# Properties

 Label 4024.2.a.f 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.

 $$\operatorname{Tr}(f)(q) =$$ $$33q - 2q^{3} + 12q^{5} + 4q^{7} + 43q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$33q - 2q^{3} + 12q^{5} + 4q^{7} + 43q^{9} + 22q^{11} + 25q^{13} - 4q^{15} + 17q^{17} + 6q^{19} + 18q^{21} + 16q^{23} + 47q^{25} - 20q^{27} + 47q^{29} - 7q^{31} - 6q^{33} + 19q^{35} + 75q^{37} + 21q^{39} + 22q^{41} - 5q^{43} + 33q^{45} + 10q^{47} + 31q^{49} + 9q^{51} + 64q^{53} - 3q^{55} + 5q^{57} + 28q^{59} + 49q^{61} - 10q^{63} + 46q^{65} - 14q^{67} + 30q^{69} + 35q^{71} + 19q^{73} - 33q^{75} + 32q^{77} - 12q^{79} + 57q^{81} + 82q^{85} - 5q^{87} + 42q^{89} - 15q^{91} + 55q^{93} + 33q^{95} + 4q^{97} + 22q^{99} + O(q^{100})$$

## 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 −3.25521 0 −2.40628 0 −4.10646 0 7.59642 0
1.2 0 −3.20822 0 2.09579 0 −1.61648 0 7.29270 0
1.3 0 −3.01200 0 3.98796 0 4.01800 0 6.07216 0
1.4 0 −2.91979 0 2.60605 0 −3.20806 0 5.52520 0
1.5 0 −2.88194 0 −1.00235 0 −1.34540 0 5.30561 0
1.6 0 −2.48200 0 −3.41370 0 3.75697 0 3.16033 0
1.7 0 −2.27275 0 2.37900 0 1.75584 0 2.16537 0
1.8 0 −2.23369 0 −1.90057 0 0.0272535 0 1.98935 0
1.9 0 −2.05859 0 3.84987 0 −4.00015 0 1.23779 0
1.10 0 −2.05780 0 −1.22779 0 1.00021 0 1.23452 0
1.11 0 −1.25385 0 2.99234 0 2.30060 0 −1.42787 0
1.12 0 −1.19201 0 −0.00829431 0 3.59968 0 −1.57911 0
1.13 0 −0.710131 0 −2.92884 0 −1.90024 0 −2.49571 0
1.14 0 −0.502871 0 −1.85170 0 1.22131 0 −2.74712 0
1.15 0 −0.254882 0 3.76559 0 −2.29648 0 −2.93504 0
1.16 0 −0.224625 0 −2.01216 0 1.98253 0 −2.94954 0
1.17 0 0.0147975 0 −2.88853 0 −1.61248 0 −2.99978 0
1.18 0 0.176667 0 2.93614 0 −0.0214699 0 −2.96879 0
1.19 0 0.220191 0 1.20642 0 −3.77303 0 −2.95152 0
1.20 0 0.368053 0 2.18272 0 4.72026 0 −2.86454 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.f 33
4.b odd 2 1 8048.2.a.y 33

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.f 33 1.a even 1 1 trivial
8048.2.a.y 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