# Properties

 Label 240.2.s.a Level $240$ Weight $2$ Character orbit 240.s Analytic conductor $1.916$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 240.s (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{2} + \zeta_{8}^{3} q^{3} + 2 q^{4} + \zeta_{8} q^{5} + ( 1 - \zeta_{8}^{2} ) q^{6} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} -\zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{2} + \zeta_{8}^{3} q^{3} + 2 q^{4} + \zeta_{8} q^{5} + ( 1 - \zeta_{8}^{2} ) q^{6} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} -\zeta_{8}^{2} q^{9} + ( -1 - \zeta_{8}^{2} ) q^{10} + 2 \zeta_{8} q^{11} + 2 \zeta_{8}^{3} q^{12} + ( -2 + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{13} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{14} - q^{15} + 4 q^{16} + ( 4 + \zeta_{8} - \zeta_{8}^{3} ) q^{17} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{18} + ( 1 - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{19} + 2 \zeta_{8} q^{20} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{21} + ( -2 - 2 \zeta_{8}^{2} ) q^{22} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{23} + ( 2 - 2 \zeta_{8}^{2} ) q^{24} + \zeta_{8}^{2} q^{25} + ( -2 + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{26} + \zeta_{8} q^{27} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{28} + ( -2 + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{29} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{30} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{31} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} -2 q^{33} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{34} + ( 2 - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{35} -2 \zeta_{8}^{2} q^{36} + ( 6 - 2 \zeta_{8} + 6 \zeta_{8}^{2} ) q^{37} + ( -4 + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{38} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{39} + ( -2 - 2 \zeta_{8}^{2} ) q^{40} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{41} + ( -2 - 4 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{42} + ( -2 + 8 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{43} + 4 \zeta_{8} q^{44} -\zeta_{8}^{3} q^{45} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{46} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{47} + 4 \zeta_{8}^{3} q^{48} + ( -5 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{49} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{50} + ( -1 + \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{51} + ( -4 + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{52} + ( -4 - 4 \zeta_{8}^{2} ) q^{53} + ( -1 - \zeta_{8}^{2} ) q^{54} + 2 \zeta_{8}^{2} q^{55} + ( 4 \zeta_{8} + 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{56} + ( \zeta_{8} + 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{57} + ( -2 + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{58} + ( -6 - 2 \zeta_{8} - 6 \zeta_{8}^{2} ) q^{59} -2 q^{60} + ( 1 - \zeta_{8}^{2} - 12 \zeta_{8}^{3} ) q^{61} + ( -8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{62} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{63} + 8 q^{64} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{65} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{66} + ( -2 + 2 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{67} + ( 8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{68} + ( 1 + 4 \zeta_{8} + \zeta_{8}^{2} ) q^{69} + ( -2 + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{70} -8 \zeta_{8}^{2} q^{71} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{72} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{73} + ( 2 - 12 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{74} -\zeta_{8} q^{75} + ( 2 - 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{76} + ( 4 - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{77} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{78} + ( -8 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{79} + 4 \zeta_{8} q^{80} - q^{81} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{82} + ( 4 - 4 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{83} + ( 4 + 4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{84} + ( 1 + 4 \zeta_{8} + \zeta_{8}^{2} ) q^{85} + ( -8 + 4 \zeta_{8} - 8 \zeta_{8}^{2} ) q^{86} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{87} + ( -4 - 4 \zeta_{8}^{2} ) q^{88} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{89} + ( -1 + \zeta_{8}^{2} ) q^{90} + 4 \zeta_{8} q^{91} + ( -2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{92} + ( -4 + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{93} + 10 q^{94} + ( 4 + \zeta_{8} - \zeta_{8}^{3} ) q^{95} + ( 4 - 4 \zeta_{8}^{2} ) q^{96} + ( -2 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{97} + ( 16 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{98} -2 \zeta_{8}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} + 4 q^{6} + O(q^{10})$$ $$4 q + 8 q^{4} + 4 q^{6} - 4 q^{10} - 8 q^{13} - 4 q^{15} + 16 q^{16} + 16 q^{17} + 4 q^{19} + 8 q^{21} - 8 q^{22} + 8 q^{24} - 8 q^{26} - 8 q^{29} - 8 q^{31} - 8 q^{33} - 8 q^{34} + 8 q^{35} + 24 q^{37} - 16 q^{38} - 8 q^{40} - 8 q^{42} - 8 q^{43} - 20 q^{49} - 4 q^{51} - 16 q^{52} - 16 q^{53} - 4 q^{54} - 8 q^{58} - 24 q^{59} - 8 q^{60} + 4 q^{61} - 32 q^{62} - 8 q^{63} + 32 q^{64} + 8 q^{65} - 8 q^{67} + 32 q^{68} + 4 q^{69} - 8 q^{70} + 8 q^{74} + 8 q^{76} + 16 q^{77} - 32 q^{79} - 4 q^{81} + 16 q^{83} + 16 q^{84} + 4 q^{85} - 32 q^{86} - 16 q^{88} - 4 q^{90} - 16 q^{93} + 40 q^{94} + 16 q^{95} + 16 q^{96} - 8 q^{97} + 64 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
−1.41421 −0.707107 0.707107i 2.00000 0.707107 0.707107i 1.00000 + 1.00000i 4.82843i −2.82843 1.00000i −1.00000 + 1.00000i
61.2 1.41421 0.707107 + 0.707107i 2.00000 −0.707107 + 0.707107i 1.00000 + 1.00000i 0.828427i 2.82843 1.00000i −1.00000 + 1.00000i
181.1 −1.41421 −0.707107 + 0.707107i 2.00000 0.707107 + 0.707107i 1.00000 1.00000i 4.82843i −2.82843 1.00000i −1.00000 1.00000i
181.2 1.41421 0.707107 0.707107i 2.00000 −0.707107 0.707107i 1.00000 1.00000i 0.828427i 2.82843 1.00000i −1.00000 1.00000i
 $$n$$: 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
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.s.a 4
3.b odd 2 1 720.2.t.a 4
4.b odd 2 1 960.2.s.a 4
8.b even 2 1 1920.2.s.a 4
8.d odd 2 1 1920.2.s.b 4
12.b even 2 1 2880.2.t.a 4
16.e even 4 1 inner 240.2.s.a 4
16.e even 4 1 1920.2.s.a 4
16.f odd 4 1 960.2.s.a 4
16.f odd 4 1 1920.2.s.b 4
48.i odd 4 1 720.2.t.a 4
48.k even 4 1 2880.2.t.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.s.a 4 1.a even 1 1 trivial
240.2.s.a 4 16.e even 4 1 inner
720.2.t.a 4 3.b odd 2 1
720.2.t.a 4 48.i odd 4 1
960.2.s.a 4 4.b odd 2 1
960.2.s.a 4 16.f odd 4 1
1920.2.s.a 4 8.b even 2 1
1920.2.s.a 4 16.e even 4 1
1920.2.s.b 4 8.d odd 2 1
1920.2.s.b 4 16.f odd 4 1
2880.2.t.a 4 12.b even 2 1
2880.2.t.a 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 24 T_{7}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$1 + T^{4}$$
$5$ $$1 + T^{4}$$
$7$ $$16 + 24 T^{2} + T^{4}$$
$11$ $$16 + T^{4}$$
$13$ $$16 + 32 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$17$ $$( 14 - 8 T + T^{2} )^{2}$$
$19$ $$196 + 56 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$23$ $$196 + 36 T^{2} + T^{4}$$
$29$ $$16 + 32 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$31$ $$( -28 + 4 T + T^{2} )^{2}$$
$37$ $$4624 - 1632 T + 288 T^{2} - 24 T^{3} + T^{4}$$
$41$ $$16 + 24 T^{2} + T^{4}$$
$43$ $$3136 - 448 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$( -50 + T^{2} )^{2}$$
$53$ $$( 32 + 8 T + T^{2} )^{2}$$
$59$ $$4624 + 1632 T + 288 T^{2} + 24 T^{3} + T^{4}$$
$61$ $$20164 + 568 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$67$ $$3136 - 448 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$71$ $$( 64 + T^{2} )^{2}$$
$73$ $$784 + 88 T^{2} + T^{4}$$
$79$ $$( 32 + 16 T + T^{2} )^{2}$$
$83$ $$4624 + 1088 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$89$ $$784 + 72 T^{2} + T^{4}$$
$97$ $$( -124 + 4 T + T^{2} )^{2}$$