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Properties

Label 1764.3.d.h
Level $1764$
Weight $3$
Character orbit 1764.d
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,3,Mod(685,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
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("1764.685");
 
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.d (of order \(2\), degree \(1\), 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: \(8\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2}) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2}) q^{5} + (\beta_{4} - \beta_1) q^{11} + ( - 2 \beta_{5} + 4 \beta_{3} + \beta_{2}) q^{13} + ( - 2 \beta_{6} - 3 \beta_{3} + 3 \beta_{2}) q^{17} + (4 \beta_{6} - \beta_{5} + \cdots - 2 \beta_{2}) q^{19}+ \cdots + ( - 14 \beta_{6} - 4 \beta_{5} + \cdots - 3 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{23} + 72 q^{25} - 80 q^{29} + 128 q^{37} - 112 q^{43} + 144 q^{53} - 240 q^{65} - 64 q^{67} - 224 q^{71} - 432 q^{79} - 96 q^{85} + 272 q^{95}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\nu^{7} - 35\nu^{5} + 126\nu^{3} - 10\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} - 12\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} - 28\nu^{5} + 91\nu^{3} - 96\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8\nu^{7} - 28\nu^{5} + 91\nu^{3} - 8\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{6} - 14\nu^{4} + 56\nu^{2} - 18 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} - 35\nu^{5} + 126\nu^{3} - 134\nu ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{7} - 2\beta_{5} + 2\beta_{4} + 6\beta_{2} ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{5} + 8\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{6} + 4\beta_{3} + 4\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13\beta_{7} - 18\beta_{5} - 18\beta_{4} + 26\beta_{2} ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 22\beta_{7} + 31\beta_{5} - 31\beta_{4} - 44\beta_{2} ) / 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
−0.662827 − 0.382683i
1.60021 − 0.923880i
0.662827 − 0.382683i
−1.60021 − 0.923880i
−1.60021 + 0.923880i
0.662827 + 0.382683i
1.60021 + 0.923880i
−0.662827 + 0.382683i
0 0 0 − 5.82798i 0 0 0 0 0
685.2 0 0 0 − 5.37964i 0 0 0 0 0
685.3 0 0 0 − 0.929003i 0 0 0 0 0
685.4 0 0 0 − 0.480662i 0 0 0 0 0
685.5 0 0 0 0.480662i 0 0 0 0 0
685.6 0 0 0 0.929003i 0 0 0 0 0
685.7 0 0 0 5.37964i 0 0 0 0 0
685.8 0 0 0 5.82798i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 685.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.d.h 8
3.b odd 2 1 588.3.d.c ✓ 8
7.b odd 2 1 inner 1764.3.d.h 8
7.c even 3 1 1764.3.z.l 8
7.c even 3 1 1764.3.z.m 8
7.d odd 6 1 1764.3.z.l 8
7.d odd 6 1 1764.3.z.m 8
12.b even 2 1 2352.3.f.j 8
21.c even 2 1 588.3.d.c ✓ 8
21.g even 6 1 588.3.m.e 8
21.g even 6 1 588.3.m.f 8
21.h odd 6 1 588.3.m.e 8
21.h odd 6 1 588.3.m.f 8
84.h odd 2 1 2352.3.f.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.d.c ✓ 8 3.b odd 2 1
588.3.d.c ✓ 8 21.c even 2 1
588.3.m.e 8 21.g even 6 1
588.3.m.e 8 21.h odd 6 1
588.3.m.f 8 21.g even 6 1
588.3.m.f 8 21.h odd 6 1
1764.3.d.h 8 1.a even 1 1 trivial
1764.3.d.h 8 7.b odd 2 1 inner
1764.3.z.l 8 7.c even 3 1
1764.3.z.l 8 7.d odd 6 1
1764.3.z.m 8 7.c even 3 1
1764.3.z.m 8 7.d odd 6 1
2352.3.f.j 8 12.b even 2 1
2352.3.f.j 8 84.h odd 2 1

Hecke kernels

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

\( T_{5}^{8} + 64T_{5}^{6} + 1052T_{5}^{4} + 1088T_{5}^{2} + 196 \) Copy content Toggle raw display
\( T_{11}^{4} - 124T_{11}^{2} + 48T_{11} + 3292 \) 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
$5$ \( T^{8} + 64 T^{6} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 124 T^{2} + \cdots + 3292)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 712 T^{6} + \cdots + 2979076 \) Copy content Toggle raw display
$17$ \( T^{8} + 768 T^{6} + \cdots + 168428484 \) Copy content Toggle raw display
$19$ \( T^{8} + 1136 T^{6} + \cdots + 285745216 \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} + \cdots - 37988)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 40 T^{3} + \cdots + 468892)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 2052452416 \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{3} + \cdots - 546236)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 26697826996036 \) Copy content Toggle raw display
$43$ \( (T^{4} + 56 T^{3} + \cdots + 192784)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 8881401308224 \) Copy content Toggle raw display
$53$ \( (T^{4} - 72 T^{3} + \cdots - 8464112)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 372481662446656 \) Copy content Toggle raw display
$61$ \( T^{8} + 13192 T^{6} + \cdots + 454276 \) Copy content Toggle raw display
$67$ \( (T^{4} + 32 T^{3} + \cdots - 553856)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 112 T^{3} + \cdots - 7722596)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 3712242251524 \) Copy content Toggle raw display
$79$ \( (T^{4} + 216 T^{3} + \cdots - 7050224)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 61585579131904 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 5315948141956 \) Copy content Toggle raw display
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