↑

Properties

Label 294.2.f.c
Level $294$
Weight $2$
Character orbit 294.f
Analytic conductor $2.348$
Analytic rank $0$
Dimension $16$
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [294,2,Mod(215,294)] 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("294.215"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{48}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{48}^{12} + \zeta_{48}^{4}) q^{2} + (\zeta_{48}^{13} - \zeta_{48}^{11} + \cdots + \zeta_{48}) q^{3} + ( - \zeta_{48}^{8} + 1) q^{4} + ( - 2 \zeta_{48}^{11} - 2 \zeta_{48}^{5}) q^{5} + ( - \zeta_{48}^{15} + \cdots + \zeta_{48}^{5}) q^{6}+ \cdots + ( - \zeta_{48}^{12} - 2 \zeta_{48}^{10} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 16 q^{9} - 8 q^{16} + 16 q^{18} - 32 q^{22} - 24 q^{25} - 16 q^{30} + 32 q^{36} + 16 q^{37} - 16 q^{39} - 32 q^{43} - 32 q^{57} + 16 q^{58} - 16 q^{64} + 16 q^{67} - 16 q^{72} + 32 q^{78}+ \cdots - 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(\zeta_{48}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.608761 + 0.793353i
0.793353 + 0.608761i
−0.793353 − 0.608761i
0.608761 − 0.793353i
−0.130526 − 0.991445i
−0.991445 + 0.130526i
0.991445 − 0.130526i
0.130526 + 0.991445i
−0.608761 − 0.793353i
0.793353 − 0.608761i
−0.793353 + 0.608761i
0.608761 + 0.793353i
−0.130526 + 0.991445i
−0.991445 − 0.130526i
0.991445 + 0.130526i
0.130526 − 0.991445i
−0.866025 − 0.500000i −1.27159 + 1.17604i 0.500000 + 0.866025i −1.84776 + 3.20041i 1.68925 − 0.382683i 0 − 1.00000i 0.233875 − 2.99087i 3.20041 − 1.84776i
215.2 −0.866025 − 0.500000i −0.806853 + 1.53264i 0.500000 + 0.866025i 0.765367 − 1.32565i 1.46508 − 0.923880i 0 − 1.00000i −1.69798 − 2.47323i −1.32565 + 0.765367i
215.3 −0.866025 − 0.500000i 0.806853 − 1.53264i 0.500000 + 0.866025i −0.765367 + 1.32565i −1.46508 + 0.923880i 0 − 1.00000i −1.69798 − 2.47323i 1.32565 − 0.765367i
215.4 −0.866025 − 0.500000i 1.27159 − 1.17604i 0.500000 + 0.866025i 1.84776 − 3.20041i −1.68925 + 0.382683i 0 − 1.00000i 0.233875 − 2.99087i −3.20041 + 1.84776i
215.5 0.866025 + 0.500000i −1.73073 − 0.0675653i 0.500000 + 0.866025i −0.765367 + 1.32565i −1.46508 − 0.923880i 0 1.00000i 2.99087 + 0.233875i −1.32565 + 0.765367i
215.6 0.866025 + 0.500000i −1.65427 + 0.513210i 0.500000 + 0.866025i 1.84776 − 3.20041i −1.68925 − 0.382683i 0 1.00000i 2.47323 − 1.69798i 3.20041 − 1.84776i
215.7 0.866025 + 0.500000i 1.65427 − 0.513210i 0.500000 + 0.866025i −1.84776 + 3.20041i 1.68925 + 0.382683i 0 1.00000i 2.47323 − 1.69798i −3.20041 + 1.84776i
215.8 0.866025 + 0.500000i 1.73073 + 0.0675653i 0.500000 + 0.866025i 0.765367 − 1.32565i 1.46508 + 0.923880i 0 1.00000i 2.99087 + 0.233875i 1.32565 − 0.765367i
227.1 −0.866025 + 0.500000i −1.27159 − 1.17604i 0.500000 − 0.866025i −1.84776 − 3.20041i 1.68925 + 0.382683i 0 1.00000i 0.233875 + 2.99087i 3.20041 + 1.84776i
227.2 −0.866025 + 0.500000i −0.806853 − 1.53264i 0.500000 − 0.866025i 0.765367 + 1.32565i 1.46508 + 0.923880i 0 1.00000i −1.69798 + 2.47323i −1.32565 − 0.765367i
227.3 −0.866025 + 0.500000i 0.806853 + 1.53264i 0.500000 − 0.866025i −0.765367 − 1.32565i −1.46508 − 0.923880i 0 1.00000i −1.69798 + 2.47323i 1.32565 + 0.765367i
227.4 −0.866025 + 0.500000i 1.27159 + 1.17604i 0.500000 − 0.866025i 1.84776 + 3.20041i −1.68925 − 0.382683i 0 1.00000i 0.233875 + 2.99087i −3.20041 − 1.84776i
227.5 0.866025 − 0.500000i −1.73073 + 0.0675653i 0.500000 − 0.866025i −0.765367 − 1.32565i −1.46508 + 0.923880i 0 − 1.00000i 2.99087 − 0.233875i −1.32565 − 0.765367i
227.6 0.866025 − 0.500000i −1.65427 − 0.513210i 0.500000 − 0.866025i 1.84776 + 3.20041i −1.68925 + 0.382683i 0 − 1.00000i 2.47323 + 1.69798i 3.20041 + 1.84776i
227.7 0.866025 − 0.500000i 1.65427 + 0.513210i 0.500000 − 0.866025i −1.84776 − 3.20041i 1.68925 − 0.382683i 0 − 1.00000i 2.47323 + 1.69798i −3.20041 − 1.84776i
227.8 0.866025 − 0.500000i 1.73073 − 0.0675653i 0.500000 − 0.866025i 0.765367 + 1.32565i 1.46508 − 0.923880i 0 − 1.00000i 2.99087 − 0.233875i 1.32565 + 0.765367i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 215.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.2.f.c 16
3.b odd 2 1 inner 294.2.f.c 16
7.b odd 2 1 inner 294.2.f.c 16
7.c even 3 1 294.2.d.b ✓ 8
7.c even 3 1 inner 294.2.f.c 16
7.d odd 6 1 294.2.d.b ✓ 8
7.d odd 6 1 inner 294.2.f.c 16
21.c even 2 1 inner 294.2.f.c 16
21.g even 6 1 294.2.d.b ✓ 8
21.g even 6 1 inner 294.2.f.c 16
21.h odd 6 1 294.2.d.b ✓ 8
21.h odd 6 1 inner 294.2.f.c 16
28.f even 6 1 2352.2.k.f 8
28.g odd 6 1 2352.2.k.f 8
84.j odd 6 1 2352.2.k.f 8
84.n even 6 1 2352.2.k.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.d.b ✓ 8 7.c even 3 1
294.2.d.b ✓ 8 7.d odd 6 1
294.2.d.b ✓ 8 21.g even 6 1
294.2.d.b ✓ 8 21.h odd 6 1
294.2.f.c 16 1.a even 1 1 trivial
294.2.f.c 16 3.b odd 2 1 inner
294.2.f.c 16 7.b odd 2 1 inner
294.2.f.c 16 7.c even 3 1 inner
294.2.f.c 16 7.d odd 6 1 inner
294.2.f.c 16 21.c even 2 1 inner
294.2.f.c 16 21.g even 6 1 inner
294.2.f.c 16 21.h odd 6 1 inner
2352.2.k.f 8 28.f even 6 1
2352.2.k.f 8 28.g odd 6 1
2352.2.k.f 8 84.j odd 6 1
2352.2.k.f 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 16T_{5}^{6} + 224T_{5}^{4} + 512T_{5}^{2} + 1024 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} - 8 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{8} + 16 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 44 T^{6} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{2} + 128)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 36 T^{6} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{2} + 64)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 24 T^{2} + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 80 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 4)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 100 T^{2} + 1250)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 46)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + 16 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 152 T^{6} + \cdots + 21381376)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 20 T^{6} + 398 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 80 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 4624)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 144 T^{2} + 3136)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 24 T^{3} + \cdots + 18496)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 116 T^{2} + 2)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 164 T^{6} + \cdots + 23059204)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 100 T^{2} + 578)^{4} \) Copy content Toggle raw display
show more
show less