Properties

Label 931.2.o.h
Level $931$
Weight $2$
Character orbit 931.o
Analytic conductor $7.434$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(227,931)]
 
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, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.227");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.o (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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.592240896.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{3} + (\beta_{6} + \beta_{4} - 2 \beta_{3} + 1) q^{4} + \beta_{2} q^{5} + (3 \beta_{7} - \beta_{5}) q^{6} + 3 \beta_{7} q^{8} + (\beta_{6} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{3} + (\beta_{6} + \beta_{4} - 2 \beta_{3} + 1) q^{4} + \beta_{2} q^{5} + (3 \beta_{7} - \beta_{5}) q^{6} + 3 \beta_{7} q^{8} + (\beta_{6} - \beta_{3}) q^{9} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{10} + ( - \beta_{6} - \beta_{4} + 1) q^{11} + (2 \beta_{6} - 5 \beta_{3}) q^{12} - 3 q^{13} - \beta_{5} q^{15} + (\beta_{6} + \beta_{3}) q^{16} + ( - \beta_{7} - 3 \beta_{5} + \beta_{2} - 3 \beta_1) q^{17} + (3 \beta_{7} - \beta_{5} - 3 \beta_{2} - \beta_1) q^{18} + (\beta_{6} + 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{19} + (\beta_{7} - \beta_{5}) q^{20} - 3 \beta_{7} q^{22} + ( - 2 \beta_{6} + 3 \beta_{3}) q^{23} + ( - 3 \beta_{5} - 3 \beta_1) q^{24} + (4 \beta_{3} - 4) q^{25} - 3 \beta_1 q^{26} + (2 \beta_{4} - 3) q^{27} + ( - 3 \beta_{7} - \beta_{5}) q^{29} + (\beta_{6} - 4 \beta_{3}) q^{30} + (3 \beta_{6} + 3 \beta_{4} - 3) q^{31} + ( - 3 \beta_{7} + \beta_{5} + 3 \beta_{2} + \beta_1) q^{32} + 3 \beta_{3} q^{33} + ( - 2 \beta_{4} - 9) q^{34} + ( - 2 \beta_{4} - 3) q^{36} + 9 \beta_{2} q^{37} + (3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + \cdots + 3) q^{38}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 6 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 6 q^{4} - 2 q^{9} + 2 q^{10} + 2 q^{11} - 16 q^{12} - 24 q^{13} + 6 q^{16} + 10 q^{19} + 8 q^{23} - 16 q^{25} - 16 q^{27} - 14 q^{30} - 6 q^{31} + 12 q^{33} - 80 q^{34} - 32 q^{36} + 18 q^{38} - 6 q^{39} - 12 q^{40} + 36 q^{41} + 8 q^{43} + 10 q^{44} - 20 q^{48} - 18 q^{52} - 16 q^{57} - 8 q^{58} - 36 q^{59} + 16 q^{64} + 60 q^{69} + 18 q^{74} + 8 q^{75} + 4 q^{76} + 28 q^{81} - 4 q^{85} + 48 q^{89} - 28 q^{90} + 76 q^{92} - 36 q^{93} - 38 q^{94} + 10 q^{95} - 40 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{6} + 40x^{4} - 63x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 97\nu ) / 120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{6} + 40\nu^{4} - 280\nu^{2} + 441 ) / 360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 57 ) / 40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{7} + 40\nu^{5} - 280\nu^{3} + 81\nu ) / 360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{6} + 160\nu^{4} - 760\nu^{2} + 1197 ) / 360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} + 40\nu^{5} - 190\nu^{3} + 81\nu ) / 270 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - 4\beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{6} - 19\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21\beta_{7} - 19\beta_{5} - 21\beta_{2} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -40\beta_{4} - 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -120\beta_{2} - 97\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(248\) \(344\)
\(\chi(n)\) \(\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
227.1
−1.99426 + 1.15139i
−1.12824 + 0.651388i
1.12824 0.651388i
1.99426 1.15139i
−1.99426 1.15139i
−1.12824 0.651388i
1.12824 + 0.651388i
1.99426 + 1.15139i
−1.99426 + 1.15139i 1.15139 1.99426i 1.65139 2.86029i −0.866025 + 0.500000i 5.30278i 0 3.00000i −1.15139 1.99426i 1.15139 1.99426i
227.2 −1.12824 + 0.651388i −0.651388 + 1.12824i −0.151388 + 0.262211i 0.866025 0.500000i 1.69722i 0 3.00000i 0.651388 + 1.12824i −0.651388 + 1.12824i
227.3 1.12824 0.651388i −0.651388 + 1.12824i −0.151388 + 0.262211i −0.866025 + 0.500000i 1.69722i 0 3.00000i 0.651388 + 1.12824i −0.651388 + 1.12824i
227.4 1.99426 1.15139i 1.15139 1.99426i 1.65139 2.86029i 0.866025 0.500000i 5.30278i 0 3.00000i −1.15139 1.99426i 1.15139 1.99426i
607.1 −1.99426 1.15139i 1.15139 + 1.99426i 1.65139 + 2.86029i −0.866025 0.500000i 5.30278i 0 3.00000i −1.15139 + 1.99426i 1.15139 + 1.99426i
607.2 −1.12824 0.651388i −0.651388 1.12824i −0.151388 0.262211i 0.866025 + 0.500000i 1.69722i 0 3.00000i 0.651388 1.12824i −0.651388 1.12824i
607.3 1.12824 + 0.651388i −0.651388 1.12824i −0.151388 0.262211i −0.866025 0.500000i 1.69722i 0 3.00000i 0.651388 1.12824i −0.651388 1.12824i
607.4 1.99426 + 1.15139i 1.15139 + 1.99426i 1.65139 + 2.86029i 0.866025 + 0.500000i 5.30278i 0 3.00000i −1.15139 + 1.99426i 1.15139 + 1.99426i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 227.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
133.c even 2 1 inner
133.o 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.o.h 8
7.b odd 2 1 931.2.o.g 8
7.c even 3 1 133.2.c.b 4
7.c even 3 1 inner 931.2.o.h 8
7.d odd 6 1 133.2.c.c yes 4
7.d odd 6 1 931.2.o.g 8
19.b odd 2 1 931.2.o.g 8
21.g even 6 1 1197.2.c.e 4
21.h odd 6 1 1197.2.c.d 4
28.f even 6 1 2128.2.m.b 4
28.g odd 6 1 2128.2.m.d 4
133.c even 2 1 inner 931.2.o.h 8
133.o even 6 1 133.2.c.b 4
133.o even 6 1 inner 931.2.o.h 8
133.r odd 6 1 133.2.c.c yes 4
133.r odd 6 1 931.2.o.g 8
399.s odd 6 1 1197.2.c.d 4
399.w even 6 1 1197.2.c.e 4
532.t even 6 1 2128.2.m.b 4
532.bh odd 6 1 2128.2.m.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.c.b 4 7.c even 3 1
133.2.c.b 4 133.o even 6 1
133.2.c.c yes 4 7.d odd 6 1
133.2.c.c yes 4 133.r odd 6 1
931.2.o.g 8 7.b odd 2 1
931.2.o.g 8 7.d odd 6 1
931.2.o.g 8 19.b odd 2 1
931.2.o.g 8 133.r odd 6 1
931.2.o.h 8 1.a even 1 1 trivial
931.2.o.h 8 7.c even 3 1 inner
931.2.o.h 8 133.c even 2 1 inner
931.2.o.h 8 133.o even 6 1 inner
1197.2.c.d 4 21.h odd 6 1
1197.2.c.d 4 399.s odd 6 1
1197.2.c.e 4 21.g even 6 1
1197.2.c.e 4 399.w even 6 1
2128.2.m.b 4 28.f even 6 1
2128.2.m.b 4 532.t even 6 1
2128.2.m.d 4 28.g odd 6 1
2128.2.m.d 4 532.bh odd 6 1

Hecke kernels

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

\( T_{2}^{8} - 7T_{2}^{6} + 40T_{2}^{4} - 63T_{2}^{2} + 81 \) Copy content Toggle raw display
\( T_{3}^{4} - T_{3}^{3} + 4T_{3}^{2} + 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{4} - T_{5}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7 T^{6} + 40 T^{4} - 63 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + 4 T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + 4 T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T + 3)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 59 T^{6} + 2640 T^{4} + \cdots + 707281 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + 50 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + 25 T^{2} + 36 T + 81)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 19 T^{2} + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3 T^{3} + 36 T^{2} - 81 T + 729)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 81 T^{2} + 6561)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T - 9)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 12)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 98 T^{6} + 9075 T^{4} + \cdots + 279841 \) Copy content Toggle raw display
$53$ \( T^{8} - 7 T^{6} + 40 T^{4} - 63 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( (T^{2} + 9 T + 81)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} - 226 T^{6} + \cdots + 57289761 \) Copy content Toggle raw display
$67$ \( T^{8} - 99 T^{6} + 9720 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$71$ \( (T^{4} + 202 T^{2} + 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 99 T^{6} + 9720 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$79$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 119 T^{2} + 2809)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T + 144)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 10 T - 27)^{4} \) Copy content Toggle raw display
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