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Properties

Label 91.4.f.a
Level $91$
Weight $4$
Character orbit 91.f
Analytic conductor $5.369$
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] = [91,4,Mod(22,91)] 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("91.22"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 91.f (of order \(3\), degree \(2\), 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: \(5.36917381052\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 5 \zeta_{6} + 5) q^{2} + (2 \zeta_{6} - 2) q^{3} - 17 \zeta_{6} q^{4} - 19 q^{5} + 10 \zeta_{6} q^{6} - 7 \zeta_{6} q^{7} - 45 q^{8} + 23 \zeta_{6} q^{9} + (95 \zeta_{6} - 95) q^{10} + ( - 50 \zeta_{6} + 50) q^{11} + \cdots + 1150 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} - 2 q^{3} - 17 q^{4} - 38 q^{5} + 10 q^{6} - 7 q^{7} - 90 q^{8} + 23 q^{9} - 95 q^{10} + 50 q^{11} + 68 q^{12} - 91 q^{13} - 70 q^{14} + 38 q^{15} - 89 q^{16} - 77 q^{17} + 230 q^{18} - 12 q^{19}+ \cdots + 2300 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.500000 + 0.866025i
0.500000 − 0.866025i
2.50000 − 4.33013i −1.00000 + 1.73205i −8.50000 − 14.7224i −19.0000 5.00000 + 8.66025i −3.50000 − 6.06218i −45.0000 11.5000 + 19.9186i −47.5000 + 82.2724i
29.1 2.50000 + 4.33013i −1.00000 − 1.73205i −8.50000 + 14.7224i −19.0000 5.00000 − 8.66025i −3.50000 + 6.06218i −45.0000 11.5000 − 19.9186i −47.5000 − 82.2724i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.4.f.a ✓ 2
13.c even 3 1 inner 91.4.f.a ✓ 2
13.c even 3 1 1183.4.a.a 1
13.e even 6 1 1183.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.4.f.a ✓ 2 1.a even 1 1 trivial
91.4.f.a ✓ 2 13.c even 3 1 inner
1183.4.a.a 1 13.c even 3 1
1183.4.a.c 1 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5T_{2} + 25 \) acting on \(S_{4}^{\mathrm{new}}(91, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( (T + 19)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} - 50T + 2500 \) Copy content Toggle raw display
$13$ \( T^{2} + 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 77T + 5929 \) Copy content Toggle raw display
$19$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$23$ \( T^{2} - 138T + 19044 \) Copy content Toggle raw display
$29$ \( T^{2} + 251T + 63001 \) Copy content Toggle raw display
$31$ \( (T - 250)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 79T + 6241 \) Copy content Toggle raw display
$41$ \( T^{2} - 219T + 47961 \) Copy content Toggle raw display
$43$ \( T^{2} + 258T + 66564 \) Copy content Toggle raw display
$47$ \( (T - 72)^{2} \) Copy content Toggle raw display
$53$ \( (T - 111)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 126T + 15876 \) Copy content Toggle raw display
$61$ \( T^{2} - 359T + 128881 \) Copy content Toggle raw display
$67$ \( T^{2} + 286T + 81796 \) Copy content Toggle raw display
$71$ \( T^{2} - 120T + 14400 \) Copy content Toggle raw display
$73$ \( (T - 89)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1030)^{2} \) Copy content Toggle raw display
$83$ \( (T - 826)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1526 T + 2328676 \) Copy content Toggle raw display
$97$ \( T^{2} + 562T + 315844 \) Copy content Toggle raw display
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