LMFDB - Embedded newform 2415.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2415,4,Mod(1,2415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2415, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2415.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2415 = 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2415.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:	$142.489612664$
Analytic rank:	$1$
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	$\chi$	$=$	2415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -5.00000 q^{5} -3.00000 q^{6} -7.00000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -5.00000 q^{5} -3.00000 q^{6} -7.00000 q^{7} -15.0000 q^{8} +9.00000 q^{9} -5.00000 q^{10} -50.0000 q^{11} +21.0000 q^{12} +46.0000 q^{13} -7.00000 q^{14} +15.0000 q^{15} +41.0000 q^{16} +12.0000 q^{17} +9.00000 q^{18} -98.0000 q^{19} +35.0000 q^{20} +21.0000 q^{21} -50.0000 q^{22} +23.0000 q^{23} +45.0000 q^{24} +25.0000 q^{25} +46.0000 q^{26} -27.0000 q^{27} +49.0000 q^{28} -160.000 q^{29} +15.0000 q^{30} +166.000 q^{31} +161.000 q^{32} +150.000 q^{33} +12.0000 q^{34} +35.0000 q^{35} -63.0000 q^{36} +242.000 q^{37} -98.0000 q^{38} -138.000 q^{39} +75.0000 q^{40} -334.000 q^{41} +21.0000 q^{42} +318.000 q^{43} +350.000 q^{44} -45.0000 q^{45} +23.0000 q^{46} +174.000 q^{47} -123.000 q^{48} +49.0000 q^{49} +25.0000 q^{50} -36.0000 q^{51} -322.000 q^{52} +238.000 q^{53} -27.0000 q^{54} +250.000 q^{55} +105.000 q^{56} +294.000 q^{57} -160.000 q^{58} +340.000 q^{59} -105.000 q^{60} +118.000 q^{61} +166.000 q^{62} -63.0000 q^{63} -167.000 q^{64} -230.000 q^{65} +150.000 q^{66} +254.000 q^{67} -84.0000 q^{68} -69.0000 q^{69} +35.0000 q^{70} -458.000 q^{71} -135.000 q^{72} +858.000 q^{73} +242.000 q^{74} -75.0000 q^{75} +686.000 q^{76} +350.000 q^{77} -138.000 q^{78} +144.000 q^{79} -205.000 q^{80} +81.0000 q^{81} -334.000 q^{82} +210.000 q^{83} -147.000 q^{84} -60.0000 q^{85} +318.000 q^{86} +480.000 q^{87} +750.000 q^{88} -1066.00 q^{89} -45.0000 q^{90} -322.000 q^{91} -161.000 q^{92} -498.000 q^{93} +174.000 q^{94} +490.000 q^{95} -483.000 q^{96} +214.000 q^{97} +49.0000 q^{98} -450.000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	1.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.00000	−0.875000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−3.00000	−0.204124
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−15.0000	−0.662913
$9$	9.00000	0.333333
$10$	−5.00000	−0.158114
$11$	−50.0000	−1.37051	−0.685253	−	0.728305i	$-0.740308\pi$
$11$	−50.0000	−1.37051	−0.685253	+	0.728305i	$0.740308\pi$
$12$	21.0000	0.505181
$13$	46.0000	0.981393	0.490696	−	0.871331i	$-0.336742\pi$
$13$	46.0000	0.981393	0.490696	+	0.871331i	$0.336742\pi$
$14$	−7.00000	−0.133631
$15$	15.0000	0.258199
$16$	41.0000	0.640625
$17$	12.0000	0.171202	0.0856008	−	0.996330i	$-0.472719\pi$
$17$	12.0000	0.171202	0.0856008	+	0.996330i	$0.472719\pi$
$18$	9.00000	0.117851
$19$	−98.0000	−1.18330	−0.591651	−	0.806194i	$-0.701524\pi$
$19$	−98.0000	−1.18330	−0.591651	+	0.806194i	$0.701524\pi$
$20$	35.0000	0.391312
$21$	21.0000	0.218218
$22$	−50.0000	−0.484547
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	45.0000	0.382733
$25$	25.0000	0.200000
$26$	46.0000	0.346975
$27$	−27.0000	−0.192450
$28$	49.0000	0.330719
$29$	−160.000	−1.02453	−0.512263	−	0.858829i	$-0.671193\pi$
$29$	−160.000	−1.02453	−0.512263	+	0.858829i	$0.671193\pi$
$30$	15.0000	0.0912871
$31$	166.000	0.961757	0.480879	−	0.876787i	$-0.340318\pi$
$31$	166.000	0.961757	0.480879	+	0.876787i	$0.340318\pi$
$32$	161.000	0.889408
$33$	150.000	0.791262
$34$	12.0000	0.0605289
$35$	35.0000	0.169031
$36$	−63.0000	−0.291667
$37$	242.000	1.07526	0.537629	−	0.843181i	$-0.319320\pi$
$37$	242.000	1.07526	0.537629	+	0.843181i	$0.319320\pi$
$38$	−98.0000	−0.418361
$39$	−138.000	−0.566607
$40$	75.0000	0.296464
$41$	−334.000	−1.27224	−0.636122	−	0.771588i	$-0.719463\pi$
$41$	−334.000	−1.27224	−0.636122	+	0.771588i	$0.719463\pi$
$42$	21.0000	0.0771517
$43$	318.000	1.12778	0.563890	−	0.825850i	$-0.309304\pi$
$43$	318.000	1.12778	0.563890	+	0.825850i	$0.309304\pi$
$44$	350.000	1.19919
$45$	−45.0000	−0.149071
$46$	23.0000	0.0737210
$47$	174.000	0.540011	0.270005	−	0.962859i	$-0.412975\pi$
$47$	174.000	0.540011	0.270005	+	0.962859i	$0.412975\pi$
$48$	−123.000	−0.369865
$49$	49.0000	0.142857
$50$	25.0000	0.0707107
$51$	−36.0000	−0.0988433
$52$	−322.000	−0.858719
$53$	238.000	0.616827	0.308413	−	0.951252i	$-0.400202\pi$
$53$	238.000	0.616827	0.308413	+	0.951252i	$0.400202\pi$
$54$	−27.0000	−0.0680414
$55$	250.000	0.612909
$56$	105.000	0.250557
$57$	294.000	0.683180
$58$	−160.000	−0.362225
$59$	340.000	0.750241	0.375121	−	0.926976i	$-0.377601\pi$
$59$	340.000	0.750241	0.375121	+	0.926976i	$0.377601\pi$
$60$	−105.000	−0.225924
$61$	118.000	0.247678	0.123839	−	0.992302i	$-0.460479\pi$
$61$	118.000	0.247678	0.123839	+	0.992302i	$0.460479\pi$
$62$	166.000	0.340033
$63$	−63.0000	−0.125988
$64$	−167.000	−0.326172
$65$	−230.000	−0.438892
$66$	150.000	0.279753
$67$	254.000	0.463150	0.231575	−	0.972817i	$-0.425612\pi$
$67$	254.000	0.463150	0.231575	+	0.972817i	$0.425612\pi$
$68$	−84.0000	−0.149801
$69$	−69.0000	−0.120386
$70$	35.0000	0.0597614
$71$	−458.000	−0.765558	−0.382779	−	0.923840i	$-0.625033\pi$
$71$	−458.000	−0.765558	−0.382779	+	0.923840i	$0.625033\pi$
$72$	−135.000	−0.220971
$73$	858.000	1.37563	0.687817	−	0.725884i	$-0.258569\pi$
$73$	858.000	1.37563	0.687817	+	0.725884i	$0.258569\pi$
$74$	242.000	0.380161
$75$	−75.0000	−0.115470
$76$	686.000	1.03539
$77$	350.000	0.518003
$78$	−138.000	−0.200326
$79$	144.000	0.205079	0.102540	−	0.994729i	$-0.467303\pi$
$79$	144.000	0.205079	0.102540	+	0.994729i	$0.467303\pi$
$80$	−205.000	−0.286496
$81$	81.0000	0.111111
$82$	−334.000	−0.449806
$83$	210.000	0.277717	0.138858	−	0.990312i	$-0.455657\pi$
$83$	210.000	0.277717	0.138858	+	0.990312i	$0.455657\pi$
$84$	−147.000	−0.190941
$85$	−60.0000	−0.0765637
$86$	318.000	0.398730
$87$	480.000	0.591510
$88$	750.000	0.908526
$89$	−1066.00	−1.26962	−0.634808	−	0.772670i	$-0.718921\pi$
$89$	−1066.00	−1.26962	−0.634808	+	0.772670i	$0.718921\pi$
$90$	−45.0000	−0.0527046
$91$	−322.000	−0.370932
$92$	−161.000	−0.182450
$93$	−498.000	−0.555271
$94$	174.000	0.190923
$95$	490.000	0.529189
$96$	−483.000	−0.513500
$97$	214.000	0.224004	0.112002	−	0.993708i	$-0.464274\pi$
$97$	214.000	0.224004	0.112002	+	0.993708i	$0.464274\pi$
$98$	49.0000	0.0505076
$99$	−450.000	−0.456835
$100$	−175.000	−0.175000
$101$	1958.00	1.92899	0.964496	−	0.264096i	$-0.0850735\pi$
$101$	1958.00	1.92899	0.964496	+	0.264096i	$0.0850735\pi$
$102$	−36.0000	−0.0349464
$103$	760.000	0.727039	0.363520	−	0.931587i	$-0.381575\pi$
$103$	760.000	0.727039	0.363520	+	0.931587i	$0.381575\pi$
$104$	−690.000	−0.650578
$105$	−105.000	−0.0975900
$106$	238.000	0.218081
$107$	672.000	0.607147	0.303573	−	0.952808i	$-0.401820\pi$
$107$	672.000	0.607147	0.303573	+	0.952808i	$0.401820\pi$
$108$	189.000	0.168394
$109$	−1162.00	−1.02110	−0.510548	−	0.859849i	$-0.670557\pi$
$109$	−1162.00	−1.02110	−0.510548	+	0.859849i	$0.670557\pi$
$110$	250.000	0.216696
$111$	−726.000	−0.620801
$112$	−287.000	−0.242133
$113$	−354.000	−0.294704	−0.147352	−	0.989084i	$-0.547075\pi$
$113$	−354.000	−0.294704	−0.147352	+	0.989084i	$0.547075\pi$
$114$	294.000	0.241541
$115$	−115.000	−0.0932505
$116$	1120.00	0.896460
$117$	414.000	0.327131
$118$	340.000	0.265250
$119$	−84.0000	−0.0647081
$120$	−225.000	−0.171163
$121$	1169.00	0.878287
$122$	118.000	0.0875674
$123$	1002.00	0.734531
$124$	−1162.00	−0.841538
$125$	−125.000	−0.0894427
$126$	−63.0000	−0.0445435
$127$	110.000	0.0768577	0.0384288	−	0.999261i	$-0.487765\pi$
$127$	110.000	0.0768577	0.0384288	+	0.999261i	$0.487765\pi$
$128$	−1455.00	−1.00473
$129$	−954.000	−0.651124
$130$	−230.000	−0.155172
$131$	−2700.00	−1.80076	−0.900382	−	0.435100i	$-0.856713\pi$
$131$	−2700.00	−1.80076	−0.900382	+	0.435100i	$0.856713\pi$
$132$	−1050.00	−0.692354
$133$	686.000	0.447246
$134$	254.000	0.163748
$135$	135.000	0.0860663
$136$	−180.000	−0.113492
$137$	−1734.00	−1.08135	−0.540677	−	0.841230i	$-0.681832\pi$
$137$	−1734.00	−1.08135	−0.540677	+	0.841230i	$0.681832\pi$
$138$	−69.0000	−0.0425628
$139$	1796.00	1.09593	0.547967	−	0.836500i	$-0.315402\pi$
$139$	1796.00	1.09593	0.547967	+	0.836500i	$0.315402\pi$
$140$	−245.000	−0.147902
$141$	−522.000	−0.311775
$142$	−458.000	−0.270666
$143$	−2300.00	−1.34500
$144$	369.000	0.213542
$145$	800.000	0.458182
$146$	858.000	0.486360
$147$	−147.000	−0.0824786
$148$	−1694.00	−0.940851
$149$	386.000	0.212231	0.106115	−	0.994354i	$-0.466159\pi$
$149$	386.000	0.212231	0.106115	+	0.994354i	$0.466159\pi$
$150$	−75.0000	−0.0408248
$151$	−2784.00	−1.50039	−0.750194	−	0.661217i	$-0.770040\pi$
$151$	−2784.00	−1.50039	−0.750194	+	0.661217i	$0.770040\pi$
$152$	1470.00	0.784426
$153$	108.000	0.0570672
$154$	350.000	0.183142
$155$	−830.000	−0.430111
$156$	966.000	0.495781
$157$	−646.000	−0.328385	−0.164192	−	0.986428i	$-0.552502\pi$
$157$	−646.000	−0.328385	−0.164192	+	0.986428i	$0.552502\pi$
$158$	144.000	0.0725065
$159$	−714.000	−0.356125
$160$	−805.000	−0.397755
$161$	−161.000	−0.0788110
$162$	81.0000	0.0392837
$163$	2616.00	1.25706	0.628530	−	0.777785i	$-0.283657\pi$
$163$	2616.00	1.25706	0.628530	+	0.777785i	$0.283657\pi$
$164$	2338.00	1.11321
$165$	−750.000	−0.353863
$166$	210.000	0.0981877
$167$	3030.00	1.40400	0.702001	−	0.712176i	$-0.252290\pi$
$167$	3030.00	1.40400	0.702001	+	0.712176i	$0.252290\pi$
$168$	−315.000	−0.144659
$169$	−81.0000	−0.0368685
$170$	−60.0000	−0.0270694
$171$	−882.000	−0.394434
$172$	−2226.00	−0.986808
$173$	−2796.00	−1.22876	−0.614381	−	0.789009i	$-0.710594\pi$
$173$	−2796.00	−1.22876	−0.614381	+	0.789009i	$0.710594\pi$
$174$	480.000	0.209130
$175$	−175.000	−0.0755929
$176$	−2050.00	−0.877980
$177$	−1020.00	−0.433152
$178$	−1066.00	−0.448877
$179$	−3876.00	−1.61847	−0.809234	−	0.587486i	$-0.800118\pi$
$179$	−3876.00	−1.61847	−0.809234	+	0.587486i	$0.800118\pi$
$180$	315.000	0.130437
$181$	−938.000	−0.385199	−0.192599	−	0.981277i	$-0.561692\pi$
$181$	−938.000	−0.385199	−0.192599	+	0.981277i	$0.561692\pi$
$182$	−322.000	−0.131144
$183$	−354.000	−0.142997
$184$	−345.000	−0.138227
$185$	−1210.00	−0.480870
$186$	−498.000	−0.196318
$187$	−600.000	−0.234633
$188$	−1218.00	−0.472509
$189$	189.000	0.0727393
$190$	490.000	0.187097
$191$	−3180.00	−1.20469	−0.602347	−	0.798234i	$-0.705768\pi$
$191$	−3180.00	−1.20469	−0.602347	+	0.798234i	$0.705768\pi$
$192$	501.000	0.188315
$193$	−854.000	−0.318509	−0.159255	−	0.987238i	$-0.550909\pi$
$193$	−854.000	−0.318509	−0.159255	+	0.987238i	$0.550909\pi$
$194$	214.000	0.0791974
$195$	690.000	0.253394
$196$	−343.000	−0.125000
$197$	−3838.00	−1.38805	−0.694026	−	0.719950i	$-0.744165\pi$
$197$	−3838.00	−1.38805	−0.694026	+	0.719950i	$0.744165\pi$
$198$	−450.000	−0.161516
$199$	4532.00	1.61440	0.807198	−	0.590280i	$-0.200983\pi$
$199$	4532.00	1.61440	0.807198	+	0.590280i	$0.200983\pi$
$200$	−375.000	−0.132583
$201$	−762.000	−0.267400
$202$	1958.00	0.682002
$203$	1120.00	0.387234
$204$	252.000	0.0864879
$205$	1670.00	0.568965
$206$	760.000	0.257047
$207$	207.000	0.0695048
$208$	1886.00	0.628705
$209$	4900.00	1.62172
$210$	−105.000	−0.0345033
$211$	108.000	0.0352371	0.0176185	−	0.999845i	$-0.494392\pi$
$211$	108.000	0.0352371	0.0176185	+	0.999845i	$0.494392\pi$
$212$	−1666.00	−0.539723
$213$	1374.00	0.441995
$214$	672.000	0.214659
$215$	−1590.00	−0.504359
$216$	405.000	0.127578
$217$	−1162.00	−0.363510
$218$	−1162.00	−0.361012
$219$	−2574.00	−0.794223
$220$	−1750.00	−0.536295
$221$	552.000	0.168016
$222$	−726.000	−0.219486
$223$	2024.00	0.607790	0.303895	−	0.952706i	$-0.401713\pi$
$223$	2024.00	0.607790	0.303895	+	0.952706i	$0.401713\pi$
$224$	−1127.00	−0.336165
$225$	225.000	0.0666667
$226$	−354.000	−0.104193
$227$	3794.00	1.10932	0.554662	−	0.832076i	$-0.312848\pi$
$227$	3794.00	1.10932	0.554662	+	0.832076i	$0.312848\pi$
$228$	−2058.00	−0.597782
$229$	−5338.00	−1.54037	−0.770186	−	0.637820i	$-0.779836\pi$
$229$	−5338.00	−1.54037	−0.770186	+	0.637820i	$0.779836\pi$
$230$	−115.000	−0.0329690
$231$	−1050.00	−0.299069
$232$	2400.00	0.679171
$233$	758.000	0.213125	0.106563	−	0.994306i	$-0.466016\pi$
$233$	758.000	0.213125	0.106563	+	0.994306i	$0.466016\pi$
$234$	414.000	0.115658
$235$	−870.000	−0.241500
$236$	−2380.00	−0.656461
$237$	−432.000	−0.118403
$238$	−84.0000	−0.0228778
$239$	5366.00	1.45229	0.726146	−	0.687541i	$-0.241310\pi$
$239$	5366.00	1.45229	0.726146	+	0.687541i	$0.241310\pi$
$240$	615.000	0.165409
$241$	−1884.00	−0.503565	−0.251782	−	0.967784i	$-0.581017\pi$
$241$	−1884.00	−0.503565	−0.251782	+	0.967784i	$0.581017\pi$
$242$	1169.00	0.310521
$243$	−243.000	−0.0641500
$244$	−826.000	−0.216718
$245$	−245.000	−0.0638877
$246$	1002.00	0.259696
$247$	−4508.00	−1.16128
$248$	−2490.00	−0.637561
$249$	−630.000	−0.160340
$250$	−125.000	−0.0316228
$251$	728.000	0.183072	0.0915358	−	0.995802i	$-0.470822\pi$
$251$	728.000	0.183072	0.0915358	+	0.995802i	$0.470822\pi$
$252$	441.000	0.110240
$253$	−1150.00	−0.285770
$254$	110.000	0.0271733
$255$	180.000	0.0442041
$256$	−119.000	−0.0290527
$257$	794.000	0.192717	0.0963587	−	0.995347i	$-0.469280\pi$
$257$	794.000	0.192717	0.0963587	+	0.995347i	$0.469280\pi$
$258$	−954.000	−0.230207
$259$	−1694.00	−0.406409
$260$	1610.00	0.384031
$261$	−1440.00	−0.341509
$262$	−2700.00	−0.636666
$263$	7040.00	1.65059	0.825295	−	0.564702i	$-0.191009\pi$
$263$	7040.00	1.65059	0.825295	+	0.564702i	$0.191009\pi$
$264$	−2250.00	−0.524538
$265$	−1190.00	−0.275853
$266$	686.000	0.158125
$267$	3198.00	0.733013
$268$	−1778.00	−0.405256
$269$	−654.000	−0.148235	−0.0741173	−	0.997250i	$-0.523614\pi$
$269$	−654.000	−0.148235	−0.0741173	+	0.997250i	$0.523614\pi$
$270$	135.000	0.0304290
$271$	−7802.00	−1.74885	−0.874424	−	0.485163i	$-0.838760\pi$
$271$	−7802.00	−1.74885	−0.874424	+	0.485163i	$0.838760\pi$
$272$	492.000	0.109676
$273$	966.000	0.214157
$274$	−1734.00	−0.382317
$275$	−1250.00	−0.274101
$276$	483.000	0.105338
$277$	−3648.00	−0.791289	−0.395645	−	0.918404i	$-0.629479\pi$
$277$	−3648.00	−0.791289	−0.395645	+	0.918404i	$0.629479\pi$
$278$	1796.00	0.387471
$279$	1494.00	0.320586
$280$	−525.000	−0.112053
$281$	−252.000	−0.0534984	−0.0267492	−	0.999642i	$-0.508516\pi$
$281$	−252.000	−0.0534984	−0.0267492	+	0.999642i	$0.508516\pi$
$282$	−522.000	−0.110229
$283$	−1672.00	−0.351202	−0.175601	−	0.984461i	$-0.556187\pi$
$283$	−1672.00	−0.351202	−0.175601	+	0.984461i	$0.556187\pi$
$284$	3206.00	0.669863
$285$	−1470.00	−0.305527
$286$	−2300.00	−0.475531
$287$	2338.00	0.480863
$288$	1449.00	0.296469
$289$	−4769.00	−0.970690
$290$	800.000	0.161992
$291$	−642.000	−0.129329
$292$	−6006.00	−1.20368
$293$	4266.00	0.850588	0.425294	−	0.905055i	$-0.360171\pi$
$293$	4266.00	0.850588	0.425294	+	0.905055i	$0.360171\pi$
$294$	−147.000	−0.0291606
$295$	−1700.00	−0.335518
$296$	−3630.00	−0.712802
$297$	1350.00	0.263754
$298$	386.000	0.0750348
$299$	1058.00	0.204635
$300$	525.000	0.101036
$301$	−2226.00	−0.426261
$302$	−2784.00	−0.530468
$303$	−5874.00	−1.11370
$304$	−4018.00	−0.758053
$305$	−590.000	−0.110765
$306$	108.000	0.0201763
$307$	−7916.00	−1.47163	−0.735814	−	0.677183i	$-0.763200\pi$
$307$	−7916.00	−1.47163	−0.735814	+	0.677183i	$0.763200\pi$
$308$	−2450.00	−0.453252
$309$	−2280.00	−0.419756
$310$	−830.000	−0.152067
$311$	−388.000	−0.0707442	−0.0353721	−	0.999374i	$-0.511262\pi$
$311$	−388.000	−0.0707442	−0.0353721	+	0.999374i	$0.511262\pi$
$312$	2070.00	0.375611
$313$	−1050.00	−0.189615	−0.0948075	−	0.995496i	$-0.530224\pi$
$313$	−1050.00	−0.189615	−0.0948075	+	0.995496i	$0.530224\pi$
$314$	−646.000	−0.116102
$315$	315.000	0.0563436
$316$	−1008.00	−0.179444
$317$	−1482.00	−0.262579	−0.131289	−	0.991344i	$-0.541912\pi$
$317$	−1482.00	−0.262579	−0.131289	+	0.991344i	$0.541912\pi$
$318$	−714.000	−0.125909
$319$	8000.00	1.40412
$320$	835.000	0.145868
$321$	−2016.00	−0.350536
$322$	−161.000	−0.0278639
$323$	−1176.00	−0.202583
$324$	−567.000	−0.0972222
$325$	1150.00	0.196279
$326$	2616.00	0.444438
$327$	3486.00	0.589530
$328$	5010.00	0.843387
$329$	−1218.00	−0.204105
$330$	−750.000	−0.125110
$331$	9260.00	1.53769	0.768845	−	0.639435i	$-0.220832\pi$
$331$	9260.00	1.53769	0.768845	+	0.639435i	$0.220832\pi$
$332$	−1470.00	−0.243002
$333$	2178.00	0.358419
$334$	3030.00	0.496390
$335$	−1270.00	−0.207127
$336$	861.000	0.139796
$337$	−4928.00	−0.796573	−0.398287	−	0.917261i	$-0.630395\pi$
$337$	−4928.00	−0.796573	−0.398287	+	0.917261i	$0.630395\pi$
$338$	−81.0000	−0.0130350
$339$	1062.00	0.170147
$340$	420.000	0.0669932
$341$	−8300.00	−1.31809
$342$	−882.000	−0.139454
$343$	−343.000	−0.0539949
$344$	−4770.00	−0.747620
$345$	345.000	0.0538382
$346$	−2796.00	−0.434433
$347$	−2292.00	−0.354585	−0.177293	−	0.984158i	$-0.556734\pi$
$347$	−2292.00	−0.354585	−0.177293	+	0.984158i	$0.556734\pi$
$348$	−3360.00	−0.517572
$349$	−10204.0	−1.56506	−0.782532	−	0.622610i	$-0.786072\pi$
$349$	−10204.0	−1.56506	−0.782532	+	0.622610i	$0.786072\pi$
$350$	−175.000	−0.0267261
$351$	−1242.00	−0.188869
$352$	−8050.00	−1.21894
$353$	−3010.00	−0.453842	−0.226921	−	0.973913i	$-0.572866\pi$
$353$	−3010.00	−0.453842	−0.226921	+	0.973913i	$0.572866\pi$
$354$	−1020.00	−0.153142
$355$	2290.00	0.342368
$356$	7462.00	1.11091
$357$	252.000	0.0373593
$358$	−3876.00	−0.572215
$359$	−3204.00	−0.471032	−0.235516	−	0.971870i	$-0.575678\pi$
$359$	−3204.00	−0.471032	−0.235516	+	0.971870i	$0.575678\pi$
$360$	675.000	0.0988212
$361$	2745.00	0.400204
$362$	−938.000	−0.136188
$363$	−3507.00	−0.507079
$364$	2254.00	0.324565
$365$	−4290.00	−0.615202
$366$	−354.000	−0.0505570
$367$	8048.00	1.14469	0.572346	−	0.820012i	$-0.306033\pi$
$367$	8048.00	1.14469	0.572346	+	0.820012i	$0.306033\pi$
$368$	943.000	0.133580
$369$	−3006.00	−0.424082
$370$	−1210.00	−0.170013
$371$	−1666.00	−0.233139
$372$	3486.00	0.485862
$373$	−7990.00	−1.10913	−0.554566	−	0.832139i	$-0.687116\pi$
$373$	−7990.00	−1.10913	−0.554566	+	0.832139i	$0.687116\pi$
$374$	−600.000	−0.0829552
$375$	375.000	0.0516398
$376$	−2610.00	−0.357980
$377$	−7360.00	−1.00546
$378$	189.000	0.0257172
$379$	2352.00	0.318771	0.159385	−	0.987216i	$-0.449049\pi$
$379$	2352.00	0.318771	0.159385	+	0.987216i	$0.449049\pi$
$380$	−3430.00	−0.463040
$381$	−330.000	−0.0443738
$382$	−3180.00	−0.425924
$383$	−176.000	−0.0234809	−0.0117404	−	0.999931i	$-0.503737\pi$
$383$	−176.000	−0.0234809	−0.0117404	+	0.999931i	$0.503737\pi$
$384$	4365.00	0.580079
$385$	−1750.00	−0.231658
$386$	−854.000	−0.112610
$387$	2862.00	0.375927
$388$	−1498.00	−0.196004
$389$	12590.0	1.64097	0.820486	−	0.571666i	$-0.193703\pi$
$389$	12590.0	1.64097	0.820486	+	0.571666i	$0.193703\pi$
$390$	690.000	0.0895885
$391$	276.000	0.0356980
$392$	−735.000	−0.0947018
$393$	8100.00	1.03967
$394$	−3838.00	−0.490750
$395$	−720.000	−0.0917143
$396$	3150.00	0.399731
$397$	−6362.00	−0.804281	−0.402141	−	0.915578i	$-0.631734\pi$
$397$	−6362.00	−0.804281	−0.402141	+	0.915578i	$0.631734\pi$
$398$	4532.00	0.570775
$399$	−2058.00	−0.258218
$400$	1025.00	0.128125
$401$	7936.00	0.988292	0.494146	−	0.869379i	$-0.335481\pi$
$401$	7936.00	0.988292	0.494146	+	0.869379i	$0.335481\pi$
$402$	−762.000	−0.0945401
$403$	7636.00	0.943862
$404$	−13706.0	−1.68787
$405$	−405.000	−0.0496904
$406$	1120.00	0.136908
$407$	−12100.0	−1.47365
$408$	540.000	0.0655245
$409$	−10334.0	−1.24935	−0.624674	−	0.780886i	$-0.714768\pi$
$409$	−10334.0	−1.24935	−0.624674	+	0.780886i	$0.714768\pi$
$410$	1670.00	0.201160
$411$	5202.00	0.624321
$412$	−5320.00	−0.636159
$413$	−2380.00	−0.283565
$414$	207.000	0.0245737
$415$	−1050.00	−0.124199
$416$	7406.00	0.872858
$417$	−5388.00	−0.632737
$418$	4900.00	0.573366
$419$	−4044.00	−0.471509	−0.235755	−	0.971813i	$-0.575756\pi$
$419$	−4044.00	−0.471509	−0.235755	+	0.971813i	$0.575756\pi$
$420$	735.000	0.0853913
$421$	−1646.00	−0.190549	−0.0952745	−	0.995451i	$-0.530373\pi$
$421$	−1646.00	−0.190549	−0.0952745	+	0.995451i	$0.530373\pi$
$422$	108.000	0.0124582
$423$	1566.00	0.180004
$424$	−3570.00	−0.408902
$425$	300.000	0.0342403
$426$	1374.00	0.156269
$427$	−826.000	−0.0936134
$428$	−4704.00	−0.531253
$429$	6900.00	0.776539
$430$	−1590.00	−0.178318
$431$	−1072.00	−0.119806	−0.0599030	−	0.998204i	$-0.519079\pi$
$431$	−1072.00	−0.119806	−0.0599030	+	0.998204i	$0.519079\pi$
$432$	−1107.00	−0.123288
$433$	9782.00	1.08566	0.542832	−	0.839841i	$-0.317352\pi$
$433$	9782.00	1.08566	0.542832	+	0.839841i	$0.317352\pi$
$434$	−1162.00	−0.128520
$435$	−2400.00	−0.264531
$436$	8134.00	0.893459
$437$	−2254.00	−0.246736
$438$	−2574.00	−0.280800
$439$	−5578.00	−0.606431	−0.303216	−	0.952922i	$-0.598060\pi$
$439$	−5578.00	−0.606431	−0.303216	+	0.952922i	$0.598060\pi$
$440$	−3750.00	−0.406305
$441$	441.000	0.0476190
$442$	552.000	0.0594026
$443$	−3244.00	−0.347917	−0.173958	−	0.984753i	$-0.555656\pi$
$443$	−3244.00	−0.347917	−0.173958	+	0.984753i	$0.555656\pi$
$444$	5082.00	0.543201
$445$	5330.00	0.567789
$446$	2024.00	0.214886
$447$	−1158.00	−0.122531
$448$	1169.00	0.123281
$449$	−1794.00	−0.188561	−0.0942807	−	0.995546i	$-0.530055\pi$
$449$	−1794.00	−0.188561	−0.0942807	+	0.995546i	$0.530055\pi$
$450$	225.000	0.0235702
$451$	16700.0	1.74362
$452$	2478.00	0.257866
$453$	8352.00	0.866250
$454$	3794.00	0.392205
$455$	1610.00	0.165886
$456$	−4410.00	−0.452889
$457$	−4472.00	−0.457749	−0.228875	−	0.973456i	$-0.573505\pi$
$457$	−4472.00	−0.457749	−0.228875	+	0.973456i	$0.573505\pi$
$458$	−5338.00	−0.544603
$459$	−324.000	−0.0329478
$460$	805.000	0.0815942
$461$	−3850.00	−0.388964	−0.194482	−	0.980906i	$-0.562303\pi$
$461$	−3850.00	−0.388964	−0.194482	+	0.980906i	$0.562303\pi$
$462$	−1050.00	−0.105737
$463$	9470.00	0.950558	0.475279	−	0.879835i	$-0.342347\pi$
$463$	9470.00	0.950558	0.475279	+	0.879835i	$0.342347\pi$
$464$	−6560.00	−0.656337
$465$	2490.00	0.248325
$466$	758.000	0.0753512
$467$	−3906.00	−0.387041	−0.193520	−	0.981096i	$-0.561991\pi$
$467$	−3906.00	−0.387041	−0.193520	+	0.981096i	$0.561991\pi$
$468$	−2898.00	−0.286240
$469$	−1778.00	−0.175054
$470$	−870.000	−0.0853832
$471$	1938.00	0.189593
$472$	−5100.00	−0.497344
$473$	−15900.0	−1.54563
$474$	−432.000	−0.0418616
$475$	−2450.00	−0.236660
$476$	588.000	0.0566196
$477$	2142.00	0.205609
$478$	5366.00	0.513463
$479$	13200.0	1.25913	0.629565	−	0.776948i	$-0.283233\pi$
$479$	13200.0	1.25913	0.629565	+	0.776948i	$0.283233\pi$
$480$	2415.00	0.229644
$481$	11132.0	1.05525
$482$	−1884.00	−0.178037
$483$	483.000	0.0455016
$484$	−8183.00	−0.768501
$485$	−1070.00	−0.100178
$486$	−243.000	−0.0226805
$487$	−1090.00	−0.101422	−0.0507111	−	0.998713i	$-0.516149\pi$
$487$	−1090.00	−0.101422	−0.0507111	+	0.998713i	$0.516149\pi$
$488$	−1770.00	−0.164189
$489$	−7848.00	−0.725764
$490$	−245.000	−0.0225877
$491$	9012.00	0.828322	0.414161	−	0.910204i	$-0.364075\pi$
$491$	9012.00	0.828322	0.414161	+	0.910204i	$0.364075\pi$
$492$	−7014.00	−0.642715
$493$	−1920.00	−0.175401
$494$	−4508.00	−0.410576
$495$	2250.00	0.204303
$496$	6806.00	0.616126
$497$	3206.00	0.289354
$498$	−630.000	−0.0566887
$499$	14068.0	1.26206	0.631032	−	0.775757i	$-0.282632\pi$
$499$	14068.0	1.26206	0.631032	+	0.775757i	$0.282632\pi$
$500$	875.000	0.0782624
$501$	−9090.00	−0.810601
$502$	728.000	0.0647256
$503$	−4572.00	−0.405279	−0.202640	−	0.979253i	$-0.564952\pi$
$503$	−4572.00	−0.405279	−0.202640	+	0.979253i	$0.564952\pi$
$504$	945.000	0.0835191
$505$	−9790.00	−0.862672
$506$	−1150.00	−0.101035
$507$	243.000	0.0212860
$508$	−770.000	−0.0672504
$509$	−11634.0	−1.01310	−0.506550	−	0.862211i	$-0.669079\pi$
$509$	−11634.0	−1.01310	−0.506550	+	0.862211i	$0.669079\pi$
$510$	180.000	0.0156285
$511$	−6006.00	−0.519941
$512$	11521.0	0.994455
$513$	2646.00	0.227727
$514$	794.000	0.0681359
$515$	−3800.00	−0.325142
$516$	6678.00	0.569734
$517$	−8700.00	−0.740088
$518$	−1694.00	−0.143687
$519$	8388.00	0.709426
$520$	3450.00	0.290947
$521$	318.000	0.0267406	0.0133703	−	0.999911i	$-0.495744\pi$
$521$	318.000	0.0267406	0.0133703	+	0.999911i	$0.495744\pi$
$522$	−1440.00	−0.120742
$523$	7544.00	0.630738	0.315369	−	0.948969i	$-0.397872\pi$
$523$	7544.00	0.630738	0.315369	+	0.948969i	$0.397872\pi$
$524$	18900.0	1.57567
$525$	525.000	0.0436436
$526$	7040.00	0.583571
$527$	1992.00	0.164654
$528$	6150.00	0.506902
$529$	529.000	0.0434783
$530$	−1190.00	−0.0975289
$531$	3060.00	0.250080
$532$	−4802.00	−0.391340
$533$	−15364.0	−1.24857
$534$	3198.00	0.259159
$535$	−3360.00	−0.271524
$536$	−3810.00	−0.307028
$537$	11628.0	0.934423
$538$	−654.000	−0.0524088
$539$	−2450.00	−0.195787
$540$	−945.000	−0.0753080
$541$	7762.00	0.616847	0.308424	−	0.951249i	$-0.400199\pi$
$541$	7762.00	0.616847	0.308424	+	0.951249i	$0.400199\pi$
$542$	−7802.00	−0.618311
$543$	2814.00	0.222395
$544$	1932.00	0.152268
$545$	5810.00	0.456648
$546$	966.000	0.0757161
$547$	4704.00	0.367694	0.183847	−	0.982955i	$-0.441145\pi$
$547$	4704.00	0.367694	0.183847	+	0.982955i	$0.441145\pi$
$548$	12138.0	0.946186
$549$	1062.00	0.0825593
$550$	−1250.00	−0.0969094
$551$	15680.0	1.21232
$552$	1035.00	0.0798053
$553$	−1008.00	−0.0775127
$554$	−3648.00	−0.279763
$555$	3630.00	0.277630
$556$	−12572.0	−0.958942
$557$	10870.0	0.826888	0.413444	−	0.910530i	$-0.364326\pi$
$557$	10870.0	0.826888	0.413444	+	0.910530i	$0.364326\pi$
$558$	1494.00	0.113344
$559$	14628.0	1.10680
$560$	1435.00	0.108285
$561$	1800.00	0.135465
$562$	−252.000	−0.0189146
$563$	4634.00	0.346891	0.173446	−	0.984843i	$-0.444510\pi$
$563$	4634.00	0.346891	0.173446	+	0.984843i	$0.444510\pi$
$564$	3654.00	0.272803
$565$	1770.00	0.131796
$566$	−1672.00	−0.124169
$567$	−567.000	−0.0419961
$568$	6870.00	0.507498
$569$	−15768.0	−1.16174	−0.580869	−	0.813997i	$-0.697287\pi$
$569$	−15768.0	−1.16174	−0.580869	+	0.813997i	$0.697287\pi$
$570$	−1470.00	−0.108020
$571$	20060.0	1.47020	0.735101	−	0.677958i	$-0.237135\pi$
$571$	20060.0	1.47020	0.735101	+	0.677958i	$0.237135\pi$
$572$	16100.0	1.17688
$573$	9540.00	0.695531
$574$	2338.00	0.170011
$575$	575.000	0.0417029
$576$	−1503.00	−0.108724
$577$	6254.00	0.451226	0.225613	−	0.974217i	$-0.427562\pi$
$577$	6254.00	0.451226	0.225613	+	0.974217i	$0.427562\pi$
$578$	−4769.00	−0.343191
$579$	2562.00	0.183891
$580$	−5600.00	−0.400909
$581$	−1470.00	−0.104967
$582$	−642.000	−0.0457247
$583$	−11900.0	−0.845365
$584$	−12870.0	−0.911925
$585$	−2070.00	−0.146297
$586$	4266.00	0.300728
$587$	−6956.00	−0.489105	−0.244553	−	0.969636i	$-0.578641\pi$
$587$	−6956.00	−0.489105	−0.244553	+	0.969636i	$0.578641\pi$
$588$	1029.00	0.0721688
$589$	−16268.0	−1.13805
$590$	−1700.00	−0.118624
$591$	11514.0	0.801392
$592$	9922.00	0.688837
$593$	−7358.00	−0.509539	−0.254770	−	0.967002i	$-0.582000\pi$
$593$	−7358.00	−0.509539	−0.254770	+	0.967002i	$0.582000\pi$
$594$	1350.00	0.0932511
$595$	420.000	0.0289384
$596$	−2702.00	−0.185702
$597$	−13596.0	−0.932072
$598$	1058.00	0.0723492
$599$	−20154.0	−1.37474	−0.687371	−	0.726307i	$-0.741235\pi$
$599$	−20154.0	−1.37474	−0.687371	+	0.726307i	$0.741235\pi$
$600$	1125.00	0.0765466
$601$	−24234.0	−1.64480	−0.822401	−	0.568909i	$-0.807366\pi$
$601$	−24234.0	−1.64480	−0.822401	+	0.568909i	$0.807366\pi$
$602$	−2226.00	−0.150706
$603$	2286.00	0.154383
$604$	19488.0	1.31284
$605$	−5845.00	−0.392782
$606$	−5874.00	−0.393754
$607$	−23644.0	−1.58102	−0.790511	−	0.612448i	$-0.790185\pi$
$607$	−23644.0	−1.58102	−0.790511	+	0.612448i	$0.790185\pi$
$608$	−15778.0	−1.05244
$609$	−3360.00	−0.223570
$610$	−590.000	−0.0391613
$611$	8004.00	0.529963
$612$	−756.000	−0.0499338
$613$	23206.0	1.52901	0.764504	−	0.644619i	$-0.222984\pi$
$613$	23206.0	1.52901	0.764504	+	0.644619i	$0.222984\pi$
$614$	−7916.00	−0.520299
$615$	−5010.00	−0.328492
$616$	−5250.00	−0.343390
$617$	4046.00	0.263996	0.131998	−	0.991250i	$-0.457861\pi$
$617$	4046.00	0.263996	0.131998	+	0.991250i	$0.457861\pi$
$618$	−2280.00	−0.148406
$619$	−7110.00	−0.461672	−0.230836	−	0.972993i	$-0.574146\pi$
$619$	−7110.00	−0.461672	−0.230836	+	0.972993i	$0.574146\pi$
$620$	5810.00	0.376347
$621$	−621.000	−0.0401286
$622$	−388.000	−0.0250119
$623$	7462.00	0.479870
$624$	−5658.00	−0.362983
$625$	625.000	0.0400000
$626$	−1050.00	−0.0670390
$627$	−14700.0	−0.936302
$628$	4522.00	0.287337
$629$	2904.00	0.184086
$630$	315.000	0.0199205
$631$	8464.00	0.533988	0.266994	−	0.963698i	$-0.413970\pi$
$631$	8464.00	0.533988	0.266994	+	0.963698i	$0.413970\pi$
$632$	−2160.00	−0.135950
$633$	−324.000	−0.0203441
$634$	−1482.00	−0.0928356
$635$	−550.000	−0.0343718
$636$	4998.00	0.311609
$637$	2254.00	0.140199
$638$	8000.00	0.496431
$639$	−4122.00	−0.255186
$640$	7275.00	0.449328
$641$	−29276.0	−1.80395	−0.901975	−	0.431787i	$-0.857883\pi$
$641$	−29276.0	−1.80395	−0.901975	+	0.431787i	$0.857883\pi$
$642$	−2016.00	−0.123933
$643$	14500.0	0.889307	0.444653	−	0.895703i	$-0.353327\pi$
$643$	14500.0	0.889307	0.444653	+	0.895703i	$0.353327\pi$
$644$	1127.00	0.0689597
$645$	4770.00	0.291192
$646$	−1176.00	−0.0716240
$647$	−6958.00	−0.422793	−0.211397	−	0.977400i	$-0.567801\pi$
$647$	−6958.00	−0.422793	−0.211397	+	0.977400i	$0.567801\pi$
$648$	−1215.00	−0.0736570
$649$	−17000.0	−1.02821
$650$	1150.00	0.0693949
$651$	3486.00	0.209873
$652$	−18312.0	−1.09993
$653$	7362.00	0.441190	0.220595	−	0.975365i	$-0.429200\pi$
$653$	7362.00	0.441190	0.220595	+	0.975365i	$0.429200\pi$
$654$	3486.00	0.208430
$655$	13500.0	0.805326
$656$	−13694.0	−0.815032
$657$	7722.00	0.458545
$658$	−1218.00	−0.0721620
$659$	−28710.0	−1.69709	−0.848545	−	0.529123i	$-0.822521\pi$
$659$	−28710.0	−1.69709	−0.848545	+	0.529123i	$0.822521\pi$
$660$	5250.00	0.309630
$661$	−1802.00	−0.106036	−0.0530179	−	0.998594i	$-0.516884\pi$
$661$	−1802.00	−0.106036	−0.0530179	+	0.998594i	$0.516884\pi$
$662$	9260.00	0.543656
$663$	−1656.00	−0.0970041
$664$	−3150.00	−0.184102
$665$	−3430.00	−0.200015
$666$	2178.00	0.126720
$667$	−3680.00	−0.213628
$668$	−21210.0	−1.22850
$669$	−6072.00	−0.350907
$670$	−1270.00	−0.0732304
$671$	−5900.00	−0.339444
$672$	3381.00	0.194085
$673$	30622.0	1.75392	0.876962	−	0.480559i	$-0.159566\pi$
$673$	30622.0	1.75392	0.876962	+	0.480559i	$0.159566\pi$
$674$	−4928.00	−0.281631
$675$	−675.000	−0.0384900
$676$	567.000	0.0322599
$677$	−8622.00	−0.489469	−0.244734	−	0.969590i	$-0.578701\pi$
$677$	−8622.00	−0.489469	−0.244734	+	0.969590i	$0.578701\pi$
$678$	1062.00	0.0601561
$679$	−1498.00	−0.0846656
$680$	900.000	0.0507550
$681$	−11382.0	−0.640469
$682$	−8300.00	−0.466017
$683$	15844.0	0.887634	0.443817	−	0.896118i	$-0.353624\pi$
$683$	15844.0	0.887634	0.443817	+	0.896118i	$0.353624\pi$
$684$	6174.00	0.345130
$685$	8670.00	0.483597
$686$	−343.000	−0.0190901
$687$	16014.0	0.889334
$688$	13038.0	0.722484
$689$	10948.0	0.605349
$690$	345.000	0.0190347
$691$	16984.0	0.935024	0.467512	−	0.883987i	$-0.345150\pi$
$691$	16984.0	0.935024	0.467512	+	0.883987i	$0.345150\pi$
$692$	19572.0	1.07517
$693$	3150.00	0.172668
$694$	−2292.00	−0.125365
$695$	−8980.00	−0.490116
$696$	−7200.00	−0.392120
$697$	−4008.00	−0.217810
$698$	−10204.0	−0.553334
$699$	−2274.00	−0.123048
$700$	1225.00	0.0661438
$701$	7758.00	0.417997	0.208998	−	0.977916i	$-0.432980\pi$
$701$	7758.00	0.417997	0.208998	+	0.977916i	$0.432980\pi$
$702$	−1242.00	−0.0667753
$703$	−23716.0	−1.27236
$704$	8350.00	0.447021
$705$	2610.00	0.139430
$706$	−3010.00	−0.160457
$707$	−13706.0	−0.729091
$708$	7140.00	0.379008
$709$	−606.000	−0.0320999	−0.0160499	−	0.999871i	$-0.505109\pi$
$709$	−606.000	−0.0320999	−0.0160499	+	0.999871i	$0.505109\pi$
$710$	2290.00	0.121045
$711$	1296.00	0.0683598
$712$	15990.0	0.841644
$713$	3818.00	0.200540
$714$	252.000	0.0132085
$715$	11500.0	0.601504
$716$	27132.0	1.41616
$717$	−16098.0	−0.838481
$718$	−3204.00	−0.166535
$719$	−16556.0	−0.858741	−0.429370	−	0.903128i	$-0.641264\pi$
$719$	−16556.0	−0.858741	−0.429370	+	0.903128i	$0.641264\pi$
$720$	−1845.00	−0.0954987
$721$	−5320.00	−0.274795
$722$	2745.00	0.141494
$723$	5652.00	0.290733
$724$	6566.00	0.337049
$725$	−4000.00	−0.204905
$726$	−3507.00	−0.179280
$727$	−27944.0	−1.42556	−0.712782	−	0.701385i	$-0.752565\pi$
$727$	−27944.0	−1.42556	−0.712782	+	0.701385i	$0.752565\pi$
$728$	4830.00	0.245895
$729$	729.000	0.0370370
$730$	−4290.00	−0.217507
$731$	3816.00	0.193078
$732$	2478.00	0.125122
$733$	4358.00	0.219599	0.109800	−	0.993954i	$-0.464979\pi$
$733$	4358.00	0.219599	0.109800	+	0.993954i	$0.464979\pi$
$734$	8048.00	0.404710
$735$	735.000	0.0368856
$736$	3703.00	0.185454
$737$	−12700.0	−0.634750
$738$	−3006.00	−0.149935
$739$	−14332.0	−0.713412	−0.356706	−	0.934217i	$-0.616100\pi$
$739$	−14332.0	−0.713412	−0.356706	+	0.934217i	$0.616100\pi$
$740$	8470.00	0.420761
$741$	13524.0	0.670468
$742$	−1666.00	−0.0824269
$743$	32528.0	1.60611	0.803053	−	0.595908i	$-0.203208\pi$
$743$	32528.0	1.60611	0.803053	+	0.595908i	$0.203208\pi$
$744$	7470.00	0.368096
$745$	−1930.00	−0.0949124
$746$	−7990.00	−0.392138
$747$	1890.00	0.0925723
$748$	4200.00	0.205304
$749$	−4704.00	−0.229480
$750$	375.000	0.0182574
$751$	7400.00	0.359560	0.179780	−	0.983707i	$-0.442461\pi$
$751$	7400.00	0.359560	0.179780	+	0.983707i	$0.442461\pi$
$752$	7134.00	0.345944
$753$	−2184.00	−0.105696
$754$	−7360.00	−0.355485
$755$	13920.0	0.670994
$756$	−1323.00	−0.0636469
$757$	21022.0	1.00932	0.504662	−	0.863317i	$-0.331617\pi$
$757$	21022.0	1.00932	0.504662	+	0.863317i	$0.331617\pi$
$758$	2352.00	0.112702
$759$	3450.00	0.164990
$760$	−7350.00	−0.350806
$761$	990.000	0.0471583	0.0235792	−	0.999722i	$-0.492494\pi$
$761$	990.000	0.0471583	0.0235792	+	0.999722i	$0.492494\pi$
$762$	−330.000	−0.0156885
$763$	8134.00	0.385938
$764$	22260.0	1.05411
$765$	−540.000	−0.0255212
$766$	−176.000	−0.00830175
$767$	15640.0	0.736281
$768$	357.000	0.0167736
$769$	27752.0	1.30138	0.650691	−	0.759343i	$-0.274479\pi$
$769$	27752.0	1.30138	0.650691	+	0.759343i	$0.274479\pi$
$770$	−1750.00	−0.0819034
$771$	−2382.00	−0.111265
$772$	5978.00	0.278696
$773$	37054.0	1.72411	0.862057	−	0.506812i	$-0.169176\pi$
$773$	37054.0	1.72411	0.862057	+	0.506812i	$0.169176\pi$
$774$	2862.00	0.132910
$775$	4150.00	0.192351
$776$	−3210.00	−0.148495
$777$	5082.00	0.234641
$778$	12590.0	0.580171
$779$	32732.0	1.50545
$780$	−4830.00	−0.221720
$781$	22900.0	1.04920
$782$	276.000	0.0126212
$783$	4320.00	0.197170
$784$	2009.00	0.0915179
$785$	3230.00	0.146858
$786$	8100.00	0.367579
$787$	12596.0	0.570520	0.285260	−	0.958450i	$-0.407920\pi$
$787$	12596.0	0.570520	0.285260	+	0.958450i	$0.407920\pi$
$788$	26866.0	1.21455
$789$	−21120.0	−0.952968
$790$	−720.000	−0.0324259
$791$	2478.00	0.111388
$792$	6750.00	0.302842
$793$	5428.00	0.243069
$794$	−6362.00	−0.284356
$795$	3570.00	0.159264
$796$	−31724.0	−1.41260
$797$	−32258.0	−1.43367	−0.716836	−	0.697242i	$-0.754410\pi$
$797$	−32258.0	−1.43367	−0.716836	+	0.697242i	$0.754410\pi$
$798$	−2058.00	−0.0912937
$799$	2088.00	0.0924507
$800$	4025.00	0.177882
$801$	−9594.00	−0.423205
$802$	7936.00	0.349414
$803$	−42900.0	−1.88532
$804$	5334.00	0.233975
$805$	805.000	0.0352454
$806$	7636.00	0.333705
$807$	1962.00	0.0855832
$808$	−29370.0	−1.27875
$809$	20826.0	0.905072	0.452536	−	0.891746i	$-0.350519\pi$
$809$	20826.0	0.905072	0.452536	+	0.891746i	$0.350519\pi$
$810$	−405.000	−0.0175682
$811$	25284.0	1.09475	0.547374	−	0.836888i	$-0.315627\pi$
$811$	25284.0	1.09475	0.547374	+	0.836888i	$0.315627\pi$
$812$	−7840.00	−0.338830
$813$	23406.0	1.00970
$814$	−12100.0	−0.521013
$815$	−13080.0	−0.562175
$816$	−1476.00	−0.0633215
$817$	−31164.0	−1.33450
$818$	−10334.0	−0.441711
$819$	−2898.00	−0.123644
$820$	−11690.0	−0.497845
$821$	−784.000	−0.0333274	−0.0166637	−	0.999861i	$-0.505304\pi$
$821$	−784.000	−0.0333274	−0.0166637	+	0.999861i	$0.505304\pi$
$822$	5202.00	0.220731
$823$	25238.0	1.06894	0.534472	−	0.845186i	$-0.320510\pi$
$823$	25238.0	1.06894	0.534472	+	0.845186i	$0.320510\pi$
$824$	−11400.0	−0.481963
$825$	3750.00	0.158252
$826$	−2380.00	−0.100255
$827$	−21088.0	−0.886701	−0.443350	−	0.896348i	$-0.646210\pi$
$827$	−21088.0	−0.886701	−0.443350	+	0.896348i	$0.646210\pi$
$828$	−1449.00	−0.0608167
$829$	−35688.0	−1.49517	−0.747585	−	0.664166i	$-0.768787\pi$
$829$	−35688.0	−1.49517	−0.747585	+	0.664166i	$0.768787\pi$
$830$	−1050.00	−0.0439109
$831$	10944.0	0.456851
$832$	−7682.00	−0.320103
$833$	588.000	0.0244574
$834$	−5388.00	−0.223706
$835$	−15150.0	−0.627889
$836$	−34300.0	−1.41901
$837$	−4482.00	−0.185090
$838$	−4044.00	−0.166704
$839$	−33312.0	−1.37075	−0.685374	−	0.728191i	$-0.740361\pi$
$839$	−33312.0	−1.37075	−0.685374	+	0.728191i	$0.740361\pi$
$840$	1575.00	0.0646936
$841$	1211.00	0.0496535
$842$	−1646.00	−0.0673692
$843$	756.000	0.0308873
$844$	−756.000	−0.0308325
$845$	405.000	0.0164881
$846$	1566.00	0.0636409
$847$	−8183.00	−0.331961
$848$	9758.00	0.395155
$849$	5016.00	0.202766
$850$	300.000	0.0121058
$851$	5566.00	0.224207
$852$	−9618.00	−0.386746
$853$	−16586.0	−0.665761	−0.332880	−	0.942969i	$-0.608021\pi$
$853$	−16586.0	−0.665761	−0.332880	+	0.942969i	$0.608021\pi$
$854$	−826.000	−0.0330974
$855$	4410.00	0.176396
$856$	−10080.0	−0.402485
$857$	16238.0	0.647234	0.323617	−	0.946188i	$-0.395101\pi$
$857$	16238.0	0.647234	0.323617	+	0.946188i	$0.395101\pi$
$858$	6900.00	0.274548
$859$	−11336.0	−0.450267	−0.225133	−	0.974328i	$-0.572282\pi$
$859$	−11336.0	−0.450267	−0.225133	+	0.974328i	$0.572282\pi$
$860$	11130.0	0.441314
$861$	−7014.00	−0.277627
$862$	−1072.00	−0.0423578
$863$	5208.00	0.205426	0.102713	−	0.994711i	$-0.467248\pi$
$863$	5208.00	0.205426	0.102713	+	0.994711i	$0.467248\pi$
$864$	−4347.00	−0.171167
$865$	13980.0	0.549519
$866$	9782.00	0.383841
$867$	14307.0	0.560428
$868$	8134.00	0.318071
$869$	−7200.00	−0.281062
$870$	−2400.00	−0.0935260
$871$	11684.0	0.454532
$872$	17430.0	0.676897
$873$	1926.00	0.0746681
$874$	−2254.00	−0.0872342
$875$	875.000	0.0338062
$876$	18018.0	0.694945
$877$	14716.0	0.566618	0.283309	−	0.959029i	$-0.408568\pi$
$877$	14716.0	0.566618	0.283309	+	0.959029i	$0.408568\pi$
$878$	−5578.00	−0.214406
$879$	−12798.0	−0.491087
$880$	10250.0	0.392645
$881$	−27862.0	−1.06549	−0.532744	−	0.846277i	$-0.678839\pi$
$881$	−27862.0	−1.06549	−0.532744	+	0.846277i	$0.678839\pi$
$882$	441.000	0.0168359
$883$	12328.0	0.469842	0.234921	−	0.972015i	$-0.424517\pi$
$883$	12328.0	0.469842	0.234921	+	0.972015i	$0.424517\pi$
$884$	−3864.00	−0.147014
$885$	5100.00	0.193711
$886$	−3244.00	−0.123007
$887$	−34270.0	−1.29726	−0.648632	−	0.761102i	$-0.724659\pi$
$887$	−34270.0	−1.29726	−0.648632	+	0.761102i	$0.724659\pi$
$888$	10890.0	0.411537
$889$	−770.000	−0.0290495
$890$	5330.00	0.200744
$891$	−4050.00	−0.152278
$892$	−14168.0	−0.531816
$893$	−17052.0	−0.638996
$894$	−1158.00	−0.0433214
$895$	19380.0	0.723801
$896$	10185.0	0.379751
$897$	−3174.00	−0.118146
$898$	−1794.00	−0.0666665
$899$	−26560.0	−0.985345
$900$	−1575.00	−0.0583333
$901$	2856.00	0.105602
$902$	16700.0	0.616463
$903$	6678.00	0.246102
$904$	5310.00	0.195363
$905$	4690.00	0.172266
$906$	8352.00	0.306266
$907$	38738.0	1.41816	0.709082	−	0.705126i	$-0.249110\pi$
$907$	38738.0	1.41816	0.709082	+	0.705126i	$0.249110\pi$
$908$	−26558.0	−0.970659
$909$	17622.0	0.642998
$910$	1610.00	0.0586494
$911$	−14240.0	−0.517884	−0.258942	−	0.965893i	$-0.583374\pi$
$911$	−14240.0	−0.517884	−0.258942	+	0.965893i	$0.583374\pi$
$912$	12054.0	0.437662
$913$	−10500.0	−0.380613
$914$	−4472.00	−0.161839
$915$	1770.00	0.0639502
$916$	37366.0	1.34782
$917$	18900.0	0.680625
$918$	−324.000	−0.0116488
$919$	−39184.0	−1.40649	−0.703243	−	0.710949i	$-0.748265\pi$
$919$	−39184.0	−1.40649	−0.703243	+	0.710949i	$0.748265\pi$
$920$	1725.00	0.0618169
$921$	23748.0	0.849645
$922$	−3850.00	−0.137520
$923$	−21068.0	−0.751313
$924$	7350.00	0.261685
$925$	6050.00	0.215052
$926$	9470.00	0.336073
$927$	6840.00	0.242346
$928$	−25760.0	−0.911221
$929$	55230.0	1.95052	0.975262	−	0.221050i	$-0.0709485\pi$
$929$	55230.0	1.95052	0.975262	+	0.221050i	$0.0709485\pi$
$930$	2490.00	0.0877960
$931$	−4802.00	−0.169043
$932$	−5306.00	−0.186485
$933$	1164.00	0.0408442
$934$	−3906.00	−0.136840
$935$	3000.00	0.104931
$936$	−6210.00	−0.216859
$937$	−9910.00	−0.345513	−0.172756	−	0.984965i	$-0.555267\pi$
$937$	−9910.00	−0.345513	−0.172756	+	0.984965i	$0.555267\pi$
$938$	−1778.00	−0.0618910
$939$	3150.00	0.109474
$940$	6090.00	0.211313
$941$	26634.0	0.922682	0.461341	−	0.887223i	$-0.347369\pi$
$941$	26634.0	0.922682	0.461341	+	0.887223i	$0.347369\pi$
$942$	1938.00	0.0670313
$943$	−7682.00	−0.265281
$944$	13940.0	0.480623
$945$	−945.000	−0.0325300
$946$	−15900.0	−0.546463
$947$	−48716.0	−1.67166	−0.835828	−	0.548992i	$-0.815012\pi$
$947$	−48716.0	−1.67166	−0.835828	+	0.548992i	$0.815012\pi$
$948$	3024.00	0.103602
$949$	39468.0	1.35004
$950$	−2450.00	−0.0836721
$951$	4446.00	0.151600
$952$	1260.00	0.0428958
$953$	−46346.0	−1.57533	−0.787667	−	0.616101i	$-0.788711\pi$
$953$	−46346.0	−1.57533	−0.787667	+	0.616101i	$0.788711\pi$
$954$	2142.00	0.0726937
$955$	15900.0	0.538756
$956$	−37562.0	−1.27076
$957$	−24000.0	−0.810669
$958$	13200.0	0.445170
$959$	12138.0	0.408714
$960$	−2505.00	−0.0842172
$961$	−2235.00	−0.0750227
$962$	11132.0	0.373087
$963$	6048.00	0.202382
$964$	13188.0	0.440619
$965$	4270.00	0.142442
$966$	483.000	0.0160872
$967$	−50266.0	−1.67161	−0.835804	−	0.549027i	$-0.814998\pi$
$967$	−50266.0	−1.67161	−0.835804	+	0.549027i	$0.814998\pi$
$968$	−17535.0	−0.582228
$969$	3528.00	0.116961
$970$	−1070.00	−0.0354182
$971$	−35016.0	−1.15728	−0.578639	−	0.815584i	$-0.696416\pi$
$971$	−35016.0	−1.15728	−0.578639	+	0.815584i	$0.696416\pi$
$972$	1701.00	0.0561313
$973$	−12572.0	−0.414224
$974$	−1090.00	−0.0358582
$975$	−3450.00	−0.113321
$976$	4838.00	0.158669
$977$	23382.0	0.765667	0.382833	−	0.923817i	$-0.374948\pi$
$977$	23382.0	0.765667	0.382833	+	0.923817i	$0.374948\pi$
$978$	−7848.00	−0.256596
$979$	53300.0	1.74002
$980$	1715.00	0.0559017
$981$	−10458.0	−0.340365
$982$	9012.00	0.292856
$983$	33216.0	1.07775	0.538874	−	0.842387i	$-0.318850\pi$
$983$	33216.0	1.07775	0.538874	+	0.842387i	$0.318850\pi$
$984$	−15030.0	−0.486930
$985$	19190.0	0.620756
$986$	−1920.00	−0.0620134
$987$	3654.00	0.117840
$988$	31556.0	1.01612
$989$	7314.00	0.235158
$990$	2250.00	0.0722320
$991$	6512.00	0.208739	0.104370	−	0.994539i	$-0.466718\pi$
$991$	6512.00	0.208739	0.104370	+	0.994539i	$0.466718\pi$
$992$	26726.0	0.855395
$993$	−27780.0	−0.887786
$994$	3206.00	0.102302
$995$	−22660.0	−0.721980
$996$	4410.00	0.140297
$997$	11462.0	0.364097	0.182049	−	0.983290i	$-0.441727\pi$
$997$	11462.0	0.364097	0.182049	+	0.983290i	$0.441727\pi$
$998$	14068.0	0.446207
$999$	−6534.00	−0.206934

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2415.4.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.4.a.b.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:	\( N \)	\(=\)	\( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2415.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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