Properties

Label 169.4.c.c
Level $169$
Weight $4$
Character orbit 169.c
Analytic conductor $9.971$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,4,Mod(22,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.22");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), not 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: \(9.97132279097\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{2} + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} + 9 q^{5} - 3 \zeta_{6} q^{6} + 15 \zeta_{6} q^{7} + 21 q^{8} + 26 \zeta_{6} q^{9} + ( - 27 \zeta_{6} + 27) q^{10} + (48 \zeta_{6} - 48) q^{11} + \cdots - 1248 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + q^{3} - q^{4} + 18 q^{5} - 3 q^{6} + 15 q^{7} + 42 q^{8} + 26 q^{9} + 27 q^{10} - 48 q^{11} - 2 q^{12} + 90 q^{14} + 9 q^{15} + 71 q^{16} - 45 q^{17} + 156 q^{18} + 6 q^{19} - 9 q^{20}+ \cdots - 2496 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.500000 + 0.866025i
0.500000 0.866025i
1.50000 2.59808i 0.500000 0.866025i −0.500000 0.866025i 9.00000 −1.50000 2.59808i 7.50000 + 12.9904i 21.0000 13.0000 + 22.5167i 13.5000 23.3827i
146.1 1.50000 + 2.59808i 0.500000 + 0.866025i −0.500000 + 0.866025i 9.00000 −1.50000 + 2.59808i 7.50000 12.9904i 21.0000 13.0000 22.5167i 13.5000 + 23.3827i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.c.c 2
13.b even 2 1 169.4.c.b 2
13.c even 3 1 169.4.a.b 1
13.c even 3 1 inner 169.4.c.c 2
13.d odd 4 2 169.4.e.d 4
13.e even 6 1 169.4.a.c 1
13.e even 6 1 169.4.c.b 2
13.f odd 12 2 13.4.b.a 2
13.f odd 12 2 169.4.e.d 4
39.h odd 6 1 1521.4.a.d 1
39.i odd 6 1 1521.4.a.i 1
39.k even 12 2 117.4.b.a 2
52.l even 12 2 208.4.f.b 2
65.o even 12 2 325.4.d.a 2
65.s odd 12 2 325.4.c.b 2
65.t even 12 2 325.4.d.b 2
104.u even 12 2 832.4.f.c 2
104.x odd 12 2 832.4.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 13.f odd 12 2
117.4.b.a 2 39.k even 12 2
169.4.a.b 1 13.c even 3 1
169.4.a.c 1 13.e even 6 1
169.4.c.b 2 13.b even 2 1
169.4.c.b 2 13.e even 6 1
169.4.c.c 2 1.a even 1 1 trivial
169.4.c.c 2 13.c even 3 1 inner
169.4.e.d 4 13.d odd 4 2
169.4.e.d 4 13.f odd 12 2
208.4.f.b 2 52.l even 12 2
325.4.c.b 2 65.s odd 12 2
325.4.d.a 2 65.o even 12 2
325.4.d.b 2 65.t even 12 2
832.4.f.c 2 104.u even 12 2
832.4.f.e 2 104.x odd 12 2
1521.4.a.d 1 39.h odd 6 1
1521.4.a.i 1 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T - 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$11$ \( T^{2} + 48T + 2304 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 45T + 2025 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 162T + 26244 \) Copy content Toggle raw display
$29$ \( T^{2} - 144T + 20736 \) Copy content Toggle raw display
$31$ \( (T + 264)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 303T + 91809 \) Copy content Toggle raw display
$41$ \( T^{2} + 192T + 36864 \) Copy content Toggle raw display
$43$ \( T^{2} + 97T + 9409 \) Copy content Toggle raw display
$47$ \( (T + 111)^{2} \) Copy content Toggle raw display
$53$ \( (T + 414)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 522T + 272484 \) Copy content Toggle raw display
$61$ \( T^{2} + 376T + 141376 \) Copy content Toggle raw display
$67$ \( T^{2} + 36T + 1296 \) Copy content Toggle raw display
$71$ \( T^{2} - 357T + 127449 \) Copy content Toggle raw display
$73$ \( (T - 1098)^{2} \) Copy content Toggle raw display
$79$ \( (T + 830)^{2} \) Copy content Toggle raw display
$83$ \( (T - 438)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 438T + 191844 \) Copy content Toggle raw display
$97$ \( T^{2} + 852T + 725904 \) Copy content Toggle raw display
show more
show less