Properties

Label 3528.1.cu.a
Level $3528$
Weight $1$
Character orbit 3528.cu
Analytic conductor $1.761$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.cu (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.21168.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{3} ) q^{5} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{3} ) q^{5} -\beta_{1} q^{11} + q^{13} -\beta_{2} q^{19} + ( 1 - \beta_{2} ) q^{25} + ( -1 + \beta_{2} ) q^{31} -\beta_{2} q^{37} -\beta_{3} q^{41} - q^{43} + ( \beta_{1} - \beta_{3} ) q^{47} + 2 q^{55} + ( -\beta_{1} + \beta_{3} ) q^{65} + ( 1 - \beta_{2} ) q^{67} -\beta_{3} q^{71} + ( 1 - \beta_{2} ) q^{73} -\beta_{2} q^{79} -\beta_{3} q^{83} + \beta_{1} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{13} - 2q^{19} + 2q^{25} - 2q^{31} - 2q^{37} - 4q^{43} + 8q^{55} + 2q^{67} + 2q^{73} - 2q^{79} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1745.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0 0 0 −1.22474 0.707107i 0 0 0 0 0
1745.2 0 0 0 1.22474 + 0.707107i 0 0 0 0 0
2321.1 0 0 0 −1.22474 + 0.707107i 0 0 0 0 0
2321.2 0 0 0 1.22474 0.707107i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.cu.a 4
3.b odd 2 1 inner 3528.1.cu.a 4
7.b odd 2 1 504.1.cu.a 4
7.c even 3 1 3528.1.d.b 2
7.c even 3 1 inner 3528.1.cu.a 4
7.d odd 6 1 504.1.cu.a 4
7.d odd 6 1 3528.1.d.a 2
21.c even 2 1 504.1.cu.a 4
21.g even 6 1 504.1.cu.a 4
21.g even 6 1 3528.1.d.a 2
21.h odd 6 1 3528.1.d.b 2
21.h odd 6 1 inner 3528.1.cu.a 4
28.d even 2 1 1008.1.dc.a 4
28.f even 6 1 1008.1.dc.a 4
84.h odd 2 1 1008.1.dc.a 4
84.j odd 6 1 1008.1.dc.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.cu.a 4 7.b odd 2 1
504.1.cu.a 4 7.d odd 6 1
504.1.cu.a 4 21.c even 2 1
504.1.cu.a 4 21.g even 6 1
1008.1.dc.a 4 28.d even 2 1
1008.1.dc.a 4 28.f even 6 1
1008.1.dc.a 4 84.h odd 2 1
1008.1.dc.a 4 84.j odd 6 1
3528.1.d.a 2 7.d odd 6 1
3528.1.d.a 2 21.g even 6 1
3528.1.d.b 2 7.c even 3 1
3528.1.d.b 2 21.h odd 6 1
3528.1.cu.a 4 1.a even 1 1 trivial
3528.1.cu.a 4 3.b odd 2 1 inner
3528.1.cu.a 4 7.c even 3 1 inner
3528.1.cu.a 4 21.h odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3528, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 4 - 2 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 4 - 2 T^{2} + T^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( T^{4} \)
$19$ \( ( 1 + T + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 1 + T + T^{2} )^{2} \)
$37$ \( ( 1 + T + T^{2} )^{2} \)
$41$ \( ( 2 + T^{2} )^{2} \)
$43$ \( ( 1 + T )^{4} \)
$47$ \( 4 - 2 T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( 1 - T + T^{2} )^{2} \)
$71$ \( ( 2 + T^{2} )^{2} \)
$73$ \( ( 1 - T + T^{2} )^{2} \)
$79$ \( ( 1 + T + T^{2} )^{2} \)
$83$ \( ( 2 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
show more
show less