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

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

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

## Hecke characteristic polynomials

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