Properties

Label 5225.2.a.l
Level $5225$
Weight $2$
Character orbit 5225.a
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $6$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7281497.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{3} q^{3} + (\beta_{5} + \beta_{4}) q^{4} + \beta_{2} q^{6} + (\beta_{3} + \beta_1 - 1) q^{7} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{8} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{3} q^{3} + (\beta_{5} + \beta_{4}) q^{4} + \beta_{2} q^{6} + (\beta_{3} + \beta_1 - 1) q^{7} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{8} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{9} - q^{11} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{12}+ \cdots + (\beta_{5} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9} - 6 q^{11} - q^{12} + 5 q^{13} - 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{18} - 6 q^{19} - 21 q^{21} - 2 q^{22} - 4 q^{23} - q^{24} - 14 q^{26} + 16 q^{27} - 10 q^{28} - 9 q^{29} - 21 q^{31} + q^{32} - q^{33} - 28 q^{36} + 3 q^{37} - 2 q^{38} + 20 q^{39} - 23 q^{41} - q^{42} - 7 q^{43} - 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} - 16 q^{51} - 13 q^{52} + 17 q^{53} + q^{54} - 2 q^{56} - q^{57} - 23 q^{58} - 29 q^{59} + 17 q^{61} - 2 q^{62} - 6 q^{63} - 18 q^{64} - 8 q^{67} + q^{68} - 38 q^{69} - 12 q^{71} - 13 q^{72} - 2 q^{73} - 37 q^{74} - 4 q^{76} + 5 q^{77} - q^{78} + 3 q^{79} - 2 q^{81} - 24 q^{82} + 11 q^{83} - 3 q^{84} - 12 q^{86} + 12 q^{87} - 12 q^{88} - 22 q^{89} - 18 q^{91} + 15 q^{92} - 18 q^{93} + 22 q^{94} - 17 q^{96} + 2 q^{97} + q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 7\nu^{2} + 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 3\nu^{4} + 3\nu^{3} - 11\nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - \beta_{4} + 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 6\beta_{4} + 9\beta_{3} + 10\beta_{2} + 22\beta _1 + 15 \) 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
−0.326248
2.59744
1.77015
−0.748369
−1.79049
0.497517
−1.73890 0.901323 1.02379 0 −1.56731 −2.22757 1.69754 −2.18762 0
1.2 −1.21244 −1.77266 −0.529980 0 2.14925 3.37010 3.06746 0.142317 0
1.3 −0.205229 3.10246 −1.95788 0 −0.636714 −2.33231 0.812271 6.62527 0
1.4 0.412130 −1.67805 −1.83015 0 −0.691575 −0.0703171 −1.57852 −0.184142 0
1.5 2.23198 1.34246 2.98176 0 2.99635 −4.13295 2.19127 −1.19780 0
1.6 2.51246 −0.895533 4.31247 0 −2.24999 0.393051 5.80998 −2.19802 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.l 6
5.b even 2 1 1045.2.a.f 6
15.d odd 2 1 9405.2.a.z 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.f 6 5.b even 2 1
5225.2.a.l 6 1.a even 1 1 trivial
9405.2.a.z 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\):

\( T_{2}^{6} - 2T_{2}^{5} - 6T_{2}^{4} + 8T_{2}^{3} + 11T_{2}^{2} - 3T_{2} - 1 \) Copy content Toggle raw display
\( T_{7}^{6} + 5T_{7}^{5} - 7T_{7}^{4} - 58T_{7}^{3} - 53T_{7}^{2} + 25T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + \cdots - 10 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 5 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 5 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{6} + T^{5} + \cdots - 1360 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 4 T^{5} + \cdots - 8912 \) Copy content Toggle raw display
$29$ \( T^{6} + 9 T^{5} + \cdots + 3074 \) Copy content Toggle raw display
$31$ \( T^{6} + 21 T^{5} + \cdots - 20 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} + \cdots - 592 \) Copy content Toggle raw display
$41$ \( T^{6} + 23 T^{5} + \cdots - 4210 \) Copy content Toggle raw display
$43$ \( T^{6} + 7 T^{5} + \cdots - 326 \) Copy content Toggle raw display
$47$ \( T^{6} - 18 T^{5} + \cdots - 620 \) Copy content Toggle raw display
$53$ \( T^{6} - 17 T^{5} + \cdots - 4084 \) Copy content Toggle raw display
$59$ \( T^{6} + 29 T^{5} + \cdots - 54968 \) Copy content Toggle raw display
$61$ \( T^{6} - 17 T^{5} + \cdots + 2294 \) Copy content Toggle raw display
$67$ \( T^{6} + 8 T^{5} + \cdots - 73006 \) Copy content Toggle raw display
$71$ \( T^{6} + 12 T^{5} + \cdots - 5912 \) Copy content Toggle raw display
$73$ \( T^{6} + 2 T^{5} + \cdots - 4000 \) Copy content Toggle raw display
$79$ \( T^{6} - 3 T^{5} + \cdots + 28264 \) Copy content Toggle raw display
$83$ \( T^{6} - 11 T^{5} + \cdots - 82582 \) Copy content Toggle raw display
$89$ \( T^{6} + 22 T^{5} + \cdots + 3286 \) Copy content Toggle raw display
$97$ \( T^{6} - 2 T^{5} + \cdots - 967268 \) Copy content Toggle raw display
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