# Properties

 Label 7098.2.a.bg 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.56155 −1.56155
−1.00000 −1.00000 1.00000 −3.56155 1.00000 −1.00000 −1.00000 1.00000 3.56155
1.2 −1.00000 −1.00000 1.00000 0.561553 1.00000 −1.00000 −1.00000 1.00000 −0.561553
 $$n$$: 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.bg 2
13.b even 2 1 7098.2.a.bv 2
13.d odd 4 2 546.2.c.e 4
39.f even 4 2 1638.2.c.h 4
52.f even 4 2 4368.2.h.n 4
91.i even 4 2 3822.2.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.e 4 13.d odd 4 2
1638.2.c.h 4 39.f even 4 2
3822.2.c.h 4 91.i even 4 2
4368.2.h.n 4 52.f even 4 2
7098.2.a.bg 2 1.a even 1 1 trivial
7098.2.a.bv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5}^{2} + 3 T_{5} - 2$$ $$T_{11}^{2} - 7 T_{11} + 8$$ $$T_{17}^{2} - T_{17} - 38$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-2 + 3 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$8 - 7 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-38 - T + T^{2}$$
$19$ $$-4 + T + T^{2}$$
$23$ $$-38 - T + T^{2}$$
$29$ $$-4 + T + T^{2}$$
$31$ $$-64 + 4 T + T^{2}$$
$37$ $$-18 - 9 T + T^{2}$$
$41$ $$( 4 + T )^{2}$$
$43$ $$68 - 17 T + T^{2}$$
$47$ $$-64 + 4 T + T^{2}$$
$53$ $$64 - 18 T + T^{2}$$
$59$ $$-52 + 8 T + T^{2}$$
$61$ $$-4 - T + T^{2}$$
$67$ $$-8 + 6 T + T^{2}$$
$71$ $$-144 - 6 T + T^{2}$$
$73$ $$86 - 19 T + T^{2}$$
$79$ $$( -16 + T )^{2}$$
$83$ $$( -2 + T )^{2}$$
$89$ $$( 8 + T )^{2}$$
$97$ $$( 10 + T )^{2}$$