Properties

Label 29.2.d.a
Level 29
Weight 2
Character orbit 29.d
Analytic conductor 0.232
Analytic rank 0
Dimension 6
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 29.d (of order \(7\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{14}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.900969 0.433884i
−0.623490 0.781831i
−0.623490 + 0.781831i
0.222521 + 0.974928i
0.222521 0.974928i
0.900969 + 0.433884i
−0.277479 1.21572i −1.62349 0.781831i 0.400969 0.193096i 0.900969 + 3.94740i −0.500000 + 2.19064i −0.623490 0.300257i −1.90097 2.38374i 0.153989 + 0.193096i 4.54892 2.19064i
16.1 0.400969 + 0.193096i −0.777479 + 0.974928i −1.12349 1.40881i −0.623490 0.300257i −0.500000 + 0.240787i 0.222521 0.279032i −0.376510 1.64960i 0.321552 + 1.40881i −0.192021 0.240787i
20.1 0.400969 0.193096i −0.777479 0.974928i −1.12349 + 1.40881i −0.623490 + 0.300257i −0.500000 0.240787i 0.222521 + 0.279032i −0.376510 + 1.64960i 0.321552 1.40881i −0.192021 + 0.240787i
23.1 −1.12349 + 1.40881i −0.0990311 + 0.433884i −0.277479 1.21572i 0.222521 0.279032i −0.500000 0.626980i 0.900969 3.94740i −1.22252 0.588735i 2.52446 + 1.21572i 0.143104 + 0.626980i
24.1 −1.12349 1.40881i −0.0990311 0.433884i −0.277479 + 1.21572i 0.222521 + 0.279032i −0.500000 + 0.626980i 0.900969 + 3.94740i −1.22252 + 0.588735i 2.52446 1.21572i 0.143104 0.626980i
25.1 −0.277479 + 1.21572i −1.62349 + 0.781831i 0.400969 + 0.193096i 0.900969 3.94740i −0.500000 2.19064i −0.623490 + 0.300257i −1.90097 + 2.38374i 0.153989 0.193096i 4.54892 + 2.19064i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.1
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(29, [\chi])\).