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 $4$

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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:

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

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