# Properties

 Label 43.3.f.a Level $43$ Weight $3$ Character orbit 43.f Analytic conductor $1.172$ Analytic rank $0$ Dimension $42$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 43.f (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.17166513675$$ Analytic rank: $$0$$ Dimension: $$42$$ Relative dimension: $$7$$ over $$\Q(\zeta_{14})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$42q - 7q^{2} - 7q^{3} + 5q^{4} - 7q^{5} - 20q^{6} + 21q^{8} - 36q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$42q - 7q^{2} - 7q^{3} + 5q^{4} - 7q^{5} - 20q^{6} + 21q^{8} - 36q^{9} - 5q^{10} - 24q^{11} - 35q^{12} - 34q^{13} + 69q^{14} + 7q^{15} - 39q^{16} + 22q^{17} - 70q^{18} - 49q^{19} + 133q^{20} + 77q^{22} + 42q^{23} - 349q^{24} + 10q^{25} + 49q^{26} - 7q^{27} + 105q^{28} + 63q^{29} - 252q^{30} - 152q^{31} + 343q^{32} + 329q^{33} + 161q^{34} + 58q^{35} + 576q^{36} - 289q^{38} + 77q^{39} - 101q^{40} + 133q^{41} - 79q^{43} + 148q^{44} + 84q^{45} - 504q^{46} + 6q^{47} - 595q^{48} - 302q^{49} + 161q^{51} - 267q^{52} - 394q^{53} - 227q^{54} - 637q^{55} + 355q^{56} - 7q^{57} + 165q^{58} - 46q^{59} - 657q^{60} - 175q^{61} - 91q^{62} + 511q^{63} + 725q^{64} + 161q^{65} - 227q^{66} - 756q^{67} - 586q^{68} + 441q^{69} + 1526q^{70} + 266q^{71} + 1078q^{72} - 252q^{73} + 204q^{74} + 112q^{75} + 994q^{76} + 791q^{77} + 94q^{78} - 178q^{79} - 428q^{81} + 245q^{82} + 238q^{83} + 66q^{84} + 365q^{86} + 426q^{87} - 119q^{88} + 252q^{89} - 926q^{90} - 224q^{91} - 764q^{92} + 133q^{94} + 11q^{95} - 2602q^{96} - 491q^{97} - 553q^{98} + 431q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −1.57834 3.27745i −1.75180 + 3.63765i −5.75657 + 7.21851i −7.35771 1.67935i 14.6871 2.47266i 18.5581 + 4.23577i −4.55228 5.70838i 6.10896 + 26.7651i
2.2 −1.52706 3.17098i 2.15990 4.48508i −5.22924 + 6.55726i 5.44765 + 1.24339i −17.5204 9.10137i 15.0532 + 3.43580i −9.83935 12.3382i −4.37614 19.1731i
2.3 −0.686008 1.42451i 0.358272 0.743960i 0.935338 1.17288i −1.02237 0.233349i −1.30556 7.62808i −8.47820 1.93509i 5.18629 + 6.50340i 0.368944 + 1.61645i
2.4 −0.468493 0.972836i −1.93369 + 4.01535i 1.76703 2.21579i 8.30688 + 1.89599i 4.81220 6.84462i −7.19423 1.64204i −6.77248 8.49242i −2.04723 8.96949i
2.5 0.311899 + 0.647664i 1.16241 2.41377i 2.17177 2.72331i −3.49769 0.798324i 1.92587 9.27538i 5.24449 + 1.19702i 1.13631 + 1.42489i −0.573878 2.51432i
2.6 1.03788 + 2.15517i −1.08703 + 2.25725i −1.07362 + 1.34627i −0.904592 0.206467i −5.99296 5.06547i 5.31261 + 1.21257i 1.69788 + 2.12908i −0.493882 2.16384i
2.7 1.68761 + 3.50435i 2.07233 4.30323i −6.93850 + 8.70061i −2.02712 0.462677i 18.5773 2.90918i −27.0314 6.16973i −8.61185 10.7989i −1.79960 7.88456i
8.1 −3.53859 0.807660i 1.86243 0.425087i 8.26544 + 3.98042i −5.65332 4.50837i −6.93370 11.0869i −14.6823 11.7087i −4.82078 + 2.32157i 16.3636 + 20.5192i
8.2 −2.63550 0.601536i −1.75981 + 0.401664i 2.98015 + 1.43516i 7.00305 + 5.58475i 4.87959 3.35098i 1.46315 + 1.16682i −5.17313 + 2.49125i −15.0971 18.9312i
8.3 −0.844666 0.192789i 4.86248 1.10983i −2.92758 1.40985i 0.831553 + 0.663142i −4.32114 0.302383i 4.91050 + 3.91600i 14.3033 6.88811i −0.574538 0.720448i
8.4 −0.811544 0.185230i −3.07466 + 0.701771i −2.97958 1.43489i −3.71827 2.96522i 2.62521 0.589128i 4.75551 + 3.79239i 0.852334 0.410462i 2.46829 + 3.09514i
8.5 1.71853 + 0.392244i 0.160301 0.0365876i −0.804371 0.387365i 5.40310 + 4.30882i 0.289833 6.65286i −6.74303 5.37738i −8.08436 + 3.89322i 7.59529 + 9.52420i
8.6 2.11328 + 0.482341i 1.61544 0.368714i 0.629407 + 0.303106i −3.78878 3.02145i 3.59172 10.9446i −5.59495 4.46183i −5.63502 + 2.71368i −6.54936 8.21264i
8.7 3.62198 + 0.826694i −3.87513 + 0.884474i 8.83146 + 4.25301i −2.82431 2.25231i −14.7669 9.33217i 16.8530 + 13.4399i 6.12565 2.94996i −8.36764 10.4927i
22.1 −1.57834 + 3.27745i −1.75180 3.63765i −5.75657 7.21851i −7.35771 + 1.67935i 14.6871 2.47266i 18.5581 4.23577i −4.55228 + 5.70838i 6.10896 26.7651i
22.2 −1.52706 + 3.17098i 2.15990 + 4.48508i −5.22924 6.55726i 5.44765 1.24339i −17.5204 9.10137i 15.0532 3.43580i −9.83935 + 12.3382i −4.37614 + 19.1731i
22.3 −0.686008 + 1.42451i 0.358272 + 0.743960i 0.935338 + 1.17288i −1.02237 + 0.233349i −1.30556 7.62808i −8.47820 + 1.93509i 5.18629 6.50340i 0.368944 1.61645i
22.4 −0.468493 + 0.972836i −1.93369 4.01535i 1.76703 + 2.21579i 8.30688 1.89599i 4.81220 6.84462i −7.19423 + 1.64204i −6.77248 + 8.49242i −2.04723 + 8.96949i
22.5 0.311899 0.647664i 1.16241 + 2.41377i 2.17177 + 2.72331i −3.49769 + 0.798324i 1.92587 9.27538i 5.24449 1.19702i 1.13631 1.42489i −0.573878 + 2.51432i
22.6 1.03788 2.15517i −1.08703 2.25725i −1.07362 1.34627i −0.904592 + 0.206467i −5.99296 5.06547i 5.31261 1.21257i 1.69788 2.12908i −0.493882 + 2.16384i
See all 42 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 39.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.f odd 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.3.f.a 42
3.b odd 2 1 387.3.w.b 42
43.f odd 14 1 inner 43.3.f.a 42
129.j even 14 1 387.3.w.b 42

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.f.a 42 1.a even 1 1 trivial
43.3.f.a 42 43.f odd 14 1 inner
387.3.w.b 42 3.b odd 2 1
387.3.w.b 42 129.j even 14 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(43, [\chi])$$.