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Properties

Label 167.5.b.a
Level $167$
Weight $5$
Character orbit 167.b
Self dual yes
Analytic conductor $17.263$
Analytic rank $0$
Dimension $11$
CM discriminant -167
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [167,5,Mod(166,167)] 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("167.166"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(167, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 167 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 167.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [11] 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: yes
Analytic conductor: \(17.2627838350\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 22x^{9} + 176x^{7} - 616x^{5} + 880x^{3} - 352x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{4}) q^{2} + (2 \beta_{6} - 3 \beta_{5}) q^{3} + (7 \beta_{8} - 2 \beta_{3} + 16) q^{4} + ( - \beta_{10} - 11 \beta_{9} + \cdots - 25 \beta_1) q^{6} + ( - 15 \beta_{9} - 2 \beta_{2}) q^{7}+ \cdots + ( - 1855 \beta_{9} + \cdots + 3182 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 176 q^{4} + 891 q^{9} + 2816 q^{16} + 6875 q^{25} + 14256 q^{36} + 36971 q^{42} + 26059 q^{44} + 4763 q^{48} + 26411 q^{49} - 25861 q^{54} - 64229 q^{62} + 45056 q^{64} - 108229 q^{72} + 72171 q^{81}+ \cdots - 202037 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 22x^{9} + 176x^{7} - 616x^{5} + 880x^{3} - 352x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 14\nu^{5} + \nu^{4} + 56\nu^{3} - 8\nu^{2} - 56\nu + 8 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 12\nu^{4} - 30\nu^{3} - 36\nu^{2} + 60\nu + 16 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} - \nu^{5} + 24\nu^{4} + 10\nu^{3} - 72\nu^{2} - 20\nu + 32 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} + 6\nu^{4} - 56\nu^{3} - 48\nu^{2} + 56\nu + 48 ) / 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} - 16\nu^{6} + 80\nu^{4} + 2\nu^{3} - 128\nu^{2} - 12\nu + 32 ) / 7 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} - 18\nu^{7} + 108\nu^{5} - 240\nu^{3} - 3\nu^{2} + 144\nu + 12 ) / 7 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} - 20\nu^{8} + 140\nu^{6} - 400\nu^{4} + 400\nu^{2} + \nu - 64 ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{4} + 8\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} + 2\beta_{5} + 10\beta_{3} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{7} - 3\beta_{6} - \beta_{5} + 12\beta_{4} + 60\beta_{2} + 160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{7} - 14\beta_{6} + 28\beta_{5} + 6\beta_{4} + 84\beta_{3} + 280\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7\beta_{8} + 112\beta_{7} - 48\beta_{6} - 16\beta_{5} + 112\beta_{4} - 2\beta_{3} + 448\beta_{2} + 1120 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7\beta_{9} - 18\beta_{7} - 144\beta_{6} + 288\beta_{5} + 108\beta_{4} + 672\beta_{3} + 3\beta_{2} + 2016\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7 \beta_{10} + 140 \beta_{8} + 960 \beta_{7} - 540 \beta_{6} - 180 \beta_{5} + 960 \beta_{4} + \cdots + 8064 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/167\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
166.1
−2.13199
2.14316
−1.53633
1.52199
2.56928
−2.57636
0.805035
−0.788677
−2.79841
2.80084
−0.00852428
−7.70256 −15.1472 43.3294 0 116.672 97.9933 −210.506 148.437 0
166.2 −7.64822 17.2685 42.4952 0 −132.073 −93.6999 −202.641 217.200 0
166.3 −5.31139 2.57010 12.2109 0 −13.6508 −94.3478 20.1256 −74.3946 0
166.4 −5.16562 −7.46972 10.6836 0 38.5858 81.8156 27.4623 −25.2032 0
166.5 −1.23389 11.7811 −14.4775 0 −14.5366 83.0588 37.6060 57.7933 0
166.6 −1.04298 −7.48522 −14.9122 0 7.80692 −63.3030 32.2408 −24.9715 0
166.7 3.23535 −18.0000 −5.53248 0 −58.2364 −65.0409 −69.6652 243.000 0
166.8 3.41080 17.2733 −4.36643 0 58.9157 39.6620 −69.4659 217.366 0
166.9 6.67740 11.7939 28.5877 0 78.7528 41.7537 84.0530 58.0967 0
166.10 6.78168 −15.1380 29.9912 0 −102.661 −12.8079 94.8835 148.158 0
166.11 7.99942 2.55324 47.9907 0 20.4244 −15.0839 255.907 −74.4810 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 166.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.b odd 2 1 CM by \(\Q(\sqrt{-167}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 167.5.b.a ✓ 11
167.b odd 2 1 CM 167.5.b.a ✓ 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
167.5.b.a ✓ 11 1.a even 1 1 trivial
167.5.b.a ✓ 11 167.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} - 176T_{2}^{9} + 11264T_{2}^{7} - 315392T_{2}^{5} + 3604480T_{2}^{3} - 11534336T_{2} - 8314961 \) acting on \(S_{5}^{\mathrm{new}}(167, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} - 176 T^{9} + \cdots - 8314961 \) Copy content Toggle raw display
$3$ \( T^{11} + \cdots + 62761269134 \) Copy content Toggle raw display
$5$ \( T^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots - 77\!\cdots\!66 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 10\!\cdots\!58 \) Copy content Toggle raw display
$13$ \( T^{11} \) Copy content Toggle raw display
$17$ \( T^{11} \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots - 24\!\cdots\!74 \) Copy content Toggle raw display
$23$ \( T^{11} \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots - 27\!\cdots\!82 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots - 89\!\cdots\!74 \) Copy content Toggle raw display
$37$ \( T^{11} \) Copy content Toggle raw display
$41$ \( T^{11} \) Copy content Toggle raw display
$43$ \( T^{11} \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 46\!\cdots\!46 \) Copy content Toggle raw display
$53$ \( T^{11} \) Copy content Toggle raw display
$59$ \( T^{11} \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 30\!\cdots\!86 \) Copy content Toggle raw display
$67$ \( T^{11} \) Copy content Toggle raw display
$71$ \( T^{11} \) Copy content Toggle raw display
$73$ \( T^{11} \) Copy content Toggle raw display
$79$ \( T^{11} \) Copy content Toggle raw display
$83$ \( T^{11} \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 11\!\cdots\!74 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 26\!\cdots\!18 \) Copy content Toggle raw display
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