LMFDB - Embedded newform 6004.2.a.g.1.13

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			1.13
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)

$f(q)$	$=$	$q-0.338800 q^{3} -3.08469 q^{5} -4.48412 q^{7} -2.88521 q^{9} +O(q^{10})$	$q-0.338800 q^{3} -3.08469 q^{5} -4.48412 q^{7} -2.88521 q^{9} +5.90726 q^{11} +1.46463 q^{13} +1.04509 q^{15} -2.53371 q^{17} +1.00000 q^{19} +1.51922 q^{21} +2.61368 q^{23} +4.51532 q^{25} +1.99391 q^{27} +2.21728 q^{29} -8.75033 q^{31} -2.00138 q^{33} +13.8321 q^{35} -1.37694 q^{37} -0.496218 q^{39} +9.51019 q^{41} +10.4231 q^{43} +8.90000 q^{45} +4.08096 q^{47} +13.1073 q^{49} +0.858421 q^{51} -0.486802 q^{53} -18.2221 q^{55} -0.338800 q^{57} -11.1934 q^{59} +2.62780 q^{61} +12.9376 q^{63} -4.51794 q^{65} -10.4774 q^{67} -0.885514 q^{69} -3.34173 q^{71} +10.8433 q^{73} -1.52979 q^{75} -26.4889 q^{77} -1.00000 q^{79} +7.98011 q^{81} +5.71898 q^{83} +7.81571 q^{85} -0.751216 q^{87} -12.0355 q^{89} -6.56759 q^{91} +2.96461 q^{93} -3.08469 q^{95} +10.8032 q^{97} -17.0437 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.338800	−0.195606	−0.0978032	−	0.995206i	$-0.531182\pi$
$3$	−0.338800	−0.195606	−0.0978032	+	0.995206i	$0.531182\pi$
$4$	0	0
$5$	−3.08469	−1.37952	−0.689758	−	0.724040i	$-0.742283\pi$
$5$	−3.08469	−1.37952	−0.689758	+	0.724040i	$0.742283\pi$
$6$	0	0
$7$	−4.48412	−1.69484	−0.847419	−	0.530925i	$-0.821845\pi$
$7$	−4.48412	−1.69484	−0.847419	+	0.530925i	$0.821845\pi$
$8$	0	0
$9$	−2.88521	−0.961738
$10$	0	0
$11$	5.90726	1.78111	0.890553	−	0.454879i	$-0.150318\pi$
$11$	5.90726	1.78111	0.890553	+	0.454879i	$0.150318\pi$
$12$	0	0
$13$	1.46463	0.406216	0.203108	−	0.979156i	$-0.434896\pi$
$13$	1.46463	0.406216	0.203108	+	0.979156i	$0.434896\pi$
$14$	0	0
$15$	1.04509	0.269842
$16$	0	0
$17$	−2.53371	−0.614514	−0.307257	−	0.951626i	$-0.599411\pi$
$17$	−2.53371	−0.614514	−0.307257	+	0.951626i	$0.599411\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.51922	0.331521
$22$	0	0
$23$	2.61368	0.544989	0.272495	−	0.962157i	$-0.412151\pi$
$23$	2.61368	0.544989	0.272495	+	0.962157i	$0.412151\pi$
$24$	0	0
$25$	4.51532	0.903065
$26$	0	0
$27$	1.99391	0.383729
$28$	0	0
$29$	2.21728	0.411739	0.205869	−	0.978579i	$-0.433998\pi$
$29$	2.21728	0.411739	0.205869	+	0.978579i	$0.433998\pi$
$30$	0	0
$31$	−8.75033	−1.57161	−0.785803	−	0.618477i	$-0.787750\pi$
$31$	−8.75033	−1.57161	−0.785803	+	0.618477i	$0.787750\pi$
$32$	0	0
$33$	−2.00138	−0.348396
$34$	0	0
$35$	13.8321	2.33806
$36$	0	0
$37$	−1.37694	−0.226368	−0.113184	−	0.993574i	$-0.536105\pi$
$37$	−1.37694	−0.226368	−0.113184	+	0.993574i	$0.536105\pi$
$38$	0	0
$39$	−0.496218	−0.0794585
$40$	0	0
$41$	9.51019	1.48524	0.742621	−	0.669712i	$-0.233582\pi$
$41$	9.51019	1.48524	0.742621	+	0.669712i	$0.233582\pi$
$42$	0	0
$43$	10.4231	1.58950	0.794752	−	0.606934i	$-0.207601\pi$
$43$	10.4231	1.58950	0.794752	+	0.606934i	$0.207601\pi$
$44$	0	0
$45$	8.90000	1.32673
$46$	0	0
$47$	4.08096	0.595269	0.297635	−	0.954680i	$-0.403802\pi$
$47$	4.08096	0.595269	0.297635	+	0.954680i	$0.403802\pi$
$48$	0	0
$49$	13.1073	1.87248
$50$	0	0
$51$	0.858421	0.120203
$52$	0	0
$53$	−0.486802	−0.0668675	−0.0334337	−	0.999441i	$-0.510644\pi$
$53$	−0.486802	−0.0668675	−0.0334337	+	0.999441i	$0.510644\pi$
$54$	0	0
$55$	−18.2221	−2.45706
$56$	0	0
$57$	−0.338800	−0.0448752
$58$	0	0
$59$	−11.1934	−1.45726	−0.728628	−	0.684909i	$-0.759842\pi$
$59$	−11.1934	−1.45726	−0.728628	+	0.684909i	$0.759842\pi$
$60$	0	0
$61$	2.62780	0.336455	0.168228	−	0.985748i	$-0.446196\pi$
$61$	2.62780	0.336455	0.168228	+	0.985748i	$0.446196\pi$
$62$	0	0
$63$	12.9376	1.62999
$64$	0	0
$65$	−4.51794	−0.560382
$66$	0	0
$67$	−10.4774	−1.28001	−0.640006	−	0.768370i	$-0.721068\pi$
$67$	−10.4774	−1.28001	−0.640006	+	0.768370i	$0.721068\pi$
$68$	0	0
$69$	−0.885514	−0.106603
$70$	0	0
$71$	−3.34173	−0.396590	−0.198295	−	0.980142i	$-0.563540\pi$
$71$	−3.34173	−0.396590	−0.198295	+	0.980142i	$0.563540\pi$
$72$	0	0
$73$	10.8433	1.26911	0.634554	−	0.772879i	$-0.281184\pi$
$73$	10.8433	1.26911	0.634554	+	0.772879i	$0.281184\pi$
$74$	0	0
$75$	−1.52979	−0.176645
$76$	0	0
$77$	−26.4889	−3.01869
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	7.98011	0.886678
$82$	0	0
$83$	5.71898	0.627740	0.313870	−	0.949466i	$-0.398374\pi$
$83$	5.71898	0.627740	0.313870	+	0.949466i	$0.398374\pi$
$84$	0	0
$85$	7.81571	0.847733
$86$	0	0
$87$	−0.751216	−0.0805388
$88$	0	0
$89$	−12.0355	−1.27576	−0.637878	−	0.770137i	$-0.720188\pi$
$89$	−12.0355	−1.27576	−0.637878	+	0.770137i	$0.720188\pi$
$90$	0	0
$91$	−6.56759	−0.688471
$92$	0	0
$93$	2.96461	0.307416
$94$	0	0
$95$	−3.08469	−0.316483
$96$	0	0
$97$	10.8032	1.09690	0.548449	−	0.836184i	$-0.315219\pi$
$97$	10.8032	1.09690	0.548449	+	0.836184i	$0.315219\pi$
$98$	0	0
$99$	−17.0437	−1.71296
$100$	0	0
$101$	13.9850	1.39156	0.695782	−	0.718253i	$-0.255058\pi$
$101$	13.9850	1.39156	0.695782	+	0.718253i	$0.255058\pi$
$102$	0	0
$103$	15.0791	1.48579	0.742894	−	0.669410i	$-0.233453\pi$
$103$	15.0791	1.48579	0.742894	+	0.669410i	$0.233453\pi$
$104$	0	0
$105$	−4.68633	−0.457339
$106$	0	0
$107$	−6.29158	−0.608230	−0.304115	−	0.952635i	$-0.598361\pi$
$107$	−6.29158	−0.608230	−0.304115	+	0.952635i	$0.598361\pi$
$108$	0	0
$109$	−20.2424	−1.93887	−0.969433	−	0.245356i	$-0.921095\pi$
$109$	−20.2424	−1.93887	−0.969433	+	0.245356i	$0.921095\pi$
$110$	0	0
$111$	0.466508	0.0442790
$112$	0	0
$113$	12.0917	1.13749	0.568747	−	0.822512i	$-0.307428\pi$
$113$	12.0917	1.13749	0.568747	+	0.822512i	$0.307428\pi$
$114$	0	0
$115$	−8.06238	−0.751821
$116$	0	0
$117$	−4.22578	−0.390674
$118$	0	0
$119$	11.3614	1.04150
$120$	0	0
$121$	23.8957	2.17234
$122$	0	0
$123$	−3.22206	−0.290523
$124$	0	0
$125$	1.49508	0.133724
$126$	0	0
$127$	−5.18235	−0.459859	−0.229930	−	0.973207i	$-0.573850\pi$
$127$	−5.18235	−0.459859	−0.229930	+	0.973207i	$0.573850\pi$
$128$	0	0
$129$	−3.53134	−0.310917
$130$	0	0
$131$	18.8011	1.64266	0.821328	−	0.570456i	$-0.193233\pi$
$131$	18.8011	1.64266	0.821328	+	0.570456i	$0.193233\pi$
$132$	0	0
$133$	−4.48412	−0.388822
$134$	0	0
$135$	−6.15060	−0.529360
$136$	0	0
$137$	−1.72878	−0.147700	−0.0738500	−	0.997269i	$-0.523529\pi$
$137$	−1.72878	−0.147700	−0.0738500	+	0.997269i	$0.523529\pi$
$138$	0	0
$139$	−17.1941	−1.45838	−0.729190	−	0.684311i	$-0.760103\pi$
$139$	−17.1941	−1.45838	−0.729190	+	0.684311i	$0.760103\pi$
$140$	0	0
$141$	−1.38263	−0.116439
$142$	0	0
$143$	8.65197	0.723514
$144$	0	0
$145$	−6.83963	−0.568000
$146$	0	0
$147$	−4.44077	−0.366268
$148$	0	0
$149$	−1.98047	−0.162247	−0.0811233	−	0.996704i	$-0.525851\pi$
$149$	−1.98047	−0.162247	−0.0811233	+	0.996704i	$0.525851\pi$
$150$	0	0
$151$	−16.5704	−1.34848	−0.674240	−	0.738512i	$-0.735529\pi$
$151$	−16.5704	−1.34848	−0.674240	+	0.738512i	$0.735529\pi$
$152$	0	0
$153$	7.31029	0.591002
$154$	0	0
$155$	26.9921	2.16806
$156$	0	0
$157$	−19.2261	−1.53441	−0.767206	−	0.641401i	$-0.778354\pi$
$157$	−19.2261	−1.53441	−0.767206	+	0.641401i	$0.778354\pi$
$158$	0	0
$159$	0.164929	0.0130797
$160$	0	0
$161$	−11.7200	−0.923668
$162$	0	0
$163$	−9.35592	−0.732813	−0.366406	−	0.930455i	$-0.619412\pi$
$163$	−9.35592	−0.732813	−0.366406	+	0.930455i	$0.619412\pi$
$164$	0	0
$165$	6.17364	0.480618
$166$	0	0
$167$	−6.03989	−0.467380	−0.233690	−	0.972311i	$-0.575080\pi$
$167$	−6.03989	−0.467380	−0.233690	+	0.972311i	$0.575080\pi$
$168$	0	0
$169$	−10.8548	−0.834988
$170$	0	0
$171$	−2.88521	−0.220638
$172$	0	0
$173$	−16.6995	−1.26964	−0.634820	−	0.772660i	$-0.718926\pi$
$173$	−16.6995	−1.26964	−0.634820	+	0.772660i	$0.718926\pi$
$174$	0	0
$175$	−20.2473	−1.53055
$176$	0	0
$177$	3.79233	0.285049
$178$	0	0
$179$	13.8403	1.03447	0.517236	−	0.855843i	$-0.326961\pi$
$179$	13.8403	1.03447	0.517236	+	0.855843i	$0.326961\pi$
$180$	0	0
$181$	−16.7466	−1.24477	−0.622384	−	0.782712i	$-0.713836\pi$
$181$	−16.7466	−1.24477	−0.622384	+	0.782712i	$0.713836\pi$
$182$	0	0
$183$	−0.890299	−0.0658128
$184$	0	0
$185$	4.24744	0.312278
$186$	0	0
$187$	−14.9673	−1.09452
$188$	0	0
$189$	−8.94094	−0.650358
$190$	0	0
$191$	−0.352402	−0.0254989	−0.0127495	−	0.999919i	$-0.504058\pi$
$191$	−0.352402	−0.0254989	−0.0127495	+	0.999919i	$0.504058\pi$
$192$	0	0
$193$	−21.1203	−1.52027	−0.760137	−	0.649762i	$-0.774868\pi$
$193$	−21.1203	−1.52027	−0.760137	+	0.649762i	$0.774868\pi$
$194$	0	0
$195$	1.53068	0.109614
$196$	0	0
$197$	−11.0954	−0.790512	−0.395256	−	0.918571i	$-0.629344\pi$
$197$	−11.0954	−0.790512	−0.395256	+	0.918571i	$0.629344\pi$
$198$	0	0
$199$	19.2457	1.36429	0.682144	−	0.731218i	$-0.261048\pi$
$199$	19.2457	1.36429	0.682144	+	0.731218i	$0.261048\pi$
$200$	0	0
$201$	3.54973	0.250379
$202$	0	0
$203$	−9.94256	−0.697831
$204$	0	0
$205$	−29.3360	−2.04892
$206$	0	0
$207$	−7.54101	−0.524137
$208$	0	0
$209$	5.90726	0.408614
$210$	0	0
$211$	−28.5102	−1.96272	−0.981361	−	0.192175i	$-0.938446\pi$
$211$	−28.5102	−1.96272	−0.981361	+	0.192175i	$0.938446\pi$
$212$	0	0
$213$	1.13218	0.0775756
$214$	0	0
$215$	−32.1520	−2.19275
$216$	0	0
$217$	39.2375	2.66362
$218$	0	0
$219$	−3.67370	−0.248246
$220$	0	0
$221$	−3.71095	−0.249626
$222$	0	0
$223$	−8.94778	−0.599188	−0.299594	−	0.954067i	$-0.596851\pi$
$223$	−8.94778	−0.599188	−0.299594	+	0.954067i	$0.596851\pi$
$224$	0	0
$225$	−13.0277	−0.868512
$226$	0	0
$227$	−1.57611	−0.104610	−0.0523052	−	0.998631i	$-0.516657\pi$
$227$	−1.57611	−0.104610	−0.0523052	+	0.998631i	$0.516657\pi$
$228$	0	0
$229$	−17.5970	−1.16284	−0.581422	−	0.813602i	$-0.697504\pi$
$229$	−17.5970	−1.16284	−0.581422	+	0.813602i	$0.697504\pi$
$230$	0	0
$231$	8.97443	0.590474
$232$	0	0
$233$	−26.6286	−1.74450	−0.872248	−	0.489063i	$-0.837339\pi$
$233$	−26.6286	−1.74450	−0.872248	+	0.489063i	$0.837339\pi$
$234$	0	0
$235$	−12.5885	−0.821184
$236$	0	0
$237$	0.338800	0.0220074
$238$	0	0
$239$	−22.4660	−1.45320	−0.726601	−	0.687059i	$-0.758901\pi$
$239$	−22.4660	−1.45320	−0.726601	+	0.687059i	$0.758901\pi$
$240$	0	0
$241$	15.2590	0.982921	0.491461	−	0.870900i	$-0.336463\pi$
$241$	15.2590	0.982921	0.491461	+	0.870900i	$0.336463\pi$
$242$	0	0
$243$	−8.68540	−0.557169
$244$	0	0
$245$	−40.4321	−2.58311
$246$	0	0
$247$	1.46463	0.0931924
$248$	0	0
$249$	−1.93759	−0.122790
$250$	0	0
$251$	10.6697	0.673463	0.336732	−	0.941601i	$-0.390679\pi$
$251$	10.6697	0.673463	0.336732	+	0.941601i	$0.390679\pi$
$252$	0	0
$253$	15.4397	0.970683
$254$	0	0
$255$	−2.64796	−0.165822
$256$	0	0
$257$	10.7725	0.671969	0.335984	−	0.941868i	$-0.390931\pi$
$257$	10.7725	0.671969	0.335984	+	0.941868i	$0.390931\pi$
$258$	0	0
$259$	6.17437	0.383656
$260$	0	0
$261$	−6.39733	−0.395985
$262$	0	0
$263$	15.7910	0.973714	0.486857	−	0.873482i	$-0.338143\pi$
$263$	15.7910	0.973714	0.486857	+	0.873482i	$0.338143\pi$
$264$	0	0
$265$	1.50164	0.0922447
$266$	0	0
$267$	4.07762	0.249546
$268$	0	0
$269$	−22.9870	−1.40154	−0.700770	−	0.713387i	$-0.747160\pi$
$269$	−22.9870	−1.40154	−0.700770	+	0.713387i	$0.747160\pi$
$270$	0	0
$271$	8.72036	0.529724	0.264862	−	0.964286i	$-0.414674\pi$
$271$	8.72036	0.529724	0.264862	+	0.964286i	$0.414674\pi$
$272$	0	0
$273$	2.22510	0.134669
$274$	0	0
$275$	26.6732	1.60845
$276$	0	0
$277$	4.41417	0.265222	0.132611	−	0.991168i	$-0.457664\pi$
$277$	4.41417	0.265222	0.132611	+	0.991168i	$0.457664\pi$
$278$	0	0
$279$	25.2466	1.51147
$280$	0	0
$281$	−13.6091	−0.811848	−0.405924	−	0.913907i	$-0.633050\pi$
$281$	−13.6091	−0.811848	−0.405924	+	0.913907i	$0.633050\pi$
$282$	0	0
$283$	−11.7878	−0.700714	−0.350357	−	0.936616i	$-0.613940\pi$
$283$	−11.7878	−0.700714	−0.350357	+	0.936616i	$0.613940\pi$
$284$	0	0
$285$	1.04509	0.0619060
$286$	0	0
$287$	−42.6448	−2.51725
$288$	0	0
$289$	−10.5803	−0.622372
$290$	0	0
$291$	−3.66012	−0.214560
$292$	0	0
$293$	18.8709	1.10245	0.551224	−	0.834357i	$-0.314161\pi$
$293$	18.8709	1.10245	0.551224	+	0.834357i	$0.314161\pi$
$294$	0	0
$295$	34.5282	2.01031
$296$	0	0
$297$	11.7786	0.683461
$298$	0	0
$299$	3.82808	0.221383
$300$	0	0
$301$	−46.7383	−2.69395
$302$	0	0
$303$	−4.73814	−0.272199
$304$	0	0
$305$	−8.10595	−0.464145
$306$	0	0
$307$	21.4752	1.22566	0.612828	−	0.790216i	$-0.290032\pi$
$307$	21.4752	1.22566	0.612828	+	0.790216i	$0.290032\pi$
$308$	0	0
$309$	−5.10880	−0.290629
$310$	0	0
$311$	0.784787	0.0445012	0.0222506	−	0.999752i	$-0.492917\pi$
$311$	0.784787	0.0445012	0.0222506	+	0.999752i	$0.492917\pi$
$312$	0	0
$313$	31.3559	1.77234	0.886169	−	0.463362i	$-0.153357\pi$
$313$	31.3559	1.77234	0.886169	+	0.463362i	$0.153357\pi$
$314$	0	0
$315$	−39.9087	−2.24860
$316$	0	0
$317$	7.62243	0.428118	0.214059	−	0.976821i	$-0.431331\pi$
$317$	7.62243	0.428118	0.214059	+	0.976821i	$0.431331\pi$
$318$	0	0
$319$	13.0981	0.733351
$320$	0	0
$321$	2.13159	0.118974
$322$	0	0
$323$	−2.53371	−0.140979
$324$	0	0
$325$	6.61330	0.366840
$326$	0	0
$327$	6.85812	0.379255
$328$	0	0
$329$	−18.2995	−1.00889
$330$	0	0
$331$	15.5149	0.852774	0.426387	−	0.904541i	$-0.359786\pi$
$331$	15.5149	0.852774	0.426387	+	0.904541i	$0.359786\pi$
$332$	0	0
$333$	3.97277	0.217706
$334$	0	0
$335$	32.3194	1.76580
$336$	0	0
$337$	19.7990	1.07852	0.539259	−	0.842140i	$-0.318704\pi$
$337$	19.7990	1.07852	0.539259	+	0.842140i	$0.318704\pi$
$338$	0	0
$339$	−4.09668	−0.222501
$340$	0	0
$341$	−51.6905	−2.79920
$342$	0	0
$343$	−27.3860	−1.47870
$344$	0	0
$345$	2.73154	0.147061
$346$	0	0
$347$	−8.86787	−0.476052	−0.238026	−	0.971259i	$-0.576500\pi$
$347$	−8.86787	−0.476052	−0.238026	+	0.971259i	$0.576500\pi$
$348$	0	0
$349$	1.19957	0.0642115	0.0321058	−	0.999484i	$-0.489779\pi$
$349$	1.19957	0.0642115	0.0321058	+	0.999484i	$0.489779\pi$
$350$	0	0
$351$	2.92035	0.155877
$352$	0	0
$353$	−36.8592	−1.96182	−0.980909	−	0.194466i	$-0.937703\pi$
$353$	−36.8592	−1.96182	−0.980909	+	0.194466i	$0.937703\pi$
$354$	0	0
$355$	10.3082	0.547103
$356$	0	0
$357$	−3.84926	−0.203725
$358$	0	0
$359$	32.8930	1.73602	0.868012	−	0.496542i	$-0.165397\pi$
$359$	32.8930	1.73602	0.868012	+	0.496542i	$0.165397\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−8.09588	−0.424923
$364$	0	0
$365$	−33.4481	−1.75075
$366$	0	0
$367$	17.9928	0.939219	0.469609	−	0.882874i	$-0.344395\pi$
$367$	17.9928	0.939219	0.469609	+	0.882874i	$0.344395\pi$
$368$	0	0
$369$	−27.4389	−1.42841
$370$	0	0
$371$	2.18288	0.113330
$372$	0	0
$373$	16.2887	0.843396	0.421698	−	0.906736i	$-0.361434\pi$
$373$	16.2887	0.843396	0.421698	+	0.906736i	$0.361434\pi$
$374$	0	0
$375$	−0.506533	−0.0261572
$376$	0	0
$377$	3.24751	0.167255
$378$	0	0
$379$	−14.1494	−0.726805	−0.363402	−	0.931632i	$-0.618385\pi$
$379$	−14.1494	−0.726805	−0.363402	+	0.931632i	$0.618385\pi$
$380$	0	0
$381$	1.75578	0.0899514
$382$	0	0
$383$	−31.8077	−1.62530	−0.812650	−	0.582752i	$-0.801976\pi$
$383$	−31.8077	−1.62530	−0.812650	+	0.582752i	$0.801976\pi$
$384$	0	0
$385$	81.7100	4.16433
$386$	0	0
$387$	−30.0728	−1.52869
$388$	0	0
$389$	−9.14502	−0.463671	−0.231835	−	0.972755i	$-0.574473\pi$
$389$	−9.14502	−0.463671	−0.231835	+	0.972755i	$0.574473\pi$
$390$	0	0
$391$	−6.62229	−0.334904
$392$	0	0
$393$	−6.36981	−0.321314
$394$	0	0
$395$	3.08469	0.155208
$396$	0	0
$397$	37.0203	1.85799	0.928997	−	0.370087i	$-0.120672\pi$
$397$	37.0203	1.85799	0.928997	+	0.370087i	$0.120672\pi$
$398$	0	0
$399$	1.51922	0.0760562
$400$	0	0
$401$	−6.64571	−0.331871	−0.165935	−	0.986137i	$-0.553064\pi$
$401$	−6.64571	−0.331871	−0.165935	+	0.986137i	$0.553064\pi$
$402$	0	0
$403$	−12.8160	−0.638412
$404$	0	0
$405$	−24.6162	−1.22319
$406$	0	0
$407$	−8.13395	−0.403185
$408$	0	0
$409$	−30.4596	−1.50613	−0.753065	−	0.657946i	$-0.771425\pi$
$409$	−30.4596	−1.50613	−0.753065	+	0.657946i	$0.771425\pi$
$410$	0	0
$411$	0.585713	0.0288911
$412$	0	0
$413$	50.1926	2.46981
$414$	0	0
$415$	−17.6413	−0.865977
$416$	0	0
$417$	5.82535	0.285269
$418$	0	0
$419$	9.79385	0.478461	0.239230	−	0.970963i	$-0.423105\pi$
$419$	9.79385	0.478461	0.239230	+	0.970963i	$0.423105\pi$
$420$	0	0
$421$	−25.3652	−1.23622	−0.618111	−	0.786091i	$-0.712102\pi$
$421$	−25.3652	−1.23622	−0.618111	+	0.786091i	$0.712102\pi$
$422$	0	0
$423$	−11.7744	−0.572493
$424$	0	0
$425$	−11.4405	−0.554946
$426$	0	0
$427$	−11.7834	−0.570237
$428$	0	0
$429$	−2.93129	−0.141524
$430$	0	0
$431$	−28.6505	−1.38005	−0.690023	−	0.723787i	$-0.742400\pi$
$431$	−28.6505	−1.38005	−0.690023	+	0.723787i	$0.742400\pi$
$432$	0	0
$433$	−8.24308	−0.396137	−0.198069	−	0.980188i	$-0.563467\pi$
$433$	−8.24308	−0.396137	−0.198069	+	0.980188i	$0.563467\pi$
$434$	0	0
$435$	2.31727	0.111105
$436$	0	0
$437$	2.61368	0.125029
$438$	0	0
$439$	25.5811	1.22092	0.610461	−	0.792047i	$-0.290984\pi$
$439$	25.5811	1.22092	0.610461	+	0.792047i	$0.290984\pi$
$440$	0	0
$441$	−37.8174	−1.80083
$442$	0	0
$443$	29.5111	1.40211	0.701056	−	0.713106i	$-0.252712\pi$
$443$	29.5111	1.40211	0.701056	+	0.713106i	$0.252712\pi$
$444$	0	0
$445$	37.1257	1.75993
$446$	0	0
$447$	0.670985	0.0317365
$448$	0	0
$449$	−18.3951	−0.868121	−0.434060	−	0.900884i	$-0.642920\pi$
$449$	−18.3951	−0.868121	−0.434060	+	0.900884i	$0.642920\pi$
$450$	0	0
$451$	56.1792	2.64537
$452$	0	0
$453$	5.61405	0.263771
$454$	0	0
$455$	20.2590	0.949757
$456$	0	0
$457$	−27.1112	−1.26821	−0.634104	−	0.773247i	$-0.718631\pi$
$457$	−27.1112	−1.26821	−0.634104	+	0.773247i	$0.718631\pi$
$458$	0	0
$459$	−5.05199	−0.235807
$460$	0	0
$461$	37.5785	1.75021	0.875103	−	0.483937i	$-0.160793\pi$
$461$	37.5785	1.75021	0.875103	+	0.483937i	$0.160793\pi$
$462$	0	0
$463$	7.58950	0.352714	0.176357	−	0.984326i	$-0.443569\pi$
$463$	7.58950	0.352714	0.176357	+	0.984326i	$0.443569\pi$
$464$	0	0
$465$	−9.14492	−0.424086
$466$	0	0
$467$	26.3208	1.21798	0.608991	−	0.793177i	$-0.291574\pi$
$467$	26.3208	1.21798	0.608991	+	0.793177i	$0.291574\pi$
$468$	0	0
$469$	46.9817	2.16941
$470$	0	0
$471$	6.51381	0.300141
$472$	0	0
$473$	61.5718	2.83108
$474$	0	0
$475$	4.51532	0.207177
$476$	0	0
$477$	1.40453	0.0643090
$478$	0	0
$479$	−19.2534	−0.879709	−0.439855	−	0.898069i	$-0.644970\pi$
$479$	−19.2534	−0.879709	−0.439855	+	0.898069i	$0.644970\pi$
$480$	0	0
$481$	−2.01671	−0.0919542
$482$	0	0
$483$	3.97075	0.180675
$484$	0	0
$485$	−33.3245	−1.51319
$486$	0	0
$487$	−7.52108	−0.340813	−0.170406	−	0.985374i	$-0.554508\pi$
$487$	−7.52108	−0.340813	−0.170406	+	0.985374i	$0.554508\pi$
$488$	0	0
$489$	3.16979	0.143343
$490$	0	0
$491$	−1.14655	−0.0517431	−0.0258715	−	0.999665i	$-0.508236\pi$
$491$	−1.14655	−0.0517431	−0.0258715	+	0.999665i	$0.508236\pi$
$492$	0	0
$493$	−5.61794	−0.253020
$494$	0	0
$495$	52.5746	2.36305
$496$	0	0
$497$	14.9847	0.672156
$498$	0	0
$499$	−23.6929	−1.06064	−0.530320	−	0.847797i	$-0.677928\pi$
$499$	−23.6929	−1.06064	−0.530320	+	0.847797i	$0.677928\pi$
$500$	0	0
$501$	2.04632	0.0914226
$502$	0	0
$503$	0.419221	0.0186921	0.00934606	−	0.999956i	$-0.497025\pi$
$503$	0.419221	0.0186921	0.00934606	+	0.999956i	$0.497025\pi$
$504$	0	0
$505$	−43.1396	−1.91968
$506$	0	0
$507$	3.67762	0.163329
$508$	0	0
$509$	17.1532	0.760302	0.380151	−	0.924924i	$-0.375872\pi$
$509$	17.1532	0.760302	0.380151	+	0.924924i	$0.375872\pi$
$510$	0	0
$511$	−48.6225	−2.15093
$512$	0	0
$513$	1.99391	0.0880334
$514$	0	0
$515$	−46.5144	−2.04967
$516$	0	0
$517$	24.1073	1.06024
$518$	0	0
$519$	5.65780	0.248350
$520$	0	0
$521$	−35.7570	−1.56654	−0.783271	−	0.621680i	$-0.786450\pi$
$521$	−35.7570	−1.56654	−0.783271	+	0.621680i	$0.786450\pi$
$522$	0	0
$523$	37.9493	1.65941	0.829703	−	0.558205i	$-0.188510\pi$
$523$	37.9493	1.65941	0.829703	+	0.558205i	$0.188510\pi$
$524$	0	0
$525$	6.85977	0.299385
$526$	0	0
$527$	22.1708	0.965775
$528$	0	0
$529$	−16.1687	−0.702987
$530$	0	0
$531$	32.2954	1.40150
$532$	0	0
$533$	13.9289	0.603330
$534$	0	0
$535$	19.4076	0.839063
$536$	0	0
$537$	−4.68909	−0.202349
$538$	0	0
$539$	77.4284	3.33508
$540$	0	0
$541$	39.4666	1.69680	0.848401	−	0.529354i	$-0.177566\pi$
$541$	39.4666	1.69680	0.848401	+	0.529354i	$0.177566\pi$
$542$	0	0
$543$	5.67377	0.243485
$544$	0	0
$545$	62.4414	2.67470
$546$	0	0
$547$	31.0747	1.32866	0.664330	−	0.747440i	$-0.268717\pi$
$547$	31.0747	1.32866	0.664330	+	0.747440i	$0.268717\pi$
$548$	0	0
$549$	−7.58176	−0.323582
$550$	0	0
$551$	2.21728	0.0944594
$552$	0	0
$553$	4.48412	0.190684
$554$	0	0
$555$	−1.43903	−0.0610835
$556$	0	0
$557$	14.3467	0.607891	0.303945	−	0.952689i	$-0.401696\pi$
$557$	14.3467	0.607891	0.303945	+	0.952689i	$0.401696\pi$
$558$	0	0
$559$	15.2660	0.645682
$560$	0	0
$561$	5.07092	0.214094
$562$	0	0
$563$	34.7398	1.46411	0.732053	−	0.681248i	$-0.238562\pi$
$563$	34.7398	1.46411	0.732053	+	0.681248i	$0.238562\pi$
$564$	0	0
$565$	−37.2993	−1.56919
$566$	0	0
$567$	−35.7837	−1.50278
$568$	0	0
$569$	0.144869	0.00607321	0.00303660	−	0.999995i	$-0.499033\pi$
$569$	0.144869	0.00607321	0.00303660	+	0.999995i	$0.499033\pi$
$570$	0	0
$571$	−5.76371	−0.241204	−0.120602	−	0.992701i	$-0.538482\pi$
$571$	−5.76371	−0.241204	−0.120602	+	0.992701i	$0.538482\pi$
$572$	0	0
$573$	0.119394	0.00498775
$574$	0	0
$575$	11.8016	0.492160
$576$	0	0
$577$	−2.07164	−0.0862434	−0.0431217	−	0.999070i	$-0.513730\pi$
$577$	−2.07164	−0.0862434	−0.0431217	+	0.999070i	$0.513730\pi$
$578$	0	0
$579$	7.15557	0.297375
$580$	0	0
$581$	−25.6446	−1.06392
$582$	0	0
$583$	−2.87567	−0.119098
$584$	0	0
$585$	13.0352	0.538941
$586$	0	0
$587$	−12.8255	−0.529367	−0.264683	−	0.964335i	$-0.585267\pi$
$587$	−12.8255	−0.529367	−0.264683	+	0.964335i	$0.585267\pi$
$588$	0	0
$589$	−8.75033	−0.360551
$590$	0	0
$591$	3.75911	0.154629
$592$	0	0
$593$	20.5701	0.844714	0.422357	−	0.906430i	$-0.361203\pi$
$593$	20.5701	0.844714	0.422357	+	0.906430i	$0.361203\pi$
$594$	0	0
$595$	−35.0466	−1.43677
$596$	0	0
$597$	−6.52043	−0.266864
$598$	0	0
$599$	5.65119	0.230901	0.115451	−	0.993313i	$-0.463169\pi$
$599$	5.65119	0.230901	0.115451	+	0.993313i	$0.463169\pi$
$600$	0	0
$601$	26.9956	1.10117	0.550586	−	0.834779i	$-0.314404\pi$
$601$	26.9956	1.10117	0.550586	+	0.834779i	$0.314404\pi$
$602$	0	0
$603$	30.2294	1.23104
$604$	0	0
$605$	−73.7109	−2.99678
$606$	0	0
$607$	22.9708	0.932356	0.466178	−	0.884691i	$-0.345631\pi$
$607$	22.9708	0.932356	0.466178	+	0.884691i	$0.345631\pi$
$608$	0	0
$609$	3.36854	0.136500
$610$	0	0
$611$	5.97711	0.241808
$612$	0	0
$613$	−12.0176	−0.485388	−0.242694	−	0.970103i	$-0.578031\pi$
$613$	−12.0176	−0.485388	−0.242694	+	0.970103i	$0.578031\pi$
$614$	0	0
$615$	9.93905	0.400781
$616$	0	0
$617$	−12.8926	−0.519038	−0.259519	−	0.965738i	$-0.583564\pi$
$617$	−12.8926	−0.519038	−0.259519	+	0.965738i	$0.583564\pi$
$618$	0	0
$619$	−10.1606	−0.408390	−0.204195	−	0.978930i	$-0.565458\pi$
$619$	−10.1606	−0.408390	−0.204195	+	0.978930i	$0.565458\pi$
$620$	0	0
$621$	5.21144	0.209128
$622$	0	0
$623$	53.9685	2.16220
$624$	0	0
$625$	−27.1885	−1.08754
$626$	0	0
$627$	−2.00138	−0.0799275
$628$	0	0
$629$	3.48877	0.139106
$630$	0	0
$631$	28.3400	1.12820	0.564098	−	0.825708i	$-0.309224\pi$
$631$	28.3400	1.12820	0.564098	+	0.825708i	$0.309224\pi$
$632$	0	0
$633$	9.65925	0.383921
$634$	0	0
$635$	15.9860	0.634383
$636$	0	0
$637$	19.1974	0.760630
$638$	0	0
$639$	9.64161	0.381416
$640$	0	0
$641$	1.40962	0.0556767	0.0278384	−	0.999612i	$-0.491138\pi$
$641$	1.40962	0.0556767	0.0278384	+	0.999612i	$0.491138\pi$
$642$	0	0
$643$	−11.9832	−0.472573	−0.236287	−	0.971683i	$-0.575930\pi$
$643$	−11.9832	−0.472573	−0.236287	+	0.971683i	$0.575930\pi$
$644$	0	0
$645$	10.8931	0.428915
$646$	0	0
$647$	−44.6259	−1.75443	−0.877213	−	0.480101i	$-0.840600\pi$
$647$	−44.6259	−1.75443	−0.877213	+	0.480101i	$0.840600\pi$
$648$	0	0
$649$	−66.1223	−2.59553
$650$	0	0
$651$	−13.2937	−0.521021
$652$	0	0
$653$	−39.2720	−1.53683	−0.768416	−	0.639951i	$-0.778955\pi$
$653$	−39.2720	−1.53683	−0.768416	+	0.639951i	$0.778955\pi$
$654$	0	0
$655$	−57.9955	−2.26607
$656$	0	0
$657$	−31.2851	−1.22055
$658$	0	0
$659$	−10.7789	−0.419884	−0.209942	−	0.977714i	$-0.567328\pi$
$659$	−10.7789	−0.419884	−0.209942	+	0.977714i	$0.567328\pi$
$660$	0	0
$661$	−23.6337	−0.919243	−0.459621	−	0.888115i	$-0.652015\pi$
$661$	−23.6337	−0.919243	−0.459621	+	0.888115i	$0.652015\pi$
$662$	0	0
$663$	1.25727	0.0488284
$664$	0	0
$665$	13.8321	0.536387
$666$	0	0
$667$	5.79526	0.224393
$668$	0	0
$669$	3.03151	0.117205
$670$	0	0
$671$	15.5231	0.599262
$672$	0	0
$673$	9.51520	0.366784	0.183392	−	0.983040i	$-0.441292\pi$
$673$	9.51520	0.366784	0.183392	+	0.983040i	$0.441292\pi$
$674$	0	0
$675$	9.00316	0.346532
$676$	0	0
$677$	−49.8809	−1.91708	−0.958539	−	0.284962i	$-0.908019\pi$
$677$	−49.8809	−1.91708	−0.958539	+	0.284962i	$0.908019\pi$
$678$	0	0
$679$	−48.4428	−1.85906
$680$	0	0
$681$	0.533988	0.0204625
$682$	0	0
$683$	−41.1571	−1.57483	−0.787417	−	0.616421i	$-0.788582\pi$
$683$	−41.1571	−1.57483	−0.787417	+	0.616421i	$0.788582\pi$
$684$	0	0
$685$	5.33277	0.203755
$686$	0	0
$687$	5.96188	0.227460
$688$	0	0
$689$	−0.712987	−0.0271627
$690$	0	0
$691$	16.6379	0.632934	0.316467	−	0.948604i	$-0.397503\pi$
$691$	16.6379	0.632934	0.316467	+	0.948604i	$0.397503\pi$
$692$	0	0
$693$	76.4260	2.90319
$694$	0	0
$695$	53.0383	2.01186
$696$	0	0
$697$	−24.0961	−0.912703
$698$	0	0
$699$	9.02177	0.341235
$700$	0	0
$701$	−23.7052	−0.895332	−0.447666	−	0.894201i	$-0.647745\pi$
$701$	−23.7052	−0.895332	−0.447666	+	0.894201i	$0.647745\pi$
$702$	0	0
$703$	−1.37694	−0.0519323
$704$	0	0
$705$	4.26499	0.160629
$706$	0	0
$707$	−62.7106	−2.35848
$708$	0	0
$709$	30.5771	1.14835	0.574173	−	0.818734i	$-0.305324\pi$
$709$	30.5771	1.14835	0.574173	+	0.818734i	$0.305324\pi$
$710$	0	0
$711$	2.88521	0.108204
$712$	0	0
$713$	−22.8705	−0.856508
$714$	0	0
$715$	−26.6887	−0.998100
$716$	0	0
$717$	7.61148	0.284256
$718$	0	0
$719$	−20.6191	−0.768963	−0.384482	−	0.923133i	$-0.625620\pi$
$719$	−20.6191	−0.768963	−0.384482	+	0.923133i	$0.625620\pi$
$720$	0	0
$721$	−67.6164	−2.51817
$722$	0	0
$723$	−5.16977	−0.192266
$724$	0	0
$725$	10.0117	0.371827
$726$	0	0
$727$	−0.908317	−0.0336876	−0.0168438	−	0.999858i	$-0.505362\pi$
$727$	−0.908317	−0.0336876	−0.0168438	+	0.999858i	$0.505362\pi$
$728$	0	0
$729$	−20.9977	−0.777693
$730$	0	0
$731$	−26.4090	−0.976773
$732$	0	0
$733$	51.9088	1.91730	0.958648	−	0.284595i	$-0.0918591\pi$
$733$	51.9088	1.91730	0.958648	+	0.284595i	$0.0918591\pi$
$734$	0	0
$735$	13.6984	0.505273
$736$	0	0
$737$	−61.8925	−2.27984
$738$	0	0
$739$	−2.64811	−0.0974123	−0.0487062	−	0.998813i	$-0.515510\pi$
$739$	−2.64811	−0.0974123	−0.0487062	+	0.998813i	$0.515510\pi$
$740$	0	0
$741$	−0.496218	−0.0182290
$742$	0	0
$743$	−33.3942	−1.22511	−0.612557	−	0.790427i	$-0.709859\pi$
$743$	−33.3942	−1.22511	−0.612557	+	0.790427i	$0.709859\pi$
$744$	0	0
$745$	6.10915	0.223822
$746$	0	0
$747$	−16.5005	−0.603721
$748$	0	0
$749$	28.2122	1.03085
$750$	0	0
$751$	−4.63282	−0.169054	−0.0845270	−	0.996421i	$-0.526938\pi$
$751$	−4.63282	−0.169054	−0.0845270	+	0.996421i	$0.526938\pi$
$752$	0	0
$753$	−3.61488	−0.131734
$754$	0	0
$755$	51.1146	1.86025
$756$	0	0
$757$	−33.5146	−1.21811	−0.609055	−	0.793128i	$-0.708451\pi$
$757$	−33.5146	−1.21811	−0.609055	+	0.793128i	$0.708451\pi$
$758$	0	0
$759$	−5.23096	−0.189872
$760$	0	0
$761$	6.94894	0.251899	0.125949	−	0.992037i	$-0.459802\pi$
$761$	6.94894	0.251899	0.125949	+	0.992037i	$0.459802\pi$
$762$	0	0
$763$	90.7692	3.28606
$764$	0	0
$765$	−22.5500	−0.815297
$766$	0	0
$767$	−16.3942	−0.591962
$768$	0	0
$769$	38.0036	1.37045	0.685223	−	0.728334i	$-0.259705\pi$
$769$	38.0036	1.37045	0.685223	+	0.728334i	$0.259705\pi$
$770$	0	0
$771$	−3.64972	−0.131441
$772$	0	0
$773$	−32.8033	−1.17985	−0.589926	−	0.807457i	$-0.700843\pi$
$773$	−32.8033	−1.17985	−0.589926	+	0.807457i	$0.700843\pi$
$774$	0	0
$775$	−39.5106	−1.41926
$776$	0	0
$777$	−2.09188	−0.0750457
$778$	0	0
$779$	9.51019	0.340738
$780$	0	0
$781$	−19.7405	−0.706369
$782$	0	0
$783$	4.42106	0.157996
$784$	0	0
$785$	59.3067	2.11675
$786$	0	0
$787$	30.8568	1.09993	0.549963	−	0.835189i	$-0.314642\pi$
$787$	30.8568	1.09993	0.549963	+	0.835189i	$0.314642\pi$
$788$	0	0
$789$	−5.34999	−0.190465
$790$	0	0
$791$	−54.2208	−1.92787
$792$	0	0
$793$	3.84876	0.136674
$794$	0	0
$795$	−0.508754	−0.0180437
$796$	0	0
$797$	−23.6793	−0.838765	−0.419382	−	0.907810i	$-0.637753\pi$
$797$	−23.6793	−0.838765	−0.419382	+	0.907810i	$0.637753\pi$
$798$	0	0
$799$	−10.3400	−0.365802
$800$	0	0
$801$	34.7249	1.22694
$802$	0	0
$803$	64.0539	2.26041
$804$	0	0
$805$	36.1527	1.27421
$806$	0	0
$807$	7.78799	0.274150
$808$	0	0
$809$	−24.7466	−0.870044	−0.435022	−	0.900420i	$-0.643259\pi$
$809$	−24.7466	−0.870044	−0.435022	+	0.900420i	$0.643259\pi$
$810$	0	0
$811$	45.6336	1.60241	0.801206	−	0.598389i	$-0.204192\pi$
$811$	45.6336	1.60241	0.801206	+	0.598389i	$0.204192\pi$
$812$	0	0
$813$	−2.95446	−0.103617
$814$	0	0
$815$	28.8601	1.01093
$816$	0	0
$817$	10.4231	0.364657
$818$	0	0
$819$	18.9489	0.662129
$820$	0	0
$821$	−47.3053	−1.65097	−0.825484	−	0.564426i	$-0.809098\pi$
$821$	−47.3053	−1.65097	−0.825484	+	0.564426i	$0.809098\pi$
$822$	0	0
$823$	14.4445	0.503505	0.251753	−	0.967792i	$-0.418993\pi$
$823$	14.4445	0.503505	0.251753	+	0.967792i	$0.418993\pi$
$824$	0	0
$825$	−9.03688	−0.314624
$826$	0	0
$827$	10.2502	0.356435	0.178218	−	0.983991i	$-0.442967\pi$
$827$	10.2502	0.356435	0.178218	+	0.983991i	$0.442967\pi$
$828$	0	0
$829$	−46.0441	−1.59918	−0.799590	−	0.600547i	$-0.794950\pi$
$829$	−46.0441	−1.59918	−0.799590	+	0.600547i	$0.794950\pi$
$830$	0	0
$831$	−1.49552	−0.0518791
$832$	0	0
$833$	−33.2101	−1.15066
$834$	0	0
$835$	18.6312	0.644759
$836$	0	0
$837$	−17.4474	−0.603070
$838$	0	0
$839$	2.67240	0.0922614	0.0461307	−	0.998935i	$-0.485311\pi$
$839$	2.67240	0.0922614	0.0461307	+	0.998935i	$0.485311\pi$
$840$	0	0
$841$	−24.0837	−0.830471
$842$	0	0
$843$	4.61075	0.158803
$844$	0	0
$845$	33.4839	1.15188
$846$	0	0
$847$	−107.151	−3.68176
$848$	0	0
$849$	3.99372	0.137064
$850$	0	0
$851$	−3.59888	−0.123368
$852$	0	0
$853$	28.0188	0.959345	0.479672	−	0.877448i	$-0.340755\pi$
$853$	28.0188	0.959345	0.479672	+	0.877448i	$0.340755\pi$
$854$	0	0
$855$	8.90000	0.304373
$856$	0	0
$857$	5.52577	0.188757	0.0943783	−	0.995536i	$-0.469914\pi$
$857$	5.52577	0.188757	0.0943783	+	0.995536i	$0.469914\pi$
$858$	0	0
$859$	−44.3461	−1.51307	−0.756534	−	0.653954i	$-0.773109\pi$
$859$	−44.3461	−1.51307	−0.756534	+	0.653954i	$0.773109\pi$
$860$	0	0
$861$	14.4481	0.492389
$862$	0	0
$863$	0.737216	0.0250951	0.0125476	−	0.999921i	$-0.496006\pi$
$863$	0.737216	0.0250951	0.0125476	+	0.999921i	$0.496006\pi$
$864$	0	0
$865$	51.5128	1.75149
$866$	0	0
$867$	3.58462	0.121740
$868$	0	0
$869$	−5.90726	−0.200390
$870$	0	0
$871$	−15.3455	−0.519962
$872$	0	0
$873$	−31.1695	−1.05493
$874$	0	0
$875$	−6.70411	−0.226640
$876$	0	0
$877$	8.07642	0.272721	0.136361	−	0.990659i	$-0.456459\pi$
$877$	8.07642	0.272721	0.136361	+	0.990659i	$0.456459\pi$
$878$	0	0
$879$	−6.39346	−0.215646
$880$	0	0
$881$	44.8201	1.51003	0.755014	−	0.655708i	$-0.227630\pi$
$881$	44.8201	1.51003	0.755014	+	0.655708i	$0.227630\pi$
$882$	0	0
$883$	−38.9580	−1.31104	−0.655521	−	0.755177i	$-0.727551\pi$
$883$	−38.9580	−1.31104	−0.655521	+	0.755177i	$0.727551\pi$
$884$	0	0
$885$	−11.6982	−0.393229
$886$	0	0
$887$	37.9365	1.27378	0.636892	−	0.770953i	$-0.280220\pi$
$887$	37.9365	1.27378	0.636892	+	0.770953i	$0.280220\pi$
$888$	0	0
$889$	23.2383	0.779387
$890$	0	0
$891$	47.1406	1.57927
$892$	0	0
$893$	4.08096	0.136564
$894$	0	0
$895$	−42.6930	−1.42707
$896$	0	0
$897$	−1.29695	−0.0433040
$898$	0	0
$899$	−19.4020	−0.647091
$900$	0	0
$901$	1.23342	0.0410910
$902$	0	0
$903$	15.8350	0.526954
$904$	0	0
$905$	51.6582	1.71718
$906$	0	0
$907$	−10.9918	−0.364975	−0.182488	−	0.983208i	$-0.558415\pi$
$907$	−10.9918	−0.364975	−0.182488	+	0.983208i	$0.558415\pi$
$908$	0	0
$909$	−40.3499	−1.33832
$910$	0	0
$911$	−4.02940	−0.133500	−0.0667500	−	0.997770i	$-0.521263\pi$
$911$	−4.02940	−0.133500	−0.0667500	+	0.997770i	$0.521263\pi$
$912$	0	0
$913$	33.7835	1.11807
$914$	0	0
$915$	2.74630	0.0907898
$916$	0	0
$917$	−84.3062	−2.78404
$918$	0	0
$919$	5.78258	0.190750	0.0953748	−	0.995441i	$-0.469595\pi$
$919$	5.78258	0.190750	0.0953748	+	0.995441i	$0.469595\pi$
$920$	0	0
$921$	−7.27581	−0.239746
$922$	0	0
$923$	−4.89441	−0.161101
$924$	0	0
$925$	−6.21733	−0.204425
$926$	0	0
$927$	−43.5064	−1.42894
$928$	0	0
$929$	16.5042	0.541484	0.270742	−	0.962652i	$-0.412731\pi$
$929$	16.5042	0.541484	0.270742	+	0.962652i	$0.412731\pi$
$930$	0	0
$931$	13.1073	0.429575
$932$	0	0
$933$	−0.265886	−0.00870472
$934$	0	0
$935$	46.1694	1.50990
$936$	0	0
$937$	−2.59864	−0.0848938	−0.0424469	−	0.999099i	$-0.513515\pi$
$937$	−2.59864	−0.0848938	−0.0424469	+	0.999099i	$0.513515\pi$
$938$	0	0
$939$	−10.6234	−0.346681
$940$	0	0
$941$	−23.2239	−0.757079	−0.378540	−	0.925585i	$-0.623574\pi$
$941$	−23.2239	−0.757079	−0.378540	+	0.925585i	$0.623574\pi$
$942$	0	0
$943$	24.8566	0.809441
$944$	0	0
$945$	27.5800	0.897179
$946$	0	0
$947$	−25.4744	−0.827808	−0.413904	−	0.910320i	$-0.635835\pi$
$947$	−25.4744	−0.827808	−0.413904	+	0.910320i	$0.635835\pi$
$948$	0	0
$949$	15.8814	0.515532
$950$	0	0
$951$	−2.58248	−0.0837427
$952$	0	0
$953$	−1.82084	−0.0589827	−0.0294914	−	0.999565i	$-0.509389\pi$
$953$	−1.82084	−0.0589827	−0.0294914	+	0.999565i	$0.509389\pi$
$954$	0	0
$955$	1.08705	0.0351762
$956$	0	0
$957$	−4.43763	−0.143448
$958$	0	0
$959$	7.75208	0.250328
$960$	0	0
$961$	45.5683	1.46995
$962$	0	0
$963$	18.1526	0.584958
$964$	0	0
$965$	65.1497	2.09724
$966$	0	0
$967$	16.9154	0.543962	0.271981	−	0.962303i	$-0.412321\pi$
$967$	16.9154	0.543962	0.271981	+	0.962303i	$0.412321\pi$
$968$	0	0
$969$	0.858421	0.0275765
$970$	0	0
$971$	−23.9082	−0.767251	−0.383625	−	0.923489i	$-0.625325\pi$
$971$	−23.9082	−0.767251	−0.383625	+	0.923489i	$0.625325\pi$
$972$	0	0
$973$	77.1002	2.47172
$974$	0	0
$975$	−2.24059	−0.0717562
$976$	0	0
$977$	5.22674	0.167218	0.0836092	−	0.996499i	$-0.473355\pi$
$977$	5.22674	0.167218	0.0836092	+	0.996499i	$0.473355\pi$
$978$	0	0
$979$	−71.0966	−2.27226
$980$	0	0
$981$	58.4035	1.86468
$982$	0	0
$983$	−28.0200	−0.893699	−0.446849	−	0.894609i	$-0.647454\pi$
$983$	−28.0200	−0.893699	−0.446849	+	0.894609i	$0.647454\pi$
$984$	0	0
$985$	34.2258	1.09052
$986$	0	0
$987$	6.19988	0.197344
$988$	0	0
$989$	27.2425	0.866262
$990$	0	0
$991$	−5.85812	−0.186089	−0.0930446	−	0.995662i	$-0.529660\pi$
$991$	−5.85812	−0.186089	−0.0930446	+	0.995662i	$0.529660\pi$
$992$	0	0
$993$	−5.25644	−0.166808
$994$	0	0
$995$	−59.3669	−1.88206
$996$	0	0
$997$	5.23537	0.165806	0.0829029	−	0.996558i	$-0.473581\pi$
$997$	5.23537	0.165806	0.0829029	+	0.996558i	$0.473581\pi$
$998$	0	0
$999$	−2.74550	−0.0868637

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.13	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-0.338800 q^{3} -3.08469 q^{5} -4.48412 q^{7} -2.88521 q^{9} +O(q^{10})\)	\(q-0.338800 q^{3} -3.08469 q^{5} -4.48412 q^{7} -2.88521 q^{9} +5.90726 q^{11} +1.46463 q^{13} +1.04509 q^{15} -2.53371 q^{17} +1.00000 q^{19} +1.51922 q^{21} +2.61368 q^{23} +4.51532 q^{25} +1.99391 q^{27} +2.21728 q^{29} -8.75033 q^{31} -2.00138 q^{33} +13.8321 q^{35} -1.37694 q^{37} -0.496218 q^{39} +9.51019 q^{41} +10.4231 q^{43} +8.90000 q^{45} +4.08096 q^{47} +13.1073 q^{49} +0.858421 q^{51} -0.486802 q^{53} -18.2221 q^{55} -0.338800 q^{57} -11.1934 q^{59} +2.62780 q^{61} +12.9376 q^{63} -4.51794 q^{65} -10.4774 q^{67} -0.885514 q^{69} -3.34173 q^{71} +10.8433 q^{73} -1.52979 q^{75} -26.4889 q^{77} -1.00000 q^{79} +7.98011 q^{81} +5.71898 q^{83} +7.81571 q^{85} -0.751216 q^{87} -12.0355 q^{89} -6.56759 q^{91} +2.96461 q^{93} -3.08469 q^{95} +10.8032 q^{97} -17.0437 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 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

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Groups

Database

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