# Properties

 Label 3136.2.a.bi Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1568) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} -3 q^{9} +O(q^{10})$$ $$q + \beta q^{5} -3 q^{9} -\beta q^{13} + 5 \beta q^{17} -3 q^{25} + 4 q^{29} + 12 q^{37} -9 \beta q^{41} -3 \beta q^{45} + 14 q^{53} + 11 \beta q^{61} -2 q^{65} + 11 \beta q^{73} + 9 q^{81} + 10 q^{85} -3 \beta q^{89} -5 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{9} + O(q^{10})$$ $$2 q - 6 q^{9} - 6 q^{25} + 8 q^{29} + 24 q^{37} + 28 q^{53} - 4 q^{65} + 18 q^{81} + 20 q^{85} + O(q^{100})$$

## 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 0 0 −1.41421 0 0 0 −3.00000 0
1.2 0 0 0 1.41421 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
7.b odd 2 1 inner
28.d even 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.bi 2
4.b odd 2 1 CM 3136.2.a.bi 2
7.b odd 2 1 inner 3136.2.a.bi 2
8.b even 2 1 1568.2.a.o 2
8.d odd 2 1 1568.2.a.o 2
28.d even 2 1 inner 3136.2.a.bi 2
56.e even 2 1 1568.2.a.o 2
56.h odd 2 1 1568.2.a.o 2
56.j odd 6 2 1568.2.i.t 4
56.k odd 6 2 1568.2.i.t 4
56.m even 6 2 1568.2.i.t 4
56.p even 6 2 1568.2.i.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.o 2 8.b even 2 1
1568.2.a.o 2 8.d odd 2 1
1568.2.a.o 2 56.e even 2 1
1568.2.a.o 2 56.h odd 2 1
1568.2.i.t 4 56.j odd 6 2
1568.2.i.t 4 56.k odd 6 2
1568.2.i.t 4 56.m even 6 2
1568.2.i.t 4 56.p even 6 2
3136.2.a.bi 2 1.a even 1 1 trivial
3136.2.a.bi 2 4.b odd 2 1 CM
3136.2.a.bi 2 7.b odd 2 1 inner
3136.2.a.bi 2 28.d 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3136))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$-2 + T^{2}$$
$17$ $$-50 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$( -12 + T )^{2}$$
$41$ $$-162 + T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -14 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$-242 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-242 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$-18 + T^{2}$$
$97$ $$-50 + T^{2}$$