Properties

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

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Newspace parameters

Level: \( N \) = \( 33 = 3 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 33.e (of order \(5\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.26350632667\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

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

\(n\) \(13\) \(23\)
\(\chi(n)\) \(\zeta_{10}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.809017 + 0.587785i
−0.309017 0.951057i
0.809017 0.587785i
−0.309017 + 0.951057i
−1.30902 0.951057i 0.309017 0.951057i 0.190983 + 0.587785i 0.309017 0.224514i −1.30902 + 0.951057i 0.927051 + 2.85317i −0.690983 + 2.12663i −0.809017 0.587785i −0.618034
16.1 −0.190983 0.587785i −0.809017 0.587785i 1.30902 0.951057i −0.809017 + 2.48990i −0.190983 + 0.587785i −2.42705 + 1.76336i −1.80902 1.31433i 0.309017 + 0.951057i 1.61803
25.1 −1.30902 + 0.951057i 0.309017 + 0.951057i 0.190983 0.587785i 0.309017 + 0.224514i −1.30902 0.951057i 0.927051 2.85317i −0.690983 2.12663i −0.809017 + 0.587785i −0.618034
31.1 −0.190983 + 0.587785i −0.809017 + 0.587785i 1.30902 + 0.951057i −0.809017 2.48990i −0.190983 0.587785i −2.42705 1.76336i −1.80902 + 1.31433i 0.309017 0.951057i 1.61803
\(n\): 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
11.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut +\mathstrut 3 T_{2}^{3} \) \(\mathstrut +\mathstrut 4 T_{2}^{2} \) \(\mathstrut +\mathstrut 2 T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(33, [\chi])\).