# Properties

 Label 345.2.m.c Level $345$ Weight $2$ Character orbit 345.m Analytic conductor $2.755$ Analytic rank $0$ Dimension $50$ 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: $$50$$ Relative dimension: $$5$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$50 q - 5 q^{3} - 4 q^{4} + 5 q^{5} - 11 q^{6} - 3 q^{7} - 5 q^{9}+O(q^{10})$$ 50 * q - 5 * q^3 - 4 * q^4 + 5 * q^5 - 11 * q^6 - 3 * q^7 - 5 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$50 q - 5 q^{3} - 4 q^{4} + 5 q^{5} - 11 q^{6} - 3 q^{7} - 5 q^{9} + 15 q^{11} - 4 q^{12} - 19 q^{13} + 55 q^{14} + 5 q^{15} + 12 q^{16} + 5 q^{17} - 11 q^{19} + 4 q^{20} + 8 q^{21} - 18 q^{22} + 14 q^{23} + 66 q^{24} - 5 q^{25} - 18 q^{26} - 5 q^{27} + 10 q^{28} - 22 q^{29} + 6 q^{31} + 33 q^{32} + 4 q^{33} + 18 q^{34} - 8 q^{35} - 15 q^{36} + 25 q^{37} - 97 q^{38} - 19 q^{39} + 22 q^{40} - 42 q^{41} - 11 q^{42} - 25 q^{43} + 25 q^{44} - 50 q^{45} - 44 q^{46} + 86 q^{47} - 10 q^{48} - 8 q^{49} - 11 q^{50} - 17 q^{51} - 67 q^{52} - 26 q^{53} - 4 q^{55} - 132 q^{56} + 22 q^{57} + 8 q^{58} - 76 q^{59} + 4 q^{60} + 13 q^{61} - 8 q^{62} + 8 q^{63} + 76 q^{64} + 8 q^{65} + 4 q^{66} + 84 q^{67} + 66 q^{68} + 25 q^{69} + 22 q^{70} + 55 q^{71} - 59 q^{73} + 17 q^{74} - 5 q^{75} + 82 q^{76} - 56 q^{77} - 7 q^{78} + 7 q^{79} + 10 q^{80} - 5 q^{81} - 150 q^{82} + 19 q^{83} + 10 q^{84} + 6 q^{85} + 44 q^{86} - 11 q^{87} - 62 q^{88} - 74 q^{89} - 56 q^{91} + 41 q^{92} + 28 q^{93} - 161 q^{94} + 11 q^{95} - 44 q^{96} - 68 q^{97} - 198 q^{98} + 4 q^{99}+O(q^{100})$$ 50 * q - 5 * q^3 - 4 * q^4 + 5 * q^5 - 11 * q^6 - 3 * q^7 - 5 * q^9 + 15 * q^11 - 4 * q^12 - 19 * q^13 + 55 * q^14 + 5 * q^15 + 12 * q^16 + 5 * q^17 - 11 * q^19 + 4 * q^20 + 8 * q^21 - 18 * q^22 + 14 * q^23 + 66 * q^24 - 5 * q^25 - 18 * q^26 - 5 * q^27 + 10 * q^28 - 22 * q^29 + 6 * q^31 + 33 * q^32 + 4 * q^33 + 18 * q^34 - 8 * q^35 - 15 * q^36 + 25 * q^37 - 97 * q^38 - 19 * q^39 + 22 * q^40 - 42 * q^41 - 11 * q^42 - 25 * q^43 + 25 * q^44 - 50 * q^45 - 44 * q^46 + 86 * q^47 - 10 * q^48 - 8 * q^49 - 11 * q^50 - 17 * q^51 - 67 * q^52 - 26 * q^53 - 4 * q^55 - 132 * q^56 + 22 * q^57 + 8 * q^58 - 76 * q^59 + 4 * q^60 + 13 * q^61 - 8 * q^62 + 8 * q^63 + 76 * q^64 + 8 * q^65 + 4 * q^66 + 84 * q^67 + 66 * q^68 + 25 * q^69 + 22 * q^70 + 55 * q^71 - 59 * q^73 + 17 * q^74 - 5 * q^75 + 82 * q^76 - 56 * q^77 - 7 * q^78 + 7 * q^79 + 10 * q^80 - 5 * q^81 - 150 * q^82 + 19 * q^83 + 10 * q^84 + 6 * q^85 + 44 * q^86 - 11 * q^87 - 62 * q^88 - 74 * q^89 - 56 * q^91 + 41 * q^92 + 28 * q^93 - 161 * q^94 + 11 * q^95 - 44 * q^96 - 68 * q^97 - 198 * 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 −1.71238 + 1.97619i 0.841254 0.540641i −0.688461 4.78835i −0.415415 0.909632i −0.372136 + 2.58826i −3.22191 + 0.946039i 6.24206 + 4.01153i 0.415415 0.909632i 2.50896 + 0.736696i
16.2 −1.32781 + 1.53237i 0.841254 0.540641i −0.300461 2.08975i −0.415415 0.909632i −0.288560 + 2.00698i 3.59545 1.05572i 0.189750 + 0.121945i 0.415415 0.909632i 1.94549 + 0.571247i
16.3 −0.615779 + 0.710647i 0.841254 0.540641i 0.158794 + 1.10444i −0.415415 0.909632i −0.133822 + 0.930750i 0.864280 0.253775i −2.46475 1.58400i 0.415415 0.909632i 0.902231 + 0.264919i
16.4 0.707484 0.816480i 0.841254 0.540641i 0.118524 + 0.824351i −0.415415 0.909632i 0.153751 1.06936i 2.39967 0.704607i 2.57463 + 1.65461i 0.415415 0.909632i −1.03660 0.304372i
16.5 1.45237 1.67613i 0.841254 0.540641i −0.415386 2.88907i −0.415415 0.909632i 0.315630 2.19526i −1.14822 + 0.337148i −1.71422 1.10166i 0.415415 0.909632i −2.12799 0.624835i
31.1 −0.387876 + 2.69773i 0.415415 + 0.909632i −5.20834 1.52931i 0.654861 + 0.755750i −2.61507 + 0.767855i −4.11911 2.64719i 3.88144 8.49917i −0.654861 + 0.755750i −2.29282 + 1.47350i
31.2 −0.290073 + 2.01750i 0.415415 + 0.909632i −2.06718 0.606979i 0.654861 + 0.755750i −1.95568 + 0.574241i 3.11544 + 2.00217i 0.130777 0.286362i −0.654861 + 0.755750i −1.71468 + 1.10196i
31.3 −0.124133 + 0.863362i 0.415415 + 0.909632i 1.18900 + 0.349122i 0.654861 + 0.755750i −0.836908 + 0.245738i −2.78196 1.78785i −1.17370 + 2.57004i −0.654861 + 0.755750i −0.733775 + 0.471568i
31.4 0.0510170 0.354831i 0.415415 + 0.909632i 1.79568 + 0.527260i 0.654861 + 0.755750i 0.343959 0.100995i 2.04891 + 1.31675i 0.576535 1.26244i −0.654861 + 0.755750i 0.301573 0.193809i
31.5 0.193334 1.34467i 0.415415 + 0.909632i 0.148229 + 0.0435239i 0.654861 + 0.755750i 1.30347 0.382733i −0.958335 0.615885i 1.21586 2.66237i −0.654861 + 0.755750i 1.14284 0.734459i
121.1 −0.831671 + 1.82111i −0.959493 0.281733i −1.31503 1.51762i −0.841254 + 0.540641i 1.31105 1.51303i 0.450906 3.13612i 0.0155624 0.00456953i 0.841254 + 0.540641i −0.284918 1.98165i
121.2 −0.168180 + 0.368263i −0.959493 0.281733i 1.20239 + 1.38763i −0.841254 + 0.540641i 0.265120 0.305964i 0.179033 1.24520i −1.49013 + 0.437542i 0.841254 + 0.540641i −0.0576160 0.400728i
121.3 0.582364 1.27520i −0.959493 0.281733i 0.0227365 + 0.0262393i −0.841254 + 0.540641i −0.918039 + 1.05947i −0.746335 + 5.19088i 2.73690 0.803626i 0.841254 + 0.540641i 0.199509 + 1.38762i
121.4 0.627141 1.37325i −0.959493 0.281733i −0.182780 0.210939i −0.841254 + 0.540641i −0.988626 + 1.14093i 0.326039 2.26765i 2.49274 0.731935i 0.841254 + 0.540641i 0.214849 + 1.49431i
121.5 1.16525 2.55155i −0.959493 0.281733i −3.84287 4.43491i −0.841254 + 0.540641i −1.83691 + 2.11991i 0.000825631 0.00574239i −10.4110 + 3.05695i 0.841254 + 0.540641i 0.399198 + 2.77649i
151.1 −1.71238 1.97619i 0.841254 + 0.540641i −0.688461 + 4.78835i −0.415415 + 0.909632i −0.372136 2.58826i −3.22191 0.946039i 6.24206 4.01153i 0.415415 + 0.909632i 2.50896 0.736696i
151.2 −1.32781 1.53237i 0.841254 + 0.540641i −0.300461 + 2.08975i −0.415415 + 0.909632i −0.288560 2.00698i 3.59545 + 1.05572i 0.189750 0.121945i 0.415415 + 0.909632i 1.94549 0.571247i
151.3 −0.615779 0.710647i 0.841254 + 0.540641i 0.158794 1.10444i −0.415415 + 0.909632i −0.133822 0.930750i 0.864280 + 0.253775i −2.46475 + 1.58400i 0.415415 + 0.909632i 0.902231 0.264919i
151.4 0.707484 + 0.816480i 0.841254 + 0.540641i 0.118524 0.824351i −0.415415 + 0.909632i 0.153751 + 1.06936i 2.39967 + 0.704607i 2.57463 1.65461i 0.415415 + 0.909632i −1.03660 + 0.304372i
151.5 1.45237 + 1.67613i 0.841254 + 0.540641i −0.415386 + 2.88907i −0.415415 + 0.909632i 0.315630 + 2.19526i −1.14822 0.337148i −1.71422 + 1.10166i 0.415415 + 0.909632i −2.12799 + 0.624835i
See all 50 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 16.5 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.c 50
23.c even 11 1 inner 345.2.m.c 50
23.c even 11 1 7935.2.a.bv 25
23.d odd 22 1 7935.2.a.bw 25

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.m.c 50 1.a even 1 1 trivial
345.2.m.c 50 23.c even 11 1 inner
7935.2.a.bv 25 23.c even 11 1
7935.2.a.bw 25 23.d odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{50} + 7 T_{2}^{48} + 54 T_{2}^{46} + 11 T_{2}^{45} + 325 T_{2}^{44} + 77 T_{2}^{43} + \cdots + 64009$$ acting on $$S_{2}^{\mathrm{new}}(345, [\chi])$$.