Properties

Label 360.2.k.f
Level 360
Weight 2
Character orbit 360.k
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.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( \beta_{2} - \beta_{3} ) q^{4} + \beta_{3} q^{5} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( \beta_{2} - \beta_{3} ) q^{4} + \beta_{3} q^{5} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{8} + \beta_{4} q^{10} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{11} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{13} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{14} + ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{16} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{19} + ( 1 - \beta_{5} ) q^{20} + ( -2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{22} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} - q^{25} + ( -4 + 2 \beta_{2} - 2 \beta_{3} ) q^{26} + ( -4 - 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{28} -2 \beta_{3} q^{29} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{31} + ( -2 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{32} + ( 2 + 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} ) q^{34} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{37} + ( 2 + 4 \beta_{3} - 2 \beta_{5} ) q^{38} + ( 1 - \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{40} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{41} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{43} + ( -2 + 4 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{5} ) q^{44} + ( -4 - 2 \beta_{2} + 2 \beta_{3} ) q^{46} + ( -3 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{47} + ( 7 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} ) q^{49} + \beta_{1} q^{50} + ( -4 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{52} + 2 \beta_{3} q^{53} + ( 2 + 2 \beta_{2} + 2 \beta_{4} ) q^{55} + ( -2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{56} -2 \beta_{4} q^{58} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{59} + ( 4 \beta_{1} + 4 \beta_{3} - 4 \beta_{5} ) q^{61} + ( -2 + 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} ) q^{62} + ( 5 + \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{64} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{65} + 4 \beta_{3} q^{67} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{68} + ( 2 + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{70} + ( -4 \beta_{2} - 4 \beta_{4} ) q^{71} -6 q^{73} + ( 2 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{74} + ( 2 - 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{76} + ( 4 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{77} + ( 5 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{79} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{80} + ( 8 - 6 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{82} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{83} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{85} + ( 4 + 8 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{86} + ( 2 - 2 \beta_{2} + 10 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{88} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{89} + ( 4 - 8 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} ) q^{91} + ( 4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{92} + ( 2 \beta_{2} + 6 \beta_{3} + 4 \beta_{5} ) q^{94} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{95} + ( 4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{97} + ( 4 - 5 \beta_{1} - 8 \beta_{3} - 4 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{2} - 2q^{4} + 4q^{7} - 8q^{8} + O(q^{10}) \) \( 6q - 2q^{2} - 2q^{4} + 4q^{7} - 8q^{8} + 16q^{14} + 10q^{16} - 12q^{17} + 4q^{20} - 20q^{22} + 8q^{23} - 6q^{25} - 28q^{26} - 28q^{28} - 12q^{31} - 12q^{32} + 12q^{34} + 8q^{38} + 6q^{40} + 20q^{41} + 4q^{44} - 20q^{46} - 8q^{47} + 30q^{49} + 2q^{50} - 8q^{52} + 8q^{55} - 4q^{56} - 4q^{62} + 22q^{64} + 16q^{68} + 8q^{70} + 8q^{71} - 36q^{73} - 12q^{74} + 12q^{76} + 36q^{79} - 16q^{80} + 28q^{82} + 16q^{86} + 12q^{88} + 28q^{89} + 24q^{92} + 4q^{94} + 8q^{95} + 36q^{97} + 6q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 3 \nu^{3} + 2 \nu - 8 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 3 \nu^{3} - 4 \nu^{2} + 2 \nu - 8 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - \nu^{3} + 2 \nu^{2} + 4 \)\()/2\)
\(\beta_{5}\)\(=\)\( -\nu^{5} + \nu^{4} - 2 \nu^{3} + 3 \nu^{2} - 2 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{1} + 1\)
\(\nu^{5}\)\(=\)\(-3 \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_{1} + 2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.40680 + 0.144584i
1.40680 0.144584i
0.264658 + 1.38923i
0.264658 1.38923i
−0.671462 + 1.24464i
−0.671462 1.24464i
−1.40680 0.144584i 0 1.95819 + 0.406803i 1.00000i 0 −3.62721 −2.69597 0.855416i 0 0.144584 1.40680i
181.2 −1.40680 + 0.144584i 0 1.95819 0.406803i 1.00000i 0 −3.62721 −2.69597 + 0.855416i 0 0.144584 + 1.40680i
181.3 −0.264658 1.38923i 0 −1.85991 + 0.735342i 1.00000i 0 0.941367 1.51380 + 2.38923i 0 −1.38923 + 0.264658i
181.4 −0.264658 + 1.38923i 0 −1.85991 0.735342i 1.00000i 0 0.941367 1.51380 2.38923i 0 −1.38923 0.264658i
181.5 0.671462 1.24464i 0 −1.09828 1.67146i 1.00000i 0 4.68585 −2.81783 + 0.244644i 0 1.24464 + 0.671462i
181.6 0.671462 + 1.24464i 0 −1.09828 + 1.67146i 1.00000i 0 4.68585 −2.81783 0.244644i 0 1.24464 0.671462i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{7}^{3} - 2 T_{7}^{2} - 16 T_{7} + 16 \)
\( T_{11}^{6} + 64 T_{11}^{4} + 1088 T_{11}^{2} + 4096 \)
\( T_{17}^{3} + 6 T_{17}^{2} - 16 T_{17} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 3 T^{2} + 6 T^{3} + 6 T^{4} + 8 T^{5} + 8 T^{6} \)
$3$ 1
$5$ \( ( 1 + T^{2} )^{3} \)
$7$ \( ( 1 - 2 T + 5 T^{2} - 12 T^{3} + 35 T^{4} - 98 T^{5} + 343 T^{6} )^{2} \)
$11$ \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 10527 T^{8} - 29282 T^{10} + 1771561 T^{12} \)
$13$ \( 1 - 22 T^{2} + 407 T^{4} - 7284 T^{6} + 68783 T^{8} - 628342 T^{10} + 4826809 T^{12} \)
$17$ \( ( 1 + 6 T + 35 T^{2} + 172 T^{3} + 595 T^{4} + 1734 T^{5} + 4913 T^{6} )^{2} \)
$19$ \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 955567 T^{8} - 9643754 T^{10} + 47045881 T^{12} \)
$23$ \( ( 1 - 4 T + 57 T^{2} - 168 T^{3} + 1311 T^{4} - 2116 T^{5} + 12167 T^{6} )^{2} \)
$29$ \( ( 1 - 54 T^{2} + 841 T^{4} )^{3} \)
$31$ \( ( 1 + 6 T + 77 T^{2} + 308 T^{3} + 2387 T^{4} + 5766 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( 1 - 86 T^{2} + 6055 T^{4} - 248372 T^{6} + 8289295 T^{8} - 161177846 T^{10} + 2565726409 T^{12} \)
$41$ \( ( 1 - 10 T + 87 T^{2} - 588 T^{3} + 3567 T^{4} - 16810 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( 1 - 114 T^{2} + 8087 T^{4} - 416540 T^{6} + 14952863 T^{8} - 389743314 T^{10} + 6321363049 T^{12} \)
$47$ \( ( 1 + 4 T + 49 T^{2} - 120 T^{3} + 2303 T^{4} + 8836 T^{5} + 103823 T^{6} )^{2} \)
$53$ \( ( 1 - 102 T^{2} + 2809 T^{4} )^{3} \)
$59$ \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 118044191 T^{8} - 3320156914 T^{10} + 42180533641 T^{12} \)
$61$ \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 40034239 T^{8} - 1523042510 T^{10} + 51520374361 T^{12} \)
$67$ \( ( 1 - 118 T^{2} + 4489 T^{4} )^{3} \)
$71$ \( ( 1 - 4 T + 101 T^{2} - 632 T^{3} + 7171 T^{4} - 20164 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( ( 1 + 6 T + 73 T^{2} )^{6} \)
$79$ \( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 25043 T^{4} - 112338 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 - 274 T^{2} + 37415 T^{4} - 3513756 T^{6} + 257751935 T^{8} - 13003579954 T^{10} + 326940373369 T^{12} \)
$89$ \( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 23407 T^{4} - 110894 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( ( 1 - 18 T + 287 T^{2} - 3164 T^{3} + 27839 T^{4} - 169362 T^{5} + 912673 T^{6} )^{2} \)
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