# Properties

 Label 4005.2.a.r Level 4005 Weight 2 Character orbit 4005.a Self dual Yes Analytic conductor 31.980 Analytic rank 0 Dimension 10 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: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{3} ) q^{8} + ( 1 - \beta_{1} ) q^{10} + ( 1 + \beta_{2} + \beta_{5} - \beta_{9} ) q^{11} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{13} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} - \beta_{9} ) q^{14} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{16} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( 1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} ) q^{20} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{22} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{23} + q^{25} + ( 2 - 2 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{26} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{28} + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{31} + ( -3 + 4 \beta_{1} - \beta_{2} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{32} + ( -3 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{34} + ( -1 - \beta_{2} - \beta_{3} - \beta_{7} ) q^{35} + ( -2 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{37} + ( -4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{38} + ( 2 - \beta_{1} - \beta_{3} ) q^{40} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{41} + ( -\beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{43} + ( 3 - 2 \beta_{1} + 5 \beta_{2} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{44} + ( 1 - 4 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{46} + ( -2 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{47} + ( 4 + 2 \beta_{1} + \beta_{3} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{49} + ( -1 + \beta_{1} ) q^{50} + ( -5 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{52} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{53} + ( -1 - \beta_{2} - \beta_{5} + \beta_{9} ) q^{55} + ( 5 - \beta_{1} + \beta_{3} + \beta_{4} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{56} + ( -5 + 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{58} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{59} + ( \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{61} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{62} + ( 1 - 4 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{9} ) q^{64} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{65} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{9} ) q^{67} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} + \beta_{7} - \beta_{9} ) q^{68} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{70} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{71} + ( -\beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{73} + ( 2 - 4 \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{74} + ( -1 + 8 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 7 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} ) q^{76} + ( 4 + 2 \beta_{2} + \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{77} + ( 2 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{79} + ( -3 + 3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{80} + ( 1 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{8} + 3 \beta_{9} ) q^{82} + ( 2 - \beta_{1} + 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{83} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{85} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{86} + ( 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{88} - q^{89} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{91} + ( -2 + 5 \beta_{1} - 2 \beta_{2} - \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{92} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{94} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{95} + ( 3 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{97} + ( 4 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 6q^{2} + 14q^{4} - 10q^{5} + 7q^{7} - 15q^{8} + O(q^{10})$$ $$10q - 6q^{2} + 14q^{4} - 10q^{5} + 7q^{7} - 15q^{8} + 6q^{10} + 10q^{11} + 7q^{13} + 7q^{14} + 22q^{16} - 11q^{17} + 10q^{19} - 14q^{20} + 8q^{22} - 6q^{23} + 10q^{25} + 14q^{26} + 16q^{28} + 3q^{29} + 12q^{31} - 21q^{32} - 6q^{34} - 7q^{35} - 19q^{37} - 6q^{38} + 15q^{40} - 13q^{41} + 9q^{43} + 26q^{44} + 8q^{46} - 21q^{47} + 53q^{49} - 6q^{50} - 43q^{52} - 7q^{53} - 10q^{55} + 53q^{56} - 42q^{58} + 19q^{59} + 4q^{61} + 28q^{62} + 5q^{64} - 7q^{65} - 6q^{67} - 2q^{68} - 7q^{70} + 6q^{71} + 6q^{73} - 2q^{76} + 40q^{77} + 25q^{79} - 22q^{80} + q^{82} + 22q^{83} + 11q^{85} + 2q^{86} + 20q^{88} - 10q^{89} - 10q^{91} - 10q^{92} + 25q^{94} - 10q^{95} + 40q^{97} + 56q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4 x^{9} - 8 x^{8} + 35 x^{7} + 29 x^{6} - 103 x^{5} - 57 x^{4} + 106 x^{3} + 29 x^{2} - 39 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu^{2} - 2 \nu + 5$$ $$\beta_{4}$$ $$=$$ $$\nu^{9} - 5 \nu^{8} - 2 \nu^{7} + 32 \nu^{6} - 6 \nu^{5} - 62 \nu^{4} + 6 \nu^{3} + 23 \nu^{2} - 5 \nu + 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{8} - 5 \nu^{7} - 2 \nu^{6} + 32 \nu^{5} - 5 \nu^{4} - 66 \nu^{3} + 4 \nu^{2} + 36 \nu - 3$$ $$\beta_{6}$$ $$=$$ $$-\nu^{9} + 4 \nu^{8} + 7 \nu^{7} - 30 \nu^{6} - 27 \nu^{5} + 71 \nu^{4} + 62 \nu^{3} - 41 \nu^{2} - 31 \nu + 7$$ $$\beta_{7}$$ $$=$$ $$\nu^{9} - 3 \nu^{8} - 12 \nu^{7} + 28 \nu^{6} + 58 \nu^{5} - 73 \nu^{4} - 122 \nu^{3} + 32 \nu^{2} + 55 \nu - 3$$ $$\beta_{8}$$ $$=$$ $$\nu^{9} - 2 \nu^{8} - 16 \nu^{7} + 21 \nu^{6} + 90 \nu^{5} - 54 \nu^{4} - 199 \nu^{3} + 6 \nu^{2} + 99 \nu - 3$$ $$\beta_{9}$$ $$=$$ $$2 \nu^{9} - 8 \nu^{8} - 14 \nu^{7} + 61 \nu^{6} + 49 \nu^{5} - 142 \nu^{4} - 101 \nu^{3} + 72 \nu^{2} + 39 \nu - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3 \beta_{2} + 5 \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 13 \beta_{2} + 9 \beta_{1} + 20$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{7} - \beta_{6} + 7 \beta_{5} + 3 \beta_{4} + 18 \beta_{3} + 44 \beta_{2} + 32 \beta_{1} + 52$$ $$\nu^{6}$$ $$=$$ $$\beta_{9} - 20 \beta_{7} - 3 \beta_{6} + 35 \beta_{5} + 15 \beta_{4} + 67 \beta_{3} + 161 \beta_{2} + 78 \beta_{1} + 189$$ $$\nu^{7}$$ $$=$$ $$5 \beta_{9} + \beta_{8} - 77 \beta_{7} - 15 \beta_{6} + 126 \beta_{5} + 51 \beta_{4} + 250 \beta_{3} + 556 \beta_{2} + 251 \beta_{1} + 596$$ $$\nu^{8}$$ $$=$$ $$27 \beta_{9} + 5 \beta_{8} - 302 \beta_{7} - 49 \beta_{6} + 487 \beta_{5} + 194 \beta_{4} + 894 \beta_{3} + 1953 \beta_{2} + 722 \beta_{1} + 2049$$ $$\nu^{9}$$ $$=$$ $$113 \beta_{9} + 27 \beta_{8} - 1110 \beta_{7} - 185 \beta_{6} + 1733 \beta_{5} + 673 \beta_{4} + 3176 \beta_{3} + 6754 \beta_{2} + 2318 \beta_{1} + 6846$$

## 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
 −1.71694 −1.53398 −1.32759 −1.05100 0.0538492 0.704585 0.770724 2.21375 2.43743 3.44917
−2.71694 0 5.38176 −1.00000 0 0.704196 −9.18805 0 2.71694
1.2 −2.53398 0 4.42108 −1.00000 0 −3.60443 −6.13497 0 2.53398
1.3 −2.32759 0 3.41766 −1.00000 0 −0.231668 −3.29972 0 2.32759
1.4 −2.05100 0 2.20661 −1.00000 0 4.21605 −0.423758 0 2.05100
1.5 −0.946151 0 −1.10480 −1.00000 0 2.86767 2.93761 0 0.946151
1.6 −0.295415 0 −1.91273 −1.00000 0 3.54012 1.15588 0 0.295415
1.7 −0.229276 0 −1.94743 −1.00000 0 −3.83075 0.905050 0 0.229276
1.8 1.21375 0 −0.526821 −1.00000 0 −4.72892 −3.06692 0 −1.21375
1.9 1.43743 0 0.0662179 −1.00000 0 3.21347 −2.77969 0 −1.43743
1.10 2.44917 0 3.99846 −1.00000 0 4.85425 4.89457 0 −2.44917
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4005))$$:

 $$T_{2}^{10} + \cdots$$ $$T_{7}^{10} - \cdots$$ $$T_{11}^{10} - \cdots$$