# Properties

 Label 240.6.f.a Level $240$ Weight $6$ Character orbit 240.f Analytic conductor $38.492$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 240.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$38.4921167551$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -9 i q^{3} + ( -55 + 10 i ) q^{5} + 4 i q^{7} -81 q^{9} +O(q^{10})$$ $$q -9 i q^{3} + ( -55 + 10 i ) q^{5} + 4 i q^{7} -81 q^{9} + 500 q^{11} + 288 i q^{13} + ( 90 + 495 i ) q^{15} + 1516 i q^{17} -1344 q^{19} + 36 q^{21} -4100 i q^{23} + ( 2925 - 1100 i ) q^{25} + 729 i q^{27} + 2646 q^{29} + 5612 q^{31} -4500 i q^{33} + ( -40 - 220 i ) q^{35} -7288 i q^{37} + 2592 q^{39} -18986 q^{41} -2404 i q^{43} + ( 4455 - 810 i ) q^{45} -8900 i q^{47} + 16791 q^{49} + 13644 q^{51} -39804 i q^{53} + ( -27500 + 5000 i ) q^{55} + 12096 i q^{57} -28300 q^{59} + 18290 q^{61} -324 i q^{63} + ( -2880 - 15840 i ) q^{65} -65956 i q^{67} -36900 q^{69} + 28800 q^{71} + 30808 i q^{73} + ( -9900 - 26325 i ) q^{75} + 2000 i q^{77} + 60228 q^{79} + 6561 q^{81} -2468 i q^{83} + ( -15160 - 83380 i ) q^{85} -23814 i q^{87} -22678 q^{89} -1152 q^{91} -50508 i q^{93} + ( 73920 - 13440 i ) q^{95} -36968 i q^{97} -40500 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 110q^{5} - 162q^{9} + O(q^{10})$$ $$2q - 110q^{5} - 162q^{9} + 1000q^{11} + 180q^{15} - 2688q^{19} + 72q^{21} + 5850q^{25} + 5292q^{29} + 11224q^{31} - 80q^{35} + 5184q^{39} - 37972q^{41} + 8910q^{45} + 33582q^{49} + 27288q^{51} - 55000q^{55} - 56600q^{59} + 36580q^{61} - 5760q^{65} - 73800q^{69} + 57600q^{71} - 19800q^{75} + 120456q^{79} + 13122q^{81} - 30320q^{85} - 45356q^{89} - 2304q^{91} + 147840q^{95} - 81000q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.00000i − 1.00000i
0 9.00000i 0 −55.0000 + 10.0000i 0 4.00000i 0 −81.0000 0
49.2 0 9.00000i 0 −55.0000 10.0000i 0 4.00000i 0 −81.0000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.6.f.a 2
3.b odd 2 1 720.6.f.g 2
4.b odd 2 1 30.6.c.a 2
5.b even 2 1 inner 240.6.f.a 2
12.b even 2 1 90.6.c.b 2
15.d odd 2 1 720.6.f.g 2
20.d odd 2 1 30.6.c.a 2
20.e even 4 1 150.6.a.f 1
20.e even 4 1 150.6.a.j 1
60.h even 2 1 90.6.c.b 2
60.l odd 4 1 450.6.a.f 1
60.l odd 4 1 450.6.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 4.b odd 2 1
30.6.c.a 2 20.d odd 2 1
90.6.c.b 2 12.b even 2 1
90.6.c.b 2 60.h even 2 1
150.6.a.f 1 20.e even 4 1
150.6.a.j 1 20.e even 4 1
240.6.f.a 2 1.a even 1 1 trivial
240.6.f.a 2 5.b even 2 1 inner
450.6.a.f 1 60.l odd 4 1
450.6.a.s 1 60.l odd 4 1
720.6.f.g 2 3.b odd 2 1
720.6.f.g 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 16$$ acting on $$S_{6}^{\mathrm{new}}(240, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$81 + T^{2}$$
$5$ $$3125 + 110 T + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( -500 + T )^{2}$$
$13$ $$82944 + T^{2}$$
$17$ $$2298256 + T^{2}$$
$19$ $$( 1344 + T )^{2}$$
$23$ $$16810000 + T^{2}$$
$29$ $$( -2646 + T )^{2}$$
$31$ $$( -5612 + T )^{2}$$
$37$ $$53114944 + T^{2}$$
$41$ $$( 18986 + T )^{2}$$
$43$ $$5779216 + T^{2}$$
$47$ $$79210000 + T^{2}$$
$53$ $$1584358416 + T^{2}$$
$59$ $$( 28300 + T )^{2}$$
$61$ $$( -18290 + T )^{2}$$
$67$ $$4350193936 + T^{2}$$
$71$ $$( -28800 + T )^{2}$$
$73$ $$949132864 + T^{2}$$
$79$ $$( -60228 + T )^{2}$$
$83$ $$6091024 + T^{2}$$
$89$ $$( 22678 + T )^{2}$$
$97$ $$1366633024 + T^{2}$$