# Properties

 Label 150.3.i.a Level $150$ Weight $3$ Character orbit 150.i Analytic conductor $4.087$ Analytic rank $0$ Dimension $80$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,3,Mod(29,150)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("150.29");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 150.i (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$4.08720396540$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$20$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 40 q^{4} + 20 q^{9}+O(q^{10})$$ 80 * q - 40 * q^4 + 20 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 40 q^{4} + 20 q^{9} + 16 q^{10} + 20 q^{12} + 32 q^{15} - 80 q^{16} + 60 q^{19} - 60 q^{21} + 40 q^{22} + 116 q^{25} - 210 q^{27} - 40 q^{28} - 68 q^{30} + 180 q^{31} - 50 q^{33} - 120 q^{34} + 40 q^{36} - 40 q^{37} + 220 q^{39} + 32 q^{40} + 468 q^{45} + 120 q^{46} - 40 q^{48} - 680 q^{49} + 20 q^{51} - 120 q^{54} - 272 q^{55} - 156 q^{60} - 200 q^{61} - 830 q^{63} - 160 q^{64} + 160 q^{66} + 500 q^{67} - 280 q^{69} - 584 q^{70} + 120 q^{73} - 138 q^{75} - 80 q^{76} + 620 q^{78} + 400 q^{79} - 420 q^{81} + 180 q^{84} + 1632 q^{85} + 750 q^{87} + 160 q^{88} + 472 q^{90} - 340 q^{91} + 160 q^{94} + 20 q^{97} - 260 q^{99}+O(q^{100})$$ 80 * q - 40 * q^4 + 20 * q^9 + 16 * q^10 + 20 * q^12 + 32 * q^15 - 80 * q^16 + 60 * q^19 - 60 * q^21 + 40 * q^22 + 116 * q^25 - 210 * q^27 - 40 * q^28 - 68 * q^30 + 180 * q^31 - 50 * q^33 - 120 * q^34 + 40 * q^36 - 40 * q^37 + 220 * q^39 + 32 * q^40 + 468 * q^45 + 120 * q^46 - 40 * q^48 - 680 * q^49 + 20 * q^51 - 120 * q^54 - 272 * q^55 - 156 * q^60 - 200 * q^61 - 830 * q^63 - 160 * q^64 + 160 * q^66 + 500 * q^67 - 280 * q^69 - 584 * q^70 + 120 * q^73 - 138 * q^75 - 80 * q^76 + 620 * q^78 + 400 * q^79 - 420 * q^81 + 180 * q^84 + 1632 * q^85 + 750 * q^87 + 160 * q^88 + 472 * q^90 - 340 * q^91 + 160 * q^94 + 20 * q^97 - 260 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −0.437016 + 1.34500i −2.95317 + 0.527988i −1.61803 1.17557i 4.87810 + 1.09731i 0.580441 4.20275i 10.6357i 2.28825 1.66251i 8.44246 3.11848i −3.60769 + 6.08149i
29.2 −0.437016 + 1.34500i −2.88702 0.815530i −1.61803 1.17557i −1.13734 + 4.86893i 2.35856 3.52664i 10.7444i 2.28825 1.66251i 7.66982 + 4.70891i −6.05166 3.65752i
29.3 −0.437016 + 1.34500i −2.70555 1.29615i −1.61803 1.17557i −2.68239 4.21957i 2.92568 3.07252i 8.88015i 2.28825 1.66251i 5.64000 + 7.01359i 6.84756 1.76378i
29.4 −0.437016 + 1.34500i −1.99435 + 2.24110i −1.61803 1.17557i 2.92277 4.05678i −2.14271 3.66180i 10.3258i 2.28825 1.66251i −1.04510 8.93911i 4.17905 + 5.70399i
29.5 −0.437016 + 1.34500i −0.0352727 2.99979i −1.61803 1.17557i −4.11716 + 2.83707i 4.05013 + 1.26352i 2.70868i 2.28825 1.66251i −8.99751 + 0.211621i −2.01659 6.77742i
29.6 −0.437016 + 1.34500i −0.0189915 + 2.99994i −1.61803 1.17557i −4.54494 2.08411i −4.02661 1.33657i 1.94466i 2.28825 1.66251i −8.99928 0.113946i 4.78934 5.20214i
29.7 −0.437016 + 1.34500i 1.95554 2.27505i −1.61803 1.17557i 4.48538 + 2.20939i 2.20534 + 3.62443i 0.464101i 2.28825 1.66251i −1.35173 8.89791i −4.93180 + 5.06728i
29.8 −0.437016 + 1.34500i 2.11142 + 2.13118i −1.61803 1.17557i −0.399233 + 4.98404i −3.78915 + 1.90849i 0.905152i 2.28825 1.66251i −0.0838289 + 8.99961i −6.52904 2.71507i
29.9 −0.437016 + 1.34500i 2.48265 + 1.68417i −1.61803 1.17557i 3.26810 3.78412i −3.35017 + 2.60315i 7.29758i 2.28825 1.66251i 3.32712 + 8.36243i 3.66142 + 6.04930i
29.10 −0.437016 + 1.34500i 2.92672 0.659017i −1.61803 1.17557i −4.08750 2.87964i −0.392649 + 4.22443i 12.3067i 2.28825 1.66251i 8.13139 3.85752i 5.65941 4.23923i
29.11 0.437016 1.34500i −2.84724 + 0.945094i −1.61803 1.17557i 4.54494 + 2.08411i 0.0268580 + 4.24256i 1.94466i −2.28825 + 1.66251i 7.21359 5.38183i 4.78934 5.20214i
29.12 0.437016 1.34500i −2.67933 1.34951i −1.61803 1.17557i 0.399233 4.98404i −2.98600 + 3.01394i 0.905152i −2.28825 + 1.66251i 5.35766 + 7.23156i −6.52904 2.71507i
29.13 0.437016 1.34500i −2.36893 1.84070i −1.61803 1.17557i −3.26810 + 3.78412i −3.51100 + 2.38178i 7.29758i −2.28825 + 1.66251i 2.22361 + 8.72098i 3.66142 + 6.04930i
29.14 0.437016 1.34500i −1.51513 + 2.58928i −1.61803 1.17557i −2.92277 + 4.05678i 2.82044 + 3.16940i 10.3258i −2.28825 + 1.66251i −4.40877 7.84619i 4.17905 + 5.70399i
29.15 0.437016 1.34500i −0.277644 2.98712i −1.61803 1.17557i 4.08750 + 2.87964i −4.13901 0.931990i 12.3067i −2.28825 + 1.66251i −8.84583 + 1.65872i 5.65941 4.23923i
29.16 0.437016 1.34500i 0.410434 + 2.97179i −1.61803 1.17557i −4.87810 1.09731i 4.17642 + 0.746688i 10.6357i −2.28825 + 1.66251i −8.66309 + 2.43945i −3.60769 + 6.08149i
29.17 0.437016 1.34500i 1.55941 2.56286i −1.61803 1.17557i −4.48538 2.20939i −2.76555 3.21741i 0.464101i −2.28825 + 1.66251i −4.13649 7.99309i −4.93180 + 5.06728i
29.18 0.437016 1.34500i 1.66776 + 2.49371i −1.61803 1.17557i 1.13734 4.86893i 4.08287 1.15333i 10.7444i −2.28825 + 1.66251i −3.43719 + 8.31780i −6.05166 3.65752i
29.19 0.437016 1.34500i 2.06877 + 2.17260i −1.61803 1.17557i 2.68239 + 4.21957i 3.82622 1.83303i 8.88015i −2.28825 + 1.66251i −0.440370 + 8.98922i 6.84756 1.76378i
29.20 0.437016 1.34500i 2.86387 0.893441i −1.61803 1.17557i 4.11716 2.83707i 0.0498831 4.24235i 2.70868i −2.28825 + 1.66251i 7.40353 5.11740i −2.01659 6.77742i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.20 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.e even 10 1 inner
75.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.3.i.a 80
3.b odd 2 1 inner 150.3.i.a 80
25.e even 10 1 inner 150.3.i.a 80
75.h odd 10 1 inner 150.3.i.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.i.a 80 1.a even 1 1 trivial
150.3.i.a 80 3.b odd 2 1 inner
150.3.i.a 80 25.e even 10 1 inner
150.3.i.a 80 75.h odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(150, [\chi])$$.