Properties

Label 5225.2.a.j
Level $5225$
Weight $2$
Character orbit 5225.a
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $5$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3} + \beta_1) q^{2} + (\beta_{4} + \beta_{2} - \beta_1 + 2) q^{3} + (\beta_{4} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} + \beta_1) q^{6} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + 3) q^{7} + (\beta_{4} + 2 \beta_{2}) q^{8} + (\beta_{4} - \beta_{3} + 3 \beta_{2} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{3} + \beta_1) q^{2} + (\beta_{4} + \beta_{2} - \beta_1 + 2) q^{3} + (\beta_{4} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} + \beta_1) q^{6} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + 3) q^{7} + (\beta_{4} + 2 \beta_{2}) q^{8} + (\beta_{4} - \beta_{3} + 3 \beta_{2} + \cdots + 3) q^{9}+ \cdots + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 3 q^{2} + 7 q^{3} + 5 q^{4} + 2 q^{6} + 11 q^{7} - 3 q^{8} + 8 q^{9} - 5 q^{11} + 7 q^{12} - q^{13} - 3 q^{16} + 3 q^{17} + 7 q^{18} + 5 q^{19} + 11 q^{21} - 3 q^{22} + 8 q^{23} + 9 q^{24} - 16 q^{26} + 10 q^{27} + 22 q^{28} + 11 q^{29} - 5 q^{31} + 2 q^{32} - 7 q^{33} + 4 q^{34} - 3 q^{36} + 9 q^{37} + 3 q^{38} - 8 q^{39} + 15 q^{41} - 11 q^{42} + 13 q^{43} - 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{48} + 20 q^{49} + 24 q^{51} - q^{52} + 5 q^{53} + 17 q^{54} + 7 q^{57} + 33 q^{58} - 17 q^{59} + 3 q^{61} - 14 q^{62} + 22 q^{63} - 17 q^{64} - 2 q^{66} + 28 q^{67} + 25 q^{68} - 2 q^{69} - 6 q^{71} + 26 q^{72} + 16 q^{73} - 21 q^{74} + 5 q^{76} - 11 q^{77} - 29 q^{78} + 3 q^{79} + q^{81} - 2 q^{82} + 33 q^{83} + 33 q^{84} + 10 q^{86} + 3 q^{88} - 16 q^{89} - 22 q^{91} + 19 q^{92} + 26 q^{93} - 10 q^{94} + 5 q^{96} + 14 q^{97} - 10 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) 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
0.284630
−1.68251
−0.830830
1.30972
1.91899
−2.22871 1.47889 2.96714 0 −3.29602 4.42518 −2.15546 −0.812880 0
1.2 −0.0881559 3.20362 −1.99223 0 −0.282418 1.95185 0.351939 7.26315 0
1.3 1.37279 1.23648 −0.115460 0 1.69742 −2.43232 −2.90407 −1.47112 0
1.4 1.54620 −1.51334 0.390736 0 −2.33992 4.16140 −2.48825 −0.709811 0
1.5 2.39788 2.59435 3.74982 0 6.22094 2.89389 4.19584 3.73066 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(1\)
\(19\) \(-1\)

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 5225.2.a.j 5
5.b even 2 1 1045.2.a.d 5
15.d odd 2 1 9405.2.a.v 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.d 5 5.b even 2 1
5225.2.a.j 5 1.a even 1 1 trivial
9405.2.a.v 5 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{5} - 3T_{2}^{4} - 3T_{2}^{3} + 15T_{2}^{2} - 10T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{5} - 11T_{7}^{4} + 33T_{7}^{3} + 22T_{7}^{2} - 231T_{7} + 253 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 3 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{5} - 7 T^{4} + \cdots + 23 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 11 T^{4} + \cdots + 253 \) Copy content Toggle raw display
$11$ \( (T + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + T^{4} + \cdots + 529 \) Copy content Toggle raw display
$17$ \( T^{5} - 3 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$19$ \( (T - 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} - 8 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{5} - 11 T^{4} + \cdots + 1441 \) Copy content Toggle raw display
$31$ \( T^{5} + 5 T^{4} + \cdots + 10649 \) Copy content Toggle raw display
$37$ \( T^{5} - 9 T^{4} + \cdots - 3917 \) Copy content Toggle raw display
$41$ \( T^{5} - 15 T^{4} + \cdots + 593 \) Copy content Toggle raw display
$43$ \( T^{5} - 13 T^{4} + \cdots - 28753 \) Copy content Toggle raw display
$47$ \( T^{5} - 20 T^{4} + \cdots + 989 \) Copy content Toggle raw display
$53$ \( T^{5} - 5 T^{4} + \cdots - 3917 \) Copy content Toggle raw display
$59$ \( T^{5} + 17 T^{4} + \cdots + 197 \) Copy content Toggle raw display
$61$ \( T^{5} - 3 T^{4} + \cdots + 241 \) Copy content Toggle raw display
$67$ \( T^{5} - 28 T^{4} + \cdots + 11881 \) Copy content Toggle raw display
$71$ \( T^{5} + 6 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$73$ \( T^{5} - 16 T^{4} + \cdots - 23 \) Copy content Toggle raw display
$79$ \( T^{5} - 3 T^{4} + \cdots - 15203 \) Copy content Toggle raw display
$83$ \( T^{5} - 33 T^{4} + \cdots + 737 \) Copy content Toggle raw display
$89$ \( T^{5} + 16 T^{4} + \cdots + 9791 \) Copy content Toggle raw display
$97$ \( T^{5} - 14 T^{4} + \cdots - 3323 \) Copy content Toggle raw display
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