# Properties

 Label 240.2.y.a Level $240$ Weight $2$ Character orbit 240.y Analytic conductor $1.916$ Analytic rank $1$ Dimension $2$ 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.y (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$-1$$ $$i$$ $$1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 − 1.00000i 1.00000i
−1.00000 + 1.00000i −1.00000 2.00000i −1.00000 + 2.00000i 1.00000 1.00000i −1.00000 + 1.00000i 2.00000 + 2.00000i 1.00000 −1.00000 3.00000i
187.1 −1.00000 1.00000i −1.00000 2.00000i −1.00000 2.00000i 1.00000 + 1.00000i −1.00000 1.00000i 2.00000 2.00000i 1.00000 −1.00000 + 3.00000i
 $$n$$: 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
80.s 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.y.a 2
3.b odd 2 1 720.2.z.b 2
4.b odd 2 1 960.2.y.b 2
5.c odd 4 1 240.2.bc.b yes 2
8.b even 2 1 1920.2.y.f 2
8.d odd 2 1 1920.2.y.b 2
15.e even 4 1 720.2.bd.c 2
16.e even 4 1 960.2.bc.c 2
16.e even 4 1 1920.2.bc.d 2
16.f odd 4 1 240.2.bc.b yes 2
16.f odd 4 1 1920.2.bc.c 2
20.e even 4 1 960.2.bc.c 2
40.i odd 4 1 1920.2.bc.c 2
40.k even 4 1 1920.2.bc.d 2
48.k even 4 1 720.2.bd.c 2
80.i odd 4 1 960.2.y.b 2
80.j even 4 1 1920.2.y.f 2
80.s even 4 1 inner 240.2.y.a 2
80.t odd 4 1 1920.2.y.b 2
240.z odd 4 1 720.2.z.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.y.a 2 1.a even 1 1 trivial
240.2.y.a 2 80.s even 4 1 inner
240.2.bc.b yes 2 5.c odd 4 1
240.2.bc.b yes 2 16.f odd 4 1
720.2.z.b 2 3.b odd 2 1
720.2.z.b 2 240.z odd 4 1
720.2.bd.c 2 15.e even 4 1
720.2.bd.c 2 48.k even 4 1
960.2.y.b 2 4.b odd 2 1
960.2.y.b 2 80.i odd 4 1
960.2.bc.c 2 16.e even 4 1
960.2.bc.c 2 20.e even 4 1
1920.2.y.b 2 8.d odd 2 1
1920.2.y.b 2 80.t odd 4 1
1920.2.y.f 2 8.b even 2 1
1920.2.y.f 2 80.j even 4 1
1920.2.bc.c 2 16.f odd 4 1
1920.2.bc.c 2 40.i odd 4 1
1920.2.bc.d 2 16.e even 4 1
1920.2.bc.d 2 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$:

 $$T_{7}^{2} + 2 T_{7} + 2$$ $$T_{11}^{2} + 6 T_{11} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 2 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$2 + 2 T + T^{2}$$
$11$ $$18 + 6 T + T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$50 + 10 T + T^{2}$$
$19$ $$50 + 10 T + T^{2}$$
$23$ $$2 + 2 T + T^{2}$$
$29$ $$50 - 10 T + T^{2}$$
$31$ $$4 + T^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$50 + 10 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$50 + 10 T + T^{2}$$
$61$ $$2 + 2 T + T^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$50 - 10 T + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$( 2 + T )^{2}$$
$97$ $$2 - 2 T + T^{2}$$