# Properties

 Label 5775.2.a.bp Level $5775$ Weight $2$ Character orbit 5775.a Self dual yes Analytic conductor $46.114$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5775.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.1136071673$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} - q^{3} + (\beta_1 + 2) q^{4} + (\beta_{2} + 1) q^{6} + q^{7} + ( - 2 \beta_1 - 1) q^{8} + q^{9}+O(q^{10})$$ q + (-b2 - 1) * q^2 - q^3 + (b1 + 2) * q^4 + (b2 + 1) * q^6 + q^7 + (-2*b1 - 1) * q^8 + q^9 $$q + ( - \beta_{2} - 1) q^{2} - q^{3} + (\beta_1 + 2) q^{4} + (\beta_{2} + 1) q^{6} + q^{7} + ( - 2 \beta_1 - 1) q^{8} + q^{9} - q^{11} + ( - \beta_1 - 2) q^{12} + ( - \beta_{2} + 3 \beta_1 + 1) q^{13} + ( - \beta_{2} - 1) q^{14} + (\beta_{2} + 2 \beta_1 - 1) q^{16} + (2 \beta_{2} - 4 \beta_1 - 2) q^{17} + ( - \beta_{2} - 1) q^{18} + ( - \beta_{2} + \beta_1 - 3) q^{19} - q^{21} + (\beta_{2} + 1) q^{22} + ( - 2 \beta_{2} - 4) q^{23} + (2 \beta_1 + 1) q^{24} + ( - 2 \beta_{2} - 5 \beta_1 - 1) q^{26} - q^{27} + (\beta_1 + 2) q^{28} + (\beta_{2} - 3 \beta_1 - 1) q^{29} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{31} + (2 \beta_{2} - \beta_1 - 2) q^{32} + q^{33} + (4 \beta_{2} + 6 \beta_1) q^{34} + (\beta_1 + 2) q^{36} + ( - 3 \beta_{2} + \beta_1 - 1) q^{37} + (2 \beta_{2} - \beta_1 + 5) q^{38} + (\beta_{2} - 3 \beta_1 - 1) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{41} + (\beta_{2} + 1) q^{42} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{43} + ( - \beta_1 - 2) q^{44} + (2 \beta_{2} + 2 \beta_1 + 10) q^{46} + ( - 3 \beta_{2} - \beta_1 - 1) q^{47} + ( - \beta_{2} - 2 \beta_1 + 1) q^{48} + q^{49} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{51} + (\beta_{2} + 6 \beta_1 + 10) q^{52} + 2 \beta_1 q^{53} + (\beta_{2} + 1) q^{54} + ( - 2 \beta_1 - 1) q^{56} + (\beta_{2} - \beta_1 + 3) q^{57} + (2 \beta_{2} + 5 \beta_1 + 1) q^{58} + (3 \beta_{2} - 3 \beta_1 + 1) q^{59} - 2 q^{61} + ( - 2 \beta_{2} + 12) q^{62} + q^{63} + (2 \beta_{2} - 4 \beta_1 - 1) q^{64} + ( - \beta_{2} - 1) q^{66} + ( - \beta_{2} + 5 \beta_1 + 1) q^{67} + ( - 8 \beta_1 - 14) q^{68} + (2 \beta_{2} + 4) q^{69} + (4 \beta_1 - 4) q^{71} + ( - 2 \beta_1 - 1) q^{72} + (\beta_{2} - 3 \beta_1 + 7) q^{73} + ( - 2 \beta_{2} + \beta_1 + 9) q^{74} + ( - \beta_{2} - 2 \beta_1 - 4) q^{76} - q^{77} + (2 \beta_{2} + 5 \beta_1 + 1) q^{78} + ( - 4 \beta_1 + 4) q^{79} + q^{81} + ( - 6 \beta_{2} - 2 \beta_1) q^{82} + (6 \beta_1 - 2) q^{83} + ( - \beta_1 - 2) q^{84} + ( - 6 \beta_{2} + 10 \beta_1 + 6) q^{86} + ( - \beta_{2} + 3 \beta_1 + 1) q^{87} + (2 \beta_1 + 1) q^{88} + (4 \beta_{2} + 10) q^{89} + ( - \beta_{2} + 3 \beta_1 + 1) q^{91} + ( - 4 \beta_{2} - 6 \beta_1 - 10) q^{92} + (4 \beta_{2} - 2 \beta_1 + 2) q^{93} + ( - 2 \beta_{2} + 5 \beta_1 + 11) q^{94} + ( - 2 \beta_{2} + \beta_1 + 2) q^{96} + ( - 4 \beta_{2} - 2 \beta_1) q^{97} + ( - \beta_{2} - 1) q^{98} - q^{99}+O(q^{100})$$ q + (-b2 - 1) * q^2 - q^3 + (b1 + 2) * q^4 + (b2 + 1) * q^6 + q^7 + (-2*b1 - 1) * q^8 + q^9 - q^11 + (-b1 - 2) * q^12 + (-b2 + 3*b1 + 1) * q^13 + (-b2 - 1) * q^14 + (b2 + 2*b1 - 1) * q^16 + (2*b2 - 4*b1 - 2) * q^17 + (-b2 - 1) * q^18 + (-b2 + b1 - 3) * q^19 - q^21 + (b2 + 1) * q^22 + (-2*b2 - 4) * q^23 + (2*b1 + 1) * q^24 + (-2*b2 - 5*b1 - 1) * q^26 - q^27 + (b1 + 2) * q^28 + (b2 - 3*b1 - 1) * q^29 + (-4*b2 + 2*b1 - 2) * q^31 + (2*b2 - b1 - 2) * q^32 + q^33 + (4*b2 + 6*b1) * q^34 + (b1 + 2) * q^36 + (-3*b2 + b1 - 1) * q^37 + (2*b2 - b1 + 5) * q^38 + (b2 - 3*b1 - 1) * q^39 + (-2*b2 + 2*b1 + 4) * q^41 + (b2 + 1) * q^42 + (-2*b2 - 4*b1 + 4) * q^43 + (-b1 - 2) * q^44 + (2*b2 + 2*b1 + 10) * q^46 + (-3*b2 - b1 - 1) * q^47 + (-b2 - 2*b1 + 1) * q^48 + q^49 + (-2*b2 + 4*b1 + 2) * q^51 + (b2 + 6*b1 + 10) * q^52 + 2*b1 * q^53 + (b2 + 1) * q^54 + (-2*b1 - 1) * q^56 + (b2 - b1 + 3) * q^57 + (2*b2 + 5*b1 + 1) * q^58 + (3*b2 - 3*b1 + 1) * q^59 - 2 * q^61 + (-2*b2 + 12) * q^62 + q^63 + (2*b2 - 4*b1 - 1) * q^64 + (-b2 - 1) * q^66 + (-b2 + 5*b1 + 1) * q^67 + (-8*b1 - 14) * q^68 + (2*b2 + 4) * q^69 + (4*b1 - 4) * q^71 + (-2*b1 - 1) * q^72 + (b2 - 3*b1 + 7) * q^73 + (-2*b2 + b1 + 9) * q^74 + (-b2 - 2*b1 - 4) * q^76 - q^77 + (2*b2 + 5*b1 + 1) * q^78 + (-4*b1 + 4) * q^79 + q^81 + (-6*b2 - 2*b1) * q^82 + (6*b1 - 2) * q^83 + (-b1 - 2) * q^84 + (-6*b2 + 10*b1 + 6) * q^86 + (-b2 + 3*b1 + 1) * q^87 + (2*b1 + 1) * q^88 + (4*b2 + 10) * q^89 + (-b2 + 3*b1 + 1) * q^91 + (-4*b2 - 6*b1 - 10) * q^92 + (4*b2 - 2*b1 + 2) * q^93 + (-2*b2 + 5*b1 + 11) * q^94 + (-2*b2 + b1 + 2) * q^96 + (-4*b2 - 2*b1) * q^97 + (-b2 - 1) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 3 q^{3} + 6 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 3 * q^3 + 6 * q^4 + 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 $$3 q - 2 q^{2} - 3 q^{3} + 6 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{11} - 6 q^{12} + 4 q^{13} - 2 q^{14} - 4 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 3 q^{21} + 2 q^{22} - 10 q^{23} + 3 q^{24} - q^{26} - 3 q^{27} + 6 q^{28} - 4 q^{29} - 2 q^{31} - 8 q^{32} + 3 q^{33} - 4 q^{34} + 6 q^{36} + 13 q^{38} - 4 q^{39} + 14 q^{41} + 2 q^{42} + 14 q^{43} - 6 q^{44} + 28 q^{46} + 4 q^{48} + 3 q^{49} + 8 q^{51} + 29 q^{52} + 2 q^{54} - 3 q^{56} + 8 q^{57} + q^{58} - 6 q^{61} + 38 q^{62} + 3 q^{63} - 5 q^{64} - 2 q^{66} + 4 q^{67} - 42 q^{68} + 10 q^{69} - 12 q^{71} - 3 q^{72} + 20 q^{73} + 29 q^{74} - 11 q^{76} - 3 q^{77} + q^{78} + 12 q^{79} + 3 q^{81} + 6 q^{82} - 6 q^{83} - 6 q^{84} + 24 q^{86} + 4 q^{87} + 3 q^{88} + 26 q^{89} + 4 q^{91} - 26 q^{92} + 2 q^{93} + 35 q^{94} + 8 q^{96} + 4 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 3 * q^3 + 6 * q^4 + 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^11 - 6 * q^12 + 4 * q^13 - 2 * q^14 - 4 * q^16 - 8 * q^17 - 2 * q^18 - 8 * q^19 - 3 * q^21 + 2 * q^22 - 10 * q^23 + 3 * q^24 - q^26 - 3 * q^27 + 6 * q^28 - 4 * q^29 - 2 * q^31 - 8 * q^32 + 3 * q^33 - 4 * q^34 + 6 * q^36 + 13 * q^38 - 4 * q^39 + 14 * q^41 + 2 * q^42 + 14 * q^43 - 6 * q^44 + 28 * q^46 + 4 * q^48 + 3 * q^49 + 8 * q^51 + 29 * q^52 + 2 * q^54 - 3 * q^56 + 8 * q^57 + q^58 - 6 * q^61 + 38 * q^62 + 3 * q^63 - 5 * q^64 - 2 * q^66 + 4 * q^67 - 42 * q^68 + 10 * q^69 - 12 * q^71 - 3 * q^72 + 20 * q^73 + 29 * q^74 - 11 * q^76 - 3 * q^77 + q^78 + 12 * q^79 + 3 * q^81 + 6 * q^82 - 6 * q^83 - 6 * q^84 + 24 * q^86 + 4 * q^87 + 3 * q^88 + 26 * q^89 + 4 * q^91 - 26 * q^92 + 2 * q^93 + 35 * q^94 + 8 * q^96 + 4 * q^97 - 2 * q^98 - 3 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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
 2.11491 −1.86081 −0.254102
−2.47283 −1.00000 4.11491 0 2.47283 1.00000 −5.22982 1.00000 0
1.2 −1.46260 −1.00000 0.139194 0 1.46260 1.00000 2.72161 1.00000 0
1.3 1.93543 −1.00000 1.74590 0 −1.93543 1.00000 −0.491797 1.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 5775.2.a.bp 3
5.b even 2 1 231.2.a.e 3
15.d odd 2 1 693.2.a.l 3
20.d odd 2 1 3696.2.a.bo 3
35.c odd 2 1 1617.2.a.t 3
55.d odd 2 1 2541.2.a.bg 3
105.g even 2 1 4851.2.a.bi 3
165.d even 2 1 7623.2.a.cd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 5.b even 2 1
693.2.a.l 3 15.d odd 2 1
1617.2.a.t 3 35.c odd 2 1
2541.2.a.bg 3 55.d odd 2 1
3696.2.a.bo 3 20.d odd 2 1
4851.2.a.bi 3 105.g even 2 1
5775.2.a.bp 3 1.a even 1 1 trivial
7623.2.a.cd 3 165.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5775))$$:

 $$T_{2}^{3} + 2T_{2}^{2} - 4T_{2} - 7$$ T2^3 + 2*T2^2 - 4*T2 - 7 $$T_{13}^{3} - 4T_{13}^{2} - 27T_{13} + 94$$ T13^3 - 4*T13^2 - 27*T13 + 94 $$T_{17}^{3} + 8T_{17}^{2} - 40T_{17} - 328$$ T17^3 + 8*T17^2 - 40*T17 - 328

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} - 4 T - 7$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$(T - 1)^{3}$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3} - 4 T^{2} - 27 T + 94$$
$17$ $$T^{3} + 8 T^{2} - 40 T - 328$$
$19$ $$T^{3} + 8 T^{2} + 15 T + 4$$
$23$ $$T^{3} + 10 T^{2} + 12 T - 64$$
$29$ $$T^{3} + 4 T^{2} - 27 T - 94$$
$31$ $$T^{3} + 2 T^{2} - 76 T - 256$$
$37$ $$T^{3} - 43T - 106$$
$41$ $$T^{3} - 14 T^{2} + 40 T + 32$$
$43$ $$T^{3} - 14 T^{2} - 44 T + 848$$
$47$ $$T^{3} - 61T - 32$$
$53$ $$T^{3} - 16T - 8$$
$59$ $$T^{3} - 57T - 52$$
$61$ $$(T + 2)^{3}$$
$67$ $$T^{3} - 4 T^{2} - 85 T + 236$$
$71$ $$T^{3} + 12 T^{2} - 16 T - 256$$
$73$ $$T^{3} - 20 T^{2} + 101 T - 134$$
$79$ $$T^{3} - 12 T^{2} - 16 T + 256$$
$83$ $$T^{3} + 6 T^{2} - 132 T - 496$$
$89$ $$T^{3} - 26 T^{2} + 140 T + 328$$
$97$ $$T^{3} - 4 T^{2} - 120 T + 232$$