LMFDB - Embedded newform 63.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,4,Mod(1,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, 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("63.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	63.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:	$3.71712033036$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	63.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.35890 q^{2} +11.0000 q^{4} +8.71780 q^{5} -7.00000 q^{7} +13.0767 q^{8} +O(q^{10})$	\(q+4.35890 q^{2} +11.0000 q^{4} +8.71780 q^{5} -7.00000 q^{7} +13.0767 q^{8} +38.0000 q^{10} -43.5890 q^{11} +82.0000 q^{13} -30.5123 q^{14} -31.0000 q^{16} -78.4602 q^{17} -20.0000 q^{19} +95.8958 q^{20} -190.000 q^{22} -130.767 q^{23} -49.0000 q^{25} +357.430 q^{26} -77.0000 q^{28} +244.098 q^{29} +156.000 q^{31} -239.739 q^{32} -342.000 q^{34} -61.0246 q^{35} +186.000 q^{37} -87.1780 q^{38} +114.000 q^{40} -165.638 q^{41} +164.000 q^{43} -479.479 q^{44} -570.000 q^{46} +470.761 q^{47} +49.0000 q^{49} -213.586 q^{50} +902.000 q^{52} -156.920 q^{53} -380.000 q^{55} -91.5369 q^{56} +1064.00 q^{58} +156.920 q^{59} +790.000 q^{61} +679.988 q^{62} -797.000 q^{64} +714.859 q^{65} -44.0000 q^{67} -863.062 q^{68} -266.000 q^{70} +444.608 q^{71} +126.000 q^{73} +810.755 q^{74} -220.000 q^{76} +305.123 q^{77} -712.000 q^{79} -270.252 q^{80} -722.000 q^{82} -1464.59 q^{83} -684.000 q^{85} +714.859 q^{86} -570.000 q^{88} -1455.87 q^{89} -574.000 q^{91} -1438.44 q^{92} +2052.00 q^{94} -174.356 q^{95} +798.000 q^{97} +213.586 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 22 q^{4} - 14 q^{7}+O(q^{10}) $ 2 * q + 22 * q^4 - 14 * q^7	$ 2 q + 22 q^{4} - 14 q^{7} + 76 q^{10} + 164 q^{13} - 62 q^{16} - 40 q^{19} - 380 q^{22} - 98 q^{25} - 154 q^{28} + 312 q^{31} - 684 q^{34} + 372 q^{37} + 228 q^{40} + 328 q^{43} - 1140 q^{46} + 98 q^{49} + 1804 q^{52} - 760 q^{55} + 2128 q^{58} + 1580 q^{61} - 1594 q^{64} - 88 q^{67} - 532 q^{70} + 252 q^{73} - 440 q^{76} - 1424 q^{79} - 1444 q^{82} - 1368 q^{85} - 1140 q^{88} - 1148 q^{91} + 4104 q^{94} + 1596 q^{97}+O(q^{100}) $ 2 * q + 22 * q^4 - 14 * q^7 + 76 * q^10 + 164 * q^13 - 62 * q^16 - 40 * q^19 - 380 * q^22 - 98 * q^25 - 154 * q^28 + 312 * q^31 - 684 * q^34 + 372 * q^37 + 228 * q^40 + 328 * q^43 - 1140 * q^46 + 98 * q^49 + 1804 * q^52 - 760 * q^55 + 2128 * q^58 + 1580 * q^61 - 1594 * q^64 - 88 * q^67 - 532 * q^70 + 252 * q^73 - 440 * q^76 - 1424 * q^79 - 1444 * q^82 - 1368 * q^85 - 1140 * q^88 - 1148 * q^91 + 4104 * q^94 + 1596 * q^97

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 $

$2$	4.35890	1.54110	0.770552	−	0.637377i	$-0.219981\pi$
$2$	4.35890	1.54110	0.770552	+	0.637377i	$0.219981\pi$
$3$	0	0
$3$	0	0
$4$	11.0000	1.37500
$5$	8.71780	0.779744	0.389872	−	0.920869i	$-0.372519\pi$
$5$	8.71780	0.779744	0.389872	+	0.920869i	$0.372519\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	13.0767	0.577914
$9$	0	0
$10$	38.0000	1.20167
$11$	−43.5890	−1.19478	−0.597390	−	0.801951i	$-0.703795\pi$
$11$	−43.5890	−1.19478	−0.597390	+	0.801951i	$0.703795\pi$
$12$	0	0
$13$	82.0000	1.74944	0.874720	−	0.484629i	$-0.161046\pi$
$13$	82.0000	1.74944	0.874720	+	0.484629i	$0.161046\pi$
$14$	−30.5123	−0.582482
$15$	0	0
$16$	−31.0000	−0.484375
$17$	−78.4602	−1.11938	−0.559688	−	0.828703i	$-0.689079\pi$
$17$	−78.4602	−1.11938	−0.559688	+	0.828703i	$0.689079\pi$
$18$	0	0
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	95.8958	1.07215
$21$	0	0
$22$	−190.000	−1.84128
$23$	−130.767	−1.18551	−0.592756	−	0.805382i	$-0.701960\pi$
$23$	−130.767	−1.18551	−0.592756	+	0.805382i	$0.701960\pi$
$24$	0	0
$25$	−49.0000	−0.392000
$26$	357.430	2.69607
$27$	0	0
$28$	−77.0000	−0.519701
$29$	244.098	1.56303	0.781516	−	0.623885i	$-0.214447\pi$
$29$	244.098	1.56303	0.781516	+	0.623885i	$0.214447\pi$
$30$	0	0
$31$	156.000	0.903820	0.451910	−	0.892063i	$-0.350743\pi$
$31$	156.000	0.903820	0.451910	+	0.892063i	$0.350743\pi$
$32$	−239.739	−1.32439
$33$	0	0
$34$	−342.000	−1.72507
$35$	−61.0246	−0.294715
$36$	0	0
$37$	186.000	0.826438	0.413219	−	0.910632i	$-0.364404\pi$
$37$	186.000	0.826438	0.413219	+	0.910632i	$0.364404\pi$
$38$	−87.1780	−0.372161
$39$	0	0
$40$	114.000	0.450625
$41$	−165.638	−0.630935	−0.315467	−	0.948936i	$-0.602161\pi$
$41$	−165.638	−0.630935	−0.315467	+	0.948936i	$0.602161\pi$
$42$	0	0
$43$	164.000	0.581622	0.290811	−	0.956780i	$-0.406075\pi$
$43$	164.000	0.581622	0.290811	+	0.956780i	$0.406075\pi$
$44$	−479.479	−1.64282
$45$	0	0
$46$	−570.000	−1.82700
$47$	470.761	1.46101	0.730506	−	0.682906i	$-0.239284\pi$
$47$	470.761	1.46101	0.730506	+	0.682906i	$0.239284\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−213.586	−0.604113
$51$	0	0
$52$	902.000	2.40548
$53$	−156.920	−0.406692	−0.203346	−	0.979107i	$-0.565182\pi$
$53$	−156.920	−0.406692	−0.203346	+	0.979107i	$0.565182\pi$
$54$	0	0
$55$	−380.000	−0.931622
$56$	−91.5369	−0.218431
$57$	0	0
$58$	1064.00	2.40879
$59$	156.920	0.346259	0.173130	−	0.984899i	$-0.444612\pi$
$59$	156.920	0.346259	0.173130	+	0.984899i	$0.444612\pi$
$60$	0	0
$61$	790.000	1.65818	0.829091	−	0.559113i	$-0.188858\pi$
$61$	790.000	1.65818	0.829091	+	0.559113i	$0.188858\pi$
$62$	679.988	1.39288
$63$	0	0
$64$	−797.000	−1.55664
$65$	714.859	1.36411
$66$	0	0
$67$	−44.0000	−0.0802307	−0.0401153	−	0.999195i	$-0.512773\pi$
$67$	−44.0000	−0.0802307	−0.0401153	+	0.999195i	$0.512773\pi$
$68$	−863.062	−1.53914
$69$	0	0
$70$	−266.000	−0.454187
$71$	444.608	0.743172	0.371586	−	0.928398i	$-0.378814\pi$
$71$	444.608	0.743172	0.371586	+	0.928398i	$0.378814\pi$
$72$	0	0
$73$	126.000	0.202016	0.101008	−	0.994886i	$-0.467793\pi$
$73$	126.000	0.202016	0.101008	+	0.994886i	$0.467793\pi$
$74$	810.755	1.27363
$75$	0	0
$76$	−220.000	−0.332049
$77$	305.123	0.451584
$78$	0	0
$79$	−712.000	−1.01400	−0.507002	−	0.861945i	$-0.669246\pi$
$79$	−712.000	−1.01400	−0.507002	+	0.861945i	$0.669246\pi$
$80$	−270.252	−0.377688
$81$	0	0
$82$	−722.000	−0.972336
$83$	−1464.59	−1.93686	−0.968432	−	0.249280i	$-0.919806\pi$
$83$	−1464.59	−1.93686	−0.968432	+	0.249280i	$0.919806\pi$
$84$	0	0
$85$	−684.000	−0.872826
$86$	714.859	0.896340
$87$	0	0
$88$	−570.000	−0.690480
$89$	−1455.87	−1.73396	−0.866978	−	0.498346i	$-0.833941\pi$
$89$	−1455.87	−1.73396	−0.866978	+	0.498346i	$0.833941\pi$
$90$	0	0
$91$	−574.000	−0.661226
$92$	−1438.44	−1.63008
$93$	0	0
$94$	2052.00	2.25157
$95$	−174.356	−0.188300
$96$	0	0
$97$	798.000	0.835305	0.417653	−	0.908607i	$-0.362853\pi$
$97$	798.000	0.835305	0.417653	+	0.908607i	$0.362853\pi$
$98$	213.586	0.220158
$99$	0	0
$100$	−539.000	−0.539000
$101$	409.737	0.403666	0.201833	−	0.979420i	$-0.435310\pi$
$101$	409.737	0.403666	0.201833	+	0.979420i	$0.435310\pi$
$102$	0	0
$103$	−916.000	−0.876273	−0.438137	−	0.898908i	$-0.644361\pi$
$103$	−916.000	−0.876273	−0.438137	+	0.898908i	$0.644361\pi$
$104$	1072.29	1.01103
$105$	0	0
$106$	−684.000	−0.626754
$107$	897.933	0.811275	0.405638	−	0.914034i	$-0.367049\pi$
$107$	897.933	0.811275	0.405638	+	0.914034i	$0.367049\pi$
$108$	0	0
$109$	−342.000	−0.300529	−0.150264	−	0.988646i	$-0.548013\pi$
$109$	−342.000	−0.300529	−0.150264	+	0.988646i	$0.548013\pi$
$110$	−1656.38	−1.43573
$111$	0	0
$112$	217.000	0.183077
$113$	−488.197	−0.406422	−0.203211	−	0.979135i	$-0.565138\pi$
$113$	−488.197	−0.406422	−0.203211	+	0.979135i	$0.565138\pi$
$114$	0	0
$115$	−1140.00	−0.924396
$116$	2685.08	2.14917
$117$	0	0
$118$	684.000	0.533621
$119$	549.221	0.423084
$120$	0	0
$121$	569.000	0.427498
$122$	3443.53	2.55543
$123$	0	0
$124$	1716.00	1.24275
$125$	−1516.90	−1.08540
$126$	0	0
$127$	456.000	0.318610	0.159305	−	0.987229i	$-0.449075\pi$
$127$	456.000	0.318610	0.159305	+	0.987229i	$0.449075\pi$
$128$	−1556.13	−1.07456
$129$	0	0
$130$	3116.00	2.10224
$131$	1499.46	1.00007	0.500033	−	0.866007i	$-0.333321\pi$
$131$	1499.46	1.00007	0.500033	+	0.866007i	$0.333321\pi$
$132$	0	0
$133$	140.000	0.0912747
$134$	−191.792	−0.123644
$135$	0	0
$136$	−1026.00	−0.646903
$137$	−889.215	−0.554531	−0.277266	−	0.960793i	$-0.589428\pi$
$137$	−889.215	−0.554531	−0.277266	+	0.960793i	$0.589428\pi$
$138$	0	0
$139$	−768.000	−0.468640	−0.234320	−	0.972160i	$-0.575286\pi$
$139$	−768.000	−0.468640	−0.234320	+	0.972160i	$0.575286\pi$
$140$	−671.270	−0.405234
$141$	0	0
$142$	1938.00	1.14531
$143$	−3574.30	−2.09019
$144$	0	0
$145$	2128.00	1.21876
$146$	549.221	0.311328
$147$	0	0
$148$	2046.00	1.13635
$149$	993.829	0.546427	0.273214	−	0.961953i	$-0.411913\pi$
$149$	993.829	0.546427	0.273214	+	0.961953i	$0.411913\pi$
$150$	0	0
$151$	−3496.00	−1.88411	−0.942054	−	0.335460i	$-0.891108\pi$
$151$	−3496.00	−1.88411	−0.942054	+	0.335460i	$0.891108\pi$
$152$	−261.534	−0.139561
$153$	0	0
$154$	1330.00	0.695938
$155$	1359.98	0.704748
$156$	0	0
$157$	−506.000	−0.257218	−0.128609	−	0.991695i	$-0.541051\pi$
$157$	−506.000	−0.257218	−0.128609	+	0.991695i	$0.541051\pi$
$158$	−3103.54	−1.56268
$159$	0	0
$160$	−2090.00	−1.03268
$161$	915.369	0.448082
$162$	0	0
$163$	−2564.00	−1.23207	−0.616037	−	0.787717i	$-0.711263\pi$
$163$	−2564.00	−1.23207	−0.616037	+	0.787717i	$0.711263\pi$
$164$	−1822.02	−0.867536
$165$	0	0
$166$	−6384.00	−2.98491
$167$	−645.117	−0.298926	−0.149463	−	0.988767i	$-0.547755\pi$
$167$	−645.117	−0.298926	−0.149463	+	0.988767i	$0.547755\pi$
$168$	0	0
$169$	4527.00	2.06054
$170$	−2981.49	−1.34512
$171$	0	0
$172$	1804.00	0.799731
$173$	3861.98	1.69723	0.848616	−	0.529009i	$-0.177436\pi$
$173$	3861.98	1.69723	0.848616	+	0.529009i	$0.177436\pi$
$174$	0	0
$175$	343.000	0.148162
$176$	1351.26	0.578721
$177$	0	0
$178$	−6346.00	−2.67221
$179$	−270.252	−0.112847	−0.0564234	−	0.998407i	$-0.517970\pi$
$179$	−270.252	−0.112847	−0.0564234	+	0.998407i	$0.517970\pi$
$180$	0	0
$181$	418.000	0.171656	0.0858279	−	0.996310i	$-0.472646\pi$
$181$	418.000	0.171656	0.0858279	+	0.996310i	$0.472646\pi$
$182$	−2502.01	−1.01902
$183$	0	0
$184$	−1710.00	−0.685124
$185$	1621.51	0.644410
$186$	0	0
$187$	3420.00	1.33741
$188$	5178.37	2.00889
$189$	0	0
$190$	−760.000	−0.290191
$191$	1525.61	0.577956	0.288978	−	0.957336i	$-0.406685\pi$
$191$	1525.61	0.577956	0.288978	+	0.957336i	$0.406685\pi$
$192$	0	0
$193$	1358.00	0.506482	0.253241	−	0.967403i	$-0.418503\pi$
$193$	1358.00	0.506482	0.253241	+	0.967403i	$0.418503\pi$
$194$	3478.40	1.28729
$195$	0	0
$196$	539.000	0.196429
$197$	3748.65	1.35574	0.677869	−	0.735183i	$-0.262904\pi$
$197$	3748.65	1.35574	0.677869	+	0.735183i	$0.262904\pi$
$198$	0	0
$199$	−1056.00	−0.376170	−0.188085	−	0.982153i	$-0.560228\pi$
$199$	−1056.00	−0.376170	−0.188085	+	0.982153i	$0.560228\pi$
$200$	−640.758	−0.226542
$201$	0	0
$202$	1786.00	0.622092
$203$	−1708.69	−0.590771
$204$	0	0
$205$	−1444.00	−0.491967
$206$	−3992.75	−1.35043
$207$	0	0
$208$	−2542.00	−0.847385
$209$	871.780	0.288528
$210$	0	0
$211$	−3620.00	−1.18110	−0.590548	−	0.807003i	$-0.701088\pi$
$211$	−3620.00	−1.18110	−0.590548	+	0.807003i	$0.701088\pi$
$212$	−1726.12	−0.559201
$213$	0	0
$214$	3914.00	1.25026
$215$	1429.72	0.453516
$216$	0	0
$217$	−1092.00	−0.341612
$218$	−1490.74	−0.463146
$219$	0	0
$220$	−4180.00	−1.28098
$221$	−6433.73	−1.95828
$222$	0	0
$223$	5368.00	1.61196	0.805982	−	0.591940i	$-0.201638\pi$
$223$	5368.00	1.61196	0.805982	+	0.591940i	$0.201638\pi$
$224$	1678.18	0.500571
$225$	0	0
$226$	−2128.00	−0.626338
$227$	1621.51	0.474112	0.237056	−	0.971496i	$-0.423818\pi$
$227$	1621.51	0.474112	0.237056	+	0.971496i	$0.423818\pi$
$228$	0	0
$229$	2186.00	0.630808	0.315404	−	0.948958i	$-0.397860\pi$
$229$	2186.00	0.630808	0.315404	+	0.948958i	$0.397860\pi$
$230$	−4969.14	−1.42459
$231$	0	0
$232$	3192.00	0.903298
$233$	−4132.24	−1.16185	−0.580927	−	0.813956i	$-0.697310\pi$
$233$	−4132.24	−1.16185	−0.580927	+	0.813956i	$0.697310\pi$
$234$	0	0
$235$	4104.00	1.13921
$236$	1726.12	0.476106
$237$	0	0
$238$	2394.00	0.652017
$239$	−4838.38	−1.30949	−0.654746	−	0.755849i	$-0.727224\pi$
$239$	−4838.38	−1.30949	−0.654746	+	0.755849i	$0.727224\pi$
$240$	0	0
$241$	1286.00	0.343728	0.171864	−	0.985121i	$-0.445021\pi$
$241$	1286.00	0.343728	0.171864	+	0.985121i	$0.445021\pi$
$242$	2480.21	0.658819
$243$	0	0
$244$	8690.00	2.28000
$245$	427.172	0.111392
$246$	0	0
$247$	−1640.00	−0.422472
$248$	2039.96	0.522330
$249$	0	0
$250$	−6612.00	−1.67272
$251$	−1795.87	−0.451610	−0.225805	−	0.974173i	$-0.572501\pi$
$251$	−1795.87	−0.451610	−0.225805	+	0.974173i	$0.572501\pi$
$252$	0	0
$253$	5700.00	1.41643
$254$	1987.66	0.491011
$255$	0	0
$256$	−407.000	−0.0993652
$257$	−1944.07	−0.471859	−0.235929	−	0.971770i	$-0.575813\pi$
$257$	−1944.07	−0.471859	−0.235929	+	0.971770i	$0.575813\pi$
$258$	0	0
$259$	−1302.00	−0.312364
$260$	7863.45	1.87566
$261$	0	0
$262$	6536.00	1.54120
$263$	5344.01	1.25295	0.626475	−	0.779442i	$-0.284497\pi$
$263$	5344.01	1.25295	0.626475	+	0.779442i	$0.284497\pi$
$264$	0	0
$265$	−1368.00	−0.317115
$266$	610.246	0.140664
$267$	0	0
$268$	−484.000	−0.110317
$269$	−4001.47	−0.906966	−0.453483	−	0.891265i	$-0.649819\pi$
$269$	−4001.47	−0.906966	−0.453483	+	0.891265i	$0.649819\pi$
$270$	0	0
$271$	2788.00	0.624941	0.312470	−	0.949928i	$-0.398844\pi$
$271$	2788.00	0.624941	0.312470	+	0.949928i	$0.398844\pi$
$272$	2432.27	0.542198
$273$	0	0
$274$	−3876.00	−0.854590
$275$	2135.86	0.468354
$276$	0	0
$277$	−4562.00	−0.989545	−0.494773	−	0.869022i	$-0.664749\pi$
$277$	−4562.00	−0.989545	−0.494773	+	0.869022i	$0.664749\pi$
$278$	−3347.63	−0.722222
$279$	0	0
$280$	−798.000	−0.170320
$281$	−1551.77	−0.329433	−0.164717	−	0.986341i	$-0.552671\pi$
$281$	−1551.77	−0.329433	−0.164717	+	0.986341i	$0.552671\pi$
$282$	0	0
$283$	−6788.00	−1.42581	−0.712906	−	0.701260i	$-0.752621\pi$
$283$	−6788.00	−1.42581	−0.712906	+	0.701260i	$0.752621\pi$
$284$	4890.68	1.02186
$285$	0	0
$286$	−15580.0	−3.22121
$287$	1159.47	0.238471
$288$	0	0
$289$	1243.00	0.253002
$290$	9275.74	1.87824
$291$	0	0
$292$	1386.00	0.277772
$293$	1142.03	0.227707	0.113854	−	0.993498i	$-0.463681\pi$
$293$	1142.03	0.227707	0.113854	+	0.993498i	$0.463681\pi$
$294$	0	0
$295$	1368.00	0.269993
$296$	2432.27	0.477610
$297$	0	0
$298$	4332.00	0.842101
$299$	−10722.9	−2.07398
$300$	0	0
$301$	−1148.00	−0.219833
$302$	−15238.7	−2.90361
$303$	0	0
$304$	620.000	0.116972
$305$	6887.06	1.29296
$306$	0	0
$307$	532.000	0.0989018	0.0494509	−	0.998777i	$-0.484253\pi$
$307$	532.000	0.0989018	0.0494509	+	0.998777i	$0.484253\pi$
$308$	3356.35	0.620928
$309$	0	0
$310$	5928.00	1.08609
$311$	6538.35	1.19214	0.596070	−	0.802932i	$-0.296728\pi$
$311$	6538.35	1.19214	0.596070	+	0.802932i	$0.296728\pi$
$312$	0	0
$313$	4994.00	0.901845	0.450923	−	0.892563i	$-0.351095\pi$
$313$	4994.00	0.901845	0.450923	+	0.892563i	$0.351095\pi$
$314$	−2205.60	−0.396399
$315$	0	0
$316$	−7832.00	−1.39425
$317$	−470.761	−0.0834088	−0.0417044	−	0.999130i	$-0.513279\pi$
$317$	−470.761	−0.0834088	−0.0417044	+	0.999130i	$0.513279\pi$
$318$	0	0
$319$	−10640.0	−1.86748
$320$	−6948.08	−1.21378
$321$	0	0
$322$	3990.00	0.690540
$323$	1569.20	0.270318
$324$	0	0
$325$	−4018.00	−0.685780
$326$	−11176.2	−1.89875
$327$	0	0
$328$	−2166.00	−0.364626
$329$	−3295.33	−0.552211
$330$	0	0
$331$	2588.00	0.429756	0.214878	−	0.976641i	$-0.431065\pi$
$331$	2588.00	0.429756	0.214878	+	0.976641i	$0.431065\pi$
$332$	−16110.5	−2.66319
$333$	0	0
$334$	−2812.00	−0.460676
$335$	−383.583	−0.0625594
$336$	0	0
$337$	238.000	0.0384709	0.0192354	−	0.999815i	$-0.493877\pi$
$337$	238.000	0.0384709	0.0192354	+	0.999815i	$0.493877\pi$
$338$	19732.7	3.17550
$339$	0	0
$340$	−7524.00	−1.20014
$341$	−6799.88	−1.07987
$342$	0	0
$343$	−343.000	−0.0539949
$344$	2144.58	0.336128
$345$	0	0
$346$	16834.0	2.61561
$347$	7052.70	1.09109	0.545546	−	0.838081i	$-0.316322\pi$
$347$	7052.70	1.09109	0.545546	+	0.838081i	$0.316322\pi$
$348$	0	0
$349$	−10850.0	−1.66415	−0.832073	−	0.554666i	$-0.812846\pi$
$349$	−10850.0	−1.66415	−0.832073	+	0.554666i	$0.812846\pi$
$350$	1495.10	0.228333
$351$	0	0
$352$	10450.0	1.58235
$353$	5291.70	0.797872	0.398936	−	0.916979i	$-0.369379\pi$
$353$	5291.70	0.797872	0.398936	+	0.916979i	$0.369379\pi$
$354$	0	0
$355$	3876.00	0.579484
$356$	−16014.6	−2.38419
$357$	0	0
$358$	−1178.00	−0.173908
$359$	−4820.94	−0.708745	−0.354373	−	0.935104i	$-0.615306\pi$
$359$	−4820.94	−0.708745	−0.354373	+	0.935104i	$0.615306\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	1822.02	0.264539
$363$	0	0
$364$	−6314.00	−0.909186
$365$	1098.44	0.157521
$366$	0	0
$367$	11712.0	1.66583	0.832917	−	0.553397i	$-0.186669\pi$
$367$	11712.0	1.66583	0.832917	+	0.553397i	$0.186669\pi$
$368$	4053.78	0.574233
$369$	0	0
$370$	7068.00	0.993102
$371$	1098.44	0.153715
$372$	0	0
$373$	−10450.0	−1.45062	−0.725309	−	0.688423i	$-0.758303\pi$
$373$	−10450.0	−1.45062	−0.725309	+	0.688423i	$0.758303\pi$
$374$	14907.4	2.06108
$375$	0	0
$376$	6156.00	0.844339
$377$	20016.1	2.73443
$378$	0	0
$379$	−756.000	−0.102462	−0.0512310	−	0.998687i	$-0.516314\pi$
$379$	−756.000	−0.102462	−0.0512310	+	0.998687i	$0.516314\pi$
$380$	−1917.92	−0.258913
$381$	0	0
$382$	6650.00	0.890690
$383$	6381.43	0.851373	0.425686	−	0.904871i	$-0.360033\pi$
$383$	6381.43	0.851373	0.425686	+	0.904871i	$0.360033\pi$
$384$	0	0
$385$	2660.00	0.352120
$386$	5919.38	0.780541
$387$	0	0
$388$	8778.00	1.14854
$389$	−418.454	−0.0545411	−0.0272705	−	0.999628i	$-0.508682\pi$
$389$	−418.454	−0.0545411	−0.0272705	+	0.999628i	$0.508682\pi$
$390$	0	0
$391$	10260.0	1.32703
$392$	640.758	0.0825591
$393$	0	0
$394$	16340.0	2.08933
$395$	−6207.07	−0.790663
$396$	0	0
$397$	−5802.00	−0.733486	−0.366743	−	0.930322i	$-0.619527\pi$
$397$	−5802.00	−0.733486	−0.366743	+	0.930322i	$0.619527\pi$
$398$	−4603.00	−0.579717
$399$	0	0
$400$	1519.00	0.189875
$401$	−4132.24	−0.514599	−0.257299	−	0.966332i	$-0.582833\pi$
$401$	−4132.24	−0.514599	−0.257299	+	0.966332i	$0.582833\pi$
$402$	0	0
$403$	12792.0	1.58118
$404$	4507.10	0.555041
$405$	0	0
$406$	−7448.00	−0.910439
$407$	−8107.55	−0.987411
$408$	0	0
$409$	−1330.00	−0.160793	−0.0803964	−	0.996763i	$-0.525619\pi$
$409$	−1330.00	−0.160793	−0.0803964	+	0.996763i	$0.525619\pi$
$410$	−6294.25	−0.758173
$411$	0	0
$412$	−10076.0	−1.20488
$413$	−1098.44	−0.130874
$414$	0	0
$415$	−12768.0	−1.51026
$416$	−19658.6	−2.31693
$417$	0	0
$418$	3800.00	0.444651
$419$	10409.1	1.21364	0.606820	−	0.794839i	$-0.292445\pi$
$419$	10409.1	1.21364	0.606820	+	0.794839i	$0.292445\pi$
$420$	0	0
$421$	−12274.0	−1.42090	−0.710449	−	0.703749i	$-0.751508\pi$
$421$	−12274.0	−1.42090	−0.710449	+	0.703749i	$0.751508\pi$
$422$	−15779.2	−1.82019
$423$	0	0
$424$	−2052.00	−0.235033
$425$	3844.55	0.438795
$426$	0	0
$427$	−5530.00	−0.626734
$428$	9877.27	1.11550
$429$	0	0
$430$	6232.00	0.698916
$431$	−4681.46	−0.523197	−0.261598	−	0.965177i	$-0.584250\pi$
$431$	−4681.46	−0.523197	−0.261598	+	0.965177i	$0.584250\pi$
$432$	0	0
$433$	5770.00	0.640389	0.320195	−	0.947352i	$-0.396252\pi$
$433$	5770.00	0.640389	0.320195	+	0.947352i	$0.396252\pi$
$434$	−4759.92	−0.526459
$435$	0	0
$436$	−3762.00	−0.413227
$437$	2615.34	0.286290
$438$	0	0
$439$	−1872.00	−0.203521	−0.101760	−	0.994809i	$-0.532448\pi$
$439$	−1872.00	−0.203521	−0.101760	+	0.994809i	$0.532448\pi$
$440$	−4969.14	−0.538397
$441$	0	0
$442$	−28044.0	−3.01791
$443$	−11115.2	−1.19210	−0.596048	−	0.802949i	$-0.703263\pi$
$443$	−11115.2	−1.19210	−0.596048	+	0.802949i	$0.703263\pi$
$444$	0	0
$445$	−12692.0	−1.35204
$446$	23398.6	2.48420
$447$	0	0
$448$	5579.00	0.588355
$449$	−7636.79	−0.802678	−0.401339	−	0.915930i	$-0.631455\pi$
$449$	−7636.79	−0.802678	−0.401339	+	0.915930i	$0.631455\pi$
$450$	0	0
$451$	7220.00	0.753828
$452$	−5370.16	−0.558830
$453$	0	0
$454$	7068.00	0.730656
$455$	−5004.02	−0.515587
$456$	0	0
$457$	15142.0	1.54992	0.774959	−	0.632011i	$-0.217770\pi$
$457$	15142.0	1.54992	0.774959	+	0.632011i	$0.217770\pi$
$458$	9528.55	0.972140
$459$	0	0
$460$	−12540.0	−1.27104
$461$	−13190.0	−1.33258	−0.666292	−	0.745691i	$-0.732119\pi$
$461$	−13190.0	−1.33258	−0.666292	+	0.745691i	$0.732119\pi$
$462$	0	0
$463$	9328.00	0.936304	0.468152	−	0.883648i	$-0.344920\pi$
$463$	9328.00	0.936304	0.468152	+	0.883648i	$0.344920\pi$
$464$	−7567.05	−0.757094
$465$	0	0
$466$	−18012.0	−1.79054
$467$	3399.94	0.336896	0.168448	−	0.985711i	$-0.446124\pi$
$467$	3399.94	0.336896	0.168448	+	0.985711i	$0.446124\pi$
$468$	0	0
$469$	308.000	0.0303243
$470$	17888.9	1.75565
$471$	0	0
$472$	2052.00	0.200108
$473$	−7148.59	−0.694911
$474$	0	0
$475$	980.000	0.0946642
$476$	6041.43	0.581741
$477$	0	0
$478$	−21090.0	−2.01806
$479$	3574.30	0.340947	0.170474	−	0.985362i	$-0.445470\pi$
$479$	3574.30	0.340947	0.170474	+	0.985362i	$0.445470\pi$
$480$	0	0
$481$	15252.0	1.44580
$482$	5605.54	0.529721
$483$	0	0
$484$	6259.00	0.587810
$485$	6956.80	0.651324
$486$	0	0
$487$	8968.00	0.834454	0.417227	−	0.908802i	$-0.363002\pi$
$487$	8968.00	0.834454	0.417227	+	0.908802i	$0.363002\pi$
$488$	10330.6	0.958287
$489$	0	0
$490$	1862.00	0.171667
$491$	5169.65	0.475159	0.237580	−	0.971368i	$-0.423646\pi$
$491$	5169.65	0.475159	0.237580	+	0.971368i	$0.423646\pi$
$492$	0	0
$493$	−19152.0	−1.74962
$494$	−7148.59	−0.651074
$495$	0	0
$496$	−4836.00	−0.437788
$497$	−3112.25	−0.280893
$498$	0	0
$499$	3940.00	0.353464	0.176732	−	0.984259i	$-0.443447\pi$
$499$	3940.00	0.353464	0.176732	+	0.984259i	$0.443447\pi$
$500$	−16685.9	−1.49243
$501$	0	0
$502$	−7828.00	−0.695978
$503$	10252.1	0.908787	0.454394	−	0.890801i	$-0.349856\pi$
$503$	10252.1	0.908787	0.454394	+	0.890801i	$0.349856\pi$
$504$	0	0
$505$	3572.00	0.314756
$506$	24845.7	2.18286
$507$	0	0
$508$	5016.00	0.438089
$509$	9772.65	0.851012	0.425506	−	0.904956i	$-0.360096\pi$
$509$	9772.65	0.851012	0.425506	+	0.904956i	$0.360096\pi$
$510$	0	0
$511$	−882.000	−0.0763550
$512$	10674.9	0.921426
$513$	0	0
$514$	−8474.00	−0.727183
$515$	−7985.50	−0.683269
$516$	0	0
$517$	−20520.0	−1.74559
$518$	−5675.29	−0.481386
$519$	0	0
$520$	9348.00	0.788340
$521$	−7401.41	−0.622383	−0.311192	−	0.950347i	$-0.600728\pi$
$521$	−7401.41	−0.622383	−0.311192	+	0.950347i	$0.600728\pi$
$522$	0	0
$523$	−2768.00	−0.231427	−0.115713	−	0.993283i	$-0.536915\pi$
$523$	−2768.00	−0.231427	−0.115713	+	0.993283i	$0.536915\pi$
$524$	16494.1	1.37509
$525$	0	0
$526$	23294.0	1.93093
$527$	−12239.8	−1.01171
$528$	0	0
$529$	4933.00	0.405441
$530$	−5962.97	−0.488708
$531$	0	0
$532$	1540.00	0.125503
$533$	−13582.3	−1.10378
$534$	0	0
$535$	7828.00	0.632587
$536$	−575.375	−0.0463664
$537$	0	0
$538$	−17442.0	−1.39773
$539$	−2135.86	−0.170683
$540$	0	0
$541$	16310.0	1.29616	0.648079	−	0.761573i	$-0.275573\pi$
$541$	16310.0	1.29616	0.648079	+	0.761573i	$0.275573\pi$
$542$	12152.6	0.963098
$543$	0	0
$544$	18810.0	1.48249
$545$	−2981.49	−0.234336
$546$	0	0
$547$	11140.0	0.870771	0.435386	−	0.900244i	$-0.356612\pi$
$547$	11140.0	0.870771	0.435386	+	0.900244i	$0.356612\pi$
$548$	−9781.37	−0.762481
$549$	0	0
$550$	9310.00	0.721781
$551$	−4881.97	−0.377457
$552$	0	0
$553$	4984.00	0.383257
$554$	−19885.3	−1.52499
$555$	0	0
$556$	−8448.00	−0.644380
$557$	22788.3	1.73352	0.866761	−	0.498723i	$-0.166197\pi$
$557$	22788.3	1.73352	0.866761	+	0.498723i	$0.166197\pi$
$558$	0	0
$559$	13448.0	1.01751
$560$	1891.76	0.142753
$561$	0	0
$562$	−6764.00	−0.507691
$563$	−11524.9	−0.862732	−0.431366	−	0.902177i	$-0.641968\pi$
$563$	−11524.9	−0.862732	−0.431366	+	0.902177i	$0.641968\pi$
$564$	0	0
$565$	−4256.00	−0.316905
$566$	−29588.2	−2.19732
$567$	0	0
$568$	5814.00	0.429489
$569$	−1691.25	−0.124606	−0.0623032	−	0.998057i	$-0.519845\pi$
$569$	−1691.25	−0.124606	−0.0623032	+	0.998057i	$0.519845\pi$
$570$	0	0
$571$	11228.0	0.822902	0.411451	−	0.911432i	$-0.365022\pi$
$571$	11228.0	0.822902	0.411451	+	0.911432i	$0.365022\pi$
$572$	−39317.3	−2.87402
$573$	0	0
$574$	5054.00	0.367509
$575$	6407.58	0.464721
$576$	0	0
$577$	2050.00	0.147907	0.0739537	−	0.997262i	$-0.476438\pi$
$577$	2050.00	0.147907	0.0739537	+	0.997262i	$0.476438\pi$
$578$	5418.11	0.389903
$579$	0	0
$580$	23408.0	1.67580
$581$	10252.1	0.732065
$582$	0	0
$583$	6840.00	0.485907
$584$	1647.66	0.116748
$585$	0	0
$586$	4978.00	0.350920
$587$	18394.6	1.29340	0.646699	−	0.762745i	$-0.276149\pi$
$587$	18394.6	1.29340	0.646699	+	0.762745i	$0.276149\pi$
$588$	0	0
$589$	−3120.00	−0.218264
$590$	5962.97	0.416088
$591$	0	0
$592$	−5766.00	−0.400306
$593$	−12632.1	−0.874769	−0.437384	−	0.899275i	$-0.644095\pi$
$593$	−12632.1	−0.874769	−0.437384	+	0.899275i	$0.644095\pi$
$594$	0	0
$595$	4788.00	0.329897
$596$	10932.1	0.751337
$597$	0	0
$598$	−46740.0	−3.19622
$599$	−9598.30	−0.654717	−0.327359	−	0.944900i	$-0.606159\pi$
$599$	−9598.30	−0.654717	−0.327359	+	0.944900i	$0.606159\pi$
$600$	0	0
$601$	−10758.0	−0.730163	−0.365082	−	0.930976i	$-0.618959\pi$
$601$	−10758.0	−0.730163	−0.365082	+	0.930976i	$0.618959\pi$
$602$	−5004.02	−0.338785
$603$	0	0
$604$	−38456.0	−2.59065
$605$	4960.43	0.333339
$606$	0	0
$607$	−21352.0	−1.42776	−0.713881	−	0.700268i	$-0.753064\pi$
$607$	−21352.0	−1.42776	−0.713881	+	0.700268i	$0.753064\pi$
$608$	4794.79	0.319826
$609$	0	0
$610$	30020.0	1.99258
$611$	38602.4	2.55595
$612$	0	0
$613$	−5714.00	−0.376487	−0.188243	−	0.982122i	$-0.560279\pi$
$613$	−5714.00	−0.376487	−0.188243	+	0.982122i	$0.560279\pi$
$614$	2318.93	0.152418
$615$	0	0
$616$	3990.00	0.260977
$617$	6747.58	0.440271	0.220135	−	0.975469i	$-0.429350\pi$
$617$	6747.58	0.440271	0.220135	+	0.975469i	$0.429350\pi$
$618$	0	0
$619$	−1880.00	−0.122074	−0.0610368	−	0.998136i	$-0.519441\pi$
$619$	−1880.00	−0.122074	−0.0610368	+	0.998136i	$0.519441\pi$
$620$	14959.7	0.969028
$621$	0	0
$622$	28500.0	1.83721
$623$	10191.1	0.655374
$624$	0	0
$625$	−7099.00	−0.454336
$626$	21768.3	1.38984
$627$	0	0
$628$	−5566.00	−0.353674
$629$	−14593.6	−0.925095
$630$	0	0
$631$	−28888.0	−1.82252	−0.911262	−	0.411826i	$-0.864891\pi$
$631$	−28888.0	−1.82252	−0.911262	+	0.411826i	$0.864891\pi$
$632$	−9310.61	−0.586006
$633$	0	0
$634$	−2052.00	−0.128542
$635$	3975.32	0.248434
$636$	0	0
$637$	4018.00	0.249920
$638$	−46378.7	−2.87798
$639$	0	0
$640$	−13566.0	−0.837880
$641$	25996.5	1.60187	0.800935	−	0.598751i	$-0.204336\pi$
$641$	25996.5	1.60187	0.800935	+	0.598751i	$0.204336\pi$
$642$	0	0
$643$	24788.0	1.52029	0.760143	−	0.649756i	$-0.225129\pi$
$643$	24788.0	1.52029	0.760143	+	0.649756i	$0.225129\pi$
$644$	10069.1	0.616112
$645$	0	0
$646$	6840.00	0.416589
$647$	28472.3	1.73008	0.865041	−	0.501702i	$-0.167292\pi$
$647$	28472.3	1.73008	0.865041	+	0.501702i	$0.167292\pi$
$648$	0	0
$649$	−6840.00	−0.413703
$650$	−17514.1	−1.05686
$651$	0	0
$652$	−28204.0	−1.69410
$653$	3243.02	0.194348	0.0971740	−	0.995267i	$-0.469020\pi$
$653$	3243.02	0.194348	0.0971740	+	0.995267i	$0.469020\pi$
$654$	0	0
$655$	13072.0	0.779794
$656$	5134.78	0.305609
$657$	0	0
$658$	−14364.0	−0.851014
$659$	1176.90	0.0695685	0.0347842	−	0.999395i	$-0.488926\pi$
$659$	1176.90	0.0695685	0.0347842	+	0.999395i	$0.488926\pi$
$660$	0	0
$661$	11590.0	0.681995	0.340998	−	0.940064i	$-0.389235\pi$
$661$	11590.0	0.681995	0.340998	+	0.940064i	$0.389235\pi$
$662$	11280.8	0.662299
$663$	0	0
$664$	−19152.0	−1.11934
$665$	1220.49	0.0711709
$666$	0	0
$667$	−31920.0	−1.85299
$668$	−7096.29	−0.411023
$669$	0	0
$670$	−1672.00	−0.0964104
$671$	−34435.3	−1.98116
$672$	0	0
$673$	23062.0	1.32091	0.660457	−	0.750864i	$-0.270363\pi$
$673$	23062.0	1.32091	0.660457	+	0.750864i	$0.270363\pi$
$674$	1037.42	0.0592876
$675$	0	0
$676$	49797.0	2.83324
$677$	−22884.2	−1.29913	−0.649566	−	0.760305i	$-0.725049\pi$
$677$	−22884.2	−1.29913	−0.649566	+	0.760305i	$0.725049\pi$
$678$	0	0
$679$	−5586.00	−0.315716
$680$	−8944.46	−0.504418
$681$	0	0
$682$	−29640.0	−1.66419
$683$	−24715.0	−1.38461	−0.692307	−	0.721603i	$-0.743406\pi$
$683$	−24715.0	−1.38461	−0.692307	+	0.721603i	$0.743406\pi$
$684$	0	0
$685$	−7752.00	−0.432392
$686$	−1495.10	−0.0832118
$687$	0	0
$688$	−5084.00	−0.281723
$689$	−12867.5	−0.711483
$690$	0	0
$691$	−10600.0	−0.583564	−0.291782	−	0.956485i	$-0.594248\pi$
$691$	−10600.0	−0.583564	−0.291782	+	0.956485i	$0.594248\pi$
$692$	42481.8	2.33369
$693$	0	0
$694$	30742.0	1.68148
$695$	−6695.27	−0.365419
$696$	0	0
$697$	12996.0	0.706253
$698$	−47294.1	−2.56462
$699$	0	0
$700$	3773.00	0.203723
$701$	12449.0	0.670746	0.335373	−	0.942085i	$-0.391138\pi$
$701$	12449.0	0.670746	0.335373	+	0.942085i	$0.391138\pi$
$702$	0	0
$703$	−3720.00	−0.199577
$704$	34740.4	1.85984
$705$	0	0
$706$	23066.0	1.22960
$707$	−2868.16	−0.152572
$708$	0	0
$709$	−13710.0	−0.726220	−0.363110	−	0.931746i	$-0.618285\pi$
$709$	−13710.0	−0.726220	−0.363110	+	0.931746i	$0.618285\pi$
$710$	16895.1	0.893044
$711$	0	0
$712$	−19038.0	−1.00208
$713$	−20399.6	−1.07149
$714$	0	0
$715$	−31160.0	−1.62982
$716$	−2972.77	−0.155164
$717$	0	0
$718$	−21014.0	−1.09225
$719$	−2510.73	−0.130228	−0.0651142	−	0.997878i	$-0.520741\pi$
$719$	−2510.73	−0.130228	−0.0651142	+	0.997878i	$0.520741\pi$
$720$	0	0
$721$	6412.00	0.331200
$722$	−28154.1	−1.45123
$723$	0	0
$724$	4598.00	0.236027
$725$	−11960.8	−0.612708
$726$	0	0
$727$	620.000	0.0316293	0.0158147	−	0.999875i	$-0.494966\pi$
$727$	620.000	0.0316293	0.0158147	+	0.999875i	$0.494966\pi$
$728$	−7506.02	−0.382132
$729$	0	0
$730$	4788.00	0.242756
$731$	−12867.5	−0.651054
$732$	0	0
$733$	−20214.0	−1.01858	−0.509291	−	0.860594i	$-0.670092\pi$
$733$	−20214.0	−1.01858	−0.509291	+	0.860594i	$0.670092\pi$
$734$	51051.4	2.56722
$735$	0	0
$736$	31350.0	1.57008
$737$	1917.92	0.0958580
$738$	0	0
$739$	−12324.0	−0.613458	−0.306729	−	0.951797i	$-0.599235\pi$
$739$	−12324.0	−0.613458	−0.306729	+	0.951797i	$0.599235\pi$
$740$	17836.6	0.886063
$741$	0	0
$742$	4788.00	0.236891
$743$	29736.4	1.46827	0.734134	−	0.679005i	$-0.237588\pi$
$743$	29736.4	1.46827	0.734134	+	0.679005i	$0.237588\pi$
$744$	0	0
$745$	8664.00	0.426073
$746$	−45550.5	−2.23555
$747$	0	0
$748$	37620.0	1.83894
$749$	−6285.53	−0.306633
$750$	0	0
$751$	19336.0	0.939522	0.469761	−	0.882794i	$-0.344340\pi$
$751$	19336.0	0.939522	0.469761	+	0.882794i	$0.344340\pi$
$752$	−14593.6	−0.707678
$753$	0	0
$754$	87248.0	4.21404
$755$	−30477.4	−1.46912
$756$	0	0
$757$	15986.0	0.767531	0.383766	−	0.923431i	$-0.374627\pi$
$757$	15986.0	0.767531	0.383766	+	0.923431i	$0.374627\pi$
$758$	−3295.33	−0.157905
$759$	0	0
$760$	−2280.00	−0.108821
$761$	37007.1	1.76282	0.881409	−	0.472354i	$-0.156596\pi$
$761$	37007.1	1.76282	0.881409	+	0.472354i	$0.156596\pi$
$762$	0	0
$763$	2394.00	0.113589
$764$	16781.8	0.794690
$765$	0	0
$766$	27816.0	1.31205
$767$	12867.5	0.605759
$768$	0	0
$769$	−36070.0	−1.69144	−0.845720	−	0.533627i	$-0.820829\pi$
$769$	−36070.0	−1.69144	−0.845720	+	0.533627i	$0.820829\pi$
$770$	11594.7	0.542653
$771$	0	0
$772$	14938.0	0.696412
$773$	−531.786	−0.0247439	−0.0123719	−	0.999923i	$-0.503938\pi$
$773$	−531.786	−0.0247439	−0.0123719	+	0.999923i	$0.503938\pi$
$774$	0	0
$775$	−7644.00	−0.354298
$776$	10435.2	0.482735
$777$	0	0
$778$	−1824.00	−0.0840534
$779$	3312.76	0.152365
$780$	0	0
$781$	−19380.0	−0.887927
$782$	44722.3	2.04510
$783$	0	0
$784$	−1519.00	−0.0691964
$785$	−4411.21	−0.200564
$786$	0	0
$787$	−1136.00	−0.0514537	−0.0257268	−	0.999669i	$-0.508190\pi$
$787$	−1136.00	−0.0514537	−0.0257268	+	0.999669i	$0.508190\pi$
$788$	41235.2	1.86414
$789$	0	0
$790$	−27056.0	−1.21849
$791$	3417.38	0.153613
$792$	0	0
$793$	64780.0	2.90089
$794$	−25290.3	−1.13038
$795$	0	0
$796$	−11616.0	−0.517234
$797$	18054.6	0.802416	0.401208	−	0.915987i	$-0.368591\pi$
$797$	18054.6	0.802416	0.401208	+	0.915987i	$0.368591\pi$
$798$	0	0
$799$	−36936.0	−1.63542
$800$	11747.2	0.519159
$801$	0	0
$802$	−18012.0	−0.793050
$803$	−5492.21	−0.241365
$804$	0	0
$805$	7980.00	0.349389
$806$	55759.0	2.43676
$807$	0	0
$808$	5358.00	0.233284
$809$	−38707.0	−1.68216	−0.841079	−	0.540912i	$-0.818079\pi$
$809$	−38707.0	−1.68216	−0.841079	+	0.540912i	$0.818079\pi$
$810$	0	0
$811$	17936.0	0.776595	0.388297	−	0.921534i	$-0.373063\pi$
$811$	17936.0	0.776595	0.388297	+	0.921534i	$0.373063\pi$
$812$	−18795.6	−0.812309
$813$	0	0
$814$	−35340.0	−1.52170
$815$	−22352.4	−0.960701
$816$	0	0
$817$	−3280.00	−0.140456
$818$	−5797.34	−0.247798
$819$	0	0
$820$	−15884.0	−0.676455
$821$	24915.5	1.05914	0.529571	−	0.848266i	$-0.322353\pi$
$821$	24915.5	1.05914	0.529571	+	0.848266i	$0.322353\pi$
$822$	0	0
$823$	23424.0	0.992113	0.496057	−	0.868290i	$-0.334781\pi$
$823$	23424.0	0.992113	0.496057	+	0.868290i	$0.334781\pi$
$824$	−11978.3	−0.506411
$825$	0	0
$826$	−4788.00	−0.201690
$827$	−26650.3	−1.12058	−0.560291	−	0.828296i	$-0.689311\pi$
$827$	−26650.3	−1.12058	−0.560291	+	0.828296i	$0.689311\pi$
$828$	0	0
$829$	−26254.0	−1.09993	−0.549963	−	0.835189i	$-0.685358\pi$
$829$	−26254.0	−1.09993	−0.549963	+	0.835189i	$0.685358\pi$
$830$	−55654.4	−2.32746
$831$	0	0
$832$	−65354.0	−2.72325
$833$	−3844.55	−0.159911
$834$	0	0
$835$	−5624.00	−0.233086
$836$	9589.58	0.396725
$837$	0	0
$838$	45372.0	1.87035
$839$	−6189.64	−0.254696	−0.127348	−	0.991858i	$-0.540647\pi$
$839$	−6189.64	−0.254696	−0.127348	+	0.991858i	$0.540647\pi$
$840$	0	0
$841$	35195.0	1.44307
$842$	−53501.1	−2.18975
$843$	0	0
$844$	−39820.0	−1.62401
$845$	39465.5	1.60669
$846$	0	0
$847$	−3983.00	−0.161579
$848$	4864.53	0.196991
$849$	0	0
$850$	16758.0	0.676229
$851$	−24322.7	−0.979753
$852$	0	0
$853$	−45322.0	−1.81922	−0.909611	−	0.415462i	$-0.863620\pi$
$853$	−45322.0	−1.81922	−0.909611	+	0.415462i	$0.863620\pi$
$854$	−24104.7	−0.965862
$855$	0	0
$856$	11742.0	0.468847
$857$	8691.64	0.346442	0.173221	−	0.984883i	$-0.444582\pi$
$857$	8691.64	0.346442	0.173221	+	0.984883i	$0.444582\pi$
$858$	0	0
$859$	−43252.0	−1.71797	−0.858987	−	0.511998i	$-0.828906\pi$
$859$	−43252.0	−1.71797	−0.858987	+	0.511998i	$0.828906\pi$
$860$	15726.9	0.623585
$861$	0	0
$862$	−20406.0	−0.806301
$863$	−29318.0	−1.15642	−0.578212	−	0.815886i	$-0.696250\pi$
$863$	−29318.0	−1.15642	−0.578212	+	0.815886i	$0.696250\pi$
$864$	0	0
$865$	33668.0	1.32341
$866$	25150.8	0.986906
$867$	0	0
$868$	−12012.0	−0.469716
$869$	31035.4	1.21151
$870$	0	0
$871$	−3608.00	−0.140359
$872$	−4472.23	−0.173680
$873$	0	0
$874$	11400.0	0.441202
$875$	10618.3	0.410244
$876$	0	0
$877$	47110.0	1.81390	0.906951	−	0.421237i	$-0.138404\pi$
$877$	47110.0	1.81390	0.906951	+	0.421237i	$0.138404\pi$
$878$	−8159.86	−0.313647
$879$	0	0
$880$	11780.0	0.451254
$881$	−42133.1	−1.61124	−0.805619	−	0.592434i	$-0.798167\pi$
$881$	−42133.1	−1.61124	−0.805619	+	0.592434i	$0.798167\pi$
$882$	0	0
$883$	−22732.0	−0.866356	−0.433178	−	0.901308i	$-0.642608\pi$
$883$	−22732.0	−0.866356	−0.433178	+	0.901308i	$0.642608\pi$
$884$	−70771.1	−2.69263
$885$	0	0
$886$	−48450.0	−1.83714
$887$	21463.2	0.812474	0.406237	−	0.913768i	$-0.366841\pi$
$887$	21463.2	0.812474	0.406237	+	0.913768i	$0.366841\pi$
$888$	0	0
$889$	−3192.00	−0.120423
$890$	−55323.1	−2.08364
$891$	0	0
$892$	59048.0	2.21645
$893$	−9415.22	−0.352820
$894$	0	0
$895$	−2356.00	−0.0879915
$896$	10892.9	0.406145
$897$	0	0
$898$	−33288.0	−1.23701
$899$	38079.3	1.41270
$900$	0	0
$901$	12312.0	0.455241
$902$	31471.3	1.16173
$903$	0	0
$904$	−6384.00	−0.234877
$905$	3644.04	0.133847
$906$	0	0
$907$	−6916.00	−0.253189	−0.126594	−	0.991955i	$-0.540405\pi$
$907$	−6916.00	−0.253189	−0.126594	+	0.991955i	$0.540405\pi$
$908$	17836.6	0.651904
$909$	0	0
$910$	−21812.0	−0.794572
$911$	38210.1	1.38963	0.694817	−	0.719186i	$-0.255485\pi$
$911$	38210.1	1.38963	0.694817	+	0.719186i	$0.255485\pi$
$912$	0	0
$913$	63840.0	2.31412
$914$	66002.4	2.38859
$915$	0	0
$916$	24046.0	0.867360
$917$	−10496.2	−0.377989
$918$	0	0
$919$	−47632.0	−1.70972	−0.854861	−	0.518857i	$-0.826358\pi$
$919$	−47632.0	−1.70972	−0.854861	+	0.518857i	$0.826358\pi$
$920$	−14907.4	−0.534221
$921$	0	0
$922$	−57494.0	−2.05365
$923$	36457.8	1.30013
$924$	0	0
$925$	−9114.00	−0.323964
$926$	40659.8	1.44294
$927$	0	0
$928$	−58520.0	−2.07006
$929$	−3304.05	−0.116687	−0.0583435	−	0.998297i	$-0.518582\pi$
$929$	−3304.05	−0.116687	−0.0583435	+	0.998297i	$0.518582\pi$
$930$	0	0
$931$	−980.000	−0.0344986
$932$	−45454.6	−1.59755
$933$	0	0
$934$	14820.0	0.519192
$935$	29814.9	1.04283
$936$	0	0
$937$	21858.0	0.762081	0.381040	−	0.924558i	$-0.375566\pi$
$937$	21858.0	0.762081	0.381040	+	0.924558i	$0.375566\pi$
$938$	1342.54	0.0467330
$939$	0	0
$940$	45144.0	1.56642
$941$	−50380.2	−1.74532	−0.872660	−	0.488328i	$-0.837607\pi$
$941$	−50380.2	−1.74532	−0.872660	+	0.488328i	$0.837607\pi$
$942$	0	0
$943$	21660.0	0.747982
$944$	−4864.53	−0.167719
$945$	0	0
$946$	−31160.0	−1.07093
$947$	31061.5	1.06585	0.532927	−	0.846161i	$-0.321092\pi$
$947$	31061.5	1.06585	0.532927	+	0.846161i	$0.321092\pi$
$948$	0	0
$949$	10332.0	0.353415
$950$	4271.72	0.145887
$951$	0	0
$952$	7182.00	0.244506
$953$	22770.9	0.773999	0.387000	−	0.922080i	$-0.373511\pi$
$953$	22770.9	0.773999	0.387000	+	0.922080i	$0.373511\pi$
$954$	0	0
$955$	13300.0	0.450657
$956$	−53222.2	−1.80055
$957$	0	0
$958$	15580.0	0.525435
$959$	6224.51	0.209593
$960$	0	0
$961$	−5455.00	−0.183109
$962$	66481.9	2.22813
$963$	0	0
$964$	14146.0	0.472627
$965$	11838.8	0.394926
$966$	0	0
$967$	−36416.0	−1.21102	−0.605512	−	0.795836i	$-0.707032\pi$
$967$	−36416.0	−1.21102	−0.605512	+	0.795836i	$0.707032\pi$
$968$	7440.64	0.247057
$969$	0	0
$970$	30324.0	1.00376
$971$	−18621.2	−0.615431	−0.307715	−	0.951478i	$-0.599564\pi$
$971$	−18621.2	−0.615431	−0.307715	+	0.951478i	$0.599564\pi$
$972$	0	0
$973$	5376.00	0.177129
$974$	39090.6	1.28598
$975$	0	0
$976$	−24490.0	−0.803182
$977$	−5806.05	−0.190125	−0.0950625	−	0.995471i	$-0.530305\pi$
$977$	−5806.05	−0.190125	−0.0950625	+	0.995471i	$0.530305\pi$
$978$	0	0
$979$	63460.0	2.07170
$980$	4698.89	0.153164
$981$	0	0
$982$	22534.0	0.732270
$983$	−3243.02	−0.105225	−0.0526126	−	0.998615i	$-0.516755\pi$
$983$	−3243.02	−0.105225	−0.0526126	+	0.998615i	$0.516755\pi$
$984$	0	0
$985$	32680.0	1.05713
$986$	−83481.6	−2.69635
$987$	0	0
$988$	−18040.0	−0.580900
$989$	−21445.8	−0.689521
$990$	0	0
$991$	49448.0	1.58503	0.792516	−	0.609851i	$-0.208771\pi$
$991$	49448.0	1.58503	0.792516	+	0.609851i	$0.208771\pi$
$992$	−37399.4	−1.19701
$993$	0	0
$994$	−13566.0	−0.432885
$995$	−9205.99	−0.293316
$996$	0	0
$997$	16294.0	0.517589	0.258794	−	0.965932i	$-0.416675\pi$
$997$	16294.0	0.517589	0.258794	+	0.965932i	$0.416675\pi$
$998$	17174.1	0.544725
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.4.a.d.1.2	yes	2
3.2	odd	2	inner	63.4.a.d.1.1	✓	2
4.3	odd	2		1008.4.a.be.1.2		2
5.4	even	2		1575.4.a.t.1.1		2
7.2	even	3		441.4.e.r.361.1		4
7.3	odd	6		441.4.e.s.226.1		4
7.4	even	3		441.4.e.r.226.1		4
7.5	odd	6		441.4.e.s.361.1		4
7.6	odd	2		441.4.a.q.1.2		2
12.11	even	2		1008.4.a.be.1.1		2
15.14	odd	2		1575.4.a.t.1.2		2
21.2	odd	6		441.4.e.r.361.2		4
21.5	even	6		441.4.e.s.361.2		4
21.11	odd	6		441.4.e.r.226.2		4
21.17	even	6		441.4.e.s.226.2		4
21.20	even	2		441.4.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.a.d.1.1	✓	2	3.2	odd	2	inner
63.4.a.d.1.2	yes	2	1.1	even	1	trivial
441.4.a.q.1.1		2	21.20	even	2
441.4.a.q.1.2		2	7.6	odd	2
441.4.e.r.226.1		4	7.4	even	3
441.4.e.r.226.2		4	21.11	odd	6
441.4.e.r.361.1		4	7.2	even	3
441.4.e.r.361.2		4	21.2	odd	6
441.4.e.s.226.1		4	7.3	odd	6
441.4.e.s.226.2		4	21.17	even	6
441.4.e.s.361.1		4	7.5	odd	6
441.4.e.s.361.2		4	21.5	even	6
1008.4.a.be.1.1		2	12.11	even	2
1008.4.a.be.1.2		2	4.3	odd	2
1575.4.a.t.1.1		2	5.4	even	2
1575.4.a.t.1.2		2	15.14	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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)

\(f(q)\)	\(=\)	\(q+4.35890 q^{2} +11.0000 q^{4} +8.71780 q^{5} -7.00000 q^{7} +13.0767 q^{8} +O(q^{10})\)	\(q+4.35890 q^{2} +11.0000 q^{4} +8.71780 q^{5} -7.00000 q^{7} +13.0767 q^{8} +38.0000 q^{10} -43.5890 q^{11} +82.0000 q^{13} -30.5123 q^{14} -31.0000 q^{16} -78.4602 q^{17} -20.0000 q^{19} +95.8958 q^{20} -190.000 q^{22} -130.767 q^{23} -49.0000 q^{25} +357.430 q^{26} -77.0000 q^{28} +244.098 q^{29} +156.000 q^{31} -239.739 q^{32} -342.000 q^{34} -61.0246 q^{35} +186.000 q^{37} -87.1780 q^{38} +114.000 q^{40} -165.638 q^{41} +164.000 q^{43} -479.479 q^{44} -570.000 q^{46} +470.761 q^{47} +49.0000 q^{49} -213.586 q^{50} +902.000 q^{52} -156.920 q^{53} -380.000 q^{55} -91.5369 q^{56} +1064.00 q^{58} +156.920 q^{59} +790.000 q^{61} +679.988 q^{62} -797.000 q^{64} +714.859 q^{65} -44.0000 q^{67} -863.062 q^{68} -266.000 q^{70} +444.608 q^{71} +126.000 q^{73} +810.755 q^{74} -220.000 q^{76} +305.123 q^{77} -712.000 q^{79} -270.252 q^{80} -722.000 q^{82} -1464.59 q^{83} -684.000 q^{85} +714.859 q^{86} -570.000 q^{88} -1455.87 q^{89} -574.000 q^{91} -1438.44 q^{92} +2052.00 q^{94} -174.356 q^{95} +798.000 q^{97} +213.586 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 22 q^{4} - 14 q^{7}+O(q^{10}) \) 2 * q + 22 * q^4 - 14 * q^7	\( 2 q + 22 q^{4} - 14 q^{7} + 76 q^{10} + 164 q^{13} - 62 q^{16} - 40 q^{19} - 380 q^{22} - 98 q^{25} - 154 q^{28} + 312 q^{31} - 684 q^{34} + 372 q^{37} + 228 q^{40} + 328 q^{43} - 1140 q^{46} + 98 q^{49} + 1804 q^{52} - 760 q^{55} + 2128 q^{58} + 1580 q^{61} - 1594 q^{64} - 88 q^{67} - 532 q^{70} + 252 q^{73} - 440 q^{76} - 1424 q^{79} - 1444 q^{82} - 1368 q^{85} - 1140 q^{88} - 1148 q^{91} + 4104 q^{94} + 1596 q^{97}+O(q^{100}) \) 2 * q + 22 * q^4 - 14 * q^7 + 76 * q^10 + 164 * q^13 - 62 * q^16 - 40 * q^19 - 380 * q^22 - 98 * q^25 - 154 * q^28 + 312 * q^31 - 684 * q^34 + 372 * q^37 + 228 * q^40 + 328 * q^43 - 1140 * q^46 + 98 * q^49 + 1804 * q^52 - 760 * q^55 + 2128 * q^58 + 1580 * q^61 - 1594 * q^64 - 88 * q^67 - 532 * q^70 + 252 * q^73 - 440 * q^76 - 1424 * q^79 - 1444 * q^82 - 1368 * q^85 - 1140 * q^88 - 1148 * q^91 + 4104 * q^94 + 1596 * q^97

Introduction

L-functions

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