Properties

Label 3600.1.bh.b
Level 3600
Weight 1
Character orbit 3600.bh
Analytic conductor 1.797
Analytic rank 0
Dimension 4
Projective image \(D_{6}\)
CM discriminant -3
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.bh (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 900)
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.450000.1

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{7} +O(q^{10})\) \( q -\beta_{1} q^{7} + \beta_{3} q^{13} + \beta_{2} q^{19} - q^{31} + \beta_{3} q^{43} + 2 \beta_{2} q^{49} - q^{61} + \beta_{1} q^{67} + 2 \beta_{2} q^{79} + 3 q^{91} -\beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{31} - 4q^{61} + 12q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2593.1
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
0 0 0 0 0 −1.22474 + 1.22474i 0 0 0
2593.2 0 0 0 0 0 1.22474 1.22474i 0 0 0
3457.1 0 0 0 0 0 −1.22474 1.22474i 0 0 0
3457.2 0 0 0 0 0 1.22474 + 1.22474i 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 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.bh.b 4
3.b odd 2 1 CM 3600.1.bh.b 4
4.b odd 2 1 900.1.l.a 4
5.b even 2 1 inner 3600.1.bh.b 4
5.c odd 4 2 inner 3600.1.bh.b 4
12.b even 2 1 900.1.l.a 4
15.d odd 2 1 inner 3600.1.bh.b 4
15.e even 4 2 inner 3600.1.bh.b 4
20.d odd 2 1 900.1.l.a 4
20.e even 4 2 900.1.l.a 4
60.h even 2 1 900.1.l.a 4
60.l odd 4 2 900.1.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.1.l.a 4 4.b odd 2 1
900.1.l.a 4 12.b even 2 1
900.1.l.a 4 20.d odd 2 1
900.1.l.a 4 20.e even 4 2
900.1.l.a 4 60.h even 2 1
900.1.l.a 4 60.l odd 4 2
3600.1.bh.b 4 1.a even 1 1 trivial
3600.1.bh.b 4 3.b odd 2 1 CM
3600.1.bh.b 4 5.b even 2 1 inner
3600.1.bh.b 4 5.c odd 4 2 inner
3600.1.bh.b 4 15.d odd 2 1 inner
3600.1.bh.b 4 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 9 \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ \( 1 - T^{4} + T^{8} \)
$11$ \( ( 1 + T^{2} )^{4} \)
$13$ \( 1 - T^{4} + T^{8} \)
$17$ \( ( 1 + T^{4} )^{2} \)
$19$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$23$ \( ( 1 + T^{4} )^{2} \)
$29$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$31$ \( ( 1 + T + T^{2} )^{4} \)
$37$ \( ( 1 + T^{4} )^{2} \)
$41$ \( ( 1 + T^{2} )^{4} \)
$43$ \( 1 - T^{4} + T^{8} \)
$47$ \( ( 1 + T^{4} )^{2} \)
$53$ \( ( 1 + T^{4} )^{2} \)
$59$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$61$ \( ( 1 + T + T^{2} )^{4} \)
$67$ \( 1 - T^{4} + T^{8} \)
$71$ \( ( 1 + T^{2} )^{4} \)
$73$ \( ( 1 + T^{4} )^{2} \)
$79$ \( ( 1 + T^{2} )^{4} \)
$83$ \( ( 1 + T^{4} )^{2} \)
$89$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$97$ \( 1 - T^{4} + T^{8} \)
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