LMFDB - Embedded newform 729.3.b.a.728.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,3,Mod(728,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("729.728");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	729.b (of order $2$, degree $1$, 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:	$19.8638112719$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			728.13
Character	$\chi$	$=$	729.728
Dual form			729.3.b.a.728.18

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.581939i q^{2} +3.66135 q^{4} -7.66608i q^{5} +1.33395 q^{7} -4.45844i q^{8} -4.46119 q^{10} +1.07486i q^{11} +20.9280 q^{13} -0.776278i q^{14} +12.0509 q^{16} +14.0720i q^{17} +19.5967 q^{19} -28.0682i q^{20} +0.625503 q^{22} -1.79211i q^{23} -33.7688 q^{25} -12.1788i q^{26} +4.88406 q^{28} -40.1577i q^{29} -17.0274 q^{31} -24.8466i q^{32} +8.18903 q^{34} -10.2262i q^{35} -3.60025 q^{37} -11.4041i q^{38} -34.1787 q^{40} +4.81131i q^{41} -17.1825 q^{43} +3.93544i q^{44} -1.04290 q^{46} -45.8144i q^{47} -47.2206 q^{49} +19.6514i q^{50} +76.6247 q^{52} +51.2852i q^{53} +8.23996 q^{55} -5.94734i q^{56} -23.3693 q^{58} +93.7084i q^{59} -22.3932 q^{61} +9.90890i q^{62} +33.7442 q^{64} -160.436i q^{65} -19.5320 q^{67} +51.5224i q^{68} -5.95101 q^{70} -86.5721i q^{71} -36.1509 q^{73} +2.09513i q^{74} +71.7502 q^{76} +1.43381i q^{77} -29.2309 q^{79} -92.3828i q^{80} +2.79989 q^{82} -118.531i q^{83} +107.877 q^{85} +9.99915i q^{86} +4.79219 q^{88} +121.110i q^{89} +27.9169 q^{91} -6.56154i q^{92} -26.6612 q^{94} -150.230i q^{95} -9.63502 q^{97} +27.4795i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67}+ \cdots - 24 q^{94}+O(q^{100}) $ 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 0.581939i	− 0.290969i	−0.989361	−	0.145485i	$-0.953526\pi$
$2$	− 0.581939i	− 0.290969i	0.989361	−	0.145485i	$-0.0464742\pi$
$3$	0	0
$3$	0	0
$4$	3.66135	0.915337
$5$	− 7.66608i	− 1.53322i	−0.642115	−	0.766608i	$-0.721943\pi$
$5$	− 7.66608i	− 1.53322i	0.642115	−	0.766608i	$-0.278057\pi$
$6$	0	0
$7$	1.33395	0.190565	0.0952823	−	0.995450i	$-0.469625\pi$
$7$	1.33395	0.190565	0.0952823	+	0.995450i	$0.469625\pi$
$8$	− 4.45844i	− 0.557304i
$9$	0	0
$10$	−4.46119	−0.446119
$11$	1.07486i	0.0977146i	0.998806	+	0.0488573i	$0.0155579\pi$
$11$	1.07486i	0.0977146i	−0.998806	+	0.0488573i	$0.984442\pi$
$12$	0	0
$13$	20.9280	1.60985	0.804923	−	0.593379i	$-0.202207\pi$
$13$	20.9280	1.60985	0.804923	+	0.593379i	$0.202207\pi$
$14$	− 0.776278i	− 0.0554485i
$15$	0	0
$16$	12.0509	0.753178
$17$	14.0720i	0.827763i	0.910331	+	0.413881i	$0.135827\pi$
$17$	14.0720i	0.827763i	−0.910331	+	0.413881i	$0.864173\pi$
$18$	0	0
$19$	19.5967	1.03140	0.515702	−	0.856768i	$-0.327531\pi$
$19$	19.5967	1.03140	0.515702	+	0.856768i	$0.327531\pi$
$20$	− 28.0682i	− 1.40341i
$21$	0	0
$22$	0.625503	0.0284319
$23$	− 1.79211i	− 0.0779178i	−0.999241	−	0.0389589i	$-0.987596\pi$
$23$	− 1.79211i	− 0.0779178i	0.999241	−	0.0389589i	$-0.0124041\pi$
$24$	0	0
$25$	−33.7688	−1.35075
$26$	− 12.1788i	− 0.468416i
$27$	0	0
$28$	4.88406	0.174431
$29$	− 40.1577i	− 1.38475i	−0.721539	−	0.692374i	$-0.756565\pi$
$29$	− 40.1577i	− 1.38475i	0.721539	−	0.692374i	$-0.243435\pi$
$30$	0	0
$31$	−17.0274	−0.549271	−0.274635	−	0.961548i	$-0.588557\pi$
$31$	−17.0274	−0.549271	−0.274635	+	0.961548i	$0.588557\pi$
$32$	− 24.8466i	− 0.776456i
$33$	0	0
$34$	8.18903	0.240854
$35$	− 10.2262i	− 0.292177i
$36$	0	0
$37$	−3.60025	−0.0973040	−0.0486520	−	0.998816i	$-0.515493\pi$
$37$	−3.60025	−0.0973040	−0.0486520	+	0.998816i	$0.515493\pi$
$38$	− 11.4041i	− 0.300107i
$39$	0	0
$40$	−34.1787	−0.854468
$41$	4.81131i	0.117349i	0.998277	+	0.0586745i	$0.0186874\pi$
$41$	4.81131i	0.117349i	−0.998277	+	0.0586745i	$0.981313\pi$
$42$	0	0
$43$	−17.1825	−0.399593	−0.199796	−	0.979837i	$-0.564028\pi$
$43$	−17.1825	−0.399593	−0.199796	+	0.979837i	$0.564028\pi$
$44$	3.93544i	0.0894417i
$45$	0	0
$46$	−1.04290	−0.0226717
$47$	− 45.8144i	− 0.974775i	−0.873186	−	0.487388i	$-0.837950\pi$
$47$	− 45.8144i	− 0.974775i	0.873186	−	0.487388i	$-0.162050\pi$
$48$	0	0
$49$	−47.2206	−0.963685
$50$	19.6514i	0.393027i
$51$	0	0
$52$	76.6247	1.47355
$53$	51.2852i	0.967645i	0.875166	+	0.483822i	$0.160752\pi$
$53$	51.2852i	0.967645i	−0.875166	+	0.483822i	$0.839248\pi$
$54$	0	0
$55$	8.23996	0.149818
$56$	− 5.94734i	− 0.106202i
$57$	0	0
$58$	−23.3693	−0.402919
$59$	93.7084i	1.58828i	0.607736	+	0.794139i	$0.292078\pi$
$59$	93.7084i	1.58828i	−0.607736	+	0.794139i	$0.707922\pi$
$60$	0	0
$61$	−22.3932	−0.367102	−0.183551	−	0.983010i	$-0.558759\pi$
$61$	−22.3932	−0.367102	−0.183551	+	0.983010i	$0.558759\pi$
$62$	9.90890i	0.159821i
$63$	0	0
$64$	33.7442	0.527253
$65$	− 160.436i	− 2.46824i
$66$	0	0
$67$	−19.5320	−0.291522	−0.145761	−	0.989320i	$-0.546563\pi$
$67$	−19.5320	−0.291522	−0.145761	+	0.989320i	$0.546563\pi$
$68$	51.5224i	0.757682i
$69$	0	0
$70$	−5.95101	−0.0850145
$71$	− 86.5721i	− 1.21933i	−0.792661	−	0.609663i	$-0.791305\pi$
$71$	− 86.5721i	− 1.21933i	0.792661	−	0.609663i	$-0.208695\pi$
$72$	0	0
$73$	−36.1509	−0.495218	−0.247609	−	0.968860i	$-0.579645\pi$
$73$	−36.1509	−0.495218	−0.247609	+	0.968860i	$0.579645\pi$
$74$	2.09513i	0.0283125i
$75$	0	0
$76$	71.7502	0.944081
$77$	1.43381i	0.0186209i
$78$	0	0
$79$	−29.2309	−0.370011	−0.185006	−	0.982737i	$-0.559230\pi$
$79$	−29.2309	−0.370011	−0.185006	+	0.982737i	$0.559230\pi$
$80$	− 92.3828i	− 1.15479i
$81$	0	0
$82$	2.79989	0.0341450
$83$	− 118.531i	− 1.42809i	−0.700101	−	0.714044i	$-0.746862\pi$
$83$	− 118.531i	− 1.42809i	0.700101	−	0.714044i	$-0.253138\pi$
$84$	0	0
$85$	107.877	1.26914
$86$	9.99915i	0.116269i
$87$	0	0
$88$	4.79219	0.0544568
$89$	121.110i	1.36079i	0.732847	+	0.680394i	$0.238191\pi$
$89$	121.110i	1.36079i	−0.732847	+	0.680394i	$0.761809\pi$
$90$	0	0
$91$	27.9169	0.306780
$92$	− 6.56154i	− 0.0713211i
$93$	0	0
$94$	−26.6612	−0.283630
$95$	− 150.230i	− 1.58136i
$96$	0	0
$97$	−9.63502	−0.0993301	−0.0496651	−	0.998766i	$-0.515815\pi$
$97$	−9.63502	−0.0993301	−0.0496651	+	0.998766i	$0.515815\pi$
$98$	27.4795i	0.280403i
$99$	0	0
$100$	−123.639	−1.23639
$101$	− 3.85630i	− 0.0381812i	−0.999818	−	0.0190906i	$-0.993923\pi$
$101$	− 3.85630i	− 0.0381812i	0.999818	−	0.0190906i	$-0.00607709\pi$
$102$	0	0
$103$	122.458	1.18892	0.594458	−	0.804127i	$-0.297367\pi$
$103$	122.458	1.18892	0.594458	+	0.804127i	$0.297367\pi$
$104$	− 93.3061i	− 0.897174i
$105$	0	0
$106$	29.8448	0.281555
$107$	80.0804i	0.748415i	0.927345	+	0.374207i	$0.122085\pi$
$107$	80.0804i	0.748415i	−0.927345	+	0.374207i	$0.877915\pi$
$108$	0	0
$109$	−96.8098	−0.888163	−0.444081	−	0.895986i	$-0.646470\pi$
$109$	−96.8098	−0.888163	−0.444081	+	0.895986i	$0.646470\pi$
$110$	− 4.79516i	− 0.0435923i
$111$	0	0
$112$	16.0753	0.143529
$113$	− 64.6366i	− 0.572005i	−0.958229	−	0.286003i	$-0.907673\pi$
$113$	− 64.6366i	− 0.572005i	0.958229	−	0.286003i	$-0.0923266\pi$
$114$	0	0
$115$	−13.7385	−0.119465
$116$	− 147.031i	− 1.26751i
$117$	0	0
$118$	54.5326	0.462140
$119$	18.7713i	0.157742i
$120$	0	0
$121$	119.845	0.990452
$122$	13.0315i	0.106816i
$123$	0	0
$124$	−62.3432	−0.502768
$125$	67.2222i	0.537778i
$126$	0	0
$127$	165.459	1.30283	0.651414	−	0.758723i	$-0.274176\pi$
$127$	165.459	1.30283	0.651414	+	0.758723i	$0.274176\pi$
$128$	− 119.023i	− 0.929871i
$129$	0	0
$130$	−93.3638	−0.718183
$131$	36.7098i	0.280227i	0.990135	+	0.140114i	$0.0447468\pi$
$131$	36.7098i	0.280227i	−0.990135	+	0.140114i	$0.955253\pi$
$132$	0	0
$133$	26.1410	0.196549
$134$	11.3664i	0.0848241i
$135$	0	0
$136$	62.7390	0.461316
$137$	116.624i	0.851269i	0.904895	+	0.425635i	$0.139949\pi$
$137$	116.624i	0.851269i	−0.904895	+	0.425635i	$0.860051\pi$
$138$	0	0
$139$	−98.5297	−0.708847	−0.354423	−	0.935085i	$-0.615323\pi$
$139$	−98.5297	−0.708847	−0.354423	+	0.935085i	$0.615323\pi$
$140$	− 37.4416i	− 0.267440i
$141$	0	0
$142$	−50.3797	−0.354786
$143$	22.4947i	0.157305i
$144$	0	0
$145$	−307.852	−2.12312
$146$	21.0376i	0.144093i
$147$	0	0
$148$	−13.1818	−0.0890660
$149$	138.960i	0.932617i	0.884622	+	0.466308i	$0.154416\pi$
$149$	138.960i	0.932617i	−0.884622	+	0.466308i	$0.845584\pi$
$150$	0	0
$151$	130.412	0.863655	0.431828	−	0.901956i	$-0.357869\pi$
$151$	130.412	0.863655	0.431828	+	0.901956i	$0.357869\pi$
$152$	− 87.3704i	− 0.574805i
$153$	0	0
$154$	0.834391	0.00541812
$155$	130.533i	0.842150i
$156$	0	0
$157$	−233.610	−1.48796	−0.743982	−	0.668200i	$-0.767065\pi$
$157$	−233.610	−1.48796	−0.743982	+	0.668200i	$0.767065\pi$
$158$	17.0106i	0.107662i
$159$	0	0
$160$	−190.476	−1.19048
$161$	− 2.39059i	− 0.0148484i
$162$	0	0
$163$	65.8701	0.404111	0.202056	−	0.979374i	$-0.435238\pi$
$163$	65.8701	0.404111	0.202056	+	0.979374i	$0.435238\pi$
$164$	17.6159i	0.107414i
$165$	0	0
$166$	−68.9780	−0.415530
$167$	− 167.568i	− 1.00340i	−0.865042	−	0.501699i	$-0.832708\pi$
$167$	− 167.568i	− 1.00340i	0.865042	−	0.501699i	$-0.167292\pi$
$168$	0	0
$169$	268.981	1.59160
$170$	− 62.7777i	− 0.369281i
$171$	0	0
$172$	−62.9110	−0.365762
$173$	265.716i	1.53593i	0.640492	+	0.767965i	$0.278731\pi$
$173$	265.716i	1.53593i	−0.640492	+	0.767965i	$0.721269\pi$
$174$	0	0
$175$	−45.0459	−0.257405
$176$	12.9530i	0.0735965i
$177$	0	0
$178$	70.4786	0.395947
$179$	246.582i	1.37755i	0.724974	+	0.688776i	$0.241852\pi$
$179$	246.582i	1.37755i	−0.724974	+	0.688776i	$0.758148\pi$
$180$	0	0
$181$	−1.34144	−0.00741126	−0.00370563	−	0.999993i	$-0.501180\pi$
$181$	−1.34144	−0.00741126	−0.00370563	+	0.999993i	$0.501180\pi$
$182$	− 16.2460i	− 0.0892635i
$183$	0	0
$184$	−7.99001	−0.0434239
$185$	27.5998i	0.149188i
$186$	0	0
$187$	−15.1254	−0.0808845
$188$	− 167.743i	− 0.892248i
$189$	0	0
$190$	−87.4244	−0.460128
$191$	46.4407i	0.243145i	0.992583	+	0.121573i	$0.0387937\pi$
$191$	46.4407i	0.243145i	−0.992583	+	0.121573i	$0.961206\pi$
$192$	0	0
$193$	−142.455	−0.738110	−0.369055	−	0.929408i	$-0.620319\pi$
$193$	−142.455	−0.738110	−0.369055	+	0.929408i	$0.620319\pi$
$194$	5.60699i	0.0289020i
$195$	0	0
$196$	−172.891	−0.882096
$197$	− 54.0910i	− 0.274574i	−0.990531	−	0.137287i	$-0.956162\pi$
$197$	− 54.0910i	− 0.274574i	0.990531	−	0.137287i	$-0.0438382\pi$
$198$	0	0
$199$	328.874	1.65263	0.826317	−	0.563205i	$-0.190432\pi$
$199$	328.874	1.65263	0.826317	+	0.563205i	$0.190432\pi$
$200$	150.556i	0.752780i
$201$	0	0
$202$	−2.24413	−0.0111095
$203$	− 53.5684i	− 0.263884i
$204$	0	0
$205$	36.8839	0.179921
$206$	− 71.2633i	− 0.345938i
$207$	0	0
$208$	252.200	1.21250
$209$	21.0637i	0.100783i
$210$	0	0
$211$	94.7684	0.449139	0.224570	−	0.974458i	$-0.427902\pi$
$211$	94.7684	0.449139	0.224570	+	0.974458i	$0.427902\pi$
$212$	187.773i	0.885721i
$213$	0	0
$214$	46.6019	0.217766
$215$	131.722i	0.612662i
$216$	0	0
$217$	−22.7137	−0.104671
$218$	56.3374i	0.258428i
$219$	0	0
$220$	30.1694	0.137134
$221$	294.498i	1.33257i
$222$	0	0
$223$	−81.2658	−0.364421	−0.182210	−	0.983260i	$-0.558325\pi$
$223$	−81.2658	−0.364421	−0.182210	+	0.983260i	$0.558325\pi$
$224$	− 33.1442i	− 0.147965i
$225$	0	0
$226$	−37.6146	−0.166436
$227$	167.793i	0.739175i	0.929196	+	0.369588i	$0.120501\pi$
$227$	167.793i	0.739175i	−0.929196	+	0.369588i	$0.879499\pi$
$228$	0	0
$229$	73.4237	0.320627	0.160314	−	0.987066i	$-0.448749\pi$
$229$	73.4237	0.320627	0.160314	+	0.987066i	$0.448749\pi$
$230$	7.99494i	0.0347606i
$231$	0	0
$232$	−179.040	−0.771726
$233$	183.645i	0.788174i	0.919073	+	0.394087i	$0.128939\pi$
$233$	183.645i	0.788174i	−0.919073	+	0.394087i	$0.871061\pi$
$234$	0	0
$235$	−351.217	−1.49454
$236$	343.099i	1.45381i
$237$	0	0
$238$	10.9238	0.0458982
$239$	− 8.84969i	− 0.0370280i	−0.999829	−	0.0185140i	$-0.994106\pi$
$239$	− 8.84969i	− 0.0370280i	0.999829	−	0.0185140i	$-0.00589352\pi$
$240$	0	0
$241$	−15.9143	−0.0660344	−0.0330172	−	0.999455i	$-0.510512\pi$
$241$	−15.9143	−0.0660344	−0.0330172	+	0.999455i	$0.510512\pi$
$242$	− 69.7423i	− 0.288191i
$243$	0	0
$244$	−81.9894	−0.336022
$245$	361.997i	1.47754i
$246$	0	0
$247$	410.119	1.66040
$248$	75.9155i	0.306111i
$249$	0	0
$250$	39.1192	0.156477
$251$	264.256i	1.05281i	0.850233	+	0.526406i	$0.176461\pi$
$251$	264.256i	1.05281i	−0.850233	+	0.526406i	$0.823539\pi$
$252$	0	0
$253$	1.92627	0.00761371
$254$	− 96.2870i	− 0.379083i
$255$	0	0
$256$	65.7125	0.256689
$257$	342.126i	1.33123i	0.746295	+	0.665615i	$0.231831\pi$
$257$	342.126i	1.33123i	−0.746295	+	0.665615i	$0.768169\pi$
$258$	0	0
$259$	−4.80256	−0.0185427
$260$	− 587.411i	− 2.25927i
$261$	0	0
$262$	21.3628	0.0815376
$263$	380.388i	1.44634i	0.690670	+	0.723170i	$0.257316\pi$
$263$	380.388i	1.44634i	−0.690670	+	0.723170i	$0.742684\pi$
$264$	0	0
$265$	393.156	1.48361
$266$	− 15.2125i	− 0.0571897i
$267$	0	0
$268$	−71.5134	−0.266841
$269$	88.9301i	0.330595i	0.986244	+	0.165297i	$0.0528584\pi$
$269$	88.9301i	0.330595i	−0.986244	+	0.165297i	$0.947142\pi$
$270$	0	0
$271$	487.123	1.79750	0.898751	−	0.438460i	$-0.144476\pi$
$271$	487.123	1.79750	0.898751	+	0.438460i	$0.144476\pi$
$272$	169.579i	0.623453i
$273$	0	0
$274$	67.8680	0.247693
$275$	− 36.2967i	− 0.131988i
$276$	0	0
$277$	−87.4657	−0.315761	−0.157880	−	0.987458i	$-0.550466\pi$
$277$	−87.4657	−0.315761	−0.157880	+	0.987458i	$0.550466\pi$
$278$	57.3383i	0.206253i
$279$	0	0
$280$	−45.5928	−0.162831
$281$	− 161.353i	− 0.574211i	−0.957899	−	0.287106i	$-0.907307\pi$
$281$	− 161.353i	− 0.574211i	0.957899	−	0.287106i	$-0.0926931\pi$
$282$	0	0
$283$	−137.224	−0.484890	−0.242445	−	0.970165i	$-0.577949\pi$
$283$	−137.224	−0.484890	−0.242445	+	0.970165i	$0.577949\pi$
$284$	− 316.971i	− 1.11609i
$285$	0	0
$286$	13.0905	0.0457711
$287$	6.41806i	0.0223626i
$288$	0	0
$289$	90.9797	0.314808
$290$	179.151i	0.617762i
$291$	0	0
$292$	−132.361	−0.453291
$293$	− 388.292i	− 1.32523i	−0.748961	−	0.662615i	$-0.769447\pi$
$293$	− 388.292i	− 1.32523i	0.748961	−	0.662615i	$-0.230553\pi$
$294$	0	0
$295$	718.376	2.43517
$296$	16.0515i	0.0542280i
$297$	0	0
$298$	80.8661	0.271363
$299$	− 37.5053i	− 0.125436i
$300$	0	0
$301$	−22.9206	−0.0761482
$302$	− 75.8918i	− 0.251297i
$303$	0	0
$304$	236.156	0.776830
$305$	171.668i	0.562847i
$306$	0	0
$307$	414.645	1.35064	0.675318	−	0.737526i	$-0.264006\pi$
$307$	414.645	1.35064	0.675318	+	0.737526i	$0.264006\pi$
$308$	5.24968i	0.0170444i
$309$	0	0
$310$	75.9624	0.245040
$311$	− 436.432i	− 1.40332i	−0.712512	−	0.701659i	$-0.752443\pi$
$311$	− 436.432i	− 1.40332i	0.712512	−	0.701659i	$-0.247557\pi$
$312$	0	0
$313$	−411.290	−1.31403	−0.657013	−	0.753880i	$-0.728180\pi$
$313$	−411.290	−1.31403	−0.657013	+	0.753880i	$0.728180\pi$
$314$	135.947i	0.432952i
$315$	0	0
$316$	−107.024	−0.338685
$317$	− 408.986i	− 1.29018i	−0.764108	−	0.645089i	$-0.776820\pi$
$317$	− 408.986i	− 1.29018i	0.764108	−	0.645089i	$-0.223180\pi$
$318$	0	0
$319$	43.1639	0.135310
$320$	− 258.686i	− 0.808393i
$321$	0	0
$322$	−1.39118	−0.00432042
$323$	275.764i	0.853757i
$324$	0	0
$325$	−706.713	−2.17450
$326$	− 38.3324i	− 0.117584i
$327$	0	0
$328$	21.4509	0.0653991
$329$	− 61.1143i	− 0.185758i
$330$	0	0
$331$	−327.640	−0.989850	−0.494925	−	0.868936i	$-0.664805\pi$
$331$	−327.640	−0.989850	−0.494925	+	0.868936i	$0.664805\pi$
$332$	− 433.984i	− 1.30718i
$333$	0	0
$334$	−97.5140	−0.291958
$335$	149.734i	0.446967i
$336$	0	0
$337$	−562.601	−1.66944	−0.834719	−	0.550676i	$-0.814370\pi$
$337$	−562.601	−1.66944	−0.834719	+	0.550676i	$0.814370\pi$
$338$	− 156.531i	− 0.463108i
$339$	0	0
$340$	394.975	1.16169
$341$	− 18.3021i	− 0.0536717i
$342$	0	0
$343$	−128.354	−0.374209
$344$	76.6070i	0.222695i
$345$	0	0
$346$	154.630	0.446909
$347$	243.920i	0.702940i	0.936199	+	0.351470i	$0.114318\pi$
$347$	243.920i	0.702940i	−0.936199	+	0.351470i	$0.885682\pi$
$348$	0	0
$349$	−369.281	−1.05811	−0.529056	−	0.848587i	$-0.677454\pi$
$349$	−369.281	−1.05811	−0.529056	+	0.848587i	$0.677454\pi$
$350$	26.2140i	0.0748971i
$351$	0	0
$352$	26.7066	0.0758711
$353$	93.9193i	0.266060i	0.991112	+	0.133030i	$0.0424707\pi$
$353$	93.9193i	0.266060i	−0.991112	+	0.133030i	$0.957529\pi$
$354$	0	0
$355$	−663.669	−1.86949
$356$	443.426i	1.24558i
$357$	0	0
$358$	143.495	0.400825
$359$	103.369i	0.287936i	0.989582	+	0.143968i	$0.0459862\pi$
$359$	103.369i	0.287936i	−0.989582	+	0.143968i	$0.954014\pi$
$360$	0	0
$361$	23.0290	0.0637923
$362$	0.780635i	0.00215645i
$363$	0	0
$364$	102.214	0.280807
$365$	277.136i	0.759276i
$366$	0	0
$367$	−90.6767	−0.247076	−0.123538	−	0.992340i	$-0.539424\pi$
$367$	−90.6767	−0.247076	−0.123538	+	0.992340i	$0.539424\pi$
$368$	− 21.5965i	− 0.0586860i
$369$	0	0
$370$	16.0614	0.0434092
$371$	68.4119i	0.184399i
$372$	0	0
$373$	−432.948	−1.16072	−0.580359	−	0.814361i	$-0.697088\pi$
$373$	−432.948	−1.16072	−0.580359	+	0.814361i	$0.697088\pi$
$374$	8.80206i	0.0235349i
$375$	0	0
$376$	−204.261	−0.543247
$377$	− 840.420i	− 2.22923i
$378$	0	0
$379$	333.700	0.880476	0.440238	−	0.897881i	$-0.354894\pi$
$379$	333.700	0.880476	0.440238	+	0.897881i	$0.354894\pi$
$380$	− 550.043i	− 1.44748i
$381$	0	0
$382$	27.0257	0.0707478
$383$	− 594.908i	− 1.55328i	−0.629942	−	0.776642i	$-0.716921\pi$
$383$	− 594.908i	− 1.55328i	0.629942	−	0.776642i	$-0.283079\pi$
$384$	0	0
$385$	10.9917	0.0285499
$386$	82.9002i	0.214767i
$387$	0	0
$388$	−35.2772	−0.0909205
$389$	310.106i	0.797188i	0.917128	+	0.398594i	$0.130502\pi$
$389$	310.106i	0.797188i	−0.917128	+	0.398594i	$0.869498\pi$
$390$	0	0
$391$	25.2185	0.0644975
$392$	210.530i	0.537066i
$393$	0	0
$394$	−31.4776	−0.0798925
$395$	224.086i	0.567307i
$396$	0	0
$397$	138.065	0.347770	0.173885	−	0.984766i	$-0.444368\pi$
$397$	138.065	0.347770	0.173885	+	0.984766i	$0.444368\pi$
$398$	− 191.385i	− 0.480866i
$399$	0	0
$400$	−406.943	−1.01736
$401$	271.096i	0.676050i	0.941137	+	0.338025i	$0.109759\pi$
$401$	271.096i	0.676050i	−0.941137	+	0.338025i	$0.890241\pi$
$402$	0	0
$403$	−356.349	−0.884241
$404$	− 14.1192i	− 0.0349486i
$405$	0	0
$406$	−31.1735	−0.0767821
$407$	− 3.86977i	− 0.00950802i
$408$	0	0
$409$	289.991	0.709025	0.354513	−	0.935051i	$-0.384647\pi$
$409$	289.991	0.709025	0.354513	+	0.935051i	$0.384647\pi$
$410$	− 21.4642i	− 0.0523516i
$411$	0	0
$412$	448.363	1.08826
$413$	125.003i	0.302670i
$414$	0	0
$415$	−908.671	−2.18957
$416$	− 519.990i	− 1.24997i
$417$	0	0
$418$	12.2578	0.0293248
$419$	335.292i	0.800218i	0.916467	+	0.400109i	$0.131028\pi$
$419$	335.292i	0.800218i	−0.916467	+	0.400109i	$0.868972\pi$
$420$	0	0
$421$	603.128	1.43261	0.716303	−	0.697789i	$-0.245833\pi$
$421$	603.128	1.43261	0.716303	+	0.697789i	$0.245833\pi$
$422$	− 55.1494i	− 0.130686i
$423$	0	0
$424$	228.652	0.539273
$425$	− 475.193i	− 1.11810i
$426$	0	0
$427$	−29.8715	−0.0699567
$428$	293.202i	0.685051i
$429$	0	0
$430$	76.6543	0.178266
$431$	773.546i	1.79477i	0.441247	+	0.897385i	$0.354536\pi$
$431$	773.546i	1.79477i	−0.441247	+	0.897385i	$0.645464\pi$
$432$	0	0
$433$	71.6603	0.165497	0.0827486	−	0.996570i	$-0.473630\pi$
$433$	71.6603	0.165497	0.0827486	+	0.996570i	$0.473630\pi$
$434$	13.2180i	0.0304562i
$435$	0	0
$436$	−354.454	−0.812968
$437$	− 35.1194i	− 0.0803647i
$438$	0	0
$439$	−684.909	−1.56016	−0.780078	−	0.625682i	$-0.784821\pi$
$439$	−684.909	−1.56016	−0.780078	+	0.625682i	$0.784821\pi$
$440$	− 36.7373i	− 0.0834940i
$441$	0	0
$442$	171.380	0.387737
$443$	− 455.322i	− 1.02782i	−0.857845	−	0.513908i	$-0.828197\pi$
$443$	− 455.322i	− 1.02782i	0.857845	−	0.513908i	$-0.171803\pi$
$444$	0	0
$445$	928.439	2.08638
$446$	47.2917i	0.106035i
$447$	0	0
$448$	45.0131	0.100476
$449$	527.340i	1.17448i	0.809414	+	0.587238i	$0.199785\pi$
$449$	527.340i	1.17448i	−0.809414	+	0.587238i	$0.800215\pi$
$450$	0	0
$451$	−5.17149	−0.0114667
$452$	− 236.657i	− 0.523578i
$453$	0	0
$454$	97.6451	0.215077
$455$	− 214.013i	− 0.470359i
$456$	0	0
$457$	−147.083	−0.321845	−0.160923	−	0.986967i	$-0.551447\pi$
$457$	−147.083	−0.321845	−0.160923	+	0.986967i	$0.551447\pi$
$458$	− 42.7281i	− 0.0932928i
$459$	0	0
$460$	−50.3013	−0.109351
$461$	− 357.986i	− 0.776543i	−0.921545	−	0.388272i	$-0.873072\pi$
$461$	− 357.986i	− 0.776543i	0.921545	−	0.388272i	$-0.126928\pi$
$462$	0	0
$463$	409.789	0.885074	0.442537	−	0.896750i	$-0.354079\pi$
$463$	409.789	0.885074	0.442537	+	0.896750i	$0.354079\pi$
$464$	− 483.934i	− 1.04296i
$465$	0	0
$466$	106.870	0.229335
$467$	− 899.449i	− 1.92602i	−0.269475	−	0.963008i	$-0.586850\pi$
$467$	− 899.449i	− 1.92602i	0.269475	−	0.963008i	$-0.413150\pi$
$468$	0	0
$469$	−26.0547	−0.0555538
$470$	204.387i	0.434866i
$471$	0	0
$472$	417.793	0.885155
$473$	− 18.4688i	− 0.0390460i
$474$	0	0
$475$	−661.755	−1.39317
$476$	68.7284i	0.144387i
$477$	0	0
$478$	−5.14998	−0.0107740
$479$	− 835.310i	− 1.74386i	−0.489628	−	0.871932i	$-0.662867\pi$
$479$	− 835.310i	− 1.74386i	0.489628	−	0.871932i	$-0.337133\pi$
$480$	0	0
$481$	−75.3460	−0.156645
$482$	9.26114i	0.0192140i
$483$	0	0
$484$	438.793	0.906597
$485$	73.8629i	0.152295i
$486$	0	0
$487$	−601.667	−1.23546	−0.617728	−	0.786392i	$-0.711947\pi$
$487$	−601.667	−1.23546	−0.617728	+	0.786392i	$0.711947\pi$
$488$	99.8388i	0.204588i
$489$	0	0
$490$	210.660	0.429918
$491$	428.745i	0.873207i	0.899654	+	0.436604i	$0.143819\pi$
$491$	428.745i	0.873207i	−0.899654	+	0.436604i	$0.856181\pi$
$492$	0	0
$493$	565.098	1.14624
$494$	− 238.664i	− 0.483126i
$495$	0	0
$496$	−205.195	−0.413699
$497$	− 115.483i	− 0.232360i
$498$	0	0
$499$	−568.702	−1.13968	−0.569841	−	0.821755i	$-0.692995\pi$
$499$	−568.702	−1.13968	−0.569841	+	0.821755i	$0.692995\pi$
$500$	246.124i	0.492248i
$501$	0	0
$502$	153.781	0.306336
$503$	− 205.861i	− 0.409266i	−0.978839	−	0.204633i	$-0.934400\pi$
$503$	− 205.861i	− 0.409266i	0.978839	−	0.204633i	$-0.0656001\pi$
$504$	0	0
$505$	−29.5627	−0.0585400
$506$	− 1.12097i	− 0.00221536i
$507$	0	0
$508$	605.803	1.19253
$509$	673.500i	1.32318i	0.749865	+	0.661591i	$0.230118\pi$
$509$	673.500i	1.32318i	−0.749865	+	0.661591i	$0.769882\pi$
$510$	0	0
$511$	−48.2236	−0.0943710
$512$	− 514.334i	− 1.00456i
$513$	0	0
$514$	199.097	0.387347
$515$	− 938.776i	− 1.82287i
$516$	0	0
$517$	49.2441	0.0952498
$518$	2.79480i	0.00539536i
$519$	0	0
$520$	−715.292	−1.37556
$521$	783.643i	1.50411i	0.659099	+	0.752056i	$0.270938\pi$
$521$	783.643i	1.50411i	−0.659099	+	0.752056i	$0.729062\pi$
$522$	0	0
$523$	462.927	0.885137	0.442569	−	0.896735i	$-0.354067\pi$
$523$	462.927	0.885137	0.442569	+	0.896735i	$0.354067\pi$
$524$	134.407i	0.256502i
$525$	0	0
$526$	221.362	0.420841
$527$	− 239.609i	− 0.454666i
$528$	0	0
$529$	525.788	0.993929
$530$	− 228.793i	− 0.431685i
$531$	0	0
$532$	95.7113	0.179908
$533$	100.691i	0.188914i
$534$	0	0
$535$	613.903	1.14748
$536$	87.0821i	0.162467i
$537$	0	0
$538$	51.7518	0.0961930
$539$	− 50.7555i	− 0.0941661i
$540$	0	0
$541$	595.277	1.10033	0.550164	−	0.835057i	$-0.314565\pi$
$541$	595.277	1.10033	0.550164	+	0.835057i	$0.314565\pi$
$542$	− 283.476i	− 0.523018i
$543$	0	0
$544$	349.641	0.642722
$545$	742.151i	1.36175i
$546$	0	0
$547$	1009.80	1.84607	0.923035	−	0.384716i	$-0.125701\pi$
$547$	1009.80	1.84607	0.923035	+	0.384716i	$0.125701\pi$
$548$	427.001i	0.779198i
$549$	0	0
$550$	−21.1225	−0.0384045
$551$	− 786.957i	− 1.42823i
$552$	0	0
$553$	−38.9926	−0.0705110
$554$	50.8997i	0.0918766i
$555$	0	0
$556$	−360.751	−0.648833
$557$	− 61.7370i	− 0.110838i	−0.998463	−	0.0554192i	$-0.982350\pi$
$557$	− 61.7370i	− 0.110838i	0.998463	−	0.0554192i	$-0.0176495\pi$
$558$	0	0
$559$	−359.595	−0.643282
$560$	− 123.234i	− 0.220061i
$561$	0	0
$562$	−93.8978	−0.167078
$563$	− 970.072i	− 1.72304i	−0.507723	−	0.861521i	$-0.669512\pi$
$563$	− 970.072i	− 1.72304i	0.507723	−	0.861521i	$-0.330488\pi$
$564$	0	0
$565$	−495.510	−0.877008
$566$	79.8559i	0.141088i
$567$	0	0
$568$	−385.976	−0.679535
$569$	− 96.0833i	− 0.168864i	−0.996429	−	0.0844318i	$-0.973093\pi$
$569$	− 96.0833i	− 0.168864i	0.996429	−	0.0844318i	$-0.0269075\pi$
$570$	0	0
$571$	−646.610	−1.13242	−0.566208	−	0.824262i	$-0.691590\pi$
$571$	−646.610	−1.13242	−0.566208	+	0.824262i	$0.691590\pi$
$572$	82.3608i	0.143987i
$573$	0	0
$574$	3.73492	0.00650682
$575$	60.5174i	0.105248i
$576$	0	0
$577$	253.985	0.440181	0.220091	−	0.975479i	$-0.429365\pi$
$577$	253.985	0.440181	0.220091	+	0.975479i	$0.429365\pi$
$578$	− 52.9446i	− 0.0915996i
$579$	0	0
$580$	−1127.15	−1.94337
$581$	− 158.115i	− 0.272143i
$582$	0	0
$583$	−55.1244	−0.0945530
$584$	161.177i	0.275987i
$585$	0	0
$586$	−225.962	−0.385601
$587$	223.570i	0.380869i	0.981700	+	0.190435i	$0.0609897\pi$
$587$	223.570i	0.380869i	−0.981700	+	0.190435i	$0.939010\pi$
$588$	0	0
$589$	−333.680	−0.566519
$590$	− 418.051i	− 0.708561i
$591$	0	0
$592$	−43.3861	−0.0732873
$593$	69.9992i	0.118042i	0.998257	+	0.0590212i	$0.0187980\pi$
$593$	69.9992i	0.118042i	−0.998257	+	0.0590212i	$0.981202\pi$
$594$	0	0
$595$	143.903	0.241853
$596$	508.780i	0.853658i
$597$	0	0
$598$	−21.8258	−0.0364979
$599$	− 418.867i	− 0.699277i	−0.936885	−	0.349638i	$-0.886304\pi$
$599$	− 418.867i	− 0.699277i	0.936885	−	0.349638i	$-0.113696\pi$
$600$	0	0
$601$	−794.668	−1.32224	−0.661122	−	0.750279i	$-0.729919\pi$
$601$	−794.668	−1.32224	−0.661122	+	0.750279i	$0.729919\pi$
$602$	13.3384i	0.0221568i
$603$	0	0
$604$	477.484	0.790536
$605$	− 918.739i	− 1.51858i
$606$	0	0
$607$	328.586	0.541328	0.270664	−	0.962674i	$-0.412757\pi$
$607$	328.586	0.541328	0.270664	+	0.962674i	$0.412757\pi$
$608$	− 486.910i	− 0.800839i
$609$	0	0
$610$	99.9005	0.163771
$611$	− 958.805i	− 1.56924i
$612$	0	0
$613$	735.328	1.19956	0.599778	−	0.800167i	$-0.295256\pi$
$613$	735.328	1.19956	0.599778	+	0.800167i	$0.295256\pi$
$614$	− 241.298i	− 0.392994i
$615$	0	0
$616$	6.39256	0.0103775
$617$	− 337.036i	− 0.546249i	−0.961979	−	0.273125i	$-0.911943\pi$
$617$	− 337.036i	− 0.546249i	0.961979	−	0.273125i	$-0.0880571\pi$
$618$	0	0
$619$	1175.92	1.89971	0.949853	−	0.312696i	$-0.101232\pi$
$619$	1175.92	1.89971	0.949853	+	0.312696i	$0.101232\pi$
$620$	477.928i	0.770851i
$621$	0	0
$622$	−253.977	−0.408323
$623$	161.555i	0.259318i
$624$	0	0
$625$	−328.889	−0.526222
$626$	239.346i	0.382341i
$627$	0	0
$628$	−855.328	−1.36199
$629$	− 50.6626i	− 0.0805447i
$630$	0	0
$631$	−214.760	−0.340349	−0.170175	−	0.985414i	$-0.554433\pi$
$631$	−214.760	−0.340349	−0.170175	+	0.985414i	$0.554433\pi$
$632$	130.324i	0.206209i
$633$	0	0
$634$	−238.005	−0.375402
$635$	− 1268.42i	− 1.99752i
$636$	0	0
$637$	−988.232	−1.55138
$638$	− 25.1188i	− 0.0393711i
$639$	0	0
$640$	−912.443	−1.42569
$641$	994.747i	1.55187i	0.630814	+	0.775934i	$0.282721\pi$
$641$	994.747i	1.55187i	−0.630814	+	0.775934i	$0.717279\pi$
$642$	0	0
$643$	−849.202	−1.32069	−0.660344	−	0.750964i	$-0.729589\pi$
$643$	−849.202	−1.32069	−0.660344	+	0.750964i	$0.729589\pi$
$644$	− 8.75277i	− 0.0135913i
$645$	0	0
$646$	160.478	0.248417
$647$	− 350.755i	− 0.542125i	−0.962562	−	0.271063i	$-0.912625\pi$
$647$	− 350.755i	− 0.542125i	0.962562	−	0.271063i	$-0.0873751\pi$
$648$	0	0
$649$	−100.723	−0.155198
$650$	411.264i	0.632713i
$651$	0	0
$652$	241.173	0.369898
$653$	− 281.525i	− 0.431125i	−0.976490	−	0.215562i	$-0.930842\pi$
$653$	− 281.525i	− 0.431125i	0.976490	−	0.215562i	$-0.0691585\pi$
$654$	0	0
$655$	281.420	0.429649
$656$	57.9804i	0.0883848i
$657$	0	0
$658$	−35.5648	−0.0540498
$659$	100.793i	0.152949i	0.997072	+	0.0764743i	$0.0243663\pi$
$659$	100.793i	0.152949i	−0.997072	+	0.0764743i	$0.975634\pi$
$660$	0	0
$661$	−724.657	−1.09630	−0.548152	−	0.836379i	$-0.684668\pi$
$661$	−724.657	−1.09630	−0.548152	+	0.836379i	$0.684668\pi$
$662$	190.667i	0.288016i
$663$	0	0
$664$	−528.464	−0.795880
$665$	− 200.399i	− 0.301352i
$666$	0	0
$667$	−71.9670	−0.107897
$668$	− 613.523i	− 0.918447i
$669$	0	0
$670$	87.1359	0.130054
$671$	− 24.0696i	− 0.0358712i
$672$	0	0
$673$	409.151	0.607950	0.303975	−	0.952680i	$-0.401686\pi$
$673$	409.151	0.607950	0.303975	+	0.952680i	$0.401686\pi$
$674$	327.399i	0.485755i
$675$	0	0
$676$	984.833	1.45685
$677$	− 338.685i	− 0.500274i	−0.968210	−	0.250137i	$-0.919524\pi$
$677$	− 338.685i	− 0.500274i	0.968210	−	0.250137i	$-0.0804756\pi$
$678$	0	0
$679$	−12.8527	−0.0189288
$680$	− 480.962i	− 0.707297i
$681$	0	0
$682$	−10.6507	−0.0156168
$683$	− 311.366i	− 0.455880i	−0.973675	−	0.227940i	$-0.926801\pi$
$683$	− 311.366i	− 0.455880i	0.973675	−	0.227940i	$-0.0731990\pi$
$684$	0	0
$685$	894.048	1.30518
$686$	74.6939i	0.108883i
$687$	0	0
$688$	−207.064	−0.300964
$689$	1073.30i	1.55776i
$690$	0	0
$691$	827.688	1.19781	0.598906	−	0.800819i	$-0.295602\pi$
$691$	827.688	1.19781	0.598906	+	0.800819i	$0.295602\pi$
$692$	972.878i	1.40589i
$693$	0	0
$694$	141.947	0.204534
$695$	755.337i	1.08682i
$696$	0	0
$697$	−67.7046	−0.0971372
$698$	214.899i	0.307878i
$699$	0	0
$700$	−164.929	−0.235613
$701$	215.718i	0.307729i	0.988092	+	0.153864i	$0.0491718\pi$
$701$	215.718i	0.307729i	−0.988092	+	0.153864i	$0.950828\pi$
$702$	0	0
$703$	−70.5529	−0.100360
$704$	36.2703i	0.0515203i
$705$	0	0
$706$	54.6553	0.0774154
$707$	− 5.14411i	− 0.00727597i
$708$	0	0
$709$	1031.81	1.45531	0.727655	−	0.685944i	$-0.240610\pi$
$709$	1031.81	1.45531	0.727655	+	0.685944i	$0.240610\pi$
$710$	386.215i	0.543964i
$711$	0	0
$712$	539.961	0.758373
$713$	30.5149i	0.0427980i
$714$	0	0
$715$	172.446	0.241183
$716$	902.821i	1.26092i
$717$	0	0
$718$	60.1545	0.0837806
$719$	− 623.485i	− 0.867155i	−0.901116	−	0.433578i	$-0.857251\pi$
$719$	− 623.485i	− 0.867155i	0.901116	−	0.433578i	$-0.142749\pi$
$720$	0	0
$721$	163.354	0.226565
$722$	− 13.4015i	− 0.0185616i
$723$	0	0
$724$	−4.91147	−0.00678380
$725$	1356.08i	1.87045i
$726$	0	0
$727$	−751.352	−1.03350	−0.516748	−	0.856138i	$-0.672858\pi$
$727$	−751.352	−1.03350	−0.516748	+	0.856138i	$0.672858\pi$
$728$	− 124.466i	− 0.170970i
$729$	0	0
$730$	161.276	0.220926
$731$	− 241.791i	− 0.330768i
$732$	0	0
$733$	750.318	1.02363	0.511813	−	0.859097i	$-0.328974\pi$
$733$	750.318	1.02363	0.511813	+	0.859097i	$0.328974\pi$
$734$	52.7683i	0.0718914i
$735$	0	0
$736$	−44.5278	−0.0604998
$737$	− 20.9942i	− 0.0284860i
$738$	0	0
$739$	−1353.19	−1.83111	−0.915557	−	0.402187i	$-0.868250\pi$
$739$	−1353.19	−1.83111	−0.915557	+	0.402187i	$0.868250\pi$
$740$	101.052i	0.136557i
$741$	0	0
$742$	39.8116	0.0536544
$743$	− 848.767i	− 1.14235i	−0.820828	−	0.571176i	$-0.806487\pi$
$743$	− 848.767i	− 1.14235i	0.820828	−	0.571176i	$-0.193513\pi$
$744$	0	0
$745$	1065.28	1.42990
$746$	251.949i	0.337733i
$747$	0	0
$748$	−55.3793	−0.0740366
$749$	106.823i	0.142621i
$750$	0	0
$751$	805.905	1.07311	0.536555	−	0.843866i	$-0.319726\pi$
$751$	805.905	1.07311	0.536555	+	0.843866i	$0.319726\pi$
$752$	− 552.103i	− 0.734180i
$753$	0	0
$754$	−489.073	−0.648638
$755$	− 999.749i	− 1.32417i
$756$	0	0
$757$	−171.631	−0.226725	−0.113363	−	0.993554i	$-0.536162\pi$
$757$	−171.631	−0.226725	−0.113363	+	0.993554i	$0.536162\pi$
$758$	− 194.193i	− 0.256191i
$759$	0	0
$760$	−669.789	−0.881301
$761$	1351.20i	1.77555i	0.460275	+	0.887776i	$0.347751\pi$
$761$	1351.20i	1.77555i	−0.460275	+	0.887776i	$0.652249\pi$
$762$	0	0
$763$	−129.140	−0.169252
$764$	170.036i	0.222560i
$765$	0	0
$766$	−346.200	−0.451958
$767$	1961.13i	2.55688i
$768$	0	0
$769$	1006.44	1.30877	0.654384	−	0.756163i	$-0.272928\pi$
$769$	1006.44	1.30877	0.654384	+	0.756163i	$0.272928\pi$
$770$	− 6.39651i	− 0.00830715i
$771$	0	0
$772$	−521.578	−0.675619
$773$	− 28.4890i	− 0.0368552i	−0.999830	−	0.0184276i	$-0.994134\pi$
$773$	− 28.4890i	− 0.0368552i	0.999830	−	0.0184276i	$-0.00586601\pi$
$774$	0	0
$775$	574.994	0.741928
$776$	42.9571i	0.0553571i
$777$	0	0
$778$	180.463	0.231957
$779$	94.2856i	0.121034i
$780$	0	0
$781$	93.0529	0.119146
$782$	− 14.6756i	− 0.0187668i
$783$	0	0
$784$	−569.048	−0.725827
$785$	1790.87i	2.28137i
$786$	0	0
$787$	199.966	0.254086	0.127043	−	0.991897i	$-0.459451\pi$
$787$	199.966	0.254086	0.127043	+	0.991897i	$0.459451\pi$
$788$	− 198.046i	− 0.251327i
$789$	0	0
$790$	130.404	0.165069
$791$	− 86.2221i	− 0.109004i
$792$	0	0
$793$	−468.646	−0.590978
$794$	− 80.3451i	− 0.101190i
$795$	0	0
$796$	1204.12	1.51272
$797$	1051.47i	1.31928i	0.751581	+	0.659641i	$0.229292\pi$
$797$	1051.47i	1.31928i	−0.751581	+	0.659641i	$0.770708\pi$
$798$	0	0
$799$	644.700	0.806883
$800$	839.040i	1.04880i
$801$	0	0
$802$	157.761	0.196710
$803$	− 38.8572i	− 0.0483900i
$804$	0	0
$805$	−18.3264	−0.0227658
$806$	207.373i	0.257287i
$807$	0	0
$808$	−17.1931	−0.0212785
$809$	482.349i	0.596229i	0.954530	+	0.298114i	$0.0963577\pi$
$809$	482.349i	0.596229i	−0.954530	+	0.298114i	$0.903642\pi$
$810$	0	0
$811$	−1202.48	−1.48272	−0.741358	−	0.671110i	$-0.765818\pi$
$811$	−1202.48	−1.48272	−0.741358	+	0.671110i	$0.765818\pi$
$812$	− 196.133i	− 0.241543i
$813$	0	0
$814$	−2.25197	−0.00276654
$815$	− 504.965i	− 0.619589i
$816$	0	0
$817$	−336.719	−0.412141
$818$	− 168.757i	− 0.206305i
$819$	0	0
$820$	135.045	0.164689
$821$	1297.77i	1.58072i	0.612645	+	0.790358i	$0.290106\pi$
$821$	1297.77i	1.58072i	−0.612645	+	0.790358i	$0.709894\pi$
$822$	0	0
$823$	1057.54	1.28499	0.642493	−	0.766292i	$-0.277900\pi$
$823$	1057.54	1.28499	0.642493	+	0.766292i	$0.277900\pi$
$824$	− 545.973i	− 0.662588i
$825$	0	0
$826$	72.7438	0.0880676
$827$	886.532i	1.07199i	0.844222	+	0.535993i	$0.180063\pi$
$827$	886.532i	1.07199i	−0.844222	+	0.535993i	$0.819937\pi$
$828$	0	0
$829$	813.250	0.981001	0.490500	−	0.871441i	$-0.336814\pi$
$829$	813.250	0.981001	0.490500	+	0.871441i	$0.336814\pi$
$830$	528.791i	0.637097i
$831$	0	0
$832$	706.199	0.848797
$833$	− 664.486i	− 0.797703i
$834$	0	0
$835$	−1284.59	−1.53843
$836$	77.1214i	0.0922505i
$837$	0	0
$838$	195.119	0.232839
$839$	1494.62i	1.78143i	0.454563	+	0.890715i	$0.349795\pi$
$839$	1494.62i	1.78143i	−0.454563	+	0.890715i	$0.650205\pi$
$840$	0	0
$841$	−771.640	−0.917527
$842$	− 350.983i	− 0.416845i
$843$	0	0
$844$	346.980	0.411114
$845$	− 2062.03i	− 2.44027i
$846$	0	0
$847$	159.867	0.188745
$848$	618.030i	0.728809i
$849$	0	0
$850$	−276.533	−0.325333
$851$	6.45204i	0.00758172i
$852$	0	0
$853$	−836.912	−0.981139	−0.490569	−	0.871402i	$-0.663211\pi$
$853$	−836.912	−0.981139	−0.490569	+	0.871402i	$0.663211\pi$
$854$	17.3834i	0.0203553i
$855$	0	0
$856$	357.033	0.417095
$857$	− 793.709i	− 0.926148i	−0.886320	−	0.463074i	$-0.846746\pi$
$857$	− 793.709i	− 0.926148i	0.886320	−	0.463074i	$-0.153254\pi$
$858$	0	0
$859$	−984.535	−1.14614	−0.573070	−	0.819506i	$-0.694248\pi$
$859$	−984.535	−1.14614	−0.573070	+	0.819506i	$0.694248\pi$
$860$	482.281i	0.560792i
$861$	0	0
$862$	450.157	0.522223
$863$	217.563i	0.252101i	0.992024	+	0.126050i	$0.0402301\pi$
$863$	217.563i	0.252101i	−0.992024	+	0.126050i	$0.959770\pi$
$864$	0	0
$865$	2037.00	2.35491
$866$	− 41.7019i	− 0.0481546i
$867$	0	0
$868$	−83.1628	−0.0958097
$869$	− 31.4191i	− 0.0361555i
$870$	0	0
$871$	−408.765	−0.469306
$872$	431.620i	0.494977i
$873$	0	0
$874$	−20.4373	−0.0233837
$875$	89.6712i	0.102481i
$876$	0	0
$877$	−310.454	−0.353995	−0.176998	−	0.984211i	$-0.556638\pi$
$877$	−310.454	−0.353995	−0.176998	+	0.984211i	$0.556638\pi$
$878$	398.575i	0.453958i
$879$	0	0
$880$	99.2986	0.112839
$881$	− 1557.70i	− 1.76811i	−0.467388	−	0.884053i	$-0.654805\pi$
$881$	− 1557.70i	− 1.76811i	0.467388	−	0.884053i	$-0.345195\pi$
$882$	0	0
$883$	−825.815	−0.935238	−0.467619	−	0.883930i	$-0.654888\pi$
$883$	−825.815	−0.935238	−0.467619	+	0.883930i	$0.654888\pi$
$884$	1078.26i	1.21975i
$885$	0	0
$886$	−264.970	−0.299063
$887$	409.129i	0.461250i	0.973043	+	0.230625i	$0.0740771\pi$
$887$	409.129i	0.461250i	−0.973043	+	0.230625i	$0.925923\pi$
$888$	0	0
$889$	220.714	0.248273
$890$	− 540.295i	− 0.607073i
$891$	0	0
$892$	−297.542	−0.333568
$893$	− 897.810i	− 1.00539i
$894$	0	0
$895$	1890.32	2.11208
$896$	− 158.772i	− 0.177200i
$897$	0	0
$898$	306.879	0.341736
$899$	683.781i	0.760601i
$900$	0	0
$901$	−721.683	−0.800980
$902$	3.00949i	0.00333646i
$903$	0	0
$904$	−288.178	−0.318781
$905$	10.2836i	0.0113631i
$906$	0	0
$907$	394.398	0.434838	0.217419	−	0.976078i	$-0.430236\pi$
$907$	394.398	0.434838	0.217419	+	0.976078i	$0.430236\pi$
$908$	614.348i	0.676594i
$909$	0	0
$910$	−124.543	−0.136860
$911$	220.477i	0.242017i	0.992651	+	0.121008i	$0.0386128\pi$
$911$	220.477i	0.242017i	−0.992651	+	0.121008i	$0.961387\pi$
$912$	0	0
$913$	127.405	0.139545
$914$	85.5935i	0.0936471i
$915$	0	0
$916$	268.830	0.293482
$917$	48.9691i	0.0534014i
$918$	0	0
$919$	−873.180	−0.950142	−0.475071	−	0.879948i	$-0.657578\pi$
$919$	−873.180	−0.950142	−0.475071	+	0.879948i	$0.657578\pi$
$920$	61.2520i	0.0665783i
$921$	0	0
$922$	−208.326	−0.225950
$923$	− 1811.78i	− 1.96293i
$924$	0	0
$925$	121.576	0.131434
$926$	− 238.472i	− 0.257529i
$927$	0	0
$928$	−997.782	−1.07520
$929$	38.6751i	0.0416309i	0.999783	+	0.0208154i	$0.00662624\pi$
$929$	38.6751i	0.0416309i	−0.999783	+	0.0208154i	$0.993374\pi$
$930$	0	0
$931$	−925.365	−0.993948
$932$	672.387i	0.721445i
$933$	0	0
$934$	−523.424	−0.560411
$935$	115.953i	0.124013i
$936$	0	0
$937$	−805.352	−0.859501	−0.429750	−	0.902948i	$-0.641398\pi$
$937$	−805.352	−0.859501	−0.429750	+	0.902948i	$0.641398\pi$
$938$	15.1623i	0.0161645i
$939$	0	0
$940$	−1285.93	−1.36801
$941$	1341.27i	1.42537i	0.701483	+	0.712686i	$0.252521\pi$
$941$	1341.27i	1.42537i	−0.701483	+	0.712686i	$0.747479\pi$
$942$	0	0
$943$	8.62240	0.00914358
$944$	1129.27i	1.19626i
$945$	0	0
$946$	−10.7477	−0.0113612
$947$	590.901i	0.623971i	0.950087	+	0.311986i	$0.100994\pi$
$947$	590.901i	0.623971i	−0.950087	+	0.311986i	$0.899006\pi$
$948$	0	0
$949$	−756.566	−0.797225
$950$	385.101i	0.405370i
$951$	0	0
$952$	83.6908	0.0879105
$953$	− 362.670i	− 0.380556i	−0.981730	−	0.190278i	$-0.939061\pi$
$953$	− 362.670i	− 0.380556i	0.981730	−	0.190278i	$-0.0609390\pi$
$954$	0	0
$955$	356.018	0.372794
$956$	− 32.4018i	− 0.0338931i
$957$	0	0
$958$	−486.100	−0.507411
$959$	155.571i	0.162222i
$960$	0	0
$961$	−671.068	−0.698302
$962$	43.8468i	0.0455788i
$963$	0	0
$964$	−58.2677	−0.0604437
$965$	1092.07i	1.13168i
$966$	0	0
$967$	29.7675	0.0307834	0.0153917	−	0.999882i	$-0.495100\pi$
$967$	29.7675	0.0307834	0.0153917	+	0.999882i	$0.495100\pi$
$968$	− 534.320i	− 0.551983i
$969$	0	0
$970$	42.9837	0.0443131
$971$	− 1095.66i	− 1.12839i	−0.825643	−	0.564193i	$-0.809188\pi$
$971$	− 1095.66i	− 1.12839i	0.825643	−	0.564193i	$-0.190812\pi$
$972$	0	0
$973$	−131.434	−0.135081
$974$	350.133i	0.359480i
$975$	0	0
$976$	−269.858	−0.276494
$977$	620.804i	0.635419i	0.948188	+	0.317710i	$0.102914\pi$
$977$	620.804i	0.635419i	−0.948188	+	0.317710i	$0.897086\pi$
$978$	0	0
$979$	−130.176	−0.132969
$980$	1325.40i	1.35244i
$981$	0	0
$982$	249.503	0.254077
$983$	− 607.462i	− 0.617967i	−0.951067	−	0.308984i	$-0.900011\pi$
$983$	− 607.462i	− 0.617967i	0.951067	−	0.308984i	$-0.0999889\pi$
$984$	0	0
$985$	−414.666	−0.420981
$986$	− 328.852i	− 0.333522i
$987$	0	0
$988$	1501.59	1.51983
$989$	30.7929i	0.0311354i
$990$	0	0
$991$	−539.632	−0.544532	−0.272266	−	0.962222i	$-0.587773\pi$
$991$	−539.632	−0.544532	−0.272266	+	0.962222i	$0.587773\pi$
$992$	423.073i	0.426485i
$993$	0	0
$994$	−67.2040	−0.0676097
$995$	− 2521.18i	− 2.53385i
$996$	0	0
$997$	663.930	0.665927	0.332964	−	0.942940i	$-0.391951\pi$
$997$	663.930	0.665927	0.332964	+	0.942940i	$0.391951\pi$
$998$	330.950i	0.331613i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.3.b.a.728.13		30
3.2	odd	2	inner	729.3.b.a.728.18		30
27.2	odd	18		243.3.f.a.53.3		30
27.4	even	9		81.3.f.a.35.3		30
27.5	odd	18		243.3.f.c.26.3		30
27.7	even	9		27.3.f.a.5.3	✓	30
27.11	odd	18		243.3.f.b.215.3		30
27.13	even	9		243.3.f.a.188.3		30
27.14	odd	18		243.3.f.d.188.3		30
27.16	even	9		243.3.f.c.215.3		30
27.20	odd	18		81.3.f.a.44.3		30
27.22	even	9		243.3.f.b.26.3		30
27.23	odd	18		27.3.f.a.11.3	yes	30
27.25	even	9		243.3.f.d.53.3		30
108.7	odd	18		432.3.bc.a.113.2		30
108.23	even	18		432.3.bc.a.65.2		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.5.3	✓	30	27.7	even	9
27.3.f.a.11.3	yes	30	27.23	odd	18
81.3.f.a.35.3		30	27.4	even	9
81.3.f.a.44.3		30	27.20	odd	18
243.3.f.a.53.3		30	27.2	odd	18
243.3.f.a.188.3		30	27.13	even	9
243.3.f.b.26.3		30	27.22	even	9
243.3.f.b.215.3		30	27.11	odd	18
243.3.f.c.26.3		30	27.5	odd	18
243.3.f.c.215.3		30	27.16	even	9
243.3.f.d.53.3		30	27.25	even	9
243.3.f.d.188.3		30	27.14	odd	18
432.3.bc.a.65.2		30	108.23	even	18
432.3.bc.a.113.2		30	108.7	odd	18
729.3.b.a.728.13		30	1.1	even	1	trivial
729.3.b.a.728.18		30	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	729.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.581939i q^{2} +3.66135 q^{4} -7.66608i q^{5} +1.33395 q^{7} -4.45844i q^{8} -4.46119 q^{10} +1.07486i q^{11} +20.9280 q^{13} -0.776278i q^{14} +12.0509 q^{16} +14.0720i q^{17} +19.5967 q^{19} -28.0682i q^{20} +0.625503 q^{22} -1.79211i q^{23} -33.7688 q^{25} -12.1788i q^{26} +4.88406 q^{28} -40.1577i q^{29} -17.0274 q^{31} -24.8466i q^{32} +8.18903 q^{34} -10.2262i q^{35} -3.60025 q^{37} -11.4041i q^{38} -34.1787 q^{40} +4.81131i q^{41} -17.1825 q^{43} +3.93544i q^{44} -1.04290 q^{46} -45.8144i q^{47} -47.2206 q^{49} +19.6514i q^{50} +76.6247 q^{52} +51.2852i q^{53} +8.23996 q^{55} -5.94734i q^{56} -23.3693 q^{58} +93.7084i q^{59} -22.3932 q^{61} +9.90890i q^{62} +33.7442 q^{64} -160.436i q^{65} -19.5320 q^{67} +51.5224i q^{68} -5.95101 q^{70} -86.5721i q^{71} -36.1509 q^{73} +2.09513i q^{74} +71.7502 q^{76} +1.43381i q^{77} -29.2309 q^{79} -92.3828i q^{80} +2.79989 q^{82} -118.531i q^{83} +107.877 q^{85} +9.99915i q^{86} +4.79219 q^{88} +121.110i q^{89} +27.9169 q^{91} -6.56154i q^{92} -26.6612 q^{94} -150.230i q^{95} -9.63502 q^{97} +27.4795i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67}+ \cdots - 24 q^{94}+O(q^{100}) \) 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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