# Properties

 Label 345.2.m.a Level $345$ Weight $2$ Character orbit 345.m Analytic conductor $2.755$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [345,2,Mod(16,345)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(345, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 0, 8]))

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

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

chi := DirichletCharacter("345.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$345 = 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 345.m (of order $$11$$, 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: $$2.75483886973$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$3$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30 q + 3 q^{3} + 2 q^{4} + 3 q^{5} + 11 q^{6} + q^{7} + 22 q^{8} - 3 q^{9}+O(q^{10})$$ 30 * q + 3 * q^3 + 2 * q^4 + 3 * q^5 + 11 * q^6 + q^7 + 22 * q^8 - 3 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$30 q + 3 q^{3} + 2 q^{4} + 3 q^{5} + 11 q^{6} + q^{7} + 22 q^{8} - 3 q^{9} - 7 q^{11} - 2 q^{12} + 7 q^{13} - 47 q^{14} - 3 q^{15} + 26 q^{16} + 15 q^{17} - 7 q^{19} - 2 q^{20} - 12 q^{21} - 18 q^{22} + 12 q^{23} - 3 q^{25} - 18 q^{26} + 3 q^{27} + 2 q^{28} - 6 q^{29} + 22 q^{31} + 33 q^{32} - 4 q^{33} - 53 q^{34} - 12 q^{35} - 9 q^{36} - 23 q^{37} + 21 q^{38} - 7 q^{39} + 11 q^{40} + 26 q^{41} + 3 q^{42} - 19 q^{43} - 47 q^{44} - 30 q^{45} - 44 q^{46} + 14 q^{47} + 7 q^{48} + 2 q^{49} - 11 q^{50} - 15 q^{51} + 55 q^{52} - 14 q^{53} - 11 q^{54} - 4 q^{55} + 6 q^{56} - 26 q^{57} - 18 q^{58} + 40 q^{59} - 9 q^{60} + 37 q^{61} + 19 q^{62} - 10 q^{63} + 14 q^{64} + 4 q^{65} - 4 q^{66} - 108 q^{67} + 54 q^{68} + 21 q^{69} - 30 q^{70} + 39 q^{71} + 22 q^{72} + 57 q^{73} - 11 q^{74} + 3 q^{75} + 22 q^{76} + 2 q^{77} + 7 q^{78} + 55 q^{79} + 7 q^{80} - 3 q^{81} - 26 q^{82} - 79 q^{83} - 2 q^{84} + 18 q^{85} - 44 q^{86} - 5 q^{87} - 8 q^{88} - 30 q^{89} + 40 q^{91} + 35 q^{92} + 44 q^{93} - 29 q^{94} + 7 q^{95} + 22 q^{96} - 52 q^{97} + 28 q^{98} + 4 q^{99}+O(q^{100})$$ 30 * q + 3 * q^3 + 2 * q^4 + 3 * q^5 + 11 * q^6 + q^7 + 22 * q^8 - 3 * q^9 - 7 * q^11 - 2 * q^12 + 7 * q^13 - 47 * q^14 - 3 * q^15 + 26 * q^16 + 15 * q^17 - 7 * q^19 - 2 * q^20 - 12 * q^21 - 18 * q^22 + 12 * q^23 - 3 * q^25 - 18 * q^26 + 3 * q^27 + 2 * q^28 - 6 * q^29 + 22 * q^31 + 33 * q^32 - 4 * q^33 - 53 * q^34 - 12 * q^35 - 9 * q^36 - 23 * q^37 + 21 * q^38 - 7 * q^39 + 11 * q^40 + 26 * q^41 + 3 * q^42 - 19 * q^43 - 47 * q^44 - 30 * q^45 - 44 * q^46 + 14 * q^47 + 7 * q^48 + 2 * q^49 - 11 * q^50 - 15 * q^51 + 55 * q^52 - 14 * q^53 - 11 * q^54 - 4 * q^55 + 6 * q^56 - 26 * q^57 - 18 * q^58 + 40 * q^59 - 9 * q^60 + 37 * q^61 + 19 * q^62 - 10 * q^63 + 14 * q^64 + 4 * q^65 - 4 * q^66 - 108 * q^67 + 54 * q^68 + 21 * q^69 - 30 * q^70 + 39 * q^71 + 22 * q^72 + 57 * q^73 - 11 * q^74 + 3 * q^75 + 22 * q^76 + 2 * q^77 + 7 * q^78 + 55 * q^79 + 7 * q^80 - 3 * q^81 - 26 * q^82 - 79 * q^83 - 2 * q^84 + 18 * q^85 - 44 * q^86 - 5 * q^87 - 8 * q^88 - 30 * q^89 + 40 * q^91 + 35 * q^92 + 44 * q^93 - 29 * q^94 + 7 * q^95 + 22 * q^96 - 52 * q^97 + 28 * q^98 + 4 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −1.52611 + 1.76122i −0.841254 + 0.540641i −0.488270 3.39599i −0.415415 0.909632i 0.331655 2.30671i 1.12603 0.330631i 2.80528 + 1.80285i 0.415415 0.909632i 2.23603 + 0.656559i
16.2 −0.758566 + 0.875431i −0.841254 + 0.540641i 0.0936713 + 0.651498i −0.415415 0.909632i 0.164852 1.14657i −1.51957 + 0.446185i −2.59035 1.66472i 0.415415 0.909632i 1.11144 + 0.326348i
16.3 0.996473 1.14999i −0.841254 + 0.540641i −0.0448918 0.312230i −0.415415 0.909632i −0.216554 + 1.50617i 2.33660 0.686088i 2.15640 + 1.38584i 0.415415 0.909632i −1.46002 0.428700i
31.1 −0.353423 + 2.45811i −0.415415 0.909632i −3.99842 1.17404i 0.654861 + 0.755750i 2.38279 0.699652i 2.47922 + 1.59330i 2.23579 4.89569i −0.654861 + 0.755750i −2.08916 + 1.34262i
31.2 −0.139065 + 0.967216i −0.415415 0.909632i 1.00282 + 0.294454i 0.654861 + 0.755750i 0.937580 0.275298i 1.72023 + 1.10553i −1.23611 + 2.70671i −0.654861 + 0.755750i −0.822041 + 0.528294i
31.3 0.101148 0.703501i −0.415415 0.909632i 1.43430 + 0.421149i 0.654861 + 0.755750i −0.681946 + 0.200237i −3.66580 2.35586i 1.03186 2.25945i −0.654861 + 0.755750i 0.597909 0.384253i
121.1 −0.883723 + 1.93508i 0.959493 + 0.281733i −1.65386 1.90866i −0.841254 + 0.540641i −1.39310 + 1.60773i −0.350847 + 2.44020i 1.07266 0.314961i 0.841254 + 0.540641i −0.302750 2.10567i
121.2 0.0328990 0.0720388i 0.959493 + 0.281733i 1.30561 + 1.50676i −0.841254 + 0.540641i 0.0518621 0.0598520i −0.274339 + 1.90807i 0.303474 0.0891079i 0.841254 + 0.540641i 0.0112707 + 0.0783895i
121.3 0.597726 1.30884i 0.959493 + 0.281733i −0.0460597 0.0531557i −0.841254 + 0.540641i 0.942257 1.08742i 0.462869 3.21932i 2.66406 0.782239i 0.841254 + 0.540641i 0.204772 + 1.42422i
151.1 −1.52611 1.76122i −0.841254 0.540641i −0.488270 + 3.39599i −0.415415 + 0.909632i 0.331655 + 2.30671i 1.12603 + 0.330631i 2.80528 1.80285i 0.415415 + 0.909632i 2.23603 0.656559i
151.2 −0.758566 0.875431i −0.841254 0.540641i 0.0936713 0.651498i −0.415415 + 0.909632i 0.164852 + 1.14657i −1.51957 0.446185i −2.59035 + 1.66472i 0.415415 + 0.909632i 1.11144 0.326348i
151.3 0.996473 + 1.14999i −0.841254 0.540641i −0.0448918 + 0.312230i −0.415415 + 0.909632i −0.216554 1.50617i 2.33660 + 0.686088i 2.15640 1.38584i 0.415415 + 0.909632i −1.46002 + 0.428700i
196.1 −2.00240 + 1.28686i 0.142315 0.989821i 1.52275 3.33435i 0.959493 0.281733i 0.988793 + 2.16515i 2.34831 + 2.71009i 0.564215 + 3.92420i −0.959493 0.281733i −1.55873 + 1.79887i
196.2 −0.766261 + 0.492446i 0.142315 0.989821i −0.486177 + 1.06458i 0.959493 0.281733i 0.378383 + 0.828544i −1.59175 1.83698i −0.410966 2.85833i −0.959493 0.281733i −0.596484 + 0.688379i
196.3 1.83027 1.17624i 0.142315 0.989821i 1.13552 2.48643i 0.959493 0.281733i −0.903797 1.97904i −0.659426 0.761019i −0.227095 1.57948i −0.959493 0.281733i 1.42475 1.64425i
211.1 −0.883723 1.93508i 0.959493 0.281733i −1.65386 + 1.90866i −0.841254 0.540641i −1.39310 1.60773i −0.350847 2.44020i 1.07266 + 0.314961i 0.841254 0.540641i −0.302750 + 2.10567i
211.2 0.0328990 + 0.0720388i 0.959493 0.281733i 1.30561 1.50676i −0.841254 0.540641i 0.0518621 + 0.0598520i −0.274339 1.90807i 0.303474 + 0.0891079i 0.841254 0.540641i 0.0112707 0.0783895i
211.3 0.597726 + 1.30884i 0.959493 0.281733i −0.0460597 + 0.0531557i −0.841254 0.540641i 0.942257 + 1.08742i 0.462869 + 3.21932i 2.66406 + 0.782239i 0.841254 0.540641i 0.204772 1.42422i
256.1 −0.353423 2.45811i −0.415415 + 0.909632i −3.99842 + 1.17404i 0.654861 0.755750i 2.38279 + 0.699652i 2.47922 1.59330i 2.23579 + 4.89569i −0.654861 0.755750i −2.08916 1.34262i
256.2 −0.139065 0.967216i −0.415415 + 0.909632i 1.00282 0.294454i 0.654861 0.755750i 0.937580 + 0.275298i 1.72023 1.10553i −1.23611 2.70671i −0.654861 0.755750i −0.822041 0.528294i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.2.m.a 30
23.c even 11 1 inner 345.2.m.a 30
23.c even 11 1 7935.2.a.bp 15
23.d odd 22 1 7935.2.a.bq 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.m.a 30 1.a even 1 1 trivial
345.2.m.a 30 23.c even 11 1 inner
7935.2.a.bp 15 23.c even 11 1
7935.2.a.bq 15 23.d odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{30} + 2 T_{2}^{28} - 22 T_{2}^{27} + 15 T_{2}^{26} - 33 T_{2}^{25} + 371 T_{2}^{24} - 330 T_{2}^{23} + 599 T_{2}^{22} - 3036 T_{2}^{21} + 6412 T_{2}^{20} - 4741 T_{2}^{19} + 22482 T_{2}^{18} - 25784 T_{2}^{17} + 69395 T_{2}^{16} + \cdots + 121$$ acting on $$S_{2}^{\mathrm{new}}(345, [\chi])$$.