# Properties

 Label 1856.2.e.g Level $1856$ Weight $2$ Character orbit 1856.e Analytic conductor $14.820$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1856.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8202346151$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 29) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 3 q^{5} + 2 q^{7} -2 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 3 q^{5} + 2 q^{7} -2 q^{9} -\beta q^{11} + q^{13} + 3 \beta q^{15} -2 \beta q^{17} + 2 \beta q^{21} + 6 q^{23} + 4 q^{25} + \beta q^{27} + ( 3 - 2 \beta ) q^{29} + 3 \beta q^{31} + 5 q^{33} + 6 q^{35} + \beta q^{39} -2 \beta q^{41} + 3 \beta q^{43} -6 q^{45} + \beta q^{47} -3 q^{49} + 10 q^{51} + 9 q^{53} -3 \beta q^{55} -6 q^{59} -6 \beta q^{61} -4 q^{63} + 3 q^{65} -8 q^{67} + 6 \beta q^{69} + 4 \beta q^{75} -2 \beta q^{77} -3 \beta q^{79} -11 q^{81} + 6 q^{83} -6 \beta q^{85} + ( 10 + 3 \beta ) q^{87} -2 \beta q^{89} + 2 q^{91} -15 q^{93} + 6 \beta q^{97} + 2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5} + 4 q^{7} - 4 q^{9} + O(q^{10})$$ $$2 q + 6 q^{5} + 4 q^{7} - 4 q^{9} + 2 q^{13} + 12 q^{23} + 8 q^{25} + 6 q^{29} + 10 q^{33} + 12 q^{35} - 12 q^{45} - 6 q^{49} + 20 q^{51} + 18 q^{53} - 12 q^{59} - 8 q^{63} + 6 q^{65} - 16 q^{67} - 22 q^{81} + 12 q^{83} + 20 q^{87} + 4 q^{91} - 30 q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times$$.

 $$n$$ $$321$$ $$581$$ $$639$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1217.1
 − 2.23607i 2.23607i
0 2.23607i 0 3.00000 0 2.00000 0 −2.00000 0
1217.2 0 2.23607i 0 3.00000 0 2.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.2.e.g 2
4.b odd 2 1 1856.2.e.f 2
8.b even 2 1 29.2.b.a 2
8.d odd 2 1 464.2.e.a 2
24.f even 2 1 4176.2.o.k 2
24.h odd 2 1 261.2.c.a 2
29.b even 2 1 inner 1856.2.e.g 2
40.f even 2 1 725.2.c.c 2
40.i odd 4 2 725.2.d.a 4
56.h odd 2 1 1421.2.b.b 2
116.d odd 2 1 1856.2.e.f 2
232.b odd 2 1 464.2.e.a 2
232.g even 2 1 29.2.b.a 2
232.l odd 4 2 841.2.a.b 2
232.o even 14 6 841.2.e.g 12
232.s even 14 6 841.2.e.g 12
232.u odd 28 12 841.2.d.h 12
696.l even 2 1 4176.2.o.k 2
696.n odd 2 1 261.2.c.a 2
696.t even 4 2 7569.2.a.i 2
1160.e even 2 1 725.2.c.c 2
1160.be odd 4 2 725.2.d.a 4
1624.i odd 2 1 1421.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.b.a 2 8.b even 2 1
29.2.b.a 2 232.g even 2 1
261.2.c.a 2 24.h odd 2 1
261.2.c.a 2 696.n odd 2 1
464.2.e.a 2 8.d odd 2 1
464.2.e.a 2 232.b odd 2 1
725.2.c.c 2 40.f even 2 1
725.2.c.c 2 1160.e even 2 1
725.2.d.a 4 40.i odd 4 2
725.2.d.a 4 1160.be odd 4 2
841.2.a.b 2 232.l odd 4 2
841.2.d.h 12 232.u odd 28 12
841.2.e.g 12 232.o even 14 6
841.2.e.g 12 232.s even 14 6
1421.2.b.b 2 56.h odd 2 1
1421.2.b.b 2 1624.i odd 2 1
1856.2.e.f 2 4.b odd 2 1
1856.2.e.f 2 116.d odd 2 1
1856.2.e.g 2 1.a even 1 1 trivial
1856.2.e.g 2 29.b even 2 1 inner
4176.2.o.k 2 24.f even 2 1
4176.2.o.k 2 696.l even 2 1
7569.2.a.i 2 696.t even 4 2

## Hecke kernels

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

 $$T_{3}^{2} + 5$$ $$T_{7} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$5 + T^{2}$$
$5$ $$( -3 + T )^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$5 + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$20 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$29 - 6 T + T^{2}$$
$31$ $$45 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$20 + T^{2}$$
$43$ $$45 + T^{2}$$
$47$ $$5 + T^{2}$$
$53$ $$( -9 + T )^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$180 + T^{2}$$
$67$ $$( 8 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$45 + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$20 + T^{2}$$
$97$ $$180 + T^{2}$$