# Properties

 Label 4005.2.a.o Level 4005 Weight 2 Character orbit 4005.a Self dual Yes Analytic conductor 31.980 Analytic rank 0 Dimension 7 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4005 = 3^{2} \cdot 5 \cdot 89$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4005.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$31.9800860095$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{3} ) q^{4} - q^{5} + ( -2 - \beta_{6} ) q^{7} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{3} ) q^{4} - q^{5} + ( -2 - \beta_{6} ) q^{7} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{8} + ( -1 + \beta_{1} ) q^{10} + ( 1 - \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( 3 \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{14} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{16} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{17} + ( -3 \beta_{1} - \beta_{2} + \beta_{6} ) q^{19} + ( -2 + \beta_{1} - \beta_{3} ) q^{20} + ( 1 - 2 \beta_{1} - \beta_{5} + \beta_{6} ) q^{22} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} + q^{25} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{26} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{28} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{29} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{31} + ( 6 + \beta_{2} + 5 \beta_{3} + 2 \beta_{4} - 5 \beta_{6} ) q^{32} + ( 5 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{34} + ( 2 + \beta_{6} ) q^{35} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{37} + ( 6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{38} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{40} + ( 2 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{41} + ( -5 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{43} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{44} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{46} + ( 2 + 3 \beta_{2} - \beta_{4} - \beta_{6} ) q^{47} + ( 2 + \beta_{2} + \beta_{3} + 4 \beta_{6} ) q^{49} + ( 1 - \beta_{1} ) q^{50} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{52} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{53} + ( -1 + \beta_{4} + \beta_{5} - \beta_{6} ) q^{55} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{56} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 6 \beta_{6} ) q^{58} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{59} + ( -1 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{6} ) q^{61} + ( -7 + \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + 5 \beta_{6} ) q^{62} + ( 7 - \beta_{1} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - 7 \beta_{6} ) q^{64} + ( 2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{65} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{67} + ( 12 + \beta_{1} + 8 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} - 7 \beta_{6} ) q^{68} + ( -3 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{70} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{71} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} ) q^{73} + ( 5 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{74} + ( 7 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{76} + ( -5 - 2 \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{77} + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{79} + ( -4 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{80} + ( 6 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{82} + ( 8 - 4 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{83} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( -8 + 5 \beta_{1} - 2 \beta_{3} ) q^{86} + ( 4 - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{88} - q^{89} + ( 5 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - \beta_{6} ) q^{91} + ( -1 - 7 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{92} + ( 8 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{94} + ( 3 \beta_{1} + \beta_{2} - \beta_{6} ) q^{95} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{97} + ( -5 - 6 \beta_{1} - 8 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 7 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10})$$ $$7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 8 x^{5} + 6 x^{4} + 19 x^{3} - 10 x^{2} - 12 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 3$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + \nu^{3} + 4 \nu^{2} - 2 \nu - 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 6 \nu^{3} + 5 \nu^{2} + 8 \nu - 5$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - \nu^{5} - 7 \nu^{4} + 5 \nu^{3} + 13 \nu^{2} - 4 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 5 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 6 \beta_{1} + 10$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} + \beta_{5} + 8 \beta_{3} + \beta_{2} + 23 \beta_{1} + 29$$

## 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
 2.26266 −1.96388 1.89340 −1.49803 −1.07810 0.885013 0.498937
−2.11962 0 2.49279 −1.00000 0 −4.83304 −1.04452 0 2.11962
1.2 −0.856822 0 −1.26586 −1.00000 0 −0.580377 2.79826 0 0.856822
1.3 −0.584976 0 −1.65780 −1.00000 0 −2.74591 2.13973 0 0.584976
1.4 0.755898 0 −1.42862 −1.00000 0 0.0498231 −2.59169 0 −0.755898
1.5 1.83770 0 1.37716 −1.00000 0 −4.72699 −1.14460 0 −1.83770
1.6 2.21675 0 2.91399 −1.00000 0 −3.75132 2.02609 0 −2.21675
1.7 2.75106 0 5.56834 −1.00000 0 0.587818 9.81674 0 −2.75106
 $$n$$: 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$89$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4005))$$:

 $$T_{2}^{7} - 4 T_{2}^{6} - 3 T_{2}^{5} + 24 T_{2}^{4} - 8 T_{2}^{3} - 29 T_{2}^{2} + 6 T_{2} + 9$$ $$T_{7}^{7} + 16 T_{7}^{6} + 94 T_{7}^{5} + 236 T_{7}^{4} + 189 T_{7}^{3} - 96 T_{7}^{2} - 76 T_{7} + 4$$ $$T_{11}^{7} - 10 T_{11}^{6} + 14 T_{11}^{5} + 149 T_{11}^{4} - 639 T_{11}^{3} + 968 T_{11}^{2} - 608 T_{11} + 128$$