LMFDB - Embedded newform 650.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,2,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, 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("650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	650.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:	$5.19027613138$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} -2.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} -3.00000 q^{22} -6.00000 q^{23} +2.00000 q^{24} -1.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} -3.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -2.00000 q^{42} -10.0000 q^{43} +3.00000 q^{44} +6.00000 q^{46} -3.00000 q^{47} -2.00000 q^{48} -6.00000 q^{49} -6.00000 q^{51} +1.00000 q^{52} +3.00000 q^{53} -4.00000 q^{54} +1.00000 q^{56} +8.00000 q^{57} +3.00000 q^{58} -15.0000 q^{59} -13.0000 q^{61} +1.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} -13.0000 q^{67} +3.00000 q^{68} +12.0000 q^{69} -1.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} -4.00000 q^{76} -3.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} -11.0000 q^{81} +15.0000 q^{83} +2.00000 q^{84} +10.0000 q^{86} +6.00000 q^{87} -3.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} -6.00000 q^{92} +2.00000 q^{93} +3.00000 q^{94} +2.00000 q^{96} -4.00000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
$2$	−1.00000	−0.707107
$3$	−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
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−2.00000	−0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	−3.00000	−0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	−1.00000	−0.196116
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	4.00000	0.648886
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−2.00000	−0.308607
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	6.00000	0.884652
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−2.00000	−0.288675
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	1.00000	0.138675
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	8.00000	1.05963
$58$	3.00000	0.393919
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	1.00000	0.127000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	3.00000	0.363803
$69$	12.0000	1.44463
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−4.00000	−0.458831
$77$	−3.00000	−0.341882
$78$	2.00000	0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	10.0000	1.07833
$87$	6.00000	0.643268
$88$	−3.00000	−0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−6.00000	−0.625543
$93$	2.00000	0.207390
$94$	3.00000	0.309426
$95$	0	0
$96$	2.00000	0.204124
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	6.00000	0.606092
$99$	3.00000	0.301511
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	6.00000	0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−3.00000	−0.291386
$107$	−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
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	−3.00000	−0.278543
$117$	1.00000	0.0924500
$118$	15.0000	1.38086
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	13.0000	1.17696
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	1.00000	0.0890871
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	20.0000	1.76090
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	−6.00000	−0.522233
$133$	4.00000	0.346844
$134$	13.0000	1.12303
$135$	0	0
$136$	−3.00000	−0.257248
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−12.0000	−1.02151
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	12.0000	0.989743
$148$	2.00000	0.164399
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	4.00000	0.324443
$153$	3.00000	0.242536
$154$	3.00000	0.241747
$155$	0	0
$156$	−2.00000	−0.160128
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	4.00000	0.318223
$159$	−6.00000	−0.475831
$160$	0	0
$161$	6.00000	0.472866
$162$	11.0000	0.864242
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	0	0
$166$	−15.0000	−1.16423
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−2.00000	−0.154303
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−10.0000	−0.762493
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	3.00000	0.226134
$177$	30.0000	2.25494
$178$	−6.00000	−0.449719
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	1.00000	0.0741249
$183$	26.0000	1.92198
$184$	6.00000	0.442326
$185$	0	0
$186$	−2.00000	−0.146647
$187$	9.00000	0.658145
$188$	−3.00000	−0.218797
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−2.00000	−0.144338
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	−3.00000	−0.213201
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	26.0000	1.83390
$202$	−9.00000	−0.633238
$203$	3.00000	0.210559
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−8.00000	−0.557386
$207$	−6.00000	−0.417029
$208$	1.00000	0.0693375
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	3.00000	0.206041
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	−4.00000	−0.272166
$217$	1.00000	0.0678844
$218$	−14.0000	−0.948200
$219$	20.0000	1.35147
$220$	0	0
$221$	3.00000	0.201802
$222$	4.00000	0.268462
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−18.0000	−1.19734
$227$	9.00000	0.597351	0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000	0.597351	0.298675	+	0.954355i	$0.403455\pi$
$228$	8.00000	0.529813
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	3.00000	0.196960
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−15.0000	−0.976417
$237$	8.00000	0.519656
$238$	3.00000	0.194461
$239$	−21.0000	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	2.00000	0.128565
$243$	10.0000	0.641500
$244$	−13.0000	−0.832240
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	1.00000	0.0635001
$249$	−30.0000	−1.90117
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−1.00000	−0.0629941
$253$	−18.0000	−1.13165
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	−20.0000	−1.24515
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−3.00000	−0.185695
$262$	6.00000	0.370681
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	−4.00000	−0.245256
$267$	−12.0000	−0.734388
$268$	−13.0000	−0.794101
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	23.0000	1.39715	0.698575	−	0.715537i	$-0.253818\pi$
$271$	23.0000	1.39715	0.698575	+	0.715537i	$0.253818\pi$
$272$	3.00000	0.181902
$273$	2.00000	0.121046
$274$	−18.0000	−1.08742
$275$	0	0
$276$	12.0000	0.722315
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−2.00000	−0.119952
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	−6.00000	−0.357295
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	0	0
$285$	0	0
$286$	−3.00000	−0.177394
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	8.00000	0.468968
$292$	−10.0000	−0.585206
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	−12.0000	−0.699854
$295$	0	0
$296$	−2.00000	−0.116248
$297$	12.0000	0.696311
$298$	−12.0000	−0.695141
$299$	−6.00000	−0.346989
$300$	0	0
$301$	10.0000	0.576390
$302$	−11.0000	−0.632979
$303$	−18.0000	−1.03407
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−3.00000	−0.171499
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	−3.00000	−0.170941
$309$	−16.0000	−0.910208
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	2.00000	0.113228
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	−5.00000	−0.282166
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	6.00000	0.336463
$319$	−9.00000	−0.503903
$320$	0	0
$321$	12.0000	0.669775
$322$	−6.00000	−0.334367
$323$	−12.0000	−0.667698
$324$	−11.0000	−0.611111
$325$	0	0
$326$	16.0000	0.886158
$327$	−28.0000	−1.54840
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	15.0000	0.823232
$333$	2.00000	0.109599
$334$	0	0
$335$	0	0
$336$	2.00000	0.109109
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−1.00000	−0.0543928
$339$	−36.0000	−1.95525
$340$	0	0
$341$	−3.00000	−0.162459
$342$	4.00000	0.216295
$343$	13.0000	0.701934
$344$	10.0000	0.539164
$345$	0	0
$346$	15.0000	0.806405
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	6.00000	0.321634
$349$	32.0000	1.71292	0.856460	−	0.516213i	$-0.172659\pi$
$349$	32.0000	1.71292	0.856460	+	0.516213i	$0.172659\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	−3.00000	−0.159901
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−30.0000	−1.59448
$355$	0	0
$356$	6.00000	0.317999
$357$	6.00000	0.317554
$358$	24.0000	1.26844
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	25.0000	1.31397
$363$	4.00000	0.209946
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	−26.0000	−1.35904
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	0	0
$371$	−3.00000	−0.155752
$372$	2.00000	0.103695
$373$	−31.0000	−1.60512	−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000	−1.60512	−0.802560	+	0.596572i	$0.796529\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	3.00000	0.154713
$377$	−3.00000	−0.154508
$378$	4.00000	0.205738
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	0	0
$381$	32.0000	1.63941
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	4.00000	0.203595
$387$	−10.0000	−0.508329
$388$	−4.00000	−0.203069
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	6.00000	0.303046
$393$	12.0000	0.605320
$394$	24.0000	1.20910
$395$	0	0
$396$	3.00000	0.150756
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−20.0000	−1.00251
$399$	−8.00000	−0.400501
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	−26.0000	−1.29676
$403$	−1.00000	−0.0498135
$404$	9.00000	0.447767
$405$	0	0
$406$	−3.00000	−0.148888
$407$	6.00000	0.297409
$408$	6.00000	0.297044
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	8.00000	0.394132
$413$	15.0000	0.738102
$414$	6.00000	0.294884
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−4.00000	−0.195881
$418$	12.0000	0.586939
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	22.0000	1.07094
$423$	−3.00000	−0.145865
$424$	−3.00000	−0.145693
$425$	0	0
$426$	0	0
$427$	13.0000	0.629114
$428$	−6.00000	−0.290021
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	4.00000	0.192450
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	−1.00000	−0.0480015
$435$	0	0
$436$	14.0000	0.670478
$437$	24.0000	1.14808
$438$	−20.0000	−0.955637
$439$	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
$441$	−6.00000	−0.285714
$442$	−3.00000	−0.142695
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	4.00000	0.189405
$447$	−24.0000	−1.13516
$448$	−1.00000	−0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	18.0000	0.846649
$453$	−22.0000	−1.03365
$454$	−9.00000	−0.422391
$455$	0	0
$456$	−8.00000	−0.374634
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−20.0000	−0.934539
$459$	12.0000	0.560112
$460$	0	0
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	−6.00000	−0.279145
$463$	29.0000	1.34774	0.673872	−	0.738848i	$-0.264630\pi$
$463$	29.0000	1.34774	0.673872	+	0.738848i	$0.264630\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−6.00000	−0.277945
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	1.00000	0.0462250
$469$	13.0000	0.600284
$470$	0	0
$471$	−10.0000	−0.460776
$472$	15.0000	0.690431
$473$	−30.0000	−1.37940
$474$	−8.00000	−0.367452
$475$	0	0
$476$	−3.00000	−0.137505
$477$	3.00000	0.137361
$478$	21.0000	0.960518
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−8.00000	−0.364390
$483$	−12.0000	−0.546019
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	−10.0000	−0.453609
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	13.0000	0.588482
$489$	32.0000	1.44709
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	4.00000	0.179969
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	30.0000	1.34433
$499$	−7.00000	−0.313363	−0.156682	−	0.987649i	$-0.550080\pi$
$499$	−7.00000	−0.313363	−0.156682	+	0.987649i	$0.550080\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	18.0000	0.800198
$507$	−2.00000	−0.0888231
$508$	−16.0000	−0.709885
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	−1.00000	−0.0441942
$513$	−16.0000	−0.706417
$514$	15.0000	0.661622
$515$	0	0
$516$	20.0000	0.880451
$517$	−9.00000	−0.395820
$518$	2.00000	0.0878750
$519$	30.0000	1.31685
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	3.00000	0.131306
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−18.0000	−0.784837
$527$	−3.00000	−0.130682
$528$	−6.00000	−0.261116
$529$	13.0000	0.565217
$530$	0	0
$531$	−15.0000	−0.650945
$532$	4.00000	0.173422
$533$	0	0
$534$	12.0000	0.519291
$535$	0	0
$536$	13.0000	0.561514
$537$	48.0000	2.07135
$538$	−21.0000	−0.905374
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	−23.0000	−0.987935
$543$	50.0000	2.14571
$544$	−3.00000	−0.128624
$545$	0	0
$546$	−2.00000	−0.0855921
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	18.0000	0.768922
$549$	−13.0000	−0.554826
$550$	0	0
$551$	12.0000	0.511217
$552$	−12.0000	−0.510754
$553$	4.00000	0.170097
$554$	22.0000	0.934690
$555$	0	0
$556$	2.00000	0.0848189
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	1.00000	0.0423334
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−18.0000	−0.759961
$562$	−24.0000	−1.01238
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	−26.0000	−1.09286
$567$	11.0000	0.461957
$568$	0	0
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	3.00000	0.125436
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	8.00000	0.332756
$579$	8.00000	0.332469
$580$	0	0
$581$	−15.0000	−0.622305
$582$	−8.00000	−0.331611
$583$	9.00000	0.372742
$584$	10.0000	0.413803
$585$	0	0
$586$	12.0000	0.495715
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	12.0000	0.494872
$589$	4.00000	0.164817
$590$	0	0
$591$	48.0000	1.97446
$592$	2.00000	0.0821995
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	−12.0000	−0.492366
$595$	0	0
$596$	12.0000	0.491539
$597$	−40.0000	−1.63709
$598$	6.00000	0.245358
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	−10.0000	−0.407570
$603$	−13.0000	−0.529401
$604$	11.0000	0.447584
$605$	0	0
$606$	18.0000	0.731200
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	4.00000	0.162221
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−3.00000	−0.121367
$612$	3.00000	0.121268
$613$	44.0000	1.77714	0.888572	−	0.458738i	$-0.151698\pi$
$613$	44.0000	1.77714	0.888572	+	0.458738i	$0.151698\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	3.00000	0.120873
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	16.0000	0.643614
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	−24.0000	−0.963087
$622$	−18.0000	−0.721734
$623$	−6.00000	−0.240385
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	19.0000	0.759393
$627$	24.0000	0.958468
$628$	5.00000	0.199522
$629$	6.00000	0.239236
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	4.00000	0.159111
$633$	44.0000	1.74884
$634$	12.0000	0.476581
$635$	0	0
$636$	−6.00000	−0.237915
$637$	−6.00000	−0.237729
$638$	9.00000	0.356313
$639$	0	0
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	−12.0000	−0.473602
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	12.0000	0.472134
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	11.0000	0.432121
$649$	−45.0000	−1.76640
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	−16.0000	−0.626608
$653$	21.0000	0.821794	0.410897	−	0.911682i	$-0.365216\pi$
$653$	21.0000	0.821794	0.410897	+	0.911682i	$0.365216\pi$
$654$	28.0000	1.09489
$655$	0	0
$656$	0	0
$657$	−10.0000	−0.390137
$658$	−3.00000	−0.116952
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	4.00000	0.155464
$663$	−6.00000	−0.233021
$664$	−15.0000	−0.582113
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	18.0000	0.696963
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	−39.0000	−1.50558
$672$	−2.00000	−0.0771517
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	1.00000	0.0384615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	36.0000	1.38257
$679$	4.00000	0.153506
$680$	0	0
$681$	−18.0000	−0.689761
$682$	3.00000	0.114876
$683$	39.0000	1.49229	0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000	1.49229	0.746147	+	0.665782i	$0.231902\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−13.0000	−0.496342
$687$	−40.0000	−1.52610
$688$	−10.0000	−0.381246
$689$	3.00000	0.114291
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−15.0000	−0.570214
$693$	−3.00000	−0.113961
$694$	6.00000	0.227757
$695$	0	0
$696$	−6.00000	−0.227429
$697$	0	0
$698$	−32.0000	−1.21122
$699$	−12.0000	−0.453882
$700$	0	0
$701$	3.00000	0.113308	0.0566542	−	0.998394i	$-0.481957\pi$
$701$	3.00000	0.113308	0.0566542	+	0.998394i	$0.481957\pi$
$702$	−4.00000	−0.150970
$703$	−8.00000	−0.301726
$704$	3.00000	0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	−9.00000	−0.338480
$708$	30.0000	1.12747
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	−6.00000	−0.224860
$713$	6.00000	0.224702
$714$	−6.00000	−0.224544
$715$	0	0
$716$	−24.0000	−0.896922
$717$	42.0000	1.56852
$718$	−9.00000	−0.335877
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	3.00000	0.111648
$723$	−16.0000	−0.595046
$724$	−25.0000	−0.929118
$725$	0	0
$726$	−4.00000	−0.148454
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	1.00000	0.0370625
$729$	13.0000	0.481481
$730$	0	0
$731$	−30.0000	−1.10959
$732$	26.0000	0.960988
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	6.00000	0.221163
$737$	−39.0000	−1.43658
$738$	0	0
$739$	−43.0000	−1.58178	−0.790890	−	0.611958i	$-0.790382\pi$
$739$	−43.0000	−1.58178	−0.790890	+	0.611958i	$0.790382\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	3.00000	0.110133
$743$	−3.00000	−0.110059	−0.0550297	−	0.998485i	$-0.517525\pi$
$743$	−3.00000	−0.110059	−0.0550297	+	0.998485i	$0.517525\pi$
$744$	−2.00000	−0.0733236
$745$	0	0
$746$	31.0000	1.13499
$747$	15.0000	0.548821
$748$	9.00000	0.329073
$749$	6.00000	0.219235
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	3.00000	0.109254
$755$	0	0
$756$	−4.00000	−0.145479
$757$	−13.0000	−0.472493	−0.236247	−	0.971693i	$-0.575917\pi$
$757$	−13.0000	−0.472493	−0.236247	+	0.971693i	$0.575917\pi$
$758$	−11.0000	−0.399538
$759$	36.0000	1.30672
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	−32.0000	−1.15924
$763$	−14.0000	−0.506834
$764$	0	0
$765$	0	0
$766$	12.0000	0.433578
$767$	−15.0000	−0.541619
$768$	−2.00000	−0.0721688
$769$	−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
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	10.0000	0.359443
$775$	0	0
$776$	4.00000	0.143592
$777$	4.00000	0.143499
$778$	30.0000	1.07555
$779$	0	0
$780$	0	0
$781$	0	0
$782$	18.0000	0.643679
$783$	−12.0000	−0.428845
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−12.0000	−0.428026
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	−24.0000	−0.854965
$789$	−36.0000	−1.28163
$790$	0	0
$791$	−18.0000	−0.640006
$792$	−3.00000	−0.106600
$793$	−13.0000	−0.461644
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	20.0000	0.708881
$797$	−33.0000	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$797$	−33.0000	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$798$	8.00000	0.283197
$799$	−9.00000	−0.318397
$800$	0	0
$801$	6.00000	0.212000
$802$	−24.0000	−0.847469
$803$	−30.0000	−1.05868
$804$	26.0000	0.916949
$805$	0	0
$806$	1.00000	0.0352235
$807$	−42.0000	−1.47847
$808$	−9.00000	−0.316619
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	3.00000	0.105279
$813$	−46.0000	−1.61329
$814$	−6.00000	−0.210300
$815$	0	0
$816$	−6.00000	−0.210042
$817$	40.0000	1.39942
$818$	−14.0000	−0.489499
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	36.0000	1.25564
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−15.0000	−0.521917
$827$	15.0000	0.521601	0.260801	−	0.965393i	$-0.416014\pi$
$827$	15.0000	0.521601	0.260801	+	0.965393i	$0.416014\pi$
$828$	−6.00000	−0.208514
$829$	17.0000	0.590434	0.295217	−	0.955430i	$-0.404608\pi$
$829$	17.0000	0.590434	0.295217	+	0.955430i	$0.404608\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	1.00000	0.0346688
$833$	−18.0000	−0.623663
$834$	4.00000	0.138509
$835$	0	0
$836$	−12.0000	−0.415029
$837$	−4.00000	−0.138260
$838$	−6.00000	−0.207267
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−20.0000	−0.689246
$843$	−48.0000	−1.65321
$844$	−22.0000	−0.757271
$845$	0	0
$846$	3.00000	0.103142
$847$	2.00000	0.0687208
$848$	3.00000	0.103020
$849$	−52.0000	−1.78464
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−13.0000	−0.444851
$855$	0	0
$856$	6.00000	0.205076
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	6.00000	0.204837
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	0	0
$861$	0	0
$862$	12.0000	0.408722
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	22.0000	0.747590
$867$	16.0000	0.543388
$868$	1.00000	0.0339422
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−13.0000	−0.440488
$872$	−14.0000	−0.474100
$873$	−4.00000	−0.135379
$874$	−24.0000	−0.811812
$875$	0	0
$876$	20.0000	0.675737
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	−8.00000	−0.269987
$879$	24.0000	0.809500
$880$	0	0
$881$	−39.0000	−1.31394	−0.656972	−	0.753915i	$-0.728163\pi$
$881$	−39.0000	−1.31394	−0.656972	+	0.753915i	$0.728163\pi$
$882$	6.00000	0.202031
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	6.00000	0.201574
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	4.00000	0.134231
$889$	16.0000	0.536623
$890$	0	0
$891$	−33.0000	−1.10554
$892$	−4.00000	−0.133930
$893$	12.0000	0.401565
$894$	24.0000	0.802680
$895$	0	0
$896$	1.00000	0.0334077
$897$	12.0000	0.400668
$898$	6.00000	0.200223
$899$	3.00000	0.100056
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	−20.0000	−0.665558
$904$	−18.0000	−0.598671
$905$	0	0
$906$	22.0000	0.730901
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	9.00000	0.298675
$909$	9.00000	0.298511
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	8.00000	0.264906
$913$	45.0000	1.48928
$914$	10.0000	0.330771
$915$	0	0
$916$	20.0000	0.660819
$917$	6.00000	0.198137
$918$	−12.0000	−0.396059
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	36.0000	1.18560
$923$	0	0
$924$	6.00000	0.197386
$925$	0	0
$926$	−29.0000	−0.952999
$927$	8.00000	0.262754
$928$	3.00000	0.0984798
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	6.00000	0.196537
$933$	−36.0000	−1.17859
$934$	−30.0000	−0.981630
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	−13.0000	−0.424465
$939$	38.0000	1.24008
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	10.0000	0.325818
$943$	0	0
$944$	−15.0000	−0.488208
$945$	0	0
$946$	30.0000	0.975384
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	8.00000	0.259828
$949$	−10.0000	−0.324614
$950$	0	0
$951$	24.0000	0.778253
$952$	3.00000	0.0972306
$953$	−3.00000	−0.0971795	−0.0485898	−	0.998819i	$-0.515473\pi$
$953$	−3.00000	−0.0971795	−0.0485898	+	0.998819i	$0.515473\pi$
$954$	−3.00000	−0.0971286
$955$	0	0
$956$	−21.0000	−0.679189
$957$	18.0000	0.581857
$958$	−27.0000	−0.872330
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−30.0000	−0.967742
$962$	−2.00000	−0.0644826
$963$	−6.00000	−0.193347
$964$	8.00000	0.257663
$965$	0	0
$966$	12.0000	0.386094
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	2.00000	0.0642824
$969$	24.0000	0.770991
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	10.0000	0.320750
$973$	−2.00000	−0.0641171
$974$	−11.0000	−0.352463
$975$	0	0
$976$	−13.0000	−0.416120
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−32.0000	−1.02325
$979$	18.0000	0.575282
$980$	0	0
$981$	14.0000	0.446986
$982$	30.0000	0.957338
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	0	0
$985$	0	0
$986$	9.00000	0.286618
$987$	−6.00000	−0.190982
$988$	−4.00000	−0.127257
$989$	60.0000	1.90789
$990$	0	0
$991$	50.0000	1.58830	0.794151	−	0.607720i	$-0.207916\pi$
$991$	50.0000	1.58830	0.794151	+	0.607720i	$0.207916\pi$
$992$	1.00000	0.0317500
$993$	8.00000	0.253872
$994$	0	0
$995$	0	0
$996$	−30.0000	−0.950586
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	7.00000	0.221581
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.2.a.b.1.1	✓	1
3.2	odd	2		5850.2.a.bm.1.1		1
4.3	odd	2		5200.2.a.bg.1.1		1
5.2	odd	4		650.2.b.h.599.1		2
5.3	odd	4		650.2.b.h.599.2		2
5.4	even	2		650.2.a.k.1.1	yes	1
13.12	even	2		8450.2.a.p.1.1		1
15.2	even	4		5850.2.e.j.5149.2		2
15.8	even	4		5850.2.e.j.5149.1		2
15.14	odd	2		5850.2.a.q.1.1		1
20.19	odd	2		5200.2.a.g.1.1		1
65.64	even	2		8450.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.a.b.1.1	✓	1	1.1	even	1	trivial
650.2.a.k.1.1	yes	1	5.4	even	2
650.2.b.h.599.1		2	5.2	odd	4
650.2.b.h.599.2		2	5.3	odd	4
5200.2.a.g.1.1		1	20.19	odd	2
5200.2.a.bg.1.1		1	4.3	odd	2
5850.2.a.q.1.1		1	15.14	odd	2
5850.2.a.bm.1.1		1	3.2	odd	2
5850.2.e.j.5149.1		2	15.8	even	4
5850.2.e.j.5149.2		2	15.2	even	4
8450.2.a.j.1.1		1	65.64	even	2
8450.2.a.p.1.1		1	13.12	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	650.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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