Properties

 Label 800.2.c.f Level $800$ Weight $2$ Character orbit 800.c Analytic conductor $6.388$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{7} -5 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{7} -5 q^{9} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} + 2 \zeta_{8}^{2} q^{13} + 2 \zeta_{8}^{2} q^{17} -8 q^{21} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} -6 q^{29} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{31} + 16 \zeta_{8}^{2} q^{33} -10 \zeta_{8}^{2} q^{37} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{39} + 2 q^{41} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{43} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{47} - q^{49} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{51} -6 \zeta_{8}^{2} q^{53} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{59} -2 q^{61} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{63} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{67} -8 q^{69} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + 6 \zeta_{8}^{2} q^{73} + 16 \zeta_{8}^{2} q^{77} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{79} + q^{81} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{83} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{87} -10 q^{89} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{91} -16 \zeta_{8}^{2} q^{93} + 2 \zeta_{8}^{2} q^{97} + ( -20 \zeta_{8} + 20 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{9} + O(q^{10})$$ $$4 q - 20 q^{9} - 32 q^{21} - 24 q^{29} + 8 q^{41} - 4 q^{49} - 8 q^{61} - 32 q^{69} + 4 q^{81} - 40 q^{89} + O(q^{100})$$

Character values

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

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 2.82843i 0 0 0 2.82843i 0 −5.00000 0
449.2 0 2.82843i 0 0 0 2.82843i 0 −5.00000 0
449.3 0 2.82843i 0 0 0 2.82843i 0 −5.00000 0
449.4 0 2.82843i 0 0 0 2.82843i 0 −5.00000 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.c.f 4
3.b odd 2 1 7200.2.f.bh 4
4.b odd 2 1 inner 800.2.c.f 4
5.b even 2 1 inner 800.2.c.f 4
5.c odd 4 1 160.2.a.c 2
5.c odd 4 1 800.2.a.m 2
8.b even 2 1 1600.2.c.n 4
8.d odd 2 1 1600.2.c.n 4
12.b even 2 1 7200.2.f.bh 4
15.d odd 2 1 7200.2.f.bh 4
15.e even 4 1 1440.2.a.o 2
15.e even 4 1 7200.2.a.cm 2
20.d odd 2 1 inner 800.2.c.f 4
20.e even 4 1 160.2.a.c 2
20.e even 4 1 800.2.a.m 2
35.f even 4 1 7840.2.a.bf 2
40.e odd 2 1 1600.2.c.n 4
40.f even 2 1 1600.2.c.n 4
40.i odd 4 1 320.2.a.g 2
40.i odd 4 1 1600.2.a.bc 2
40.k even 4 1 320.2.a.g 2
40.k even 4 1 1600.2.a.bc 2
60.h even 2 1 7200.2.f.bh 4
60.l odd 4 1 1440.2.a.o 2
60.l odd 4 1 7200.2.a.cm 2
80.i odd 4 1 1280.2.d.l 4
80.j even 4 1 1280.2.d.l 4
80.s even 4 1 1280.2.d.l 4
80.t odd 4 1 1280.2.d.l 4
120.q odd 4 1 2880.2.a.bk 2
120.w even 4 1 2880.2.a.bk 2
140.j odd 4 1 7840.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 5.c odd 4 1
160.2.a.c 2 20.e even 4 1
320.2.a.g 2 40.i odd 4 1
320.2.a.g 2 40.k even 4 1
800.2.a.m 2 5.c odd 4 1
800.2.a.m 2 20.e even 4 1
800.2.c.f 4 1.a even 1 1 trivial
800.2.c.f 4 4.b odd 2 1 inner
800.2.c.f 4 5.b even 2 1 inner
800.2.c.f 4 20.d odd 2 1 inner
1280.2.d.l 4 80.i odd 4 1
1280.2.d.l 4 80.j even 4 1
1280.2.d.l 4 80.s even 4 1
1280.2.d.l 4 80.t odd 4 1
1440.2.a.o 2 15.e even 4 1
1440.2.a.o 2 60.l odd 4 1
1600.2.a.bc 2 40.i odd 4 1
1600.2.a.bc 2 40.k even 4 1
1600.2.c.n 4 8.b even 2 1
1600.2.c.n 4 8.d odd 2 1
1600.2.c.n 4 40.e odd 2 1
1600.2.c.n 4 40.f even 2 1
2880.2.a.bk 2 120.q odd 4 1
2880.2.a.bk 2 120.w even 4 1
7200.2.a.cm 2 15.e even 4 1
7200.2.a.cm 2 60.l odd 4 1
7200.2.f.bh 4 3.b odd 2 1
7200.2.f.bh 4 12.b even 2 1
7200.2.f.bh 4 15.d odd 2 1
7200.2.f.bh 4 60.h even 2 1
7840.2.a.bf 2 35.f even 4 1
7840.2.a.bf 2 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} + 8$$ $$T_{11}^{2} - 32$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 8 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 8 + T^{2} )^{2}$$
$11$ $$( -32 + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( 4 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( 8 + T^{2} )^{2}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$( -32 + T^{2} )^{2}$$
$37$ $$( 100 + T^{2} )^{2}$$
$41$ $$( -2 + T )^{4}$$
$43$ $$( 72 + T^{2} )^{2}$$
$47$ $$( 8 + T^{2} )^{2}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$( -128 + T^{2} )^{2}$$
$61$ $$( 2 + T )^{4}$$
$67$ $$( 8 + T^{2} )^{2}$$
$71$ $$( -32 + T^{2} )^{2}$$
$73$ $$( 36 + T^{2} )^{2}$$
$79$ $$( -128 + T^{2} )^{2}$$
$83$ $$( 8 + T^{2} )^{2}$$
$89$ $$( 10 + T )^{4}$$
$97$ $$( 4 + T^{2} )^{2}$$