Properties

Label 1133.1.s.a
Level $1133$
Weight $1$
Character orbit 1133.s
Analytic conductor $0.565$
Analytic rank $0$
Dimension $16$
Projective image $D_{17}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1133 = 11 \cdot 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1133.s (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.565440659313\)
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: \(D_{17}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{17} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{34}^{5} - \zeta_{34}^{9} ) q^{3} -\zeta_{34}^{11} q^{4} + ( -\zeta_{34}^{3} + \zeta_{34}^{10} ) q^{5} + ( -\zeta_{34} + \zeta_{34}^{10} + \zeta_{34}^{14} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{34}^{5} - \zeta_{34}^{9} ) q^{3} -\zeta_{34}^{11} q^{4} + ( -\zeta_{34}^{3} + \zeta_{34}^{10} ) q^{5} + ( -\zeta_{34} + \zeta_{34}^{10} + \zeta_{34}^{14} ) q^{9} + \zeta_{34}^{14} q^{11} + ( -\zeta_{34}^{3} + \zeta_{34}^{16} ) q^{12} + ( \zeta_{34}^{2} + \zeta_{34}^{8} + \zeta_{34}^{12} - \zeta_{34}^{15} ) q^{15} -\zeta_{34}^{5} q^{16} + ( \zeta_{34}^{4} + \zeta_{34}^{14} ) q^{20} + ( -\zeta_{34} - \zeta_{34}^{5} ) q^{23} + ( -\zeta_{34}^{3} + \zeta_{34}^{6} - \zeta_{34}^{13} ) q^{25} + ( \zeta_{34}^{2} + \zeta_{34}^{6} + \zeta_{34}^{10} - \zeta_{34}^{15} ) q^{27} + ( -\zeta_{34}^{3} - \zeta_{34}^{7} ) q^{31} + ( \zeta_{34}^{2} + \zeta_{34}^{6} ) q^{33} + ( \zeta_{34}^{4} + \zeta_{34}^{8} + \zeta_{34}^{12} ) q^{36} + ( 1 + \zeta_{34}^{2} ) q^{37} + \zeta_{34}^{8} q^{44} + ( 1 - \zeta_{34}^{3} + \zeta_{34}^{4} - \zeta_{34}^{7} - \zeta_{34}^{11} - \zeta_{34}^{13} ) q^{45} + ( \zeta_{34}^{4} - \zeta_{34}^{13} ) q^{47} + ( \zeta_{34}^{10} + \zeta_{34}^{14} ) q^{48} -\zeta_{34} q^{49} + ( \zeta_{34}^{4} + \zeta_{34}^{16} ) q^{53} + ( 1 - \zeta_{34}^{7} ) q^{55} + ( -\zeta_{34}^{7} - \zeta_{34}^{9} ) q^{59} + ( \zeta_{34}^{2} + \zeta_{34}^{6} - \zeta_{34}^{9} - \zeta_{34}^{13} ) q^{60} + \zeta_{34}^{16} q^{64} + ( 1 + \zeta_{34}^{16} ) q^{67} + ( \zeta_{34}^{6} + 2 \zeta_{34}^{10} + \zeta_{34}^{14} ) q^{69} + ( -\zeta_{34}^{9} + \zeta_{34}^{12} ) q^{71} + ( -\zeta_{34} - \zeta_{34}^{5} + \zeta_{34}^{8} - \zeta_{34}^{11} + \zeta_{34}^{12} - \zeta_{34}^{15} ) q^{75} + ( \zeta_{34}^{8} - \zeta_{34}^{15} ) q^{80} + ( \zeta_{34}^{2} - \zeta_{34}^{3} - \zeta_{34}^{7} - \zeta_{34}^{11} - \zeta_{34}^{15} ) q^{81} + ( -\zeta_{34} - \zeta_{34}^{11} ) q^{89} + ( \zeta_{34}^{12} + \zeta_{34}^{16} ) q^{92} + ( \zeta_{34}^{8} + 2 \zeta_{34}^{12} + \zeta_{34}^{16} ) q^{93} + ( -\zeta_{34}^{9} + \zeta_{34}^{16} ) q^{97} + ( -\zeta_{34}^{7} - \zeta_{34}^{11} - \zeta_{34}^{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 2q^{3} - q^{4} - 2q^{5} - 3q^{9} + O(q^{10}) \) \( 16q - 2q^{3} - q^{4} - 2q^{5} - 3q^{9} - q^{11} - 2q^{12} - 4q^{15} - q^{16} - 2q^{20} - 2q^{23} - 3q^{25} - 4q^{27} - 2q^{31} - 2q^{33} - 3q^{36} + 15q^{37} - q^{44} + 11q^{45} - 2q^{47} - 2q^{48} - q^{49} - 2q^{53} + 15q^{55} - 2q^{59} - 4q^{60} - q^{64} + 15q^{67} - 4q^{69} - 2q^{71} - 6q^{75} - 2q^{80} - 5q^{81} - 2q^{89} - 2q^{92} - 4q^{93} - 2q^{97} - 3q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1133\mathbb{Z}\right)^\times\).

\(n\) \(310\) \(1035\)
\(\chi(n)\) \(-1\) \(-\zeta_{34}^{13}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0.850217 + 0.526432i
0.850217 0.526432i
−0.0922684 0.995734i
−0.445738 + 0.895163i
0.273663 + 0.961826i
−0.932472 0.361242i
0.602635 0.798017i
−0.0922684 + 0.995734i
−0.932472 + 0.361242i
0.982973 0.183750i
0.982973 + 0.183750i
−0.445738 0.895163i
0.273663 0.961826i
−0.739009 0.673696i
0.602635 + 0.798017i
−0.739009 + 0.673696i
0 0.658809 + 0.600584i −0.982973 + 0.183750i 0.831277 1.66943i 0 0 0 −0.0189399 0.204394i 0
164.1 0 0.658809 0.600584i −0.982973 0.183750i 0.831277 + 1.66943i 0 0 0 −0.0189399 + 0.204394i 0
175.1 0 1.18475 + 1.56886i −0.850217 0.526432i −0.876298 0.163808i 0 0 0 −0.784029 + 2.75558i 0
219.1 0 −0.111208 + 1.20013i 0.932472 + 0.361242i −0.890705 + 1.17948i 0 0 0 −0.444966 0.0831786i 0
285.1 0 −1.58561 + 0.614268i 0.0922684 0.995734i 1.67148 + 1.03494i 0 0 0 1.39782 1.27428i 0
318.1 0 −1.25664 + 0.778076i −0.602635 0.798017i −0.404479 + 0.368731i 0 0 0 0.527993 1.06035i 0
373.1 0 0.538007 0.100571i 0.739009 0.673696i −0.0505009 + 0.177492i 0 0 0 −0.653136 + 0.253026i 0
505.1 0 1.18475 1.56886i −0.850217 + 0.526432i −0.876298 + 0.163808i 0 0 0 −0.784029 2.75558i 0
538.1 0 −1.25664 0.778076i −0.602635 + 0.798017i −0.404479 0.368731i 0 0 0 0.527993 + 1.06035i 0
549.1 0 −0.510366 + 1.79375i 0.445738 + 0.895163i −1.12388 0.435393i 0 0 0 −2.10685 1.30451i 0
615.1 0 −0.510366 1.79375i 0.445738 0.895163i −1.12388 + 0.435393i 0 0 0 −2.10685 + 1.30451i 0
626.1 0 −0.111208 1.20013i 0.932472 0.361242i −0.890705 1.17948i 0 0 0 −0.444966 + 0.0831786i 0
648.1 0 −1.58561 0.614268i 0.0922684 + 0.995734i 1.67148 1.03494i 0 0 0 1.39782 + 1.27428i 0
802.1 0 0.0822551 0.165190i −0.273663 + 0.961826i −0.156896 + 1.69318i 0 0 0 0.582113 + 0.770842i 0
890.1 0 0.538007 + 0.100571i 0.739009 + 0.673696i −0.0505009 0.177492i 0 0 0 −0.653136 0.253026i 0
1044.1 0 0.0822551 + 0.165190i −0.273663 0.961826i −0.156896 1.69318i 0 0 0 0.582113 0.770842i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1044.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
103.e even 17 1 inner
1133.s odd 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1133.1.s.a 16
11.b odd 2 1 CM 1133.1.s.a 16
103.e even 17 1 inner 1133.1.s.a 16
1133.s odd 34 1 inner 1133.1.s.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1133.1.s.a 16 1.a even 1 1 trivial
1133.1.s.a 16 11.b odd 2 1 CM
1133.1.s.a 16 103.e even 17 1 inner
1133.1.s.a 16 1133.s odd 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1133, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 1 - 8 T + 47 T^{2} - 104 T^{3} + 67 T^{4} + 8 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} + 9 T^{9} + 47 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$5$ \( 1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
$13$ \( T^{16} \)
$17$ \( T^{16} \)
$19$ \( T^{16} \)
$23$ \( 1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$29$ \( T^{16} \)
$31$ \( 1 + 9 T + 30 T^{2} + 15 T^{3} + 50 T^{4} - 94 T^{5} - 47 T^{6} - 15 T^{7} + 120 T^{8} + 60 T^{9} + 30 T^{10} - 36 T^{11} - 18 T^{12} - 9 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$37$ \( 1 - 8 T + 64 T^{2} - 308 T^{3} + 1036 T^{4} - 2576 T^{5} + 4900 T^{6} - 7274 T^{7} + 8518 T^{8} - 7896 T^{9} + 5776 T^{10} - 3300 T^{11} + 1444 T^{12} - 468 T^{13} + 106 T^{14} - 15 T^{15} + T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( ( 1 - 4 T - 10 T^{2} + 10 T^{3} + 15 T^{4} - 6 T^{5} - 7 T^{6} + T^{7} + T^{8} )^{2} \)
$53$ \( 1 - 8 T + 13 T^{2} + 15 T^{3} + 118 T^{4} + 59 T^{5} + 72 T^{6} + 2 T^{7} + T^{8} + 60 T^{9} + 30 T^{10} + 15 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$59$ \( 1 - 8 T + 30 T^{2} - 2 T^{3} - T^{4} + 59 T^{5} + 140 T^{6} + 70 T^{7} + 35 T^{8} - 25 T^{9} - 4 T^{10} - 2 T^{11} - T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$61$ \( T^{16} \)
$67$ \( 1 - 8 T + 64 T^{2} - 308 T^{3} + 1036 T^{4} - 2576 T^{5} + 4900 T^{6} - 7274 T^{7} + 8518 T^{8} - 7896 T^{9} + 5776 T^{10} - 3300 T^{11} + 1444 T^{12} - 468 T^{13} + 106 T^{14} - 15 T^{15} + T^{16} \)
$71$ \( 1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$73$ \( T^{16} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( 1 - 8 T + 30 T^{2} - 2 T^{3} - T^{4} + 59 T^{5} + 140 T^{6} + 70 T^{7} + 35 T^{8} - 25 T^{9} - 4 T^{10} - 2 T^{11} - T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$97$ \( 1 + 9 T + 30 T^{2} + 15 T^{3} + 50 T^{4} - 94 T^{5} - 47 T^{6} - 15 T^{7} + 120 T^{8} + 60 T^{9} + 30 T^{10} - 36 T^{11} - 18 T^{12} - 9 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
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