LMFDB - Embedded newform 3234.2.a.o.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3234,2,Mod(1,3234)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3234.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,1,1,3,-1,0,-1,1,-3,-1,1,6,0,3,1,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.8236200137$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3234.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +3.00000 q^{20} +1.00000 q^{22} +5.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} -3.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +5.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} +5.00000 q^{41} -10.0000 q^{43} -1.00000 q^{44} +3.00000 q^{45} -5.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -5.00000 q^{51} +6.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} -3.00000 q^{55} +6.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +3.00000 q^{60} -5.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +18.0000 q^{65} +1.00000 q^{66} +5.00000 q^{67} -5.00000 q^{68} +5.00000 q^{69} +4.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} +2.00000 q^{74} +4.00000 q^{75} +6.00000 q^{76} -6.00000 q^{78} -1.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -5.00000 q^{82} +1.00000 q^{83} -15.0000 q^{85} +10.0000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} -3.00000 q^{90} +5.00000 q^{92} +4.00000 q^{93} -9.00000 q^{94} +18.0000 q^{95} -1.00000 q^{96} +9.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	1.00000	0.213201
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	−6.00000	−1.17670
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−3.00000	−0.547723
$31$	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
$33$	−1.00000	−0.174078
$34$	5.00000	0.857493
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−6.00000	−0.973329
$39$	6.00000	0.960769
$40$	−3.00000	−0.474342
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−1.00000	−0.150756
$45$	3.00000	0.447214
$46$	−5.00000	−0.737210
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−5.00000	−0.700140
$52$	6.00000	0.832050
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−1.00000	−0.136083
$55$	−3.00000	−0.404520
$56$	0	0
$57$	6.00000	0.794719
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	3.00000	0.387298
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	18.0000	2.23263
$66$	1.00000	0.123091
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	−5.00000	−0.606339
$69$	5.00000	0.601929
$70$	0	0
$71$	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
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	2.00000	0.232495
$75$	4.00000	0.461880
$76$	6.00000	0.688247
$77$	0	0
$78$	−6.00000	−0.679366
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	−5.00000	−0.552158
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	0	0
$85$	−15.0000	−1.62698
$86$	10.0000	1.07833
$87$	−6.00000	−0.643268
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	5.00000	0.521286
$93$	4.00000	0.414781
$94$	−9.00000	−0.928279
$95$	18.0000	1.84676
$96$	−1.00000	−0.102062
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	4.00000	0.400000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	5.00000	0.495074
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−13.0000	−1.25676	−0.628379	−	0.777908i	$-0.716281\pi$
$107$	−13.0000	−1.25676	−0.628379	+	0.777908i	$0.716281\pi$
$108$	1.00000	0.0962250
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	3.00000	0.286039
$111$	−2.00000	−0.189832
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−6.00000	−0.561951
$115$	15.0000	1.39876
$116$	−6.00000	−0.557086
$117$	6.00000	0.554700
$118$	12.0000	1.10469
$119$	0	0
$120$	−3.00000	−0.273861
$121$	1.00000	0.0909091
$122$	5.00000	0.452679
$123$	5.00000	0.450835
$124$	4.00000	0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	−18.0000	−1.57870
$131$	−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
$133$	0	0
$134$	−5.00000	−0.431934
$135$	3.00000	0.258199
$136$	5.00000	0.428746
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−5.00000	−0.425628
$139$	22.0000	1.86602	0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000	1.86602	0.933008	+	0.359856i	$0.117174\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	−4.00000	−0.335673
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	−18.0000	−1.49482
$146$	−12.0000	−0.993127
$147$	0	0
$148$	−2.00000	−0.164399
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−4.00000	−0.326599
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	−6.00000	−0.486664
$153$	−5.00000	−0.404226
$154$	0	0
$155$	12.0000	0.963863
$156$	6.00000	0.480384
$157$	−24.0000	−1.91541	−0.957704	−	0.287754i	$-0.907091\pi$
$157$	−24.0000	−1.91541	−0.957704	+	0.287754i	$0.907091\pi$
$158$	1.00000	0.0795557
$159$	2.00000	0.158610
$160$	−3.00000	−0.237171
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	5.00000	0.390434
$165$	−3.00000	−0.233550
$166$	−1.00000	−0.0776151
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	15.0000	1.15045
$171$	6.00000	0.458831
$172$	−10.0000	−0.762493
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−12.0000	−0.901975
$178$	−6.00000	−0.449719
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	3.00000	0.223607
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	−5.00000	−0.369611
$184$	−5.00000	−0.368605
$185$	−6.00000	−0.441129
$186$	−4.00000	−0.293294
$187$	5.00000	0.365636
$188$	9.00000	0.656392
$189$	0	0
$190$	−18.0000	−1.30586
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−9.00000	−0.646162
$195$	18.0000	1.28901
$196$	0	0
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	1.00000	0.0710669
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	−4.00000	−0.282843
$201$	5.00000	0.352673
$202$	8.00000	0.562878
$203$	0	0
$204$	−5.00000	−0.350070
$205$	15.0000	1.04765
$206$	2.00000	0.139347
$207$	5.00000	0.347524
$208$	6.00000	0.416025
$209$	−6.00000	−0.415029
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	2.00000	0.137361
$213$	4.00000	0.274075
$214$	13.0000	0.888662
$215$	−30.0000	−2.04598
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	11.0000	0.745014
$219$	12.0000	0.810885
$220$	−3.00000	−0.202260
$221$	−30.0000	−2.01802
$222$	2.00000	0.134231
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	−10.0000	−0.665190
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	6.00000	0.397360
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	−15.0000	−0.989071
$231$	0	0
$232$	6.00000	0.393919
$233$	−29.0000	−1.89985	−0.949927	−	0.312473i	$-0.898843\pi$
$233$	−29.0000	−1.89985	−0.949927	+	0.312473i	$0.898843\pi$
$234$	−6.00000	−0.392232
$235$	27.0000	1.76129
$236$	−12.0000	−0.781133
$237$	−1.00000	−0.0649570
$238$	0	0
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	3.00000	0.193649
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	−5.00000	−0.320092
$245$	0	0
$246$	−5.00000	−0.318788
$247$	36.0000	2.29063
$248$	−4.00000	−0.254000
$249$	1.00000	0.0633724
$250$	3.00000	0.189737
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	−5.00000	−0.313728
$255$	−15.0000	−0.939336
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	10.0000	0.622573
$259$	0	0
$260$	18.0000	1.11631
$261$	−6.00000	−0.371391
$262$	4.00000	0.247121
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	1.00000	0.0615457
$265$	6.00000	0.368577
$266$	0	0
$267$	6.00000	0.367194
$268$	5.00000	0.305424
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	−3.00000	−0.182574
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	2.00000	0.120824
$275$	−4.00000	−0.241209
$276$	5.00000	0.300965
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−22.0000	−1.31947
$279$	4.00000	0.239474
$280$	0	0
$281$	11.0000	0.656205	0.328102	−	0.944642i	$-0.393591\pi$
$281$	11.0000	0.656205	0.328102	+	0.944642i	$0.393591\pi$
$282$	−9.00000	−0.535942
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	4.00000	0.237356
$285$	18.0000	1.06623
$286$	6.00000	0.354787
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	8.00000	0.470588
$290$	18.0000	1.05700
$291$	9.00000	0.527589
$292$	12.0000	0.702247
$293$	8.00000	0.467365	0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000	0.467365	0.233682	+	0.972313i	$0.424922\pi$
$294$	0	0
$295$	−36.0000	−2.09600
$296$	2.00000	0.116248
$297$	−1.00000	−0.0580259
$298$	−18.0000	−1.04271
$299$	30.0000	1.73494
$300$	4.00000	0.230940
$301$	0	0
$302$	−9.00000	−0.517892
$303$	−8.00000	−0.459588
$304$	6.00000	0.344124
$305$	−15.0000	−0.858898
$306$	5.00000	0.285831
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	0	0
$309$	−2.00000	−0.113776
$310$	−12.0000	−0.681554
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	−6.00000	−0.339683
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	24.0000	1.35440
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	−2.00000	−0.112154
$319$	6.00000	0.335936
$320$	3.00000	0.167705
$321$	−13.0000	−0.725589
$322$	0	0
$323$	−30.0000	−1.66924
$324$	1.00000	0.0555556
$325$	24.0000	1.33128
$326$	1.00000	0.0553849
$327$	−11.0000	−0.608301
$328$	−5.00000	−0.276079
$329$	0	0
$330$	3.00000	0.165145
$331$	35.0000	1.92377	0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000	1.92377	0.961887	+	0.273447i	$0.0881639\pi$
$332$	1.00000	0.0548821
$333$	−2.00000	−0.109599
$334$	24.0000	1.31322
$335$	15.0000	0.819538
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−23.0000	−1.25104
$339$	10.0000	0.543125
$340$	−15.0000	−0.813489
$341$	−4.00000	−0.216612
$342$	−6.00000	−0.324443
$343$	0	0
$344$	10.0000	0.539164
$345$	15.0000	0.807573
$346$	12.0000	0.645124
$347$	7.00000	0.375780	0.187890	−	0.982190i	$-0.439835\pi$
$347$	7.00000	0.375780	0.187890	+	0.982190i	$0.439835\pi$
$348$	−6.00000	−0.321634
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	1.00000	0.0533002
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	12.0000	0.637793
$355$	12.0000	0.636894
$356$	6.00000	0.317999
$357$	0	0
$358$	8.00000	0.422813
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	−3.00000	−0.158114
$361$	17.0000	0.894737
$362$	22.0000	1.15629
$363$	1.00000	0.0524864
$364$	0	0
$365$	36.0000	1.88433
$366$	5.00000	0.261354
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	5.00000	0.260643
$369$	5.00000	0.260290
$370$	6.00000	0.311925
$371$	0	0
$372$	4.00000	0.207390
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	−5.00000	−0.258544
$375$	−3.00000	−0.154919
$376$	−9.00000	−0.464140
$377$	−36.0000	−1.85409
$378$	0	0
$379$	13.0000	0.667765	0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000	0.667765	0.333883	+	0.942615i	$0.391641\pi$
$380$	18.0000	0.923381
$381$	5.00000	0.256158
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−10.0000	−0.508329
$388$	9.00000	0.456906
$389$	−21.0000	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$389$	−21.0000	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$390$	−18.0000	−0.911465
$391$	−25.0000	−1.26430
$392$	0	0
$393$	−4.00000	−0.201773
$394$	24.0000	1.20910
$395$	−3.00000	−0.150946
$396$	−1.00000	−0.0502519
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	−10.0000	−0.501255
$399$	0	0
$400$	4.00000	0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−5.00000	−0.249377
$403$	24.0000	1.19553
$404$	−8.00000	−0.398015
$405$	3.00000	0.149071
$406$	0	0
$407$	2.00000	0.0991363
$408$	5.00000	0.247537
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−15.0000	−0.740797
$411$	−2.00000	−0.0986527
$412$	−2.00000	−0.0985329
$413$	0	0
$414$	−5.00000	−0.245737
$415$	3.00000	0.147264
$416$	−6.00000	−0.294174
$417$	22.0000	1.07734
$418$	6.00000	0.293470
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−14.0000	−0.681509
$423$	9.00000	0.437595
$424$	−2.00000	−0.0971286
$425$	−20.0000	−0.970143
$426$	−4.00000	−0.193801
$427$	0	0
$428$	−13.0000	−0.628379
$429$	−6.00000	−0.289683
$430$	30.0000	1.44673
$431$	22.0000	1.05970	0.529851	−	0.848091i	$-0.322248\pi$
$431$	22.0000	1.05970	0.529851	+	0.848091i	$0.322248\pi$
$432$	1.00000	0.0481125
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	−18.0000	−0.863034
$436$	−11.0000	−0.526804
$437$	30.0000	1.43509
$438$	−12.0000	−0.573382
$439$	−5.00000	−0.238637	−0.119318	−	0.992856i	$-0.538071\pi$
$439$	−5.00000	−0.238637	−0.119318	+	0.992856i	$0.538071\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	30.0000	1.42695
$443$	2.00000	0.0950229	0.0475114	−	0.998871i	$-0.484871\pi$
$443$	2.00000	0.0950229	0.0475114	+	0.998871i	$0.484871\pi$
$444$	−2.00000	−0.0949158
$445$	18.0000	0.853282
$446$	−16.0000	−0.757622
$447$	18.0000	0.851371
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−4.00000	−0.188562
$451$	−5.00000	−0.235441
$452$	10.0000	0.470360
$453$	9.00000	0.422857
$454$	−3.00000	−0.140797
$455$	0	0
$456$	−6.00000	−0.280976
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	−26.0000	−1.21490
$459$	−5.00000	−0.233380
$460$	15.0000	0.699379
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	28.0000	1.30127	0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000	1.30127	0.650635	+	0.759390i	$0.274503\pi$
$464$	−6.00000	−0.278543
$465$	12.0000	0.556487
$466$	29.0000	1.34340
$467$	16.0000	0.740392	0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000	0.740392	0.370196	+	0.928954i	$0.379291\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	−27.0000	−1.24542
$471$	−24.0000	−1.10586
$472$	12.0000	0.552345
$473$	10.0000	0.459800
$474$	1.00000	0.0459315
$475$	24.0000	1.10120
$476$	0	0
$477$	2.00000	0.0915737
$478$	−26.0000	−1.18921
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	−3.00000	−0.136931
$481$	−12.0000	−0.547153
$482$	26.0000	1.18427
$483$	0	0
$484$	1.00000	0.0454545
$485$	27.0000	1.22601
$486$	−1.00000	−0.0453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	5.00000	0.226339
$489$	−1.00000	−0.0452216
$490$	0	0
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	5.00000	0.225417
$493$	30.0000	1.35113
$494$	−36.0000	−1.61972
$495$	−3.00000	−0.134840
$496$	4.00000	0.179605
$497$	0	0
$498$	−1.00000	−0.0448111
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	−3.00000	−0.134164
$501$	−24.0000	−1.07224
$502$	−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	$0.885813\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	5.00000	0.222277
$507$	23.0000	1.02147
$508$	5.00000	0.221839
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	15.0000	0.664211
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	6.00000	0.264906
$514$	14.0000	0.617514
$515$	−6.00000	−0.264392
$516$	−10.0000	−0.440225
$517$	−9.00000	−0.395820
$518$	0	0
$519$	−12.0000	−0.526742
$520$	−18.0000	−0.789352
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	6.00000	0.262613
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−20.0000	−0.871214
$528$	−1.00000	−0.0435194
$529$	2.00000	0.0869565
$530$	−6.00000	−0.260623
$531$	−12.0000	−0.520756
$532$	0	0
$533$	30.0000	1.29944
$534$	−6.00000	−0.259645
$535$	−39.0000	−1.68612
$536$	−5.00000	−0.215967
$537$	−8.00000	−0.345225
$538$	9.00000	0.388018
$539$	0	0
$540$	3.00000	0.129099
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	24.0000	1.03089
$543$	−22.0000	−0.944110
$544$	5.00000	0.214373
$545$	−33.0000	−1.41356
$546$	0	0
$547$	−42.0000	−1.79579	−0.897895	−	0.440209i	$-0.854904\pi$
$547$	−42.0000	−1.79579	−0.897895	+	0.440209i	$0.854904\pi$
$548$	−2.00000	−0.0854358
$549$	−5.00000	−0.213395
$550$	4.00000	0.170561
$551$	−36.0000	−1.53365
$552$	−5.00000	−0.212814
$553$	0	0
$554$	−18.0000	−0.764747
$555$	−6.00000	−0.254686
$556$	22.0000	0.933008
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	−4.00000	−0.169334
$559$	−60.0000	−2.53773
$560$	0	0
$561$	5.00000	0.211100
$562$	−11.0000	−0.464007
$563$	−16.0000	−0.674320	−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000	−0.674320	−0.337160	+	0.941447i	$0.609466\pi$
$564$	9.00000	0.378968
$565$	30.0000	1.26211
$566$	6.00000	0.252199
$567$	0	0
$568$	−4.00000	−0.167836
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−18.0000	−0.753937
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	0	0
$575$	20.0000	0.834058
$576$	1.00000	0.0416667
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	−8.00000	−0.332756
$579$	6.00000	0.249351
$580$	−18.0000	−0.747409
$581$	0	0
$582$	−9.00000	−0.373062
$583$	−2.00000	−0.0828315
$584$	−12.0000	−0.496564
$585$	18.0000	0.744208
$586$	−8.00000	−0.330477
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	36.0000	1.48210
$591$	−24.0000	−0.987228
$592$	−2.00000	−0.0821995
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	18.0000	0.737309
$597$	10.0000	0.409273
$598$	−30.0000	−1.22679
$599$	−3.00000	−0.122577	−0.0612883	−	0.998120i	$-0.519521\pi$
$599$	−3.00000	−0.122577	−0.0612883	+	0.998120i	$0.519521\pi$
$600$	−4.00000	−0.163299
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	0	0
$603$	5.00000	0.203616
$604$	9.00000	0.366205
$605$	3.00000	0.121967
$606$	8.00000	0.324978
$607$	33.0000	1.33943	0.669714	−	0.742619i	$-0.266417\pi$
$607$	33.0000	1.33943	0.669714	+	0.742619i	$0.266417\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	15.0000	0.607332
$611$	54.0000	2.18461
$612$	−5.00000	−0.202113
$613$	−41.0000	−1.65597	−0.827987	−	0.560747i	$-0.810514\pi$
$613$	−41.0000	−1.65597	−0.827987	+	0.560747i	$0.810514\pi$
$614$	−18.0000	−0.726421
$615$	15.0000	0.604858
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	2.00000	0.0804518
$619$	−29.0000	−1.16561	−0.582804	−	0.812613i	$-0.698045\pi$
$619$	−29.0000	−1.16561	−0.582804	+	0.812613i	$0.698045\pi$
$620$	12.0000	0.481932
$621$	5.00000	0.200643
$622$	21.0000	0.842023
$623$	0	0
$624$	6.00000	0.240192
$625$	−29.0000	−1.16000
$626$	6.00000	0.239808
$627$	−6.00000	−0.239617
$628$	−24.0000	−0.957704
$629$	10.0000	0.398726
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	1.00000	0.0397779
$633$	14.0000	0.556450
$634$	−9.00000	−0.357436
$635$	15.0000	0.595257
$636$	2.00000	0.0793052
$637$	0	0
$638$	−6.00000	−0.237542
$639$	4.00000	0.158238
$640$	−3.00000	−0.118585
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	13.0000	0.513069
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	−30.0000	−1.18125
$646$	30.0000	1.18033
$647$	17.0000	0.668339	0.334169	−	0.942513i	$-0.391544\pi$
$647$	17.0000	0.668339	0.334169	+	0.942513i	$0.391544\pi$
$648$	−1.00000	−0.0392837
$649$	12.0000	0.471041
$650$	−24.0000	−0.941357
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	11.0000	0.430134
$655$	−12.0000	−0.468879
$656$	5.00000	0.195217
$657$	12.0000	0.468165
$658$	0	0
$659$	31.0000	1.20759	0.603794	−	0.797140i	$-0.293655\pi$
$659$	31.0000	1.20759	0.603794	+	0.797140i	$0.293655\pi$
$660$	−3.00000	−0.116775
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	−35.0000	−1.36031
$663$	−30.0000	−1.16510
$664$	−1.00000	−0.0388075
$665$	0	0
$666$	2.00000	0.0774984
$667$	−30.0000	−1.16160
$668$	−24.0000	−0.928588
$669$	16.0000	0.618596
$670$	−15.0000	−0.579501
$671$	5.00000	0.193023
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	2.00000	0.0770371
$675$	4.00000	0.153960
$676$	23.0000	0.884615
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−10.0000	−0.384048
$679$	0	0
$680$	15.0000	0.575224
$681$	3.00000	0.114960
$682$	4.00000	0.153168
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	6.00000	0.229416
$685$	−6.00000	−0.229248
$686$	0	0
$687$	26.0000	0.991962
$688$	−10.0000	−0.381246
$689$	12.0000	0.457164
$690$	−15.0000	−0.571040
$691$	−27.0000	−1.02713	−0.513564	−	0.858051i	$-0.671675\pi$
$691$	−27.0000	−1.02713	−0.513564	+	0.858051i	$0.671675\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	−7.00000	−0.265716
$695$	66.0000	2.50352
$696$	6.00000	0.227429
$697$	−25.0000	−0.946943
$698$	19.0000	0.719161
$699$	−29.0000	−1.09688
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	−6.00000	−0.226455
$703$	−12.0000	−0.452589
$704$	−1.00000	−0.0376889
$705$	27.0000	1.01688
$706$	−24.0000	−0.903252
$707$	0	0
$708$	−12.0000	−0.450988
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	−12.0000	−0.450352
$711$	−1.00000	−0.0375029
$712$	−6.00000	−0.224860
$713$	20.0000	0.749006
$714$	0	0
$715$	−18.0000	−0.673162
$716$	−8.00000	−0.298974
$717$	26.0000	0.970988
$718$	10.0000	0.373197
$719$	37.0000	1.37987	0.689934	−	0.723873i	$-0.257640\pi$
$719$	37.0000	1.37987	0.689934	+	0.723873i	$0.257640\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−17.0000	−0.632674
$723$	−26.0000	−0.966950
$724$	−22.0000	−0.817624
$725$	−24.0000	−0.891338
$726$	−1.00000	−0.0371135
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−36.0000	−1.33242
$731$	50.0000	1.84932
$732$	−5.00000	−0.184805
$733$	−15.0000	−0.554038	−0.277019	−	0.960864i	$-0.589346\pi$
$733$	−15.0000	−0.554038	−0.277019	+	0.960864i	$0.589346\pi$
$734$	14.0000	0.516749
$735$	0	0
$736$	−5.00000	−0.184302
$737$	−5.00000	−0.184177
$738$	−5.00000	−0.184053
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	−6.00000	−0.220564
$741$	36.0000	1.32249
$742$	0	0
$743$	22.0000	0.807102	0.403551	−	0.914957i	$-0.367776\pi$
$743$	22.0000	0.807102	0.403551	+	0.914957i	$0.367776\pi$
$744$	−4.00000	−0.146647
$745$	54.0000	1.97841
$746$	7.00000	0.256288
$747$	1.00000	0.0365881
$748$	5.00000	0.182818
$749$	0	0
$750$	3.00000	0.109545
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	9.00000	0.328196
$753$	12.0000	0.437304
$754$	36.0000	1.31104
$755$	27.0000	0.982631
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−13.0000	−0.472181
$759$	−5.00000	−0.181489
$760$	−18.0000	−0.652929
$761$	47.0000	1.70375	0.851874	−	0.523746i	$-0.175466\pi$
$761$	47.0000	1.70375	0.851874	+	0.523746i	$0.175466\pi$
$762$	−5.00000	−0.181131
$763$	0	0
$764$	0	0
$765$	−15.0000	−0.542326
$766$	−8.00000	−0.289052
$767$	−72.0000	−2.59977
$768$	1.00000	0.0360844
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	6.00000	0.215945
$773$	11.0000	0.395643	0.197821	−	0.980238i	$-0.436613\pi$
$773$	11.0000	0.395643	0.197821	+	0.980238i	$0.436613\pi$
$774$	10.0000	0.359443
$775$	16.0000	0.574737
$776$	−9.00000	−0.323081
$777$	0	0
$778$	21.0000	0.752886
$779$	30.0000	1.07486
$780$	18.0000	0.644503
$781$	−4.00000	−0.143131
$782$	25.0000	0.893998
$783$	−6.00000	−0.214423
$784$	0	0
$785$	−72.0000	−2.56979
$786$	4.00000	0.142675
$787$	−50.0000	−1.78231	−0.891154	−	0.453701i	$-0.850103\pi$
$787$	−50.0000	−1.78231	−0.891154	+	0.453701i	$0.850103\pi$
$788$	−24.0000	−0.854965
$789$	24.0000	0.854423
$790$	3.00000	0.106735
$791$	0	0
$792$	1.00000	0.0355335
$793$	−30.0000	−1.06533
$794$	−26.0000	−0.922705
$795$	6.00000	0.212798
$796$	10.0000	0.354441
$797$	−53.0000	−1.87736	−0.938678	−	0.344795i	$-0.887949\pi$
$797$	−53.0000	−1.87736	−0.938678	+	0.344795i	$0.887949\pi$
$798$	0	0
$799$	−45.0000	−1.59199
$800$	−4.00000	−0.141421
$801$	6.00000	0.212000
$802$	6.00000	0.211867
$803$	−12.0000	−0.423471
$804$	5.00000	0.176336
$805$	0	0
$806$	−24.0000	−0.845364
$807$	−9.00000	−0.316815
$808$	8.00000	0.281439
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	−3.00000	−0.105409
$811$	42.0000	1.47482	0.737410	−	0.675446i	$-0.236049\pi$
$811$	42.0000	1.47482	0.737410	+	0.675446i	$0.236049\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	−2.00000	−0.0701000
$815$	−3.00000	−0.105085
$816$	−5.00000	−0.175035
$817$	−60.0000	−2.09913
$818$	−10.0000	−0.349642
$819$	0	0
$820$	15.0000	0.523823
$821$	−8.00000	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$821$	−8.00000	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$822$	2.00000	0.0697580
$823$	10.0000	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$823$	10.0000	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$824$	2.00000	0.0696733
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	5.00000	0.173762
$829$	−54.0000	−1.87550	−0.937749	−	0.347314i	$-0.887094\pi$
$829$	−54.0000	−1.87550	−0.937749	+	0.347314i	$0.887094\pi$
$830$	−3.00000	−0.104132
$831$	18.0000	0.624413
$832$	6.00000	0.208013
$833$	0	0
$834$	−22.0000	−0.761798
$835$	−72.0000	−2.49166
$836$	−6.00000	−0.207514
$837$	4.00000	0.138260
$838$	20.0000	0.690889
$839$	−9.00000	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$839$	−9.00000	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−8.00000	−0.275698
$843$	11.0000	0.378860
$844$	14.0000	0.481900
$845$	69.0000	2.37367
$846$	−9.00000	−0.309426
$847$	0	0
$848$	2.00000	0.0686803
$849$	−6.00000	−0.205919
$850$	20.0000	0.685994
$851$	−10.0000	−0.342796
$852$	4.00000	0.137038
$853$	15.0000	0.513590	0.256795	−	0.966466i	$-0.417333\pi$
$853$	15.0000	0.513590	0.256795	+	0.966466i	$0.417333\pi$
$854$	0	0
$855$	18.0000	0.615587
$856$	13.0000	0.444331
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	6.00000	0.204837
$859$	−45.0000	−1.53538	−0.767690	−	0.640821i	$-0.778594\pi$
$859$	−45.0000	−1.53538	−0.767690	+	0.640821i	$0.778594\pi$
$860$	−30.0000	−1.02299
$861$	0	0
$862$	−22.0000	−0.749323
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	−1.00000	−0.0340207
$865$	−36.0000	−1.22404
$866$	−11.0000	−0.373795
$867$	8.00000	0.271694
$868$	0	0
$869$	1.00000	0.0339227
$870$	18.0000	0.610257
$871$	30.0000	1.01651
$872$	11.0000	0.372507
$873$	9.00000	0.304604
$874$	−30.0000	−1.01477
$875$	0	0
$876$	12.0000	0.405442
$877$	−3.00000	−0.101303	−0.0506514	−	0.998716i	$-0.516130\pi$
$877$	−3.00000	−0.101303	−0.0506514	+	0.998716i	$0.516130\pi$
$878$	5.00000	0.168742
$879$	8.00000	0.269833
$880$	−3.00000	−0.101130
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	−13.0000	−0.437485	−0.218742	−	0.975783i	$-0.570195\pi$
$883$	−13.0000	−0.437485	−0.218742	+	0.975783i	$0.570195\pi$
$884$	−30.0000	−1.00901
$885$	−36.0000	−1.21013
$886$	−2.00000	−0.0671913
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	−18.0000	−0.603361
$891$	−1.00000	−0.0335013
$892$	16.0000	0.535720
$893$	54.0000	1.80704
$894$	−18.0000	−0.602010
$895$	−24.0000	−0.802232
$896$	0	0
$897$	30.0000	1.00167
$898$	−2.00000	−0.0667409
$899$	−24.0000	−0.800445
$900$	4.00000	0.133333
$901$	−10.0000	−0.333148
$902$	5.00000	0.166482
$903$	0	0
$904$	−10.0000	−0.332595
$905$	−66.0000	−2.19391
$906$	−9.00000	−0.299005
$907$	23.0000	0.763702	0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000	0.763702	0.381851	+	0.924224i	$0.375287\pi$
$908$	3.00000	0.0995585
$909$	−8.00000	−0.265343
$910$	0	0
$911$	51.0000	1.68971	0.844853	−	0.534999i	$-0.179688\pi$
$911$	51.0000	1.68971	0.844853	+	0.534999i	$0.179688\pi$
$912$	6.00000	0.198680
$913$	−1.00000	−0.0330952
$914$	26.0000	0.860004
$915$	−15.0000	−0.495885
$916$	26.0000	0.859064
$917$	0	0
$918$	5.00000	0.165025
$919$	−17.0000	−0.560778	−0.280389	−	0.959886i	$-0.590464\pi$
$919$	−17.0000	−0.560778	−0.280389	+	0.959886i	$0.590464\pi$
$920$	−15.0000	−0.494535
$921$	18.0000	0.593120
$922$	10.0000	0.329332
$923$	24.0000	0.789970
$924$	0	0
$925$	−8.00000	−0.263038
$926$	−28.0000	−0.920137
$927$	−2.00000	−0.0656886
$928$	6.00000	0.196960
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	−12.0000	−0.393496
$931$	0	0
$932$	−29.0000	−0.949927
$933$	−21.0000	−0.687509
$934$	−16.0000	−0.523536
$935$	15.0000	0.490552
$936$	−6.00000	−0.196116
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	27.0000	0.880643
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	24.0000	0.781962
$943$	25.0000	0.814112
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−10.0000	−0.325128
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	−1.00000	−0.0324785
$949$	72.0000	2.33722
$950$	−24.0000	−0.778663
$951$	9.00000	0.291845
$952$	0	0
$953$	−21.0000	−0.680257	−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000	−0.680257	−0.340128	+	0.940379i	$0.610471\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	26.0000	0.840900
$957$	6.00000	0.193952
$958$	−12.0000	−0.387702
$959$	0	0
$960$	3.00000	0.0968246
$961$	−15.0000	−0.483871
$962$	12.0000	0.386896
$963$	−13.0000	−0.418919
$964$	−26.0000	−0.837404
$965$	18.0000	0.579441
$966$	0	0
$967$	−21.0000	−0.675314	−0.337657	−	0.941269i	$-0.609634\pi$
$967$	−21.0000	−0.675314	−0.337657	+	0.941269i	$0.609634\pi$
$968$	−1.00000	−0.0321412
$969$	−30.0000	−0.963739
$970$	−27.0000	−0.866918
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	4.00000	0.128168
$975$	24.0000	0.768615
$976$	−5.00000	−0.160046
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	1.00000	0.0319765
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−11.0000	−0.351203
$982$	3.00000	0.0957338
$983$	−59.0000	−1.88181	−0.940904	−	0.338674i	$-0.890022\pi$
$983$	−59.0000	−1.88181	−0.940904	+	0.338674i	$0.890022\pi$
$984$	−5.00000	−0.159394
$985$	−72.0000	−2.29411
$986$	−30.0000	−0.955395
$987$	0	0
$988$	36.0000	1.14531
$989$	−50.0000	−1.58991
$990$	3.00000	0.0953463
$991$	22.0000	0.698853	0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000	0.698853	0.349427	+	0.936964i	$0.386376\pi$
$992$	−4.00000	−0.127000
$993$	35.0000	1.11069
$994$	0	0
$995$	30.0000	0.951064
$996$	1.00000	0.0316862
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−12.0000	−0.379853
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.o.1.1		1
3.2	odd	2		9702.2.a.be.1.1		1
7.2	even	3		462.2.i.a.67.1	✓	2
7.4	even	3		462.2.i.a.331.1	yes	2
7.6	odd	2		3234.2.a.a.1.1		1
21.2	odd	6		1386.2.k.j.991.1		2
21.11	odd	6		1386.2.k.j.793.1		2
21.20	even	2		9702.2.a.ce.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.a.67.1	✓	2	7.2	even	3
462.2.i.a.331.1	yes	2	7.4	even	3
1386.2.k.j.793.1		2	21.11	odd	6
1386.2.k.j.991.1		2	21.2	odd	6
3234.2.a.a.1.1		1	7.6	odd	2
3234.2.a.o.1.1		1	1.1	even	1	trivial
9702.2.a.be.1.1		1	3.2	odd	2
9702.2.a.ce.1.1		1	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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