LMFDB - Embedded newform 4650.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4650.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:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4650.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -5.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -5.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +3.00000 q^{19} -5.00000 q^{21} -1.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{27} -5.00000 q^{28} +6.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} +3.00000 q^{38} +2.00000 q^{41} -5.00000 q^{42} +1.00000 q^{43} -1.00000 q^{44} +1.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} +18.0000 q^{49} +4.00000 q^{51} -3.00000 q^{53} +1.00000 q^{54} -5.00000 q^{56} +3.00000 q^{57} +6.00000 q^{58} -14.0000 q^{59} +14.0000 q^{61} -1.00000 q^{62} -5.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -10.0000 q^{67} +4.00000 q^{68} +1.00000 q^{69} +9.00000 q^{71} +1.00000 q^{72} +7.00000 q^{73} +4.00000 q^{74} +3.00000 q^{76} +5.00000 q^{77} +15.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +10.0000 q^{83} -5.00000 q^{84} +1.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} -1.00000 q^{89} +1.00000 q^{92} -1.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} -10.0000 q^{97} +18.0000 q^{98} -1.00000 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
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−5.00000	−1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.00000	0.235702
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−5.00000	−1.09109
$22$	−1.00000	−0.213201
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	−5.00000	−0.944911
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	4.00000	0.685994
$35$	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	$0.393356\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−5.00000	−0.771517
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	1.00000	0.147442
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	1.00000	0.144338
$49$	18.0000	2.57143
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−5.00000	−0.668153
$57$	3.00000	0.397360
$58$	6.00000	0.787839
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−1.00000	−0.127000
$63$	−5.00000	−0.629941
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	4.00000	0.485071
$69$	1.00000	0.120386
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	1.00000	0.117851
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	3.00000	0.344124
$77$	5.00000	0.569803
$78$	0	0
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	−5.00000	−0.545545
$85$	0	0
$86$	1.00000	0.107833
$87$	6.00000	0.643268
$88$	−1.00000	−0.106600
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	0	0
$92$	1.00000	0.104257
$93$	−1.00000	−0.103695
$94$	−4.00000	−0.412568
$95$	0	0
$96$	1.00000	0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	18.0000	1.81827
$99$	−1.00000	−0.100504
$100$	0	0
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	4.00000	0.396059
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	−3.00000	−0.291386
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	1.00000	0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−5.00000	−0.472456
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	−14.0000	−1.28880
$119$	−20.0000	−1.83340
$120$	0	0
$121$	−10.0000	−0.909091
$122$	14.0000	1.26750
$123$	2.00000	0.180334
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−5.00000	−0.445435
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	1.00000	0.0880451
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	−1.00000	−0.0870388
$133$	−15.0000	−1.30066
$134$	−10.0000	−0.863868
$135$	0	0
$136$	4.00000	0.342997
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	1.00000	0.0851257
$139$	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
$141$	−4.00000	−0.336861
$142$	9.00000	0.755263
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	7.00000	0.579324
$147$	18.0000	1.48461
$148$	4.00000	0.328798
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	3.00000	0.243332
$153$	4.00000	0.323381
$154$	5.00000	0.402911
$155$	0	0
$156$	0	0
$157$	−1.00000	−0.0798087	−0.0399043	−	0.999204i	$-0.512705\pi$
$157$	−1.00000	−0.0798087	−0.0399043	+	0.999204i	$0.512705\pi$
$158$	15.0000	1.19334
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−5.00000	−0.394055
$162$	1.00000	0.0785674
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	10.0000	0.776151
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	−5.00000	−0.385758
$169$	−13.0000	−1.00000
$170$	0	0
$171$	3.00000	0.229416
$172$	1.00000	0.0762493
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−14.0000	−1.05230
$178$	−1.00000	−0.0749532
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	1.00000	0.0737210
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	−4.00000	−0.292509
$188$	−4.00000	−0.291730
$189$	−5.00000	−0.363696
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000	0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	18.0000	1.28571
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−1.00000	−0.0710669
$199$	−9.00000	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	7.00000	0.492518
$203$	−30.0000	−2.10559
$204$	4.00000	0.280056
$205$	0	0
$206$	8.00000	0.557386
$207$	1.00000	0.0695048
$208$	0	0
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−21.0000	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$211$	−21.0000	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$212$	−3.00000	−0.206041
$213$	9.00000	0.616670
$214$	−9.00000	−0.615227
$215$	0	0
$216$	1.00000	0.0680414
$217$	5.00000	0.339422
$218$	10.0000	0.677285
$219$	7.00000	0.473016
$220$	0	0
$221$	0	0
$222$	4.00000	0.268462
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−5.00000	−0.334077
$225$	0	0
$226$	9.00000	0.598671
$227$	−29.0000	−1.92480	−0.962399	−	0.271640i	$-0.912434\pi$
$227$	−29.0000	−1.92480	−0.962399	+	0.271640i	$0.912434\pi$
$228$	3.00000	0.198680
$229$	23.0000	1.51988	0.759941	−	0.649992i	$-0.225228\pi$
$229$	23.0000	1.51988	0.759941	+	0.649992i	$0.225228\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	6.00000	0.393919
$233$	−13.0000	−0.851658	−0.425829	−	0.904804i	$-0.640018\pi$
$233$	−13.0000	−0.851658	−0.425829	+	0.904804i	$0.640018\pi$
$234$	0	0
$235$	0	0
$236$	−14.0000	−0.911322
$237$	15.0000	0.974355
$238$	−20.0000	−1.29641
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	−10.0000	−0.642824
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	0	0
$246$	2.00000	0.127515
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	10.0000	0.633724
$250$	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
$253$	−1.00000	−0.0628695
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	1.00000	0.0622573
$259$	−20.0000	−1.24274
$260$	0	0
$261$	6.00000	0.371391
$262$	18.0000	1.11204
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	−15.0000	−0.919709
$267$	−1.00000	−0.0611990
$268$	−10.0000	−0.610847
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	1.00000	0.0601929
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	4.00000	0.239904
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	−4.00000	−0.238197
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	9.00000	0.534052
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	7.00000	0.409644
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	18.0000	1.04978
$295$	0	0
$296$	4.00000	0.232495
$297$	−1.00000	−0.0580259
$298$	1.00000	0.0579284
$299$	0	0
$300$	0	0
$301$	−5.00000	−0.288195
$302$	−8.00000	−0.460348
$303$	7.00000	0.402139
$304$	3.00000	0.172062
$305$	0	0
$306$	4.00000	0.228665
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	5.00000	0.284901
$309$	8.00000	0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−1.00000	−0.0564333
$315$	0	0
$316$	15.0000	0.843816
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	−3.00000	−0.168232
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−9.00000	−0.502331
$322$	−5.00000	−0.278639
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	0	0
$326$	8.00000	0.443079
$327$	10.0000	0.553001
$328$	2.00000	0.110432
$329$	20.0000	1.10264
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	10.0000	0.548821
$333$	4.00000	0.219199
$334$	9.00000	0.492458
$335$	0	0
$336$	−5.00000	−0.272772
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−13.0000	−0.707107
$339$	9.00000	0.488813
$340$	0	0
$341$	1.00000	0.0541530
$342$	3.00000	0.162221
$343$	−55.0000	−2.96972
$344$	1.00000	0.0539164
$345$	0	0
$346$	2.00000	0.107521
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	6.00000	0.321634
$349$	−32.0000	−1.71292	−0.856460	−	0.516213i	$-0.827341\pi$
$349$	−32.0000	−1.71292	−0.856460	+	0.516213i	$0.827341\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	−14.0000	−0.744092
$355$	0	0
$356$	−1.00000	−0.0529999
$357$	−20.0000	−1.05851
$358$	−8.00000	−0.422813
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	11.0000	0.578147
$363$	−10.0000	−0.524864
$364$	0	0
$365$	0	0
$366$	14.0000	0.731792
$367$	−26.0000	−1.35719	−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000	−1.35719	−0.678594	+	0.734513i	$0.737411\pi$
$368$	1.00000	0.0521286
$369$	2.00000	0.104116
$370$	0	0
$371$	15.0000	0.778761
$372$	−1.00000	−0.0518476
$373$	3.00000	0.155334	0.0776671	−	0.996979i	$-0.475253\pi$
$373$	3.00000	0.155334	0.0776671	+	0.996979i	$0.475253\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−4.00000	−0.206284
$377$	0	0
$378$	−5.00000	−0.257172
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	24.0000	1.22795
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	1.00000	0.0508329
$388$	−10.0000	−0.507673
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	18.0000	0.909137
$393$	18.0000	0.907980
$394$	2.00000	0.100759
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	−17.0000	−0.853206	−0.426603	−	0.904439i	$-0.640290\pi$
$397$	−17.0000	−0.853206	−0.426603	+	0.904439i	$0.640290\pi$
$398$	−9.00000	−0.451129
$399$	−15.0000	−0.750939
$400$	0	0
$401$	−29.0000	−1.44819	−0.724095	−	0.689700i	$-0.757743\pi$
$401$	−29.0000	−1.44819	−0.724095	+	0.689700i	$0.757743\pi$
$402$	−10.0000	−0.498755
$403$	0	0
$404$	7.00000	0.348263
$405$	0	0
$406$	−30.0000	−1.48888
$407$	−4.00000	−0.198273
$408$	4.00000	0.198030
$409$	−16.0000	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	8.00000	0.394132
$413$	70.0000	3.44447
$414$	1.00000	0.0491473
$415$	0	0
$416$	0	0
$417$	4.00000	0.195881
$418$	−3.00000	−0.146735
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−21.0000	−1.02226
$423$	−4.00000	−0.194487
$424$	−3.00000	−0.145693
$425$	0	0
$426$	9.00000	0.436051
$427$	−70.0000	−3.38754
$428$	−9.00000	−0.435031
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	1.00000	0.0481125
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	5.00000	0.240008
$435$	0	0
$436$	10.0000	0.478913
$437$	3.00000	0.143509
$438$	7.00000	0.334473
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	0	0
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	−8.00000	−0.378811
$447$	1.00000	0.0472984
$448$	−5.00000	−0.236228
$449$	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
$451$	−2.00000	−0.0941763
$452$	9.00000	0.423324
$453$	−8.00000	−0.375873
$454$	−29.0000	−1.36104
$455$	0	0
$456$	3.00000	0.140488
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	23.0000	1.07472
$459$	4.00000	0.186704
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	5.00000	0.232621
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−13.0000	−0.602213
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	50.0000	2.30879
$470$	0	0
$471$	−1.00000	−0.0460776
$472$	−14.0000	−0.644402
$473$	−1.00000	−0.0459800
$474$	15.0000	0.688973
$475$	0	0
$476$	−20.0000	−0.916698
$477$	−3.00000	−0.137361
$478$	−22.0000	−1.00626
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	0	0
$481$	0	0
$482$	4.00000	0.182195
$483$	−5.00000	−0.227508
$484$	−10.0000	−0.454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	−36.0000	−1.63132	−0.815658	−	0.578535i	$-0.803625\pi$
$487$	−36.0000	−1.63132	−0.815658	+	0.578535i	$0.803625\pi$
$488$	14.0000	0.633750
$489$	8.00000	0.361773
$490$	0	0
$491$	27.0000	1.21849	0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000	1.21849	0.609246	+	0.792981i	$0.291472\pi$
$492$	2.00000	0.0901670
$493$	24.0000	1.08091
$494$	0	0
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−45.0000	−2.01853
$498$	10.0000	0.448111
$499$	−42.0000	−1.88018	−0.940089	−	0.340929i	$-0.889258\pi$
$499$	−42.0000	−1.88018	−0.940089	+	0.340929i	$0.889258\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	−12.0000	−0.535586
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	−5.00000	−0.222718
$505$	0	0
$506$	−1.00000	−0.0444554
$507$	−13.0000	−0.577350
$508$	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
$511$	−35.0000	−1.54831
$512$	1.00000	0.0441942
$513$	3.00000	0.132453
$514$	−3.00000	−0.132324
$515$	0	0
$516$	1.00000	0.0440225
$517$	4.00000	0.175920
$518$	−20.0000	−0.878750
$519$	2.00000	0.0877903
$520$	0	0
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	6.00000	0.262613
$523$	41.0000	1.79280	0.896402	−	0.443241i	$-0.146171\pi$
$523$	41.0000	1.79280	0.896402	+	0.443241i	$0.146171\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	−8.00000	−0.348817
$527$	−4.00000	−0.174243
$528$	−1.00000	−0.0435194
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−14.0000	−0.607548
$532$	−15.0000	−0.650332
$533$	0	0
$534$	−1.00000	−0.0432742
$535$	0	0
$536$	−10.0000	−0.431934
$537$	−8.00000	−0.345225
$538$	30.0000	1.29339
$539$	−18.0000	−0.775315
$540$	0	0
$541$	4.00000	0.171973	0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000	0.171973	0.0859867	+	0.996296i	$0.472596\pi$
$542$	13.0000	0.558398
$543$	11.0000	0.472055
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	6.00000	0.256307
$549$	14.0000	0.597505
$550$	0	0
$551$	18.0000	0.766826
$552$	1.00000	0.0425628
$553$	−75.0000	−3.18932
$554$	−26.0000	−1.10463
$555$	0	0
$556$	4.00000	0.169638
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	−1.00000	−0.0423334
$559$	0	0
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	−22.0000	−0.924729
$567$	−5.00000	−0.209980
$568$	9.00000	0.377632
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	0	0
$571$	34.0000	1.42286	0.711428	−	0.702759i	$-0.248049\pi$
$571$	34.0000	1.42286	0.711428	+	0.702759i	$0.248049\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	−10.0000	−0.417392
$575$	0	0
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−1.00000	−0.0415945
$579$	10.0000	0.415586
$580$	0	0
$581$	−50.0000	−2.07435
$582$	−10.0000	−0.414513
$583$	3.00000	0.124247
$584$	7.00000	0.289662
$585$	0	0
$586$	−30.0000	−1.23929
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	18.0000	0.742307
$589$	−3.00000	−0.123613
$590$	0	0
$591$	2.00000	0.0822690
$592$	4.00000	0.164399
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	1.00000	0.0409616
$597$	−9.00000	−0.368345
$598$	0	0
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	−5.00000	−0.203785
$603$	−10.0000	−0.407231
$604$	−8.00000	−0.325515
$605$	0	0
$606$	7.00000	0.284356
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	3.00000	0.121666
$609$	−30.0000	−1.21566
$610$	0	0
$611$	0	0
$612$	4.00000	0.161690
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	5.00000	0.201456
$617$	13.0000	0.523360	0.261680	−	0.965155i	$-0.415723\pi$
$617$	13.0000	0.523360	0.261680	+	0.965155i	$0.415723\pi$
$618$	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
$621$	1.00000	0.0401286
$622$	0	0
$623$	5.00000	0.200321
$624$	0	0
$625$	0	0
$626$	−26.0000	−1.03917
$627$	−3.00000	−0.119808
$628$	−1.00000	−0.0399043
$629$	16.0000	0.637962
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	15.0000	0.596668
$633$	−21.0000	−0.834675
$634$	22.0000	0.873732
$635$	0	0
$636$	−3.00000	−0.118958
$637$	0	0
$638$	−6.00000	−0.237542
$639$	9.00000	0.356034
$640$	0	0
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	−9.00000	−0.355202
$643$	3.00000	0.118308	0.0591542	−	0.998249i	$-0.481160\pi$
$643$	3.00000	0.118308	0.0591542	+	0.998249i	$0.481160\pi$
$644$	−5.00000	−0.197028
$645$	0	0
$646$	12.0000	0.472134
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	1.00000	0.0392837
$649$	14.0000	0.549548
$650$	0	0
$651$	5.00000	0.195965
$652$	8.00000	0.313304
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	2.00000	0.0780869
$657$	7.00000	0.273096
$658$	20.0000	0.779681
$659$	10.0000	0.389545	0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000	0.389545	0.194772	+	0.980848i	$0.437603\pi$
$660$	0	0
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	10.0000	0.388075
$665$	0	0
$666$	4.00000	0.154997
$667$	6.00000	0.232321
$668$	9.00000	0.348220
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−14.0000	−0.540464
$672$	−5.00000	−0.192879
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−11.0000	−0.422764	−0.211382	−	0.977403i	$-0.567796\pi$
$677$	−11.0000	−0.422764	−0.211382	+	0.977403i	$0.567796\pi$
$678$	9.00000	0.345643
$679$	50.0000	1.91882
$680$	0	0
$681$	−29.0000	−1.11128
$682$	1.00000	0.0382920
$683$	31.0000	1.18618	0.593091	−	0.805135i	$-0.297907\pi$
$683$	31.0000	1.18618	0.593091	+	0.805135i	$0.297907\pi$
$684$	3.00000	0.114708
$685$	0	0
$686$	−55.0000	−2.09991
$687$	23.0000	0.877505
$688$	1.00000	0.0381246
$689$	0	0
$690$	0	0
$691$	−5.00000	−0.190209	−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000	−0.190209	−0.0951045	+	0.995467i	$0.530319\pi$
$692$	2.00000	0.0760286
$693$	5.00000	0.189934
$694$	−30.0000	−1.13878
$695$	0	0
$696$	6.00000	0.227429
$697$	8.00000	0.303022
$698$	−32.0000	−1.21122
$699$	−13.0000	−0.491705
$700$	0	0
$701$	35.0000	1.32193	0.660966	−	0.750416i	$-0.270147\pi$
$701$	35.0000	1.32193	0.660966	+	0.750416i	$0.270147\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	8.00000	0.301084
$707$	−35.0000	−1.31631
$708$	−14.0000	−0.526152
$709$	21.0000	0.788672	0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000	0.788672	0.394336	+	0.918966i	$0.370975\pi$
$710$	0	0
$711$	15.0000	0.562544
$712$	−1.00000	−0.0374766
$713$	−1.00000	−0.0374503
$714$	−20.0000	−0.748481
$715$	0	0
$716$	−8.00000	−0.298974
$717$	−22.0000	−0.821605
$718$	−15.0000	−0.559795
$719$	50.0000	1.86469	0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000	1.86469	0.932343	+	0.361576i	$0.117761\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	−10.0000	−0.372161
$723$	4.00000	0.148762
$724$	11.0000	0.408812
$725$	0	0
$726$	−10.0000	−0.371135
$727$	27.0000	1.00137	0.500687	−	0.865628i	$-0.333081\pi$
$727$	27.0000	1.00137	0.500687	+	0.865628i	$0.333081\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	14.0000	0.517455
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−26.0000	−0.959678
$735$	0	0
$736$	1.00000	0.0368605
$737$	10.0000	0.368355
$738$	2.00000	0.0736210
$739$	46.0000	1.69214	0.846069	−	0.533074i	$-0.178963\pi$
$739$	46.0000	1.69214	0.846069	+	0.533074i	$0.178963\pi$
$740$	0	0
$741$	0	0
$742$	15.0000	0.550667
$743$	−21.0000	−0.770415	−0.385208	−	0.922830i	$-0.625870\pi$
$743$	−21.0000	−0.770415	−0.385208	+	0.922830i	$0.625870\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	3.00000	0.109838
$747$	10.0000	0.365881
$748$	−4.00000	−0.146254
$749$	45.0000	1.64426
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	−4.00000	−0.145865
$753$	−12.0000	−0.437304
$754$	0	0
$755$	0	0
$756$	−5.00000	−0.181848
$757$	4.00000	0.145382	0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000	0.145382	0.0726912	+	0.997354i	$0.476841\pi$
$758$	−19.0000	−0.690111
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	16.0000	0.579619
$763$	−50.0000	−1.81012
$764$	24.0000	0.868290
$765$	0	0
$766$	20.0000	0.722629
$767$	0	0
$768$	1.00000	0.0360844
$769$	−27.0000	−0.973645	−0.486822	−	0.873501i	$-0.661844\pi$
$769$	−27.0000	−0.973645	−0.486822	+	0.873501i	$0.661844\pi$
$770$	0	0
$771$	−3.00000	−0.108042
$772$	10.0000	0.359908
$773$	51.0000	1.83434	0.917171	−	0.398493i	$-0.130467\pi$
$773$	51.0000	1.83434	0.917171	+	0.398493i	$0.130467\pi$
$774$	1.00000	0.0359443
$775$	0	0
$776$	−10.0000	−0.358979
$777$	−20.0000	−0.717496
$778$	−16.0000	−0.573628
$779$	6.00000	0.214972
$780$	0	0
$781$	−9.00000	−0.322045
$782$	4.00000	0.143040
$783$	6.00000	0.214423
$784$	18.0000	0.642857
$785$	0	0
$786$	18.0000	0.642039
$787$	25.0000	0.891154	0.445577	−	0.895244i	$-0.352999\pi$
$787$	25.0000	0.891154	0.445577	+	0.895244i	$0.352999\pi$
$788$	2.00000	0.0712470
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−45.0000	−1.60002
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	−17.0000	−0.603307
$795$	0	0
$796$	−9.00000	−0.318997
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	−15.0000	−0.530994
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−1.00000	−0.0353333
$802$	−29.0000	−1.02403
$803$	−7.00000	−0.247025
$804$	−10.0000	−0.352673
$805$	0	0
$806$	0	0
$807$	30.0000	1.05605
$808$	7.00000	0.246259
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	−17.0000	−0.596951	−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	−30.0000	−1.05279
$813$	13.0000	0.455930
$814$	−4.00000	−0.140200
$815$	0	0
$816$	4.00000	0.140028
$817$	3.00000	0.104957
$818$	−16.0000	−0.559427
$819$	0	0
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	6.00000	0.209274
$823$	−18.0000	−0.627441	−0.313720	−	0.949515i	$-0.601575\pi$
$823$	−18.0000	−0.627441	−0.313720	+	0.949515i	$0.601575\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	70.0000	2.43561
$827$	34.0000	1.18230	0.591148	−	0.806563i	$-0.298675\pi$
$827$	34.0000	1.18230	0.591148	+	0.806563i	$0.298675\pi$
$828$	1.00000	0.0347524
$829$	39.0000	1.35453	0.677263	−	0.735741i	$-0.263166\pi$
$829$	39.0000	1.35453	0.677263	+	0.735741i	$0.263166\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	72.0000	2.49465
$834$	4.00000	0.138509
$835$	0	0
$836$	−3.00000	−0.103757
$837$	−1.00000	−0.0345651
$838$	26.0000	0.898155
$839$	−35.0000	−1.20833	−0.604167	−	0.796858i	$-0.706494\pi$
$839$	−35.0000	−1.20833	−0.604167	+	0.796858i	$0.706494\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	22.0000	0.758170
$843$	0	0
$844$	−21.0000	−0.722850
$845$	0	0
$846$	−4.00000	−0.137523
$847$	50.0000	1.71802
$848$	−3.00000	−0.103020
$849$	−22.0000	−0.755038
$850$	0	0
$851$	4.00000	0.137118
$852$	9.00000	0.308335
$853$	−49.0000	−1.67773	−0.838864	−	0.544341i	$-0.816780\pi$
$853$	−49.0000	−1.67773	−0.838864	+	0.544341i	$0.816780\pi$
$854$	−70.0000	−2.39535
$855$	0	0
$856$	−9.00000	−0.307614
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	−10.0000	−0.340799
$862$	8.00000	0.272481
$863$	−57.0000	−1.94030	−0.970151	−	0.242500i	$-0.922032\pi$
$863$	−57.0000	−1.94030	−0.970151	+	0.242500i	$0.922032\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−11.0000	−0.373795
$867$	−1.00000	−0.0339618
$868$	5.00000	0.169711
$869$	−15.0000	−0.508840
$870$	0	0
$871$	0	0
$872$	10.0000	0.338643
$873$	−10.0000	−0.338449
$874$	3.00000	0.101477
$875$	0	0
$876$	7.00000	0.236508
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	36.0000	1.21494
$879$	−30.0000	−1.01187
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	18.0000	0.606092
$883$	39.0000	1.31245	0.656227	−	0.754563i	$-0.272151\pi$
$883$	39.0000	1.31245	0.656227	+	0.754563i	$0.272151\pi$
$884$	0	0
$885$	0	0
$886$	−15.0000	−0.503935
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	4.00000	0.134231
$889$	−80.0000	−2.68311
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−8.00000	−0.267860
$893$	−12.0000	−0.401565
$894$	1.00000	0.0334450
$895$	0	0
$896$	−5.00000	−0.167038
$897$	0	0
$898$	6.00000	0.200223
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−12.0000	−0.399778
$902$	−2.00000	−0.0665927
$903$	−5.00000	−0.166390
$904$	9.00000	0.299336
$905$	0	0
$906$	−8.00000	−0.265782
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	−29.0000	−0.962399
$909$	7.00000	0.232175
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	3.00000	0.0993399
$913$	−10.0000	−0.330952
$914$	10.0000	0.330771
$915$	0	0
$916$	23.0000	0.759941
$917$	−90.0000	−2.97206
$918$	4.00000	0.132020
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	24.0000	0.790398
$923$	0	0
$924$	5.00000	0.164488
$925$	0	0
$926$	6.00000	0.197172
$927$	8.00000	0.262754
$928$	6.00000	0.196960
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	0	0
$931$	54.0000	1.76978
$932$	−13.0000	−0.425829
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	50.0000	1.63256
$939$	−26.0000	−0.848478
$940$	0	0
$941$	−52.0000	−1.69515	−0.847576	−	0.530674i	$-0.821939\pi$
$941$	−52.0000	−1.69515	−0.847576	+	0.530674i	$0.821939\pi$
$942$	−1.00000	−0.0325818
$943$	2.00000	0.0651290
$944$	−14.0000	−0.455661
$945$	0	0
$946$	−1.00000	−0.0325128
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	15.0000	0.487177
$949$	0	0
$950$	0	0
$951$	22.0000	0.713399
$952$	−20.0000	−0.648204
$953$	−4.00000	−0.129573	−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000	−0.129573	−0.0647864	+	0.997899i	$0.520637\pi$
$954$	−3.00000	−0.0971286
$955$	0	0
$956$	−22.0000	−0.711531
$957$	−6.00000	−0.193952
$958$	29.0000	0.936947
$959$	−30.0000	−0.968751
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−9.00000	−0.290021
$964$	4.00000	0.128831
$965$	0	0
$966$	−5.00000	−0.160872
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	−10.0000	−0.321412
$969$	12.0000	0.385496
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	1.00000	0.0320750
$973$	−20.0000	−0.641171
$974$	−36.0000	−1.15351
$975$	0	0
$976$	14.0000	0.448129
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	8.00000	0.255812
$979$	1.00000	0.0319601
$980$	0	0
$981$	10.0000	0.319275
$982$	27.0000	0.861605
$983$	−28.0000	−0.893061	−0.446531	−	0.894768i	$-0.647341\pi$
$983$	−28.0000	−0.893061	−0.446531	+	0.894768i	$0.647341\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	24.0000	0.764316
$987$	20.0000	0.636607
$988$	0	0
$989$	1.00000	0.0317982
$990$	0	0
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	−1.00000	−0.0317500
$993$	28.0000	0.888553
$994$	−45.0000	−1.42731
$995$	0	0
$996$	10.0000	0.316862
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	−42.0000	−1.32949
$999$	4.00000	0.126554

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.bi.1.1		1
5.2	odd	4		930.2.d.c.559.2	yes	2
5.3	odd	4		930.2.d.c.559.1	✓	2
5.4	even	2		4650.2.a.m.1.1		1
15.2	even	4		2790.2.d.e.559.1		2
15.8	even	4		2790.2.d.e.559.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.d.c.559.1	✓	2	5.3	odd	4
930.2.d.c.559.2	yes	2	5.2	odd	4
2790.2.d.e.559.1		2	15.2	even	4
2790.2.d.e.559.2		2	15.8	even	4
4650.2.a.m.1.1		1	5.4	even	2
4650.2.a.bi.1.1		1	1.1	even	1	trivial

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4650.2.a.bi.1.1

Level	$4650$
Weight	$2$
Character	4650.1
Self dual	yes
Analytic conductor	$37.130$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$