# Properties

 Label 1805.2.b.m Level $1805$ Weight $2$ Character orbit 1805.b Analytic conductor $14.413$ 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] = [1805,2,Mod(1084,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("1805.1084");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.b (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: $$14.4129975648$$ Analytic rank: $$0$$ Dimension: $$40$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 48 q^{4} + 6 q^{5} + 20 q^{6} - 52 q^{9}+O(q^{10})$$ 40 * q - 48 * q^4 + 6 * q^5 + 20 * q^6 - 52 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 48 q^{4} + 6 q^{5} + 20 q^{6} - 52 q^{9} + 20 q^{11} + 40 q^{16} - 18 q^{20} - 92 q^{24} - 26 q^{25} + 76 q^{26} + 40 q^{30} + 4 q^{35} + 156 q^{36} - 80 q^{39} - 48 q^{44} - 22 q^{45} - 72 q^{49} - 32 q^{54} - 40 q^{55} + 80 q^{61} - 72 q^{64} + 16 q^{66} - 100 q^{74} - 66 q^{80} + 40 q^{81} + 44 q^{85} + 380 q^{96} - 128 q^{99}+O(q^{100})$$ 40 * q - 48 * q^4 + 6 * q^5 + 20 * q^6 - 52 * q^9 + 20 * q^11 + 40 * q^16 - 18 * q^20 - 92 * q^24 - 26 * q^25 + 76 * q^26 + 40 * q^30 + 4 * q^35 + 156 * q^36 - 80 * q^39 - 48 * q^44 - 22 * q^45 - 72 * q^49 - 32 * q^54 - 40 * q^55 + 80 * q^61 - 72 * q^64 + 16 * q^66 - 100 * q^74 - 66 * q^80 + 40 * q^81 + 44 * q^85 + 380 * q^96 - 128 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1084.1 2.66900i 1.76244i −5.12357 −0.705325 2.12191i 4.70394 2.20993i 8.33681i −0.106178 −5.66339 + 1.88251i
1084.2 2.66900i 1.76244i −5.12357 −0.705325 + 2.12191i 4.70394 2.20993i 8.33681i −0.106178 5.66339 + 1.88251i
1084.3 2.55697i 3.05876i −4.53809 1.22552 1.87032i 7.82114 1.35232i 6.48981i −6.35598 −4.78235 3.13361i
1084.4 2.55697i 3.05876i −4.53809 1.22552 + 1.87032i 7.82114 1.35232i 6.48981i −6.35598 4.78235 3.13361i
1084.5 2.42726i 1.05699i −3.89157 0.123305 2.23267i −2.56557 4.52506i 4.59133i 1.88278 −5.41925 0.299292i
1084.6 2.42726i 1.05699i −3.89157 0.123305 + 2.23267i −2.56557 4.52506i 4.59133i 1.88278 5.41925 0.299292i
1084.7 1.96724i 3.10308i −1.87002 −2.04586 0.902473i −6.10450 2.84392i 0.255698i −6.62912 −1.77538 + 4.02469i
1084.8 1.96724i 3.10308i −1.87002 −2.04586 + 0.902473i −6.10450 2.84392i 0.255698i −6.62912 1.77538 + 4.02469i
1084.9 1.74138i 0.766290i −1.03240 2.19703 0.415991i −1.33440 2.32579i 1.68496i 2.41280 −0.724398 3.82586i
1084.10 1.74138i 0.766290i −1.03240 2.19703 + 0.415991i −1.33440 2.32579i 1.68496i 2.41280 0.724398 3.82586i
1084.11 1.47577i 2.52656i −0.177886 2.11161 0.735611i −3.72861 4.13170i 2.68901i −3.38349 −1.08559 3.11624i
1084.12 1.47577i 2.52656i −0.177886 2.11161 + 0.735611i −3.72861 4.13170i 2.68901i −3.38349 1.08559 3.11624i
1084.13 1.40041i 2.85260i 0.0388655 0.846513 2.06964i 3.99479 4.34514i 2.85524i −5.13731 −2.89834 1.18546i
1084.14 1.40041i 2.85260i 0.0388655 0.846513 + 2.06964i 3.99479 4.34514i 2.85524i −5.13731 2.89834 1.18546i
1084.15 0.937092i 1.03012i 1.12186 −1.84457 1.26394i 0.965321 1.65832i 2.92547i 1.93884 −1.18443 + 1.72854i
1084.16 0.937092i 1.03012i 1.12186 −1.84457 + 1.26394i 0.965321 1.65832i 2.92547i 1.93884 1.18443 + 1.72854i
1084.17 0.717155i 1.56975i 1.48569 0.811198 2.08374i 1.12576 2.51917i 2.49978i 0.535883 −1.49436 0.581755i
1084.18 0.717155i 1.56975i 1.48569 0.811198 + 2.08374i 1.12576 2.51917i 2.49978i 0.535883 1.49436 0.581755i
1084.19 0.113485i 1.07621i 1.98712 −1.21941 1.87431i 0.122133 1.50555i 0.452479i 1.84178 −0.212706 + 0.138385i
1084.20 0.113485i 1.07621i 1.98712 −1.21941 + 1.87431i 0.122133 1.50555i 0.452479i 1.84178 0.212706 + 0.138385i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1084.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.b odd 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.b.m 40
5.b even 2 1 inner 1805.2.b.m 40
5.c odd 4 2 9025.2.a.cv 40
19.b odd 2 1 inner 1805.2.b.m 40
95.d odd 2 1 inner 1805.2.b.m 40
95.g even 4 2 9025.2.a.cv 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.b.m 40 1.a even 1 1 trivial
1805.2.b.m 40 5.b even 2 1 inner
1805.2.b.m 40 19.b odd 2 1 inner
1805.2.b.m 40 95.d odd 2 1 inner
9025.2.a.cv 40 5.c odd 4 2
9025.2.a.cv 40 95.g even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1805, [\chi])$$:

 $$T_{2}^{20} + 32 T_{2}^{18} + 431 T_{2}^{16} + 3186 T_{2}^{14} + 14144 T_{2}^{12} + 38798 T_{2}^{10} + \cdots + 80$$ T2^20 + 32*T2^18 + 431*T2^16 + 3186*T2^14 + 14144*T2^12 + 38798*T2^10 + 65129*T2^8 + 63724*T2^6 + 32511*T2^4 + 6620*T2^2 + 80 $$T_{29}^{20} - 300 T_{29}^{18} + 37770 T_{29}^{16} - 2590500 T_{29}^{14} + 105058275 T_{29}^{12} + \cdots + 299880050000$$ T29^20 - 300*T29^18 + 37770*T29^16 - 2590500*T29^14 + 105058275*T29^12 - 2555438000*T29^10 + 36006903250*T29^8 - 266964147500*T29^6 + 853235780625*T29^4 - 997363212500*T29^2 + 299880050000