Properties

Label 91.2.f.b
Level $91$
Weight $2$
Character orbit 91.f
Analytic conductor $0.727$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,2,Mod(22,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.f (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + (\beta_1 - 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{6} + (\beta_1 - 1) q^{7} - \beta_{3} q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + (\beta_1 - 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{6} + (\beta_1 - 1) q^{7} - \beta_{3} q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 1) q^{9}+ \cdots + (5 \beta_{3} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{4} - 6 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{4} - 6 q^{6} - 2 q^{7} - 2 q^{9} - 6 q^{10} - 6 q^{11} + 4 q^{12} + 4 q^{13} - 6 q^{15} + 10 q^{16} + 12 q^{17} + 24 q^{18} - 4 q^{19} + 4 q^{21} + 6 q^{22} - 6 q^{23} + 6 q^{24} - 8 q^{25} - 18 q^{26} + 16 q^{27} - 2 q^{28} + 6 q^{29} - 6 q^{30} - 4 q^{31} - 12 q^{34} - 2 q^{36} + 14 q^{37} - 14 q^{39} - 12 q^{40} - 6 q^{42} - 10 q^{43} + 12 q^{44} - 12 q^{45} - 6 q^{46} + 24 q^{47} + 10 q^{48} - 2 q^{49} - 36 q^{51} - 8 q^{52} - 12 q^{53} + 6 q^{55} + 8 q^{57} - 18 q^{59} + 12 q^{60} + 20 q^{61} - 18 q^{62} - 2 q^{63} + 4 q^{64} + 18 q^{65} + 24 q^{66} + 2 q^{67} + 12 q^{68} - 12 q^{69} + 12 q^{70} - 12 q^{71} + 12 q^{72} + 8 q^{73} + 4 q^{75} - 4 q^{76} + 12 q^{77} - 24 q^{78} + 44 q^{79} - 2 q^{81} + 18 q^{82} + 12 q^{83} - 2 q^{84} + 6 q^{85} - 36 q^{86} + 6 q^{87} - 6 q^{88} - 12 q^{89} + 12 q^{90} - 8 q^{91} + 12 q^{92} - 16 q^{93} - 24 q^{94} - 36 q^{96} + 8 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i −1.36603 + 2.36603i −0.500000 0.866025i 1.73205 −2.36603 4.09808i −0.500000 0.866025i −1.73205 −2.23205 3.86603i −1.50000 + 2.59808i
22.2 0.866025 1.50000i 0.366025 0.633975i −0.500000 0.866025i −1.73205 −0.633975 1.09808i −0.500000 0.866025i 1.73205 1.23205 + 2.13397i −1.50000 + 2.59808i
29.1 −0.866025 1.50000i −1.36603 2.36603i −0.500000 + 0.866025i 1.73205 −2.36603 + 4.09808i −0.500000 + 0.866025i −1.73205 −2.23205 + 3.86603i −1.50000 2.59808i
29.2 0.866025 + 1.50000i 0.366025 + 0.633975i −0.500000 + 0.866025i −1.73205 −0.633975 + 1.09808i −0.500000 + 0.866025i 1.73205 1.23205 2.13397i −1.50000 2.59808i
\(n\): 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 91.2.f.b 4
3.b odd 2 1 819.2.o.b 4
4.b odd 2 1 1456.2.s.o 4
7.b odd 2 1 637.2.f.d 4
7.c even 3 1 637.2.g.e 4
7.c even 3 1 637.2.h.d 4
7.d odd 6 1 637.2.g.d 4
7.d odd 6 1 637.2.h.e 4
13.c even 3 1 inner 91.2.f.b 4
13.c even 3 1 1183.2.a.f 2
13.e even 6 1 1183.2.a.e 2
13.f odd 12 2 1183.2.c.e 4
39.i odd 6 1 819.2.o.b 4
52.j odd 6 1 1456.2.s.o 4
91.g even 3 1 637.2.h.d 4
91.h even 3 1 637.2.g.e 4
91.m odd 6 1 637.2.h.e 4
91.n odd 6 1 637.2.f.d 4
91.n odd 6 1 8281.2.a.r 2
91.t odd 6 1 8281.2.a.t 2
91.v odd 6 1 637.2.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 1.a even 1 1 trivial
91.2.f.b 4 13.c even 3 1 inner
637.2.f.d 4 7.b odd 2 1
637.2.f.d 4 91.n odd 6 1
637.2.g.d 4 7.d odd 6 1
637.2.g.d 4 91.v odd 6 1
637.2.g.e 4 7.c even 3 1
637.2.g.e 4 91.h even 3 1
637.2.h.d 4 7.c even 3 1
637.2.h.d 4 91.g even 3 1
637.2.h.e 4 7.d odd 6 1
637.2.h.e 4 91.m odd 6 1
819.2.o.b 4 3.b odd 2 1
819.2.o.b 4 39.i odd 6 1
1183.2.a.e 2 13.e even 6 1
1183.2.a.f 2 13.c even 3 1
1183.2.c.e 4 13.f odd 12 2
1456.2.s.o 4 4.b odd 2 1
1456.2.s.o 4 52.j odd 6 1
8281.2.a.r 2 91.n odd 6 1
8281.2.a.t 2 91.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 12 T^{3} + \cdots + 1089 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( (T^{2} - 12 T - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 39)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + \cdots + 6084 \) Copy content Toggle raw display
$61$ \( T^{4} - 20 T^{3} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 23)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 22 T + 94)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
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