Properties

Label 17.2.d.a
Level 17
Weight 2
Character orbit 17.d
Analytic conductor 0.136
Analytic rank 0
Dimension 4
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 17.d (of order \(8\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.135745683436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{2} \) \( + ( -1 - \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} \) \( + ( 2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{4} \) \( + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} \) \( + ( 1 - \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} \) \( + ( -1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} \) \( + ( 1 - 3 \zeta_{8} + \zeta_{8}^{2} ) q^{8} \) \( + ( 2 + \zeta_{8} + 2 \zeta_{8}^{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{2} \) \( + ( -1 - \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} \) \( + ( 2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{4} \) \( + ( -\zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} \) \( + ( 1 - \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} \) \( + ( -1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{7} \) \( + ( 1 - 3 \zeta_{8} + \zeta_{8}^{2} ) q^{8} \) \( + ( 2 + \zeta_{8} + 2 \zeta_{8}^{2} ) q^{9} \) \( + ( 1 + \zeta_{8}^{3} ) q^{10} \) \( + ( -1 + \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{11} \) \( + ( -1 + \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{12} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{13} \) \( + ( 1 + \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{14} \) \( + ( -2 + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{15} \) \( -3 q^{16} \) \( + ( -2 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{17} \) \( + ( -3 + \zeta_{8} - \zeta_{8}^{3} ) q^{18} \) \( + ( 2 - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{19} \) \( + ( 1 + \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{20} \) \( + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} \) \( + ( 3 - 3 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{22} \) \( + ( 1 + 3 \zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{23} \) \( + ( 3 + \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{24} \) \( + ( -1 + 3 \zeta_{8} - \zeta_{8}^{2} ) q^{25} \) \( + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{26} \) \( + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{27} \) \( + ( 1 - 5 \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{28} \) \( + ( -1 + \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{29} \) \( -2 \zeta_{8}^{2} q^{30} \) \( + ( -3 - 3 \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{31} \) \( + ( 1 - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{32} \) \( + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{33} \) \( + ( -5 + 3 \zeta_{8} - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{34} \) \( + 2 q^{35} \) \( + 7 \zeta_{8}^{3} q^{36} \) \( + ( -5 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{37} \) \( + 2 \zeta_{8}^{2} q^{38} \) \( + ( 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{39} \) \( + ( -2 + \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{40} \) \( + ( -1 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{41} \) \( + ( -2 + 4 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{42} \) \( + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{43} \) \( + ( -5 + \zeta_{8} + \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{44} \) \( + ( 3 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{45} \) \( + ( 5 - 5 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{46} \) \( + ( 2 \zeta_{8} + 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{47} \) \( + ( 3 + 3 \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{48} \) \( + ( 2 - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{49} \) \( + ( 5 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{50} \) \( + ( 7 + \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{51} \) \( + ( 4 - \zeta_{8} + \zeta_{8}^{3} ) q^{52} \) \( + ( -1 + \zeta_{8}^{2} ) q^{53} \) \( + ( -2 - 2 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{54} \) \( + 2 \zeta_{8}^{2} q^{55} \) \( + ( -5 + 5 \zeta_{8} - 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{56} \) \( + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{57} \) \( + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{58} \) \( -6 \zeta_{8} q^{59} \) \( + ( -2 - 6 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{60} \) \( + ( 5 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{61} \) \( + ( -3 + 9 \zeta_{8} - 9 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{62} \) \( + ( -3 + 3 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{63} \) \( + ( 2 \zeta_{8} - 7 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{64} \) \( + ( -1 - \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{65} \) \( + ( 2 - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{66} \) \( + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{67} \) \( + ( 3 - 4 \zeta_{8} - 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{68} \) \( + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{69} \) \( + ( -2 + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{70} \) \( + ( 5 + 5 \zeta_{8} + 5 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{71} \) \( + ( -5 \zeta_{8} + \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{72} \) \( + ( -7 + 7 \zeta_{8} ) q^{73} \) \( + ( 5 - 10 \zeta_{8} + 10 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{74} \) \( + ( -3 - \zeta_{8} - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{75} \) \( + ( 2 + 6 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{76} \) \( + ( 2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{77} \) \( + ( -2 - 2 \zeta_{8}^{3} ) q^{78} \) \( + ( -1 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{79} \) \( + ( 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{80} \) \( + ( -2 \zeta_{8} - 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{81} \) \( + ( 1 + \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{82} \) \( + ( 4 - 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{83} \) \( + ( 8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{84} \) \( + ( 1 + 5 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{85} \) \( + 2 q^{86} \) \( + ( 4 - 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{87} \) \( + ( 3 + 3 \zeta_{8} - 5 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{88} \) \( + ( -\zeta_{8} - 8 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{89} \) \( + ( 1 - \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{90} \) \( + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{91} \) \( + ( -7 + 7 \zeta_{8} + 7 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{92} \) \( + ( 6 + 6 \zeta_{8}^{2} ) q^{93} \) \( + ( -6 + 4 \zeta_{8} - 6 \zeta_{8}^{2} ) q^{94} \) \( + ( -2 - 4 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{95} \) \( + ( -5 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{96} \) \( + ( 6 - 6 \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{97} \) \( + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{98} \) \( + ( -1 - \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 4q^{12} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 8q^{15} \) \(\mathstrut -\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 12q^{18} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 4q^{20} \) \(\mathstrut +\mathstrut 12q^{22} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 12q^{24} \) \(\mathstrut -\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 4q^{32} \) \(\mathstrut -\mathstrut 20q^{34} \) \(\mathstrut +\mathstrut 8q^{35} \) \(\mathstrut -\mathstrut 8q^{40} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 8q^{42} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 20q^{44} \) \(\mathstrut +\mathstrut 12q^{45} \) \(\mathstrut +\mathstrut 20q^{46} \) \(\mathstrut +\mathstrut 12q^{48} \) \(\mathstrut +\mathstrut 8q^{49} \) \(\mathstrut +\mathstrut 20q^{50} \) \(\mathstrut +\mathstrut 28q^{51} \) \(\mathstrut +\mathstrut 16q^{52} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut -\mathstrut 8q^{54} \) \(\mathstrut -\mathstrut 20q^{56} \) \(\mathstrut -\mathstrut 24q^{57} \) \(\mathstrut -\mathstrut 8q^{60} \) \(\mathstrut -\mathstrut 12q^{62} \) \(\mathstrut -\mathstrut 12q^{63} \) \(\mathstrut -\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 8q^{66} \) \(\mathstrut +\mathstrut 16q^{67} \) \(\mathstrut +\mathstrut 12q^{68} \) \(\mathstrut -\mathstrut 32q^{69} \) \(\mathstrut -\mathstrut 8q^{70} \) \(\mathstrut +\mathstrut 20q^{71} \) \(\mathstrut -\mathstrut 28q^{73} \) \(\mathstrut +\mathstrut 20q^{74} \) \(\mathstrut -\mathstrut 12q^{75} \) \(\mathstrut +\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 8q^{77} \) \(\mathstrut -\mathstrut 8q^{78} \) \(\mathstrut -\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut 4q^{82} \) \(\mathstrut +\mathstrut 16q^{83} \) \(\mathstrut +\mathstrut 32q^{84} \) \(\mathstrut +\mathstrut 4q^{85} \) \(\mathstrut +\mathstrut 8q^{86} \) \(\mathstrut +\mathstrut 16q^{87} \) \(\mathstrut +\mathstrut 12q^{88} \) \(\mathstrut +\mathstrut 4q^{90} \) \(\mathstrut -\mathstrut 28q^{92} \) \(\mathstrut +\mathstrut 24q^{93} \) \(\mathstrut -\mathstrut 24q^{94} \) \(\mathstrut -\mathstrut 8q^{95} \) \(\mathstrut -\mathstrut 20q^{96} \) \(\mathstrut +\mathstrut 24q^{97} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\zeta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.292893 0.292893i −2.41421 + 1.00000i 1.82843i 0.707107 + 1.70711i 1.00000 + 0.414214i 0.414214 1.00000i −1.12132 + 1.12132i 2.70711 2.70711i 0.292893 0.707107i
8.1 −1.70711 + 1.70711i 0.414214 1.00000i 3.82843i −0.707107 0.292893i 1.00000 + 2.41421i −2.41421 + 1.00000i 3.12132 + 3.12132i 1.29289 + 1.29289i 1.70711 0.707107i
9.1 −0.292893 + 0.292893i −2.41421 1.00000i 1.82843i 0.707107 1.70711i 1.00000 0.414214i 0.414214 + 1.00000i −1.12132 1.12132i 2.70711 + 2.70711i 0.292893 + 0.707107i
15.1 −1.70711 1.70711i 0.414214 + 1.00000i 3.82843i −0.707107 + 0.292893i 1.00000 2.41421i −2.41421 1.00000i 3.12132 3.12132i 1.29289 1.29289i 1.70711 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
17.d Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(17, [\chi])\).