# Properties

 Label 6003.2.a.v Level 6003 Weight 2 Character orbit 6003.a Self dual yes Analytic conductor 47.934 Analytic rank 0 Dimension 30 CM no Inner twists 1

# Related objects

## 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: $$0$$ Dimension: $$30$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q - q^{2} + 37q^{4} + 10q^{7} - 6q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - q^{2} + 37q^{4} + 10q^{7} - 6q^{8} + 8q^{10} + 36q^{13} - 7q^{14} + 47q^{16} - 18q^{17} + 16q^{19} + 25q^{22} + 30q^{23} + 56q^{25} - 11q^{26} + 27q^{28} - 30q^{29} + 14q^{31} + 7q^{32} + 3q^{34} + 22q^{35} + 40q^{37} - 6q^{38} + 30q^{40} - 14q^{41} + 34q^{43} - 5q^{44} - q^{46} + 2q^{47} + 74q^{49} + 21q^{50} + 71q^{52} - 16q^{53} + 22q^{55} - 14q^{56} + q^{58} + 32q^{59} + 46q^{61} - 20q^{62} + 68q^{64} - 12q^{65} + 14q^{67} - 27q^{68} + 32q^{71} + 50q^{73} + 26q^{74} + 56q^{76} - 34q^{77} + 16q^{79} - 2q^{80} + 38q^{82} + 14q^{83} + 38q^{85} - 10q^{86} + 40q^{88} + 2q^{89} + 32q^{91} + 37q^{92} + 29q^{94} + 28q^{95} + 56q^{97} - 8q^{98} + O(q^{100})$$

## 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 −2.78139 0 5.73612 −0.00312667 0 2.66771 −10.3916 0 0.00869647
1.2 −2.66815 0 5.11905 −1.38302 0 0.919018 −8.32211 0 3.69010
1.3 −2.53344 0 4.41832 −4.44002 0 −2.94217 −6.12666 0 11.2485
1.4 −2.40738 0 3.79549 3.50117 0 5.09845 −4.32244 0 −8.42865
1.5 −2.27733 0 3.18622 2.98462 0 −2.44710 −2.70140 0 −6.79696
1.6 −1.99322 0 1.97291 0.735105 0 −3.83374 0.0540029 0 −1.46522
1.7 −1.96558 0 1.86349 −2.31356 0 5.13967 0.268317 0 4.54748
1.8 −1.89039 0 1.57357 −2.79891 0 1.18730 0.806117 0 5.29102
1.9 −1.58386 0 0.508607 2.90489 0 0.0968084 2.36216 0 −4.60094
1.10 −1.33909 0 −0.206831 0.887651 0 −3.28671 2.95515 0 −1.18865
1.11 −0.959410 0 −1.07953 2.76105 0 2.95320 2.95453 0 −2.64898
1.12 −0.809520 0 −1.34468 −2.87436 0 0.883619 2.70758 0 2.32685
1.13 −0.784780 0 −1.38412 −2.15464 0 −2.00724 2.65579 0 1.69092
1.14 −0.473233 0 −1.77605 1.04483 0 3.83278 1.78695 0 −0.494446
1.15 −0.0795257 0 −1.99368 −3.50033 0 0.586662 0.317600 0 0.278366
1.16 0.395940 0 −1.84323 −0.579308 0 −2.59556 −1.52169 0 −0.229371
1.17 0.480943 0 −1.76869 1.51137 0 3.82960 −1.81253 0 0.726880
1.18 0.534050 0 −1.71479 2.64563 0 −5.04348 −1.98388 0 1.41290
1.19 0.619961 0 −1.61565 0.522930 0 −1.16804 −2.24156 0 0.324196
1.20 1.12036 0 −0.744798 4.16877 0 2.81319 −3.07516 0 4.67051
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.30 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.v 30
3.b odd 2 1 6003.2.a.w yes 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6003.2.a.v 30 1.a even 1 1 trivial
6003.2.a.w yes 30 3.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{30} + \cdots$$ $$T_{5}^{30} - \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database