LMFDB - Embedded newform 1122.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1122,2,Mod(1,1122)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1122, 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("1122.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1122 = 2 \cdot 3 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1122.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:	$8.95921510679$
Analytic rank:	$0$
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	$\chi$	$=$	1122.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +4.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +2.00000 q^{45} -4.00000 q^{46} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} +4.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} -4.00000 q^{69} +8.00000 q^{70} +4.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} -2.00000 q^{78} +4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +4.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -8.00000 q^{91} -4.00000 q^{92} -4.00000 q^{93} -8.00000 q^{95} +1.00000 q^{96} -6.00000 q^{97} +9.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	4.00000	1.06904
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	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
$22$	−1.00000	−0.213201
$23$	−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
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	8.00000	1.35225
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	2.00000	0.316228
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	4.00000	0.617213
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	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
$51$	1.00000	0.140028
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	−2.00000	−0.269680
$56$	4.00000	0.534522
$57$	−4.00000	−0.529813
$58$	2.00000	0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	2.00000	0.258199
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−4.00000	−0.508001
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−1.00000	−0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	1.00000	0.121268
$69$	−4.00000	−0.481543
$70$	8.00000	0.956183
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−6.00000	−0.697486
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	−2.00000	−0.226455
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	4.00000	0.436436
$85$	2.00000	0.216930
$86$	−4.00000	−0.431331
$87$	2.00000	0.214423
$88$	−1.00000	−0.106600
$89$	−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
$91$	−8.00000	−0.838628
$92$	−4.00000	−0.417029
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−8.00000	−0.820783
$96$	1.00000	0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	9.00000	0.909137
$99$	−1.00000	−0.100504
$100$	−1.00000	−0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	1.00000	0.0990148
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−2.00000	−0.196116
$105$	8.00000	0.780720
$106$	6.00000	0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	1.00000	0.0962250
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	−2.00000	−0.190693
$111$	−6.00000	−0.569495
$112$	4.00000	0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−4.00000	−0.374634
$115$	−8.00000	−0.746004
$116$	2.00000	0.185695
$117$	−2.00000	−0.184900
$118$	4.00000	0.368230
$119$	4.00000	0.366679
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	−6.00000	−0.543214
$123$	−6.00000	−0.541002
$124$	−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
$129$	−4.00000	−0.352180
$130$	−4.00000	−0.350823
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−1.00000	−0.0870388
$133$	−16.0000	−1.38738
$134$	−4.00000	−0.345547
$135$	2.00000	0.172133
$136$	1.00000	0.0857493
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−4.00000	−0.340503
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	8.00000	0.676123
$141$	0	0
$142$	4.00000	0.335673
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	10.0000	0.827606
$147$	9.00000	0.742307
$148$	−6.00000	−0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−4.00000	−0.324443
$153$	1.00000	0.0808452
$154$	−4.00000	−0.322329
$155$	−8.00000	−0.642575
$156$	−2.00000	−0.160128
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	4.00000	0.318223
$159$	6.00000	0.475831
$160$	2.00000	0.158114
$161$	−16.0000	−1.26098
$162$	1.00000	0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−6.00000	−0.468521
$165$	−2.00000	−0.155700
$166$	12.0000	0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	4.00000	0.308607
$169$	−9.00000	−0.692308
$170$	2.00000	0.153393
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	2.00000	0.151620
$175$	−4.00000	−0.302372
$176$	−1.00000	−0.0753778
$177$	4.00000	0.300658
$178$	−6.00000	−0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	2.00000	0.149071
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	−8.00000	−0.592999
$183$	−6.00000	−0.443533
$184$	−4.00000	−0.294884
$185$	−12.0000	−0.882258
$186$	−4.00000	−0.293294
$187$	−1.00000	−0.0731272
$188$	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
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−6.00000	−0.430775
$195$	−4.00000	−0.286446
$196$	9.00000	0.642857
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−1.00000	−0.0710669
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	14.0000	0.985037
$203$	8.00000	0.561490
$204$	1.00000	0.0700140
$205$	−12.0000	−0.838116
$206$	16.0000	1.11477
$207$	−4.00000	−0.278019
$208$	−2.00000	−0.138675
$209$	4.00000	0.276686
$210$	8.00000	0.552052
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000	0.412082
$213$	4.00000	0.274075
$214$	4.00000	0.273434
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	−16.0000	−1.08615
$218$	18.0000	1.21911
$219$	10.0000	0.675737
$220$	−2.00000	−0.134840
$221$	−2.00000	−0.134535
$222$	−6.00000	−0.402694
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	4.00000	0.267261
$225$	−1.00000	−0.0666667
$226$	−6.00000	−0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−4.00000	−0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−8.00000	−0.527504
$231$	−4.00000	−0.263181
$232$	2.00000	0.131306
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	4.00000	0.260378
$237$	4.00000	0.259828
$238$	4.00000	0.259281
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	2.00000	0.129099
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	18.0000	1.14998
$246$	−6.00000	−0.382546
$247$	8.00000	0.509028
$248$	−4.00000	−0.254000
$249$	12.0000	0.760469
$250$	−12.0000	−0.758947
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	4.00000	0.251976
$253$	4.00000	0.251478
$254$	−8.00000	−0.501965
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−4.00000	−0.249029
$259$	−24.0000	−1.49129
$260$	−4.00000	−0.248069
$261$	2.00000	0.123797
$262$	−4.00000	−0.247121
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	−1.00000	−0.0615457
$265$	12.0000	0.737154
$266$	−16.0000	−0.981023
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	2.00000	0.121716
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	1.00000	0.0606339
$273$	−8.00000	−0.484182
$274$	−6.00000	−0.362473
$275$	1.00000	0.0603023
$276$	−4.00000	−0.240772
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−12.0000	−0.719712
$279$	−4.00000	−0.239474
$280$	8.00000	0.478091
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	4.00000	0.237356
$285$	−8.00000	−0.473879
$286$	2.00000	0.118262
$287$	−24.0000	−1.41668
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	4.00000	0.234888
$291$	−6.00000	−0.351726
$292$	10.0000	0.585206
$293$	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
$295$	8.00000	0.465778
$296$	−6.00000	−0.348743
$297$	−1.00000	−0.0580259
$298$	6.00000	0.347571
$299$	8.00000	0.462652
$300$	−1.00000	−0.0577350
$301$	−16.0000	−0.922225
$302$	−16.0000	−0.920697
$303$	14.0000	0.804279
$304$	−4.00000	−0.229416
$305$	−12.0000	−0.687118
$306$	1.00000	0.0571662
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	−4.00000	−0.227921
$309$	16.0000	0.910208
$310$	−8.00000	−0.454369
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−2.00000	−0.113228
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−2.00000	−0.112867
$315$	8.00000	0.450749
$316$	4.00000	0.225018
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	6.00000	0.336463
$319$	−2.00000	−0.111979
$320$	2.00000	0.111803
$321$	4.00000	0.223258
$322$	−16.0000	−0.891645
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−12.0000	−0.664619
$327$	18.0000	0.995402
$328$	−6.00000	−0.331295
$329$	0	0
$330$	−2.00000	−0.110096
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	12.0000	0.658586
$333$	−6.00000	−0.328798
$334$	−12.0000	−0.656611
$335$	−8.00000	−0.437087
$336$	4.00000	0.218218
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−9.00000	−0.489535
$339$	−6.00000	−0.325875
$340$	2.00000	0.108465
$341$	4.00000	0.216612
$342$	−4.00000	−0.216295
$343$	8.00000	0.431959
$344$	−4.00000	−0.215666
$345$	−8.00000	−0.430706
$346$	18.0000	0.967686
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	2.00000	0.107211
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−4.00000	−0.213809
$351$	−2.00000	−0.106752
$352$	−1.00000	−0.0533002
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	4.00000	0.212598
$355$	8.00000	0.424596
$356$	−6.00000	−0.317999
$357$	4.00000	0.211702
$358$	20.0000	1.05703
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	2.00000	0.105409
$361$	−3.00000	−0.157895
$362$	26.0000	1.36653
$363$	1.00000	0.0524864
$364$	−8.00000	−0.419314
$365$	20.0000	1.04685
$366$	−6.00000	−0.313625
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	−4.00000	−0.208514
$369$	−6.00000	−0.312348
$370$	−12.0000	−0.623850
$371$	24.0000	1.24602
$372$	−4.00000	−0.207390
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−1.00000	−0.0517088
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−4.00000	−0.206010
$378$	4.00000	0.205738
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	−8.00000	−0.410391
$381$	−8.00000	−0.409852
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	1.00000	0.0510310
$385$	−8.00000	−0.407718
$386$	−22.0000	−1.11977
$387$	−4.00000	−0.203331
$388$	−6.00000	−0.304604
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	−4.00000	−0.202548
$391$	−4.00000	−0.202289
$392$	9.00000	0.454569
$393$	−4.00000	−0.201773
$394$	18.0000	0.906827
$395$	8.00000	0.402524
$396$	−1.00000	−0.0502519
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−20.0000	−1.00251
$399$	−16.0000	−0.801002
$400$	−1.00000	−0.0500000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	−4.00000	−0.199502
$403$	8.00000	0.398508
$404$	14.0000	0.696526
$405$	2.00000	0.0993808
$406$	8.00000	0.397033
$407$	6.00000	0.297409
$408$	1.00000	0.0495074
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	−12.0000	−0.592638
$411$	−6.00000	−0.295958
$412$	16.0000	0.788263
$413$	16.0000	0.787309
$414$	−4.00000	−0.196589
$415$	24.0000	1.17811
$416$	−2.00000	−0.0980581
$417$	−12.0000	−0.587643
$418$	4.00000	0.195646
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	8.00000	0.390360
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	6.00000	0.291386
$425$	−1.00000	−0.0485071
$426$	4.00000	0.193801
$427$	−24.0000	−1.16144
$428$	4.00000	0.193347
$429$	2.00000	0.0965609
$430$	−8.00000	−0.385794
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−16.0000	−0.768025
$435$	4.00000	0.191785
$436$	18.0000	0.862044
$437$	16.0000	0.765384
$438$	10.0000	0.477818
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−2.00000	−0.0953463
$441$	9.00000	0.428571
$442$	−2.00000	−0.0951303
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	−6.00000	−0.284747
$445$	−12.0000	−0.568855
$446$	−24.0000	−1.13643
$447$	6.00000	0.283790
$448$	4.00000	0.188982
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−1.00000	−0.0471405
$451$	6.00000	0.282529
$452$	−6.00000	−0.282216
$453$	−16.0000	−0.751746
$454$	−12.0000	−0.563188
$455$	−16.0000	−0.750092
$456$	−4.00000	−0.187317
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−10.0000	−0.467269
$459$	1.00000	0.0466760
$460$	−8.00000	−0.373002
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	−4.00000	−0.186097
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	2.00000	0.0928477
$465$	−8.00000	−0.370991
$466$	18.0000	0.833834
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−0.0924500
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	4.00000	0.184115
$473$	4.00000	0.183920
$474$	4.00000	0.183726
$475$	4.00000	0.183533
$476$	4.00000	0.183340
$477$	6.00000	0.274721
$478$	24.0000	1.09773
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	2.00000	0.0912871
$481$	12.0000	0.547153
$482$	26.0000	1.18427
$483$	−16.0000	−0.728025
$484$	1.00000	0.0454545
$485$	−12.0000	−0.544892
$486$	1.00000	0.0453609
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	−6.00000	−0.271607
$489$	−12.0000	−0.542659
$490$	18.0000	0.813157
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−6.00000	−0.270501
$493$	2.00000	0.0900755
$494$	8.00000	0.359937
$495$	−2.00000	−0.0898933
$496$	−4.00000	−0.179605
$497$	16.0000	0.717698
$498$	12.0000	0.537733
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−12.0000	−0.536656
$501$	−12.0000	−0.536120
$502$	−20.0000	−0.892644
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	4.00000	0.178174
$505$	28.0000	1.24598
$506$	4.00000	0.177822
$507$	−9.00000	−0.399704
$508$	−8.00000	−0.354943
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	2.00000	0.0885615
$511$	40.0000	1.76950
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	2.00000	0.0882162
$515$	32.0000	1.41009
$516$	−4.00000	−0.176090
$517$	0	0
$518$	−24.0000	−1.05450
$519$	18.0000	0.790112
$520$	−4.00000	−0.175412
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	2.00000	0.0875376
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−4.00000	−0.174741
$525$	−4.00000	−0.174574
$526$	−16.0000	−0.697633
$527$	−4.00000	−0.174243
$528$	−1.00000	−0.0435194
$529$	−7.00000	−0.304348
$530$	12.0000	0.521247
$531$	4.00000	0.173585
$532$	−16.0000	−0.693688
$533$	12.0000	0.519778
$534$	−6.00000	−0.259645
$535$	8.00000	0.345870
$536$	−4.00000	−0.172774
$537$	20.0000	0.863064
$538$	10.0000	0.431131
$539$	−9.00000	−0.387657
$540$	2.00000	0.0860663
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	−8.00000	−0.343629
$543$	26.0000	1.11577
$544$	1.00000	0.0428746
$545$	36.0000	1.54207
$546$	−8.00000	−0.342368
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−6.00000	−0.256307
$549$	−6.00000	−0.256074
$550$	1.00000	0.0426401
$551$	−8.00000	−0.340811
$552$	−4.00000	−0.170251
$553$	16.0000	0.680389
$554$	10.0000	0.424859
$555$	−12.0000	−0.509372
$556$	−12.0000	−0.508913
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	−4.00000	−0.169334
$559$	8.00000	0.338364
$560$	8.00000	0.338062
$561$	−1.00000	−0.0422200
$562$	10.0000	0.421825
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	20.0000	0.840663
$567$	4.00000	0.167984
$568$	4.00000	0.167836
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	−8.00000	−0.335083
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	−24.0000	−1.00174
$575$	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
$579$	−22.0000	−0.914289
$580$	4.00000	0.166091
$581$	48.0000	1.99138
$582$	−6.00000	−0.248708
$583$	−6.00000	−0.248495
$584$	10.0000	0.413803
$585$	−4.00000	−0.165380
$586$	6.00000	0.247858
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	9.00000	0.371154
$589$	16.0000	0.659269
$590$	8.00000	0.329355
$591$	18.0000	0.740421
$592$	−6.00000	−0.246598
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	−1.00000	−0.0410305
$595$	8.00000	0.327968
$596$	6.00000	0.245770
$597$	−20.0000	−0.818546
$598$	8.00000	0.327144
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	−1.00000	−0.0408248
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−16.0000	−0.652111
$603$	−4.00000	−0.162893
$604$	−16.0000	−0.651031
$605$	2.00000	0.0813116
$606$	14.0000	0.568711
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	−4.00000	−0.162221
$609$	8.00000	0.324176
$610$	−12.0000	−0.485866
$611$	0	0
$612$	1.00000	0.0404226
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	4.00000	0.161427
$615$	−12.0000	−0.483887
$616$	−4.00000	−0.161165
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	16.0000	0.643614
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	−8.00000	−0.321288
$621$	−4.00000	−0.160514
$622$	12.0000	0.481156
$623$	−24.0000	−0.961540
$624$	−2.00000	−0.0800641
$625$	−19.0000	−0.760000
$626$	−14.0000	−0.559553
$627$	4.00000	0.159745
$628$	−2.00000	−0.0798087
$629$	−6.00000	−0.239236
$630$	8.00000	0.318728
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	4.00000	0.159111
$633$	−4.00000	−0.158986
$634$	−30.0000	−1.19145
$635$	−16.0000	−0.634941
$636$	6.00000	0.237915
$637$	−18.0000	−0.713186
$638$	−2.00000	−0.0791808
$639$	4.00000	0.158238
$640$	2.00000	0.0790569
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	4.00000	0.157867
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	−16.0000	−0.630488
$645$	−8.00000	−0.315000
$646$	−4.00000	−0.157378
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	1.00000	0.0392837
$649$	−4.00000	−0.157014
$650$	2.00000	0.0784465
$651$	−16.0000	−0.627089
$652$	−12.0000	−0.469956
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	18.0000	0.703856
$655$	−8.00000	−0.312586
$656$	−6.00000	−0.234261
$657$	10.0000	0.390137
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	−2.00000	−0.0778499
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	−20.0000	−0.777322
$663$	−2.00000	−0.0776736
$664$	12.0000	0.465690
$665$	−32.0000	−1.24091
$666$	−6.00000	−0.232495
$667$	−8.00000	−0.309761
$668$	−12.0000	−0.464294
$669$	−24.0000	−0.927894
$670$	−8.00000	−0.309067
$671$	6.00000	0.231627
$672$	4.00000	0.154303
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	2.00000	0.0770371
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	50.0000	1.92166	0.960828	−	0.277145i	$-0.0893883\pi$
$677$	50.0000	1.92166	0.960828	+	0.277145i	$0.0893883\pi$
$678$	−6.00000	−0.230429
$679$	−24.0000	−0.921035
$680$	2.00000	0.0766965
$681$	−12.0000	−0.459841
$682$	4.00000	0.153168
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	−4.00000	−0.152944
$685$	−12.0000	−0.458496
$686$	8.00000	0.305441
$687$	−10.0000	−0.381524
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	−8.00000	−0.304555
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	18.0000	0.684257
$693$	−4.00000	−0.151947
$694$	12.0000	0.455514
$695$	−24.0000	−0.910372
$696$	2.00000	0.0758098
$697$	−6.00000	−0.227266
$698$	14.0000	0.529908
$699$	18.0000	0.680823
$700$	−4.00000	−0.151186
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	−2.00000	−0.0754851
$703$	24.0000	0.905177
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	18.0000	0.677439
$707$	56.0000	2.10610
$708$	4.00000	0.150329
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	8.00000	0.300235
$711$	4.00000	0.150012
$712$	−6.00000	−0.224860
$713$	16.0000	0.599205
$714$	4.00000	0.149696
$715$	4.00000	0.149592
$716$	20.0000	0.747435
$717$	24.0000	0.896296
$718$	−8.00000	−0.298557
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	2.00000	0.0745356
$721$	64.0000	2.38348
$722$	−3.00000	−0.111648
$723$	26.0000	0.966950
$724$	26.0000	0.966282
$725$	−2.00000	−0.0742781
$726$	1.00000	0.0371135
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	20.0000	0.740233
$731$	−4.00000	−0.147945
$732$	−6.00000	−0.221766
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	−20.0000	−0.738213
$735$	18.0000	0.663940
$736$	−4.00000	−0.147442
$737$	4.00000	0.147342
$738$	−6.00000	−0.220863
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−12.0000	−0.441129
$741$	8.00000	0.293887
$742$	24.0000	0.881068
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	−4.00000	−0.146647
$745$	12.0000	0.439646
$746$	−10.0000	−0.366126
$747$	12.0000	0.439057
$748$	−1.00000	−0.0365636
$749$	16.0000	0.584627
$750$	−12.0000	−0.438178
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	−20.0000	−0.728841
$754$	−4.00000	−0.145671
$755$	−32.0000	−1.16460
$756$	4.00000	0.145479
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	28.0000	1.01701
$759$	4.00000	0.145191
$760$	−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
$763$	72.0000	2.60658
$764$	0	0
$765$	2.00000	0.0723102
$766$	16.0000	0.578103
$767$	−8.00000	−0.288863
$768$	1.00000	0.0360844
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	−8.00000	−0.288300
$771$	2.00000	0.0720282
$772$	−22.0000	−0.791797
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−4.00000	−0.143777
$775$	4.00000	0.143684
$776$	−6.00000	−0.215387
$777$	−24.0000	−0.860995
$778$	−18.0000	−0.645331
$779$	24.0000	0.859889
$780$	−4.00000	−0.143223
$781$	−4.00000	−0.143131
$782$	−4.00000	−0.143040
$783$	2.00000	0.0714742
$784$	9.00000	0.321429
$785$	−4.00000	−0.142766
$786$	−4.00000	−0.142675
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	18.0000	0.641223
$789$	−16.0000	−0.569615
$790$	8.00000	0.284627
$791$	−24.0000	−0.853342
$792$	−1.00000	−0.0355335
$793$	12.0000	0.426132
$794$	−14.0000	−0.496841
$795$	12.0000	0.425596
$796$	−20.0000	−0.708881
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	−16.0000	−0.566394
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	34.0000	1.20058
$803$	−10.0000	−0.352892
$804$	−4.00000	−0.141069
$805$	−32.0000	−1.12785
$806$	8.00000	0.281788
$807$	10.0000	0.352017
$808$	14.0000	0.492518
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	2.00000	0.0702728
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	8.00000	0.280745
$813$	−8.00000	−0.280572
$814$	6.00000	0.210300
$815$	−24.0000	−0.840683
$816$	1.00000	0.0350070
$817$	16.0000	0.559769
$818$	−6.00000	−0.209785
$819$	−8.00000	−0.279543
$820$	−12.0000	−0.419058
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	−6.00000	−0.209274
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	16.0000	0.557386
$825$	1.00000	0.0348155
$826$	16.0000	0.556711
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	−4.00000	−0.139010
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	24.0000	0.833052
$831$	10.0000	0.346896
$832$	−2.00000	−0.0693375
$833$	9.00000	0.311832
$834$	−12.0000	−0.415526
$835$	−24.0000	−0.830554
$836$	4.00000	0.138343
$837$	−4.00000	−0.138260
$838$	4.00000	0.138178
$839$	−44.0000	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$840$	8.00000	0.276026
$841$	−25.0000	−0.862069
$842$	14.0000	0.482472
$843$	10.0000	0.344418
$844$	−4.00000	−0.137686
$845$	−18.0000	−0.619219
$846$	0	0
$847$	4.00000	0.137442
$848$	6.00000	0.206041
$849$	20.0000	0.686398
$850$	−1.00000	−0.0342997
$851$	24.0000	0.822709
$852$	4.00000	0.137038
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−24.0000	−0.821263
$855$	−8.00000	−0.273594
$856$	4.00000	0.136717
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	2.00000	0.0682789
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−8.00000	−0.272798
$861$	−24.0000	−0.817918
$862$	−4.00000	−0.136241
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	1.00000	0.0340207
$865$	36.0000	1.22404
$866$	2.00000	0.0679628
$867$	1.00000	0.0339618
$868$	−16.0000	−0.543075
$869$	−4.00000	−0.135691
$870$	4.00000	0.135613
$871$	8.00000	0.271070
$872$	18.0000	0.609557
$873$	−6.00000	−0.203069
$874$	16.0000	0.541208
$875$	−48.0000	−1.62270
$876$	10.0000	0.337869
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	20.0000	0.674967
$879$	6.00000	0.202375
$880$	−2.00000	−0.0674200
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	9.00000	0.303046
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	−2.00000	−0.0672673
$885$	8.00000	0.268917
$886$	28.0000	0.940678
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−6.00000	−0.201347
$889$	−32.0000	−1.07325
$890$	−12.0000	−0.402241
$891$	−1.00000	−0.0335013
$892$	−24.0000	−0.803579
$893$	0	0
$894$	6.00000	0.200670
$895$	40.0000	1.33705
$896$	4.00000	0.133631
$897$	8.00000	0.267112
$898$	2.00000	0.0667409
$899$	−8.00000	−0.266815
$900$	−1.00000	−0.0333333
$901$	6.00000	0.199889
$902$	6.00000	0.199778
$903$	−16.0000	−0.532447
$904$	−6.00000	−0.199557
$905$	52.0000	1.72854
$906$	−16.0000	−0.531564
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	−12.0000	−0.398234
$909$	14.0000	0.464351
$910$	−16.0000	−0.530395
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	−4.00000	−0.132453
$913$	−12.0000	−0.397142
$914$	10.0000	0.330771
$915$	−12.0000	−0.396708
$916$	−10.0000	−0.330409
$917$	−16.0000	−0.528367
$918$	1.00000	0.0330049
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−8.00000	−0.263752
$921$	4.00000	0.131804
$922$	6.00000	0.197599
$923$	−8.00000	−0.263323
$924$	−4.00000	−0.131590
$925$	6.00000	0.197279
$926$	−40.0000	−1.31448
$927$	16.0000	0.525509
$928$	2.00000	0.0656532
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−8.00000	−0.262330
$931$	−36.0000	−1.17985
$932$	18.0000	0.589610
$933$	12.0000	0.392862
$934$	−12.0000	−0.392652
$935$	−2.00000	−0.0654070
$936$	−2.00000	−0.0653720
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	−16.0000	−0.522419
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−2.00000	−0.0651635
$943$	24.0000	0.781548
$944$	4.00000	0.130189
$945$	8.00000	0.260240
$946$	4.00000	0.130051
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	4.00000	0.129914
$949$	−20.0000	−0.649227
$950$	4.00000	0.129777
$951$	−30.0000	−0.972817
$952$	4.00000	0.129641
$953$	−38.0000	−1.23094	−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000	−1.23094	−0.615470	+	0.788160i	$0.711034\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	−2.00000	−0.0646508
$958$	−12.0000	−0.387702
$959$	−24.0000	−0.775000
$960$	2.00000	0.0645497
$961$	−15.0000	−0.483871
$962$	12.0000	0.386896
$963$	4.00000	0.128898
$964$	26.0000	0.837404
$965$	−44.0000	−1.41641
$966$	−16.0000	−0.514792
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	1.00000	0.0321412
$969$	−4.00000	−0.128499
$970$	−12.0000	−0.385297
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	1.00000	0.0320750
$973$	−48.0000	−1.53881
$974$	4.00000	0.128168
$975$	2.00000	0.0640513
$976$	−6.00000	−0.192055
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−12.0000	−0.383718
$979$	6.00000	0.191761
$980$	18.0000	0.574989
$981$	18.0000	0.574696
$982$	−20.0000	−0.638226
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	−6.00000	−0.191273
$985$	36.0000	1.14706
$986$	2.00000	0.0636930
$987$	0	0
$988$	8.00000	0.254514
$989$	16.0000	0.508770
$990$	−2.00000	−0.0635642
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	−4.00000	−0.127000
$993$	−20.0000	−0.634681
$994$	16.0000	0.507489
$995$	−40.0000	−1.26809
$996$	12.0000	0.380235
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	−20.0000	−0.633089
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1122.2.a.m.1.1	✓	1
3.2	odd	2		3366.2.a.e.1.1		1
4.3	odd	2		8976.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1122.2.a.m.1.1	✓	1	1.1	even	1	trivial
3366.2.a.e.1.1		1	3.2	odd	2
8976.2.a.p.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:	\( N \)	\(=\)	\( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1122.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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