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Properties

Label 279.2.h.c
Level $279$
Weight $2$
Character orbit 279.h
Analytic conductor $2.228$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [279,2,Mod(118,279)] 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("279.118"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(279, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 279 = 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 279.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.22782621639\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + (2 \beta_{3} + 1) q^{4} + (\beta_{2} + 1) q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} + 3) q^{8} + (\beta_{2} - \beta_1 + 1) q^{10} + ( - \beta_{2} + 3 \beta_1 - 1) q^{11}+ \cdots - 2 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} + 12 q^{8} + 2 q^{10} - 2 q^{11} + 2 q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 6 q^{19} + 2 q^{20} - 14 q^{22} + 16 q^{23} + 8 q^{25} - 6 q^{26} - 10 q^{28}+ \cdots + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(218\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
0.707107 − 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 − 1.22474i
−0.414214 0 −1.82843 0.500000 − 0.866025i 0 0.207107 + 0.358719i 1.58579 0 −0.207107 + 0.358719i
118.2 2.41421 0 3.82843 0.500000 − 0.866025i 0 −1.20711 − 2.09077i 4.41421 0 1.20711 − 2.09077i
253.1 −0.414214 0 −1.82843 0.500000 + 0.866025i 0 0.207107 − 0.358719i 1.58579 0 −0.207107 − 0.358719i
253.2 2.41421 0 3.82843 0.500000 + 0.866025i 0 −1.20711 + 2.09077i 4.41421 0 1.20711 + 2.09077i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 279.2.h.c 4
3.b odd 2 1 31.2.c.a ✓ 4
12.b even 2 1 496.2.i.h 4
15.d odd 2 1 775.2.e.e 4
15.e even 4 2 775.2.o.d 8
31.c even 3 1 inner 279.2.h.c 4
31.c even 3 1 8649.2.a.l 2
31.e odd 6 1 8649.2.a.k 2
93.c even 2 1 961.2.c.a 4
93.g even 6 1 961.2.a.c 2
93.g even 6 1 961.2.c.a 4
93.h odd 6 1 31.2.c.a ✓ 4
93.h odd 6 1 961.2.a.a 2
93.k even 10 4 961.2.g.r 16
93.l odd 10 4 961.2.g.o 16
93.o odd 30 4 961.2.d.l 8
93.o odd 30 4 961.2.g.o 16
93.p even 30 4 961.2.d.i 8
93.p even 30 4 961.2.g.r 16
372.p even 6 1 496.2.i.h 4
465.u odd 6 1 775.2.e.e 4
465.be even 12 2 775.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a ✓ 4 3.b odd 2 1
31.2.c.a ✓ 4 93.h odd 6 1
279.2.h.c 4 1.a even 1 1 trivial
279.2.h.c 4 31.c even 3 1 inner
496.2.i.h 4 12.b even 2 1
496.2.i.h 4 372.p even 6 1
775.2.e.e 4 15.d odd 2 1
775.2.e.e 4 465.u odd 6 1
775.2.o.d 8 15.e even 4 2
775.2.o.d 8 465.be even 12 2
961.2.a.a 2 93.h odd 6 1
961.2.a.c 2 93.g even 6 1
961.2.c.a 4 93.c even 2 1
961.2.c.a 4 93.g even 6 1
961.2.d.i 8 93.p even 30 4
961.2.d.l 8 93.o odd 30 4
961.2.g.o 16 93.l odd 10 4
961.2.g.o 16 93.o odd 30 4
961.2.g.r 16 93.k even 10 4
961.2.g.r 16 93.p even 30 4
8649.2.a.k 2 31.e odd 6 1
8649.2.a.l 2 31.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(279, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 31)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 2 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + \cdots + 9409 \) Copy content Toggle raw display
$47$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$71$ \( T^{4} + 14 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$79$ \( T^{4} - 22 T^{3} + \cdots + 10609 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 56)^{2} \) Copy content Toggle raw display
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