Properties

Label 450.6.a.v
Level $450$
Weight $6$
Character orbit 450.a
Self dual yes
Analytic conductor $72.173$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} + 16q^{4} + 148q^{7} + 64q^{8} + O(q^{10}) \) \( q + 4q^{2} + 16q^{4} + 148q^{7} + 64q^{8} + 384q^{11} + 334q^{13} + 592q^{14} + 256q^{16} - 576q^{17} - 664q^{19} + 1536q^{22} + 3840q^{23} + 1336q^{26} + 2368q^{28} + 96q^{29} - 4564q^{31} + 1024q^{32} - 2304q^{34} - 5798q^{37} - 2656q^{38} - 6720q^{41} + 14872q^{43} + 6144q^{44} + 15360q^{46} + 19200q^{47} + 5097q^{49} + 5344q^{52} - 7776q^{53} + 9472q^{56} + 384q^{58} - 13056q^{59} + 42782q^{61} - 18256q^{62} + 4096q^{64} - 36656q^{67} - 9216q^{68} + 64512q^{71} + 16810q^{73} - 23192q^{74} - 10624q^{76} + 56832q^{77} + 28076q^{79} - 26880q^{82} + 66432q^{83} + 59488q^{86} + 24576q^{88} - 81792q^{89} + 49432q^{91} + 61440q^{92} + 76800q^{94} + 29938q^{97} + 20388q^{98} + O(q^{100}) \)

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
4.00000 0 16.0000 0 0 148.000 64.0000 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 450.6.a.v 1
3.b odd 2 1 450.6.a.k 1
5.b even 2 1 18.6.a.a 1
5.c odd 4 2 450.6.c.m 2
15.d odd 2 1 18.6.a.c yes 1
15.e even 4 2 450.6.c.c 2
20.d odd 2 1 144.6.a.a 1
35.c odd 2 1 882.6.a.k 1
40.e odd 2 1 576.6.a.bi 1
40.f even 2 1 576.6.a.bh 1
45.h odd 6 2 162.6.c.a 2
45.j even 6 2 162.6.c.l 2
60.h even 2 1 144.6.a.l 1
105.g even 2 1 882.6.a.l 1
120.i odd 2 1 576.6.a.a 1
120.m even 2 1 576.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.a.a 1 5.b even 2 1
18.6.a.c yes 1 15.d odd 2 1
144.6.a.a 1 20.d odd 2 1
144.6.a.l 1 60.h even 2 1
162.6.c.a 2 45.h odd 6 2
162.6.c.l 2 45.j even 6 2
450.6.a.k 1 3.b odd 2 1
450.6.a.v 1 1.a even 1 1 trivial
450.6.c.c 2 15.e even 4 2
450.6.c.m 2 5.c odd 4 2
576.6.a.a 1 120.i odd 2 1
576.6.a.b 1 120.m even 2 1
576.6.a.bh 1 40.f even 2 1
576.6.a.bi 1 40.e odd 2 1
882.6.a.k 1 35.c odd 2 1
882.6.a.l 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} - 148 \)
\( T_{11} - 384 \)
\( T_{17} + 576 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -148 + T \)
$11$ \( -384 + T \)
$13$ \( -334 + T \)
$17$ \( 576 + T \)
$19$ \( 664 + T \)
$23$ \( -3840 + T \)
$29$ \( -96 + T \)
$31$ \( 4564 + T \)
$37$ \( 5798 + T \)
$41$ \( 6720 + T \)
$43$ \( -14872 + T \)
$47$ \( -19200 + T \)
$53$ \( 7776 + T \)
$59$ \( 13056 + T \)
$61$ \( -42782 + T \)
$67$ \( 36656 + T \)
$71$ \( -64512 + T \)
$73$ \( -16810 + T \)
$79$ \( -28076 + T \)
$83$ \( -66432 + T \)
$89$ \( 81792 + T \)
$97$ \( -29938 + T \)
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