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Properties

Label 1764.3.z.k
Level $1764$
Weight $3$
Character orbit 1764.z
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,3,Mod(325,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 7^{4} \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{6} + \beta_{2} - \beta_1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{6} + \beta_{2} - \beta_1) q^{5} + (\beta_{7} - 6 \beta_{4}) q^{11} + (4 \beta_{6} + 4 \beta_{3} - 3 \beta_1) q^{13} - 3 \beta_{3} q^{17} + ( - \beta_{6} - 5 \beta_{2} + 5 \beta_1) q^{19} + ( - 26 \beta_{4} + 26) q^{23} + (2 \beta_{7} + 27 \beta_{4}) q^{25} + ( - 2 \beta_{5} - 8) q^{29} + (8 \beta_{3} - 2 \beta_{2}) q^{31} + ( - 32 \beta_{4} + 32) q^{37} + (5 \beta_{6} + 5 \beta_{3} + 3 \beta_1) q^{41} + ( - 7 \beta_{5} + 10) q^{43} + ( - 2 \beta_{6} - 12 \beta_{2} + 12 \beta_1) q^{47} + ( - 2 \beta_{7} + 6 \beta_{4}) q^{53} + ( - 6 \beta_{6} - 6 \beta_{3} + 14 \beta_1) q^{55} + (3 \beta_{3} + 11 \beta_{2}) q^{59} + ( - 8 \beta_{6} + 9 \beta_{2} - 9 \beta_1) q^{61} + ( - 14 \beta_{7} - 14 \beta_{5} + \cdots + 24) q^{65}+ \cdots + ( - 33 \beta_{6} - 33 \beta_{3} - 22 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{11} + 104 q^{23} + 108 q^{25} - 64 q^{29} + 128 q^{37} + 80 q^{43} + 24 q^{53} + 96 q^{65} - 112 q^{67} - 448 q^{71} - 240 q^{79} - 432 q^{85} - 248 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 8\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 41\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 16 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -12\nu^{7} + 49\nu^{5} - 168\nu^{3} + 96\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{6} + 14\nu^{4} - 42\nu^{2} + 24 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{5} - 14\beta_{4} + 14 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} - 42\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{6} - 24\beta_{2} + 24\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{5} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{3} - 82\beta_{2} ) / 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{4}\)

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.

comment: embeddings in the coefficient field
 
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
0.662827 + 0.382683i
−1.60021 − 0.923880i
1.60021 + 0.923880i
−0.662827 − 0.382683i
0.662827 − 0.382683i
−1.60021 + 0.923880i
1.60021 − 0.923880i
−0.662827 + 0.382683i
0 0 0 −7.33820 + 4.23671i 0 0 0 0 0
325.2 0 0 0 −4.91434 + 2.83730i 0 0 0 0 0
325.3 0 0 0 4.91434 − 2.83730i 0 0 0 0 0
325.4 0 0 0 7.33820 − 4.23671i 0 0 0 0 0
901.1 0 0 0 −7.33820 − 4.23671i 0 0 0 0 0
901.2 0 0 0 −4.91434 − 2.83730i 0 0 0 0 0
901.3 0 0 0 4.91434 + 2.83730i 0 0 0 0 0
901.4 0 0 0 7.33820 + 4.23671i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 325.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
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.k 8
3.b odd 2 1 196.3.h.c 8
7.b odd 2 1 inner 1764.3.z.k 8
7.c even 3 1 1764.3.d.e 4
7.c even 3 1 inner 1764.3.z.k 8
7.d odd 6 1 1764.3.d.e 4
7.d odd 6 1 inner 1764.3.z.k 8
12.b even 2 1 784.3.s.g 8
21.c even 2 1 196.3.h.c 8
21.g even 6 1 196.3.b.b ✓ 4
21.g even 6 1 196.3.h.c 8
21.h odd 6 1 196.3.b.b ✓ 4
21.h odd 6 1 196.3.h.c 8
84.h odd 2 1 784.3.s.g 8
84.j odd 6 1 784.3.c.d 4
84.j odd 6 1 784.3.s.g 8
84.n even 6 1 784.3.c.d 4
84.n even 6 1 784.3.s.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.3.b.b ✓ 4 21.g even 6 1
196.3.b.b ✓ 4 21.h odd 6 1
196.3.h.c 8 3.b odd 2 1
196.3.h.c 8 21.c even 2 1
196.3.h.c 8 21.g even 6 1
196.3.h.c 8 21.h odd 6 1
784.3.c.d 4 84.j odd 6 1
784.3.c.d 4 84.n even 6 1
784.3.s.g 8 12.b even 2 1
784.3.s.g 8 84.h odd 2 1
784.3.s.g 8 84.j odd 6 1
784.3.s.g 8 84.n even 6 1
1764.3.d.e 4 7.c even 3 1
1764.3.d.e 4 7.d odd 6 1
1764.3.z.k 8 1.a even 1 1 trivial
1764.3.z.k 8 7.b odd 2 1 inner
1764.3.z.k 8 7.c even 3 1 inner
1764.3.z.k 8 7.d odd 6 1 inner

Hecke kernels

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

\( T_{5}^{8} - 104T_{5}^{6} + 8504T_{5}^{4} - 240448T_{5}^{2} + 5345344 \) Copy content Toggle raw display
\( T_{11}^{4} + 12T_{11}^{3} + 206T_{11}^{2} - 744T_{11} + 3844 \) Copy content Toggle raw display
\( T_{13}^{4} + 776T_{13}^{2} + 150152 \) Copy content Toggle raw display
\( T_{19}^{8} - 1060T_{19}^{6} + 1077998T_{19}^{4} - 48338120T_{19}^{2} + 2079542404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 104 T^{6} + \cdots + 5345344 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 12 T^{3} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 776 T^{2} + 150152)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 180 T^{6} + \cdots + 26244 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 2079542404 \) Copy content Toggle raw display
$23$ \( (T^{2} - 26 T + 676)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 16 T - 328)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 94450499584 \) Copy content Toggle raw display
$37$ \( (T^{2} - 32 T + 1024)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 740 T^{2} + 132098)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 20 T - 4702)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 1420787713024 \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + \cdots + 126736)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 111931731844 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 8688298598464 \) Copy content Toggle raw display
$67$ \( (T^{4} + 56 T^{3} + \cdots + 7529536)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 112 T + 1568)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 2755075464964 \) Copy content Toggle raw display
$79$ \( (T^{4} + 120 T^{3} + \cdots + 10291264)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 25108 T^{2} + 102330818)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 65\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{4} + 35332 T^{2} + 310652738)^{2} \) Copy content Toggle raw display
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