LMFDB - Embedded newform 6004.2.a.e.1.11

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:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-1.34737 q^{3} -0.232989 q^{5} +3.14804 q^{7} -1.18461 q^{9} +O(q^{10})$	$q-1.34737 q^{3} -0.232989 q^{5} +3.14804 q^{7} -1.18461 q^{9} -0.322722 q^{11} -1.02856 q^{13} +0.313922 q^{15} +3.42192 q^{17} +1.00000 q^{19} -4.24156 q^{21} +4.22435 q^{23} -4.94572 q^{25} +5.63819 q^{27} -5.73494 q^{29} +10.0444 q^{31} +0.434824 q^{33} -0.733459 q^{35} +0.754270 q^{37} +1.38585 q^{39} -3.84593 q^{41} -1.66759 q^{43} +0.276000 q^{45} +2.33683 q^{47} +2.91014 q^{49} -4.61057 q^{51} +2.39261 q^{53} +0.0751907 q^{55} -1.34737 q^{57} -5.15705 q^{59} -10.5404 q^{61} -3.72918 q^{63} +0.239643 q^{65} -8.64855 q^{67} -5.69174 q^{69} +7.12861 q^{71} +8.06209 q^{73} +6.66369 q^{75} -1.01594 q^{77} +1.00000 q^{79} -4.04289 q^{81} +12.5580 q^{83} -0.797270 q^{85} +7.72707 q^{87} +17.7037 q^{89} -3.23794 q^{91} -13.5335 q^{93} -0.232989 q^{95} +14.5916 q^{97} +0.382298 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * 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
$3$	−1.34737	−0.777902	−0.388951	−	0.921258i	$-0.627162\pi$
$3$	−1.34737	−0.777902	−0.388951	+	0.921258i	$0.627162\pi$
$4$	0	0
$5$	−0.232989	−0.104196	−0.0520980	−	0.998642i	$-0.516591\pi$
$5$	−0.232989	−0.104196	−0.0520980	+	0.998642i	$0.516591\pi$
$6$	0	0
$7$	3.14804	1.18985	0.594923	−	0.803782i	$-0.297182\pi$
$7$	3.14804	1.18985	0.594923	+	0.803782i	$0.297182\pi$
$8$	0	0
$9$	−1.18461	−0.394869
$10$	0	0
$11$	−0.322722	−0.0973043	−0.0486521	−	0.998816i	$-0.515493\pi$
$11$	−0.322722	−0.0973043	−0.0486521	+	0.998816i	$0.515493\pi$
$12$	0	0
$13$	−1.02856	−0.285271	−0.142635	−	0.989775i	$-0.545558\pi$
$13$	−1.02856	−0.285271	−0.142635	+	0.989775i	$0.545558\pi$
$14$	0	0
$15$	0.313922	0.0810542
$16$	0	0
$17$	3.42192	0.829937	0.414969	−	0.909836i	$-0.363793\pi$
$17$	3.42192	0.829937	0.414969	+	0.909836i	$0.363793\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.24156	−0.925584
$22$	0	0
$23$	4.22435	0.880838	0.440419	−	0.897792i	$-0.354830\pi$
$23$	4.22435	0.880838	0.440419	+	0.897792i	$0.354830\pi$
$24$	0	0
$25$	−4.94572	−0.989143
$26$	0	0
$27$	5.63819	1.08507
$28$	0	0
$29$	−5.73494	−1.06495	−0.532476	−	0.846445i	$-0.678738\pi$
$29$	−5.73494	−1.06495	−0.532476	+	0.846445i	$0.678738\pi$
$30$	0	0
$31$	10.0444	1.80403	0.902015	−	0.431705i	$-0.142088\pi$
$31$	10.0444	1.80403	0.902015	+	0.431705i	$0.142088\pi$
$32$	0	0
$33$	0.434824	0.0756932
$34$	0	0
$35$	−0.733459	−0.123977
$36$	0	0
$37$	0.754270	0.124001	0.0620006	−	0.998076i	$-0.480252\pi$
$37$	0.754270	0.124001	0.0620006	+	0.998076i	$0.480252\pi$
$38$	0	0
$39$	1.38585	0.221913
$40$	0	0
$41$	−3.84593	−0.600633	−0.300317	−	0.953840i	$-0.597092\pi$
$41$	−3.84593	−0.600633	−0.300317	+	0.953840i	$0.597092\pi$
$42$	0	0
$43$	−1.66759	−0.254306	−0.127153	−	0.991883i	$-0.540584\pi$
$43$	−1.66759	−0.254306	−0.127153	+	0.991883i	$0.540584\pi$
$44$	0	0
$45$	0.276000	0.0411437
$46$	0	0
$47$	2.33683	0.340861	0.170431	−	0.985370i	$-0.445484\pi$
$47$	2.33683	0.340861	0.170431	+	0.985370i	$0.445484\pi$
$48$	0	0
$49$	2.91014	0.415735
$50$	0	0
$51$	−4.61057	−0.645610
$52$	0	0
$53$	2.39261	0.328650	0.164325	−	0.986406i	$-0.447455\pi$
$53$	2.39261	0.328650	0.164325	+	0.986406i	$0.447455\pi$
$54$	0	0
$55$	0.0751907	0.0101387
$56$	0	0
$57$	−1.34737	−0.178463
$58$	0	0
$59$	−5.15705	−0.671391	−0.335695	−	0.941971i	$-0.608971\pi$
$59$	−5.15705	−0.671391	−0.335695	+	0.941971i	$0.608971\pi$
$60$	0	0
$61$	−10.5404	−1.34956	−0.674780	−	0.738019i	$-0.735762\pi$
$61$	−10.5404	−1.34956	−0.674780	+	0.738019i	$0.735762\pi$
$62$	0	0
$63$	−3.72918	−0.469833
$64$	0	0
$65$	0.239643	0.0297241
$66$	0	0
$67$	−8.64855	−1.05659	−0.528294	−	0.849061i	$-0.677168\pi$
$67$	−8.64855	−1.05659	−0.528294	+	0.849061i	$0.677168\pi$
$68$	0	0
$69$	−5.69174	−0.685205
$70$	0	0
$71$	7.12861	0.846010	0.423005	−	0.906127i	$-0.360975\pi$
$71$	7.12861	0.846010	0.423005	+	0.906127i	$0.360975\pi$
$72$	0	0
$73$	8.06209	0.943596	0.471798	−	0.881707i	$-0.343605\pi$
$73$	8.06209	0.943596	0.471798	+	0.881707i	$0.343605\pi$
$74$	0	0
$75$	6.66369	0.769456
$76$	0	0
$77$	−1.01594	−0.115777
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−4.04289	−0.449210
$82$	0	0
$83$	12.5580	1.37842	0.689210	−	0.724561i	$-0.257958\pi$
$83$	12.5580	1.37842	0.689210	+	0.724561i	$0.257958\pi$
$84$	0	0
$85$	−0.797270	−0.0864761
$86$	0	0
$87$	7.72707	0.828429
$88$	0	0
$89$	17.7037	1.87659	0.938294	−	0.345839i	$-0.112406\pi$
$89$	17.7037	1.87659	0.938294	+	0.345839i	$0.112406\pi$
$90$	0	0
$91$	−3.23794	−0.339429
$92$	0	0
$93$	−13.5335	−1.40336
$94$	0	0
$95$	−0.232989	−0.0239042
$96$	0	0
$97$	14.5916	1.48155	0.740774	−	0.671754i	$-0.234459\pi$
$97$	14.5916	1.48155	0.740774	+	0.671754i	$0.234459\pi$
$98$	0	0
$99$	0.382298	0.0384224
$100$	0	0
$101$	1.19894	0.119299	0.0596496	−	0.998219i	$-0.481002\pi$
$101$	1.19894	0.119299	0.0596496	+	0.998219i	$0.481002\pi$
$102$	0	0
$103$	8.50447	0.837970	0.418985	−	0.907993i	$-0.362386\pi$
$103$	8.50447	0.837970	0.418985	+	0.907993i	$0.362386\pi$
$104$	0	0
$105$	0.988237	0.0964421
$106$	0	0
$107$	−16.2959	−1.57538	−0.787691	−	0.616070i	$-0.788724\pi$
$107$	−16.2959	−1.57538	−0.787691	+	0.616070i	$0.788724\pi$
$108$	0	0
$109$	5.10072	0.488560	0.244280	−	0.969705i	$-0.421448\pi$
$109$	5.10072	0.488560	0.244280	+	0.969705i	$0.421448\pi$
$110$	0	0
$111$	−1.01628	−0.0964608
$112$	0	0
$113$	8.03022	0.755420	0.377710	−	0.925924i	$-0.376712\pi$
$113$	8.03022	0.755420	0.377710	+	0.925924i	$0.376712\pi$
$114$	0	0
$115$	−0.984228	−0.0917797
$116$	0	0
$117$	1.21844	0.112645
$118$	0	0
$119$	10.7723	0.987498
$120$	0	0
$121$	−10.8959	−0.990532
$122$	0	0
$123$	5.18187	0.467234
$124$	0	0
$125$	2.31724	0.207261
$126$	0	0
$127$	−19.2264	−1.70607	−0.853033	−	0.521858i	$-0.825239\pi$
$127$	−19.2264	−1.70607	−0.853033	+	0.521858i	$0.825239\pi$
$128$	0	0
$129$	2.24686	0.197825
$130$	0	0
$131$	6.27228	0.548012	0.274006	−	0.961728i	$-0.411651\pi$
$131$	6.27228	0.548012	0.274006	+	0.961728i	$0.411651\pi$
$132$	0	0
$133$	3.14804	0.272970
$134$	0	0
$135$	−1.31364	−0.113060
$136$	0	0
$137$	2.94770	0.251839	0.125919	−	0.992040i	$-0.459812\pi$
$137$	2.94770	0.251839	0.125919	+	0.992040i	$0.459812\pi$
$138$	0	0
$139$	−11.9459	−1.01324	−0.506619	−	0.862170i	$-0.669105\pi$
$139$	−11.9459	−1.01324	−0.506619	+	0.862170i	$0.669105\pi$
$140$	0	0
$141$	−3.14856	−0.265157
$142$	0	0
$143$	0.331938	0.0277581
$144$	0	0
$145$	1.33618	0.110964
$146$	0	0
$147$	−3.92103	−0.323401
$148$	0	0
$149$	−19.3263	−1.58327	−0.791636	−	0.610993i	$-0.790770\pi$
$149$	−19.3263	−1.58327	−0.791636	+	0.610993i	$0.790770\pi$
$150$	0	0
$151$	8.50303	0.691967	0.345983	−	0.938241i	$-0.387545\pi$
$151$	8.50303	0.691967	0.345983	+	0.938241i	$0.387545\pi$
$152$	0	0
$153$	−4.05362	−0.327716
$154$	0	0
$155$	−2.34024	−0.187973
$156$	0	0
$157$	3.30739	0.263958	0.131979	−	0.991252i	$-0.457867\pi$
$157$	3.30739	0.263958	0.131979	+	0.991252i	$0.457867\pi$
$158$	0	0
$159$	−3.22372	−0.255657
$160$	0	0
$161$	13.2984	1.04806
$162$	0	0
$163$	4.70004	0.368136	0.184068	−	0.982914i	$-0.441073\pi$
$163$	4.70004	0.368136	0.184068	+	0.982914i	$0.441073\pi$
$164$	0	0
$165$	−0.101309	−0.00788692
$166$	0	0
$167$	−8.08862	−0.625916	−0.312958	−	0.949767i	$-0.601320\pi$
$167$	−8.08862	−0.625916	−0.312958	+	0.949767i	$0.601320\pi$
$168$	0	0
$169$	−11.9421	−0.918620
$170$	0	0
$171$	−1.18461	−0.0905891
$172$	0	0
$173$	10.4338	0.793270	0.396635	−	0.917976i	$-0.370178\pi$
$173$	10.4338	0.793270	0.396635	+	0.917976i	$0.370178\pi$
$174$	0	0
$175$	−15.5693	−1.17693
$176$	0	0
$177$	6.94843	0.522276
$178$	0	0
$179$	7.26376	0.542919	0.271459	−	0.962450i	$-0.412494\pi$
$179$	7.26376	0.542919	0.271459	+	0.962450i	$0.412494\pi$
$180$	0	0
$181$	13.7936	1.02527	0.512635	−	0.858607i	$-0.328669\pi$
$181$	13.7936	1.02527	0.512635	+	0.858607i	$0.328669\pi$
$182$	0	0
$183$	14.2018	1.04983
$184$	0	0
$185$	−0.175737	−0.0129204
$186$	0	0
$187$	−1.10433	−0.0807564
$188$	0	0
$189$	17.7492	1.29107
$190$	0	0
$191$	−15.1331	−1.09499	−0.547496	−	0.836808i	$-0.684419\pi$
$191$	−15.1331	−1.09499	−0.547496	+	0.836808i	$0.684419\pi$
$192$	0	0
$193$	0.933077	0.0671644	0.0335822	−	0.999436i	$-0.489308\pi$
$193$	0.933077	0.0671644	0.0335822	+	0.999436i	$0.489308\pi$
$194$	0	0
$195$	−0.322887	−0.0231224
$196$	0	0
$197$	13.0152	0.927291	0.463646	−	0.886021i	$-0.346541\pi$
$197$	13.0152	0.927291	0.463646	+	0.886021i	$0.346541\pi$
$198$	0	0
$199$	−12.3945	−0.878622	−0.439311	−	0.898335i	$-0.644777\pi$
$199$	−12.3945	−0.878622	−0.439311	+	0.898335i	$0.644777\pi$
$200$	0	0
$201$	11.6528	0.821922
$202$	0	0
$203$	−18.0538	−1.26713
$204$	0	0
$205$	0.896060	0.0625835
$206$	0	0
$207$	−5.00419	−0.347815
$208$	0	0
$209$	−0.322722	−0.0223231
$210$	0	0
$211$	8.12144	0.559103	0.279552	−	0.960131i	$-0.409814\pi$
$211$	8.12144	0.559103	0.279552	+	0.960131i	$0.409814\pi$
$212$	0	0
$213$	−9.60484	−0.658113
$214$	0	0
$215$	0.388531	0.0264976
$216$	0	0
$217$	31.6202	2.14652
$218$	0	0
$219$	−10.8626	−0.734025
$220$	0	0
$221$	−3.51964	−0.236757
$222$	0	0
$223$	8.01183	0.536512	0.268256	−	0.963348i	$-0.413553\pi$
$223$	8.01183	0.536512	0.268256	+	0.963348i	$0.413553\pi$
$224$	0	0
$225$	5.85872	0.390582
$226$	0	0
$227$	21.9033	1.45377	0.726887	−	0.686757i	$-0.240966\pi$
$227$	21.9033	1.45377	0.726887	+	0.686757i	$0.240966\pi$
$228$	0	0
$229$	−2.28773	−0.151177	−0.0755886	−	0.997139i	$-0.524084\pi$
$229$	−2.28773	−0.151177	−0.0755886	+	0.997139i	$0.524084\pi$
$230$	0	0
$231$	1.36884	0.0900633
$232$	0	0
$233$	16.8849	1.10617	0.553084	−	0.833125i	$-0.313451\pi$
$233$	16.8849	1.10617	0.553084	+	0.833125i	$0.313451\pi$
$234$	0	0
$235$	−0.544456	−0.0355164
$236$	0	0
$237$	−1.34737	−0.0875208
$238$	0	0
$239$	−6.25724	−0.404747	−0.202374	−	0.979308i	$-0.564866\pi$
$239$	−6.25724	−0.404747	−0.202374	+	0.979308i	$0.564866\pi$
$240$	0	0
$241$	−5.65868	−0.364508	−0.182254	−	0.983252i	$-0.558339\pi$
$241$	−5.65868	−0.364508	−0.182254	+	0.983252i	$0.558339\pi$
$242$	0	0
$243$	−11.4673	−0.735630
$244$	0	0
$245$	−0.678032	−0.0433179
$246$	0	0
$247$	−1.02856	−0.0654456
$248$	0	0
$249$	−16.9202	−1.07228
$250$	0	0
$251$	30.0337	1.89571	0.947855	−	0.318701i	$-0.103246\pi$
$251$	30.0337	1.89571	0.947855	+	0.318701i	$0.103246\pi$
$252$	0	0
$253$	−1.36329	−0.0857093
$254$	0	0
$255$	1.07421	0.0672699
$256$	0	0
$257$	5.12370	0.319608	0.159804	−	0.987149i	$-0.448914\pi$
$257$	5.12370	0.319608	0.159804	+	0.987149i	$0.448914\pi$
$258$	0	0
$259$	2.37447	0.147543
$260$	0	0
$261$	6.79365	0.420516
$262$	0	0
$263$	15.7623	0.971948	0.485974	−	0.873973i	$-0.338465\pi$
$263$	15.7623	0.971948	0.485974	+	0.873973i	$0.338465\pi$
$264$	0	0
$265$	−0.557452	−0.0342440
$266$	0	0
$267$	−23.8533	−1.45980
$268$	0	0
$269$	−2.00423	−0.122200	−0.0611000	−	0.998132i	$-0.519461\pi$
$269$	−2.00423	−0.122200	−0.0611000	+	0.998132i	$0.519461\pi$
$270$	0	0
$271$	10.9723	0.666523	0.333261	−	0.942834i	$-0.391851\pi$
$271$	10.9723	0.666523	0.333261	+	0.942834i	$0.391851\pi$
$272$	0	0
$273$	4.36269	0.264042
$274$	0	0
$275$	1.59609	0.0962479
$276$	0	0
$277$	21.9486	1.31877	0.659383	−	0.751807i	$-0.270818\pi$
$277$	21.9486	1.31877	0.659383	+	0.751807i	$0.270818\pi$
$278$	0	0
$279$	−11.8987	−0.712355
$280$	0	0
$281$	−8.82673	−0.526559	−0.263279	−	0.964720i	$-0.584804\pi$
$281$	−8.82673	−0.526559	−0.263279	+	0.964720i	$0.584804\pi$
$282$	0	0
$283$	2.75319	0.163660	0.0818300	−	0.996646i	$-0.473924\pi$
$283$	2.75319	0.163660	0.0818300	+	0.996646i	$0.473924\pi$
$284$	0	0
$285$	0.313922	0.0185951
$286$	0	0
$287$	−12.1071	−0.714661
$288$	0	0
$289$	−5.29048	−0.311205
$290$	0	0
$291$	−19.6602	−1.15250
$292$	0	0
$293$	11.4079	0.666455	0.333227	−	0.942847i	$-0.391862\pi$
$293$	11.4079	0.666455	0.333227	+	0.942847i	$0.391862\pi$
$294$	0	0
$295$	1.20154	0.0699562
$296$	0	0
$297$	−1.81957	−0.105582
$298$	0	0
$299$	−4.34499	−0.251277
$300$	0	0
$301$	−5.24965	−0.302585
$302$	0	0
$303$	−1.61541	−0.0928031
$304$	0	0
$305$	2.45580	0.140619
$306$	0	0
$307$	2.29462	0.130961	0.0654806	−	0.997854i	$-0.479142\pi$
$307$	2.29462	0.130961	0.0654806	+	0.997854i	$0.479142\pi$
$308$	0	0
$309$	−11.4586	−0.651859
$310$	0	0
$311$	−7.43725	−0.421728	−0.210864	−	0.977515i	$-0.567628\pi$
$311$	−7.43725	−0.421728	−0.210864	+	0.977515i	$0.567628\pi$
$312$	0	0
$313$	1.13571	0.0641939	0.0320969	−	0.999485i	$-0.489781\pi$
$313$	1.13571	0.0641939	0.0320969	+	0.999485i	$0.489781\pi$
$314$	0	0
$315$	0.868860	0.0489547
$316$	0	0
$317$	23.5014	1.31997	0.659986	−	0.751278i	$-0.270562\pi$
$317$	23.5014	1.31997	0.659986	+	0.751278i	$0.270562\pi$
$318$	0	0
$319$	1.85079	0.103624
$320$	0	0
$321$	21.9565	1.22549
$322$	0	0
$323$	3.42192	0.190401
$324$	0	0
$325$	5.08696	0.282174
$326$	0	0
$327$	−6.87253	−0.380052
$328$	0	0
$329$	7.35642	0.405573
$330$	0	0
$331$	35.8100	1.96830	0.984148	−	0.177351i	$-0.0567528\pi$
$331$	35.8100	1.96830	0.984148	+	0.177351i	$0.0567528\pi$
$332$	0	0
$333$	−0.893513	−0.0489642
$334$	0	0
$335$	2.01502	0.110092
$336$	0	0
$337$	14.3781	0.783226	0.391613	−	0.920130i	$-0.371917\pi$
$337$	14.3781	0.783226	0.391613	+	0.920130i	$0.371917\pi$
$338$	0	0
$339$	−10.8196	−0.587643
$340$	0	0
$341$	−3.24155	−0.175540
$342$	0	0
$343$	−12.8750	−0.695186
$344$	0	0
$345$	1.32611	0.0713956
$346$	0	0
$347$	16.6993	0.896465	0.448233	−	0.893917i	$-0.352054\pi$
$347$	16.6993	0.896465	0.448233	+	0.893917i	$0.352054\pi$
$348$	0	0
$349$	−0.950807	−0.0508955	−0.0254478	−	0.999676i	$-0.508101\pi$
$349$	−0.950807	−0.0508955	−0.0254478	+	0.999676i	$0.508101\pi$
$350$	0	0
$351$	−5.79922	−0.309539
$352$	0	0
$353$	1.37329	0.0730931	0.0365465	−	0.999332i	$-0.488364\pi$
$353$	1.37329	0.0730931	0.0365465	+	0.999332i	$0.488364\pi$
$354$	0	0
$355$	−1.66089	−0.0881508
$356$	0	0
$357$	−14.5143	−0.768176
$358$	0	0
$359$	3.73622	0.197190	0.0985952	−	0.995128i	$-0.468565\pi$
$359$	3.73622	0.197190	0.0985952	+	0.995128i	$0.468565\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	14.6807	0.770537
$364$	0	0
$365$	−1.87838	−0.0983188
$366$	0	0
$367$	35.3532	1.84542	0.922711	−	0.385493i	$-0.125969\pi$
$367$	35.3532	1.84542	0.922711	+	0.385493i	$0.125969\pi$
$368$	0	0
$369$	4.55591	0.237171
$370$	0	0
$371$	7.53202	0.391043
$372$	0	0
$373$	24.7348	1.28072	0.640359	−	0.768076i	$-0.278786\pi$
$373$	24.7348	1.28072	0.640359	+	0.768076i	$0.278786\pi$
$374$	0	0
$375$	−3.12218	−0.161228
$376$	0	0
$377$	5.89873	0.303800
$378$	0	0
$379$	−30.0578	−1.54397	−0.771984	−	0.635642i	$-0.780735\pi$
$379$	−30.0578	−1.54397	−0.771984	+	0.635642i	$0.780735\pi$
$380$	0	0
$381$	25.9050	1.32715
$382$	0	0
$383$	28.5169	1.45715	0.728573	−	0.684968i	$-0.240184\pi$
$383$	28.5169	1.45715	0.728573	+	0.684968i	$0.240184\pi$
$384$	0	0
$385$	0.236703	0.0120635
$386$	0	0
$387$	1.97544	0.100417
$388$	0	0
$389$	−1.35348	−0.0686240	−0.0343120	−	0.999411i	$-0.510924\pi$
$389$	−1.35348	−0.0686240	−0.0343120	+	0.999411i	$0.510924\pi$
$390$	0	0
$391$	14.4554	0.731040
$392$	0	0
$393$	−8.45105	−0.426299
$394$	0	0
$395$	−0.232989	−0.0117230
$396$	0	0
$397$	−30.2075	−1.51607	−0.758036	−	0.652212i	$-0.773841\pi$
$397$	−30.2075	−1.51607	−0.758036	+	0.652212i	$0.773841\pi$
$398$	0	0
$399$	−4.24156	−0.212344
$400$	0	0
$401$	11.4179	0.570184	0.285092	−	0.958500i	$-0.407976\pi$
$401$	11.4179	0.570184	0.285092	+	0.958500i	$0.407976\pi$
$402$	0	0
$403$	−10.3313	−0.514637
$404$	0	0
$405$	0.941950	0.0468059
$406$	0	0
$407$	−0.243419	−0.0120659
$408$	0	0
$409$	10.4882	0.518609	0.259305	−	0.965796i	$-0.416507\pi$
$409$	10.4882	0.518609	0.259305	+	0.965796i	$0.416507\pi$
$410$	0	0
$411$	−3.97162	−0.195906
$412$	0	0
$413$	−16.2346	−0.798852
$414$	0	0
$415$	−2.92588	−0.143626
$416$	0	0
$417$	16.0955	0.788199
$418$	0	0
$419$	10.8948	0.532244	0.266122	−	0.963939i	$-0.414258\pi$
$419$	10.8948	0.532244	0.266122	+	0.963939i	$0.414258\pi$
$420$	0	0
$421$	23.6369	1.15199	0.575997	−	0.817452i	$-0.304614\pi$
$421$	23.6369	1.15199	0.575997	+	0.817452i	$0.304614\pi$
$422$	0	0
$423$	−2.76822	−0.134595
$424$	0	0
$425$	−16.9238	−0.820927
$426$	0	0
$427$	−33.1816	−1.60577
$428$	0	0
$429$	−0.447242	−0.0215931
$430$	0	0
$431$	16.5391	0.796660	0.398330	−	0.917242i	$-0.369590\pi$
$431$	16.5391	0.796660	0.398330	+	0.917242i	$0.369590\pi$
$432$	0	0
$433$	36.6611	1.76182	0.880910	−	0.473284i	$-0.156932\pi$
$433$	36.6611	1.76182	0.880910	+	0.473284i	$0.156932\pi$
$434$	0	0
$435$	−1.80032	−0.0863189
$436$	0	0
$437$	4.22435	0.202078
$438$	0	0
$439$	−12.2143	−0.582956	−0.291478	−	0.956578i	$-0.594147\pi$
$439$	−12.2143	−0.582956	−0.291478	+	0.956578i	$0.594147\pi$
$440$	0	0
$441$	−3.44737	−0.164161
$442$	0	0
$443$	−2.00086	−0.0950637	−0.0475319	−	0.998870i	$-0.515136\pi$
$443$	−2.00086	−0.0950637	−0.0475319	+	0.998870i	$0.515136\pi$
$444$	0	0
$445$	−4.12477	−0.195533
$446$	0	0
$447$	26.0396	1.23163
$448$	0	0
$449$	−18.4972	−0.872936	−0.436468	−	0.899720i	$-0.643771\pi$
$449$	−18.4972	−0.872936	−0.436468	+	0.899720i	$0.643771\pi$
$450$	0	0
$451$	1.24116	0.0584442
$452$	0	0
$453$	−11.4567	−0.538282
$454$	0	0
$455$	0.754406	0.0353671
$456$	0	0
$457$	−9.45855	−0.442452	−0.221226	−	0.975223i	$-0.571006\pi$
$457$	−9.45855	−0.442452	−0.221226	+	0.975223i	$0.571006\pi$
$458$	0	0
$459$	19.2934	0.900541
$460$	0	0
$461$	−4.85606	−0.226169	−0.113085	−	0.993585i	$-0.536073\pi$
$461$	−4.85606	−0.226169	−0.113085	+	0.993585i	$0.536073\pi$
$462$	0	0
$463$	19.4164	0.902355	0.451178	−	0.892434i	$-0.351004\pi$
$463$	19.4164	0.902355	0.451178	+	0.892434i	$0.351004\pi$
$464$	0	0
$465$	3.15316	0.146224
$466$	0	0
$467$	14.6746	0.679057	0.339529	−	0.940596i	$-0.389732\pi$
$467$	14.6746	0.679057	0.339529	+	0.940596i	$0.389732\pi$
$468$	0	0
$469$	−27.2260	−1.25718
$470$	0	0
$471$	−4.45626	−0.205334
$472$	0	0
$473$	0.538169	0.0247450
$474$	0	0
$475$	−4.94572	−0.226925
$476$	0	0
$477$	−2.83430	−0.129774
$478$	0	0
$479$	21.6351	0.988531	0.494265	−	0.869311i	$-0.335437\pi$
$479$	21.6351	0.988531	0.494265	+	0.869311i	$0.335437\pi$
$480$	0	0
$481$	−0.775812	−0.0353740
$482$	0	0
$483$	−17.9178	−0.815289
$484$	0	0
$485$	−3.39967	−0.154371
$486$	0	0
$487$	−16.4371	−0.744835	−0.372418	−	0.928065i	$-0.621471\pi$
$487$	−16.4371	−0.744835	−0.372418	+	0.928065i	$0.621471\pi$
$488$	0	0
$489$	−6.33267	−0.286373
$490$	0	0
$491$	35.5249	1.60322	0.801608	−	0.597850i	$-0.203978\pi$
$491$	35.5249	1.60322	0.801608	+	0.597850i	$0.203978\pi$
$492$	0	0
$493$	−19.6245	−0.883844
$494$	0	0
$495$	−0.0890713	−0.00400346
$496$	0	0
$497$	22.4411	1.00662
$498$	0	0
$499$	22.0330	0.986333	0.493167	−	0.869935i	$-0.335839\pi$
$499$	22.0330	0.986333	0.493167	+	0.869935i	$0.335839\pi$
$500$	0	0
$501$	10.8983	0.486901
$502$	0	0
$503$	22.6756	1.01105	0.505527	−	0.862811i	$-0.331298\pi$
$503$	22.6756	1.01105	0.505527	+	0.862811i	$0.331298\pi$
$504$	0	0
$505$	−0.279341	−0.0124305
$506$	0	0
$507$	16.0903	0.714597
$508$	0	0
$509$	−10.6798	−0.473373	−0.236686	−	0.971586i	$-0.576061\pi$
$509$	−10.6798	−0.473373	−0.236686	+	0.971586i	$0.576061\pi$
$510$	0	0
$511$	25.3798	1.12273
$512$	0	0
$513$	5.63819	0.248932
$514$	0	0
$515$	−1.98145	−0.0873131
$516$	0	0
$517$	−0.754145	−0.0331673
$518$	0	0
$519$	−14.0582	−0.617087
$520$	0	0
$521$	−19.8073	−0.867775	−0.433887	−	0.900967i	$-0.642858\pi$
$521$	−19.8073	−0.867775	−0.433887	+	0.900967i	$0.642858\pi$
$522$	0	0
$523$	−12.1917	−0.533107	−0.266553	−	0.963820i	$-0.585885\pi$
$523$	−12.1917	−0.533107	−0.266553	+	0.963820i	$0.585885\pi$
$524$	0	0
$525$	20.9775	0.915535
$526$	0	0
$527$	34.3712	1.49723
$528$	0	0
$529$	−5.15488	−0.224125
$530$	0	0
$531$	6.10907	0.265111
$532$	0	0
$533$	3.95576	0.171343
$534$	0	0
$535$	3.79676	0.164148
$536$	0	0
$537$	−9.78694	−0.422338
$538$	0	0
$539$	−0.939167	−0.0404528
$540$	0	0
$541$	15.6236	0.671713	0.335856	−	0.941913i	$-0.390974\pi$
$541$	15.6236	0.671713	0.335856	+	0.941913i	$0.390974\pi$
$542$	0	0
$543$	−18.5850	−0.797560
$544$	0	0
$545$	−1.18841	−0.0509060
$546$	0	0
$547$	18.4759	0.789972	0.394986	−	0.918687i	$-0.370749\pi$
$547$	18.4759	0.789972	0.394986	+	0.918687i	$0.370749\pi$
$548$	0	0
$549$	12.4862	0.532899
$550$	0	0
$551$	−5.73494	−0.244317
$552$	0	0
$553$	3.14804	0.133868
$554$	0	0
$555$	0.236782	0.0100508
$556$	0	0
$557$	13.5866	0.575682	0.287841	−	0.957678i	$-0.407063\pi$
$557$	13.5866	0.575682	0.287841	+	0.957678i	$0.407063\pi$
$558$	0	0
$559$	1.71522	0.0725460
$560$	0	0
$561$	1.48793	0.0628206
$562$	0	0
$563$	−33.4714	−1.41065	−0.705325	−	0.708884i	$-0.749199\pi$
$563$	−33.4714	−1.41065	−0.705325	+	0.708884i	$0.749199\pi$
$564$	0	0
$565$	−1.87096	−0.0787117
$566$	0	0
$567$	−12.7272	−0.534491
$568$	0	0
$569$	37.3911	1.56752	0.783758	−	0.621066i	$-0.213300\pi$
$569$	37.3911	1.56752	0.783758	+	0.621066i	$0.213300\pi$
$570$	0	0
$571$	−15.5009	−0.648691	−0.324346	−	0.945939i	$-0.605144\pi$
$571$	−15.5009	−0.648691	−0.324346	+	0.945939i	$0.605144\pi$
$572$	0	0
$573$	20.3898	0.851797
$574$	0	0
$575$	−20.8924	−0.871275
$576$	0	0
$577$	31.9912	1.33181	0.665905	−	0.746037i	$-0.268046\pi$
$577$	31.9912	1.33181	0.665905	+	0.746037i	$0.268046\pi$
$578$	0	0
$579$	−1.25720	−0.0522473
$580$	0	0
$581$	39.5331	1.64011
$582$	0	0
$583$	−0.772147	−0.0319791
$584$	0	0
$585$	−0.283883	−0.0117371
$586$	0	0
$587$	−27.6678	−1.14197	−0.570986	−	0.820960i	$-0.693439\pi$
$587$	−27.6678	−1.14197	−0.570986	+	0.820960i	$0.693439\pi$
$588$	0	0
$589$	10.0444	0.413873
$590$	0	0
$591$	−17.5362	−0.721342
$592$	0	0
$593$	17.3213	0.711300	0.355650	−	0.934619i	$-0.384260\pi$
$593$	17.3213	0.711300	0.355650	+	0.934619i	$0.384260\pi$
$594$	0	0
$595$	−2.50984	−0.102893
$596$	0	0
$597$	16.6999	0.683482
$598$	0	0
$599$	3.30210	0.134920	0.0674601	−	0.997722i	$-0.478510\pi$
$599$	3.30210	0.134920	0.0674601	+	0.997722i	$0.478510\pi$
$600$	0	0
$601$	19.9190	0.812513	0.406256	−	0.913759i	$-0.366834\pi$
$601$	19.9190	0.812513	0.406256	+	0.913759i	$0.366834\pi$
$602$	0	0
$603$	10.2451	0.417214
$604$	0	0
$605$	2.53862	0.103209
$606$	0	0
$607$	13.7996	0.560108	0.280054	−	0.959984i	$-0.409648\pi$
$607$	13.7996	0.560108	0.280054	+	0.959984i	$0.409648\pi$
$608$	0	0
$609$	24.3251	0.985703
$610$	0	0
$611$	−2.40357	−0.0972379
$612$	0	0
$613$	−20.8542	−0.842294	−0.421147	−	0.906992i	$-0.638372\pi$
$613$	−20.8542	−0.842294	−0.421147	+	0.906992i	$0.638372\pi$
$614$	0	0
$615$	−1.20732	−0.0486838
$616$	0	0
$617$	−0.559629	−0.0225298	−0.0112649	−	0.999937i	$-0.503586\pi$
$617$	−0.559629	−0.0225298	−0.0112649	+	0.999937i	$0.503586\pi$
$618$	0	0
$619$	−34.5618	−1.38916	−0.694578	−	0.719417i	$-0.744409\pi$
$619$	−34.5618	−1.38916	−0.694578	+	0.719417i	$0.744409\pi$
$620$	0	0
$621$	23.8177	0.955771
$622$	0	0
$623$	55.7319	2.23285
$624$	0	0
$625$	24.1887	0.967547
$626$	0	0
$627$	0.434824	0.0173652
$628$	0	0
$629$	2.58105	0.102913
$630$	0	0
$631$	−35.1655	−1.39991	−0.699957	−	0.714185i	$-0.746798\pi$
$631$	−35.1655	−1.39991	−0.699957	+	0.714185i	$0.746798\pi$
$632$	0	0
$633$	−10.9426	−0.434927
$634$	0	0
$635$	4.47954	0.177765
$636$	0	0
$637$	−2.99325	−0.118597
$638$	0	0
$639$	−8.44459	−0.334063
$640$	0	0
$641$	28.8632	1.14003	0.570015	−	0.821634i	$-0.306937\pi$
$641$	28.8632	1.14003	0.570015	+	0.821634i	$0.306937\pi$
$642$	0	0
$643$	−25.2059	−0.994024	−0.497012	−	0.867744i	$-0.665570\pi$
$643$	−25.2059	−0.994024	−0.497012	+	0.867744i	$0.665570\pi$
$644$	0	0
$645$	−0.523493	−0.0206125
$646$	0	0
$647$	−37.6804	−1.48137	−0.740685	−	0.671853i	$-0.765499\pi$
$647$	−37.6804	−1.48137	−0.740685	+	0.671853i	$0.765499\pi$
$648$	0	0
$649$	1.66429	0.0653292
$650$	0	0
$651$	−42.6040	−1.66978
$652$	0	0
$653$	−22.5084	−0.880822	−0.440411	−	0.897796i	$-0.645167\pi$
$653$	−22.5084	−0.880822	−0.440411	+	0.897796i	$0.645167\pi$
$654$	0	0
$655$	−1.46137	−0.0571006
$656$	0	0
$657$	−9.55039	−0.372596
$658$	0	0
$659$	30.9916	1.20726	0.603631	−	0.797264i	$-0.293720\pi$
$659$	30.9916	1.20726	0.603631	+	0.797264i	$0.293720\pi$
$660$	0	0
$661$	−24.5687	−0.955611	−0.477806	−	0.878466i	$-0.658568\pi$
$661$	−24.5687	−0.955611	−0.477806	+	0.878466i	$0.658568\pi$
$662$	0	0
$663$	4.74225	0.184174
$664$	0	0
$665$	−0.733459	−0.0284423
$666$	0	0
$667$	−24.2264	−0.938050
$668$	0	0
$669$	−10.7949	−0.417354
$670$	0	0
$671$	3.40162	0.131318
$672$	0	0
$673$	−10.2970	−0.396921	−0.198460	−	0.980109i	$-0.563594\pi$
$673$	−10.2970	−0.396921	−0.198460	+	0.980109i	$0.563594\pi$
$674$	0	0
$675$	−27.8849	−1.07329
$676$	0	0
$677$	−16.3697	−0.629140	−0.314570	−	0.949234i	$-0.601860\pi$
$677$	−16.3697	−0.629140	−0.314570	+	0.949234i	$0.601860\pi$
$678$	0	0
$679$	45.9348	1.76282
$680$	0	0
$681$	−29.5118	−1.13089
$682$	0	0
$683$	−21.1930	−0.810926	−0.405463	−	0.914112i	$-0.632890\pi$
$683$	−21.1930	−0.810926	−0.405463	+	0.914112i	$0.632890\pi$
$684$	0	0
$685$	−0.686781	−0.0262406
$686$	0	0
$687$	3.08240	0.117601
$688$	0	0
$689$	−2.46094	−0.0937543
$690$	0	0
$691$	−45.1333	−1.71695	−0.858476	−	0.512853i	$-0.828589\pi$
$691$	−45.1333	−1.71695	−0.858476	+	0.512853i	$0.828589\pi$
$692$	0	0
$693$	1.20349	0.0457168
$694$	0	0
$695$	2.78326	0.105575
$696$	0	0
$697$	−13.1605	−0.498488
$698$	0	0
$699$	−22.7502	−0.860491
$700$	0	0
$701$	−9.95788	−0.376104	−0.188052	−	0.982159i	$-0.560217\pi$
$701$	−9.95788	−0.376104	−0.188052	+	0.982159i	$0.560217\pi$
$702$	0	0
$703$	0.754270	0.0284478
$704$	0	0
$705$	0.733581	0.0276283
$706$	0	0
$707$	3.77432	0.141948
$708$	0	0
$709$	−12.1901	−0.457809	−0.228905	−	0.973449i	$-0.573514\pi$
$709$	−12.1901	−0.457809	−0.228905	+	0.973449i	$0.573514\pi$
$710$	0	0
$711$	−1.18461	−0.0444262
$712$	0	0
$713$	42.4311	1.58906
$714$	0	0
$715$	−0.0773380	−0.00289228
$716$	0	0
$717$	8.43079	0.314854
$718$	0	0
$719$	−18.8387	−0.702566	−0.351283	−	0.936269i	$-0.614254\pi$
$719$	−18.8387	−0.702566	−0.351283	+	0.936269i	$0.614254\pi$
$720$	0	0
$721$	26.7724	0.997056
$722$	0	0
$723$	7.62431	0.283551
$724$	0	0
$725$	28.3634	1.05339
$726$	0	0
$727$	−36.9150	−1.36910	−0.684551	−	0.728965i	$-0.740002\pi$
$727$	−36.9150	−1.36910	−0.684551	+	0.728965i	$0.740002\pi$
$728$	0	0
$729$	27.5794	1.02146
$730$	0	0
$731$	−5.70637	−0.211058
$732$	0	0
$733$	36.9505	1.36480	0.682399	−	0.730980i	$-0.260937\pi$
$733$	36.9505	1.36480	0.682399	+	0.730980i	$0.260937\pi$
$734$	0	0
$735$	0.913557	0.0336971
$736$	0	0
$737$	2.79107	0.102811
$738$	0	0
$739$	−15.8419	−0.582755	−0.291378	−	0.956608i	$-0.594114\pi$
$739$	−15.8419	−0.582755	−0.291378	+	0.956608i	$0.594114\pi$
$740$	0	0
$741$	1.38585	0.0509103
$742$	0	0
$743$	−37.4003	−1.37209	−0.686043	−	0.727561i	$-0.740654\pi$
$743$	−37.4003	−1.37209	−0.686043	+	0.727561i	$0.740654\pi$
$744$	0	0
$745$	4.50282	0.164970
$746$	0	0
$747$	−14.8763	−0.544295
$748$	0	0
$749$	−51.3001	−1.87446
$750$	0	0
$751$	−18.9871	−0.692849	−0.346424	−	0.938078i	$-0.612604\pi$
$751$	−18.9871	−0.692849	−0.346424	+	0.938078i	$0.612604\pi$
$752$	0	0
$753$	−40.4664	−1.47468
$754$	0	0
$755$	−1.98111	−0.0721001
$756$	0	0
$757$	−50.7130	−1.84320	−0.921598	−	0.388147i	$-0.873115\pi$
$757$	−50.7130	−1.84320	−0.921598	+	0.388147i	$0.873115\pi$
$758$	0	0
$759$	1.83685	0.0666734
$760$	0	0
$761$	46.6972	1.69277	0.846386	−	0.532570i	$-0.178774\pi$
$761$	46.6972	1.69277	0.846386	+	0.532570i	$0.178774\pi$
$762$	0	0
$763$	16.0572	0.581311
$764$	0	0
$765$	0.944451	0.0341467
$766$	0	0
$767$	5.30433	0.191528
$768$	0	0
$769$	−49.3961	−1.78127	−0.890635	−	0.454719i	$-0.849740\pi$
$769$	−49.3961	−1.78127	−0.890635	+	0.454719i	$0.849740\pi$
$770$	0	0
$771$	−6.90350	−0.248623
$772$	0	0
$773$	42.0971	1.51413	0.757064	−	0.653341i	$-0.226633\pi$
$773$	42.0971	1.51413	0.757064	+	0.653341i	$0.226633\pi$
$774$	0	0
$775$	−49.6768	−1.78444
$776$	0	0
$777$	−3.19928	−0.114774
$778$	0	0
$779$	−3.84593	−0.137795
$780$	0	0
$781$	−2.30056	−0.0823204
$782$	0	0
$783$	−32.3347	−1.15555
$784$	0	0
$785$	−0.770586	−0.0275034
$786$	0	0
$787$	−41.5545	−1.48126	−0.740630	−	0.671914i	$-0.765473\pi$
$787$	−41.5545	−1.48126	−0.740630	+	0.671914i	$0.765473\pi$
$788$	0	0
$789$	−21.2376	−0.756080
$790$	0	0
$791$	25.2794	0.898834
$792$	0	0
$793$	10.8414	0.384990
$794$	0	0
$795$	0.751091	0.0266385
$796$	0	0
$797$	−19.0599	−0.675136	−0.337568	−	0.941301i	$-0.609604\pi$
$797$	−19.0599	−0.675136	−0.337568	+	0.941301i	$0.609604\pi$
$798$	0	0
$799$	7.99643	0.282894
$800$	0	0
$801$	−20.9719	−0.741006
$802$	0	0
$803$	−2.60181	−0.0918159
$804$	0	0
$805$	−3.09839	−0.109204
$806$	0	0
$807$	2.70043	0.0950595
$808$	0	0
$809$	15.4302	0.542498	0.271249	−	0.962509i	$-0.412563\pi$
$809$	15.4302	0.542498	0.271249	+	0.962509i	$0.412563\pi$
$810$	0	0
$811$	−19.9798	−0.701585	−0.350793	−	0.936453i	$-0.614088\pi$
$811$	−19.9798	−0.701585	−0.350793	+	0.936453i	$0.614088\pi$
$812$	0	0
$813$	−14.7838	−0.518489
$814$	0	0
$815$	−1.09506	−0.0383582
$816$	0	0
$817$	−1.66759	−0.0583417
$818$	0	0
$819$	3.83569	0.134030
$820$	0	0
$821$	−5.14401	−0.179527	−0.0897636	−	0.995963i	$-0.528611\pi$
$821$	−5.14401	−0.179527	−0.0897636	+	0.995963i	$0.528611\pi$
$822$	0	0
$823$	−35.0765	−1.22269	−0.611345	−	0.791364i	$-0.709371\pi$
$823$	−35.0765	−1.22269	−0.611345	+	0.791364i	$0.709371\pi$
$824$	0	0
$825$	−2.15052	−0.0748714
$826$	0	0
$827$	−12.6039	−0.438282	−0.219141	−	0.975693i	$-0.570325\pi$
$827$	−12.6039	−0.438282	−0.219141	+	0.975693i	$0.570325\pi$
$828$	0	0
$829$	22.0936	0.767344	0.383672	−	0.923469i	$-0.374659\pi$
$829$	22.0936	0.767344	0.383672	+	0.923469i	$0.374659\pi$
$830$	0	0
$831$	−29.5728	−1.02587
$832$	0	0
$833$	9.95827	0.345034
$834$	0	0
$835$	1.88456	0.0652179
$836$	0	0
$837$	56.6323	1.95750
$838$	0	0
$839$	27.2722	0.941542	0.470771	−	0.882255i	$-0.343976\pi$
$839$	27.2722	0.941542	0.470771	+	0.882255i	$0.343976\pi$
$840$	0	0
$841$	3.88959	0.134124
$842$	0	0
$843$	11.8928	0.409611
$844$	0	0
$845$	2.78237	0.0957165
$846$	0	0
$847$	−34.3006	−1.17858
$848$	0	0
$849$	−3.70955	−0.127311
$850$	0	0
$851$	3.18630	0.109225
$852$	0	0
$853$	14.3414	0.491041	0.245520	−	0.969391i	$-0.421041\pi$
$853$	14.3414	0.491041	0.245520	+	0.969391i	$0.421041\pi$
$854$	0	0
$855$	0.276000	0.00943901
$856$	0	0
$857$	−11.3287	−0.386983	−0.193491	−	0.981102i	$-0.561981\pi$
$857$	−11.3287	−0.386983	−0.193491	+	0.981102i	$0.561981\pi$
$858$	0	0
$859$	9.36261	0.319448	0.159724	−	0.987162i	$-0.448940\pi$
$859$	9.36261	0.319448	0.159724	+	0.987162i	$0.448940\pi$
$860$	0	0
$861$	16.3127	0.555936
$862$	0	0
$863$	−43.1045	−1.46729	−0.733646	−	0.679531i	$-0.762183\pi$
$863$	−43.1045	−1.46729	−0.733646	+	0.679531i	$0.762183\pi$
$864$	0	0
$865$	−2.43097	−0.0826555
$866$	0	0
$867$	7.12821	0.242087
$868$	0	0
$869$	−0.322722	−0.0109476
$870$	0	0
$871$	8.89554	0.301414
$872$	0	0
$873$	−17.2852	−0.585017
$874$	0	0
$875$	7.29477	0.246608
$876$	0	0
$877$	3.33426	0.112590	0.0562950	−	0.998414i	$-0.482071\pi$
$877$	3.33426	0.112590	0.0562950	+	0.998414i	$0.482071\pi$
$878$	0	0
$879$	−15.3706	−0.518436
$880$	0	0
$881$	41.6277	1.40247	0.701237	−	0.712928i	$-0.252632\pi$
$881$	41.6277	1.40247	0.701237	+	0.712928i	$0.252632\pi$
$882$	0	0
$883$	54.2805	1.82668	0.913342	−	0.407193i	$-0.133492\pi$
$883$	54.2805	1.82668	0.913342	+	0.407193i	$0.133492\pi$
$884$	0	0
$885$	−1.61891	−0.0544191
$886$	0	0
$887$	20.8398	0.699732	0.349866	−	0.936800i	$-0.386227\pi$
$887$	20.8398	0.699732	0.349866	+	0.936800i	$0.386227\pi$
$888$	0	0
$889$	−60.5254	−2.02996
$890$	0	0
$891$	1.30473	0.0437101
$892$	0	0
$893$	2.33683	0.0781990
$894$	0	0
$895$	−1.69238	−0.0565699
$896$	0	0
$897$	5.85429	0.195469
$898$	0	0
$899$	−57.6041	−1.92121
$900$	0	0
$901$	8.18731	0.272759
$902$	0	0
$903$	7.07319	0.235381
$904$	0	0
$905$	−3.21376	−0.106829
$906$	0	0
$907$	−10.3353	−0.343177	−0.171589	−	0.985169i	$-0.554890\pi$
$907$	−10.3353	−0.343177	−0.171589	+	0.985169i	$0.554890\pi$
$908$	0	0
$909$	−1.42027	−0.0471075
$910$	0	0
$911$	0.0843414	0.00279435	0.00139718	−	0.999999i	$-0.499555\pi$
$911$	0.0843414	0.00279435	0.00139718	+	0.999999i	$0.499555\pi$
$912$	0	0
$913$	−4.05274	−0.134126
$914$	0	0
$915$	−3.30886	−0.109388
$916$	0	0
$917$	19.7454	0.652050
$918$	0	0
$919$	−56.3730	−1.85957	−0.929787	−	0.368098i	$-0.880009\pi$
$919$	−56.3730	−1.85957	−0.929787	+	0.368098i	$0.880009\pi$
$920$	0	0
$921$	−3.09170	−0.101875
$922$	0	0
$923$	−7.33220	−0.241342
$924$	0	0
$925$	−3.73041	−0.122655
$926$	0	0
$927$	−10.0744	−0.330888
$928$	0	0
$929$	8.62326	0.282920	0.141460	−	0.989944i	$-0.454820\pi$
$929$	8.62326	0.282920	0.141460	+	0.989944i	$0.454820\pi$
$930$	0	0
$931$	2.91014	0.0953761
$932$	0	0
$933$	10.0207	0.328063
$934$	0	0
$935$	0.257296	0.00841449
$936$	0	0
$937$	−15.0249	−0.490842	−0.245421	−	0.969417i	$-0.578926\pi$
$937$	−15.0249	−0.490842	−0.245421	+	0.969417i	$0.578926\pi$
$938$	0	0
$939$	−1.53021	−0.0499365
$940$	0	0
$941$	−21.3881	−0.697233	−0.348617	−	0.937265i	$-0.613348\pi$
$941$	−21.3881	−0.697233	−0.348617	+	0.937265i	$0.613348\pi$
$942$	0	0
$943$	−16.2465	−0.529060
$944$	0	0
$945$	−4.13538	−0.134524
$946$	0	0
$947$	21.2193	0.689533	0.344767	−	0.938688i	$-0.387958\pi$
$947$	21.2193	0.689533	0.344767	+	0.938688i	$0.387958\pi$
$948$	0	0
$949$	−8.29233	−0.269180
$950$	0	0
$951$	−31.6650	−1.02681
$952$	0	0
$953$	36.1378	1.17062	0.585309	−	0.810810i	$-0.300973\pi$
$953$	36.1378	1.17062	0.585309	+	0.810810i	$0.300973\pi$
$954$	0	0
$955$	3.52585	0.114094
$956$	0	0
$957$	−2.49369	−0.0806096
$958$	0	0
$959$	9.27946	0.299649
$960$	0	0
$961$	69.8902	2.25452
$962$	0	0
$963$	19.3042	0.622069
$964$	0	0
$965$	−0.217397	−0.00699825
$966$	0	0
$967$	−47.4184	−1.52487	−0.762437	−	0.647063i	$-0.775997\pi$
$967$	−47.4184	−1.52487	−0.762437	+	0.647063i	$0.775997\pi$
$968$	0	0
$969$	−4.61057	−0.148113
$970$	0	0
$971$	−21.2639	−0.682391	−0.341195	−	0.939992i	$-0.610832\pi$
$971$	−21.2639	−0.682391	−0.341195	+	0.939992i	$0.610832\pi$
$972$	0	0
$973$	−37.6061	−1.20560
$974$	0	0
$975$	−6.85400	−0.219504
$976$	0	0
$977$	−28.7079	−0.918446	−0.459223	−	0.888321i	$-0.651872\pi$
$977$	−28.7079	−0.918446	−0.459223	+	0.888321i	$0.651872\pi$
$978$	0	0
$979$	−5.71337	−0.182600
$980$	0	0
$981$	−6.04234	−0.192917
$982$	0	0
$983$	4.25463	0.135702	0.0678508	−	0.997695i	$-0.478386\pi$
$983$	4.25463	0.135702	0.0678508	+	0.997695i	$0.478386\pi$
$984$	0	0
$985$	−3.03239	−0.0966200
$986$	0	0
$987$	−9.91179	−0.315496
$988$	0	0
$989$	−7.04449	−0.224002
$990$	0	0
$991$	25.9941	0.825730	0.412865	−	0.910792i	$-0.364528\pi$
$991$	25.9941	0.825730	0.412865	+	0.910792i	$0.364528\pi$
$992$	0	0
$993$	−48.2492	−1.53114
$994$	0	0
$995$	2.88778	0.0915488
$996$	0	0
$997$	−16.8968	−0.535128	−0.267564	−	0.963540i	$-0.586219\pi$
$997$	−16.8968	−0.535128	−0.267564	+	0.963540i	$0.586219\pi$
$998$	0	0
$999$	4.25272	0.134550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.11	✓	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.34737 q^{3} -0.232989 q^{5} +3.14804 q^{7} -1.18461 q^{9} +O(q^{10})\)	\(q-1.34737 q^{3} -0.232989 q^{5} +3.14804 q^{7} -1.18461 q^{9} -0.322722 q^{11} -1.02856 q^{13} +0.313922 q^{15} +3.42192 q^{17} +1.00000 q^{19} -4.24156 q^{21} +4.22435 q^{23} -4.94572 q^{25} +5.63819 q^{27} -5.73494 q^{29} +10.0444 q^{31} +0.434824 q^{33} -0.733459 q^{35} +0.754270 q^{37} +1.38585 q^{39} -3.84593 q^{41} -1.66759 q^{43} +0.276000 q^{45} +2.33683 q^{47} +2.91014 q^{49} -4.61057 q^{51} +2.39261 q^{53} +0.0751907 q^{55} -1.34737 q^{57} -5.15705 q^{59} -10.5404 q^{61} -3.72918 q^{63} +0.239643 q^{65} -8.64855 q^{67} -5.69174 q^{69} +7.12861 q^{71} +8.06209 q^{73} +6.66369 q^{75} -1.01594 q^{77} +1.00000 q^{79} -4.04289 q^{81} +12.5580 q^{83} -0.797270 q^{85} +7.72707 q^{87} +17.7037 q^{89} -3.23794 q^{91} -13.5335 q^{93} -0.232989 q^{95} +14.5916 q^{97} +0.382298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

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