# Properties

 Label 7098.2.a.cu Level $7098$ Weight $2$ Character orbit 7098.a Self dual yes Analytic conductor $56.678$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.6148961.1 Defining polynomial: $$x^{6} - 2 x^{5} - 11 x^{4} + 12 x^{3} + 32 x^{2} - 16 x - 29$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 11 x^{4} + 12 x^{3} + 32 x^{2} - 16 x - 29$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - 2 \nu^{3} - 9 \nu^{2} + 7 \nu + 15$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} - 9 \nu^{3} + 9 \nu^{2} + 13 \nu - 6$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{4} - 3 \nu^{3} - 19 \nu^{2} + 8 \nu + 29$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - 12 \nu^{3} - 12 \nu^{2} + 23 \nu + 31$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + 3 \beta_{4} + \beta_{3} - 4 \beta_{2} + 7 \beta_{1} + 5$$ $$\nu^{4}$$ $$=$$ $$-11 \beta_{5} + 15 \beta_{4} + 11 \beta_{3} - 6 \beta_{2} + 16 \beta_{1} + 31$$ $$\nu^{5}$$ $$=$$ $$-22 \beta_{5} + 48 \beta_{4} + 24 \beta_{3} - 48 \beta_{2} + 73 \beta_{1} + 77$$

## 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
 −1.07530 1.49793 −1.33906 3.58604 −2.29987 1.63026
1.00000 1.00000 1.00000 −0.937626 1.00000 −1.00000 1.00000 1.00000 −0.937626
1.2 1.00000 1.00000 1.00000 −0.867888 1.00000 −1.00000 1.00000 1.00000 −0.867888
1.3 1.00000 1.00000 1.00000 0.404061 1.00000 −1.00000 1.00000 1.00000 0.404061
1.4 1.00000 1.00000 1.00000 2.59594 1.00000 −1.00000 1.00000 1.00000 2.59594
1.5 1.00000 1.00000 1.00000 3.86789 1.00000 −1.00000 1.00000 1.00000 3.86789
1.6 1.00000 1.00000 1.00000 3.93763 1.00000 −1.00000 1.00000 1.00000 3.93763
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.cu yes 6
13.b even 2 1 7098.2.a.cq 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7098.2.a.cq 6 13.b even 2 1
7098.2.a.cu yes 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7098))$$:

 $$T_{5}^{6} - 9 T_{5}^{5} + 21 T_{5}^{4} + 9 T_{5}^{3} - 49 T_{5}^{2} - 15 T_{5} + 13$$ $$T_{11}^{6} - 10 T_{11}^{5} - 9 T_{11}^{4} + 296 T_{11}^{3} - 342 T_{11}^{2} - 1630 T_{11} + 937$$ $$T_{17}^{6} - 4 T_{17}^{5} - 55 T_{17}^{4} + 256 T_{17}^{3} + 452 T_{17}^{2} - 2486 T_{17} + 337$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{6}$$
$3$ $$( -1 + T )^{6}$$
$5$ $$13 - 15 T - 49 T^{2} + 9 T^{3} + 21 T^{4} - 9 T^{5} + T^{6}$$
$7$ $$( 1 + T )^{6}$$
$11$ $$937 - 1630 T - 342 T^{2} + 296 T^{3} - 9 T^{4} - 10 T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$337 - 2486 T + 452 T^{2} + 256 T^{3} - 55 T^{4} - 4 T^{5} + T^{6}$$
$19$ $$-1 + 54 T + 116 T^{2} + 26 T^{3} - 25 T^{4} - 2 T^{5} + T^{6}$$
$23$ $$547 - 1442 T + 934 T^{2} + 14 T^{3} - 67 T^{4} + T^{6}$$
$29$ $$64 + 176 T + 88 T^{2} - 67 T^{3} - 27 T^{4} + 4 T^{5} + T^{6}$$
$31$ $$-5447 - 12287 T + 1201 T^{2} + 845 T^{3} - 95 T^{4} - 9 T^{5} + T^{6}$$
$37$ $$-211 + 601 T - 525 T^{2} + 149 T^{3} + 7 T^{4} - 9 T^{5} + T^{6}$$
$41$ $$-111007 + 93411 T - 25963 T^{2} + 2361 T^{3} + 91 T^{4} - 25 T^{5} + T^{6}$$
$43$ $$8728 + 6112 T - 9310 T^{2} + 1755 T^{3} - 19 T^{5} + T^{6}$$
$47$ $$-392 + 8232 T - 4214 T^{2} + 385 T^{3} + 105 T^{4} - 21 T^{5} + T^{6}$$
$53$ $$6728 + 13224 T + 5786 T^{2} - 487 T^{3} - 153 T^{4} + 4 T^{5} + T^{6}$$
$59$ $$2456 + 15276 T - 14812 T^{2} + 3407 T^{3} - 91 T^{4} - 20 T^{5} + T^{6}$$
$61$ $$7288 - 8336 T + 2286 T^{2} + 227 T^{3} - 111 T^{4} - 3 T^{5} + T^{6}$$
$67$ $$87688 + 26100 T - 22750 T^{2} + 3267 T^{3} + 7 T^{4} - 24 T^{5} + T^{6}$$
$71$ $$203624 - 168468 T + 13574 T^{2} + 3025 T^{3} - 244 T^{4} - 13 T^{5} + T^{6}$$
$73$ $$-14792 - 18276 T - 8344 T^{2} - 1647 T^{3} - 98 T^{4} + 9 T^{5} + T^{6}$$
$79$ $$-169000 + 100100 T - 19720 T^{2} + 791 T^{3} + 215 T^{4} - 28 T^{5} + T^{6}$$
$83$ $$106952 - 50432 T - 2826 T^{2} + 2679 T^{3} - 162 T^{4} - 15 T^{5} + T^{6}$$
$89$ $$1163 - 2277 T - 143 T^{2} + 363 T^{3} - 19 T^{4} - 11 T^{5} + T^{6}$$
$97$ $$-569192 - 5356 T + 26936 T^{2} + 149 T^{3} - 343 T^{4} - 2 T^{5} + T^{6}$$