Properties

Label 1020.1.cl.a
Level $1020$
Weight $1$
Character orbit 1020.cl
Analytic conductor $0.509$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1020 = 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1020.cl (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.509046312886\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{16})\)
Coefficient field: \(\Q(\zeta_{32})\)
Defining polynomial: \(x^{16} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{32}^{11} q^{2} -\zeta_{32}^{15} q^{3} -\zeta_{32}^{6} q^{4} -\zeta_{32}^{3} q^{5} -\zeta_{32}^{10} q^{6} -\zeta_{32} q^{8} -\zeta_{32}^{14} q^{9} +O(q^{10})\) \( q -\zeta_{32}^{11} q^{2} -\zeta_{32}^{15} q^{3} -\zeta_{32}^{6} q^{4} -\zeta_{32}^{3} q^{5} -\zeta_{32}^{10} q^{6} -\zeta_{32} q^{8} -\zeta_{32}^{14} q^{9} + \zeta_{32}^{14} q^{10} -\zeta_{32}^{5} q^{12} -\zeta_{32}^{2} q^{15} + \zeta_{32}^{12} q^{16} + \zeta_{32}^{7} q^{17} -\zeta_{32}^{9} q^{18} + ( -\zeta_{32}^{8} - \zeta_{32}^{12} ) q^{19} + \zeta_{32}^{9} q^{20} + ( -\zeta_{32}^{5} + \zeta_{32}^{13} ) q^{23} - q^{24} + \zeta_{32}^{6} q^{25} -\zeta_{32}^{13} q^{27} + \zeta_{32}^{13} q^{30} + ( \zeta_{32}^{4} + \zeta_{32}^{10} ) q^{31} + \zeta_{32}^{7} q^{32} + \zeta_{32}^{2} q^{34} -\zeta_{32}^{4} q^{36} + ( -\zeta_{32}^{3} - \zeta_{32}^{7} ) q^{38} + \zeta_{32}^{4} q^{40} -\zeta_{32} q^{45} + ( -1 + \zeta_{32}^{8} ) q^{46} + ( -\zeta_{32}^{9} + \zeta_{32}^{15} ) q^{47} + \zeta_{32}^{11} q^{48} + \zeta_{32}^{10} q^{49} + \zeta_{32} q^{50} + \zeta_{32}^{6} q^{51} + ( \zeta_{32} + \zeta_{32}^{3} ) q^{53} -\zeta_{32}^{8} q^{54} + ( -\zeta_{32}^{7} - \zeta_{32}^{11} ) q^{57} + \zeta_{32}^{8} q^{60} + ( \zeta_{32}^{2} + \zeta_{32}^{8} ) q^{61} + ( \zeta_{32}^{5} - \zeta_{32}^{15} ) q^{62} + \zeta_{32}^{2} q^{64} -\zeta_{32}^{13} q^{68} + ( -\zeta_{32}^{4} + \zeta_{32}^{12} ) q^{69} + \zeta_{32}^{15} q^{72} + \zeta_{32}^{5} q^{75} + ( -\zeta_{32}^{2} + \zeta_{32}^{14} ) q^{76} + ( -\zeta_{32}^{4} - \zeta_{32}^{14} ) q^{79} -\zeta_{32}^{15} q^{80} -\zeta_{32}^{12} q^{81} + ( \zeta_{32}^{9} - \zeta_{32}^{11} ) q^{83} -\zeta_{32}^{10} q^{85} + \zeta_{32}^{12} q^{90} + ( \zeta_{32}^{3} + \zeta_{32}^{11} ) q^{92} + ( \zeta_{32}^{3} + \zeta_{32}^{9} ) q^{93} + ( -\zeta_{32}^{4} + \zeta_{32}^{10} ) q^{94} + ( \zeta_{32}^{11} + \zeta_{32}^{15} ) q^{95} + \zeta_{32}^{6} q^{96} + \zeta_{32}^{5} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 16q^{24} - 16q^{46} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1020\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(341\) \(511\) \(817\)
\(\chi(n)\) \(\zeta_{32}^{14}\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
−0.555570 0.831470i
0.555570 + 0.831470i
−0.980785 0.195090i
0.980785 + 0.195090i
−0.195090 0.980785i
0.195090 + 0.980785i
−0.555570 + 0.831470i
0.555570 0.831470i
0.831470 + 0.555570i
−0.831470 0.555570i
−0.980785 + 0.195090i
0.980785 0.195090i
−0.195090 + 0.980785i
0.195090 0.980785i
0.831470 0.555570i
−0.831470 + 0.555570i
−0.195090 0.980785i −0.555570 + 0.831470i −0.923880 + 0.382683i −0.980785 + 0.195090i 0.923880 + 0.382683i 0 0.555570 + 0.831470i −0.382683 0.923880i 0.382683 + 0.923880i
299.2 0.195090 + 0.980785i 0.555570 0.831470i −0.923880 + 0.382683i 0.980785 0.195090i 0.923880 + 0.382683i 0 −0.555570 0.831470i −0.382683 0.923880i 0.382683 + 0.923880i
419.1 −0.555570 + 0.831470i −0.980785 + 0.195090i −0.382683 0.923880i 0.831470 + 0.555570i 0.382683 0.923880i 0 0.980785 + 0.195090i 0.923880 0.382683i −0.923880 + 0.382683i
419.2 0.555570 0.831470i 0.980785 0.195090i −0.382683 0.923880i −0.831470 0.555570i 0.382683 0.923880i 0 −0.980785 0.195090i 0.923880 0.382683i −0.923880 + 0.382683i
479.1 −0.831470 + 0.555570i −0.195090 + 0.980785i 0.382683 0.923880i −0.555570 0.831470i −0.382683 0.923880i 0 0.195090 + 0.980785i −0.923880 0.382683i 0.923880 + 0.382683i
479.2 0.831470 0.555570i 0.195090 0.980785i 0.382683 0.923880i 0.555570 + 0.831470i −0.382683 0.923880i 0 −0.195090 0.980785i −0.923880 0.382683i 0.923880 + 0.382683i
539.1 −0.195090 + 0.980785i −0.555570 0.831470i −0.923880 0.382683i −0.980785 0.195090i 0.923880 0.382683i 0 0.555570 0.831470i −0.382683 + 0.923880i 0.382683 0.923880i
539.2 0.195090 0.980785i 0.555570 + 0.831470i −0.923880 0.382683i 0.980785 + 0.195090i 0.923880 0.382683i 0 −0.555570 + 0.831470i −0.382683 + 0.923880i 0.382683 0.923880i
719.1 −0.980785 0.195090i 0.831470 0.555570i 0.923880 + 0.382683i 0.195090 0.980785i −0.923880 + 0.382683i 0 −0.831470 0.555570i 0.382683 0.923880i −0.382683 + 0.923880i
719.2 0.980785 + 0.195090i −0.831470 + 0.555570i 0.923880 + 0.382683i −0.195090 + 0.980785i −0.923880 + 0.382683i 0 0.831470 + 0.555570i 0.382683 0.923880i −0.382683 + 0.923880i
779.1 −0.555570 0.831470i −0.980785 0.195090i −0.382683 + 0.923880i 0.831470 0.555570i 0.382683 + 0.923880i 0 0.980785 0.195090i 0.923880 + 0.382683i −0.923880 0.382683i
779.2 0.555570 + 0.831470i 0.980785 + 0.195090i −0.382683 + 0.923880i −0.831470 + 0.555570i 0.382683 + 0.923880i 0 −0.980785 + 0.195090i 0.923880 + 0.382683i −0.923880 0.382683i
839.1 −0.831470 0.555570i −0.195090 0.980785i 0.382683 + 0.923880i −0.555570 + 0.831470i −0.382683 + 0.923880i 0 0.195090 0.980785i −0.923880 + 0.382683i 0.923880 0.382683i
839.2 0.831470 + 0.555570i 0.195090 + 0.980785i 0.382683 + 0.923880i 0.555570 0.831470i −0.382683 + 0.923880i 0 −0.195090 + 0.980785i −0.923880 + 0.382683i 0.923880 0.382683i
959.1 −0.980785 + 0.195090i 0.831470 + 0.555570i 0.923880 0.382683i 0.195090 + 0.980785i −0.923880 0.382683i 0 −0.831470 + 0.555570i 0.382683 + 0.923880i −0.382683 0.923880i
959.2 0.980785 0.195090i −0.831470 0.555570i 0.923880 0.382683i −0.195090 0.980785i −0.923880 0.382683i 0 0.831470 0.555570i 0.382683 + 0.923880i −0.382683 0.923880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 959.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
68.i even 16 1 inner
204.t odd 16 1 inner
340.bg even 16 1 inner
1020.cl odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1020.1.cl.a 16
3.b odd 2 1 inner 1020.1.cl.a 16
4.b odd 2 1 1020.1.cl.b yes 16
5.b even 2 1 inner 1020.1.cl.a 16
12.b even 2 1 1020.1.cl.b yes 16
15.d odd 2 1 CM 1020.1.cl.a 16
17.e odd 16 1 1020.1.cl.b yes 16
20.d odd 2 1 1020.1.cl.b yes 16
51.i even 16 1 1020.1.cl.b yes 16
60.h even 2 1 1020.1.cl.b yes 16
68.i even 16 1 inner 1020.1.cl.a 16
85.p odd 16 1 1020.1.cl.b yes 16
204.t odd 16 1 inner 1020.1.cl.a 16
255.be even 16 1 1020.1.cl.b yes 16
340.bg even 16 1 inner 1020.1.cl.a 16
1020.cl odd 16 1 inner 1020.1.cl.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1020.1.cl.a 16 1.a even 1 1 trivial
1020.1.cl.a 16 3.b odd 2 1 inner
1020.1.cl.a 16 5.b even 2 1 inner
1020.1.cl.a 16 15.d odd 2 1 CM
1020.1.cl.a 16 68.i even 16 1 inner
1020.1.cl.a 16 204.t odd 16 1 inner
1020.1.cl.a 16 340.bg even 16 1 inner
1020.1.cl.a 16 1020.cl odd 16 1 inner
1020.1.cl.b yes 16 4.b odd 2 1
1020.1.cl.b yes 16 12.b even 2 1
1020.1.cl.b yes 16 17.e odd 16 1
1020.1.cl.b yes 16 20.d odd 2 1
1020.1.cl.b yes 16 51.i even 16 1
1020.1.cl.b yes 16 60.h even 2 1
1020.1.cl.b yes 16 85.p odd 16 1
1020.1.cl.b yes 16 255.be even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{4} + 2 T_{19}^{2} + 4 T_{19} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1020, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{16} \)
$3$ \( 1 + T^{16} \)
$5$ \( 1 + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( T^{16} \)
$17$ \( 1 + T^{16} \)
$19$ \( ( 2 + 4 T + 2 T^{2} + T^{4} )^{4} \)
$23$ \( 256 + T^{16} \)
$29$ \( T^{16} \)
$31$ \( ( 2 - 8 T + 20 T^{2} - 16 T^{3} + 2 T^{4} + T^{8} )^{2} \)
$37$ \( T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( 4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16} \)
$53$ \( 4 - 32 T^{2} + 128 T^{4} + 192 T^{6} + 140 T^{8} + 16 T^{10} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( ( 2 - 8 T + 4 T^{2} + 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} )^{2} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( T^{16} \)
$79$ \( ( 2 - 8 T + 12 T^{2} + 2 T^{4} + 8 T^{5} + T^{8} )^{2} \)
$83$ \( 4 - 32 T^{2} + 128 T^{4} + 192 T^{6} + 140 T^{8} + 16 T^{10} + T^{16} \)
$89$ \( T^{16} \)
$97$ \( T^{16} \)
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