# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 240.w (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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{3} + ( -2 + \zeta_{8}^{2} ) q^{5} + 4 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{8} q^{3} + ( -2 + \zeta_{8}^{2} ) q^{5} + 4 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} + ( 3 - 3 \zeta_{8}^{2} ) q^{13} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{15} + ( -1 - \zeta_{8}^{2} ) q^{17} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{19} -4 q^{21} -4 \zeta_{8} q^{23} + ( 3 - 4 \zeta_{8}^{2} ) q^{25} + \zeta_{8}^{3} q^{27} + 4 \zeta_{8}^{2} q^{29} + ( -4 + 4 \zeta_{8}^{2} ) q^{33} + ( -4 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{35} + ( 5 + 5 \zeta_{8}^{2} ) q^{37} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{39} -4 \zeta_{8} q^{43} + ( -1 - 2 \zeta_{8}^{2} ) q^{45} + 4 \zeta_{8}^{3} q^{47} -9 \zeta_{8}^{2} q^{49} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{51} + ( 1 - \zeta_{8}^{2} ) q^{53} + ( -12 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{55} + ( 4 + 4 \zeta_{8}^{2} ) q^{57} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{59} + 4 q^{61} -4 \zeta_{8} q^{63} + ( -3 + 9 \zeta_{8}^{2} ) q^{65} -4 \zeta_{8}^{3} q^{67} -4 \zeta_{8}^{2} q^{69} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + ( -3 + 3 \zeta_{8}^{2} ) q^{73} + ( 3 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{75} + ( -16 - 16 \zeta_{8}^{2} ) q^{77} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{79} - q^{81} -4 \zeta_{8} q^{83} + ( 3 + \zeta_{8}^{2} ) q^{85} + 4 \zeta_{8}^{3} q^{87} + 8 \zeta_{8}^{2} q^{89} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{91} + ( -4 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{95} + ( -3 - 3 \zeta_{8}^{2} ) q^{97} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{5} + O(q^{10})$$ $$4 q - 8 q^{5} + 12 q^{13} - 4 q^{17} - 16 q^{21} + 12 q^{25} - 16 q^{33} + 20 q^{37} - 4 q^{45} + 4 q^{53} + 16 q^{57} + 16 q^{61} - 12 q^{65} - 12 q^{73} - 64 q^{77} - 4 q^{81} + 12 q^{85} - 12 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −0.707107 0.707107i 0 −2.00000 + 1.00000i 0 2.82843 2.82843i 0 1.00000i 0
127.2 0 0.707107 + 0.707107i 0 −2.00000 + 1.00000i 0 −2.82843 + 2.82843i 0 1.00000i 0
223.1 0 −0.707107 + 0.707107i 0 −2.00000 1.00000i 0 2.82843 + 2.82843i 0 1.00000i 0
223.2 0 0.707107 0.707107i 0 −2.00000 1.00000i 0 −2.82843 2.82843i 0 1.00000i 0
 $$n$$: 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
4.b odd 2 1 inner
5.c odd 4 1 inner
20.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.w.a 4
3.b odd 2 1 720.2.x.e 4
4.b odd 2 1 inner 240.2.w.a 4
5.b even 2 1 1200.2.w.a 4
5.c odd 4 1 inner 240.2.w.a 4
5.c odd 4 1 1200.2.w.a 4
8.b even 2 1 960.2.w.c 4
8.d odd 2 1 960.2.w.c 4
12.b even 2 1 720.2.x.e 4
15.d odd 2 1 3600.2.x.d 4
15.e even 4 1 720.2.x.e 4
15.e even 4 1 3600.2.x.d 4
20.d odd 2 1 1200.2.w.a 4
20.e even 4 1 inner 240.2.w.a 4
20.e even 4 1 1200.2.w.a 4
40.i odd 4 1 960.2.w.c 4
40.k even 4 1 960.2.w.c 4
60.h even 2 1 3600.2.x.d 4
60.l odd 4 1 720.2.x.e 4
60.l odd 4 1 3600.2.x.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.w.a 4 1.a even 1 1 trivial
240.2.w.a 4 4.b odd 2 1 inner
240.2.w.a 4 5.c odd 4 1 inner
240.2.w.a 4 20.e even 4 1 inner
720.2.x.e 4 3.b odd 2 1
720.2.x.e 4 12.b even 2 1
720.2.x.e 4 15.e even 4 1
720.2.x.e 4 60.l odd 4 1
960.2.w.c 4 8.b even 2 1
960.2.w.c 4 8.d odd 2 1
960.2.w.c 4 40.i odd 4 1
960.2.w.c 4 40.k even 4 1
1200.2.w.a 4 5.b even 2 1
1200.2.w.a 4 5.c odd 4 1
1200.2.w.a 4 20.d odd 2 1
1200.2.w.a 4 20.e even 4 1
3600.2.x.d 4 15.d odd 2 1
3600.2.x.d 4 15.e even 4 1
3600.2.x.d 4 60.h even 2 1
3600.2.x.d 4 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 256$$ acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + T^{4}$$
$5$ $$( 5 + 4 T + T^{2} )^{2}$$
$7$ $$256 + T^{4}$$
$11$ $$( 32 + T^{2} )^{2}$$
$13$ $$( 18 - 6 T + T^{2} )^{2}$$
$17$ $$( 2 + 2 T + T^{2} )^{2}$$
$19$ $$( -32 + T^{2} )^{2}$$
$23$ $$256 + T^{4}$$
$29$ $$( 16 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 50 - 10 T + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$256 + T^{4}$$
$47$ $$256 + T^{4}$$
$53$ $$( 2 - 2 T + T^{2} )^{2}$$
$59$ $$( -128 + T^{2} )^{2}$$
$61$ $$( -4 + T )^{4}$$
$67$ $$256 + T^{4}$$
$71$ $$( 32 + T^{2} )^{2}$$
$73$ $$( 18 + 6 T + T^{2} )^{2}$$
$79$ $$( -32 + T^{2} )^{2}$$
$83$ $$256 + T^{4}$$
$89$ $$( 64 + T^{2} )^{2}$$
$97$ $$( 18 + 6 T + T^{2} )^{2}$$