Properties

 Label 1600.1.bd.a Level $1600$ Weight $1$ Character orbit 1600.bd Analytic conductor $0.799$ Analytic rank $0$ Dimension $8$ Projective image $A_{5}$ CM/RM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1600.bd (of order $$10$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.798504020213$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{5}$$ Projective field: Galois closure of 5.1.25000000.3

$q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + (\zeta_{20}^{6} - 1) q^{3} - \zeta_{20}^{3} q^{5} + \zeta_{20}^{5} q^{7} + ( - \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{9} +O(q^{10})$$ q + (z^6 - 1) * q^3 - z^3 * q^5 + z^5 * q^7 + (-z^6 - z^2 + 1) * q^9 $$q + (\zeta_{20}^{6} - 1) q^{3} - \zeta_{20}^{3} q^{5} + \zeta_{20}^{5} q^{7} + ( - \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{9} - \zeta_{20}^{4} q^{11} + \zeta_{20} q^{13} + ( - \zeta_{20}^{9} + \zeta_{20}^{3}) q^{15} + ( - \zeta_{20}^{4} - 1) q^{17} + (\zeta_{20}^{8} - \zeta_{20}^{6}) q^{19} + ( - \zeta_{20}^{5} - \zeta_{20}) q^{21} - \zeta_{20}^{9} q^{23} + \zeta_{20}^{6} q^{25} + ( - \zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{2} - 1) q^{27} + \zeta_{20}^{3} q^{29} - \zeta_{20}^{7} q^{31} + (\zeta_{20}^{4} + 1) q^{33} - \zeta_{20}^{8} q^{35} + ( - \zeta_{20}^{7} + \zeta_{20}^{5}) q^{37} + (\zeta_{20}^{7} - \zeta_{20}) q^{39} - \zeta_{20}^{6} q^{41} + q^{43} + (\zeta_{20}^{9} + \zeta_{20}^{5} - \zeta_{20}^{3}) q^{45} + \zeta_{20}^{3} q^{47} + ( - \zeta_{20}^{6} + \zeta_{20}^{4} + 1) q^{51} + \zeta_{20}^{7} q^{55} + ( - \zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{57} + ( - \zeta_{20}^{7} + \zeta_{20}) q^{61} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}) q^{63} - \zeta_{20}^{4} q^{65} + \zeta_{20}^{2} q^{67} + (\zeta_{20}^{9} + \zeta_{20}^{5}) q^{69} - \zeta_{20}^{3} q^{71} + ( - \zeta_{20}^{6} - \zeta_{20}^{2}) q^{75} - \zeta_{20}^{9} q^{77} + (\zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{2} + 1) q^{81} - \zeta_{20}^{2} q^{83} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{85} + (\zeta_{20}^{9} - \zeta_{20}^{3}) q^{87} + \zeta_{20}^{6} q^{91} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{93} + (\zeta_{20}^{9} + \zeta_{20}) q^{95} + (\zeta_{20}^{4} - \zeta_{20}^{2}) q^{97} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{99} +O(q^{100})$$ q + (z^6 - 1) * q^3 - z^3 * q^5 + z^5 * q^7 + (-z^6 - z^2 + 1) * q^9 - z^4 * q^11 + z * q^13 + (-z^9 + z^3) * q^15 + (-z^4 - 1) * q^17 + (z^8 - z^6) * q^19 + (-z^5 - z) * q^21 - z^9 * q^23 + z^6 * q^25 + (-z^8 + z^6 - z^2 - 1) * q^27 + z^3 * q^29 - z^7 * q^31 + (z^4 + 1) * q^33 - z^8 * q^35 + (-z^7 + z^5) * q^37 + (z^7 - z) * q^39 - z^6 * q^41 + q^43 + (z^9 + z^5 - z^3) * q^45 + z^3 * q^47 + (-z^6 + z^4 + 1) * q^51 + z^7 * q^55 + (-z^8 + z^6 - z^4 + z^2) * q^57 + (-z^7 + z) * q^61 + (-z^7 + z^5 + z) * q^63 - z^4 * q^65 + z^2 * q^67 + (z^9 + z^5) * q^69 - z^3 * q^71 + (-z^6 - z^2) * q^75 - z^9 * q^77 + (z^8 - z^6 + z^4 + z^2 + 1) * q^81 - z^2 * q^83 + (z^7 + z^3) * q^85 + (z^9 - z^3) * q^87 + z^6 * q^91 + (z^7 + z^3) * q^93 + (z^9 + z) * q^95 + (z^4 - z^2) * q^97 + (z^6 - z^4 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{3} + 4 q^{9}+O(q^{10})$$ 8 * q - 6 * q^3 + 4 * q^9 $$8 q - 6 q^{3} + 4 q^{9} + 2 q^{11} - 6 q^{17} - 4 q^{19} + 2 q^{25} - 2 q^{27} + 6 q^{33} + 2 q^{35} - 2 q^{41} + 8 q^{43} + 12 q^{51} + 8 q^{57} + 2 q^{65} + 2 q^{67} - 4 q^{75} - 2 q^{83} + 2 q^{91} - 4 q^{97} - 4 q^{99}+O(q^{100})$$ 8 * q - 6 * q^3 + 4 * q^9 + 2 * q^11 - 6 * q^17 - 4 * q^19 + 2 * q^25 - 2 * q^27 + 6 * q^33 + 2 * q^35 - 2 * q^41 + 8 * q^43 + 12 * q^51 + 8 * q^57 + 2 * q^65 + 2 * q^67 - 4 * q^75 - 2 * q^83 + 2 * q^91 - 4 * q^97 - 4 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$\zeta_{20}^{8}$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 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 −1.30902 + 0.951057i 0 −0.587785 0.809017i 0 1.00000i 0 0.500000 1.53884i 0
31.2 0 −1.30902 + 0.951057i 0 0.587785 + 0.809017i 0 1.00000i 0 0.500000 1.53884i 0
671.1 0 −1.30902 0.951057i 0 −0.587785 + 0.809017i 0 1.00000i 0 0.500000 + 1.53884i 0
671.2 0 −1.30902 0.951057i 0 0.587785 0.809017i 0 1.00000i 0 0.500000 + 1.53884i 0
991.1 0 −0.190983 0.587785i 0 −0.951057 + 0.309017i 0 1.00000i 0 0.500000 0.363271i 0
991.2 0 −0.190983 0.587785i 0 0.951057 0.309017i 0 1.00000i 0 0.500000 0.363271i 0
1311.1 0 −0.190983 + 0.587785i 0 −0.951057 0.309017i 0 1.00000i 0 0.500000 + 0.363271i 0
1311.2 0 −0.190983 + 0.587785i 0 0.951057 + 0.309017i 0 1.00000i 0 0.500000 + 0.363271i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1311.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
25.d even 5 1 inner
200.n odd 10 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.1.bd.a 8
4.b odd 2 1 1600.1.bd.b yes 8
8.b even 2 1 1600.1.bd.b yes 8
8.d odd 2 1 inner 1600.1.bd.a 8
25.d even 5 1 inner 1600.1.bd.a 8
100.j odd 10 1 1600.1.bd.b yes 8
200.n odd 10 1 inner 1600.1.bd.a 8
200.t even 10 1 1600.1.bd.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.1.bd.a 8 1.a even 1 1 trivial
1600.1.bd.a 8 8.d odd 2 1 inner
1600.1.bd.a 8 25.d even 5 1 inner
1600.1.bd.a 8 200.n odd 10 1 inner
1600.1.bd.b yes 8 4.b odd 2 1
1600.1.bd.b yes 8 8.b even 2 1
1600.1.bd.b yes 8 100.j odd 10 1
1600.1.bd.b yes 8 200.t even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3T_{3}^{3} + 4T_{3}^{2} + 2T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1600, [\chi])$$.

Hecke characteristic polynomials

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