# Properties

 Label 5700.2.f.q Level $5700$ Weight $2$ Character orbit 5700.f Analytic conductor $45.515$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.5089536.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 16 x^{2} - 24 x + 18$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + ( \beta_{1} - \beta_{4} ) q^{7} - q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} + ( \beta_{1} - \beta_{4} ) q^{7} - q^{9} + ( 2 - \beta_{2} - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{4} ) q^{13} + ( -2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{17} + q^{19} + ( 1 - \beta_{3} ) q^{21} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} ) q^{23} -\beta_{4} q^{27} + ( -\beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{31} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{33} + ( -3 \beta_{1} + \beta_{4} ) q^{37} + ( -1 - \beta_{3} ) q^{39} + ( 4 + \beta_{2} + \beta_{3} ) q^{41} + ( \beta_{1} - 3 \beta_{4} - 2 \beta_{5} ) q^{43} + ( -2 \beta_{1} - 3 \beta_{4} - \beta_{5} ) q^{47} + ( \beta_{2} + 4 \beta_{3} ) q^{49} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 \beta_{1} + 3 \beta_{4} + 3 \beta_{5} ) q^{53} + \beta_{4} q^{57} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 5 - \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{4} ) q^{63} + ( 4 \beta_{1} + 4 \beta_{4} ) q^{67} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{69} + ( 4 + 2 \beta_{2} - 2 \beta_{3} ) q^{71} + ( -4 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 6 \beta_{1} - 5 \beta_{4} - \beta_{5} ) q^{77} + ( 6 - 2 \beta_{2} ) q^{79} + q^{81} + ( \beta_{4} + \beta_{5} ) q^{83} + ( -\beta_{1} + \beta_{5} ) q^{87} + ( -\beta_{2} - \beta_{3} ) q^{89} + ( -5 + \beta_{2} + 2 \beta_{3} ) q^{91} + ( 2 \beta_{1} + \beta_{4} + \beta_{5} ) q^{93} + ( -\beta_{1} + 3 \beta_{4} ) q^{97} + ( -2 + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{9} + O(q^{10})$$ $$6 q - 6 q^{9} + 12 q^{11} + 6 q^{19} + 8 q^{21} - 4 q^{39} + 24 q^{41} - 6 q^{49} + 4 q^{51} + 8 q^{59} + 28 q^{61} - 12 q^{69} + 32 q^{71} + 32 q^{79} + 6 q^{81} - 32 q^{91} - 12 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 16 x^{2} - 24 x + 18$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-5 \nu^{5} - 11 \nu^{4} + 101 \nu^{3} - 136 \nu^{2} + 292 \nu - 147$$$$)/393$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{5} + 48 \nu^{4} - 12 \nu^{3} - 2 \nu^{2} + 12 \nu + 701$$$$)/131$$ $$\beta_{3}$$ $$=$$ $$($$$$-6 \nu^{5} + 13 \nu^{4} - 36 \nu^{3} - 6 \nu^{2} + 36 \nu + 7$$$$)/131$$ $$\beta_{4}$$ $$=$$ $$($$$$23 \nu^{5} - 28 \nu^{4} + 7 \nu^{3} + 154 \nu^{2} + 386 \nu - 267$$$$)/393$$ $$\beta_{5}$$ $$=$$ $$($$$$-161 \nu^{5} + 196 \nu^{4} - 49 \nu^{3} - 292 \nu^{2} - 2702 \nu + 1869$$$$)/393$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 7 \beta_{4}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + 5 \beta_{4} - 4 \beta_{3} + \beta_{2} + 4 \beta_{1} - 5$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{3} + 3 \beta_{2} - 16$$ $$\nu^{5}$$ $$=$$ $$($$$$-7 \beta_{5} - 31 \beta_{4} - 18 \beta_{3} + 7 \beta_{2} - 18 \beta_{1} - 31$$$$)/2$$

## 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
 1.66044 − 1.66044i 0.675970 − 0.675970i −1.33641 + 1.33641i −1.33641 − 1.33641i 0.675970 + 0.675970i 1.66044 + 1.66044i
0 1.00000i 0 0 0 1.32088i 0 −1.00000 0
3649.2 0 1.00000i 0 0 0 0.648061i 0 −1.00000 0
3649.3 0 1.00000i 0 0 0 4.67282i 0 −1.00000 0
3649.4 0 1.00000i 0 0 0 4.67282i 0 −1.00000 0
3649.5 0 1.00000i 0 0 0 0.648061i 0 −1.00000 0
3649.6 0 1.00000i 0 0 0 1.32088i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3649.6 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.q 6
5.b even 2 1 inner 5700.2.f.q 6
5.c odd 4 1 1140.2.a.g 3
5.c odd 4 1 5700.2.a.w 3
15.e even 4 1 3420.2.a.m 3
20.e even 4 1 4560.2.a.br 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.g 3 5.c odd 4 1
3420.2.a.m 3 15.e even 4 1
4560.2.a.br 3 20.e even 4 1
5700.2.a.w 3 5.c odd 4 1
5700.2.f.q 6 1.a even 1 1 trivial
5700.2.f.q 6 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 $$S_{2}^{\mathrm{new}}(5700, [\chi])$$:

 $$T_{7}^{6} + 24 T_{7}^{4} + 48 T_{7}^{2} + 16$$ $$T_{11}^{3} - 6 T_{11}^{2} - 12 T_{11} + 76$$ $$T_{13}^{6} + 20 T_{13}^{4} + 112 T_{13}^{2} + 144$$ $$T_{17}^{6} + 92 T_{17}^{4} + 2224 T_{17}^{2} + 5184$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$T^{6}$$
$7$ $$16 + 48 T^{2} + 24 T^{4} + T^{6}$$
$11$ $$( 76 - 12 T - 6 T^{2} + T^{3} )^{2}$$
$13$ $$144 + 112 T^{2} + 20 T^{4} + T^{6}$$
$17$ $$5184 + 2224 T^{2} + 92 T^{4} + T^{6}$$
$19$ $$( -1 + T )^{6}$$
$23$ $$118336 + 7728 T^{2} + 156 T^{4} + T^{6}$$
$29$ $$( 36 - 24 T + T^{3} )^{2}$$
$31$ $$( -208 - 72 T + T^{3} )^{2}$$
$37$ $$16 + 5136 T^{2} + 180 T^{4} + T^{6}$$
$41$ $$( -4 + 24 T - 12 T^{2} + T^{3} )^{2}$$
$43$ $$219024 + 11088 T^{2} + 184 T^{4} + T^{6}$$
$47$ $$5184 + 2736 T^{2} + 156 T^{4} + T^{6}$$
$53$ $$2214144 + 52032 T^{2} + 400 T^{4} + T^{6}$$
$59$ $$( 864 - 144 T - 4 T^{2} + T^{3} )^{2}$$
$61$ $$( 8 + 44 T - 14 T^{2} + T^{3} )^{2}$$
$67$ $$589824 + 28672 T^{2} + 320 T^{4} + T^{6}$$
$71$ $$( 1312 - 64 T - 16 T^{2} + T^{3} )^{2}$$
$73$ $$107584 + 9264 T^{2} + 204 T^{4} + T^{6}$$
$79$ $$( 384 - 16 T^{2} + T^{3} )^{2}$$
$83$ $$576 + 496 T^{2} + 44 T^{4} + T^{6}$$
$89$ $$( 36 - 24 T + T^{3} )^{2}$$
$97$ $$144 + 336 T^{2} + 52 T^{4} + T^{6}$$