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Properties

Label 931.2.o.f
Level $931$
Weight $2$
Character orbit 931.o
Analytic conductor $7.434$
Analytic rank $0$
Dimension $6$
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(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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.6967728.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 8\nu^{4} + 64\nu^{3} - 50\nu^{2} + 7\nu - 56 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 56\nu^{5} - 55\nu^{4} + 440\nu^{3} + 344\nu^{2} + 2750\nu - 385 ) / 393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 70\nu^{5} - 36\nu^{4} + 550\nu^{3} + 561\nu^{2} + 3634\nu + 534 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 77\nu^{5} - 92\nu^{4} + 605\nu^{3} + 211\nu^{2} + 3683\nu - 1430 ) / 393 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} - \beta_{4} + 4\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 7\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} + 16\beta_{4} - 31\beta_{3} - 6\beta _1 - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 28\beta_{5} + 14\beta_{4} - 48\beta_{3} - 55\beta_{2} - 55\beta _1 + 28 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
1.56632 − 2.71294i
0.0702177 − 0.121621i
−1.13654 + 1.96854i
1.56632 + 2.71294i
0.0702177 + 0.121621i
−1.13654 − 1.96854i
1.50000 − 0.866025i −1.06632 + 1.84692i 0.500000 − 0.866025i 2.90671 − 1.67819i 3.69384i 0 1.73205i −0.774071 − 1.34073i 2.90671 − 5.03457i
227.2 1.50000 − 0.866025i 0.429782 − 0.744405i 0.500000 − 0.866025i −1.99014 + 1.14901i − 1.48881i 0 1.73205i 1.13057 + 1.95821i −1.99014 + 3.44702i
227.3 1.50000 − 0.866025i 1.63654 − 2.83456i 0.500000 − 0.866025i 0.583430 − 0.336844i − 5.66913i 0 1.73205i −3.85650 − 6.67966i 0.583430 − 1.01053i
607.1 1.50000 + 0.866025i −1.06632 − 1.84692i 0.500000 + 0.866025i 2.90671 + 1.67819i − 3.69384i 0 − 1.73205i −0.774071 + 1.34073i 2.90671 + 5.03457i
607.2 1.50000 + 0.866025i 0.429782 + 0.744405i 0.500000 + 0.866025i −1.99014 − 1.14901i 1.48881i 0 − 1.73205i 1.13057 − 1.95821i −1.99014 − 3.44702i
607.3 1.50000 + 0.866025i 1.63654 + 2.83456i 0.500000 + 0.866025i 0.583430 + 0.336844i 5.66913i 0 − 1.73205i −3.85650 + 6.67966i 0.583430 + 1.01053i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 227.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.f 6
7.b odd 2 1 133.2.o.e yes 6
7.c even 3 1 133.2.o.d ✓ 6
7.c even 3 1 931.2.c.d 6
7.d odd 6 1 931.2.c.e 6
7.d odd 6 1 931.2.o.e 6
19.b odd 2 1 931.2.o.e 6
133.c even 2 1 133.2.o.d ✓ 6
133.o even 6 1 931.2.c.d 6
133.o even 6 1 inner 931.2.o.f 6
133.r odd 6 1 133.2.o.e yes 6
133.r odd 6 1 931.2.c.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.o.d ✓ 6 7.c even 3 1
133.2.o.d ✓ 6 133.c even 2 1
133.2.o.e yes 6 7.b odd 2 1
133.2.o.e yes 6 133.r odd 6 1
931.2.c.d 6 7.c even 3 1
931.2.c.d 6 133.o even 6 1
931.2.c.e 6 7.d odd 6 1
931.2.c.e 6 133.r odd 6 1
931.2.o.e 6 7.d odd 6 1
931.2.o.e 6 19.b odd 2 1
931.2.o.f 6 1.a even 1 1 trivial
931.2.o.f 6 133.o 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 \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{2} - 3T_{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{6} - 2T_{3}^{5} + 10T_{3}^{4} + 48T_{3}^{2} - 36T_{3} + 36 \) Copy content Toggle raw display
\( T_{5}^{6} - 3T_{5}^{5} - 4T_{5}^{4} + 21T_{5}^{3} + 40T_{5}^{2} - 63T_{5} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3 T + 3)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 5 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{3} - 4 T^{2} - 12 T - 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 12 T^{5} + \cdots + 48 \) Copy content Toggle raw display
$19$ \( T^{6} - 11 T^{5} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + T^{5} + \cdots + 441 \) Copy content Toggle raw display
$29$ \( T^{6} + 108 T^{4} + \cdots + 38988 \) Copy content Toggle raw display
$31$ \( T^{6} + 16 T^{5} + \cdots + 12996 \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + \cdots + 972 \) Copy content Toggle raw display
$41$ \( (T^{3} - 6 T^{2} + \cdots + 216)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 5 T^{2} + \cdots + 167)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 27 T^{5} + \cdots + 6627 \) Copy content Toggle raw display
$53$ \( T^{6} + 6 T^{5} + \cdots + 428652 \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T + 36)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} + 15 T^{5} + \cdots + 3 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots + 1728 \) Copy content Toggle raw display
$71$ \( T^{6} + 108 T^{4} + \cdots + 108 \) Copy content Toggle raw display
$73$ \( T^{6} - 3 T^{5} + \cdots + 11907 \) Copy content Toggle raw display
$79$ \( T^{6} - 198 T^{4} + \cdots + 38988 \) Copy content Toggle raw display
$83$ \( T^{6} + 341 T^{4} + \cdots + 1232643 \) Copy content Toggle raw display
$89$ \( T^{6} + 48 T^{4} + \cdots + 15876 \) Copy content Toggle raw display
$97$ \( (T^{3} + 4 T^{2} + \cdots - 288)^{2} \) Copy content Toggle raw display
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