# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + ( 1 + \beta ) q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + ( 1 + \beta ) q^{7} + q^{8} + q^{9} + q^{10} + ( 1 - 2 \beta ) q^{11} + q^{12} + ( 1 + \beta ) q^{14} + q^{15} + q^{16} + 2 \beta q^{17} + q^{18} + ( -1 - \beta ) q^{19} + q^{20} + ( 1 + \beta ) q^{21} + ( 1 - 2 \beta ) q^{22} -3 \beta q^{23} + q^{24} + q^{25} + q^{27} + ( 1 + \beta ) q^{28} + ( 4 + \beta ) q^{29} + q^{30} + ( -2 + 3 \beta ) q^{31} + q^{32} + ( 1 - 2 \beta ) q^{33} + 2 \beta q^{34} + ( 1 + \beta ) q^{35} + q^{36} + ( -1 + 2 \beta ) q^{37} + ( -1 - \beta ) q^{38} + q^{40} + ( 6 - 4 \beta ) q^{41} + ( 1 + \beta ) q^{42} + ( 2 + \beta ) q^{43} + ( 1 - 2 \beta ) q^{44} + q^{45} -3 \beta q^{46} + 7 q^{47} + q^{48} + ( -2 + 3 \beta ) q^{49} + q^{50} + 2 \beta q^{51} + ( 7 - \beta ) q^{53} + q^{54} + ( 1 - 2 \beta ) q^{55} + ( 1 + \beta ) q^{56} + ( -1 - \beta ) q^{57} + ( 4 + \beta ) q^{58} + ( 8 + \beta ) q^{59} + q^{60} + 6 q^{61} + ( -2 + 3 \beta ) q^{62} + ( 1 + \beta ) q^{63} + q^{64} + ( 1 - 2 \beta ) q^{66} + ( 4 + 4 \beta ) q^{67} + 2 \beta q^{68} -3 \beta q^{69} + ( 1 + \beta ) q^{70} + ( -10 + 2 \beta ) q^{71} + q^{72} -6 \beta q^{73} + ( -1 + 2 \beta ) q^{74} + q^{75} + ( -1 - \beta ) q^{76} + ( -7 - 3 \beta ) q^{77} + ( 10 - \beta ) q^{79} + q^{80} + q^{81} + ( 6 - 4 \beta ) q^{82} + ( -4 + 2 \beta ) q^{83} + ( 1 + \beta ) q^{84} + 2 \beta q^{85} + ( 2 + \beta ) q^{86} + ( 4 + \beta ) q^{87} + ( 1 - 2 \beta ) q^{88} + ( -11 + 5 \beta ) q^{89} + q^{90} -3 \beta q^{92} + ( -2 + 3 \beta ) q^{93} + 7 q^{94} + ( -1 - \beta ) q^{95} + q^{96} + ( -4 + 2 \beta ) q^{97} + ( -2 + 3 \beta ) q^{98} + ( 1 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 3q^{14} + 2q^{15} + 2q^{16} + 2q^{17} + 2q^{18} - 3q^{19} + 2q^{20} + 3q^{21} - 3q^{23} + 2q^{24} + 2q^{25} + 2q^{27} + 3q^{28} + 9q^{29} + 2q^{30} - q^{31} + 2q^{32} + 2q^{34} + 3q^{35} + 2q^{36} - 3q^{38} + 2q^{40} + 8q^{41} + 3q^{42} + 5q^{43} + 2q^{45} - 3q^{46} + 14q^{47} + 2q^{48} - q^{49} + 2q^{50} + 2q^{51} + 13q^{53} + 2q^{54} + 3q^{56} - 3q^{57} + 9q^{58} + 17q^{59} + 2q^{60} + 12q^{61} - q^{62} + 3q^{63} + 2q^{64} + 12q^{67} + 2q^{68} - 3q^{69} + 3q^{70} - 18q^{71} + 2q^{72} - 6q^{73} + 2q^{75} - 3q^{76} - 17q^{77} + 19q^{79} + 2q^{80} + 2q^{81} + 8q^{82} - 6q^{83} + 3q^{84} + 2q^{85} + 5q^{86} + 9q^{87} - 17q^{89} + 2q^{90} - 3q^{92} - q^{93} + 14q^{94} - 3q^{95} + 2q^{96} - 6q^{97} - q^{98} + O(q^{100})$$

## 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
 −1.56155 2.56155
1.00000 1.00000 1.00000 1.00000 1.00000 −0.561553 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 3.56155 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.bi 2
13.b even 2 1 5070.2.a.bb 2
13.c even 3 2 390.2.i.g 4
13.d odd 4 2 5070.2.b.r 4
39.i odd 6 2 1170.2.i.o 4
65.n even 6 2 1950.2.i.bi 4
65.q odd 12 4 1950.2.z.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g 4 13.c even 3 2
1170.2.i.o 4 39.i odd 6 2
1950.2.i.bi 4 65.n even 6 2
1950.2.z.n 8 65.q odd 12 4
5070.2.a.bb 2 13.b even 2 1
5070.2.a.bi 2 1.a even 1 1 trivial
5070.2.b.r 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5070))$$:

 $$T_{7}^{2} - 3 T_{7} - 2$$ $$T_{11}^{2} - 17$$ $$T_{17}^{2} - 2 T_{17} - 16$$ $$T_{31}^{2} + T_{31} - 38$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-2 - 3 T + T^{2}$$
$11$ $$-17 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-16 - 2 T + T^{2}$$
$19$ $$-2 + 3 T + T^{2}$$
$23$ $$-36 + 3 T + T^{2}$$
$29$ $$16 - 9 T + T^{2}$$
$31$ $$-38 + T + T^{2}$$
$37$ $$-17 + T^{2}$$
$41$ $$-52 - 8 T + T^{2}$$
$43$ $$2 - 5 T + T^{2}$$
$47$ $$( -7 + T )^{2}$$
$53$ $$38 - 13 T + T^{2}$$
$59$ $$68 - 17 T + T^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$-32 - 12 T + T^{2}$$
$71$ $$64 + 18 T + T^{2}$$
$73$ $$-144 + 6 T + T^{2}$$
$79$ $$86 - 19 T + T^{2}$$
$83$ $$-8 + 6 T + T^{2}$$
$89$ $$-34 + 17 T + T^{2}$$
$97$ $$-8 + 6 T + T^{2}$$
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