Properties

Label 900.1.c.b
Level $900$
Weight $1$
Character orbit 900.c
Self dual yes
Analytic conductor $0.449$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -15, 60
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 900.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.449158511370\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{15})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.13500.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{8} + q^{16} - 2 q^{17} + q^{32} - 2 q^{34} + q^{49} - 2 q^{53} - 2 q^{61} + q^{64} - 2 q^{68} + q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0
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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
60.h even 2 1 RM by \(\Q(\sqrt{15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.1.c.b 1
3.b odd 2 1 900.1.c.a 1
4.b odd 2 1 CM 900.1.c.b 1
5.b even 2 1 900.1.c.a 1
5.c odd 4 2 180.1.f.a 2
12.b even 2 1 900.1.c.a 1
15.d odd 2 1 CM 900.1.c.b 1
15.e even 4 2 180.1.f.a 2
20.d odd 2 1 900.1.c.a 1
20.e even 4 2 180.1.f.a 2
40.i odd 4 2 2880.1.j.b 2
40.k even 4 2 2880.1.j.b 2
45.k odd 12 4 1620.1.p.b 4
45.l even 12 4 1620.1.p.b 4
60.h even 2 1 RM 900.1.c.b 1
60.l odd 4 2 180.1.f.a 2
120.q odd 4 2 2880.1.j.b 2
120.w even 4 2 2880.1.j.b 2
180.v odd 12 4 1620.1.p.b 4
180.x even 12 4 1620.1.p.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.f.a 2 5.c odd 4 2
180.1.f.a 2 15.e even 4 2
180.1.f.a 2 20.e even 4 2
180.1.f.a 2 60.l odd 4 2
900.1.c.a 1 3.b odd 2 1
900.1.c.a 1 5.b even 2 1
900.1.c.a 1 12.b even 2 1
900.1.c.a 1 20.d odd 2 1
900.1.c.b 1 1.a even 1 1 trivial
900.1.c.b 1 4.b odd 2 1 CM
900.1.c.b 1 15.d odd 2 1 CM
900.1.c.b 1 60.h even 2 1 RM
1620.1.p.b 4 45.k odd 12 4
1620.1.p.b 4 45.l even 12 4
1620.1.p.b 4 180.v odd 12 4
1620.1.p.b 4 180.x even 12 4
2880.1.j.b 2 40.i odd 4 2
2880.1.j.b 2 40.k even 4 2
2880.1.j.b 2 120.q odd 4 2
2880.1.j.b 2 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17} + 2 \) acting on \(S_{1}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

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