Properties

Label 117.3.bd.b
Level $117$
Weight $3$
Character orbit 117.bd
Analytic conductor $3.188$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,3,Mod(19,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.bd (of order \(12\), degree \(4\), 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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} + 1) q^{2} + ( - \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{4} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots + 3) q^{5}+ \cdots + (4 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + \cdots + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} + 1) q^{2} + ( - \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{4} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots + 3) q^{5}+ \cdots + ( - 19 \zeta_{12}^{3} + 31 \zeta_{12}^{2} + \cdots - 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 6 q^{4} + 14 q^{5} + 16 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 6 q^{4} + 14 q^{5} + 16 q^{7} + 6 q^{8} + 24 q^{10} - 4 q^{11} - 26 q^{13} + 40 q^{14} - 2 q^{16} + 12 q^{17} + 10 q^{19} - 26 q^{20} - 4 q^{22} - 18 q^{23} - 52 q^{26} - 44 q^{28} - 2 q^{29} - 20 q^{31} + 20 q^{32} - 18 q^{34} - 32 q^{35} - 68 q^{37} + 72 q^{40} - 100 q^{41} + 180 q^{43} - 88 q^{44} - 30 q^{46} + 68 q^{47} - 72 q^{49} + 46 q^{50} - 128 q^{53} - 100 q^{55} + 84 q^{56} - 40 q^{58} + 164 q^{59} - 124 q^{61} - 6 q^{62} - 52 q^{65} + 118 q^{67} - 72 q^{68} + 164 q^{70} + 86 q^{71} + 58 q^{73} - 68 q^{74} + 14 q^{76} - 40 q^{79} + 140 q^{80} + 24 q^{82} + 188 q^{83} + 96 q^{85} + 180 q^{86} - 204 q^{88} + 110 q^{89} + 52 q^{91} + 156 q^{92} + 26 q^{94} + 78 q^{95} + 178 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\zeta_{12}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0.500000 + 1.86603i 0 0.232051 0.133975i 4.36603 4.36603i 0 2.26795 8.46410i 5.83013 + 5.83013i 0 10.3301 + 5.96410i
28.1 0.500000 + 0.133975i 0 −3.23205 1.86603i 2.63397 2.63397i 0 5.73205 1.53590i −2.83013 2.83013i 0 1.66987 0.964102i
37.1 0.500000 1.86603i 0 0.232051 + 0.133975i 4.36603 + 4.36603i 0 2.26795 + 8.46410i 5.83013 5.83013i 0 10.3301 5.96410i
46.1 0.500000 0.133975i 0 −3.23205 + 1.86603i 2.63397 + 2.63397i 0 5.73205 + 1.53590i −2.83013 + 2.83013i 0 1.66987 + 0.964102i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.3.bd.b 4
3.b odd 2 1 13.3.f.a 4
12.b even 2 1 208.3.bd.d 4
13.f odd 12 1 inner 117.3.bd.b 4
15.d odd 2 1 325.3.t.a 4
15.e even 4 1 325.3.w.a 4
15.e even 4 1 325.3.w.b 4
39.d odd 2 1 169.3.f.b 4
39.f even 4 1 169.3.f.a 4
39.f even 4 1 169.3.f.c 4
39.h odd 6 1 169.3.d.c 4
39.h odd 6 1 169.3.f.a 4
39.i odd 6 1 169.3.d.a 4
39.i odd 6 1 169.3.f.c 4
39.k even 12 1 13.3.f.a 4
39.k even 12 1 169.3.d.a 4
39.k even 12 1 169.3.d.c 4
39.k even 12 1 169.3.f.b 4
156.v odd 12 1 208.3.bd.d 4
195.bc odd 12 1 325.3.w.a 4
195.bh even 12 1 325.3.t.a 4
195.bn odd 12 1 325.3.w.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.3.f.a 4 3.b odd 2 1
13.3.f.a 4 39.k even 12 1
117.3.bd.b 4 1.a even 1 1 trivial
117.3.bd.b 4 13.f odd 12 1 inner
169.3.d.a 4 39.i odd 6 1
169.3.d.a 4 39.k even 12 1
169.3.d.c 4 39.h odd 6 1
169.3.d.c 4 39.k even 12 1
169.3.f.a 4 39.f even 4 1
169.3.f.a 4 39.h odd 6 1
169.3.f.b 4 39.d odd 2 1
169.3.f.b 4 39.k even 12 1
169.3.f.c 4 39.f even 4 1
169.3.f.c 4 39.i odd 6 1
208.3.bd.d 4 12.b even 2 1
208.3.bd.d 4 156.v odd 12 1
325.3.t.a 4 15.d odd 2 1
325.3.t.a 4 195.bh even 12 1
325.3.w.a 4 15.e even 4 1
325.3.w.a 4 195.bc odd 12 1
325.3.w.b 4 15.e even 4 1
325.3.w.b 4 195.bn odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} - 4T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 14 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{4} - 16 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 12 T^{3} + \cdots + 45369 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$23$ \( T^{4} + 18 T^{3} + \cdots + 39204 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + \cdots + 11449 \) Copy content Toggle raw display
$31$ \( T^{4} + 20 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$37$ \( T^{4} + 68 T^{3} + \cdots + 1868689 \) Copy content Toggle raw display
$41$ \( T^{4} + 100 T^{3} + \cdots + 833569 \) Copy content Toggle raw display
$43$ \( (T^{2} - 90 T + 2700)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 68 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$53$ \( (T^{2} + 64 T - 1163)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 164 T^{3} + \cdots + 16613776 \) Copy content Toggle raw display
$61$ \( T^{4} + 124 T^{3} + \cdots + 6355441 \) Copy content Toggle raw display
$67$ \( T^{4} - 118 T^{3} + \cdots + 9721924 \) Copy content Toggle raw display
$71$ \( T^{4} - 86 T^{3} + \cdots + 2208196 \) Copy content Toggle raw display
$73$ \( T^{4} - 58 T^{3} + \cdots + 3463321 \) Copy content Toggle raw display
$79$ \( (T^{2} + 20 T - 5192)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 188 T^{3} + \cdots + 11587216 \) Copy content Toggle raw display
$89$ \( T^{4} - 110 T^{3} + \cdots + 8702500 \) Copy content Toggle raw display
$97$ \( T^{4} - 178 T^{3} + \cdots + 18028516 \) Copy content Toggle raw display
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