# Properties

 Label 230.2.j.a Level $230$ Weight $2$ Character orbit 230.j Analytic conductor $1.837$ Analytic rank $0$ Dimension $120$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [230,2,Mod(9,230)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(230, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 10]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("230.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 230.j (of order $$22$$, degree $$10$$, 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: $$1.83655924649$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$12$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$120 q + 12 q^{4} - 4 q^{6} + 8 q^{9}+O(q^{10})$$ 120 * q + 12 * q^4 - 4 * q^6 + 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$120 q + 12 q^{4} - 4 q^{6} + 8 q^{9} + 8 q^{11} - 6 q^{15} - 12 q^{16} - 16 q^{19} - 22 q^{20} + 4 q^{24} - 52 q^{25} - 4 q^{26} - 8 q^{29} - 44 q^{30} + 12 q^{31} + 16 q^{35} - 8 q^{36} - 36 q^{39} - 28 q^{41} - 8 q^{44} + 16 q^{45} - 4 q^{46} - 58 q^{49} + 12 q^{50} - 24 q^{51} - 6 q^{54} - 36 q^{55} + 22 q^{56} - 102 q^{59} - 38 q^{60} + 72 q^{61} + 12 q^{64} - 138 q^{65} + 80 q^{66} - 212 q^{69} - 108 q^{70} + 176 q^{71} - 88 q^{74} - 100 q^{75} + 16 q^{76} - 104 q^{79} - 22 q^{80} - 28 q^{81} - 22 q^{84} + 2 q^{85} + 62 q^{86} + 48 q^{89} + 24 q^{90} - 56 q^{91} + 24 q^{94} + 18 q^{95} - 4 q^{96} + 188 q^{99}+O(q^{100})$$ 120 * q + 12 * q^4 - 4 * q^6 + 8 * q^9 + 8 * q^11 - 6 * q^15 - 12 * q^16 - 16 * q^19 - 22 * q^20 + 4 * q^24 - 52 * q^25 - 4 * q^26 - 8 * q^29 - 44 * q^30 + 12 * q^31 + 16 * q^35 - 8 * q^36 - 36 * q^39 - 28 * q^41 - 8 * q^44 + 16 * q^45 - 4 * q^46 - 58 * q^49 + 12 * q^50 - 24 * q^51 - 6 * q^54 - 36 * q^55 + 22 * q^56 - 102 * q^59 - 38 * q^60 + 72 * q^61 + 12 * q^64 - 138 * q^65 + 80 * q^66 - 212 * q^69 - 108 * q^70 + 176 * q^71 - 88 * q^74 - 100 * q^75 + 16 * q^76 - 104 * q^79 - 22 * q^80 - 28 * q^81 - 22 * q^84 + 2 * q^85 + 62 * q^86 + 48 * q^89 + 24 * q^90 - 56 * q^91 + 24 * q^94 + 18 * q^95 - 4 * q^96 + 188 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −0.281733 0.959493i −1.95972 + 1.69811i −0.841254 + 0.540641i −0.0903172 2.23424i 2.18144 + 1.40193i 0.748675 + 0.341908i 0.755750 + 0.654861i 0.529991 3.68617i −2.11830 + 0.716118i
9.2 −0.281733 0.959493i −1.62154 + 1.40507i −0.841254 + 0.540641i 1.91000 + 1.16271i 1.80499 + 1.16000i −4.06165 1.85489i 0.755750 + 0.654861i 0.228216 1.58728i 0.577502 2.16021i
9.3 −0.281733 0.959493i −0.0697998 + 0.0604819i −0.841254 + 0.540641i −0.948149 + 2.02510i 0.0776969 + 0.0499327i 1.04108 + 0.475445i 0.755750 + 0.654861i −0.425731 + 2.96102i 2.21019 + 0.339207i
9.4 −0.281733 0.959493i 0.951117 0.824147i −0.841254 + 0.540641i −1.90749 1.16682i −1.05872 0.680401i −4.71597 2.15371i 0.755750 + 0.654861i −0.201540 + 1.40174i −0.582157 + 2.15896i
9.5 −0.281733 0.959493i 1.78070 1.54299i −0.841254 + 0.540641i −0.352110 2.20817i −1.98217 1.27386i 3.91611 + 1.78843i 0.755750 + 0.654861i 0.363147 2.52574i −2.01952 + 0.959961i
9.6 −0.281733 0.959493i 2.36951 2.05319i −0.841254 + 0.540641i 1.30136 + 1.81837i −2.63759 1.69508i −1.55660 0.710876i 0.755750 + 0.654861i 0.972037 6.76066i 1.37808 1.76094i
9.7 0.281733 + 0.959493i −2.36951 + 2.05319i −0.841254 + 0.540641i −0.736350 + 2.11135i −2.63759 1.69508i 1.55660 + 0.710876i −0.755750 0.654861i 0.972037 6.76066i −2.23328 0.111687i
9.8 0.281733 + 0.959493i −1.78070 + 1.54299i −0.841254 + 0.540641i −0.284267 2.21793i −1.98217 1.27386i −3.91611 1.78843i −0.755750 0.654861i 0.363147 2.52574i 2.04800 0.897613i
9.9 0.281733 + 0.959493i −0.951117 + 0.824147i −0.841254 + 0.540641i 1.50149 1.65696i −1.05872 0.680401i 4.71597 + 2.15371i −0.755750 0.654861i −0.201540 + 1.40174i 2.01286 + 0.973851i
9.10 0.281733 + 0.959493i 0.0697998 0.0604819i −0.841254 + 0.540641i 1.48028 + 1.67594i 0.0776969 + 0.0499327i −1.04108 0.475445i −0.755750 0.654861i −0.425731 + 2.96102i −1.19101 + 1.89248i
9.11 0.281733 + 0.959493i 1.62154 1.40507i −0.841254 + 0.540641i −1.50506 + 1.65372i 1.80499 + 1.16000i 4.06165 + 1.85489i −0.755750 0.654861i 0.228216 1.58728i −2.01076 0.978188i
9.12 0.281733 + 0.959493i 1.95972 1.69811i −0.841254 + 0.540641i −0.542800 2.16919i 2.18144 + 1.40193i −0.748675 0.341908i −0.755750 0.654861i 0.529991 3.68617i 1.92839 1.13194i
29.1 −0.909632 0.415415i −0.768680 + 2.61788i 0.654861 + 0.755750i 1.33202 + 1.79603i 1.78672 2.06199i 1.27046 + 0.182665i −0.281733 0.959493i −3.73869 2.40271i −0.465548 2.18707i
29.2 −0.909632 0.415415i −0.642787 + 2.18913i 0.654861 + 0.755750i −1.99872 1.00255i 1.49410 1.72428i −2.70319 0.388660i −0.281733 0.959493i −1.85536 1.19237i 1.40163 + 1.74225i
29.3 −0.909632 0.415415i 0.0359844 0.122552i 0.654861 + 0.755750i −1.96836 + 1.06092i −0.0836423 + 0.0965284i −0.277941 0.0399619i −0.281733 0.959493i 2.51004 + 1.61310i 2.23121 0.147361i
29.4 −0.909632 0.415415i 0.192005 0.653908i 0.654861 + 0.755750i 2.23607 + 0.00163718i −0.446297 + 0.515054i −0.838442 0.120550i −0.281733 0.959493i 2.13303 + 1.37082i −2.03332 0.930385i
29.5 −0.909632 0.415415i 0.537451 1.83039i 0.654861 + 0.755750i −0.575877 2.16064i −1.24925 + 1.44172i 2.93634 + 0.422181i −0.281733 0.959493i −0.537711 0.345566i −0.373726 + 2.20462i
29.6 −0.909632 0.415415i 0.880099 2.99734i 0.654861 + 0.755750i −1.33842 + 1.79127i −2.04571 + 2.36087i −3.80587 0.547202i −0.281733 0.959493i −5.68573 3.65400i 1.96159 1.07339i
29.7 0.909632 + 0.415415i −0.880099 + 2.99734i 0.654861 + 0.755750i −2.18539 + 0.473349i −2.04571 + 2.36087i 3.80587 + 0.547202i 0.281733 + 0.959493i −5.68573 3.65400i −2.18454 0.477271i
29.8 0.909632 + 0.415415i −0.537451 + 1.83039i 0.654861 + 0.755750i 1.72616 + 1.42140i −1.24925 + 1.44172i −2.93634 0.422181i 0.281733 + 0.959493i −0.537711 0.345566i 0.979700 + 2.01002i
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 9.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.2.j.a 120
5.b even 2 1 inner 230.2.j.a 120
23.c even 11 1 inner 230.2.j.a 120
115.j even 22 1 inner 230.2.j.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.j.a 120 1.a even 1 1 trivial
230.2.j.a 120 5.b even 2 1 inner
230.2.j.a 120 23.c even 11 1 inner
230.2.j.a 120 115.j even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(230, [\chi])$$.