Properties

 Label 7098.2.a.cb 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.

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

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
 1.80194 0.445042 −1.24698
−1.00000 −1.00000 1.00000 −4.04892 1.00000 1.00000 −1.00000 1.00000 4.04892
1.2 −1.00000 −1.00000 1.00000 0.356896 1.00000 1.00000 −1.00000 1.00000 −0.356896
1.3 −1.00000 −1.00000 1.00000 0.692021 1.00000 1.00000 −1.00000 1.00000 −0.692021
 $$n$$: 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.cb 3
13.b even 2 1 7098.2.a.ck yes 3

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

 $$T_{5}^{3} + 3 T_{5}^{2} - 4 T_{5} + 1$$ $$T_{11}^{3} - 3 T_{11}^{2} - 18 T_{11} + 13$$ $$T_{17}^{3} + 9 T_{17}^{2} + 6 T_{17} - 29$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$1 - 4 T + 3 T^{2} + T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$13 - 18 T - 3 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-29 + 6 T + 9 T^{2} + T^{3}$$
$19$ $$-211 - 4 T + 11 T^{2} + T^{3}$$
$23$ $$-13 + 24 T - 11 T^{2} + T^{3}$$
$29$ $$64 - 16 T - 8 T^{2} + T^{3}$$
$31$ $$-211 - 4 T + 11 T^{2} + T^{3}$$
$37$ $$-433 - 130 T + 3 T^{2} + T^{3}$$
$41$ $$-203 + 126 T - 21 T^{2} + T^{3}$$
$43$ $$328 - 32 T - 10 T^{2} + T^{3}$$
$47$ $$392 - 14 T^{2} + T^{3}$$
$53$ $$-328 - 120 T - 2 T^{2} + T^{3}$$
$59$ $$1112 + 332 T + 32 T^{2} + T^{3}$$
$61$ $$-1112 - 184 T + 6 T^{2} + T^{3}$$
$67$ $$-104 - 4 T + 10 T^{2} + T^{3}$$
$71$ $$2359 - 70 T - 21 T^{2} + T^{3}$$
$73$ $$-8 + 20 T + 16 T^{2} + T^{3}$$
$79$ $$8 - 4 T - 4 T^{2} + T^{3}$$
$83$ $$328 + 152 T + 22 T^{2} + T^{3}$$
$89$ $$49 + 98 T - 21 T^{2} + T^{3}$$
$97$ $$1448 + 404 T + 36 T^{2} + T^{3}$$