# Properties

 Label 6025.2.a.q Level $6025$ Weight $2$ Character orbit 6025.a Self dual yes Analytic conductor $48.110$ Analytic rank $0$ Dimension $66$ 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: $$0$$ Dimension: $$66$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$66q + 78q^{4} + 16q^{6} + 90q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$66q + 78q^{4} + 16q^{6} + 90q^{9} + 48q^{11} + 30q^{14} + 98q^{16} + 12q^{19} + 18q^{21} + 42q^{24} + 48q^{26} + 56q^{29} + 48q^{31} + 8q^{34} + 158q^{36} + 84q^{39} + 56q^{41} + 144q^{44} + 36q^{46} + 98q^{49} + 44q^{51} + 86q^{54} + 104q^{56} + 108q^{59} + 22q^{61} + 136q^{64} + 74q^{66} + 20q^{69} + 212q^{71} + 84q^{74} + 6q^{76} + 66q^{79} + 162q^{81} - 52q^{84} + 100q^{86} + 54q^{89} + 72q^{91} - 96q^{94} + 122q^{96} + 112q^{99} + O(q^{100})$$

## 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 −2.81437 0.650611 5.92068 0 −1.83106 −2.61686 −11.0342 −2.57671 0
1.2 −2.73687 −1.37229 5.49044 0 3.75578 −0.639768 −9.55288 −1.11681 0
1.3 −2.70326 −3.35528 5.30763 0 9.07021 3.43095 −8.94141 8.25791 0
1.4 −2.60935 2.94245 4.80870 0 −7.67788 2.28979 −7.32887 5.65803 0
1.5 −2.55975 2.19931 4.55230 0 −5.62967 −4.06530 −6.53325 1.83695 0
1.6 −2.48606 −2.54168 4.18051 0 6.31876 −3.51286 −5.42088 3.46011 0
1.7 −2.48219 −1.89252 4.16124 0 4.69758 −3.72710 −5.36461 0.581628 0
1.8 −2.37391 0.703313 3.63543 0 −1.66960 4.47866 −3.88235 −2.50535 0
1.9 −2.33352 1.59052 3.44531 0 −3.71151 −3.91727 −3.37267 −0.470249 0
1.10 −2.31942 −2.66924 3.37972 0 6.19110 0.0358936 −3.20016 4.12486 0
1.11 −2.20997 −2.26852 2.88397 0 5.01336 0.372373 −1.95355 2.14617 0
1.12 −2.06213 2.32228 2.25239 0 −4.78886 3.49367 −0.520471 2.39299 0
1.13 −1.90752 0.479938 1.63863 0 −0.915491 1.00580 0.689327 −2.76966 0
1.14 −1.85162 0.305536 1.42851 0 −0.565737 −2.51443 1.05818 −2.90665 0
1.15 −1.85071 2.57343 1.42511 0 −4.76266 −3.52351 1.06395 3.62254 0
1.16 −1.67685 3.17567 0.811815 0 −5.32511 0.164609 1.99240 7.08488 0
1.17 −1.64329 −3.10101 0.700400 0 5.09585 −0.459168 2.13562 6.61624 0
1.18 −1.54552 −0.0372080 0.388631 0 0.0575057 5.06287 2.49040 −2.99862 0
1.19 −1.47177 −1.07414 0.166115 0 1.58089 −3.64779 2.69906 −1.84623 0
1.20 −1.38861 −0.0366779 −0.0717484 0 0.0509315 3.69082 2.87686 −2.99865 0
See all 66 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.66 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.q 66
5.b even 2 1 inner 6025.2.a.q 66
5.c odd 4 2 1205.2.b.d 66

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.d 66 5.c odd 4 2
6025.2.a.q 66 1.a even 1 1 trivial
6025.2.a.q 66 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6025))$$:

 $$35\!\cdots\!81$$$$T_{2}^{46} -$$$$18\!\cdots\!53$$$$T_{2}^{44} +$$$$80\!\cdots\!51$$$$T_{2}^{42} -$$$$29\!\cdots\!53$$$$T_{2}^{40} +$$$$95\!\cdots\!74$$$$T_{2}^{38} -$$$$25\!\cdots\!99$$$$T_{2}^{36} +$$$$59\!\cdots\!38$$$$T_{2}^{34} -$$$$11\!\cdots\!39$$$$T_{2}^{32} +$$$$19\!\cdots\!05$$$$T_{2}^{30} -$$$$26\!\cdots\!55$$$$T_{2}^{28} +$$$$30\!\cdots\!19$$$$T_{2}^{26} -$$$$29\!\cdots\!22$$$$T_{2}^{24} +$$$$23\!\cdots\!98$$$$T_{2}^{22} -$$$$14\!\cdots\!86$$$$T_{2}^{20} +$$$$73\!\cdots\!27$$$$T_{2}^{18} -$$$$28\!\cdots\!10$$$$T_{2}^{16} +$$$$85\!\cdots\!89$$$$T_{2}^{14} -$$$$18\!\cdots\!24$$$$T_{2}^{12} +$$$$30\!\cdots\!62$$$$T_{2}^{10} -$$$$33\!\cdots\!31$$$$T_{2}^{8} + 246600093167 T_{2}^{6} - 11048999599 T_{2}^{4} + 253747673 T_{2}^{2} - 1841449$$">$$T_{2}^{66} - \cdots$$ $$88\!\cdots\!21$$$$T_{3}^{48} +$$$$71\!\cdots\!90$$$$T_{3}^{46} -$$$$49\!\cdots\!46$$$$T_{3}^{44} +$$$$28\!\cdots\!67$$$$T_{3}^{42} -$$$$13\!\cdots\!25$$$$T_{3}^{40} +$$$$56\!\cdots\!45$$$$T_{3}^{38} -$$$$19\!\cdots\!48$$$$T_{3}^{36} +$$$$57\!\cdots\!33$$$$T_{3}^{34} -$$$$14\!\cdots\!76$$$$T_{3}^{32} +$$$$28\!\cdots\!31$$$$T_{3}^{30} -$$$$48\!\cdots\!79$$$$T_{3}^{28} +$$$$67\!\cdots\!17$$$$T_{3}^{26} -$$$$75\!\cdots\!17$$$$T_{3}^{24} +$$$$68\!\cdots\!46$$$$T_{3}^{22} -$$$$48\!\cdots\!20$$$$T_{3}^{20} +$$$$26\!\cdots\!32$$$$T_{3}^{18} -$$$$10\!\cdots\!07$$$$T_{3}^{16} +$$$$32\!\cdots\!23$$$$T_{3}^{14} -$$$$66\!\cdots\!68$$$$T_{3}^{12} +$$$$89\!\cdots\!84$$$$T_{3}^{10} -$$$$69\!\cdots\!00$$$$T_{3}^{8} +$$$$25\!\cdots\!48$$$$T_{3}^{6} - 205383271424 T_{3}^{4} + 427458560 T_{3}^{2} - 262144$$">$$T_{3}^{66} - \cdots$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database