Properties

Label 927.1.v.a
Level $927$
Weight $1$
Character orbit 927.v
Analytic conductor $0.463$
Analytic rank $0$
Dimension $16$
Projective image $D_{34}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 927.v (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.462633266711\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
Defining polynomial: \(x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{34}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{34} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{34}^{11} q^{4} + ( \zeta_{34}^{2} + \zeta_{34}^{16} ) q^{7} +O(q^{10})\) \( q + \zeta_{34}^{11} q^{4} + ( \zeta_{34}^{2} + \zeta_{34}^{16} ) q^{7} + ( \zeta_{34}^{3} + \zeta_{34}^{15} ) q^{13} -\zeta_{34}^{5} q^{16} + ( \zeta_{34}^{8} + \zeta_{34}^{12} ) q^{19} -\zeta_{34}^{13} q^{25} + ( -\zeta_{34}^{10} + \zeta_{34}^{13} ) q^{28} + ( -\zeta_{34}^{4} + \zeta_{34}^{6} ) q^{31} + ( \zeta_{34}^{9} + \zeta_{34}^{10} ) q^{37} + ( -\zeta_{34}^{3} - \zeta_{34}^{12} ) q^{43} + ( -\zeta_{34} + \zeta_{34}^{4} - \zeta_{34}^{15} ) q^{49} + ( -\zeta_{34}^{9} + \zeta_{34}^{14} ) q^{52} + ( \zeta_{34} - \zeta_{34}^{8} ) q^{61} -\zeta_{34}^{16} q^{64} + ( \zeta_{34}^{5} - \zeta_{34}^{11} ) q^{67} + ( -\zeta_{34}^{7} - \zeta_{34}^{14} ) q^{73} + ( -\zeta_{34}^{2} - \zeta_{34}^{6} ) q^{76} + ( -\zeta_{34}^{6} + \zeta_{34}^{7} ) q^{79} + ( -1 - \zeta_{34}^{2} + \zeta_{34}^{5} - \zeta_{34}^{14} ) q^{91} + ( -1 - \zeta_{34}^{8} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + q^{4} - 2q^{7} + O(q^{10}) \) \( 16q + q^{4} - 2q^{7} + 2q^{13} - q^{16} - 2q^{19} - q^{25} + 2q^{28} - 3q^{49} - 2q^{52} + 2q^{61} + q^{64} + 2q^{76} + 2q^{79} - 13q^{91} - 15q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).

\(n\) \(722\) \(829\)
\(\chi(n)\) \(1\) \(\zeta_{34}^{13}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−0.0922684 + 0.995734i
0.602635 + 0.798017i
0.273663 0.961826i
0.273663 + 0.961826i
0.850217 0.526432i
0.982973 0.183750i
0.982973 + 0.183750i
0.602635 0.798017i
−0.932472 + 0.361242i
−0.739009 + 0.673696i
−0.739009 0.673696i
−0.0922684 0.995734i
−0.932472 0.361242i
0.850217 + 0.526432i
−0.445738 + 0.895163i
−0.445738 0.895163i
0 0 0.850217 0.526432i 0 0 −0.890705 + 0.811985i 0 0 0
37.1 0 0 −0.739009 0.673696i 0 0 −0.876298 + 1.75984i 0 0 0
73.1 0 0 −0.0922684 0.995734i 0 0 −1.12388 1.48826i 0 0 0
127.1 0 0 −0.0922684 + 0.995734i 0 0 −1.12388 + 1.48826i 0 0 0
145.1 0 0 0.982973 + 0.183750i 0 0 −0.404479 1.42160i 0 0 0
172.1 0 0 −0.445738 0.895163i 0 0 −0.0505009 0.544991i 0 0 0
415.1 0 0 −0.445738 + 0.895163i 0 0 −0.0505009 + 0.544991i 0 0 0
451.1 0 0 −0.739009 + 0.673696i 0 0 −0.876298 1.75984i 0 0 0
595.1 0 0 0.602635 0.798017i 0 0 1.67148 0.312454i 0 0 0
604.1 0 0 0.273663 + 0.961826i 0 0 0.831277 0.322039i 0 0 0
640.1 0 0 0.273663 0.961826i 0 0 0.831277 + 0.322039i 0 0 0
649.1 0 0 0.850217 + 0.526432i 0 0 −0.890705 0.811985i 0 0 0
712.1 0 0 0.602635 + 0.798017i 0 0 1.67148 + 0.312454i 0 0 0
748.1 0 0 0.982973 0.183750i 0 0 −0.404479 + 1.42160i 0 0 0
811.1 0 0 −0.932472 0.361242i 0 0 −0.156896 + 0.0971461i 0 0 0
919.1 0 0 −0.932472 + 0.361242i 0 0 −0.156896 0.0971461i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 919.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
103.f odd 34 1 inner
309.k even 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.1.v.a 16
3.b odd 2 1 CM 927.1.v.a 16
103.f odd 34 1 inner 927.1.v.a 16
309.k even 34 1 inner 927.1.v.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.1.v.a 16 1.a even 1 1 trivial
927.1.v.a 16 3.b odd 2 1 CM
927.1.v.a 16 103.f odd 34 1 inner
927.1.v.a 16 309.k even 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(927, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( 1 + 9 T + 30 T^{2} + 15 T^{3} + 50 T^{4} - 94 T^{5} - 47 T^{6} - 15 T^{7} + 120 T^{8} + 60 T^{9} + 30 T^{10} - 36 T^{11} - 18 T^{12} - 9 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$11$ \( T^{16} \)
$13$ \( 1 - 9 T + 47 T^{2} - 83 T^{3} + 50 T^{4} - 25 T^{5} + 21 T^{6} + 100 T^{7} - 16 T^{8} + 8 T^{9} - 4 T^{10} + 2 T^{11} + 16 T^{12} - 8 T^{13} + 4 T^{14} - 2 T^{15} + T^{16} \)
$17$ \( T^{16} \)
$19$ \( 1 - 8 T + 47 T^{2} - 104 T^{3} + 67 T^{4} + 8 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} + 9 T^{9} + 47 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( 17 + 102 T + 255 T^{2} + 238 T^{3} + 51 T^{4} + 17 T^{8} - 85 T^{9} + 17 T^{10} + T^{16} \)
$37$ \( 17 - 17 T + 85 T^{2} + 102 T^{3} + 17 T^{4} - 255 T^{5} + 238 T^{7} - 51 T^{9} + T^{16} \)
$41$ \( T^{16} \)
$43$ \( 17 - 34 T + 17 T^{2} + 221 T^{3} - 85 T^{5} + 68 T^{6} + 119 T^{8} - 17 T^{11} + T^{16} \)
$47$ \( T^{16} \)
$53$ \( T^{16} \)
$59$ \( T^{16} \)
$61$ \( 1 - 9 T + 30 T^{2} - 15 T^{3} + 50 T^{4} + 94 T^{5} - 47 T^{6} + 15 T^{7} + 120 T^{8} - 60 T^{9} + 30 T^{10} + 36 T^{11} - 18 T^{12} + 9 T^{13} + 4 T^{14} - 2 T^{15} + T^{16} \)
$67$ \( 17 - 119 T + 442 T^{2} - 935 T^{3} + 1122 T^{4} - 714 T^{5} + 204 T^{6} - 17 T^{7} + T^{16} \)
$71$ \( T^{16} \)
$73$ \( 17 - 119 T + 442 T^{2} - 935 T^{3} + 1122 T^{4} - 714 T^{5} + 204 T^{6} - 17 T^{7} + T^{16} \)
$79$ \( 1 + 8 T + 13 T^{2} - 15 T^{3} + 118 T^{4} - 59 T^{5} + 72 T^{6} - 2 T^{7} + T^{8} - 60 T^{9} + 30 T^{10} - 15 T^{11} + 16 T^{12} - 8 T^{13} + 4 T^{14} - 2 T^{15} + T^{16} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( 1 + 8 T + 64 T^{2} + 308 T^{3} + 1036 T^{4} + 2576 T^{5} + 4900 T^{6} + 7274 T^{7} + 8518 T^{8} + 7896 T^{9} + 5776 T^{10} + 3300 T^{11} + 1444 T^{12} + 468 T^{13} + 106 T^{14} + 15 T^{15} + T^{16} \)
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