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Properties

Label 9522.2.a.cf
Level $9522$
Weight $2$
Character orbit 9522.a
Self dual yes
Analytic conductor $76.034$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9522,2,Mod(1,9522)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9522, 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("9522.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9522.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: \(76.0335528047\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.546984493056.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 20x^{6} + 129x^{4} - 320x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (\beta_{4} + \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + (\beta_{4} + \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{8} + (\beta_{4} + \beta_1) q^{10} + (\beta_{5} + \beta_{4}) q^{11} + (\beta_{6} + 2) q^{13} + ( - \beta_{2} + \beta_1) q^{14} + q^{16} + ( - \beta_{5} - 2 \beta_{4} + \cdots + \beta_1) q^{17}+ \cdots + ( - \beta_{7} + \beta_{6} - \beta_{3} - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} + 12 q^{26} - 12 q^{29} - 12 q^{31} + 8 q^{32} + 36 q^{35} + 24 q^{41} + 48 q^{47} - 16 q^{49} + 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58} + 36 q^{59} - 12 q^{62} + 8 q^{64} + 36 q^{70} + 24 q^{71} + 12 q^{73} + 12 q^{77} + 24 q^{82} + 12 q^{85} + 48 q^{94} + 72 q^{95} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 20x^{6} + 129x^{4} - 320x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 20\nu^{5} + 129\nu^{3} - 256\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 20\nu^{4} + 113\nu^{2} - 160 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 84\nu^{5} - 389\nu^{3} + 496\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 100\nu^{5} - 581\nu^{3} + 896\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 16\nu^{4} + 65\nu^{2} - 64 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} - 16\nu^{4} + 69\nu^{2} - 84 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 5\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{7} - 11\beta_{6} - 4\beta_{3} + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16\beta_{5} - 4\beta_{4} + 60\beta_{2} + 47\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 127\beta_{7} - 107\beta_{6} - 64\beta_{3} + 315 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 191\beta_{5} - 80\beta_{4} + 619\beta_{2} + 422\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.27550
−1.25511
−3.18696
−1.75786
1.75786
3.18696
1.25511
2.27550
1.00000 0 1.00000 −3.68971 0 −1.75786 1.00000 0 −3.68971
1.2 1.00000 0 1.00000 −2.66933 0 −3.18696 1.00000 0 −2.66933
1.3 1.00000 0 1.00000 −1.77275 0 −1.25511 1.00000 0 −1.77275
1.4 1.00000 0 1.00000 −0.343645 0 −2.27550 1.00000 0 −0.343645
1.5 1.00000 0 1.00000 0.343645 0 2.27550 1.00000 0 0.343645
1.6 1.00000 0 1.00000 1.77275 0 1.25511 1.00000 0 1.77275
1.7 1.00000 0 1.00000 2.66933 0 3.18696 1.00000 0 2.66933
1.8 1.00000 0 1.00000 3.68971 0 1.75786 1.00000 0 3.68971
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9522.2.a.cf yes 8
3.b odd 2 1 9522.2.a.cd ✓ 8
23.b odd 2 1 inner 9522.2.a.cf yes 8
69.c even 2 1 9522.2.a.cd ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9522.2.a.cd ✓ 8 3.b odd 2 1
9522.2.a.cd ✓ 8 69.c even 2 1
9522.2.a.cf yes 8 1.a even 1 1 trivial
9522.2.a.cf yes 8 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9522))\):

\( T_{5}^{8} - 24T_{5}^{6} + 165T_{5}^{4} - 324T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{8} - 20T_{7}^{6} + 129T_{7}^{4} - 320T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{8} - 60T_{11}^{6} + 1161T_{11}^{4} - 8640T_{11}^{2} + 20736 \) Copy content Toggle raw display
\( T_{29}^{4} + 6T_{29}^{3} - 6T_{29}^{2} - 42T_{29} + 33 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 24 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{8} - 20 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{8} - 60 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} - 3 T^{2} + \cdots - 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 84 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( T^{8} - 80 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} - 6 T^{2} + \cdots + 33)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 260 T^{6} + \cdots + 4096576 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + \cdots - 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 272 T^{6} + \cdots + 1982464 \) Copy content Toggle raw display
$47$ \( (T^{4} - 24 T^{3} + \cdots - 8832)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 384 T^{6} + \cdots + 27910089 \) Copy content Toggle raw display
$59$ \( (T^{4} - 18 T^{3} + \cdots + 96)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 260 T^{6} + \cdots + 9048064 \) Copy content Toggle raw display
$67$ \( T^{8} - 176 T^{6} + \cdots + 495616 \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + \cdots - 768)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + \cdots + 8781)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 356 T^{6} + \cdots + 40347904 \) Copy content Toggle raw display
$83$ \( T^{8} - 396 T^{6} + \cdots + 5760000 \) Copy content Toggle raw display
$89$ \( T^{8} - 564 T^{6} + \cdots + 283450896 \) Copy content Toggle raw display
$97$ \( T^{8} - 440 T^{6} + \cdots + 1329409 \) Copy content Toggle raw display
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