Properties

Label 2475.4.a.j
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
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] = [2475,4,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(146.029727264\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 8 q^{4} + 21 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 8 q^{4} + 21 q^{7} - 11 q^{11} - 68 q^{13} + 84 q^{14} - 64 q^{16} - 21 q^{17} + 125 q^{19} - 44 q^{22} - 137 q^{23} - 272 q^{26} + 168 q^{28} + 150 q^{29} + 292 q^{31} - 256 q^{32} - 84 q^{34} - 349 q^{37} + 500 q^{38} - 497 q^{41} - 208 q^{43} - 88 q^{44} - 548 q^{46} + 369 q^{47} + 98 q^{49} - 544 q^{52} - 542 q^{53} + 600 q^{58} - 235 q^{59} + 482 q^{61} + 1168 q^{62} - 512 q^{64} - 734 q^{67} - 168 q^{68} - 587 q^{71} - 518 q^{73} - 1396 q^{74} + 1000 q^{76} - 231 q^{77} - 1045 q^{79} - 1988 q^{82} + 608 q^{83} - 832 q^{86} + 770 q^{89} - 1428 q^{91} - 1096 q^{92} + 1476 q^{94} + 1541 q^{97} + 392 q^{98}+O(q^{100}) \) 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
4.00000 0 8.00000 0 0 21.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

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 2475.4.a.j 1
3.b odd 2 1 825.4.a.b 1
5.b even 2 1 2475.4.a.c 1
15.d odd 2 1 825.4.a.h yes 1
15.e even 4 2 825.4.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.b 1 3.b odd 2 1
825.4.a.h yes 1 15.d odd 2 1
825.4.c.c 2 15.e even 4 2
2475.4.a.c 1 5.b even 2 1
2475.4.a.j 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2} - 4 \) Copy content Toggle raw display
\( T_{7} - 21 \) Copy content Toggle raw display
\( T_{29} - 150 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 21 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 68 \) Copy content Toggle raw display
$17$ \( T + 21 \) Copy content Toggle raw display
$19$ \( T - 125 \) Copy content Toggle raw display
$23$ \( T + 137 \) Copy content Toggle raw display
$29$ \( T - 150 \) Copy content Toggle raw display
$31$ \( T - 292 \) Copy content Toggle raw display
$37$ \( T + 349 \) Copy content Toggle raw display
$41$ \( T + 497 \) Copy content Toggle raw display
$43$ \( T + 208 \) Copy content Toggle raw display
$47$ \( T - 369 \) Copy content Toggle raw display
$53$ \( T + 542 \) Copy content Toggle raw display
$59$ \( T + 235 \) Copy content Toggle raw display
$61$ \( T - 482 \) Copy content Toggle raw display
$67$ \( T + 734 \) Copy content Toggle raw display
$71$ \( T + 587 \) Copy content Toggle raw display
$73$ \( T + 518 \) Copy content Toggle raw display
$79$ \( T + 1045 \) Copy content Toggle raw display
$83$ \( T - 608 \) Copy content Toggle raw display
$89$ \( T - 770 \) Copy content Toggle raw display
$97$ \( T - 1541 \) Copy content Toggle raw display
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