LMFDB - Embedded newform 2023.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2023,4,Mod(1,2023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2023, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2023.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2023 = 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2023.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:	$119.360863942$
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	$\chi$	$=$	2023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +6.00000 q^{5} -3.00000 q^{6} -7.00000 q^{7} -21.0000 q^{8} -26.0000 q^{9} +O(q^{10})$

\(q+3.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +6.00000 q^{5} -3.00000 q^{6} -7.00000 q^{7} -21.0000 q^{8} -26.0000 q^{9} +18.0000 q^{10} +30.0000 q^{11} -1.00000 q^{12} -22.0000 q^{13} -21.0000 q^{14} -6.00000 q^{15} -71.0000 q^{16} -78.0000 q^{18} +83.0000 q^{19} +6.00000 q^{20} +7.00000 q^{21} +90.0000 q^{22} +48.0000 q^{23} +21.0000 q^{24} -89.0000 q^{25} -66.0000 q^{26} +53.0000 q^{27} -7.00000 q^{28} -15.0000 q^{29} -18.0000 q^{30} +7.00000 q^{31} -45.0000 q^{32} -30.0000 q^{33} -42.0000 q^{35} -26.0000 q^{36} -50.0000 q^{37} +249.000 q^{38} +22.0000 q^{39} -126.000 q^{40} -360.000 q^{41} +21.0000 q^{42} +68.0000 q^{43} +30.0000 q^{44} -156.000 q^{45} +144.000 q^{46} -27.0000 q^{47} +71.0000 q^{48} +49.0000 q^{49} -267.000 q^{50} -22.0000 q^{52} +213.000 q^{53} +159.000 q^{54} +180.000 q^{55} +147.000 q^{56} -83.0000 q^{57} -45.0000 q^{58} +189.000 q^{59} -6.00000 q^{60} -314.000 q^{61} +21.0000 q^{62} +182.000 q^{63} +433.000 q^{64} -132.000 q^{65} -90.0000 q^{66} +314.000 q^{67} -48.0000 q^{69} -126.000 q^{70} +804.000 q^{71} +546.000 q^{72} +448.000 q^{73} -150.000 q^{74} +89.0000 q^{75} +83.0000 q^{76} -210.000 q^{77} +66.0000 q^{78} +1060.00 q^{79} -426.000 q^{80} +649.000 q^{81} -1080.00 q^{82} +873.000 q^{83} +7.00000 q^{84} +204.000 q^{86} +15.0000 q^{87} -630.000 q^{88} +270.000 q^{89} -468.000 q^{90} +154.000 q^{91} +48.0000 q^{92} -7.00000 q^{93} -81.0000 q^{94} +498.000 q^{95} +45.0000 q^{96} -1130.00 q^{97} +147.000 q^{98} -780.000 q^{99} +O(q^{100})\)

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 $

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	−1.00000	−0.192450	−0.0962250	−	0.995360i	$-0.530677\pi$
$3$	−1.00000	−0.192450	−0.0962250	+	0.995360i	$0.530677\pi$
$4$	1.00000	0.125000
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	−3.00000	−0.204124
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−21.0000	−0.928078
$9$	−26.0000	−0.962963
$10$	18.0000	0.569210
$11$	30.0000	0.822304	0.411152	−	0.911567i	$-0.365127\pi$
$11$	30.0000	0.822304	0.411152	+	0.911567i	$0.365127\pi$
$12$	−1.00000	−0.0240563
$13$	−22.0000	−0.469362	−0.234681	−	0.972072i	$-0.575405\pi$
$13$	−22.0000	−0.469362	−0.234681	+	0.972072i	$0.575405\pi$
$14$	−21.0000	−0.400892
$15$	−6.00000	−0.103280
$16$	−71.0000	−1.10938
$17$	0	0
$17$	0	0
$18$	−78.0000	−1.02138
$19$	83.0000	1.00218	0.501092	−	0.865394i	$-0.332932\pi$
$19$	83.0000	1.00218	0.501092	+	0.865394i	$0.332932\pi$
$20$	6.00000	0.0670820
$21$	7.00000	0.0727393
$22$	90.0000	0.872185
$23$	48.0000	0.435161	0.217580	−	0.976042i	$-0.430184\pi$
$23$	48.0000	0.435161	0.217580	+	0.976042i	$0.430184\pi$
$24$	21.0000	0.178609
$25$	−89.0000	−0.712000
$26$	−66.0000	−0.497833
$27$	53.0000	0.377772
$28$	−7.00000	−0.0472456
$29$	−15.0000	−0.0960493	−0.0480247	−	0.998846i	$-0.515293\pi$
$29$	−15.0000	−0.0960493	−0.0480247	+	0.998846i	$0.515293\pi$
$30$	−18.0000	−0.109545
$31$	7.00000	0.0405560	0.0202780	−	0.999794i	$-0.493545\pi$
$31$	7.00000	0.0405560	0.0202780	+	0.999794i	$0.493545\pi$
$32$	−45.0000	−0.248592
$33$	−30.0000	−0.158252
$34$	0	0
$35$	−42.0000	−0.202837
$36$	−26.0000	−0.120370
$37$	−50.0000	−0.222161	−0.111080	−	0.993811i	$-0.535431\pi$
$37$	−50.0000	−0.222161	−0.111080	+	0.993811i	$0.535431\pi$
$38$	249.000	1.06298
$39$	22.0000	0.0903287
$40$	−126.000	−0.498059
$41$	−360.000	−1.37128	−0.685641	−	0.727940i	$-0.740478\pi$
$41$	−360.000	−1.37128	−0.685641	+	0.727940i	$0.740478\pi$
$42$	21.0000	0.0771517
$43$	68.0000	0.241161	0.120580	−	0.992704i	$-0.461524\pi$
$43$	68.0000	0.241161	0.120580	+	0.992704i	$0.461524\pi$
$44$	30.0000	0.102788
$45$	−156.000	−0.516780
$46$	144.000	0.461557
$47$	−27.0000	−0.0837948	−0.0418974	−	0.999122i	$-0.513340\pi$
$47$	−27.0000	−0.0837948	−0.0418974	+	0.999122i	$0.513340\pi$
$48$	71.0000	0.213499
$49$	49.0000	0.142857
$50$	−267.000	−0.755190
$51$	0	0
$52$	−22.0000	−0.0586702
$53$	213.000	0.552034	0.276017	−	0.961153i	$-0.410985\pi$
$53$	213.000	0.552034	0.276017	+	0.961153i	$0.410985\pi$
$54$	159.000	0.400688
$55$	180.000	0.441294
$56$	147.000	0.350780
$57$	−83.0000	−0.192871
$58$	−45.0000	−0.101876
$59$	189.000	0.417046	0.208523	−	0.978017i	$-0.433134\pi$
$59$	189.000	0.417046	0.208523	+	0.978017i	$0.433134\pi$
$60$	−6.00000	−0.0129099
$61$	−314.000	−0.659075	−0.329538	−	0.944142i	$-0.606893\pi$
$61$	−314.000	−0.659075	−0.329538	+	0.944142i	$0.606893\pi$
$62$	21.0000	0.0430162
$63$	182.000	0.363966
$64$	433.000	0.845703
$65$	−132.000	−0.251886
$66$	−90.0000	−0.167852
$67$	314.000	0.572555	0.286278	−	0.958147i	$-0.407582\pi$
$67$	314.000	0.572555	0.286278	+	0.958147i	$0.407582\pi$
$68$	0	0
$69$	−48.0000	−0.0837467
$70$	−126.000	−0.215141
$71$	804.000	1.34390	0.671952	−	0.740594i	$-0.265456\pi$
$71$	804.000	1.34390	0.671952	+	0.740594i	$0.265456\pi$
$72$	546.000	0.893704
$73$	448.000	0.718280	0.359140	−	0.933284i	$-0.383070\pi$
$73$	448.000	0.718280	0.359140	+	0.933284i	$0.383070\pi$
$74$	−150.000	−0.235637
$75$	89.0000	0.137024
$76$	83.0000	0.125273
$77$	−210.000	−0.310802
$78$	66.0000	0.0958081
$79$	1060.00	1.50961	0.754806	−	0.655948i	$-0.227731\pi$
$79$	1060.00	1.50961	0.754806	+	0.655948i	$0.227731\pi$
$80$	−426.000	−0.595353
$81$	649.000	0.890261
$82$	−1080.00	−1.45446
$83$	873.000	1.15451	0.577254	−	0.816564i	$-0.304124\pi$
$83$	873.000	1.15451	0.577254	+	0.816564i	$0.304124\pi$
$84$	7.00000	0.00909241
$85$	0	0
$86$	204.000	0.255789
$87$	15.0000	0.0184847
$88$	−630.000	−0.763162
$89$	270.000	0.321572	0.160786	−	0.986989i	$-0.448597\pi$
$89$	270.000	0.321572	0.160786	+	0.986989i	$0.448597\pi$
$90$	−468.000	−0.548128
$91$	154.000	0.177402
$92$	48.0000	0.0543951
$93$	−7.00000	−0.00780501
$94$	−81.0000	−0.0888778
$95$	498.000	0.537829
$96$	45.0000	0.0478416
$97$	−1130.00	−1.18283	−0.591413	−	0.806369i	$-0.701430\pi$
$97$	−1130.00	−1.18283	−0.591413	+	0.806369i	$0.701430\pi$
$98$	147.000	0.151523
$99$	−780.000	−0.791848
$100$	−89.0000	−0.0890000
$101$	1668.00	1.64329	0.821645	−	0.570000i	$-0.193057\pi$
$101$	1668.00	1.64329	0.821645	+	0.570000i	$0.193057\pi$
$102$	0	0
$103$	575.000	0.550062	0.275031	−	0.961435i	$-0.411312\pi$
$103$	575.000	0.550062	0.275031	+	0.961435i	$0.411312\pi$
$104$	462.000	0.435604
$105$	42.0000	0.0390360
$106$	639.000	0.585520
$107$	972.000	0.878194	0.439097	−	0.898440i	$-0.355298\pi$
$107$	972.000	0.878194	0.439097	+	0.898440i	$0.355298\pi$
$108$	53.0000	0.0472215
$109$	−407.000	−0.357647	−0.178824	−	0.983881i	$-0.557229\pi$
$109$	−407.000	−0.357647	−0.178824	+	0.983881i	$0.557229\pi$
$110$	540.000	0.468063
$111$	50.0000	0.0427549
$112$	497.000	0.419304
$113$	1002.00	0.834161	0.417081	−	0.908869i	$-0.363053\pi$
$113$	1002.00	0.834161	0.417081	+	0.908869i	$0.363053\pi$
$114$	−249.000	−0.204570
$115$	288.000	0.233532
$116$	−15.0000	−0.0120062
$117$	572.000	0.451978
$118$	567.000	0.442344
$119$	0	0
$120$	126.000	0.0958514
$121$	−431.000	−0.323817
$122$	−942.000	−0.699055
$123$	360.000	0.263903
$124$	7.00000	0.00506950
$125$	−1284.00	−0.918756
$126$	546.000	0.386044
$127$	1022.00	0.714077	0.357039	−	0.934090i	$-0.383786\pi$
$127$	1022.00	0.714077	0.357039	+	0.934090i	$0.383786\pi$
$128$	1659.00	1.14560
$129$	−68.0000	−0.0464114
$130$	−396.000	−0.267165
$131$	−2685.00	−1.79076	−0.895380	−	0.445303i	$-0.853096\pi$
$131$	−2685.00	−1.79076	−0.895380	+	0.445303i	$0.853096\pi$
$132$	−30.0000	−0.0197816
$133$	−581.000	−0.378790
$134$	942.000	0.607287
$135$	318.000	0.202734
$136$	0	0
$137$	−2547.00	−1.58836	−0.794178	−	0.607685i	$-0.792098\pi$
$137$	−2547.00	−1.58836	−0.794178	+	0.607685i	$0.792098\pi$
$138$	−144.000	−0.0888268
$139$	628.000	0.383211	0.191605	−	0.981472i	$-0.438631\pi$
$139$	628.000	0.383211	0.191605	+	0.981472i	$0.438631\pi$
$140$	−42.0000	−0.0253546
$141$	27.0000	0.0161263
$142$	2412.00	1.42543
$143$	−660.000	−0.385958
$144$	1846.00	1.06829
$145$	−90.0000	−0.0515455
$146$	1344.00	0.761851
$147$	−49.0000	−0.0274929
$148$	−50.0000	−0.0277701
$149$	−435.000	−0.239172	−0.119586	−	0.992824i	$-0.538157\pi$
$149$	−435.000	−0.239172	−0.119586	+	0.992824i	$0.538157\pi$
$150$	267.000	0.145336
$151$	1826.00	0.984091	0.492046	−	0.870569i	$-0.336249\pi$
$151$	1826.00	0.984091	0.492046	+	0.870569i	$0.336249\pi$
$152$	−1743.00	−0.930105
$153$	0	0
$154$	−630.000	−0.329655
$155$	42.0000	0.0217647
$156$	22.0000	0.0112911
$157$	3332.00	1.69377	0.846887	−	0.531773i	$-0.178474\pi$
$157$	3332.00	1.69377	0.846887	+	0.531773i	$0.178474\pi$
$158$	3180.00	1.60118
$159$	−213.000	−0.106239
$160$	−270.000	−0.133409
$161$	−336.000	−0.164475
$162$	1947.00	0.944264
$163$	988.000	0.474762	0.237381	−	0.971417i	$-0.423711\pi$
$163$	988.000	0.474762	0.237381	+	0.971417i	$0.423711\pi$
$164$	−360.000	−0.171410
$165$	−180.000	−0.0849272
$166$	2619.00	1.22454
$167$	2211.00	1.02450	0.512252	−	0.858835i	$-0.328811\pi$
$167$	2211.00	1.02450	0.512252	+	0.858835i	$0.328811\pi$
$168$	−147.000	−0.0675077
$169$	−1713.00	−0.779700
$170$	0	0
$171$	−2158.00	−0.965067
$172$	68.0000	0.0301451
$173$	2388.00	1.04946	0.524729	−	0.851269i	$-0.324167\pi$
$173$	2388.00	1.04946	0.524729	+	0.851269i	$0.324167\pi$
$174$	45.0000	0.0196060
$175$	623.000	0.269111
$176$	−2130.00	−0.912243
$177$	−189.000	−0.0802605
$178$	810.000	0.341079
$179$	1968.00	0.821761	0.410881	−	0.911689i	$-0.365221\pi$
$179$	1968.00	0.821761	0.410881	+	0.911689i	$0.365221\pi$
$180$	−156.000	−0.0645975
$181$	652.000	0.267750	0.133875	−	0.990998i	$-0.457258\pi$
$181$	652.000	0.267750	0.133875	+	0.990998i	$0.457258\pi$
$182$	462.000	0.188163
$183$	314.000	0.126839
$184$	−1008.00	−0.403863
$185$	−300.000	−0.119224
$186$	−21.0000	−0.00827847
$187$	0	0
$188$	−27.0000	−0.0104743
$189$	−371.000	−0.142785
$190$	1494.00	0.570453
$191$	4134.00	1.56610	0.783052	−	0.621957i	$-0.213662\pi$
$191$	4134.00	1.56610	0.783052	+	0.621957i	$0.213662\pi$
$192$	−433.000	−0.162756
$193$	−1259.00	−0.469559	−0.234779	−	0.972049i	$-0.575437\pi$
$193$	−1259.00	−0.469559	−0.234779	+	0.972049i	$0.575437\pi$
$194$	−3390.00	−1.25458
$195$	132.000	0.0484755
$196$	49.0000	0.0178571
$197$	−1293.00	−0.467627	−0.233813	−	0.972282i	$-0.575120\pi$
$197$	−1293.00	−0.467627	−0.233813	+	0.972282i	$0.575120\pi$
$198$	−2340.00	−0.839882
$199$	340.000	0.121115	0.0605577	−	0.998165i	$-0.480712\pi$
$199$	340.000	0.121115	0.0605577	+	0.998165i	$0.480712\pi$
$200$	1869.00	0.660791
$201$	−314.000	−0.110188
$202$	5004.00	1.74297
$203$	105.000	0.0363032
$204$	0	0
$205$	−2160.00	−0.735907
$206$	1725.00	0.583429
$207$	−1248.00	−0.419043
$208$	1562.00	0.520698
$209$	2490.00	0.824100
$210$	126.000	0.0414039
$211$	790.000	0.257753	0.128876	−	0.991661i	$-0.458863\pi$
$211$	790.000	0.257753	0.128876	+	0.991661i	$0.458863\pi$
$212$	213.000	0.0690042
$213$	−804.000	−0.258635
$214$	2916.00	0.931466
$215$	408.000	0.129420
$216$	−1113.00	−0.350602
$217$	−49.0000	−0.0153287
$218$	−1221.00	−0.379342
$219$	−448.000	−0.138233
$220$	180.000	0.0551618
$221$	0	0
$222$	150.000	0.0453484
$223$	−97.0000	−0.0291283	−0.0145641	−	0.999894i	$-0.504636\pi$
$223$	−97.0000	−0.0291283	−0.0145641	+	0.999894i	$0.504636\pi$
$224$	315.000	0.0939590
$225$	2314.00	0.685630
$226$	3006.00	0.884762
$227$	−369.000	−0.107892	−0.0539458	−	0.998544i	$-0.517180\pi$
$227$	−369.000	−0.107892	−0.0539458	+	0.998544i	$0.517180\pi$
$228$	−83.0000	−0.0241088
$229$	−1888.00	−0.544815	−0.272407	−	0.962182i	$-0.587820\pi$
$229$	−1888.00	−0.544815	−0.272407	+	0.962182i	$0.587820\pi$
$230$	864.000	0.247698
$231$	210.000	0.0598138
$232$	315.000	0.0891412
$233$	−3633.00	−1.02148	−0.510742	−	0.859734i	$-0.670629\pi$
$233$	−3633.00	−1.02148	−0.510742	+	0.859734i	$0.670629\pi$
$234$	1716.00	0.479395
$235$	−162.000	−0.0449690
$236$	189.000	0.0521307
$237$	−1060.00	−0.290525
$238$	0	0
$239$	−1728.00	−0.467678	−0.233839	−	0.972275i	$-0.575129\pi$
$239$	−1728.00	−0.467678	−0.233839	+	0.972275i	$0.575129\pi$
$240$	426.000	0.114576
$241$	5704.00	1.52459	0.762297	−	0.647228i	$-0.224072\pi$
$241$	5704.00	1.52459	0.762297	+	0.647228i	$0.224072\pi$
$242$	−1293.00	−0.343459
$243$	−2080.00	−0.549103
$244$	−314.000	−0.0823844
$245$	294.000	0.0766652
$246$	1080.00	0.279912
$247$	−1826.00	−0.470387
$248$	−147.000	−0.0376392
$249$	−873.000	−0.222185
$250$	−3852.00	−0.974487
$251$	2928.00	0.736310	0.368155	−	0.929765i	$-0.379990\pi$
$251$	2928.00	0.736310	0.368155	+	0.929765i	$0.379990\pi$
$252$	182.000	0.0454957
$253$	1440.00	0.357834
$254$	3066.00	0.757394
$255$	0	0
$256$	1513.00	0.369385
$257$	−5586.00	−1.35582	−0.677909	−	0.735146i	$-0.737114\pi$
$257$	−5586.00	−1.35582	−0.677909	+	0.735146i	$0.737114\pi$
$258$	−204.000	−0.0492267
$259$	350.000	0.0839689
$260$	−132.000	−0.0314857
$261$	390.000	0.0924919
$262$	−8055.00	−1.89939
$263$	−432.000	−0.101286	−0.0506431	−	0.998717i	$-0.516127\pi$
$263$	−432.000	−0.101286	−0.0506431	+	0.998717i	$0.516127\pi$
$264$	630.000	0.146871
$265$	1278.00	0.296253
$266$	−1743.00	−0.401768
$267$	−270.000	−0.0618866
$268$	314.000	0.0715694
$269$	228.000	0.0516781	0.0258390	−	0.999666i	$-0.491774\pi$
$269$	228.000	0.0516781	0.0258390	+	0.999666i	$0.491774\pi$
$270$	954.000	0.215032
$271$	−3457.00	−0.774900	−0.387450	−	0.921891i	$-0.626644\pi$
$271$	−3457.00	−0.774900	−0.387450	+	0.921891i	$0.626644\pi$
$272$	0	0
$273$	−154.000	−0.0341410
$274$	−7641.00	−1.68471
$275$	−2670.00	−0.585480
$276$	−48.0000	−0.0104683
$277$	2947.00	0.639235	0.319617	−	0.947547i	$-0.396446\pi$
$277$	2947.00	0.639235	0.319617	+	0.947547i	$0.396446\pi$
$278$	1884.00	0.406456
$279$	−182.000	−0.0390540
$280$	882.000	0.188249
$281$	6210.00	1.31835	0.659177	−	0.751988i	$-0.270905\pi$
$281$	6210.00	1.31835	0.659177	+	0.751988i	$0.270905\pi$
$282$	81.0000	0.0171045
$283$	−6935.00	−1.45669	−0.728344	−	0.685211i	$-0.759710\pi$
$283$	−6935.00	−1.45669	−0.728344	+	0.685211i	$0.759710\pi$
$284$	804.000	0.167988
$285$	−498.000	−0.103505
$286$	−1980.00	−0.409370
$287$	2520.00	0.518296
$288$	1170.00	0.239385
$289$	0	0
$290$	−270.000	−0.0546722
$291$	1130.00	0.227635
$292$	448.000	0.0897850
$293$	1614.00	0.321812	0.160906	−	0.986970i	$-0.448558\pi$
$293$	1614.00	0.321812	0.160906	+	0.986970i	$0.448558\pi$
$294$	−147.000	−0.0291606
$295$	1134.00	0.223810
$296$	1050.00	0.206182
$297$	1590.00	0.310644
$298$	−1305.00	−0.253680
$299$	−1056.00	−0.204248
$300$	89.0000	0.0171281
$301$	−476.000	−0.0911501
$302$	5478.00	1.04379
$303$	−1668.00	−0.316251
$304$	−5893.00	−1.11180
$305$	−1884.00	−0.353697
$306$	0	0
$307$	10001.0	1.85924	0.929621	−	0.368517i	$-0.120134\pi$
$307$	10001.0	1.85924	0.929621	+	0.368517i	$0.120134\pi$
$308$	−210.000	−0.0388502
$309$	−575.000	−0.105860
$310$	126.000	0.0230849
$311$	5703.00	1.03983	0.519916	−	0.854218i	$-0.325963\pi$
$311$	5703.00	1.03983	0.519916	+	0.854218i	$0.325963\pi$
$312$	−462.000	−0.0838320
$313$	−10904.0	−1.96911	−0.984554	−	0.175084i	$-0.943980\pi$
$313$	−10904.0	−1.96911	−0.984554	+	0.175084i	$0.943980\pi$
$314$	9996.00	1.79652
$315$	1092.00	0.195325
$316$	1060.00	0.188701
$317$	−9981.00	−1.76842	−0.884209	−	0.467091i	$-0.845302\pi$
$317$	−9981.00	−1.76842	−0.884209	+	0.467091i	$0.845302\pi$
$318$	−639.000	−0.112683
$319$	−450.000	−0.0789817
$320$	2598.00	0.453852
$321$	−972.000	−0.169009
$322$	−1008.00	−0.174452
$323$	0	0
$324$	649.000	0.111283
$325$	1958.00	0.334186
$326$	2964.00	0.503561
$327$	407.000	0.0688292
$328$	7560.00	1.27266
$329$	189.000	0.0316715
$330$	−540.000	−0.0900789
$331$	10118.0	1.68017	0.840084	−	0.542456i	$-0.182506\pi$
$331$	10118.0	1.68017	0.840084	+	0.542456i	$0.182506\pi$
$332$	873.000	0.144314
$333$	1300.00	0.213933
$334$	6633.00	1.08665
$335$	1884.00	0.307265
$336$	−497.000	−0.0806952
$337$	7855.00	1.26970	0.634850	−	0.772635i	$-0.281062\pi$
$337$	7855.00	1.26970	0.634850	+	0.772635i	$0.281062\pi$
$338$	−5139.00	−0.826996
$339$	−1002.00	−0.160534
$340$	0	0
$341$	210.000	0.0333494
$342$	−6474.00	−1.02361
$343$	−343.000	−0.0539949
$344$	−1428.00	−0.223816
$345$	−288.000	−0.0449432
$346$	7164.00	1.11312
$347$	6714.00	1.03869	0.519346	−	0.854564i	$-0.326175\pi$
$347$	6714.00	1.03869	0.519346	+	0.854564i	$0.326175\pi$
$348$	15.0000	0.00231059
$349$	2306.00	0.353689	0.176844	−	0.984239i	$-0.443411\pi$
$349$	2306.00	0.353689	0.176844	+	0.984239i	$0.443411\pi$
$350$	1869.00	0.285435
$351$	−1166.00	−0.177312
$352$	−1350.00	−0.204418
$353$	2490.00	0.375437	0.187719	−	0.982223i	$-0.439891\pi$
$353$	2490.00	0.375437	0.187719	+	0.982223i	$0.439891\pi$
$354$	−567.000	−0.0851291
$355$	4824.00	0.721215
$356$	270.000	0.0401965
$357$	0	0
$358$	5904.00	0.871609
$359$	−4194.00	−0.616576	−0.308288	−	0.951293i	$-0.599756\pi$
$359$	−4194.00	−0.616576	−0.308288	+	0.951293i	$0.599756\pi$
$360$	3276.00	0.479612
$361$	30.0000	0.00437382
$362$	1956.00	0.283992
$363$	431.000	0.0623185
$364$	154.000	0.0221753
$365$	2688.00	0.385469
$366$	942.000	0.134533
$367$	−1223.00	−0.173951	−0.0869756	−	0.996210i	$-0.527720\pi$
$367$	−1223.00	−0.173951	−0.0869756	+	0.996210i	$0.527720\pi$
$368$	−3408.00	−0.482756
$369$	9360.00	1.32049
$370$	−900.000	−0.126456
$371$	−1491.00	−0.208649
$372$	−7.00000	−0.000975627 0
$373$	1577.00	0.218911	0.109456	−	0.993992i	$-0.465089\pi$
$373$	1577.00	0.218911	0.109456	+	0.993992i	$0.465089\pi$
$374$	0	0
$375$	1284.00	0.176815
$376$	567.000	0.0777681
$377$	330.000	0.0450819
$378$	−1113.00	−0.151446
$379$	5620.00	0.761689	0.380844	−	0.924639i	$-0.375633\pi$
$379$	5620.00	0.761689	0.380844	+	0.924639i	$0.375633\pi$
$380$	498.000	0.0672286
$381$	−1022.00	−0.137424
$382$	12402.0	1.66110
$383$	5967.00	0.796082	0.398041	−	0.917368i	$-0.369690\pi$
$383$	5967.00	0.796082	0.398041	+	0.917368i	$0.369690\pi$
$384$	−1659.00	−0.220470
$385$	−1260.00	−0.166794
$386$	−3777.00	−0.498042
$387$	−1768.00	−0.232229
$388$	−1130.00	−0.147853
$389$	−5181.00	−0.675288	−0.337644	−	0.941274i	$-0.609630\pi$
$389$	−5181.00	−0.675288	−0.337644	+	0.941274i	$0.609630\pi$
$390$	396.000	0.0514160
$391$	0	0
$392$	−1029.00	−0.132583
$393$	2685.00	0.344632
$394$	−3879.00	−0.495993
$395$	6360.00	0.810143
$396$	−780.000	−0.0989810
$397$	11050.0	1.39694	0.698468	−	0.715641i	$-0.253865\pi$
$397$	11050.0	1.39694	0.698468	+	0.715641i	$0.253865\pi$
$398$	1020.00	0.128462
$399$	581.000	0.0728982
$400$	6319.00	0.789875
$401$	−8265.00	−1.02926	−0.514631	−	0.857412i	$-0.672071\pi$
$401$	−8265.00	−1.02926	−0.514631	+	0.857412i	$0.672071\pi$
$402$	−942.000	−0.116872
$403$	−154.000	−0.0190355
$404$	1668.00	0.205411
$405$	3894.00	0.477764
$406$	315.000	0.0385054
$407$	−1500.00	−0.182684
$408$	0	0
$409$	3038.00	0.367285	0.183642	−	0.982993i	$-0.441211\pi$
$409$	3038.00	0.367285	0.183642	+	0.982993i	$0.441211\pi$
$410$	−6480.00	−0.780547
$411$	2547.00	0.305679
$412$	575.000	0.0687578
$413$	−1323.00	−0.157629
$414$	−3744.00	−0.444463
$415$	5238.00	0.619574
$416$	990.000	0.116680
$417$	−628.000	−0.0737489
$418$	7470.00	0.874090
$419$	−5628.00	−0.656195	−0.328098	−	0.944644i	$-0.606407\pi$
$419$	−5628.00	−0.656195	−0.328098	+	0.944644i	$0.606407\pi$
$420$	42.0000	0.00487950
$421$	−13855.0	−1.60392	−0.801961	−	0.597376i	$-0.796210\pi$
$421$	−13855.0	−1.60392	−0.801961	+	0.597376i	$0.796210\pi$
$422$	2370.00	0.273388
$423$	702.000	0.0806913
$424$	−4473.00	−0.512330
$425$	0	0
$426$	−2412.00	−0.274323
$427$	2198.00	0.249107
$428$	972.000	0.109774
$429$	660.000	0.0742776
$430$	1224.00	0.137271
$431$	−3714.00	−0.415074	−0.207537	−	0.978227i	$-0.566545\pi$
$431$	−3714.00	−0.415074	−0.207537	+	0.978227i	$0.566545\pi$
$432$	−3763.00	−0.419091
$433$	−13246.0	−1.47012	−0.735060	−	0.678002i	$-0.762846\pi$
$433$	−13246.0	−1.47012	−0.735060	+	0.678002i	$0.762846\pi$
$434$	−147.000	−0.0162586
$435$	90.0000	0.00991993
$436$	−407.000	−0.0447059
$437$	3984.00	0.436111
$438$	−1344.00	−0.146618
$439$	16453.0	1.78874	0.894372	−	0.447323i	$-0.147623\pi$
$439$	16453.0	1.78874	0.894372	+	0.447323i	$0.147623\pi$
$440$	−3780.00	−0.409556
$441$	−1274.00	−0.137566
$442$	0	0
$443$	8088.00	0.867432	0.433716	−	0.901050i	$-0.357202\pi$
$443$	8088.00	0.867432	0.433716	+	0.901050i	$0.357202\pi$
$444$	50.0000	0.00534436
$445$	1620.00	0.172574
$446$	−291.000	−0.0308952
$447$	435.000	0.0460286
$448$	−3031.00	−0.319646
$449$	−5199.00	−0.546450	−0.273225	−	0.961950i	$-0.588090\pi$
$449$	−5199.00	−0.546450	−0.273225	+	0.961950i	$0.588090\pi$
$450$	6942.00	0.727220
$451$	−10800.0	−1.12761
$452$	1002.00	0.104270
$453$	−1826.00	−0.189388
$454$	−1107.00	−0.114436
$455$	924.000	0.0952039
$456$	1743.00	0.178999
$457$	11459.0	1.17293	0.586465	−	0.809974i	$-0.300519\pi$
$457$	11459.0	1.17293	0.586465	+	0.809974i	$0.300519\pi$
$458$	−5664.00	−0.577863
$459$	0	0
$460$	288.000	0.0291915
$461$	1242.00	0.125479	0.0627394	−	0.998030i	$-0.480016\pi$
$461$	1242.00	0.125479	0.0627394	+	0.998030i	$0.480016\pi$
$462$	630.000	0.0634421
$463$	−11596.0	−1.16396	−0.581978	−	0.813204i	$-0.697721\pi$
$463$	−11596.0	−1.16396	−0.581978	+	0.813204i	$0.697721\pi$
$464$	1065.00	0.106555
$465$	−42.0000	−0.00418861
$466$	−10899.0	−1.08345
$467$	−2652.00	−0.262784	−0.131392	−	0.991331i	$-0.541945\pi$
$467$	−2652.00	−0.262784	−0.131392	+	0.991331i	$0.541945\pi$
$468$	572.000	0.0564972
$469$	−2198.00	−0.216406
$470$	−486.000	−0.0476968
$471$	−3332.00	−0.325967
$472$	−3969.00	−0.387051
$473$	2040.00	0.198307
$474$	−3180.00	−0.308148
$475$	−7387.00	−0.713555
$476$	0	0
$477$	−5538.00	−0.531588
$478$	−5184.00	−0.496047
$479$	−1668.00	−0.159108	−0.0795541	−	0.996831i	$-0.525350\pi$
$479$	−1668.00	−0.159108	−0.0795541	+	0.996831i	$0.525350\pi$
$480$	270.000	0.0256745
$481$	1100.00	0.104274
$482$	17112.0	1.61708
$483$	336.000	0.0316533
$484$	−431.000	−0.0404771
$485$	−6780.00	−0.634771
$486$	−6240.00	−0.582412
$487$	−18158.0	−1.68956	−0.844782	−	0.535111i	$-0.820270\pi$
$487$	−18158.0	−1.68956	−0.844782	+	0.535111i	$0.820270\pi$
$488$	6594.00	0.611673
$489$	−988.000	−0.0913679
$490$	882.000	0.0813157
$491$	17022.0	1.56455	0.782273	−	0.622936i	$-0.214060\pi$
$491$	17022.0	1.56455	0.782273	+	0.622936i	$0.214060\pi$
$492$	360.000	0.0329879
$493$	0	0
$494$	−5478.00	−0.498921
$495$	−4680.00	−0.424950
$496$	−497.000	−0.0449919
$497$	−5628.00	−0.507948
$498$	−2619.00	−0.235663
$499$	−9530.00	−0.854953	−0.427476	−	0.904027i	$-0.640597\pi$
$499$	−9530.00	−0.854953	−0.427476	+	0.904027i	$0.640597\pi$
$500$	−1284.00	−0.114844
$501$	−2211.00	−0.197166
$502$	8784.00	0.780974
$503$	−10875.0	−0.964001	−0.482000	−	0.876171i	$-0.660090\pi$
$503$	−10875.0	−0.964001	−0.482000	+	0.876171i	$0.660090\pi$
$504$	−3822.00	−0.337789
$505$	10008.0	0.881881
$506$	4320.00	0.379540
$507$	1713.00	0.150053
$508$	1022.00	0.0892597
$509$	−15798.0	−1.37571	−0.687853	−	0.725850i	$-0.741447\pi$
$509$	−15798.0	−1.37571	−0.687853	+	0.725850i	$0.741447\pi$
$510$	0	0
$511$	−3136.00	−0.271484
$512$	−8733.00	−0.753804
$513$	4399.00	0.378598
$514$	−16758.0	−1.43806
$515$	3450.00	0.295194
$516$	−68.0000	−0.00580142
$517$	−810.000	−0.0689048
$518$	1050.00	0.0890625
$519$	−2388.00	−0.201968
$520$	2772.00	0.233770
$521$	−6018.00	−0.506053	−0.253026	−	0.967459i	$-0.581426\pi$
$521$	−6018.00	−0.506053	−0.253026	+	0.967459i	$0.581426\pi$
$522$	1170.00	0.0981025
$523$	−11896.0	−0.994600	−0.497300	−	0.867579i	$-0.665675\pi$
$523$	−11896.0	−0.994600	−0.497300	+	0.867579i	$0.665675\pi$
$524$	−2685.00	−0.223845
$525$	−623.000	−0.0517904
$526$	−1296.00	−0.107430
$527$	0	0
$528$	2130.00	0.175561
$529$	−9863.00	−0.810635
$530$	3834.00	0.314223
$531$	−4914.00	−0.401600
$532$	−581.000	−0.0473488
$533$	7920.00	0.643627
$534$	−810.000	−0.0656407
$535$	5832.00	0.471288
$536$	−6594.00	−0.531376
$537$	−1968.00	−0.158148
$538$	684.000	0.0548129
$539$	1470.00	0.117472
$540$	318.000	0.0253417
$541$	2167.00	0.172212	0.0861059	−	0.996286i	$-0.472558\pi$
$541$	2167.00	0.172212	0.0861059	+	0.996286i	$0.472558\pi$
$542$	−10371.0	−0.821905
$543$	−652.000	−0.0515285
$544$	0	0
$545$	−2442.00	−0.191934
$546$	−462.000	−0.0362120
$547$	−6218.00	−0.486037	−0.243019	−	0.970022i	$-0.578138\pi$
$547$	−6218.00	−0.486037	−0.243019	+	0.970022i	$0.578138\pi$
$548$	−2547.00	−0.198545
$549$	8164.00	0.634665
$550$	−8010.00	−0.620996
$551$	−1245.00	−0.0962591
$552$	1008.00	0.0777234
$553$	−7420.00	−0.570580
$554$	8841.00	0.678011
$555$	300.000	0.0229447
$556$	628.000	0.0479013
$557$	17130.0	1.30309	0.651545	−	0.758610i	$-0.274121\pi$
$557$	17130.0	1.30309	0.651545	+	0.758610i	$0.274121\pi$
$558$	−546.000	−0.0414230
$559$	−1496.00	−0.113192
$560$	2982.00	0.225022
$561$	0	0
$562$	18630.0	1.39833
$563$	−14859.0	−1.11231	−0.556156	−	0.831078i	$-0.687724\pi$
$563$	−14859.0	−1.11231	−0.556156	+	0.831078i	$0.687724\pi$
$564$	27.0000	0.00201579
$565$	6012.00	0.447658
$566$	−20805.0	−1.54505
$567$	−4543.00	−0.336487
$568$	−16884.0	−1.24725
$569$	2601.00	0.191634	0.0958169	−	0.995399i	$-0.469454\pi$
$569$	2601.00	0.191634	0.0958169	+	0.995399i	$0.469454\pi$
$570$	−1494.00	−0.109784
$571$	−6404.00	−0.469350	−0.234675	−	0.972074i	$-0.575403\pi$
$571$	−6404.00	−0.469350	−0.234675	+	0.972074i	$0.575403\pi$
$572$	−660.000	−0.0482447
$573$	−4134.00	−0.301397
$574$	7560.00	0.549736
$575$	−4272.00	−0.309834
$576$	−11258.0	−0.814381
$577$	4166.00	0.300577	0.150288	−	0.988642i	$-0.451980\pi$
$577$	4166.00	0.300577	0.150288	+	0.988642i	$0.451980\pi$
$578$	0	0
$579$	1259.00	0.0903666
$580$	−90.0000	−0.00644318
$581$	−6111.00	−0.436363
$582$	3390.00	0.241443
$583$	6390.00	0.453940
$584$	−9408.00	−0.666620
$585$	3432.00	0.242557
$586$	4842.00	0.341333
$587$	25653.0	1.80377	0.901885	−	0.431977i	$-0.142184\pi$
$587$	25653.0	1.80377	0.901885	+	0.431977i	$0.142184\pi$
$588$	−49.0000	−0.00343661
$589$	581.000	0.0406446
$590$	3402.00	0.237387
$591$	1293.00	0.0899948
$592$	3550.00	0.246460
$593$	2268.00	0.157058	0.0785292	−	0.996912i	$-0.474978\pi$
$593$	2268.00	0.157058	0.0785292	+	0.996912i	$0.474978\pi$
$594$	4770.00	0.329487
$595$	0	0
$596$	−435.000	−0.0298965
$597$	−340.000	−0.0233087
$598$	−3168.00	−0.216637
$599$	−4938.00	−0.336830	−0.168415	−	0.985716i	$-0.553865\pi$
$599$	−4938.00	−0.336830	−0.168415	+	0.985716i	$0.553865\pi$
$600$	−1869.00	−0.127169
$601$	−8042.00	−0.545824	−0.272912	−	0.962039i	$-0.587987\pi$
$601$	−8042.00	−0.545824	−0.272912	+	0.962039i	$0.587987\pi$
$602$	−1428.00	−0.0966793
$603$	−8164.00	−0.551350
$604$	1826.00	0.123011
$605$	−2586.00	−0.173778
$606$	−5004.00	−0.335435
$607$	10753.0	0.719029	0.359515	−	0.933139i	$-0.382942\pi$
$607$	10753.0	0.719029	0.359515	+	0.933139i	$0.382942\pi$
$608$	−3735.00	−0.249135
$609$	−105.000	−0.00698656
$610$	−5652.00	−0.375152
$611$	594.000	0.0393301
$612$	0	0
$613$	−17218.0	−1.13447	−0.567234	−	0.823557i	$-0.691986\pi$
$613$	−17218.0	−1.13447	−0.567234	+	0.823557i	$0.691986\pi$
$614$	30003.0	1.97202
$615$	2160.00	0.141625
$616$	4410.00	0.288448
$617$	−5097.00	−0.332573	−0.166286	−	0.986077i	$-0.553178\pi$
$617$	−5097.00	−0.332573	−0.166286	+	0.986077i	$0.553178\pi$
$618$	−1725.00	−0.112281
$619$	11095.0	0.720429	0.360215	−	0.932869i	$-0.382703\pi$
$619$	11095.0	0.720429	0.360215	+	0.932869i	$0.382703\pi$
$620$	42.0000	0.00272058
$621$	2544.00	0.164392
$622$	17109.0	1.10291
$623$	−1890.00	−0.121543
$624$	−1562.00	−0.100208
$625$	3421.00	0.218944
$626$	−32712.0	−2.08855
$627$	−2490.00	−0.158598
$628$	3332.00	0.211722
$629$	0	0
$630$	3276.00	0.207173
$631$	21458.0	1.35377	0.676885	−	0.736088i	$-0.263329\pi$
$631$	21458.0	1.35377	0.676885	+	0.736088i	$0.263329\pi$
$632$	−22260.0	−1.40104
$633$	−790.000	−0.0496046
$634$	−29943.0	−1.87569
$635$	6132.00	0.383214
$636$	−213.000	−0.0132799
$637$	−1078.00	−0.0670517
$638$	−1350.00	−0.0837727
$639$	−20904.0	−1.29413
$640$	9954.00	0.614791
$641$	−10161.0	−0.626108	−0.313054	−	0.949735i	$-0.601352\pi$
$641$	−10161.0	−0.626108	−0.313054	+	0.949735i	$0.601352\pi$
$642$	−2916.00	−0.179261
$643$	17524.0	1.07477	0.537387	−	0.843336i	$-0.319412\pi$
$643$	17524.0	1.07477	0.537387	+	0.843336i	$0.319412\pi$
$644$	−336.000	−0.0205594
$645$	−408.000	−0.0249070
$646$	0	0
$647$	4896.00	0.297499	0.148749	−	0.988875i	$-0.452475\pi$
$647$	4896.00	0.297499	0.148749	+	0.988875i	$0.452475\pi$
$648$	−13629.0	−0.826231
$649$	5670.00	0.342938
$650$	5874.00	0.354457
$651$	49.0000	0.00295002
$652$	988.000	0.0593452
$653$	10929.0	0.654954	0.327477	−	0.944859i	$-0.393802\pi$
$653$	10929.0	0.654954	0.327477	+	0.944859i	$0.393802\pi$
$654$	1221.00	0.0730044
$655$	−16110.0	−0.961023
$656$	25560.0	1.52127
$657$	−11648.0	−0.691677
$658$	567.000	0.0335926
$659$	−28632.0	−1.69248	−0.846240	−	0.532802i	$-0.821139\pi$
$659$	−28632.0	−1.69248	−0.846240	+	0.532802i	$0.821139\pi$
$660$	−180.000	−0.0106159
$661$	7982.00	0.469688	0.234844	−	0.972033i	$-0.424542\pi$
$661$	7982.00	0.469688	0.234844	+	0.972033i	$0.424542\pi$
$662$	30354.0	1.78209
$663$	0	0
$664$	−18333.0	−1.07147
$665$	−3486.00	−0.203280
$666$	3900.00	0.226910
$667$	−720.000	−0.0417969
$668$	2211.00	0.128063
$669$	97.0000	0.00560573
$670$	5652.00	0.325904
$671$	−9420.00	−0.541960
$672$	−315.000	−0.0180824
$673$	1015.00	0.0581358	0.0290679	−	0.999577i	$-0.490746\pi$
$673$	1015.00	0.0581358	0.0290679	+	0.999577i	$0.490746\pi$
$674$	23565.0	1.34672
$675$	−4717.00	−0.268974
$676$	−1713.00	−0.0974624
$677$	−264.000	−0.0149872	−0.00749361	−	0.999972i	$-0.502385\pi$
$677$	−264.000	−0.0149872	−0.00749361	+	0.999972i	$0.502385\pi$
$678$	−3006.00	−0.170272
$679$	7910.00	0.447066
$680$	0	0
$681$	369.000	0.0207637
$682$	630.000	0.0353724
$683$	−10176.0	−0.570093	−0.285047	−	0.958514i	$-0.592009\pi$
$683$	−10176.0	−0.570093	−0.285047	+	0.958514i	$0.592009\pi$
$684$	−2158.00	−0.120633
$685$	−15282.0	−0.852402
$686$	−1029.00	−0.0572703
$687$	1888.00	0.104850
$688$	−4828.00	−0.267537
$689$	−4686.00	−0.259104
$690$	−864.000	−0.0476694
$691$	27625.0	1.52085	0.760423	−	0.649428i	$-0.224992\pi$
$691$	27625.0	1.52085	0.760423	+	0.649428i	$0.224992\pi$
$692$	2388.00	0.131182
$693$	5460.00	0.299290
$694$	20142.0	1.10170
$695$	3768.00	0.205652
$696$	−315.000	−0.0171552
$697$	0	0
$698$	6918.00	0.375143
$699$	3633.00	0.196585
$700$	623.000	0.0336388
$701$	5187.00	0.279473	0.139736	−	0.990189i	$-0.455375\pi$
$701$	5187.00	0.279473	0.139736	+	0.990189i	$0.455375\pi$
$702$	−3498.00	−0.188068
$703$	−4150.00	−0.222646
$704$	12990.0	0.695425
$705$	162.000	0.00865429
$706$	7470.00	0.398211
$707$	−11676.0	−0.621105
$708$	−189.000	−0.0100326
$709$	23650.0	1.25274	0.626371	−	0.779525i	$-0.284539\pi$
$709$	23650.0	1.25274	0.626371	+	0.779525i	$0.284539\pi$
$710$	14472.0	0.764964
$711$	−27560.0	−1.45370
$712$	−5670.00	−0.298444
$713$	336.000	0.0176484
$714$	0	0
$715$	−3960.00	−0.207127
$716$	1968.00	0.102720
$717$	1728.00	0.0900047
$718$	−12582.0	−0.653978
$719$	−4272.00	−0.221584	−0.110792	−	0.993844i	$-0.535339\pi$
$719$	−4272.00	−0.221584	−0.110792	+	0.993844i	$0.535339\pi$
$720$	11076.0	0.573303
$721$	−4025.00	−0.207904
$722$	90.0000	0.00463913
$723$	−5704.00	−0.293408
$724$	652.000	0.0334688
$725$	1335.00	0.0683871
$726$	1293.00	0.0660988
$727$	−10504.0	−0.535862	−0.267931	−	0.963438i	$-0.586340\pi$
$727$	−10504.0	−0.535862	−0.267931	+	0.963438i	$0.586340\pi$
$728$	−3234.00	−0.164643
$729$	−15443.0	−0.784586
$730$	8064.00	0.408852
$731$	0	0
$732$	314.000	0.0158549
$733$	36992.0	1.86403	0.932013	−	0.362425i	$-0.118051\pi$
$733$	36992.0	1.86403	0.932013	+	0.362425i	$0.118051\pi$
$734$	−3669.00	−0.184503
$735$	−294.000	−0.0147542
$736$	−2160.00	−0.108178
$737$	9420.00	0.470814
$738$	28080.0	1.40059
$739$	−35566.0	−1.77039	−0.885194	−	0.465222i	$-0.845974\pi$
$739$	−35566.0	−1.77039	−0.885194	+	0.465222i	$0.845974\pi$
$740$	−300.000	−0.0149030
$741$	1826.00	0.0905260
$742$	−4473.00	−0.221306
$743$	−35982.0	−1.77665	−0.888325	−	0.459215i	$-0.848131\pi$
$743$	−35982.0	−1.77665	−0.888325	+	0.459215i	$0.848131\pi$
$744$	147.000	0.00724366
$745$	−2610.00	−0.128353
$746$	4731.00	0.232191
$747$	−22698.0	−1.11175
$748$	0	0
$749$	−6804.00	−0.331926
$750$	3852.00	0.187540
$751$	8230.00	0.399889	0.199945	−	0.979807i	$-0.435924\pi$
$751$	8230.00	0.399889	0.199945	+	0.979807i	$0.435924\pi$
$752$	1917.00	0.0929598
$753$	−2928.00	−0.141703
$754$	990.000	0.0478165
$755$	10956.0	0.528119
$756$	−371.000	−0.0178481
$757$	12887.0	0.618740	0.309370	−	0.950942i	$-0.399882\pi$
$757$	12887.0	0.618740	0.309370	+	0.950942i	$0.399882\pi$
$758$	16860.0	0.807893
$759$	−1440.00	−0.0688652
$760$	−10458.0	−0.499147
$761$	−36042.0	−1.71685	−0.858424	−	0.512941i	$-0.828556\pi$
$761$	−36042.0	−1.71685	−0.858424	+	0.512941i	$0.828556\pi$
$762$	−3066.00	−0.145760
$763$	2849.00	0.135178
$764$	4134.00	0.195763
$765$	0	0
$766$	17901.0	0.844373
$767$	−4158.00	−0.195745
$768$	−1513.00	−0.0710881
$769$	3170.00	0.148652	0.0743258	−	0.997234i	$-0.476320\pi$
$769$	3170.00	0.148652	0.0743258	+	0.997234i	$0.476320\pi$
$770$	−3780.00	−0.176911
$771$	5586.00	0.260927
$772$	−1259.00	−0.0586948
$773$	−21390.0	−0.995271	−0.497636	−	0.867386i	$-0.665798\pi$
$773$	−21390.0	−0.995271	−0.497636	+	0.867386i	$0.665798\pi$
$774$	−5304.00	−0.246316
$775$	−623.000	−0.0288759
$776$	23730.0	1.09775
$777$	−350.000	−0.0161598
$778$	−15543.0	−0.716251
$779$	−29880.0	−1.37428
$780$	132.000	0.00605943
$781$	24120.0	1.10510
$782$	0	0
$783$	−795.000	−0.0362848
$784$	−3479.00	−0.158482
$785$	19992.0	0.908975
$786$	8055.00	0.365537
$787$	31708.0	1.43617	0.718086	−	0.695954i	$-0.245018\pi$
$787$	31708.0	1.43617	0.718086	+	0.695954i	$0.245018\pi$
$788$	−1293.00	−0.0584533
$789$	432.000	0.0194925
$790$	19080.0	0.859286
$791$	−7014.00	−0.315283
$792$	16380.0	0.734896
$793$	6908.00	0.309345
$794$	33150.0	1.48167
$795$	−1278.00	−0.0570138
$796$	340.000	0.0151394
$797$	12114.0	0.538394	0.269197	−	0.963085i	$-0.413242\pi$
$797$	12114.0	0.538394	0.269197	+	0.963085i	$0.413242\pi$
$798$	1743.00	0.0773202
$799$	0	0
$800$	4005.00	0.176998
$801$	−7020.00	−0.309662
$802$	−24795.0	−1.09170
$803$	13440.0	0.590644
$804$	−314.000	−0.0137735
$805$	−2016.00	−0.0882667
$806$	−462.000	−0.0201901
$807$	−228.000	−0.00994545
$808$	−35028.0	−1.52510
$809$	−37086.0	−1.61171	−0.805856	−	0.592112i	$-0.798294\pi$
$809$	−37086.0	−1.61171	−0.805856	+	0.592112i	$0.798294\pi$
$810$	11682.0	0.506745
$811$	−20063.0	−0.868690	−0.434345	−	0.900747i	$-0.643020\pi$
$811$	−20063.0	−0.868690	−0.434345	+	0.900747i	$0.643020\pi$
$812$	105.000	0.00453790
$813$	3457.00	0.149129
$814$	−4500.00	−0.193765
$815$	5928.00	0.254784
$816$	0	0
$817$	5644.00	0.241687
$818$	9114.00	0.389564
$819$	−4004.00	−0.170832
$820$	−2160.00	−0.0919884
$821$	18642.0	0.792461	0.396230	−	0.918151i	$-0.370318\pi$
$821$	18642.0	0.792461	0.396230	+	0.918151i	$0.370318\pi$
$822$	7641.00	0.324222
$823$	10606.0	0.449213	0.224606	−	0.974450i	$-0.427890\pi$
$823$	10606.0	0.449213	0.224606	+	0.974450i	$0.427890\pi$
$824$	−12075.0	−0.510501
$825$	2670.00	0.112676
$826$	−3969.00	−0.167190
$827$	32154.0	1.35200	0.676000	−	0.736902i	$-0.263712\pi$
$827$	32154.0	1.35200	0.676000	+	0.736902i	$0.263712\pi$
$828$	−1248.00	−0.0523804
$829$	32198.0	1.34895	0.674477	−	0.738296i	$-0.264369\pi$
$829$	32198.0	1.34895	0.674477	+	0.738296i	$0.264369\pi$
$830$	15714.0	0.657158
$831$	−2947.00	−0.123021
$832$	−9526.00	−0.396941
$833$	0	0
$834$	−1884.00	−0.0782225
$835$	13266.0	0.549807
$836$	2490.00	0.103013
$837$	371.000	0.0153210
$838$	−16884.0	−0.696000
$839$	−18627.0	−0.766478	−0.383239	−	0.923649i	$-0.625192\pi$
$839$	−18627.0	−0.766478	−0.383239	+	0.923649i	$0.625192\pi$
$840$	−882.000	−0.0362284
$841$	−24164.0	−0.990775
$842$	−41565.0	−1.70122
$843$	−6210.00	−0.253717
$844$	790.000	0.0322191
$845$	−10278.0	−0.418431
$846$	2106.00	0.0855860
$847$	3017.00	0.122391
$848$	−15123.0	−0.612413
$849$	6935.00	0.280340
$850$	0	0
$851$	−2400.00	−0.0966756
$852$	−804.000	−0.0323293
$853$	−6668.00	−0.267653	−0.133826	−	0.991005i	$-0.542726\pi$
$853$	−6668.00	−0.267653	−0.133826	+	0.991005i	$0.542726\pi$
$854$	6594.00	0.264218
$855$	−12948.0	−0.517909
$856$	−20412.0	−0.815032
$857$	2328.00	0.0927923	0.0463961	−	0.998923i	$-0.485226\pi$
$857$	2328.00	0.0927923	0.0463961	+	0.998923i	$0.485226\pi$
$858$	1980.00	0.0787833
$859$	33761.0	1.34099	0.670495	−	0.741914i	$-0.266082\pi$
$859$	33761.0	1.34099	0.670495	+	0.741914i	$0.266082\pi$
$860$	408.000	0.0161775
$861$	−2520.00	−0.0997461
$862$	−11142.0	−0.440253
$863$	19740.0	0.778630	0.389315	−	0.921105i	$-0.372712\pi$
$863$	19740.0	0.778630	0.389315	+	0.921105i	$0.372712\pi$
$864$	−2385.00	−0.0939113
$865$	14328.0	0.563198
$866$	−39738.0	−1.55930
$867$	0	0
$868$	−49.0000	−0.00191609
$869$	31800.0	1.24136
$870$	270.000	0.0105217
$871$	−6908.00	−0.268736
$872$	8547.00	0.331924
$873$	29380.0	1.13902
$874$	11952.0	0.462566
$875$	8988.00	0.347257
$876$	−448.000	−0.0172791
$877$	−28946.0	−1.11452	−0.557262	−	0.830337i	$-0.688148\pi$
$877$	−28946.0	−1.11452	−0.557262	+	0.830337i	$0.688148\pi$
$878$	49359.0	1.89725
$879$	−1614.00	−0.0619327
$880$	−12780.0	−0.489561
$881$	−4806.00	−0.183789	−0.0918946	−	0.995769i	$-0.529292\pi$
$881$	−4806.00	−0.183789	−0.0918946	+	0.995769i	$0.529292\pi$
$882$	−3822.00	−0.145911
$883$	12638.0	0.481656	0.240828	−	0.970568i	$-0.422581\pi$
$883$	12638.0	0.481656	0.240828	+	0.970568i	$0.422581\pi$
$884$	0	0
$885$	−1134.00	−0.0430723
$886$	24264.0	0.920051
$887$	40092.0	1.51765	0.758826	−	0.651293i	$-0.225773\pi$
$887$	40092.0	1.51765	0.758826	+	0.651293i	$0.225773\pi$
$888$	−1050.00	−0.0396798
$889$	−7154.00	−0.269896
$890$	4860.00	0.183042
$891$	19470.0	0.732065
$892$	−97.0000	−0.00364103
$893$	−2241.00	−0.0839778
$894$	1305.00	0.0488207
$895$	11808.0	0.441003
$896$	−11613.0	−0.432995
$897$	1056.00	0.0393075
$898$	−15597.0	−0.579598
$899$	−105.000	−0.00389538
$900$	2314.00	0.0857037
$901$	0	0
$902$	−32400.0	−1.19601
$903$	476.000	0.0175418
$904$	−21042.0	−0.774166
$905$	3912.00	0.143690
$906$	−5478.00	−0.200877
$907$	−39344.0	−1.44035	−0.720174	−	0.693793i	$-0.755938\pi$
$907$	−39344.0	−1.44035	−0.720174	+	0.693793i	$0.755938\pi$
$908$	−369.000	−0.0134864
$909$	−43368.0	−1.58243
$910$	2772.00	0.100979
$911$	−3342.00	−0.121543	−0.0607714	−	0.998152i	$-0.519356\pi$
$911$	−3342.00	−0.121543	−0.0607714	+	0.998152i	$0.519356\pi$
$912$	5893.00	0.213966
$913$	26190.0	0.949357
$914$	34377.0	1.24408
$915$	1884.00	0.0680690
$916$	−1888.00	−0.0681018
$917$	18795.0	0.676844
$918$	0	0
$919$	−21454.0	−0.770079	−0.385039	−	0.922900i	$-0.625812\pi$
$919$	−21454.0	−0.770079	−0.385039	+	0.922900i	$0.625812\pi$
$920$	−6048.00	−0.216735
$921$	−10001.0	−0.357811
$922$	3726.00	0.133090
$923$	−17688.0	−0.630777
$924$	210.000	0.00747672
$925$	4450.00	0.158178
$926$	−34788.0	−1.23456
$927$	−14950.0	−0.529690
$928$	675.000	0.0238771
$929$	46992.0	1.65959	0.829794	−	0.558070i	$-0.188458\pi$
$929$	46992.0	1.65959	0.829794	+	0.558070i	$0.188458\pi$
$930$	−126.000	−0.00444269
$931$	4067.00	0.143169
$932$	−3633.00	−0.127685
$933$	−5703.00	−0.200116
$934$	−7956.00	−0.278724
$935$	0	0
$936$	−12012.0	−0.419471
$937$	33584.0	1.17091	0.585454	−	0.810705i	$-0.300916\pi$
$937$	33584.0	1.17091	0.585454	+	0.810705i	$0.300916\pi$
$938$	−6594.00	−0.229533
$939$	10904.0	0.378955
$940$	−162.000	−0.00562112
$941$	−46176.0	−1.59968	−0.799838	−	0.600216i	$-0.795081\pi$
$941$	−46176.0	−1.59968	−0.799838	+	0.600216i	$0.795081\pi$
$942$	−9996.00	−0.345740
$943$	−17280.0	−0.596728
$944$	−13419.0	−0.462660
$945$	−2226.00	−0.0766262
$946$	6120.00	0.210337
$947$	906.000	0.0310887	0.0155444	−	0.999879i	$-0.495052\pi$
$947$	906.000	0.0310887	0.0155444	+	0.999879i	$0.495052\pi$
$948$	−1060.00	−0.0363156
$949$	−9856.00	−0.337133
$950$	−22161.0	−0.756840
$951$	9981.00	0.340332
$952$	0	0
$953$	−35937.0	−1.22153	−0.610763	−	0.791814i	$-0.709137\pi$
$953$	−35937.0	−1.22153	−0.610763	+	0.791814i	$0.709137\pi$
$954$	−16614.0	−0.563834
$955$	24804.0	0.840459
$956$	−1728.00	−0.0584597
$957$	450.000	0.0152000
$958$	−5004.00	−0.168760
$959$	17829.0	0.600342
$960$	−2598.00	−0.0873438
$961$	−29742.0	−0.998355
$962$	3300.00	0.110599
$963$	−25272.0	−0.845669
$964$	5704.00	0.190574
$965$	−7554.00	−0.251992
$966$	1008.00	0.0335734
$967$	29990.0	0.997325	0.498663	−	0.866796i	$-0.333825\pi$
$967$	29990.0	0.997325	0.498663	+	0.866796i	$0.333825\pi$
$968$	9051.00	0.300527
$969$	0	0
$970$	−20340.0	−0.673276
$971$	34896.0	1.15331	0.576656	−	0.816987i	$-0.304357\pi$
$971$	34896.0	1.15331	0.576656	+	0.816987i	$0.304357\pi$
$972$	−2080.00	−0.0686379
$973$	−4396.00	−0.144840
$974$	−54474.0	−1.79205
$975$	−1958.00	−0.0643140
$976$	22294.0	0.731161
$977$	−14931.0	−0.488930	−0.244465	−	0.969658i	$-0.578612\pi$
$977$	−14931.0	−0.488930	−0.244465	+	0.969658i	$0.578612\pi$
$978$	−2964.00	−0.0969103
$979$	8100.00	0.264430
$980$	294.000	0.00958315
$981$	10582.0	0.344401
$982$	51066.0	1.65945
$983$	−56109.0	−1.82055	−0.910274	−	0.414006i	$-0.864129\pi$
$983$	−56109.0	−1.82055	−0.910274	+	0.414006i	$0.864129\pi$
$984$	−7560.00	−0.244923
$985$	−7758.00	−0.250955
$986$	0	0
$987$	−189.000	−0.00609517
$988$	−1826.00	−0.0587984
$989$	3264.00	0.104944
$990$	−14040.0	−0.450728
$991$	−5762.00	−0.184698	−0.0923491	−	0.995727i	$-0.529438\pi$
$991$	−5762.00	−0.184698	−0.0923491	+	0.995727i	$0.529438\pi$
$992$	−315.000	−0.0100819
$993$	−10118.0	−0.323348
$994$	−16884.0	−0.538761
$995$	2040.00	0.0649973
$996$	−873.000	−0.0277732
$997$	27604.0	0.876858	0.438429	−	0.898766i	$-0.355535\pi$
$997$	27604.0	0.876858	0.438429	+	0.898766i	$0.355535\pi$
$998$	−28590.0	−0.906814
$999$	−2650.00	−0.0839262

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2023.4.a.c.1.1	✓	1
17.16	even	2		2023.4.a.d.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2023.4.a.c.1.1	✓	1	1.1	even	1	trivial
2023.4.a.d.1.1	yes	1	17.16	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:	\( N \)	\(=\)	\( 2023 = 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2023.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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