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

Related objects

Downloads

Learn more about

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.

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

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 no
633.m Odd 1 no