# Properties

 Label 4016.2.a.m Level 4016 Weight 2 Character orbit 4016.a Self dual yes Analytic conductor 32.068 Analytic rank 0 Dimension 23 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: $$23$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 2008) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} - 8q^{11} + 8q^{13} - 7q^{15} + 19q^{17} + 9q^{19} + 9q^{21} - 21q^{23} + 65q^{25} - 5q^{27} + 10q^{29} + 9q^{31} + 34q^{33} - 12q^{35} + 11q^{37} + 9q^{39} + 35q^{41} + 9q^{43} + 29q^{45} - 37q^{47} + 77q^{49} + 17q^{51} + 38q^{53} + 20q^{55} + 51q^{57} - 17q^{59} - 22q^{63} + 41q^{65} - 9q^{67} + 8q^{69} - 13q^{71} + 41q^{73} - 25q^{75} + 36q^{77} + 36q^{79} + 127q^{81} - 29q^{83} + 34q^{85} - 10q^{87} + 36q^{89} + 6q^{91} + 36q^{93} - 25q^{95} + 40q^{97} - 19q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 0 −3.32587 0 −0.316897 0 −0.268063 0 8.06143 0
1.2 0 −3.26165 0 −4.10224 0 1.61221 0 7.63839 0
1.3 0 −3.15136 0 4.04418 0 −3.52087 0 6.93109 0
1.4 0 −2.67736 0 4.29688 0 4.01755 0 4.16827 0
1.5 0 −2.53334 0 1.50187 0 −0.478852 0 3.41783 0
1.6 0 −2.51727 0 −0.472191 0 −4.16870 0 3.33662 0
1.7 0 −1.96174 0 0.807414 0 −3.34299 0 0.848432 0
1.8 0 −1.63778 0 2.87336 0 2.53418 0 −0.317671 0
1.9 0 −1.39383 0 −3.54571 0 3.34862 0 −1.05724 0
1.10 0 −0.452441 0 1.91039 0 −1.96124 0 −2.79530 0
1.11 0 −0.259421 0 −1.96371 0 4.76268 0 −2.93270 0
1.12 0 0.0812979 0 4.08000 0 −4.55548 0 −2.99339 0
1.13 0 0.201914 0 −1.79819 0 −3.34181 0 −2.95923 0
1.14 0 0.347951 0 −1.66668 0 −3.92094 0 −2.87893 0
1.15 0 1.15579 0 −4.09592 0 0.621313 0 −1.66414 0
1.16 0 1.19399 0 −1.76720 0 4.20476 0 −1.57440 0
1.17 0 1.28390 0 3.72143 0 0.978625 0 −1.35160 0
1.18 0 2.13165 0 0.257423 0 −0.578263 0 1.54395 0
1.19 0 2.21536 0 3.09704 0 4.67984 0 1.90784 0
1.20 0 2.74485 0 2.14962 0 3.54966 0 4.53421 0
See all 23 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.23 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.m 23
4.b odd 2 1 2008.2.a.d 23

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.2.a.d 23 4.b odd 2 1
4016.2.a.m 23 1.a even 1 1 trivial

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{23} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4016))$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database