Properties

Label 3600.1.bf.b
Level $3600$
Weight $1$
Character orbit 3600.bf
Analytic conductor $1.797$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.bf (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.153600.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + q^{8} + q^{16} + ( -1 + i ) q^{17} + ( 1 + i ) q^{19} + ( 1 - i ) q^{23} + q^{32} + ( -1 + i ) q^{34} + ( 1 + i ) q^{38} + ( 1 - i ) q^{46} + ( -1 + i ) q^{47} -i q^{49} -2 i q^{53} + ( 1 - i ) q^{61} + q^{64} + ( -1 + i ) q^{68} + ( 1 + i ) q^{76} + 2 i q^{79} -2 q^{83} + ( 1 - i ) q^{92} + ( -1 + i ) q^{94} -i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{8} + 2q^{16} - 2q^{17} + 2q^{19} + 2q^{23} + 2q^{32} - 2q^{34} + 2q^{38} + 2q^{46} - 2q^{47} + 2q^{61} + 2q^{64} - 2q^{68} + 2q^{76} - 4q^{83} + 2q^{92} - 2q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-i\) \(i\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2557.1
1.00000i
1.00000i
1.00000 0 1.00000 0 0 0 1.00000 0 0
3493.1 1.00000 0 1.00000 0 0 0 1.00000 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 \(\Q(\sqrt{-15}) \)
80.t odd 4 1 inner
240.bb 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.bf.b yes 2
3.b odd 2 1 3600.1.bf.a yes 2
5.b even 2 1 3600.1.bf.a yes 2
5.c odd 4 1 3600.1.bb.a 2
5.c odd 4 1 3600.1.bb.b yes 2
15.d odd 2 1 CM 3600.1.bf.b yes 2
15.e even 4 1 3600.1.bb.a 2
15.e even 4 1 3600.1.bb.b yes 2
16.e even 4 1 3600.1.bb.a 2
48.i odd 4 1 3600.1.bb.b yes 2
80.i odd 4 1 3600.1.bf.a yes 2
80.q even 4 1 3600.1.bb.b yes 2
80.t odd 4 1 inner 3600.1.bf.b yes 2
240.bb even 4 1 inner 3600.1.bf.b yes 2
240.bf even 4 1 3600.1.bf.a yes 2
240.bm odd 4 1 3600.1.bb.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.bb.a 2 5.c odd 4 1
3600.1.bb.a 2 15.e even 4 1
3600.1.bb.a 2 16.e even 4 1
3600.1.bb.a 2 240.bm odd 4 1
3600.1.bb.b yes 2 5.c odd 4 1
3600.1.bb.b yes 2 15.e even 4 1
3600.1.bb.b yes 2 48.i odd 4 1
3600.1.bb.b yes 2 80.q even 4 1
3600.1.bf.a yes 2 3.b odd 2 1
3600.1.bf.a yes 2 5.b even 2 1
3600.1.bf.a yes 2 80.i odd 4 1
3600.1.bf.a yes 2 240.bf even 4 1
3600.1.bf.b yes 2 1.a even 1 1 trivial
3600.1.bf.b yes 2 15.d odd 2 1 CM
3600.1.bf.b yes 2 80.t odd 4 1 inner
3600.1.bf.b yes 2 240.bb even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + 2 T_{17} + 2 \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\).

Hecke characteristic polynomials

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