# Properties

 Label 2240.2.n.i Level $2240$ Weight $2$ Character orbit 2240.n Analytic conductor $17.886$ Analytic rank $0$ Dimension $16$ CM discriminant -56 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: 16.0.101415451701035401216.7 Defining polynomial: $$x^{16} - 18 x^{12} + 145 x^{8} - 72 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{22}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{6} - \beta_{7} ) q^{3} -\beta_{11} q^{5} -\beta_{3} q^{7} + ( 3 - \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{6} - \beta_{7} ) q^{3} -\beta_{11} q^{5} -\beta_{3} q^{7} + ( 3 - \beta_{1} ) q^{9} + ( \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{13} + ( -\beta_{2} + \beta_{3} - \beta_{5} - \beta_{8} ) q^{15} + ( -\beta_{14} - \beta_{15} ) q^{19} + ( -\beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{21} + ( -2 \beta_{2} - \beta_{5} ) q^{23} + ( -\beta_{1} - \beta_{4} ) q^{25} + ( \beta_{6} - \beta_{7} + \beta_{14} - \beta_{15} ) q^{27} + ( -2 \beta_{6} - \beta_{14} ) q^{35} + ( 5 \beta_{5} - 4 \beta_{8} ) q^{39} + ( \beta_{10} - 3 \beta_{11} + 2 \beta_{12} ) q^{45} + 7 q^{49} + ( -\beta_{1} + 2 \beta_{4} - 2 \beta_{9} ) q^{57} + ( -4 \beta_{6} - 4 \beta_{7} + \beta_{14} + \beta_{15} ) q^{59} + ( -2 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{61} + ( 2 \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{63} + ( -3 + 2 \beta_{1} - 3 \beta_{4} + \beta_{9} ) q^{65} + ( -\beta_{10} - 6 \beta_{11} + 6 \beta_{12} + \beta_{13} ) q^{69} -2 \beta_{8} q^{71} + ( -\beta_{6} - 4 \beta_{7} + 2 \beta_{14} - \beta_{15} ) q^{75} + 6 \beta_{8} q^{79} + ( -1 - 3 \beta_{1} ) q^{81} + ( \beta_{6} - \beta_{7} + 2 \beta_{14} - 2 \beta_{15} ) q^{83} + ( 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{14} - 2 \beta_{15} ) q^{91} + ( \beta_{2} + 3 \beta_{3} - 4 \beta_{5} + 2 \beta_{8} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 48q^{9} + O(q^{10})$$ $$16q + 48q^{9} + 112q^{49} - 48q^{65} - 16q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 18 x^{12} + 145 x^{8} - 72 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{12} + 10 \nu^{8} - 5 \nu^{4} - 508$$$$)/180$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{14} + 20 \nu^{10} - 185 \nu^{6} + 442 \nu^{2}$$$$)/60$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{14} - 130 \nu^{10} + 1079 \nu^{6} - 884 \nu^{2}$$$$)/144$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{12} + 190 \nu^{8} - 1495 \nu^{4} + 212$$$$)/120$$ $$\beta_{5}$$ $$=$$ $$($$$$-13 \nu^{14} + 230 \nu^{10} - 1805 \nu^{6} + 316 \nu^{2}$$$$)/120$$ $$\beta_{6}$$ $$=$$ $$($$$$-35 \nu^{15} - 76 \nu^{13} + 650 \nu^{11} + 1360 \nu^{9} - 5395 \nu^{7} - 10820 \nu^{5} + 5140 \nu^{3} + 2912 \nu$$$$)/1440$$ $$\beta_{7}$$ $$=$$ $$($$$$-16 \nu^{15} + 5 \nu^{13} + 280 \nu^{11} - 90 \nu^{9} - 2180 \nu^{7} + 745 \nu^{5} + 92 \nu^{3} - 540 \nu$$$$)/240$$ $$\beta_{8}$$ $$=$$ $$($$$$11 \nu^{14} - 194 \nu^{10} + 1531 \nu^{6} - 268 \nu^{2}$$$$)/72$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{12} - 18 \nu^{8} + 141 \nu^{4} - 36$$$$)/4$$ $$\beta_{10}$$ $$=$$ $$($$$$35 \nu^{15} - 76 \nu^{13} - 650 \nu^{11} + 1360 \nu^{9} + 5395 \nu^{7} - 10820 \nu^{5} - 5140 \nu^{3} + 2912 \nu$$$$)/720$$ $$\beta_{11}$$ $$=$$ $$($$$$-29 \nu^{15} + 16 \nu^{13} + 518 \nu^{11} - 280 \nu^{9} - 4141 \nu^{7} + 2168 \nu^{5} + 1564 \nu^{3} - 176 \nu$$$$)/288$$ $$\beta_{12}$$ $$=$$ $$($$$$85 \nu^{15} - 17 \nu^{13} - 1510 \nu^{11} + 290 \nu^{9} + 11945 \nu^{7} - 2245 \nu^{5} - 2960 \nu^{3} - 836 \nu$$$$)/720$$ $$\beta_{13}$$ $$=$$ $$($$$$-16 \nu^{15} - 5 \nu^{13} + 280 \nu^{11} + 90 \nu^{9} - 2180 \nu^{7} - 745 \nu^{5} + 92 \nu^{3} + 540 \nu$$$$)/120$$ $$\beta_{14}$$ $$=$$ $$($$$$305 \nu^{15} - 8 \nu^{13} - 5390 \nu^{11} + 200 \nu^{9} + 42385 \nu^{7} - 1840 \nu^{5} - 6700 \nu^{3} + 6256 \nu$$$$)/1440$$ $$\beta_{15}$$ $$=$$ $$($$$$97 \nu^{15} + 95 \nu^{13} - 1750 \nu^{11} - 1670 \nu^{9} + 14165 \nu^{7} + 13075 \nu^{5} - 7544 \nu^{3} - 2500 \nu$$$$)/720$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - \beta_{10} - \beta_{7} - 3 \beta_{6}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{8} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{15} + 3 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} - 2 \beta_{11} - 5 \beta_{10} + 9 \beta_{7} + 7 \beta_{6}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{9} - 6 \beta_{4} + 9 \beta_{1} + 18$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{15} + 13 \beta_{14} - 7 \beta_{13} - 26 \beta_{12} - 22 \beta_{11} - 3 \beta_{10} + 25 \beta_{7} - 19 \beta_{6}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$25 \beta_{8} + 40 \beta_{5} + 14 \beta_{3} + 10 \beta_{2}$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$41 \beta_{15} + 3 \beta_{14} + 45 \beta_{13} + 6 \beta_{12} - 82 \beta_{11} - 61 \beta_{10} + 49 \beta_{7} + 119 \beta_{6}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-36 \beta_{9} - 102 \beta_{4} + 63 \beta_{1} + 34$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-31 \beta_{15} + 157 \beta_{14} - 191 \beta_{13} - 314 \beta_{12} - 62 \beta_{11} + 113 \beta_{10} + 413 \beta_{7} + 69 \beta_{6}$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$319 \beta_{8} + 422 \beta_{5} - 82 \beta_{3} - 58 \beta_{2}$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$553 \beta_{15} - 345 \beta_{14} + 65 \beta_{13} - 690 \beta_{12} - 1106 \beta_{11} - 437 \beta_{10} - 423 \beta_{7} + 1219 \beta_{6}$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$-350 \beta_{9} - 990 \beta_{4} - 135 \beta_{1} - 1782$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$973 \beta_{15} + 997 \beta_{14} - 2383 \beta_{13} - 1994 \beta_{12} + 1946 \beta_{11} + 2373 \beta_{10} + 3793 \beta_{7} + 3749 \beta_{6}$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$2197 \beta_{8} + 1924 \beta_{5} - 3346 \beta_{3} - 2366 \beta_{2}$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$4097 \beta_{15} - 6429 \beta_{14} - 4995 \beta_{13} - 12858 \beta_{12} - 8194 \beta_{11} + 635 \beta_{10} - 14087 \beta_{7} + 5159 \beta_{6}$$$$)/8$$

## 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
 0.137538 − 0.752908i 0.137538 + 0.752908i −0.752908 + 0.137538i −0.752908 − 0.137538i 0.332046 − 1.81768i 0.332046 + 1.81768i 1.81768 − 0.332046i 1.81768 + 0.332046i −1.81768 − 0.332046i −1.81768 + 0.332046i −0.332046 − 1.81768i −0.332046 + 1.81768i 0.752908 + 0.137538i 0.752908 − 0.137538i −0.137538 − 0.752908i −0.137538 + 0.752908i
0 −2.97127 0 −2.14973 0.615370i 0 −2.64575 0 5.82843 0
1119.2 0 −2.97127 0 −2.14973 + 0.615370i 0 −2.64575 0 5.82843 0
1119.3 0 −2.97127 0 2.14973 0.615370i 0 2.64575 0 5.82843 0
1119.4 0 −2.97127 0 2.14973 + 0.615370i 0 2.64575 0 5.82843 0
1119.5 0 −1.78089 0 −0.615370 2.14973i 0 2.64575 0 0.171573 0
1119.6 0 −1.78089 0 −0.615370 + 2.14973i 0 2.64575 0 0.171573 0
1119.7 0 −1.78089 0 0.615370 2.14973i 0 −2.64575 0 0.171573 0
1119.8 0 −1.78089 0 0.615370 + 2.14973i 0 −2.64575 0 0.171573 0
1119.9 0 1.78089 0 −0.615370 2.14973i 0 −2.64575 0 0.171573 0
1119.10 0 1.78089 0 −0.615370 + 2.14973i 0 −2.64575 0 0.171573 0
1119.11 0 1.78089 0 0.615370 2.14973i 0 2.64575 0 0.171573 0
1119.12 0 1.78089 0 0.615370 + 2.14973i 0 2.64575 0 0.171573 0
1119.13 0 2.97127 0 −2.14973 0.615370i 0 2.64575 0 5.82843 0
1119.14 0 2.97127 0 −2.14973 + 0.615370i 0 2.64575 0 5.82843 0
1119.15 0 2.97127 0 2.14973 0.615370i 0 −2.64575 0 5.82843 0
1119.16 0 2.97127 0 2.14973 + 0.615370i 0 −2.64575 0 5.82843 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
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
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
35.c odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner
56.e even 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.i 16
4.b odd 2 1 inner 2240.2.n.i 16
5.b even 2 1 inner 2240.2.n.i 16
7.b odd 2 1 inner 2240.2.n.i 16
8.b even 2 1 inner 2240.2.n.i 16
8.d odd 2 1 inner 2240.2.n.i 16
20.d odd 2 1 inner 2240.2.n.i 16
28.d even 2 1 inner 2240.2.n.i 16
35.c odd 2 1 inner 2240.2.n.i 16
40.e odd 2 1 inner 2240.2.n.i 16
40.f even 2 1 inner 2240.2.n.i 16
56.e even 2 1 inner 2240.2.n.i 16
56.h odd 2 1 CM 2240.2.n.i 16
140.c even 2 1 inner 2240.2.n.i 16
280.c odd 2 1 inner 2240.2.n.i 16
280.n even 2 1 inner 2240.2.n.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.n.i 16 1.a even 1 1 trivial
2240.2.n.i 16 4.b odd 2 1 inner
2240.2.n.i 16 5.b even 2 1 inner
2240.2.n.i 16 7.b odd 2 1 inner
2240.2.n.i 16 8.b even 2 1 inner
2240.2.n.i 16 8.d odd 2 1 inner
2240.2.n.i 16 20.d odd 2 1 inner
2240.2.n.i 16 28.d even 2 1 inner
2240.2.n.i 16 35.c odd 2 1 inner
2240.2.n.i 16 40.e odd 2 1 inner
2240.2.n.i 16 40.f even 2 1 inner
2240.2.n.i 16 56.e even 2 1 inner
2240.2.n.i 16 56.h odd 2 1 CM
2240.2.n.i 16 140.c even 2 1 inner
2240.2.n.i 16 280.c odd 2 1 inner
2240.2.n.i 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 $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{4} - 12 T_{3}^{2} + 28$$ $$T_{11}$$ $$T_{17}$$ $$T_{23}^{2} - 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 28 - 12 T^{2} + T^{4} )^{4}$$
$5$ $$( 625 - 22 T^{4} + T^{8} )^{2}$$
$7$ $$( -7 + T^{2} )^{8}$$
$11$ $$T^{16}$$
$13$ $$( 28 + 52 T^{2} + T^{4} )^{4}$$
$17$ $$T^{16}$$
$19$ $$( 1372 + 76 T^{2} + T^{4} )^{4}$$
$23$ $$( -56 + T^{2} )^{8}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$T^{16}$$
$41$ $$T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$( 8092 + 236 T^{2} + T^{4} )^{4}$$
$61$ $$( 14812 - 244 T^{2} + T^{4} )^{4}$$
$67$ $$T^{16}$$
$71$ $$( 32 + T^{2} )^{8}$$
$73$ $$T^{16}$$
$79$ $$( 288 + T^{2} )^{8}$$
$83$ $$( 26908 - 332 T^{2} + T^{4} )^{4}$$
$89$ $$T^{16}$$
$97$ $$T^{16}$$