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Properties

Label 169.8.b.c
Level $169$
Weight $8$
Character orbit 169.b
Analytic conductor $52.793$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [169,8,Mod(168,169)] 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("169.168"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(52.7930693068\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 709x^{6} + 165860x^{4} + 15215296x^{2} + 437311744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{4} + \beta_1) q^{2} + (\beta_{3} + \beta_{2} + 20) q^{3} + (\beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 65) q^{4} + ( - 6 \beta_{7} - \beta_{6} + \cdots - 10 \beta_1) q^{5} + (11 \beta_{7} - 4 \beta_{6} + \cdots + 4 \beta_1) q^{6}+ \cdots + ( - 136548 \beta_{7} + \cdots - 383060 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 160 q^{3} - 506 q^{4} + 6988 q^{9} + 8990 q^{10} + 7310 q^{12} - 36570 q^{14} - 72318 q^{16} - 23628 q^{17} + 119860 q^{22} - 345840 q^{23} - 438828 q^{25} + 349600 q^{27} + 266688 q^{29} + 1249390 q^{30}+ \cdots - 25016304 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 709x^{6} + 165860x^{4} + 15215296x^{2} + 437311744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 1349\nu^{4} - 372148\nu^{2} - 20786528 ) / 328536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -205\nu^{6} - 112277\nu^{4} - 15675448\nu^{2} - 504100544 ) / 657072 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -497\nu^{7} - 341917\nu^{5} - 68327276\nu^{3} - 3670822624\nu ) / 3435172416 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\nu^{6} + 12775\nu^{4} + 1860920\nu^{2} + 67091608 ) / 36504 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -12770\nu^{7} - 7165315\nu^{5} - 1073178497\nu^{3} - 42156112840\nu ) / 429396552 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -55105\nu^{7} - 30422333\nu^{5} - 4345106044\nu^{3} - 147575290208\nu ) / 1145057472 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} + 2\beta_{2} - 177 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 15\beta_{7} - 26\beta_{6} + 355\beta_{4} - 240\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -369\beta_{5} - 734\beta_{3} - 1148\beta_{2} + 42441 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6319\beta_{7} + 11218\beta_{6} - 204035\beta_{4} + 68904\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 125633\beta_{5} + 245870\beta_{3} + 475820\beta_{2} - 12169241 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2285039\beta_{7} - 4143090\beta_{6} + 84651267\beta_{4} - 21794216\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
168.1
− 12.6802i
− 12.1994i
− 18.6191i
− 7.26058i
7.26058i
18.6191i
12.1994i
12.6802i
− 16.6802i −6.69280 −150.230 − 406.835i 111.637i − 315.324i 370.802i −2142.21 −6786.11
168.2 − 16.1994i 71.3782 −134.421 532.467i − 1156.29i − 6.33667i 104.021i 2907.85 8625.66
168.3 − 14.6191i −51.4405 −85.7173 123.659i 752.012i − 559.667i − 618.134i 459.121 1807.78
168.4 − 3.26058i 66.7551 117.369 259.973i − 217.660i − 1453.99i − 800.043i 2269.24 847.662
168.5 3.26058i 66.7551 117.369 − 259.973i 217.660i 1453.99i 800.043i 2269.24 847.662
168.6 14.6191i −51.4405 −85.7173 − 123.659i − 752.012i 559.667i 618.134i 459.121 1807.78
168.7 16.1994i 71.3782 −134.421 − 532.467i 1156.29i 6.33667i − 104.021i 2907.85 8625.66
168.8 16.6802i −6.69280 −150.230 406.835i − 111.637i 315.324i − 370.802i −2142.21 −6786.11
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 168.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.8.b.c 8
13.b even 2 1 inner 169.8.b.c 8
13.d odd 4 1 13.8.a.c ✓ 4
13.d odd 4 1 169.8.a.c 4
39.f even 4 1 117.8.a.e 4
52.f even 4 1 208.8.a.k 4
65.g odd 4 1 325.8.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.a.c ✓ 4 13.d odd 4 1
117.8.a.e 4 39.f even 4 1
169.8.a.c 4 13.d odd 4 1
169.8.b.c 8 1.a even 1 1 trivial
169.8.b.c 8 13.b even 2 1 inner
208.8.a.k 4 52.f even 4 1
325.8.a.c 4 65.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 765T_{2}^{6} + 196580T_{2}^{4} + 17608896T_{2}^{2} + 165894400 \) acting on \(S_{8}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 765 T^{6} + \cdots + 165894400 \) Copy content Toggle raw display
$3$ \( (T^{4} - 80 T^{3} + \cdots + 1640448)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 17\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 32\!\cdots\!40)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 70\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 11\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 28\!\cdots\!64)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 63\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
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