# Properties

 Label 7098.2.a.bo Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.00000 1.00000 1.00000 −0.732051 −1.00000 −1.00000 −1.00000 1.00000 0.732051
1.2 −1.00000 1.00000 1.00000 2.73205 −1.00000 −1.00000 −1.00000 1.00000 −2.73205
 $$n$$: 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.bo 2
13.b even 2 1 7098.2.a.by 2
13.f odd 12 2 546.2.s.b 4
39.k even 12 2 1638.2.bj.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.b 4 13.f odd 12 2
1638.2.bj.a 4 39.k even 12 2
7098.2.a.bo 2 1.a even 1 1 trivial
7098.2.a.by 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5}^{2} - 2 T_{5} - 2$$ $$T_{11}^{2} - 27$$ $$T_{17}^{2} + 8 T_{17} + 13$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-2 - 2 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-27 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$13 + 8 T + T^{2}$$
$19$ $$-3 - 6 T + T^{2}$$
$23$ $$4 + 8 T + T^{2}$$
$29$ $$37 + 14 T + T^{2}$$
$31$ $$22 - 10 T + T^{2}$$
$37$ $$22 - 10 T + T^{2}$$
$41$ $$13 - 10 T + T^{2}$$
$43$ $$46 + 14 T + T^{2}$$
$47$ $$-39 + 6 T + T^{2}$$
$53$ $$-39 + 6 T + T^{2}$$
$59$ $$-44 + 4 T + T^{2}$$
$61$ $$1 - 4 T + T^{2}$$
$67$ $$( 10 + T )^{2}$$
$71$ $$78 - 18 T + T^{2}$$
$73$ $$-8 - 4 T + T^{2}$$
$79$ $$( -9 + T )^{2}$$
$83$ $$118 + 22 T + T^{2}$$
$89$ $$-183 + 6 T + T^{2}$$
$97$ $$54 + 18 T + T^{2}$$