# Properties

 Label 7098.2.a.bf Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
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.bf 1
13.b even 2 1 7098.2.a.h 1
13.e even 6 2 546.2.l.d 2
39.h odd 6 2 1638.2.r.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.d 2 13.e even 6 2
1638.2.r.k 2 39.h odd 6 2
7098.2.a.h 1 13.b even 2 1
7098.2.a.bf 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5} - 2$$ $$T_{11} - 4$$ $$T_{17} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$-2 + T$$
$7$ $$-1 + T$$
$11$ $$-4 + T$$
$13$ $$T$$
$17$ $$-7 + T$$
$19$ $$2 + T$$
$23$ $$1 + T$$
$29$ $$2 + T$$
$31$ $$-9 + T$$
$37$ $$-2 + T$$
$41$ $$2 + T$$
$43$ $$5 + T$$
$47$ $$6 + T$$
$53$ $$-3 + T$$
$59$ $$15 + T$$
$61$ $$7 + T$$
$67$ $$-5 + T$$
$71$ $$1 + T$$
$73$ $$12 + T$$
$79$ $$4 + T$$
$83$ $$-1 + T$$
$89$ $$3 + T$$
$97$ $$-16 + T$$