LMFDB - Embedded newform 3234.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

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

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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	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	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	−1.00000	−0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	−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	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	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	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−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	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	3.00000	0.387298
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	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	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	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	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	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	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	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	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	4.00000	0.400000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	5.00000	0.495074
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	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	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	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	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	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	$-0.0929087\pi$
$157$	24.0000	1.91541	0.957704	+	0.287754i	$0.0929087\pi$
$158$	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	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	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	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	−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	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	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	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	−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	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	−10.0000	−0.665190
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	6.00000	0.397360
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	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	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	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	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	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	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	−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	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−3.00000	−0.182574
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	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	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	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	$-0.575078\pi$
$293$	−8.00000	−0.467365	−0.233682	+	0.972313i	$0.575078\pi$
$294$	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	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	0	0
$309$	−2.00000	−0.113776
$310$	−12.0000	−0.681554
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	−6.00000	−0.339683
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−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	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	1.00000	0.0533002
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	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	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	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	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	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	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	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	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−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	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	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	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	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	$-0.461929\pi$
$439$	5.00000	0.238637	0.119318	+	0.992856i	$0.461929\pi$
$440$	−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	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	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	$-0.620709\pi$
$467$	−16.0000	−0.740392	−0.370196	+	0.928954i	$0.620709\pi$
$468$	−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	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	−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	$-0.114187\pi$
$503$	42.0000	1.87269	0.936344	+	0.351085i	$0.114187\pi$
$504$	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	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	−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	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	6.00000	0.262613
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	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	$-0.390534\pi$
$563$	16.0000	0.674320	0.337160	+	0.941447i	$0.390534\pi$
$564$	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	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	−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	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	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	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	−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	$-0.145454\pi$
$601$	44.0000	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$602$	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	$-0.733583\pi$
$607$	−33.0000	−1.33943	−0.669714	+	0.742619i	$0.733583\pi$
$608$	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	$-0.301955\pi$
$619$	29.0000	1.16561	0.582804	+	0.812613i	$0.301955\pi$
$620$	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	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	−30.0000	−1.18125
$646$	30.0000	1.18033
$647$	−17.0000	−0.668339	−0.334169	−	0.942513i	$-0.608456\pi$
$647$	−17.0000	−0.668339	−0.334169	+	0.942513i	$0.608456\pi$
$648$	−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	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	−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	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	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	$-0.328325\pi$
$691$	27.0000	1.02713	0.513564	+	0.858051i	$0.328325\pi$
$692$	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	$-0.742360\pi$
$719$	−37.0000	−1.37987	−0.689934	+	0.723873i	$0.742360\pi$
$720$	−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	$-0.633762\pi$
$727$	−22.0000	−0.815935	−0.407967	+	0.912996i	$0.633762\pi$
$728$	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	$-0.410654\pi$
$733$	15.0000	0.554038	0.277019	+	0.960864i	$0.410654\pi$
$734$	−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	$-0.824534\pi$
$761$	−47.0000	−1.70375	−0.851874	+	0.523746i	$0.824534\pi$
$762$	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	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	6.00000	0.215945
$773$	−11.0000	−0.395643	−0.197821	−	0.980238i	$-0.563387\pi$
$773$	−11.0000	−0.395643	−0.197821	+	0.980238i	$0.563387\pi$
$774$	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	$-0.149897\pi$
$787$	50.0000	1.78231	0.891154	+	0.453701i	$0.149897\pi$
$788$	−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	$-0.112051\pi$
$797$	53.0000	1.87736	0.938678	+	0.344795i	$0.112051\pi$
$798$	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	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	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	$-0.112906\pi$
$829$	54.0000	1.87550	0.937749	+	0.347314i	$0.112906\pi$
$830$	−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	$-0.450347\pi$
$839$	9.00000	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$840$	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	$-0.582667\pi$
$853$	−15.0000	−0.513590	−0.256795	+	0.966466i	$0.582667\pi$
$854$	0	0
$855$	18.0000	0.615587
$856$	13.0000	0.444331
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	6.00000	0.204837
$859$	45.0000	1.53538	0.767690	−	0.640821i	$-0.221406\pi$
$859$	45.0000	1.53538	0.767690	+	0.640821i	$0.221406\pi$
$860$	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	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	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	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−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	$-0.359741\pi$
$929$	26.0000	0.853032	0.426516	+	0.904480i	$0.359741\pi$
$930$	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	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	27.0000	0.880643
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	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	$-0.316319\pi$
$971$	34.0000	1.09111	0.545556	+	0.838074i	$0.316319\pi$
$972$	−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	$-0.109978\pi$
$983$	59.0000	1.88181	0.940904	+	0.338674i	$0.109978\pi$
$984$	−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	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−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.a.1.1		1
3.2	odd	2		9702.2.a.ce.1.1		1
7.3	odd	6		462.2.i.a.331.1	yes	2
7.5	odd	6		462.2.i.a.67.1	✓	2
7.6	odd	2		3234.2.a.o.1.1		1
21.5	even	6		1386.2.k.j.991.1		2
21.17	even	6		1386.2.k.j.793.1		2
21.20	even	2		9702.2.a.be.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.a.67.1	✓	2	7.5	odd	6
462.2.i.a.331.1	yes	2	7.3	odd	6
1386.2.k.j.793.1		2	21.17	even	6
1386.2.k.j.991.1		2	21.5	even	6
3234.2.a.a.1.1		1	1.1	even	1	trivial
3234.2.a.o.1.1		1	7.6	odd	2
9702.2.a.be.1.1		1	21.20	even	2
9702.2.a.ce.1.1		1	3.2	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:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
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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