LMFDB - Embedded newform 15.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,6,Mod(1,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("15.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	15.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:	$2.40575729719$
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	$\chi$	$=$	15.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+7.00000 q^{2} +9.00000 q^{3} +17.0000 q^{4} -25.0000 q^{5} +63.0000 q^{6} +12.0000 q^{7} -105.000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+7.00000 q^{2} +9.00000 q^{3} +17.0000 q^{4} -25.0000 q^{5} +63.0000 q^{6} +12.0000 q^{7} -105.000 q^{8} +81.0000 q^{9} -175.000 q^{10} +112.000 q^{11} +153.000 q^{12} -974.000 q^{13} +84.0000 q^{14} -225.000 q^{15} -1279.00 q^{16} +2182.00 q^{17} +567.000 q^{18} +1420.00 q^{19} -425.000 q^{20} +108.000 q^{21} +784.000 q^{22} +3216.00 q^{23} -945.000 q^{24} +625.000 q^{25} -6818.00 q^{26} +729.000 q^{27} +204.000 q^{28} -4150.00 q^{29} -1575.00 q^{30} -5688.00 q^{31} -5593.00 q^{32} +1008.00 q^{33} +15274.0 q^{34} -300.000 q^{35} +1377.00 q^{36} +6482.00 q^{37} +9940.00 q^{38} -8766.00 q^{39} +2625.00 q^{40} +5402.00 q^{41} +756.000 q^{42} -21764.0 q^{43} +1904.00 q^{44} -2025.00 q^{45} +22512.0 q^{46} -368.000 q^{47} -11511.0 q^{48} -16663.0 q^{49} +4375.00 q^{50} +19638.0 q^{51} -16558.0 q^{52} +12586.0 q^{53} +5103.00 q^{54} -2800.00 q^{55} -1260.00 q^{56} +12780.0 q^{57} -29050.0 q^{58} -25520.0 q^{59} -3825.00 q^{60} +11782.0 q^{61} -39816.0 q^{62} +972.000 q^{63} +1777.00 q^{64} +24350.0 q^{65} +7056.00 q^{66} -13188.0 q^{67} +37094.0 q^{68} +28944.0 q^{69} -2100.00 q^{70} -35968.0 q^{71} -8505.00 q^{72} +73186.0 q^{73} +45374.0 q^{74} +5625.00 q^{75} +24140.0 q^{76} +1344.00 q^{77} -61362.0 q^{78} -52440.0 q^{79} +31975.0 q^{80} +6561.00 q^{81} +37814.0 q^{82} +69036.0 q^{83} +1836.00 q^{84} -54550.0 q^{85} -152348. q^{86} -37350.0 q^{87} -11760.0 q^{88} -33870.0 q^{89} -14175.0 q^{90} -11688.0 q^{91} +54672.0 q^{92} -51192.0 q^{93} -2576.00 q^{94} -35500.0 q^{95} -50337.0 q^{96} +143042. q^{97} -116641. q^{98} +9072.00 q^{99} +O(q^{100})\)

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 $

$2$	7.00000	1.23744	0.618718	−	0.785613i	$-0.287652\pi$
$2$	7.00000	1.23744	0.618718	+	0.785613i	$0.287652\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	17.0000	0.531250
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	63.0000	0.714435
$7$	12.0000	0.0925627	0.0462814	−	0.998928i	$-0.485263\pi$
$7$	12.0000	0.0925627	0.0462814	+	0.998928i	$0.485263\pi$
$8$	−105.000	−0.580049
$9$	81.0000	0.333333
$10$	−175.000	−0.553399
$11$	112.000	0.279085	0.139542	−	0.990216i	$-0.455437\pi$
$11$	112.000	0.279085	0.139542	+	0.990216i	$0.455437\pi$
$12$	153.000	0.306717
$13$	−974.000	−1.59846	−0.799228	−	0.601028i	$-0.794758\pi$
$13$	−974.000	−1.59846	−0.799228	+	0.601028i	$0.794758\pi$
$14$	84.0000	0.114541
$15$	−225.000	−0.258199
$16$	−1279.00	−1.24902
$17$	2182.00	1.83119	0.915593	−	0.402106i	$-0.131722\pi$
$17$	2182.00	1.83119	0.915593	+	0.402106i	$0.131722\pi$
$18$	567.000	0.412479
$19$	1420.00	0.902411	0.451205	−	0.892420i	$-0.350994\pi$
$19$	1420.00	0.902411	0.451205	+	0.892420i	$0.350994\pi$
$20$	−425.000	−0.237582
$21$	108.000	0.0534411
$22$	784.000	0.345350
$23$	3216.00	1.26764	0.633821	−	0.773480i	$-0.281486\pi$
$23$	3216.00	1.26764	0.633821	+	0.773480i	$0.281486\pi$
$24$	−945.000	−0.334891
$25$	625.000	0.200000
$26$	−6818.00	−1.97799
$27$	729.000	0.192450
$28$	204.000	0.0491739
$29$	−4150.00	−0.916333	−0.458166	−	0.888867i	$-0.651494\pi$
$29$	−4150.00	−0.916333	−0.458166	+	0.888867i	$0.651494\pi$
$30$	−1575.00	−0.319505
$31$	−5688.00	−1.06305	−0.531527	−	0.847041i	$-0.678382\pi$
$31$	−5688.00	−1.06305	−0.531527	+	0.847041i	$0.678382\pi$
$32$	−5593.00	−0.965539
$33$	1008.00	0.161130
$34$	15274.0	2.26598
$35$	−300.000	−0.0413953
$36$	1377.00	0.177083
$37$	6482.00	0.778403	0.389202	−	0.921153i	$-0.372751\pi$
$37$	6482.00	0.778403	0.389202	+	0.921153i	$0.372751\pi$
$38$	9940.00	1.11668
$39$	−8766.00	−0.922869
$40$	2625.00	0.259406
$41$	5402.00	0.501874	0.250937	−	0.968003i	$-0.419261\pi$
$41$	5402.00	0.501874	0.250937	+	0.968003i	$0.419261\pi$
$42$	756.000	0.0661300
$43$	−21764.0	−1.79501	−0.897506	−	0.441001i	$-0.854623\pi$
$43$	−21764.0	−1.79501	−0.897506	+	0.441001i	$0.854623\pi$
$44$	1904.00	0.148264
$45$	−2025.00	−0.149071
$46$	22512.0	1.56863
$47$	−368.000	−0.0242998	−0.0121499	−	0.999926i	$-0.503868\pi$
$47$	−368.000	−0.0242998	−0.0121499	+	0.999926i	$0.503868\pi$
$48$	−11511.0	−0.721124
$49$	−16663.0	−0.991432
$50$	4375.00	0.247487
$51$	19638.0	1.05724
$52$	−16558.0	−0.849180
$53$	12586.0	0.615457	0.307729	−	0.951474i	$-0.400431\pi$
$53$	12586.0	0.615457	0.307729	+	0.951474i	$0.400431\pi$
$54$	5103.00	0.238145
$55$	−2800.00	−0.124811
$56$	−1260.00	−0.0536909
$57$	12780.0	0.521007
$58$	−29050.0	−1.13390
$59$	−25520.0	−0.954444	−0.477222	−	0.878783i	$-0.658356\pi$
$59$	−25520.0	−0.954444	−0.477222	+	0.878783i	$0.658356\pi$
$60$	−3825.00	−0.137168
$61$	11782.0	0.405410	0.202705	−	0.979240i	$-0.435027\pi$
$61$	11782.0	0.405410	0.202705	+	0.979240i	$0.435027\pi$
$62$	−39816.0	−1.31546
$63$	972.000	0.0308542
$64$	1777.00	0.0542297
$65$	24350.0	0.714851
$66$	7056.00	0.199388
$67$	−13188.0	−0.358915	−0.179458	−	0.983766i	$-0.557434\pi$
$67$	−13188.0	−0.358915	−0.179458	+	0.983766i	$0.557434\pi$
$68$	37094.0	0.972818
$69$	28944.0	0.731873
$70$	−2100.00	−0.0512241
$71$	−35968.0	−0.846780	−0.423390	−	0.905948i	$-0.639160\pi$
$71$	−35968.0	−0.846780	−0.423390	+	0.905948i	$0.639160\pi$
$72$	−8505.00	−0.193350
$73$	73186.0	1.60739	0.803694	−	0.595042i	$-0.202865\pi$
$73$	73186.0	1.60739	0.803694	+	0.595042i	$0.202865\pi$
$74$	45374.0	0.963225
$75$	5625.00	0.115470
$76$	24140.0	0.479406
$77$	1344.00	0.0258329
$78$	−61362.0	−1.14199
$79$	−52440.0	−0.945355	−0.472678	−	0.881235i	$-0.656712\pi$
$79$	−52440.0	−0.945355	−0.472678	+	0.881235i	$0.656712\pi$
$80$	31975.0	0.558580
$81$	6561.00	0.111111
$82$	37814.0	0.621038
$83$	69036.0	1.09997	0.549984	−	0.835175i	$-0.314634\pi$
$83$	69036.0	1.09997	0.549984	+	0.835175i	$0.314634\pi$
$84$	1836.00	0.0283906
$85$	−54550.0	−0.818931
$86$	−152348.	−2.22122
$87$	−37350.0	−0.529045
$88$	−11760.0	−0.161883
$89$	−33870.0	−0.453252	−0.226626	−	0.973982i	$-0.572770\pi$
$89$	−33870.0	−0.453252	−0.226626	+	0.973982i	$0.572770\pi$
$90$	−14175.0	−0.184466
$91$	−11688.0	−0.147957
$92$	54672.0	0.673435
$93$	−51192.0	−0.613755
$94$	−2576.00	−0.0300695
$95$	−35500.0	−0.403570
$96$	−50337.0	−0.557454
$97$	143042.	1.54360	0.771799	−	0.635867i	$-0.219357\pi$
$97$	143042.	1.54360	0.771799	+	0.635867i	$0.219357\pi$
$98$	−116641.	−1.22683
$99$	9072.00	0.0930283
$100$	10625.0	0.106250
$101$	63042.0	0.614931	0.307466	−	0.951559i	$-0.400519\pi$
$101$	63042.0	0.614931	0.307466	+	0.951559i	$0.400519\pi$
$102$	137466.	1.30826
$103$	38636.0	0.358839	0.179419	−	0.983773i	$-0.442578\pi$
$103$	38636.0	0.358839	0.179419	+	0.983773i	$0.442578\pi$
$104$	102270.	0.927182
$105$	−2700.00	−0.0238996
$106$	88102.0	0.761590
$107$	−69228.0	−0.584551	−0.292275	−	0.956334i	$-0.594412\pi$
$107$	−69228.0	−0.584551	−0.292275	+	0.956334i	$0.594412\pi$
$108$	12393.0	0.102239
$109$	51590.0	0.415910	0.207955	−	0.978138i	$-0.433319\pi$
$109$	51590.0	0.415910	0.207955	+	0.978138i	$0.433319\pi$
$110$	−19600.0	−0.154445
$111$	58338.0	0.449411
$112$	−15348.0	−0.115613
$113$	20206.0	0.148862	0.0744311	−	0.997226i	$-0.476286\pi$
$113$	20206.0	0.148862	0.0744311	+	0.997226i	$0.476286\pi$
$114$	89460.0	0.644714
$115$	−80400.0	−0.566907
$116$	−70550.0	−0.486802
$117$	−78894.0	−0.532819
$118$	−178640.	−1.18106
$119$	26184.0	0.169500
$120$	23625.0	0.149768
$121$	−148507.	−0.922112
$122$	82474.0	0.501669
$123$	48618.0	0.289757
$124$	−96696.0	−0.564747
$125$	−15625.0	−0.0894427
$126$	6804.00	0.0381802
$127$	−198908.	−1.09432	−0.547158	−	0.837029i	$-0.684290\pi$
$127$	−198908.	−1.09432	−0.547158	+	0.837029i	$0.684290\pi$
$128$	191415.	1.03264
$129$	−195876.	−1.03635
$130$	170450.	0.884583
$131$	150672.	0.767104	0.383552	−	0.923519i	$-0.374701\pi$
$131$	150672.	0.767104	0.383552	+	0.923519i	$0.374701\pi$
$132$	17136.0	0.0856002
$133$	17040.0	0.0835296
$134$	−92316.0	−0.444135
$135$	−18225.0	−0.0860663
$136$	−229110.	−1.06218
$137$	−19098.0	−0.0869334	−0.0434667	−	0.999055i	$-0.513840\pi$
$137$	−19098.0	−0.0869334	−0.0434667	+	0.999055i	$0.513840\pi$
$138$	202608.	0.905647
$139$	196460.	0.862456	0.431228	−	0.902243i	$-0.358080\pi$
$139$	196460.	0.862456	0.431228	+	0.902243i	$0.358080\pi$
$140$	−5100.00	−0.0219913
$141$	−3312.00	−0.0140295
$142$	−251776.	−1.04784
$143$	−109088.	−0.446105
$144$	−103599.	−0.416341
$145$	103750.	0.409796
$146$	512302.	1.98904
$147$	−149967.	−0.572404
$148$	110194.	0.413527
$149$	−362710.	−1.33842	−0.669212	−	0.743071i	$-0.733368\pi$
$149$	−362710.	−1.33842	−0.669212	+	0.743071i	$0.733368\pi$
$150$	39375.0	0.142887
$151$	76072.0	0.271508	0.135754	−	0.990743i	$-0.456654\pi$
$151$	76072.0	0.271508	0.135754	+	0.990743i	$0.456654\pi$
$152$	−149100.	−0.523442
$153$	176742.	0.610395
$154$	9408.00	0.0319665
$155$	142200.	0.475412
$156$	−149022.	−0.490274
$157$	252722.	0.818265	0.409132	−	0.912475i	$-0.365831\pi$
$157$	252722.	0.818265	0.409132	+	0.912475i	$0.365831\pi$
$158$	−367080.	−1.16982
$159$	113274.	0.355335
$160$	139825.	0.431802
$161$	38592.0	0.117336
$162$	45927.0	0.137493
$163$	85916.0	0.253282	0.126641	−	0.991949i	$-0.459580\pi$
$163$	85916.0	0.253282	0.126641	+	0.991949i	$0.459580\pi$
$164$	91834.0	0.266621
$165$	−25200.0	−0.0720594
$166$	483252.	1.36114
$167$	316272.	0.877545	0.438773	−	0.898598i	$-0.355413\pi$
$167$	316272.	0.877545	0.438773	+	0.898598i	$0.355413\pi$
$168$	−11340.0	−0.0309984
$169$	577383.	1.55506
$170$	−381850.	−1.01338
$171$	115020.	0.300804
$172$	−369988.	−0.953601
$173$	−597534.	−1.51791	−0.758957	−	0.651140i	$-0.774291\pi$
$173$	−597534.	−1.51791	−0.758957	+	0.651140i	$0.774291\pi$
$174$	−261450.	−0.654660
$175$	7500.00	0.0185125
$176$	−143248.	−0.348584
$177$	−229680.	−0.551049
$178$	−237090.	−0.560871
$179$	−282680.	−0.659421	−0.329710	−	0.944082i	$-0.606951\pi$
$179$	−282680.	−0.659421	−0.329710	+	0.944082i	$0.606951\pi$
$180$	−34425.0	−0.0791941
$181$	−294658.	−0.668531	−0.334266	−	0.942479i	$-0.608488\pi$
$181$	−294658.	−0.668531	−0.334266	+	0.942479i	$0.608488\pi$
$182$	−81816.0	−0.183088
$183$	106038.	0.234064
$184$	−337680.	−0.735294
$185$	−162050.	−0.348113
$186$	−358344.	−0.759483
$187$	244384.	0.511056
$188$	−6256.00	−0.0129093
$189$	8748.00	0.0178137
$190$	−248500.	−0.499393
$191$	723272.	1.43456	0.717279	−	0.696786i	$-0.245387\pi$
$191$	723272.	1.43456	0.717279	+	0.696786i	$0.245387\pi$
$192$	15993.0	0.0313096
$193$	80426.0	0.155419	0.0777093	−	0.996976i	$-0.475239\pi$
$193$	80426.0	0.155419	0.0777093	+	0.996976i	$0.475239\pi$
$194$	1.00129e6	1.91011
$195$	219150.	0.412719
$196$	−283271.	−0.526698
$197$	509802.	0.935914	0.467957	−	0.883751i	$-0.344990\pi$
$197$	509802.	0.935914	0.467957	+	0.883751i	$0.344990\pi$
$198$	63504.0	0.115117
$199$	−435320.	−0.779248	−0.389624	−	0.920974i	$-0.627395\pi$
$199$	−435320.	−0.779248	−0.389624	+	0.920974i	$0.627395\pi$
$200$	−65625.0	−0.116010
$201$	−118692.	−0.207220
$202$	441294.	0.760939
$203$	−49800.0	−0.0848182
$204$	333846.	0.561656
$205$	−135050.	−0.224445
$206$	270452.	0.444040
$207$	260496.	0.422547
$208$	1.24575e6	1.99651
$209$	159040.	0.251849
$210$	−18900.0	−0.0295742
$211$	−1.12275e6	−1.73611	−0.868053	−	0.496472i	$-0.834629\pi$
$211$	−1.12275e6	−1.73611	−0.868053	+	0.496472i	$0.834629\pi$
$212$	213962.	0.326962
$213$	−323712.	−0.488888
$214$	−484596.	−0.723345
$215$	544100.	0.802754
$216$	−76545.0	−0.111630
$217$	−68256.0	−0.0983992
$218$	361130.	0.514662
$219$	658674.	0.928026
$220$	−47600.0	−0.0663056
$221$	−2.12527e6	−2.92707
$222$	408366.	0.556118
$223$	−325084.	−0.437757	−0.218879	−	0.975752i	$-0.570240\pi$
$223$	−325084.	−0.437757	−0.218879	+	0.975752i	$0.570240\pi$
$224$	−67116.0	−0.0893729
$225$	50625.0	0.0666667
$226$	141442.	0.184207
$227$	−311308.	−0.400983	−0.200491	−	0.979695i	$-0.564254\pi$
$227$	−311308.	−0.400983	−0.200491	+	0.979695i	$0.564254\pi$
$228$	217260.	0.276785
$229$	−615450.	−0.775540	−0.387770	−	0.921756i	$-0.626754\pi$
$229$	−615450.	−0.775540	−0.387770	+	0.921756i	$0.626754\pi$
$230$	−562800.	−0.701511
$231$	12096.0	0.0149146
$232$	435750.	0.531517
$233$	−304434.	−0.367370	−0.183685	−	0.982985i	$-0.558803\pi$
$233$	−304434.	−0.367370	−0.183685	+	0.982985i	$0.558803\pi$
$234$	−552258.	−0.659329
$235$	9200.00	0.0108672
$236$	−433840.	−0.507049
$237$	−471960.	−0.545801
$238$	183288.	0.209745
$239$	780760.	0.884144	0.442072	−	0.896980i	$-0.354244\pi$
$239$	780760.	0.884144	0.442072	+	0.896980i	$0.354244\pi$
$240$	287775.	0.322496
$241$	635842.	0.705191	0.352595	−	0.935776i	$-0.385299\pi$
$241$	635842.	0.705191	0.352595	+	0.935776i	$0.385299\pi$
$242$	−1.03955e6	−1.14105
$243$	59049.0	0.0641500
$244$	200294.	0.215374
$245$	416575.	0.443382
$246$	340326.	0.358556
$247$	−1.38308e6	−1.44246
$248$	597240.	0.616623
$249$	621324.	0.635067
$250$	−109375.	−0.110680
$251$	1.20559e6	1.20786	0.603929	−	0.797038i	$-0.293601\pi$
$251$	1.20559e6	1.20786	0.603929	+	0.797038i	$0.293601\pi$
$252$	16524.0	0.0163913
$253$	360192.	0.353780
$254$	−1.39236e6	−1.35415
$255$	−490950.	−0.472810
$256$	1.28304e6	1.22360
$257$	1.96702e6	1.85770	0.928852	−	0.370452i	$-0.120797\pi$
$257$	1.96702e6	1.85770	0.928852	+	0.370452i	$0.120797\pi$
$258$	−1.37113e6	−1.28242
$259$	77784.0	0.0720511
$260$	413950.	0.379765
$261$	−336150.	−0.305444
$262$	1.05470e6	0.949243
$263$	−625264.	−0.557409	−0.278705	−	0.960377i	$-0.589905\pi$
$263$	−625264.	−0.557409	−0.278705	+	0.960377i	$0.589905\pi$
$264$	−105840.	−0.0934631
$265$	−314650.	−0.275241
$266$	119280.	0.103363
$267$	−304830.	−0.261685
$268$	−224196.	−0.190674
$269$	527050.	0.444090	0.222045	−	0.975036i	$-0.428727\pi$
$269$	527050.	0.444090	0.222045	+	0.975036i	$0.428727\pi$
$270$	−127575.	−0.106502
$271$	2.10923e6	1.74462	0.872311	−	0.488952i	$-0.162621\pi$
$271$	2.10923e6	1.74462	0.872311	+	0.488952i	$0.162621\pi$
$272$	−2.79078e6	−2.28719
$273$	−105192.	−0.0854233
$274$	−133686.	−0.107575
$275$	70000.0	0.0558170
$276$	492048.	0.388808
$277$	−267438.	−0.209423	−0.104711	−	0.994503i	$-0.533392\pi$
$277$	−267438.	−0.209423	−0.104711	+	0.994503i	$0.533392\pi$
$278$	1.37522e6	1.06724
$279$	−460728.	−0.354351
$280$	31500.0	0.0240113
$281$	838002.	0.633110	0.316555	−	0.948574i	$-0.397474\pi$
$281$	838002.	0.633110	0.316555	+	0.948574i	$0.397474\pi$
$282$	−23184.0	−0.0173606
$283$	−2.23772e6	−1.66089	−0.830444	−	0.557102i	$-0.811913\pi$
$283$	−2.23772e6	−1.66089	−0.830444	+	0.557102i	$0.811913\pi$
$284$	−611456.	−0.449852
$285$	−319500.	−0.233001
$286$	−763616.	−0.552027
$287$	64824.0	0.0464549
$288$	−453033.	−0.321846
$289$	3.34127e6	2.35324
$290$	726250.	0.507097
$291$	1.28738e6	0.891197
$292$	1.24416e6	0.853925
$293$	785706.	0.534676	0.267338	−	0.963603i	$-0.413856\pi$
$293$	785706.	0.534676	0.267338	+	0.963603i	$0.413856\pi$
$294$	−1.04977e6	−0.708313
$295$	638000.	0.426841
$296$	−680610.	−0.451512
$297$	81648.0	0.0537099
$298$	−2.53897e6	−1.65622
$299$	−3.13238e6	−2.02627
$300$	95625.0	0.0613435
$301$	−261168.	−0.166151
$302$	532504.	0.335974
$303$	567378.	0.355031
$304$	−1.81618e6	−1.12713
$305$	−294550.	−0.181305
$306$	1.23719e6	0.755326
$307$	−2.94955e6	−1.78612	−0.893058	−	0.449942i	$-0.851445\pi$
$307$	−2.94955e6	−1.78612	−0.893058	+	0.449942i	$0.851445\pi$
$308$	22848.0	0.0137237
$309$	347724.	0.207176
$310$	995400.	0.588293
$311$	−3.07757e6	−1.80429	−0.902146	−	0.431431i	$-0.858009\pi$
$311$	−3.07757e6	−1.80429	−0.902146	+	0.431431i	$0.858009\pi$
$312$	920430.	0.535309
$313$	1.61367e6	0.931007	0.465503	−	0.885046i	$-0.345873\pi$
$313$	1.61367e6	0.931007	0.465503	+	0.885046i	$0.345873\pi$
$314$	1.76905e6	1.01255
$315$	−24300.0	−0.0137984
$316$	−891480.	−0.502220
$317$	−2.00496e6	−1.12062	−0.560308	−	0.828284i	$-0.689317\pi$
$317$	−2.00496e6	−1.12062	−0.560308	+	0.828284i	$0.689317\pi$
$318$	792918.	0.439704
$319$	−464800.	−0.255735
$320$	−44425.0	−0.0242523
$321$	−623052.	−0.337491
$322$	270144.	0.145196
$323$	3.09844e6	1.65248
$324$	111537.	0.0590278
$325$	−608750.	−0.319691
$326$	601412.	0.313421
$327$	464310.	0.240126
$328$	−567210.	−0.291111
$329$	−4416.00	−0.00224926
$330$	−176400.	−0.0891690
$331$	470772.	0.236179	0.118089	−	0.993003i	$-0.462323\pi$
$331$	470772.	0.236179	0.118089	+	0.993003i	$0.462323\pi$
$332$	1.17361e6	0.584358
$333$	525042.	0.259468
$334$	2.21390e6	1.08591
$335$	329700.	0.160512
$336$	−138132.	−0.0667492
$337$	2.31548e6	1.11062	0.555311	−	0.831642i	$-0.312599\pi$
$337$	2.31548e6	1.11062	0.555311	+	0.831642i	$0.312599\pi$
$338$	4.04168e6	1.92429
$339$	181854.	0.0859456
$340$	−927350.	−0.435057
$341$	−637056.	−0.296682
$342$	805140.	0.372226
$343$	−401640.	−0.184332
$344$	2.28522e6	1.04119
$345$	−723600.	−0.327304
$346$	−4.18274e6	−1.87832
$347$	1.25393e6	0.559050	0.279525	−	0.960138i	$-0.409823\pi$
$347$	1.25393e6	0.559050	0.279525	+	0.960138i	$0.409823\pi$
$348$	−634950.	−0.281055
$349$	616390.	0.270889	0.135445	−	0.990785i	$-0.456754\pi$
$349$	616390.	0.270889	0.135445	+	0.990785i	$0.456754\pi$
$350$	52500.0	0.0229081
$351$	−710046.	−0.307623
$352$	−626416.	−0.269467
$353$	−281274.	−0.120141	−0.0600707	−	0.998194i	$-0.519133\pi$
$353$	−281274.	−0.120141	−0.0600707	+	0.998194i	$0.519133\pi$
$354$	−1.60776e6	−0.681888
$355$	899200.	0.378691
$356$	−575790.	−0.240790
$357$	235656.	0.0978606
$358$	−1.97876e6	−0.815991
$359$	−1.19148e6	−0.487922	−0.243961	−	0.969785i	$-0.578447\pi$
$359$	−1.19148e6	−0.487922	−0.243961	+	0.969785i	$0.578447\pi$
$360$	212625.	0.0864685
$361$	−459699.	−0.185655
$362$	−2.06261e6	−0.827265
$363$	−1.33656e6	−0.532381
$364$	−198696.	−0.0786024
$365$	−1.82965e6	−0.718846
$366$	742266.	0.289639
$367$	−793068.	−0.307359	−0.153679	−	0.988121i	$-0.549112\pi$
$367$	−793068.	−0.307359	−0.153679	+	0.988121i	$0.549112\pi$
$368$	−4.11326e6	−1.58331
$369$	437562.	0.167291
$370$	−1.13435e6	−0.430767
$371$	151032.	0.0569684
$372$	−870264.	−0.326057
$373$	635626.	0.236554	0.118277	−	0.992981i	$-0.462263\pi$
$373$	635626.	0.236554	0.118277	+	0.992981i	$0.462263\pi$
$374$	1.71069e6	0.632400
$375$	−140625.	−0.0516398
$376$	38640.0	0.0140951
$377$	4.04210e6	1.46472
$378$	61236.0	0.0220433
$379$	−2.12834e6	−0.761102	−0.380551	−	0.924760i	$-0.624266\pi$
$379$	−2.12834e6	−0.761102	−0.380551	+	0.924760i	$0.624266\pi$
$380$	−603500.	−0.214397
$381$	−1.79017e6	−0.631804
$382$	5.06290e6	1.77518
$383$	−4.88174e6	−1.70051	−0.850253	−	0.526375i	$-0.823551\pi$
$383$	−4.88174e6	−1.70051	−0.850253	+	0.526375i	$0.823551\pi$
$384$	1.72273e6	0.596198
$385$	−33600.0	−0.0115528
$386$	562982.	0.192321
$387$	−1.76288e6	−0.598338
$388$	2.43171e6	0.820036
$389$	−2.30607e6	−0.772678	−0.386339	−	0.922357i	$-0.626260\pi$
$389$	−2.30607e6	−0.772678	−0.386339	+	0.922357i	$0.626260\pi$
$390$	1.53405e6	0.510714
$391$	7.01731e6	2.32129
$392$	1.74962e6	0.575079
$393$	1.35605e6	0.442888
$394$	3.56861e6	1.15813
$395$	1.31100e6	0.422776
$396$	154224.	0.0494213
$397$	−423398.	−0.134826	−0.0674128	−	0.997725i	$-0.521474\pi$
$397$	−423398.	−0.134826	−0.0674128	+	0.997725i	$0.521474\pi$
$398$	−3.04724e6	−0.964271
$399$	153360.	0.0482258
$400$	−799375.	−0.249805
$401$	−2.60756e6	−0.809791	−0.404896	−	0.914363i	$-0.632692\pi$
$401$	−2.60756e6	−0.809791	−0.404896	+	0.914363i	$0.632692\pi$
$402$	−830844.	−0.256421
$403$	5.54011e6	1.69924
$404$	1.07171e6	0.326682
$405$	−164025.	−0.0496904
$406$	−348600.	−0.104957
$407$	725984.	0.217241
$408$	−2.06199e6	−0.613248
$409$	−4.80871e6	−1.42141	−0.710707	−	0.703489i	$-0.751625\pi$
$409$	−4.80871e6	−1.42141	−0.710707	+	0.703489i	$0.751625\pi$
$410$	−945350.	−0.277737
$411$	−171882.	−0.0501910
$412$	656812.	0.190633
$413$	−306240.	−0.0883460
$414$	1.82347e6	0.522875
$415$	−1.72590e6	−0.491921
$416$	5.44758e6	1.54337
$417$	1.76814e6	0.497939
$418$	1.11328e6	0.311648
$419$	139760.	0.0388909	0.0194454	−	0.999811i	$-0.493810\pi$
$419$	139760.	0.0388909	0.0194454	+	0.999811i	$0.493810\pi$
$420$	−45900.0	−0.0126967
$421$	−3.00310e6	−0.825780	−0.412890	−	0.910781i	$-0.635481\pi$
$421$	−3.00310e6	−0.825780	−0.412890	+	0.910781i	$0.635481\pi$
$422$	−7.85924e6	−2.14832
$423$	−29808.0	−0.00809994
$424$	−1.32153e6	−0.356995
$425$	1.36375e6	0.366237
$426$	−2.26598e6	−0.604969
$427$	141384.	0.0375259
$428$	−1.17688e6	−0.310543
$429$	−981792.	−0.257559
$430$	3.80870e6	0.993358
$431$	5.97955e6	1.55051	0.775257	−	0.631646i	$-0.217621\pi$
$431$	5.97955e6	1.55051	0.775257	+	0.631646i	$0.217621\pi$
$432$	−932391.	−0.240375
$433$	1.75235e6	0.449159	0.224580	−	0.974456i	$-0.427899\pi$
$433$	1.75235e6	0.449159	0.224580	+	0.974456i	$0.427899\pi$
$434$	−477792.	−0.121763
$435$	933750.	0.236596
$436$	877030.	0.220952
$437$	4.56672e6	1.14393
$438$	4.61072e6	1.14837
$439$	−4.72556e6	−1.17029	−0.585143	−	0.810930i	$-0.698962\pi$
$439$	−4.72556e6	−1.17029	−0.585143	+	0.810930i	$0.698962\pi$
$440$	294000.	0.0723962
$441$	−1.34970e6	−0.330477
$442$	−1.48769e7	−3.62206
$443$	2.48584e6	0.601815	0.300908	−	0.953653i	$-0.402710\pi$
$443$	2.48584e6	0.601815	0.300908	+	0.953653i	$0.402710\pi$
$444$	991746.	0.238750
$445$	846750.	0.202701
$446$	−2.27559e6	−0.541697
$447$	−3.26439e6	−0.772740
$448$	21324.0	0.00501965
$449$	1.46233e6	0.342318	0.171159	−	0.985243i	$-0.445249\pi$
$449$	1.46233e6	0.342318	0.171159	+	0.985243i	$0.445249\pi$
$450$	354375.	0.0824958
$451$	605024.	0.140066
$452$	343502.	0.0790830
$453$	684648.	0.156755
$454$	−2.17916e6	−0.496191
$455$	292200.	0.0661686
$456$	−1.34190e6	−0.302209
$457$	−1.45684e6	−0.326303	−0.163151	−	0.986601i	$-0.552166\pi$
$457$	−1.45684e6	−0.326303	−0.163151	+	0.986601i	$0.552166\pi$
$458$	−4.30815e6	−0.959681
$459$	1.59068e6	0.352412
$460$	−1.36680e6	−0.301169
$461$	4.32280e6	0.947356	0.473678	−	0.880698i	$-0.342926\pi$
$461$	4.32280e6	0.947356	0.473678	+	0.880698i	$0.342926\pi$
$462$	84672.0	0.0184559
$463$	1.07848e6	0.233807	0.116904	−	0.993143i	$-0.462703\pi$
$463$	1.07848e6	0.233807	0.116904	+	0.993143i	$0.462703\pi$
$464$	5.30785e6	1.14452
$465$	1.27980e6	0.274479
$466$	−2.13104e6	−0.454597
$467$	−3.40023e6	−0.721466	−0.360733	−	0.932669i	$-0.617473\pi$
$467$	−3.40023e6	−0.721466	−0.360733	+	0.932669i	$0.617473\pi$
$468$	−1.34120e6	−0.283060
$469$	−158256.	−0.0332222
$470$	64400.0	0.0134475
$471$	2.27450e6	0.472425
$472$	2.67960e6	0.553624
$473$	−2.43757e6	−0.500961
$474$	−3.30372e6	−0.675394
$475$	887500.	0.180482
$476$	445128.	0.0900466
$477$	1.01947e6	0.205152
$478$	5.46532e6	1.09407
$479$	2.75268e6	0.548172	0.274086	−	0.961705i	$-0.411625\pi$
$479$	2.75268e6	0.548172	0.274086	+	0.961705i	$0.411625\pi$
$480$	1.25842e6	0.249301
$481$	−6.31347e6	−1.24424
$482$	4.45089e6	0.872629
$483$	347328.	0.0677442
$484$	−2.52462e6	−0.489872
$485$	−3.57605e6	−0.690318
$486$	413343.	0.0793816
$487$	3.56997e6	0.682091	0.341046	−	0.940047i	$-0.389219\pi$
$487$	3.56997e6	0.682091	0.341046	+	0.940047i	$0.389219\pi$
$488$	−1.23711e6	−0.235157
$489$	773244.	0.146233
$490$	2.91602e6	0.548657
$491$	1.96455e6	0.367756	0.183878	−	0.982949i	$-0.441135\pi$
$491$	1.96455e6	0.367756	0.183878	+	0.982949i	$0.441135\pi$
$492$	826506.	0.153934
$493$	−9.05530e6	−1.67798
$494$	−9.68156e6	−1.78496
$495$	−226800.	−0.0416035
$496$	7.27495e6	1.32778
$497$	−431616.	−0.0783802
$498$	4.34927e6	0.785856
$499$	5.14798e6	0.925519	0.462760	−	0.886484i	$-0.346859\pi$
$499$	5.14798e6	0.925519	0.462760	+	0.886484i	$0.346859\pi$
$500$	−265625.	−0.0475164
$501$	2.84645e6	0.506651
$502$	8.43914e6	1.49465
$503$	1.97502e6	0.348057	0.174029	−	0.984741i	$-0.444321\pi$
$503$	1.97502e6	0.348057	0.174029	+	0.984741i	$0.444321\pi$
$504$	−102060.	−0.0178970
$505$	−1.57605e6	−0.275006
$506$	2.52134e6	0.437780
$507$	5.19645e6	0.897814
$508$	−3.38144e6	−0.581356
$509$	−3.32447e6	−0.568759	−0.284379	−	0.958712i	$-0.591787\pi$
$509$	−3.32447e6	−0.568759	−0.284379	+	0.958712i	$0.591787\pi$
$510$	−3.43665e6	−0.585073
$511$	878232.	0.148784
$512$	2.85601e6	0.481487
$513$	1.03518e6	0.173669
$514$	1.37692e7	2.29879
$515$	−965900.	−0.160477
$516$	−3.32989e6	−0.550562
$517$	−41216.0	−0.00678171
$518$	544488.	0.0891587
$519$	−5.37781e6	−0.876368
$520$	−2.55675e6	−0.414648
$521$	−2.97960e6	−0.480910	−0.240455	−	0.970660i	$-0.577297\pi$
$521$	−2.97960e6	−0.480910	−0.240455	+	0.970660i	$0.577297\pi$
$522$	−2.35305e6	−0.377968
$523$	−6.19108e6	−0.989720	−0.494860	−	0.868973i	$-0.664781\pi$
$523$	−6.19108e6	−0.989720	−0.494860	+	0.868973i	$0.664781\pi$
$524$	2.56142e6	0.407524
$525$	67500.0	0.0106882
$526$	−4.37685e6	−0.689759
$527$	−1.24112e7	−1.94665
$528$	−1.28923e6	−0.201255
$529$	3.90631e6	0.606915
$530$	−2.20255e6	−0.340593
$531$	−2.06712e6	−0.318148
$532$	289680.	0.0443751
$533$	−5.26155e6	−0.802224
$534$	−2.13381e6	−0.323819
$535$	1.73070e6	0.261419
$536$	1.38474e6	0.208188
$537$	−2.54412e6	−0.380717
$538$	3.68935e6	0.549533
$539$	−1.86626e6	−0.276694
$540$	−309825.	−0.0457227
$541$	−8.55398e6	−1.25654	−0.628268	−	0.777997i	$-0.716236\pi$
$541$	−8.55398e6	−1.25654	−0.628268	+	0.777997i	$0.716236\pi$
$542$	1.47646e7	2.15886
$543$	−2.65192e6	−0.385977
$544$	−1.22039e7	−1.76808
$545$	−1.28975e6	−0.186001
$546$	−736344.	−0.105706
$547$	−2.54371e6	−0.363495	−0.181748	−	0.983345i	$-0.558175\pi$
$547$	−2.54371e6	−0.363495	−0.181748	+	0.983345i	$0.558175\pi$
$548$	−324666.	−0.0461833
$549$	954342.	0.135137
$550$	490000.	0.0690700
$551$	−5.89300e6	−0.826908
$552$	−3.03912e6	−0.424522
$553$	−629280.	−0.0875046
$554$	−1.87207e6	−0.259147
$555$	−1.45845e6	−0.200983
$556$	3.33982e6	0.458180
$557$	7.58704e6	1.03618	0.518089	−	0.855327i	$-0.326644\pi$
$557$	7.58704e6	1.03618	0.518089	+	0.855327i	$0.326644\pi$
$558$	−3.22510e6	−0.438487
$559$	2.11981e7	2.86925
$560$	383700.	0.0517037
$561$	2.19946e6	0.295058
$562$	5.86601e6	0.783434
$563$	7.50940e6	0.998468	0.499234	−	0.866467i	$-0.333615\pi$
$563$	7.50940e6	0.998468	0.499234	+	0.866467i	$0.333615\pi$
$564$	−56304.0	−0.00745318
$565$	−505150.	−0.0665732
$566$	−1.56641e7	−2.05524
$567$	78732.0	0.0102847
$568$	3.77664e6	0.491173
$569$	1.38890e7	1.79842	0.899209	−	0.437519i	$-0.144143\pi$
$569$	1.38890e7	1.79842	0.899209	+	0.437519i	$0.144143\pi$
$570$	−2.23650e6	−0.288325
$571$	−1.93539e6	−0.248415	−0.124207	−	0.992256i	$-0.539639\pi$
$571$	−1.93539e6	−0.248415	−0.124207	+	0.992256i	$0.539639\pi$
$572$	−1.85450e6	−0.236993
$573$	6.50945e6	0.828243
$574$	453768.	0.0574849
$575$	2.01000e6	0.253528
$576$	143937.	0.0180766
$577$	−4.89408e6	−0.611972	−0.305986	−	0.952036i	$-0.598986\pi$
$577$	−4.89408e6	−0.611972	−0.305986	+	0.952036i	$0.598986\pi$
$578$	2.33889e7	2.91199
$579$	723834.	0.0897310
$580$	1.76375e6	0.217704
$581$	828432.	0.101816
$582$	9.01165e6	1.10280
$583$	1.40963e6	0.171765
$584$	−7.68453e6	−0.932363
$585$	1.97235e6	0.238284
$586$	5.49994e6	0.661628
$587$	−6.43883e6	−0.771279	−0.385640	−	0.922650i	$-0.626019\pi$
$587$	−6.43883e6	−0.771279	−0.385640	+	0.922650i	$0.626019\pi$
$588$	−2.54944e6	−0.304089
$589$	−8.07696e6	−0.959312
$590$	4.46600e6	0.528188
$591$	4.58822e6	0.540350
$592$	−8.29048e6	−0.972244
$593$	4.30365e6	0.502574	0.251287	−	0.967913i	$-0.419146\pi$
$593$	4.30365e6	0.502574	0.251287	+	0.967913i	$0.419146\pi$
$594$	571536.	0.0664626
$595$	−654600.	−0.0758025
$596$	−6.16607e6	−0.711038
$597$	−3.91788e6	−0.449899
$598$	−2.19267e7	−2.50738
$599$	−4.50988e6	−0.513568	−0.256784	−	0.966469i	$-0.582663\pi$
$599$	−4.50988e6	−0.513568	−0.256784	+	0.966469i	$0.582663\pi$
$600$	−590625.	−0.0669782
$601$	−5.11596e6	−0.577751	−0.288876	−	0.957367i	$-0.593281\pi$
$601$	−5.11596e6	−0.577751	−0.288876	+	0.957367i	$0.593281\pi$
$602$	−1.82818e6	−0.205602
$603$	−1.06823e6	−0.119638
$604$	1.29322e6	0.144239
$605$	3.71267e6	0.412381
$606$	3.97165e6	0.439328
$607$	1.61925e7	1.78378	0.891891	−	0.452250i	$-0.149378\pi$
$607$	1.61925e7	1.78378	0.891891	+	0.452250i	$0.149378\pi$
$608$	−7.94206e6	−0.871313
$609$	−448200.	−0.0489698
$610$	−2.06185e6	−0.224353
$611$	358432.	0.0388422
$612$	3.00461e6	0.324273
$613$	1.55525e7	1.67166	0.835830	−	0.548988i	$-0.184987\pi$
$613$	1.55525e7	1.67166	0.835830	+	0.548988i	$0.184987\pi$
$614$	−2.06468e7	−2.21021
$615$	−1.21545e6	−0.129583
$616$	−141120.	−0.0149843
$617$	−1.46710e7	−1.55148	−0.775740	−	0.631053i	$-0.782623\pi$
$617$	−1.46710e7	−1.55148	−0.775740	+	0.631053i	$0.782623\pi$
$618$	2.43407e6	0.256367
$619$	−9.66826e6	−1.01420	−0.507098	−	0.861889i	$-0.669282\pi$
$619$	−9.66826e6	−1.01420	−0.507098	+	0.861889i	$0.669282\pi$
$620$	2.41740e6	0.252563
$621$	2.34446e6	0.243958
$622$	−2.15430e7	−2.23270
$623$	−406440.	−0.0419543
$624$	1.12117e7	1.15268
$625$	390625.	0.0400000
$626$	1.12957e7	1.15206
$627$	1.43136e6	0.145405
$628$	4.29627e6	0.434703
$629$	1.41437e7	1.42540
$630$	−170100.	−0.0170747
$631$	−1.16557e6	−0.116537	−0.0582686	−	0.998301i	$-0.518558\pi$
$631$	−1.16557e6	−0.116537	−0.0582686	+	0.998301i	$0.518558\pi$
$632$	5.50620e6	0.548352
$633$	−1.01047e7	−1.00234
$634$	−1.40347e7	−1.38669
$635$	4.97270e6	0.489393
$636$	1.92566e6	0.188771
$637$	1.62298e7	1.58476
$638$	−3.25360e6	−0.316455
$639$	−2.91341e6	−0.282260
$640$	−4.78538e6	−0.461813
$641$	1.95088e6	0.187537	0.0937683	−	0.995594i	$-0.470109\pi$
$641$	1.95088e6	0.187537	0.0937683	+	0.995594i	$0.470109\pi$
$642$	−4.36136e6	−0.417623
$643$	−1.17387e7	−1.11968	−0.559839	−	0.828601i	$-0.689137\pi$
$643$	−1.17387e7	−1.11968	−0.559839	+	0.828601i	$0.689137\pi$
$644$	656064.	0.0623349
$645$	4.89690e6	0.463470
$646$	2.16891e7	2.04484
$647$	−1.01369e7	−0.952015	−0.476008	−	0.879441i	$-0.657917\pi$
$647$	−1.01369e7	−0.952015	−0.476008	+	0.879441i	$0.657917\pi$
$648$	−688905.	−0.0644498
$649$	−2.85824e6	−0.266371
$650$	−4.26125e6	−0.395598
$651$	−614304.	−0.0568108
$652$	1.46057e6	0.134556
$653$	2.47095e6	0.226767	0.113384	−	0.993551i	$-0.463831\pi$
$653$	2.47095e6	0.226767	0.113384	+	0.993551i	$0.463831\pi$
$654$	3.25017e6	0.297140
$655$	−3.76680e6	−0.343059
$656$	−6.90916e6	−0.626853
$657$	5.92807e6	0.535796
$658$	−30912.0	−0.00278332
$659$	1.62242e7	1.45529	0.727644	−	0.685955i	$-0.240616\pi$
$659$	1.62242e7	1.45529	0.727644	+	0.685955i	$0.240616\pi$
$660$	−428400.	−0.0382816
$661$	1.54679e7	1.37698	0.688490	−	0.725246i	$-0.258274\pi$
$661$	1.54679e7	1.37698	0.688490	+	0.725246i	$0.258274\pi$
$662$	3.29540e6	0.292256
$663$	−1.91274e7	−1.68994
$664$	−7.24878e6	−0.638035
$665$	−426000.	−0.0373556
$666$	3.67529e6	0.321075
$667$	−1.33464e7	−1.16158
$668$	5.37662e6	0.466196
$669$	−2.92576e6	−0.252739
$670$	2.30790e6	0.198623
$671$	1.31958e6	0.113144
$672$	−604044.	−0.0515995
$673$	−1.94441e7	−1.65482	−0.827410	−	0.561598i	$-0.810187\pi$
$673$	−1.94441e7	−1.65482	−0.827410	+	0.561598i	$0.810187\pi$
$674$	1.62084e7	1.37433
$675$	455625.	0.0384900
$676$	9.81551e6	0.826126
$677$	643242.	0.0539390	0.0269695	−	0.999636i	$-0.491414\pi$
$677$	643242.	0.0539390	0.0269695	+	0.999636i	$0.491414\pi$
$678$	1.27298e6	0.106352
$679$	1.71650e6	0.142880
$680$	5.72775e6	0.475020
$681$	−2.80177e6	−0.231507
$682$	−4.45939e6	−0.367126
$683$	1.14412e6	0.0938465	0.0469233	−	0.998898i	$-0.485058\pi$
$683$	1.14412e6	0.0938465	0.0469233	+	0.998898i	$0.485058\pi$
$684$	1.95534e6	0.159802
$685$	477450.	0.0388778
$686$	−2.81148e6	−0.228100
$687$	−5.53905e6	−0.447758
$688$	2.78362e7	2.24201
$689$	−1.22588e7	−0.983781
$690$	−5.06520e6	−0.405018
$691$	1.63625e6	0.130363	0.0651816	−	0.997873i	$-0.479237\pi$
$691$	1.63625e6	0.130363	0.0651816	+	0.997873i	$0.479237\pi$
$692$	−1.01581e7	−0.806392
$693$	108864.	0.00861095
$694$	8.77752e6	0.691789
$695$	−4.91150e6	−0.385702
$696$	3.92175e6	0.306872
$697$	1.17872e7	0.919025
$698$	4.31473e6	0.335209
$699$	−2.73991e6	−0.212101
$700$	127500.	0.00983479
$701$	1.58303e7	1.21673	0.608364	−	0.793658i	$-0.291826\pi$
$701$	1.58303e7	1.21673	0.608364	+	0.793658i	$0.291826\pi$
$702$	−4.97032e6	−0.380664
$703$	9.20444e6	0.702440
$704$	199024.	0.0151347
$705$	82800.0	0.00627419
$706$	−1.96892e6	−0.148667
$707$	756504.	0.0569197
$708$	−3.90456e6	−0.292745
$709$	910870.	0.0680520	0.0340260	−	0.999421i	$-0.489167\pi$
$709$	910870.	0.0680520	0.0340260	+	0.999421i	$0.489167\pi$
$710$	6.29440e6	0.468607
$711$	−4.24764e6	−0.315118
$712$	3.55635e6	0.262908
$713$	−1.82926e7	−1.34757
$714$	1.64959e6	0.121096
$715$	2.72720e6	0.199504
$716$	−4.80556e6	−0.350317
$717$	7.02684e6	0.510461
$718$	−8.34036e6	−0.603773
$719$	−1.52246e7	−1.09831	−0.549155	−	0.835721i	$-0.685050\pi$
$719$	−1.52246e7	−1.09831	−0.549155	+	0.835721i	$0.685050\pi$
$720$	2.58998e6	0.186193
$721$	463632.	0.0332151
$722$	−3.21789e6	−0.229736
$723$	5.72258e6	0.407142
$724$	−5.00919e6	−0.355157
$725$	−2.59375e6	−0.183267
$726$	−9.35594e6	−0.658788
$727$	1.81793e7	1.27567	0.637837	−	0.770171i	$-0.279829\pi$
$727$	1.81793e7	1.27567	0.637837	+	0.770171i	$0.279829\pi$
$728$	1.22724e6	0.0858225
$729$	531441.	0.0370370
$730$	−1.28076e7	−0.889527
$731$	−4.74890e7	−3.28700
$732$	1.80265e6	0.124346
$733$	2.08512e7	1.43341	0.716707	−	0.697374i	$-0.245648\pi$
$733$	2.08512e7	1.43341	0.716707	+	0.697374i	$0.245648\pi$
$734$	−5.55148e6	−0.380337
$735$	3.74918e6	0.255987
$736$	−1.79871e7	−1.22396
$737$	−1.47706e6	−0.100168
$738$	3.06293e6	0.207013
$739$	−2.12513e7	−1.43144	−0.715722	−	0.698385i	$-0.753902\pi$
$739$	−2.12513e7	−1.43144	−0.715722	+	0.698385i	$0.753902\pi$
$740$	−2.75485e6	−0.184935
$741$	−1.24477e7	−0.832807
$742$	1.05722e6	0.0704948
$743$	7.12262e6	0.473334	0.236667	−	0.971591i	$-0.423945\pi$
$743$	7.12262e6	0.473334	0.236667	+	0.971591i	$0.423945\pi$
$744$	5.37516e6	0.356007
$745$	9.06775e6	0.598562
$746$	4.44938e6	0.292720
$747$	5.59192e6	0.366656
$748$	4.15453e6	0.271499
$749$	−830736.	−0.0541076
$750$	−984375.	−0.0639010
$751$	−1.00277e7	−0.648785	−0.324393	−	0.945923i	$-0.605160\pi$
$751$	−1.00277e7	−0.648785	−0.324393	+	0.945923i	$0.605160\pi$
$752$	470672.	0.0303511
$753$	1.08503e7	0.697357
$754$	2.82947e7	1.81249
$755$	−1.90180e6	−0.121422
$756$	148716.	0.00946353
$757$	−2.18303e7	−1.38459	−0.692294	−	0.721616i	$-0.743400\pi$
$757$	−2.18303e7	−1.38459	−0.692294	+	0.721616i	$0.743400\pi$
$758$	−1.48984e7	−0.941816
$759$	3.24173e6	0.204255
$760$	3.72750e6	0.234090
$761$	2.56780e7	1.60731	0.803655	−	0.595096i	$-0.202886\pi$
$761$	2.56780e7	1.60731	0.803655	+	0.595096i	$0.202886\pi$
$762$	−1.25312e7	−0.781817
$763$	619080.	0.0384978
$764$	1.22956e7	0.762109
$765$	−4.41855e6	−0.272977
$766$	−3.41722e7	−2.10427
$767$	2.48565e7	1.52564
$768$	1.15474e7	0.706448
$769$	5.81453e6	0.354567	0.177284	−	0.984160i	$-0.443269\pi$
$769$	5.81453e6	0.354567	0.177284	+	0.984160i	$0.443269\pi$
$770$	−235200.	−0.0142959
$771$	1.77032e7	1.07255
$772$	1.36724e6	0.0825662
$773$	1.55507e7	0.936057	0.468029	−	0.883713i	$-0.344964\pi$
$773$	1.55507e7	0.936057	0.468029	+	0.883713i	$0.344964\pi$
$774$	−1.23402e7	−0.740405
$775$	−3.55500e6	−0.212611
$776$	−1.50194e7	−0.895362
$777$	700056.	0.0415987
$778$	−1.61425e7	−0.956140
$779$	7.67084e6	0.452897
$780$	3.72555e6	0.219257
$781$	−4.02842e6	−0.236323
$782$	4.91212e7	2.87245
$783$	−3.02535e6	−0.176348
$784$	2.13120e7	1.23832
$785$	−6.31805e6	−0.365939
$786$	9.49234e6	0.548046
$787$	2.35987e7	1.35816	0.679079	−	0.734065i	$-0.262379\pi$
$787$	2.35987e7	1.35816	0.679079	+	0.734065i	$0.262379\pi$
$788$	8.66663e6	0.497204
$789$	−5.62738e6	−0.321820
$790$	9.17700e6	0.523158
$791$	242472.	0.0137791
$792$	−952560.	−0.0539609
$793$	−1.14757e7	−0.648030
$794$	−2.96379e6	−0.166838
$795$	−2.83185e6	−0.158910
$796$	−7.40044e6	−0.413976
$797$	−2.01127e7	−1.12157	−0.560783	−	0.827963i	$-0.689500\pi$
$797$	−2.01127e7	−1.12157	−0.560783	+	0.827963i	$0.689500\pi$
$798$	1.07352e6	0.0596764
$799$	−802976.	−0.0444975
$800$	−3.49562e6	−0.193108
$801$	−2.74347e6	−0.151084
$802$	−1.82529e7	−1.00207
$803$	8.19683e6	0.448598
$804$	−2.01776e6	−0.110086
$805$	−964800.	−0.0524744
$806$	3.87808e7	2.10271
$807$	4.74345e6	0.256396
$808$	−6.61941e6	−0.356690
$809$	−4.24111e6	−0.227829	−0.113914	−	0.993491i	$-0.536339\pi$
$809$	−4.24111e6	−0.227829	−0.113914	+	0.993491i	$0.536339\pi$
$810$	−1.14817e6	−0.0614887
$811$	6.04321e6	0.322638	0.161319	−	0.986902i	$-0.448425\pi$
$811$	6.04321e6	0.322638	0.161319	+	0.986902i	$0.448425\pi$
$812$	−846600.	−0.0450597
$813$	1.89831e7	1.00726
$814$	5.08189e6	0.268822
$815$	−2.14790e6	−0.113271
$816$	−2.51170e7	−1.32051
$817$	−3.09049e7	−1.61984
$818$	−3.36610e7	−1.75891
$819$	−946728.	−0.0493191
$820$	−2.29585e6	−0.119236
$821$	1.66230e7	0.860702	0.430351	−	0.902662i	$-0.358390\pi$
$821$	1.66230e7	0.860702	0.430351	+	0.902662i	$0.358390\pi$
$822$	−1.20317e6	−0.0621082
$823$	−2.59172e7	−1.33380	−0.666898	−	0.745149i	$-0.732378\pi$
$823$	−2.59172e7	−1.33380	−0.666898	+	0.745149i	$0.732378\pi$
$824$	−4.05678e6	−0.208144
$825$	630000.	0.0322259
$826$	−2.14368e6	−0.109323
$827$	1.67704e7	0.852668	0.426334	−	0.904566i	$-0.359805\pi$
$827$	1.67704e7	0.852668	0.426334	+	0.904566i	$0.359805\pi$
$828$	4.42843e6	0.224478
$829$	6.15999e6	0.311310	0.155655	−	0.987811i	$-0.450251\pi$
$829$	6.15999e6	0.311310	0.155655	+	0.987811i	$0.450251\pi$
$830$	−1.20813e7	−0.608721
$831$	−2.40694e6	−0.120910
$832$	−1.73080e6	−0.0866838
$833$	−3.63587e7	−1.81550
$834$	1.23770e7	0.616169
$835$	−7.90680e6	−0.392450
$836$	2.70368e6	0.133795
$837$	−4.14655e6	−0.204585
$838$	978320.	0.0481250
$839$	2.80172e6	0.137410	0.0687052	−	0.997637i	$-0.478113\pi$
$839$	2.80172e6	0.137410	0.0687052	+	0.997637i	$0.478113\pi$
$840$	283500.	0.0138629
$841$	−3.28865e6	−0.160335
$842$	−2.10217e7	−1.02185
$843$	7.54202e6	0.365526
$844$	−1.90867e7	−0.922306
$845$	−1.44346e7	−0.695444
$846$	−208656.	−0.0100232
$847$	−1.78208e6	−0.0853532
$848$	−1.60975e7	−0.768721
$849$	−2.01395e7	−0.958914
$850$	9.54625e6	0.453195
$851$	2.08461e7	0.986736
$852$	−5.50310e6	−0.259722
$853$	2.54991e7	1.19992	0.599960	−	0.800030i	$-0.295183\pi$
$853$	2.54991e7	1.19992	0.599960	+	0.800030i	$0.295183\pi$
$854$	989688.	0.0464359
$855$	−2.87550e6	−0.134523
$856$	7.26894e6	0.339068
$857$	−1.19499e7	−0.555794	−0.277897	−	0.960611i	$-0.589637\pi$
$857$	−1.19499e7	−0.555794	−0.277897	+	0.960611i	$0.589637\pi$
$858$	−6.87254e6	−0.318713
$859$	−1.01568e7	−0.469651	−0.234825	−	0.972038i	$-0.575452\pi$
$859$	−1.01568e7	−0.469651	−0.234825	+	0.972038i	$0.575452\pi$
$860$	9.24970e6	0.426463
$861$	583416.	0.0268207
$862$	4.18569e7	1.91866
$863$	3.66497e7	1.67511	0.837556	−	0.546351i	$-0.183984\pi$
$863$	3.66497e7	1.67511	0.837556	+	0.546351i	$0.183984\pi$
$864$	−4.07730e6	−0.185818
$865$	1.49384e7	0.678832
$866$	1.22664e7	0.555806
$867$	3.00714e7	1.35864
$868$	−1.16035e6	−0.0522746
$869$	−5.87328e6	−0.263834
$870$	6.53625e6	0.292773
$871$	1.28451e7	0.573710
$872$	−5.41695e6	−0.241248
$873$	1.15864e7	0.514533
$874$	3.19670e7	1.41555
$875$	−187500.	−0.00827906
$876$	1.11975e7	0.493014
$877$	−8.08232e6	−0.354844	−0.177422	−	0.984135i	$-0.556776\pi$
$877$	−8.08232e6	−0.354844	−0.177422	+	0.984135i	$0.556776\pi$
$878$	−3.30789e7	−1.44815
$879$	7.07135e6	0.308696
$880$	3.58120e6	0.155891
$881$	288202.	0.0125100	0.00625500	−	0.999980i	$-0.498009\pi$
$881$	288202.	0.0125100	0.00625500	+	0.999980i	$0.498009\pi$
$882$	−9.44792e6	−0.408945
$883$	6.20688e6	0.267899	0.133950	−	0.990988i	$-0.457234\pi$
$883$	6.20688e6	0.267899	0.133950	+	0.990988i	$0.457234\pi$
$884$	−3.61296e7	−1.55501
$885$	5.74200e6	0.246436
$886$	1.74009e7	0.744708
$887$	−1.49976e7	−0.640050	−0.320025	−	0.947409i	$-0.603691\pi$
$887$	−1.49976e7	−0.640050	−0.320025	+	0.947409i	$0.603691\pi$
$888$	−6.12549e6	−0.260680
$889$	−2.38690e6	−0.101293
$890$	5.92725e6	0.250829
$891$	734832.	0.0310094
$892$	−5.52643e6	−0.232559
$893$	−522560.	−0.0219284
$894$	−2.28507e7	−0.956217
$895$	7.06700e6	0.294902
$896$	2.29698e6	0.0955844
$897$	−2.81915e7	−1.16987
$898$	1.02363e7	0.423597
$899$	2.36052e7	0.974111
$900$	860625.	0.0354167
$901$	2.74627e7	1.12702
$902$	4.23517e6	0.173322
$903$	−2.35051e6	−0.0959275
$904$	−2.12163e6	−0.0863473
$905$	7.36645e6	0.298976
$906$	4.79254e6	0.193975
$907$	−3.92150e7	−1.58283	−0.791415	−	0.611279i	$-0.790655\pi$
$907$	−3.92150e7	−1.58283	−0.791415	+	0.611279i	$0.790655\pi$
$908$	−5.29224e6	−0.213022
$909$	5.10640e6	0.204977
$910$	2.04540e6	0.0818794
$911$	−3.72997e7	−1.48905	−0.744526	−	0.667594i	$-0.767324\pi$
$911$	−3.72997e7	−1.48905	−0.744526	+	0.667594i	$0.767324\pi$
$912$	−1.63456e7	−0.650750
$913$	7.73203e6	0.306985
$914$	−1.01979e7	−0.403779
$915$	−2.65095e6	−0.104676
$916$	−1.04626e7	−0.412005
$917$	1.80806e6	0.0710052
$918$	1.11347e7	0.436087
$919$	5.78776e6	0.226059	0.113029	−	0.993592i	$-0.463945\pi$
$919$	5.78776e6	0.226059	0.113029	+	0.993592i	$0.463945\pi$
$920$	8.44200e6	0.328833
$921$	−2.65459e7	−1.03121
$922$	3.02596e7	1.17229
$923$	3.50328e7	1.35354
$924$	205632.	0.00792339
$925$	4.05125e6	0.155681
$926$	7.54933e6	0.289322
$927$	3.12952e6	0.119613
$928$	2.32110e7	0.884755
$929$	1.62700e7	0.618513	0.309256	−	0.950979i	$-0.399920\pi$
$929$	1.62700e7	0.618513	0.309256	+	0.950979i	$0.399920\pi$
$930$	8.95860e6	0.339651
$931$	−2.36615e7	−0.894679
$932$	−5.17538e6	−0.195165
$933$	−2.76981e7	−1.04171
$934$	−2.38016e7	−0.892769
$935$	−6.10960e6	−0.228551
$936$	8.28387e6	0.309061
$937$	−1.20396e7	−0.447983	−0.223992	−	0.974591i	$-0.571909\pi$
$937$	−1.20396e7	−0.447983	−0.223992	+	0.974591i	$0.571909\pi$
$938$	−1.10779e6	−0.0411103
$939$	1.45230e7	0.537517
$940$	156400.	0.00577321
$941$	3.10171e7	1.14190	0.570949	−	0.820985i	$-0.306575\pi$
$941$	3.10171e7	1.14190	0.570949	+	0.820985i	$0.306575\pi$
$942$	1.59215e7	0.584597
$943$	1.73728e7	0.636197
$944$	3.26401e7	1.19212
$945$	−218700.	−0.00796653
$946$	−1.70630e7	−0.619908
$947$	3.27325e6	0.118605	0.0593027	−	0.998240i	$-0.481112\pi$
$947$	3.27325e6	0.118605	0.0593027	+	0.998240i	$0.481112\pi$
$948$	−8.02332e6	−0.289957
$949$	−7.12832e7	−2.56934
$950$	6.21250e6	0.223335
$951$	−1.80446e7	−0.646988
$952$	−2.74932e6	−0.0983180
$953$	−2.62021e7	−0.934552	−0.467276	−	0.884112i	$-0.654765\pi$
$953$	−2.62021e7	−0.934552	−0.467276	+	0.884112i	$0.654765\pi$
$954$	7.13626e6	0.253863
$955$	−1.80818e7	−0.641554
$956$	1.32729e7	0.469701
$957$	−4.18320e6	−0.147648
$958$	1.92688e7	0.678328
$959$	−229176.	−0.00804679
$960$	−399825.	−0.0140021
$961$	3.72419e6	0.130084
$962$	−4.41943e7	−1.53967
$963$	−5.60747e6	−0.194850
$964$	1.08093e7	0.374633
$965$	−2.01065e6	−0.0695053
$966$	2.43130e6	0.0838291
$967$	−2.01481e7	−0.692897	−0.346449	−	0.938069i	$-0.612613\pi$
$967$	−2.01481e7	−0.692897	−0.346449	+	0.938069i	$0.612613\pi$
$968$	1.55932e7	0.534869
$969$	2.78860e7	0.954061
$970$	−2.50323e7	−0.854225
$971$	−1.57046e7	−0.534537	−0.267269	−	0.963622i	$-0.586121\pi$
$971$	−1.57046e7	−0.534537	−0.267269	+	0.963622i	$0.586121\pi$
$972$	1.00383e6	0.0340797
$973$	2.35752e6	0.0798313
$974$	2.49898e7	0.844045
$975$	−5.47875e6	−0.184574
$976$	−1.50692e7	−0.506367
$977$	2.84554e7	0.953736	0.476868	−	0.878975i	$-0.341772\pi$
$977$	2.84554e7	0.953736	0.476868	+	0.878975i	$0.341772\pi$
$978$	5.41271e6	0.180954
$979$	−3.79344e6	−0.126496
$980$	7.08178e6	0.235547
$981$	4.17879e6	0.138637
$982$	1.37519e7	0.455075
$983$	−7.50074e6	−0.247583	−0.123791	−	0.992308i	$-0.539505\pi$
$983$	−7.50074e6	−0.247583	−0.123791	+	0.992308i	$0.539505\pi$
$984$	−5.10489e6	−0.168073
$985$	−1.27450e7	−0.418553
$986$	−6.33871e7	−2.07639
$987$	−39744.0	−0.00129861
$988$	−2.35124e7	−0.766309
$989$	−6.99930e7	−2.27543
$990$	−1.58760e6	−0.0514817
$991$	3.22184e7	1.04212	0.521062	−	0.853519i	$-0.325536\pi$
$991$	3.22184e7	1.04212	0.521062	+	0.853519i	$0.325536\pi$
$992$	3.18130e7	1.02642
$993$	4.23695e6	0.136358
$994$	−3.02131e6	−0.0969906
$995$	1.08830e7	0.348490
$996$	1.05625e7	0.337380
$997$	1.22112e7	0.389065	0.194532	−	0.980896i	$-0.437681\pi$
$997$	1.22112e7	0.389065	0.194532	+	0.980896i	$0.437681\pi$
$998$	3.60359e7	1.14527
$999$	4.72538e6	0.149804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.6.a.b.1.1	✓	1
3.2	odd	2		45.6.a.a.1.1		1
4.3	odd	2		240.6.a.b.1.1		1
5.2	odd	4		75.6.b.a.49.2		2
5.3	odd	4		75.6.b.a.49.1		2
5.4	even	2		75.6.a.a.1.1		1
7.6	odd	2		735.6.a.b.1.1		1
8.3	odd	2		960.6.a.x.1.1		1
8.5	even	2		960.6.a.k.1.1		1
12.11	even	2		720.6.a.q.1.1		1
15.2	even	4		225.6.b.a.199.1		2
15.8	even	4		225.6.b.a.199.2		2
15.14	odd	2		225.6.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.6.a.b.1.1	✓	1	1.1	even	1	trivial
45.6.a.a.1.1		1	3.2	odd	2
75.6.a.a.1.1		1	5.4	even	2
75.6.b.a.49.1		2	5.3	odd	4
75.6.b.a.49.2		2	5.2	odd	4
225.6.a.h.1.1		1	15.14	odd	2
225.6.b.a.199.1		2	15.2	even	4
225.6.b.a.199.2		2	15.8	even	4
240.6.a.b.1.1		1	4.3	odd	2
720.6.a.q.1.1		1	12.11	even	2
735.6.a.b.1.1		1	7.6	odd	2
960.6.a.k.1.1		1	8.5	even	2
960.6.a.x.1.1		1	8.3	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:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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