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Properties

Label 9675.2.a.cv
Level $9675$
Weight $2$
Character orbit 9675.a
Self dual yes
Analytic conductor $77.255$
Analytic rank $1$
Dimension $10$
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] = [9675,2,Mod(1,9675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9675, 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("9675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9675 = 3^{2} \cdot 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9675.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: \(77.2552639556\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 13x^{8} + 58x^{6} - 103x^{4} + 65x^{2} - 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{5} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{5} q^{7} + \beta_{3} q^{8} - \beta_{3} q^{11} + (\beta_{5} + \beta_{2}) q^{13} - \beta_{6} q^{14} + (\beta_{4} - 1) q^{16} + ( - \beta_{8} + \beta_{6}) q^{17} + (\beta_{5} + \beta_{4} - 1) q^{19} + ( - \beta_{4} - 2 \beta_{2} - 1) q^{22} + ( - \beta_{9} - \beta_{6} + \cdots - \beta_1) q^{23}+ \cdots + ( - \beta_{9} + \beta_{6} + \cdots - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 6 q^{4} - 4 q^{13} - 10 q^{16} - 10 q^{19} - 2 q^{22} - 16 q^{31} - 8 q^{34} + 10 q^{37} + 10 q^{43} - 28 q^{46} - 26 q^{49} + 36 q^{52} - 16 q^{58} - 32 q^{61} - 28 q^{64} - 14 q^{67} + 4 q^{73} - 24 q^{76} - 18 q^{79} + 20 q^{82} - 56 q^{88} - 44 q^{91} - 12 q^{94} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 13x^{8} + 58x^{6} - 103x^{4} + 65x^{2} - 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - 12\nu^{6} + 47\nu^{4} - 65\nu^{2} + 21 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{9} - 12\nu^{7} + 47\nu^{5} - 65\nu^{3} + 21\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{8} + 13\nu^{6} - 56\nu^{4} + 87\nu^{2} - 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\nu^{9} + 13\nu^{7} - 56\nu^{5} + 87\nu^{3} - 32\nu \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 2\nu^{9} - 25\nu^{7} + 104\nu^{5} - 158\nu^{3} + 59\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{6} + 6\beta_{3} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} + \beta_{5} + 9\beta_{4} + 32\beta_{2} + 62 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9\beta_{9} + 10\beta_{8} - 8\beta_{6} + 32\beta_{3} + 85\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12\beta_{7} + 13\beta_{5} + 61\beta_{4} + 167\beta_{2} + 307 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 61\beta_{9} + 73\beta_{8} - 48\beta_{6} + 167\beta_{3} + 413\beta_1 \) 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
−2.31027
−2.06582
−1.55196
−0.756861
−0.643165
0.643165
0.756861
1.55196
2.06582
2.31027
−2.31027 0 3.33734 0 0 0.0587517 −3.08960 0 0
1.2 −2.06582 0 2.26760 0 0 1.39822 −0.552809 0 0
1.3 −1.55196 0 0.408567 0 0 −3.08237 2.46983 0 0
1.4 −0.756861 0 −1.42716 0 0 2.95974 2.59388 0 0
1.5 −0.643165 0 −1.58634 0 0 −1.33434 2.30661 0 0
1.6 0.643165 0 −1.58634 0 0 −1.33434 −2.30661 0 0
1.7 0.756861 0 −1.42716 0 0 2.95974 −2.59388 0 0
1.8 1.55196 0 0.408567 0 0 −3.08237 −2.46983 0 0
1.9 2.06582 0 2.26760 0 0 1.39822 0.552809 0 0
1.10 2.31027 0 3.33734 0 0 0.0587517 3.08960 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(43\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9675.2.a.cv ✓ 10
3.b odd 2 1 inner 9675.2.a.cv ✓ 10
5.b even 2 1 9675.2.a.cw yes 10
15.d odd 2 1 9675.2.a.cw yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9675.2.a.cv ✓ 10 1.a even 1 1 trivial
9675.2.a.cv ✓ 10 3.b odd 2 1 inner
9675.2.a.cw yes 10 5.b even 2 1
9675.2.a.cw yes 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9675))\):

\( T_{2}^{10} - 13T_{2}^{8} + 58T_{2}^{6} - 103T_{2}^{4} + 65T_{2}^{2} - 13 \) Copy content Toggle raw display
\( T_{7}^{5} - 11T_{7}^{3} + T_{7}^{2} + 17T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{10} - 28T_{11}^{8} + 291T_{11}^{6} - 1348T_{11}^{4} + 2470T_{11}^{2} - 637 \) Copy content Toggle raw display
\( T_{13}^{5} + 2T_{13}^{4} - 19T_{13}^{3} + 3T_{13}^{2} + 39T_{13} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 13 T^{8} + \cdots - 13 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} - 11 T^{3} + T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} - 28 T^{8} + \cdots - 637 \) Copy content Toggle raw display
$13$ \( (T^{5} + 2 T^{4} - 19 T^{3} + \cdots + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} - 69 T^{8} + \cdots - 77077 \) Copy content Toggle raw display
$19$ \( (T^{5} + 5 T^{4} - 7 T^{3} + \cdots + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 111 T^{8} + \cdots - 1675453 \) Copy content Toggle raw display
$29$ \( T^{10} - 148 T^{8} + \cdots - 3289117 \) Copy content Toggle raw display
$31$ \( (T^{5} + 8 T^{4} + \cdots + 241)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 5 T^{4} + \cdots - 2033)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 161 T^{8} + \cdots - 1070797 \) Copy content Toggle raw display
$43$ \( (T - 1)^{10} \) Copy content Toggle raw display
$47$ \( T^{10} - 276 T^{8} + \cdots - 13711477 \) Copy content Toggle raw display
$53$ \( T^{10} - 269 T^{8} + \cdots - 2217397 \) Copy content Toggle raw display
$59$ \( T^{10} - 270 T^{8} + \cdots - 1638325 \) Copy content Toggle raw display
$61$ \( (T^{5} + 16 T^{4} + \cdots + 65)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 7 T^{4} + \cdots + 90809)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} - 332 T^{8} + \cdots - 2437357 \) Copy content Toggle raw display
$73$ \( (T^{5} - 2 T^{4} + \cdots - 9713)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 9 T^{4} + \cdots + 875)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 24899877925 \) Copy content Toggle raw display
$89$ \( T^{10} - 192 T^{8} + \cdots - 2551237 \) Copy content Toggle raw display
$97$ \( (T^{5} - 2 T^{4} + \cdots + 91931)^{2} \) Copy content Toggle raw display
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