# Properties

 Label 1856.4.a.m Level $1856$ Weight $4$ Character orbit 1856.a Self dual yes Analytic conductor $109.508$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.507544971$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{22})$$ Defining polynomial: $$x^{2} - 22$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 116) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{22}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 5 + \beta ) q^{3} -15 q^{5} + 2 \beta q^{7} + ( 20 + 10 \beta ) q^{9} +O(q^{10})$$ $$q + ( 5 + \beta ) q^{3} -15 q^{5} + 2 \beta q^{7} + ( 20 + 10 \beta ) q^{9} + ( -23 - 5 \beta ) q^{11} + ( 5 + 10 \beta ) q^{13} + ( -75 - 15 \beta ) q^{15} + ( 70 - 14 \beta ) q^{17} + ( -30 - 20 \beta ) q^{19} + ( 44 + 10 \beta ) q^{21} + ( -30 - 12 \beta ) q^{23} + 100 q^{25} + ( 185 + 43 \beta ) q^{27} -29 q^{29} + ( 29 - 65 \beta ) q^{31} + ( -225 - 48 \beta ) q^{33} -30 \beta q^{35} + ( 60 + 8 \beta ) q^{37} + ( 245 + 55 \beta ) q^{39} + ( 140 + 50 \beta ) q^{41} + ( -75 - 77 \beta ) q^{43} + ( -300 - 150 \beta ) q^{45} + ( 275 - 37 \beta ) q^{47} -255 q^{49} + 42 q^{51} + ( -245 + 34 \beta ) q^{53} + ( 345 + 75 \beta ) q^{55} + ( -590 - 130 \beta ) q^{57} + ( -638 - 20 \beta ) q^{59} + ( -198 - 70 \beta ) q^{61} + ( 440 + 40 \beta ) q^{63} + ( -75 - 150 \beta ) q^{65} + ( -40 + 108 \beta ) q^{67} + ( -414 - 90 \beta ) q^{69} + ( 862 + 10 \beta ) q^{71} + ( 100 + 32 \beta ) q^{73} + ( 500 + 100 \beta ) q^{75} + ( -220 - 46 \beta ) q^{77} + ( -827 - 15 \beta ) q^{79} + ( 1331 + 130 \beta ) q^{81} + ( -750 + 60 \beta ) q^{83} + ( -1050 + 210 \beta ) q^{85} + ( -145 - 29 \beta ) q^{87} + ( -108 + 90 \beta ) q^{89} + ( 440 + 10 \beta ) q^{91} + ( -1285 - 296 \beta ) q^{93} + ( 450 + 300 \beta ) q^{95} + ( 940 - 82 \beta ) q^{97} + ( -1560 - 330 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{3} - 30 q^{5} + 40 q^{9} + O(q^{10})$$ $$2 q + 10 q^{3} - 30 q^{5} + 40 q^{9} - 46 q^{11} + 10 q^{13} - 150 q^{15} + 140 q^{17} - 60 q^{19} + 88 q^{21} - 60 q^{23} + 200 q^{25} + 370 q^{27} - 58 q^{29} + 58 q^{31} - 450 q^{33} + 120 q^{37} + 490 q^{39} + 280 q^{41} - 150 q^{43} - 600 q^{45} + 550 q^{47} - 510 q^{49} + 84 q^{51} - 490 q^{53} + 690 q^{55} - 1180 q^{57} - 1276 q^{59} - 396 q^{61} + 880 q^{63} - 150 q^{65} - 80 q^{67} - 828 q^{69} + 1724 q^{71} + 200 q^{73} + 1000 q^{75} - 440 q^{77} - 1654 q^{79} + 2662 q^{81} - 1500 q^{83} - 2100 q^{85} - 290 q^{87} - 216 q^{89} + 880 q^{91} - 2570 q^{93} + 900 q^{95} + 1880 q^{97} - 3120 q^{99} + O(q^{100})$$

## 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
 −4.69042 4.69042
0 0.309584 0 −15.0000 0 −9.38083 0 −26.9042 0
1.2 0 9.69042 0 −15.0000 0 9.38083 0 66.9042 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 1856.4.a.m 2
4.b odd 2 1 1856.4.a.g 2
8.b even 2 1 464.4.a.c 2
8.d odd 2 1 116.4.a.b 2
24.f even 2 1 1044.4.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.4.a.b 2 8.d odd 2 1
464.4.a.c 2 8.b even 2 1
1044.4.a.b 2 24.f even 2 1
1856.4.a.g 2 4.b odd 2 1
1856.4.a.m 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{2} - 10 T_{3} + 3$$ $$T_{5} + 15$$ $$T_{7}^{2} - 88$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 10 T + T^{2}$$
$5$ $$( 15 + T )^{2}$$
$7$ $$-88 + T^{2}$$
$11$ $$-21 + 46 T + T^{2}$$
$13$ $$-2175 - 10 T + T^{2}$$
$17$ $$588 - 140 T + T^{2}$$
$19$ $$-7900 + 60 T + T^{2}$$
$23$ $$-2268 + 60 T + T^{2}$$
$29$ $$( 29 + T )^{2}$$
$31$ $$-92109 - 58 T + T^{2}$$
$37$ $$2192 - 120 T + T^{2}$$
$41$ $$-35400 - 280 T + T^{2}$$
$43$ $$-124813 + 150 T + T^{2}$$
$47$ $$45507 - 550 T + T^{2}$$
$53$ $$34593 + 490 T + T^{2}$$
$59$ $$398244 + 1276 T + T^{2}$$
$61$ $$-68596 + 396 T + T^{2}$$
$67$ $$-255008 + 80 T + T^{2}$$
$71$ $$740844 - 1724 T + T^{2}$$
$73$ $$-12528 - 200 T + T^{2}$$
$79$ $$678979 + 1654 T + T^{2}$$
$83$ $$483300 + 1500 T + T^{2}$$
$89$ $$-166536 + 216 T + T^{2}$$
$97$ $$735672 - 1880 T + T^{2}$$
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