Properties

Label 91.2.k.a
Level $91$
Weight $2$
Character orbit 91.k
Analytic conductor $0.727$
Analytic rank $0$
Dimension $2$
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(4,91)]
 
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([4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.k (of order \(6\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
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: $\mathrm{SU}(2)[C_{6}]$

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

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 1) q^{2} + \zeta_{6} q^{3} - q^{4} + ( - \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 2) q^{6} + (2 \zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 2 \zeta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 1) q^{2} + \zeta_{6} q^{3} - q^{4} + ( - \zeta_{6} + 2) q^{5} + ( - \zeta_{6} + 2) q^{6} + (2 \zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 2 \zeta_{6} + 2) q^{9} - 3 \zeta_{6} q^{10} + (3 \zeta_{6} - 6) q^{11} - \zeta_{6} q^{12} + (4 \zeta_{6} - 3) q^{13} + (4 \zeta_{6} + 1) q^{14} + (\zeta_{6} + 1) q^{15} - 5 q^{16} + 6 q^{17} + ( - 2 \zeta_{6} - 2) q^{18} + ( - \zeta_{6} - 1) q^{19} + (\zeta_{6} - 2) q^{20} + ( - \zeta_{6} - 2) q^{21} + 9 \zeta_{6} q^{22} + ( - \zeta_{6} + 2) q^{24} + (2 \zeta_{6} - 2) q^{25} + (2 \zeta_{6} + 5) q^{26} + 5 q^{27} + ( - 2 \zeta_{6} + 3) q^{28} + (3 \zeta_{6} - 3) q^{29} + ( - 3 \zeta_{6} + 3) q^{30} + ( - \zeta_{6} - 1) q^{31} + (6 \zeta_{6} - 3) q^{32} + ( - 3 \zeta_{6} - 3) q^{33} + ( - 12 \zeta_{6} + 6) q^{34} + (5 \zeta_{6} - 4) q^{35} + (2 \zeta_{6} - 2) q^{36} + (3 \zeta_{6} - 3) q^{38} + (\zeta_{6} - 4) q^{39} - 3 \zeta_{6} q^{40} + ( - 3 \zeta_{6} - 3) q^{41} + (5 \zeta_{6} - 4) q^{42} - 11 \zeta_{6} q^{43} + ( - 3 \zeta_{6} + 6) q^{44} + ( - 4 \zeta_{6} + 2) q^{45} + ( - 5 \zeta_{6} + 10) q^{47} - 5 \zeta_{6} q^{48} + ( - 8 \zeta_{6} + 5) q^{49} + (2 \zeta_{6} + 2) q^{50} + 6 \zeta_{6} q^{51} + ( - 4 \zeta_{6} + 3) q^{52} + ( - 9 \zeta_{6} + 9) q^{53} + ( - 10 \zeta_{6} + 5) q^{54} + (9 \zeta_{6} - 9) q^{55} + (4 \zeta_{6} + 1) q^{56} + ( - 2 \zeta_{6} + 1) q^{57} + (3 \zeta_{6} + 3) q^{58} + (4 \zeta_{6} - 2) q^{59} + ( - \zeta_{6} - 1) q^{60} + (7 \zeta_{6} - 7) q^{61} + (3 \zeta_{6} - 3) q^{62} + (6 \zeta_{6} - 2) q^{63} - q^{64} + (7 \zeta_{6} - 2) q^{65} + (9 \zeta_{6} - 9) q^{66} + ( - 5 \zeta_{6} + 10) q^{67} - 6 q^{68} + (3 \zeta_{6} + 6) q^{70} + ( - \zeta_{6} + 2) q^{71} + ( - 2 \zeta_{6} - 2) q^{72} + (5 \zeta_{6} + 5) q^{73} - 2 q^{75} + (\zeta_{6} + 1) q^{76} + ( - 15 \zeta_{6} + 12) q^{77} + (7 \zeta_{6} - 2) q^{78} + 5 \zeta_{6} q^{79} + (5 \zeta_{6} - 10) q^{80} - \zeta_{6} q^{81} + (9 \zeta_{6} - 9) q^{82} + ( - 4 \zeta_{6} + 2) q^{83} + (\zeta_{6} + 2) q^{84} + ( - 6 \zeta_{6} + 12) q^{85} + (11 \zeta_{6} - 22) q^{86} - 3 q^{87} + 9 \zeta_{6} q^{88} + ( - 8 \zeta_{6} + 4) q^{89} - 6 q^{90} + ( - 10 \zeta_{6} + 1) q^{91} + ( - 2 \zeta_{6} + 1) q^{93} - 15 \zeta_{6} q^{94} - 3 q^{95} + (3 \zeta_{6} - 6) q^{96} + (3 \zeta_{6} - 6) q^{97} + ( - 2 \zeta_{6} - 11) q^{98} + (12 \zeta_{6} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{4} + 3 q^{5} + 3 q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 2 q^{4} + 3 q^{5} + 3 q^{6} - 4 q^{7} + 2 q^{9} - 3 q^{10} - 9 q^{11} - q^{12} - 2 q^{13} + 6 q^{14} + 3 q^{15} - 10 q^{16} + 12 q^{17} - 6 q^{18} - 3 q^{19} - 3 q^{20} - 5 q^{21} + 9 q^{22} + 3 q^{24} - 2 q^{25} + 12 q^{26} + 10 q^{27} + 4 q^{28} - 3 q^{29} + 3 q^{30} - 3 q^{31} - 9 q^{33} - 3 q^{35} - 2 q^{36} - 3 q^{38} - 7 q^{39} - 3 q^{40} - 9 q^{41} - 3 q^{42} - 11 q^{43} + 9 q^{44} + 15 q^{47} - 5 q^{48} + 2 q^{49} + 6 q^{50} + 6 q^{51} + 2 q^{52} + 9 q^{53} - 9 q^{55} + 6 q^{56} + 9 q^{58} - 3 q^{60} - 7 q^{61} - 3 q^{62} + 2 q^{63} - 2 q^{64} + 3 q^{65} - 9 q^{66} + 15 q^{67} - 12 q^{68} + 15 q^{70} + 3 q^{71} - 6 q^{72} + 15 q^{73} - 4 q^{75} + 3 q^{76} + 9 q^{77} + 3 q^{78} + 5 q^{79} - 15 q^{80} - q^{81} - 9 q^{82} + 5 q^{84} + 18 q^{85} - 33 q^{86} - 6 q^{87} + 9 q^{88} - 12 q^{90} - 8 q^{91} - 15 q^{94} - 6 q^{95} - 9 q^{96} - 9 q^{97} - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 0.866025i
0.500000 0.866025i
1.73205i 0.500000 + 0.866025i −1.00000 1.50000 0.866025i 1.50000 0.866025i −2.00000 + 1.73205i 1.73205i 1.00000 1.73205i −1.50000 2.59808i
23.1 1.73205i 0.500000 0.866025i −1.00000 1.50000 + 0.866025i 1.50000 + 0.866025i −2.00000 1.73205i 1.73205i 1.00000 + 1.73205i −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
91.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.k.a 2
3.b odd 2 1 819.2.bm.a 2
7.b odd 2 1 637.2.k.b 2
7.c even 3 1 91.2.u.a yes 2
7.c even 3 1 637.2.q.c 2
7.d odd 6 1 637.2.q.b 2
7.d odd 6 1 637.2.u.a 2
13.e even 6 1 91.2.u.a yes 2
13.f odd 12 2 1183.2.e.e 4
21.h odd 6 1 819.2.do.c 2
39.h odd 6 1 819.2.do.c 2
91.k even 6 1 inner 91.2.k.a 2
91.l odd 6 1 637.2.k.b 2
91.p odd 6 1 637.2.q.b 2
91.t odd 6 1 637.2.u.a 2
91.u even 6 1 637.2.q.c 2
91.x odd 12 2 8281.2.a.s 2
91.ba even 12 2 8281.2.a.w 2
91.bd odd 12 2 1183.2.e.e 4
273.bp odd 6 1 819.2.bm.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 1.a even 1 1 trivial
91.2.k.a 2 91.k even 6 1 inner
91.2.u.a yes 2 7.c even 3 1
91.2.u.a yes 2 13.e even 6 1
637.2.k.b 2 7.b odd 2 1
637.2.k.b 2 91.l odd 6 1
637.2.q.b 2 7.d odd 6 1
637.2.q.b 2 91.p odd 6 1
637.2.q.c 2 7.c even 3 1
637.2.q.c 2 91.u even 6 1
637.2.u.a 2 7.d odd 6 1
637.2.u.a 2 91.t odd 6 1
819.2.bm.a 2 3.b odd 2 1
819.2.bm.a 2 273.bp odd 6 1
819.2.do.c 2 21.h odd 6 1
819.2.do.c 2 39.h odd 6 1
1183.2.e.e 4 13.f odd 12 2
1183.2.e.e 4 91.bd odd 12 2
8281.2.a.s 2 91.x odd 12 2
8281.2.a.w 2 91.ba even 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 13 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$43$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$47$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$73$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$79$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
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