# Properties

 Label 4005.2.a.q Level 4005 Weight 2 Character orbit 4005.a Self dual yes Analytic conductor 31.980 Analytic rank 1 Dimension 9 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: $$1$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 4 x^{8} - 6 x^{7} + 31 x^{6} + 13 x^{5} - 75 x^{4} - 17 x^{3} + 52 x^{2} + 11 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 4 \nu + 2$$ $$\beta_{4}$$ $$=$$ $$\nu^{6} - 3 \nu^{5} - 4 \nu^{4} + 13 \nu^{3} + 3 \nu^{2} - 12 \nu$$ $$\beta_{5}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} + 4 \nu^{4} - 12 \nu^{3} - 5 \nu^{2} + 8 \nu + 3$$ $$\beta_{6}$$ $$=$$ $$\nu^{8} - 5 \nu^{7} + \nu^{6} + 23 \nu^{5} - 16 \nu^{4} - 28 \nu^{3} + 12 \nu^{2} + 10 \nu + 1$$ $$\beta_{7}$$ $$=$$ $$\nu^{8} - 5 \nu^{7} + 27 \nu^{5} - 15 \nu^{4} - 44 \nu^{3} + 19 \nu^{2} + 22 \nu - 2$$ $$\beta_{8}$$ $$=$$ $$-\nu^{8} + 6 \nu^{7} - 5 \nu^{6} - 25 \nu^{5} + 35 \nu^{4} + 24 \nu^{3} - 35 \nu^{2} - 5 \nu + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + 2 \beta_{2} + 6 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 8 \beta_{2} + 12 \beta_{1} + 16$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} - \beta_{6} + 9 \beta_{5} + 10 \beta_{4} + 3 \beta_{3} + 20 \beta_{2} + 44 \beta_{1} + 30$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{7} - 3 \beta_{6} + 22 \beta_{5} + 26 \beta_{4} + 13 \beta_{3} + 63 \beta_{2} + 111 \beta_{1} + 106$$ $$\nu^{7}$$ $$=$$ $$\beta_{8} + 14 \beta_{7} - 13 \beta_{6} + 72 \beta_{5} + 90 \beta_{4} + 39 \beta_{3} + 171 \beta_{2} + 346 \beta_{1} + 256$$ $$\nu^{8}$$ $$=$$ $$5 \beta_{8} + 44 \beta_{7} - 38 \beta_{6} + 191 \beta_{5} + 254 \beta_{4} + 129 \beta_{3} + 504 \beta_{2} + 945 \beta_{1} + 787$$

## 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
 −1.72546 −1.70432 −1.02867 −0.286244 0.0690386 1.09195 1.89144 2.81142 2.88084
−2.72546 0 5.42813 1.00000 0 −4.89499 −9.34325 0 −2.72546
1.2 −2.70432 0 5.31333 1.00000 0 3.91511 −8.96030 0 −2.70432
1.3 −2.02867 0 2.11549 1.00000 0 2.64395 −0.234298 0 −2.02867
1.4 −1.28624 0 −0.345575 1.00000 0 −0.646623 3.01698 0 −1.28624
1.5 −0.930961 0 −1.13331 1.00000 0 −1.89313 2.91699 0 −0.930961
1.6 0.0919478 0 −1.99155 1.00000 0 −4.79397 −0.367014 0 0.0919478
1.7 0.891444 0 −1.20533 1.00000 0 3.64096 −2.85737 0 0.891444
1.8 1.81142 0 1.28123 1.00000 0 −0.549078 −1.30199 0 1.81142
1.9 1.88084 0 1.53757 1.00000 0 −0.422231 −0.869761 0 1.88084
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 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.q 9
3.b odd 2 1 1335.2.a.h 9
15.d odd 2 1 6675.2.a.x 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1335.2.a.h 9 3.b odd 2 1
4005.2.a.q 9 1.a even 1 1 trivial
6675.2.a.x 9 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4005))$$:

 $$T_{2}^{9} + \cdots$$ $$T_{7}^{9} + \cdots$$ $$T_{11}^{9} + \cdots$$