LMFDB - Embedded newform 6400.2.a.cg.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6400,2,Mod(1,6400)] 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("6400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6400 = 2^{8} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,0,0,-4,0,6,0,0,0,0,0,6,0,-2,0,0,0,0,0,0,0,-10,0, 0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	yes
Analytic conductor:	$51.1042572936$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1600)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.46410 q^{7} -2.00000 q^{9} +3.00000 q^{11} +3.46410 q^{13} +3.00000 q^{17} -1.00000 q^{19} +3.46410 q^{21} -5.00000 q^{27} +10.3923 q^{29} +6.92820 q^{31} +3.00000 q^{33} -10.3923 q^{37} +3.46410 q^{39} +9.00000 q^{41} +4.00000 q^{43} -10.3923 q^{47} +5.00000 q^{49} +3.00000 q^{51} -1.00000 q^{57} -12.0000 q^{59} -3.46410 q^{61} -6.92820 q^{63} +11.0000 q^{67} -10.3923 q^{71} -7.00000 q^{73} +10.3923 q^{77} -10.3923 q^{79} +1.00000 q^{81} +15.0000 q^{83} +10.3923 q^{87} +3.00000 q^{89} +12.0000 q^{91} +6.92820 q^{93} +14.0000 q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{9} + 6 q^{11} + 6 q^{17} - 2 q^{19} - 10 q^{27} + 6 q^{33} + 18 q^{41} + 8 q^{43} + 10 q^{49} + 6 q^{51} - 2 q^{57} - 24 q^{59} + 22 q^{67} - 14 q^{73} + 2 q^{81} + 30 q^{83} + 6 q^{89}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^9 + 6 * q^11 + 6 * q^17 - 2 * q^19 - 10 * q^27 + 6 * q^33 + 18 * q^41 + 8 * q^43 + 10 * q^49 + 6 * q^51 - 2 * q^57 - 24 * q^59 + 22 * q^67 - 14 * q^73 + 2 * q^81 + 30 * q^83 + 6 * q^89 + 24 * q^91 + 28 * q^97 - 12 * q^99

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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	3.46410	0.755929
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	10.3923	1.92980	0.964901	−	0.262613i	$-0.0845842\pi$
$29$	10.3923	1.92980	0.964901	+	0.262613i	$0.0845842\pi$
$30$	0	0
$31$	6.92820	1.24434	0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.3923	−1.70848	−0.854242	−	0.519875i	$-0.825978\pi$
$37$	−10.3923	−1.70848	−0.854242	+	0.519875i	$0.825978\pi$
$38$	0	0
$39$	3.46410	0.554700
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.3923	−1.51587	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923	−1.51587	−0.757937	+	0.652328i	$0.773792\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−3.46410	−0.443533	−0.221766	−	0.975100i	$-0.571182\pi$
$61$	−3.46410	−0.443533	−0.221766	+	0.975100i	$0.571182\pi$
$62$	0	0
$63$	−6.92820	−0.872872
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.3923	−1.23334	−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923	−1.23334	−0.616670	+	0.787222i	$0.711519\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3923	1.18431
$78$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.3923	1.11417
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	0	0
$93$	6.92820	0.718421
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	−6.00000	−0.603023

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	6400.2.a.cg.1.2		2
4.3	odd	2		6400.2.a.ba.1.1		2
5.4	even	2		6400.2.a.bb.1.1		2
8.3	odd	2	inner	6400.2.a.cg.1.1		2
8.5	even	2		6400.2.a.ba.1.2		2
16.3	odd	4		1600.2.d.f.801.2	yes	4
16.5	even	4		1600.2.d.f.801.1	yes	4
16.11	odd	4		1600.2.d.f.801.4	yes	4
16.13	even	4		1600.2.d.f.801.3	yes	4
20.19	odd	2		6400.2.a.cf.1.2		2
40.19	odd	2		6400.2.a.bb.1.2		2
40.29	even	2		6400.2.a.cf.1.1		2
80.3	even	4		1600.2.f.g.1249.2		4
80.13	odd	4		1600.2.f.c.1249.3		4
80.19	odd	4		1600.2.d.e.801.3	yes	4
80.27	even	4		1600.2.f.g.1249.3		4
80.29	even	4		1600.2.d.e.801.2	yes	4
80.37	odd	4		1600.2.f.c.1249.2		4
80.43	even	4		1600.2.f.c.1249.1		4
80.53	odd	4		1600.2.f.g.1249.4		4
80.59	odd	4		1600.2.d.e.801.1	✓	4
80.67	even	4		1600.2.f.c.1249.4		4
80.69	even	4		1600.2.d.e.801.4	yes	4
80.77	odd	4		1600.2.f.g.1249.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.2.d.e.801.1	✓	4	80.59	odd	4
1600.2.d.e.801.2	yes	4	80.29	even	4
1600.2.d.e.801.3	yes	4	80.19	odd	4
1600.2.d.e.801.4	yes	4	80.69	even	4
1600.2.d.f.801.1	yes	4	16.5	even	4
1600.2.d.f.801.2	yes	4	16.3	odd	4
1600.2.d.f.801.3	yes	4	16.13	even	4
1600.2.d.f.801.4	yes	4	16.11	odd	4
1600.2.f.c.1249.1		4	80.43	even	4
1600.2.f.c.1249.2		4	80.37	odd	4
1600.2.f.c.1249.3		4	80.13	odd	4
1600.2.f.c.1249.4		4	80.67	even	4
1600.2.f.g.1249.1		4	80.77	odd	4
1600.2.f.g.1249.2		4	80.3	even	4
1600.2.f.g.1249.3		4	80.27	even	4
1600.2.f.g.1249.4		4	80.53	odd	4
6400.2.a.ba.1.1		2	4.3	odd	2
6400.2.a.ba.1.2		2	8.5	even	2
6400.2.a.bb.1.1		2	5.4	even	2
6400.2.a.bb.1.2		2	40.19	odd	2
6400.2.a.cf.1.1		2	40.29	even	2
6400.2.a.cf.1.2		2	20.19	odd	2
6400.2.a.cg.1.1		2	8.3	odd	2	inner
6400.2.a.cg.1.2		2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.46410 q^{7} -2.00000 q^{9} +3.00000 q^{11} +3.46410 q^{13} +3.00000 q^{17} -1.00000 q^{19} +3.46410 q^{21} -5.00000 q^{27} +10.3923 q^{29} +6.92820 q^{31} +3.00000 q^{33} -10.3923 q^{37} +3.46410 q^{39} +9.00000 q^{41} +4.00000 q^{43} -10.3923 q^{47} +5.00000 q^{49} +3.00000 q^{51} -1.00000 q^{57} -12.0000 q^{59} -3.46410 q^{61} -6.92820 q^{63} +11.0000 q^{67} -10.3923 q^{71} -7.00000 q^{73} +10.3923 q^{77} -10.3923 q^{79} +1.00000 q^{81} +15.0000 q^{83} +10.3923 q^{87} +3.00000 q^{89} +12.0000 q^{91} +6.92820 q^{93} +14.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{9} + 6 q^{11} + 6 q^{17} - 2 q^{19} - 10 q^{27} + 6 q^{33} + 18 q^{41} + 8 q^{43} + 10 q^{49} + 6 q^{51} - 2 q^{57} - 24 q^{59} + 22 q^{67} - 14 q^{73} + 2 q^{81} + 30 q^{83} + 6 q^{89}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^9 + 6 * q^11 + 6 * q^17 - 2 * q^19 - 10 * q^27 + 6 * q^33 + 18 * q^41 + 8 * q^43 + 10 * q^49 + 6 * q^51 - 2 * q^57 - 24 * q^59 + 22 * q^67 - 14 * q^73 + 2 * q^81 + 30 * q^83 + 6 * q^89 + 24 * q^91 + 28 * q^97 - 12 * q^99

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