# Properties

 Label 19.4.e.a Level 19 Weight 4 Character orbit 19.e Analytic conductor 1.121 Analytic rank 0 Dimension 24 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 19.e (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.12103629011$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{2} - 3q^{3} - 24q^{4} - 6q^{5} + 42q^{6} + 3q^{7} - 75q^{8} - 51q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 6q^{2} - 3q^{3} - 24q^{4} - 6q^{5} + 42q^{6} + 3q^{7} - 75q^{8} - 51q^{9} + 75q^{10} + 39q^{11} - 219q^{12} - 156q^{13} + 93q^{14} - 192q^{15} + 504q^{16} + 12q^{17} + 264q^{18} + 546q^{19} - 198q^{20} + 453q^{21} - 6q^{22} + 6q^{23} + 192q^{24} - 498q^{25} - 639q^{26} - 870q^{27} - 1368q^{28} - 630q^{29} - 522q^{30} - 591q^{31} + 147q^{32} + 1506q^{33} - 408q^{34} + 2001q^{35} + 1059q^{36} - 72q^{37} + 2934q^{38} + 336q^{39} + 2886q^{40} - 477q^{41} + 237q^{42} + 588q^{43} - 3423q^{44} - 1569q^{45} - 1728q^{46} - 1242q^{47} - 4599q^{48} - 639q^{49} - 1788q^{50} + 9q^{51} + 2733q^{52} - 300q^{53} + 3777q^{54} + 315q^{55} + 4638q^{56} + 3342q^{57} - 2820q^{58} + 2097q^{59} + 1116q^{60} - 2316q^{61} - 1320q^{62} - 2979q^{63} - 1785q^{64} - 2433q^{65} - 1590q^{66} + 57q^{67} - 438q^{68} - 1767q^{69} - 213q^{70} - 792q^{71} - 1686q^{72} + 4068q^{73} + 4287q^{74} + 1332q^{75} + 5538q^{76} + 3786q^{77} + 2121q^{78} + 1824q^{79} - 2739q^{80} + 1536q^{81} + 2205q^{82} + 1071q^{83} - 1437q^{84} - 2394q^{85} - 5256q^{86} + 759q^{87} + 1101q^{88} - 3006q^{89} - 3822q^{90} - 3285q^{91} - 1452q^{92} - 135q^{93} - 1086q^{94} - 3078q^{95} - 1590q^{96} - 2535q^{97} - 2403q^{98} + 492q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −5.03941 1.83419i 0.944789 + 5.35816i 15.9030 + 13.3442i −7.64679 + 6.41642i 5.06673 28.7349i −9.43070 + 16.3344i −34.2145 59.2612i −2.44557 + 0.890116i 50.3042 18.3092i
4.2 −2.16911 0.789492i −0.930433 5.27675i −2.04660 1.71731i 6.28846 5.27664i −2.14774 + 12.1804i −1.44586 + 2.50430i 12.3168 + 21.3333i −1.60664 + 0.584768i −17.8062 + 6.48094i
4.3 0.579114 + 0.210780i 1.63522 + 9.27381i −5.83741 4.89817i 9.88078 8.29096i −1.00776 + 5.71527i 6.05695 10.4909i −4.81321 8.33672i −57.9579 + 21.0950i 7.46967 2.71874i
4.4 2.98397 + 1.08608i −0.341646 1.93757i 1.59618 + 1.33936i −6.63193 + 5.56485i 1.08489 6.15271i −4.36312 + 7.55715i −9.39359 16.2702i 21.7342 7.91062i −25.8334 + 9.40257i
5.1 −5.03941 + 1.83419i 0.944789 5.35816i 15.9030 13.3442i −7.64679 6.41642i 5.06673 + 28.7349i −9.43070 16.3344i −34.2145 + 59.2612i −2.44557 0.890116i 50.3042 + 18.3092i
5.2 −2.16911 + 0.789492i −0.930433 + 5.27675i −2.04660 + 1.71731i 6.28846 + 5.27664i −2.14774 12.1804i −1.44586 2.50430i 12.3168 21.3333i −1.60664 0.584768i −17.8062 6.48094i
5.3 0.579114 0.210780i 1.63522 9.27381i −5.83741 + 4.89817i 9.88078 + 8.29096i −1.00776 5.71527i 6.05695 + 10.4909i −4.81321 + 8.33672i −57.9579 21.0950i 7.46967 + 2.71874i
5.4 2.98397 1.08608i −0.341646 + 1.93757i 1.59618 1.33936i −6.63193 5.56485i 1.08489 + 6.15271i −4.36312 7.55715i −9.39359 + 16.2702i 21.7342 + 7.91062i −25.8334 9.40257i
6.1 −2.26193 + 1.89799i −8.82799 3.21312i 0.124797 0.707761i −0.0925462 0.524856i 26.0668 9.48752i −11.8949 + 20.6025i −10.7499 18.6194i 46.9260 + 39.3756i 1.20550 + 1.01154i
6.2 −1.74757 + 1.46638i 4.00345 + 1.45714i −0.485473 + 2.75325i 1.27381 + 7.22413i −9.13303 + 3.32415i 13.1019 22.6932i −12.3141 21.3286i −6.77881 5.68809i −12.8194 10.7568i
6.3 1.38101 1.15881i 2.58415 + 0.940555i −0.824823 + 4.67780i −3.13553 17.7825i 4.65867 1.69562i −14.1277 + 24.4699i 11.4927 + 19.9060i −14.8900 12.4942i −24.9367 20.9244i
6.4 2.98693 2.50633i −3.72413 1.35547i 1.25086 7.09397i 2.58856 + 14.6804i −14.5210 + 5.28519i 5.35146 9.26900i 1.55302 + 2.68992i −8.65137 7.25936i 44.5258 + 37.3616i
9.1 −0.851642 4.82990i 0.458552 0.384771i −15.0851 + 5.49052i 6.94640 + 2.52828i −2.24893 1.88707i 14.5751 25.2448i 19.7481 + 34.2048i −4.62628 + 26.2369i 6.29551 35.7036i
9.2 −0.163781 0.928850i 1.56059 1.30949i 6.68160 2.43190i −3.55727 1.29474i −1.47192 1.23509i −11.7083 + 20.2794i −7.12592 12.3424i −3.96782 + 22.5026i −0.620006 + 3.51623i
9.3 0.477474 + 2.70789i −4.62264 + 3.87886i 0.412858 0.150268i 5.11043 + 1.86004i −12.7107 10.6656i 11.7392 20.3328i 11.6027 + 20.0964i 1.63479 9.27135i −2.59670 + 14.7266i
9.4 0.824938 + 4.67846i 5.76007 4.83328i −13.6899 + 4.98271i −14.0244 5.10445i 27.3640 + 22.9611i 3.64598 6.31503i −15.6022 27.0238i 5.12940 29.0903i 12.3117 69.8233i
16.1 −2.26193 1.89799i −8.82799 + 3.21312i 0.124797 + 0.707761i −0.0925462 + 0.524856i 26.0668 + 9.48752i −11.8949 20.6025i −10.7499 + 18.6194i 46.9260 39.3756i 1.20550 1.01154i
16.2 −1.74757 1.46638i 4.00345 1.45714i −0.485473 2.75325i 1.27381 7.22413i −9.13303 3.32415i 13.1019 + 22.6932i −12.3141 + 21.3286i −6.77881 + 5.68809i −12.8194 + 10.7568i
16.3 1.38101 + 1.15881i 2.58415 0.940555i −0.824823 4.67780i −3.13553 + 17.7825i 4.65867 + 1.69562i −14.1277 24.4699i 11.4927 19.9060i −14.8900 + 12.4942i −24.9367 + 20.9244i
16.4 2.98693 + 2.50633i −3.72413 + 1.35547i 1.25086 + 7.09397i 2.58856 14.6804i −14.5210 5.28519i 5.35146 + 9.26900i 1.55302 2.68992i −8.65137 + 7.25936i 44.5258 37.3616i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.4.e.a 24
3.b odd 2 1 171.4.u.b 24
19.e even 9 1 inner 19.4.e.a 24
19.e even 9 1 361.4.a.n 12
19.f odd 18 1 361.4.a.m 12
57.l odd 18 1 171.4.u.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.e.a 24 1.a even 1 1 trivial
19.4.e.a 24 19.e even 9 1 inner
171.4.u.b 24 3.b odd 2 1
171.4.u.b 24 57.l odd 18 1
361.4.a.m 12 19.f odd 18 1
361.4.a.n 12 19.e even 9 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(19, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database