# Properties

 Label 6025.2.a.p Level $6025$ Weight $2$ Character orbit 6025.a Self dual yes Analytic conductor $48.110$ Analytic rank $1$ Dimension $46$ CM no Inner twists $2$

# Related objects

## 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: $$46$$ Twist minimal: no (minimal twist has level 1205) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$46q + 34q^{4} - 4q^{6} + 34q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$46q + 34q^{4} - 4q^{6} + 34q^{9} - 64q^{11} - 66q^{14} + 22q^{16} - 14q^{21} - 50q^{24} - 60q^{26} - 36q^{29} - 36q^{31} + 12q^{34} - 34q^{36} - 88q^{39} - 76q^{41} - 100q^{44} - 12q^{46} + 22q^{49} - 112q^{51} - 26q^{54} - 120q^{56} - 84q^{59} - 78q^{61} + 28q^{64} - 2q^{66} - 24q^{69} - 172q^{71} - 16q^{74} - 18q^{76} - 54q^{79} - 42q^{81} + 44q^{84} - 80q^{86} - 86q^{89} - 88q^{91} + 4q^{94} - 122q^{96} - 148q^{99} + O(q^{100})$$

## 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 −2.68673 2.58072 5.21850 0 −6.93370 1.24945 −8.64724 3.66013 0
1.2 −2.67602 −0.199289 5.16108 0 0.533301 2.87374 −8.45912 −2.96028 0
1.3 −2.49572 −1.66439 4.22863 0 4.15385 4.24911 −5.56203 −0.229807 0
1.4 −2.44838 0.992209 3.99457 0 −2.42931 −0.744704 −4.88345 −2.01552 0
1.5 −2.36215 2.05253 3.57974 0 −4.84838 3.07728 −3.73158 1.21289 0
1.6 −1.95888 0.350465 1.83721 0 −0.686519 1.14869 0.318879 −2.87717 0
1.7 −1.92263 0.557567 1.69651 0 −1.07200 −0.976687 0.583491 −2.68912 0
1.8 −1.88286 −2.55671 1.54517 0 4.81392 3.51799 0.856390 3.53675 0
1.9 −1.84209 −1.69681 1.39329 0 3.12567 1.54055 1.11761 −0.120844 0
1.10 −1.83307 −2.40621 1.36013 0 4.41073 −0.302396 1.17293 2.78983 0
1.11 −1.70963 −1.38336 0.922819 0 2.36503 −5.05937 1.84158 −1.08631 0
1.12 −1.49203 3.12646 0.226139 0 −4.66475 4.66024 2.64665 6.77474 0
1.13 −1.45267 2.41257 0.110263 0 −3.50468 −1.25057 2.74517 2.82049 0
1.14 −1.30874 −1.88426 −0.287190 0 2.46601 1.28779 2.99335 0.550432 0
1.15 −1.09514 −0.223927 −0.800662 0 0.245232 −2.26365 3.06713 −2.94986 0
1.16 −0.930197 1.32621 −1.13473 0 −1.23364 −4.33497 2.91592 −1.24116 0
1.17 −0.779313 −2.96208 −1.39267 0 2.30839 −0.937135 2.64395 5.77395 0
1.18 −0.760721 1.12938 −1.42130 0 −0.859145 2.53853 2.60266 −1.72450 0
1.19 −0.636788 2.24729 −1.59450 0 −1.43105 1.14493 2.28894 2.05030 0
1.20 −0.434323 −1.75836 −1.81136 0 0.763695 2.56007 1.65536 0.0918227 0
See all 46 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6025.2.a.p 46
5.b even 2 1 inner 6025.2.a.p 46
5.c odd 4 2 1205.2.b.c 46

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.c 46 5.c odd 4 2
6025.2.a.p 46 1.a even 1 1 trivial
6025.2.a.p 46 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{2}^{46} - \cdots$$ $$14\!\cdots\!52$$$$T_{3}^{20} +$$$$20\!\cdots\!59$$$$T_{3}^{18} -$$$$22\!\cdots\!17$$$$T_{3}^{16} +$$$$18\!\cdots\!56$$$$T_{3}^{14} -$$$$10\!\cdots\!11$$$$T_{3}^{12} + 445144634428 T_{3}^{10} - 120475489301 T_{3}^{8} + 19726497117 T_{3}^{6} - 1753417481 T_{3}^{4} + 73674451 T_{3}^{2} - 1136356$$">$$T_{3}^{46} - \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database