# Properties

 Label 5700.2.f.n Level $5700$ Weight $2$ Character orbit 5700.f Analytic conductor $45.515$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5700.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$45.5147291521$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1140) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{7} - q^{9}+O(q^{10})$$ q - b1 * q^3 + (b2 - b1) * q^7 - q^9 $$q - \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{7} - q^{9} + ( - \beta_{3} - 1) q^{11} + (\beta_{2} + \beta_1) q^{13} - 2 \beta_1 q^{17} + q^{19} + (\beta_{3} - 1) q^{21} - 2 \beta_1 q^{23} + \beta_1 q^{27} + ( - \beta_{3} + 1) q^{29} + 4 q^{31} + (\beta_{2} + \beta_1) q^{33} + (\beta_{2} - 7 \beta_1) q^{37} + (\beta_{3} + 1) q^{39} + (\beta_{3} + 3) q^{41} + (\beta_{2} + 7 \beta_1) q^{43} - 6 \beta_1 q^{47} + (2 \beta_{3} - 7) q^{49} - 2 q^{51} - \beta_1 q^{57} + ( - 2 \beta_{3} + 2) q^{59} - 2 \beta_{3} q^{61} + ( - \beta_{2} + \beta_1) q^{63} - 4 \beta_1 q^{67} - 2 q^{69} + ( - 2 \beta_{3} - 2) q^{71} + 6 \beta_1 q^{73} - 12 \beta_1 q^{77} - 8 q^{79} + q^{81} + ( - 2 \beta_{2} - 4 \beta_1) q^{83} + (\beta_{2} - \beta_1) q^{87} + ( - \beta_{3} - 3) q^{89} - 12 q^{91} - 4 \beta_1 q^{93} + ( - \beta_{2} - 13 \beta_1) q^{97} + (\beta_{3} + 1) q^{99}+O(q^{100})$$ q - b1 * q^3 + (b2 - b1) * q^7 - q^9 + (-b3 - 1) * q^11 + (b2 + b1) * q^13 - 2*b1 * q^17 + q^19 + (b3 - 1) * q^21 - 2*b1 * q^23 + b1 * q^27 + (-b3 + 1) * q^29 + 4 * q^31 + (b2 + b1) * q^33 + (b2 - 7*b1) * q^37 + (b3 + 1) * q^39 + (b3 + 3) * q^41 + (b2 + 7*b1) * q^43 - 6*b1 * q^47 + (2*b3 - 7) * q^49 - 2 * q^51 - b1 * q^57 + (-2*b3 + 2) * q^59 - 2*b3 * q^61 + (-b2 + b1) * q^63 - 4*b1 * q^67 - 2 * q^69 + (-2*b3 - 2) * q^71 + 6*b1 * q^73 - 12*b1 * q^77 - 8 * q^79 + q^81 + (-2*b2 - 4*b1) * q^83 + (b2 - b1) * q^87 + (-b3 - 3) * q^89 - 12 * q^91 - 4*b1 * q^93 + (-b2 - 13*b1) * q^97 + (b3 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} + 4 q^{29} + 16 q^{31} + 4 q^{39} + 12 q^{41} - 28 q^{49} - 8 q^{51} + 8 q^{59} - 8 q^{69} - 8 q^{71} - 32 q^{79} + 4 q^{81} - 12 q^{89} - 48 q^{91} + 4 q^{99}+O(q^{100})$$ 4 * q - 4 * q^9 - 4 * q^11 + 4 * q^19 - 4 * q^21 + 4 * q^29 + 16 * q^31 + 4 * q^39 + 12 * q^41 - 28 * q^49 - 8 * q^51 + 8 * q^59 - 8 * q^69 - 8 * q^71 - 32 * q^79 + 4 * q^81 - 12 * q^89 - 48 * q^91 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 3$$ (v^3 + 4*v) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 10\nu ) / 3$$ (v^3 + 10*v) / 3 $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 7$$ 2*v^2 + 7
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 7 ) / 2$$ (b3 - 7) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{2} + 5\beta_1$$ -2*b2 + 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1901$$ $$2851$$ $$3877$$ $$4201$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3649.1
 − 2.30278i 1.30278i − 1.30278i 2.30278i
0 1.00000i 0 0 0 4.60555i 0 −1.00000 0
3649.2 0 1.00000i 0 0 0 2.60555i 0 −1.00000 0
3649.3 0 1.00000i 0 0 0 2.60555i 0 −1.00000 0
3649.4 0 1.00000i 0 0 0 4.60555i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{7}^{4} + 28T_{7}^{2} + 144$$ T7^4 + 28*T7^2 + 144 $$T_{11}^{2} + 2T_{11} - 12$$ T11^2 + 2*T11 - 12 $$T_{13}^{4} + 28T_{13}^{2} + 144$$ T13^4 + 28*T13^2 + 144 $$T_{17}^{2} + 4$$ T17^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 28T^{2} + 144$$
$11$ $$(T^{2} + 2 T - 12)^{2}$$
$13$ $$T^{4} + 28T^{2} + 144$$
$17$ $$(T^{2} + 4)^{2}$$
$19$ $$(T - 1)^{4}$$
$23$ $$(T^{2} + 4)^{2}$$
$29$ $$(T^{2} - 2 T - 12)^{2}$$
$31$ $$(T - 4)^{4}$$
$37$ $$T^{4} + 124T^{2} + 1296$$
$41$ $$(T^{2} - 6 T - 4)^{2}$$
$43$ $$T^{4} + 124T^{2} + 1296$$
$47$ $$(T^{2} + 36)^{2}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} - 4 T - 48)^{2}$$
$61$ $$(T^{2} - 52)^{2}$$
$67$ $$(T^{2} + 16)^{2}$$
$71$ $$(T^{2} + 4 T - 48)^{2}$$
$73$ $$(T^{2} + 36)^{2}$$
$79$ $$(T + 8)^{4}$$
$83$ $$T^{4} + 136T^{2} + 1296$$
$89$ $$(T^{2} + 6 T - 4)^{2}$$
$97$ $$T^{4} + 364 T^{2} + 24336$$