# Properties

 Label 50.3.f.b Level $50$ Weight $3$ Character orbit 50.f Analytic conductor $1.362$ Analytic rank $0$ Dimension $24$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,3,Mod(3,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([7]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("50.3");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 50.f (of order $$20$$, degree $$8$$, 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: $$1.36240132180$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$3$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 6 q^{2} + 2 q^{3} - 4 q^{6} - 2 q^{7} - 12 q^{8} + 40 q^{9}+O(q^{10})$$ 24 * q + 6 * q^2 + 2 * q^3 - 4 * q^6 - 2 * q^7 - 12 * q^8 + 40 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 6 q^{2} + 2 q^{3} - 4 q^{6} - 2 q^{7} - 12 q^{8} + 40 q^{9} - 32 q^{11} + 4 q^{12} + 2 q^{13} + 30 q^{14} - 20 q^{15} + 24 q^{16} - 92 q^{17} - 136 q^{18} - 230 q^{19} - 20 q^{20} + 68 q^{21} - 48 q^{22} - 18 q^{23} + 40 q^{25} + 36 q^{26} + 260 q^{27} + 44 q^{28} + 100 q^{29} + 120 q^{30} - 132 q^{31} + 96 q^{32} + 364 q^{33} + 150 q^{34} + 50 q^{35} - 108 q^{36} - 192 q^{37} + 20 q^{38} - 80 q^{39} + 20 q^{40} + 168 q^{41} - 8 q^{42} - 78 q^{43} - 40 q^{44} - 310 q^{45} + 26 q^{46} - 22 q^{47} - 8 q^{48} - 30 q^{50} + 168 q^{51} + 4 q^{52} - 108 q^{53} - 80 q^{54} - 40 q^{55} - 48 q^{56} + 280 q^{57} + 40 q^{58} + 450 q^{59} - 100 q^{60} - 492 q^{61} - 458 q^{62} - 558 q^{63} + 120 q^{65} + 202 q^{66} - 572 q^{67} - 136 q^{68} - 670 q^{69} - 260 q^{70} - 2 q^{71} + 128 q^{72} + 262 q^{73} + 140 q^{75} - 40 q^{76} + 496 q^{77} - 62 q^{78} - 360 q^{79} - 80 q^{80} - 46 q^{81} + 272 q^{82} + 772 q^{83} + 620 q^{84} + 490 q^{85} - 264 q^{86} + 210 q^{87} - 84 q^{88} + 900 q^{89} + 1110 q^{90} + 798 q^{91} - 16 q^{92} + 294 q^{93} - 190 q^{94} + 16 q^{96} + 378 q^{97} + 106 q^{98}+O(q^{100})$$ 24 * q + 6 * q^2 + 2 * q^3 - 4 * q^6 - 2 * q^7 - 12 * q^8 + 40 * q^9 - 32 * q^11 + 4 * q^12 + 2 * q^13 + 30 * q^14 - 20 * q^15 + 24 * q^16 - 92 * q^17 - 136 * q^18 - 230 * q^19 - 20 * q^20 + 68 * q^21 - 48 * q^22 - 18 * q^23 + 40 * q^25 + 36 * q^26 + 260 * q^27 + 44 * q^28 + 100 * q^29 + 120 * q^30 - 132 * q^31 + 96 * q^32 + 364 * q^33 + 150 * q^34 + 50 * q^35 - 108 * q^36 - 192 * q^37 + 20 * q^38 - 80 * q^39 + 20 * q^40 + 168 * q^41 - 8 * q^42 - 78 * q^43 - 40 * q^44 - 310 * q^45 + 26 * q^46 - 22 * q^47 - 8 * q^48 - 30 * q^50 + 168 * q^51 + 4 * q^52 - 108 * q^53 - 80 * q^54 - 40 * q^55 - 48 * q^56 + 280 * q^57 + 40 * q^58 + 450 * q^59 - 100 * q^60 - 492 * q^61 - 458 * q^62 - 558 * q^63 + 120 * q^65 + 202 * q^66 - 572 * q^67 - 136 * q^68 - 670 * q^69 - 260 * q^70 - 2 * q^71 + 128 * q^72 + 262 * q^73 + 140 * q^75 - 40 * q^76 + 496 * q^77 - 62 * q^78 - 360 * q^79 - 80 * q^80 - 46 * q^81 + 272 * q^82 + 772 * q^83 + 620 * q^84 + 490 * q^85 - 264 * q^86 + 210 * q^87 - 84 * q^88 + 900 * q^89 + 1110 * q^90 + 798 * q^91 - 16 * q^92 + 294 * q^93 - 190 * q^94 + 16 * q^96 + 378 * q^97 + 106 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 0.642040 + 1.26007i −0.828443 5.23058i −1.17557 + 1.61803i 4.96105 + 0.622901i 6.05902 4.40214i 4.28629 4.28629i −2.79360 0.442463i −18.1132 + 5.88532i 2.40029 + 6.65121i
3.2 0.642040 + 1.26007i 0.304004 + 1.91941i −1.17557 + 1.61803i −3.29532 + 3.76044i −2.22341 + 1.61540i 6.96775 6.96775i −2.79360 0.442463i 4.96781 1.61414i −6.85415 1.73799i
3.3 0.642040 + 1.26007i 0.382399 + 2.41437i −1.17557 + 1.61803i 3.59048 3.47972i −2.79677 + 2.03197i −7.03135 + 7.03135i −2.79360 0.442463i 2.87654 0.934645i 6.68993 + 2.29016i
13.1 1.39680 0.221232i −2.14176 + 4.20343i 1.90211 0.618034i −1.43099 + 4.79085i −2.06168 + 6.34519i 8.41873 8.41873i 2.52015 1.28408i −7.79165 10.7243i −0.938916 + 7.00845i
13.2 1.39680 0.221232i −0.299213 + 0.587238i 1.90211 0.618034i 3.77205 3.28201i −0.288025 + 0.886451i −5.08008 + 5.08008i 2.52015 1.28408i 5.03475 + 6.92973i 4.54273 5.41882i
13.3 1.39680 0.221232i 1.54417 3.03060i 1.90211 0.618034i −4.51960 + 2.13851i 1.48643 4.57476i −2.71827 + 2.71827i 2.52015 1.28408i −1.51000 2.07833i −5.83988 + 3.98695i
17.1 0.642040 1.26007i −0.828443 + 5.23058i −1.17557 1.61803i 4.96105 0.622901i 6.05902 + 4.40214i 4.28629 + 4.28629i −2.79360 + 0.442463i −18.1132 5.88532i 2.40029 6.65121i
17.2 0.642040 1.26007i 0.304004 1.91941i −1.17557 1.61803i −3.29532 3.76044i −2.22341 1.61540i 6.96775 + 6.96775i −2.79360 + 0.442463i 4.96781 + 1.61414i −6.85415 + 1.73799i
17.3 0.642040 1.26007i 0.382399 2.41437i −1.17557 1.61803i 3.59048 + 3.47972i −2.79677 2.03197i −7.03135 7.03135i −2.79360 + 0.442463i 2.87654 + 0.934645i 6.68993 2.29016i
23.1 0.221232 1.39680i −5.16469 + 2.63154i −1.90211 0.618034i −4.87769 1.09913i 2.53315 + 7.79623i −2.21400 + 2.21400i −1.28408 + 2.52015i 14.4589 19.9010i −2.61437 + 6.57001i
23.2 0.221232 1.39680i 0.687579 0.350339i −1.90211 0.618034i 2.80598 4.13842i −0.337240 1.03792i 2.38638 2.38638i −1.28408 + 2.52015i −4.94004 + 6.79938i −5.15978 4.83494i
23.3 0.221232 1.39680i 4.75588 2.42324i −1.90211 0.618034i −2.45795 + 4.35413i −2.33264 7.17912i −6.88294 + 6.88294i −1.28408 + 2.52015i 11.4562 15.7681i 5.53808 + 4.39655i
27.1 1.39680 + 0.221232i −2.14176 4.20343i 1.90211 + 0.618034i −1.43099 4.79085i −2.06168 6.34519i 8.41873 + 8.41873i 2.52015 + 1.28408i −7.79165 + 10.7243i −0.938916 7.00845i
27.2 1.39680 + 0.221232i −0.299213 0.587238i 1.90211 + 0.618034i 3.77205 + 3.28201i −0.288025 0.886451i −5.08008 5.08008i 2.52015 + 1.28408i 5.03475 6.92973i 4.54273 + 5.41882i
27.3 1.39680 + 0.221232i 1.54417 + 3.03060i 1.90211 + 0.618034i −4.51960 2.13851i 1.48643 + 4.57476i −2.71827 2.71827i 2.52015 + 1.28408i −1.51000 + 2.07833i −5.83988 3.98695i
33.1 −1.26007 0.642040i −3.62730 0.574508i 1.17557 + 1.61803i 0.654751 + 4.95694i 4.20181 + 3.05279i −8.92599 + 8.92599i −0.442463 2.79360i 4.26773 + 1.38667i 2.35752 6.66649i
33.2 −1.26007 0.642040i 0.185739 + 0.0294182i 1.17557 + 1.61803i 4.66951 1.78765i −0.215157 0.156321i 7.34278 7.34278i −0.442463 2.79360i −8.52588 2.77022i −7.03167 0.745442i
33.3 −1.26007 0.642040i 5.20163 + 0.823858i 1.17557 + 1.61803i −3.87227 + 3.16315i −6.02549 4.37778i 2.45070 2.45070i −0.442463 2.79360i 17.8187 + 5.78966i 6.91021 1.49965i
37.1 0.221232 + 1.39680i −5.16469 2.63154i −1.90211 + 0.618034i −4.87769 + 1.09913i 2.53315 7.79623i −2.21400 2.21400i −1.28408 2.52015i 14.4589 + 19.9010i −2.61437 6.57001i
37.2 0.221232 + 1.39680i 0.687579 + 0.350339i −1.90211 + 0.618034i 2.80598 + 4.13842i −0.337240 + 1.03792i 2.38638 + 2.38638i −1.28408 2.52015i −4.94004 6.79938i −5.15978 + 4.83494i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.3.f.b 24
4.b odd 2 1 400.3.bg.b 24
5.b even 2 1 250.3.f.e 24
5.c odd 4 1 250.3.f.d 24
5.c odd 4 1 250.3.f.f 24
25.d even 5 1 250.3.f.f 24
25.e even 10 1 250.3.f.d 24
25.f odd 20 1 inner 50.3.f.b 24
25.f odd 20 1 250.3.f.e 24
100.l even 20 1 400.3.bg.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.3.f.b 24 1.a even 1 1 trivial
50.3.f.b 24 25.f odd 20 1 inner
250.3.f.d 24 5.c odd 4 1
250.3.f.d 24 25.e even 10 1
250.3.f.e 24 5.b even 2 1
250.3.f.e 24 25.f odd 20 1
250.3.f.f 24 5.c odd 4 1
250.3.f.f 24 25.d even 5 1
400.3.bg.b 24 4.b odd 2 1
400.3.bg.b 24 100.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} - 2 T_{3}^{23} - 18 T_{3}^{22} - 96 T_{3}^{21} + 104 T_{3}^{20} + 2798 T_{3}^{19} + \cdots + 533794816$$ acting on $$S_{3}^{\mathrm{new}}(50, [\chi])$$.