# Properties

 Label 3136.2.a.bl Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3136 = 2^{6} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3136.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.0410860739$$ Analytic rank: $$1$$ 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 1568) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{9}+O(q^{10})$$ q + b * q^3 - q^9 $$q + \beta q^{3} - q^{9} - 2 q^{11} + 2 \beta q^{13} - 3 \beta q^{17} - 3 \beta q^{19} + 8 q^{23} - 5 q^{25} - 4 \beta q^{27} - 6 q^{29} - 6 \beta q^{31} - 2 \beta q^{33} + 2 q^{37} + 4 q^{39} + 3 \beta q^{41} - 6 q^{43} + 2 \beta q^{47} - 6 q^{51} - 6 q^{53} - 6 q^{57} + 9 \beta q^{59} - 4 \beta q^{61} - 12 q^{67} + 8 \beta q^{69} - 4 q^{71} + \beta q^{73} - 5 \beta q^{75} + 12 q^{79} - 5 q^{81} - 7 \beta q^{83} - 6 \beta q^{87} - 3 \beta q^{89} - 12 q^{93} + 13 \beta q^{97} + 2 q^{99} +O(q^{100})$$ q + b * q^3 - q^9 - 2 * q^11 + 2*b * q^13 - 3*b * q^17 - 3*b * q^19 + 8 * q^23 - 5 * q^25 - 4*b * q^27 - 6 * q^29 - 6*b * q^31 - 2*b * q^33 + 2 * q^37 + 4 * q^39 + 3*b * q^41 - 6 * q^43 + 2*b * q^47 - 6 * q^51 - 6 * q^53 - 6 * q^57 + 9*b * q^59 - 4*b * q^61 - 12 * q^67 + 8*b * q^69 - 4 * q^71 + b * q^73 - 5*b * q^75 + 12 * q^79 - 5 * q^81 - 7*b * q^83 - 6*b * q^87 - 3*b * q^89 - 12 * q^93 + 13*b * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 4 q^{11} + 16 q^{23} - 10 q^{25} - 12 q^{29} + 4 q^{37} + 8 q^{39} - 12 q^{43} - 12 q^{51} - 12 q^{53} - 12 q^{57} - 24 q^{67} - 8 q^{71} + 24 q^{79} - 10 q^{81} - 24 q^{93} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 4 * q^11 + 16 * q^23 - 10 * q^25 - 12 * q^29 + 4 * q^37 + 8 * q^39 - 12 * q^43 - 12 * q^51 - 12 * q^53 - 12 * q^57 - 24 * q^67 - 8 * q^71 + 24 * q^79 - 10 * q^81 - 24 * q^93 + 4 * q^99

## 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 0 0 −1.00000 0
1.2 0 1.41421 0 0 0 0 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.a.bl 2
4.b odd 2 1 3136.2.a.bo 2
7.b odd 2 1 inner 3136.2.a.bl 2
8.b even 2 1 1568.2.a.r yes 2
8.d odd 2 1 1568.2.a.q 2
28.d even 2 1 3136.2.a.bo 2
56.e even 2 1 1568.2.a.q 2
56.h odd 2 1 1568.2.a.r yes 2
56.j odd 6 2 1568.2.i.q 4
56.k odd 6 2 1568.2.i.r 4
56.m even 6 2 1568.2.i.r 4
56.p even 6 2 1568.2.i.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.q 2 8.d odd 2 1
1568.2.a.q 2 56.e even 2 1
1568.2.a.r yes 2 8.b even 2 1
1568.2.a.r yes 2 56.h odd 2 1
1568.2.i.q 4 56.j odd 6 2
1568.2.i.q 4 56.p even 6 2
1568.2.i.r 4 56.k odd 6 2
1568.2.i.r 4 56.m even 6 2
3136.2.a.bl 2 1.a even 1 1 trivial
3136.2.a.bl 2 7.b odd 2 1 inner
3136.2.a.bo 2 4.b odd 2 1
3136.2.a.bo 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3136))$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{5}$$ T5 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} - 8$$
$17$ $$T^{2} - 18$$
$19$ $$T^{2} - 18$$
$23$ $$(T - 8)^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} - 72$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 18$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 162$$
$61$ $$T^{2} - 32$$
$67$ $$(T + 12)^{2}$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} - 2$$
$79$ $$(T - 12)^{2}$$
$83$ $$T^{2} - 98$$
$89$ $$T^{2} - 18$$
$97$ $$T^{2} - 338$$