Properties

Label 91.2.f.a
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: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1 - 1) q^{2} + (\beta_{3} + \beta_1 + 1) q^{3} + (3 \beta_{2} + 3 \beta_1) q^{4} + ( - \beta_{2} + 1) q^{5} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{6} + \beta_{3} q^{7} + ( - 4 \beta_{2} + 1) q^{8} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1 - 1) q^{2} + (\beta_{3} + \beta_1 + 1) q^{3} + (3 \beta_{2} + 3 \beta_1) q^{4} + ( - \beta_{2} + 1) q^{5} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{6} + \beta_{3} q^{7} + ( - 4 \beta_{2} + 1) q^{8} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{9} + ( - 2 \beta_{3} - 3 \beta_1 - 2) q^{10} + (3 \beta_{3} - 3 \beta_1 + 3) q^{11} + (6 \beta_{2} - 3) q^{12} + ( - 3 \beta_{3} - 4) q^{13} + ( - \beta_{2} + 1) q^{14} + (2 \beta_{3} + 3 \beta_1 + 2) q^{15} + ( - 5 \beta_{3} - 3 \beta_1 - 5) q^{16} + (5 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{17} + ( - 5 \beta_{2} + 2) q^{18} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{19} + (3 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{20} + (\beta_{2} - 1) q^{21} + (3 \beta_{2} + 3 \beta_1) q^{22} + ( - 2 \beta_{3} + 4 \beta_1 - 2) q^{23} + (5 \beta_{3} + 9 \beta_1 + 5) q^{24} + ( - 3 \beta_{2} - 3) q^{25} + (4 \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 1) q^{26} + (2 \beta_{2} + 1) q^{27} - 3 \beta_1 q^{28} + (\beta_{3} - 5 \beta_1 + 1) q^{29} + ( - 5 \beta_{3} - 8 \beta_{2} - 8 \beta_1) q^{30} + (6 \beta_{2} + 5) q^{31} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{32} + ( - 3 \beta_{2} - 3 \beta_1) q^{33} + (3 \beta_{2} + 1) q^{34} + (\beta_{3} + \beta_{2} + \beta_1) q^{35} + ( - 9 \beta_{3} - 6 \beta_1 - 9) q^{36} + ( - 4 \beta_{3} - 4) q^{37} + 3 \beta_{2} q^{38} + ( - 4 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 1) q^{39} + ( - 9 \beta_{2} + 5) q^{40} + ( - 4 \beta_{3} + 2 \beta_1 - 4) q^{41} + (2 \beta_{3} + 3 \beta_1 + 2) q^{42} + ( - 2 \beta_{3} + 9 \beta_{2} + 9 \beta_1) q^{43} + 9 q^{44} + (2 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{45} + ( - 2 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{46} + (2 \beta_{2} + 1) q^{47} + ( - 8 \beta_{3} - 11 \beta_{2} - 11 \beta_1) q^{48} + ( - \beta_{3} - 1) q^{49} - 3 \beta_1 q^{50} + ( - 3 \beta_{2} - 1) q^{51} + ( - 12 \beta_{2} - 3 \beta_1) q^{52} + (2 \beta_{2} + 7) q^{53} + (\beta_{3} + 3 \beta_1 + 1) q^{54} - 3 \beta_1 q^{55} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{56} - 3 \beta_{2} q^{57} + (4 \beta_{3} + 9 \beta_{2} + 9 \beta_1) q^{58} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{59} + (15 \beta_{2} - 9) q^{60} - 6 \beta_{3} q^{61} + (\beta_{3} + 7 \beta_1 + 1) q^{62} + (\beta_{3} - 3 \beta_1 + 1) q^{63} + ( - 6 \beta_{2} - 1) q^{64} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 4) q^{65} + (6 \beta_{2} - 3) q^{66} + (3 \beta_{3} + 6 \beta_1 + 3) q^{67} + (12 \beta_{3} - 3 \beta_1 + 12) q^{68} + (2 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{69} + ( - 3 \beta_{2} + 2) q^{70} + (2 \beta_{3} - 10 \beta_{2} - 10 \beta_1) q^{71} + (11 \beta_{3} + 11 \beta_{2} + 11 \beta_1) q^{72} - 2 q^{73} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{74} + 3 \beta_1 q^{75} + (9 \beta_{3} + 9) q^{76} + ( - 3 \beta_{2} - 3) q^{77} + (2 \beta_{3} + 12 \beta_{2} + 3 \beta_1 - 6) q^{78} + 4 q^{79} + ( - 8 \beta_{3} - 11 \beta_1 - 8) q^{80} + ( - 4 \beta_{3} + 6 \beta_1 - 4) q^{81} + 2 \beta_{3} q^{82} + ( - 6 \beta_{2} - 3) q^{83} + ( - 3 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{84} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{85} + ( - 16 \beta_{2} + 7) q^{86} + ( - 4 \beta_{3} - 9 \beta_{2} - 9 \beta_1) q^{87} + ( - 9 \beta_{3} - 3 \beta_1 - 9) q^{88} + ( - 13 \beta_{3} + 5 \beta_1 - 13) q^{89} + ( - 12 \beta_{2} + 7) q^{90} + ( - \beta_{3} + 3) q^{91} + (6 \beta_{2} - 12) q^{92} + ( - \beta_{3} - 7 \beta_1 - 1) q^{93} + (\beta_{3} + 3 \beta_1 + 1) q^{94} + ( - 3 \beta_{2} - 3 \beta_1) q^{95} + (12 \beta_{2} - 9) q^{96} + (14 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{97} + (\beta_{3} + \beta_{2} + \beta_1) q^{98} + (3 \beta_{2} + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 3 q^{3} - 3 q^{4} + 6 q^{5} + 7 q^{6} - 2 q^{7} + 12 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} + 3 q^{3} - 3 q^{4} + 6 q^{5} + 7 q^{6} - 2 q^{7} + 12 q^{8} - q^{9} - 7 q^{10} + 3 q^{11} - 24 q^{12} - 10 q^{13} + 6 q^{14} + 7 q^{15} - 13 q^{16} - 6 q^{17} + 18 q^{18} - 3 q^{19} - 12 q^{20} - 6 q^{21} - 3 q^{22} + 19 q^{24} - 6 q^{25} - 6 q^{26} - 3 q^{28} - 3 q^{29} + 18 q^{30} + 8 q^{31} - 15 q^{32} + 3 q^{33} - 2 q^{34} - 3 q^{35} - 24 q^{36} - 8 q^{37} - 6 q^{38} + 6 q^{39} + 38 q^{40} - 6 q^{41} + 7 q^{42} - 5 q^{43} + 36 q^{44} - 9 q^{45} + 10 q^{46} + 27 q^{48} - 2 q^{49} - 3 q^{50} + 2 q^{51} + 21 q^{52} + 24 q^{53} + 5 q^{54} - 3 q^{55} - 6 q^{56} + 6 q^{57} - 17 q^{58} - 66 q^{60} + 12 q^{61} + 9 q^{62} - q^{63} + 8 q^{64} - 15 q^{65} - 24 q^{66} + 12 q^{67} + 21 q^{68} - 10 q^{69} + 14 q^{70} + 6 q^{71} - 33 q^{72} - 8 q^{73} - 12 q^{74} + 3 q^{75} + 18 q^{76} - 6 q^{77} - 49 q^{78} + 16 q^{79} - 27 q^{80} - 2 q^{81} - 4 q^{82} + 12 q^{84} + q^{85} + 60 q^{86} + 17 q^{87} - 21 q^{88} - 21 q^{89} + 52 q^{90} + 14 q^{91} - 60 q^{92} - 9 q^{93} + 5 q^{94} + 3 q^{95} - 60 q^{96} - 31 q^{97} - 3 q^{98} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(66\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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
0.809017 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 0.535233i
−1.30902 + 2.26728i 1.30902 2.26728i −2.42705 4.20378i 2.61803 3.42705 + 5.93583i −0.500000 0.866025i 7.47214 −1.92705 3.33775i −3.42705 + 5.93583i
22.2 −0.190983 + 0.330792i 0.190983 0.330792i 0.927051 + 1.60570i 0.381966 0.0729490 + 0.126351i −0.500000 0.866025i −1.47214 1.42705 + 2.47172i −0.0729490 + 0.126351i
29.1 −1.30902 2.26728i 1.30902 + 2.26728i −2.42705 + 4.20378i 2.61803 3.42705 5.93583i −0.500000 + 0.866025i 7.47214 −1.92705 + 3.33775i −3.42705 5.93583i
29.2 −0.190983 0.330792i 0.190983 + 0.330792i 0.927051 1.60570i 0.381966 0.0729490 0.126351i −0.500000 + 0.866025i −1.47214 1.42705 2.47172i −0.0729490 0.126351i
\(n\): 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.a 4
3.b odd 2 1 819.2.o.c 4
4.b odd 2 1 1456.2.s.h 4
7.b odd 2 1 637.2.f.c 4
7.c even 3 1 637.2.g.b 4
7.c even 3 1 637.2.h.g 4
7.d odd 6 1 637.2.g.c 4
7.d odd 6 1 637.2.h.f 4
13.c even 3 1 inner 91.2.f.a 4
13.c even 3 1 1183.2.a.g 2
13.e even 6 1 1183.2.a.c 2
13.f odd 12 2 1183.2.c.c 4
39.i odd 6 1 819.2.o.c 4
52.j odd 6 1 1456.2.s.h 4
91.g even 3 1 637.2.h.g 4
91.h even 3 1 637.2.g.b 4
91.m odd 6 1 637.2.h.f 4
91.n odd 6 1 637.2.f.c 4
91.n odd 6 1 8281.2.a.bb 2
91.t odd 6 1 8281.2.a.n 2
91.v odd 6 1 637.2.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 1.a even 1 1 trivial
91.2.f.a 4 13.c even 3 1 inner
637.2.f.c 4 7.b odd 2 1
637.2.f.c 4 91.n odd 6 1
637.2.g.b 4 7.c even 3 1
637.2.g.b 4 91.h even 3 1
637.2.g.c 4 7.d odd 6 1
637.2.g.c 4 91.v odd 6 1
637.2.h.f 4 7.d odd 6 1
637.2.h.f 4 91.m odd 6 1
637.2.h.g 4 7.c even 3 1
637.2.h.g 4 91.g even 3 1
819.2.o.c 4 3.b odd 2 1
819.2.o.c 4 39.i odd 6 1
1183.2.a.c 2 13.e even 6 1
1183.2.a.g 2 13.c even 3 1
1183.2.c.c 4 13.f odd 12 2
1456.2.s.h 4 4.b odd 2 1
1456.2.s.h 4 52.j odd 6 1
8281.2.a.n 2 91.t odd 6 1
8281.2.a.bb 2 91.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{3} + 8T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + 8 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + 8 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + 47 T^{2} - 66 T + 121 \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} + 18 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$23$ \( T^{4} + 20T^{2} + 400 \) Copy content Toggle raw display
$29$ \( T^{4} + 3 T^{3} + 38 T^{2} - 87 T + 841 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 41)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + 32 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + 120 T^{2} + \cdots + 9025 \) Copy content Toggle raw display
$47$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 31)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 5T^{2} + 25 \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + 153 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + 152 T^{2} + \cdots + 13456 \) Copy content Toggle raw display
$73$ \( (T + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 21 T^{3} + 362 T^{2} + \cdots + 6241 \) Copy content Toggle raw display
$97$ \( T^{4} + 31 T^{3} + 732 T^{2} + \cdots + 52441 \) Copy content Toggle raw display
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