# Properties

 Label 1856.2.a.r Level $1856$ Weight $2$ Character orbit 1856.a Self dual yes Analytic conductor $14.820$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 29) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{3} + q^{5} + 2 \beta q^{7} -2 \beta q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{3} + q^{5} + 2 \beta q^{7} -2 \beta q^{9} + ( -1 - \beta ) q^{11} + ( 1 - 2 \beta ) q^{13} + ( -1 + \beta ) q^{15} + ( -2 - 2 \beta ) q^{17} -6 q^{19} + ( 4 - 2 \beta ) q^{21} + ( -2 - 4 \beta ) q^{23} -4 q^{25} + ( -1 - \beta ) q^{27} - q^{29} + ( 3 - 5 \beta ) q^{31} - q^{33} + 2 \beta q^{35} + 4 q^{37} + ( -5 + 3 \beta ) q^{39} + ( 4 + 6 \beta ) q^{41} + ( -5 - \beta ) q^{43} -2 \beta q^{45} + ( 1 + 3 \beta ) q^{47} + q^{49} -2 q^{51} + ( -1 + 6 \beta ) q^{53} + ( -1 - \beta ) q^{55} + ( 6 - 6 \beta ) q^{57} + ( -2 - 4 \beta ) q^{59} + ( 2 - 2 \beta ) q^{61} -8 q^{63} + ( 1 - 2 \beta ) q^{65} + 4 \beta q^{67} + ( -6 + 2 \beta ) q^{69} + ( -6 + 2 \beta ) q^{71} + 4 q^{73} + ( 4 - 4 \beta ) q^{75} + ( -4 - 2 \beta ) q^{77} + ( -1 + \beta ) q^{79} + ( -1 + 6 \beta ) q^{81} + ( -2 + 4 \beta ) q^{83} + ( -2 - 2 \beta ) q^{85} + ( 1 - \beta ) q^{87} + ( -4 + 6 \beta ) q^{89} + ( -8 + 2 \beta ) q^{91} + ( -13 + 8 \beta ) q^{93} -6 q^{95} + ( -4 - 6 \beta ) q^{97} + ( 4 + 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + O(q^{10})$$ $$2 q - 2 q^{3} + 2 q^{5} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 4 q^{17} - 12 q^{19} + 8 q^{21} - 4 q^{23} - 8 q^{25} - 2 q^{27} - 2 q^{29} + 6 q^{31} - 2 q^{33} + 8 q^{37} - 10 q^{39} + 8 q^{41} - 10 q^{43} + 2 q^{47} + 2 q^{49} - 4 q^{51} - 2 q^{53} - 2 q^{55} + 12 q^{57} - 4 q^{59} + 4 q^{61} - 16 q^{63} + 2 q^{65} - 12 q^{69} - 12 q^{71} + 8 q^{73} + 8 q^{75} - 8 q^{77} - 2 q^{79} - 2 q^{81} - 4 q^{83} - 4 q^{85} + 2 q^{87} - 8 q^{89} - 16 q^{91} - 26 q^{93} - 12 q^{95} - 8 q^{97} + 8 q^{99} + O(q^{100})$$

## 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 −2.41421 0 1.00000 0 −2.82843 0 2.82843 0
1.2 0 0.414214 0 1.00000 0 2.82843 0 −2.82843 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.r 2
4.b odd 2 1 1856.2.a.w 2
8.b even 2 1 29.2.a.a 2
8.d odd 2 1 464.2.a.h 2
24.f even 2 1 4176.2.a.bq 2
24.h odd 2 1 261.2.a.d 2
40.f even 2 1 725.2.a.b 2
40.i odd 4 2 725.2.b.b 4
56.h odd 2 1 1421.2.a.j 2
88.b odd 2 1 3509.2.a.j 2
104.e even 2 1 4901.2.a.g 2
120.i odd 2 1 6525.2.a.o 2
136.h even 2 1 8381.2.a.e 2
232.g even 2 1 841.2.a.d 2
232.l odd 4 2 841.2.b.a 4
232.o even 14 6 841.2.d.f 12
232.s even 14 6 841.2.d.j 12
232.u odd 28 12 841.2.e.k 24
696.n odd 2 1 7569.2.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 8.b even 2 1
261.2.a.d 2 24.h odd 2 1
464.2.a.h 2 8.d odd 2 1
725.2.a.b 2 40.f even 2 1
725.2.b.b 4 40.i odd 4 2
841.2.a.d 2 232.g even 2 1
841.2.b.a 4 232.l odd 4 2
841.2.d.f 12 232.o even 14 6
841.2.d.j 12 232.s even 14 6
841.2.e.k 24 232.u odd 28 12
1421.2.a.j 2 56.h odd 2 1
1856.2.a.r 2 1.a even 1 1 trivial
1856.2.a.w 2 4.b odd 2 1
3509.2.a.j 2 88.b odd 2 1
4176.2.a.bq 2 24.f even 2 1
4901.2.a.g 2 104.e even 2 1
6525.2.a.o 2 120.i odd 2 1
7569.2.a.c 2 696.n odd 2 1
8381.2.a.e 2 136.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{2} + 2 T_{3} - 1$$ $$T_{5} - 1$$ $$T_{17}^{2} + 4 T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 + 2 T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$-1 + 2 T + T^{2}$$
$13$ $$-7 - 2 T + T^{2}$$
$17$ $$-4 + 4 T + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$-28 + 4 T + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$-41 - 6 T + T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$-56 - 8 T + T^{2}$$
$43$ $$23 + 10 T + T^{2}$$
$47$ $$-17 - 2 T + T^{2}$$
$53$ $$-71 + 2 T + T^{2}$$
$59$ $$-28 + 4 T + T^{2}$$
$61$ $$-4 - 4 T + T^{2}$$
$67$ $$-32 + T^{2}$$
$71$ $$28 + 12 T + T^{2}$$
$73$ $$( -4 + T )^{2}$$
$79$ $$-1 + 2 T + T^{2}$$
$83$ $$-28 + 4 T + T^{2}$$
$89$ $$-56 + 8 T + T^{2}$$
$97$ $$-56 + 8 T + T^{2}$$