# Properties

 Label 4024.2.a.g Level 4024 Weight 2 Character orbit 4024.a Self dual yes Analytic conductor 32.132 Analytic rank 0 Dimension 33 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4024 = 2^{3} \cdot 503$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4024.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1318017734$$ Analytic rank: $$0$$ Dimension: $$33$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$33q + 10q^{3} + 12q^{7} + 47q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$33q + 10q^{3} + 12q^{7} + 47q^{9} + 22q^{11} - 17q^{13} + 22q^{15} + 9q^{17} + 16q^{19} + 6q^{21} + 36q^{23} + 47q^{25} + 34q^{27} + 13q^{29} + 21q^{31} + 14q^{33} + 33q^{35} - 55q^{37} + 37q^{39} + 42q^{41} + 23q^{43} + 5q^{45} + 20q^{47} + 55q^{49} + 53q^{51} - 32q^{53} + 35q^{55} + 21q^{57} + 20q^{59} - 15q^{61} + 48q^{63} + 34q^{65} + 66q^{67} - 4q^{69} + 61q^{71} + 19q^{73} + 59q^{75} + 2q^{77} + 62q^{79} + 77q^{81} + 36q^{83} - 14q^{85} + 43q^{87} + 34q^{89} + 41q^{91} - 11q^{93} + 61q^{95} - 8q^{97} + 98q^{99} + O(q^{100})$$

## 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 −3.28143 0 −3.08568 0 0.442605 0 7.76776 0
1.2 0 −2.94941 0 3.18881 0 1.60634 0 5.69904 0
1.3 0 −2.71136 0 −1.51865 0 −2.20669 0 4.35148 0
1.4 0 −2.52320 0 −3.79071 0 1.89288 0 3.36655 0
1.5 0 −2.39336 0 0.556589 0 4.72926 0 2.72817 0
1.6 0 −2.15460 0 −0.0166844 0 −4.51155 0 1.64231 0
1.7 0 −2.14143 0 2.09880 0 3.79183 0 1.58570 0
1.8 0 −2.13657 0 −1.27952 0 1.02417 0 1.56495 0
1.9 0 −1.65797 0 −0.641790 0 −1.04800 0 −0.251130 0
1.10 0 −1.02127 0 0.309055 0 −3.45255 0 −1.95700 0
1.11 0 −0.950380 0 3.01565 0 0.279316 0 −2.09678 0
1.12 0 −0.741660 0 −2.54437 0 −0.0722362 0 −2.44994 0
1.13 0 −0.648270 0 2.57860 0 −1.91468 0 −2.57975 0
1.14 0 −0.105343 0 −1.39521 0 −1.93153 0 −2.98890 0
1.15 0 0.221515 0 4.40155 0 5.16895 0 −2.95093 0
1.16 0 0.281703 0 −0.580455 0 2.82149 0 −2.92064 0
1.17 0 0.410265 0 −4.21207 0 3.36092 0 −2.83168 0
1.18 0 0.649284 0 −2.92377 0 −1.90965 0 −2.57843 0
1.19 0 0.885832 0 −0.282055 0 −2.95093 0 −2.21530 0
1.20 0 0.940347 0 2.53897 0 0.182218 0 −2.11575 0
See all 33 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.33 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 4024.2.a.g 33
4.b odd 2 1 8048.2.a.x 33

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.g 33 1.a even 1 1 trivial
8048.2.a.x 33 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$503$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4024))$$:

 $$T_{3}^{33} - \cdots$$ $$T_{5}^{33} - \cdots$$ $$T_{7}^{33} - \cdots$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database