# Properties

 Label 2240.2.n.j Level $2240$ Weight $2$ Character orbit 2240.n Analytic conductor $17.886$ Analytic rank $0$ Dimension $16$ CM discriminant -35 Inner twists $16$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: 16.0.968265199641600000000.1 Defining polynomial: $$x^{16} + 9 x^{14} + 44 x^{12} + 261 x^{10} + 1029 x^{8} + 1044 x^{6} + 704 x^{4} + 576 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{24}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} -\beta_{7} q^{5} + ( \beta_{2} - \beta_{9} ) q^{7} + ( 3 + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} -\beta_{7} q^{5} + ( \beta_{2} - \beta_{9} ) q^{7} + ( 3 + \beta_{5} ) q^{9} + ( -\beta_{6} + \beta_{13} ) q^{11} + ( \beta_{1} - 2 \beta_{7} ) q^{13} + ( -\beta_{11} + \beta_{12} ) q^{15} + \beta_{14} q^{17} + \beta_{8} q^{21} -5 q^{25} + ( 5 \beta_{3} + \beta_{4} ) q^{27} + ( -\beta_{8} - 2 \beta_{10} ) q^{29} + ( -3 \beta_{14} - \beta_{15} ) q^{33} + ( \beta_{6} + 2 \beta_{13} ) q^{35} -\beta_{11} q^{39} + ( -5 \beta_{1} - \beta_{7} ) q^{45} + ( 4 \beta_{2} + \beta_{9} ) q^{47} -7 q^{49} + ( 5 \beta_{6} - 3 \beta_{13} ) q^{51} + ( -4 \beta_{2} + \beta_{9} ) q^{55} + 7 \beta_{2} q^{63} + ( -8 + \beta_{5} ) q^{65} -3 \beta_{12} q^{71} + ( 2 \beta_{14} - \beta_{15} ) q^{73} -5 \beta_{3} q^{75} + ( 3 \beta_{1} + 2 \beta_{7} ) q^{77} + ( -3 \beta_{11} + \beta_{12} ) q^{79} + ( 17 + 4 \beta_{5} ) q^{81} + ( -2 \beta_{3} - \beta_{4} ) q^{83} + ( \beta_{8} - 2 \beta_{10} ) q^{85} + ( -6 \beta_{2} + 5 \beta_{9} ) q^{87} + ( 5 \beta_{6} + 3 \beta_{13} ) q^{91} + ( -\beta_{14} + 2 \beta_{15} ) q^{97} + ( -10 \beta_{6} + 8 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 56q^{9} + O(q^{10})$$ $$16q + 56q^{9} - 80q^{25} - 112q^{49} - 120q^{65} + 304q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 9 x^{14} + 44 x^{12} + 261 x^{10} + 1029 x^{8} + 1044 x^{6} + 704 x^{4} + 576 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$497 \nu^{15} + 2509 \nu^{13} + 3120 \nu^{11} + 29877 \nu^{9} - 70311 \nu^{7} - 1854840 \nu^{5} - 3412832 \nu^{3} - 2926144 \nu$$$$)/979200$$ $$\beta_{2}$$ $$=$$ $$($$$$2149 \nu^{14} + 19933 \nu^{12} + 101340 \nu^{10} + 602409 \nu^{8} + 2424393 \nu^{6} + 3235620 \nu^{4} + 3583856 \nu^{2} + 1643072$$$$)/734400$$ $$\beta_{3}$$ $$=$$ $$($$$$-3461 \nu^{14} - 31577 \nu^{12} - 154680 \nu^{10} - 906681 \nu^{8} - 3598197 \nu^{6} - 3654480 \nu^{4} - 924544 \nu^{2} - 478528$$$$)/587520$$ $$\beta_{4}$$ $$=$$ $$($$$$1129 \nu^{14} + 10413 \nu^{12} + 50680 \nu^{10} + 296749 \nu^{8} + 1193593 \nu^{6} + 1214320 \nu^{4} + 304896 \nu^{2} + 1033792$$$$)/97920$$ $$\beta_{5}$$ $$=$$ $$($$$$-211 \nu^{14} - 1947 \nu^{12} - 9540 \nu^{10} - 55791 \nu^{8} - 222927 \nu^{6} - 226620 \nu^{4} - 57104 \nu^{2} - 62208$$$$)/16320$$ $$\beta_{6}$$ $$=$$ $$($$$$281 \nu^{15} + 2173 \nu^{13} + 9400 \nu^{11} + 59773 \nu^{9} + 206089 \nu^{7} - 12368 \nu^{5} + 56672 \nu^{3} + 80448 \nu$$$$)/39168$$ $$\beta_{7}$$ $$=$$ $$($$$$1433 \nu^{15} + 12397 \nu^{13} + 57960 \nu^{11} + 347133 \nu^{9} + 1322937 \nu^{7} + 850080 \nu^{5} - 24608 \nu^{3} + 381248 \nu$$$$)/195840$$ $$\beta_{8}$$ $$=$$ $$($$$$4869 \nu^{14} + 42713 \nu^{12} + 208440 \nu^{10} + 1251129 \nu^{8} + 4861173 \nu^{6} + 4791120 \nu^{4} + 5233536 \nu^{2} + 2665792$$$$)/326400$$ $$\beta_{9}$$ $$=$$ $$($$$$11621 \nu^{14} + 95837 \nu^{12} + 439260 \nu^{10} + 2705961 \nu^{8} + 9941577 \nu^{6} + 4710180 \nu^{4} + 4845424 \nu^{2} + 3563008$$$$)/734400$$ $$\beta_{10}$$ $$=$$ $$($$$$6087 \nu^{14} + 49379 \nu^{12} + 222120 \nu^{10} + 1376067 \nu^{8} + 4966359 \nu^{6} + 1523760 \nu^{4} + 1465728 \nu^{2} + 1471936$$$$)/326400$$ $$\beta_{11}$$ $$=$$ $$($$$$-23527 \nu^{15} - 211299 \nu^{13} - 1027720 \nu^{11} - 6097507 \nu^{9} - 24001879 \nu^{7} - 23516560 \nu^{5} - 14834688 \nu^{3} - 19911616 \nu$$$$)/1958400$$ $$\beta_{12}$$ $$=$$ $$($$$$-1127 \nu^{15} - 9779 \nu^{13} - 45720 \nu^{11} - 273507 \nu^{9} - 1043559 \nu^{7} - 670560 \nu^{5} + 19712 \nu^{3} - 300736 \nu$$$$)/86400$$ $$\beta_{13}$$ $$=$$ $$($$$$1121 \nu^{15} + 9662 \nu^{13} + 45885 \nu^{11} + 276741 \nu^{9} + 1055262 \nu^{7} + 813825 \nu^{5} + 624184 \nu^{3} + 284368 \nu$$$$)/73440$$ $$\beta_{14}$$ $$=$$ $$($$$$-2257 \nu^{15} - 19829 \nu^{13} - 95480 \nu^{11} - 571957 \nu^{9} - 2210849 \nu^{7} - 1960240 \nu^{5} - 1404928 \nu^{3} - 558656 \nu$$$$)/130560$$ $$\beta_{15}$$ $$=$$ $$($$$$6439 \nu^{15} + 52163 \nu^{13} + 235560 \nu^{11} + 1462179 \nu^{9} + 5282103 \nu^{7} + 1775280 \nu^{5} + 2249216 \nu^{3} + 1757632 \nu$$$$)/195840$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - 2 \beta_{13} + \beta_{12} - 2 \beta_{11} - 2 \beta_{6} + 2 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{10} - 3 \beta_{9} + \beta_{8} - \beta_{5} + 3 \beta_{3} - 4$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{14} + 4 \beta_{13} - 5 \beta_{12} - 10 \beta_{7} + 2 \beta_{6}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$7 \beta_{10} - 9 \beta_{9} - 2 \beta_{8} + 9 \beta_{5} + 3 \beta_{4} - 15 \beta_{3} + 12 \beta_{2} - 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-22 \beta_{15} - 11 \beta_{14} + 5 \beta_{13} + 17 \beta_{12} - 5 \beta_{11} + 22 \beta_{7} + 63 \beta_{6} + 11 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-35 \beta_{10} + 45 \beta_{9} + 5 \beta_{8} + 9 \beta_{4} + 18 \beta_{3} - 45 \beta_{2} - 81$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$38 \beta_{15} + 93 \beta_{14} + 91 \beta_{13} + 13 \beta_{12} + 91 \beta_{11} + 186 \beta_{7} - 143 \beta_{6} - 203 \beta_{1}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-127 \beta_{10} + 255 \beta_{9} - 190 \beta_{8} - 63 \beta_{5} - 99 \beta_{4} - 57 \beta_{3} + 396 \beta_{2} + 760$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$323 \beta_{15} - 646 \beta_{14} - 1156 \beta_{13} + 95 \beta_{12} + 170 \beta_{7} - 578 \beta_{6}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$347 \beta_{10} - 891 \beta_{9} + 952 \beta_{8} - 605 \beta_{5} - 561 \beta_{4} + 231 \beta_{3} - 2244 \beta_{2} + 3808$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$1664 \beta_{15} + 2635 \beta_{14} + 2047 \beta_{13} - 899 \beta_{12} - 2047 \beta_{11} - 5270 \beta_{7} - 5643 \beta_{6} + 4577 \beta_{1}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$5670 \beta_{10} - 7245 \beta_{9} - 810 \beta_{8} + 846 \beta_{4} + 1692 \beta_{3} + 7245 \beta_{2} - 7567$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-15880 \beta_{15} - 10413 \beta_{14} + 233 \beta_{13} - 11875 \beta_{12} + 233 \beta_{11} - 20826 \beta_{7} + 47267 \beta_{6} - 521 \beta_{1}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-20851 \beta_{10} + 21957 \beta_{9} + 13456 \beta_{8} + 34307 \beta_{5} + 13689 \beta_{4} - 49335 \beta_{3} - 54756 \beta_{2} - 53824$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$-30349 \beta_{15} + 60698 \beta_{14} + 108580 \beta_{13} + 93775 \beta_{12} + 167750 \beta_{7} + 54290 \beta_{6}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1119.1
 0.369600 + 2.25820i −0.369600 + 2.25820i −0.369600 − 2.25820i 0.369600 − 2.25820i 0.141174 + 0.862555i −0.141174 + 0.862555i −0.141174 − 0.862555i 0.141174 − 0.862555i 1.77086 − 1.44918i −1.77086 − 1.44918i −1.77086 + 1.44918i 1.77086 + 1.44918i 0.676408 − 0.553538i −0.676408 − 0.553538i −0.676408 + 0.553538i 0.676408 + 0.553538i
0 −3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.2 0 −3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.3 0 −3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.4 0 −3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.5 0 −1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.6 0 −1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.7 0 −1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.8 0 −1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.9 0 1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.10 0 1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.11 0 1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.12 0 1.17325 0 2.23607i 0 2.64575i 0 −1.62348 0
1119.13 0 3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.14 0 3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.15 0 3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
1119.16 0 3.40932 0 2.23607i 0 2.64575i 0 8.62348 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1119.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner
140.c even 2 1 inner
280.c odd 2 1 inner
280.n 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.n.j 16
4.b odd 2 1 inner 2240.2.n.j 16
5.b even 2 1 inner 2240.2.n.j 16
7.b odd 2 1 inner 2240.2.n.j 16
8.b even 2 1 inner 2240.2.n.j 16
8.d odd 2 1 inner 2240.2.n.j 16
20.d odd 2 1 inner 2240.2.n.j 16
28.d even 2 1 inner 2240.2.n.j 16
35.c odd 2 1 CM 2240.2.n.j 16
40.e odd 2 1 inner 2240.2.n.j 16
40.f even 2 1 inner 2240.2.n.j 16
56.e even 2 1 inner 2240.2.n.j 16
56.h odd 2 1 inner 2240.2.n.j 16
140.c even 2 1 inner 2240.2.n.j 16
280.c odd 2 1 inner 2240.2.n.j 16
280.n even 2 1 inner 2240.2.n.j 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.n.j 16 1.a even 1 1 trivial
2240.2.n.j 16 4.b odd 2 1 inner
2240.2.n.j 16 5.b even 2 1 inner
2240.2.n.j 16 7.b odd 2 1 inner
2240.2.n.j 16 8.b even 2 1 inner
2240.2.n.j 16 8.d odd 2 1 inner
2240.2.n.j 16 20.d odd 2 1 inner
2240.2.n.j 16 28.d even 2 1 inner
2240.2.n.j 16 35.c odd 2 1 CM
2240.2.n.j 16 40.e odd 2 1 inner
2240.2.n.j 16 40.f even 2 1 inner
2240.2.n.j 16 56.e even 2 1 inner
2240.2.n.j 16 56.h odd 2 1 inner
2240.2.n.j 16 140.c even 2 1 inner
2240.2.n.j 16 280.c odd 2 1 inner
2240.2.n.j 16 280.n even 2 1 inner

## Hecke kernels

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

 $$T_{3}^{4} - 13 T_{3}^{2} + 16$$ $$T_{11}^{4} - 31 T_{11}^{2} + 4$$ $$T_{17}^{4} - 39 T_{17}^{2} + 144$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 16 - 13 T^{2} + T^{4} )^{4}$$
$5$ $$( 5 + T^{2} )^{8}$$
$7$ $$( 7 + T^{2} )^{8}$$
$11$ $$( 4 - 31 T^{2} + T^{4} )^{4}$$
$13$ $$( 36 + 33 T^{2} + T^{4} )^{4}$$
$17$ $$( 144 - 39 T^{2} + T^{4} )^{4}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$( 2704 + 139 T^{2} + T^{4} )^{4}$$
$31$ $$T^{16}$$
$37$ $$T^{16}$$
$41$ $$T^{16}$$
$43$ $$T^{16}$$
$47$ $$( 6084 + 219 T^{2} + T^{4} )^{4}$$
$53$ $$T^{16}$$
$59$ $$T^{16}$$
$61$ $$T^{16}$$
$67$ $$T^{16}$$
$71$ $$( 144 + T^{2} )^{8}$$
$73$ $$( -112 + T^{2} )^{8}$$
$79$ $$( 55696 + 473 T^{2} + T^{4} )^{4}$$
$83$ $$( -80 + T^{2} )^{8}$$
$89$ $$T^{16}$$
$97$ $$( 2704 - 239 T^{2} + T^{4} )^{4}$$