LMFDB - Embedded newform 4225.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{12} -1.00000 q^{16} +1.00000 q^{18} +6.00000 q^{19} -2.00000 q^{22} +6.00000 q^{23} +6.00000 q^{24} +4.00000 q^{27} -6.00000 q^{29} +6.00000 q^{31} +5.00000 q^{32} +4.00000 q^{33} -1.00000 q^{36} +6.00000 q^{37} +6.00000 q^{38} -8.00000 q^{41} -6.00000 q^{43} +2.00000 q^{44} +6.00000 q^{46} -8.00000 q^{47} +2.00000 q^{48} -7.00000 q^{49} +12.0000 q^{53} +4.00000 q^{54} -12.0000 q^{57} -6.00000 q^{58} +2.00000 q^{59} +6.00000 q^{61} +6.00000 q^{62} +7.00000 q^{64} +4.00000 q^{66} -12.0000 q^{67} -12.0000 q^{69} -2.00000 q^{71} -3.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} -6.00000 q^{76} -11.0000 q^{81} -8.00000 q^{82} -4.00000 q^{83} -6.00000 q^{86} +12.0000 q^{87} +6.00000 q^{88} -8.00000 q^{89} -6.00000 q^{92} -12.0000 q^{93} -8.00000 q^{94} -10.0000 q^{96} +6.00000 q^{97} -7.00000 q^{98} -2.00000 q^{99} +O(q^{100})$

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	−2.00000	−0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	6.00000	1.22474
$25$	0	0
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	5.00000	0.883883
$33$	4.00000	0.696311
$34$	0	0
$35$	0	0
$36$	−1.00000	−0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	2.00000	0.288675
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	0	0
$57$	−12.0000	−1.58944
$58$	−6.00000	−0.787839
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−12.0000	−1.44463
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−3.00000	−0.353553
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−8.00000	−0.883452
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0	0
$86$	−6.00000	−0.646997
$87$	12.0000	1.28654
$88$	6.00000	0.639602
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−12.0000	−1.24434
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−10.0000	−1.02062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−7.00000	−0.707107
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−4.00000	−0.384900
$109$	−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
$111$	−12.0000	−1.13899
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−12.0000	−1.12390
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	2.00000	0.184115
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	6.00000	0.543214
$123$	16.0000	1.44267
$124$	−6.00000	−0.538816
$125$	0	0
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−3.00000	−0.265165
$129$	12.0000	1.05654
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−12.0000	−1.02151
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	16.0000	1.34744
$142$	−2.00000	−0.167836
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	14.0000	1.15470
$148$	−6.00000	−0.493197
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	−18.0000	−1.45999
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	−24.0000	−1.90332
$160$	0	0
$161$	0	0
$162$	−11.0000	−0.864242
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	6.00000	0.458831
$172$	6.00000	0.457496
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	2.00000	0.150756
$177$	−4.00000	−0.300658
$178$	−8.00000	−0.599625
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−12.0000	−0.887066
$184$	−18.0000	−1.32698
$185$	0	0
$186$	−12.0000	−0.879883
$187$	0	0
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−14.0000	−1.01036
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	7.00000	0.500000
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−2.00000	−0.142134
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	24.0000	1.69283
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	0	0
$206$	6.00000	0.418040
$207$	6.00000	0.417029
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−12.0000	−0.824163
$213$	4.00000	0.274075
$214$	−6.00000	−0.410152
$215$	0	0
$216$	−12.0000	−0.816497
$217$	0	0
$218$	−12.0000	−0.812743
$219$	−12.0000	−0.810885
$220$	0	0
$221$	0	0
$222$	−12.0000	−0.805387
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	12.0000	0.794719
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	0	0
$231$	0	0
$232$	18.0000	1.18176
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	−2.00000	−0.130189
$237$	0	0
$238$	0	0
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	−7.00000	−0.449977
$243$	10.0000	0.641500
$244$	−6.00000	−0.384111
$245$	0	0
$246$	16.0000	1.02012
$247$	0	0
$248$	−18.0000	−1.14300
$249$	8.00000	0.506979
$250$	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
$253$	−12.0000	−0.754434
$254$	2.00000	0.125491
$255$	0	0
$256$	−17.0000	−1.06250
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	12.0000	0.747087
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	−12.0000	−0.741362
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	0	0
$267$	16.0000	0.979184
$268$	12.0000	0.733017
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	0	0
$273$	0	0
$274$	2.00000	0.120824
$275$	0	0
$276$	12.0000	0.722315
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	4.00000	0.239904
$279$	6.00000	0.359211
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	16.0000	0.952786
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	0	0
$287$	0	0
$288$	5.00000	0.294628
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−12.0000	−0.703452
$292$	−6.00000	−0.351123
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	14.0000	0.816497
$295$	0	0
$296$	−18.0000	−1.04623
$297$	−8.00000	−0.464207
$298$	−20.0000	−1.15857
$299$	0	0
$300$	0	0
$301$	0	0
$302$	18.0000	1.03578
$303$	12.0000	0.689382
$304$	−6.00000	−0.344124
$305$	0	0
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−24.0000	−1.34585
$319$	12.0000	0.671871
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	11.0000	0.611111
$325$	0	0
$326$	−12.0000	−0.664619
$327$	24.0000	1.32720
$328$	24.0000	1.32518
$329$	0	0
$330$	0	0
$331$	−30.0000	−1.64895	−0.824475	−	0.565899i	$-0.808529\pi$
$331$	−30.0000	−1.64895	−0.824475	+	0.565899i	$0.808529\pi$
$332$	4.00000	0.219529
$333$	6.00000	0.328798
$334$	−16.0000	−0.875481
$335$	0	0
$336$	0	0
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.0000	−0.649836
$342$	6.00000	0.324443
$343$	0	0
$344$	18.0000	0.970495
$345$	0	0
$346$	12.0000	0.645124
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	−12.0000	−0.643268
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	0	0
$351$	0	0
$352$	−10.0000	−0.533002
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	8.00000	0.423999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	2.00000	0.105118
$363$	14.0000	0.734809
$364$	0	0
$365$	0	0
$366$	−12.0000	−0.627250
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−6.00000	−0.312772
$369$	−8.00000	−0.416463
$370$	0	0
$371$	0	0
$372$	12.0000	0.622171
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	0	0
$376$	24.0000	1.23771
$377$	0	0
$378$	0	0
$379$	−18.0000	−0.924598	−0.462299	−	0.886724i	$-0.652975\pi$
$379$	−18.0000	−0.924598	−0.462299	+	0.886724i	$0.652975\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	6.00000	0.306186
$385$	0	0
$386$	6.00000	0.305392
$387$	−6.00000	−0.304997
$388$	−6.00000	−0.304604
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	21.0000	1.06066
$393$	24.0000	1.21064
$394$	2.00000	0.100759
$395$	0	0
$396$	2.00000	0.100504
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−24.0000	−1.20301
$399$	0	0
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	24.0000	1.19701
$403$	0	0
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	−6.00000	−0.295599
$413$	0	0
$414$	6.00000	0.294884
$415$	0	0
$416$	0	0
$417$	−8.00000	−0.391762
$418$	−12.0000	−0.586939
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	36.0000	1.75453	0.877266	−	0.480004i	$-0.159365\pi$
$421$	36.0000	1.75453	0.877266	+	0.480004i	$0.159365\pi$
$422$	−12.0000	−0.584151
$423$	−8.00000	−0.388973
$424$	−36.0000	−1.74831
$425$	0	0
$426$	4.00000	0.193801
$427$	0	0
$428$	6.00000	0.290021
$429$	0	0
$430$	0	0
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	−4.00000	−0.192450
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	0	0
$436$	12.0000	0.574696
$437$	36.0000	1.72211
$438$	−12.0000	−0.573382
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	12.0000	0.569495
$445$	0	0
$446$	24.0000	1.13643
$447$	40.0000	1.89194
$448$	0	0
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	−36.0000	−1.69143
$454$	4.00000	0.187729
$455$	0	0
$456$	36.0000	1.68585
$457$	30.0000	1.40334	0.701670	−	0.712502i	$-0.252438\pi$
$457$	30.0000	1.40334	0.701670	+	0.712502i	$0.252438\pi$
$458$	−12.0000	−0.560723
$459$	0	0
$460$	0	0
$461$	−4.00000	−0.186299	−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000	−0.186299	−0.0931493	+	0.995652i	$0.529693\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−24.0000	−1.11178
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−24.0000	−1.10586
$472$	−6.00000	−0.276172
$473$	12.0000	0.551761
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.0000	0.549442
$478$	10.0000	0.457389
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	10.0000	0.453609
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−18.0000	−0.814822
$489$	24.0000	1.08532
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−16.0000	−0.721336
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−6.00000	−0.269408
$497$	0	0
$498$	8.00000	0.358489
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	32.0000	1.42965
$502$	−12.0000	−0.535586
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	24.0000	1.05963
$514$	0	0
$515$	0	0
$516$	−12.0000	−0.528271
$517$	16.0000	0.703679
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−6.00000	−0.262613
$523$	−42.0000	−1.83653	−0.918266	−	0.395964i	$-0.870410\pi$
$523$	−42.0000	−1.83653	−0.918266	+	0.395964i	$0.870410\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−6.00000	−0.261612
$527$	0	0
$528$	−4.00000	−0.174078
$529$	13.0000	0.565217
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	0	0
$534$	16.0000	0.692388
$535$	0	0
$536$	36.0000	1.55496
$537$	24.0000	1.03568
$538$	18.0000	0.776035
$539$	14.0000	0.603023
$540$	0	0
$541$	−12.0000	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$541$	−12.0000	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$542$	−6.00000	−0.257722
$543$	−4.00000	−0.171656
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.0000	0.769624	0.384812	−	0.922995i	$-0.374266\pi$
$547$	18.0000	0.769624	0.384812	+	0.922995i	$0.374266\pi$
$548$	−2.00000	−0.0854358
$549$	6.00000	0.256074
$550$	0	0
$551$	−36.0000	−1.53365
$552$	36.0000	1.53226
$553$	0	0
$554$	12.0000	0.509831
$555$	0	0
$556$	−4.00000	−0.169638
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	6.00000	0.254000
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−8.00000	−0.337460
$563$	−30.0000	−1.26435	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−30.0000	−1.26435	−0.632175	+	0.774826i	$0.717837\pi$
$564$	−16.0000	−0.673722
$565$	0	0
$566$	−22.0000	−0.924729
$567$	0	0
$568$	6.00000	0.251754
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	−17.0000	−0.707107
$579$	−12.0000	−0.498703
$580$	0	0
$581$	0	0
$582$	−12.0000	−0.497416
$583$	−24.0000	−0.993978
$584$	−18.0000	−0.744845
$585$	0	0
$586$	26.0000	1.07405
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	−14.0000	−0.577350
$589$	36.0000	1.48335
$590$	0	0
$591$	−4.00000	−0.164538
$592$	−6.00000	−0.246598
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	−8.00000	−0.328244
$595$	0	0
$596$	20.0000	0.819232
$597$	48.0000	1.96451
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	−18.0000	−0.732410
$605$	0	0
$606$	12.0000	0.487467
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	30.0000	1.21666
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	−12.0000	−0.482711
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	0	0
$621$	24.0000	0.963087
$622$	−24.0000	−0.962312
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−8.00000	−0.319744
$627$	24.0000	0.958468
$628$	−12.0000	−0.478852
$629$	0	0
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	0	0
$633$	24.0000	0.953914
$634$	2.00000	0.0794301
$635$	0	0
$636$	24.0000	0.951662
$637$	0	0
$638$	12.0000	0.475085
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	12.0000	0.473602
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	33.0000	1.29636
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	12.0000	0.469956
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	24.0000	0.938474
$655$	0	0
$656$	8.00000	0.312348
$657$	6.00000	0.234082
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	−30.0000	−1.16598
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	6.00000	0.232495
$667$	−36.0000	−1.39393
$668$	16.0000	0.619059
$669$	−48.0000	−1.85579
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	48.0000	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$673$	48.0000	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$674$	−32.0000	−1.23259
$675$	0	0
$676$	0	0
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8.00000	−0.306561
$682$	−12.0000	−0.459504
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	0	0
$687$	24.0000	0.915657
$688$	6.00000	0.228748
$689$	0	0
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	−6.00000	−0.227757
$695$	0	0
$696$	−36.0000	−1.36458
$697$	0	0
$698$	12.0000	0.454207
$699$	48.0000	1.81553
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	36.0000	1.35777
$704$	−14.0000	−0.527645
$705$	0	0
$706$	−14.0000	−0.526897
$707$	0	0
$708$	4.00000	0.150329
$709$	−12.0000	−0.450669	−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000	−0.450669	−0.225335	+	0.974281i	$0.572348\pi$
$710$	0	0
$711$	0	0
$712$	24.0000	0.899438
$713$	36.0000	1.34821
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	−20.0000	−0.746914
$718$	−2.00000	−0.0746393
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	17.0000	0.632674
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	14.0000	0.519589
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	12.0000	0.443533
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	−18.0000	−0.664392
$735$	0	0
$736$	30.0000	1.10581
$737$	24.0000	0.884051
$738$	−8.00000	−0.294484
$739$	−6.00000	−0.220714	−0.110357	−	0.993892i	$-0.535199\pi$
$739$	−6.00000	−0.220714	−0.110357	+	0.993892i	$0.535199\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	36.0000	1.31982
$745$	0	0
$746$	−4.00000	−0.146450
$747$	−4.00000	−0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	8.00000	0.291730
$753$	24.0000	0.874609
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	−18.0000	−0.653789
$759$	24.0000	0.871145
$760$	0	0
$761$	40.0000	1.45000	0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000	1.45000	0.724999	+	0.688749i	$0.241840\pi$
$762$	−4.00000	−0.144905
$763$	0	0
$764$	0	0
$765$	0	0
$766$	8.00000	0.289052
$767$	0	0
$768$	34.0000	1.22687
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	−6.00000	−0.215945
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	−6.00000	−0.215666
$775$	0	0
$776$	−18.0000	−0.646162
$777$	0	0
$778$	6.00000	0.215110
$779$	−48.0000	−1.71978
$780$	0	0
$781$	4.00000	0.143131
$782$	0	0
$783$	−24.0000	−0.857690
$784$	7.00000	0.250000
$785$	0	0
$786$	24.0000	0.856052
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−2.00000	−0.0712470
$789$	12.0000	0.427211
$790$	0	0
$791$	0	0
$792$	6.00000	0.213201
$793$	0	0
$794$	−18.0000	−0.638796
$795$	0	0
$796$	24.0000	0.850657
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−8.00000	−0.282666
$802$	−16.0000	−0.564980
$803$	−12.0000	−0.423471
$804$	−24.0000	−0.846415
$805$	0	0
$806$	0	0
$807$	−36.0000	−1.26726
$808$	18.0000	0.633238
$809$	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
$811$	−30.0000	−1.05344	−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000	−1.05344	−0.526721	+	0.850038i	$0.676579\pi$
$812$	0	0
$813$	12.0000	0.420858
$814$	−12.0000	−0.420600
$815$	0	0
$816$	0	0
$817$	−36.0000	−1.25948
$818$	24.0000	0.839140
$819$	0	0
$820$	0	0
$821$	20.0000	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$822$	−4.00000	−0.139516
$823$	−42.0000	−1.46403	−0.732014	−	0.681290i	$-0.761419\pi$
$823$	−42.0000	−1.46403	−0.732014	+	0.681290i	$0.761419\pi$
$824$	−18.0000	−0.627060
$825$	0	0
$826$	0	0
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	−6.00000	−0.208514
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	−24.0000	−0.832551
$832$	0	0
$833$	0	0
$834$	−8.00000	−0.277017
$835$	0	0
$836$	12.0000	0.415029
$837$	24.0000	0.829561
$838$	12.0000	0.414533
$839$	46.0000	1.58810	0.794048	−	0.607855i	$-0.207970\pi$
$839$	46.0000	1.58810	0.794048	+	0.607855i	$0.207970\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	36.0000	1.24064
$843$	16.0000	0.551069
$844$	12.0000	0.413057
$845$	0	0
$846$	−8.00000	−0.275046
$847$	0	0
$848$	−12.0000	−0.412082
$849$	44.0000	1.51008
$850$	0	0
$851$	36.0000	1.23406
$852$	−4.00000	−0.137038
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	0	0
$855$	0	0
$856$	18.0000	0.615227
$857$	−24.0000	−0.819824	−0.409912	−	0.912125i	$-0.634441\pi$
$857$	−24.0000	−0.819824	−0.409912	+	0.912125i	$0.634441\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	0	0
$862$	−10.0000	−0.340601
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	20.0000	0.680414
$865$	0	0
$866$	16.0000	0.543702
$867$	34.0000	1.15470
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	36.0000	1.21911
$873$	6.00000	0.203069
$874$	36.0000	1.21772
$875$	0	0
$876$	12.0000	0.405442
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	8.00000	0.269987
$879$	−52.0000	−1.75392
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−7.00000	−0.235702
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	0	0
$886$	6.00000	0.201574
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	36.0000	1.20808
$889$	0	0
$890$	0	0
$891$	22.0000	0.737028
$892$	−24.0000	−0.803579
$893$	−48.0000	−1.60626
$894$	40.0000	1.33780
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−16.0000	−0.533927
$899$	−36.0000	−1.20067
$900$	0	0
$901$	0	0
$902$	16.0000	0.532742
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−36.0000	−1.19602
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−4.00000	−0.132745
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	12.0000	0.397360
$913$	8.00000	0.264761
$914$	30.0000	0.992312
$915$	0	0
$916$	12.0000	0.396491
$917$	0	0
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	−4.00000	−0.131733
$923$	0	0
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	6.00000	0.197066
$928$	−30.0000	−0.984798
$929$	−16.0000	−0.524943	−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000	−0.524943	−0.262471	+	0.964940i	$0.584538\pi$
$930$	0	0
$931$	−42.0000	−1.37649
$932$	24.0000	0.786146
$933$	48.0000	1.57145
$934$	−18.0000	−0.588978
$935$	0	0
$936$	0	0
$937$	−56.0000	−1.82944	−0.914720	−	0.404088i	$-0.867589\pi$
$937$	−56.0000	−1.82944	−0.914720	+	0.404088i	$0.867589\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	−24.0000	−0.781962
$943$	−48.0000	−1.56310
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	12.0000	0.390154
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−4.00000	−0.129709
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	12.0000	0.388514
$955$	0	0
$956$	−10.0000	−0.323423
$957$	−24.0000	−0.775810
$958$	22.0000	0.710788
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	21.0000	0.674966
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−10.0000	−0.320750
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−6.00000	−0.192055
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	24.0000	0.767435
$979$	16.0000	0.511362
$980$	0	0
$981$	−12.0000	−0.383131
$982$	−12.0000	−0.382935
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	−48.0000	−1.53018
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−36.0000	−1.14473
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	30.0000	0.952501
$993$	60.0000	1.90404
$994$	0	0
$995$	0	0
$996$	−8.00000	−0.253490
$997$	−60.0000	−1.90022	−0.950110	−	0.311916i	$-0.899029\pi$
$997$	−60.0000	−1.90022	−0.950110	+	0.311916i	$0.899029\pi$
$998$	6.00000	0.189927
$999$	24.0000	0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.k.1.1		1
5.2	odd	4		845.2.b.b.339.2		2
5.3	odd	4		845.2.b.b.339.1		2
5.4	even	2		4225.2.a.h.1.1		1
13.5	odd	4		325.2.c.b.51.1		2
13.8	odd	4		325.2.c.b.51.2		2
13.12	even	2		4225.2.a.e.1.1		1
65.2	even	12		845.2.l.a.654.1		4
65.3	odd	12		845.2.n.b.529.2		4
65.7	even	12		845.2.l.b.699.2		4
65.8	even	4		65.2.d.b.64.1	yes	2
65.12	odd	4		845.2.b.a.339.1		2
65.17	odd	12		845.2.n.a.484.1		4
65.18	even	4		65.2.d.a.64.1	✓	2
65.22	odd	12		845.2.n.b.484.2		4
65.23	odd	12		845.2.n.a.529.1		4
65.28	even	12		845.2.l.b.654.2		4
65.32	even	12		845.2.l.a.699.2		4
65.33	even	12		845.2.l.a.699.1		4
65.34	odd	4		325.2.c.e.51.1		2
65.37	even	12		845.2.l.b.654.1		4
65.38	odd	4		845.2.b.a.339.2		2
65.42	odd	12		845.2.n.b.529.1		4
65.43	odd	12		845.2.n.a.484.2		4
65.44	odd	4		325.2.c.e.51.2		2
65.47	even	4		65.2.d.a.64.2	yes	2
65.48	odd	12		845.2.n.b.484.1		4
65.57	even	4		65.2.d.b.64.2	yes	2
65.58	even	12		845.2.l.b.699.1		4
65.62	odd	12		845.2.n.a.529.2		4
65.63	even	12		845.2.l.a.654.2		4
65.64	even	2		4225.2.a.m.1.1		1
195.8	odd	4		585.2.h.b.64.1		2
195.47	odd	4		585.2.h.c.64.1		2
195.83	odd	4		585.2.h.c.64.2		2
195.122	odd	4		585.2.h.b.64.2		2
260.47	odd	4		1040.2.f.a.129.1		2
260.83	odd	4		1040.2.f.a.129.2		2
260.187	odd	4		1040.2.f.b.129.1		2
260.203	odd	4		1040.2.f.b.129.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.d.a.64.1	✓	2	65.18	even	4
65.2.d.a.64.2	yes	2	65.47	even	4
65.2.d.b.64.1	yes	2	65.8	even	4
65.2.d.b.64.2	yes	2	65.57	even	4
325.2.c.b.51.1		2	13.5	odd	4
325.2.c.b.51.2		2	13.8	odd	4
325.2.c.e.51.1		2	65.34	odd	4
325.2.c.e.51.2		2	65.44	odd	4
585.2.h.b.64.1		2	195.8	odd	4
585.2.h.b.64.2		2	195.122	odd	4
585.2.h.c.64.1		2	195.47	odd	4
585.2.h.c.64.2		2	195.83	odd	4
845.2.b.a.339.1		2	65.12	odd	4
845.2.b.a.339.2		2	65.38	odd	4
845.2.b.b.339.1		2	5.3	odd	4
845.2.b.b.339.2		2	5.2	odd	4
845.2.l.a.654.1		4	65.2	even	12
845.2.l.a.654.2		4	65.63	even	12
845.2.l.a.699.1		4	65.33	even	12
845.2.l.a.699.2		4	65.32	even	12
845.2.l.b.654.1		4	65.37	even	12
845.2.l.b.654.2		4	65.28	even	12
845.2.l.b.699.1		4	65.58	even	12
845.2.l.b.699.2		4	65.7	even	12
845.2.n.a.484.1		4	65.17	odd	12
845.2.n.a.484.2		4	65.43	odd	12
845.2.n.a.529.1		4	65.23	odd	12
845.2.n.a.529.2		4	65.62	odd	12
845.2.n.b.484.1		4	65.48	odd	12
845.2.n.b.484.2		4	65.22	odd	12
845.2.n.b.529.1		4	65.42	odd	12
845.2.n.b.529.2		4	65.3	odd	12
1040.2.f.a.129.1		2	260.47	odd	4
1040.2.f.a.129.2		2	260.83	odd	4
1040.2.f.b.129.1		2	260.187	odd	4
1040.2.f.b.129.2		2	260.203	odd	4
4225.2.a.e.1.1		1	13.12	even	2
4225.2.a.h.1.1		1	5.4	even	2
4225.2.a.k.1.1		1	1.1	even	1	trivial
4225.2.a.m.1.1		1	65.64	even	2

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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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