# Properties

 Label 984.1.cj.b Level $984$ Weight $1$ Character orbit 984.cj Analytic conductor $0.491$ Analytic rank $0$ Dimension $16$ Projective image $D_{40}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$984 = 2^{3} \cdot 3 \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 984.cj (of order $$40$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.491079972431$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{40})$$ Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{40}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{40} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{40}^{13} q^{2} + \zeta_{40}^{17} q^{3} -\zeta_{40}^{6} q^{4} -\zeta_{40}^{10} q^{6} -\zeta_{40}^{19} q^{8} -\zeta_{40}^{14} q^{9} +O(q^{10})$$ $$q + \zeta_{40}^{13} q^{2} + \zeta_{40}^{17} q^{3} -\zeta_{40}^{6} q^{4} -\zeta_{40}^{10} q^{6} -\zeta_{40}^{19} q^{8} -\zeta_{40}^{14} q^{9} + ( \zeta_{40}^{15} + \zeta_{40}^{18} ) q^{11} + \zeta_{40}^{3} q^{12} + \zeta_{40}^{12} q^{16} + ( -\zeta_{40}^{4} + \zeta_{40}^{19} ) q^{17} + \zeta_{40}^{7} q^{18} + ( -\zeta_{40}^{9} + \zeta_{40}^{10} ) q^{19} + ( -\zeta_{40}^{8} - \zeta_{40}^{11} ) q^{22} + \zeta_{40}^{16} q^{24} -\zeta_{40}^{2} q^{25} + \zeta_{40}^{11} q^{27} -\zeta_{40}^{5} q^{32} + ( -\zeta_{40}^{12} - \zeta_{40}^{15} ) q^{33} + ( -\zeta_{40}^{12} - \zeta_{40}^{17} ) q^{34} - q^{36} + ( \zeta_{40}^{2} - \zeta_{40}^{3} ) q^{38} -\zeta_{40}^{11} q^{41} + ( -\zeta_{40} - \zeta_{40}^{5} ) q^{43} + ( \zeta_{40} + \zeta_{40}^{4} ) q^{44} -\zeta_{40}^{9} q^{48} + \zeta_{40}^{9} q^{49} -\zeta_{40}^{15} q^{50} + ( \zeta_{40} - \zeta_{40}^{16} ) q^{51} -\zeta_{40}^{4} q^{54} + ( \zeta_{40}^{6} - \zeta_{40}^{7} ) q^{57} + ( \zeta_{40}^{3} - \zeta_{40}^{13} ) q^{59} -\zeta_{40}^{18} q^{64} + ( \zeta_{40}^{5} + \zeta_{40}^{8} ) q^{66} + ( -1 - \zeta_{40}^{7} ) q^{67} + ( \zeta_{40}^{5} + \zeta_{40}^{10} ) q^{68} -\zeta_{40}^{13} q^{72} + ( -\zeta_{40}^{3} + \zeta_{40}^{7} ) q^{73} -\zeta_{40}^{19} q^{75} + ( \zeta_{40}^{15} - \zeta_{40}^{16} ) q^{76} -\zeta_{40}^{8} q^{81} + \zeta_{40}^{4} q^{82} + ( \zeta_{40}^{6} + \zeta_{40}^{14} ) q^{83} + ( -\zeta_{40}^{14} - \zeta_{40}^{18} ) q^{86} + ( \zeta_{40}^{14} + \zeta_{40}^{17} ) q^{88} + ( -\zeta_{40} + \zeta_{40}^{8} ) q^{89} + \zeta_{40}^{2} q^{96} + ( \zeta_{40}^{13} + \zeta_{40}^{14} ) q^{97} -\zeta_{40}^{2} q^{98} + ( \zeta_{40}^{9} + \zeta_{40}^{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 4q^{16} - 4q^{17} + 4q^{22} - 4q^{24} - 4q^{33} - 4q^{34} - 16q^{36} + 4q^{44} + 4q^{51} - 4q^{54} - 4q^{66} - 16q^{67} + 4q^{76} + 4q^{81} + 4q^{82} - 4q^{89} + 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/984\mathbb{Z}\right)^\times$$.

 $$n$$ $$247$$ $$329$$ $$457$$ $$493$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{40}^{11}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.453990 + 0.891007i −0.156434 + 0.987688i −0.453990 − 0.891007i −0.987688 + 0.156434i −0.891007 + 0.453990i 0.891007 − 0.453990i 0.987688 − 0.156434i 0.453990 + 0.891007i 0.156434 − 0.987688i 0.453990 − 0.891007i 0.987688 + 0.156434i −0.156434 − 0.987688i −0.891007 − 0.453990i 0.891007 + 0.453990i 0.156434 + 0.987688i −0.987688 − 0.156434i
0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 0 1.00000i 0 −0.453990 0.891007i 0.951057 + 0.309017i 0
35.1 −0.891007 0.453990i −0.453990 0.891007i 0.587785 + 0.809017i 0 1.00000i 0 −0.156434 0.987688i −0.587785 + 0.809017i 0
179.1 0.156434 0.987688i −0.987688 + 0.156434i −0.951057 0.309017i 0 1.00000i 0 −0.453990 + 0.891007i 0.951057 0.309017i 0
227.1 0.453990 + 0.891007i 0.891007 + 0.453990i −0.587785 + 0.809017i 0 1.00000i 0 −0.987688 0.156434i 0.587785 + 0.809017i 0
275.1 −0.987688 0.156434i 0.156434 + 0.987688i 0.951057 + 0.309017i 0 1.00000i 0 −0.891007 0.453990i −0.951057 + 0.309017i 0
299.1 0.987688 + 0.156434i −0.156434 0.987688i 0.951057 + 0.309017i 0 1.00000i 0 0.891007 + 0.453990i −0.951057 + 0.309017i 0
347.1 −0.453990 0.891007i −0.891007 0.453990i −0.587785 + 0.809017i 0 1.00000i 0 0.987688 + 0.156434i 0.587785 + 0.809017i 0
395.1 −0.156434 + 0.987688i 0.987688 0.156434i −0.951057 0.309017i 0 1.00000i 0 0.453990 0.891007i 0.951057 0.309017i 0
539.1 0.891007 + 0.453990i 0.453990 + 0.891007i 0.587785 + 0.809017i 0 1.00000i 0 0.156434 + 0.987688i −0.587785 + 0.809017i 0
563.1 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 0 1.00000i 0 0.453990 + 0.891007i 0.951057 + 0.309017i 0
587.1 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 0.809017i 0 1.00000i 0 0.987688 0.156434i 0.587785 0.809017i 0
731.1 −0.891007 + 0.453990i −0.453990 + 0.891007i 0.587785 0.809017i 0 1.00000i 0 −0.156434 + 0.987688i −0.587785 0.809017i 0
755.1 −0.987688 + 0.156434i 0.156434 0.987688i 0.951057 0.309017i 0 1.00000i 0 −0.891007 + 0.453990i −0.951057 0.309017i 0
803.1 0.987688 0.156434i −0.156434 + 0.987688i 0.951057 0.309017i 0 1.00000i 0 0.891007 0.453990i −0.951057 0.309017i 0
827.1 0.891007 0.453990i 0.453990 0.891007i 0.587785 0.809017i 0 1.00000i 0 0.156434 0.987688i −0.587785 0.809017i 0
971.1 0.453990 0.891007i 0.891007 0.453990i −0.587785 0.809017i 0 1.00000i 0 −0.987688 + 0.156434i 0.587785 0.809017i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 971.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
123.o even 40 1 inner
984.cj odd 40 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 984.1.cj.b yes 16
3.b odd 2 1 984.1.cj.a 16
4.b odd 2 1 3936.1.fx.a 16
8.b even 2 1 3936.1.fx.a 16
8.d odd 2 1 CM 984.1.cj.b yes 16
12.b even 2 1 3936.1.fx.b 16
24.f even 2 1 984.1.cj.a 16
24.h odd 2 1 3936.1.fx.b 16
41.h odd 40 1 984.1.cj.a 16
123.o even 40 1 inner 984.1.cj.b yes 16
164.o even 40 1 3936.1.fx.b 16
328.bd even 40 1 984.1.cj.a 16
328.bf odd 40 1 3936.1.fx.b 16
492.be odd 40 1 3936.1.fx.a 16
984.ch even 40 1 3936.1.fx.a 16
984.cj odd 40 1 inner 984.1.cj.b yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
984.1.cj.a 16 3.b odd 2 1
984.1.cj.a 16 24.f even 2 1
984.1.cj.a 16 41.h odd 40 1
984.1.cj.a 16 328.bd even 40 1
984.1.cj.b yes 16 1.a even 1 1 trivial
984.1.cj.b yes 16 8.d odd 2 1 CM
984.1.cj.b yes 16 123.o even 40 1 inner
984.1.cj.b yes 16 984.cj odd 40 1 inner
3936.1.fx.a 16 4.b odd 2 1
3936.1.fx.a 16 8.b even 2 1
3936.1.fx.a 16 492.be odd 40 1
3936.1.fx.a 16 984.ch even 40 1
3936.1.fx.b 16 12.b even 2 1
3936.1.fx.b 16 24.h odd 2 1
3936.1.fx.b 16 164.o even 40 1
3936.1.fx.b 16 328.bf odd 40 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{16} - \cdots$$ acting on $$S_{1}^{\mathrm{new}}(984, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$3$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$T^{16}$$
$11$ $$1 + 12 T + 68 T^{2} + 144 T^{3} + 222 T^{4} + 92 T^{5} + 118 T^{6} + 72 T^{7} - 80 T^{8} - 8 T^{9} + 6 T^{10} - 16 T^{11} + 7 T^{12} + 4 T^{13} - 2 T^{14} + T^{16}$$
$13$ $$T^{16}$$
$17$ $$16 + 32 T + 16 T^{2} - 32 T^{3} - 56 T^{4} - 112 T^{5} - 128 T^{6} - 16 T^{7} + 156 T^{8} + 160 T^{9} + 112 T^{10} + 64 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16}$$
$19$ $$1 + 8 T - 12 T^{2} - 104 T^{3} + 242 T^{4} - 132 T^{5} + 88 T^{6} - 112 T^{7} + 120 T^{8} - 52 T^{9} + 66 T^{10} - 24 T^{11} + 27 T^{12} - 4 T^{13} + 8 T^{14} + T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$T^{16}$$
$41$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$43$ $$625 - 500 T^{4} + 150 T^{8} + 5 T^{12} + T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$( 16 - 8 T^{2} + 4 T^{4} - 2 T^{6} + T^{8} )^{2}$$
$61$ $$T^{16}$$
$67$ $$1 + 8 T + 76 T^{2} + 392 T^{3} + 1394 T^{4} + 3632 T^{5} + 7112 T^{6} + 10656 T^{7} + 12376 T^{8} + 11220 T^{9} + 7942 T^{10} + 4356 T^{11} + 1819 T^{12} + 560 T^{13} + 120 T^{14} + 16 T^{15} + T^{16}$$
$71$ $$T^{16}$$
$73$ $$( 1 + 7 T^{4} + T^{8} )^{2}$$
$79$ $$T^{16}$$
$83$ $$( 1 + 3 T^{2} + T^{4} )^{4}$$
$89$ $$1 - 8 T + 46 T^{2} - 32 T^{3} - 131 T^{4} - 72 T^{5} + 162 T^{6} + 304 T^{7} + 256 T^{8} + 160 T^{9} + 82 T^{10} + 44 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16}$$
$97$ $$1 + 12 T + 58 T^{2} + 104 T^{3} + 237 T^{4} + 112 T^{5} - 42 T^{6} - 208 T^{7} - 80 T^{8} - 8 T^{9} + 86 T^{10} + 24 T^{11} + 2 T^{12} - 16 T^{13} - 2 T^{14} + T^{16}$$