Properties

Label 117.2.q.b
Level $117$
Weight $2$
Character orbit 117.q
Analytic conductor $0.934$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 6) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 6) q^{7} + (3 \zeta_{6} - 4) q^{13} + (4 \zeta_{6} - 4) q^{16} + (2 \zeta_{6} - 4) q^{19} + 5 q^{25} + ( - 6 \zeta_{6} - 6) q^{28} + (10 \zeta_{6} - 5) q^{31} + ( - 4 \zeta_{6} - 4) q^{37} + 13 \zeta_{6} q^{43} + ( - 20 \zeta_{6} + 20) q^{49} + (2 \zeta_{6} + 6) q^{52} - 13 \zeta_{6} q^{61} + 8 q^{64} + ( - 7 \zeta_{6} - 7) q^{67} + (2 \zeta_{6} - 1) q^{73} + (4 \zeta_{6} + 4) q^{76} + 13 q^{79} + (21 \zeta_{6} - 15) q^{91} + (11 \zeta_{6} - 22) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 9 q^{7} - 5 q^{13} - 4 q^{16} - 6 q^{19} + 10 q^{25} - 18 q^{28} - 12 q^{37} + 13 q^{43} + 20 q^{49} + 14 q^{52} - 13 q^{61} + 16 q^{64} - 21 q^{67} + 12 q^{76} + 26 q^{79} - 9 q^{91} - 33 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\zeta_{6}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 −1.00000 + 1.73205i 0 0 4.50000 + 2.59808i 0 0 0
82.1 0 0 −1.00000 1.73205i 0 0 4.50000 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.e even 6 1 inner
39.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.q.b 2
3.b odd 2 1 CM 117.2.q.b 2
4.b odd 2 1 1872.2.by.a 2
12.b even 2 1 1872.2.by.a 2
13.c even 3 1 1521.2.b.d 2
13.e even 6 1 inner 117.2.q.b 2
13.e even 6 1 1521.2.b.d 2
13.f odd 12 2 1521.2.a.i 2
39.h odd 6 1 inner 117.2.q.b 2
39.h odd 6 1 1521.2.b.d 2
39.i odd 6 1 1521.2.b.d 2
39.k even 12 2 1521.2.a.i 2
52.i odd 6 1 1872.2.by.a 2
156.r even 6 1 1872.2.by.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.q.b 2 1.a even 1 1 trivial
117.2.q.b 2 3.b odd 2 1 CM
117.2.q.b 2 13.e even 6 1 inner
117.2.q.b 2 39.h odd 6 1 inner
1521.2.a.i 2 13.f odd 12 2
1521.2.a.i 2 39.k even 12 2
1521.2.b.d 2 13.c even 3 1
1521.2.b.d 2 13.e even 6 1
1521.2.b.d 2 39.h odd 6 1
1521.2.b.d 2 39.i odd 6 1
1872.2.by.a 2 4.b odd 2 1
1872.2.by.a 2 12.b even 2 1
1872.2.by.a 2 52.i odd 6 1
1872.2.by.a 2 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 75 \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( (T - 13)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 33T + 363 \) Copy content Toggle raw display
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