Properties

Label 931.2.s.c
Level $931$
Weight $2$
Character orbit 931.s
Analytic conductor $7.434$
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] = [931,2,Mod(31,931)]
 
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([1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.s (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
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 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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(248\) \(344\)
\(\chi(n)\) \(1 - \beta_{2}\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.895644 + 1.09445i
1.39564 0.228425i
−0.895644 1.09445i
1.39564 + 0.228425i
−1.89564 1.09445i −2.79129 1.39564 + 2.41733i 3.39564 + 1.96048i 5.29129 + 3.05493i 0 1.73205i 4.79129 −4.29129 7.43273i
31.2 0.395644 + 0.228425i 1.79129 −0.895644 1.55130i 1.10436 + 0.637600i 0.708712 + 0.409175i 0 1.73205i 0.208712 0.291288 + 0.504525i
901.1 −1.89564 + 1.09445i −2.79129 1.39564 2.41733i 3.39564 1.96048i 5.29129 3.05493i 0 1.73205i 4.79129 −4.29129 + 7.43273i
901.2 0.395644 0.228425i 1.79129 −0.895644 + 1.55130i 1.10436 0.637600i 0.708712 0.409175i 0 1.73205i 0.208712 0.291288 0.504525i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.s.c 4
7.b odd 2 1 133.2.s.c yes 4
7.c even 3 1 133.2.i.c 4
7.c even 3 1 931.2.p.f 4
7.d odd 6 1 931.2.i.d 4
7.d odd 6 1 931.2.p.e 4
19.d odd 6 1 931.2.i.d 4
133.i even 6 1 931.2.p.f 4
133.j odd 6 1 931.2.p.e 4
133.n odd 6 1 133.2.s.c yes 4
133.p even 6 1 133.2.i.c 4
133.s even 6 1 inner 931.2.s.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.i.c 4 7.c even 3 1
133.2.i.c 4 133.p even 6 1
133.2.s.c yes 4 7.b odd 2 1
133.2.s.c yes 4 133.n odd 6 1
931.2.i.d 4 7.d odd 6 1
931.2.i.d 4 19.d odd 6 1
931.2.p.e 4 7.d odd 6 1
931.2.p.e 4 133.j odd 6 1
931.2.p.f 4 7.c even 3 1
931.2.p.f 4 133.i even 6 1
931.2.s.c 4 1.a even 1 1 trivial
931.2.s.c 4 133.s even 6 1 inner

Hecke kernels

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

\( T_{2}^{4} + 3T_{2}^{3} + 2T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} + T_{3} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + 2 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T - 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 9 T^{3} + 32 T^{2} - 45 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 3 T^{3} + 12 T^{2} + 9 T + 9 \) Copy content Toggle raw display
$17$ \( T^{4} + 20T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 19)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T - 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} - 13 T^{2} + 150 T + 625 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 21 T^{3} + 168 T^{2} + \cdots + 441 \) Copy content Toggle raw display
$41$ \( T^{4} + 3 T^{3} + 54 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$43$ \( T^{4} - 9 T^{3} + 66 T^{2} - 135 T + 225 \) Copy content Toggle raw display
$47$ \( T^{4} + 38T^{2} + 25 \) Copy content Toggle raw display
$53$ \( T^{4} - 9 T^{3} + 32 T^{2} - 45 T + 25 \) Copy content Toggle raw display
$59$ \( (T^{2} + 15 T + 51)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 45T^{2} + 81 \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} - 13 T^{2} + 150 T + 625 \) Copy content Toggle raw display
$73$ \( T^{4} + 132T^{2} + 3600 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 209T^{2} + 4489 \) Copy content Toggle raw display
$89$ \( (T^{2} - 9 T - 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
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