Properties

Label 9522.2.a.bz
Level $9522$
Weight $2$
Character orbit 9522.a
Self dual yes
Analytic conductor $76.034$
Analytic rank $1$
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] = [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: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 46)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( - \beta_{4} + \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + ( - \beta_{4} + \beta_1) q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{7} + q^{8} + ( - \beta_{4} + \beta_1) q^{10} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{11} + (2 \beta_{4} + 2 \beta_{3} + \beta_1) q^{13} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{14} + q^{16} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{17} + (\beta_{4} - 2 \beta_{2} - \beta_1 - 2) q^{19} + ( - \beta_{4} + \beta_1) q^{20} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{22} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{25} + (2 \beta_{4} + 2 \beta_{3} + \beta_1) q^{26} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{28} + (2 \beta_{4} - \beta_{2} + \beta_1 - 2) q^{29} + (4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 3) q^{31} + q^{32} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{34} + (2 \beta_{4} - \beta_{2} - \beta_1 + 1) q^{35} + (3 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{37} + (\beta_{4} - 2 \beta_{2} - \beta_1 - 2) q^{38} + ( - \beta_{4} + \beta_1) q^{40} + ( - 3 \beta_{4} - \beta_{2} - \beta_1) q^{41} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 6) q^{43} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{44} + ( - 4 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 1) q^{47} + (\beta_{4} + 3 \beta_{3} - 6 \beta_{2} - \beta_1 + 1) q^{49} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{50} + (2 \beta_{4} + 2 \beta_{3} + \beta_1) q^{52} + (3 \beta_1 - 1) q^{53} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{55} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{56} + (2 \beta_{4} - \beta_{2} + \beta_1 - 2) q^{58} + (4 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 7) q^{59} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - 1) q^{61} + (4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 3) q^{62} + q^{64} + (5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 1) q^{65} + ( - 4 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 9) q^{67} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{68} + (2 \beta_{4} - \beta_{2} - \beta_1 + 1) q^{70} + (\beta_{3} - 4 \beta_{2} - 1) q^{71} + ( - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{73} + (3 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{74} + (\beta_{4} - 2 \beta_{2} - \beta_1 - 2) q^{76} + (4 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} - 2) q^{77} + ( - 7 \beta_{4} + \beta_{3} - 5 \beta_{2} + \beta_1 - 6) q^{79} + ( - \beta_{4} + \beta_1) q^{80} + ( - 3 \beta_{4} - \beta_{2} - \beta_1) q^{82} + (8 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 6) q^{83} + ( - \beta_{4} - 2 \beta_{2} - 2) q^{85} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 6) q^{86} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{88} + (4 \beta_{4} + 3 \beta_{2} - 7 \beta_1) q^{89} + ( - 3 \beta_{4} - 2 \beta_{3} + 3 \beta_1 - 6) q^{91} + ( - 4 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 1) q^{94} + (2 \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1 - 4) q^{95} + ( - 5 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 12) q^{97} + (\beta_{4} + 3 \beta_{3} - 6 \beta_{2} - \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} + 5 q^{4} + 2 q^{5} - 9 q^{7} + 5 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} + 5 q^{4} + 2 q^{5} - 9 q^{7} + 5 q^{8} + 2 q^{10} + q^{11} + q^{13} - 9 q^{14} + 5 q^{16} - q^{17} - 10 q^{19} + 2 q^{20} + q^{22} - 11 q^{25} + q^{26} - 9 q^{28} - 10 q^{29} + 5 q^{32} - q^{34} + 3 q^{35} - 20 q^{37} - 10 q^{38} + 2 q^{40} + 3 q^{41} - 27 q^{43} + q^{44} + 11 q^{47} + 12 q^{49} - 11 q^{50} + q^{52} - 2 q^{53} - 4 q^{55} - 9 q^{56} - 10 q^{58} + 17 q^{59} - 7 q^{61} + 5 q^{64} - 15 q^{65} - 28 q^{67} - q^{68} + 3 q^{70} + 8 q^{73} - 20 q^{74} - 10 q^{76} - 26 q^{77} - 16 q^{79} + 2 q^{80} + 3 q^{82} + 18 q^{83} - 7 q^{85} - 27 q^{86} + q^{88} - 14 q^{89} - 26 q^{91} + 11 q^{94} - 26 q^{95} - 47 q^{97} + 12 q^{98}+O(q^{100}) \) 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
−0.830830
−1.68251
1.91899
1.30972
1.00000 0 1.00000 −1.39788 0 −4.29177 1.00000 0 −1.39788
1.2 1.00000 0 1.00000 −0.546200 0 −4.70760 1.00000 0 −0.546200
1.3 1.00000 0 1.00000 −0.372786 0 2.05954 1.00000 0 −0.372786
1.4 1.00000 0 1.00000 1.08816 0 −0.863693 1.00000 0 1.08816
1.5 1.00000 0 1.00000 3.22871 0 −1.19647 1.00000 0 3.22871
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

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 9522.2.a.bz 5
3.b odd 2 1 1058.2.a.j 5
12.b even 2 1 8464.2.a.bu 5
23.b odd 2 1 9522.2.a.bw 5
23.c even 11 2 414.2.i.c 10
69.c even 2 1 1058.2.a.k 5
69.h odd 22 2 46.2.c.b 10
276.h odd 2 1 8464.2.a.bv 5
276.o even 22 2 368.2.m.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.c.b 10 69.h odd 22 2
368.2.m.a 10 276.o even 22 2
414.2.i.c 10 23.c even 11 2
1058.2.a.j 5 3.b odd 2 1
1058.2.a.k 5 69.c even 2 1
8464.2.a.bu 5 12.b even 2 1
8464.2.a.bv 5 276.h odd 2 1
9522.2.a.bw 5 23.b odd 2 1
9522.2.a.bz 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9522))\):

\( T_{5}^{5} - 2T_{5}^{4} - 5T_{5}^{3} + 2T_{5}^{2} + 4T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{5} + 9T_{7}^{4} + 17T_{7}^{3} - 31T_{7}^{2} - 84T_{7} - 43 \) Copy content Toggle raw display
\( T_{11}^{5} - T_{11}^{4} - 15T_{11}^{3} - 19T_{11}^{2} - 8T_{11} - 1 \) Copy content Toggle raw display
\( T_{29}^{5} + 10T_{29}^{4} + 7T_{29}^{3} - 129T_{29}^{2} - 239T_{29} - 23 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 2 T^{4} - 5 T^{3} + 2 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{5} + 9 T^{4} + 17 T^{3} - 31 T^{2} + \cdots - 43 \) Copy content Toggle raw display
$11$ \( T^{5} - T^{4} - 15 T^{3} - 19 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{5} - T^{4} - 48 T^{3} + 47 T^{2} + \cdots - 661 \) Copy content Toggle raw display
$17$ \( T^{5} + T^{4} - 37 T^{3} - 58 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{5} + 10 T^{4} + 7 T^{3} - 140 T^{2} + \cdots + 373 \) Copy content Toggle raw display
$23$ \( T^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 10 T^{4} + 7 T^{3} - 129 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$31$ \( T^{5} - 66 T^{3} - 33 T^{2} + \cdots + 1441 \) Copy content Toggle raw display
$37$ \( T^{5} + 20 T^{4} + 105 T^{3} + \cdots - 3827 \) Copy content Toggle raw display
$41$ \( T^{5} - 3 T^{4} - 47 T^{3} + \cdots - 1409 \) Copy content Toggle raw display
$43$ \( T^{5} + 27 T^{4} + 219 T^{3} + \cdots - 7303 \) Copy content Toggle raw display
$47$ \( T^{5} - 11 T^{4} - 66 T^{3} + \cdots - 3883 \) Copy content Toggle raw display
$53$ \( T^{5} + 2 T^{4} - 38 T^{3} - 35 T^{2} + \cdots + 43 \) Copy content Toggle raw display
$59$ \( T^{5} - 17 T^{4} - 34 T^{3} + \cdots - 2881 \) Copy content Toggle raw display
$61$ \( T^{5} + 7 T^{4} - 9 T^{3} - 72 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$67$ \( T^{5} + 28 T^{4} + 186 T^{3} + \cdots - 397 \) Copy content Toggle raw display
$71$ \( T^{5} - 66 T^{3} - 33 T^{2} + \cdots + 1441 \) Copy content Toggle raw display
$73$ \( T^{5} - 8 T^{4} - 91 T^{3} + \cdots + 5897 \) Copy content Toggle raw display
$79$ \( T^{5} + 16 T^{4} - 100 T^{3} + \cdots + 10891 \) Copy content Toggle raw display
$83$ \( T^{5} - 18 T^{4} - 141 T^{3} + \cdots + 30383 \) Copy content Toggle raw display
$89$ \( T^{5} + 14 T^{4} - 113 T^{3} + \cdots + 617 \) Copy content Toggle raw display
$97$ \( T^{5} + 47 T^{4} + 811 T^{3} + \cdots + 28073 \) Copy content Toggle raw display
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