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Properties

Label 351.2.q.e
Level $351$
Weight $2$
Character orbit 351.q
Analytic conductor $2.803$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [351,2,Mod(82,351)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("351.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,3,0,1,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 2) q^{5} + ( - 2 \zeta_{6} + 1) q^{8} + ( - 6 \zeta_{6} + 6) q^{10} + (3 \zeta_{6} + 3) q^{11} + ( - 3 \zeta_{6} - 1) q^{13} + ( - 5 \zeta_{6} + 5) q^{16}+ \cdots + (7 \zeta_{6} - 14) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + q^{4} + 6 q^{10} + 9 q^{11} - 5 q^{13} + 5 q^{16} + 6 q^{17} - 6 q^{19} + 6 q^{20} + 9 q^{22} + 3 q^{23} - 14 q^{25} - 3 q^{26} - 6 q^{29} + 9 q^{32} - 12 q^{37} - 12 q^{38} - 12 q^{40}+ \cdots - 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(326\)
\(\chi(n)\) \(\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
0.500000 + 0.866025i
0.500000 − 0.866025i
1.50000 + 0.866025i 0 0.500000 + 0.866025i − 3.46410i 0 0 − 1.73205i 0 3.00000 − 5.19615i
244.1 1.50000 − 0.866025i 0 0.500000 − 0.866025i 3.46410i 0 0 1.73205i 0 3.00000 + 5.19615i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 351.2.q.e yes 2
3.b odd 2 1 351.2.q.b ✓ 2
13.e even 6 1 inner 351.2.q.e yes 2
13.f odd 12 2 4563.2.a.o 2
39.h odd 6 1 351.2.q.b ✓ 2
39.k even 12 2 4563.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
351.2.q.b ✓ 2 3.b odd 2 1
351.2.q.b ✓ 2 39.h odd 6 1
351.2.q.e yes 2 1.a even 1 1 trivial
351.2.q.e yes 2 13.e even 6 1 inner
4563.2.a.n 2 39.k even 12 2
4563.2.a.o 2 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(351, [\chi])\):

\( T_{2}^{2} - 3T_{2} + 3 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} + 3 \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 75 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$97$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
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