Properties

Label 5376.2.c.m
Level $5376$
Weight $2$
Character orbit 5376.c
Analytic conductor $42.928$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5376,2,Mod(2689,5376)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5376, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("5376.2689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.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: \(42.9275761266\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 672)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{3} + 2 i q^{5} - q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} + 2 i q^{5} - q^{7} - q^{9} - 4 i q^{11} - 6 i q^{13} + 2 q^{15} - 2 q^{17} - 4 i q^{19} + i q^{21} - 4 q^{23} + q^{25} + i q^{27} - 2 i q^{29} - 8 q^{31} - 4 q^{33} - 2 i q^{35} + 10 i q^{37} - 6 q^{39} + 2 q^{41} + 8 i q^{43} - 2 i q^{45} + q^{49} + 2 i q^{51} + 10 i q^{53} + 8 q^{55} - 4 q^{57} - 12 i q^{59} + 10 i q^{61} + q^{63} + 12 q^{65} + 8 i q^{67} + 4 i q^{69} + 12 q^{71} - 2 q^{73} - i q^{75} + 4 i q^{77} + q^{81} - 12 i q^{83} - 4 i q^{85} - 2 q^{87} - 6 q^{89} + 6 i q^{91} + 8 i q^{93} + 8 q^{95} + 2 q^{97} + 4 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} - 2 q^{9} + 4 q^{15} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 16 q^{31} - 8 q^{33} - 12 q^{39} + 4 q^{41} + 2 q^{49} + 16 q^{55} - 8 q^{57} + 2 q^{63} + 24 q^{65} + 24 q^{71} - 4 q^{73} + 2 q^{81} - 4 q^{87} - 12 q^{89} + 16 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times\).

\(n\) \(1793\) \(2815\) \(4609\) \(5125\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2689.1
1.00000i
1.00000i
0 1.00000i 0 2.00000i 0 −1.00000 0 −1.00000 0
2689.2 0 1.00000i 0 2.00000i 0 −1.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5376.2.c.m 2
4.b odd 2 1 5376.2.c.s 2
8.b even 2 1 inner 5376.2.c.m 2
8.d odd 2 1 5376.2.c.s 2
16.e even 4 1 672.2.a.b 1
16.e even 4 1 1344.2.a.r 1
16.f odd 4 1 672.2.a.f yes 1
16.f odd 4 1 1344.2.a.h 1
48.i odd 4 1 2016.2.a.l 1
48.i odd 4 1 4032.2.a.n 1
48.k even 4 1 2016.2.a.k 1
48.k even 4 1 4032.2.a.f 1
112.j even 4 1 4704.2.a.m 1
112.j even 4 1 9408.2.a.cb 1
112.l odd 4 1 4704.2.a.be 1
112.l odd 4 1 9408.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.b 1 16.e even 4 1
672.2.a.f yes 1 16.f odd 4 1
1344.2.a.h 1 16.f odd 4 1
1344.2.a.r 1 16.e even 4 1
2016.2.a.k 1 48.k even 4 1
2016.2.a.l 1 48.i odd 4 1
4032.2.a.f 1 48.k even 4 1
4032.2.a.n 1 48.i odd 4 1
4704.2.a.m 1 112.j even 4 1
4704.2.a.be 1 112.l odd 4 1
5376.2.c.m 2 1.a even 1 1 trivial
5376.2.c.m 2 8.b even 2 1 inner
5376.2.c.s 2 4.b odd 2 1
5376.2.c.s 2 8.d odd 2 1
9408.2.a.g 1 112.l odd 4 1
9408.2.a.cb 1 112.j even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5376, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} + 36 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display
\( T_{31} + 8 \) Copy content Toggle raw display
\( T_{47} \) Copy content Toggle raw display
\( T_{71} - 12 \) Copy content Toggle raw display
\( T_{79} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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