Properties

Label 3600.1.bo.a
Level $3600$
Weight $1$
Character orbit 3600.bo
Analytic conductor $1.797$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -15
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.bo (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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 720)
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.92160.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{8} +O(q^{10})\) \( q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{8} - q^{16} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{17} + ( -1 + \zeta_{8}^{2} ) q^{19} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{23} + 2 \zeta_{8}^{2} q^{31} + \zeta_{8} q^{32} + ( 1 + \zeta_{8}^{2} ) q^{34} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{38} + ( 1 + \zeta_{8}^{2} ) q^{46} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{47} - q^{49} + ( -1 - \zeta_{8}^{2} ) q^{61} -2 \zeta_{8}^{3} q^{62} -\zeta_{8}^{2} q^{64} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{68} + ( -1 - \zeta_{8}^{2} ) q^{76} + 2 \zeta_{8}^{3} q^{83} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{92} + ( -1 + \zeta_{8}^{2} ) q^{94} + \zeta_{8} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{16} - 4q^{19} + 4q^{34} + 4q^{46} - 4q^{49} - 4q^{61} - 4q^{76} - 4q^{94} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i 0 1.00000i 0 0 0 0.707107 0.707107i 0 0
451.2 0.707107 + 0.707107i 0 1.00000i 0 0 0 −0.707107 + 0.707107i 0 0
2251.1 −0.707107 + 0.707107i 0 1.00000i 0 0 0 0.707107 + 0.707107i 0 0
2251.2 0.707107 0.707107i 0 1.00000i 0 0 0 −0.707107 0.707107i 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner
80.k odd 4 1 inner
240.t even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.bo.a 4
3.b odd 2 1 inner 3600.1.bo.a 4
5.b even 2 1 inner 3600.1.bo.a 4
5.c odd 4 2 720.1.r.a 4
15.d odd 2 1 CM 3600.1.bo.a 4
15.e even 4 2 720.1.r.a 4
16.f odd 4 1 inner 3600.1.bo.a 4
20.e even 4 2 2880.1.r.a 4
48.k even 4 1 inner 3600.1.bo.a 4
60.l odd 4 2 2880.1.r.a 4
80.i odd 4 1 2880.1.r.a 4
80.j even 4 1 720.1.r.a 4
80.k odd 4 1 inner 3600.1.bo.a 4
80.s even 4 1 720.1.r.a 4
80.t odd 4 1 2880.1.r.a 4
240.t even 4 1 inner 3600.1.bo.a 4
240.z odd 4 1 720.1.r.a 4
240.bb even 4 1 2880.1.r.a 4
240.bd odd 4 1 720.1.r.a 4
240.bf even 4 1 2880.1.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.r.a 4 5.c odd 4 2
720.1.r.a 4 15.e even 4 2
720.1.r.a 4 80.j even 4 1
720.1.r.a 4 80.s even 4 1
720.1.r.a 4 240.z odd 4 1
720.1.r.a 4 240.bd odd 4 1
2880.1.r.a 4 20.e even 4 2
2880.1.r.a 4 60.l odd 4 2
2880.1.r.a 4 80.i odd 4 1
2880.1.r.a 4 80.t odd 4 1
2880.1.r.a 4 240.bb even 4 1
2880.1.r.a 4 240.bf even 4 1
3600.1.bo.a 4 1.a even 1 1 trivial
3600.1.bo.a 4 3.b odd 2 1 inner
3600.1.bo.a 4 5.b even 2 1 inner
3600.1.bo.a 4 15.d odd 2 1 CM
3600.1.bo.a 4 16.f odd 4 1 inner
3600.1.bo.a 4 48.k even 4 1 inner
3600.1.bo.a 4 80.k odd 4 1 inner
3600.1.bo.a 4 240.t even 4 1 inner

Hecke kernels

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