LMFDB - Embedded newform 6160.2.a.bh.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:	$1$
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 1540)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ 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)

$f(q)$	$=$	$q-1.67513 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.193937 q^{9} +O(q^{10})$	$q-1.67513 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.193937 q^{9} -1.00000 q^{11} -1.67513 q^{13} +1.67513 q^{15} -6.24965 q^{17} +3.73813 q^{19} -1.67513 q^{21} +8.73084 q^{23} +1.00000 q^{25} +5.35026 q^{27} -4.38787 q^{29} -7.44358 q^{31} +1.67513 q^{33} -1.00000 q^{35} +11.4314 q^{37} +2.80606 q^{39} +4.79384 q^{41} +10.5442 q^{43} +0.193937 q^{45} -6.89938 q^{47} +1.00000 q^{49} +10.4690 q^{51} -6.80606 q^{53} +1.00000 q^{55} -6.26187 q^{57} +3.13093 q^{59} -4.40597 q^{61} -0.193937 q^{63} +1.67513 q^{65} -2.46898 q^{67} -14.6253 q^{69} +8.31265 q^{71} -4.45088 q^{73} -1.67513 q^{75} -1.00000 q^{77} -0.231548 q^{79} -8.38058 q^{81} -16.2374 q^{83} +6.24965 q^{85} +7.35026 q^{87} -1.29948 q^{89} -1.67513 q^{91} +12.4690 q^{93} -3.73813 q^{95} -9.14903 q^{97} +0.193937 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 3 q^{25} + 6 q^{27} - 14 q^{29} - 6 q^{31} - 3 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} + 22 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} - 20 q^{53} + 3 q^{55} - 28 q^{57} + 14 q^{59} + 14 q^{61} - q^{63} + 24 q^{67} - 2 q^{69} + 4 q^{71} - 10 q^{73} - 3 q^{77} - 12 q^{79} - 13 q^{81} - 6 q^{83} + 2 q^{85} + 12 q^{87} - 24 q^{89} + 6 q^{93} - 2 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 - q^9 - 3 * q^11 - 2 * q^17 + 2 * q^19 + 4 * q^23 + 3 * q^25 + 6 * q^27 - 14 * q^29 - 6 * q^31 - 3 * q^35 - 8 * q^37 + 8 * q^39 - 12 * q^41 + 22 * q^43 + q^45 - 14 * q^47 + 3 * q^49 - 20 * q^53 + 3 * q^55 - 28 * q^57 + 14 * q^59 + 14 * q^61 - q^63 + 24 * q^67 - 2 * q^69 + 4 * q^71 - 10 * q^73 - 3 * q^77 - 12 * q^79 - 13 * q^81 - 6 * q^83 + 2 * q^85 + 12 * q^87 - 24 * q^89 + 6 * q^93 - 2 * q^95 - 4 * 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
$3$	−1.67513	−0.967137	−0.483569	−	0.875306i	$-0.660660\pi$
$3$	−1.67513	−0.967137	−0.483569	+	0.875306i	$0.660660\pi$
$4$	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
$9$	−0.193937	−0.0646455
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.67513	−0.464598	−0.232299	−	0.972644i	$-0.574625\pi$
$13$	−1.67513	−0.464598	−0.232299	+	0.972644i	$0.574625\pi$
$14$	0	0
$15$	1.67513	0.432517
$16$	0	0
$17$	−6.24965	−1.51576	−0.757881	−	0.652393i	$-0.773765\pi$
$17$	−6.24965	−1.51576	−0.757881	+	0.652393i	$0.773765\pi$
$18$	0	0
$19$	3.73813	0.857587	0.428793	−	0.903403i	$-0.358939\pi$
$19$	3.73813	0.857587	0.428793	+	0.903403i	$0.358939\pi$
$20$	0	0
$21$	−1.67513	−0.365544
$22$	0	0
$23$	8.73084	1.82051	0.910253	−	0.414052i	$-0.135887\pi$
$23$	8.73084	1.82051	0.910253	+	0.414052i	$0.135887\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.35026	1.02966
$28$	0	0
$29$	−4.38787	−0.814808	−0.407404	−	0.913248i	$-0.633566\pi$
$29$	−4.38787	−0.814808	−0.407404	+	0.913248i	$0.633566\pi$
$30$	0	0
$31$	−7.44358	−1.33691	−0.668453	−	0.743754i	$-0.733043\pi$
$31$	−7.44358	−1.33691	−0.668453	+	0.743754i	$0.733043\pi$
$32$	0	0
$33$	1.67513	0.291603
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	11.4314	1.87930	0.939652	−	0.342131i	$-0.111149\pi$
$37$	11.4314	1.87930	0.939652	+	0.342131i	$0.111149\pi$
$38$	0	0
$39$	2.80606	0.449330
$40$	0	0
$41$	4.79384	0.748673	0.374336	−	0.927293i	$-0.377871\pi$
$41$	4.79384	0.748673	0.374336	+	0.927293i	$0.377871\pi$
$42$	0	0
$43$	10.5442	1.60798	0.803988	−	0.594646i	$-0.202708\pi$
$43$	10.5442	1.60798	0.803988	+	0.594646i	$0.202708\pi$
$44$	0	0
$45$	0.193937	0.0289104
$46$	0	0
$47$	−6.89938	−1.00638	−0.503189	−	0.864176i	$-0.667840\pi$
$47$	−6.89938	−1.00638	−0.503189	+	0.864176i	$0.667840\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	10.4690	1.46595
$52$	0	0
$53$	−6.80606	−0.934885	−0.467442	−	0.884024i	$-0.654824\pi$
$53$	−6.80606	−0.934885	−0.467442	+	0.884024i	$0.654824\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−6.26187	−0.829404
$58$	0	0
$59$	3.13093	0.407613	0.203806	−	0.979011i	$-0.434669\pi$
$59$	3.13093	0.407613	0.203806	+	0.979011i	$0.434669\pi$
$60$	0	0
$61$	−4.40597	−0.564127	−0.282063	−	0.959396i	$-0.591019\pi$
$61$	−4.40597	−0.564127	−0.282063	+	0.959396i	$0.591019\pi$
$62$	0	0
$63$	−0.193937	−0.0244337
$64$	0	0
$65$	1.67513	0.207774
$66$	0	0
$67$	−2.46898	−0.301633	−0.150817	−	0.988562i	$-0.548190\pi$
$67$	−2.46898	−0.301633	−0.150817	+	0.988562i	$0.548190\pi$
$68$	0	0
$69$	−14.6253	−1.76068
$70$	0	0
$71$	8.31265	0.986530	0.493265	−	0.869879i	$-0.335803\pi$
$71$	8.31265	0.986530	0.493265	+	0.869879i	$0.335803\pi$
$72$	0	0
$73$	−4.45088	−0.520936	−0.260468	−	0.965483i	$-0.583877\pi$
$73$	−4.45088	−0.520936	−0.260468	+	0.965483i	$0.583877\pi$
$74$	0	0
$75$	−1.67513	−0.193427
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−0.231548	−0.0260512	−0.0130256	−	0.999915i	$-0.504146\pi$
$79$	−0.231548	−0.0260512	−0.0130256	+	0.999915i	$0.504146\pi$
$80$	0	0
$81$	−8.38058	−0.931175
$82$	0	0
$83$	−16.2374	−1.78229	−0.891145	−	0.453719i	$-0.850097\pi$
$83$	−16.2374	−1.78229	−0.891145	+	0.453719i	$0.850097\pi$
$84$	0	0
$85$	6.24965	0.677869
$86$	0	0
$87$	7.35026	0.788031
$88$	0	0
$89$	−1.29948	−0.137744	−0.0688721	−	0.997625i	$-0.521940\pi$
$89$	−1.29948	−0.137744	−0.0688721	+	0.997625i	$0.521940\pi$
$90$	0	0
$91$	−1.67513	−0.175601
$92$	0	0
$93$	12.4690	1.29297
$94$	0	0
$95$	−3.73813	−0.383525
$96$	0	0
$97$	−9.14903	−0.928943	−0.464472	−	0.885588i	$-0.653756\pi$
$97$	−9.14903	−0.928943	−0.464472	+	0.885588i	$0.653756\pi$
$98$	0	0
$99$	0.193937	0.0194914
$100$	0	0
$101$	12.1441	1.20838	0.604192	−	0.796839i	$-0.293496\pi$
$101$	12.1441	1.20838	0.604192	+	0.796839i	$0.293496\pi$
$102$	0	0
$103$	14.9502	1.47308	0.736542	−	0.676392i	$-0.236457\pi$
$103$	14.9502	1.47308	0.736542	+	0.676392i	$0.236457\pi$
$104$	0	0
$105$	1.67513	0.163476
$106$	0	0
$107$	−1.08840	−0.105219	−0.0526096	−	0.998615i	$-0.516754\pi$
$107$	−1.08840	−0.105219	−0.0526096	+	0.998615i	$0.516754\pi$
$108$	0	0
$109$	2.12601	0.203635	0.101817	−	0.994803i	$-0.467534\pi$
$109$	2.12601	0.203635	0.101817	+	0.994803i	$0.467534\pi$
$110$	0	0
$111$	−19.1490	−1.81755
$112$	0	0
$113$	−12.0508	−1.13364	−0.566821	−	0.823841i	$-0.691827\pi$
$113$	−12.0508	−1.13364	−0.566821	+	0.823841i	$0.691827\pi$
$114$	0	0
$115$	−8.73084	−0.814155
$116$	0	0
$117$	0.324869	0.0300342
$118$	0	0
$119$	−6.24965	−0.572904
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.03032	−0.724069
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.73813	−0.154235	−0.0771173	−	0.997022i	$-0.524572\pi$
$127$	−1.73813	−0.154235	−0.0771173	+	0.997022i	$0.524572\pi$
$128$	0	0
$129$	−17.6629	−1.55513
$130$	0	0
$131$	8.62530	0.753596	0.376798	−	0.926295i	$-0.377025\pi$
$131$	8.62530	0.753596	0.376798	+	0.926295i	$0.377025\pi$
$132$	0	0
$133$	3.73813	0.324137
$134$	0	0
$135$	−5.35026	−0.460477
$136$	0	0
$137$	−1.19394	−0.102005	−0.0510024	−	0.998699i	$-0.516242\pi$
$137$	−1.19394	−0.102005	−0.0510024	+	0.998699i	$0.516242\pi$
$138$	0	0
$139$	19.1998	1.62851	0.814254	−	0.580509i	$-0.197147\pi$
$139$	19.1998	1.62851	0.814254	+	0.580509i	$0.197147\pi$
$140$	0	0
$141$	11.5574	0.973306
$142$	0	0
$143$	1.67513	0.140081
$144$	0	0
$145$	4.38787	0.364393
$146$	0	0
$147$	−1.67513	−0.138162
$148$	0	0
$149$	13.7235	1.12428	0.562138	−	0.827043i	$-0.309979\pi$
$149$	13.7235	1.12428	0.562138	+	0.827043i	$0.309979\pi$
$150$	0	0
$151$	−1.96968	−0.160291	−0.0801453	−	0.996783i	$-0.525538\pi$
$151$	−1.96968	−0.160291	−0.0801453	+	0.996783i	$0.525538\pi$
$152$	0	0
$153$	1.21203	0.0979872
$154$	0	0
$155$	7.44358	0.597883
$156$	0	0
$157$	22.6761	1.80975	0.904874	−	0.425679i	$-0.139965\pi$
$157$	22.6761	1.80975	0.904874	+	0.425679i	$0.139965\pi$
$158$	0	0
$159$	11.4010	0.904162
$160$	0	0
$161$	8.73084	0.688087
$162$	0	0
$163$	−17.6326	−1.38109	−0.690546	−	0.723289i	$-0.742630\pi$
$163$	−17.6326	−1.38109	−0.690546	+	0.723289i	$0.742630\pi$
$164$	0	0
$165$	−1.67513	−0.130409
$166$	0	0
$167$	−12.0508	−0.932518	−0.466259	−	0.884648i	$-0.654398\pi$
$167$	−12.0508	−0.932518	−0.466259	+	0.884648i	$0.654398\pi$
$168$	0	0
$169$	−10.1939	−0.784149
$170$	0	0
$171$	−0.724961	−0.0554392
$172$	0	0
$173$	−15.9878	−1.21553	−0.607764	−	0.794118i	$-0.707933\pi$
$173$	−15.9878	−1.21553	−0.607764	+	0.794118i	$0.707933\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−5.24472	−0.394217
$178$	0	0
$179$	−4.31265	−0.322343	−0.161171	−	0.986926i	$-0.551527\pi$
$179$	−4.31265	−0.322343	−0.161171	+	0.986926i	$0.551527\pi$
$180$	0	0
$181$	0.337088	0.0250555	0.0125278	−	0.999922i	$-0.496012\pi$
$181$	0.337088	0.0250555	0.0125278	+	0.999922i	$0.496012\pi$
$182$	0	0
$183$	7.38058	0.545588
$184$	0	0
$185$	−11.4314	−0.840451
$186$	0	0
$187$	6.24965	0.457019
$188$	0	0
$189$	5.35026	0.389174
$190$	0	0
$191$	−18.1114	−1.31050	−0.655248	−	0.755414i	$-0.727436\pi$
$191$	−18.1114	−1.31050	−0.655248	+	0.755414i	$0.727436\pi$
$192$	0	0
$193$	−10.0205	−0.721289	−0.360645	−	0.932703i	$-0.617443\pi$
$193$	−10.0205	−0.721289	−0.360645	+	0.932703i	$0.617443\pi$
$194$	0	0
$195$	−2.80606	−0.200946
$196$	0	0
$197$	−21.7685	−1.55094	−0.775469	−	0.631386i	$-0.782487\pi$
$197$	−21.7685	−1.55094	−0.775469	+	0.631386i	$0.782487\pi$
$198$	0	0
$199$	−23.3684	−1.65654	−0.828270	−	0.560330i	$-0.810674\pi$
$199$	−23.3684	−1.65654	−0.828270	+	0.560330i	$0.810674\pi$
$200$	0	0
$201$	4.13586	0.291721
$202$	0	0
$203$	−4.38787	−0.307968
$204$	0	0
$205$	−4.79384	−0.334817
$206$	0	0
$207$	−1.69323	−0.117688
$208$	0	0
$209$	−3.73813	−0.258572
$210$	0	0
$211$	7.66291	0.527537	0.263768	−	0.964586i	$-0.415035\pi$
$211$	7.66291	0.527537	0.263768	+	0.964586i	$0.415035\pi$
$212$	0	0
$213$	−13.9248	−0.954110
$214$	0	0
$215$	−10.5442	−0.719108
$216$	0	0
$217$	−7.44358	−0.505303
$218$	0	0
$219$	7.45580	0.503816
$220$	0	0
$221$	10.4690	0.704220
$222$	0	0
$223$	−18.4509	−1.23556	−0.617781	−	0.786350i	$-0.711968\pi$
$223$	−18.4509	−1.23556	−0.617781	+	0.786350i	$0.711968\pi$
$224$	0	0
$225$	−0.193937	−0.0129291
$226$	0	0
$227$	21.9248	1.45520	0.727599	−	0.686002i	$-0.240636\pi$
$227$	21.9248	1.45520	0.727599	+	0.686002i	$0.240636\pi$
$228$	0	0
$229$	17.6629	1.16720	0.583599	−	0.812042i	$-0.301644\pi$
$229$	17.6629	1.16720	0.583599	+	0.812042i	$0.301644\pi$
$230$	0	0
$231$	1.67513	0.110216
$232$	0	0
$233$	−12.3430	−0.808615	−0.404307	−	0.914623i	$-0.632487\pi$
$233$	−12.3430	−0.808615	−0.404307	+	0.914623i	$0.632487\pi$
$234$	0	0
$235$	6.89938	0.450066
$236$	0	0
$237$	0.387873	0.0251951
$238$	0	0
$239$	28.2882	1.82981	0.914906	−	0.403667i	$-0.132264\pi$
$239$	28.2882	1.82981	0.914906	+	0.403667i	$0.132264\pi$
$240$	0	0
$241$	−4.81828	−0.310373	−0.155186	−	0.987885i	$-0.549598\pi$
$241$	−4.81828	−0.310373	−0.155186	+	0.987885i	$0.549598\pi$
$242$	0	0
$243$	−2.01222	−0.129084
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−6.26187	−0.398433
$248$	0	0
$249$	27.1998	1.72372
$250$	0	0
$251$	2.54183	0.160439	0.0802194	−	0.996777i	$-0.474438\pi$
$251$	2.54183	0.160439	0.0802194	+	0.996777i	$0.474438\pi$
$252$	0	0
$253$	−8.73084	−0.548903
$254$	0	0
$255$	−10.4690	−0.655593
$256$	0	0
$257$	2.88717	0.180096	0.0900482	−	0.995937i	$-0.471298\pi$
$257$	2.88717	0.180096	0.0900482	+	0.995937i	$0.471298\pi$
$258$	0	0
$259$	11.4314	0.710310
$260$	0	0
$261$	0.850969	0.0526737
$262$	0	0
$263$	−1.48612	−0.0916380	−0.0458190	−	0.998950i	$-0.514590\pi$
$263$	−1.48612	−0.0916380	−0.0458190	+	0.998950i	$0.514590\pi$
$264$	0	0
$265$	6.80606	0.418093
$266$	0	0
$267$	2.17679	0.133218
$268$	0	0
$269$	−25.1246	−1.53187	−0.765937	−	0.642916i	$-0.777724\pi$
$269$	−25.1246	−1.53187	−0.765937	+	0.642916i	$0.777724\pi$
$270$	0	0
$271$	−19.7235	−1.19812	−0.599060	−	0.800704i	$-0.704459\pi$
$271$	−19.7235	−1.19812	−0.599060	+	0.800704i	$0.704459\pi$
$272$	0	0
$273$	2.80606	0.169831
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	19.3561	1.16300	0.581499	−	0.813547i	$-0.302466\pi$
$277$	19.3561	1.16300	0.581499	+	0.813547i	$0.302466\pi$
$278$	0	0
$279$	1.44358	0.0864250
$280$	0	0
$281$	7.14903	0.426475	0.213238	−	0.977000i	$-0.431599\pi$
$281$	7.14903	0.426475	0.213238	+	0.977000i	$0.431599\pi$
$282$	0	0
$283$	−25.7137	−1.52852	−0.764260	−	0.644908i	$-0.776896\pi$
$283$	−25.7137	−1.52852	−0.764260	+	0.644908i	$0.776896\pi$
$284$	0	0
$285$	6.26187	0.370921
$286$	0	0
$287$	4.79384	0.282972
$288$	0	0
$289$	22.0581	1.29753
$290$	0	0
$291$	15.3258	0.898416
$292$	0	0
$293$	−21.2628	−1.24219	−0.621094	−	0.783736i	$-0.713311\pi$
$293$	−21.2628	−1.24219	−0.621094	+	0.783736i	$0.713311\pi$
$294$	0	0
$295$	−3.13093	−0.182290
$296$	0	0
$297$	−5.35026	−0.310454
$298$	0	0
$299$	−14.6253	−0.845803
$300$	0	0
$301$	10.5442	0.607757
$302$	0	0
$303$	−20.3430	−1.16867
$304$	0	0
$305$	4.40597	0.252285
$306$	0	0
$307$	−2.71037	−0.154689	−0.0773446	−	0.997004i	$-0.524644\pi$
$307$	−2.71037	−0.154689	−0.0773446	+	0.997004i	$0.524644\pi$
$308$	0	0
$309$	−25.0435	−1.42467
$310$	0	0
$311$	−5.07030	−0.287510	−0.143755	−	0.989613i	$-0.545918\pi$
$311$	−5.07030	−0.287510	−0.143755	+	0.989613i	$0.545918\pi$
$312$	0	0
$313$	6.83638	0.386415	0.193208	−	0.981158i	$-0.438111\pi$
$313$	6.83638	0.386415	0.193208	+	0.981158i	$0.438111\pi$
$314$	0	0
$315$	0.193937	0.0109271
$316$	0	0
$317$	−33.8496	−1.90118	−0.950590	−	0.310449i	$-0.899521\pi$
$317$	−33.8496	−1.90118	−0.950590	+	0.310449i	$0.899521\pi$
$318$	0	0
$319$	4.38787	0.245674
$320$	0	0
$321$	1.82321	0.101761
$322$	0	0
$323$	−23.3620	−1.29990
$324$	0	0
$325$	−1.67513	−0.0929195
$326$	0	0
$327$	−3.56134	−0.196943
$328$	0	0
$329$	−6.89938	−0.380375
$330$	0	0
$331$	−23.9756	−1.31782	−0.658908	−	0.752223i	$-0.728981\pi$
$331$	−23.9756	−1.31782	−0.658908	+	0.752223i	$0.728981\pi$
$332$	0	0
$333$	−2.21696	−0.121489
$334$	0	0
$335$	2.46898	0.134895
$336$	0	0
$337$	−18.0205	−0.981637	−0.490819	−	0.871262i	$-0.663302\pi$
$337$	−18.0205	−0.981637	−0.490819	+	0.871262i	$0.663302\pi$
$338$	0	0
$339$	20.1866	1.09639
$340$	0	0
$341$	7.44358	0.403093
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	14.6253	0.787400
$346$	0	0
$347$	23.6932	1.27192	0.635960	−	0.771722i	$-0.280604\pi$
$347$	23.6932	1.27192	0.635960	+	0.771722i	$0.280604\pi$
$348$	0	0
$349$	16.9053	0.904918	0.452459	−	0.891785i	$-0.350547\pi$
$349$	16.9053	0.904918	0.452459	+	0.891785i	$0.350547\pi$
$350$	0	0
$351$	−8.96239	−0.478377
$352$	0	0
$353$	−10.8364	−0.576762	−0.288381	−	0.957516i	$-0.593117\pi$
$353$	−10.8364	−0.576762	−0.288381	+	0.957516i	$0.593117\pi$
$354$	0	0
$355$	−8.31265	−0.441190
$356$	0	0
$357$	10.4690	0.554077
$358$	0	0
$359$	9.63259	0.508389	0.254194	−	0.967153i	$-0.418190\pi$
$359$	9.63259	0.508389	0.254194	+	0.967153i	$0.418190\pi$
$360$	0	0
$361$	−5.02635	−0.264545
$362$	0	0
$363$	−1.67513	−0.0879216
$364$	0	0
$365$	4.45088	0.232970
$366$	0	0
$367$	13.7866	0.719652	0.359826	−	0.933019i	$-0.382836\pi$
$367$	13.7866	0.719652	0.359826	+	0.933019i	$0.382836\pi$
$368$	0	0
$369$	−0.929702	−0.0483983
$370$	0	0
$371$	−6.80606	−0.353353
$372$	0	0
$373$	−6.85685	−0.355034	−0.177517	−	0.984118i	$-0.556807\pi$
$373$	−6.85685	−0.355034	−0.177517	+	0.984118i	$0.556807\pi$
$374$	0	0
$375$	1.67513	0.0865034
$376$	0	0
$377$	7.35026	0.378558
$378$	0	0
$379$	−30.8627	−1.58531	−0.792656	−	0.609669i	$-0.791302\pi$
$379$	−30.8627	−1.58531	−0.792656	+	0.609669i	$0.791302\pi$
$380$	0	0
$381$	2.91160	0.149166
$382$	0	0
$383$	−35.2628	−1.80185	−0.900923	−	0.433979i	$-0.857109\pi$
$383$	−35.2628	−1.80185	−0.900923	+	0.433979i	$0.857109\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−2.04491	−0.103948
$388$	0	0
$389$	−1.09825	−0.0556833	−0.0278416	−	0.999612i	$-0.508863\pi$
$389$	−1.09825	−0.0556833	−0.0278416	+	0.999612i	$0.508863\pi$
$390$	0	0
$391$	−54.5647	−2.75945
$392$	0	0
$393$	−14.4485	−0.728831
$394$	0	0
$395$	0.231548	0.0116504
$396$	0	0
$397$	1.11283	0.0558515	0.0279258	−	0.999610i	$-0.491110\pi$
$397$	1.11283	0.0558515	0.0279258	+	0.999610i	$0.491110\pi$
$398$	0	0
$399$	−6.26187	−0.313485
$400$	0	0
$401$	18.2520	0.911462	0.455731	−	0.890118i	$-0.349378\pi$
$401$	18.2520	0.911462	0.455731	+	0.890118i	$0.349378\pi$
$402$	0	0
$403$	12.4690	0.621124
$404$	0	0
$405$	8.38058	0.416434
$406$	0	0
$407$	−11.4314	−0.566632
$408$	0	0
$409$	−15.6956	−0.776097	−0.388049	−	0.921639i	$-0.626851\pi$
$409$	−15.6956	−0.776097	−0.388049	+	0.921639i	$0.626851\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	3.13093	0.154063
$414$	0	0
$415$	16.2374	0.797064
$416$	0	0
$417$	−32.1622	−1.57499
$418$	0	0
$419$	−24.4568	−1.19479	−0.597395	−	0.801947i	$-0.703798\pi$
$419$	−24.4568	−1.19479	−0.597395	+	0.801947i	$0.703798\pi$
$420$	0	0
$421$	4.78163	0.233042	0.116521	−	0.993188i	$-0.462826\pi$
$421$	4.78163	0.233042	0.116521	+	0.993188i	$0.462826\pi$
$422$	0	0
$423$	1.33804	0.0650579
$424$	0	0
$425$	−6.24965	−0.303152
$426$	0	0
$427$	−4.40597	−0.213220
$428$	0	0
$429$	−2.80606	−0.135478
$430$	0	0
$431$	0.917483	0.0441936	0.0220968	−	0.999756i	$-0.492966\pi$
$431$	0.917483	0.0441936	0.0220968	+	0.999756i	$0.492966\pi$
$432$	0	0
$433$	−18.7513	−0.901130	−0.450565	−	0.892744i	$-0.648778\pi$
$433$	−18.7513	−0.901130	−0.450565	+	0.892744i	$0.648778\pi$
$434$	0	0
$435$	−7.35026	−0.352418
$436$	0	0
$437$	32.6371	1.56124
$438$	0	0
$439$	−6.36344	−0.303710	−0.151855	−	0.988403i	$-0.548525\pi$
$439$	−6.36344	−0.303710	−0.151855	+	0.988403i	$0.548525\pi$
$440$	0	0
$441$	−0.193937	−0.00923507
$442$	0	0
$443$	13.4460	0.638836	0.319418	−	0.947614i	$-0.396513\pi$
$443$	13.4460	0.638836	0.319418	+	0.947614i	$0.396513\pi$
$444$	0	0
$445$	1.29948	0.0616011
$446$	0	0
$447$	−22.9887	−1.08733
$448$	0	0
$449$	−13.9189	−0.656873	−0.328437	−	0.944526i	$-0.606522\pi$
$449$	−13.9189	−0.656873	−0.328437	+	0.944526i	$0.606522\pi$
$450$	0	0
$451$	−4.79384	−0.225733
$452$	0	0
$453$	3.29948	0.155023
$454$	0	0
$455$	1.67513	0.0785313
$456$	0	0
$457$	−27.9452	−1.30722	−0.653612	−	0.756830i	$-0.726747\pi$
$457$	−27.9452	−1.30722	−0.653612	+	0.756830i	$0.726747\pi$
$458$	0	0
$459$	−33.4372	−1.56072
$460$	0	0
$461$	−40.2047	−1.87252	−0.936261	−	0.351306i	$-0.885738\pi$
$461$	−40.2047	−1.87252	−0.936261	+	0.351306i	$0.885738\pi$
$462$	0	0
$463$	−24.0205	−1.11633	−0.558163	−	0.829731i	$-0.688494\pi$
$463$	−24.0205	−1.11633	−0.558163	+	0.829731i	$0.688494\pi$
$464$	0	0
$465$	−12.4690	−0.578235
$466$	0	0
$467$	−3.77479	−0.174677	−0.0873383	−	0.996179i	$-0.527836\pi$
$467$	−3.77479	−0.174677	−0.0873383	+	0.996179i	$0.527836\pi$
$468$	0	0
$469$	−2.46898	−0.114007
$470$	0	0
$471$	−37.9854	−1.75028
$472$	0	0
$473$	−10.5442	−0.484823
$474$	0	0
$475$	3.73813	0.171517
$476$	0	0
$477$	1.31994	0.0604361
$478$	0	0
$479$	−34.6009	−1.58095	−0.790477	−	0.612492i	$-0.790167\pi$
$479$	−34.6009	−1.58095	−0.790477	+	0.612492i	$0.790167\pi$
$480$	0	0
$481$	−19.1490	−0.873121
$482$	0	0
$483$	−14.6253	−0.665474
$484$	0	0
$485$	9.14903	0.415436
$486$	0	0
$487$	39.8046	1.80372	0.901860	−	0.432028i	$-0.142202\pi$
$487$	39.8046	1.80372	0.901860	+	0.432028i	$0.142202\pi$
$488$	0	0
$489$	29.5369	1.33570
$490$	0	0
$491$	6.15633	0.277831	0.138916	−	0.990304i	$-0.455638\pi$
$491$	6.15633	0.277831	0.138916	+	0.990304i	$0.455638\pi$
$492$	0	0
$493$	27.4227	1.23505
$494$	0	0
$495$	−0.193937	−0.00871680
$496$	0	0
$497$	8.31265	0.372873
$498$	0	0
$499$	5.14903	0.230502	0.115251	−	0.993336i	$-0.463233\pi$
$499$	5.14903	0.230502	0.115251	+	0.993336i	$0.463233\pi$
$500$	0	0
$501$	20.1866	0.901873
$502$	0	0
$503$	−39.1900	−1.74739	−0.873697	−	0.486470i	$-0.838284\pi$
$503$	−39.1900	−1.74739	−0.873697	+	0.486470i	$0.838284\pi$
$504$	0	0
$505$	−12.1441	−0.540406
$506$	0	0
$507$	17.0762	0.758380
$508$	0	0
$509$	−9.97556	−0.442159	−0.221080	−	0.975256i	$-0.570958\pi$
$509$	−9.97556	−0.442159	−0.221080	+	0.975256i	$0.570958\pi$
$510$	0	0
$511$	−4.45088	−0.196895
$512$	0	0
$513$	20.0000	0.883022
$514$	0	0
$515$	−14.9502	−0.658783
$516$	0	0
$517$	6.89938	0.303435
$518$	0	0
$519$	26.7816	1.17558
$520$	0	0
$521$	−10.2619	−0.449580	−0.224790	−	0.974407i	$-0.572170\pi$
$521$	−10.2619	−0.449580	−0.224790	+	0.974407i	$0.572170\pi$
$522$	0	0
$523$	28.1114	1.22923	0.614613	−	0.788829i	$-0.289312\pi$
$523$	28.1114	1.22923	0.614613	+	0.788829i	$0.289312\pi$
$524$	0	0
$525$	−1.67513	−0.0731087
$526$	0	0
$527$	46.5198	2.02643
$528$	0	0
$529$	53.2276	2.31424
$530$	0	0
$531$	−0.607202	−0.0263503
$532$	0	0
$533$	−8.03032	−0.347832
$534$	0	0
$535$	1.08840	0.0470555
$536$	0	0
$537$	7.22425	0.311750
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	25.8251	1.11031	0.555154	−	0.831747i	$-0.312659\pi$
$541$	25.8251	1.11031	0.555154	+	0.831747i	$0.312659\pi$
$542$	0	0
$543$	−0.564666	−0.0242322
$544$	0	0
$545$	−2.12601	−0.0910682
$546$	0	0
$547$	38.4241	1.64289	0.821447	−	0.570284i	$-0.193167\pi$
$547$	38.4241	1.64289	0.821447	+	0.570284i	$0.193167\pi$
$548$	0	0
$549$	0.854479	0.0364683
$550$	0	0
$551$	−16.4025	−0.698768
$552$	0	0
$553$	−0.231548	−0.00984642
$554$	0	0
$555$	19.1490	0.812831
$556$	0	0
$557$	−34.6820	−1.46952	−0.734761	−	0.678326i	$-0.762706\pi$
$557$	−34.6820	−1.46952	−0.734761	+	0.678326i	$0.762706\pi$
$558$	0	0
$559$	−17.6629	−0.747062
$560$	0	0
$561$	−10.4690	−0.442000
$562$	0	0
$563$	−6.51388	−0.274527	−0.137264	−	0.990535i	$-0.543831\pi$
$563$	−6.51388	−0.274527	−0.137264	+	0.990535i	$0.543831\pi$
$564$	0	0
$565$	12.0508	0.506980
$566$	0	0
$567$	−8.38058	−0.351951
$568$	0	0
$569$	−6.49929	−0.272465	−0.136232	−	0.990677i	$-0.543499\pi$
$569$	−6.49929	−0.272465	−0.136232	+	0.990677i	$0.543499\pi$
$570$	0	0
$571$	−13.4372	−0.562331	−0.281166	−	0.959659i	$-0.590721\pi$
$571$	−13.4372	−0.562331	−0.281166	+	0.959659i	$0.590721\pi$
$572$	0	0
$573$	30.3390	1.26743
$574$	0	0
$575$	8.73084	0.364101
$576$	0	0
$577$	−22.8627	−0.951788	−0.475894	−	0.879503i	$-0.657875\pi$
$577$	−22.8627	−0.951788	−0.475894	+	0.879503i	$0.657875\pi$
$578$	0	0
$579$	16.7856	0.697586
$580$	0	0
$581$	−16.2374	−0.673642
$582$	0	0
$583$	6.80606	0.281878
$584$	0	0
$585$	−0.324869	−0.0134317
$586$	0	0
$587$	21.8373	0.901323	0.450662	−	0.892695i	$-0.351188\pi$
$587$	21.8373	0.901323	0.450662	+	0.892695i	$0.351188\pi$
$588$	0	0
$589$	−27.8251	−1.14651
$590$	0	0
$591$	36.4650	1.49997
$592$	0	0
$593$	−6.19886	−0.254557	−0.127278	−	0.991867i	$-0.540624\pi$
$593$	−6.19886	−0.254557	−0.127278	+	0.991867i	$0.540624\pi$
$594$	0	0
$595$	6.24965	0.256211
$596$	0	0
$597$	39.1451	1.60210
$598$	0	0
$599$	−23.9511	−0.978616	−0.489308	−	0.872111i	$-0.662751\pi$
$599$	−23.9511	−0.978616	−0.489308	+	0.872111i	$0.662751\pi$
$600$	0	0
$601$	27.4337	1.11905	0.559523	−	0.828815i	$-0.310984\pi$
$601$	27.4337	1.11905	0.559523	+	0.828815i	$0.310984\pi$
$602$	0	0
$603$	0.478825	0.0194992
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−40.1524	−1.62973	−0.814867	−	0.579648i	$-0.803190\pi$
$607$	−40.1524	−1.62973	−0.814867	+	0.579648i	$0.803190\pi$
$608$	0	0
$609$	7.35026	0.297848
$610$	0	0
$611$	11.5574	0.467561
$612$	0	0
$613$	−24.0713	−0.972229	−0.486114	−	0.873895i	$-0.661586\pi$
$613$	−24.0713	−0.972229	−0.486114	+	0.873895i	$0.661586\pi$
$614$	0	0
$615$	8.03032	0.323814
$616$	0	0
$617$	19.8046	0.797305	0.398653	−	0.917102i	$-0.369478\pi$
$617$	19.8046	0.797305	0.398653	+	0.917102i	$0.369478\pi$
$618$	0	0
$619$	30.3209	1.21870	0.609350	−	0.792901i	$-0.291430\pi$
$619$	30.3209	1.21870	0.609350	+	0.792901i	$0.291430\pi$
$620$	0	0
$621$	46.7123	1.87450
$622$	0	0
$623$	−1.29948	−0.0520624
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.26187	0.250075
$628$	0	0
$629$	−71.4420	−2.84858
$630$	0	0
$631$	18.1768	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$631$	18.1768	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$632$	0	0
$633$	−12.8364	−0.510200
$634$	0	0
$635$	1.73813	0.0689758
$636$	0	0
$637$	−1.67513	−0.0663711
$638$	0	0
$639$	−1.61213	−0.0637748
$640$	0	0
$641$	37.2155	1.46993	0.734963	−	0.678107i	$-0.237200\pi$
$641$	37.2155	1.46993	0.734963	+	0.678107i	$0.237200\pi$
$642$	0	0
$643$	−4.01222	−0.158226	−0.0791132	−	0.996866i	$-0.525209\pi$
$643$	−4.01222	−0.158226	−0.0791132	+	0.996866i	$0.525209\pi$
$644$	0	0
$645$	17.6629	0.695476
$646$	0	0
$647$	35.7161	1.40414	0.702072	−	0.712106i	$-0.252259\pi$
$647$	35.7161	1.40414	0.702072	+	0.712106i	$0.252259\pi$
$648$	0	0
$649$	−3.13093	−0.122900
$650$	0	0
$651$	12.4690	0.488698
$652$	0	0
$653$	−29.3503	−1.14856	−0.574282	−	0.818657i	$-0.694719\pi$
$653$	−29.3503	−1.14856	−0.574282	+	0.818657i	$0.694719\pi$
$654$	0	0
$655$	−8.62530	−0.337018
$656$	0	0
$657$	0.863188	0.0336762
$658$	0	0
$659$	47.1187	1.83548	0.917742	−	0.397176i	$-0.130010\pi$
$659$	47.1187	1.83548	0.917742	+	0.397176i	$0.130010\pi$
$660$	0	0
$661$	−33.9365	−1.31998	−0.659989	−	0.751275i	$-0.729439\pi$
$661$	−33.9365	−1.31998	−0.659989	+	0.751275i	$0.729439\pi$
$662$	0	0
$663$	−17.5369	−0.681077
$664$	0	0
$665$	−3.73813	−0.144959
$666$	0	0
$667$	−38.3098	−1.48336
$668$	0	0
$669$	30.9076	1.19496
$670$	0	0
$671$	4.40597	0.170091
$672$	0	0
$673$	9.40693	0.362610	0.181305	−	0.983427i	$-0.441968\pi$
$673$	9.40693	0.362610	0.181305	+	0.983427i	$0.441968\pi$
$674$	0	0
$675$	5.35026	0.205932
$676$	0	0
$677$	36.4119	1.39942	0.699711	−	0.714426i	$-0.253312\pi$
$677$	36.4119	1.39942	0.699711	+	0.714426i	$0.253312\pi$
$678$	0	0
$679$	−9.14903	−0.351108
$680$	0	0
$681$	−36.7269	−1.40738
$682$	0	0
$683$	26.7210	1.02245	0.511225	−	0.859447i	$-0.329192\pi$
$683$	26.7210	1.02245	0.511225	+	0.859447i	$0.329192\pi$
$684$	0	0
$685$	1.19394	0.0456180
$686$	0	0
$687$	−29.5877	−1.12884
$688$	0	0
$689$	11.4010	0.434345
$690$	0	0
$691$	−5.44358	−0.207084	−0.103542	−	0.994625i	$-0.533018\pi$
$691$	−5.44358	−0.207084	−0.103542	+	0.994625i	$0.533018\pi$
$692$	0	0
$693$	0.193937	0.00736704
$694$	0	0
$695$	−19.1998	−0.728291
$696$	0	0
$697$	−29.9598	−1.13481
$698$	0	0
$699$	20.6761	0.782041
$700$	0	0
$701$	−31.7381	−1.19873	−0.599366	−	0.800475i	$-0.704581\pi$
$701$	−31.7381	−1.19873	−0.599366	+	0.800475i	$0.704581\pi$
$702$	0	0
$703$	42.7320	1.61167
$704$	0	0
$705$	−11.5574	−0.435276
$706$	0	0
$707$	12.1441	0.456726
$708$	0	0
$709$	33.4763	1.25723	0.628614	−	0.777718i	$-0.283623\pi$
$709$	33.4763	1.25723	0.628614	+	0.777718i	$0.283623\pi$
$710$	0	0
$711$	0.0449056	0.00168409
$712$	0	0
$713$	−64.9887	−2.43385
$714$	0	0
$715$	−1.67513	−0.0626463
$716$	0	0
$717$	−47.3865	−1.76968
$718$	0	0
$719$	19.2062	0.716269	0.358134	−	0.933670i	$-0.383413\pi$
$719$	19.2062	0.716269	0.358134	+	0.933670i	$0.383413\pi$
$720$	0	0
$721$	14.9502	0.556773
$722$	0	0
$723$	8.07125	0.300173
$724$	0	0
$725$	−4.38787	−0.162962
$726$	0	0
$727$	16.8651	0.625492	0.312746	−	0.949837i	$-0.398751\pi$
$727$	16.8651	0.625492	0.312746	+	0.949837i	$0.398751\pi$
$728$	0	0
$729$	28.5125	1.05602
$730$	0	0
$731$	−65.8975	−2.43731
$732$	0	0
$733$	−12.9502	−0.478325	−0.239163	−	0.970980i	$-0.576873\pi$
$733$	−12.9502	−0.478325	−0.239163	+	0.970980i	$0.576873\pi$
$734$	0	0
$735$	1.67513	0.0617881
$736$	0	0
$737$	2.46898	0.0909459
$738$	0	0
$739$	23.1754	0.852520	0.426260	−	0.904601i	$-0.359831\pi$
$739$	23.1754	0.852520	0.426260	+	0.904601i	$0.359831\pi$
$740$	0	0
$741$	10.4894	0.385339
$742$	0	0
$743$	5.08840	0.186675	0.0933376	−	0.995635i	$-0.470246\pi$
$743$	5.08840	0.186675	0.0933376	+	0.995635i	$0.470246\pi$
$744$	0	0
$745$	−13.7235	−0.502792
$746$	0	0
$747$	3.14903	0.115217
$748$	0	0
$749$	−1.08840	−0.0397691
$750$	0	0
$751$	54.0019	1.97056	0.985279	−	0.170955i	$-0.0546853\pi$
$751$	54.0019	1.97056	0.985279	+	0.170955i	$0.0546853\pi$
$752$	0	0
$753$	−4.25790	−0.155166
$754$	0	0
$755$	1.96968	0.0716841
$756$	0	0
$757$	−48.2130	−1.75233	−0.876165	−	0.482010i	$-0.839907\pi$
$757$	−48.2130	−1.75233	−0.876165	+	0.482010i	$0.839907\pi$
$758$	0	0
$759$	14.6253	0.530865
$760$	0	0
$761$	−10.7285	−0.388907	−0.194453	−	0.980912i	$-0.562293\pi$
$761$	−10.7285	−0.388907	−0.194453	+	0.980912i	$0.562293\pi$
$762$	0	0
$763$	2.12601	0.0769667
$764$	0	0
$765$	−1.21203	−0.0438212
$766$	0	0
$767$	−5.24472	−0.189376
$768$	0	0
$769$	20.2946	0.731840	0.365920	−	0.930646i	$-0.380754\pi$
$769$	20.2946	0.731840	0.365920	+	0.930646i	$0.380754\pi$
$770$	0	0
$771$	−4.83638	−0.174178
$772$	0	0
$773$	3.22425	0.115968	0.0579842	−	0.998318i	$-0.481533\pi$
$773$	3.22425	0.115968	0.0579842	+	0.998318i	$0.481533\pi$
$774$	0	0
$775$	−7.44358	−0.267381
$776$	0	0
$777$	−19.1490	−0.686968
$778$	0	0
$779$	17.9200	0.642052
$780$	0	0
$781$	−8.31265	−0.297450
$782$	0	0
$783$	−23.4763	−0.838973
$784$	0	0
$785$	−22.6761	−0.809344
$786$	0	0
$787$	−38.0625	−1.35678	−0.678392	−	0.734701i	$-0.737323\pi$
$787$	−38.0625	−1.35678	−0.678392	+	0.734701i	$0.737323\pi$
$788$	0	0
$789$	2.48944	0.0886265
$790$	0	0
$791$	−12.0508	−0.428477
$792$	0	0
$793$	7.38058	0.262092
$794$	0	0
$795$	−11.4010	−0.404353
$796$	0	0
$797$	9.02302	0.319612	0.159806	−	0.987148i	$-0.448913\pi$
$797$	9.02302	0.319612	0.159806	+	0.987148i	$0.448913\pi$
$798$	0	0
$799$	43.1187	1.52543
$800$	0	0
$801$	0.252016	0.00890455
$802$	0	0
$803$	4.45088	0.157068
$804$	0	0
$805$	−8.73084	−0.307722
$806$	0	0
$807$	42.0870	1.48153
$808$	0	0
$809$	−49.2360	−1.73105	−0.865523	−	0.500869i	$-0.833014\pi$
$809$	−49.2360	−1.73105	−0.865523	+	0.500869i	$0.833014\pi$
$810$	0	0
$811$	−19.1002	−0.670697	−0.335349	−	0.942094i	$-0.608854\pi$
$811$	−19.1002	−0.670697	−0.335349	+	0.942094i	$0.608854\pi$
$812$	0	0
$813$	33.0395	1.15875
$814$	0	0
$815$	17.6326	0.617643
$816$	0	0
$817$	39.4156	1.37898
$818$	0	0
$819$	0.324869	0.0113518
$820$	0	0
$821$	−13.7842	−0.481071	−0.240536	−	0.970640i	$-0.577323\pi$
$821$	−13.7842	−0.481071	−0.240536	+	0.970640i	$0.577323\pi$
$822$	0	0
$823$	29.8554	1.04070	0.520348	−	0.853955i	$-0.325802\pi$
$823$	29.8554	1.04070	0.520348	+	0.853955i	$0.325802\pi$
$824$	0	0
$825$	1.67513	0.0583206
$826$	0	0
$827$	−18.2433	−0.634382	−0.317191	−	0.948362i	$-0.602740\pi$
$827$	−18.2433	−0.634382	−0.317191	+	0.948362i	$0.602740\pi$
$828$	0	0
$829$	24.4485	0.849132	0.424566	−	0.905397i	$-0.360427\pi$
$829$	24.4485	0.849132	0.424566	+	0.905397i	$0.360427\pi$
$830$	0	0
$831$	−32.4241	−1.12478
$832$	0	0
$833$	−6.24965	−0.216537
$834$	0	0
$835$	12.0508	0.417035
$836$	0	0
$837$	−39.8251	−1.37656
$838$	0	0
$839$	−9.41915	−0.325185	−0.162593	−	0.986693i	$-0.551986\pi$
$839$	−9.41915	−0.325185	−0.162593	+	0.986693i	$0.551986\pi$
$840$	0	0
$841$	−9.74657	−0.336089
$842$	0	0
$843$	−11.9756	−0.412460
$844$	0	0
$845$	10.1939	0.350682
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	43.0738	1.47829
$850$	0	0
$851$	99.8054	3.42129
$852$	0	0
$853$	−20.3757	−0.697649	−0.348825	−	0.937188i	$-0.613419\pi$
$853$	−20.3757	−0.697649	−0.348825	+	0.937188i	$0.613419\pi$
$854$	0	0
$855$	0.724961	0.0247931
$856$	0	0
$857$	−29.5393	−1.00904	−0.504521	−	0.863399i	$-0.668331\pi$
$857$	−29.5393	−1.00904	−0.504521	+	0.863399i	$0.668331\pi$
$858$	0	0
$859$	−19.9184	−0.679608	−0.339804	−	0.940496i	$-0.610361\pi$
$859$	−19.9184	−0.679608	−0.339804	+	0.940496i	$0.610361\pi$
$860$	0	0
$861$	−8.03032	−0.273672
$862$	0	0
$863$	−21.5066	−0.732093	−0.366046	−	0.930597i	$-0.619289\pi$
$863$	−21.5066	−0.732093	−0.366046	+	0.930597i	$0.619289\pi$
$864$	0	0
$865$	15.9878	0.543601
$866$	0	0
$867$	−36.9502	−1.25489
$868$	0	0
$869$	0.231548	0.00785473
$870$	0	0
$871$	4.13586	0.140138
$872$	0	0
$873$	1.77433	0.0600520
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−41.5936	−1.40452	−0.702258	−	0.711923i	$-0.747824\pi$
$877$	−41.5936	−1.40452	−0.702258	+	0.711923i	$0.747824\pi$
$878$	0	0
$879$	35.6180	1.20137
$880$	0	0
$881$	−2.51388	−0.0846948	−0.0423474	−	0.999103i	$-0.513484\pi$
$881$	−2.51388	−0.0846948	−0.0423474	+	0.999103i	$0.513484\pi$
$882$	0	0
$883$	43.8094	1.47430	0.737152	−	0.675727i	$-0.236170\pi$
$883$	43.8094	1.47430	0.737152	+	0.675727i	$0.236170\pi$
$884$	0	0
$885$	5.24472	0.176299
$886$	0	0
$887$	6.42407	0.215699	0.107850	−	0.994167i	$-0.465604\pi$
$887$	6.42407	0.215699	0.107850	+	0.994167i	$0.465604\pi$
$888$	0	0
$889$	−1.73813	−0.0582952
$890$	0	0
$891$	8.38058	0.280760
$892$	0	0
$893$	−25.7908	−0.863057
$894$	0	0
$895$	4.31265	0.144156
$896$	0	0
$897$	24.4993	0.818008
$898$	0	0
$899$	32.6615	1.08932
$900$	0	0
$901$	42.5355	1.41706
$902$	0	0
$903$	−17.6629	−0.587785
$904$	0	0
$905$	−0.337088	−0.0112052
$906$	0	0
$907$	28.0567	0.931606	0.465803	−	0.884888i	$-0.345765\pi$
$907$	28.0567	0.931606	0.465803	+	0.884888i	$0.345765\pi$
$908$	0	0
$909$	−2.35519	−0.0781166
$910$	0	0
$911$	−3.68735	−0.122167	−0.0610837	−	0.998133i	$-0.519456\pi$
$911$	−3.68735	−0.122167	−0.0610837	+	0.998133i	$0.519456\pi$
$912$	0	0
$913$	16.2374	0.537380
$914$	0	0
$915$	−7.38058	−0.243994
$916$	0	0
$917$	8.62530	0.284833
$918$	0	0
$919$	−4.96239	−0.163694	−0.0818470	−	0.996645i	$-0.526082\pi$
$919$	−4.96239	−0.163694	−0.0818470	+	0.996645i	$0.526082\pi$
$920$	0	0
$921$	4.54023	0.149606
$922$	0	0
$923$	−13.9248	−0.458340
$924$	0	0
$925$	11.4314	0.375861
$926$	0	0
$927$	−2.89938	−0.0952283
$928$	0	0
$929$	41.3160	1.35553	0.677767	−	0.735277i	$-0.262948\pi$
$929$	41.3160	1.35553	0.677767	+	0.735277i	$0.262948\pi$
$930$	0	0
$931$	3.73813	0.122512
$932$	0	0
$933$	8.49341	0.278062
$934$	0	0
$935$	−6.24965	−0.204385
$936$	0	0
$937$	−34.3004	−1.12055	−0.560273	−	0.828308i	$-0.689304\pi$
$937$	−34.3004	−1.12055	−0.560273	+	0.828308i	$0.689304\pi$
$938$	0	0
$939$	−11.4518	−0.373716
$940$	0	0
$941$	16.1178	0.525424	0.262712	−	0.964874i	$-0.415383\pi$
$941$	16.1178	0.525424	0.262712	+	0.964874i	$0.415383\pi$
$942$	0	0
$943$	41.8543	1.36296
$944$	0	0
$945$	−5.35026	−0.174044
$946$	0	0
$947$	−16.6702	−0.541709	−0.270854	−	0.962620i	$-0.587306\pi$
$947$	−16.6702	−0.541709	−0.270854	+	0.962620i	$0.587306\pi$
$948$	0	0
$949$	7.45580	0.242026
$950$	0	0
$951$	56.7024	1.83870
$952$	0	0
$953$	16.5052	0.534655	0.267327	−	0.963606i	$-0.413859\pi$
$953$	16.5052	0.534655	0.267327	+	0.963606i	$0.413859\pi$
$954$	0	0
$955$	18.1114	0.586072
$956$	0	0
$957$	−7.35026	−0.237600
$958$	0	0
$959$	−1.19394	−0.0385542
$960$	0	0
$961$	24.4069	0.787320
$962$	0	0
$963$	0.211080	0.00680195
$964$	0	0
$965$	10.0205	0.322570
$966$	0	0
$967$	−41.6786	−1.34029	−0.670147	−	0.742228i	$-0.733769\pi$
$967$	−41.6786	−1.34029	−0.670147	+	0.742228i	$0.733769\pi$
$968$	0	0
$969$	39.1344	1.25718
$970$	0	0
$971$	11.4290	0.366774	0.183387	−	0.983041i	$-0.441294\pi$
$971$	11.4290	0.366774	0.183387	+	0.983041i	$0.441294\pi$
$972$	0	0
$973$	19.1998	0.615518
$974$	0	0
$975$	2.80606	0.0898660
$976$	0	0
$977$	−11.6775	−0.373596	−0.186798	−	0.982398i	$-0.559811\pi$
$977$	−11.6775	−0.373596	−0.186798	+	0.982398i	$0.559811\pi$
$978$	0	0
$979$	1.29948	0.0415314
$980$	0	0
$981$	−0.412311	−0.0131641
$982$	0	0
$983$	−30.4969	−0.972701	−0.486350	−	0.873764i	$-0.661672\pi$
$983$	−30.4969	−0.972701	−0.486350	+	0.873764i	$0.661672\pi$
$984$	0	0
$985$	21.7685	0.693601
$986$	0	0
$987$	11.5574	0.367875
$988$	0	0
$989$	92.0597	2.92733
$990$	0	0
$991$	−7.28963	−0.231563	−0.115781	−	0.993275i	$-0.536937\pi$
$991$	−7.28963	−0.231563	−0.115781	+	0.993275i	$0.536937\pi$
$992$	0	0
$993$	40.1622	1.27451
$994$	0	0
$995$	23.3684	0.740827
$996$	0	0
$997$	−32.4217	−1.02681	−0.513403	−	0.858148i	$-0.671615\pi$
$997$	−32.4217	−1.02681	−0.513403	+	0.858148i	$0.671615\pi$
$998$	0	0
$999$	61.1608	1.93504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bh.1.1		3
4.3	odd	2		1540.2.a.g.1.3	✓	3
20.3	even	4		7700.2.e.r.1849.5		6
20.7	even	4		7700.2.e.r.1849.2		6
20.19	odd	2		7700.2.a.x.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1540.2.a.g.1.3	✓	3	4.3	odd	2
6160.2.a.bh.1.1		3	1.1	even	1	trivial
7700.2.a.x.1.1		3	20.19	odd	2
7700.2.e.r.1849.2		6	20.7	even	4
7700.2.e.r.1849.5		6	20.3	even	4

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)

\(f(q)\)	\(=\)	\(q-1.67513 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.193937 q^{9} +O(q^{10})\)	\(q-1.67513 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.193937 q^{9} -1.00000 q^{11} -1.67513 q^{13} +1.67513 q^{15} -6.24965 q^{17} +3.73813 q^{19} -1.67513 q^{21} +8.73084 q^{23} +1.00000 q^{25} +5.35026 q^{27} -4.38787 q^{29} -7.44358 q^{31} +1.67513 q^{33} -1.00000 q^{35} +11.4314 q^{37} +2.80606 q^{39} +4.79384 q^{41} +10.5442 q^{43} +0.193937 q^{45} -6.89938 q^{47} +1.00000 q^{49} +10.4690 q^{51} -6.80606 q^{53} +1.00000 q^{55} -6.26187 q^{57} +3.13093 q^{59} -4.40597 q^{61} -0.193937 q^{63} +1.67513 q^{65} -2.46898 q^{67} -14.6253 q^{69} +8.31265 q^{71} -4.45088 q^{73} -1.67513 q^{75} -1.00000 q^{77} -0.231548 q^{79} -8.38058 q^{81} -16.2374 q^{83} +6.24965 q^{85} +7.35026 q^{87} -1.29948 q^{89} -1.67513 q^{91} +12.4690 q^{93} -3.73813 q^{95} -9.14903 q^{97} +0.193937 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7 - q^9	\( 3 q - 3 q^{5} + 3 q^{7} - q^{9} - 3 q^{11} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 3 q^{25} + 6 q^{27} - 14 q^{29} - 6 q^{31} - 3 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} + 22 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} - 20 q^{53} + 3 q^{55} - 28 q^{57} + 14 q^{59} + 14 q^{61} - q^{63} + 24 q^{67} - 2 q^{69} + 4 q^{71} - 10 q^{73} - 3 q^{77} - 12 q^{79} - 13 q^{81} - 6 q^{83} + 2 q^{85} + 12 q^{87} - 24 q^{89} + 6 q^{93} - 2 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 - q^9 - 3 * q^11 - 2 * q^17 + 2 * q^19 + 4 * q^23 + 3 * q^25 + 6 * q^27 - 14 * q^29 - 6 * q^31 - 3 * q^35 - 8 * q^37 + 8 * q^39 - 12 * q^41 + 22 * q^43 + q^45 - 14 * q^47 + 3 * q^49 - 20 * q^53 + 3 * q^55 - 28 * q^57 + 14 * q^59 + 14 * q^61 - q^63 + 24 * q^67 - 2 * q^69 + 4 * q^71 - 10 * q^73 - 3 * q^77 - 12 * q^79 - 13 * q^81 - 6 * q^83 + 2 * q^85 + 12 * q^87 - 24 * q^89 + 6 * q^93 - 2 * q^95 - 4 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more