LMFDB - Embedded newform 6004.2.a.h.1.17

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:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+0.246226 q^{3} +2.74327 q^{5} +2.47378 q^{7} -2.93937 q^{9} +O(q^{10})$	$q+0.246226 q^{3} +2.74327 q^{5} +2.47378 q^{7} -2.93937 q^{9} +5.71848 q^{11} -5.08134 q^{13} +0.675466 q^{15} +1.37133 q^{17} -1.00000 q^{19} +0.609111 q^{21} +6.90720 q^{23} +2.52554 q^{25} -1.46243 q^{27} -4.25029 q^{29} +3.24932 q^{31} +1.40804 q^{33} +6.78626 q^{35} -2.82720 q^{37} -1.25116 q^{39} +1.21862 q^{41} +5.15173 q^{43} -8.06350 q^{45} +10.4384 q^{47} -0.880391 q^{49} +0.337658 q^{51} -4.71005 q^{53} +15.6874 q^{55} -0.246226 q^{57} -3.36320 q^{59} +3.42423 q^{61} -7.27137 q^{63} -13.9395 q^{65} +11.9986 q^{67} +1.70073 q^{69} +3.75182 q^{71} +10.0381 q^{73} +0.621856 q^{75} +14.1463 q^{77} -1.00000 q^{79} +8.45803 q^{81} +3.09006 q^{83} +3.76193 q^{85} -1.04653 q^{87} -2.31935 q^{89} -12.5701 q^{91} +0.800068 q^{93} -2.74327 q^{95} +2.86172 q^{97} -16.8087 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * 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
$3$	0.246226	0.142159	0.0710794	−	0.997471i	$-0.477356\pi$
$3$	0.246226	0.142159	0.0710794	+	0.997471i	$0.477356\pi$
$4$	0	0
$5$	2.74327	1.22683	0.613414	−	0.789761i	$-0.289796\pi$
$5$	2.74327	1.22683	0.613414	+	0.789761i	$0.289796\pi$
$6$	0	0
$7$	2.47378	0.935003	0.467501	−	0.883992i	$-0.345154\pi$
$7$	2.47378	0.935003	0.467501	+	0.883992i	$0.345154\pi$
$8$	0	0
$9$	−2.93937	−0.979791
$10$	0	0
$11$	5.71848	1.72419	0.862094	−	0.506749i	$-0.169153\pi$
$11$	5.71848	1.72419	0.862094	+	0.506749i	$0.169153\pi$
$12$	0	0
$13$	−5.08134	−1.40931	−0.704654	−	0.709551i	$-0.748898\pi$
$13$	−5.08134	−1.40931	−0.704654	+	0.709551i	$0.748898\pi$
$14$	0	0
$15$	0.675466	0.174405
$16$	0	0
$17$	1.37133	0.332596	0.166298	−	0.986076i	$-0.446819\pi$
$17$	1.37133	0.332596	0.166298	+	0.986076i	$0.446819\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.609111	0.132919
$22$	0	0
$23$	6.90720	1.44025	0.720125	−	0.693844i	$-0.244084\pi$
$23$	6.90720	1.44025	0.720125	+	0.693844i	$0.244084\pi$
$24$	0	0
$25$	2.52554	0.505109
$26$	0	0
$27$	−1.46243	−0.281445
$28$	0	0
$29$	−4.25029	−0.789259	−0.394629	−	0.918840i	$-0.629127\pi$
$29$	−4.25029	−0.789259	−0.394629	+	0.918840i	$0.629127\pi$
$30$	0	0
$31$	3.24932	0.583595	0.291798	−	0.956480i	$-0.405747\pi$
$31$	3.24932	0.583595	0.291798	+	0.956480i	$0.405747\pi$
$32$	0	0
$33$	1.40804	0.245108
$34$	0	0
$35$	6.78626	1.14709
$36$	0	0
$37$	−2.82720	−0.464788	−0.232394	−	0.972622i	$-0.574656\pi$
$37$	−2.82720	−0.464788	−0.232394	+	0.972622i	$0.574656\pi$
$38$	0	0
$39$	−1.25116	−0.200346
$40$	0	0
$41$	1.21862	0.190316	0.0951580	−	0.995462i	$-0.469664\pi$
$41$	1.21862	0.190316	0.0951580	+	0.995462i	$0.469664\pi$
$42$	0	0
$43$	5.15173	0.785631	0.392816	−	0.919617i	$-0.371501\pi$
$43$	5.15173	0.785631	0.392816	+	0.919617i	$0.371501\pi$
$44$	0	0
$45$	−8.06350	−1.20204
$46$	0	0
$47$	10.4384	1.52260	0.761300	−	0.648399i	$-0.224561\pi$
$47$	10.4384	1.52260	0.761300	+	0.648399i	$0.224561\pi$
$48$	0	0
$49$	−0.880391	−0.125770
$50$	0	0
$51$	0.337658	0.0472815
$52$	0	0
$53$	−4.71005	−0.646975	−0.323488	−	0.946232i	$-0.604855\pi$
$53$	−4.71005	−0.646975	−0.323488	+	0.946232i	$0.604855\pi$
$54$	0	0
$55$	15.6874	2.11528
$56$	0	0
$57$	−0.246226	−0.0326135
$58$	0	0
$59$	−3.36320	−0.437851	−0.218925	−	0.975742i	$-0.570255\pi$
$59$	−3.36320	−0.437851	−0.218925	+	0.975742i	$0.570255\pi$
$60$	0	0
$61$	3.42423	0.438428	0.219214	−	0.975677i	$-0.429651\pi$
$61$	3.42423	0.438428	0.219214	+	0.975677i	$0.429651\pi$
$62$	0	0
$63$	−7.27137	−0.916107
$64$	0	0
$65$	−13.9395	−1.72898
$66$	0	0
$67$	11.9986	1.46587	0.732933	−	0.680301i	$-0.238151\pi$
$67$	11.9986	1.46587	0.732933	+	0.680301i	$0.238151\pi$
$68$	0	0
$69$	1.70073	0.204744
$70$	0	0
$71$	3.75182	0.445259	0.222630	−	0.974903i	$-0.428536\pi$
$71$	3.75182	0.445259	0.222630	+	0.974903i	$0.428536\pi$
$72$	0	0
$73$	10.0381	1.17487	0.587437	−	0.809270i	$-0.300137\pi$
$73$	10.0381	1.17487	0.587437	+	0.809270i	$0.300137\pi$
$74$	0	0
$75$	0.621856	0.0718057
$76$	0	0
$77$	14.1463	1.61212
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	8.45803	0.939781
$82$	0	0
$83$	3.09006	0.339178	0.169589	−	0.985515i	$-0.445756\pi$
$83$	3.09006	0.339178	0.169589	+	0.985515i	$0.445756\pi$
$84$	0	0
$85$	3.76193	0.408039
$86$	0	0
$87$	−1.04653	−0.112200
$88$	0	0
$89$	−2.31935	−0.245851	−0.122925	−	0.992416i	$-0.539228\pi$
$89$	−2.31935	−0.245851	−0.122925	+	0.992416i	$0.539228\pi$
$90$	0	0
$91$	−12.5701	−1.31771
$92$	0	0
$93$	0.800068	0.0829632
$94$	0	0
$95$	−2.74327	−0.281454
$96$	0	0
$97$	2.86172	0.290563	0.145282	−	0.989390i	$-0.453591\pi$
$97$	2.86172	0.290563	0.145282	+	0.989390i	$0.453591\pi$
$98$	0	0
$99$	−16.8087	−1.68934
$100$	0	0
$101$	−8.36724	−0.832571	−0.416286	−	0.909234i	$-0.636668\pi$
$101$	−8.36724	−0.832571	−0.416286	+	0.909234i	$0.636668\pi$
$102$	0	0
$103$	14.9002	1.46816	0.734079	−	0.679064i	$-0.237614\pi$
$103$	14.9002	1.46816	0.734079	+	0.679064i	$0.237614\pi$
$104$	0	0
$105$	1.67096	0.163069
$106$	0	0
$107$	−16.0182	−1.54854	−0.774270	−	0.632855i	$-0.781883\pi$
$107$	−16.0182	−1.54854	−0.774270	+	0.632855i	$0.781883\pi$
$108$	0	0
$109$	1.03102	0.0987541	0.0493771	−	0.998780i	$-0.484276\pi$
$109$	1.03102	0.0987541	0.0493771	+	0.998780i	$0.484276\pi$
$110$	0	0
$111$	−0.696130	−0.0660738
$112$	0	0
$113$	2.22017	0.208856	0.104428	−	0.994532i	$-0.466699\pi$
$113$	2.22017	0.208856	0.104428	+	0.994532i	$0.466699\pi$
$114$	0	0
$115$	18.9483	1.76694
$116$	0	0
$117$	14.9359	1.38083
$118$	0	0
$119$	3.39237	0.310978
$120$	0	0
$121$	21.7010	1.97282
$122$	0	0
$123$	0.300056	0.0270551
$124$	0	0
$125$	−6.78811	−0.607147
$126$	0	0
$127$	15.4009	1.36661	0.683305	−	0.730133i	$-0.260542\pi$
$127$	15.4009	1.36661	0.683305	+	0.730133i	$0.260542\pi$
$128$	0	0
$129$	1.26849	0.111684
$130$	0	0
$131$	12.5921	1.10018	0.550090	−	0.835105i	$-0.314593\pi$
$131$	12.5921	1.10018	0.550090	+	0.835105i	$0.314593\pi$
$132$	0	0
$133$	−2.47378	−0.214504
$134$	0	0
$135$	−4.01184	−0.345285
$136$	0	0
$137$	2.21596	0.189322	0.0946610	−	0.995510i	$-0.469823\pi$
$137$	2.21596	0.189322	0.0946610	+	0.995510i	$0.469823\pi$
$138$	0	0
$139$	1.39044	0.117936	0.0589678	−	0.998260i	$-0.481219\pi$
$139$	1.39044	0.117936	0.0589678	+	0.998260i	$0.481219\pi$
$140$	0	0
$141$	2.57022	0.216451
$142$	0	0
$143$	−29.0575	−2.42991
$144$	0	0
$145$	−11.6597	−0.968285
$146$	0	0
$147$	−0.216775	−0.0178793
$148$	0	0
$149$	−2.53559	−0.207724	−0.103862	−	0.994592i	$-0.533120\pi$
$149$	−2.53559	−0.207724	−0.103862	+	0.994592i	$0.533120\pi$
$150$	0	0
$151$	−4.33069	−0.352426	−0.176213	−	0.984352i	$-0.556385\pi$
$151$	−4.33069	−0.352426	−0.176213	+	0.984352i	$0.556385\pi$
$152$	0	0
$153$	−4.03085	−0.325875
$154$	0	0
$155$	8.91377	0.715971
$156$	0	0
$157$	2.74379	0.218978	0.109489	−	0.993988i	$-0.465079\pi$
$157$	2.74379	0.218978	0.109489	+	0.993988i	$0.465079\pi$
$158$	0	0
$159$	−1.15974	−0.0919733
$160$	0	0
$161$	17.0869	1.34664
$162$	0	0
$163$	−10.5052	−0.822830	−0.411415	−	0.911448i	$-0.634965\pi$
$163$	−10.5052	−0.822830	−0.411415	+	0.911448i	$0.634965\pi$
$164$	0	0
$165$	3.86264	0.300706
$166$	0	0
$167$	15.0796	1.16689	0.583446	−	0.812152i	$-0.301704\pi$
$167$	15.0796	1.16689	0.583446	+	0.812152i	$0.301704\pi$
$168$	0	0
$169$	12.8200	0.986152
$170$	0	0
$171$	2.93937	0.224779
$172$	0	0
$173$	2.20662	0.167766	0.0838832	−	0.996476i	$-0.473268\pi$
$173$	2.20662	0.167766	0.0838832	+	0.996476i	$0.473268\pi$
$174$	0	0
$175$	6.24765	0.472278
$176$	0	0
$177$	−0.828108	−0.0622444
$178$	0	0
$179$	−22.9899	−1.71834	−0.859171	−	0.511688i	$-0.829020\pi$
$179$	−22.9899	−1.71834	−0.859171	+	0.511688i	$0.829020\pi$
$180$	0	0
$181$	−16.6713	−1.23917	−0.619586	−	0.784929i	$-0.712699\pi$
$181$	−16.6713	−1.23917	−0.619586	+	0.784929i	$0.712699\pi$
$182$	0	0
$183$	0.843136	0.0623264
$184$	0	0
$185$	−7.75577	−0.570216
$186$	0	0
$187$	7.84192	0.573458
$188$	0	0
$189$	−3.61774	−0.263152
$190$	0	0
$191$	−22.0587	−1.59611	−0.798055	−	0.602585i	$-0.794137\pi$
$191$	−22.0587	−1.59611	−0.798055	+	0.602585i	$0.794137\pi$
$192$	0	0
$193$	−5.53356	−0.398315	−0.199157	−	0.979968i	$-0.563820\pi$
$193$	−5.53356	−0.398315	−0.199157	+	0.979968i	$0.563820\pi$
$194$	0	0
$195$	−3.43227	−0.245790
$196$	0	0
$197$	3.36822	0.239976	0.119988	−	0.992775i	$-0.461714\pi$
$197$	3.36822	0.239976	0.119988	+	0.992775i	$0.461714\pi$
$198$	0	0
$199$	17.8449	1.26499	0.632495	−	0.774564i	$-0.282031\pi$
$199$	17.8449	1.26499	0.632495	+	0.774564i	$0.282031\pi$
$200$	0	0
$201$	2.95438	0.208386
$202$	0	0
$203$	−10.5143	−0.737959
$204$	0	0
$205$	3.34300	0.233485
$206$	0	0
$207$	−20.3028	−1.41114
$208$	0	0
$209$	−5.71848	−0.395556
$210$	0	0
$211$	−3.16899	−0.218163	−0.109081	−	0.994033i	$-0.534791\pi$
$211$	−3.16899	−0.218163	−0.109081	+	0.994033i	$0.534791\pi$
$212$	0	0
$213$	0.923797	0.0632975
$214$	0	0
$215$	14.1326	0.963835
$216$	0	0
$217$	8.03812	0.545663
$218$	0	0
$219$	2.47165	0.167019
$220$	0	0
$221$	−6.96819	−0.468731
$222$	0	0
$223$	−16.3211	−1.09294	−0.546472	−	0.837477i	$-0.684030\pi$
$223$	−16.3211	−1.09294	−0.546472	+	0.837477i	$0.684030\pi$
$224$	0	0
$225$	−7.42351	−0.494901
$226$	0	0
$227$	18.3374	1.21710	0.608548	−	0.793517i	$-0.291752\pi$
$227$	18.3374	1.21710	0.608548	+	0.793517i	$0.291752\pi$
$228$	0	0
$229$	−6.09324	−0.402652	−0.201326	−	0.979524i	$-0.564525\pi$
$229$	−6.09324	−0.402652	−0.201326	+	0.979524i	$0.564525\pi$
$230$	0	0
$231$	3.48319	0.229177
$232$	0	0
$233$	−17.8456	−1.16910	−0.584552	−	0.811356i	$-0.698730\pi$
$233$	−17.8456	−1.16910	−0.584552	+	0.811356i	$0.698730\pi$
$234$	0	0
$235$	28.6354	1.86797
$236$	0	0
$237$	−0.246226	−0.0159941
$238$	0	0
$239$	16.4043	1.06111	0.530554	−	0.847651i	$-0.321984\pi$
$239$	16.4043	1.06111	0.530554	+	0.847651i	$0.321984\pi$
$240$	0	0
$241$	14.5617	0.938003	0.469001	−	0.883197i	$-0.344614\pi$
$241$	14.5617	0.938003	0.469001	+	0.883197i	$0.344614\pi$
$242$	0	0
$243$	6.46988	0.415043
$244$	0	0
$245$	−2.41515	−0.154298
$246$	0	0
$247$	5.08134	0.323318
$248$	0	0
$249$	0.760854	0.0482171
$250$	0	0
$251$	−20.4209	−1.28896	−0.644479	−	0.764622i	$-0.722926\pi$
$251$	−20.4209	−1.28896	−0.644479	+	0.764622i	$0.722926\pi$
$252$	0	0
$253$	39.4987	2.48326
$254$	0	0
$255$	0.926287	0.0580063
$256$	0	0
$257$	−4.21332	−0.262820	−0.131410	−	0.991328i	$-0.541950\pi$
$257$	−4.21332	−0.262820	−0.131410	+	0.991328i	$0.541950\pi$
$258$	0	0
$259$	−6.99388	−0.434578
$260$	0	0
$261$	12.4932	0.773308
$262$	0	0
$263$	1.69734	0.104662	0.0523311	−	0.998630i	$-0.483335\pi$
$263$	1.69734	0.104662	0.0523311	+	0.998630i	$0.483335\pi$
$264$	0	0
$265$	−12.9210	−0.793728
$266$	0	0
$267$	−0.571085	−0.0349499
$268$	0	0
$269$	14.4572	0.881471	0.440736	−	0.897637i	$-0.354718\pi$
$269$	14.4572	0.881471	0.440736	+	0.897637i	$0.354718\pi$
$270$	0	0
$271$	1.21915	0.0740583	0.0370291	−	0.999314i	$-0.488211\pi$
$271$	1.21915	0.0740583	0.0370291	+	0.999314i	$0.488211\pi$
$272$	0	0
$273$	−3.09510	−0.187324
$274$	0	0
$275$	14.4423	0.870902
$276$	0	0
$277$	−18.0091	−1.08206	−0.541030	−	0.841003i	$-0.681966\pi$
$277$	−18.0091	−1.08206	−0.541030	+	0.841003i	$0.681966\pi$
$278$	0	0
$279$	−9.55096	−0.571801
$280$	0	0
$281$	−17.8432	−1.06444	−0.532219	−	0.846606i	$-0.678642\pi$
$281$	−17.8432	−1.06444	−0.532219	+	0.846606i	$0.678642\pi$
$282$	0	0
$283$	10.3874	0.617468	0.308734	−	0.951148i	$-0.400095\pi$
$283$	10.3874	0.617468	0.308734	+	0.951148i	$0.400095\pi$
$284$	0	0
$285$	−0.675466	−0.0400112
$286$	0	0
$287$	3.01459	0.177946
$288$	0	0
$289$	−15.1195	−0.889380
$290$	0	0
$291$	0.704630	0.0413061
$292$	0	0
$293$	−30.0717	−1.75681	−0.878404	−	0.477919i	$-0.841391\pi$
$293$	−30.0717	−1.75681	−0.878404	+	0.477919i	$0.841391\pi$
$294$	0	0
$295$	−9.22616	−0.537168
$296$	0	0
$297$	−8.36288	−0.485264
$298$	0	0
$299$	−35.0978	−2.02976
$300$	0	0
$301$	12.7443	0.734567
$302$	0	0
$303$	−2.06024	−0.118357
$304$	0	0
$305$	9.39360	0.537876
$306$	0	0
$307$	−3.82190	−0.218128	−0.109064	−	0.994035i	$-0.534785\pi$
$307$	−3.82190	−0.218128	−0.109064	+	0.994035i	$0.534785\pi$
$308$	0	0
$309$	3.66882	0.208712
$310$	0	0
$311$	19.5513	1.10865	0.554326	−	0.832300i	$-0.312976\pi$
$311$	19.5513	1.10865	0.554326	+	0.832300i	$0.312976\pi$
$312$	0	0
$313$	−27.2027	−1.53759	−0.768794	−	0.639496i	$-0.779143\pi$
$313$	−27.2027	−1.53759	−0.768794	+	0.639496i	$0.779143\pi$
$314$	0	0
$315$	−19.9474	−1.12391
$316$	0	0
$317$	9.62476	0.540580	0.270290	−	0.962779i	$-0.412880\pi$
$317$	9.62476	0.540580	0.270290	+	0.962779i	$0.412880\pi$
$318$	0	0
$319$	−24.3052	−1.36083
$320$	0	0
$321$	−3.94411	−0.220139
$322$	0	0
$323$	−1.37133	−0.0763028
$324$	0	0
$325$	−12.8331	−0.711854
$326$	0	0
$327$	0.253865	0.0140388
$328$	0	0
$329$	25.8224	1.42364
$330$	0	0
$331$	−6.40777	−0.352203	−0.176101	−	0.984372i	$-0.556349\pi$
$331$	−6.40777	−0.352203	−0.176101	+	0.984372i	$0.556349\pi$
$332$	0	0
$333$	8.31018	0.455395
$334$	0	0
$335$	32.9155	1.79837
$336$	0	0
$337$	12.7077	0.692231	0.346116	−	0.938192i	$-0.387500\pi$
$337$	12.7077	0.692231	0.346116	+	0.938192i	$0.387500\pi$
$338$	0	0
$339$	0.546665	0.0296908
$340$	0	0
$341$	18.5812	1.00623
$342$	0	0
$343$	−19.4944	−1.05260
$344$	0	0
$345$	4.66558	0.251186
$346$	0	0
$347$	−17.2368	−0.925319	−0.462659	−	0.886536i	$-0.653105\pi$
$347$	−17.2368	−0.925319	−0.462659	+	0.886536i	$0.653105\pi$
$348$	0	0
$349$	15.7217	0.841562	0.420781	−	0.907162i	$-0.361756\pi$
$349$	15.7217	0.841562	0.420781	+	0.907162i	$0.361756\pi$
$350$	0	0
$351$	7.43110	0.396643
$352$	0	0
$353$	20.0744	1.06845	0.534226	−	0.845342i	$-0.320603\pi$
$353$	20.0744	1.06845	0.534226	+	0.845342i	$0.320603\pi$
$354$	0	0
$355$	10.2923	0.546257
$356$	0	0
$357$	0.835292	0.0442083
$358$	0	0
$359$	0.341012	0.0179979	0.00899897	−	0.999960i	$-0.497136\pi$
$359$	0.341012	0.0179979	0.00899897	+	0.999960i	$0.497136\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.34337	0.280454
$364$	0	0
$365$	27.5373	1.44137
$366$	0	0
$367$	20.5884	1.07470	0.537352	−	0.843358i	$-0.319425\pi$
$367$	20.5884	1.07470	0.537352	+	0.843358i	$0.319425\pi$
$368$	0	0
$369$	−3.58197	−0.186470
$370$	0	0
$371$	−11.6517	−0.604924
$372$	0	0
$373$	6.31995	0.327235	0.163617	−	0.986524i	$-0.447684\pi$
$373$	6.31995	0.327235	0.163617	+	0.986524i	$0.447684\pi$
$374$	0	0
$375$	−1.67141	−0.0863113
$376$	0	0
$377$	21.5971	1.11231
$378$	0	0
$379$	20.7143	1.06402	0.532012	−	0.846737i	$-0.321436\pi$
$379$	20.7143	1.06402	0.532012	+	0.846737i	$0.321436\pi$
$380$	0	0
$381$	3.79211	0.194276
$382$	0	0
$383$	14.8241	0.757478	0.378739	−	0.925504i	$-0.376358\pi$
$383$	14.8241	0.757478	0.378739	+	0.925504i	$0.376358\pi$
$384$	0	0
$385$	38.8071	1.97779
$386$	0	0
$387$	−15.1428	−0.769754
$388$	0	0
$389$	11.9066	0.603687	0.301844	−	0.953357i	$-0.402398\pi$
$389$	11.9066	0.603687	0.301844	+	0.953357i	$0.402398\pi$
$390$	0	0
$391$	9.47204	0.479022
$392$	0	0
$393$	3.10052	0.156400
$394$	0	0
$395$	−2.74327	−0.138029
$396$	0	0
$397$	4.34805	0.218223	0.109111	−	0.994030i	$-0.465200\pi$
$397$	4.34805	0.218223	0.109111	+	0.994030i	$0.465200\pi$
$398$	0	0
$399$	−0.609111	−0.0304937
$400$	0	0
$401$	35.6008	1.77782	0.888910	−	0.458081i	$-0.151463\pi$
$401$	35.6008	1.77782	0.888910	+	0.458081i	$0.151463\pi$
$402$	0	0
$403$	−16.5109	−0.822466
$404$	0	0
$405$	23.2027	1.15295
$406$	0	0
$407$	−16.1673	−0.801382
$408$	0	0
$409$	9.52442	0.470952	0.235476	−	0.971880i	$-0.424335\pi$
$409$	9.52442	0.470952	0.235476	+	0.971880i	$0.424335\pi$
$410$	0	0
$411$	0.545627	0.0269138
$412$	0	0
$413$	−8.31982	−0.409392
$414$	0	0
$415$	8.47687	0.416113
$416$	0	0
$417$	0.342363	0.0167656
$418$	0	0
$419$	−35.7588	−1.74693	−0.873465	−	0.486887i	$-0.838132\pi$
$419$	−35.7588	−1.74693	−0.873465	+	0.486887i	$0.838132\pi$
$420$	0	0
$421$	−25.5984	−1.24759	−0.623795	−	0.781588i	$-0.714410\pi$
$421$	−25.5984	−1.24759	−0.623795	+	0.781588i	$0.714410\pi$
$422$	0	0
$423$	−30.6824	−1.49183
$424$	0	0
$425$	3.46335	0.167997
$426$	0	0
$427$	8.47081	0.409931
$428$	0	0
$429$	−7.15473	−0.345434
$430$	0	0
$431$	11.5368	0.555708	0.277854	−	0.960623i	$-0.410377\pi$
$431$	11.5368	0.555708	0.277854	+	0.960623i	$0.410377\pi$
$432$	0	0
$433$	13.2950	0.638916	0.319458	−	0.947600i	$-0.396499\pi$
$433$	13.2950	0.638916	0.319458	+	0.947600i	$0.396499\pi$
$434$	0	0
$435$	−2.87093	−0.137650
$436$	0	0
$437$	−6.90720	−0.330416
$438$	0	0
$439$	−11.6301	−0.555074	−0.277537	−	0.960715i	$-0.589518\pi$
$439$	−11.6301	−0.555074	−0.277537	+	0.960715i	$0.589518\pi$
$440$	0	0
$441$	2.58780	0.123228
$442$	0	0
$443$	−7.06488	−0.335663	−0.167831	−	0.985816i	$-0.553676\pi$
$443$	−7.06488	−0.335663	−0.167831	+	0.985816i	$0.553676\pi$
$444$	0	0
$445$	−6.36261	−0.301617
$446$	0	0
$447$	−0.624329	−0.0295298
$448$	0	0
$449$	18.8558	0.889862	0.444931	−	0.895565i	$-0.353228\pi$
$449$	18.8558	0.889862	0.444931	+	0.895565i	$0.353228\pi$
$450$	0	0
$451$	6.96864	0.328140
$452$	0	0
$453$	−1.06633	−0.0501005
$454$	0	0
$455$	−34.4833	−1.61660
$456$	0	0
$457$	−17.9351	−0.838970	−0.419485	−	0.907762i	$-0.637789\pi$
$457$	−17.9351	−0.838970	−0.419485	+	0.907762i	$0.637789\pi$
$458$	0	0
$459$	−2.00547	−0.0936075
$460$	0	0
$461$	−7.49715	−0.349177	−0.174589	−	0.984641i	$-0.555860\pi$
$461$	−7.49715	−0.349177	−0.174589	+	0.984641i	$0.555860\pi$
$462$	0	0
$463$	−36.7780	−1.70922	−0.854610	−	0.519270i	$-0.826204\pi$
$463$	−36.7780	−1.70922	−0.854610	+	0.519270i	$0.826204\pi$
$464$	0	0
$465$	2.19481	0.101782
$466$	0	0
$467$	−13.8029	−0.638724	−0.319362	−	0.947633i	$-0.603469\pi$
$467$	−13.8029	−0.638724	−0.319362	+	0.947633i	$0.603469\pi$
$468$	0	0
$469$	29.6820	1.37059
$470$	0	0
$471$	0.675593	0.0311297
$472$	0	0
$473$	29.4601	1.35458
$474$	0	0
$475$	−2.52554	−0.115880
$476$	0	0
$477$	13.8446	0.633900
$478$	0	0
$479$	−33.4312	−1.52751	−0.763755	−	0.645507i	$-0.776646\pi$
$479$	−33.4312	−1.52751	−0.763755	+	0.645507i	$0.776646\pi$
$480$	0	0
$481$	14.3659	0.655030
$482$	0	0
$483$	4.20725	0.191436
$484$	0	0
$485$	7.85047	0.356471
$486$	0	0
$487$	−22.8676	−1.03623	−0.518114	−	0.855311i	$-0.673366\pi$
$487$	−22.8676	−1.03623	−0.518114	+	0.855311i	$0.673366\pi$
$488$	0	0
$489$	−2.58666	−0.116973
$490$	0	0
$491$	38.7038	1.74668	0.873338	−	0.487114i	$-0.161951\pi$
$491$	38.7038	1.74668	0.873338	+	0.487114i	$0.161951\pi$
$492$	0	0
$493$	−5.82855	−0.262505
$494$	0	0
$495$	−46.1110	−2.07253
$496$	0	0
$497$	9.28119	0.416318
$498$	0	0
$499$	23.5822	1.05568	0.527841	−	0.849343i	$-0.323002\pi$
$499$	23.5822	1.05568	0.527841	+	0.849343i	$0.323002\pi$
$500$	0	0
$501$	3.71299	0.165884
$502$	0	0
$503$	28.0680	1.25149	0.625745	−	0.780028i	$-0.284795\pi$
$503$	28.0680	1.25149	0.625745	+	0.780028i	$0.284795\pi$
$504$	0	0
$505$	−22.9536	−1.02142
$506$	0	0
$507$	3.15662	0.140190
$508$	0	0
$509$	−4.31080	−0.191073	−0.0955365	−	0.995426i	$-0.530457\pi$
$509$	−4.31080	−0.191073	−0.0955365	+	0.995426i	$0.530457\pi$
$510$	0	0
$511$	24.8322	1.09851
$512$	0	0
$513$	1.46243	0.0645679
$514$	0	0
$515$	40.8752	1.80118
$516$	0	0
$517$	59.6919	2.62525
$518$	0	0
$519$	0.543329	0.0238495
$520$	0	0
$521$	4.30627	0.188661	0.0943305	−	0.995541i	$-0.469929\pi$
$521$	4.30627	0.188661	0.0943305	+	0.995541i	$0.469929\pi$
$522$	0	0
$523$	9.50025	0.415417	0.207708	−	0.978191i	$-0.433399\pi$
$523$	9.50025	0.415417	0.207708	+	0.978191i	$0.433399\pi$
$524$	0	0
$525$	1.53834	0.0671385
$526$	0	0
$527$	4.45589	0.194102
$528$	0	0
$529$	24.7094	1.07432
$530$	0	0
$531$	9.88569	0.429002
$532$	0	0
$533$	−6.19220	−0.268214
$534$	0	0
$535$	−43.9424	−1.89979
$536$	0	0
$537$	−5.66071	−0.244278
$538$	0	0
$539$	−5.03450	−0.216851
$540$	0	0
$541$	38.4354	1.65246	0.826232	−	0.563329i	$-0.190480\pi$
$541$	38.4354	1.65246	0.826232	+	0.563329i	$0.190480\pi$
$542$	0	0
$543$	−4.10493	−0.176159
$544$	0	0
$545$	2.82838	0.121154
$546$	0	0
$547$	8.36393	0.357616	0.178808	−	0.983884i	$-0.442776\pi$
$547$	8.36393	0.357616	0.178808	+	0.983884i	$0.442776\pi$
$548$	0	0
$549$	−10.0651	−0.429568
$550$	0	0
$551$	4.25029	0.181068
$552$	0	0
$553$	−2.47378	−0.105196
$554$	0	0
$555$	−1.90968	−0.0810612
$556$	0	0
$557$	−19.9001	−0.843194	−0.421597	−	0.906783i	$-0.638530\pi$
$557$	−19.9001	−0.843194	−0.421597	+	0.906783i	$0.638530\pi$
$558$	0	0
$559$	−26.1777	−1.10720
$560$	0	0
$561$	1.93089	0.0815222
$562$	0	0
$563$	−19.2779	−0.812467	−0.406234	−	0.913769i	$-0.633158\pi$
$563$	−19.2779	−0.812467	−0.406234	+	0.913769i	$0.633158\pi$
$564$	0	0
$565$	6.09053	0.256231
$566$	0	0
$567$	20.9233	0.878698
$568$	0	0
$569$	−19.9237	−0.835247	−0.417623	−	0.908620i	$-0.637137\pi$
$569$	−19.9237	−0.835247	−0.417623	+	0.908620i	$0.637137\pi$
$570$	0	0
$571$	5.80172	0.242795	0.121397	−	0.992604i	$-0.461263\pi$
$571$	5.80172	0.242795	0.121397	+	0.992604i	$0.461263\pi$
$572$	0	0
$573$	−5.43143	−0.226901
$574$	0	0
$575$	17.4444	0.727483
$576$	0	0
$577$	−34.6013	−1.44047	−0.720235	−	0.693730i	$-0.755966\pi$
$577$	−34.6013	−1.44047	−0.720235	+	0.693730i	$0.755966\pi$
$578$	0	0
$579$	−1.36251	−0.0566240
$580$	0	0
$581$	7.64414	0.317132
$582$	0	0
$583$	−26.9343	−1.11551
$584$	0	0
$585$	40.9733	1.69404
$586$	0	0
$587$	29.8621	1.23254	0.616270	−	0.787535i	$-0.288643\pi$
$587$	29.8621	1.23254	0.616270	+	0.787535i	$0.288643\pi$
$588$	0	0
$589$	−3.24932	−0.133886
$590$	0	0
$591$	0.829344	0.0341147
$592$	0	0
$593$	−10.5799	−0.434464	−0.217232	−	0.976120i	$-0.569703\pi$
$593$	−10.5799	−0.434464	−0.217232	+	0.976120i	$0.569703\pi$
$594$	0	0
$595$	9.30621	0.381517
$596$	0	0
$597$	4.39388	0.179830
$598$	0	0
$599$	−0.743018	−0.0303589	−0.0151794	−	0.999885i	$-0.504832\pi$
$599$	−0.743018	−0.0303589	−0.0151794	+	0.999885i	$0.504832\pi$
$600$	0	0
$601$	−24.6384	−1.00502	−0.502510	−	0.864571i	$-0.667590\pi$
$601$	−24.6384	−1.00502	−0.502510	+	0.864571i	$0.667590\pi$
$602$	0	0
$603$	−35.2684	−1.43624
$604$	0	0
$605$	59.5318	2.42031
$606$	0	0
$607$	−12.8413	−0.521212	−0.260606	−	0.965445i	$-0.583922\pi$
$607$	−12.8413	−0.521212	−0.260606	+	0.965445i	$0.583922\pi$
$608$	0	0
$609$	−2.58890	−0.104907
$610$	0	0
$611$	−53.0411	−2.14581
$612$	0	0
$613$	16.4828	0.665735	0.332867	−	0.942974i	$-0.391984\pi$
$613$	16.4828	0.665735	0.332867	+	0.942974i	$0.391984\pi$
$614$	0	0
$615$	0.823134	0.0331920
$616$	0	0
$617$	28.3098	1.13971	0.569854	−	0.821746i	$-0.307000\pi$
$617$	28.3098	1.13971	0.569854	+	0.821746i	$0.307000\pi$
$618$	0	0
$619$	8.89760	0.357625	0.178812	−	0.983883i	$-0.442774\pi$
$619$	8.89760	0.357625	0.178812	+	0.983883i	$0.442774\pi$
$620$	0	0
$621$	−10.1013	−0.405351
$622$	0	0
$623$	−5.73757	−0.229871
$624$	0	0
$625$	−31.2493	−1.24997
$626$	0	0
$627$	−1.40804	−0.0562317
$628$	0	0
$629$	−3.87702	−0.154587
$630$	0	0
$631$	11.2168	0.446535	0.223267	−	0.974757i	$-0.428328\pi$
$631$	11.2168	0.446535	0.223267	+	0.974757i	$0.428328\pi$
$632$	0	0
$633$	−0.780290	−0.0310137
$634$	0	0
$635$	42.2489	1.67660
$636$	0	0
$637$	4.47356	0.177249
$638$	0	0
$639$	−11.0280	−0.436261
$640$	0	0
$641$	22.1904	0.876468	0.438234	−	0.898861i	$-0.355604\pi$
$641$	22.1904	0.876468	0.438234	+	0.898861i	$0.355604\pi$
$642$	0	0
$643$	13.3608	0.526900	0.263450	−	0.964673i	$-0.415140\pi$
$643$	13.3608	0.526900	0.263450	+	0.964673i	$0.415140\pi$
$644$	0	0
$645$	3.47982	0.137018
$646$	0	0
$647$	−42.8997	−1.68656	−0.843280	−	0.537474i	$-0.819379\pi$
$647$	−42.8997	−1.68656	−0.843280	+	0.537474i	$0.819379\pi$
$648$	0	0
$649$	−19.2324	−0.754937
$650$	0	0
$651$	1.97920	0.0775708
$652$	0	0
$653$	−43.2378	−1.69203	−0.846013	−	0.533163i	$-0.821003\pi$
$653$	−43.2378	−1.69203	−0.846013	+	0.533163i	$0.821003\pi$
$654$	0	0
$655$	34.5437	1.34973
$656$	0	0
$657$	−29.5058	−1.15113
$658$	0	0
$659$	44.2009	1.72182	0.860912	−	0.508754i	$-0.169894\pi$
$659$	44.2009	1.72182	0.860912	+	0.508754i	$0.169894\pi$
$660$	0	0
$661$	−20.3779	−0.792609	−0.396304	−	0.918119i	$-0.629707\pi$
$661$	−20.3779	−0.792609	−0.396304	+	0.918119i	$0.629707\pi$
$662$	0	0
$663$	−1.71575	−0.0666343
$664$	0	0
$665$	−6.78626	−0.263160
$666$	0	0
$667$	−29.3576	−1.13673
$668$	0	0
$669$	−4.01870	−0.155372
$670$	0	0
$671$	19.5814	0.755932
$672$	0	0
$673$	−6.08123	−0.234414	−0.117207	−	0.993107i	$-0.537394\pi$
$673$	−6.08123	−0.234414	−0.117207	+	0.993107i	$0.537394\pi$
$674$	0	0
$675$	−3.69343	−0.142160
$676$	0	0
$677$	38.2478	1.46998	0.734991	−	0.678077i	$-0.237186\pi$
$677$	38.2478	1.46998	0.734991	+	0.678077i	$0.237186\pi$
$678$	0	0
$679$	7.07927	0.271677
$680$	0	0
$681$	4.51515	0.173021
$682$	0	0
$683$	47.5064	1.81778	0.908891	−	0.417033i	$-0.136930\pi$
$683$	47.5064	1.81778	0.908891	+	0.417033i	$0.136930\pi$
$684$	0	0
$685$	6.07897	0.232266
$686$	0	0
$687$	−1.50032	−0.0572406
$688$	0	0
$689$	23.9334	0.911788
$690$	0	0
$691$	−32.0723	−1.22009	−0.610044	−	0.792367i	$-0.708848\pi$
$691$	−32.0723	−1.22009	−0.610044	+	0.792367i	$0.708848\pi$
$692$	0	0
$693$	−41.5812	−1.57954
$694$	0	0
$695$	3.81436	0.144687
$696$	0	0
$697$	1.67113	0.0632984
$698$	0	0
$699$	−4.39406	−0.166199
$700$	0	0
$701$	45.4765	1.71762	0.858812	−	0.512291i	$-0.171203\pi$
$701$	45.4765	1.71762	0.858812	+	0.512291i	$0.171203\pi$
$702$	0	0
$703$	2.82720	0.106630
$704$	0	0
$705$	7.05080	0.265549
$706$	0	0
$707$	−20.6987	−0.778456
$708$	0	0
$709$	−18.5976	−0.698448	−0.349224	−	0.937039i	$-0.613555\pi$
$709$	−18.5976	−0.698448	−0.349224	+	0.937039i	$0.613555\pi$
$710$	0	0
$711$	2.93937	0.110235
$712$	0	0
$713$	22.4437	0.840523
$714$	0	0
$715$	−79.7127	−2.98109
$716$	0	0
$717$	4.03918	0.150846
$718$	0	0
$719$	−37.2824	−1.39040	−0.695200	−	0.718816i	$-0.744684\pi$
$719$	−37.2824	−1.39040	−0.695200	+	0.718816i	$0.744684\pi$
$720$	0	0
$721$	36.8598	1.37273
$722$	0	0
$723$	3.58548	0.133345
$724$	0	0
$725$	−10.7343	−0.398661
$726$	0	0
$727$	−23.3865	−0.867357	−0.433679	−	0.901068i	$-0.642785\pi$
$727$	−23.3865	−0.867357	−0.433679	+	0.901068i	$0.642785\pi$
$728$	0	0
$729$	−23.7810	−0.880779
$730$	0	0
$731$	7.06472	0.261298
$732$	0	0
$733$	−40.3911	−1.49188	−0.745939	−	0.666014i	$-0.767999\pi$
$733$	−40.3911	−1.49188	−0.745939	+	0.666014i	$0.767999\pi$
$734$	0	0
$735$	−0.594674	−0.0219349
$736$	0	0
$737$	68.6139	2.52743
$738$	0	0
$739$	−5.03810	−0.185330	−0.0926648	−	0.995697i	$-0.529538\pi$
$739$	−5.03810	−0.185330	−0.0926648	+	0.995697i	$0.529538\pi$
$740$	0	0
$741$	1.25116	0.0459625
$742$	0	0
$743$	1.50227	0.0551129	0.0275564	−	0.999620i	$-0.491227\pi$
$743$	1.50227	0.0551129	0.0275564	+	0.999620i	$0.491227\pi$
$744$	0	0
$745$	−6.95581	−0.254841
$746$	0	0
$747$	−9.08283	−0.332323
$748$	0	0
$749$	−39.6256	−1.44789
$750$	0	0
$751$	−30.8720	−1.12653	−0.563267	−	0.826275i	$-0.690456\pi$
$751$	−30.8720	−1.12653	−0.563267	+	0.826275i	$0.690456\pi$
$752$	0	0
$753$	−5.02818	−0.183237
$754$	0	0
$755$	−11.8803	−0.432367
$756$	0	0
$757$	50.9868	1.85315	0.926574	−	0.376113i	$-0.122740\pi$
$757$	50.9868	1.85315	0.926574	+	0.376113i	$0.122740\pi$
$758$	0	0
$759$	9.72562	0.353018
$760$	0	0
$761$	−15.1304	−0.548478	−0.274239	−	0.961662i	$-0.588426\pi$
$761$	−15.1304	−0.548478	−0.274239	+	0.961662i	$0.588426\pi$
$762$	0	0
$763$	2.55053	0.0923354
$764$	0	0
$765$	−11.0577	−0.399793
$766$	0	0
$767$	17.0895	0.617067
$768$	0	0
$769$	19.4007	0.699609	0.349804	−	0.936823i	$-0.386248\pi$
$769$	19.4007	0.699609	0.349804	+	0.936823i	$0.386248\pi$
$770$	0	0
$771$	−1.03743	−0.0373622
$772$	0	0
$773$	−29.7206	−1.06898	−0.534488	−	0.845176i	$-0.679496\pi$
$773$	−29.7206	−1.06898	−0.534488	+	0.845176i	$0.679496\pi$
$774$	0	0
$775$	8.20630	0.294779
$776$	0	0
$777$	−1.72208	−0.0617792
$778$	0	0
$779$	−1.21862	−0.0436615
$780$	0	0
$781$	21.4547	0.767710
$782$	0	0
$783$	6.21575	0.222133
$784$	0	0
$785$	7.52696	0.268649
$786$	0	0
$787$	47.9506	1.70926	0.854628	−	0.519242i	$-0.173786\pi$
$787$	47.9506	1.70926	0.854628	+	0.519242i	$0.173786\pi$
$788$	0	0
$789$	0.417929	0.0148787
$790$	0	0
$791$	5.49223	0.195281
$792$	0	0
$793$	−17.3997	−0.617880
$794$	0	0
$795$	−3.18148	−0.112835
$796$	0	0
$797$	−29.3490	−1.03959	−0.519797	−	0.854290i	$-0.673992\pi$
$797$	−29.3490	−1.03959	−0.519797	+	0.854290i	$0.673992\pi$
$798$	0	0
$799$	14.3145	0.506411
$800$	0	0
$801$	6.81744	0.240882
$802$	0	0
$803$	57.4029	2.02570
$804$	0	0
$805$	46.8741	1.65209
$806$	0	0
$807$	3.55975	0.125309
$808$	0	0
$809$	−15.4098	−0.541780	−0.270890	−	0.962610i	$-0.587318\pi$
$809$	−15.4098	−0.541780	−0.270890	+	0.962610i	$0.587318\pi$
$810$	0	0
$811$	8.00411	0.281062	0.140531	−	0.990076i	$-0.455119\pi$
$811$	8.00411	0.281062	0.140531	+	0.990076i	$0.455119\pi$
$812$	0	0
$813$	0.300188	0.0105280
$814$	0	0
$815$	−28.8186	−1.00947
$816$	0	0
$817$	−5.15173	−0.180236
$818$	0	0
$819$	36.9483	1.29108
$820$	0	0
$821$	3.08837	0.107785	0.0538923	−	0.998547i	$-0.482837\pi$
$821$	3.08837	0.107785	0.0538923	+	0.998547i	$0.482837\pi$
$822$	0	0
$823$	−26.2977	−0.916681	−0.458341	−	0.888777i	$-0.651556\pi$
$823$	−26.2977	−0.916681	−0.458341	+	0.888777i	$0.651556\pi$
$824$	0	0
$825$	3.55607	0.123806
$826$	0	0
$827$	35.6709	1.24040	0.620200	−	0.784444i	$-0.287051\pi$
$827$	35.6709	1.24040	0.620200	+	0.784444i	$0.287051\pi$
$828$	0	0
$829$	−10.1162	−0.351350	−0.175675	−	0.984448i	$-0.556211\pi$
$829$	−10.1162	−0.351350	−0.175675	+	0.984448i	$0.556211\pi$
$830$	0	0
$831$	−4.43431	−0.153825
$832$	0	0
$833$	−1.20731	−0.0418307
$834$	0	0
$835$	41.3673	1.43158
$836$	0	0
$837$	−4.75190	−0.164250
$838$	0	0
$839$	1.87591	0.0647635	0.0323817	−	0.999476i	$-0.489691\pi$
$839$	1.87591	0.0647635	0.0323817	+	0.999476i	$0.489691\pi$
$840$	0	0
$841$	−10.9351	−0.377071
$842$	0	0
$843$	−4.39348	−0.151319
$844$	0	0
$845$	35.1687	1.20984
$846$	0	0
$847$	53.6837	1.84459
$848$	0	0
$849$	2.55766	0.0877785
$850$	0	0
$851$	−19.5280	−0.669411
$852$	0	0
$853$	29.5938	1.01327	0.506636	−	0.862160i	$-0.330889\pi$
$853$	29.5938	1.01327	0.506636	+	0.862160i	$0.330889\pi$
$854$	0	0
$855$	8.06350	0.275766
$856$	0	0
$857$	−38.3202	−1.30899	−0.654496	−	0.756066i	$-0.727119\pi$
$857$	−38.3202	−1.30899	−0.654496	+	0.756066i	$0.727119\pi$
$858$	0	0
$859$	−34.4679	−1.17603	−0.588014	−	0.808851i	$-0.700090\pi$
$859$	−34.4679	−1.17603	−0.588014	+	0.808851i	$0.700090\pi$
$860$	0	0
$861$	0.742273	0.0252966
$862$	0	0
$863$	−27.1224	−0.923256	−0.461628	−	0.887074i	$-0.652735\pi$
$863$	−27.1224	−0.923256	−0.461628	+	0.887074i	$0.652735\pi$
$864$	0	0
$865$	6.05337	0.205821
$866$	0	0
$867$	−3.72281	−0.126433
$868$	0	0
$869$	−5.71848	−0.193986
$870$	0	0
$871$	−60.9690	−2.06586
$872$	0	0
$873$	−8.41165	−0.284691
$874$	0	0
$875$	−16.7923	−0.567684
$876$	0	0
$877$	9.29726	0.313946	0.156973	−	0.987603i	$-0.449826\pi$
$877$	9.29726	0.313946	0.156973	+	0.987603i	$0.449826\pi$
$878$	0	0
$879$	−7.40445	−0.249746
$880$	0	0
$881$	−8.82921	−0.297464	−0.148732	−	0.988878i	$-0.547519\pi$
$881$	−8.82921	−0.297464	−0.148732	+	0.988878i	$0.547519\pi$
$882$	0	0
$883$	14.3319	0.482305	0.241153	−	0.970487i	$-0.422475\pi$
$883$	14.3319	0.482305	0.241153	+	0.970487i	$0.422475\pi$
$884$	0	0
$885$	−2.27172	−0.0763632
$886$	0	0
$887$	14.9481	0.501909	0.250954	−	0.967999i	$-0.419256\pi$
$887$	14.9481	0.501909	0.250954	+	0.967999i	$0.419256\pi$
$888$	0	0
$889$	38.0986	1.27778
$890$	0	0
$891$	48.3671	1.62036
$892$	0	0
$893$	−10.4384	−0.349309
$894$	0	0
$895$	−63.0674	−2.10811
$896$	0	0
$897$	−8.64200	−0.288548
$898$	0	0
$899$	−13.8105	−0.460607
$900$	0	0
$901$	−6.45903	−0.215182
$902$	0	0
$903$	3.13797	0.104425
$904$	0	0
$905$	−45.7340	−1.52025
$906$	0	0
$907$	−46.6026	−1.54742	−0.773708	−	0.633543i	$-0.781600\pi$
$907$	−46.6026	−1.54742	−0.773708	+	0.633543i	$0.781600\pi$
$908$	0	0
$909$	24.5944	0.815746
$910$	0	0
$911$	41.0986	1.36166	0.680828	−	0.732443i	$-0.261620\pi$
$911$	41.0986	1.36166	0.680828	+	0.732443i	$0.261620\pi$
$912$	0	0
$913$	17.6704	0.584806
$914$	0	0
$915$	2.31295	0.0764638
$916$	0	0
$917$	31.1502	1.02867
$918$	0	0
$919$	−20.5025	−0.676315	−0.338158	−	0.941089i	$-0.609804\pi$
$919$	−20.5025	−0.676315	−0.338158	+	0.941089i	$0.609804\pi$
$920$	0	0
$921$	−0.941054	−0.0310088
$922$	0	0
$923$	−19.0643	−0.627508
$924$	0	0
$925$	−7.14021	−0.234769
$926$	0	0
$927$	−43.7972	−1.43849
$928$	0	0
$929$	−0.828985	−0.0271981	−0.0135991	−	0.999908i	$-0.504329\pi$
$929$	−0.828985	−0.0271981	−0.0135991	+	0.999908i	$0.504329\pi$
$930$	0	0
$931$	0.880391	0.0288536
$932$	0	0
$933$	4.81404	0.157605
$934$	0	0
$935$	21.5125	0.703535
$936$	0	0
$937$	−22.1192	−0.722604	−0.361302	−	0.932449i	$-0.617668\pi$
$937$	−22.1192	−0.722604	−0.361302	+	0.932449i	$0.617668\pi$
$938$	0	0
$939$	−6.69803	−0.218582
$940$	0	0
$941$	−17.5624	−0.572518	−0.286259	−	0.958152i	$-0.592412\pi$
$941$	−17.5624	−0.572518	−0.286259	+	0.958152i	$0.592412\pi$
$942$	0	0
$943$	8.41722	0.274102
$944$	0	0
$945$	−9.92444	−0.322842
$946$	0	0
$947$	−18.4064	−0.598127	−0.299064	−	0.954233i	$-0.596674\pi$
$947$	−18.4064	−0.598127	−0.299064	+	0.954233i	$0.596674\pi$
$948$	0	0
$949$	−51.0071	−1.65576
$950$	0	0
$951$	2.36987	0.0768483
$952$	0	0
$953$	−28.0503	−0.908640	−0.454320	−	0.890839i	$-0.650118\pi$
$953$	−28.0503	−0.908640	−0.454320	+	0.890839i	$0.650118\pi$
$954$	0	0
$955$	−60.5129	−1.95815
$956$	0	0
$957$	−5.98458	−0.193454
$958$	0	0
$959$	5.48180	0.177017
$960$	0	0
$961$	−20.4419	−0.659417
$962$	0	0
$963$	47.0835	1.51725
$964$	0	0
$965$	−15.1801	−0.488664
$966$	0	0
$967$	7.84642	0.252324	0.126162	−	0.992010i	$-0.459734\pi$
$967$	7.84642	0.252324	0.126162	+	0.992010i	$0.459734\pi$
$968$	0	0
$969$	−0.337658	−0.0108471
$970$	0	0
$971$	3.33468	0.107015	0.0535075	−	0.998567i	$-0.482960\pi$
$971$	3.33468	0.107015	0.0535075	+	0.998567i	$0.482960\pi$
$972$	0	0
$973$	3.43965	0.110270
$974$	0	0
$975$	−3.15986	−0.101196
$976$	0	0
$977$	−11.8553	−0.379286	−0.189643	−	0.981853i	$-0.560733\pi$
$977$	−11.8553	−0.379286	−0.189643	+	0.981853i	$0.560733\pi$
$978$	0	0
$979$	−13.2632	−0.423893
$980$	0	0
$981$	−3.03056	−0.0967584
$982$	0	0
$983$	−19.3584	−0.617438	−0.308719	−	0.951153i	$-0.599900\pi$
$983$	−19.3584	−0.617438	−0.308719	+	0.951153i	$0.599900\pi$
$984$	0	0
$985$	9.23994	0.294409
$986$	0	0
$987$	6.35816	0.202382
$988$	0	0
$989$	35.5840	1.13151
$990$	0	0
$991$	23.3742	0.742506	0.371253	−	0.928532i	$-0.378928\pi$
$991$	23.3742	0.742506	0.371253	+	0.928532i	$0.378928\pi$
$992$	0	0
$993$	−1.57776	−0.0500688
$994$	0	0
$995$	48.9534	1.55193
$996$	0	0
$997$	−5.29543	−0.167708	−0.0838539	−	0.996478i	$-0.526723\pi$
$997$	−5.29543	−0.167708	−0.0838539	+	0.996478i	$0.526723\pi$
$998$	0	0
$999$	4.13458	0.130812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.17	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.17	✓	31	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)

\(f(q)\)	\(=\)	\(q+0.246226 q^{3} +2.74327 q^{5} +2.47378 q^{7} -2.93937 q^{9} +O(q^{10})\)	\(q+0.246226 q^{3} +2.74327 q^{5} +2.47378 q^{7} -2.93937 q^{9} +5.71848 q^{11} -5.08134 q^{13} +0.675466 q^{15} +1.37133 q^{17} -1.00000 q^{19} +0.609111 q^{21} +6.90720 q^{23} +2.52554 q^{25} -1.46243 q^{27} -4.25029 q^{29} +3.24932 q^{31} +1.40804 q^{33} +6.78626 q^{35} -2.82720 q^{37} -1.25116 q^{39} +1.21862 q^{41} +5.15173 q^{43} -8.06350 q^{45} +10.4384 q^{47} -0.880391 q^{49} +0.337658 q^{51} -4.71005 q^{53} +15.6874 q^{55} -0.246226 q^{57} -3.36320 q^{59} +3.42423 q^{61} -7.27137 q^{63} -13.9395 q^{65} +11.9986 q^{67} +1.70073 q^{69} +3.75182 q^{71} +10.0381 q^{73} +0.621856 q^{75} +14.1463 q^{77} -1.00000 q^{79} +8.45803 q^{81} +3.09006 q^{83} +3.76193 q^{85} -1.04653 q^{87} -2.31935 q^{89} -12.5701 q^{91} +0.800068 q^{93} -2.74327 q^{95} +2.86172 q^{97} -16.8087 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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