Properties

Label 117.2.e.a
Level $117$
Weight $2$
Character orbit 117.e
Analytic conductor $0.934$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(40,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.40");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.e (of order \(3\), degree \(2\), 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: \(0.934249703649\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + (\zeta_{6} - 2) q^{3} - 2 \zeta_{6} q^{4} - 4 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 2) q^{6} + (2 \zeta_{6} - 2) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + (\zeta_{6} - 2) q^{3} - 2 \zeta_{6} q^{4} - 4 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 2) q^{6} + (2 \zeta_{6} - 2) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + 8 q^{10} + (2 \zeta_{6} - 2) q^{11} + (2 \zeta_{6} + 2) q^{12} - \zeta_{6} q^{13} - 4 \zeta_{6} q^{14} + (4 \zeta_{6} + 4) q^{15} + ( - 4 \zeta_{6} + 4) q^{16} - 5 q^{17} + 6 \zeta_{6} q^{18} - 6 q^{19} + (8 \zeta_{6} - 8) q^{20} + ( - 4 \zeta_{6} + 2) q^{21} - 4 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} + 2 q^{26} + (6 \zeta_{6} - 3) q^{27} + 4 q^{28} + ( - 2 \zeta_{6} + 2) q^{29} + (8 \zeta_{6} - 16) q^{30} - 4 \zeta_{6} q^{31} + 8 \zeta_{6} q^{32} + ( - 4 \zeta_{6} + 2) q^{33} + ( - 10 \zeta_{6} + 10) q^{34} + 8 q^{35} - 6 q^{36} - 2 q^{37} + ( - 12 \zeta_{6} + 12) q^{38} + (\zeta_{6} + 1) q^{39} + 6 \zeta_{6} q^{41} + (4 \zeta_{6} + 4) q^{42} + ( - 3 \zeta_{6} + 3) q^{43} + 4 q^{44} - 12 q^{45} - 6 q^{46} + (6 \zeta_{6} - 6) q^{47} + (8 \zeta_{6} - 4) q^{48} + 3 \zeta_{6} q^{49} - 22 \zeta_{6} q^{50} + ( - 5 \zeta_{6} + 10) q^{51} + (2 \zeta_{6} - 2) q^{52} + 3 q^{53} + ( - 6 \zeta_{6} - 6) q^{54} + 8 q^{55} + ( - 6 \zeta_{6} + 12) q^{57} + 4 \zeta_{6} q^{58} - 12 \zeta_{6} q^{59} + ( - 16 \zeta_{6} + 8) q^{60} + ( - 5 \zeta_{6} + 5) q^{61} + 8 q^{62} + 6 \zeta_{6} q^{63} - 8 q^{64} + (4 \zeta_{6} - 4) q^{65} + (4 \zeta_{6} + 4) q^{66} - 4 \zeta_{6} q^{67} + 10 \zeta_{6} q^{68} + ( - 3 \zeta_{6} - 3) q^{69} + (16 \zeta_{6} - 16) q^{70} + 12 q^{71} - 4 q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + ( - 22 \zeta_{6} + 11) q^{75} + 12 \zeta_{6} q^{76} - 4 \zeta_{6} q^{77} + (2 \zeta_{6} - 4) q^{78} + (5 \zeta_{6} - 5) q^{79} - 16 q^{80} - 9 \zeta_{6} q^{81} - 12 q^{82} + (2 \zeta_{6} - 2) q^{83} + (4 \zeta_{6} - 8) q^{84} + 20 \zeta_{6} q^{85} + 6 \zeta_{6} q^{86} + (4 \zeta_{6} - 2) q^{87} - 10 q^{89} + ( - 24 \zeta_{6} + 24) q^{90} + 2 q^{91} + ( - 6 \zeta_{6} + 6) q^{92} + (4 \zeta_{6} + 4) q^{93} - 12 \zeta_{6} q^{94} + 24 \zeta_{6} q^{95} + ( - 8 \zeta_{6} - 8) q^{96} + ( - 8 \zeta_{6} + 8) q^{97} - 6 q^{98} + 6 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 4 q^{5} - 2 q^{7} + 3 q^{9} + 16 q^{10} - 2 q^{11} + 6 q^{12} - q^{13} - 4 q^{14} + 12 q^{15} + 4 q^{16} - 10 q^{17} + 6 q^{18} - 12 q^{19} - 8 q^{20} - 4 q^{22} + 3 q^{23} - 11 q^{25} + 4 q^{26} + 8 q^{28} + 2 q^{29} - 24 q^{30} - 4 q^{31} + 8 q^{32} + 10 q^{34} + 16 q^{35} - 12 q^{36} - 4 q^{37} + 12 q^{38} + 3 q^{39} + 6 q^{41} + 12 q^{42} + 3 q^{43} + 8 q^{44} - 24 q^{45} - 12 q^{46} - 6 q^{47} + 3 q^{49} - 22 q^{50} + 15 q^{51} - 2 q^{52} + 6 q^{53} - 18 q^{54} + 16 q^{55} + 18 q^{57} + 4 q^{58} - 12 q^{59} + 5 q^{61} + 16 q^{62} + 6 q^{63} - 16 q^{64} - 4 q^{65} + 12 q^{66} - 4 q^{67} + 10 q^{68} - 9 q^{69} - 16 q^{70} + 24 q^{71} - 8 q^{73} + 4 q^{74} + 12 q^{76} - 4 q^{77} - 6 q^{78} - 5 q^{79} - 32 q^{80} - 9 q^{81} - 24 q^{82} - 2 q^{83} - 12 q^{84} + 20 q^{85} + 6 q^{86} - 20 q^{89} + 24 q^{90} + 4 q^{91} + 6 q^{92} + 12 q^{93} - 12 q^{94} + 24 q^{95} - 24 q^{96} + 8 q^{97} - 12 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 1.73205i −1.50000 0.866025i −1.00000 + 1.73205i −2.00000 + 3.46410i 3.46410i −1.00000 1.73205i 0 1.50000 + 2.59808i 8.00000
79.1 −1.00000 + 1.73205i −1.50000 + 0.866025i −1.00000 1.73205i −2.00000 3.46410i 3.46410i −1.00000 + 1.73205i 0 1.50000 2.59808i 8.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.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.e.a 2
3.b odd 2 1 351.2.e.a 2
9.c even 3 1 inner 117.2.e.a 2
9.c even 3 1 1053.2.a.d 1
9.d odd 6 1 351.2.e.a 2
9.d odd 6 1 1053.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.e.a 2 1.a even 1 1 trivial
117.2.e.a 2 9.c even 3 1 inner
351.2.e.a 2 3.b odd 2 1
351.2.e.a 2 9.d odd 6 1
1053.2.a.a 1 9.d odd 6 1
1053.2.a.d 1 9.c even 3 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 16 \) 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
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( (T + 5)^{2} \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$83$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
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