.03798	0.518988	−	0.854782i	$-0.326309\pi$
$491$	23.0000	1.03798	0.518988	+	0.854782i	$0.326309\pi$
$492$	− 36.0000i	− 1.62301i
$493$	3.00000i	0.135113i
$494$	36.0000	1.61972
$495$	0	0
$496$	−24.0000	−1.07763
$497$	0	0
$498$	− 72.0000i	− 3.22640i
$499$	27.0000	1.20869	0.604343	−	0.796724i	$-0.293436\pi$
$499$	27.0000	1.20869	0.604343	+	0.796724i	$0.293436\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	− 24.0000i	− 1.07117i
$503$	9.00000i	0.401290i	0.979664	+	0.200645i	$0.0643038\pi$
$503$	9.00000i	0.401290i	−0.979664	+	0.200645i	$0.935696\pi$
$504$	0	0
$505$	0	0
$506$	8.00000	0.355643
$507$	12.0000i	0.532939i
$508$	− 4.00000i	− 0.177471i
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	0	0
$512$	32.0000i	1.41421i
$513$	54.0000i	2.38416i
$514$	36.0000	1.58789
$515$	0	0
$516$	−36.0000	−1.58481
$517$	9.00000i	0.395820i
$518$	0	0
$519$	9.00000	0.395056
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	− 12.0000i	− 0.525226i
$523$	24.0000i	1.04945i	0.851273	+	0.524723i	$0.175831\pi$
$523$	24.0000i	1.04945i	−0.851273	+	0.524723i	$0.824169\pi$
$524$	−24.0000	−1.04844
$525$	0	0
$526$	20.0000	0.872041
$527$	18.0000i	0.784092i
$528$	− 12.0000i	− 0.522233i
$529$	7.00000	0.304348
$530$	0	0
$531$	−36.0000	−1.56227
$532$	0	0
$533$	18.0000i	0.779667i
$534$	72.0000	3.11574
$535$	0	0
$536$	0	0
$537$	− 12.0000i	− 0.517838i
$538$	− 48.0000i	− 2.06943i
$539$	0	0
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	48.0000i	2.06178i
$543$	− 18.0000i	− 0.772454i
$544$	24.0000	1.02899
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	16.0000i	0.683486i
$549$	0	0
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	0	0
$554$	−56.0000	−2.37921
$555$	0	0
$556$	0	0
$557$	− 4.00000i	− 0.169485i	−0.996403	−	0.0847427i	$-0.972993\pi$
$557$	− 4.00000i	− 0.169485i	0.996403	−	0.0847427i	$-0.0270068\pi$
$558$	− 72.0000i	− 3.04800i
$559$	18.0000	0.761319
$560$	0	0
$561$	−9.00000	−0.379980
$562$	− 10.0000i	− 0.421825i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	54.0000	2.27381
$565$	0	0
$566$	−42.0000	−1.76539
$567$	0	0
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	− 6.00000i	− 0.250873i
$573$	51.0000i	2.13056i
$574$	0	0
$575$	0	0
$576$	−48.0000	−2.00000
$577$	− 15.0000i	− 0.624458i	−0.950007	−	0.312229i	$-0.898924\pi$
$577$	− 15.0000i	− 0.624458i	0.950007	−	0.312229i	$-0.101076\pi$
$578$	16.0000i	0.665512i
$579$	−36.0000	−1.49611
$580$	0	0
$581$	0	0
$582$	− 90.0000i	− 3.73062i
$583$	− 10.0000i	− 0.414158i
$584$	0	0
$585$	0	0
$586$	−18.0000	−0.743573
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	− 45.0000i	− 1.84793i	−0.382479	−	0.923964i	$-0.624930\pi$
$593$	− 45.0000i	− 1.84793i	0.382479	−	0.923964i	$-0.375070\pi$
$594$	18.0000	0.738549
$595$	0	0
$596$	20.0000	0.819232
$597$	− 18.0000i	− 0.736691i
$598$	24.0000i	0.981433i
$599$	−17.0000	−0.694601	−0.347301	−	0.937754i	$-0.612902\pi$
$599$	−17.0000	−0.694601	−0.347301	+	0.937754i	$0.612902\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	− 84.0000i	− 3.42074i
$604$	−30.0000	−1.22068
$605$	0	0
$606$	108.000	4.38720
$607$	− 21.0000i	− 0.852364i	−0.904638	−	0.426182i	$-0.859858\pi$
$607$	− 21.0000i	− 0.852364i	0.904638	−	0.426182i	$-0.140142\pi$
$608$	48.0000i	1.94666i
$609$	0	0
$610$	0	0
$611$	−27.0000	−1.09230
$612$	36.0000i	1.45521i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	−6.00000	−0.242140
$615$	0	0
$616$	0	0
$617$	− 4.00000i	− 0.161034i	−0.996753	−	0.0805170i	$-0.974343\pi$
$617$	− 4.00000i	− 0.161034i	0.996753	−	0.0805170i	$-0.0256571\pi$
$618$	54.0000i	2.17220i
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	−36.0000	−1.44463
$622$	− 12.0000i	− 0.481156i
$623$	0	0
$624$	36.0000	1.44115
$625$	0	0
$626$	−6.00000	−0.239808
$627$	− 18.0000i	− 0.718851i
$628$	36.0000i	1.43656i
$629$	0	0
$630$	0	0
$631$	−11.0000	−0.437903	−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000	−0.437903	−0.218952	+	0.975736i	$0.570264\pi$
$632$	0	0
$633$	45.0000i	1.78859i
$634$	−44.0000	−1.74746
$635$	0	0
$636$	−60.0000	−2.37915
$637$	0	0
$638$	2.00000i	0.0791808i
$639$	48.0000	1.89885
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	− 12.0000i	− 0.473602i
$643$	27.0000i	1.06478i	0.846500	+	0.532388i	$0.178705\pi$
$643$	27.0000i	1.06478i	−0.846500	+	0.532388i	$0.821295\pi$
$644$	0	0
$645$	0	0
$646$	36.0000	1.41640
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	0	0
$652$	− 32.0000i	− 1.25322i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	−90.0000	−3.51928
$655$	0	0
$656$	−24.0000	−0.937043
$657$	− 36.0000i	− 1.40449i
$658$	0	0
$659$	19.0000	0.740135	0.370067	−	0.929005i	$-0.379335\pi$
$659$	19.0000	0.740135	0.370067	+	0.929005i	$0.379335\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	24.0000i	0.932786i
$663$	− 27.0000i	− 1.04859i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.00000i	− 0.154881i
$668$	6.00000i	0.232147i
$669$	−9.00000	−0.347960
$670$	0	0
$671$	0	0
$672$	0	0
$673$	12.0000i	0.462566i	0.972887	+	0.231283i	$0.0742923\pi$
$673$	12.0000i	0.462566i	−0.972887	+	0.231283i	$0.925708\pi$
$674$	−48.0000	−1.84889
$675$	0	0
$676$	−8.00000	−0.307692
$677$	9.00000i	0.345898i	0.984931	+	0.172949i	$0.0553296\pi$
$677$	9.00000i	0.345898i	−0.984931	+	0.172949i	$0.944670\pi$
$678$	− 48.0000i	− 1.84343i
$679$	0	0
$680$	0	0
$681$	9.00000	0.344881
$682$	12.0000i	0.459504i
$683$	8.00000i	0.306111i	0.988218	+	0.153056i	$0.0489114\pi$
$683$	8.00000i	0.306111i	−0.988218	+	0.153056i	$0.951089\pi$
$684$	−72.0000	−2.75299
$685$	0	0
$686$	0	0
$687$	18.0000i	0.686743i
$688$	24.0000i	0.914991i
$689$	30.0000	1.14291
$690$	0	0
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	32.0000	1.21470
$695$	0	0
$696$	0	0
$697$	18.0000i	0.681799i
$698$	60.0000i	2.27103i
$699$	24.0000	0.907763
$700$	0	0
$701$	47.0000	1.77517	0.887583	−	0.460648i	$-0.152383\pi$
$701$	47.0000	1.77517	0.887583	+	0.460648i	$0.152383\pi$
$702$	54.0000i	2.03810i
$703$	0	0
$704$	8.00000	0.301511
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	− 36.0000i	− 1.35296i
$709$	33.0000	1.23934	0.619671	−	0.784862i	$-0.287266\pi$
$709$	33.0000	1.23934	0.619671	+	0.784862i	$0.287266\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	− 24.0000i	− 0.898807i
$714$	0	0
$715$	0	0
$716$	8.00000	0.298974
$717$	− 21.0000i	− 0.784259i
$718$	− 64.0000i	− 2.38846i
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	34.0000i	1.26535i
$723$	− 72.0000i	− 2.67771i
$724$	12.0000	0.445976
$725$	0	0
$726$	60.0000	2.22681
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	18.0000	0.665754
$732$	0	0
$733$	15.0000i	0.554038i	0.960864	+	0.277019i	$0.0893464\pi$
$733$	15.0000i	0.554038i	−0.960864	+	0.277019i	$0.910654\pi$
$734$	−66.0000	−2.43610
$735$	0	0
$736$	−32.0000	−1.17954
$737$	14.0000i	0.515697i
$738$	− 72.0000i	− 2.65036i
$739$	43.0000	1.58178	0.790890	−	0.611958i	$-0.209618\pi$
$739$	43.0000	1.58178	0.790890	+	0.611958i	$0.209618\pi$
$740$	0	0
$741$	54.0000	1.98374
$742$	0	0
$743$	− 22.0000i	− 0.807102i	−0.914957	−	0.403551i	$-0.867776\pi$
$743$	− 22.0000i	− 0.807102i	0.914957	−	0.403551i	$-0.132224\pi$
$744$	0	0
$745$	0	0
$746$	−8.00000	−0.292901
$747$	− 72.0000i	− 2.63434i
$748$	− 6.00000i	− 0.219382i
$749$	0	0
$750$	0	0
$751$	15.0000	0.547358	0.273679	−	0.961821i	$-0.411759\pi$
$751$	15.0000	0.547358	0.273679	+	0.961821i	$0.411759\pi$
$752$	− 36.0000i	− 1.31278i
$753$	− 36.0000i	− 1.31191i
$754$	−6.00000	−0.218507
$755$	0	0
$756$	0	0
$757$	− 48.0000i	− 1.74459i	−0.488980	−	0.872295i	$-0.662631\pi$
$757$	− 48.0000i	− 1.74459i	0.488980	−	0.872295i	$-0.337369\pi$
$758$	48.0000i	1.74344i
$759$	12.0000	0.435572
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	− 12.0000i	− 0.434714i
$763$	0	0
$764$	−34.0000	−1.23008
$765$	0	0
$766$	0	0
$767$	18.0000i	0.649942i
$768$	48.0000i	1.73205i
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	54.0000	1.94476
$772$	− 24.0000i	− 0.863779i
$773$	3.00000i	0.107903i	0.998544	+	0.0539513i	$0.0171816\pi$
$773$	3.00000i	0.107903i	−0.998544	+	0.0539513i	$0.982818\pi$
$774$	−72.0000	−2.58799
$775$	0	0
$776$	0	0
$777$	0	0
$778$	− 26.0000i	− 0.932145i
$779$	−36.0000	−1.28983
$780$	0	0
$781$	−8.00000	−0.286263
$782$	24.0000i	0.858238i
$783$	− 9.00000i	− 0.321634i
$784$	0	0
$785$	0	0
$786$	−72.0000	−2.56815
$787$	− 39.0000i	− 1.39020i	−0.718913	−	0.695100i	$-0.755360\pi$
$787$	− 39.0000i	− 1.39020i	0.718913	−	0.695100i	$-0.244640\pi$
$788$	− 4.00000i	− 0.142494i
$789$	30.0000	1.06803
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	18.0000	0.638796
$795$	0	0
$796$	12.0000	0.425329
$797$	27.0000i	0.956389i	0.878254	+	0.478195i	$0.158709\pi$
$797$	27.0000i	0.956389i	−0.878254	+	0.478195i	$0.841291\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	0	0
$801$	72.0000	2.54399
$802$	− 38.0000i	− 1.34183i
$803$	6.00000i	0.211735i
$804$	84.0000	2.96245
$805$	0	0
$806$	−36.0000	−1.26805
$807$	− 72.0000i	− 2.53452i
$808$	0	0
$809$	35.0000	1.23053	0.615267	−	0.788319i	$-0.289048\pi$
$809$	35.0000	1.23053	0.615267	+	0.788319i	$0.289048\pi$
$810$	0	0
$811$	48.0000	1.68551	0.842754	−	0.538299i	$-0.180933\pi$
$811$	48.0000	1.68551	0.842754	+	0.538299i	$0.180933\pi$
$812$	0	0
$813$	72.0000i	2.52515i
$814$	0	0
$815$	0	0
$816$	36.0000	1.26025
$817$	36.0000i	1.25948i
$818$	− 60.0000i	− 2.09785i
$819$	0	0
$820$	0	0
$821$	−23.0000	−0.802706	−0.401353	−	0.915924i	$-0.631460\pi$
$821$	−23.0000	−0.802706	−0.401353	+	0.915924i	$0.631460\pi$
$822$	48.0000i	1.67419i
$823$	50.0000i	1.74289i	0.490493	+	0.871445i	$0.336817\pi$
$823$	50.0000i	1.74289i	−0.490493	+	0.871445i	$0.663183\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.0000i	0.556375i	0.960527	+	0.278187i	$0.0897336\pi$
$827$	16.0000i	0.556375i	−0.960527	+	0.278187i	$0.910266\pi$
$828$	− 48.0000i	− 1.66812i
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	−84.0000	−2.91393
$832$	24.0000i	0.832050i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	12.0000	0.415029
$837$	− 54.0000i	− 1.86651i
$838$	12.0000i	0.414533i
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	2.00000i	0.0689246i
$843$	− 15.0000i	− 0.516627i
$844$	−30.0000	−1.03264
$845$	0	0
$846$	108.000	3.71312
$847$	0	0
$848$	40.0000i	1.37361i
$849$	−63.0000	−2.16215
$850$	0	0
$851$	0	0
$852$	48.0000i	1.64445i
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 54.0000i	− 1.84460i	−0.386469	−	0.922302i	$-0.626305\pi$
$857$	− 54.0000i	− 1.84460i	0.386469	−	0.922302i	$-0.373695\pi$
$858$	− 18.0000i	− 0.614510i
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	0	0
$861$	0	0
$862$	10.0000i	0.340601i
$863$	− 34.0000i	− 1.15737i	−0.815550	−	0.578687i	$-0.803565\pi$
$863$	− 34.0000i	− 1.15737i	0.815550	−	0.578687i	$-0.196435\pi$
$864$	−72.0000	−2.44949
$865$	0	0
$866$	60.0000	2.03888
$867$	24.0000i	0.815083i
$868$	0	0
$869$	1.00000	0.0339227
$870$	0	0
$871$	−42.0000	−1.42312
$872$	0	0
$873$	− 90.0000i	− 3.04604i
$874$	−48.0000	−1.62362
$875$	0	0
$876$	36.0000	1.21633
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	− 24.0000i	− 0.809961i
$879$	−27.0000	−0.910687
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	18.0000	0.605406
$885$	0	0
$886$	−68.0000	−2.28450
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.00000	0.301511
$892$	− 6.00000i	− 0.200895i
$893$	− 54.0000i	− 1.80704i
$894$	60.0000	2.00670
$895$	0	0
$896$	0	0
$897$	36.0000i	1.20201i
$898$	− 46.0000i	− 1.53504i
$899$	6.00000	0.200111
$900$	0	0
$901$	30.0000	0.999445
$902$	12.0000i	0.399556i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−90.0000	−2.99005
$907$	− 24.0000i	− 0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$	− 24.0000i	− 0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$	6.00000i	0.199117i
$909$	108.000	3.58213
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	72.0000i	2.38416i
$913$	12.0000i	0.397142i
$914$	12.0000	0.396925
$915$	0	0
$916$	−12.0000	−0.396491
$917$	0	0
$918$	54.0000i	1.78227i
$919$	9.00000	0.296883	0.148441	−	0.988921i	$-0.452574\pi$
$919$	9.00000	0.296883	0.148441	+	0.988921i	$0.452574\pi$
$920$	0	0
$921$	−9.00000	−0.296560
$922$	72.0000i	2.37119i
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	8.00000	0.262896
$927$	54.0000i	1.77359i
$928$	− 8.00000i	− 0.262613i
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	0	0
$932$	16.0000i	0.524097i
$933$	− 18.0000i	− 0.589294i
$934$	−30.0000	−0.981630
$935$	0	0
$936$	0	0
$937$	27.0000i	0.882052i	0.897494	+	0.441026i	$0.145385\pi$
$937$	27.0000i	0.882052i	−0.897494	+	0.441026i	$0.854615\pi$
$938$	0	0
$939$	−9.00000	−0.293704
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	108.000i	3.51883i
$943$	− 24.0000i	− 0.781548i
$944$	−24.0000	−0.781133
$945$	0	0
$946$	12.0000	0.390154
$947$	2.00000i	0.0649913i	0.999472	+	0.0324956i	$0.0103455\pi$
$947$	2.00000i	0.0649913i	−0.999472	+	0.0324956i	$0.989654\pi$
$948$	− 6.00000i	− 0.194871i
$949$	−18.0000	−0.584305
$950$	0	0
$951$	−66.0000	−2.14020
$952$	0	0
$953$	− 38.0000i	− 1.23094i	−0.788160	−	0.615470i	$-0.788966\pi$
$953$	− 38.0000i	− 1.23094i	0.788160	−	0.615470i	$-0.211034\pi$
$954$	−120.000	−3.88514
$955$	0	0
$956$	14.0000	0.452792
$957$	3.00000i	0.0969762i
$958$	48.0000i	1.55081i
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	48.0000	1.54598
$965$	0	0
$966$	0	0
$967$	22.0000i	0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$	22.0000i	0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$	0	0
$969$	54.0000	1.73473
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	0	0
$974$	72.0000	2.30703
$975$	0	0
$976$	0	0
$977$	− 34.0000i	− 1.08776i	−0.839164	−	0.543878i	$-0.816955\pi$
$977$	− 34.0000i	− 1.08776i	0.839164	−	0.543878i	$-0.183045\pi$
$978$	− 96.0000i	− 3.06974i
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−90.0000	−2.87348
$982$	46.0000i	1.46792i
$983$	45.0000i	1.43528i	0.696416	+	0.717639i	$0.254777\pi$
$983$	45.0000i	1.43528i	−0.696416	+	0.717639i	$0.745223\pi$
$984$	0	0
$985$	0	0
$986$	−6.00000	−0.191079
$987$	0	0
$988$	36.0000i	1.14531i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	− 48.0000i	− 1.52400i
$993$	36.0000i	1.14243i
$994$	0	0
$995$	0	0
$996$	72.0000	2.28141
$997$	3.00000i	0.0950110i	0.998871	+	0.0475055i	$0.0151272\pi$
$997$	3.00000i	0.0950110i	−0.998871	+	0.0475055i	$0.984873\pi$
$998$	54.0000i	1.70934i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.b.a.99.2		2
5.2	odd	4		245.2.a.b.1.1	yes	1
5.3	odd	4		1225.2.a.h.1.1		1
5.4	even	2	inner	1225.2.b.a.99.1		2
7.6	odd	2		1225.2.b.b.99.2		2
15.2	even	4		2205.2.a.l.1.1		1
20.7	even	4		3920.2.a.a.1.1		1
35.2	odd	12		245.2.e.c.116.1		2
35.12	even	12		245.2.e.d.116.1		2
35.13	even	4		1225.2.a.j.1.1		1
35.17	even	12		245.2.e.d.226.1		2
35.27	even	4		245.2.a.a.1.1	✓	1
35.32	odd	12		245.2.e.c.226.1		2
35.34	odd	2		1225.2.b.b.99.1		2
105.62	odd	4		2205.2.a.j.1.1		1
140.27	odd	4		3920.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.a.a.1.1	✓	1	35.27	even	4
245.2.a.b.1.1	yes	1	5.2	odd	4
245.2.e.c.116.1		2	35.2	odd	12
245.2.e.c.226.1		2	35.32	odd	12
245.2.e.d.116.1		2	35.12	even	12
245.2.e.d.226.1		2	35.17	even	12
1225.2.a.h.1.1		1	5.3	odd	4
1225.2.a.j.1.1		1	35.13	even	4
1225.2.b.a.99.1		2	5.4	even	2	inner
1225.2.b.a.99.2		2	1.1	even	1	trivial
1225.2.b.b.99.1		2	35.34	odd	2
1225.2.b.b.99.2		2	7.6	odd	2
2205.2.a.j.1.1		1	105.62	odd	4
2205.2.a.l.1.1		1	15.2	even	4
3920.2.a.a.1.1		1	20.7	even	4
3920.2.a.bj.1.1		1	140.27	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +3.00000i q^{3} -2.00000 q^{4} -6.00000 q^{6} -6.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} +3.00000i q^{3} -2.00000 q^{4} -6.00000 q^{6} -6.00000 q^{9} +1.00000 q^{11} -6.00000i q^{12} +3.00000i q^{13} -4.00000 q^{16} +3.00000i q^{17} -12.0000i q^{18} -6.00000 q^{19} +2.00000i q^{22} -4.00000i q^{23} -6.00000 q^{26} -9.00000i q^{27} +1.00000 q^{29} +6.00000 q^{31} -8.00000i q^{32} +3.00000i q^{33} -6.00000 q^{34} +12.0000 q^{36} -12.0000i q^{38} -9.00000 q^{39} +6.00000 q^{41} -6.00000i q^{43} -2.00000 q^{44} +8.00000 q^{46} +9.00000i q^{47} -12.0000i q^{48} -9.00000 q^{51} -6.00000i q^{52} -10.0000i q^{53} +18.0000 q^{54} -18.0000i q^{57} +2.00000i q^{58} +6.00000 q^{59} +12.0000i q^{62} +8.00000 q^{64} -6.00000 q^{66} +14.0000i q^{67} -6.00000i q^{68} +12.0000 q^{69} -8.00000 q^{71} +6.00000i q^{73} +12.0000 q^{76} -18.0000i q^{78} +1.00000 q^{79} +9.00000 q^{81} +12.0000i q^{82} +12.0000i q^{83} +12.0000 q^{86} +3.00000i q^{87} -12.0000 q^{89} +8.00000i q^{92} +18.0000i q^{93} -18.0000 q^{94} +24.0000 q^{96} +15.0000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 12 q^{6} - 12 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 - 12 * q^6 - 12 * q^9	\( 2 q - 4 q^{4} - 12 q^{6} - 12 q^{9} + 2 q^{11} - 8 q^{16} - 12 q^{19} - 12 q^{26} + 2 q^{29} + 12 q^{31} - 12 q^{34} + 24 q^{36} - 18 q^{39} + 12 q^{41} - 4 q^{44} + 16 q^{46} - 18 q^{51} + 36 q^{54} + 12 q^{59} + 16 q^{64} - 12 q^{66} + 24 q^{69} - 16 q^{71} + 24 q^{76} + 2 q^{79} + 18 q^{81} + 24 q^{86} - 24 q^{89} - 36 q^{94} + 48 q^{96} - 12 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 - 12 * q^6 - 12 * q^9 + 2 * q^11 - 8 * q^16 - 12 * q^19 - 12 * q^26 + 2 * q^29 + 12 * q^31 - 12 * q^34 + 24 * q^36 - 18 * q^39 + 12 * q^41 - 4 * q^44 + 16 * q^46 - 18 * q^51 + 36 * q^54 + 12 * q^59 + 16 * q^64 - 12 * q^66 + 24 * q^69 - 16 * q^71 + 24 * q^76 + 2 * q^79 + 18 * q^81 + 24 * q^86 - 24 * q^89 - 36 * q^94 + 48 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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