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Properties

Label 1764.2.l.a
Level $1764$
Weight $2$
Character orbit 1764.l
Analytic conductor $14.086$
Analytic rank $1$
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] = [1764,2,Mod(949,1764)] 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, 2, 4])) 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("1764.949"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.l (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} - 1) q^{3} - 3 q^{5} + 3 \zeta_{6} q^{9} + 3 q^{11} + (\zeta_{6} - 1) q^{13} + (3 \zeta_{6} + 3) q^{15} + ( - 6 \zeta_{6} + 6) q^{17} - 4 \zeta_{6} q^{19} - 3 q^{23} + 4 q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + \cdots + 9 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 6 q^{5} + 3 q^{9} + 6 q^{11} - q^{13} + 9 q^{15} + 6 q^{17} - 4 q^{19} - 6 q^{23} + 8 q^{25} - 3 q^{29} + 5 q^{31} - 9 q^{33} - 2 q^{37} + 3 q^{39} + 3 q^{41} + q^{43} - 9 q^{45} - 9 q^{47}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −1.50000 − 0.866025i 0 −3.00000 0 0 0 1.50000 + 2.59808i 0
961.1 0 −1.50000 + 0.866025i 0 −3.00000 0 0 0 1.50000 − 2.59808i 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
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.l.a 2
3.b odd 2 1 5292.2.l.c 2
7.b odd 2 1 1764.2.l.c 2
7.c even 3 1 1764.2.i.c 2
7.c even 3 1 1764.2.j.b 2
7.d odd 6 1 36.2.e.a ✓ 2
7.d odd 6 1 1764.2.i.a 2
9.c even 3 1 1764.2.i.c 2
9.d odd 6 1 5292.2.i.a 2
21.c even 2 1 5292.2.l.a 2
21.g even 6 1 108.2.e.a 2
21.g even 6 1 5292.2.i.c 2
21.h odd 6 1 5292.2.i.a 2
21.h odd 6 1 5292.2.j.a 2
28.f even 6 1 144.2.i.a 2
35.i odd 6 1 900.2.i.b 2
35.k even 12 2 900.2.s.b 4
56.j odd 6 1 576.2.i.f 2
56.m even 6 1 576.2.i.e 2
63.g even 3 1 inner 1764.2.l.a 2
63.h even 3 1 1764.2.j.b 2
63.i even 6 1 108.2.e.a 2
63.j odd 6 1 5292.2.j.a 2
63.k odd 6 1 324.2.a.c 1
63.k odd 6 1 1764.2.l.c 2
63.l odd 6 1 1764.2.i.a 2
63.n odd 6 1 5292.2.l.c 2
63.o even 6 1 5292.2.i.c 2
63.s even 6 1 324.2.a.a 1
63.s even 6 1 5292.2.l.a 2
63.t odd 6 1 36.2.e.a ✓ 2
84.j odd 6 1 432.2.i.c 2
105.p even 6 1 2700.2.i.b 2
105.w odd 12 2 2700.2.s.b 4
168.ba even 6 1 1728.2.i.d 2
168.be odd 6 1 1728.2.i.c 2
252.n even 6 1 1296.2.a.k 1
252.r odd 6 1 432.2.i.c 2
252.bj even 6 1 144.2.i.a 2
252.bn odd 6 1 1296.2.a.b 1
315.q odd 6 1 900.2.i.b 2
315.u even 6 1 8100.2.a.g 1
315.bn odd 6 1 8100.2.a.j 1
315.bq even 6 1 2700.2.i.b 2
315.bs even 12 2 900.2.s.b 4
315.bu odd 12 2 2700.2.s.b 4
315.bw odd 12 2 8100.2.d.c 2
315.cg even 12 2 8100.2.d.h 2
504.u odd 6 1 5184.2.a.bb 1
504.y even 6 1 5184.2.a.ba 1
504.bf even 6 1 576.2.i.e 2
504.bp odd 6 1 576.2.i.f 2
504.ca even 6 1 1728.2.i.d 2
504.cm odd 6 1 1728.2.i.c 2
504.cw odd 6 1 5184.2.a.e 1
504.cz even 6 1 5184.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a ✓ 2 7.d odd 6 1
36.2.e.a ✓ 2 63.t odd 6 1
108.2.e.a 2 21.g even 6 1
108.2.e.a 2 63.i even 6 1
144.2.i.a 2 28.f even 6 1
144.2.i.a 2 252.bj even 6 1
324.2.a.a 1 63.s even 6 1
324.2.a.c 1 63.k odd 6 1
432.2.i.c 2 84.j odd 6 1
432.2.i.c 2 252.r odd 6 1
576.2.i.e 2 56.m even 6 1
576.2.i.e 2 504.bf even 6 1
576.2.i.f 2 56.j odd 6 1
576.2.i.f 2 504.bp odd 6 1
900.2.i.b 2 35.i odd 6 1
900.2.i.b 2 315.q odd 6 1
900.2.s.b 4 35.k even 12 2
900.2.s.b 4 315.bs even 12 2
1296.2.a.b 1 252.bn odd 6 1
1296.2.a.k 1 252.n even 6 1
1728.2.i.c 2 168.be odd 6 1
1728.2.i.c 2 504.cm odd 6 1
1728.2.i.d 2 168.ba even 6 1
1728.2.i.d 2 504.ca even 6 1
1764.2.i.a 2 7.d odd 6 1
1764.2.i.a 2 63.l odd 6 1
1764.2.i.c 2 7.c even 3 1
1764.2.i.c 2 9.c even 3 1
1764.2.j.b 2 7.c even 3 1
1764.2.j.b 2 63.h even 3 1
1764.2.l.a 2 1.a even 1 1 trivial
1764.2.l.a 2 63.g even 3 1 inner
1764.2.l.c 2 7.b odd 2 1
1764.2.l.c 2 63.k odd 6 1
2700.2.i.b 2 105.p even 6 1
2700.2.i.b 2 315.bq even 6 1
2700.2.s.b 4 105.w odd 12 2
2700.2.s.b 4 315.bu odd 12 2
5184.2.a.e 1 504.cw odd 6 1
5184.2.a.f 1 504.cz even 6 1
5184.2.a.ba 1 504.y even 6 1
5184.2.a.bb 1 504.u odd 6 1
5292.2.i.a 2 9.d odd 6 1
5292.2.i.a 2 21.h odd 6 1
5292.2.i.c 2 21.g even 6 1
5292.2.i.c 2 63.o even 6 1
5292.2.j.a 2 21.h odd 6 1
5292.2.j.a 2 63.j odd 6 1
5292.2.l.a 2 21.c even 2 1
5292.2.l.a 2 63.s even 6 1
5292.2.l.c 2 3.b odd 2 1
5292.2.l.c 2 63.n odd 6 1
8100.2.a.g 1 315.u even 6 1
8100.2.a.j 1 315.bn odd 6 1
8100.2.d.c 2 315.bw odd 12 2
8100.2.d.h 2 315.cg even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( (T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$83$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
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