Properties

Label 180.3.f.f
Level $180$
Weight $3$
Character orbit 180.f
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{15})\)
Defining polynomial: \(x^{4} - 7 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 4 + \beta_{3} ) q^{4} + 5 \beta_{2} q^{5} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 4 + \beta_{3} ) q^{4} + 5 \beta_{2} q^{5} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{8} + 5 \beta_{3} q^{10} + ( 12 + 7 \beta_{3} ) q^{16} + 14 \beta_{2} q^{17} + ( -8 - 16 \beta_{3} ) q^{19} + ( -5 \beta_{1} + 20 \beta_{2} ) q^{20} + ( 16 \beta_{1} - 8 \beta_{2} ) q^{23} -25 q^{25} + ( -16 - 32 \beta_{3} ) q^{31} + ( 5 \beta_{1} + 28 \beta_{2} ) q^{32} + 14 \beta_{3} q^{34} + ( 8 \beta_{1} - 64 \beta_{2} ) q^{38} + ( -20 + 15 \beta_{3} ) q^{40} + ( 64 + 8 \beta_{3} ) q^{46} + ( -48 \beta_{1} + 24 \beta_{2} ) q^{47} -49 q^{49} -25 \beta_{1} q^{50} -86 \beta_{2} q^{53} + 118 q^{61} + ( 16 \beta_{1} - 128 \beta_{2} ) q^{62} + ( 20 + 33 \beta_{3} ) q^{64} + ( -14 \beta_{1} + 56 \beta_{2} ) q^{68} + ( 32 - 56 \beta_{3} ) q^{76} + ( 32 + 64 \beta_{3} ) q^{79} + ( -35 \beta_{1} + 60 \beta_{2} ) q^{80} + ( 32 \beta_{1} - 16 \beta_{2} ) q^{83} -70 q^{85} + ( 56 \beta_{1} + 32 \beta_{2} ) q^{92} + ( -192 - 24 \beta_{3} ) q^{94} + ( 80 \beta_{1} - 40 \beta_{2} ) q^{95} -49 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 14q^{4} + O(q^{10}) \) \( 4q + 14q^{4} - 10q^{10} + 34q^{16} - 100q^{25} - 28q^{34} - 110q^{40} + 240q^{46} - 196q^{49} + 472q^{61} + 14q^{64} + 240q^{76} - 280q^{85} - 720q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 7 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 3 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(4 \beta_{2} + 3 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.93649 0.500000i
−1.93649 + 0.500000i
1.93649 0.500000i
1.93649 + 0.500000i
−1.93649 0.500000i 0 3.50000 + 1.93649i 5.00000i 0 0 −5.80948 5.50000i 0 −2.50000 + 9.68246i
19.2 −1.93649 + 0.500000i 0 3.50000 1.93649i 5.00000i 0 0 −5.80948 + 5.50000i 0 −2.50000 9.68246i
19.3 1.93649 0.500000i 0 3.50000 1.93649i 5.00000i 0 0 5.80948 5.50000i 0 −2.50000 9.68246i
19.4 1.93649 + 0.500000i 0 3.50000 + 1.93649i 5.00000i 0 0 5.80948 + 5.50000i 0 −2.50000 + 9.68246i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.f.f 4
3.b odd 2 1 inner 180.3.f.f 4
4.b odd 2 1 inner 180.3.f.f 4
5.b even 2 1 inner 180.3.f.f 4
5.c odd 4 1 900.3.c.g 2
5.c odd 4 1 900.3.c.i 2
12.b even 2 1 inner 180.3.f.f 4
15.d odd 2 1 CM 180.3.f.f 4
15.e even 4 1 900.3.c.g 2
15.e even 4 1 900.3.c.i 2
20.d odd 2 1 inner 180.3.f.f 4
20.e even 4 1 900.3.c.g 2
20.e even 4 1 900.3.c.i 2
60.h even 2 1 inner 180.3.f.f 4
60.l odd 4 1 900.3.c.g 2
60.l odd 4 1 900.3.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.f.f 4 1.a even 1 1 trivial
180.3.f.f 4 3.b odd 2 1 inner
180.3.f.f 4 4.b odd 2 1 inner
180.3.f.f 4 5.b even 2 1 inner
180.3.f.f 4 12.b even 2 1 inner
180.3.f.f 4 15.d odd 2 1 CM
180.3.f.f 4 20.d odd 2 1 inner
180.3.f.f 4 60.h even 2 1 inner
900.3.c.g 2 5.c odd 4 1
900.3.c.g 2 15.e even 4 1
900.3.c.g 2 20.e even 4 1
900.3.c.g 2 60.l odd 4 1
900.3.c.i 2 5.c odd 4 1
900.3.c.i 2 15.e even 4 1
900.3.c.i 2 20.e even 4 1
900.3.c.i 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\):

\( T_{7} \)
\( T_{13} \)
\( T_{23}^{2} - 960 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 7 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 25 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 196 + T^{2} )^{2} \)
$19$ \( ( 960 + T^{2} )^{2} \)
$23$ \( ( -960 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 3840 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( -8640 + T^{2} )^{2} \)
$53$ \( ( 7396 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( -118 + T )^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 15360 + T^{2} )^{2} \)
$83$ \( ( -3840 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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