# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.2580992.1 Defining polynomial: $$x^{6} - 2 x^{5} + x^{4} + 2 x^{2} - 8 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( \beta_{2} + \beta_{5} ) q^{4} - q^{5} + \beta_{2} q^{7} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{4} q^{2} + ( \beta_{2} + \beta_{5} ) q^{4} - q^{5} + \beta_{2} q^{7} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{8} -\beta_{4} q^{10} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{13} + \beta_{3} q^{14} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{16} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{17} + ( 3 - \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{19} + ( -\beta_{2} - \beta_{5} ) q^{20} + ( -4 + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{22} + ( -1 + \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{23} + q^{25} + ( -4 + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{26} + ( -1 + \beta_{1} ) q^{28} + ( 2 + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{31} + ( -4 + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{32} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{34} -\beta_{2} q^{35} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{37} + ( 4 + 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{38} + ( -\beta_{2} - \beta_{3} + \beta_{5} ) q^{40} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{41} + ( -2 - 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{43} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{44} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{46} + ( -1 - \beta_{1} - 3 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{47} + 5 q^{49} + \beta_{4} q^{50} + ( -1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{52} + ( -2 + 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{55} + ( -3 - \beta_{1} - 2 \beta_{4} ) q^{56} + ( -2 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} ) q^{58} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{59} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{61} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{62} + ( 1 - \beta_{1} - 3 \beta_{2} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{64} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{65} + ( -2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{67} + ( 8 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{68} -\beta_{3} q^{70} + ( 8 - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{71} + ( -2 + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{73} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{74} + ( 6 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{76} + ( 2 + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{77} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{80} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{82} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} ) q^{83} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{85} + ( 6 + 2 \beta_{1} + 4 \beta_{5} ) q^{86} + ( 1 + 3 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{88} + ( -3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{89} + ( -2 + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{91} + ( 6 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{92} + ( 6 - 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{5} ) q^{94} + ( -3 + \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{95} + ( 2 \beta_{1} + 4 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} ) q^{97} + 5 \beta_{4} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{2} + 2q^{4} - 6q^{5} - 2q^{8} + O(q^{10})$$ $$6q - 2q^{2} + 2q^{4} - 6q^{5} - 2q^{8} + 2q^{10} - 6q^{16} + 16q^{19} - 2q^{20} - 20q^{22} - 4q^{23} + 6q^{25} - 20q^{26} - 8q^{28} + 12q^{29} - 22q^{32} - 4q^{34} + 20q^{38} + 2q^{40} - 16q^{43} + 12q^{44} - 8q^{46} - 8q^{47} + 30q^{49} - 2q^{50} - 4q^{52} - 8q^{53} - 12q^{56} - 20q^{58} - 4q^{62} + 14q^{64} + 44q^{68} + 48q^{71} - 12q^{73} - 4q^{74} - 4q^{76} + 12q^{77} + 6q^{80} + 16q^{82} + 40q^{86} - 8q^{88} - 12q^{91} + 32q^{92} + 44q^{94} - 16q^{95} + 4q^{97} - 10q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + \nu^{3} + 2 \nu^{2} + 6 \nu - 8$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + \nu^{3} - 2 \nu^{2} + 2 \nu - 4$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} - \nu^{3} - 2 \nu^{2} + 2 \nu + 4$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} + \nu^{3} + 2 \nu - 8$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{5} + 2 \nu^{4} + \nu^{3} - 6 \nu + 16$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} - 2 \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_{1} - 3$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_{1} + 7$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.38078 − 0.305697i 1.38078 + 0.305697i 0.681664 − 1.23909i 0.681664 + 1.23909i −1.06244 − 0.933389i −1.06244 + 0.933389i
−1.38078 0.305697i 0 1.81310 + 0.844199i −1.00000 0 1.41421i −2.24542 1.71991i 0 1.38078 + 0.305697i
251.2 −1.38078 + 0.305697i 0 1.81310 0.844199i −1.00000 0 1.41421i −2.24542 + 1.71991i 0 1.38078 0.305697i
251.3 −0.681664 1.23909i 0 −1.07067 + 1.68928i −1.00000 0 1.41421i 2.82300 + 0.175128i 0 0.681664 + 1.23909i
251.4 −0.681664 + 1.23909i 0 −1.07067 1.68928i −1.00000 0 1.41421i 2.82300 0.175128i 0 0.681664 1.23909i
251.5 1.06244 0.933389i 0 0.257569 1.98335i −1.00000 0 1.41421i −1.57758 2.34760i 0 −1.06244 + 0.933389i
251.6 1.06244 + 0.933389i 0 0.257569 + 1.98335i −1.00000 0 1.41421i −1.57758 + 2.34760i 0 −1.06244 0.933389i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.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.b.c 6
3.b odd 2 1 360.2.b.d yes 6
4.b odd 2 1 1440.2.b.c 6
5.b even 2 1 1800.2.b.e 6
5.c odd 4 2 1800.2.m.d 12
8.b even 2 1 1440.2.b.d 6
8.d odd 2 1 360.2.b.d yes 6
12.b even 2 1 1440.2.b.d 6
15.d odd 2 1 1800.2.b.d 6
15.e even 4 2 1800.2.m.e 12
20.d odd 2 1 7200.2.b.d 6
20.e even 4 2 7200.2.m.d 12
24.f even 2 1 inner 360.2.b.c 6
24.h odd 2 1 1440.2.b.c 6
40.e odd 2 1 1800.2.b.d 6
40.f even 2 1 7200.2.b.e 6
40.i odd 4 2 7200.2.m.e 12
40.k even 4 2 1800.2.m.e 12
60.h even 2 1 7200.2.b.e 6
60.l odd 4 2 7200.2.m.e 12
120.i odd 2 1 7200.2.b.d 6
120.m even 2 1 1800.2.b.e 6
120.q odd 4 2 1800.2.m.d 12
120.w even 4 2 7200.2.m.d 12

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

## Hecke kernels

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

 $$T_{7}^{2} + 2$$ $$T_{23}^{3} + 2 T_{23}^{2} - 32 T_{23} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$8 + 8 T + 2 T^{2} + T^{4} + 2 T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$( 1 + T )^{6}$$
$7$ $$( 2 + T^{2} )^{3}$$
$11$ $$8 + 220 T^{2} + 46 T^{4} + T^{6}$$
$13$ $$2312 + 604 T^{2} + 46 T^{4} + T^{6}$$
$17$ $$15488 + 2000 T^{2} + 80 T^{4} + T^{6}$$
$19$ $$( 16 - 4 T - 8 T^{2} + T^{3} )^{2}$$
$23$ $$( -32 - 32 T + 2 T^{2} + T^{3} )^{2}$$
$29$ $$( 8 - 28 T - 6 T^{2} + T^{3} )^{2}$$
$31$ $$15488 + 2000 T^{2} + 80 T^{4} + T^{6}$$
$37$ $$5000 + 1244 T^{2} + 78 T^{4} + T^{6}$$
$41$ $$49928 + 4748 T^{2} + 134 T^{4} + T^{6}$$
$43$ $$( 64 - 56 T + 8 T^{2} + T^{3} )^{2}$$
$47$ $$( -352 - 92 T + 4 T^{2} + T^{3} )^{2}$$
$53$ $$( -352 - 72 T + 4 T^{2} + T^{3} )^{2}$$
$59$ $$8 + 92 T^{2} + 110 T^{4} + T^{6}$$
$61$ $$512 + 3520 T^{2} + 184 T^{4} + T^{6}$$
$67$ $$( 64 - 40 T + T^{3} )^{2}$$
$71$ $$( -128 + 152 T - 24 T^{2} + T^{3} )^{2}$$
$73$ $$( -824 - 148 T + 6 T^{2} + T^{3} )^{2}$$
$79$ $$881792 + 31184 T^{2} + 336 T^{4} + T^{6}$$
$83$ $$320000 + 19904 T^{2} + 312 T^{4} + T^{6}$$
$89$ $$200 + 1356 T^{2} + 118 T^{4} + T^{6}$$
$97$ $$( 1208 - 204 T - 2 T^{2} + T^{3} )^{2}$$