Properties

Label 4275.2.a.e
Level $4275$
Weight $2$
Character orbit 4275.a
Self dual yes
Analytic conductor $34.136$
Analytic rank $0$
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] = [4275,2,Mod(1,4275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4275, 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("4275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.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: \(34.1360468641\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} - 2 q^{7} + 3 q^{8} + 4 q^{11} + 2 q^{13} + 2 q^{14} - q^{16} + 4 q^{17} + q^{19} - 4 q^{22} - 6 q^{23} - 2 q^{26} + 2 q^{28} + 6 q^{29} - 4 q^{31} - 5 q^{32} - 4 q^{34} + 10 q^{37} - q^{38} + 10 q^{41} - 2 q^{43} - 4 q^{44} + 6 q^{46} - 6 q^{47} - 3 q^{49} - 2 q^{52} + 10 q^{53} - 6 q^{56} - 6 q^{58} + 2 q^{61} + 4 q^{62} + 7 q^{64} - 8 q^{67} - 4 q^{68} - 4 q^{71} - 4 q^{73} - 10 q^{74} - q^{76} - 8 q^{77} + 4 q^{79} - 10 q^{82} - 18 q^{83} + 2 q^{86} + 12 q^{88} + 2 q^{89} - 4 q^{91} + 6 q^{92} + 6 q^{94} - 6 q^{97} + 3 q^{98}+O(q^{100}) \) 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
−1.00000 0 −1.00000 0 0 −2.00000 3.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(19\) \(-1\)

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 4275.2.a.e 1
3.b odd 2 1 475.2.a.c 1
5.b even 2 1 4275.2.a.p 1
5.c odd 4 2 855.2.c.b 2
12.b even 2 1 7600.2.a.l 1
15.d odd 2 1 475.2.a.a 1
15.e even 4 2 95.2.b.a 2
57.d even 2 1 9025.2.a.c 1
60.h even 2 1 7600.2.a.i 1
60.l odd 4 2 1520.2.d.b 2
285.b even 2 1 9025.2.a.h 1
285.j odd 4 2 1805.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.a 2 15.e even 4 2
475.2.a.a 1 15.d odd 2 1
475.2.a.c 1 3.b odd 2 1
855.2.c.b 2 5.c odd 4 2
1520.2.d.b 2 60.l odd 4 2
1805.2.b.c 2 285.j odd 4 2
4275.2.a.e 1 1.a even 1 1 trivial
4275.2.a.p 1 5.b even 2 1
7600.2.a.i 1 60.h even 2 1
7600.2.a.l 1 12.b even 2 1
9025.2.a.c 1 57.d even 2 1
9025.2.a.h 1 285.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4275))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 4 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T + 2 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T - 10 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 8 \) Copy content Toggle raw display
$71$ \( T + 4 \) Copy content Toggle raw display
$73$ \( T + 4 \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T + 18 \) Copy content Toggle raw display
$89$ \( T - 2 \) Copy content Toggle raw display
$97$ \( T + 6 \) Copy content Toggle raw display
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