Properties

Label 117.2.r.a
Level $117$
Weight $2$
Character orbit 117.r
Analytic conductor $0.934$
Analytic rank $1$
Dimension $2$
CM no
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(43,117)]
 
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([4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.r (of order \(6\), 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 2) q^{2} + (\zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{4} + (\zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} + 3) q^{6} + ( - 2 \zeta_{6} + 1) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 2) q^{2} + (\zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{4} + (\zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} + 3) q^{6} + ( - 2 \zeta_{6} + 1) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + ( - 3 \zeta_{6} + 3) q^{10} + (2 \zeta_{6} - 4) q^{11} + (2 \zeta_{6} - 1) q^{12} + ( - 3 \zeta_{6} - 1) q^{13} + 3 \zeta_{6} q^{14} + ( - 3 \zeta_{6} + 3) q^{15} + 5 \zeta_{6} q^{16} - 3 \zeta_{6} q^{17} + (6 \zeta_{6} - 3) q^{18} + (\zeta_{6} - 2) q^{19} + (2 \zeta_{6} - 1) q^{20} + 3 \zeta_{6} q^{21} + ( - 6 \zeta_{6} + 6) q^{22} - 3 q^{23} + 3 \zeta_{6} q^{24} + (2 \zeta_{6} - 2) q^{25} + (2 \zeta_{6} + 5) q^{26} + (6 \zeta_{6} - 3) q^{27} + ( - \zeta_{6} - 1) q^{28} + 6 \zeta_{6} q^{29} + (6 \zeta_{6} - 3) q^{30} + (5 \zeta_{6} - 10) q^{31} + ( - 3 \zeta_{6} - 3) q^{32} + ( - 6 \zeta_{6} + 6) q^{33} + (3 \zeta_{6} + 3) q^{34} + 3 \zeta_{6} q^{35} - 3 \zeta_{6} q^{36} + ( - 3 \zeta_{6} - 3) q^{37} + ( - 3 \zeta_{6} + 3) q^{38} + (2 \zeta_{6} + 5) q^{39} + 3 \zeta_{6} q^{40} + ( - 14 \zeta_{6} + 7) q^{41} + ( - 3 \zeta_{6} - 3) q^{42} - q^{43} + (4 \zeta_{6} - 2) q^{44} + (6 \zeta_{6} - 3) q^{45} + ( - 3 \zeta_{6} + 6) q^{46} + ( - 3 \zeta_{6} - 3) q^{47} + ( - 5 \zeta_{6} - 5) q^{48} + 4 q^{49} + ( - 4 \zeta_{6} + 2) q^{50} + (3 \zeta_{6} + 3) q^{51} + (\zeta_{6} - 4) q^{52} + 6 q^{53} - 9 \zeta_{6} q^{54} + ( - 6 \zeta_{6} + 6) q^{55} - 3 q^{56} + ( - 3 \zeta_{6} + 3) q^{57} + ( - 6 \zeta_{6} - 6) q^{58} + (2 \zeta_{6} + 2) q^{59} - 3 \zeta_{6} q^{60} - 5 q^{61} + ( - 15 \zeta_{6} + 15) q^{62} + ( - 3 \zeta_{6} - 3) q^{63} - q^{64} + (2 \zeta_{6} + 5) q^{65} + (12 \zeta_{6} - 6) q^{66} + (14 \zeta_{6} - 7) q^{67} - 3 q^{68} + ( - 3 \zeta_{6} + 6) q^{69} + ( - 3 \zeta_{6} - 3) q^{70} + (5 \zeta_{6} - 10) q^{71} + ( - 3 \zeta_{6} - 3) q^{72} + (8 \zeta_{6} - 4) q^{73} + 9 q^{74} + ( - 4 \zeta_{6} + 2) q^{75} + (2 \zeta_{6} - 1) q^{76} + 6 \zeta_{6} q^{77} + (3 \zeta_{6} - 12) q^{78} + ( - 11 \zeta_{6} + 11) q^{79} + ( - 5 \zeta_{6} - 5) q^{80} - 9 \zeta_{6} q^{81} + 21 \zeta_{6} q^{82} + ( - 3 \zeta_{6} - 3) q^{83} + 3 q^{84} + (3 \zeta_{6} + 3) q^{85} + ( - \zeta_{6} + 2) q^{86} + ( - 6 \zeta_{6} - 6) q^{87} + 6 \zeta_{6} q^{88} + (9 \zeta_{6} + 9) q^{89} - 9 \zeta_{6} q^{90} + (5 \zeta_{6} - 7) q^{91} + (3 \zeta_{6} - 3) q^{92} + ( - 15 \zeta_{6} + 15) q^{93} + 9 q^{94} + ( - 3 \zeta_{6} + 3) q^{95} + 9 q^{96} + ( - 18 \zeta_{6} + 9) q^{97} + (4 \zeta_{6} - 8) q^{98} + (12 \zeta_{6} - 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 3 q^{3} + q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 3 q^{3} + q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{9} + 3 q^{10} - 6 q^{11} - 5 q^{13} + 3 q^{14} + 3 q^{15} + 5 q^{16} - 3 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{22} - 6 q^{23} + 3 q^{24} - 2 q^{25} + 12 q^{26} - 3 q^{28} + 6 q^{29} - 15 q^{31} - 9 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{35} - 3 q^{36} - 9 q^{37} + 3 q^{38} + 12 q^{39} + 3 q^{40} - 9 q^{42} - 2 q^{43} + 9 q^{46} - 9 q^{47} - 15 q^{48} + 8 q^{49} + 9 q^{51} - 7 q^{52} + 12 q^{53} - 9 q^{54} + 6 q^{55} - 6 q^{56} + 3 q^{57} - 18 q^{58} + 6 q^{59} - 3 q^{60} - 10 q^{61} + 15 q^{62} - 9 q^{63} - 2 q^{64} + 12 q^{65} - 6 q^{68} + 9 q^{69} - 9 q^{70} - 15 q^{71} - 9 q^{72} + 18 q^{74} + 6 q^{77} - 21 q^{78} + 11 q^{79} - 15 q^{80} - 9 q^{81} + 21 q^{82} - 9 q^{83} + 6 q^{84} + 9 q^{85} + 3 q^{86} - 18 q^{87} + 6 q^{88} + 27 q^{89} - 9 q^{90} - 9 q^{91} - 3 q^{92} + 15 q^{93} + 18 q^{94} + 3 q^{95} + 18 q^{96} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.50000 + 0.866025i −1.50000 + 0.866025i 0.500000 0.866025i −1.50000 + 0.866025i 1.50000 2.59808i 1.73205i 1.73205i 1.50000 2.59808i 1.50000 2.59808i
49.1 −1.50000 0.866025i −1.50000 0.866025i 0.500000 + 0.866025i −1.50000 0.866025i 1.50000 + 2.59808i 1.73205i 1.73205i 1.50000 + 2.59808i 1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.r 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.r.a yes 2
3.b odd 2 1 351.2.r.a 2
9.c even 3 1 117.2.l.a 2
9.d odd 6 1 351.2.l.a 2
13.e even 6 1 117.2.l.a 2
39.h odd 6 1 351.2.l.a 2
117.m odd 6 1 351.2.r.a 2
117.r even 6 1 inner 117.2.r.a yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.l.a 2 9.c even 3 1
117.2.l.a 2 13.e even 6 1
117.2.r.a yes 2 1.a even 1 1 trivial
117.2.r.a yes 2 117.r even 6 1 inner
351.2.l.a 2 9.d odd 6 1
351.2.l.a 2 39.h odd 6 1
351.2.r.a 2 3.b odd 2 1
351.2.r.a 2 117.m odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 3 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$23$ \( (T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$37$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$41$ \( T^{2} + 147 \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$61$ \( (T + 5)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 147 \) Copy content Toggle raw display
$71$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$73$ \( T^{2} + 48 \) Copy content Toggle raw display
$79$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$89$ \( T^{2} - 27T + 243 \) Copy content Toggle raw display
$97$ \( T^{2} + 243 \) Copy content Toggle raw display
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