# Properties

 Label 6003.2.a.u Level $6003$ Weight $2$ Character orbit 6003.a Self dual yes Analytic conductor $47.934$ Analytic rank $1$ Dimension $22$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$6003 = 3^{2} \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6003.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$47.9341963334$$ Analytic rank: $$1$$ Dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22q + 3q^{2} + 17q^{4} - 6q^{7} + 6q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q + 3q^{2} + 17q^{4} - 6q^{7} + 6q^{8} - 12q^{10} - 28q^{13} + q^{14} + 3q^{16} + 10q^{17} - 8q^{19} - 11q^{22} - 22q^{23} - 11q^{26} - 21q^{28} - 22q^{29} - 18q^{31} - 5q^{32} - 33q^{34} - 2q^{35} - 28q^{37} - 14q^{38} - 30q^{40} + 10q^{41} - 14q^{43} - 37q^{44} - 3q^{46} + 18q^{47} + 2q^{49} - 7q^{50} - 57q^{52} - 20q^{53} - 42q^{55} + 2q^{56} - 3q^{58} + 20q^{59} - 38q^{61} - 4q^{62} - 24q^{64} - 12q^{65} - 50q^{67} - 11q^{68} - 48q^{70} - 12q^{71} - 46q^{73} + 6q^{74} - 16q^{76} + 14q^{77} - 20q^{79} + 58q^{80} - 42q^{82} - 22q^{83} - 66q^{85} - 22q^{86} - 68q^{88} + 14q^{89} - 16q^{91} - 17q^{92} - 27q^{94} + 20q^{95} - 48q^{97} + 28q^{98} + O(q^{100})$$

## 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 −2.60337 0 4.77753 2.74879 0 −2.14366 −7.23092 0 −7.15611
1.2 −2.31658 0 3.36653 2.07685 0 1.30399 −3.16567 0 −4.81118
1.3 −2.08661 0 2.35393 −1.43041 0 −0.274571 −0.738503 0 2.98470
1.4 −1.93234 0 1.73393 −2.82056 0 −3.97268 0.514145 0 5.45026
1.5 −1.60427 0 0.573689 1.49095 0 2.64378 2.28819 0 −2.39188
1.6 −1.16689 0 −0.638362 1.66410 0 −0.970636 3.07868 0 −1.94183
1.7 −1.14621 0 −0.686192 −0.461250 0 −3.03232 3.07895 0 0.528691
1.8 −1.05077 0 −0.895876 −3.10634 0 3.07400 3.04291 0 3.26406
1.9 −0.289055 0 −1.91645 0.894239 0 3.31166 1.13207 0 −0.258484
1.10 −0.183492 0 −1.96633 −0.926834 0 −0.385808 0.727789 0 0.170066
1.11 −0.144466 0 −1.97913 4.08805 0 −0.828594 0.574849 0 −0.590584
1.12 0.230149 0 −1.94703 −2.59479 0 −4.77655 −0.908406 0 −0.597188
1.13 0.424349 0 −1.81993 2.43761 0 3.41293 −1.62098 0 1.03440
1.14 1.06922 0 −0.856774 −0.420836 0 1.13415 −3.05451 0 −0.449966
1.15 1.24737 0 −0.444068 1.99313 0 −4.26394 −3.04866 0 2.48617
1.16 1.35991 0 −0.150655 −4.20064 0 0.870260 −2.92469 0 −5.71248
1.17 1.78641 0 1.19125 0.326188 0 1.17242 −1.44475 0 0.582705
1.18 1.96070 0 1.84436 2.84151 0 −2.21458 −0.305158 0 5.57136
1.19 2.18238 0 2.76278 −0.541552 0 0.936842 1.66467 0 −1.18187
1.20 2.24483 0 3.03928 −2.70888 0 −1.81542 2.33301 0 −6.08099
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$23$$ $$1$$
$$29$$ $$1$$

## 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 6003.2.a.u yes 22
3.b odd 2 1 6003.2.a.t 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6003.2.a.t 22 3.b odd 2 1
6003.2.a.u yes 22 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6003))$$:

 $$T_{2}^{22} - \cdots$$ $$T_{5}^{22} - \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database