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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This is the first weight $1$ newform with projective image $A_5$.

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.315908152997\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{5}\)
Projective field: Galois closure of 5.1.17839074969.1

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
211.d even 5 1 inner
633.m odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.1.m.b 8
3.b odd 2 1 inner 633.1.m.b 8
211.d even 5 1 inner 633.1.m.b 8
633.m odd 10 1 inner 633.1.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.1.m.b 8 1.a even 1 1 trivial
633.1.m.b 8 3.b odd 2 1 inner
633.1.m.b 8 211.d even 5 1 inner
633.1.m.b 8 633.m odd 10 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \) Copy content Toggle raw display
$31$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \) Copy content Toggle raw display
$43$ \( (T + 1)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \) Copy content Toggle raw display
$73$ \( (T + 1)^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \) Copy content Toggle raw display
$89$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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