LMFDB - Embedded newform 6004.2.a.g.1.15

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:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q+0.267543 q^{3} +3.98483 q^{5} -3.08625 q^{7} -2.92842 q^{9} +O(q^{10})$	$q+0.267543 q^{3} +3.98483 q^{5} -3.08625 q^{7} -2.92842 q^{9} +5.66644 q^{11} -4.10489 q^{13} +1.06611 q^{15} +0.737930 q^{17} +1.00000 q^{19} -0.825706 q^{21} -6.58466 q^{23} +10.8789 q^{25} -1.58611 q^{27} -7.16235 q^{29} -7.97038 q^{31} +1.51602 q^{33} -12.2982 q^{35} +4.21298 q^{37} -1.09824 q^{39} -10.2411 q^{41} -0.0742958 q^{43} -11.6693 q^{45} -6.74831 q^{47} +2.52497 q^{49} +0.197428 q^{51} +6.32796 q^{53} +22.5798 q^{55} +0.267543 q^{57} +4.35950 q^{59} -15.4906 q^{61} +9.03785 q^{63} -16.3573 q^{65} +1.03751 q^{67} -1.76168 q^{69} -2.60621 q^{71} -1.11620 q^{73} +2.91057 q^{75} -17.4881 q^{77} -1.00000 q^{79} +8.36091 q^{81} +17.0678 q^{83} +2.94052 q^{85} -1.91624 q^{87} -5.80665 q^{89} +12.6687 q^{91} -2.13242 q^{93} +3.98483 q^{95} -8.59304 q^{97} -16.5937 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	0.267543	0.154466	0.0772330	−	0.997013i	$-0.475391\pi$
$3$	0.267543	0.154466	0.0772330	+	0.997013i	$0.475391\pi$
$4$	0	0
$5$	3.98483	1.78207	0.891035	−	0.453934i	$-0.149980\pi$
$5$	3.98483	1.78207	0.891035	+	0.453934i	$0.149980\pi$
$6$	0	0
$7$	−3.08625	−1.16649	−0.583247	−	0.812295i	$-0.698218\pi$
$7$	−3.08625	−1.16649	−0.583247	+	0.812295i	$0.698218\pi$
$8$	0	0
$9$	−2.92842	−0.976140
$10$	0	0
$11$	5.66644	1.70850	0.854248	−	0.519866i	$-0.174018\pi$
$11$	5.66644	1.70850	0.854248	+	0.519866i	$0.174018\pi$
$12$	0	0
$13$	−4.10489	−1.13849	−0.569246	−	0.822167i	$-0.692765\pi$
$13$	−4.10489	−1.13849	−0.569246	+	0.822167i	$0.692765\pi$
$14$	0	0
$15$	1.06611	0.275269
$16$	0	0
$17$	0.737930	0.178974	0.0894871	−	0.995988i	$-0.471477\pi$
$17$	0.737930	0.178974	0.0894871	+	0.995988i	$0.471477\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.825706	−0.180184
$22$	0	0
$23$	−6.58466	−1.37300	−0.686498	−	0.727132i	$-0.740853\pi$
$23$	−6.58466	−1.37300	−0.686498	+	0.727132i	$0.740853\pi$
$24$	0	0
$25$	10.8789	2.17578
$26$	0	0
$27$	−1.58611	−0.305247
$28$	0	0
$29$	−7.16235	−1.33002	−0.665008	−	0.746837i	$-0.731572\pi$
$29$	−7.16235	−1.33002	−0.665008	+	0.746837i	$0.731572\pi$
$30$	0	0
$31$	−7.97038	−1.43152	−0.715761	−	0.698345i	$-0.753920\pi$
$31$	−7.97038	−1.43152	−0.715761	+	0.698345i	$0.753920\pi$
$32$	0	0
$33$	1.51602	0.263904
$34$	0	0
$35$	−12.2982	−2.07878
$36$	0	0
$37$	4.21298	0.692610	0.346305	−	0.938122i	$-0.387436\pi$
$37$	4.21298	0.692610	0.346305	+	0.938122i	$0.387436\pi$
$38$	0	0
$39$	−1.09824	−0.175858
$40$	0	0
$41$	−10.2411	−1.59939	−0.799697	−	0.600404i	$-0.795006\pi$
$41$	−10.2411	−1.59939	−0.799697	+	0.600404i	$0.795006\pi$
$42$	0	0
$43$	−0.0742958	−0.0113300	−0.00566501	−	0.999984i	$-0.501803\pi$
$43$	−0.0742958	−0.0113300	−0.00566501	+	0.999984i	$0.501803\pi$
$44$	0	0
$45$	−11.6693	−1.73955
$46$	0	0
$47$	−6.74831	−0.984342	−0.492171	−	0.870499i	$-0.663796\pi$
$47$	−6.74831	−0.984342	−0.492171	+	0.870499i	$0.663796\pi$
$48$	0	0
$49$	2.52497	0.360710
$50$	0	0
$51$	0.197428	0.0276454
$52$	0	0
$53$	6.32796	0.869212	0.434606	−	0.900621i	$-0.356888\pi$
$53$	6.32796	0.869212	0.434606	+	0.900621i	$0.356888\pi$
$54$	0	0
$55$	22.5798	3.04466
$56$	0	0
$57$	0.267543	0.0354369
$58$	0	0
$59$	4.35950	0.567558	0.283779	−	0.958890i	$-0.408412\pi$
$59$	4.35950	0.567558	0.283779	+	0.958890i	$0.408412\pi$
$60$	0	0
$61$	−15.4906	−1.98337	−0.991685	−	0.128691i	$-0.958922\pi$
$61$	−15.4906	−1.98337	−0.991685	+	0.128691i	$0.958922\pi$
$62$	0	0
$63$	9.03785	1.13866
$64$	0	0
$65$	−16.3573	−2.02887
$66$	0	0
$67$	1.03751	0.126752	0.0633759	−	0.997990i	$-0.479813\pi$
$67$	1.03751	0.126752	0.0633759	+	0.997990i	$0.479813\pi$
$68$	0	0
$69$	−1.76168	−0.212081
$70$	0	0
$71$	−2.60621	−0.309300	−0.154650	−	0.987969i	$-0.549425\pi$
$71$	−2.60621	−0.309300	−0.154650	+	0.987969i	$0.549425\pi$
$72$	0	0
$73$	−1.11620	−0.130642	−0.0653208	−	0.997864i	$-0.520807\pi$
$73$	−1.11620	−0.130642	−0.0653208	+	0.997864i	$0.520807\pi$
$74$	0	0
$75$	2.91057	0.336083
$76$	0	0
$77$	−17.4881	−1.99295
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	8.36091	0.928990
$82$	0	0
$83$	17.0678	1.87343	0.936717	−	0.350086i	$-0.113848\pi$
$83$	17.0678	1.87343	0.936717	+	0.350086i	$0.113848\pi$
$84$	0	0
$85$	2.94052	0.318945
$86$	0	0
$87$	−1.91624	−0.205442
$88$	0	0
$89$	−5.80665	−0.615503	−0.307752	−	0.951467i	$-0.599577\pi$
$89$	−5.80665	−0.615503	−0.307752	+	0.951467i	$0.599577\pi$
$90$	0	0
$91$	12.6687	1.32805
$92$	0	0
$93$	−2.13242	−0.221122
$94$	0	0
$95$	3.98483	0.408835
$96$	0	0
$97$	−8.59304	−0.872491	−0.436246	−	0.899828i	$-0.643692\pi$
$97$	−8.59304	−0.872491	−0.436246	+	0.899828i	$0.643692\pi$
$98$	0	0
$99$	−16.5937	−1.66773
$100$	0	0
$101$	−3.24212	−0.322603	−0.161302	−	0.986905i	$-0.551569\pi$
$101$	−3.24212	−0.322603	−0.161302	+	0.986905i	$0.551569\pi$
$102$	0	0
$103$	−3.50418	−0.345277	−0.172638	−	0.984985i	$-0.555229\pi$
$103$	−3.50418	−0.345277	−0.172638	+	0.984985i	$0.555229\pi$
$104$	0	0
$105$	−3.29030	−0.321100
$106$	0	0
$107$	16.3533	1.58094	0.790468	−	0.612504i	$-0.209837\pi$
$107$	16.3533	1.58094	0.790468	+	0.612504i	$0.209837\pi$
$108$	0	0
$109$	−4.32019	−0.413799	−0.206900	−	0.978362i	$-0.566337\pi$
$109$	−4.32019	−0.413799	−0.206900	+	0.978362i	$0.566337\pi$
$110$	0	0
$111$	1.12715	0.106985
$112$	0	0
$113$	−1.94549	−0.183016	−0.0915082	−	0.995804i	$-0.529169\pi$
$113$	−1.94549	−0.183016	−0.0915082	+	0.995804i	$0.529169\pi$
$114$	0	0
$115$	−26.2387	−2.44678
$116$	0	0
$117$	12.0209	1.11133
$118$	0	0
$119$	−2.27744	−0.208772
$120$	0	0
$121$	21.1085	1.91896
$122$	0	0
$123$	−2.73994	−0.247052
$124$	0	0
$125$	23.4263	2.09531
$126$	0	0
$127$	−14.4542	−1.28261	−0.641303	−	0.767288i	$-0.721606\pi$
$127$	−14.4542	−1.28261	−0.641303	+	0.767288i	$0.721606\pi$
$128$	0	0
$129$	−0.0198773	−0.00175010
$130$	0	0
$131$	9.52369	0.832089	0.416045	−	0.909344i	$-0.363416\pi$
$131$	9.52369	0.832089	0.416045	+	0.909344i	$0.363416\pi$
$132$	0	0
$133$	−3.08625	−0.267612
$134$	0	0
$135$	−6.32037	−0.543971
$136$	0	0
$137$	−8.32265	−0.711052	−0.355526	−	0.934666i	$-0.615698\pi$
$137$	−8.32265	−0.711052	−0.355526	+	0.934666i	$0.615698\pi$
$138$	0	0
$139$	8.68277	0.736463	0.368231	−	0.929734i	$-0.379963\pi$
$139$	8.68277	0.736463	0.368231	+	0.929734i	$0.379963\pi$
$140$	0	0
$141$	−1.80546	−0.152047
$142$	0	0
$143$	−23.2601	−1.94511
$144$	0	0
$145$	−28.5408	−2.37018
$146$	0	0
$147$	0.675537	0.0557174
$148$	0	0
$149$	−11.1694	−0.915035	−0.457518	−	0.889201i	$-0.651261\pi$
$149$	−11.1694	−0.915035	−0.457518	+	0.889201i	$0.651261\pi$
$150$	0	0
$151$	−10.0572	−0.818445	−0.409222	−	0.912435i	$-0.634200\pi$
$151$	−10.0572	−0.818445	−0.409222	+	0.912435i	$0.634200\pi$
$152$	0	0
$153$	−2.16097	−0.174704
$154$	0	0
$155$	−31.7606	−2.55107
$156$	0	0
$157$	−19.1717	−1.53007	−0.765033	−	0.643990i	$-0.777278\pi$
$157$	−19.1717	−1.53007	−0.765033	+	0.643990i	$0.777278\pi$
$158$	0	0
$159$	1.69300	0.134264
$160$	0	0
$161$	20.3219	1.60159
$162$	0	0
$163$	−12.2144	−0.956702	−0.478351	−	0.878169i	$-0.658765\pi$
$163$	−12.2144	−0.956702	−0.478351	+	0.878169i	$0.658765\pi$
$164$	0	0
$165$	6.04107	0.470296
$166$	0	0
$167$	−10.5405	−0.815645	−0.407822	−	0.913061i	$-0.633712\pi$
$167$	−10.5405	−0.815645	−0.407822	+	0.913061i	$0.633712\pi$
$168$	0	0
$169$	3.85015	0.296165
$170$	0	0
$171$	−2.92842	−0.223942
$172$	0	0
$173$	9.79404	0.744627	0.372314	−	0.928107i	$-0.378565\pi$
$173$	9.79404	0.744627	0.372314	+	0.928107i	$0.378565\pi$
$174$	0	0
$175$	−33.5750	−2.53803
$176$	0	0
$177$	1.16635	0.0876685
$178$	0	0
$179$	14.6238	1.09303	0.546515	−	0.837449i	$-0.315954\pi$
$179$	14.6238	1.09303	0.546515	+	0.837449i	$0.315954\pi$
$180$	0	0
$181$	6.12088	0.454962	0.227481	−	0.973783i	$-0.426951\pi$
$181$	6.12088	0.454962	0.227481	+	0.973783i	$0.426951\pi$
$182$	0	0
$183$	−4.14441	−0.306363
$184$	0	0
$185$	16.7880	1.23428
$186$	0	0
$187$	4.18143	0.305777
$188$	0	0
$189$	4.89513	0.356068
$190$	0	0
$191$	0.507669	0.0367336	0.0183668	−	0.999831i	$-0.494153\pi$
$191$	0.507669	0.0367336	0.0183668	+	0.999831i	$0.494153\pi$
$192$	0	0
$193$	1.57498	0.113370	0.0566849	−	0.998392i	$-0.481947\pi$
$193$	1.57498	0.113370	0.0566849	+	0.998392i	$0.481947\pi$
$194$	0	0
$195$	−4.37628	−0.313392
$196$	0	0
$197$	−5.76214	−0.410535	−0.205268	−	0.978706i	$-0.565807\pi$
$197$	−5.76214	−0.410535	−0.205268	+	0.978706i	$0.565807\pi$
$198$	0	0
$199$	13.5143	0.958005	0.479003	−	0.877814i	$-0.340998\pi$
$199$	13.5143	0.958005	0.479003	+	0.877814i	$0.340998\pi$
$200$	0	0
$201$	0.277578	0.0195788
$202$	0	0
$203$	22.1048	1.55146
$204$	0	0
$205$	−40.8091	−2.85023
$206$	0	0
$207$	19.2827	1.34024
$208$	0	0
$209$	5.66644	0.391956
$210$	0	0
$211$	−5.33274	−0.367121	−0.183561	−	0.983008i	$-0.558762\pi$
$211$	−5.33274	−0.367121	−0.183561	+	0.983008i	$0.558762\pi$
$212$	0	0
$213$	−0.697272	−0.0477763
$214$	0	0
$215$	−0.296056	−0.0201909
$216$	0	0
$217$	24.5986	1.66986
$218$	0	0
$219$	−0.298632	−0.0201797
$220$	0	0
$221$	−3.02912	−0.203761
$222$	0	0
$223$	−7.27422	−0.487118	−0.243559	−	0.969886i	$-0.578315\pi$
$223$	−7.27422	−0.487118	−0.243559	+	0.969886i	$0.578315\pi$
$224$	0	0
$225$	−31.8579	−2.12386
$226$	0	0
$227$	−16.7390	−1.11101	−0.555503	−	0.831514i	$-0.687474\pi$
$227$	−16.7390	−1.11101	−0.555503	+	0.831514i	$0.687474\pi$
$228$	0	0
$229$	25.4085	1.67904	0.839522	−	0.543326i	$-0.182835\pi$
$229$	25.4085	1.67904	0.839522	+	0.543326i	$0.182835\pi$
$230$	0	0
$231$	−4.67881	−0.307843
$232$	0	0
$233$	23.9450	1.56869	0.784344	−	0.620326i	$-0.213000\pi$
$233$	23.9450	1.56869	0.784344	+	0.620326i	$0.213000\pi$
$234$	0	0
$235$	−26.8909	−1.75417
$236$	0	0
$237$	−0.267543	−0.0173788
$238$	0	0
$239$	10.8368	0.700976	0.350488	−	0.936567i	$-0.386016\pi$
$239$	10.8368	0.700976	0.350488	+	0.936567i	$0.386016\pi$
$240$	0	0
$241$	−17.8010	−1.14666	−0.573331	−	0.819324i	$-0.694349\pi$
$241$	−17.8010	−1.14666	−0.573331	+	0.819324i	$0.694349\pi$
$242$	0	0
$243$	6.99523	0.448744
$244$	0	0
$245$	10.0616	0.642810
$246$	0	0
$247$	−4.10489	−0.261188
$248$	0	0
$249$	4.56637	0.289382
$250$	0	0
$251$	8.05985	0.508733	0.254367	−	0.967108i	$-0.418133\pi$
$251$	8.05985	0.508733	0.254367	+	0.967108i	$0.418133\pi$
$252$	0	0
$253$	−37.3116	−2.34576
$254$	0	0
$255$	0.786717	0.0492661
$256$	0	0
$257$	31.1377	1.94232	0.971159	−	0.238433i	$-0.0766339\pi$
$257$	31.1377	1.94232	0.971159	+	0.238433i	$0.0766339\pi$
$258$	0	0
$259$	−13.0023	−0.807926
$260$	0	0
$261$	20.9744	1.29828
$262$	0	0
$263$	−1.28299	−0.0791123	−0.0395561	−	0.999217i	$-0.512594\pi$
$263$	−1.28299	−0.0791123	−0.0395561	+	0.999217i	$0.512594\pi$
$264$	0	0
$265$	25.2159	1.54900
$266$	0	0
$267$	−1.55353	−0.0950744
$268$	0	0
$269$	−10.2559	−0.625316	−0.312658	−	0.949866i	$-0.601219\pi$
$269$	−10.2559	−0.625316	−0.312658	+	0.949866i	$0.601219\pi$
$270$	0	0
$271$	−2.81545	−0.171026	−0.0855132	−	0.996337i	$-0.527253\pi$
$271$	−2.81545	−0.171026	−0.0855132	+	0.996337i	$0.527253\pi$
$272$	0	0
$273$	3.38943	0.205138
$274$	0	0
$275$	61.6445	3.71730
$276$	0	0
$277$	−26.4165	−1.58722	−0.793608	−	0.608429i	$-0.791800\pi$
$277$	−26.4165	−1.58722	−0.793608	+	0.608429i	$0.791800\pi$
$278$	0	0
$279$	23.3406	1.39737
$280$	0	0
$281$	−11.1123	−0.662905	−0.331452	−	0.943472i	$-0.607539\pi$
$281$	−11.1123	−0.662905	−0.331452	+	0.943472i	$0.607539\pi$
$282$	0	0
$283$	17.0505	1.01354	0.506772	−	0.862080i	$-0.330838\pi$
$283$	17.0505	1.01354	0.506772	+	0.862080i	$0.330838\pi$
$284$	0	0
$285$	1.06611	0.0631511
$286$	0	0
$287$	31.6067	1.86568
$288$	0	0
$289$	−16.4555	−0.967968
$290$	0	0
$291$	−2.29901	−0.134770
$292$	0	0
$293$	−15.0465	−0.879027	−0.439514	−	0.898236i	$-0.644849\pi$
$293$	−15.0465	−0.879027	−0.439514	+	0.898236i	$0.644849\pi$
$294$	0	0
$295$	17.3719	1.01143
$296$	0	0
$297$	−8.98758	−0.521512
$298$	0	0
$299$	27.0293	1.56315
$300$	0	0
$301$	0.229296	0.0132164
$302$	0	0
$303$	−0.867408	−0.0498313
$304$	0	0
$305$	−61.7275	−3.53450
$306$	0	0
$307$	−10.5847	−0.604103	−0.302051	−	0.953292i	$-0.597671\pi$
$307$	−10.5847	−0.604103	−0.302051	+	0.953292i	$0.597671\pi$
$308$	0	0
$309$	−0.937518	−0.0533335
$310$	0	0
$311$	19.3012	1.09447	0.547236	−	0.836979i	$-0.315680\pi$
$311$	19.3012	1.09447	0.547236	+	0.836979i	$0.315680\pi$
$312$	0	0
$313$	−19.1934	−1.08487	−0.542437	−	0.840097i	$-0.682498\pi$
$313$	−19.1934	−1.08487	−0.542437	+	0.840097i	$0.682498\pi$
$314$	0	0
$315$	36.0143	2.02918
$316$	0	0
$317$	5.46787	0.307106	0.153553	−	0.988140i	$-0.450928\pi$
$317$	5.46787	0.307106	0.153553	+	0.988140i	$0.450928\pi$
$318$	0	0
$319$	−40.5850	−2.27232
$320$	0	0
$321$	4.37522	0.244201
$322$	0	0
$323$	0.737930	0.0410595
$324$	0	0
$325$	−44.6566	−2.47710
$326$	0	0
$327$	−1.15584	−0.0639180
$328$	0	0
$329$	20.8270	1.14823
$330$	0	0
$331$	−18.4884	−1.01622	−0.508108	−	0.861293i	$-0.669655\pi$
$331$	−18.4884	−1.01622	−0.508108	+	0.861293i	$0.669655\pi$
$332$	0	0
$333$	−12.3374	−0.676085
$334$	0	0
$335$	4.13429	0.225881
$336$	0	0
$337$	9.87107	0.537711	0.268856	−	0.963180i	$-0.413355\pi$
$337$	9.87107	0.537711	0.268856	+	0.963180i	$0.413355\pi$
$338$	0	0
$339$	−0.520503	−0.0282698
$340$	0	0
$341$	−45.1636	−2.44575
$342$	0	0
$343$	13.8111	0.745729
$344$	0	0
$345$	−7.02000	−0.377944
$346$	0	0
$347$	5.72655	0.307417	0.153709	−	0.988116i	$-0.450878\pi$
$347$	5.72655	0.307417	0.153709	+	0.988116i	$0.450878\pi$
$348$	0	0
$349$	−15.5689	−0.833382	−0.416691	−	0.909048i	$-0.636810\pi$
$349$	−15.5689	−0.833382	−0.416691	+	0.909048i	$0.636810\pi$
$350$	0	0
$351$	6.51080	0.347521
$352$	0	0
$353$	1.82597	0.0971865	0.0485933	−	0.998819i	$-0.484526\pi$
$353$	1.82597	0.0971865	0.0485933	+	0.998819i	$0.484526\pi$
$354$	0	0
$355$	−10.3853	−0.551194
$356$	0	0
$357$	−0.609313	−0.0322483
$358$	0	0
$359$	−13.8860	−0.732876	−0.366438	−	0.930442i	$-0.619423\pi$
$359$	−13.8860	−0.732876	−0.366438	+	0.930442i	$0.619423\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.64744	0.296413
$364$	0	0
$365$	−4.44788	−0.232813
$366$	0	0
$367$	24.2946	1.26817	0.634085	−	0.773264i	$-0.281377\pi$
$367$	24.2946	1.26817	0.634085	+	0.773264i	$0.281377\pi$
$368$	0	0
$369$	29.9903	1.56123
$370$	0	0
$371$	−19.5297	−1.01393
$372$	0	0
$373$	−25.6746	−1.32938	−0.664690	−	0.747119i	$-0.731437\pi$
$373$	−25.6746	−1.32938	−0.664690	+	0.747119i	$0.731437\pi$
$374$	0	0
$375$	6.26755	0.323655
$376$	0	0
$377$	29.4007	1.51421
$378$	0	0
$379$	−29.5409	−1.51741	−0.758706	−	0.651433i	$-0.774168\pi$
$379$	−29.5409	−1.51741	−0.758706	+	0.651433i	$0.774168\pi$
$380$	0	0
$381$	−3.86713	−0.198119
$382$	0	0
$383$	19.3035	0.986364	0.493182	−	0.869926i	$-0.335834\pi$
$383$	19.3035	0.986364	0.493182	+	0.869926i	$0.335834\pi$
$384$	0	0
$385$	−69.6870	−3.55158
$386$	0	0
$387$	0.217570	0.0110597
$388$	0	0
$389$	−7.82362	−0.396673	−0.198337	−	0.980134i	$-0.563554\pi$
$389$	−7.82362	−0.396673	−0.198337	+	0.980134i	$0.563554\pi$
$390$	0	0
$391$	−4.85901	−0.245731
$392$	0	0
$393$	2.54800	0.128530
$394$	0	0
$395$	−3.98483	−0.200499
$396$	0	0
$397$	−15.2982	−0.767793	−0.383897	−	0.923376i	$-0.625418\pi$
$397$	−15.2982	−0.767793	−0.383897	+	0.923376i	$0.625418\pi$
$398$	0	0
$399$	−0.825706	−0.0413370
$400$	0	0
$401$	3.46073	0.172821	0.0864104	−	0.996260i	$-0.472460\pi$
$401$	3.46073	0.172821	0.0864104	+	0.996260i	$0.472460\pi$
$402$	0	0
$403$	32.7175	1.62978
$404$	0	0
$405$	33.3168	1.65553
$406$	0	0
$407$	23.8726	1.18332
$408$	0	0
$409$	33.6841	1.66557	0.832785	−	0.553596i	$-0.186745\pi$
$409$	33.6841	1.66557	0.832785	+	0.553596i	$0.186745\pi$
$410$	0	0
$411$	−2.22667	−0.109833
$412$	0	0
$413$	−13.4545	−0.662053
$414$	0	0
$415$	68.0123	3.33859
$416$	0	0
$417$	2.32301	0.113758
$418$	0	0
$419$	−12.6607	−0.618516	−0.309258	−	0.950978i	$-0.600081\pi$
$419$	−12.6607	−0.618516	−0.309258	+	0.950978i	$0.600081\pi$
$420$	0	0
$421$	24.9508	1.21603	0.608013	−	0.793927i	$-0.291967\pi$
$421$	24.9508	1.21603	0.608013	+	0.793927i	$0.291967\pi$
$422$	0	0
$423$	19.7619	0.960855
$424$	0	0
$425$	8.02784	0.389408
$426$	0	0
$427$	47.8080	2.31359
$428$	0	0
$429$	−6.22308	−0.300453
$430$	0	0
$431$	5.25871	0.253303	0.126652	−	0.991947i	$-0.459577\pi$
$431$	5.25871	0.253303	0.126652	+	0.991947i	$0.459577\pi$
$432$	0	0
$433$	29.1341	1.40010	0.700048	−	0.714096i	$-0.253162\pi$
$433$	29.1341	1.40010	0.700048	+	0.714096i	$0.253162\pi$
$434$	0	0
$435$	−7.63588	−0.366112
$436$	0	0
$437$	−6.58466	−0.314987
$438$	0	0
$439$	−14.5518	−0.694522	−0.347261	−	0.937769i	$-0.612888\pi$
$439$	−14.5518	−0.694522	−0.347261	+	0.937769i	$0.612888\pi$
$440$	0	0
$441$	−7.39417	−0.352103
$442$	0	0
$443$	30.3678	1.44282	0.721409	−	0.692509i	$-0.243495\pi$
$443$	30.3678	1.44282	0.721409	+	0.692509i	$0.243495\pi$
$444$	0	0
$445$	−23.1385	−1.09687
$446$	0	0
$447$	−2.98830	−0.141342
$448$	0	0
$449$	−26.7498	−1.26240	−0.631201	−	0.775619i	$-0.717438\pi$
$449$	−26.7498	−1.26240	−0.631201	+	0.775619i	$0.717438\pi$
$450$	0	0
$451$	−58.0306	−2.73256
$452$	0	0
$453$	−2.69074	−0.126422
$454$	0	0
$455$	50.4828	2.36667
$456$	0	0
$457$	5.00356	0.234057	0.117028	−	0.993129i	$-0.462663\pi$
$457$	5.00356	0.234057	0.117028	+	0.993129i	$0.462663\pi$
$458$	0	0
$459$	−1.17044	−0.0546313
$460$	0	0
$461$	14.3969	0.670530	0.335265	−	0.942124i	$-0.391174\pi$
$461$	14.3969	0.670530	0.335265	+	0.942124i	$0.391174\pi$
$462$	0	0
$463$	20.9455	0.973418	0.486709	−	0.873564i	$-0.338197\pi$
$463$	20.9455	0.973418	0.486709	+	0.873564i	$0.338197\pi$
$464$	0	0
$465$	−8.49733	−0.394054
$466$	0	0
$467$	17.3177	0.801368	0.400684	−	0.916216i	$-0.368773\pi$
$467$	17.3177	0.801368	0.400684	+	0.916216i	$0.368773\pi$
$468$	0	0
$469$	−3.20201	−0.147855
$470$	0	0
$471$	−5.12925	−0.236343
$472$	0	0
$473$	−0.420993	−0.0193573
$474$	0	0
$475$	10.8789	0.499157
$476$	0	0
$477$	−18.5309	−0.848473
$478$	0	0
$479$	−4.75877	−0.217434	−0.108717	−	0.994073i	$-0.534674\pi$
$479$	−4.75877	−0.217434	−0.108717	+	0.994073i	$0.534674\pi$
$480$	0	0
$481$	−17.2938	−0.788532
$482$	0	0
$483$	5.43699	0.247392
$484$	0	0
$485$	−34.2418	−1.55484
$486$	0	0
$487$	−22.0088	−0.997315	−0.498658	−	0.866799i	$-0.666173\pi$
$487$	−22.0088	−0.997315	−0.498658	+	0.866799i	$0.666173\pi$
$488$	0	0
$489$	−3.26787	−0.147778
$490$	0	0
$491$	13.3411	0.602075	0.301037	−	0.953612i	$-0.402667\pi$
$491$	13.3411	0.602075	0.301037	+	0.953612i	$0.402667\pi$
$492$	0	0
$493$	−5.28531	−0.238038
$494$	0	0
$495$	−66.1231	−2.97201
$496$	0	0
$497$	8.04342	0.360797
$498$	0	0
$499$	17.3197	0.775336	0.387668	−	0.921799i	$-0.373281\pi$
$499$	17.3197	0.775336	0.387668	+	0.921799i	$0.373281\pi$
$500$	0	0
$501$	−2.82003	−0.125989
$502$	0	0
$503$	22.4785	1.00227	0.501134	−	0.865369i	$-0.332916\pi$
$503$	22.4785	1.00227	0.501134	+	0.865369i	$0.332916\pi$
$504$	0	0
$505$	−12.9193	−0.574902
$506$	0	0
$507$	1.03008	0.0457475
$508$	0	0
$509$	−26.6450	−1.18102	−0.590510	−	0.807030i	$-0.701074\pi$
$509$	−26.6450	−1.18102	−0.590510	+	0.807030i	$0.701074\pi$
$510$	0	0
$511$	3.44489	0.152393
$512$	0	0
$513$	−1.58611	−0.0700284
$514$	0	0
$515$	−13.9636	−0.615308
$516$	0	0
$517$	−38.2389	−1.68174
$518$	0	0
$519$	2.62033	0.115020
$520$	0	0
$521$	−24.9282	−1.09213	−0.546063	−	0.837744i	$-0.683874\pi$
$521$	−24.9282	−1.09213	−0.546063	+	0.837744i	$0.683874\pi$
$522$	0	0
$523$	−35.9077	−1.57013	−0.785067	−	0.619411i	$-0.787372\pi$
$523$	−35.9077	−1.57013	−0.785067	+	0.619411i	$0.787372\pi$
$524$	0	0
$525$	−8.98275	−0.392039
$526$	0	0
$527$	−5.88158	−0.256206
$528$	0	0
$529$	20.3577	0.885119
$530$	0	0
$531$	−12.7664	−0.554016
$532$	0	0
$533$	42.0387	1.82090
$534$	0	0
$535$	65.1652	2.81734
$536$	0	0
$537$	3.91248	0.168836
$538$	0	0
$539$	14.3076	0.616271
$540$	0	0
$541$	−44.0737	−1.89488	−0.947438	−	0.319940i	$-0.896337\pi$
$541$	−44.0737	−1.89488	−0.947438	+	0.319940i	$0.896337\pi$
$542$	0	0
$543$	1.63760	0.0702761
$544$	0	0
$545$	−17.2152	−0.737420
$546$	0	0
$547$	−22.8305	−0.976164	−0.488082	−	0.872798i	$-0.662303\pi$
$547$	−22.8305	−0.976164	−0.488082	+	0.872798i	$0.662303\pi$
$548$	0	0
$549$	45.3630	1.93605
$550$	0	0
$551$	−7.16235	−0.305126
$552$	0	0
$553$	3.08625	0.131241
$554$	0	0
$555$	4.49152	0.190654
$556$	0	0
$557$	−38.0115	−1.61060	−0.805298	−	0.592870i	$-0.797995\pi$
$557$	−38.0115	−1.61060	−0.805298	+	0.592870i	$0.797995\pi$
$558$	0	0
$559$	0.304977	0.0128991
$560$	0	0
$561$	1.11871	0.0472321
$562$	0	0
$563$	12.3346	0.519842	0.259921	−	0.965630i	$-0.416304\pi$
$563$	12.3346	0.519842	0.259921	+	0.965630i	$0.416304\pi$
$564$	0	0
$565$	−7.75245	−0.326148
$566$	0	0
$567$	−25.8039	−1.08366
$568$	0	0
$569$	34.3615	1.44051	0.720254	−	0.693711i	$-0.244025\pi$
$569$	34.3615	1.44051	0.720254	+	0.693711i	$0.244025\pi$
$570$	0	0
$571$	−20.2425	−0.847122	−0.423561	−	0.905867i	$-0.639220\pi$
$571$	−20.2425	−0.847122	−0.423561	+	0.905867i	$0.639220\pi$
$572$	0	0
$573$	0.135823	0.00567410
$574$	0	0
$575$	−71.6337	−2.98733
$576$	0	0
$577$	−21.6444	−0.901069	−0.450535	−	0.892759i	$-0.648767\pi$
$577$	−21.6444	−0.901069	−0.450535	+	0.892759i	$0.648767\pi$
$578$	0	0
$579$	0.421376	0.0175118
$580$	0	0
$581$	−52.6756	−2.18535
$582$	0	0
$583$	35.8570	1.48504
$584$	0	0
$585$	47.9011	1.98047
$586$	0	0
$587$	−33.0794	−1.36533	−0.682666	−	0.730730i	$-0.739180\pi$
$587$	−33.0794	−1.36533	−0.682666	+	0.730730i	$0.739180\pi$
$588$	0	0
$589$	−7.97038	−0.328414
$590$	0	0
$591$	−1.54162	−0.0634138
$592$	0	0
$593$	36.4590	1.49719	0.748595	−	0.663028i	$-0.230729\pi$
$593$	36.4590	1.49719	0.748595	+	0.663028i	$0.230729\pi$
$594$	0	0
$595$	−9.07521	−0.372047
$596$	0	0
$597$	3.61566	0.147979
$598$	0	0
$599$	22.1215	0.903862	0.451931	−	0.892053i	$-0.350735\pi$
$599$	22.1215	0.903862	0.451931	+	0.892053i	$0.350735\pi$
$600$	0	0
$601$	−17.6738	−0.720930	−0.360465	−	0.932773i	$-0.617382\pi$
$601$	−17.6738	−0.720930	−0.360465	+	0.932773i	$0.617382\pi$
$602$	0	0
$603$	−3.03826	−0.123727
$604$	0	0
$605$	84.1138	3.41971
$606$	0	0
$607$	28.6143	1.16142	0.580708	−	0.814112i	$-0.302776\pi$
$607$	28.6143	1.16142	0.580708	+	0.814112i	$0.302776\pi$
$608$	0	0
$609$	5.91400	0.239647
$610$	0	0
$611$	27.7011	1.12067
$612$	0	0
$613$	16.9460	0.684444	0.342222	−	0.939619i	$-0.388820\pi$
$613$	16.9460	0.684444	0.342222	+	0.939619i	$0.388820\pi$
$614$	0	0
$615$	−10.9182	−0.440264
$616$	0	0
$617$	−29.5611	−1.19009	−0.595043	−	0.803694i	$-0.702865\pi$
$617$	−29.5611	−1.19009	−0.595043	+	0.803694i	$0.702865\pi$
$618$	0	0
$619$	−20.8148	−0.836617	−0.418309	−	0.908305i	$-0.637377\pi$
$619$	−20.8148	−0.836617	−0.418309	+	0.908305i	$0.637377\pi$
$620$	0	0
$621$	10.4440	0.419102
$622$	0	0
$623$	17.9208	0.717981
$624$	0	0
$625$	38.9555	1.55822
$626$	0	0
$627$	1.51602	0.0605438
$628$	0	0
$629$	3.10889	0.123959
$630$	0	0
$631$	−39.1313	−1.55779	−0.778896	−	0.627153i	$-0.784220\pi$
$631$	−39.1313	−1.55779	−0.778896	+	0.627153i	$0.784220\pi$
$632$	0	0
$633$	−1.42674	−0.0567077
$634$	0	0
$635$	−57.5977	−2.28569
$636$	0	0
$637$	−10.3647	−0.410665
$638$	0	0
$639$	7.63207	0.301920
$640$	0	0
$641$	−45.6850	−1.80445	−0.902224	−	0.431268i	$-0.858066\pi$
$641$	−45.6850	−1.80445	−0.902224	+	0.431268i	$0.858066\pi$
$642$	0	0
$643$	−10.2820	−0.405484	−0.202742	−	0.979232i	$-0.564985\pi$
$643$	−10.2820	−0.405484	−0.202742	+	0.979232i	$0.564985\pi$
$644$	0	0
$645$	−0.0792078	−0.00311881
$646$	0	0
$647$	−6.89390	−0.271027	−0.135514	−	0.990775i	$-0.543268\pi$
$647$	−6.89390	−0.271027	−0.135514	+	0.990775i	$0.543268\pi$
$648$	0	0
$649$	24.7028	0.969670
$650$	0	0
$651$	6.58119	0.257937
$652$	0	0
$653$	11.8571	0.464004	0.232002	−	0.972715i	$-0.425472\pi$
$653$	11.8571	0.464004	0.232002	+	0.972715i	$0.425472\pi$
$654$	0	0
$655$	37.9503	1.48284
$656$	0	0
$657$	3.26871	0.127525
$658$	0	0
$659$	7.88681	0.307227	0.153613	−	0.988131i	$-0.450909\pi$
$659$	7.88681	0.307227	0.153613	+	0.988131i	$0.450909\pi$
$660$	0	0
$661$	−5.94556	−0.231256	−0.115628	−	0.993293i	$-0.536888\pi$
$661$	−5.94556	−0.231256	−0.115628	+	0.993293i	$0.536888\pi$
$662$	0	0
$663$	−0.810421	−0.0314741
$664$	0	0
$665$	−12.2982	−0.476904
$666$	0	0
$667$	47.1616	1.82611
$668$	0	0
$669$	−1.94617	−0.0752432
$670$	0	0
$671$	−87.7766	−3.38858
$672$	0	0
$673$	−49.8614	−1.92202	−0.961008	−	0.276520i	$-0.910819\pi$
$673$	−49.8614	−1.92202	−0.961008	+	0.276520i	$0.910819\pi$
$674$	0	0
$675$	−17.2551	−0.664148
$676$	0	0
$677$	−8.28532	−0.318431	−0.159215	−	0.987244i	$-0.550896\pi$
$677$	−8.28532	−0.318431	−0.159215	+	0.987244i	$0.550896\pi$
$678$	0	0
$679$	26.5203	1.01776
$680$	0	0
$681$	−4.47840	−0.171613
$682$	0	0
$683$	31.1620	1.19238	0.596190	−	0.802844i	$-0.296681\pi$
$683$	31.1620	1.19238	0.596190	+	0.802844i	$0.296681\pi$
$684$	0	0
$685$	−33.1644	−1.26714
$686$	0	0
$687$	6.79788	0.259355
$688$	0	0
$689$	−25.9756	−0.989592
$690$	0	0
$691$	5.78987	0.220257	0.110128	−	0.993917i	$-0.464874\pi$
$691$	5.78987	0.220257	0.110128	+	0.993917i	$0.464874\pi$
$692$	0	0
$693$	51.2124	1.94540
$694$	0	0
$695$	34.5994	1.31243
$696$	0	0
$697$	−7.55722	−0.286250
$698$	0	0
$699$	6.40631	0.242309
$700$	0	0
$701$	13.7565	0.519576	0.259788	−	0.965666i	$-0.416347\pi$
$701$	13.7565	0.519576	0.259788	+	0.965666i	$0.416347\pi$
$702$	0	0
$703$	4.21298	0.158896
$704$	0	0
$705$	−7.19446	−0.270959
$706$	0	0
$707$	10.0060	0.376315
$708$	0	0
$709$	−0.784026	−0.0294447	−0.0147224	−	0.999892i	$-0.504686\pi$
$709$	−0.784026	−0.0294447	−0.0147224	+	0.999892i	$0.504686\pi$
$710$	0	0
$711$	2.92842	0.109824
$712$	0	0
$713$	52.4822	1.96547
$714$	0	0
$715$	−92.6876	−3.46632
$716$	0	0
$717$	2.89932	0.108277
$718$	0	0
$719$	38.4772	1.43496	0.717479	−	0.696581i	$-0.245296\pi$
$719$	38.4772	1.43496	0.717479	+	0.696581i	$0.245296\pi$
$720$	0	0
$721$	10.8148	0.402763
$722$	0	0
$723$	−4.76253	−0.177120
$724$	0	0
$725$	−77.9183	−2.89381
$726$	0	0
$727$	40.6305	1.50690	0.753451	−	0.657504i	$-0.228388\pi$
$727$	40.6305	1.50690	0.753451	+	0.657504i	$0.228388\pi$
$728$	0	0
$729$	−23.2112	−0.859674
$730$	0	0
$731$	−0.0548251	−0.00202778
$732$	0	0
$733$	35.4159	1.30812	0.654059	−	0.756444i	$-0.273065\pi$
$733$	35.4159	1.30812	0.654059	+	0.756444i	$0.273065\pi$
$734$	0	0
$735$	2.69190	0.0992923
$736$	0	0
$737$	5.87897	0.216555
$738$	0	0
$739$	27.6195	1.01600	0.508000	−	0.861357i	$-0.330385\pi$
$739$	27.6195	1.01600	0.508000	+	0.861357i	$0.330385\pi$
$740$	0	0
$741$	−1.09824	−0.0403447
$742$	0	0
$743$	−52.1845	−1.91446	−0.957232	−	0.289323i	$-0.906570\pi$
$743$	−52.1845	−1.91446	−0.957232	+	0.289323i	$0.906570\pi$
$744$	0	0
$745$	−44.5083	−1.63066
$746$	0	0
$747$	−49.9817	−1.82874
$748$	0	0
$749$	−50.4705	−1.84415
$750$	0	0
$751$	−4.68961	−0.171126	−0.0855632	−	0.996333i	$-0.527269\pi$
$751$	−4.68961	−0.171126	−0.0855632	+	0.996333i	$0.527269\pi$
$752$	0	0
$753$	2.15636	0.0785820
$754$	0	0
$755$	−40.0763	−1.45853
$756$	0	0
$757$	13.5813	0.493620	0.246810	−	0.969064i	$-0.420618\pi$
$757$	13.5813	0.493620	0.246810	+	0.969064i	$0.420618\pi$
$758$	0	0
$759$	−9.98245	−0.362340
$760$	0	0
$761$	23.4668	0.850671	0.425336	−	0.905036i	$-0.360156\pi$
$761$	23.4668	0.850671	0.425336	+	0.905036i	$0.360156\pi$
$762$	0	0
$763$	13.3332	0.482695
$764$	0	0
$765$	−8.61109	−0.311335
$766$	0	0
$767$	−17.8953	−0.646161
$768$	0	0
$769$	−37.3675	−1.34751	−0.673753	−	0.738956i	$-0.735319\pi$
$769$	−37.3675	−1.34751	−0.673753	+	0.738956i	$0.735319\pi$
$770$	0	0
$771$	8.33068	0.300022
$772$	0	0
$773$	53.4358	1.92195	0.960977	−	0.276630i	$-0.0892176\pi$
$773$	53.4358	1.92195	0.960977	+	0.276630i	$0.0892176\pi$
$774$	0	0
$775$	−86.7087	−3.11467
$776$	0	0
$777$	−3.47869	−0.124797
$778$	0	0
$779$	−10.2411	−0.366926
$780$	0	0
$781$	−14.7679	−0.528437
$782$	0	0
$783$	11.3603	0.405983
$784$	0	0
$785$	−76.3959	−2.72669
$786$	0	0
$787$	41.9471	1.49525	0.747626	−	0.664120i	$-0.231194\pi$
$787$	41.9471	1.49525	0.747626	+	0.664120i	$0.231194\pi$
$788$	0	0
$789$	−0.343254	−0.0122202
$790$	0	0
$791$	6.00428	0.213488
$792$	0	0
$793$	63.5873	2.25805
$794$	0	0
$795$	6.74633	0.239268
$796$	0	0
$797$	20.7217	0.734001	0.367000	−	0.930221i	$-0.380385\pi$
$797$	20.7217	0.734001	0.367000	+	0.930221i	$0.380385\pi$
$798$	0	0
$799$	−4.97978	−0.176172
$800$	0	0
$801$	17.0043	0.600818
$802$	0	0
$803$	−6.32489	−0.223201
$804$	0	0
$805$	80.9795	2.85415
$806$	0	0
$807$	−2.74391	−0.0965901
$808$	0	0
$809$	−25.2923	−0.889229	−0.444614	−	0.895722i	$-0.646659\pi$
$809$	−25.2923	−0.889229	−0.444614	+	0.895722i	$0.646659\pi$
$810$	0	0
$811$	−10.4491	−0.366919	−0.183460	−	0.983027i	$-0.558730\pi$
$811$	−10.4491	−0.366919	−0.183460	+	0.983027i	$0.558730\pi$
$812$	0	0
$813$	−0.753254	−0.0264178
$814$	0	0
$815$	−48.6721	−1.70491
$816$	0	0
$817$	−0.0742958	−0.00259928
$818$	0	0
$819$	−37.0994	−1.29636
$820$	0	0
$821$	7.67738	0.267942	0.133971	−	0.990985i	$-0.457227\pi$
$821$	7.67738	0.267942	0.133971	+	0.990985i	$0.457227\pi$
$822$	0	0
$823$	−28.4103	−0.990320	−0.495160	−	0.868802i	$-0.664891\pi$
$823$	−28.4103	−0.990320	−0.495160	+	0.868802i	$0.664891\pi$
$824$	0	0
$825$	16.4925	0.574197
$826$	0	0
$827$	6.35233	0.220892	0.110446	−	0.993882i	$-0.464772\pi$
$827$	6.35233	0.220892	0.110446	+	0.993882i	$0.464772\pi$
$828$	0	0
$829$	−40.9232	−1.42132	−0.710660	−	0.703535i	$-0.751604\pi$
$829$	−40.9232	−1.42132	−0.710660	+	0.703535i	$0.751604\pi$
$830$	0	0
$831$	−7.06756	−0.245171
$832$	0	0
$833$	1.86325	0.0645577
$834$	0	0
$835$	−42.0019	−1.45354
$836$	0	0
$837$	12.6419	0.436967
$838$	0	0
$839$	14.5133	0.501056	0.250528	−	0.968109i	$-0.419396\pi$
$839$	14.5133	0.501056	0.250528	+	0.968109i	$0.419396\pi$
$840$	0	0
$841$	22.2993	0.768940
$842$	0	0
$843$	−2.97302	−0.102396
$844$	0	0
$845$	15.3422	0.527787
$846$	0	0
$847$	−65.1462	−2.23845
$848$	0	0
$849$	4.56173	0.156558
$850$	0	0
$851$	−27.7411	−0.950951
$852$	0	0
$853$	28.0533	0.960526	0.480263	−	0.877125i	$-0.340541\pi$
$853$	28.0533	0.960526	0.480263	+	0.877125i	$0.340541\pi$
$854$	0	0
$855$	−11.6693	−0.399080
$856$	0	0
$857$	−16.1110	−0.550342	−0.275171	−	0.961395i	$-0.588734\pi$
$857$	−16.1110	−0.550342	−0.275171	+	0.961395i	$0.588734\pi$
$858$	0	0
$859$	−27.4236	−0.935682	−0.467841	−	0.883813i	$-0.654968\pi$
$859$	−27.4236	−0.935682	−0.467841	+	0.883813i	$0.654968\pi$
$860$	0	0
$861$	8.45615	0.288185
$862$	0	0
$863$	52.5151	1.78764	0.893818	−	0.448430i	$-0.148017\pi$
$863$	52.5151	1.78764	0.893818	+	0.448430i	$0.148017\pi$
$864$	0	0
$865$	39.0276	1.32698
$866$	0	0
$867$	−4.40254	−0.149518
$868$	0	0
$869$	−5.66644	−0.192221
$870$	0	0
$871$	−4.25886	−0.144306
$872$	0	0
$873$	25.1640	0.851674
$874$	0	0
$875$	−72.2996	−2.44417
$876$	0	0
$877$	−31.6799	−1.06975	−0.534876	−	0.844930i	$-0.679642\pi$
$877$	−31.6799	−1.06975	−0.534876	+	0.844930i	$0.679642\pi$
$878$	0	0
$879$	−4.02559	−0.135780
$880$	0	0
$881$	−1.08430	−0.0365308	−0.0182654	−	0.999833i	$-0.505814\pi$
$881$	−1.08430	−0.0365308	−0.0182654	+	0.999833i	$0.505814\pi$
$882$	0	0
$883$	−14.6799	−0.494018	−0.247009	−	0.969013i	$-0.579448\pi$
$883$	−14.6799	−0.494018	−0.247009	+	0.969013i	$0.579448\pi$
$884$	0	0
$885$	4.64772	0.156231
$886$	0	0
$887$	−34.9533	−1.17362	−0.586809	−	0.809725i	$-0.699616\pi$
$887$	−34.9533	−1.17362	−0.586809	+	0.809725i	$0.699616\pi$
$888$	0	0
$889$	44.6094	1.49615
$890$	0	0
$891$	47.3766	1.58717
$892$	0	0
$893$	−6.74831	−0.225823
$894$	0	0
$895$	58.2732	1.94786
$896$	0	0
$897$	7.23151	0.241453
$898$	0	0
$899$	57.0866	1.90395
$900$	0	0
$901$	4.66959	0.155567
$902$	0	0
$903$	0.0613465	0.00204148
$904$	0	0
$905$	24.3907	0.810774
$906$	0	0
$907$	48.8899	1.62336	0.811682	−	0.584100i	$-0.198552\pi$
$907$	48.8899	1.62336	0.811682	+	0.584100i	$0.198552\pi$
$908$	0	0
$909$	9.49431	0.314906
$910$	0	0
$911$	−27.9939	−0.927479	−0.463739	−	0.885972i	$-0.653493\pi$
$911$	−27.9939	−0.927479	−0.463739	+	0.885972i	$0.653493\pi$
$912$	0	0
$913$	96.7136	3.20075
$914$	0	0
$915$	−16.5148	−0.545961
$916$	0	0
$917$	−29.3925	−0.970627
$918$	0	0
$919$	38.2174	1.26068	0.630338	−	0.776321i	$-0.282916\pi$
$919$	38.2174	1.26068	0.630338	+	0.776321i	$0.282916\pi$
$920$	0	0
$921$	−2.83187	−0.0933134
$922$	0	0
$923$	10.6982	0.352136
$924$	0	0
$925$	45.8325	1.50696
$926$	0	0
$927$	10.2617	0.337039
$928$	0	0
$929$	26.9191	0.883187	0.441594	−	0.897215i	$-0.354413\pi$
$929$	26.9191	0.883187	0.441594	+	0.897215i	$0.354413\pi$
$930$	0	0
$931$	2.52497	0.0827525
$932$	0	0
$933$	5.16390	0.169059
$934$	0	0
$935$	16.6623	0.544915
$936$	0	0
$937$	−45.1360	−1.47453	−0.737265	−	0.675604i	$-0.763883\pi$
$937$	−45.1360	−1.47453	−0.737265	+	0.675604i	$0.763883\pi$
$938$	0	0
$939$	−5.13505	−0.167576
$940$	0	0
$941$	−23.2948	−0.759387	−0.379694	−	0.925112i	$-0.623971\pi$
$941$	−23.2948	−0.759387	−0.379694	+	0.925112i	$0.623971\pi$
$942$	0	0
$943$	67.4342	2.19596
$944$	0	0
$945$	19.5063	0.634539
$946$	0	0
$947$	−2.33510	−0.0758807	−0.0379403	−	0.999280i	$-0.512080\pi$
$947$	−2.33510	−0.0758807	−0.0379403	+	0.999280i	$0.512080\pi$
$948$	0	0
$949$	4.58189	0.148735
$950$	0	0
$951$	1.46289	0.0474375
$952$	0	0
$953$	−33.1929	−1.07522	−0.537612	−	0.843192i	$-0.680673\pi$
$953$	−33.1929	−1.07522	−0.537612	+	0.843192i	$0.680673\pi$
$954$	0	0
$955$	2.02297	0.0654619
$956$	0	0
$957$	−10.8582	−0.350997
$958$	0	0
$959$	25.6858	0.829438
$960$	0	0
$961$	32.5269	1.04926
$962$	0	0
$963$	−47.8894	−1.54321
$964$	0	0
$965$	6.27604	0.202033
$966$	0	0
$967$	46.3742	1.49129	0.745647	−	0.666342i	$-0.232141\pi$
$967$	46.3742	1.49129	0.745647	+	0.666342i	$0.232141\pi$
$968$	0	0
$969$	0.197428	0.00634230
$970$	0	0
$971$	−38.2017	−1.22595	−0.612975	−	0.790102i	$-0.710028\pi$
$971$	−38.2017	−1.22595	−0.612975	+	0.790102i	$0.710028\pi$
$972$	0	0
$973$	−26.7972	−0.859080
$974$	0	0
$975$	−11.9476	−0.382628
$976$	0	0
$977$	5.15462	0.164911	0.0824554	−	0.996595i	$-0.473724\pi$
$977$	5.15462	0.164911	0.0824554	+	0.996595i	$0.473724\pi$
$978$	0	0
$979$	−32.9030	−1.05158
$980$	0	0
$981$	12.6513	0.403926
$982$	0	0
$983$	41.7510	1.33165	0.665826	−	0.746107i	$-0.268079\pi$
$983$	41.7510	1.33165	0.665826	+	0.746107i	$0.268079\pi$
$984$	0	0
$985$	−22.9612	−0.731603
$986$	0	0
$987$	5.57212	0.177362
$988$	0	0
$989$	0.489213	0.0155561
$990$	0	0
$991$	40.0015	1.27069	0.635344	−	0.772229i	$-0.280858\pi$
$991$	40.0015	1.27069	0.635344	+	0.772229i	$0.280858\pi$
$992$	0	0
$993$	−4.94645	−0.156971
$994$	0	0
$995$	53.8523	1.70723
$996$	0	0
$997$	−33.6269	−1.06497	−0.532487	−	0.846438i	$-0.678743\pi$
$997$	−33.6269	−1.06497	−0.532487	+	0.846438i	$0.678743\pi$
$998$	0	0
$999$	−6.68225	−0.211417

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.15	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.15	✓	27	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)

\(f(q)\)	\(=\)	\(q+0.267543 q^{3} +3.98483 q^{5} -3.08625 q^{7} -2.92842 q^{9} +O(q^{10})\)	\(q+0.267543 q^{3} +3.98483 q^{5} -3.08625 q^{7} -2.92842 q^{9} +5.66644 q^{11} -4.10489 q^{13} +1.06611 q^{15} +0.737930 q^{17} +1.00000 q^{19} -0.825706 q^{21} -6.58466 q^{23} +10.8789 q^{25} -1.58611 q^{27} -7.16235 q^{29} -7.97038 q^{31} +1.51602 q^{33} -12.2982 q^{35} +4.21298 q^{37} -1.09824 q^{39} -10.2411 q^{41} -0.0742958 q^{43} -11.6693 q^{45} -6.74831 q^{47} +2.52497 q^{49} +0.197428 q^{51} +6.32796 q^{53} +22.5798 q^{55} +0.267543 q^{57} +4.35950 q^{59} -15.4906 q^{61} +9.03785 q^{63} -16.3573 q^{65} +1.03751 q^{67} -1.76168 q^{69} -2.60621 q^{71} -1.11620 q^{73} +2.91057 q^{75} -17.4881 q^{77} -1.00000 q^{79} +8.36091 q^{81} +17.0678 q^{83} +2.94052 q^{85} -1.91624 q^{87} -5.80665 q^{89} +12.6687 q^{91} -2.13242 q^{93} +3.98483 q^{95} -8.59304 q^{97} -16.5937 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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