LMFDB - Embedded newform 6160.2.a.bj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 385)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21432 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.90321 q^{9} +O(q^{10})$	$q+2.21432 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.90321 q^{9} -1.00000 q^{11} -2.21432 q^{13} +2.21432 q^{15} +4.21432 q^{17} -7.80642 q^{19} -2.21432 q^{21} -2.90321 q^{23} +1.00000 q^{25} -2.42864 q^{27} +0.755569 q^{29} -7.11753 q^{31} -2.21432 q^{33} -1.00000 q^{35} -6.28100 q^{37} -4.90321 q^{39} -5.54617 q^{41} +4.14764 q^{43} +1.90321 q^{45} -1.03011 q^{47} +1.00000 q^{49} +9.33185 q^{51} -6.57628 q^{53} -1.00000 q^{55} -17.2859 q^{57} -2.68889 q^{59} +8.79060 q^{61} -1.90321 q^{63} -2.21432 q^{65} -1.52543 q^{67} -6.42864 q^{69} -7.61285 q^{71} -12.2143 q^{73} +2.21432 q^{75} +1.00000 q^{77} -3.52543 q^{79} -11.0874 q^{81} +2.13335 q^{83} +4.21432 q^{85} +1.67307 q^{87} +17.2859 q^{89} +2.21432 q^{91} -15.7605 q^{93} -7.80642 q^{95} +5.28592 q^{97} -1.90321 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q + 3 * q^5 - 3 * q^7 - q^9	$ 3 q + 3 q^{5} - 3 q^{7} - q^{9} - 3 q^{11} + 6 q^{17} - 10 q^{19} - 2 q^{23} + 3 q^{25} + 6 q^{27} + 2 q^{29} - 8 q^{31} - 3 q^{35} - 12 q^{37} - 8 q^{39} + 10 q^{41} + 6 q^{43} - q^{45} - 10 q^{47} + 3 q^{49} + 8 q^{51} - 3 q^{55} - 12 q^{57} - 8 q^{59} + q^{63} + 2 q^{67} - 6 q^{69} + 4 q^{71} - 30 q^{73} + 3 q^{77} - 4 q^{79} - 13 q^{81} + 6 q^{83} + 6 q^{85} - 8 q^{87} + 12 q^{89} - 14 q^{93} - 10 q^{95} - 24 q^{97} + q^{99}+O(q^{100}) $ 3 * q + 3 * q^5 - 3 * q^7 - q^9 - 3 * q^11 + 6 * q^17 - 10 * q^19 - 2 * q^23 + 3 * q^25 + 6 * q^27 + 2 * q^29 - 8 * q^31 - 3 * q^35 - 12 * q^37 - 8 * q^39 + 10 * q^41 + 6 * q^43 - q^45 - 10 * q^47 + 3 * q^49 + 8 * q^51 - 3 * q^55 - 12 * q^57 - 8 * q^59 + q^63 + 2 * q^67 - 6 * q^69 + 4 * q^71 - 30 * q^73 + 3 * q^77 - 4 * q^79 - 13 * q^81 + 6 * q^83 + 6 * q^85 - 8 * q^87 + 12 * q^89 - 14 * q^93 - 10 * q^95 - 24 * q^97 + 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
$3$	2.21432	1.27844	0.639219	−	0.769025i	$-0.279258\pi$
$3$	2.21432	1.27844	0.639219	+	0.769025i	$0.279258\pi$
$4$	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
$9$	1.90321	0.634404
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.21432	−0.614142	−0.307071	−	0.951687i	$-0.599349\pi$
$13$	−2.21432	−0.614142	−0.307071	+	0.951687i	$0.599349\pi$
$14$	0	0
$15$	2.21432	0.571735
$16$	0	0
$17$	4.21432	1.02212	0.511061	−	0.859544i	$-0.329252\pi$
$17$	4.21432	1.02212	0.511061	+	0.859544i	$0.329252\pi$
$18$	0	0
$19$	−7.80642	−1.79092	−0.895458	−	0.445146i	$-0.853152\pi$
$19$	−7.80642	−1.79092	−0.895458	+	0.445146i	$0.853152\pi$
$20$	0	0
$21$	−2.21432	−0.483204
$22$	0	0
$23$	−2.90321	−0.605362	−0.302681	−	0.953092i	$-0.597882\pi$
$23$	−2.90321	−0.605362	−0.302681	+	0.953092i	$0.597882\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.42864	−0.467392
$28$	0	0
$29$	0.755569	0.140306	0.0701528	−	0.997536i	$-0.477651\pi$
$29$	0.755569	0.140306	0.0701528	+	0.997536i	$0.477651\pi$
$30$	0	0
$31$	−7.11753	−1.27835	−0.639173	−	0.769063i	$-0.720723\pi$
$31$	−7.11753	−1.27835	−0.639173	+	0.769063i	$0.720723\pi$
$32$	0	0
$33$	−2.21432	−0.385464
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.28100	−1.03259	−0.516295	−	0.856411i	$-0.672689\pi$
$37$	−6.28100	−1.03259	−0.516295	+	0.856411i	$0.672689\pi$
$38$	0	0
$39$	−4.90321	−0.785142
$40$	0	0
$41$	−5.54617	−0.866166	−0.433083	−	0.901354i	$-0.642574\pi$
$41$	−5.54617	−0.866166	−0.433083	+	0.901354i	$0.642574\pi$
$42$	0	0
$43$	4.14764	0.632510	0.316255	−	0.948674i	$-0.397575\pi$
$43$	4.14764	0.632510	0.316255	+	0.948674i	$0.397575\pi$
$44$	0	0
$45$	1.90321	0.283714
$46$	0	0
$47$	−1.03011	−0.150257	−0.0751286	−	0.997174i	$-0.523937\pi$
$47$	−1.03011	−0.150257	−0.0751286	+	0.997174i	$0.523937\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.33185	1.30672
$52$	0	0
$53$	−6.57628	−0.903322	−0.451661	−	0.892190i	$-0.649168\pi$
$53$	−6.57628	−0.903322	−0.451661	+	0.892190i	$0.649168\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−17.2859	−2.28958
$58$	0	0
$59$	−2.68889	−0.350064	−0.175032	−	0.984563i	$-0.556003\pi$
$59$	−2.68889	−0.350064	−0.175032	+	0.984563i	$0.556003\pi$
$60$	0	0
$61$	8.79060	1.12552	0.562761	−	0.826620i	$-0.309739\pi$
$61$	8.79060	1.12552	0.562761	+	0.826620i	$0.309739\pi$
$62$	0	0
$63$	−1.90321	−0.239782
$64$	0	0
$65$	−2.21432	−0.274653
$66$	0	0
$67$	−1.52543	−0.186361	−0.0931803	−	0.995649i	$-0.529703\pi$
$67$	−1.52543	−0.186361	−0.0931803	+	0.995649i	$0.529703\pi$
$68$	0	0
$69$	−6.42864	−0.773917
$70$	0	0
$71$	−7.61285	−0.903479	−0.451739	−	0.892150i	$-0.649196\pi$
$71$	−7.61285	−0.903479	−0.451739	+	0.892150i	$0.649196\pi$
$72$	0	0
$73$	−12.2143	−1.42958	−0.714789	−	0.699340i	$-0.753477\pi$
$73$	−12.2143	−1.42958	−0.714789	+	0.699340i	$0.753477\pi$
$74$	0	0
$75$	2.21432	0.255688
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−3.52543	−0.396642	−0.198321	−	0.980137i	$-0.563549\pi$
$79$	−3.52543	−0.396642	−0.198321	+	0.980137i	$0.563549\pi$
$80$	0	0
$81$	−11.0874	−1.23194
$82$	0	0
$83$	2.13335	0.234166	0.117083	−	0.993122i	$-0.462646\pi$
$83$	2.13335	0.234166	0.117083	+	0.993122i	$0.462646\pi$
$84$	0	0
$85$	4.21432	0.457107
$86$	0	0
$87$	1.67307	0.179372
$88$	0	0
$89$	17.2859	1.83230	0.916152	−	0.400831i	$-0.131279\pi$
$89$	17.2859	1.83230	0.916152	+	0.400831i	$0.131279\pi$
$90$	0	0
$91$	2.21432	0.232124
$92$	0	0
$93$	−15.7605	−1.63429
$94$	0	0
$95$	−7.80642	−0.800922
$96$	0	0
$97$	5.28592	0.536704	0.268352	−	0.963321i	$-0.413521\pi$
$97$	5.28592	0.536704	0.268352	+	0.963321i	$0.413521\pi$
$98$	0	0
$99$	−1.90321	−0.191280
$100$	0	0
$101$	18.1082	1.80183	0.900915	−	0.433996i	$-0.142897\pi$
$101$	18.1082	1.80183	0.900915	+	0.433996i	$0.142897\pi$
$102$	0	0
$103$	−16.7447	−1.64990	−0.824951	−	0.565205i	$-0.808797\pi$
$103$	−16.7447	−1.64990	−0.824951	+	0.565205i	$0.808797\pi$
$104$	0	0
$105$	−2.21432	−0.216095
$106$	0	0
$107$	0.561993	0.0543299	0.0271649	−	0.999631i	$-0.491352\pi$
$107$	0.561993	0.0543299	0.0271649	+	0.999631i	$0.491352\pi$
$108$	0	0
$109$	−3.93978	−0.377362	−0.188681	−	0.982038i	$-0.560421\pi$
$109$	−3.93978	−0.377362	−0.188681	+	0.982038i	$0.560421\pi$
$110$	0	0
$111$	−13.9081	−1.32010
$112$	0	0
$113$	11.2859	1.06169	0.530845	−	0.847469i	$-0.321875\pi$
$113$	11.2859	1.06169	0.530845	+	0.847469i	$0.321875\pi$
$114$	0	0
$115$	−2.90321	−0.270726
$116$	0	0
$117$	−4.21432	−0.389614
$118$	0	0
$119$	−4.21432	−0.386326
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.2810	−1.10734
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.9906	1.15273	0.576366	−	0.817192i	$-0.304470\pi$
$127$	12.9906	1.15273	0.576366	+	0.817192i	$0.304470\pi$
$128$	0	0
$129$	9.18421	0.808624
$130$	0	0
$131$	8.47013	0.740038	0.370019	−	0.929024i	$-0.379351\pi$
$131$	8.47013	0.740038	0.370019	+	0.929024i	$0.379351\pi$
$132$	0	0
$133$	7.80642	0.676903
$134$	0	0
$135$	−2.42864	−0.209024
$136$	0	0
$137$	−11.1383	−0.951607	−0.475804	−	0.879552i	$-0.657843\pi$
$137$	−11.1383	−0.951607	−0.475804	+	0.879552i	$0.657843\pi$
$138$	0	0
$139$	8.04149	0.682070	0.341035	−	0.940051i	$-0.389223\pi$
$139$	8.04149	0.682070	0.341035	+	0.940051i	$0.389223\pi$
$140$	0	0
$141$	−2.28100	−0.192095
$142$	0	0
$143$	2.21432	0.185171
$144$	0	0
$145$	0.755569	0.0627466
$146$	0	0
$147$	2.21432	0.182634
$148$	0	0
$149$	−6.13335	−0.502464	−0.251232	−	0.967927i	$-0.580836\pi$
$149$	−6.13335	−0.502464	−0.251232	+	0.967927i	$0.580836\pi$
$150$	0	0
$151$	−10.1476	−0.825803	−0.412902	−	0.910776i	$-0.635485\pi$
$151$	−10.1476	−0.825803	−0.412902	+	0.910776i	$0.635485\pi$
$152$	0	0
$153$	8.02074	0.648439
$154$	0	0
$155$	−7.11753	−0.571694
$156$	0	0
$157$	−21.7146	−1.73301	−0.866505	−	0.499168i	$-0.833639\pi$
$157$	−21.7146	−1.73301	−0.866505	+	0.499168i	$0.833639\pi$
$158$	0	0
$159$	−14.5620	−1.15484
$160$	0	0
$161$	2.90321	0.228805
$162$	0	0
$163$	8.01429	0.627728	0.313864	−	0.949468i	$-0.398376\pi$
$163$	8.01429	0.627728	0.313864	+	0.949468i	$0.398376\pi$
$164$	0	0
$165$	−2.21432	−0.172385
$166$	0	0
$167$	−15.1240	−1.17033	−0.585165	−	0.810915i	$-0.698970\pi$
$167$	−15.1240	−1.17033	−0.585165	+	0.810915i	$0.698970\pi$
$168$	0	0
$169$	−8.09679	−0.622830
$170$	0	0
$171$	−14.8573	−1.13616
$172$	0	0
$173$	19.6938	1.49729	0.748646	−	0.662969i	$-0.230704\pi$
$173$	19.6938	1.49729	0.748646	+	0.662969i	$0.230704\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−5.95407	−0.447535
$178$	0	0
$179$	−5.53972	−0.414058	−0.207029	−	0.978335i	$-0.566379\pi$
$179$	−5.53972	−0.414058	−0.207029	+	0.978335i	$0.566379\pi$
$180$	0	0
$181$	−19.9081	−1.47976	−0.739880	−	0.672739i	$-0.765118\pi$
$181$	−19.9081	−1.47976	−0.739880	+	0.672739i	$0.765118\pi$
$182$	0	0
$183$	19.4652	1.43891
$184$	0	0
$185$	−6.28100	−0.461788
$186$	0	0
$187$	−4.21432	−0.308182
$188$	0	0
$189$	2.42864	0.176658
$190$	0	0
$191$	20.8988	1.51218	0.756091	−	0.654467i	$-0.227107\pi$
$191$	20.8988	1.51218	0.756091	+	0.654467i	$0.227107\pi$
$192$	0	0
$193$	−13.6271	−0.980903	−0.490451	−	0.871469i	$-0.663168\pi$
$193$	−13.6271	−0.980903	−0.490451	+	0.871469i	$0.663168\pi$
$194$	0	0
$195$	−4.90321	−0.351126
$196$	0	0
$197$	15.7003	1.11860	0.559299	−	0.828966i	$-0.311070\pi$
$197$	15.7003	1.11860	0.559299	+	0.828966i	$0.311070\pi$
$198$	0	0
$199$	23.9146	1.69526	0.847630	−	0.530588i	$-0.178029\pi$
$199$	23.9146	1.69526	0.847630	+	0.530588i	$0.178029\pi$
$200$	0	0
$201$	−3.37778	−0.238251
$202$	0	0
$203$	−0.755569	−0.0530305
$204$	0	0
$205$	−5.54617	−0.387361
$206$	0	0
$207$	−5.52543	−0.384044
$208$	0	0
$209$	7.80642	0.539982
$210$	0	0
$211$	−11.4795	−0.790281	−0.395141	−	0.918621i	$-0.629304\pi$
$211$	−11.4795	−0.790281	−0.395141	+	0.918621i	$0.629304\pi$
$212$	0	0
$213$	−16.8573	−1.15504
$214$	0	0
$215$	4.14764	0.282867
$216$	0	0
$217$	7.11753	0.483170
$218$	0	0
$219$	−27.0464	−1.82763
$220$	0	0
$221$	−9.33185	−0.627728
$222$	0	0
$223$	−19.7669	−1.32369	−0.661846	−	0.749640i	$-0.730227\pi$
$223$	−19.7669	−1.32369	−0.661846	+	0.749640i	$0.730227\pi$
$224$	0	0
$225$	1.90321	0.126881
$226$	0	0
$227$	−3.61285	−0.239793	−0.119897	−	0.992786i	$-0.538256\pi$
$227$	−3.61285	−0.239793	−0.119897	+	0.992786i	$0.538256\pi$
$228$	0	0
$229$	−18.7239	−1.23731	−0.618656	−	0.785662i	$-0.712322\pi$
$229$	−18.7239	−1.23731	−0.618656	+	0.785662i	$0.712322\pi$
$230$	0	0
$231$	2.21432	0.145692
$232$	0	0
$233$	7.00492	0.458908	0.229454	−	0.973320i	$-0.426306\pi$
$233$	7.00492	0.458908	0.229454	+	0.973320i	$0.426306\pi$
$234$	0	0
$235$	−1.03011	−0.0671971
$236$	0	0
$237$	−7.80642	−0.507082
$238$	0	0
$239$	−20.9906	−1.35777	−0.678886	−	0.734244i	$-0.737537\pi$
$239$	−20.9906	−1.35777	−0.678886	+	0.734244i	$0.737537\pi$
$240$	0	0
$241$	10.7491	0.692411	0.346206	−	0.938159i	$-0.387470\pi$
$241$	10.7491	0.692411	0.346206	+	0.938159i	$0.387470\pi$
$242$	0	0
$243$	−17.2652	−1.10756
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	17.2859	1.09988
$248$	0	0
$249$	4.72393	0.299367
$250$	0	0
$251$	6.30174	0.397762	0.198881	−	0.980024i	$-0.436269\pi$
$251$	6.30174	0.397762	0.198881	+	0.980024i	$0.436269\pi$
$252$	0	0
$253$	2.90321	0.182523
$254$	0	0
$255$	9.33185	0.584383
$256$	0	0
$257$	26.2766	1.63909	0.819543	−	0.573018i	$-0.194227\pi$
$257$	26.2766	1.63909	0.819543	+	0.573018i	$0.194227\pi$
$258$	0	0
$259$	6.28100	0.390282
$260$	0	0
$261$	1.43801	0.0890104
$262$	0	0
$263$	−12.7239	−0.784591	−0.392295	−	0.919839i	$-0.628319\pi$
$263$	−12.7239	−0.784591	−0.392295	+	0.919839i	$0.628319\pi$
$264$	0	0
$265$	−6.57628	−0.403978
$266$	0	0
$267$	38.2766	2.34249
$268$	0	0
$269$	−14.1017	−0.859796	−0.429898	−	0.902877i	$-0.641451\pi$
$269$	−14.1017	−0.859796	−0.429898	+	0.902877i	$0.641451\pi$
$270$	0	0
$271$	5.28592	0.321097	0.160548	−	0.987028i	$-0.448674\pi$
$271$	5.28592	0.321097	0.160548	+	0.987028i	$0.448674\pi$
$272$	0	0
$273$	4.90321	0.296756
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	15.1798	0.912064	0.456032	−	0.889963i	$-0.349270\pi$
$277$	15.1798	0.912064	0.456032	+	0.889963i	$0.349270\pi$
$278$	0	0
$279$	−13.5462	−0.810988
$280$	0	0
$281$	32.8671	1.96069	0.980344	−	0.197295i	$-0.0632158\pi$
$281$	32.8671	1.96069	0.980344	+	0.197295i	$0.0632158\pi$
$282$	0	0
$283$	8.94914	0.531971	0.265986	−	0.963977i	$-0.414303\pi$
$283$	8.94914	0.531971	0.265986	+	0.963977i	$0.414303\pi$
$284$	0	0
$285$	−17.2859	−1.02393
$286$	0	0
$287$	5.54617	0.327380
$288$	0	0
$289$	0.760491	0.0447348
$290$	0	0
$291$	11.7047	0.686142
$292$	0	0
$293$	−22.9195	−1.33897	−0.669486	−	0.742825i	$-0.733486\pi$
$293$	−22.9195	−1.33897	−0.669486	+	0.742825i	$0.733486\pi$
$294$	0	0
$295$	−2.68889	−0.156553
$296$	0	0
$297$	2.42864	0.140924
$298$	0	0
$299$	6.42864	0.371778
$300$	0	0
$301$	−4.14764	−0.239066
$302$	0	0
$303$	40.0973	2.30353
$304$	0	0
$305$	8.79060	0.503348
$306$	0	0
$307$	18.2953	1.04417	0.522084	−	0.852894i	$-0.325155\pi$
$307$	18.2953	1.04417	0.522084	+	0.852894i	$0.325155\pi$
$308$	0	0
$309$	−37.0781	−2.10930
$310$	0	0
$311$	5.54617	0.314495	0.157247	−	0.987559i	$-0.449738\pi$
$311$	5.54617	0.314495	0.157247	+	0.987559i	$0.449738\pi$
$312$	0	0
$313$	−23.8064	−1.34562	−0.672809	−	0.739816i	$-0.734913\pi$
$313$	−23.8064	−1.34562	−0.672809	+	0.739816i	$0.734913\pi$
$314$	0	0
$315$	−1.90321	−0.107234
$316$	0	0
$317$	10.5906	0.594826	0.297413	−	0.954749i	$-0.403876\pi$
$317$	10.5906	0.594826	0.297413	+	0.954749i	$0.403876\pi$
$318$	0	0
$319$	−0.755569	−0.0423037
$320$	0	0
$321$	1.24443	0.0694574
$322$	0	0
$323$	−32.8988	−1.83054
$324$	0	0
$325$	−2.21432	−0.122828
$326$	0	0
$327$	−8.72393	−0.482434
$328$	0	0
$329$	1.03011	0.0567919
$330$	0	0
$331$	−30.1146	−1.65525	−0.827625	−	0.561282i	$-0.810308\pi$
$331$	−30.1146	−1.65525	−0.827625	+	0.561282i	$0.810308\pi$
$332$	0	0
$333$	−11.9541	−0.655079
$334$	0	0
$335$	−1.52543	−0.0833430
$336$	0	0
$337$	−30.4558	−1.65904	−0.829518	−	0.558481i	$-0.811385\pi$
$337$	−30.4558	−1.65904	−0.829518	+	0.558481i	$0.811385\pi$
$338$	0	0
$339$	24.9906	1.35730
$340$	0	0
$341$	7.11753	0.385436
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−6.42864	−0.346106
$346$	0	0
$347$	28.3511	1.52196	0.760982	−	0.648772i	$-0.224717\pi$
$347$	28.3511	1.52196	0.760982	+	0.648772i	$0.224717\pi$
$348$	0	0
$349$	−32.5654	−1.74319	−0.871593	−	0.490231i	$-0.836912\pi$
$349$	−32.5654	−1.74319	−0.871593	+	0.490231i	$0.836912\pi$
$350$	0	0
$351$	5.37778	0.287045
$352$	0	0
$353$	−24.2766	−1.29211	−0.646055	−	0.763291i	$-0.723582\pi$
$353$	−24.2766	−1.29211	−0.646055	+	0.763291i	$0.723582\pi$
$354$	0	0
$355$	−7.61285	−0.404048
$356$	0	0
$357$	−9.33185	−0.493894
$358$	0	0
$359$	−16.4746	−0.869495	−0.434747	−	0.900552i	$-0.643162\pi$
$359$	−16.4746	−0.869495	−0.434747	+	0.900552i	$0.643162\pi$
$360$	0	0
$361$	41.9403	2.20738
$362$	0	0
$363$	2.21432	0.116222
$364$	0	0
$365$	−12.2143	−0.639327
$366$	0	0
$367$	−6.71609	−0.350577	−0.175289	−	0.984517i	$-0.556086\pi$
$367$	−6.71609	−0.350577	−0.175289	+	0.984517i	$0.556086\pi$
$368$	0	0
$369$	−10.5555	−0.549499
$370$	0	0
$371$	6.57628	0.341424
$372$	0	0
$373$	10.5477	0.546139	0.273070	−	0.961994i	$-0.411961\pi$
$373$	10.5477	0.546139	0.273070	+	0.961994i	$0.411961\pi$
$374$	0	0
$375$	2.21432	0.114347
$376$	0	0
$377$	−1.67307	−0.0861675
$378$	0	0
$379$	33.0607	1.69821	0.849107	−	0.528221i	$-0.177141\pi$
$379$	33.0607	1.69821	0.849107	+	0.528221i	$0.177141\pi$
$380$	0	0
$381$	28.7654	1.47370
$382$	0	0
$383$	17.3067	0.884329	0.442165	−	0.896934i	$-0.354211\pi$
$383$	17.3067	0.884329	0.442165	+	0.896934i	$0.354211\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	7.89384	0.401267
$388$	0	0
$389$	−33.1842	−1.68251	−0.841253	−	0.540642i	$-0.818182\pi$
$389$	−33.1842	−1.68251	−0.841253	+	0.540642i	$0.818182\pi$
$390$	0	0
$391$	−12.2351	−0.618754
$392$	0	0
$393$	18.7556	0.946093
$394$	0	0
$395$	−3.52543	−0.177384
$396$	0	0
$397$	10.6637	0.535196	0.267598	−	0.963531i	$-0.413770\pi$
$397$	10.6637	0.535196	0.267598	+	0.963531i	$0.413770\pi$
$398$	0	0
$399$	17.2859	0.865378
$400$	0	0
$401$	−10.5906	−0.528868	−0.264434	−	0.964404i	$-0.585185\pi$
$401$	−10.5906	−0.528868	−0.264434	+	0.964404i	$0.585185\pi$
$402$	0	0
$403$	15.7605	0.785086
$404$	0	0
$405$	−11.0874	−0.550938
$406$	0	0
$407$	6.28100	0.311337
$408$	0	0
$409$	11.8829	0.587574	0.293787	−	0.955871i	$-0.405084\pi$
$409$	11.8829	0.587574	0.293787	+	0.955871i	$0.405084\pi$
$410$	0	0
$411$	−24.6637	−1.21657
$412$	0	0
$413$	2.68889	0.132312
$414$	0	0
$415$	2.13335	0.104722
$416$	0	0
$417$	17.8064	0.871984
$418$	0	0
$419$	2.61576	0.127788	0.0638942	−	0.997957i	$-0.479648\pi$
$419$	2.61576	0.127788	0.0638942	+	0.997957i	$0.479648\pi$
$420$	0	0
$421$	−21.5254	−1.04909	−0.524543	−	0.851384i	$-0.675764\pi$
$421$	−21.5254	−1.04909	−0.524543	+	0.851384i	$0.675764\pi$
$422$	0	0
$423$	−1.96052	−0.0953238
$424$	0	0
$425$	4.21432	0.204425
$426$	0	0
$427$	−8.79060	−0.425407
$428$	0	0
$429$	4.90321	0.236729
$430$	0	0
$431$	19.0464	0.917433	0.458717	−	0.888583i	$-0.348309\pi$
$431$	19.0464	0.917433	0.458717	+	0.888583i	$0.348309\pi$
$432$	0	0
$433$	18.6953	0.898441	0.449220	−	0.893421i	$-0.351702\pi$
$433$	18.6953	0.898441	0.449220	+	0.893421i	$0.351702\pi$
$434$	0	0
$435$	1.67307	0.0802176
$436$	0	0
$437$	22.6637	1.08415
$438$	0	0
$439$	0.990632	0.0472803	0.0236401	−	0.999721i	$-0.492474\pi$
$439$	0.990632	0.0472803	0.0236401	+	0.999721i	$0.492474\pi$
$440$	0	0
$441$	1.90321	0.0906291
$442$	0	0
$443$	−20.3511	−0.966908	−0.483454	−	0.875370i	$-0.660618\pi$
$443$	−20.3511	−0.966908	−0.483454	+	0.875370i	$0.660618\pi$
$444$	0	0
$445$	17.2859	0.819431
$446$	0	0
$447$	−13.5812	−0.642369
$448$	0	0
$449$	−29.1383	−1.37512	−0.687560	−	0.726127i	$-0.741318\pi$
$449$	−29.1383	−1.37512	−0.687560	+	0.726127i	$0.741318\pi$
$450$	0	0
$451$	5.54617	0.261159
$452$	0	0
$453$	−22.4701	−1.05574
$454$	0	0
$455$	2.21432	0.103809
$456$	0	0
$457$	31.3131	1.46477	0.732383	−	0.680893i	$-0.238408\pi$
$457$	31.3131	1.46477	0.732383	+	0.680893i	$0.238408\pi$
$458$	0	0
$459$	−10.2351	−0.477732
$460$	0	0
$461$	11.9017	0.554317	0.277158	−	0.960824i	$-0.410607\pi$
$461$	11.9017	0.554317	0.277158	+	0.960824i	$0.410607\pi$
$462$	0	0
$463$	−3.89384	−0.180962	−0.0904811	−	0.995898i	$-0.528840\pi$
$463$	−3.89384	−0.180962	−0.0904811	+	0.995898i	$0.528840\pi$
$464$	0	0
$465$	−15.7605	−0.730875
$466$	0	0
$467$	21.3985	0.990206	0.495103	−	0.868834i	$-0.335130\pi$
$467$	21.3985	0.990206	0.495103	+	0.868834i	$0.335130\pi$
$468$	0	0
$469$	1.52543	0.0704377
$470$	0	0
$471$	−48.0830	−2.21555
$472$	0	0
$473$	−4.14764	−0.190709
$474$	0	0
$475$	−7.80642	−0.358183
$476$	0	0
$477$	−12.5161	−0.573071
$478$	0	0
$479$	11.9813	0.547438	0.273719	−	0.961810i	$-0.411746\pi$
$479$	11.9813	0.547438	0.273719	+	0.961810i	$0.411746\pi$
$480$	0	0
$481$	13.9081	0.634156
$482$	0	0
$483$	6.42864	0.292513
$484$	0	0
$485$	5.28592	0.240021
$486$	0	0
$487$	16.5892	0.751728	0.375864	−	0.926675i	$-0.377346\pi$
$487$	16.5892	0.751728	0.375864	+	0.926675i	$0.377346\pi$
$488$	0	0
$489$	17.7462	0.802511
$490$	0	0
$491$	41.2400	1.86113	0.930567	−	0.366121i	$-0.119314\pi$
$491$	41.2400	1.86113	0.930567	+	0.366121i	$0.119314\pi$
$492$	0	0
$493$	3.18421	0.143410
$494$	0	0
$495$	−1.90321	−0.0855430
$496$	0	0
$497$	7.61285	0.341483
$498$	0	0
$499$	32.3783	1.44945	0.724725	−	0.689038i	$-0.241967\pi$
$499$	32.3783	1.44945	0.724725	+	0.689038i	$0.241967\pi$
$500$	0	0
$501$	−33.4893	−1.49619
$502$	0	0
$503$	−33.4795	−1.49278	−0.746388	−	0.665511i	$-0.768214\pi$
$503$	−33.4795	−1.49278	−0.746388	+	0.665511i	$0.768214\pi$
$504$	0	0
$505$	18.1082	0.805803
$506$	0	0
$507$	−17.9289	−0.796249
$508$	0	0
$509$	−8.79706	−0.389923	−0.194961	−	0.980811i	$-0.562458\pi$
$509$	−8.79706	−0.389923	−0.194961	+	0.980811i	$0.562458\pi$
$510$	0	0
$511$	12.2143	0.540330
$512$	0	0
$513$	18.9590	0.837060
$514$	0	0
$515$	−16.7447	−0.737858
$516$	0	0
$517$	1.03011	0.0453043
$518$	0	0
$519$	43.6084	1.91420
$520$	0	0
$521$	28.4415	1.24605	0.623023	−	0.782203i	$-0.285904\pi$
$521$	28.4415	1.24605	0.623023	+	0.782203i	$0.285904\pi$
$522$	0	0
$523$	3.80642	0.166443	0.0832216	−	0.996531i	$-0.473479\pi$
$523$	3.80642	0.166443	0.0832216	+	0.996531i	$0.473479\pi$
$524$	0	0
$525$	−2.21432	−0.0966408
$526$	0	0
$527$	−29.9956	−1.30663
$528$	0	0
$529$	−14.5714	−0.633537
$530$	0	0
$531$	−5.11753	−0.222082
$532$	0	0
$533$	12.2810	0.531949
$534$	0	0
$535$	0.561993	0.0242971
$536$	0	0
$537$	−12.2667	−0.529347
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	32.2351	1.38589	0.692947	−	0.720989i	$-0.256312\pi$
$541$	32.2351	1.38589	0.692947	+	0.720989i	$0.256312\pi$
$542$	0	0
$543$	−44.0830	−1.89178
$544$	0	0
$545$	−3.93978	−0.168762
$546$	0	0
$547$	−19.3876	−0.828955	−0.414478	−	0.910060i	$-0.636036\pi$
$547$	−19.3876	−0.828955	−0.414478	+	0.910060i	$0.636036\pi$
$548$	0	0
$549$	16.7304	0.714035
$550$	0	0
$551$	−5.89829	−0.251276
$552$	0	0
$553$	3.52543	0.149916
$554$	0	0
$555$	−13.9081	−0.590367
$556$	0	0
$557$	−1.58565	−0.0671862	−0.0335931	−	0.999436i	$-0.510695\pi$
$557$	−1.58565	−0.0671862	−0.0335931	+	0.999436i	$0.510695\pi$
$558$	0	0
$559$	−9.18421	−0.388451
$560$	0	0
$561$	−9.33185	−0.393991
$562$	0	0
$563$	−28.8069	−1.21407	−0.607033	−	0.794677i	$-0.707640\pi$
$563$	−28.8069	−1.21407	−0.607033	+	0.794677i	$0.707640\pi$
$564$	0	0
$565$	11.2859	0.474802
$566$	0	0
$567$	11.0874	0.465628
$568$	0	0
$569$	−14.6953	−0.616061	−0.308030	−	0.951376i	$-0.599670\pi$
$569$	−14.6953	−0.616061	−0.308030	+	0.951376i	$0.599670\pi$
$570$	0	0
$571$	12.3368	0.516278	0.258139	−	0.966108i	$-0.416891\pi$
$571$	12.3368	0.516278	0.258139	+	0.966108i	$0.416891\pi$
$572$	0	0
$573$	46.2766	1.93323
$574$	0	0
$575$	−2.90321	−0.121072
$576$	0	0
$577$	−8.13335	−0.338596	−0.169298	−	0.985565i	$-0.554150\pi$
$577$	−8.13335	−0.338596	−0.169298	+	0.985565i	$0.554150\pi$
$578$	0	0
$579$	−30.1748	−1.25402
$580$	0	0
$581$	−2.13335	−0.0885064
$582$	0	0
$583$	6.57628	0.272362
$584$	0	0
$585$	−4.21432	−0.174241
$586$	0	0
$587$	−2.30619	−0.0951865	−0.0475932	−	0.998867i	$-0.515155\pi$
$587$	−2.30619	−0.0951865	−0.0475932	+	0.998867i	$0.515155\pi$
$588$	0	0
$589$	55.5625	2.28941
$590$	0	0
$591$	34.7654	1.43006
$592$	0	0
$593$	−25.7067	−1.05565	−0.527824	−	0.849354i	$-0.676992\pi$
$593$	−25.7067	−1.05565	−0.527824	+	0.849354i	$0.676992\pi$
$594$	0	0
$595$	−4.21432	−0.172770
$596$	0	0
$597$	52.9545	2.16729
$598$	0	0
$599$	−27.5210	−1.12448	−0.562238	−	0.826975i	$-0.690060\pi$
$599$	−27.5210	−1.12448	−0.562238	+	0.826975i	$0.690060\pi$
$600$	0	0
$601$	17.1175	0.698239	0.349119	−	0.937078i	$-0.386481\pi$
$601$	17.1175	0.698239	0.349119	+	0.937078i	$0.386481\pi$
$602$	0	0
$603$	−2.90321	−0.118228
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	21.4291	0.869781	0.434890	−	0.900483i	$-0.356787\pi$
$607$	21.4291	0.869781	0.434890	+	0.900483i	$0.356787\pi$
$608$	0	0
$609$	−1.67307	−0.0677962
$610$	0	0
$611$	2.28100	0.0922792
$612$	0	0
$613$	−40.3037	−1.62785	−0.813927	−	0.580968i	$-0.802674\pi$
$613$	−40.3037	−1.62785	−0.813927	+	0.580968i	$0.802674\pi$
$614$	0	0
$615$	−12.2810	−0.495218
$616$	0	0
$617$	−15.0553	−0.606104	−0.303052	−	0.952974i	$-0.598006\pi$
$617$	−15.0553	−0.606104	−0.303052	+	0.952974i	$0.598006\pi$
$618$	0	0
$619$	−14.6702	−0.589643	−0.294822	−	0.955552i	$-0.595260\pi$
$619$	−14.6702	−0.589643	−0.294822	+	0.955552i	$0.595260\pi$
$620$	0	0
$621$	7.05086	0.282941
$622$	0	0
$623$	−17.2859	−0.692546
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	17.2859	0.690333
$628$	0	0
$629$	−26.4701	−1.05543
$630$	0	0
$631$	16.2034	0.645048	0.322524	−	0.946561i	$-0.395469\pi$
$631$	16.2034	0.645048	0.322524	+	0.946561i	$0.395469\pi$
$632$	0	0
$633$	−25.4193	−1.01033
$634$	0	0
$635$	12.9906	0.515518
$636$	0	0
$637$	−2.21432	−0.0877345
$638$	0	0
$639$	−14.4889	−0.573171
$640$	0	0
$641$	−28.7225	−1.13447	−0.567236	−	0.823555i	$-0.691987\pi$
$641$	−28.7225	−1.13447	−0.567236	+	0.823555i	$0.691987\pi$
$642$	0	0
$643$	7.87755	0.310660	0.155330	−	0.987863i	$-0.450356\pi$
$643$	7.87755	0.310660	0.155330	+	0.987863i	$0.450356\pi$
$644$	0	0
$645$	9.18421	0.361628
$646$	0	0
$647$	−29.7540	−1.16975	−0.584876	−	0.811123i	$-0.698857\pi$
$647$	−29.7540	−1.16975	−0.584876	+	0.811123i	$0.698857\pi$
$648$	0	0
$649$	2.68889	0.105548
$650$	0	0
$651$	15.7605	0.617702
$652$	0	0
$653$	16.7141	0.654073	0.327036	−	0.945012i	$-0.393950\pi$
$653$	16.7141	0.654073	0.327036	+	0.945012i	$0.393950\pi$
$654$	0	0
$655$	8.47013	0.330955
$656$	0	0
$657$	−23.2464	−0.906930
$658$	0	0
$659$	−1.38223	−0.0538440	−0.0269220	−	0.999638i	$-0.508571\pi$
$659$	−1.38223	−0.0538440	−0.0269220	+	0.999638i	$0.508571\pi$
$660$	0	0
$661$	0.723926	0.0281575	0.0140787	−	0.999901i	$-0.495518\pi$
$661$	0.723926	0.0281575	0.0140787	+	0.999901i	$0.495518\pi$
$662$	0	0
$663$	−20.6637	−0.802512
$664$	0	0
$665$	7.80642	0.302720
$666$	0	0
$667$	−2.19358	−0.0849356
$668$	0	0
$669$	−43.7703	−1.69226
$670$	0	0
$671$	−8.79060	−0.339357
$672$	0	0
$673$	27.5254	1.06103	0.530514	−	0.847676i	$-0.321999\pi$
$673$	27.5254	1.06103	0.530514	+	0.847676i	$0.321999\pi$
$674$	0	0
$675$	−2.42864	−0.0934784
$676$	0	0
$677$	−20.4079	−0.784339	−0.392170	−	0.919893i	$-0.628275\pi$
$677$	−20.4079	−0.784339	−0.392170	+	0.919893i	$0.628275\pi$
$678$	0	0
$679$	−5.28592	−0.202855
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	38.5575	1.47536	0.737682	−	0.675149i	$-0.235920\pi$
$683$	38.5575	1.47536	0.737682	+	0.675149i	$0.235920\pi$
$684$	0	0
$685$	−11.1383	−0.425572
$686$	0	0
$687$	−41.4608	−1.58183
$688$	0	0
$689$	14.5620	0.554768
$690$	0	0
$691$	−9.59655	−0.365070	−0.182535	−	0.983199i	$-0.558430\pi$
$691$	−9.59655	−0.365070	−0.182535	+	0.983199i	$0.558430\pi$
$692$	0	0
$693$	1.90321	0.0722970
$694$	0	0
$695$	8.04149	0.305031
$696$	0	0
$697$	−23.3733	−0.885328
$698$	0	0
$699$	15.5111	0.586685
$700$	0	0
$701$	−30.0701	−1.13573	−0.567865	−	0.823121i	$-0.692231\pi$
$701$	−30.0701	−1.13573	−0.567865	+	0.823121i	$0.692231\pi$
$702$	0	0
$703$	49.0321	1.84928
$704$	0	0
$705$	−2.28100	−0.0859073
$706$	0	0
$707$	−18.1082	−0.681028
$708$	0	0
$709$	−4.10171	−0.154043	−0.0770215	−	0.997029i	$-0.524541\pi$
$709$	−4.10171	−0.154043	−0.0770215	+	0.997029i	$0.524541\pi$
$710$	0	0
$711$	−6.70964	−0.251631
$712$	0	0
$713$	20.6637	0.773862
$714$	0	0
$715$	2.21432	0.0828109
$716$	0	0
$717$	−46.4800	−1.73583
$718$	0	0
$719$	−19.2696	−0.718636	−0.359318	−	0.933215i	$-0.616991\pi$
$719$	−19.2696	−0.718636	−0.359318	+	0.933215i	$0.616991\pi$
$720$	0	0
$721$	16.7447	0.623604
$722$	0	0
$723$	23.8020	0.885205
$724$	0	0
$725$	0.755569	0.0280611
$726$	0	0
$727$	37.4084	1.38740	0.693700	−	0.720264i	$-0.255979\pi$
$727$	37.4084	1.38740	0.693700	+	0.720264i	$0.255979\pi$
$728$	0	0
$729$	−4.96836	−0.184013
$730$	0	0
$731$	17.4795	0.646502
$732$	0	0
$733$	4.60147	0.169959	0.0849796	−	0.996383i	$-0.472917\pi$
$733$	4.60147	0.169959	0.0849796	+	0.996383i	$0.472917\pi$
$734$	0	0
$735$	2.21432	0.0816764
$736$	0	0
$737$	1.52543	0.0561898
$738$	0	0
$739$	−35.0420	−1.28904	−0.644520	−	0.764588i	$-0.722943\pi$
$739$	−35.0420	−1.28904	−0.644520	+	0.764588i	$0.722943\pi$
$740$	0	0
$741$	38.2766	1.40612
$742$	0	0
$743$	−40.7654	−1.49554	−0.747769	−	0.663959i	$-0.768875\pi$
$743$	−40.7654	−1.49554	−0.747769	+	0.663959i	$0.768875\pi$
$744$	0	0
$745$	−6.13335	−0.224709
$746$	0	0
$747$	4.06022	0.148556
$748$	0	0
$749$	−0.561993	−0.0205348
$750$	0	0
$751$	−24.4286	−0.891414	−0.445707	−	0.895179i	$-0.647048\pi$
$751$	−24.4286	−0.891414	−0.445707	+	0.895179i	$0.647048\pi$
$752$	0	0
$753$	13.9541	0.508514
$754$	0	0
$755$	−10.1476	−0.369311
$756$	0	0
$757$	36.9086	1.34147	0.670733	−	0.741699i	$-0.265980\pi$
$757$	36.9086	1.34147	0.670733	+	0.741699i	$0.265980\pi$
$758$	0	0
$759$	6.42864	0.233345
$760$	0	0
$761$	−3.96190	−0.143619	−0.0718094	−	0.997418i	$-0.522877\pi$
$761$	−3.96190	−0.143619	−0.0718094	+	0.997418i	$0.522877\pi$
$762$	0	0
$763$	3.93978	0.142630
$764$	0	0
$765$	8.02074	0.289991
$766$	0	0
$767$	5.95407	0.214989
$768$	0	0
$769$	2.29190	0.0826479	0.0413239	−	0.999146i	$-0.486842\pi$
$769$	2.29190	0.0826479	0.0413239	+	0.999146i	$0.486842\pi$
$770$	0	0
$771$	58.1847	2.09547
$772$	0	0
$773$	−11.3047	−0.406600	−0.203300	−	0.979116i	$-0.565167\pi$
$773$	−11.3047	−0.406600	−0.203300	+	0.979116i	$0.565167\pi$
$774$	0	0
$775$	−7.11753	−0.255669
$776$	0	0
$777$	13.9081	0.498952
$778$	0	0
$779$	43.2958	1.55123
$780$	0	0
$781$	7.61285	0.272409
$782$	0	0
$783$	−1.83500	−0.0655777
$784$	0	0
$785$	−21.7146	−0.775026
$786$	0	0
$787$	−20.2163	−0.720634	−0.360317	−	0.932830i	$-0.617332\pi$
$787$	−20.2163	−0.720634	−0.360317	+	0.932830i	$0.617332\pi$
$788$	0	0
$789$	−28.1748	−1.00305
$790$	0	0
$791$	−11.2859	−0.401281
$792$	0	0
$793$	−19.4652	−0.691230
$794$	0	0
$795$	−14.5620	−0.516461
$796$	0	0
$797$	50.4514	1.78708	0.893540	−	0.448984i	$-0.148214\pi$
$797$	50.4514	1.78708	0.893540	+	0.448984i	$0.148214\pi$
$798$	0	0
$799$	−4.34122	−0.153581
$800$	0	0
$801$	32.8988	1.16242
$802$	0	0
$803$	12.2143	0.431034
$804$	0	0
$805$	2.90321	0.102325
$806$	0	0
$807$	−31.2257	−1.09920
$808$	0	0
$809$	17.9911	0.632534	0.316267	−	0.948670i	$-0.397571\pi$
$809$	17.9911	0.632534	0.316267	+	0.948670i	$0.397571\pi$
$810$	0	0
$811$	38.3082	1.34518	0.672591	−	0.740014i	$-0.265181\pi$
$811$	38.3082	1.34518	0.672591	+	0.740014i	$0.265181\pi$
$812$	0	0
$813$	11.7047	0.410502
$814$	0	0
$815$	8.01429	0.280728
$816$	0	0
$817$	−32.3783	−1.13277
$818$	0	0
$819$	4.21432	0.147260
$820$	0	0
$821$	22.3082	0.778561	0.389281	−	0.921119i	$-0.372724\pi$
$821$	22.3082	0.778561	0.389281	+	0.921119i	$0.372724\pi$
$822$	0	0
$823$	−38.3541	−1.33694	−0.668470	−	0.743739i	$-0.733051\pi$
$823$	−38.3541	−1.33694	−0.668470	+	0.743739i	$0.733051\pi$
$824$	0	0
$825$	−2.21432	−0.0770927
$826$	0	0
$827$	30.5446	1.06214	0.531071	−	0.847328i	$-0.321790\pi$
$827$	30.5446	1.06214	0.531071	+	0.847328i	$0.321790\pi$
$828$	0	0
$829$	−52.9215	−1.83804	−0.919020	−	0.394211i	$-0.871018\pi$
$829$	−52.9215	−1.83804	−0.919020	+	0.394211i	$0.871018\pi$
$830$	0	0
$831$	33.6128	1.16602
$832$	0	0
$833$	4.21432	0.146018
$834$	0	0
$835$	−15.1240	−0.523387
$836$	0	0
$837$	17.2859	0.597489
$838$	0	0
$839$	−38.6987	−1.33603	−0.668014	−	0.744148i	$-0.732856\pi$
$839$	−38.6987	−1.33603	−0.668014	+	0.744148i	$0.732856\pi$
$840$	0	0
$841$	−28.4291	−0.980314
$842$	0	0
$843$	72.7783	2.50662
$844$	0	0
$845$	−8.09679	−0.278538
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	19.8163	0.680093
$850$	0	0
$851$	18.2351	0.625090
$852$	0	0
$853$	29.3352	1.00442	0.502210	−	0.864746i	$-0.332521\pi$
$853$	29.3352	1.00442	0.502210	+	0.864746i	$0.332521\pi$
$854$	0	0
$855$	−14.8573	−0.508108
$856$	0	0
$857$	−21.0627	−0.719488	−0.359744	−	0.933051i	$-0.617136\pi$
$857$	−21.0627	−0.719488	−0.359744	+	0.933051i	$0.617136\pi$
$858$	0	0
$859$	−16.9525	−0.578413	−0.289207	−	0.957267i	$-0.593391\pi$
$859$	−16.9525	−0.578413	−0.289207	+	0.957267i	$0.593391\pi$
$860$	0	0
$861$	12.2810	0.418535
$862$	0	0
$863$	49.8336	1.69636	0.848178	−	0.529711i	$-0.177700\pi$
$863$	49.8336	1.69636	0.848178	+	0.529711i	$0.177700\pi$
$864$	0	0
$865$	19.6938	0.669610
$866$	0	0
$867$	1.68397	0.0571906
$868$	0	0
$869$	3.52543	0.119592
$870$	0	0
$871$	3.37778	0.114452
$872$	0	0
$873$	10.0602	0.340487
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	1.29036	0.0435725	0.0217863	−	0.999763i	$-0.493065\pi$
$877$	1.29036	0.0435725	0.0217863	+	0.999763i	$0.493065\pi$
$878$	0	0
$879$	−50.7511	−1.71179
$880$	0	0
$881$	23.4380	0.789647	0.394823	−	0.918757i	$-0.370806\pi$
$881$	23.4380	0.789647	0.394823	+	0.918757i	$0.370806\pi$
$882$	0	0
$883$	31.0366	1.04446	0.522232	−	0.852804i	$-0.325100\pi$
$883$	31.0366	1.04446	0.522232	+	0.852804i	$0.325100\pi$
$884$	0	0
$885$	−5.95407	−0.200144
$886$	0	0
$887$	1.40636	0.0472211	0.0236106	−	0.999721i	$-0.492484\pi$
$887$	1.40636	0.0472211	0.0236106	+	0.999721i	$0.492484\pi$
$888$	0	0
$889$	−12.9906	−0.435692
$890$	0	0
$891$	11.0874	0.371443
$892$	0	0
$893$	8.04149	0.269098
$894$	0	0
$895$	−5.53972	−0.185172
$896$	0	0
$897$	14.2351	0.475295
$898$	0	0
$899$	−5.37778	−0.179359
$900$	0	0
$901$	−27.7146	−0.923306
$902$	0	0
$903$	−9.18421	−0.305631
$904$	0	0
$905$	−19.9081	−0.661769
$906$	0	0
$907$	−5.86220	−0.194651	−0.0973256	−	0.995253i	$-0.531029\pi$
$907$	−5.86220	−0.194651	−0.0973256	+	0.995253i	$0.531029\pi$
$908$	0	0
$909$	34.4637	1.14309
$910$	0	0
$911$	19.2257	0.636976	0.318488	−	0.947927i	$-0.396825\pi$
$911$	19.2257	0.636976	0.318488	+	0.947927i	$0.396825\pi$
$912$	0	0
$913$	−2.13335	−0.0706037
$914$	0	0
$915$	19.4652	0.643500
$916$	0	0
$917$	−8.47013	−0.279708
$918$	0	0
$919$	−0.520505	−0.0171699	−0.00858494	−	0.999963i	$-0.502733\pi$
$919$	−0.520505	−0.0171699	−0.00858494	+	0.999963i	$0.502733\pi$
$920$	0	0
$921$	40.5116	1.33490
$922$	0	0
$923$	16.8573	0.554864
$924$	0	0
$925$	−6.28100	−0.206518
$926$	0	0
$927$	−31.8687	−1.04670
$928$	0	0
$929$	19.6414	0.644414	0.322207	−	0.946669i	$-0.395575\pi$
$929$	19.6414	0.644414	0.322207	+	0.946669i	$0.395575\pi$
$930$	0	0
$931$	−7.80642	−0.255845
$932$	0	0
$933$	12.2810	0.402062
$934$	0	0
$935$	−4.21432	−0.137823
$936$	0	0
$937$	−54.2973	−1.77382	−0.886908	−	0.461947i	$-0.847151\pi$
$937$	−54.2973	−1.77382	−0.886908	+	0.461947i	$0.847151\pi$
$938$	0	0
$939$	−52.7150	−1.72029
$940$	0	0
$941$	1.95206	0.0636353	0.0318177	−	0.999494i	$-0.489870\pi$
$941$	1.95206	0.0636353	0.0318177	+	0.999494i	$0.489870\pi$
$942$	0	0
$943$	16.1017	0.524344
$944$	0	0
$945$	2.42864	0.0790036
$946$	0	0
$947$	9.39207	0.305201	0.152601	−	0.988288i	$-0.451235\pi$
$947$	9.39207	0.305201	0.152601	+	0.988288i	$0.451235\pi$
$948$	0	0
$949$	27.0464	0.877964
$950$	0	0
$951$	23.4509	0.760448
$952$	0	0
$953$	−11.2716	−0.365124	−0.182562	−	0.983194i	$-0.558439\pi$
$953$	−11.2716	−0.365124	−0.182562	+	0.983194i	$0.558439\pi$
$954$	0	0
$955$	20.8988	0.676268
$956$	0	0
$957$	−1.67307	−0.0540827
$958$	0	0
$959$	11.1383	0.359674
$960$	0	0
$961$	19.6593	0.634170
$962$	0	0
$963$	1.06959	0.0344671
$964$	0	0
$965$	−13.6271	−0.438673
$966$	0	0
$967$	−17.3002	−0.556337	−0.278169	−	0.960532i	$-0.589727\pi$
$967$	−17.3002	−0.556337	−0.278169	+	0.960532i	$0.589727\pi$
$968$	0	0
$969$	−72.8484	−2.34023
$970$	0	0
$971$	−28.2928	−0.907961	−0.453980	−	0.891012i	$-0.649996\pi$
$971$	−28.2928	−0.907961	−0.453980	+	0.891012i	$0.649996\pi$
$972$	0	0
$973$	−8.04149	−0.257798
$974$	0	0
$975$	−4.90321	−0.157028
$976$	0	0
$977$	−45.9782	−1.47097	−0.735486	−	0.677539i	$-0.763046\pi$
$977$	−45.9782	−1.47097	−0.735486	+	0.677539i	$0.763046\pi$
$978$	0	0
$979$	−17.2859	−0.552460
$980$	0	0
$981$	−7.49823	−0.239400
$982$	0	0
$983$	−1.41726	−0.0452037	−0.0226018	−	0.999745i	$-0.507195\pi$
$983$	−1.41726	−0.0452037	−0.0226018	+	0.999745i	$0.507195\pi$
$984$	0	0
$985$	15.7003	0.500252
$986$	0	0
$987$	2.28100	0.0726049
$988$	0	0
$989$	−12.0415	−0.382897
$990$	0	0
$991$	28.9906	0.920918	0.460459	−	0.887681i	$-0.347685\pi$
$991$	28.9906	0.920918	0.460459	+	0.887681i	$0.347685\pi$
$992$	0	0
$993$	−66.6834	−2.11613
$994$	0	0
$995$	23.9146	0.758143
$996$	0	0
$997$	−5.80843	−0.183955	−0.0919774	−	0.995761i	$-0.529319\pi$
$997$	−5.80843	−0.183955	−0.0919774	+	0.995761i	$0.529319\pi$
$998$	0	0
$999$	15.2543	0.482624

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bj.1.3		3
4.3	odd	2		385.2.a.g.1.2	✓	3
12.11	even	2		3465.2.a.ba.1.2		3
20.3	even	4		1925.2.b.o.1849.3		6
20.7	even	4		1925.2.b.o.1849.4		6
20.19	odd	2		1925.2.a.u.1.2		3
28.27	even	2		2695.2.a.i.1.2		3
44.43	even	2		4235.2.a.o.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.g.1.2	✓	3	4.3	odd	2
1925.2.a.u.1.2		3	20.19	odd	2
1925.2.b.o.1849.3		6	20.3	even	4
1925.2.b.o.1849.4		6	20.7	even	4
2695.2.a.i.1.2		3	28.27	even	2
3465.2.a.ba.1.2		3	12.11	even	2
4235.2.a.o.1.2		3	44.43	even	2
6160.2.a.bj.1.3		3	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+2.21432 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.90321 q^{9} +O(q^{10})\)	\(q+2.21432 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.90321 q^{9} -1.00000 q^{11} -2.21432 q^{13} +2.21432 q^{15} +4.21432 q^{17} -7.80642 q^{19} -2.21432 q^{21} -2.90321 q^{23} +1.00000 q^{25} -2.42864 q^{27} +0.755569 q^{29} -7.11753 q^{31} -2.21432 q^{33} -1.00000 q^{35} -6.28100 q^{37} -4.90321 q^{39} -5.54617 q^{41} +4.14764 q^{43} +1.90321 q^{45} -1.03011 q^{47} +1.00000 q^{49} +9.33185 q^{51} -6.57628 q^{53} -1.00000 q^{55} -17.2859 q^{57} -2.68889 q^{59} +8.79060 q^{61} -1.90321 q^{63} -2.21432 q^{65} -1.52543 q^{67} -6.42864 q^{69} -7.61285 q^{71} -12.2143 q^{73} +2.21432 q^{75} +1.00000 q^{77} -3.52543 q^{79} -11.0874 q^{81} +2.13335 q^{83} +4.21432 q^{85} +1.67307 q^{87} +17.2859 q^{89} +2.21432 q^{91} -15.7605 q^{93} -7.80642 q^{95} +5.28592 q^{97} -1.90321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q + 3 * q^5 - 3 * q^7 - q^9	\( 3 q + 3 q^{5} - 3 q^{7} - q^{9} - 3 q^{11} + 6 q^{17} - 10 q^{19} - 2 q^{23} + 3 q^{25} + 6 q^{27} + 2 q^{29} - 8 q^{31} - 3 q^{35} - 12 q^{37} - 8 q^{39} + 10 q^{41} + 6 q^{43} - q^{45} - 10 q^{47} + 3 q^{49} + 8 q^{51} - 3 q^{55} - 12 q^{57} - 8 q^{59} + q^{63} + 2 q^{67} - 6 q^{69} + 4 q^{71} - 30 q^{73} + 3 q^{77} - 4 q^{79} - 13 q^{81} + 6 q^{83} + 6 q^{85} - 8 q^{87} + 12 q^{89} - 14 q^{93} - 10 q^{95} - 24 q^{97} + q^{99}+O(q^{100}) \) 3 * q + 3 * q^5 - 3 * q^7 - q^9 - 3 * q^11 + 6 * q^17 - 10 * q^19 - 2 * q^23 + 3 * q^25 + 6 * q^27 + 2 * q^29 - 8 * q^31 - 3 * q^35 - 12 * q^37 - 8 * q^39 + 10 * q^41 + 6 * q^43 - q^45 - 10 * q^47 + 3 * q^49 + 8 * q^51 - 3 * q^55 - 12 * q^57 - 8 * q^59 + q^63 + 2 * q^67 - 6 * q^69 + 4 * q^71 - 30 * q^73 + 3 * q^77 - 4 * q^79 - 13 * q^81 + 6 * q^83 + 6 * q^85 - 8 * q^87 + 12 * q^89 - 14 * q^93 - 10 * q^95 - 24 * q^97 + q^99

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