# Properties

 Label 7098.2.a.ck Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.6778153547$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

## 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
 −1.24698 0.445042 1.80194
1.00000 −1.00000 1.00000 −0.692021 −1.00000 −1.00000 1.00000 1.00000 −0.692021
1.2 1.00000 −1.00000 1.00000 −0.356896 −1.00000 −1.00000 1.00000 1.00000 −0.356896
1.3 1.00000 −1.00000 1.00000 4.04892 −1.00000 −1.00000 1.00000 1.00000 4.04892
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$
$$13$$ $$1$$

## 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 7098.2.a.ck yes 3
13.b even 2 1 7098.2.a.cb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.cb 3 13.b even 2 1
7098.2.a.ck yes 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5}^{3} - 3 T_{5}^{2} - 4 T_{5} - 1$$ $$T_{11}^{3} + 3 T_{11}^{2} - 18 T_{11} - 13$$ $$T_{17}^{3} + 9 T_{17}^{2} + 6 T_{17} - 29$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$-1 - 4 T - 3 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-13 - 18 T + 3 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-29 + 6 T + 9 T^{2} + T^{3}$$
$19$ $$211 - 4 T - 11 T^{2} + T^{3}$$
$23$ $$-13 + 24 T - 11 T^{2} + T^{3}$$
$29$ $$64 - 16 T - 8 T^{2} + T^{3}$$
$31$ $$211 - 4 T - 11 T^{2} + T^{3}$$
$37$ $$433 - 130 T - 3 T^{2} + T^{3}$$
$41$ $$203 + 126 T + 21 T^{2} + T^{3}$$
$43$ $$328 - 32 T - 10 T^{2} + T^{3}$$
$47$ $$-392 + 14 T^{2} + T^{3}$$
$53$ $$-328 - 120 T - 2 T^{2} + T^{3}$$
$59$ $$-1112 + 332 T - 32 T^{2} + T^{3}$$
$61$ $$-1112 - 184 T + 6 T^{2} + T^{3}$$
$67$ $$104 - 4 T - 10 T^{2} + T^{3}$$
$71$ $$-2359 - 70 T + 21 T^{2} + T^{3}$$
$73$ $$8 + 20 T - 16 T^{2} + T^{3}$$
$79$ $$8 - 4 T - 4 T^{2} + T^{3}$$
$83$ $$-328 + 152 T - 22 T^{2} + T^{3}$$
$89$ $$-49 + 98 T + 21 T^{2} + T^{3}$$
$97$ $$-1448 + 404 T - 36 T^{2} + T^{3}$$
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