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Properties

Label 931.2.c.e
Level $931$
Weight $2$
Character orbit 931.c
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(930,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.930");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.c (of order \(2\), degree \(1\), 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: \(6\)
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: \( 2^{3} \)
Twist minimal: no (minimal twist has level 133)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_1 + 1) q^{3} - q^{4} - \beta_{4} q^{5} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{6} - \beta_{3} q^{8} + ( - \beta_{2} + \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_1 + 1) q^{3} - q^{4} - \beta_{4} q^{5} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{6} - \beta_{3} q^{8} + ( - \beta_{2} + \beta_1 + 3) q^{9} + ( - \beta_{2} - \beta_1 + 1) q^{10} + ( - \beta_1 - 2) q^{11} + ( - \beta_1 - 1) q^{12} + ( - \beta_{2} - 1) q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{15} - 5 q^{16} + (\beta_{4} - \beta_{3}) q^{17} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{18} + (\beta_{5} + \beta_{3} + \beta_1 + 1) q^{19} + \beta_{4} q^{20} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{22} + \beta_{2} q^{23} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{24} + 2 \beta_1 q^{25} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{26} + ( - 2 \beta_{2} + 2 \beta_1 + 6) q^{27} + ( - \beta_{5} - 2 \beta_{4}) q^{29} + (3 \beta_1 - 3) q^{30} + (\beta_1 - 5) q^{31} + 3 \beta_{3} q^{32} + (\beta_{2} - 2 \beta_1 - 7) q^{33} + (\beta_{2} + \beta_1 - 4) q^{34} + (\beta_{2} - \beta_1 - 3) q^{36} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{37} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{38}+ \cdots + (3 \beta_{2} - 6 \beta_1 - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 6 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 6 q^{4} + 14 q^{9} + 6 q^{10} - 10 q^{11} - 4 q^{12} - 8 q^{13} - 30 q^{16} + 4 q^{19} + 2 q^{23} - 4 q^{25} + 28 q^{27} - 24 q^{30} - 32 q^{31} - 36 q^{33} - 24 q^{34} - 14 q^{36} + 18 q^{38} - 4 q^{39} + 6 q^{40} - 12 q^{41} + 10 q^{43} + 10 q^{44} - 20 q^{48} + 8 q^{52} + 32 q^{57} + 12 q^{58} - 36 q^{59} - 6 q^{64} - 24 q^{74} + 56 q^{75} - 4 q^{76} + 38 q^{81} + 40 q^{85} + 54 q^{90} - 2 q^{92} + 8 q^{93} + 54 q^{94} - 2 q^{95} + 8 q^{97} - 54 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 8\nu^{4} + 64\nu^{3} - 50\nu^{2} + 7\nu - 56 ) / 393 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{5} + 64\nu^{4} - 119\nu^{3} + 400\nu^{2} - 56\nu + 2020 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 112\nu^{5} - 110\nu^{4} + 880\nu^{3} + 688\nu^{2} + 5500\nu - 377 ) / 393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 147\nu^{5} - 128\nu^{4} + 1155\nu^{3} + 772\nu^{2} + 7317\nu - 503 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 148\nu^{5} - 136\nu^{4} + 1219\nu^{3} + 722\nu^{2} + 8110\nu - 559 ) / 393 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 8\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{5} + 30\beta_{4} - 31\beta_{3} + 8\beta_{2} + 14\beta _1 - 39 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -55\beta_{5} + 97\beta_{4} - 48\beta_{3} - 14\beta_{2} - 69\beta _1 + 62 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(248\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
930.1
1.56632 + 2.71294i
0.0702177 + 0.121621i
−1.13654 − 1.96854i
1.56632 − 2.71294i
0.0702177 − 0.121621i
−1.13654 + 1.96854i
− 1.73205i −2.13264 −1.00000 3.35638i 3.69384i 0 − 1.73205i 1.54814 5.81342
930.2 − 1.73205i 0.859565 −1.00000 − 2.29801i − 1.48881i 0 − 1.73205i −2.26115 −3.98028
930.3 − 1.73205i 3.27307 −1.00000 0.673687i − 5.66913i 0 − 1.73205i 7.71301 1.16686
930.4 1.73205i −2.13264 −1.00000 − 3.35638i − 3.69384i 0 1.73205i 1.54814 5.81342
930.5 1.73205i 0.859565 −1.00000 2.29801i 1.48881i 0 1.73205i −2.26115 −3.98028
930.6 1.73205i 3.27307 −1.00000 − 0.673687i 5.66913i 0 1.73205i 7.71301 1.16686
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 930.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.c.e 6
7.b odd 2 1 931.2.c.d 6
7.c even 3 1 133.2.o.e yes 6
7.c even 3 1 931.2.o.e 6
7.d odd 6 1 133.2.o.d ✓ 6
7.d odd 6 1 931.2.o.f 6
19.b odd 2 1 931.2.c.d 6
133.c even 2 1 inner 931.2.c.e 6
133.o even 6 1 133.2.o.e yes 6
133.o even 6 1 931.2.o.e 6
133.r odd 6 1 133.2.o.d ✓ 6
133.r odd 6 1 931.2.o.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.o.d ✓ 6 7.d odd 6 1
133.2.o.d ✓ 6 133.r odd 6 1
133.2.o.e yes 6 7.c even 3 1
133.2.o.e yes 6 133.o even 6 1
931.2.c.d 6 7.b odd 2 1
931.2.c.d 6 19.b odd 2 1
931.2.c.e 6 1.a even 1 1 trivial
931.2.c.e 6 133.c even 2 1 inner
931.2.o.e 6 7.c even 3 1
931.2.o.e 6 133.o even 6 1
931.2.o.f 6 7.d odd 6 1
931.2.o.f 6 133.r odd 6 1

Hecke kernels

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

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{3} - 2T_{3}^{2} - 6T_{3} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3)^{3} \) Copy content Toggle raw display
$3$ \( (T^{3} - 2 T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 17 T^{4} + \cdots + 27 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} + T - 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 4 T^{2} - 12 T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 32 T^{4} + \cdots + 48 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( (T^{3} - T^{2} - 17 T - 21)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 108 T^{4} + \cdots + 38988 \) Copy content Toggle raw display
$31$ \( (T^{3} + 16 T^{2} + \cdots + 114)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 120 T^{4} + \cdots + 972 \) 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
$47$ \( T^{6} + 113 T^{4} + \cdots + 6627 \) Copy content Toggle raw display
$53$ \( T^{6} + 276 T^{4} + \cdots + 428652 \) Copy content Toggle raw display
$59$ \( (T + 6)^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 41 T^{4} + \cdots + 3 \) Copy content Toggle raw display
$67$ \( T^{6} + 180 T^{4} + \cdots + 1728 \) Copy content Toggle raw display
$71$ \( T^{6} + 108 T^{4} + \cdots + 108 \) Copy content Toggle raw display
$73$ \( T^{6} + 177 T^{4} + \cdots + 11907 \) Copy content Toggle raw display
$79$ \( T^{6} + 396 T^{4} + \cdots + 38988 \) Copy content Toggle raw display
$83$ \( T^{6} + 341 T^{4} + \cdots + 1232643 \) Copy content Toggle raw display
$89$ \( (T^{3} - 48 T + 126)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 4 T^{2} + \cdots + 288)^{2} \) Copy content Toggle raw display
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