Properties

Label 360.2.d.a
Level $360$
Weight $2$
Character orbit 360.d
Analytic conductor $2.875$
Analytic rank $0$
Dimension $4$
CM discriminant -120
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -2 q^{4} + \beta_{2} q^{5} + 2 \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} -2 q^{4} + \beta_{2} q^{5} + 2 \beta_{1} q^{8} + \beta_{3} q^{10} + 2 \beta_{2} q^{11} + 2 \beta_{3} q^{13} + 4 q^{16} -2 \beta_{1} q^{17} -2 \beta_{2} q^{20} + 2 \beta_{3} q^{22} + 4 \beta_{1} q^{23} -5 q^{25} -4 \beta_{2} q^{26} + 2 \beta_{2} q^{29} + 2 q^{31} -4 \beta_{1} q^{32} -4 q^{34} + 2 \beta_{3} q^{37} -2 \beta_{3} q^{40} -4 \beta_{3} q^{43} -4 \beta_{2} q^{44} + 8 q^{46} -8 \beta_{1} q^{47} + 7 q^{49} + 5 \beta_{1} q^{50} -4 \beta_{3} q^{52} -10 q^{55} + 2 \beta_{3} q^{58} + 2 \beta_{2} q^{59} -2 \beta_{1} q^{62} -8 q^{64} + 10 \beta_{1} q^{65} -4 \beta_{3} q^{67} + 4 \beta_{1} q^{68} -4 \beta_{2} q^{74} + 14 q^{79} + 4 \beta_{2} q^{80} + 2 \beta_{3} q^{85} + 8 \beta_{2} q^{86} -4 \beta_{3} q^{88} -8 \beta_{1} q^{92} -16 q^{94} -7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 16q^{16} - 20q^{25} + 8q^{31} - 16q^{34} + 32q^{46} + 28q^{49} - 40q^{55} - 32q^{64} + 56q^{79} - 64q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 7 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} + 7 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.58114 + 0.707107i
1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 0.707107i
1.41421i 0 −2.00000 2.23607i 0 0 2.82843i 0 −3.16228
109.2 1.41421i 0 −2.00000 2.23607i 0 0 2.82843i 0 3.16228
109.3 1.41421i 0 −2.00000 2.23607i 0 0 2.82843i 0 3.16228
109.4 1.41421i 0 −2.00000 2.23607i 0 0 2.82843i 0 −3.16228
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
120.i odd 2 1 CM by \(\Q(\sqrt{-30}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.d.a 4
3.b odd 2 1 inner 360.2.d.a 4
4.b odd 2 1 1440.2.d.a 4
5.b even 2 1 inner 360.2.d.a 4
5.c odd 4 2 1800.2.k.o 4
8.b even 2 1 inner 360.2.d.a 4
8.d odd 2 1 1440.2.d.a 4
12.b even 2 1 1440.2.d.a 4
15.d odd 2 1 inner 360.2.d.a 4
15.e even 4 2 1800.2.k.o 4
20.d odd 2 1 1440.2.d.a 4
20.e even 4 2 7200.2.k.m 4
24.f even 2 1 1440.2.d.a 4
24.h odd 2 1 inner 360.2.d.a 4
40.e odd 2 1 1440.2.d.a 4
40.f even 2 1 inner 360.2.d.a 4
40.i odd 4 2 1800.2.k.o 4
40.k even 4 2 7200.2.k.m 4
60.h even 2 1 1440.2.d.a 4
60.l odd 4 2 7200.2.k.m 4
120.i odd 2 1 CM 360.2.d.a 4
120.m even 2 1 1440.2.d.a 4
120.q odd 4 2 7200.2.k.m 4
120.w even 4 2 1800.2.k.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.d.a 4 1.a even 1 1 trivial
360.2.d.a 4 3.b odd 2 1 inner
360.2.d.a 4 5.b even 2 1 inner
360.2.d.a 4 8.b even 2 1 inner
360.2.d.a 4 15.d odd 2 1 inner
360.2.d.a 4 24.h odd 2 1 inner
360.2.d.a 4 40.f even 2 1 inner
360.2.d.a 4 120.i odd 2 1 CM
1440.2.d.a 4 4.b odd 2 1
1440.2.d.a 4 8.d odd 2 1
1440.2.d.a 4 12.b even 2 1
1440.2.d.a 4 20.d odd 2 1
1440.2.d.a 4 24.f even 2 1
1440.2.d.a 4 40.e odd 2 1
1440.2.d.a 4 60.h even 2 1
1440.2.d.a 4 120.m even 2 1
1800.2.k.o 4 5.c odd 4 2
1800.2.k.o 4 15.e even 4 2
1800.2.k.o 4 40.i odd 4 2
1800.2.k.o 4 120.w even 4 2
7200.2.k.m 4 20.e even 4 2
7200.2.k.m 4 40.k even 4 2
7200.2.k.m 4 60.l odd 4 2
7200.2.k.m 4 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\):

\( T_{7} \)
\( T_{11}^{2} + 20 \)
\( T_{13}^{2} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 20 + T^{2} )^{2} \)
$13$ \( ( -40 + T^{2} )^{2} \)
$17$ \( ( 8 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( 32 + T^{2} )^{2} \)
$29$ \( ( 20 + T^{2} )^{2} \)
$31$ \( ( -2 + T )^{4} \)
$37$ \( ( -40 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -160 + T^{2} )^{2} \)
$47$ \( ( 128 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( ( 20 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( -160 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( -14 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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