# Properties

 Label 240.2.bc.d Level $240$ Weight $2$ Character orbit 240.bc Analytic conductor $1.916$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( \beta_{2} - \beta_{3} ) q^{4} + ( 2 - \beta_{3} ) q^{5} -\beta_{4} q^{6} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( \beta_{2} - \beta_{3} ) q^{4} + ( 2 - \beta_{3} ) q^{5} -\beta_{4} q^{6} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{8} - q^{9} + ( -2 \beta_{1} - \beta_{4} ) q^{10} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{11} + ( -1 + \beta_{5} ) q^{12} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{14} + ( -1 - 2 \beta_{3} ) q^{15} + ( 1 + \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{16} + ( -1 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{17} + \beta_{1} q^{18} + ( 2 - \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{19} + ( -1 + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{20} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{21} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{22} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} + ( -1 + \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{24} + ( 3 - 4 \beta_{3} ) q^{25} + \beta_{3} q^{27} + ( 3 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{28} + ( 1 - \beta_{3} ) q^{29} + ( \beta_{1} - 2 \beta_{4} ) q^{30} + ( -2 \beta_{1} + 2 \beta_{5} ) q^{31} + ( -2 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{32} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{33} + ( 6 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{34} + ( \beta_{1} + 3 \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{35} + ( -\beta_{2} + \beta_{3} ) q^{36} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{37} + ( -6 - \beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{38} + ( -5 + 3 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{40} + 4 \beta_{3} q^{41} + ( -2 + \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{42} + ( 2 - 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{43} + ( 3 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{44} + ( -2 + \beta_{3} ) q^{45} + ( -\beta_{1} - 2 \beta_{2} + 6 \beta_{3} - \beta_{4} ) q^{46} + ( -4 + 5 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{47} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{48} + ( 2 - 6 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{49} + ( -3 \beta_{1} - 4 \beta_{4} ) q^{50} + ( 1 - 2 \beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{51} + ( -2 + 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{53} + \beta_{4} q^{54} + ( \beta_{1} + 3 \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{55} + ( -5 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{56} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{57} + ( -\beta_{1} - \beta_{4} ) q^{58} + ( 4 - 5 \beta_{1} + \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{59} + ( -2 - \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{60} + ( -3 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{61} + ( -2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{62} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{63} + ( 5 + \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{64} + ( -2 + \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{66} + ( -6 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{67} + ( -1 - 4 \beta_{1} - \beta_{2} - 7 \beta_{3} + \beta_{5} ) q^{68} + ( -2 + \beta_{1} - \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{69} + ( -2 + 3 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{70} + ( -4 - 4 \beta_{2} - 4 \beta_{4} ) q^{71} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{72} + ( 3 + 4 \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{73} + ( 8 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{74} + ( -4 - 3 \beta_{3} ) q^{75} + ( -1 + 6 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{76} + ( 2 - 6 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{77} + ( -4 - 4 \beta_{2} - 4 \beta_{4} ) q^{79} + ( 4 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{80} + q^{81} + 4 \beta_{4} q^{82} + ( 4 \beta_{1} + 8 \beta_{3} - 4 \beta_{5} ) q^{83} + ( 3 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{84} + ( -1 - 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{85} + ( 6 - 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 2 \beta_{5} ) q^{86} + ( -1 - \beta_{3} ) q^{87} + ( -5 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{88} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{89} + ( 2 \beta_{1} + \beta_{4} ) q^{90} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{92} + ( -2 \beta_{2} - 2 \beta_{4} ) q^{93} + ( 4 + \beta_{1} - 2 \beta_{2} + 6 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{94} + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 7 \beta_{4} + 3 \beta_{5} ) q^{95} + ( 3 - 3 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{96} + ( -3 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{97} + ( -8 + 4 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} ) q^{98} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} - 2 q^{4} + 12 q^{5} - 2 q^{7} - 8 q^{8} - 6 q^{9} + O(q^{10})$$ $$6 q - 2 q^{2} - 2 q^{4} + 12 q^{5} - 2 q^{7} - 8 q^{8} - 6 q^{9} - 4 q^{10} - 2 q^{11} - 4 q^{12} + 6 q^{14} - 6 q^{15} + 10 q^{16} - 2 q^{17} + 2 q^{18} + 10 q^{19} - 8 q^{20} - 2 q^{21} + 6 q^{22} + 10 q^{23} - 6 q^{24} + 18 q^{25} + 14 q^{28} + 6 q^{29} + 2 q^{30} - 12 q^{32} - 2 q^{33} + 26 q^{34} - 6 q^{35} + 2 q^{36} + 8 q^{37} - 34 q^{38} - 22 q^{40} - 14 q^{42} + 4 q^{43} + 14 q^{44} - 12 q^{45} + 2 q^{46} - 10 q^{47} + 16 q^{48} - 6 q^{50} - 2 q^{51} - 6 q^{55} - 34 q^{56} + 10 q^{57} - 2 q^{58} + 6 q^{59} - 6 q^{60} - 14 q^{61} - 20 q^{62} + 2 q^{63} + 22 q^{64} - 14 q^{66} - 36 q^{67} - 10 q^{68} - 10 q^{69} - 2 q^{70} - 16 q^{71} + 8 q^{72} + 10 q^{73} + 32 q^{74} - 24 q^{75} - 2 q^{76} - 16 q^{79} + 36 q^{80} + 6 q^{81} + 26 q^{84} - 6 q^{85} + 24 q^{86} - 6 q^{87} - 34 q^{88} + 28 q^{89} + 4 q^{90} + 10 q^{92} + 4 q^{93} + 38 q^{94} + 30 q^{95} + 10 q^{96} - 10 q^{97} - 56 q^{98} + 2 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 3 \nu^{3} + 2 \nu - 8$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 3 \nu^{3} - 4 \nu^{2} + 2 \nu - 8$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} - \nu^{3} + 2 \nu^{2} + 4$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$-\nu^{5} + \nu^{4} - 2 \nu^{3} + 3 \nu^{2} - 2 \nu + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_{1} + 2$$

## Character values

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

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$ $$1$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 1.40680 − 0.144584i 0.264658 + 1.38923i −0.671462 − 1.24464i 1.40680 + 0.144584i 0.264658 − 1.38923i −0.671462 + 1.24464i
−1.40680 + 0.144584i 1.00000i 1.95819 0.406803i 2.00000 + 1.00000i −0.144584 1.40680i 2.10278 2.10278i −2.69597 + 0.855416i −1.00000 −2.95819 1.11763i
43.2 −0.264658 1.38923i 1.00000i −1.85991 + 0.735342i 2.00000 + 1.00000i 1.38923 0.264658i −3.24914 + 3.24914i 1.51380 + 2.38923i −1.00000 0.859912 3.04312i
43.3 0.671462 + 1.24464i 1.00000i −1.09828 + 1.67146i 2.00000 + 1.00000i −1.24464 + 0.671462i 0.146365 0.146365i −2.81783 0.244644i −1.00000 0.0982788 + 3.16075i
67.1 −1.40680 0.144584i 1.00000i 1.95819 + 0.406803i 2.00000 1.00000i −0.144584 + 1.40680i 2.10278 + 2.10278i −2.69597 0.855416i −1.00000 −2.95819 + 1.11763i
67.2 −0.264658 + 1.38923i 1.00000i −1.85991 0.735342i 2.00000 1.00000i 1.38923 + 0.264658i −3.24914 3.24914i 1.51380 2.38923i −1.00000 0.859912 + 3.04312i
67.3 0.671462 1.24464i 1.00000i −1.09828 1.67146i 2.00000 1.00000i −1.24464 0.671462i 0.146365 + 0.146365i −2.81783 + 0.244644i −1.00000 0.0982788 3.16075i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.j 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.bc.d yes 6
3.b odd 2 1 720.2.bd.e 6
4.b odd 2 1 960.2.bc.d 6
5.c odd 4 1 240.2.y.d 6
8.b even 2 1 1920.2.bc.g 6
8.d odd 2 1 1920.2.bc.h 6
15.e even 4 1 720.2.z.e 6
16.e even 4 1 960.2.y.d 6
16.e even 4 1 1920.2.y.h 6
16.f odd 4 1 240.2.y.d 6
16.f odd 4 1 1920.2.y.g 6
20.e even 4 1 960.2.y.d 6
40.i odd 4 1 1920.2.y.g 6
40.k even 4 1 1920.2.y.h 6
48.k even 4 1 720.2.z.e 6
80.i odd 4 1 1920.2.bc.h 6
80.j even 4 1 inner 240.2.bc.d yes 6
80.s even 4 1 1920.2.bc.g 6
80.t odd 4 1 960.2.bc.d 6
240.bd odd 4 1 720.2.bd.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.y.d 6 5.c odd 4 1
240.2.y.d 6 16.f odd 4 1
240.2.bc.d yes 6 1.a even 1 1 trivial
240.2.bc.d yes 6 80.j even 4 1 inner
720.2.z.e 6 15.e even 4 1
720.2.z.e 6 48.k even 4 1
720.2.bd.e 6 3.b odd 2 1
720.2.bd.e 6 240.bd odd 4 1
960.2.y.d 6 16.e even 4 1
960.2.y.d 6 20.e even 4 1
960.2.bc.d 6 4.b odd 2 1
960.2.bc.d 6 80.t odd 4 1
1920.2.y.g 6 16.f odd 4 1
1920.2.y.g 6 40.i odd 4 1
1920.2.y.h 6 16.e even 4 1
1920.2.y.h 6 40.k even 4 1
1920.2.bc.g 6 8.b even 2 1
1920.2.bc.g 6 80.s even 4 1
1920.2.bc.h 6 8.d odd 2 1
1920.2.bc.h 6 80.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(240, [\chi])$$:

 $$T_{7}^{6} + 2 T_{7}^{5} + 2 T_{7}^{4} - 32 T_{7}^{3} + 196 T_{7}^{2} - 56 T_{7} + 8$$ $$T_{11}^{6} + 2 T_{11}^{5} + 2 T_{11}^{4} - 32 T_{11}^{3} + 196 T_{11}^{2} - 56 T_{11} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$8 + 8 T + 6 T^{2} + 6 T^{3} + 3 T^{4} + 2 T^{5} + T^{6}$$
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$( 5 - 4 T + T^{2} )^{3}$$
$7$ $$8 - 56 T + 196 T^{2} - 32 T^{3} + 2 T^{4} + 2 T^{5} + T^{6}$$
$11$ $$8 - 56 T + 196 T^{2} - 32 T^{3} + 2 T^{4} + 2 T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$8 + 136 T + 1156 T^{2} - 64 T^{3} + 2 T^{4} + 2 T^{5} + T^{6}$$
$19$ $$14792 - 2408 T + 196 T^{2} - 32 T^{3} + 50 T^{4} - 10 T^{5} + T^{6}$$
$23$ $$14792 - 2408 T + 196 T^{2} - 32 T^{3} + 50 T^{4} - 10 T^{5} + T^{6}$$
$29$ $$( 2 - 2 T + T^{2} )^{3}$$
$31$ $$64 + 752 T^{2} + 60 T^{4} + T^{6}$$
$37$ $$( 128 - 64 T - 4 T^{2} + T^{3} )^{2}$$
$41$ $$( 16 + T^{2} )^{3}$$
$43$ $$( -136 - 60 T - 2 T^{2} + T^{3} )^{2}$$
$47$ $$412232 + 85352 T + 8836 T^{2} - 32 T^{3} + 50 T^{4} + 10 T^{5} + T^{6}$$
$53$ $$118336 + 14128 T^{2} + 236 T^{4} + T^{6}$$
$59$ $$35912 + 29480 T + 12100 T^{2} + 928 T^{3} + 18 T^{4} - 6 T^{5} + T^{6}$$
$61$ $$4232 + 184 T + 4 T^{2} + 64 T^{3} + 98 T^{4} + 14 T^{5} + T^{6}$$
$67$ $$( -1208 - 28 T + 18 T^{2} + T^{3} )^{2}$$
$71$ $$( -512 - 96 T + 8 T^{2} + T^{3} )^{2}$$
$73$ $$42632 - 35624 T + 14884 T^{2} + 928 T^{3} + 50 T^{4} - 10 T^{5} + T^{6}$$
$79$ $$( -512 - 96 T + 8 T^{2} + T^{3} )^{2}$$
$83$ $$495616 + 28416 T^{2} + 368 T^{4} + T^{6}$$
$89$ $$( 184 - 4 T - 14 T^{2} + T^{3} )^{2}$$
$97$ $$1338248 + 356648 T + 47524 T^{2} - 544 T^{3} + 50 T^{4} + 10 T^{5} + T^{6}$$