# Properties

 Label 4016.2.a.g Level 4016 Weight 2 Character orbit 4016.a Self dual yes Analytic conductor 32.068 Analytic rank 0 Dimension 7 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4016 = 2^{4} \cdot 251$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4016.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0679214517$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1004) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{5} q^{5} + ( 1 + \beta_{6} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{5} q^{5} + ( 1 + \beta_{6} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{9} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{11} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{15} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{17} + ( 2 + \beta_{2} + \beta_{4} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{23} + ( -1 - \beta_{3} ) q^{25} + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{27} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{29} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{31} + ( \beta_{1} + 2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{33} + ( 2 - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{35} + ( -\beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{37} + ( 2 + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{39} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} ) q^{41} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{43} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{45} + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{47} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{49} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{51} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{53} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{57} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{61} + ( -1 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{63} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{65} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{67} + ( 2 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{69} + ( 3 - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{71} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{73} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} ) q^{77} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{79} + ( 1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + \beta_{6} ) q^{81} + ( -2 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{83} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{85} + ( 1 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{87} + ( 1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{89} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{91} + ( 3 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{93} + ( 1 - 2 \beta_{2} - \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{95} + ( -3 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{97} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 3q^{3} - 2q^{5} + 6q^{7} + O(q^{10})$$ $$7q + 3q^{3} - 2q^{5} + 6q^{7} + 5q^{11} - q^{13} + 6q^{15} - 8q^{17} + 15q^{19} - 3q^{21} + 5q^{23} - 9q^{25} + 9q^{27} + 21q^{31} + 7q^{35} - q^{37} + 23q^{39} - 10q^{41} + 23q^{43} - 4q^{45} + 10q^{47} - 13q^{49} + 20q^{51} - q^{53} + 23q^{55} - 6q^{57} + 4q^{59} + 3q^{61} + 4q^{63} + 4q^{65} + 28q^{67} + 18q^{69} + 18q^{71} - 7q^{73} - 11q^{75} + 6q^{77} + 30q^{79} - 5q^{81} - 13q^{83} + q^{85} + 7q^{87} + 18q^{91} + 36q^{93} - 2q^{97} + 10q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 6 x^{5} + 18 x^{4} + 8 x^{3} - 17 x^{2} - 9 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{5} - 8 \nu^{4} + 13 \nu^{3} + 15 \nu^{2} - 17 \nu - 5$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} + 6 \nu^{4} - 18 \nu^{3} - 7 \nu^{2} + 17 \nu + 5$$ $$\beta_{4}$$ $$=$$ $$\nu^{6} - 3 \nu^{5} - 6 \nu^{4} + 18 \nu^{3} + 8 \nu^{2} - 18 \nu - 7$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{6} - 16 \nu^{5} - 25 \nu^{4} + 92 \nu^{3} + 9 \nu^{2} - 73 \nu - 16$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{6} - 23 \nu^{5} - 35 \nu^{4} + 136 \nu^{3} + 15 \nu^{2} - 125 \nu - 29$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 6 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + 6 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 10$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 16 \beta_{3} + 12 \beta_{2} + 38 \beta_{1} + 9$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{6} + 3 \beta_{5} + 41 \beta_{4} + 76 \beta_{3} + 30 \beta_{2} + 64 \beta_{1} + 60$$

## 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
 −2.18229 −0.844838 −0.358013 −0.164919 1.40474 2.29157 2.85375
0 −2.18229 0 −1.66714 0 1.81734 0 1.76240 0
1.2 0 −0.844838 0 −2.47034 0 0.972158 0 −2.28625 0
1.3 0 −0.358013 0 2.25358 0 4.66949 0 −2.87183 0
1.4 0 −0.164919 0 −1.38175 0 −1.87004 0 −2.97280 0
1.5 0 1.40474 0 2.59588 0 −1.10443 0 −1.02671 0
1.6 0 2.29157 0 −1.78468 0 −0.671050 0 2.25130 0
1.7 0 2.85375 0 0.454452 0 2.18653 0 5.14389 0
 $$n$$: 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.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4016.2.a.g 7
4.b odd 2 1 1004.2.a.a 7
12.b even 2 1 9036.2.a.i 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1004.2.a.a 7 4.b odd 2 1
4016.2.a.g 7 1.a even 1 1 trivial
9036.2.a.i 7 12.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$251$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{7} - 3 T_{3}^{6} - 6 T_{3}^{5} + 18 T_{3}^{4} + 8 T_{3}^{3} - 17 T_{3}^{2} - 9 T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4016))$$.