# Properties

 Label 4012.2.a.j Level 4012 Weight 2 Character orbit 4012.a Self dual yes Analytic conductor 32.036 Analytic rank 0 Dimension 21 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$4012 = 2^{2} \cdot 17 \cdot 59$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4012.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0359812909$$ Analytic rank: $$0$$ Dimension: $$21$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$21q + 9q^{3} + q^{5} + 11q^{7} + 26q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$21q + 9q^{3} + q^{5} + 11q^{7} + 26q^{9} + 12q^{11} - 4q^{13} + 9q^{15} + 21q^{17} + 4q^{19} + 8q^{21} + 19q^{23} + 26q^{25} + 33q^{27} - q^{29} + 13q^{31} + 11q^{33} + 15q^{35} - 4q^{37} + 18q^{39} + 9q^{41} + 7q^{43} + 7q^{45} + 33q^{47} + 36q^{49} + 9q^{51} + 7q^{53} + 12q^{55} + 26q^{57} + 21q^{59} + 3q^{61} + 51q^{63} + q^{65} + 8q^{69} + 55q^{71} + 22q^{73} + 14q^{75} - 15q^{77} + 28q^{79} + 25q^{81} + 54q^{83} + q^{85} + 34q^{87} + 30q^{89} + 35q^{91} - 5q^{93} + 42q^{95} + 9q^{97} + 20q^{99} + O(q^{100})$$

## 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 −2.75827 0 0.760103 0 0.427165 0 4.60804 0
1.2 0 −2.52360 0 −3.46979 0 1.96229 0 3.36854 0
1.3 0 −2.50436 0 2.93974 0 −0.515336 0 3.27181 0
1.4 0 −1.96950 0 −1.93705 0 −2.26754 0 0.878948 0
1.5 0 −1.89300 0 −2.37849 0 4.40428 0 0.583446 0
1.6 0 −1.58435 0 2.56834 0 3.40444 0 −0.489844 0
1.7 0 −1.03698 0 0.719057 0 −1.71880 0 −1.92468 0
1.8 0 0.0134244 0 3.34756 0 −3.13615 0 −2.99982 0
1.9 0 0.188519 0 −3.21916 0 −0.179540 0 −2.96446 0
1.10 0 0.210829 0 0.593138 0 −1.60715 0 −2.95555 0
1.11 0 0.461334 0 1.69592 0 4.42760 0 −2.78717 0
1.12 0 0.765447 0 −3.22340 0 −4.55619 0 −2.41409 0
1.13 0 0.967697 0 −3.19923 0 3.75414 0 −2.06356 0
1.14 0 1.90340 0 1.03137 0 0.216287 0 0.622916 0
1.15 0 1.97354 0 3.25764 0 2.36478 0 0.894844 0
1.16 0 2.22137 0 −1.27900 0 0.396716 0 1.93446 0
1.17 0 2.29806 0 4.01491 0 4.07651 0 2.28106 0
1.18 0 2.45518 0 −1.72896 0 −4.96392 0 3.02793 0
1.19 0 3.11094 0 3.12050 0 −1.63406 0 6.67797 0
1.20 0 3.31779 0 −0.252330 0 4.50552 0 8.00775 0
See all 21 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.21 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 4012.2.a.j 21

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.a.j 21 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$17$$ $$-1$$
$$59$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4012))$$:

 $$T_{3}^{21} - \cdots$$ $$T_{5}^{21} - \cdots$$