Properties

Label 960.1.cl.a
Level $960$
Weight $1$
Character orbit 960.cl
Analytic conductor $0.479$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 960.cl (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.479102412128\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{32}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.980785 0.195090i
0.980785 + 0.195090i
0.195090 + 0.980785i
−0.195090 0.980785i
0.831470 0.555570i
−0.831470 + 0.555570i
0.831470 + 0.555570i
−0.831470 0.555570i
0.195090 0.980785i
−0.195090 + 0.980785i
−0.980785 + 0.195090i
0.980785 0.195090i
0.555570 + 0.831470i
−0.555570 0.831470i
0.555570 0.831470i
−0.555570 + 0.831470i
−0.831470 + 0.555570i −0.980785 0.195090i 0.382683 0.923880i 0.555570 0.831470i 0.923880 0.382683i 0 0.195090 + 0.980785i 0.923880 + 0.382683i 1.00000i
29.2 0.831470 0.555570i 0.980785 + 0.195090i 0.382683 0.923880i −0.555570 + 0.831470i 0.923880 0.382683i 0 −0.195090 0.980785i 0.923880 + 0.382683i 1.00000i
149.1 −0.555570 + 0.831470i 0.195090 + 0.980785i −0.382683 0.923880i −0.831470 + 0.555570i −0.923880 0.382683i 0 0.980785 + 0.195090i −0.923880 + 0.382683i 1.00000i
149.2 0.555570 0.831470i −0.195090 0.980785i −0.382683 0.923880i 0.831470 0.555570i −0.923880 0.382683i 0 −0.980785 0.195090i −0.923880 + 0.382683i 1.00000i
269.1 −0.195090 + 0.980785i 0.831470 0.555570i −0.923880 0.382683i 0.980785 0.195090i 0.382683 + 0.923880i 0 0.555570 0.831470i 0.382683 0.923880i 1.00000i
269.2 0.195090 0.980785i −0.831470 + 0.555570i −0.923880 0.382683i −0.980785 + 0.195090i 0.382683 + 0.923880i 0 −0.555570 + 0.831470i 0.382683 0.923880i 1.00000i
389.1 −0.195090 0.980785i 0.831470 + 0.555570i −0.923880 + 0.382683i 0.980785 + 0.195090i 0.382683 0.923880i 0 0.555570 + 0.831470i 0.382683 + 0.923880i 1.00000i
389.2 0.195090 + 0.980785i −0.831470 0.555570i −0.923880 + 0.382683i −0.980785 0.195090i 0.382683 0.923880i 0 −0.555570 0.831470i 0.382683 + 0.923880i 1.00000i
509.1 −0.555570 0.831470i 0.195090 0.980785i −0.382683 + 0.923880i −0.831470 0.555570i −0.923880 + 0.382683i 0 0.980785 0.195090i −0.923880 0.382683i 1.00000i
509.2 0.555570 + 0.831470i −0.195090 + 0.980785i −0.382683 + 0.923880i 0.831470 + 0.555570i −0.923880 + 0.382683i 0 −0.980785 + 0.195090i −0.923880 0.382683i 1.00000i
629.1 −0.831470 0.555570i −0.980785 + 0.195090i 0.382683 + 0.923880i 0.555570 + 0.831470i 0.923880 + 0.382683i 0 0.195090 0.980785i 0.923880 0.382683i 1.00000i
629.2 0.831470 + 0.555570i 0.980785 0.195090i 0.382683 + 0.923880i −0.555570 0.831470i 0.923880 + 0.382683i 0 −0.195090 + 0.980785i 0.923880 0.382683i 1.00000i
749.1 −0.980785 0.195090i 0.555570 + 0.831470i 0.923880 + 0.382683i −0.195090 0.980785i −0.382683 0.923880i 0 −0.831470 0.555570i −0.382683 + 0.923880i 1.00000i
749.2 0.980785 + 0.195090i −0.555570 0.831470i 0.923880 + 0.382683i 0.195090 + 0.980785i −0.382683 0.923880i 0 0.831470 + 0.555570i −0.382683 + 0.923880i 1.00000i
869.1 −0.980785 + 0.195090i 0.555570 0.831470i 0.923880 0.382683i −0.195090 + 0.980785i −0.382683 + 0.923880i 0 −0.831470 + 0.555570i −0.382683 0.923880i 1.00000i
869.2 0.980785 0.195090i −0.555570 + 0.831470i 0.923880 0.382683i 0.195090 0.980785i −0.382683 + 0.923880i 0 0.831470 0.555570i −0.382683 0.923880i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 869.2
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
64.i even 16 1 inner
192.q odd 16 1 inner
320.bf even 16 1 inner
960.cl odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.1.cl.a 16
3.b odd 2 1 inner 960.1.cl.a 16
4.b odd 2 1 3840.1.cl.a 16
5.b even 2 1 inner 960.1.cl.a 16
12.b even 2 1 3840.1.cl.a 16
15.d odd 2 1 CM 960.1.cl.a 16
20.d odd 2 1 3840.1.cl.a 16
60.h even 2 1 3840.1.cl.a 16
64.i even 16 1 inner 960.1.cl.a 16
64.j odd 16 1 3840.1.cl.a 16
192.q odd 16 1 inner 960.1.cl.a 16
192.s even 16 1 3840.1.cl.a 16
320.bf even 16 1 inner 960.1.cl.a 16
320.bh odd 16 1 3840.1.cl.a 16
960.cl odd 16 1 inner 960.1.cl.a 16
960.cp even 16 1 3840.1.cl.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.1.cl.a 16 1.a even 1 1 trivial
960.1.cl.a 16 3.b odd 2 1 inner
960.1.cl.a 16 5.b even 2 1 inner
960.1.cl.a 16 15.d odd 2 1 CM
960.1.cl.a 16 64.i even 16 1 inner
960.1.cl.a 16 192.q odd 16 1 inner
960.1.cl.a 16 320.bf even 16 1 inner
960.1.cl.a 16 960.cl odd 16 1 inner
3840.1.cl.a 16 4.b odd 2 1
3840.1.cl.a 16 12.b even 2 1
3840.1.cl.a 16 20.d odd 2 1
3840.1.cl.a 16 60.h even 2 1
3840.1.cl.a 16 64.j odd 16 1
3840.1.cl.a 16 192.s even 16 1
3840.1.cl.a 16 320.bh odd 16 1
3840.1.cl.a 16 960.cp even 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(960, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{16} \)
$3$ \( 1 + T^{16} \)
$5$ \( 1 + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( T^{16} \)
$17$ \( 4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16} \)
$19$ \( ( 2 + 8 T + 12 T^{2} + 2 T^{4} - 8 T^{5} + T^{8} )^{2} \)
$23$ \( 4 - 32 T^{2} + 128 T^{4} + 192 T^{6} + 140 T^{8} + 16 T^{10} + T^{16} \)
$29$ \( T^{16} \)
$31$ \( ( 2 + 4 T^{2} + T^{4} )^{4} \)
$37$ \( T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( 4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16} \)
$53$ \( 16 + 64 T^{4} + 128 T^{8} - 16 T^{12} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( ( 2 + 8 T + 4 T^{2} - 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} )^{2} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( T^{16} \)
$79$ \( ( 2 + 2 T + T^{2} )^{8} \)
$83$ \( 16 - 64 T^{4} + 128 T^{8} + 16 T^{12} + T^{16} \)
$89$ \( T^{16} \)
$97$ \( T^{16} \)
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