LMFDB - Embedded newform 1287.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, 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("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.78165$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.95594 q^{2} +1.82571 q^{4} +3.78165 q^{5} +1.70316 q^{7} +0.340899 q^{8} +O(q^{10})$	\(q-1.95594 q^{2} +1.82571 q^{4} +3.78165 q^{5} +1.70316 q^{7} +0.340899 q^{8} -7.39670 q^{10} +1.00000 q^{11} -1.00000 q^{13} -3.33128 q^{14} -4.31820 q^{16} +4.36226 q^{17} +3.86015 q^{19} +6.90421 q^{20} -1.95594 q^{22} +6.73760 q^{23} +9.30090 q^{25} +1.95594 q^{26} +3.10947 q^{28} +3.83339 q^{29} -5.78165 q^{31} +7.76435 q^{32} -8.53233 q^{34} +6.44075 q^{35} -10.0041 q^{37} -7.55023 q^{38} +1.28916 q^{40} +4.29684 q^{41} -2.95594 q^{43} +1.82571 q^{44} -13.1784 q^{46} -8.59774 q^{47} -4.09925 q^{49} -18.1920 q^{50} -1.82571 q^{52} -13.8511 q^{53} +3.78165 q^{55} +0.580605 q^{56} -7.49789 q^{58} +9.31820 q^{59} +14.9463 q^{61} +11.3086 q^{62} -6.55023 q^{64} -3.78165 q^{65} +0.130232 q^{67} +7.96422 q^{68} -12.5977 q^{70} -13.8125 q^{71} +5.26647 q^{73} +19.5674 q^{74} +7.04752 q^{76} +1.70316 q^{77} -12.3015 q^{79} -16.3299 q^{80} -8.40438 q^{82} +0.245106 q^{83} +16.4966 q^{85} +5.78165 q^{86} +0.340899 q^{88} +3.16467 q^{89} -1.70316 q^{91} +12.3009 q^{92} +16.8167 q^{94} +14.5977 q^{95} +9.64348 q^{97} +8.01790 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 6 * q^7 + 12 * q^8	$ 4 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 6 q^{7} + 12 q^{8} + 4 q^{11} - 4 q^{13} + 4 q^{14} + 4 q^{16} + 6 q^{17} - 2 q^{19} + 20 q^{20} + 2 q^{22} + 8 q^{23} - 2 q^{26} + 10 q^{28} + 4 q^{29} - 14 q^{31} + 6 q^{32} - 10 q^{34} + 6 q^{35} - 2 q^{37} - 8 q^{38} + 18 q^{40} + 18 q^{41} - 2 q^{43} + 8 q^{44} - 14 q^{46} + 2 q^{47} - 24 q^{50} - 8 q^{52} + 4 q^{53} + 6 q^{55} - 24 q^{58} + 16 q^{59} + 22 q^{61} - 4 q^{62} - 4 q^{64} - 6 q^{65} - 10 q^{67} - 4 q^{68} - 14 q^{70} - 2 q^{71} + 2 q^{73} + 22 q^{74} + 14 q^{76} + 6 q^{77} + 2 q^{79} + 6 q^{80} + 8 q^{82} + 4 q^{83} + 6 q^{85} + 14 q^{86} + 12 q^{88} - 16 q^{89} - 6 q^{91} + 12 q^{92} + 26 q^{94} + 22 q^{95} - 2 q^{97} + 34 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 6 * q^7 + 12 * q^8 + 4 * q^11 - 4 * q^13 + 4 * q^14 + 4 * q^16 + 6 * q^17 - 2 * q^19 + 20 * q^20 + 2 * q^22 + 8 * q^23 - 2 * q^26 + 10 * q^28 + 4 * q^29 - 14 * q^31 + 6 * q^32 - 10 * q^34 + 6 * q^35 - 2 * q^37 - 8 * q^38 + 18 * q^40 + 18 * q^41 - 2 * q^43 + 8 * q^44 - 14 * q^46 + 2 * q^47 - 24 * q^50 - 8 * q^52 + 4 * q^53 + 6 * q^55 - 24 * q^58 + 16 * q^59 + 22 * q^61 - 4 * q^62 - 4 * q^64 - 6 * q^65 - 10 * q^67 - 4 * q^68 - 14 * q^70 - 2 * q^71 + 2 * q^73 + 22 * q^74 + 14 * q^76 + 6 * q^77 + 2 * q^79 + 6 * q^80 + 8 * q^82 + 4 * q^83 + 6 * q^85 + 14 * q^86 + 12 * q^88 - 16 * q^89 - 6 * q^91 + 12 * q^92 + 26 * q^94 + 22 * q^95 - 2 * q^97 + 34 * 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 $

$2$	−1.95594	−1.38306	−0.691530	−	0.722348i	$-0.743063\pi$
$2$	−1.95594	−1.38306	−0.691530	+	0.722348i	$0.743063\pi$
$3$	0	0
$3$	0	0
$4$	1.82571	0.912855
$5$	3.78165	1.69121	0.845603	−	0.533812i	$-0.179241\pi$
$5$	3.78165	1.69121	0.845603	+	0.533812i	$0.179241\pi$
$6$	0	0
$7$	1.70316	0.643733	0.321867	−	0.946785i	$-0.395690\pi$
$7$	1.70316	0.643733	0.321867	+	0.946785i	$0.395690\pi$
$8$	0.340899	0.120526
$9$	0	0
$10$	−7.39670	−2.33904
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.33128	−0.890322
$15$	0	0
$16$	−4.31820	−1.07955
$17$	4.36226	1.05800	0.529002	−	0.848621i	$-0.322567\pi$
$17$	4.36226	1.05800	0.529002	+	0.848621i	$0.322567\pi$
$18$	0	0
$19$	3.86015	0.885579	0.442789	−	0.896626i	$-0.353989\pi$
$19$	3.86015	0.885579	0.442789	+	0.896626i	$0.353989\pi$
$20$	6.90421	1.54383
$21$	0	0
$22$	−1.95594	−0.417008
$23$	6.73760	1.40489	0.702443	−	0.711740i	$-0.252093\pi$
$23$	6.73760	1.40489	0.702443	+	0.711740i	$0.252093\pi$
$24$	0	0
$25$	9.30090	1.86018
$26$	1.95594	0.383592
$27$	0	0
$28$	3.10947	0.587635
$29$	3.83339	0.711843	0.355921	−	0.934516i	$-0.384167\pi$
$29$	3.83339	0.711843	0.355921	+	0.934516i	$0.384167\pi$
$30$	0	0
$31$	−5.78165	−1.03842	−0.519208	−	0.854648i	$-0.673773\pi$
$31$	−5.78165	−1.03842	−0.519208	+	0.854648i	$0.673773\pi$
$32$	7.76435	1.37256
$33$	0	0
$34$	−8.53233	−1.46328
$35$	6.44075	1.08869
$36$	0	0
$37$	−10.0041	−1.64466	−0.822329	−	0.569013i	$-0.807326\pi$
$37$	−10.0041	−1.64466	−0.822329	+	0.569013i	$0.807326\pi$
$38$	−7.55023	−1.22481
$39$	0	0
$40$	1.28916	0.203835
$41$	4.29684	0.671054	0.335527	−	0.942031i	$-0.391086\pi$
$41$	4.29684	0.671054	0.335527	+	0.942031i	$0.391086\pi$
$42$	0	0
$43$	−2.95594	−0.450777	−0.225389	−	0.974269i	$-0.572365\pi$
$43$	−2.95594	−0.450777	−0.225389	+	0.974269i	$0.572365\pi$
$44$	1.82571	0.275236
$45$	0	0
$46$	−13.1784	−1.94304
$47$	−8.59774	−1.25411	−0.627055	−	0.778975i	$-0.715740\pi$
$47$	−8.59774	−1.25411	−0.627055	+	0.778975i	$0.715740\pi$
$48$	0	0
$49$	−4.09925	−0.585607
$50$	−18.1920	−2.57274
$51$	0	0
$52$	−1.82571	−0.253181
$53$	−13.8511	−1.90260	−0.951300	−	0.308268i	$-0.900251\pi$
$53$	−13.8511	−1.90260	−0.951300	+	0.308268i	$0.900251\pi$
$54$	0	0
$55$	3.78165	0.509918
$56$	0.580605	0.0775867
$57$	0	0
$58$	−7.49789	−0.984521
$59$	9.31820	1.21313	0.606563	−	0.795035i	$-0.292548\pi$
$59$	9.31820	1.21313	0.606563	+	0.795035i	$0.292548\pi$
$60$	0	0
$61$	14.9463	1.91368	0.956840	−	0.290614i	$-0.0938597\pi$
$61$	14.9463	1.91368	0.956840	+	0.290614i	$0.0938597\pi$
$62$	11.3086	1.43619
$63$	0	0
$64$	−6.55023	−0.818779
$65$	−3.78165	−0.469056
$66$	0	0
$67$	0.130232	0.0159103	0.00795516	−	0.999968i	$-0.497468\pi$
$67$	0.130232	0.0159103	0.00795516	+	0.999968i	$0.497468\pi$
$68$	7.96422	0.965804
$69$	0	0
$70$	−12.5977	−1.50572
$71$	−13.8125	−1.63924	−0.819620	−	0.572908i	$-0.805815\pi$
$71$	−13.8125	−1.63924	−0.819620	+	0.572908i	$0.805815\pi$
$72$	0	0
$73$	5.26647	0.616393	0.308197	−	0.951323i	$-0.400275\pi$
$73$	5.26647	0.616393	0.308197	+	0.951323i	$0.400275\pi$
$74$	19.5674	2.27466
$75$	0	0
$76$	7.04752	0.808406
$77$	1.70316	0.194093
$78$	0	0
$79$	−12.3015	−1.38403	−0.692014	−	0.721884i	$-0.743276\pi$
$79$	−12.3015	−1.38403	−0.692014	+	0.721884i	$0.743276\pi$
$80$	−16.3299	−1.82574
$81$	0	0
$82$	−8.40438	−0.928108
$83$	0.245106	0.0269038	0.0134519	−	0.999910i	$-0.495718\pi$
$83$	0.245106	0.0269038	0.0134519	+	0.999910i	$0.495718\pi$
$84$	0	0
$85$	16.4966	1.78930
$86$	5.78165	0.623452
$87$	0	0
$88$	0.340899	0.0363400
$89$	3.16467	0.335454	0.167727	−	0.985833i	$-0.446357\pi$
$89$	3.16467	0.335454	0.167727	+	0.985833i	$0.446357\pi$
$90$	0	0
$91$	−1.70316	−0.178539
$92$	12.3009	1.28246
$93$	0	0
$94$	16.8167	1.73451
$95$	14.5977	1.49770
$96$	0	0
$97$	9.64348	0.979147	0.489573	−	0.871962i	$-0.337152\pi$
$97$	9.64348	0.979147	0.489573	+	0.871962i	$0.337152\pi$
$98$	8.01790	0.809930
$99$	0	0
$100$	16.9808	1.69808
$101$	−15.9529	−1.58738	−0.793688	−	0.608325i	$-0.791842\pi$
$101$	−15.9529	−1.58738	−0.793688	+	0.608325i	$0.791842\pi$
$102$	0	0
$103$	5.23009	0.515336	0.257668	−	0.966234i	$-0.417046\pi$
$103$	5.23009	0.515336	0.257668	+	0.966234i	$0.417046\pi$
$104$	−0.340899	−0.0334279
$105$	0	0
$106$	27.0920	2.63141
$107$	−0.464055	−0.0448619	−0.0224309	−	0.999748i	$-0.507141\pi$
$107$	−0.464055	−0.0448619	−0.0224309	+	0.999748i	$0.507141\pi$
$108$	0	0
$109$	−6.55730	−0.628076	−0.314038	−	0.949410i	$-0.601682\pi$
$109$	−6.55730	−0.628076	−0.314038	+	0.949410i	$0.601682\pi$
$110$	−7.39670	−0.705247
$111$	0	0
$112$	−7.35458	−0.694943
$113$	12.2725	1.15450	0.577248	−	0.816569i	$-0.304126\pi$
$113$	12.2725	1.15450	0.577248	+	0.816569i	$0.304126\pi$
$114$	0	0
$115$	25.4793	2.37595
$116$	6.99866	0.649809
$117$	0	0
$118$	−18.2259	−1.67783
$119$	7.42962	0.681072
$120$	0	0
$121$	1.00000	0.0909091
$122$	−29.2341	−2.64674
$123$	0	0
$124$	−10.5556	−0.947923
$125$	16.2645	1.45474
$126$	0	0
$127$	−13.1170	−1.16395	−0.581973	−	0.813208i	$-0.697719\pi$
$127$	−13.1170	−1.16395	−0.581973	+	0.813208i	$0.697719\pi$
$128$	−2.71684	−0.240137
$129$	0	0
$130$	7.39670	0.648733
$131$	2.86615	0.250417	0.125208	−	0.992130i	$-0.460040\pi$
$131$	2.86615	0.250417	0.125208	+	0.992130i	$0.460040\pi$
$132$	0	0
$133$	6.57444	0.570077
$134$	−0.254725	−0.0220049
$135$	0	0
$136$	1.48709	0.127517
$137$	22.0308	1.88222	0.941110	−	0.338101i	$-0.109784\pi$
$137$	22.0308	1.88222	0.941110	+	0.338101i	$0.109784\pi$
$138$	0	0
$139$	2.27008	0.192546	0.0962730	−	0.995355i	$-0.469308\pi$
$139$	2.27008	0.192546	0.0962730	+	0.995355i	$0.469308\pi$
$140$	11.7590	0.993813
$141$	0	0
$142$	27.0164	2.26717
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	14.4966	1.20387
$146$	−10.3009	−0.852509
$147$	0	0
$148$	−18.2645	−1.50133
$149$	−10.2776	−0.841974	−0.420987	−	0.907067i	$-0.638316\pi$
$149$	−10.2776	−0.841974	−0.420987	+	0.907067i	$0.638316\pi$
$150$	0	0
$151$	6.03250	0.490918	0.245459	−	0.969407i	$-0.421061\pi$
$151$	6.03250	0.490918	0.245459	+	0.969407i	$0.421061\pi$
$152$	1.31592	0.106735
$153$	0	0
$154$	−3.33128	−0.268442
$155$	−21.8642	−1.75618
$156$	0	0
$157$	4.89459	0.390631	0.195315	−	0.980741i	$-0.437427\pi$
$157$	4.89459	0.390631	0.195315	+	0.980741i	$0.437427\pi$
$158$	24.0610	1.91419
$159$	0	0
$160$	29.3621	2.32128
$161$	11.4752	0.904372
$162$	0	0
$163$	−11.6134	−0.909629	−0.454815	−	0.890586i	$-0.650294\pi$
$163$	−11.6134	−0.909629	−0.454815	+	0.890586i	$0.650294\pi$
$164$	7.84479	0.612575
$165$	0	0
$166$	−0.479412	−0.0372096
$167$	14.9504	1.15690	0.578448	−	0.815719i	$-0.303659\pi$
$167$	14.9504	1.15690	0.578448	+	0.815719i	$0.303659\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−32.2663	−2.47471
$171$	0	0
$172$	−5.39670	−0.411494
$173$	5.54963	0.421930	0.210965	−	0.977494i	$-0.432339\pi$
$173$	5.54963	0.421930	0.210965	+	0.977494i	$0.432339\pi$
$174$	0	0
$175$	15.8409	1.19746
$176$	−4.31820	−0.325497
$177$	0	0
$178$	−6.18991	−0.463953
$179$	−10.0558	−0.751606	−0.375803	−	0.926700i	$-0.622633\pi$
$179$	−10.0558	−0.751606	−0.375803	+	0.926700i	$0.622633\pi$
$180$	0	0
$181$	5.07082	0.376911	0.188455	−	0.982082i	$-0.439652\pi$
$181$	5.07082	0.376911	0.188455	+	0.982082i	$0.439652\pi$
$182$	3.33128	0.246931
$183$	0	0
$184$	2.29684	0.169325
$185$	−37.8319	−2.78146
$186$	0	0
$187$	4.36226	0.319000
$188$	−15.6970	−1.14482
$189$	0	0
$190$	−28.5523	−2.07141
$191$	−2.75067	−0.199032	−0.0995159	−	0.995036i	$-0.531729\pi$
$191$	−2.75067	−0.199032	−0.0995159	+	0.995036i	$0.531729\pi$
$192$	0	0
$193$	9.39609	0.676346	0.338173	−	0.941084i	$-0.390191\pi$
$193$	9.39609	0.676346	0.338173	+	0.941084i	$0.390191\pi$
$194$	−18.8621	−1.35422
$195$	0	0
$196$	−7.48405	−0.534575
$197$	1.09412	0.0779526	0.0389763	−	0.999240i	$-0.487590\pi$
$197$	1.09412	0.0779526	0.0389763	+	0.999240i	$0.487590\pi$
$198$	0	0
$199$	−13.8499	−0.981795	−0.490898	−	0.871217i	$-0.663331\pi$
$199$	−13.8499	−0.981795	−0.490898	+	0.871217i	$0.663331\pi$
$200$	3.17067	0.224200
$201$	0	0
$202$	31.2030	2.19544
$203$	6.52887	0.458237
$204$	0	0
$205$	16.2492	1.13489
$206$	−10.2297	−0.712740
$207$	0	0
$208$	4.31820	0.299413
$209$	3.86015	0.267012
$210$	0	0
$211$	−3.60314	−0.248051	−0.124025	−	0.992279i	$-0.539580\pi$
$211$	−3.60314	−0.248051	−0.124025	+	0.992279i	$0.539580\pi$
$212$	−25.2882	−1.73680
$213$	0	0
$214$	0.907665	0.0620467
$215$	−11.1784	−0.762357
$216$	0	0
$217$	−9.84707	−0.668463
$218$	12.8257	0.868667
$219$	0	0
$220$	6.90421	0.465481
$221$	−4.36226	−0.293437
$222$	0	0
$223$	−6.34918	−0.425173	−0.212586	−	0.977142i	$-0.568189\pi$
$223$	−6.34918	−0.425173	−0.212586	+	0.977142i	$0.568189\pi$
$224$	13.2239	0.883561
$225$	0	0
$226$	−24.0042	−1.59674
$227$	24.4346	1.62178	0.810890	−	0.585198i	$-0.198983\pi$
$227$	24.4346	1.62178	0.810890	+	0.585198i	$0.198983\pi$
$228$	0	0
$229$	12.7893	0.845143	0.422571	−	0.906330i	$-0.361128\pi$
$229$	12.7893	0.845143	0.422571	+	0.906330i	$0.361128\pi$
$230$	−49.8360	−3.28609
$231$	0	0
$232$	1.30680	0.0857956
$233$	29.6226	1.94064	0.970319	−	0.241827i	$-0.0777468\pi$
$233$	29.6226	1.94064	0.970319	+	0.241827i	$0.0777468\pi$
$234$	0	0
$235$	−32.5137	−2.12096
$236$	17.0123	1.10741
$237$	0	0
$238$	−14.5319	−0.941963
$239$	2.82977	0.183043	0.0915214	−	0.995803i	$-0.470827\pi$
$239$	2.82977	0.183043	0.0915214	+	0.995803i	$0.470827\pi$
$240$	0	0
$241$	−23.3110	−1.50159	−0.750796	−	0.660535i	$-0.770330\pi$
$241$	−23.3110	−1.50159	−0.750796	+	0.660535i	$0.770330\pi$
$242$	−1.95594	−0.125733
$243$	0	0
$244$	27.2877	1.74691
$245$	−15.5020	−0.990383
$246$	0	0
$247$	−3.86015	−0.245615
$248$	−1.97096	−0.125156
$249$	0	0
$250$	−31.8125	−2.01200
$251$	4.19971	0.265083	0.132542	−	0.991177i	$-0.457686\pi$
$251$	4.19971	0.265083	0.132542	+	0.991177i	$0.457686\pi$
$252$	0	0
$253$	6.73760	0.423589
$254$	25.6561	1.60981
$255$	0	0
$256$	18.4144	1.15090
$257$	−8.84691	−0.551855	−0.275928	−	0.961178i	$-0.588985\pi$
$257$	−8.84691	−0.551855	−0.275928	+	0.961178i	$0.588985\pi$
$258$	0	0
$259$	−17.0385	−1.05872
$260$	−6.90421	−0.428181
$261$	0	0
$262$	−5.60603	−0.346341
$263$	0.307064	0.0189344	0.00946720	−	0.999955i	$-0.496986\pi$
$263$	0.307064	0.0189344	0.00946720	+	0.999955i	$0.496986\pi$
$264$	0	0
$265$	−52.3802	−3.21769
$266$	−12.8592	−0.788450
$267$	0	0
$268$	0.237765	0.0145238
$269$	−16.8280	−1.02602	−0.513010	−	0.858382i	$-0.671470\pi$
$269$	−16.8280	−1.02602	−0.513010	+	0.858382i	$0.671470\pi$
$270$	0	0
$271$	−8.20873	−0.498645	−0.249322	−	0.968421i	$-0.580208\pi$
$271$	−8.20873	−0.498645	−0.249322	+	0.968421i	$0.580208\pi$
$272$	−18.8371	−1.14217
$273$	0	0
$274$	−43.0910	−2.60322
$275$	9.30090	0.560866
$276$	0	0
$277$	0.838789	0.0503980	0.0251990	−	0.999682i	$-0.491978\pi$
$277$	0.838789	0.0503980	0.0251990	+	0.999682i	$0.491978\pi$
$278$	−4.44015	−0.266303
$279$	0	0
$280$	2.19565	0.131215
$281$	17.6566	1.05330	0.526651	−	0.850082i	$-0.323447\pi$
$281$	17.6566	1.05330	0.526651	+	0.850082i	$0.323447\pi$
$282$	0	0
$283$	−29.4809	−1.75246	−0.876230	−	0.481894i	$-0.839949\pi$
$283$	−29.4809	−1.75246	−0.876230	+	0.481894i	$0.839949\pi$
$284$	−25.2176	−1.49639
$285$	0	0
$286$	1.95594	0.115657
$287$	7.31820	0.431980
$288$	0	0
$289$	2.02930	0.119371
$290$	−28.3544	−1.66503
$291$	0	0
$292$	9.61504	0.562678
$293$	2.31220	0.135080	0.0675401	−	0.997717i	$-0.478485\pi$
$293$	2.31220	0.135080	0.0675401	+	0.997717i	$0.478485\pi$
$294$	0	0
$295$	35.2382	2.05165
$296$	−3.41038	−0.198224
$297$	0	0
$298$	20.1024	1.16450
$299$	−6.73760	−0.389645
$300$	0	0
$301$	−5.03444	−0.290180
$302$	−11.7992	−0.678969
$303$	0	0
$304$	−16.6689	−0.956027
$305$	56.5218	3.23643
$306$	0	0
$307$	29.6959	1.69484	0.847418	−	0.530926i	$-0.178156\pi$
$307$	29.6959	1.69484	0.847418	+	0.530926i	$0.178156\pi$
$308$	3.10947	0.177179
$309$	0	0
$310$	42.7651	2.42890
$311$	−28.3424	−1.60715	−0.803575	−	0.595203i	$-0.797072\pi$
$311$	−28.3424	−1.60715	−0.803575	+	0.595203i	$0.797072\pi$
$312$	0	0
$313$	29.8462	1.68701	0.843503	−	0.537125i	$-0.180490\pi$
$313$	29.8462	1.68701	0.843503	+	0.537125i	$0.180490\pi$
$314$	−9.57353	−0.540266
$315$	0	0
$316$	−22.4590	−1.26342
$317$	3.83232	0.215244	0.107622	−	0.994192i	$-0.465676\pi$
$317$	3.83232	0.215244	0.107622	+	0.994192i	$0.465676\pi$
$318$	0	0
$319$	3.83339	0.214629
$320$	−24.7707	−1.38472
$321$	0	0
$322$	−22.4448	−1.25080
$323$	16.8390	0.936945
$324$	0	0
$325$	−9.30090	−0.515921
$326$	22.7151	1.25807
$327$	0	0
$328$	1.46479	0.0808795
$329$	−14.6433	−0.807312
$330$	0	0
$331$	−21.2194	−1.16632	−0.583162	−	0.812356i	$-0.698185\pi$
$331$	−21.2194	−1.16632	−0.583162	+	0.812356i	$0.698185\pi$
$332$	0.447492	0.0245593
$333$	0	0
$334$	−29.2421	−1.60006
$335$	0.492490	0.0269076
$336$	0	0
$337$	−15.5026	−0.844478	−0.422239	−	0.906485i	$-0.638756\pi$
$337$	−15.5026	−0.844478	−0.422239	+	0.906485i	$0.638756\pi$
$338$	−1.95594	−0.106389
$339$	0	0
$340$	30.1179	1.63337
$341$	−5.78165	−0.313094
$342$	0	0
$343$	−18.9038	−1.02071
$344$	−1.00768	−0.0543304
$345$	0	0
$346$	−10.8547	−0.583555
$347$	−21.4672	−1.15242	−0.576211	−	0.817301i	$-0.695469\pi$
$347$	−21.4672	−1.15242	−0.576211	+	0.817301i	$0.695469\pi$
$348$	0	0
$349$	−16.3140	−0.873267	−0.436634	−	0.899639i	$-0.643829\pi$
$349$	−16.3140	−0.873267	−0.436634	+	0.899639i	$0.643829\pi$
$350$	−30.9839	−1.65616
$351$	0	0
$352$	7.76435	0.413842
$353$	20.4788	1.08998	0.544989	−	0.838444i	$-0.316534\pi$
$353$	20.4788	1.08998	0.544989	+	0.838444i	$0.316534\pi$
$354$	0	0
$355$	−52.2340	−2.77229
$356$	5.77777	0.306221
$357$	0	0
$358$	19.6686	1.03952
$359$	12.9940	0.685797	0.342898	−	0.939372i	$-0.388591\pi$
$359$	12.9940	0.685797	0.342898	+	0.939372i	$0.388591\pi$
$360$	0	0
$361$	−4.09925	−0.215750
$362$	−9.91823	−0.521290
$363$	0	0
$364$	−3.10947	−0.162981
$365$	19.9159	1.04245
$366$	0	0
$367$	29.8854	1.56001	0.780003	−	0.625776i	$-0.215218\pi$
$367$	29.8854	1.56001	0.780003	+	0.625776i	$0.215218\pi$
$368$	−29.0943	−1.51665
$369$	0	0
$370$	73.9970	3.84692
$371$	−23.5907	−1.22477
$372$	0	0
$373$	−11.1420	−0.576909	−0.288455	−	0.957494i	$-0.593141\pi$
$373$	−11.1420	−0.576909	−0.288455	+	0.957494i	$0.593141\pi$
$374$	−8.53233	−0.441196
$375$	0	0
$376$	−2.93097	−0.151153
$377$	−3.83339	−0.197430
$378$	0	0
$379$	−17.2681	−0.887005	−0.443502	−	0.896273i	$-0.646264\pi$
$379$	−17.2681	−0.887005	−0.443502	+	0.896273i	$0.646264\pi$
$380$	26.6513	1.36718
$381$	0	0
$382$	5.38016	0.275273
$383$	−15.4063	−0.787226	−0.393613	−	0.919276i	$-0.628775\pi$
$383$	−15.4063	−0.787226	−0.393613	+	0.919276i	$0.628775\pi$
$384$	0	0
$385$	6.44075	0.328251
$386$	−18.3782	−0.935427
$387$	0	0
$388$	17.6062	0.893820
$389$	−16.0842	−0.815503	−0.407751	−	0.913093i	$-0.633687\pi$
$389$	−16.0842	−0.815503	−0.407751	+	0.913093i	$0.633687\pi$
$390$	0	0
$391$	29.3911	1.48637
$392$	−1.39743	−0.0705810
$393$	0	0
$394$	−2.14003	−0.107813
$395$	−46.5200	−2.34068
$396$	0	0
$397$	30.1418	1.51277	0.756387	−	0.654124i	$-0.226963\pi$
$397$	30.1418	1.51277	0.756387	+	0.654124i	$0.226963\pi$
$398$	27.0897	1.35788
$399$	0	0
$400$	−40.1632	−2.00816
$401$	11.4938	0.573974	0.286987	−	0.957934i	$-0.407346\pi$
$401$	11.4938	0.573974	0.286987	+	0.957934i	$0.407346\pi$
$402$	0	0
$403$	5.78165	0.288005
$404$	−29.1254	−1.44904
$405$	0	0
$406$	−12.7701	−0.633769
$407$	−10.0041	−0.495883
$408$	0	0
$409$	8.35880	0.413316	0.206658	−	0.978413i	$-0.433741\pi$
$409$	8.35880	0.413316	0.206658	+	0.978413i	$0.433741\pi$
$410$	−31.7824	−1.56962
$411$	0	0
$412$	9.54863	0.470427
$413$	15.8704	0.780930
$414$	0	0
$415$	0.926904	0.0454999
$416$	−7.76435	−0.380679
$417$	0	0
$418$	−7.55023	−0.369294
$419$	2.67564	0.130713	0.0653567	−	0.997862i	$-0.479181\pi$
$419$	2.67564	0.130713	0.0653567	+	0.997862i	$0.479181\pi$
$420$	0	0
$421$	−7.24424	−0.353063	−0.176531	−	0.984295i	$-0.556488\pi$
$421$	−7.24424	−0.353063	−0.176531	+	0.984295i	$0.556488\pi$
$422$	7.04754	0.343069
$423$	0	0
$424$	−4.72184	−0.229313
$425$	40.5729	1.96808
$426$	0	0
$427$	25.4560	1.23190
$428$	−0.847230	−0.0409524
$429$	0	0
$430$	21.8642	1.05439
$431$	−25.2036	−1.21402	−0.607008	−	0.794696i	$-0.707630\pi$
$431$	−25.2036	−1.21402	−0.607008	+	0.794696i	$0.707630\pi$
$432$	0	0
$433$	−7.70829	−0.370437	−0.185218	−	0.982697i	$-0.559299\pi$
$433$	−7.70829	−0.370437	−0.185218	+	0.982697i	$0.559299\pi$
$434$	19.2603	0.924524
$435$	0	0
$436$	−11.9717	−0.573343
$437$	26.0081	1.24414
$438$	0	0
$439$	−34.9175	−1.66652	−0.833260	−	0.552882i	$-0.813528\pi$
$439$	−34.9175	−1.66652	−0.833260	+	0.552882i	$0.813528\pi$
$440$	1.28916	0.0614584
$441$	0	0
$442$	8.53233	0.405841
$443$	−33.5868	−1.59576	−0.797878	−	0.602819i	$-0.794044\pi$
$443$	−33.5868	−1.59576	−0.797878	+	0.602819i	$0.794044\pi$
$444$	0	0
$445$	11.9677	0.567323
$446$	12.4186	0.588039
$447$	0	0
$448$	−11.1561	−0.527075
$449$	−8.82015	−0.416249	−0.208124	−	0.978102i	$-0.566736\pi$
$449$	−8.82015	−0.416249	−0.208124	+	0.978102i	$0.566736\pi$
$450$	0	0
$451$	4.29684	0.202330
$452$	22.4060	1.05389
$453$	0	0
$454$	−47.7927	−2.24302
$455$	−6.44075	−0.301947
$456$	0	0
$457$	5.32544	0.249113	0.124557	−	0.992212i	$-0.460249\pi$
$457$	5.32544	0.249113	0.124557	+	0.992212i	$0.460249\pi$
$458$	−25.0152	−1.16888
$459$	0	0
$460$	46.5178	2.16890
$461$	24.6108	1.14624	0.573120	−	0.819471i	$-0.305733\pi$
$461$	24.6108	1.14624	0.573120	+	0.819471i	$0.305733\pi$
$462$	0	0
$463$	−34.7515	−1.61504	−0.807519	−	0.589842i	$-0.799190\pi$
$463$	−34.7515	−1.61504	−0.807519	+	0.589842i	$0.799190\pi$
$464$	−16.5533	−0.768470
$465$	0	0
$466$	−57.9400	−2.68402
$467$	28.4437	1.31622	0.658110	−	0.752922i	$-0.271356\pi$
$467$	28.4437	1.31622	0.658110	+	0.752922i	$0.271356\pi$
$468$	0	0
$469$	0.221805	0.0102420
$470$	63.5949	2.93341
$471$	0	0
$472$	3.17657	0.146213
$473$	−2.95594	−0.135914
$474$	0	0
$475$	35.9029	1.64734
$476$	13.5643	0.621720
$477$	0	0
$478$	−5.53487	−0.253159
$479$	2.41232	0.110222	0.0551108	−	0.998480i	$-0.482449\pi$
$479$	2.41232	0.110222	0.0551108	+	0.998480i	$0.482449\pi$
$480$	0	0
$481$	10.0041	0.456146
$482$	45.5949	2.07679
$483$	0	0
$484$	1.82571	0.0829869
$485$	36.4683	1.65594
$486$	0	0
$487$	−6.91621	−0.313403	−0.156702	−	0.987646i	$-0.550086\pi$
$487$	−6.91621	−0.313403	−0.156702	+	0.987646i	$0.550086\pi$
$488$	5.09519	0.230648
$489$	0	0
$490$	30.3209	1.36976
$491$	25.9121	1.16940	0.584698	−	0.811251i	$-0.301213\pi$
$491$	25.9121	1.16940	0.584698	+	0.811251i	$0.301213\pi$
$492$	0	0
$493$	16.7222	0.753132
$494$	7.55023	0.339701
$495$	0	0
$496$	24.9663	1.12102
$497$	−23.5248	−1.05523
$498$	0	0
$499$	−11.9278	−0.533964	−0.266982	−	0.963702i	$-0.586026\pi$
$499$	−11.9278	−0.533964	−0.266982	+	0.963702i	$0.586026\pi$
$500$	29.6943	1.32797
$501$	0	0
$502$	−8.21439	−0.366626
$503$	−13.6322	−0.607829	−0.303914	−	0.952699i	$-0.598294\pi$
$503$	−13.6322	−0.607829	−0.303914	+	0.952699i	$0.598294\pi$
$504$	0	0
$505$	−60.3284	−2.68458
$506$	−13.1784	−0.585849
$507$	0	0
$508$	−23.9478	−1.06251
$509$	11.3421	0.502730	0.251365	−	0.967892i	$-0.419121\pi$
$509$	11.3421	0.502730	0.251365	+	0.967892i	$0.419121\pi$
$510$	0	0
$511$	8.96962	0.396793
$512$	−30.5839	−1.35163
$513$	0	0
$514$	17.3040	0.763249
$515$	19.7784	0.871539
$516$	0	0
$517$	−8.59774	−0.378128
$518$	33.3263	1.46427
$519$	0	0
$520$	−1.28916	−0.0565335
$521$	−9.44904	−0.413970	−0.206985	−	0.978344i	$-0.566365\pi$
$521$	−9.44904	−0.413970	−0.206985	+	0.978344i	$0.566365\pi$
$522$	0	0
$523$	10.9286	0.477873	0.238937	−	0.971035i	$-0.423201\pi$
$523$	10.9286	0.477873	0.238937	+	0.971035i	$0.423201\pi$
$524$	5.23276	0.228594
$525$	0	0
$526$	−0.600600	−0.0261874
$527$	−25.2211	−1.09865
$528$	0	0
$529$	22.3952	0.973704
$530$	102.453	4.45026
$531$	0	0
$532$	12.0030	0.520398
$533$	−4.29684	−0.186117
$534$	0	0
$535$	−1.75489	−0.0758707
$536$	0.0443958	0.00191761
$537$	0	0
$538$	32.9146	1.41905
$539$	−4.09925	−0.176567
$540$	0	0
$541$	−2.85535	−0.122761	−0.0613806	−	0.998114i	$-0.519550\pi$
$541$	−2.85535	−0.122761	−0.0613806	+	0.998114i	$0.519550\pi$
$542$	16.0558	0.689656
$543$	0	0
$544$	33.8701	1.45217
$545$	−24.7975	−1.06221
$546$	0	0
$547$	15.6226	0.667973	0.333986	−	0.942578i	$-0.391606\pi$
$547$	15.6226	0.667973	0.333986	+	0.942578i	$0.391606\pi$
$548$	40.2219	1.71819
$549$	0	0
$550$	−18.1920	−0.775711
$551$	14.7975	0.630393
$552$	0	0
$553$	−20.9514	−0.890945
$554$	−1.64062	−0.0697034
$555$	0	0
$556$	4.14452	0.175767
$557$	36.0791	1.52872	0.764360	−	0.644789i	$-0.223055\pi$
$557$	36.0791	1.52872	0.764360	+	0.644789i	$0.223055\pi$
$558$	0	0
$559$	2.95594	0.125023
$560$	−27.8125	−1.17529
$561$	0	0
$562$	−34.5352	−1.45678
$563$	−23.5825	−0.993886	−0.496943	−	0.867783i	$-0.665544\pi$
$563$	−23.5825	−0.993886	−0.496943	+	0.867783i	$0.665544\pi$
$564$	0	0
$565$	46.4102	1.95249
$566$	57.6630	2.42376
$567$	0	0
$568$	−4.70866	−0.197571
$569$	−34.7806	−1.45808	−0.729039	−	0.684472i	$-0.760033\pi$
$569$	−34.7806	−1.45808	−0.729039	+	0.684472i	$0.760033\pi$
$570$	0	0
$571$	42.0179	1.75840	0.879198	−	0.476457i	$-0.158079\pi$
$571$	42.0179	1.75840	0.879198	+	0.476457i	$0.158079\pi$
$572$	−1.82571	−0.0763368
$573$	0	0
$574$	−14.3140	−0.597454
$575$	62.6657	2.61334
$576$	0	0
$577$	−44.8015	−1.86511	−0.932556	−	0.361026i	$-0.882427\pi$
$577$	−44.8015	−1.86511	−0.932556	+	0.361026i	$0.882427\pi$
$578$	−3.96920	−0.165097
$579$	0	0
$580$	26.4665	1.09896
$581$	0.417453	0.0173189
$582$	0	0
$583$	−13.8511	−0.573655
$584$	1.79533	0.0742915
$585$	0	0
$586$	−4.52253	−0.186824
$587$	−19.1813	−0.791699	−0.395849	−	0.918316i	$-0.629550\pi$
$587$	−19.1813	−0.791699	−0.395849	+	0.918316i	$0.629550\pi$
$588$	0	0
$589$	−22.3180	−0.919599
$590$	−68.9239	−2.83755
$591$	0	0
$592$	43.1996	1.77549
$593$	−27.8764	−1.14475	−0.572373	−	0.819993i	$-0.693977\pi$
$593$	−27.8764	−1.14475	−0.572373	+	0.819993i	$0.693977\pi$
$594$	0	0
$595$	28.0962	1.15183
$596$	−18.7639	−0.768601
$597$	0	0
$598$	13.1784	0.538903
$599$	5.18013	0.211654	0.105827	−	0.994385i	$-0.466251\pi$
$599$	5.18013	0.211654	0.105827	+	0.994385i	$0.466251\pi$
$600$	0	0
$601$	−17.8204	−0.726910	−0.363455	−	0.931612i	$-0.618403\pi$
$601$	−17.8204	−0.726910	−0.363455	+	0.931612i	$0.618403\pi$
$602$	9.84707	0.401337
$603$	0	0
$604$	11.0136	0.448137
$605$	3.78165	0.153746
$606$	0	0
$607$	33.7949	1.37169	0.685847	−	0.727746i	$-0.259432\pi$
$607$	33.7949	1.37169	0.685847	+	0.727746i	$0.259432\pi$
$608$	29.9716	1.21551
$609$	0	0
$610$	−110.553	−4.47618
$611$	8.59774	0.347828
$612$	0	0
$613$	−25.4349	−1.02731	−0.513654	−	0.857998i	$-0.671708\pi$
$613$	−25.4349	−1.02731	−0.513654	+	0.857998i	$0.671708\pi$
$614$	−58.0835	−2.34406
$615$	0	0
$616$	0.580605	0.0233933
$617$	17.7709	0.715428	0.357714	−	0.933831i	$-0.383556\pi$
$617$	17.7709	0.715428	0.357714	+	0.933831i	$0.383556\pi$
$618$	0	0
$619$	34.0702	1.36940	0.684698	−	0.728827i	$-0.259934\pi$
$619$	34.0702	1.36940	0.684698	+	0.728827i	$0.259934\pi$
$620$	−39.9177	−1.60313
$621$	0	0
$622$	55.4361	2.22279
$623$	5.38993	0.215943
$624$	0	0
$625$	15.0023	0.600091
$626$	−58.3774	−2.33323
$627$	0	0
$628$	8.93610	0.356589
$629$	−43.6403	−1.74005
$630$	0	0
$631$	9.04316	0.360003	0.180001	−	0.983666i	$-0.442390\pi$
$631$	9.04316	0.360003	0.180001	+	0.983666i	$0.442390\pi$
$632$	−4.19357	−0.166811
$633$	0	0
$634$	−7.49579	−0.297696
$635$	−49.6039	−1.96847
$636$	0	0
$637$	4.09925	0.162418
$638$	−7.49789	−0.296844
$639$	0	0
$640$	−10.2741	−0.406121
$641$	11.3802	0.449489	0.224745	−	0.974418i	$-0.427845\pi$
$641$	11.3802	0.449489	0.224745	+	0.974418i	$0.427845\pi$
$642$	0	0
$643$	−47.2114	−1.86184	−0.930919	−	0.365226i	$-0.880992\pi$
$643$	−47.2114	−1.86184	−0.930919	+	0.365226i	$0.880992\pi$
$644$	20.9504	0.825561
$645$	0	0
$646$	−32.9361	−1.29585
$647$	−10.6030	−0.416847	−0.208424	−	0.978039i	$-0.566833\pi$
$647$	−10.6030	−0.416847	−0.208424	+	0.978039i	$0.566833\pi$
$648$	0	0
$649$	9.31820	0.365771
$650$	18.1920	0.713550
$651$	0	0
$652$	−21.2027	−0.830360
$653$	24.1257	0.944113	0.472057	−	0.881568i	$-0.343512\pi$
$653$	24.1257	0.944113	0.472057	+	0.881568i	$0.343512\pi$
$654$	0	0
$655$	10.8388	0.423507
$656$	−18.5546	−0.724437
$657$	0	0
$658$	28.6415	1.11656
$659$	−15.2107	−0.592523	−0.296262	−	0.955107i	$-0.595740\pi$
$659$	−15.2107	−0.592523	−0.296262	+	0.955107i	$0.595740\pi$
$660$	0	0
$661$	32.4114	1.26066	0.630329	−	0.776328i	$-0.282920\pi$
$661$	32.4114	1.26066	0.630329	+	0.776328i	$0.282920\pi$
$662$	41.5039	1.61310
$663$	0	0
$664$	0.0835563	0.00324261
$665$	24.8623	0.964117
$666$	0	0
$667$	25.8278	1.00006
$668$	27.2951	1.05608
$669$	0	0
$670$	−0.963283	−0.0372149
$671$	14.9463	0.576996
$672$	0	0
$673$	−25.4065	−0.979348	−0.489674	−	0.871905i	$-0.662884\pi$
$673$	−25.4065	−0.979348	−0.489674	+	0.871905i	$0.662884\pi$
$674$	30.3221	1.16796
$675$	0	0
$676$	1.82571	0.0702197
$677$	−9.53833	−0.366588	−0.183294	−	0.983058i	$-0.558676\pi$
$677$	−9.53833	−0.366588	−0.183294	+	0.983058i	$0.558676\pi$
$678$	0	0
$679$	16.4244	0.630309
$680$	5.62366	0.215658
$681$	0	0
$682$	11.3086	0.433028
$683$	−38.1277	−1.45891	−0.729457	−	0.684026i	$-0.760227\pi$
$683$	−38.1277	−1.45891	−0.729457	+	0.684026i	$0.760227\pi$
$684$	0	0
$685$	83.3129	3.18322
$686$	36.9747	1.41170
$687$	0	0
$688$	12.7644	0.486636
$689$	13.8511	0.527686
$690$	0	0
$691$	−42.9358	−1.63336	−0.816678	−	0.577094i	$-0.804187\pi$
$691$	−42.9358	−1.63336	−0.816678	+	0.577094i	$0.804187\pi$
$692$	10.1320	0.385161
$693$	0	0
$694$	41.9887	1.59387
$695$	8.58467	0.325635
$696$	0	0
$697$	18.7439	0.709977
$698$	31.9092	1.20778
$699$	0	0
$700$	28.9209	1.09311
$701$	−25.4269	−0.960361	−0.480181	−	0.877170i	$-0.659429\pi$
$701$	−25.4269	−0.960361	−0.480181	+	0.877170i	$0.659429\pi$
$702$	0	0
$703$	−38.6172	−1.45647
$704$	−6.55023	−0.246871
$705$	0	0
$706$	−40.0554	−1.50750
$707$	−27.1704	−1.02185
$708$	0	0
$709$	−44.2573	−1.66212	−0.831059	−	0.556185i	$-0.812265\pi$
$709$	−44.2573	−1.66212	−0.831059	+	0.556185i	$0.812265\pi$
$710$	102.167	3.83425
$711$	0	0
$712$	1.07883	0.0404310
$713$	−38.9544	−1.45886
$714$	0	0
$715$	−3.78165	−0.141426
$716$	−18.3590	−0.686107
$717$	0	0
$718$	−25.4155	−0.948498
$719$	16.5065	0.615587	0.307793	−	0.951453i	$-0.400409\pi$
$719$	16.5065	0.615587	0.307793	+	0.951453i	$0.400409\pi$
$720$	0	0
$721$	8.90766	0.331739
$722$	8.01790	0.298395
$723$	0	0
$724$	9.25785	0.344065
$725$	35.6540	1.32416
$726$	0	0
$727$	−13.4252	−0.497914	−0.248957	−	0.968514i	$-0.580088\pi$
$727$	−13.4252	−0.497914	−0.248957	+	0.968514i	$0.580088\pi$
$728$	−0.580605	−0.0215187
$729$	0	0
$730$	−38.9544	−1.44177
$731$	−12.8946	−0.476924
$732$	0	0
$733$	33.2018	1.22634	0.613169	−	0.789952i	$-0.289895\pi$
$733$	33.2018	1.22634	0.613169	+	0.789952i	$0.289895\pi$
$734$	−58.4541	−2.15758
$735$	0	0
$736$	52.3131	1.92829
$737$	0.130232	0.00479714
$738$	0	0
$739$	−0.292960	−0.0107767	−0.00538835	−	0.999985i	$-0.501715\pi$
$739$	−0.292960	−0.0107767	−0.00538835	+	0.999985i	$0.501715\pi$
$740$	−69.0701	−2.53907
$741$	0	0
$742$	46.1420	1.69393
$743$	0.374734	0.0137477	0.00687383	−	0.999976i	$-0.497812\pi$
$743$	0.374734	0.0137477	0.00687383	+	0.999976i	$0.497812\pi$
$744$	0	0
$745$	−38.8663	−1.42395
$746$	21.7931	0.797900
$747$	0	0
$748$	7.96422	0.291201
$749$	−0.790359	−0.0288791
$750$	0	0
$751$	24.4568	0.892442	0.446221	−	0.894923i	$-0.352770\pi$
$751$	24.4568	0.892442	0.446221	+	0.894923i	$0.352770\pi$
$752$	37.1268	1.35387
$753$	0	0
$754$	7.49789	0.273057
$755$	22.8128	0.830243
$756$	0	0
$757$	−8.15083	−0.296247	−0.148123	−	0.988969i	$-0.547323\pi$
$757$	−8.15083	−0.296247	−0.148123	+	0.988969i	$0.547323\pi$
$758$	33.7755	1.22678
$759$	0	0
$760$	4.97636	0.180512
$761$	38.1441	1.38272	0.691361	−	0.722509i	$-0.257011\pi$
$761$	38.1441	1.38272	0.691361	+	0.722509i	$0.257011\pi$
$762$	0	0
$763$	−11.1681	−0.404313
$764$	−5.02194	−0.181687
$765$	0	0
$766$	30.1339	1.08878
$767$	−9.31820	−0.336461
$768$	0	0
$769$	−17.1994	−0.620225	−0.310113	−	0.950700i	$-0.600367\pi$
$769$	−17.1994	−0.620225	−0.310113	+	0.950700i	$0.600367\pi$
$770$	−12.5977	−0.453991
$771$	0	0
$772$	17.1546	0.617406
$773$	17.6690	0.635511	0.317755	−	0.948173i	$-0.397071\pi$
$773$	17.6690	0.635511	0.317755	+	0.948173i	$0.397071\pi$
$774$	0	0
$775$	−53.7746	−1.93164
$776$	3.28746	0.118013
$777$	0	0
$778$	31.4598	1.12789
$779$	16.5864	0.594271
$780$	0	0
$781$	−13.8125	−0.494249
$782$	−57.4874	−2.05574
$783$	0	0
$784$	17.7014	0.632193
$785$	18.5096	0.660637
$786$	0	0
$787$	22.1775	0.790542	0.395271	−	0.918565i	$-0.370651\pi$
$787$	22.1775	0.790542	0.395271	+	0.918565i	$0.370651\pi$
$788$	1.99754	0.0711595
$789$	0	0
$790$	90.9905	3.23730
$791$	20.9020	0.743188
$792$	0	0
$793$	−14.9463	−0.530759
$794$	−58.9557	−2.09226
$795$	0	0
$796$	−25.2860	−0.896237
$797$	7.64330	0.270740	0.135370	−	0.990795i	$-0.456778\pi$
$797$	7.64330	0.270740	0.135370	+	0.990795i	$0.456778\pi$
$798$	0	0
$799$	−37.5056	−1.32685
$800$	72.2155	2.55320
$801$	0	0
$802$	−22.4813	−0.793841
$803$	5.26647	0.185850
$804$	0	0
$805$	43.3952	1.52948
$806$	−11.3086	−0.398328
$807$	0	0
$808$	−5.43834	−0.191320
$809$	−9.55298	−0.335865	−0.167932	−	0.985799i	$-0.553709\pi$
$809$	−9.55298	−0.335865	−0.167932	+	0.985799i	$0.553709\pi$
$810$	0	0
$811$	34.8653	1.22428	0.612142	−	0.790748i	$-0.290308\pi$
$811$	34.8653	1.22428	0.612142	+	0.790748i	$0.290308\pi$
$812$	11.9198	0.418304
$813$	0	0
$814$	19.5674	0.685836
$815$	−43.9177	−1.53837
$816$	0	0
$817$	−11.4104	−0.399199
$818$	−16.3493	−0.571641
$819$	0	0
$820$	29.6663	1.03599
$821$	14.9294	0.521038	0.260519	−	0.965469i	$-0.416106\pi$
$821$	14.9294	0.521038	0.260519	+	0.965469i	$0.416106\pi$
$822$	0	0
$823$	−33.3002	−1.16077	−0.580385	−	0.814342i	$-0.697098\pi$
$823$	−33.3002	−1.16077	−0.580385	+	0.814342i	$0.697098\pi$
$824$	1.78293	0.0621114
$825$	0	0
$826$	−31.0415	−1.08007
$827$	11.6770	0.406049	0.203025	−	0.979174i	$-0.434923\pi$
$827$	11.6770	0.406049	0.203025	+	0.979174i	$0.434923\pi$
$828$	0	0
$829$	−27.0966	−0.941103	−0.470551	−	0.882373i	$-0.655945\pi$
$829$	−27.0966	−0.941103	−0.470551	+	0.882373i	$0.655945\pi$
$830$	−1.81297	−0.0629291
$831$	0	0
$832$	6.55023	0.227088
$833$	−17.8820	−0.619575
$834$	0	0
$835$	56.5372	1.95655
$836$	7.04752	0.243743
$837$	0	0
$838$	−5.23339	−0.180785
$839$	41.3913	1.42899	0.714493	−	0.699642i	$-0.246657\pi$
$839$	41.3913	1.42899	0.714493	+	0.699642i	$0.246657\pi$
$840$	0	0
$841$	−14.3051	−0.493280
$842$	14.1693	0.488307
$843$	0	0
$844$	−6.57830	−0.226434
$845$	3.78165	0.130093
$846$	0	0
$847$	1.70316	0.0585212
$848$	59.8120	2.05395
$849$	0	0
$850$	−79.3583	−2.72197
$851$	−67.4033	−2.31056
$852$	0	0
$853$	−2.90766	−0.0995565	−0.0497783	−	0.998760i	$-0.515851\pi$
$853$	−2.90766	−0.0995565	−0.0497783	+	0.998760i	$0.515851\pi$
$854$	−49.7904	−1.70379
$855$	0	0
$856$	−0.158196	−0.00540703
$857$	11.8881	0.406090	0.203045	−	0.979169i	$-0.434916\pi$
$857$	11.8881	0.406090	0.203045	+	0.979169i	$0.434916\pi$
$858$	0	0
$859$	13.2920	0.453519	0.226759	−	0.973951i	$-0.427187\pi$
$859$	13.2920	0.453519	0.226759	+	0.973951i	$0.427187\pi$
$860$	−20.4084	−0.695922
$861$	0	0
$862$	49.2968	1.67906
$863$	17.1693	0.584451	0.292225	−	0.956349i	$-0.405604\pi$
$863$	17.1693	0.584451	0.292225	+	0.956349i	$0.405604\pi$
$864$	0	0
$865$	20.9868	0.713571
$866$	15.0770	0.512337
$867$	0	0
$868$	−17.9779	−0.610210
$869$	−12.3015	−0.417300
$870$	0	0
$871$	−0.130232	−0.00441273
$872$	−2.23538	−0.0756995
$873$	0	0
$874$	−50.8704	−1.72072
$875$	27.7011	0.936467
$876$	0	0
$877$	30.6992	1.03664	0.518320	−	0.855187i	$-0.326558\pi$
$877$	30.6992	1.03664	0.518320	+	0.855187i	$0.326558\pi$
$878$	68.2965	2.30490
$879$	0	0
$880$	−16.3299	−0.550482
$881$	−0.107036	−0.00360613	−0.00180307	−	0.999998i	$-0.500574\pi$
$881$	−0.107036	−0.00360613	−0.00180307	+	0.999998i	$0.500574\pi$
$882$	0	0
$883$	−55.5483	−1.86935	−0.934675	−	0.355504i	$-0.884309\pi$
$883$	−55.5483	−1.86935	−0.934675	+	0.355504i	$0.884309\pi$
$884$	−7.96422	−0.267866
$885$	0	0
$886$	65.6938	2.20703
$887$	2.21577	0.0743984	0.0371992	−	0.999308i	$-0.488156\pi$
$887$	2.21577	0.0743984	0.0371992	+	0.999308i	$0.488156\pi$
$888$	0	0
$889$	−22.3403	−0.749270
$890$	−23.4081	−0.784641
$891$	0	0
$892$	−11.5918	−0.388121
$893$	−33.1886	−1.11061
$894$	0	0
$895$	−38.0275	−1.27112
$896$	−4.62721	−0.154584
$897$	0	0
$898$	17.2517	0.575697
$899$	−22.1633	−0.739188
$900$	0	0
$901$	−60.4222	−2.01296
$902$	−8.40438	−0.279835
$903$	0	0
$904$	4.18368	0.139147
$905$	19.1761	0.637434
$906$	0	0
$907$	−10.7164	−0.355832	−0.177916	−	0.984046i	$-0.556936\pi$
$907$	−10.7164	−0.355832	−0.177916	+	0.984046i	$0.556936\pi$
$908$	44.6105	1.48045
$909$	0	0
$910$	12.5977	0.417611
$911$	−22.0093	−0.729202	−0.364601	−	0.931164i	$-0.618795\pi$
$911$	−22.0093	−0.729202	−0.364601	+	0.931164i	$0.618795\pi$
$912$	0	0
$913$	0.245106	0.00811181
$914$	−10.4162	−0.344539
$915$	0	0
$916$	23.3496	0.771493
$917$	4.88151	0.161202
$918$	0	0
$919$	4.61866	0.152356	0.0761778	−	0.997094i	$-0.475728\pi$
$919$	4.61866	0.152356	0.0761778	+	0.997094i	$0.475728\pi$
$920$	8.68586	0.286364
$921$	0	0
$922$	−48.1374	−1.58532
$923$	13.8125	0.454643
$924$	0	0
$925$	−93.0468	−3.05936
$926$	67.9718	2.23369
$927$	0	0
$928$	29.7638	0.977045
$929$	15.2840	0.501453	0.250726	−	0.968058i	$-0.419331\pi$
$929$	15.2840	0.501453	0.250726	+	0.968058i	$0.419331\pi$
$930$	0	0
$931$	−15.8237	−0.518602
$932$	54.0822	1.77152
$933$	0	0
$934$	−55.6343	−1.82041
$935$	16.4966	0.539495
$936$	0	0
$937$	14.1956	0.463752	0.231876	−	0.972745i	$-0.425514\pi$
$937$	14.1956	0.463752	0.231876	+	0.972745i	$0.425514\pi$
$938$	−0.433838	−0.0141653
$939$	0	0
$940$	−59.3606	−1.93613
$941$	−50.6843	−1.65226	−0.826130	−	0.563480i	$-0.809462\pi$
$941$	−50.6843	−1.65226	−0.826130	+	0.563480i	$0.809462\pi$
$942$	0	0
$943$	28.9504	0.942754
$944$	−40.2379	−1.30963
$945$	0	0
$946$	5.78165	0.187978
$947$	37.0780	1.20487	0.602437	−	0.798166i	$-0.294196\pi$
$947$	37.0780	1.20487	0.602437	+	0.798166i	$0.294196\pi$
$948$	0	0
$949$	−5.26647	−0.170957
$950$	−70.2239	−2.27837
$951$	0	0
$952$	2.53275	0.0820869
$953$	−28.1375	−0.911462	−0.455731	−	0.890118i	$-0.650622\pi$
$953$	−28.1375	−0.911462	−0.455731	+	0.890118i	$0.650622\pi$
$954$	0	0
$955$	−10.4021	−0.336604
$956$	5.16635	0.167092
$957$	0	0
$958$	−4.71836	−0.152443
$959$	37.5220	1.21165
$960$	0	0
$961$	2.42752	0.0783070
$962$	−19.5674	−0.630877
$963$	0	0
$964$	−42.5591	−1.37074
$965$	35.5328	1.14384
$966$	0	0
$967$	3.65085	0.117403	0.0587016	−	0.998276i	$-0.481304\pi$
$967$	3.65085	0.117403	0.0587016	+	0.998276i	$0.481304\pi$
$968$	0.340899	0.0109569
$969$	0	0
$970$	−71.3299	−2.29026
$971$	−48.8874	−1.56887	−0.784435	−	0.620211i	$-0.787047\pi$
$971$	−48.8874	−1.56887	−0.784435	+	0.620211i	$0.787047\pi$
$972$	0	0
$973$	3.86631	0.123948
$974$	13.5277	0.433456
$975$	0	0
$976$	−64.5412	−2.06591
$977$	6.66634	0.213275	0.106637	−	0.994298i	$-0.465992\pi$
$977$	6.66634	0.213275	0.106637	+	0.994298i	$0.465992\pi$
$978$	0	0
$979$	3.16467	0.101143
$980$	−28.3021	−0.904077
$981$	0	0
$982$	−50.6825	−1.61734
$983$	−35.7738	−1.14101	−0.570504	−	0.821295i	$-0.693252\pi$
$983$	−35.7738	−1.14101	−0.570504	+	0.821295i	$0.693252\pi$
$984$	0	0
$985$	4.13757	0.131834
$986$	−32.7077	−1.04163
$987$	0	0
$988$	−7.04752	−0.224211
$989$	−19.9159	−0.633290
$990$	0	0
$991$	29.2695	0.929775	0.464887	−	0.885370i	$-0.346095\pi$
$991$	29.2695	0.929775	0.464887	+	0.885370i	$0.346095\pi$
$992$	−44.8908	−1.42528
$993$	0	0
$994$	46.0132	1.45945
$995$	−52.3756	−1.66042
$996$	0	0
$997$	−60.2352	−1.90767	−0.953834	−	0.300335i	$-0.902902\pi$
$997$	−60.2352	−1.90767	−0.953834	+	0.300335i	$0.902902\pi$
$998$	23.3302	0.738504
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.n.1.1	yes	4
3.2	odd	2		1287.2.a.l.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1287.2.a.l.1.4	✓	4	3.2	odd	2
1287.2.a.n.1.1	yes	4	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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q-1.95594 q^{2} +1.82571 q^{4} +3.78165 q^{5} +1.70316 q^{7} +0.340899 q^{8} +O(q^{10})\)	\(q-1.95594 q^{2} +1.82571 q^{4} +3.78165 q^{5} +1.70316 q^{7} +0.340899 q^{8} -7.39670 q^{10} +1.00000 q^{11} -1.00000 q^{13} -3.33128 q^{14} -4.31820 q^{16} +4.36226 q^{17} +3.86015 q^{19} +6.90421 q^{20} -1.95594 q^{22} +6.73760 q^{23} +9.30090 q^{25} +1.95594 q^{26} +3.10947 q^{28} +3.83339 q^{29} -5.78165 q^{31} +7.76435 q^{32} -8.53233 q^{34} +6.44075 q^{35} -10.0041 q^{37} -7.55023 q^{38} +1.28916 q^{40} +4.29684 q^{41} -2.95594 q^{43} +1.82571 q^{44} -13.1784 q^{46} -8.59774 q^{47} -4.09925 q^{49} -18.1920 q^{50} -1.82571 q^{52} -13.8511 q^{53} +3.78165 q^{55} +0.580605 q^{56} -7.49789 q^{58} +9.31820 q^{59} +14.9463 q^{61} +11.3086 q^{62} -6.55023 q^{64} -3.78165 q^{65} +0.130232 q^{67} +7.96422 q^{68} -12.5977 q^{70} -13.8125 q^{71} +5.26647 q^{73} +19.5674 q^{74} +7.04752 q^{76} +1.70316 q^{77} -12.3015 q^{79} -16.3299 q^{80} -8.40438 q^{82} +0.245106 q^{83} +16.4966 q^{85} +5.78165 q^{86} +0.340899 q^{88} +3.16467 q^{89} -1.70316 q^{91} +12.3009 q^{92} +16.8167 q^{94} +14.5977 q^{95} +9.64348 q^{97} +8.01790 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 6 * q^7 + 12 * q^8	\( 4 q + 2 q^{2} + 8 q^{4} + 6 q^{5} + 6 q^{7} + 12 q^{8} + 4 q^{11} - 4 q^{13} + 4 q^{14} + 4 q^{16} + 6 q^{17} - 2 q^{19} + 20 q^{20} + 2 q^{22} + 8 q^{23} - 2 q^{26} + 10 q^{28} + 4 q^{29} - 14 q^{31} + 6 q^{32} - 10 q^{34} + 6 q^{35} - 2 q^{37} - 8 q^{38} + 18 q^{40} + 18 q^{41} - 2 q^{43} + 8 q^{44} - 14 q^{46} + 2 q^{47} - 24 q^{50} - 8 q^{52} + 4 q^{53} + 6 q^{55} - 24 q^{58} + 16 q^{59} + 22 q^{61} - 4 q^{62} - 4 q^{64} - 6 q^{65} - 10 q^{67} - 4 q^{68} - 14 q^{70} - 2 q^{71} + 2 q^{73} + 22 q^{74} + 14 q^{76} + 6 q^{77} + 2 q^{79} + 6 q^{80} + 8 q^{82} + 4 q^{83} + 6 q^{85} + 14 q^{86} + 12 q^{88} - 16 q^{89} - 6 q^{91} + 12 q^{92} + 26 q^{94} + 22 q^{95} - 2 q^{97} + 34 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 6 * q^5 + 6 * q^7 + 12 * q^8 + 4 * q^11 - 4 * q^13 + 4 * q^14 + 4 * q^16 + 6 * q^17 - 2 * q^19 + 20 * q^20 + 2 * q^22 + 8 * q^23 - 2 * q^26 + 10 * q^28 + 4 * q^29 - 14 * q^31 + 6 * q^32 - 10 * q^34 + 6 * q^35 - 2 * q^37 - 8 * q^38 + 18 * q^40 + 18 * q^41 - 2 * q^43 + 8 * q^44 - 14 * q^46 + 2 * q^47 - 24 * q^50 - 8 * q^52 + 4 * q^53 + 6 * q^55 - 24 * q^58 + 16 * q^59 + 22 * q^61 - 4 * q^62 - 4 * q^64 - 6 * q^65 - 10 * q^67 - 4 * q^68 - 14 * q^70 - 2 * q^71 + 2 * q^73 + 22 * q^74 + 14 * q^76 + 6 * q^77 + 2 * q^79 + 6 * q^80 + 8 * q^82 + 4 * q^83 + 6 * q^85 + 14 * q^86 + 12 * q^88 - 16 * q^89 - 6 * q^91 + 12 * q^92 + 26 * q^94 + 22 * q^95 - 2 * q^97 + 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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