# Properties

 Label 800.2.y.a Level $800$ Weight $2$ Character orbit 800.y Analytic conductor $6.388$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.y (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{3} -2 q^{4} + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( -1 - \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} - \zeta_{8}^{2} ) q^{3} -2 q^{4} + ( 1 + \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{6} + ( -1 - \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( -1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} + ( -2 + 2 \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{11} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{12} + ( -1 + \zeta_{8}^{3} ) q^{13} -2 \zeta_{8}^{3} q^{14} + 4 q^{16} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{17} + ( 1 - 2 \zeta_{8} + \zeta_{8}^{2} ) q^{18} + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{19} + ( -1 + \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{21} + ( -1 - \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{22} + ( -3 + 3 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{24} + ( -1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{26} + ( -3 - 3 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( 2 + 2 \zeta_{8}^{2} ) q^{28} + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{29} -4 q^{31} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{33} -4 q^{34} + ( 2 - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{36} + ( -1 - \zeta_{8} ) q^{37} + ( 1 - 5 \zeta_{8} - 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{38} + ( 1 + 2 \zeta_{8} + \zeta_{8}^{2} ) q^{39} + ( -3 + 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{41} + ( -2 - 2 \zeta_{8} ) q^{42} + ( -4 + 4 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{43} + ( 4 - 4 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{44} + ( 4 - 6 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{46} + ( -4 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{47} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{48} -5 \zeta_{8}^{2} q^{49} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{51} + ( 2 - 2 \zeta_{8}^{3} ) q^{52} + ( -1 + \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{53} + ( 4 - 4 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{54} + 4 \zeta_{8}^{3} q^{56} + ( 5 + 4 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{57} + ( -1 - 3 \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{58} + ( -4 - 4 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{59} + ( 1 + \zeta_{8}^{3} ) q^{61} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{62} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{63} -8 q^{64} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{66} + ( 2 - 3 \zeta_{8} - 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{67} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{68} + ( -1 - \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{69} + ( -3 + 4 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{71} + ( -2 + 4 \zeta_{8} - 2 \zeta_{8}^{2} ) q^{72} + ( -7 + 7 \zeta_{8}^{2} ) q^{73} + ( 1 - \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{74} + ( 4 + 6 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{76} + ( 1 - 3 \zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{77} + ( -2 + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{78} -6 \zeta_{8}^{2} q^{79} + ( 5 \zeta_{8} + 3 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{81} + ( -4 - 6 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{82} + ( -4 + \zeta_{8} - \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{83} + ( 2 - 2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{84} + ( -7 - 7 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{86} + ( 1 - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{87} + ( 2 + 2 \zeta_{8} - 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{88} + ( 3 - 8 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{89} + ( 1 + \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{91} + ( 6 - 6 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{92} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{93} + ( 8 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{94} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{96} + ( 10 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{97} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{98} + ( 5 - 2 \zeta_{8} - 2 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} + 4 q^{6} - 4 q^{7} - 4 q^{9} + O(q^{10})$$ $$4 q - 8 q^{4} + 4 q^{6} - 4 q^{7} - 4 q^{9} - 8 q^{11} - 4 q^{13} + 16 q^{16} + 4 q^{18} - 8 q^{19} - 4 q^{21} - 4 q^{22} - 12 q^{23} - 8 q^{24} - 4 q^{26} - 12 q^{27} + 8 q^{28} - 4 q^{29} - 16 q^{31} - 8 q^{33} - 16 q^{34} + 8 q^{36} - 4 q^{37} + 4 q^{38} + 4 q^{39} - 12 q^{41} - 8 q^{42} - 16 q^{43} + 16 q^{44} + 16 q^{46} + 8 q^{51} + 8 q^{52} - 4 q^{53} + 16 q^{54} + 20 q^{57} - 4 q^{58} - 16 q^{59} + 4 q^{61} + 8 q^{63} - 32 q^{64} + 8 q^{67} - 4 q^{69} - 12 q^{71} - 8 q^{72} - 28 q^{73} + 4 q^{74} + 16 q^{76} + 4 q^{77} - 8 q^{78} - 16 q^{82} - 16 q^{83} + 8 q^{84} - 28 q^{86} + 4 q^{87} + 8 q^{88} + 12 q^{89} + 4 q^{91} + 24 q^{92} + 32 q^{94} + 16 q^{96} + 40 q^{97} + 20 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$\zeta_{8}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
1.41421i −0.707107 1.70711i −2.00000 0 2.41421 1.00000i −1.00000 1.00000i 2.82843i −0.292893 + 0.292893i 0
301.1 1.41421i −0.707107 + 1.70711i −2.00000 0 2.41421 + 1.00000i −1.00000 + 1.00000i 2.82843i −0.292893 0.292893i 0
501.1 1.41421i 0.707107 0.292893i −2.00000 0 −0.414214 1.00000i −1.00000 1.00000i 2.82843i −1.70711 + 1.70711i 0
701.1 1.41421i 0.707107 + 0.292893i −2.00000 0 −0.414214 + 1.00000i −1.00000 + 1.00000i 2.82843i −1.70711 1.70711i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.y.a 4
5.b even 2 1 32.2.g.a 4
5.c odd 4 1 800.2.ba.a 4
5.c odd 4 1 800.2.ba.b 4
15.d odd 2 1 288.2.v.a 4
20.d odd 2 1 128.2.g.a 4
32.g even 8 1 inner 800.2.y.a 4
40.e odd 2 1 256.2.g.a 4
40.f even 2 1 256.2.g.b 4
60.h even 2 1 1152.2.v.a 4
80.k odd 4 1 512.2.g.b 4
80.k odd 4 1 512.2.g.c 4
80.q even 4 1 512.2.g.a 4
80.q even 4 1 512.2.g.d 4
160.v odd 8 1 800.2.ba.a 4
160.y odd 8 1 128.2.g.a 4
160.y odd 8 1 256.2.g.a 4
160.y odd 8 1 512.2.g.b 4
160.y odd 8 1 512.2.g.c 4
160.z even 8 1 32.2.g.a 4
160.z even 8 1 256.2.g.b 4
160.z even 8 1 512.2.g.a 4
160.z even 8 1 512.2.g.d 4
160.bb odd 8 1 800.2.ba.b 4
320.bf even 16 2 4096.2.a.e 4
320.bh odd 16 2 4096.2.a.f 4
480.bs even 8 1 1152.2.v.a 4
480.bu odd 8 1 288.2.v.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a 4 5.b even 2 1
32.2.g.a 4 160.z even 8 1
128.2.g.a 4 20.d odd 2 1
128.2.g.a 4 160.y odd 8 1
256.2.g.a 4 40.e odd 2 1
256.2.g.a 4 160.y odd 8 1
256.2.g.b 4 40.f even 2 1
256.2.g.b 4 160.z even 8 1
288.2.v.a 4 15.d odd 2 1
288.2.v.a 4 480.bu odd 8 1
512.2.g.a 4 80.q even 4 1
512.2.g.a 4 160.z even 8 1
512.2.g.b 4 80.k odd 4 1
512.2.g.b 4 160.y odd 8 1
512.2.g.c 4 80.k odd 4 1
512.2.g.c 4 160.y odd 8 1
512.2.g.d 4 80.q even 4 1
512.2.g.d 4 160.z even 8 1
800.2.y.a 4 1.a even 1 1 trivial
800.2.y.a 4 32.g even 8 1 inner
800.2.ba.a 4 5.c odd 4 1
800.2.ba.a 4 160.v odd 8 1
800.2.ba.b 4 5.c odd 4 1
800.2.ba.b 4 160.bb odd 8 1
1152.2.v.a 4 60.h even 2 1
1152.2.v.a 4 480.bs even 8 1
4096.2.a.e 4 320.bf even 16 2
4096.2.a.f 4 320.bh odd 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 2 T_{3}^{2} - 4 T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(800, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$2 - 4 T + 2 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 2 + 2 T + T^{2} )^{2}$$
$11$ $$2 - 4 T + 18 T^{2} + 8 T^{3} + T^{4}$$
$13$ $$2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$( 8 + T^{2} )^{2}$$
$19$ $$578 + 68 T + 18 T^{2} + 8 T^{3} + T^{4}$$
$23$ $$4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$98 + 28 T + 6 T^{2} + 4 T^{3} + T^{4}$$
$31$ $$( 4 + T )^{4}$$
$37$ $$2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4}$$
$41$ $$4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$43$ $$1922 + 868 T + 162 T^{2} + 16 T^{3} + T^{4}$$
$47$ $$16 + 136 T^{2} + T^{4}$$
$53$ $$98 - 140 T + 54 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$1058 + 460 T + 114 T^{2} + 16 T^{3} + T^{4}$$
$61$ $$2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4}$$
$67$ $$578 - 68 T + 18 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$73$ $$( 98 + 14 T + T^{2} )^{2}$$
$79$ $$( 36 + T^{2} )^{2}$$
$83$ $$1058 + 460 T + 114 T^{2} + 16 T^{3} + T^{4}$$
$89$ $$2116 + 552 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$97$ $$( 28 - 20 T + T^{2} )^{2}$$