# Properties

 Label 7098.2.a.ca 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: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + \beta q^{5} + q^{6} + q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + \beta q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + \beta q^{10} -4 q^{11} + q^{12} + q^{14} + \beta q^{15} + q^{16} + 3 q^{17} + q^{18} + ( -2 + 2 \beta ) q^{19} + \beta q^{20} + q^{21} -4 q^{22} + ( -1 - \beta ) q^{23} + q^{24} + ( 9 + \beta ) q^{25} + q^{27} + q^{28} + ( 4 + \beta ) q^{29} + \beta q^{30} + ( 3 - \beta ) q^{31} + q^{32} -4 q^{33} + 3 q^{34} + \beta q^{35} + q^{36} + ( -4 + \beta ) q^{37} + ( -2 + 2 \beta ) q^{38} + \beta q^{40} + ( 4 + \beta ) q^{41} + q^{42} + ( 1 - 3 \beta ) q^{43} -4 q^{44} + \beta q^{45} + ( -1 - \beta ) q^{46} + ( 2 - 2 \beta ) q^{47} + q^{48} + q^{49} + ( 9 + \beta ) q^{50} + 3 q^{51} + ( 5 - 2 \beta ) q^{53} + q^{54} -4 \beta q^{55} + q^{56} + ( -2 + 2 \beta ) q^{57} + ( 4 + \beta ) q^{58} + ( 1 - 3 \beta ) q^{59} + \beta q^{60} -9 q^{61} + ( 3 - \beta ) q^{62} + q^{63} + q^{64} -4 q^{66} + ( -7 + \beta ) q^{67} + 3 q^{68} + ( -1 - \beta ) q^{69} + \beta q^{70} + ( 1 + \beta ) q^{71} + q^{72} + ( 10 - \beta ) q^{73} + ( -4 + \beta ) q^{74} + ( 9 + \beta ) q^{75} + ( -2 + 2 \beta ) q^{76} -4 q^{77} -8 q^{79} + \beta q^{80} + q^{81} + ( 4 + \beta ) q^{82} + ( 1 - 3 \beta ) q^{83} + q^{84} + 3 \beta q^{85} + ( 1 - 3 \beta ) q^{86} + ( 4 + \beta ) q^{87} -4 q^{88} + ( 5 - \beta ) q^{89} + \beta q^{90} + ( -1 - \beta ) q^{92} + ( 3 - \beta ) q^{93} + ( 2 - 2 \beta ) q^{94} + 28 q^{95} + q^{96} + ( 8 - 2 \beta ) q^{97} + q^{98} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + q^{5} + 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 2q^{4} + q^{5} + 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + q^{10} - 8q^{11} + 2q^{12} + 2q^{14} + q^{15} + 2q^{16} + 6q^{17} + 2q^{18} - 2q^{19} + q^{20} + 2q^{21} - 8q^{22} - 3q^{23} + 2q^{24} + 19q^{25} + 2q^{27} + 2q^{28} + 9q^{29} + q^{30} + 5q^{31} + 2q^{32} - 8q^{33} + 6q^{34} + q^{35} + 2q^{36} - 7q^{37} - 2q^{38} + q^{40} + 9q^{41} + 2q^{42} - q^{43} - 8q^{44} + q^{45} - 3q^{46} + 2q^{47} + 2q^{48} + 2q^{49} + 19q^{50} + 6q^{51} + 8q^{53} + 2q^{54} - 4q^{55} + 2q^{56} - 2q^{57} + 9q^{58} - q^{59} + q^{60} - 18q^{61} + 5q^{62} + 2q^{63} + 2q^{64} - 8q^{66} - 13q^{67} + 6q^{68} - 3q^{69} + q^{70} + 3q^{71} + 2q^{72} + 19q^{73} - 7q^{74} + 19q^{75} - 2q^{76} - 8q^{77} - 16q^{79} + q^{80} + 2q^{81} + 9q^{82} - q^{83} + 2q^{84} + 3q^{85} - q^{86} + 9q^{87} - 8q^{88} + 9q^{89} + q^{90} - 3q^{92} + 5q^{93} + 2q^{94} + 56q^{95} + 2q^{96} + 14q^{97} + 2q^{98} - 8q^{99} + O(q^{100})$$

## 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
 −3.27492 4.27492
1.00000 1.00000 1.00000 −3.27492 1.00000 1.00000 1.00000 1.00000 −3.27492
1.2 1.00000 1.00000 1.00000 4.27492 1.00000 1.00000 1.00000 1.00000 4.27492
 $$n$$: 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.ca 2
13.b even 2 1 7098.2.a.bm 2
13.c even 3 2 546.2.l.j 4
39.i odd 6 2 1638.2.r.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.j 4 13.c even 3 2
1638.2.r.x 4 39.i odd 6 2
7098.2.a.bm 2 13.b even 2 1
7098.2.a.ca 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5}^{2} - T_{5} - 14$$ $$T_{11} + 4$$ $$T_{17} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-14 - T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$-56 + 2 T + T^{2}$$
$23$ $$-12 + 3 T + T^{2}$$
$29$ $$6 - 9 T + T^{2}$$
$31$ $$-8 - 5 T + T^{2}$$
$37$ $$-2 + 7 T + T^{2}$$
$41$ $$6 - 9 T + T^{2}$$
$43$ $$-128 + T + T^{2}$$
$47$ $$-56 - 2 T + T^{2}$$
$53$ $$-41 - 8 T + T^{2}$$
$59$ $$-128 + T + T^{2}$$
$61$ $$( 9 + T )^{2}$$
$67$ $$28 + 13 T + T^{2}$$
$71$ $$-12 - 3 T + T^{2}$$
$73$ $$76 - 19 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$-128 + T + T^{2}$$
$89$ $$6 - 9 T + T^{2}$$
$97$ $$-8 - 14 T + T^{2}$$