# Properties

 Label 8040.2.a.t Level 8040 Weight 2 Character orbit 8040.a Self dual Yes Analytic conductor 64.200 Analytic rank 0 Dimension 7 CM No Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8040.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$64.1997232251$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ q^{3}$$ $$- q^{5}$$ $$+ ( 1 + \beta_{3} ) q^{7}$$ $$+ q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ q^{3}$$ $$- q^{5}$$ $$+ ( 1 + \beta_{3} ) q^{7}$$ $$+ q^{9}$$ $$+ ( -\beta_{1} - \beta_{6} ) q^{11}$$ $$+ ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{13}$$ $$- q^{15}$$ $$-\beta_{2} q^{17}$$ $$+ ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{19}$$ $$+ ( 1 + \beta_{3} ) q^{21}$$ $$+ ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{23}$$ $$+ q^{25}$$ $$+ q^{27}$$ $$+ ( -\beta_{5} - \beta_{6} ) q^{29}$$ $$+ ( 1 - \beta_{4} ) q^{31}$$ $$+ ( -\beta_{1} - \beta_{6} ) q^{33}$$ $$+ ( -1 - \beta_{3} ) q^{35}$$ $$+ ( 2 + \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{37}$$ $$+ ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{39}$$ $$+ ( 1 + \beta_{4} ) q^{41}$$ $$+ ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{43}$$ $$- q^{45}$$ $$+ ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{47}$$ $$+ ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{49}$$ $$-\beta_{2} q^{51}$$ $$+ ( 3 - \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{53}$$ $$+ ( \beta_{1} + \beta_{6} ) q^{55}$$ $$+ ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{57}$$ $$+ ( -\beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{59}$$ $$+ ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{61}$$ $$+ ( 1 + \beta_{3} ) q^{63}$$ $$+ ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{65}$$ $$+ q^{67}$$ $$+ ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{69}$$ $$+ ( 2 \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{71}$$ $$+ ( 1 + 3 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{73}$$ $$+ q^{75}$$ $$+ ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{77}$$ $$+ ( 4 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{79}$$ $$+ q^{81}$$ $$+ ( -5 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{83}$$ $$+ \beta_{2} q^{85}$$ $$+ ( -\beta_{5} - \beta_{6} ) q^{87}$$ $$+ ( 2 - 4 \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{89}$$ $$+ ( 5 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 4 \beta_{6} ) q^{91}$$ $$+ ( 1 - \beta_{4} ) q^{93}$$ $$+ ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{95}$$ $$+ ( 6 + \beta_{3} - 4 \beta_{5} - \beta_{6} ) q^{97}$$ $$+ ( -\beta_{1} - \beta_{6} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q$$ $$\mathstrut +\mathstrut 7q^{3}$$ $$\mathstrut -\mathstrut 7q^{5}$$ $$\mathstrut +\mathstrut 10q^{7}$$ $$\mathstrut +\mathstrut 7q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$7q$$ $$\mathstrut +\mathstrut 7q^{3}$$ $$\mathstrut -\mathstrut 7q^{5}$$ $$\mathstrut +\mathstrut 10q^{7}$$ $$\mathstrut +\mathstrut 7q^{9}$$ $$\mathstrut -\mathstrut q^{13}$$ $$\mathstrut -\mathstrut 7q^{15}$$ $$\mathstrut -\mathstrut 2q^{17}$$ $$\mathstrut +\mathstrut 9q^{19}$$ $$\mathstrut +\mathstrut 10q^{21}$$ $$\mathstrut +\mathstrut 2q^{23}$$ $$\mathstrut +\mathstrut 7q^{25}$$ $$\mathstrut +\mathstrut 7q^{27}$$ $$\mathstrut -\mathstrut q^{29}$$ $$\mathstrut +\mathstrut 9q^{31}$$ $$\mathstrut -\mathstrut 10q^{35}$$ $$\mathstrut +\mathstrut 23q^{37}$$ $$\mathstrut -\mathstrut q^{39}$$ $$\mathstrut +\mathstrut 5q^{41}$$ $$\mathstrut -\mathstrut 3q^{43}$$ $$\mathstrut -\mathstrut 7q^{45}$$ $$\mathstrut +\mathstrut 11q^{47}$$ $$\mathstrut +\mathstrut 13q^{49}$$ $$\mathstrut -\mathstrut 2q^{51}$$ $$\mathstrut +\mathstrut 13q^{53}$$ $$\mathstrut +\mathstrut 9q^{57}$$ $$\mathstrut +\mathstrut q^{59}$$ $$\mathstrut +\mathstrut 4q^{61}$$ $$\mathstrut +\mathstrut 10q^{63}$$ $$\mathstrut +\mathstrut q^{65}$$ $$\mathstrut +\mathstrut 7q^{67}$$ $$\mathstrut +\mathstrut 2q^{69}$$ $$\mathstrut +\mathstrut q^{71}$$ $$\mathstrut +\mathstrut 14q^{73}$$ $$\mathstrut +\mathstrut 7q^{75}$$ $$\mathstrut +\mathstrut 18q^{77}$$ $$\mathstrut +\mathstrut 25q^{79}$$ $$\mathstrut +\mathstrut 7q^{81}$$ $$\mathstrut -\mathstrut 29q^{83}$$ $$\mathstrut +\mathstrut 2q^{85}$$ $$\mathstrut -\mathstrut q^{87}$$ $$\mathstrut +\mathstrut 7q^{89}$$ $$\mathstrut +\mathstrut 27q^{91}$$ $$\mathstrut +\mathstrut 9q^{93}$$ $$\mathstrut -\mathstrut 9q^{95}$$ $$\mathstrut +\mathstrut 38q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7}\mathstrut -\mathstrut$$ $$x^{6}\mathstrut -\mathstrut$$ $$15$$ $$x^{5}\mathstrut +\mathstrut$$ $$3$$ $$x^{4}\mathstrut +\mathstrut$$ $$43$$ $$x^{3}\mathstrut -\mathstrut$$ $$6$$ $$x^{2}\mathstrut -\mathstrut$$ $$29$$ $$x\mathstrut +\mathstrut$$ $$6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$6 \nu^{6} - 4 \nu^{5} - 73 \nu^{4} - 43 \nu^{3} + 42 \nu^{2} + 143 \nu + 112$$$$)/55$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{6} - 23 \nu^{5} - 76 \nu^{4} + 234 \nu^{3} + 104 \nu^{2} - 484 \nu + 39$$$$)/55$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{6} - 29 \nu^{5} - 213 \nu^{4} + 197 \nu^{3} + 497 \nu^{2} - 242 \nu - 123$$$$)/55$$ $$\beta_{6}$$ $$=$$ $$($$$$-24 \nu^{6} + 16 \nu^{5} + 347 \nu^{4} + 62 \nu^{3} - 773 \nu^{2} - 132 \nu + 267$$$$)/55$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$4$$ $$\nu^{3}$$ $$=$$ $$-$$$$\beta_{6}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$5$$ $$\nu^{4}$$ $$=$$ $$-$$$$\beta_{6}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$23$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$41$$ $$\nu^{5}$$ $$=$$ $$-$$$$16$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$33$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$30$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$25$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$134$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$105$$ $$\nu^{6}$$ $$=$$ $$-$$$$30$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$85$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$83$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$19$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$175$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$410$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$558$$

## 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
 −1.70642 1.51692 3.95767 −1.05266 0.211675 −2.84959 0.922407
0 1.00000 0 −1.00000 0 −2.80016 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −0.274078 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 −0.212435 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 0.670274 0 1.00000 0
1.5 0 1.00000 0 −1.00000 0 3.61083 0 1.00000 0
1.6 0 1.00000 0 −1.00000 0 4.47701 0 1.00000 0
1.7 0 1.00000 0 −1.00000 0 4.52856 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$67$$ $$-1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8040))$$:

 $$T_{7}^{7}$$ $$\mathstrut -\mathstrut 10 T_{7}^{6}$$ $$\mathstrut +\mathstrut 19 T_{7}^{5}$$ $$\mathstrut +\mathstrut 74 T_{7}^{4}$$ $$\mathstrut -\mathstrut 223 T_{7}^{3}$$ $$\mathstrut +\mathstrut 17 T_{7}^{2}$$ $$\mathstrut +\mathstrut 52 T_{7}$$ $$\mathstrut +\mathstrut 8$$ $$T_{11}^{7}$$ $$\mathstrut -\mathstrut 41 T_{11}^{5}$$ $$\mathstrut +\mathstrut 18 T_{11}^{4}$$ $$\mathstrut +\mathstrut 525 T_{11}^{3}$$ $$\mathstrut -\mathstrut 377 T_{11}^{2}$$ $$\mathstrut -\mathstrut 2000 T_{11}$$ $$\mathstrut +\mathstrut 1530$$ $$T_{13}^{7}$$ $$\mathstrut +\mathstrut T_{13}^{6}$$ $$\mathstrut -\mathstrut 55 T_{13}^{5}$$ $$\mathstrut -\mathstrut 69 T_{13}^{4}$$ $$\mathstrut +\mathstrut 971 T_{13}^{3}$$ $$\mathstrut +\mathstrut 1494 T_{13}^{2}$$ $$\mathstrut -\mathstrut 5580 T_{13}$$ $$\mathstrut -\mathstrut 10152$$ $$T_{17}^{7}$$ $$\mathstrut +\mathstrut 2 T_{17}^{6}$$ $$\mathstrut -\mathstrut 76 T_{17}^{5}$$ $$\mathstrut +\mathstrut 51 T_{17}^{4}$$ $$\mathstrut +\mathstrut 1653 T_{17}^{3}$$ $$\mathstrut -\mathstrut 4766 T_{17}^{2}$$ $$\mathstrut +\mathstrut 1834 T_{17}$$ $$\mathstrut +\mathstrut 3332$$