Properties

Label 5225.2.a.n
Level $5225$
Weight $2$
Character orbit 5225.a
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + \beta_{2} + 2) q^{4} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + \beta_{2} + 2) q^{4} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - 1) q^{6}+ \cdots + (\beta_{6} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} - 2 q^{3} + 15 q^{4} - 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} - 2 q^{3} + 15 q^{4} - 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9} - 7 q^{11} + 16 q^{12} + 4 q^{13} + 6 q^{14} + 27 q^{16} - 2 q^{17} - 9 q^{18} + 7 q^{19} - 14 q^{21} - q^{22} - 10 q^{23} - 2 q^{24} - 8 q^{26} + 4 q^{27} - 26 q^{28} - 18 q^{29} + 24 q^{31} + 49 q^{32} + 2 q^{33} - 6 q^{34} + 29 q^{36} + q^{38} + 24 q^{39} - 12 q^{41} + 44 q^{42} - 2 q^{43} - 15 q^{44} - 4 q^{46} - 8 q^{47} + 72 q^{48} + 17 q^{49} - 24 q^{51} + 60 q^{52} - 2 q^{53} - 52 q^{54} + 26 q^{56} - 2 q^{57} + 8 q^{58} - 10 q^{59} + 14 q^{61} - 14 q^{62} + 55 q^{64} + 2 q^{66} - 8 q^{67} + 18 q^{68} - 6 q^{69} + 10 q^{71} - 53 q^{72} + 6 q^{73} + 26 q^{74} + 15 q^{76} + 10 q^{77} - 22 q^{78} + 52 q^{79} - q^{81} - 24 q^{82} + 10 q^{83} - 6 q^{84} + 8 q^{86} - 6 q^{87} - 9 q^{88} + 12 q^{91} - 2 q^{93} + 24 q^{94} + 6 q^{96} + 24 q^{97} - 19 q^{98} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 7\nu^{2} - 2\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 9\nu^{2} + 2\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 9\nu^{3} + 14\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 10\nu^{4} + 23\nu^{2} - 4\nu - 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 12\nu^{4} + 4\nu^{3} - 41\nu^{2} - 20\nu + 34 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{3} + 9\beta_{2} + 2\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{6} + 9\beta_{5} + 2\beta_{4} + 9\beta_{3} + 31\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{5} + 47\beta_{3} + 67\beta_{2} + 24\beta _1 + 158 \) 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
−2.55401
−2.03821
−1.45416
0.456669
1.19313
2.61330
2.78328
−2.55401 2.99825 4.52299 0 −7.65758 −4.42321 −6.44376 5.98952 0
1.2 −2.03821 −1.87275 2.15429 0 3.81704 −1.92338 −0.314472 0.507178 0
1.3 −1.45416 −1.71116 0.114592 0 2.48830 2.00933 2.74169 −0.0719365 0
1.4 0.456669 0.835165 −1.79145 0 0.381394 −4.69915 −1.73144 −2.30250 0
1.5 1.19313 −3.16232 −0.576442 0 −3.77306 1.30958 −3.07403 7.00029 0
1.6 2.61330 −1.19599 4.82936 0 −3.12549 −3.61829 7.39397 −1.56960 0
1.7 2.78328 2.10880 5.74667 0 5.86939 1.34513 10.4280 1.44705 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.n 7
5.b even 2 1 209.2.a.d 7
15.d odd 2 1 1881.2.a.p 7
20.d odd 2 1 3344.2.a.ba 7
55.d odd 2 1 2299.2.a.q 7
95.d odd 2 1 3971.2.a.i 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.d 7 5.b even 2 1
1881.2.a.p 7 15.d odd 2 1
2299.2.a.q 7 55.d odd 2 1
3344.2.a.ba 7 20.d odd 2 1
3971.2.a.i 7 95.d odd 2 1
5225.2.a.n 7 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{7} - T_{2}^{6} - 14T_{2}^{5} + 10T_{2}^{4} + 59T_{2}^{3} - 27T_{2}^{2} - 66T_{2} + 30 \) Copy content Toggle raw display
\( T_{7}^{7} + 10T_{7}^{6} + 17T_{7}^{5} - 86T_{7}^{4} - 185T_{7}^{3} + 316T_{7}^{2} + 394T_{7} - 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} + \cdots + 30 \) Copy content Toggle raw display
$3$ \( T^{7} + 2 T^{6} + \cdots - 64 \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + 10 T^{6} + \cdots - 512 \) Copy content Toggle raw display
$11$ \( (T + 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} - 4 T^{6} + \cdots + 5716 \) Copy content Toggle raw display
$17$ \( T^{7} + 2 T^{6} + \cdots + 17088 \) Copy content Toggle raw display
$19$ \( (T - 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} + 10 T^{6} + \cdots - 1920 \) Copy content Toggle raw display
$29$ \( T^{7} + 18 T^{6} + \cdots - 276 \) Copy content Toggle raw display
$31$ \( T^{7} - 24 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$37$ \( T^{7} - 121 T^{5} + \cdots + 8992 \) Copy content Toggle raw display
$41$ \( T^{7} + 12 T^{6} + \cdots - 1824 \) Copy content Toggle raw display
$43$ \( T^{7} + 2 T^{6} + \cdots - 4976 \) Copy content Toggle raw display
$47$ \( T^{7} + 8 T^{6} + \cdots - 79872 \) Copy content Toggle raw display
$53$ \( T^{7} + 2 T^{6} + \cdots - 768 \) Copy content Toggle raw display
$59$ \( T^{7} + 10 T^{6} + \cdots - 6552192 \) Copy content Toggle raw display
$61$ \( T^{7} - 14 T^{6} + \cdots - 36544 \) Copy content Toggle raw display
$67$ \( T^{7} + 8 T^{6} + \cdots - 13544 \) Copy content Toggle raw display
$71$ \( T^{7} - 10 T^{6} + \cdots + 39756 \) Copy content Toggle raw display
$73$ \( T^{7} - 6 T^{6} + \cdots - 67328 \) Copy content Toggle raw display
$79$ \( T^{7} - 52 T^{6} + \cdots - 203264 \) Copy content Toggle raw display
$83$ \( T^{7} - 10 T^{6} + \cdots - 576936 \) Copy content Toggle raw display
$89$ \( T^{7} - 401 T^{5} + \cdots - 8199552 \) Copy content Toggle raw display
$97$ \( T^{7} - 24 T^{6} + \cdots + 17393056 \) Copy content Toggle raw display
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