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Properties

Label 13.3.f.a
Level $13$
Weight $3$
Character orbit 13.f
Analytic conductor $0.354$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [13,3,Mod(2,13)] 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("13.2"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 13.f (of order \(12\), degree \(4\), 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: \(0.354224343668\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - 1) q^{2} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{4} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} + \cdots - 3) q^{5}+ \cdots + (26 \zeta_{12}^{3} + 52 \zeta_{12}^{2} + \cdots + 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 6 q^{4} - 14 q^{5} + 10 q^{6} + 16 q^{7} - 6 q^{8} + 10 q^{9} + 24 q^{10} + 4 q^{11} - 26 q^{13} - 40 q^{14} - 14 q^{15} - 2 q^{16} - 12 q^{17} - 2 q^{18} + 10 q^{19} + 26 q^{20}+ \cdots + 208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
−0.500000 − 0.133975i 0.366025 + 0.633975i −3.23205 − 1.86603i −2.63397 + 2.63397i −0.0980762 − 0.366025i 5.73205 − 1.53590i 2.83013 + 2.83013i 4.23205 − 7.33013i 1.66987 − 0.964102i
6.1 −0.500000 − 1.86603i −1.36603 + 2.36603i 0.232051 − 0.133975i −4.36603 + 4.36603i 5.09808 + 1.36603i 2.26795 − 8.46410i −5.83013 − 5.83013i 0.767949 + 1.33013i 10.3301 + 5.96410i
7.1 −0.500000 + 0.133975i 0.366025 − 0.633975i −3.23205 + 1.86603i −2.63397 − 2.63397i −0.0980762 + 0.366025i 5.73205 + 1.53590i 2.83013 − 2.83013i 4.23205 + 7.33013i 1.66987 + 0.964102i
11.1 −0.500000 + 1.86603i −1.36603 − 2.36603i 0.232051 + 0.133975i −4.36603 − 4.36603i 5.09808 − 1.36603i 2.26795 + 8.46410i −5.83013 + 5.83013i 0.767949 − 1.33013i 10.3301 − 5.96410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.3.f.a ✓ 4
3.b odd 2 1 117.3.bd.b 4
4.b odd 2 1 208.3.bd.d 4
5.b even 2 1 325.3.t.a 4
5.c odd 4 1 325.3.w.a 4
5.c odd 4 1 325.3.w.b 4
13.b even 2 1 169.3.f.b 4
13.c even 3 1 169.3.d.a 4
13.c even 3 1 169.3.f.c 4
13.d odd 4 1 169.3.f.a 4
13.d odd 4 1 169.3.f.c 4
13.e even 6 1 169.3.d.c 4
13.e even 6 1 169.3.f.a 4
13.f odd 12 1 inner 13.3.f.a ✓ 4
13.f odd 12 1 169.3.d.a 4
13.f odd 12 1 169.3.d.c 4
13.f odd 12 1 169.3.f.b 4
39.k even 12 1 117.3.bd.b 4
52.l even 12 1 208.3.bd.d 4
65.o even 12 1 325.3.w.b 4
65.s odd 12 1 325.3.t.a 4
65.t even 12 1 325.3.w.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.3.f.a ✓ 4 1.a even 1 1 trivial
13.3.f.a ✓ 4 13.f odd 12 1 inner
117.3.bd.b 4 3.b odd 2 1
117.3.bd.b 4 39.k even 12 1
169.3.d.a 4 13.c even 3 1
169.3.d.a 4 13.f odd 12 1
169.3.d.c 4 13.e even 6 1
169.3.d.c 4 13.f odd 12 1
169.3.f.a 4 13.d odd 4 1
169.3.f.a 4 13.e even 6 1
169.3.f.b 4 13.b even 2 1
169.3.f.b 4 13.f odd 12 1
169.3.f.c 4 13.c even 3 1
169.3.f.c 4 13.d odd 4 1
208.3.bd.d 4 4.b odd 2 1
208.3.bd.d 4 52.l even 12 1
325.3.t.a 4 5.b even 2 1
325.3.t.a 4 65.s odd 12 1
325.3.w.a 4 5.c odd 4 1
325.3.w.a 4 65.t even 12 1
325.3.w.b 4 5.c odd 4 1
325.3.w.b 4 65.o even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{4} + 14 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{4} - 16 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 12 T^{3} + \cdots + 45369 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{3} + \cdots + 39204 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + \cdots + 11449 \) Copy content Toggle raw display
$31$ \( T^{4} + 20 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$37$ \( T^{4} + 68 T^{3} + \cdots + 1868689 \) Copy content Toggle raw display
$41$ \( T^{4} - 100 T^{3} + \cdots + 833569 \) Copy content Toggle raw display
$43$ \( (T^{2} - 90 T + 2700)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 68 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$53$ \( (T^{2} - 64 T - 1163)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 164 T^{3} + \cdots + 16613776 \) Copy content Toggle raw display
$61$ \( T^{4} + 124 T^{3} + \cdots + 6355441 \) Copy content Toggle raw display
$67$ \( T^{4} - 118 T^{3} + \cdots + 9721924 \) Copy content Toggle raw display
$71$ \( T^{4} + 86 T^{3} + \cdots + 2208196 \) Copy content Toggle raw display
$73$ \( T^{4} - 58 T^{3} + \cdots + 3463321 \) Copy content Toggle raw display
$79$ \( (T^{2} + 20 T - 5192)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 188 T^{3} + \cdots + 11587216 \) Copy content Toggle raw display
$89$ \( T^{4} + 110 T^{3} + \cdots + 8702500 \) Copy content Toggle raw display
$97$ \( T^{4} - 178 T^{3} + \cdots + 18028516 \) Copy content Toggle raw display
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