Properties

Label 117.2.t.b
Level $117$
Weight $2$
Character orbit 117.t
Analytic conductor $0.934$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.t (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + (2 \zeta_{6} + 2) q^{5} + ( - 2 \zeta_{6} + 4) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + (2 \zeta_{6} + 2) q^{5} + ( - 2 \zeta_{6} + 4) q^{7} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 4) q^{11} + ( - 2 \zeta_{6} + 4) q^{12} + (\zeta_{6} + 3) q^{13} - 6 \zeta_{6} q^{15} - 4 \zeta_{6} q^{16} - 3 q^{17} + ( - 4 \zeta_{6} + 2) q^{19} + (4 \zeta_{6} - 8) q^{20} - 6 q^{21} + (3 \zeta_{6} - 3) q^{23} + 7 \zeta_{6} q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + (8 \zeta_{6} - 4) q^{28} - 6 \zeta_{6} q^{29} + ( - 2 \zeta_{6} - 2) q^{31} + 6 q^{33} + 12 q^{35} - 6 q^{36} + ( - 8 \zeta_{6} + 4) q^{37} + ( - 5 \zeta_{6} - 2) q^{39} + (4 \zeta_{6} + 4) q^{41} + \zeta_{6} q^{43} + ( - 8 \zeta_{6} + 4) q^{44} + (12 \zeta_{6} - 6) q^{45} + ( - 4 \zeta_{6} + 8) q^{47} + (8 \zeta_{6} - 4) q^{48} + ( - 5 \zeta_{6} + 5) q^{49} + (3 \zeta_{6} + 3) q^{51} + (6 \zeta_{6} - 8) q^{52} - 9 q^{53} - 12 q^{55} + (6 \zeta_{6} - 6) q^{57} + ( - 2 \zeta_{6} - 2) q^{59} + 12 q^{60} - 7 \zeta_{6} q^{61} + (6 \zeta_{6} + 6) q^{63} + 8 q^{64} + (10 \zeta_{6} + 4) q^{65} + ( - 6 \zeta_{6} + 6) q^{68} + ( - 3 \zeta_{6} + 6) q^{69} + (12 \zeta_{6} - 6) q^{73} + ( - 14 \zeta_{6} + 7) q^{75} + (4 \zeta_{6} + 4) q^{76} + (12 \zeta_{6} - 12) q^{77} - \zeta_{6} q^{79} + ( - 16 \zeta_{6} + 8) q^{80} + (9 \zeta_{6} - 9) q^{81} + (4 \zeta_{6} - 8) q^{83} + ( - 12 \zeta_{6} + 12) q^{84} + ( - 6 \zeta_{6} - 6) q^{85} + (12 \zeta_{6} - 6) q^{87} + ( - 4 \zeta_{6} + 2) q^{89} + ( - 4 \zeta_{6} + 14) q^{91} - 6 \zeta_{6} q^{92} + 6 \zeta_{6} q^{93} + ( - 12 \zeta_{6} + 12) q^{95} + (4 \zeta_{6} - 8) q^{97} + ( - 6 \zeta_{6} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 2 q^{4} + 6 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 2 q^{4} + 6 q^{5} + 6 q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{12} + 7 q^{13} - 6 q^{15} - 4 q^{16} - 6 q^{17} - 12 q^{20} - 12 q^{21} - 3 q^{23} + 7 q^{25} - 6 q^{29} - 6 q^{31} + 12 q^{33} + 24 q^{35} - 12 q^{36} - 9 q^{39} + 12 q^{41} + q^{43} + 12 q^{47} + 5 q^{49} + 9 q^{51} - 10 q^{52} - 18 q^{53} - 24 q^{55} - 6 q^{57} - 6 q^{59} + 24 q^{60} - 7 q^{61} + 18 q^{63} + 16 q^{64} + 18 q^{65} + 6 q^{68} + 9 q^{69} + 12 q^{76} - 12 q^{77} - q^{79} - 9 q^{81} - 12 q^{83} + 12 q^{84} - 18 q^{85} + 24 q^{91} - 6 q^{92} + 6 q^{93} + 12 q^{95} - 12 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 + 0.866025i −1.00000 1.73205i 3.00000 1.73205i 0 3.00000 + 1.73205i 0 1.50000 2.59808i 0
103.1 0 −1.50000 0.866025i −1.00000 + 1.73205i 3.00000 + 1.73205i 0 3.00000 1.73205i 0 1.50000 + 2.59808i 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
117.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.t.b yes 2
3.b odd 2 1 351.2.t.a 2
9.c even 3 1 117.2.t.a 2
9.c even 3 1 1053.2.b.e 2
9.d odd 6 1 351.2.t.b 2
9.d odd 6 1 1053.2.b.f 2
13.b even 2 1 117.2.t.a 2
39.d odd 2 1 351.2.t.b 2
117.n odd 6 1 351.2.t.a 2
117.n odd 6 1 1053.2.b.f 2
117.t even 6 1 inner 117.2.t.b yes 2
117.t even 6 1 1053.2.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.t.a 2 9.c even 3 1
117.2.t.a 2 13.b even 2 1
117.2.t.b yes 2 1.a even 1 1 trivial
117.2.t.b yes 2 117.t even 6 1 inner
351.2.t.a 2 3.b odd 2 1
351.2.t.a 2 117.n odd 6 1
351.2.t.b 2 9.d odd 6 1
351.2.t.b 2 39.d odd 2 1
1053.2.b.e 2 9.c even 3 1
1053.2.b.e 2 117.t even 6 1
1053.2.b.f 2 9.d odd 6 1
1053.2.b.f 2 117.n odd 6 1

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{2} - 6T_{5} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 13 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 48 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 108 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$89$ \( T^{2} + 12 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
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