Properties

Label 900.3.c.d
Level $900$
Weight $3$
Character orbit 900.c
Self dual yes
Analytic conductor $24.523$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 4q^{4} + 8q^{8} + O(q^{10}) \) \( q + 2q^{2} + 4q^{4} + 8q^{8} + 24q^{13} + 16q^{16} - 16q^{17} + 48q^{26} + 42q^{29} + 32q^{32} - 32q^{34} - 24q^{37} + 18q^{41} + 49q^{49} + 96q^{52} + 56q^{53} + 84q^{58} + 22q^{61} + 64q^{64} - 64q^{68} - 96q^{73} - 48q^{74} + 36q^{82} - 78q^{89} - 144q^{97} + 98q^{98} + O(q^{100}) \)

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
2.00000 0 4.00000 0 0 0 8.00000 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 \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.d 1
3.b odd 2 1 100.3.b.a 1
4.b odd 2 1 CM 900.3.c.d 1
5.b even 2 1 900.3.c.a 1
5.c odd 4 2 180.3.f.c 2
12.b even 2 1 100.3.b.a 1
15.d odd 2 1 100.3.b.b 1
15.e even 4 2 20.3.d.c 2
20.d odd 2 1 900.3.c.a 1
20.e even 4 2 180.3.f.c 2
24.f even 2 1 1600.3.b.a 1
24.h odd 2 1 1600.3.b.a 1
60.h even 2 1 100.3.b.b 1
60.l odd 4 2 20.3.d.c 2
120.i odd 2 1 1600.3.b.c 1
120.m even 2 1 1600.3.b.c 1
120.q odd 4 2 320.3.h.d 2
120.w even 4 2 320.3.h.d 2
240.z odd 4 2 1280.3.e.a 2
240.bb even 4 2 1280.3.e.a 2
240.bd odd 4 2 1280.3.e.d 2
240.bf even 4 2 1280.3.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.c 2 15.e even 4 2
20.3.d.c 2 60.l odd 4 2
100.3.b.a 1 3.b odd 2 1
100.3.b.a 1 12.b even 2 1
100.3.b.b 1 15.d odd 2 1
100.3.b.b 1 60.h even 2 1
180.3.f.c 2 5.c odd 4 2
180.3.f.c 2 20.e even 4 2
320.3.h.d 2 120.q odd 4 2
320.3.h.d 2 120.w even 4 2
900.3.c.a 1 5.b even 2 1
900.3.c.a 1 20.d odd 2 1
900.3.c.d 1 1.a even 1 1 trivial
900.3.c.d 1 4.b odd 2 1 CM
1280.3.e.a 2 240.z odd 4 2
1280.3.e.a 2 240.bb even 4 2
1280.3.e.d 2 240.bd odd 4 2
1280.3.e.d 2 240.bf even 4 2
1600.3.b.a 1 24.f even 2 1
1600.3.b.a 1 24.h odd 2 1
1600.3.b.c 1 120.i odd 2 1
1600.3.b.c 1 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7} \)
\( T_{13} - 24 \)
\( T_{17} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -24 + T \)
$17$ \( 16 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -42 + T \)
$31$ \( T \)
$37$ \( 24 + T \)
$41$ \( -18 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( -56 + T \)
$59$ \( T \)
$61$ \( -22 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( 96 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( 78 + T \)
$97$ \( 144 + T \)
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