# Properties

 Label 240.3.j.b Level $240$ Weight $3$ Character orbit 240.j Analytic conductor $6.540$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 240.j (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53952634465$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - 2 x^{3} - x^{2} + 2 x + 22$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{5} + 2 \beta_{1} q^{7} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{5} + 2 \beta_{1} q^{7} + 3 q^{9} -2 \beta_{3} q^{11} -2 \beta_{2} q^{13} + ( -2 \beta_{1} - \beta_{3} ) q^{15} + 2 \beta_{2} q^{17} -4 \beta_{3} q^{19} + 6 q^{21} + 16 \beta_{1} q^{23} + ( -17 - 4 \beta_{2} ) q^{25} + 3 \beta_{1} q^{27} -8 q^{29} + 6 \beta_{2} q^{33} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{35} -10 \beta_{2} q^{37} + 2 \beta_{3} q^{39} + 50 q^{41} + 36 \beta_{1} q^{43} + ( -6 + 3 \beta_{2} ) q^{45} -28 \beta_{1} q^{47} -37 q^{49} -2 \beta_{3} q^{51} + 6 \beta_{2} q^{53} + ( -42 \beta_{1} + 4 \beta_{3} ) q^{55} + 12 \beta_{2} q^{57} + 2 \beta_{3} q^{59} -26 q^{61} + 6 \beta_{1} q^{63} + ( 42 + 4 \beta_{2} ) q^{65} -32 \beta_{1} q^{67} + 48 q^{69} + 12 \beta_{3} q^{71} -28 \beta_{2} q^{73} + ( -17 \beta_{1} + 4 \beta_{3} ) q^{75} + 12 \beta_{2} q^{77} + 16 \beta_{3} q^{79} + 9 q^{81} + 76 \beta_{1} q^{83} + ( -42 - 4 \beta_{2} ) q^{85} -8 \beta_{1} q^{87} -86 q^{89} + 4 \beta_{3} q^{91} + ( -84 \beta_{1} + 8 \beta_{3} ) q^{95} -24 \beta_{2} q^{97} -6 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{5} + 12 q^{9} + O(q^{10})$$ $$4 q - 8 q^{5} + 12 q^{9} + 24 q^{21} - 68 q^{25} - 32 q^{29} + 200 q^{41} - 24 q^{45} - 148 q^{49} - 104 q^{61} + 168 q^{65} + 192 q^{69} + 36 q^{81} - 168 q^{85} - 344 q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - x^{2} + 2 x + 22$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} + 3 \nu^{2} + 13 \nu - 7$$$$)/19$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + \nu + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$12 \nu^{3} - 18 \nu^{2} + 36 \nu - 15$$$$)/19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 6 \beta_{1} + 3$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6 \beta_{2} + 6 \beta_{1} + 9$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$8 \beta_{3} - 9 \beta_{2} - 9 \beta_{1} + 12$$$$)/6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −1.23205 − 1.32288i −1.23205 + 1.32288i 2.23205 + 1.32288i 2.23205 − 1.32288i
0 −1.73205 0 −2.00000 4.58258i 0 −3.46410 0 3.00000 0
79.2 0 −1.73205 0 −2.00000 + 4.58258i 0 −3.46410 0 3.00000 0
79.3 0 1.73205 0 −2.00000 4.58258i 0 3.46410 0 3.00000 0
79.4 0 1.73205 0 −2.00000 + 4.58258i 0 3.46410 0 3.00000 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.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.j.b 4
3.b odd 2 1 720.3.j.f 4
4.b odd 2 1 inner 240.3.j.b 4
5.b even 2 1 inner 240.3.j.b 4
5.c odd 4 2 1200.3.e.l 4
8.b even 2 1 960.3.j.c 4
8.d odd 2 1 960.3.j.c 4
12.b even 2 1 720.3.j.f 4
15.d odd 2 1 720.3.j.f 4
15.e even 4 2 3600.3.e.be 4
20.d odd 2 1 inner 240.3.j.b 4
20.e even 4 2 1200.3.e.l 4
40.e odd 2 1 960.3.j.c 4
40.f even 2 1 960.3.j.c 4
60.h even 2 1 720.3.j.f 4
60.l odd 4 2 3600.3.e.be 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.j.b 4 1.a even 1 1 trivial
240.3.j.b 4 4.b odd 2 1 inner
240.3.j.b 4 5.b even 2 1 inner
240.3.j.b 4 20.d odd 2 1 inner
720.3.j.f 4 3.b odd 2 1
720.3.j.f 4 12.b even 2 1
720.3.j.f 4 15.d odd 2 1
720.3.j.f 4 60.h even 2 1
960.3.j.c 4 8.b even 2 1
960.3.j.c 4 8.d odd 2 1
960.3.j.c 4 40.e odd 2 1
960.3.j.c 4 40.f even 2 1
1200.3.e.l 4 5.c odd 4 2
1200.3.e.l 4 20.e even 4 2
3600.3.e.be 4 15.e even 4 2
3600.3.e.be 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(240, [\chi])$$:

 $$T_{7}^{2} - 12$$ $$T_{11}^{2} + 252$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$( 25 + 4 T + T^{2} )^{2}$$
$7$ $$( -12 + T^{2} )^{2}$$
$11$ $$( 252 + T^{2} )^{2}$$
$13$ $$( 84 + T^{2} )^{2}$$
$17$ $$( 84 + T^{2} )^{2}$$
$19$ $$( 1008 + T^{2} )^{2}$$
$23$ $$( -768 + T^{2} )^{2}$$
$29$ $$( 8 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$( 2100 + T^{2} )^{2}$$
$41$ $$( -50 + T )^{4}$$
$43$ $$( -3888 + T^{2} )^{2}$$
$47$ $$( -2352 + T^{2} )^{2}$$
$53$ $$( 756 + T^{2} )^{2}$$
$59$ $$( 252 + T^{2} )^{2}$$
$61$ $$( 26 + T )^{4}$$
$67$ $$( -3072 + T^{2} )^{2}$$
$71$ $$( 9072 + T^{2} )^{2}$$
$73$ $$( 16464 + T^{2} )^{2}$$
$79$ $$( 16128 + T^{2} )^{2}$$
$83$ $$( -17328 + T^{2} )^{2}$$
$89$ $$( 86 + T )^{4}$$
$97$ $$( 12096 + T^{2} )^{2}$$