Properties

Label 931.2.f.g
Level $931$
Weight $2$
Character orbit 931.f
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(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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - 3 \beta_{2} - 3) q^{5} + (\beta_{3} + 2) q^{6} - 3 q^{8} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - 3 \beta_{2} - 3) q^{5} + (\beta_{3} + 2) q^{6} - 3 q^{8} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{9} + ( - 3 \beta_{3} - 3 \beta_1 + 3) q^{10} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{11} + (\beta_{2} + 1) q^{12} + ( - 2 \beta_{3} + 3) q^{13} + (3 \beta_{3} + 3) q^{15} + (2 \beta_{2} - \beta_1 + 2) q^{16} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{3} + 9 \beta_{2} - \beta_1 + 1) q^{18} + (\beta_{2} + 1) q^{19} + ( - 3 \beta_{3} + 6) q^{20} + ( - 2 \beta_{3} - 1) q^{22} + (3 \beta_{2} + 3) q^{23} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 3) q^{24} + 4 \beta_{2} q^{25} + (6 \beta_{2} + 3 \beta_1 + 6) q^{26} + (4 \beta_{3} + 7) q^{27} + ( - 3 \beta_{3} + 6) q^{29} + ( - 9 \beta_{2} + 3 \beta_1 - 9) q^{30} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{31} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{32} + (9 \beta_{2} - 4 \beta_1 + 9) q^{33} + (4 \beta_{3} - 7) q^{34} + (2 \beta_{3} - 7) q^{36} + (\beta_{2} - 2 \beta_1 + 1) q^{37} + (\beta_{3} + \beta_1 - 1) q^{38} + (\beta_{3} - 4 \beta_{2} + \beta_1 - 1) q^{39} + (9 \beta_{2} + 9) q^{40} + ( - \beta_{3} - 2) q^{41} - 10 q^{43} + \beta_1 q^{44} + ( - 9 \beta_{3} + 12 \beta_{2} - 9 \beta_1 + 9) q^{45} + (3 \beta_{3} + 3 \beta_1 - 3) q^{46} + ( - 3 \beta_{2} + 4 \beta_1 - 3) q^{47} + ( - 3 \beta_{3} - 4) q^{48} + (4 \beta_{3} - 4) q^{50} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{51} + (5 \beta_{3} + 7 \beta_{2} + 5 \beta_1 - 5) q^{52} + ( - 3 \beta_{3} - 3 \beta_1 + 3) q^{53} + ( - 12 \beta_{2} + 7 \beta_1 - 12) q^{54} + ( - 3 \beta_{3} - 6) q^{55} + ( - \beta_{3} - 1) q^{57} + (9 \beta_{2} + 6 \beta_1 + 9) q^{58} + (4 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 4) q^{59} - 3 \beta_{2} q^{60} + (5 \beta_{2} - 4 \beta_1 + 5) q^{61} + 3 q^{62} + (6 \beta_{3} - 5) q^{64} + ( - 3 \beta_{2} - 6 \beta_1 - 3) q^{65} + (5 \beta_{3} - 12 \beta_{2} + 5 \beta_1 - 5) q^{66} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 3) q^{67} + ( - 6 \beta_{2} - 5 \beta_1 - 6) q^{68} + ( - 3 \beta_{3} - 3) q^{69} + ( - 4 \beta_{3} + 7) q^{71} + (12 \beta_{2} - 9 \beta_1 + 12) q^{72} + ( - 5 \beta_{3} + 10 \beta_{2} - 5 \beta_1 + 5) q^{73} + ( - \beta_{3} - 6 \beta_{2} - \beta_1 + 1) q^{74} + ( - 8 \beta_{2} + 4 \beta_1 - 8) q^{75} + (\beta_{3} - 2) q^{76} - 3 \beta_{3} q^{78} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{79} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 3) q^{80} + ( - 6 \beta_{3} + 22 \beta_{2} - 6 \beta_1 + 6) q^{81} + (3 \beta_{2} - 2 \beta_1 + 3) q^{82} + ( - 3 \beta_{3} + 9) q^{83} + ( - 3 \beta_{3} + 12) q^{85} - 10 \beta_1 q^{86} - 3 \beta_{2} q^{87} + ( - 3 \beta_{3} + 9 \beta_{2} - 3 \beta_1 + 3) q^{88} + ( - 6 \beta_{2} - 2 \beta_1 - 6) q^{89} + (3 \beta_{3} + 24) q^{90} + (3 \beta_{3} - 6) q^{92} + ( - 5 \beta_{2} + 2 \beta_1 - 5) q^{93} + (\beta_{3} + 12 \beta_{2} + \beta_1 - 1) q^{94} - 3 \beta_{2} q^{95} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{96} + (2 \beta_{3} - 7) q^{97} + ( - 10 \beta_{3} - 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 3 q^{3} - 3 q^{4} - 6 q^{5} + 10 q^{6} - 12 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - 3 q^{3} - 3 q^{4} - 6 q^{5} + 10 q^{6} - 12 q^{8} - 5 q^{9} + 3 q^{10} + 5 q^{11} + 2 q^{12} + 8 q^{13} + 18 q^{15} + 3 q^{16} - 7 q^{17} - 17 q^{18} + 2 q^{19} + 18 q^{20} - 8 q^{22} + 6 q^{23} + 9 q^{24} - 8 q^{25} + 15 q^{26} + 36 q^{27} + 18 q^{29} - 15 q^{30} - q^{31} - 7 q^{32} + 14 q^{33} - 20 q^{34} - 24 q^{36} - q^{38} + 7 q^{39} + 18 q^{40} - 10 q^{41} - 40 q^{43} + q^{44} - 15 q^{45} - 3 q^{46} - 2 q^{47} - 22 q^{48} - 8 q^{50} - 4 q^{51} - 19 q^{52} + 3 q^{53} - 17 q^{54} - 30 q^{55} - 6 q^{57} + 24 q^{58} + 2 q^{59} + 6 q^{60} + 6 q^{61} + 12 q^{62} - 8 q^{64} - 12 q^{65} + 19 q^{66} - 7 q^{67} - 17 q^{68} - 18 q^{69} + 20 q^{71} + 15 q^{72} - 15 q^{73} + 13 q^{74} - 12 q^{75} - 6 q^{76} - 6 q^{78} - 8 q^{79} + 9 q^{80} - 38 q^{81} + 4 q^{82} + 30 q^{83} + 42 q^{85} - 10 q^{86} + 6 q^{87} - 15 q^{88} - 14 q^{89} + 102 q^{90} - 18 q^{92} - 8 q^{93} - 25 q^{94} + 6 q^{95} - 4 q^{96} - 24 q^{97} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

Character values

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

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

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
−0.651388 + 1.12824i
1.15139 1.99426i
−0.651388 1.12824i
1.15139 + 1.99426i
−0.651388 + 1.12824i −1.65139 2.86029i 0.151388 + 0.262211i −1.50000 + 2.59808i 4.30278 0 −3.00000 −3.95416 + 6.84881i −1.95416 3.38471i
324.2 1.15139 1.99426i 0.151388 + 0.262211i −1.65139 2.86029i −1.50000 + 2.59808i 0.697224 0 −3.00000 1.45416 2.51868i 3.45416 + 5.98279i
704.1 −0.651388 1.12824i −1.65139 + 2.86029i 0.151388 0.262211i −1.50000 2.59808i 4.30278 0 −3.00000 −3.95416 6.84881i −1.95416 + 3.38471i
704.2 1.15139 + 1.99426i 0.151388 0.262211i −1.65139 + 2.86029i −1.50000 2.59808i 0.697224 0 −3.00000 1.45416 + 2.51868i 3.45416 5.98279i
\(n\): 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.g 4
7.b odd 2 1 931.2.f.h 4
7.c even 3 1 931.2.a.g 2
7.c even 3 1 inner 931.2.f.g 4
7.d odd 6 1 133.2.a.b 2
7.d odd 6 1 931.2.f.h 4
21.g even 6 1 1197.2.a.h 2
21.h odd 6 1 8379.2.a.bf 2
28.f even 6 1 2128.2.a.l 2
35.i odd 6 1 3325.2.a.n 2
56.j odd 6 1 8512.2.a.bh 2
56.m even 6 1 8512.2.a.l 2
133.o even 6 1 2527.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.a.b 2 7.d odd 6 1
931.2.a.g 2 7.c even 3 1
931.2.f.g 4 1.a even 1 1 trivial
931.2.f.g 4 7.c even 3 1 inner
931.2.f.h 4 7.b odd 2 1
931.2.f.h 4 7.d odd 6 1
1197.2.a.h 2 21.g even 6 1
2128.2.a.l 2 28.f even 6 1
2527.2.a.d 2 133.o even 6 1
3325.2.a.n 2 35.i odd 6 1
8379.2.a.bf 2 21.h odd 6 1
8512.2.a.l 2 56.m even 6 1
8512.2.a.bh 2 56.j odd 6 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{4} + 3T_{3}^{3} + 10T_{3}^{2} - 3T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + 4 T^{2} + 3 T + 9 \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + 10 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 5 T^{3} + 22 T^{2} - 15 T + 9 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 7 T^{3} + 40 T^{2} + 63 T + 81 \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 9 T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} + 4 T^{2} - 3 T + 9 \) Copy content Toggle raw display
$37$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$41$ \( (T^{2} + 5 T + 3)^{2} \) Copy content Toggle raw display
$43$ \( (T + 10)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} + 55 T^{2} + \cdots + 2601 \) Copy content Toggle raw display
$53$ \( T^{4} - 3 T^{3} + 36 T^{2} + 81 T + 729 \) Copy content Toggle raw display
$59$ \( T^{4} - 2 T^{3} + 55 T^{2} + \cdots + 2601 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + 79 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$67$ \( T^{4} + 7 T^{3} + 66 T^{2} - 119 T + 289 \) Copy content Toggle raw display
$71$ \( (T^{2} - 10 T - 27)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 15 T^{3} + 250 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + 100 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$83$ \( (T^{2} - 15 T + 27)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 14 T^{3} + 160 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$97$ \( (T^{2} + 12 T + 23)^{2} \) Copy content Toggle raw display
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