# Properties

 Label 3136.2.a.r Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $1$ Dimension $1$ CM discriminant -4 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1568) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4 q^{5} - 3 q^{9}+O(q^{10})$$ q + 4 * q^5 - 3 * q^9 $$q + 4 q^{5} - 3 q^{9} - 4 q^{13} - 8 q^{17} + 11 q^{25} - 10 q^{29} - 2 q^{37} - 8 q^{41} - 12 q^{45} - 14 q^{53} - 12 q^{61} - 16 q^{65} + 16 q^{73} + 9 q^{81} - 32 q^{85} + 16 q^{89} + 8 q^{97}+O(q^{100})$$ q + 4 * q^5 - 3 * q^9 - 4 * q^13 - 8 * q^17 + 11 * q^25 - 10 * q^29 - 2 * q^37 - 8 * q^41 - 12 * q^45 - 14 * q^53 - 12 * q^61 - 16 * q^65 + 16 * q^73 + 9 * q^81 - 32 * q^85 + 16 * q^89 + 8 * 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
 0
0 0 0 4.00000 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 $$\Q(\sqrt{-1})$$

## Twists

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

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

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{5} - 4$$ T5 - 4 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 4$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T + 8$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 10$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T + 8$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T + 14$$
$59$ $$T$$
$61$ $$T + 12$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T - 16$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 16$$
$97$ $$T - 8$$