Properties

Label 690.2.m.b
Level $690$
Weight $2$
Character orbit 690.m
Analytic conductor $5.510$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.m (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Defining polynomial: \(x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{22}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{22}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.415415 + 0.909632i
0.959493 0.281733i
−0.841254 + 0.540641i
0.959493 + 0.281733i
0.654861 + 0.755750i
0.142315 0.989821i
0.654861 0.755750i
−0.841254 0.540641i
0.142315 + 0.989821i
−0.415415 0.909632i
0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i 0.959493 0.281733i −2.31329 1.48666i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.841254 + 0.540641i
121.1 −0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i 0.654861 0.755750i −0.0867074 + 0.603063i 0.959493 0.281733i 0.841254 + 0.540641i 0.142315 + 0.989821i
151.1 0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i 0.142315 + 0.989821i 1.88745 + 0.554206i −0.841254 + 0.540641i 0.415415 + 0.909632i 0.959493 0.281733i
211.1 −0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i 0.654861 + 0.755750i −0.0867074 0.603063i 0.959493 + 0.281733i 0.841254 0.540641i 0.142315 0.989821i
271.1 0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i −0.841254 + 0.540641i 1.24302 + 2.72183i 0.654861 + 0.755750i −0.142315 0.989821i −0.415415 + 0.909632i
301.1 −0.841254 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.415415 + 0.909632i −0.730471 + 0.843008i 0.142315 0.989821i −0.959493 + 0.281733i 0.654861 + 0.755750i
331.1 0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i −0.841254 0.540641i 1.24302 2.72183i 0.654861 0.755750i −0.142315 + 0.989821i −0.415415 0.909632i
361.1 0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i 0.142315 0.989821i 1.88745 0.554206i −0.841254 0.540641i 0.415415 0.909632i 0.959493 + 0.281733i
541.1 −0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −0.415415 0.909632i −0.730471 0.843008i 0.142315 + 0.989821i −0.959493 0.281733i 0.654861 0.755750i
601.1 0.142315 + 0.989821i 0.415415 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 0.959493 + 0.281733i −2.31329 + 1.48666i −0.415415 0.909632i −0.654861 0.755750i −0.841254 0.540641i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.m.b 10
23.c even 11 1 inner 690.2.m.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.b 10 1.a even 1 1 trivial
690.2.m.b 10 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 11 T_{7}^{7} + 22 T_{7}^{6} - 143 T_{7}^{5} + 121 T_{7}^{3} + 363 T_{7}^{2} + 121 T_{7} + 121 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$5$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
$7$ \( 121 + 121 T + 363 T^{2} + 121 T^{3} - 143 T^{5} + 22 T^{6} + 11 T^{7} + T^{10} \)
$11$ \( 1 + 3 T + 20 T^{2} + 5 T^{3} + 70 T^{4} + T^{5} - 30 T^{6} - 13 T^{7} + 5 T^{8} + 4 T^{9} + T^{10} \)
$13$ \( 4489 + 1809 T + 4612 T^{2} - 1679 T^{3} + 1043 T^{4} - 824 T^{5} + 170 T^{6} + 25 T^{7} - 7 T^{8} - 2 T^{9} + T^{10} \)
$17$ \( 529 + 644 T + 146 T^{2} + 7 T^{3} + 669 T^{4} + 175 T^{5} + 60 T^{6} + 63 T^{7} + 15 T^{8} + 2 T^{9} + T^{10} \)
$19$ \( 121 + 121 T + 121 T^{3} + 242 T^{4} + 143 T^{5} + 66 T^{6} + 33 T^{7} + 11 T^{8} + T^{10} \)
$23$ \( 6436343 + 279841 T - 255507 T^{2} - 69299 T^{3} + 12167 T^{4} + 3279 T^{5} + 529 T^{6} - 131 T^{7} - 21 T^{8} + T^{9} + T^{10} \)
$29$ \( 4489 + 8241 T + 9574 T^{2} + 7488 T^{3} + 4174 T^{4} + 1583 T^{5} + 405 T^{6} + 62 T^{7} + 14 T^{8} + 6 T^{9} + T^{10} \)
$31$ \( 223729 + 62436 T + 32307 T^{2} + 18271 T^{3} + 16093 T^{4} - 5104 T^{5} + 2706 T^{6} - 946 T^{7} + 198 T^{8} - 22 T^{9} + T^{10} \)
$37$ \( 175561 + 302099 T + 335426 T^{2} - 145446 T^{3} + 45417 T^{4} - 20054 T^{5} + 7067 T^{6} - 1444 T^{7} + 202 T^{8} - 20 T^{9} + T^{10} \)
$41$ \( 982081 + 627303 T + 246865 T^{2} + 73333 T^{3} + 34461 T^{4} + 9547 T^{5} + 170 T^{6} - 47 T^{7} + 48 T^{8} - 9 T^{9} + T^{10} \)
$43$ \( 4489 + 22043 T + 32517 T^{2} + 10064 T^{3} + 3488 T^{4} - 815 T^{5} + 1431 T^{6} - 408 T^{7} + 89 T^{8} - 12 T^{9} + T^{10} \)
$47$ \( ( 1541 + 438 T - 159 T^{2} - 42 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$53$ \( 3179089 + 2499766 T + 1323226 T^{2} + 347362 T^{3} + 50510 T^{4} + 25972 T^{5} + 9289 T^{6} + 1147 T^{7} + 191 T^{8} + 20 T^{9} + T^{10} \)
$59$ \( 4695889 + 2812766 T + 18546275 T^{2} + 687159 T^{3} + 127050 T^{4} + 90409 T^{5} + 9119 T^{6} + 473 T^{7} + 154 T^{8} + 22 T^{9} + T^{10} \)
$61$ \( 157226521 + 16112615 T + 14720028 T^{2} - 4747493 T^{3} + 2343874 T^{4} - 713945 T^{5} + 128511 T^{6} - 14799 T^{7} + 1103 T^{8} - 49 T^{9} + T^{10} \)
$67$ \( 6285049 - 3063554 T + 11100816 T^{2} - 1493271 T^{3} + 558108 T^{4} - 234002 T^{5} + 40195 T^{6} - 1627 T^{7} - 32 T^{8} - 10 T^{9} + T^{10} \)
$71$ \( 47100769 - 43566324 T + 33418551 T^{2} - 9215777 T^{3} + 1103158 T^{4} - 95732 T^{5} + 9571 T^{6} + T^{7} - 10 T^{8} + T^{9} + T^{10} \)
$73$ \( 1849 + 1505 T + 13457 T^{2} + 9254 T^{3} + 29694 T^{4} - 2056 T^{5} - 772 T^{6} + 169 T^{7} + 69 T^{8} - 17 T^{9} + T^{10} \)
$79$ \( 20948929 - 13277877 T + 10130239 T^{2} - 764363 T^{3} - 20511 T^{4} + 41966 T^{5} - 6300 T^{6} + 86 T^{7} + 159 T^{8} - 18 T^{9} + T^{10} \)
$83$ \( 1102040809 + 92852009 T - 12386464 T^{2} + 7349579 T^{3} + 5443541 T^{4} - 113983 T^{5} + 34037 T^{6} - 4816 T^{7} + 181 T^{8} - 15 T^{9} + T^{10} \)
$89$ \( 11881 - 6976 T + 19837 T^{2} + 10134 T^{3} + 6616 T^{4} + 4610 T^{5} + 1307 T^{6} + 103 T^{7} + 3 T^{8} + 5 T^{9} + T^{10} \)
$97$ \( 1 + 23 T + 210 T^{2} + 573 T^{3} + 4555 T^{4} + 7107 T^{5} + 3807 T^{6} + 606 T^{7} + 67 T^{8} + T^{9} + T^{10} \)
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