# Properties

 Label 240.2.v.e Level $240$ Weight $2$ Character orbit 240.v 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.v (of order $$4$$, degree $$2$$, not 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 1 - \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( 1 - \zeta_{8}^{2} ) q^{7} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{11} + ( -2 - \zeta_{8} - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{15} + 2 \zeta_{8} q^{17} + 4 \zeta_{8}^{2} q^{19} + ( 2 - \zeta_{8} - \zeta_{8}^{3} ) q^{21} + 4 \zeta_{8}^{3} q^{23} + ( -4 + 3 \zeta_{8}^{2} ) q^{25} + ( 1 + 5 \zeta_{8} - \zeta_{8}^{2} ) q^{27} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{29} + 2 q^{31} + ( -1 - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{33} + ( -3 \zeta_{8} + \zeta_{8}^{3} ) q^{35} + ( 6 - 6 \zeta_{8}^{2} ) q^{37} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + ( -6 - 6 \zeta_{8}^{2} ) q^{43} + ( -2 + \zeta_{8} - 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{45} + 5 \zeta_{8}^{2} q^{49} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{51} + 4 \zeta_{8}^{3} q^{53} + ( -3 + \zeta_{8}^{2} ) q^{55} + ( -4 + 4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{57} + ( -7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{59} -6 q^{61} + ( 1 + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} + ( 4 - 4 \zeta_{8}^{2} ) q^{67} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{69} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{71} + ( -5 - 5 \zeta_{8}^{2} ) q^{73} + ( -7 + 3 \zeta_{8} - \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{75} -2 \zeta_{8} q^{77} + 6 \zeta_{8}^{2} q^{79} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} -12 \zeta_{8}^{3} q^{83} + ( 2 - 4 \zeta_{8}^{2} ) q^{85} + ( -5 - 10 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{87} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{89} + ( 2 + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{93} + ( 4 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{95} + ( 3 - 3 \zeta_{8}^{2} ) q^{97} + ( \zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{7} + O(q^{10})$$ $$4 q + 4 q^{3} + 4 q^{7} - 8 q^{15} + 8 q^{21} - 16 q^{25} + 4 q^{27} + 8 q^{31} - 4 q^{33} + 24 q^{37} - 24 q^{43} - 8 q^{45} + 8 q^{51} - 12 q^{55} - 16 q^{57} - 24 q^{61} + 4 q^{63} + 16 q^{67} - 20 q^{73} - 28 q^{75} + 28 q^{81} + 8 q^{85} - 20 q^{87} + 8 q^{93} + 12 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0 0.292893 1.70711i 0 0.707107 2.12132i 0 1.00000 + 1.00000i 0 −2.82843 1.00000i 0
17.2 0 1.70711 0.292893i 0 −0.707107 + 2.12132i 0 1.00000 + 1.00000i 0 2.82843 1.00000i 0
113.1 0 0.292893 + 1.70711i 0 0.707107 + 2.12132i 0 1.00000 1.00000i 0 −2.82843 + 1.00000i 0
113.2 0 1.70711 + 0.292893i 0 −0.707107 2.12132i 0 1.00000 1.00000i 0 2.82843 + 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.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.v.e 4
3.b odd 2 1 inner 240.2.v.e 4
4.b odd 2 1 30.2.e.a 4
5.b even 2 1 1200.2.v.b 4
5.c odd 4 1 inner 240.2.v.e 4
5.c odd 4 1 1200.2.v.b 4
8.b even 2 1 960.2.v.c 4
8.d odd 2 1 960.2.v.k 4
12.b even 2 1 30.2.e.a 4
15.d odd 2 1 1200.2.v.b 4
15.e even 4 1 inner 240.2.v.e 4
15.e even 4 1 1200.2.v.b 4
20.d odd 2 1 150.2.e.a 4
20.e even 4 1 30.2.e.a 4
20.e even 4 1 150.2.e.a 4
24.f even 2 1 960.2.v.k 4
24.h odd 2 1 960.2.v.c 4
36.f odd 6 2 810.2.m.f 8
36.h even 6 2 810.2.m.f 8
40.i odd 4 1 960.2.v.c 4
40.k even 4 1 960.2.v.k 4
60.h even 2 1 150.2.e.a 4
60.l odd 4 1 30.2.e.a 4
60.l odd 4 1 150.2.e.a 4
120.q odd 4 1 960.2.v.k 4
120.w even 4 1 960.2.v.c 4
180.v odd 12 2 810.2.m.f 8
180.x even 12 2 810.2.m.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.e.a 4 4.b odd 2 1
30.2.e.a 4 12.b even 2 1
30.2.e.a 4 20.e even 4 1
30.2.e.a 4 60.l odd 4 1
150.2.e.a 4 20.d odd 2 1
150.2.e.a 4 20.e even 4 1
150.2.e.a 4 60.h even 2 1
150.2.e.a 4 60.l odd 4 1
240.2.v.e 4 1.a even 1 1 trivial
240.2.v.e 4 3.b odd 2 1 inner
240.2.v.e 4 5.c odd 4 1 inner
240.2.v.e 4 15.e even 4 1 inner
810.2.m.f 8 36.f odd 6 2
810.2.m.f 8 36.h even 6 2
810.2.m.f 8 180.v odd 12 2
810.2.m.f 8 180.x even 12 2
960.2.v.c 4 8.b even 2 1
960.2.v.c 4 24.h odd 2 1
960.2.v.c 4 40.i odd 4 1
960.2.v.c 4 120.w even 4 1
960.2.v.k 4 8.d odd 2 1
960.2.v.k 4 24.f even 2 1
960.2.v.k 4 40.k even 4 1
960.2.v.k 4 120.q odd 4 1
1200.2.v.b 4 5.b even 2 1
1200.2.v.b 4 5.c odd 4 1
1200.2.v.b 4 15.d odd 2 1
1200.2.v.b 4 15.e even 4 1

## Hecke kernels

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

 $$T_{7}^{2} - 2 T_{7} + 2$$ $$T_{11}^{2} + 2$$ $$T_{17}^{4} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 12 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$5$ $$25 + 8 T^{2} + T^{4}$$
$7$ $$( 2 - 2 T + T^{2} )^{2}$$
$11$ $$( 2 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$16 + T^{4}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$256 + T^{4}$$
$29$ $$( -50 + T^{2} )^{2}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$( 72 - 12 T + T^{2} )^{2}$$
$41$ $$( 32 + T^{2} )^{2}$$
$43$ $$( 72 + 12 T + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$256 + T^{4}$$
$59$ $$( -98 + T^{2} )^{2}$$
$61$ $$( 6 + T )^{4}$$
$67$ $$( 32 - 8 T + T^{2} )^{2}$$
$71$ $$( 200 + T^{2} )^{2}$$
$73$ $$( 50 + 10 T + T^{2} )^{2}$$
$79$ $$( 36 + T^{2} )^{2}$$
$83$ $$20736 + T^{4}$$
$89$ $$( -8 + T^{2} )^{2}$$
$97$ $$( 18 - 6 T + T^{2} )^{2}$$