LMFDB - Embedded newform 6020.2.a.f.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 12x^{6} + 26x^{5} + 55x^{4} - 52x^{3} - 82x^{2} + 22x + 27 $ x^8 - 3x^7 - 12x^6 + 26x^5 + 55x^4 - 52x^3 - 82x^2 + 22*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.53600$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53600 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.43127 q^{9} +O(q^{10})$	$q+2.53600 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.43127 q^{9} -0.621492 q^{11} -0.0537593 q^{13} -2.53600 q^{15} -1.09838 q^{17} -6.96727 q^{19} +2.53600 q^{21} -6.28735 q^{23} +1.00000 q^{25} +1.09371 q^{27} -2.29015 q^{29} -5.52869 q^{31} -1.57610 q^{33} -1.00000 q^{35} -0.885819 q^{37} -0.136333 q^{39} -4.72429 q^{41} -1.00000 q^{43} -3.43127 q^{45} +8.27633 q^{47} +1.00000 q^{49} -2.78548 q^{51} -10.4142 q^{53} +0.621492 q^{55} -17.6690 q^{57} +4.44764 q^{59} -11.6761 q^{61} +3.43127 q^{63} +0.0537593 q^{65} -7.87244 q^{67} -15.9447 q^{69} +12.5490 q^{71} +4.96642 q^{73} +2.53600 q^{75} -0.621492 q^{77} +10.1247 q^{79} -7.52018 q^{81} -13.7250 q^{83} +1.09838 q^{85} -5.80781 q^{87} +7.01855 q^{89} -0.0537593 q^{91} -14.0207 q^{93} +6.96727 q^{95} -4.58792 q^{97} -2.13251 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9	$ 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9} + 6 q^{11} - 13 q^{13} + 5 q^{15} - 2 q^{17} - 14 q^{19} - 5 q^{21} + 6 q^{23} + 8 q^{25} - 11 q^{27} - 7 q^{29} - 18 q^{31} - q^{33} - 8 q^{35} + 2 q^{37} + 9 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - q^{47} + 8 q^{49} - 19 q^{51} + 5 q^{53} - 6 q^{55} - 4 q^{57} - 12 q^{59} - 23 q^{61} + 11 q^{63} + 13 q^{65} + 8 q^{67} - 18 q^{69} + 20 q^{71} - 4 q^{73} - 5 q^{75} + 6 q^{77} + 24 q^{79} - 8 q^{81} - 14 q^{83} + 2 q^{85} + 10 q^{87} - 21 q^{89} - 13 q^{91} + q^{93} + 14 q^{95} + 7 q^{97} + 4 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9 + 6 * q^11 - 13 * q^13 + 5 * q^15 - 2 * q^17 - 14 * q^19 - 5 * q^21 + 6 * q^23 + 8 * q^25 - 11 * q^27 - 7 * q^29 - 18 * q^31 - q^33 - 8 * q^35 + 2 * q^37 + 9 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - q^47 + 8 * q^49 - 19 * q^51 + 5 * q^53 - 6 * q^55 - 4 * q^57 - 12 * q^59 - 23 * q^61 + 11 * q^63 + 13 * q^65 + 8 * q^67 - 18 * q^69 + 20 * q^71 - 4 * q^73 - 5 * q^75 + 6 * q^77 + 24 * q^79 - 8 * q^81 - 14 * q^83 + 2 * q^85 + 10 * q^87 - 21 * q^89 - 13 * q^91 + q^93 + 14 * q^95 + 7 * q^97 + 4 * 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
$3$	2.53600	1.46416	0.732079	−	0.681220i	$-0.238550\pi$
$3$	2.53600	1.46416	0.732079	+	0.681220i	$0.238550\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.43127	1.14376
$10$	0	0
$11$	−0.621492	−0.187387	−0.0936934	−	0.995601i	$-0.529867\pi$
$11$	−0.621492	−0.187387	−0.0936934	+	0.995601i	$0.529867\pi$
$12$	0	0
$13$	−0.0537593	−0.0149102	−0.00745508	−	0.999972i	$-0.502373\pi$
$13$	−0.0537593	−0.0149102	−0.00745508	+	0.999972i	$0.502373\pi$
$14$	0	0
$15$	−2.53600	−0.654791
$16$	0	0
$17$	−1.09838	−0.266396	−0.133198	−	0.991089i	$-0.542525\pi$
$17$	−1.09838	−0.266396	−0.133198	+	0.991089i	$0.542525\pi$
$18$	0	0
$19$	−6.96727	−1.59840	−0.799201	−	0.601064i	$-0.794743\pi$
$19$	−6.96727	−1.59840	−0.799201	+	0.601064i	$0.794743\pi$
$20$	0	0
$21$	2.53600	0.553400
$22$	0	0
$23$	−6.28735	−1.31100	−0.655501	−	0.755194i	$-0.727543\pi$
$23$	−6.28735	−1.31100	−0.655501	+	0.755194i	$0.727543\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.09371	0.210484
$28$	0	0
$29$	−2.29015	−0.425270	−0.212635	−	0.977132i	$-0.568205\pi$
$29$	−2.29015	−0.425270	−0.212635	+	0.977132i	$0.568205\pi$
$30$	0	0
$31$	−5.52869	−0.992981	−0.496491	−	0.868042i	$-0.665378\pi$
$31$	−5.52869	−0.992981	−0.496491	+	0.868042i	$0.665378\pi$
$32$	0	0
$33$	−1.57610	−0.274364
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−0.885819	−0.145628	−0.0728139	−	0.997346i	$-0.523198\pi$
$37$	−0.885819	−0.145628	−0.0728139	+	0.997346i	$0.523198\pi$
$38$	0	0
$39$	−0.136333	−0.0218308
$40$	0	0
$41$	−4.72429	−0.737809	−0.368905	−	0.929467i	$-0.620267\pi$
$41$	−4.72429	−0.737809	−0.368905	+	0.929467i	$0.620267\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−3.43127	−0.511504
$46$	0	0
$47$	8.27633	1.20723	0.603614	−	0.797277i	$-0.293727\pi$
$47$	8.27633	1.20723	0.603614	+	0.797277i	$0.293727\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.78548	−0.390045
$52$	0	0
$53$	−10.4142	−1.43050	−0.715249	−	0.698869i	$-0.753687\pi$
$53$	−10.4142	−1.43050	−0.715249	+	0.698869i	$0.753687\pi$
$54$	0	0
$55$	0.621492	0.0838019
$56$	0	0
$57$	−17.6690	−2.34031
$58$	0	0
$59$	4.44764	0.579033	0.289517	−	0.957173i	$-0.406505\pi$
$59$	4.44764	0.579033	0.289517	+	0.957173i	$0.406505\pi$
$60$	0	0
$61$	−11.6761	−1.49497	−0.747487	−	0.664276i	$-0.768740\pi$
$61$	−11.6761	−1.49497	−0.747487	+	0.664276i	$0.768740\pi$
$62$	0	0
$63$	3.43127	0.432300
$64$	0	0
$65$	0.0537593	0.00666802
$66$	0	0
$67$	−7.87244	−0.961772	−0.480886	−	0.876783i	$-0.659685\pi$
$67$	−7.87244	−0.961772	−0.480886	+	0.876783i	$0.659685\pi$
$68$	0	0
$69$	−15.9447	−1.91951
$70$	0	0
$71$	12.5490	1.48929	0.744645	−	0.667461i	$-0.232619\pi$
$71$	12.5490	1.48929	0.744645	+	0.667461i	$0.232619\pi$
$72$	0	0
$73$	4.96642	0.581276	0.290638	−	0.956833i	$-0.406132\pi$
$73$	4.96642	0.581276	0.290638	+	0.956833i	$0.406132\pi$
$74$	0	0
$75$	2.53600	0.292832
$76$	0	0
$77$	−0.621492	−0.0708255
$78$	0	0
$79$	10.1247	1.13912	0.569560	−	0.821950i	$-0.307114\pi$
$79$	10.1247	1.13912	0.569560	+	0.821950i	$0.307114\pi$
$80$	0	0
$81$	−7.52018	−0.835576
$82$	0	0
$83$	−13.7250	−1.50651	−0.753257	−	0.657726i	$-0.771518\pi$
$83$	−13.7250	−1.50651	−0.753257	+	0.657726i	$0.771518\pi$
$84$	0	0
$85$	1.09838	0.119136
$86$	0	0
$87$	−5.80781	−0.622663
$88$	0	0
$89$	7.01855	0.743965	0.371982	−	0.928240i	$-0.378678\pi$
$89$	7.01855	0.743965	0.371982	+	0.928240i	$0.378678\pi$
$90$	0	0
$91$	−0.0537593	−0.00563551
$92$	0	0
$93$	−14.0207	−1.45388
$94$	0	0
$95$	6.96727	0.714827
$96$	0	0
$97$	−4.58792	−0.465833	−0.232917	−	0.972497i	$-0.574827\pi$
$97$	−4.58792	−0.465833	−0.232917	+	0.972497i	$0.574827\pi$
$98$	0	0
$99$	−2.13251	−0.214325
$100$	0	0
$101$	−10.9507	−1.08964	−0.544820	−	0.838553i	$-0.683402\pi$
$101$	−10.9507	−1.08964	−0.544820	+	0.838553i	$0.683402\pi$
$102$	0	0
$103$	11.7751	1.16023	0.580117	−	0.814533i	$-0.303007\pi$
$103$	11.7751	1.16023	0.580117	+	0.814533i	$0.303007\pi$
$104$	0	0
$105$	−2.53600	−0.247488
$106$	0	0
$107$	−16.7840	−1.62257	−0.811285	−	0.584651i	$-0.801231\pi$
$107$	−16.7840	−1.62257	−0.811285	+	0.584651i	$0.801231\pi$
$108$	0	0
$109$	−5.91550	−0.566602	−0.283301	−	0.959031i	$-0.591430\pi$
$109$	−5.91550	−0.566602	−0.283301	+	0.959031i	$0.591430\pi$
$110$	0	0
$111$	−2.24643	−0.213222
$112$	0	0
$113$	19.0830	1.79518	0.897591	−	0.440830i	$-0.145316\pi$
$113$	19.0830	1.79518	0.897591	+	0.440830i	$0.145316\pi$
$114$	0	0
$115$	6.28735	0.586298
$116$	0	0
$117$	−0.184463	−0.0170536
$118$	0	0
$119$	−1.09838	−0.100688
$120$	0	0
$121$	−10.6137	−0.964886
$122$	0	0
$123$	−11.9808	−1.08027
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.98942	−0.176532	−0.0882661	−	0.996097i	$-0.528133\pi$
$127$	−1.98942	−0.176532	−0.0882661	+	0.996097i	$0.528133\pi$
$128$	0	0
$129$	−2.53600	−0.223282
$130$	0	0
$131$	7.28091	0.636136	0.318068	−	0.948068i	$-0.396966\pi$
$131$	7.28091	0.636136	0.318068	+	0.948068i	$0.396966\pi$
$132$	0	0
$133$	−6.96727	−0.604139
$134$	0	0
$135$	−1.09371	−0.0941313
$136$	0	0
$137$	6.83908	0.584302	0.292151	−	0.956372i	$-0.405629\pi$
$137$	6.83908	0.584302	0.292151	+	0.956372i	$0.405629\pi$
$138$	0	0
$139$	−15.6699	−1.32910	−0.664552	−	0.747242i	$-0.731377\pi$
$139$	−15.6699	−1.32910	−0.664552	+	0.747242i	$0.731377\pi$
$140$	0	0
$141$	20.9887	1.76757
$142$	0	0
$143$	0.0334110	0.00279397
$144$	0	0
$145$	2.29015	0.190187
$146$	0	0
$147$	2.53600	0.209165
$148$	0	0
$149$	14.8649	1.21778	0.608890	−	0.793254i	$-0.291615\pi$
$149$	14.8649	1.21778	0.608890	+	0.793254i	$0.291615\pi$
$150$	0	0
$151$	12.8008	1.04171	0.520856	−	0.853644i	$-0.325613\pi$
$151$	12.8008	1.04171	0.520856	+	0.853644i	$0.325613\pi$
$152$	0	0
$153$	−3.76883	−0.304692
$154$	0	0
$155$	5.52869	0.444075
$156$	0	0
$157$	9.90582	0.790571	0.395285	−	0.918558i	$-0.370646\pi$
$157$	9.90582	0.790571	0.395285	+	0.918558i	$0.370646\pi$
$158$	0	0
$159$	−26.4103	−2.09448
$160$	0	0
$161$	−6.28735	−0.495512
$162$	0	0
$163$	23.1501	1.81325	0.906627	−	0.421933i	$-0.138648\pi$
$163$	23.1501	1.81325	0.906627	+	0.421933i	$0.138648\pi$
$164$	0	0
$165$	1.57610	0.122699
$166$	0	0
$167$	−13.1720	−1.01928	−0.509642	−	0.860387i	$-0.670222\pi$
$167$	−13.1720	−1.01928	−0.509642	+	0.860387i	$0.670222\pi$
$168$	0	0
$169$	−12.9971	−0.999778
$170$	0	0
$171$	−23.9066	−1.82818
$172$	0	0
$173$	17.1998	1.30767	0.653837	−	0.756635i	$-0.273158\pi$
$173$	17.1998	1.30767	0.653837	+	0.756635i	$0.273158\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	11.2792	0.847796
$178$	0	0
$179$	9.63634	0.720254	0.360127	−	0.932903i	$-0.382733\pi$
$179$	9.63634	0.720254	0.360127	+	0.932903i	$0.382733\pi$
$180$	0	0
$181$	18.5481	1.37867	0.689335	−	0.724442i	$-0.257903\pi$
$181$	18.5481	1.37867	0.689335	+	0.724442i	$0.257903\pi$
$182$	0	0
$183$	−29.6106	−2.18888
$184$	0	0
$185$	0.885819	0.0651267
$186$	0	0
$187$	0.682632	0.0499190
$188$	0	0
$189$	1.09371	0.0795555
$190$	0	0
$191$	22.6670	1.64012	0.820062	−	0.572274i	$-0.193939\pi$
$191$	22.6670	1.64012	0.820062	+	0.572274i	$0.193939\pi$
$192$	0	0
$193$	−9.78088	−0.704043	−0.352022	−	0.935992i	$-0.614506\pi$
$193$	−9.78088	−0.704043	−0.352022	+	0.935992i	$0.614506\pi$
$194$	0	0
$195$	0.136333	0.00976304
$196$	0	0
$197$	−5.68729	−0.405203	−0.202601	−	0.979261i	$-0.564940\pi$
$197$	−5.68729	−0.405203	−0.202601	+	0.979261i	$0.564940\pi$
$198$	0	0
$199$	−14.5775	−1.03337	−0.516685	−	0.856176i	$-0.672834\pi$
$199$	−14.5775	−1.03337	−0.516685	+	0.856176i	$0.672834\pi$
$200$	0	0
$201$	−19.9645	−1.40819
$202$	0	0
$203$	−2.29015	−0.160737
$204$	0	0
$205$	4.72429	0.329958
$206$	0	0
$207$	−21.5736	−1.49947
$208$	0	0
$209$	4.33010	0.299519
$210$	0	0
$211$	−7.70849	−0.530674	−0.265337	−	0.964156i	$-0.585483\pi$
$211$	−7.70849	−0.530674	−0.265337	+	0.964156i	$0.585483\pi$
$212$	0	0
$213$	31.8241	2.18055
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−5.52869	−0.375312
$218$	0	0
$219$	12.5948	0.851080
$220$	0	0
$221$	0.0590480	0.00397200
$222$	0	0
$223$	−8.20463	−0.549422	−0.274711	−	0.961527i	$-0.588582\pi$
$223$	−8.20463	−0.549422	−0.274711	+	0.961527i	$0.588582\pi$
$224$	0	0
$225$	3.43127	0.228752
$226$	0	0
$227$	−25.6256	−1.70083	−0.850414	−	0.526114i	$-0.823649\pi$
$227$	−25.6256	−1.70083	−0.850414	+	0.526114i	$0.823649\pi$
$228$	0	0
$229$	−20.2332	−1.33705	−0.668525	−	0.743690i	$-0.733074\pi$
$229$	−20.2332	−1.33705	−0.668525	+	0.743690i	$0.733074\pi$
$230$	0	0
$231$	−1.57610	−0.103700
$232$	0	0
$233$	−4.45616	−0.291933	−0.145966	−	0.989290i	$-0.546629\pi$
$233$	−4.45616	−0.291933	−0.145966	+	0.989290i	$0.546629\pi$
$234$	0	0
$235$	−8.27633	−0.539888
$236$	0	0
$237$	25.6762	1.66785
$238$	0	0
$239$	−21.8715	−1.41475	−0.707375	−	0.706838i	$-0.750121\pi$
$239$	−21.8715	−1.41475	−0.707375	+	0.706838i	$0.750121\pi$
$240$	0	0
$241$	24.1477	1.55549	0.777745	−	0.628579i	$-0.216363\pi$
$241$	24.1477	1.55549	0.777745	+	0.628579i	$0.216363\pi$
$242$	0	0
$243$	−22.3523	−1.43390
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0.374556	0.0238324
$248$	0	0
$249$	−34.8065	−2.20577
$250$	0	0
$251$	−5.61654	−0.354513	−0.177257	−	0.984165i	$-0.556722\pi$
$251$	−5.61654	−0.354513	−0.177257	+	0.984165i	$0.556722\pi$
$252$	0	0
$253$	3.90753	0.245665
$254$	0	0
$255$	2.78548	0.174434
$256$	0	0
$257$	13.7070	0.855022	0.427511	−	0.904010i	$-0.359391\pi$
$257$	13.7070	0.855022	0.427511	+	0.904010i	$0.359391\pi$
$258$	0	0
$259$	−0.885819	−0.0550421
$260$	0	0
$261$	−7.85814	−0.486406
$262$	0	0
$263$	6.16001	0.379843	0.189921	−	0.981799i	$-0.439177\pi$
$263$	6.16001	0.379843	0.189921	+	0.981799i	$0.439177\pi$
$264$	0	0
$265$	10.4142	0.639739
$266$	0	0
$267$	17.7990	1.08928
$268$	0	0
$269$	9.64684	0.588178	0.294089	−	0.955778i	$-0.404984\pi$
$269$	9.64684	0.588178	0.294089	+	0.955778i	$0.404984\pi$
$270$	0	0
$271$	9.30486	0.565230	0.282615	−	0.959233i	$-0.408798\pi$
$271$	9.30486	0.565230	0.282615	+	0.959233i	$0.408798\pi$
$272$	0	0
$273$	−0.136333	−0.00825127
$274$	0	0
$275$	−0.621492	−0.0374774
$276$	0	0
$277$	−17.8485	−1.07241	−0.536206	−	0.844087i	$-0.680143\pi$
$277$	−17.8485	−1.07241	−0.536206	+	0.844087i	$0.680143\pi$
$278$	0	0
$279$	−18.9704	−1.13573
$280$	0	0
$281$	2.62313	0.156483	0.0782415	−	0.996934i	$-0.475069\pi$
$281$	2.62313	0.156483	0.0782415	+	0.996934i	$0.475069\pi$
$282$	0	0
$283$	15.5604	0.924967	0.462484	−	0.886628i	$-0.346958\pi$
$283$	15.5604	0.924967	0.462484	+	0.886628i	$0.346958\pi$
$284$	0	0
$285$	17.6690	1.04662
$286$	0	0
$287$	−4.72429	−0.278866
$288$	0	0
$289$	−15.7936	−0.929033
$290$	0	0
$291$	−11.6350	−0.682053
$292$	0	0
$293$	7.78918	0.455049	0.227524	−	0.973772i	$-0.426937\pi$
$293$	7.78918	0.455049	0.227524	+	0.973772i	$0.426937\pi$
$294$	0	0
$295$	−4.44764	−0.258952
$296$	0	0
$297$	−0.679730	−0.0394419
$298$	0	0
$299$	0.338004	0.0195472
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	−27.7710	−1.59540
$304$	0	0
$305$	11.6761	0.668573
$306$	0	0
$307$	−29.0983	−1.66073	−0.830363	−	0.557223i	$-0.811867\pi$
$307$	−29.0983	−1.66073	−0.830363	+	0.557223i	$0.811867\pi$
$308$	0	0
$309$	29.8616	1.69877
$310$	0	0
$311$	23.3146	1.32205	0.661025	−	0.750364i	$-0.270122\pi$
$311$	23.3146	1.32205	0.661025	+	0.750364i	$0.270122\pi$
$312$	0	0
$313$	−23.2712	−1.31537	−0.657684	−	0.753294i	$-0.728464\pi$
$313$	−23.2712	−1.31537	−0.657684	+	0.753294i	$0.728464\pi$
$314$	0	0
$315$	−3.43127	−0.193330
$316$	0	0
$317$	−1.57245	−0.0883176	−0.0441588	−	0.999025i	$-0.514061\pi$
$317$	−1.57245	−0.0883176	−0.0441588	+	0.999025i	$0.514061\pi$
$318$	0	0
$319$	1.42331	0.0796901
$320$	0	0
$321$	−42.5641	−2.37570
$322$	0	0
$323$	7.65269	0.425807
$324$	0	0
$325$	−0.0537593	−0.00298203
$326$	0	0
$327$	−15.0017	−0.829595
$328$	0	0
$329$	8.27633	0.456289
$330$	0	0
$331$	17.6319	0.969140	0.484570	−	0.874753i	$-0.338976\pi$
$331$	17.6319	0.969140	0.484570	+	0.874753i	$0.338976\pi$
$332$	0	0
$333$	−3.03949	−0.166563
$334$	0	0
$335$	7.87244	0.430117
$336$	0	0
$337$	−0.374091	−0.0203780	−0.0101890	−	0.999948i	$-0.503243\pi$
$337$	−0.374091	−0.0203780	−0.0101890	+	0.999948i	$0.503243\pi$
$338$	0	0
$339$	48.3945	2.62843
$340$	0	0
$341$	3.43603	0.186072
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	15.9447	0.858433
$346$	0	0
$347$	−13.6060	−0.730409	−0.365204	−	0.930927i	$-0.619001\pi$
$347$	−13.6060	−0.730409	−0.365204	+	0.930927i	$0.619001\pi$
$348$	0	0
$349$	−26.3459	−1.41026	−0.705131	−	0.709078i	$-0.749112\pi$
$349$	−26.3459	−1.41026	−0.705131	+	0.709078i	$0.749112\pi$
$350$	0	0
$351$	−0.0587969	−0.00313835
$352$	0	0
$353$	−2.71257	−0.144375	−0.0721877	−	0.997391i	$-0.522998\pi$
$353$	−2.71257	−0.144375	−0.0721877	+	0.997391i	$0.522998\pi$
$354$	0	0
$355$	−12.5490	−0.666030
$356$	0	0
$357$	−2.78548	−0.147423
$358$	0	0
$359$	−5.48293	−0.289378	−0.144689	−	0.989477i	$-0.546218\pi$
$359$	−5.48293	−0.289378	−0.144689	+	0.989477i	$0.546218\pi$
$360$	0	0
$361$	29.5428	1.55489
$362$	0	0
$363$	−26.9164	−1.41275
$364$	0	0
$365$	−4.96642	−0.259955
$366$	0	0
$367$	6.58698	0.343838	0.171919	−	0.985111i	$-0.445003\pi$
$367$	6.58698	0.343838	0.171919	+	0.985111i	$0.445003\pi$
$368$	0	0
$369$	−16.2103	−0.843875
$370$	0	0
$371$	−10.4142	−0.540678
$372$	0	0
$373$	16.7726	0.868452	0.434226	−	0.900804i	$-0.357022\pi$
$373$	16.7726	0.868452	0.434226	+	0.900804i	$0.357022\pi$
$374$	0	0
$375$	−2.53600	−0.130958
$376$	0	0
$377$	0.123117	0.00634085
$378$	0	0
$379$	−1.93947	−0.0996241	−0.0498121	−	0.998759i	$-0.515862\pi$
$379$	−1.93947	−0.0996241	−0.0498121	+	0.998759i	$0.515862\pi$
$380$	0	0
$381$	−5.04515	−0.258471
$382$	0	0
$383$	−26.0081	−1.32895	−0.664477	−	0.747309i	$-0.731346\pi$
$383$	−26.0081	−1.32895	−0.664477	+	0.747309i	$0.731346\pi$
$384$	0	0
$385$	0.621492	0.0316741
$386$	0	0
$387$	−3.43127	−0.174421
$388$	0	0
$389$	9.74554	0.494118	0.247059	−	0.969000i	$-0.420536\pi$
$389$	9.74554	0.494118	0.247059	+	0.969000i	$0.420536\pi$
$390$	0	0
$391$	6.90588	0.349245
$392$	0	0
$393$	18.4643	0.931403
$394$	0	0
$395$	−10.1247	−0.509430
$396$	0	0
$397$	−31.7449	−1.59323	−0.796615	−	0.604486i	$-0.793378\pi$
$397$	−31.7449	−1.59323	−0.796615	+	0.604486i	$0.793378\pi$
$398$	0	0
$399$	−17.6690	−0.884554
$400$	0	0
$401$	−6.96695	−0.347913	−0.173957	−	0.984753i	$-0.555655\pi$
$401$	−6.96695	−0.347913	−0.173957	+	0.984753i	$0.555655\pi$
$402$	0	0
$403$	0.297218	0.0148055
$404$	0	0
$405$	7.52018	0.373681
$406$	0	0
$407$	0.550529	0.0272887
$408$	0	0
$409$	−31.6802	−1.56649	−0.783244	−	0.621715i	$-0.786436\pi$
$409$	−31.6802	−1.56649	−0.783244	+	0.621715i	$0.786436\pi$
$410$	0	0
$411$	17.3439	0.855511
$412$	0	0
$413$	4.44764	0.218854
$414$	0	0
$415$	13.7250	0.673734
$416$	0	0
$417$	−39.7388	−1.94602
$418$	0	0
$419$	9.70562	0.474151	0.237075	−	0.971491i	$-0.423811\pi$
$419$	9.70562	0.474151	0.237075	+	0.971491i	$0.423811\pi$
$420$	0	0
$421$	−12.2995	−0.599441	−0.299721	−	0.954027i	$-0.596894\pi$
$421$	−12.2995	−0.599441	−0.299721	+	0.954027i	$0.596894\pi$
$422$	0	0
$423$	28.3984	1.38078
$424$	0	0
$425$	−1.09838	−0.0532791
$426$	0	0
$427$	−11.6761	−0.565047
$428$	0	0
$429$	0.0847301	0.00409081
$430$	0	0
$431$	1.73528	0.0835857	0.0417928	−	0.999126i	$-0.486693\pi$
$431$	1.73528	0.0835857	0.0417928	+	0.999126i	$0.486693\pi$
$432$	0	0
$433$	−30.2101	−1.45181	−0.725903	−	0.687797i	$-0.758578\pi$
$433$	−30.2101	−1.45181	−0.725903	+	0.687797i	$0.758578\pi$
$434$	0	0
$435$	5.80781	0.278463
$436$	0	0
$437$	43.8056	2.09551
$438$	0	0
$439$	−8.23598	−0.393082	−0.196541	−	0.980496i	$-0.562971\pi$
$439$	−8.23598	−0.393082	−0.196541	+	0.980496i	$0.562971\pi$
$440$	0	0
$441$	3.43127	0.163394
$442$	0	0
$443$	−33.1126	−1.57323	−0.786614	−	0.617445i	$-0.788168\pi$
$443$	−33.1126	−1.57323	−0.786614	+	0.617445i	$0.788168\pi$
$444$	0	0
$445$	−7.01855	−0.332711
$446$	0	0
$447$	37.6973	1.78302
$448$	0	0
$449$	−41.7742	−1.97145	−0.985723	−	0.168377i	$-0.946147\pi$
$449$	−41.7742	−1.97145	−0.985723	+	0.168377i	$0.946147\pi$
$450$	0	0
$451$	2.93610	0.138256
$452$	0	0
$453$	32.4627	1.52523
$454$	0	0
$455$	0.0537593	0.00252028
$456$	0	0
$457$	41.3443	1.93401	0.967003	−	0.254766i	$-0.0819984\pi$
$457$	41.3443	1.93401	0.967003	+	0.254766i	$0.0819984\pi$
$458$	0	0
$459$	−1.20130	−0.0560720
$460$	0	0
$461$	20.7025	0.964211	0.482106	−	0.876113i	$-0.339872\pi$
$461$	20.7025	0.964211	0.482106	+	0.876113i	$0.339872\pi$
$462$	0	0
$463$	2.38368	0.110779	0.0553895	−	0.998465i	$-0.482360\pi$
$463$	2.38368	0.110779	0.0553895	+	0.998465i	$0.482360\pi$
$464$	0	0
$465$	14.0207	0.650195
$466$	0	0
$467$	27.3951	1.26769	0.633847	−	0.773458i	$-0.281475\pi$
$467$	27.3951	1.26769	0.633847	+	0.773458i	$0.281475\pi$
$468$	0	0
$469$	−7.87244	−0.363515
$470$	0	0
$471$	25.1211	1.15752
$472$	0	0
$473$	0.621492	0.0285762
$474$	0	0
$475$	−6.96727	−0.319680
$476$	0	0
$477$	−35.7339	−1.63614
$478$	0	0
$479$	18.7714	0.857685	0.428843	−	0.903379i	$-0.358921\pi$
$479$	18.7714	0.857685	0.428843	+	0.903379i	$0.358921\pi$
$480$	0	0
$481$	0.0476210	0.00217133
$482$	0	0
$483$	−15.9447	−0.725508
$484$	0	0
$485$	4.58792	0.208327
$486$	0	0
$487$	6.83973	0.309938	0.154969	−	0.987919i	$-0.450472\pi$
$487$	6.83973	0.309938	0.154969	+	0.987919i	$0.450472\pi$
$488$	0	0
$489$	58.7085	2.65489
$490$	0	0
$491$	−8.66023	−0.390831	−0.195415	−	0.980721i	$-0.562606\pi$
$491$	−8.66023	−0.390831	−0.195415	+	0.980721i	$0.562606\pi$
$492$	0	0
$493$	2.51545	0.113290
$494$	0	0
$495$	2.13251	0.0958491
$496$	0	0
$497$	12.5490	0.562898
$498$	0	0
$499$	25.5297	1.14287	0.571434	−	0.820648i	$-0.306387\pi$
$499$	25.5297	1.14287	0.571434	+	0.820648i	$0.306387\pi$
$500$	0	0
$501$	−33.4042	−1.49239
$502$	0	0
$503$	15.6924	0.699691	0.349845	−	0.936807i	$-0.386234\pi$
$503$	15.6924	0.699691	0.349845	+	0.936807i	$0.386234\pi$
$504$	0	0
$505$	10.9507	0.487302
$506$	0	0
$507$	−32.9606	−1.46383
$508$	0	0
$509$	−21.5754	−0.956311	−0.478155	−	0.878275i	$-0.658694\pi$
$509$	−21.5754	−0.956311	−0.478155	+	0.878275i	$0.658694\pi$
$510$	0	0
$511$	4.96642	0.219702
$512$	0	0
$513$	−7.62015	−0.336438
$514$	0	0
$515$	−11.7751	−0.518872
$516$	0	0
$517$	−5.14367	−0.226218
$518$	0	0
$519$	43.6185	1.91464
$520$	0	0
$521$	22.2793	0.976074	0.488037	−	0.872823i	$-0.337713\pi$
$521$	22.2793	0.976074	0.488037	+	0.872823i	$0.337713\pi$
$522$	0	0
$523$	21.2327	0.928439	0.464220	−	0.885720i	$-0.346335\pi$
$523$	21.2327	0.928439	0.464220	+	0.885720i	$0.346335\pi$
$524$	0	0
$525$	2.53600	0.110680
$526$	0	0
$527$	6.07258	0.264526
$528$	0	0
$529$	16.5307	0.718728
$530$	0	0
$531$	15.2611	0.662274
$532$	0	0
$533$	0.253974	0.0110009
$534$	0	0
$535$	16.7840	0.725635
$536$	0	0
$537$	24.4377	1.05456
$538$	0	0
$539$	−0.621492	−0.0267695
$540$	0	0
$541$	8.67910	0.373143	0.186572	−	0.982441i	$-0.440262\pi$
$541$	8.67910	0.373143	0.186572	+	0.982441i	$0.440262\pi$
$542$	0	0
$543$	47.0379	2.01859
$544$	0	0
$545$	5.91550	0.253392
$546$	0	0
$547$	−8.85475	−0.378602	−0.189301	−	0.981919i	$-0.560622\pi$
$547$	−8.85475	−0.378602	−0.189301	+	0.981919i	$0.560622\pi$
$548$	0	0
$549$	−40.0640	−1.70989
$550$	0	0
$551$	15.9561	0.679753
$552$	0	0
$553$	10.1247	0.430547
$554$	0	0
$555$	2.24643	0.0953558
$556$	0	0
$557$	−44.0665	−1.86716	−0.933579	−	0.358370i	$-0.883333\pi$
$557$	−44.0665	−1.86716	−0.933579	+	0.358370i	$0.883333\pi$
$558$	0	0
$559$	0.0537593	0.00227378
$560$	0	0
$561$	1.73115	0.0730893
$562$	0	0
$563$	−13.8400	−0.583287	−0.291644	−	0.956527i	$-0.594202\pi$
$563$	−13.8400	−0.583287	−0.291644	+	0.956527i	$0.594202\pi$
$564$	0	0
$565$	−19.0830	−0.802830
$566$	0	0
$567$	−7.52018	−0.315818
$568$	0	0
$569$	34.7694	1.45761	0.728805	−	0.684721i	$-0.240076\pi$
$569$	34.7694	1.45761	0.728805	+	0.684721i	$0.240076\pi$
$570$	0	0
$571$	−17.6955	−0.740535	−0.370267	−	0.928925i	$-0.620734\pi$
$571$	−17.6955	−0.740535	−0.370267	+	0.928925i	$0.620734\pi$
$572$	0	0
$573$	57.4833	2.40140
$574$	0	0
$575$	−6.28735	−0.262201
$576$	0	0
$577$	−35.9809	−1.49791	−0.748953	−	0.662623i	$-0.769443\pi$
$577$	−35.9809	−1.49791	−0.748953	+	0.662623i	$0.769443\pi$
$578$	0	0
$579$	−24.8043	−1.03083
$580$	0	0
$581$	−13.7250	−0.569409
$582$	0	0
$583$	6.47233	0.268057
$584$	0	0
$585$	0.184463	0.00762660
$586$	0	0
$587$	−39.5260	−1.63141	−0.815705	−	0.578468i	$-0.803651\pi$
$587$	−39.5260	−1.63141	−0.815705	+	0.578468i	$0.803651\pi$
$588$	0	0
$589$	38.5198	1.58718
$590$	0	0
$591$	−14.4229	−0.593281
$592$	0	0
$593$	40.6959	1.67118	0.835590	−	0.549353i	$-0.185126\pi$
$593$	40.6959	1.67118	0.835590	+	0.549353i	$0.185126\pi$
$594$	0	0
$595$	1.09838	0.0450291
$596$	0	0
$597$	−36.9684	−1.51302
$598$	0	0
$599$	−5.10271	−0.208491	−0.104245	−	0.994552i	$-0.533243\pi$
$599$	−5.10271	−0.208491	−0.104245	+	0.994552i	$0.533243\pi$
$600$	0	0
$601$	−6.93750	−0.282987	−0.141493	−	0.989939i	$-0.545190\pi$
$601$	−6.93750	−0.282987	−0.141493	+	0.989939i	$0.545190\pi$
$602$	0	0
$603$	−27.0125	−1.10003
$604$	0	0
$605$	10.6137	0.431510
$606$	0	0
$607$	2.69905	0.109551	0.0547756	−	0.998499i	$-0.482556\pi$
$607$	2.69905	0.109551	0.0547756	+	0.998499i	$0.482556\pi$
$608$	0	0
$609$	−5.80781	−0.235345
$610$	0	0
$611$	−0.444930	−0.0179999
$612$	0	0
$613$	20.8401	0.841722	0.420861	−	0.907125i	$-0.361728\pi$
$613$	20.8401	0.841722	0.420861	+	0.907125i	$0.361728\pi$
$614$	0	0
$615$	11.9808	0.483111
$616$	0	0
$617$	−33.9602	−1.36719	−0.683593	−	0.729864i	$-0.739584\pi$
$617$	−33.9602	−1.36719	−0.683593	+	0.729864i	$0.739584\pi$
$618$	0	0
$619$	18.4723	0.742464	0.371232	−	0.928540i	$-0.378936\pi$
$619$	18.4723	0.742464	0.371232	+	0.928540i	$0.378936\pi$
$620$	0	0
$621$	−6.87652	−0.275945
$622$	0	0
$623$	7.01855	0.281192
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.9811	0.438543
$628$	0	0
$629$	0.972964	0.0387946
$630$	0	0
$631$	−36.1744	−1.44008	−0.720040	−	0.693933i	$-0.755877\pi$
$631$	−36.1744	−1.44008	−0.720040	+	0.693933i	$0.755877\pi$
$632$	0	0
$633$	−19.5487	−0.776991
$634$	0	0
$635$	1.98942	0.0789476
$636$	0	0
$637$	−0.0537593	−0.00213002
$638$	0	0
$639$	43.0590	1.70339
$640$	0	0
$641$	16.0971	0.635797	0.317899	−	0.948125i	$-0.397023\pi$
$641$	16.0971	0.635797	0.317899	+	0.948125i	$0.397023\pi$
$642$	0	0
$643$	−1.02877	−0.0405706	−0.0202853	−	0.999794i	$-0.506457\pi$
$643$	−1.02877	−0.0405706	−0.0202853	+	0.999794i	$0.506457\pi$
$644$	0	0
$645$	2.53600	0.0998547
$646$	0	0
$647$	39.8441	1.56643	0.783217	−	0.621748i	$-0.213577\pi$
$647$	39.8441	1.56643	0.783217	+	0.621748i	$0.213577\pi$
$648$	0	0
$649$	−2.76417	−0.108503
$650$	0	0
$651$	−14.0207	−0.549515
$652$	0	0
$653$	14.7218	0.576109	0.288054	−	0.957614i	$-0.406992\pi$
$653$	14.7218	0.576109	0.288054	+	0.957614i	$0.406992\pi$
$654$	0	0
$655$	−7.28091	−0.284489
$656$	0	0
$657$	17.0412	0.664839
$658$	0	0
$659$	17.9018	0.697354	0.348677	−	0.937243i	$-0.386631\pi$
$659$	17.9018	0.697354	0.348677	+	0.937243i	$0.386631\pi$
$660$	0	0
$661$	15.8813	0.617712	0.308856	−	0.951109i	$-0.400054\pi$
$661$	15.8813	0.617712	0.308856	+	0.951109i	$0.400054\pi$
$662$	0	0
$663$	0.149746	0.00581563
$664$	0	0
$665$	6.96727	0.270179
$666$	0	0
$667$	14.3990	0.557531
$668$	0	0
$669$	−20.8069	−0.804441
$670$	0	0
$671$	7.25661	0.280138
$672$	0	0
$673$	13.7929	0.531676	0.265838	−	0.964018i	$-0.414351\pi$
$673$	13.7929	0.531676	0.265838	+	0.964018i	$0.414351\pi$
$674$	0	0
$675$	1.09371	0.0420968
$676$	0	0
$677$	36.1731	1.39024	0.695122	−	0.718892i	$-0.255350\pi$
$677$	36.1731	1.39024	0.695122	+	0.718892i	$0.255350\pi$
$678$	0	0
$679$	−4.58792	−0.176068
$680$	0	0
$681$	−64.9863	−2.49028
$682$	0	0
$683$	29.6602	1.13491	0.567457	−	0.823403i	$-0.307927\pi$
$683$	29.6602	1.13491	0.567457	+	0.823403i	$0.307927\pi$
$684$	0	0
$685$	−6.83908	−0.261308
$686$	0	0
$687$	−51.3114	−1.95765
$688$	0	0
$689$	0.559860	0.0213290
$690$	0	0
$691$	−18.4479	−0.701792	−0.350896	−	0.936415i	$-0.614123\pi$
$691$	−18.4479	−0.701792	−0.350896	+	0.936415i	$0.614123\pi$
$692$	0	0
$693$	−2.13251	−0.0810073
$694$	0	0
$695$	15.6699	0.594393
$696$	0	0
$697$	5.18905	0.196549
$698$	0	0
$699$	−11.3008	−0.427435
$700$	0	0
$701$	23.8584	0.901120	0.450560	−	0.892746i	$-0.351224\pi$
$701$	23.8584	0.901120	0.450560	+	0.892746i	$0.351224\pi$
$702$	0	0
$703$	6.17174	0.232772
$704$	0	0
$705$	−20.9887	−0.790482
$706$	0	0
$707$	−10.9507	−0.411845
$708$	0	0
$709$	−1.07195	−0.0402580	−0.0201290	−	0.999797i	$-0.506408\pi$
$709$	−1.07195	−0.0402580	−0.0201290	+	0.999797i	$0.506408\pi$
$710$	0	0
$711$	34.7407	1.30288
$712$	0	0
$713$	34.7608	1.30180
$714$	0	0
$715$	−0.0334110	−0.00124950
$716$	0	0
$717$	−55.4660	−2.07142
$718$	0	0
$719$	−43.5390	−1.62373	−0.811866	−	0.583844i	$-0.801548\pi$
$719$	−43.5390	−1.62373	−0.811866	+	0.583844i	$0.801548\pi$
$720$	0	0
$721$	11.7751	0.438527
$722$	0	0
$723$	61.2385	2.27748
$724$	0	0
$725$	−2.29015	−0.0850541
$726$	0	0
$727$	6.88244	0.255256	0.127628	−	0.991822i	$-0.459264\pi$
$727$	6.88244	0.255256	0.127628	+	0.991822i	$0.459264\pi$
$728$	0	0
$729$	−34.1247	−1.26388
$730$	0	0
$731$	1.09838	0.0406250
$732$	0	0
$733$	44.3849	1.63939	0.819697	−	0.572798i	$-0.194142\pi$
$733$	44.3849	1.63939	0.819697	+	0.572798i	$0.194142\pi$
$734$	0	0
$735$	−2.53600	−0.0935416
$736$	0	0
$737$	4.89266	0.180223
$738$	0	0
$739$	−3.48856	−0.128329	−0.0641644	−	0.997939i	$-0.520438\pi$
$739$	−3.48856	−0.128329	−0.0641644	+	0.997939i	$0.520438\pi$
$740$	0	0
$741$	0.949871	0.0348944
$742$	0	0
$743$	37.1989	1.36469	0.682347	−	0.731028i	$-0.260959\pi$
$743$	37.1989	1.36469	0.682347	+	0.731028i	$0.260959\pi$
$744$	0	0
$745$	−14.8649	−0.544608
$746$	0	0
$747$	−47.0942	−1.72309
$748$	0	0
$749$	−16.7840	−0.613274
$750$	0	0
$751$	−18.4711	−0.674021	−0.337010	−	0.941501i	$-0.609416\pi$
$751$	−18.4711	−0.674021	−0.337010	+	0.941501i	$0.609416\pi$
$752$	0	0
$753$	−14.2435	−0.519063
$754$	0	0
$755$	−12.8008	−0.465868
$756$	0	0
$757$	−52.3313	−1.90201	−0.951007	−	0.309170i	$-0.899949\pi$
$757$	−52.3313	−1.90201	−0.951007	+	0.309170i	$0.899949\pi$
$758$	0	0
$759$	9.90949	0.359692
$760$	0	0
$761$	43.3233	1.57047	0.785235	−	0.619198i	$-0.212542\pi$
$761$	43.3233	1.57047	0.785235	+	0.619198i	$0.212542\pi$
$762$	0	0
$763$	−5.91550	−0.214155
$764$	0	0
$765$	3.76883	0.136262
$766$	0	0
$767$	−0.239102	−0.00863347
$768$	0	0
$769$	1.01551	0.0366204	0.0183102	−	0.999832i	$-0.494171\pi$
$769$	1.01551	0.0366204	0.0183102	+	0.999832i	$0.494171\pi$
$770$	0	0
$771$	34.7610	1.25189
$772$	0	0
$773$	−46.8023	−1.68336	−0.841680	−	0.539976i	$-0.818433\pi$
$773$	−46.8023	−1.68336	−0.841680	+	0.539976i	$0.818433\pi$
$774$	0	0
$775$	−5.52869	−0.198596
$776$	0	0
$777$	−2.24643	−0.0805903
$778$	0	0
$779$	32.9154	1.17932
$780$	0	0
$781$	−7.79908	−0.279073
$782$	0	0
$783$	−2.50475	−0.0895126
$784$	0	0
$785$	−9.90582	−0.353554
$786$	0	0
$787$	3.88041	0.138322	0.0691608	−	0.997606i	$-0.477968\pi$
$787$	3.88041	0.138322	0.0691608	+	0.997606i	$0.477968\pi$
$788$	0	0
$789$	15.6218	0.556150
$790$	0	0
$791$	19.0830	0.678515
$792$	0	0
$793$	0.627700	0.0222903
$794$	0	0
$795$	26.4103	0.936678
$796$	0	0
$797$	19.0767	0.675733	0.337866	−	0.941194i	$-0.390295\pi$
$797$	19.0767	0.675733	0.337866	+	0.941194i	$0.390295\pi$
$798$	0	0
$799$	−9.09054	−0.321600
$800$	0	0
$801$	24.0826	0.850915
$802$	0	0
$803$	−3.08659	−0.108923
$804$	0	0
$805$	6.28735	0.221600
$806$	0	0
$807$	24.4643	0.861186
$808$	0	0
$809$	−43.2519	−1.52066	−0.760328	−	0.649539i	$-0.774962\pi$
$809$	−43.2519	−1.52066	−0.760328	+	0.649539i	$0.774962\pi$
$810$	0	0
$811$	−28.3954	−0.997099	−0.498549	−	0.866861i	$-0.666134\pi$
$811$	−28.3954	−0.997099	−0.498549	+	0.866861i	$0.666134\pi$
$812$	0	0
$813$	23.5971	0.827586
$814$	0	0
$815$	−23.1501	−0.810912
$816$	0	0
$817$	6.96727	0.243754
$818$	0	0
$819$	−0.184463	−0.00644566
$820$	0	0
$821$	43.5132	1.51862	0.759310	−	0.650729i	$-0.225537\pi$
$821$	43.5132	1.51862	0.759310	+	0.650729i	$0.225537\pi$
$822$	0	0
$823$	17.0338	0.593762	0.296881	−	0.954914i	$-0.404054\pi$
$823$	17.0338	0.593762	0.296881	+	0.954914i	$0.404054\pi$
$824$	0	0
$825$	−1.57610	−0.0548728
$826$	0	0
$827$	38.9990	1.35613	0.678064	−	0.735003i	$-0.262819\pi$
$827$	38.9990	1.35613	0.678064	+	0.735003i	$0.262819\pi$
$828$	0	0
$829$	11.6034	0.403004	0.201502	−	0.979488i	$-0.435418\pi$
$829$	11.6034	0.403004	0.201502	+	0.979488i	$0.435418\pi$
$830$	0	0
$831$	−45.2637	−1.57018
$832$	0	0
$833$	−1.09838	−0.0380565
$834$	0	0
$835$	13.1720	0.455837
$836$	0	0
$837$	−6.04676	−0.209007
$838$	0	0
$839$	20.7739	0.717195	0.358598	−	0.933492i	$-0.383255\pi$
$839$	20.7739	0.717195	0.358598	+	0.933492i	$0.383255\pi$
$840$	0	0
$841$	−23.7552	−0.819145
$842$	0	0
$843$	6.65226	0.229116
$844$	0	0
$845$	12.9971	0.447114
$846$	0	0
$847$	−10.6137	−0.364693
$848$	0	0
$849$	39.4610	1.35430
$850$	0	0
$851$	5.56945	0.190918
$852$	0	0
$853$	−9.07269	−0.310643	−0.155322	−	0.987864i	$-0.549641\pi$
$853$	−9.07269	−0.310643	−0.155322	+	0.987864i	$0.549641\pi$
$854$	0	0
$855$	23.9066	0.817589
$856$	0	0
$857$	−38.6539	−1.32039	−0.660196	−	0.751093i	$-0.729527\pi$
$857$	−38.6539	−1.32039	−0.660196	+	0.751093i	$0.729527\pi$
$858$	0	0
$859$	−7.08444	−0.241718	−0.120859	−	0.992670i	$-0.538565\pi$
$859$	−7.08444	−0.241718	−0.120859	+	0.992670i	$0.538565\pi$
$860$	0	0
$861$	−11.9808	−0.408303
$862$	0	0
$863$	40.8739	1.39136	0.695682	−	0.718350i	$-0.255102\pi$
$863$	40.8739	1.39136	0.695682	+	0.718350i	$0.255102\pi$
$864$	0	0
$865$	−17.1998	−0.584810
$866$	0	0
$867$	−40.0524	−1.36025
$868$	0	0
$869$	−6.29243	−0.213456
$870$	0	0
$871$	0.423217	0.0143402
$872$	0	0
$873$	−15.7424	−0.532800
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	4.52120	0.152670	0.0763350	−	0.997082i	$-0.475678\pi$
$877$	4.52120	0.152670	0.0763350	+	0.997082i	$0.475678\pi$
$878$	0	0
$879$	19.7533	0.666263
$880$	0	0
$881$	−47.2293	−1.59120	−0.795598	−	0.605825i	$-0.792843\pi$
$881$	−47.2293	−1.59120	−0.795598	+	0.605825i	$0.792843\pi$
$882$	0	0
$883$	−5.18262	−0.174409	−0.0872045	−	0.996190i	$-0.527793\pi$
$883$	−5.18262	−0.174409	−0.0872045	+	0.996190i	$0.527793\pi$
$884$	0	0
$885$	−11.2792	−0.379146
$886$	0	0
$887$	−41.2325	−1.38445	−0.692226	−	0.721681i	$-0.743370\pi$
$887$	−41.2325	−1.38445	−0.692226	+	0.721681i	$0.743370\pi$
$888$	0	0
$889$	−1.98942	−0.0667229
$890$	0	0
$891$	4.67373	0.156576
$892$	0	0
$893$	−57.6634	−1.92963
$894$	0	0
$895$	−9.63634	−0.322107
$896$	0	0
$897$	0.857175	0.0286203
$898$	0	0
$899$	12.6615	0.422286
$900$	0	0
$901$	11.4387	0.381079
$902$	0	0
$903$	−2.53600	−0.0843926
$904$	0	0
$905$	−18.5481	−0.616560
$906$	0	0
$907$	−13.5416	−0.449642	−0.224821	−	0.974400i	$-0.572180\pi$
$907$	−13.5416	−0.449642	−0.224821	+	0.974400i	$0.572180\pi$
$908$	0	0
$909$	−37.5750	−1.24628
$910$	0	0
$911$	12.9764	0.429926	0.214963	−	0.976622i	$-0.431037\pi$
$911$	12.9764	0.429926	0.214963	+	0.976622i	$0.431037\pi$
$912$	0	0
$913$	8.52997	0.282301
$914$	0	0
$915$	29.6106	0.978896
$916$	0	0
$917$	7.28091	0.240437
$918$	0	0
$919$	38.6171	1.27386	0.636931	−	0.770921i	$-0.280204\pi$
$919$	38.6171	1.27386	0.636931	+	0.770921i	$0.280204\pi$
$920$	0	0
$921$	−73.7931	−2.43157
$922$	0	0
$923$	−0.674624	−0.0222055
$924$	0	0
$925$	−0.885819	−0.0291255
$926$	0	0
$927$	40.4036	1.32703
$928$	0	0
$929$	−10.0472	−0.329637	−0.164819	−	0.986324i	$-0.552704\pi$
$929$	−10.0472	−0.329637	−0.164819	+	0.986324i	$0.552704\pi$
$930$	0	0
$931$	−6.96727	−0.228343
$932$	0	0
$933$	59.1257	1.93569
$934$	0	0
$935$	−0.682632	−0.0223245
$936$	0	0
$937$	−46.0685	−1.50499	−0.752496	−	0.658596i	$-0.771150\pi$
$937$	−46.0685	−1.50499	−0.752496	+	0.658596i	$0.771150\pi$
$938$	0	0
$939$	−59.0158	−1.92591
$940$	0	0
$941$	33.0657	1.07791	0.538955	−	0.842334i	$-0.318819\pi$
$941$	33.0657	1.07791	0.538955	+	0.842334i	$0.318819\pi$
$942$	0	0
$943$	29.7032	0.967270
$944$	0	0
$945$	−1.09371	−0.0355783
$946$	0	0
$947$	−11.4601	−0.372405	−0.186202	−	0.982511i	$-0.559618\pi$
$947$	−11.4601	−0.372405	−0.186202	+	0.982511i	$0.559618\pi$
$948$	0	0
$949$	−0.266992	−0.00866691
$950$	0	0
$951$	−3.98773	−0.129311
$952$	0	0
$953$	−20.7378	−0.671764	−0.335882	−	0.941904i	$-0.609034\pi$
$953$	−20.7378	−0.671764	−0.335882	+	0.941904i	$0.609034\pi$
$954$	0	0
$955$	−22.6670	−0.733486
$956$	0	0
$957$	3.60951	0.116679
$958$	0	0
$959$	6.83908	0.220846
$960$	0	0
$961$	−0.433637	−0.0139883
$962$	0	0
$963$	−57.5904	−1.85583
$964$	0	0
$965$	9.78088	0.314858
$966$	0	0
$967$	−38.9870	−1.25374	−0.626869	−	0.779124i	$-0.715664\pi$
$967$	−38.9870	−1.25374	−0.626869	+	0.779124i	$0.715664\pi$
$968$	0	0
$969$	19.4072	0.623449
$970$	0	0
$971$	45.2186	1.45113	0.725567	−	0.688152i	$-0.241578\pi$
$971$	45.2186	1.45113	0.725567	+	0.688152i	$0.241578\pi$
$972$	0	0
$973$	−15.6699	−0.502354
$974$	0	0
$975$	−0.136333	−0.00436616
$976$	0	0
$977$	−9.93376	−0.317809	−0.158905	−	0.987294i	$-0.550796\pi$
$977$	−9.93376	−0.317809	−0.158905	+	0.987294i	$0.550796\pi$
$978$	0	0
$979$	−4.36197	−0.139409
$980$	0	0
$981$	−20.2977	−0.648056
$982$	0	0
$983$	30.8532	0.984064	0.492032	−	0.870577i	$-0.336254\pi$
$983$	30.8532	0.984064	0.492032	+	0.870577i	$0.336254\pi$
$984$	0	0
$985$	5.68729	0.181212
$986$	0	0
$987$	20.9887	0.668079
$988$	0	0
$989$	6.28735	0.199926
$990$	0	0
$991$	−14.6696	−0.465996	−0.232998	−	0.972477i	$-0.574854\pi$
$991$	−14.6696	−0.465996	−0.232998	+	0.972477i	$0.574854\pi$
$992$	0	0
$993$	44.7145	1.41897
$994$	0	0
$995$	14.5775	0.462137
$996$	0	0
$997$	−14.7576	−0.467379	−0.233690	−	0.972311i	$-0.575080\pi$
$997$	−14.7576	−0.467379	−0.233690	+	0.972311i	$0.575080\pi$
$998$	0	0
$999$	−0.968826	−0.0306523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.f.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.f.1.8	✓	8	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:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53600 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.43127 q^{9} +O(q^{10})\)	\(q+2.53600 q^{3} -1.00000 q^{5} +1.00000 q^{7} +3.43127 q^{9} -0.621492 q^{11} -0.0537593 q^{13} -2.53600 q^{15} -1.09838 q^{17} -6.96727 q^{19} +2.53600 q^{21} -6.28735 q^{23} +1.00000 q^{25} +1.09371 q^{27} -2.29015 q^{29} -5.52869 q^{31} -1.57610 q^{33} -1.00000 q^{35} -0.885819 q^{37} -0.136333 q^{39} -4.72429 q^{41} -1.00000 q^{43} -3.43127 q^{45} +8.27633 q^{47} +1.00000 q^{49} -2.78548 q^{51} -10.4142 q^{53} +0.621492 q^{55} -17.6690 q^{57} +4.44764 q^{59} -11.6761 q^{61} +3.43127 q^{63} +0.0537593 q^{65} -7.87244 q^{67} -15.9447 q^{69} +12.5490 q^{71} +4.96642 q^{73} +2.53600 q^{75} -0.621492 q^{77} +10.1247 q^{79} -7.52018 q^{81} -13.7250 q^{83} +1.09838 q^{85} -5.80781 q^{87} +7.01855 q^{89} -0.0537593 q^{91} -14.0207 q^{93} +6.96727 q^{95} -4.58792 q^{97} -2.13251 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9	\( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9} + 6 q^{11} - 13 q^{13} + 5 q^{15} - 2 q^{17} - 14 q^{19} - 5 q^{21} + 6 q^{23} + 8 q^{25} - 11 q^{27} - 7 q^{29} - 18 q^{31} - q^{33} - 8 q^{35} + 2 q^{37} + 9 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - q^{47} + 8 q^{49} - 19 q^{51} + 5 q^{53} - 6 q^{55} - 4 q^{57} - 12 q^{59} - 23 q^{61} + 11 q^{63} + 13 q^{65} + 8 q^{67} - 18 q^{69} + 20 q^{71} - 4 q^{73} - 5 q^{75} + 6 q^{77} + 24 q^{79} - 8 q^{81} - 14 q^{83} + 2 q^{85} + 10 q^{87} - 21 q^{89} - 13 q^{91} + q^{93} + 14 q^{95} + 7 q^{97} + 4 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9 + 6 * q^11 - 13 * q^13 + 5 * q^15 - 2 * q^17 - 14 * q^19 - 5 * q^21 + 6 * q^23 + 8 * q^25 - 11 * q^27 - 7 * q^29 - 18 * q^31 - q^33 - 8 * q^35 + 2 * q^37 + 9 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - q^47 + 8 * q^49 - 19 * q^51 + 5 * q^53 - 6 * q^55 - 4 * q^57 - 12 * q^59 - 23 * q^61 + 11 * q^63 + 13 * q^65 + 8 * q^67 - 18 * q^69 + 20 * q^71 - 4 * q^73 - 5 * q^75 + 6 * q^77 + 24 * q^79 - 8 * q^81 - 14 * q^83 + 2 * q^85 + 10 * q^87 - 21 * q^89 - 13 * q^91 + q^93 + 14 * q^95 + 7 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more