# Properties

 Label 150.3.j.a Level $150$ Weight $3$ Character orbit 150.j 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(11,150)]

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, 8]))

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

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

chi := DirichletCharacter("150.11");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 150.j (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.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 4 q^{3} + 40 q^{4} - 8 q^{7} + 20 q^{9}+O(q^{10})$$ 80 * q - 4 * q^3 + 40 * q^4 - 8 * q^7 + 20 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 4 q^{3} + 40 q^{4} - 8 q^{7} + 20 q^{9} - 12 q^{12} + 40 q^{13} - 60 q^{15} - 80 q^{16} - 32 q^{18} + 60 q^{19} + 60 q^{21} - 8 q^{22} - 180 q^{25} + 182 q^{27} - 24 q^{28} + 20 q^{30} - 180 q^{31} + 26 q^{33} - 120 q^{34} - 40 q^{36} + 128 q^{37} + 220 q^{39} + 128 q^{42} - 56 q^{43} - 100 q^{45} - 120 q^{46} + 24 q^{48} + 440 q^{49} - 20 q^{51} - 80 q^{52} - 120 q^{54} - 792 q^{57} - 100 q^{60} + 200 q^{61} - 574 q^{63} + 160 q^{64} - 160 q^{66} - 732 q^{67} - 220 q^{69} + 280 q^{70} + 64 q^{72} - 400 q^{73} + 590 q^{75} + 80 q^{76} - 260 q^{78} + 400 q^{79} + 140 q^{81} + 448 q^{82} + 180 q^{84} - 200 q^{85} + 310 q^{87} - 64 q^{88} + 440 q^{90} + 340 q^{91} + 348 q^{93} + 160 q^{94} - 276 q^{97} + 180 q^{99}+O(q^{100})$$ 80 * q - 4 * q^3 + 40 * q^4 - 8 * q^7 + 20 * q^9 - 12 * q^12 + 40 * q^13 - 60 * q^15 - 80 * q^16 - 32 * q^18 + 60 * q^19 + 60 * q^21 - 8 * q^22 - 180 * q^25 + 182 * q^27 - 24 * q^28 + 20 * q^30 - 180 * q^31 + 26 * q^33 - 120 * q^34 - 40 * q^36 + 128 * q^37 + 220 * q^39 + 128 * q^42 - 56 * q^43 - 100 * q^45 - 120 * q^46 + 24 * q^48 + 440 * q^49 - 20 * q^51 - 80 * q^52 - 120 * q^54 - 792 * q^57 - 100 * q^60 + 200 * q^61 - 574 * q^63 + 160 * q^64 - 160 * q^66 - 732 * q^67 - 220 * q^69 + 280 * q^70 + 64 * q^72 - 400 * q^73 + 590 * q^75 + 80 * q^76 - 260 * q^78 + 400 * q^79 + 140 * q^81 + 448 * q^82 + 180 * q^84 - 200 * q^85 + 310 * q^87 - 64 * q^88 + 440 * q^90 + 340 * q^91 + 348 * q^93 + 160 * q^94 - 276 * q^97 + 180 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −0.831254 1.14412i −2.89929 + 0.770808i −0.618034 + 1.90211i 4.91294 0.928983i 3.29194 + 2.67640i −5.64982 2.68999 0.874032i 7.81171 4.46959i −5.14677 4.84879i
11.2 −0.831254 1.14412i −2.74953 1.20004i −0.618034 + 1.90211i −3.75827 3.29779i 0.912562 + 4.14334i −3.79877 2.68999 0.874032i 6.11980 + 6.59909i −0.648993 + 7.04122i
11.3 −0.831254 1.14412i −2.21254 2.02600i −0.618034 + 1.90211i 1.16748 + 4.86179i −0.478809 + 4.21554i 6.03242 2.68999 0.874032i 0.790661 + 8.96520i 4.59201 5.37712i
11.4 −0.831254 1.14412i −1.90427 + 2.31814i −0.618034 + 1.90211i −4.93889 + 0.779331i 4.23516 + 0.251761i 8.16986 2.68999 0.874032i −1.74750 8.82872i 4.99712 + 5.00288i
11.5 −0.831254 1.14412i −0.513013 + 2.95581i −0.618034 + 1.90211i 2.36498 4.40532i 3.80825 1.87008i 1.96623 2.68999 0.874032i −8.47364 3.03274i −7.00613 + 0.956120i
11.6 −0.831254 1.14412i 0.571625 2.94504i −0.618034 + 1.90211i 0.970509 4.90491i −3.84465 + 1.79406i 12.2853 2.68999 0.874032i −8.34649 3.36692i −6.41856 + 2.96684i
11.7 −0.831254 1.14412i 0.888659 + 2.86536i −0.618034 + 1.90211i 1.35845 + 4.81192i 2.53962 3.39858i −9.92920 2.68999 0.874032i −7.42057 + 5.09266i 4.37621 5.55417i
11.8 −0.831254 1.14412i 2.70308 1.30129i −0.618034 + 1.90211i 4.40886 + 2.35838i −3.73578 2.01095i 0.752203 2.68999 0.874032i 5.61328 7.03499i −0.966599 7.00469i
11.9 −0.831254 1.14412i 2.78863 + 1.10613i −0.618034 + 1.90211i −4.36319 + 2.44184i −1.05251 4.11002i 4.66363 2.68999 0.874032i 6.55293 + 6.16920i 6.42068 + 2.96224i
11.10 −0.831254 1.14412i 2.82664 1.00504i −0.618034 + 1.90211i −1.09537 4.87854i −3.49954 2.39858i −13.2558 2.68999 0.874032i 6.97980 5.68176i −4.67112 + 5.30855i
11.11 0.831254 + 1.14412i −2.90622 0.744235i −0.618034 + 1.90211i 4.36319 2.44184i −1.56431 3.94372i 4.66363 −2.68999 + 0.874032i 7.89223 + 4.32582i 6.42068 + 2.96224i
11.12 0.831254 + 1.14412i −2.40316 + 1.79578i −0.618034 + 1.90211i −1.35845 4.81192i −4.05223 1.25675i −9.92920 −2.68999 + 0.874032i 2.55032 8.63110i 4.37621 5.55417i
11.13 0.831254 + 1.14412i −1.69605 2.47455i −0.618034 + 1.90211i 1.09537 + 4.87854i 1.42134 3.99747i −13.2558 −2.68999 + 0.874032i −3.24680 + 8.39394i −4.67112 + 5.30855i
11.14 0.831254 + 1.14412i −1.42196 2.64160i −0.618034 + 1.90211i −4.40886 2.35838i 1.84031 3.82273i 0.752203 −2.68999 + 0.874032i −4.95608 + 7.51247i −0.966599 7.00469i
11.15 0.831254 + 1.14412i −1.32235 + 2.69284i −0.618034 + 1.90211i −2.36498 + 4.40532i −4.18015 + 0.725510i 1.96623 −2.68999 + 0.874032i −5.50280 7.12174i −7.00613 + 0.956120i
11.16 0.831254 + 1.14412i 0.178022 + 2.99471i −0.618034 + 1.90211i 4.93889 0.779331i −3.27834 + 2.69305i 8.16986 −2.68999 + 0.874032i −8.93662 + 1.06625i 4.99712 + 5.00288i
11.17 0.831254 + 1.14412i 1.26859 2.71858i −0.618034 + 1.90211i −0.970509 + 4.90491i 4.16491 0.808400i 12.2853 −2.68999 + 0.874032i −5.78133 6.89755i −6.41856 + 2.96684i
11.18 0.831254 + 1.14412i 1.89250 + 2.32775i −0.618034 + 1.90211i −4.91294 + 0.928983i −1.09009 + 4.10021i −5.64982 −2.68999 + 0.874032i −1.83688 + 8.81055i −5.14677 4.84879i
11.19 0.831254 + 1.14412i 2.92978 + 0.645279i −0.618034 + 1.90211i 3.75827 + 3.29779i 1.69711 + 3.88842i −3.79877 −2.68999 + 0.874032i 8.16723 + 3.78105i −0.648993 + 7.04122i
11.20 0.831254 + 1.14412i 2.98083 0.338569i −0.618034 + 1.90211i −1.16748 4.86179i 2.86519 + 3.12900i 6.03242 −2.68999 + 0.874032i 8.77074 2.01844i 4.59201 5.37712i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.20 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.d even 5 1 inner
75.j 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.j.a 80
3.b odd 2 1 inner 150.3.j.a 80
25.d even 5 1 inner 150.3.j.a 80
75.j odd 10 1 inner 150.3.j.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.j.a 80 1.a even 1 1 trivial
150.3.j.a 80 3.b odd 2 1 inner
150.3.j.a 80 25.d even 5 1 inner
150.3.j.a 80 75.j odd 10 1 inner

## Hecke kernels

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