Properties

Label 1028.1.l.a
Level $1028$
Weight $1$
Character orbit 1028.l
Analytic conductor $0.513$
Analytic rank $0$
Dimension $16$
Projective image $D_{32}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1028 = 2^{2} \cdot 257 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1028.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.513038832987\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{32})\)
Defining polynomial: \(x^{16} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1028\mathbb{Z}\right)^\times\).

\(n\) \(515\) \(517\)
\(\chi(n)\) \(-1\) \(\zeta_{32}^{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−0.980785 + 0.195090i
0.831470 0.555570i
−0.555570 0.831470i
0.555570 + 0.831470i
−0.831470 + 0.555570i
0.980785 0.195090i
−0.195090 0.980785i
−0.195090 + 0.980785i
0.980785 + 0.195090i
−0.831470 0.555570i
−0.555570 + 0.831470i
0.555570 0.831470i
0.831470 + 0.555570i
−0.980785 0.195090i
0.195090 0.980785i
0.195090 + 0.980785i
1.00000i 0 −1.00000 1.75535 + 0.172887i 0 0 1.00000i 0.195090 + 0.980785i −0.172887 + 1.75535i
223.1 1.00000i 0 −1.00000 0.577774 1.90466i 0 0 1.00000i 0.555570 + 0.831470i −1.90466 0.577774i
227.1 1.00000i 0 −1.00000 −1.36347 + 0.728789i 0 0 1.00000i 0.831470 0.555570i 0.728789 + 1.36347i
287.1 1.00000i 0 −1.00000 0.598102 + 1.11897i 0 0 1.00000i −0.831470 + 0.555570i 1.11897 0.598102i
291.1 1.00000i 0 −1.00000 0.187593 + 0.0569057i 0 0 1.00000i −0.555570 0.831470i 0.0569057 0.187593i
499.1 1.00000i 0 −1.00000 0.0924099 0.938254i 0 0 1.00000i −0.195090 0.980785i 0.938254 + 0.0924099i
531.1 1.00000i 0 −1.00000 −1.47945 + 1.21415i 0 0 1.00000i −0.980785 + 0.195090i −1.21415 1.47945i
635.1 1.00000i 0 −1.00000 −1.47945 1.21415i 0 0 1.00000i −0.980785 0.195090i −1.21415 + 1.47945i
651.1 1.00000i 0 −1.00000 0.0924099 + 0.938254i 0 0 1.00000i −0.195090 + 0.980785i 0.938254 0.0924099i
703.1 1.00000i 0 −1.00000 0.187593 0.0569057i 0 0 1.00000i −0.555570 + 0.831470i 0.0569057 + 0.187593i
711.1 1.00000i 0 −1.00000 −1.36347 0.728789i 0 0 1.00000i 0.831470 + 0.555570i 0.728789 1.36347i
831.1 1.00000i 0 −1.00000 0.598102 1.11897i 0 0 1.00000i −0.831470 0.555570i 1.11897 + 0.598102i
839.1 1.00000i 0 −1.00000 0.577774 + 1.90466i 0 0 1.00000i 0.555570 0.831470i −1.90466 + 0.577774i
891.1 1.00000i 0 −1.00000 1.75535 0.172887i 0 0 1.00000i 0.195090 0.980785i −0.172887 1.75535i
907.1 1.00000i 0 −1.00000 −0.368309 + 0.448786i 0 0 1.00000i 0.980785 + 0.195090i 0.448786 + 0.368309i
1011.1 1.00000i 0 −1.00000 −0.368309 0.448786i 0 0 1.00000i 0.980785 0.195090i 0.448786 0.368309i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1011.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
257.f even 32 1 inner
1028.l odd 32 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1028.1.l.a 16
4.b odd 2 1 CM 1028.1.l.a 16
257.f even 32 1 inner 1028.1.l.a 16
1028.l odd 32 1 inner 1028.1.l.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1028.1.l.a 16 1.a even 1 1 trivial
1028.1.l.a 16 4.b odd 2 1 CM
1028.1.l.a 16 257.f even 32 1 inner
1028.1.l.a 16 1028.l odd 32 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1028, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( T^{16} \)
$5$ \( 2 - 16 T + 24 T^{2} + 32 T^{3} + 148 T^{4} + 176 T^{6} + 2 T^{8} + 16 T^{9} - 32 T^{11} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( ( 2 - 8 T + 20 T^{2} - 16 T^{3} + 2 T^{4} + T^{8} )^{2} \)
$17$ \( 4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( 256 + T^{16} \)
$31$ \( T^{16} \)
$37$ \( 2 + 16 T + 72 T^{2} + 80 T^{3} + 4 T^{4} + 56 T^{6} - 160 T^{7} + 6 T^{8} + 16 T^{11} + 4 T^{12} + T^{16} \)
$41$ \( 2 + 16 T + 40 T^{2} + 140 T^{4} - 48 T^{5} + 192 T^{7} + 2 T^{8} + 88 T^{10} + 16 T^{13} + T^{16} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( 2 - 16 T + 24 T^{2} + 32 T^{3} + 148 T^{4} + 176 T^{6} + 2 T^{8} + 16 T^{9} - 32 T^{11} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( 16 + 64 T^{4} + 128 T^{8} - 16 T^{12} + T^{16} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( 4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( ( 2 - 8 T + 20 T^{2} - 16 T^{3} + 2 T^{4} + T^{8} )^{2} \)
$97$ \( 2 + 16 T + 56 T^{2} + 16 T^{3} + 4 T^{4} - 160 T^{5} + 72 T^{6} + 6 T^{8} + 80 T^{9} + 4 T^{12} + T^{16} \)
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