# Properties

 Label 12.8.a.a Level 12 Weight 8 Character orbit 12.a Self dual yes Analytic conductor 3.749 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 12.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.74862030581$$ 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

 $$f(q)$$ $$=$$ $$q - 27q^{3} - 378q^{5} - 832q^{7} + 729q^{9} + O(q^{10})$$ $$q - 27q^{3} - 378q^{5} - 832q^{7} + 729q^{9} - 2484q^{11} + 14870q^{13} + 10206q^{15} - 22302q^{17} - 16300q^{19} + 22464q^{21} - 115128q^{23} + 64759q^{25} - 19683q^{27} + 157086q^{29} - 16456q^{31} + 67068q^{33} + 314496q^{35} - 149266q^{37} - 401490q^{39} - 241110q^{41} - 443188q^{43} - 275562q^{45} + 922752q^{47} - 131319q^{49} + 602154q^{51} - 697626q^{53} + 938952q^{55} + 440100q^{57} + 870156q^{59} + 2067062q^{61} - 606528q^{63} - 5620860q^{65} - 1680748q^{67} + 3108456q^{69} - 1070280q^{71} - 2403334q^{73} - 1748493q^{75} + 2066688q^{77} + 2301512q^{79} + 531441q^{81} + 4708692q^{83} + 8430156q^{85} - 4241322q^{87} + 4143690q^{89} - 12371840q^{91} + 444312q^{93} + 6161400q^{95} - 1622974q^{97} - 1810836q^{99} + O(q^{100})$$

## 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 −27.0000 0 −378.000 0 −832.000 0 729.000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 12.8.a.a 1
3.b odd 2 1 36.8.a.c 1
4.b odd 2 1 48.8.a.e 1
5.b even 2 1 300.8.a.g 1
5.c odd 4 2 300.8.d.c 2
7.b odd 2 1 588.8.a.d 1
7.c even 3 2 588.8.i.h 2
7.d odd 6 2 588.8.i.a 2
8.b even 2 1 192.8.a.o 1
8.d odd 2 1 192.8.a.g 1
9.c even 3 2 324.8.e.f 2
9.d odd 6 2 324.8.e.a 2
12.b even 2 1 144.8.a.j 1
24.f even 2 1 576.8.a.e 1
24.h odd 2 1 576.8.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.8.a.a 1 1.a even 1 1 trivial
36.8.a.c 1 3.b odd 2 1
48.8.a.e 1 4.b odd 2 1
144.8.a.j 1 12.b even 2 1
192.8.a.g 1 8.d odd 2 1
192.8.a.o 1 8.b even 2 1
300.8.a.g 1 5.b even 2 1
300.8.d.c 2 5.c odd 4 2
324.8.e.a 2 9.d odd 6 2
324.8.e.f 2 9.c even 3 2
576.8.a.d 1 24.h odd 2 1
576.8.a.e 1 24.f even 2 1
588.8.a.d 1 7.b odd 2 1
588.8.i.a 2 7.d odd 6 2
588.8.i.h 2 7.c even 3 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 378$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(12))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 27 T$$
$5$ $$1 + 378 T + 78125 T^{2}$$
$7$ $$1 + 832 T + 823543 T^{2}$$
$11$ $$1 + 2484 T + 19487171 T^{2}$$
$13$ $$1 - 14870 T + 62748517 T^{2}$$
$17$ $$1 + 22302 T + 410338673 T^{2}$$
$19$ $$1 + 16300 T + 893871739 T^{2}$$
$23$ $$1 + 115128 T + 3404825447 T^{2}$$
$29$ $$1 - 157086 T + 17249876309 T^{2}$$
$31$ $$1 + 16456 T + 27512614111 T^{2}$$
$37$ $$1 + 149266 T + 94931877133 T^{2}$$
$41$ $$1 + 241110 T + 194754273881 T^{2}$$
$43$ $$1 + 443188 T + 271818611107 T^{2}$$
$47$ $$1 - 922752 T + 506623120463 T^{2}$$
$53$ $$1 + 697626 T + 1174711139837 T^{2}$$
$59$ $$1 - 870156 T + 2488651484819 T^{2}$$
$61$ $$1 - 2067062 T + 3142742836021 T^{2}$$
$67$ $$1 + 1680748 T + 6060711605323 T^{2}$$
$71$ $$1 + 1070280 T + 9095120158391 T^{2}$$
$73$ $$1 + 2403334 T + 11047398519097 T^{2}$$
$79$ $$1 - 2301512 T + 19203908986159 T^{2}$$
$83$ $$1 - 4708692 T + 27136050989627 T^{2}$$
$89$ $$1 - 4143690 T + 44231334895529 T^{2}$$
$97$ $$1 + 1622974 T + 80798284478113 T^{2}$$