# Properties

 Label 4005.2.a.n Level 4005 Weight 2 Character orbit 4005.a Self dual yes Analytic conductor 31.980 Analytic rank 0 Dimension 6 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: $$6$$ Coefficient field: 6.6.10407557.1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1335) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{2} + ( 2 + \beta_{2} + \beta_{3} ) q^{4} + q^{5} -\beta_{4} q^{7} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{2} + ( 2 + \beta_{2} + \beta_{3} ) q^{4} + q^{5} -\beta_{4} q^{7} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{8} + ( 1 + \beta_{2} ) q^{10} + \beta_{1} q^{13} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{14} + ( 2 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{16} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + 2 q^{19} + ( 2 + \beta_{2} + \beta_{3} ) q^{20} + ( 3 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{23} + q^{25} + \beta_{4} q^{26} + ( \beta_{1} - 3 \beta_{4} + 2 \beta_{5} ) q^{28} + ( \beta_{1} - \beta_{5} ) q^{29} + ( 2 + 2 \beta_{4} ) q^{31} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{32} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{34} -\beta_{4} q^{35} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} + ( 2 + 2 \beta_{2} ) q^{38} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{40} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{41} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{43} + ( 4 \beta_{2} - 2 \beta_{4} ) q^{46} + ( 2 - 2 \beta_{3} + \beta_{4} ) q^{47} + ( 1 + \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{49} + ( 1 + \beta_{2} ) q^{50} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{52} + ( 4 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{53} + ( 3 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{56} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{58} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{59} + ( 2 - 2 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{62} + ( 1 + 6 \beta_{2} + 4 \beta_{4} - 3 \beta_{5} ) q^{64} + \beta_{1} q^{65} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{67} + ( 8 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{68} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{70} + ( 2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{71} + ( -1 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{74} + ( 4 + 2 \beta_{2} + 2 \beta_{3} ) q^{76} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{79} + ( 2 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{80} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{82} + ( 3 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{83} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{85} + ( 3 + \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} ) q^{86} + q^{89} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{91} + ( 8 + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{92} + ( 3 - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{94} + 2 q^{95} + ( -5 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{97} + ( 5 + 2 \beta_{1} - 3 \beta_{2} - 5 \beta_{4} + 4 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 4q^{2} + 8q^{4} + 6q^{5} + q^{7} + 9q^{8} + O(q^{10})$$ $$6q + 4q^{2} + 8q^{4} + 6q^{5} + q^{7} + 9q^{8} + 4q^{10} - 3q^{13} + 5q^{14} + 12q^{16} + 13q^{17} + 12q^{19} + 8q^{20} + 19q^{23} + 6q^{25} - q^{26} - 6q^{28} + 10q^{31} + 17q^{32} - 2q^{34} + q^{35} - q^{37} + 8q^{38} + 9q^{40} + 4q^{41} - 7q^{43} - 6q^{46} + 15q^{47} - q^{49} + 4q^{50} + q^{52} + 27q^{53} + 14q^{56} - 6q^{58} + 4q^{59} + 8q^{61} - 2q^{62} - q^{64} - 3q^{65} - 11q^{67} + 47q^{68} + 5q^{70} + 16q^{71} + q^{73} + 10q^{74} + 16q^{76} + 12q^{80} + q^{82} + 17q^{83} + 13q^{85} + 20q^{86} + 6q^{89} - 18q^{91} + 36q^{92} + 17q^{94} + 12q^{95} - 29q^{97} + 23q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 5 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 6 \nu^{3} + 7 \nu^{2} + 10 \nu$$ $$\beta_{5}$$ $$=$$ $$2 \nu^{5} - 4 \nu^{4} - 11 \nu^{3} + 14 \nu^{2} + 18 \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + 3 \beta_{2} - \beta_{1} + 12$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{5} - 6 \beta_{4} - \beta_{2} - \beta_{1} + 6$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{5} - 22 \beta_{4} + 4 \beta_{3} + 13 \beta_{2} - 3 \beta_{1} + 60$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$41 \beta_{5} - 78 \beta_{4} + 8 \beta_{3} + 3 \beta_{2} - \beta_{1} + 108$$$$)/4$$

## 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
 0.191235 −0.763968 2.09854 −1.43605 2.63450 −1.72426
−2.15466 0 2.64258 1.00000 0 −2.12396 −1.38454 0 −2.15466
1.2 −0.652386 0 −1.57439 1.00000 0 1.82035 2.33188 0 −0.652386
1.3 0.305330 0 −1.90677 1.00000 0 1.72660 −1.19286 0 0.305330
1.4 1.49828 0 0.244835 1.00000 0 −3.23110 −2.62972 0 1.49828
1.5 2.30610 0 3.31809 1.00000 0 4.21574 3.03965 0 2.30610
1.6 2.69734 0 5.27566 1.00000 0 −1.40763 8.83558 0 2.69734
 $$n$$: 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

## 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 4005.2.a.n 6
3.b odd 2 1 1335.2.a.g 6
15.d odd 2 1 6675.2.a.u 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1335.2.a.g 6 3.b odd 2 1
4005.2.a.n 6 1.a even 1 1 trivial
6675.2.a.u 6 15.d odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$89$$ $$-1$$

## Hecke kernels

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

 $$T_{2}^{6} - 4 T_{2}^{5} - 2 T_{2}^{4} + 21 T_{2}^{3} - 13 T_{2}^{2} - 11 T_{2} + 4$$ $$T_{7}^{6} - T_{7}^{5} - 20 T_{7}^{4} + 7 T_{7}^{3} + 96 T_{7}^{2} - 16 T_{7} - 128$$ $$T_{11}$$