LMFDB - Embedded newform 1225.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(1,1225)]

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([0, 0]))

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

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

chi := DirichletCharacter("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.82843 q^{3} -1.00000 q^{4} -2.82843 q^{6} -3.00000 q^{8} +5.00000 q^{9} +2.82843 q^{12} -4.24264 q^{13} -1.00000 q^{16} -4.24264 q^{17} +5.00000 q^{18} -2.82843 q^{19} +4.00000 q^{23} +8.48528 q^{24} -4.24264 q^{26} -5.65685 q^{27} +5.65685 q^{31} +5.00000 q^{32} -4.24264 q^{34} -5.00000 q^{36} +6.00000 q^{37} -2.82843 q^{38} +12.0000 q^{39} +4.24264 q^{41} +4.00000 q^{46} +2.82843 q^{48} +12.0000 q^{51} +4.24264 q^{52} -8.00000 q^{53} -5.65685 q^{54} +8.00000 q^{57} +8.48528 q^{59} +9.89949 q^{61} +5.65685 q^{62} +7.00000 q^{64} -12.0000 q^{67} +4.24264 q^{68} -11.3137 q^{69} +12.0000 q^{71} -15.0000 q^{72} +12.7279 q^{73} +6.00000 q^{74} +2.82843 q^{76} +12.0000 q^{78} +12.0000 q^{79} +1.00000 q^{81} +4.24264 q^{82} +8.48528 q^{83} -4.24264 q^{89} -4.00000 q^{92} -16.0000 q^{93} -14.1421 q^{96} +4.24264 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 10 q^{9} - 2 q^{16} + 10 q^{18} + 8 q^{23} + 10 q^{32} - 10 q^{36} + 12 q^{37} + 24 q^{39} + 8 q^{46} + 24 q^{51} - 16 q^{53} + 16 q^{57} + 14 q^{64} - 24 q^{67} + 24 q^{71}+ \cdots - 32 q^{93}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 10 * q^9 - 2 * q^16 + 10 * q^18 + 8 * q^23 + 10 * q^32 - 10 * q^36 + 12 * q^37 + 24 * q^39 + 8 * q^46 + 24 * q^51 - 16 * q^53 + 16 * q^57 + 14 * q^64 - 24 * q^67 + 24 * q^71 - 30 * q^72 + 12 * q^74 + 24 * q^78 + 24 * q^79 + 2 * q^81 - 8 * q^92 - 32 * q^93

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.82843	−1.15470
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	5.00000	1.66667
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	2.82843	0.816497
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.24264	−1.02899	−0.514496	−	0.857493i	$-0.672021\pi$
$17$	−4.24264	−1.02899	−0.514496	+	0.857493i	$0.672021\pi$
$18$	5.00000	1.17851
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	8.48528	1.73205
$25$	0	0
$26$	−4.24264	−0.832050
$27$	−5.65685	−1.08866
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−4.24264	−0.727607
$35$	0	0
$36$	−5.00000	−0.833333
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−2.82843	−0.458831
$39$	12.0000	1.92154
$40$	0	0
$41$	4.24264	0.662589	0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264	0.662589	0.331295	+	0.943527i	$0.392515\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	2.82843	0.408248
$49$	0	0
$50$	0	0
$51$	12.0000	1.68034
$52$	4.24264	0.588348
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	−5.65685	−0.769800
$55$	0	0
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	9.89949	1.26750	0.633750	−	0.773538i	$-0.281515\pi$
$61$	9.89949	1.26750	0.633750	+	0.773538i	$0.281515\pi$
$62$	5.65685	0.718421
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	4.24264	0.514496
$69$	−11.3137	−1.36201
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−15.0000	−1.76777
$73$	12.7279	1.48969	0.744845	−	0.667237i	$-0.232523\pi$
$73$	12.7279	1.48969	0.744845	+	0.667237i	$0.232523\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	2.82843	0.324443
$77$	0	0
$78$	12.0000	1.35873
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.24264	0.468521
$83$	8.48528	0.931381	0.465690	−	0.884948i	$-0.345806\pi$
$83$	8.48528	0.931381	0.465690	+	0.884948i	$0.345806\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.24264	−0.449719	−0.224860	−	0.974391i	$-0.572192\pi$
$89$	−4.24264	−0.449719	−0.224860	+	0.974391i	$0.572192\pi$
$90$	0	0
$91$	0	0
$92$	−4.00000	−0.417029
$93$	−16.0000	−1.65912
$94$	0	0
$95$	0	0
$96$	−14.1421	−1.44338
$97$	4.24264	0.430775	0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264	0.430775	0.215387	+	0.976529i	$0.430899\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.24264	−0.422159	−0.211079	−	0.977469i	$-0.567698\pi$
$101$	−4.24264	−0.422159	−0.211079	+	0.977469i	$0.567698\pi$
$102$	12.0000	1.18818
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	12.7279	1.24808
$105$	0	0
$106$	−8.00000	−0.777029
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	5.65685	0.544331
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	−16.9706	−1.61077
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	0	0
$117$	−21.2132	−1.96116
$118$	8.48528	0.781133
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	9.89949	0.896258
$123$	−12.0000	−1.08200
$124$	−5.65685	−0.508001
$125$	0	0
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	0	0
$131$	−8.48528	−0.741362	−0.370681	−	0.928760i	$-0.620876\pi$
$131$	−8.48528	−0.741362	−0.370681	+	0.928760i	$0.620876\pi$
$132$	0	0
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	12.7279	1.09141
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	−11.3137	−0.963087
$139$	2.82843	0.239904	0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843	0.239904	0.119952	+	0.992780i	$0.461726\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	0	0
$144$	−5.00000	−0.416667
$145$	0	0
$146$	12.7279	1.05337
$147$	0	0
$148$	−6.00000	−0.493197
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	8.48528	0.688247
$153$	−21.2132	−1.71499
$154$	0	0
$155$	0	0
$156$	−12.0000	−0.960769
$157$	4.24264	0.338600	0.169300	−	0.985565i	$-0.445849\pi$
$157$	4.24264	0.338600	0.169300	+	0.985565i	$0.445849\pi$
$158$	12.0000	0.954669
$159$	22.6274	1.79447
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	−4.24264	−0.331295
$165$	0	0
$166$	8.48528	0.658586
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	−14.1421	−1.08148
$172$	0	0
$173$	−4.24264	−0.322562	−0.161281	−	0.986909i	$-0.551563\pi$
$173$	−4.24264	−0.322562	−0.161281	+	0.986909i	$0.551563\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−24.0000	−1.80395
$178$	−4.24264	−0.317999
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−18.3848	−1.36653	−0.683265	−	0.730171i	$-0.739441\pi$
$181$	−18.3848	−1.36653	−0.683265	+	0.730171i	$0.739441\pi$
$182$	0	0
$183$	−28.0000	−2.06982
$184$	−12.0000	−0.884652
$185$	0	0
$186$	−16.0000	−1.17318
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−19.7990	−1.42887
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	4.24264	0.304604
$195$	0	0
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	−11.3137	−0.802008	−0.401004	−	0.916076i	$-0.631339\pi$
$199$	−11.3137	−0.802008	−0.401004	+	0.916076i	$0.631339\pi$
$200$	0	0
$201$	33.9411	2.39402
$202$	−4.24264	−0.298511
$203$	0	0
$204$	−12.0000	−0.840168
$205$	0	0
$206$	0	0
$207$	20.0000	1.39010
$208$	4.24264	0.294174
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	8.00000	0.549442
$213$	−33.9411	−2.32561
$214$	4.00000	0.273434
$215$	0	0
$216$	16.9706	1.15470
$217$	0	0
$218$	−16.0000	−1.08366
$219$	−36.0000	−2.43265
$220$	0	0
$221$	18.0000	1.21081
$222$	−16.9706	−1.13899
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	8.00000	0.532152
$227$	2.82843	0.187729	0.0938647	−	0.995585i	$-0.470078\pi$
$227$	2.82843	0.187729	0.0938647	+	0.995585i	$0.470078\pi$
$228$	−8.00000	−0.529813
$229$	7.07107	0.467269	0.233635	−	0.972324i	$-0.424938\pi$
$229$	7.07107	0.467269	0.233635	+	0.972324i	$0.424938\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	−21.2132	−1.38675
$235$	0	0
$236$	−8.48528	−0.552345
$237$	−33.9411	−2.20471
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−9.89949	−0.637683	−0.318841	−	0.947808i	$-0.603294\pi$
$241$	−9.89949	−0.637683	−0.318841	+	0.947808i	$0.603294\pi$
$242$	−11.0000	−0.707107
$243$	14.1421	0.907218
$244$	−9.89949	−0.633750
$245$	0	0
$246$	−12.0000	−0.765092
$247$	12.0000	0.763542
$248$	−16.9706	−1.07763
$249$	−24.0000	−1.52094
$250$	0	0
$251$	25.4558	1.60676	0.803379	−	0.595468i	$-0.203033\pi$
$251$	25.4558	1.60676	0.803379	+	0.595468i	$0.203033\pi$
$252$	0	0
$253$	0	0
$254$	−12.0000	−0.752947
$255$	0	0
$256$	−17.0000	−1.06250
$257$	26.8701	1.67611	0.838054	−	0.545587i	$-0.183693\pi$
$257$	26.8701	1.67611	0.838054	+	0.545587i	$0.183693\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−8.48528	−0.524222
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	12.0000	0.734388
$268$	12.0000	0.733017
$269$	−21.2132	−1.29339	−0.646696	−	0.762748i	$-0.723850\pi$
$269$	−21.2132	−1.29339	−0.646696	+	0.762748i	$0.723850\pi$
$270$	0	0
$271$	−5.65685	−0.343629	−0.171815	−	0.985129i	$-0.554963\pi$
$271$	−5.65685	−0.343629	−0.171815	+	0.985129i	$0.554963\pi$
$272$	4.24264	0.257248
$273$	0	0
$274$	−14.0000	−0.845771
$275$	0	0
$276$	11.3137	0.681005
$277$	24.0000	1.44202	0.721010	−	0.692925i	$-0.243678\pi$
$277$	24.0000	1.44202	0.721010	+	0.692925i	$0.243678\pi$
$278$	2.82843	0.169638
$279$	28.2843	1.69334
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	25.4558	1.51319	0.756596	−	0.653882i	$-0.226861\pi$
$283$	25.4558	1.51319	0.756596	+	0.653882i	$0.226861\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	0	0
$288$	25.0000	1.47314
$289$	1.00000	0.0588235
$290$	0	0
$291$	−12.0000	−0.703452
$292$	−12.7279	−0.744845
$293$	−1.41421	−0.0826192	−0.0413096	−	0.999146i	$-0.513153\pi$
$293$	−1.41421	−0.0826192	−0.0413096	+	0.999146i	$0.513153\pi$
$294$	0	0
$295$	0	0
$296$	−18.0000	−1.04623
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−16.9706	−0.981433
$300$	0	0
$301$	0	0
$302$	0	0
$303$	12.0000	0.689382
$304$	2.82843	0.162221
$305$	0	0
$306$	−21.2132	−1.21268
$307$	25.4558	1.45284	0.726421	−	0.687250i	$-0.241182\pi$
$307$	25.4558	1.45284	0.726421	+	0.687250i	$0.241182\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−33.9411	−1.92462	−0.962312	−	0.271947i	$-0.912333\pi$
$311$	−33.9411	−1.92462	−0.962312	+	0.271947i	$0.912333\pi$
$312$	−36.0000	−2.03810
$313$	−21.2132	−1.19904	−0.599521	−	0.800359i	$-0.704642\pi$
$313$	−21.2132	−1.19904	−0.599521	+	0.800359i	$0.704642\pi$
$314$	4.24264	0.239426
$315$	0	0
$316$	−12.0000	−0.675053
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	22.6274	1.26888
$319$	0	0
$320$	0	0
$321$	−11.3137	−0.631470
$322$	0	0
$323$	12.0000	0.667698
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	24.0000	1.32924
$327$	45.2548	2.50260
$328$	−12.7279	−0.702782
$329$	0	0
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−8.48528	−0.465690
$333$	30.0000	1.64399
$334$	11.3137	0.619059
$335$	0	0
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	5.00000	0.271964
$339$	−22.6274	−1.22895
$340$	0	0
$341$	0	0
$342$	−14.1421	−0.764719
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−4.24264	−0.228086
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	1.41421	0.0757011	0.0378506	−	0.999283i	$-0.487949\pi$
$349$	1.41421	0.0757011	0.0378506	+	0.999283i	$0.487949\pi$
$350$	0	0
$351$	24.0000	1.28103
$352$	0	0
$353$	15.5563	0.827981	0.413990	−	0.910281i	$-0.364135\pi$
$353$	15.5563	0.827981	0.413990	+	0.910281i	$0.364135\pi$
$354$	−24.0000	−1.27559
$355$	0	0
$356$	4.24264	0.224860
$357$	0	0
$358$	12.0000	0.634220
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	−18.3848	−0.966282
$363$	31.1127	1.63299
$364$	0	0
$365$	0	0
$366$	−28.0000	−1.46358
$367$	16.9706	0.885856	0.442928	−	0.896557i	$-0.353940\pi$
$367$	16.9706	0.885856	0.442928	+	0.896557i	$0.353940\pi$
$368$	−4.00000	−0.208514
$369$	21.2132	1.10432
$370$	0	0
$371$	0	0
$372$	16.0000	0.829561
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	33.9411	1.73886
$382$	12.0000	0.613973
$383$	33.9411	1.73431	0.867155	−	0.498038i	$-0.165946\pi$
$383$	33.9411	1.73431	0.867155	+	0.498038i	$0.165946\pi$
$384$	8.48528	0.433013
$385$	0	0
$386$	24.0000	1.22157
$387$	0	0
$388$	−4.24264	−0.215387
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−16.9706	−0.858238
$392$	0	0
$393$	24.0000	1.21064
$394$	−8.00000	−0.403034
$395$	0	0
$396$	0	0
$397$	4.24264	0.212932	0.106466	−	0.994316i	$-0.466046\pi$
$397$	4.24264	0.212932	0.106466	+	0.994316i	$0.466046\pi$
$398$	−11.3137	−0.567105
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	33.9411	1.69283
$403$	−24.0000	−1.19553
$404$	4.24264	0.211079
$405$	0	0
$406$	0	0
$407$	0	0
$408$	−36.0000	−1.78227
$409$	18.3848	0.909069	0.454534	−	0.890729i	$-0.349806\pi$
$409$	18.3848	0.909069	0.454534	+	0.890729i	$0.349806\pi$
$410$	0	0
$411$	39.5980	1.95322
$412$	0	0
$413$	0	0
$414$	20.0000	0.982946
$415$	0	0
$416$	−21.2132	−1.04006
$417$	−8.00000	−0.391762
$418$	0	0
$419$	8.48528	0.414533	0.207267	−	0.978285i	$-0.433543\pi$
$419$	8.48528	0.414533	0.207267	+	0.978285i	$0.433543\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	24.0000	1.16554
$425$	0	0
$426$	−33.9411	−1.64445
$427$	0	0
$428$	−4.00000	−0.193347
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	5.65685	0.272166
$433$	4.24264	0.203888	0.101944	−	0.994790i	$-0.467494\pi$
$433$	4.24264	0.203888	0.101944	+	0.994790i	$0.467494\pi$
$434$	0	0
$435$	0	0
$436$	16.0000	0.766261
$437$	−11.3137	−0.541208
$438$	−36.0000	−1.72015
$439$	28.2843	1.34993	0.674967	−	0.737848i	$-0.264158\pi$
$439$	28.2843	1.34993	0.674967	+	0.737848i	$0.264158\pi$
$440$	0	0
$441$	0	0
$442$	18.0000	0.856173
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	16.9706	0.805387
$445$	0	0
$446$	0	0
$447$	16.9706	0.802680
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	−8.00000	−0.376288
$453$	0	0
$454$	2.82843	0.132745
$455$	0	0
$456$	−24.0000	−1.12390
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	7.07107	0.330409
$459$	24.0000	1.12022
$460$	0	0
$461$	21.2132	0.987997	0.493999	−	0.869463i	$-0.335535\pi$
$461$	21.2132	0.987997	0.493999	+	0.869463i	$0.335535\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	0	0
$466$	−14.0000	−0.648537
$467$	−31.1127	−1.43972	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	−31.1127	−1.43972	−0.719862	+	0.694117i	$0.755795\pi$
$468$	21.2132	0.980581
$469$	0	0
$470$	0	0
$471$	−12.0000	−0.552931
$472$	−25.4558	−1.17170
$473$	0	0
$474$	−33.9411	−1.55897
$475$	0	0
$476$	0	0
$477$	−40.0000	−1.83147
$478$	24.0000	1.09773
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−25.4558	−1.16069
$482$	−9.89949	−0.450910
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	14.1421	0.641500
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−29.6985	−1.34439
$489$	−67.8823	−3.06974
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	12.0000	0.541002
$493$	0	0
$494$	12.0000	0.539906
$495$	0	0
$496$	−5.65685	−0.254000
$497$	0	0
$498$	−24.0000	−1.07547
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−32.0000	−1.42965
$502$	25.4558	1.13615
$503$	16.9706	0.756680	0.378340	−	0.925667i	$-0.376495\pi$
$503$	16.9706	0.756680	0.378340	+	0.925667i	$0.376495\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−14.1421	−0.628074
$508$	12.0000	0.532414
$509$	−12.7279	−0.564155	−0.282078	−	0.959392i	$-0.591024\pi$
$509$	−12.7279	−0.564155	−0.282078	+	0.959392i	$0.591024\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	16.0000	0.706417
$514$	26.8701	1.18519
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	29.6985	1.30111	0.650557	−	0.759457i	$-0.274535\pi$
$521$	29.6985	1.30111	0.650557	+	0.759457i	$0.274535\pi$
$522$	0	0
$523$	−8.48528	−0.371035	−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528	−0.371035	−0.185518	+	0.982641i	$0.559396\pi$
$524$	8.48528	0.370681
$525$	0	0
$526$	−16.0000	−0.697633
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	42.4264	1.84115
$532$	0	0
$533$	−18.0000	−0.779667
$534$	12.0000	0.519291
$535$	0	0
$536$	36.0000	1.55496
$537$	−33.9411	−1.46467
$538$	−21.2132	−0.914566
$539$	0	0
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	−5.65685	−0.242983
$543$	52.0000	2.23153
$544$	−21.2132	−0.909509
$545$	0	0
$546$	0	0
$547$	−24.0000	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$548$	14.0000	0.598050
$549$	49.4975	2.11250
$550$	0	0
$551$	0	0
$552$	33.9411	1.44463
$553$	0	0
$554$	24.0000	1.01966
$555$	0	0
$556$	−2.82843	−0.119952
$557$	40.0000	1.69485	0.847427	−	0.530912i	$-0.178150\pi$
$557$	40.0000	1.69485	0.847427	+	0.530912i	$0.178150\pi$
$558$	28.2843	1.19737
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.4558	−1.07284	−0.536418	−	0.843952i	$-0.680223\pi$
$563$	−25.4558	−1.07284	−0.536418	+	0.843952i	$0.680223\pi$
$564$	0	0
$565$	0	0
$566$	25.4558	1.06999
$567$	0	0
$568$	−36.0000	−1.51053
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	−33.9411	−1.41791
$574$	0	0
$575$	0	0
$576$	35.0000	1.45833
$577$	38.1838	1.58961	0.794805	−	0.606864i	$-0.207573\pi$
$577$	38.1838	1.58961	0.794805	+	0.606864i	$0.207573\pi$
$578$	1.00000	0.0415945
$579$	−67.8823	−2.82109
$580$	0	0
$581$	0	0
$582$	−12.0000	−0.497416
$583$	0	0
$584$	−38.1838	−1.58006
$585$	0	0
$586$	−1.41421	−0.0584206
$587$	−42.4264	−1.75113	−0.875563	−	0.483105i	$-0.839509\pi$
$587$	−42.4264	−1.75113	−0.875563	+	0.483105i	$0.839509\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	22.6274	0.930768
$592$	−6.00000	−0.246598
$593$	−7.07107	−0.290374	−0.145187	−	0.989404i	$-0.546378\pi$
$593$	−7.07107	−0.290374	−0.145187	+	0.989404i	$0.546378\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	32.0000	1.30967
$598$	−16.9706	−0.693978
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−41.0122	−1.67292	−0.836461	−	0.548026i	$-0.815379\pi$
$601$	−41.0122	−1.67292	−0.836461	+	0.548026i	$0.815379\pi$
$602$	0	0
$603$	−60.0000	−2.44339
$604$	0	0
$605$	0	0
$606$	12.0000	0.487467
$607$	−33.9411	−1.37763	−0.688814	−	0.724938i	$-0.741868\pi$
$607$	−33.9411	−1.37763	−0.688814	+	0.724938i	$0.741868\pi$
$608$	−14.1421	−0.573539
$609$	0	0
$610$	0	0
$611$	0	0
$612$	21.2132	0.857493
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	25.4558	1.02731
$615$	0	0
$616$	0	0
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	0	0
$619$	−19.7990	−0.795789	−0.397894	−	0.917431i	$-0.630259\pi$
$619$	−19.7990	−0.795789	−0.397894	+	0.917431i	$0.630259\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	−33.9411	−1.36092
$623$	0	0
$624$	−12.0000	−0.480384
$625$	0	0
$626$	−21.2132	−0.847850
$627$	0	0
$628$	−4.24264	−0.169300
$629$	−25.4558	−1.01499
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	−36.0000	−1.43200
$633$	−11.3137	−0.449680
$634$	8.00000	0.317721
$635$	0	0
$636$	−22.6274	−0.897235
$637$	0	0
$638$	0	0
$639$	60.0000	2.37356
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	−11.3137	−0.446516
$643$	8.48528	0.334627	0.167313	−	0.985904i	$-0.446491\pi$
$643$	8.48528	0.334627	0.167313	+	0.985904i	$0.446491\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	−5.65685	−0.222394	−0.111197	−	0.993798i	$-0.535468\pi$
$647$	−5.65685	−0.222394	−0.111197	+	0.993798i	$0.535468\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−24.0000	−0.939913
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	45.2548	1.76960
$655$	0	0
$656$	−4.24264	−0.165647
$657$	63.6396	2.48282
$658$	0	0
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	−15.5563	−0.605072	−0.302536	−	0.953138i	$-0.597833\pi$
$661$	−15.5563	−0.605072	−0.302536	+	0.953138i	$0.597833\pi$
$662$	−8.00000	−0.310929
$663$	−50.9117	−1.97725
$664$	−25.4558	−0.987878
$665$	0	0
$666$	30.0000	1.16248
$667$	0	0
$668$	−11.3137	−0.437741
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	−5.00000	−0.192308
$677$	9.89949	0.380468	0.190234	−	0.981739i	$-0.439075\pi$
$677$	9.89949	0.380468	0.190234	+	0.981739i	$0.439075\pi$
$678$	−22.6274	−0.869001
$679$	0	0
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	14.1421	0.540738
$685$	0	0
$686$	0	0
$687$	−20.0000	−0.763048
$688$	0	0
$689$	33.9411	1.29305
$690$	0	0
$691$	2.82843	0.107598	0.0537992	−	0.998552i	$-0.482867\pi$
$691$	2.82843	0.107598	0.0537992	+	0.998552i	$0.482867\pi$
$692$	4.24264	0.161281
$693$	0	0
$694$	8.00000	0.303676
$695$	0	0
$696$	0	0
$697$	−18.0000	−0.681799
$698$	1.41421	0.0535288
$699$	39.5980	1.49773
$700$	0	0
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	24.0000	0.905822
$703$	−16.9706	−0.640057
$704$	0	0
$705$	0	0
$706$	15.5563	0.585471
$707$	0	0
$708$	24.0000	0.901975
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	60.0000	2.25018
$712$	12.7279	0.476999
$713$	22.6274	0.847403
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−67.8823	−2.53511
$718$	24.0000	0.895672
$719$	−16.9706	−0.632895	−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706	−0.632895	−0.316448	+	0.948610i	$0.602490\pi$
$720$	0	0
$721$	0	0
$722$	−11.0000	−0.409378
$723$	28.0000	1.04133
$724$	18.3848	0.683265
$725$	0	0
$726$	31.1127	1.15470
$727$	−33.9411	−1.25881	−0.629403	−	0.777079i	$-0.716701\pi$
$727$	−33.9411	−1.25881	−0.629403	+	0.777079i	$0.716701\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	0	0
$732$	28.0000	1.03491
$733$	4.24264	0.156706	0.0783528	−	0.996926i	$-0.475034\pi$
$733$	4.24264	0.156706	0.0783528	+	0.996926i	$0.475034\pi$
$734$	16.9706	0.626395
$735$	0	0
$736$	20.0000	0.737210
$737$	0	0
$738$	21.2132	0.780869
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	−33.9411	−1.24686
$742$	0	0
$743$	28.0000	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$743$	28.0000	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$744$	48.0000	1.75977
$745$	0	0
$746$	−24.0000	−0.878702
$747$	42.4264	1.55230
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	−72.0000	−2.62383
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	0	0
$761$	−29.6985	−1.07657	−0.538285	−	0.842763i	$-0.680927\pi$
$761$	−29.6985	−1.07657	−0.538285	+	0.842763i	$0.680927\pi$
$762$	33.9411	1.22956
$763$	0	0
$764$	−12.0000	−0.434145
$765$	0	0
$766$	33.9411	1.22634
$767$	−36.0000	−1.29988
$768$	48.0833	1.73506
$769$	24.0416	0.866963	0.433482	−	0.901162i	$-0.357285\pi$
$769$	24.0416	0.866963	0.433482	+	0.901162i	$0.357285\pi$
$770$	0	0
$771$	−76.0000	−2.73707
$772$	−24.0000	−0.863779
$773$	21.2132	0.762986	0.381493	−	0.924372i	$-0.375410\pi$
$773$	21.2132	0.762986	0.381493	+	0.924372i	$0.375410\pi$
$774$	0	0
$775$	0	0
$776$	−12.7279	−0.456906
$777$	0	0
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	−16.9706	−0.606866
$783$	0	0
$784$	0	0
$785$	0	0
$786$	24.0000	0.856052
$787$	−8.48528	−0.302468	−0.151234	−	0.988498i	$-0.548325\pi$
$787$	−8.48528	−0.302468	−0.151234	+	0.988498i	$0.548325\pi$
$788$	8.00000	0.284988
$789$	45.2548	1.61111
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−42.0000	−1.49146
$794$	4.24264	0.150566
$795$	0	0
$796$	11.3137	0.401004
$797$	4.24264	0.150282	0.0751410	−	0.997173i	$-0.476059\pi$
$797$	4.24264	0.150282	0.0751410	+	0.997173i	$0.476059\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−21.2132	−0.749532
$802$	24.0000	0.847469
$803$	0	0
$804$	−33.9411	−1.19701
$805$	0	0
$806$	−24.0000	−0.845364
$807$	60.0000	2.11210
$808$	12.7279	0.447767
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	14.1421	0.496598	0.248299	−	0.968683i	$-0.420129\pi$
$811$	14.1421	0.496598	0.248299	+	0.968683i	$0.420129\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	0	0
$816$	−12.0000	−0.420084
$817$	0	0
$818$	18.3848	0.642809
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	39.5980	1.38114
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	52.0000	1.80822	0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000	1.80822	0.904109	+	0.427303i	$0.140536\pi$
$828$	−20.0000	−0.695048
$829$	24.0416	0.835000	0.417500	−	0.908677i	$-0.362906\pi$
$829$	24.0416	0.835000	0.417500	+	0.908677i	$0.362906\pi$
$830$	0	0
$831$	−67.8823	−2.35481
$832$	−29.6985	−1.02961
$833$	0	0
$834$	−8.00000	−0.277017
$835$	0	0
$836$	0	0
$837$	−32.0000	−1.10608
$838$	8.48528	0.293119
$839$	16.9706	0.585889	0.292944	−	0.956129i	$-0.405365\pi$
$839$	16.9706	0.585889	0.292944	+	0.956129i	$0.405365\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−6.00000	−0.206774
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	8.00000	0.274721
$849$	−72.0000	−2.47103
$850$	0	0
$851$	24.0000	0.822709
$852$	33.9411	1.16280
$853$	12.7279	0.435796	0.217898	−	0.975972i	$-0.430080\pi$
$853$	12.7279	0.435796	0.217898	+	0.975972i	$0.430080\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	55.1543	1.88404	0.942018	−	0.335562i	$-0.108926\pi$
$857$	55.1543	1.88404	0.942018	+	0.335562i	$0.108926\pi$
$858$	0	0
$859$	−31.1127	−1.06155	−0.530776	−	0.847512i	$-0.678099\pi$
$859$	−31.1127	−1.06155	−0.530776	+	0.847512i	$0.678099\pi$
$860$	0	0
$861$	0	0
$862$	24.0000	0.817443
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−28.2843	−0.962250
$865$	0	0
$866$	4.24264	0.144171
$867$	−2.82843	−0.0960584
$868$	0	0
$869$	0	0
$870$	0	0
$871$	50.9117	1.72508
$872$	48.0000	1.62549
$873$	21.2132	0.717958
$874$	−11.3137	−0.382692
$875$	0	0
$876$	36.0000	1.21633
$877$	−30.0000	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$878$	28.2843	0.954548
$879$	4.00000	0.134917
$880$	0	0
$881$	46.6690	1.57232	0.786160	−	0.618023i	$-0.212066\pi$
$881$	46.6690	1.57232	0.786160	+	0.618023i	$0.212066\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−18.0000	−0.605406
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−16.9706	−0.569816	−0.284908	−	0.958555i	$-0.591963\pi$
$887$	−16.9706	−0.569816	−0.284908	+	0.958555i	$0.591963\pi$
$888$	50.9117	1.70848
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	16.9706	0.567581
$895$	0	0
$896$	0	0
$897$	48.0000	1.60267
$898$	−18.0000	−0.600668
$899$	0	0
$900$	0	0
$901$	33.9411	1.13074
$902$	0	0
$903$	0	0
$904$	−24.0000	−0.798228
$905$	0	0
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−2.82843	−0.0938647
$909$	−21.2132	−0.703598
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	−8.00000	−0.264906
$913$	0	0
$914$	−24.0000	−0.793849
$915$	0	0
$916$	−7.07107	−0.233635
$917$	0	0
$918$	24.0000	0.792118
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	−72.0000	−2.37248
$922$	21.2132	0.698620
$923$	−50.9117	−1.67578
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	0	0
$928$	0	0
$929$	−12.7279	−0.417590	−0.208795	−	0.977959i	$-0.566954\pi$
$929$	−12.7279	−0.417590	−0.208795	+	0.977959i	$0.566954\pi$
$930$	0	0
$931$	0	0
$932$	14.0000	0.458585
$933$	96.0000	3.14290
$934$	−31.1127	−1.01804
$935$	0	0
$936$	63.6396	2.08013
$937$	−12.7279	−0.415803	−0.207902	−	0.978150i	$-0.566663\pi$
$937$	−12.7279	−0.415803	−0.207902	+	0.978150i	$0.566663\pi$
$938$	0	0
$939$	60.0000	1.95803
$940$	0	0
$941$	−21.2132	−0.691531	−0.345765	−	0.938321i	$-0.612381\pi$
$941$	−21.2132	−0.691531	−0.345765	+	0.938321i	$0.612381\pi$
$942$	−12.0000	−0.390981
$943$	16.9706	0.552638
$944$	−8.48528	−0.276172
$945$	0	0
$946$	0	0
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	33.9411	1.10236
$949$	−54.0000	−1.75291
$950$	0	0
$951$	−22.6274	−0.733744
$952$	0	0
$953$	32.0000	1.03658	0.518291	−	0.855204i	$-0.326568\pi$
$953$	32.0000	1.03658	0.518291	+	0.855204i	$0.326568\pi$
$954$	−40.0000	−1.29505
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	−25.4558	−0.820729
$963$	20.0000	0.644491
$964$	9.89949	0.318841
$965$	0	0
$966$	0	0
$967$	36.0000	1.15768	0.578841	−	0.815440i	$-0.303505\pi$
$967$	36.0000	1.15768	0.578841	+	0.815440i	$0.303505\pi$
$968$	33.0000	1.06066
$969$	−33.9411	−1.09035
$970$	0	0
$971$	8.48528	0.272306	0.136153	−	0.990688i	$-0.456526\pi$
$971$	8.48528	0.272306	0.136153	+	0.990688i	$0.456526\pi$
$972$	−14.1421	−0.453609
$973$	0	0
$974$	−12.0000	−0.384505
$975$	0	0
$976$	−9.89949	−0.316875
$977$	−10.0000	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$977$	−10.0000	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$978$	−67.8823	−2.17064
$979$	0	0
$980$	0	0
$981$	−80.0000	−2.55420
$982$	−12.0000	−0.382935
$983$	28.2843	0.902128	0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843	0.902128	0.451064	+	0.892492i	$0.351045\pi$
$984$	36.0000	1.14764
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−12.0000	−0.381771
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	28.2843	0.898027
$993$	22.6274	0.718059
$994$	0	0
$995$	0	0
$996$	24.0000	0.760469
$997$	−55.1543	−1.74676	−0.873378	−	0.487044i	$-0.838075\pi$
$997$	−55.1543	−1.74676	−0.873378	+	0.487044i	$0.838075\pi$
$998$	−12.0000	−0.379853
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.a.v.1.1		2
5.2	odd	4		245.2.b.f.99.4	yes	4
5.3	odd	4		245.2.b.f.99.1	✓	4
5.4	even	2		1225.2.a.l.1.2		2
7.6	odd	2	inner	1225.2.a.v.1.2		2
15.2	even	4		2205.2.d.k.1324.2		4
15.8	even	4		2205.2.d.k.1324.4		4
35.2	odd	12		245.2.j.g.214.3		8
35.3	even	12		245.2.j.g.79.4		8
35.12	even	12		245.2.j.g.214.4		8
35.13	even	4		245.2.b.f.99.2	yes	4
35.17	even	12		245.2.j.g.79.1		8
35.18	odd	12		245.2.j.g.79.3		8
35.23	odd	12		245.2.j.g.214.2		8
35.27	even	4		245.2.b.f.99.3	yes	4
35.32	odd	12		245.2.j.g.79.2		8
35.33	even	12		245.2.j.g.214.1		8
35.34	odd	2		1225.2.a.l.1.1		2
105.62	odd	4		2205.2.d.k.1324.1		4
105.83	odd	4		2205.2.d.k.1324.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.b.f.99.1	✓	4	5.3	odd	4
245.2.b.f.99.2	yes	4	35.13	even	4
245.2.b.f.99.3	yes	4	35.27	even	4
245.2.b.f.99.4	yes	4	5.2	odd	4
245.2.j.g.79.1		8	35.17	even	12
245.2.j.g.79.2		8	35.32	odd	12
245.2.j.g.79.3		8	35.18	odd	12
245.2.j.g.79.4		8	35.3	even	12
245.2.j.g.214.1		8	35.33	even	12
245.2.j.g.214.2		8	35.23	odd	12
245.2.j.g.214.3		8	35.2	odd	12
245.2.j.g.214.4		8	35.12	even	12
1225.2.a.l.1.1		2	35.34	odd	2
1225.2.a.l.1.2		2	5.4	even	2
1225.2.a.v.1.1		2	1.1	even	1	trivial
1225.2.a.v.1.2		2	7.6	odd	2	inner
2205.2.d.k.1324.1		4	105.62	odd	4
2205.2.d.k.1324.2		4	15.2	even	4
2205.2.d.k.1324.3		4	105.83	odd	4
2205.2.d.k.1324.4		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.82843 q^{3} -1.00000 q^{4} -2.82843 q^{6} -3.00000 q^{8} +5.00000 q^{9} +2.82843 q^{12} -4.24264 q^{13} -1.00000 q^{16} -4.24264 q^{17} +5.00000 q^{18} -2.82843 q^{19} +4.00000 q^{23} +8.48528 q^{24} -4.24264 q^{26} -5.65685 q^{27} +5.65685 q^{31} +5.00000 q^{32} -4.24264 q^{34} -5.00000 q^{36} +6.00000 q^{37} -2.82843 q^{38} +12.0000 q^{39} +4.24264 q^{41} +4.00000 q^{46} +2.82843 q^{48} +12.0000 q^{51} +4.24264 q^{52} -8.00000 q^{53} -5.65685 q^{54} +8.00000 q^{57} +8.48528 q^{59} +9.89949 q^{61} +5.65685 q^{62} +7.00000 q^{64} -12.0000 q^{67} +4.24264 q^{68} -11.3137 q^{69} +12.0000 q^{71} -15.0000 q^{72} +12.7279 q^{73} +6.00000 q^{74} +2.82843 q^{76} +12.0000 q^{78} +12.0000 q^{79} +1.00000 q^{81} +4.24264 q^{82} +8.48528 q^{83} -4.24264 q^{89} -4.00000 q^{92} -16.0000 q^{93} -14.1421 q^{96} +4.24264 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 10 q^{9} - 2 q^{16} + 10 q^{18} + 8 q^{23} + 10 q^{32} - 10 q^{36} + 12 q^{37} + 24 q^{39} + 8 q^{46} + 24 q^{51} - 16 q^{53} + 16 q^{57} + 14 q^{64} - 24 q^{67} + 24 q^{71}+ \cdots - 32 q^{93}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 10 * q^9 - 2 * q^16 + 10 * q^18 + 8 * q^23 + 10 * q^32 - 10 * q^36 + 12 * q^37 + 24 * q^39 + 8 * q^46 + 24 * q^51 - 16 * q^53 + 16 * q^57 + 14 * q^64 - 24 * q^67 + 24 * q^71 - 30 * q^72 + 12 * q^74 + 24 * q^78 + 24 * q^79 + 2 * q^81 - 8 * q^92 - 32 * q^93

Introduction

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