Properties

Label 360.2.d.e
Level $360$
Weight $2$
Character orbit 360.d
Analytic conductor $2.875$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Coefficient field: 6.0.839056.1
Defining polynomial: \(x^{6} + 6 x^{4} + 8 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( -\beta_{1} - \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( -\beta_{1} - \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{8} + ( -2 - \beta_{1} - \beta_{4} ) q^{10} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{11} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{13} + ( 2 + 2 \beta_{2} ) q^{14} + ( -\beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{16} + ( -\beta_{1} - \beta_{3} + \beta_{4} ) q^{17} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{20} + ( -2 - 2 \beta_{5} ) q^{22} + ( 2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{23} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{25} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{26} + ( -2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{28} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{29} + ( -2 + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} + ( -4 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{32} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} ) q^{34} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{35} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{40} + ( 2 + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{41} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{43} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{44} + ( -2 \beta_{1} - 2 \beta_{2} + 4 \beta_{5} ) q^{46} + ( 2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{47} + ( -1 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{49} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{50} + ( -4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{52} + ( 4 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{53} + ( -2 + \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{55} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{56} + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{5} ) q^{58} + ( 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{59} + ( 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{62} + ( -4 - \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{64} + ( 2 - \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{65} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{67} + ( -4 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{68} + ( 6 + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{70} + ( -4 - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{71} + ( -4 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{73} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{74} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{76} + 4 q^{77} + ( 2 - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{79} + ( 4 + \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{80} + ( 8 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{82} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{83} + ( 2 + \beta_{1} - 5 \beta_{3} + \beta_{4} ) q^{85} + ( -4 - 4 \beta_{4} ) q^{86} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{88} + ( 2 - 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{89} + ( -2 \beta_{1} - 6 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 8 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{92} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{94} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{95} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{97} + ( 4 - \beta_{3} + 4 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} + q^{4} - q^{8} + O(q^{10}) \) \( 6q - q^{2} + q^{4} - q^{8} - 11q^{10} + 8q^{13} + 10q^{14} + q^{16} - 9q^{20} - 10q^{22} + 2q^{25} + 14q^{26} + 2q^{28} - 16q^{31} - 21q^{32} + 12q^{34} - 4q^{35} + 16q^{37} - 2q^{38} - 3q^{40} + 4q^{41} - 22q^{44} - 2q^{46} - 6q^{49} + 15q^{50} - 26q^{52} + 24q^{53} - 8q^{55} - 26q^{56} - 12q^{58} + 28q^{62} - 23q^{64} + 12q^{65} - 24q^{68} + 38q^{70} - 16q^{71} - 18q^{74} + 6q^{76} + 24q^{77} + 16q^{79} + 27q^{80} + 50q^{82} - 16q^{83} + 16q^{85} - 20q^{86} - 18q^{88} + 20q^{89} + 46q^{92} - 2q^{94} - 32q^{95} + 21q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 6 x^{4} + 8 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + \nu^{4} + 5 \nu^{3} + 3 \nu^{2} + 3 \nu - 1 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 5 \nu^{3} + 5 \nu^{2} - 5 \nu + 3 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 7 \nu^{3} + 5 \nu^{2} - 9 \nu + 3 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{5} + \nu^{4} + 17 \nu^{3} + 5 \nu^{2} + 19 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - \beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_{1} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{4} + 2 \beta_{3} - \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{5} + \beta_{4} - 3 \beta_{3} + 10 \beta_{2} + 5 \beta_{1} + 14\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{5} + 16 \beta_{4} - 17 \beta_{3} + 5 \beta_{1}\)\()/2\)

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
1.32132i
1.32132i
2.02852i
2.02852i
0.373087i
0.373087i
−1.34067 0.450129i 0 1.59477 + 1.20695i −0.254102 2.22158i 0 2.64265i −1.59477 2.33596i 0 −0.659335 + 3.09278i
109.2 −1.34067 + 0.450129i 0 1.59477 1.20695i −0.254102 + 2.22158i 0 2.64265i −1.59477 + 2.33596i 0 −0.659335 3.09278i
109.3 −0.321037 1.37729i 0 −1.79387 + 0.884323i 2.11491 0.726062i 0 4.05705i 1.79387 + 2.18678i 0 −1.67896 2.67975i
109.4 −0.321037 + 1.37729i 0 −1.79387 0.884323i 2.11491 + 0.726062i 0 4.05705i 1.79387 2.18678i 0 −1.67896 + 2.67975i
109.5 1.16170 0.806504i 0 0.699104 1.87383i −1.86081 1.23992i 0 0.746175i −0.699104 2.74067i 0 −3.16170 + 0.0603290i
109.6 1.16170 + 0.806504i 0 0.699104 + 1.87383i −1.86081 + 1.23992i 0 0.746175i −0.699104 + 2.74067i 0 −3.16170 0.0603290i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.e 6
3.b odd 2 1 120.2.d.b yes 6
4.b odd 2 1 1440.2.d.f 6
5.b even 2 1 360.2.d.f 6
5.c odd 4 2 1800.2.k.u 12
8.b even 2 1 360.2.d.f 6
8.d odd 2 1 1440.2.d.e 6
12.b even 2 1 480.2.d.b 6
15.d odd 2 1 120.2.d.a 6
15.e even 4 2 600.2.k.f 12
20.d odd 2 1 1440.2.d.e 6
20.e even 4 2 7200.2.k.u 12
24.f even 2 1 480.2.d.a 6
24.h odd 2 1 120.2.d.a 6
40.e odd 2 1 1440.2.d.f 6
40.f even 2 1 inner 360.2.d.e 6
40.i odd 4 2 1800.2.k.u 12
40.k even 4 2 7200.2.k.u 12
48.i odd 4 2 3840.2.f.l 12
48.k even 4 2 3840.2.f.m 12
60.h even 2 1 480.2.d.a 6
60.l odd 4 2 2400.2.k.f 12
120.i odd 2 1 120.2.d.b yes 6
120.m even 2 1 480.2.d.b 6
120.q odd 4 2 2400.2.k.f 12
120.w even 4 2 600.2.k.f 12
240.t even 4 2 3840.2.f.m 12
240.bm odd 4 2 3840.2.f.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 15.d odd 2 1
120.2.d.a 6 24.h odd 2 1
120.2.d.b yes 6 3.b odd 2 1
120.2.d.b yes 6 120.i odd 2 1
360.2.d.e 6 1.a even 1 1 trivial
360.2.d.e 6 40.f even 2 1 inner
360.2.d.f 6 5.b even 2 1
360.2.d.f 6 8.b even 2 1
480.2.d.a 6 24.f even 2 1
480.2.d.a 6 60.h even 2 1
480.2.d.b 6 12.b even 2 1
480.2.d.b 6 120.m even 2 1
600.2.k.f 12 15.e even 4 2
600.2.k.f 12 120.w even 4 2
1440.2.d.e 6 8.d odd 2 1
1440.2.d.e 6 20.d odd 2 1
1440.2.d.f 6 4.b odd 2 1
1440.2.d.f 6 40.e odd 2 1
1800.2.k.u 12 5.c odd 4 2
1800.2.k.u 12 40.i odd 4 2
2400.2.k.f 12 60.l odd 4 2
2400.2.k.f 12 120.q odd 4 2
3840.2.f.l 12 48.i odd 4 2
3840.2.f.l 12 240.bm odd 4 2
3840.2.f.m 12 48.k even 4 2
3840.2.f.m 12 240.t even 4 2
7200.2.k.u 12 20.e even 4 2
7200.2.k.u 12 40.k even 4 2

Hecke kernels

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

\( T_{7}^{6} + 24 T_{7}^{4} + 128 T_{7}^{2} + 64 \)
\( T_{11}^{6} + 32 T_{11}^{4} + 96 T_{11}^{2} + 64 \)
\( T_{13}^{3} - 4 T_{13}^{2} - 16 T_{13} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + 4 T + T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( 125 - 5 T^{2} - 8 T^{3} - T^{4} + T^{6} \)
$7$ \( 64 + 128 T^{2} + 24 T^{4} + T^{6} \)
$11$ \( 64 + 96 T^{2} + 32 T^{4} + T^{6} \)
$13$ \( ( 56 - 16 T - 4 T^{2} + T^{3} )^{2} \)
$17$ \( 1024 + 368 T^{2} + 36 T^{4} + T^{6} \)
$19$ \( 1024 + 512 T^{2} + 60 T^{4} + T^{6} \)
$23$ \( 16384 + 2304 T^{2} + 92 T^{4} + T^{6} \)
$29$ \( 12544 + 3120 T^{2} + 108 T^{4} + T^{6} \)
$31$ \( ( -64 - 4 T + 8 T^{2} + T^{3} )^{2} \)
$37$ \( ( 8 - 8 T^{2} + T^{3} )^{2} \)
$41$ \( ( -56 - 100 T - 2 T^{2} + T^{3} )^{2} \)
$43$ \( ( -64 - 64 T + T^{3} )^{2} \)
$47$ \( 1024 + 512 T^{2} + 60 T^{4} + T^{6} \)
$53$ \( ( 8 + 32 T - 12 T^{2} + T^{3} )^{2} \)
$59$ \( 179776 + 9888 T^{2} + 176 T^{4} + T^{6} \)
$61$ \( 65536 + 7168 T^{2} + 176 T^{4} + T^{6} \)
$67$ \( ( 64 - 64 T + T^{3} )^{2} \)
$71$ \( ( -128 - 80 T + 8 T^{2} + T^{3} )^{2} \)
$73$ \( 16384 + 34560 T^{2} + 384 T^{4} + T^{6} \)
$79$ \( ( 64 - 4 T - 8 T^{2} + T^{3} )^{2} \)
$83$ \( ( -448 - 64 T + 8 T^{2} + T^{3} )^{2} \)
$89$ \( ( 1384 - 164 T - 10 T^{2} + T^{3} )^{2} \)
$97$ \( 262144 + 28416 T^{2} + 336 T^{4} + T^{6} \)
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