# Properties

 Label 4030.2.a.n Level 4030 Weight 2 Character orbit 4030.a Self dual Yes Analytic conductor 32.180 Analytic rank 0 Dimension 8 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4030 = 2 \cdot 5 \cdot 13 \cdot 31$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4030.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$32.1797120146$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} -\beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} + ( -\beta_{1} + \beta_{6} ) q^{7} + q^{8} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + q^{2} -\beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} + ( -\beta_{1} + \beta_{6} ) q^{7} + q^{8} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} - q^{10} + ( 1 - \beta_{5} ) q^{11} -\beta_{1} q^{12} - q^{13} + ( -\beta_{1} + \beta_{6} ) q^{14} + \beta_{1} q^{15} + q^{16} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} ) q^{17} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{18} + \beta_{2} q^{19} - q^{20} + ( 2 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{21} + ( 1 - \beta_{5} ) q^{22} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} ) q^{23} -\beta_{1} q^{24} + q^{25} - q^{26} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{27} + ( -\beta_{1} + \beta_{6} ) q^{28} + ( 2 - \beta_{2} - \beta_{3} ) q^{29} + \beta_{1} q^{30} - q^{31} + q^{32} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} ) q^{33} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} ) q^{34} + ( \beta_{1} - \beta_{6} ) q^{35} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{36} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{37} + \beta_{2} q^{38} + \beta_{1} q^{39} - q^{40} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{41} + ( 2 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{42} + ( 3 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{43} + ( 1 - \beta_{5} ) q^{44} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{45} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} ) q^{46} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{47} -\beta_{1} q^{48} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{49} + q^{50} + ( 1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{51} - q^{52} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{53} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{54} + ( -1 + \beta_{5} ) q^{55} + ( -\beta_{1} + \beta_{6} ) q^{56} + ( -1 + \beta_{2} - \beta_{5} + \beta_{7} ) q^{57} + ( 2 - \beta_{2} - \beta_{3} ) q^{58} + ( 4 + 2 \beta_{1} - 3 \beta_{2} + \beta_{5} - 2 \beta_{6} ) q^{59} + \beta_{1} q^{60} + ( -2 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{61} - q^{62} + ( 3 - 3 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{63} + q^{64} + q^{65} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} ) q^{66} + ( 3 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{67} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} ) q^{68} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{69} + ( \beta_{1} - \beta_{6} ) q^{70} + ( 1 + \beta_{1} + 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{71} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{72} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{73} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{74} -\beta_{1} q^{75} + \beta_{2} q^{76} + ( -3 + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{77} + \beta_{1} q^{78} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{79} - q^{80} + ( 4 + \beta_{1} - 4 \beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{81} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{82} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{83} + ( 2 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{84} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} ) q^{85} + ( 3 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{86} + ( 1 - 2 \beta_{2} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{87} + ( 1 - \beta_{5} ) q^{88} + ( 3 - 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{89} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{90} + ( \beta_{1} - \beta_{6} ) q^{91} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} ) q^{92} + \beta_{1} q^{93} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{94} -\beta_{2} q^{95} -\beta_{1} q^{96} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{97} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{98} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{2} - q^{3} + 8q^{4} - 8q^{5} - q^{6} + q^{7} + 8q^{8} + 9q^{9} + O(q^{10})$$ $$8q + 8q^{2} - q^{3} + 8q^{4} - 8q^{5} - q^{6} + q^{7} + 8q^{8} + 9q^{9} - 8q^{10} + 4q^{11} - q^{12} - 8q^{13} + q^{14} + q^{15} + 8q^{16} - 5q^{17} + 9q^{18} + 2q^{19} - 8q^{20} + 17q^{21} + 4q^{22} + 4q^{23} - q^{24} + 8q^{25} - 8q^{26} + 11q^{27} + q^{28} + 11q^{29} + q^{30} - 8q^{31} + 8q^{32} + 10q^{33} - 5q^{34} - q^{35} + 9q^{36} + 19q^{37} + 2q^{38} + q^{39} - 8q^{40} + 10q^{41} + 17q^{42} + 19q^{43} + 4q^{44} - 9q^{45} + 4q^{46} + 11q^{47} - q^{48} + 11q^{49} + 8q^{50} + 7q^{51} - 8q^{52} + 8q^{53} + 11q^{54} - 4q^{55} + q^{56} - 11q^{57} + 11q^{58} + 28q^{59} + q^{60} - 12q^{61} - 8q^{62} + 20q^{63} + 8q^{64} + 8q^{65} + 10q^{66} + 24q^{67} - 5q^{68} + 30q^{69} - q^{70} + 18q^{71} + 9q^{72} - 3q^{73} + 19q^{74} - q^{75} + 2q^{76} - 7q^{77} + q^{78} + 22q^{79} - 8q^{80} + 24q^{81} + 10q^{82} + 17q^{83} + 17q^{84} + 5q^{85} + 19q^{86} + 11q^{87} + 4q^{88} + 17q^{89} - 9q^{90} - q^{91} + 4q^{92} + q^{93} + 11q^{94} - 2q^{95} - q^{96} - 24q^{97} + 11q^{98} + 23q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 16 x^{6} + 18 x^{5} + 64 x^{4} - 84 x^{3} - 19 x^{2} + 22 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{7} + 7 \nu^{6} - 44 \nu^{5} - 112 \nu^{4} + 186 \nu^{3} + 426 \nu^{2} - 319 \nu - 108$$$$)/58$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} + 5 \nu^{6} + 39 \nu^{5} - 80 \nu^{4} - 211 \nu^{3} + 325 \nu^{2} + 203 \nu - 102$$$$)/29$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{7} - 3 \nu^{6} - 122 \nu^{5} + 48 \nu^{4} + 608 \nu^{3} - 166 \nu^{2} - 725 \nu - 136$$$$)/58$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{7} + 20 \nu^{6} + 127 \nu^{5} - 320 \nu^{4} - 467 \nu^{3} + 1271 \nu^{2} - 203 \nu - 205$$$$)/29$$ $$\beta_{6}$$ $$=$$ $$($$$$-25 \nu^{7} + 19 \nu^{6} + 386 \nu^{5} - 362 \nu^{4} - 1434 \nu^{3} + 1786 \nu^{2} + 145 \nu - 318$$$$)/58$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + 16 \nu^{5} - 18 \nu^{4} - 62 \nu^{3} + 82 \nu^{2} + 3 \nu - 8$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} - \beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} + 8 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - 2 \beta_{6} + \beta_{5} + 10 \beta_{4} + 9 \beta_{3} - 13 \beta_{2} + \beta_{1} + 31$$ $$\nu^{5}$$ $$=$$ $$13 \beta_{7} - 13 \beta_{6} - \beta_{5} + 12 \beta_{4} + 3 \beta_{3} - 12 \beta_{2} + 69 \beta_{1} - 10$$ $$\nu^{6}$$ $$=$$ $$12 \beta_{7} - 28 \beta_{6} + 17 \beta_{5} + 94 \beta_{4} + 81 \beta_{3} - 138 \beta_{2} + 20 \beta_{1} + 263$$ $$\nu^{7}$$ $$=$$ $$138 \beta_{7} - 138 \beta_{6} - 17 \beta_{5} + 126 \beta_{4} + 49 \beta_{3} - 116 \beta_{2} + 613 \beta_{1} - 73$$

## 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
 3.08271 2.04806 1.54913 0.635022 −0.178838 −0.447568 −2.63528 −3.05324
1.00000 −3.08271 1.00000 −1.00000 −3.08271 −2.62527 1.00000 6.50312 −1.00000
1.2 1.00000 −2.04806 1.00000 −1.00000 −2.04806 3.38208 1.00000 1.19454 −1.00000
1.3 1.00000 −1.54913 1.00000 −1.00000 −1.54913 −2.44745 1.00000 −0.600186 −1.00000
1.4 1.00000 −0.635022 1.00000 −1.00000 −0.635022 1.23180 1.00000 −2.59675 −1.00000
1.5 1.00000 0.178838 1.00000 −1.00000 0.178838 −4.63233 1.00000 −2.96802 −1.00000
1.6 1.00000 0.447568 1.00000 −1.00000 0.447568 1.86515 1.00000 −2.79968 −1.00000
1.7 1.00000 2.63528 1.00000 −1.00000 2.63528 0.203142 1.00000 3.94469 −1.00000
1.8 1.00000 3.05324 1.00000 −1.00000 3.05324 4.02288 1.00000 6.32229 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$
$$31$$ $$1$$

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4030))$$.