Properties

Label 180.3.b.a
Level $180$
Weight $3$
Character orbit 180.b
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{23})\)
Defining polynomial: \( x^{4} + 24x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{5} - \beta_{2} q^{7} + 7 \beta_1 q^{11} - 3 \beta_{2} q^{13} + 4 \beta_{3} q^{17} + 12 q^{19} + 2 \beta_{3} q^{23} + ( - 2 \beta_{2} + 21) q^{25} - 6 \beta_1 q^{29} - 38 q^{31} + ( - 2 \beta_{3} - 23 \beta_1) q^{35} - \beta_{2} q^{37} + 49 \beta_1 q^{41} + 10 \beta_{2} q^{43} - 16 \beta_{3} q^{47} + 3 q^{49} + (7 \beta_{2} + 14) q^{55} - 59 \beta_1 q^{59} - 70 q^{61} + ( - 6 \beta_{3} - 69 \beta_1) q^{65} + 16 \beta_{2} q^{67} + 84 \beta_1 q^{71} - 2 \beta_{2} q^{73} + 14 \beta_{3} q^{77} + 30 q^{79} - 28 \beta_{3} q^{83} + ( - 4 \beta_{2} + 92) q^{85} + 23 \beta_1 q^{89} - 138 q^{91} + (12 \beta_{3} - 12 \beta_1) q^{95} - 14 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 48 q^{19} + 84 q^{25} - 152 q^{31} + 12 q^{49} + 56 q^{55} - 280 q^{61} + 120 q^{79} + 368 q^{85} - 552 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 24x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 13\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 35\nu ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{2} + 35\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
4.09827i
4.09827i
2.68406i
2.68406i
0 0 0 −4.79583 1.41421i 0 6.78233i 0 0 0
89.2 0 0 0 −4.79583 + 1.41421i 0 6.78233i 0 0 0
89.3 0 0 0 4.79583 1.41421i 0 6.78233i 0 0 0
89.4 0 0 0 4.79583 + 1.41421i 0 6.78233i 0 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.b.a 4
3.b odd 2 1 inner 180.3.b.a 4
4.b odd 2 1 720.3.c.b 4
5.b even 2 1 inner 180.3.b.a 4
5.c odd 4 2 900.3.g.c 4
8.b even 2 1 2880.3.c.c 4
8.d odd 2 1 2880.3.c.f 4
9.c even 3 2 1620.3.t.c 8
9.d odd 6 2 1620.3.t.c 8
12.b even 2 1 720.3.c.b 4
15.d odd 2 1 inner 180.3.b.a 4
15.e even 4 2 900.3.g.c 4
20.d odd 2 1 720.3.c.b 4
20.e even 4 2 3600.3.l.r 4
24.f even 2 1 2880.3.c.f 4
24.h odd 2 1 2880.3.c.c 4
40.e odd 2 1 2880.3.c.f 4
40.f even 2 1 2880.3.c.c 4
45.h odd 6 2 1620.3.t.c 8
45.j even 6 2 1620.3.t.c 8
60.h even 2 1 720.3.c.b 4
60.l odd 4 2 3600.3.l.r 4
120.i odd 2 1 2880.3.c.c 4
120.m even 2 1 2880.3.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.b.a 4 1.a even 1 1 trivial
180.3.b.a 4 3.b odd 2 1 inner
180.3.b.a 4 5.b even 2 1 inner
180.3.b.a 4 15.d odd 2 1 inner
720.3.c.b 4 4.b odd 2 1
720.3.c.b 4 12.b even 2 1
720.3.c.b 4 20.d odd 2 1
720.3.c.b 4 60.h even 2 1
900.3.g.c 4 5.c odd 4 2
900.3.g.c 4 15.e even 4 2
1620.3.t.c 8 9.c even 3 2
1620.3.t.c 8 9.d odd 6 2
1620.3.t.c 8 45.h odd 6 2
1620.3.t.c 8 45.j even 6 2
2880.3.c.c 4 8.b even 2 1
2880.3.c.c 4 24.h odd 2 1
2880.3.c.c 4 40.f even 2 1
2880.3.c.c 4 120.i odd 2 1
2880.3.c.f 4 8.d odd 2 1
2880.3.c.f 4 24.f even 2 1
2880.3.c.f 4 40.e odd 2 1
2880.3.c.f 4 120.m even 2 1
3600.3.l.r 4 20.e even 4 2
3600.3.l.r 4 60.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(180, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 42T^{2} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 46)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 414)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 368)^{2} \) Copy content Toggle raw display
$19$ \( (T - 12)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 92)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$31$ \( (T + 38)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 46)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4802)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 5888)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 6962)^{2} \) Copy content Toggle raw display
$61$ \( (T + 70)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 11776)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 14112)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 184)^{2} \) Copy content Toggle raw display
$79$ \( (T - 30)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 18032)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1058)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 9016)^{2} \) Copy content Toggle raw display
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