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Properties

Label 525.2.m.c
Level $525$
Weight $2$
Character orbit 525.m
Analytic conductor $4.192$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(118,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.m (of order \(4\), degree \(2\), 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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{11} + 24 q^{16} + 12 q^{21} - 24 q^{36} - 120 q^{46} + 48 q^{51} - 96 q^{56} - 96 q^{71} - 24 q^{81} - 120 q^{86} + 108 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1 −1.68361 + 1.68361i −0.707107 + 0.707107i − 3.66908i 0 − 2.38098i 0.901937 + 2.48727i 2.81008 + 2.81008i − 1.00000i 0
118.2 −1.68361 + 1.68361i 0.707107 − 0.707107i − 3.66908i 0 2.38098i −2.48727 − 0.901937i 2.81008 + 2.81008i − 1.00000i 0
118.3 −1.11266 + 1.11266i −0.707107 + 0.707107i − 0.476024i 0 − 1.57354i −1.62049 − 2.09141i −1.69567 − 1.69567i − 1.00000i 0
118.4 −1.11266 + 1.11266i 0.707107 − 0.707107i − 0.476024i 0 1.57354i 2.09141 + 1.62049i −1.69567 − 1.69567i − 1.00000i 0
118.5 −0.653796 + 0.653796i −0.707107 + 0.707107i 1.14510i 0 − 0.924607i 0.741188 − 2.53981i −2.05625 − 2.05625i − 1.00000i 0
118.6 −0.653796 + 0.653796i 0.707107 − 0.707107i 1.14510i 0 0.924607i 2.53981 − 0.741188i −2.05625 − 2.05625i − 1.00000i 0
118.7 0.653796 − 0.653796i −0.707107 + 0.707107i 1.14510i 0 0.924607i −2.53981 + 0.741188i 2.05625 + 2.05625i − 1.00000i 0
118.8 0.653796 − 0.653796i 0.707107 − 0.707107i 1.14510i 0 − 0.924607i −0.741188 + 2.53981i 2.05625 + 2.05625i − 1.00000i 0
118.9 1.11266 − 1.11266i −0.707107 + 0.707107i − 0.476024i 0 1.57354i −2.09141 − 1.62049i 1.69567 + 1.69567i − 1.00000i 0
118.10 1.11266 − 1.11266i 0.707107 − 0.707107i − 0.476024i 0 − 1.57354i 1.62049 + 2.09141i 1.69567 + 1.69567i − 1.00000i 0
118.11 1.68361 − 1.68361i −0.707107 + 0.707107i − 3.66908i 0 2.38098i 2.48727 + 0.901937i −2.81008 − 2.81008i − 1.00000i 0
118.12 1.68361 − 1.68361i 0.707107 − 0.707107i − 3.66908i 0 − 2.38098i −0.901937 − 2.48727i −2.81008 − 2.81008i − 1.00000i 0
307.1 −1.68361 − 1.68361i −0.707107 − 0.707107i 3.66908i 0 2.38098i 0.901937 − 2.48727i 2.81008 − 2.81008i 1.00000i 0
307.2 −1.68361 − 1.68361i 0.707107 + 0.707107i 3.66908i 0 − 2.38098i −2.48727 + 0.901937i 2.81008 − 2.81008i 1.00000i 0
307.3 −1.11266 − 1.11266i −0.707107 − 0.707107i 0.476024i 0 1.57354i −1.62049 + 2.09141i −1.69567 + 1.69567i 1.00000i 0
307.4 −1.11266 − 1.11266i 0.707107 + 0.707107i 0.476024i 0 − 1.57354i 2.09141 − 1.62049i −1.69567 + 1.69567i 1.00000i 0
307.5 −0.653796 − 0.653796i −0.707107 − 0.707107i − 1.14510i 0 0.924607i 0.741188 + 2.53981i −2.05625 + 2.05625i 1.00000i 0
307.6 −0.653796 − 0.653796i 0.707107 + 0.707107i − 1.14510i 0 − 0.924607i 2.53981 + 0.741188i −2.05625 + 2.05625i 1.00000i 0
307.7 0.653796 + 0.653796i −0.707107 − 0.707107i − 1.14510i 0 − 0.924607i −2.53981 − 0.741188i 2.05625 − 2.05625i 1.00000i 0
307.8 0.653796 + 0.653796i 0.707107 + 0.707107i − 1.14510i 0 0.924607i −0.741188 − 2.53981i 2.05625 − 2.05625i 1.00000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.m.c ✓ 24
5.b even 2 1 inner 525.2.m.c ✓ 24
5.c odd 4 2 inner 525.2.m.c ✓ 24
7.b odd 2 1 inner 525.2.m.c ✓ 24
35.c odd 2 1 inner 525.2.m.c ✓ 24
35.f even 4 2 inner 525.2.m.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.m.c ✓ 24 1.a even 1 1 trivial
525.2.m.c ✓ 24 5.b even 2 1 inner
525.2.m.c ✓ 24 5.c odd 4 2 inner
525.2.m.c ✓ 24 7.b odd 2 1 inner
525.2.m.c ✓ 24 35.c odd 2 1 inner
525.2.m.c ✓ 24 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 39T_{2}^{8} + 225T_{2}^{4} + 144 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display