# Properties

 Label 7098.2.a.bw Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.56155 −1.56155
1.00000 1.00000 1.00000 −3.56155 1.00000 −1.00000 1.00000 1.00000 −3.56155
1.2 1.00000 1.00000 1.00000 0.561553 1.00000 −1.00000 1.00000 1.00000 0.561553
 $$n$$: 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.bw 2
13.b even 2 1 7098.2.a.bq 2
13.c even 3 2 546.2.l.i 4
39.i odd 6 2 1638.2.r.z 4

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-2 + 3 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-4 - T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 5 + T )^{2}$$
$19$ $$-36 - 3 T + T^{2}$$
$23$ $$8 + 10 T + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$-16 - 2 T + T^{2}$$
$37$ $$-106 - T + T^{2}$$
$41$ $$-17 + T^{2}$$
$43$ $$32 + 14 T + T^{2}$$
$47$ $$-36 + 3 T + T^{2}$$
$53$ $$47 + 16 T + T^{2}$$
$59$ $$64 - 18 T + T^{2}$$
$61$ $$( 1 + T )^{2}$$
$67$ $$-8 + 6 T + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$16 + 9 T + T^{2}$$
$79$ $$8 - 7 T + T^{2}$$
$83$ $$-64 + 4 T + T^{2}$$
$89$ $$-26 + 7 T + T^{2}$$
$97$ $$-52 - 8 T + T^{2}$$