LMFDB - Embedded newform 6160.2.a.bn.1.1

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:	$0$
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			1.1
Root			$2.17009$ 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)

$f(q)$	$=$	$q-1.17009 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.63090 q^{9} +O(q^{10})$	$q-1.17009 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.63090 q^{9} +1.00000 q^{11} -0.0917087 q^{13} +1.17009 q^{15} -5.51026 q^{17} -0.921622 q^{19} -1.17009 q^{21} +5.70928 q^{23} +1.00000 q^{25} +5.41855 q^{27} +1.41855 q^{29} -0.879362 q^{31} -1.17009 q^{33} -1.00000 q^{35} -8.78765 q^{37} +0.107307 q^{39} -1.61757 q^{41} -3.86603 q^{43} +1.63090 q^{45} +5.90829 q^{47} +1.00000 q^{49} +6.44748 q^{51} -10.0494 q^{53} -1.00000 q^{55} +1.07838 q^{57} +2.14116 q^{59} -3.03612 q^{61} -1.63090 q^{63} +0.0917087 q^{65} +1.52586 q^{67} -6.68035 q^{69} -4.09890 q^{71} +14.1906 q^{73} -1.17009 q^{75} +1.00000 q^{77} -14.5464 q^{79} -1.44748 q^{81} +8.52359 q^{83} +5.51026 q^{85} -1.65983 q^{87} -2.83710 q^{89} -0.0917087 q^{91} +1.02893 q^{93} +0.921622 q^{95} +14.2557 q^{97} -1.63090 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - q^9	$ 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - q^{9} + 3 q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{19} + 2 q^{21} + 10 q^{23} + 3 q^{25} + 2 q^{27} - 10 q^{29} + 10 q^{31} + 2 q^{33} - 3 q^{35} - 16 q^{37} + 12 q^{39} + 2 q^{43} + q^{45} + 20 q^{47} + 3 q^{49} + 20 q^{51} - 12 q^{53} - 3 q^{55} - 14 q^{59} + 10 q^{61} - q^{63} - 2 q^{65} + 2 q^{67} + 2 q^{69} + 24 q^{71} + 4 q^{73} + 2 q^{75} + 3 q^{77} - 8 q^{79} - 5 q^{81} + 10 q^{83} - 16 q^{87} + 20 q^{89} + 2 q^{91} + 18 q^{93} + 6 q^{95} - q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - q^9 + 3 * q^11 + 2 * q^13 - 2 * q^15 - 6 * q^19 + 2 * q^21 + 10 * q^23 + 3 * q^25 + 2 * q^27 - 10 * q^29 + 10 * q^31 + 2 * q^33 - 3 * q^35 - 16 * q^37 + 12 * q^39 + 2 * q^43 + q^45 + 20 * q^47 + 3 * q^49 + 20 * q^51 - 12 * q^53 - 3 * q^55 - 14 * q^59 + 10 * q^61 - q^63 - 2 * q^65 + 2 * q^67 + 2 * q^69 + 24 * q^71 + 4 * q^73 + 2 * q^75 + 3 * q^77 - 8 * q^79 - 5 * q^81 + 10 * q^83 - 16 * q^87 + 20 * q^89 + 2 * q^91 + 18 * q^93 + 6 * q^95 - 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
$3$	−1.17009	−0.675550	−0.337775	−	0.941227i	$-0.609674\pi$
$3$	−1.17009	−0.675550	−0.337775	+	0.941227i	$0.609674\pi$
$4$	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
$9$	−1.63090	−0.543633
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.0917087	−0.0254354	−0.0127177	−	0.999919i	$-0.504048\pi$
$13$	−0.0917087	−0.0254354	−0.0127177	+	0.999919i	$0.504048\pi$
$14$	0	0
$15$	1.17009	0.302115
$16$	0	0
$17$	−5.51026	−1.33643	−0.668217	−	0.743966i	$-0.732942\pi$
$17$	−5.51026	−1.33643	−0.668217	+	0.743966i	$0.732942\pi$
$18$	0	0
$19$	−0.921622	−0.211435	−0.105717	−	0.994396i	$-0.533714\pi$
$19$	−0.921622	−0.211435	−0.105717	+	0.994396i	$0.533714\pi$
$20$	0	0
$21$	−1.17009	−0.255334
$22$	0	0
$23$	5.70928	1.19047	0.595233	−	0.803553i	$-0.297060\pi$
$23$	5.70928	1.19047	0.595233	+	0.803553i	$0.297060\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.41855	1.04280
$28$	0	0
$29$	1.41855	0.263418	0.131709	−	0.991288i	$-0.457954\pi$
$29$	1.41855	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$30$	0	0
$31$	−0.879362	−0.157938	−0.0789690	−	0.996877i	$-0.525163\pi$
$31$	−0.879362	−0.157938	−0.0789690	+	0.996877i	$0.525163\pi$
$32$	0	0
$33$	−1.17009	−0.203686
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−8.78765	−1.44468	−0.722341	−	0.691537i	$-0.756934\pi$
$37$	−8.78765	−1.44468	−0.722341	+	0.691537i	$0.756934\pi$
$38$	0	0
$39$	0.107307	0.0171829
$40$	0	0
$41$	−1.61757	−0.252621	−0.126311	−	0.991991i	$-0.540314\pi$
$41$	−1.61757	−0.252621	−0.126311	+	0.991991i	$0.540314\pi$
$42$	0	0
$43$	−3.86603	−0.589564	−0.294782	−	0.955565i	$-0.595247\pi$
$43$	−3.86603	−0.589564	−0.294782	+	0.955565i	$0.595247\pi$
$44$	0	0
$45$	1.63090	0.243120
$46$	0	0
$47$	5.90829	0.861813	0.430906	−	0.902397i	$-0.358194\pi$
$47$	5.90829	0.861813	0.430906	+	0.902397i	$0.358194\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.44748	0.902828
$52$	0	0
$53$	−10.0494	−1.38040	−0.690199	−	0.723620i	$-0.742477\pi$
$53$	−10.0494	−1.38040	−0.690199	+	0.723620i	$0.742477\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	1.07838	0.142835
$58$	0	0
$59$	2.14116	0.278755	0.139377	−	0.990239i	$-0.455490\pi$
$59$	2.14116	0.278755	0.139377	+	0.990239i	$0.455490\pi$
$60$	0	0
$61$	−3.03612	−0.388735	−0.194367	−	0.980929i	$-0.562265\pi$
$61$	−3.03612	−0.388735	−0.194367	+	0.980929i	$0.562265\pi$
$62$	0	0
$63$	−1.63090	−0.205474
$64$	0	0
$65$	0.0917087	0.0113751
$66$	0	0
$67$	1.52586	0.186413	0.0932066	−	0.995647i	$-0.470288\pi$
$67$	1.52586	0.186413	0.0932066	+	0.995647i	$0.470288\pi$
$68$	0	0
$69$	−6.68035	−0.804219
$70$	0	0
$71$	−4.09890	−0.486450	−0.243225	−	0.969970i	$-0.578205\pi$
$71$	−4.09890	−0.486450	−0.243225	+	0.969970i	$0.578205\pi$
$72$	0	0
$73$	14.1906	1.66088	0.830442	−	0.557105i	$-0.188088\pi$
$73$	14.1906	1.66088	0.830442	+	0.557105i	$0.188088\pi$
$74$	0	0
$75$	−1.17009	−0.135110
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−14.5464	−1.63660	−0.818298	−	0.574795i	$-0.805082\pi$
$79$	−14.5464	−1.63660	−0.818298	+	0.574795i	$0.805082\pi$
$80$	0	0
$81$	−1.44748	−0.160831
$82$	0	0
$83$	8.52359	0.935586	0.467793	−	0.883838i	$-0.345049\pi$
$83$	8.52359	0.935586	0.467793	+	0.883838i	$0.345049\pi$
$84$	0	0
$85$	5.51026	0.597672
$86$	0	0
$87$	−1.65983	−0.177952
$88$	0	0
$89$	−2.83710	−0.300732	−0.150366	−	0.988630i	$-0.548045\pi$
$89$	−2.83710	−0.300732	−0.150366	+	0.988630i	$0.548045\pi$
$90$	0	0
$91$	−0.0917087	−0.00961369
$92$	0	0
$93$	1.02893	0.106695
$94$	0	0
$95$	0.921622	0.0945564
$96$	0	0
$97$	14.2557	1.44744	0.723721	−	0.690093i	$-0.242430\pi$
$97$	14.2557	1.44744	0.723721	+	0.690093i	$0.242430\pi$
$98$	0	0
$99$	−1.63090	−0.163911
$100$	0	0
$101$	9.03612	0.899127	0.449564	−	0.893248i	$-0.351579\pi$
$101$	9.03612	0.899127	0.449564	+	0.893248i	$0.351579\pi$
$102$	0	0
$103$	−3.32684	−0.327803	−0.163902	−	0.986477i	$-0.552408\pi$
$103$	−3.32684	−0.327803	−0.163902	+	0.986477i	$0.552408\pi$
$104$	0	0
$105$	1.17009	0.114189
$106$	0	0
$107$	−8.09890	−0.782950	−0.391475	−	0.920189i	$-0.628035\pi$
$107$	−8.09890	−0.782950	−0.391475	+	0.920189i	$0.628035\pi$
$108$	0	0
$109$	15.1773	1.45372	0.726860	−	0.686786i	$-0.240979\pi$
$109$	15.1773	1.45372	0.726860	+	0.686786i	$0.240979\pi$
$110$	0	0
$111$	10.2823	0.975954
$112$	0	0
$113$	−7.07838	−0.665878	−0.332939	−	0.942948i	$-0.608040\pi$
$113$	−7.07838	−0.665878	−0.332939	+	0.942948i	$0.608040\pi$
$114$	0	0
$115$	−5.70928	−0.532393
$116$	0	0
$117$	0.149568	0.0138275
$118$	0	0
$119$	−5.51026	−0.505125
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.89269	0.170658
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.65983	0.857171	0.428586	−	0.903501i	$-0.359012\pi$
$127$	9.65983	0.857171	0.428586	+	0.903501i	$0.359012\pi$
$128$	0	0
$129$	4.52359	0.398280
$130$	0	0
$131$	−4.68035	−0.408924	−0.204462	−	0.978875i	$-0.565544\pi$
$131$	−4.68035	−0.408924	−0.204462	+	0.978875i	$0.565544\pi$
$132$	0	0
$133$	−0.921622	−0.0799148
$134$	0	0
$135$	−5.41855	−0.466355
$136$	0	0
$137$	8.88655	0.759229	0.379615	−	0.925145i	$-0.376057\pi$
$137$	8.88655	0.759229	0.379615	+	0.925145i	$0.376057\pi$
$138$	0	0
$139$	15.0205	1.27402	0.637012	−	0.770854i	$-0.280170\pi$
$139$	15.0205	1.27402	0.637012	+	0.770854i	$0.280170\pi$
$140$	0	0
$141$	−6.91321	−0.582197
$142$	0	0
$143$	−0.0917087	−0.00766907
$144$	0	0
$145$	−1.41855	−0.117804
$146$	0	0
$147$	−1.17009	−0.0965071
$148$	0	0
$149$	−13.7009	−1.12242	−0.561209	−	0.827674i	$-0.689664\pi$
$149$	−13.7009	−1.12242	−0.561209	+	0.827674i	$0.689664\pi$
$150$	0	0
$151$	−1.05559	−0.0859028	−0.0429514	−	0.999077i	$-0.513676\pi$
$151$	−1.05559	−0.0859028	−0.0429514	+	0.999077i	$0.513676\pi$
$152$	0	0
$153$	8.98667	0.726529
$154$	0	0
$155$	0.879362	0.0706320
$156$	0	0
$157$	−17.7587	−1.41730	−0.708650	−	0.705560i	$-0.750696\pi$
$157$	−17.7587	−1.41730	−0.708650	+	0.705560i	$0.750696\pi$
$158$	0	0
$159$	11.7587	0.932527
$160$	0	0
$161$	5.70928	0.449954
$162$	0	0
$163$	11.4680	0.898243	0.449122	−	0.893471i	$-0.351737\pi$
$163$	11.4680	0.898243	0.449122	+	0.893471i	$0.351737\pi$
$164$	0	0
$165$	1.17009	0.0910911
$166$	0	0
$167$	5.60197	0.433493	0.216747	−	0.976228i	$-0.430455\pi$
$167$	5.60197	0.433493	0.216747	+	0.976228i	$0.430455\pi$
$168$	0	0
$169$	−12.9916	−0.999353
$170$	0	0
$171$	1.50307	0.114943
$172$	0	0
$173$	21.6092	1.64291	0.821457	−	0.570271i	$-0.193162\pi$
$173$	21.6092	1.64291	0.821457	+	0.570271i	$0.193162\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−2.50534	−0.188313
$178$	0	0
$179$	2.05786	0.153812	0.0769058	−	0.997038i	$-0.475496\pi$
$179$	2.05786	0.153812	0.0769058	+	0.997038i	$0.475496\pi$
$180$	0	0
$181$	20.2823	1.50757	0.753786	−	0.657120i	$-0.228225\pi$
$181$	20.2823	1.50757	0.753786	+	0.657120i	$0.228225\pi$
$182$	0	0
$183$	3.55252	0.262610
$184$	0	0
$185$	8.78765	0.646081
$186$	0	0
$187$	−5.51026	−0.402950
$188$	0	0
$189$	5.41855	0.394142
$190$	0	0
$191$	20.2823	1.46758	0.733788	−	0.679378i	$-0.237750\pi$
$191$	20.2823	1.46758	0.733788	+	0.679378i	$0.237750\pi$
$192$	0	0
$193$	−24.3051	−1.74952	−0.874760	−	0.484557i	$-0.838981\pi$
$193$	−24.3051	−1.74952	−0.874760	+	0.484557i	$0.838981\pi$
$194$	0	0
$195$	−0.107307	−0.00768443
$196$	0	0
$197$	−14.1483	−1.00803	−0.504014	−	0.863696i	$-0.668144\pi$
$197$	−14.1483	−1.00803	−0.504014	+	0.863696i	$0.668144\pi$
$198$	0	0
$199$	−10.4813	−0.743002	−0.371501	−	0.928433i	$-0.621157\pi$
$199$	−10.4813	−0.743002	−0.371501	+	0.928433i	$0.621157\pi$
$200$	0	0
$201$	−1.78539	−0.125931
$202$	0	0
$203$	1.41855	0.0995627
$204$	0	0
$205$	1.61757	0.112976
$206$	0	0
$207$	−9.31124	−0.647176
$208$	0	0
$209$	−0.921622	−0.0637499
$210$	0	0
$211$	2.65368	0.182687	0.0913436	−	0.995819i	$-0.470884\pi$
$211$	2.65368	0.182687	0.0913436	+	0.995819i	$0.470884\pi$
$212$	0	0
$213$	4.79606	0.328621
$214$	0	0
$215$	3.86603	0.263661
$216$	0	0
$217$	−0.879362	−0.0596950
$218$	0	0
$219$	−16.6042	−1.12201
$220$	0	0
$221$	0.505339	0.0339928
$222$	0	0
$223$	8.67316	0.580798	0.290399	−	0.956906i	$-0.406212\pi$
$223$	8.67316	0.580798	0.290399	+	0.956906i	$0.406212\pi$
$224$	0	0
$225$	−1.63090	−0.108727
$226$	0	0
$227$	−9.67420	−0.642099	−0.321050	−	0.947062i	$-0.604036\pi$
$227$	−9.67420	−0.642099	−0.321050	+	0.947062i	$0.604036\pi$
$228$	0	0
$229$	−13.5486	−0.895320	−0.447660	−	0.894204i	$-0.647742\pi$
$229$	−13.5486	−0.895320	−0.447660	+	0.894204i	$0.647742\pi$
$230$	0	0
$231$	−1.17009	−0.0769860
$232$	0	0
$233$	8.38962	0.549622	0.274811	−	0.961498i	$-0.411385\pi$
$233$	8.38962	0.549622	0.274811	+	0.961498i	$0.411385\pi$
$234$	0	0
$235$	−5.90829	−0.385414
$236$	0	0
$237$	17.0205	1.10560
$238$	0	0
$239$	29.4908	1.90760	0.953800	−	0.300442i	$-0.0971341\pi$
$239$	29.4908	1.90760	0.953800	+	0.300442i	$0.0971341\pi$
$240$	0	0
$241$	3.64423	0.234745	0.117373	−	0.993088i	$-0.462553\pi$
$241$	3.64423	0.234745	0.117373	+	0.993088i	$0.462553\pi$
$242$	0	0
$243$	−14.5620	−0.934151
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0.0845208	0.00537793
$248$	0	0
$249$	−9.97334	−0.632035
$250$	0	0
$251$	−23.1350	−1.46027	−0.730135	−	0.683303i	$-0.760543\pi$
$251$	−23.1350	−1.46027	−0.730135	+	0.683303i	$0.760543\pi$
$252$	0	0
$253$	5.70928	0.358939
$254$	0	0
$255$	−6.44748	−0.403757
$256$	0	0
$257$	20.8104	1.29812	0.649060	−	0.760737i	$-0.275162\pi$
$257$	20.8104	1.29812	0.649060	+	0.760737i	$0.275162\pi$
$258$	0	0
$259$	−8.78765	−0.546038
$260$	0	0
$261$	−2.31351	−0.143203
$262$	0	0
$263$	23.7009	1.46146	0.730729	−	0.682668i	$-0.239180\pi$
$263$	23.7009	1.46146	0.730729	+	0.682668i	$0.239180\pi$
$264$	0	0
$265$	10.0494	0.617333
$266$	0	0
$267$	3.31965	0.203160
$268$	0	0
$269$	−3.50307	−0.213586	−0.106793	−	0.994281i	$-0.534058\pi$
$269$	−3.50307	−0.213586	−0.106793	+	0.994281i	$0.534058\pi$
$270$	0	0
$271$	8.49693	0.516152	0.258076	−	0.966125i	$-0.416912\pi$
$271$	8.49693	0.516152	0.258076	+	0.966125i	$0.416912\pi$
$272$	0	0
$273$	0.107307	0.00649452
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−25.9649	−1.56008	−0.780041	−	0.625729i	$-0.784802\pi$
$277$	−25.9649	−1.56008	−0.780041	+	0.625729i	$0.784802\pi$
$278$	0	0
$279$	1.43415	0.0858603
$280$	0	0
$281$	11.6742	0.696425	0.348212	−	0.937416i	$-0.386789\pi$
$281$	11.6742	0.696425	0.348212	+	0.937416i	$0.386789\pi$
$282$	0	0
$283$	14.2557	0.847411	0.423705	−	0.905800i	$-0.360729\pi$
$283$	14.2557	0.847411	0.423705	+	0.905800i	$0.360729\pi$
$284$	0	0
$285$	−1.07838	−0.0638776
$286$	0	0
$287$	−1.61757	−0.0954819
$288$	0	0
$289$	13.3630	0.786056
$290$	0	0
$291$	−16.6803	−0.977819
$292$	0	0
$293$	25.1122	1.46707	0.733536	−	0.679651i	$-0.237869\pi$
$293$	25.1122	1.46707	0.733536	+	0.679651i	$0.237869\pi$
$294$	0	0
$295$	−2.14116	−0.124663
$296$	0	0
$297$	5.41855	0.314416
$298$	0	0
$299$	−0.523590	−0.0302800
$300$	0	0
$301$	−3.86603	−0.222834
$302$	0	0
$303$	−10.5730	−0.607405
$304$	0	0
$305$	3.03612	0.173848
$306$	0	0
$307$	−8.02666	−0.458106	−0.229053	−	0.973414i	$-0.573563\pi$
$307$	−8.02666	−0.458106	−0.229053	+	0.973414i	$0.573563\pi$
$308$	0	0
$309$	3.89269	0.221448
$310$	0	0
$311$	26.3968	1.49683	0.748413	−	0.663233i	$-0.230816\pi$
$311$	26.3968	1.49683	0.748413	+	0.663233i	$0.230816\pi$
$312$	0	0
$313$	25.7321	1.45446	0.727231	−	0.686393i	$-0.240807\pi$
$313$	25.7321	1.45446	0.727231	+	0.686393i	$0.240807\pi$
$314$	0	0
$315$	1.63090	0.0918907
$316$	0	0
$317$	6.31351	0.354602	0.177301	−	0.984157i	$-0.443263\pi$
$317$	6.31351	0.354602	0.177301	+	0.984157i	$0.443263\pi$
$318$	0	0
$319$	1.41855	0.0794236
$320$	0	0
$321$	9.47641	0.528922
$322$	0	0
$323$	5.07838	0.282568
$324$	0	0
$325$	−0.0917087	−0.00508709
$326$	0	0
$327$	−17.7587	−0.982060
$328$	0	0
$329$	5.90829	0.325735
$330$	0	0
$331$	−3.50307	−0.192546	−0.0962731	−	0.995355i	$-0.530692\pi$
$331$	−3.50307	−0.192546	−0.0962731	+	0.995355i	$0.530692\pi$
$332$	0	0
$333$	14.3318	0.785376
$334$	0	0
$335$	−1.52586	−0.0833665
$336$	0	0
$337$	7.57918	0.412864	0.206432	−	0.978461i	$-0.433815\pi$
$337$	7.57918	0.412864	0.206432	+	0.978461i	$0.433815\pi$
$338$	0	0
$339$	8.28231	0.449834
$340$	0	0
$341$	−0.879362	−0.0476201
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	6.68035	0.359658
$346$	0	0
$347$	35.4824	1.90479	0.952397	−	0.304861i	$-0.0986100\pi$
$347$	35.4824	1.90479	0.952397	+	0.304861i	$0.0986100\pi$
$348$	0	0
$349$	−13.6586	−0.731128	−0.365564	−	0.930786i	$-0.619124\pi$
$349$	−13.6586	−0.731128	−0.365564	+	0.930786i	$0.619124\pi$
$350$	0	0
$351$	−0.496928	−0.0265241
$352$	0	0
$353$	26.4657	1.40863	0.704314	−	0.709888i	$-0.251255\pi$
$353$	26.4657	1.40863	0.704314	+	0.709888i	$0.251255\pi$
$354$	0	0
$355$	4.09890	0.217547
$356$	0	0
$357$	6.44748	0.341237
$358$	0	0
$359$	15.3958	0.812557	0.406279	−	0.913749i	$-0.366826\pi$
$359$	15.3958	0.812557	0.406279	+	0.913749i	$0.366826\pi$
$360$	0	0
$361$	−18.1506	−0.955295
$362$	0	0
$363$	−1.17009	−0.0614136
$364$	0	0
$365$	−14.1906	−0.742770
$366$	0	0
$367$	34.6875	1.81067	0.905337	−	0.424693i	$-0.139618\pi$
$367$	34.6875	1.81067	0.905337	+	0.424693i	$0.139618\pi$
$368$	0	0
$369$	2.63809	0.137333
$370$	0	0
$371$	−10.0494	−0.521741
$372$	0	0
$373$	36.3584	1.88257	0.941284	−	0.337616i	$-0.109621\pi$
$373$	36.3584	1.88257	0.941284	+	0.337616i	$0.109621\pi$
$374$	0	0
$375$	1.17009	0.0604230
$376$	0	0
$377$	−0.130094	−0.00670016
$378$	0	0
$379$	33.1461	1.70260	0.851300	−	0.524680i	$-0.175815\pi$
$379$	33.1461	1.70260	0.851300	+	0.524680i	$0.175815\pi$
$380$	0	0
$381$	−11.3028	−0.579062
$382$	0	0
$383$	34.2628	1.75075	0.875375	−	0.483445i	$-0.160615\pi$
$383$	34.2628	1.75075	0.875375	+	0.483445i	$0.160615\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	6.30510	0.320506
$388$	0	0
$389$	23.2762	1.18015	0.590074	−	0.807349i	$-0.299098\pi$
$389$	23.2762	1.18015	0.590074	+	0.807349i	$0.299098\pi$
$390$	0	0
$391$	−31.4596	−1.59098
$392$	0	0
$393$	5.47641	0.276248
$394$	0	0
$395$	14.5464	0.731908
$396$	0	0
$397$	−29.8576	−1.49851	−0.749256	−	0.662281i	$-0.769588\pi$
$397$	−29.8576	−1.49851	−0.749256	+	0.662281i	$0.769588\pi$
$398$	0	0
$399$	1.07838	0.0539864
$400$	0	0
$401$	−5.51745	−0.275528	−0.137764	−	0.990465i	$-0.543992\pi$
$401$	−5.51745	−0.275528	−0.137764	+	0.990465i	$0.543992\pi$
$402$	0	0
$403$	0.0806452	0.00401722
$404$	0	0
$405$	1.44748	0.0719259
$406$	0	0
$407$	−8.78765	−0.435588
$408$	0	0
$409$	−3.43415	−0.169808	−0.0849039	−	0.996389i	$-0.527058\pi$
$409$	−3.43415	−0.169808	−0.0849039	+	0.996389i	$0.527058\pi$
$410$	0	0
$411$	−10.3980	−0.512897
$412$	0	0
$413$	2.14116	0.105359
$414$	0	0
$415$	−8.52359	−0.418407
$416$	0	0
$417$	−17.5753	−0.860666
$418$	0	0
$419$	−18.2134	−0.889782	−0.444891	−	0.895585i	$-0.646758\pi$
$419$	−18.2134	−0.889782	−0.444891	+	0.895585i	$0.646758\pi$
$420$	0	0
$421$	10.6576	0.519418	0.259709	−	0.965687i	$-0.416373\pi$
$421$	10.6576	0.519418	0.259709	+	0.965687i	$0.416373\pi$
$422$	0	0
$423$	−9.63582	−0.468510
$424$	0	0
$425$	−5.51026	−0.267287
$426$	0	0
$427$	−3.03612	−0.146928
$428$	0	0
$429$	0.107307	0.00518084
$430$	0	0
$431$	7.61038	0.366579	0.183290	−	0.983059i	$-0.441325\pi$
$431$	7.61038	0.366579	0.183290	+	0.983059i	$0.441325\pi$
$432$	0	0
$433$	33.5318	1.61144	0.805718	−	0.592299i	$-0.201780\pi$
$433$	33.5318	1.61144	0.805718	+	0.592299i	$0.201780\pi$
$434$	0	0
$435$	1.65983	0.0795826
$436$	0	0
$437$	−5.26180	−0.251706
$438$	0	0
$439$	13.7587	0.656668	0.328334	−	0.944562i	$-0.393513\pi$
$439$	13.7587	0.656668	0.328334	+	0.944562i	$0.393513\pi$
$440$	0	0
$441$	−1.63090	−0.0776618
$442$	0	0
$443$	11.1689	0.530649	0.265324	−	0.964159i	$-0.414521\pi$
$443$	11.1689	0.530649	0.265324	+	0.964159i	$0.414521\pi$
$444$	0	0
$445$	2.83710	0.134492
$446$	0	0
$447$	16.0312	0.758250
$448$	0	0
$449$	−19.0700	−0.899967	−0.449984	−	0.893037i	$-0.648570\pi$
$449$	−19.0700	−0.899967	−0.449984	+	0.893037i	$0.648570\pi$
$450$	0	0
$451$	−1.61757	−0.0761682
$452$	0	0
$453$	1.23513	0.0580316
$454$	0	0
$455$	0.0917087	0.00429937
$456$	0	0
$457$	−0.787653	−0.0368449	−0.0184224	−	0.999830i	$-0.505864\pi$
$457$	−0.787653	−0.0368449	−0.0184224	+	0.999830i	$0.505864\pi$
$458$	0	0
$459$	−29.8576	−1.39363
$460$	0	0
$461$	25.5864	1.19168	0.595838	−	0.803105i	$-0.296820\pi$
$461$	25.5864	1.19168	0.595838	+	0.803105i	$0.296820\pi$
$462$	0	0
$463$	14.2595	0.662696	0.331348	−	0.943509i	$-0.392497\pi$
$463$	14.2595	0.662696	0.331348	+	0.943509i	$0.392497\pi$
$464$	0	0
$465$	−1.02893	−0.0477155
$466$	0	0
$467$	−18.3474	−0.849015	−0.424507	−	0.905425i	$-0.639553\pi$
$467$	−18.3474	−0.849015	−0.424507	+	0.905425i	$0.639553\pi$
$468$	0	0
$469$	1.52586	0.0704576
$470$	0	0
$471$	20.7792	0.957457
$472$	0	0
$473$	−3.86603	−0.177760
$474$	0	0
$475$	−0.921622	−0.0422869
$476$	0	0
$477$	16.3896	0.750429
$478$	0	0
$479$	−12.1711	−0.556113	−0.278057	−	0.960565i	$-0.589690\pi$
$479$	−12.1711	−0.556113	−0.278057	+	0.960565i	$0.589690\pi$
$480$	0	0
$481$	0.805905	0.0367461
$482$	0	0
$483$	−6.68035	−0.303966
$484$	0	0
$485$	−14.2557	−0.647316
$486$	0	0
$487$	4.23287	0.191809	0.0959047	−	0.995391i	$-0.469426\pi$
$487$	4.23287	0.191809	0.0959047	+	0.995391i	$0.469426\pi$
$488$	0	0
$489$	−13.4186	−0.606808
$490$	0	0
$491$	−22.0183	−0.993670	−0.496835	−	0.867845i	$-0.665505\pi$
$491$	−22.0183	−0.993670	−0.496835	+	0.867845i	$0.665505\pi$
$492$	0	0
$493$	−7.81658	−0.352041
$494$	0	0
$495$	1.63090	0.0733034
$496$	0	0
$497$	−4.09890	−0.183861
$498$	0	0
$499$	−32.9939	−1.47701	−0.738504	−	0.674249i	$-0.764467\pi$
$499$	−32.9939	−1.47701	−0.738504	+	0.674249i	$0.764467\pi$
$500$	0	0
$501$	−6.55479	−0.292846
$502$	0	0
$503$	29.0349	1.29460	0.647301	−	0.762235i	$-0.275898\pi$
$503$	29.0349	1.29460	0.647301	+	0.762235i	$0.275898\pi$
$504$	0	0
$505$	−9.03612	−0.402102
$506$	0	0
$507$	15.2013	0.675113
$508$	0	0
$509$	−21.5031	−0.953107	−0.476553	−	0.879146i	$-0.658114\pi$
$509$	−21.5031	−0.953107	−0.476553	+	0.879146i	$0.658114\pi$
$510$	0	0
$511$	14.1906	0.627755
$512$	0	0
$513$	−4.99386	−0.220484
$514$	0	0
$515$	3.32684	0.146598
$516$	0	0
$517$	5.90829	0.259846
$518$	0	0
$519$	−25.2846	−1.10987
$520$	0	0
$521$	24.5958	1.07756	0.538781	−	0.842446i	$-0.318885\pi$
$521$	24.5958	1.07756	0.538781	+	0.842446i	$0.318885\pi$
$522$	0	0
$523$	−38.0677	−1.66458	−0.832292	−	0.554337i	$-0.812972\pi$
$523$	−38.0677	−1.66458	−0.832292	+	0.554337i	$0.812972\pi$
$524$	0	0
$525$	−1.17009	−0.0510668
$526$	0	0
$527$	4.84551	0.211074
$528$	0	0
$529$	9.59583	0.417210
$530$	0	0
$531$	−3.49201	−0.151540
$532$	0	0
$533$	0.148345	0.00642554
$534$	0	0
$535$	8.09890	0.350146
$536$	0	0
$537$	−2.40787	−0.103907
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	2.13009	0.0915799	0.0457899	−	0.998951i	$-0.485420\pi$
$541$	2.13009	0.0915799	0.0457899	+	0.998951i	$0.485420\pi$
$542$	0	0
$543$	−23.7321	−1.01844
$544$	0	0
$545$	−15.1773	−0.650123
$546$	0	0
$547$	9.13170	0.390443	0.195222	−	0.980759i	$-0.437457\pi$
$547$	9.13170	0.390443	0.195222	+	0.980759i	$0.437457\pi$
$548$	0	0
$549$	4.95160	0.211329
$550$	0	0
$551$	−1.30737	−0.0556957
$552$	0	0
$553$	−14.5464	−0.618575
$554$	0	0
$555$	−10.2823	−0.436460
$556$	0	0
$557$	−11.7093	−0.496138	−0.248069	−	0.968742i	$-0.579796\pi$
$557$	−11.7093	−0.496138	−0.248069	+	0.968742i	$0.579796\pi$
$558$	0	0
$559$	0.354549	0.0149958
$560$	0	0
$561$	6.44748	0.272213
$562$	0	0
$563$	−12.5958	−0.530851	−0.265425	−	0.964131i	$-0.585512\pi$
$563$	−12.5958	−0.530851	−0.265425	+	0.964131i	$0.585512\pi$
$564$	0	0
$565$	7.07838	0.297790
$566$	0	0
$567$	−1.44748	−0.0607885
$568$	0	0
$569$	7.54411	0.316266	0.158133	−	0.987418i	$-0.449453\pi$
$569$	7.54411	0.316266	0.158133	+	0.987418i	$0.449453\pi$
$570$	0	0
$571$	−36.8104	−1.54047	−0.770234	−	0.637761i	$-0.779861\pi$
$571$	−36.8104	−1.54047	−0.770234	+	0.637761i	$0.779861\pi$
$572$	0	0
$573$	−23.7321	−0.991421
$574$	0	0
$575$	5.70928	0.238093
$576$	0	0
$577$	−39.5174	−1.64513	−0.822566	−	0.568669i	$-0.807459\pi$
$577$	−39.5174	−1.64513	−0.822566	+	0.568669i	$0.807459\pi$
$578$	0	0
$579$	28.4391	1.18189
$580$	0	0
$581$	8.52359	0.353618
$582$	0	0
$583$	−10.0494	−0.416206
$584$	0	0
$585$	−0.149568	−0.00618386
$586$	0	0
$587$	−1.56812	−0.0647232	−0.0323616	−	0.999476i	$-0.510303\pi$
$587$	−1.56812	−0.0647232	−0.0323616	+	0.999476i	$0.510303\pi$
$588$	0	0
$589$	0.810439	0.0333936
$590$	0	0
$591$	16.5548	0.680973
$592$	0	0
$593$	4.95547	0.203497	0.101748	−	0.994810i	$-0.467556\pi$
$593$	4.95547	0.203497	0.101748	+	0.994810i	$0.467556\pi$
$594$	0	0
$595$	5.51026	0.225899
$596$	0	0
$597$	12.2641	0.501935
$598$	0	0
$599$	12.3668	0.505295	0.252648	−	0.967558i	$-0.418699\pi$
$599$	12.3668	0.505295	0.252648	+	0.967558i	$0.418699\pi$
$600$	0	0
$601$	24.9516	1.01780	0.508898	−	0.860827i	$-0.330053\pi$
$601$	24.9516	1.01780	0.508898	+	0.860827i	$0.330053\pi$
$602$	0	0
$603$	−2.48852	−0.101340
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	19.2762	0.782396	0.391198	−	0.920307i	$-0.372061\pi$
$607$	19.2762	0.782396	0.391198	+	0.920307i	$0.372061\pi$
$608$	0	0
$609$	−1.65983	−0.0672596
$610$	0	0
$611$	−0.541842	−0.0219206
$612$	0	0
$613$	−42.6986	−1.72458	−0.862290	−	0.506415i	$-0.830971\pi$
$613$	−42.6986	−1.72458	−0.862290	+	0.506415i	$0.830971\pi$
$614$	0	0
$615$	−1.89269	−0.0763207
$616$	0	0
$617$	31.7770	1.27929	0.639646	−	0.768669i	$-0.279081\pi$
$617$	31.7770	1.27929	0.639646	+	0.768669i	$0.279081\pi$
$618$	0	0
$619$	19.8420	0.797518	0.398759	−	0.917056i	$-0.369441\pi$
$619$	19.8420	0.797518	0.398759	+	0.917056i	$0.369441\pi$
$620$	0	0
$621$	30.9360	1.24142
$622$	0	0
$623$	−2.83710	−0.113666
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.07838	0.0430663
$628$	0	0
$629$	48.4222	1.93072
$630$	0	0
$631$	3.63317	0.144634	0.0723170	−	0.997382i	$-0.476961\pi$
$631$	3.63317	0.144634	0.0723170	+	0.997382i	$0.476961\pi$
$632$	0	0
$633$	−3.10504	−0.123414
$634$	0	0
$635$	−9.65983	−0.383339
$636$	0	0
$637$	−0.0917087	−0.00363363
$638$	0	0
$639$	6.68488	0.264450
$640$	0	0
$641$	37.5402	1.48275	0.741375	−	0.671091i	$-0.234174\pi$
$641$	37.5402	1.48275	0.741375	+	0.671091i	$0.234174\pi$
$642$	0	0
$643$	34.8710	1.37518	0.687588	−	0.726101i	$-0.258670\pi$
$643$	34.8710	1.37518	0.687588	+	0.726101i	$0.258670\pi$
$644$	0	0
$645$	−4.52359	−0.178116
$646$	0	0
$647$	−30.6342	−1.20436	−0.602178	−	0.798362i	$-0.705700\pi$
$647$	−30.6342	−1.20436	−0.602178	+	0.798362i	$0.705700\pi$
$648$	0	0
$649$	2.14116	0.0840478
$650$	0	0
$651$	1.02893	0.0403269
$652$	0	0
$653$	5.40417	0.211482	0.105741	−	0.994394i	$-0.466279\pi$
$653$	5.40417	0.211482	0.105741	+	0.994394i	$0.466279\pi$
$654$	0	0
$655$	4.68035	0.182876
$656$	0	0
$657$	−23.1434	−0.902911
$658$	0	0
$659$	19.4101	0.756112	0.378056	−	0.925783i	$-0.376593\pi$
$659$	19.4101	0.756112	0.378056	+	0.925783i	$0.376593\pi$
$660$	0	0
$661$	−26.3090	−1.02330	−0.511650	−	0.859194i	$-0.670966\pi$
$661$	−26.3090	−1.02330	−0.511650	+	0.859194i	$0.670966\pi$
$662$	0	0
$663$	−0.591290	−0.0229638
$664$	0	0
$665$	0.921622	0.0357390
$666$	0	0
$667$	8.09890	0.313591
$668$	0	0
$669$	−10.1483	−0.392358
$670$	0	0
$671$	−3.03612	−0.117208
$672$	0	0
$673$	−15.7938	−0.608806	−0.304403	−	0.952543i	$-0.598457\pi$
$673$	−15.7938	−0.608806	−0.304403	+	0.952543i	$0.598457\pi$
$674$	0	0
$675$	5.41855	0.208560
$676$	0	0
$677$	−15.7081	−0.603710	−0.301855	−	0.953354i	$-0.597606\pi$
$677$	−15.7081	−0.603710	−0.301855	+	0.953354i	$0.597606\pi$
$678$	0	0
$679$	14.2557	0.547082
$680$	0	0
$681$	11.3197	0.433770
$682$	0	0
$683$	44.0326	1.68486	0.842431	−	0.538805i	$-0.181124\pi$
$683$	44.0326	1.68486	0.842431	+	0.538805i	$0.181124\pi$
$684$	0	0
$685$	−8.88655	−0.339538
$686$	0	0
$687$	15.8531	0.604833
$688$	0	0
$689$	0.921622	0.0351110
$690$	0	0
$691$	−4.63809	−0.176441	−0.0882205	−	0.996101i	$-0.528118\pi$
$691$	−4.63809	−0.176441	−0.0882205	+	0.996101i	$0.528118\pi$
$692$	0	0
$693$	−1.63090	−0.0619527
$694$	0	0
$695$	−15.0205	−0.569761
$696$	0	0
$697$	8.91321	0.337612
$698$	0	0
$699$	−9.81658	−0.371297
$700$	0	0
$701$	−14.6491	−0.553291	−0.276645	−	0.960972i	$-0.589223\pi$
$701$	−14.6491	−0.553291	−0.276645	+	0.960972i	$0.589223\pi$
$702$	0	0
$703$	8.09890	0.305456
$704$	0	0
$705$	6.91321	0.260367
$706$	0	0
$707$	9.03612	0.339838
$708$	0	0
$709$	−25.5174	−0.958328	−0.479164	−	0.877725i	$-0.659060\pi$
$709$	−25.5174	−0.958328	−0.479164	+	0.877725i	$0.659060\pi$
$710$	0	0
$711$	23.7237	0.889706
$712$	0	0
$713$	−5.02052	−0.188020
$714$	0	0
$715$	0.0917087	0.00342971
$716$	0	0
$717$	−34.5068	−1.28868
$718$	0	0
$719$	−39.2918	−1.46534	−0.732668	−	0.680586i	$-0.761725\pi$
$719$	−39.2918	−1.46534	−0.732668	+	0.680586i	$0.761725\pi$
$720$	0	0
$721$	−3.32684	−0.123898
$722$	0	0
$723$	−4.26406	−0.158582
$724$	0	0
$725$	1.41855	0.0526837
$726$	0	0
$727$	37.7081	1.39851	0.699257	−	0.714870i	$-0.253514\pi$
$727$	37.7081	1.39851	0.699257	+	0.714870i	$0.253514\pi$
$728$	0	0
$729$	21.3812	0.791897
$730$	0	0
$731$	21.3028	0.787914
$732$	0	0
$733$	−4.34736	−0.160573	−0.0802867	−	0.996772i	$-0.525584\pi$
$733$	−4.34736	−0.160573	−0.0802867	+	0.996772i	$0.525584\pi$
$734$	0	0
$735$	1.17009	0.0431593
$736$	0	0
$737$	1.52586	0.0562057
$738$	0	0
$739$	−38.1568	−1.40362	−0.701809	−	0.712365i	$-0.747624\pi$
$739$	−38.1568	−1.40362	−0.701809	+	0.712365i	$0.747624\pi$
$740$	0	0
$741$	−0.0988967	−0.00363306
$742$	0	0
$743$	29.2618	1.07351	0.536756	−	0.843738i	$-0.319650\pi$
$743$	29.2618	1.07351	0.536756	+	0.843738i	$0.319650\pi$
$744$	0	0
$745$	13.7009	0.501961
$746$	0	0
$747$	−13.9011	−0.508615
$748$	0	0
$749$	−8.09890	−0.295927
$750$	0	0
$751$	−41.6886	−1.52124	−0.760619	−	0.649199i	$-0.775104\pi$
$751$	−41.6886	−1.52124	−0.760619	+	0.649199i	$0.775104\pi$
$752$	0	0
$753$	27.0700	0.986484
$754$	0	0
$755$	1.05559	0.0384169
$756$	0	0
$757$	39.7419	1.44444	0.722222	−	0.691661i	$-0.243121\pi$
$757$	39.7419	1.44444	0.722222	+	0.691661i	$0.243121\pi$
$758$	0	0
$759$	−6.68035	−0.242481
$760$	0	0
$761$	−36.4112	−1.31990	−0.659952	−	0.751308i	$-0.729423\pi$
$761$	−36.4112	−1.31990	−0.659952	+	0.751308i	$0.729423\pi$
$762$	0	0
$763$	15.1773	0.549454
$764$	0	0
$765$	−8.98667	−0.324914
$766$	0	0
$767$	−0.196363	−0.00709025
$768$	0	0
$769$	−10.5347	−0.379889	−0.189945	−	0.981795i	$-0.560831\pi$
$769$	−10.5347	−0.379889	−0.189945	+	0.981795i	$0.560831\pi$
$770$	0	0
$771$	−24.3500	−0.876944
$772$	0	0
$773$	−1.52198	−0.0547419	−0.0273709	−	0.999625i	$-0.508714\pi$
$773$	−1.52198	−0.0547419	−0.0273709	+	0.999625i	$0.508714\pi$
$774$	0	0
$775$	−0.879362	−0.0315876
$776$	0	0
$777$	10.2823	0.368876
$778$	0	0
$779$	1.49079	0.0534129
$780$	0	0
$781$	−4.09890	−0.146670
$782$	0	0
$783$	7.68649	0.274693
$784$	0	0
$785$	17.7587	0.633836
$786$	0	0
$787$	15.6020	0.556150	0.278075	−	0.960559i	$-0.410304\pi$
$787$	15.6020	0.556150	0.278075	+	0.960559i	$0.410304\pi$
$788$	0	0
$789$	−27.7321	−0.987288
$790$	0	0
$791$	−7.07838	−0.251678
$792$	0	0
$793$	0.278438	0.00988764
$794$	0	0
$795$	−11.7587	−0.417039
$796$	0	0
$797$	12.6491	0.448056	0.224028	−	0.974583i	$-0.428079\pi$
$797$	12.6491	0.448056	0.224028	+	0.974583i	$0.428079\pi$
$798$	0	0
$799$	−32.5562	−1.15176
$800$	0	0
$801$	4.62702	0.163488
$802$	0	0
$803$	14.1906	0.500776
$804$	0	0
$805$	−5.70928	−0.201226
$806$	0	0
$807$	4.09890	0.144288
$808$	0	0
$809$	−49.9299	−1.75544	−0.877720	−	0.479174i	$-0.840936\pi$
$809$	−49.9299	−1.75544	−0.877720	+	0.479174i	$0.840936\pi$
$810$	0	0
$811$	−7.95896	−0.279477	−0.139738	−	0.990188i	$-0.544626\pi$
$811$	−7.95896	−0.279477	−0.139738	+	0.990188i	$0.544626\pi$
$812$	0	0
$813$	−9.94214	−0.348686
$814$	0	0
$815$	−11.4680	−0.401706
$816$	0	0
$817$	3.56302	0.124654
$818$	0	0
$819$	0.149568	0.00522631
$820$	0	0
$821$	4.92162	0.171766	0.0858829	−	0.996305i	$-0.472629\pi$
$821$	4.92162	0.171766	0.0858829	+	0.996305i	$0.472629\pi$
$822$	0	0
$823$	4.04945	0.141155	0.0705774	−	0.997506i	$-0.477516\pi$
$823$	4.04945	0.141155	0.0705774	+	0.997506i	$0.477516\pi$
$824$	0	0
$825$	−1.17009	−0.0407372
$826$	0	0
$827$	−33.3256	−1.15885	−0.579423	−	0.815027i	$-0.696722\pi$
$827$	−33.3256	−1.15885	−0.579423	+	0.815027i	$0.696722\pi$
$828$	0	0
$829$	−0.156755	−0.00544434	−0.00272217	−	0.999996i	$-0.500866\pi$
$829$	−0.156755	−0.00544434	−0.00272217	+	0.999996i	$0.500866\pi$
$830$	0	0
$831$	30.3812	1.05391
$832$	0	0
$833$	−5.51026	−0.190919
$834$	0	0
$835$	−5.60197	−0.193864
$836$	0	0
$837$	−4.76487	−0.164698
$838$	0	0
$839$	49.6775	1.71506	0.857529	−	0.514435i	$-0.171998\pi$
$839$	49.6775	1.71506	0.857529	+	0.514435i	$0.171998\pi$
$840$	0	0
$841$	−26.9877	−0.930611
$842$	0	0
$843$	−13.6598	−0.470469
$844$	0	0
$845$	12.9916	0.446924
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−16.6803	−0.572468
$850$	0	0
$851$	−50.1711	−1.71984
$852$	0	0
$853$	39.4257	1.34991	0.674956	−	0.737858i	$-0.264163\pi$
$853$	39.4257	1.34991	0.674956	+	0.737858i	$0.264163\pi$
$854$	0	0
$855$	−1.50307	−0.0514040
$856$	0	0
$857$	33.2423	1.13554	0.567768	−	0.823189i	$-0.307807\pi$
$857$	33.2423	1.13554	0.567768	+	0.823189i	$0.307807\pi$
$858$	0	0
$859$	−7.71646	−0.263282	−0.131641	−	0.991297i	$-0.542025\pi$
$859$	−7.71646	−0.263282	−0.131641	+	0.991297i	$0.542025\pi$
$860$	0	0
$861$	1.89269	0.0645028
$862$	0	0
$863$	24.6453	0.838935	0.419467	−	0.907770i	$-0.362217\pi$
$863$	24.6453	0.838935	0.419467	+	0.907770i	$0.362217\pi$
$864$	0	0
$865$	−21.6092	−0.734733
$866$	0	0
$867$	−15.6358	−0.531020
$868$	0	0
$869$	−14.5464	−0.493452
$870$	0	0
$871$	−0.139935	−0.00474150
$872$	0	0
$873$	−23.2495	−0.786877
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	6.17954	0.208668	0.104334	−	0.994542i	$-0.466729\pi$
$877$	6.17954	0.208668	0.104334	+	0.994542i	$0.466729\pi$
$878$	0	0
$879$	−29.3835	−0.991080
$880$	0	0
$881$	−49.3295	−1.66195	−0.830976	−	0.556308i	$-0.812218\pi$
$881$	−49.3295	−1.66195	−0.830976	+	0.556308i	$0.812218\pi$
$882$	0	0
$883$	22.1529	0.745504	0.372752	−	0.927931i	$-0.378414\pi$
$883$	22.1529	0.745504	0.372752	+	0.927931i	$0.378414\pi$
$884$	0	0
$885$	2.50534	0.0842160
$886$	0	0
$887$	−38.7358	−1.30062	−0.650310	−	0.759669i	$-0.725361\pi$
$887$	−38.7358	−1.30062	−0.650310	+	0.759669i	$0.725361\pi$
$888$	0	0
$889$	9.65983	0.323980
$890$	0	0
$891$	−1.44748	−0.0484924
$892$	0	0
$893$	−5.44521	−0.182217
$894$	0	0
$895$	−2.05786	−0.0687866
$896$	0	0
$897$	0.612646	0.0204557
$898$	0	0
$899$	−1.24742	−0.0416038
$900$	0	0
$901$	55.3751	1.84481
$902$	0	0
$903$	4.52359	0.150536
$904$	0	0
$905$	−20.2823	−0.674207
$906$	0	0
$907$	−18.4352	−0.612131	−0.306065	−	0.952011i	$-0.599013\pi$
$907$	−18.4352	−0.612131	−0.306065	+	0.952011i	$0.599013\pi$
$908$	0	0
$909$	−14.7370	−0.488795
$910$	0	0
$911$	11.9011	0.394301	0.197151	−	0.980373i	$-0.436831\pi$
$911$	11.9011	0.394301	0.197151	+	0.980373i	$0.436831\pi$
$912$	0	0
$913$	8.52359	0.282090
$914$	0	0
$915$	−3.55252	−0.117443
$916$	0	0
$917$	−4.68035	−0.154559
$918$	0	0
$919$	−56.3812	−1.85984	−0.929922	−	0.367756i	$-0.880126\pi$
$919$	−56.3812	−1.85984	−0.929922	+	0.367756i	$0.880126\pi$
$920$	0	0
$921$	9.39189	0.309473
$922$	0	0
$923$	0.375905	0.0123731
$924$	0	0
$925$	−8.78765	−0.288936
$926$	0	0
$927$	5.42574	0.178205
$928$	0	0
$929$	19.2351	0.631084	0.315542	−	0.948912i	$-0.397814\pi$
$929$	19.2351	0.631084	0.315542	+	0.948912i	$0.397814\pi$
$930$	0	0
$931$	−0.921622	−0.0302049
$932$	0	0
$933$	−30.8865	−1.01118
$934$	0	0
$935$	5.51026	0.180205
$936$	0	0
$937$	17.6358	0.576137	0.288069	−	0.957610i	$-0.406987\pi$
$937$	17.6358	0.576137	0.288069	+	0.957610i	$0.406987\pi$
$938$	0	0
$939$	−30.1087	−0.982562
$940$	0	0
$941$	−20.1990	−0.658469	−0.329235	−	0.944248i	$-0.606791\pi$
$941$	−20.1990	−0.658469	−0.329235	+	0.944248i	$0.606791\pi$
$942$	0	0
$943$	−9.23513	−0.300737
$944$	0	0
$945$	−5.41855	−0.176265
$946$	0	0
$947$	−17.6925	−0.574928	−0.287464	−	0.957792i	$-0.592812\pi$
$947$	−17.6925	−0.574928	−0.287464	+	0.957792i	$0.592812\pi$
$948$	0	0
$949$	−1.30140	−0.0422453
$950$	0	0
$951$	−7.38735	−0.239551
$952$	0	0
$953$	39.7093	1.28631	0.643155	−	0.765736i	$-0.277625\pi$
$953$	39.7093	1.28631	0.643155	+	0.765736i	$0.277625\pi$
$954$	0	0
$955$	−20.2823	−0.656320
$956$	0	0
$957$	−1.65983	−0.0536546
$958$	0	0
$959$	8.88655	0.286962
$960$	0	0
$961$	−30.2267	−0.975056
$962$	0	0
$963$	13.2085	0.425637
$964$	0	0
$965$	24.3051	0.782409
$966$	0	0
$967$	−3.01664	−0.0970087	−0.0485044	−	0.998823i	$-0.515445\pi$
$967$	−3.01664	−0.0970087	−0.0485044	+	0.998823i	$0.515445\pi$
$968$	0	0
$969$	−5.94214	−0.190889
$970$	0	0
$971$	18.2134	0.584496	0.292248	−	0.956343i	$-0.405597\pi$
$971$	18.2134	0.584496	0.292248	+	0.956343i	$0.405597\pi$
$972$	0	0
$973$	15.0205	0.481536
$974$	0	0
$975$	0.107307	0.00343658
$976$	0	0
$977$	−30.0845	−0.962489	−0.481245	−	0.876586i	$-0.659815\pi$
$977$	−30.0845	−0.962489	−0.481245	+	0.876586i	$0.659815\pi$
$978$	0	0
$979$	−2.83710	−0.0906742
$980$	0	0
$981$	−24.7526	−0.790289
$982$	0	0
$983$	−5.92267	−0.188904	−0.0944519	−	0.995529i	$-0.530110\pi$
$983$	−5.92267	−0.188904	−0.0944519	+	0.995529i	$0.530110\pi$
$984$	0	0
$985$	14.1483	0.450804
$986$	0	0
$987$	−6.91321	−0.220050
$988$	0	0
$989$	−22.0722	−0.701856
$990$	0	0
$991$	−9.24742	−0.293754	−0.146877	−	0.989155i	$-0.546922\pi$
$991$	−9.24742	−0.293754	−0.146877	+	0.989155i	$0.546922\pi$
$992$	0	0
$993$	4.09890	0.130075
$994$	0	0
$995$	10.4813	0.332281
$996$	0	0
$997$	32.1496	1.01819	0.509094	−	0.860711i	$-0.329981\pi$
$997$	32.1496	1.01819	0.509094	+	0.860711i	$0.329981\pi$
$998$	0	0
$999$	−47.6163	−1.50651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bn.1.1		3
4.3	odd	2		385.2.a.f.1.2	✓	3
12.11	even	2		3465.2.a.bh.1.2		3
20.3	even	4		1925.2.b.n.1849.5		6
20.7	even	4		1925.2.b.n.1849.2		6
20.19	odd	2		1925.2.a.v.1.2		3
28.27	even	2		2695.2.a.g.1.2		3
44.43	even	2		4235.2.a.q.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.f.1.2	✓	3	4.3	odd	2
1925.2.a.v.1.2		3	20.19	odd	2
1925.2.b.n.1849.2		6	20.7	even	4
1925.2.b.n.1849.5		6	20.3	even	4
2695.2.a.g.1.2		3	28.27	even	2
3465.2.a.bh.1.2		3	12.11	even	2
4235.2.a.q.1.2		3	44.43	even	2
6160.2.a.bn.1.1		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)

\(f(q)\)	\(=\)	\(q-1.17009 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.63090 q^{9} +O(q^{10})\)	\(q-1.17009 q^{3} -1.00000 q^{5} +1.00000 q^{7} -1.63090 q^{9} +1.00000 q^{11} -0.0917087 q^{13} +1.17009 q^{15} -5.51026 q^{17} -0.921622 q^{19} -1.17009 q^{21} +5.70928 q^{23} +1.00000 q^{25} +5.41855 q^{27} +1.41855 q^{29} -0.879362 q^{31} -1.17009 q^{33} -1.00000 q^{35} -8.78765 q^{37} +0.107307 q^{39} -1.61757 q^{41} -3.86603 q^{43} +1.63090 q^{45} +5.90829 q^{47} +1.00000 q^{49} +6.44748 q^{51} -10.0494 q^{53} -1.00000 q^{55} +1.07838 q^{57} +2.14116 q^{59} -3.03612 q^{61} -1.63090 q^{63} +0.0917087 q^{65} +1.52586 q^{67} -6.68035 q^{69} -4.09890 q^{71} +14.1906 q^{73} -1.17009 q^{75} +1.00000 q^{77} -14.5464 q^{79} -1.44748 q^{81} +8.52359 q^{83} +5.51026 q^{85} -1.65983 q^{87} -2.83710 q^{89} -0.0917087 q^{91} +1.02893 q^{93} +0.921622 q^{95} +14.2557 q^{97} -1.63090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - q^9	\( 3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - q^{9} + 3 q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{19} + 2 q^{21} + 10 q^{23} + 3 q^{25} + 2 q^{27} - 10 q^{29} + 10 q^{31} + 2 q^{33} - 3 q^{35} - 16 q^{37} + 12 q^{39} + 2 q^{43} + q^{45} + 20 q^{47} + 3 q^{49} + 20 q^{51} - 12 q^{53} - 3 q^{55} - 14 q^{59} + 10 q^{61} - q^{63} - 2 q^{65} + 2 q^{67} + 2 q^{69} + 24 q^{71} + 4 q^{73} + 2 q^{75} + 3 q^{77} - 8 q^{79} - 5 q^{81} + 10 q^{83} - 16 q^{87} + 20 q^{89} + 2 q^{91} + 18 q^{93} + 6 q^{95} - q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^5 + 3 * q^7 - q^9 + 3 * q^11 + 2 * q^13 - 2 * q^15 - 6 * q^19 + 2 * q^21 + 10 * q^23 + 3 * q^25 + 2 * q^27 - 10 * q^29 + 10 * q^31 + 2 * q^33 - 3 * q^35 - 16 * q^37 + 12 * q^39 + 2 * q^43 + q^45 + 20 * q^47 + 3 * q^49 + 20 * q^51 - 12 * q^53 - 3 * q^55 - 14 * q^59 + 10 * q^61 - q^63 - 2 * q^65 + 2 * q^67 + 2 * q^69 + 24 * q^71 + 4 * q^73 + 2 * q^75 + 3 * q^77 - 8 * q^79 - 5 * q^81 + 10 * q^83 - 16 * q^87 + 20 * q^89 + 2 * q^91 + 18 * q^93 + 6 * q^95 - q^99

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