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Properties

Label 528.4.b.d
Level $528$
Weight $4$
Character orbit 528.b
Analytic conductor $31.153$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,4,Mod(65,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 528.b (of order \(2\), degree \(1\), 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: \(31.1530084830\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 21x^{4} + 114x^{3} - 567x^{2} - 729x + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 66)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{5} q^{5} + ( - \beta_{5} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{4} - \beta_{3} + 7) q^{9} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{11} + (4 \beta_{3} + 5 \beta_{2} + \beta_1) q^{13}+ \cdots + ( - 69 \beta_{5} + 8 \beta_{4} + \cdots - 280) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + 43 q^{9} + 6 q^{11} + 7 q^{15} + 156 q^{17} + 36 q^{21} - 204 q^{25} + 278 q^{27} - 144 q^{29} + 114 q^{31} + 157 q^{33} - 588 q^{35} - 54 q^{37} + 48 q^{39} - 408 q^{41} + 551 q^{45} + 606 q^{49}+ \cdots - 1753 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 21x^{4} + 114x^{3} - 567x^{2} - 729x + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 21\nu^{3} - 114\nu^{2} + 567\nu + 729 ) / 729 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{5} - 34\nu^{4} + 123\nu^{3} + 393\nu^{2} - 7776\nu + 24057 ) / 1458 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 4\nu^{4} + 9\nu^{3} + 69\nu^{2} - 504\nu + 2511 ) / 162 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} - 88\nu^{4} + 177\nu^{3} + 1527\nu^{2} - 9558\nu + 44469 ) / 1458 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{5} - 92\nu^{4} + 93\nu^{3} + 525\nu^{2} - 10368\nu + 51759 ) / 1458 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 5\beta _1 + 21 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12\beta_{5} + 3\beta_{4} + 16\beta_{3} + 11\beta_{2} + 55\beta _1 - 150 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -75\beta_{5} - 15\beta_{4} + 4\beta_{3} + 104\beta_{2} - 83\beta _1 + 1425 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15\beta_{5} - 294\beta_{4} + 793\beta_{3} - 118\beta_{2} + 22\beta _1 - 1932 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
4.85873 − 1.84194i
4.85873 + 1.84194i
0.832570 − 5.12902i
0.832570 + 5.12902i
−5.19130 − 0.224541i
−5.19130 + 0.224541i
0 −4.85873 − 1.84194i 0 − 16.6397i 0 3.25743i 0 20.2145 + 17.8990i 0
65.2 0 −4.85873 + 1.84194i 0 16.6397i 0 − 3.25743i 0 20.2145 − 17.8990i 0
65.3 0 −0.832570 − 5.12902i 0 6.09929i 0 1.16981i 0 −25.6137 + 8.54053i 0
65.4 0 −0.832570 + 5.12902i 0 − 6.09929i 0 − 1.16981i 0 −25.6137 − 8.54053i 0
65.5 0 5.19130 − 0.224541i 0 12.7640i 0 26.7212i 0 26.8992 − 2.33132i 0
65.6 0 5.19130 + 0.224541i 0 − 12.7640i 0 − 26.7212i 0 26.8992 + 2.33132i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.4.b.d 6
3.b odd 2 1 528.4.b.c 6
4.b odd 2 1 66.4.b.a ✓ 6
11.b odd 2 1 528.4.b.c 6
12.b even 2 1 66.4.b.b yes 6
33.d even 2 1 inner 528.4.b.d 6
44.c even 2 1 66.4.b.b yes 6
132.d odd 2 1 66.4.b.a ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.b.a ✓ 6 4.b odd 2 1
66.4.b.a ✓ 6 132.d odd 2 1
66.4.b.b yes 6 12.b even 2 1
66.4.b.b yes 6 44.c even 2 1
528.4.b.c 6 3.b odd 2 1
528.4.b.c 6 11.b odd 2 1
528.4.b.d 6 1.a even 1 1 trivial
528.4.b.d 6 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(528, [\chi])\):

\( T_{5}^{6} + 477T_{5}^{4} + 61470T_{5}^{2} + 1678112 \) Copy content Toggle raw display
\( T_{17}^{3} - 78T_{17}^{2} - 1608T_{17} + 133056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{6} + 477 T^{4} + \cdots + 1678112 \) Copy content Toggle raw display
$7$ \( T^{6} + 726 T^{4} + \cdots + 10368 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 2357947691 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 38459847168 \) Copy content Toggle raw display
$17$ \( (T^{3} - 78 T^{2} + \cdots + 133056)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 184987680768 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 68188290632 \) Copy content Toggle raw display
$29$ \( (T^{3} + 72 T^{2} + \cdots - 2511360)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 57 T^{2} + \cdots - 722848)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 27 T^{2} + \cdots + 188452)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 204 T^{2} + \cdots - 4045104)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 338585981755392 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 27115367339552 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 52\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 2025875153408 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{3} + 33 T^{2} + \cdots + 62825648)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 965647255339008 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 183824512639488 \) Copy content Toggle raw display
$83$ \( (T^{3} + 696 T^{2} + \cdots - 455107968)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 54\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( (T^{3} - 627 T^{2} + \cdots - 9054752)^{2} \) Copy content Toggle raw display
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