LMFDB - Embedded newform 2450.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.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:	$19.5633484952$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 490)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} -4.00000 q^{11} +4.24264 q^{13} +1.00000 q^{16} -4.24264 q^{17} -3.00000 q^{18} -5.65685 q^{19} -4.00000 q^{22} +4.24264 q^{26} -4.00000 q^{29} +5.65685 q^{31} +1.00000 q^{32} -4.24264 q^{34} -3.00000 q^{36} -6.00000 q^{37} -5.65685 q^{38} +1.41421 q^{41} -12.0000 q^{43} -4.00000 q^{44} +4.24264 q^{52} -12.0000 q^{53} -4.00000 q^{58} +11.3137 q^{59} -7.07107 q^{61} +5.65685 q^{62} +1.00000 q^{64} -12.0000 q^{67} -4.24264 q^{68} +8.00000 q^{71} -3.00000 q^{72} -4.24264 q^{73} -6.00000 q^{74} -5.65685 q^{76} +9.00000 q^{81} +1.41421 q^{82} +16.9706 q^{83} -12.0000 q^{86} -4.00000 q^{88} +4.24264 q^{89} +4.24264 q^{97} +12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 6 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 6 q^{9} - 8 q^{11} + 2 q^{16} - 6 q^{18} - 8 q^{22} - 8 q^{29} + 2 q^{32} - 6 q^{36} - 12 q^{37} - 24 q^{43} - 8 q^{44} - 24 q^{53} - 8 q^{58} + 2 q^{64} - 24 q^{67} + 16 q^{71} - 6 q^{72} - 12 q^{74} + 18 q^{81} - 24 q^{86} - 8 q^{88} + 24 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 6 * q^9 - 8 * q^11 + 2 * q^16 - 6 * q^18 - 8 * q^22 - 8 * q^29 + 2 * q^32 - 6 * q^36 - 12 * q^37 - 24 * q^43 - 8 * q^44 - 24 * q^53 - 8 * q^58 + 2 * q^64 - 24 * q^67 + 16 * q^71 - 6 * q^72 - 12 * q^74 + 18 * q^81 - 24 * q^86 - 8 * q^88 + 24 * q^99

Display 100 coefficients 10 coefficients

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
$2$	1.00000	0.707107
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	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	$0.672021\pi$
$18$	−3.00000	−0.707107
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	4.24264	0.832050
$27$	0	0
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.24264	−0.727607
$35$	0	0
$36$	−3.00000	−0.500000
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−5.65685	−0.917663
$39$	0	0
$40$	0	0
$41$	1.41421	0.220863	0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421	0.220863	0.110432	+	0.993884i	$0.464777\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	4.24264	0.588348
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−4.00000	−0.525226
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	−7.07107	−0.905357	−0.452679	−	0.891674i	$-0.649532\pi$
$61$	−7.07107	−0.905357	−0.452679	+	0.891674i	$0.649532\pi$
$62$	5.65685	0.718421
$63$	0	0
$64$	1.00000	0.125000
$65$	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
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−3.00000	−0.353553
$73$	−4.24264	−0.496564	−0.248282	−	0.968688i	$-0.579866\pi$
$73$	−4.24264	−0.496564	−0.248282	+	0.968688i	$0.579866\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−5.65685	−0.648886
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	1.41421	0.156174
$83$	16.9706	1.86276	0.931381	−	0.364047i	$-0.118605\pi$
$83$	16.9706	1.86276	0.931381	+	0.364047i	$0.118605\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	−4.00000	−0.426401
$89$	4.24264	0.449719	0.224860	−	0.974391i	$-0.427808\pi$
$89$	4.24264	0.449719	0.224860	+	0.974391i	$0.427808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	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
$99$	12.0000	1.20605
$100$	0	0
$101$	−9.89949	−0.985037	−0.492518	−	0.870302i	$-0.663924\pi$
$101$	−9.89949	−0.985037	−0.492518	+	0.870302i	$0.663924\pi$
$102$	0	0
$103$	−16.9706	−1.67216	−0.836080	−	0.548608i	$-0.815158\pi$
$103$	−16.9706	−1.67216	−0.836080	+	0.548608i	$0.815158\pi$
$104$	4.24264	0.416025
$105$	0	0
$106$	−12.0000	−1.16554
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	−4.00000	−0.371391
$117$	−12.7279	−1.17670
$118$	11.3137	1.04151
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	−7.07107	−0.640184
$123$	0	0
$124$	5.65685	0.508001
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	−4.24264	−0.363803
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−5.65685	−0.479808	−0.239904	−	0.970797i	$-0.577116\pi$
$139$	−5.65685	−0.479808	−0.239904	+	0.970797i	$0.577116\pi$
$140$	0	0
$141$	0	0
$142$	8.00000	0.671345
$143$	−16.9706	−1.41915
$144$	−3.00000	−0.250000
$145$	0	0
$146$	−4.24264	−0.351123
$147$	0	0
$148$	−6.00000	−0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−5.65685	−0.458831
$153$	12.7279	1.02899
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.24264	−0.338600	−0.169300	−	0.985565i	$-0.554151\pi$
$157$	−4.24264	−0.338600	−0.169300	+	0.985565i	$0.554151\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	9.00000	0.707107
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	1.41421	0.110432
$165$	0	0
$166$	16.9706	1.31717
$167$	16.9706	1.31322	0.656611	−	0.754230i	$-0.271989\pi$
$167$	16.9706	1.31322	0.656611	+	0.754230i	$0.271989\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	16.9706	1.29777
$172$	−12.0000	−0.914991
$173$	4.24264	0.322562	0.161281	−	0.986909i	$-0.448437\pi$
$173$	4.24264	0.322562	0.161281	+	0.986909i	$0.448437\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	4.24264	0.317999
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	15.5563	1.15629	0.578147	−	0.815933i	$-0.303776\pi$
$181$	15.5563	1.15629	0.578147	+	0.815933i	$0.303776\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.9706	1.24101
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	4.24264	0.304604
$195$	0	0
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	12.0000	0.852803
$199$	−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
$201$	0	0
$202$	−9.89949	−0.696526
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−16.9706	−1.18240
$207$	0	0
$208$	4.24264	0.294174
$209$	22.6274	1.56517
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−12.0000	−0.812743
$219$	0	0
$220$	0	0
$221$	−18.0000	−1.21081
$222$	0	0
$223$	−16.9706	−1.13643	−0.568216	−	0.822879i	$-0.692366\pi$
$223$	−16.9706	−1.13643	−0.568216	+	0.822879i	$0.692366\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.9706	1.12638	0.563188	−	0.826329i	$-0.309575\pi$
$227$	16.9706	1.12638	0.563188	+	0.826329i	$0.309575\pi$
$228$	0	0
$229$	−15.5563	−1.02799	−0.513996	−	0.857792i	$-0.671835\pi$
$229$	−15.5563	−1.02799	−0.513996	+	0.857792i	$0.671835\pi$
$230$	0	0
$231$	0	0
$232$	−4.00000	−0.262613
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−12.7279	−0.832050
$235$	0	0
$236$	11.3137	0.736460
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	15.5563	1.00207	0.501036	−	0.865426i	$-0.332952\pi$
$241$	15.5563	1.00207	0.501036	+	0.865426i	$0.332952\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	−7.07107	−0.452679
$245$	0	0
$246$	0	0
$247$	−24.0000	−1.52708
$248$	5.65685	0.359211
$249$	0	0
$250$	0	0
$251$	5.65685	0.357057	0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685	0.357057	0.178529	+	0.983935i	$0.442866\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.7279	−0.793946	−0.396973	−	0.917830i	$-0.629939\pi$
$257$	−12.7279	−0.793946	−0.396973	+	0.917830i	$0.629939\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	12.0000	0.742781
$262$	−16.9706	−1.04844
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−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
$271$	11.3137	0.687259	0.343629	−	0.939105i	$-0.388344\pi$
$271$	11.3137	0.687259	0.343629	+	0.939105i	$0.388344\pi$
$272$	−4.24264	−0.257248
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	−5.65685	−0.339276
$279$	−16.9706	−1.01600
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−16.9706	−1.00349
$287$	0	0
$288$	−3.00000	−0.176777
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	−4.24264	−0.248282
$293$	−4.24264	−0.247858	−0.123929	−	0.992291i	$-0.539549\pi$
$293$	−4.24264	−0.247858	−0.123929	+	0.992291i	$0.539549\pi$
$294$	0	0
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	10.0000	0.579284
$299$	0	0
$300$	0	0
$301$	0	0
$302$	8.00000	0.460348
$303$	0	0
$304$	−5.65685	−0.324443
$305$	0	0
$306$	12.7279	0.727607
$307$	−16.9706	−0.968561	−0.484281	−	0.874913i	$-0.660919\pi$
$307$	−16.9706	−0.968561	−0.484281	+	0.874913i	$0.660919\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.9706	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$311$	−16.9706	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$312$	0	0
$313$	12.7279	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	12.7279	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	−4.24264	−0.239426
$315$	0	0
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0000	1.33540
$324$	9.00000	0.500000
$325$	0	0
$326$	−12.0000	−0.664619
$327$	0	0
$328$	1.41421	0.0780869
$329$	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	$0.393022\pi$
$332$	16.9706	0.931381
$333$	18.0000	0.986394
$334$	16.9706	0.928588
$335$	0	0
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	0	0
$341$	−22.6274	−1.22534
$342$	16.9706	0.917663
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	4.24264	0.228086
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	24.0416	1.28692	0.643459	−	0.765480i	$-0.277498\pi$
$349$	24.0416	1.28692	0.643459	+	0.765480i	$0.277498\pi$
$350$	0	0
$351$	0	0
$352$	−4.00000	−0.213201
$353$	21.2132	1.12906	0.564532	−	0.825411i	$-0.309057\pi$
$353$	21.2132	1.12906	0.564532	+	0.825411i	$0.309057\pi$
$354$	0	0
$355$	0	0
$356$	4.24264	0.224860
$357$	0	0
$358$	20.0000	1.05703
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	15.5563	0.817624
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−16.9706	−0.885856	−0.442928	−	0.896557i	$-0.646060\pi$
$367$	−16.9706	−0.885856	−0.442928	+	0.896557i	$0.646060\pi$
$368$	0	0
$369$	−4.24264	−0.220863
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	16.9706	0.877527
$375$	0	0
$376$	0	0
$377$	−16.9706	−0.874028
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	8.00000	0.409316
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−24.0000	−1.22157
$387$	36.0000	1.82998
$388$	4.24264	0.215387
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−12.0000	−0.604551
$395$	0	0
$396$	12.0000	0.603023
$397$	29.6985	1.49052	0.745262	−	0.666772i	$-0.232324\pi$
$397$	29.6985	1.49052	0.745262	+	0.666772i	$0.232324\pi$
$398$	−11.3137	−0.567105
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	−9.89949	−0.492518
$405$	0	0
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	9.89949	0.489499	0.244749	−	0.969586i	$-0.421294\pi$
$409$	9.89949	0.489499	0.244749	+	0.969586i	$0.421294\pi$
$410$	0	0
$411$	0	0
$412$	−16.9706	−0.836080
$413$	0	0
$414$	0	0
$415$	0	0
$416$	4.24264	0.208013
$417$	0	0
$418$	22.6274	1.10674
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−29.6985	−1.42722	−0.713609	−	0.700544i	$-0.752941\pi$
$433$	−29.6985	−1.42722	−0.713609	+	0.700544i	$0.752941\pi$
$434$	0	0
$435$	0	0
$436$	−12.0000	−0.574696
$437$	0	0
$438$	0	0
$439$	39.5980	1.88991	0.944954	−	0.327203i	$-0.106106\pi$
$439$	39.5980	1.88991	0.944954	+	0.327203i	$0.106106\pi$
$440$	0	0
$441$	0	0
$442$	−18.0000	−0.856173
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	−16.9706	−0.803579
$447$	0	0
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−5.65685	−0.266371
$452$	0	0
$453$	0	0
$454$	16.9706	0.796468
$455$	0	0
$456$	0	0
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	−15.5563	−0.726900
$459$	0	0
$460$	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	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	6.00000	0.277945
$467$	−16.9706	−0.785304	−0.392652	−	0.919687i	$-0.628442\pi$
$467$	−16.9706	−0.785304	−0.392652	+	0.919687i	$0.628442\pi$
$468$	−12.7279	−0.588348
$469$	0	0
$470$	0	0
$471$	0	0
$472$	11.3137	0.520756
$473$	48.0000	2.20704
$474$	0	0
$475$	0	0
$476$	0	0
$477$	36.0000	1.64833
$478$	−8.00000	−0.365911
$479$	16.9706	0.775405	0.387702	−	0.921785i	$-0.373269\pi$
$479$	16.9706	0.775405	0.387702	+	0.921785i	$0.373269\pi$
$480$	0	0
$481$	−25.4558	−1.16069
$482$	15.5563	0.708572
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−7.07107	−0.320092
$489$	0	0
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	16.9706	0.764316
$494$	−24.0000	−1.07981
$495$	0	0
$496$	5.65685	0.254000
$497$	0	0
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	5.65685	0.252478
$503$	−33.9411	−1.51336	−0.756680	−	0.653785i	$-0.773180\pi$
$503$	−33.9411	−1.51336	−0.756680	+	0.653785i	$0.773180\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−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
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.7279	−0.561405
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12.7279	−0.557620	−0.278810	−	0.960346i	$-0.589940\pi$
$521$	−12.7279	−0.557620	−0.278810	+	0.960346i	$0.589940\pi$
$522$	12.0000	0.525226
$523$	33.9411	1.48414	0.742071	−	0.670321i	$-0.233844\pi$
$523$	33.9411	1.48414	0.742071	+	0.670321i	$0.233844\pi$
$524$	−16.9706	−0.741362
$525$	0	0
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−33.9411	−1.47292
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	0	0
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−21.2132	−0.914566
$539$	0	0
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	11.3137	0.485965
$543$	0	0
$544$	−4.24264	−0.181902
$545$	0	0
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	6.00000	0.256307
$549$	21.2132	0.905357
$550$	0	0
$551$	22.6274	0.963960
$552$	0	0
$553$	0	0
$554$	12.0000	0.509831
$555$	0	0
$556$	−5.65685	−0.239904
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	−16.9706	−0.718421
$559$	−50.9117	−2.15333
$560$	0	0
$561$	0	0
$562$	16.0000	0.674919
$563$	33.9411	1.43045	0.715224	−	0.698895i	$-0.246325\pi$
$563$	33.9411	1.43045	0.715224	+	0.698895i	$0.246325\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	8.00000	0.335673
$569$	−8.00000	−0.335377	−0.167689	−	0.985840i	$-0.553630\pi$
$569$	−8.00000	−0.335377	−0.167689	+	0.985840i	$0.553630\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−16.9706	−0.709575
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−3.00000	−0.125000
$577$	21.2132	0.883117	0.441559	−	0.897232i	$-0.354426\pi$
$577$	21.2132	0.883117	0.441559	+	0.897232i	$0.354426\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	48.0000	1.98796
$584$	−4.24264	−0.175562
$585$	0	0
$586$	−4.24264	−0.175262
$587$	−16.9706	−0.700450	−0.350225	−	0.936666i	$-0.613895\pi$
$587$	−16.9706	−0.700450	−0.350225	+	0.936666i	$0.613895\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	−6.00000	−0.246598
$593$	−12.7279	−0.522673	−0.261337	−	0.965248i	$-0.584163\pi$
$593$	−12.7279	−0.522673	−0.261337	+	0.965248i	$0.584163\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−26.8701	−1.09605	−0.548026	−	0.836461i	$-0.684621\pi$
$601$	−26.8701	−1.09605	−0.548026	+	0.836461i	$0.684621\pi$
$602$	0	0
$603$	36.0000	1.46603
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	16.9706	0.688814	0.344407	−	0.938820i	$-0.388080\pi$
$607$	16.9706	0.688814	0.344407	+	0.938820i	$0.388080\pi$
$608$	−5.65685	−0.229416
$609$	0	0
$610$	0	0
$611$	0	0
$612$	12.7279	0.514496
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−16.9706	−0.684876
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	45.2548	1.81895	0.909473	−	0.415764i	$-0.136486\pi$
$619$	45.2548	1.81895	0.909473	+	0.415764i	$0.136486\pi$
$620$	0	0
$621$	0	0
$622$	−16.9706	−0.680458
$623$	0	0
$624$	0	0
$625$	0	0
$626$	12.7279	0.508710
$627$	0	0
$628$	−4.24264	−0.169300
$629$	25.4558	1.01499
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	0	0
$633$	0	0
$634$	12.0000	0.476581
$635$	0	0
$636$	0	0
$637$	0	0
$638$	16.0000	0.633446
$639$	−24.0000	−0.949425
$640$	0	0
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	16.9706	0.667182	0.333591	−	0.942718i	$-0.391740\pi$
$647$	16.9706	0.667182	0.333591	+	0.942718i	$0.391740\pi$
$648$	9.00000	0.353553
$649$	−45.2548	−1.77641
$650$	0	0
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	0	0
$656$	1.41421	0.0552158
$657$	12.7279	0.496564
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	24.0416	0.935111	0.467556	−	0.883964i	$-0.345135\pi$
$661$	24.0416	0.935111	0.467556	+	0.883964i	$0.345135\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	16.9706	0.658586
$665$	0	0
$666$	18.0000	0.697486
$667$	0	0
$668$	16.9706	0.656611
$669$	0	0
$670$	0	0
$671$	28.2843	1.09190
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	5.00000	0.192308
$677$	−21.2132	−0.815290	−0.407645	−	0.913141i	$-0.633650\pi$
$677$	−21.2132	−0.815290	−0.407645	+	0.913141i	$0.633650\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−22.6274	−0.866449
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	16.9706	0.648886
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−12.0000	−0.457496
$689$	−50.9117	−1.93958
$690$	0	0
$691$	5.65685	0.215197	0.107598	−	0.994194i	$-0.465684\pi$
$691$	5.65685	0.215197	0.107598	+	0.994194i	$0.465684\pi$
$692$	4.24264	0.161281
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	−6.00000	−0.227266
$698$	24.0416	0.909989
$699$	0	0
$700$	0	0
$701$	−44.0000	−1.66186	−0.830929	−	0.556379i	$-0.812190\pi$
$701$	−44.0000	−1.66186	−0.830929	+	0.556379i	$0.812190\pi$
$702$	0	0
$703$	33.9411	1.28011
$704$	−4.00000	−0.150756
$705$	0	0
$706$	21.2132	0.798369
$707$	0	0
$708$	0	0
$709$	−44.0000	−1.65245	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−44.0000	−1.65245	−0.826227	+	0.563337i	$0.809517\pi$
$710$	0	0
$711$	0	0
$712$	4.24264	0.159000
$713$	0	0
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	0	0
$718$	16.0000	0.597115
$719$	5.65685	0.210965	0.105483	−	0.994421i	$-0.466361\pi$
$719$	5.65685	0.210965	0.105483	+	0.994421i	$0.466361\pi$
$720$	0	0
$721$	0	0
$722$	13.0000	0.483810
$723$	0	0
$724$	15.5563	0.578147
$725$	0	0
$726$	0	0
$727$	−16.9706	−0.629403	−0.314702	−	0.949191i	$-0.601904\pi$
$727$	−16.9706	−0.629403	−0.314702	+	0.949191i	$0.601904\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	50.9117	1.88304
$732$	0	0
$733$	12.7279	0.470117	0.235058	−	0.971981i	$-0.424472\pi$
$733$	12.7279	0.470117	0.235058	+	0.971981i	$0.424472\pi$
$734$	−16.9706	−0.626395
$735$	0	0
$736$	0	0
$737$	48.0000	1.76810
$738$	−4.24264	−0.156174
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	12.0000	0.439351
$747$	−50.9117	−1.86276
$748$	16.9706	0.620505
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	−16.9706	−0.618031
$755$	0	0
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	0	0
$761$	46.6690	1.69175	0.845876	−	0.533380i	$-0.179078\pi$
$761$	46.6690	1.69175	0.845876	+	0.533380i	$0.179078\pi$
$762$	0	0
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	0	0
$767$	48.0000	1.73318
$768$	0	0
$769$	−7.07107	−0.254989	−0.127495	−	0.991839i	$-0.540694\pi$
$769$	−7.07107	−0.254989	−0.127495	+	0.991839i	$0.540694\pi$
$770$	0	0
$771$	0	0
$772$	−24.0000	−0.863779
$773$	−38.1838	−1.37337	−0.686687	−	0.726953i	$-0.740936\pi$
$773$	−38.1838	−1.37337	−0.686687	+	0.726953i	$0.740936\pi$
$774$	36.0000	1.29399
$775$	0	0
$776$	4.24264	0.152302
$777$	0	0
$778$	−20.0000	−0.717035
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−50.9117	−1.81481	−0.907403	−	0.420262i	$-0.861938\pi$
$787$	−50.9117	−1.81481	−0.907403	+	0.420262i	$0.861938\pi$
$788$	−12.0000	−0.427482
$789$	0	0
$790$	0	0
$791$	0	0
$792$	12.0000	0.426401
$793$	−30.0000	−1.06533
$794$	29.6985	1.05396
$795$	0	0
$796$	−11.3137	−0.401004
$797$	−4.24264	−0.150282	−0.0751410	−	0.997173i	$-0.523941\pi$
$797$	−4.24264	−0.150282	−0.0751410	+	0.997173i	$0.523941\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−12.7279	−0.449719
$802$	−8.00000	−0.282490
$803$	16.9706	0.598878
$804$	0	0
$805$	0	0
$806$	24.0000	0.845364
$807$	0	0
$808$	−9.89949	−0.348263
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−45.2548	−1.58911	−0.794556	−	0.607191i	$-0.792296\pi$
$811$	−45.2548	−1.58911	−0.794556	+	0.607191i	$0.792296\pi$
$812$	0	0
$813$	0	0
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	67.8823	2.37490
$818$	9.89949	0.346128
$819$	0	0
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	−16.9706	−0.591198
$825$	0	0
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−15.5563	−0.540294	−0.270147	−	0.962819i	$-0.587072\pi$
$829$	−15.5563	−0.540294	−0.270147	+	0.962819i	$0.587072\pi$
$830$	0	0
$831$	0	0
$832$	4.24264	0.147087
$833$	0	0
$834$	0	0
$835$	0	0
$836$	22.6274	0.782586
$837$	0	0
$838$	0	0
$839$	−22.6274	−0.781185	−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274	−0.781185	−0.390593	+	0.920564i	$0.627730\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−6.00000	−0.206774
$843$	0	0
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−12.0000	−0.412082
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−46.6690	−1.59792	−0.798959	−	0.601386i	$-0.794616\pi$
$853$	−46.6690	−1.59792	−0.798959	+	0.601386i	$0.794616\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	−12.7279	−0.434778	−0.217389	−	0.976085i	$-0.569754\pi$
$857$	−12.7279	−0.434778	−0.217389	+	0.976085i	$0.569754\pi$
$858$	0	0
$859$	−5.65685	−0.193009	−0.0965047	−	0.995333i	$-0.530766\pi$
$859$	−5.65685	−0.193009	−0.0965047	+	0.995333i	$0.530766\pi$
$860$	0	0
$861$	0	0
$862$	16.0000	0.544962
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	0	0
$866$	−29.6985	−1.00920
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−50.9117	−1.72508
$872$	−12.0000	−0.406371
$873$	−12.7279	−0.430775
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	39.5980	1.33637
$879$	0	0
$880$	0	0
$881$	32.5269	1.09586	0.547930	−	0.836524i	$-0.315416\pi$
$881$	32.5269	1.09586	0.547930	+	0.836524i	$0.315416\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	−18.0000	−0.605406
$885$	0	0
$886$	−12.0000	−0.403148
$887$	16.9706	0.569816	0.284908	−	0.958555i	$-0.408037\pi$
$887$	16.9706	0.569816	0.284908	+	0.958555i	$0.408037\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−36.0000	−1.20605
$892$	−16.9706	−0.568216
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	−22.6274	−0.754667
$900$	0	0
$901$	50.9117	1.69611
$902$	−5.65685	−0.188353
$903$	0	0
$904$	0	0
$905$	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	$0.563843\pi$
$908$	16.9706	0.563188
$909$	29.6985	0.985037
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	−67.8823	−2.24657
$914$	−24.0000	−0.793849
$915$	0	0
$916$	−15.5563	−0.513996
$917$	0	0
$918$	0	0
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	0	0
$921$	0	0
$922$	21.2132	0.698620
$923$	33.9411	1.11719
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	50.9117	1.67216
$928$	−4.00000	−0.131306
$929$	29.6985	0.974376	0.487188	−	0.873297i	$-0.338023\pi$
$929$	29.6985	0.974376	0.487188	+	0.873297i	$0.338023\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	0	0
$934$	−16.9706	−0.555294
$935$	0	0
$936$	−12.7279	−0.416025
$937$	4.24264	0.138601	0.0693005	−	0.997596i	$-0.477923\pi$
$937$	4.24264	0.138601	0.0693005	+	0.997596i	$0.477923\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−49.4975	−1.61357	−0.806786	−	0.590844i	$-0.798795\pi$
$941$	−49.4975	−1.61357	−0.806786	+	0.590844i	$0.798795\pi$
$942$	0	0
$943$	0	0
$944$	11.3137	0.368230
$945$	0	0
$946$	48.0000	1.56061
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	36.0000	1.16554
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	16.9706	0.548294
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	−25.4558	−0.820729
$963$	−36.0000	−1.16008
$964$	15.5563	0.501036
$965$	0	0
$966$	0	0
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	0	0
$971$	−45.2548	−1.45230	−0.726148	−	0.687538i	$-0.758691\pi$
$971$	−45.2548	−1.45230	−0.726148	+	0.687538i	$0.758691\pi$
$972$	0	0
$973$	0	0
$974$	24.0000	0.769010
$975$	0	0
$976$	−7.07107	−0.226339
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−16.9706	−0.542382
$980$	0	0
$981$	36.0000	1.14939
$982$	−4.00000	−0.127645
$983$	−16.9706	−0.541277	−0.270638	−	0.962681i	$-0.587235\pi$
$983$	−16.9706	−0.541277	−0.270638	+	0.962681i	$0.587235\pi$
$984$	0	0
$985$	0	0
$986$	16.9706	0.540453
$987$	0	0
$988$	−24.0000	−0.763542
$989$	0	0
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	5.65685	0.179605
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	38.1838	1.20929	0.604646	−	0.796494i	$-0.293315\pi$
$997$	38.1838	1.20929	0.604646	+	0.796494i	$0.293315\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.bo.1.2		2
5.2	odd	4		490.2.c.g.99.4	yes	4
5.3	odd	4		490.2.c.g.99.2	yes	4
5.4	even	2		2450.2.a.bi.1.1		2
7.6	odd	2	inner	2450.2.a.bo.1.1		2
35.2	odd	12		490.2.i.d.459.3		8
35.3	even	12		490.2.i.d.79.4		8
35.12	even	12		490.2.i.d.459.4		8
35.13	even	4		490.2.c.g.99.1	✓	4
35.17	even	12		490.2.i.d.79.1		8
35.18	odd	12		490.2.i.d.79.3		8
35.23	odd	12		490.2.i.d.459.2		8
35.27	even	4		490.2.c.g.99.3	yes	4
35.32	odd	12		490.2.i.d.79.2		8
35.33	even	12		490.2.i.d.459.1		8
35.34	odd	2		2450.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.c.g.99.1	✓	4	35.13	even	4
490.2.c.g.99.2	yes	4	5.3	odd	4
490.2.c.g.99.3	yes	4	35.27	even	4
490.2.c.g.99.4	yes	4	5.2	odd	4
490.2.i.d.79.1		8	35.17	even	12
490.2.i.d.79.2		8	35.32	odd	12
490.2.i.d.79.3		8	35.18	odd	12
490.2.i.d.79.4		8	35.3	even	12
490.2.i.d.459.1		8	35.33	even	12
490.2.i.d.459.2		8	35.23	odd	12
490.2.i.d.459.3		8	35.2	odd	12
490.2.i.d.459.4		8	35.12	even	12
2450.2.a.bi.1.1		2	5.4	even	2
2450.2.a.bi.1.2		2	35.34	odd	2
2450.2.a.bo.1.1		2	7.6	odd	2	inner
2450.2.a.bo.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} -4.00000 q^{11} +4.24264 q^{13} +1.00000 q^{16} -4.24264 q^{17} -3.00000 q^{18} -5.65685 q^{19} -4.00000 q^{22} +4.24264 q^{26} -4.00000 q^{29} +5.65685 q^{31} +1.00000 q^{32} -4.24264 q^{34} -3.00000 q^{36} -6.00000 q^{37} -5.65685 q^{38} +1.41421 q^{41} -12.0000 q^{43} -4.00000 q^{44} +4.24264 q^{52} -12.0000 q^{53} -4.00000 q^{58} +11.3137 q^{59} -7.07107 q^{61} +5.65685 q^{62} +1.00000 q^{64} -12.0000 q^{67} -4.24264 q^{68} +8.00000 q^{71} -3.00000 q^{72} -4.24264 q^{73} -6.00000 q^{74} -5.65685 q^{76} +9.00000 q^{81} +1.41421 q^{82} +16.9706 q^{83} -12.0000 q^{86} -4.00000 q^{88} +4.24264 q^{89} +4.24264 q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 6 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 6 q^{9} - 8 q^{11} + 2 q^{16} - 6 q^{18} - 8 q^{22} - 8 q^{29} + 2 q^{32} - 6 q^{36} - 12 q^{37} - 24 q^{43} - 8 q^{44} - 24 q^{53} - 8 q^{58} + 2 q^{64} - 24 q^{67} + 16 q^{71} - 6 q^{72} - 12 q^{74} + 18 q^{81} - 24 q^{86} - 8 q^{88} + 24 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 6 * q^9 - 8 * q^11 + 2 * q^16 - 6 * q^18 - 8 * q^22 - 8 * q^29 + 2 * q^32 - 6 * q^36 - 12 * q^37 - 24 * q^43 - 8 * q^44 - 24 * q^53 - 8 * q^58 + 2 * q^64 - 24 * q^67 + 16 * q^71 - 6 * q^72 - 12 * q^74 + 18 * q^81 - 24 * q^86 - 8 * q^88 + 24 * q^99

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