# Properties

 Label 4851.2.a.bd Level $4851$ Weight $2$ Character orbit 4851.a Self dual yes Analytic conductor $38.735$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - 2 q^{5} + (\beta + 3) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 - 2 * q^5 + (b + 3) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - 2 q^{5} + (\beta + 3) q^{8} + ( - 2 \beta - 2) q^{10} - q^{11} + (2 \beta - 2) q^{13} + 3 q^{16} + ( - \beta - 3) q^{17} + ( - 3 \beta - 3) q^{19} + ( - 4 \beta - 2) q^{20} + ( - \beta - 1) q^{22} + 7 q^{23} - q^{25} + 2 q^{26} + ( - 3 \beta + 1) q^{29} - 4 \beta q^{31} + (\beta - 3) q^{32} + ( - 4 \beta - 5) q^{34} + ( - 6 \beta - 1) q^{37} + ( - 6 \beta - 9) q^{38} + ( - 2 \beta - 6) q^{40} + (2 \beta - 4) q^{41} + (3 \beta - 7) q^{43} + ( - 2 \beta - 1) q^{44} + (7 \beta + 7) q^{46} + (2 \beta + 7) q^{47} + ( - \beta - 1) q^{50} + ( - 2 \beta + 6) q^{52} + ( - 2 \beta + 10) q^{53} + 2 q^{55} + ( - 2 \beta - 5) q^{58} + ( - 4 \beta - 3) q^{59} + 4 q^{61} + ( - 4 \beta - 8) q^{62} + ( - 2 \beta - 7) q^{64} + ( - 4 \beta + 4) q^{65} + (2 \beta - 6) q^{67} + ( - 7 \beta - 7) q^{68} + ( - 2 \beta + 7) q^{71} + (4 \beta - 6) q^{73} + ( - 7 \beta - 13) q^{74} + ( - 9 \beta - 15) q^{76} + (8 \beta + 2) q^{79} - 6 q^{80} - 2 \beta q^{82} - 2 \beta q^{83} + (2 \beta + 6) q^{85} + ( - 4 \beta - 1) q^{86} + ( - \beta - 3) q^{88} - 10 \beta q^{89} + (14 \beta + 7) q^{92} + (9 \beta + 11) q^{94} + (6 \beta + 6) q^{95} + (6 \beta + 3) q^{97} +O(q^{100})$$ q + (b + 1) * q^2 + (2*b + 1) * q^4 - 2 * q^5 + (b + 3) * q^8 + (-2*b - 2) * q^10 - q^11 + (2*b - 2) * q^13 + 3 * q^16 + (-b - 3) * q^17 + (-3*b - 3) * q^19 + (-4*b - 2) * q^20 + (-b - 1) * q^22 + 7 * q^23 - q^25 + 2 * q^26 + (-3*b + 1) * q^29 - 4*b * q^31 + (b - 3) * q^32 + (-4*b - 5) * q^34 + (-6*b - 1) * q^37 + (-6*b - 9) * q^38 + (-2*b - 6) * q^40 + (2*b - 4) * q^41 + (3*b - 7) * q^43 + (-2*b - 1) * q^44 + (7*b + 7) * q^46 + (2*b + 7) * q^47 + (-b - 1) * q^50 + (-2*b + 6) * q^52 + (-2*b + 10) * q^53 + 2 * q^55 + (-2*b - 5) * q^58 + (-4*b - 3) * q^59 + 4 * q^61 + (-4*b - 8) * q^62 + (-2*b - 7) * q^64 + (-4*b + 4) * q^65 + (2*b - 6) * q^67 + (-7*b - 7) * q^68 + (-2*b + 7) * q^71 + (4*b - 6) * q^73 + (-7*b - 13) * q^74 + (-9*b - 15) * q^76 + (8*b + 2) * q^79 - 6 * q^80 - 2*b * q^82 - 2*b * q^83 + (2*b + 6) * q^85 + (-4*b - 1) * q^86 + (-b - 3) * q^88 - 10*b * q^89 + (14*b + 7) * q^92 + (9*b + 11) * q^94 + (6*b + 6) * q^95 + (6*b + 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 6 q^{8} - 4 q^{10} - 2 q^{11} - 4 q^{13} + 6 q^{16} - 6 q^{17} - 6 q^{19} - 4 q^{20} - 2 q^{22} + 14 q^{23} - 2 q^{25} + 4 q^{26} + 2 q^{29} - 6 q^{32} - 10 q^{34} - 2 q^{37} - 18 q^{38} - 12 q^{40} - 8 q^{41} - 14 q^{43} - 2 q^{44} + 14 q^{46} + 14 q^{47} - 2 q^{50} + 12 q^{52} + 20 q^{53} + 4 q^{55} - 10 q^{58} - 6 q^{59} + 8 q^{61} - 16 q^{62} - 14 q^{64} + 8 q^{65} - 12 q^{67} - 14 q^{68} + 14 q^{71} - 12 q^{73} - 26 q^{74} - 30 q^{76} + 4 q^{79} - 12 q^{80} + 12 q^{85} - 2 q^{86} - 6 q^{88} + 14 q^{92} + 22 q^{94} + 12 q^{95} + 6 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 6 * q^8 - 4 * q^10 - 2 * q^11 - 4 * q^13 + 6 * q^16 - 6 * q^17 - 6 * q^19 - 4 * q^20 - 2 * q^22 + 14 * q^23 - 2 * q^25 + 4 * q^26 + 2 * q^29 - 6 * q^32 - 10 * q^34 - 2 * q^37 - 18 * q^38 - 12 * q^40 - 8 * q^41 - 14 * q^43 - 2 * q^44 + 14 * q^46 + 14 * q^47 - 2 * q^50 + 12 * q^52 + 20 * q^53 + 4 * q^55 - 10 * q^58 - 6 * q^59 + 8 * q^61 - 16 * q^62 - 14 * q^64 + 8 * q^65 - 12 * q^67 - 14 * q^68 + 14 * q^71 - 12 * q^73 - 26 * q^74 - 30 * q^76 + 4 * q^79 - 12 * q^80 + 12 * q^85 - 2 * q^86 - 6 * q^88 + 14 * q^92 + 22 * q^94 + 12 * q^95 + 6 * 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
 −1.41421 1.41421
−0.414214 0 −1.82843 −2.00000 0 0 1.58579 0 0.828427
1.2 2.41421 0 3.82843 −2.00000 0 0 4.41421 0 −4.82843
 $$n$$: 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.bd 2
3.b odd 2 1 1617.2.a.m 2
7.b odd 2 1 4851.2.a.be 2
7.d odd 6 2 693.2.i.f 4
21.c even 2 1 1617.2.a.n 2
21.g even 6 2 231.2.i.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.d 4 21.g even 6 2
693.2.i.f 4 7.d odd 6 2
1617.2.a.m 2 3.b odd 2 1
1617.2.a.n 2 21.c even 2 1
4851.2.a.bd 2 1.a even 1 1 trivial
4851.2.a.be 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 2T_{2} - 1$$ T2^2 - 2*T2 - 1 $$T_{5} + 2$$ T5 + 2 $$T_{13}^{2} + 4T_{13} - 4$$ T13^2 + 4*T13 - 4 $$T_{17}^{2} + 6T_{17} + 7$$ T17^2 + 6*T17 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2}$$
$5$ $$(T + 2)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 4T - 4$$
$17$ $$T^{2} + 6T + 7$$
$19$ $$T^{2} + 6T - 9$$
$23$ $$(T - 7)^{2}$$
$29$ $$T^{2} - 2T - 17$$
$31$ $$T^{2} - 32$$
$37$ $$T^{2} + 2T - 71$$
$41$ $$T^{2} + 8T + 8$$
$43$ $$T^{2} + 14T + 31$$
$47$ $$T^{2} - 14T + 41$$
$53$ $$T^{2} - 20T + 92$$
$59$ $$T^{2} + 6T - 23$$
$61$ $$(T - 4)^{2}$$
$67$ $$T^{2} + 12T + 28$$
$71$ $$T^{2} - 14T + 41$$
$73$ $$T^{2} + 12T + 4$$
$79$ $$T^{2} - 4T - 124$$
$83$ $$T^{2} - 8$$
$89$ $$T^{2} - 200$$
$97$ $$T^{2} - 6T - 63$$