Properties

Label 360.2.q.d
Level $360$
Weight $2$
Character orbit 360.q
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.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Defining polynomial: \(x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{7} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -1 - \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{7} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} + ( \beta_{2} + \beta_{4} - \beta_{5} ) q^{11} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{13} -\beta_{4} q^{15} + ( 2 - \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{17} + ( -2 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{21} + ( 4 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{23} + \beta_{3} q^{25} + ( 2 - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{27} + ( 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{5} ) q^{29} + ( 6 - \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{31} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{33} + ( -2 + \beta_{1} + \beta_{2} - \beta_{5} ) q^{35} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{37} + ( -6 + 2 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{39} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{41} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{43} + ( -1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{45} + ( \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{47} + ( -6 + 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{49} + ( 2 + \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{51} + ( -4 - 2 \beta_{2} + 2 \beta_{4} ) q^{53} + ( -\beta_{1} - \beta_{4} + \beta_{5} ) q^{55} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{57} + ( 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{59} + ( -4 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{61} + ( 6 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} ) q^{63} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{65} + ( -2 + 5 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{67} + ( -4 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 7 \beta_{5} ) q^{69} + ( -6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{71} + ( 4 - 4 \beta_{2} + 4 \beta_{4} ) q^{73} + ( \beta_{1} + \beta_{4} ) q^{75} + ( 6 - 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{77} + ( 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 5 + 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{81} + ( -\beta_{1} + \beta_{2} - 6 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{83} + ( -2 + 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{85} + ( 2 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{87} -11 q^{89} + ( -12 + 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{91} + ( 2 + \beta_{2} + 4 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{93} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{95} + ( -2 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{97} + ( -2 + 5 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{3} - 3q^{5} + 5q^{7} + 5q^{9} + O(q^{10}) \) \( 6q - q^{3} - 3q^{5} + 5q^{7} + 5q^{9} + 2q^{11} - q^{15} + 4q^{17} - 8q^{19} + 12q^{21} + 7q^{23} - 3q^{25} + 2q^{27} + 7q^{29} + 16q^{31} - 20q^{33} - 10q^{35} + 4q^{37} - 18q^{39} + q^{41} - 2q^{43} + 2q^{45} + 13q^{47} - 10q^{49} - 20q^{53} - 4q^{55} - 14q^{57} + 6q^{59} + 11q^{61} + 27q^{63} - q^{67} - 33q^{69} - 28q^{71} + 32q^{73} + 2q^{75} + 12q^{77} + 6q^{79} + 29q^{81} + 21q^{83} - 2q^{85} + 2q^{87} - 66q^{89} - 60q^{91} + 2q^{93} + 4q^{95} - 30q^{97} - 14q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 5 \nu^{4} + \nu^{3} + 9 \nu^{2} - 6 \nu - 45 \)\()/27\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 9 \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 2 \nu^{4} - 2 \nu^{3} - 6 \nu - 18 \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( 4 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 18 \nu^{2} - 24 \nu - 72 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 2\)
\(\nu^{4}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 5\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + \beta_{4} - 10 \beta_{3} - 4 \beta_{2} + 6 \beta_{1} + 4\)

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.71903 + 0.211943i
0.403374 1.68443i
−1.62241 + 0.606458i
1.71903 0.211943i
0.403374 + 1.68443i
−1.62241 0.606458i
0 −1.71903 0.211943i 0 −0.500000 + 0.866025i 0 −0.719035 1.24540i 0 2.91016 + 0.728674i 0
121.2 0 −0.403374 + 1.68443i 0 −0.500000 + 0.866025i 0 0.596626 + 1.03339i 0 −2.67458 1.35891i 0
121.3 0 1.62241 0.606458i 0 −0.500000 + 0.866025i 0 2.62241 + 4.54214i 0 2.26442 1.96784i 0
241.1 0 −1.71903 + 0.211943i 0 −0.500000 0.866025i 0 −0.719035 + 1.24540i 0 2.91016 0.728674i 0
241.2 0 −0.403374 1.68443i 0 −0.500000 0.866025i 0 0.596626 1.03339i 0 −2.67458 + 1.35891i 0
241.3 0 1.62241 + 0.606458i 0 −0.500000 0.866025i 0 2.62241 4.54214i 0 2.26442 + 1.96784i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.q.d 6
3.b odd 2 1 1080.2.q.d 6
4.b odd 2 1 720.2.q.j 6
9.c even 3 1 inner 360.2.q.d 6
9.c even 3 1 3240.2.a.r 3
9.d odd 6 1 1080.2.q.d 6
9.d odd 6 1 3240.2.a.q 3
12.b even 2 1 2160.2.q.j 6
36.f odd 6 1 720.2.q.j 6
36.f odd 6 1 6480.2.a.bx 3
36.h even 6 1 2160.2.q.j 6
36.h even 6 1 6480.2.a.bu 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.d 6 1.a even 1 1 trivial
360.2.q.d 6 9.c even 3 1 inner
720.2.q.j 6 4.b odd 2 1
720.2.q.j 6 36.f odd 6 1
1080.2.q.d 6 3.b odd 2 1
1080.2.q.d 6 9.d odd 6 1
2160.2.q.j 6 12.b even 2 1
2160.2.q.j 6 36.h even 6 1
3240.2.a.q 3 9.d odd 6 1
3240.2.a.r 3 9.c even 3 1
6480.2.a.bu 3 36.h even 6 1
6480.2.a.bx 3 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 5 T_{7}^{5} + 28 T_{7}^{4} - 3 T_{7}^{3} + 54 T_{7}^{2} - 27 T_{7} + 81 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 27 + 9 T - 6 T^{2} - 3 T^{3} - 2 T^{4} + T^{5} + T^{6} \)
$5$ \( ( 1 + T + T^{2} )^{3} \)
$7$ \( 81 - 27 T + 54 T^{2} - 3 T^{3} + 28 T^{4} - 5 T^{5} + T^{6} \)
$11$ \( 144 - 96 T + 88 T^{2} - 8 T^{3} + 12 T^{4} - 2 T^{5} + T^{6} \)
$13$ \( 1296 + 864 T + 576 T^{2} + 72 T^{3} + 24 T^{4} + T^{6} \)
$17$ \( ( 108 - 36 T - 2 T^{2} + T^{3} )^{2} \)
$19$ \( ( -4 - 4 T + 4 T^{2} + T^{3} )^{2} \)
$23$ \( 91809 - 15453 T + 4722 T^{2} - 249 T^{3} + 100 T^{4} - 7 T^{5} + T^{6} \)
$29$ \( 729 - 135 T + 214 T^{2} - 19 T^{3} + 54 T^{4} - 7 T^{5} + T^{6} \)
$31$ \( 11664 - 8208 T + 4048 T^{2} - 1000 T^{3} + 180 T^{4} - 16 T^{5} + T^{6} \)
$37$ \( ( 12 - 8 T - 2 T^{2} + T^{3} )^{2} \)
$41$ \( 9 + 27 T + 78 T^{2} + 15 T^{3} + 10 T^{4} - T^{5} + T^{6} \)
$43$ \( 11664 + 3888 T + 1512 T^{2} + 144 T^{3} + 40 T^{4} + 2 T^{5} + T^{6} \)
$47$ \( 1261129 - 95455 T + 21824 T^{2} - 1141 T^{3} + 254 T^{4} - 13 T^{5} + T^{6} \)
$53$ \( ( -24 + 12 T + 10 T^{2} + T^{3} )^{2} \)
$59$ \( 5184 + 4320 T + 3168 T^{2} + 504 T^{3} + 96 T^{4} - 6 T^{5} + T^{6} \)
$61$ \( 269361 - 21279 T + 7390 T^{2} - 587 T^{3} + 162 T^{4} - 11 T^{5} + T^{6} \)
$67$ \( 12769 - 15029 T + 17576 T^{2} - 359 T^{3} + 134 T^{4} + T^{5} + T^{6} \)
$71$ \( ( -36 - 20 T + 14 T^{2} + T^{3} )^{2} \)
$73$ \( ( 384 - 16 T^{2} + T^{3} )^{2} \)
$79$ \( 10816 + 8736 T + 6432 T^{2} + 712 T^{3} + 120 T^{4} - 6 T^{5} + T^{6} \)
$83$ \( 59049 - 31347 T + 11538 T^{2} - 2223 T^{3} + 312 T^{4} - 21 T^{5} + T^{6} \)
$89$ \( ( 11 + T )^{6} \)
$97$ \( 1201216 - 170976 T + 57216 T^{2} + 6872 T^{3} + 744 T^{4} + 30 T^{5} + T^{6} \)
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