# Properties

 Label 5070.2.a.ca Level $5070$ Weight $2$ Character orbit 5070.a Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.131472.2 Defining polynomial: $$x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 52$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + \beta_{1} q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + \beta_{1} q^{7} + q^{8} + q^{9} - q^{10} + ( -1 - \beta_{3} ) q^{11} + q^{12} + \beta_{1} q^{14} - q^{15} + q^{16} + 4 q^{17} + q^{18} + ( \beta_{1} + \beta_{3} ) q^{19} - q^{20} + \beta_{1} q^{21} + ( -1 - \beta_{3} ) q^{22} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} + q^{24} + q^{25} + q^{27} + \beta_{1} q^{28} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{29} - q^{30} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{31} + q^{32} + ( -1 - \beta_{3} ) q^{33} + 4 q^{34} -\beta_{1} q^{35} + q^{36} + ( -2 - 3 \beta_{2} + \beta_{3} ) q^{37} + ( \beta_{1} + \beta_{3} ) q^{38} - q^{40} + ( 2 - 2 \beta_{1} ) q^{41} + \beta_{1} q^{42} + ( 3 + \beta_{1} - 4 \beta_{2} ) q^{43} + ( -1 - \beta_{3} ) q^{44} - q^{45} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{46} + ( 3 - 2 \beta_{1} - 2 \beta_{2} ) q^{47} + q^{48} + ( 3 + \beta_{1} + 4 \beta_{2} ) q^{49} + q^{50} + 4 q^{51} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{53} + q^{54} + ( 1 + \beta_{3} ) q^{55} + \beta_{1} q^{56} + ( \beta_{1} + \beta_{3} ) q^{57} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{58} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{59} - q^{60} + ( 4 + 2 \beta_{2} ) q^{61} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{62} + \beta_{1} q^{63} + q^{64} + ( -1 - \beta_{3} ) q^{66} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{67} + 4 q^{68} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{69} -\beta_{1} q^{70} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{71} + q^{72} + ( 2 + 2 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -2 - 3 \beta_{2} + \beta_{3} ) q^{74} + q^{75} + ( \beta_{1} + \beta_{3} ) q^{76} + ( -2 - \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{77} + ( -1 - 3 \beta_{1} ) q^{79} - q^{80} + q^{81} + ( 2 - 2 \beta_{1} ) q^{82} + ( 6 - 2 \beta_{1} + 2 \beta_{3} ) q^{83} + \beta_{1} q^{84} -4 q^{85} + ( 3 + \beta_{1} - 4 \beta_{2} ) q^{86} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{87} + ( -1 - \beta_{3} ) q^{88} + ( -\beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{89} - q^{90} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{92} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{93} + ( 3 - 2 \beta_{1} - 2 \beta_{2} ) q^{94} + ( -\beta_{1} - \beta_{3} ) q^{95} + q^{96} + ( -8 - 2 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 3 + \beta_{1} + 4 \beta_{2} ) q^{98} + ( -1 - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 2q^{7} + 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 2q^{7} + 4q^{8} + 4q^{9} - 4q^{10} - 2q^{11} + 4q^{12} + 2q^{14} - 4q^{15} + 4q^{16} + 16q^{17} + 4q^{18} - 4q^{20} + 2q^{21} - 2q^{22} + 4q^{23} + 4q^{24} + 4q^{25} + 4q^{27} + 2q^{28} + 8q^{29} - 4q^{30} + 4q^{31} + 4q^{32} - 2q^{33} + 16q^{34} - 2q^{35} + 4q^{36} - 10q^{37} - 4q^{40} + 4q^{41} + 2q^{42} + 14q^{43} - 2q^{44} - 4q^{45} + 4q^{46} + 8q^{47} + 4q^{48} + 14q^{49} + 4q^{50} + 16q^{51} + 8q^{53} + 4q^{54} + 2q^{55} + 2q^{56} + 8q^{58} - 6q^{59} - 4q^{60} + 16q^{61} + 4q^{62} + 2q^{63} + 4q^{64} - 2q^{66} + 12q^{67} + 16q^{68} + 4q^{69} - 2q^{70} + 16q^{71} + 4q^{72} + 12q^{73} - 10q^{74} + 4q^{75} - 8q^{77} - 10q^{79} - 4q^{80} + 4q^{81} + 4q^{82} + 16q^{83} + 2q^{84} - 16q^{85} + 14q^{86} + 8q^{87} - 2q^{88} - 4q^{89} - 4q^{90} + 4q^{92} + 4q^{93} + 8q^{94} + 4q^{96} - 36q^{97} + 14q^{98} - 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 52$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 10$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 14 \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$4 \beta_{2} + \beta_{1} + 10$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{3} + 4 \beta_{2} + 15 \beta_{1} + 10$$

## 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
 −3.64466 −1.32258 2.32258 4.64466
1.00000 1.00000 1.00000 −1.00000 1.00000 −3.64466 1.00000 1.00000 −1.00000
1.2 1.00000 1.00000 1.00000 −1.00000 1.00000 −1.32258 1.00000 1.00000 −1.00000
1.3 1.00000 1.00000 1.00000 −1.00000 1.00000 2.32258 1.00000 1.00000 −1.00000
1.4 1.00000 1.00000 1.00000 −1.00000 1.00000 4.64466 1.00000 1.00000 −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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$-1$$

## 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 5070.2.a.ca 4
13.b even 2 1 5070.2.a.bz 4
13.d odd 4 2 5070.2.b.ba 8
13.f odd 12 2 390.2.bb.c 8
39.k even 12 2 1170.2.bs.f 8
65.o even 12 2 1950.2.y.j 8
65.s odd 12 2 1950.2.bc.g 8
65.t even 12 2 1950.2.y.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c 8 13.f odd 12 2
1170.2.bs.f 8 39.k even 12 2
1950.2.y.j 8 65.o even 12 2
1950.2.y.k 8 65.t even 12 2
1950.2.bc.g 8 65.s odd 12 2
5070.2.a.bz 4 13.b even 2 1
5070.2.a.ca 4 1.a even 1 1 trivial
5070.2.b.ba 8 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7}^{4} - 2 T_{7}^{3} - 19 T_{7}^{2} + 20 T_{7} + 52$$ $$T_{11}^{4} + 2 T_{11}^{3} - 34 T_{11}^{2} - 62 T_{11} + 181$$ $$T_{17} - 4$$ $$T_{31}^{4} - 4 T_{31}^{3} - 79 T_{31}^{2} + 556 T_{31} - 956$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$52 + 20 T - 19 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$181 - 62 T - 34 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -4 + T )^{4}$$
$19$ $$468 - 72 T - 63 T^{2} + T^{4}$$
$23$ $$52 + 16 T - 43 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$376 + 148 T - 39 T^{2} - 8 T^{3} + T^{4}$$
$31$ $$-956 + 556 T - 79 T^{2} - 4 T^{3} + T^{4}$$
$37$ $$-803 - 430 T - 34 T^{2} + 10 T^{3} + T^{4}$$
$41$ $$832 + 160 T - 76 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$-572 + 836 T - 43 T^{2} - 14 T^{3} + T^{4}$$
$47$ $$-1103 + 776 T - 82 T^{2} - 8 T^{3} + T^{4}$$
$53$ $$52 + 4 T - 75 T^{2} - 8 T^{3} + T^{4}$$
$59$ $$-284 - 216 T - 31 T^{2} + 6 T^{3} + T^{4}$$
$61$ $$( 4 - 8 T + T^{2} )^{2}$$
$67$ $$352 + 144 T - 52 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$-5024 + 1936 T - 100 T^{2} - 16 T^{3} + T^{4}$$
$73$ $$-4544 + 1728 T - 124 T^{2} - 12 T^{3} + T^{4}$$
$79$ $$3508 - 860 T - 147 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$-5024 + 1936 T - 100 T^{2} - 16 T^{3} + T^{4}$$
$89$ $$-1004 - 784 T - 115 T^{2} + 4 T^{3} + T^{4}$$
$97$ $$-5408 + 624 T + 380 T^{2} + 36 T^{3} + T^{4}$$