# Properties

 Label 240.3.bg.a Level $240$ Weight $3$ Character orbit 240.bg Analytic conductor $6.540$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.53952634465$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -1 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{5} + ( -1 + \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -1 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{5} + ( -1 + \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( -4 - 3 \beta_{1} + 3 \beta_{3} ) q^{11} + ( -8 + 2 \beta_{1} - 8 \beta_{2} ) q^{13} + ( -6 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{15} + ( -10 + 10 \beta_{2} + 6 \beta_{3} ) q^{17} + ( 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{19} + ( -6 - \beta_{1} + \beta_{3} ) q^{21} + ( -14 + 2 \beta_{1} - 14 \beta_{2} ) q^{23} + ( 4 + 14 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{25} + 3 \beta_{3} q^{27} + ( -7 \beta_{1} - 18 \beta_{2} - 7 \beta_{3} ) q^{29} + ( 4 - 6 \beta_{1} + 6 \beta_{3} ) q^{31} + ( -9 - 4 \beta_{1} - 9 \beta_{2} ) q^{33} + ( 10 + 5 \beta_{1} - 10 \beta_{2} - 5 \beta_{3} ) q^{35} + ( 16 - 16 \beta_{2} + 18 \beta_{3} ) q^{37} + ( -8 \beta_{1} + 6 \beta_{2} - 8 \beta_{3} ) q^{39} + ( -14 + 6 \beta_{1} - 6 \beta_{3} ) q^{41} + ( 2 + 20 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 9 - 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{45} + ( -32 + 32 \beta_{2} + 10 \beta_{3} ) q^{47} + ( -4 \beta_{1} + 35 \beta_{2} - 4 \beta_{3} ) q^{49} + ( -18 - 10 \beta_{1} + 10 \beta_{3} ) q^{51} + ( 14 - 12 \beta_{1} + 14 \beta_{2} ) q^{53} + ( 31 + 16 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -18 + 18 \beta_{2} - 6 \beta_{3} ) q^{57} + ( -31 \beta_{1} + 36 \beta_{2} - 31 \beta_{3} ) q^{59} + ( 50 - 18 \beta_{1} + 18 \beta_{3} ) q^{61} + ( -3 - 6 \beta_{1} - 3 \beta_{2} ) q^{63} + ( -28 + 22 \beta_{1} + 26 \beta_{2} - 14 \beta_{3} ) q^{65} + ( 50 - 50 \beta_{2} + 4 \beta_{3} ) q^{67} + ( -14 \beta_{1} + 6 \beta_{2} - 14 \beta_{3} ) q^{69} + 68 q^{71} + ( 19 + 48 \beta_{1} + 19 \beta_{2} ) q^{73} + ( -6 + 4 \beta_{1} + 42 \beta_{2} - 3 \beta_{3} ) q^{75} + ( 22 - 22 \beta_{2} - 14 \beta_{3} ) q^{77} + ( -10 \beta_{1} - 10 \beta_{3} ) q^{79} -9 q^{81} + ( 4 - 14 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 58 + 8 \beta_{1} - 16 \beta_{2} - 36 \beta_{3} ) q^{85} + ( 21 - 21 \beta_{2} - 18 \beta_{3} ) q^{87} + ( -36 \beta_{1} + 6 \beta_{2} - 36 \beta_{3} ) q^{89} + ( 4 + 14 \beta_{1} - 14 \beta_{3} ) q^{91} + ( -18 + 4 \beta_{1} - 18 \beta_{2} ) q^{93} + ( -36 + 24 \beta_{1} - 48 \beta_{2} - 18 \beta_{3} ) q^{95} + ( -5 + 5 \beta_{2} + 16 \beta_{3} ) q^{97} + ( -9 \beta_{1} - 12 \beta_{2} - 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} - 4q^{7} + O(q^{10})$$ $$4q - 4q^{5} - 4q^{7} - 16q^{11} - 32q^{13} - 24q^{15} - 40q^{17} - 24q^{21} - 56q^{23} + 16q^{25} + 16q^{31} - 36q^{33} + 40q^{35} + 64q^{37} - 56q^{41} + 8q^{43} + 36q^{45} - 128q^{47} - 72q^{51} + 56q^{53} + 124q^{55} - 72q^{57} + 200q^{61} - 12q^{63} - 112q^{65} + 200q^{67} + 272q^{71} + 76q^{73} - 24q^{75} + 88q^{77} - 36q^{81} + 16q^{83} + 232q^{85} + 84q^{87} + 16q^{91} - 72q^{93} - 144q^{95} - 20q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

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

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 2.67423 + 4.22474i 0 1.44949 + 1.44949i 0 3.00000i 0
97.2 0 1.22474 1.22474i 0 −4.67423 + 1.77526i 0 −3.44949 3.44949i 0 3.00000i 0
193.1 0 −1.22474 1.22474i 0 2.67423 4.22474i 0 1.44949 1.44949i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 −4.67423 1.77526i 0 −3.44949 + 3.44949i 0 3.00000i 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.bg.a 4
3.b odd 2 1 720.3.bh.k 4
4.b odd 2 1 15.3.f.a 4
5.b even 2 1 1200.3.bg.k 4
5.c odd 4 1 inner 240.3.bg.a 4
5.c odd 4 1 1200.3.bg.k 4
8.b even 2 1 960.3.bg.h 4
8.d odd 2 1 960.3.bg.i 4
12.b even 2 1 45.3.g.b 4
15.e even 4 1 720.3.bh.k 4
20.d odd 2 1 75.3.f.c 4
20.e even 4 1 15.3.f.a 4
20.e even 4 1 75.3.f.c 4
36.f odd 6 2 405.3.l.h 8
36.h even 6 2 405.3.l.f 8
40.i odd 4 1 960.3.bg.h 4
40.k even 4 1 960.3.bg.i 4
60.h even 2 1 225.3.g.a 4
60.l odd 4 1 45.3.g.b 4
60.l odd 4 1 225.3.g.a 4
180.v odd 12 2 405.3.l.f 8
180.x even 12 2 405.3.l.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a 4 4.b odd 2 1
15.3.f.a 4 20.e even 4 1
45.3.g.b 4 12.b even 2 1
45.3.g.b 4 60.l odd 4 1
75.3.f.c 4 20.d odd 2 1
75.3.f.c 4 20.e even 4 1
225.3.g.a 4 60.h even 2 1
225.3.g.a 4 60.l odd 4 1
240.3.bg.a 4 1.a even 1 1 trivial
240.3.bg.a 4 5.c odd 4 1 inner
405.3.l.f 8 36.h even 6 2
405.3.l.f 8 180.v odd 12 2
405.3.l.h 8 36.f odd 6 2
405.3.l.h 8 180.x even 12 2
720.3.bh.k 4 3.b odd 2 1
720.3.bh.k 4 15.e even 4 1
960.3.bg.h 4 8.b even 2 1
960.3.bg.h 4 40.i odd 4 1
960.3.bg.i 4 8.d odd 2 1
960.3.bg.i 4 40.k even 4 1
1200.3.bg.k 4 5.b even 2 1
1200.3.bg.k 4 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 4 T_{7}^{3} + 8 T_{7}^{2} - 40 T_{7} + 100$$ acting on $$S_{3}^{\mathrm{new}}(240, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$625 + 100 T + 4 T^{3} + T^{4}$$
$7$ $$100 - 40 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$( -38 + 8 T + T^{2} )^{2}$$
$13$ $$13456 + 3712 T + 512 T^{2} + 32 T^{3} + T^{4}$$
$17$ $$8464 + 3680 T + 800 T^{2} + 40 T^{3} + T^{4}$$
$19$ $$32400 + 504 T^{2} + T^{4}$$
$23$ $$144400 + 21280 T + 1568 T^{2} + 56 T^{3} + T^{4}$$
$29$ $$900 + 1236 T^{2} + T^{4}$$
$31$ $$( -200 - 8 T + T^{2} )^{2}$$
$37$ $$211600 + 29440 T + 2048 T^{2} - 64 T^{3} + T^{4}$$
$41$ $$( -20 + 28 T + T^{2} )^{2}$$
$43$ $$1420864 + 9536 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$3055504 + 223744 T + 8192 T^{2} + 128 T^{3} + T^{4}$$
$53$ $$1600 + 2240 T + 1568 T^{2} - 56 T^{3} + T^{4}$$
$59$ $$19980900 + 14124 T^{2} + T^{4}$$
$61$ $$( 556 - 100 T + T^{2} )^{2}$$
$67$ $$24522304 - 990400 T + 20000 T^{2} - 200 T^{3} + T^{4}$$
$71$ $$( -68 + T )^{4}$$
$73$ $$38316100 + 470440 T + 2888 T^{2} - 76 T^{3} + T^{4}$$
$79$ $$( 600 + T^{2} )^{2}$$
$83$ $$309136 + 8896 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$89$ $$59907600 + 15624 T^{2} + T^{4}$$
$97$ $$515524 - 14360 T + 200 T^{2} + 20 T^{3} + T^{4}$$