Properties

Label 450.2.a.f
Level $450$
Weight $2$
Character orbit 450.a
Self dual yes
Analytic conductor $3.593$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + 2q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + 2q^{7} + q^{8} - 2q^{11} + 6q^{13} + 2q^{14} + q^{16} - 2q^{17} - 2q^{22} + 4q^{23} + 6q^{26} + 2q^{28} - 8q^{31} + q^{32} - 2q^{34} + 2q^{37} - 2q^{41} - 4q^{43} - 2q^{44} + 4q^{46} + 8q^{47} - 3q^{49} + 6q^{52} - 6q^{53} + 2q^{56} - 10q^{59} + 2q^{61} - 8q^{62} + q^{64} - 8q^{67} - 2q^{68} - 12q^{71} - 4q^{73} + 2q^{74} - 4q^{77} - 2q^{82} + 4q^{83} - 4q^{86} - 2q^{88} + 10q^{89} + 12q^{91} + 4q^{92} + 8q^{94} - 8q^{97} - 3q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 0 1.00000 0 0 2.00000 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.a.f 1
3.b odd 2 1 150.2.a.a 1
4.b odd 2 1 3600.2.a.o 1
5.b even 2 1 450.2.a.b 1
5.c odd 4 2 90.2.c.a 2
12.b even 2 1 1200.2.a.m 1
15.d odd 2 1 150.2.a.c 1
15.e even 4 2 30.2.c.a 2
20.d odd 2 1 3600.2.a.bg 1
20.e even 4 2 720.2.f.f 2
21.c even 2 1 7350.2.a.bg 1
24.f even 2 1 4800.2.a.m 1
24.h odd 2 1 4800.2.a.cg 1
40.i odd 4 2 2880.2.f.e 2
40.k even 4 2 2880.2.f.c 2
45.k odd 12 4 810.2.i.b 4
45.l even 12 4 810.2.i.e 4
60.h even 2 1 1200.2.a.g 1
60.l odd 4 2 240.2.f.a 2
105.g even 2 1 7350.2.a.cc 1
105.k odd 4 2 1470.2.g.g 2
105.w odd 12 4 1470.2.n.a 4
105.x even 12 4 1470.2.n.h 4
120.i odd 2 1 4800.2.a.l 1
120.m even 2 1 4800.2.a.cj 1
120.q odd 4 2 960.2.f.i 2
120.w even 4 2 960.2.f.h 2
240.z odd 4 2 3840.2.d.j 2
240.bb even 4 2 3840.2.d.y 2
240.bd odd 4 2 3840.2.d.x 2
240.bf even 4 2 3840.2.d.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 15.e even 4 2
90.2.c.a 2 5.c odd 4 2
150.2.a.a 1 3.b odd 2 1
150.2.a.c 1 15.d odd 2 1
240.2.f.a 2 60.l odd 4 2
450.2.a.b 1 5.b even 2 1
450.2.a.f 1 1.a even 1 1 trivial
720.2.f.f 2 20.e even 4 2
810.2.i.b 4 45.k odd 12 4
810.2.i.e 4 45.l even 12 4
960.2.f.h 2 120.w even 4 2
960.2.f.i 2 120.q odd 4 2
1200.2.a.g 1 60.h even 2 1
1200.2.a.m 1 12.b even 2 1
1470.2.g.g 2 105.k odd 4 2
1470.2.n.a 4 105.w odd 12 4
1470.2.n.h 4 105.x even 12 4
2880.2.f.c 2 40.k even 4 2
2880.2.f.e 2 40.i odd 4 2
3600.2.a.o 1 4.b odd 2 1
3600.2.a.bg 1 20.d odd 2 1
3840.2.d.g 2 240.bf even 4 2
3840.2.d.j 2 240.z odd 4 2
3840.2.d.x 2 240.bd odd 4 2
3840.2.d.y 2 240.bb even 4 2
4800.2.a.l 1 120.i odd 2 1
4800.2.a.m 1 24.f even 2 1
4800.2.a.cg 1 24.h odd 2 1
4800.2.a.cj 1 120.m even 2 1
7350.2.a.bg 1 21.c even 2 1
7350.2.a.cc 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} - 2 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -2 + T \)
$11$ \( 2 + T \)
$13$ \( -6 + T \)
$17$ \( 2 + T \)
$19$ \( T \)
$23$ \( -4 + T \)
$29$ \( T \)
$31$ \( 8 + T \)
$37$ \( -2 + T \)
$41$ \( 2 + T \)
$43$ \( 4 + T \)
$47$ \( -8 + T \)
$53$ \( 6 + T \)
$59$ \( 10 + T \)
$61$ \( -2 + T \)
$67$ \( 8 + T \)
$71$ \( 12 + T \)
$73$ \( 4 + T \)
$79$ \( T \)
$83$ \( -4 + T \)
$89$ \( -10 + T \)
$97$ \( 8 + T \)
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