LMFDB - Embedded newform 1003.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1003,2,Mod(1,1003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1003, 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("1003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1003 = 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1003.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$8.00899532273$
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	$\chi$	$=$	1003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +3.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} -2.00000 q^{18} +7.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} +4.00000 q^{22} +2.00000 q^{23} -3.00000 q^{24} -4.00000 q^{25} -5.00000 q^{27} -3.00000 q^{28} +3.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +5.00000 q^{32} +4.00000 q^{33} +1.00000 q^{34} +3.00000 q^{35} +2.00000 q^{36} +7.00000 q^{38} -3.00000 q^{40} -3.00000 q^{41} +3.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +2.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -4.00000 q^{50} +1.00000 q^{51} -9.00000 q^{53} -5.00000 q^{54} +4.00000 q^{55} -9.00000 q^{56} +7.00000 q^{57} +3.00000 q^{58} -1.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} -6.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} +2.00000 q^{69} +3.00000 q^{70} -4.00000 q^{71} +6.00000 q^{72} -2.00000 q^{73} -4.00000 q^{75} -7.00000 q^{76} +12.0000 q^{77} +5.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -3.00000 q^{82} +16.0000 q^{83} -3.00000 q^{84} +1.00000 q^{85} +4.00000 q^{86} +3.00000 q^{87} -12.0000 q^{88} -2.00000 q^{90} -2.00000 q^{92} +4.00000 q^{93} +6.00000 q^{94} +7.00000 q^{95} +5.00000 q^{96} -8.00000 q^{97} +2.00000 q^{98} -8.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	1.00000	0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−3.00000	−1.06066
$9$	−2.00000	−0.666667
$10$	1.00000	0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	3.00000	0.801784
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.00000	−0.471405
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	−1.00000	−0.223607
$21$	3.00000	0.654654
$22$	4.00000	0.852803
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−3.00000	−0.612372
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	−3.00000	−0.566947
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	1.00000	0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.00000	0.883883
$33$	4.00000	0.696311
$34$	1.00000	0.171499
$35$	3.00000	0.507093
$36$	2.00000	0.333333
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	7.00000	1.13555
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	3.00000	0.462910
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	−2.00000	−0.298142
$46$	2.00000	0.294884
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	−4.00000	−0.565685
$51$	1.00000	0.140028
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−5.00000	−0.680414
$55$	4.00000	0.539360
$56$	−9.00000	−1.20268
$57$	7.00000	0.927173
$58$	3.00000	0.393919
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−1.00000	−0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	−6.00000	−0.755929
$64$	7.00000	0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−1.00000	−0.121268
$69$	2.00000	0.240772
$70$	3.00000	0.358569
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	6.00000	0.707107
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	−7.00000	−0.802955
$77$	12.0000	1.36753
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−3.00000	−0.331295
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	−3.00000	−0.327327
$85$	1.00000	0.108465
$86$	4.00000	0.431331
$87$	3.00000	0.321634
$88$	−12.0000	−1.27920
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	−2.00000	−0.208514
$93$	4.00000	0.414781
$94$	6.00000	0.618853
$95$	7.00000	0.718185
$96$	5.00000	0.510310
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	2.00000	0.202031
$99$	−8.00000	−0.804030
$100$	4.00000	0.400000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	1.00000	0.0990148
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	−9.00000	−0.874157
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	5.00000	0.481125
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−3.00000	−0.283473
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	7.00000	0.655610
$115$	2.00000	0.186501
$116$	−3.00000	−0.278543
$117$	0	0
$118$	−1.00000	−0.0920575
$119$	3.00000	0.275010
$120$	−3.00000	−0.273861
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	−3.00000	−0.270501
$124$	−4.00000	−0.359211
$125$	−9.00000	−0.804984
$126$	−6.00000	−0.534522
$127$	15.0000	1.33103	0.665517	−	0.746382i	$-0.268211\pi$
$127$	15.0000	1.33103	0.665517	+	0.746382i	$0.268211\pi$
$128$	−3.00000	−0.265165
$129$	4.00000	0.352180
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	−4.00000	−0.348155
$133$	21.0000	1.82093
$134$	−4.00000	−0.345547
$135$	−5.00000	−0.430331
$136$	−3.00000	−0.257248
$137$	11.0000	0.939793	0.469897	−	0.882721i	$-0.344291\pi$
$137$	11.0000	0.939793	0.469897	+	0.882721i	$0.344291\pi$
$138$	2.00000	0.170251
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−3.00000	−0.253546
$141$	6.00000	0.505291
$142$	−4.00000	−0.335673
$143$	0	0
$144$	2.00000	0.166667
$145$	3.00000	0.249136
$146$	−2.00000	−0.165521
$147$	2.00000	0.164957
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−4.00000	−0.326599
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−21.0000	−1.70332
$153$	−2.00000	−0.161690
$154$	12.0000	0.966988
$155$	4.00000	0.321288
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	5.00000	0.397779
$159$	−9.00000	−0.713746
$160$	5.00000	0.395285
$161$	6.00000	0.472866
$162$	1.00000	0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	3.00000	0.234261
$165$	4.00000	0.311400
$166$	16.0000	1.24184
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	−9.00000	−0.694365
$169$	−13.0000	−1.00000
$170$	1.00000	0.0766965
$171$	−14.0000	−1.07061
$172$	−4.00000	−0.304997
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	3.00000	0.227429
$175$	−12.0000	−0.907115
$176$	−4.00000	−0.301511
$177$	−1.00000	−0.0751646
$178$	0	0
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	2.00000	0.149071
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	−6.00000	−0.442326
$185$	0	0
$186$	4.00000	0.293294
$187$	4.00000	0.292509
$188$	−6.00000	−0.437595
$189$	−15.0000	−1.09109
$190$	7.00000	0.507833
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	7.00000	0.505181
$193$	−15.0000	−1.07972	−0.539862	−	0.841754i	$-0.681524\pi$
$193$	−15.0000	−1.07972	−0.539862	+	0.841754i	$0.681524\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−2.00000	−0.142857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−8.00000	−0.568535
$199$	−9.00000	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$200$	12.0000	0.848528
$201$	−4.00000	−0.282138
$202$	−8.00000	−0.562878
$203$	9.00000	0.631676
$204$	−1.00000	−0.0700140
$205$	−3.00000	−0.209529
$206$	−16.0000	−1.11477
$207$	−4.00000	−0.278019
$208$	0	0
$209$	28.0000	1.93680
$210$	3.00000	0.207020
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	9.00000	0.618123
$213$	−4.00000	−0.274075
$214$	9.00000	0.615227
$215$	4.00000	0.272798
$216$	15.0000	1.02062
$217$	12.0000	0.814613
$218$	−10.0000	−0.677285
$219$	−2.00000	−0.135147
$220$	−4.00000	−0.269680
$221$	0	0
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	15.0000	1.00223
$225$	8.00000	0.533333
$226$	−14.0000	−0.931266
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	−7.00000	−0.463586
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	2.00000	0.131876
$231$	12.0000	0.789542
$232$	−9.00000	−0.590879
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	1.00000	0.0650945
$237$	5.00000	0.324785
$238$	3.00000	0.194461
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	−1.00000	−0.0645497
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	5.00000	0.321412
$243$	16.0000	1.02640
$244$	10.0000	0.640184
$245$	2.00000	0.127775
$246$	−3.00000	−0.191273
$247$	0	0
$248$	−12.0000	−0.762001
$249$	16.0000	1.01396
$250$	−9.00000	−0.569210
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	6.00000	0.377964
$253$	8.00000	0.502956
$254$	15.0000	0.941184
$255$	1.00000	0.0626224
$256$	−17.0000	−1.06250
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	−18.0000	−1.11204
$263$	−11.0000	−0.678289	−0.339145	−	0.940734i	$-0.610138\pi$
$263$	−11.0000	−0.678289	−0.339145	+	0.940734i	$0.610138\pi$
$264$	−12.0000	−0.738549
$265$	−9.00000	−0.552866
$266$	21.0000	1.28759
$267$	0	0
$268$	4.00000	0.244339
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	−5.00000	−0.304290
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	11.0000	0.664534
$275$	−16.0000	−0.964836
$276$	−2.00000	−0.120386
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	4.00000	0.239904
$279$	−8.00000	−0.478947
$280$	−9.00000	−0.537853
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	6.00000	0.357295
$283$	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
$285$	7.00000	0.414644
$286$	0	0
$287$	−9.00000	−0.531253
$288$	−10.0000	−0.589256
$289$	1.00000	0.0588235
$290$	3.00000	0.176166
$291$	−8.00000	−0.468968
$292$	2.00000	0.117041
$293$	−3.00000	−0.175262	−0.0876309	−	0.996153i	$-0.527930\pi$
$293$	−3.00000	−0.175262	−0.0876309	+	0.996153i	$0.527930\pi$
$294$	2.00000	0.116642
$295$	−1.00000	−0.0582223
$296$	0	0
$297$	−20.0000	−1.16052
$298$	−10.0000	−0.579284
$299$	0	0
$300$	4.00000	0.230940
$301$	12.0000	0.691669
$302$	8.00000	0.460348
$303$	−8.00000	−0.459588
$304$	−7.00000	−0.401478
$305$	−10.0000	−0.572598
$306$	−2.00000	−0.114332
$307$	−31.0000	−1.76926	−0.884632	−	0.466290i	$-0.845590\pi$
$307$	−31.0000	−1.76926	−0.884632	+	0.466290i	$0.845590\pi$
$308$	−12.0000	−0.683763
$309$	−16.0000	−0.910208
$310$	4.00000	0.227185
$311$	7.00000	0.396934	0.198467	−	0.980108i	$-0.436404\pi$
$311$	7.00000	0.396934	0.198467	+	0.980108i	$0.436404\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	2.00000	0.112867
$315$	−6.00000	−0.338062
$316$	−5.00000	−0.281272
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−9.00000	−0.504695
$319$	12.0000	0.671871
$320$	7.00000	0.391312
$321$	9.00000	0.502331
$322$	6.00000	0.334367
$323$	7.00000	0.389490
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−16.0000	−0.886158
$327$	−10.0000	−0.553001
$328$	9.00000	0.496942
$329$	18.0000	0.992372
$330$	4.00000	0.220193
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	−16.0000	−0.878114
$333$	0	0
$334$	15.0000	0.820763
$335$	−4.00000	−0.218543
$336$	−3.00000	−0.163663
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	−13.0000	−0.707107
$339$	−14.0000	−0.760376
$340$	−1.00000	−0.0542326
$341$	16.0000	0.866449
$342$	−14.0000	−0.757033
$343$	−15.0000	−0.809924
$344$	−12.0000	−0.646997
$345$	2.00000	0.107676
$346$	−24.0000	−1.29025
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−3.00000	−0.160817
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−12.0000	−0.641427
$351$	0	0
$352$	20.0000	1.06600
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	−1.00000	−0.0531494
$355$	−4.00000	−0.212298
$356$	0	0
$357$	3.00000	0.158777
$358$	−14.0000	−0.739923
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	6.00000	0.316228
$361$	30.0000	1.57895
$362$	−5.00000	−0.262794
$363$	5.00000	0.262432
$364$	0	0
$365$	−2.00000	−0.104685
$366$	−10.0000	−0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−2.00000	−0.104257
$369$	6.00000	0.312348
$370$	0	0
$371$	−27.0000	−1.40177
$372$	−4.00000	−0.207390
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	4.00000	0.206835
$375$	−9.00000	−0.464758
$376$	−18.0000	−0.928279
$377$	0	0
$378$	−15.0000	−0.771517
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	−7.00000	−0.359092
$381$	15.0000	0.768473
$382$	2.00000	0.102329
$383$	−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
$385$	12.0000	0.611577
$386$	−15.0000	−0.763480
$387$	−8.00000	−0.406663
$388$	8.00000	0.406138
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	−6.00000	−0.303046
$393$	−18.0000	−0.907980
$394$	2.00000	0.100759
$395$	5.00000	0.251577
$396$	8.00000	0.402015
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−9.00000	−0.451129
$399$	21.0000	1.05131
$400$	4.00000	0.200000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	8.00000	0.398015
$405$	1.00000	0.0496904
$406$	9.00000	0.446663
$407$	0	0
$408$	−3.00000	−0.148522
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−3.00000	−0.148159
$411$	11.0000	0.542590
$412$	16.0000	0.788263
$413$	−3.00000	−0.147620
$414$	−4.00000	−0.196589
$415$	16.0000	0.785409
$416$	0	0
$417$	4.00000	0.195881
$418$	28.0000	1.36952
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	−3.00000	−0.146385
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	12.0000	0.584151
$423$	−12.0000	−0.583460
$424$	27.0000	1.31124
$425$	−4.00000	−0.194029
$426$	−4.00000	−0.193801
$427$	−30.0000	−1.45180
$428$	−9.00000	−0.435031
$429$	0	0
$430$	4.00000	0.192897
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	5.00000	0.240563
$433$	−41.0000	−1.97033	−0.985167	−	0.171598i	$-0.945107\pi$
$433$	−41.0000	−1.97033	−0.985167	+	0.171598i	$0.945107\pi$
$434$	12.0000	0.576018
$435$	3.00000	0.143839
$436$	10.0000	0.478913
$437$	14.0000	0.669711
$438$	−2.00000	−0.0955637
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−12.0000	−0.572078
$441$	−4.00000	−0.190476
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	0	0
$446$	−24.0000	−1.13643
$447$	−10.0000	−0.472984
$448$	21.0000	0.992157
$449$	3.00000	0.141579	0.0707894	−	0.997491i	$-0.477448\pi$
$449$	3.00000	0.141579	0.0707894	+	0.997491i	$0.477448\pi$
$450$	8.00000	0.377124
$451$	−12.0000	−0.565058
$452$	14.0000	0.658505
$453$	8.00000	0.375873
$454$	14.0000	0.657053
$455$	0	0
$456$	−21.0000	−0.983415
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	26.0000	1.21490
$459$	−5.00000	−0.233380
$460$	−2.00000	−0.0932505
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	12.0000	0.558291
$463$	30.0000	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$463$	30.0000	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$464$	−3.00000	−0.139272
$465$	4.00000	0.185496
$466$	8.00000	0.370593
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	6.00000	0.276759
$471$	2.00000	0.0921551
$472$	3.00000	0.138086
$473$	16.0000	0.735681
$474$	5.00000	0.229658
$475$	−28.0000	−1.28473
$476$	−3.00000	−0.137505
$477$	18.0000	0.824163
$478$	21.0000	0.960518
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	5.00000	0.228218
$481$	0	0
$482$	7.00000	0.318841
$483$	6.00000	0.273009
$484$	−5.00000	−0.227273
$485$	−8.00000	−0.363261
$486$	16.0000	0.725775
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	30.0000	1.35804
$489$	−16.0000	−0.723545
$490$	2.00000	0.0903508
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	3.00000	0.135250
$493$	3.00000	0.135113
$494$	0	0
$495$	−8.00000	−0.359573
$496$	−4.00000	−0.179605
$497$	−12.0000	−0.538274
$498$	16.0000	0.716977
$499$	−23.0000	−1.02962	−0.514811	−	0.857304i	$-0.672138\pi$
$499$	−23.0000	−1.02962	−0.514811	+	0.857304i	$0.672138\pi$
$500$	9.00000	0.402492
$501$	15.0000	0.670151
$502$	−15.0000	−0.669483
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	18.0000	0.801784
$505$	−8.00000	−0.355995
$506$	8.00000	0.355643
$507$	−13.0000	−0.577350
$508$	−15.0000	−0.665517
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	1.00000	0.0442807
$511$	−6.00000	−0.265424
$512$	−11.0000	−0.486136
$513$	−35.0000	−1.54529
$514$	−19.0000	−0.838054
$515$	−16.0000	−0.705044
$516$	−4.00000	−0.176090
$517$	24.0000	1.05552
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	−6.00000	−0.262613
$523$	33.0000	1.44299	0.721495	−	0.692420i	$-0.243455\pi$
$523$	33.0000	1.44299	0.721495	+	0.692420i	$0.243455\pi$
$524$	18.0000	0.786334
$525$	−12.0000	−0.523723
$526$	−11.0000	−0.479623
$527$	4.00000	0.174243
$528$	−4.00000	−0.174078
$529$	−19.0000	−0.826087
$530$	−9.00000	−0.390935
$531$	2.00000	0.0867926
$532$	−21.0000	−0.910465
$533$	0	0
$534$	0	0
$535$	9.00000	0.389104
$536$	12.0000	0.518321
$537$	−14.0000	−0.604145
$538$	−2.00000	−0.0862261
$539$	8.00000	0.344584
$540$	5.00000	0.215166
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	11.0000	0.472490
$543$	−5.00000	−0.214571
$544$	5.00000	0.214373
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−11.0000	−0.469897
$549$	20.0000	0.853579
$550$	−16.0000	−0.682242
$551$	21.0000	0.894630
$552$	−6.00000	−0.255377
$553$	15.0000	0.637865
$554$	13.0000	0.552317
$555$	0	0
$556$	−4.00000	−0.169638
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	−8.00000	−0.338667
$559$	0	0
$560$	−3.00000	−0.126773
$561$	4.00000	0.168880
$562$	15.0000	0.632737
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	−6.00000	−0.252646
$565$	−14.0000	−0.588984
$566$	6.00000	0.252199
$567$	3.00000	0.125988
$568$	12.0000	0.503509
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	7.00000	0.293198
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	2.00000	0.0835512
$574$	−9.00000	−0.375653
$575$	−8.00000	−0.333623
$576$	−14.0000	−0.583333
$577$	−23.0000	−0.957503	−0.478751	−	0.877951i	$-0.658910\pi$
$577$	−23.0000	−0.957503	−0.478751	+	0.877951i	$0.658910\pi$
$578$	1.00000	0.0415945
$579$	−15.0000	−0.623379
$580$	−3.00000	−0.124568
$581$	48.0000	1.99138
$582$	−8.00000	−0.331611
$583$	−36.0000	−1.49097
$584$	6.00000	0.248282
$585$	0	0
$586$	−3.00000	−0.123929
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−2.00000	−0.0824786
$589$	28.0000	1.15372
$590$	−1.00000	−0.0411693
$591$	2.00000	0.0822690
$592$	0	0
$593$	−27.0000	−1.10876	−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000	−1.10876	−0.554379	+	0.832265i	$0.687044\pi$
$594$	−20.0000	−0.820610
$595$	3.00000	0.122988
$596$	10.0000	0.409616
$597$	−9.00000	−0.368345
$598$	0	0
$599$	−37.0000	−1.51178	−0.755890	−	0.654699i	$-0.772795\pi$
$599$	−37.0000	−1.51178	−0.755890	+	0.654699i	$0.772795\pi$
$600$	12.0000	0.489898
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	12.0000	0.489083
$603$	8.00000	0.325785
$604$	−8.00000	−0.325515
$605$	5.00000	0.203279
$606$	−8.00000	−0.324978
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	35.0000	1.41944
$609$	9.00000	0.364698
$610$	−10.0000	−0.404888
$611$	0	0
$612$	2.00000	0.0808452
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	−31.0000	−1.25106
$615$	−3.00000	−0.120972
$616$	−36.0000	−1.45048
$617$	−21.0000	−0.845428	−0.422714	−	0.906263i	$-0.638923\pi$
$617$	−21.0000	−0.845428	−0.422714	+	0.906263i	$0.638923\pi$
$618$	−16.0000	−0.643614
$619$	−29.0000	−1.16561	−0.582804	−	0.812613i	$-0.698045\pi$
$619$	−29.0000	−1.16561	−0.582804	+	0.812613i	$0.698045\pi$
$620$	−4.00000	−0.160644
$621$	−10.0000	−0.401286
$622$	7.00000	0.280674
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	22.0000	0.879297
$627$	28.0000	1.11821
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	−6.00000	−0.239046
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	−15.0000	−0.596668
$633$	12.0000	0.476957
$634$	2.00000	0.0794301
$635$	15.0000	0.595257
$636$	9.00000	0.356873
$637$	0	0
$638$	12.0000	0.475085
$639$	8.00000	0.316475
$640$	−3.00000	−0.118585
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	9.00000	0.355202
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	−6.00000	−0.236433
$645$	4.00000	0.157500
$646$	7.00000	0.275411
$647$	−15.0000	−0.589711	−0.294855	−	0.955542i	$-0.595271\pi$
$647$	−15.0000	−0.589711	−0.294855	+	0.955542i	$0.595271\pi$
$648$	−3.00000	−0.117851
$649$	−4.00000	−0.157014
$650$	0	0
$651$	12.0000	0.470317
$652$	16.0000	0.626608
$653$	13.0000	0.508729	0.254365	−	0.967108i	$-0.418134\pi$
$653$	13.0000	0.508729	0.254365	+	0.967108i	$0.418134\pi$
$654$	−10.0000	−0.391031
$655$	−18.0000	−0.703318
$656$	3.00000	0.117130
$657$	4.00000	0.156055
$658$	18.0000	0.701713
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	−4.00000	−0.155700
$661$	−29.0000	−1.12797	−0.563985	−	0.825785i	$-0.690732\pi$
$661$	−29.0000	−1.12797	−0.563985	+	0.825785i	$0.690732\pi$
$662$	−25.0000	−0.971653
$663$	0	0
$664$	−48.0000	−1.86276
$665$	21.0000	0.814345
$666$	0	0
$667$	6.00000	0.232321
$668$	−15.0000	−0.580367
$669$	−24.0000	−0.927894
$670$	−4.00000	−0.154533
$671$	−40.0000	−1.54418
$672$	15.0000	0.578638
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	30.0000	1.15556
$675$	20.0000	0.769800
$676$	13.0000	0.500000
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	−14.0000	−0.537667
$679$	−24.0000	−0.921035
$680$	−3.00000	−0.115045
$681$	14.0000	0.536481
$682$	16.0000	0.612672
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	14.0000	0.535303
$685$	11.0000	0.420288
$686$	−15.0000	−0.572703
$687$	26.0000	0.991962
$688$	−4.00000	−0.152499
$689$	0	0
$690$	2.00000	0.0761387
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	24.0000	0.912343
$693$	−24.0000	−0.911685
$694$	12.0000	0.455514
$695$	4.00000	0.151729
$696$	−9.00000	−0.341144
$697$	−3.00000	−0.113633
$698$	−14.0000	−0.529908
$699$	8.00000	0.302588
$700$	12.0000	0.453557
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	0	0
$703$	0	0
$704$	28.0000	1.05529
$705$	6.00000	0.225973
$706$	24.0000	0.903252
$707$	−24.0000	−0.902613
$708$	1.00000	0.0375823
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	−4.00000	−0.150117
$711$	−10.0000	−0.375029
$712$	0	0
$713$	8.00000	0.299602
$714$	3.00000	0.112272
$715$	0	0
$716$	14.0000	0.523205
$717$	21.0000	0.784259
$718$	3.00000	0.111959
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	2.00000	0.0745356
$721$	−48.0000	−1.78761
$722$	30.0000	1.11648
$723$	7.00000	0.260333
$724$	5.00000	0.185824
$725$	−12.0000	−0.445669
$726$	5.00000	0.185567
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−2.00000	−0.0740233
$731$	4.00000	0.147945
$732$	10.0000	0.369611
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−8.00000	−0.295285
$735$	2.00000	0.0737711
$736$	10.0000	0.368605
$737$	−16.0000	−0.589368
$738$	6.00000	0.220863
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	0	0
$742$	−27.0000	−0.991201
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	−12.0000	−0.439941
$745$	−10.0000	−0.366372
$746$	−14.0000	−0.512576
$747$	−32.0000	−1.17082
$748$	−4.00000	−0.146254
$749$	27.0000	0.986559
$750$	−9.00000	−0.328634
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	−6.00000	−0.218797
$753$	−15.0000	−0.546630
$754$	0	0
$755$	8.00000	0.291150
$756$	15.0000	0.545545
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	19.0000	0.690111
$759$	8.00000	0.290382
$760$	−21.0000	−0.761750
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	15.0000	0.543393
$763$	−30.0000	−1.08607
$764$	−2.00000	−0.0723575
$765$	−2.00000	−0.0723102
$766$	−8.00000	−0.289052
$767$	0	0
$768$	−17.0000	−0.613435
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	12.0000	0.432450
$771$	−19.0000	−0.684268
$772$	15.0000	0.539862
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−8.00000	−0.287554
$775$	−16.0000	−0.574737
$776$	24.0000	0.861550
$777$	0	0
$778$	34.0000	1.21896
$779$	−21.0000	−0.752403
$780$	0	0
$781$	−16.0000	−0.572525
$782$	2.00000	0.0715199
$783$	−15.0000	−0.536056
$784$	−2.00000	−0.0714286
$785$	2.00000	0.0713831
$786$	−18.0000	−0.642039
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	−2.00000	−0.0712470
$789$	−11.0000	−0.391610
$790$	5.00000	0.177892
$791$	−42.0000	−1.49335
$792$	24.0000	0.852803
$793$	0	0
$794$	−22.0000	−0.780751
$795$	−9.00000	−0.319197
$796$	9.00000	0.318997
$797$	36.0000	1.27519	0.637593	−	0.770374i	$-0.279930\pi$
$797$	36.0000	1.27519	0.637593	+	0.770374i	$0.279930\pi$
$798$	21.0000	0.743392
$799$	6.00000	0.212265
$800$	−20.0000	−0.707107
$801$	0	0
$802$	−24.0000	−0.847469
$803$	−8.00000	−0.282314
$804$	4.00000	0.141069
$805$	6.00000	0.211472
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	24.0000	0.844317
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	1.00000	0.0351364
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	−9.00000	−0.315838
$813$	11.0000	0.385787
$814$	0	0
$815$	−16.0000	−0.560456
$816$	−1.00000	−0.0350070
$817$	28.0000	0.979596
$818$	−4.00000	−0.139857
$819$	0	0
$820$	3.00000	0.104765
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	11.0000	0.383669
$823$	−30.0000	−1.04573	−0.522867	−	0.852414i	$-0.675138\pi$
$823$	−30.0000	−1.04573	−0.522867	+	0.852414i	$0.675138\pi$
$824$	48.0000	1.67216
$825$	−16.0000	−0.557048
$826$	−3.00000	−0.104383
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	4.00000	0.139010
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	16.0000	0.555368
$831$	13.0000	0.450965
$832$	0	0
$833$	2.00000	0.0692959
$834$	4.00000	0.138509
$835$	15.0000	0.519096
$836$	−28.0000	−0.968400
$837$	−20.0000	−0.691301
$838$	−4.00000	−0.138178
$839$	−50.0000	−1.72619	−0.863096	−	0.505040i	$-0.831478\pi$
$839$	−50.0000	−1.72619	−0.863096	+	0.505040i	$0.831478\pi$
$840$	−9.00000	−0.310530
$841$	−20.0000	−0.689655
$842$	−20.0000	−0.689246
$843$	15.0000	0.516627
$844$	−12.0000	−0.413057
$845$	−13.0000	−0.447214
$846$	−12.0000	−0.412568
$847$	15.0000	0.515406
$848$	9.00000	0.309061
$849$	6.00000	0.205919
$850$	−4.00000	−0.137199
$851$	0	0
$852$	4.00000	0.137038
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	−30.0000	−1.02658
$855$	−14.0000	−0.478790
$856$	−27.0000	−0.922841
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	−4.00000	−0.136399
$861$	−9.00000	−0.306719
$862$	8.00000	0.272481
$863$	34.0000	1.15737	0.578687	−	0.815550i	$-0.303565\pi$
$863$	34.0000	1.15737	0.578687	+	0.815550i	$0.303565\pi$
$864$	−25.0000	−0.850517
$865$	−24.0000	−0.816024
$866$	−41.0000	−1.39324
$867$	1.00000	0.0339618
$868$	−12.0000	−0.407307
$869$	20.0000	0.678454
$870$	3.00000	0.101710
$871$	0	0
$872$	30.0000	1.01593
$873$	16.0000	0.541518
$874$	14.0000	0.473557
$875$	−27.0000	−0.912767
$876$	2.00000	0.0675737
$877$	−33.0000	−1.11433	−0.557165	−	0.830402i	$-0.688111\pi$
$877$	−33.0000	−1.11433	−0.557165	+	0.830402i	$0.688111\pi$
$878$	8.00000	0.269987
$879$	−3.00000	−0.101187
$880$	−4.00000	−0.134840
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	−4.00000	−0.134687
$883$	−35.0000	−1.17784	−0.588922	−	0.808190i	$-0.700447\pi$
$883$	−35.0000	−1.17784	−0.588922	+	0.808190i	$0.700447\pi$
$884$	0	0
$885$	−1.00000	−0.0336146
$886$	24.0000	0.806296
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	45.0000	1.50925
$890$	0	0
$891$	4.00000	0.134005
$892$	24.0000	0.803579
$893$	42.0000	1.40548
$894$	−10.0000	−0.334450
$895$	−14.0000	−0.467968
$896$	−9.00000	−0.300669
$897$	0	0
$898$	3.00000	0.100111
$899$	12.0000	0.400222
$900$	−8.00000	−0.266667
$901$	−9.00000	−0.299833
$902$	−12.0000	−0.399556
$903$	12.0000	0.399335
$904$	42.0000	1.39690
$905$	−5.00000	−0.166206
$906$	8.00000	0.265782
$907$	−43.0000	−1.42779	−0.713896	−	0.700252i	$-0.753071\pi$
$907$	−43.0000	−1.42779	−0.713896	+	0.700252i	$0.753071\pi$
$908$	−14.0000	−0.464606
$909$	16.0000	0.530687
$910$	0	0
$911$	−37.0000	−1.22586	−0.612932	−	0.790135i	$-0.710010\pi$
$911$	−37.0000	−1.22586	−0.612932	+	0.790135i	$0.710010\pi$
$912$	−7.00000	−0.231793
$913$	64.0000	2.11809
$914$	32.0000	1.05847
$915$	−10.0000	−0.330590
$916$	−26.0000	−0.859064
$917$	−54.0000	−1.78324
$918$	−5.00000	−0.165025
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	−6.00000	−0.197814
$921$	−31.0000	−1.02148
$922$	10.0000	0.329332
$923$	0	0
$924$	−12.0000	−0.394771
$925$	0	0
$926$	30.0000	0.985861
$927$	32.0000	1.05102
$928$	15.0000	0.492399
$929$	−28.0000	−0.918650	−0.459325	−	0.888268i	$-0.651909\pi$
$929$	−28.0000	−0.918650	−0.459325	+	0.888268i	$0.651909\pi$
$930$	4.00000	0.131165
$931$	14.0000	0.458831
$932$	−8.00000	−0.262049
$933$	7.00000	0.229170
$934$	6.00000	0.196326
$935$	4.00000	0.130814
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−12.0000	−0.391814
$939$	22.0000	0.717943
$940$	−6.00000	−0.195698
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	2.00000	0.0651635
$943$	−6.00000	−0.195387
$944$	1.00000	0.0325472
$945$	−15.0000	−0.487950
$946$	16.0000	0.520205
$947$	−35.0000	−1.13735	−0.568674	−	0.822563i	$-0.692543\pi$
$947$	−35.0000	−1.13735	−0.568674	+	0.822563i	$0.692543\pi$
$948$	−5.00000	−0.162392
$949$	0	0
$950$	−28.0000	−0.908440
$951$	2.00000	0.0648544
$952$	−9.00000	−0.291692
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	18.0000	0.582772
$955$	2.00000	0.0647185
$956$	−21.0000	−0.679189
$957$	12.0000	0.387905
$958$	32.0000	1.03387
$959$	33.0000	1.06563
$960$	7.00000	0.225924
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−18.0000	−0.580042
$964$	−7.00000	−0.225455
$965$	−15.0000	−0.482867
$966$	6.00000	0.193047
$967$	−10.0000	−0.321578	−0.160789	−	0.986989i	$-0.551404\pi$
$967$	−10.0000	−0.321578	−0.160789	+	0.986989i	$0.551404\pi$
$968$	−15.0000	−0.482118
$969$	7.00000	0.224872
$970$	−8.00000	−0.256865
$971$	61.0000	1.95758	0.978792	−	0.204859i	$-0.0656735\pi$
$971$	61.0000	1.95758	0.978792	+	0.204859i	$0.0656735\pi$
$972$	−16.0000	−0.513200
$973$	12.0000	0.384702
$974$	−1.00000	−0.0320421
$975$	0	0
$976$	10.0000	0.320092
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	−16.0000	−0.511624
$979$	0	0
$980$	−2.00000	−0.0638877
$981$	20.0000	0.638551
$982$	33.0000	1.05307
$983$	−10.0000	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$983$	−10.0000	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$984$	9.00000	0.286910
$985$	2.00000	0.0637253
$986$	3.00000	0.0955395
$987$	18.0000	0.572946
$988$	0	0
$989$	8.00000	0.254385
$990$	−8.00000	−0.254257
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	20.0000	0.635001
$993$	−25.0000	−0.793351
$994$	−12.0000	−0.380617
$995$	−9.00000	−0.285319
$996$	−16.0000	−0.506979
$997$	31.0000	0.981780	0.490890	−	0.871222i	$-0.336672\pi$
$997$	31.0000	0.981780	0.490890	+	0.871222i	$0.336672\pi$
$998$	−23.0000	−0.728052
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1003.2.a.c.1.1	✓	1
3.2	odd	2		9027.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1003.2.a.c.1.1	✓	1	1.1	even	1	trivial
9027.2.a.b.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:	\( N \)	\(=\)	\( 1003 = 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1003.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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