# Properties

 Label 116.4.a.a Level $116$ Weight $4$ Character orbit 116.a Self dual yes Analytic conductor $6.844$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$116 = 2^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 116.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.84422156067$$ Analytic rank: $$1$$ 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: yes 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 116.4.a.a 2
3.b odd 2 1 1044.4.a.d 2
4.b odd 2 1 464.4.a.d 2
8.b even 2 1 1856.4.a.j 2
8.d odd 2 1 1856.4.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.4.a.a 2 1.a even 1 1 trivial
464.4.a.d 2 4.b odd 2 1
1044.4.a.d 2 3.b odd 2 1
1856.4.a.j 2 8.b even 2 1
1856.4.a.k 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 13$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(116))$$.

## Hecke characteristic polynomials

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