# Properties

 Label 5070.2.b.r Level $5070$ Weight $2$ Character orbit 5070.b Analytic conductor $40.484$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + q^{3} - q^{4} + \beta_{2} q^{5} + \beta_{2} q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} -\beta_{2} q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + q^{3} - q^{4} + \beta_{2} q^{5} + \beta_{2} q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} -\beta_{2} q^{8} + q^{9} - q^{10} + ( -2 \beta_{1} - \beta_{2} ) q^{11} - q^{12} + ( 2 - \beta_{3} ) q^{14} + \beta_{2} q^{15} + q^{16} + ( -2 + 2 \beta_{3} ) q^{17} + \beta_{2} q^{18} + ( \beta_{1} - \beta_{2} ) q^{19} -\beta_{2} q^{20} + ( \beta_{1} - \beta_{2} ) q^{21} + ( -1 + 2 \beta_{3} ) q^{22} + ( 3 - 3 \beta_{3} ) q^{23} -\beta_{2} q^{24} - q^{25} + q^{27} + ( -\beta_{1} + \beta_{2} ) q^{28} + ( 5 - \beta_{3} ) q^{29} - q^{30} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{31} + \beta_{2} q^{32} + ( -2 \beta_{1} - \beta_{2} ) q^{33} + 2 \beta_{1} q^{34} + ( 2 - \beta_{3} ) q^{35} - q^{36} + ( 2 \beta_{1} + \beta_{2} ) q^{37} + ( 2 - \beta_{3} ) q^{38} + q^{40} + ( 4 \beta_{1} + 6 \beta_{2} ) q^{41} + ( 2 - \beta_{3} ) q^{42} + ( -3 + \beta_{3} ) q^{43} + ( 2 \beta_{1} + \beta_{2} ) q^{44} + \beta_{2} q^{45} -3 \beta_{1} q^{46} -7 \beta_{2} q^{47} + q^{48} + ( -1 + 3 \beta_{3} ) q^{49} -\beta_{2} q^{50} + ( -2 + 2 \beta_{3} ) q^{51} + ( 6 + \beta_{3} ) q^{53} + \beta_{2} q^{54} + ( -1 + 2 \beta_{3} ) q^{55} + ( -2 + \beta_{3} ) q^{56} + ( \beta_{1} - \beta_{2} ) q^{57} + ( -\beta_{1} + 4 \beta_{2} ) q^{58} + ( \beta_{1} - 8 \beta_{2} ) q^{59} -\beta_{2} q^{60} + 6 q^{61} + ( -1 + 3 \beta_{3} ) q^{62} + ( \beta_{1} - \beta_{2} ) q^{63} - q^{64} + ( -1 + 2 \beta_{3} ) q^{66} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 2 - 2 \beta_{3} ) q^{68} + ( 3 - 3 \beta_{3} ) q^{69} + ( -\beta_{1} + \beta_{2} ) q^{70} + ( -2 \beta_{1} - 10 \beta_{2} ) q^{71} -\beta_{2} q^{72} -6 \beta_{1} q^{73} + ( 1 - 2 \beta_{3} ) q^{74} - q^{75} + ( -\beta_{1} + \beta_{2} ) q^{76} + ( 10 - 3 \beta_{3} ) q^{77} + ( 9 + \beta_{3} ) q^{79} + \beta_{2} q^{80} + q^{81} + ( -2 - 4 \beta_{3} ) q^{82} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{83} + ( -\beta_{1} + \beta_{2} ) q^{84} + 2 \beta_{1} q^{85} + ( \beta_{1} - 2 \beta_{2} ) q^{86} + ( 5 - \beta_{3} ) q^{87} + ( 1 - 2 \beta_{3} ) q^{88} + ( 5 \beta_{1} + 11 \beta_{2} ) q^{89} - q^{90} + ( -3 + 3 \beta_{3} ) q^{92} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{93} + 7 q^{94} + ( 2 - \beta_{3} ) q^{95} + \beta_{2} q^{96} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{97} + ( 3 \beta_{1} + 2 \beta_{2} ) q^{98} + ( -2 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 4q^{4} + 4q^{9} - 4q^{10} - 4q^{12} + 6q^{14} + 4q^{16} - 4q^{17} + 6q^{23} - 4q^{25} + 4q^{27} + 18q^{29} - 4q^{30} + 6q^{35} - 4q^{36} + 6q^{38} + 4q^{40} + 6q^{42} - 10q^{43} + 4q^{48} + 2q^{49} - 4q^{51} + 26q^{53} - 6q^{56} + 24q^{61} + 2q^{62} - 4q^{64} + 4q^{68} + 6q^{69} - 4q^{75} + 34q^{77} + 38q^{79} + 4q^{81} - 16q^{82} + 18q^{87} - 4q^{90} - 6q^{92} + 28q^{94} + 6q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 5 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 1.56155i 2.56155i − 2.56155i 1.56155i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.561553i 1.00000i 1.00000 −1.00000
1351.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.56155i 1.00000i 1.00000 −1.00000
1351.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.56155i 1.00000i 1.00000 −1.00000
1351.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.561553i 1.00000i 1.00000 −1.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.b.r 4
13.b even 2 1 inner 5070.2.b.r 4
13.d odd 4 1 5070.2.a.bb 2
13.d odd 4 1 5070.2.a.bi 2
13.f odd 12 2 390.2.i.g 4
39.k even 12 2 1170.2.i.o 4
65.o even 12 2 1950.2.z.n 8
65.s odd 12 2 1950.2.i.bi 4
65.t even 12 2 1950.2.z.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g 4 13.f odd 12 2
1170.2.i.o 4 39.k even 12 2
1950.2.i.bi 4 65.s odd 12 2
1950.2.z.n 8 65.o even 12 2
1950.2.z.n 8 65.t even 12 2
5070.2.a.bb 2 13.d odd 4 1
5070.2.a.bi 2 13.d odd 4 1
5070.2.b.r 4 1.a even 1 1 trivial
5070.2.b.r 4 13.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 $$S_{2}^{\mathrm{new}}(5070, [\chi])$$:

 $$T_{7}^{4} + 13 T_{7}^{2} + 4$$ $$T_{11}^{2} + 17$$ $$T_{17}^{2} + 2 T_{17} - 16$$ $$T_{31}^{4} + 77 T_{31}^{2} + 1444$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$4 + 13 T^{2} + T^{4}$$
$11$ $$( 17 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -16 + 2 T + T^{2} )^{2}$$
$19$ $$4 + 13 T^{2} + T^{4}$$
$23$ $$( -36 - 3 T + T^{2} )^{2}$$
$29$ $$( 16 - 9 T + T^{2} )^{2}$$
$31$ $$1444 + 77 T^{2} + T^{4}$$
$37$ $$( 17 + T^{2} )^{2}$$
$41$ $$2704 + 168 T^{2} + T^{4}$$
$43$ $$( 2 + 5 T + T^{2} )^{2}$$
$47$ $$( 49 + T^{2} )^{2}$$
$53$ $$( 38 - 13 T + T^{2} )^{2}$$
$59$ $$4624 + 153 T^{2} + T^{4}$$
$61$ $$( -6 + T )^{4}$$
$67$ $$1024 + 208 T^{2} + T^{4}$$
$71$ $$4096 + 196 T^{2} + T^{4}$$
$73$ $$20736 + 324 T^{2} + T^{4}$$
$79$ $$( 86 - 19 T + T^{2} )^{2}$$
$83$ $$64 + 52 T^{2} + T^{4}$$
$89$ $$1156 + 357 T^{2} + T^{4}$$
$97$ $$64 + 52 T^{2} + T^{4}$$
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