Properties

Label 633.1.m.b
Level 633
Weight 1
Character orbit 633.m
Analytic conductor 0.316
Analytic rank 0
Dimension 8
Projective image \(A_{5}\)
CM/RM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 633 = 3 \cdot 211 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 633.m (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.315908152997\)
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: \( 1 \)
Projective image \(A_{5}\)
Projective field Galois closure of 5.1.17839074969.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/633\mathbb{Z}\right)^\times\).

\(n\) \(212\) \(424\)
\(\chi(n)\) \(-1\) \(\zeta_{20}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
0.951057 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
0.587785 0.809017i
−0.587785 + 0.809017i
−0.951057 1.30902i −0.309017 0.951057i −0.500000 + 1.53884i −0.951057 0.309017i −0.951057 + 1.30902i −0.809017 0.587785i 0.951057 0.309017i −0.809017 + 0.587785i 0.500000 + 1.53884i
71.2 0.951057 + 1.30902i −0.309017 0.951057i −0.500000 + 1.53884i 0.951057 + 0.309017i 0.951057 1.30902i −0.809017 0.587785i −0.951057 + 0.309017i −0.809017 + 0.587785i 0.500000 + 1.53884i
107.1 −0.951057 + 1.30902i −0.309017 + 0.951057i −0.500000 1.53884i −0.951057 + 0.309017i −0.951057 1.30902i −0.809017 + 0.587785i 0.951057 + 0.309017i −0.809017 0.587785i 0.500000 1.53884i
107.2 0.951057 1.30902i −0.309017 + 0.951057i −0.500000 1.53884i 0.951057 0.309017i 0.951057 + 1.30902i −0.809017 + 0.587785i −0.951057 0.309017i −0.809017 0.587785i 0.500000 1.53884i
188.1 −0.587785 0.190983i 0.809017 0.587785i −0.500000 0.363271i −0.587785 + 0.809017i −0.587785 + 0.190983i 0.309017 + 0.951057i 0.587785 + 0.809017i 0.309017 0.951057i 0.500000 0.363271i
188.2 0.587785 + 0.190983i 0.809017 0.587785i −0.500000 0.363271i 0.587785 0.809017i 0.587785 0.190983i 0.309017 + 0.951057i −0.587785 0.809017i 0.309017 0.951057i 0.500000 0.363271i
266.1 −0.587785 + 0.190983i 0.809017 + 0.587785i −0.500000 + 0.363271i −0.587785 0.809017i −0.587785 0.190983i 0.309017 0.951057i 0.587785 0.809017i 0.309017 + 0.951057i 0.500000 + 0.363271i
266.2 0.587785 0.190983i 0.809017 + 0.587785i −0.500000 + 0.363271i 0.587785 + 0.809017i 0.587785 + 0.190983i 0.309017 0.951057i −0.587785 + 0.809017i 0.309017 + 0.951057i 0.500000 + 0.363271i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 266.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
211.d Even 1 yes
633.m Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{8} + T_{2}^{6} + 6 T_{2}^{4} - 4 T_{2}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(633, [\chi])\).