# Properties

 Label 7168.2.a.k Level $7168$ Weight $2$ Character orbit 7168.a Self dual yes Analytic conductor $57.237$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7168 = 2^{10} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7168.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.2367681689$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 112) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} + q^{7} - 3 q^{9}+O(q^{10})$$ q + 2*b * q^5 + q^7 - 3 * q^9 $$q + 2 \beta q^{5} + q^{7} - 3 q^{9} + \beta q^{11} + 2 q^{17} - 2 \beta q^{19} - 6 q^{23} + 3 q^{25} - 7 \beta q^{29} + 8 q^{31} + 2 \beta q^{35} - 5 \beta q^{37} - 10 q^{41} - \beta q^{43} - 6 \beta q^{45} - 12 q^{47} + q^{49} + \beta q^{53} + 4 q^{55} - 8 \beta q^{59} - 6 \beta q^{61} - 3 q^{63} + 3 \beta q^{67} + 6 q^{73} + \beta q^{77} + 10 q^{79} + 9 q^{81} + 10 \beta q^{83} + 4 \beta q^{85} - 14 q^{89} - 8 q^{95} - 2 q^{97} - 3 \beta q^{99} +O(q^{100})$$ q + 2*b * q^5 + q^7 - 3 * q^9 + b * q^11 + 2 * q^17 - 2*b * q^19 - 6 * q^23 + 3 * q^25 - 7*b * q^29 + 8 * q^31 + 2*b * q^35 - 5*b * q^37 - 10 * q^41 - b * q^43 - 6*b * q^45 - 12 * q^47 + q^49 + b * q^53 + 4 * q^55 - 8*b * q^59 - 6*b * q^61 - 3 * q^63 + 3*b * q^67 + 6 * q^73 + b * q^77 + 10 * q^79 + 9 * q^81 + 10*b * q^83 + 4*b * q^85 - 14 * q^89 - 8 * q^95 - 2 * q^97 - 3*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^7 - 6 * q^9 $$2 q + 2 q^{7} - 6 q^{9} + 4 q^{17} - 12 q^{23} + 6 q^{25} + 16 q^{31} - 20 q^{41} - 24 q^{47} + 2 q^{49} + 8 q^{55} - 6 q^{63} + 12 q^{73} + 20 q^{79} + 18 q^{81} - 28 q^{89} - 16 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^7 - 6 * q^9 + 4 * q^17 - 12 * q^23 + 6 * q^25 + 16 * q^31 - 20 * q^41 - 24 * q^47 + 2 * q^49 + 8 * q^55 - 6 * q^63 + 12 * q^73 + 20 * q^79 + 18 * q^81 - 28 * q^89 - 16 * q^95 - 4 * q^97

## 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
 −1.41421 1.41421
0 0 0 −2.82843 0 1.00000 0 −3.00000 0
1.2 0 0 0 2.82843 0 1.00000 0 −3.00000 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
$$2$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7168.2.a.k 2
4.b odd 2 1 7168.2.a.b 2
8.b even 2 1 inner 7168.2.a.k 2
8.d odd 2 1 7168.2.a.b 2
32.g even 8 2 448.2.m.a 2
32.g even 8 2 896.2.m.c 2
32.h odd 8 2 112.2.m.b 2
32.h odd 8 2 896.2.m.b 2
224.x even 8 2 784.2.m.a 2
224.be even 24 4 784.2.x.e 4
224.bf odd 24 4 784.2.x.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.b 2 32.h odd 8 2
448.2.m.a 2 32.g even 8 2
784.2.m.a 2 224.x even 8 2
784.2.x.d 4 224.bf odd 24 4
784.2.x.e 4 224.be even 24 4
896.2.m.b 2 32.h odd 8 2
896.2.m.c 2 32.g even 8 2
7168.2.a.b 2 4.b odd 2 1
7168.2.a.b 2 8.d odd 2 1
7168.2.a.k 2 1.a even 1 1 trivial
7168.2.a.k 2 8.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7168))$$:

 $$T_{3}$$ T3 $$T_{5}^{2} - 8$$ T5^2 - 8 $$T_{11}^{2} - 2$$ T11^2 - 2 $$T_{13}$$ T13 $$T_{17} - 2$$ T17 - 2 $$T_{23} + 6$$ T23 + 6 $$T_{31} - 8$$ T31 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 8$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 2$$
$13$ $$T^{2}$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} - 8$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} - 98$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} - 50$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} - 2$$
$47$ $$(T + 12)^{2}$$
$53$ $$T^{2} - 2$$
$59$ $$T^{2} - 128$$
$61$ $$T^{2} - 72$$
$67$ $$T^{2} - 18$$
$71$ $$T^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} - 200$$
$89$ $$(T + 14)^{2}$$
$97$ $$(T + 2)^{2}$$