Properties

Label 1216.2.a.e
Level $1216$
Weight $2$
Character orbit 1216.a
Self dual yes
Analytic conductor $9.710$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, 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("1216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(9.70980888579\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} - q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{7} - 2 q^{9} + 6 q^{11} - 5 q^{13} + 3 q^{17} - q^{19} + q^{21} + 3 q^{23} - 5 q^{25} + 5 q^{27} - 9 q^{29} - 4 q^{31} - 6 q^{33} - 2 q^{37} + 5 q^{39} - 8 q^{43} - 6 q^{49} - 3 q^{51} + 3 q^{53} + q^{57} - 9 q^{59} + 10 q^{61} + 2 q^{63} - 5 q^{67} - 3 q^{69} - 6 q^{71} - 7 q^{73} + 5 q^{75} - 6 q^{77} - 10 q^{79} + q^{81} + 6 q^{83} + 9 q^{87} - 12 q^{89} + 5 q^{91} + 4 q^{93} - 10 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −1.00000 0 0 0 −1.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.a.e 1
4.b odd 2 1 1216.2.a.m 1
8.b even 2 1 38.2.a.a 1
8.d odd 2 1 304.2.a.c 1
24.f even 2 1 2736.2.a.n 1
24.h odd 2 1 342.2.a.e 1
40.e odd 2 1 7600.2.a.n 1
40.f even 2 1 950.2.a.d 1
40.i odd 4 2 950.2.b.b 2
56.h odd 2 1 1862.2.a.b 1
88.b odd 2 1 4598.2.a.p 1
104.e even 2 1 6422.2.a.h 1
120.i odd 2 1 8550.2.a.m 1
152.b even 2 1 5776.2.a.m 1
152.g odd 2 1 722.2.a.e 1
152.l odd 6 2 722.2.c.c 2
152.p even 6 2 722.2.c.e 2
152.s odd 18 6 722.2.e.e 6
152.t even 18 6 722.2.e.f 6
456.p even 2 1 6498.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 8.b even 2 1
304.2.a.c 1 8.d odd 2 1
342.2.a.e 1 24.h odd 2 1
722.2.a.e 1 152.g odd 2 1
722.2.c.c 2 152.l odd 6 2
722.2.c.e 2 152.p even 6 2
722.2.e.e 6 152.s odd 18 6
722.2.e.f 6 152.t even 18 6
950.2.a.d 1 40.f even 2 1
950.2.b.b 2 40.i odd 4 2
1216.2.a.e 1 1.a even 1 1 trivial
1216.2.a.m 1 4.b odd 2 1
1862.2.a.b 1 56.h odd 2 1
2736.2.a.n 1 24.f even 2 1
4598.2.a.p 1 88.b odd 2 1
5776.2.a.m 1 152.b even 2 1
6422.2.a.h 1 104.e even 2 1
6498.2.a.f 1 456.p even 2 1
7600.2.a.n 1 40.e odd 2 1
8550.2.a.m 1 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T + 9 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 3 \) Copy content Toggle raw display
$59$ \( T + 9 \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T + 5 \) Copy content Toggle raw display
$71$ \( T + 6 \) Copy content Toggle raw display
$73$ \( T + 7 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T + 12 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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