# Properties

 Label 2240.2.g.p Level $2240$ Weight $2$ Character orbit 2240.g 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.g (of order $$2$$, degree $$1$$, not 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: $$x^{12} + 20 x^{10} + 148 x^{8} + 494 x^{6} + 708 x^{4} + 304 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{9} q^{3} -\beta_{10} q^{5} -\beta_{5} q^{7} + ( -1 - \beta_{3} + \beta_{10} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{9} q^{3} -\beta_{10} q^{5} -\beta_{5} q^{7} + ( -1 - \beta_{3} + \beta_{10} - \beta_{11} ) q^{9} -\beta_{2} q^{11} + ( \beta_{4} + \beta_{10} + \beta_{11} ) q^{13} + ( \beta_{1} + \beta_{2} + \beta_{5} - \beta_{9} ) q^{15} + ( \beta_{4} - 2 \beta_{6} ) q^{17} + ( -2 \beta_{1} + \beta_{7} + \beta_{8} ) q^{19} -\beta_{3} q^{21} + ( -2 \beta_{5} + 2 \beta_{9} ) q^{23} + ( -\beta_{3} + \beta_{6} - 2 \beta_{11} ) q^{25} + ( 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{27} + ( 4 - \beta_{3} - \beta_{10} + \beta_{11} ) q^{29} + ( 2 \beta_{7} + 2 \beta_{8} ) q^{31} + ( \beta_{4} - 2 \beta_{10} - 2 \beta_{11} ) q^{33} + \beta_{7} q^{35} + ( -2 \beta_{6} + 2 \beta_{10} + 2 \beta_{11} ) q^{37} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{7} + 2 \beta_{8} ) q^{39} + ( 2 + 2 \beta_{3} ) q^{41} + ( -4 \beta_{5} - 2 \beta_{9} ) q^{43} + ( -4 - \beta_{4} + \beta_{10} + 2 \beta_{11} ) q^{45} + ( 2 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{47} - q^{49} + ( \beta_{2} + 4 \beta_{7} + 4 \beta_{8} ) q^{51} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{53} + ( -2 \beta_{1} + \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{55} + ( -2 \beta_{4} + 2 \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{57} + ( 2 \beta_{1} - \beta_{7} - \beta_{8} ) q^{59} + ( -4 + 2 \beta_{3} - \beta_{10} + \beta_{11} ) q^{61} + ( \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{63} + ( 5 - 2 \beta_{3} - 3 \beta_{6} + \beta_{10} + \beta_{11} ) q^{65} + ( 4 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{67} + ( 8 - 2 \beta_{10} + 2 \beta_{11} ) q^{69} + 2 \beta_{1} q^{71} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{73} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{7} + 3 \beta_{9} ) q^{75} + \beta_{4} q^{77} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{7} - 2 \beta_{8} ) q^{79} + ( -1 + 2 \beta_{3} + 2 \beta_{10} - 2 \beta_{11} ) q^{81} + ( -2 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{83} + ( -2 - 5 \beta_{3} + 2 \beta_{4} + \beta_{10} + \beta_{11} ) q^{85} + ( 6 \beta_{5} - \beta_{7} + \beta_{8} - 5 \beta_{9} ) q^{87} -6 q^{89} + ( \beta_{2} - \beta_{7} - \beta_{8} ) q^{91} + ( -4 \beta_{4} + 4 \beta_{6} ) q^{93} + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{5} + 4 \beta_{8} - \beta_{9} ) q^{95} + ( \beta_{4} + 2 \beta_{6} ) q^{97} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{7} + 2 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 16q^{9} + O(q^{10})$$ $$12q - 16q^{9} - 4q^{21} - 4q^{25} + 44q^{29} + 32q^{41} - 48q^{45} - 12q^{49} - 40q^{61} + 52q^{65} + 96q^{69} - 4q^{81} - 44q^{85} - 72q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 20 x^{10} + 148 x^{8} + 494 x^{6} + 708 x^{4} + 304 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{10} + 67 \nu^{8} + 382 \nu^{6} + 874 \nu^{4} + 751 \nu^{2} + 116$$$$)/62$$ $$\beta_{2}$$ $$=$$ $$($$$$-9 \nu^{10} - 143 \nu^{8} - 689 \nu^{6} - 773 \nu^{4} + 1077 \nu^{2} + 607$$$$)/124$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{10} + 58 \nu^{8} + 395 \nu^{6} + 1105 \nu^{4} + 1098 \nu^{2} + 211$$$$)/62$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} + 71 \nu^{9} + 979 \nu^{7} + 5039 \nu^{5} + 9945 \nu^{3} + 5113 \nu$$$$)/186$$ $$\beta_{5}$$ $$=$$ $$($$$$11 \nu^{11} + 223 \nu^{9} + 1655 \nu^{7} + 5395 \nu^{5} + 7095 \nu^{3} + 2303 \nu$$$$)/372$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{11} - 125 \nu^{9} - 808 \nu^{7} - 2351 \nu^{5} - 3213 \nu^{3} - 1660 \nu$$$$)/186$$ $$\beta_{7}$$ $$=$$ $$($$$$-16 \nu^{11} - 12 \nu^{10} - 299 \nu^{9} - 201 \nu^{8} - 1993 \nu^{7} - 1146 \nu^{6} - 5666 \nu^{5} - 2436 \nu^{4} - 6321 \nu^{3} - 951 \nu^{2} - 1549 \nu + 768$$$$)/372$$ $$\beta_{8}$$ $$=$$ $$($$$$16 \nu^{11} - 12 \nu^{10} + 299 \nu^{9} - 201 \nu^{8} + 1993 \nu^{7} - 1146 \nu^{6} + 5666 \nu^{5} - 2436 \nu^{4} + 6321 \nu^{3} - 951 \nu^{2} + 1549 \nu + 768$$$$)/372$$ $$\beta_{9}$$ $$=$$ $$($$$$25 \nu^{11} + 473 \nu^{9} + 3271 \nu^{7} + 10097 \nu^{5} + 13521 \nu^{3} + 6367 \nu$$$$)/372$$ $$\beta_{10}$$ $$=$$ $$($$$$-20 \nu^{11} - 15 \nu^{10} - 397 \nu^{9} - 228 \nu^{8} - 2933 \nu^{7} - 1107 \nu^{6} - 9826 \nu^{5} - 1743 \nu^{4} - 13923 \nu^{3} - 282 \nu^{2} - 4889 \nu - 249$$$$)/372$$ $$\beta_{11}$$ $$=$$ $$($$$$-20 \nu^{11} + 15 \nu^{10} - 397 \nu^{9} + 228 \nu^{8} - 2933 \nu^{7} + 1107 \nu^{6} - 9826 \nu^{5} + 1743 \nu^{4} - 13923 \nu^{3} + 282 \nu^{2} - 4889 \nu + 249$$$$)/372$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{6} - \beta_{5}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{3} - \beta_{1} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - 4 \beta_{9} - \beta_{8} + \beta_{7} - 6 \beta_{6} + 8 \beta_{5}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-7 \beta_{11} + 7 \beta_{10} - 5 \beta_{8} - 5 \beta_{7} - 7 \beta_{3} + 9 \beta_{1} + 37$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-7 \beta_{11} - 7 \beta_{10} + 21 \beta_{9} + 11 \beta_{8} - 11 \beta_{7} + 35 \beta_{6} - 61 \beta_{5} + 2 \beta_{4}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$24 \beta_{11} - 24 \beta_{10} + 12 \beta_{8} + 12 \beta_{7} + 23 \beta_{3} + 2 \beta_{2} - 33 \beta_{1} - 108$$ $$\nu^{7}$$ $$=$$ $$($$$$40 \beta_{11} + 40 \beta_{10} - 121 \beta_{9} - 96 \beta_{8} + 96 \beta_{7} - 213 \beta_{6} + 433 \beta_{5} - 24 \beta_{4}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-335 \beta_{11} + 335 \beta_{10} - 115 \beta_{8} - 115 \beta_{7} - 291 \beta_{3} - 56 \beta_{2} + 459 \beta_{1} + 1329$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-209 \beta_{11} - 209 \beta_{10} + 730 \beta_{9} + 765 \beta_{8} - 765 \beta_{7} + 1344 \beta_{6} - 2974 \beta_{5} + 220 \beta_{4}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$2369 \beta_{11} - 2369 \beta_{10} + 539 \beta_{8} + 539 \beta_{7} + 1823 \beta_{3} + 556 \beta_{2} - 3133 \beta_{1} - 8461$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$1007 \beta_{11} + 1007 \beta_{10} - 4523 \beta_{9} - 5815 \beta_{8} + 5815 \beta_{7} - 8705 \beta_{6} + 20179 \beta_{5} - 1830 \beta_{4}$$$$)/2$$

## 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
 1.67709i 0.178681i 0.810591i 2.36682i 1.97020i − 2.64856i 2.64856i − 1.97020i − 2.36682i − 0.810591i − 0.178681i − 1.67709i
0 2.85577i 0 0.649832 2.13956i 0 1.00000i 0 −5.15544 0
449.2 0 2.85577i 0 0.649832 + 2.13956i 0 1.00000i 0 −5.15544 0
449.3 0 2.17741i 0 1.45926 1.69427i 0 1.00000i 0 −1.74111 0
449.4 0 2.17741i 0 1.45926 + 1.69427i 0 1.00000i 0 −1.74111 0
449.5 0 0.321637i 0 −2.10909 0.742782i 0 1.00000i 0 2.89655 0
449.6 0 0.321637i 0 −2.10909 + 0.742782i 0 1.00000i 0 2.89655 0
449.7 0 0.321637i 0 −2.10909 0.742782i 0 1.00000i 0 2.89655 0
449.8 0 0.321637i 0 −2.10909 + 0.742782i 0 1.00000i 0 2.89655 0
449.9 0 2.17741i 0 1.45926 1.69427i 0 1.00000i 0 −1.74111 0
449.10 0 2.17741i 0 1.45926 + 1.69427i 0 1.00000i 0 −1.74111 0
449.11 0 2.85577i 0 0.649832 2.13956i 0 1.00000i 0 −5.15544 0
449.12 0 2.85577i 0 0.649832 + 2.13956i 0 1.00000i 0 −5.15544 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.12 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
5.b even 2 1 inner
20.d odd 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.p 12
4.b odd 2 1 inner 2240.2.g.p 12
5.b even 2 1 inner 2240.2.g.p 12
8.b even 2 1 1120.2.g.d 12
8.d odd 2 1 1120.2.g.d 12
20.d odd 2 1 inner 2240.2.g.p 12
40.e odd 2 1 1120.2.g.d 12
40.f even 2 1 1120.2.g.d 12
40.i odd 4 1 5600.2.a.by 6
40.i odd 4 1 5600.2.a.bz 6
40.k even 4 1 5600.2.a.by 6
40.k even 4 1 5600.2.a.bz 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.d 12 8.b even 2 1
1120.2.g.d 12 8.d odd 2 1
1120.2.g.d 12 40.e odd 2 1
1120.2.g.d 12 40.f even 2 1
2240.2.g.p 12 1.a even 1 1 trivial
2240.2.g.p 12 4.b odd 2 1 inner
2240.2.g.p 12 5.b even 2 1 inner
2240.2.g.p 12 20.d odd 2 1 inner
5600.2.a.by 6 40.i odd 4 1
5600.2.a.by 6 40.k even 4 1
5600.2.a.bz 6 40.i odd 4 1
5600.2.a.bz 6 40.k even 4 1

## Hecke kernels

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

 $$T_{3}^{6} + 13 T_{3}^{4} + 40 T_{3}^{2} + 4$$ $$T_{11}^{6} - 45 T_{11}^{4} + 592 T_{11}^{2} - 1856$$ $$T_{19}^{6} - 104 T_{19}^{4} + 2708 T_{19}^{2} - 4176$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( 4 + 40 T^{2} + 13 T^{4} + T^{6} )^{2}$$
$5$ $$( 125 + 5 T^{2} + 16 T^{3} + T^{4} + T^{6} )^{2}$$
$7$ $$( 1 + T^{2} )^{6}$$
$11$ $$( -1856 + 592 T^{2} - 45 T^{4} + T^{6} )^{2}$$
$13$ $$( 116 + 1056 T^{2} + 65 T^{4} + T^{6} )^{2}$$
$17$ $$( 37584 + 3608 T^{2} + 109 T^{4} + T^{6} )^{2}$$
$19$ $$( -4176 + 2708 T^{2} - 104 T^{4} + T^{6} )^{2}$$
$23$ $$( 1024 + 656 T^{2} + 56 T^{4} + T^{6} )^{2}$$
$29$ $$( 12 + 16 T - 11 T^{2} + T^{3} )^{4}$$
$31$ $$( -29696 + 4416 T^{2} - 128 T^{4} + T^{6} )^{2}$$
$37$ $$( 118784 + 8080 T^{2} + 168 T^{4} + T^{6} )^{2}$$
$41$ $$( 48 - 4 T - 8 T^{2} + T^{3} )^{4}$$
$43$ $$( 256 + 2048 T^{2} + 116 T^{4} + T^{6} )^{2}$$
$47$ $$( 602176 + 21680 T^{2} + 257 T^{4} + T^{6} )^{2}$$
$53$ $$( 1856 + 8176 T^{2} + 204 T^{4} + T^{6} )^{2}$$
$59$ $$( -4176 + 2708 T^{2} - 104 T^{4} + T^{6} )^{2}$$
$61$ $$( -124 + 2 T + 10 T^{2} + T^{3} )^{4}$$
$67$ $$( 65536 + 7168 T^{2} + 196 T^{4} + T^{6} )^{2}$$
$71$ $$( -7424 + 1680 T^{2} - 104 T^{4} + T^{6} )^{2}$$
$73$ $$( 1856 + 8176 T^{2} + 204 T^{4} + T^{6} )^{2}$$
$79$ $$( -464 + 648 T^{2} - 149 T^{4} + T^{6} )^{2}$$
$83$ $$( 419904 + 34084 T^{2} + 372 T^{4} + T^{6} )^{2}$$
$89$ $$( 6 + T )^{12}$$
$97$ $$( 464 + 664 T^{2} + 189 T^{4} + T^{6} )^{2}$$