LMFDB - Embedded newform 4225.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4225,2,Mod(1,4225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [1,-1,-2,-1,0,2,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	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 325)
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)

$f(q)$

$=$

\(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} -5.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} +5.00000 q^{14} -1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +10.0000 q^{21} +3.00000 q^{22} -4.00000 q^{23} -6.00000 q^{24} +4.00000 q^{27} +5.00000 q^{28} -1.00000 q^{29} -1.00000 q^{31} -5.00000 q^{32} +6.00000 q^{33} +5.00000 q^{34} -1.00000 q^{36} +4.00000 q^{37} -4.00000 q^{38} +8.00000 q^{41} -10.0000 q^{42} +4.00000 q^{43} +3.00000 q^{44} +4.00000 q^{46} -7.00000 q^{47} +2.00000 q^{48} +18.0000 q^{49} +10.0000 q^{51} -3.00000 q^{53} -4.00000 q^{54} -15.0000 q^{56} -8.00000 q^{57} +1.00000 q^{58} +3.00000 q^{59} +1.00000 q^{61} +1.00000 q^{62} -5.00000 q^{63} +7.00000 q^{64} -6.00000 q^{66} -3.00000 q^{67} +5.00000 q^{68} +8.00000 q^{69} -8.00000 q^{71} +3.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} -4.00000 q^{76} +15.0000 q^{77} +10.0000 q^{79} -11.0000 q^{81} -8.00000 q^{82} +9.00000 q^{83} -10.0000 q^{84} -4.00000 q^{86} +2.00000 q^{87} -9.00000 q^{88} +18.0000 q^{89} +4.00000 q^{92} +2.00000 q^{93} +7.00000 q^{94} +10.0000 q^{96} +14.0000 q^{97} -18.0000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	5.00000	1.33631
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	10.0000	2.18218
$22$	3.00000	0.639602
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−6.00000	−1.22474
$25$	0	0
$26$	0	0
$27$	4.00000	0.769800
$28$	5.00000	0.944911
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	−5.00000	−0.883883
$33$	6.00000	1.04447
$34$	5.00000	0.857493
$35$	0	0
$36$	−1.00000	−0.166667
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	−10.0000	−1.54303
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	4.00000	0.589768
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	2.00000	0.288675
$49$	18.0000	2.57143
$50$	0	0
$51$	10.0000	1.40028
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−15.0000	−2.00446
$57$	−8.00000	−1.05963
$58$	1.00000	0.131306
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	1.00000	0.127000
$63$	−5.00000	−0.629941
$64$	7.00000	0.875000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	5.00000	0.606339
$69$	8.00000	0.963087
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	3.00000	0.353553
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−4.00000	−0.458831
$77$	15.0000	1.70941
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−8.00000	−0.883452
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−10.0000	−1.09109
$85$	0	0
$86$	−4.00000	−0.431331
$87$	2.00000	0.214423
$88$	−9.00000	−0.959403
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	2.00000	0.207390
$94$	7.00000	0.721995
$95$	0	0
$96$	10.0000	1.02062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−18.0000	−1.81827
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−1.00000	−0.0995037	−0.0497519	−	0.998762i	$-0.515843\pi$
$101$	−1.00000	−0.0995037	−0.0497519	+	0.998762i	$0.515843\pi$
$102$	−10.0000	−0.990148
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	3.00000	0.291386
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	−4.00000	−0.384900
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	5.00000	0.472456
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	1.00000	0.0928477
$117$	0	0
$118$	−3.00000	−0.276172
$119$	25.0000	2.29175
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−1.00000	−0.0905357
$123$	−16.0000	−1.44267
$124$	1.00000	0.0898027
$125$	0	0
$126$	5.00000	0.445435
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	3.00000	0.265165
$129$	−8.00000	−0.704361
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	−6.00000	−0.522233
$133$	−20.0000	−1.73422
$134$	3.00000	0.259161
$135$	0	0
$136$	−15.0000	−1.28624
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−8.00000	−0.681005
$139$	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
$141$	14.0000	1.17901
$142$	8.00000	0.671345
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	−36.0000	−2.96923
$148$	−4.00000	−0.328798
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	12.0000	0.973329
$153$	−5.00000	−0.404226
$154$	−15.0000	−1.20873
$155$	0	0
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	−10.0000	−0.795557
$159$	6.00000	0.475831
$160$	0	0
$161$	20.0000	1.57622
$162$	11.0000	0.864242
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	−9.00000	−0.698535
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	30.0000	2.31455
$169$	0	0
$170$	0	0
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	3.00000	0.226134
$177$	−6.00000	−0.450988
$178$	−18.0000	−1.34916
$179$	−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
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	−12.0000	−0.884652
$185$	0	0
$186$	−2.00000	−0.146647
$187$	15.0000	1.09691
$188$	7.00000	0.510527
$189$	−20.0000	−1.45479
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	−14.0000	−1.01036
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−18.0000	−1.28571
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	3.00000	0.213201
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	1.00000	0.0703598
$203$	5.00000	0.350931
$204$	−10.0000	−0.700140
$205$	0	0
$206$	4.00000	0.278693
$207$	−4.00000	−0.278019
$208$	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	$0.635539\pi$
$212$	3.00000	0.206041
$213$	16.0000	1.09630
$214$	16.0000	1.09374
$215$	0	0
$216$	12.0000	0.816497
$217$	5.00000	0.339422
$218$	−2.00000	−0.135457
$219$	−8.00000	−0.540590
$220$	0	0
$221$	0	0
$222$	8.00000	0.536925
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	25.0000	1.67038
$225$	0	0
$226$	−10.0000	−0.665190
$227$	−9.00000	−0.597351	−0.298675	−	0.954355i	$-0.596545\pi$
$227$	−9.00000	−0.597351	−0.298675	+	0.954355i	$0.596545\pi$
$228$	8.00000	0.529813
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−30.0000	−1.97386
$232$	−3.00000	−0.196960
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	0	0
$236$	−3.00000	−0.195283
$237$	−20.0000	−1.29914
$238$	−25.0000	−1.62051
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	2.00000	0.128565
$243$	10.0000	0.641500
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	16.0000	1.02012
$247$	0	0
$248$	−3.00000	−0.190500
$249$	−18.0000	−1.14070
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	5.00000	0.314970
$253$	12.0000	0.754434
$254$	18.0000	1.12942
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	8.00000	0.498058
$259$	−20.0000	−1.24274
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−18.0000	−1.11204
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	18.0000	1.10782
$265$	0	0
$266$	20.0000	1.22628
$267$	−36.0000	−2.20316
$268$	3.00000	0.183254
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	0	0
$271$	−29.0000	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−29.0000	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−8.00000	−0.481543
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−4.00000	−0.239904
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−14.0000	−0.833688
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	−40.0000	−2.36113
$288$	−5.00000	−0.294628
$289$	8.00000	0.470588
$290$	0	0
$291$	−28.0000	−1.64139
$292$	−4.00000	−0.234082
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	36.0000	2.09956
$295$	0	0
$296$	12.0000	0.697486
$297$	−12.0000	−0.696311
$298$	10.0000	0.579284
$299$	0	0
$300$	0	0
$301$	−20.0000	−1.15278
$302$	13.0000	0.748066
$303$	2.00000	0.114897
$304$	−4.00000	−0.229416
$305$	0	0
$306$	5.00000	0.285831
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	−15.0000	−0.854704
$309$	8.00000	0.455104
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	17.0000	0.960897	0.480448	−	0.877023i	$-0.340474\pi$
$313$	17.0000	0.960897	0.480448	+	0.877023i	$0.340474\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	−6.00000	−0.336463
$319$	3.00000	0.167968
$320$	0	0
$321$	32.0000	1.78607
$322$	−20.0000	−1.11456
$323$	−20.0000	−1.11283
$324$	11.0000	0.611111
$325$	0	0
$326$	−12.0000	−0.664619
$327$	−4.00000	−0.221201
$328$	24.0000	1.32518
$329$	35.0000	1.92961
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−9.00000	−0.493939
$333$	4.00000	0.219199
$334$	−16.0000	−0.875481
$335$	0	0
$336$	−10.0000	−0.545545
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	0	0
$339$	−20.0000	−1.08625
$340$	0	0
$341$	3.00000	0.162459
$342$	−4.00000	−0.216295
$343$	−55.0000	−2.96972
$344$	12.0000	0.646997
$345$	0	0
$346$	13.0000	0.698884
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	−2.00000	−0.107211
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	15.0000	0.799503
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−18.0000	−0.953998
$357$	−50.0000	−2.64628
$358$	12.0000	0.634220
$359$	−23.0000	−1.21389	−0.606947	−	0.794742i	$-0.707606\pi$
$359$	−23.0000	−1.21389	−0.606947	+	0.794742i	$0.707606\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	3.00000	0.157676
$363$	4.00000	0.209946
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	4.00000	0.208514
$369$	8.00000	0.416463
$370$	0	0
$371$	15.0000	0.778761
$372$	−2.00000	−0.103695
$373$	31.0000	1.60512	0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000	1.60512	0.802560	+	0.596572i	$0.203471\pi$
$374$	−15.0000	−0.775632
$375$	0	0
$376$	−21.0000	−1.08299
$377$	0	0
$378$	20.0000	1.02869
$379$	−27.0000	−1.38690	−0.693448	−	0.720506i	$-0.743909\pi$
$379$	−27.0000	−1.38690	−0.693448	+	0.720506i	$0.743909\pi$
$380$	0	0
$381$	36.0000	1.84434
$382$	−10.0000	−0.511645
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	−6.00000	−0.306186
$385$	0	0
$386$	−4.00000	−0.203595
$387$	4.00000	0.203331
$388$	−14.0000	−0.710742
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	54.0000	2.72741
$393$	−36.0000	−1.81596
$394$	−8.00000	−0.403034
$395$	0	0
$396$	3.00000	0.150756
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	−6.00000	−0.300753
$399$	40.0000	2.00250
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	−6.00000	−0.299253
$403$	0	0
$404$	1.00000	0.0497519
$405$	0	0
$406$	−5.00000	−0.248146
$407$	−12.0000	−0.594818
$408$	30.0000	1.48522
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	4.00000	0.197066
$413$	−15.0000	−0.738102
$414$	4.00000	0.196589
$415$	0	0
$416$	0	0
$417$	−8.00000	−0.391762
$418$	12.0000	0.586939
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	12.0000	0.584151
$423$	−7.00000	−0.340352
$424$	−9.00000	−0.437079
$425$	0	0
$426$	−16.0000	−0.775203
$427$	−5.00000	−0.241967
$428$	16.0000	0.773389
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−4.00000	−0.192450
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	−16.0000	−0.765384
$438$	8.00000	0.382255
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	−16.0000	−0.757622
$447$	20.0000	0.945968
$448$	−35.0000	−1.65359
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	−10.0000	−0.470360
$453$	26.0000	1.22159
$454$	9.00000	0.422391
$455$	0	0
$456$	−24.0000	−1.12390
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−2.00000	−0.0934539
$459$	−20.0000	−0.933520
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	30.0000	1.39573
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	14.0000	0.648537
$467$	−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
$469$	15.0000	0.692636
$470$	0	0
$471$	26.0000	1.19802
$472$	9.00000	0.414259
$473$	−12.0000	−0.551761
$474$	20.0000	0.918630
$475$	0	0
$476$	−25.0000	−1.14587
$477$	−3.00000	−0.137361
$478$	5.00000	0.228695
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−40.0000	−1.82006
$484$	2.00000	0.0909091
$485$	0	0
$486$	−10.0000	−0.453609
$487$	15.0000	0.679715	0.339857	−	0.940477i	$-0.389621\pi$
$487$	15.0000	0.679715	0.339857	+	0.940477i	$0.389621\pi$
$488$	3.00000	0.135804
$489$	−24.0000	−1.08532
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	16.0000	0.721336
$493$	5.00000	0.225189
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	40.0000	1.79425
$498$	18.0000	0.806599
$499$	19.0000	0.850557	0.425278	−	0.905063i	$-0.360176\pi$
$499$	19.0000	0.850557	0.425278	+	0.905063i	$0.360176\pi$
$500$	0	0
$501$	−32.0000	−1.42965
$502$	12.0000	0.535586
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	−15.0000	−0.668153
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	18.0000	0.798621
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	11.0000	0.486136
$513$	16.0000	0.706417
$514$	15.0000	0.661622
$515$	0	0
$516$	8.00000	0.352180
$517$	21.0000	0.923579
$518$	20.0000	0.878750
$519$	26.0000	1.14127
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	1.00000	0.0437688
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	6.00000	0.261612
$527$	5.00000	0.217803
$528$	−6.00000	−0.261116
$529$	−7.00000	−0.304348
$530$	0	0
$531$	3.00000	0.130189
$532$	20.0000	0.867110
$533$	0	0
$534$	36.0000	1.55787
$535$	0	0
$536$	−9.00000	−0.388741
$537$	24.0000	1.03568
$538$	17.0000	0.732922
$539$	−54.0000	−2.32594
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	29.0000	1.24566
$543$	6.00000	0.257485
$544$	25.0000	1.07187
$545$	0	0
$546$	0	0
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	−18.0000	−0.768922
$549$	1.00000	0.0426790
$550$	0	0
$551$	−4.00000	−0.170406
$552$	24.0000	1.02151
$553$	−50.0000	−2.12622
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	1.00000	0.0423334
$559$	0	0
$560$	0	0
$561$	−30.0000	−1.26660
$562$	−18.0000	−0.759284
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	−14.0000	−0.589506
$565$	0	0
$566$	−28.0000	−1.17693
$567$	55.0000	2.30978
$568$	−24.0000	−1.00702
$569$	33.0000	1.38343	0.691716	−	0.722170i	$-0.256855\pi$
$569$	33.0000	1.38343	0.691716	+	0.722170i	$0.256855\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	0	0
$573$	−20.0000	−0.835512
$574$	40.0000	1.66957
$575$	0	0
$576$	7.00000	0.291667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−8.00000	−0.332756
$579$	−8.00000	−0.332469
$580$	0	0
$581$	−45.0000	−1.86691
$582$	28.0000	1.16064
$583$	9.00000	0.372742
$584$	12.0000	0.496564
$585$	0	0
$586$	6.00000	0.247858
$587$	−25.0000	−1.03186	−0.515930	−	0.856631i	$-0.672554\pi$
$587$	−25.0000	−1.03186	−0.515930	+	0.856631i	$0.672554\pi$
$588$	36.0000	1.48461
$589$	−4.00000	−0.164817
$590$	0	0
$591$	−16.0000	−0.658152
$592$	−4.00000	−0.164399
$593$	8.00000	0.328521	0.164260	−	0.986417i	$-0.447476\pi$
$593$	8.00000	0.328521	0.164260	+	0.986417i	$0.447476\pi$
$594$	12.0000	0.492366
$595$	0	0
$596$	10.0000	0.409616
$597$	−12.0000	−0.491127
$598$	0	0
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	0	0
$601$	29.0000	1.18293	0.591467	−	0.806329i	$-0.298549\pi$
$601$	29.0000	1.18293	0.591467	+	0.806329i	$0.298549\pi$
$602$	20.0000	0.815139
$603$	−3.00000	−0.122169
$604$	13.0000	0.528962
$605$	0	0
$606$	−2.00000	−0.0812444
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−20.0000	−0.811107
$609$	−10.0000	−0.405220
$610$	0	0
$611$	0	0
$612$	5.00000	0.202113
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	45.0000	1.81310
$617$	4.00000	0.161034	0.0805170	−	0.996753i	$-0.474343\pi$
$617$	4.00000	0.161034	0.0805170	+	0.996753i	$0.474343\pi$
$618$	−8.00000	−0.321807
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	4.00000	0.160385
$623$	−90.0000	−3.60577
$624$	0	0
$625$	0	0
$626$	−17.0000	−0.679457
$627$	24.0000	0.958468
$628$	13.0000	0.518756
$629$	−20.0000	−0.797452
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	30.0000	1.19334
$633$	24.0000	0.953914
$634$	22.0000	0.873732
$635$	0	0
$636$	−6.00000	−0.237915
$637$	0	0
$638$	−3.00000	−0.118771
$639$	−8.00000	−0.316475
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	−32.0000	−1.26294
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−20.0000	−0.788110
$645$	0	0
$646$	20.0000	0.786889
$647$	14.0000	0.550397	0.275198	−	0.961387i	$-0.411256\pi$
$647$	14.0000	0.550397	0.275198	+	0.961387i	$0.411256\pi$
$648$	−33.0000	−1.29636
$649$	−9.00000	−0.353281
$650$	0	0
$651$	−10.0000	−0.391931
$652$	−12.0000	−0.469956
$653$	31.0000	1.21312	0.606562	−	0.795036i	$-0.292548\pi$
$653$	31.0000	1.21312	0.606562	+	0.795036i	$0.292548\pi$
$654$	4.00000	0.156412
$655$	0	0
$656$	−8.00000	−0.312348
$657$	4.00000	0.156055
$658$	−35.0000	−1.36444
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	27.0000	1.04780
$665$	0	0
$666$	−4.00000	−0.154997
$667$	4.00000	0.154881
$668$	−16.0000	−0.619059
$669$	−32.0000	−1.23719
$670$	0	0
$671$	−3.00000	−0.115814
$672$	−50.0000	−1.92879
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	−3.00000	−0.115556
$675$	0	0
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	20.0000	0.768095
$679$	−70.0000	−2.68635
$680$	0	0
$681$	18.0000	0.689761
$682$	−3.00000	−0.114876
$683$	41.0000	1.56882	0.784411	−	0.620242i	$-0.212966\pi$
$683$	41.0000	1.56882	0.784411	+	0.620242i	$0.212966\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	55.0000	2.09991
$687$	−4.00000	−0.152610
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	−17.0000	−0.646710	−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000	−0.646710	−0.323355	+	0.946278i	$0.604811\pi$
$692$	13.0000	0.494186
$693$	15.0000	0.569803
$694$	−4.00000	−0.151838
$695$	0	0
$696$	6.00000	0.227429
$697$	−40.0000	−1.51511
$698$	2.00000	0.0757011
$699$	28.0000	1.05906
$700$	0	0
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	−21.0000	−0.791467
$705$	0	0
$706$	6.00000	0.225813
$707$	5.00000	0.188044
$708$	6.00000	0.225494
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	54.0000	2.02374
$713$	4.00000	0.149801
$714$	50.0000	1.87120
$715$	0	0
$716$	12.0000	0.448461
$717$	10.0000	0.373457
$718$	23.0000	0.858352
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	3.00000	0.111648
$723$	0	0
$724$	3.00000	0.111494
$725$	0	0
$726$	−4.00000	−0.148454
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−20.0000	−0.739727
$732$	2.00000	0.0739221
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	18.0000	0.664392
$735$	0	0
$736$	20.0000	0.737210
$737$	9.00000	0.331519
$738$	−8.00000	−0.294484
$739$	−9.00000	−0.331070	−0.165535	−	0.986204i	$-0.552935\pi$
$739$	−9.00000	−0.331070	−0.165535	+	0.986204i	$0.552935\pi$
$740$	0	0
$741$	0	0
$742$	−15.0000	−0.550667
$743$	41.0000	1.50414	0.752072	−	0.659081i	$-0.229055\pi$
$743$	41.0000	1.50414	0.752072	+	0.659081i	$0.229055\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	−31.0000	−1.13499
$747$	9.00000	0.329293
$748$	−15.0000	−0.548454
$749$	80.0000	2.92314
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	7.00000	0.255264
$753$	24.0000	0.874609
$754$	0	0
$755$	0	0
$756$	20.0000	0.727393
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	27.0000	0.980684
$759$	−24.0000	−0.871145
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	−36.0000	−1.30414
$763$	−10.0000	−0.362024
$764$	−10.0000	−0.361787
$765$	0	0
$766$	−32.0000	−1.15621
$767$	0	0
$768$	34.0000	1.22687
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	−4.00000	−0.143963
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	42.0000	1.50771
$777$	40.0000	1.43499
$778$	34.0000	1.21896
$779$	32.0000	1.14652
$780$	0	0
$781$	24.0000	0.858788
$782$	−20.0000	−0.715199
$783$	−4.00000	−0.142948
$784$	−18.0000	−0.642857
$785$	0	0
$786$	36.0000	1.28408
$787$	43.0000	1.53278	0.766392	−	0.642373i	$-0.222050\pi$
$787$	43.0000	1.53278	0.766392	+	0.642373i	$0.222050\pi$
$788$	−8.00000	−0.284988
$789$	12.0000	0.427211
$790$	0	0
$791$	−50.0000	−1.77780
$792$	−9.00000	−0.319801
$793$	0	0
$794$	−8.00000	−0.283909
$795$	0	0
$796$	−6.00000	−0.212664
$797$	53.0000	1.87736	0.938678	−	0.344795i	$-0.112051\pi$
$797$	53.0000	1.87736	0.938678	+	0.344795i	$0.112051\pi$
$798$	−40.0000	−1.41598
$799$	35.0000	1.23821
$800$	0	0
$801$	18.0000	0.635999
$802$	−26.0000	−0.918092
$803$	−12.0000	−0.423471
$804$	−6.00000	−0.211604
$805$	0	0
$806$	0	0
$807$	34.0000	1.19686
$808$	−3.00000	−0.105540
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−25.0000	−0.877869	−0.438934	−	0.898519i	$-0.644644\pi$
$811$	−25.0000	−0.877869	−0.438934	+	0.898519i	$0.644644\pi$
$812$	−5.00000	−0.175466
$813$	58.0000	2.03415
$814$	12.0000	0.420600
$815$	0	0
$816$	−10.0000	−0.350070
$817$	16.0000	0.559769
$818$	−6.00000	−0.209785
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	36.0000	1.25564
$823$	−12.0000	−0.418294	−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000	−0.418294	−0.209147	+	0.977884i	$0.567069\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	15.0000	0.521917
$827$	−19.0000	−0.660695	−0.330347	−	0.943859i	$-0.607166\pi$
$827$	−19.0000	−0.660695	−0.330347	+	0.943859i	$0.607166\pi$
$828$	4.00000	0.139010
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	0	0
$833$	−90.0000	−3.11832
$834$	8.00000	0.277017
$835$	0	0
$836$	12.0000	0.415029
$837$	−4.00000	−0.138260
$838$	18.0000	0.621800
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−4.00000	−0.137849
$843$	−36.0000	−1.23991
$844$	12.0000	0.413057
$845$	0	0
$846$	7.00000	0.240665
$847$	10.0000	0.343604
$848$	3.00000	0.103020
$849$	−56.0000	−1.92192
$850$	0	0
$851$	−16.0000	−0.548473
$852$	−16.0000	−0.548151
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	5.00000	0.171096
$855$	0	0
$856$	−48.0000	−1.64061
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	80.0000	2.72639
$862$	0	0
$863$	33.0000	1.12333	0.561667	−	0.827364i	$-0.310160\pi$
$863$	33.0000	1.12333	0.561667	+	0.827364i	$0.310160\pi$
$864$	−20.0000	−0.680414
$865$	0	0
$866$	14.0000	0.475739
$867$	−16.0000	−0.543388
$868$	−5.00000	−0.169711
$869$	−30.0000	−1.01768
$870$	0	0
$871$	0	0
$872$	6.00000	0.203186
$873$	14.0000	0.473828
$874$	16.0000	0.541208
$875$	0	0
$876$	8.00000	0.270295
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	22.0000	0.742464
$879$	12.0000	0.404750
$880$	0	0
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	−18.0000	−0.606092
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	0	0
$886$	24.0000	0.806296
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	−24.0000	−0.805387
$889$	90.0000	3.01850
$890$	0	0
$891$	33.0000	1.10554
$892$	−16.0000	−0.535720
$893$	−28.0000	−0.936984
$894$	−20.0000	−0.668900
$895$	0	0
$896$	−15.0000	−0.501115
$897$	0	0
$898$	−6.00000	−0.200223
$899$	1.00000	0.0333519
$900$	0	0
$901$	15.0000	0.499722
$902$	24.0000	0.799113
$903$	40.0000	1.33112
$904$	30.0000	0.997785
$905$	0	0
$906$	−26.0000	−0.863792
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	9.00000	0.298675
$909$	−1.00000	−0.0331679
$910$	0	0
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	8.00000	0.264906
$913$	−27.0000	−0.893570
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	−90.0000	−2.97206
$918$	20.0000	0.660098
$919$	46.0000	1.51740	0.758700	−	0.651440i	$-0.225835\pi$
$919$	46.0000	1.51740	0.758700	+	0.651440i	$0.225835\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	26.0000	0.856264
$923$	0	0
$924$	30.0000	0.986928
$925$	0	0
$926$	19.0000	0.624379
$927$	−4.00000	−0.131377
$928$	5.00000	0.164133
$929$	56.0000	1.83730	0.918650	−	0.395072i	$-0.129280\pi$
$929$	56.0000	1.83730	0.918650	+	0.395072i	$0.129280\pi$
$930$	0	0
$931$	72.0000	2.35970
$932$	14.0000	0.458585
$933$	8.00000	0.261908
$934$	18.0000	0.588978
$935$	0	0
$936$	0	0
$937$	−1.00000	−0.0326686	−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000	−0.0326686	−0.0163343	+	0.999867i	$0.505200\pi$
$938$	−15.0000	−0.489767
$939$	−34.0000	−1.10955
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	−26.0000	−0.847126
$943$	−32.0000	−1.04206
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	12.0000	0.390154
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	20.0000	0.649570
$949$	0	0
$950$	0	0
$951$	44.0000	1.42680
$952$	75.0000	2.43076
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	5.00000	0.161712
$957$	−6.00000	−0.193952
$958$	−3.00000	−0.0969256
$959$	−90.0000	−2.90625
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−16.0000	−0.515593
$964$	0	0
$965$	0	0
$966$	40.0000	1.28698
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	−6.00000	−0.192847
$969$	40.0000	1.28499
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	−10.0000	−0.320750
$973$	−20.0000	−0.641171
$974$	−15.0000	−0.480631
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	24.0000	0.767435
$979$	−54.0000	−1.72585
$980$	0	0
$981$	2.00000	0.0638551
$982$	−28.0000	−0.893516
$983$	11.0000	0.350846	0.175423	−	0.984493i	$-0.443871\pi$
$983$	11.0000	0.350846	0.175423	+	0.984493i	$0.443871\pi$
$984$	−48.0000	−1.53018
$985$	0	0
$986$	−5.00000	−0.159232
$987$	−70.0000	−2.22812
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	5.00000	0.158750
$993$	−40.0000	−1.26936
$994$	−40.0000	−1.26872
$995$	0	0
$996$	18.0000	0.570352
$997$	45.0000	1.42516	0.712582	−	0.701589i	$-0.247526\pi$
$997$	45.0000	1.42516	0.712582	+	0.701589i	$0.247526\pi$
$998$	−19.0000	−0.601434
$999$	16.0000	0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.d.1.1		1
5.4	even	2		4225.2.a.n.1.1		1
13.5	odd	4		325.2.c.a.51.2	yes	2
13.8	odd	4		325.2.c.a.51.1	✓	2
13.12	even	2		4225.2.a.l.1.1		1
65.8	even	4		325.2.d.b.324.1		2
65.18	even	4		325.2.d.c.324.1		2
65.34	odd	4		325.2.c.f.51.2	yes	2
65.44	odd	4		325.2.c.f.51.1	yes	2
65.47	even	4		325.2.d.c.324.2		2
65.57	even	4		325.2.d.b.324.2		2
65.64	even	2		4225.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.c.a.51.1	✓	2	13.8	odd	4
325.2.c.a.51.2	yes	2	13.5	odd	4
325.2.c.f.51.1	yes	2	65.44	odd	4
325.2.c.f.51.2	yes	2	65.34	odd	4
325.2.d.b.324.1		2	65.8	even	4
325.2.d.b.324.2		2	65.57	even	4
325.2.d.c.324.1		2	65.18	even	4
325.2.d.c.324.2		2	65.47	even	4
4225.2.a.d.1.1		1	1.1	even	1	trivial
4225.2.a.f.1.1		1	65.64	even	2
4225.2.a.l.1.1		1	13.12	even	2
4225.2.a.n.1.1		1	5.4	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 $\F_{q}$
Belyi maps

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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