Properties

Label 548.1.k.a
Level $548$
Weight $1$
Character orbit 548.k
Analytic conductor $0.273$
Analytic rank $0$
Dimension $16$
Projective image $D_{17}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 548 = 2^{2} \cdot 137 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 548.k (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.273487626923\)
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: \(\Z[a_1, a_2]\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/548\mathbb{Z}\right)^\times\).

\(n\) \(275\) \(277\)
\(\chi(n)\) \(-1\) \(-\zeta_{34}^{15}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−0.739009 0.673696i
−0.0922684 + 0.995734i
0.273663 + 0.961826i
−0.932472 0.361242i
−0.445738 + 0.895163i
0.273663 0.961826i
0.850217 0.526432i
0.850217 + 0.526432i
0.982973 0.183750i
0.982973 + 0.183750i
−0.445738 0.895163i
0.602635 0.798017i
−0.0922684 0.995734i
0.602635 + 0.798017i
−0.739009 + 0.673696i
−0.932472 + 0.361242i
0.445738 0.895163i 0 −0.602635 0.798017i −0.243964 0.489946i 0 0 −0.982973 + 0.183750i 0.0922684 0.995734i −0.547326
115.1 −0.602635 + 0.798017i 0 −0.273663 0.961826i 1.02474 + 1.35698i 0 0 0.932472 + 0.361242i −0.982973 + 0.183750i −1.70043
119.1 0.932472 0.361242i 0 0.739009 0.673696i 0.172075 + 0.0666624i 0 0 0.445738 0.895163i −0.850217 0.526432i 0.184537
123.1 −0.850217 + 0.526432i 0 0.445738 0.895163i 1.02474 + 0.634493i 0 0 0.0922684 + 0.995734i 0.739009 0.673696i −1.20527
171.1 0.0922684 0.995734i 0 −0.982973 0.183750i 0.172075 + 1.85699i 0 0 −0.273663 + 0.961826i −0.602635 + 0.798017i 1.86494
175.1 0.932472 + 0.361242i 0 0.739009 + 0.673696i 0.172075 0.0666624i 0 0 0.445738 + 0.895163i −0.850217 + 0.526432i 0.184537
187.1 0.739009 0.673696i 0 0.0922684 0.995734i −1.45285 1.32445i 0 0 −0.602635 0.798017i 0.445738 + 0.895163i −1.96595
211.1 0.739009 + 0.673696i 0 0.0922684 + 0.995734i −1.45285 + 1.32445i 0 0 −0.602635 + 0.798017i 0.445738 0.895163i −1.96595
259.1 −0.273663 + 0.961826i 0 −0.850217 0.526432i −0.243964 0.857445i 0 0 0.739009 0.673696i 0.932472 + 0.361242i 0.891477
347.1 −0.273663 0.961826i 0 −0.850217 + 0.526432i −0.243964 + 0.857445i 0 0 0.739009 + 0.673696i 0.932472 0.361242i 0.891477
407.1 0.0922684 + 0.995734i 0 −0.982973 + 0.183750i 0.172075 1.85699i 0 0 −0.273663 0.961826i −0.602635 0.798017i 1.86494
427.1 −0.982973 + 0.183750i 0 0.932472 0.361242i −1.45285 0.271585i 0 0 −0.850217 + 0.526432i −0.273663 + 0.961826i 1.47802
467.1 −0.602635 0.798017i 0 −0.273663 + 0.961826i 1.02474 1.35698i 0 0 0.932472 0.361242i −0.982973 0.183750i −1.70043
471.1 −0.982973 0.183750i 0 0.932472 + 0.361242i −1.45285 + 0.271585i 0 0 −0.850217 0.526432i −0.273663 0.961826i 1.47802
483.1 0.445738 + 0.895163i 0 −0.602635 + 0.798017i −0.243964 + 0.489946i 0 0 −0.982973 0.183750i 0.0922684 + 0.995734i −0.547326
499.1 −0.850217 0.526432i 0 0.445738 + 0.895163i 1.02474 0.634493i 0 0 0.0922684 0.995734i 0.739009 + 0.673696i −1.20527
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 499.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
137.e even 17 1 inner
548.k odd 34 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 548.1.k.a 16
4.b odd 2 1 CM 548.1.k.a 16
137.e even 17 1 inner 548.1.k.a 16
548.k odd 34 1 inner 548.1.k.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
548.1.k.a 16 1.a even 1 1 trivial
548.1.k.a 16 4.b odd 2 1 CM
548.1.k.a 16 137.e even 17 1 inner
548.1.k.a 16 548.k odd 34 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(548, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 1 - 8 T + 13 T^{2} + 15 T^{3} + 118 T^{4} + 59 T^{5} + 72 T^{6} + 2 T^{7} + T^{8} + 60 T^{9} + 30 T^{10} + 15 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( 1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$17$ \( 1 - 8 T + 64 T^{2} - 308 T^{3} + 1036 T^{4} - 2576 T^{5} + 4900 T^{6} - 7274 T^{7} + 8518 T^{8} - 7896 T^{9} + 5776 T^{10} - 3300 T^{11} + 1444 T^{12} - 468 T^{13} + 106 T^{14} - 15 T^{15} + T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( 1 - 8 T + 47 T^{2} - 104 T^{3} + 67 T^{4} + 8 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} + 9 T^{9} + 47 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$31$ \( T^{16} \)
$37$ \( ( 1 - 4 T - 10 T^{2} + 10 T^{3} + 15 T^{4} - 6 T^{5} - 7 T^{6} + T^{7} + T^{8} )^{2} \)
$41$ \( ( 1 - 4 T - 10 T^{2} + 10 T^{3} + 15 T^{4} - 6 T^{5} - 7 T^{6} + T^{7} + T^{8} )^{2} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( 1 + 9 T + 64 T^{2} + 253 T^{3} + 594 T^{4} + 858 T^{5} + 786 T^{6} + 495 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$59$ \( T^{16} \)
$61$ \( 1 + 9 T + 30 T^{2} + 15 T^{3} + 50 T^{4} - 94 T^{5} - 47 T^{6} - 15 T^{7} + 120 T^{8} + 60 T^{9} + 30 T^{10} - 36 T^{11} - 18 T^{12} - 9 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( 1 - 8 T + 13 T^{2} + 15 T^{3} + 118 T^{4} + 59 T^{5} + 72 T^{6} + 2 T^{7} + T^{8} + 60 T^{9} + 30 T^{10} + 15 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( 1 + 9 T + 47 T^{2} + 83 T^{3} + 50 T^{4} + 25 T^{5} + 21 T^{6} - 100 T^{7} - 16 T^{8} - 8 T^{9} - 4 T^{10} - 2 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
$97$ \( 65536 + 32768 T + 16384 T^{2} + 8192 T^{3} + 4096 T^{4} + 2048 T^{5} + 1024 T^{6} + 512 T^{7} + 256 T^{8} + 128 T^{9} + 64 T^{10} + 32 T^{11} + 16 T^{12} + 8 T^{13} + 4 T^{14} + 2 T^{15} + T^{16} \)
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