# Properties

 Label 350.2.m.a Level 350 Weight 2 Character orbit 350.m Analytic conductor 2.795 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.m (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{10})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{4} + 10q^{5} + 2q^{6} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 6q^{4} + 10q^{5} + 2q^{6} + 8q^{9} + 2q^{11} + 10q^{12} - 6q^{14} + 20q^{15} - 6q^{16} - 22q^{19} - 2q^{21} - 10q^{22} - 10q^{23} + 8q^{24} - 10q^{25} - 4q^{26} - 30q^{27} - 12q^{29} - 10q^{30} + 20q^{33} - 8q^{36} + 10q^{37} - 10q^{38} - 48q^{39} + 10q^{40} + 42q^{41} - 2q^{44} - 40q^{45} + 10q^{46} + 30q^{47} + 10q^{48} - 24q^{49} + 20q^{50} - 52q^{51} + 10q^{53} + 4q^{54} + 10q^{55} + 6q^{56} - 20q^{58} - 10q^{60} + 46q^{61} - 20q^{63} + 6q^{64} + 10q^{65} - 10q^{66} + 10q^{67} + 32q^{71} + 30q^{73} - 28q^{74} - 10q^{75} - 48q^{76} + 20q^{77} - 20q^{78} - 44q^{79} + 76q^{81} + 50q^{83} + 2q^{84} - 50q^{85} - 6q^{86} - 20q^{87} - 20q^{88} - 4q^{89} + 50q^{90} - 6q^{91} + 30q^{92} - 6q^{94} - 60q^{95} + 2q^{96} + 30q^{97} + 56q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −0.951057 0.309017i −0.840443 + 1.15677i 0.809017 + 0.587785i −1.82116 + 1.29744i 1.15677 0.840443i 1.00000i −0.587785 0.809017i 0.295277 + 0.908768i 2.13296 0.671171i
29.2 −0.951057 0.309017i 0.310301 0.427092i 0.809017 + 0.587785i 1.73700 + 1.40813i −0.427092 + 0.310301i 1.00000i −0.587785 0.809017i 0.840930 + 2.58812i −1.21685 1.87597i
29.3 −0.951057 0.309017i 1.06039 1.45950i 0.809017 + 0.587785i 0.717612 2.11779i −1.45950 + 1.06039i 1.00000i −0.587785 0.809017i −0.0786699 0.242121i −1.33692 + 1.79238i
29.4 0.951057 + 0.309017i −0.974986 + 1.34195i 0.809017 + 0.587785i 2.09432 + 0.783465i −1.34195 + 0.974986i 1.00000i 0.587785 + 0.809017i 0.0768108 + 0.236399i 1.74971 + 1.39230i
29.5 0.951057 + 0.309017i 1.26372 1.73935i 0.809017 + 0.587785i 0.915083 + 2.04025i 1.73935 1.26372i 1.00000i 0.587785 + 0.809017i −0.501328 1.54293i 0.239823 + 2.22317i
29.6 0.951057 + 0.309017i 1.41709 1.95046i 0.809017 + 0.587785i −0.0248168 2.23593i 1.95046 1.41709i 1.00000i 0.587785 + 0.809017i −0.869087 2.67478i 0.667338 2.13416i
169.1 −0.951057 + 0.309017i −0.840443 1.15677i 0.809017 0.587785i −1.82116 1.29744i 1.15677 + 0.840443i 1.00000i −0.587785 + 0.809017i 0.295277 0.908768i 2.13296 + 0.671171i
169.2 −0.951057 + 0.309017i 0.310301 + 0.427092i 0.809017 0.587785i 1.73700 1.40813i −0.427092 0.310301i 1.00000i −0.587785 + 0.809017i 0.840930 2.58812i −1.21685 + 1.87597i
169.3 −0.951057 + 0.309017i 1.06039 + 1.45950i 0.809017 0.587785i 0.717612 + 2.11779i −1.45950 1.06039i 1.00000i −0.587785 + 0.809017i −0.0786699 + 0.242121i −1.33692 1.79238i
169.4 0.951057 0.309017i −0.974986 1.34195i 0.809017 0.587785i 2.09432 0.783465i −1.34195 0.974986i 1.00000i 0.587785 0.809017i 0.0768108 0.236399i 1.74971 1.39230i
169.5 0.951057 0.309017i 1.26372 + 1.73935i 0.809017 0.587785i 0.915083 2.04025i 1.73935 + 1.26372i 1.00000i 0.587785 0.809017i −0.501328 + 1.54293i 0.239823 2.22317i
169.6 0.951057 0.309017i 1.41709 + 1.95046i 0.809017 0.587785i −0.0248168 + 2.23593i 1.95046 + 1.41709i 1.00000i 0.587785 0.809017i −0.869087 + 2.67478i 0.667338 + 2.13416i
239.1 −0.587785 0.809017i −2.21568 0.719918i −0.309017 + 0.951057i 0.562587 + 2.16414i 0.719918 + 2.21568i 1.00000i 0.951057 0.309017i 1.96390 + 1.42686i 1.42014 1.72719i
239.2 −0.587785 0.809017i 0.195447 + 0.0635047i −0.309017 + 0.951057i 0.0419655 2.23567i −0.0635047 0.195447i 1.00000i 0.951057 0.309017i −2.39288 1.73853i −1.83337 + 1.28015i
239.3 −0.587785 0.809017i 1.85325 + 0.602159i −0.309017 + 0.951057i 1.98854 + 1.02259i −0.602159 1.85325i 1.00000i 0.951057 0.309017i 0.644904 + 0.468550i −0.341542 2.20983i
239.4 0.587785 + 0.809017i −3.21572 1.04485i −0.309017 + 0.951057i −2.16379 0.563933i −1.04485 3.21572i 1.00000i −0.951057 + 0.309017i 6.82209 + 4.95654i −0.815611 2.08201i
239.5 0.587785 + 0.809017i −0.331395 0.107677i −0.309017 + 0.951057i −1.25842 + 1.84834i −0.107677 0.331395i 1.00000i −0.951057 + 0.309017i −2.32882 1.69199i −2.23502 + 0.0683444i
239.6 0.587785 + 0.809017i 1.47802 + 0.480239i −0.309017 + 0.951057i 2.21108 0.333354i 0.480239 + 1.47802i 1.00000i −0.951057 + 0.309017i −0.473123 0.343744i 1.56933 + 1.59286i
309.1 −0.587785 + 0.809017i −2.21568 + 0.719918i −0.309017 0.951057i 0.562587 2.16414i 0.719918 2.21568i 1.00000i 0.951057 + 0.309017i 1.96390 1.42686i 1.42014 + 1.72719i
309.2 −0.587785 + 0.809017i 0.195447 0.0635047i −0.309017 0.951057i 0.0419655 + 2.23567i −0.0635047 + 0.195447i 1.00000i 0.951057 + 0.309017i −2.39288 + 1.73853i −1.83337 1.28015i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 309.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.m.a 24
25.e even 10 1 inner 350.2.m.a 24
25.f odd 20 1 8750.2.a.z 12
25.f odd 20 1 8750.2.a.bb 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.m.a 24 1.a even 1 1 trivial
350.2.m.a 24 25.e even 10 1 inner
8750.2.a.z 12 25.f odd 20 1
8750.2.a.bb 12 25.f odd 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database