# Properties

 Label 240.3.c.d Level $240$ Weight $3$ Character orbit 240.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53952634465$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 60) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( -3 \beta_{2} + 3 \beta_{3} ) q^{13} + ( -5 + \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{15} + ( 7 \beta_{2} + 7 \beta_{3} ) q^{17} + 6 q^{19} + ( -16 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{21} + ( \beta_{2} + \beta_{3} ) q^{23} + ( -15 - 5 \beta_{2} + 5 \beta_{3} ) q^{25} + ( -9 \beta_{2} + 2 \beta_{3} ) q^{27} + ( 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{29} -34 q^{31} + ( -9 \beta_{2} + \beta_{3} ) q^{33} + ( 4 \beta_{1} + 10 \beta_{2} + 6 \beta_{3} ) q^{35} + ( 11 \beta_{2} - 11 \beta_{3} ) q^{37} + ( -24 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{39} + ( -4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -7 \beta_{2} + 7 \beta_{3} ) q^{43} + ( 40 + \beta_{1} + 5 \beta_{2} - 6 \beta_{3} ) q^{45} + ( \beta_{2} + \beta_{3} ) q^{47} -15 q^{49} + ( 70 - 14 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} ) q^{51} + ( -9 \beta_{2} - 9 \beta_{3} ) q^{53} + ( 40 + 5 \beta_{2} - 5 \beta_{3} ) q^{55} + 6 \beta_{3} q^{57} + ( 22 \beta_{1} + 11 \beta_{2} - 11 \beta_{3} ) q^{59} + 74 q^{61} + ( -18 \beta_{2} - 14 \beta_{3} ) q^{63} + ( 6 \beta_{1} + 15 \beta_{2} + 9 \beta_{3} ) q^{65} + ( 23 \beta_{2} - 23 \beta_{3} ) q^{67} + ( 10 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{69} + ( 12 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{71} + ( -14 \beta_{2} + 14 \beta_{3} ) q^{73} + ( -40 - 10 \beta_{1} - 5 \beta_{2} - 10 \beta_{3} ) q^{75} + ( -16 \beta_{2} - 16 \beta_{3} ) q^{77} + 78 q^{79} + ( -79 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{81} + ( -23 \beta_{2} - 23 \beta_{3} ) q^{83} + ( -70 + 35 \beta_{2} - 35 \beta_{3} ) q^{85} + ( 27 \beta_{2} - 3 \beta_{3} ) q^{87} + ( 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{89} -96 q^{91} -34 \beta_{3} q^{93} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{95} + ( -8 \beta_{2} + 8 \beta_{3} ) q^{97} + ( -80 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} - 20q^{15} + 24q^{19} - 64q^{21} - 60q^{25} - 136q^{31} - 96q^{39} + 160q^{45} - 60q^{49} + 280q^{51} + 160q^{55} + 296q^{61} + 40q^{69} - 160q^{75} + 312q^{79} - 316q^{81} - 280q^{85} - 384q^{91} - 320q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 2 \nu^{2} + 4 \nu + 3$$ $$\beta_{3}$$ $$=$$ $$-2 \nu^{3} + 2 \nu^{2} - 4 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - 2 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 − 1.61803i 1.61803i 0.618034i − 0.618034i
0 −2.23607 2.00000i 0 2.23607 4.47214i 0 8.00000i 0 1.00000 + 8.94427i 0
209.2 0 −2.23607 + 2.00000i 0 2.23607 + 4.47214i 0 8.00000i 0 1.00000 8.94427i 0
209.3 0 2.23607 2.00000i 0 −2.23607 + 4.47214i 0 8.00000i 0 1.00000 8.94427i 0
209.4 0 2.23607 + 2.00000i 0 −2.23607 4.47214i 0 8.00000i 0 1.00000 + 8.94427i 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.b even 2 1 inner
15.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.c.d 4
3.b odd 2 1 inner 240.3.c.d 4
4.b odd 2 1 60.3.b.a 4
5.b even 2 1 inner 240.3.c.d 4
5.c odd 4 1 1200.3.l.h 2
5.c odd 4 1 1200.3.l.q 2
8.b even 2 1 960.3.c.g 4
8.d odd 2 1 960.3.c.h 4
12.b even 2 1 60.3.b.a 4
15.d odd 2 1 inner 240.3.c.d 4
15.e even 4 1 1200.3.l.h 2
15.e even 4 1 1200.3.l.q 2
20.d odd 2 1 60.3.b.a 4
20.e even 4 1 300.3.g.e 2
20.e even 4 1 300.3.g.h 2
24.f even 2 1 960.3.c.h 4
24.h odd 2 1 960.3.c.g 4
36.f odd 6 2 1620.3.t.b 8
36.h even 6 2 1620.3.t.b 8
40.e odd 2 1 960.3.c.h 4
40.f even 2 1 960.3.c.g 4
60.h even 2 1 60.3.b.a 4
60.l odd 4 1 300.3.g.e 2
60.l odd 4 1 300.3.g.h 2
120.i odd 2 1 960.3.c.g 4
120.m even 2 1 960.3.c.h 4
180.n even 6 2 1620.3.t.b 8
180.p odd 6 2 1620.3.t.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.b.a 4 4.b odd 2 1
60.3.b.a 4 12.b even 2 1
60.3.b.a 4 20.d odd 2 1
60.3.b.a 4 60.h even 2 1
240.3.c.d 4 1.a even 1 1 trivial
240.3.c.d 4 3.b odd 2 1 inner
240.3.c.d 4 5.b even 2 1 inner
240.3.c.d 4 15.d odd 2 1 inner
300.3.g.e 2 20.e even 4 1
300.3.g.e 2 60.l odd 4 1
300.3.g.h 2 20.e even 4 1
300.3.g.h 2 60.l odd 4 1
960.3.c.g 4 8.b even 2 1
960.3.c.g 4 24.h odd 2 1
960.3.c.g 4 40.f even 2 1
960.3.c.g 4 120.i odd 2 1
960.3.c.h 4 8.d odd 2 1
960.3.c.h 4 24.f even 2 1
960.3.c.h 4 40.e odd 2 1
960.3.c.h 4 120.m even 2 1
1200.3.l.h 2 5.c odd 4 1
1200.3.l.h 2 15.e even 4 1
1200.3.l.q 2 5.c odd 4 1
1200.3.l.q 2 15.e even 4 1
1620.3.t.b 8 36.f odd 6 2
1620.3.t.b 8 36.h even 6 2
1620.3.t.b 8 180.n even 6 2
1620.3.t.b 8 180.p odd 6 2

## Hecke kernels

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

 $$T_{7}^{2} + 64$$ $$T_{17}^{2} - 980$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 - 2 T^{2} + T^{4}$$
$5$ $$625 + 30 T^{2} + T^{4}$$
$7$ $$( 64 + T^{2} )^{2}$$
$11$ $$( 80 + T^{2} )^{2}$$
$13$ $$( 144 + T^{2} )^{2}$$
$17$ $$( -980 + T^{2} )^{2}$$
$19$ $$( -6 + T )^{4}$$
$23$ $$( -20 + T^{2} )^{2}$$
$29$ $$( 720 + T^{2} )^{2}$$
$31$ $$( 34 + T )^{4}$$
$37$ $$( 1936 + T^{2} )^{2}$$
$41$ $$( 320 + T^{2} )^{2}$$
$43$ $$( 784 + T^{2} )^{2}$$
$47$ $$( -20 + T^{2} )^{2}$$
$53$ $$( -1620 + T^{2} )^{2}$$
$59$ $$( 9680 + T^{2} )^{2}$$
$61$ $$( -74 + T )^{4}$$
$67$ $$( 8464 + T^{2} )^{2}$$
$71$ $$( 2880 + T^{2} )^{2}$$
$73$ $$( 3136 + T^{2} )^{2}$$
$79$ $$( -78 + T )^{4}$$
$83$ $$( -10580 + T^{2} )^{2}$$
$89$ $$( 320 + T^{2} )^{2}$$
$97$ $$( 1024 + T^{2} )^{2}$$