# Properties

 Label 7098.2.a.bd 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

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

## 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.bd 1
13.b even 2 1 7098.2.a.i 1
13.e even 6 2 546.2.l.c 2
39.h odd 6 2 1638.2.r.l 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$-2 + T$$
$7$ $$-1 + T$$
$11$ $$3 + T$$
$13$ $$T$$
$17$ $$7 + T$$
$19$ $$-5 + T$$
$23$ $$-6 + T$$
$29$ $$-5 + T$$
$31$ $$-2 + T$$
$37$ $$-2 + T$$
$41$ $$-5 + T$$
$43$ $$-2 + T$$
$47$ $$-1 + T$$
$53$ $$-3 + T$$
$59$ $$-6 + T$$
$61$ $$-7 + T$$
$67$ $$2 + T$$
$71$ $$8 + T$$
$73$ $$12 + T$$
$79$ $$-3 + T$$
$83$ $$-8 + T$$
$89$ $$-11 + T$$
$97$ $$-2 + T$$