LMFDB - Embedded newform 3234.2.a.r.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:	yes
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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} -1.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} -4.00000 q^{39} +2.00000 q^{40} -2.00000 q^{41} -12.0000 q^{43} +1.00000 q^{44} +2.00000 q^{45} -6.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} -6.00000 q^{51} +4.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -2.00000 q^{57} -6.00000 q^{58} +8.00000 q^{59} -2.00000 q^{60} +4.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +8.00000 q^{65} -1.00000 q^{66} +6.00000 q^{68} -8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} -4.00000 q^{78} +12.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} +12.0000 q^{85} -12.0000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -12.0000 q^{89} +2.00000 q^{90} -2.00000 q^{93} -6.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +1.00000 q^{99} +O(q^{100})\)

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
$7$	0	0
$7$	0	0
$8$	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
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	4.00000	0.784465
$27$	−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	$0.688081\pi$
$30$	−2.00000	−0.365148
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	−4.00000	−0.640513
$40$	2.00000	0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	1.00000	0.150756
$45$	2.00000	0.298142
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	−6.00000	−0.840168
$52$	4.00000	0.554700
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	−1.00000	−0.136083
$55$	2.00000	0.269680
$56$	0	0
$57$	−2.00000	−0.264906
$58$	−6.00000	−0.787839
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	−2.00000	−0.258199
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	−1.00000	−0.123091
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	2.00000	0.229416
$77$	0	0
$78$	−4.00000	−0.452911
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	−12.0000	−1.29399
$87$	6.00000	0.643268
$88$	1.00000	0.106600
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	0	0
$93$	−2.00000	−0.207390
$94$	−6.00000	−0.618853
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−1.00000	−0.100000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	−6.00000	−0.594089
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	10.0000	0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	2.00000	0.190693
$111$	−2.00000	−0.189832
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−6.00000	−0.557086
$117$	4.00000	0.369800
$118$	8.00000	0.736460
$119$	0	0
$120$	−2.00000	−0.182574
$121$	1.00000	0.0909091
$122$	4.00000	0.362143
$123$	2.00000	0.180334
$124$	2.00000	0.179605
$125$	−12.0000	−1.07331
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	1.00000	0.0883883
$129$	12.0000	1.05654
$130$	8.00000	0.701646
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	0	0
$135$	−2.00000	−0.172133
$136$	6.00000	0.514496
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	−8.00000	−0.671345
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	−12.0000	−0.996546
$146$	6.00000	0.496564
$147$	0	0
$148$	2.00000	0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	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	$0.725663\pi$
$152$	2.00000	0.162221
$153$	6.00000	0.485071
$154$	0	0
$155$	4.00000	0.321288
$156$	−4.00000	−0.320256
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	12.0000	0.954669
$159$	−10.0000	−0.793052
$160$	2.00000	0.158114
$161$	0	0
$162$	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	$0.655730\pi$
$164$	−2.00000	−0.156174
$165$	−2.00000	−0.155700
$166$	6.00000	0.465690
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	12.0000	0.920358
$171$	2.00000	0.152944
$172$	−12.0000	−0.914991
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	1.00000	0.0753778
$177$	−8.00000	−0.601317
$178$	−12.0000	−0.899438
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	2.00000	0.149071
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	4.00000	0.294086
$186$	−2.00000	−0.146647
$187$	6.00000	0.438763
$188$	−6.00000	−0.437595
$189$	0	0
$190$	4.00000	0.290191
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	−8.00000	−0.572892
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	1.00000	0.0710669
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	12.0000	0.844317
$203$	0	0
$204$	−6.00000	−0.420084
$205$	−4.00000	−0.279372
$206$	6.00000	0.418040
$207$	0	0
$208$	4.00000	0.277350
$209$	2.00000	0.138343
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	10.0000	0.686803
$213$	8.00000	0.548151
$214$	−12.0000	−0.820303
$215$	−24.0000	−1.63679
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−6.00000	−0.405442
$220$	2.00000	0.134840
$221$	24.0000	1.61441
$222$	−2.00000	−0.134231
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	14.0000	0.931266
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	−2.00000	−0.132453
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	4.00000	0.261488
$235$	−12.0000	−0.782794
$236$	8.00000	0.520756
$237$	−12.0000	−0.779484
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	−2.00000	−0.129099
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	2.00000	0.127515
$247$	8.00000	0.509028
$248$	2.00000	0.127000
$249$	−6.00000	−0.380235
$250$	−12.0000	−0.758947
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	4.00000	0.250982
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	12.0000	0.747087
$259$	0	0
$260$	8.00000	0.496139
$261$	−6.00000	−0.371391
$262$	2.00000	0.123560
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−1.00000	−0.0615457
$265$	20.0000	1.22859
$266$	0	0
$267$	12.0000	0.734388
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−2.00000	−0.121716
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	10.0000	0.604122
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−10.0000	−0.599760
$279$	2.00000	0.119737
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	6.00000	0.357295
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−8.00000	−0.474713
$285$	−4.00000	−0.236940
$286$	4.00000	0.236525
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	−12.0000	−0.704664
$291$	0	0
$292$	6.00000	0.351123
$293$	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
$295$	16.0000	0.931556
$296$	2.00000	0.116248
$297$	−1.00000	−0.0580259
$298$	10.0000	0.579284
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−12.0000	−0.689382
$304$	2.00000	0.114708
$305$	8.00000	0.458079
$306$	6.00000	0.342997
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	4.00000	0.227185
$311$	−34.0000	−1.92796	−0.963982	−	0.265969i	$-0.914308\pi$
$311$	−34.0000	−1.92796	−0.963982	+	0.265969i	$0.914308\pi$
$312$	−4.00000	−0.226455
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	12.0000	0.675053
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−10.0000	−0.560772
$319$	−6.00000	−0.335936
$320$	2.00000	0.111803
$321$	12.0000	0.669775
$322$	0	0
$323$	12.0000	0.667698
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	−12.0000	−0.664619
$327$	10.0000	0.553001
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−2.00000	−0.110096
$331$	−24.0000	−1.31916	−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000	−1.31916	−0.659580	+	0.751635i	$0.729266\pi$
$332$	6.00000	0.329293
$333$	2.00000	0.109599
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	3.00000	0.163178
$339$	−14.0000	−0.760376
$340$	12.0000	0.650791
$341$	2.00000	0.108306
$342$	2.00000	0.108148
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−24.0000	−1.29025
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	6.00000	0.321634
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	1.00000	0.0533002
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	−8.00000	−0.425195
$355$	−16.0000	−0.849192
$356$	−12.0000	−0.635999
$357$	0	0
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	2.00000	0.105409
$361$	−15.0000	−0.789474
$362$	18.0000	0.946059
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	12.0000	0.628109
$366$	−4.00000	−0.209083
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	4.00000	0.207950
$371$	0	0
$372$	−2.00000	−0.103695
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	6.00000	0.310253
$375$	12.0000	0.619677
$376$	−6.00000	−0.309426
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	4.00000	0.205196
$381$	−4.00000	−0.204926
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−12.0000	−0.609994
$388$	0	0
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	−8.00000	−0.405096
$391$	0	0
$392$	0	0
$393$	−2.00000	−0.100887
$394$	−18.0000	−0.906827
$395$	24.0000	1.20757
$396$	1.00000	0.0502519
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	12.0000	0.597022
$405$	2.00000	0.0993808
$406$	0	0
$407$	2.00000	0.0991363
$408$	−6.00000	−0.297044
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	−4.00000	−0.197546
$411$	−10.0000	−0.493264
$412$	6.00000	0.295599
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	4.00000	0.196116
$417$	10.0000	0.489702
$418$	2.00000	0.0978232
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	20.0000	0.973585
$423$	−6.00000	−0.291730
$424$	10.0000	0.485643
$425$	−6.00000	−0.291043
$426$	8.00000	0.387601
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−4.00000	−0.193122
$430$	−24.0000	−1.15738
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−1.00000	−0.0481125
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	−10.0000	−0.478913
$437$	0	0
$438$	−6.00000	−0.286691
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	24.0000	1.14156
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−2.00000	−0.0949158
$445$	−24.0000	−1.13771
$446$	−14.0000	−0.662919
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−1.00000	−0.0471405
$451$	−2.00000	−0.0941763
$452$	14.0000	0.658505
$453$	16.0000	0.751746
$454$	−6.00000	−0.281594
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	−6.00000	−0.280362
$459$	−6.00000	−0.280056
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−6.00000	−0.278543
$465$	−4.00000	−0.185496
$466$	26.0000	1.20443
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	−12.0000	−0.553519
$471$	−14.0000	−0.645086
$472$	8.00000	0.368230
$473$	−12.0000	−0.551761
$474$	−12.0000	−0.551178
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	10.0000	0.457869
$478$	−16.0000	−0.731823
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−2.00000	−0.0912871
$481$	8.00000	0.364769
$482$	22.0000	1.00207
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	4.00000	0.181071
$489$	12.0000	0.542659
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	2.00000	0.0901670
$493$	−36.0000	−1.62136
$494$	8.00000	0.359937
$495$	2.00000	0.0898933
$496$	2.00000	0.0898027
$497$	0	0
$498$	−6.00000	−0.268866
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	−12.0000	−0.536656
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	−3.00000	−0.133235
$508$	4.00000	0.177471
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	−12.0000	−0.531369
$511$	0	0
$512$	1.00000	0.0441942
$513$	−2.00000	−0.0883022
$514$	20.0000	0.882162
$515$	12.0000	0.528783
$516$	12.0000	0.528271
$517$	−6.00000	−0.263880
$518$	0	0
$519$	24.0000	1.05348
$520$	8.00000	0.350823
$521$	28.0000	1.22670	0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000	1.22670	0.613351	+	0.789810i	$0.289821\pi$
$522$	−6.00000	−0.262613
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	2.00000	0.0873704
$525$	0	0
$526$	4.00000	0.174408
$527$	12.0000	0.522728
$528$	−1.00000	−0.0435194
$529$	−23.0000	−1.00000
$530$	20.0000	0.868744
$531$	8.00000	0.347170
$532$	0	0
$533$	−8.00000	−0.346518
$534$	12.0000	0.519291
$535$	−24.0000	−1.03761
$536$	0	0
$537$	0	0
$538$	−6.00000	−0.258678
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−28.0000	−1.20270
$543$	−18.0000	−0.772454
$544$	6.00000	0.257248
$545$	−20.0000	−0.856706
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	10.0000	0.427179
$549$	4.00000	0.170716
$550$	−1.00000	−0.0426401
$551$	−12.0000	−0.511217
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	−4.00000	−0.169791
$556$	−10.0000	−0.424094
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	2.00000	0.0846668
$559$	−48.0000	−2.03018
$560$	0	0
$561$	−6.00000	−0.253320
$562$	2.00000	0.0843649
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	6.00000	0.252646
$565$	28.0000	1.17797
$566$	14.0000	0.588464
$567$	0	0
$568$	−8.00000	−0.335673
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	−4.00000	−0.167542
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	19.0000	0.790296
$579$	14.0000	0.581820
$580$	−12.0000	−0.498273
$581$	0	0
$582$	0	0
$583$	10.0000	0.414158
$584$	6.00000	0.248282
$585$	8.00000	0.330759
$586$	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
$589$	4.00000	0.164817
$590$	16.0000	0.658710
$591$	18.0000	0.740421
$592$	2.00000	0.0821995
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	10.0000	0.409616
$597$	−14.0000	−0.572982
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	1.00000	0.0408248
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	0	0
$604$	−16.0000	−0.651031
$605$	2.00000	0.0813116
$606$	−12.0000	−0.487467
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	8.00000	0.323911
$611$	−24.0000	−0.970936
$612$	6.00000	0.242536
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−26.0000	−1.04927
$615$	4.00000	0.161296
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−6.00000	−0.241355
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	−34.0000	−1.36328
$623$	0	0
$624$	−4.00000	−0.160128
$625$	−19.0000	−0.760000
$626$	−20.0000	−0.799361
$627$	−2.00000	−0.0798723
$628$	14.0000	0.558661
$629$	12.0000	0.478471
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	12.0000	0.477334
$633$	−20.0000	−0.794929
$634$	18.0000	0.714871
$635$	8.00000	0.317470
$636$	−10.0000	−0.396526
$637$	0	0
$638$	−6.00000	−0.237542
$639$	−8.00000	−0.316475
$640$	2.00000	0.0790569
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	12.0000	0.473602
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	12.0000	0.472134
$647$	−10.0000	−0.393141	−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000	−0.393141	−0.196570	+	0.980490i	$0.562980\pi$
$648$	1.00000	0.0392837
$649$	8.00000	0.314027
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	10.0000	0.391031
$655$	4.00000	0.156293
$656$	−2.00000	−0.0780869
$657$	6.00000	0.234082
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	−2.00000	−0.0778499
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	−24.0000	−0.932786
$663$	−24.0000	−0.932083
$664$	6.00000	0.232845
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	12.0000	0.464294
$669$	14.0000	0.541271
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	−26.0000	−1.00148
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−14.0000	−0.537667
$679$	0	0
$680$	12.0000	0.460179
$681$	6.00000	0.229920
$682$	2.00000	0.0765840
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	2.00000	0.0764719
$685$	20.0000	0.764161
$686$	0	0
$687$	6.00000	0.228914
$688$	−12.0000	−0.457496
$689$	40.0000	1.52388
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	−24.0000	−0.912343
$693$	0	0
$694$	20.0000	0.759190
$695$	−20.0000	−0.758643
$696$	6.00000	0.227429
$697$	−12.0000	−0.454532
$698$	−16.0000	−0.605609
$699$	−26.0000	−0.983410
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	−4.00000	−0.150970
$703$	4.00000	0.150863
$704$	1.00000	0.0376889
$705$	12.0000	0.451946
$706$	−12.0000	−0.451626
$707$	0	0
$708$	−8.00000	−0.300658
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	−16.0000	−0.600469
$711$	12.0000	0.450035
$712$	−12.0000	−0.449719
$713$	0	0
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	16.0000	0.597531
$718$	20.0000	0.746393
$719$	46.0000	1.71551	0.857755	−	0.514058i	$-0.171858\pi$
$719$	46.0000	1.71551	0.857755	+	0.514058i	$0.171858\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	−15.0000	−0.558242
$723$	−22.0000	−0.818189
$724$	18.0000	0.668965
$725$	6.00000	0.222834
$726$	−1.00000	−0.0371135
$727$	−30.0000	−1.11264	−0.556319	−	0.830969i	$-0.687787\pi$
$727$	−30.0000	−1.11264	−0.556319	+	0.830969i	$0.687787\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	−72.0000	−2.66302
$732$	−4.00000	−0.147844
$733$	12.0000	0.443230	0.221615	−	0.975134i	$-0.428867\pi$
$733$	12.0000	0.443230	0.221615	+	0.975134i	$0.428867\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	0	0
$737$	0	0
$738$	−2.00000	−0.0736210
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	4.00000	0.147043
$741$	−8.00000	−0.293887
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−2.00000	−0.0733236
$745$	20.0000	0.732743
$746$	6.00000	0.219676
$747$	6.00000	0.219529
$748$	6.00000	0.219382
$749$	0	0
$750$	12.0000	0.438178
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−24.0000	−0.874028
$755$	−32.0000	−1.16460
$756$	0	0
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	4.00000	0.145095
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	−4.00000	−0.144905
$763$	0	0
$764$	0	0
$765$	12.0000	0.433861
$766$	6.00000	0.216789
$767$	32.0000	1.15545
$768$	−1.00000	−0.0360844
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−20.0000	−0.720282
$772$	−14.0000	−0.503871
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	−12.0000	−0.431331
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	22.0000	0.788738
$779$	−4.00000	−0.143315
$780$	−8.00000	−0.286446
$781$	−8.00000	−0.286263
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	28.0000	0.999363
$786$	−2.00000	−0.0713376
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	−18.0000	−0.641223
$789$	−4.00000	−0.142404
$790$	24.0000	0.853882
$791$	0	0
$792$	1.00000	0.0355335
$793$	16.0000	0.568177
$794$	−34.0000	−1.20661
$795$	−20.0000	−0.709327
$796$	14.0000	0.496217
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	−1.00000	−0.0353553
$801$	−12.0000	−0.423999
$802$	−30.0000	−1.05934
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	6.00000	0.211210
$808$	12.0000	0.422159
$809$	14.0000	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$810$	2.00000	0.0702728
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	0	0
$813$	28.0000	0.982003
$814$	2.00000	0.0701000
$815$	−24.0000	−0.840683
$816$	−6.00000	−0.210042
$817$	−24.0000	−0.839654
$818$	2.00000	0.0699284
$819$	0	0
$820$	−4.00000	−0.139686
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	−10.0000	−0.348790
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	6.00000	0.209020
$825$	1.00000	0.0348155
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	12.0000	0.416526
$831$	−10.0000	−0.346896
$832$	4.00000	0.138675
$833$	0	0
$834$	10.0000	0.346272
$835$	24.0000	0.830554
$836$	2.00000	0.0691714
$837$	−2.00000	−0.0691301
$838$	12.0000	0.414533
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	14.0000	0.482472
$843$	−2.00000	−0.0688837
$844$	20.0000	0.688428
$845$	6.00000	0.206406
$846$	−6.00000	−0.206284
$847$	0	0
$848$	10.0000	0.343401
$849$	−14.0000	−0.480479
$850$	−6.00000	−0.205798
$851$	0	0
$852$	8.00000	0.274075
$853$	36.0000	1.23262	0.616308	−	0.787505i	$-0.288628\pi$
$853$	36.0000	1.23262	0.616308	+	0.787505i	$0.288628\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	−12.0000	−0.410152
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	−4.00000	−0.136558
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	−24.0000	−0.818393
$861$	0	0
$862$	12.0000	0.408722
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	−1.00000	−0.0340207
$865$	−48.0000	−1.63205
$866$	8.00000	0.271851
$867$	−19.0000	−0.645274
$868$	0	0
$869$	12.0000	0.407072
$870$	12.0000	0.406838
$871$	0	0
$872$	−10.0000	−0.338643
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−6.00000	−0.202721
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	28.0000	0.944954
$879$	−8.00000	−0.269833
$880$	2.00000	0.0674200
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	24.0000	0.807207
$885$	−16.0000	−0.537834
$886$	24.0000	0.806296
$887$	40.0000	1.34307	0.671534	−	0.740973i	$-0.265636\pi$
$887$	40.0000	1.34307	0.671534	+	0.740973i	$0.265636\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	−24.0000	−0.804482
$891$	1.00000	0.0335013
$892$	−14.0000	−0.468755
$893$	−12.0000	−0.401565
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−12.0000	−0.400222
$900$	−1.00000	−0.0333333
$901$	60.0000	1.99889
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	14.0000	0.465633
$905$	36.0000	1.19668
$906$	16.0000	0.531564
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	−6.00000	−0.199117
$909$	12.0000	0.398015
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	−2.00000	−0.0662266
$913$	6.00000	0.198571
$914$	−6.00000	−0.198462
$915$	−8.00000	−0.264472
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−6.00000	−0.198030
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	26.0000	0.856729
$922$	12.0000	0.395199
$923$	−32.0000	−1.05329
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	16.0000	0.525793
$927$	6.00000	0.197066
$928$	−6.00000	−0.196960
$929$	−44.0000	−1.44359	−0.721797	−	0.692105i	$-0.756683\pi$
$929$	−44.0000	−1.44359	−0.721797	+	0.692105i	$0.756683\pi$
$930$	−4.00000	−0.131165
$931$	0	0
$932$	26.0000	0.851658
$933$	34.0000	1.11311
$934$	28.0000	0.916188
$935$	12.0000	0.392442
$936$	4.00000	0.130744
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	−12.0000	−0.391397
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	−14.0000	−0.456145
$943$	0	0
$944$	8.00000	0.260378
$945$	0	0
$946$	−12.0000	−0.390154
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−12.0000	−0.389742
$949$	24.0000	0.779073
$950$	−2.00000	−0.0648886
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	−16.0000	−0.517477
$957$	6.00000	0.193952
$958$	0	0
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	−27.0000	−0.870968
$962$	8.00000	0.257930
$963$	−12.0000	−0.386695
$964$	22.0000	0.708572
$965$	−28.0000	−0.901352
$966$	0	0
$967$	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
$969$	−12.0000	−0.385496
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	4.00000	0.128103
$976$	4.00000	0.128037
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	12.0000	0.383718
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−20.0000	−0.638226
$983$	−38.0000	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$983$	−38.0000	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$984$	2.00000	0.0637577
$985$	−36.0000	−1.14706
$986$	−36.0000	−1.14647
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	2.00000	0.0635642
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	2.00000	0.0635001
$993$	24.0000	0.761617
$994$	0	0
$995$	28.0000	0.887660
$996$	−6.00000	−0.190117
$997$	−8.00000	−0.253363	−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000	−0.253363	−0.126681	+	0.991943i	$0.540433\pi$
$998$	−40.0000	−1.26618
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.r.1.1	✓	1
3.2	odd	2		9702.2.a.g.1.1		1
7.6	odd	2		3234.2.a.u.1.1	yes	1
21.20	even	2		9702.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.r.1.1	✓	1	1.1	even	1	trivial
3234.2.a.u.1.1	yes	1	7.6	odd	2
9702.2.a.g.1.1		1	3.2	odd	2
9702.2.a.q.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 $\F_{q}$

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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