# Properties

 Label 363.4.d.d Level $363$ Weight $4$ Character orbit 363.d Analytic conductor $21.418$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(362,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("363.362");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.d (of order $$2$$, degree $$1$$, 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: $$21.4176933321$$ Analytic rank: $$0$$ Dimension: $$40$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 8 q^{3} + 132 q^{4} - 60 q^{9}+O(q^{10})$$ 40 * q - 8 * q^3 + 132 * q^4 - 60 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 8 q^{3} + 132 q^{4} - 60 q^{9} - 90 q^{12} - 212 q^{15} + 460 q^{16} + 296 q^{25} + 280 q^{27} - 320 q^{31} - 188 q^{34} - 2254 q^{36} + 796 q^{37} + 1240 q^{42} - 1662 q^{45} - 486 q^{48} + 1536 q^{49} + 2268 q^{58} - 6636 q^{60} - 1512 q^{64} + 4556 q^{67} - 1140 q^{69} - 4276 q^{70} + 3462 q^{75} - 7640 q^{78} + 3572 q^{81} + 9408 q^{82} - 816 q^{91} - 11516 q^{93} - 7916 q^{97}+O(q^{100})$$ 40 * q - 8 * q^3 + 132 * q^4 - 60 * q^9 - 90 * q^12 - 212 * q^15 + 460 * q^16 + 296 * q^25 + 280 * q^27 - 320 * q^31 - 188 * q^34 - 2254 * q^36 + 796 * q^37 + 1240 * q^42 - 1662 * q^45 - 486 * q^48 + 1536 * q^49 + 2268 * q^58 - 6636 * q^60 - 1512 * q^64 + 4556 * q^67 - 1140 * q^69 - 4276 * q^70 + 3462 * q^75 - 7640 * q^78 + 3572 * q^81 + 9408 * q^82 - 816 * q^91 - 11516 * q^93 - 7916 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
362.1 −5.40291 −0.843368 + 5.12725i 21.1915 7.39188i 4.55664 27.7021i 14.1673i −71.2724 −25.5775 8.64832i 39.9377i
362.2 −5.40291 −0.843368 5.12725i 21.1915 7.39188i 4.55664 + 27.7021i 14.1673i −71.2724 −25.5775 + 8.64832i 39.9377i
362.3 −4.73579 −3.47601 + 3.86230i 14.4277 3.60263i 16.4617 18.2910i 29.9863i −30.4401 −2.83468 26.8508i 17.0613i
362.4 −4.73579 −3.47601 3.86230i 14.4277 3.60263i 16.4617 + 18.2910i 29.9863i −30.4401 −2.83468 + 26.8508i 17.0613i
362.5 −4.13050 2.49192 + 4.55964i 9.06107 0.648224i −10.2929 18.8336i 14.4739i −4.38274 −14.5806 + 22.7245i 2.67749i
362.6 −4.13050 2.49192 4.55964i 9.06107 0.648224i −10.2929 + 18.8336i 14.4739i −4.38274 −14.5806 22.7245i 2.67749i
362.7 −4.08863 5.14531 + 0.725123i 8.71694 6.78488i −21.0373 2.96476i 18.8082i −2.93129 25.9484 + 7.46197i 27.7409i
362.8 −4.08863 5.14531 0.725123i 8.71694 6.78488i −21.0373 + 2.96476i 18.8082i −2.93129 25.9484 7.46197i 27.7409i
362.9 −3.67560 −1.47344 + 4.98287i 5.51006 17.2142i 5.41579 18.3150i 16.8899i 9.15204 −22.6579 14.6839i 63.2724i
362.10 −3.67560 −1.47344 4.98287i 5.51006 17.2142i 5.41579 + 18.3150i 16.8899i 9.15204 −22.6579 + 14.6839i 63.2724i
362.11 −2.85553 −4.97590 + 1.49680i 0.154061 5.17068i 14.2088 4.27416i 16.0801i 22.4043 22.5192 14.8959i 14.7651i
362.12 −2.85553 −4.97590 1.49680i 0.154061 5.17068i 14.2088 + 4.27416i 16.0801i 22.4043 22.5192 + 14.8959i 14.7651i
362.13 −1.60481 −4.98378 1.47035i −5.42459 18.2497i 7.99801 + 2.35963i 4.32675i 21.5439 22.6761 + 14.6558i 29.2872i
362.14 −1.60481 −4.98378 + 1.47035i −5.42459 18.2497i 7.99801 2.35963i 4.32675i 21.5439 22.6761 14.6558i 29.2872i
362.15 −1.45083 4.65440 2.31010i −5.89509 15.3838i −6.75274 + 3.35155i 13.5090i 20.1594 16.3269 21.5042i 22.3193i
362.16 −1.45083 4.65440 + 2.31010i −5.89509 15.3838i −6.75274 3.35155i 13.5090i 20.1594 16.3269 + 21.5042i 22.3193i
362.17 −1.05192 −1.20902 + 5.05354i −6.89347 12.2895i 1.27179 5.31590i 22.4006i 15.6667 −24.0765 12.2197i 12.9275i
362.18 −1.05192 −1.20902 5.05354i −6.89347 12.2895i 1.27179 + 5.31590i 22.4006i 15.6667 −24.0765 + 12.2197i 12.9275i
362.19 −0.389720 2.66989 4.45777i −7.84812 4.25677i −1.04051 + 1.73728i 11.6631i 6.17632 −12.7434 23.8035i 1.65895i
362.20 −0.389720 2.66989 + 4.45777i −7.84812 4.25677i −1.04051 1.73728i 11.6631i 6.17632 −12.7434 + 23.8035i 1.65895i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 362.40 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
11.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.4.d.d 40
3.b odd 2 1 inner 363.4.d.d 40
11.b odd 2 1 inner 363.4.d.d 40
11.c even 5 1 33.4.f.a 40
11.d odd 10 1 33.4.f.a 40
33.d even 2 1 inner 363.4.d.d 40
33.f even 10 1 33.4.f.a 40
33.h odd 10 1 33.4.f.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.f.a 40 11.c even 5 1
33.4.f.a 40 11.d odd 10 1
33.4.f.a 40 33.f even 10 1
33.4.f.a 40 33.h odd 10 1
363.4.d.d 40 1.a even 1 1 trivial
363.4.d.d 40 3.b odd 2 1 inner
363.4.d.d 40 11.b odd 2 1 inner
363.4.d.d 40 33.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} - 113 T_{2}^{18} + 5291 T_{2}^{16} - 133194 T_{2}^{14} + 1954379 T_{2}^{12} - 16940617 T_{2}^{10} + 84188449 T_{2}^{8} - 225011796 T_{2}^{6} + 298938496 T_{2}^{4} - 163887680 T_{2}^{2} + \cdots + 18740480$$ acting on $$S_{4}^{\mathrm{new}}(363, [\chi])$$.