Properties

Label 3600.1.bk.a
Level 3600
Weight 1
Character orbit 3600.bk
Analytic conductor 1.797
Analytic rank 0
Dimension 4
Projective image \(D_{4}\)
CM discriminant -4
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.bk (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 720)
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.13500.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + ( -1 - \zeta_{8}^{2} ) q^{13} -2 \zeta_{8} q^{17} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + ( 1 - \zeta_{8}^{2} ) q^{37} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} -\zeta_{8}^{2} q^{49} + 2 \zeta_{8}^{3} q^{53} + ( 1 + \zeta_{8}^{2} ) q^{73} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{89} + ( 1 - \zeta_{8}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{13} + 4q^{37} + 4q^{73} + 4q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 0 0 0 0 0 0 0 0
143.2 0 0 0 0 0 0 0 0 0
1007.1 0 0 0 0 0 0 0 0 0
1007.2 0 0 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.bk.a 4
3.b odd 2 1 inner 3600.1.bk.a 4
4.b odd 2 1 CM 3600.1.bk.a 4
5.b even 2 1 720.1.bk.a 4
5.c odd 4 1 720.1.bk.a 4
5.c odd 4 1 inner 3600.1.bk.a 4
12.b even 2 1 inner 3600.1.bk.a 4
15.d odd 2 1 720.1.bk.a 4
15.e even 4 1 720.1.bk.a 4
15.e even 4 1 inner 3600.1.bk.a 4
20.d odd 2 1 720.1.bk.a 4
20.e even 4 1 720.1.bk.a 4
20.e even 4 1 inner 3600.1.bk.a 4
40.e odd 2 1 2880.1.bk.a 4
40.f even 2 1 2880.1.bk.a 4
40.i odd 4 1 2880.1.bk.a 4
40.k even 4 1 2880.1.bk.a 4
60.h even 2 1 720.1.bk.a 4
60.l odd 4 1 720.1.bk.a 4
60.l odd 4 1 inner 3600.1.bk.a 4
120.i odd 2 1 2880.1.bk.a 4
120.m even 2 1 2880.1.bk.a 4
120.q odd 4 1 2880.1.bk.a 4
120.w even 4 1 2880.1.bk.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bk.a 4 5.b even 2 1
720.1.bk.a 4 5.c odd 4 1
720.1.bk.a 4 15.d odd 2 1
720.1.bk.a 4 15.e even 4 1
720.1.bk.a 4 20.d odd 2 1
720.1.bk.a 4 20.e even 4 1
720.1.bk.a 4 60.h even 2 1
720.1.bk.a 4 60.l odd 4 1
2880.1.bk.a 4 40.e odd 2 1
2880.1.bk.a 4 40.f even 2 1
2880.1.bk.a 4 40.i odd 4 1
2880.1.bk.a 4 40.k even 4 1
2880.1.bk.a 4 120.i odd 2 1
2880.1.bk.a 4 120.m even 2 1
2880.1.bk.a 4 120.q odd 4 1
2880.1.bk.a 4 120.w even 4 1
3600.1.bk.a 4 1.a even 1 1 trivial
3600.1.bk.a 4 3.b odd 2 1 inner
3600.1.bk.a 4 4.b odd 2 1 CM
3600.1.bk.a 4 5.c odd 4 1 inner
3600.1.bk.a 4 12.b even 2 1 inner
3600.1.bk.a 4 15.e even 4 1 inner
3600.1.bk.a 4 20.e even 4 1 inner
3600.1.bk.a 4 60.l odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3600, [\chi])\).

Hecke characteristic polynomials

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