LMFDB - Embedded newform 99.3.b.a.89.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,3,Mod(89,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("99.89");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	99.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:	$2.69755461717$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.65306824704.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 $ x^8 - 4x^7 - 2x^6 + 20x^5 + x^4 - 40x^3 + 36x^2 - 12x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.1
Root			$2.75726 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	99.89
Dual form			99.3.b.a.89.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.89935i q^{2} -11.2049 q^{4} -6.10332i q^{5} +2.61086 q^{7} +28.0945i q^{8} +O(q^{10})$	\(q-3.89935i q^{2} -11.2049 q^{4} -6.10332i q^{5} +2.61086 q^{7} +28.0945i q^{8} -23.7990 q^{10} +3.31662i q^{11} -7.69890 q^{13} -10.1807i q^{14} +64.7307 q^{16} -27.5859i q^{17} +3.63254 q^{19} +68.3873i q^{20} +12.9327 q^{22} -22.4470i q^{23} -12.2506 q^{25} +30.0207i q^{26} -29.2545 q^{28} -16.9284i q^{29} +3.26034 q^{31} -140.030i q^{32} -107.567 q^{34} -15.9349i q^{35} +15.0843 q^{37} -14.1645i q^{38} +171.470 q^{40} +40.2542i q^{41} +69.4624 q^{43} -37.1625i q^{44} -87.5288 q^{46} +21.7183i q^{47} -42.1834 q^{49} +47.7693i q^{50} +86.2656 q^{52} +12.1729i q^{53} +20.2424 q^{55} +73.3509i q^{56} -66.0098 q^{58} +34.0467i q^{59} +61.3863 q^{61} -12.7132i q^{62} -287.101 q^{64} +46.9889i q^{65} +54.6101 q^{67} +309.098i q^{68} -62.1359 q^{70} -11.5967i q^{71} +41.7131 q^{73} -58.8188i q^{74} -40.7023 q^{76} +8.65924i q^{77} +96.9294 q^{79} -395.073i q^{80} +156.965 q^{82} -89.8729i q^{83} -168.366 q^{85} -270.858i q^{86} -93.1790 q^{88} +117.316i q^{89} -20.1007 q^{91} +251.517i q^{92} +84.6871 q^{94} -22.1705i q^{95} -7.47681 q^{97} +164.488i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 + 16 * q^7	$ 8 q - 16 q^{4} + 16 q^{7} - 48 q^{10} - 8 q^{13} + 104 q^{16} + 40 q^{19} - 112 q^{25} - 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} + 432 q^{40} - 104 q^{43} + 24 q^{46} - 96 q^{49} + 280 q^{52} - 432 q^{58} - 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} + 448 q^{73} - 344 q^{76} + 448 q^{79} + 504 q^{82} + 48 q^{85} - 264 q^{88} - 544 q^{91} + 360 q^{94} - 152 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 16 * q^7 - 48 * q^10 - 8 * q^13 + 104 * q^16 + 40 * q^19 - 112 * q^25 - 32 * q^28 - 56 * q^31 - 216 * q^34 + 136 * q^37 + 432 * q^40 - 104 * q^43 + 24 * q^46 - 96 * q^49 + 280 * q^52 - 432 * q^58 - 8 * q^61 - 592 * q^64 + 112 * q^67 + 168 * q^70 + 448 * q^73 - 344 * q^76 + 448 * q^79 + 504 * q^82 + 48 * q^85 - 264 * q^88 - 544 * q^91 + 360 * q^94 - 152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 3.89935i	− 1.94967i	−0.222917	−	0.974837i	$-0.571558\pi$
$2$	− 3.89935i	− 1.94967i	0.222917	−	0.974837i	$-0.428442\pi$
$3$	0	0
$3$	0	0
$4$	−11.2049	−2.80123
$5$	− 6.10332i	− 1.22066i	−0.792145	−	0.610332i	$-0.791036\pi$
$5$	− 6.10332i	− 1.22066i	0.792145	−	0.610332i	$-0.208964\pi$
$6$	0	0
$7$	2.61086	0.372980	0.186490	−	0.982457i	$-0.440289\pi$
$7$	2.61086	0.372980	0.186490	+	0.982457i	$0.440289\pi$
$8$	28.0945i	3.51182i
$9$	0	0
$10$	−23.7990	−2.37990
$11$	3.31662i	0.301511i
$11$	3.31662i	0.301511i
$12$	0	0
$13$	−7.69890	−0.592223	−0.296111	−	0.955153i	$-0.595690\pi$
$13$	−7.69890	−0.592223	−0.296111	+	0.955153i	$0.595690\pi$
$14$	− 10.1807i	− 0.727190i
$15$	0	0
$16$	64.7307	4.04567
$17$	− 27.5859i	− 1.62270i	−0.584559	−	0.811351i	$-0.698732\pi$
$17$	− 27.5859i	− 1.62270i	0.584559	−	0.811351i	$-0.301268\pi$
$18$	0	0
$19$	3.63254	0.191186	0.0955930	−	0.995420i	$-0.469525\pi$
$19$	3.63254	0.191186	0.0955930	+	0.995420i	$0.469525\pi$
$20$	68.3873i	3.41937i
$21$	0	0
$22$	12.9327	0.587849
$23$	− 22.4470i	− 0.975957i	−0.872856	−	0.487979i	$-0.837734\pi$
$23$	− 22.4470i	− 0.975957i	0.872856	−	0.487979i	$-0.162266\pi$
$24$	0	0
$25$	−12.2506	−0.490023
$26$	30.0207i	1.15464i
$27$	0	0
$28$	−29.2545	−1.04480
$29$	− 16.9284i	− 0.583738i	−0.956458	−	0.291869i	$-0.905723\pi$
$29$	− 16.9284i	− 0.583738i	0.956458	−	0.291869i	$-0.0942771\pi$
$30$	0	0
$31$	3.26034	0.105172	0.0525862	−	0.998616i	$-0.483254\pi$
$31$	3.26034	0.105172	0.0525862	+	0.998616i	$0.483254\pi$
$32$	− 140.030i	− 4.37592i
$33$	0	0
$34$	−107.567	−3.16374
$35$	− 15.9349i	− 0.455284i
$36$	0	0
$37$	15.0843	0.407683	0.203841	−	0.979004i	$-0.434657\pi$
$37$	15.0843	0.407683	0.203841	+	0.979004i	$0.434657\pi$
$38$	− 14.1645i	− 0.372751i
$39$	0	0
$40$	171.470	4.28675
$41$	40.2542i	0.981809i	0.871214	+	0.490904i	$0.163334\pi$
$41$	40.2542i	0.981809i	−0.871214	+	0.490904i	$0.836666\pi$
$42$	0	0
$43$	69.4624	1.61540	0.807702	−	0.589590i	$-0.200711\pi$
$43$	69.4624	1.61540	0.807702	+	0.589590i	$0.200711\pi$
$44$	− 37.1625i	− 0.844603i
$45$	0	0
$46$	−87.5288	−1.90280
$47$	21.7183i	0.462091i	0.972943	+	0.231045i	$0.0742146\pi$
$47$	21.7183i	0.462091i	−0.972943	+	0.231045i	$0.925785\pi$
$48$	0	0
$49$	−42.1834	−0.860886
$50$	47.7693i	0.955385i
$51$	0	0
$52$	86.2656	1.65895
$53$	12.1729i	0.229677i	0.993384	+	0.114839i	$0.0366351\pi$
$53$	12.1729i	0.229677i	−0.993384	+	0.114839i	$0.963365\pi$
$54$	0	0
$55$	20.2424	0.368044
$56$	73.3509i	1.30984i
$57$	0	0
$58$	−66.0098	−1.13810
$59$	34.0467i	0.577063i	0.957470	+	0.288532i	$0.0931670\pi$
$59$	34.0467i	0.577063i	−0.957470	+	0.288532i	$0.906833\pi$
$60$	0	0
$61$	61.3863	1.00633	0.503166	−	0.864190i	$-0.332168\pi$
$61$	61.3863	1.00633	0.503166	+	0.864190i	$0.332168\pi$
$62$	− 12.7132i	− 0.205052i
$63$	0	0
$64$	−287.101	−4.48596
$65$	46.9889i	0.722906i
$66$	0	0
$67$	54.6101	0.815076	0.407538	−	0.913188i	$-0.366387\pi$
$67$	54.6101	0.815076	0.407538	+	0.913188i	$0.366387\pi$
$68$	309.098i	4.54557i
$69$	0	0
$70$	−62.1359	−0.887655
$71$	− 11.5967i	− 0.163334i	−0.996660	−	0.0816670i	$-0.973976\pi$
$71$	− 11.5967i	− 0.163334i	0.996660	−	0.0816670i	$-0.0260244\pi$
$72$	0	0
$73$	41.7131	0.571412	0.285706	−	0.958317i	$-0.407772\pi$
$73$	41.7131	0.571412	0.285706	+	0.958317i	$0.407772\pi$
$74$	− 58.8188i	− 0.794849i
$75$	0	0
$76$	−40.7023	−0.535557
$77$	8.65924i	0.112458i
$78$	0	0
$79$	96.9294	1.22695	0.613477	−	0.789712i	$-0.289770\pi$
$79$	96.9294	1.22695	0.613477	+	0.789712i	$0.289770\pi$
$80$	− 395.073i	− 4.93841i
$81$	0	0
$82$	156.965	1.91421
$83$	− 89.8729i	− 1.08281i	−0.840763	−	0.541403i	$-0.817893\pi$
$83$	− 89.8729i	− 1.08281i	0.840763	−	0.541403i	$-0.182107\pi$
$84$	0	0
$85$	−168.366	−1.98078
$86$	− 270.858i	− 3.14951i
$87$	0	0
$88$	−93.1790	−1.05885
$89$	117.316i	1.31816i	0.752074	+	0.659078i	$0.229053\pi$
$89$	117.316i	1.31816i	−0.752074	+	0.659078i	$0.770947\pi$
$90$	0	0
$91$	−20.1007	−0.220887
$92$	251.517i	2.73388i
$93$	0	0
$94$	84.6871	0.900927
$95$	− 22.1705i	− 0.233374i
$96$	0	0
$97$	−7.47681	−0.0770806	−0.0385403	−	0.999257i	$-0.512271\pi$
$97$	−7.47681	−0.0770806	−0.0385403	+	0.999257i	$0.512271\pi$
$98$	164.488i	1.67845i
$99$	0	0
$100$	137.267	1.37267
$101$	− 55.4534i	− 0.549044i	−0.961581	−	0.274522i	$-0.911480\pi$
$101$	− 55.4534i	− 0.549044i	0.961581	−	0.274522i	$-0.0885196\pi$
$102$	0	0
$103$	−167.207	−1.62337	−0.811683	−	0.584098i	$-0.801448\pi$
$103$	−167.207	−1.62337	−0.811683	+	0.584098i	$0.801448\pi$
$104$	− 216.297i	− 2.07978i
$105$	0	0
$106$	47.4664	0.447796
$107$	− 37.0923i	− 0.346657i	−0.984864	−	0.173329i	$-0.944548\pi$
$107$	− 37.0923i	− 0.346657i	0.984864	−	0.173329i	$-0.0554523\pi$
$108$	0	0
$109$	−97.9327	−0.898465	−0.449232	−	0.893415i	$-0.648303\pi$
$109$	−97.9327	−0.898465	−0.449232	+	0.893415i	$0.648303\pi$
$110$	− 78.9323i	− 0.717567i
$111$	0	0
$112$	169.003	1.50895
$113$	207.091i	1.83267i	0.400416	+	0.916334i	$0.368866\pi$
$113$	207.091i	1.83267i	−0.400416	+	0.916334i	$0.631134\pi$
$114$	0	0
$115$	−137.001	−1.19132
$116$	189.682i	1.63519i
$117$	0	0
$118$	132.760	1.12509
$119$	− 72.0230i	− 0.605235i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	− 239.367i	− 1.96202i
$123$	0	0
$124$	−36.5319	−0.294612
$125$	− 77.8139i	− 0.622511i
$126$	0	0
$127$	182.507	1.43707	0.718533	−	0.695493i	$-0.244814\pi$
$127$	182.507	1.43707	0.718533	+	0.695493i	$0.244814\pi$
$128$	559.391i	4.37024i
$129$	0	0
$130$	183.226	1.40943
$131$	− 163.461i	− 1.24780i	−0.781506	−	0.623898i	$-0.785548\pi$
$131$	− 163.461i	− 1.24780i	0.781506	−	0.623898i	$-0.214452\pi$
$132$	0	0
$133$	9.48404	0.0713086
$134$	− 212.944i	− 1.58913i
$135$	0	0
$136$	775.014	5.69863
$137$	199.563i	1.45666i	0.685224	+	0.728332i	$0.259704\pi$
$137$	199.563i	1.45666i	−0.685224	+	0.728332i	$0.740296\pi$
$138$	0	0
$139$	2.75894	0.0198485	0.00992425	−	0.999951i	$-0.496841\pi$
$139$	2.75894	0.0198485	0.00992425	+	0.999951i	$0.496841\pi$
$140$	178.550i	1.27536i
$141$	0	0
$142$	−45.2196	−0.318448
$143$	− 25.5344i	− 0.178562i
$144$	0	0
$145$	−103.320	−0.712549
$146$	− 162.654i	− 1.11407i
$147$	0	0
$148$	−169.018	−1.14201
$149$	− 79.7092i	− 0.534961i	−0.963563	−	0.267481i	$-0.913809\pi$
$149$	− 79.7092i	− 0.534961i	0.963563	−	0.267481i	$-0.0861911\pi$
$150$	0	0
$151$	−149.776	−0.991894	−0.495947	−	0.868353i	$-0.665179\pi$
$151$	−149.776	−0.991894	−0.495947	+	0.868353i	$0.665179\pi$
$152$	102.054i	0.671411i
$153$	0	0
$154$	33.7654	0.219256
$155$	− 19.8989i	− 0.128380i
$156$	0	0
$157$	−273.723	−1.74346	−0.871730	−	0.489986i	$-0.837002\pi$
$157$	−273.723	−1.74346	−0.871730	+	0.489986i	$0.837002\pi$
$158$	− 377.962i	− 2.39216i
$159$	0	0
$160$	−854.646	−5.34154
$161$	− 58.6060i	− 0.364012i
$162$	0	0
$163$	289.639	1.77693	0.888464	−	0.458946i	$-0.151773\pi$
$163$	289.639	1.77693	0.888464	+	0.458946i	$0.151773\pi$
$164$	− 451.045i	− 2.75027i
$165$	0	0
$166$	−350.446	−2.11112
$167$	47.9894i	0.287362i	0.989624	+	0.143681i	$0.0458939\pi$
$167$	47.9894i	0.287362i	−0.989624	+	0.143681i	$0.954106\pi$
$168$	0	0
$169$	−109.727	−0.649272
$170$	656.518i	3.86187i
$171$	0	0
$172$	−778.321	−4.52512
$173$	− 35.0195i	− 0.202425i	−0.994865	−	0.101212i	$-0.967728\pi$
$173$	− 35.0195i	− 0.202425i	0.994865	−	0.101212i	$-0.0322722\pi$
$174$	0	0
$175$	−31.9845	−0.182769
$176$	214.688i	1.21982i
$177$	0	0
$178$	457.456	2.56998
$179$	190.174i	1.06242i	0.847239	+	0.531212i	$0.178263\pi$
$179$	190.174i	1.06242i	−0.847239	+	0.531212i	$0.821737\pi$
$180$	0	0
$181$	−78.2856	−0.432517	−0.216258	−	0.976336i	$-0.569385\pi$
$181$	−78.2856	−0.432517	−0.216258	+	0.976336i	$0.569385\pi$
$182$	78.3798i	0.430658i
$183$	0	0
$184$	630.639	3.42738
$185$	− 92.0642i	− 0.497644i
$186$	0	0
$187$	91.4922	0.489263
$188$	− 243.352i	− 1.29442i
$189$	0	0
$190$	−86.4507	−0.455004
$191$	99.6608i	0.521784i	0.965368	+	0.260892i	$0.0840167\pi$
$191$	99.6608i	0.521784i	−0.965368	+	0.260892i	$0.915983\pi$
$192$	0	0
$193$	29.9880	0.155378	0.0776891	−	0.996978i	$-0.475246\pi$
$193$	29.9880	0.155378	0.0776891	+	0.996978i	$0.475246\pi$
$194$	29.1547i	0.150282i
$195$	0	0
$196$	472.662	2.41154
$197$	− 82.2920i	− 0.417726i	−0.977945	−	0.208863i	$-0.933024\pi$
$197$	− 82.2920i	− 0.417726i	0.977945	−	0.208863i	$-0.0669763\pi$
$198$	0	0
$199$	217.671	1.09383	0.546913	−	0.837189i	$-0.315803\pi$
$199$	217.671	1.09383	0.546913	+	0.837189i	$0.315803\pi$
$200$	− 344.174i	− 1.72087i
$201$	0	0
$202$	−216.232	−1.07046
$203$	− 44.1977i	− 0.217723i
$204$	0	0
$205$	245.684	1.19846
$206$	651.997i	3.16504i
$207$	0	0
$208$	−498.355	−2.39594
$209$	12.0478i	0.0576448i
$210$	0	0
$211$	−105.008	−0.497669	−0.248834	−	0.968546i	$-0.580048\pi$
$211$	−105.008	−0.497669	−0.248834	+	0.968546i	$0.580048\pi$
$212$	− 136.396i	− 0.643380i
$213$	0	0
$214$	−144.636	−0.675869
$215$	− 423.952i	− 1.97187i
$216$	0	0
$217$	8.51230	0.0392272
$218$	381.874i	1.75171i
$219$	0	0
$220$	−226.815	−1.03098
$221$	212.381i	0.961002i
$222$	0	0
$223$	238.589	1.06991	0.534953	−	0.844882i	$-0.320329\pi$
$223$	238.589	1.06991	0.534953	+	0.844882i	$0.320329\pi$
$224$	− 365.598i	− 1.63213i
$225$	0	0
$226$	807.522	3.57310
$227$	326.217i	1.43708i	0.695485	+	0.718540i	$0.255190\pi$
$227$	326.217i	1.43708i	−0.695485	+	0.718540i	$0.744810\pi$
$228$	0	0
$229$	233.208	1.01837	0.509187	−	0.860656i	$-0.329946\pi$
$229$	233.208	1.01837	0.509187	+	0.860656i	$0.329946\pi$
$230$	534.216i	2.32268i
$231$	0	0
$232$	475.596	2.04998
$233$	− 65.6551i	− 0.281781i	−0.990025	−	0.140891i	$-0.955003\pi$
$233$	− 65.6551i	− 0.281781i	0.990025	−	0.140891i	$-0.0449966\pi$
$234$	0	0
$235$	132.554	0.564058
$236$	− 381.491i	− 1.61649i
$237$	0	0
$238$	−280.843	−1.18001
$239$	214.400i	0.897072i	0.893765	+	0.448536i	$0.148054\pi$
$239$	214.400i	0.897072i	−0.893765	+	0.448536i	$0.851946\pi$
$240$	0	0
$241$	−404.958	−1.68032	−0.840161	−	0.542337i	$-0.817540\pi$
$241$	−404.958	−1.68032	−0.840161	+	0.542337i	$0.817540\pi$
$242$	42.8928i	0.177243i
$243$	0	0
$244$	−687.829	−2.81897
$245$	257.459i	1.05085i
$246$	0	0
$247$	−27.9665	−0.113225
$248$	91.5978i	0.369346i
$249$	0	0
$250$	−303.424	−1.21369
$251$	− 300.918i	− 1.19888i	−0.800420	−	0.599439i	$-0.795390\pi$
$251$	− 300.918i	− 1.19888i	0.800420	−	0.599439i	$-0.204610\pi$
$252$	0	0
$253$	74.4483	0.294262
$254$	− 711.660i	− 2.80181i
$255$	0	0
$256$	1032.85	4.03458
$257$	424.503i	1.65176i	0.563844	+	0.825881i	$0.309322\pi$
$257$	424.503i	1.65176i	−0.563844	+	0.825881i	$0.690678\pi$
$258$	0	0
$259$	39.3829	0.152058
$260$	− 526.507i	− 2.02503i
$261$	0	0
$262$	−637.393	−2.43280
$263$	306.813i	1.16659i	0.812261	+	0.583294i	$0.198236\pi$
$263$	306.813i	1.16659i	−0.812261	+	0.583294i	$0.801764\pi$
$264$	0	0
$265$	74.2951	0.280359
$266$	− 36.9816i	− 0.139029i
$267$	0	0
$268$	−611.902	−2.28322
$269$	− 248.773i	− 0.924809i	−0.886669	−	0.462404i	$-0.846987\pi$
$269$	− 248.773i	− 0.924809i	0.886669	−	0.462404i	$-0.153013\pi$
$270$	0	0
$271$	230.792	0.851631	0.425816	−	0.904810i	$-0.359987\pi$
$271$	230.792	0.851631	0.425816	+	0.904810i	$0.359987\pi$
$272$	− 1785.66i	− 6.56492i
$273$	0	0
$274$	778.166	2.84002
$275$	− 40.6306i	− 0.147747i
$276$	0	0
$277$	442.379	1.59703	0.798517	−	0.601972i	$-0.205618\pi$
$277$	442.379	1.59703	0.798517	+	0.601972i	$0.205618\pi$
$278$	− 10.7581i	− 0.0386981i
$279$	0	0
$280$	447.684	1.59887
$281$	− 219.013i	− 0.779407i	−0.920940	−	0.389704i	$-0.872577\pi$
$281$	− 219.013i	− 0.779407i	0.920940	−	0.389704i	$-0.127423\pi$
$282$	0	0
$283$	−144.620	−0.511023	−0.255512	−	0.966806i	$-0.582244\pi$
$283$	−144.620	−0.511023	−0.255512	+	0.966806i	$0.582244\pi$
$284$	129.940i	0.457537i
$285$	0	0
$286$	−99.5674	−0.348138
$287$	105.098i	0.366195i
$288$	0	0
$289$	−471.984	−1.63316
$290$	402.879i	1.38924i
$291$	0	0
$292$	−467.392	−1.60066
$293$	− 103.257i	− 0.352412i	−0.984353	−	0.176206i	$-0.943618\pi$
$293$	− 103.257i	− 0.352412i	0.984353	−	0.176206i	$-0.0563824\pi$
$294$	0	0
$295$	207.798	0.704401
$296$	423.786i	1.43171i
$297$	0	0
$298$	−310.814	−1.04300
$299$	172.817i	0.577984i
$300$	0	0
$301$	181.357	0.602514
$302$	584.029i	1.93387i
$303$	0	0
$304$	235.137	0.773476
$305$	− 374.660i	− 1.22839i
$306$	0	0
$307$	332.933	1.08447	0.542237	−	0.840226i	$-0.317578\pi$
$307$	332.933	1.08447	0.542237	+	0.840226i	$0.317578\pi$
$308$	− 97.0262i	− 0.315020i
$309$	0	0
$310$	−77.5929	−0.250300
$311$	− 534.673i	− 1.71921i	−0.510962	−	0.859603i	$-0.670711\pi$
$311$	− 534.673i	− 1.71921i	0.510962	−	0.859603i	$-0.329289\pi$
$312$	0	0
$313$	−312.885	−0.999631	−0.499816	−	0.866132i	$-0.666599\pi$
$313$	−312.885	−0.999631	−0.499816	+	0.866132i	$0.666599\pi$
$314$	1067.34i	3.39918i
$315$	0	0
$316$	−1086.09	−3.43699
$317$	− 260.003i	− 0.820200i	−0.912041	−	0.410100i	$-0.865494\pi$
$317$	− 260.003i	− 0.820200i	0.912041	−	0.410100i	$-0.134506\pi$
$318$	0	0
$319$	56.1452	0.176004
$320$	1752.27i	5.47585i
$321$	0	0
$322$	−228.525	−0.709706
$323$	− 100.207i	− 0.310238i
$324$	0	0
$325$	94.3159	0.290203
$326$	− 1129.41i	− 3.46443i
$327$	0	0
$328$	−1130.92	−3.44793
$329$	56.7034i	0.172351i
$330$	0	0
$331$	−545.194	−1.64711	−0.823555	−	0.567236i	$-0.808013\pi$
$331$	−545.194	−1.64711	−0.823555	+	0.567236i	$0.808013\pi$
$332$	1007.02i	3.03319i
$333$	0	0
$334$	187.128	0.560262
$335$	− 333.303i	− 0.994934i
$336$	0	0
$337$	321.319	0.953468	0.476734	−	0.879048i	$-0.341820\pi$
$337$	321.319	0.953468	0.476734	+	0.879048i	$0.341820\pi$
$338$	427.864i	1.26587i
$339$	0	0
$340$	1886.53	5.54861
$341$	10.8133i	0.0317107i
$342$	0	0
$343$	−238.067	−0.694073
$344$	1951.51i	5.67301i
$345$	0	0
$346$	−136.553	−0.394662
$347$	− 307.133i	− 0.885111i	−0.896741	−	0.442555i	$-0.854072\pi$
$347$	− 307.133i	− 0.885111i	0.896741	−	0.442555i	$-0.145928\pi$
$348$	0	0
$349$	483.011	1.38399	0.691993	−	0.721904i	$-0.256733\pi$
$349$	483.011	1.38399	0.691993	+	0.721904i	$0.256733\pi$
$350$	124.719i	0.356340i
$351$	0	0
$352$	464.426	1.31939
$353$	137.108i	0.388408i	0.980961	+	0.194204i	$0.0622124\pi$
$353$	137.108i	0.388408i	−0.980961	+	0.194204i	$0.937788\pi$
$354$	0	0
$355$	−70.7785	−0.199376
$356$	− 1314.52i	− 3.69246i
$357$	0	0
$358$	741.555	2.07138
$359$	236.029i	0.657463i	0.944423	+	0.328731i	$0.106621\pi$
$359$	236.029i	0.657463i	−0.944423	+	0.328731i	$0.893379\pi$
$360$	0	0
$361$	−347.805	−0.963448
$362$	305.263i	0.843268i
$363$	0	0
$364$	225.227	0.618757
$365$	− 254.589i	− 0.697503i
$366$	0	0
$367$	−332.474	−0.905924	−0.452962	−	0.891530i	$-0.649633\pi$
$367$	−332.474	−0.905924	−0.452962	+	0.891530i	$0.649633\pi$
$368$	− 1453.01i	− 3.94840i
$369$	0	0
$370$	−358.990	−0.970244
$371$	31.7817i	0.0856650i
$372$	0	0
$373$	−191.412	−0.513169	−0.256585	−	0.966522i	$-0.582597\pi$
$373$	−191.412	−0.513169	−0.256585	+	0.966522i	$0.582597\pi$
$374$	− 356.760i	− 0.953904i
$375$	0	0
$376$	−610.165	−1.62278
$377$	130.330i	0.345703i
$378$	0	0
$379$	−387.997	−1.02374	−0.511870	−	0.859063i	$-0.671047\pi$
$379$	−387.997	−1.02374	−0.511870	+	0.859063i	$0.671047\pi$
$380$	248.419i	0.653735i
$381$	0	0
$382$	388.612	1.01731
$383$	− 119.163i	− 0.311131i	−0.987826	−	0.155566i	$-0.950280\pi$
$383$	− 119.163i	− 0.311131i	0.987826	−	0.155566i	$-0.0497200\pi$
$384$	0	0
$385$	52.8502	0.137273
$386$	− 116.934i	− 0.302937i
$387$	0	0
$388$	83.7772	0.215921
$389$	204.409i	0.525473i	0.964868	+	0.262736i	$0.0846249\pi$
$389$	204.409i	0.525473i	−0.964868	+	0.262736i	$0.915375\pi$
$390$	0	0
$391$	−619.222	−1.58369
$392$	− 1185.12i	− 3.02327i
$393$	0	0
$394$	−320.885	−0.814430
$395$	− 591.592i	− 1.49770i
$396$	0	0
$397$	543.945	1.37014	0.685069	−	0.728478i	$-0.259772\pi$
$397$	543.945	1.37014	0.685069	+	0.728478i	$0.259772\pi$
$398$	− 848.777i	− 2.13261i
$399$	0	0
$400$	−792.988	−1.98247
$401$	− 112.010i	− 0.279326i	−0.990199	−	0.139663i	$-0.955398\pi$
$401$	− 112.010i	− 0.279326i	0.990199	−	0.139663i	$-0.0446020\pi$
$402$	0	0
$403$	−25.1010	−0.0622855
$404$	621.352i	1.53800i
$405$	0	0
$406$	−172.342	−0.424488
$407$	50.0289i	0.122921i
$408$	0	0
$409$	−156.376	−0.382337	−0.191168	−	0.981557i	$-0.561228\pi$
$409$	−156.376	−0.382337	−0.191168	+	0.981557i	$0.561228\pi$
$410$	− 958.009i	− 2.33661i
$411$	0	0
$412$	1873.54	4.54742
$413$	88.8912i	0.215233i
$414$	0	0
$415$	−548.524	−1.32174
$416$	1078.07i	2.59152i
$417$	0	0
$418$	46.9784	0.112389
$419$	618.122i	1.47523i	0.675221	+	0.737616i	$0.264048\pi$
$419$	618.122i	1.47523i	−0.675221	+	0.737616i	$0.735952\pi$
$420$	0	0
$421$	289.351	0.687295	0.343647	−	0.939099i	$-0.388338\pi$
$421$	289.351	0.687295	0.343647	+	0.939099i	$0.388338\pi$
$422$	409.464i	0.970293i
$423$	0	0
$424$	−341.992	−0.806585
$425$	337.944i	0.795161i
$426$	0	0
$427$	160.271	0.375342
$428$	415.617i	0.971068i
$429$	0	0
$430$	−1653.14	−3.84450
$431$	608.054i	1.41080i	0.708811	+	0.705399i	$0.249232\pi$
$431$	608.054i	1.41080i	−0.708811	+	0.705399i	$0.750768\pi$
$432$	0	0
$433$	−39.7517	−0.0918054	−0.0459027	−	0.998946i	$-0.514616\pi$
$433$	−39.7517	−0.0918054	−0.0459027	+	0.998946i	$0.514616\pi$
$434$	− 33.1924i	− 0.0764803i
$435$	0	0
$436$	1097.33	2.51681
$437$	− 81.5396i	− 0.186589i
$438$	0	0
$439$	−474.376	−1.08058	−0.540292	−	0.841478i	$-0.681686\pi$
$439$	−474.376	−1.08058	−0.540292	+	0.841478i	$0.681686\pi$
$440$	568.702i	1.29250i
$441$	0	0
$442$	828.149	1.87364
$443$	140.746i	0.317711i	0.987302	+	0.158855i	$0.0507804\pi$
$443$	140.746i	0.317711i	−0.987302	+	0.158855i	$0.949220\pi$
$444$	0	0
$445$	716.017	1.60903
$446$	− 930.343i	− 2.08597i
$447$	0	0
$448$	−749.582	−1.67317
$449$	193.508i	0.430976i	0.976506	+	0.215488i	$0.0691342\pi$
$449$	193.508i	0.430976i	−0.976506	+	0.215488i	$0.930866\pi$
$450$	0	0
$451$	−133.508	−0.296026
$452$	− 2320.44i	− 5.13373i
$453$	0	0
$454$	1272.04	2.80184
$455$	122.681i	0.269629i
$456$	0	0
$457$	−31.5272	−0.0689874	−0.0344937	−	0.999405i	$-0.510982\pi$
$457$	−31.5272	−0.0689874	−0.0344937	+	0.999405i	$0.510982\pi$
$458$	− 909.358i	− 1.98550i
$459$	0	0
$460$	1535.09	3.33715
$461$	− 502.003i	− 1.08894i	−0.838779	−	0.544471i	$-0.816730\pi$
$461$	− 502.003i	− 1.08894i	0.838779	−	0.544471i	$-0.183270\pi$
$462$	0	0
$463$	786.087	1.69781	0.848906	−	0.528544i	$-0.177262\pi$
$463$	786.087	1.69781	0.848906	+	0.528544i	$0.177262\pi$
$464$	− 1095.79i	− 2.36161i
$465$	0	0
$466$	−256.012	−0.549382
$467$	538.345i	1.15277i	0.817177	+	0.576387i	$0.195538\pi$
$467$	538.345i	1.15277i	−0.817177	+	0.576387i	$0.804462\pi$
$468$	0	0
$469$	142.579	0.304007
$470$	− 516.873i	− 1.09973i
$471$	0	0
$472$	−956.527	−2.02654
$473$	230.381i	0.487063i
$474$	0	0
$475$	−44.5006	−0.0936856
$476$	807.013i	1.69541i
$477$	0	0
$478$	836.022	1.74900
$479$	54.9462i	0.114710i	0.998354	+	0.0573551i	$0.0182667\pi$
$479$	54.9462i	0.114710i	−0.998354	+	0.0573551i	$0.981733\pi$
$480$	0	0
$481$	−116.132	−0.241439
$482$	1579.07i	3.27608i
$483$	0	0
$484$	123.254	0.254657
$485$	45.6334i	0.0940895i
$486$	0	0
$487$	281.751	0.578543	0.289272	−	0.957247i	$-0.406587\pi$
$487$	281.751	0.578543	0.289272	+	0.957247i	$0.406587\pi$
$488$	1724.62i	3.53406i
$489$	0	0
$490$	1003.92	2.04882
$491$	575.037i	1.17116i	0.810616	+	0.585578i	$0.199132\pi$
$491$	575.037i	1.17116i	−0.810616	+	0.585578i	$0.800868\pi$
$492$	0	0
$493$	−466.986	−0.947233
$494$	109.051i	0.220752i
$495$	0	0
$496$	211.044	0.425493
$497$	− 30.2774i	− 0.0609203i
$498$	0	0
$499$	−173.598	−0.347892	−0.173946	−	0.984755i	$-0.555652\pi$
$499$	−173.598	−0.347892	−0.173946	+	0.984755i	$0.555652\pi$
$500$	871.899i	1.74380i
$501$	0	0
$502$	−1173.39	−2.33742
$503$	195.789i	0.389242i	0.980879	+	0.194621i	$0.0623477\pi$
$503$	195.789i	0.389242i	−0.980879	+	0.194621i	$0.937652\pi$
$504$	0	0
$505$	−338.450	−0.670199
$506$	− 290.300i	− 0.573716i
$507$	0	0
$508$	−2044.98	−4.02555
$509$	176.019i	0.345813i	0.984938	+	0.172906i	$0.0553158\pi$
$509$	176.019i	0.345813i	−0.984938	+	0.172906i	$0.944684\pi$
$510$	0	0
$511$	108.907	0.213125
$512$	− 1789.90i	− 3.49589i
$513$	0	0
$514$	1655.29	3.22040
$515$	1020.52i	1.98159i
$516$	0	0
$517$	−72.0314	−0.139326
$518$	− 153.568i	− 0.296463i
$519$	0	0
$520$	−1320.13	−2.53871
$521$	− 907.916i	− 1.74264i	−0.490714	−	0.871321i	$-0.663264\pi$
$521$	− 907.916i	− 1.74264i	0.490714	−	0.871321i	$-0.336736\pi$
$522$	0	0
$523$	−439.027	−0.839441	−0.419720	−	0.907654i	$-0.637872\pi$
$523$	−439.027	−0.839441	−0.419720	+	0.907654i	$0.637872\pi$
$524$	1831.57i	3.49537i
$525$	0	0
$526$	1196.37	2.27447
$527$	− 89.9396i	− 0.170663i
$528$	0	0
$529$	25.1315	0.0475076
$530$	− 289.703i	− 0.546609i
$531$	0	0
$532$	−106.268	−0.199752
$533$	− 309.913i	− 0.581450i
$534$	0	0
$535$	−226.387	−0.423152
$536$	1534.24i	2.86240i
$537$	0	0
$538$	−970.055	−1.80308
$539$	− 139.907i	− 0.259567i
$540$	0	0
$541$	−40.4729	−0.0748113	−0.0374057	−	0.999300i	$-0.511909\pi$
$541$	−40.4729	−0.0748113	−0.0374057	+	0.999300i	$0.511909\pi$
$542$	− 899.939i	− 1.66040i
$543$	0	0
$544$	−3862.85	−7.10082
$545$	597.715i	1.09672i
$546$	0	0
$547$	178.824	0.326918	0.163459	−	0.986550i	$-0.447735\pi$
$547$	178.824	0.326918	0.163459	+	0.986550i	$0.447735\pi$
$548$	− 2236.09i	− 4.08046i
$549$	0	0
$550$	−158.433	−0.288060
$551$	− 61.4930i	− 0.111603i
$552$	0	0
$553$	253.069	0.457630
$554$	− 1724.99i	− 3.11370i
$555$	0	0
$556$	−30.9137	−0.0556003
$557$	− 358.907i	− 0.644357i	−0.946679	−	0.322179i	$-0.895585\pi$
$557$	− 358.907i	− 0.644357i	0.946679	−	0.322179i	$-0.104415\pi$
$558$	0	0
$559$	−534.784	−0.956680
$560$	− 1031.48i	− 1.84193i
$561$	0	0
$562$	−854.010	−1.51959
$563$	− 205.504i	− 0.365016i	−0.983204	−	0.182508i	$-0.941578\pi$
$563$	− 205.504i	− 0.365016i	0.983204	−	0.182508i	$-0.0584216\pi$
$564$	0	0
$565$	1263.95	2.23707
$566$	563.922i	0.996329i
$567$	0	0
$568$	325.804	0.573599
$569$	63.9111i	0.112322i	0.998422	+	0.0561609i	$0.0178860\pi$
$569$	63.9111i	0.112322i	−0.998422	+	0.0561609i	$0.982114\pi$
$570$	0	0
$571$	645.953	1.13127	0.565633	−	0.824657i	$-0.308632\pi$
$571$	645.953	1.13127	0.565633	+	0.824657i	$0.308632\pi$
$572$	286.111i	0.500193i
$573$	0	0
$574$	409.814	0.713961
$575$	274.989i	0.478241i
$576$	0	0
$577$	−30.5030	−0.0528648	−0.0264324	−	0.999651i	$-0.508415\pi$
$577$	−30.5030	−0.0528648	−0.0264324	+	0.999651i	$0.508415\pi$
$578$	1840.43i	3.18414i
$579$	0	0
$580$	1157.69	1.99601
$581$	− 234.646i	− 0.403865i
$582$	0	0
$583$	−40.3729	−0.0692503
$584$	1171.91i	2.00670i
$585$	0	0
$586$	−402.634	−0.687088
$587$	− 433.560i	− 0.738604i	−0.929309	−	0.369302i	$-0.879597\pi$
$587$	− 433.560i	− 0.738604i	0.929309	−	0.369302i	$-0.120403\pi$
$588$	0	0
$589$	11.8433	0.0201075
$590$	− 810.278i	− 1.37335i
$591$	0	0
$592$	976.415	1.64935
$593$	− 583.923i	− 0.984693i	−0.870399	−	0.492347i	$-0.836139\pi$
$593$	− 583.923i	− 0.984693i	0.870399	−	0.492347i	$-0.163861\pi$
$594$	0	0
$595$	−439.580	−0.738790
$596$	893.136i	1.49855i
$597$	0	0
$598$	673.875	1.12688
$599$	62.6790i	0.104639i	0.998630	+	0.0523197i	$0.0166615\pi$
$599$	62.6790i	0.104639i	−0.998630	+	0.0523197i	$0.983339\pi$
$600$	0	0
$601$	−456.794	−0.760057	−0.380028	−	0.924975i	$-0.624086\pi$
$601$	−456.794	−0.760057	−0.380028	+	0.924975i	$0.624086\pi$
$602$	− 707.173i	− 1.17471i
$603$	0	0
$604$	1678.23	2.77852
$605$	67.1366i	0.110970i
$606$	0	0
$607$	849.069	1.39880	0.699398	−	0.714732i	$-0.253451\pi$
$607$	849.069	1.39880	0.699398	+	0.714732i	$0.253451\pi$
$608$	− 508.662i	− 0.836616i
$609$	0	0
$610$	−1460.93	−2.39497
$611$	− 167.207i	− 0.273661i
$612$	0	0
$613$	217.267	0.354433	0.177216	−	0.984172i	$-0.443291\pi$
$613$	217.267	0.354433	0.177216	+	0.984172i	$0.443291\pi$
$614$	− 1298.22i	− 2.11437i
$615$	0	0
$616$	−243.277	−0.394931
$617$	417.676i	0.676947i	0.940976	+	0.338473i	$0.109910\pi$
$617$	417.676i	0.676947i	−0.940976	+	0.338473i	$0.890090\pi$
$618$	0	0
$619$	542.307	0.876101	0.438051	−	0.898950i	$-0.355669\pi$
$619$	542.307	0.876101	0.438051	+	0.898950i	$0.355669\pi$
$620$	222.966i	0.359623i
$621$	0	0
$622$	−2084.88	−3.35189
$623$	306.295i	0.491646i
$624$	0	0
$625$	−781.188	−1.24990
$626$	1220.05i	1.94896i
$627$	0	0
$628$	3067.05	4.88384
$629$	− 416.114i	− 0.661548i
$630$	0	0
$631$	−660.757	−1.04716	−0.523579	−	0.851977i	$-0.675404\pi$
$631$	−660.757	−1.04716	−0.523579	+	0.851977i	$0.675404\pi$
$632$	2723.19i	4.30884i
$633$	0	0
$634$	−1013.84	−1.59912
$635$	− 1113.90i	− 1.75418i
$636$	0	0
$637$	324.766	0.509836
$638$	− 218.930i	− 0.343150i
$639$	0	0
$640$	3414.14	5.33460
$641$	− 656.483i	− 1.02416i	−0.858939	−	0.512078i	$-0.828876\pi$
$641$	− 656.483i	− 1.02416i	0.858939	−	0.512078i	$-0.171124\pi$
$642$	0	0
$643$	193.650	0.301166	0.150583	−	0.988597i	$-0.451885\pi$
$643$	193.650	0.301166	0.150583	+	0.988597i	$0.451885\pi$
$644$	656.676i	1.01968i
$645$	0	0
$646$	−390.742	−0.604863
$647$	484.835i	0.749359i	0.927154	+	0.374679i	$0.122247\pi$
$647$	484.835i	0.749359i	−0.927154	+	0.374679i	$0.877753\pi$
$648$	0	0
$649$	−112.920	−0.173991
$650$	− 367.771i	− 0.565801i
$651$	0	0
$652$	−3245.39	−4.97759
$653$	530.535i	0.812457i	0.913771	+	0.406229i	$0.133156\pi$
$653$	530.535i	0.812457i	−0.913771	+	0.406229i	$0.866844\pi$
$654$	0	0
$655$	−997.658	−1.52314
$656$	2605.68i	3.97207i
$657$	0	0
$658$	221.106	0.336028
$659$	513.861i	0.779759i	0.920866	+	0.389880i	$0.127483\pi$
$659$	513.861i	0.779759i	−0.920866	+	0.389880i	$0.872517\pi$
$660$	0	0
$661$	356.601	0.539487	0.269743	−	0.962932i	$-0.413061\pi$
$661$	356.601	0.539487	0.269743	+	0.962932i	$0.413061\pi$
$662$	2125.90i	3.21133i
$663$	0	0
$664$	2524.94	3.80262
$665$	− 57.8842i	− 0.0870439i
$666$	0	0
$667$	−379.992	−0.569703
$668$	− 537.718i	− 0.804967i
$669$	0	0
$670$	−1299.66	−1.93980
$671$	203.595i	0.303421i
$672$	0	0
$673$	−551.643	−0.819677	−0.409839	−	0.912158i	$-0.634415\pi$
$673$	−551.643	−0.819677	−0.409839	+	0.912158i	$0.634415\pi$
$674$	− 1252.93i	− 1.85895i
$675$	0	0
$676$	1229.48	1.81876
$677$	− 673.629i	− 0.995021i	−0.867458	−	0.497510i	$-0.834248\pi$
$677$	− 673.629i	− 0.995021i	0.867458	−	0.497510i	$-0.165752\pi$
$678$	0	0
$679$	−19.5209	−0.0287495
$680$	− 4730.16i	− 6.95612i
$681$	0	0
$682$	42.1650	0.0618255
$683$	22.2083i	0.0325158i	0.999868	+	0.0162579i	$0.00517528\pi$
$683$	22.2083i	0.0325158i	−0.999868	+	0.0162579i	$0.994825\pi$
$684$	0	0
$685$	1218.00	1.77810
$686$	928.307i	1.35322i
$687$	0	0
$688$	4496.35	6.53539
$689$	− 93.7179i	− 0.136020i
$690$	0	0
$691$	−932.929	−1.35011	−0.675057	−	0.737766i	$-0.735881\pi$
$691$	−932.929	−1.35011	−0.675057	+	0.737766i	$0.735881\pi$
$692$	392.391i	0.567039i
$693$	0	0
$694$	−1197.62	−1.72568
$695$	− 16.8387i	− 0.0242284i
$696$	0	0
$697$	1110.45	1.59318
$698$	− 1883.43i	− 2.69832i
$699$	0	0
$700$	358.384	0.511978
$701$	− 690.617i	− 0.985188i	−0.870259	−	0.492594i	$-0.836049\pi$
$701$	− 690.617i	− 0.985188i	0.870259	−	0.492594i	$-0.163951\pi$
$702$	0	0
$703$	54.7941	0.0779433
$704$	− 952.208i	− 1.35257i
$705$	0	0
$706$	534.633	0.757270
$707$	− 144.781i	− 0.204782i
$708$	0	0
$709$	−432.626	−0.610192	−0.305096	−	0.952322i	$-0.598689\pi$
$709$	−432.626	−0.610192	−0.305096	+	0.952322i	$0.598689\pi$
$710$	275.990i	0.388719i
$711$	0	0
$712$	−3295.94	−4.62912
$713$	− 73.1850i	− 0.102644i
$714$	0	0
$715$	−155.844	−0.217964
$716$	− 2130.89i	− 2.97610i
$717$	0	0
$718$	920.360	1.28184
$719$	− 1085.00i	− 1.50904i	−0.656274	−	0.754522i	$-0.727869\pi$
$719$	− 1085.00i	− 1.50904i	0.656274	−	0.754522i	$-0.272131\pi$
$720$	0	0
$721$	−436.553	−0.605483
$722$	1356.21i	1.87841i
$723$	0	0
$724$	877.184	1.21158
$725$	207.383i	0.286045i
$726$	0	0
$727$	688.568	0.947137	0.473568	−	0.880757i	$-0.342966\pi$
$727$	688.568	0.947137	0.473568	+	0.880757i	$0.342966\pi$
$728$	− 564.721i	− 0.775716i
$729$	0	0
$730$	−992.730	−1.35990
$731$	− 1916.19i	− 2.62132i
$732$	0	0
$733$	−259.491	−0.354012	−0.177006	−	0.984210i	$-0.556641\pi$
$733$	−259.491	−0.354012	−0.177006	+	0.984210i	$0.556641\pi$
$734$	1296.43i	1.76626i
$735$	0	0
$736$	−3143.25	−4.27071
$737$	181.121i	0.245755i
$738$	0	0
$739$	195.434	0.264457	0.132228	−	0.991219i	$-0.457787\pi$
$739$	195.434	0.264457	0.132228	+	0.991219i	$0.457787\pi$
$740$	1031.57i	1.39402i
$741$	0	0
$742$	123.928	0.167019
$743$	322.257i	0.433725i	0.976202	+	0.216862i	$0.0695823\pi$
$743$	322.257i	0.433725i	−0.976202	+	0.216862i	$0.930418\pi$
$744$	0	0
$745$	−486.491	−0.653008
$746$	746.383i	1.00051i
$747$	0	0
$748$	−1025.16	−1.37054
$749$	− 96.8429i	− 0.129296i
$750$	0	0
$751$	407.900	0.543142	0.271571	−	0.962418i	$-0.412457\pi$
$751$	407.900	0.543142	0.271571	+	0.962418i	$0.412457\pi$
$752$	1405.84i	1.86947i
$753$	0	0
$754$	508.202	0.674009
$755$	914.131i	1.21077i
$756$	0	0
$757$	596.605	0.788118	0.394059	−	0.919085i	$-0.371071\pi$
$757$	596.605	0.788118	0.394059	+	0.919085i	$0.371071\pi$
$758$	1512.94i	1.99596i
$759$	0	0
$760$	622.871	0.819567
$761$	− 671.334i	− 0.882174i	−0.897464	−	0.441087i	$-0.854593\pi$
$761$	− 671.334i	− 0.882174i	0.897464	−	0.441087i	$-0.145407\pi$
$762$	0	0
$763$	−255.689	−0.335109
$764$	− 1116.69i	− 1.46164i
$765$	0	0
$766$	−464.659	−0.606605
$767$	− 262.122i	− 0.341750i
$768$	0	0
$769$	947.641	1.23230	0.616151	−	0.787628i	$-0.288691\pi$
$769$	947.641	1.23230	0.616151	+	0.787628i	$0.288691\pi$
$770$	− 206.081i	− 0.267638i
$771$	0	0
$772$	−336.013	−0.435251
$773$	− 1038.33i	− 1.34325i	−0.740893	−	0.671623i	$-0.765597\pi$
$773$	− 1038.33i	− 1.34325i	0.740893	−	0.671623i	$-0.234403\pi$
$774$	0	0
$775$	−39.9411	−0.0515369
$776$	− 210.058i	− 0.270693i
$777$	0	0
$778$	797.062	1.02450
$779$	146.225i	0.187708i
$780$	0	0
$781$	38.4620	0.0492471
$782$	2414.56i	3.08768i
$783$	0	0
$784$	−2730.56	−3.48286
$785$	1670.62i	2.12818i
$786$	0	0
$787$	−12.9336	−0.0164340	−0.00821702	−	0.999966i	$-0.502616\pi$
$787$	−12.9336	−0.0164340	−0.00821702	+	0.999966i	$0.502616\pi$
$788$	922.076i	1.17015i
$789$	0	0
$790$	−2306.82	−2.92003
$791$	540.687i	0.683548i
$792$	0	0
$793$	−472.607	−0.595973
$794$	− 2121.03i	− 2.67133i
$795$	0	0
$796$	−2438.99	−3.06406
$797$	− 459.455i	− 0.576480i	−0.957558	−	0.288240i	$-0.906930\pi$
$797$	− 459.455i	− 0.576480i	0.957558	−	0.288240i	$-0.0930701\pi$
$798$	0	0
$799$	599.119	0.749836
$800$	1715.44i	2.14430i
$801$	0	0
$802$	−436.765	−0.544595
$803$	138.347i	0.172287i
$804$	0	0
$805$	−357.692	−0.444337
$806$	97.8778i	0.121436i
$807$	0	0
$808$	1557.94	1.92814
$809$	1546.11i	1.91113i	0.294773	+	0.955567i	$0.404756\pi$
$809$	1546.11i	1.91113i	−0.294773	+	0.955567i	$0.595244\pi$
$810$	0	0
$811$	1461.97	1.80268	0.901340	−	0.433112i	$-0.142585\pi$
$811$	1461.97	1.80268	0.901340	+	0.433112i	$0.142585\pi$
$812$	495.232i	0.609892i
$813$	0	0
$814$	195.080	0.239656
$815$	− 1767.76i	− 2.16903i
$816$	0	0
$817$	252.325	0.308843
$818$	609.764i	0.745433i
$819$	0	0
$820$	−2752.87	−3.35716
$821$	− 795.413i	− 0.968834i	−0.874837	−	0.484417i	$-0.839032\pi$
$821$	− 795.413i	− 0.968834i	0.874837	−	0.484417i	$-0.160968\pi$
$822$	0	0
$823$	−73.3941	−0.0891788	−0.0445894	−	0.999005i	$-0.514198\pi$
$823$	−73.3941	−0.0891788	−0.0445894	+	0.999005i	$0.514198\pi$
$824$	− 4697.59i	− 5.70096i
$825$	0	0
$826$	346.618	0.419634
$827$	858.149i	1.03766i	0.854876	+	0.518832i	$0.173633\pi$
$827$	858.149i	1.03766i	−0.854876	+	0.518832i	$0.826367\pi$
$828$	0	0
$829$	116.737	0.140816	0.0704081	−	0.997518i	$-0.477570\pi$
$829$	116.737	0.140816	0.0704081	+	0.997518i	$0.477570\pi$
$830$	2138.89i	2.57697i
$831$	0	0
$832$	2210.36	2.65669
$833$	1163.67i	1.39696i
$834$	0	0
$835$	292.895	0.350772
$836$	− 134.994i	− 0.161476i
$837$	0	0
$838$	2410.27	2.87622
$839$	− 119.850i	− 0.142849i	−0.997446	−	0.0714243i	$-0.977246\pi$
$839$	− 119.850i	− 0.142849i	0.997446	−	0.0714243i	$-0.0227544\pi$
$840$	0	0
$841$	554.429	0.659250
$842$	− 1128.28i	− 1.34000i
$843$	0	0
$844$	1176.61	1.39409
$845$	669.699i	0.792544i
$846$	0	0
$847$	−28.7195	−0.0339073
$848$	787.960i	0.929199i
$849$	0	0
$850$	1317.76	1.55031
$851$	− 338.597i	− 0.397881i
$852$	0	0
$853$	−480.900	−0.563775	−0.281887	−	0.959447i	$-0.590960\pi$
$853$	−480.900	−0.563775	−0.281887	+	0.959447i	$0.590960\pi$
$854$	− 624.952i	− 0.731794i
$855$	0	0
$856$	1042.09	1.21740
$857$	898.929i	1.04893i	0.851433	+	0.524463i	$0.175734\pi$
$857$	898.929i	1.04893i	−0.851433	+	0.524463i	$0.824266\pi$
$858$	0	0
$859$	−892.879	−1.03944	−0.519720	−	0.854337i	$-0.673964\pi$
$859$	−892.879	−1.03944	−0.519720	+	0.854337i	$0.673964\pi$
$860$	4750.35i	5.52366i
$861$	0	0
$862$	2371.01	2.75060
$863$	1385.04i	1.60491i	0.596712	+	0.802455i	$0.296473\pi$
$863$	1385.04i	1.60491i	−0.596712	+	0.802455i	$0.703527\pi$
$864$	0	0
$865$	−213.735	−0.247093
$866$	155.006i	0.178991i
$867$	0	0
$868$	−95.3797	−0.109884
$869$	321.479i	0.369941i
$870$	0	0
$871$	−420.437	−0.482706
$872$	− 2751.37i	− 3.15524i
$873$	0	0
$874$	−317.951	−0.363789
$875$	− 203.161i	− 0.232184i
$876$	0	0
$877$	−8.82028	−0.0100573	−0.00502867	−	0.999987i	$-0.501601\pi$
$877$	−8.82028	−0.0100573	−0.00502867	+	0.999987i	$0.501601\pi$
$878$	1849.76i	2.10679i
$879$	0	0
$880$	1310.31	1.48899
$881$	871.048i	0.988704i	0.869262	+	0.494352i	$0.164595\pi$
$881$	871.048i	0.988704i	−0.869262	+	0.494352i	$0.835405\pi$
$882$	0	0
$883$	−426.150	−0.482616	−0.241308	−	0.970449i	$-0.577576\pi$
$883$	−426.150	−0.482616	−0.241308	+	0.970449i	$0.577576\pi$
$884$	− 2379.72i	− 2.69199i
$885$	0	0
$886$	548.818	0.619433
$887$	− 1032.71i	− 1.16427i	−0.813091	−	0.582136i	$-0.802217\pi$
$887$	− 1032.71i	− 1.16427i	0.813091	−	0.582136i	$-0.197783\pi$
$888$	0	0
$889$	476.501	0.535997
$890$	− 2792.00i	− 3.13708i
$891$	0	0
$892$	−2673.38	−2.99706
$893$	78.8924i	0.0883454i
$894$	0	0
$895$	1160.69	1.29686
$896$	1460.49i	1.63001i
$897$	0	0
$898$	754.557	0.840264
$899$	− 55.1924i	− 0.0613931i
$900$	0	0
$901$	335.801	0.372698
$902$	520.594i	0.577155i
$903$	0	0
$904$	−5818.14	−6.43599
$905$	477.802i	0.527958i
$906$	0	0
$907$	1283.21	1.41479	0.707393	−	0.706820i	$-0.249871\pi$
$907$	1283.21	1.41479	0.707393	+	0.706820i	$0.249871\pi$
$908$	− 3655.24i	− 4.02560i
$909$	0	0
$910$	478.378	0.525690
$911$	1125.40i	1.23535i	0.786433	+	0.617675i	$0.211925\pi$
$911$	1125.40i	1.23535i	−0.786433	+	0.617675i	$0.788075\pi$
$912$	0	0
$913$	298.075	0.326478
$914$	122.936i	0.134503i
$915$	0	0
$916$	−2613.07	−2.85270
$917$	− 426.775i	− 0.465403i
$918$	0	0
$919$	−1149.37	−1.25067	−0.625336	−	0.780356i	$-0.715038\pi$
$919$	−1149.37	−1.25067	−0.625336	+	0.780356i	$0.715038\pi$
$920$	− 3848.99i	− 4.18369i
$921$	0	0
$922$	−1957.48	−2.12308
$923$	89.2819i	0.0967301i
$924$	0	0
$925$	−184.791	−0.199774
$926$	− 3065.23i	− 3.31018i
$927$	0	0
$928$	−2370.48	−2.55439
$929$	250.860i	0.270032i	0.990843	+	0.135016i	$0.0431087\pi$
$929$	250.860i	0.270032i	−0.990843	+	0.135016i	$0.956891\pi$
$930$	0	0
$931$	−153.233	−0.164589
$932$	735.661i	0.789335i
$933$	0	0
$934$	2099.20	2.24753
$935$	− 558.407i	− 0.597226i
$936$	0	0
$937$	−817.848	−0.872837	−0.436418	−	0.899744i	$-0.643753\pi$
$937$	−817.848	−0.872837	−0.436418	+	0.899744i	$0.643753\pi$
$938$	− 555.966i	− 0.592715i
$939$	0	0
$940$	−1485.25	−1.58006
$941$	1289.31i	1.37015i	0.728472	+	0.685076i	$0.240231\pi$
$941$	1289.31i	1.37015i	−0.728472	+	0.685076i	$0.759769\pi$
$942$	0	0
$943$	903.586	0.958203
$944$	2203.87i	2.33461i
$945$	0	0
$946$	898.335	0.949614
$947$	16.5616i	0.0174885i	0.999962	+	0.00874423i	$0.00278341\pi$
$947$	16.5616i	0.0174885i	−0.999962	+	0.00874423i	$0.997217\pi$
$948$	0	0
$949$	−321.145	−0.338403
$950$	173.524i	0.182656i
$951$	0	0
$952$	2023.45	2.12548
$953$	905.921i	0.950599i	0.879824	+	0.475300i	$0.157660\pi$
$953$	905.921i	0.950599i	−0.879824	+	0.475300i	$0.842340\pi$
$954$	0	0
$955$	608.262	0.636924
$956$	− 2402.34i	− 2.51291i
$957$	0	0
$958$	214.254	0.223648
$959$	521.031i	0.543307i
$960$	0	0
$961$	−950.370	−0.988939
$962$	452.840i	0.470728i
$963$	0	0
$964$	4537.52	4.70697
$965$	− 183.027i	− 0.189665i
$966$	0	0
$967$	−1034.62	−1.06993	−0.534965	−	0.844874i	$-0.679675\pi$
$967$	−1034.62	−1.06993	−0.534965	+	0.844874i	$0.679675\pi$
$968$	− 309.040i	− 0.319256i
$969$	0	0
$970$	177.941	0.183444
$971$	1733.52i	1.78529i	0.450757	+	0.892647i	$0.351154\pi$
$971$	1733.52i	1.78529i	−0.450757	+	0.892647i	$0.648846\pi$
$972$	0	0
$973$	7.20321	0.00740309
$974$	− 1098.64i	− 1.12797i
$975$	0	0
$976$	3973.58	4.07129
$977$	− 835.320i	− 0.854985i	−0.904019	−	0.427492i	$-0.859397\pi$
$977$	− 835.320i	− 0.854985i	0.904019	−	0.427492i	$-0.140603\pi$
$978$	0	0
$979$	−389.093	−0.397439
$980$	− 2884.81i	− 2.94368i
$981$	0	0
$982$	2242.27	2.28337
$983$	− 1505.21i	− 1.53124i	−0.643294	−	0.765619i	$-0.722433\pi$
$983$	− 1505.21i	− 1.53124i	0.643294	−	0.765619i	$-0.277567\pi$
$984$	0	0
$985$	−502.255	−0.509904
$986$	1820.94i	1.84680i
$987$	0	0
$988$	313.363	0.317169
$989$	− 1559.22i	− 1.57657i
$990$	0	0
$991$	−141.776	−0.143064	−0.0715318	−	0.997438i	$-0.522789\pi$
$991$	−141.776	−0.143064	−0.0715318	+	0.997438i	$0.522789\pi$
$992$	− 456.544i	− 0.460226i
$993$	0	0
$994$	−118.062	−0.118775
$995$	− 1328.52i	− 1.33520i
$996$	0	0
$997$	−1566.45	−1.57116	−0.785579	−	0.618761i	$-0.787635\pi$
$997$	−1566.45	−1.57116	−0.785579	+	0.618761i	$0.787635\pi$
$998$	676.919i	0.678276i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.3.b.a.89.1	✓	8
3.2	odd	2	inner	99.3.b.a.89.8	yes	8
4.3	odd	2		1584.3.i.b.881.3		8
11.10	odd	2		1089.3.b.g.485.8		8
12.11	even	2		1584.3.i.b.881.6		8
33.32	even	2		1089.3.b.g.485.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.b.a.89.1	✓	8	1.1	even	1	trivial
99.3.b.a.89.8	yes	8	3.2	odd	2	inner
1089.3.b.g.485.1		8	33.32	even	2
1089.3.b.g.485.8		8	11.10	odd	2
1584.3.i.b.881.3		8	4.3	odd	2
1584.3.i.b.881.6		8	12.11	even	2

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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	99.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.89935i q^{2} -11.2049 q^{4} -6.10332i q^{5} +2.61086 q^{7} +28.0945i q^{8} +O(q^{10})\)	\(q-3.89935i q^{2} -11.2049 q^{4} -6.10332i q^{5} +2.61086 q^{7} +28.0945i q^{8} -23.7990 q^{10} +3.31662i q^{11} -7.69890 q^{13} -10.1807i q^{14} +64.7307 q^{16} -27.5859i q^{17} +3.63254 q^{19} +68.3873i q^{20} +12.9327 q^{22} -22.4470i q^{23} -12.2506 q^{25} +30.0207i q^{26} -29.2545 q^{28} -16.9284i q^{29} +3.26034 q^{31} -140.030i q^{32} -107.567 q^{34} -15.9349i q^{35} +15.0843 q^{37} -14.1645i q^{38} +171.470 q^{40} +40.2542i q^{41} +69.4624 q^{43} -37.1625i q^{44} -87.5288 q^{46} +21.7183i q^{47} -42.1834 q^{49} +47.7693i q^{50} +86.2656 q^{52} +12.1729i q^{53} +20.2424 q^{55} +73.3509i q^{56} -66.0098 q^{58} +34.0467i q^{59} +61.3863 q^{61} -12.7132i q^{62} -287.101 q^{64} +46.9889i q^{65} +54.6101 q^{67} +309.098i q^{68} -62.1359 q^{70} -11.5967i q^{71} +41.7131 q^{73} -58.8188i q^{74} -40.7023 q^{76} +8.65924i q^{77} +96.9294 q^{79} -395.073i q^{80} +156.965 q^{82} -89.8729i q^{83} -168.366 q^{85} -270.858i q^{86} -93.1790 q^{88} +117.316i q^{89} -20.1007 q^{91} +251.517i q^{92} +84.6871 q^{94} -22.1705i q^{95} -7.47681 q^{97} +164.488i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 + 16 * q^7	\( 8 q - 16 q^{4} + 16 q^{7} - 48 q^{10} - 8 q^{13} + 104 q^{16} + 40 q^{19} - 112 q^{25} - 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} + 432 q^{40} - 104 q^{43} + 24 q^{46} - 96 q^{49} + 280 q^{52} - 432 q^{58} - 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} + 448 q^{73} - 344 q^{76} + 448 q^{79} + 504 q^{82} + 48 q^{85} - 264 q^{88} - 544 q^{91} + 360 q^{94} - 152 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 16 * q^7 - 48 * q^10 - 8 * q^13 + 104 * q^16 + 40 * q^19 - 112 * q^25 - 32 * q^28 - 56 * q^31 - 216 * q^34 + 136 * q^37 + 432 * q^40 - 104 * q^43 + 24 * q^46 - 96 * q^49 + 280 * q^52 - 432 * q^58 - 8 * q^61 - 592 * q^64 + 112 * q^67 + 168 * q^70 + 448 * q^73 - 344 * q^76 + 448 * q^79 + 504 * q^82 + 48 * q^85 - 264 * q^88 - 544 * q^91 + 360 * q^94 - 152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more