LMFDB - Embedded newform 6003.2.a.w.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.60849 q^{2} +4.80424 q^{4} -1.19758 q^{5} -4.49655 q^{7} -7.31485 q^{8} +O(q^{10})$	\(q-2.60849 q^{2} +4.80424 q^{4} -1.19758 q^{5} -4.49655 q^{7} -7.31485 q^{8} +3.12388 q^{10} -5.34458 q^{11} +2.49925 q^{13} +11.7292 q^{14} +9.47226 q^{16} -3.03331 q^{17} +2.47521 q^{19} -5.75346 q^{20} +13.9413 q^{22} -1.00000 q^{23} -3.56580 q^{25} -6.51928 q^{26} -21.6025 q^{28} +1.00000 q^{29} -10.2808 q^{31} -10.0786 q^{32} +7.91238 q^{34} +5.38498 q^{35} +9.08423 q^{37} -6.45656 q^{38} +8.76011 q^{40} -0.132467 q^{41} +2.26810 q^{43} -25.6767 q^{44} +2.60849 q^{46} -6.82328 q^{47} +13.2190 q^{49} +9.30138 q^{50} +12.0070 q^{52} -3.33618 q^{53} +6.40056 q^{55} +32.8916 q^{56} -2.60849 q^{58} -9.14960 q^{59} -7.63195 q^{61} +26.8175 q^{62} +7.34555 q^{64} -2.99305 q^{65} -14.0368 q^{67} -14.5728 q^{68} -14.0467 q^{70} -15.9191 q^{71} -1.85706 q^{73} -23.6962 q^{74} +11.8915 q^{76} +24.0322 q^{77} -9.29193 q^{79} -11.3438 q^{80} +0.345538 q^{82} -7.74213 q^{83} +3.63263 q^{85} -5.91632 q^{86} +39.0948 q^{88} -12.3300 q^{89} -11.2380 q^{91} -4.80424 q^{92} +17.7985 q^{94} -2.96426 q^{95} -10.8139 q^{97} -34.4817 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.60849	−1.84448	−0.922242	−	0.386613i	$-0.873645\pi$
$2$	−2.60849	−1.84448	−0.922242	+	0.386613i	$0.873645\pi$
$3$	0	0
$3$	0	0
$4$	4.80424	2.40212
$5$	−1.19758	−0.535574	−0.267787	−	0.963478i	$-0.586292\pi$
$5$	−1.19758	−0.535574	−0.267787	+	0.963478i	$0.586292\pi$
$6$	0	0
$7$	−4.49655	−1.69954	−0.849769	−	0.527155i	$-0.823259\pi$
$7$	−4.49655	−1.69954	−0.849769	+	0.527155i	$0.823259\pi$
$8$	−7.31485	−2.58619
$9$	0	0
$10$	3.12388	0.987857
$11$	−5.34458	−1.61145	−0.805726	−	0.592289i	$-0.798224\pi$
$11$	−5.34458	−1.61145	−0.805726	+	0.592289i	$0.798224\pi$
$12$	0	0
$13$	2.49925	0.693167	0.346584	−	0.938019i	$-0.387342\pi$
$13$	2.49925	0.693167	0.346584	+	0.938019i	$0.387342\pi$
$14$	11.7292	3.13477
$15$	0	0
$16$	9.47226	2.36807
$17$	−3.03331	−0.735686	−0.367843	−	0.929888i	$-0.619904\pi$
$17$	−3.03331	−0.735686	−0.367843	+	0.929888i	$0.619904\pi$
$18$	0	0
$19$	2.47521	0.567852	0.283926	−	0.958846i	$-0.408363\pi$
$19$	2.47521	0.567852	0.283926	+	0.958846i	$0.408363\pi$
$20$	−5.75346	−1.28651
$21$	0	0
$22$	13.9413	2.97230
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.56580	−0.713161
$26$	−6.51928	−1.27854
$27$	0	0
$28$	−21.6025	−4.08250
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−10.2808	−1.84649	−0.923245	−	0.384212i	$-0.874473\pi$
$31$	−10.2808	−1.84649	−0.923245	+	0.384212i	$0.874473\pi$
$32$	−10.0786	−1.78167
$33$	0	0
$34$	7.91238	1.35696
$35$	5.38498	0.910228
$36$	0	0
$37$	9.08423	1.49344	0.746719	−	0.665139i	$-0.231628\pi$
$37$	9.08423	1.49344	0.746719	+	0.665139i	$0.231628\pi$
$38$	−6.45656	−1.04739
$39$	0	0
$40$	8.76011	1.38510
$41$	−0.132467	−0.0206878	−0.0103439	−	0.999947i	$-0.503293\pi$
$41$	−0.132467	−0.0206878	−0.0103439	+	0.999947i	$0.503293\pi$
$42$	0	0
$43$	2.26810	0.345881	0.172941	−	0.984932i	$-0.444673\pi$
$43$	2.26810	0.345881	0.172941	+	0.984932i	$0.444673\pi$
$44$	−25.6767	−3.87090
$45$	0	0
$46$	2.60849	0.384602
$47$	−6.82328	−0.995277	−0.497639	−	0.867384i	$-0.665799\pi$
$47$	−6.82328	−0.995277	−0.497639	+	0.867384i	$0.665799\pi$
$48$	0	0
$49$	13.2190	1.88843
$50$	9.30138	1.31541
$51$	0	0
$52$	12.0070	1.66507
$53$	−3.33618	−0.458259	−0.229130	−	0.973396i	$-0.573588\pi$
$53$	−3.33618	−0.458259	−0.229130	+	0.973396i	$0.573588\pi$
$54$	0	0
$55$	6.40056	0.863051
$56$	32.8916	4.39533
$57$	0	0
$58$	−2.60849	−0.342512
$59$	−9.14960	−1.19118	−0.595588	−	0.803290i	$-0.703081\pi$
$59$	−9.14960	−1.19118	−0.595588	+	0.803290i	$0.703081\pi$
$60$	0	0
$61$	−7.63195	−0.977171	−0.488585	−	0.872516i	$-0.662487\pi$
$61$	−7.63195	−0.977171	−0.488585	+	0.872516i	$0.662487\pi$
$62$	26.8175	3.40582
$63$	0	0
$64$	7.34555	0.918194
$65$	−2.99305	−0.371242
$66$	0	0
$67$	−14.0368	−1.71487	−0.857434	−	0.514594i	$-0.827943\pi$
$67$	−14.0368	−1.71487	−0.857434	+	0.514594i	$0.827943\pi$
$68$	−14.5728	−1.76721
$69$	0	0
$70$	−14.0467	−1.67890
$71$	−15.9191	−1.88924	−0.944622	−	0.328159i	$-0.893572\pi$
$71$	−15.9191	−1.88924	−0.944622	+	0.328159i	$0.893572\pi$
$72$	0	0
$73$	−1.85706	−0.217353	−0.108676	−	0.994077i	$-0.534661\pi$
$73$	−1.85706	−0.217353	−0.108676	+	0.994077i	$0.534661\pi$
$74$	−23.6962	−2.75462
$75$	0	0
$76$	11.8915	1.36405
$77$	24.0322	2.73872
$78$	0	0
$79$	−9.29193	−1.04542	−0.522712	−	0.852509i	$-0.675080\pi$
$79$	−9.29193	−1.04542	−0.522712	+	0.852509i	$0.675080\pi$
$80$	−11.3438	−1.26827
$81$	0	0
$82$	0.345538	0.0381583
$83$	−7.74213	−0.849809	−0.424905	−	0.905238i	$-0.639692\pi$
$83$	−7.74213	−0.849809	−0.424905	+	0.905238i	$0.639692\pi$
$84$	0	0
$85$	3.63263	0.394014
$86$	−5.91632	−0.637973
$87$	0	0
$88$	39.0948	4.16752
$89$	−12.3300	−1.30698	−0.653488	−	0.756937i	$-0.726695\pi$
$89$	−12.3300	−1.30698	−0.653488	+	0.756937i	$0.726695\pi$
$90$	0	0
$91$	−11.2380	−1.17806
$92$	−4.80424	−0.500877
$93$	0	0
$94$	17.7985	1.83577
$95$	−2.96426	−0.304126
$96$	0	0
$97$	−10.8139	−1.09798	−0.548992	−	0.835827i	$-0.684988\pi$
$97$	−10.8139	−1.09798	−0.548992	+	0.835827i	$0.684988\pi$
$98$	−34.4817	−3.48318
$99$	0	0
$100$	−17.1310	−1.71310
$101$	−16.1829	−1.61026	−0.805128	−	0.593101i	$-0.797903\pi$
$101$	−16.1829	−1.61026	−0.805128	+	0.593101i	$0.797903\pi$
$102$	0	0
$103$	−5.59817	−0.551604	−0.275802	−	0.961214i	$-0.588943\pi$
$103$	−5.59817	−0.551604	−0.275802	+	0.961214i	$0.588943\pi$
$104$	−18.2816	−1.79266
$105$	0	0
$106$	8.70240	0.845252
$107$	−11.6469	−1.12595	−0.562975	−	0.826474i	$-0.690343\pi$
$107$	−11.6469	−1.12595	−0.562975	+	0.826474i	$0.690343\pi$
$108$	0	0
$109$	−7.53500	−0.721722	−0.360861	−	0.932620i	$-0.617517\pi$
$109$	−7.53500	−0.721722	−0.360861	+	0.932620i	$0.617517\pi$
$110$	−16.6958	−1.59188
$111$	0	0
$112$	−42.5925	−4.02462
$113$	15.5070	1.45878	0.729388	−	0.684101i	$-0.239805\pi$
$113$	15.5070	1.45878	0.729388	+	0.684101i	$0.239805\pi$
$114$	0	0
$115$	1.19758	0.111675
$116$	4.80424	0.446063
$117$	0	0
$118$	23.8667	2.19711
$119$	13.6395	1.25033
$120$	0	0
$121$	17.5645	1.59678
$122$	19.9079	1.80238
$123$	0	0
$124$	−49.3916	−4.43549
$125$	10.2582	0.917524
$126$	0	0
$127$	8.60549	0.763614	0.381807	−	0.924242i	$-0.375302\pi$
$127$	8.60549	0.763614	0.381807	+	0.924242i	$0.375302\pi$
$128$	0.996455	0.0880750
$129$	0	0
$130$	7.80735	0.684750
$131$	−4.64890	−0.406177	−0.203088	−	0.979160i	$-0.565098\pi$
$131$	−4.64890	−0.406177	−0.203088	+	0.979160i	$0.565098\pi$
$132$	0	0
$133$	−11.1299	−0.965085
$134$	36.6149	3.16305
$135$	0	0
$136$	22.1882	1.90262
$137$	19.1758	1.63830	0.819149	−	0.573581i	$-0.194446\pi$
$137$	19.1758	1.63830	0.819149	+	0.573581i	$0.194446\pi$
$138$	0	0
$139$	4.91686	0.417042	0.208521	−	0.978018i	$-0.433135\pi$
$139$	4.91686	0.417042	0.208521	+	0.978018i	$0.433135\pi$
$140$	25.8707	2.18648
$141$	0	0
$142$	41.5248	3.48468
$143$	−13.3574	−1.11701
$144$	0	0
$145$	−1.19758	−0.0994535
$146$	4.84413	0.400903
$147$	0	0
$148$	43.6429	3.58742
$149$	−1.07325	−0.0879240	−0.0439620	−	0.999033i	$-0.513998\pi$
$149$	−1.07325	−0.0879240	−0.0439620	+	0.999033i	$0.513998\pi$
$150$	0	0
$151$	−16.8065	−1.36770	−0.683849	−	0.729624i	$-0.739695\pi$
$151$	−16.8065	−1.36770	−0.683849	+	0.729624i	$0.739695\pi$
$152$	−18.1058	−1.46857
$153$	0	0
$154$	−62.6878	−5.05153
$155$	12.3121	0.988931
$156$	0	0
$157$	22.8732	1.82548	0.912739	−	0.408543i	$-0.133963\pi$
$157$	22.8732	1.82548	0.912739	+	0.408543i	$0.133963\pi$
$158$	24.2380	1.92827
$159$	0	0
$160$	12.0700	0.954215
$161$	4.49655	0.354378
$162$	0	0
$163$	−12.1364	−0.950595	−0.475298	−	0.879825i	$-0.657660\pi$
$163$	−12.1364	−0.950595	−0.475298	+	0.879825i	$0.657660\pi$
$164$	−0.636401	−0.0496946
$165$	0	0
$166$	20.1953	1.56746
$167$	11.5829	0.896313	0.448157	−	0.893955i	$-0.352081\pi$
$167$	11.5829	0.896313	0.448157	+	0.893955i	$0.352081\pi$
$168$	0	0
$169$	−6.75375	−0.519519
$170$	−9.47570	−0.726753
$171$	0	0
$172$	10.8965	0.830849
$173$	−17.4525	−1.32689	−0.663445	−	0.748225i	$-0.730906\pi$
$173$	−17.4525	−1.32689	−0.663445	+	0.748225i	$0.730906\pi$
$174$	0	0
$175$	16.0338	1.21204
$176$	−50.6253	−3.81602
$177$	0	0
$178$	32.1627	2.41070
$179$	4.23008	0.316171	0.158086	−	0.987425i	$-0.449468\pi$
$179$	4.23008	0.316171	0.158086	+	0.987425i	$0.449468\pi$
$180$	0	0
$181$	11.3584	0.844265	0.422132	−	0.906534i	$-0.361282\pi$
$181$	11.3584	0.844265	0.422132	+	0.906534i	$0.361282\pi$
$182$	29.3143	2.17292
$183$	0	0
$184$	7.31485	0.539258
$185$	−10.8791	−0.799846
$186$	0	0
$187$	16.2118	1.18552
$188$	−32.7807	−2.39078
$189$	0	0
$190$	7.73225	0.560956
$191$	12.8165	0.927373	0.463686	−	0.885999i	$-0.346526\pi$
$191$	12.8165	0.927373	0.463686	+	0.885999i	$0.346526\pi$
$192$	0	0
$193$	−12.6943	−0.913755	−0.456878	−	0.889530i	$-0.651032\pi$
$193$	−12.6943	−0.913755	−0.456878	+	0.889530i	$0.651032\pi$
$194$	28.2080	2.02522
$195$	0	0
$196$	63.5073	4.53624
$197$	−22.2193	−1.58306	−0.791529	−	0.611132i	$-0.790715\pi$
$197$	−22.2193	−1.58306	−0.791529	+	0.611132i	$0.790715\pi$
$198$	0	0
$199$	−17.1739	−1.21743	−0.608714	−	0.793390i	$-0.708314\pi$
$199$	−17.1739	−1.21743	−0.608714	+	0.793390i	$0.708314\pi$
$200$	26.0833	1.84437
$201$	0	0
$202$	42.2129	2.97009
$203$	−4.49655	−0.315596
$204$	0	0
$205$	0.158639	0.0110798
$206$	14.6028	1.01742
$207$	0	0
$208$	23.6736	1.64147
$209$	−13.2289	−0.915065
$210$	0	0
$211$	−17.7661	−1.22307	−0.611534	−	0.791218i	$-0.709447\pi$
$211$	−17.7661	−1.22307	−0.611534	+	0.791218i	$0.709447\pi$
$212$	−16.0278	−1.10079
$213$	0	0
$214$	30.3809	2.07680
$215$	−2.71622	−0.185245
$216$	0	0
$217$	46.2283	3.13818
$218$	19.6550	1.33120
$219$	0	0
$220$	30.7498	2.07315
$221$	−7.58101	−0.509954
$222$	0	0
$223$	24.9622	1.67159	0.835796	−	0.549040i	$-0.185007\pi$
$223$	24.9622	1.67159	0.835796	+	0.549040i	$0.185007\pi$
$224$	45.3192	3.02801
$225$	0	0
$226$	−40.4499	−2.69069
$227$	21.9072	1.45403	0.727016	−	0.686620i	$-0.240907\pi$
$227$	21.9072	1.45403	0.727016	+	0.686620i	$0.240907\pi$
$228$	0	0
$229$	13.3388	0.881451	0.440725	−	0.897642i	$-0.354721\pi$
$229$	13.3388	0.881451	0.440725	+	0.897642i	$0.354721\pi$
$230$	−3.12388	−0.205982
$231$	0	0
$232$	−7.31485	−0.480243
$233$	−20.6670	−1.35394	−0.676970	−	0.736011i	$-0.736707\pi$
$233$	−20.6670	−1.35394	−0.676970	+	0.736011i	$0.736707\pi$
$234$	0	0
$235$	8.17141	0.533044
$236$	−43.9569	−2.86135
$237$	0	0
$238$	−35.5784	−2.30621
$239$	7.20394	0.465984	0.232992	−	0.972479i	$-0.425148\pi$
$239$	7.20394	0.465984	0.232992	+	0.972479i	$0.425148\pi$
$240$	0	0
$241$	−4.35951	−0.280821	−0.140410	−	0.990093i	$-0.544842\pi$
$241$	−4.35951	−0.280821	−0.140410	+	0.990093i	$0.544842\pi$
$242$	−45.8170	−2.94523
$243$	0	0
$244$	−36.6657	−2.34728
$245$	−15.8308	−1.01139
$246$	0	0
$247$	6.18616	0.393616
$248$	75.2027	4.77537
$249$	0	0
$250$	−26.7585	−1.69236
$251$	10.4224	0.657857	0.328929	−	0.944355i	$-0.393312\pi$
$251$	10.4224	0.657857	0.328929	+	0.944355i	$0.393312\pi$
$252$	0	0
$253$	5.34458	0.336011
$254$	−22.4474	−1.40847
$255$	0	0
$256$	−17.2903	−1.08065
$257$	9.13033	0.569534	0.284767	−	0.958597i	$-0.408084\pi$
$257$	9.13033	0.569534	0.284767	+	0.958597i	$0.408084\pi$
$258$	0	0
$259$	−40.8478	−2.53816
$260$	−14.3793	−0.891769
$261$	0	0
$262$	12.1266	0.749186
$263$	19.0260	1.17319	0.586597	−	0.809879i	$-0.300467\pi$
$263$	19.0260	1.17319	0.586597	+	0.809879i	$0.300467\pi$
$264$	0	0
$265$	3.99533	0.245431
$266$	29.0323	1.78008
$267$	0	0
$268$	−67.4362	−4.11932
$269$	4.33575	0.264355	0.132178	−	0.991226i	$-0.457803\pi$
$269$	4.33575	0.264355	0.132178	+	0.991226i	$0.457803\pi$
$270$	0	0
$271$	−0.332628	−0.0202057	−0.0101029	−	0.999949i	$-0.503216\pi$
$271$	−0.332628	−0.0202057	−0.0101029	+	0.999949i	$0.503216\pi$
$272$	−28.7323	−1.74215
$273$	0	0
$274$	−50.0199	−3.02182
$275$	19.0577	1.14922
$276$	0	0
$277$	12.6762	0.761641	0.380821	−	0.924649i	$-0.375642\pi$
$277$	12.6762	0.761641	0.380821	+	0.924649i	$0.375642\pi$
$278$	−12.8256	−0.769228
$279$	0	0
$280$	−39.3903	−2.35402
$281$	8.82758	0.526610	0.263305	−	0.964713i	$-0.415188\pi$
$281$	8.82758	0.526610	0.263305	+	0.964713i	$0.415188\pi$
$282$	0	0
$283$	−23.7545	−1.41206	−0.706030	−	0.708182i	$-0.749515\pi$
$283$	−23.7545	−1.41206	−0.706030	+	0.708182i	$0.749515\pi$
$284$	−76.4790	−4.53820
$285$	0	0
$286$	34.8428	2.06030
$287$	0.595643	0.0351597
$288$	0	0
$289$	−7.79902	−0.458766
$290$	3.12388	0.183440
$291$	0	0
$292$	−8.92177	−0.522107
$293$	30.3038	1.77037	0.885185	−	0.465240i	$-0.154032\pi$
$293$	30.3038	1.77037	0.885185	+	0.465240i	$0.154032\pi$
$294$	0	0
$295$	10.9574	0.637963
$296$	−66.4498	−3.86232
$297$	0	0
$298$	2.79956	0.162174
$299$	−2.49925	−0.144535
$300$	0	0
$301$	−10.1986	−0.587839
$302$	43.8398	2.52270
$303$	0	0
$304$	23.4458	1.34471
$305$	9.13986	0.523347
$306$	0	0
$307$	−12.8966	−0.736047	−0.368023	−	0.929817i	$-0.619965\pi$
$307$	−12.8966	−0.736047	−0.368023	+	0.929817i	$0.619965\pi$
$308$	115.457	6.57874
$309$	0	0
$310$	−32.1160	−1.82407
$311$	−5.28307	−0.299575	−0.149788	−	0.988718i	$-0.547859\pi$
$311$	−5.28307	−0.299575	−0.149788	+	0.988718i	$0.547859\pi$
$312$	0	0
$313$	27.8852	1.57616	0.788082	−	0.615570i	$-0.211074\pi$
$313$	27.8852	1.57616	0.788082	+	0.615570i	$0.211074\pi$
$314$	−59.6645	−3.36707
$315$	0	0
$316$	−44.6407	−2.51124
$317$	−27.4934	−1.54418	−0.772092	−	0.635511i	$-0.780789\pi$
$317$	−27.4934	−1.54418	−0.772092	+	0.635511i	$0.780789\pi$
$318$	0	0
$319$	−5.34458	−0.299239
$320$	−8.79687	−0.491760
$321$	0	0
$322$	−11.7292	−0.653645
$323$	−7.50808	−0.417760
$324$	0	0
$325$	−8.91184	−0.494340
$326$	31.6577	1.75336
$327$	0	0
$328$	0.968973	0.0535026
$329$	30.6812	1.69151
$330$	0	0
$331$	25.8102	1.41866	0.709329	−	0.704878i	$-0.248998\pi$
$331$	25.8102	1.41866	0.709329	+	0.704878i	$0.248998\pi$
$332$	−37.1951	−2.04135
$333$	0	0
$334$	−30.2140	−1.65324
$335$	16.8102	0.918438
$336$	0	0
$337$	−16.4658	−0.896951	−0.448476	−	0.893795i	$-0.648033\pi$
$337$	−16.4658	−0.896951	−0.448476	+	0.893795i	$0.648033\pi$
$338$	17.6171	0.958245
$339$	0	0
$340$	17.4520	0.946470
$341$	54.9467	2.97553
$342$	0	0
$343$	−27.9641	−1.50992
$344$	−16.5908	−0.894515
$345$	0	0
$346$	45.5248	2.44743
$347$	17.9553	0.963891	0.481945	−	0.876201i	$-0.339930\pi$
$347$	17.9553	0.963891	0.481945	+	0.876201i	$0.339930\pi$
$348$	0	0
$349$	−25.4297	−1.36122	−0.680610	−	0.732646i	$-0.738285\pi$
$349$	−25.4297	−1.36122	−0.680610	+	0.732646i	$0.738285\pi$
$350$	−41.8242	−2.23560
$351$	0	0
$352$	53.8661	2.87107
$353$	−4.07263	−0.216764	−0.108382	−	0.994109i	$-0.534567\pi$
$353$	−4.07263	−0.216764	−0.108382	+	0.994109i	$0.534567\pi$
$354$	0	0
$355$	19.0643	1.01183
$356$	−59.2363	−3.13952
$357$	0	0
$358$	−11.0341	−0.583173
$359$	0.0517594	0.00273176	0.00136588	−	0.999999i	$-0.499565\pi$
$359$	0.0517594	0.00273176	0.00136588	+	0.999999i	$0.499565\pi$
$360$	0	0
$361$	−12.8733	−0.677545
$362$	−29.6284	−1.55723
$363$	0	0
$364$	−53.9902	−2.82985
$365$	2.22398	0.116408
$366$	0	0
$367$	0.0785274	0.00409910	0.00204955	−	0.999998i	$-0.499348\pi$
$367$	0.0785274	0.00409910	0.00204955	+	0.999998i	$0.499348\pi$
$368$	−9.47226	−0.493776
$369$	0	0
$370$	28.3780	1.47530
$371$	15.0013	0.778829
$372$	0	0
$373$	−4.50105	−0.233055	−0.116528	−	0.993187i	$-0.537176\pi$
$373$	−4.50105	−0.233055	−0.116528	+	0.993187i	$0.537176\pi$
$374$	−42.2883	−2.18668
$375$	0	0
$376$	49.9113	2.57398
$377$	2.49925	0.128718
$378$	0	0
$379$	−5.39781	−0.277267	−0.138634	−	0.990344i	$-0.544271\pi$
$379$	−5.39781	−0.277267	−0.138634	+	0.990344i	$0.544271\pi$
$380$	−14.2410	−0.730548
$381$	0	0
$382$	−33.4319	−1.71052
$383$	0.622108	0.0317882	0.0158941	−	0.999874i	$-0.494941\pi$
$383$	0.622108	0.0317882	0.0158941	+	0.999874i	$0.494941\pi$
$384$	0	0
$385$	−28.7805	−1.46679
$386$	33.1130	1.68541
$387$	0	0
$388$	−51.9526	−2.63749
$389$	−22.8627	−1.15919	−0.579593	−	0.814906i	$-0.696788\pi$
$389$	−22.8627	−1.15919	−0.579593	+	0.814906i	$0.696788\pi$
$390$	0	0
$391$	3.03331	0.153401
$392$	−96.6951	−4.88384
$393$	0	0
$394$	57.9589	2.91993
$395$	11.1278	0.559902
$396$	0	0
$397$	−18.2389	−0.915382	−0.457691	−	0.889111i	$-0.651323\pi$
$397$	−18.2389	−0.915382	−0.457691	+	0.889111i	$0.651323\pi$
$398$	44.7981	2.24553
$399$	0	0
$400$	−33.7762	−1.68881
$401$	21.1481	1.05609	0.528043	−	0.849217i	$-0.322926\pi$
$401$	21.1481	1.05609	0.528043	+	0.849217i	$0.322926\pi$
$402$	0	0
$403$	−25.6943	−1.27993
$404$	−77.7464	−3.86803
$405$	0	0
$406$	11.7292	0.582112
$407$	−48.5514	−2.40660
$408$	0	0
$409$	−5.25709	−0.259946	−0.129973	−	0.991518i	$-0.541489\pi$
$409$	−5.25709	−0.259946	−0.129973	+	0.991518i	$0.541489\pi$
$410$	−0.413809	−0.0204366
$411$	0	0
$412$	−26.8950	−1.32502
$413$	41.1417	2.02445
$414$	0	0
$415$	9.27181	0.455135
$416$	−25.1890	−1.23499
$417$	0	0
$418$	34.5076	1.68782
$419$	10.7913	0.527189	0.263594	−	0.964634i	$-0.415092\pi$
$419$	10.7913	0.527189	0.263594	+	0.964634i	$0.415092\pi$
$420$	0	0
$421$	−30.3399	−1.47868	−0.739339	−	0.673334i	$-0.764862\pi$
$421$	−30.3399	−1.47868	−0.739339	+	0.673334i	$0.764862\pi$
$422$	46.3427	2.25593
$423$	0	0
$424$	24.4036	1.18515
$425$	10.8162	0.524663
$426$	0	0
$427$	34.3175	1.66074
$428$	−55.9546	−2.70467
$429$	0	0
$430$	7.08526	0.341681
$431$	30.4135	1.46497	0.732483	−	0.680786i	$-0.238361\pi$
$431$	30.4135	1.46497	0.732483	+	0.680786i	$0.238361\pi$
$432$	0	0
$433$	−13.6761	−0.657230	−0.328615	−	0.944464i	$-0.606582\pi$
$433$	−13.6761	−0.657230	−0.328615	+	0.944464i	$0.606582\pi$
$434$	−120.586	−5.78832
$435$	0	0
$436$	−36.2000	−1.73366
$437$	−2.47521	−0.118405
$438$	0	0
$439$	25.3312	1.20899	0.604496	−	0.796608i	$-0.293374\pi$
$439$	25.3312	1.20899	0.604496	+	0.796608i	$0.293374\pi$
$440$	−46.8191	−2.23201
$441$	0	0
$442$	19.7750	0.940601
$443$	−13.7648	−0.653986	−0.326993	−	0.945027i	$-0.606035\pi$
$443$	−13.7648	−0.653986	−0.326993	+	0.945027i	$0.606035\pi$
$444$	0	0
$445$	14.7661	0.699982
$446$	−65.1137	−3.08322
$447$	0	0
$448$	−33.0297	−1.56050
$449$	32.7228	1.54428	0.772141	−	0.635451i	$-0.219186\pi$
$449$	32.7228	1.54428	0.772141	+	0.635451i	$0.219186\pi$
$450$	0	0
$451$	0.707978	0.0333374
$452$	74.4994	3.50415
$453$	0	0
$454$	−57.1448	−2.68194
$455$	13.4584	0.630940
$456$	0	0
$457$	11.5766	0.541529	0.270765	−	0.962646i	$-0.412723\pi$
$457$	11.5766	0.541529	0.270765	+	0.962646i	$0.412723\pi$
$458$	−34.7941	−1.62582
$459$	0	0
$460$	5.75346	0.268256
$461$	−14.5927	−0.679648	−0.339824	−	0.940489i	$-0.610367\pi$
$461$	−14.5927	−0.679648	−0.339824	+	0.940489i	$0.610367\pi$
$462$	0	0
$463$	−15.0410	−0.699017	−0.349508	−	0.936933i	$-0.613651\pi$
$463$	−15.0410	−0.699017	−0.349508	+	0.936933i	$0.613651\pi$
$464$	9.47226	0.439739
$465$	0	0
$466$	53.9097	2.49732
$467$	8.04893	0.372460	0.186230	−	0.982506i	$-0.440373\pi$
$467$	8.04893	0.372460	0.186230	+	0.982506i	$0.440373\pi$
$468$	0	0
$469$	63.1172	2.91448
$470$	−21.3151	−0.983191
$471$	0	0
$472$	66.9280	3.08061
$473$	−12.1220	−0.557371
$474$	0	0
$475$	−8.82611	−0.404970
$476$	65.5272	3.00344
$477$	0	0
$478$	−18.7914	−0.859501
$479$	−29.4150	−1.34401	−0.672003	−	0.740548i	$-0.734566\pi$
$479$	−29.4150	−1.34401	−0.672003	+	0.740548i	$0.734566\pi$
$480$	0	0
$481$	22.7038	1.03520
$482$	11.3718	0.517969
$483$	0	0
$484$	84.3843	3.83565
$485$	12.9505	0.588052
$486$	0	0
$487$	−0.307974	−0.0139556	−0.00697781	−	0.999976i	$-0.502221\pi$
$487$	−0.307974	−0.0139556	−0.00697781	+	0.999976i	$0.502221\pi$
$488$	55.8266	2.52715
$489$	0	0
$490$	41.2946	1.86550
$491$	23.1998	1.04699	0.523497	−	0.852028i	$-0.324627\pi$
$491$	23.1998	1.04699	0.523497	+	0.852028i	$0.324627\pi$
$492$	0	0
$493$	−3.03331	−0.136613
$494$	−16.1366	−0.726019
$495$	0	0
$496$	−97.3826	−4.37261
$497$	71.5809	3.21084
$498$	0	0
$499$	−23.1509	−1.03638	−0.518188	−	0.855267i	$-0.673393\pi$
$499$	−23.1509	−1.03638	−0.518188	+	0.855267i	$0.673393\pi$
$500$	49.2830	2.20400
$501$	0	0
$502$	−27.1868	−1.21341
$503$	20.7305	0.924329	0.462165	−	0.886794i	$-0.347073\pi$
$503$	20.7305	0.924329	0.462165	+	0.886794i	$0.347073\pi$
$504$	0	0
$505$	19.3803	0.862411
$506$	−13.9413	−0.619767
$507$	0	0
$508$	41.3429	1.83429
$509$	16.5758	0.734708	0.367354	−	0.930081i	$-0.380264\pi$
$509$	16.5758	0.734708	0.367354	+	0.930081i	$0.380264\pi$
$510$	0	0
$511$	8.35038	0.369399
$512$	43.1089	1.90516
$513$	0	0
$514$	−23.8164	−1.05050
$515$	6.70425	0.295425
$516$	0	0
$517$	36.4675	1.60384
$518$	106.551	4.68159
$519$	0	0
$520$	21.8937	0.960103
$521$	4.66201	0.204246	0.102123	−	0.994772i	$-0.467436\pi$
$521$	4.66201	0.204246	0.102123	+	0.994772i	$0.467436\pi$
$522$	0	0
$523$	−33.4379	−1.46214	−0.731068	−	0.682305i	$-0.760978\pi$
$523$	−33.4379	−1.46214	−0.731068	+	0.682305i	$0.760978\pi$
$524$	−22.3345	−0.975685
$525$	0	0
$526$	−49.6292	−2.16394
$527$	31.1849	1.35844
$528$	0	0
$529$	1.00000	0.0434783
$530$	−10.4218	−0.452694
$531$	0	0
$532$	−53.4708	−2.31825
$533$	−0.331067	−0.0143401
$534$	0	0
$535$	13.9481	0.603029
$536$	102.677	4.43497
$537$	0	0
$538$	−11.3098	−0.487599
$539$	−70.6500	−3.04311
$540$	0	0
$541$	23.1171	0.993882	0.496941	−	0.867784i	$-0.334457\pi$
$541$	23.1171	0.993882	0.496941	+	0.867784i	$0.334457\pi$
$542$	0.867659	0.0372691
$543$	0	0
$544$	30.5717	1.31075
$545$	9.02375	0.386535
$546$	0	0
$547$	7.44561	0.318351	0.159176	−	0.987250i	$-0.449116\pi$
$547$	7.44561	0.318351	0.159176	+	0.987250i	$0.449116\pi$
$548$	92.1251	3.93539
$549$	0	0
$550$	−49.7120	−2.11973
$551$	2.47521	0.105447
$552$	0	0
$553$	41.7817	1.77674
$554$	−33.0659	−1.40484
$555$	0	0
$556$	23.6218	1.00179
$557$	−8.95937	−0.379621	−0.189810	−	0.981821i	$-0.560787\pi$
$557$	−8.95937	−0.379621	−0.189810	+	0.981821i	$0.560787\pi$
$558$	0	0
$559$	5.66854	0.239754
$560$	51.0079	2.15548
$561$	0	0
$562$	−23.0267	−0.971323
$563$	16.9537	0.714511	0.357256	−	0.934007i	$-0.383712\pi$
$563$	16.9537	0.714511	0.357256	+	0.934007i	$0.383712\pi$
$564$	0	0
$565$	−18.5708	−0.781281
$566$	61.9635	2.60452
$567$	0	0
$568$	116.446	4.88595
$569$	−7.47467	−0.313354	−0.156677	−	0.987650i	$-0.550078\pi$
$569$	−7.47467	−0.313354	−0.156677	+	0.987650i	$0.550078\pi$
$570$	0	0
$571$	−43.5452	−1.82231	−0.911155	−	0.412064i	$-0.864808\pi$
$571$	−43.5452	−1.82231	−0.911155	+	0.412064i	$0.864808\pi$
$572$	−64.1724	−2.68318
$573$	0	0
$574$	−1.55373	−0.0648515
$575$	3.56580	0.148704
$576$	0	0
$577$	13.6577	0.568579	0.284290	−	0.958738i	$-0.408242\pi$
$577$	13.6577	0.568579	0.284290	+	0.958738i	$0.408242\pi$
$578$	20.3437	0.846186
$579$	0	0
$580$	−5.75346	−0.238899
$581$	34.8129	1.44428
$582$	0	0
$583$	17.8305	0.738462
$584$	13.5841	0.562115
$585$	0	0
$586$	−79.0474	−3.26542
$587$	−26.5424	−1.09552	−0.547760	−	0.836635i	$-0.684519\pi$
$587$	−26.5424	−1.09552	−0.547760	+	0.836635i	$0.684519\pi$
$588$	0	0
$589$	−25.4472	−1.04853
$590$	−28.5822	−1.17671
$591$	0	0
$592$	86.0482	3.53656
$593$	−41.3882	−1.69961	−0.849805	−	0.527097i	$-0.823281\pi$
$593$	−41.3882	−1.69961	−0.849805	+	0.527097i	$0.823281\pi$
$594$	0	0
$595$	−16.3343	−0.669642
$596$	−5.15615	−0.211204
$597$	0	0
$598$	6.51928	0.266593
$599$	−16.2009	−0.661952	−0.330976	−	0.943639i	$-0.607378\pi$
$599$	−16.2009	−0.661952	−0.330976	+	0.943639i	$0.607378\pi$
$600$	0	0
$601$	−15.4648	−0.630823	−0.315411	−	0.948955i	$-0.602143\pi$
$601$	−15.4648	−0.630823	−0.315411	+	0.948955i	$0.602143\pi$
$602$	26.6030	1.08426
$603$	0	0
$604$	−80.7427	−3.28537
$605$	−21.0349	−0.855191
$606$	0	0
$607$	−33.0567	−1.34173	−0.670865	−	0.741579i	$-0.734077\pi$
$607$	−33.0567	−1.34173	−0.670865	+	0.741579i	$0.734077\pi$
$608$	−24.9467	−1.01172
$609$	0	0
$610$	−23.8413	−0.965305
$611$	−17.0531	−0.689894
$612$	0	0
$613$	10.8047	0.436399	0.218199	−	0.975904i	$-0.429982\pi$
$613$	10.8047	0.436399	0.218199	+	0.975904i	$0.429982\pi$
$614$	33.6407	1.35763
$615$	0	0
$616$	−175.792	−7.08286
$617$	3.09048	0.124418	0.0622090	−	0.998063i	$-0.480185\pi$
$617$	3.09048	0.124418	0.0622090	+	0.998063i	$0.480185\pi$
$618$	0	0
$619$	−1.50813	−0.0606170	−0.0303085	−	0.999541i	$-0.509649\pi$
$619$	−1.50813	−0.0606170	−0.0303085	+	0.999541i	$0.509649\pi$
$620$	59.1503	2.37553
$621$	0	0
$622$	13.7808	0.552562
$623$	55.4425	2.22126
$624$	0	0
$625$	5.54399	0.221759
$626$	−72.7384	−2.90721
$627$	0	0
$628$	109.888	4.38502
$629$	−27.5553	−1.09870
$630$	0	0
$631$	27.2305	1.08403	0.542014	−	0.840369i	$-0.317662\pi$
$631$	27.2305	1.08403	0.542014	+	0.840369i	$0.317662\pi$
$632$	67.9691	2.70367
$633$	0	0
$634$	71.7164	2.84822
$635$	−10.3058	−0.408971
$636$	0	0
$637$	33.0376	1.30900
$638$	13.9413	0.551942
$639$	0	0
$640$	−1.19333	−0.0471706
$641$	5.32710	0.210408	0.105204	−	0.994451i	$-0.466450\pi$
$641$	5.32710	0.210408	0.105204	+	0.994451i	$0.466450\pi$
$642$	0	0
$643$	−30.8242	−1.21559	−0.607794	−	0.794095i	$-0.707945\pi$
$643$	−30.8242	−1.21559	−0.607794	+	0.794095i	$0.707945\pi$
$644$	21.6025	0.851259
$645$	0	0
$646$	19.5848	0.770553
$647$	2.69886	0.106103	0.0530515	−	0.998592i	$-0.483105\pi$
$647$	2.69886	0.106103	0.0530515	+	0.998592i	$0.483105\pi$
$648$	0	0
$649$	48.9008	1.91952
$650$	23.2465	0.911802
$651$	0	0
$652$	−58.3062	−2.28345
$653$	−24.0326	−0.940469	−0.470235	−	0.882541i	$-0.655831\pi$
$653$	−24.0326	−0.940469	−0.470235	+	0.882541i	$0.655831\pi$
$654$	0	0
$655$	5.56743	0.217537
$656$	−1.25476	−0.0489901
$657$	0	0
$658$	−80.0318	−3.11997
$659$	−27.4102	−1.06775	−0.533874	−	0.845564i	$-0.679264\pi$
$659$	−27.4102	−1.06775	−0.533874	+	0.845564i	$0.679264\pi$
$660$	0	0
$661$	−19.3644	−0.753190	−0.376595	−	0.926378i	$-0.622905\pi$
$661$	−19.3644	−0.753190	−0.376595	+	0.926378i	$0.622905\pi$
$662$	−67.3258	−2.61669
$663$	0	0
$664$	56.6325	2.19777
$665$	13.3289	0.516874
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	55.6471	2.15305
$669$	0	0
$670$	−43.8492	−1.69404
$671$	40.7896	1.57466
$672$	0	0
$673$	5.10607	0.196825	0.0984123	−	0.995146i	$-0.468624\pi$
$673$	5.10607	0.196825	0.0984123	+	0.995146i	$0.468624\pi$
$674$	42.9510	1.65441
$675$	0	0
$676$	−32.4466	−1.24795
$677$	−11.9634	−0.459791	−0.229896	−	0.973215i	$-0.573838\pi$
$677$	−11.9634	−0.459791	−0.229896	+	0.973215i	$0.573838\pi$
$678$	0	0
$679$	48.6253	1.86607
$680$	−26.5721	−1.01900
$681$	0	0
$682$	−143.328	−5.48832
$683$	0.298274	0.0114132	0.00570658	−	0.999984i	$-0.498184\pi$
$683$	0.298274	0.0114132	0.00570658	+	0.999984i	$0.498184\pi$
$684$	0	0
$685$	−22.9645	−0.877429
$686$	72.9442	2.78502
$687$	0	0
$688$	21.4840	0.819070
$689$	−8.33794	−0.317650
$690$	0	0
$691$	0.290440	0.0110489	0.00552444	−	0.999985i	$-0.498242\pi$
$691$	0.290440	0.0110489	0.00552444	+	0.999985i	$0.498242\pi$
$692$	−83.8461	−3.18735
$693$	0	0
$694$	−46.8363	−1.77788
$695$	−5.88833	−0.223357
$696$	0	0
$697$	0.401812	0.0152197
$698$	66.3331	2.51075
$699$	0	0
$700$	77.0304	2.91148
$701$	11.9054	0.449661	0.224830	−	0.974398i	$-0.427817\pi$
$701$	11.9054	0.449661	0.224830	+	0.974398i	$0.427817\pi$
$702$	0	0
$703$	22.4854	0.848051
$704$	−39.2589	−1.47962
$705$	0	0
$706$	10.6234	0.399818
$707$	72.7672	2.73669
$708$	0	0
$709$	−51.7847	−1.94482	−0.972408	−	0.233289i	$-0.925051\pi$
$709$	−51.7847	−1.94482	−0.972408	+	0.233289i	$0.925051\pi$
$710$	−49.7292	−1.86630
$711$	0	0
$712$	90.1920	3.38009
$713$	10.2808	0.385020
$714$	0	0
$715$	15.9966	0.598239
$716$	20.3223	0.759482
$717$	0	0
$718$	−0.135014	−0.00503868
$719$	7.79945	0.290870	0.145435	−	0.989368i	$-0.453542\pi$
$719$	7.79945	0.290870	0.145435	+	0.989368i	$0.453542\pi$
$720$	0	0
$721$	25.1725	0.937472
$722$	33.5801	1.24972
$723$	0	0
$724$	54.5686	2.02803
$725$	−3.56580	−0.132431
$726$	0	0
$727$	33.3363	1.23637	0.618187	−	0.786031i	$-0.287867\pi$
$727$	33.3363	1.23637	0.618187	+	0.786031i	$0.287867\pi$
$728$	82.2044	3.04670
$729$	0	0
$730$	−5.80123	−0.214713
$731$	−6.87984	−0.254460
$732$	0	0
$733$	25.7238	0.950132	0.475066	−	0.879950i	$-0.342424\pi$
$733$	25.7238	0.950132	0.475066	+	0.879950i	$0.342424\pi$
$734$	−0.204838	−0.00756072
$735$	0	0
$736$	10.0786	0.371504
$737$	75.0208	2.76343
$738$	0	0
$739$	10.2327	0.376416	0.188208	−	0.982129i	$-0.439732\pi$
$739$	10.2327	0.376416	0.188208	+	0.982129i	$0.439732\pi$
$740$	−52.2658	−1.92133
$741$	0	0
$742$	−39.1308	−1.43654
$743$	−1.77612	−0.0651597	−0.0325798	−	0.999469i	$-0.510372\pi$
$743$	−1.77612	−0.0651597	−0.0325798	+	0.999469i	$0.510372\pi$
$744$	0	0
$745$	1.28530	0.0470898
$746$	11.7410	0.429867
$747$	0	0
$748$	77.8853	2.84777
$749$	52.3710	1.91360
$750$	0	0
$751$	25.6224	0.934974	0.467487	−	0.884000i	$-0.345160\pi$
$751$	25.6224	0.934974	0.467487	+	0.884000i	$0.345160\pi$
$752$	−64.6319	−2.35688
$753$	0	0
$754$	−6.51928	−0.237418
$755$	20.1272	0.732502
$756$	0	0
$757$	24.1155	0.876493	0.438247	−	0.898855i	$-0.355600\pi$
$757$	24.1155	0.876493	0.438247	+	0.898855i	$0.355600\pi$
$758$	14.0802	0.511415
$759$	0	0
$760$	21.6831	0.786528
$761$	−3.47156	−0.125844	−0.0629220	−	0.998018i	$-0.520042\pi$
$761$	−3.47156	−0.125844	−0.0629220	+	0.998018i	$0.520042\pi$
$762$	0	0
$763$	33.8815	1.22659
$764$	61.5738	2.22766
$765$	0	0
$766$	−1.62276	−0.0586329
$767$	−22.8671	−0.825685
$768$	0	0
$769$	28.8408	1.04003	0.520013	−	0.854158i	$-0.325927\pi$
$769$	28.8408	1.04003	0.520013	+	0.854158i	$0.325927\pi$
$770$	75.0736	2.70547
$771$	0	0
$772$	−60.9865	−2.19495
$773$	24.3024	0.874097	0.437049	−	0.899438i	$-0.356024\pi$
$773$	24.3024	0.874097	0.437049	+	0.899438i	$0.356024\pi$
$774$	0	0
$775$	36.6594	1.31684
$776$	79.1020	2.83960
$777$	0	0
$778$	59.6373	2.13810
$779$	−0.327882	−0.0117476
$780$	0	0
$781$	85.0807	3.04443
$782$	−7.91238	−0.282946
$783$	0	0
$784$	125.214	4.47192
$785$	−27.3924	−0.977678
$786$	0	0
$787$	37.4733	1.33578	0.667890	−	0.744260i	$-0.267198\pi$
$787$	37.4733	1.33578	0.667890	+	0.744260i	$0.267198\pi$
$788$	−106.747	−3.80270
$789$	0	0
$790$	−29.0269	−1.03273
$791$	−69.7280	−2.47924
$792$	0	0
$793$	−19.0742	−0.677343
$794$	47.5760	1.68841
$795$	0	0
$796$	−82.5077	−2.92441
$797$	−14.4477	−0.511762	−0.255881	−	0.966708i	$-0.582366\pi$
$797$	−14.4477	−0.511762	−0.255881	+	0.966708i	$0.582366\pi$
$798$	0	0
$799$	20.6971	0.732212
$800$	35.9385	1.27062
$801$	0	0
$802$	−55.1647	−1.94794
$803$	9.92521	0.350253
$804$	0	0
$805$	−5.38498	−0.189796
$806$	67.0236	2.36080
$807$	0	0
$808$	118.375	4.16443
$809$	−17.2514	−0.606529	−0.303264	−	0.952906i	$-0.598076\pi$
$809$	−17.2514	−0.606529	−0.303264	+	0.952906i	$0.598076\pi$
$810$	0	0
$811$	−53.1459	−1.86621	−0.933103	−	0.359609i	$-0.882910\pi$
$811$	−53.1459	−1.86621	−0.933103	+	0.359609i	$0.882910\pi$
$812$	−21.6025	−0.758101
$813$	0	0
$814$	126.646	4.43894
$815$	14.5343	0.509114
$816$	0	0
$817$	5.61401	0.196409
$818$	13.7131	0.479467
$819$	0	0
$820$	0.762141	0.0266151
$821$	−50.2329	−1.75314	−0.876569	−	0.481276i	$-0.840174\pi$
$821$	−50.2329	−1.75314	−0.876569	+	0.481276i	$0.840174\pi$
$822$	0	0
$823$	11.3933	0.397145	0.198572	−	0.980086i	$-0.436370\pi$
$823$	11.3933	0.397145	0.198572	+	0.980086i	$0.436370\pi$
$824$	40.9498	1.42655
$825$	0	0
$826$	−107.318	−3.73406
$827$	37.6213	1.30822	0.654111	−	0.756399i	$-0.273043\pi$
$827$	37.6213	1.30822	0.654111	+	0.756399i	$0.273043\pi$
$828$	0	0
$829$	47.3525	1.64462	0.822310	−	0.569040i	$-0.192685\pi$
$829$	47.3525	1.64462	0.822310	+	0.569040i	$0.192685\pi$
$830$	−24.1855	−0.839490
$831$	0	0
$832$	18.3584	0.636462
$833$	−40.0974	−1.38929
$834$	0	0
$835$	−13.8715	−0.480042
$836$	−63.5551	−2.19810
$837$	0	0
$838$	−28.1490	−0.972391
$839$	−12.0869	−0.417285	−0.208642	−	0.977992i	$-0.566904\pi$
$839$	−12.0869	−0.417285	−0.208642	+	0.977992i	$0.566904\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	79.1415	2.72740
$843$	0	0
$844$	−85.3526	−2.93796
$845$	8.08815	0.278241
$846$	0	0
$847$	−78.9799	−2.71378
$848$	−31.6011	−1.08519
$849$	0	0
$850$	−28.2140	−0.967732
$851$	−9.08423	−0.311404
$852$	0	0
$853$	23.7407	0.812866	0.406433	−	0.913681i	$-0.366772\pi$
$853$	23.7407	0.812866	0.406433	+	0.913681i	$0.366772\pi$
$854$	−89.5169	−3.06321
$855$	0	0
$856$	85.1955	2.91192
$857$	−22.4271	−0.766094	−0.383047	−	0.923729i	$-0.625125\pi$
$857$	−22.4271	−0.766094	−0.383047	+	0.923729i	$0.625125\pi$
$858$	0	0
$859$	−30.2852	−1.03332	−0.516660	−	0.856191i	$-0.672825\pi$
$859$	−30.2852	−1.03332	−0.516660	+	0.856191i	$0.672825\pi$
$860$	−13.0494	−0.444981
$861$	0	0
$862$	−79.3334	−2.70211
$863$	23.8885	0.813176	0.406588	−	0.913612i	$-0.366719\pi$
$863$	23.8885	0.813176	0.406588	+	0.913612i	$0.366719\pi$
$864$	0	0
$865$	20.9008	0.710647
$866$	35.6740	1.21225
$867$	0	0
$868$	222.092	7.53829
$869$	49.6615	1.68465
$870$	0	0
$871$	−35.0815	−1.18869
$872$	55.1174	1.86651
$873$	0	0
$874$	6.45656	0.218397
$875$	−46.1267	−1.55937
$876$	0	0
$877$	17.7700	0.600051	0.300026	−	0.953931i	$-0.403005\pi$
$877$	17.7700	0.600051	0.300026	+	0.953931i	$0.403005\pi$
$878$	−66.0763	−2.22997
$879$	0	0
$880$	60.6277	2.04376
$881$	7.94362	0.267628	0.133814	−	0.991006i	$-0.457278\pi$
$881$	7.94362	0.267628	0.133814	+	0.991006i	$0.457278\pi$
$882$	0	0
$883$	39.4201	1.32659	0.663295	−	0.748358i	$-0.269157\pi$
$883$	39.4201	1.32659	0.663295	+	0.748358i	$0.269157\pi$
$884$	−36.4210	−1.22497
$885$	0	0
$886$	35.9054	1.20627
$887$	4.47672	0.150313	0.0751567	−	0.997172i	$-0.476054\pi$
$887$	4.47672	0.150313	0.0751567	+	0.997172i	$0.476054\pi$
$888$	0	0
$889$	−38.6951	−1.29779
$890$	−38.5174	−1.29111
$891$	0	0
$892$	119.924	4.01537
$893$	−16.8890	−0.565170
$894$	0	0
$895$	−5.06586	−0.169333
$896$	−4.48061	−0.149687
$897$	0	0
$898$	−85.3571	−2.84840
$899$	−10.2808	−0.342885
$900$	0	0
$901$	10.1197	0.337135
$902$	−1.84676	−0.0614903
$903$	0	0
$904$	−113.431	−3.77267
$905$	−13.6026	−0.452166
$906$	0	0
$907$	18.5410	0.615643	0.307821	−	0.951444i	$-0.400400\pi$
$907$	18.5410	0.615643	0.307821	+	0.951444i	$0.400400\pi$
$908$	105.248	3.49276
$909$	0	0
$910$	−35.1062	−1.16376
$911$	−10.1420	−0.336019	−0.168010	−	0.985785i	$-0.553734\pi$
$911$	−10.1420	−0.336019	−0.168010	+	0.985785i	$0.553734\pi$
$912$	0	0
$913$	41.3784	1.36943
$914$	−30.1974	−0.998842
$915$	0	0
$916$	64.0827	2.11735
$917$	20.9040	0.690312
$918$	0	0
$919$	−23.8028	−0.785183	−0.392592	−	0.919713i	$-0.628421\pi$
$919$	−23.8028	−0.785183	−0.392592	+	0.919713i	$0.628421\pi$
$920$	−8.76011	−0.288812
$921$	0	0
$922$	38.0649	1.25360
$923$	−39.7857	−1.30956
$924$	0	0
$925$	−32.3926	−1.06506
$926$	39.2345	1.28933
$927$	0	0
$928$	−10.0786	−0.330848
$929$	49.0084	1.60791	0.803957	−	0.594687i	$-0.202724\pi$
$929$	49.0084	1.60791	0.803957	+	0.594687i	$0.202724\pi$
$930$	0	0
$931$	32.7198	1.07235
$932$	−99.2893	−3.25233
$933$	0	0
$934$	−20.9956	−0.686997
$935$	−19.4149	−0.634935
$936$	0	0
$937$	51.1577	1.67125	0.835624	−	0.549302i	$-0.185106\pi$
$937$	51.1577	1.67125	0.835624	+	0.549302i	$0.185106\pi$
$938$	−164.641	−5.37572
$939$	0	0
$940$	39.2574	1.28044
$941$	15.8642	0.517158	0.258579	−	0.965990i	$-0.416746\pi$
$941$	15.8642	0.517158	0.258579	+	0.965990i	$0.416746\pi$
$942$	0	0
$943$	0.132467	0.00431370
$944$	−86.6674	−2.82078
$945$	0	0
$946$	31.6202	1.02806
$947$	35.9214	1.16729	0.583645	−	0.812009i	$-0.301626\pi$
$947$	35.9214	1.16729	0.583645	+	0.812009i	$0.301626\pi$
$948$	0	0
$949$	−4.64126	−0.150662
$950$	23.0228	0.746960
$951$	0	0
$952$	−99.7706	−3.23358
$953$	−5.95020	−0.192746	−0.0963729	−	0.995345i	$-0.530724\pi$
$953$	−5.95020	−0.192746	−0.0963729	+	0.995345i	$0.530724\pi$
$954$	0	0
$955$	−15.3488	−0.496676
$956$	34.6095	1.11935
$957$	0	0
$958$	76.7289	2.47900
$959$	−86.2250	−2.78435
$960$	0	0
$961$	74.6953	2.40952
$962$	−59.2227	−1.90942
$963$	0	0
$964$	−20.9441	−0.674565
$965$	15.2024	0.489383
$966$	0	0
$967$	−50.3771	−1.62002	−0.810010	−	0.586417i	$-0.800538\pi$
$967$	−50.3771	−1.62002	−0.810010	+	0.586417i	$0.800538\pi$
$968$	−128.482	−4.12957
$969$	0	0
$970$	−33.7813	−1.08465
$971$	22.2396	0.713704	0.356852	−	0.934161i	$-0.383850\pi$
$971$	22.2396	0.713704	0.356852	+	0.934161i	$0.383850\pi$
$972$	0	0
$973$	−22.1089	−0.708780
$974$	0.803347	0.0257409
$975$	0	0
$976$	−72.2918	−2.31400
$977$	−15.2575	−0.488130	−0.244065	−	0.969759i	$-0.578481\pi$
$977$	−15.2575	−0.488130	−0.244065	+	0.969759i	$0.578481\pi$
$978$	0	0
$979$	65.8986	2.10613
$980$	−76.0550	−2.42949
$981$	0	0
$982$	−60.5166	−1.93116
$983$	−7.58332	−0.241870	−0.120935	−	0.992660i	$-0.538589\pi$
$983$	−7.58332	−0.241870	−0.120935	+	0.992660i	$0.538589\pi$
$984$	0	0
$985$	26.6093	0.847844
$986$	7.91238	0.251981
$987$	0	0
$988$	29.7198	0.945514
$989$	−2.26810	−0.0721213
$990$	0	0
$991$	3.66287	0.116355	0.0581775	−	0.998306i	$-0.481471\pi$
$991$	3.66287	0.116355	0.0581775	+	0.998306i	$0.481471\pi$
$992$	103.617	3.28983
$993$	0	0
$994$	−186.718	−5.92235
$995$	20.5671	0.652022
$996$	0	0
$997$	2.06945	0.0655403	0.0327701	−	0.999463i	$-0.489567\pi$
$997$	2.06945	0.0655403	0.0327701	+	0.999463i	$0.489567\pi$
$998$	60.3889	1.91158
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.3	yes	30
3.2	odd	2		6003.2.a.v.1.28	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.28	✓	30	3.2	odd	2
6003.2.a.w.1.3	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)

\(f(q)\)	\(=\)	\(q-2.60849 q^{2} +4.80424 q^{4} -1.19758 q^{5} -4.49655 q^{7} -7.31485 q^{8} +O(q^{10})\)	\(q-2.60849 q^{2} +4.80424 q^{4} -1.19758 q^{5} -4.49655 q^{7} -7.31485 q^{8} +3.12388 q^{10} -5.34458 q^{11} +2.49925 q^{13} +11.7292 q^{14} +9.47226 q^{16} -3.03331 q^{17} +2.47521 q^{19} -5.75346 q^{20} +13.9413 q^{22} -1.00000 q^{23} -3.56580 q^{25} -6.51928 q^{26} -21.6025 q^{28} +1.00000 q^{29} -10.2808 q^{31} -10.0786 q^{32} +7.91238 q^{34} +5.38498 q^{35} +9.08423 q^{37} -6.45656 q^{38} +8.76011 q^{40} -0.132467 q^{41} +2.26810 q^{43} -25.6767 q^{44} +2.60849 q^{46} -6.82328 q^{47} +13.2190 q^{49} +9.30138 q^{50} +12.0070 q^{52} -3.33618 q^{53} +6.40056 q^{55} +32.8916 q^{56} -2.60849 q^{58} -9.14960 q^{59} -7.63195 q^{61} +26.8175 q^{62} +7.34555 q^{64} -2.99305 q^{65} -14.0368 q^{67} -14.5728 q^{68} -14.0467 q^{70} -15.9191 q^{71} -1.85706 q^{73} -23.6962 q^{74} +11.8915 q^{76} +24.0322 q^{77} -9.29193 q^{79} -11.3438 q^{80} +0.345538 q^{82} -7.74213 q^{83} +3.63263 q^{85} -5.91632 q^{86} +39.0948 q^{88} -12.3300 q^{89} -11.2380 q^{91} -4.80424 q^{92} +17.7985 q^{94} -2.96426 q^{95} -10.8139 q^{97} -34.4817 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 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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