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Properties

Label 91.2.g.a
Level $91$
Weight $2$
Character orbit 91.g
Analytic conductor $0.727$
Analytic rank $1$
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] = [91,2,Mod(9,91)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("91.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.g (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: \(0.726638658394\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} - 3 q^{3} + ( - \zeta_{6} + 1) q^{4} + (3 \zeta_{6} - 3) q^{5} + 3 \zeta_{6} q^{6} + (\zeta_{6} - 3) q^{7} - 3 q^{8} + 6 q^{9} + 3 q^{10} - 3 q^{11} + (3 \zeta_{6} - 3) q^{12} + ( - 4 \zeta_{6} + 1) q^{13} + \cdots - 18 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 6 q^{3} + q^{4} - 3 q^{5} + 3 q^{6} - 5 q^{7} - 6 q^{8} + 12 q^{9} + 6 q^{10} - 6 q^{11} - 3 q^{12} - 2 q^{13} + 4 q^{14} + 9 q^{15} + q^{16} + 2 q^{17} - 6 q^{18} - 2 q^{19} + 3 q^{20}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−0.500000 − 0.866025i −3.00000 0.500000 − 0.866025i −1.50000 + 2.59808i 1.50000 + 2.59808i −2.50000 + 0.866025i −3.00000 6.00000 3.00000
81.1 −0.500000 + 0.866025i −3.00000 0.500000 + 0.866025i −1.50000 − 2.59808i 1.50000 − 2.59808i −2.50000 − 0.866025i −3.00000 6.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.g.a ✓ 2
3.b odd 2 1 819.2.n.c 2
7.b odd 2 1 637.2.g.a 2
7.c even 3 1 91.2.h.a yes 2
7.c even 3 1 637.2.f.b 2
7.d odd 6 1 637.2.f.a 2
7.d odd 6 1 637.2.h.a 2
13.c even 3 1 91.2.h.a yes 2
13.c even 3 1 1183.2.e.a 2
13.e even 6 1 1183.2.e.c 2
21.h odd 6 1 819.2.s.a 2
39.i odd 6 1 819.2.s.a 2
91.g even 3 1 inner 91.2.g.a ✓ 2
91.g even 3 1 8281.2.a.i 1
91.h even 3 1 637.2.f.b 2
91.h even 3 1 1183.2.e.a 2
91.k even 6 1 1183.2.e.c 2
91.m odd 6 1 637.2.g.a 2
91.m odd 6 1 8281.2.a.j 1
91.n odd 6 1 637.2.h.a 2
91.p odd 6 1 8281.2.a.g 1
91.u even 6 1 8281.2.a.c 1
91.v odd 6 1 637.2.f.a 2
273.bm odd 6 1 819.2.n.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a ✓ 2 1.a even 1 1 trivial
91.2.g.a ✓ 2 91.g even 3 1 inner
91.2.h.a yes 2 7.c even 3 1
91.2.h.a yes 2 13.c even 3 1
637.2.f.a 2 7.d odd 6 1
637.2.f.a 2 91.v odd 6 1
637.2.f.b 2 7.c even 3 1
637.2.f.b 2 91.h even 3 1
637.2.g.a 2 7.b odd 2 1
637.2.g.a 2 91.m odd 6 1
637.2.h.a 2 7.d odd 6 1
637.2.h.a 2 91.n odd 6 1
819.2.n.c 2 3.b odd 2 1
819.2.n.c 2 273.bm odd 6 1
819.2.s.a 2 21.h odd 6 1
819.2.s.a 2 39.i odd 6 1
1183.2.e.a 2 13.c even 3 1
1183.2.e.a 2 91.h even 3 1
1183.2.e.c 2 13.e even 6 1
1183.2.e.c 2 91.k even 6 1
8281.2.a.c 1 91.u even 6 1
8281.2.a.g 1 91.p odd 6 1
8281.2.a.i 1 91.g even 3 1
8281.2.a.j 1 91.m odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 7 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$47$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$61$ \( (T + 13)^{2} \) Copy content Toggle raw display
$67$ \( (T + 3)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$73$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
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