LMFDB - Embedded newform 1002.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1002,2,Mod(1,1002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1002, 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("1002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1002 = 2 \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1002.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:	$8.00101028253$
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	$\chi$	$=$	1002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +4.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{28} -2.00000 q^{29} -2.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +4.00000 q^{34} -8.00000 q^{35} +1.00000 q^{36} -12.0000 q^{37} +4.00000 q^{38} -2.00000 q^{40} +12.0000 q^{41} +4.00000 q^{42} -8.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +4.00000 q^{46} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +14.0000 q^{53} -1.00000 q^{54} -8.00000 q^{55} +4.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +2.00000 q^{59} +2.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} -4.00000 q^{68} -4.00000 q^{69} +8.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +12.0000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +16.0000 q^{77} -14.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} +6.00000 q^{83} -4.00000 q^{84} -8.00000 q^{85} +8.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -4.00000 q^{92} +4.00000 q^{93} -8.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−1.00000	−0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	4.00000	1.06904
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−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	$0.651732\pi$
$20$	2.00000	0.447214
$21$	−4.00000	−0.872872
$22$	4.00000	0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−2.00000	−0.365148
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	4.00000	0.685994
$35$	−8.00000	−1.35225
$36$	1.00000	0.166667
$37$	−12.0000	−1.97279	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−12.0000	−1.97279	−0.986394	+	0.164399i	$0.947432\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	−2.00000	−0.316228
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	4.00000	0.617213
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−4.00000	−0.603023
$45$	2.00000	0.298142
$46$	4.00000	0.589768
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	−4.00000	−0.560112
$52$	0	0
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	−1.00000	−0.136083
$55$	−8.00000	−1.07872
$56$	4.00000	0.534522
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	2.00000	0.258199
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000	−0.508001
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−4.00000	−0.485071
$69$	−4.00000	−0.481543
$70$	8.00000	0.956183
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	12.0000	1.39497
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	16.0000	1.82337
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−4.00000	−0.436436
$85$	−8.00000	−0.867722
$86$	8.00000	0.862662
$87$	−2.00000	−0.214423
$88$	4.00000	0.426401
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	−4.00000	−0.417029
$93$	4.00000	0.414781
$94$	0	0
$95$	−8.00000	−0.820783
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−9.00000	−0.909137
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	4.00000	0.396059
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	−14.0000	−1.35980
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	1.00000	0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	8.00000	0.762770
$111$	−12.0000	−1.13899
$112$	−4.00000	−0.377964
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	4.00000	0.374634
$115$	−8.00000	−0.746004
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−2.00000	−0.184115
$119$	16.0000	1.46672
$120$	−2.00000	−0.182574
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	12.0000	1.08200
$124$	4.00000	0.359211
$125$	−12.0000	−1.07331
$126$	4.00000	0.356348
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−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	$0.211977\pi$
$132$	−4.00000	−0.348155
$133$	16.0000	1.38738
$134$	4.00000	0.345547
$135$	2.00000	0.172133
$136$	4.00000	0.342997
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	4.00000	0.340503
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−8.00000	−0.676123
$141$	0	0
$142$	−8.00000	−0.671345
$143$	0	0
$144$	1.00000	0.0833333
$145$	−4.00000	−0.332182
$146$	−6.00000	−0.496564
$147$	9.00000	0.742307
$148$	−12.0000	−0.986394
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	1.00000	0.0816497
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	4.00000	0.324443
$153$	−4.00000	−0.323381
$154$	−16.0000	−1.28932
$155$	8.00000	0.642575
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	14.0000	1.11378
$159$	14.0000	1.11027
$160$	−2.00000	−0.158114
$161$	16.0000	1.26098
$162$	−1.00000	−0.0785674
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	12.0000	0.937043
$165$	−8.00000	−0.622799
$166$	−6.00000	−0.465690
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	4.00000	0.308607
$169$	−13.0000	−1.00000
$170$	8.00000	0.613572
$171$	−4.00000	−0.305888
$172$	−8.00000	−0.609994
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	2.00000	0.151620
$175$	4.00000	0.302372
$176$	−4.00000	−0.301511
$177$	2.00000	0.150329
$178$	6.00000	0.449719
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	2.00000	0.149071
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	4.00000	0.294884
$185$	−24.0000	−1.76452
$186$	−4.00000	−0.293294
$187$	16.0000	1.17004
$188$	0	0
$189$	−4.00000	−0.290957
$190$	8.00000	0.580381
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	4.00000	0.284268
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	−4.00000	−0.282138
$202$	2.00000	0.140720
$203$	8.00000	0.561490
$204$	−4.00000	−0.280056
$205$	24.0000	1.67623
$206$	6.00000	0.418040
$207$	−4.00000	−0.278019
$208$	0	0
$209$	16.0000	1.10674
$210$	8.00000	0.552052
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	14.0000	0.961524
$213$	8.00000	0.548151
$214$	16.0000	1.09374
$215$	−16.0000	−1.09119
$216$	−1.00000	−0.0680414
$217$	−16.0000	−1.08615
$218$	16.0000	1.08366
$219$	6.00000	0.405442
$220$	−8.00000	−0.539360
$221$	0	0
$222$	12.0000	0.805387
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	4.00000	0.267261
$225$	−1.00000	−0.0666667
$226$	4.00000	0.266076
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	−4.00000	−0.264906
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	8.00000	0.527504
$231$	16.0000	1.05272
$232$	2.00000	0.131306
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	2.00000	0.130189
$237$	−14.0000	−0.909398
$238$	−16.0000	−1.03713
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	2.00000	0.129099
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	18.0000	1.14998
$246$	−12.0000	−0.765092
$247$	0	0
$248$	−4.00000	−0.254000
$249$	6.00000	0.380235
$250$	12.0000	0.758947
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	−4.00000	−0.251976
$253$	16.0000	1.00591
$254$	8.00000	0.501965
$255$	−8.00000	−0.500979
$256$	1.00000	0.0625000
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	8.00000	0.498058
$259$	48.0000	2.98257
$260$	0	0
$261$	−2.00000	−0.123797
$262$	−18.0000	−1.11204
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	4.00000	0.246183
$265$	28.0000	1.72003
$266$	−16.0000	−0.981023
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−2.00000	−0.121716
$271$	6.00000	0.364474	0.182237	−	0.983255i	$-0.441666\pi$
$271$	6.00000	0.364474	0.182237	+	0.983255i	$0.441666\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	18.0000	1.08742
$275$	4.00000	0.241209
$276$	−4.00000	−0.240772
$277$	24.0000	1.44202	0.721010	−	0.692925i	$-0.243678\pi$
$277$	24.0000	1.44202	0.721010	+	0.692925i	$0.243678\pi$
$278$	16.0000	0.959616
$279$	4.00000	0.239474
$280$	8.00000	0.478091
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	8.00000	0.474713
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−48.0000	−2.83335
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	4.00000	0.234888
$291$	14.0000	0.820695
$292$	6.00000	0.351123
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−9.00000	−0.524891
$295$	4.00000	0.232889
$296$	12.0000	0.697486
$297$	−4.00000	−0.232104
$298$	−18.0000	−1.04271
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	32.0000	1.84445
$302$	−2.00000	−0.115087
$303$	−2.00000	−0.114897
$304$	−4.00000	−0.229416
$305$	−4.00000	−0.229039
$306$	4.00000	0.228665
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	16.0000	0.911685
$309$	−6.00000	−0.341328
$310$	−8.00000	−0.454369
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−6.00000	−0.338600
$315$	−8.00000	−0.450749
$316$	−14.0000	−0.787562
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−14.0000	−0.785081
$319$	8.00000	0.447914
$320$	2.00000	0.111803
$321$	−16.0000	−0.893033
$322$	−16.0000	−0.891645
$323$	16.0000	0.890264
$324$	1.00000	0.0555556
$325$	0	0
$326$	−24.0000	−1.32924
$327$	−16.0000	−0.884802
$328$	−12.0000	−0.662589
$329$	0	0
$330$	8.00000	0.440386
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	6.00000	0.329293
$333$	−12.0000	−0.657596
$334$	−1.00000	−0.0547176
$335$	−8.00000	−0.437087
$336$	−4.00000	−0.218218
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	13.0000	0.707107
$339$	−4.00000	−0.217250
$340$	−8.00000	−0.433861
$341$	−16.0000	−0.866449
$342$	4.00000	0.216295
$343$	−8.00000	−0.431959
$344$	8.00000	0.431331
$345$	−8.00000	−0.430706
$346$	−2.00000	−0.107521
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	−2.00000	−0.107211
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	4.00000	0.213201
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	−2.00000	−0.106299
$355$	16.0000	0.849192
$356$	−6.00000	−0.317999
$357$	16.0000	0.846810
$358$	4.00000	0.211407
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	−14.0000	−0.735824
$363$	5.00000	0.262432
$364$	0	0
$365$	12.0000	0.628109
$366$	2.00000	0.104542
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	−4.00000	−0.208514
$369$	12.0000	0.624695
$370$	24.0000	1.24770
$371$	−56.0000	−2.90738
$372$	4.00000	0.207390
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	−16.0000	−0.827340
$375$	−12.0000	−0.619677
$376$	0	0
$377$	0	0
$378$	4.00000	0.205738
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−8.00000	−0.410391
$381$	−8.00000	−0.409852
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	−1.00000	−0.0510310
$385$	32.0000	1.63087
$386$	2.00000	0.101797
$387$	−8.00000	−0.406663
$388$	14.0000	0.710742
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	−9.00000	−0.454569
$393$	18.0000	0.907980
$394$	18.0000	0.906827
$395$	−28.0000	−1.40883
$396$	−4.00000	−0.201008
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	−16.0000	−0.802008
$399$	16.0000	0.801002
$400$	−1.00000	−0.0500000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	2.00000	0.0993808
$406$	−8.00000	−0.397033
$407$	48.0000	2.37927
$408$	4.00000	0.198030
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−24.0000	−1.18528
$411$	−18.0000	−0.887875
$412$	−6.00000	−0.295599
$413$	−8.00000	−0.393654
$414$	4.00000	0.196589
$415$	12.0000	0.589057
$416$	0	0
$417$	−16.0000	−0.783523
$418$	−16.0000	−0.782586
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	−8.00000	−0.390360
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	−14.0000	−0.679900
$425$	4.00000	0.194029
$426$	−8.00000	−0.387601
$427$	8.00000	0.387147
$428$	−16.0000	−0.773389
$429$	0	0
$430$	16.0000	0.771589
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000	0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	16.0000	0.768025
$435$	−4.00000	−0.191785
$436$	−16.0000	−0.766261
$437$	16.0000	0.765384
$438$	−6.00000	−0.286691
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	8.00000	0.381385
$441$	9.00000	0.428571
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	−12.0000	−0.569495
$445$	−12.0000	−0.568855
$446$	4.00000	0.189405
$447$	18.0000	0.851371
$448$	−4.00000	−0.188982
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	1.00000	0.0471405
$451$	−48.0000	−2.26023
$452$	−4.00000	−0.188144
$453$	2.00000	0.0939682
$454$	2.00000	0.0938647
$455$	0	0
$456$	4.00000	0.187317
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	−10.0000	−0.467269
$459$	−4.00000	−0.186704
$460$	−8.00000	−0.373002
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	−16.0000	−0.744387
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−2.00000	−0.0928477
$465$	8.00000	0.370991
$466$	−6.00000	−0.277945
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	6.00000	0.276465
$472$	−2.00000	−0.0920575
$473$	32.0000	1.47136
$474$	14.0000	0.643041
$475$	4.00000	0.183533
$476$	16.0000	0.733359
$477$	14.0000	0.641016
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	−2.00000	−0.0912871
$481$	0	0
$482$	−18.0000	−0.819878
$483$	16.0000	0.728025
$484$	5.00000	0.227273
$485$	28.0000	1.27141
$486$	−1.00000	−0.0453609
$487$	−10.0000	−0.453143	−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000	−0.453143	−0.226572	+	0.973995i	$0.572752\pi$
$488$	2.00000	0.0905357
$489$	24.0000	1.08532
$490$	−18.0000	−0.813157
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	12.0000	0.541002
$493$	8.00000	0.360302
$494$	0	0
$495$	−8.00000	−0.359573
$496$	4.00000	0.179605
$497$	−32.0000	−1.43540
$498$	−6.00000	−0.268866
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	−12.0000	−0.536656
$501$	1.00000	0.0446767
$502$	16.0000	0.714115
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	4.00000	0.178174
$505$	−4.00000	−0.177998
$506$	−16.0000	−0.711287
$507$	−13.0000	−0.577350
$508$	−8.00000	−0.354943
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	8.00000	0.354246
$511$	−24.0000	−1.06170
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	16.0000	0.705730
$515$	−12.0000	−0.528783
$516$	−8.00000	−0.352180
$517$	0	0
$518$	−48.0000	−2.10900
$519$	2.00000	0.0877903
$520$	0	0
$521$	−32.0000	−1.40195	−0.700973	−	0.713188i	$-0.747251\pi$
$521$	−32.0000	−1.40195	−0.700973	+	0.713188i	$0.747251\pi$
$522$	2.00000	0.0875376
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	18.0000	0.786334
$525$	4.00000	0.174574
$526$	−24.0000	−1.04645
$527$	−16.0000	−0.696971
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	−28.0000	−1.21624
$531$	2.00000	0.0867926
$532$	16.0000	0.693688
$533$	0	0
$534$	6.00000	0.259645
$535$	−32.0000	−1.38348
$536$	4.00000	0.172774
$537$	−4.00000	−0.172613
$538$	−6.00000	−0.258678
$539$	−36.0000	−1.55063
$540$	2.00000	0.0860663
$541$	36.0000	1.54776	0.773880	−	0.633332i	$-0.218313\pi$
$541$	36.0000	1.54776	0.773880	+	0.633332i	$0.218313\pi$
$542$	−6.00000	−0.257722
$543$	14.0000	0.600798
$544$	4.00000	0.171499
$545$	−32.0000	−1.37073
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−18.0000	−0.768922
$549$	−2.00000	−0.0853579
$550$	−4.00000	−0.170561
$551$	8.00000	0.340811
$552$	4.00000	0.170251
$553$	56.0000	2.38136
$554$	−24.0000	−1.01966
$555$	−24.0000	−1.01874
$556$	−16.0000	−0.678551
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	−8.00000	−0.338062
$561$	16.0000	0.675521
$562$	−6.00000	−0.253095
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	20.0000	0.840663
$567$	−4.00000	−0.167984
$568$	−8.00000	−0.335673
$569$	−28.0000	−1.17382	−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000	−1.17382	−0.586911	+	0.809652i	$0.699656\pi$
$570$	8.00000	0.335083
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	0	0
$574$	48.0000	2.00348
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	1.00000	0.0415945
$579$	−2.00000	−0.0831172
$580$	−4.00000	−0.166091
$581$	−24.0000	−0.995688
$582$	−14.0000	−0.580319
$583$	−56.0000	−2.31928
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−6.00000	−0.247858
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	9.00000	0.371154
$589$	−16.0000	−0.659269
$590$	−4.00000	−0.164677
$591$	−18.0000	−0.740421
$592$	−12.0000	−0.493197
$593$	−20.0000	−0.821302	−0.410651	−	0.911793i	$-0.634698\pi$
$593$	−20.0000	−0.821302	−0.410651	+	0.911793i	$0.634698\pi$
$594$	4.00000	0.164122
$595$	32.0000	1.31187
$596$	18.0000	0.737309
$597$	16.0000	0.654836
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	1.00000	0.0408248
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	−32.0000	−1.30422
$603$	−4.00000	−0.162893
$604$	2.00000	0.0813788
$605$	10.0000	0.406558
$606$	2.00000	0.0812444
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	4.00000	0.162221
$609$	8.00000	0.324176
$610$	4.00000	0.161955
$611$	0	0
$612$	−4.00000	−0.161690
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−12.0000	−0.484281
$615$	24.0000	0.967773
$616$	−16.0000	−0.644658
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	6.00000	0.241355
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	8.00000	0.321288
$621$	−4.00000	−0.160514
$622$	−24.0000	−0.962312
$623$	24.0000	0.961540
$624$	0	0
$625$	−19.0000	−0.760000
$626$	26.0000	1.03917
$627$	16.0000	0.638978
$628$	6.00000	0.239426
$629$	48.0000	1.91389
$630$	8.00000	0.318728
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	14.0000	0.556890
$633$	−20.0000	−0.794929
$634$	6.00000	0.238290
$635$	−16.0000	−0.634941
$636$	14.0000	0.555136
$637$	0	0
$638$	−8.00000	−0.316723
$639$	8.00000	0.316475
$640$	−2.00000	−0.0790569
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	16.0000	0.631470
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	16.0000	0.630488
$645$	−16.0000	−0.629999
$646$	−16.0000	−0.629512
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−1.00000	−0.0392837
$649$	−8.00000	−0.314027
$650$	0	0
$651$	−16.0000	−0.627089
$652$	24.0000	0.939913
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	16.0000	0.625650
$655$	36.0000	1.40664
$656$	12.0000	0.468521
$657$	6.00000	0.234082
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	−8.00000	−0.311400
$661$	28.0000	1.08907	0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000	1.08907	0.544537	+	0.838737i	$0.316705\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	−6.00000	−0.232845
$665$	32.0000	1.24091
$666$	12.0000	0.464991
$667$	8.00000	0.309761
$668$	1.00000	0.0386912
$669$	−4.00000	−0.154649
$670$	8.00000	0.309067
$671$	8.00000	0.308837
$672$	4.00000	0.154303
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	22.0000	0.847408
$675$	−1.00000	−0.0384900
$676$	−13.0000	−0.500000
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	4.00000	0.153619
$679$	−56.0000	−2.14908
$680$	8.00000	0.306786
$681$	−2.00000	−0.0766402
$682$	16.0000	0.612672
$683$	34.0000	1.30097	0.650487	−	0.759517i	$-0.274565\pi$
$683$	34.0000	1.30097	0.650487	+	0.759517i	$0.274565\pi$
$684$	−4.00000	−0.152944
$685$	−36.0000	−1.37549
$686$	8.00000	0.305441
$687$	10.0000	0.381524
$688$	−8.00000	−0.304997
$689$	0	0
$690$	8.00000	0.304555
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	2.00000	0.0760286
$693$	16.0000	0.607790
$694$	6.00000	0.227757
$695$	−32.0000	−1.21383
$696$	2.00000	0.0758098
$697$	−48.0000	−1.81813
$698$	16.0000	0.605609
$699$	6.00000	0.226941
$700$	4.00000	0.151186
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	48.0000	1.81035
$704$	−4.00000	−0.150756
$705$	0	0
$706$	18.0000	0.677439
$707$	8.00000	0.300871
$708$	2.00000	0.0751646
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	−16.0000	−0.600469
$711$	−14.0000	−0.525041
$712$	6.00000	0.224860
$713$	−16.0000	−0.599205
$714$	−16.0000	−0.598785
$715$	0	0
$716$	−4.00000	−0.149487
$717$	0	0
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	2.00000	0.0745356
$721$	24.0000	0.893807
$722$	3.00000	0.111648
$723$	18.0000	0.669427
$724$	14.0000	0.520306
$725$	2.00000	0.0742781
$726$	−5.00000	−0.185567
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	32.0000	1.18356
$732$	−2.00000	−0.0739221
$733$	50.0000	1.84679	0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000	1.84679	0.923396	+	0.383849i	$0.125402\pi$
$734$	−12.0000	−0.442928
$735$	18.0000	0.663940
$736$	4.00000	0.147442
$737$	16.0000	0.589368
$738$	−12.0000	−0.441726
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−24.0000	−0.882258
$741$	0	0
$742$	56.0000	2.05582
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−4.00000	−0.146647
$745$	36.0000	1.31894
$746$	32.0000	1.17160
$747$	6.00000	0.219529
$748$	16.0000	0.585018
$749$	64.0000	2.33851
$750$	12.0000	0.438178
$751$	−14.0000	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	0	0
$755$	4.00000	0.145575
$756$	−4.00000	−0.145479
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	20.0000	0.726433
$759$	16.0000	0.580763
$760$	8.00000	0.290191
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	8.00000	0.289809
$763$	64.0000	2.31696
$764$	0	0
$765$	−8.00000	−0.289241
$766$	8.00000	0.289052
$767$	0	0
$768$	1.00000	0.0360844
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	−32.0000	−1.15320
$771$	−16.0000	−0.576226
$772$	−2.00000	−0.0719816
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	8.00000	0.287554
$775$	−4.00000	−0.143684
$776$	−14.0000	−0.502571
$777$	48.0000	1.72199
$778$	−34.0000	−1.21896
$779$	−48.0000	−1.71978
$780$	0	0
$781$	−32.0000	−1.14505
$782$	−16.0000	−0.572159
$783$	−2.00000	−0.0714742
$784$	9.00000	0.321429
$785$	12.0000	0.428298
$786$	−18.0000	−0.642039
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−18.0000	−0.641223
$789$	24.0000	0.854423
$790$	28.0000	0.996195
$791$	16.0000	0.568895
$792$	4.00000	0.142134
$793$	0	0
$794$	−10.0000	−0.354887
$795$	28.0000	0.993058
$796$	16.0000	0.567105
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	−16.0000	−0.566394
$799$	0	0
$800$	1.00000	0.0353553
$801$	−6.00000	−0.212000
$802$	−24.0000	−0.847469
$803$	−24.0000	−0.846942
$804$	−4.00000	−0.141069
$805$	32.0000	1.12785
$806$	0	0
$807$	6.00000	0.211210
$808$	2.00000	0.0703598
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	−2.00000	−0.0702728
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	8.00000	0.280745
$813$	6.00000	0.210429
$814$	−48.0000	−1.68240
$815$	48.0000	1.68137
$816$	−4.00000	−0.140028
$817$	32.0000	1.11954
$818$	−26.0000	−0.909069
$819$	0	0
$820$	24.0000	0.838116
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	18.0000	0.627822
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	6.00000	0.209020
$825$	4.00000	0.139262
$826$	8.00000	0.278356
$827$	−54.0000	−1.87776	−0.938882	−	0.344239i	$-0.888137\pi$
$827$	−54.0000	−1.87776	−0.938882	+	0.344239i	$0.888137\pi$
$828$	−4.00000	−0.139010
$829$	44.0000	1.52818	0.764092	−	0.645108i	$-0.223188\pi$
$829$	44.0000	1.52818	0.764092	+	0.645108i	$0.223188\pi$
$830$	−12.0000	−0.416526
$831$	24.0000	0.832551
$832$	0	0
$833$	−36.0000	−1.24733
$834$	16.0000	0.554035
$835$	2.00000	0.0692129
$836$	16.0000	0.553372
$837$	4.00000	0.138260
$838$	−32.0000	−1.10542
$839$	32.0000	1.10476	0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000	1.10476	0.552381	+	0.833592i	$0.313719\pi$
$840$	8.00000	0.276026
$841$	−25.0000	−0.862069
$842$	10.0000	0.344623
$843$	6.00000	0.206651
$844$	−20.0000	−0.688428
$845$	−26.0000	−0.894427
$846$	0	0
$847$	−20.0000	−0.687208
$848$	14.0000	0.480762
$849$	−20.0000	−0.686398
$850$	−4.00000	−0.137199
$851$	48.0000	1.64542
$852$	8.00000	0.274075
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	−8.00000	−0.273754
$855$	−8.00000	−0.273594
$856$	16.0000	0.546869
$857$	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
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−16.0000	−0.545595
$861$	−48.0000	−1.63584
$862$	−24.0000	−0.817443
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	−1.00000	−0.0340207
$865$	4.00000	0.136004
$866$	2.00000	0.0679628
$867$	−1.00000	−0.0339618
$868$	−16.0000	−0.543075
$869$	56.0000	1.89967
$870$	4.00000	0.135613
$871$	0	0
$872$	16.0000	0.541828
$873$	14.0000	0.473828
$874$	−16.0000	−0.541208
$875$	48.0000	1.62270
$876$	6.00000	0.202721
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	26.0000	0.877457
$879$	6.00000	0.202375
$880$	−8.00000	−0.269680
$881$	−4.00000	−0.134763	−0.0673817	−	0.997727i	$-0.521465\pi$
$881$	−4.00000	−0.134763	−0.0673817	+	0.997727i	$0.521465\pi$
$882$	−9.00000	−0.303046
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	18.0000	0.604722
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	12.0000	0.402694
$889$	32.0000	1.07325
$890$	12.0000	0.402241
$891$	−4.00000	−0.134005
$892$	−4.00000	−0.133930
$893$	0	0
$894$	−18.0000	−0.602010
$895$	−8.00000	−0.267411
$896$	4.00000	0.133631
$897$	0	0
$898$	10.0000	0.333704
$899$	−8.00000	−0.266815
$900$	−1.00000	−0.0333333
$901$	−56.0000	−1.86563
$902$	48.0000	1.59823
$903$	32.0000	1.06489
$904$	4.00000	0.133038
$905$	28.0000	0.930751
$906$	−2.00000	−0.0664455
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−2.00000	−0.0663723
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−4.00000	−0.132453
$913$	−24.0000	−0.794284
$914$	14.0000	0.463079
$915$	−4.00000	−0.132236
$916$	10.0000	0.330409
$917$	−72.0000	−2.37765
$918$	4.00000	0.132020
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	8.00000	0.263752
$921$	12.0000	0.395413
$922$	−18.0000	−0.592798
$923$	0	0
$924$	16.0000	0.526361
$925$	12.0000	0.394558
$926$	22.0000	0.722965
$927$	−6.00000	−0.197066
$928$	2.00000	0.0656532
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	−8.00000	−0.262330
$931$	−36.0000	−1.17985
$932$	6.00000	0.196537
$933$	24.0000	0.785725
$934$	−4.00000	−0.130884
$935$	32.0000	1.04651
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	−16.0000	−0.522419
$939$	−26.0000	−0.848478
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−6.00000	−0.195491
$943$	−48.0000	−1.56310
$944$	2.00000	0.0650945
$945$	−8.00000	−0.260240
$946$	−32.0000	−1.04041
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	−14.0000	−0.454699
$949$	0	0
$950$	−4.00000	−0.129777
$951$	−6.00000	−0.194563
$952$	−16.0000	−0.518563
$953$	−52.0000	−1.68445	−0.842223	−	0.539130i	$-0.818753\pi$
$953$	−52.0000	−1.68445	−0.842223	+	0.539130i	$0.818753\pi$
$954$	−14.0000	−0.453267
$955$	0	0
$956$	0	0
$957$	8.00000	0.258603
$958$	36.0000	1.16311
$959$	72.0000	2.32500
$960$	2.00000	0.0645497
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−16.0000	−0.515593
$964$	18.0000	0.579741
$965$	−4.00000	−0.128765
$966$	−16.0000	−0.514792
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−5.00000	−0.160706
$969$	16.0000	0.513994
$970$	−28.0000	−0.899026
$971$	−2.00000	−0.0641831	−0.0320915	−	0.999485i	$-0.510217\pi$
$971$	−2.00000	−0.0641831	−0.0320915	+	0.999485i	$0.510217\pi$
$972$	1.00000	0.0320750
$973$	64.0000	2.05175
$974$	10.0000	0.320421
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	−24.0000	−0.767435
$979$	24.0000	0.767043
$980$	18.0000	0.574989
$981$	−16.0000	−0.510841
$982$	28.0000	0.893516
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	−12.0000	−0.382546
$985$	−36.0000	−1.14706
$986$	−8.00000	−0.254772
$987$	0	0
$988$	0	0
$989$	32.0000	1.01754
$990$	8.00000	0.254257
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	−4.00000	−0.127000
$993$	8.00000	0.253872
$994$	32.0000	1.01498
$995$	32.0000	1.01447
$996$	6.00000	0.190117
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	0	0
$999$	−12.0000	−0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1002.2.a.d.1.1	✓	1
3.2	odd	2		3006.2.a.d.1.1		1
4.3	odd	2		8016.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.d.1.1	✓	1	1.1	even	1	trivial
3006.2.a.d.1.1		1	3.2	odd	2
8016.2.a.f.1.1		1	4.3	odd	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:	\( N \)	\(=\)	\( 1002 = 2 \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1002.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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