# Properties

 Label 1656.2.a.c Level $1656$ Weight $2$ Character orbit 1656.a Self dual yes Analytic conductor $13.223$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1656 = 2^{3} \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1656.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.2232265747$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 184) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{7} + O(q^{10})$$ $$q - 2 q^{7} - 5 q^{13} + 6 q^{17} + 6 q^{19} - q^{23} - 5 q^{25} - 9 q^{29} + 3 q^{31} - 8 q^{37} - 3 q^{41} - 8 q^{43} - 7 q^{47} - 3 q^{49} + 2 q^{53} - 4 q^{59} - 10 q^{61} + 8 q^{67} - 7 q^{71} + 9 q^{73} - 6 q^{79} + 14 q^{83} - 16 q^{89} + 10 q^{91} + 6 q^{97} + O(q^{100})$$

## 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
 0
0 0 0 0 0 −2.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1656.2.a.c 1
3.b odd 2 1 184.2.a.d 1
4.b odd 2 1 3312.2.a.i 1
12.b even 2 1 368.2.a.a 1
15.d odd 2 1 4600.2.a.a 1
15.e even 4 2 4600.2.e.a 2
21.c even 2 1 9016.2.a.b 1
24.f even 2 1 1472.2.a.m 1
24.h odd 2 1 1472.2.a.a 1
60.h even 2 1 9200.2.a.bj 1
69.c even 2 1 4232.2.a.j 1
276.h odd 2 1 8464.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.d 1 3.b odd 2 1
368.2.a.a 1 12.b even 2 1
1472.2.a.a 1 24.h odd 2 1
1472.2.a.m 1 24.f even 2 1
1656.2.a.c 1 1.a even 1 1 trivial
3312.2.a.i 1 4.b odd 2 1
4232.2.a.j 1 69.c even 2 1
4600.2.a.a 1 15.d odd 2 1
4600.2.e.a 2 15.e even 4 2
8464.2.a.b 1 276.h odd 2 1
9016.2.a.b 1 21.c even 2 1
9200.2.a.bj 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1656))$$:

 $$T_{5}$$ $$T_{7} + 2$$ $$T_{11}$$ $$T_{13} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$2 + T$$
$11$ $$T$$
$13$ $$5 + T$$
$17$ $$-6 + T$$
$19$ $$-6 + T$$
$23$ $$1 + T$$
$29$ $$9 + T$$
$31$ $$-3 + T$$
$37$ $$8 + T$$
$41$ $$3 + T$$
$43$ $$8 + T$$
$47$ $$7 + T$$
$53$ $$-2 + T$$
$59$ $$4 + T$$
$61$ $$10 + T$$
$67$ $$-8 + T$$
$71$ $$7 + T$$
$73$ $$-9 + T$$
$79$ $$6 + T$$
$83$ $$-14 + T$$
$89$ $$16 + T$$
$97$ $$-6 + T$$