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Properties

Label 169.2.c.b
Level $169$
Weight $2$
Character orbit 169.c
Analytic conductor $1.349$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) 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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{4} - 1) q^{2} + ( - \beta_{5} - \beta_1 + 1) q^{3} + ( - \beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + (\beta_{4} + \beta_{3}) q^{6} + (2 \beta_{4} + 2 \beta_{3} + \cdots + \beta_1) q^{7}+ \cdots + ( - 4 \beta_{3} - 2 \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 2 q^{3} + 8 q^{5} - q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9} - 5 q^{10} - 8 q^{11} - 10 q^{14} - 2 q^{15} - 2 q^{16} + 2 q^{17} - 18 q^{18} - 4 q^{19} - 4 q^{21} - 3 q^{22} + 5 q^{23} + 9 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{5}\)

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
0.222521 − 0.385418i
0.900969 − 1.56052i
−0.623490 + 1.07992i
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.623490 − 1.07992i
−1.12349 + 1.94594i 0.277479 − 0.480608i −1.52446 − 2.64044i 1.44504 0.623490 + 1.07992i 1.02446 + 1.77441i 2.35690 1.34601 + 2.33136i −1.62349 + 2.81197i
22.2 −0.277479 + 0.480608i −0.400969 + 0.694498i 0.846011 + 1.46533i 2.80194 −0.222521 − 0.385418i −1.34601 − 2.33136i −2.04892 1.17845 + 2.04113i −0.777479 + 1.34663i
22.3 0.400969 − 0.694498i 1.12349 − 1.94594i 0.678448 + 1.17511i −0.246980 −0.900969 − 1.56052i −1.17845 − 2.04113i 2.69202 −1.02446 − 1.77441i −0.0990311 + 0.171527i
146.1 −1.12349 − 1.94594i 0.277479 + 0.480608i −1.52446 + 2.64044i 1.44504 0.623490 − 1.07992i 1.02446 − 1.77441i 2.35690 1.34601 − 2.33136i −1.62349 − 2.81197i
146.2 −0.277479 − 0.480608i −0.400969 − 0.694498i 0.846011 − 1.46533i 2.80194 −0.222521 + 0.385418i −1.34601 + 2.33136i −2.04892 1.17845 − 2.04113i −0.777479 − 1.34663i
146.3 0.400969 + 0.694498i 1.12349 + 1.94594i 0.678448 − 1.17511i −0.246980 −0.900969 + 1.56052i −1.17845 + 2.04113i 2.69202 −1.02446 + 1.77441i −0.0990311 − 0.171527i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.3
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 169.2.c.b 6
13.b even 2 1 169.2.c.c 6
13.c even 3 1 169.2.a.c yes 3
13.c even 3 1 inner 169.2.c.b 6
13.d odd 4 2 169.2.e.b 12
13.e even 6 1 169.2.a.b ✓ 3
13.e even 6 1 169.2.c.c 6
13.f odd 12 2 169.2.b.b 6
13.f odd 12 2 169.2.e.b 12
39.h odd 6 1 1521.2.a.r 3
39.i odd 6 1 1521.2.a.o 3
39.k even 12 2 1521.2.b.l 6
52.i odd 6 1 2704.2.a.z 3
52.j odd 6 1 2704.2.a.ba 3
52.l even 12 2 2704.2.f.o 6
65.l even 6 1 4225.2.a.bg 3
65.n even 6 1 4225.2.a.bb 3
91.n odd 6 1 8281.2.a.bj 3
91.t odd 6 1 8281.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b ✓ 3 13.e even 6 1
169.2.a.c yes 3 13.c even 3 1
169.2.b.b 6 13.f odd 12 2
169.2.c.b 6 1.a even 1 1 trivial
169.2.c.b 6 13.c even 3 1 inner
169.2.c.c 6 13.b even 2 1
169.2.c.c 6 13.e even 6 1
169.2.e.b 12 13.d odd 4 2
169.2.e.b 12 13.f odd 12 2
1521.2.a.o 3 39.i odd 6 1
1521.2.a.r 3 39.h odd 6 1
1521.2.b.l 6 39.k even 12 2
2704.2.a.z 3 52.i odd 6 1
2704.2.a.ba 3 52.j odd 6 1
2704.2.f.o 6 52.l even 12 2
4225.2.a.bb 3 65.n even 6 1
4225.2.a.bg 3 65.l even 6 1
8281.2.a.bf 3 91.t odd 6 1
8281.2.a.bj 3 91.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 2T_{2}^{5} + 5T_{2}^{4} + 3T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{3} - 4 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{6} + 8 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$19$ \( T^{6} + 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{6} - 5 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} + \cdots + 6889 \) Copy content Toggle raw display
$31$ \( (T^{3} - 5 T^{2} + \cdots + 167)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$41$ \( T^{6} + 7 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$43$ \( T^{6} + 13 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( (T^{3} - 18 T^{2} + \cdots - 167)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - T^{2} - 86 T + 337)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 19 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + \cdots + 57121 \) Copy content Toggle raw display
$67$ \( T^{6} + T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$71$ \( T^{6} + 27 T^{5} + \cdots + 299209 \) Copy content Toggle raw display
$73$ \( (T^{3} + 9 T^{2} + \cdots - 911)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} + \cdots + 127)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 7 T^{2} + \cdots - 203)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 11 T^{5} + \cdots + 78961 \) Copy content Toggle raw display
$97$ \( T^{6} - 7 T^{5} + \cdots + 90601 \) Copy content Toggle raw display
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