# Properties

 Label 8028.2.a.j Level 8028 Weight 2 Character orbit 8028.a Self dual Yes Analytic conductor 64.104 Analytic rank 0 Dimension 7 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8028 = 2^{2} \cdot 3^{2} \cdot 223$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8028.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$64.1039027427$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 1 + \beta_{1} + \beta_{6} ) q^{5}$$ $$+ ( \beta_{3} + \beta_{6} ) q^{7}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 1 + \beta_{1} + \beta_{6} ) q^{5}$$ $$+ ( \beta_{3} + \beta_{6} ) q^{7}$$ $$+ ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{11}$$ $$+ ( -1 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{13}$$ $$+ ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{17}$$ $$+ ( -1 + \beta_{2} + \beta_{6} ) q^{19}$$ $$+ ( 2 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{23}$$ $$+ ( 1 + \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{25}$$ $$+ ( 3 - \beta_{2} - \beta_{5} ) q^{29}$$ $$+ ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{31}$$ $$+ ( 3 - 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{35}$$ $$+ ( -2 + \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{37}$$ $$+ ( -1 + \beta_{1} + \beta_{3} + \beta_{6} ) q^{41}$$ $$+ ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{43}$$ $$+ ( 3 - \beta_{2} + 2 \beta_{5} ) q^{47}$$ $$+ ( 1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{49}$$ $$+ ( 6 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{53}$$ $$+ ( 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{55}$$ $$+ ( 4 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{59}$$ $$+ ( -3 + \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{61}$$ $$+ ( -4 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{65}$$ $$+ ( -3 + \beta_{3} - \beta_{5} ) q^{67}$$ $$+ ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{71}$$ $$+ ( 4 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{73}$$ $$+ ( 1 + \beta_{2} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{77}$$ $$+ ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{79}$$ $$+ ( 6 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{83}$$ $$+ ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{85}$$ $$+ ( 4 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{89}$$ $$+ ( \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{91}$$ $$+ ( 1 - 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{95}$$ $$+ ( 5 + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{97}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q$$ $$\mathstrut +\mathstrut 7q^{5}$$ $$\mathstrut -\mathstrut q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$7q$$ $$\mathstrut +\mathstrut 7q^{5}$$ $$\mathstrut -\mathstrut q^{7}$$ $$\mathstrut +\mathstrut 12q^{11}$$ $$\mathstrut -\mathstrut 9q^{13}$$ $$\mathstrut -\mathstrut 8q^{17}$$ $$\mathstrut -\mathstrut 12q^{19}$$ $$\mathstrut +\mathstrut 12q^{23}$$ $$\mathstrut +\mathstrut 12q^{25}$$ $$\mathstrut +\mathstrut 24q^{29}$$ $$\mathstrut +\mathstrut q^{31}$$ $$\mathstrut +\mathstrut 15q^{35}$$ $$\mathstrut -\mathstrut 13q^{37}$$ $$\mathstrut -\mathstrut 5q^{41}$$ $$\mathstrut -\mathstrut 13q^{43}$$ $$\mathstrut +\mathstrut 21q^{47}$$ $$\mathstrut +\mathstrut 4q^{49}$$ $$\mathstrut +\mathstrut 35q^{53}$$ $$\mathstrut +\mathstrut q^{55}$$ $$\mathstrut +\mathstrut 23q^{59}$$ $$\mathstrut -\mathstrut 17q^{61}$$ $$\mathstrut -\mathstrut 18q^{67}$$ $$\mathstrut +\mathstrut 4q^{71}$$ $$\mathstrut +\mathstrut 23q^{73}$$ $$\mathstrut -\mathstrut 3q^{77}$$ $$\mathstrut +\mathstrut 4q^{79}$$ $$\mathstrut +\mathstrut 44q^{83}$$ $$\mathstrut +\mathstrut 20q^{85}$$ $$\mathstrut +\mathstrut 2q^{91}$$ $$\mathstrut +\mathstrut 12q^{95}$$ $$\mathstrut +\mathstrut 32q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7}\mathstrut -\mathstrut$$ $$3$$ $$x^{6}\mathstrut -\mathstrut$$ $$11$$ $$x^{5}\mathstrut +\mathstrut$$ $$24$$ $$x^{4}\mathstrut +\mathstrut$$ $$38$$ $$x^{3}\mathstrut -\mathstrut$$ $$46$$ $$x^{2}\mathstrut -\mathstrut$$ $$36$$ $$x\mathstrut +\mathstrut$$ $$9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 4 \nu^{5} - 3 \nu^{4} + 15 \nu^{3} - 5 \nu^{2} + 11 \nu - 3$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{5} - 13 \nu^{4} + 15 \nu^{3} + 33 \nu^{2} - 13 \nu + 15$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 3 \nu^{4} - 10 \nu^{3} + 19 \nu^{2} + 26 \nu - 15$$$$)/6$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{5} - 13 \nu^{4} + 15 \nu^{3} + 45 \nu^{2} - 37 \nu - 33$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{5} + 7 \nu^{4} - 29 \nu^{3} - 13 \nu^{2} + 41 \nu + 3$$$$)/6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$4$$ $$\nu^{3}$$ $$=$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$4$$ $$\nu^{4}$$ $$=$$ $$5$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$15$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$7$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$31$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$32$$ $$\nu^{5}$$ $$=$$ $$25$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$56$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$35$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$129$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$75$$ $$\nu^{6}$$ $$=$$ $$100$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$229$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$131$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$74$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$57$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$458$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$359$$

## 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
 −0.828592 −2.20362 2.33527 1.49109 −1.80533 0.206062 3.80512
0 0 0 −3.49299 0 −0.758389 0 0 0
1.2 0 0 0 −0.778137 0 −3.30215 0 0 0
1.3 0 0 0 0.389930 0 −3.58741 0 0 0
1.4 0 0 0 1.18873 0 2.92270 0 0 0
1.5 0 0 0 2.57376 0 2.76633 0 0 0
1.6 0 0 0 2.98220 0 2.92860 0 0 0
1.7 0 0 0 4.13650 0 −1.96968 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$223$$ $$-1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8028))$$:

 $$T_{5}^{7}$$ $$\mathstrut -\mathstrut 7 T_{5}^{6}$$ $$\mathstrut +\mathstrut T_{5}^{5}$$ $$\mathstrut +\mathstrut 83 T_{5}^{4}$$ $$\mathstrut -\mathstrut 171 T_{5}^{3}$$ $$\mathstrut +\mathstrut 30 T_{5}^{2}$$ $$\mathstrut +\mathstrut 112 T_{5}$$ $$\mathstrut -\mathstrut 40$$ $$T_{7}^{7}$$ $$\mathstrut +\mathstrut T_{7}^{6}$$ $$\mathstrut -\mathstrut 26 T_{7}^{5}$$ $$\mathstrut -\mathstrut 20 T_{7}^{4}$$ $$\mathstrut +\mathstrut 218 T_{7}^{3}$$ $$\mathstrut +\mathstrut 141 T_{7}^{2}$$ $$\mathstrut -\mathstrut 571 T_{7}$$ $$\mathstrut -\mathstrut 419$$