LMFDB - Embedded newform 1521.4.a.s.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1521,4,Mod(1,1521)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1521.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1521, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,14,24,0,0,54,0,-32,-44,0,0,56,0,-30,-164] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.74166$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.74166 q^{2} -0.483315 q^{4} +19.4833 q^{5} -7.48331 q^{7} +23.2583 q^{8} -53.4166 q^{10} +22.8999 q^{11} +20.5167 q^{14} -59.8999 q^{16} -67.0334 q^{17} -16.5167 q^{19} -9.41657 q^{20} -62.7836 q^{22} +175.600 q^{23} +254.600 q^{25} +3.61680 q^{28} -291.800 q^{29} -117.283 q^{31} -21.8418 q^{32} +183.783 q^{34} -145.800 q^{35} +154.766 q^{37} +45.2831 q^{38} +453.150 q^{40} -251.716 q^{41} -502.566 q^{43} -11.0679 q^{44} -481.434 q^{46} -281.733 q^{47} -287.000 q^{49} -698.025 q^{50} -366.999 q^{53} +446.166 q^{55} -174.049 q^{56} +800.015 q^{58} -79.6663 q^{59} -194.865 q^{61} +321.550 q^{62} +539.082 q^{64} -400.082 q^{67} +32.3982 q^{68} +399.733 q^{70} +528.299 q^{71} +734.366 q^{73} -424.316 q^{74} +7.98276 q^{76} -171.367 q^{77} +113.266 q^{79} -1167.05 q^{80} +690.118 q^{82} -933.466 q^{83} -1306.03 q^{85} +1377.86 q^{86} +532.613 q^{88} +1190.91 q^{89} -84.8699 q^{92} +772.415 q^{94} -321.800 q^{95} -557.165 q^{97} +786.856 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 14 q^{4} + 24 q^{5} + 54 q^{8} - 32 q^{10} - 44 q^{11} + 56 q^{14} - 30 q^{16} - 164 q^{17} - 48 q^{19} + 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 112 q^{28} - 404 q^{29} - 40 q^{31}+ \cdots - 574 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 14 * q^4 + 24 * q^5 + 54 * q^8 - 32 * q^10 - 44 * q^11 + 56 * q^14 - 30 * q^16 - 164 * q^17 - 48 * q^19 + 56 * q^20 - 380 * q^22 - 8 * q^23 + 150 * q^25 + 112 * q^28 - 404 * q^29 - 40 * q^31 - 126 * q^32 - 276 * q^34 - 112 * q^35 + 100 * q^37 - 104 * q^38 + 592 * q^40 + 200 * q^41 - 616 * q^43 - 980 * q^44 - 1352 * q^46 - 324 * q^47 - 574 * q^49 - 1194 * q^50 + 164 * q^53 + 144 * q^55 + 56 * q^56 + 268 * q^58 + 140 * q^59 + 628 * q^61 + 688 * q^62 - 194 * q^64 + 472 * q^67 - 1372 * q^68 + 560 * q^70 + 428 * q^71 + 900 * q^73 - 684 * q^74 - 448 * q^76 - 672 * q^77 - 432 * q^79 - 1032 * q^80 + 2832 * q^82 - 1388 * q^83 - 1744 * q^85 + 840 * q^86 - 1524 * q^88 + 960 * q^89 - 2744 * q^92 + 572 * q^94 - 464 * q^95 + 532 * q^97 - 574 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	−2.74166	−0.969322	−0.484661	−	0.874702i	$-0.661057\pi$
$2$	−2.74166	−0.969322	−0.484661	+	0.874702i	$0.661057\pi$
$3$	0	0
$3$	0	0
$4$	−0.483315	−0.0604143
$5$	19.4833	1.74264	0.871320	−	0.490715i	$-0.163264\pi$
$5$	19.4833	1.74264	0.871320	+	0.490715i	$0.163264\pi$
$6$	0	0
$7$	−7.48331	−0.404061	−0.202031	−	0.979379i	$-0.564754\pi$
$7$	−7.48331	−0.404061	−0.202031	+	0.979379i	$0.564754\pi$
$8$	23.2583	1.02788
$9$	0	0
$10$	−53.4166	−1.68918
$11$	22.8999	0.627689	0.313844	−	0.949474i	$-0.398383\pi$
$11$	22.8999	0.627689	0.313844	+	0.949474i	$0.398383\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	20.5167	0.391665
$15$	0	0
$16$	−59.8999	−0.935936
$17$	−67.0334	−0.956352	−0.478176	−	0.878264i	$-0.658702\pi$
$17$	−67.0334	−0.956352	−0.478176	+	0.878264i	$0.658702\pi$
$18$	0	0
$19$	−16.5167	−0.199431	−0.0997155	−	0.995016i	$-0.531793\pi$
$19$	−16.5167	−0.199431	−0.0997155	+	0.995016i	$0.531793\pi$
$20$	−9.41657	−0.105280
$21$	0	0
$22$	−62.7836	−0.608433
$23$	175.600	1.59196	0.795979	−	0.605324i	$-0.206956\pi$
$23$	175.600	1.59196	0.795979	+	0.605324i	$0.206956\pi$
$24$	0	0
$25$	254.600	2.03680
$26$	0	0
$27$	0	0
$28$	3.61680	0.0244111
$29$	−291.800	−1.86848	−0.934239	−	0.356648i	$-0.883920\pi$
$29$	−291.800	−1.86848	−0.934239	+	0.356648i	$0.883920\pi$
$30$	0	0
$31$	−117.283	−0.679505	−0.339753	−	0.940515i	$-0.610343\pi$
$31$	−117.283	−0.679505	−0.339753	+	0.940515i	$0.610343\pi$
$32$	−21.8418	−0.120660
$33$	0	0
$34$	183.783	0.927013
$35$	−145.800	−0.704133
$36$	0	0
$37$	154.766	0.687661	0.343830	−	0.939032i	$-0.388276\pi$
$37$	154.766	0.687661	0.343830	+	0.939032i	$0.388276\pi$
$38$	45.2831	0.193313
$39$	0	0
$40$	453.150	1.79123
$41$	−251.716	−0.958815	−0.479407	−	0.877592i	$-0.659148\pi$
$41$	−251.716	−0.958815	−0.479407	+	0.877592i	$0.659148\pi$
$42$	0	0
$43$	−502.566	−1.78234	−0.891170	−	0.453669i	$-0.850115\pi$
$43$	−502.566	−1.78234	−0.891170	+	0.453669i	$0.850115\pi$
$44$	−11.0679	−0.0379214
$45$	0	0
$46$	−481.434	−1.54312
$47$	−281.733	−0.874361	−0.437181	−	0.899374i	$-0.644023\pi$
$47$	−281.733	−0.874361	−0.437181	+	0.899374i	$0.644023\pi$
$48$	0	0
$49$	−287.000	−0.836735
$50$	−698.025	−1.97431
$51$	0	0
$52$	0	0
$53$	−366.999	−0.951154	−0.475577	−	0.879674i	$-0.657761\pi$
$53$	−366.999	−0.951154	−0.475577	+	0.879674i	$0.657761\pi$
$54$	0	0
$55$	446.166	1.09384
$56$	−174.049	−0.415328
$57$	0	0
$58$	800.015	1.81116
$59$	−79.6663	−0.175791	−0.0878955	−	0.996130i	$-0.528014\pi$
$59$	−79.6663	−0.175791	−0.0878955	+	0.996130i	$0.528014\pi$
$60$	0	0
$61$	−194.865	−0.409016	−0.204508	−	0.978865i	$-0.565559\pi$
$61$	−194.865	−0.409016	−0.204508	+	0.978865i	$0.565559\pi$
$62$	321.550	0.658660
$63$	0	0
$64$	539.082	1.05289
$65$	0	0
$66$	0	0
$67$	−400.082	−0.729519	−0.364759	−	0.931102i	$-0.618849\pi$
$67$	−400.082	−0.729519	−0.364759	+	0.931102i	$0.618849\pi$
$68$	32.3982	0.0577774
$69$	0	0
$70$	399.733	0.682532
$71$	528.299	0.883065	0.441532	−	0.897245i	$-0.354435\pi$
$71$	528.299	0.883065	0.441532	+	0.897245i	$0.354435\pi$
$72$	0	0
$73$	734.366	1.17741	0.588706	−	0.808347i	$-0.299638\pi$
$73$	734.366	1.17741	0.588706	+	0.808347i	$0.299638\pi$
$74$	−424.316	−0.666565
$75$	0	0
$76$	7.98276	0.0120485
$77$	−171.367	−0.253625
$78$	0	0
$79$	113.266	0.161309	0.0806545	−	0.996742i	$-0.474299\pi$
$79$	113.266	0.161309	0.0806545	+	0.996742i	$0.474299\pi$
$80$	−1167.05	−1.63100
$81$	0	0
$82$	690.118	0.929400
$83$	−933.466	−1.23447	−0.617236	−	0.786778i	$-0.711748\pi$
$83$	−933.466	−1.23447	−0.617236	+	0.786778i	$0.711748\pi$
$84$	0	0
$85$	−1306.03	−1.66658
$86$	1377.86	1.72766
$87$	0	0
$88$	532.613	0.645191
$89$	1190.91	1.41839	0.709195	−	0.705012i	$-0.249059\pi$
$89$	1190.91	1.41839	0.709195	+	0.705012i	$0.249059\pi$
$90$	0	0
$91$	0	0
$92$	−84.8699	−0.0961771
$93$	0	0
$94$	772.415	0.847538
$95$	−321.800	−0.347536
$96$	0	0
$97$	−557.165	−0.583211	−0.291606	−	0.956539i	$-0.594189\pi$
$97$	−557.165	−0.583211	−0.291606	+	0.956539i	$0.594189\pi$
$98$	786.856	0.811066
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.s.1.1		2
3.2	odd	2		507.4.a.f.1.2		2
13.12	even	2		117.4.a.c.1.2		2
39.5	even	4		507.4.b.f.337.2		4
39.8	even	4		507.4.b.f.337.3		4
39.38	odd	2		39.4.a.b.1.1	✓	2
52.51	odd	2		1872.4.a.t.1.1		2
156.155	even	2		624.4.a.r.1.2		2
195.194	odd	2		975.4.a.j.1.2		2
273.272	even	2		1911.4.a.h.1.1		2
312.77	odd	2		2496.4.a.bc.1.1		2
312.155	even	2		2496.4.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.1	✓	2	39.38	odd	2
117.4.a.c.1.2		2	13.12	even	2
507.4.a.f.1.2		2	3.2	odd	2
507.4.b.f.337.2		4	39.5	even	4
507.4.b.f.337.3		4	39.8	even	4
624.4.a.r.1.2		2	156.155	even	2
975.4.a.j.1.2		2	195.194	odd	2
1521.4.a.s.1.1		2	1.1	even	1	trivial
1872.4.a.t.1.1		2	52.51	odd	2
1911.4.a.h.1.1		2	273.272	even	2
2496.4.a.s.1.1		2	312.155	even	2
2496.4.a.bc.1.1		2	312.77	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74166 q^{2} -0.483315 q^{4} +19.4833 q^{5} -7.48331 q^{7} +23.2583 q^{8} -53.4166 q^{10} +22.8999 q^{11} +20.5167 q^{14} -59.8999 q^{16} -67.0334 q^{17} -16.5167 q^{19} -9.41657 q^{20} -62.7836 q^{22} +175.600 q^{23} +254.600 q^{25} +3.61680 q^{28} -291.800 q^{29} -117.283 q^{31} -21.8418 q^{32} +183.783 q^{34} -145.800 q^{35} +154.766 q^{37} +45.2831 q^{38} +453.150 q^{40} -251.716 q^{41} -502.566 q^{43} -11.0679 q^{44} -481.434 q^{46} -281.733 q^{47} -287.000 q^{49} -698.025 q^{50} -366.999 q^{53} +446.166 q^{55} -174.049 q^{56} +800.015 q^{58} -79.6663 q^{59} -194.865 q^{61} +321.550 q^{62} +539.082 q^{64} -400.082 q^{67} +32.3982 q^{68} +399.733 q^{70} +528.299 q^{71} +734.366 q^{73} -424.316 q^{74} +7.98276 q^{76} -171.367 q^{77} +113.266 q^{79} -1167.05 q^{80} +690.118 q^{82} -933.466 q^{83} -1306.03 q^{85} +1377.86 q^{86} +532.613 q^{88} +1190.91 q^{89} -84.8699 q^{92} +772.415 q^{94} -321.800 q^{95} -557.165 q^{97} +786.856 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 14 q^{4} + 24 q^{5} + 54 q^{8} - 32 q^{10} - 44 q^{11} + 56 q^{14} - 30 q^{16} - 164 q^{17} - 48 q^{19} + 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 112 q^{28} - 404 q^{29} - 40 q^{31}+ \cdots - 574 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 14 * q^4 + 24 * q^5 + 54 * q^8 - 32 * q^10 - 44 * q^11 + 56 * q^14 - 30 * q^16 - 164 * q^17 - 48 * q^19 + 56 * q^20 - 380 * q^22 - 8 * q^23 + 150 * q^25 + 112 * q^28 - 404 * q^29 - 40 * q^31 - 126 * q^32 - 276 * q^34 - 112 * q^35 + 100 * q^37 - 104 * q^38 + 592 * q^40 + 200 * q^41 - 616 * q^43 - 980 * q^44 - 1352 * q^46 - 324 * q^47 - 574 * q^49 - 1194 * q^50 + 164 * q^53 + 144 * q^55 + 56 * q^56 + 268 * q^58 + 140 * q^59 + 628 * q^61 + 688 * q^62 - 194 * q^64 + 472 * q^67 - 1372 * q^68 + 560 * q^70 + 428 * q^71 + 900 * q^73 - 684 * q^74 - 448 * q^76 - 672 * q^77 - 432 * q^79 - 1032 * q^80 + 2832 * q^82 - 1388 * q^83 - 1744 * q^85 + 840 * q^86 - 1524 * q^88 + 960 * q^89 - 2744 * q^92 + 572 * q^94 - 464 * q^95 + 532 * q^97 - 574 * q^98

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