Properties

Label 33.2.f.a
Level 33
Weight 2
Character orbit 33.f
Analytic conductor 0.264
Analytic rank 0
Dimension 8
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 33 = 3 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 33.f (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.26350632667\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(\zeta_{20}^{2}\) \(-1\)

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
−0.951057 0.309017i
0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
−0.951057 + 0.309017i
0.951057 0.309017i
0.587785 0.809017i
−0.587785 + 0.809017i
−0.587785 + 1.80902i −1.67229 + 0.451057i −1.30902 0.951057i 2.48990 0.809017i 0.166977 3.29032i 0.427051 0.587785i −0.587785 + 0.427051i 2.59310 1.50859i 4.97980i
2.2 0.587785 1.80902i −0.945746 + 1.45106i −1.30902 0.951057i −2.48990 + 0.809017i 2.06909 + 2.56378i 0.427051 0.587785i 0.587785 0.427051i −1.21113 2.74466i 4.97980i
8.1 −0.951057 + 0.690983i 1.34786 1.08779i −0.190983 + 0.587785i −0.224514 + 0.309017i −0.530249 + 1.96589i −2.92705 0.951057i −0.951057 2.92705i 0.633446 2.93236i 0.449028i
8.2 0.951057 0.690983i −1.72982 0.0877853i −0.190983 + 0.587785i 0.224514 0.309017i −1.70582 + 1.11179i −2.92705 0.951057i 0.951057 + 2.92705i 2.98459 + 0.303706i 0.449028i
17.1 −0.587785 1.80902i −1.67229 0.451057i −1.30902 + 0.951057i 2.48990 + 0.809017i 0.166977 + 3.29032i 0.427051 + 0.587785i −0.587785 0.427051i 2.59310 + 1.50859i 4.97980i
17.2 0.587785 + 1.80902i −0.945746 1.45106i −1.30902 + 0.951057i −2.48990 0.809017i 2.06909 2.56378i 0.427051 + 0.587785i 0.587785 + 0.427051i −1.21113 + 2.74466i 4.97980i
29.1 −0.951057 0.690983i 1.34786 + 1.08779i −0.190983 0.587785i −0.224514 0.309017i −0.530249 1.96589i −2.92705 + 0.951057i −0.951057 + 2.92705i 0.633446 + 2.93236i 0.449028i
29.2 0.951057 + 0.690983i −1.72982 + 0.0877853i −0.190983 0.587785i 0.224514 + 0.309017i −1.70582 1.11179i −2.92705 + 0.951057i 0.951057 2.92705i 2.98459 0.303706i 0.449028i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
11.d Odd 1 yes
33.f Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(33, [\chi])\).