# Properties

 Label 418.2.s.a Level $418$ Weight $2$ Character orbit 418.s Analytic conductor $3.338$ Analytic rank $0$ Dimension $80$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.s (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$10$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 10 q^{2} - 3 q^{3} + 10 q^{4} - 2 q^{5} + 7 q^{6} - 10 q^{7} + 20 q^{8} - 11 q^{9}+O(q^{10})$$ 80 * q - 10 * q^2 - 3 * q^3 + 10 * q^4 - 2 * q^5 + 7 * q^6 - 10 * q^7 + 20 * q^8 - 11 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 10 q^{2} - 3 q^{3} + 10 q^{4} - 2 q^{5} + 7 q^{6} - 10 q^{7} + 20 q^{8} - 11 q^{9} + 2 q^{10} - q^{11} + 5 q^{13} - 4 q^{14} - 27 q^{15} + 10 q^{16} - 6 q^{17} - 17 q^{18} - 2 q^{19} + 4 q^{20} + 24 q^{21} - 2 q^{22} - 6 q^{23} - 7 q^{24} - 10 q^{26} - 45 q^{27} + 6 q^{28} - 65 q^{29} + 30 q^{30} + 40 q^{32} + 3 q^{33} + 24 q^{34} - 13 q^{35} - q^{36} + 22 q^{38} - 30 q^{39} - 3 q^{40} - 14 q^{41} - 14 q^{42} + 12 q^{43} + 24 q^{44} - 12 q^{45} - 2 q^{46} - q^{47} + 3 q^{48} + 32 q^{49} + 30 q^{50} - 28 q^{51} - 5 q^{52} - q^{53} - 27 q^{54} - 23 q^{55} + 28 q^{57} - 10 q^{58} + 56 q^{59} - 28 q^{60} + 28 q^{61} + 15 q^{62} + 88 q^{63} - 20 q^{64} + 8 q^{65} - 57 q^{66} - 27 q^{67} - 60 q^{69} + 17 q^{70} + 2 q^{71} + 11 q^{72} - q^{73} + 12 q^{74} - 35 q^{75} - 11 q^{76} - 8 q^{77} - 6 q^{79} + 3 q^{80} + 43 q^{81} - 16 q^{82} - 25 q^{83} + 52 q^{84} - 33 q^{85} - 43 q^{86} - 9 q^{88} - 36 q^{89} + 74 q^{90} + 38 q^{91} - 11 q^{92} + 15 q^{93} - 2 q^{94} - 61 q^{95} - 24 q^{97} - 44 q^{98} + 37 q^{99}+O(q^{100})$$ 80 * q - 10 * q^2 - 3 * q^3 + 10 * q^4 - 2 * q^5 + 7 * q^6 - 10 * q^7 + 20 * q^8 - 11 * q^9 + 2 * q^10 - q^11 + 5 * q^13 - 4 * q^14 - 27 * q^15 + 10 * q^16 - 6 * q^17 - 17 * q^18 - 2 * q^19 + 4 * q^20 + 24 * q^21 - 2 * q^22 - 6 * q^23 - 7 * q^24 - 10 * q^26 - 45 * q^27 + 6 * q^28 - 65 * q^29 + 30 * q^30 + 40 * q^32 + 3 * q^33 + 24 * q^34 - 13 * q^35 - q^36 + 22 * q^38 - 30 * q^39 - 3 * q^40 - 14 * q^41 - 14 * q^42 + 12 * q^43 + 24 * q^44 - 12 * q^45 - 2 * q^46 - q^47 + 3 * q^48 + 32 * q^49 + 30 * q^50 - 28 * q^51 - 5 * q^52 - q^53 - 27 * q^54 - 23 * q^55 + 28 * q^57 - 10 * q^58 + 56 * q^59 - 28 * q^60 + 28 * q^61 + 15 * q^62 + 88 * q^63 - 20 * q^64 + 8 * q^65 - 57 * q^66 - 27 * q^67 - 60 * q^69 + 17 * q^70 + 2 * q^71 + 11 * q^72 - q^73 + 12 * q^74 - 35 * q^75 - 11 * q^76 - 8 * q^77 - 6 * q^79 + 3 * q^80 + 43 * q^81 - 16 * q^82 - 25 * q^83 + 52 * q^84 - 33 * q^85 - 43 * q^86 - 9 * q^88 - 36 * q^89 + 74 * q^90 + 38 * q^91 - 11 * q^92 + 15 * q^93 - 2 * q^94 - 61 * q^95 - 24 * q^97 - 44 * q^98 + 37 * 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −0.913545 0.406737i −1.94271 1.74922i 0.669131 + 0.743145i 0.100715 0.958243i 1.06328 + 2.38816i −0.0448322 0.0145669i −0.309017 0.951057i 0.400750 + 3.81288i −0.481761 + 0.834434i
107.2 −0.913545 0.406737i −1.85282 1.66828i 0.669131 + 0.743145i −0.211376 + 2.01110i 1.01408 + 2.27766i 0.508834 + 0.165330i −0.309017 0.951057i 0.336173 + 3.19847i 1.01109 1.75126i
107.3 −0.913545 0.406737i −1.47425 1.32742i 0.669131 + 0.743145i 0.214664 2.04239i 0.806884 + 1.81229i −4.92396 1.59989i −0.309017 0.951057i 0.0977827 + 0.930340i −1.02682 + 1.77850i
107.4 −0.913545 0.406737i −0.808456 0.727937i 0.669131 + 0.743145i 0.425216 4.04566i 0.442482 + 0.993832i 4.38637 + 1.42522i −0.309017 0.951057i −0.189877 1.80656i −2.03397 + 3.52294i
107.5 −0.913545 0.406737i −0.370496 0.333596i 0.669131 + 0.743145i −0.348800 + 3.31861i 0.202779 + 0.455450i 2.27900 + 0.740493i −0.309017 0.951057i −0.287604 2.73637i 1.66844 2.88983i
107.6 −0.913545 0.406737i 0.0608391 + 0.0547798i 0.669131 + 0.743145i −0.122891 + 1.16923i −0.0332984 0.0747894i −0.576821 0.187421i −0.309017 0.951057i −0.312885 2.97690i 0.587836 1.01816i
107.7 −0.913545 0.406737i 0.116360 + 0.104771i 0.669131 + 0.743145i 0.258207 2.45667i −0.0636859 0.143041i −0.618608 0.200998i −0.309017 0.951057i −0.311023 2.95918i −1.23510 + 2.13926i
107.8 −0.913545 0.406737i 1.34081 + 1.20727i 0.669131 + 0.743145i 0.264977 2.52108i −0.733850 1.64825i −0.512875 0.166643i −0.309017 0.951057i 0.0266838 + 0.253879i −1.26749 + 2.19535i
107.9 −0.913545 0.406737i 2.10303 + 1.89358i 0.669131 + 0.743145i −0.360171 + 3.42680i −1.15103 2.58525i 3.62558 + 1.17802i −0.309017 0.951057i 0.523516 + 4.98092i 1.72284 2.98404i
107.10 −0.913545 0.406737i 2.22316 + 2.00174i 0.669131 + 0.743145i −0.155938 + 1.48365i −1.21677 2.73292i −4.91364 1.59654i −0.309017 0.951057i 0.621881 + 5.91681i 0.745913 1.29196i
145.1 0.978148 0.207912i −1.24186 + 2.78926i 0.913545 0.406737i −1.24858 + 1.38669i −0.634802 + 2.98651i −0.788641 + 1.08547i 0.809017 0.587785i −4.23038 4.69832i −0.932989 + 1.61598i
145.2 0.978148 0.207912i −0.879298 + 1.97493i 0.913545 0.406737i 0.155783 0.173014i −0.449471 + 2.11459i 2.76336 3.80344i 0.809017 0.587785i −1.11981 1.24368i 0.116407 0.201622i
145.3 0.978148 0.207912i −0.766319 + 1.72118i 0.913545 0.406737i 1.42537 1.58303i −0.391719 + 1.84289i 0.188148 0.258963i 0.809017 0.587785i −0.367824 0.408510i 1.06509 1.84479i
145.4 0.978148 0.207912i −0.222075 + 0.498789i 0.913545 0.406737i −0.572934 + 0.636307i −0.113518 + 0.534061i −2.17900 + 2.99913i 0.809017 0.587785i 1.80792 + 2.00790i −0.428118 + 0.741522i
145.5 0.978148 0.207912i −0.167101 + 0.375316i 0.913545 0.406737i 1.82285 2.02448i −0.0854173 + 0.401857i 0.891945 1.22766i 0.809017 0.587785i 1.89445 + 2.10400i 1.36210 2.35923i
145.6 0.978148 0.207912i 0.100523 0.225779i 0.913545 0.406737i −2.22917 + 2.47574i 0.0513844 0.241745i −1.25716 + 1.73034i 0.809017 0.587785i 1.96652 + 2.18404i −1.66572 + 2.88511i
145.7 0.978148 0.207912i 0.411849 0.925027i 0.913545 0.406737i 2.71323 3.01335i 0.210525 0.990442i −2.18824 + 3.01185i 0.809017 0.587785i 1.32134 + 1.46749i 2.02743 3.51161i
145.8 0.978148 0.207912i 0.588176 1.32107i 0.913545 0.406737i −1.28088 + 1.42257i 0.300658 1.41449i 1.62246 2.23312i 0.809017 0.587785i 0.608128 + 0.675395i −0.957126 + 1.65779i
145.9 0.978148 0.207912i 1.00011 2.24629i 0.913545 0.406737i 0.0731199 0.0812078i 0.511227 2.40514i 1.06798 1.46995i 0.809017 0.587785i −2.03819 2.26364i 0.0546380 0.0946357i
145.10 0.978148 0.207912i 1.34512 3.02120i 0.913545 0.406737i 0.223897 0.248663i 0.687587 3.23484i −2.45911 + 3.38468i 0.809017 0.587785i −5.31088 5.89833i 0.167304 0.289780i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.t even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.s.a 80
11.d odd 10 1 418.2.s.b yes 80
19.d odd 6 1 418.2.s.b yes 80
209.t even 30 1 inner 418.2.s.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.s.a 80 1.a even 1 1 trivial
418.2.s.a 80 209.t even 30 1 inner
418.2.s.b yes 80 11.d odd 10 1
418.2.s.b yes 80 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{80} + 3 T_{3}^{79} + 25 T_{3}^{78} + 91 T_{3}^{77} + 289 T_{3}^{76} + 940 T_{3}^{75} + 941 T_{3}^{74} + 939 T_{3}^{73} - 18414 T_{3}^{72} - 92240 T_{3}^{71} - 370701 T_{3}^{70} - 1290637 T_{3}^{69} - 3491539 T_{3}^{68} + \cdots + 429981696$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.