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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [169,4,Mod(22,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 4 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{4} + (\beta_{3} + 2) q^{5} + ( - \beta_{3} + 12 \beta_{2} + \cdots - 12) q^{6} + ( - 11 \beta_{3} + 10 \beta_{2} + \cdots - 10) q^{7}+ \cdots + ( - 390 \beta_{3} - 130) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 5 q^{3} + 7 q^{4} + 6 q^{5} - 23 q^{6} - 9 q^{7} + 6 q^{8} - 35 q^{9} - 7 q^{10} + 80 q^{11} - 86 q^{12} + 178 q^{14} - 33 q^{15} + 39 q^{16} - 19 q^{17} + 220 q^{18} - 84 q^{19} + 19 q^{20}+ \cdots + 260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display

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

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

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{2}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−0.780776 + 1.35234i
1.28078 − 2.21837i
−0.780776 − 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i −4.34233 + 7.52113i 2.78078 + 4.81645i 3.56155 −6.78078 − 11.7446i −13.5885 − 23.5360i −21.1771 −24.2116 − 41.9358i −2.78078 + 4.81645i
22.2 1.28078 − 2.21837i 1.84233 − 3.19101i 0.719224 + 1.24573i −0.561553 −4.71922 − 8.17394i 9.08854 + 15.7418i 24.1771 6.71165 + 11.6249i −0.719224 + 1.24573i
146.1 −0.780776 − 1.35234i −4.34233 − 7.52113i 2.78078 − 4.81645i 3.56155 −6.78078 + 11.7446i −13.5885 + 23.5360i −21.1771 −24.2116 + 41.9358i −2.78078 − 4.81645i
146.2 1.28078 + 2.21837i 1.84233 + 3.19101i 0.719224 − 1.24573i −0.561553 −4.71922 + 8.17394i 9.08854 − 15.7418i 24.1771 6.71165 − 11.6249i −0.719224 − 1.24573i
\(n\): 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.j 4
13.b even 2 1 169.4.c.g 4
13.c even 3 1 169.4.a.g 2
13.c even 3 1 inner 169.4.c.j 4
13.d odd 4 2 169.4.e.f 8
13.e even 6 1 13.4.a.b ✓ 2
13.e even 6 1 169.4.c.g 4
13.f odd 12 2 169.4.b.f 4
13.f odd 12 2 169.4.e.f 8
39.h odd 6 1 117.4.a.d 2
39.i odd 6 1 1521.4.a.r 2
52.i odd 6 1 208.4.a.h 2
65.l even 6 1 325.4.a.f 2
65.r odd 12 2 325.4.b.e 4
91.t odd 6 1 637.4.a.b 2
104.p odd 6 1 832.4.a.z 2
104.s even 6 1 832.4.a.s 2
143.i odd 6 1 1573.4.a.b 2
156.r even 6 1 1872.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b ✓ 2 13.e even 6 1
117.4.a.d 2 39.h odd 6 1
169.4.a.g 2 13.c even 3 1
169.4.b.f 4 13.f odd 12 2
169.4.c.g 4 13.b even 2 1
169.4.c.g 4 13.e even 6 1
169.4.c.j 4 1.a even 1 1 trivial
169.4.c.j 4 13.c even 3 1 inner
169.4.e.f 8 13.d odd 4 2
169.4.e.f 8 13.f odd 12 2
208.4.a.h 2 52.i odd 6 1
325.4.a.f 2 65.l even 6 1
325.4.b.e 4 65.r odd 12 2
637.4.a.b 2 91.t odd 6 1
832.4.a.s 2 104.s even 6 1
832.4.a.z 2 104.p odd 6 1
1521.4.a.r 2 39.i odd 6 1
1573.4.a.b 2 143.i odd 6 1
1872.4.a.bb 2 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 5 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 9 T^{3} + \cdots + 244036 \) Copy content Toggle raw display
$11$ \( T^{4} - 80 T^{3} + \cdots + 976144 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 19 T^{3} + \cdots + 1295044 \) Copy content Toggle raw display
$19$ \( T^{4} + 84 T^{3} + \cdots + 6697744 \) Copy content Toggle raw display
$23$ \( T^{4} + 196 T^{3} + \cdots + 80856064 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 1496451856 \) Copy content Toggle raw display
$31$ \( (T^{2} - 86 T - 3064)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 209 T^{3} + \cdots + 116942596 \) Copy content Toggle raw display
$41$ \( T^{4} + 230 T^{3} + \cdots + 124724224 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 4397811856 \) Copy content Toggle raw display
$47$ \( (T^{2} + 435 T - 14918)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 118 T - 344)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 368 T^{3} + \cdots + 991746064 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 15981005056 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 51799939216 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 49503580036 \) Copy content Toggle raw display
$73$ \( (T^{2} + 456 T - 235316)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1008 T + 247216)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1958 T + 817664)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 260316284944 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 776999938576 \) Copy content Toggle raw display
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