Properties

Label 2475.4.a.i
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
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] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{2} + q^{4} - 7 q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + q^{4} - 7 q^{7} - 21 q^{8} - 11 q^{11} - 16 q^{13} - 21 q^{14} - 71 q^{16} - 21 q^{17} + 125 q^{19} - 33 q^{22} + 81 q^{23} - 48 q^{26} - 7 q^{28} - 186 q^{29} - 58 q^{31} - 45 q^{32} - 63 q^{34} - 253 q^{37} + 375 q^{38} - 63 q^{41} - 100 q^{43} - 11 q^{44} + 243 q^{46} + 219 q^{47} - 294 q^{49} - 16 q^{52} + 192 q^{53} + 147 q^{56} - 558 q^{58} - 249 q^{59} - 64 q^{61} - 174 q^{62} + 433 q^{64} + 272 q^{67} - 21 q^{68} + 645 q^{71} - 112 q^{73} - 759 q^{74} + 125 q^{76} + 77 q^{77} + 509 q^{79} - 189 q^{82} + 1254 q^{83} - 300 q^{86} + 231 q^{88} - 756 q^{89} + 112 q^{91} + 81 q^{92} + 657 q^{94} + 839 q^{97} - 882 q^{98}+O(q^{100}) \) 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
3.00000 0 1.00000 0 0 −7.00000 −21.0000 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.i 1
3.b odd 2 1 825.4.a.c 1
5.b even 2 1 2475.4.a.d 1
15.d odd 2 1 825.4.a.g yes 1
15.e even 4 2 825.4.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.c 1 3.b odd 2 1
825.4.a.g yes 1 15.d odd 2 1
825.4.c.d 2 15.e even 4 2
2475.4.a.d 1 5.b even 2 1
2475.4.a.i 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{7} + 7 \) Copy content Toggle raw display
\( T_{29} + 186 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 16 \) Copy content Toggle raw display
$17$ \( T + 21 \) Copy content Toggle raw display
$19$ \( T - 125 \) Copy content Toggle raw display
$23$ \( T - 81 \) Copy content Toggle raw display
$29$ \( T + 186 \) Copy content Toggle raw display
$31$ \( T + 58 \) Copy content Toggle raw display
$37$ \( T + 253 \) Copy content Toggle raw display
$41$ \( T + 63 \) Copy content Toggle raw display
$43$ \( T + 100 \) Copy content Toggle raw display
$47$ \( T - 219 \) Copy content Toggle raw display
$53$ \( T - 192 \) Copy content Toggle raw display
$59$ \( T + 249 \) Copy content Toggle raw display
$61$ \( T + 64 \) Copy content Toggle raw display
$67$ \( T - 272 \) Copy content Toggle raw display
$71$ \( T - 645 \) Copy content Toggle raw display
$73$ \( T + 112 \) Copy content Toggle raw display
$79$ \( T - 509 \) Copy content Toggle raw display
$83$ \( T - 1254 \) Copy content Toggle raw display
$89$ \( T + 756 \) Copy content Toggle raw display
$97$ \( T - 839 \) Copy content Toggle raw display
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