Properties

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

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.b (of order \(2\), degree \(1\), 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: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - q^{4} - 2 \beta q^{7} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - q^{4} - 2 \beta q^{7} - \beta q^{8} + 2 \beta q^{11} + (2 \beta - 1) q^{13} - 6 q^{14} - 5 q^{16} + 6 q^{17} + 2 \beta q^{19} + 6 q^{22} + 5 q^{25} + (\beta + 6) q^{26} + 2 \beta q^{28} - 6 q^{29} - 2 \beta q^{31} + 3 \beta q^{32} - 6 \beta q^{34} + 4 \beta q^{37} + 6 q^{38} + 4 \beta q^{41} - 4 q^{43} - 2 \beta q^{44} - 2 \beta q^{47} - 5 q^{49} - 5 \beta q^{50} + ( - 2 \beta + 1) q^{52} - 6 q^{53} - 6 q^{56} + 6 \beta q^{58} - 6 \beta q^{59} - 2 q^{61} - 6 q^{62} - q^{64} - 6 \beta q^{67} - 6 q^{68} - 2 \beta q^{71} + 12 q^{74} - 2 \beta q^{76} + 12 q^{77} - 8 q^{79} + 12 q^{82} + 2 \beta q^{83} + 4 \beta q^{86} + 6 q^{88} + 4 \beta q^{89} + (2 \beta + 12) q^{91} - 6 q^{94} - 8 \beta q^{97} + 5 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{13} - 12 q^{14} - 10 q^{16} + 12 q^{17} + 12 q^{22} + 10 q^{25} + 12 q^{26} - 12 q^{29} + 12 q^{38} - 8 q^{43} - 10 q^{49} + 2 q^{52} - 12 q^{53} - 12 q^{56} - 4 q^{61} - 12 q^{62} - 2 q^{64} - 12 q^{68} + 24 q^{74} + 24 q^{77} - 16 q^{79} + 24 q^{82} + 12 q^{88} + 24 q^{91} - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.500000 + 0.866025i
0.500000 0.866025i
1.73205i 0 −1.00000 0 0 3.46410i 1.73205i 0 0
64.2 1.73205i 0 −1.00000 0 0 3.46410i 1.73205i 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.b.a 2
3.b odd 2 1 39.2.b.a 2
4.b odd 2 1 1872.2.c.e 2
12.b even 2 1 624.2.c.e 2
13.b even 2 1 inner 117.2.b.a 2
13.d odd 4 2 1521.2.a.l 2
15.d odd 2 1 975.2.b.d 2
15.e even 4 2 975.2.h.f 4
21.c even 2 1 1911.2.c.d 2
24.f even 2 1 2496.2.c.d 2
24.h odd 2 1 2496.2.c.k 2
39.d odd 2 1 39.2.b.a 2
39.f even 4 2 507.2.a.f 2
39.h odd 6 1 507.2.j.a 2
39.h odd 6 1 507.2.j.c 2
39.i odd 6 1 507.2.j.a 2
39.i odd 6 1 507.2.j.c 2
39.k even 12 4 507.2.e.e 4
52.b odd 2 1 1872.2.c.e 2
156.h even 2 1 624.2.c.e 2
156.l odd 4 2 8112.2.a.bv 2
195.e odd 2 1 975.2.b.d 2
195.s even 4 2 975.2.h.f 4
273.g even 2 1 1911.2.c.d 2
312.b odd 2 1 2496.2.c.k 2
312.h even 2 1 2496.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 3.b odd 2 1
39.2.b.a 2 39.d odd 2 1
117.2.b.a 2 1.a even 1 1 trivial
117.2.b.a 2 13.b even 2 1 inner
507.2.a.f 2 39.f even 4 2
507.2.e.e 4 39.k even 12 4
507.2.j.a 2 39.h odd 6 1
507.2.j.a 2 39.i odd 6 1
507.2.j.c 2 39.h odd 6 1
507.2.j.c 2 39.i odd 6 1
624.2.c.e 2 12.b even 2 1
624.2.c.e 2 156.h even 2 1
975.2.b.d 2 15.d odd 2 1
975.2.b.d 2 195.e odd 2 1
975.2.h.f 4 15.e even 4 2
975.2.h.f 4 195.s even 4 2
1521.2.a.l 2 13.d odd 4 2
1872.2.c.e 2 4.b odd 2 1
1872.2.c.e 2 52.b odd 2 1
1911.2.c.d 2 21.c even 2 1
1911.2.c.d 2 273.g even 2 1
2496.2.c.d 2 24.f even 2 1
2496.2.c.d 2 312.h even 2 1
2496.2.c.k 2 24.h odd 2 1
2496.2.c.k 2 312.b odd 2 1
8112.2.a.bv 2 156.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 13 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 48 \) Copy content Toggle raw display
$41$ \( T^{2} + 48 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 108 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 108 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 192 \) Copy content Toggle raw display
show more
show less