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Properties

Label 1764.4.k.z
Level $1764$
Weight $4$
Character orbit 1764.k
Analytic conductor $104.079$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

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 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);
 
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")
 
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 = [4,0,0,0,3,0,0,0,0,0,-51,0,-122,0,0,0,-24] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 84)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1 + 1) q^{5} + (\beta_{3} - \beta_{2} + 25 \beta_1 - 1) q^{11} + ( - 5 \beta_{2} - 33) q^{13} + (8 \beta_{3} - 8 \beta_{2} + 8 \beta_1 - 8) q^{17} + (\beta_{3} + 84 \beta_1 + 84) q^{19}+ \cdots + (21 \beta_{2} - 410) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{5} - 51 q^{11} - 122 q^{13} - 24 q^{17} + 169 q^{19} - 192 q^{23} - 11 q^{25} + 78 q^{29} - 92 q^{31} + 173 q^{37} + 348 q^{41} - 994 q^{43} - 180 q^{47} + 285 q^{53} - 666 q^{55} + 1269 q^{59}+ \cdots - 1682 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{3} + 3\nu^{2} + 27\nu + 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{3} + 8\nu^{2} + 52\nu - 145 ) / 20 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 5\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 41\beta _1 + 42 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 57 ) / 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−1.63746 − 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 − 0.656712i
0 0 0 −4.91238 − 8.50848i 0 0 0 0 0
361.2 0 0 0 6.41238 + 11.1066i 0 0 0 0 0
1549.1 0 0 0 −4.91238 + 8.50848i 0 0 0 0 0
1549.2 0 0 0 6.41238 − 11.1066i 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.z 4
3.b odd 2 1 588.4.i.i 4
7.b odd 2 1 252.4.k.d 4
7.c even 3 1 1764.4.a.p 2
7.c even 3 1 inner 1764.4.k.z 4
7.d odd 6 1 252.4.k.d 4
7.d odd 6 1 1764.4.a.x 2
21.c even 2 1 84.4.i.b ✓ 4
21.g even 6 1 84.4.i.b ✓ 4
21.g even 6 1 588.4.a.g 2
21.h odd 6 1 588.4.a.h 2
21.h odd 6 1 588.4.i.i 4
84.h odd 2 1 336.4.q.h 4
84.j odd 6 1 336.4.q.h 4
84.j odd 6 1 2352.4.a.cb 2
84.n even 6 1 2352.4.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b ✓ 4 21.c even 2 1
84.4.i.b ✓ 4 21.g even 6 1
252.4.k.d 4 7.b odd 2 1
252.4.k.d 4 7.d odd 6 1
336.4.q.h 4 84.h odd 2 1
336.4.q.h 4 84.j odd 6 1
588.4.a.g 2 21.g even 6 1
588.4.a.h 2 21.h odd 6 1
588.4.i.i 4 3.b odd 2 1
588.4.i.i 4 21.h odd 6 1
1764.4.a.p 2 7.c even 3 1
1764.4.a.x 2 7.d odd 6 1
1764.4.k.z 4 1.a even 1 1 trivial
1764.4.k.z 4 7.c even 3 1 inner
2352.4.a.bp 2 84.n even 6 1
2352.4.a.cb 2 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 \(S_{4}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} - 3T_{5}^{3} + 135T_{5}^{2} + 378T_{5} + 15876 \) Copy content Toggle raw display
\( T_{11}^{4} + 51T_{11}^{3} + 2079T_{11}^{2} + 26622T_{11} + 272484 \) Copy content Toggle raw display
\( T_{13}^{2} + 61T_{13} - 2276 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 15876 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 51 T^{3} + \cdots + 272484 \) Copy content Toggle raw display
$13$ \( (T^{2} + 61 T - 2276)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 24 T^{3} + \cdots + 65028096 \) Copy content Toggle raw display
$19$ \( T^{4} - 169 T^{3} + \cdots + 49168144 \) Copy content Toggle raw display
$23$ \( (T^{2} + 96 T + 9216)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 39 T - 36684)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 92 T^{3} + \cdots + 114682681 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 1506681856 \) Copy content Toggle raw display
$41$ \( (T^{2} - 174 T - 140688)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 497 T + 15454)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 180 T^{3} + \cdots + 611671824 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 5356483344 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 161150862096 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 19408947856 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 32767516324 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1404 T + 361476)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 165260136484 \) Copy content Toggle raw display
$79$ \( T^{4} - 182 T^{3} + \cdots + 38800441 \) Copy content Toggle raw display
$83$ \( (T^{2} + 399 T - 197334)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 11416495104 \) Copy content Toggle raw display
$97$ \( (T^{2} + 841 T + 120262)^{2} \) Copy content Toggle raw display
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