# Properties

 Label 240.2.w.b Level $240$ Weight $2$ Character orbit 240.w Analytic conductor $1.916$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} - \zeta_{24}^{5} ) q^{3} + ( 1 + \zeta_{24} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( 1 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{7} -\zeta_{24}^{6} q^{9} +O(q^{10})$$ $$q + ( \zeta_{24} - \zeta_{24}^{5} ) q^{3} + ( 1 + \zeta_{24} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{5} + ( 1 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{7} -\zeta_{24}^{6} q^{9} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{11} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{13} + ( \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{15} + ( 2 - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{17} + ( -2 \zeta_{24} - 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{19} + ( 2 - \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{21} + ( 2 + 2 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{23} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{25} -\zeta_{24}^{3} q^{27} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{29} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{31} + ( -1 - \zeta_{24}^{6} ) q^{33} + ( -2 + 5 \zeta_{24} - 4 \zeta_{24}^{2} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 5 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{35} + ( -4 - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{37} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{39} + ( -6 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{41} + ( -2 - 4 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{43} + ( -1 + \zeta_{24}^{3} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{45} + ( -4 + 8 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{47} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{49} + ( -2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{51} + ( -2 - 2 \zeta_{24}^{6} ) q^{53} + ( 1 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{55} + ( -2 - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{57} + ( -7 \zeta_{24} - 7 \zeta_{24}^{3} + 7 \zeta_{24}^{5} ) q^{59} + ( -2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{61} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{63} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{65} + ( 2 - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{67} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{69} + ( 4 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{71} + ( 3 + 3 \zeta_{24}^{6} ) q^{73} + ( 2 + 4 \zeta_{24}^{2} + \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{75} + ( 2 - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{77} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{79} - q^{81} + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{83} + ( -6 + 4 \zeta_{24} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{85} + ( 3 - 6 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{87} -2 \zeta_{24}^{6} q^{89} + ( -4 + 6 \zeta_{24} - 6 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 6 \zeta_{24}^{5} ) q^{91} + ( 6 + 6 \zeta_{24}^{6} ) q^{93} + ( -4 \zeta_{24} - 8 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{95} + ( 3 - 3 \zeta_{24}^{6} ) q^{97} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{5} + O(q^{10})$$ $$8 q + 8 q^{5} + 16 q^{17} + 16 q^{21} - 8 q^{33} - 32 q^{37} - 48 q^{41} - 8 q^{45} - 16 q^{53} - 16 q^{57} - 16 q^{61} + 24 q^{73} + 16 q^{77} - 8 q^{81} - 48 q^{85} + 48 q^{93} + 24 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{24}^{3}$$ $$1$$ $$1$$

## 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
 −0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 + 0.258819i
0 −0.707107 0.707107i 0 −0.224745 + 2.22474i 0 −3.14626 + 3.14626i 0 1.00000i 0
127.2 0 −0.707107 0.707107i 0 2.22474 0.224745i 0 0.317837 0.317837i 0 1.00000i 0
127.3 0 0.707107 + 0.707107i 0 −0.224745 + 2.22474i 0 3.14626 3.14626i 0 1.00000i 0
127.4 0 0.707107 + 0.707107i 0 2.22474 0.224745i 0 −0.317837 + 0.317837i 0 1.00000i 0
223.1 0 −0.707107 + 0.707107i 0 −0.224745 2.22474i 0 −3.14626 3.14626i 0 1.00000i 0
223.2 0 −0.707107 + 0.707107i 0 2.22474 + 0.224745i 0 0.317837 + 0.317837i 0 1.00000i 0
223.3 0 0.707107 0.707107i 0 −0.224745 2.22474i 0 3.14626 + 3.14626i 0 1.00000i 0
223.4 0 0.707107 0.707107i 0 2.22474 + 0.224745i 0 −0.317837 0.317837i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.4 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.b 8
3.b odd 2 1 720.2.x.f 8
4.b odd 2 1 inner 240.2.w.b 8
5.b even 2 1 1200.2.w.b 8
5.c odd 4 1 inner 240.2.w.b 8
5.c odd 4 1 1200.2.w.b 8
8.b even 2 1 960.2.w.d 8
8.d odd 2 1 960.2.w.d 8
12.b even 2 1 720.2.x.f 8
15.d odd 2 1 3600.2.x.n 8
15.e even 4 1 720.2.x.f 8
15.e even 4 1 3600.2.x.n 8
20.d odd 2 1 1200.2.w.b 8
20.e even 4 1 inner 240.2.w.b 8
20.e even 4 1 1200.2.w.b 8
40.i odd 4 1 960.2.w.d 8
40.k even 4 1 960.2.w.d 8
60.h even 2 1 3600.2.x.n 8
60.l odd 4 1 720.2.x.f 8
60.l odd 4 1 3600.2.x.n 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 392 T_{7}^{4} + 16$$ acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 + T^{4} )^{2}$$
$5$ $$( 25 - 20 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$7$ $$16 + 392 T^{4} + T^{8}$$
$11$ $$( 2 + T^{2} )^{4}$$
$13$ $$( 144 + T^{4} )^{2}$$
$17$ $$( 16 + 32 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$19$ $$( 16 - 40 T^{2} + T^{4} )^{2}$$
$23$ $$160000 + 2336 T^{4} + T^{8}$$
$29$ $$( 2500 + 116 T^{2} + T^{4} )^{2}$$
$31$ $$( 72 + T^{2} )^{4}$$
$37$ $$( 400 + 320 T + 128 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$41$ $$( 12 + 12 T + T^{2} )^{4}$$
$43$ $$4096 + 6272 T^{4} + T^{8}$$
$47$ $$71639296 + 23072 T^{4} + T^{8}$$
$53$ $$( 8 + 4 T + T^{2} )^{4}$$
$59$ $$( -98 + T^{2} )^{4}$$
$61$ $$( -20 + 4 T + T^{2} )^{4}$$
$67$ $$4096 + 6272 T^{4} + T^{8}$$
$71$ $$( 256 + 160 T^{2} + T^{4} )^{2}$$
$73$ $$( 18 - 6 T + T^{2} )^{4}$$
$79$ $$( 1600 - 112 T^{2} + T^{4} )^{2}$$
$83$ $$( 16 + T^{4} )^{2}$$
$89$ $$( 4 + T^{2} )^{4}$$
$97$ $$( 18 - 6 T + T^{2} )^{4}$$