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Properties

Label 162.5.b.c
Level $162$
Weight $5$
Character orbit 162.b
Analytic conductor $16.746$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,5,Mod(161,162)] 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("162.161"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 162.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-64,0,0,52,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.7459340196\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.221456830464.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 8 q^{4} - \beta_{4} q^{5} + (\beta_1 + 6) q^{7} - 8 \beta_{2} q^{8} + ( - \beta_{3} - \beta_1 + 1) q^{10} + ( - \beta_{7} + \beta_{6} + 11 \beta_{2}) q^{11} + (2 \beta_{3} - 3 \beta_1 - 2) q^{13}+ \cdots + ( - 19 \beta_{7} + \cdots - 601 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4} + 52 q^{7} - 20 q^{13} + 512 q^{16} + 100 q^{19} - 672 q^{22} - 1588 q^{25} - 416 q^{28} + 2956 q^{31} + 192 q^{34} - 32 q^{37} + 136 q^{43} + 2112 q^{46} - 4884 q^{49} + 160 q^{52} - 3996 q^{55}+ \cdots - 62672 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{4} + 6\nu^{3} - 33\nu^{2} + 30\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -38\nu^{7} + 133\nu^{6} - 1025\nu^{5} + 2230\nu^{4} - 6077\nu^{3} + 6952\nu^{2} - 1749\nu - 213 ) / 4230 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{4} - 30\nu^{3} + 273\nu^{2} - 258\nu + 865 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -132\nu^{7} + 462\nu^{6} - 4080\nu^{5} + 9045\nu^{4} - 35358\nu^{3} + 44223\nu^{2} - 85926\nu + 35883 ) / 470 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 27\nu^{6} - 81\nu^{5} + 765\nu^{4} - 1395\nu^{3} + 5877\nu^{2} - 5193\nu + 11260 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 216\nu^{7} - 756\nu^{6} + 7830\nu^{5} - 17685\nu^{4} + 78624\nu^{3} - 100629\nu^{2} + 198288\nu - 82944 ) / 235 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1994 \nu^{7} + 6979 \nu^{6} - 57125 \nu^{5} + 125365 \nu^{4} - 443111 \nu^{3} + 542791 \nu^{2} + \cdots + 338466 ) / 2115 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} - \beta_{6} - 12\beta_{4} - 42\beta_{2} + 81 ) / 162 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - \beta_{6} - 12\beta_{4} + 3\beta_{3} - 42\beta_{2} + 15\beta _1 - 1224 ) / 162 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -19\beta_{7} + 4\beta_{6} + 58\beta_{4} + 3\beta_{3} + 590\beta_{2} + 15\beta _1 - 1251 ) / 108 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -60\beta_{7} + 13\beta_{6} + 186\beta_{4} - 24\beta_{3} + 1812\beta_{2} - 228\beta _1 + 10575 ) / 162 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 573\beta_{7} - 34\beta_{6} - 1374\beta_{4} - 135\beta_{3} - 20658\beta_{2} - 1215\beta _1 + 59157 ) / 324 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 337\beta_{7} - 28\beta_{6} + 4\beta_{5} - 844\beta_{4} + 19\beta_{3} - 11846\beta_{2} + 845\beta _1 - 31133 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2742 \beta_{7} - 169 \beta_{6} + 42 \beta_{5} + 5028 \beta_{4} + 441 \beta_{3} + 95250 \beta_{2} + \cdots - 432624 ) / 162 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.500000 − 1.74753i
0.500000 − 2.20403i
0.500000 + 3.61825i
0.500000 + 3.16175i
0.500000 − 3.16175i
0.500000 − 3.61825i
0.500000 + 2.20403i
0.500000 + 1.74753i
− 2.82843i 0 −8.00000 − 42.7277i 0 39.6847 22.6274i 0 −120.852
161.2 − 2.82843i 0 −8.00000 − 2.20976i 0 43.9826 22.6274i 0 −6.25016
161.3 − 2.82843i 0 −8.00000 7.40592i 0 −60.3765 22.6274i 0 20.9471
161.4 − 2.82843i 0 −8.00000 37.5315i 0 2.70916 22.6274i 0 106.155
161.5 2.82843i 0 −8.00000 − 37.5315i 0 2.70916 − 22.6274i 0 106.155
161.6 2.82843i 0 −8.00000 − 7.40592i 0 −60.3765 − 22.6274i 0 20.9471
161.7 2.82843i 0 −8.00000 2.20976i 0 43.9826 − 22.6274i 0 −6.25016
161.8 2.82843i 0 −8.00000 42.7277i 0 39.6847 − 22.6274i 0 −120.852
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.5.b.c 8
3.b odd 2 1 inner 162.5.b.c 8
4.b odd 2 1 1296.5.e.e 8
9.c even 3 1 18.5.d.a ✓ 8
9.c even 3 1 54.5.d.a 8
9.d odd 6 1 18.5.d.a ✓ 8
9.d odd 6 1 54.5.d.a 8
12.b even 2 1 1296.5.e.e 8
36.f odd 6 1 144.5.q.b 8
36.f odd 6 1 432.5.q.b 8
36.h even 6 1 144.5.q.b 8
36.h even 6 1 432.5.q.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.5.d.a ✓ 8 9.c even 3 1
18.5.d.a ✓ 8 9.d odd 6 1
54.5.d.a 8 9.c even 3 1
54.5.d.a 8 9.d odd 6 1
144.5.q.b 8 36.f odd 6 1
144.5.q.b 8 36.h even 6 1
162.5.b.c 8 1.a even 1 1 trivial
162.5.b.c 8 3.b odd 2 1 inner
432.5.q.b 8 36.f odd 6 1
432.5.q.b 8 36.h even 6 1
1296.5.e.e 8 4.b odd 2 1
1296.5.e.e 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 3294T_{5}^{6} + 2765097T_{5}^{4} + 154472184T_{5}^{2} + 688747536 \) acting on \(S_{5}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 3294 T^{6} + \cdots + 688747536 \) Copy content Toggle raw display
$7$ \( (T^{4} - 26 T^{3} + \cdots - 285500)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 53\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( (T^{4} + 10 T^{3} + \cdots + 738398896)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} - 50 T^{3} + \cdots + 8154234820)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( (T^{4} - 1478 T^{3} + \cdots - 727726946432)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots - 305304165104)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 80\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{4} - 68 T^{3} + \cdots + 150128761105)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 317147769769024)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 27967816823305)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 13267734129308)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 851414599190300)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 799415366370601)^{2} \) Copy content Toggle raw display
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