# Properties

 Label 240.2.y.b Level $240$ Weight $2$ Character orbit 240.y Analytic conductor $1.916$ Analytic rank $0$ 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: $$0$$ 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.

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

## 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 3.00000 3.00000i 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 3.00000 + 3.00000i 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.b 2
3.b odd 2 1 720.2.z.c 2
4.b odd 2 1 960.2.y.a 2
5.c odd 4 1 240.2.bc.a yes 2
8.b even 2 1 1920.2.y.c 2
8.d odd 2 1 1920.2.y.e 2
15.e even 4 1 720.2.bd.d 2
16.e even 4 1 960.2.bc.a 2
16.e even 4 1 1920.2.bc.b 2
16.f odd 4 1 240.2.bc.a yes 2
16.f odd 4 1 1920.2.bc.e 2
20.e even 4 1 960.2.bc.a 2
40.i odd 4 1 1920.2.bc.e 2
40.k even 4 1 1920.2.bc.b 2
48.k even 4 1 720.2.bd.d 2
80.i odd 4 1 960.2.y.a 2
80.j even 4 1 1920.2.y.c 2
80.s even 4 1 inner 240.2.y.b 2
80.t odd 4 1 1920.2.y.e 2
240.z odd 4 1 720.2.z.c 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + 2 T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$5 + 2 T + T^{2}$$
$7$ $$18 - 6 T + T^{2}$$
$11$ $$2 - 2 T + T^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$18 - 6 T + T^{2}$$
$19$ $$18 - 6 T + T^{2}$$
$23$ $$2 + 2 T + T^{2}$$
$29$ $$18 + 6 T + T^{2}$$
$31$ $$100 + T^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$50 + 10 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$98 - 14 T + T^{2}$$
$61$ $$162 + 18 T + T^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$50 - 10 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$98 + 14 T + T^{2}$$