# Properties

 Label 5700.2.a.u Level $5700$ Weight $2$ Character orbit 5700.a Self dual yes Analytic conductor $45.515$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5700.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$45.5147291521$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1140) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( - \beta - 1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 + (-b - 1) * q^7 + q^9 $$q + q^{3} + ( - \beta - 1) q^{7} + q^{9} + (\beta - 1) q^{11} + (\beta - 1) q^{13} - 2 q^{17} - q^{19} + ( - \beta - 1) q^{21} + 2 q^{23} + q^{27} + ( - \beta - 1) q^{29} + 4 q^{31} + (\beta - 1) q^{33} + ( - \beta - 7) q^{37} + (\beta - 1) q^{39} + ( - \beta + 3) q^{41} + (\beta - 7) q^{43} - 6 q^{47} + (2 \beta + 7) q^{49} - 2 q^{51} - q^{57} + ( - 2 \beta - 2) q^{59} + 2 \beta q^{61} + ( - \beta - 1) q^{63} - 4 q^{67} + 2 q^{69} + (2 \beta - 2) q^{71} - 6 q^{73} - 12 q^{77} + 8 q^{79} + q^{81} + ( - 2 \beta + 4) q^{83} + ( - \beta - 1) q^{87} + ( - \beta + 3) q^{89} - 12 q^{91} + 4 q^{93} + (\beta - 13) q^{97} + (\beta - 1) q^{99}+O(q^{100})$$ q + q^3 + (-b - 1) * q^7 + q^9 + (b - 1) * q^11 + (b - 1) * q^13 - 2 * q^17 - q^19 + (-b - 1) * q^21 + 2 * q^23 + q^27 + (-b - 1) * q^29 + 4 * q^31 + (b - 1) * q^33 + (-b - 7) * q^37 + (b - 1) * q^39 + (-b + 3) * q^41 + (b - 7) * q^43 - 6 * q^47 + (2*b + 7) * q^49 - 2 * q^51 - q^57 + (-2*b - 2) * q^59 + 2*b * q^61 + (-b - 1) * q^63 - 4 * q^67 + 2 * q^69 + (2*b - 2) * q^71 - 6 * q^73 - 12 * q^77 + 8 * q^79 + q^81 + (-2*b + 4) * q^83 + (-b - 1) * q^87 + (-b + 3) * q^89 - 12 * q^91 + 4 * q^93 + (b - 13) * q^97 + (b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{27} - 2 q^{29} + 8 q^{31} - 2 q^{33} - 14 q^{37} - 2 q^{39} + 6 q^{41} - 14 q^{43} - 12 q^{47} + 14 q^{49} - 4 q^{51} - 2 q^{57} - 4 q^{59} - 2 q^{63} - 8 q^{67} + 4 q^{69} - 4 q^{71} - 12 q^{73} - 24 q^{77} + 16 q^{79} + 2 q^{81} + 8 q^{83} - 2 q^{87} + 6 q^{89} - 24 q^{91} + 8 q^{93} - 26 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 - 4 * q^17 - 2 * q^19 - 2 * q^21 + 4 * q^23 + 2 * q^27 - 2 * q^29 + 8 * q^31 - 2 * q^33 - 14 * q^37 - 2 * q^39 + 6 * q^41 - 14 * q^43 - 12 * q^47 + 14 * q^49 - 4 * q^51 - 2 * q^57 - 4 * q^59 - 2 * q^63 - 8 * q^67 + 4 * q^69 - 4 * q^71 - 12 * q^73 - 24 * q^77 + 16 * q^79 + 2 * q^81 + 8 * q^83 - 2 * q^87 + 6 * q^89 - 24 * q^91 + 8 * q^93 - 26 * q^97 - 2 * 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 1.00000 0 0 0 −4.60555 0 1.00000 0
1.2 0 1.00000 0 0 0 2.60555 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$1$$

## 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 5700.2.a.u 2
5.b even 2 1 1140.2.a.e 2
5.c odd 4 2 5700.2.f.n 4
15.d odd 2 1 3420.2.a.i 2
20.d odd 2 1 4560.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.e 2 5.b even 2 1
3420.2.a.i 2 15.d odd 2 1
4560.2.a.bl 2 20.d odd 2 1
5700.2.a.u 2 1.a even 1 1 trivial
5700.2.f.n 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5700))$$:

 $$T_{7}^{2} + 2T_{7} - 12$$ T7^2 + 2*T7 - 12 $$T_{11}^{2} + 2T_{11} - 12$$ T11^2 + 2*T11 - 12 $$T_{13}^{2} + 2T_{13} - 12$$ T13^2 + 2*T13 - 12 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T - 12$$
$11$ $$T^{2} + 2T - 12$$
$13$ $$T^{2} + 2T - 12$$
$17$ $$(T + 2)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$(T - 2)^{2}$$
$29$ $$T^{2} + 2T - 12$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 14T + 36$$
$41$ $$T^{2} - 6T - 4$$
$43$ $$T^{2} + 14T + 36$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 4T - 48$$
$61$ $$T^{2} - 52$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} + 4T - 48$$
$73$ $$(T + 6)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 8T - 36$$
$89$ $$T^{2} - 6T - 4$$
$97$ $$T^{2} + 26T + 156$$