Properties

 Label 7098.2.a.ce Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ Analytic rank $0$ 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: $$0$$ 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.

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

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
 −1.24698 0.445042 1.80194
−1.00000 −1.00000 1.00000 −0.246980 1.00000 −1.00000 −1.00000 1.00000 0.246980
1.2 −1.00000 −1.00000 1.00000 1.44504 1.00000 −1.00000 −1.00000 1.00000 −1.44504
1.3 −1.00000 −1.00000 1.00000 2.80194 1.00000 −1.00000 −1.00000 1.00000 −2.80194
 $$n$$: 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.ce 3
13.b even 2 1 7098.2.a.ch yes 3

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

 $$T_{5}^{3} - 4 T_{5}^{2} + 3 T_{5} + 1$$ $$T_{11}^{3} + 3 T_{11}^{2} - 18 T_{11} - 27$$ $$T_{17}^{3} + 3 T_{17}^{2} - 4 T_{17} + 1$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$1 + 3 T - 4 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-27 - 18 T + 3 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$1 - 4 T + 3 T^{2} + T^{3}$$
$19$ $$-7 + 14 T - 7 T^{2} + T^{3}$$
$23$ $$43 - 36 T + 5 T^{2} + T^{3}$$
$29$ $$-127 - 43 T + 2 T^{2} + T^{3}$$
$31$ $$13 + 3 T - 10 T^{2} + T^{3}$$
$37$ $$-7 - 7 T + T^{3}$$
$41$ $$-1 + 3 T + 4 T^{2} + T^{3}$$
$43$ $$203 - 28 T - 7 T^{2} + T^{3}$$
$47$ $$83 - 25 T - 3 T^{2} + T^{3}$$
$53$ $$-113 - 71 T + 2 T^{2} + T^{3}$$
$59$ $$13 - 29 T + 2 T^{2} + T^{3}$$
$61$ $$( 7 + T )^{3}$$
$67$ $$461 - 79 T - 6 T^{2} + T^{3}$$
$71$ $$-8 - 36 T + 2 T^{2} + T^{3}$$
$73$ $$29 + 24 T - 11 T^{2} + T^{3}$$
$79$ $$377 + 159 T + 22 T^{2} + T^{3}$$
$83$ $$-1079 + 318 T - 31 T^{2} + T^{3}$$
$89$ $$889 - 273 T + T^{3}$$
$97$ $$-679 + 245 T - 28 T^{2} + T^{3}$$