Properties

Label 873.1.br.a
Level $873$
Weight $1$
Character orbit 873.br
Analytic conductor $0.436$
Analytic rank $0$
Dimension $16$
Projective image $D_{32}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 873 = 3^{2} \cdot 97 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 873.br (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.435683756029\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{32})\)
Defining polynomial: \(x^{16} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{32}^{6} q^{4} + ( \zeta_{32} + \zeta_{32}^{4} ) q^{7} +O(q^{10})\) \( q + \zeta_{32}^{6} q^{4} + ( \zeta_{32} + \zeta_{32}^{4} ) q^{7} + ( -\zeta_{32}^{5} + \zeta_{32}^{14} ) q^{13} + \zeta_{32}^{12} q^{16} + ( -\zeta_{32}^{3} - \zeta_{32}^{8} ) q^{19} + \zeta_{32}^{11} q^{25} + ( \zeta_{32}^{7} + \zeta_{32}^{10} ) q^{28} + ( \zeta_{32}^{3} - \zeta_{32}^{7} ) q^{31} + ( -\zeta_{32}^{12} - \zeta_{32}^{13} ) q^{37} + ( \zeta_{32}^{13} - \zeta_{32}^{15} ) q^{43} + ( \zeta_{32}^{2} + \zeta_{32}^{5} + \zeta_{32}^{8} ) q^{49} + ( -\zeta_{32}^{4} - \zeta_{32}^{11} ) q^{52} + ( -\zeta_{32} + \zeta_{32}^{15} ) q^{61} -\zeta_{32}^{2} q^{64} + ( \zeta_{32}^{9} - \zeta_{32}^{10} ) q^{67} + ( -\zeta_{32}^{6} - \zeta_{32}^{14} ) q^{73} + ( -\zeta_{32}^{9} - \zeta_{32}^{14} ) q^{76} + ( -1 - \zeta_{32}^{10} ) q^{79} + ( -\zeta_{32}^{2} - \zeta_{32}^{6} - \zeta_{32}^{9} + \zeta_{32}^{15} ) q^{91} -\zeta_{32}^{11} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 16q^{79} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/873\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(389\)
\(\chi(n)\) \(-\zeta_{32}^{11}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.980785 + 0.195090i
−0.831470 0.555570i
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
0.555570 0.831470i
−0.555570 0.831470i
−0.195090 + 0.980785i
−0.195090 0.980785i
−0.980785 + 0.195090i
0.831470 + 0.555570i
−0.980785 0.195090i
0 0 0.382683 + 0.923880i 0 0 1.68789 + 0.902197i 0 0 0
28.1 0 0 −0.923880 0.382683i 0 0 −1.53858 + 0.151537i 0 0 0
46.1 0 0 0.382683 0.923880i 0 0 1.68789 0.902197i 0 0 0
55.1 0 0 −0.382683 + 0.923880i 0 0 0.902197 + 0.273678i 0 0 0
127.1 0 0 −0.382683 0.923880i 0 0 0.902197 0.273678i 0 0 0
271.1 0 0 0.923880 0.382683i 0 0 −0.151537 + 0.124363i 0 0 0
325.1 0 0 0.923880 + 0.382683i 0 0 −1.26268 + 1.53858i 0 0 0
343.1 0 0 −0.923880 + 0.382683i 0 0 −1.53858 0.151537i 0 0 0
433.1 0 0 −0.923880 + 0.382683i 0 0 0.124363 1.26268i 0 0 0
451.1 0 0 0.923880 + 0.382683i 0 0 −0.151537 0.124363i 0 0 0
505.1 0 0 0.923880 0.382683i 0 0 −1.26268 1.53858i 0 0 0
649.1 0 0 −0.382683 0.923880i 0 0 0.512016 + 1.68789i 0 0 0
721.1 0 0 −0.382683 + 0.923880i 0 0 0.512016 1.68789i 0 0 0
730.1 0 0 0.382683 0.923880i 0 0 −0.273678 0.512016i 0 0 0
748.1 0 0 −0.923880 0.382683i 0 0 0.124363 + 1.26268i 0 0 0
757.1 0 0 0.382683 + 0.923880i 0 0 −0.273678 + 0.512016i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
97.j odd 32 1 inner
291.s even 32 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 873.1.br.a 16
3.b odd 2 1 CM 873.1.br.a 16
97.j odd 32 1 inner 873.1.br.a 16
291.s even 32 1 inner 873.1.br.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
873.1.br.a 16 1.a even 1 1 trivial
873.1.br.a 16 3.b odd 2 1 CM
873.1.br.a 16 97.j odd 32 1 inner
873.1.br.a 16 291.s even 32 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(873, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( 2 + 16 T + 56 T^{2} + 16 T^{3} + 4 T^{4} - 160 T^{5} + 72 T^{6} + 6 T^{8} + 80 T^{9} + 4 T^{12} + T^{16} \)
$11$ \( T^{16} \)
$13$ \( 2 - 16 T + 88 T^{2} - 192 T^{3} + 140 T^{4} - 16 T^{5} + 2 T^{8} + 48 T^{9} + 40 T^{10} + T^{16} \)
$17$ \( T^{16} \)
$19$ \( 2 - 16 T + 8 T^{2} + 112 T^{3} + 28 T^{4} - 112 T^{5} + 56 T^{6} + 16 T^{7} + 70 T^{8} + 56 T^{10} + 28 T^{12} + 8 T^{14} + T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( 16 + 64 T^{4} + 128 T^{8} - 16 T^{12} + T^{16} \)
$37$ \( 2 + 16 T + 72 T^{2} + 80 T^{3} + 4 T^{4} + 56 T^{6} - 160 T^{7} + 6 T^{8} + 16 T^{11} + 4 T^{12} + T^{16} \)
$41$ \( T^{16} \)
$43$ \( 4 + 32 T^{2} + 128 T^{4} - 192 T^{6} + 140 T^{8} - 16 T^{10} + T^{16} \)
$47$ \( T^{16} \)
$53$ \( T^{16} \)
$59$ \( T^{16} \)
$61$ \( ( 2 - 16 T^{2} + 20 T^{4} - 8 T^{6} + T^{8} )^{2} \)
$67$ \( 2 - 16 T + 24 T^{2} + 32 T^{3} + 148 T^{4} + 176 T^{6} + 2 T^{8} + 16 T^{9} - 32 T^{11} + T^{16} \)
$71$ \( T^{16} \)
$73$ \( ( 16 + T^{8} )^{2} \)
$79$ \( ( 2 + 8 T + 28 T^{2} + 56 T^{3} + 70 T^{4} + 56 T^{5} + 28 T^{6} + 8 T^{7} + T^{8} )^{2} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( 1 + T^{16} \)
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