Properties

Label 931.2.f.b
Level $931$
Weight $2$
Character orbit 931.f
Analytic conductor $7.434$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(324,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.f (of order \(3\), degree \(2\), not 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: \(7.43407242818\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + 3 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + 3 \zeta_{6} q^{5} - \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + 4 \zeta_{6} q^{12} + 4 q^{13} - 6 q^{15} - 4 \zeta_{6} q^{16} + (3 \zeta_{6} - 3) q^{17} + \zeta_{6} q^{19} + 6 q^{20} + (4 \zeta_{6} - 4) q^{25} - 4 q^{27} + 6 q^{29} + (4 \zeta_{6} - 4) q^{31} - 6 \zeta_{6} q^{33} - 2 q^{36} - 2 \zeta_{6} q^{37} + (8 \zeta_{6} - 8) q^{39} + 6 q^{41} - q^{43} + 6 \zeta_{6} q^{44} + ( - 3 \zeta_{6} + 3) q^{45} - 3 \zeta_{6} q^{47} + 8 q^{48} - 6 \zeta_{6} q^{51} + ( - 8 \zeta_{6} + 8) q^{52} + (12 \zeta_{6} - 12) q^{53} - 9 q^{55} - 2 q^{57} + (6 \zeta_{6} - 6) q^{59} + (12 \zeta_{6} - 12) q^{60} - \zeta_{6} q^{61} - 8 q^{64} + 12 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} + 6 \zeta_{6} q^{68} + 6 q^{71} + (7 \zeta_{6} - 7) q^{73} - 8 \zeta_{6} q^{75} + 2 q^{76} - 8 \zeta_{6} q^{79} + ( - 12 \zeta_{6} + 12) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} - 12 q^{83} - 9 q^{85} + (12 \zeta_{6} - 12) q^{87} + 12 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} + (3 \zeta_{6} - 3) q^{95} - 8 q^{97} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 3 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} + 3 q^{5} - q^{9} - 3 q^{11} + 4 q^{12} + 8 q^{13} - 12 q^{15} - 4 q^{16} - 3 q^{17} + q^{19} + 12 q^{20} - 4 q^{25} - 8 q^{27} + 12 q^{29} - 4 q^{31} - 6 q^{33} - 4 q^{36} - 2 q^{37} - 8 q^{39} + 12 q^{41} - 2 q^{43} + 6 q^{44} + 3 q^{45} - 3 q^{47} + 16 q^{48} - 6 q^{51} + 8 q^{52} - 12 q^{53} - 18 q^{55} - 4 q^{57} - 6 q^{59} - 12 q^{60} - q^{61} - 16 q^{64} + 12 q^{65} + 4 q^{67} + 6 q^{68} + 12 q^{71} - 7 q^{73} - 8 q^{75} + 4 q^{76} - 8 q^{79} + 12 q^{80} + 11 q^{81} - 24 q^{83} - 18 q^{85} - 12 q^{87} + 12 q^{89} - 8 q^{93} - 3 q^{95} - 16 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/931\mathbb{Z}\right)^\times\).

\(n\) \(248\) \(344\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 1.73205i 1.00000 + 1.73205i 1.50000 2.59808i 0 0 0 −0.500000 + 0.866025i 0
704.1 0 −1.00000 + 1.73205i 1.00000 1.73205i 1.50000 + 2.59808i 0 0 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.f.b 2
7.b odd 2 1 931.2.f.c 2
7.c even 3 1 931.2.a.a 1
7.c even 3 1 inner 931.2.f.b 2
7.d odd 6 1 19.2.a.a 1
7.d odd 6 1 931.2.f.c 2
21.g even 6 1 171.2.a.b 1
21.h odd 6 1 8379.2.a.j 1
28.f even 6 1 304.2.a.f 1
35.i odd 6 1 475.2.a.b 1
35.k even 12 2 475.2.b.a 2
56.j odd 6 1 1216.2.a.o 1
56.m even 6 1 1216.2.a.b 1
77.i even 6 1 2299.2.a.b 1
84.j odd 6 1 2736.2.a.c 1
91.s odd 6 1 3211.2.a.a 1
105.p even 6 1 4275.2.a.i 1
119.h odd 6 1 5491.2.a.b 1
133.i even 6 1 361.2.c.a 2
133.k odd 6 1 361.2.c.c 2
133.o even 6 1 361.2.a.b 1
133.s even 6 1 361.2.c.a 2
133.t odd 6 1 361.2.c.c 2
133.x odd 18 3 361.2.e.d 6
133.z odd 18 3 361.2.e.d 6
133.bb even 18 3 361.2.e.e 6
133.bf even 18 3 361.2.e.e 6
140.s even 6 1 7600.2.a.c 1
399.s odd 6 1 3249.2.a.d 1
532.bh odd 6 1 5776.2.a.c 1
665.y even 6 1 9025.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 7.d odd 6 1
171.2.a.b 1 21.g even 6 1
304.2.a.f 1 28.f even 6 1
361.2.a.b 1 133.o even 6 1
361.2.c.a 2 133.i even 6 1
361.2.c.a 2 133.s even 6 1
361.2.c.c 2 133.k odd 6 1
361.2.c.c 2 133.t odd 6 1
361.2.e.d 6 133.x odd 18 3
361.2.e.d 6 133.z odd 18 3
361.2.e.e 6 133.bb even 18 3
361.2.e.e 6 133.bf even 18 3
475.2.a.b 1 35.i odd 6 1
475.2.b.a 2 35.k even 12 2
931.2.a.a 1 7.c even 3 1
931.2.f.b 2 1.a even 1 1 trivial
931.2.f.b 2 7.c even 3 1 inner
931.2.f.c 2 7.b odd 2 1
931.2.f.c 2 7.d odd 6 1
1216.2.a.b 1 56.m even 6 1
1216.2.a.o 1 56.j odd 6 1
2299.2.a.b 1 77.i even 6 1
2736.2.a.c 1 84.j odd 6 1
3211.2.a.a 1 91.s odd 6 1
3249.2.a.d 1 399.s odd 6 1
4275.2.a.i 1 105.p even 6 1
5491.2.a.b 1 119.h odd 6 1
5776.2.a.c 1 532.bh odd 6 1
7600.2.a.c 1 140.s even 6 1
8379.2.a.j 1 21.h odd 6 1
9025.2.a.d 1 665.y even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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