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Properties

Label 7225.2.a.by
Level $7225$
Weight $2$
Character orbit 7225.a
Self dual yes
Analytic conductor $57.692$
Analytic rank $1$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7225,2,Mod(1,7225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7225 = 5^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7225.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: \(57.6919154604\)
Analytic rank: \(1\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} - 8 q^{9} - 8 q^{16} - 16 q^{19} - 32 q^{21} - 48 q^{26} + 8 q^{36} - 56 q^{49} - 64 q^{59} - 104 q^{64} - 16 q^{66} - 32 q^{69} - 48 q^{76} - 88 q^{81} - 96 q^{84} - 112 q^{89} - 128 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.30578 −1.87767 3.31660 0 4.32949 1.54574 −3.03579 0.525655 0
1.2 −2.30578 1.87767 3.31660 0 −4.32949 −1.54574 −3.03579 0.525655 0
1.3 −2.10495 −1.80557 2.43082 0 3.80063 2.68338 −0.906851 0.260076 0
1.4 −2.10495 1.80557 2.43082 0 −3.80063 −2.68338 −0.906851 0.260076 0
1.5 −1.74231 −0.598698 1.03565 0 1.04312 1.03798 1.68020 −2.64156 0
1.6 −1.74231 0.598698 1.03565 0 −1.04312 −1.03798 1.68020 −2.64156 0
1.7 −0.951739 −2.47483 −1.09419 0 2.35540 0.533377 2.94486 3.12480 0
1.8 −0.951739 2.47483 −1.09419 0 −2.35540 −0.533377 2.94486 3.12480 0
1.9 −0.499161 −0.171281 −1.75084 0 0.0854968 3.87301 1.87227 −2.97066 0
1.10 −0.499161 0.171281 −1.75084 0 −0.0854968 −3.87301 1.87227 −2.97066 0
1.11 −0.248918 −1.64368 −1.93804 0 0.409143 −1.43109 0.980251 −0.298307 0
1.12 −0.248918 1.64368 −1.93804 0 −0.409143 1.43109 0.980251 −0.298307 0
1.13 0.248918 −1.64368 −1.93804 0 −0.409143 −1.43109 −0.980251 −0.298307 0
1.14 0.248918 1.64368 −1.93804 0 0.409143 1.43109 −0.980251 −0.298307 0
1.15 0.499161 −0.171281 −1.75084 0 −0.0854968 3.87301 −1.87227 −2.97066 0
1.16 0.499161 0.171281 −1.75084 0 0.0854968 −3.87301 −1.87227 −2.97066 0
1.17 0.951739 −2.47483 −1.09419 0 −2.35540 0.533377 −2.94486 3.12480 0
1.18 0.951739 2.47483 −1.09419 0 2.35540 −0.533377 −2.94486 3.12480 0
1.19 1.74231 −0.598698 1.03565 0 −1.04312 1.03798 −1.68020 −2.64156 0
1.20 1.74231 0.598698 1.03565 0 1.04312 −1.03798 −1.68020 −2.64156 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(17\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.b even 2 1 inner
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7225.2.a.by 24
5.b even 2 1 inner 7225.2.a.by 24
5.c odd 4 2 1445.2.b.i 24
17.b even 2 1 inner 7225.2.a.by 24
17.e odd 16 2 425.2.m.e 24
85.c even 2 1 inner 7225.2.a.by 24
85.g odd 4 2 1445.2.b.i 24
85.o even 16 2 85.2.m.a ✓ 24
85.p odd 16 2 425.2.m.e 24
85.r even 16 2 85.2.m.a ✓ 24
255.bc odd 16 2 765.2.bh.b 24
255.bj odd 16 2 765.2.bh.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.m.a ✓ 24 85.o even 16 2
85.2.m.a ✓ 24 85.r even 16 2
425.2.m.e 24 17.e odd 16 2
425.2.m.e 24 85.p odd 16 2
765.2.bh.b 24 255.bc odd 16 2
765.2.bh.b 24 255.bj odd 16 2
1445.2.b.i 24 5.c odd 4 2
1445.2.b.i 24 85.g odd 4 2
7225.2.a.by 24 1.a even 1 1 trivial
7225.2.a.by 24 5.b even 2 1 inner
7225.2.a.by 24 17.b even 2 1 inner
7225.2.a.by 24 85.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7225))\):

\( T_{2}^{12} - 14T_{2}^{10} + 69T_{2}^{8} - 140T_{2}^{6} + 103T_{2}^{4} - 22T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{12} - 16T_{3}^{10} + 94T_{3}^{8} - 248T_{3}^{6} + 274T_{3}^{4} - 76T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{12} - 28T_{7}^{10} + 248T_{7}^{8} - 884T_{7}^{6} + 1394T_{7}^{4} - 900T_{7}^{2} + 162 \) Copy content Toggle raw display
\( T_{11}^{12} - 60T_{11}^{10} + 1270T_{11}^{8} - 11252T_{11}^{6} + 38776T_{11}^{4} - 27936T_{11}^{2} + 162 \) Copy content Toggle raw display