# Properties

 Label 3136.2.a.n Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $0$ Dimension $1$ CM discriminant -7 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{9}+O(q^{10})$$ q - 3 * q^9 $$q - 3 q^{9} - 4 q^{11} + 8 q^{23} - 5 q^{25} - 2 q^{29} + 6 q^{37} + 12 q^{43} + 10 q^{53} - 4 q^{67} + 16 q^{71} + 8 q^{79} + 9 q^{81} + 12 q^{99}+O(q^{100})$$ q - 3 * q^9 - 4 * q^11 + 8 * q^23 - 5 * q^25 - 2 * q^29 + 6 * q^37 + 12 * q^43 + 10 * q^53 - 4 * q^67 + 16 * q^71 + 8 * q^79 + 9 * q^81 + 12 * 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
 0
0 0 0 0 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.a.n 1
4.b odd 2 1 3136.2.a.o 1
7.b odd 2 1 CM 3136.2.a.n 1
8.b even 2 1 49.2.a.a 1
8.d odd 2 1 784.2.a.f 1
24.f even 2 1 7056.2.a.bg 1
24.h odd 2 1 441.2.a.c 1
28.d even 2 1 3136.2.a.o 1
40.f even 2 1 1225.2.a.c 1
40.i odd 4 2 1225.2.b.c 2
56.e even 2 1 784.2.a.f 1
56.h odd 2 1 49.2.a.a 1
56.j odd 6 2 49.2.c.a 2
56.k odd 6 2 784.2.i.f 2
56.m even 6 2 784.2.i.f 2
56.p even 6 2 49.2.c.a 2
88.b odd 2 1 5929.2.a.c 1
104.e even 2 1 8281.2.a.d 1
168.e odd 2 1 7056.2.a.bg 1
168.i even 2 1 441.2.a.c 1
168.s odd 6 2 441.2.e.d 2
168.ba even 6 2 441.2.e.d 2
280.c odd 2 1 1225.2.a.c 1
280.s even 4 2 1225.2.b.c 2
616.o even 2 1 5929.2.a.c 1
728.l odd 2 1 8281.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 8.b even 2 1
49.2.a.a 1 56.h odd 2 1
49.2.c.a 2 56.j odd 6 2
49.2.c.a 2 56.p even 6 2
441.2.a.c 1 24.h odd 2 1
441.2.a.c 1 168.i even 2 1
441.2.e.d 2 168.s odd 6 2
441.2.e.d 2 168.ba even 6 2
784.2.a.f 1 8.d odd 2 1
784.2.a.f 1 56.e even 2 1
784.2.i.f 2 56.k odd 6 2
784.2.i.f 2 56.m even 6 2
1225.2.a.c 1 40.f even 2 1
1225.2.a.c 1 280.c odd 2 1
1225.2.b.c 2 40.i odd 4 2
1225.2.b.c 2 280.s even 4 2
3136.2.a.n 1 1.a even 1 1 trivial
3136.2.a.n 1 7.b odd 2 1 CM
3136.2.a.o 1 4.b odd 2 1
3136.2.a.o 1 28.d even 2 1
5929.2.a.c 1 88.b odd 2 1
5929.2.a.c 1 616.o even 2 1
7056.2.a.bg 1 24.f even 2 1
7056.2.a.bg 1 168.e odd 2 1
8281.2.a.d 1 104.e even 2 1
8281.2.a.d 1 728.l odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 8$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T - 6$$
$41$ $$T$$
$43$ $$T - 12$$
$47$ $$T$$
$53$ $$T - 10$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T + 4$$
$71$ $$T - 16$$
$73$ $$T$$
$79$ $$T - 8$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$