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Properties

Label 167.7.b.a
Level $167$
Weight $7$
Character orbit 167.b
Self dual yes
Analytic conductor $38.419$
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] = [167,7,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, 7); 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, 7, names="a")
 
Level: \( N \) \(=\) \( 167 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(38.4190319645\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{5} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3}) q^{2} + ( - 2 \beta_{8} - 5 \beta_{6}) q^{3} + ( - 14 \beta_{10} + \beta_1 + 64) q^{4} + ( - 10 \beta_{9} - 69 \beta_{7} + \cdots - 47 \beta_{2}) q^{6} + (5 \beta_{9} + 137 \beta_{7}) q^{7}+ \cdots + (35721 \beta_{10} + \cdots - 137052 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 704 q^{4} + 8019 q^{9} + 45056 q^{16} + 171875 q^{25} + 513216 q^{36} - 1458061 q^{42} - 376277 q^{44} + 1461251 q^{48} + 1294139 q^{49} + 3418283 q^{54} + 4580059 q^{62} + 2883584 q^{64} + 3800819 q^{72}+ \cdots - 8922749 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( 5\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} + 6\nu^{4} - 56\nu^{3} - 48\nu^{2} + 56\nu + 48 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} + 5\nu^{5} - 36\nu^{4} - 50\nu^{3} + 108\nu^{2} + 100\nu - 48 ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{6} + 5\nu^{5} + 48\nu^{4} - 50\nu^{3} - 144\nu^{2} + 100\nu + 64 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} - 70\nu^{5} + 5\nu^{4} + 280\nu^{3} - 40\nu^{2} - 280\nu + 40 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} - 18\nu^{7} + 108\nu^{5} - 240\nu^{3} - 3\nu^{2} + 144\nu + 12 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} - 16\nu^{6} + 80\nu^{4} + 2\nu^{3} - 128\nu^{2} - 12\nu + 32 ) / 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 18\nu^{7} + 108\nu^{5} - 240\nu^{3} + 32\nu^{2} + 144\nu - 128 ) / 7 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{8} + 32\nu^{6} - 160\nu^{4} + 31\nu^{3} + 256\nu^{2} - 186\nu - 64 ) / 7 \) 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 \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{6} + 20 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + 2\beta_{7} + 6\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{8} - 8\beta_{6} + \beta_{5} + 5\beta_{2} + 120 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10\beta_{9} + 20\beta_{7} + 3\beta_{4} + 4\beta_{3} + 40\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 60\beta_{8} - 60\beta_{6} + 12\beta_{5} - 5\beta_{4} + 5\beta_{3} + 60\beta_{2} + 800 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 84\beta_{9} + 168\beta_{7} + 6\beta_{5} + 42\beta_{4} + 56\beta_{3} - 5\beta_{2} + 280\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2 \beta_{9} + 448 \beta_{8} + 31 \beta_{7} - 448 \beta_{6} + 112 \beta_{5} - 80 \beta_{4} + \cdots + 5600 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 672 \beta_{9} + 3 \beta_{8} + 1344 \beta_{7} + 32 \beta_{6} + 108 \beta_{5} + 432 \beta_{4} + \cdots + 2016 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 35 \beta_{10} - 40 \beta_{9} + 3360 \beta_{8} + 620 \beta_{7} - 3360 \beta_{6} + 960 \beta_{5} + \cdots + 40320 ) / 5 \) 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
−0.00852428
2.56928
−2.57636
2.14316
−2.13199
−0.788677
0.805035
−2.79841
2.80084
−1.53633
1.52199
−15.9974 29.2268 191.916 0 −467.553 583.537 −2046.33 125.208 0
166.2 −13.6143 15.1768 121.348 0 −206.621 −458.288 −780.755 −498.665 0
166.3 −13.3014 −53.4558 112.928 0 711.039 −661.511 −650.819 2128.52 0
166.4 −6.90872 −49.1042 −16.2696 0 339.247 295.912 554.560 1682.22 0
166.5 −6.38239 40.7854 −23.2651 0 −260.308 685.894 556.960 934.446 0
166.6 1.99030 49.1360 −60.0387 0 97.7955 −109.562 −246.874 1685.35 0
166.7 2.56303 0.0383257 −57.4309 0 0.0982298 −654.709 −311.231 −728.999 0
166.8 10.2574 −15.2503 41.2145 0 −156.429 −85.6638 −233.720 −496.428 0
166.9 10.6947 −40.8356 50.3767 0 −436.724 570.484 −145.697 938.543 0
166.10 15.2679 −29.1624 169.108 0 −445.247 273.950 1604.77 121.443 0
166.11 15.4309 53.4449 174.112 0 824.702 −440.042 1699.13 2127.36 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 167.7.b.a ✓ 11
167.b odd 2 1 CM 167.7.b.a ✓ 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
167.7.b.a ✓ 11 1.a even 1 1 trivial
167.7.b.a ✓ 11 167.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} - 704T_{2}^{9} + 180224T_{2}^{7} - 20185088T_{2}^{5} + 922746880T_{2}^{3} - 11811160064T_{2} + 16841125159 \) acting on \(S_{7}^{\mathrm{new}}(167, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + \cdots + 16841125159 \) Copy content Toggle raw display
$3$ \( T^{11} + \cdots + 86799227101046 \) Copy content Toggle raw display
$5$ \( T^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots - 15\!\cdots\!86 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots - 44\!\cdots\!62 \) Copy content Toggle raw display
$13$ \( T^{11} \) Copy content Toggle raw display
$17$ \( T^{11} \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots + 24\!\cdots\!82 \) Copy content Toggle raw display
$23$ \( T^{11} \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots - 29\!\cdots\!78 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 20\!\cdots\!18 \) 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
$47$ \( T^{11} + \cdots + 20\!\cdots\!54 \) Copy content Toggle raw display
$53$ \( T^{11} \) Copy content Toggle raw display
$59$ \( T^{11} \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots - 13\!\cdots\!62 \) 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
$89$ \( T^{11} + \cdots + 20\!\cdots\!62 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 28\!\cdots\!54 \) Copy content Toggle raw display
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