LMFDB - Embedded newform 6003.2.a.w.1.17

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:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+0.473233 q^{2} -1.77605 q^{4} -1.04483 q^{5} +3.83278 q^{7} -1.78695 q^{8} +O(q^{10})$	\(q+0.473233 q^{2} -1.77605 q^{4} -1.04483 q^{5} +3.83278 q^{7} -1.78695 q^{8} -0.494446 q^{10} +2.56396 q^{11} +2.32837 q^{13} +1.81380 q^{14} +2.70646 q^{16} +6.35999 q^{17} +7.47559 q^{19} +1.85566 q^{20} +1.21335 q^{22} -1.00000 q^{23} -3.90834 q^{25} +1.10186 q^{26} -6.80721 q^{28} +1.00000 q^{29} +8.99906 q^{31} +4.85469 q^{32} +3.00976 q^{34} -4.00459 q^{35} -5.17971 q^{37} +3.53770 q^{38} +1.86705 q^{40} +0.118500 q^{41} -2.97677 q^{43} -4.55372 q^{44} -0.473233 q^{46} -10.7617 q^{47} +7.69021 q^{49} -1.84956 q^{50} -4.13531 q^{52} -5.29192 q^{53} -2.67889 q^{55} -6.84899 q^{56} +0.473233 q^{58} -9.96740 q^{59} +3.51664 q^{61} +4.25865 q^{62} -3.11551 q^{64} -2.43274 q^{65} +14.1207 q^{67} -11.2957 q^{68} -1.89510 q^{70} +10.5253 q^{71} -3.05985 q^{73} -2.45121 q^{74} -13.2770 q^{76} +9.82709 q^{77} -0.286819 q^{79} -2.82777 q^{80} +0.0560781 q^{82} +4.82285 q^{83} -6.64508 q^{85} -1.40870 q^{86} -4.58167 q^{88} -1.13200 q^{89} +8.92414 q^{91} +1.77605 q^{92} -5.09277 q^{94} -7.81069 q^{95} -2.76909 q^{97} +3.63926 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * 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 $

$2$	0.473233	0.334626	0.167313	−	0.985904i	$-0.446491\pi$
$2$	0.473233	0.334626	0.167313	+	0.985904i	$0.446491\pi$
$3$	0	0
$3$	0	0
$4$	−1.77605	−0.888025
$5$	−1.04483	−0.467260	−0.233630	−	0.972326i	$-0.575060\pi$
$5$	−1.04483	−0.467260	−0.233630	+	0.972326i	$0.575060\pi$
$6$	0	0
$7$	3.83278	1.44866	0.724328	−	0.689456i	$-0.242150\pi$
$7$	3.83278	1.44866	0.724328	+	0.689456i	$0.242150\pi$
$8$	−1.78695	−0.631783
$9$	0	0
$10$	−0.494446	−0.156357
$11$	2.56396	0.773063	0.386531	−	0.922276i	$-0.373673\pi$
$11$	2.56396	0.773063	0.386531	+	0.922276i	$0.373673\pi$
$12$	0	0
$13$	2.32837	0.645774	0.322887	−	0.946437i	$-0.395347\pi$
$13$	2.32837	0.645774	0.322887	+	0.946437i	$0.395347\pi$
$14$	1.81380	0.484758
$15$	0	0
$16$	2.70646	0.676614
$17$	6.35999	1.54253	0.771263	−	0.636517i	$-0.219626\pi$
$17$	6.35999	1.54253	0.771263	+	0.636517i	$0.219626\pi$
$18$	0	0
$19$	7.47559	1.71502	0.857509	−	0.514469i	$-0.172011\pi$
$19$	7.47559	1.71502	0.857509	+	0.514469i	$0.172011\pi$
$20$	1.85566	0.414939
$21$	0	0
$22$	1.21335	0.258687
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.90834	−0.781668
$26$	1.10186	0.216093
$27$	0	0
$28$	−6.80721	−1.28644
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	8.99906	1.61628	0.808139	−	0.588991i	$-0.200475\pi$
$31$	8.99906	1.61628	0.808139	+	0.588991i	$0.200475\pi$
$32$	4.85469	0.858196
$33$	0	0
$34$	3.00976	0.516169
$35$	−4.00459	−0.676899
$36$	0	0
$37$	−5.17971	−0.851539	−0.425769	−	0.904832i	$-0.639996\pi$
$37$	−5.17971	−0.851539	−0.425769	+	0.904832i	$0.639996\pi$
$38$	3.53770	0.573890
$39$	0	0
$40$	1.86705	0.295207
$41$	0.118500	0.0185066	0.00925330	−	0.999957i	$-0.497055\pi$
$41$	0.118500	0.0185066	0.00925330	+	0.999957i	$0.497055\pi$
$42$	0	0
$43$	−2.97677	−0.453953	−0.226976	−	0.973900i	$-0.572884\pi$
$43$	−2.97677	−0.453953	−0.226976	+	0.973900i	$0.572884\pi$
$44$	−4.55372	−0.686499
$45$	0	0
$46$	−0.473233	−0.0697744
$47$	−10.7617	−1.56975	−0.784875	−	0.619654i	$-0.787273\pi$
$47$	−10.7617	−1.56975	−0.784875	+	0.619654i	$0.787273\pi$
$48$	0	0
$49$	7.69021	1.09860
$50$	−1.84956	−0.261567
$51$	0	0
$52$	−4.13531	−0.573464
$53$	−5.29192	−0.726901	−0.363451	−	0.931613i	$-0.618401\pi$
$53$	−5.29192	−0.726901	−0.363451	+	0.931613i	$0.618401\pi$
$54$	0	0
$55$	−2.67889	−0.361221
$56$	−6.84899	−0.915235
$57$	0	0
$58$	0.473233	0.0621385
$59$	−9.96740	−1.29764	−0.648822	−	0.760940i	$-0.724738\pi$
$59$	−9.96740	−1.29764	−0.648822	+	0.760940i	$0.724738\pi$
$60$	0	0
$61$	3.51664	0.450260	0.225130	−	0.974329i	$-0.427719\pi$
$61$	3.51664	0.450260	0.225130	+	0.974329i	$0.427719\pi$
$62$	4.25865	0.540849
$63$	0	0
$64$	−3.11551	−0.389439
$65$	−2.43274	−0.301745
$66$	0	0
$67$	14.1207	1.72512	0.862561	−	0.505952i	$-0.168859\pi$
$67$	14.1207	1.72512	0.862561	+	0.505952i	$0.168859\pi$
$68$	−11.2957	−1.36980
$69$	0	0
$70$	−1.89510	−0.226508
$71$	10.5253	1.24912	0.624560	−	0.780977i	$-0.285278\pi$
$71$	10.5253	1.24912	0.624560	+	0.780977i	$0.285278\pi$
$72$	0	0
$73$	−3.05985	−0.358129	−0.179064	−	0.983837i	$-0.557307\pi$
$73$	−3.05985	−0.358129	−0.179064	+	0.983837i	$0.557307\pi$
$74$	−2.45121	−0.284947
$75$	0	0
$76$	−13.2770	−1.52298
$77$	9.82709	1.11990
$78$	0	0
$79$	−0.286819	−0.0322696	−0.0161348	−	0.999870i	$-0.505136\pi$
$79$	−0.286819	−0.0322696	−0.0161348	+	0.999870i	$0.505136\pi$
$80$	−2.82777	−0.316155
$81$	0	0
$82$	0.0560781	0.00619279
$83$	4.82285	0.529376	0.264688	−	0.964334i	$-0.414731\pi$
$83$	4.82285	0.529376	0.264688	+	0.964334i	$0.414731\pi$
$84$	0	0
$85$	−6.64508	−0.720760
$86$	−1.40870	−0.151904
$87$	0	0
$88$	−4.58167	−0.488408
$89$	−1.13200	−0.119992	−0.0599961	−	0.998199i	$-0.519109\pi$
$89$	−1.13200	−0.119992	−0.0599961	+	0.998199i	$0.519109\pi$
$90$	0	0
$91$	8.92414	0.935504
$92$	1.77605	0.185166
$93$	0	0
$94$	−5.09277	−0.525279
$95$	−7.81069	−0.801360
$96$	0	0
$97$	−2.76909	−0.281158	−0.140579	−	0.990069i	$-0.544896\pi$
$97$	−2.76909	−0.281158	−0.140579	+	0.990069i	$0.544896\pi$
$98$	3.63926	0.367621
$99$	0	0
$100$	6.94141	0.694141
$101$	−7.62918	−0.759132	−0.379566	−	0.925165i	$-0.623927\pi$
$101$	−7.62918	−0.759132	−0.379566	+	0.925165i	$0.623927\pi$
$102$	0	0
$103$	−1.31505	−0.129575	−0.0647877	−	0.997899i	$-0.520637\pi$
$103$	−1.31505	−0.129575	−0.0647877	+	0.997899i	$0.520637\pi$
$104$	−4.16069	−0.407989
$105$	0	0
$106$	−2.50431	−0.243240
$107$	−3.09134	−0.298852	−0.149426	−	0.988773i	$-0.547742\pi$
$107$	−3.09134	−0.298852	−0.149426	+	0.988773i	$0.547742\pi$
$108$	0	0
$109$	−10.6005	−1.01534	−0.507672	−	0.861551i	$-0.669494\pi$
$109$	−10.6005	−1.01534	−0.507672	+	0.861551i	$0.669494\pi$
$110$	−1.26774	−0.120874
$111$	0	0
$112$	10.3733	0.980181
$113$	2.36263	0.222257	0.111129	−	0.993806i	$-0.464553\pi$
$113$	2.36263	0.222257	0.111129	+	0.993806i	$0.464553\pi$
$114$	0	0
$115$	1.04483	0.0974304
$116$	−1.77605	−0.164902
$117$	0	0
$118$	−4.71690	−0.434226
$119$	24.3765	2.23459
$120$	0	0
$121$	−4.42611	−0.402374
$122$	1.66419	0.150669
$123$	0	0
$124$	−15.9828	−1.43530
$125$	9.30766	0.832502
$126$	0	0
$127$	14.1438	1.25506	0.627528	−	0.778594i	$-0.284067\pi$
$127$	14.1438	1.25506	0.627528	+	0.778594i	$0.284067\pi$
$128$	−11.1837	−0.988512
$129$	0	0
$130$	−1.15125	−0.100972
$131$	17.7401	1.54996	0.774979	−	0.631986i	$-0.217760\pi$
$131$	17.7401	1.54996	0.774979	+	0.631986i	$0.217760\pi$
$132$	0	0
$133$	28.6523	2.48447
$134$	6.68240	0.577271
$135$	0	0
$136$	−11.3650	−0.974541
$137$	8.12457	0.694129	0.347064	−	0.937841i	$-0.387179\pi$
$137$	8.12457	0.694129	0.347064	+	0.937841i	$0.387179\pi$
$138$	0	0
$139$	3.00074	0.254519	0.127260	−	0.991869i	$-0.459382\pi$
$139$	3.00074	0.254519	0.127260	+	0.991869i	$0.459382\pi$
$140$	7.11235	0.601103
$141$	0	0
$142$	4.98091	0.417989
$143$	5.96985	0.499224
$144$	0	0
$145$	−1.04483	−0.0867680
$146$	−1.44802	−0.119839
$147$	0	0
$148$	9.19942	0.756188
$149$	15.2424	1.24870	0.624352	−	0.781143i	$-0.285363\pi$
$149$	15.2424	1.24870	0.624352	+	0.781143i	$0.285363\pi$
$150$	0	0
$151$	7.08563	0.576620	0.288310	−	0.957537i	$-0.406907\pi$
$151$	7.08563	0.576620	0.288310	+	0.957537i	$0.406907\pi$
$152$	−13.3585	−1.08352
$153$	0	0
$154$	4.65051	0.374748
$155$	−9.40244	−0.755222
$156$	0	0
$157$	−1.33644	−0.106659	−0.0533297	−	0.998577i	$-0.516983\pi$
$157$	−1.33644	−0.106659	−0.0533297	+	0.998577i	$0.516983\pi$
$158$	−0.135732	−0.0107983
$159$	0	0
$160$	−5.07230	−0.401001
$161$	−3.83278	−0.302065
$162$	0	0
$163$	9.55986	0.748786	0.374393	−	0.927270i	$-0.377851\pi$
$163$	9.55986	0.748786	0.374393	+	0.927270i	$0.377851\pi$
$164$	−0.210462	−0.0164343
$165$	0	0
$166$	2.28233	0.177143
$167$	−12.1300	−0.938648	−0.469324	−	0.883026i	$-0.655502\pi$
$167$	−12.1300	−0.938648	−0.469324	+	0.883026i	$0.655502\pi$
$168$	0	0
$169$	−7.57868	−0.582975
$170$	−3.14467	−0.241185
$171$	0	0
$172$	5.28689	0.403121
$173$	−23.8506	−1.81333	−0.906664	−	0.421853i	$-0.861380\pi$
$173$	−23.8506	−1.81333	−0.906664	+	0.421853i	$0.861380\pi$
$174$	0	0
$175$	−14.9798	−1.13237
$176$	6.93924	0.523065
$177$	0	0
$178$	−0.535702	−0.0401525
$179$	−20.3159	−1.51848	−0.759240	−	0.650811i	$-0.774429\pi$
$179$	−20.3159	−1.51848	−0.759240	+	0.650811i	$0.774429\pi$
$180$	0	0
$181$	15.6061	1.15999	0.579997	−	0.814619i	$-0.303054\pi$
$181$	15.6061	1.15999	0.579997	+	0.814619i	$0.303054\pi$
$182$	4.22320	0.313044
$183$	0	0
$184$	1.78695	0.131736
$185$	5.41189	0.397890
$186$	0	0
$187$	16.3068	1.19247
$188$	19.1133	1.39398
$189$	0	0
$190$	−3.69627	−0.268156
$191$	−19.5710	−1.41611	−0.708055	−	0.706157i	$-0.750427\pi$
$191$	−19.5710	−1.41611	−0.708055	+	0.706157i	$0.750427\pi$
$192$	0	0
$193$	5.09645	0.366850	0.183425	−	0.983034i	$-0.441282\pi$
$193$	5.09645	0.366850	0.183425	+	0.983034i	$0.441282\pi$
$194$	−1.31042	−0.0940829
$195$	0	0
$196$	−13.6582	−0.975586
$197$	23.2485	1.65639	0.828195	−	0.560440i	$-0.189368\pi$
$197$	23.2485	1.65639	0.828195	+	0.560440i	$0.189368\pi$
$198$	0	0
$199$	24.2684	1.72034	0.860170	−	0.510007i	$-0.170357\pi$
$199$	24.2684	1.72034	0.860170	+	0.510007i	$0.170357\pi$
$200$	6.98402	0.493844
$201$	0	0
$202$	−3.61038	−0.254025
$203$	3.83278	0.269009
$204$	0	0
$205$	−0.123812	−0.00864739
$206$	−0.622324	−0.0433594
$207$	0	0
$208$	6.30164	0.436940
$209$	19.1671	1.32582
$210$	0	0
$211$	−13.0565	−0.898847	−0.449423	−	0.893319i	$-0.648371\pi$
$211$	−13.0565	−0.898847	−0.449423	+	0.893319i	$0.648371\pi$
$212$	9.39872	0.645507
$213$	0	0
$214$	−1.46293	−0.100004
$215$	3.11020	0.212114
$216$	0	0
$217$	34.4914	2.34143
$218$	−5.01651	−0.339761
$219$	0	0
$220$	4.75784	0.320774
$221$	14.8084	0.996123
$222$	0	0
$223$	21.0983	1.41285	0.706424	−	0.707789i	$-0.250307\pi$
$223$	21.0983	1.41285	0.706424	+	0.707789i	$0.250307\pi$
$224$	18.6070	1.24323
$225$	0	0
$226$	1.11807	0.0743731
$227$	28.6355	1.90060	0.950301	−	0.311333i	$-0.100775\pi$
$227$	28.6355	1.90060	0.950301	+	0.311333i	$0.100775\pi$
$228$	0	0
$229$	−26.5257	−1.75287	−0.876433	−	0.481524i	$-0.840084\pi$
$229$	−26.5257	−1.75287	−0.876433	+	0.481524i	$0.840084\pi$
$230$	0.494446	0.0326028
$231$	0	0
$232$	−1.78695	−0.117319
$233$	−2.34122	−0.153378	−0.0766892	−	0.997055i	$-0.524435\pi$
$233$	−2.34122	−0.153378	−0.0766892	+	0.997055i	$0.524435\pi$
$234$	0	0
$235$	11.2441	0.733481
$236$	17.7026	1.15234
$237$	0	0
$238$	11.5357	0.747751
$239$	3.07650	0.199002	0.0995011	−	0.995037i	$-0.468275\pi$
$239$	3.07650	0.199002	0.0995011	+	0.995037i	$0.468275\pi$
$240$	0	0
$241$	−19.8645	−1.27958	−0.639791	−	0.768549i	$-0.720979\pi$
$241$	−19.8645	−1.27958	−0.639791	+	0.768549i	$0.720979\pi$
$242$	−2.09458	−0.134645
$243$	0	0
$244$	−6.24573	−0.399842
$245$	−8.03493	−0.513333
$246$	0	0
$247$	17.4060	1.10751
$248$	−16.0809	−1.02114
$249$	0	0
$250$	4.40469	0.278577
$251$	−12.3435	−0.779116	−0.389558	−	0.921002i	$-0.627372\pi$
$251$	−12.3435	−0.779116	−0.389558	+	0.921002i	$0.627372\pi$
$252$	0	0
$253$	−2.56396	−0.161195
$254$	6.69329	0.419975
$255$	0	0
$256$	0.938515	0.0586572
$257$	0.764367	0.0476799	0.0238399	−	0.999716i	$-0.492411\pi$
$257$	0.764367	0.0476799	0.0238399	+	0.999716i	$0.492411\pi$
$258$	0	0
$259$	−19.8527	−1.23359
$260$	4.32067	0.267957
$261$	0	0
$262$	8.39520	0.518657
$263$	−12.4337	−0.766693	−0.383346	−	0.923605i	$-0.625228\pi$
$263$	−12.4337	−0.766693	−0.383346	+	0.923605i	$0.625228\pi$
$264$	0	0
$265$	5.52913	0.339652
$266$	13.5592	0.831369
$267$	0	0
$268$	−25.0791	−1.53195
$269$	17.2895	1.05416	0.527080	−	0.849816i	$-0.323287\pi$
$269$	17.2895	1.05416	0.527080	+	0.849816i	$0.323287\pi$
$270$	0	0
$271$	12.2849	0.746254	0.373127	−	0.927780i	$-0.378286\pi$
$271$	12.2849	0.746254	0.373127	+	0.927780i	$0.378286\pi$
$272$	17.2130	1.04369
$273$	0	0
$274$	3.84481	0.232274
$275$	−10.0208	−0.604279
$276$	0	0
$277$	24.0877	1.44729	0.723645	−	0.690172i	$-0.242465\pi$
$277$	24.0877	1.44729	0.723645	+	0.690172i	$0.242465\pi$
$278$	1.42005	0.0851688
$279$	0	0
$280$	7.15600	0.427653
$281$	−25.8168	−1.54010	−0.770050	−	0.637983i	$-0.779769\pi$
$281$	−25.8168	−1.54010	−0.770050	+	0.637983i	$0.779769\pi$
$282$	0	0
$283$	−1.67500	−0.0995687	−0.0497843	−	0.998760i	$-0.515853\pi$
$283$	−1.67500	−0.0995687	−0.0497843	+	0.998760i	$0.515853\pi$
$284$	−18.6934	−1.10925
$285$	0	0
$286$	2.82513	0.167053
$287$	0.454185	0.0268097
$288$	0	0
$289$	23.4495	1.37938
$290$	−0.494446	−0.0290349
$291$	0	0
$292$	5.43445	0.318027
$293$	−3.20666	−0.187335	−0.0936674	−	0.995604i	$-0.529859\pi$
$293$	−3.20666	−0.187335	−0.0936674	+	0.995604i	$0.529859\pi$
$294$	0	0
$295$	10.4142	0.606337
$296$	9.25589	0.537987
$297$	0	0
$298$	7.21319	0.417849
$299$	−2.32837	−0.134653
$300$	0	0
$301$	−11.4093	−0.657621
$302$	3.35315	0.192952
$303$	0	0
$304$	20.2324	1.16041
$305$	−3.67427	−0.210388
$306$	0	0
$307$	−10.2790	−0.586655	−0.293328	−	0.956012i	$-0.594763\pi$
$307$	−10.2790	−0.586655	−0.293328	+	0.956012i	$0.594763\pi$
$308$	−17.4534	−0.994501
$309$	0	0
$310$	−4.44955	−0.252717
$311$	−8.33747	−0.472775	−0.236387	−	0.971659i	$-0.575963\pi$
$311$	−8.33747	−0.472775	−0.236387	+	0.971659i	$0.575963\pi$
$312$	0	0
$313$	−0.141232	−0.00798291	−0.00399146	−	0.999992i	$-0.501271\pi$
$313$	−0.141232	−0.00798291	−0.00399146	+	0.999992i	$0.501271\pi$
$314$	−0.632447	−0.0356911
$315$	0	0
$316$	0.509404	0.0286562
$317$	−7.06189	−0.396635	−0.198318	−	0.980138i	$-0.563548\pi$
$317$	−7.06189	−0.396635	−0.198318	+	0.980138i	$0.563548\pi$
$318$	0	0
$319$	2.56396	0.143554
$320$	3.25517	0.181969
$321$	0	0
$322$	−1.81380	−0.101079
$323$	47.5447	2.64546
$324$	0	0
$325$	−9.10007	−0.504781
$326$	4.52404	0.250563
$327$	0	0
$328$	−0.211754	−0.0116922
$329$	−41.2471	−2.27403
$330$	0	0
$331$	25.4463	1.39866	0.699329	−	0.714800i	$-0.253482\pi$
$331$	25.4463	1.39866	0.699329	+	0.714800i	$0.253482\pi$
$332$	−8.56562	−0.470099
$333$	0	0
$334$	−5.74032	−0.314096
$335$	−14.7537	−0.806081
$336$	0	0
$337$	−15.8723	−0.864617	−0.432308	−	0.901726i	$-0.642301\pi$
$337$	−15.8723	−0.864617	−0.432308	+	0.901726i	$0.642301\pi$
$338$	−3.58648	−0.195079
$339$	0	0
$340$	11.8020	0.640053
$341$	23.0732	1.24948
$342$	0	0
$343$	2.64543	0.142840
$344$	5.31934	0.286800
$345$	0	0
$346$	−11.2869	−0.606787
$347$	−3.77366	−0.202581	−0.101290	−	0.994857i	$-0.532297\pi$
$347$	−3.77366	−0.202581	−0.101290	+	0.994857i	$0.532297\pi$
$348$	0	0
$349$	−8.83978	−0.473183	−0.236591	−	0.971609i	$-0.576030\pi$
$349$	−8.83978	−0.473183	−0.236591	+	0.971609i	$0.576030\pi$
$350$	−7.08894	−0.378920
$351$	0	0
$352$	12.4472	0.663439
$353$	−26.2225	−1.39568	−0.697840	−	0.716253i	$-0.745856\pi$
$353$	−26.2225	−1.39568	−0.697840	+	0.716253i	$0.745856\pi$
$354$	0	0
$355$	−10.9971	−0.583664
$356$	2.01050	0.106556
$357$	0	0
$358$	−9.61414	−0.508123
$359$	−17.3443	−0.915398	−0.457699	−	0.889107i	$-0.651326\pi$
$359$	−17.3443	−0.915398	−0.457699	+	0.889107i	$0.651326\pi$
$360$	0	0
$361$	36.8845	1.94129
$362$	7.38533	0.388164
$363$	0	0
$364$	−15.8497	−0.830751
$365$	3.19701	0.167339
$366$	0	0
$367$	−10.0965	−0.527032	−0.263516	−	0.964655i	$-0.584882\pi$
$367$	−10.0965	−0.527032	−0.263516	+	0.964655i	$0.584882\pi$
$368$	−2.70646	−0.141084
$369$	0	0
$370$	2.56108	0.133144
$371$	−20.2828	−1.05303
$372$	0	0
$373$	−13.0445	−0.675421	−0.337710	−	0.941250i	$-0.609652\pi$
$373$	−13.0445	−0.675421	−0.337710	+	0.941250i	$0.609652\pi$
$374$	7.71690	0.399031
$375$	0	0
$376$	19.2306	0.991741
$377$	2.32837	0.119917
$378$	0	0
$379$	7.81802	0.401585	0.200792	−	0.979634i	$-0.435648\pi$
$379$	7.81802	0.401585	0.200792	+	0.979634i	$0.435648\pi$
$380$	13.8722	0.711628
$381$	0	0
$382$	−9.26166	−0.473868
$383$	−3.73452	−0.190825	−0.0954126	−	0.995438i	$-0.530417\pi$
$383$	−3.73452	−0.190825	−0.0954126	+	0.995438i	$0.530417\pi$
$384$	0	0
$385$	−10.2676	−0.523285
$386$	2.41181	0.122758
$387$	0	0
$388$	4.91804	0.249675
$389$	−22.7578	−1.15387	−0.576933	−	0.816792i	$-0.695751\pi$
$389$	−22.7578	−1.15387	−0.576933	+	0.816792i	$0.695751\pi$
$390$	0	0
$391$	−6.35999	−0.321639
$392$	−13.7420	−0.694078
$393$	0	0
$394$	11.0020	0.554272
$395$	0.299675	0.0150783
$396$	0	0
$397$	14.6267	0.734093	0.367046	−	0.930203i	$-0.380369\pi$
$397$	14.6267	0.734093	0.367046	+	0.930203i	$0.380369\pi$
$398$	11.4846	0.575671
$399$	0	0
$400$	−10.5778	−0.528888
$401$	28.2873	1.41260	0.706301	−	0.707912i	$-0.250363\pi$
$401$	28.2873	1.41260	0.706301	+	0.707912i	$0.250363\pi$
$402$	0	0
$403$	20.9532	1.04375
$404$	13.5498	0.674128
$405$	0	0
$406$	1.81380	0.0900173
$407$	−13.2806	−0.658293
$408$	0	0
$409$	−15.9885	−0.790581	−0.395291	−	0.918556i	$-0.629356\pi$
$409$	−15.9885	−0.790581	−0.395291	+	0.918556i	$0.629356\pi$
$410$	−0.0585918	−0.00289364
$411$	0	0
$412$	2.33559	0.115066
$413$	−38.2028	−1.87984
$414$	0	0
$415$	−5.03903	−0.247356
$416$	11.3035	0.554201
$417$	0	0
$418$	9.07051	0.443653
$419$	−20.1167	−0.982766	−0.491383	−	0.870943i	$-0.663509\pi$
$419$	−20.1167	−0.982766	−0.491383	+	0.870943i	$0.663509\pi$
$420$	0	0
$421$	6.27858	0.305999	0.153000	−	0.988226i	$-0.451107\pi$
$421$	6.27858	0.305999	0.153000	+	0.988226i	$0.451107\pi$
$422$	−6.17877	−0.300778
$423$	0	0
$424$	9.45641	0.459244
$425$	−24.8570	−1.20574
$426$	0	0
$427$	13.4785	0.652271
$428$	5.49038	0.265388
$429$	0	0
$430$	1.47185	0.0709789
$431$	12.1897	0.587159	0.293579	−	0.955935i	$-0.405153\pi$
$431$	12.1897	0.587159	0.293579	+	0.955935i	$0.405153\pi$
$432$	0	0
$433$	25.3379	1.21766	0.608830	−	0.793301i	$-0.291639\pi$
$433$	25.3379	1.21766	0.608830	+	0.793301i	$0.291639\pi$
$434$	16.3225	0.783504
$435$	0	0
$436$	18.8270	0.901651
$437$	−7.47559	−0.357606
$438$	0	0
$439$	3.21937	0.153652	0.0768260	−	0.997045i	$-0.475521\pi$
$439$	3.21937	0.153652	0.0768260	+	0.997045i	$0.475521\pi$
$440$	4.78705	0.228213
$441$	0	0
$442$	7.00784	0.333329
$443$	20.6170	0.979541	0.489771	−	0.871851i	$-0.337081\pi$
$443$	20.6170	0.979541	0.489771	+	0.871851i	$0.337081\pi$
$444$	0	0
$445$	1.18275	0.0560675
$446$	9.98442	0.472776
$447$	0	0
$448$	−11.9411	−0.564163
$449$	−32.3690	−1.52759	−0.763793	−	0.645461i	$-0.776665\pi$
$449$	−32.3690	−1.52759	−0.763793	+	0.645461i	$0.776665\pi$
$450$	0	0
$451$	0.303829	0.0143068
$452$	−4.19614	−0.197370
$453$	0	0
$454$	13.5512	0.635991
$455$	−9.32417	−0.437124
$456$	0	0
$457$	0.711685	0.0332912	0.0166456	−	0.999861i	$-0.494701\pi$
$457$	0.711685	0.0332912	0.0166456	+	0.999861i	$0.494701\pi$
$458$	−12.5528	−0.586555
$459$	0	0
$460$	−1.85566	−0.0865207
$461$	13.5927	0.633076	0.316538	−	0.948580i	$-0.397480\pi$
$461$	13.5927	0.633076	0.316538	+	0.948580i	$0.397480\pi$
$462$	0	0
$463$	39.6700	1.84362	0.921811	−	0.387640i	$-0.126710\pi$
$463$	39.6700	1.84362	0.921811	+	0.387640i	$0.126710\pi$
$464$	2.70646	0.125644
$465$	0	0
$466$	−1.10794	−0.0513244
$467$	3.81127	0.176365	0.0881824	−	0.996104i	$-0.471894\pi$
$467$	3.81127	0.176365	0.0881824	+	0.996104i	$0.471894\pi$
$468$	0	0
$469$	54.1217	2.49911
$470$	5.32106	0.245442
$471$	0	0
$472$	17.8113	0.819829
$473$	−7.63231	−0.350934
$474$	0	0
$475$	−29.2172	−1.34058
$476$	−43.2938	−1.98437
$477$	0	0
$478$	1.45590	0.0665914
$479$	24.6634	1.12690	0.563450	−	0.826150i	$-0.309474\pi$
$479$	24.6634	1.12690	0.563450	+	0.826150i	$0.309474\pi$
$480$	0	0
$481$	−12.0603	−0.549902
$482$	−9.40051	−0.428182
$483$	0	0
$484$	7.86100	0.357318
$485$	2.89321	0.131374
$486$	0	0
$487$	29.2115	1.32370	0.661850	−	0.749636i	$-0.269771\pi$
$487$	29.2115	1.32370	0.661850	+	0.749636i	$0.269771\pi$
$488$	−6.28407	−0.284466
$489$	0	0
$490$	−3.80239	−0.171775
$491$	7.93069	0.357907	0.178953	−	0.983858i	$-0.442729\pi$
$491$	7.93069	0.357907	0.178953	+	0.983858i	$0.442729\pi$
$492$	0	0
$493$	6.35999	0.286440
$494$	8.23708	0.370604
$495$	0	0
$496$	24.3556	1.09360
$497$	40.3411	1.80954
$498$	0	0
$499$	36.4840	1.63325	0.816623	−	0.577171i	$-0.195843\pi$
$499$	36.4840	1.63325	0.816623	+	0.577171i	$0.195843\pi$
$500$	−16.5309	−0.739283
$501$	0	0
$502$	−5.84136	−0.260713
$503$	26.4478	1.17925	0.589624	−	0.807678i	$-0.299276\pi$
$503$	26.4478	1.17925	0.589624	+	0.807678i	$0.299276\pi$
$504$	0	0
$505$	7.97116	0.354712
$506$	−1.21335	−0.0539400
$507$	0	0
$508$	−25.1200	−1.11452
$509$	−34.7697	−1.54114	−0.770571	−	0.637354i	$-0.780029\pi$
$509$	−34.7697	−1.54114	−0.770571	+	0.637354i	$0.780029\pi$
$510$	0	0
$511$	−11.7278	−0.518805
$512$	22.8116	1.00814
$513$	0	0
$514$	0.361724	0.0159549
$515$	1.37399	0.0605454
$516$	0	0
$517$	−27.5925	−1.21352
$518$	−9.39495	−0.412790
$519$	0	0
$520$	4.34719	0.190637
$521$	−18.9963	−0.832244	−0.416122	−	0.909309i	$-0.636611\pi$
$521$	−18.9963	−0.832244	−0.416122	+	0.909309i	$0.636611\pi$
$522$	0	0
$523$	4.84802	0.211989	0.105995	−	0.994367i	$-0.466197\pi$
$523$	4.84802	0.211989	0.105995	+	0.994367i	$0.466197\pi$
$524$	−31.5073	−1.37640
$525$	0	0
$526$	−5.88402	−0.256556
$527$	57.2340	2.49315
$528$	0	0
$529$	1.00000	0.0434783
$530$	2.61657	0.113656
$531$	0	0
$532$	−50.8880	−2.20627
$533$	0.275912	0.0119511
$534$	0	0
$535$	3.22991	0.139641
$536$	−25.2331	−1.08990
$537$	0	0
$538$	8.18197	0.352750
$539$	19.7174	0.849288
$540$	0	0
$541$	8.52437	0.366491	0.183246	−	0.983067i	$-0.441340\pi$
$541$	8.52437	0.366491	0.183246	+	0.983067i	$0.441340\pi$
$542$	5.81361	0.249716
$543$	0	0
$544$	30.8758	1.32379
$545$	11.0757	0.474429
$546$	0	0
$547$	40.4348	1.72887	0.864433	−	0.502748i	$-0.167678\pi$
$547$	40.4348	1.72887	0.864433	+	0.502748i	$0.167678\pi$
$548$	−14.4296	−0.616404
$549$	0	0
$550$	−4.74219	−0.202207
$551$	7.47559	0.318471
$552$	0	0
$553$	−1.09931	−0.0467475
$554$	11.3991	0.484301
$555$	0	0
$556$	−5.32946	−0.226019
$557$	−0.156519	−0.00663191	−0.00331595	−	0.999995i	$-0.501056\pi$
$557$	−0.156519	−0.00663191	−0.00331595	+	0.999995i	$0.501056\pi$
$558$	0	0
$559$	−6.93102	−0.293151
$560$	−10.8382	−0.457999
$561$	0	0
$562$	−12.2174	−0.515358
$563$	−9.14136	−0.385262	−0.192631	−	0.981271i	$-0.561702\pi$
$563$	−9.14136	−0.385262	−0.192631	+	0.981271i	$0.561702\pi$
$564$	0	0
$565$	−2.46853	−0.103852
$566$	−0.792667	−0.0333183
$567$	0	0
$568$	−18.8082	−0.789173
$569$	−43.2851	−1.81461	−0.907304	−	0.420476i	$-0.861863\pi$
$569$	−43.2851	−1.81461	−0.907304	+	0.420476i	$0.861863\pi$
$570$	0	0
$571$	26.9946	1.12969	0.564844	−	0.825198i	$-0.308937\pi$
$571$	26.9946	1.12969	0.564844	+	0.825198i	$0.308937\pi$
$572$	−10.6028	−0.443324
$573$	0	0
$574$	0.214935	0.00897122
$575$	3.90834	0.162989
$576$	0	0
$577$	15.2683	0.635626	0.317813	−	0.948153i	$-0.397052\pi$
$577$	15.2683	0.635626	0.317813	+	0.948153i	$0.397052\pi$
$578$	11.0971	0.461578
$579$	0	0
$580$	1.85566	0.0770522
$581$	18.4849	0.766884
$582$	0	0
$583$	−13.5683	−0.561940
$584$	5.46781	0.226260
$585$	0	0
$586$	−1.51750	−0.0626872
$587$	−13.8956	−0.573534	−0.286767	−	0.958000i	$-0.592581\pi$
$587$	−13.8956	−0.573534	−0.286767	+	0.958000i	$0.592581\pi$
$588$	0	0
$589$	67.2733	2.77195
$590$	4.92834	0.202896
$591$	0	0
$592$	−14.0187	−0.576163
$593$	37.4682	1.53864	0.769318	−	0.638866i	$-0.220596\pi$
$593$	37.4682	1.53864	0.769318	+	0.638866i	$0.220596\pi$
$594$	0	0
$595$	−25.4691	−1.04413
$596$	−27.0712	−1.10888
$597$	0	0
$598$	−1.10186	−0.0450585
$599$	38.3625	1.56745	0.783725	−	0.621108i	$-0.213317\pi$
$599$	38.3625	1.56745	0.783725	+	0.621108i	$0.213317\pi$
$600$	0	0
$601$	−40.5272	−1.65314	−0.826570	−	0.562834i	$-0.809711\pi$
$601$	−40.5272	−1.65314	−0.826570	+	0.562834i	$0.809711\pi$
$602$	−5.39926	−0.220057
$603$	0	0
$604$	−12.5844	−0.512054
$605$	4.62451	0.188013
$606$	0	0
$607$	0.348810	0.0141578	0.00707888	−	0.999975i	$-0.497747\pi$
$607$	0.348810	0.0141578	0.00707888	+	0.999975i	$0.497747\pi$
$608$	36.2917	1.47182
$609$	0	0
$610$	−1.73879	−0.0704015
$611$	−25.0572	−1.01370
$612$	0	0
$613$	−11.9753	−0.483679	−0.241839	−	0.970316i	$-0.577751\pi$
$613$	−11.9753	−0.483679	−0.241839	+	0.970316i	$0.577751\pi$
$614$	−4.86438	−0.196310
$615$	0	0
$616$	−17.5605	−0.707534
$617$	−4.83441	−0.194626	−0.0973131	−	0.995254i	$-0.531025\pi$
$617$	−4.83441	−0.194626	−0.0973131	+	0.995254i	$0.531025\pi$
$618$	0	0
$619$	−10.0325	−0.403239	−0.201620	−	0.979464i	$-0.564621\pi$
$619$	−10.0325	−0.403239	−0.201620	+	0.979464i	$0.564621\pi$
$620$	16.6992	0.670657
$621$	0	0
$622$	−3.94557	−0.158203
$623$	−4.33872	−0.173827
$624$	0	0
$625$	9.81683	0.392673
$626$	−0.0668357	−0.00267129
$627$	0	0
$628$	2.37358	0.0947163
$629$	−32.9429	−1.31352
$630$	0	0
$631$	11.9638	0.476271	0.238136	−	0.971232i	$-0.423464\pi$
$631$	11.9638	0.476271	0.238136	+	0.971232i	$0.423464\pi$
$632$	0.512531	0.0203874
$633$	0	0
$634$	−3.34192	−0.132724
$635$	−14.7778	−0.586437
$636$	0	0
$637$	17.9057	0.709449
$638$	1.21335	0.0480370
$639$	0	0
$640$	11.6851	0.461892
$641$	−34.0018	−1.34299	−0.671494	−	0.741010i	$-0.734347\pi$
$641$	−34.0018	−1.34299	−0.671494	+	0.741010i	$0.734347\pi$
$642$	0	0
$643$	−4.25288	−0.167717	−0.0838587	−	0.996478i	$-0.526724\pi$
$643$	−4.25288	−0.167717	−0.0838587	+	0.996478i	$0.526724\pi$
$644$	6.80721	0.268242
$645$	0	0
$646$	22.4997	0.885240
$647$	−46.7915	−1.83956	−0.919781	−	0.392431i	$-0.871634\pi$
$647$	−46.7915	−1.83956	−0.919781	+	0.392431i	$0.871634\pi$
$648$	0	0
$649$	−25.5560	−1.00316
$650$	−4.30645	−0.168913
$651$	0	0
$652$	−16.9788	−0.664941
$653$	36.0850	1.41212	0.706058	−	0.708154i	$-0.250472\pi$
$653$	36.0850	1.41212	0.706058	+	0.708154i	$0.250472\pi$
$654$	0	0
$655$	−18.5353	−0.724234
$656$	0.320715	0.0125218
$657$	0	0
$658$	−19.5195	−0.760949
$659$	−2.53082	−0.0985867	−0.0492934	−	0.998784i	$-0.515697\pi$
$659$	−2.53082	−0.0985867	−0.0492934	+	0.998784i	$0.515697\pi$
$660$	0	0
$661$	−13.0031	−0.505763	−0.252882	−	0.967497i	$-0.581378\pi$
$661$	−13.0031	−0.505763	−0.252882	+	0.967497i	$0.581378\pi$
$662$	12.0421	0.468028
$663$	0	0
$664$	−8.61819	−0.334451
$665$	−29.9367	−1.16089
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	21.5435	0.833543
$669$	0	0
$670$	−6.98194	−0.269736
$671$	9.01652	0.348079
$672$	0	0
$673$	−36.1647	−1.39405	−0.697023	−	0.717049i	$-0.745493\pi$
$673$	−36.1647	−1.39405	−0.697023	+	0.717049i	$0.745493\pi$
$674$	−7.51128	−0.289324
$675$	0	0
$676$	13.4601	0.517697
$677$	−15.7179	−0.604089	−0.302044	−	0.953294i	$-0.597669\pi$
$677$	−15.7179	−0.604089	−0.302044	+	0.953294i	$0.597669\pi$
$678$	0	0
$679$	−10.6133	−0.407301
$680$	11.8744	0.455364
$681$	0	0
$682$	10.9190	0.418110
$683$	−39.2207	−1.50074	−0.750370	−	0.661018i	$-0.770125\pi$
$683$	−39.2207	−1.50074	−0.750370	+	0.661018i	$0.770125\pi$
$684$	0	0
$685$	−8.48875	−0.324339
$686$	1.25191	0.0477980
$687$	0	0
$688$	−8.05649	−0.307151
$689$	−12.3216	−0.469414
$690$	0	0
$691$	−48.3185	−1.83812	−0.919060	−	0.394117i	$-0.871050\pi$
$691$	−48.3185	−1.83812	−0.919060	+	0.394117i	$0.871050\pi$
$692$	42.3599	1.61028
$693$	0	0
$694$	−1.78582	−0.0677888
$695$	−3.13524	−0.118927
$696$	0	0
$697$	0.753660	0.0285469
$698$	−4.18328	−0.158339
$699$	0	0
$700$	26.6049	1.00557
$701$	25.3649	0.958019	0.479010	−	0.877810i	$-0.340996\pi$
$701$	25.3649	0.958019	0.479010	+	0.877810i	$0.340996\pi$
$702$	0	0
$703$	−38.7214	−1.46040
$704$	−7.98805	−0.301061
$705$	0	0
$706$	−12.4093	−0.467031
$707$	−29.2410	−1.09972
$708$	0	0
$709$	21.1340	0.793703	0.396852	−	0.917883i	$-0.370103\pi$
$709$	21.1340	0.793703	0.396852	+	0.917883i	$0.370103\pi$
$710$	−5.20418	−0.195309
$711$	0	0
$712$	2.02284	0.0758090
$713$	−8.99906	−0.337017
$714$	0	0
$715$	−6.23745	−0.233267
$716$	36.0820	1.34845
$717$	0	0
$718$	−8.20790	−0.306316
$719$	41.6848	1.55458	0.777291	−	0.629142i	$-0.216593\pi$
$719$	41.6848	1.55458	0.777291	+	0.629142i	$0.216593\pi$
$720$	0	0
$721$	−5.04029	−0.187710
$722$	17.4550	0.649606
$723$	0	0
$724$	−27.7173	−1.03010
$725$	−3.90834	−0.145152
$726$	0	0
$727$	10.1346	0.375870	0.187935	−	0.982181i	$-0.439821\pi$
$727$	10.1346	0.375870	0.187935	+	0.982181i	$0.439821\pi$
$728$	−15.9470	−0.591036
$729$	0	0
$730$	1.51293	0.0559961
$731$	−18.9322	−0.700233
$732$	0	0
$733$	−41.3568	−1.52755	−0.763775	−	0.645483i	$-0.776656\pi$
$733$	−41.3568	−1.52755	−0.763775	+	0.645483i	$0.776656\pi$
$734$	−4.77798	−0.176359
$735$	0	0
$736$	−4.85469	−0.178946
$737$	36.2050	1.33363
$738$	0	0
$739$	31.4713	1.15769	0.578845	−	0.815438i	$-0.303504\pi$
$739$	31.4713	1.15769	0.578845	+	0.815438i	$0.303504\pi$
$740$	−9.61179	−0.353336
$741$	0	0
$742$	−9.59848	−0.352371
$743$	10.4909	0.384873	0.192436	−	0.981309i	$-0.438361\pi$
$743$	10.4909	0.384873	0.192436	+	0.981309i	$0.438361\pi$
$744$	0	0
$745$	−15.9256	−0.583469
$746$	−6.17311	−0.226014
$747$	0	0
$748$	−28.9616	−1.05894
$749$	−11.8484	−0.432933
$750$	0	0
$751$	−32.4188	−1.18298	−0.591489	−	0.806313i	$-0.701460\pi$
$751$	−32.4188	−1.18298	−0.591489	+	0.806313i	$0.701460\pi$
$752$	−29.1260	−1.06211
$753$	0	0
$754$	1.10186	0.0401275
$755$	−7.40324	−0.269432
$756$	0	0
$757$	−46.3828	−1.68581	−0.842906	−	0.538061i	$-0.819157\pi$
$757$	−46.3828	−1.68581	−0.842906	+	0.538061i	$0.819157\pi$
$758$	3.69975	0.134381
$759$	0	0
$760$	13.9573	0.506285
$761$	12.2143	0.442768	0.221384	−	0.975187i	$-0.428943\pi$
$761$	12.2143	0.442768	0.221384	+	0.975187i	$0.428943\pi$
$762$	0	0
$763$	−40.6294	−1.47088
$764$	34.7591	1.25754
$765$	0	0
$766$	−1.76730	−0.0638551
$767$	−23.2078	−0.837985
$768$	0	0
$769$	−2.07966	−0.0749946	−0.0374973	−	0.999297i	$-0.511939\pi$
$769$	−2.07966	−0.0749946	−0.0374973	+	0.999297i	$0.511939\pi$
$770$	−4.85896	−0.175105
$771$	0	0
$772$	−9.05155	−0.325772
$773$	−6.91282	−0.248637	−0.124318	−	0.992242i	$-0.539674\pi$
$773$	−6.91282	−0.248637	−0.124318	+	0.992242i	$0.539674\pi$
$774$	0	0
$775$	−35.1714	−1.26339
$776$	4.94822	0.177631
$777$	0	0
$778$	−10.7697	−0.386114
$779$	0.885858	0.0317392
$780$	0	0
$781$	26.9864	0.965649
$782$	−3.00976	−0.107629
$783$	0	0
$784$	20.8132	0.743329
$785$	1.39635	0.0498377
$786$	0	0
$787$	−12.8364	−0.457570	−0.228785	−	0.973477i	$-0.573475\pi$
$787$	−12.8364	−0.457570	−0.228785	+	0.973477i	$0.573475\pi$
$788$	−41.2906	−1.47092
$789$	0	0
$790$	0.141816	0.00504559
$791$	9.05543	0.321974
$792$	0	0
$793$	8.18805	0.290766
$794$	6.92183	0.245647
$795$	0	0
$796$	−43.1019	−1.52771
$797$	−20.2940	−0.718849	−0.359425	−	0.933174i	$-0.617027\pi$
$797$	−20.2940	−0.718849	−0.359425	+	0.933174i	$0.617027\pi$
$798$	0	0
$799$	−68.4441	−2.42138
$800$	−18.9738	−0.670824
$801$	0	0
$802$	13.3865	0.472694
$803$	−7.84534	−0.276856
$804$	0	0
$805$	4.00459	0.141143
$806$	9.91573	0.349267
$807$	0	0
$808$	13.6330	0.479606
$809$	−34.7953	−1.22334	−0.611669	−	0.791114i	$-0.709502\pi$
$809$	−34.7953	−1.22334	−0.611669	+	0.791114i	$0.709502\pi$
$810$	0	0
$811$	23.1603	0.813268	0.406634	−	0.913591i	$-0.366702\pi$
$811$	23.1603	0.813268	0.406634	+	0.913591i	$0.366702\pi$
$812$	−6.80721	−0.238886
$813$	0	0
$814$	−6.28480	−0.220282
$815$	−9.98838	−0.349878
$816$	0	0
$817$	−22.2531	−0.778537
$818$	−7.56629	−0.264549
$819$	0	0
$820$	0.219896	0.00767910
$821$	49.5815	1.73041	0.865203	−	0.501422i	$-0.167190\pi$
$821$	49.5815	1.73041	0.865203	+	0.501422i	$0.167190\pi$
$822$	0	0
$823$	−25.0778	−0.874159	−0.437079	−	0.899423i	$-0.643987\pi$
$823$	−25.0778	−0.874159	−0.437079	+	0.899423i	$0.643987\pi$
$824$	2.34993	0.0818636
$825$	0	0
$826$	−18.0788	−0.629044
$827$	35.7445	1.24296	0.621480	−	0.783430i	$-0.286532\pi$
$827$	35.7445	1.24296	0.621480	+	0.783430i	$0.286532\pi$
$828$	0	0
$829$	43.5954	1.51413	0.757066	−	0.653339i	$-0.226632\pi$
$829$	43.5954	1.51413	0.757066	+	0.653339i	$0.226632\pi$
$830$	−2.38464	−0.0827719
$831$	0	0
$832$	−7.25408	−0.251490
$833$	48.9097	1.69462
$834$	0	0
$835$	12.6737	0.438593
$836$	−34.0418	−1.17736
$837$	0	0
$838$	−9.51990	−0.328859
$839$	−18.4127	−0.635679	−0.317839	−	0.948145i	$-0.602957\pi$
$839$	−18.4127	−0.635679	−0.317839	+	0.948145i	$0.602957\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	2.97123	0.102395
$843$	0	0
$844$	23.1890	0.798199
$845$	7.91840	0.272401
$846$	0	0
$847$	−16.9643	−0.582901
$848$	−14.3224	−0.491832
$849$	0	0
$850$	−11.7632	−0.403473
$851$	5.17971	0.177558
$852$	0	0
$853$	−18.4481	−0.631651	−0.315826	−	0.948817i	$-0.602281\pi$
$853$	−18.4481	−0.631651	−0.315826	+	0.948817i	$0.602281\pi$
$854$	6.37848	0.218267
$855$	0	0
$856$	5.52408	0.188809
$857$	2.81664	0.0962147	0.0481074	−	0.998842i	$-0.484681\pi$
$857$	2.81664	0.0962147	0.0481074	+	0.998842i	$0.484681\pi$
$858$	0	0
$859$	−45.9546	−1.56795	−0.783975	−	0.620793i	$-0.786811\pi$
$859$	−45.9546	−1.56795	−0.783975	+	0.620793i	$0.786811\pi$
$860$	−5.52387	−0.188363
$861$	0	0
$862$	5.76859	0.196479
$863$	−18.0449	−0.614254	−0.307127	−	0.951669i	$-0.599368\pi$
$863$	−18.0449	−0.614254	−0.307127	+	0.951669i	$0.599368\pi$
$864$	0	0
$865$	24.9197	0.847296
$866$	11.9907	0.407461
$867$	0	0
$868$	−61.2585	−2.07925
$869$	−0.735391	−0.0249464
$870$	0	0
$871$	32.8783	1.11404
$872$	18.9426	0.641477
$873$	0	0
$874$	−3.53770	−0.119664
$875$	35.6742	1.20601
$876$	0	0
$877$	32.0156	1.08109	0.540545	−	0.841315i	$-0.318218\pi$
$877$	32.0156	1.08109	0.540545	+	0.841315i	$0.318218\pi$
$878$	1.52351	0.0514160
$879$	0	0
$880$	−7.25030	−0.244407
$881$	48.5137	1.63447	0.817234	−	0.576306i	$-0.195506\pi$
$881$	48.5137	1.63447	0.817234	+	0.576306i	$0.195506\pi$
$882$	0	0
$883$	13.9176	0.468364	0.234182	−	0.972193i	$-0.424759\pi$
$883$	13.9176	0.468364	0.234182	+	0.972193i	$0.424759\pi$
$884$	−26.3005	−0.884583
$885$	0	0
$886$	9.75662	0.327780
$887$	56.2498	1.88868	0.944341	−	0.328968i	$-0.106701\pi$
$887$	56.2498	1.88868	0.944341	+	0.328968i	$0.106701\pi$
$888$	0	0
$889$	54.2099	1.81814
$890$	0.559714	0.0187617
$891$	0	0
$892$	−37.4717	−1.25464
$893$	−80.4498	−2.69215
$894$	0	0
$895$	21.2265	0.709525
$896$	−42.8648	−1.43201
$897$	0	0
$898$	−15.3181	−0.511170
$899$	8.99906	0.300135
$900$	0	0
$901$	−33.6566	−1.12126
$902$	0.143782	0.00478742
$903$	0	0
$904$	−4.22190	−0.140418
$905$	−16.3057	−0.542019
$906$	0	0
$907$	24.0052	0.797080	0.398540	−	0.917151i	$-0.369517\pi$
$907$	24.0052	0.797080	0.398540	+	0.917151i	$0.369517\pi$
$908$	−50.8580	−1.68778
$909$	0	0
$910$	−4.41250	−0.146273
$911$	27.2733	0.903605	0.451803	−	0.892118i	$-0.350781\pi$
$911$	27.2733	0.903605	0.451803	+	0.892118i	$0.350781\pi$
$912$	0	0
$913$	12.3656	0.409241
$914$	0.336793	0.0111401
$915$	0	0
$916$	47.1109	1.55659
$917$	67.9939	2.24536
$918$	0	0
$919$	−54.4520	−1.79621	−0.898103	−	0.439784i	$-0.855055\pi$
$919$	−54.4520	−1.79621	−0.898103	+	0.439784i	$0.855055\pi$
$920$	−1.86705	−0.0615549
$921$	0	0
$922$	6.43252	0.211844
$923$	24.5068	0.806650
$924$	0	0
$925$	20.2441	0.665621
$926$	18.7732	0.616924
$927$	0	0
$928$	4.85469	0.159363
$929$	−28.7869	−0.944468	−0.472234	−	0.881473i	$-0.656552\pi$
$929$	−28.7869	−0.944468	−0.472234	+	0.881473i	$0.656552\pi$
$930$	0	0
$931$	57.4889	1.88412
$932$	4.15812	0.136204
$933$	0	0
$934$	1.80362	0.0590163
$935$	−17.0377	−0.557193
$936$	0	0
$937$	−10.2809	−0.335863	−0.167932	−	0.985799i	$-0.553709\pi$
$937$	−10.2809	−0.335863	−0.167932	+	0.985799i	$0.553709\pi$
$938$	25.6122	0.836267
$939$	0	0
$940$	−19.9700	−0.651350
$941$	−5.08436	−0.165745	−0.0828726	−	0.996560i	$-0.526409\pi$
$941$	−5.08436	−0.165745	−0.0828726	+	0.996560i	$0.526409\pi$
$942$	0	0
$943$	−0.118500	−0.00385889
$944$	−26.9763	−0.878005
$945$	0	0
$946$	−3.61186	−0.117432
$947$	33.5032	1.08871	0.544354	−	0.838856i	$-0.316775\pi$
$947$	33.5032	1.08871	0.544354	+	0.838856i	$0.316775\pi$
$948$	0	0
$949$	−7.12448	−0.231270
$950$	−13.8265	−0.448592
$951$	0	0
$952$	−43.5596	−1.41177
$953$	−51.6427	−1.67287	−0.836435	−	0.548065i	$-0.815364\pi$
$953$	−51.6427	−1.67287	−0.836435	+	0.548065i	$0.815364\pi$
$954$	0	0
$955$	20.4483	0.661692
$956$	−5.46402	−0.176719
$957$	0	0
$958$	11.6715	0.377090
$959$	31.1397	1.00555
$960$	0	0
$961$	49.9831	1.61236
$962$	−5.70733	−0.184012
$963$	0	0
$964$	35.2803	1.13630
$965$	−5.32490	−0.171414
$966$	0	0
$967$	−30.0050	−0.964897	−0.482448	−	0.875924i	$-0.660252\pi$
$967$	−30.0050	−0.964897	−0.482448	+	0.875924i	$0.660252\pi$
$968$	7.90925	0.254213
$969$	0	0
$970$	1.36916	0.0439612
$971$	−22.9615	−0.736869	−0.368434	−	0.929654i	$-0.620106\pi$
$971$	−22.9615	−0.736869	−0.368434	+	0.929654i	$0.620106\pi$
$972$	0	0
$973$	11.5012	0.368710
$974$	13.8239	0.442945
$975$	0	0
$976$	9.51764	0.304652
$977$	−39.9218	−1.27721	−0.638606	−	0.769534i	$-0.720488\pi$
$977$	−39.9218	−1.27721	−0.638606	+	0.769534i	$0.720488\pi$
$978$	0	0
$979$	−2.90241	−0.0927615
$980$	14.2704	0.455852
$981$	0	0
$982$	3.75306	0.119765
$983$	−16.8911	−0.538742	−0.269371	−	0.963036i	$-0.586816\pi$
$983$	−16.8911	−0.538742	−0.269371	+	0.963036i	$0.586816\pi$
$984$	0	0
$985$	−24.2907	−0.773965
$986$	3.00976	0.0958503
$987$	0	0
$988$	−30.9139	−0.983501
$989$	2.97677	0.0946557
$990$	0	0
$991$	−37.1299	−1.17947	−0.589734	−	0.807597i	$-0.700768\pi$
$991$	−37.1299	−1.17947	−0.589734	+	0.807597i	$0.700768\pi$
$992$	43.6876	1.38708
$993$	0	0
$994$	19.0907	0.605521
$995$	−25.3562	−0.803846
$996$	0	0
$997$	14.6652	0.464452	0.232226	−	0.972662i	$-0.425399\pi$
$997$	14.6652	0.464452	0.232226	+	0.972662i	$0.425399\pi$
$998$	17.2654	0.546527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.17	yes	30
3.2	odd	2		6003.2.a.v.1.14	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.14	✓	30	3.2	odd	2
6003.2.a.w.1.17	yes	30	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+0.473233 q^{2} -1.77605 q^{4} -1.04483 q^{5} +3.83278 q^{7} -1.78695 q^{8} +O(q^{10})\)	\(q+0.473233 q^{2} -1.77605 q^{4} -1.04483 q^{5} +3.83278 q^{7} -1.78695 q^{8} -0.494446 q^{10} +2.56396 q^{11} +2.32837 q^{13} +1.81380 q^{14} +2.70646 q^{16} +6.35999 q^{17} +7.47559 q^{19} +1.85566 q^{20} +1.21335 q^{22} -1.00000 q^{23} -3.90834 q^{25} +1.10186 q^{26} -6.80721 q^{28} +1.00000 q^{29} +8.99906 q^{31} +4.85469 q^{32} +3.00976 q^{34} -4.00459 q^{35} -5.17971 q^{37} +3.53770 q^{38} +1.86705 q^{40} +0.118500 q^{41} -2.97677 q^{43} -4.55372 q^{44} -0.473233 q^{46} -10.7617 q^{47} +7.69021 q^{49} -1.84956 q^{50} -4.13531 q^{52} -5.29192 q^{53} -2.67889 q^{55} -6.84899 q^{56} +0.473233 q^{58} -9.96740 q^{59} +3.51664 q^{61} +4.25865 q^{62} -3.11551 q^{64} -2.43274 q^{65} +14.1207 q^{67} -11.2957 q^{68} -1.89510 q^{70} +10.5253 q^{71} -3.05985 q^{73} -2.45121 q^{74} -13.2770 q^{76} +9.82709 q^{77} -0.286819 q^{79} -2.82777 q^{80} +0.0560781 q^{82} +4.82285 q^{83} -6.64508 q^{85} -1.40870 q^{86} -4.58167 q^{88} -1.13200 q^{89} +8.92414 q^{91} +1.77605 q^{92} -5.09277 q^{94} -7.81069 q^{95} -2.76909 q^{97} +3.63926 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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