# Properties

 Label 25.3.f.a Level 25 Weight 3 Character orbit 25.f Analytic conductor 0.681 Analytic rank 0 Dimension 32 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 25.f (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.681200660901$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$4$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 10q^{2} - 10q^{3} - 10q^{4} - 10q^{5} - 6q^{6} - 10q^{7} - 10q^{8} - 10q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 10q^{2} - 10q^{3} - 10q^{4} - 10q^{5} - 6q^{6} - 10q^{7} - 10q^{8} - 10q^{9} - 10q^{10} - 6q^{11} - 10q^{12} - 10q^{13} - 10q^{14} - 10q^{15} + 2q^{16} + 60q^{17} + 140q^{18} + 90q^{19} + 130q^{20} - 6q^{21} + 70q^{22} + 10q^{23} - 40q^{25} + 4q^{26} - 100q^{27} - 250q^{28} - 110q^{29} - 250q^{30} - 6q^{31} - 290q^{32} - 190q^{33} - 260q^{34} - 120q^{35} - 58q^{36} + 50q^{37} + 320q^{38} + 390q^{39} + 440q^{40} - 86q^{41} + 690q^{42} + 230q^{43} + 340q^{44} + 310q^{45} - 6q^{46} + 70q^{47} + 160q^{48} - 100q^{50} - 16q^{51} - 320q^{52} - 190q^{53} - 660q^{54} - 250q^{55} - 70q^{56} - 650q^{57} - 640q^{58} - 260q^{59} - 550q^{60} + 114q^{61} + 60q^{62} - 20q^{63} + 340q^{64} + 360q^{65} + 138q^{66} + 270q^{67} + 710q^{68} + 340q^{69} + 310q^{70} - 66q^{71} + 360q^{72} + 30q^{73} - 90q^{75} - 80q^{76} - 250q^{77} - 500q^{78} - 210q^{79} - 850q^{80} + 62q^{81} + 30q^{82} - 10q^{84} + 600q^{85} - 6q^{86} + 300q^{87} + 190q^{88} - 10q^{89} + 380q^{90} - 6q^{91} - 30q^{92} + 520q^{93} + 790q^{94} + 310q^{95} + 174q^{96} + 270q^{97} + 170q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −3.57427 0.566108i 1.61679 + 3.17313i 8.65068 + 2.81078i −0.872190 + 4.92334i −3.98250 12.2569i −0.574149 0.574149i −16.4311 8.37205i −2.16466 + 2.97940i 5.90458 17.1036i
2.2 −1.86717 0.295731i −2.19472 4.30737i −0.405347 0.131705i 4.99561 + 0.209511i 2.82409 + 8.69166i −3.57009 3.57009i 7.45551 + 3.79877i −8.44662 + 11.6258i −9.26571 1.86855i
2.3 0.287585 + 0.0455490i 1.72787 + 3.39113i −3.72360 1.20987i 2.36408 4.40581i 0.342446 + 1.05394i −2.38950 2.38950i −2.05348 1.04630i −3.22416 + 4.43767i 0.880552 1.15936i
2.4 1.80600 + 0.286042i −0.665351 1.30583i −0.624420 0.202886i −3.20727 + 3.83580i −0.828102 2.54863i 3.62927 + 3.62927i −7.58652 3.86553i 4.02758 5.54349i −6.88953 + 6.01004i
3.1 −1.69523 3.32707i −0.0858318 0.541921i −5.84445 + 8.04419i 2.26962 4.45520i −1.65750 + 1.20425i 1.68463 1.68463i 21.9189 + 3.47161i 8.27320 2.68812i −18.6703 + 0.00137996i
3.2 −0.395527 0.776265i −0.296456 1.87175i 1.90500 2.62200i 1.22928 + 4.84653i −1.33572 + 0.970456i −5.60844 + 5.60844i −6.23083 0.986866i 5.14394 1.67137i 3.27598 2.87118i
3.3 −0.259330 0.508965i 0.838638 + 5.29495i 2.15935 2.97209i −3.73307 3.32629i 2.47746 1.79998i 1.66138 1.66138i −4.32944 0.685716i −18.7737 + 6.09994i −0.724866 + 2.76261i
3.4 1.29583 + 2.54321i −0.363254 2.29349i −2.43759 + 3.35505i −4.45624 2.26758i 5.36211 3.89580i −3.40272 + 3.40272i −0.414625 0.0656701i 3.43135 1.11491i −0.00760491 14.2715i
8.1 −2.38234 1.21387i −3.57679 0.566508i 1.85096 + 2.54762i −4.45026 2.27929i 7.83348 + 5.69136i 6.54971 6.54971i 0.355933 + 2.24727i 3.91299 + 1.27141i 7.83532 + 10.8321i
8.2 −1.61837 0.824603i 3.42034 + 0.541729i −0.411975 0.567034i 4.00059 2.99921i −5.08868 3.69715i −8.06323 + 8.06323i 1.33571 + 8.43332i 2.84577 + 0.924645i −8.94762 + 1.55494i
8.3 0.972743 + 0.495637i 0.872241 + 0.138149i −1.65057 2.27181i −2.66494 + 4.23062i 0.779995 + 0.566699i 1.62783 1.62783i −1.16272 7.34115i −7.81779 2.54015i −4.68915 + 2.79446i
8.4 2.70026 + 1.37585i −4.42692 0.701156i 3.04732 + 4.19427i 1.95091 4.60369i −10.9892 7.98410i −4.77540 + 4.77540i 0.561510 + 3.54523i 10.5465 + 3.42677i 11.6020 9.74701i
12.1 −0.513943 3.24491i 2.81033 + 1.43193i −6.46108 + 2.09933i −4.99960 0.0628765i 3.20215 9.85519i 7.51823 + 7.51823i 4.16668 + 8.17758i 0.557438 + 0.767248i 2.36548 + 16.2556i
12.2 −0.312579 1.97355i −4.02069 2.04864i 0.00704800 0.00229003i 4.93389 + 0.810386i −2.78631 + 8.57538i 3.91191 + 3.91191i −3.63528 7.13464i 6.67894 + 9.19277i 0.0571038 9.99057i
12.3 0.0933465 + 0.589367i 0.210730 + 0.107372i 3.46559 1.12604i −3.31432 + 3.74370i −0.0436108 + 0.134220i −7.64532 7.64532i 2.07076 + 4.06409i −5.25719 7.23590i −2.51579 1.60389i
12.4 0.463000 + 2.92327i −0.866921 0.441718i −4.52691 + 1.47088i 0.953911 4.90816i 0.889877 2.73876i 4.44588 + 4.44588i −1.02103 2.00389i −4.73363 6.51528i 14.7895 + 0.516059i
13.1 −3.57427 + 0.566108i 1.61679 3.17313i 8.65068 2.81078i −0.872190 4.92334i −3.98250 + 12.2569i −0.574149 + 0.574149i −16.4311 + 8.37205i −2.16466 2.97940i 5.90458 + 17.1036i
13.2 −1.86717 + 0.295731i −2.19472 + 4.30737i −0.405347 + 0.131705i 4.99561 0.209511i 2.82409 8.69166i −3.57009 + 3.57009i 7.45551 3.79877i −8.44662 11.6258i −9.26571 + 1.86855i
13.3 0.287585 0.0455490i 1.72787 3.39113i −3.72360 + 1.20987i 2.36408 + 4.40581i 0.342446 1.05394i −2.38950 + 2.38950i −2.05348 + 1.04630i −3.22416 4.43767i 0.880552 + 1.15936i
13.4 1.80600 0.286042i −0.665351 + 1.30583i −0.624420 + 0.202886i −3.20727 3.83580i −0.828102 + 2.54863i 3.62927 3.62927i −7.58652 + 3.86553i 4.02758 + 5.54349i −6.88953 6.01004i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.3.f.a 32
3.b odd 2 1 225.3.r.a 32
4.b odd 2 1 400.3.bg.c 32
5.b even 2 1 125.3.f.c 32
5.c odd 4 1 125.3.f.a 32
5.c odd 4 1 125.3.f.b 32
25.d even 5 1 125.3.f.a 32
25.e even 10 1 125.3.f.b 32
25.f odd 20 1 inner 25.3.f.a 32
25.f odd 20 1 125.3.f.c 32
75.l even 20 1 225.3.r.a 32
100.l even 20 1 400.3.bg.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.3.f.a 32 1.a even 1 1 trivial
25.3.f.a 32 25.f odd 20 1 inner
125.3.f.a 32 5.c odd 4 1
125.3.f.a 32 25.d even 5 1
125.3.f.b 32 5.c odd 4 1
125.3.f.b 32 25.e even 10 1
125.3.f.c 32 5.b even 2 1
125.3.f.c 32 25.f odd 20 1
225.3.r.a 32 3.b odd 2 1
225.3.r.a 32 75.l even 20 1
400.3.bg.c 32 4.b odd 2 1
400.3.bg.c 32 100.l even 20 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database