Properties

 Label 8.8.a.a Level $8$ Weight $8$ Character orbit 8.a Self dual yes Analytic conductor $2.499$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 8.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$2.49908020387$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 84q^{3} - 82q^{5} - 456q^{7} + 4869q^{9} + O(q^{10})$$ $$q - 84q^{3} - 82q^{5} - 456q^{7} + 4869q^{9} - 2524q^{11} - 10778q^{13} + 6888q^{15} - 11150q^{17} + 4124q^{19} + 38304q^{21} + 81704q^{23} - 71401q^{25} - 225288q^{27} + 99798q^{29} - 40480q^{31} + 212016q^{33} + 37392q^{35} - 419442q^{37} + 905352q^{39} + 141402q^{41} - 690428q^{43} - 399258q^{45} - 682032q^{47} - 615607q^{49} + 936600q^{51} + 1813118q^{53} + 206968q^{55} - 346416q^{57} - 966028q^{59} + 1887670q^{61} - 2220264q^{63} + 883796q^{65} + 2965868q^{67} - 6863136q^{69} - 2548232q^{71} - 1680326q^{73} + 5997684q^{75} + 1150944q^{77} + 4038064q^{79} + 8275689q^{81} - 5385764q^{83} + 914300q^{85} - 8383032q^{87} - 6473046q^{89} + 4914768q^{91} + 3400320q^{93} - 338168q^{95} - 6065758q^{97} - 12289356q^{99} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −84.0000 0 −82.0000 0 −456.000 0 4869.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.8.a.a 1
3.b odd 2 1 72.8.a.d 1
4.b odd 2 1 16.8.a.c 1
5.b even 2 1 200.8.a.i 1
5.c odd 4 2 200.8.c.a 2
7.b odd 2 1 392.8.a.d 1
8.b even 2 1 64.8.a.g 1
8.d odd 2 1 64.8.a.a 1
12.b even 2 1 144.8.a.g 1
16.e even 4 2 256.8.b.e 2
16.f odd 4 2 256.8.b.c 2
20.d odd 2 1 400.8.a.b 1
20.e even 4 2 400.8.c.b 2
24.f even 2 1 576.8.a.k 1
24.h odd 2 1 576.8.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 1.a even 1 1 trivial
16.8.a.c 1 4.b odd 2 1
64.8.a.a 1 8.d odd 2 1
64.8.a.g 1 8.b even 2 1
72.8.a.d 1 3.b odd 2 1
144.8.a.g 1 12.b even 2 1
200.8.a.i 1 5.b even 2 1
200.8.c.a 2 5.c odd 4 2
256.8.b.c 2 16.f odd 4 2
256.8.b.e 2 16.e even 4 2
392.8.a.d 1 7.b odd 2 1
400.8.a.b 1 20.d odd 2 1
400.8.c.b 2 20.e even 4 2
576.8.a.j 1 24.h odd 2 1
576.8.a.k 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 84$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(8))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$84 + T$$
$5$ $$82 + T$$
$7$ $$456 + T$$
$11$ $$2524 + T$$
$13$ $$10778 + T$$
$17$ $$11150 + T$$
$19$ $$-4124 + T$$
$23$ $$-81704 + T$$
$29$ $$-99798 + T$$
$31$ $$40480 + T$$
$37$ $$419442 + T$$
$41$ $$-141402 + T$$
$43$ $$690428 + T$$
$47$ $$682032 + T$$
$53$ $$-1813118 + T$$
$59$ $$966028 + T$$
$61$ $$-1887670 + T$$
$67$ $$-2965868 + T$$
$71$ $$2548232 + T$$
$73$ $$1680326 + T$$
$79$ $$-4038064 + T$$
$83$ $$5385764 + T$$
$89$ $$6473046 + T$$
$97$ $$6065758 + T$$