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Properties

Label 2496.2.c.i
Level $2496$
Weight $2$
Character orbit 2496.c
Analytic conductor $19.931$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2496,2,Mod(961,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(19.9306603445\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + 2 \beta q^{7} + q^{9} + 3 \beta q^{11} + ( - \beta - 3) q^{13} + \beta q^{15} - 2 q^{17} + 2 \beta q^{21} + 8 q^{23} + q^{25} + q^{27} - 2 q^{29} - 4 \beta q^{31} + 3 \beta q^{33} + \cdots + 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9} - 6 q^{13} - 4 q^{17} + 16 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 16 q^{35} - 6 q^{39} - 16 q^{43} - 18 q^{49} - 4 q^{51} - 12 q^{53} - 24 q^{55} - 4 q^{61} + 8 q^{65} + 16 q^{69}+ \cdots + 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(769\) \(833\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
− 1.00000i
1.00000i
0 1.00000 0 − 2.00000i 0 − 4.00000i 0 1.00000 0
961.2 0 1.00000 0 2.00000i 0 4.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.2.c.i 2
4.b odd 2 1 2496.2.c.b 2
8.b even 2 1 624.2.c.d 2
8.d odd 2 1 156.2.b.b ✓ 2
13.b even 2 1 inner 2496.2.c.i 2
24.f even 2 1 468.2.b.c 2
24.h odd 2 1 1872.2.c.h 2
40.e odd 2 1 3900.2.c.a 2
40.k even 4 1 3900.2.j.b 2
40.k even 4 1 3900.2.j.e 2
52.b odd 2 1 2496.2.c.b 2
56.e even 2 1 7644.2.e.b 2
104.e even 2 1 624.2.c.d 2
104.h odd 2 1 156.2.b.b ✓ 2
104.j odd 4 1 8112.2.a.d 1
104.j odd 4 1 8112.2.a.l 1
104.m even 4 1 2028.2.a.d 1
104.m even 4 1 2028.2.a.f 1
104.n odd 6 2 2028.2.q.e 4
104.p odd 6 2 2028.2.q.e 4
104.u even 12 2 2028.2.i.a 2
104.u even 12 2 2028.2.i.d 2
312.b odd 2 1 1872.2.c.h 2
312.h even 2 1 468.2.b.c 2
312.w odd 4 1 6084.2.a.d 1
312.w odd 4 1 6084.2.a.n 1
520.b odd 2 1 3900.2.c.a 2
520.bc even 4 1 3900.2.j.b 2
520.bc even 4 1 3900.2.j.e 2
728.b even 2 1 7644.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.b.b ✓ 2 8.d odd 2 1
156.2.b.b ✓ 2 104.h odd 2 1
468.2.b.c 2 24.f even 2 1
468.2.b.c 2 312.h even 2 1
624.2.c.d 2 8.b even 2 1
624.2.c.d 2 104.e even 2 1
1872.2.c.h 2 24.h odd 2 1
1872.2.c.h 2 312.b odd 2 1
2028.2.a.d 1 104.m even 4 1
2028.2.a.f 1 104.m even 4 1
2028.2.i.a 2 104.u even 12 2
2028.2.i.d 2 104.u even 12 2
2028.2.q.e 4 104.n odd 6 2
2028.2.q.e 4 104.p odd 6 2
2496.2.c.b 2 4.b odd 2 1
2496.2.c.b 2 52.b odd 2 1
2496.2.c.i 2 1.a even 1 1 trivial
2496.2.c.i 2 13.b even 2 1 inner
3900.2.c.a 2 40.e odd 2 1
3900.2.c.a 2 520.b odd 2 1
3900.2.j.b 2 40.k even 4 1
3900.2.j.b 2 520.bc even 4 1
3900.2.j.e 2 40.k even 4 1
3900.2.j.e 2 520.bc even 4 1
6084.2.a.d 1 312.w odd 4 1
6084.2.a.n 1 312.w odd 4 1
7644.2.e.b 2 56.e even 2 1
7644.2.e.b 2 728.b even 2 1
8112.2.a.d 1 104.j odd 4 1
8112.2.a.l 1 104.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2496, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 36 \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display
\( T_{43} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 196 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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