# 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

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: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{2} + \zeta_{20}^{6} q^{3} + ( -1 - \zeta_{20}^{4} - \zeta_{20}^{8} ) q^{4} + \zeta_{20}^{9} q^{5} + ( -\zeta_{20} - \zeta_{20}^{5} ) q^{6} + \zeta_{20}^{8} q^{7} + ( \zeta_{20}^{3} - \zeta_{20}^{5} + \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{8} -\zeta_{20}^{2} q^{9} +O(q^{10})$$ $$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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{8} + T_{2}^{6} + 6 T_{2}^{4} - 4 T_{2}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(633, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$7$ $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
$11$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
$13$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$17$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$19$ $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
$23$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$29$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
$31$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$37$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$41$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
$43$ $$( 1 + T + T^{2} )^{8}$$
$47$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
$53$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$59$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$61$ $$( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$67$ $$( 1 + T^{2} )^{8}$$
$71$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$73$ $$( 1 + T + T^{2} )^{8}$$
$79$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$83$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$89$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$97$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$