# Properties

 Label 2070.2.a.z Level $2070$ Weight $2$ Character orbit 2070.a Self dual yes Analytic conductor $16.529$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.5290332184$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9 x + 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} - q^{8} - q^{10} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} ) q^{14} + q^{16} + ( 2 + \beta_{1} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} ) q^{19} + q^{20} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{22} + q^{23} + q^{25} + ( 1 - \beta_{1} + \beta_{2} ) q^{26} + ( 1 - \beta_{1} - \beta_{2} ) q^{28} + ( 2 - 2 \beta_{1} ) q^{29} + ( -1 - \beta_{1} + \beta_{2} ) q^{31} - q^{32} + ( -2 - \beta_{1} ) q^{34} + ( 1 - \beta_{1} - \beta_{2} ) q^{35} + 2 \beta_{2} q^{37} + ( -1 + \beta_{1} + \beta_{2} ) q^{38} - q^{40} + ( \beta_{1} + 2 \beta_{2} ) q^{41} + 8 q^{43} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{44} - q^{46} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 11 - \beta_{1} + 2 \beta_{2} ) q^{49} - q^{50} + ( -1 + \beta_{1} - \beta_{2} ) q^{52} + 6 q^{53} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} ) q^{56} + ( -2 + 2 \beta_{1} ) q^{58} + ( -4 - 2 \beta_{1} ) q^{59} + ( 2 - \beta_{1} + 4 \beta_{2} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} ) q^{62} + q^{64} + ( -1 + \beta_{1} - \beta_{2} ) q^{65} + ( 4 + 4 \beta_{2} ) q^{67} + ( 2 + \beta_{1} ) q^{68} + ( -1 + \beta_{1} + \beta_{2} ) q^{70} + ( -4 + \beta_{1} ) q^{71} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{73} -2 \beta_{2} q^{74} + ( 1 - \beta_{1} - \beta_{2} ) q^{76} + ( 5 + 8 \beta_{1} - \beta_{2} ) q^{77} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{79} + q^{80} + ( -\beta_{1} - 2 \beta_{2} ) q^{82} + ( -2 + 2 \beta_{2} ) q^{83} + ( 2 + \beta_{1} ) q^{85} -8 q^{86} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{88} + ( -4 - 4 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -2 + 5 \beta_{1} - 2 \beta_{2} ) q^{91} + q^{92} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 1 - \beta_{1} - \beta_{2} ) q^{95} + ( -10 - 3 \beta_{1} ) q^{97} + ( -11 + \beta_{1} - 2 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} - 3q^{8} + O(q^{10})$$ $$3q - 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} - 3q^{8} - 3q^{10} - 3q^{11} - q^{13} - 3q^{14} + 3q^{16} + 7q^{17} + 3q^{19} + 3q^{20} + 3q^{22} + 3q^{23} + 3q^{25} + q^{26} + 3q^{28} + 4q^{29} - 5q^{31} - 3q^{32} - 7q^{34} + 3q^{35} - 2q^{37} - 3q^{38} - 3q^{40} - q^{41} + 24q^{43} - 3q^{44} - 3q^{46} + 14q^{47} + 30q^{49} - 3q^{50} - q^{52} + 18q^{53} - 3q^{55} - 3q^{56} - 4q^{58} - 14q^{59} + q^{61} + 5q^{62} + 3q^{64} - q^{65} + 8q^{67} + 7q^{68} - 3q^{70} - 11q^{71} - 8q^{73} + 2q^{74} + 3q^{76} + 24q^{77} - 4q^{79} + 3q^{80} + q^{82} - 8q^{83} + 7q^{85} - 24q^{86} + 3q^{88} - 18q^{89} + q^{91} + 3q^{92} - 14q^{94} + 3q^{95} - 33q^{97} - 30q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 7$$

## 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
 2.68740 1.43163 −3.11903
−1.00000 0 1.00000 1.00000 0 −4.59692 −1.00000 0 −1.00000
1.2 −1.00000 0 1.00000 1.00000 0 3.08719 −1.00000 0 −1.00000
1.3 −1.00000 0 1.00000 1.00000 0 4.50973 −1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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 2070.2.a.z 3
3.b odd 2 1 230.2.a.d 3
12.b even 2 1 1840.2.a.r 3
15.d odd 2 1 1150.2.a.q 3
15.e even 4 2 1150.2.b.j 6
24.f even 2 1 7360.2.a.ce 3
24.h odd 2 1 7360.2.a.bz 3
60.h even 2 1 9200.2.a.cf 3
69.c even 2 1 5290.2.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 3.b odd 2 1
1150.2.a.q 3 15.d odd 2 1
1150.2.b.j 6 15.e even 4 2
1840.2.a.r 3 12.b even 2 1
2070.2.a.z 3 1.a even 1 1 trivial
5290.2.a.r 3 69.c even 2 1
7360.2.a.bz 3 24.h odd 2 1
7360.2.a.ce 3 24.f even 2 1
9200.2.a.cf 3 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2070))$$:

 $$T_{7}^{3} - 3 T_{7}^{2} - 21 T_{7} + 64$$ $$T_{11}^{3} + 3 T_{11}^{2} - 39 T_{11} - 144$$ $$T_{13}^{3} + T_{13}^{2} - 15 T_{13} - 18$$ $$T_{17}^{3} - 7 T_{17}^{2} + 7 T_{17} + 18$$ $$T_{29}^{3} - 4 T_{29}^{2} - 32 T_{29} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$64 - 21 T - 3 T^{2} + T^{3}$$
$11$ $$-144 - 39 T + 3 T^{2} + T^{3}$$
$13$ $$-18 - 15 T + T^{2} + T^{3}$$
$17$ $$18 + 7 T - 7 T^{2} + T^{3}$$
$19$ $$64 - 21 T - 3 T^{2} + T^{3}$$
$23$ $$( -1 + T )^{3}$$
$29$ $$-24 - 32 T - 4 T^{2} + T^{3}$$
$31$ $$-8 - 7 T + 5 T^{2} + T^{3}$$
$37$ $$-32 - 40 T + 2 T^{2} + T^{3}$$
$41$ $$-186 - 59 T + T^{2} + T^{3}$$
$43$ $$( -8 + T )^{3}$$
$47$ $$288 + 4 T - 14 T^{2} + T^{3}$$
$53$ $$( -6 + T )^{3}$$
$59$ $$-144 + 28 T + 14 T^{2} + T^{3}$$
$61$ $$526 - 157 T - T^{2} + T^{3}$$
$67$ $$384 - 144 T - 8 T^{2} + T^{3}$$
$71$ $$24 + 31 T + 11 T^{2} + T^{3}$$
$73$ $$-248 - 40 T + 8 T^{2} + T^{3}$$
$79$ $$-1152 - 240 T + 4 T^{2} + T^{3}$$
$83$ $$-96 - 20 T + 8 T^{2} + T^{3}$$
$89$ $$-1152 - 48 T + 18 T^{2} + T^{3}$$
$97$ $$166 + 279 T + 33 T^{2} + T^{3}$$