# Properties

 Label 6912.2.a.bd Level 6912 Weight 2 Character orbit 6912.a Self dual yes Analytic conductor 55.193 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$6912 = 2^{8} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6912.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$55.1925978771$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 216) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + q^{7} +O(q^{10})$$ $$q - q^{5} + q^{7} + 3 q^{11} + \beta q^{13} + \beta q^{17} + \beta q^{19} + \beta q^{23} -4 q^{25} + 6 q^{29} + 7 q^{31} - q^{35} -\beta q^{37} + \beta q^{41} -2 \beta q^{43} -6 q^{49} + 9 q^{53} -3 q^{55} -4 q^{59} -\beta q^{65} -3 \beta q^{71} -3 q^{73} + 3 q^{77} -4 q^{79} + 7 q^{83} -\beta q^{85} -2 \beta q^{89} + \beta q^{91} -\beta q^{95} + 7 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} + 2q^{7} + O(q^{10})$$ $$2q - 2q^{5} + 2q^{7} + 6q^{11} - 8q^{25} + 12q^{29} + 14q^{31} - 2q^{35} - 12q^{49} + 18q^{53} - 6q^{55} - 8q^{59} - 6q^{73} + 6q^{77} - 8q^{79} + 14q^{83} + 14q^{97} + O(q^{100})$$

## 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
 −2.64575 2.64575
0 0 0 −1.00000 0 1.00000 0 0 0
1.2 0 0 0 −1.00000 0 1.00000 0 0 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
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6912.2.a.bd 2
3.b odd 2 1 6912.2.a.bv 2
4.b odd 2 1 6912.2.a.bc 2
8.b even 2 1 6912.2.a.bv 2
8.d odd 2 1 6912.2.a.bu 2
12.b even 2 1 6912.2.a.bu 2
16.e even 4 2 864.2.d.a 4
16.f odd 4 2 216.2.d.b 4
24.f even 2 1 6912.2.a.bc 2
24.h odd 2 1 inner 6912.2.a.bd 2
48.i odd 4 2 864.2.d.a 4
48.k even 4 2 216.2.d.b 4
144.u even 12 4 648.2.n.n 8
144.v odd 12 4 648.2.n.n 8
144.w odd 12 4 2592.2.r.p 8
144.x even 12 4 2592.2.r.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b 4 16.f odd 4 2
216.2.d.b 4 48.k even 4 2
648.2.n.n 8 144.u even 12 4
648.2.n.n 8 144.v odd 12 4
864.2.d.a 4 16.e even 4 2
864.2.d.a 4 48.i odd 4 2
2592.2.r.p 8 144.w odd 12 4
2592.2.r.p 8 144.x even 12 4
6912.2.a.bc 2 4.b odd 2 1
6912.2.a.bc 2 24.f even 2 1
6912.2.a.bd 2 1.a even 1 1 trivial
6912.2.a.bd 2 24.h odd 2 1 inner
6912.2.a.bu 2 8.d odd 2 1
6912.2.a.bu 2 12.b even 2 1
6912.2.a.bv 2 3.b odd 2 1
6912.2.a.bv 2 8.b even 2 1

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6912))$$:

 $$T_{5} + 1$$ $$T_{7} - 1$$ $$T_{11} - 3$$ $$T_{13}^{2} - 28$$ $$T_{17}^{2} - 28$$ $$T_{19}^{2} - 28$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + T + 5 T^{2} )^{2}$$
$7$ $$( 1 - T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 3 T + 11 T^{2} )^{2}$$
$13$ $$1 - 2 T^{2} + 169 T^{4}$$
$17$ $$1 + 6 T^{2} + 289 T^{4}$$
$19$ $$1 + 10 T^{2} + 361 T^{4}$$
$23$ $$1 + 18 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{2}$$
$37$ $$1 + 46 T^{2} + 1369 T^{4}$$
$41$ $$1 + 54 T^{2} + 1681 T^{4}$$
$43$ $$1 - 26 T^{2} + 1849 T^{4}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 - 9 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$( 1 + 67 T^{2} )^{2}$$
$71$ $$1 - 110 T^{2} + 5041 T^{4}$$
$73$ $$( 1 + 3 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 7 T + 83 T^{2} )^{2}$$
$89$ $$1 + 66 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 7 T + 97 T^{2} )^{2}$$
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