LMFDB - Embedded newform 6024.2.a.q.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 57 x^{16} + 51 x^{15} + 1328 x^{14} - 1116 x^{13} - 16275 x^{12} + 13699 x^{11} + \cdots + 44032 $ x^18 - x^17 - 57x^16 + 51x^15 + 1328x^14 - 1116x^13 - 16275x^12 + 13699x^11 + 112394x^10 - 101250x^9 - 432956x^8 + 439806x^7 + 844117x^6 - 1010785x^5 - 593552x^4 + 965980x^3 - 88040x^2 - 184992x + 44032
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$3.41379$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.41379 q^{5} -0.715150 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.41379 q^{5} -0.715150 q^{7} +1.00000 q^{9} +1.11046 q^{11} +4.44880 q^{13} +3.41379 q^{15} -4.71176 q^{17} +1.96808 q^{19} -0.715150 q^{21} +8.82854 q^{23} +6.65397 q^{25} +1.00000 q^{27} +2.02705 q^{29} +2.28911 q^{31} +1.11046 q^{33} -2.44137 q^{35} +1.64487 q^{37} +4.44880 q^{39} -6.66762 q^{41} +2.36115 q^{43} +3.41379 q^{45} -1.81730 q^{47} -6.48856 q^{49} -4.71176 q^{51} +1.29546 q^{53} +3.79087 q^{55} +1.96808 q^{57} -11.9503 q^{59} +2.13919 q^{61} -0.715150 q^{63} +15.1873 q^{65} +3.53746 q^{67} +8.82854 q^{69} -2.09839 q^{71} -13.5964 q^{73} +6.65397 q^{75} -0.794143 q^{77} +12.0446 q^{79} +1.00000 q^{81} +3.91809 q^{83} -16.0850 q^{85} +2.02705 q^{87} +7.46061 q^{89} -3.18156 q^{91} +2.28911 q^{93} +6.71861 q^{95} -13.4967 q^{97} +1.11046 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.41379	1.52669	0.763347	−	0.645989i	$-0.223555\pi$
$5$	3.41379	1.52669	0.763347	+	0.645989i	$0.223555\pi$
$6$	0	0
$7$	−0.715150	−0.270301	−0.135151	−	0.990825i	$-0.543152\pi$
$7$	−0.715150	−0.270301	−0.135151	+	0.990825i	$0.543152\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.11046	0.334815	0.167408	−	0.985888i	$-0.446460\pi$
$11$	1.11046	0.334815	0.167408	+	0.985888i	$0.446460\pi$
$12$	0	0
$13$	4.44880	1.23387	0.616937	−	0.787013i	$-0.288373\pi$
$13$	4.44880	1.23387	0.616937	+	0.787013i	$0.288373\pi$
$14$	0	0
$15$	3.41379	0.881437
$16$	0	0
$17$	−4.71176	−1.14277	−0.571385	−	0.820682i	$-0.693594\pi$
$17$	−4.71176	−1.14277	−0.571385	+	0.820682i	$0.693594\pi$
$18$	0	0
$19$	1.96808	0.451509	0.225754	−	0.974184i	$-0.427515\pi$
$19$	1.96808	0.451509	0.225754	+	0.974184i	$0.427515\pi$
$20$	0	0
$21$	−0.715150	−0.156058
$22$	0	0
$23$	8.82854	1.84088	0.920439	−	0.390886i	$-0.127831\pi$
$23$	8.82854	1.84088	0.920439	+	0.390886i	$0.127831\pi$
$24$	0	0
$25$	6.65397	1.33079
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.02705	0.376414	0.188207	−	0.982129i	$-0.439732\pi$
$29$	2.02705	0.376414	0.188207	+	0.982129i	$0.439732\pi$
$30$	0	0
$31$	2.28911	0.411136	0.205568	−	0.978643i	$-0.434096\pi$
$31$	2.28911	0.411136	0.205568	+	0.978643i	$0.434096\pi$
$32$	0	0
$33$	1.11046	0.193306
$34$	0	0
$35$	−2.44137	−0.412667
$36$	0	0
$37$	1.64487	0.270415	0.135208	−	0.990817i	$-0.456830\pi$
$37$	1.64487	0.270415	0.135208	+	0.990817i	$0.456830\pi$
$38$	0	0
$39$	4.44880	0.712378
$40$	0	0
$41$	−6.66762	−1.04131	−0.520654	−	0.853768i	$-0.674312\pi$
$41$	−6.66762	−1.04131	−0.520654	+	0.853768i	$0.674312\pi$
$42$	0	0
$43$	2.36115	0.360071	0.180036	−	0.983660i	$-0.442379\pi$
$43$	2.36115	0.360071	0.180036	+	0.983660i	$0.442379\pi$
$44$	0	0
$45$	3.41379	0.508898
$46$	0	0
$47$	−1.81730	−0.265080	−0.132540	−	0.991178i	$-0.542313\pi$
$47$	−1.81730	−0.265080	−0.132540	+	0.991178i	$0.542313\pi$
$48$	0	0
$49$	−6.48856	−0.926937
$50$	0	0
$51$	−4.71176	−0.659778
$52$	0	0
$53$	1.29546	0.177945	0.0889727	−	0.996034i	$-0.471642\pi$
$53$	1.29546	0.177945	0.0889727	+	0.996034i	$0.471642\pi$
$54$	0	0
$55$	3.79087	0.511161
$56$	0	0
$57$	1.96808	0.260679
$58$	0	0
$59$	−11.9503	−1.55579	−0.777896	−	0.628393i	$-0.783713\pi$
$59$	−11.9503	−1.55579	−0.777896	+	0.628393i	$0.783713\pi$
$60$	0	0
$61$	2.13919	0.273896	0.136948	−	0.990578i	$-0.456271\pi$
$61$	2.13919	0.273896	0.136948	+	0.990578i	$0.456271\pi$
$62$	0	0
$63$	−0.715150	−0.0901004
$64$	0	0
$65$	15.1873	1.88375
$66$	0	0
$67$	3.53746	0.432169	0.216085	−	0.976375i	$-0.430671\pi$
$67$	3.53746	0.432169	0.216085	+	0.976375i	$0.430671\pi$
$68$	0	0
$69$	8.82854	1.06283
$70$	0	0
$71$	−2.09839	−0.249033	−0.124516	−	0.992218i	$-0.539738\pi$
$71$	−2.09839	−0.249033	−0.124516	+	0.992218i	$0.539738\pi$
$72$	0	0
$73$	−13.5964	−1.59134	−0.795670	−	0.605731i	$-0.792881\pi$
$73$	−13.5964	−1.59134	−0.795670	+	0.605731i	$0.792881\pi$
$74$	0	0
$75$	6.65397	0.768334
$76$	0	0
$77$	−0.794143	−0.0905010
$78$	0	0
$79$	12.0446	1.35512	0.677560	−	0.735467i	$-0.263037\pi$
$79$	12.0446	1.35512	0.677560	+	0.735467i	$0.263037\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.91809	0.430066	0.215033	−	0.976607i	$-0.431014\pi$
$83$	3.91809	0.430066	0.215033	+	0.976607i	$0.431014\pi$
$84$	0	0
$85$	−16.0850	−1.74466
$86$	0	0
$87$	2.02705	0.217323
$88$	0	0
$89$	7.46061	0.790823	0.395411	−	0.918504i	$-0.370602\pi$
$89$	7.46061	0.790823	0.395411	+	0.918504i	$0.370602\pi$
$90$	0	0
$91$	−3.18156	−0.333518
$92$	0	0
$93$	2.28911	0.237369
$94$	0	0
$95$	6.71861	0.689315
$96$	0	0
$97$	−13.4967	−1.37038	−0.685191	−	0.728363i	$-0.740281\pi$
$97$	−13.4967	−1.37038	−0.685191	+	0.728363i	$0.740281\pi$
$98$	0	0
$99$	1.11046	0.111605
$100$	0	0
$101$	−1.54001	−0.153237	−0.0766186	−	0.997060i	$-0.524412\pi$
$101$	−1.54001	−0.153237	−0.0766186	+	0.997060i	$0.524412\pi$
$102$	0	0
$103$	10.6285	1.04726	0.523628	−	0.851947i	$-0.324578\pi$
$103$	10.6285	1.04726	0.523628	+	0.851947i	$0.324578\pi$
$104$	0	0
$105$	−2.44137	−0.238254
$106$	0	0
$107$	13.6804	1.32254	0.661268	−	0.750150i	$-0.270019\pi$
$107$	13.6804	1.32254	0.661268	+	0.750150i	$0.270019\pi$
$108$	0	0
$109$	−8.95620	−0.857849	−0.428924	−	0.903340i	$-0.641107\pi$
$109$	−8.95620	−0.857849	−0.428924	+	0.903340i	$0.641107\pi$
$110$	0	0
$111$	1.64487	0.156124
$112$	0	0
$113$	−5.67879	−0.534215	−0.267108	−	0.963667i	$-0.586068\pi$
$113$	−5.67879	−0.534215	−0.267108	+	0.963667i	$0.586068\pi$
$114$	0	0
$115$	30.1388	2.81046
$116$	0	0
$117$	4.44880	0.411291
$118$	0	0
$119$	3.36961	0.308892
$120$	0	0
$121$	−9.76689	−0.887899
$122$	0	0
$123$	−6.66762	−0.601199
$124$	0	0
$125$	5.64631	0.505022
$126$	0	0
$127$	11.3888	1.01059	0.505295	−	0.862947i	$-0.331384\pi$
$127$	11.3888	1.01059	0.505295	+	0.862947i	$0.331384\pi$
$128$	0	0
$129$	2.36115	0.207887
$130$	0	0
$131$	11.7623	1.02767	0.513837	−	0.857888i	$-0.328224\pi$
$131$	11.7623	1.02767	0.513837	+	0.857888i	$0.328224\pi$
$132$	0	0
$133$	−1.40747	−0.122043
$134$	0	0
$135$	3.41379	0.293812
$136$	0	0
$137$	11.8277	1.01051	0.505253	−	0.862971i	$-0.331399\pi$
$137$	11.8277	1.01051	0.505253	+	0.862971i	$0.331399\pi$
$138$	0	0
$139$	−7.85003	−0.665831	−0.332915	−	0.942957i	$-0.608032\pi$
$139$	−7.85003	−0.665831	−0.332915	+	0.942957i	$0.608032\pi$
$140$	0	0
$141$	−1.81730	−0.153044
$142$	0	0
$143$	4.94020	0.413120
$144$	0	0
$145$	6.91992	0.574668
$146$	0	0
$147$	−6.48856	−0.535167
$148$	0	0
$149$	6.21605	0.509238	0.254619	−	0.967041i	$-0.418050\pi$
$149$	6.21605	0.509238	0.254619	+	0.967041i	$0.418050\pi$
$150$	0	0
$151$	9.79218	0.796876	0.398438	−	0.917195i	$-0.369552\pi$
$151$	9.79218	0.796876	0.398438	+	0.917195i	$0.369552\pi$
$152$	0	0
$153$	−4.71176	−0.380923
$154$	0	0
$155$	7.81453	0.627678
$156$	0	0
$157$	15.2324	1.21568	0.607838	−	0.794061i	$-0.292037\pi$
$157$	15.2324	1.21568	0.607838	+	0.794061i	$0.292037\pi$
$158$	0	0
$159$	1.29546	0.102737
$160$	0	0
$161$	−6.31373	−0.497592
$162$	0	0
$163$	−22.5350	−1.76508	−0.882538	−	0.470241i	$-0.844167\pi$
$163$	−22.5350	−1.76508	−0.882538	+	0.470241i	$0.844167\pi$
$164$	0	0
$165$	3.79087	0.295119
$166$	0	0
$167$	−14.2348	−1.10152	−0.550762	−	0.834662i	$-0.685663\pi$
$167$	−14.2348	−1.10152	−0.550762	+	0.834662i	$0.685663\pi$
$168$	0	0
$169$	6.79179	0.522445
$170$	0	0
$171$	1.96808	0.150503
$172$	0	0
$173$	−5.14693	−0.391314	−0.195657	−	0.980672i	$-0.562684\pi$
$173$	−5.14693	−0.391314	−0.195657	+	0.980672i	$0.562684\pi$
$174$	0	0
$175$	−4.75859	−0.359715
$176$	0	0
$177$	−11.9503	−0.898237
$178$	0	0
$179$	−6.23048	−0.465688	−0.232844	−	0.972514i	$-0.574803\pi$
$179$	−6.23048	−0.465688	−0.232844	+	0.972514i	$0.574803\pi$
$180$	0	0
$181$	−15.3965	−1.14441	−0.572205	−	0.820111i	$-0.693912\pi$
$181$	−15.3965	−1.14441	−0.572205	+	0.820111i	$0.693912\pi$
$182$	0	0
$183$	2.13919	0.158134
$184$	0	0
$185$	5.61525	0.412841
$186$	0	0
$187$	−5.23221	−0.382617
$188$	0	0
$189$	−0.715150	−0.0520195
$190$	0	0
$191$	−10.9278	−0.790704	−0.395352	−	0.918530i	$-0.629377\pi$
$191$	−10.9278	−0.790704	−0.395352	+	0.918530i	$0.629377\pi$
$192$	0	0
$193$	22.5568	1.62367	0.811837	−	0.583884i	$-0.198468\pi$
$193$	22.5568	1.62367	0.811837	+	0.583884i	$0.198468\pi$
$194$	0	0
$195$	15.1873	1.08758
$196$	0	0
$197$	−4.17034	−0.297124	−0.148562	−	0.988903i	$-0.547464\pi$
$197$	−4.17034	−0.297124	−0.148562	+	0.988903i	$0.547464\pi$
$198$	0	0
$199$	9.92437	0.703520	0.351760	−	0.936090i	$-0.385583\pi$
$199$	9.92437	0.703520	0.351760	+	0.936090i	$0.385583\pi$
$200$	0	0
$201$	3.53746	0.249513
$202$	0	0
$203$	−1.44964	−0.101745
$204$	0	0
$205$	−22.7619	−1.58976
$206$	0	0
$207$	8.82854	0.613626
$208$	0	0
$209$	2.18547	0.151172
$210$	0	0
$211$	−20.1989	−1.39055	−0.695276	−	0.718743i	$-0.744718\pi$
$211$	−20.1989	−1.39055	−0.695276	+	0.718743i	$0.744718\pi$
$212$	0	0
$213$	−2.09839	−0.143779
$214$	0	0
$215$	8.06046	0.549719
$216$	0	0
$217$	−1.63705	−0.111130
$218$	0	0
$219$	−13.5964	−0.918760
$220$	0	0
$221$	−20.9617	−1.41003
$222$	0	0
$223$	4.15398	0.278171	0.139085	−	0.990280i	$-0.455584\pi$
$223$	4.15398	0.278171	0.139085	+	0.990280i	$0.455584\pi$
$224$	0	0
$225$	6.65397	0.443598
$226$	0	0
$227$	17.3280	1.15010	0.575051	−	0.818118i	$-0.304982\pi$
$227$	17.3280	1.15010	0.575051	+	0.818118i	$0.304982\pi$
$228$	0	0
$229$	23.3004	1.53973	0.769866	−	0.638205i	$-0.220323\pi$
$229$	23.3004	1.53973	0.769866	+	0.638205i	$0.220323\pi$
$230$	0	0
$231$	−0.794143	−0.0522508
$232$	0	0
$233$	−7.09727	−0.464958	−0.232479	−	0.972601i	$-0.574684\pi$
$233$	−7.09727	−0.464958	−0.232479	+	0.972601i	$0.574684\pi$
$234$	0	0
$235$	−6.20387	−0.404696
$236$	0	0
$237$	12.0446	0.782379
$238$	0	0
$239$	13.1224	0.848814	0.424407	−	0.905471i	$-0.360483\pi$
$239$	13.1224	0.848814	0.424407	+	0.905471i	$0.360483\pi$
$240$	0	0
$241$	−7.33870	−0.472727	−0.236364	−	0.971665i	$-0.575956\pi$
$241$	−7.33870	−0.472727	−0.236364	+	0.971665i	$0.575956\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−22.1506	−1.41515
$246$	0	0
$247$	8.75559	0.557105
$248$	0	0
$249$	3.91809	0.248299
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	9.80372	0.616354
$254$	0	0
$255$	−16.0850	−1.00728
$256$	0	0
$257$	11.5343	0.719493	0.359746	−	0.933050i	$-0.382863\pi$
$257$	11.5343	0.719493	0.359746	+	0.933050i	$0.382863\pi$
$258$	0	0
$259$	−1.17633	−0.0730936
$260$	0	0
$261$	2.02705	0.125471
$262$	0	0
$263$	4.28866	0.264450	0.132225	−	0.991220i	$-0.457788\pi$
$263$	4.28866	0.264450	0.132225	+	0.991220i	$0.457788\pi$
$264$	0	0
$265$	4.42244	0.271668
$266$	0	0
$267$	7.46061	0.456582
$268$	0	0
$269$	4.87146	0.297018	0.148509	−	0.988911i	$-0.452553\pi$
$269$	4.87146	0.297018	0.148509	+	0.988911i	$0.452553\pi$
$270$	0	0
$271$	9.34646	0.567757	0.283878	−	0.958860i	$-0.408379\pi$
$271$	9.34646	0.567757	0.283878	+	0.958860i	$0.408379\pi$
$272$	0	0
$273$	−3.18156	−0.192557
$274$	0	0
$275$	7.38895	0.445570
$276$	0	0
$277$	−25.1150	−1.50901	−0.754506	−	0.656293i	$-0.772124\pi$
$277$	−25.1150	−1.50901	−0.754506	+	0.656293i	$0.772124\pi$
$278$	0	0
$279$	2.28911	0.137045
$280$	0	0
$281$	3.63659	0.216940	0.108470	−	0.994100i	$-0.465405\pi$
$281$	3.63659	0.216940	0.108470	+	0.994100i	$0.465405\pi$
$282$	0	0
$283$	21.5230	1.27941	0.639704	−	0.768621i	$-0.279057\pi$
$283$	21.5230	1.27941	0.639704	+	0.768621i	$0.279057\pi$
$284$	0	0
$285$	6.71861	0.397976
$286$	0	0
$287$	4.76835	0.281467
$288$	0	0
$289$	5.20067	0.305922
$290$	0	0
$291$	−13.4967	−0.791191
$292$	0	0
$293$	−31.4060	−1.83476	−0.917378	−	0.398017i	$-0.869699\pi$
$293$	−31.4060	−1.83476	−0.917378	+	0.398017i	$0.869699\pi$
$294$	0	0
$295$	−40.7957	−2.37522
$296$	0	0
$297$	1.11046	0.0644353
$298$	0	0
$299$	39.2764	2.27141
$300$	0	0
$301$	−1.68857	−0.0973277
$302$	0	0
$303$	−1.54001	−0.0884715
$304$	0	0
$305$	7.30276	0.418155
$306$	0	0
$307$	8.03693	0.458692	0.229346	−	0.973345i	$-0.426341\pi$
$307$	8.03693	0.458692	0.229346	+	0.973345i	$0.426341\pi$
$308$	0	0
$309$	10.6285	0.604633
$310$	0	0
$311$	−7.62605	−0.432434	−0.216217	−	0.976345i	$-0.569372\pi$
$311$	−7.62605	−0.432434	−0.216217	+	0.976345i	$0.569372\pi$
$312$	0	0
$313$	12.5672	0.710340	0.355170	−	0.934802i	$-0.384423\pi$
$313$	12.5672	0.710340	0.355170	+	0.934802i	$0.384423\pi$
$314$	0	0
$315$	−2.44137	−0.137556
$316$	0	0
$317$	9.16710	0.514876	0.257438	−	0.966295i	$-0.417122\pi$
$317$	9.16710	0.514876	0.257438	+	0.966295i	$0.417122\pi$
$318$	0	0
$319$	2.25095	0.126029
$320$	0	0
$321$	13.6804	0.763566
$322$	0	0
$323$	−9.27312	−0.515970
$324$	0	0
$325$	29.6022	1.64203
$326$	0	0
$327$	−8.95620	−0.495279
$328$	0	0
$329$	1.29964	0.0716514
$330$	0	0
$331$	−33.9752	−1.86745	−0.933724	−	0.357994i	$-0.883461\pi$
$331$	−33.9752	−1.86745	−0.933724	+	0.357994i	$0.883461\pi$
$332$	0	0
$333$	1.64487	0.0901384
$334$	0	0
$335$	12.0761	0.659790
$336$	0	0
$337$	−21.6022	−1.17675	−0.588373	−	0.808590i	$-0.700231\pi$
$337$	−21.6022	−1.17675	−0.588373	+	0.808590i	$0.700231\pi$
$338$	0	0
$339$	−5.67879	−0.308429
$340$	0	0
$341$	2.54195	0.137655
$342$	0	0
$343$	9.64634	0.520854
$344$	0	0
$345$	30.1388	1.62262
$346$	0	0
$347$	−32.9301	−1.76778	−0.883889	−	0.467696i	$-0.845084\pi$
$347$	−32.9301	−1.76778	−0.883889	+	0.467696i	$0.845084\pi$
$348$	0	0
$349$	7.28273	0.389836	0.194918	−	0.980820i	$-0.437556\pi$
$349$	7.28273	0.389836	0.194918	+	0.980820i	$0.437556\pi$
$350$	0	0
$351$	4.44880	0.237459
$352$	0	0
$353$	3.81080	0.202829	0.101414	−	0.994844i	$-0.467663\pi$
$353$	3.81080	0.202829	0.101414	+	0.994844i	$0.467663\pi$
$354$	0	0
$355$	−7.16346	−0.380197
$356$	0	0
$357$	3.36961	0.178339
$358$	0	0
$359$	27.9976	1.47766	0.738828	−	0.673894i	$-0.235379\pi$
$359$	27.9976	1.47766	0.738828	+	0.673894i	$0.235379\pi$
$360$	0	0
$361$	−15.1267	−0.796140
$362$	0	0
$363$	−9.76689	−0.512629
$364$	0	0
$365$	−46.4153	−2.42949
$366$	0	0
$367$	29.0632	1.51709	0.758543	−	0.651623i	$-0.225911\pi$
$367$	29.0632	1.51709	0.758543	+	0.651623i	$0.225911\pi$
$368$	0	0
$369$	−6.66762	−0.347103
$370$	0	0
$371$	−0.926449	−0.0480988
$372$	0	0
$373$	−21.3447	−1.10518	−0.552592	−	0.833452i	$-0.686361\pi$
$373$	−21.3447	−1.10518	−0.552592	+	0.833452i	$0.686361\pi$
$374$	0	0
$375$	5.64631	0.291574
$376$	0	0
$377$	9.01793	0.464447
$378$	0	0
$379$	8.11677	0.416931	0.208465	−	0.978030i	$-0.433153\pi$
$379$	8.11677	0.416931	0.208465	+	0.978030i	$0.433153\pi$
$380$	0	0
$381$	11.3888	0.583465
$382$	0	0
$383$	−31.6491	−1.61719	−0.808597	−	0.588363i	$-0.799773\pi$
$383$	−31.6491	−1.61719	−0.808597	+	0.588363i	$0.799773\pi$
$384$	0	0
$385$	−2.71104	−0.138167
$386$	0	0
$387$	2.36115	0.120024
$388$	0	0
$389$	−20.4470	−1.03670	−0.518351	−	0.855168i	$-0.673454\pi$
$389$	−20.4470	−1.03670	−0.518351	+	0.855168i	$0.673454\pi$
$390$	0	0
$391$	−41.5980	−2.10370
$392$	0	0
$393$	11.7623	0.593328
$394$	0	0
$395$	41.1177	2.06885
$396$	0	0
$397$	36.5908	1.83644	0.918220	−	0.396072i	$-0.129627\pi$
$397$	36.5908	1.83644	0.918220	+	0.396072i	$0.129627\pi$
$398$	0	0
$399$	−1.40747	−0.0704617
$400$	0	0
$401$	16.8009	0.838999	0.419500	−	0.907756i	$-0.362206\pi$
$401$	16.8009	0.838999	0.419500	+	0.907756i	$0.362206\pi$
$402$	0	0
$403$	10.1838	0.507290
$404$	0	0
$405$	3.41379	0.169633
$406$	0	0
$407$	1.82656	0.0905392
$408$	0	0
$409$	−6.68959	−0.330779	−0.165390	−	0.986228i	$-0.552888\pi$
$409$	−6.68959	−0.330779	−0.165390	+	0.986228i	$0.552888\pi$
$410$	0	0
$411$	11.8277	0.583416
$412$	0	0
$413$	8.54623	0.420533
$414$	0	0
$415$	13.3755	0.656580
$416$	0	0
$417$	−7.85003	−0.384418
$418$	0	0
$419$	−19.9385	−0.974057	−0.487029	−	0.873386i	$-0.661919\pi$
$419$	−19.9385	−0.974057	−0.487029	+	0.873386i	$0.661919\pi$
$420$	0	0
$421$	17.0389	0.830425	0.415213	−	0.909724i	$-0.363707\pi$
$421$	17.0389	0.830425	0.415213	+	0.909724i	$0.363707\pi$
$422$	0	0
$423$	−1.81730	−0.0883599
$424$	0	0
$425$	−31.3519	−1.52079
$426$	0	0
$427$	−1.52984	−0.0740343
$428$	0	0
$429$	4.94020	0.238515
$430$	0	0
$431$	−11.3591	−0.547148	−0.273574	−	0.961851i	$-0.588206\pi$
$431$	−11.3591	−0.547148	−0.273574	+	0.961851i	$0.588206\pi$
$432$	0	0
$433$	20.4443	0.982490	0.491245	−	0.871021i	$-0.336542\pi$
$433$	20.4443	0.982490	0.491245	+	0.871021i	$0.336542\pi$
$434$	0	0
$435$	6.91992	0.331785
$436$	0	0
$437$	17.3753	0.831172
$438$	0	0
$439$	−25.0672	−1.19639	−0.598197	−	0.801349i	$-0.704116\pi$
$439$	−25.0672	−1.19639	−0.598197	+	0.801349i	$0.704116\pi$
$440$	0	0
$441$	−6.48856	−0.308979
$442$	0	0
$443$	−18.2339	−0.866317	−0.433158	−	0.901318i	$-0.642601\pi$
$443$	−18.2339	−0.866317	−0.433158	+	0.901318i	$0.642601\pi$
$444$	0	0
$445$	25.4690	1.20734
$446$	0	0
$447$	6.21605	0.294009
$448$	0	0
$449$	35.1458	1.65863	0.829316	−	0.558780i	$-0.188730\pi$
$449$	35.1458	1.65863	0.829316	+	0.558780i	$0.188730\pi$
$450$	0	0
$451$	−7.40411	−0.348646
$452$	0	0
$453$	9.79218	0.460077
$454$	0	0
$455$	−10.8612	−0.509179
$456$	0	0
$457$	−25.1628	−1.17706	−0.588532	−	0.808474i	$-0.700294\pi$
$457$	−25.1628	−1.17706	−0.588532	+	0.808474i	$0.700294\pi$
$458$	0	0
$459$	−4.71176	−0.219926
$460$	0	0
$461$	−25.1628	−1.17195	−0.585975	−	0.810329i	$-0.699288\pi$
$461$	−25.1628	−1.17195	−0.585975	+	0.810329i	$0.699288\pi$
$462$	0	0
$463$	−4.36455	−0.202838	−0.101419	−	0.994844i	$-0.532338\pi$
$463$	−4.36455	−0.202838	−0.101419	+	0.994844i	$0.532338\pi$
$464$	0	0
$465$	7.81453	0.362390
$466$	0	0
$467$	−13.0952	−0.605973	−0.302986	−	0.952995i	$-0.597984\pi$
$467$	−13.0952	−0.605973	−0.302986	+	0.952995i	$0.597984\pi$
$468$	0	0
$469$	−2.52981	−0.116816
$470$	0	0
$471$	15.2324	0.701870
$472$	0	0
$473$	2.62195	0.120557
$474$	0	0
$475$	13.0955	0.600865
$476$	0	0
$477$	1.29546	0.0593151
$478$	0	0
$479$	−2.39927	−0.109625	−0.0548126	−	0.998497i	$-0.517456\pi$
$479$	−2.39927	−0.109625	−0.0548126	+	0.998497i	$0.517456\pi$
$480$	0	0
$481$	7.31770	0.333658
$482$	0	0
$483$	−6.31373	−0.287285
$484$	0	0
$485$	−46.0749	−2.09215
$486$	0	0
$487$	21.7893	0.987369	0.493685	−	0.869641i	$-0.335650\pi$
$487$	21.7893	0.987369	0.493685	+	0.869641i	$0.335650\pi$
$488$	0	0
$489$	−22.5350	−1.01907
$490$	0	0
$491$	−40.6045	−1.83245	−0.916227	−	0.400659i	$-0.868781\pi$
$491$	−40.6045	−1.83245	−0.916227	+	0.400659i	$0.868781\pi$
$492$	0	0
$493$	−9.55097	−0.430154
$494$	0	0
$495$	3.79087	0.170387
$496$	0	0
$497$	1.50066	0.0673139
$498$	0	0
$499$	3.94719	0.176701	0.0883503	−	0.996089i	$-0.471840\pi$
$499$	3.94719	0.176701	0.0883503	+	0.996089i	$0.471840\pi$
$500$	0	0
$501$	−14.2348	−0.635965
$502$	0	0
$503$	2.61190	0.116459	0.0582294	−	0.998303i	$-0.481455\pi$
$503$	2.61190	0.116459	0.0582294	+	0.998303i	$0.481455\pi$
$504$	0	0
$505$	−5.25729	−0.233946
$506$	0	0
$507$	6.79179	0.301634
$508$	0	0
$509$	−22.7240	−1.00722	−0.503611	−	0.863930i	$-0.667996\pi$
$509$	−22.7240	−1.00722	−0.503611	+	0.863930i	$0.667996\pi$
$510$	0	0
$511$	9.72347	0.430141
$512$	0	0
$513$	1.96808	0.0868929
$514$	0	0
$515$	36.2834	1.59884
$516$	0	0
$517$	−2.01803	−0.0887528
$518$	0	0
$519$	−5.14693	−0.225925
$520$	0	0
$521$	12.5680	0.550616	0.275308	−	0.961356i	$-0.411220\pi$
$521$	12.5680	0.550616	0.275308	+	0.961356i	$0.411220\pi$
$522$	0	0
$523$	−18.2968	−0.800065	−0.400033	−	0.916501i	$-0.631001\pi$
$523$	−18.2968	−0.800065	−0.400033	+	0.916501i	$0.631001\pi$
$524$	0	0
$525$	−4.75859	−0.207682
$526$	0	0
$527$	−10.7857	−0.469833
$528$	0	0
$529$	54.9431	2.38883
$530$	0	0
$531$	−11.9503	−0.518597
$532$	0	0
$533$	−29.6629	−1.28484
$534$	0	0
$535$	46.7021	2.01911
$536$	0	0
$537$	−6.23048	−0.268865
$538$	0	0
$539$	−7.20527	−0.310353
$540$	0	0
$541$	−10.5846	−0.455067	−0.227534	−	0.973770i	$-0.573066\pi$
$541$	−10.5846	−0.455067	−0.227534	+	0.973770i	$0.573066\pi$
$542$	0	0
$543$	−15.3965	−0.660725
$544$	0	0
$545$	−30.5746	−1.30967
$546$	0	0
$547$	13.6960	0.585599	0.292799	−	0.956174i	$-0.405413\pi$
$547$	13.6960	0.585599	0.292799	+	0.956174i	$0.405413\pi$
$548$	0	0
$549$	2.13919	0.0912986
$550$	0	0
$551$	3.98940	0.169954
$552$	0	0
$553$	−8.61367	−0.366291
$554$	0	0
$555$	5.61525	0.238354
$556$	0	0
$557$	17.1034	0.724695	0.362347	−	0.932043i	$-0.381975\pi$
$557$	17.1034	0.724695	0.362347	+	0.932043i	$0.381975\pi$
$558$	0	0
$559$	10.5043	0.444283
$560$	0	0
$561$	−5.23221	−0.220904
$562$	0	0
$563$	−30.2111	−1.27324	−0.636622	−	0.771176i	$-0.719669\pi$
$563$	−30.2111	−1.27324	−0.636622	+	0.771176i	$0.719669\pi$
$564$	0	0
$565$	−19.3862	−0.815583
$566$	0	0
$567$	−0.715150	−0.0300335
$568$	0	0
$569$	−8.78561	−0.368312	−0.184156	−	0.982897i	$-0.558955\pi$
$569$	−8.78561	−0.368312	−0.184156	+	0.982897i	$0.558955\pi$
$570$	0	0
$571$	6.44098	0.269547	0.134773	−	0.990876i	$-0.456969\pi$
$571$	6.44098	0.269547	0.134773	+	0.990876i	$0.456969\pi$
$572$	0	0
$573$	−10.9278	−0.456513
$574$	0	0
$575$	58.7449	2.44983
$576$	0	0
$577$	−11.8116	−0.491724	−0.245862	−	0.969305i	$-0.579071\pi$
$577$	−11.8116	−0.491724	−0.245862	+	0.969305i	$0.579071\pi$
$578$	0	0
$579$	22.5568	0.937429
$580$	0	0
$581$	−2.80202	−0.116248
$582$	0	0
$583$	1.43855	0.0595788
$584$	0	0
$585$	15.1873	0.627916
$586$	0	0
$587$	−15.3410	−0.633193	−0.316596	−	0.948560i	$-0.602540\pi$
$587$	−15.3410	−0.633193	−0.316596	+	0.948560i	$0.602540\pi$
$588$	0	0
$589$	4.50514	0.185631
$590$	0	0
$591$	−4.17034	−0.171545
$592$	0	0
$593$	2.27264	0.0933259	0.0466630	−	0.998911i	$-0.485141\pi$
$593$	2.27264	0.0933259	0.0466630	+	0.998911i	$0.485141\pi$
$594$	0	0
$595$	11.5032	0.471583
$596$	0	0
$597$	9.92437	0.406177
$598$	0	0
$599$	9.28533	0.379388	0.189694	−	0.981843i	$-0.439250\pi$
$599$	9.28533	0.379388	0.189694	+	0.981843i	$0.439250\pi$
$600$	0	0
$601$	28.4071	1.15875	0.579375	−	0.815061i	$-0.303297\pi$
$601$	28.4071	1.15875	0.579375	+	0.815061i	$0.303297\pi$
$602$	0	0
$603$	3.53746	0.144056
$604$	0	0
$605$	−33.3421	−1.35555
$606$	0	0
$607$	−2.96344	−0.120282	−0.0601411	−	0.998190i	$-0.519155\pi$
$607$	−2.96344	−0.120282	−0.0601411	+	0.998190i	$0.519155\pi$
$608$	0	0
$609$	−1.44964	−0.0587425
$610$	0	0
$611$	−8.08478	−0.327075
$612$	0	0
$613$	0.378546	0.0152893	0.00764467	−	0.999971i	$-0.497567\pi$
$613$	0.378546	0.0152893	0.00764467	+	0.999971i	$0.497567\pi$
$614$	0	0
$615$	−22.7619	−0.917847
$616$	0	0
$617$	−1.56730	−0.0630971	−0.0315486	−	0.999502i	$-0.510044\pi$
$617$	−1.56730	−0.0630971	−0.0315486	+	0.999502i	$0.510044\pi$
$618$	0	0
$619$	13.1974	0.530448	0.265224	−	0.964187i	$-0.414554\pi$
$619$	13.1974	0.530448	0.265224	+	0.964187i	$0.414554\pi$
$620$	0	0
$621$	8.82854	0.354277
$622$	0	0
$623$	−5.33545	−0.213760
$624$	0	0
$625$	−13.9945	−0.559781
$626$	0	0
$627$	2.18547	0.0872792
$628$	0	0
$629$	−7.75024	−0.309022
$630$	0	0
$631$	20.7916	0.827701	0.413850	−	0.910345i	$-0.364184\pi$
$631$	20.7916	0.827701	0.413850	+	0.910345i	$0.364184\pi$
$632$	0	0
$633$	−20.1989	−0.802835
$634$	0	0
$635$	38.8789	1.54286
$636$	0	0
$637$	−28.8663	−1.14372
$638$	0	0
$639$	−2.09839	−0.0830109
$640$	0	0
$641$	−10.7280	−0.423731	−0.211865	−	0.977299i	$-0.567954\pi$
$641$	−10.7280	−0.423731	−0.211865	+	0.977299i	$0.567954\pi$
$642$	0	0
$643$	49.7476	1.96185	0.980927	−	0.194378i	$-0.0622690\pi$
$643$	49.7476	1.96185	0.980927	+	0.194378i	$0.0622690\pi$
$644$	0	0
$645$	8.06046	0.317380
$646$	0	0
$647$	6.51404	0.256093	0.128047	−	0.991768i	$-0.459129\pi$
$647$	6.51404	0.256093	0.128047	+	0.991768i	$0.459129\pi$
$648$	0	0
$649$	−13.2703	−0.520903
$650$	0	0
$651$	−1.63705	−0.0641612
$652$	0	0
$653$	−21.8390	−0.854627	−0.427314	−	0.904104i	$-0.640540\pi$
$653$	−21.8390	−0.854627	−0.427314	+	0.904104i	$0.640540\pi$
$654$	0	0
$655$	40.1539	1.56894
$656$	0	0
$657$	−13.5964	−0.530447
$658$	0	0
$659$	0.760997	0.0296442	0.0148221	−	0.999890i	$-0.495282\pi$
$659$	0.760997	0.0296442	0.0148221	+	0.999890i	$0.495282\pi$
$660$	0	0
$661$	3.80909	0.148157	0.0740783	−	0.997252i	$-0.476399\pi$
$661$	3.80909	0.148157	0.0740783	+	0.997252i	$0.476399\pi$
$662$	0	0
$663$	−20.9617	−0.814083
$664$	0	0
$665$	−4.80482	−0.186323
$666$	0	0
$667$	17.8959	0.692932
$668$	0	0
$669$	4.15398	0.160602
$670$	0	0
$671$	2.37548	0.0917045
$672$	0	0
$673$	−31.1467	−1.20062	−0.600308	−	0.799769i	$-0.704955\pi$
$673$	−31.1467	−1.20062	−0.600308	+	0.799769i	$0.704955\pi$
$674$	0	0
$675$	6.65397	0.256111
$676$	0	0
$677$	49.4503	1.90053	0.950264	−	0.311446i	$-0.100813\pi$
$677$	49.4503	1.90053	0.950264	+	0.311446i	$0.100813\pi$
$678$	0	0
$679$	9.65217	0.370416
$680$	0	0
$681$	17.3280	0.664011
$682$	0	0
$683$	34.0456	1.30272	0.651359	−	0.758770i	$-0.274199\pi$
$683$	34.0456	1.30272	0.651359	+	0.758770i	$0.274199\pi$
$684$	0	0
$685$	40.3772	1.54273
$686$	0	0
$687$	23.3004	0.888965
$688$	0	0
$689$	5.76325	0.219562
$690$	0	0
$691$	36.8060	1.40017	0.700083	−	0.714061i	$-0.253146\pi$
$691$	36.8060	1.40017	0.700083	+	0.714061i	$0.253146\pi$
$692$	0	0
$693$	−0.794143	−0.0301670
$694$	0	0
$695$	−26.7984	−1.01652
$696$	0	0
$697$	31.4162	1.18997
$698$	0	0
$699$	−7.09727	−0.268444
$700$	0	0
$701$	−16.2689	−0.614469	−0.307235	−	0.951634i	$-0.599404\pi$
$701$	−16.2689	−0.614469	−0.307235	+	0.951634i	$0.599404\pi$
$702$	0	0
$703$	3.23724	0.122095
$704$	0	0
$705$	−6.20387	−0.233651
$706$	0	0
$707$	1.10134	0.0414202
$708$	0	0
$709$	16.8658	0.633408	0.316704	−	0.948524i	$-0.397424\pi$
$709$	16.8658	0.633408	0.316704	+	0.948524i	$0.397424\pi$
$710$	0	0
$711$	12.0446	0.451707
$712$	0	0
$713$	20.2095	0.756851
$714$	0	0
$715$	16.8648	0.630708
$716$	0	0
$717$	13.1224	0.490063
$718$	0	0
$719$	−20.6348	−0.769548	−0.384774	−	0.923011i	$-0.625721\pi$
$719$	−20.6348	−0.769548	−0.384774	+	0.923011i	$0.625721\pi$
$720$	0	0
$721$	−7.60096	−0.283074
$722$	0	0
$723$	−7.33870	−0.272929
$724$	0	0
$725$	13.4879	0.500929
$726$	0	0
$727$	−1.86346	−0.0691119	−0.0345559	−	0.999403i	$-0.511002\pi$
$727$	−1.86346	−0.0691119	−0.0345559	+	0.999403i	$0.511002\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.1252	−0.411479
$732$	0	0
$733$	46.9226	1.73312	0.866562	−	0.499069i	$-0.166325\pi$
$733$	46.9226	1.73312	0.866562	+	0.499069i	$0.166325\pi$
$734$	0	0
$735$	−22.1506	−0.817037
$736$	0	0
$737$	3.92819	0.144697
$738$	0	0
$739$	3.30454	0.121560	0.0607798	−	0.998151i	$-0.480641\pi$
$739$	3.30454	0.121560	0.0607798	+	0.998151i	$0.480641\pi$
$740$	0	0
$741$	8.75559	0.321645
$742$	0	0
$743$	−15.2552	−0.559660	−0.279830	−	0.960050i	$-0.590278\pi$
$743$	−15.2552	−0.559660	−0.279830	+	0.960050i	$0.590278\pi$
$744$	0	0
$745$	21.2203	0.777451
$746$	0	0
$747$	3.91809	0.143355
$748$	0	0
$749$	−9.78355	−0.357483
$750$	0	0
$751$	20.5565	0.750117	0.375059	−	0.927001i	$-0.377623\pi$
$751$	20.5565	0.750117	0.375059	+	0.927001i	$0.377623\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	33.4285	1.21659
$756$	0	0
$757$	−17.0985	−0.621455	−0.310728	−	0.950499i	$-0.600573\pi$
$757$	−17.0985	−0.621455	−0.310728	+	0.950499i	$0.600573\pi$
$758$	0	0
$759$	9.80372	0.355852
$760$	0	0
$761$	−20.2924	−0.735599	−0.367799	−	0.929905i	$-0.619889\pi$
$761$	−20.2924	−0.735599	−0.367799	+	0.929905i	$0.619889\pi$
$762$	0	0
$763$	6.40503	0.231878
$764$	0	0
$765$	−16.0850	−0.581553
$766$	0	0
$767$	−53.1643	−1.91965
$768$	0	0
$769$	51.7562	1.86638	0.933188	−	0.359389i	$-0.117015\pi$
$769$	51.7562	1.86638	0.933188	+	0.359389i	$0.117015\pi$
$770$	0	0
$771$	11.5343	0.415399
$772$	0	0
$773$	−42.7643	−1.53813	−0.769063	−	0.639173i	$-0.779277\pi$
$773$	−42.7643	−1.53813	−0.769063	+	0.639173i	$0.779277\pi$
$774$	0	0
$775$	15.2316	0.547137
$776$	0	0
$777$	−1.17633	−0.0422006
$778$	0	0
$779$	−13.1224	−0.470159
$780$	0	0
$781$	−2.33017	−0.0833800
$782$	0	0
$783$	2.02705	0.0724408
$784$	0	0
$785$	52.0001	1.85596
$786$	0	0
$787$	−23.4268	−0.835076	−0.417538	−	0.908660i	$-0.637107\pi$
$787$	−23.4268	−0.835076	−0.417538	+	0.908660i	$0.637107\pi$
$788$	0	0
$789$	4.28866	0.152680
$790$	0	0
$791$	4.06118	0.144399
$792$	0	0
$793$	9.51684	0.337953
$794$	0	0
$795$	4.42244	0.156848
$796$	0	0
$797$	−10.9730	−0.388685	−0.194342	−	0.980934i	$-0.562257\pi$
$797$	−10.9730	−0.388685	−0.194342	+	0.980934i	$0.562257\pi$
$798$	0	0
$799$	8.56266	0.302925
$800$	0	0
$801$	7.46061	0.263608
$802$	0	0
$803$	−15.0982	−0.532805
$804$	0	0
$805$	−21.5538	−0.759670
$806$	0	0
$807$	4.87146	0.171483
$808$	0	0
$809$	−36.5360	−1.28454	−0.642269	−	0.766479i	$-0.722007\pi$
$809$	−36.5360	−1.28454	−0.642269	+	0.766479i	$0.722007\pi$
$810$	0	0
$811$	49.3996	1.73465	0.867327	−	0.497739i	$-0.165836\pi$
$811$	49.3996	1.73465	0.867327	+	0.497739i	$0.165836\pi$
$812$	0	0
$813$	9.34646	0.327795
$814$	0	0
$815$	−76.9297	−2.69473
$816$	0	0
$817$	4.64692	0.162575
$818$	0	0
$819$	−3.18156	−0.111173
$820$	0	0
$821$	−29.7400	−1.03793	−0.518967	−	0.854794i	$-0.673683\pi$
$821$	−29.7400	−1.03793	−0.518967	+	0.854794i	$0.673683\pi$
$822$	0	0
$823$	−24.5348	−0.855229	−0.427614	−	0.903961i	$-0.640646\pi$
$823$	−24.5348	−0.855229	−0.427614	+	0.903961i	$0.640646\pi$
$824$	0	0
$825$	7.38895	0.257250
$826$	0	0
$827$	−23.7516	−0.825923	−0.412962	−	0.910748i	$-0.635506\pi$
$827$	−23.7516	−0.825923	−0.412962	+	0.910748i	$0.635506\pi$
$828$	0	0
$829$	43.5513	1.51260	0.756299	−	0.654226i	$-0.227006\pi$
$829$	43.5513	1.51260	0.756299	+	0.654226i	$0.227006\pi$
$830$	0	0
$831$	−25.1150	−0.871229
$832$	0	0
$833$	30.5725	1.05928
$834$	0	0
$835$	−48.5947	−1.68169
$836$	0	0
$837$	2.28911	0.0791231
$838$	0	0
$839$	53.5988	1.85044	0.925219	−	0.379434i	$-0.123881\pi$
$839$	53.5988	1.85044	0.925219	+	0.379434i	$0.123881\pi$
$840$	0	0
$841$	−24.8911	−0.858313
$842$	0	0
$843$	3.63659	0.125251
$844$	0	0
$845$	23.1857	0.797614
$846$	0	0
$847$	6.98479	0.240000
$848$	0	0
$849$	21.5230	0.738667
$850$	0	0
$851$	14.5218	0.497802
$852$	0	0
$853$	−32.9008	−1.12650	−0.563251	−	0.826286i	$-0.690449\pi$
$853$	−32.9008	−1.12650	−0.563251	+	0.826286i	$0.690449\pi$
$854$	0	0
$855$	6.71861	0.229772
$856$	0	0
$857$	−43.8731	−1.49868	−0.749339	−	0.662187i	$-0.769629\pi$
$857$	−43.8731	−1.49868	−0.749339	+	0.662187i	$0.769629\pi$
$858$	0	0
$859$	50.8218	1.73402	0.867009	−	0.498292i	$-0.166039\pi$
$859$	50.8218	1.73402	0.867009	+	0.498292i	$0.166039\pi$
$860$	0	0
$861$	4.76835	0.162505
$862$	0	0
$863$	4.35997	0.148415	0.0742076	−	0.997243i	$-0.476357\pi$
$863$	4.35997	0.148415	0.0742076	+	0.997243i	$0.476357\pi$
$864$	0	0
$865$	−17.5706	−0.597417
$866$	0	0
$867$	5.20067	0.176624
$868$	0	0
$869$	13.3750	0.453715
$870$	0	0
$871$	15.7374	0.533242
$872$	0	0
$873$	−13.4967	−0.456794
$874$	0	0
$875$	−4.03796	−0.136508
$876$	0	0
$877$	−27.5618	−0.930695	−0.465348	−	0.885128i	$-0.654071\pi$
$877$	−27.5618	−0.930695	−0.465348	+	0.885128i	$0.654071\pi$
$878$	0	0
$879$	−31.4060	−1.05930
$880$	0	0
$881$	39.8945	1.34408	0.672040	−	0.740515i	$-0.265418\pi$
$881$	39.8945	1.34408	0.672040	+	0.740515i	$0.265418\pi$
$882$	0	0
$883$	24.4518	0.822869	0.411435	−	0.911439i	$-0.365028\pi$
$883$	24.4518	0.822869	0.411435	+	0.911439i	$0.365028\pi$
$884$	0	0
$885$	−40.7957	−1.37133
$886$	0	0
$887$	−0.180412	−0.00605764	−0.00302882	−	0.999995i	$-0.500964\pi$
$887$	−0.180412	−0.00605764	−0.00302882	+	0.999995i	$0.500964\pi$
$888$	0	0
$889$	−8.14468	−0.273164
$890$	0	0
$891$	1.11046	0.0372017
$892$	0	0
$893$	−3.57658	−0.119686
$894$	0	0
$895$	−21.2696	−0.710963
$896$	0	0
$897$	39.2764	1.31140
$898$	0	0
$899$	4.64013	0.154757
$900$	0	0
$901$	−6.10390	−0.203351
$902$	0	0
$903$	−1.68857	−0.0561922
$904$	0	0
$905$	−52.5603	−1.74716
$906$	0	0
$907$	19.0326	0.631967	0.315983	−	0.948765i	$-0.397666\pi$
$907$	19.0326	0.631967	0.315983	+	0.948765i	$0.397666\pi$
$908$	0	0
$909$	−1.54001	−0.0510791
$910$	0	0
$911$	−26.0111	−0.861785	−0.430892	−	0.902403i	$-0.641801\pi$
$911$	−26.0111	−0.861785	−0.430892	+	0.902403i	$0.641801\pi$
$912$	0	0
$913$	4.35087	0.143993
$914$	0	0
$915$	7.30276	0.241422
$916$	0	0
$917$	−8.41178	−0.277781
$918$	0	0
$919$	−26.9689	−0.889622	−0.444811	−	0.895625i	$-0.646729\pi$
$919$	−26.9689	−0.889622	−0.444811	+	0.895625i	$0.646729\pi$
$920$	0	0
$921$	8.03693	0.264826
$922$	0	0
$923$	−9.33530	−0.307275
$924$	0	0
$925$	10.9449	0.359867
$926$	0	0
$927$	10.6285	0.349085
$928$	0	0
$929$	−30.3212	−0.994805	−0.497402	−	0.867520i	$-0.665713\pi$
$929$	−30.3212	−0.994805	−0.497402	+	0.867520i	$0.665713\pi$
$930$	0	0
$931$	−12.7700	−0.418520
$932$	0	0
$933$	−7.62605	−0.249666
$934$	0	0
$935$	−17.8617	−0.584139
$936$	0	0
$937$	−8.04341	−0.262767	−0.131383	−	0.991332i	$-0.541942\pi$
$937$	−8.04341	−0.262767	−0.131383	+	0.991332i	$0.541942\pi$
$938$	0	0
$939$	12.5672	0.410115
$940$	0	0
$941$	−42.7053	−1.39215	−0.696077	−	0.717967i	$-0.745073\pi$
$941$	−42.7053	−1.39215	−0.696077	+	0.717967i	$0.745073\pi$
$942$	0	0
$943$	−58.8654	−1.91692
$944$	0	0
$945$	−2.44137	−0.0794178
$946$	0	0
$947$	−41.0647	−1.33442	−0.667211	−	0.744869i	$-0.732512\pi$
$947$	−41.0647	−1.33442	−0.667211	+	0.744869i	$0.732512\pi$
$948$	0	0
$949$	−60.4877	−1.96351
$950$	0	0
$951$	9.16710	0.297264
$952$	0	0
$953$	−44.2337	−1.43287	−0.716435	−	0.697654i	$-0.754227\pi$
$953$	−44.2337	−1.43287	−0.716435	+	0.697654i	$0.754227\pi$
$954$	0	0
$955$	−37.3051	−1.20716
$956$	0	0
$957$	2.25095	0.0727629
$958$	0	0
$959$	−8.45855	−0.273141
$960$	0	0
$961$	−25.7600	−0.830967
$962$	0	0
$963$	13.6804	0.440845
$964$	0	0
$965$	77.0042	2.47885
$966$	0	0
$967$	−30.1629	−0.969973	−0.484986	−	0.874522i	$-0.661175\pi$
$967$	−30.1629	−0.969973	−0.484986	+	0.874522i	$0.661175\pi$
$968$	0	0
$969$	−9.27312	−0.297895
$970$	0	0
$971$	3.84762	0.123476	0.0617380	−	0.998092i	$-0.480336\pi$
$971$	3.84762	0.123476	0.0617380	+	0.998092i	$0.480336\pi$
$972$	0	0
$973$	5.61395	0.179975
$974$	0	0
$975$	29.6022	0.948028
$976$	0	0
$977$	0.835311	0.0267240	0.0133620	−	0.999911i	$-0.495747\pi$
$977$	0.835311	0.0267240	0.0133620	+	0.999911i	$0.495747\pi$
$978$	0	0
$979$	8.28469	0.264780
$980$	0	0
$981$	−8.95620	−0.285950
$982$	0	0
$983$	37.0313	1.18111	0.590557	−	0.806996i	$-0.298908\pi$
$983$	37.0313	1.18111	0.590557	+	0.806996i	$0.298908\pi$
$984$	0	0
$985$	−14.2367	−0.453618
$986$	0	0
$987$	1.29964	0.0413680
$988$	0	0
$989$	20.8455	0.662847
$990$	0	0
$991$	21.1773	0.672718	0.336359	−	0.941734i	$-0.390804\pi$
$991$	21.1773	0.672718	0.336359	+	0.941734i	$0.390804\pi$
$992$	0	0
$993$	−33.9752	−1.07817
$994$	0	0
$995$	33.8797	1.07406
$996$	0	0
$997$	29.4589	0.932973	0.466486	−	0.884528i	$-0.345520\pi$
$997$	29.4589	0.932973	0.466486	+	0.884528i	$0.345520\pi$
$998$	0	0
$999$	1.64487	0.0520415

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.q.1.16	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.q.1.16	✓	18	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.41379 q^{5} -0.715150 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.41379 q^{5} -0.715150 q^{7} +1.00000 q^{9} +1.11046 q^{11} +4.44880 q^{13} +3.41379 q^{15} -4.71176 q^{17} +1.96808 q^{19} -0.715150 q^{21} +8.82854 q^{23} +6.65397 q^{25} +1.00000 q^{27} +2.02705 q^{29} +2.28911 q^{31} +1.11046 q^{33} -2.44137 q^{35} +1.64487 q^{37} +4.44880 q^{39} -6.66762 q^{41} +2.36115 q^{43} +3.41379 q^{45} -1.81730 q^{47} -6.48856 q^{49} -4.71176 q^{51} +1.29546 q^{53} +3.79087 q^{55} +1.96808 q^{57} -11.9503 q^{59} +2.13919 q^{61} -0.715150 q^{63} +15.1873 q^{65} +3.53746 q^{67} +8.82854 q^{69} -2.09839 q^{71} -13.5964 q^{73} +6.65397 q^{75} -0.794143 q^{77} +12.0446 q^{79} +1.00000 q^{81} +3.91809 q^{83} -16.0850 q^{85} +2.02705 q^{87} +7.46061 q^{89} -3.18156 q^{91} +2.28911 q^{93} +6.71861 q^{95} -13.4967 q^{97} +1.11046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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