# Properties

 Label 6025.2.a.l Level $6025$ Weight $2$ Character orbit 6025.a Self dual yes Analytic conductor $48.110$ Analytic rank $1$ Dimension $40$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$6025 = 5^{2} \cdot 241$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.1098672178$$ Analytic rank: $$1$$ Dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 11q^{2} - 8q^{3} + 41q^{4} + 3q^{6} - 16q^{7} - 33q^{8} + 38q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 11q^{2} - 8q^{3} + 41q^{4} + 3q^{6} - 16q^{7} - 33q^{8} + 38q^{9} + q^{11} - 14q^{12} - 9q^{13} - q^{14} + 43q^{16} - 12q^{17} - 42q^{18} + 2q^{21} - 5q^{22} - 77q^{23} - 2q^{24} + 2q^{26} - 38q^{27} - 42q^{28} + 2q^{29} + q^{31} - 72q^{32} - 20q^{33} + 5q^{34} + 32q^{36} - 28q^{37} - 23q^{38} - 2q^{39} - 2q^{41} - 37q^{42} - 31q^{43} + 3q^{44} + 14q^{46} - 96q^{47} - 13q^{48} + 40q^{49} - 10q^{51} - 42q^{52} - 54q^{53} + 4q^{54} - 15q^{56} - 37q^{57} - 27q^{58} + q^{59} + 5q^{61} - 39q^{62} - 70q^{63} + 65q^{64} - 52q^{66} - 34q^{67} - 52q^{68} + 21q^{69} - 9q^{71} - 70q^{72} - 25q^{73} + 22q^{74} - 47q^{76} - 54q^{77} - 58q^{78} + 13q^{79} + 12q^{81} + 5q^{82} - 63q^{83} + 95q^{84} - 18q^{86} - 47q^{87} - 13q^{88} + 19q^{89} - 31q^{91} - 137q^{92} - 52q^{93} + 120q^{94} - 49q^{96} - 36q^{97} - 64q^{98} + 16q^{99} + O(q^{100})$$

## 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 −2.77437 2.60911 5.69713 0 −7.23863 1.40482 −10.2572 3.80744 0
1.2 −2.74359 −0.0877195 5.52727 0 0.240666 −1.38863 −9.67736 −2.99231 0
1.3 −2.67635 −2.41547 5.16285 0 6.46465 −0.976502 −8.46490 2.83451 0
1.4 −2.60048 −3.03019 4.76251 0 7.87996 −1.49919 −7.18385 6.18207 0
1.5 −2.50972 −0.136405 4.29867 0 0.342337 −4.91163 −5.76901 −2.98139 0
1.6 −2.49478 2.89336 4.22391 0 −7.21828 0.185663 −5.54817 5.37152 0
1.7 −2.40563 −2.30634 3.78704 0 5.54820 4.72754 −4.29895 2.31922 0
1.8 −2.17490 0.995168 2.73018 0 −2.16439 3.57474 −1.58807 −2.00964 0
1.9 −2.06779 0.977473 2.27574 0 −2.02121 −1.48726 −0.570172 −2.04455 0
1.10 −1.92684 −1.97893 1.71272 0 3.81308 −3.74943 0.553547 0.916158 0
1.11 −1.60847 −3.19065 0.587161 0 5.13204 −4.87921 2.27250 7.18022 0
1.12 −1.59175 0.0342835 0.533671 0 −0.0545707 −1.01795 2.33403 −2.99882 0
1.13 −1.55382 2.23966 0.414344 0 −3.48002 1.09916 2.46382 2.01608 0
1.14 −1.26445 0.637501 −0.401165 0 −0.806088 3.31632 3.03615 −2.59359 0
1.15 −1.23402 −3.02124 −0.477193 0 3.72827 0.346766 3.05691 6.12789 0
1.16 −1.19733 −1.42352 −0.566401 0 1.70442 1.72539 3.07283 −0.973586 0
1.17 −1.14940 0.361587 −0.678878 0 −0.415609 −3.63700 3.07910 −2.86925 0
1.18 −0.679522 2.20024 −1.53825 0 −1.49511 −5.10754 2.40432 1.84107 0
1.19 −0.667714 2.21224 −1.55416 0 −1.47714 −0.972247 2.37316 1.89399 0
1.20 −0.599456 2.83593 −1.64065 0 −1.70002 0.360434 2.18241 5.04251 0
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$241$$ $$1$$

## 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 6025.2.a.l 40
5.b even 2 1 6025.2.a.o yes 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6025.2.a.l 40 1.a even 1 1 trivial
6025.2.a.o yes 40 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6025))$$:

 $$T_{2}^{40} + \cdots$$ $$T_{3}^{40} + \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database