LMFDB - Embedded newform 6004.2.a.g.1.21

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			1.21
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)

$f(q)$	$=$	$q+1.41553 q^{3} +0.642687 q^{5} +1.33052 q^{7} -0.996264 q^{9} +O(q^{10})$	$q+1.41553 q^{3} +0.642687 q^{5} +1.33052 q^{7} -0.996264 q^{9} +1.93346 q^{11} -5.21530 q^{13} +0.909745 q^{15} -1.65264 q^{17} +1.00000 q^{19} +1.88340 q^{21} +1.45774 q^{23} -4.58695 q^{25} -5.65685 q^{27} -0.0964612 q^{29} -5.73913 q^{31} +2.73688 q^{33} +0.855108 q^{35} -7.59822 q^{37} -7.38243 q^{39} +5.41539 q^{41} -6.60481 q^{43} -0.640286 q^{45} +1.50562 q^{47} -5.22971 q^{49} -2.33937 q^{51} +0.0801188 q^{53} +1.24261 q^{55} +1.41553 q^{57} -7.20139 q^{59} +10.8769 q^{61} -1.32555 q^{63} -3.35181 q^{65} -14.7930 q^{67} +2.06348 q^{69} -8.01632 q^{71} +10.6504 q^{73} -6.49299 q^{75} +2.57251 q^{77} -1.00000 q^{79} -5.01867 q^{81} +7.39572 q^{83} -1.06213 q^{85} -0.136544 q^{87} -4.64376 q^{89} -6.93907 q^{91} -8.12394 q^{93} +0.642687 q^{95} -5.13964 q^{97} -1.92624 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.41553	0.817259	0.408629	−	0.912700i	$-0.366007\pi$
$3$	1.41553	0.817259	0.408629	+	0.912700i	$0.366007\pi$
$4$	0	0
$5$	0.642687	0.287418	0.143709	−	0.989620i	$-0.454097\pi$
$5$	0.642687	0.287418	0.143709	+	0.989620i	$0.454097\pi$
$6$	0	0
$7$	1.33052	0.502890	0.251445	−	0.967872i	$-0.419094\pi$
$7$	1.33052	0.502890	0.251445	+	0.967872i	$0.419094\pi$
$8$	0	0
$9$	−0.996264	−0.332088
$10$	0	0
$11$	1.93346	0.582960	0.291480	−	0.956577i	$-0.405852\pi$
$11$	1.93346	0.582960	0.291480	+	0.956577i	$0.405852\pi$
$12$	0	0
$13$	−5.21530	−1.44646	−0.723232	−	0.690605i	$-0.757344\pi$
$13$	−5.21530	−1.44646	−0.723232	+	0.690605i	$0.757344\pi$
$14$	0	0
$15$	0.909745	0.234895
$16$	0	0
$17$	−1.65264	−0.400824	−0.200412	−	0.979712i	$-0.564228\pi$
$17$	−1.65264	−0.400824	−0.200412	+	0.979712i	$0.564228\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.88340	0.410991
$22$	0	0
$23$	1.45774	0.303960	0.151980	−	0.988384i	$-0.451435\pi$
$23$	1.45774	0.303960	0.151980	+	0.988384i	$0.451435\pi$
$24$	0	0
$25$	−4.58695	−0.917391
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−0.0964612	−0.0179124	−0.00895619	−	0.999960i	$-0.502851\pi$
$29$	−0.0964612	−0.0179124	−0.00895619	+	0.999960i	$0.502851\pi$
$30$	0	0
$31$	−5.73913	−1.03078	−0.515389	−	0.856956i	$-0.672353\pi$
$31$	−5.73913	−1.03078	−0.515389	+	0.856956i	$0.672353\pi$
$32$	0	0
$33$	2.73688	0.476430
$34$	0	0
$35$	0.855108	0.144540
$36$	0	0
$37$	−7.59822	−1.24914	−0.624570	−	0.780969i	$-0.714726\pi$
$37$	−7.59822	−1.24914	−0.624570	+	0.780969i	$0.714726\pi$
$38$	0	0
$39$	−7.38243	−1.18214
$40$	0	0
$41$	5.41539	0.845741	0.422871	−	0.906190i	$-0.361022\pi$
$41$	5.41539	0.845741	0.422871	+	0.906190i	$0.361022\pi$
$42$	0	0
$43$	−6.60481	−1.00722	−0.503612	−	0.863930i	$-0.667996\pi$
$43$	−6.60481	−1.00722	−0.503612	+	0.863930i	$0.667996\pi$
$44$	0	0
$45$	−0.640286	−0.0954482
$46$	0	0
$47$	1.50562	0.219617	0.109809	−	0.993953i	$-0.464976\pi$
$47$	1.50562	0.219617	0.109809	+	0.993953i	$0.464976\pi$
$48$	0	0
$49$	−5.22971	−0.747102
$50$	0	0
$51$	−2.33937	−0.327577
$52$	0	0
$53$	0.0801188	0.0110052	0.00550258	−	0.999985i	$-0.498248\pi$
$53$	0.0801188	0.0110052	0.00550258	+	0.999985i	$0.498248\pi$
$54$	0	0
$55$	1.24261	0.167554
$56$	0	0
$57$	1.41553	0.187492
$58$	0	0
$59$	−7.20139	−0.937540	−0.468770	−	0.883320i	$-0.655303\pi$
$59$	−7.20139	−0.937540	−0.468770	+	0.883320i	$0.655303\pi$
$60$	0	0
$61$	10.8769	1.39265	0.696324	−	0.717728i	$-0.254818\pi$
$61$	10.8769	1.39265	0.696324	+	0.717728i	$0.254818\pi$
$62$	0	0
$63$	−1.32555	−0.167004
$64$	0	0
$65$	−3.35181	−0.415740
$66$	0	0
$67$	−14.7930	−1.80726	−0.903629	−	0.428315i	$-0.859107\pi$
$67$	−14.7930	−1.80726	−0.903629	+	0.428315i	$0.859107\pi$
$68$	0	0
$69$	2.06348	0.248414
$70$	0	0
$71$	−8.01632	−0.951362	−0.475681	−	0.879618i	$-0.657798\pi$
$71$	−8.01632	−0.951362	−0.475681	+	0.879618i	$0.657798\pi$
$72$	0	0
$73$	10.6504	1.24654	0.623268	−	0.782008i	$-0.285804\pi$
$73$	10.6504	1.24654	0.623268	+	0.782008i	$0.285804\pi$
$74$	0	0
$75$	−6.49299	−0.749746
$76$	0	0
$77$	2.57251	0.293165
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−5.01867	−0.557630
$82$	0	0
$83$	7.39572	0.811785	0.405893	−	0.913921i	$-0.366961\pi$
$83$	7.39572	0.811785	0.405893	+	0.913921i	$0.366961\pi$
$84$	0	0
$85$	−1.06213	−0.115204
$86$	0	0
$87$	−0.136544	−0.0146391
$88$	0	0
$89$	−4.64376	−0.492238	−0.246119	−	0.969240i	$-0.579155\pi$
$89$	−4.64376	−0.492238	−0.246119	+	0.969240i	$0.579155\pi$
$90$	0	0
$91$	−6.93907	−0.727412
$92$	0	0
$93$	−8.12394	−0.842413
$94$	0	0
$95$	0.642687	0.0659383
$96$	0	0
$97$	−5.13964	−0.521851	−0.260926	−	0.965359i	$-0.584028\pi$
$97$	−5.13964	−0.521851	−0.260926	+	0.965359i	$0.584028\pi$
$98$	0	0
$99$	−1.92624	−0.193594
$100$	0	0
$101$	11.1457	1.10904	0.554520	−	0.832170i	$-0.312902\pi$
$101$	11.1457	1.10904	0.554520	+	0.832170i	$0.312902\pi$
$102$	0	0
$103$	−7.15830	−0.705329	−0.352664	−	0.935750i	$-0.614724\pi$
$103$	−7.15830	−0.705329	−0.352664	+	0.935750i	$0.614724\pi$
$104$	0	0
$105$	1.21043	0.118126
$106$	0	0
$107$	13.4227	1.29762	0.648809	−	0.760951i	$-0.275267\pi$
$107$	13.4227	1.29762	0.648809	+	0.760951i	$0.275267\pi$
$108$	0	0
$109$	−8.85953	−0.848589	−0.424294	−	0.905524i	$-0.639478\pi$
$109$	−8.85953	−0.848589	−0.424294	+	0.905524i	$0.639478\pi$
$110$	0	0
$111$	−10.7555	−1.02087
$112$	0	0
$113$	17.2233	1.62024	0.810118	−	0.586267i	$-0.199403\pi$
$113$	17.2233	1.62024	0.810118	+	0.586267i	$0.199403\pi$
$114$	0	0
$115$	0.936872	0.0873638
$116$	0	0
$117$	5.19582	0.480353
$118$	0	0
$119$	−2.19887	−0.201570
$120$	0	0
$121$	−7.26173	−0.660157
$122$	0	0
$123$	7.66566	0.691190
$124$	0	0
$125$	−6.16141	−0.551093
$126$	0	0
$127$	−11.8815	−1.05431	−0.527157	−	0.849768i	$-0.676742\pi$
$127$	−11.8815	−1.05431	−0.527157	+	0.849768i	$0.676742\pi$
$128$	0	0
$129$	−9.34933	−0.823163
$130$	0	0
$131$	−16.7271	−1.46145	−0.730726	−	0.682671i	$-0.760818\pi$
$131$	−16.7271	−1.46145	−0.730726	+	0.682671i	$0.760818\pi$
$132$	0	0
$133$	1.33052	0.115371
$134$	0	0
$135$	−3.63558	−0.312901
$136$	0	0
$137$	−15.3185	−1.30875	−0.654373	−	0.756172i	$-0.727067\pi$
$137$	−15.3185	−1.30875	−0.654373	+	0.756172i	$0.727067\pi$
$138$	0	0
$139$	7.32048	0.620915	0.310457	−	0.950587i	$-0.399518\pi$
$139$	7.32048	0.620915	0.310457	+	0.950587i	$0.399518\pi$
$140$	0	0
$141$	2.13126	0.179484
$142$	0	0
$143$	−10.0836	−0.843231
$144$	0	0
$145$	−0.0619943	−0.00514835
$146$	0	0
$147$	−7.40284	−0.610576
$148$	0	0
$149$	17.4219	1.42726	0.713631	−	0.700522i	$-0.247049\pi$
$149$	17.4219	1.42726	0.713631	+	0.700522i	$0.247049\pi$
$150$	0	0
$151$	8.53489	0.694559	0.347280	−	0.937762i	$-0.387105\pi$
$151$	8.53489	0.694559	0.347280	+	0.937762i	$0.387105\pi$
$152$	0	0
$153$	1.64647	0.133109
$154$	0	0
$155$	−3.68847	−0.296265
$156$	0	0
$157$	22.6908	1.81093	0.905463	−	0.424425i	$-0.139524\pi$
$157$	22.6908	1.81093	0.905463	+	0.424425i	$0.139524\pi$
$158$	0	0
$159$	0.113411	0.00899407
$160$	0	0
$161$	1.93956	0.152859
$162$	0	0
$163$	−16.3631	−1.28166	−0.640828	−	0.767684i	$-0.721409\pi$
$163$	−16.3631	−1.28166	−0.640828	+	0.767684i	$0.721409\pi$
$164$	0	0
$165$	1.75896	0.136935
$166$	0	0
$167$	−23.4032	−1.81099	−0.905497	−	0.424353i	$-0.860501\pi$
$167$	−23.4032	−1.81099	−0.905497	+	0.424353i	$0.860501\pi$
$168$	0	0
$169$	14.1994	1.09226
$170$	0	0
$171$	−0.996264	−0.0761862
$172$	0	0
$173$	−22.0490	−1.67635	−0.838175	−	0.545401i	$-0.816377\pi$
$173$	−22.0490	−1.67635	−0.838175	+	0.545401i	$0.816377\pi$
$174$	0	0
$175$	−6.10304	−0.461346
$176$	0	0
$177$	−10.1938	−0.766213
$178$	0	0
$179$	8.45284	0.631795	0.315897	−	0.948793i	$-0.397694\pi$
$179$	8.45284	0.631795	0.315897	+	0.948793i	$0.397694\pi$
$180$	0	0
$181$	−17.2964	−1.28563	−0.642817	−	0.766020i	$-0.722234\pi$
$181$	−17.2964	−1.28563	−0.642817	+	0.766020i	$0.722234\pi$
$182$	0	0
$183$	15.3967	1.13815
$184$	0	0
$185$	−4.88328	−0.359026
$186$	0	0
$187$	−3.19532	−0.233665
$188$	0	0
$189$	−7.52655	−0.547476
$190$	0	0
$191$	11.6517	0.843089	0.421544	−	0.906808i	$-0.361488\pi$
$191$	11.6517	0.843089	0.421544	+	0.906808i	$0.361488\pi$
$192$	0	0
$193$	10.3297	0.743547	0.371773	−	0.928324i	$-0.378750\pi$
$193$	10.3297	0.743547	0.371773	+	0.928324i	$0.378750\pi$
$194$	0	0
$195$	−4.74459	−0.339767
$196$	0	0
$197$	−5.98534	−0.426437	−0.213219	−	0.977004i	$-0.568395\pi$
$197$	−5.98534	−0.426437	−0.213219	+	0.977004i	$0.568395\pi$
$198$	0	0
$199$	−4.70869	−0.333790	−0.166895	−	0.985975i	$-0.553374\pi$
$199$	−4.70869	−0.333790	−0.166895	+	0.985975i	$0.553374\pi$
$200$	0	0
$201$	−20.9401	−1.47700
$202$	0	0
$203$	−0.128344	−0.00900795
$204$	0	0
$205$	3.48040	0.243082
$206$	0	0
$207$	−1.45230	−0.100942
$208$	0	0
$209$	1.93346	0.133740
$210$	0	0
$211$	17.0324	1.17256	0.586278	−	0.810110i	$-0.300593\pi$
$211$	17.0324	1.17256	0.586278	+	0.810110i	$0.300593\pi$
$212$	0	0
$213$	−11.3474	−0.777509
$214$	0	0
$215$	−4.24483	−0.289495
$216$	0	0
$217$	−7.63603	−0.518368
$218$	0	0
$219$	15.0760	1.01874
$220$	0	0
$221$	8.61902	0.579778
$222$	0	0
$223$	17.4741	1.17015	0.585075	−	0.810979i	$-0.301065\pi$
$223$	17.4741	1.17015	0.585075	+	0.810979i	$0.301065\pi$
$224$	0	0
$225$	4.56982	0.304654
$226$	0	0
$227$	20.2456	1.34375	0.671874	−	0.740665i	$-0.265489\pi$
$227$	20.2456	1.34375	0.671874	+	0.740665i	$0.265489\pi$
$228$	0	0
$229$	−15.8439	−1.04700	−0.523499	−	0.852026i	$-0.675373\pi$
$229$	−15.8439	−1.04700	−0.523499	+	0.852026i	$0.675373\pi$
$230$	0	0
$231$	3.64147	0.239591
$232$	0	0
$233$	25.8263	1.69194	0.845969	−	0.533232i	$-0.179023\pi$
$233$	25.8263	1.69194	0.845969	+	0.533232i	$0.179023\pi$
$234$	0	0
$235$	0.967643	0.0631221
$236$	0	0
$237$	−1.41553	−0.0919488
$238$	0	0
$239$	−12.4909	−0.807972	−0.403986	−	0.914765i	$-0.632376\pi$
$239$	−12.4909	−0.807972	−0.403986	+	0.914765i	$0.632376\pi$
$240$	0	0
$241$	−12.8050	−0.824840	−0.412420	−	0.910994i	$-0.635316\pi$
$241$	−12.8050	−0.824840	−0.412420	+	0.910994i	$0.635316\pi$
$242$	0	0
$243$	9.86645	0.632933
$244$	0	0
$245$	−3.36107	−0.214731
$246$	0	0
$247$	−5.21530	−0.331842
$248$	0	0
$249$	10.4689	0.663439
$250$	0	0
$251$	−0.527481	−0.0332943	−0.0166472	−	0.999861i	$-0.505299\pi$
$251$	−0.527481	−0.0332943	−0.0166472	+	0.999861i	$0.505299\pi$
$252$	0	0
$253$	2.81849	0.177197
$254$	0	0
$255$	−1.50348	−0.0941517
$256$	0	0
$257$	−3.54045	−0.220847	−0.110424	−	0.993885i	$-0.535221\pi$
$257$	−3.54045	−0.220847	−0.110424	+	0.993885i	$0.535221\pi$
$258$	0	0
$259$	−10.1096	−0.628179
$260$	0	0
$261$	0.0961008	0.00594849
$262$	0	0
$263$	18.3694	1.13270	0.566352	−	0.824163i	$-0.308354\pi$
$263$	18.3694	1.13270	0.566352	+	0.824163i	$0.308354\pi$
$264$	0	0
$265$	0.0514913	0.00316309
$266$	0	0
$267$	−6.57340	−0.402286
$268$	0	0
$269$	−11.8957	−0.725295	−0.362647	−	0.931926i	$-0.618127\pi$
$269$	−11.8957	−0.725295	−0.362647	+	0.931926i	$0.618127\pi$
$270$	0	0
$271$	−7.58733	−0.460898	−0.230449	−	0.973084i	$-0.574019\pi$
$271$	−7.58733	−0.460898	−0.230449	+	0.973084i	$0.574019\pi$
$272$	0	0
$273$	−9.82248	−0.594484
$274$	0	0
$275$	−8.86870	−0.534802
$276$	0	0
$277$	15.1353	0.909389	0.454695	−	0.890647i	$-0.349748\pi$
$277$	15.1353	0.909389	0.454695	+	0.890647i	$0.349748\pi$
$278$	0	0
$279$	5.71769	0.342309
$280$	0	0
$281$	−7.86338	−0.469090	−0.234545	−	0.972105i	$-0.575360\pi$
$281$	−7.86338	−0.469090	−0.234545	+	0.972105i	$0.575360\pi$
$282$	0	0
$283$	20.3069	1.20712	0.603560	−	0.797318i	$-0.293748\pi$
$283$	20.3069	1.20712	0.603560	+	0.797318i	$0.293748\pi$
$284$	0	0
$285$	0.909745	0.0538887
$286$	0	0
$287$	7.20528	0.425314
$288$	0	0
$289$	−14.2688	−0.839340
$290$	0	0
$291$	−7.27533	−0.426488
$292$	0	0
$293$	−11.2096	−0.654873	−0.327436	−	0.944873i	$-0.606185\pi$
$293$	−11.2096	−0.654873	−0.327436	+	0.944873i	$0.606185\pi$
$294$	0	0
$295$	−4.62824	−0.269466
$296$	0	0
$297$	−10.9373	−0.634646
$298$	0	0
$299$	−7.60257	−0.439668
$300$	0	0
$301$	−8.78784	−0.506523
$302$	0	0
$303$	15.7771	0.906373
$304$	0	0
$305$	6.99046	0.400272
$306$	0	0
$307$	8.11820	0.463330	0.231665	−	0.972796i	$-0.425583\pi$
$307$	8.11820	0.463330	0.231665	+	0.972796i	$0.425583\pi$
$308$	0	0
$309$	−10.1328	−0.576436
$310$	0	0
$311$	−3.83714	−0.217584	−0.108792	−	0.994065i	$-0.534698\pi$
$311$	−3.83714	−0.217584	−0.108792	+	0.994065i	$0.534698\pi$
$312$	0	0
$313$	−4.88873	−0.276327	−0.138164	−	0.990409i	$-0.544120\pi$
$313$	−4.88873	−0.276327	−0.138164	+	0.990409i	$0.544120\pi$
$314$	0	0
$315$	−0.851914	−0.0479999
$316$	0	0
$317$	9.36341	0.525901	0.262951	−	0.964809i	$-0.415304\pi$
$317$	9.36341	0.525901	0.262951	+	0.964809i	$0.415304\pi$
$318$	0	0
$319$	−0.186504	−0.0104422
$320$	0	0
$321$	19.0002	1.06049
$322$	0	0
$323$	−1.65264	−0.0919554
$324$	0	0
$325$	23.9223	1.32697
$326$	0	0
$327$	−12.5410	−0.693517
$328$	0	0
$329$	2.00326	0.110443
$330$	0	0
$331$	1.04474	0.0574239	0.0287120	−	0.999588i	$-0.490859\pi$
$331$	1.04474	0.0574239	0.0287120	+	0.999588i	$0.490859\pi$
$332$	0	0
$333$	7.56983	0.414824
$334$	0	0
$335$	−9.50730	−0.519439
$336$	0	0
$337$	30.1550	1.64265	0.821324	−	0.570462i	$-0.193236\pi$
$337$	30.1550	1.64265	0.821324	+	0.570462i	$0.193236\pi$
$338$	0	0
$339$	24.3802	1.32415
$340$	0	0
$341$	−11.0964	−0.600903
$342$	0	0
$343$	−16.2719	−0.878599
$344$	0	0
$345$	1.32617	0.0713988
$346$	0	0
$347$	−20.7269	−1.11268	−0.556340	−	0.830955i	$-0.687795\pi$
$347$	−20.7269	−1.11268	−0.556340	+	0.830955i	$0.687795\pi$
$348$	0	0
$349$	−26.1358	−1.39902	−0.699509	−	0.714624i	$-0.746598\pi$
$349$	−26.1358	−1.39902	−0.699509	+	0.714624i	$0.746598\pi$
$350$	0	0
$351$	29.5022	1.57471
$352$	0	0
$353$	5.10887	0.271918	0.135959	−	0.990714i	$-0.456588\pi$
$353$	5.10887	0.271918	0.135959	+	0.990714i	$0.456588\pi$
$354$	0	0
$355$	−5.15198	−0.273439
$356$	0	0
$357$	−3.11258	−0.164735
$358$	0	0
$359$	11.7863	0.622059	0.311029	−	0.950400i	$-0.399326\pi$
$359$	11.7863	0.622059	0.311029	+	0.950400i	$0.399326\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−10.2792	−0.539519
$364$	0	0
$365$	6.84488	0.358277
$366$	0	0
$367$	−13.5919	−0.709494	−0.354747	−	0.934962i	$-0.615433\pi$
$367$	−13.5919	−0.709494	−0.354747	+	0.934962i	$0.615433\pi$
$368$	0	0
$369$	−5.39515	−0.280861
$370$	0	0
$371$	0.106600	0.00553438
$372$	0	0
$373$	0.174924	0.00905722	0.00452861	−	0.999990i	$-0.498558\pi$
$373$	0.174924	0.00905722	0.00452861	+	0.999990i	$0.498558\pi$
$374$	0	0
$375$	−8.72168	−0.450386
$376$	0	0
$377$	0.503074	0.0259096
$378$	0	0
$379$	6.78000	0.348265	0.174133	−	0.984722i	$-0.444288\pi$
$379$	6.78000	0.348265	0.174133	+	0.984722i	$0.444288\pi$
$380$	0	0
$381$	−16.8187	−0.861647
$382$	0	0
$383$	−11.7147	−0.598591	−0.299296	−	0.954160i	$-0.596752\pi$
$383$	−11.7147	−0.598591	−0.299296	+	0.954160i	$0.596752\pi$
$384$	0	0
$385$	1.65332	0.0842609
$386$	0	0
$387$	6.58014	0.334487
$388$	0	0
$389$	25.3448	1.28503	0.642515	−	0.766273i	$-0.277891\pi$
$389$	25.3448	1.28503	0.642515	+	0.766273i	$0.277891\pi$
$390$	0	0
$391$	−2.40913	−0.121835
$392$	0	0
$393$	−23.6778	−1.19438
$394$	0	0
$395$	−0.642687	−0.0323371
$396$	0	0
$397$	−38.5061	−1.93257	−0.966284	−	0.257480i	$-0.917108\pi$
$397$	−38.5061	−1.93257	−0.966284	+	0.257480i	$0.917108\pi$
$398$	0	0
$399$	1.88340	0.0942878
$400$	0	0
$401$	−34.1188	−1.70381	−0.851905	−	0.523697i	$-0.824552\pi$
$401$	−34.1188	−1.70381	−0.851905	+	0.523697i	$0.824552\pi$
$402$	0	0
$403$	29.9313	1.49098
$404$	0	0
$405$	−3.22543	−0.160273
$406$	0	0
$407$	−14.6909	−0.728199
$408$	0	0
$409$	14.6675	0.725260	0.362630	−	0.931933i	$-0.381879\pi$
$409$	14.6675	0.725260	0.362630	+	0.931933i	$0.381879\pi$
$410$	0	0
$411$	−21.6838	−1.06958
$412$	0	0
$413$	−9.58159	−0.471479
$414$	0	0
$415$	4.75313	0.233322
$416$	0	0
$417$	10.3624	0.507448
$418$	0	0
$419$	−23.1940	−1.13310	−0.566550	−	0.824027i	$-0.691722\pi$
$419$	−23.1940	−1.13310	−0.566550	+	0.824027i	$0.691722\pi$
$420$	0	0
$421$	−15.7906	−0.769588	−0.384794	−	0.923002i	$-0.625728\pi$
$421$	−15.7906	−0.769588	−0.384794	+	0.923002i	$0.625728\pi$
$422$	0	0
$423$	−1.50000	−0.0729323
$424$	0	0
$425$	7.58059	0.367713
$426$	0	0
$427$	14.4720	0.700348
$428$	0	0
$429$	−14.2736	−0.689138
$430$	0	0
$431$	3.55450	0.171214	0.0856072	−	0.996329i	$-0.472717\pi$
$431$	3.55450	0.171214	0.0856072	+	0.996329i	$0.472717\pi$
$432$	0	0
$433$	5.44653	0.261743	0.130872	−	0.991399i	$-0.458222\pi$
$433$	5.44653	0.261743	0.130872	+	0.991399i	$0.458222\pi$
$434$	0	0
$435$	−0.0877551	−0.00420753
$436$	0	0
$437$	1.45774	0.0697333
$438$	0	0
$439$	17.1089	0.816565	0.408283	−	0.912856i	$-0.366128\pi$
$439$	17.1089	0.816565	0.408283	+	0.912856i	$0.366128\pi$
$440$	0	0
$441$	5.21018	0.248104
$442$	0	0
$443$	−13.1017	−0.622482	−0.311241	−	0.950331i	$-0.600745\pi$
$443$	−13.1017	−0.622482	−0.311241	+	0.950331i	$0.600745\pi$
$444$	0	0
$445$	−2.98449	−0.141478
$446$	0	0
$447$	24.6614	1.16644
$448$	0	0
$449$	−27.0013	−1.27427	−0.637135	−	0.770752i	$-0.719881\pi$
$449$	−27.0013	−1.27427	−0.637135	+	0.770752i	$0.719881\pi$
$450$	0	0
$451$	10.4704	0.493034
$452$	0	0
$453$	12.0814	0.567635
$454$	0	0
$455$	−4.45965	−0.209071
$456$	0	0
$457$	25.2455	1.18093	0.590467	−	0.807062i	$-0.298944\pi$
$457$	25.2455	1.18093	0.590467	+	0.807062i	$0.298944\pi$
$458$	0	0
$459$	9.34874	0.436362
$460$	0	0
$461$	−33.3158	−1.55167	−0.775837	−	0.630934i	$-0.782672\pi$
$461$	−33.3158	−1.55167	−0.775837	+	0.630934i	$0.782672\pi$
$462$	0	0
$463$	16.6210	0.772444	0.386222	−	0.922406i	$-0.373780\pi$
$463$	16.6210	0.772444	0.386222	+	0.922406i	$0.373780\pi$
$464$	0	0
$465$	−5.22115	−0.242125
$466$	0	0
$467$	2.87012	0.132813	0.0664067	−	0.997793i	$-0.478847\pi$
$467$	2.87012	0.132813	0.0664067	+	0.997793i	$0.478847\pi$
$468$	0	0
$469$	−19.6825	−0.908852
$470$	0	0
$471$	32.1197	1.48000
$472$	0	0
$473$	−12.7701	−0.587172
$474$	0	0
$475$	−4.58695	−0.210464
$476$	0	0
$477$	−0.0798195	−0.00365468
$478$	0	0
$479$	−1.75270	−0.0800828	−0.0400414	−	0.999198i	$-0.512749\pi$
$479$	−1.75270	−0.0800828	−0.0400414	+	0.999198i	$0.512749\pi$
$480$	0	0
$481$	39.6270	1.80684
$482$	0	0
$483$	2.74551	0.124925
$484$	0	0
$485$	−3.30318	−0.149990
$486$	0	0
$487$	−14.1357	−0.640551	−0.320276	−	0.947324i	$-0.603776\pi$
$487$	−14.1357	−0.640551	−0.320276	+	0.947324i	$0.603776\pi$
$488$	0	0
$489$	−23.1625	−1.04745
$490$	0	0
$491$	−20.8121	−0.939236	−0.469618	−	0.882870i	$-0.655608\pi$
$491$	−20.8121	−0.939236	−0.469618	+	0.882870i	$0.655608\pi$
$492$	0	0
$493$	0.159416	0.00717972
$494$	0	0
$495$	−1.23797	−0.0556425
$496$	0	0
$497$	−10.6659	−0.478430
$498$	0	0
$499$	43.2900	1.93793	0.968963	−	0.247205i	$-0.0795121\pi$
$499$	43.2900	1.93793	0.968963	+	0.247205i	$0.0795121\pi$
$500$	0	0
$501$	−33.1280	−1.48005
$502$	0	0
$503$	32.5372	1.45076	0.725381	−	0.688348i	$-0.241664\pi$
$503$	32.5372	1.45076	0.725381	+	0.688348i	$0.241664\pi$
$504$	0	0
$505$	7.16321	0.318759
$506$	0	0
$507$	20.0997	0.892658
$508$	0	0
$509$	14.3895	0.637804	0.318902	−	0.947788i	$-0.396686\pi$
$509$	14.3895	0.637804	0.318902	+	0.947788i	$0.396686\pi$
$510$	0	0
$511$	14.1706	0.626870
$512$	0	0
$513$	−5.65685	−0.249756
$514$	0	0
$515$	−4.60055	−0.202724
$516$	0	0
$517$	2.91106	0.128028
$518$	0	0
$519$	−31.2110	−1.37001
$520$	0	0
$521$	−18.0622	−0.791319	−0.395659	−	0.918397i	$-0.629484\pi$
$521$	−18.0622	−0.791319	−0.395659	+	0.918397i	$0.629484\pi$
$522$	0	0
$523$	−16.9305	−0.740319	−0.370160	−	0.928968i	$-0.620697\pi$
$523$	−16.9305	−0.740319	−0.370160	+	0.928968i	$0.620697\pi$
$524$	0	0
$525$	−8.63906	−0.377039
$526$	0	0
$527$	9.48473	0.413161
$528$	0	0
$529$	−20.8750	−0.907608
$530$	0	0
$531$	7.17448	0.311346
$532$	0	0
$533$	−28.2429	−1.22333
$534$	0	0
$535$	8.62657	0.372959
$536$	0	0
$537$	11.9653	0.516340
$538$	0	0
$539$	−10.1114	−0.435531
$540$	0	0
$541$	−11.7390	−0.504700	−0.252350	−	0.967636i	$-0.581203\pi$
$541$	−11.7390	−0.504700	−0.252350	+	0.967636i	$0.581203\pi$
$542$	0	0
$543$	−24.4837	−1.05070
$544$	0	0
$545$	−5.69390	−0.243900
$546$	0	0
$547$	−14.8322	−0.634177	−0.317089	−	0.948396i	$-0.602705\pi$
$547$	−14.8322	−0.634177	−0.317089	+	0.948396i	$0.602705\pi$
$548$	0	0
$549$	−10.8363	−0.462481
$550$	0	0
$551$	−0.0964612	−0.00410938
$552$	0	0
$553$	−1.33052	−0.0565795
$554$	0	0
$555$	−6.91244	−0.293417
$556$	0	0
$557$	39.0955	1.65653	0.828264	−	0.560338i	$-0.189329\pi$
$557$	39.0955	1.65653	0.828264	+	0.560338i	$0.189329\pi$
$558$	0	0
$559$	34.4461	1.45691
$560$	0	0
$561$	−4.52308	−0.190965
$562$	0	0
$563$	−1.43413	−0.0604413	−0.0302207	−	0.999543i	$-0.509621\pi$
$563$	−1.43413	−0.0604413	−0.0302207	+	0.999543i	$0.509621\pi$
$564$	0	0
$565$	11.0692	0.465685
$566$	0	0
$567$	−6.67744	−0.280426
$568$	0	0
$569$	−36.7102	−1.53897	−0.769486	−	0.638664i	$-0.779487\pi$
$569$	−36.7102	−1.53897	−0.769486	+	0.638664i	$0.779487\pi$
$570$	0	0
$571$	−39.6097	−1.65762	−0.828808	−	0.559534i	$-0.810980\pi$
$571$	−39.6097	−1.65762	−0.828808	+	0.559534i	$0.810980\pi$
$572$	0	0
$573$	16.4934	0.689022
$574$	0	0
$575$	−6.68660	−0.278850
$576$	0	0
$577$	13.8492	0.576548	0.288274	−	0.957548i	$-0.406919\pi$
$577$	13.8492	0.576548	0.288274	+	0.957548i	$0.406919\pi$
$578$	0	0
$579$	14.6220	0.607670
$580$	0	0
$581$	9.84015	0.408238
$582$	0	0
$583$	0.154907	0.00641557
$584$	0	0
$585$	3.33928	0.138062
$586$	0	0
$587$	33.1456	1.36806	0.684032	−	0.729452i	$-0.260225\pi$
$587$	33.1456	1.36806	0.684032	+	0.729452i	$0.260225\pi$
$588$	0	0
$589$	−5.73913	−0.236477
$590$	0	0
$591$	−8.47244	−0.348510
$592$	0	0
$593$	−23.5972	−0.969022	−0.484511	−	0.874785i	$-0.661002\pi$
$593$	−23.5972	−0.969022	−0.484511	+	0.874785i	$0.661002\pi$
$594$	0	0
$595$	−1.41319	−0.0579350
$596$	0	0
$597$	−6.66531	−0.272793
$598$	0	0
$599$	−41.6611	−1.70223	−0.851114	−	0.524981i	$-0.824072\pi$
$599$	−41.6611	−1.70223	−0.851114	+	0.524981i	$0.824072\pi$
$600$	0	0
$601$	−6.57740	−0.268298	−0.134149	−	0.990961i	$-0.542830\pi$
$601$	−6.57740	−0.268298	−0.134149	+	0.990961i	$0.542830\pi$
$602$	0	0
$603$	14.7378	0.600169
$604$	0	0
$605$	−4.66702	−0.189741
$606$	0	0
$607$	28.3654	1.15132	0.575658	−	0.817690i	$-0.304746\pi$
$607$	28.3654	1.15132	0.575658	+	0.817690i	$0.304746\pi$
$608$	0	0
$609$	−0.181675	−0.00736183
$610$	0	0
$611$	−7.85226	−0.317669
$612$	0	0
$613$	7.06554	0.285374	0.142687	−	0.989768i	$-0.454426\pi$
$613$	7.06554	0.285374	0.142687	+	0.989768i	$0.454426\pi$
$614$	0	0
$615$	4.92662	0.198661
$616$	0	0
$617$	44.3501	1.78547	0.892733	−	0.450586i	$-0.148785\pi$
$617$	44.3501	1.78547	0.892733	+	0.450586i	$0.148785\pi$
$618$	0	0
$619$	18.5118	0.744050	0.372025	−	0.928223i	$-0.378664\pi$
$619$	18.5118	0.744050	0.372025	+	0.928223i	$0.378664\pi$
$620$	0	0
$621$	−8.24623	−0.330910
$622$	0	0
$623$	−6.17862	−0.247541
$624$	0	0
$625$	18.9749	0.758996
$626$	0	0
$627$	2.73688	0.109300
$628$	0	0
$629$	12.5571	0.500686
$630$	0	0
$631$	11.6945	0.465550	0.232775	−	0.972531i	$-0.425219\pi$
$631$	11.6945	0.465550	0.232775	+	0.972531i	$0.425219\pi$
$632$	0	0
$633$	24.1099	0.958282
$634$	0	0
$635$	−7.63609	−0.303029
$636$	0	0
$637$	27.2745	1.08066
$638$	0	0
$639$	7.98637	0.315936
$640$	0	0
$641$	33.8803	1.33819	0.669095	−	0.743177i	$-0.266682\pi$
$641$	33.8803	1.33819	0.669095	+	0.743177i	$0.266682\pi$
$642$	0	0
$643$	43.1523	1.70176	0.850881	−	0.525359i	$-0.176069\pi$
$643$	43.1523	1.70176	0.850881	+	0.525359i	$0.176069\pi$
$644$	0	0
$645$	−6.00869	−0.236592
$646$	0	0
$647$	−35.4426	−1.39339	−0.696697	−	0.717366i	$-0.745348\pi$
$647$	−35.4426	−1.39339	−0.696697	+	0.717366i	$0.745348\pi$
$648$	0	0
$649$	−13.9236	−0.546549
$650$	0	0
$651$	−10.8091	−0.423641
$652$	0	0
$653$	39.0569	1.52841	0.764207	−	0.644971i	$-0.223130\pi$
$653$	39.0569	1.52841	0.764207	+	0.644971i	$0.223130\pi$
$654$	0	0
$655$	−10.7503	−0.420048
$656$	0	0
$657$	−10.6106	−0.413960
$658$	0	0
$659$	31.2655	1.21793	0.608965	−	0.793197i	$-0.291585\pi$
$659$	31.2655	1.21793	0.608965	+	0.793197i	$0.291585\pi$
$660$	0	0
$661$	20.8381	0.810510	0.405255	−	0.914204i	$-0.367183\pi$
$661$	20.8381	0.810510	0.405255	+	0.914204i	$0.367183\pi$
$662$	0	0
$663$	12.2005	0.473829
$664$	0	0
$665$	0.855108	0.0331597
$666$	0	0
$667$	−0.140616	−0.00544466
$668$	0	0
$669$	24.7351	0.956315
$670$	0	0
$671$	21.0301	0.811858
$672$	0	0
$673$	−39.2747	−1.51393	−0.756964	−	0.653456i	$-0.773318\pi$
$673$	−39.2747	−1.51393	−0.756964	+	0.653456i	$0.773318\pi$
$674$	0	0
$675$	25.9477	0.998727
$676$	0	0
$677$	3.38317	0.130026	0.0650129	−	0.997884i	$-0.479291\pi$
$677$	3.38317	0.130026	0.0650129	+	0.997884i	$0.479291\pi$
$678$	0	0
$679$	−6.83840	−0.262434
$680$	0	0
$681$	28.6584	1.09819
$682$	0	0
$683$	21.2944	0.814809	0.407405	−	0.913248i	$-0.366434\pi$
$683$	21.2944	0.814809	0.407405	+	0.913248i	$0.366434\pi$
$684$	0	0
$685$	−9.84499	−0.376158
$686$	0	0
$687$	−22.4276	−0.855668
$688$	0	0
$689$	−0.417844	−0.0159186
$690$	0	0
$691$	−38.0941	−1.44917	−0.724584	−	0.689187i	$-0.757968\pi$
$691$	−38.0941	−1.44917	−0.724584	+	0.689187i	$0.757968\pi$
$692$	0	0
$693$	−2.56290	−0.0973565
$694$	0	0
$695$	4.70478	0.178462
$696$	0	0
$697$	−8.94969	−0.338994
$698$	0	0
$699$	36.5580	1.38275
$700$	0	0
$701$	−4.91662	−0.185698	−0.0928490	−	0.995680i	$-0.529597\pi$
$701$	−4.91662	−0.185698	−0.0928490	+	0.995680i	$0.529597\pi$
$702$	0	0
$703$	−7.59822	−0.286572
$704$	0	0
$705$	1.36973	0.0515871
$706$	0	0
$707$	14.8296	0.557725
$708$	0	0
$709$	−26.0735	−0.979212	−0.489606	−	0.871944i	$-0.662859\pi$
$709$	−26.0735	−0.979212	−0.489606	+	0.871944i	$0.662859\pi$
$710$	0	0
$711$	0.996264	0.0373628
$712$	0	0
$713$	−8.36618	−0.313316
$714$	0	0
$715$	−6.48059	−0.242360
$716$	0	0
$717$	−17.6813	−0.660322
$718$	0	0
$719$	4.73114	0.176442	0.0882209	−	0.996101i	$-0.471882\pi$
$719$	4.73114	0.176442	0.0882209	+	0.996101i	$0.471882\pi$
$720$	0	0
$721$	−9.52427	−0.354702
$722$	0	0
$723$	−18.1259	−0.674108
$724$	0	0
$725$	0.442463	0.0164327
$726$	0	0
$727$	−45.3031	−1.68020	−0.840099	−	0.542434i	$-0.817503\pi$
$727$	−45.3031	−1.68020	−0.840099	+	0.542434i	$0.817503\pi$
$728$	0	0
$729$	29.0223	1.07490
$730$	0	0
$731$	10.9154	0.403720
$732$	0	0
$733$	−52.7436	−1.94813	−0.974065	−	0.226268i	$-0.927347\pi$
$733$	−52.7436	−1.94813	−0.974065	+	0.226268i	$0.927347\pi$
$734$	0	0
$735$	−4.75771	−0.175491
$736$	0	0
$737$	−28.6018	−1.05356
$738$	0	0
$739$	−18.6112	−0.684622	−0.342311	−	0.939587i	$-0.611210\pi$
$739$	−18.6112	−0.684622	−0.342311	+	0.939587i	$0.611210\pi$
$740$	0	0
$741$	−7.38243	−0.271201
$742$	0	0
$743$	0.405748	0.0148855	0.00744273	−	0.999972i	$-0.497631\pi$
$743$	0.405748	0.0148855	0.00744273	+	0.999972i	$0.497631\pi$
$744$	0	0
$745$	11.1969	0.410221
$746$	0	0
$747$	−7.36808	−0.269584
$748$	0	0
$749$	17.8591	0.652558
$750$	0	0
$751$	17.4223	0.635749	0.317874	−	0.948133i	$-0.397031\pi$
$751$	17.4223	0.635749	0.317874	+	0.948133i	$0.397031\pi$
$752$	0	0
$753$	−0.746668	−0.0272101
$754$	0	0
$755$	5.48526	0.199629
$756$	0	0
$757$	−13.1639	−0.478450	−0.239225	−	0.970964i	$-0.576893\pi$
$757$	−13.1639	−0.478450	−0.239225	+	0.970964i	$0.576893\pi$
$758$	0	0
$759$	3.98967	0.144816
$760$	0	0
$761$	28.4807	1.03242	0.516212	−	0.856461i	$-0.327342\pi$
$761$	28.4807	1.03242	0.516212	+	0.856461i	$0.327342\pi$
$762$	0	0
$763$	−11.7878	−0.426747
$764$	0	0
$765$	1.05816	0.0382580
$766$	0	0
$767$	37.5574	1.35612
$768$	0	0
$769$	40.7191	1.46837	0.734184	−	0.678951i	$-0.237565\pi$
$769$	40.7191	1.46837	0.734184	+	0.678951i	$0.237565\pi$
$770$	0	0
$771$	−5.01163	−0.180489
$772$	0	0
$773$	25.7812	0.927286	0.463643	−	0.886022i	$-0.346542\pi$
$773$	25.7812	0.927286	0.463643	+	0.886022i	$0.346542\pi$
$774$	0	0
$775$	26.3251	0.945627
$776$	0	0
$777$	−14.3105	−0.513385
$778$	0	0
$779$	5.41539	0.194026
$780$	0	0
$781$	−15.4992	−0.554606
$782$	0	0
$783$	0.545666	0.0195005
$784$	0	0
$785$	14.5831	0.520493
$786$	0	0
$787$	−5.38688	−0.192021	−0.0960107	−	0.995380i	$-0.530608\pi$
$787$	−5.38688	−0.192021	−0.0960107	+	0.995380i	$0.530608\pi$
$788$	0	0
$789$	26.0025	0.925713
$790$	0	0
$791$	22.9160	0.814800
$792$	0	0
$793$	−56.7264	−2.01441
$794$	0	0
$795$	0.0728877	0.00258506
$796$	0	0
$797$	−20.2002	−0.715526	−0.357763	−	0.933812i	$-0.616460\pi$
$797$	−20.2002	−0.715526	−0.357763	+	0.933812i	$0.616460\pi$
$798$	0	0
$799$	−2.48825	−0.0880280
$800$	0	0
$801$	4.62641	0.163466
$802$	0	0
$803$	20.5922	0.726682
$804$	0	0
$805$	1.24653	0.0439343
$806$	0	0
$807$	−16.8388	−0.592754
$808$	0	0
$809$	5.95846	0.209488	0.104744	−	0.994499i	$-0.466598\pi$
$809$	5.95846	0.209488	0.104744	+	0.994499i	$0.466598\pi$
$810$	0	0
$811$	−15.1558	−0.532193	−0.266096	−	0.963946i	$-0.585734\pi$
$811$	−15.1558	−0.532193	−0.266096	+	0.963946i	$0.585734\pi$
$812$	0	0
$813$	−10.7401	−0.376673
$814$	0	0
$815$	−10.5163	−0.368372
$816$	0	0
$817$	−6.60481	−0.231073
$818$	0	0
$819$	6.91314	0.241565
$820$	0	0
$821$	10.2340	0.357169	0.178584	−	0.983925i	$-0.442848\pi$
$821$	10.2340	0.357169	0.178584	+	0.983925i	$0.442848\pi$
$822$	0	0
$823$	−39.1399	−1.36433	−0.682165	−	0.731199i	$-0.738961\pi$
$823$	−39.1399	−1.36433	−0.682165	+	0.731199i	$0.738961\pi$
$824$	0	0
$825$	−12.5539	−0.437072
$826$	0	0
$827$	−42.1583	−1.46599	−0.732994	−	0.680235i	$-0.761878\pi$
$827$	−42.1583	−1.46599	−0.732994	+	0.680235i	$0.761878\pi$
$828$	0	0
$829$	37.4950	1.30226	0.651128	−	0.758968i	$-0.274296\pi$
$829$	37.4950	1.30226	0.651128	+	0.758968i	$0.274296\pi$
$830$	0	0
$831$	21.4245	0.743207
$832$	0	0
$833$	8.64284	0.299457
$834$	0	0
$835$	−15.0409	−0.520513
$836$	0	0
$837$	32.4654	1.12217
$838$	0	0
$839$	−19.5279	−0.674178	−0.337089	−	0.941473i	$-0.609442\pi$
$839$	−19.5279	−0.674178	−0.337089	+	0.941473i	$0.609442\pi$
$840$	0	0
$841$	−28.9907	−0.999679
$842$	0	0
$843$	−11.1309	−0.383368
$844$	0	0
$845$	9.12574	0.313935
$846$	0	0
$847$	−9.66188	−0.331986
$848$	0	0
$849$	28.7451	0.986529
$850$	0	0
$851$	−11.0763	−0.379689
$852$	0	0
$853$	30.6923	1.05088	0.525442	−	0.850829i	$-0.323900\pi$
$853$	30.6923	1.05088	0.525442	+	0.850829i	$0.323900\pi$
$854$	0	0
$855$	−0.640286	−0.0218973
$856$	0	0
$857$	−12.2942	−0.419960	−0.209980	−	0.977706i	$-0.567340\pi$
$857$	−12.2942	−0.419960	−0.209980	+	0.977706i	$0.567340\pi$
$858$	0	0
$859$	24.4783	0.835190	0.417595	−	0.908633i	$-0.362873\pi$
$859$	24.4783	0.835190	0.417595	+	0.908633i	$0.362873\pi$
$860$	0	0
$861$	10.1993	0.347592
$862$	0	0
$863$	−18.1764	−0.618731	−0.309366	−	0.950943i	$-0.600117\pi$
$863$	−18.1764	−0.618731	−0.309366	+	0.950943i	$0.600117\pi$
$864$	0	0
$865$	−14.1706	−0.481814
$866$	0	0
$867$	−20.1979	−0.685958
$868$	0	0
$869$	−1.93346	−0.0655882
$870$	0	0
$871$	77.1502	2.61413
$872$	0	0
$873$	5.12044	0.173301
$874$	0	0
$875$	−8.19788	−0.277139
$876$	0	0
$877$	32.2526	1.08909	0.544547	−	0.838730i	$-0.316702\pi$
$877$	32.2526	1.08909	0.544547	+	0.838730i	$0.316702\pi$
$878$	0	0
$879$	−15.8676	−0.535201
$880$	0	0
$881$	−45.3656	−1.52841	−0.764204	−	0.644975i	$-0.776868\pi$
$881$	−45.3656	−1.52841	−0.764204	+	0.644975i	$0.776868\pi$
$882$	0	0
$883$	−20.3669	−0.685402	−0.342701	−	0.939444i	$-0.611342\pi$
$883$	−20.3669	−0.685402	−0.342701	+	0.939444i	$0.611342\pi$
$884$	0	0
$885$	−6.55142	−0.220224
$886$	0	0
$887$	−54.0111	−1.81351	−0.906757	−	0.421654i	$-0.861450\pi$
$887$	−54.0111	−1.81351	−0.906757	+	0.421654i	$0.861450\pi$
$888$	0	0
$889$	−15.8086	−0.530203
$890$	0	0
$891$	−9.70340	−0.325076
$892$	0	0
$893$	1.50562	0.0503837
$894$	0	0
$895$	5.43253	0.181589
$896$	0	0
$897$	−10.7617	−0.359322
$898$	0	0
$899$	0.553603	0.0184637
$900$	0	0
$901$	−0.132408	−0.00441114
$902$	0	0
$903$	−12.4395	−0.413960
$904$	0	0
$905$	−11.1162	−0.369515
$906$	0	0
$907$	8.58408	0.285030	0.142515	−	0.989793i	$-0.454481\pi$
$907$	8.58408	0.285030	0.142515	+	0.989793i	$0.454481\pi$
$908$	0	0
$909$	−11.1041	−0.368299
$910$	0	0
$911$	−55.7805	−1.84809	−0.924045	−	0.382283i	$-0.875138\pi$
$911$	−55.7805	−1.84809	−0.924045	+	0.382283i	$0.875138\pi$
$912$	0	0
$913$	14.2993	0.473239
$914$	0	0
$915$	9.89523	0.327126
$916$	0	0
$917$	−22.2557	−0.734949
$918$	0	0
$919$	23.2970	0.768497	0.384248	−	0.923230i	$-0.374461\pi$
$919$	23.2970	0.768497	0.384248	+	0.923230i	$0.374461\pi$
$920$	0	0
$921$	11.4916	0.378660
$922$	0	0
$923$	41.8075	1.37611
$924$	0	0
$925$	34.8527	1.14595
$926$	0	0
$927$	7.13156	0.234231
$928$	0	0
$929$	30.7079	1.00749	0.503746	−	0.863852i	$-0.331955\pi$
$929$	30.7079	1.00749	0.503746	+	0.863852i	$0.331955\pi$
$930$	0	0
$931$	−5.22971	−0.171397
$932$	0	0
$933$	−5.43161	−0.177823
$934$	0	0
$935$	−2.05359	−0.0671596
$936$	0	0
$937$	9.54260	0.311743	0.155872	−	0.987777i	$-0.450181\pi$
$937$	9.54260	0.311743	0.155872	+	0.987777i	$0.450181\pi$
$938$	0	0
$939$	−6.92017	−0.225831
$940$	0	0
$941$	17.1446	0.558899	0.279450	−	0.960160i	$-0.409848\pi$
$941$	17.1446	0.558899	0.279450	+	0.960160i	$0.409848\pi$
$942$	0	0
$943$	7.89424	0.257072
$944$	0	0
$945$	−4.83722	−0.157355
$946$	0	0
$947$	27.6790	0.899446	0.449723	−	0.893168i	$-0.351523\pi$
$947$	27.6790	0.899446	0.449723	+	0.893168i	$0.351523\pi$
$948$	0	0
$949$	−55.5451	−1.80307
$950$	0	0
$951$	13.2542	0.429798
$952$	0	0
$953$	46.7690	1.51500	0.757498	−	0.652837i	$-0.226421\pi$
$953$	46.7690	1.51500	0.757498	+	0.652837i	$0.226421\pi$
$954$	0	0
$955$	7.48841	0.242319
$956$	0	0
$957$	−0.264003	−0.00853399
$958$	0	0
$959$	−20.3816	−0.658155
$960$	0	0
$961$	1.93764	0.0625045
$962$	0	0
$963$	−13.3725	−0.430923
$964$	0	0
$965$	6.63875	0.213709
$966$	0	0
$967$	24.6617	0.793066	0.396533	−	0.918021i	$-0.370213\pi$
$967$	24.6617	0.793066	0.396533	+	0.918021i	$0.370213\pi$
$968$	0	0
$969$	−2.33937	−0.0751514
$970$	0	0
$971$	−34.9613	−1.12196	−0.560981	−	0.827828i	$-0.689576\pi$
$971$	−34.9613	−1.12196	−0.560981	+	0.827828i	$0.689576\pi$
$972$	0	0
$973$	9.74005	0.312252
$974$	0	0
$975$	33.8629	1.08448
$976$	0	0
$977$	−39.8874	−1.27611	−0.638056	−	0.769990i	$-0.720261\pi$
$977$	−39.8874	−1.27611	−0.638056	+	0.769990i	$0.720261\pi$
$978$	0	0
$979$	−8.97854	−0.286955
$980$	0	0
$981$	8.82643	0.281806
$982$	0	0
$983$	25.4845	0.812831	0.406415	−	0.913688i	$-0.366779\pi$
$983$	25.4845	0.812831	0.406415	+	0.913688i	$0.366779\pi$
$984$	0	0
$985$	−3.84670	−0.122566
$986$	0	0
$987$	2.83568	0.0902607
$988$	0	0
$989$	−9.62812	−0.306156
$990$	0	0
$991$	−10.3321	−0.328211	−0.164106	−	0.986443i	$-0.552474\pi$
$991$	−10.3321	−0.328211	−0.164106	+	0.986443i	$0.552474\pi$
$992$	0	0
$993$	1.47886	0.0469302
$994$	0	0
$995$	−3.02622	−0.0959375
$996$	0	0
$997$	−34.1889	−1.08277	−0.541387	−	0.840774i	$-0.682100\pi$
$997$	−34.1889	−1.08277	−0.541387	+	0.840774i	$0.682100\pi$
$998$	0	0
$999$	42.9820	1.35989

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.21	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.21	✓	27	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.41553 q^{3} +0.642687 q^{5} +1.33052 q^{7} -0.996264 q^{9} +O(q^{10})\)	\(q+1.41553 q^{3} +0.642687 q^{5} +1.33052 q^{7} -0.996264 q^{9} +1.93346 q^{11} -5.21530 q^{13} +0.909745 q^{15} -1.65264 q^{17} +1.00000 q^{19} +1.88340 q^{21} +1.45774 q^{23} -4.58695 q^{25} -5.65685 q^{27} -0.0964612 q^{29} -5.73913 q^{31} +2.73688 q^{33} +0.855108 q^{35} -7.59822 q^{37} -7.38243 q^{39} +5.41539 q^{41} -6.60481 q^{43} -0.640286 q^{45} +1.50562 q^{47} -5.22971 q^{49} -2.33937 q^{51} +0.0801188 q^{53} +1.24261 q^{55} +1.41553 q^{57} -7.20139 q^{59} +10.8769 q^{61} -1.32555 q^{63} -3.35181 q^{65} -14.7930 q^{67} +2.06348 q^{69} -8.01632 q^{71} +10.6504 q^{73} -6.49299 q^{75} +2.57251 q^{77} -1.00000 q^{79} -5.01867 q^{81} +7.39572 q^{83} -1.06213 q^{85} -0.136544 q^{87} -4.64376 q^{89} -6.93907 q^{91} -8.12394 q^{93} +0.642687 q^{95} -5.13964 q^{97} -1.92624 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 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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