# Properties

 Label 2240.2.l.a Level $2240$ Weight $2$ Character orbit 2240.l Analytic conductor $17.886$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 3 x^{10} + 11 x^{8} + 22 x^{6} + 99 x^{4} + 243 x^{2} + 729$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{3} -\beta_{4} q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{4} - \beta_{5} - \beta_{10} ) q^{9} +O(q^{10})$$ $$q + \beta_{9} q^{3} -\beta_{4} q^{5} + \beta_{3} q^{7} + ( 1 + \beta_{4} - \beta_{5} - \beta_{10} ) q^{9} + ( -\beta_{2} - \beta_{7} - \beta_{8} ) q^{11} + ( 2 + \beta_{10} ) q^{13} + ( -\beta_{3} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{15} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{17} + ( -\beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{19} + \beta_{1} q^{21} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} ) q^{23} + ( -2 + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{25} + ( -\beta_{2} + \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{27} + ( -\beta_{1} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{29} + 2 \beta_{9} q^{31} + ( \beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{33} + \beta_{7} q^{35} + ( 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{10} ) q^{37} + ( \beta_{2} - \beta_{7} + \beta_{9} ) q^{39} + 2 q^{41} + ( -2 \beta_{2} + 2 \beta_{7} + 2 \beta_{11} ) q^{43} + ( -2 - 2 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} + \beta_{10} ) q^{45} + ( -\beta_{2} + 2 \beta_{3} - \beta_{7} + \beta_{8} ) q^{47} - q^{49} + ( -\beta_{2} + 2 \beta_{3} - \beta_{7} - \beta_{8} ) q^{51} + 2 \beta_{10} q^{53} + ( -\beta_{2} - 4 \beta_{3} + \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{55} + 2 \beta_{6} q^{57} + ( \beta_{2} - 2 \beta_{3} + \beta_{7} - 2 \beta_{8} ) q^{59} + ( 4 \beta_{1} - \beta_{4} - \beta_{5} ) q^{61} + ( -\beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} ) q^{63} + ( -1 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{65} + ( -2 \beta_{2} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{67} + ( -2 \beta_{1} - 4 \beta_{6} ) q^{69} + ( 2 \beta_{9} - 2 \beta_{11} ) q^{71} + ( -2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{73} + ( -4 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{75} + ( -\beta_{4} + \beta_{5} - \beta_{10} ) q^{77} + ( -\beta_{2} + \beta_{7} - 5 \beta_{9} ) q^{79} + ( -1 + 4 \beta_{4} - 4 \beta_{5} ) q^{81} + ( -\beta_{2} + \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{83} + ( -6 - \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{10} ) q^{85} + ( -5 \beta_{2} - 6 \beta_{3} - 5 \beta_{7} + \beta_{8} ) q^{87} + ( 6 + 2 \beta_{10} ) q^{89} + ( 2 \beta_{3} - \beta_{8} ) q^{91} + ( 8 + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{10} ) q^{93} + ( \beta_{2} - 3 \beta_{3} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{95} + ( -\beta_{1} - 3 \beta_{4} - 3 \beta_{5} ) q^{97} + ( -4 \beta_{2} - 2 \beta_{3} - 4 \beta_{7} + 2 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 4q^{5} + 24q^{9} + O(q^{10})$$ $$12q - 4q^{5} + 24q^{9} + 20q^{13} - 12q^{25} + 24q^{37} + 24q^{41} - 48q^{45} - 12q^{49} - 8q^{53} - 12q^{65} - 4q^{77} + 20q^{81} - 56q^{85} + 64q^{89} + 120q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 3 x^{10} + 11 x^{8} + 22 x^{6} + 99 x^{4} + 243 x^{2} + 729$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{10} - 51 \nu^{8} - 82 \nu^{6} - 587 \nu^{4} - 1773 \nu^{2} - 3888$$$$)/2268$$ $$\beta_{2}$$ $$=$$ $$($$$$-10 \nu^{11} - 39 \nu^{9} + 25 \nu^{7} - 1048 \nu^{5} - 1593 \nu^{3} - 2673 \nu$$$$)/6804$$ $$\beta_{3}$$ $$=$$ $$($$$$-23 \nu^{11} + 30 \nu^{9} - 37 \nu^{7} + 97 \nu^{5} + 954 \nu^{3} + 2187 \nu$$$$)/13608$$ $$\beta_{4}$$ $$=$$ $$($$$$-23 \nu^{10} - 96 \nu^{8} - 415 \nu^{6} - 155 \nu^{4} - 2952 \nu^{2} - 3483$$$$)/4536$$ $$\beta_{5}$$ $$=$$ $$($$$$-23 \nu^{10} + 30 \nu^{8} - 37 \nu^{6} + 97 \nu^{4} - 3582 \nu^{2} - 2349$$$$)/4536$$ $$\beta_{6}$$ $$=$$ $$($$$$10 \nu^{10} + 39 \nu^{8} - 25 \nu^{6} - 86 \nu^{4} + 459 \nu^{2} + 1539$$$$)/1134$$ $$\beta_{7}$$ $$=$$ $$($$$$25 \nu^{11} + 129 \nu^{9} + 32 \nu^{7} + 415 \nu^{5} + 423 \nu^{3} + 4698 \nu$$$$)/6804$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{11} + 3 \nu^{9} + 11 \nu^{7} + 22 \nu^{5} + 99 \nu^{3} + 486 \nu$$$$)/243$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{11} + 3 \nu^{9} + 11 \nu^{7} + 22 \nu^{5} + 99 \nu^{3}$$$$)/243$$ $$\beta_{10}$$ $$=$$ $$($$$$4 \nu^{10} + 3 \nu^{8} + 17 \nu^{6} + 70 \nu^{4} + 117 \nu^{2} + 243$$$$)/324$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{11} + 7 \nu^{7} + 11 \nu^{5} - 96 \nu^{3} - 45 \nu$$$$)/216$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{9} + \beta_{8}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} - 2 \beta_{5} - \beta_{1} - 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{9} - \beta_{7} + 3 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{10} - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 5 \beta_{1} - 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{11} - \beta_{9} - \beta_{8} - 2 \beta_{7} - 8 \beta_{3} - 14 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$\beta_{10} - 5 \beta_{6} + 3 \beta_{5} - 11 \beta_{4} + 4 \beta_{1} + 6$$ $$\nu^{7}$$ $$=$$ $$($$$$16 \beta_{11} + 21 \beta_{9} + 11 \beta_{8} - 14 \beta_{7} - 12 \beta_{3} + 18 \beta_{2}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-17 \beta_{10} + 34 \beta_{6} + 40 \beta_{5} - 14 \beta_{4} - 19 \beta_{1} - 56$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$3 \beta_{11} + 15 \beta_{9} - 20 \beta_{8} + 54 \beta_{7} + 77 \beta_{3} + 23 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$($$$$143 \beta_{10} + 52 \beta_{6} - 32 \beta_{5} + 34 \beta_{4} + 97 \beta_{1} - 2$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-84 \beta_{11} - 11 \beta_{9} + 21 \beta_{8} + 72 \beta_{7} - 748 \beta_{3} - 28 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$$.

 $$n$$ $$897$$ $$1471$$ $$1541$$ $$1921$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1569.1
 1.55635 − 0.760115i 1.55635 + 0.760115i 1.02957 − 1.39283i 1.02957 + 1.39283i 0.517456 − 1.65295i 0.517456 + 1.65295i −0.517456 + 1.65295i −0.517456 − 1.65295i −1.02957 + 1.39283i −1.02957 − 1.39283i −1.55635 + 0.760115i −1.55635 − 0.760115i
0 −3.11270 0 −2.08433 0.809661i 0 1.00000i 0 6.68890 0
1569.2 0 −3.11270 0 −2.08433 + 0.809661i 0 1.00000i 0 6.68890 0
1569.3 0 −2.05914 0 1.27280 1.83847i 0 1.00000i 0 1.24006 0
1569.4 0 −2.05914 0 1.27280 + 1.83847i 0 1.00000i 0 1.24006 0
1569.5 0 −1.03491 0 −0.188470 2.22811i 0 1.00000i 0 −1.92896 0
1569.6 0 −1.03491 0 −0.188470 + 2.22811i 0 1.00000i 0 −1.92896 0
1569.7 0 1.03491 0 −0.188470 2.22811i 0 1.00000i 0 −1.92896 0
1569.8 0 1.03491 0 −0.188470 + 2.22811i 0 1.00000i 0 −1.92896 0
1569.9 0 2.05914 0 1.27280 1.83847i 0 1.00000i 0 1.24006 0
1569.10 0 2.05914 0 1.27280 + 1.83847i 0 1.00000i 0 1.24006 0
1569.11 0 3.11270 0 −2.08433 0.809661i 0 1.00000i 0 6.68890 0
1569.12 0 3.11270 0 −2.08433 + 0.809661i 0 1.00000i 0 6.68890 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1569.12 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
40.e odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.l.a 12
4.b odd 2 1 inner 2240.2.l.a 12
5.b even 2 1 2240.2.l.b yes 12
8.b even 2 1 2240.2.l.b yes 12
8.d odd 2 1 2240.2.l.b yes 12
20.d odd 2 1 2240.2.l.b yes 12
40.e odd 2 1 inner 2240.2.l.a 12
40.f even 2 1 inner 2240.2.l.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.l.a 12 1.a even 1 1 trivial
2240.2.l.a 12 4.b odd 2 1 inner
2240.2.l.a 12 40.e odd 2 1 inner
2240.2.l.a 12 40.f even 2 1 inner
2240.2.l.b yes 12 5.b even 2 1
2240.2.l.b yes 12 8.b even 2 1
2240.2.l.b yes 12 8.d odd 2 1
2240.2.l.b yes 12 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{6} - 15 T_{3}^{4} + 56 T_{3}^{2} - 44$$ $$T_{13}^{3} - 5 T_{13}^{2} - 2 T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( -44 + 56 T^{2} - 15 T^{4} + T^{6} )^{2}$$
$5$ $$( 125 + 50 T + 25 T^{2} + 16 T^{3} + 5 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$7$ $$( 1 + T^{2} )^{6}$$
$11$ $$( 2704 + 680 T^{2} + 49 T^{4} + T^{6} )^{2}$$
$13$ $$( 2 - 2 T - 5 T^{2} + T^{3} )^{4}$$
$17$ $$( 176 + 152 T^{2} + 35 T^{4} + T^{6} )^{2}$$
$19$ $$( 256 + 164 T^{2} + 28 T^{4} + T^{6} )^{2}$$
$23$ $$( 3136 + 2160 T^{2} + 92 T^{4} + T^{6} )^{2}$$
$29$ $$( 85184 + 6160 T^{2} + 139 T^{4} + T^{6} )^{2}$$
$31$ $$( -2816 + 896 T^{2} - 60 T^{4} + T^{6} )^{2}$$
$37$ $$( 32 - 64 T - 6 T^{2} + T^{3} )^{4}$$
$41$ $$( -2 + T )^{12}$$
$43$ $$( -45056 + 12048 T^{2} - 216 T^{4} + T^{6} )^{2}$$
$47$ $$( 256 + 352 T^{2} + 65 T^{4} + T^{6} )^{2}$$
$53$ $$( -112 - 40 T + 2 T^{2} + T^{3} )^{4}$$
$59$ $$( 38416 + 4116 T^{2} + 128 T^{4} + T^{6} )^{2}$$
$61$ $$( 180224 + 12644 T^{2} + 212 T^{4} + T^{6} )^{2}$$
$67$ $$( -1301696 + 36976 T^{2} - 340 T^{4} + T^{6} )^{2}$$
$71$ $$( -704 + 2608 T^{2} - 196 T^{4} + T^{6} )^{2}$$
$73$ $$( 45056 + 12048 T^{2} + 216 T^{4} + T^{6} )^{2}$$
$79$ $$( -790064 + 30616 T^{2} - 331 T^{4} + T^{6} )^{2}$$
$83$ $$( -240944 + 15156 T^{2} - 232 T^{4} + T^{6} )^{2}$$
$89$ $$( -16 + 44 T - 16 T^{2} + T^{3} )^{4}$$
$97$ $$( 1655984 + 46616 T^{2} + 387 T^{4} + T^{6} )^{2}$$