Properties

Label 450.4.a.h
Level 450
Weight 4
Character orbit 450.a
Self dual yes
Analytic conductor 26.551
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5508595026\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{4} + 16q^{7} - 8q^{8} + O(q^{10}) \) \( q - 2q^{2} + 4q^{4} + 16q^{7} - 8q^{8} - 12q^{11} - 38q^{13} - 32q^{14} + 16q^{16} - 126q^{17} + 20q^{19} + 24q^{22} + 168q^{23} + 76q^{26} + 64q^{28} - 30q^{29} - 88q^{31} - 32q^{32} + 252q^{34} - 254q^{37} - 40q^{38} - 42q^{41} + 52q^{43} - 48q^{44} - 336q^{46} - 96q^{47} - 87q^{49} - 152q^{52} + 198q^{53} - 128q^{56} + 60q^{58} + 660q^{59} - 538q^{61} + 176q^{62} + 64q^{64} - 884q^{67} - 504q^{68} - 792q^{71} - 218q^{73} + 508q^{74} + 80q^{76} - 192q^{77} - 520q^{79} + 84q^{82} - 492q^{83} - 104q^{86} + 96q^{88} - 810q^{89} - 608q^{91} + 672q^{92} + 192q^{94} - 1154q^{97} + 174q^{98} + O(q^{100}) \)

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
−2.00000 0 4.00000 0 0 16.0000 −8.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 450.4.a.h 1
3.b odd 2 1 150.4.a.i 1
5.b even 2 1 18.4.a.a 1
5.c odd 4 2 450.4.c.e 2
12.b even 2 1 1200.4.a.b 1
15.d odd 2 1 6.4.a.a 1
15.e even 4 2 150.4.c.d 2
20.d odd 2 1 144.4.a.c 1
35.c odd 2 1 882.4.a.n 1
35.i odd 6 2 882.4.g.f 2
35.j even 6 2 882.4.g.i 2
40.e odd 2 1 576.4.a.r 1
40.f even 2 1 576.4.a.q 1
45.h odd 6 2 162.4.c.f 2
45.j even 6 2 162.4.c.c 2
55.d odd 2 1 2178.4.a.e 1
60.h even 2 1 48.4.a.c 1
60.l odd 4 2 1200.4.f.j 2
105.g even 2 1 294.4.a.e 1
105.o odd 6 2 294.4.e.h 2
105.p even 6 2 294.4.e.g 2
120.i odd 2 1 192.4.a.i 1
120.m even 2 1 192.4.a.c 1
165.d even 2 1 726.4.a.f 1
195.e odd 2 1 1014.4.a.g 1
195.n even 4 2 1014.4.b.d 2
240.t even 4 2 768.4.d.c 2
240.bm odd 4 2 768.4.d.n 2
255.h odd 2 1 1734.4.a.d 1
285.b even 2 1 2166.4.a.i 1
420.o odd 2 1 2352.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 15.d odd 2 1
18.4.a.a 1 5.b even 2 1
48.4.a.c 1 60.h even 2 1
144.4.a.c 1 20.d odd 2 1
150.4.a.i 1 3.b odd 2 1
150.4.c.d 2 15.e even 4 2
162.4.c.c 2 45.j even 6 2
162.4.c.f 2 45.h odd 6 2
192.4.a.c 1 120.m even 2 1
192.4.a.i 1 120.i odd 2 1
294.4.a.e 1 105.g even 2 1
294.4.e.g 2 105.p even 6 2
294.4.e.h 2 105.o odd 6 2
450.4.a.h 1 1.a even 1 1 trivial
450.4.c.e 2 5.c odd 4 2
576.4.a.q 1 40.f even 2 1
576.4.a.r 1 40.e odd 2 1
726.4.a.f 1 165.d even 2 1
768.4.d.c 2 240.t even 4 2
768.4.d.n 2 240.bm odd 4 2
882.4.a.n 1 35.c odd 2 1
882.4.g.f 2 35.i odd 6 2
882.4.g.i 2 35.j even 6 2
1014.4.a.g 1 195.e odd 2 1
1014.4.b.d 2 195.n even 4 2
1200.4.a.b 1 12.b even 2 1
1200.4.f.j 2 60.l odd 4 2
1734.4.a.d 1 255.h odd 2 1
2166.4.a.i 1 285.b even 2 1
2178.4.a.e 1 55.d odd 2 1
2352.4.a.e 1 420.o odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} - 16 \)
\( T_{11} + 12 \)
\( T_{17} + 126 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ 1
$5$ 1
$7$ \( 1 - 16 T + 343 T^{2} \)
$11$ \( 1 + 12 T + 1331 T^{2} \)
$13$ \( 1 + 38 T + 2197 T^{2} \)
$17$ \( 1 + 126 T + 4913 T^{2} \)
$19$ \( 1 - 20 T + 6859 T^{2} \)
$23$ \( 1 - 168 T + 12167 T^{2} \)
$29$ \( 1 + 30 T + 24389 T^{2} \)
$31$ \( 1 + 88 T + 29791 T^{2} \)
$37$ \( 1 + 254 T + 50653 T^{2} \)
$41$ \( 1 + 42 T + 68921 T^{2} \)
$43$ \( 1 - 52 T + 79507 T^{2} \)
$47$ \( 1 + 96 T + 103823 T^{2} \)
$53$ \( 1 - 198 T + 148877 T^{2} \)
$59$ \( 1 - 660 T + 205379 T^{2} \)
$61$ \( 1 + 538 T + 226981 T^{2} \)
$67$ \( 1 + 884 T + 300763 T^{2} \)
$71$ \( 1 + 792 T + 357911 T^{2} \)
$73$ \( 1 + 218 T + 389017 T^{2} \)
$79$ \( 1 + 520 T + 493039 T^{2} \)
$83$ \( 1 + 492 T + 571787 T^{2} \)
$89$ \( 1 + 810 T + 704969 T^{2} \)
$97$ \( 1 + 1154 T + 912673 T^{2} \)
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