Properties

Label 180.4.a.a
Level $180$
Weight $4$
Character orbit 180.a
Self dual yes
Analytic conductor $10.620$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 180.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6203438010\)
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)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} - 16 q^{7} + 60 q^{11} + 86 q^{13} - 18 q^{17} + 44 q^{19} - 48 q^{23} + 25 q^{25} + 186 q^{29} + 176 q^{31} + 80 q^{35} + 254 q^{37} - 186 q^{41} - 100 q^{43} - 168 q^{47} - 87 q^{49} + 498 q^{53} - 300 q^{55} + 252 q^{59} - 58 q^{61} - 430 q^{65} - 1036 q^{67} - 168 q^{71} + 506 q^{73} - 960 q^{77} + 272 q^{79} - 948 q^{83} + 90 q^{85} + 1014 q^{89} - 1376 q^{91} - 220 q^{95} - 766 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
0 0 0 −5.00000 0 −16.0000 0 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 180.4.a.a 1
3.b odd 2 1 20.4.a.a 1
4.b odd 2 1 720.4.a.k 1
5.b even 2 1 900.4.a.m 1
5.c odd 4 2 900.4.d.k 2
9.c even 3 2 1620.4.i.j 2
9.d odd 6 2 1620.4.i.d 2
12.b even 2 1 80.4.a.c 1
15.d odd 2 1 100.4.a.a 1
15.e even 4 2 100.4.c.a 2
21.c even 2 1 980.4.a.c 1
21.g even 6 2 980.4.i.n 2
21.h odd 6 2 980.4.i.e 2
24.f even 2 1 320.4.a.k 1
24.h odd 2 1 320.4.a.d 1
33.d even 2 1 2420.4.a.d 1
48.i odd 4 2 1280.4.d.n 2
48.k even 4 2 1280.4.d.c 2
60.h even 2 1 400.4.a.o 1
60.l odd 4 2 400.4.c.j 2
120.i odd 2 1 1600.4.a.bl 1
120.m even 2 1 1600.4.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 3.b odd 2 1
80.4.a.c 1 12.b even 2 1
100.4.a.a 1 15.d odd 2 1
100.4.c.a 2 15.e even 4 2
180.4.a.a 1 1.a even 1 1 trivial
320.4.a.d 1 24.h odd 2 1
320.4.a.k 1 24.f even 2 1
400.4.a.o 1 60.h even 2 1
400.4.c.j 2 60.l odd 4 2
720.4.a.k 1 4.b odd 2 1
900.4.a.m 1 5.b even 2 1
900.4.d.k 2 5.c odd 4 2
980.4.a.c 1 21.c even 2 1
980.4.i.e 2 21.h odd 6 2
980.4.i.n 2 21.g even 6 2
1280.4.d.c 2 48.k even 4 2
1280.4.d.n 2 48.i odd 4 2
1600.4.a.p 1 120.m even 2 1
1600.4.a.bl 1 120.i odd 2 1
1620.4.i.d 2 9.d odd 6 2
1620.4.i.j 2 9.c even 3 2
2420.4.a.d 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(180))\):

\( T_{7} + 16 \) Copy content Toggle raw display
\( T_{11} - 60 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 16 \) Copy content Toggle raw display
$11$ \( T - 60 \) Copy content Toggle raw display
$13$ \( T - 86 \) Copy content Toggle raw display
$17$ \( T + 18 \) Copy content Toggle raw display
$19$ \( T - 44 \) Copy content Toggle raw display
$23$ \( T + 48 \) Copy content Toggle raw display
$29$ \( T - 186 \) Copy content Toggle raw display
$31$ \( T - 176 \) Copy content Toggle raw display
$37$ \( T - 254 \) Copy content Toggle raw display
$41$ \( T + 186 \) Copy content Toggle raw display
$43$ \( T + 100 \) Copy content Toggle raw display
$47$ \( T + 168 \) Copy content Toggle raw display
$53$ \( T - 498 \) Copy content Toggle raw display
$59$ \( T - 252 \) Copy content Toggle raw display
$61$ \( T + 58 \) Copy content Toggle raw display
$67$ \( T + 1036 \) Copy content Toggle raw display
$71$ \( T + 168 \) Copy content Toggle raw display
$73$ \( T - 506 \) Copy content Toggle raw display
$79$ \( T - 272 \) Copy content Toggle raw display
$83$ \( T + 948 \) Copy content Toggle raw display
$89$ \( T - 1014 \) Copy content Toggle raw display
$97$ \( T + 766 \) Copy content Toggle raw display
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