LMFDB - Embedded newform 1134.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1134,2,Mod(1,1134)] 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("1134.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1134, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1134.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.05503558921$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1134.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.267949 q^{5} -1.00000 q^{7} -1.00000 q^{8} -0.267949 q^{10} +6.19615 q^{11} -6.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +0.732051 q^{19} +0.267949 q^{20} -6.19615 q^{22} -4.19615 q^{23} -4.92820 q^{25} +6.46410 q^{26} -1.00000 q^{28} +1.53590 q^{29} +8.19615 q^{31} -1.00000 q^{32} -7.00000 q^{34} -0.267949 q^{35} +10.6603 q^{37} -0.732051 q^{38} -0.267949 q^{40} +2.53590 q^{41} -1.46410 q^{43} +6.19615 q^{44} +4.19615 q^{46} +4.73205 q^{47} +1.00000 q^{49} +4.92820 q^{50} -6.46410 q^{52} -9.46410 q^{53} +1.66025 q^{55} +1.00000 q^{56} -1.53590 q^{58} +4.19615 q^{59} +3.92820 q^{61} -8.19615 q^{62} +1.00000 q^{64} -1.73205 q^{65} -6.73205 q^{67} +7.00000 q^{68} +0.267949 q^{70} +6.53590 q^{71} +8.26795 q^{73} -10.6603 q^{74} +0.732051 q^{76} -6.19615 q^{77} -9.12436 q^{79} +0.267949 q^{80} -2.53590 q^{82} +16.5885 q^{83} +1.87564 q^{85} +1.46410 q^{86} -6.19615 q^{88} +9.92820 q^{89} +6.46410 q^{91} -4.19615 q^{92} -4.73205 q^{94} +0.196152 q^{95} +10.9282 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{7} - 2 q^{8} - 4 q^{10} + 2 q^{11} - 6 q^{13} + 2 q^{14} + 2 q^{16} + 14 q^{17} - 2 q^{19} + 4 q^{20} - 2 q^{22} + 2 q^{23} + 4 q^{25} + 6 q^{26} - 2 q^{28} + 10 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^7 - 2 * q^8 - 4 * q^10 + 2 * q^11 - 6 * q^13 + 2 * q^14 + 2 * q^16 + 14 * q^17 - 2 * q^19 + 4 * q^20 - 2 * q^22 + 2 * q^23 + 4 * q^25 + 6 * q^26 - 2 * q^28 + 10 * q^29 + 6 * q^31 - 2 * q^32 - 14 * q^34 - 4 * q^35 + 4 * q^37 + 2 * q^38 - 4 * q^40 + 12 * q^41 + 4 * q^43 + 2 * q^44 - 2 * q^46 + 6 * q^47 + 2 * q^49 - 4 * q^50 - 6 * q^52 - 12 * q^53 - 14 * q^55 + 2 * q^56 - 10 * q^58 - 2 * q^59 - 6 * q^61 - 6 * q^62 + 2 * q^64 - 10 * q^67 + 14 * q^68 + 4 * q^70 + 20 * q^71 + 20 * q^73 - 4 * q^74 - 2 * q^76 - 2 * q^77 + 6 * q^79 + 4 * q^80 - 12 * q^82 + 2 * q^83 + 28 * q^85 - 4 * q^86 - 2 * q^88 + 6 * q^89 + 6 * q^91 + 2 * q^92 - 6 * q^94 - 10 * q^95 + 8 * q^97 - 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.267949	0.119831	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	0.267949	0.119831	0.0599153	+	0.998203i	$0.480917\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.267949	−0.0847330
$11$	6.19615	1.86821	0.934105	−	0.356998i	$-0.116200\pi$
$11$	6.19615	1.86821	0.934105	+	0.356998i	$0.116200\pi$
$12$	0	0
$13$	−6.46410	−1.79282	−0.896410	−	0.443227i	$-0.853834\pi$
$13$	−6.46410	−1.79282	−0.896410	+	0.443227i	$0.853834\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	0.732051	0.167944	0.0839720	−	0.996468i	$-0.473239\pi$
$19$	0.732051	0.167944	0.0839720	+	0.996468i	$0.473239\pi$
$20$	0.267949	0.0599153
$21$	0	0
$22$	−6.19615	−1.32102
$23$	−4.19615	−0.874958	−0.437479	−	0.899229i	$-0.644129\pi$
$23$	−4.19615	−0.874958	−0.437479	+	0.899229i	$0.644129\pi$
$24$	0	0
$25$	−4.92820	−0.985641
$26$	6.46410	1.26771
$27$	0	0
$28$	−1.00000	−0.188982
$29$	1.53590	0.285209	0.142605	−	0.989780i	$-0.454452\pi$
$29$	1.53590	0.285209	0.142605	+	0.989780i	$0.454452\pi$
$30$	0	0
$31$	8.19615	1.47207	0.736036	−	0.676942i	$-0.236695\pi$
$31$	8.19615	1.47207	0.736036	+	0.676942i	$0.236695\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−7.00000	−1.20049
$35$	−0.267949	−0.0452917
$36$	0	0
$37$	10.6603	1.75253	0.876267	−	0.481825i	$-0.160026\pi$
$37$	10.6603	1.75253	0.876267	+	0.481825i	$0.160026\pi$
$38$	−0.732051	−0.118754
$39$	0	0
$40$	−0.267949	−0.0423665
$41$	2.53590	0.396041	0.198020	−	0.980198i	$-0.436549\pi$
$41$	2.53590	0.396041	0.198020	+	0.980198i	$0.436549\pi$
$42$	0	0
$43$	−1.46410	−0.223273	−0.111637	−	0.993749i	$-0.535609\pi$
$43$	−1.46410	−0.223273	−0.111637	+	0.993749i	$0.535609\pi$
$44$	6.19615	0.934105
$45$	0	0
$46$	4.19615	0.618689
$47$	4.73205	0.690241	0.345120	−	0.938558i	$-0.387838\pi$
$47$	4.73205	0.690241	0.345120	+	0.938558i	$0.387838\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.92820	0.696953
$51$	0	0
$52$	−6.46410	−0.896410
$53$	−9.46410	−1.29999	−0.649997	−	0.759937i	$-0.725230\pi$
$53$	−9.46410	−1.29999	−0.649997	+	0.759937i	$0.725230\pi$
$54$	0	0
$55$	1.66025	0.223869
$56$	1.00000	0.133631
$57$	0	0
$58$	−1.53590	−0.201673
$59$	4.19615	0.546293	0.273146	−	0.961973i	$-0.411936\pi$
$59$	4.19615	0.546293	0.273146	+	0.961973i	$0.411936\pi$
$60$	0	0
$61$	3.92820	0.502955	0.251477	−	0.967863i	$-0.419084\pi$
$61$	3.92820	0.502955	0.251477	+	0.967863i	$0.419084\pi$
$62$	−8.19615	−1.04091
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.73205	−0.214834
$66$	0	0
$67$	−6.73205	−0.822451	−0.411225	−	0.911534i	$-0.634899\pi$
$67$	−6.73205	−0.822451	−0.411225	+	0.911534i	$0.634899\pi$
$68$	7.00000	0.848875
$69$	0	0
$70$	0.267949	0.0320261
$71$	6.53590	0.775668	0.387834	−	0.921729i	$-0.373223\pi$
$71$	6.53590	0.775668	0.387834	+	0.921729i	$0.373223\pi$
$72$	0	0
$73$	8.26795	0.967690	0.483845	−	0.875154i	$-0.339240\pi$
$73$	8.26795	0.967690	0.483845	+	0.875154i	$0.339240\pi$
$74$	−10.6603	−1.23923
$75$	0	0
$76$	0.732051	0.0839720
$77$	−6.19615	−0.706117
$78$	0	0
$79$	−9.12436	−1.02657	−0.513285	−	0.858218i	$-0.671572\pi$
$79$	−9.12436	−1.02657	−0.513285	+	0.858218i	$0.671572\pi$
$80$	0.267949	0.0299576
$81$	0	0
$82$	−2.53590	−0.280043
$83$	16.5885	1.82082	0.910410	−	0.413707i	$-0.135766\pi$
$83$	16.5885	1.82082	0.910410	+	0.413707i	$0.135766\pi$
$84$	0	0
$85$	1.87564	0.203442
$86$	1.46410	0.157878
$87$	0	0
$88$	−6.19615	−0.660512
$89$	9.92820	1.05239	0.526194	−	0.850365i	$-0.323619\pi$
$89$	9.92820	1.05239	0.526194	+	0.850365i	$0.323619\pi$
$90$	0	0
$91$	6.46410	0.677622
$92$	−4.19615	−0.437479
$93$	0	0
$94$	−4.73205	−0.488074
$95$	0.196152	0.0201248
$96$	0	0
$97$	10.9282	1.10959	0.554795	−	0.831987i	$-0.312797\pi$
$97$	10.9282	1.10959	0.554795	+	0.831987i	$0.312797\pi$
$98$	−1.00000	−0.101015
$99$	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	1134.2.a.l.1.1	✓	2
3.2	odd	2		1134.2.a.m.1.2	yes	2
4.3	odd	2		9072.2.a.bp.1.1		2
7.6	odd	2		7938.2.a.bg.1.2		2
9.2	odd	6		1134.2.f.r.757.1		4
9.4	even	3		1134.2.f.s.379.2		4
9.5	odd	6		1134.2.f.r.379.1		4
9.7	even	3		1134.2.f.s.757.2		4
12.11	even	2		9072.2.a.y.1.2		2
21.20	even	2		7938.2.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.a.l.1.1	✓	2	1.1	even	1	trivial
1134.2.a.m.1.2	yes	2	3.2	odd	2
1134.2.f.r.379.1		4	9.5	odd	6
1134.2.f.r.757.1		4	9.2	odd	6
1134.2.f.s.379.2		4	9.4	even	3
1134.2.f.s.757.2		4	9.7	even	3
7938.2.a.bg.1.2		2	7.6	odd	2
7938.2.a.bt.1.1		2	21.20	even	2
9072.2.a.y.1.2		2	12.11	even	2
9072.2.a.bp.1.1		2	4.3	odd	2

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Elliptic curves over $\Q$ over number fields
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Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.267949 q^{5} -1.00000 q^{7} -1.00000 q^{8} -0.267949 q^{10} +6.19615 q^{11} -6.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +0.732051 q^{19} +0.267949 q^{20} -6.19615 q^{22} -4.19615 q^{23} -4.92820 q^{25} +6.46410 q^{26} -1.00000 q^{28} +1.53590 q^{29} +8.19615 q^{31} -1.00000 q^{32} -7.00000 q^{34} -0.267949 q^{35} +10.6603 q^{37} -0.732051 q^{38} -0.267949 q^{40} +2.53590 q^{41} -1.46410 q^{43} +6.19615 q^{44} +4.19615 q^{46} +4.73205 q^{47} +1.00000 q^{49} +4.92820 q^{50} -6.46410 q^{52} -9.46410 q^{53} +1.66025 q^{55} +1.00000 q^{56} -1.53590 q^{58} +4.19615 q^{59} +3.92820 q^{61} -8.19615 q^{62} +1.00000 q^{64} -1.73205 q^{65} -6.73205 q^{67} +7.00000 q^{68} +0.267949 q^{70} +6.53590 q^{71} +8.26795 q^{73} -10.6603 q^{74} +0.732051 q^{76} -6.19615 q^{77} -9.12436 q^{79} +0.267949 q^{80} -2.53590 q^{82} +16.5885 q^{83} +1.87564 q^{85} +1.46410 q^{86} -6.19615 q^{88} +9.92820 q^{89} +6.46410 q^{91} -4.19615 q^{92} -4.73205 q^{94} +0.196152 q^{95} +10.9282 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{7} - 2 q^{8} - 4 q^{10} + 2 q^{11} - 6 q^{13} + 2 q^{14} + 2 q^{16} + 14 q^{17} - 2 q^{19} + 4 q^{20} - 2 q^{22} + 2 q^{23} + 4 q^{25} + 6 q^{26} - 2 q^{28} + 10 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^7 - 2 * q^8 - 4 * q^10 + 2 * q^11 - 6 * q^13 + 2 * q^14 + 2 * q^16 + 14 * q^17 - 2 * q^19 + 4 * q^20 - 2 * q^22 + 2 * q^23 + 4 * q^25 + 6 * q^26 - 2 * q^28 + 10 * q^29 + 6 * q^31 - 2 * q^32 - 14 * q^34 - 4 * q^35 + 4 * q^37 + 2 * q^38 - 4 * q^40 + 12 * q^41 + 4 * q^43 + 2 * q^44 - 2 * q^46 + 6 * q^47 + 2 * q^49 - 4 * q^50 - 6 * q^52 - 12 * q^53 - 14 * q^55 + 2 * q^56 - 10 * q^58 - 2 * q^59 - 6 * q^61 - 6 * q^62 + 2 * q^64 - 10 * q^67 + 14 * q^68 + 4 * q^70 + 20 * q^71 + 20 * q^73 - 4 * q^74 - 2 * q^76 - 2 * q^77 + 6 * q^79 + 4 * q^80 - 12 * q^82 + 2 * q^83 + 28 * q^85 - 4 * q^86 - 2 * q^88 + 6 * q^89 + 6 * q^91 + 2 * q^92 - 6 * q^94 - 10 * q^95 + 8 * q^97 - 2 * q^98

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