Properties

 Label 4851.2.a.ba Level $4851$ Weight $2$ Character orbit 4851.a Self dual yes Analytic conductor $38.735$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4851 = 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4851.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$38.7354300205$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta + 3) q^{4} + 3 q^{5} + (2 \beta + 5) q^{8}+O(q^{10})$$ q + b * q^2 + (b + 3) * q^4 + 3 * q^5 + (2*b + 5) * q^8 $$q + \beta q^{2} + (\beta + 3) q^{4} + 3 q^{5} + (2 \beta + 5) q^{8} + 3 \beta q^{10} + q^{11} - q^{13} + (5 \beta + 4) q^{16} + ( - 2 \beta + 4) q^{17} + ( - 2 \beta + 3) q^{19} + (3 \beta + 9) q^{20} + \beta q^{22} + ( - 2 \beta + 2) q^{23} + 4 q^{25} - \beta q^{26} + ( - 4 \beta + 1) q^{29} + 2 \beta q^{31} + (5 \beta + 15) q^{32} + (2 \beta - 10) q^{34} + q^{37} + (\beta - 10) q^{38} + (6 \beta + 15) q^{40} + (4 \beta - 4) q^{41} + ( - 2 \beta - 2) q^{43} + (\beta + 3) q^{44} - 10 q^{46} + (2 \beta + 5) q^{47} + 4 \beta q^{50} + ( - \beta - 3) q^{52} + ( - 2 \beta + 6) q^{53} + 3 q^{55} + ( - 3 \beta - 20) q^{58} + ( - 2 \beta + 1) q^{59} - 10 q^{61} + (2 \beta + 10) q^{62} + (10 \beta + 17) q^{64} - 3 q^{65} + ( - 2 \beta + 5) q^{67} + ( - 4 \beta + 2) q^{68} + (4 \beta - 4) q^{71} - 7 q^{73} + \beta q^{74} + ( - 5 \beta - 1) q^{76} - 4 \beta q^{79} + (15 \beta + 12) q^{80} + 20 q^{82} + (2 \beta - 8) q^{83} + ( - 6 \beta + 12) q^{85} + ( - 4 \beta - 10) q^{86} + (2 \beta + 5) q^{88} + ( - 4 \beta + 2) q^{89} + ( - 6 \beta - 4) q^{92} + (7 \beta + 10) q^{94} + ( - 6 \beta + 9) q^{95} + (2 \beta + 6) q^{97} +O(q^{100})$$ q + b * q^2 + (b + 3) * q^4 + 3 * q^5 + (2*b + 5) * q^8 + 3*b * q^10 + q^11 - q^13 + (5*b + 4) * q^16 + (-2*b + 4) * q^17 + (-2*b + 3) * q^19 + (3*b + 9) * q^20 + b * q^22 + (-2*b + 2) * q^23 + 4 * q^25 - b * q^26 + (-4*b + 1) * q^29 + 2*b * q^31 + (5*b + 15) * q^32 + (2*b - 10) * q^34 + q^37 + (b - 10) * q^38 + (6*b + 15) * q^40 + (4*b - 4) * q^41 + (-2*b - 2) * q^43 + (b + 3) * q^44 - 10 * q^46 + (2*b + 5) * q^47 + 4*b * q^50 + (-b - 3) * q^52 + (-2*b + 6) * q^53 + 3 * q^55 + (-3*b - 20) * q^58 + (-2*b + 1) * q^59 - 10 * q^61 + (2*b + 10) * q^62 + (10*b + 17) * q^64 - 3 * q^65 + (-2*b + 5) * q^67 + (-4*b + 2) * q^68 + (4*b - 4) * q^71 - 7 * q^73 + b * q^74 + (-5*b - 1) * q^76 - 4*b * q^79 + (15*b + 12) * q^80 + 20 * q^82 + (2*b - 8) * q^83 + (-6*b + 12) * q^85 + (-4*b - 10) * q^86 + (2*b + 5) * q^88 + (-4*b + 2) * q^89 + (-6*b - 4) * q^92 + (7*b + 10) * q^94 + (-6*b + 9) * q^95 + (2*b + 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 7 q^{4} + 6 q^{5} + 12 q^{8}+O(q^{10})$$ 2 * q + q^2 + 7 * q^4 + 6 * q^5 + 12 * q^8 $$2 q + q^{2} + 7 q^{4} + 6 q^{5} + 12 q^{8} + 3 q^{10} + 2 q^{11} - 2 q^{13} + 13 q^{16} + 6 q^{17} + 4 q^{19} + 21 q^{20} + q^{22} + 2 q^{23} + 8 q^{25} - q^{26} - 2 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 2 q^{37} - 19 q^{38} + 36 q^{40} - 4 q^{41} - 6 q^{43} + 7 q^{44} - 20 q^{46} + 12 q^{47} + 4 q^{50} - 7 q^{52} + 10 q^{53} + 6 q^{55} - 43 q^{58} - 20 q^{61} + 22 q^{62} + 44 q^{64} - 6 q^{65} + 8 q^{67} - 4 q^{71} - 14 q^{73} + q^{74} - 7 q^{76} - 4 q^{79} + 39 q^{80} + 40 q^{82} - 14 q^{83} + 18 q^{85} - 24 q^{86} + 12 q^{88} - 14 q^{92} + 27 q^{94} + 12 q^{95} + 14 q^{97}+O(q^{100})$$ 2 * q + q^2 + 7 * q^4 + 6 * q^5 + 12 * q^8 + 3 * q^10 + 2 * q^11 - 2 * q^13 + 13 * q^16 + 6 * q^17 + 4 * q^19 + 21 * q^20 + q^22 + 2 * q^23 + 8 * q^25 - q^26 - 2 * q^29 + 2 * q^31 + 35 * q^32 - 18 * q^34 + 2 * q^37 - 19 * q^38 + 36 * q^40 - 4 * q^41 - 6 * q^43 + 7 * q^44 - 20 * q^46 + 12 * q^47 + 4 * q^50 - 7 * q^52 + 10 * q^53 + 6 * q^55 - 43 * q^58 - 20 * q^61 + 22 * q^62 + 44 * q^64 - 6 * q^65 + 8 * q^67 - 4 * q^71 - 14 * q^73 + q^74 - 7 * q^76 - 4 * q^79 + 39 * q^80 + 40 * q^82 - 14 * q^83 + 18 * q^85 - 24 * q^86 + 12 * q^88 - 14 * q^92 + 27 * q^94 + 12 * q^95 + 14 * q^97

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
 −1.79129 2.79129
−1.79129 0 1.20871 3.00000 0 0 1.41742 0 −5.37386
1.2 2.79129 0 5.79129 3.00000 0 0 10.5826 0 8.37386
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

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 4851.2.a.ba 2
3.b odd 2 1 1617.2.a.o 2
7.b odd 2 1 693.2.a.j 2
21.c even 2 1 231.2.a.b 2
77.b even 2 1 7623.2.a.bf 2
84.h odd 2 1 3696.2.a.bl 2
105.g even 2 1 5775.2.a.bn 2
231.h odd 2 1 2541.2.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 21.c even 2 1
693.2.a.j 2 7.b odd 2 1
1617.2.a.o 2 3.b odd 2 1
2541.2.a.z 2 231.h odd 2 1
3696.2.a.bl 2 84.h odd 2 1
4851.2.a.ba 2 1.a even 1 1 trivial
5775.2.a.bn 2 105.g even 2 1
7623.2.a.bf 2 77.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4851))$$:

 $$T_{2}^{2} - T_{2} - 5$$ T2^2 - T2 - 5 $$T_{5} - 3$$ T5 - 3 $$T_{13} + 1$$ T13 + 1 $$T_{17}^{2} - 6T_{17} - 12$$ T17^2 - 6*T17 - 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 5$$
$3$ $$T^{2}$$
$5$ $$(T - 3)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 6T - 12$$
$19$ $$T^{2} - 4T - 17$$
$23$ $$T^{2} - 2T - 20$$
$29$ $$T^{2} + 2T - 83$$
$31$ $$T^{2} - 2T - 20$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 4T - 80$$
$43$ $$T^{2} + 6T - 12$$
$47$ $$T^{2} - 12T + 15$$
$53$ $$T^{2} - 10T + 4$$
$59$ $$T^{2} - 21$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} - 8T - 5$$
$71$ $$T^{2} + 4T - 80$$
$73$ $$(T + 7)^{2}$$
$79$ $$T^{2} + 4T - 80$$
$83$ $$T^{2} + 14T + 28$$
$89$ $$T^{2} - 84$$
$97$ $$T^{2} - 14T + 28$$