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Properties

Label 12.9.c.b
Level $12$
Weight $9$
Character orbit 12.c
Analytic conductor $4.889$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [12,9,Mod(5,12)] 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("12.5"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 12.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.88854332073\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-110}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 110 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(\beta = 6\sqrt{-110}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 51) q^{3} - 18 \beta q^{5} - 3094 q^{7} + ( - 102 \beta - 1359) q^{9} - 18 \beta q^{11} - 7294 q^{13} + (918 \beta + 71280) q^{15} + 936 \beta q^{17} - 80326 q^{19} + ( - 3094 \beta + 157794) q^{21} + \cdots + (24462 \beta - 7270560) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 102 q^{3} - 6188 q^{7} - 2718 q^{9} - 14588 q^{13} + 142560 q^{15} - 160652 q^{19} + 315588 q^{21} - 1784830 q^{25} + 946458 q^{27} + 871828 q^{31} + 142560 q^{33} + 2318596 q^{37} + 743988 q^{39}+ \cdots - 14541120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
− 10.4881i
10.4881i
0 −51.0000 − 62.9285i 0 1132.71i 0 −3094.00 0 −1359.00 + 6418.71i 0
5.2 0 −51.0000 + 62.9285i 0 − 1132.71i 0 −3094.00 0 −1359.00 − 6418.71i 0
\(n\): 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 12.9.c.b ✓ 2
3.b odd 2 1 inner 12.9.c.b ✓ 2
4.b odd 2 1 48.9.e.c 2
5.b even 2 1 300.9.g.d 2
5.c odd 4 2 300.9.b.c 4
8.b even 2 1 192.9.e.g 2
8.d odd 2 1 192.9.e.d 2
9.c even 3 2 324.9.g.f 4
9.d odd 6 2 324.9.g.f 4
12.b even 2 1 48.9.e.c 2
15.d odd 2 1 300.9.g.d 2
15.e even 4 2 300.9.b.c 4
24.f even 2 1 192.9.e.d 2
24.h odd 2 1 192.9.e.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.b ✓ 2 1.a even 1 1 trivial
12.9.c.b ✓ 2 3.b odd 2 1 inner
48.9.e.c 2 4.b odd 2 1
48.9.e.c 2 12.b even 2 1
192.9.e.d 2 8.d odd 2 1
192.9.e.d 2 24.f even 2 1
192.9.e.g 2 8.b even 2 1
192.9.e.g 2 24.h odd 2 1
300.9.b.c 4 5.c odd 4 2
300.9.b.c 4 15.e even 4 2
300.9.g.d 2 5.b even 2 1
300.9.g.d 2 15.d odd 2 1
324.9.g.f 4 9.c even 3 2
324.9.g.f 4 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 1283040 \) acting on \(S_{9}^{\mathrm{new}}(12, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 102T + 6561 \) Copy content Toggle raw display
$5$ \( T^{2} + 1283040 \) Copy content Toggle raw display
$7$ \( (T + 3094)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1283040 \) Copy content Toggle raw display
$13$ \( (T + 7294)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3469340160 \) Copy content Toggle raw display
$19$ \( (T + 80326)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9489363840 \) Copy content Toggle raw display
$29$ \( T^{2} + 746946113760 \) Copy content Toggle raw display
$31$ \( (T - 435914)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1159298)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 7377998348160 \) Copy content Toggle raw display
$43$ \( (T - 990266)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 44905188810240 \) Copy content Toggle raw display
$53$ \( T^{2} + 101424159318240 \) Copy content Toggle raw display
$59$ \( T^{2} + 2554431279840 \) Copy content Toggle raw display
$61$ \( (T - 19369154)^{2} \) Copy content Toggle raw display
$67$ \( (T + 28024294)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 11\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( (T + 25230142)^{2} \) Copy content Toggle raw display
$79$ \( (T + 63401398)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 22\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{2} + 61\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( (T - 19550306)^{2} \) Copy content Toggle raw display
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