# Properties

 Label 1856.4.a.k Level $1856$ Weight $4$ Character orbit 1856.a Self dual yes Analytic conductor $109.508$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + ( - 2 \beta + 5) q^{5} + ( - 4 \beta + 10) q^{7} - 14 q^{9} +O(q^{10})$$ q - b * q^3 + (-2*b + 5) * q^5 + (-4*b + 10) * q^7 - 14 * q^9 $$q - \beta q^{3} + ( - 2 \beta + 5) q^{5} + ( - 4 \beta + 10) q^{7} - 14 q^{9} + (\beta - 16) q^{11} + (6 \beta + 27) q^{13} + ( - 5 \beta + 26) q^{15} + ( - 22 \beta - 22) q^{17} + ( - 12 \beta - 16) q^{19} + ( - 10 \beta + 52) q^{21} + ( - 10 \beta + 18) q^{23} + ( - 20 \beta - 48) q^{25} + 41 \beta q^{27} + 29 q^{29} + ( - 3 \beta - 10) q^{31} + (16 \beta - 13) q^{33} + ( - 40 \beta + 154) q^{35} + ( - 48 \beta + 72) q^{37} + ( - 27 \beta - 78) q^{39} + ( - 2 \beta + 48) q^{41} + (9 \beta + 120) q^{43} + (28 \beta - 70) q^{45} + (49 \beta - 298) q^{47} + ( - 80 \beta - 35) q^{49} + (22 \beta + 286) q^{51} + ( - 54 \beta + 17) q^{53} + (37 \beta - 106) q^{55} + (16 \beta + 156) q^{57} + ( - 70 \beta + 362) q^{59} + ( - 102 \beta + 306) q^{61} + (56 \beta - 140) q^{63} + ( - 24 \beta - 21) q^{65} + ( - 204 \beta + 264) q^{67} + ( - 18 \beta + 130) q^{69} + ( - 166 \beta + 52) q^{71} + (168 \beta - 436) q^{73} + (48 \beta + 260) q^{75} + (74 \beta - 212) q^{77} + ( - 165 \beta + 410) q^{79} - 155 q^{81} + (90 \beta - 114) q^{83} + ( - 66 \beta + 462) q^{85} - 29 \beta q^{87} + ( - 382 \beta - 16) q^{89} + ( - 48 \beta - 42) q^{91} + (10 \beta + 39) q^{93} + ( - 28 \beta + 232) q^{95} + ( - 42 \beta - 948) q^{97} + ( - 14 \beta + 224) q^{99} +O(q^{100})$$ q - b * q^3 + (-2*b + 5) * q^5 + (-4*b + 10) * q^7 - 14 * q^9 + (b - 16) * q^11 + (6*b + 27) * q^13 + (-5*b + 26) * q^15 + (-22*b - 22) * q^17 + (-12*b - 16) * q^19 + (-10*b + 52) * q^21 + (-10*b + 18) * q^23 + (-20*b - 48) * q^25 + 41*b * q^27 + 29 * q^29 + (-3*b - 10) * q^31 + (16*b - 13) * q^33 + (-40*b + 154) * q^35 + (-48*b + 72) * q^37 + (-27*b - 78) * q^39 + (-2*b + 48) * q^41 + (9*b + 120) * q^43 + (28*b - 70) * q^45 + (49*b - 298) * q^47 + (-80*b - 35) * q^49 + (22*b + 286) * q^51 + (-54*b + 17) * q^53 + (37*b - 106) * q^55 + (16*b + 156) * q^57 + (-70*b + 362) * q^59 + (-102*b + 306) * q^61 + (56*b - 140) * q^63 + (-24*b - 21) * q^65 + (-204*b + 264) * q^67 + (-18*b + 130) * q^69 + (-166*b + 52) * q^71 + (168*b - 436) * q^73 + (48*b + 260) * q^75 + (74*b - 212) * q^77 + (-165*b + 410) * q^79 - 155 * q^81 + (90*b - 114) * q^83 + (-66*b + 462) * q^85 - 29*b * q^87 + (-382*b - 16) * q^89 + (-48*b - 42) * q^91 + (10*b + 39) * q^93 + (-28*b + 232) * q^95 + (-42*b - 948) * q^97 + (-14*b + 224) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{5} + 20 q^{7} - 28 q^{9}+O(q^{10})$$ 2 * q + 10 * q^5 + 20 * q^7 - 28 * q^9 $$2 q + 10 q^{5} + 20 q^{7} - 28 q^{9} - 32 q^{11} + 54 q^{13} + 52 q^{15} - 44 q^{17} - 32 q^{19} + 104 q^{21} + 36 q^{23} - 96 q^{25} + 58 q^{29} - 20 q^{31} - 26 q^{33} + 308 q^{35} + 144 q^{37} - 156 q^{39} + 96 q^{41} + 240 q^{43} - 140 q^{45} - 596 q^{47} - 70 q^{49} + 572 q^{51} + 34 q^{53} - 212 q^{55} + 312 q^{57} + 724 q^{59} + 612 q^{61} - 280 q^{63} - 42 q^{65} + 528 q^{67} + 260 q^{69} + 104 q^{71} - 872 q^{73} + 520 q^{75} - 424 q^{77} + 820 q^{79} - 310 q^{81} - 228 q^{83} + 924 q^{85} - 32 q^{89} - 84 q^{91} + 78 q^{93} + 464 q^{95} - 1896 q^{97} + 448 q^{99}+O(q^{100})$$ 2 * q + 10 * q^5 + 20 * q^7 - 28 * q^9 - 32 * q^11 + 54 * q^13 + 52 * q^15 - 44 * q^17 - 32 * q^19 + 104 * q^21 + 36 * q^23 - 96 * q^25 + 58 * q^29 - 20 * q^31 - 26 * q^33 + 308 * q^35 + 144 * q^37 - 156 * q^39 + 96 * q^41 + 240 * q^43 - 140 * q^45 - 596 * q^47 - 70 * q^49 + 572 * q^51 + 34 * q^53 - 212 * q^55 + 312 * q^57 + 724 * q^59 + 612 * q^61 - 280 * q^63 - 42 * q^65 + 528 * q^67 + 260 * q^69 + 104 * q^71 - 872 * q^73 + 520 * q^75 - 424 * q^77 + 820 * q^79 - 310 * q^81 - 228 * q^83 + 924 * q^85 - 32 * q^89 - 84 * q^91 + 78 * q^93 + 464 * q^95 - 1896 * q^97 + 448 * 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
 2.30278 −1.30278
0 −3.60555 0 −2.21110 0 −4.42221 0 −14.0000 0
1.2 0 3.60555 0 12.2111 0 24.4222 0 −14.0000 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.k 2
4.b odd 2 1 1856.4.a.j 2
8.b even 2 1 464.4.a.d 2
8.d odd 2 1 116.4.a.a 2
24.f even 2 1 1044.4.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.4.a.a 2 8.d odd 2 1
464.4.a.d 2 8.b even 2 1
1044.4.a.d 2 24.f even 2 1
1856.4.a.j 2 4.b odd 2 1
1856.4.a.k 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{2} - 13$$ T3^2 - 13 $$T_{5}^{2} - 10T_{5} - 27$$ T5^2 - 10*T5 - 27 $$T_{7}^{2} - 20T_{7} - 108$$ T7^2 - 20*T7 - 108

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 13$$
$5$ $$T^{2} - 10T - 27$$
$7$ $$T^{2} - 20T - 108$$
$11$ $$T^{2} + 32T + 243$$
$13$ $$T^{2} - 54T + 261$$
$17$ $$T^{2} + 44T - 5808$$
$19$ $$T^{2} + 32T - 1616$$
$23$ $$T^{2} - 36T - 976$$
$29$ $$(T - 29)^{2}$$
$31$ $$T^{2} + 20T - 17$$
$37$ $$T^{2} - 144T - 24768$$
$41$ $$T^{2} - 96T + 2252$$
$43$ $$T^{2} - 240T + 13347$$
$47$ $$T^{2} + 596T + 57591$$
$53$ $$T^{2} - 34T - 37619$$
$59$ $$T^{2} - 724T + 67344$$
$61$ $$T^{2} - 612T - 41616$$
$67$ $$T^{2} - 528T - 471312$$
$71$ $$T^{2} - 104T - 355524$$
$73$ $$T^{2} + 872T - 176816$$
$79$ $$T^{2} - 820T - 185825$$
$83$ $$T^{2} + 228T - 92304$$
$89$ $$T^{2} + 32T - 1896756$$
$97$ $$T^{2} + 1896 T + 875772$$