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Properties

Label 117.4.q.b
Level $117$
Weight $4$
Character orbit 117.q
Analytic conductor $6.903$
Analytic rank $0$
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] = [117,4,Mod(10,117)]
 
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([0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.10");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.q (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: \(6.90322347067\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q - 8 \zeta_{6} q^{4} + (6 \zeta_{6} - 3) q^{5} + ( - 6 \zeta_{6} + 12) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 8 \zeta_{6} q^{4} + (6 \zeta_{6} - 3) q^{5} + ( - 6 \zeta_{6} + 12) q^{7} + ( - 30 \zeta_{6} - 30) q^{11} + ( - 39 \zeta_{6} - 13) q^{13} + (64 \zeta_{6} - 64) q^{16} - 117 \zeta_{6} q^{17} + (14 \zeta_{6} - 28) q^{19} + ( - 24 \zeta_{6} + 48) q^{20} + ( - 18 \zeta_{6} + 18) q^{23} + 98 q^{25} + ( - 48 \zeta_{6} - 48) q^{28} + (99 \zeta_{6} - 99) q^{29} + ( - 224 \zeta_{6} + 112) q^{31} + 54 \zeta_{6} q^{35} + (65 \zeta_{6} + 65) q^{37} + (21 \zeta_{6} + 21) q^{41} + 82 \zeta_{6} q^{43} + (480 \zeta_{6} - 240) q^{44} + ( - 84 \zeta_{6} + 42) q^{47} + (235 \zeta_{6} - 235) q^{49} + (416 \zeta_{6} - 312) q^{52} + 261 q^{53} + ( - 270 \zeta_{6} + 270) q^{55} + ( - 456 \zeta_{6} + 912) q^{59} + 719 \zeta_{6} q^{61} + 512 q^{64} + ( - 195 \zeta_{6} + 273) q^{65} + ( - 406 \zeta_{6} - 406) q^{67} + (936 \zeta_{6} - 936) q^{68} + ( - 270 \zeta_{6} + 540) q^{71} + ( - 790 \zeta_{6} + 395) q^{73} + (112 \zeta_{6} + 112) q^{76} - 540 q^{77} - 440 q^{79} + ( - 192 \zeta_{6} - 192) q^{80} + (1380 \zeta_{6} - 690) q^{83} + ( - 351 \zeta_{6} + 702) q^{85} + ( - 876 \zeta_{6} - 876) q^{89} + ( - 156 \zeta_{6} - 390) q^{91} - 144 q^{92} - 126 \zeta_{6} q^{95} + (668 \zeta_{6} - 1336) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 18 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 18 q^{7} - 90 q^{11} - 65 q^{13} - 64 q^{16} - 117 q^{17} - 42 q^{19} + 72 q^{20} + 18 q^{23} + 196 q^{25} - 144 q^{28} - 99 q^{29} + 54 q^{35} + 195 q^{37} + 63 q^{41} + 82 q^{43} - 235 q^{49} - 208 q^{52} + 522 q^{53} + 270 q^{55} + 1368 q^{59} + 719 q^{61} + 1024 q^{64} + 351 q^{65} - 1218 q^{67} - 936 q^{68} + 810 q^{71} + 336 q^{76} - 1080 q^{77} - 880 q^{79} - 576 q^{80} + 1053 q^{85} - 2628 q^{89} - 936 q^{91} - 288 q^{92} - 126 q^{95} - 2004 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 −4.00000 + 6.92820i − 5.19615i 0 9.00000 + 5.19615i 0 0 0
82.1 0 0 −4.00000 − 6.92820i 5.19615i 0 9.00000 − 5.19615i 0 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.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.q.b 2
3.b odd 2 1 39.4.j.a ✓ 2
12.b even 2 1 624.4.bv.a 2
13.e even 6 1 inner 117.4.q.b 2
13.f odd 12 2 1521.4.a.m 2
39.h odd 6 1 39.4.j.a ✓ 2
39.h odd 6 1 507.4.b.a 2
39.i odd 6 1 507.4.b.a 2
39.k even 12 2 507.4.a.g 2
156.r even 6 1 624.4.bv.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.a ✓ 2 3.b odd 2 1
39.4.j.a ✓ 2 39.h odd 6 1
117.4.q.b 2 1.a even 1 1 trivial
117.4.q.b 2 13.e even 6 1 inner
507.4.a.g 2 39.k even 12 2
507.4.b.a 2 39.h odd 6 1
507.4.b.a 2 39.i odd 6 1
624.4.bv.a 2 12.b even 2 1
624.4.bv.a 2 156.r even 6 1
1521.4.a.m 2 13.f odd 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 27 \) Copy content Toggle raw display
$7$ \( T^{2} - 18T + 108 \) Copy content Toggle raw display
$11$ \( T^{2} + 90T + 2700 \) Copy content Toggle raw display
$13$ \( T^{2} + 65T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 117T + 13689 \) Copy content Toggle raw display
$19$ \( T^{2} + 42T + 588 \) Copy content Toggle raw display
$23$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$29$ \( T^{2} + 99T + 9801 \) Copy content Toggle raw display
$31$ \( T^{2} + 37632 \) Copy content Toggle raw display
$37$ \( T^{2} - 195T + 12675 \) Copy content Toggle raw display
$41$ \( T^{2} - 63T + 1323 \) Copy content Toggle raw display
$43$ \( T^{2} - 82T + 6724 \) Copy content Toggle raw display
$47$ \( T^{2} + 5292 \) Copy content Toggle raw display
$53$ \( (T - 261)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 1368 T + 623808 \) Copy content Toggle raw display
$61$ \( T^{2} - 719T + 516961 \) Copy content Toggle raw display
$67$ \( T^{2} + 1218 T + 494508 \) Copy content Toggle raw display
$71$ \( T^{2} - 810T + 218700 \) Copy content Toggle raw display
$73$ \( T^{2} + 468075 \) Copy content Toggle raw display
$79$ \( (T + 440)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1428300 \) Copy content Toggle raw display
$89$ \( T^{2} + 2628 T + 2302128 \) Copy content Toggle raw display
$97$ \( T^{2} + 2004 T + 1338672 \) Copy content Toggle raw display
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