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Properties

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

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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 425)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{9} - 16 q^{13} + 24 q^{16} - 40 q^{18} - 16 q^{21} - 16 q^{26} - 56 q^{32} - 48 q^{33} + 24 q^{36} - 48 q^{38} - 32 q^{43} - 88 q^{47} + 16 q^{49} - 48 q^{52} - 48 q^{53} - 8 q^{59} + 72 q^{64} + 32 q^{66} - 40 q^{67} - 48 q^{69} - 120 q^{72} + 32 q^{76} - 120 q^{77} - 24 q^{81} - 104 q^{83} + 40 q^{84} - 16 q^{86} - 64 q^{87} + 16 q^{89} + 72 q^{93} + 112 q^{94} - 48 q^{98}+O(q^{100}) \) 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 −2.71078 −1.05855 5.34834 0 2.86951 −1.42176 −9.07661 −1.87946 0
1.2 −2.71078 1.05855 5.34834 0 −2.86951 1.42176 −9.07661 −1.87946 0
1.3 −2.58407 −3.08178 4.67741 0 7.96353 2.81582 −6.91861 6.49736 0
1.4 −2.58407 3.08178 4.67741 0 −7.96353 −2.81582 −6.91861 6.49736 0
1.5 −2.16291 −2.77489 2.67820 0 6.00186 −3.14227 −1.46688 4.70004 0
1.6 −2.16291 2.77489 2.67820 0 −6.00186 3.14227 −1.46688 4.70004 0
1.7 −1.38987 −0.110824 −0.0682683 0 0.154031 −1.71589 2.87462 −2.98772 0
1.8 −1.38987 0.110824 −0.0682683 0 −0.154031 1.71589 2.87462 −2.98772 0
1.9 −1.30287 −0.334695 −0.302521 0 0.436066 −3.90464 2.99989 −2.88798 0
1.10 −1.30287 0.334695 −0.302521 0 −0.436066 3.90464 2.99989 −2.88798 0
1.11 −0.903848 −2.88161 −1.18306 0 2.60454 −0.726991 2.87700 5.30368 0
1.12 −0.903848 2.88161 −1.18306 0 −2.60454 0.726991 2.87700 5.30368 0
1.13 −0.265267 −1.80920 −1.92963 0 0.479920 4.92737 1.04240 0.273187 0
1.14 −0.265267 1.80920 −1.92963 0 −0.479920 −4.92737 1.04240 0.273187 0
1.15 0.555780 −2.50339 −1.69111 0 −1.39134 4.03587 −2.05144 3.26698 0
1.16 0.555780 2.50339 −1.69111 0 1.39134 −4.03587 −2.05144 3.26698 0
1.17 1.15016 −2.05557 −0.677141 0 −2.36423 0.375463 −3.07913 1.22538 0
1.18 1.15016 2.05557 −0.677141 0 2.36423 −0.375463 −3.07913 1.22538 0
1.19 1.29691 −1.83794 −0.318036 0 −2.38363 −3.50681 −3.00627 0.378015 0
1.20 1.29691 1.83794 −0.318036 0 2.38363 3.50681 −3.00627 0.378015 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
17.b 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.bx 24
5.b even 2 1 7225.2.a.cb 24
17.b even 2 1 inner 7225.2.a.bx 24
17.e odd 16 2 425.2.m.c ✓ 24
85.c even 2 1 7225.2.a.cb 24
85.o even 16 2 425.2.n.e 24
85.p odd 16 2 425.2.m.d yes 24
85.r even 16 2 425.2.n.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.m.c ✓ 24 17.e odd 16 2
425.2.m.d yes 24 85.p odd 16 2
425.2.n.d 24 85.r even 16 2
425.2.n.e 24 85.o even 16 2
7225.2.a.bx 24 1.a even 1 1 trivial
7225.2.a.bx 24 17.b even 2 1 inner
7225.2.a.cb 24 5.b even 2 1
7225.2.a.cb 24 85.c even 2 1

Hecke kernels

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

\( T_{2}^{12} + 4 T_{2}^{11} - 10 T_{2}^{10} - 52 T_{2}^{9} + 21 T_{2}^{8} + 232 T_{2}^{7} + 44 T_{2}^{6} + \cdots - 25 \) Copy content Toggle raw display
\( T_{3}^{24} - 48 T_{3}^{22} + 996 T_{3}^{20} - 11720 T_{3}^{18} + 86219 T_{3}^{16} - 412068 T_{3}^{14} + \cdots + 529 \) Copy content Toggle raw display
\( T_{7}^{24} - 92 T_{7}^{22} + 3526 T_{7}^{20} - 73340 T_{7}^{18} + 902299 T_{7}^{16} - 6705908 T_{7}^{14} + \cdots + 97969 \) Copy content Toggle raw display
\( T_{11}^{24} - 136 T_{11}^{22} + 7732 T_{11}^{20} - 238912 T_{11}^{18} + 4359900 T_{11}^{16} + \cdots + 3844 \) Copy content Toggle raw display