# Properties

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

# Related objects

## 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: $$2$$ Coefficient field: $$\Q(\sqrt{129})$$ Defining polynomial: $$x^{2} - x - 32$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{129})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −5.17891 6.17891
−1.00000 −1.00000 1.00000 2.00000 1.00000 1.00000 −1.00000 1.00000 −2.00000
1.2 −1.00000 −1.00000 1.00000 2.00000 1.00000 1.00000 −1.00000 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.bk 2
13.b even 2 1 7098.2.a.br 2
13.e even 6 2 546.2.l.k 4
39.h odd 6 2 1638.2.r.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.k 4 13.e even 6 2
1638.2.r.ba 4 39.h odd 6 2
7098.2.a.bk 2 1.a even 1 1 trivial
7098.2.a.br 2 13.b even 2 1

## Hecke kernels

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

 $$T_{5} - 2$$ $$T_{11}^{2} + T_{11} - 32$$ $$T_{17} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( -2 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-32 + T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -5 + T )^{2}$$
$19$ $$-26 - 5 T + T^{2}$$
$23$ $$-26 + 5 T + T^{2}$$
$29$ $$-26 - 5 T + T^{2}$$
$31$ $$-30 - 3 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$-30 - 3 T + T^{2}$$
$43$ $$10 - 13 T + T^{2}$$
$47$ $$-30 - 3 T + T^{2}$$
$53$ $$( 3 + T )^{2}$$
$59$ $$10 - 13 T + T^{2}$$
$61$ $$( 3 + T )^{2}$$
$67$ $$58 - 19 T + T^{2}$$
$71$ $$-20 + 7 T + T^{2}$$
$73$ $$( 4 + T )^{2}$$
$79$ $$24 - 15 T + T^{2}$$
$83$ $$-32 + T + T^{2}$$
$89$ $$( 9 + T )^{2}$$
$97$ $$-128 + 2 T + T^{2}$$