# Properties

 Label 7098.2.a.cm Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.80194 −1.24698 0.445042
1.00000 1.00000 1.00000 −2.24698 1.00000 −1.00000 1.00000 1.00000 −2.24698
1.2 1.00000 1.00000 1.00000 −0.554958 1.00000 −1.00000 1.00000 1.00000 −0.554958
1.3 1.00000 1.00000 1.00000 0.801938 1.00000 −1.00000 1.00000 1.00000 0.801938
 $$n$$: 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.cm yes 3
13.b even 2 1 7098.2.a.cf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.cf 3 13.b even 2 1
7098.2.a.cm yes 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5}^{3} + 2 T_{5}^{2} - T_{5} - 1$$ $$T_{11}^{3} + 9 T_{11}^{2} + 20 T_{11} + 13$$ $$T_{17}^{3} - 5 T_{17}^{2} - 8 T_{17} + 41$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-1 - T + 2 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$13 + 20 T + 9 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$41 - 8 T - 5 T^{2} + T^{3}$$
$19$ $$-7 + 7 T^{2} + T^{3}$$
$23$ $$13 - 16 T + T^{2} + T^{3}$$
$29$ $$-13 + 5 T + 6 T^{2} + T^{3}$$
$31$ $$13 - 15 T + 2 T^{2} + T^{3}$$
$37$ $$7 + 49 T + 14 T^{2} + T^{3}$$
$41$ $$13 + 19 T + 8 T^{2} + T^{3}$$
$43$ $$127 - 60 T + 3 T^{2} + T^{3}$$
$47$ $$49 - 49 T + 7 T^{2} + T^{3}$$
$53$ $$41 + 17 T - 10 T^{2} + T^{3}$$
$59$ $$-1 + 5 T + 8 T^{2} + T^{3}$$
$61$ $$( 1 + T )^{3}$$
$67$ $$7 - 7 T + T^{3}$$
$71$ $$104 + 124 T + 22 T^{2} + T^{3}$$
$73$ $$49 + 98 T + 21 T^{2} + T^{3}$$
$79$ $$13 - 99 T + 2 T^{2} + T^{3}$$
$83$ $$-349 - 256 T + 3 T^{2} + T^{3}$$
$89$ $$351 + 171 T + 24 T^{2} + T^{3}$$
$97$ $$-727 - 183 T - 2 T^{2} + T^{3}$$