Properties

 Label 548.1.j.a Level $548$ Weight $1$ Character orbit 548.j Analytic conductor $0.273$ Analytic rank $0$ Dimension $16$ Projective image $D_{34}$ CM discriminant -4 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$548 = 2^{2} \cdot 137$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 548.j (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: $$D_{34}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{34} - \cdots)$$

$q$-expansion

The $$q$$-expansion and trace form are shown below.

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

Character values

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

 $$n$$ $$275$$ $$277$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{34}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0.982973 + 0.183750i 0.850217 − 0.526432i 0.850217 + 0.526432i 0.273663 + 0.961826i −0.445738 − 0.895163i −0.932472 + 0.361242i 0.273663 − 0.961826i −0.0922684 − 0.995734i −0.739009 + 0.673696i −0.932472 − 0.361242i −0.739009 − 0.673696i 0.602635 − 0.798017i −0.0922684 + 0.995734i 0.602635 + 0.798017i −0.445738 + 0.895163i 0.982973 − 0.183750i
−0.273663 + 0.961826i 0 −0.850217 0.526432i −1.72198 + 0.489946i 0 0 0.739009 0.673696i −0.932472 0.361242i 1.79033i
63.1 0.739009 + 0.673696i 0 0.0922684 + 0.995734i −0.247582 0.271585i 0 0 −0.602635 + 0.798017i −0.445738 + 0.895163i 0.367499i
87.1 0.739009 0.673696i 0 0.0922684 0.995734i −0.247582 + 0.271585i 0 0 −0.602635 0.798017i −0.445738 0.895163i 0.367499i
99.1 0.932472 + 0.361242i 0 0.739009 + 0.673696i −0.719401 1.85699i 0 0 0.445738 + 0.895163i 0.850217 0.526432i 1.99147i
103.1 0.0922684 0.995734i 0 −0.982973 0.183750i 0.719401 0.0666624i 0 0 −0.273663 + 0.961826i 0.602635 0.798017i 0.722483i
151.1 −0.850217 + 0.526432i 0 0.445738 0.895163i 0.840204 1.35698i 0 0 0.0922684 + 0.995734i −0.739009 + 0.673696i 1.59603i
155.1 0.932472 0.361242i 0 0.739009 0.673696i −0.719401 + 1.85699i 0 0 0.445738 0.895163i 0.850217 + 0.526432i 1.99147i
159.1 −0.602635 + 0.798017i 0 −0.273663 0.961826i −0.840204 + 0.634493i 0 0 0.932472 + 0.361242i 0.982973 0.183750i 1.05286i
215.1 0.445738 0.895163i 0 −0.602635 0.798017i 1.72198 0.857445i 0 0 −0.982973 + 0.183750i −0.0922684 + 0.995734i 1.92365i
323.1 −0.850217 0.526432i 0 0.445738 + 0.895163i 0.840204 + 1.35698i 0 0 0.0922684 0.995734i −0.739009 0.673696i 1.59603i
339.1 0.445738 + 0.895163i 0 −0.602635 + 0.798017i 1.72198 + 0.857445i 0 0 −0.982973 0.183750i −0.0922684 0.995734i 1.92365i
351.1 −0.982973 0.183750i 0 0.932472 + 0.361242i 0.247582 + 1.32445i 0 0 −0.850217 0.526432i 0.273663 + 0.961826i 1.34739i
355.1 −0.602635 0.798017i 0 −0.273663 + 0.961826i −0.840204 0.634493i 0 0 0.932472 0.361242i 0.982973 + 0.183750i 1.05286i
395.1 −0.982973 + 0.183750i 0 0.932472 0.361242i 0.247582 1.32445i 0 0 −0.850217 + 0.526432i 0.273663 0.961826i 1.34739i
415.1 0.0922684 + 0.995734i 0 −0.982973 + 0.183750i 0.719401 + 0.0666624i 0 0 −0.273663 0.961826i 0.602635 + 0.798017i 0.722483i
475.1 −0.273663 0.961826i 0 −0.850217 + 0.526432i −1.72198 0.489946i 0 0 0.739009 + 0.673696i −0.932472 + 0.361242i 1.79033i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 475.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
137.f even 34 1 inner
548.j 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.j.a 16
4.b odd 2 1 CM 548.1.j.a 16
137.f even 34 1 inner 548.1.j.a 16
548.j odd 34 1 inner 548.1.j.a 16

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

Hecke kernels

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

Hecke characteristic polynomials

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