Properties

Label 117.2.g.b
Level $117$
Weight $2$
Character orbit 117.g
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.g (of order \(3\), 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + q^{5} - 2 \zeta_{6} q^{7} + 3 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + q^{5} - 2 \zeta_{6} q^{7} + 3 q^{8} + ( - \zeta_{6} + 1) q^{10} + (2 \zeta_{6} - 2) q^{11} + ( - \zeta_{6} - 3) q^{13} - 2 q^{14} + ( - \zeta_{6} + 1) q^{16} - 7 \zeta_{6} q^{17} + 6 \zeta_{6} q^{19} + \zeta_{6} q^{20} + 2 \zeta_{6} q^{22} + (6 \zeta_{6} - 6) q^{23} - 4 q^{25} + (3 \zeta_{6} - 4) q^{26} + ( - 2 \zeta_{6} + 2) q^{28} + (\zeta_{6} - 1) q^{29} + 4 q^{31} + 5 \zeta_{6} q^{32} - 7 q^{34} - 2 \zeta_{6} q^{35} + (\zeta_{6} - 1) q^{37} + 6 q^{38} + 3 q^{40} + ( - 9 \zeta_{6} + 9) q^{41} - 6 \zeta_{6} q^{43} - 2 q^{44} + 6 \zeta_{6} q^{46} - 6 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} + (4 \zeta_{6} - 4) q^{50} + ( - 4 \zeta_{6} + 1) q^{52} + 9 q^{53} + (2 \zeta_{6} - 2) q^{55} - 6 \zeta_{6} q^{56} + \zeta_{6} q^{58} - \zeta_{6} q^{61} + ( - 4 \zeta_{6} + 4) q^{62} + 7 q^{64} + ( - \zeta_{6} - 3) q^{65} + ( - 2 \zeta_{6} + 2) q^{67} + ( - 7 \zeta_{6} + 7) q^{68} - 2 q^{70} + 6 \zeta_{6} q^{71} + 11 q^{73} + \zeta_{6} q^{74} + (6 \zeta_{6} - 6) q^{76} + 4 q^{77} - 4 q^{79} + ( - \zeta_{6} + 1) q^{80} - 9 \zeta_{6} q^{82} + 14 q^{83} - 7 \zeta_{6} q^{85} - 6 q^{86} + (6 \zeta_{6} - 6) q^{88} + (14 \zeta_{6} - 14) q^{89} + (8 \zeta_{6} - 2) q^{91} - 6 q^{92} + (6 \zeta_{6} - 6) q^{94} + 6 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} - 3 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + q^{10} - 2 q^{11} - 7 q^{13} - 4 q^{14} + q^{16} - 7 q^{17} + 6 q^{19} + q^{20} + 2 q^{22} - 6 q^{23} - 8 q^{25} - 5 q^{26} + 2 q^{28} - q^{29} + 8 q^{31} + 5 q^{32} - 14 q^{34} - 2 q^{35} - q^{37} + 12 q^{38} + 6 q^{40} + 9 q^{41} - 6 q^{43} - 4 q^{44} + 6 q^{46} - 12 q^{47} + 3 q^{49} - 4 q^{50} - 2 q^{52} + 18 q^{53} - 2 q^{55} - 6 q^{56} + q^{58} - q^{61} + 4 q^{62} + 14 q^{64} - 7 q^{65} + 2 q^{67} + 7 q^{68} - 4 q^{70} + 6 q^{71} + 22 q^{73} + q^{74} - 6 q^{76} + 8 q^{77} - 8 q^{79} + q^{80} - 9 q^{82} + 28 q^{83} - 7 q^{85} - 12 q^{86} - 6 q^{88} - 14 q^{89} + 4 q^{91} - 12 q^{92} - 6 q^{94} + 6 q^{95} + 2 q^{97} - 3 q^{98}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 0 −1.00000 + 1.73205i 3.00000 0 0.500000 + 0.866025i
100.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 0 −1.00000 1.73205i 3.00000 0 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.g.b 2
3.b odd 2 1 39.2.e.a 2
4.b odd 2 1 1872.2.t.j 2
12.b even 2 1 624.2.q.c 2
13.c even 3 1 inner 117.2.g.b 2
13.c even 3 1 1521.2.a.a 1
13.e even 6 1 1521.2.a.d 1
13.f odd 12 2 1521.2.b.c 2
15.d odd 2 1 975.2.i.f 2
15.e even 4 2 975.2.bb.d 4
39.d odd 2 1 507.2.e.c 2
39.f even 4 2 507.2.j.d 4
39.h odd 6 1 507.2.a.b 1
39.h odd 6 1 507.2.e.c 2
39.i odd 6 1 39.2.e.a 2
39.i odd 6 1 507.2.a.c 1
39.k even 12 2 507.2.b.b 2
39.k even 12 2 507.2.j.d 4
52.j odd 6 1 1872.2.t.j 2
156.p even 6 1 624.2.q.c 2
156.p even 6 1 8112.2.a.w 1
156.r even 6 1 8112.2.a.bc 1
195.x odd 6 1 975.2.i.f 2
195.bl even 12 2 975.2.bb.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 3.b odd 2 1
39.2.e.a 2 39.i odd 6 1
117.2.g.b 2 1.a even 1 1 trivial
117.2.g.b 2 13.c even 3 1 inner
507.2.a.b 1 39.h odd 6 1
507.2.a.c 1 39.i odd 6 1
507.2.b.b 2 39.k even 12 2
507.2.e.c 2 39.d odd 2 1
507.2.e.c 2 39.h odd 6 1
507.2.j.d 4 39.f even 4 2
507.2.j.d 4 39.k even 12 2
624.2.q.c 2 12.b even 2 1
624.2.q.c 2 156.p even 6 1
975.2.i.f 2 15.d odd 2 1
975.2.i.f 2 195.x odd 6 1
975.2.bb.d 4 15.e even 4 2
975.2.bb.d 4 195.bl even 12 2
1521.2.a.a 1 13.c even 3 1
1521.2.a.d 1 13.e even 6 1
1521.2.b.c 2 13.f odd 12 2
1872.2.t.j 2 4.b odd 2 1
1872.2.t.j 2 52.j odd 6 1
8112.2.a.w 1 156.p even 6 1
8112.2.a.bc 1 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 14)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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