# Properties

 Label 2240.2.g.n Level $2240$ Weight $2$ Character orbit 2240.g Analytic conductor $17.886$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 13 x^{8} + 56 x^{6} + 97 x^{4} + 61 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{4} q^{5} -\beta_{5} q^{7} + ( -1 - \beta_{3} - \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{4} q^{5} -\beta_{5} q^{7} + ( -1 - \beta_{3} - \beta_{8} + \beta_{9} ) q^{9} + ( -1 + \beta_{2} - \beta_{8} + \beta_{9} ) q^{11} + ( 2 \beta_{1} - \beta_{5} - \beta_{7} ) q^{13} + ( 1 + \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{15} + ( -\beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{17} + ( 2 - \beta_{4} + \beta_{6} ) q^{19} + \beta_{3} q^{21} + ( -\beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{23} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} ) q^{25} + ( -\beta_{1} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{27} + ( -2 - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{29} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{31} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{33} + \beta_{8} q^{35} + ( -\beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{37} + ( -5 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{39} -2 \beta_{2} q^{41} + ( 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{43} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{45} + ( -\beta_{1} - 2 \beta_{5} + \beta_{8} + \beta_{9} ) q^{47} - q^{49} + ( -3 - \beta_{2} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{51} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{53} + ( 2 - \beta_{1} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{55} + ( 4 \beta_{1} - \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{57} + ( 4 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{59} + ( 2 - 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{61} + ( \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{63} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{9} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} ) q^{67} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{69} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{71} + ( -2 \beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{73} + ( -6 + \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} + 3 \beta_{9} ) q^{75} + ( -\beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{77} + ( -7 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{79} + ( -3 + 2 \beta_{2} + 6 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} ) q^{83} + ( 2 - 2 \beta_{1} - \beta_{3} + 4 \beta_{5} - 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{85} + ( -3 \beta_{1} + 2 \beta_{5} + 3 \beta_{8} + 3 \beta_{9} ) q^{87} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{89} + ( -1 + \beta_{2} + 2 \beta_{3} ) q^{91} + ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{93} + ( 6 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{7} - \beta_{9} ) q^{95} + ( -2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{97} + ( 6 + 2 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 2q^{5} - 14q^{9} + O(q^{10})$$ $$10q + 2q^{5} - 14q^{9} - 8q^{11} + 4q^{15} + 24q^{19} + 4q^{21} + 6q^{25} - 24q^{29} + 24q^{31} - 64q^{39} - 4q^{41} - 10q^{45} - 10q^{49} - 24q^{51} + 16q^{55} + 32q^{59} + 20q^{61} - 8q^{65} + 8q^{69} + 8q^{71} - 64q^{75} - 64q^{79} + 2q^{81} + 12q^{85} - 4q^{89} + 60q^{95} + 80q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 13 x^{8} + 56 x^{6} + 97 x^{4} + 61 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{9} + 12 \nu^{7} + 44 \nu^{5} + 55 \nu^{3} + 20 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{8} - 34 \nu^{6} - 112 \nu^{4} - 107 \nu^{2} - 2$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{8} - 34 \nu^{6} - 112 \nu^{4} - 111 \nu^{2} - 12$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{9} - 3 \nu^{8} - 34 \nu^{7} - 34 \nu^{6} - 110 \nu^{5} - 110 \nu^{4} - 93 \nu^{3} - 97 \nu^{2} + 16 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{9} - 23 \nu^{7} - 78 \nu^{5} - 82 \nu^{3} - 13 \nu$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{9} + 3 \nu^{8} - 34 \nu^{7} + 34 \nu^{6} - 110 \nu^{5} + 110 \nu^{4} - 93 \nu^{3} + 97 \nu^{2} + 16 \nu$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{9} + 35 \nu^{7} + 122 \nu^{5} + 137 \nu^{3} + 29 \nu$$$$)/2$$ $$\beta_{8}$$ $$=$$ $$($$$$5 \nu^{9} + 5 \nu^{8} + 56 \nu^{7} + 58 \nu^{6} + 178 \nu^{5} + 198 \nu^{4} + 151 \nu^{3} + 203 \nu^{2} - 14 \nu + 20$$$$)/4$$ $$\beta_{9}$$ $$=$$ $$($$$$5 \nu^{9} - 5 \nu^{8} + 56 \nu^{7} - 58 \nu^{6} + 178 \nu^{5} - 198 \nu^{4} + 151 \nu^{3} - 203 \nu^{2} - 14 \nu - 20$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{5} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{9} + \beta_{8} + 4 \beta_{7} + \beta_{6} + 6 \beta_{5} + \beta_{4} - 2 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{6} + 2 \beta_{4} + 5 \beta_{3} - 7 \beta_{2} + 23$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-8 \beta_{9} - 8 \beta_{8} - 21 \beta_{7} - 6 \beta_{6} - 41 \beta_{5} - 6 \beta_{4} + 3 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-3 \beta_{9} + 3 \beta_{8} + 17 \beta_{6} - 17 \beta_{4} - 24 \beta_{3} + 46 \beta_{2} - 128$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$52 \beta_{9} + 52 \beta_{8} + 125 \beta_{7} + 32 \beta_{6} + 273 \beta_{5} + 32 \beta_{4} + 7 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$34 \beta_{9} - 34 \beta_{8} - 118 \beta_{6} + 118 \beta_{4} + 121 \beta_{3} - 297 \beta_{2} + 769$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-327 \beta_{9} - 327 \beta_{8} - 776 \beta_{7} - 175 \beta_{6} - 1782 \beta_{5} - 175 \beta_{4} - 122 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.84576i − 0.271831i 2.52064i − 1.28447i − 1.23118i 1.23118i 1.28447i − 2.52064i 0.271831i 1.84576i
0 3.25260i 0 1.49436 1.66340i 0 1.00000i 0 −7.57939 0
449.2 0 2.19794i 0 −1.64514 + 1.51444i 0 1.00000i 0 −1.83094 0
449.3 0 1.83297i 0 −1.86302 + 1.23660i 0 1.00000i 0 −0.359777 0
449.4 0 1.63460i 0 2.23081 + 0.153266i 0 1.00000i 0 0.328072 0
449.5 0 0.746976i 0 0.782984 + 2.09450i 0 1.00000i 0 2.44203 0
449.6 0 0.746976i 0 0.782984 2.09450i 0 1.00000i 0 2.44203 0
449.7 0 1.63460i 0 2.23081 0.153266i 0 1.00000i 0 0.328072 0
449.8 0 1.83297i 0 −1.86302 1.23660i 0 1.00000i 0 −0.359777 0
449.9 0 2.19794i 0 −1.64514 1.51444i 0 1.00000i 0 −1.83094 0
449.10 0 3.25260i 0 1.49436 + 1.66340i 0 1.00000i 0 −7.57939 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.10 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.b 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.g.n 10
4.b odd 2 1 2240.2.g.o 10
5.b even 2 1 inner 2240.2.g.n 10
8.b even 2 1 1120.2.g.c yes 10
8.d odd 2 1 1120.2.g.b 10
20.d odd 2 1 2240.2.g.o 10
40.e odd 2 1 1120.2.g.b 10
40.f even 2 1 1120.2.g.c yes 10
40.i odd 4 1 5600.2.a.bv 5
40.i odd 4 1 5600.2.a.bx 5
40.k even 4 1 5600.2.a.bu 5
40.k even 4 1 5600.2.a.bw 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.b 10 8.d odd 2 1
1120.2.g.b 10 40.e odd 2 1
1120.2.g.c yes 10 8.b even 2 1
1120.2.g.c yes 10 40.f even 2 1
2240.2.g.n 10 1.a even 1 1 trivial
2240.2.g.n 10 5.b even 2 1 inner
2240.2.g.o 10 4.b odd 2 1
2240.2.g.o 10 20.d odd 2 1
5600.2.a.bu 5 40.k even 4 1
5600.2.a.bv 5 40.i odd 4 1
5600.2.a.bw 5 40.k even 4 1
5600.2.a.bx 5 40.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 $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{10} + 22 T_{3}^{8} + 165 T_{3}^{6} + 532 T_{3}^{4} + 708 T_{3}^{2} + 256$$ $$T_{11}^{5} + 4 T_{11}^{4} - 25 T_{11}^{3} - 152 T_{11}^{2} - 248 T_{11} - 128$$ $$T_{19}^{5} - 12 T_{19}^{4} + 30 T_{19}^{3} + 72 T_{19}^{2} - 184 T_{19} - 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$256 + 708 T^{2} + 532 T^{4} + 165 T^{6} + 22 T^{8} + T^{10}$$
$5$ $$3125 - 1250 T - 125 T^{2} + 100 T^{3} + 100 T^{4} - 116 T^{5} + 20 T^{6} + 4 T^{7} - T^{8} - 2 T^{9} + T^{10}$$
$7$ $$( 1 + T^{2} )^{5}$$
$11$ $$( -128 - 248 T - 152 T^{2} - 25 T^{3} + 4 T^{4} + T^{5} )^{2}$$
$13$ $$678976 + 274724 T^{2} + 41932 T^{4} + 2965 T^{6} + 94 T^{8} + T^{10}$$
$17$ $$1024 + 4496 T^{2} + 3944 T^{4} + 937 T^{6} + 66 T^{8} + T^{10}$$
$19$ $$( -256 - 184 T + 72 T^{2} + 30 T^{3} - 12 T^{4} + T^{5} )^{2}$$
$23$ $$262144 + 151552 T^{2} + 31488 T^{4} + 2832 T^{6} + 104 T^{8} + T^{10}$$
$29$ $$( 4744 - 124 T - 462 T^{2} - 15 T^{3} + 12 T^{4} + T^{5} )^{2}$$
$31$ $$( 4096 - 4480 T + 1184 T^{2} - 56 T^{3} - 12 T^{4} + T^{5} )^{2}$$
$37$ $$65536 + 196608 T^{2} + 73728 T^{4} + 5648 T^{6} + 136 T^{8} + T^{10}$$
$41$ $$( -128 - 1056 T - 632 T^{2} - 100 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$43$ $$228130816 + 45581312 T^{2} + 2136320 T^{4} + 41040 T^{6} + 344 T^{8} + T^{10}$$
$47$ $$4096 + 358976 T^{2} + 69392 T^{4} + 4617 T^{6} + 122 T^{8} + T^{10}$$
$53$ $$3444736 + 2001152 T^{2} + 333312 T^{4} + 13920 T^{6} + 208 T^{8} + T^{10}$$
$59$ $$( -58112 + 2232 T + 2048 T^{2} - 106 T^{3} - 16 T^{4} + T^{5} )^{2}$$
$61$ $$( -1648 + 72 T + 468 T^{2} - 46 T^{3} - 10 T^{4} + T^{5} )^{2}$$
$67$ $$262144 + 299008 T^{2} + 101888 T^{4} + 10768 T^{6} + 232 T^{8} + T^{10}$$
$71$ $$( 1024 - 2240 T + 1344 T^{2} - 228 T^{3} - 4 T^{4} + T^{5} )^{2}$$
$73$ $$4194304 + 2052352 T^{2} + 306688 T^{4} + 13792 T^{6} + 224 T^{8} + T^{10}$$
$79$ $$( 3904 + 4500 T + 1884 T^{2} + 363 T^{3} + 32 T^{4} + T^{5} )^{2}$$
$83$ $$750321664 + 192013312 T^{2} + 12747648 T^{4} + 155812 T^{6} + 684 T^{8} + T^{10}$$
$89$ $$( 3104 + 1936 T - 400 T^{2} - 168 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$97$ $$11723776 + 14907024 T^{2} + 1074792 T^{4} + 26681 T^{6} + 274 T^{8} + T^{10}$$