# Properties

 Label 3136.2.a.bo 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: $$x^{2} - 2$$ 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.

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

## 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
 −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.bo 2
4.b odd 2 1 3136.2.a.bl 2
7.b odd 2 1 inner 3136.2.a.bo 2
8.b even 2 1 1568.2.a.q 2
8.d odd 2 1 1568.2.a.r yes 2
28.d even 2 1 3136.2.a.bl 2
56.e even 2 1 1568.2.a.r yes 2
56.h odd 2 1 1568.2.a.q 2
56.j odd 6 2 1568.2.i.r 4
56.k odd 6 2 1568.2.i.q 4
56.m even 6 2 1568.2.i.q 4
56.p even 6 2 1568.2.i.r 4

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

## Hecke kernels

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

 $$T_{3}^{2} - 2$$ $$T_{5}$$ $$T_{11} - 2$$

## Hecke characteristic polynomials

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