LMFDB - Embedded newform 6003.2.a.t.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.60427 q^{2} +0.573689 q^{4} -1.49095 q^{5} +2.64378 q^{7} -2.28819 q^{8} +O(q^{10})$	\(q+1.60427 q^{2} +0.573689 q^{4} -1.49095 q^{5} +2.64378 q^{7} -2.28819 q^{8} -2.39188 q^{10} +0.452028 q^{11} -4.89824 q^{13} +4.24134 q^{14} -4.81826 q^{16} +1.33628 q^{17} +4.58359 q^{19} -0.855340 q^{20} +0.725176 q^{22} +1.00000 q^{23} -2.77708 q^{25} -7.85811 q^{26} +1.51671 q^{28} +1.00000 q^{29} +3.84793 q^{31} -3.15342 q^{32} +2.14375 q^{34} -3.94174 q^{35} +5.27307 q^{37} +7.35332 q^{38} +3.41157 q^{40} -2.68885 q^{41} -10.1738 q^{43} +0.259324 q^{44} +1.60427 q^{46} +12.2984 q^{47} -0.0104188 q^{49} -4.45519 q^{50} -2.81007 q^{52} -9.46067 q^{53} -0.673950 q^{55} -6.04948 q^{56} +1.60427 q^{58} -5.58963 q^{59} -12.7333 q^{61} +6.17312 q^{62} +4.57758 q^{64} +7.30302 q^{65} -10.7367 q^{67} +0.766606 q^{68} -6.32362 q^{70} -5.99397 q^{71} -4.80854 q^{73} +8.45943 q^{74} +2.62955 q^{76} +1.19506 q^{77} -14.2832 q^{79} +7.18377 q^{80} -4.31365 q^{82} -2.48262 q^{83} -1.99232 q^{85} -16.3215 q^{86} -1.03433 q^{88} +1.82218 q^{89} -12.9499 q^{91} +0.573689 q^{92} +19.7301 q^{94} -6.83388 q^{95} +5.67663 q^{97} -0.0167146 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

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 $

$2$	1.60427	1.13439	0.567196	−	0.823583i	$-0.308028\pi$
$2$	1.60427	1.13439	0.567196	+	0.823583i	$0.308028\pi$
$3$	0	0
$3$	0	0
$4$	0.573689	0.286844
$5$	−1.49095	−0.666772	−0.333386	−	0.942790i	$-0.608191\pi$
$5$	−1.49095	−0.666772	−0.333386	+	0.942790i	$0.608191\pi$
$6$	0	0
$7$	2.64378	0.999256	0.499628	−	0.866240i	$-0.333470\pi$
$7$	2.64378	0.999256	0.499628	+	0.866240i	$0.333470\pi$
$8$	−2.28819	−0.808998
$9$	0	0
$10$	−2.39188	−0.756380
$11$	0.452028	0.136292	0.0681458	−	0.997675i	$-0.478292\pi$
$11$	0.452028	0.136292	0.0681458	+	0.997675i	$0.478292\pi$
$12$	0	0
$13$	−4.89824	−1.35853	−0.679264	−	0.733894i	$-0.737701\pi$
$13$	−4.89824	−1.35853	−0.679264	+	0.733894i	$0.737701\pi$
$14$	4.24134	1.13355
$15$	0	0
$16$	−4.81826	−1.20456
$17$	1.33628	0.324094	0.162047	−	0.986783i	$-0.448190\pi$
$17$	1.33628	0.324094	0.162047	+	0.986783i	$0.448190\pi$
$18$	0	0
$19$	4.58359	1.05155	0.525773	−	0.850625i	$-0.323776\pi$
$19$	4.58359	1.05155	0.525773	+	0.850625i	$0.323776\pi$
$20$	−0.855340	−0.191260
$21$	0	0
$22$	0.725176	0.154608
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.77708	−0.555415
$26$	−7.85811	−1.54110
$27$	0	0
$28$	1.51671	0.286631
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	3.84793	0.691108	0.345554	−	0.938399i	$-0.387691\pi$
$31$	3.84793	0.691108	0.345554	+	0.938399i	$0.387691\pi$
$32$	−3.15342	−0.557450
$33$	0	0
$34$	2.14375	0.367650
$35$	−3.94174	−0.666275
$36$	0	0
$37$	5.27307	0.866887	0.433443	−	0.901181i	$-0.357298\pi$
$37$	5.27307	0.866887	0.433443	+	0.901181i	$0.357298\pi$
$38$	7.35332	1.19287
$39$	0	0
$40$	3.41157	0.539417
$41$	−2.68885	−0.419928	−0.209964	−	0.977709i	$-0.567335\pi$
$41$	−2.68885	−0.419928	−0.209964	+	0.977709i	$0.567335\pi$
$42$	0	0
$43$	−10.1738	−1.55149	−0.775745	−	0.631047i	$-0.782626\pi$
$43$	−10.1738	−1.55149	−0.775745	+	0.631047i	$0.782626\pi$
$44$	0.259324	0.0390945
$45$	0	0
$46$	1.60427	0.236537
$47$	12.2984	1.79391	0.896956	−	0.442119i	$-0.145773\pi$
$47$	12.2984	1.79391	0.896956	+	0.442119i	$0.145773\pi$
$48$	0	0
$49$	−0.0104188	−0.00148840
$50$	−4.45519	−0.630059
$51$	0	0
$52$	−2.81007	−0.389686
$53$	−9.46067	−1.29952	−0.649762	−	0.760138i	$-0.725131\pi$
$53$	−9.46067	−1.29952	−0.649762	+	0.760138i	$0.725131\pi$
$54$	0	0
$55$	−0.673950	−0.0908754
$56$	−6.04948	−0.808395
$57$	0	0
$58$	1.60427	0.210651
$59$	−5.58963	−0.727708	−0.363854	−	0.931456i	$-0.618539\pi$
$59$	−5.58963	−0.727708	−0.363854	+	0.931456i	$0.618539\pi$
$60$	0	0
$61$	−12.7333	−1.63033	−0.815164	−	0.579230i	$-0.803353\pi$
$61$	−12.7333	−1.63033	−0.815164	+	0.579230i	$0.803353\pi$
$62$	6.17312	0.783987
$63$	0	0
$64$	4.57758	0.572198
$65$	7.30302	0.905828
$66$	0	0
$67$	−10.7367	−1.31170	−0.655851	−	0.754891i	$-0.727690\pi$
$67$	−10.7367	−1.31170	−0.655851	+	0.754891i	$0.727690\pi$
$68$	0.766606	0.0929647
$69$	0	0
$70$	−6.32362	−0.755817
$71$	−5.99397	−0.711353	−0.355677	−	0.934609i	$-0.615750\pi$
$71$	−5.99397	−0.711353	−0.355677	+	0.934609i	$0.615750\pi$
$72$	0	0
$73$	−4.80854	−0.562797	−0.281398	−	0.959591i	$-0.590798\pi$
$73$	−4.80854	−0.562797	−0.281398	+	0.959591i	$0.590798\pi$
$74$	8.45943	0.983389
$75$	0	0
$76$	2.62955	0.301630
$77$	1.19506	0.136190
$78$	0	0
$79$	−14.2832	−1.60698	−0.803490	−	0.595318i	$-0.797026\pi$
$79$	−14.2832	−1.60698	−0.803490	+	0.595318i	$0.797026\pi$
$80$	7.18377	0.803170
$81$	0	0
$82$	−4.31365	−0.476363
$83$	−2.48262	−0.272503	−0.136251	−	0.990674i	$-0.543505\pi$
$83$	−2.48262	−0.272503	−0.136251	+	0.990674i	$0.543505\pi$
$84$	0	0
$85$	−1.99232	−0.216097
$86$	−16.3215	−1.76000
$87$	0	0
$88$	−1.03433	−0.110260
$89$	1.82218	0.193150	0.0965752	−	0.995326i	$-0.469211\pi$
$89$	1.82218	0.193150	0.0965752	+	0.995326i	$0.469211\pi$
$90$	0	0
$91$	−12.9499	−1.35752
$92$	0.573689	0.0598112
$93$	0	0
$94$	19.7301	2.03500
$95$	−6.83388	−0.701142
$96$	0	0
$97$	5.67663	0.576374	0.288187	−	0.957574i	$-0.406947\pi$
$97$	5.67663	0.576374	0.288187	+	0.957574i	$0.406947\pi$
$98$	−0.0167146	−0.00168843
$99$	0	0
$100$	−1.59318	−0.159318
$101$	−13.5455	−1.34782	−0.673912	−	0.738811i	$-0.735387\pi$
$101$	−13.5455	−1.34782	−0.673912	+	0.738811i	$0.735387\pi$
$102$	0	0
$103$	−11.7141	−1.15422	−0.577112	−	0.816665i	$-0.695820\pi$
$103$	−11.7141	−1.15422	−0.577112	+	0.816665i	$0.695820\pi$
$104$	11.2081	1.09905
$105$	0	0
$106$	−15.1775	−1.47417
$107$	6.38608	0.617365	0.308683	−	0.951165i	$-0.400112\pi$
$107$	6.38608	0.617365	0.308683	+	0.951165i	$0.400112\pi$
$108$	0	0
$109$	−10.6768	−1.02265	−0.511325	−	0.859387i	$-0.670845\pi$
$109$	−10.6768	−1.02265	−0.511325	+	0.859387i	$0.670845\pi$
$110$	−1.08120	−0.103088
$111$	0	0
$112$	−12.7384	−1.20367
$113$	16.7781	1.57835	0.789174	−	0.614170i	$-0.210509\pi$
$113$	16.7781	1.57835	0.789174	+	0.614170i	$0.210509\pi$
$114$	0	0
$115$	−1.49095	−0.139032
$116$	0.573689	0.0532657
$117$	0	0
$118$	−8.96729	−0.825506
$119$	3.53282	0.323853
$120$	0	0
$121$	−10.7957	−0.981425
$122$	−20.4276	−1.84943
$123$	0	0
$124$	2.20751	0.198241
$125$	11.5952	1.03711
$126$	0	0
$127$	−13.4329	−1.19198	−0.595990	−	0.802992i	$-0.703240\pi$
$127$	−13.4329	−1.19198	−0.595990	+	0.802992i	$0.703240\pi$
$128$	13.6505	1.20655
$129$	0	0
$130$	11.7160	1.02756
$131$	−14.2198	−1.24239	−0.621196	−	0.783655i	$-0.713353\pi$
$131$	−14.2198	−1.24239	−0.621196	+	0.783655i	$0.713353\pi$
$132$	0	0
$133$	12.1180	1.05076
$134$	−17.2246	−1.48798
$135$	0	0
$136$	−3.05765	−0.262192
$137$	10.7302	0.916740	0.458370	−	0.888762i	$-0.348434\pi$
$137$	10.7302	0.916740	0.458370	+	0.888762i	$0.348434\pi$
$138$	0	0
$139$	18.1469	1.53920	0.769598	−	0.638529i	$-0.220457\pi$
$139$	18.1469	1.53920	0.769598	+	0.638529i	$0.220457\pi$
$140$	−2.26133	−0.191117
$141$	0	0
$142$	−9.61596	−0.806953
$143$	−2.21414	−0.185156
$144$	0	0
$145$	−1.49095	−0.123816
$146$	−7.71420	−0.638432
$147$	0	0
$148$	3.02510	0.248662
$149$	−9.55513	−0.782787	−0.391394	−	0.920223i	$-0.628007\pi$
$149$	−9.55513	−0.782787	−0.391394	+	0.920223i	$0.628007\pi$
$150$	0	0
$151$	3.72077	0.302792	0.151396	−	0.988473i	$-0.451623\pi$
$151$	3.72077	0.302792	0.151396	+	0.988473i	$0.451623\pi$
$152$	−10.4881	−0.850699
$153$	0	0
$154$	1.91721	0.154493
$155$	−5.73705	−0.460811
$156$	0	0
$157$	9.77367	0.780024	0.390012	−	0.920810i	$-0.372471\pi$
$157$	9.77367	0.780024	0.390012	+	0.920810i	$0.372471\pi$
$158$	−22.9141	−1.82294
$159$	0	0
$160$	4.70158	0.371692
$161$	2.64378	0.208359
$162$	0	0
$163$	10.0749	0.789128	0.394564	−	0.918869i	$-0.370896\pi$
$163$	10.0749	0.789128	0.394564	+	0.918869i	$0.370896\pi$
$164$	−1.54256	−0.120454
$165$	0	0
$166$	−3.98279	−0.309125
$167$	9.78226	0.756974	0.378487	−	0.925607i	$-0.376444\pi$
$167$	9.78226	0.756974	0.378487	+	0.925607i	$0.376444\pi$
$168$	0	0
$169$	10.9928	0.845597
$170$	−3.19622	−0.245139
$171$	0	0
$172$	−5.83659	−0.445036
$173$	−0.699661	−0.0531942	−0.0265971	−	0.999646i	$-0.508467\pi$
$173$	−0.699661	−0.0531942	−0.0265971	+	0.999646i	$0.508467\pi$
$174$	0	0
$175$	−7.34199	−0.555002
$176$	−2.17799	−0.164172
$177$	0	0
$178$	2.92327	0.219108
$179$	5.88690	0.440007	0.220004	−	0.975499i	$-0.429393\pi$
$179$	5.88690	0.440007	0.220004	+	0.975499i	$0.429393\pi$
$180$	0	0
$181$	−8.06805	−0.599694	−0.299847	−	0.953987i	$-0.596936\pi$
$181$	−8.06805	−0.599694	−0.299847	+	0.953987i	$0.596936\pi$
$182$	−20.7751	−1.53996
$183$	0	0
$184$	−2.28819	−0.168688
$185$	−7.86186	−0.578015
$186$	0	0
$187$	0.604034	0.0441714
$188$	7.05548	0.514574
$189$	0	0
$190$	−10.9634	−0.795369
$191$	−8.20237	−0.593503	−0.296751	−	0.954955i	$-0.595903\pi$
$191$	−8.20237	−0.593503	−0.296751	+	0.954955i	$0.595903\pi$
$192$	0	0
$193$	6.48762	0.466989	0.233495	−	0.972358i	$-0.424984\pi$
$193$	6.48762	0.466989	0.233495	+	0.972358i	$0.424984\pi$
$194$	9.10686	0.653834
$195$	0	0
$196$	−0.00597716	−0.000426940 0
$197$	−4.31234	−0.307241	−0.153621	−	0.988130i	$-0.549093\pi$
$197$	−4.31234	−0.307241	−0.153621	+	0.988130i	$0.549093\pi$
$198$	0	0
$199$	−20.3079	−1.43959	−0.719795	−	0.694187i	$-0.755764\pi$
$199$	−20.3079	−1.43959	−0.719795	+	0.694187i	$0.755764\pi$
$200$	6.35448	0.449330
$201$	0	0
$202$	−21.7306	−1.52896
$203$	2.64378	0.185557
$204$	0	0
$205$	4.00894	0.279996
$206$	−18.7926	−1.30934
$207$	0	0
$208$	23.6010	1.63643
$209$	2.07191	0.143317
$210$	0	0
$211$	−14.7670	−1.01660	−0.508302	−	0.861179i	$-0.669727\pi$
$211$	−14.7670	−1.01660	−0.508302	+	0.861179i	$0.669727\pi$
$212$	−5.42748	−0.372761
$213$	0	0
$214$	10.2450	0.700334
$215$	15.1686	1.03449
$216$	0	0
$217$	10.1731	0.690594
$218$	−17.1285	−1.16009
$219$	0	0
$220$	−0.386638	−0.0260671
$221$	−6.54540	−0.440291
$222$	0	0
$223$	4.22162	0.282701	0.141350	−	0.989960i	$-0.454856\pi$
$223$	4.22162	0.282701	0.141350	+	0.989960i	$0.454856\pi$
$224$	−8.33694	−0.557035
$225$	0	0
$226$	26.9166	1.79046
$227$	−13.0065	−0.863270	−0.431635	−	0.902048i	$-0.642063\pi$
$227$	−13.0065	−0.863270	−0.431635	+	0.902048i	$0.642063\pi$
$228$	0	0
$229$	6.50323	0.429745	0.214873	−	0.976642i	$-0.431066\pi$
$229$	6.50323	0.429745	0.214873	+	0.976642i	$0.431066\pi$
$230$	−2.39188	−0.157716
$231$	0	0
$232$	−2.28819	−0.150227
$233$	−20.9064	−1.36963	−0.684813	−	0.728719i	$-0.740116\pi$
$233$	−20.9064	−1.36963	−0.684813	+	0.728719i	$0.740116\pi$
$234$	0	0
$235$	−18.3363	−1.19613
$236$	−3.20671	−0.208739
$237$	0	0
$238$	5.66761	0.367376
$239$	−17.6537	−1.14192	−0.570962	−	0.820976i	$-0.693430\pi$
$239$	−17.6537	−1.14192	−0.570962	+	0.820976i	$0.693430\pi$
$240$	0	0
$241$	4.04271	0.260414	0.130207	−	0.991487i	$-0.458436\pi$
$241$	4.04271	0.260414	0.130207	+	0.991487i	$0.458436\pi$
$242$	−17.3192	−1.11332
$243$	0	0
$244$	−7.30494	−0.467651
$245$	0.0155339	0.000992425 0
$246$	0	0
$247$	−22.4515	−1.42856
$248$	−8.80479	−0.559105
$249$	0	0
$250$	18.6019	1.17649
$251$	−19.5029	−1.23101	−0.615507	−	0.788132i	$-0.711049\pi$
$251$	−19.5029	−1.23101	−0.615507	+	0.788132i	$0.711049\pi$
$252$	0	0
$253$	0.452028	0.0284188
$254$	−21.5501	−1.35217
$255$	0	0
$256$	12.7440	0.796499
$257$	5.61448	0.350222	0.175111	−	0.984549i	$-0.443972\pi$
$257$	5.61448	0.350222	0.175111	+	0.984549i	$0.443972\pi$
$258$	0	0
$259$	13.9408	0.866241
$260$	4.18966	0.259832
$261$	0	0
$262$	−22.8125	−1.40936
$263$	6.89759	0.425324	0.212662	−	0.977126i	$-0.431787\pi$
$263$	6.89759	0.425324	0.212662	+	0.977126i	$0.431787\pi$
$264$	0	0
$265$	14.1054	0.866485
$266$	19.4406	1.19198
$267$	0	0
$268$	−6.15955	−0.376254
$269$	20.8808	1.27313	0.636563	−	0.771225i	$-0.280356\pi$
$269$	20.8808	1.27313	0.636563	+	0.771225i	$0.280356\pi$
$270$	0	0
$271$	23.2030	1.40948	0.704740	−	0.709466i	$-0.251064\pi$
$271$	23.2030	1.40948	0.704740	+	0.709466i	$0.251064\pi$
$272$	−6.43852	−0.390393
$273$	0	0
$274$	17.2141	1.03994
$275$	−1.25532	−0.0756985
$276$	0	0
$277$	14.5240	0.872663	0.436331	−	0.899786i	$-0.356278\pi$
$277$	14.5240	0.872663	0.436331	+	0.899786i	$0.356278\pi$
$278$	29.1125	1.74605
$279$	0	0
$280$	9.01945	0.539015
$281$	24.9248	1.48689	0.743445	−	0.668797i	$-0.233191\pi$
$281$	24.9248	1.48689	0.743445	+	0.668797i	$0.233191\pi$
$282$	0	0
$283$	−19.0099	−1.13002	−0.565010	−	0.825084i	$-0.691128\pi$
$283$	−19.0099	−1.13002	−0.565010	+	0.825084i	$0.691128\pi$
$284$	−3.43867	−0.204048
$285$	0	0
$286$	−3.55209	−0.210039
$287$	−7.10874	−0.419616
$288$	0	0
$289$	−15.2144	−0.894963
$290$	−2.39188	−0.140456
$291$	0	0
$292$	−2.75860	−0.161435
$293$	−20.5871	−1.20271	−0.601356	−	0.798982i	$-0.705372\pi$
$293$	−20.5871	−1.20271	−0.601356	+	0.798982i	$0.705372\pi$
$294$	0	0
$295$	8.33384	0.485215
$296$	−12.0658	−0.701309
$297$	0	0
$298$	−15.3290	−0.887987
$299$	−4.89824	−0.283273
$300$	0	0
$301$	−26.8973	−1.55033
$302$	5.96913	0.343485
$303$	0	0
$304$	−22.0849	−1.26666
$305$	18.9846	1.08706
$306$	0	0
$307$	−27.4167	−1.56475	−0.782377	−	0.622805i	$-0.785993\pi$
$307$	−27.4167	−1.56475	−0.782377	+	0.622805i	$0.785993\pi$
$308$	0.685595	0.0390654
$309$	0	0
$310$	−9.20380	−0.522741
$311$	10.1383	0.574892	0.287446	−	0.957797i	$-0.407194\pi$
$311$	10.1383	0.574892	0.287446	+	0.957797i	$0.407194\pi$
$312$	0	0
$313$	−2.20484	−0.124625	−0.0623125	−	0.998057i	$-0.519848\pi$
$313$	−2.20484	−0.124625	−0.0623125	+	0.998057i	$0.519848\pi$
$314$	15.6796	0.884853
$315$	0	0
$316$	−8.19408	−0.460953
$317$	−7.39664	−0.415437	−0.207718	−	0.978189i	$-0.566604\pi$
$317$	−7.39664	−0.415437	−0.207718	+	0.978189i	$0.566604\pi$
$318$	0	0
$319$	0.452028	0.0253087
$320$	−6.82493	−0.381525
$321$	0	0
$322$	4.24134	0.236361
$323$	6.12493	0.340800
$324$	0	0
$325$	13.6028	0.754547
$326$	16.1629	0.895180
$327$	0	0
$328$	6.15261	0.339721
$329$	32.5144	1.79258
$330$	0	0
$331$	−12.9076	−0.709468	−0.354734	−	0.934967i	$-0.615429\pi$
$331$	−12.9076	−0.709468	−0.354734	+	0.934967i	$0.615429\pi$
$332$	−1.42425	−0.0781659
$333$	0	0
$334$	15.6934	0.858705
$335$	16.0079	0.874605
$336$	0	0
$337$	4.95912	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$337$	4.95912	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$338$	17.6354	0.959239
$339$	0	0
$340$	−1.14297	−0.0619862
$341$	1.73937	0.0941922
$342$	0	0
$343$	−18.5340	−1.00074
$344$	23.2796	1.25515
$345$	0	0
$346$	−1.12245	−0.0603431
$347$	31.3082	1.68071	0.840357	−	0.542034i	$-0.182346\pi$
$347$	31.3082	1.68071	0.840357	+	0.542034i	$0.182346\pi$
$348$	0	0
$349$	−3.96837	−0.212422	−0.106211	−	0.994344i	$-0.533872\pi$
$349$	−3.96837	−0.212422	−0.106211	+	0.994344i	$0.533872\pi$
$350$	−11.7785	−0.629590
$351$	0	0
$352$	−1.42543	−0.0759758
$353$	3.84478	0.204637	0.102318	−	0.994752i	$-0.467374\pi$
$353$	3.84478	0.204637	0.102318	+	0.994752i	$0.467374\pi$
$354$	0	0
$355$	8.93669	0.474310
$356$	1.04536	0.0554041
$357$	0	0
$358$	9.44419	0.499141
$359$	−0.721752	−0.0380926	−0.0190463	−	0.999819i	$-0.506063\pi$
$359$	−0.721752	−0.0380926	−0.0190463	+	0.999819i	$0.506063\pi$
$360$	0	0
$361$	2.00926	0.105750
$362$	−12.9433	−0.680287
$363$	0	0
$364$	−7.42920	−0.389396
$365$	7.16928	0.375257
$366$	0	0
$367$	23.9020	1.24768	0.623838	−	0.781554i	$-0.285573\pi$
$367$	23.9020	1.24768	0.623838	+	0.781554i	$0.285573\pi$
$368$	−4.81826	−0.251169
$369$	0	0
$370$	−12.6126	−0.655696
$371$	−25.0120	−1.29856
$372$	0	0
$373$	17.5517	0.908794	0.454397	−	0.890799i	$-0.349855\pi$
$373$	17.5517	0.908794	0.454397	+	0.890799i	$0.349855\pi$
$374$	0.969035	0.0501076
$375$	0	0
$376$	−28.1412	−1.45127
$377$	−4.89824	−0.252272
$378$	0	0
$379$	−24.5915	−1.26318	−0.631590	−	0.775303i	$-0.717597\pi$
$379$	−24.5915	−1.26318	−0.631590	+	0.775303i	$0.717597\pi$
$380$	−3.92052	−0.201119
$381$	0	0
$382$	−13.1588	−0.673264
$383$	5.49843	0.280957	0.140478	−	0.990084i	$-0.455136\pi$
$383$	5.49843	0.280957	0.140478	+	0.990084i	$0.455136\pi$
$384$	0	0
$385$	−1.78178	−0.0908077
$386$	10.4079	0.529749
$387$	0	0
$388$	3.25662	0.165330
$389$	24.3699	1.23561	0.617803	−	0.786333i	$-0.288023\pi$
$389$	24.3699	1.23561	0.617803	+	0.786333i	$0.288023\pi$
$390$	0	0
$391$	1.33628	0.0675784
$392$	0.0238402	0.00120411
$393$	0	0
$394$	−6.91816	−0.348532
$395$	21.2954	1.07149
$396$	0	0
$397$	−24.1489	−1.21200	−0.606000	−	0.795464i	$-0.707227\pi$
$397$	−24.1489	−1.21200	−0.606000	+	0.795464i	$0.707227\pi$
$398$	−32.5794	−1.63306
$399$	0	0
$400$	13.3807	0.669034
$401$	−32.4595	−1.62095	−0.810475	−	0.585774i	$-0.800791\pi$
$401$	−32.4595	−1.62095	−0.810475	+	0.585774i	$0.800791\pi$
$402$	0	0
$403$	−18.8481	−0.938889
$404$	−7.77088	−0.386616
$405$	0	0
$406$	4.24134	0.210494
$407$	2.38357	0.118149
$408$	0	0
$409$	19.6683	0.972534	0.486267	−	0.873810i	$-0.338358\pi$
$409$	19.6683	0.972534	0.486267	+	0.873810i	$0.338358\pi$
$410$	6.43142	0.317625
$411$	0	0
$412$	−6.72025	−0.331083
$413$	−14.7778	−0.727166
$414$	0	0
$415$	3.70145	0.181697
$416$	15.4462	0.757312
$417$	0	0
$418$	3.32391	0.162578
$419$	−27.2402	−1.33077	−0.665385	−	0.746501i	$-0.731732\pi$
$419$	−27.2402	−1.33077	−0.665385	+	0.746501i	$0.731732\pi$
$420$	0	0
$421$	4.89033	0.238340	0.119170	−	0.992874i	$-0.461977\pi$
$421$	4.89033	0.238340	0.119170	+	0.992874i	$0.461977\pi$
$422$	−23.6903	−1.15323
$423$	0	0
$424$	21.6478	1.05131
$425$	−3.71094	−0.180007
$426$	0	0
$427$	−33.6640	−1.62911
$428$	3.66362	0.177088
$429$	0	0
$430$	24.3345	1.17352
$431$	19.6984	0.948838	0.474419	−	0.880299i	$-0.342658\pi$
$431$	19.6984	0.948838	0.474419	+	0.880299i	$0.342658\pi$
$432$	0	0
$433$	10.5432	0.506672	0.253336	−	0.967378i	$-0.418472\pi$
$433$	10.5432	0.506672	0.253336	+	0.967378i	$0.418472\pi$
$434$	16.3204	0.783404
$435$	0	0
$436$	−6.12515	−0.293342
$437$	4.58359	0.219263
$438$	0	0
$439$	−22.3066	−1.06464	−0.532319	−	0.846544i	$-0.678679\pi$
$439$	−22.3066	−1.06464	−0.532319	+	0.846544i	$0.678679\pi$
$440$	1.54213	0.0735180
$441$	0	0
$442$	−10.5006	−0.499463
$443$	36.5774	1.73785	0.868923	−	0.494947i	$-0.164813\pi$
$443$	36.5774	1.73785	0.868923	+	0.494947i	$0.164813\pi$
$444$	0	0
$445$	−2.71677	−0.128787
$446$	6.77263	0.320693
$447$	0	0
$448$	12.1021	0.571772
$449$	−11.7639	−0.555172	−0.277586	−	0.960701i	$-0.589534\pi$
$449$	−11.7639	−0.555172	−0.277586	+	0.960701i	$0.589534\pi$
$450$	0	0
$451$	−1.21544	−0.0572327
$452$	9.62539	0.452740
$453$	0	0
$454$	−20.8659	−0.979286
$455$	19.3076	0.905153
$456$	0	0
$457$	−2.47480	−0.115766	−0.0578832	−	0.998323i	$-0.518435\pi$
$457$	−2.47480	−0.115766	−0.0578832	+	0.998323i	$0.518435\pi$
$458$	10.4329	0.487499
$459$	0	0
$460$	−0.855340	−0.0398804
$461$	32.6980	1.52290	0.761449	−	0.648225i	$-0.224489\pi$
$461$	32.6980	1.52290	0.761449	+	0.648225i	$0.224489\pi$
$462$	0	0
$463$	−11.8696	−0.551628	−0.275814	−	0.961211i	$-0.588947\pi$
$463$	−11.8696	−0.551628	−0.275814	+	0.961211i	$0.588947\pi$
$464$	−4.81826	−0.223682
$465$	0	0
$466$	−33.5396	−1.55369
$467$	2.70620	0.125228	0.0626141	−	0.998038i	$-0.480056\pi$
$467$	2.70620	0.125228	0.0626141	+	0.998038i	$0.480056\pi$
$468$	0	0
$469$	−28.3856	−1.31072
$470$	−29.4165	−1.35688
$471$	0	0
$472$	12.7901	0.588714
$473$	−4.59884	−0.211455
$474$	0	0
$475$	−12.7290	−0.584045
$476$	2.02674	0.0928955
$477$	0	0
$478$	−28.3214	−1.29539
$479$	8.28008	0.378327	0.189163	−	0.981946i	$-0.439422\pi$
$479$	8.28008	0.378327	0.189163	+	0.981946i	$0.439422\pi$
$480$	0	0
$481$	−25.8287	−1.17769
$482$	6.48561	0.295412
$483$	0	0
$484$	−6.19336	−0.281516
$485$	−8.46355	−0.384310
$486$	0	0
$487$	−23.3641	−1.05873	−0.529363	−	0.848395i	$-0.677569\pi$
$487$	−23.3641	−1.05873	−0.529363	+	0.848395i	$0.677569\pi$
$488$	29.1362	1.31893
$489$	0	0
$490$	0.0249206	0.00112580
$491$	27.8118	1.25513	0.627565	−	0.778564i	$-0.284052\pi$
$491$	27.8118	1.25513	0.627565	+	0.778564i	$0.284052\pi$
$492$	0	0
$493$	1.33628	0.0601828
$494$	−36.0183	−1.62054
$495$	0	0
$496$	−18.5403	−0.832484
$497$	−15.8467	−0.710824
$498$	0	0
$499$	2.79228	0.124999	0.0624997	−	0.998045i	$-0.480093\pi$
$499$	2.79228	0.124999	0.0624997	+	0.998045i	$0.480093\pi$
$500$	6.65204	0.297488
$501$	0	0
$502$	−31.2880	−1.39645
$503$	−14.7701	−0.658567	−0.329283	−	0.944231i	$-0.606807\pi$
$503$	−14.7701	−0.658567	−0.329283	+	0.944231i	$0.606807\pi$
$504$	0	0
$505$	20.1956	0.898691
$506$	0.725176	0.0322380
$507$	0	0
$508$	−7.70633	−0.341913
$509$	−18.7434	−0.830786	−0.415393	−	0.909642i	$-0.636356\pi$
$509$	−18.7434	−0.830786	−0.415393	+	0.909642i	$0.636356\pi$
$510$	0	0
$511$	−12.7127	−0.562378
$512$	−6.85622	−0.303005
$513$	0	0
$514$	9.00716	0.397289
$515$	17.4651	0.769604
$516$	0	0
$517$	5.55924	0.244495
$518$	22.3649	0.982657
$519$	0	0
$520$	−16.7107	−0.732813
$521$	5.71918	0.250562	0.125281	−	0.992121i	$-0.460017\pi$
$521$	5.71918	0.250562	0.125281	+	0.992121i	$0.460017\pi$
$522$	0	0
$523$	−23.2442	−1.01640	−0.508199	−	0.861240i	$-0.669689\pi$
$523$	−23.2442	−1.01640	−0.508199	+	0.861240i	$0.669689\pi$
$524$	−8.15775	−0.356373
$525$	0	0
$526$	11.0656	0.482484
$527$	5.14189	0.223984
$528$	0	0
$529$	1.00000	0.0434783
$530$	22.6288	0.982934
$531$	0	0
$532$	6.95196	0.301406
$533$	13.1706	0.570484
$534$	0	0
$535$	−9.52130	−0.411642
$536$	24.5677	1.06116
$537$	0	0
$538$	33.4985	1.44422
$539$	−0.00470960	−0.000202857 0
$540$	0	0
$541$	37.5964	1.61640	0.808198	−	0.588911i	$-0.200443\pi$
$541$	37.5964	1.61640	0.808198	+	0.588911i	$0.200443\pi$
$542$	37.2239	1.59890
$543$	0	0
$544$	−4.21383	−0.180667
$545$	15.9185	0.681874
$546$	0	0
$547$	38.5226	1.64711	0.823554	−	0.567237i	$-0.191988\pi$
$547$	38.5226	1.64711	0.823554	+	0.567237i	$0.191988\pi$
$548$	6.15577	0.262962
$549$	0	0
$550$	−2.01387	−0.0858717
$551$	4.58359	0.195267
$552$	0	0
$553$	−37.7615	−1.60578
$554$	23.3005	0.989941
$555$	0	0
$556$	10.4106	0.441510
$557$	−2.46258	−0.104343	−0.0521715	−	0.998638i	$-0.516614\pi$
$557$	−2.46258	−0.104343	−0.0521715	+	0.998638i	$0.516614\pi$
$558$	0	0
$559$	49.8337	2.10774
$560$	18.9923	0.802572
$561$	0	0
$562$	39.9862	1.68672
$563$	−40.2244	−1.69525	−0.847627	−	0.530592i	$-0.821970\pi$
$563$	−40.2244	−1.69525	−0.847627	+	0.530592i	$0.821970\pi$
$564$	0	0
$565$	−25.0152	−1.05240
$566$	−30.4970	−1.28189
$567$	0	0
$568$	13.7153	0.575483
$569$	28.8951	1.21134	0.605672	−	0.795714i	$-0.292904\pi$
$569$	28.8951	1.21134	0.605672	+	0.795714i	$0.292904\pi$
$570$	0	0
$571$	−9.67958	−0.405078	−0.202539	−	0.979274i	$-0.564919\pi$
$571$	−9.67958	−0.405078	−0.202539	+	0.979274i	$0.564919\pi$
$572$	−1.27023	−0.0531109
$573$	0	0
$574$	−11.4044	−0.476008
$575$	−2.77708	−0.115812
$576$	0	0
$577$	−4.84531	−0.201713	−0.100856	−	0.994901i	$-0.532158\pi$
$577$	−4.84531	−0.201713	−0.100856	+	0.994901i	$0.532158\pi$
$578$	−24.4080	−1.01524
$579$	0	0
$580$	−0.855340	−0.0355160
$581$	−6.56350	−0.272300
$582$	0	0
$583$	−4.27649	−0.177114
$584$	11.0029	0.455301
$585$	0	0
$586$	−33.0273	−1.36435
$587$	38.9485	1.60758	0.803788	−	0.594916i	$-0.202815\pi$
$587$	38.9485	1.60758	0.803788	+	0.594916i	$0.202815\pi$
$588$	0	0
$589$	17.6373	0.726732
$590$	13.3697	0.550424
$591$	0	0
$592$	−25.4070	−1.04422
$593$	−7.84842	−0.322296	−0.161148	−	0.986930i	$-0.551520\pi$
$593$	−7.84842	−0.322296	−0.161148	+	0.986930i	$0.551520\pi$
$594$	0	0
$595$	−5.26725	−0.215936
$596$	−5.48167	−0.224538
$597$	0	0
$598$	−7.85811	−0.321342
$599$	20.9165	0.854626	0.427313	−	0.904104i	$-0.359460\pi$
$599$	20.9165	0.854626	0.427313	+	0.904104i	$0.359460\pi$
$600$	0	0
$601$	5.37909	0.219418	0.109709	−	0.993964i	$-0.465008\pi$
$601$	5.37909	0.219418	0.109709	+	0.993964i	$0.465008\pi$
$602$	−43.1506	−1.75869
$603$	0	0
$604$	2.13456	0.0868542
$605$	16.0958	0.654386
$606$	0	0
$607$	34.3675	1.39493	0.697467	−	0.716617i	$-0.254310\pi$
$607$	34.3675	1.39493	0.697467	+	0.716617i	$0.254310\pi$
$608$	−14.4540	−0.586185
$609$	0	0
$610$	30.4565	1.23315
$611$	−60.2407	−2.43708
$612$	0	0
$613$	−33.6862	−1.36057	−0.680286	−	0.732947i	$-0.738144\pi$
$613$	−33.6862	−1.36057	−0.680286	+	0.732947i	$0.738144\pi$
$614$	−43.9839	−1.77504
$615$	0	0
$616$	−2.73453	−0.110178
$617$	−9.80268	−0.394641	−0.197320	−	0.980339i	$-0.563224\pi$
$617$	−9.80268	−0.394641	−0.197320	+	0.980339i	$0.563224\pi$
$618$	0	0
$619$	14.2483	0.572688	0.286344	−	0.958127i	$-0.407560\pi$
$619$	14.2483	0.572688	0.286344	+	0.958127i	$0.407560\pi$
$620$	−3.29128	−0.132181
$621$	0	0
$622$	16.2646	0.652153
$623$	4.81744	0.193007
$624$	0	0
$625$	−3.40245	−0.136098
$626$	−3.53717	−0.141374
$627$	0	0
$628$	5.60705	0.223746
$629$	7.04627	0.280953
$630$	0	0
$631$	8.76698	0.349008	0.174504	−	0.984656i	$-0.444168\pi$
$631$	8.76698	0.349008	0.174504	+	0.984656i	$0.444168\pi$
$632$	32.6826	1.30004
$633$	0	0
$634$	−11.8662	−0.471268
$635$	20.0278	0.794779
$636$	0	0
$637$	0.0510339	0.00202204
$638$	0.725176	0.0287100
$639$	0	0
$640$	−20.3522	−0.804491
$641$	33.6529	1.32921	0.664605	−	0.747195i	$-0.268600\pi$
$641$	33.6529	1.32921	0.664605	+	0.747195i	$0.268600\pi$
$642$	0	0
$643$	−17.7464	−0.699851	−0.349926	−	0.936777i	$-0.613793\pi$
$643$	−17.7464	−0.699851	−0.349926	+	0.936777i	$0.613793\pi$
$644$	1.51671	0.0597667
$645$	0	0
$646$	9.82606	0.386601
$647$	−13.9050	−0.546663	−0.273332	−	0.961920i	$-0.588126\pi$
$647$	−13.9050	−0.546663	−0.273332	+	0.961920i	$0.588126\pi$
$648$	0	0
$649$	−2.52667	−0.0991805
$650$	21.8226	0.855952
$651$	0	0
$652$	5.77986	0.226357
$653$	−20.1732	−0.789440	−0.394720	−	0.918801i	$-0.629158\pi$
$653$	−20.1732	−0.789440	−0.394720	+	0.918801i	$0.629158\pi$
$654$	0	0
$655$	21.2010	0.828392
$656$	12.9556	0.505831
$657$	0	0
$658$	52.1619	2.03348
$659$	46.8861	1.82642	0.913212	−	0.407485i	$-0.133594\pi$
$659$	46.8861	1.82642	0.913212	+	0.407485i	$0.133594\pi$
$660$	0	0
$661$	−47.0205	−1.82888	−0.914442	−	0.404716i	$-0.867370\pi$
$661$	−47.0205	−1.82888	−0.914442	+	0.404716i	$0.867370\pi$
$662$	−20.7074	−0.804815
$663$	0	0
$664$	5.68070	0.220454
$665$	−18.0673	−0.700620
$666$	0	0
$667$	1.00000	0.0387202
$668$	5.61198	0.217134
$669$	0	0
$670$	25.6810	0.992145
$671$	−5.75580	−0.222200
$672$	0	0
$673$	25.5133	0.983466	0.491733	−	0.870746i	$-0.336364\pi$
$673$	25.5133	0.983466	0.491733	+	0.870746i	$0.336364\pi$
$674$	7.95578	0.306445
$675$	0	0
$676$	6.30643	0.242555
$677$	45.7948	1.76004	0.880019	−	0.474939i	$-0.157530\pi$
$677$	45.7948	1.76004	0.880019	+	0.474939i	$0.157530\pi$
$678$	0	0
$679$	15.0078	0.575945
$680$	4.55880	0.174822
$681$	0	0
$682$	2.79043	0.106851
$683$	−29.2313	−1.11850	−0.559252	−	0.828998i	$-0.688911\pi$
$683$	−29.2313	−1.11850	−0.559252	+	0.828998i	$0.688911\pi$
$684$	0	0
$685$	−15.9981	−0.611256
$686$	−29.7336	−1.13523
$687$	0	0
$688$	49.0200	1.86887
$689$	46.3407	1.76544
$690$	0	0
$691$	−13.4405	−0.511302	−0.255651	−	0.966769i	$-0.582290\pi$
$691$	−13.4405	−0.511302	−0.255651	+	0.966769i	$0.582290\pi$
$692$	−0.401388	−0.0152585
$693$	0	0
$694$	50.2269	1.90659
$695$	−27.0560	−1.02629
$696$	0	0
$697$	−3.59305	−0.136096
$698$	−6.36634	−0.240969
$699$	0	0
$700$	−4.21202	−0.159199
$701$	−34.4809	−1.30233	−0.651164	−	0.758937i	$-0.725719\pi$
$701$	−34.4809	−1.30233	−0.651164	+	0.758937i	$0.725719\pi$
$702$	0	0
$703$	24.1695	0.911572
$704$	2.06920	0.0779857
$705$	0	0
$706$	6.16807	0.232138
$707$	−35.8113	−1.34682
$708$	0	0
$709$	−2.67200	−0.100349	−0.0501746	−	0.998740i	$-0.515978\pi$
$709$	−2.67200	−0.100349	−0.0501746	+	0.998740i	$0.515978\pi$
$710$	14.3369	0.538054
$711$	0	0
$712$	−4.16949	−0.156258
$713$	3.84793	0.144106
$714$	0	0
$715$	3.30117	0.123457
$716$	3.37725	0.126214
$717$	0	0
$718$	−1.15789	−0.0432119
$719$	−2.13450	−0.0796034	−0.0398017	−	0.999208i	$-0.512673\pi$
$719$	−2.13450	−0.0796034	−0.0398017	+	0.999208i	$0.512673\pi$
$720$	0	0
$721$	−30.9695	−1.15336
$722$	3.22339	0.119962
$723$	0	0
$724$	−4.62855	−0.172019
$725$	−2.77708	−0.103138
$726$	0	0
$727$	29.8021	1.10530	0.552650	−	0.833414i	$-0.313617\pi$
$727$	29.8021	1.10530	0.552650	+	0.833414i	$0.313617\pi$
$728$	29.6318	1.09823
$729$	0	0
$730$	11.5015	0.425688
$731$	−13.5950	−0.502829
$732$	0	0
$733$	21.7312	0.802660	0.401330	−	0.915934i	$-0.368548\pi$
$733$	21.7312	0.802660	0.401330	+	0.915934i	$0.368548\pi$
$734$	38.3453	1.41535
$735$	0	0
$736$	−3.15342	−0.116236
$737$	−4.85331	−0.178774
$738$	0	0
$739$	−0.292924	−0.0107754	−0.00538769	−	0.999985i	$-0.501715\pi$
$739$	−0.292924	−0.0107754	−0.00538769	+	0.999985i	$0.501715\pi$
$740$	−4.51026	−0.165801
$741$	0	0
$742$	−40.1260	−1.47307
$743$	37.9029	1.39052	0.695261	−	0.718757i	$-0.255289\pi$
$743$	37.9029	1.39052	0.695261	+	0.718757i	$0.255289\pi$
$744$	0	0
$745$	14.2462	0.521940
$746$	28.1578	1.03093
$747$	0	0
$748$	0.346528	0.0126703
$749$	16.8834	0.616906
$750$	0	0
$751$	37.0898	1.35343	0.676713	−	0.736247i	$-0.263404\pi$
$751$	37.0898	1.35343	0.676713	+	0.736247i	$0.263404\pi$
$752$	−59.2571	−2.16088
$753$	0	0
$754$	−7.85811	−0.286176
$755$	−5.54747	−0.201893
$756$	0	0
$757$	14.6336	0.531867	0.265934	−	0.963991i	$-0.414320\pi$
$757$	14.6336	0.531867	0.265934	+	0.963991i	$0.414320\pi$
$758$	−39.4514	−1.43294
$759$	0	0
$760$	15.6372	0.567222
$761$	15.1704	0.549926	0.274963	−	0.961455i	$-0.411334\pi$
$761$	15.1704	0.549926	0.274963	+	0.961455i	$0.411334\pi$
$762$	0	0
$763$	−28.2271	−1.02189
$764$	−4.70561	−0.170243
$765$	0	0
$766$	8.82098	0.318715
$767$	27.3794	0.988611
$768$	0	0
$769$	34.0412	1.22756	0.613779	−	0.789478i	$-0.289649\pi$
$769$	34.0412	1.22756	0.613779	+	0.789478i	$0.289649\pi$
$770$	−2.85845	−0.103012
$771$	0	0
$772$	3.72188	0.133953
$773$	26.8567	0.965967	0.482983	−	0.875630i	$-0.339553\pi$
$773$	26.8567	0.965967	0.482983	+	0.875630i	$0.339553\pi$
$774$	0	0
$775$	−10.6860	−0.383852
$776$	−12.9892	−0.466285
$777$	0	0
$778$	39.0960	1.40166
$779$	−12.3246	−0.441574
$780$	0	0
$781$	−2.70944	−0.0969515
$782$	2.14375	0.0766603
$783$	0	0
$784$	0.0502006	0.00179288
$785$	−14.5720	−0.520098
$786$	0	0
$787$	−28.3930	−1.01210	−0.506050	−	0.862504i	$-0.668895\pi$
$787$	−28.3930	−1.01210	−0.506050	+	0.862504i	$0.668895\pi$
$788$	−2.47394	−0.0881305
$789$	0	0
$790$	34.1636	1.21549
$791$	44.3576	1.57717
$792$	0	0
$793$	62.3706	2.21485
$794$	−38.7415	−1.37488
$795$	0	0
$796$	−11.6504	−0.412938
$797$	23.8372	0.844358	0.422179	−	0.906513i	$-0.361265\pi$
$797$	23.8372	0.844358	0.422179	+	0.906513i	$0.361265\pi$
$798$	0	0
$799$	16.4341	0.581397
$800$	8.75728	0.309617
$801$	0	0
$802$	−52.0738	−1.83879
$803$	−2.17359	−0.0767045
$804$	0	0
$805$	−3.94174	−0.138928
$806$	−30.2374	−1.06507
$807$	0	0
$808$	30.9946	1.09039
$809$	15.5078	0.545224	0.272612	−	0.962124i	$-0.412113\pi$
$809$	15.5078	0.545224	0.272612	+	0.962124i	$0.412113\pi$
$810$	0	0
$811$	10.1480	0.356345	0.178173	−	0.983999i	$-0.442981\pi$
$811$	10.1480	0.356345	0.178173	+	0.983999i	$0.442981\pi$
$812$	1.51671	0.0532260
$813$	0	0
$814$	3.82390	0.134028
$815$	−15.0212	−0.526168
$816$	0	0
$817$	−46.6325	−1.63146
$818$	31.5533	1.10323
$819$	0	0
$820$	2.29988	0.0803154
$821$	41.5099	1.44870	0.724352	−	0.689430i	$-0.242139\pi$
$821$	41.5099	1.44870	0.724352	+	0.689430i	$0.242139\pi$
$822$	0	0
$823$	−32.8195	−1.14402	−0.572008	−	0.820248i	$-0.693835\pi$
$823$	−32.8195	−1.14402	−0.572008	+	0.820248i	$0.693835\pi$
$824$	26.8041	0.933765
$825$	0	0
$826$	−23.7075	−0.824891
$827$	−26.0045	−0.904264	−0.452132	−	0.891951i	$-0.649336\pi$
$827$	−26.0045	−0.904264	−0.452132	+	0.891951i	$0.649336\pi$
$828$	0	0
$829$	49.0052	1.70202	0.851010	−	0.525150i	$-0.175991\pi$
$829$	49.0052	1.70202	0.851010	+	0.525150i	$0.175991\pi$
$830$	5.93814	0.206116
$831$	0	0
$832$	−22.4221	−0.777346
$833$	−0.0139224	−0.000482383 0
$834$	0	0
$835$	−14.5848	−0.504729
$836$	1.18863	0.0411097
$837$	0	0
$838$	−43.7006	−1.50961
$839$	−10.6093	−0.366273	−0.183137	−	0.983087i	$-0.558625\pi$
$839$	−10.6093	−0.366273	−0.183137	+	0.983087i	$0.558625\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	7.84542	0.270371
$843$	0	0
$844$	−8.47168	−0.291607
$845$	−16.3896	−0.563820
$846$	0	0
$847$	−28.5414	−0.980694
$848$	45.5840	1.56536
$849$	0	0
$850$	−5.95336	−0.204199
$851$	5.27307	0.180758
$852$	0	0
$853$	−44.0684	−1.50887	−0.754436	−	0.656374i	$-0.772089\pi$
$853$	−44.0684	−1.50887	−0.754436	+	0.656374i	$0.772089\pi$
$854$	−54.0062	−1.84805
$855$	0	0
$856$	−14.6126	−0.499447
$857$	26.3167	0.898961	0.449480	−	0.893290i	$-0.351609\pi$
$857$	26.3167	0.898961	0.449480	+	0.893290i	$0.351609\pi$
$858$	0	0
$859$	7.80205	0.266202	0.133101	−	0.991102i	$-0.457506\pi$
$859$	7.80205	0.266202	0.133101	+	0.991102i	$0.457506\pi$
$860$	8.70205	0.296737
$861$	0	0
$862$	31.6016	1.07635
$863$	−4.04966	−0.137852	−0.0689260	−	0.997622i	$-0.521957\pi$
$863$	−4.04966	−0.137852	−0.0689260	+	0.997622i	$0.521957\pi$
$864$	0	0
$865$	1.04316	0.0354684
$866$	16.9141	0.574764
$867$	0	0
$868$	5.83618	0.198093
$869$	−6.45639	−0.219018
$870$	0	0
$871$	52.5911	1.78198
$872$	24.4305	0.827322
$873$	0	0
$874$	7.35332	0.248730
$875$	30.6552	1.03633
$876$	0	0
$877$	−48.5571	−1.63966	−0.819828	−	0.572610i	$-0.805931\pi$
$877$	−48.5571	−1.63966	−0.819828	+	0.572610i	$0.805931\pi$
$878$	−35.7859	−1.20772
$879$	0	0
$880$	3.24727	0.109465
$881$	−29.7511	−1.00234	−0.501170	−	0.865349i	$-0.667097\pi$
$881$	−29.7511	−1.00234	−0.501170	+	0.865349i	$0.667097\pi$
$882$	0	0
$883$	−38.6702	−1.30136	−0.650678	−	0.759353i	$-0.725515\pi$
$883$	−38.6702	−1.30136	−0.650678	+	0.759353i	$0.725515\pi$
$884$	−3.75502	−0.126295
$885$	0	0
$886$	58.6802	1.97140
$887$	19.8178	0.665417	0.332709	−	0.943030i	$-0.392037\pi$
$887$	19.8178	0.665417	0.332709	+	0.943030i	$0.392037\pi$
$888$	0	0
$889$	−35.5138	−1.19109
$890$	−4.35844	−0.146095
$891$	0	0
$892$	2.42190	0.0810911
$893$	56.3710	1.88638
$894$	0	0
$895$	−8.77705	−0.293385
$896$	36.0890	1.20565
$897$	0	0
$898$	−18.8725	−0.629783
$899$	3.84793	0.128336
$900$	0	0
$901$	−12.6421	−0.421168
$902$	−1.94989	−0.0649243
$903$	0	0
$904$	−38.3914	−1.27688
$905$	12.0290	0.399859
$906$	0	0
$907$	43.5513	1.44610	0.723048	−	0.690798i	$-0.242741\pi$
$907$	43.5513	1.44610	0.723048	+	0.690798i	$0.242741\pi$
$908$	−7.46167	−0.247624
$909$	0	0
$910$	30.9746	1.02680
$911$	−18.3499	−0.607958	−0.303979	−	0.952679i	$-0.598315\pi$
$911$	−18.3499	−0.607958	−0.303979	+	0.952679i	$0.598315\pi$
$912$	0	0
$913$	−1.12221	−0.0371398
$914$	−3.97026	−0.131325
$915$	0	0
$916$	3.73083	0.123270
$917$	−37.5941	−1.24147
$918$	0	0
$919$	18.0400	0.595084	0.297542	−	0.954709i	$-0.403833\pi$
$919$	18.0400	0.595084	0.297542	+	0.954709i	$0.403833\pi$
$920$	3.41157	0.112476
$921$	0	0
$922$	52.4565	1.72756
$923$	29.3599	0.966393
$924$	0	0
$925$	−14.6437	−0.481482
$926$	−19.0421	−0.625762
$927$	0	0
$928$	−3.15342	−0.103516
$929$	−8.78537	−0.288239	−0.144119	−	0.989560i	$-0.546035\pi$
$929$	−8.78537	−0.288239	−0.144119	+	0.989560i	$0.546035\pi$
$930$	0	0
$931$	−0.0477555	−0.00156512
$932$	−11.9938	−0.392869
$933$	0	0
$934$	4.34149	0.142058
$935$	−0.900583	−0.0294522
$936$	0	0
$937$	−11.6752	−0.381411	−0.190706	−	0.981647i	$-0.561078\pi$
$937$	−11.6752	−0.381411	−0.190706	+	0.981647i	$0.561078\pi$
$938$	−45.5382	−1.48687
$939$	0	0
$940$	−10.5193	−0.343103
$941$	7.98863	0.260422	0.130211	−	0.991486i	$-0.458435\pi$
$941$	7.98863	0.260422	0.130211	+	0.991486i	$0.458435\pi$
$942$	0	0
$943$	−2.68885	−0.0875611
$944$	26.9323	0.876571
$945$	0	0
$946$	−7.37779	−0.239873
$947$	−24.1149	−0.783628	−0.391814	−	0.920044i	$-0.628152\pi$
$947$	−24.1149	−0.783628	−0.391814	+	0.920044i	$0.628152\pi$
$948$	0	0
$949$	23.5534	0.764575
$950$	−20.4207	−0.662536
$951$	0	0
$952$	−8.08377	−0.261996
$953$	31.7749	1.02929	0.514645	−	0.857404i	$-0.327924\pi$
$953$	31.7749	1.02929	0.514645	+	0.857404i	$0.327924\pi$
$954$	0	0
$955$	12.2293	0.395731
$956$	−10.1277	−0.327555
$957$	0	0
$958$	13.2835	0.429171
$959$	28.3682	0.916057
$960$	0	0
$961$	−16.1935	−0.522370
$962$	−41.4363	−1.33596
$963$	0	0
$964$	2.31926	0.0746983
$965$	−9.67270	−0.311375
$966$	0	0
$967$	−35.4212	−1.13907	−0.569535	−	0.821967i	$-0.692877\pi$
$967$	−35.4212	−1.13907	−0.569535	+	0.821967i	$0.692877\pi$
$968$	24.7026	0.793970
$969$	0	0
$970$	−13.5778	−0.435958
$971$	−29.7556	−0.954904	−0.477452	−	0.878658i	$-0.658440\pi$
$971$	−29.7556	−0.954904	−0.477452	+	0.878658i	$0.658440\pi$
$972$	0	0
$973$	47.9763	1.53805
$974$	−37.4823	−1.20101
$975$	0	0
$976$	61.3522	1.96384
$977$	23.6394	0.756290	0.378145	−	0.925746i	$-0.376562\pi$
$977$	23.6394	0.756290	0.378145	+	0.925746i	$0.376562\pi$
$978$	0	0
$979$	0.823675	0.0263248
$980$	0.00891163	0.000284672 0
$981$	0	0
$982$	44.6177	1.42381
$983$	−38.8234	−1.23827	−0.619137	−	0.785283i	$-0.712517\pi$
$983$	−38.8234	−1.23827	−0.619137	+	0.785283i	$0.712517\pi$
$984$	0	0
$985$	6.42947	0.204860
$986$	2.14375	0.0682709
$987$	0	0
$988$	−12.8802	−0.409773
$989$	−10.1738	−0.323508
$990$	0	0
$991$	38.4439	1.22121	0.610605	−	0.791935i	$-0.290926\pi$
$991$	38.4439	1.22121	0.610605	+	0.791935i	$0.290926\pi$
$992$	−12.1341	−0.385258
$993$	0	0
$994$	−25.4225	−0.806352
$995$	30.2780	0.959878
$996$	0	0
$997$	22.2531	0.704761	0.352381	−	0.935857i	$-0.385372\pi$
$997$	22.2531	0.704761	0.352381	+	0.935857i	$0.385372\pi$
$998$	4.47957	0.141798
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.18	✓	22
3.2	odd	2		6003.2.a.u.1.5	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.18	✓	22	1.1	even	1	trivial
6003.2.a.u.1.5	yes	22	3.2	odd	2

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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+1.60427 q^{2} +0.573689 q^{4} -1.49095 q^{5} +2.64378 q^{7} -2.28819 q^{8} +O(q^{10})\)	\(q+1.60427 q^{2} +0.573689 q^{4} -1.49095 q^{5} +2.64378 q^{7} -2.28819 q^{8} -2.39188 q^{10} +0.452028 q^{11} -4.89824 q^{13} +4.24134 q^{14} -4.81826 q^{16} +1.33628 q^{17} +4.58359 q^{19} -0.855340 q^{20} +0.725176 q^{22} +1.00000 q^{23} -2.77708 q^{25} -7.85811 q^{26} +1.51671 q^{28} +1.00000 q^{29} +3.84793 q^{31} -3.15342 q^{32} +2.14375 q^{34} -3.94174 q^{35} +5.27307 q^{37} +7.35332 q^{38} +3.41157 q^{40} -2.68885 q^{41} -10.1738 q^{43} +0.259324 q^{44} +1.60427 q^{46} +12.2984 q^{47} -0.0104188 q^{49} -4.45519 q^{50} -2.81007 q^{52} -9.46067 q^{53} -0.673950 q^{55} -6.04948 q^{56} +1.60427 q^{58} -5.58963 q^{59} -12.7333 q^{61} +6.17312 q^{62} +4.57758 q^{64} +7.30302 q^{65} -10.7367 q^{67} +0.766606 q^{68} -6.32362 q^{70} -5.99397 q^{71} -4.80854 q^{73} +8.45943 q^{74} +2.62955 q^{76} +1.19506 q^{77} -14.2832 q^{79} +7.18377 q^{80} -4.31365 q^{82} -2.48262 q^{83} -1.99232 q^{85} -16.3215 q^{86} -1.03433 q^{88} +1.82218 q^{89} -12.9499 q^{91} +0.573689 q^{92} +19.7301 q^{94} -6.83388 q^{95} +5.67663 q^{97} -0.0167146 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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