Properties

Label 921.1.p.a
Level $921$
Weight $1$
Character orbit 921.p
Analytic conductor $0.460$
Analytic rank $0$
Dimension $16$
Projective image $D_{17}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 921 = 3 \cdot 307 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 921.p (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.459638876635\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{17}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{17} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{34}^{12} q^{3} -\zeta_{34}^{3} q^{4} + ( 1 - \zeta_{34} ) q^{7} -\zeta_{34}^{7} q^{9} +O(q^{10})\) \( q + \zeta_{34}^{12} q^{3} -\zeta_{34}^{3} q^{4} + ( 1 - \zeta_{34} ) q^{7} -\zeta_{34}^{7} q^{9} -\zeta_{34}^{15} q^{12} + ( \zeta_{34}^{4} + \zeta_{34}^{8} ) q^{13} + \zeta_{34}^{6} q^{16} + ( \zeta_{34}^{2} + \zeta_{34}^{10} ) q^{19} + ( \zeta_{34}^{12} - \zeta_{34}^{13} ) q^{21} -\zeta_{34}^{11} q^{25} + \zeta_{34}^{2} q^{27} + ( -\zeta_{34}^{3} + \zeta_{34}^{4} ) q^{28} + ( -\zeta_{34}^{13} + \zeta_{34}^{14} ) q^{31} + \zeta_{34}^{10} q^{36} + ( \zeta_{34}^{6} + \zeta_{34}^{12} ) q^{37} + ( -\zeta_{34}^{3} + \zeta_{34}^{16} ) q^{39} + ( \zeta_{34}^{10} + \zeta_{34}^{16} ) q^{43} -\zeta_{34} q^{48} + ( 1 - \zeta_{34} + \zeta_{34}^{2} ) q^{49} + ( -\zeta_{34}^{7} - \zeta_{34}^{11} ) q^{52} + ( -\zeta_{34}^{5} + \zeta_{34}^{14} ) q^{57} + ( -\zeta_{34}^{11} - \zeta_{34}^{15} ) q^{61} + ( -\zeta_{34}^{7} + \zeta_{34}^{8} ) q^{63} -\zeta_{34}^{9} q^{64} + ( \zeta_{34}^{4} - \zeta_{34}^{9} ) q^{67} + ( \zeta_{34}^{14} - \zeta_{34}^{15} ) q^{73} + \zeta_{34}^{6} q^{75} + ( -\zeta_{34}^{5} - \zeta_{34}^{13} ) q^{76} + ( -\zeta_{34}^{9} + \zeta_{34}^{16} ) q^{79} + \zeta_{34}^{14} q^{81} + ( -\zeta_{34}^{15} + \zeta_{34}^{16} ) q^{84} + ( \zeta_{34}^{4} - \zeta_{34}^{5} + \zeta_{34}^{8} - \zeta_{34}^{9} ) q^{91} + ( \zeta_{34}^{8} - \zeta_{34}^{9} ) q^{93} + ( -\zeta_{34}^{9} - \zeta_{34}^{13} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - q^{3} - q^{4} + 15q^{7} - q^{9} + O(q^{10}) \) \( 16q - q^{3} - q^{4} + 15q^{7} - q^{9} - q^{12} - 2q^{13} - q^{16} - 2q^{19} - 2q^{21} - q^{25} - q^{27} - 2q^{28} - 2q^{31} - q^{36} - 2q^{37} - 2q^{39} - 2q^{43} - q^{48} + 14q^{49} - 2q^{52} - 2q^{57} - 2q^{61} - 2q^{63} - q^{64} - 2q^{67} - 2q^{73} - q^{75} - 2q^{76} - 2q^{79} - q^{81} - 2q^{84} - 4q^{91} - 2q^{93} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/921\mathbb{Z}\right)^\times\).

\(n\) \(308\) \(619\)
\(\chi(n)\) \(-1\) \(-\zeta_{34}^{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
−0.932472 + 0.361242i
0.602635 0.798017i
−0.0922684 + 0.995734i
0.982973 + 0.183750i
−0.0922684 0.995734i
−0.445738 0.895163i
−0.739009 0.673696i
0.273663 + 0.961826i
0.850217 0.526432i
0.982973 0.183750i
−0.445738 + 0.895163i
0.273663 0.961826i
−0.932472 0.361242i
0.602635 + 0.798017i
−0.739009 + 0.673696i
0.850217 + 0.526432i
0 −0.273663 + 0.961826i 0.445738 0.895163i 0 0 1.93247 0.361242i 0 −0.850217 0.526432i 0
272.1 0 0.0922684 + 0.995734i 0.932472 + 0.361242i 0 0 0.397365 + 0.798017i 0 −0.982973 + 0.183750i 0
299.1 0 0.445738 + 0.895163i −0.273663 + 0.961826i 0 0 1.09227 0.995734i 0 −0.602635 + 0.798017i 0
371.1 0 −0.602635 + 0.798017i −0.850217 0.526432i 0 0 0.0170269 0.183750i 0 −0.273663 0.961826i 0
422.1 0 0.445738 0.895163i −0.273663 0.961826i 0 0 1.09227 + 0.995734i 0 −0.602635 0.798017i 0
542.1 0 0.739009 + 0.673696i −0.982973 0.183750i 0 0 1.44574 + 0.895163i 0 0.0922684 + 0.995734i 0
587.1 0 −0.850217 + 0.526432i −0.602635 + 0.798017i 0 0 1.73901 + 0.673696i 0 0.445738 0.895163i 0
611.1 0 −0.982973 + 0.183750i 0.739009 + 0.673696i 0 0 0.726337 0.961826i 0 0.932472 0.361242i 0
623.1 0 0.932472 0.361242i 0.0922684 + 0.995734i 0 0 0.149783 + 0.526432i 0 0.739009 0.673696i 0
638.1 0 −0.602635 0.798017i −0.850217 + 0.526432i 0 0 0.0170269 + 0.183750i 0 −0.273663 + 0.961826i 0
695.1 0 0.739009 0.673696i −0.982973 + 0.183750i 0 0 1.44574 0.895163i 0 0.0922684 0.995734i 0
716.1 0 −0.982973 0.183750i 0.739009 0.673696i 0 0 0.726337 + 0.961826i 0 0.932472 + 0.361242i 0
719.1 0 −0.273663 0.961826i 0.445738 + 0.895163i 0 0 1.93247 + 0.361242i 0 −0.850217 + 0.526432i 0
728.1 0 0.0922684 0.995734i 0.932472 0.361242i 0 0 0.397365 0.798017i 0 −0.982973 0.183750i 0
830.1 0 −0.850217 0.526432i −0.602635 0.798017i 0 0 1.73901 0.673696i 0 0.445738 + 0.895163i 0
887.1 0 0.932472 + 0.361242i 0.0922684 0.995734i 0 0 0.149783 0.526432i 0 0.739009 + 0.673696i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 887.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
307.f even 17 1 inner
921.p odd 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 921.1.p.a 16
3.b odd 2 1 CM 921.1.p.a 16
307.f even 17 1 inner 921.1.p.a 16
921.p odd 34 1 inner 921.1.p.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
921.1.p.a 16 1.a even 1 1 trivial
921.1.p.a 16 3.b odd 2 1 CM
921.1.p.a 16 307.f even 17 1 inner
921.1.p.a 16 921.p odd 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(921, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( 1 - 8 T + 64 T^{2} - 308 T^{3} + 1036 T^{4} - 2576 T^{5} + 4900 T^{6} - 7274 T^{7} + 8518 T^{8} - 7896 T^{9} + 5776 T^{10} - 3300 T^{11} + 1444 T^{12} - 468 T^{13} + 106 T^{14} - 15 T^{15} + T^{16} \)
$11$ \( T^{16} \)
$13$ \( 1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$17$ \( T^{16} \)
$19$ \( 1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( 1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$37$ \( 1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$41$ \( T^{16} \)
$43$ \( 1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$47$ \( T^{16} \)
$53$ \( T^{16} \)
$59$ \( T^{16} \)
$61$ \( 1 + 9 T + 13 T^{2} - 36 T^{3} + 33 T^{4} + 25 T^{5} + 140 T^{6} + 70 T^{7} + 154 T^{8} + 77 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$67$ \( 1 + 9 T + 30 T^{2} + 15 T^{3} + 50 T^{4} - 94 T^{5} - 47 T^{6} - 15 T^{7} + 120 T^{8} + 60 T^{9} + 30 T^{10} - 36 T^{11} - 18 T^{12} - 9 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$71$ \( T^{16} \)
$73$ \( 1 - 8 T + 47 T^{2} - 104 T^{3} + 67 T^{4} + 8 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} + 9 T^{9} + 47 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$79$ \( 1 + 9 T + 30 T^{2} + 15 T^{3} + 50 T^{4} - 94 T^{5} - 47 T^{6} - 15 T^{7} + 120 T^{8} + 60 T^{9} + 30 T^{10} - 36 T^{11} - 18 T^{12} - 9 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( 1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
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