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Properties

Label 84.11.p.a
Level $84$
Weight $11$
Character orbit 84.p
Analytic conductor $53.370$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [84,11,Mod(53,84)] 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("84.53"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 84.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: \(53.3700092246\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + 243 \zeta_{6} q^{3} + (3725 \zeta_{6} - 18357) q^{7} + (59049 \zeta_{6} - 59049) q^{9} - 141961 q^{13} + ( - 2024677 \zeta_{6} + 2024677) q^{19} + ( - 3555576 \zeta_{6} - 905175) q^{21} - 9765625 \zeta_{6} q^{25} + \cdots + 884916482 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 243 q^{3} - 32989 q^{7} - 59049 q^{9} - 283922 q^{13} + 2024677 q^{19} - 5365926 q^{21} - 9765625 q^{25} - 28697814 q^{27} - 516899 q^{31} + 40895593 q^{37} - 34496523 q^{39} + 143344486 q^{43} + 523323623 q^{49}+ \cdots + 1769832964 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(-1\) \(1\) \(-1 + \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 121.500 − 210.444i 0 0 0 −16494.5 − 3225.94i 0 −29524.5 − 51137.9i 0
65.1 0 121.500 + 210.444i 0 0 0 −16494.5 + 3225.94i 0 −29524.5 + 51137.9i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 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 84.11.p.a ✓ 2
3.b odd 2 1 CM 84.11.p.a ✓ 2
7.c even 3 1 inner 84.11.p.a ✓ 2
21.h odd 6 1 inner 84.11.p.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.11.p.a ✓ 2 1.a even 1 1 trivial
84.11.p.a ✓ 2 3.b odd 2 1 CM
84.11.p.a ✓ 2 7.c even 3 1 inner
84.11.p.a ✓ 2 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{11}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 243T + 59049 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 32989 T + 282475249 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 141961)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 4099316954329 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 267184576201 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 71672243)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 72\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 47\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 884916482)^{2} \) Copy content Toggle raw display
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