LMFDB - Embedded newform 736.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [736,2,Mod(1,736)] 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("736.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 736 = 2^{5} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	736.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.87698958877$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	736.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -0.732051 q^{5} +4.73205 q^{7} +1.26795 q^{11} -6.46410 q^{13} +1.26795 q^{15} +4.19615 q^{17} +3.46410 q^{19} -8.19615 q^{21} -1.00000 q^{23} -4.46410 q^{25} +5.19615 q^{27} +8.46410 q^{29} +4.26795 q^{31} -2.19615 q^{33} -3.46410 q^{35} +6.19615 q^{37} +11.1962 q^{39} -1.53590 q^{41} -6.92820 q^{43} +11.1962 q^{47} +15.3923 q^{49} -7.26795 q^{51} +8.92820 q^{53} -0.928203 q^{55} -6.00000 q^{57} +9.46410 q^{59} +2.00000 q^{61} +4.73205 q^{65} -3.80385 q^{67} +1.73205 q^{69} +10.2679 q^{71} -0.464102 q^{73} +7.73205 q^{75} +6.00000 q^{77} -10.3923 q^{79} -9.00000 q^{81} +2.19615 q^{83} -3.07180 q^{85} -14.6603 q^{87} -9.85641 q^{89} -30.5885 q^{91} -7.39230 q^{93} -2.53590 q^{95} -1.26795 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 6 q^{7} + 6 q^{11} - 6 q^{13} + 6 q^{15} - 2 q^{17} - 6 q^{21} - 2 q^{23} - 2 q^{25} + 10 q^{29} + 12 q^{31} + 6 q^{33} + 2 q^{37} + 12 q^{39} - 10 q^{41} + 12 q^{47} + 10 q^{49} - 18 q^{51}+ \cdots - 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 6 * q^7 + 6 * q^11 - 6 * q^13 + 6 * q^15 - 2 * q^17 - 6 * q^21 - 2 * q^23 - 2 * q^25 + 10 * q^29 + 12 * q^31 + 6 * q^33 + 2 * q^37 + 12 * q^39 - 10 * q^41 + 12 * q^47 + 10 * q^49 - 18 * q^51 + 4 * q^53 + 12 * q^55 - 12 * q^57 + 12 * q^59 + 4 * q^61 + 6 * q^65 - 18 * q^67 + 24 * q^71 + 6 * q^73 + 12 * q^75 + 12 * q^77 - 18 * q^81 - 6 * q^83 - 20 * q^85 - 12 * q^87 + 8 * q^89 - 30 * q^91 + 6 * q^93 - 12 * q^95 - 6 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	−1.73205	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.73205	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	−0.732051	−0.327383	−0.163692	−	0.986512i	$-0.552340\pi$
$5$	−0.732051	−0.327383	−0.163692	+	0.986512i	$0.552340\pi$
$6$	0	0
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	−6.46410	−1.79282	−0.896410	−	0.443227i	$-0.853834\pi$
$13$	−6.46410	−1.79282	−0.896410	+	0.443227i	$0.853834\pi$
$14$	0	0
$15$	1.26795	0.327383
$16$	0	0
$17$	4.19615	1.01772	0.508858	−	0.860850i	$-0.330068\pi$
$17$	4.19615	1.01772	0.508858	+	0.860850i	$0.330068\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	−8.19615	−1.78855
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	5.19615	1.00000
$28$	0	0
$29$	8.46410	1.57174	0.785872	−	0.618389i	$-0.212214\pi$
$29$	8.46410	1.57174	0.785872	+	0.618389i	$0.212214\pi$
$30$	0	0
$31$	4.26795	0.766546	0.383273	−	0.923635i	$-0.374797\pi$
$31$	4.26795	0.766546	0.383273	+	0.923635i	$0.374797\pi$
$32$	0	0
$33$	−2.19615	−0.382301
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	6.19615	1.01864	0.509321	−	0.860577i	$-0.329897\pi$
$37$	6.19615	1.01864	0.509321	+	0.860577i	$0.329897\pi$
$38$	0	0
$39$	11.1962	1.79282
$40$	0	0
$41$	−1.53590	−0.239867	−0.119934	−	0.992782i	$-0.538268\pi$
$41$	−1.53590	−0.239867	−0.119934	+	0.992782i	$0.538268\pi$
$42$	0	0
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.1962	1.63313	0.816563	−	0.577256i	$-0.195876\pi$
$47$	11.1962	1.63313	0.816563	+	0.577256i	$0.195876\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	−7.26795	−1.01772
$52$	0	0
$53$	8.92820	1.22638	0.613192	−	0.789934i	$-0.289885\pi$
$53$	8.92820	1.22638	0.613192	+	0.789934i	$0.289885\pi$
$54$	0	0
$55$	−0.928203	−0.125159
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	9.46410	1.23212	0.616061	−	0.787699i	$-0.288728\pi$
$59$	9.46410	1.23212	0.616061	+	0.787699i	$0.288728\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.73205	0.586939
$66$	0	0
$67$	−3.80385	−0.464714	−0.232357	−	0.972631i	$-0.574644\pi$
$67$	−3.80385	−0.464714	−0.232357	+	0.972631i	$0.574644\pi$
$68$	0	0
$69$	1.73205	0.208514
$70$	0	0
$71$	10.2679	1.21858	0.609291	−	0.792947i	$-0.291454\pi$
$71$	10.2679	1.21858	0.609291	+	0.792947i	$0.291454\pi$
$72$	0	0
$73$	−0.464102	−0.0543190	−0.0271595	−	0.999631i	$-0.508646\pi$
$73$	−0.464102	−0.0543190	−0.0271595	+	0.999631i	$0.508646\pi$
$74$	0	0
$75$	7.73205	0.892820
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−10.3923	−1.16923	−0.584613	−	0.811312i	$-0.698754\pi$
$79$	−10.3923	−1.16923	−0.584613	+	0.811312i	$0.698754\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	2.19615	0.241059	0.120530	−	0.992710i	$-0.461541\pi$
$83$	2.19615	0.241059	0.120530	+	0.992710i	$0.461541\pi$
$84$	0	0
$85$	−3.07180	−0.333183
$86$	0	0
$87$	−14.6603	−1.57174
$88$	0	0
$89$	−9.85641	−1.04478	−0.522388	−	0.852708i	$-0.674959\pi$
$89$	−9.85641	−1.04478	−0.522388	+	0.852708i	$0.674959\pi$
$90$	0	0
$91$	−30.5885	−3.20654
$92$	0	0
$93$	−7.39230	−0.766546
$94$	0	0
$95$	−2.53590	−0.260178
$96$	0	0
$97$	−1.26795	−0.128741	−0.0643704	−	0.997926i	$-0.520504\pi$
$97$	−1.26795	−0.128741	−0.0643704	+	0.997926i	$0.520504\pi$
$98$	0	0
$99$	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	736.2.a.c.1.1	yes	2
3.2	odd	2		6624.2.a.n.1.2		2
4.3	odd	2		736.2.a.b.1.2	✓	2
8.3	odd	2		1472.2.a.q.1.1		2
8.5	even	2		1472.2.a.r.1.2		2
12.11	even	2		6624.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.2.a.b.1.2	✓	2	4.3	odd	2
736.2.a.c.1.1	yes	2	1.1	even	1	trivial
1472.2.a.q.1.1		2	8.3	odd	2
1472.2.a.r.1.2		2	8.5	even	2
6624.2.a.k.1.2		2	12.11	even	2
6624.2.a.n.1.2		2	3.2	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	736.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -0.732051 q^{5} +4.73205 q^{7} +1.26795 q^{11} -6.46410 q^{13} +1.26795 q^{15} +4.19615 q^{17} +3.46410 q^{19} -8.19615 q^{21} -1.00000 q^{23} -4.46410 q^{25} +5.19615 q^{27} +8.46410 q^{29} +4.26795 q^{31} -2.19615 q^{33} -3.46410 q^{35} +6.19615 q^{37} +11.1962 q^{39} -1.53590 q^{41} -6.92820 q^{43} +11.1962 q^{47} +15.3923 q^{49} -7.26795 q^{51} +8.92820 q^{53} -0.928203 q^{55} -6.00000 q^{57} +9.46410 q^{59} +2.00000 q^{61} +4.73205 q^{65} -3.80385 q^{67} +1.73205 q^{69} +10.2679 q^{71} -0.464102 q^{73} +7.73205 q^{75} +6.00000 q^{77} -10.3923 q^{79} -9.00000 q^{81} +2.19615 q^{83} -3.07180 q^{85} -14.6603 q^{87} -9.85641 q^{89} -30.5885 q^{91} -7.39230 q^{93} -2.53590 q^{95} -1.26795 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 6 q^{7} + 6 q^{11} - 6 q^{13} + 6 q^{15} - 2 q^{17} - 6 q^{21} - 2 q^{23} - 2 q^{25} + 10 q^{29} + 12 q^{31} + 6 q^{33} + 2 q^{37} + 12 q^{39} - 10 q^{41} + 12 q^{47} + 10 q^{49} - 18 q^{51}+ \cdots - 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 6 * q^7 + 6 * q^11 - 6 * q^13 + 6 * q^15 - 2 * q^17 - 6 * q^21 - 2 * q^23 - 2 * q^25 + 10 * q^29 + 12 * q^31 + 6 * q^33 + 2 * q^37 + 12 * q^39 - 10 * q^41 + 12 * q^47 + 10 * q^49 - 18 * q^51 + 4 * q^53 + 12 * q^55 - 12 * q^57 + 12 * q^59 + 4 * q^61 + 6 * q^65 - 18 * q^67 + 24 * q^71 + 6 * q^73 + 12 * q^75 + 12 * q^77 - 18 * q^81 - 6 * q^83 - 20 * q^85 - 12 * q^87 + 8 * q^89 - 30 * q^91 + 6 * q^93 - 12 * q^95 - 6 * q^97

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