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Properties

Label 1764.3.z.g
Level $1764$
Weight $3$
Character orbit 1764.z
Analytic conductor $48.066$
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] = [1764,3,Mod(325,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.325"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,12,0,0,0,0,0,18,0,0,0,0,0,24,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(48.0655186332\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + ( - 4 \zeta_{6} + 8) q^{5} + ( - 18 \zeta_{6} + 18) q^{11} + ( - 24 \zeta_{6} + 12) q^{13} + (8 \zeta_{6} + 8) q^{17} + ( - 12 \zeta_{6} + 24) q^{19} + 18 \zeta_{6} q^{23} + ( - 23 \zeta_{6} + 23) q^{25}+ \cdots + (192 \zeta_{6} - 96) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{5} + 18 q^{11} + 24 q^{17} + 36 q^{19} + 18 q^{23} + 23 q^{25} - 36 q^{29} - 72 q^{31} - 10 q^{37} - 76 q^{43} + 48 q^{47} + 18 q^{53} + 12 q^{59} - 36 q^{61} - 144 q^{65} - 26 q^{67} - 36 q^{71}+ \cdots + 144 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 0 6.00000 − 3.46410i 0 0 0 0 0
901.1 0 0 0 6.00000 + 3.46410i 0 0 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
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.z.g 2
3.b odd 2 1 588.3.m.c 2
7.b odd 2 1 1764.3.z.a 2
7.c even 3 1 252.3.d.a 2
7.c even 3 1 1764.3.z.a 2
7.d odd 6 1 252.3.d.a 2
7.d odd 6 1 inner 1764.3.z.g 2
21.c even 2 1 588.3.m.b 2
21.g even 6 1 84.3.d.a ✓ 2
21.g even 6 1 588.3.m.c 2
21.h odd 6 1 84.3.d.a ✓ 2
21.h odd 6 1 588.3.m.b 2
28.f even 6 1 1008.3.f.f 2
28.g odd 6 1 1008.3.f.f 2
84.j odd 6 1 336.3.f.b 2
84.n even 6 1 336.3.f.b 2
105.o odd 6 1 2100.3.j.c 2
105.p even 6 1 2100.3.j.c 2
105.w odd 12 2 2100.3.p.b 4
105.x even 12 2 2100.3.p.b 4
168.s odd 6 1 1344.3.f.a 2
168.v even 6 1 1344.3.f.d 2
168.ba even 6 1 1344.3.f.a 2
168.be odd 6 1 1344.3.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.d.a ✓ 2 21.g even 6 1
84.3.d.a ✓ 2 21.h odd 6 1
252.3.d.a 2 7.c even 3 1
252.3.d.a 2 7.d odd 6 1
336.3.f.b 2 84.j odd 6 1
336.3.f.b 2 84.n even 6 1
588.3.m.b 2 21.c even 2 1
588.3.m.b 2 21.h odd 6 1
588.3.m.c 2 3.b odd 2 1
588.3.m.c 2 21.g even 6 1
1008.3.f.f 2 28.f even 6 1
1008.3.f.f 2 28.g odd 6 1
1344.3.f.a 2 168.s odd 6 1
1344.3.f.a 2 168.ba even 6 1
1344.3.f.d 2 168.v even 6 1
1344.3.f.d 2 168.be odd 6 1
1764.3.z.a 2 7.b odd 2 1
1764.3.z.a 2 7.c even 3 1
1764.3.z.g 2 1.a even 1 1 trivial
1764.3.z.g 2 7.d odd 6 1 inner
2100.3.j.c 2 105.o odd 6 1
2100.3.j.c 2 105.p even 6 1
2100.3.p.b 4 105.w odd 12 2
2100.3.p.b 4 105.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} - 12T_{5} + 48 \) Copy content Toggle raw display
\( T_{11}^{2} - 18T_{11} + 324 \) Copy content Toggle raw display
\( T_{13}^{2} + 432 \) Copy content Toggle raw display
\( T_{19}^{2} - 36T_{19} + 432 \) 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
$5$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$13$ \( T^{2} + 432 \) Copy content Toggle raw display
$17$ \( T^{2} - 24T + 192 \) Copy content Toggle raw display
$19$ \( T^{2} - 36T + 432 \) Copy content Toggle raw display
$23$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$29$ \( (T + 18)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 72T + 1728 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$41$ \( T^{2} + 3072 \) Copy content Toggle raw display
$43$ \( (T + 38)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 48T + 768 \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$61$ \( T^{2} + 36T + 432 \) Copy content Toggle raw display
$67$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$71$ \( (T + 18)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 72T + 1728 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$83$ \( T^{2} + 1200 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 27648 \) Copy content Toggle raw display
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