Properties

Label 360.2.m.a
Level $360$
Weight $2$
Character orbit 360.m
Analytic conductor $2.875$
Analytic rank $0$
Dimension $4$
CM discriminant -40
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.m (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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -2 q^{4} + \beta_{2} q^{5} + ( -2 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} -2 q^{4} + \beta_{2} q^{5} + ( -2 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} + \beta_{3} q^{10} + ( -\beta_{1} - 2 \beta_{2} ) q^{11} + ( 4 + \beta_{3} ) q^{13} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{14} + 4 q^{16} + 2 \beta_{3} q^{19} -2 \beta_{2} q^{20} + ( -2 - 2 \beta_{3} ) q^{22} + 2 \beta_{2} q^{23} -5 q^{25} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{26} + ( 4 - 2 \beta_{3} ) q^{28} -4 \beta_{1} q^{32} + ( 5 \beta_{1} - 2 \beta_{2} ) q^{35} + ( -8 + \beta_{3} ) q^{37} -4 \beta_{2} q^{38} -2 \beta_{3} q^{40} + ( -\beta_{1} + 4 \beta_{2} ) q^{41} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{44} + 2 \beta_{3} q^{46} -2 \beta_{1} q^{47} + ( 7 - 4 \beta_{3} ) q^{49} + 5 \beta_{1} q^{50} + ( -8 - 2 \beta_{3} ) q^{52} + 4 \beta_{1} q^{53} + ( 10 + \beta_{3} ) q^{55} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{56} + ( -7 \beta_{1} - 2 \beta_{2} ) q^{59} -8 q^{64} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{65} + ( 10 - 2 \beta_{3} ) q^{70} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{74} -4 \beta_{3} q^{76} + ( -8 \beta_{1} + 2 \beta_{2} ) q^{77} + 4 \beta_{2} q^{80} + ( -2 + 4 \beta_{3} ) q^{82} + ( 4 + 4 \beta_{3} ) q^{88} + ( -7 \beta_{1} + 4 \beta_{2} ) q^{89} + ( 2 + 2 \beta_{3} ) q^{91} -4 \beta_{2} q^{92} -4 q^{94} + 10 \beta_{1} q^{95} + ( -7 \beta_{1} + 8 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} - 8q^{7} + O(q^{10}) \) \( 4q - 8q^{4} - 8q^{7} + 16q^{13} + 16q^{16} - 8q^{22} - 20q^{25} + 16q^{28} - 32q^{37} + 28q^{49} - 32q^{52} + 40q^{55} - 32q^{64} + 40q^{70} - 8q^{82} + 16q^{88} + 8q^{91} - 16q^{94} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
−1.58114 + 0.707107i
1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 0.707107i
1.41421i 0 −2.00000 2.23607i 0 −5.16228 2.82843i 0 −3.16228
179.2 1.41421i 0 −2.00000 2.23607i 0 1.16228 2.82843i 0 3.16228
179.3 1.41421i 0 −2.00000 2.23607i 0 1.16228 2.82843i 0 3.16228
179.4 1.41421i 0 −2.00000 2.23607i 0 −5.16228 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
3.b odd 2 1 inner
120.m 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.m.a 4
3.b odd 2 1 inner 360.2.m.a 4
4.b odd 2 1 1440.2.m.b 4
5.b even 2 1 360.2.m.b yes 4
5.c odd 4 2 1800.2.b.f 8
8.b even 2 1 1440.2.m.a 4
8.d odd 2 1 360.2.m.b yes 4
12.b even 2 1 1440.2.m.b 4
15.d odd 2 1 360.2.m.b yes 4
15.e even 4 2 1800.2.b.f 8
20.d odd 2 1 1440.2.m.a 4
20.e even 4 2 7200.2.b.f 8
24.f even 2 1 360.2.m.b yes 4
24.h odd 2 1 1440.2.m.a 4
40.e odd 2 1 CM 360.2.m.a 4
40.f even 2 1 1440.2.m.b 4
40.i odd 4 2 7200.2.b.f 8
40.k even 4 2 1800.2.b.f 8
60.h even 2 1 1440.2.m.a 4
60.l odd 4 2 7200.2.b.f 8
120.i odd 2 1 1440.2.m.b 4
120.m even 2 1 inner 360.2.m.a 4
120.q odd 4 2 1800.2.b.f 8
120.w even 4 2 7200.2.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.m.a 4 1.a even 1 1 trivial
360.2.m.a 4 3.b odd 2 1 inner
360.2.m.a 4 40.e odd 2 1 CM
360.2.m.a 4 120.m even 2 1 inner
360.2.m.b yes 4 5.b even 2 1
360.2.m.b yes 4 8.d odd 2 1
360.2.m.b yes 4 15.d odd 2 1
360.2.m.b yes 4 24.f even 2 1
1440.2.m.a 4 8.b even 2 1
1440.2.m.a 4 20.d odd 2 1
1440.2.m.a 4 24.h odd 2 1
1440.2.m.a 4 60.h even 2 1
1440.2.m.b 4 4.b odd 2 1
1440.2.m.b 4 12.b even 2 1
1440.2.m.b 4 40.f even 2 1
1440.2.m.b 4 120.i odd 2 1
1800.2.b.f 8 5.c odd 4 2
1800.2.b.f 8 15.e even 4 2
1800.2.b.f 8 40.k even 4 2
1800.2.b.f 8 120.q odd 4 2
7200.2.b.f 8 20.e even 4 2
7200.2.b.f 8 40.i odd 4 2
7200.2.b.f 8 60.l odd 4 2
7200.2.b.f 8 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4 T_{7} - 6 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( -6 + 4 T + T^{2} )^{2} \)
$11$ \( 324 + 44 T^{2} + T^{4} \)
$13$ \( ( 6 - 8 T + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( -40 + T^{2} )^{2} \)
$23$ \( ( 20 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 54 + 16 T + T^{2} )^{2} \)
$41$ \( 6084 + 164 T^{2} + T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( 8 + T^{2} )^{2} \)
$53$ \( ( 32 + T^{2} )^{2} \)
$59$ \( 6084 + 236 T^{2} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( 324 + 356 T^{2} + T^{4} \)
$97$ \( T^{4} \)
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