LMFDB - Embedded newform 4020.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, 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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 21x^{5} - 12x^{4} + 93x^{3} + 18x^{2} - 120x + 48 $ x^7 - 21x^5 - 12x^4 + 93x^3 + 18x^2 - 120*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.35097$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +0.772018 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +0.772018 q^{7} +1.00000 q^{9} +3.49287 q^{11} -3.80921 q^{13} +1.00000 q^{15} -1.47309 q^{17} +5.60907 q^{19} +0.772018 q^{21} +4.75097 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.97895 q^{29} -0.171874 q^{31} +3.49287 q^{33} +0.772018 q^{35} +5.95097 q^{37} -3.80921 q^{39} -6.79989 q^{41} +2.90121 q^{43} +1.00000 q^{45} +0.666440 q^{47} -6.40399 q^{49} -1.47309 q^{51} +2.18036 q^{53} +3.49287 q^{55} +5.60907 q^{57} +10.2262 q^{59} +8.97755 q^{61} +0.772018 q^{63} -3.80921 q^{65} -1.00000 q^{67} +4.75097 q^{69} -9.65305 q^{71} -2.54263 q^{73} +1.00000 q^{75} +2.69656 q^{77} +6.88724 q^{79} +1.00000 q^{81} +12.9668 q^{83} -1.47309 q^{85} -1.97895 q^{87} -15.2122 q^{89} -2.94078 q^{91} -0.171874 q^{93} +5.60907 q^{95} +1.06150 q^{97} +3.49287 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.772018	0.291795	0.145898	−	0.989300i	$-0.453393\pi$
$7$	0.772018	0.291795	0.145898	+	0.989300i	$0.453393\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.49287	1.05314	0.526570	−	0.850132i	$-0.323478\pi$
$11$	3.49287	1.05314	0.526570	+	0.850132i	$0.323478\pi$
$12$	0	0
$13$	−3.80921	−1.05649	−0.528243	−	0.849093i	$-0.677149\pi$
$13$	−3.80921	−1.05649	−0.528243	+	0.849093i	$0.677149\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.47309	−0.357276	−0.178638	−	0.983915i	$-0.557169\pi$
$17$	−1.47309	−0.357276	−0.178638	+	0.983915i	$0.557169\pi$
$18$	0	0
$19$	5.60907	1.28681	0.643404	−	0.765526i	$-0.277521\pi$
$19$	5.60907	1.28681	0.643404	+	0.765526i	$0.277521\pi$
$20$	0	0
$21$	0.772018	0.168468
$22$	0	0
$23$	4.75097	0.990646	0.495323	−	0.868709i	$-0.335050\pi$
$23$	4.75097	0.990646	0.495323	+	0.868709i	$0.335050\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.97895	−0.367482	−0.183741	−	0.982975i	$-0.558821\pi$
$29$	−1.97895	−0.367482	−0.183741	+	0.982975i	$0.558821\pi$
$30$	0	0
$31$	−0.171874	−0.0308695	−0.0154348	−	0.999881i	$-0.504913\pi$
$31$	−0.171874	−0.0308695	−0.0154348	+	0.999881i	$0.504913\pi$
$32$	0	0
$33$	3.49287	0.608031
$34$	0	0
$35$	0.772018	0.130495
$36$	0	0
$37$	5.95097	0.978333	0.489167	−	0.872190i	$-0.337301\pi$
$37$	5.95097	0.978333	0.489167	+	0.872190i	$0.337301\pi$
$38$	0	0
$39$	−3.80921	−0.609962
$40$	0	0
$41$	−6.79989	−1.06197	−0.530983	−	0.847383i	$-0.678177\pi$
$41$	−6.79989	−1.06197	−0.530983	+	0.847383i	$0.678177\pi$
$42$	0	0
$43$	2.90121	0.442430	0.221215	−	0.975225i	$-0.428998\pi$
$43$	2.90121	0.442430	0.221215	+	0.975225i	$0.428998\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0.666440	0.0972103	0.0486051	−	0.998818i	$-0.484522\pi$
$47$	0.666440	0.0972103	0.0486051	+	0.998818i	$0.484522\pi$
$48$	0	0
$49$	−6.40399	−0.914856
$50$	0	0
$51$	−1.47309	−0.206273
$52$	0	0
$53$	2.18036	0.299495	0.149748	−	0.988724i	$-0.452154\pi$
$53$	2.18036	0.299495	0.149748	+	0.988724i	$0.452154\pi$
$54$	0	0
$55$	3.49287	0.470979
$56$	0	0
$57$	5.60907	0.742940
$58$	0	0
$59$	10.2262	1.33134	0.665668	−	0.746248i	$-0.268147\pi$
$59$	10.2262	1.33134	0.665668	+	0.746248i	$0.268147\pi$
$60$	0	0
$61$	8.97755	1.14946	0.574729	−	0.818344i	$-0.305108\pi$
$61$	8.97755	1.14946	0.574729	+	0.818344i	$0.305108\pi$
$62$	0	0
$63$	0.772018	0.0972651
$64$	0	0
$65$	−3.80921	−0.472475
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	4.75097	0.571950
$70$	0	0
$71$	−9.65305	−1.14561	−0.572803	−	0.819693i	$-0.694144\pi$
$71$	−9.65305	−1.14561	−0.572803	+	0.819693i	$0.694144\pi$
$72$	0	0
$73$	−2.54263	−0.297592	−0.148796	−	0.988868i	$-0.547540\pi$
$73$	−2.54263	−0.297592	−0.148796	+	0.988868i	$0.547540\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.69656	0.307301
$78$	0	0
$79$	6.88724	0.774875	0.387437	−	0.921896i	$-0.373360\pi$
$79$	6.88724	0.774875	0.387437	+	0.921896i	$0.373360\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9668	1.42329	0.711647	−	0.702537i	$-0.247949\pi$
$83$	12.9668	1.42329	0.711647	+	0.702537i	$0.247949\pi$
$84$	0	0
$85$	−1.47309	−0.159779
$86$	0	0
$87$	−1.97895	−0.212166
$88$	0	0
$89$	−15.2122	−1.61249	−0.806246	−	0.591580i	$-0.798504\pi$
$89$	−15.2122	−1.61249	−0.806246	+	0.591580i	$0.798504\pi$
$90$	0	0
$91$	−2.94078	−0.308277
$92$	0	0
$93$	−0.171874	−0.0178225
$94$	0	0
$95$	5.60907	0.575478
$96$	0	0
$97$	1.06150	0.107779	0.0538896	−	0.998547i	$-0.482838\pi$
$97$	1.06150	0.107779	0.0538896	+	0.998547i	$0.482838\pi$
$98$	0	0
$99$	3.49287	0.351047
$100$	0	0
$101$	6.08976	0.605954	0.302977	−	0.952998i	$-0.402020\pi$
$101$	6.08976	0.605954	0.302977	+	0.952998i	$0.402020\pi$
$102$	0	0
$103$	14.9586	1.47392	0.736959	−	0.675938i	$-0.236261\pi$
$103$	14.9586	1.47392	0.736959	+	0.675938i	$0.236261\pi$
$104$	0	0
$105$	0.772018	0.0753412
$106$	0	0
$107$	8.67550	0.838692	0.419346	−	0.907826i	$-0.362259\pi$
$107$	8.67550	0.838692	0.419346	+	0.907826i	$0.362259\pi$
$108$	0	0
$109$	−4.42094	−0.423449	−0.211725	−	0.977329i	$-0.567908\pi$
$109$	−4.42094	−0.423449	−0.211725	+	0.977329i	$0.567908\pi$
$110$	0	0
$111$	5.95097	0.564841
$112$	0	0
$113$	18.6417	1.75367	0.876834	−	0.480794i	$-0.159651\pi$
$113$	18.6417	1.75367	0.876834	+	0.480794i	$0.159651\pi$
$114$	0	0
$115$	4.75097	0.443030
$116$	0	0
$117$	−3.80921	−0.352162
$118$	0	0
$119$	−1.13725	−0.104251
$120$	0	0
$121$	1.20015	0.109104
$122$	0	0
$123$	−6.79989	−0.613126
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.47439	0.574509	0.287255	−	0.957854i	$-0.407257\pi$
$127$	6.47439	0.574509	0.287255	+	0.957854i	$0.407257\pi$
$128$	0	0
$129$	2.90121	0.255437
$130$	0	0
$131$	−17.5320	−1.53177	−0.765887	−	0.642975i	$-0.777700\pi$
$131$	−17.5320	−1.53177	−0.765887	+	0.642975i	$0.777700\pi$
$132$	0	0
$133$	4.33030	0.375485
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−14.2289	−1.21566	−0.607828	−	0.794069i	$-0.707959\pi$
$137$	−14.2289	−1.21566	−0.607828	+	0.794069i	$0.707959\pi$
$138$	0	0
$139$	−22.1873	−1.88191	−0.940953	−	0.338537i	$-0.890068\pi$
$139$	−22.1873	−1.88191	−0.940953	+	0.338537i	$0.890068\pi$
$140$	0	0
$141$	0.666440	0.0561244
$142$	0	0
$143$	−13.3051	−1.11263
$144$	0	0
$145$	−1.97895	−0.164343
$146$	0	0
$147$	−6.40399	−0.528192
$148$	0	0
$149$	19.2572	1.57761	0.788805	−	0.614644i	$-0.210700\pi$
$149$	19.2572	1.57761	0.788805	+	0.614644i	$0.210700\pi$
$150$	0	0
$151$	−19.3020	−1.57077	−0.785386	−	0.619006i	$-0.787536\pi$
$151$	−19.3020	−1.57077	−0.785386	+	0.619006i	$0.787536\pi$
$152$	0	0
$153$	−1.47309	−0.119092
$154$	0	0
$155$	−0.171874	−0.0138053
$156$	0	0
$157$	3.80650	0.303792	0.151896	−	0.988396i	$-0.451462\pi$
$157$	3.80650	0.303792	0.151896	+	0.988396i	$0.451462\pi$
$158$	0	0
$159$	2.18036	0.172914
$160$	0	0
$161$	3.66783	0.289066
$162$	0	0
$163$	−18.0767	−1.41588	−0.707938	−	0.706274i	$-0.750375\pi$
$163$	−18.0767	−1.41588	−0.707938	+	0.706274i	$0.750375\pi$
$164$	0	0
$165$	3.49287	0.271920
$166$	0	0
$167$	19.5459	1.51251	0.756254	−	0.654278i	$-0.227027\pi$
$167$	19.5459	1.51251	0.756254	+	0.654278i	$0.227027\pi$
$168$	0	0
$169$	1.51010	0.116161
$170$	0	0
$171$	5.60907	0.428936
$172$	0	0
$173$	21.3975	1.62682	0.813412	−	0.581689i	$-0.197608\pi$
$173$	21.3975	1.62682	0.813412	+	0.581689i	$0.197608\pi$
$174$	0	0
$175$	0.772018	0.0583591
$176$	0	0
$177$	10.2262	0.768648
$178$	0	0
$179$	0.270379	0.0202091	0.0101046	−	0.999949i	$-0.496784\pi$
$179$	0.270379	0.0202091	0.0101046	+	0.999949i	$0.496784\pi$
$180$	0	0
$181$	2.43492	0.180986	0.0904930	−	0.995897i	$-0.471156\pi$
$181$	2.43492	0.180986	0.0904930	+	0.995897i	$0.471156\pi$
$182$	0	0
$183$	8.97755	0.663639
$184$	0	0
$185$	5.95097	0.437524
$186$	0	0
$187$	−5.14530	−0.376262
$188$	0	0
$189$	0.772018	0.0561560
$190$	0	0
$191$	−11.3834	−0.823671	−0.411835	−	0.911258i	$-0.635112\pi$
$191$	−11.3834	−0.823671	−0.411835	+	0.911258i	$0.635112\pi$
$192$	0	0
$193$	7.72764	0.556248	0.278124	−	0.960545i	$-0.410287\pi$
$193$	7.72764	0.556248	0.278124	+	0.960545i	$0.410287\pi$
$194$	0	0
$195$	−3.80921	−0.272783
$196$	0	0
$197$	−11.9110	−0.848624	−0.424312	−	0.905516i	$-0.639484\pi$
$197$	−11.9110	−0.848624	−0.424312	+	0.905516i	$0.639484\pi$
$198$	0	0
$199$	−0.345159	−0.0244677	−0.0122338	−	0.999925i	$-0.503894\pi$
$199$	−0.345159	−0.0244677	−0.0122338	+	0.999925i	$0.503894\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−1.52779	−0.107230
$204$	0	0
$205$	−6.79989	−0.474925
$206$	0	0
$207$	4.75097	0.330215
$208$	0	0
$209$	19.5918	1.35519
$210$	0	0
$211$	19.5809	1.34800	0.674001	−	0.738730i	$-0.264574\pi$
$211$	19.5809	1.34800	0.674001	+	0.738730i	$0.264574\pi$
$212$	0	0
$213$	−9.65305	−0.661416
$214$	0	0
$215$	2.90121	0.197861
$216$	0	0
$217$	−0.132690	−0.00900758
$218$	0	0
$219$	−2.54263	−0.171815
$220$	0	0
$221$	5.61130	0.377457
$222$	0	0
$223$	−3.57499	−0.239399	−0.119700	−	0.992810i	$-0.538193\pi$
$223$	−3.57499	−0.239399	−0.119700	+	0.992810i	$0.538193\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−18.8309	−1.24985	−0.624925	−	0.780685i	$-0.714870\pi$
$227$	−18.8309	−1.24985	−0.624925	+	0.780685i	$0.714870\pi$
$228$	0	0
$229$	−9.08710	−0.600492	−0.300246	−	0.953862i	$-0.597069\pi$
$229$	−9.08710	−0.600492	−0.300246	+	0.953862i	$0.597069\pi$
$230$	0	0
$231$	2.69656	0.177421
$232$	0	0
$233$	−0.280150	−0.0183532	−0.00917660	−	0.999958i	$-0.502921\pi$
$233$	−0.280150	−0.0183532	−0.00917660	+	0.999958i	$0.502921\pi$
$234$	0	0
$235$	0.666440	0.0434738
$236$	0	0
$237$	6.88724	0.447374
$238$	0	0
$239$	21.4729	1.38897	0.694483	−	0.719509i	$-0.255633\pi$
$239$	21.4729	1.38897	0.694483	+	0.719509i	$0.255633\pi$
$240$	0	0
$241$	−8.38206	−0.539936	−0.269968	−	0.962869i	$-0.587013\pi$
$241$	−8.38206	−0.539936	−0.269968	+	0.962869i	$0.587013\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.40399	−0.409136
$246$	0	0
$247$	−21.3661	−1.35949
$248$	0	0
$249$	12.9668	0.821739
$250$	0	0
$251$	16.7462	1.05701	0.528506	−	0.848930i	$-0.322753\pi$
$251$	16.7462	1.05701	0.528506	+	0.848930i	$0.322753\pi$
$252$	0	0
$253$	16.5945	1.04329
$254$	0	0
$255$	−1.47309	−0.0922483
$256$	0	0
$257$	15.0090	0.936233	0.468117	−	0.883667i	$-0.344933\pi$
$257$	15.0090	0.936233	0.468117	+	0.883667i	$0.344933\pi$
$258$	0	0
$259$	4.59425	0.285473
$260$	0	0
$261$	−1.97895	−0.122494
$262$	0	0
$263$	4.86798	0.300172	0.150086	−	0.988673i	$-0.452045\pi$
$263$	4.86798	0.300172	0.150086	+	0.988673i	$0.452045\pi$
$264$	0	0
$265$	2.18036	0.133938
$266$	0	0
$267$	−15.2122	−0.930973
$268$	0	0
$269$	11.1049	0.677079	0.338540	−	0.940952i	$-0.390067\pi$
$269$	11.1049	0.677079	0.338540	+	0.940952i	$0.390067\pi$
$270$	0	0
$271$	17.4350	1.05910	0.529549	−	0.848279i	$-0.322361\pi$
$271$	17.4350	1.05910	0.529549	+	0.848279i	$0.322361\pi$
$272$	0	0
$273$	−2.94078	−0.177984
$274$	0	0
$275$	3.49287	0.210628
$276$	0	0
$277$	−26.6148	−1.59913	−0.799564	−	0.600581i	$-0.794936\pi$
$277$	−26.6148	−1.59913	−0.799564	+	0.600581i	$0.794936\pi$
$278$	0	0
$279$	−0.171874	−0.0102898
$280$	0	0
$281$	0.417408	0.0249005	0.0124502	−	0.999922i	$-0.496037\pi$
$281$	0.417408	0.0249005	0.0124502	+	0.999922i	$0.496037\pi$
$282$	0	0
$283$	−15.0097	−0.892232	−0.446116	−	0.894975i	$-0.647193\pi$
$283$	−15.0097	−0.892232	−0.446116	+	0.894975i	$0.647193\pi$
$284$	0	0
$285$	5.60907	0.332253
$286$	0	0
$287$	−5.24964	−0.309876
$288$	0	0
$289$	−14.8300	−0.872354
$290$	0	0
$291$	1.06150	0.0622264
$292$	0	0
$293$	29.6270	1.73083	0.865413	−	0.501059i	$-0.167056\pi$
$293$	29.6270	1.73083	0.865413	+	0.501059i	$0.167056\pi$
$294$	0	0
$295$	10.2262	0.595392
$296$	0	0
$297$	3.49287	0.202677
$298$	0	0
$299$	−18.0975	−1.04660
$300$	0	0
$301$	2.23979	0.129099
$302$	0	0
$303$	6.08976	0.349848
$304$	0	0
$305$	8.97755	0.514053
$306$	0	0
$307$	−23.9483	−1.36680	−0.683402	−	0.730042i	$-0.739500\pi$
$307$	−23.9483	−1.36680	−0.683402	+	0.730042i	$0.739500\pi$
$308$	0	0
$309$	14.9586	0.850967
$310$	0	0
$311$	3.17570	0.180078	0.0900388	−	0.995938i	$-0.471301\pi$
$311$	3.17570	0.180078	0.0900388	+	0.995938i	$0.471301\pi$
$312$	0	0
$313$	0.796458	0.0450185	0.0225092	−	0.999747i	$-0.492834\pi$
$313$	0.796458	0.0450185	0.0225092	+	0.999747i	$0.492834\pi$
$314$	0	0
$315$	0.772018	0.0434983
$316$	0	0
$317$	−7.80372	−0.438301	−0.219150	−	0.975691i	$-0.570329\pi$
$317$	−7.80372	−0.438301	−0.219150	+	0.975691i	$0.570329\pi$
$318$	0	0
$319$	−6.91223	−0.387011
$320$	0	0
$321$	8.67550	0.484219
$322$	0	0
$323$	−8.26265	−0.459746
$324$	0	0
$325$	−3.80921	−0.211297
$326$	0	0
$327$	−4.42094	−0.244478
$328$	0	0
$329$	0.514504	0.0283655
$330$	0	0
$331$	30.2554	1.66299	0.831494	−	0.555534i	$-0.187486\pi$
$331$	30.2554	1.66299	0.831494	+	0.555534i	$0.187486\pi$
$332$	0	0
$333$	5.95097	0.326111
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	3.93587	0.214401	0.107200	−	0.994237i	$-0.465811\pi$
$337$	3.93587	0.214401	0.107200	+	0.994237i	$0.465811\pi$
$338$	0	0
$339$	18.6417	1.01248
$340$	0	0
$341$	−0.600335	−0.0325099
$342$	0	0
$343$	−10.3481	−0.558746
$344$	0	0
$345$	4.75097	0.255784
$346$	0	0
$347$	−15.0702	−0.809009	−0.404505	−	0.914536i	$-0.632556\pi$
$347$	−15.0702	−0.809009	−0.404505	+	0.914536i	$0.632556\pi$
$348$	0	0
$349$	30.7710	1.64713	0.823567	−	0.567219i	$-0.191981\pi$
$349$	30.7710	1.64713	0.823567	+	0.567219i	$0.191981\pi$
$350$	0	0
$351$	−3.80921	−0.203321
$352$	0	0
$353$	−3.83282	−0.204000	−0.102000	−	0.994784i	$-0.532524\pi$
$353$	−3.83282	−0.204000	−0.102000	+	0.994784i	$0.532524\pi$
$354$	0	0
$355$	−9.65305	−0.512331
$356$	0	0
$357$	−1.13725	−0.0601896
$358$	0	0
$359$	16.6946	0.881109	0.440554	−	0.897726i	$-0.354782\pi$
$359$	16.6946	0.881109	0.440554	+	0.897726i	$0.354782\pi$
$360$	0	0
$361$	12.4617	0.655877
$362$	0	0
$363$	1.20015	0.0629914
$364$	0	0
$365$	−2.54263	−0.133087
$366$	0	0
$367$	−13.2371	−0.690969	−0.345484	−	0.938425i	$-0.612285\pi$
$367$	−13.2371	−0.690969	−0.345484	+	0.938425i	$0.612285\pi$
$368$	0	0
$369$	−6.79989	−0.353988
$370$	0	0
$371$	1.68327	0.0873913
$372$	0	0
$373$	8.71691	0.451345	0.225672	−	0.974203i	$-0.427542\pi$
$373$	8.71691	0.451345	0.225672	+	0.974203i	$0.427542\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	7.53825	0.388240
$378$	0	0
$379$	−32.6794	−1.67863	−0.839314	−	0.543647i	$-0.817043\pi$
$379$	−32.6794	−1.67863	−0.839314	+	0.543647i	$0.817043\pi$
$380$	0	0
$381$	6.47439	0.331693
$382$	0	0
$383$	−8.33406	−0.425851	−0.212925	−	0.977068i	$-0.568299\pi$
$383$	−8.33406	−0.425851	−0.212925	+	0.977068i	$0.568299\pi$
$384$	0	0
$385$	2.69656	0.137429
$386$	0	0
$387$	2.90121	0.147477
$388$	0	0
$389$	−32.4917	−1.64740	−0.823698	−	0.567029i	$-0.808093\pi$
$389$	−32.4917	−1.64740	−0.823698	+	0.567029i	$0.808093\pi$
$390$	0	0
$391$	−6.99859	−0.353934
$392$	0	0
$393$	−17.5320	−0.884370
$394$	0	0
$395$	6.88724	0.346534
$396$	0	0
$397$	3.60257	0.180808	0.0904038	−	0.995905i	$-0.471184\pi$
$397$	3.60257	0.180808	0.0904038	+	0.995905i	$0.471184\pi$
$398$	0	0
$399$	4.33030	0.216786
$400$	0	0
$401$	−5.11208	−0.255285	−0.127642	−	0.991820i	$-0.540741\pi$
$401$	−5.11208	−0.255285	−0.127642	+	0.991820i	$0.540741\pi$
$402$	0	0
$403$	0.654705	0.0326132
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	20.7860	1.03032
$408$	0	0
$409$	−15.8869	−0.785554	−0.392777	−	0.919634i	$-0.628486\pi$
$409$	−15.8869	−0.785554	−0.392777	+	0.919634i	$0.628486\pi$
$410$	0	0
$411$	−14.2289	−0.701859
$412$	0	0
$413$	7.89480	0.388478
$414$	0	0
$415$	12.9668	0.636517
$416$	0	0
$417$	−22.1873	−1.08652
$418$	0	0
$419$	−6.54463	−0.319726	−0.159863	−	0.987139i	$-0.551105\pi$
$419$	−6.54463	−0.319726	−0.159863	+	0.987139i	$0.551105\pi$
$420$	0	0
$421$	−33.0622	−1.61135	−0.805675	−	0.592357i	$-0.798197\pi$
$421$	−33.0622	−1.61135	−0.805675	+	0.592357i	$0.798197\pi$
$422$	0	0
$423$	0.666440	0.0324034
$424$	0	0
$425$	−1.47309	−0.0714552
$426$	0	0
$427$	6.93083	0.335406
$428$	0	0
$429$	−13.3051	−0.642376
$430$	0	0
$431$	−4.91115	−0.236562	−0.118281	−	0.992980i	$-0.537738\pi$
$431$	−4.91115	−0.236562	−0.118281	+	0.992980i	$0.537738\pi$
$432$	0	0
$433$	7.45669	0.358346	0.179173	−	0.983818i	$-0.442658\pi$
$433$	7.45669	0.358346	0.179173	+	0.983818i	$0.442658\pi$
$434$	0	0
$435$	−1.97895	−0.0948836
$436$	0	0
$437$	26.6485	1.27477
$438$	0	0
$439$	−26.1708	−1.24907	−0.624533	−	0.780999i	$-0.714710\pi$
$439$	−26.1708	−1.24907	−0.624533	+	0.780999i	$0.714710\pi$
$440$	0	0
$441$	−6.40399	−0.304952
$442$	0	0
$443$	16.3544	0.777019	0.388509	−	0.921445i	$-0.372990\pi$
$443$	16.3544	0.777019	0.388509	+	0.921445i	$0.372990\pi$
$444$	0	0
$445$	−15.2122	−0.721128
$446$	0	0
$447$	19.2572	0.910834
$448$	0	0
$449$	−38.8516	−1.83352	−0.916760	−	0.399438i	$-0.869205\pi$
$449$	−38.8516	−1.83352	−0.916760	+	0.399438i	$0.869205\pi$
$450$	0	0
$451$	−23.7512	−1.11840
$452$	0	0
$453$	−19.3020	−0.906886
$454$	0	0
$455$	−2.94078	−0.137866
$456$	0	0
$457$	0.434246	0.0203132	0.0101566	−	0.999948i	$-0.496767\pi$
$457$	0.434246	0.0203132	0.0101566	+	0.999948i	$0.496767\pi$
$458$	0	0
$459$	−1.47309	−0.0687578
$460$	0	0
$461$	30.2045	1.40676	0.703382	−	0.710812i	$-0.251672\pi$
$461$	30.2045	1.40676	0.703382	+	0.710812i	$0.251672\pi$
$462$	0	0
$463$	−19.8061	−0.920469	−0.460235	−	0.887797i	$-0.652235\pi$
$463$	−19.8061	−0.920469	−0.460235	+	0.887797i	$0.652235\pi$
$464$	0	0
$465$	−0.171874	−0.00797048
$466$	0	0
$467$	−23.4621	−1.08570	−0.542848	−	0.839831i	$-0.682654\pi$
$467$	−23.4621	−1.08570	−0.542848	+	0.839831i	$0.682654\pi$
$468$	0	0
$469$	−0.772018	−0.0356485
$470$	0	0
$471$	3.80650	0.175394
$472$	0	0
$473$	10.1336	0.465941
$474$	0	0
$475$	5.60907	0.257362
$476$	0	0
$477$	2.18036	0.0998317
$478$	0	0
$479$	9.57726	0.437596	0.218798	−	0.975770i	$-0.429786\pi$
$479$	9.57726	0.437596	0.218798	+	0.975770i	$0.429786\pi$
$480$	0	0
$481$	−22.6685	−1.03359
$482$	0	0
$483$	3.66783	0.166892
$484$	0	0
$485$	1.06150	0.0482003
$486$	0	0
$487$	−21.0897	−0.955664	−0.477832	−	0.878451i	$-0.658577\pi$
$487$	−21.0897	−0.955664	−0.477832	+	0.878451i	$0.658577\pi$
$488$	0	0
$489$	−18.0767	−0.817457
$490$	0	0
$491$	−5.41768	−0.244497	−0.122248	−	0.992500i	$-0.539010\pi$
$491$	−5.41768	−0.244497	−0.122248	+	0.992500i	$0.539010\pi$
$492$	0	0
$493$	2.91517	0.131293
$494$	0	0
$495$	3.49287	0.156993
$496$	0	0
$497$	−7.45233	−0.334283
$498$	0	0
$499$	36.7696	1.64603	0.823017	−	0.568016i	$-0.192289\pi$
$499$	36.7696	1.64603	0.823017	+	0.568016i	$0.192289\pi$
$500$	0	0
$501$	19.5459	0.873247
$502$	0	0
$503$	37.4158	1.66829	0.834145	−	0.551546i	$-0.185962\pi$
$503$	37.4158	1.66829	0.834145	+	0.551546i	$0.185962\pi$
$504$	0	0
$505$	6.08976	0.270991
$506$	0	0
$507$	1.51010	0.0670657
$508$	0	0
$509$	0.820242	0.0363566	0.0181783	−	0.999835i	$-0.494213\pi$
$509$	0.820242	0.0363566	0.0181783	+	0.999835i	$0.494213\pi$
$510$	0	0
$511$	−1.96295	−0.0868360
$512$	0	0
$513$	5.60907	0.247647
$514$	0	0
$515$	14.9586	0.659156
$516$	0	0
$517$	2.32779	0.102376
$518$	0	0
$519$	21.3975	0.939247
$520$	0	0
$521$	21.0460	0.922042	0.461021	−	0.887389i	$-0.347483\pi$
$521$	21.0460	0.922042	0.461021	+	0.887389i	$0.347483\pi$
$522$	0	0
$523$	25.9849	1.13624	0.568120	−	0.822946i	$-0.307671\pi$
$523$	25.9849	1.13624	0.568120	+	0.822946i	$0.307671\pi$
$524$	0	0
$525$	0.772018	0.0336936
$526$	0	0
$527$	0.253186	0.0110289
$528$	0	0
$529$	−0.428270	−0.0186204
$530$	0	0
$531$	10.2262	0.443779
$532$	0	0
$533$	25.9022	1.12195
$534$	0	0
$535$	8.67550	0.375075
$536$	0	0
$537$	0.270379	0.0116677
$538$	0	0
$539$	−22.3683	−0.963471
$540$	0	0
$541$	9.27257	0.398659	0.199330	−	0.979933i	$-0.436124\pi$
$541$	9.27257	0.398659	0.199330	+	0.979933i	$0.436124\pi$
$542$	0	0
$543$	2.43492	0.104492
$544$	0	0
$545$	−4.42094	−0.189372
$546$	0	0
$547$	19.6994	0.842284	0.421142	−	0.906995i	$-0.361629\pi$
$547$	19.6994	0.842284	0.421142	+	0.906995i	$0.361629\pi$
$548$	0	0
$549$	8.97755	0.383152
$550$	0	0
$551$	−11.1001	−0.472880
$552$	0	0
$553$	5.31707	0.226105
$554$	0	0
$555$	5.95097	0.252605
$556$	0	0
$557$	−39.2284	−1.66216	−0.831081	−	0.556151i	$-0.812278\pi$
$557$	−39.2284	−1.66216	−0.831081	+	0.556151i	$0.812278\pi$
$558$	0	0
$559$	−11.0513	−0.467421
$560$	0	0
$561$	−5.14530	−0.217235
$562$	0	0
$563$	−19.5646	−0.824552	−0.412276	−	0.911059i	$-0.635266\pi$
$563$	−19.5646	−0.824552	−0.412276	+	0.911059i	$0.635266\pi$
$564$	0	0
$565$	18.6417	0.784264
$566$	0	0
$567$	0.772018	0.0324217
$568$	0	0
$569$	−45.3427	−1.90087	−0.950433	−	0.310928i	$-0.899360\pi$
$569$	−45.3427	−1.90087	−0.950433	+	0.310928i	$0.899360\pi$
$570$	0	0
$571$	−25.8531	−1.08192	−0.540958	−	0.841049i	$-0.681938\pi$
$571$	−25.8531	−1.08192	−0.540958	+	0.841049i	$0.681938\pi$
$572$	0	0
$573$	−11.3834	−0.475547
$574$	0	0
$575$	4.75097	0.198129
$576$	0	0
$577$	27.9223	1.16242	0.581210	−	0.813754i	$-0.302580\pi$
$577$	27.9223	1.16242	0.581210	+	0.813754i	$0.302580\pi$
$578$	0	0
$579$	7.72764	0.321150
$580$	0	0
$581$	10.0106	0.415311
$582$	0	0
$583$	7.61571	0.315410
$584$	0	0
$585$	−3.80921	−0.157492
$586$	0	0
$587$	10.9344	0.451310	0.225655	−	0.974207i	$-0.427548\pi$
$587$	10.9344	0.451310	0.225655	+	0.974207i	$0.427548\pi$
$588$	0	0
$589$	−0.964055	−0.0397232
$590$	0	0
$591$	−11.9110	−0.489953
$592$	0	0
$593$	35.0801	1.44057	0.720284	−	0.693680i	$-0.244012\pi$
$593$	35.0801	1.44057	0.720284	+	0.693680i	$0.244012\pi$
$594$	0	0
$595$	−1.13725	−0.0466227
$596$	0	0
$597$	−0.345159	−0.0141264
$598$	0	0
$599$	1.23267	0.0503656	0.0251828	−	0.999683i	$-0.491983\pi$
$599$	1.23267	0.0503656	0.0251828	+	0.999683i	$0.491983\pi$
$600$	0	0
$601$	−20.1162	−0.820556	−0.410278	−	0.911960i	$-0.634568\pi$
$601$	−20.1162	−0.820556	−0.410278	+	0.911960i	$0.634568\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	1.20015	0.0487929
$606$	0	0
$607$	−39.9333	−1.62084	−0.810421	−	0.585848i	$-0.800761\pi$
$607$	−39.9333	−1.62084	−0.810421	+	0.585848i	$0.800761\pi$
$608$	0	0
$609$	−1.52779	−0.0619091
$610$	0	0
$611$	−2.53861	−0.102701
$612$	0	0
$613$	−14.0246	−0.566450	−0.283225	−	0.959053i	$-0.591404\pi$
$613$	−14.0246	−0.566450	−0.283225	+	0.959053i	$0.591404\pi$
$614$	0	0
$615$	−6.79989	−0.274198
$616$	0	0
$617$	5.58743	0.224942	0.112471	−	0.993655i	$-0.464124\pi$
$617$	5.58743	0.224942	0.112471	+	0.993655i	$0.464124\pi$
$618$	0	0
$619$	13.1025	0.526635	0.263317	−	0.964709i	$-0.415183\pi$
$619$	13.1025	0.526635	0.263317	+	0.964709i	$0.415183\pi$
$620$	0	0
$621$	4.75097	0.190650
$622$	0	0
$623$	−11.7441	−0.470518
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	19.5918	0.782419
$628$	0	0
$629$	−8.76629	−0.349535
$630$	0	0
$631$	−33.1488	−1.31963	−0.659816	−	0.751427i	$-0.729366\pi$
$631$	−33.1488	−1.31963	−0.659816	+	0.751427i	$0.729366\pi$
$632$	0	0
$633$	19.5809	0.778270
$634$	0	0
$635$	6.47439	0.256928
$636$	0	0
$637$	24.3941	0.966531
$638$	0	0
$639$	−9.65305	−0.381869
$640$	0	0
$641$	−6.08990	−0.240537	−0.120268	−	0.992741i	$-0.538375\pi$
$641$	−6.08990	−0.240537	−0.120268	+	0.992741i	$0.538375\pi$
$642$	0	0
$643$	8.23162	0.324624	0.162312	−	0.986740i	$-0.448105\pi$
$643$	8.23162	0.324624	0.162312	+	0.986740i	$0.448105\pi$
$644$	0	0
$645$	2.90121	0.114235
$646$	0	0
$647$	−43.8263	−1.72299	−0.861495	−	0.507767i	$-0.830471\pi$
$647$	−43.8263	−1.72299	−0.861495	+	0.507767i	$0.830471\pi$
$648$	0	0
$649$	35.7188	1.40208
$650$	0	0
$651$	−0.132690	−0.00520053
$652$	0	0
$653$	−8.04576	−0.314855	−0.157428	−	0.987531i	$-0.550320\pi$
$653$	−8.04576	−0.314855	−0.157428	+	0.987531i	$0.550320\pi$
$654$	0	0
$655$	−17.5320	−0.685030
$656$	0	0
$657$	−2.54263	−0.0991974
$658$	0	0
$659$	18.9610	0.738614	0.369307	−	0.929307i	$-0.379595\pi$
$659$	18.9610	0.738614	0.369307	+	0.929307i	$0.379595\pi$
$660$	0	0
$661$	−45.5682	−1.77240	−0.886199	−	0.463304i	$-0.846664\pi$
$661$	−45.5682	−1.77240	−0.886199	+	0.463304i	$0.846664\pi$
$662$	0	0
$663$	5.61130	0.217925
$664$	0	0
$665$	4.33030	0.167922
$666$	0	0
$667$	−9.40195	−0.364045
$668$	0	0
$669$	−3.57499	−0.138217
$670$	0	0
$671$	31.3574	1.21054
$672$	0	0
$673$	14.4226	0.555951	0.277976	−	0.960588i	$-0.410337\pi$
$673$	14.4226	0.555951	0.277976	+	0.960588i	$0.410337\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	4.31691	0.165912	0.0829561	−	0.996553i	$-0.473564\pi$
$677$	4.31691	0.165912	0.0829561	+	0.996553i	$0.473564\pi$
$678$	0	0
$679$	0.819498	0.0314495
$680$	0	0
$681$	−18.8309	−0.721601
$682$	0	0
$683$	−3.53346	−0.135204	−0.0676020	−	0.997712i	$-0.521535\pi$
$683$	−3.53346	−0.135204	−0.0676020	+	0.997712i	$0.521535\pi$
$684$	0	0
$685$	−14.2289	−0.543658
$686$	0	0
$687$	−9.08710	−0.346694
$688$	0	0
$689$	−8.30544	−0.316412
$690$	0	0
$691$	−11.1252	−0.423224	−0.211612	−	0.977354i	$-0.567871\pi$
$691$	−11.1252	−0.423224	−0.211612	+	0.977354i	$0.567871\pi$
$692$	0	0
$693$	2.69656	0.102434
$694$	0	0
$695$	−22.1873	−0.841614
$696$	0	0
$697$	10.0168	0.379415
$698$	0	0
$699$	−0.280150	−0.0105962
$700$	0	0
$701$	44.2127	1.66989	0.834946	−	0.550332i	$-0.185499\pi$
$701$	44.2127	1.66989	0.834946	+	0.550332i	$0.185499\pi$
$702$	0	0
$703$	33.3794	1.25893
$704$	0	0
$705$	0.666440	0.0250996
$706$	0	0
$707$	4.70140	0.176814
$708$	0	0
$709$	−3.44566	−0.129405	−0.0647023	−	0.997905i	$-0.520610\pi$
$709$	−3.44566	−0.129405	−0.0647023	+	0.997905i	$0.520610\pi$
$710$	0	0
$711$	6.88724	0.258292
$712$	0	0
$713$	−0.816570	−0.0305808
$714$	0	0
$715$	−13.3051	−0.497582
$716$	0	0
$717$	21.4729	0.801920
$718$	0	0
$719$	−5.51182	−0.205556	−0.102778	−	0.994704i	$-0.532773\pi$
$719$	−5.51182	−0.205556	−0.102778	+	0.994704i	$0.532773\pi$
$720$	0	0
$721$	11.5483	0.430082
$722$	0	0
$723$	−8.38206	−0.311732
$724$	0	0
$725$	−1.97895	−0.0734965
$726$	0	0
$727$	25.0730	0.929905	0.464953	−	0.885336i	$-0.346071\pi$
$727$	25.0730	0.929905	0.464953	+	0.885336i	$0.346071\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.27373	−0.158070
$732$	0	0
$733$	24.4533	0.903204	0.451602	−	0.892220i	$-0.350853\pi$
$733$	24.4533	0.903204	0.451602	+	0.892220i	$0.350853\pi$
$734$	0	0
$735$	−6.40399	−0.236215
$736$	0	0
$737$	−3.49287	−0.128662
$738$	0	0
$739$	−35.2777	−1.29771	−0.648856	−	0.760911i	$-0.724752\pi$
$739$	−35.2777	−1.29771	−0.648856	+	0.760911i	$0.724752\pi$
$740$	0	0
$741$	−21.3661	−0.784905
$742$	0	0
$743$	−12.1041	−0.444056	−0.222028	−	0.975040i	$-0.571268\pi$
$743$	−12.1041	−0.444056	−0.222028	+	0.975040i	$0.571268\pi$
$744$	0	0
$745$	19.2572	0.705529
$746$	0	0
$747$	12.9668	0.474431
$748$	0	0
$749$	6.69764	0.244726
$750$	0	0
$751$	−37.5005	−1.36841	−0.684206	−	0.729289i	$-0.739851\pi$
$751$	−37.5005	−1.36841	−0.684206	+	0.729289i	$0.739851\pi$
$752$	0	0
$753$	16.7462	0.610266
$754$	0	0
$755$	−19.3020	−0.702471
$756$	0	0
$757$	−33.8803	−1.23140	−0.615700	−	0.787980i	$-0.711127\pi$
$757$	−33.8803	−1.23140	−0.615700	+	0.787980i	$0.711127\pi$
$758$	0	0
$759$	16.5945	0.602343
$760$	0	0
$761$	−41.3139	−1.49763	−0.748814	−	0.662780i	$-0.769377\pi$
$761$	−41.3139	−1.49763	−0.748814	+	0.662780i	$0.769377\pi$
$762$	0	0
$763$	−3.41304	−0.123560
$764$	0	0
$765$	−1.47309	−0.0532596
$766$	0	0
$767$	−38.9537	−1.40654
$768$	0	0
$769$	26.0009	0.937615	0.468807	−	0.883300i	$-0.344684\pi$
$769$	26.0009	0.937615	0.468807	+	0.883300i	$0.344684\pi$
$770$	0	0
$771$	15.0090	0.540535
$772$	0	0
$773$	−5.08729	−0.182977	−0.0914886	−	0.995806i	$-0.529163\pi$
$773$	−5.08729	−0.182977	−0.0914886	+	0.995806i	$0.529163\pi$
$774$	0	0
$775$	−0.171874	−0.00617391
$776$	0	0
$777$	4.59425	0.164818
$778$	0	0
$779$	−38.1411	−1.36655
$780$	0	0
$781$	−33.7169	−1.20648
$782$	0	0
$783$	−1.97895	−0.0707220
$784$	0	0
$785$	3.80650	0.135860
$786$	0	0
$787$	−47.8019	−1.70395	−0.851977	−	0.523580i	$-0.824596\pi$
$787$	−47.8019	−1.70395	−0.851977	+	0.523580i	$0.824596\pi$
$788$	0	0
$789$	4.86798	0.173305
$790$	0	0
$791$	14.3918	0.511712
$792$	0	0
$793$	−34.1974	−1.21438
$794$	0	0
$795$	2.18036	0.0773293
$796$	0	0
$797$	−10.8864	−0.385617	−0.192808	−	0.981236i	$-0.561760\pi$
$797$	−10.8864	−0.385617	−0.192808	+	0.981236i	$0.561760\pi$
$798$	0	0
$799$	−0.981724	−0.0347309
$800$	0	0
$801$	−15.2122	−0.537497
$802$	0	0
$803$	−8.88107	−0.313406
$804$	0	0
$805$	3.66783	0.129274
$806$	0	0
$807$	11.1049	0.390912
$808$	0	0
$809$	−22.8878	−0.804693	−0.402346	−	0.915488i	$-0.631805\pi$
$809$	−22.8878	−0.804693	−0.402346	+	0.915488i	$0.631805\pi$
$810$	0	0
$811$	−44.4387	−1.56045	−0.780227	−	0.625496i	$-0.784896\pi$
$811$	−44.4387	−1.56045	−0.780227	+	0.625496i	$0.784896\pi$
$812$	0	0
$813$	17.4350	0.611471
$814$	0	0
$815$	−18.0767	−0.633199
$816$	0	0
$817$	16.2731	0.569324
$818$	0	0
$819$	−2.94078	−0.102759
$820$	0	0
$821$	7.47421	0.260852	0.130426	−	0.991458i	$-0.458366\pi$
$821$	7.47421	0.260852	0.130426	+	0.991458i	$0.458366\pi$
$822$	0	0
$823$	−34.8092	−1.21337	−0.606686	−	0.794942i	$-0.707501\pi$
$823$	−34.8092	−1.21337	−0.606686	+	0.794942i	$0.707501\pi$
$824$	0	0
$825$	3.49287	0.121606
$826$	0	0
$827$	7.05105	0.245189	0.122595	−	0.992457i	$-0.460879\pi$
$827$	7.05105	0.245189	0.122595	+	0.992457i	$0.460879\pi$
$828$	0	0
$829$	−14.8798	−0.516796	−0.258398	−	0.966038i	$-0.583195\pi$
$829$	−14.8798	−0.516796	−0.258398	+	0.966038i	$0.583195\pi$
$830$	0	0
$831$	−26.6148	−0.923257
$832$	0	0
$833$	9.43363	0.326856
$834$	0	0
$835$	19.5459	0.676414
$836$	0	0
$837$	−0.171874	−0.00594084
$838$	0	0
$839$	9.99371	0.345021	0.172511	−	0.985008i	$-0.444812\pi$
$839$	9.99371	0.345021	0.172511	+	0.985008i	$0.444812\pi$
$840$	0	0
$841$	−25.0837	−0.864957
$842$	0	0
$843$	0.417408	0.0143763
$844$	0	0
$845$	1.51010	0.0519489
$846$	0	0
$847$	0.926535	0.0318361
$848$	0	0
$849$	−15.0097	−0.515130
$850$	0	0
$851$	28.2729	0.969182
$852$	0	0
$853$	−18.8227	−0.644477	−0.322239	−	0.946658i	$-0.604435\pi$
$853$	−18.8227	−0.644477	−0.322239	+	0.946658i	$0.604435\pi$
$854$	0	0
$855$	5.60907	0.191826
$856$	0	0
$857$	12.8679	0.439560	0.219780	−	0.975549i	$-0.429466\pi$
$857$	12.8679	0.439560	0.219780	+	0.975549i	$0.429466\pi$
$858$	0	0
$859$	−35.0845	−1.19707	−0.598534	−	0.801097i	$-0.704250\pi$
$859$	−35.0845	−1.19707	−0.598534	+	0.801097i	$0.704250\pi$
$860$	0	0
$861$	−5.24964	−0.178907
$862$	0	0
$863$	38.6283	1.31492	0.657461	−	0.753489i	$-0.271631\pi$
$863$	38.6283	1.31492	0.657461	+	0.753489i	$0.271631\pi$
$864$	0	0
$865$	21.3975	0.727537
$866$	0	0
$867$	−14.8300	−0.503654
$868$	0	0
$869$	24.0562	0.816052
$870$	0	0
$871$	3.80921	0.129070
$872$	0	0
$873$	1.06150	0.0359264
$874$	0	0
$875$	0.772018	0.0260990
$876$	0	0
$877$	2.11218	0.0713232	0.0356616	−	0.999364i	$-0.488646\pi$
$877$	2.11218	0.0713232	0.0356616	+	0.999364i	$0.488646\pi$
$878$	0	0
$879$	29.6270	0.999293
$880$	0	0
$881$	−18.0148	−0.606935	−0.303468	−	0.952842i	$-0.598144\pi$
$881$	−18.0148	−0.606935	−0.303468	+	0.952842i	$0.598144\pi$
$882$	0	0
$883$	−10.3114	−0.347005	−0.173502	−	0.984833i	$-0.555508\pi$
$883$	−10.3114	−0.347005	−0.173502	+	0.984833i	$0.555508\pi$
$884$	0	0
$885$	10.2262	0.343750
$886$	0	0
$887$	46.9685	1.57705	0.788524	−	0.615003i	$-0.210845\pi$
$887$	46.9685	1.57705	0.788524	+	0.615003i	$0.210845\pi$
$888$	0	0
$889$	4.99834	0.167639
$890$	0	0
$891$	3.49287	0.117016
$892$	0	0
$893$	3.73811	0.125091
$894$	0	0
$895$	0.270379	0.00903779
$896$	0	0
$897$	−18.0975	−0.604257
$898$	0	0
$899$	0.340131	0.0113440
$900$	0	0
$901$	−3.21186	−0.107002
$902$	0	0
$903$	2.23979	0.0745354
$904$	0	0
$905$	2.43492	0.0809394
$906$	0	0
$907$	−59.7785	−1.98491	−0.992457	−	0.122593i	$-0.960879\pi$
$907$	−59.7785	−1.98491	−0.992457	+	0.122593i	$0.960879\pi$
$908$	0	0
$909$	6.08976	0.201985
$910$	0	0
$911$	−23.1299	−0.766329	−0.383164	−	0.923680i	$-0.625166\pi$
$911$	−23.1299	−0.766329	−0.383164	+	0.923680i	$0.625166\pi$
$912$	0	0
$913$	45.2915	1.49893
$914$	0	0
$915$	8.97755	0.296789
$916$	0	0
$917$	−13.5350	−0.446965
$918$	0	0
$919$	15.7603	0.519886	0.259943	−	0.965624i	$-0.416296\pi$
$919$	15.7603	0.519886	0.259943	+	0.965624i	$0.416296\pi$
$920$	0	0
$921$	−23.9483	−0.789125
$922$	0	0
$923$	36.7705	1.21032
$924$	0	0
$925$	5.95097	0.195667
$926$	0	0
$927$	14.9586	0.491306
$928$	0	0
$929$	35.4115	1.16181	0.580907	−	0.813970i	$-0.302698\pi$
$929$	35.4115	1.16181	0.580907	+	0.813970i	$0.302698\pi$
$930$	0	0
$931$	−35.9204	−1.17724
$932$	0	0
$933$	3.17570	0.103968
$934$	0	0
$935$	−5.14530	−0.168269
$936$	0	0
$937$	−11.5869	−0.378528	−0.189264	−	0.981926i	$-0.560610\pi$
$937$	−11.5869	−0.378528	−0.189264	+	0.981926i	$0.560610\pi$
$938$	0	0
$939$	0.796458	0.0259914
$940$	0	0
$941$	−28.3078	−0.922808	−0.461404	−	0.887190i	$-0.652654\pi$
$941$	−28.3078	−0.922808	−0.461404	+	0.887190i	$0.652654\pi$
$942$	0	0
$943$	−32.3061	−1.05203
$944$	0	0
$945$	0.772018	0.0251137
$946$	0	0
$947$	49.8955	1.62139	0.810693	−	0.585472i	$-0.199091\pi$
$947$	49.8955	1.62139	0.810693	+	0.585472i	$0.199091\pi$
$948$	0	0
$949$	9.68541	0.314402
$950$	0	0
$951$	−7.80372	−0.253053
$952$	0	0
$953$	13.1134	0.424784	0.212392	−	0.977184i	$-0.431875\pi$
$953$	13.1134	0.424784	0.212392	+	0.977184i	$0.431875\pi$
$954$	0	0
$955$	−11.3834	−0.368357
$956$	0	0
$957$	−6.91223	−0.223441
$958$	0	0
$959$	−10.9850	−0.354723
$960$	0	0
$961$	−30.9705	−0.999047
$962$	0	0
$963$	8.67550	0.279564
$964$	0	0
$965$	7.72764	0.248762
$966$	0	0
$967$	−38.5613	−1.24005	−0.620023	−	0.784583i	$-0.712877\pi$
$967$	−38.5613	−1.24005	−0.620023	+	0.784583i	$0.712877\pi$
$968$	0	0
$969$	−8.26265	−0.265434
$970$	0	0
$971$	−48.8434	−1.56746	−0.783730	−	0.621102i	$-0.786685\pi$
$971$	−48.8434	−1.56746	−0.783730	+	0.621102i	$0.786685\pi$
$972$	0	0
$973$	−17.1290	−0.549131
$974$	0	0
$975$	−3.80921	−0.121992
$976$	0	0
$977$	−2.47404	−0.0791517	−0.0395758	−	0.999217i	$-0.512601\pi$
$977$	−2.47404	−0.0791517	−0.0395758	+	0.999217i	$0.512601\pi$
$978$	0	0
$979$	−53.1343	−1.69818
$980$	0	0
$981$	−4.42094	−0.141150
$982$	0	0
$983$	55.1656	1.75951	0.879754	−	0.475429i	$-0.157707\pi$
$983$	55.1656	1.75951	0.879754	+	0.475429i	$0.157707\pi$
$984$	0	0
$985$	−11.9110	−0.379516
$986$	0	0
$987$	0.514504	0.0163768
$988$	0	0
$989$	13.7836	0.438292
$990$	0	0
$991$	−2.86512	−0.0910136	−0.0455068	−	0.998964i	$-0.514490\pi$
$991$	−2.86512	−0.0910136	−0.0455068	+	0.998964i	$0.514490\pi$
$992$	0	0
$993$	30.2554	0.960127
$994$	0	0
$995$	−0.345159	−0.0109423
$996$	0	0
$997$	−9.19937	−0.291347	−0.145673	−	0.989333i	$-0.546535\pi$
$997$	−9.19937	−0.291347	−0.145673	+	0.989333i	$0.546535\pi$
$998$	0	0
$999$	5.95097	0.188280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.i.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.i.1.4	✓	7	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +0.772018 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +0.772018 q^{7} +1.00000 q^{9} +3.49287 q^{11} -3.80921 q^{13} +1.00000 q^{15} -1.47309 q^{17} +5.60907 q^{19} +0.772018 q^{21} +4.75097 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.97895 q^{29} -0.171874 q^{31} +3.49287 q^{33} +0.772018 q^{35} +5.95097 q^{37} -3.80921 q^{39} -6.79989 q^{41} +2.90121 q^{43} +1.00000 q^{45} +0.666440 q^{47} -6.40399 q^{49} -1.47309 q^{51} +2.18036 q^{53} +3.49287 q^{55} +5.60907 q^{57} +10.2262 q^{59} +8.97755 q^{61} +0.772018 q^{63} -3.80921 q^{65} -1.00000 q^{67} +4.75097 q^{69} -9.65305 q^{71} -2.54263 q^{73} +1.00000 q^{75} +2.69656 q^{77} +6.88724 q^{79} +1.00000 q^{81} +12.9668 q^{83} -1.47309 q^{85} -1.97895 q^{87} -15.2122 q^{89} -2.94078 q^{91} -0.171874 q^{93} +5.60907 q^{95} +1.06150 q^{97} +3.49287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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