LMFDB - Embedded newform 6004.2.a.f.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.19435 q^{3} +3.68128 q^{5} +2.98419 q^{7} -1.57353 q^{9} +O(q^{10})$	$q-1.19435 q^{3} +3.68128 q^{5} +2.98419 q^{7} -1.57353 q^{9} -1.41939 q^{11} -5.78342 q^{13} -4.39674 q^{15} -4.18559 q^{17} -1.00000 q^{19} -3.56416 q^{21} -2.06768 q^{23} +8.55184 q^{25} +5.46239 q^{27} +5.06484 q^{29} -1.85777 q^{31} +1.69525 q^{33} +10.9856 q^{35} +9.02650 q^{37} +6.90742 q^{39} -1.92730 q^{41} -8.62224 q^{43} -5.79261 q^{45} -0.451340 q^{47} +1.90539 q^{49} +4.99906 q^{51} -3.51069 q^{53} -5.22518 q^{55} +1.19435 q^{57} -7.36720 q^{59} -6.74856 q^{61} -4.69571 q^{63} -21.2904 q^{65} -1.14821 q^{67} +2.46953 q^{69} +1.49758 q^{71} -9.92315 q^{73} -10.2139 q^{75} -4.23573 q^{77} +1.00000 q^{79} -1.80341 q^{81} +2.21563 q^{83} -15.4084 q^{85} -6.04918 q^{87} +5.93537 q^{89} -17.2588 q^{91} +2.21883 q^{93} -3.68128 q^{95} +6.14248 q^{97} +2.23346 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * 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
$3$	−1.19435	−0.689558	−0.344779	−	0.938684i	$-0.612046\pi$
$3$	−1.19435	−0.689558	−0.344779	+	0.938684i	$0.612046\pi$
$4$	0	0
$5$	3.68128	1.64632	0.823160	−	0.567810i	$-0.192209\pi$
$5$	3.68128	1.64632	0.823160	+	0.567810i	$0.192209\pi$
$6$	0	0
$7$	2.98419	1.12792	0.563959	−	0.825803i	$-0.309278\pi$
$7$	2.98419	1.12792	0.563959	+	0.825803i	$0.309278\pi$
$8$	0	0
$9$	−1.57353	−0.524510
$10$	0	0
$11$	−1.41939	−0.427963	−0.213981	−	0.976838i	$-0.568643\pi$
$11$	−1.41939	−0.427963	−0.213981	+	0.976838i	$0.568643\pi$
$12$	0	0
$13$	−5.78342	−1.60403	−0.802015	−	0.597303i	$-0.796239\pi$
$13$	−5.78342	−1.60403	−0.802015	+	0.597303i	$0.796239\pi$
$14$	0	0
$15$	−4.39674	−1.13523
$16$	0	0
$17$	−4.18559	−1.01516	−0.507578	−	0.861606i	$-0.669459\pi$
$17$	−4.18559	−1.01516	−0.507578	+	0.861606i	$0.669459\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.56416	−0.777765
$22$	0	0
$23$	−2.06768	−0.431141	−0.215570	−	0.976488i	$-0.569161\pi$
$23$	−2.06768	−0.431141	−0.215570	+	0.976488i	$0.569161\pi$
$24$	0	0
$25$	8.55184	1.71037
$26$	0	0
$27$	5.46239	1.05124
$28$	0	0
$29$	5.06484	0.940517	0.470258	−	0.882529i	$-0.344161\pi$
$29$	5.06484	0.940517	0.470258	+	0.882529i	$0.344161\pi$
$30$	0	0
$31$	−1.85777	−0.333665	−0.166833	−	0.985985i	$-0.553354\pi$
$31$	−1.85777	−0.333665	−0.166833	+	0.985985i	$0.553354\pi$
$32$	0	0
$33$	1.69525	0.295105
$34$	0	0
$35$	10.9856	1.85691
$36$	0	0
$37$	9.02650	1.48395	0.741974	−	0.670429i	$-0.233890\pi$
$37$	9.02650	1.48395	0.741974	+	0.670429i	$0.233890\pi$
$38$	0	0
$39$	6.90742	1.10607
$40$	0	0
$41$	−1.92730	−0.300993	−0.150497	−	0.988611i	$-0.548087\pi$
$41$	−1.92730	−0.300993	−0.150497	+	0.988611i	$0.548087\pi$
$42$	0	0
$43$	−8.62224	−1.31488	−0.657440	−	0.753507i	$-0.728361\pi$
$43$	−8.62224	−1.31488	−0.657440	+	0.753507i	$0.728361\pi$
$44$	0	0
$45$	−5.79261	−0.863511
$46$	0	0
$47$	−0.451340	−0.0658347	−0.0329174	−	0.999458i	$-0.510480\pi$
$47$	−0.451340	−0.0658347	−0.0329174	+	0.999458i	$0.510480\pi$
$48$	0	0
$49$	1.90539	0.272199
$50$	0	0
$51$	4.99906	0.700008
$52$	0	0
$53$	−3.51069	−0.482230	−0.241115	−	0.970497i	$-0.577513\pi$
$53$	−3.51069	−0.482230	−0.241115	+	0.970497i	$0.577513\pi$
$54$	0	0
$55$	−5.22518	−0.704563
$56$	0	0
$57$	1.19435	0.158195
$58$	0	0
$59$	−7.36720	−0.959128	−0.479564	−	0.877507i	$-0.659205\pi$
$59$	−7.36720	−0.959128	−0.479564	+	0.877507i	$0.659205\pi$
$60$	0	0
$61$	−6.74856	−0.864064	−0.432032	−	0.901858i	$-0.642203\pi$
$61$	−6.74856	−0.864064	−0.432032	+	0.901858i	$0.642203\pi$
$62$	0	0
$63$	−4.69571	−0.591604
$64$	0	0
$65$	−21.2904	−2.64075
$66$	0	0
$67$	−1.14821	−0.140277	−0.0701383	−	0.997537i	$-0.522344\pi$
$67$	−1.14821	−0.140277	−0.0701383	+	0.997537i	$0.522344\pi$
$68$	0	0
$69$	2.46953	0.297296
$70$	0	0
$71$	1.49758	0.177730	0.0888649	−	0.996044i	$-0.471676\pi$
$71$	1.49758	0.177730	0.0888649	+	0.996044i	$0.471676\pi$
$72$	0	0
$73$	−9.92315	−1.16142	−0.580708	−	0.814112i	$-0.697224\pi$
$73$	−9.92315	−1.16142	−0.580708	+	0.814112i	$0.697224\pi$
$74$	0	0
$75$	−10.2139	−1.17940
$76$	0	0
$77$	−4.23573	−0.482707
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−1.80341	−0.200379
$82$	0	0
$83$	2.21563	0.243197	0.121599	−	0.992579i	$-0.461198\pi$
$83$	2.21563	0.243197	0.121599	+	0.992579i	$0.461198\pi$
$84$	0	0
$85$	−15.4084	−1.67127
$86$	0	0
$87$	−6.04918	−0.648541
$88$	0	0
$89$	5.93537	0.629148	0.314574	−	0.949233i	$-0.398138\pi$
$89$	5.93537	0.629148	0.314574	+	0.949233i	$0.398138\pi$
$90$	0	0
$91$	−17.2588	−1.80922
$92$	0	0
$93$	2.21883	0.230082
$94$	0	0
$95$	−3.68128	−0.377692
$96$	0	0
$97$	6.14248	0.623674	0.311837	−	0.950136i	$-0.399056\pi$
$97$	6.14248	0.623674	0.311837	+	0.950136i	$0.399056\pi$
$98$	0	0
$99$	2.23346	0.224471
$100$	0	0
$101$	−13.4119	−1.33453	−0.667266	−	0.744819i	$-0.732536\pi$
$101$	−13.4119	−1.33453	−0.667266	+	0.744819i	$0.732536\pi$
$102$	0	0
$103$	−13.9232	−1.37190	−0.685948	−	0.727651i	$-0.740612\pi$
$103$	−13.9232	−1.37190	−0.685948	+	0.727651i	$0.740612\pi$
$104$	0	0
$105$	−13.1207	−1.28045
$106$	0	0
$107$	5.32704	0.514984	0.257492	−	0.966280i	$-0.417104\pi$
$107$	5.32704	0.514984	0.257492	+	0.966280i	$0.417104\pi$
$108$	0	0
$109$	−12.7900	−1.22506	−0.612530	−	0.790448i	$-0.709848\pi$
$109$	−12.7900	−1.22506	−0.612530	+	0.790448i	$0.709848\pi$
$110$	0	0
$111$	−10.7808	−1.02327
$112$	0	0
$113$	−4.09917	−0.385618	−0.192809	−	0.981236i	$-0.561760\pi$
$113$	−4.09917	−0.385618	−0.192809	+	0.981236i	$0.561760\pi$
$114$	0	0
$115$	−7.61171	−0.709795
$116$	0	0
$117$	9.10038	0.841330
$118$	0	0
$119$	−12.4906	−1.14501
$120$	0	0
$121$	−8.98533	−0.816848
$122$	0	0
$123$	2.30186	0.207552
$124$	0	0
$125$	13.0753	1.16949
$126$	0	0
$127$	9.21528	0.817724	0.408862	−	0.912596i	$-0.365926\pi$
$127$	9.21528	0.817724	0.408862	+	0.912596i	$0.365926\pi$
$128$	0	0
$129$	10.2980	0.906686
$130$	0	0
$131$	2.38646	0.208506	0.104253	−	0.994551i	$-0.466755\pi$
$131$	2.38646	0.208506	0.104253	+	0.994551i	$0.466755\pi$
$132$	0	0
$133$	−2.98419	−0.258762
$134$	0	0
$135$	20.1086	1.73067
$136$	0	0
$137$	−19.7273	−1.68542	−0.842708	−	0.538370i	$-0.819040\pi$
$137$	−19.7273	−1.68542	−0.842708	+	0.538370i	$0.819040\pi$
$138$	0	0
$139$	15.1743	1.28707	0.643533	−	0.765418i	$-0.277468\pi$
$139$	15.1743	1.28707	0.643533	+	0.765418i	$0.277468\pi$
$140$	0	0
$141$	0.539058	0.0453968
$142$	0	0
$143$	8.20893	0.686465
$144$	0	0
$145$	18.6451	1.54839
$146$	0	0
$147$	−2.27570	−0.187697
$148$	0	0
$149$	−23.0603	−1.88918	−0.944588	−	0.328260i	$-0.893538\pi$
$149$	−23.0603	−1.88918	−0.944588	+	0.328260i	$0.893538\pi$
$150$	0	0
$151$	19.1677	1.55984	0.779921	−	0.625877i	$-0.215259\pi$
$151$	19.1677	1.55984	0.779921	+	0.625877i	$0.215259\pi$
$152$	0	0
$153$	6.58616	0.532459
$154$	0	0
$155$	−6.83898	−0.549320
$156$	0	0
$157$	18.8866	1.50732	0.753659	−	0.657266i	$-0.228287\pi$
$157$	18.8866	1.50732	0.753659	+	0.657266i	$0.228287\pi$
$158$	0	0
$159$	4.19299	0.332525
$160$	0	0
$161$	−6.17034	−0.486291
$162$	0	0
$163$	−19.4276	−1.52168	−0.760842	−	0.648937i	$-0.775214\pi$
$163$	−19.4276	−1.52168	−0.760842	+	0.648937i	$0.775214\pi$
$164$	0	0
$165$	6.24069	0.485837
$166$	0	0
$167$	1.99572	0.154433	0.0772165	−	0.997014i	$-0.475397\pi$
$167$	1.99572	0.154433	0.0772165	+	0.997014i	$0.475397\pi$
$168$	0	0
$169$	20.4479	1.57292
$170$	0	0
$171$	1.57353	0.120331
$172$	0	0
$173$	−16.2076	−1.23224	−0.616120	−	0.787652i	$-0.711296\pi$
$173$	−16.2076	−1.23224	−0.616120	+	0.787652i	$0.711296\pi$
$174$	0	0
$175$	25.5203	1.92916
$176$	0	0
$177$	8.79901	0.661374
$178$	0	0
$179$	3.60729	0.269621	0.134811	−	0.990871i	$-0.456957\pi$
$179$	3.60729	0.269621	0.134811	+	0.990871i	$0.456957\pi$
$180$	0	0
$181$	−9.75813	−0.725316	−0.362658	−	0.931922i	$-0.618131\pi$
$181$	−9.75813	−0.725316	−0.362658	+	0.931922i	$0.618131\pi$
$182$	0	0
$183$	8.06013	0.595822
$184$	0	0
$185$	33.2291	2.44305
$186$	0	0
$187$	5.94100	0.434449
$188$	0	0
$189$	16.3008	1.18571
$190$	0	0
$191$	10.8019	0.781596	0.390798	−	0.920477i	$-0.372199\pi$
$191$	10.8019	0.781596	0.390798	+	0.920477i	$0.372199\pi$
$192$	0	0
$193$	6.94447	0.499874	0.249937	−	0.968262i	$-0.419590\pi$
$193$	6.94447	0.499874	0.249937	+	0.968262i	$0.419590\pi$
$194$	0	0
$195$	25.4282	1.82095
$196$	0	0
$197$	−2.49931	−0.178068	−0.0890342	−	0.996029i	$-0.528378\pi$
$197$	−2.49931	−0.178068	−0.0890342	+	0.996029i	$0.528378\pi$
$198$	0	0
$199$	−20.7005	−1.46742	−0.733711	−	0.679462i	$-0.762213\pi$
$199$	−20.7005	−1.46742	−0.733711	+	0.679462i	$0.762213\pi$
$200$	0	0
$201$	1.37137	0.0967288
$202$	0	0
$203$	15.1144	1.06083
$204$	0	0
$205$	−7.09492	−0.495531
$206$	0	0
$207$	3.25355	0.226138
$208$	0	0
$209$	1.41939	0.0981814
$210$	0	0
$211$	−24.5657	−1.69117	−0.845586	−	0.533839i	$-0.820749\pi$
$211$	−24.5657	−1.69117	−0.845586	+	0.533839i	$0.820749\pi$
$212$	0	0
$213$	−1.78863	−0.122555
$214$	0	0
$215$	−31.7409	−2.16471
$216$	0	0
$217$	−5.54394	−0.376347
$218$	0	0
$219$	11.8517	0.800864
$220$	0	0
$221$	24.2070	1.62834
$222$	0	0
$223$	5.09440	0.341146	0.170573	−	0.985345i	$-0.445438\pi$
$223$	5.09440	0.341146	0.170573	+	0.985345i	$0.445438\pi$
$224$	0	0
$225$	−13.4566	−0.897106
$226$	0	0
$227$	23.5191	1.56101	0.780507	−	0.625147i	$-0.214961\pi$
$227$	23.5191	1.56101	0.780507	+	0.625147i	$0.214961\pi$
$228$	0	0
$229$	−10.5144	−0.694812	−0.347406	−	0.937715i	$-0.612937\pi$
$229$	−10.5144	−0.694812	−0.347406	+	0.937715i	$0.612937\pi$
$230$	0	0
$231$	5.05895	0.332854
$232$	0	0
$233$	−25.5441	−1.67345	−0.836724	−	0.547625i	$-0.815532\pi$
$233$	−25.5441	−1.67345	−0.836724	+	0.547625i	$0.815532\pi$
$234$	0	0
$235$	−1.66151	−0.108385
$236$	0	0
$237$	−1.19435	−0.0775813
$238$	0	0
$239$	−9.17210	−0.593294	−0.296647	−	0.954987i	$-0.595868\pi$
$239$	−9.17210	−0.593294	−0.296647	+	0.954987i	$0.595868\pi$
$240$	0	0
$241$	−0.870504	−0.0560741	−0.0280370	−	0.999607i	$-0.508926\pi$
$241$	−0.870504	−0.0560741	−0.0280370	+	0.999607i	$0.508926\pi$
$242$	0	0
$243$	−14.2333	−0.913065
$244$	0	0
$245$	7.01428	0.448126
$246$	0	0
$247$	5.78342	0.367990
$248$	0	0
$249$	−2.64624	−0.167699
$250$	0	0
$251$	14.0104	0.884330	0.442165	−	0.896934i	$-0.354211\pi$
$251$	14.0104	0.884330	0.442165	+	0.896934i	$0.354211\pi$
$252$	0	0
$253$	2.93484	0.184512
$254$	0	0
$255$	18.4030	1.15244
$256$	0	0
$257$	−8.08226	−0.504157	−0.252079	−	0.967707i	$-0.581114\pi$
$257$	−8.08226	−0.504157	−0.252079	+	0.967707i	$0.581114\pi$
$258$	0	0
$259$	26.9368	1.67377
$260$	0	0
$261$	−7.96967	−0.493310
$262$	0	0
$263$	−13.0579	−0.805183	−0.402591	−	0.915380i	$-0.631890\pi$
$263$	−13.0579	−0.805183	−0.402591	+	0.915380i	$0.631890\pi$
$264$	0	0
$265$	−12.9238	−0.793905
$266$	0	0
$267$	−7.08891	−0.433834
$268$	0	0
$269$	1.43062	0.0872263	0.0436132	−	0.999048i	$-0.486113\pi$
$269$	1.43062	0.0872263	0.0436132	+	0.999048i	$0.486113\pi$
$270$	0	0
$271$	14.5815	0.885763	0.442882	−	0.896580i	$-0.353956\pi$
$271$	14.5815	0.885763	0.442882	+	0.896580i	$0.353956\pi$
$272$	0	0
$273$	20.6130	1.24756
$274$	0	0
$275$	−12.1384	−0.731974
$276$	0	0
$277$	21.3667	1.28380	0.641899	−	0.766789i	$-0.278147\pi$
$277$	21.3667	1.28380	0.641899	+	0.766789i	$0.278147\pi$
$278$	0	0
$279$	2.92326	0.175011
$280$	0	0
$281$	−12.3723	−0.738068	−0.369034	−	0.929416i	$-0.620311\pi$
$281$	−12.3723	−0.738068	−0.369034	+	0.929416i	$0.620311\pi$
$282$	0	0
$283$	−0.863567	−0.0513337	−0.0256669	−	0.999671i	$-0.508171\pi$
$283$	−0.863567	−0.0513337	−0.0256669	+	0.999671i	$0.508171\pi$
$284$	0	0
$285$	4.39674	0.260440
$286$	0	0
$287$	−5.75142	−0.339495
$288$	0	0
$289$	0.519199	0.0305411
$290$	0	0
$291$	−7.33626	−0.430059
$292$	0	0
$293$	29.5487	1.72625	0.863127	−	0.504986i	$-0.168503\pi$
$293$	29.5487	1.72625	0.863127	+	0.504986i	$0.168503\pi$
$294$	0	0
$295$	−27.1207	−1.57903
$296$	0	0
$297$	−7.75327	−0.449891
$298$	0	0
$299$	11.9582	0.691563
$300$	0	0
$301$	−25.7304	−1.48308
$302$	0	0
$303$	16.0185	0.920237
$304$	0	0
$305$	−24.8434	−1.42253
$306$	0	0
$307$	7.05055	0.402396	0.201198	−	0.979551i	$-0.435516\pi$
$307$	7.05055	0.402396	0.201198	+	0.979551i	$0.435516\pi$
$308$	0	0
$309$	16.6292	0.946002
$310$	0	0
$311$	10.2382	0.580556	0.290278	−	0.956942i	$-0.406252\pi$
$311$	10.2382	0.580556	0.290278	+	0.956942i	$0.406252\pi$
$312$	0	0
$313$	−10.0446	−0.567753	−0.283877	−	0.958861i	$-0.591621\pi$
$313$	−10.0446	−0.567753	−0.283877	+	0.958861i	$0.591621\pi$
$314$	0	0
$315$	−17.2863	−0.973970
$316$	0	0
$317$	−17.6179	−0.989520	−0.494760	−	0.869030i	$-0.664744\pi$
$317$	−17.6179	−0.989520	−0.494760	+	0.869030i	$0.664744\pi$
$318$	0	0
$319$	−7.18899	−0.402506
$320$	0	0
$321$	−6.36234	−0.355111
$322$	0	0
$323$	4.18559	0.232893
$324$	0	0
$325$	−49.4589	−2.74348
$326$	0	0
$327$	15.2757	0.844749
$328$	0	0
$329$	−1.34688	−0.0742561
$330$	0	0
$331$	−31.2364	−1.71691	−0.858455	−	0.512889i	$-0.828575\pi$
$331$	−31.2364	−1.71691	−0.858455	+	0.512889i	$0.828575\pi$
$332$	0	0
$333$	−14.2035	−0.778346
$334$	0	0
$335$	−4.22690	−0.230940
$336$	0	0
$337$	−25.5847	−1.39369	−0.696845	−	0.717222i	$-0.745413\pi$
$337$	−25.5847	−1.39369	−0.696845	+	0.717222i	$0.745413\pi$
$338$	0	0
$339$	4.89584	0.265906
$340$	0	0
$341$	2.63690	0.142796
$342$	0	0
$343$	−15.2033	−0.820900
$344$	0	0
$345$	9.09103	0.489445
$346$	0	0
$347$	28.7946	1.54577	0.772887	−	0.634543i	$-0.218812\pi$
$347$	28.7946	1.54577	0.772887	+	0.634543i	$0.218812\pi$
$348$	0	0
$349$	−36.2365	−1.93969	−0.969846	−	0.243717i	$-0.921633\pi$
$349$	−36.2365	−1.93969	−0.969846	+	0.243717i	$0.921633\pi$
$350$	0	0
$351$	−31.5913	−1.68622
$352$	0	0
$353$	−4.03895	−0.214972	−0.107486	−	0.994207i	$-0.534280\pi$
$353$	−4.03895	−0.214972	−0.107486	+	0.994207i	$0.534280\pi$
$354$	0	0
$355$	5.51301	0.292600
$356$	0	0
$357$	14.9181	0.789552
$358$	0	0
$359$	−7.34651	−0.387734	−0.193867	−	0.981028i	$-0.562103\pi$
$359$	−7.34651	−0.387734	−0.193867	+	0.981028i	$0.562103\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	10.7316	0.563264
$364$	0	0
$365$	−36.5299	−1.91206
$366$	0	0
$367$	20.7398	1.08261	0.541305	−	0.840827i	$-0.317931\pi$
$367$	20.7398	1.08261	0.541305	+	0.840827i	$0.317931\pi$
$368$	0	0
$369$	3.03266	0.157874
$370$	0	0
$371$	−10.4766	−0.543916
$372$	0	0
$373$	20.7955	1.07675	0.538375	−	0.842706i	$-0.319038\pi$
$373$	20.7955	1.07675	0.538375	+	0.842706i	$0.319038\pi$
$374$	0	0
$375$	−15.6165	−0.806434
$376$	0	0
$377$	−29.2921	−1.50862
$378$	0	0
$379$	2.49615	0.128218	0.0641092	−	0.997943i	$-0.479579\pi$
$379$	2.49615	0.128218	0.0641092	+	0.997943i	$0.479579\pi$
$380$	0	0
$381$	−11.0063	−0.563868
$382$	0	0
$383$	35.2425	1.80081	0.900403	−	0.435058i	$-0.143272\pi$
$383$	35.2425	1.80081	0.900403	+	0.435058i	$0.143272\pi$
$384$	0	0
$385$	−15.5929	−0.794690
$386$	0	0
$387$	13.5674	0.689668
$388$	0	0
$389$	27.6164	1.40021	0.700104	−	0.714041i	$-0.253137\pi$
$389$	27.6164	1.40021	0.700104	+	0.714041i	$0.253137\pi$
$390$	0	0
$391$	8.65446	0.437675
$392$	0	0
$393$	−2.85026	−0.143777
$394$	0	0
$395$	3.68128	0.185225
$396$	0	0
$397$	−19.9941	−1.00347	−0.501737	−	0.865020i	$-0.667305\pi$
$397$	−19.9941	−1.00347	−0.501737	+	0.865020i	$0.667305\pi$
$398$	0	0
$399$	3.56416	0.178431
$400$	0	0
$401$	−12.3774	−0.618097	−0.309049	−	0.951046i	$-0.600011\pi$
$401$	−12.3774	−0.618097	−0.309049	+	0.951046i	$0.600011\pi$
$402$	0	0
$403$	10.7443	0.535210
$404$	0	0
$405$	−6.63887	−0.329888
$406$	0	0
$407$	−12.8121	−0.635074
$408$	0	0
$409$	5.40427	0.267224	0.133612	−	0.991034i	$-0.457342\pi$
$409$	5.40427	0.267224	0.133612	+	0.991034i	$0.457342\pi$
$410$	0	0
$411$	23.5613	1.16219
$412$	0	0
$413$	−21.9851	−1.08182
$414$	0	0
$415$	8.15638	0.400381
$416$	0	0
$417$	−18.1234	−0.887507
$418$	0	0
$419$	29.1987	1.42645	0.713224	−	0.700936i	$-0.247234\pi$
$419$	29.1987	1.42645	0.713224	+	0.700936i	$0.247234\pi$
$420$	0	0
$421$	13.2414	0.645347	0.322673	−	0.946510i	$-0.395418\pi$
$421$	13.2414	0.645347	0.322673	+	0.946510i	$0.395418\pi$
$422$	0	0
$423$	0.710197	0.0345310
$424$	0	0
$425$	−35.7945	−1.73629
$426$	0	0
$427$	−20.1390	−0.974594
$428$	0	0
$429$	−9.80433	−0.473358
$430$	0	0
$431$	26.4269	1.27294	0.636470	−	0.771301i	$-0.280394\pi$
$431$	26.4269	1.27294	0.636470	+	0.771301i	$0.280394\pi$
$432$	0	0
$433$	−8.33285	−0.400451	−0.200226	−	0.979750i	$-0.564168\pi$
$433$	−8.33285	−0.400451	−0.200226	+	0.979750i	$0.564168\pi$
$434$	0	0
$435$	−22.2688	−1.06771
$436$	0	0
$437$	2.06768	0.0989104
$438$	0	0
$439$	−9.78129	−0.466836	−0.233418	−	0.972377i	$-0.574991\pi$
$439$	−9.78129	−0.466836	−0.233418	+	0.972377i	$0.574991\pi$
$440$	0	0
$441$	−2.99819	−0.142771
$442$	0	0
$443$	−12.7286	−0.604754	−0.302377	−	0.953188i	$-0.597780\pi$
$443$	−12.7286	−0.604754	−0.302377	+	0.953188i	$0.597780\pi$
$444$	0	0
$445$	21.8498	1.03578
$446$	0	0
$447$	27.5421	1.30270
$448$	0	0
$449$	−5.17396	−0.244174	−0.122087	−	0.992519i	$-0.538959\pi$
$449$	−5.17396	−0.244174	−0.122087	+	0.992519i	$0.538959\pi$
$450$	0	0
$451$	2.73559	0.128814
$452$	0	0
$453$	−22.8929	−1.07560
$454$	0	0
$455$	−63.5346	−2.97855
$456$	0	0
$457$	20.0146	0.936245	0.468123	−	0.883663i	$-0.344931\pi$
$457$	20.0146	0.936245	0.468123	+	0.883663i	$0.344931\pi$
$458$	0	0
$459$	−22.8634	−1.06717
$460$	0	0
$461$	−18.0998	−0.842990	−0.421495	−	0.906831i	$-0.638494\pi$
$461$	−18.0998	−0.842990	−0.421495	+	0.906831i	$0.638494\pi$
$462$	0	0
$463$	−12.2739	−0.570419	−0.285209	−	0.958465i	$-0.592063\pi$
$463$	−12.2739	−0.570419	−0.285209	+	0.958465i	$0.592063\pi$
$464$	0	0
$465$	8.16813	0.378788
$466$	0	0
$467$	−5.11397	−0.236646	−0.118323	−	0.992975i	$-0.537752\pi$
$467$	−5.11397	−0.236646	−0.118323	+	0.992975i	$0.537752\pi$
$468$	0	0
$469$	−3.42649	−0.158220
$470$	0	0
$471$	−22.5572	−1.03938
$472$	0	0
$473$	12.2383	0.562720
$474$	0	0
$475$	−8.55184	−0.392386
$476$	0	0
$477$	5.52417	0.252935
$478$	0	0
$479$	−14.0353	−0.641291	−0.320645	−	0.947199i	$-0.603900\pi$
$479$	−14.0353	−0.641291	−0.320645	+	0.947199i	$0.603900\pi$
$480$	0	0
$481$	−52.2040	−2.38030
$482$	0	0
$483$	7.36954	0.335326
$484$	0	0
$485$	22.6122	1.02677
$486$	0	0
$487$	34.3065	1.55458	0.777288	−	0.629145i	$-0.216595\pi$
$487$	34.3065	1.55458	0.777288	+	0.629145i	$0.216595\pi$
$488$	0	0
$489$	23.2033	1.04929
$490$	0	0
$491$	−3.63304	−0.163957	−0.0819783	−	0.996634i	$-0.526124\pi$
$491$	−3.63304	−0.163957	−0.0819783	+	0.996634i	$0.526124\pi$
$492$	0	0
$493$	−21.1994	−0.954771
$494$	0	0
$495$	8.22198	0.369551
$496$	0	0
$497$	4.46906	0.200465
$498$	0	0
$499$	35.0160	1.56753	0.783765	−	0.621058i	$-0.213297\pi$
$499$	35.0160	1.56753	0.783765	+	0.621058i	$0.213297\pi$
$500$	0	0
$501$	−2.38358	−0.106491
$502$	0	0
$503$	9.61885	0.428884	0.214442	−	0.976737i	$-0.431207\pi$
$503$	9.61885	0.428884	0.214442	+	0.976737i	$0.431207\pi$
$504$	0	0
$505$	−49.3729	−2.19707
$506$	0	0
$507$	−24.4219	−1.08462
$508$	0	0
$509$	25.8931	1.14769	0.573847	−	0.818963i	$-0.305451\pi$
$509$	25.8931	1.14769	0.573847	+	0.818963i	$0.305451\pi$
$510$	0	0
$511$	−29.6126	−1.30998
$512$	0	0
$513$	−5.46239	−0.241170
$514$	0	0
$515$	−51.2553	−2.25858
$516$	0	0
$517$	0.640628	0.0281748
$518$	0	0
$519$	19.3575	0.849700
$520$	0	0
$521$	22.8420	1.00072	0.500362	−	0.865816i	$-0.333200\pi$
$521$	22.8420	1.00072	0.500362	+	0.865816i	$0.333200\pi$
$522$	0	0
$523$	17.5968	0.769454	0.384727	−	0.923030i	$-0.374296\pi$
$523$	17.5968	0.769454	0.384727	+	0.923030i	$0.374296\pi$
$524$	0	0
$525$	−30.4802	−1.33026
$526$	0	0
$527$	7.77587	0.338722
$528$	0	0
$529$	−18.7247	−0.814118
$530$	0	0
$531$	11.5925	0.503072
$532$	0	0
$533$	11.1464	0.482802
$534$	0	0
$535$	19.6103	0.847829
$536$	0	0
$537$	−4.30836	−0.185919
$538$	0	0
$539$	−2.70450	−0.116491
$540$	0	0
$541$	28.9780	1.24586	0.622930	−	0.782277i	$-0.285942\pi$
$541$	28.9780	1.24586	0.622930	+	0.782277i	$0.285942\pi$
$542$	0	0
$543$	11.6546	0.500147
$544$	0	0
$545$	−47.0836	−2.01684
$546$	0	0
$547$	−28.9847	−1.23930	−0.619649	−	0.784879i	$-0.712725\pi$
$547$	−28.9847	−1.23930	−0.619649	+	0.784879i	$0.712725\pi$
$548$	0	0
$549$	10.6191	0.453211
$550$	0	0
$551$	−5.06484	−0.215769
$552$	0	0
$553$	2.98419	0.126901
$554$	0	0
$555$	−39.6872	−1.68463
$556$	0	0
$557$	−18.5707	−0.786865	−0.393432	−	0.919354i	$-0.628712\pi$
$557$	−18.5707	−0.786865	−0.393432	+	0.919354i	$0.628712\pi$
$558$	0	0
$559$	49.8660	2.10911
$560$	0	0
$561$	−7.09562	−0.299578
$562$	0	0
$563$	−12.3303	−0.519660	−0.259830	−	0.965654i	$-0.583667\pi$
$563$	−12.3303	−0.519660	−0.259830	+	0.965654i	$0.583667\pi$
$564$	0	0
$565$	−15.0902	−0.634850
$566$	0	0
$567$	−5.38172	−0.226011
$568$	0	0
$569$	−46.3090	−1.94137	−0.970687	−	0.240346i	$-0.922739\pi$
$569$	−46.3090	−1.94137	−0.970687	+	0.240346i	$0.922739\pi$
$570$	0	0
$571$	27.6019	1.15510	0.577551	−	0.816355i	$-0.304008\pi$
$571$	27.6019	1.15510	0.577551	+	0.816355i	$0.304008\pi$
$572$	0	0
$573$	−12.9012	−0.538955
$574$	0	0
$575$	−17.6825	−0.737409
$576$	0	0
$577$	15.2049	0.632990	0.316495	−	0.948594i	$-0.397494\pi$
$577$	15.2049	0.632990	0.316495	+	0.948594i	$0.397494\pi$
$578$	0	0
$579$	−8.29412	−0.344692
$580$	0	0
$581$	6.61187	0.274307
$582$	0	0
$583$	4.98304	0.206376
$584$	0	0
$585$	33.5011	1.38510
$586$	0	0
$587$	−13.2140	−0.545402	−0.272701	−	0.962099i	$-0.587917\pi$
$587$	−13.2140	−0.545402	−0.272701	+	0.962099i	$0.587917\pi$
$588$	0	0
$589$	1.85777	0.0765481
$590$	0	0
$591$	2.98505	0.122788
$592$	0	0
$593$	21.3995	0.878771	0.439385	−	0.898299i	$-0.355196\pi$
$593$	21.3995	0.878771	0.439385	+	0.898299i	$0.355196\pi$
$594$	0	0
$595$	−45.9815	−1.88506
$596$	0	0
$597$	24.7237	1.01187
$598$	0	0
$599$	0.847357	0.0346221	0.0173110	−	0.999850i	$-0.494489\pi$
$599$	0.847357	0.0346221	0.0173110	+	0.999850i	$0.494489\pi$
$600$	0	0
$601$	22.3810	0.912941	0.456470	−	0.889739i	$-0.349113\pi$
$601$	22.3810	0.912941	0.456470	+	0.889739i	$0.349113\pi$
$602$	0	0
$603$	1.80675	0.0735765
$604$	0	0
$605$	−33.0775	−1.34479
$606$	0	0
$607$	29.9975	1.21756	0.608780	−	0.793339i	$-0.291659\pi$
$607$	29.9975	1.21756	0.608780	+	0.793339i	$0.291659\pi$
$608$	0	0
$609$	−18.0519	−0.731500
$610$	0	0
$611$	2.61029	0.105601
$612$	0	0
$613$	−30.8985	−1.24798	−0.623990	−	0.781433i	$-0.714489\pi$
$613$	−30.8985	−1.24798	−0.623990	+	0.781433i	$0.714489\pi$
$614$	0	0
$615$	8.47381	0.341697
$616$	0	0
$617$	−35.9594	−1.44767	−0.723835	−	0.689974i	$-0.757622\pi$
$617$	−35.9594	−1.44767	−0.723835	+	0.689974i	$0.757622\pi$
$618$	0	0
$619$	19.6009	0.787827	0.393914	−	0.919147i	$-0.371121\pi$
$619$	19.6009	0.787827	0.393914	+	0.919147i	$0.371121\pi$
$620$	0	0
$621$	−11.2945	−0.453231
$622$	0	0
$623$	17.7123	0.709627
$624$	0	0
$625$	5.37482	0.214993
$626$	0	0
$627$	−1.69525	−0.0677017
$628$	0	0
$629$	−37.7813	−1.50644
$630$	0	0
$631$	−14.6986	−0.585142	−0.292571	−	0.956244i	$-0.594511\pi$
$631$	−14.6986	−0.585142	−0.292571	+	0.956244i	$0.594511\pi$
$632$	0	0
$633$	29.3400	1.16616
$634$	0	0
$635$	33.9241	1.34624
$636$	0	0
$637$	−11.0197	−0.436615
$638$	0	0
$639$	−2.35648	−0.0932211
$640$	0	0
$641$	−12.4223	−0.490651	−0.245326	−	0.969441i	$-0.578895\pi$
$641$	−12.4223	−0.490651	−0.245326	+	0.969441i	$0.578895\pi$
$642$	0	0
$643$	8.76264	0.345565	0.172782	−	0.984960i	$-0.444724\pi$
$643$	8.76264	0.345565	0.172782	+	0.984960i	$0.444724\pi$
$644$	0	0
$645$	37.9097	1.49269
$646$	0	0
$647$	−9.69207	−0.381035	−0.190517	−	0.981684i	$-0.561017\pi$
$647$	−9.69207	−0.381035	−0.190517	+	0.981684i	$0.561017\pi$
$648$	0	0
$649$	10.4569	0.410471
$650$	0	0
$651$	6.62140	0.259513
$652$	0	0
$653$	1.39263	0.0544977	0.0272488	−	0.999629i	$-0.491325\pi$
$653$	1.39263	0.0544977	0.0272488	+	0.999629i	$0.491325\pi$
$654$	0	0
$655$	8.78523	0.343267
$656$	0	0
$657$	15.6144	0.609175
$658$	0	0
$659$	1.79462	0.0699086	0.0349543	−	0.999389i	$-0.488871\pi$
$659$	1.79462	0.0699086	0.0349543	+	0.999389i	$0.488871\pi$
$660$	0	0
$661$	25.9497	1.00933	0.504664	−	0.863316i	$-0.331616\pi$
$661$	25.9497	1.00933	0.504664	+	0.863316i	$0.331616\pi$
$662$	0	0
$663$	−28.9116	−1.12284
$664$	0	0
$665$	−10.9856	−0.426005
$666$	0	0
$667$	−10.4724	−0.405495
$668$	0	0
$669$	−6.08450	−0.235240
$670$	0	0
$671$	9.57885	0.369787
$672$	0	0
$673$	−21.6167	−0.833261	−0.416630	−	0.909076i	$-0.636789\pi$
$673$	−21.6167	−0.833261	−0.416630	+	0.909076i	$0.636789\pi$
$674$	0	0
$675$	46.7135	1.79800
$676$	0	0
$677$	17.3950	0.668545	0.334272	−	0.942476i	$-0.391509\pi$
$677$	17.3950	0.668545	0.334272	+	0.942476i	$0.391509\pi$
$678$	0	0
$679$	18.3303	0.703453
$680$	0	0
$681$	−28.0900	−1.07641
$682$	0	0
$683$	−38.5686	−1.47579	−0.737894	−	0.674917i	$-0.764180\pi$
$683$	−38.5686	−1.47579	−0.737894	+	0.674917i	$0.764180\pi$
$684$	0	0
$685$	−72.6218	−2.77474
$686$	0	0
$687$	12.5579	0.479113
$688$	0	0
$689$	20.3038	0.773512
$690$	0	0
$691$	18.6617	0.709923	0.354962	−	0.934881i	$-0.384494\pi$
$691$	18.6617	0.709923	0.354962	+	0.934881i	$0.384494\pi$
$692$	0	0
$693$	6.66506	0.253185
$694$	0	0
$695$	55.8609	2.11892
$696$	0	0
$697$	8.06688	0.305555
$698$	0	0
$699$	30.5085	1.15394
$700$	0	0
$701$	−1.68564	−0.0636656	−0.0318328	−	0.999493i	$-0.510134\pi$
$701$	−1.68564	−0.0636656	−0.0318328	+	0.999493i	$0.510134\pi$
$702$	0	0
$703$	−9.02650	−0.340441
$704$	0	0
$705$	1.98442	0.0747377
$706$	0	0
$707$	−40.0236	−1.50524
$708$	0	0
$709$	42.9465	1.61289	0.806444	−	0.591310i	$-0.201389\pi$
$709$	42.9465	1.61289	0.806444	+	0.591310i	$0.201389\pi$
$710$	0	0
$711$	−1.57353	−0.0590120
$712$	0	0
$713$	3.84127	0.143857
$714$	0	0
$715$	30.2194	1.13014
$716$	0	0
$717$	10.9547	0.409110
$718$	0	0
$719$	−46.7823	−1.74468	−0.872342	−	0.488896i	$-0.837400\pi$
$719$	−46.7823	−1.74468	−0.872342	+	0.488896i	$0.837400\pi$
$720$	0	0
$721$	−41.5495	−1.54739
$722$	0	0
$723$	1.03969	0.0386663
$724$	0	0
$725$	43.3137	1.60863
$726$	0	0
$727$	−7.03665	−0.260975	−0.130487	−	0.991450i	$-0.541654\pi$
$727$	−7.03665	−0.260975	−0.130487	+	0.991450i	$0.541654\pi$
$728$	0	0
$729$	22.4097	0.829990
$730$	0	0
$731$	36.0892	1.33481
$732$	0	0
$733$	4.76293	0.175923	0.0879614	−	0.996124i	$-0.471965\pi$
$733$	4.76293	0.175923	0.0879614	+	0.996124i	$0.471965\pi$
$734$	0	0
$735$	−8.37750	−0.309009
$736$	0	0
$737$	1.62976	0.0600331
$738$	0	0
$739$	−34.9694	−1.28637	−0.643186	−	0.765710i	$-0.722388\pi$
$739$	−34.9694	−1.28637	−0.643186	+	0.765710i	$0.722388\pi$
$740$	0	0
$741$	−6.90742	−0.253750
$742$	0	0
$743$	−9.82172	−0.360324	−0.180162	−	0.983637i	$-0.557662\pi$
$743$	−9.82172	−0.360324	−0.180162	+	0.983637i	$0.557662\pi$
$744$	0	0
$745$	−84.8916	−3.11019
$746$	0	0
$747$	−3.48637	−0.127560
$748$	0	0
$749$	15.8969	0.580860
$750$	0	0
$751$	32.4570	1.18437	0.592187	−	0.805801i	$-0.298265\pi$
$751$	32.4570	1.18437	0.592187	+	0.805801i	$0.298265\pi$
$752$	0	0
$753$	−16.7333	−0.609797
$754$	0	0
$755$	70.5616	2.56800
$756$	0	0
$757$	13.2648	0.482116	0.241058	−	0.970511i	$-0.422506\pi$
$757$	13.2648	0.482116	0.241058	+	0.970511i	$0.422506\pi$
$758$	0	0
$759$	−3.50523	−0.127232
$760$	0	0
$761$	29.2641	1.06082	0.530412	−	0.847740i	$-0.322037\pi$
$761$	29.2641	1.06082	0.530412	+	0.847740i	$0.322037\pi$
$762$	0	0
$763$	−38.1678	−1.38177
$764$	0	0
$765$	24.2455	0.876598
$766$	0	0
$767$	42.6076	1.53847
$768$	0	0
$769$	−34.8836	−1.25794	−0.628968	−	0.777431i	$-0.716522\pi$
$769$	−34.8836	−1.25794	−0.628968	+	0.777431i	$0.716522\pi$
$770$	0	0
$771$	9.65304	0.347646
$772$	0	0
$773$	11.1820	0.402189	0.201095	−	0.979572i	$-0.435550\pi$
$773$	11.1820	0.402189	0.201095	+	0.979572i	$0.435550\pi$
$774$	0	0
$775$	−15.8874	−0.570691
$776$	0	0
$777$	−32.1719	−1.15416
$778$	0	0
$779$	1.92730	0.0690525
$780$	0	0
$781$	−2.12565	−0.0760617
$782$	0	0
$783$	27.6661	0.988707
$784$	0	0
$785$	69.5270	2.48153
$786$	0	0
$787$	−41.8206	−1.49074	−0.745372	−	0.666649i	$-0.767728\pi$
$787$	−41.8206	−1.49074	−0.745372	+	0.666649i	$0.767728\pi$
$788$	0	0
$789$	15.5957	0.555220
$790$	0	0
$791$	−12.2327	−0.434945
$792$	0	0
$793$	39.0297	1.38599
$794$	0	0
$795$	15.4356	0.547443
$796$	0	0
$797$	2.74250	0.0971443	0.0485721	−	0.998820i	$-0.484533\pi$
$797$	2.74250	0.0971443	0.0485721	+	0.998820i	$0.484533\pi$
$798$	0	0
$799$	1.88913	0.0668325
$800$	0	0
$801$	−9.33949	−0.329995
$802$	0	0
$803$	14.0848	0.497043
$804$	0	0
$805$	−22.7148	−0.800591
$806$	0	0
$807$	−1.70866	−0.0601476
$808$	0	0
$809$	0.429455	0.0150988	0.00754942	−	0.999972i	$-0.497597\pi$
$809$	0.429455	0.0150988	0.00754942	+	0.999972i	$0.497597\pi$
$810$	0	0
$811$	−47.2392	−1.65879	−0.829396	−	0.558661i	$-0.811315\pi$
$811$	−47.2392	−1.65879	−0.829396	+	0.558661i	$0.811315\pi$
$812$	0	0
$813$	−17.4154	−0.610785
$814$	0	0
$815$	−71.5184	−2.50518
$816$	0	0
$817$	8.62224	0.301654
$818$	0	0
$819$	27.1573	0.948952
$820$	0	0
$821$	35.2705	1.23095	0.615474	−	0.788157i	$-0.288965\pi$
$821$	35.2705	1.23095	0.615474	+	0.788157i	$0.288965\pi$
$822$	0	0
$823$	10.5525	0.367837	0.183918	−	0.982942i	$-0.441122\pi$
$823$	10.5525	0.367837	0.183918	+	0.982942i	$0.441122\pi$
$824$	0	0
$825$	14.4975	0.504738
$826$	0	0
$827$	3.49059	0.121380	0.0606898	−	0.998157i	$-0.480670\pi$
$827$	3.49059	0.121380	0.0606898	+	0.998157i	$0.480670\pi$
$828$	0	0
$829$	5.38885	0.187163	0.0935813	−	0.995612i	$-0.470168\pi$
$829$	5.38885	0.187163	0.0935813	+	0.995612i	$0.470168\pi$
$830$	0	0
$831$	−25.5192	−0.885253
$832$	0	0
$833$	−7.97519	−0.276324
$834$	0	0
$835$	7.34679	0.254246
$836$	0	0
$837$	−10.1479	−0.350762
$838$	0	0
$839$	2.99465	0.103387	0.0516934	−	0.998663i	$-0.483538\pi$
$839$	2.99465	0.103387	0.0516934	+	0.998663i	$0.483538\pi$
$840$	0	0
$841$	−3.34743	−0.115429
$842$	0	0
$843$	14.7768	0.508941
$844$	0	0
$845$	75.2745	2.58952
$846$	0	0
$847$	−26.8139	−0.921337
$848$	0	0
$849$	1.03140	0.0353976
$850$	0	0
$851$	−18.6639	−0.639790
$852$	0	0
$853$	−16.1864	−0.554211	−0.277105	−	0.960840i	$-0.589375\pi$
$853$	−16.1864	−0.554211	−0.277105	+	0.960840i	$0.589375\pi$
$854$	0	0
$855$	5.79261	0.198103
$856$	0	0
$857$	−20.4364	−0.698095	−0.349048	−	0.937105i	$-0.613495\pi$
$857$	−20.4364	−0.698095	−0.349048	+	0.937105i	$0.613495\pi$
$858$	0	0
$859$	−3.44376	−0.117499	−0.0587497	−	0.998273i	$-0.518711\pi$
$859$	−3.44376	−0.117499	−0.0587497	+	0.998273i	$0.518711\pi$
$860$	0	0
$861$	6.86920	0.234102
$862$	0	0
$863$	38.6166	1.31453	0.657263	−	0.753661i	$-0.271714\pi$
$863$	38.6166	1.31453	0.657263	+	0.753661i	$0.271714\pi$
$864$	0	0
$865$	−59.6647	−2.02866
$866$	0	0
$867$	−0.620104	−0.0210598
$868$	0	0
$869$	−1.41939	−0.0481496
$870$	0	0
$871$	6.64060	0.225008
$872$	0	0
$873$	−9.66538	−0.327123
$874$	0	0
$875$	39.0193	1.31909
$876$	0	0
$877$	2.14854	0.0725511	0.0362755	−	0.999342i	$-0.488451\pi$
$877$	2.14854	0.0725511	0.0362755	+	0.999342i	$0.488451\pi$
$878$	0	0
$879$	−35.2915	−1.19035
$880$	0	0
$881$	−37.8490	−1.27516	−0.637582	−	0.770382i	$-0.720065\pi$
$881$	−37.8490	−1.27516	−0.637582	+	0.770382i	$0.720065\pi$
$882$	0	0
$883$	−41.0723	−1.38219	−0.691096	−	0.722763i	$-0.742872\pi$
$883$	−41.0723	−1.38219	−0.691096	+	0.722763i	$0.742872\pi$
$884$	0	0
$885$	32.3916	1.08883
$886$	0	0
$887$	−28.1981	−0.946799	−0.473400	−	0.880848i	$-0.656973\pi$
$887$	−28.1981	−0.946799	−0.473400	+	0.880848i	$0.656973\pi$
$888$	0	0
$889$	27.5002	0.922326
$890$	0	0
$891$	2.55975	0.0857548
$892$	0	0
$893$	0.451340	0.0151035
$894$	0	0
$895$	13.2794	0.443883
$896$	0	0
$897$	−14.2823	−0.476872
$898$	0	0
$899$	−9.40930	−0.313818
$900$	0	0
$901$	14.6943	0.489539
$902$	0	0
$903$	30.7311	1.02267
$904$	0	0
$905$	−35.9224	−1.19410
$906$	0	0
$907$	50.1166	1.66410	0.832048	−	0.554704i	$-0.187168\pi$
$907$	50.1166	1.66410	0.832048	+	0.554704i	$0.187168\pi$
$908$	0	0
$909$	21.1040	0.699976
$910$	0	0
$911$	19.7130	0.653120	0.326560	−	0.945176i	$-0.394110\pi$
$911$	19.7130	0.653120	0.326560	+	0.945176i	$0.394110\pi$
$912$	0	0
$913$	−3.14485	−0.104079
$914$	0	0
$915$	29.6716	0.980914
$916$	0	0
$917$	7.12164	0.235177
$918$	0	0
$919$	6.56086	0.216423	0.108211	−	0.994128i	$-0.465488\pi$
$919$	6.56086	0.216423	0.108211	+	0.994128i	$0.465488\pi$
$920$	0	0
$921$	−8.42082	−0.277475
$922$	0	0
$923$	−8.66111	−0.285084
$924$	0	0
$925$	77.1933	2.53810
$926$	0	0
$927$	21.9086	0.719573
$928$	0	0
$929$	−21.7646	−0.714073	−0.357036	−	0.934090i	$-0.616213\pi$
$929$	−21.7646	−0.714073	−0.357036	+	0.934090i	$0.616213\pi$
$930$	0	0
$931$	−1.90539	−0.0624467
$932$	0	0
$933$	−12.2280	−0.400327
$934$	0	0
$935$	21.8705	0.715242
$936$	0	0
$937$	17.5976	0.574889	0.287444	−	0.957797i	$-0.407194\pi$
$937$	17.5976	0.574889	0.287444	+	0.957797i	$0.407194\pi$
$938$	0	0
$939$	11.9967	0.391499
$940$	0	0
$941$	−2.63862	−0.0860166	−0.0430083	−	0.999075i	$-0.513694\pi$
$941$	−2.63862	−0.0860166	−0.0430083	+	0.999075i	$0.513694\pi$
$942$	0	0
$943$	3.98503	0.129770
$944$	0	0
$945$	60.0079	1.95206
$946$	0	0
$947$	3.44204	0.111851	0.0559256	−	0.998435i	$-0.482189\pi$
$947$	3.44204	0.111851	0.0559256	+	0.998435i	$0.482189\pi$
$948$	0	0
$949$	57.3897	1.86295
$950$	0	0
$951$	21.0419	0.682331
$952$	0	0
$953$	12.7196	0.412028	0.206014	−	0.978549i	$-0.433951\pi$
$953$	12.7196	0.412028	0.206014	+	0.978549i	$0.433951\pi$
$954$	0	0
$955$	39.7647	1.28676
$956$	0	0
$957$	8.58616	0.277551
$958$	0	0
$959$	−58.8700	−1.90101
$960$	0	0
$961$	−27.5487	−0.888667
$962$	0	0
$963$	−8.38225	−0.270114
$964$	0	0
$965$	25.5645	0.822952
$966$	0	0
$967$	−14.4973	−0.466203	−0.233101	−	0.972452i	$-0.574887\pi$
$967$	−14.4973	−0.466203	−0.233101	+	0.972452i	$0.574887\pi$
$968$	0	0
$969$	−4.99906	−0.160593
$970$	0	0
$971$	−13.8198	−0.443500	−0.221750	−	0.975104i	$-0.571177\pi$
$971$	−13.8198	−0.443500	−0.221750	+	0.975104i	$0.571177\pi$
$972$	0	0
$973$	45.2830	1.45171
$974$	0	0
$975$	59.0712	1.89179
$976$	0	0
$977$	34.7581	1.11201	0.556005	−	0.831179i	$-0.312333\pi$
$977$	34.7581	1.11201	0.556005	+	0.831179i	$0.312333\pi$
$978$	0	0
$979$	−8.42462	−0.269252
$980$	0	0
$981$	20.1254	0.642556
$982$	0	0
$983$	32.3466	1.03170	0.515848	−	0.856680i	$-0.327477\pi$
$983$	32.3466	1.03170	0.515848	+	0.856680i	$0.327477\pi$
$984$	0	0
$985$	−9.20067	−0.293158
$986$	0	0
$987$	1.60865	0.0512039
$988$	0	0
$989$	17.8280	0.566898
$990$	0	0
$991$	47.3792	1.50505	0.752524	−	0.658565i	$-0.228836\pi$
$991$	47.3792	1.50505	0.752524	+	0.658565i	$0.228836\pi$
$992$	0	0
$993$	37.3072	1.18391
$994$	0	0
$995$	−76.2045	−2.41585
$996$	0	0
$997$	27.0146	0.855560	0.427780	−	0.903883i	$-0.359296\pi$
$997$	27.0146	0.855560	0.427780	+	0.903883i	$0.359296\pi$
$998$	0	0
$999$	49.3063	1.55998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.8	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.8	✓	25	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.19435 q^{3} +3.68128 q^{5} +2.98419 q^{7} -1.57353 q^{9} +O(q^{10})\)	\(q-1.19435 q^{3} +3.68128 q^{5} +2.98419 q^{7} -1.57353 q^{9} -1.41939 q^{11} -5.78342 q^{13} -4.39674 q^{15} -4.18559 q^{17} -1.00000 q^{19} -3.56416 q^{21} -2.06768 q^{23} +8.55184 q^{25} +5.46239 q^{27} +5.06484 q^{29} -1.85777 q^{31} +1.69525 q^{33} +10.9856 q^{35} +9.02650 q^{37} +6.90742 q^{39} -1.92730 q^{41} -8.62224 q^{43} -5.79261 q^{45} -0.451340 q^{47} +1.90539 q^{49} +4.99906 q^{51} -3.51069 q^{53} -5.22518 q^{55} +1.19435 q^{57} -7.36720 q^{59} -6.74856 q^{61} -4.69571 q^{63} -21.2904 q^{65} -1.14821 q^{67} +2.46953 q^{69} +1.49758 q^{71} -9.92315 q^{73} -10.2139 q^{75} -4.23573 q^{77} +1.00000 q^{79} -1.80341 q^{81} +2.21563 q^{83} -15.4084 q^{85} -6.04918 q^{87} +5.93537 q^{89} -17.2588 q^{91} +2.21883 q^{93} -3.68128 q^{95} +6.14248 q^{97} +2.23346 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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