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Properties

Label 528.2.bn.d
Level $528$
Weight $2$
Character orbit 528.bn
Analytic conductor $4.216$
Analytic rank $0$
Dimension $16$
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] = [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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} - 12 x^{13} + 23 x^{12} - 72 x^{11} + 146 x^{10} - 176 x^{9} + 223 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{3} - \beta_{15} q^{5} + (\beta_{13} + \beta_{12} + \beta_{9} + \cdots - 1) q^{7}+ \cdots + (\beta_{15} + \beta_{13} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{3} - \beta_{15} q^{5} + (\beta_{13} + \beta_{12} + \beta_{9} + \cdots - 1) q^{7}+ \cdots + (\beta_{15} - \beta_{14} + 2 \beta_{13} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{3} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{3} + 5 q^{9} - 9 q^{15} + 30 q^{19} - 12 q^{25} - q^{27} + 10 q^{31} - 41 q^{33} - 24 q^{37} + 35 q^{39} + 2 q^{45} + 12 q^{49} + 15 q^{51} - 62 q^{55} + 35 q^{57} + 40 q^{61} - 55 q^{63} - 44 q^{67} + 46 q^{69} + 10 q^{73} - 21 q^{75} - 20 q^{79} + 57 q^{81} - 60 q^{85} + 60 q^{91} + 7 q^{93} - 36 q^{97} + 101 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} - 12 x^{13} + 23 x^{12} - 72 x^{11} + 146 x^{10} - 176 x^{9} + 223 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 520 \nu^{15} + 8455 \nu^{14} - 47858 \nu^{13} + 36585 \nu^{12} - 10430 \nu^{11} + \cdots + 49929210 ) / 5677452 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 671 \nu^{15} + 10466 \nu^{14} - 30367 \nu^{13} + 17631 \nu^{12} - 9997 \nu^{11} + \cdots + 35110098 ) / 2838726 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2147 \nu^{15} + 13318 \nu^{14} - 48254 \nu^{13} + 32439 \nu^{12} - 47441 \nu^{11} + \cdots + 49898592 ) / 5677452 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2125 \nu^{15} + 14566 \nu^{14} - 10115 \nu^{13} - 510 \nu^{12} - 38201 \nu^{11} + \cdots - 1137240 ) / 5677452 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 353 \nu^{15} + 1777 \nu^{14} - 5697 \nu^{13} + 3637 \nu^{12} - 5695 \nu^{11} + 30562 \nu^{10} + \cdots + 4406076 ) / 630828 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3448 \nu^{15} + 10615 \nu^{14} - 11591 \nu^{13} - 9897 \nu^{12} - 46571 \nu^{11} + \cdots + 12109419 ) / 5677452 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5107 \nu^{15} - 12985 \nu^{14} + 2687 \nu^{13} - 12729 \nu^{12} + 61517 \nu^{11} - 179478 \nu^{10} + \cdots + 6561000 ) / 5677452 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6044 \nu^{15} - 20999 \nu^{14} + 32359 \nu^{13} - 21255 \nu^{12} + 106279 \nu^{11} + \cdots - 34819227 ) / 5677452 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2051 \nu^{15} + 14671 \nu^{14} - 11259 \nu^{13} + 6019 \nu^{12} - 58549 \nu^{11} + \cdots + 1469664 ) / 1892484 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7660 \nu^{15} + 27859 \nu^{14} - 48887 \nu^{13} + 43509 \nu^{12} - 130523 \nu^{11} + \cdots + 29824119 ) / 5677452 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3615 \nu^{15} + 5434 \nu^{14} - 9502 \nu^{13} + 18449 \nu^{12} - 48651 \nu^{11} + \cdots + 10879596 ) / 1892484 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12002 \nu^{15} - 39119 \nu^{14} + 42127 \nu^{13} - 72210 \nu^{12} + 234340 \nu^{11} + \cdots - 12859560 ) / 5677452 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3049 \nu^{15} + 8629 \nu^{14} - 11778 \nu^{13} + 13879 \nu^{12} - 49972 \nu^{11} + \cdots + 11528649 ) / 630828 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10683 \nu^{15} + 30416 \nu^{14} - 45593 \nu^{13} + 48700 \nu^{12} - 161595 \nu^{11} + \cdots + 38289996 ) / 1892484 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{14} - 3 \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{9} + 2 \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + 2 \beta_{12} + \beta_{11} - 5 \beta_{9} + 6 \beta_{8} - 5 \beta_{7} + \beta_{6} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{13} + \beta_{12} - 4 \beta_{11} - \beta_{9} + 3 \beta_{8} - 5 \beta_{7} + 6 \beta_{6} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{15} - 6 \beta_{13} + 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{8} + 6 \beta_{7} + 9 \beta_{5} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7 \beta_{15} - 7 \beta_{14} + 16 \beta_{13} + 19 \beta_{12} + 3 \beta_{11} + \beta_{10} + 8 \beta_{9} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 20 \beta_{14} - 21 \beta_{13} - 26 \beta_{12} + 20 \beta_{11} - 20 \beta_{10} - 54 \beta_{9} + \cdots - 59 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 20 \beta_{15} - 24 \beta_{14} - 20 \beta_{13} + 16 \beta_{12} + 82 \beta_{11} - 20 \beta_{10} + \cdots + 24 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 24 \beta_{15} + 38 \beta_{14} + 36 \beta_{13} + 36 \beta_{12} - 26 \beta_{11} - 8 \beta_{9} + \cdots - 119 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 12 \beta_{15} + 60 \beta_{13} + 80 \beta_{11} - 96 \beta_{10} - 218 \beta_{8} - 60 \beta_{7} + \cdots + 114 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 49 \beta_{15} + 49 \beta_{14} - 13 \beta_{13} + 220 \beta_{12} + 73 \beta_{11} - 68 \beta_{10} + \cdots + 345 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 129 \beta_{14} + 147 \beta_{13} + 2 \beta_{12} + 129 \beta_{11} - 129 \beta_{10} - 494 \beta_{9} + \cdots - 1009 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 74 \beta_{15} - 589 \beta_{14} - 74 \beta_{13} - 276 \beta_{12} + 1211 \beta_{11} - 74 \beta_{10} + \cdots + 589 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 959 \beta_{15} + 548 \beta_{14} - 737 \beta_{13} - 737 \beta_{12} + 1164 \beta_{11} - 747 \beta_{9} + \cdots - 921 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(1 + \beta_{5} + \beta_{8} + \beta_{9}\) \(-1\) \(1\)

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
1.63346 − 0.576037i
−0.982909 − 1.42615i
0.762305 + 1.55528i
−1.53089 + 0.810175i
−1.12228 − 1.31928i
0.601787 − 1.62415i
1.73062 − 0.0704449i
0.907906 + 1.47503i
−1.12228 + 1.31928i
0.601787 + 1.62415i
1.73062 + 0.0704449i
0.907906 − 1.47503i
1.63346 + 0.576037i
−0.982909 + 1.42615i
0.762305 − 1.55528i
−1.53089 − 0.810175i
0 −1.66008 + 0.494099i 0 −0.393890 − 0.127982i 0 −1.02762 − 1.41440i 0 2.51173 − 1.64049i 0
17.2 0 −0.0430774 − 1.73152i 0 0.393890 + 0.127982i 0 −1.02762 − 1.41440i 0 −2.99629 + 0.149178i 0
17.3 0 0.297452 + 1.70632i 0 2.90910 + 0.945225i 0 2.14565 + 2.95324i 0 −2.82304 + 1.01510i 0
17.4 0 1.71472 − 0.244388i 0 −2.90910 − 0.945225i 0 2.14565 + 2.95324i 0 2.88055 − 0.838115i 0
161.1 0 −1.60151 − 0.659669i 0 −1.45030 − 1.99617i 0 −3.40640 + 1.10681i 0 2.12967 + 2.11293i 0
161.2 0 −1.35869 + 1.07422i 0 2.06847 + 2.84701i 0 2.28837 − 0.743535i 0 0.692091 − 2.91908i 0
161.3 0 0.467793 + 1.66768i 0 −2.06847 − 2.84701i 0 2.28837 − 0.743535i 0 −2.56234 + 1.56026i 0
161.4 0 1.68339 + 0.407661i 0 1.45030 + 1.99617i 0 −3.40640 + 1.10681i 0 2.66763 + 1.37251i 0
305.1 0 −1.60151 + 0.659669i 0 −1.45030 + 1.99617i 0 −3.40640 − 1.10681i 0 2.12967 − 2.11293i 0
305.2 0 −1.35869 − 1.07422i 0 2.06847 − 2.84701i 0 2.28837 + 0.743535i 0 0.692091 + 2.91908i 0
305.3 0 0.467793 − 1.66768i 0 −2.06847 + 2.84701i 0 2.28837 + 0.743535i 0 −2.56234 − 1.56026i 0
305.4 0 1.68339 − 0.407661i 0 1.45030 − 1.99617i 0 −3.40640 − 1.10681i 0 2.66763 − 1.37251i 0
497.1 0 −1.66008 − 0.494099i 0 −0.393890 + 0.127982i 0 −1.02762 + 1.41440i 0 2.51173 + 1.64049i 0
497.2 0 −0.0430774 + 1.73152i 0 0.393890 − 0.127982i 0 −1.02762 + 1.41440i 0 −2.99629 − 0.149178i 0
497.3 0 0.297452 − 1.70632i 0 2.90910 − 0.945225i 0 2.14565 − 2.95324i 0 −2.82304 − 1.01510i 0
497.4 0 1.71472 + 0.244388i 0 −2.90910 + 0.945225i 0 2.14565 − 2.95324i 0 2.88055 + 0.838115i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
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.d 16
3.b odd 2 1 inner 528.2.bn.d 16
4.b odd 2 1 132.2.p.a ✓ 16
11.d odd 10 1 inner 528.2.bn.d 16
12.b even 2 1 132.2.p.a ✓ 16
33.f even 10 1 inner 528.2.bn.d 16
44.g even 10 1 132.2.p.a ✓ 16
44.g even 10 1 1452.2.b.e 16
44.h odd 10 1 1452.2.b.e 16
132.n odd 10 1 132.2.p.a ✓ 16
132.n odd 10 1 1452.2.b.e 16
132.o even 10 1 1452.2.b.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.2.p.a ✓ 16 4.b odd 2 1
132.2.p.a ✓ 16 12.b even 2 1
132.2.p.a ✓ 16 44.g even 10 1
132.2.p.a ✓ 16 132.n odd 10 1
528.2.bn.d 16 1.a even 1 1 trivial
528.2.bn.d 16 3.b odd 2 1 inner
528.2.bn.d 16 11.d odd 10 1 inner
528.2.bn.d 16 33.f even 10 1 inner
1452.2.b.e 16 44.g even 10 1
1452.2.b.e 16 44.h odd 10 1
1452.2.b.e 16 132.n odd 10 1
1452.2.b.e 16 132.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 4 T_{5}^{14} + 135 T_{5}^{12} - 1496 T_{5}^{10} + 12254 T_{5}^{8} - 14036 T_{5}^{6} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$7$ \( (T^{8} - 10 T^{6} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} + 10 T^{6} + \cdots + 3025)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 35 T^{14} + \cdots + 9150625 \) Copy content Toggle raw display
$19$ \( (T^{8} - 15 T^{7} + \cdots + 24025)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 94 T^{6} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 2342560000 \) Copy content Toggle raw display
$31$ \( (T^{8} - 5 T^{7} + 18 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 12 T^{7} + 56 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 133974300625 \) Copy content Toggle raw display
$43$ \( (T^{8} + 170 T^{6} + \cdots + 48400)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 177410282401 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 918609150481 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 7611645084241 \) Copy content Toggle raw display
$61$ \( (T^{8} - 20 T^{7} + \cdots + 93025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 11 T^{3} + \cdots + 1024)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} - 416 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$73$ \( (T^{8} - 5 T^{7} + \cdots + 36966400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 10 T^{7} + \cdots + 9579025)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 270 T^{14} + \cdots + 9150625 \) Copy content Toggle raw display
$89$ \( (T^{8} + 558 T^{6} + \cdots + 203233536)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 18 T^{7} + \cdots + 271441)^{2} \) Copy content Toggle raw display
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