# Properties

 Label 150.2.l.a Level 150 Weight 2 Character orbit 150.l Analytic conductor 1.198 Analytic rank 0 Dimension 80 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 150.l (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.19775603032$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$10$$ over $$\Q(\zeta_{20})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 4q^{3} + 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 4q^{3} + 4q^{7} - 4q^{10} - 4q^{12} - 8q^{15} + 20q^{16} - 8q^{18} - 40q^{19} - 36q^{22} - 104q^{25} + 4q^{27} - 16q^{28} + 12q^{30} + 4q^{33} - 40q^{34} - 24q^{37} - 40q^{39} - 8q^{40} - 4q^{42} - 24q^{43} - 72q^{45} - 4q^{48} - 12q^{55} - 64q^{57} + 20q^{58} + 24q^{60} + 64q^{63} + 96q^{67} + 140q^{69} + 76q^{70} + 8q^{72} + 100q^{73} + 132q^{75} + 100q^{78} + 80q^{79} - 40q^{81} + 96q^{82} + 60q^{84} + 32q^{85} + 80q^{87} + 4q^{88} + 52q^{90} + 12q^{93} - 32q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −0.891007 0.453990i −1.64867 0.530925i 0.587785 + 0.809017i 1.35266 + 1.78054i 1.22794 + 1.22154i −0.152718 0.152718i −0.156434 0.987688i 2.43624 + 1.75064i −0.396880 2.20056i
17.2 −0.891007 0.453990i −0.873670 + 1.49556i 0.587785 + 0.809017i −2.21391 + 0.314018i 1.45742 0.935916i −2.72680 2.72680i −0.156434 0.987688i −1.47340 2.61325i 2.11517 + 0.725301i
17.3 −0.891007 0.453990i −0.322239 1.70181i 0.587785 + 0.809017i −0.0831552 2.23452i −0.485490 + 1.66262i 0.0556476 + 0.0556476i −0.156434 0.987688i −2.79232 + 1.09678i −0.940360 + 2.02872i
17.4 −0.891007 0.453990i 0.983013 + 1.42607i 0.587785 + 0.809017i 1.04600 1.97633i −0.228447 1.71692i 0.462249 + 0.462249i −0.156434 0.987688i −1.06737 + 2.80370i −1.82923 + 1.28605i
17.5 −0.891007 0.453990i 1.71953 + 0.207905i 0.587785 + 0.809017i −1.62298 + 1.53816i −1.43772 0.965894i 2.58285 + 2.58285i −0.156434 0.987688i 2.91355 + 0.714995i 2.14440 0.633691i
17.6 0.891007 + 0.453990i −1.69961 0.333634i 0.587785 + 0.809017i 1.62298 1.53816i −1.36290 1.06888i 2.58285 + 2.58285i 0.156434 + 0.987688i 2.77738 + 1.13410i 2.14440 0.633691i
17.7 0.891007 + 0.453990i −1.37558 + 1.05251i 0.587785 + 0.809017i −1.04600 + 1.97633i −1.70348 + 0.313291i 0.462249 + 0.462249i 0.156434 + 0.987688i 0.784450 2.89562i −1.82923 + 1.28605i
17.8 0.891007 + 0.453990i 0.368756 + 1.69234i 0.587785 + 0.809017i 2.21391 0.314018i −0.439743 + 1.67530i −2.72680 2.72680i 0.156434 + 0.987688i −2.72804 + 1.24812i 2.11517 + 0.725301i
17.9 0.891007 + 0.453990i 0.832356 1.51894i 0.587785 + 0.809017i 0.0831552 + 2.23452i 1.43122 0.975505i 0.0556476 + 0.0556476i 0.156434 + 0.987688i −1.61437 2.52860i −0.940360 + 2.02872i
17.10 0.891007 + 0.453990i 1.73204 + 0.00452789i 0.587785 + 0.809017i −1.35266 1.78054i 1.54121 + 0.790366i −0.152718 0.152718i 0.156434 + 0.987688i 2.99996 + 0.0156850i −0.396880 2.20056i
23.1 −0.987688 0.156434i −1.58737 + 0.693000i 0.951057 + 0.309017i −0.197931 2.22729i 1.67624 0.436149i −1.43195 + 1.43195i −0.891007 0.453990i 2.03950 2.20010i −0.152931 + 2.23083i
23.2 −0.987688 0.156434i −1.36724 1.06332i 0.951057 + 0.309017i −1.48885 + 1.66833i 1.18407 + 1.26411i −1.08662 + 1.08662i −0.891007 0.453990i 0.738706 + 2.90763i 1.73150 1.41488i
23.3 −0.987688 0.156434i 0.487767 1.66195i 0.951057 + 0.309017i 2.22772 0.193062i −0.741748 + 1.56519i −0.104631 + 0.104631i −0.891007 0.453990i −2.52417 1.62129i −2.23049 0.157807i
23.4 −0.987688 0.156434i 1.10553 + 1.33334i 0.951057 + 0.309017i −0.334535 2.21090i −0.883341 1.48987i 2.78015 2.78015i −0.891007 0.453990i −0.555594 + 2.94810i −0.0154450 + 2.23601i
23.5 −0.987688 0.156434i 1.64008 + 0.556890i 0.951057 + 0.309017i 0.545143 + 2.16860i −1.53277 0.806599i −1.41702 + 1.41702i −0.891007 0.453990i 2.37975 + 1.82669i −0.199188 2.22718i
23.6 0.987688 + 0.156434i −1.49368 + 0.876876i 0.951057 + 0.309017i 0.197931 + 2.22729i −1.61247 + 0.632416i −1.43195 + 1.43195i 0.891007 + 0.453990i 1.46218 2.61955i −0.152931 + 2.23083i
23.7 0.987688 + 0.156434i −0.428879 1.67811i 0.951057 + 0.309017i 0.334535 + 2.21090i −0.161084 1.72454i 2.78015 2.78015i 0.891007 + 0.453990i −2.63213 + 1.43942i −0.0154450 + 2.23601i
23.8 0.987688 + 0.156434i 0.0565979 + 1.73113i 0.951057 + 0.309017i 1.48885 1.66833i −0.214907 + 1.71867i −1.08662 + 1.08662i 0.891007 + 0.453990i −2.99359 + 0.195956i 1.73150 1.41488i
23.9 0.987688 + 0.156434i 0.513483 1.65419i 0.951057 + 0.309017i −0.545143 2.16860i 0.765933 1.55349i −1.41702 + 1.41702i 0.891007 + 0.453990i −2.47267 1.69880i −0.199188 2.22718i
23.10 0.987688 + 0.156434i 1.63125 + 0.582259i 0.951057 + 0.309017i −2.22772 + 0.193062i 1.52008 + 0.830274i −0.104631 + 0.104631i 0.891007 + 0.453990i 2.32195 + 1.89962i −2.23049 0.157807i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 137.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.2.l.a 80
3.b odd 2 1 inner 150.2.l.a 80
5.b even 2 1 750.2.l.b 80
5.c odd 4 1 750.2.l.a 80
5.c odd 4 1 750.2.l.c 80
15.d odd 2 1 750.2.l.b 80
15.e even 4 1 750.2.l.a 80
15.e even 4 1 750.2.l.c 80
25.d even 5 1 750.2.l.c 80
25.e even 10 1 750.2.l.a 80
25.f odd 20 1 inner 150.2.l.a 80
25.f odd 20 1 750.2.l.b 80
75.h odd 10 1 750.2.l.a 80
75.j odd 10 1 750.2.l.c 80
75.l even 20 1 inner 150.2.l.a 80
75.l even 20 1 750.2.l.b 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.l.a 80 1.a even 1 1 trivial
150.2.l.a 80 3.b odd 2 1 inner
150.2.l.a 80 25.f odd 20 1 inner
150.2.l.a 80 75.l even 20 1 inner
750.2.l.a 80 5.c odd 4 1
750.2.l.a 80 15.e even 4 1
750.2.l.a 80 25.e even 10 1
750.2.l.a 80 75.h odd 10 1
750.2.l.b 80 5.b even 2 1
750.2.l.b 80 15.d odd 2 1
750.2.l.b 80 25.f odd 20 1
750.2.l.b 80 75.l even 20 1
750.2.l.c 80 5.c odd 4 1
750.2.l.c 80 15.e even 4 1
750.2.l.c 80 25.d even 5 1
750.2.l.c 80 75.j odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(150, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database