# Properties

 Label 1856.2.a.p Level $1856$ Weight $2$ Character orbit 1856.a Self dual yes Analytic conductor $14.820$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.8202346151$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 3 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 + 3 * q^5 - 2 * q^7 + 6 * q^9 $$q + 3 q^{3} + 3 q^{5} - 2 q^{7} + 6 q^{9} + q^{11} - 3 q^{13} + 9 q^{15} - 4 q^{17} + 8 q^{19} - 6 q^{21} + 4 q^{25} + 9 q^{27} + q^{29} + 3 q^{31} + 3 q^{33} - 6 q^{35} + 8 q^{37} - 9 q^{39} - 2 q^{41} - 7 q^{43} + 18 q^{45} + 11 q^{47} - 3 q^{49} - 12 q^{51} - q^{53} + 3 q^{55} + 24 q^{57} + 4 q^{59} - 4 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} - 2 q^{71} - 12 q^{73} + 12 q^{75} - 2 q^{77} - 7 q^{79} + 9 q^{81} - 12 q^{85} + 3 q^{87} - 6 q^{89} + 6 q^{91} + 9 q^{93} + 24 q^{95} - 6 q^{97} + 6 q^{99}+O(q^{100})$$ q + 3 * q^3 + 3 * q^5 - 2 * q^7 + 6 * q^9 + q^11 - 3 * q^13 + 9 * q^15 - 4 * q^17 + 8 * q^19 - 6 * q^21 + 4 * q^25 + 9 * q^27 + q^29 + 3 * q^31 + 3 * q^33 - 6 * q^35 + 8 * q^37 - 9 * q^39 - 2 * q^41 - 7 * q^43 + 18 * q^45 + 11 * q^47 - 3 * q^49 - 12 * q^51 - q^53 + 3 * q^55 + 24 * q^57 + 4 * q^59 - 4 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 - 2 * q^71 - 12 * q^73 + 12 * q^75 - 2 * q^77 - 7 * q^79 + 9 * q^81 - 12 * q^85 + 3 * q^87 - 6 * q^89 + 6 * q^91 + 9 * q^93 + 24 * q^95 - 6 * q^97 + 6 * 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 3.00000 0 3.00000 0 −2.00000 0 6.00000 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
$$2$$ $$1$$
$$29$$ $$-1$$

## 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 1856.2.a.p 1
4.b odd 2 1 1856.2.a.b 1
8.b even 2 1 58.2.a.a 1
8.d odd 2 1 464.2.a.f 1
24.f even 2 1 4176.2.a.bh 1
24.h odd 2 1 522.2.a.k 1
40.f even 2 1 1450.2.a.i 1
40.i odd 4 2 1450.2.b.f 2
56.h odd 2 1 2842.2.a.d 1
88.b odd 2 1 7018.2.a.c 1
104.e even 2 1 9802.2.a.d 1
232.g even 2 1 1682.2.a.j 1
232.l odd 4 2 1682.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.a 1 8.b even 2 1
464.2.a.f 1 8.d odd 2 1
522.2.a.k 1 24.h odd 2 1
1450.2.a.i 1 40.f even 2 1
1450.2.b.f 2 40.i odd 4 2
1682.2.a.j 1 232.g even 2 1
1682.2.b.e 2 232.l odd 4 2
1856.2.a.b 1 4.b odd 2 1
1856.2.a.p 1 1.a even 1 1 trivial
2842.2.a.d 1 56.h odd 2 1
4176.2.a.bh 1 24.f even 2 1
7018.2.a.c 1 88.b odd 2 1
9802.2.a.d 1 104.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3} - 3$$ T3 - 3 $$T_{5} - 3$$ T5 - 3 $$T_{17} + 4$$ T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 3$$
$7$ $$T + 2$$
$11$ $$T - 1$$
$13$ $$T + 3$$
$17$ $$T + 4$$
$19$ $$T - 8$$
$23$ $$T$$
$29$ $$T - 1$$
$31$ $$T - 3$$
$37$ $$T - 8$$
$41$ $$T + 2$$
$43$ $$T + 7$$
$47$ $$T - 11$$
$53$ $$T + 1$$
$59$ $$T - 4$$
$61$ $$T + 4$$
$67$ $$T - 4$$
$71$ $$T + 2$$
$73$ $$T + 12$$
$79$ $$T + 7$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T + 6$$