# Properties

 Label 7098.2.a.bz 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ 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.

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

## 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
 −1.73205 1.73205
1.00000 1.00000 1.00000 −3.46410 1.00000 −1.00000 1.00000 1.00000 −3.46410
1.2 1.00000 1.00000 1.00000 3.46410 1.00000 −1.00000 1.00000 1.00000 3.46410
 $$n$$: 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.bz 2
13.b even 2 1 7098.2.a.bn 2
13.f odd 12 2 546.2.s.c 4
39.k even 12 2 1638.2.bj.e 4

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

 $$T_{5}^{2} - 12$$ $$T_{11} + 4$$ $$T_{17}^{2} - 6 T_{17} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-12 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$-3 - 6 T + T^{2}$$
$19$ $$4 + 8 T + T^{2}$$
$23$ $$13 - 8 T + T^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$-3 - 6 T + T^{2}$$
$37$ $$-44 + 4 T + T^{2}$$
$41$ $$-48 + T^{2}$$
$43$ $$-11 - 2 T + T^{2}$$
$47$ $$4 - 8 T + T^{2}$$
$53$ $$97 + 20 T + T^{2}$$
$59$ $$61 + 16 T + T^{2}$$
$61$ $$-27 + T^{2}$$
$67$ $$-23 - 4 T + T^{2}$$
$71$ $$-23 - 10 T + T^{2}$$
$73$ $$-32 - 8 T + T^{2}$$
$79$ $$24 + 12 T + T^{2}$$
$83$ $$-3 + T^{2}$$
$89$ $$97 - 20 T + T^{2}$$
$97$ $$-104 + 4 T + T^{2}$$