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Properties

Label 1764.4.k.b
Level $1764$
Weight $4$
Character orbit 1764.k
Analytic conductor $104.079$
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] = [1764,4,Mod(361,1764)] 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, 4])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.361"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-18,0,0,0,0,0,36,0,-20,0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(104.079369250\)
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 12)
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.

\(f(q)\) \(=\) \( q - 18 \zeta_{6} q^{5} + ( - 36 \zeta_{6} + 36) q^{11} - 10 q^{13} + ( - 18 \zeta_{6} + 18) q^{17} + 100 \zeta_{6} q^{19} + 72 \zeta_{6} q^{23} + (199 \zeta_{6} - 199) q^{25} + 234 q^{29} + ( - 16 \zeta_{6} + 16) q^{31} + \cdots - 1054 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{5} + 36 q^{11} - 20 q^{13} + 18 q^{17} + 100 q^{19} + 72 q^{23} - 199 q^{25} + 468 q^{29} + 16 q^{31} + 226 q^{37} - 180 q^{41} + 904 q^{43} + 432 q^{47} + 414 q^{53} - 1296 q^{55} - 684 q^{59}+ \cdots - 2108 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 0 −9.00000 − 15.5885i 0 0 0 0 0
1549.1 0 0 0 −9.00000 + 15.5885i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.k.b 2
3.b odd 2 1 588.4.i.d 2
7.b odd 2 1 1764.4.k.o 2
7.c even 3 1 36.4.a.a 1
7.c even 3 1 inner 1764.4.k.b 2
7.d odd 6 1 1764.4.a.b 1
7.d odd 6 1 1764.4.k.o 2
21.c even 2 1 588.4.i.e 2
21.g even 6 1 588.4.a.c 1
21.g even 6 1 588.4.i.e 2
21.h odd 6 1 12.4.a.a ✓ 1
21.h odd 6 1 588.4.i.d 2
28.g odd 6 1 144.4.a.g 1
35.j even 6 1 900.4.a.g 1
35.l odd 12 2 900.4.d.c 2
56.k odd 6 1 576.4.a.a 1
56.p even 6 1 576.4.a.b 1
63.g even 3 1 324.4.e.a 2
63.h even 3 1 324.4.e.a 2
63.j odd 6 1 324.4.e.h 2
63.n odd 6 1 324.4.e.h 2
84.j odd 6 1 2352.4.a.bk 1
84.n even 6 1 48.4.a.a 1
105.o odd 6 1 300.4.a.b 1
105.x even 12 2 300.4.d.e 2
168.s odd 6 1 192.4.a.f 1
168.v even 6 1 192.4.a.l 1
231.l even 6 1 1452.4.a.d 1
273.w odd 6 1 2028.4.a.c 1
273.cd even 12 2 2028.4.b.c 2
336.bt odd 12 2 768.4.d.g 2
336.bu even 12 2 768.4.d.j 2
420.ba even 6 1 1200.4.a.be 1
420.bp odd 12 2 1200.4.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a ✓ 1 21.h odd 6 1
36.4.a.a 1 7.c even 3 1
48.4.a.a 1 84.n even 6 1
144.4.a.g 1 28.g odd 6 1
192.4.a.f 1 168.s odd 6 1
192.4.a.l 1 168.v even 6 1
300.4.a.b 1 105.o odd 6 1
300.4.d.e 2 105.x even 12 2
324.4.e.a 2 63.g even 3 1
324.4.e.a 2 63.h even 3 1
324.4.e.h 2 63.j odd 6 1
324.4.e.h 2 63.n odd 6 1
576.4.a.a 1 56.k odd 6 1
576.4.a.b 1 56.p even 6 1
588.4.a.c 1 21.g even 6 1
588.4.i.d 2 3.b odd 2 1
588.4.i.d 2 21.h odd 6 1
588.4.i.e 2 21.c even 2 1
588.4.i.e 2 21.g even 6 1
768.4.d.g 2 336.bt odd 12 2
768.4.d.j 2 336.bu even 12 2
900.4.a.g 1 35.j even 6 1
900.4.d.c 2 35.l odd 12 2
1200.4.a.be 1 420.ba even 6 1
1200.4.f.d 2 420.bp odd 12 2
1452.4.a.d 1 231.l even 6 1
1764.4.a.b 1 7.d odd 6 1
1764.4.k.b 2 1.a even 1 1 trivial
1764.4.k.b 2 7.c even 3 1 inner
1764.4.k.o 2 7.b odd 2 1
1764.4.k.o 2 7.d odd 6 1
2028.4.a.c 1 273.w odd 6 1
2028.4.b.c 2 273.cd even 12 2
2352.4.a.bk 1 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{2} + 18T_{5} + 324 \) Copy content Toggle raw display
\( T_{11}^{2} - 36T_{11} + 1296 \) Copy content Toggle raw display
\( T_{13} + 10 \) 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
$5$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 36T + 1296 \) Copy content Toggle raw display
$13$ \( (T + 10)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 18T + 324 \) Copy content Toggle raw display
$19$ \( T^{2} - 100T + 10000 \) Copy content Toggle raw display
$23$ \( T^{2} - 72T + 5184 \) Copy content Toggle raw display
$29$ \( (T - 234)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$37$ \( T^{2} - 226T + 51076 \) Copy content Toggle raw display
$41$ \( (T + 90)^{2} \) Copy content Toggle raw display
$43$ \( (T - 452)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 432T + 186624 \) Copy content Toggle raw display
$53$ \( T^{2} - 414T + 171396 \) Copy content Toggle raw display
$59$ \( T^{2} + 684T + 467856 \) Copy content Toggle raw display
$61$ \( T^{2} + 422T + 178084 \) Copy content Toggle raw display
$67$ \( T^{2} + 332T + 110224 \) Copy content Toggle raw display
$71$ \( (T - 360)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$79$ \( T^{2} + 512T + 262144 \) Copy content Toggle raw display
$83$ \( (T - 1188)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 630T + 396900 \) Copy content Toggle raw display
$97$ \( (T + 1054)^{2} \) Copy content Toggle raw display
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