Properties

Label 820.1.by.a
Level $820$
Weight $1$
Character orbit 820.by
Analytic conductor $0.409$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 820.by (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.409233310359\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q -\zeta_{40}^{6} q^{2} + \zeta_{40}^{12} q^{4} + \zeta_{40} q^{5} -\zeta_{40}^{18} q^{8} -\zeta_{40}^{15} q^{9} +O(q^{10})\) \( q -\zeta_{40}^{6} q^{2} + \zeta_{40}^{12} q^{4} + \zeta_{40} q^{5} -\zeta_{40}^{18} q^{8} -\zeta_{40}^{15} q^{9} -\zeta_{40}^{7} q^{10} + ( \zeta_{40}^{8} + \zeta_{40}^{19} ) q^{13} -\zeta_{40}^{4} q^{16} + ( \zeta_{40}^{4} + \zeta_{40}^{17} ) q^{17} -\zeta_{40} q^{18} + \zeta_{40}^{13} q^{20} + \zeta_{40}^{2} q^{25} + ( \zeta_{40}^{5} - \zeta_{40}^{14} ) q^{26} + ( -\zeta_{40}^{12} - \zeta_{40}^{17} ) q^{29} + \zeta_{40}^{10} q^{32} + ( \zeta_{40}^{3} - \zeta_{40}^{10} ) q^{34} + \zeta_{40}^{7} q^{36} + ( -\zeta_{40}^{5} - \zeta_{40}^{9} ) q^{37} -\zeta_{40}^{19} q^{40} + \zeta_{40}^{6} q^{41} -\zeta_{40}^{16} q^{45} -\zeta_{40}^{3} q^{49} -\zeta_{40}^{8} q^{50} + ( -1 - \zeta_{40}^{11} ) q^{52} + ( -\zeta_{40}^{8} + \zeta_{40}^{11} ) q^{53} + ( -\zeta_{40}^{3} + \zeta_{40}^{18} ) q^{58} + ( -\zeta_{40}^{2} + \zeta_{40}^{16} ) q^{61} -\zeta_{40}^{16} q^{64} + ( -1 + \zeta_{40}^{9} ) q^{65} + ( -\zeta_{40}^{9} + \zeta_{40}^{16} ) q^{68} -\zeta_{40}^{13} q^{72} + ( \zeta_{40}^{7} - \zeta_{40}^{13} ) q^{73} + ( \zeta_{40}^{11} + \zeta_{40}^{15} ) q^{74} -\zeta_{40}^{5} q^{80} -\zeta_{40}^{10} q^{81} -\zeta_{40}^{12} q^{82} + ( \zeta_{40}^{5} + \zeta_{40}^{18} ) q^{85} + ( \zeta_{40}^{14} - \zeta_{40}^{19} ) q^{89} -\zeta_{40}^{2} q^{90} + ( -\zeta_{40}^{14} + \zeta_{40}^{15} ) q^{97} + \zeta_{40}^{9} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{4} + O(q^{10}) \) \( 16q + 4q^{4} - 4q^{13} - 4q^{16} + 4q^{17} - 4q^{29} + 4q^{45} + 4q^{50} - 16q^{52} + 4q^{53} - 4q^{61} + 4q^{64} - 16q^{65} - 4q^{68} - 4q^{82} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/820\mathbb{Z}\right)^\times\).

\(n\) \(411\) \(621\) \(657\)
\(\chi(n)\) \(-1\) \(\zeta_{40}^{7}\) \(-\zeta_{40}^{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.891007 0.453990i
0.156434 0.987688i
−0.987688 + 0.156434i
−0.453990 0.891007i
0.453990 + 0.891007i
0.891007 0.453990i
0.453990 0.891007i
0.987688 0.156434i
−0.987688 0.156434i
−0.156434 0.987688i
−0.156434 + 0.987688i
0.891007 + 0.453990i
0.156434 + 0.987688i
0.987688 + 0.156434i
−0.453990 + 0.891007i
−0.891007 + 0.453990i
0.951057 0.309017i 0 0.809017 0.587785i −0.891007 0.453990i 0 0 0.587785 0.809017i 0.707107 + 0.707107i −0.987688 0.156434i
67.1 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 0.156434 0.987688i 0 0 −0.951057 + 0.309017i 0.707107 + 0.707107i 0.891007 0.453990i
147.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i −0.987688 + 0.156434i 0 0 0.951057 + 0.309017i −0.707107 0.707107i 0.453990 0.891007i
227.1 −0.951057 0.309017i 0 0.809017 + 0.587785i −0.453990 0.891007i 0 0 −0.587785 0.809017i −0.707107 0.707107i 0.156434 + 0.987688i
347.1 −0.951057 0.309017i 0 0.809017 + 0.587785i 0.453990 + 0.891007i 0 0 −0.587785 0.809017i 0.707107 + 0.707107i −0.156434 0.987688i
403.1 0.951057 + 0.309017i 0 0.809017 + 0.587785i 0.891007 0.453990i 0 0 0.587785 + 0.809017i −0.707107 + 0.707107i 0.987688 0.156434i
423.1 −0.951057 + 0.309017i 0 0.809017 0.587785i 0.453990 0.891007i 0 0 −0.587785 + 0.809017i 0.707107 0.707107i −0.156434 + 0.987688i
427.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i 0.987688 0.156434i 0 0 0.951057 + 0.309017i 0.707107 + 0.707107i −0.453990 + 0.891007i
463.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i −0.987688 0.156434i 0 0 0.951057 0.309017i −0.707107 + 0.707107i 0.453990 + 0.891007i
503.1 0.587785 0.809017i 0 −0.309017 0.951057i −0.156434 0.987688i 0 0 −0.951057 0.309017i −0.707107 + 0.707107i −0.891007 0.453990i
507.1 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −0.156434 + 0.987688i 0 0 −0.951057 + 0.309017i −0.707107 0.707107i −0.891007 + 0.453990i
527.1 0.951057 0.309017i 0 0.809017 0.587785i 0.891007 + 0.453990i 0 0 0.587785 0.809017i −0.707107 0.707107i 0.987688 + 0.156434i
563.1 0.587785 0.809017i 0 −0.309017 0.951057i 0.156434 + 0.987688i 0 0 −0.951057 0.309017i 0.707107 0.707107i 0.891007 + 0.453990i
603.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i 0.987688 + 0.156434i 0 0 0.951057 0.309017i 0.707107 0.707107i −0.453990 0.891007i
643.1 −0.951057 + 0.309017i 0 0.809017 0.587785i −0.453990 + 0.891007i 0 0 −0.587785 + 0.809017i −0.707107 + 0.707107i 0.156434 0.987688i
663.1 0.951057 + 0.309017i 0 0.809017 + 0.587785i −0.891007 + 0.453990i 0 0 0.587785 + 0.809017i 0.707107 0.707107i −0.987688 + 0.156434i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 663.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
205.bb even 40 1 inner
820.by odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.1.by.a 16
4.b odd 2 1 CM 820.1.by.a 16
5.c odd 4 1 820.1.bz.a yes 16
20.e even 4 1 820.1.bz.a yes 16
41.h odd 40 1 820.1.bz.a yes 16
164.o even 40 1 820.1.bz.a yes 16
205.bb even 40 1 inner 820.1.by.a 16
820.by odd 40 1 inner 820.1.by.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.1.by.a 16 1.a even 1 1 trivial
820.1.by.a 16 4.b odd 2 1 CM
820.1.by.a 16 205.bb even 40 1 inner
820.1.by.a 16 820.by odd 40 1 inner
820.1.bz.a yes 16 5.c odd 4 1
820.1.bz.a yes 16 20.e even 4 1
820.1.bz.a yes 16 41.h odd 40 1
820.1.bz.a yes 16 164.o even 40 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(820, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( 1 - 8 T - 4 T^{2} + 68 T^{3} + 269 T^{4} + 328 T^{5} + 382 T^{6} + 284 T^{7} + 156 T^{8} + 60 T^{9} + 32 T^{10} + 44 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16} \)
$17$ \( 1 + 8 T - 4 T^{2} - 68 T^{3} + 269 T^{4} - 328 T^{5} + 382 T^{6} - 284 T^{7} + 156 T^{8} - 60 T^{9} + 32 T^{10} - 44 T^{11} + 34 T^{12} - 20 T^{13} + 10 T^{14} - 4 T^{15} + T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( 16 + 32 T + 16 T^{2} - 32 T^{3} - 56 T^{4} - 112 T^{5} - 128 T^{6} - 16 T^{7} + 156 T^{8} + 160 T^{9} + 112 T^{10} + 64 T^{11} + 34 T^{12} + 20 T^{13} + 10 T^{14} + 4 T^{15} + T^{16} \)
$31$ \( T^{16} \)
$37$ \( 625 - 500 T^{4} + 150 T^{8} + 5 T^{12} + T^{16} \)
$41$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( 1 + 8 T + 46 T^{2} + 32 T^{3} - 131 T^{4} + 72 T^{5} + 162 T^{6} - 304 T^{7} + 256 T^{8} - 160 T^{9} + 82 T^{10} - 44 T^{11} + 34 T^{12} - 20 T^{13} + 10 T^{14} - 4 T^{15} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( ( 1 + 6 T + 13 T^{2} + 10 T^{3} + 16 T^{4} + 10 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( ( 1 - 12 T^{2} + 19 T^{4} - 8 T^{6} + T^{8} )^{2} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( 16 - 32 T + 48 T^{2} - 64 T^{3} + 72 T^{4} - 112 T^{5} + 128 T^{6} - 112 T^{7} + 60 T^{8} + 48 T^{9} + 16 T^{10} - 24 T^{11} + 2 T^{12} - 4 T^{13} - 2 T^{14} + T^{16} \)
$97$ \( 1 + 12 T + 68 T^{2} + 144 T^{3} + 222 T^{4} + 92 T^{5} + 118 T^{6} + 72 T^{7} - 80 T^{8} - 8 T^{9} + 6 T^{10} - 16 T^{11} + 7 T^{12} + 4 T^{13} - 2 T^{14} + T^{16} \)
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