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Properties

Label 9.9.b.a
Level $9$
Weight $9$
Character orbit 9.b
Analytic conductor $3.666$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,9,Mod(8,9)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9.8"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 9.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.66640749055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 238 q^{4} + 233 \beta q^{5} + 1652 q^{7} + 494 \beta q^{8} - 4194 q^{10} - 5036 \beta q^{11} - 26272 q^{13} + 1652 \beta q^{14} + 52036 q^{16} + 3579 \beta q^{17} + 46640 q^{19} + 55454 \beta q^{20} + \cdots - 3035697 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 476 q^{4} + 3304 q^{7} - 8388 q^{10} - 52544 q^{13} + 104072 q^{16} + 93280 q^{19} + 181296 q^{22} - 1173154 q^{25} + 786352 q^{28} - 392888 q^{31} - 128844 q^{34} + 5638828 q^{37} - 4143672 q^{40}+ \cdots - 48903488 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
− 1.41421i
1.41421i
− 4.24264i 0 238.000 − 988.535i 0 1652.00 − 2095.86i 0 −4194.00
8.2 4.24264i 0 238.000 988.535i 0 1652.00 2095.86i 0 −4194.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 9.9.b.a ✓ 2
3.b odd 2 1 inner 9.9.b.a ✓ 2
4.b odd 2 1 144.9.e.a 2
5.b even 2 1 225.9.c.a 2
5.c odd 4 2 225.9.d.a 4
9.c even 3 2 81.9.d.b 4
9.d odd 6 2 81.9.d.b 4
12.b even 2 1 144.9.e.a 2
15.d odd 2 1 225.9.c.a 2
15.e even 4 2 225.9.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.9.b.a ✓ 2 1.a even 1 1 trivial
9.9.b.a ✓ 2 3.b odd 2 1 inner
81.9.d.b 4 9.c even 3 2
81.9.d.b 4 9.d odd 6 2
144.9.e.a 2 4.b odd 2 1
144.9.e.a 2 12.b even 2 1
225.9.c.a 2 5.b even 2 1
225.9.c.a 2 15.d odd 2 1
225.9.d.a 4 5.c odd 4 2
225.9.d.a 4 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 18 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 977202 \) Copy content Toggle raw display
$7$ \( (T - 1652)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 456503328 \) Copy content Toggle raw display
$13$ \( (T + 26272)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 230566338 \) Copy content Toggle raw display
$19$ \( (T - 46640)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 107644288032 \) Copy content Toggle raw display
$29$ \( T^{2} + 378204699762 \) Copy content Toggle raw display
$31$ \( (T + 196444)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2819414)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 499042458882 \) Copy content Toggle raw display
$43$ \( (T + 2213464)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2666553329952 \) Copy content Toggle raw display
$53$ \( T^{2} + 26922373529202 \) Copy content Toggle raw display
$59$ \( T^{2} + 137939140326528 \) Copy content Toggle raw display
$61$ \( (T + 17405302)^{2} \) Copy content Toggle raw display
$67$ \( (T + 14322664)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 244785425278752 \) Copy content Toggle raw display
$73$ \( (T + 8906992)^{2} \) Copy content Toggle raw display
$79$ \( (T - 32758844)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 72\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{2} + 34\!\cdots\!38 \) Copy content Toggle raw display
$97$ \( (T + 24451744)^{2} \) Copy content Toggle raw display
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