# Properties

 Label 7168.2.a.e Level $7168$ Weight $2$ Character orbit 7168.a Self dual yes Analytic conductor $57.237$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3584) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{7} - q^{9}+O(q^{10})$$ q + b * q^3 - q^7 - q^9 $$q + \beta q^{3} - q^{7} - q^{9} - 2 \beta q^{11} + 2 \beta q^{13} + 6 q^{17} + 3 \beta q^{19} - \beta q^{21} - 5 q^{25} - 4 \beta q^{27} - \beta q^{29} + 10 q^{31} - 4 q^{33} - 3 \beta q^{37} + 4 q^{39} - 10 q^{41} + 6 \beta q^{43} - 2 q^{47} + q^{49} + 6 \beta q^{51} + 9 \beta q^{53} + 6 q^{57} + \beta q^{59} + q^{63} - 8 \beta q^{67} + 12 q^{71} + 2 q^{73} - 5 \beta q^{75} + 2 \beta q^{77} - 4 q^{79} - 5 q^{81} + 5 \beta q^{83} - 2 q^{87} - 2 q^{89} - 2 \beta q^{91} + 10 \beta q^{93} - 2 q^{97} + 2 \beta q^{99} +O(q^{100})$$ q + b * q^3 - q^7 - q^9 - 2*b * q^11 + 2*b * q^13 + 6 * q^17 + 3*b * q^19 - b * q^21 - 5 * q^25 - 4*b * q^27 - b * q^29 + 10 * q^31 - 4 * q^33 - 3*b * q^37 + 4 * q^39 - 10 * q^41 + 6*b * q^43 - 2 * q^47 + q^49 + 6*b * q^51 + 9*b * q^53 + 6 * q^57 + b * q^59 + q^63 - 8*b * q^67 + 12 * q^71 + 2 * q^73 - 5*b * q^75 + 2*b * q^77 - 4 * q^79 - 5 * q^81 + 5*b * q^83 - 2 * q^87 - 2 * q^89 - 2*b * q^91 + 10*b * q^93 - 2 * q^97 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^7 - 2 * q^9 $$2 q - 2 q^{7} - 2 q^{9} + 12 q^{17} - 10 q^{25} + 20 q^{31} - 8 q^{33} + 8 q^{39} - 20 q^{41} - 4 q^{47} + 2 q^{49} + 12 q^{57} + 2 q^{63} + 24 q^{71} + 4 q^{73} - 8 q^{79} - 10 q^{81} - 4 q^{87} - 4 q^{89} - 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 - 2 * q^9 + 12 * q^17 - 10 * q^25 + 20 * q^31 - 8 * q^33 + 8 * q^39 - 20 * q^41 - 4 * q^47 + 2 * q^49 + 12 * q^57 + 2 * q^63 + 24 * q^71 + 4 * q^73 - 8 * q^79 - 10 * q^81 - 4 * q^87 - 4 * q^89 - 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 −1.41421 0 0 0 −1.00000 0 −1.00000 0
1.2 0 1.41421 0 0 0 −1.00000 0 −1.00000 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.e 2
4.b odd 2 1 7168.2.a.n 2
8.b even 2 1 inner 7168.2.a.e 2
8.d odd 2 1 7168.2.a.n 2
32.g even 8 2 3584.2.m.h yes 2
32.g even 8 2 3584.2.m.u yes 2
32.h odd 8 2 3584.2.m.g 2
32.h odd 8 2 3584.2.m.v yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.m.g 2 32.h odd 8 2
3584.2.m.h yes 2 32.g even 8 2
3584.2.m.u yes 2 32.g even 8 2
3584.2.m.v yes 2 32.h odd 8 2
7168.2.a.e 2 1.a even 1 1 trivial
7168.2.a.e 2 8.b even 2 1 inner
7168.2.a.n 2 4.b odd 2 1
7168.2.a.n 2 8.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{5}$$ T5 $$T_{11}^{2} - 8$$ T11^2 - 8 $$T_{13}^{2} - 8$$ T13^2 - 8 $$T_{17} - 6$$ T17 - 6 $$T_{23}$$ T23 $$T_{31} - 10$$ T31 - 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 8$$
$13$ $$T^{2} - 8$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} - 18$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 2$$
$31$ $$(T - 10)^{2}$$
$37$ $$T^{2} - 18$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} - 72$$
$47$ $$(T + 2)^{2}$$
$53$ $$T^{2} - 162$$
$59$ $$T^{2} - 2$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 128$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} - 50$$
$89$ $$(T + 2)^{2}$$
$97$ $$(T + 2)^{2}$$