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Properties

Label 336.2.h.b
Level $336$
Weight $2$
Character orbit 336.h
Analytic conductor $2.683$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,2,Mod(239,336)] 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("336.239"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.h (of order \(2\), degree \(1\), 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: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.56070144.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{5} + \beta_{3}) q^{5} + \beta_{4} q^{7} + (\beta_{7} - \beta_{5} + \beta_{3} + 2) q^{9} + 2 \beta_1 q^{11} + (3 \beta_{5} - 3 \beta_{3} - 4) q^{13} + ( - \beta_{6} + 5 \beta_{4} + \cdots + \beta_1) q^{15}+ \cdots + ( - 2 \beta_{6} - 6 \beta_{4} + 6 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} - 8 q^{13} - 4 q^{21} - 24 q^{25} - 8 q^{33} + 64 q^{37} - 8 q^{45} - 8 q^{49} - 16 q^{57} - 24 q^{61} + 64 q^{69} - 64 q^{73} - 8 q^{81} + 16 q^{85} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{4} + 2\nu^{3} - 7\nu^{2} + 6\nu - 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} - \nu^{6} - 25\nu^{5} - 46\nu^{4} - 5\nu^{3} - 95\nu^{2} - 9\nu + 19 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 372\nu - 218 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{7} + 21\nu^{6} - 67\nu^{5} + 115\nu^{4} - 117\nu^{3} + 71\nu^{2} + 41\nu - 29 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 29\nu^{6} - 89\nu^{5} + 261\nu^{4} - 373\nu^{3} + 498\nu^{2} - 257\nu + 152 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} - 36\nu^{6} + 136\nu^{5} - 361\nu^{4} + 634\nu^{3} - 756\nu^{2} + 564\nu - 167 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 42\nu^{7} - 147\nu^{6} + 543\nu^{5} - 990\nu^{4} + 1559\nu^{3} - 1422\nu^{2} + 971\nu - 278 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{5} + 7\beta_{4} - 4\beta_{3} + 3\beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - 7\beta_{6} - 10\beta_{5} + 15\beta_{4} - 2\beta_{3} - \beta_{2} - 2\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{7} - 5\beta_{6} - 7\beta_{5} - 21\beta_{4} + 18\beta_{3} - 20\beta_{2} - 5\beta _1 + 46 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -17\beta_{7} + 37\beta_{6} + 41\beta_{5} - 101\beta_{4} + 23\beta_{3} - 23\beta_{2} + 7\beta _1 - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{7} + 63\beta_{6} + 80\beta_{5} + \beta_{4} - 74\beta_{3} + 77\beta_{2} + 42\beta _1 - 237 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.500000 − 2.19293i
0.500000 + 2.19293i
0.500000 + 1.56488i
0.500000 − 1.56488i
0.500000 − 0.564882i
0.500000 + 0.564882i
0.500000 + 1.19293i
0.500000 − 1.19293i
0 −1.69293 − 0.366025i 0 − 3.38587i 0 1.00000i 0 2.73205 + 1.23931i 0
239.2 0 −1.69293 + 0.366025i 0 3.38587i 0 − 1.00000i 0 2.73205 − 1.23931i 0
239.3 0 −1.06488 − 1.36603i 0 2.12976i 0 − 1.00000i 0 −0.732051 + 2.90931i 0
239.4 0 −1.06488 + 1.36603i 0 − 2.12976i 0 1.00000i 0 −0.732051 − 2.90931i 0
239.5 0 1.06488 − 1.36603i 0 − 2.12976i 0 − 1.00000i 0 −0.732051 − 2.90931i 0
239.6 0 1.06488 + 1.36603i 0 2.12976i 0 1.00000i 0 −0.732051 + 2.90931i 0
239.7 0 1.69293 − 0.366025i 0 3.38587i 0 1.00000i 0 2.73205 − 1.23931i 0
239.8 0 1.69293 + 0.366025i 0 − 3.38587i 0 − 1.00000i 0 2.73205 + 1.23931i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 239.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.h.b ✓ 8
3.b odd 2 1 inner 336.2.h.b ✓ 8
4.b odd 2 1 inner 336.2.h.b ✓ 8
7.b odd 2 1 2352.2.h.o 8
8.b even 2 1 1344.2.h.g 8
8.d odd 2 1 1344.2.h.g 8
12.b even 2 1 inner 336.2.h.b ✓ 8
21.c even 2 1 2352.2.h.o 8
24.f even 2 1 1344.2.h.g 8
24.h odd 2 1 1344.2.h.g 8
28.d even 2 1 2352.2.h.o 8
84.h odd 2 1 2352.2.h.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.h.b ✓ 8 1.a even 1 1 trivial
336.2.h.b ✓ 8 3.b odd 2 1 inner
336.2.h.b ✓ 8 4.b odd 2 1 inner
336.2.h.b ✓ 8 12.b even 2 1 inner
1344.2.h.g 8 8.b even 2 1
1344.2.h.g 8 8.d odd 2 1
1344.2.h.g 8 24.f even 2 1
1344.2.h.g 8 24.h odd 2 1
2352.2.h.o 8 7.b odd 2 1
2352.2.h.o 8 21.c even 2 1
2352.2.h.o 8 28.d even 2 1
2352.2.h.o 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 16T_{5}^{2} + 52 \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 16 T^{2} + 52)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 40 T^{2} + 208)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 26)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 40 T^{2} + 208)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 64 T^{2} + 832)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 40 T^{2} + 208)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 96 T^{2} + 576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 16 T + 52)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 88 T^{2} + 208)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 88 T^{2} + 208)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 160 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 6)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 88 T^{2} + 208)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16 T + 52)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 160 T^{2} + 6292)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 376 T^{2} + 35152)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 44)^{4} \) Copy content Toggle raw display
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