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Properties

Label 528.2.bn.b
Level $528$
Weight $2$
Character orbit 528.bn
Analytic conductor $4.216$
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] = [528,2,Mod(17,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.bn (of order \(10\), degree \(4\), 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: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.185640625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{6} - \beta_{5} - 3 \beta_{4} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + ( - 2 \beta_{7} + 3 \beta_{6} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 2 q^{9} - q^{11} + 21 q^{15} + 10 q^{17} + 15 q^{19} - 12 q^{21} - 6 q^{25} - 20 q^{27} - 23 q^{29} - 13 q^{31} + 20 q^{33} - 13 q^{35} + 6 q^{37} + 8 q^{39} - 2 q^{41} - 8 q^{45} + 10 q^{47} - 18 q^{49} - 15 q^{51} - 15 q^{53} + 14 q^{55} + 15 q^{57} - 25 q^{59} - 10 q^{61} + 6 q^{63} + 32 q^{65} + 2 q^{67} + 24 q^{69} + 40 q^{71} + 5 q^{73} - q^{75} + 12 q^{77} - 10 q^{79} - 26 q^{81} - 21 q^{83} + 30 q^{85} + 42 q^{87} - 24 q^{91} + 27 q^{93} + 20 q^{95} + 9 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{5} + 4\nu^{4} + 16\nu^{3} + 51\nu^{2} - 54\nu - 27 ) / 216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - \nu^{5} - 4\nu^{4} - 16\nu^{3} + 21\nu^{2} - 18\nu + 27 ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{7} + 6\nu^{6} + 7\nu^{5} + 16\nu^{4} + 28\nu^{3} - 69\nu^{2} - 216\nu + 351 ) / 216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + \nu^{5} + \nu^{4} + 4\nu^{3} + 3\nu^{2} + 9\nu - 81 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{7} - 30\nu^{6} - 35\nu^{5} - 8\nu^{4} + 76\nu^{3} + 165\nu^{2} + 360\nu - 1107 ) / 216 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 2\nu^{6} - 3\nu^{5} - 8\nu^{4} + 4\nu^{3} + 13\nu^{2} + 60\nu - 99 ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{4} - \beta_{3} + 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{7} + 2\beta_{6} - \beta_{5} - 6\beta_{4} - 4\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 6\beta_{6} + 6\beta_{5} + 12\beta_{2} + 6\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{6} - 8\beta_{5} - 6\beta_{4} - 8\beta_{3} + 12\beta_{2} + \beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 24\beta_{4} - 19\beta_{3} + 3\beta_{2} - 19\beta _1 + 22 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(1 - \beta_{2} - \beta_{4} + \beta_{6}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.245684 − 1.71454i
1.55470 + 0.763481i
1.71634 − 0.232753i
−1.52536 + 0.820539i
1.71634 + 0.232753i
−1.52536 − 0.820539i
−0.245684 + 1.71454i
1.55470 − 0.763481i
0 −1.70654 − 0.296161i 0 −2.01556 − 0.654895i 0 2.01556 + 2.77418i 0 2.82458 + 1.01082i 0
17.2 0 1.20654 − 1.24268i 0 0.897526 + 0.291624i 0 −0.897526 − 1.23534i 0 −0.0885088 − 2.99869i 0
161.1 0 −1.25174 + 1.19714i 0 −0.442723 − 0.609356i 0 0.442723 − 0.143849i 0 0.133706 − 2.99702i 0
161.2 0 0.751740 − 1.56041i 0 1.56076 + 2.14820i 0 −1.56076 + 0.507121i 0 −1.86977 − 2.34605i 0
305.1 0 −1.25174 − 1.19714i 0 −0.442723 + 0.609356i 0 0.442723 + 0.143849i 0 0.133706 + 2.99702i 0
305.2 0 0.751740 + 1.56041i 0 1.56076 − 2.14820i 0 −1.56076 − 0.507121i 0 −1.86977 + 2.34605i 0
497.1 0 −1.70654 + 0.296161i 0 −2.01556 + 0.654895i 0 2.01556 − 2.77418i 0 2.82458 − 1.01082i 0
497.2 0 1.20654 + 1.24268i 0 0.897526 − 0.291624i 0 −0.897526 + 1.23534i 0 −0.0885088 + 2.99869i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.bn.b 8
3.b odd 2 1 528.2.bn.a 8
4.b odd 2 1 66.2.h.a ✓ 8
11.d odd 10 1 528.2.bn.a 8
12.b even 2 1 66.2.h.b yes 8
33.f even 10 1 inner 528.2.bn.b 8
44.c even 2 1 726.2.h.j 8
44.g even 10 1 66.2.h.b yes 8
44.g even 10 1 726.2.b.c 8
44.g even 10 1 726.2.h.f 8
44.g even 10 1 726.2.h.h 8
44.h odd 10 1 726.2.b.e 8
44.h odd 10 1 726.2.h.a 8
44.h odd 10 1 726.2.h.c 8
44.h odd 10 1 726.2.h.d 8
132.d odd 2 1 726.2.h.d 8
132.n odd 10 1 66.2.h.a ✓ 8
132.n odd 10 1 726.2.b.e 8
132.n odd 10 1 726.2.h.a 8
132.n odd 10 1 726.2.h.c 8
132.o even 10 1 726.2.b.c 8
132.o even 10 1 726.2.h.f 8
132.o even 10 1 726.2.h.h 8
132.o even 10 1 726.2.h.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.h.a ✓ 8 4.b odd 2 1
66.2.h.a ✓ 8 132.n odd 10 1
66.2.h.b yes 8 12.b even 2 1
66.2.h.b yes 8 44.g even 10 1
528.2.bn.a 8 3.b odd 2 1
528.2.bn.a 8 11.d odd 10 1
528.2.bn.b 8 1.a even 1 1 trivial
528.2.bn.b 8 33.f even 10 1 inner
726.2.b.c 8 44.g even 10 1
726.2.b.c 8 132.o even 10 1
726.2.b.e 8 44.h odd 10 1
726.2.b.e 8 132.n odd 10 1
726.2.h.a 8 44.h odd 10 1
726.2.h.a 8 132.n odd 10 1
726.2.h.c 8 44.h odd 10 1
726.2.h.c 8 132.n odd 10 1
726.2.h.d 8 44.h odd 10 1
726.2.h.d 8 132.d odd 2 1
726.2.h.f 8 44.g even 10 1
726.2.h.f 8 132.o even 10 1
726.2.h.h 8 44.g even 10 1
726.2.h.h 8 132.o even 10 1
726.2.h.j 8 44.c even 2 1
726.2.h.j 8 132.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2T_{5}^{6} + 15T_{5}^{5} + 19T_{5}^{4} - 30T_{5}^{3} - 8T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{8} - 10 T^{7} + \cdots + 144400 \) Copy content Toggle raw display
$19$ \( T^{8} - 15 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{8} + 88 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$29$ \( T^{8} + 23 T^{7} + \cdots + 15376 \) Copy content Toggle raw display
$31$ \( T^{8} + 13 T^{7} + \cdots + 1175056 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 30976 \) Copy content Toggle raw display
$41$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( T^{8} + 113 T^{6} + \cdots + 430336 \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( T^{8} + 15 T^{7} + \cdots + 2310400 \) Copy content Toggle raw display
$59$ \( T^{8} + 25 T^{7} + \cdots + 844561 \) Copy content Toggle raw display
$61$ \( T^{8} + 10 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( (T^{4} - T^{3} - 149 T^{2} + \cdots + 3076)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 20 T^{3} + \cdots + 80)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 5 T^{7} + \cdots + 24025 \) Copy content Toggle raw display
$79$ \( T^{8} + 10 T^{7} + \cdots + 55696 \) Copy content Toggle raw display
$83$ \( T^{8} + 21 T^{7} + \cdots + 737881 \) Copy content Toggle raw display
$89$ \( T^{8} + 513 T^{6} + \cdots + 12702096 \) Copy content Toggle raw display
$97$ \( T^{8} - 9 T^{7} + \cdots + 2070721 \) Copy content Toggle raw display
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