# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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