LMFDB - Embedded newform 6034.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6034,2,Mod(1,6034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6034, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6034 = 2 \cdot 7 \cdot 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6034.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:	$48.1817325796$
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	$\chi$	$=$	6034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +3.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +2.00000 q^{10} +3.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} +6.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{18} -7.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} +6.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} +9.00000 q^{27} -1.00000 q^{28} +5.00000 q^{29} +6.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -2.00000 q^{35} +6.00000 q^{36} +8.00000 q^{37} -7.00000 q^{38} +3.00000 q^{39} +2.00000 q^{40} -3.00000 q^{42} -9.00000 q^{43} +12.0000 q^{45} +6.00000 q^{46} +6.00000 q^{47} +3.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +1.00000 q^{52} +10.0000 q^{53} +9.00000 q^{54} -1.00000 q^{56} -21.0000 q^{57} +5.00000 q^{58} +4.00000 q^{59} +6.00000 q^{60} +8.00000 q^{61} +2.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +1.00000 q^{67} -2.00000 q^{68} +18.0000 q^{69} -2.00000 q^{70} -15.0000 q^{71} +6.00000 q^{72} +9.00000 q^{73} +8.00000 q^{74} -3.00000 q^{75} -7.00000 q^{76} +3.00000 q^{78} +1.00000 q^{79} +2.00000 q^{80} +9.00000 q^{81} +8.00000 q^{83} -3.00000 q^{84} -4.00000 q^{85} -9.00000 q^{86} +15.0000 q^{87} -10.0000 q^{89} +12.0000 q^{90} -1.00000 q^{91} +6.00000 q^{92} +6.00000 q^{93} +6.00000 q^{94} -14.0000 q^{95} +3.00000 q^{96} +16.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	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	$0.352416\pi$
$6$	3.00000	1.22474
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	3.00000	0.866025
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.00000	−0.267261
$15$	6.00000	1.54919
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	6.00000	1.41421
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	2.00000	0.447214
$21$	−3.00000	−0.654654
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	3.00000	0.612372
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	9.00000	1.73205
$28$	−1.00000	−0.188982
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	6.00000	1.09545
$31$	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
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−2.00000	−0.338062
$36$	6.00000	1.00000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−7.00000	−1.13555
$39$	3.00000	0.480384
$40$	2.00000	0.316228
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−3.00000	−0.462910
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	12.0000	1.78885
$46$	6.00000	0.884652
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	3.00000	0.433013
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−6.00000	−0.840168
$52$	1.00000	0.138675
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	9.00000	1.22474
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−21.0000	−2.78152
$58$	5.00000	0.656532
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	6.00000	0.774597
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	2.00000	0.254000
$63$	−6.00000	−0.755929
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	−2.00000	−0.242536
$69$	18.0000	2.16695
$70$	−2.00000	−0.239046
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	6.00000	0.707107
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	8.00000	0.929981
$75$	−3.00000	−0.346410
$76$	−7.00000	−0.802955
$77$	0	0
$78$	3.00000	0.339683
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	2.00000	0.223607
$81$	9.00000	1.00000
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−3.00000	−0.327327
$85$	−4.00000	−0.433861
$86$	−9.00000	−0.970495
$87$	15.0000	1.60817
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	12.0000	1.26491
$91$	−1.00000	−0.104828
$92$	6.00000	0.625543
$93$	6.00000	0.622171
$94$	6.00000	0.618853
$95$	−14.0000	−1.43637
$96$	3.00000	0.306186
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	−6.00000	−0.594089
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	1.00000	0.0980581
$105$	−6.00000	−0.585540
$106$	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	$0.696964\pi$
$108$	9.00000	0.866025
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	24.0000	2.27798
$112$	−1.00000	−0.0944911
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−21.0000	−1.96683
$115$	12.0000	1.11901
$116$	5.00000	0.464238
$117$	6.00000	0.554700
$118$	4.00000	0.368230
$119$	2.00000	0.183340
$120$	6.00000	0.547723
$121$	−11.0000	−1.00000
$122$	8.00000	0.724286
$123$	0	0
$124$	2.00000	0.179605
$125$	−12.0000	−1.07331
$126$	−6.00000	−0.534522
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	−27.0000	−2.37722
$130$	2.00000	0.175412
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	7.00000	0.606977
$134$	1.00000	0.0863868
$135$	18.0000	1.54919
$136$	−2.00000	−0.171499
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	18.0000	1.53226
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−2.00000	−0.169031
$141$	18.0000	1.51587
$142$	−15.0000	−1.25877
$143$	0	0
$144$	6.00000	0.500000
$145$	10.0000	0.830455
$146$	9.00000	0.744845
$147$	3.00000	0.247436
$148$	8.00000	0.657596
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	−3.00000	−0.244949
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−7.00000	−0.567775
$153$	−12.0000	−0.970143
$154$	0	0
$155$	4.00000	0.321288
$156$	3.00000	0.240192
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	1.00000	0.0795557
$159$	30.0000	2.37915
$160$	2.00000	0.158114
$161$	−6.00000	−0.472866
$162$	9.00000	0.707107
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	0	0
$166$	8.00000	0.620920
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	−3.00000	−0.231455
$169$	−12.0000	−0.923077
$170$	−4.00000	−0.306786
$171$	−42.0000	−3.21182
$172$	−9.00000	−0.686244
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	15.0000	1.13715
$175$	1.00000	0.0755929
$176$	0	0
$177$	12.0000	0.901975
$178$	−10.0000	−0.749532
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	12.0000	0.894427
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	−1.00000	−0.0741249
$183$	24.0000	1.77413
$184$	6.00000	0.442326
$185$	16.0000	1.17634
$186$	6.00000	0.439941
$187$	0	0
$188$	6.00000	0.437595
$189$	−9.00000	−0.654654
$190$	−14.0000	−1.01567
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	3.00000	0.216506
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	16.0000	1.14873
$195$	6.00000	0.429669
$196$	1.00000	0.0714286
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	−1.00000	−0.0707107
$201$	3.00000	0.211604
$202$	−15.0000	−1.05540
$203$	−5.00000	−0.350931
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−4.00000	−0.278693
$207$	36.0000	2.50217
$208$	1.00000	0.0693375
$209$	0	0
$210$	−6.00000	−0.414039
$211$	−29.0000	−1.99644	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−29.0000	−1.99644	−0.998221	+	0.0596196i	$0.981011\pi$
$212$	10.0000	0.686803
$213$	−45.0000	−3.08335
$214$	−12.0000	−0.820303
$215$	−18.0000	−1.22759
$216$	9.00000	0.612372
$217$	−2.00000	−0.135769
$218$	−5.00000	−0.338643
$219$	27.0000	1.82449
$220$	0	0
$221$	−2.00000	−0.134535
$222$	24.0000	1.61077
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	−1.00000	−0.0668153
$225$	−6.00000	−0.400000
$226$	10.0000	0.665190
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	−21.0000	−1.39076
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	5.00000	0.328266
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	6.00000	0.392232
$235$	12.0000	0.782794
$236$	4.00000	0.260378
$237$	3.00000	0.194871
$238$	2.00000	0.129641
$239$	17.0000	1.09964	0.549819	−	0.835284i	$-0.314697\pi$
$239$	17.0000	1.09964	0.549819	+	0.835284i	$0.314697\pi$
$240$	6.00000	0.387298
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	8.00000	0.512148
$245$	2.00000	0.127775
$246$	0	0
$247$	−7.00000	−0.445399
$248$	2.00000	0.127000
$249$	24.0000	1.52094
$250$	−12.0000	−0.758947
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	−6.00000	−0.377964
$253$	0	0
$254$	12.0000	0.752947
$255$	−12.0000	−0.751469
$256$	1.00000	0.0625000
$257$	−17.0000	−1.06043	−0.530215	−	0.847863i	$-0.677889\pi$
$257$	−17.0000	−1.06043	−0.530215	+	0.847863i	$0.677889\pi$
$258$	−27.0000	−1.68095
$259$	−8.00000	−0.497096
$260$	2.00000	0.124035
$261$	30.0000	1.85695
$262$	−10.0000	−0.617802
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	20.0000	1.22859
$266$	7.00000	0.429198
$267$	−30.0000	−1.83597
$268$	1.00000	0.0610847
$269$	−11.0000	−0.670682	−0.335341	−	0.942097i	$-0.608852\pi$
$269$	−11.0000	−0.670682	−0.335341	+	0.942097i	$0.608852\pi$
$270$	18.0000	1.09545
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−2.00000	−0.121268
$273$	−3.00000	−0.181568
$274$	−12.0000	−0.724947
$275$	0	0
$276$	18.0000	1.08347
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	4.00000	0.239904
$279$	12.0000	0.718421
$280$	−2.00000	−0.119523
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	18.0000	1.07188
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−15.0000	−0.890086
$285$	−42.0000	−2.48787
$286$	0	0
$287$	0	0
$288$	6.00000	0.353553
$289$	−13.0000	−0.764706
$290$	10.0000	0.587220
$291$	48.0000	2.81381
$292$	9.00000	0.526685
$293$	23.0000	1.34367	0.671837	−	0.740699i	$-0.265505\pi$
$293$	23.0000	1.34367	0.671837	+	0.740699i	$0.265505\pi$
$294$	3.00000	0.174964
$295$	8.00000	0.465778
$296$	8.00000	0.464991
$297$	0	0
$298$	−15.0000	−0.868927
$299$	6.00000	0.346989
$300$	−3.00000	−0.173205
$301$	9.00000	0.518751
$302$	−10.0000	−0.575435
$303$	−45.0000	−2.58518
$304$	−7.00000	−0.401478
$305$	16.0000	0.916157
$306$	−12.0000	−0.685994
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	4.00000	0.227185
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	3.00000	0.169842
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−4.00000	−0.225733
$315$	−12.0000	−0.676123
$316$	1.00000	0.0562544
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	30.0000	1.68232
$319$	0	0
$320$	2.00000	0.111803
$321$	−36.0000	−2.00932
$322$	−6.00000	−0.334367
$323$	14.0000	0.778981
$324$	9.00000	0.500000
$325$	−1.00000	−0.0554700
$326$	−18.0000	−0.996928
$327$	−15.0000	−0.829502
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	8.00000	0.439057
$333$	48.0000	2.63038
$334$	−24.0000	−1.31322
$335$	2.00000	0.109272
$336$	−3.00000	−0.163663
$337$	19.0000	1.03500	0.517498	−	0.855684i	$-0.326864\pi$
$337$	19.0000	1.03500	0.517498	+	0.855684i	$0.326864\pi$
$338$	−12.0000	−0.652714
$339$	30.0000	1.62938
$340$	−4.00000	−0.216930
$341$	0	0
$342$	−42.0000	−2.27110
$343$	−1.00000	−0.0539949
$344$	−9.00000	−0.485247
$345$	36.0000	1.93817
$346$	0	0
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	15.0000	0.804084
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	1.00000	0.0534522
$351$	9.00000	0.480384
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	12.0000	0.637793
$355$	−30.0000	−1.59223
$356$	−10.0000	−0.529999
$357$	6.00000	0.317554
$358$	−12.0000	−0.634220
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	12.0000	0.632456
$361$	30.0000	1.57895
$362$	1.00000	0.0525588
$363$	−33.0000	−1.73205
$364$	−1.00000	−0.0524142
$365$	18.0000	0.942163
$366$	24.0000	1.25450
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	16.0000	0.831800
$371$	−10.0000	−0.519174
$372$	6.00000	0.311086
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	0	0
$375$	−36.0000	−1.85903
$376$	6.00000	0.309426
$377$	5.00000	0.257513
$378$	−9.00000	−0.462910
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	−14.0000	−0.718185
$381$	36.0000	1.84434
$382$	3.00000	0.153493
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	4.00000	0.203595
$387$	−54.0000	−2.74497
$388$	16.0000	0.812277
$389$	−7.00000	−0.354914	−0.177457	−	0.984129i	$-0.556787\pi$
$389$	−7.00000	−0.354914	−0.177457	+	0.984129i	$0.556787\pi$
$390$	6.00000	0.303822
$391$	−12.0000	−0.606866
$392$	1.00000	0.0505076
$393$	−30.0000	−1.51330
$394$	−9.00000	−0.453413
$395$	2.00000	0.100631
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	−10.0000	−0.501255
$399$	21.0000	1.05131
$400$	−1.00000	−0.0500000
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	3.00000	0.149626
$403$	2.00000	0.0996271
$404$	−15.0000	−0.746278
$405$	18.0000	0.894427
$406$	−5.00000	−0.248146
$407$	0	0
$408$	−6.00000	−0.297044
$409$	−35.0000	−1.73064	−0.865319	−	0.501221i	$-0.832884\pi$
$409$	−35.0000	−1.73064	−0.865319	+	0.501221i	$0.832884\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	−4.00000	−0.197066
$413$	−4.00000	−0.196827
$414$	36.0000	1.76930
$415$	16.0000	0.785409
$416$	1.00000	0.0490290
$417$	12.0000	0.587643
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	−6.00000	−0.292770
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−29.0000	−1.41170
$423$	36.0000	1.75038
$424$	10.0000	0.485643
$425$	2.00000	0.0970143
$426$	−45.0000	−2.18026
$427$	−8.00000	−0.387147
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−18.0000	−0.868037
$431$	−1.00000	−0.0481683
$431$	−1.00000	−0.0481683
$432$	9.00000	0.433013
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−2.00000	−0.0960031
$435$	30.0000	1.43839
$436$	−5.00000	−0.239457
$437$	−42.0000	−2.00913
$438$	27.0000	1.29011
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	−2.00000	−0.0951303
$443$	−38.0000	−1.80543	−0.902717	−	0.430234i	$-0.858431\pi$
$443$	−38.0000	−1.80543	−0.902717	+	0.430234i	$0.858431\pi$
$444$	24.0000	1.13899
$445$	−20.0000	−0.948091
$446$	12.0000	0.568216
$447$	−45.0000	−2.12843
$448$	−1.00000	−0.0472456
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	−6.00000	−0.282843
$451$	0	0
$452$	10.0000	0.470360
$453$	−30.0000	−1.40952
$454$	7.00000	0.328526
$455$	−2.00000	−0.0937614
$456$	−21.0000	−0.983415
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	14.0000	0.654177
$459$	−18.0000	−0.840168
$460$	12.0000	0.559503
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	5.00000	0.232119
$465$	12.0000	0.556487
$466$	−18.0000	−0.833834
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	6.00000	0.277350
$469$	−1.00000	−0.0461757
$470$	12.0000	0.553519
$471$	−12.0000	−0.552931
$472$	4.00000	0.184115
$473$	0	0
$474$	3.00000	0.137795
$475$	7.00000	0.321182
$476$	2.00000	0.0916698
$477$	60.0000	2.74721
$478$	17.0000	0.777562
$479$	−25.0000	−1.14228	−0.571140	−	0.820853i	$-0.693499\pi$
$479$	−25.0000	−1.14228	−0.571140	+	0.820853i	$0.693499\pi$
$480$	6.00000	0.273861
$481$	8.00000	0.364769
$482$	1.00000	0.0455488
$483$	−18.0000	−0.819028
$484$	−11.0000	−0.500000
$485$	32.0000	1.45305
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	8.00000	0.362143
$489$	−54.0000	−2.44196
$490$	2.00000	0.0903508
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	−10.0000	−0.450377
$494$	−7.00000	−0.314945
$495$	0	0
$496$	2.00000	0.0898027
$497$	15.0000	0.672842
$498$	24.0000	1.07547
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−12.0000	−0.536656
$501$	−72.0000	−3.21672
$502$	−20.0000	−0.892644
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	−6.00000	−0.267261
$505$	−30.0000	−1.33498
$506$	0	0
$507$	−36.0000	−1.59882
$508$	12.0000	0.532414
$509$	1.00000	0.0443242	0.0221621	−	0.999754i	$-0.492945\pi$
$509$	1.00000	0.0443242	0.0221621	+	0.999754i	$0.492945\pi$
$510$	−12.0000	−0.531369
$511$	−9.00000	−0.398137
$512$	1.00000	0.0441942
$513$	−63.0000	−2.78152
$514$	−17.0000	−0.749838
$515$	−8.00000	−0.352522
$516$	−27.0000	−1.18861
$517$	0	0
$518$	−8.00000	−0.351500
$519$	0	0
$520$	2.00000	0.0877058
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	30.0000	1.31306
$523$	−1.00000	−0.0437269	−0.0218635	−	0.999761i	$-0.506960\pi$
$523$	−1.00000	−0.0437269	−0.0218635	+	0.999761i	$0.506960\pi$
$524$	−10.0000	−0.436852
$525$	3.00000	0.130931
$526$	6.00000	0.261612
$527$	−4.00000	−0.174243
$528$	0	0
$529$	13.0000	0.565217
$530$	20.0000	0.868744
$531$	24.0000	1.04151
$532$	7.00000	0.303488
$533$	0	0
$534$	−30.0000	−1.29823
$535$	−24.0000	−1.03761
$536$	1.00000	0.0431934
$537$	−36.0000	−1.55351
$538$	−11.0000	−0.474244
$539$	0	0
$540$	18.0000	0.774597
$541$	21.0000	0.902861	0.451430	−	0.892306i	$-0.350914\pi$
$541$	21.0000	0.902861	0.451430	+	0.892306i	$0.350914\pi$
$542$	−20.0000	−0.859074
$543$	3.00000	0.128742
$544$	−2.00000	−0.0857493
$545$	−10.0000	−0.428353
$546$	−3.00000	−0.128388
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−12.0000	−0.512615
$549$	48.0000	2.04859
$550$	0	0
$551$	−35.0000	−1.49105
$552$	18.0000	0.766131
$553$	−1.00000	−0.0425243
$554$	2.00000	0.0849719
$555$	48.0000	2.03749
$556$	4.00000	0.169638
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	12.0000	0.508001
$559$	−9.00000	−0.380659
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	18.0000	0.759284
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	18.0000	0.757937
$565$	20.0000	0.841406
$566$	4.00000	0.168133
$567$	−9.00000	−0.377964
$568$	−15.0000	−0.629386
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	−42.0000	−1.75919
$571$	3.00000	0.125546	0.0627730	−	0.998028i	$-0.480006\pi$
$571$	3.00000	0.125546	0.0627730	+	0.998028i	$0.480006\pi$
$572$	0	0
$573$	9.00000	0.375980
$574$	0	0
$575$	−6.00000	−0.250217
$576$	6.00000	0.250000
$577$	21.0000	0.874241	0.437121	−	0.899403i	$-0.355998\pi$
$577$	21.0000	0.874241	0.437121	+	0.899403i	$0.355998\pi$
$578$	−13.0000	−0.540729
$579$	12.0000	0.498703
$580$	10.0000	0.415227
$581$	−8.00000	−0.331896
$582$	48.0000	1.98966
$583$	0	0
$584$	9.00000	0.372423
$585$	12.0000	0.496139
$586$	23.0000	0.950121
$587$	44.0000	1.81607	0.908037	−	0.418890i	$-0.137581\pi$
$587$	44.0000	1.81607	0.908037	+	0.418890i	$0.137581\pi$
$588$	3.00000	0.123718
$589$	−14.0000	−0.576860
$590$	8.00000	0.329355
$591$	−27.0000	−1.11063
$592$	8.00000	0.328798
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−15.0000	−0.614424
$597$	−30.0000	−1.22782
$598$	6.00000	0.245358
$599$	35.0000	1.43006	0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000	1.43006	0.715031	+	0.699093i	$0.246413\pi$
$600$	−3.00000	−0.122474
$601$	−33.0000	−1.34610	−0.673049	−	0.739598i	$-0.735016\pi$
$601$	−33.0000	−1.34610	−0.673049	+	0.739598i	$0.735016\pi$
$602$	9.00000	0.366813
$603$	6.00000	0.244339
$604$	−10.0000	−0.406894
$605$	−22.0000	−0.894427
$606$	−45.0000	−1.82800
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	−7.00000	−0.283887
$609$	−15.0000	−0.607831
$610$	16.0000	0.647821
$611$	6.00000	0.242734
$612$	−12.0000	−0.485071
$613$	9.00000	0.363507	0.181753	−	0.983344i	$-0.441823\pi$
$613$	9.00000	0.363507	0.181753	+	0.983344i	$0.441823\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	0	0
$617$	−28.0000	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$618$	−12.0000	−0.482711
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	4.00000	0.160644
$621$	54.0000	2.16695
$622$	12.0000	0.481156
$623$	10.0000	0.400642
$624$	3.00000	0.120096
$625$	−19.0000	−0.760000
$626$	6.00000	0.239808
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−16.0000	−0.637962
$630$	−12.0000	−0.478091
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	1.00000	0.0397779
$633$	−87.0000	−3.45794
$634$	4.00000	0.158860
$635$	24.0000	0.952411
$636$	30.0000	1.18958
$637$	1.00000	0.0396214
$638$	0	0
$639$	−90.0000	−3.56034
$640$	2.00000	0.0790569
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−36.0000	−1.42081
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	−6.00000	−0.236433
$645$	−54.0000	−2.12625
$646$	14.0000	0.550823
$647$	33.0000	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$648$	9.00000	0.353553
$649$	0	0
$650$	−1.00000	−0.0392232
$651$	−6.00000	−0.235159
$652$	−18.0000	−0.704934
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−15.0000	−0.586546
$655$	−20.0000	−0.781465
$656$	0	0
$657$	54.0000	2.10674
$658$	−6.00000	−0.233904
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	8.00000	0.311164	0.155582	−	0.987823i	$-0.450275\pi$
$661$	8.00000	0.311164	0.155582	+	0.987823i	$0.450275\pi$
$662$	28.0000	1.08825
$663$	−6.00000	−0.233021
$664$	8.00000	0.310460
$665$	14.0000	0.542897
$666$	48.0000	1.85996
$667$	30.0000	1.16160
$668$	−24.0000	−0.928588
$669$	36.0000	1.39184
$670$	2.00000	0.0772667
$671$	0	0
$672$	−3.00000	−0.115728
$673$	−23.0000	−0.886585	−0.443292	−	0.896377i	$-0.646190\pi$
$673$	−23.0000	−0.886585	−0.443292	+	0.896377i	$0.646190\pi$
$674$	19.0000	0.731853
$675$	−9.00000	−0.346410
$676$	−12.0000	−0.461538
$677$	40.0000	1.53732	0.768662	−	0.639655i	$-0.220923\pi$
$677$	40.0000	1.53732	0.768662	+	0.639655i	$0.220923\pi$
$678$	30.0000	1.15214
$679$	−16.0000	−0.614024
$680$	−4.00000	−0.153393
$681$	21.0000	0.804722
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	−42.0000	−1.60591
$685$	−24.0000	−0.916993
$686$	−1.00000	−0.0381802
$687$	42.0000	1.60240
$688$	−9.00000	−0.343122
$689$	10.0000	0.380970
$690$	36.0000	1.37050
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	0	0
$694$	28.0000	1.06287
$695$	8.00000	0.303457
$696$	15.0000	0.568574
$697$	0	0
$698$	−10.0000	−0.378506
$699$	−54.0000	−2.04247
$700$	1.00000	0.0377964
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	9.00000	0.339683
$703$	−56.0000	−2.11208
$704$	0	0
$705$	36.0000	1.35584
$706$	8.00000	0.301084
$707$	15.0000	0.564133
$708$	12.0000	0.450988
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	−30.0000	−1.12588
$711$	6.00000	0.225018
$712$	−10.0000	−0.374766
$713$	12.0000	0.449404
$714$	6.00000	0.224544
$715$	0	0
$716$	−12.0000	−0.448461
$717$	51.0000	1.90463
$718$	−21.0000	−0.783713
$719$	43.0000	1.60363	0.801815	−	0.597573i	$-0.203868\pi$
$719$	43.0000	1.60363	0.801815	+	0.597573i	$0.203868\pi$
$720$	12.0000	0.447214
$721$	4.00000	0.148968
$722$	30.0000	1.11648
$723$	3.00000	0.111571
$724$	1.00000	0.0371647
$725$	−5.00000	−0.185695
$726$	−33.0000	−1.22474
$727$	−14.0000	−0.519231	−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000	−0.519231	−0.259616	+	0.965712i	$0.583596\pi$
$728$	−1.00000	−0.0370625
$729$	−27.0000	−1.00000
$730$	18.0000	0.666210
$731$	18.0000	0.665754
$732$	24.0000	0.887066
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	−18.0000	−0.664392
$735$	6.00000	0.221313
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	16.0000	0.588172
$741$	−21.0000	−0.771454
$742$	−10.0000	−0.367112
$743$	41.0000	1.50414	0.752072	−	0.659081i	$-0.229055\pi$
$743$	41.0000	1.50414	0.752072	+	0.659081i	$0.229055\pi$
$744$	6.00000	0.219971
$745$	−30.0000	−1.09911
$746$	12.0000	0.439351
$747$	48.0000	1.75623
$748$	0	0
$749$	12.0000	0.438470
$750$	−36.0000	−1.31453
$751$	46.0000	1.67856	0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000	1.67856	0.839282	+	0.543696i	$0.182976\pi$
$752$	6.00000	0.218797
$753$	−60.0000	−2.18652
$754$	5.00000	0.182089
$755$	−20.0000	−0.727875
$756$	−9.00000	−0.327327
$757$	47.0000	1.70824	0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000	1.70824	0.854122	+	0.520073i	$0.174095\pi$
$758$	−30.0000	−1.08965
$759$	0	0
$760$	−14.0000	−0.507833
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	36.0000	1.30414
$763$	5.00000	0.181012
$764$	3.00000	0.108536
$765$	−24.0000	−0.867722
$766$	−8.00000	−0.289052
$767$	4.00000	0.144432
$768$	3.00000	0.108253
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	−51.0000	−1.83672
$772$	4.00000	0.143963
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	−54.0000	−1.94099
$775$	−2.00000	−0.0718421
$776$	16.0000	0.574367
$777$	−24.0000	−0.860995
$778$	−7.00000	−0.250962
$779$	0	0
$780$	6.00000	0.214834
$781$	0	0
$782$	−12.0000	−0.429119
$783$	45.0000	1.60817
$784$	1.00000	0.0357143
$785$	−8.00000	−0.285532
$786$	−30.0000	−1.07006
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	−9.00000	−0.320612
$789$	18.0000	0.640817
$790$	2.00000	0.0711568
$791$	−10.0000	−0.355559
$792$	0	0
$793$	8.00000	0.284088
$794$	34.0000	1.20661
$795$	60.0000	2.12798
$796$	−10.0000	−0.354441
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	21.0000	0.743392
$799$	−12.0000	−0.424529
$800$	−1.00000	−0.0353553
$801$	−60.0000	−2.12000
$802$	−2.00000	−0.0706225
$803$	0	0
$804$	3.00000	0.105802
$805$	−12.0000	−0.422944
$806$	2.00000	0.0704470
$807$	−33.0000	−1.16166
$808$	−15.0000	−0.527698
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	18.0000	0.632456
$811$	−11.0000	−0.386262	−0.193131	−	0.981173i	$-0.561864\pi$
$811$	−11.0000	−0.386262	−0.193131	+	0.981173i	$0.561864\pi$
$812$	−5.00000	−0.175466
$813$	−60.0000	−2.10429
$814$	0	0
$815$	−36.0000	−1.26102
$816$	−6.00000	−0.210042
$817$	63.0000	2.20409
$818$	−35.0000	−1.22375
$819$	−6.00000	−0.209657
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−36.0000	−1.25564
$823$	−26.0000	−0.906303	−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000	−0.906303	−0.453152	+	0.891434i	$0.649700\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−50.0000	−1.73867	−0.869335	−	0.494223i	$-0.835453\pi$
$827$	−50.0000	−1.73867	−0.869335	+	0.494223i	$0.835453\pi$
$828$	36.0000	1.25109
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	16.0000	0.555368
$831$	6.00000	0.208138
$832$	1.00000	0.0346688
$833$	−2.00000	−0.0692959
$834$	12.0000	0.415526
$835$	−48.0000	−1.66111
$836$	0	0
$837$	18.0000	0.622171
$838$	20.0000	0.690889
$839$	50.0000	1.72619	0.863096	−	0.505040i	$-0.168522\pi$
$839$	50.0000	1.72619	0.863096	+	0.505040i	$0.168522\pi$
$840$	−6.00000	−0.207020
$841$	−4.00000	−0.137931
$842$	−10.0000	−0.344623
$843$	54.0000	1.85986
$844$	−29.0000	−0.998221
$845$	−24.0000	−0.825625
$846$	36.0000	1.23771
$847$	11.0000	0.377964
$848$	10.0000	0.343401
$849$	12.0000	0.411839
$850$	2.00000	0.0685994
$851$	48.0000	1.64542
$852$	−45.0000	−1.54167
$853$	58.0000	1.98588	0.992941	−	0.118609i	$-0.0378434\pi$
$853$	58.0000	1.98588	0.992941	+	0.118609i	$0.0378434\pi$
$854$	−8.00000	−0.273754
$855$	−84.0000	−2.87274
$856$	−12.0000	−0.410152
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−18.0000	−0.613795
$861$	0	0
$862$	−1.00000	−0.0340601
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	9.00000	0.306186
$865$	0	0
$866$	6.00000	0.203888
$867$	−39.0000	−1.32451
$868$	−2.00000	−0.0678844
$869$	0	0
$870$	30.0000	1.01710
$871$	1.00000	0.0338837
$872$	−5.00000	−0.169321
$873$	96.0000	3.24911
$874$	−42.0000	−1.42067
$875$	12.0000	0.405674
$876$	27.0000	0.912245
$877$	−7.00000	−0.236373	−0.118187	−	0.992991i	$-0.537708\pi$
$877$	−7.00000	−0.236373	−0.118187	+	0.992991i	$0.537708\pi$
$878$	8.00000	0.269987
$879$	69.0000	2.32731
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	6.00000	0.202031
$883$	33.0000	1.11054	0.555269	−	0.831671i	$-0.312615\pi$
$883$	33.0000	1.11054	0.555269	+	0.831671i	$0.312615\pi$
$884$	−2.00000	−0.0672673
$885$	24.0000	0.806751
$886$	−38.0000	−1.27663
$887$	−56.0000	−1.88030	−0.940148	−	0.340766i	$-0.889313\pi$
$887$	−56.0000	−1.88030	−0.940148	+	0.340766i	$0.889313\pi$
$888$	24.0000	0.805387
$889$	−12.0000	−0.402467
$890$	−20.0000	−0.670402
$891$	0	0
$892$	12.0000	0.401790
$893$	−42.0000	−1.40548
$894$	−45.0000	−1.50503
$895$	−24.0000	−0.802232
$896$	−1.00000	−0.0334077
$897$	18.0000	0.601003
$898$	27.0000	0.901002
$899$	10.0000	0.333519
$900$	−6.00000	−0.200000
$901$	−20.0000	−0.666297
$902$	0	0
$903$	27.0000	0.898504
$904$	10.0000	0.332595
$905$	2.00000	0.0664822
$906$	−30.0000	−0.996683
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	7.00000	0.232303
$909$	−90.0000	−2.98511
$910$	−2.00000	−0.0662994
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	−21.0000	−0.695379
$913$	0	0
$914$	2.00000	0.0661541
$915$	48.0000	1.58683
$916$	14.0000	0.462573
$917$	10.0000	0.330229
$918$	−18.0000	−0.594089
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	12.0000	0.395628
$921$	21.0000	0.691974
$922$	−24.0000	−0.790398
$923$	−15.0000	−0.493731
$924$	0	0
$925$	−8.00000	−0.263038
$926$	−26.0000	−0.854413
$927$	−24.0000	−0.788263
$928$	5.00000	0.164133
$929$	5.00000	0.164045	0.0820223	−	0.996630i	$-0.473862\pi$
$929$	5.00000	0.164045	0.0820223	+	0.996630i	$0.473862\pi$
$930$	12.0000	0.393496
$931$	−7.00000	−0.229416
$932$	−18.0000	−0.589610
$933$	36.0000	1.17859
$934$	20.0000	0.654420
$935$	0	0
$936$	6.00000	0.196116
$937$	56.0000	1.82944	0.914720	−	0.404088i	$-0.132411\pi$
$937$	56.0000	1.82944	0.914720	+	0.404088i	$0.132411\pi$
$938$	−1.00000	−0.0326512
$939$	18.0000	0.587408
$940$	12.0000	0.391397
$941$	−5.00000	−0.162995	−0.0814977	−	0.996674i	$-0.525970\pi$
$941$	−5.00000	−0.162995	−0.0814977	+	0.996674i	$0.525970\pi$
$942$	−12.0000	−0.390981
$943$	0	0
$944$	4.00000	0.130189
$945$	−18.0000	−0.585540
$946$	0	0
$947$	57.0000	1.85225	0.926126	−	0.377215i	$-0.123118\pi$
$947$	57.0000	1.85225	0.926126	+	0.377215i	$0.123118\pi$
$948$	3.00000	0.0974355
$949$	9.00000	0.292152
$950$	7.00000	0.227110
$951$	12.0000	0.389127
$952$	2.00000	0.0648204
$953$	19.0000	0.615470	0.307735	−	0.951472i	$-0.400429\pi$
$953$	19.0000	0.615470	0.307735	+	0.951472i	$0.400429\pi$
$954$	60.0000	1.94257
$955$	6.00000	0.194155
$956$	17.0000	0.549819
$957$	0	0
$958$	−25.0000	−0.807713
$959$	12.0000	0.387500
$960$	6.00000	0.193649
$961$	−27.0000	−0.870968
$962$	8.00000	0.257930
$963$	−72.0000	−2.32017
$964$	1.00000	0.0322078
$965$	8.00000	0.257529
$966$	−18.0000	−0.579141
$967$	23.0000	0.739630	0.369815	−	0.929105i	$-0.379421\pi$
$967$	23.0000	0.739630	0.369815	+	0.929105i	$0.379421\pi$
$968$	−11.0000	−0.353553
$969$	42.0000	1.34923
$970$	32.0000	1.02746
$971$	23.0000	0.738105	0.369053	−	0.929409i	$-0.379682\pi$
$971$	23.0000	0.738105	0.369053	+	0.929409i	$0.379682\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	11.0000	0.352463
$975$	−3.00000	−0.0960769
$976$	8.00000	0.256074
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	−54.0000	−1.72673
$979$	0	0
$980$	2.00000	0.0638877
$981$	−30.0000	−0.957826
$982$	−36.0000	−1.14881
$983$	3.00000	0.0956851	0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000	0.0956851	0.0478426	+	0.998855i	$0.484765\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	−10.0000	−0.318465
$987$	−18.0000	−0.572946
$988$	−7.00000	−0.222700
$989$	−54.0000	−1.71710
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	2.00000	0.0635001
$993$	84.0000	2.66566
$994$	15.0000	0.475771
$995$	−20.0000	−0.634043
$996$	24.0000	0.760469
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	16.0000	0.506471
$999$	72.0000	2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6034.2.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6034.2.a.f.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6034 = 2 \cdot 7 \cdot 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6034.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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