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Properties

Label 468.2.l.d
Level $468$
Weight $2$
Character orbit 468.l
Analytic conductor $3.737$
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] = [468,2,Mod(217,468)] 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("468.217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(468, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,6,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.73699881460\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{5} + 4 \zeta_{6} q^{7} + ( - \zeta_{6} - 3) q^{13} + 3 \zeta_{6} q^{17} - 2 \zeta_{6} q^{19} + (6 \zeta_{6} - 6) q^{23} + 4 q^{25} + ( - 9 \zeta_{6} + 9) q^{29} + 2 q^{31} + 12 \zeta_{6} q^{35} + \cdots - 14 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + 4 q^{7} - 7 q^{13} + 3 q^{17} - 2 q^{19} - 6 q^{23} + 8 q^{25} + 9 q^{29} + 4 q^{31} + 12 q^{35} + 7 q^{37} + 3 q^{41} + 4 q^{43} + 12 q^{47} - 9 q^{49} - 18 q^{53} - 5 q^{61} - 21 q^{65}+ \cdots - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 0 3.00000 0 2.00000 + 3.46410i 0 0 0
289.1 0 0 0 3.00000 0 2.00000 − 3.46410i 0 0 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.l.d 2
3.b odd 2 1 52.2.e.b ✓ 2
4.b odd 2 1 1872.2.t.m 2
12.b even 2 1 208.2.i.a 2
13.c even 3 1 inner 468.2.l.d 2
13.c even 3 1 6084.2.a.o 1
13.e even 6 1 6084.2.a.c 1
13.f odd 12 2 6084.2.b.k 2
15.d odd 2 1 1300.2.i.b 2
15.e even 4 2 1300.2.bb.d 4
21.c even 2 1 2548.2.k.a 2
21.g even 6 1 2548.2.i.b 2
21.g even 6 1 2548.2.l.g 2
21.h odd 6 1 2548.2.i.g 2
21.h odd 6 1 2548.2.l.b 2
24.f even 2 1 832.2.i.i 2
24.h odd 2 1 832.2.i.c 2
39.d odd 2 1 676.2.e.d 2
39.f even 4 2 676.2.h.d 4
39.h odd 6 1 676.2.a.b 1
39.h odd 6 1 676.2.e.d 2
39.i odd 6 1 52.2.e.b ✓ 2
39.i odd 6 1 676.2.a.a 1
39.k even 12 2 676.2.d.a 2
39.k even 12 2 676.2.h.d 4
52.j odd 6 1 1872.2.t.m 2
156.p even 6 1 208.2.i.a 2
156.p even 6 1 2704.2.a.l 1
156.r even 6 1 2704.2.a.m 1
156.v odd 12 2 2704.2.f.i 2
195.x odd 6 1 1300.2.i.b 2
195.bl even 12 2 1300.2.bb.d 4
273.r even 6 1 2548.2.l.g 2
273.s odd 6 1 2548.2.l.b 2
273.bf even 6 1 2548.2.i.b 2
273.bm odd 6 1 2548.2.i.g 2
273.bn even 6 1 2548.2.k.a 2
312.bh odd 6 1 832.2.i.c 2
312.bn even 6 1 832.2.i.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.e.b ✓ 2 3.b odd 2 1
52.2.e.b ✓ 2 39.i odd 6 1
208.2.i.a 2 12.b even 2 1
208.2.i.a 2 156.p even 6 1
468.2.l.d 2 1.a even 1 1 trivial
468.2.l.d 2 13.c even 3 1 inner
676.2.a.a 1 39.i odd 6 1
676.2.a.b 1 39.h odd 6 1
676.2.d.a 2 39.k even 12 2
676.2.e.d 2 39.d odd 2 1
676.2.e.d 2 39.h odd 6 1
676.2.h.d 4 39.f even 4 2
676.2.h.d 4 39.k even 12 2
832.2.i.c 2 24.h odd 2 1
832.2.i.c 2 312.bh odd 6 1
832.2.i.i 2 24.f even 2 1
832.2.i.i 2 312.bn even 6 1
1300.2.i.b 2 15.d odd 2 1
1300.2.i.b 2 195.x odd 6 1
1300.2.bb.d 4 15.e even 4 2
1300.2.bb.d 4 195.bl even 12 2
1872.2.t.m 2 4.b odd 2 1
1872.2.t.m 2 52.j odd 6 1
2548.2.i.b 2 21.g even 6 1
2548.2.i.b 2 273.bf even 6 1
2548.2.i.g 2 21.h odd 6 1
2548.2.i.g 2 273.bm odd 6 1
2548.2.k.a 2 21.c even 2 1
2548.2.k.a 2 273.bn even 6 1
2548.2.l.b 2 21.h odd 6 1
2548.2.l.b 2 273.s odd 6 1
2548.2.l.g 2 21.g even 6 1
2548.2.l.g 2 273.r even 6 1
2704.2.a.l 1 156.p even 6 1
2704.2.a.m 1 156.r even 6 1
2704.2.f.i 2 156.v odd 12 2
6084.2.a.c 1 13.e even 6 1
6084.2.a.o 1 13.c even 3 1
6084.2.b.k 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 \(S_{2}^{\mathrm{new}}(468, [\chi])\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
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