LMFDB - Embedded newform 1089.2.d.g.1088.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1088,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1089.1088");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.d (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:	$8.69570878012$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 $ x^16 + 2x^14 - 16x^12 - 72x^10 + 26x^8 + 360x^6 + 725x^4 + 1000*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1088.10
Root			$-0.752864 + 0.902863i$ of defining polynomial
Character	$\chi$	$=$	1089.1088
Dual form			1089.2.d.g.1088.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.253675 q^{2} -1.93565 q^{4} +2.92173i q^{5} -2.71893i q^{7} -0.998377 q^{8} +O(q^{10})$	$q+0.253675 q^{2} -1.93565 q^{4} +2.92173i q^{5} -2.71893i q^{7} -0.998377 q^{8} +0.741170i q^{10} -2.85596i q^{13} -0.689726i q^{14} +3.61803 q^{16} -4.46218 q^{17} -1.91115i q^{19} -5.65544i q^{20} -0.816370i q^{23} -3.53649 q^{25} -0.724487i q^{26} +5.26290i q^{28} +9.54660 q^{29} -5.98605 q^{31} +2.91456 q^{32} -1.13195 q^{34} +7.94398 q^{35} +5.94628 q^{37} -0.484812i q^{38} -2.91698i q^{40} +8.43334 q^{41} -11.8763i q^{43} -0.207093i q^{46} -7.71816i q^{47} -0.392594 q^{49} -0.897120 q^{50} +5.52814i q^{52} -10.4556i q^{53} +2.71452i q^{56} +2.42174 q^{58} -0.0932996i q^{59} -2.14405i q^{61} -1.51851 q^{62} -6.49672 q^{64} +8.34434 q^{65} -1.40778 q^{67} +8.63722 q^{68} +2.01519 q^{70} +3.60916i q^{71} +1.75950i q^{73} +1.50842 q^{74} +3.69932i q^{76} -7.13289i q^{79} +10.5709i q^{80} +2.13933 q^{82} +7.83430 q^{83} -13.0373i q^{85} -3.01273i q^{86} +2.06830i q^{89} -7.76517 q^{91} +1.58021i q^{92} -1.95791i q^{94} +5.58387 q^{95} +1.67044 q^{97} -0.0995913 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{4}+O(q^{10}) $ 16 * q + 16 * q^4	$ 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91}+O(q^{100}) $ 16 * q + 16 * q^4 + 40 * q^16 - 32 * q^25 + 16 * q^31 + 40 * q^34 + 8 * q^37 + 16 * q^49 + 32 * q^58 - 104 * q^64 + 96 * q^67 - 64 * q^70 + 88 * q^82 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.253675	0.179376	0.0896878	−	0.995970i	$-0.471413\pi$
$2$	0.253675	0.179376	0.0896878	+	0.995970i	$0.471413\pi$
$3$	0	0
$3$	0	0
$4$	−1.93565	−0.967824
$5$	2.92173i	1.30664i	0.757084	+	0.653318i	$0.226624\pi$
$5$	2.92173i	1.30664i	−0.757084	+	0.653318i	$0.773376\pi$
$6$	0	0
$7$	− 2.71893i	− 1.02766i	−0.857892	−	0.513830i	$-0.828226\pi$
$7$	− 2.71893i	− 1.02766i	0.857892	−	0.513830i	$-0.171774\pi$
$8$	−0.998377	−0.352980
$9$	0	0
$10$	0.741170i	0.234379i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 2.85596i	− 0.792102i	−0.918229	−	0.396051i	$-0.870380\pi$
$13$	− 2.85596i	− 0.792102i	0.918229	−	0.396051i	$-0.129620\pi$
$14$	− 0.689726i	− 0.184337i
$15$	0	0
$16$	3.61803	0.904508
$17$	−4.46218	−1.08224	−0.541119	−	0.840946i	$-0.681999\pi$
$17$	−4.46218	−1.08224	−0.541119	+	0.840946i	$0.681999\pi$
$18$	0	0
$19$	− 1.91115i	− 0.438449i	−0.975674	−	0.219224i	$-0.929647\pi$
$19$	− 1.91115i	− 0.438449i	0.975674	−	0.219224i	$-0.0703527\pi$
$20$	− 5.65544i	− 1.26459i
$21$	0	0
$22$	0	0
$23$	− 0.816370i	− 0.170225i	−0.996371	−	0.0851124i	$-0.972875\pi$
$23$	− 0.816370i	− 0.170225i	0.996371	−	0.0851124i	$-0.0271249\pi$
$24$	0	0
$25$	−3.53649	−0.707297
$26$	− 0.724487i	− 0.142084i
$27$	0	0
$28$	5.26290i	0.994594i
$29$	9.54660	1.77276	0.886380	−	0.462959i	$-0.153212\pi$
$29$	9.54660	1.77276	0.886380	+	0.462959i	$0.153212\pi$
$30$	0	0
$31$	−5.98605	−1.07513	−0.537563	−	0.843224i	$-0.680655\pi$
$31$	−5.98605	−1.07513	−0.537563	+	0.843224i	$0.680655\pi$
$32$	2.91456	0.515226
$33$	0	0
$34$	−1.13195	−0.194127
$35$	7.94398	1.34278
$36$	0	0
$37$	5.94628	0.977562	0.488781	−	0.872407i	$-0.337442\pi$
$37$	5.94628	0.977562	0.488781	+	0.872407i	$0.337442\pi$
$38$	− 0.484812i	− 0.0786470i
$39$	0	0
$40$	− 2.91698i	− 0.461216i
$41$	8.43334	1.31707	0.658533	−	0.752552i	$-0.271177\pi$
$41$	8.43334	1.31707	0.658533	+	0.752552i	$0.271177\pi$
$42$	0	0
$43$	− 11.8763i	− 1.81112i	−0.424214	−	0.905562i	$-0.639450\pi$
$43$	− 11.8763i	− 1.81112i	0.424214	−	0.905562i	$-0.360550\pi$
$44$	0	0
$45$	0	0
$46$	− 0.207093i	− 0.0305342i
$47$	− 7.71816i	− 1.12581i	−0.826522	−	0.562905i	$-0.809684\pi$
$47$	− 7.71816i	− 1.12581i	0.826522	−	0.562905i	$-0.190316\pi$
$48$	0	0
$49$	−0.392594	−0.0560848
$50$	−0.897120	−0.126872
$51$	0	0
$52$	5.52814i	0.766615i
$53$	− 10.4556i	− 1.43618i	−0.695948	−	0.718092i	$-0.745016\pi$
$53$	− 10.4556i	− 1.43618i	0.695948	−	0.718092i	$-0.254984\pi$
$54$	0	0
$55$	0	0
$56$	2.71452i	0.362743i
$57$	0	0
$58$	2.42174	0.317990
$59$	− 0.0932996i	− 0.0121466i	−0.999982	−	0.00607329i	$-0.998067\pi$
$59$	− 0.0932996i	− 0.0121466i	0.999982	−	0.00607329i	$-0.00193320\pi$
$60$	0	0
$61$	− 2.14405i	− 0.274517i	−0.990535	−	0.137259i	$-0.956171\pi$
$61$	− 2.14405i	− 0.274517i	0.990535	−	0.137259i	$-0.0438291\pi$
$62$	−1.51851	−0.192851
$63$	0	0
$64$	−6.49672	−0.812089
$65$	8.34434	1.03499
$66$	0	0
$67$	−1.40778	−0.171988	−0.0859941	−	0.996296i	$-0.527407\pi$
$67$	−1.40778	−0.171988	−0.0859941	+	0.996296i	$0.527407\pi$
$68$	8.63722	1.04742
$69$	0	0
$70$	2.01519	0.240861
$71$	3.60916i	0.428328i	0.976798	+	0.214164i	$0.0687028\pi$
$71$	3.60916i	0.428328i	−0.976798	+	0.214164i	$0.931297\pi$
$72$	0	0
$73$	1.75950i	0.205933i	0.994685	+	0.102967i	$0.0328335\pi$
$73$	1.75950i	0.205933i	−0.994685	+	0.102967i	$0.967167\pi$
$74$	1.50842	0.175351
$75$	0	0
$76$	3.69932i	0.424341i
$77$	0	0
$78$	0	0
$79$	− 7.13289i	− 0.802512i	−0.915966	−	0.401256i	$-0.868574\pi$
$79$	− 7.13289i	− 0.802512i	0.915966	−	0.401256i	$-0.131426\pi$
$80$	10.5709i	1.18186i
$81$	0	0
$82$	2.13933	0.236249
$83$	7.83430	0.859926	0.429963	−	0.902847i	$-0.358527\pi$
$83$	7.83430	0.859926	0.429963	+	0.902847i	$0.358527\pi$
$84$	0	0
$85$	− 13.0373i	− 1.41409i
$86$	− 3.01273i	− 0.324871i
$87$	0	0
$88$	0	0
$89$	2.06830i	0.219240i	0.993974	+	0.109620i	$0.0349633\pi$
$89$	2.06830i	0.219240i	−0.993974	+	0.109620i	$0.965037\pi$
$90$	0	0
$91$	−7.76517	−0.814011
$92$	1.58021i	0.164748i
$93$	0	0
$94$	− 1.95791i	− 0.201943i
$95$	5.58387	0.572893
$96$	0	0
$97$	1.67044	0.169607	0.0848036	−	0.996398i	$-0.472974\pi$
$97$	1.67044	0.169607	0.0848036	+	0.996398i	$0.472974\pi$
$98$	−0.0995913	−0.0100602
$99$	0	0
$100$	6.84540	0.684540
$101$	−9.04767	−0.900277	−0.450139	−	0.892959i	$-0.648625\pi$
$101$	−9.04767	−0.900277	−0.450139	+	0.892959i	$0.648625\pi$
$102$	0	0
$103$	−16.5729	−1.63298	−0.816490	−	0.577360i	$-0.804083\pi$
$103$	−16.5729	−1.63298	−0.816490	+	0.577360i	$0.804083\pi$
$104$	2.85133i	0.279596i
$105$	0	0
$106$	− 2.65232i	− 0.257616i
$107$	−7.11298	−0.687638	−0.343819	−	0.939036i	$-0.611721\pi$
$107$	−7.11298	−0.687638	−0.343819	+	0.939036i	$0.611721\pi$
$108$	0	0
$109$	8.62045i	0.825690i	0.910801	+	0.412845i	$0.135465\pi$
$109$	8.62045i	0.825690i	−0.910801	+	0.412845i	$0.864535\pi$
$110$	0	0
$111$	0	0
$112$	− 9.83719i	− 0.929527i
$113$	− 11.4079i	− 1.07316i	−0.843848	−	0.536582i	$-0.819715\pi$
$113$	− 11.4079i	− 1.07316i	0.843848	−	0.536582i	$-0.180285\pi$
$114$	0	0
$115$	2.38521	0.222422
$116$	−18.4789	−1.71572
$117$	0	0
$118$	− 0.0236678i	− 0.00217880i
$119$	12.1324i	1.11217i
$120$	0	0
$121$	0	0
$122$	− 0.543892i	− 0.0492417i
$123$	0	0
$124$	11.5869	1.04053
$125$	4.27598i	0.382456i
$126$	0	0
$127$	2.88308i	0.255832i	0.991785	+	0.127916i	$0.0408288\pi$
$127$	2.88308i	0.255832i	−0.991785	+	0.127916i	$0.959171\pi$
$128$	−7.47718	−0.660895
$129$	0	0
$130$	2.11675	0.185652
$131$	−2.10155	−0.183613	−0.0918067	−	0.995777i	$-0.529264\pi$
$131$	−2.10155	−0.183613	−0.0918067	+	0.995777i	$0.529264\pi$
$132$	0	0
$133$	−5.19630	−0.450576
$134$	−0.357120	−0.0308505
$135$	0	0
$136$	4.45494	0.382008
$137$	5.07404i	0.433504i	0.976227	+	0.216752i	$0.0695464\pi$
$137$	5.07404i	0.433504i	−0.976227	+	0.216752i	$0.930454\pi$
$138$	0	0
$139$	− 17.4839i	− 1.48297i	−0.670970	−	0.741484i	$-0.734122\pi$
$139$	− 17.4839i	− 1.48297i	0.670970	−	0.741484i	$-0.265878\pi$
$140$	−15.3768	−1.29957
$141$	0	0
$142$	0.915555i	0.0768317i
$143$	0	0
$144$	0	0
$145$	27.8926i	2.31635i
$146$	0.446341i	0.0369394i
$147$	0	0
$148$	−11.5099	−0.946108
$149$	5.65654	0.463402	0.231701	−	0.972787i	$-0.425571\pi$
$149$	5.65654	0.463402	0.231701	+	0.972787i	$0.425571\pi$
$150$	0	0
$151$	9.17548i	0.746690i	0.927693	+	0.373345i	$0.121789\pi$
$151$	9.17548i	0.746690i	−0.927693	+	0.373345i	$0.878211\pi$
$152$	1.90805i	0.154763i
$153$	0	0
$154$	0	0
$155$	− 17.4896i	− 1.40480i
$156$	0	0
$157$	−3.05372	−0.243714	−0.121857	−	0.992548i	$-0.538885\pi$
$157$	−3.05372	−0.243714	−0.121857	+	0.992548i	$0.538885\pi$
$158$	− 1.80944i	− 0.143951i
$159$	0	0
$160$	8.51555i	0.673213i
$161$	−2.21965	−0.174933
$162$	0	0
$163$	7.90450	0.619128	0.309564	−	0.950879i	$-0.399817\pi$
$163$	7.90450	0.619128	0.309564	+	0.950879i	$0.399817\pi$
$164$	−16.3240	−1.27469
$165$	0	0
$166$	1.98737	0.154250
$167$	−16.4478	−1.27277	−0.636384	−	0.771372i	$-0.719571\pi$
$167$	−16.4478	−1.27277	−0.636384	+	0.771372i	$0.719571\pi$
$168$	0	0
$169$	4.84347	0.372575
$170$	− 3.30724i	− 0.253653i
$171$	0	0
$172$	22.9884i	1.75285i
$173$	15.5224	1.18015	0.590074	−	0.807349i	$-0.299099\pi$
$173$	15.5224	1.18015	0.590074	+	0.807349i	$0.299099\pi$
$174$	0	0
$175$	9.61547i	0.726861i
$176$	0	0
$177$	0	0
$178$	0.524677i	0.0393262i
$179$	5.18936i	0.387871i	0.981014	+	0.193936i	$0.0621253\pi$
$179$	5.18936i	0.387871i	−0.981014	+	0.193936i	$0.937875\pi$
$180$	0	0
$181$	5.31229	0.394859	0.197430	−	0.980317i	$-0.436741\pi$
$181$	5.31229	0.394859	0.197430	+	0.980317i	$0.436741\pi$
$182$	−1.96983	−0.146014
$183$	0	0
$184$	0.815045i	0.0600859i
$185$	17.3734i	1.27732i
$186$	0	0
$187$	0	0
$188$	14.9396i	1.08959i
$189$	0	0
$190$	1.41649	0.102763
$191$	− 10.8996i	− 0.788669i	−0.918967	−	0.394335i	$-0.870975\pi$
$191$	− 10.8996i	− 0.788669i	0.918967	−	0.394335i	$-0.129025\pi$
$192$	0	0
$193$	− 15.3664i	− 1.10610i	−0.833148	−	0.553050i	$-0.813464\pi$
$193$	− 15.3664i	− 1.10610i	0.833148	−	0.553050i	$-0.186536\pi$
$194$	0.423749	0.0304234
$195$	0	0
$196$	0.759923	0.0542802
$197$	−7.24149	−0.515935	−0.257967	−	0.966154i	$-0.583053\pi$
$197$	−7.24149	−0.515935	−0.257967	+	0.966154i	$0.583053\pi$
$198$	0	0
$199$	−11.3726	−0.806181	−0.403090	−	0.915160i	$-0.632064\pi$
$199$	−11.3726	−0.806181	−0.403090	+	0.915160i	$0.632064\pi$
$200$	3.53075	0.249662
$201$	0	0
$202$	−2.29517	−0.161488
$203$	− 25.9566i	− 1.82179i
$204$	0	0
$205$	24.6399i	1.72093i
$206$	−4.20414	−0.292917
$207$	0	0
$208$	− 10.3330i	− 0.716463i
$209$	0	0
$210$	0	0
$211$	− 3.69822i	− 0.254596i	−0.991864	−	0.127298i	$-0.959369\pi$
$211$	− 3.69822i	− 0.254596i	0.991864	−	0.127298i	$-0.0406305\pi$
$212$	20.2383i	1.38997i
$213$	0	0
$214$	−1.80439	−0.123345
$215$	34.6994	2.36648
$216$	0	0
$217$	16.2757i	1.10486i
$218$	2.18680i	0.148109i
$219$	0	0
$220$	0	0
$221$	12.7438i	0.857243i
$222$	0	0
$223$	−0.283223	−0.0189660	−0.00948302	−	0.999955i	$-0.503019\pi$
$223$	−0.283223	−0.0189660	−0.00948302	+	0.999955i	$0.503019\pi$
$224$	− 7.92449i	− 0.529477i
$225$	0	0
$226$	− 2.89390i	− 0.192499i
$227$	−18.2560	−1.21169	−0.605846	−	0.795582i	$-0.707165\pi$
$227$	−18.2560	−1.21169	−0.605846	+	0.795582i	$0.707165\pi$
$228$	0	0
$229$	26.3758	1.74296	0.871482	−	0.490428i	$-0.163160\pi$
$229$	26.3758	1.74296	0.871482	+	0.490428i	$0.163160\pi$
$230$	0.605069	0.0398971
$231$	0	0
$232$	−9.53111	−0.625748
$233$	−10.8022	−0.707675	−0.353838	−	0.935307i	$-0.615123\pi$
$233$	−10.8022	−0.707675	−0.353838	+	0.935307i	$0.615123\pi$
$234$	0	0
$235$	22.5504	1.47102
$236$	0.180595i	0.0117558i
$237$	0	0
$238$	3.07768i	0.199497i
$239$	7.42332	0.480175	0.240087	−	0.970751i	$-0.422824\pi$
$239$	7.42332	0.480175	0.240087	+	0.970751i	$0.422824\pi$
$240$	0	0
$241$	2.66873i	0.171908i	0.996299	+	0.0859539i	$0.0273938\pi$
$241$	2.66873i	0.171908i	−0.996299	+	0.0859539i	$0.972606\pi$
$242$	0	0
$243$	0	0
$244$	4.15012i	0.265684i
$245$	− 1.14705i	− 0.0732824i
$246$	0	0
$247$	−5.45818	−0.347296
$248$	5.97633	0.379498
$249$	0	0
$250$	1.08471i	0.0686032i
$251$	24.2656i	1.53163i	0.643060	+	0.765816i	$0.277665\pi$
$251$	24.2656i	1.53163i	−0.643060	+	0.765816i	$0.722335\pi$
$252$	0	0
$253$	0	0
$254$	0.731367i	0.0458901i
$255$	0	0
$256$	11.0967	0.693541
$257$	16.7561i	1.04522i	0.852573	+	0.522608i	$0.175041\pi$
$257$	16.7561i	1.04522i	−0.852573	+	0.522608i	$0.824959\pi$
$258$	0	0
$259$	− 16.1675i	− 1.00460i
$260$	−16.1517	−1.00169
$261$	0	0
$262$	−0.533112	−0.0329357
$263$	10.9146	0.673021	0.336511	−	0.941680i	$-0.390753\pi$
$263$	10.9146	0.673021	0.336511	+	0.941680i	$0.390753\pi$
$264$	0	0
$265$	30.5484	1.87657
$266$	−1.31817	−0.0808223
$267$	0	0
$268$	2.72498	0.166454
$269$	− 15.6867i	− 0.956435i	−0.878241	−	0.478218i	$-0.841283\pi$
$269$	− 15.6867i	− 0.956435i	0.878241	−	0.478218i	$-0.158717\pi$
$270$	0	0
$271$	2.90507i	0.176471i	0.996100	+	0.0882353i	$0.0281227\pi$
$271$	2.90507i	0.176471i	−0.996100	+	0.0882353i	$0.971877\pi$
$272$	−16.1443	−0.978894
$273$	0	0
$274$	1.28716i	0.0777601i
$275$	0	0
$276$	0	0
$277$	21.1150i	1.26868i	0.773056	+	0.634338i	$0.218727\pi$
$277$	21.1150i	1.26868i	−0.773056	+	0.634338i	$0.781273\pi$
$278$	− 4.43524i	− 0.266008i
$279$	0	0
$280$	−7.93109	−0.473973
$281$	3.09036	0.184355	0.0921777	−	0.995743i	$-0.470617\pi$
$281$	3.09036	0.184355	0.0921777	+	0.995743i	$0.470617\pi$
$282$	0	0
$283$	− 22.4247i	− 1.33301i	−0.745500	−	0.666505i	$-0.767789\pi$
$283$	− 22.4247i	− 1.33301i	0.745500	−	0.666505i	$-0.232211\pi$
$284$	− 6.98607i	− 0.414547i
$285$	0	0
$286$	0	0
$287$	− 22.9297i	− 1.35350i
$288$	0	0
$289$	2.91107	0.171239
$290$	7.07565i	0.415497i
$291$	0	0
$292$	− 3.40576i	− 0.199307i
$293$	−20.7409	−1.21170	−0.605849	−	0.795579i	$-0.707167\pi$
$293$	−20.7409	−1.21170	−0.605849	+	0.795579i	$0.707167\pi$
$294$	0	0
$295$	0.272596	0.0158712
$296$	−5.93663	−0.345059
$297$	0	0
$298$	1.43492	0.0831229
$299$	−2.33152	−0.134835
$300$	0	0
$301$	−32.2909	−1.86122
$302$	2.32759i	0.133938i
$303$	0	0
$304$	− 6.91462i	− 0.396581i
$305$	6.26432	0.358694
$306$	0	0
$307$	− 15.0077i	− 0.856533i	−0.903652	−	0.428267i	$-0.859124\pi$
$307$	− 15.0077i	− 0.856533i	0.903652	−	0.428267i	$-0.140876\pi$
$308$	0	0
$309$	0	0
$310$	− 4.43668i	− 0.251986i
$311$	2.05901i	0.116756i	0.998295	+	0.0583780i	$0.0185929\pi$
$311$	2.05901i	0.116756i	−0.998295	+	0.0583780i	$0.981407\pi$
$312$	0	0
$313$	−24.8725	−1.40588	−0.702939	−	0.711250i	$-0.748130\pi$
$313$	−24.8725	−1.40588	−0.702939	+	0.711250i	$0.748130\pi$
$314$	−0.774654	−0.0437163
$315$	0	0
$316$	13.8068i	0.776691i
$317$	− 0.432728i	− 0.0243044i	−0.999926	−	0.0121522i	$-0.996132\pi$
$317$	− 0.432728i	− 0.0243044i	0.999926	−	0.0121522i	$-0.00386827\pi$
$318$	0	0
$319$	0	0
$320$	− 18.9816i	− 1.06111i
$321$	0	0
$322$	−0.563072	−0.0313787
$323$	8.52791i	0.474506i
$324$	0	0
$325$	10.1001i	0.560251i
$326$	2.00518	0.111056
$327$	0	0
$328$	−8.41965	−0.464897
$329$	−20.9852	−1.15695
$330$	0	0
$331$	−17.3090	−0.951391	−0.475696	−	0.879610i	$-0.657804\pi$
$331$	−17.3090	−0.951391	−0.475696	+	0.879610i	$0.657804\pi$
$332$	−15.1644	−0.832257
$333$	0	0
$334$	−4.17240	−0.228304
$335$	− 4.11316i	− 0.224726i
$336$	0	0
$337$	− 25.0649i	− 1.36537i	−0.730712	−	0.682686i	$-0.760812\pi$
$337$	− 25.0649i	− 1.36537i	0.730712	−	0.682686i	$-0.239188\pi$
$338$	1.22867	0.0668308
$339$	0	0
$340$	25.2356i	1.36859i
$341$	0	0
$342$	0	0
$343$	− 17.9651i	− 0.970024i
$344$	11.8571i	0.639290i
$345$	0	0
$346$	3.93765	0.211690
$347$	−20.8730	−1.12052	−0.560260	−	0.828317i	$-0.689299\pi$
$347$	−20.8730	−1.12052	−0.560260	+	0.828317i	$0.689299\pi$
$348$	0	0
$349$	− 18.0205i	− 0.964617i	−0.876002	−	0.482308i	$-0.839798\pi$
$349$	− 18.0205i	− 0.964617i	0.876002	−	0.482308i	$-0.160202\pi$
$350$	2.43921i	0.130381i
$351$	0	0
$352$	0	0
$353$	16.2953i	0.867313i	0.901078	+	0.433656i	$0.142777\pi$
$353$	16.2953i	0.867313i	−0.901078	+	0.433656i	$0.857223\pi$
$354$	0	0
$355$	−10.5450	−0.559669
$356$	− 4.00351i	− 0.212185i
$357$	0	0
$358$	1.31641i	0.0695746i
$359$	24.7842	1.30806	0.654031	−	0.756468i	$-0.273077\pi$
$359$	24.7842	1.30806	0.654031	+	0.756468i	$0.273077\pi$
$360$	0	0
$361$	15.3475	0.807763
$362$	1.34760	0.0708281
$363$	0	0
$364$	15.0306	0.787820
$365$	−5.14076	−0.269080
$366$	0	0
$367$	8.95366	0.467377	0.233689	−	0.972311i	$-0.424920\pi$
$367$	8.95366	0.467377	0.233689	+	0.972311i	$0.424920\pi$
$368$	− 2.95365i	− 0.153970i
$369$	0	0
$370$	4.40720i	0.229120i
$371$	−28.4280	−1.47591
$372$	0	0
$373$	16.9007i	0.875088i	0.899197	+	0.437544i	$0.144152\pi$
$373$	16.9007i	0.875088i	−0.899197	+	0.437544i	$0.855848\pi$
$374$	0	0
$375$	0	0
$376$	7.70564i	0.397388i
$377$	− 27.2647i	− 1.40421i
$378$	0	0
$379$	19.7308	1.01350	0.506751	−	0.862092i	$-0.330846\pi$
$379$	19.7308	1.01350	0.506751	+	0.862092i	$0.330846\pi$
$380$	−10.8084	−0.554460
$381$	0	0
$382$	− 2.76497i	− 0.141468i
$383$	− 4.99993i	− 0.255485i	−0.991807	−	0.127742i	$-0.959227\pi$
$383$	− 4.99993i	− 0.255485i	0.991807	−	0.127742i	$-0.0407730\pi$
$384$	0	0
$385$	0	0
$386$	− 3.89808i	− 0.198407i
$387$	0	0
$388$	−3.23338	−0.164150
$389$	− 9.85341i	− 0.499588i	−0.968299	−	0.249794i	$-0.919637\pi$
$389$	− 9.85341i	− 0.499588i	0.968299	−	0.249794i	$-0.0803629\pi$
$390$	0	0
$391$	3.64279i	0.184224i
$392$	0.391956	0.0197968
$393$	0	0
$394$	−1.83699	−0.0925461
$395$	20.8403	1.04859
$396$	0	0
$397$	−19.0245	−0.954812	−0.477406	−	0.878683i	$-0.658423\pi$
$397$	−19.0245	−0.954812	−0.477406	+	0.878683i	$0.658423\pi$
$398$	−2.88494	−0.144609
$399$	0	0
$400$	−12.7951	−0.639757
$401$	30.7055i	1.53336i	0.642030	+	0.766680i	$0.278093\pi$
$401$	30.7055i	1.53336i	−0.642030	+	0.766680i	$0.721907\pi$
$402$	0	0
$403$	17.0959i	0.851609i
$404$	17.5131	0.871310
$405$	0	0
$406$	− 6.58454i	− 0.326785i
$407$	0	0
$408$	0	0
$409$	− 11.8735i	− 0.587105i	−0.955943	−	0.293552i	$-0.905162\pi$
$409$	− 11.8735i	− 0.587105i	0.955943	−	0.293552i	$-0.0948375\pi$
$410$	6.25054i	0.308692i
$411$	0	0
$412$	32.0794	1.58044
$413$	−0.253675	−0.0124825
$414$	0	0
$415$	22.8897i	1.12361i
$416$	− 8.32388i	− 0.408112i
$417$	0	0
$418$	0	0
$419$	10.0629i	0.491606i	0.969320	+	0.245803i	$0.0790516\pi$
$419$	10.0629i	0.491606i	−0.969320	+	0.245803i	$0.920948\pi$
$420$	0	0
$421$	−6.25651	−0.304923	−0.152462	−	0.988309i	$-0.548720\pi$
$421$	−6.25651	−0.304923	−0.152462	+	0.988309i	$0.548720\pi$
$422$	− 0.938148i	− 0.0456684i
$423$	0	0
$424$	10.4386i	0.506944i
$425$	15.7804	0.765464
$426$	0	0
$427$	−5.82952	−0.282110
$428$	13.7682	0.665512
$429$	0	0
$430$	8.80238	0.424489
$431$	19.6392	0.945988	0.472994	−	0.881066i	$-0.343173\pi$
$431$	19.6392	0.945988	0.472994	+	0.881066i	$0.343173\pi$
$432$	0	0
$433$	31.2889	1.50365	0.751825	−	0.659362i	$-0.229174\pi$
$433$	31.2889	1.50365	0.751825	+	0.659362i	$0.229174\pi$
$434$	4.12873i	0.198186i
$435$	0	0
$436$	− 16.6862i	− 0.799123i
$437$	−1.56021	−0.0746349
$438$	0	0
$439$	− 20.1935i	− 0.963782i	−0.876231	−	0.481891i	$-0.839950\pi$
$439$	− 20.1935i	− 0.963782i	0.876231	−	0.481891i	$-0.160050\pi$
$440$	0	0
$441$	0	0
$442$	3.23279i	0.153768i
$443$	24.3679i	1.15775i	0.815415	+	0.578876i	$0.196509\pi$
$443$	24.3679i	1.15775i	−0.815415	+	0.578876i	$0.803491\pi$
$444$	0	0
$445$	−6.04301	−0.286466
$446$	−0.0718468	−0.00340205
$447$	0	0
$448$	17.6641i	0.834552i
$449$	3.30192i	0.155827i	0.996960	+	0.0779136i	$0.0248258\pi$
$449$	3.30192i	0.155827i	−0.996960	+	0.0779136i	$0.975174\pi$
$450$	0	0
$451$	0	0
$452$	22.0817i	1.03863i
$453$	0	0
$454$	−4.63109	−0.217348
$455$	− 22.6877i	− 1.06362i
$456$	0	0
$457$	15.6174i	0.730551i	0.930899	+	0.365276i	$0.119025\pi$
$457$	15.6174i	0.730551i	−0.930899	+	0.365276i	$0.880975\pi$
$458$	6.69090	0.312645
$459$	0	0
$460$	−4.61693	−0.215265
$461$	−36.5685	−1.70316	−0.851582	−	0.524221i	$-0.824356\pi$
$461$	−36.5685	−1.70316	−0.851582	+	0.524221i	$0.824356\pi$
$462$	0	0
$463$	4.45531	0.207056	0.103528	−	0.994627i	$-0.466987\pi$
$463$	4.45531	0.207056	0.103528	+	0.994627i	$0.466987\pi$
$464$	34.5399	1.60348
$465$	0	0
$466$	−2.74025	−0.126940
$467$	41.6268i	1.92626i	0.269037	+	0.963130i	$0.413294\pi$
$467$	41.6268i	1.92626i	−0.269037	+	0.963130i	$0.586706\pi$
$468$	0	0
$469$	3.82767i	0.176745i
$470$	5.72047	0.263866
$471$	0	0
$472$	0.0931482i	0.00428749i
$473$	0	0
$474$	0	0
$475$	6.75877i	0.310114i
$476$	− 23.4840i	− 1.07639i
$477$	0	0
$478$	1.88311	0.0861316
$479$	−3.52150	−0.160902	−0.0804508	−	0.996759i	$-0.525636\pi$
$479$	−3.52150	−0.160902	−0.0804508	+	0.996759i	$0.525636\pi$
$480$	0	0
$481$	− 16.9823i	− 0.774328i
$482$	0.676990i	0.0308360i
$483$	0	0
$484$	0	0
$485$	4.88056i	0.221615i
$486$	0	0
$487$	31.9004	1.44554	0.722772	−	0.691087i	$-0.242868\pi$
$487$	31.9004	1.44554	0.722772	+	0.691087i	$0.242868\pi$
$488$	2.14057i	0.0968990i
$489$	0	0
$490$	− 0.290979i	− 0.0131451i
$491$	24.4192	1.10202	0.551012	−	0.834497i	$-0.314242\pi$
$491$	24.4192	1.10202	0.551012	+	0.834497i	$0.314242\pi$
$492$	0	0
$493$	−42.5987	−1.91855
$494$	−1.38461	−0.0622964
$495$	0	0
$496$	−21.6577	−0.972460
$497$	9.81306	0.440176
$498$	0	0
$499$	−23.2692	−1.04167	−0.520836	−	0.853657i	$-0.674380\pi$
$499$	−23.2692	−1.04167	−0.520836	+	0.853657i	$0.674380\pi$
$500$	− 8.27680i	− 0.370150i
$501$	0	0
$502$	6.15559i	0.274737i
$503$	36.5209	1.62839	0.814193	−	0.580595i	$-0.197180\pi$
$503$	36.5209	1.62839	0.814193	+	0.580595i	$0.197180\pi$
$504$	0	0
$505$	− 26.4348i	− 1.17633i
$506$	0	0
$507$	0	0
$508$	− 5.58064i	− 0.247601i
$509$	− 24.8510i	− 1.10150i	−0.834670	−	0.550751i	$-0.814341\pi$
$509$	− 24.8510i	− 1.10150i	0.834670	−	0.550751i	$-0.185659\pi$
$510$	0	0
$511$	4.78395	0.211629
$512$	17.7693	0.785300
$513$	0	0
$514$	4.25060i	0.187486i
$515$	− 48.4216i	− 2.13371i
$516$	0	0
$517$	0	0
$518$	− 4.10130i	− 0.180201i
$519$	0	0
$520$	−8.33080	−0.365330
$521$	− 9.74005i	− 0.426719i	−0.976974	−	0.213360i	$-0.931559\pi$
$521$	− 9.74005i	− 0.426719i	0.976974	−	0.213360i	$-0.0684406\pi$
$522$	0	0
$523$	− 33.2649i	− 1.45457i	−0.686335	−	0.727285i	$-0.740782\pi$
$523$	− 33.2649i	− 1.45457i	0.686335	−	0.727285i	$-0.259218\pi$
$524$	4.06786	0.177705
$525$	0	0
$526$	2.76876	0.120724
$527$	26.7108	1.16354
$528$	0	0
$529$	22.3335	0.971023
$530$	7.74936	0.336611
$531$	0	0
$532$	10.0582	0.436079
$533$	− 24.0853i	− 1.04325i
$534$	0	0
$535$	− 20.7822i	− 0.898492i
$536$	1.40550	0.0607084
$537$	0	0
$538$	− 3.97933i	− 0.171561i
$539$	0	0
$540$	0	0
$541$	− 11.6552i	− 0.501096i	−0.968104	−	0.250548i	$-0.919389\pi$
$541$	− 11.6552i	− 0.501096i	0.968104	−	0.250548i	$-0.0806108\pi$
$542$	0.736945i	0.0316545i
$543$	0	0
$544$	−13.0053	−0.557598
$545$	−25.1866	−1.07888
$546$	0	0
$547$	0.812785i	0.0347522i	0.999849	+	0.0173761i	$0.00553126\pi$
$547$	0.812785i	0.0347522i	−0.999849	+	0.0173761i	$0.994469\pi$
$548$	− 9.82156i	− 0.419556i
$549$	0	0
$550$	0	0
$551$	− 18.2450i	− 0.777264i
$552$	0	0
$553$	−19.3938	−0.824710
$554$	5.35635i	0.227569i
$555$	0	0
$556$	33.8428i	1.43525i
$557$	6.95247	0.294586	0.147293	−	0.989093i	$-0.452944\pi$
$557$	6.95247	0.294586	0.147293	+	0.989093i	$0.452944\pi$
$558$	0	0
$559$	−33.9184	−1.43459
$560$	28.7416	1.21455
$561$	0	0
$562$	0.783948	0.0330689
$563$	28.7576	1.21199	0.605994	−	0.795469i	$-0.292776\pi$
$563$	28.7576	1.21199	0.605994	+	0.795469i	$0.292776\pi$
$564$	0	0
$565$	33.3307	1.40223
$566$	− 5.68859i	− 0.239109i
$567$	0	0
$568$	− 3.60330i	− 0.151191i
$569$	−10.4489	−0.438042	−0.219021	−	0.975720i	$-0.570286\pi$
$569$	−10.4489	−0.438042	−0.219021	+	0.975720i	$0.570286\pi$
$570$	0	0
$571$	21.2530i	0.889409i	0.895677	+	0.444704i	$0.146691\pi$
$571$	21.2530i	0.889409i	−0.895677	+	0.444704i	$0.853309\pi$
$572$	0	0
$573$	0	0
$574$	− 5.81669i	− 0.242784i
$575$	2.88708i	0.120400i
$576$	0	0
$577$	15.6803	0.652779	0.326390	−	0.945235i	$-0.394168\pi$
$577$	15.6803	0.652779	0.326390	+	0.945235i	$0.394168\pi$
$578$	0.738466	0.0307162
$579$	0	0
$580$	− 53.9902i	− 2.24182i
$581$	− 21.3009i	− 0.883711i
$582$	0	0
$583$	0	0
$584$	− 1.75664i	− 0.0726903i
$585$	0	0
$586$	−5.26146	−0.217349
$587$	20.1556i	0.831912i	0.909385	+	0.415956i	$0.136553\pi$
$587$	20.1556i	0.831912i	−0.909385	+	0.415956i	$0.863447\pi$
$588$	0	0
$589$	11.4403i	0.471387i
$590$	0.0691509	0.00284690
$591$	0	0
$592$	21.5138	0.884213
$593$	40.5694	1.66599	0.832993	−	0.553284i	$-0.186626\pi$
$593$	40.5694	1.66599	0.832993	+	0.553284i	$0.186626\pi$
$594$	0	0
$595$	−35.4475	−1.45320
$596$	−10.9491	−0.448491
$597$	0	0
$598$	−0.591450	−0.0241862
$599$	10.9375i	0.446894i	0.974716	+	0.223447i	$0.0717309\pi$
$599$	10.9375i	0.446894i	−0.974716	+	0.223447i	$0.928269\pi$
$600$	0	0
$601$	− 16.3562i	− 0.667182i	−0.942718	−	0.333591i	$-0.891740\pi$
$601$	− 16.3562i	− 0.667182i	0.942718	−	0.333591i	$-0.108260\pi$
$602$	−8.19142	−0.333857
$603$	0	0
$604$	− 17.7605i	− 0.722665i
$605$	0	0
$606$	0	0
$607$	− 23.8646i	− 0.968633i	−0.874893	−	0.484317i	$-0.839068\pi$
$607$	− 23.8646i	− 0.968633i	0.874893	−	0.484317i	$-0.160932\pi$
$608$	− 5.57017i	− 0.225900i
$609$	0	0
$610$	1.58910	0.0643410
$611$	−22.0428	−0.891756
$612$	0	0
$613$	0.0485285i	0.00196005i	1.00000		0.000980023i	$0.000311951\pi$
$613$	0.0485285i	0.00196005i	−1.00000		0.000980023i	$0.999688\pi$
$614$	− 3.80708i	− 0.153641i
$615$	0	0
$616$	0	0
$617$	8.24561i	0.331956i	0.986129	+	0.165978i	$0.0530780\pi$
$617$	8.24561i	0.331956i	−0.986129	+	0.165978i	$0.946922\pi$
$618$	0	0
$619$	−1.73927	−0.0699072	−0.0349536	−	0.999389i	$-0.511128\pi$
$619$	−1.73927	−0.0699072	−0.0349536	+	0.999389i	$0.511128\pi$
$620$	33.8537i	1.35960i
$621$	0	0
$622$	0.522321i	0.0209432i
$623$	5.62357	0.225304
$624$	0	0
$625$	−30.1757	−1.20703
$626$	−6.30955	−0.252180
$627$	0	0
$628$	5.91094	0.235872
$629$	−26.5334	−1.05795
$630$	0	0
$631$	−26.3475	−1.04888	−0.524439	−	0.851448i	$-0.675725\pi$
$631$	−26.3475	−1.04888	−0.524439	+	0.851448i	$0.675725\pi$
$632$	7.12131i	0.283270i
$633$	0	0
$634$	− 0.109772i	− 0.00435962i
$635$	−8.42358	−0.334280
$636$	0	0
$637$	1.12123i	0.0444249i
$638$	0	0
$639$	0	0
$640$	− 21.8463i	− 0.863550i
$641$	− 8.69577i	− 0.343462i	−0.985144	−	0.171731i	$-0.945064\pi$
$641$	− 8.69577i	− 0.343462i	0.985144	−	0.171731i	$-0.0549361\pi$
$642$	0	0
$643$	−21.6159	−0.852450	−0.426225	−	0.904617i	$-0.640157\pi$
$643$	−21.6159	−0.852450	−0.426225	+	0.904617i	$0.640157\pi$
$644$	4.29647	0.169305
$645$	0	0
$646$	2.16332i	0.0851147i
$647$	− 29.3673i	− 1.15455i	−0.816551	−	0.577273i	$-0.804117\pi$
$647$	− 29.3673i	− 1.15455i	0.816551	−	0.577273i	$-0.195883\pi$
$648$	0	0
$649$	0	0
$650$	2.56214i	0.100495i
$651$	0	0
$652$	−15.3003	−0.599207
$653$	23.6935i	0.927199i	0.886045	+	0.463599i	$0.153442\pi$
$653$	23.6935i	0.927199i	−0.886045	+	0.463599i	$0.846558\pi$
$654$	0	0
$655$	− 6.14016i	− 0.239916i
$656$	30.5121	1.19130
$657$	0	0
$658$	−5.32342	−0.207528
$659$	34.5941	1.34759	0.673797	−	0.738916i	$-0.264662\pi$
$659$	34.5941	1.34759	0.673797	+	0.738916i	$0.264662\pi$
$660$	0	0
$661$	35.4913	1.38045	0.690225	−	0.723595i	$-0.257512\pi$
$661$	35.4913	1.38045	0.690225	+	0.723595i	$0.257512\pi$
$662$	−4.39088	−0.170656
$663$	0	0
$664$	−7.82158	−0.303536
$665$	− 15.1822i	− 0.588739i
$666$	0	0
$667$	− 7.79356i	− 0.301768i
$668$	31.8372	1.23182
$669$	0	0
$670$	− 1.04341i	− 0.0403104i
$671$	0	0
$672$	0	0
$673$	− 20.9013i	− 0.805684i	−0.915270	−	0.402842i	$-0.868022\pi$
$673$	− 20.9013i	− 0.805684i	0.915270	−	0.402842i	$-0.131978\pi$
$674$	− 6.35835i	− 0.244914i
$675$	0	0
$676$	−9.37527	−0.360587
$677$	−24.2380	−0.931543	−0.465772	−	0.884905i	$-0.654223\pi$
$677$	−24.2380	−0.931543	−0.465772	+	0.884905i	$0.654223\pi$
$678$	0	0
$679$	− 4.54181i	− 0.174299i
$680$	13.0161i	0.499145i
$681$	0	0
$682$	0	0
$683$	− 17.6311i	− 0.674636i	−0.941391	−	0.337318i	$-0.890480\pi$
$683$	− 17.6311i	− 0.674636i	0.941391	−	0.337318i	$-0.109520\pi$
$684$	0	0
$685$	−14.8250	−0.566433
$686$	− 4.55730i	− 0.173999i
$687$	0	0
$688$	− 42.9690i	− 1.63818i
$689$	−29.8608	−1.13760
$690$	0	0
$691$	29.1363	1.10840	0.554199	−	0.832384i	$-0.313024\pi$
$691$	29.1363	1.10840	0.554199	+	0.832384i	$0.313024\pi$
$692$	−30.0459	−1.14218
$693$	0	0
$694$	−5.29496	−0.200994
$695$	51.0833	1.93770
$696$	0	0
$697$	−37.6311	−1.42538
$698$	− 4.57136i	− 0.173029i
$699$	0	0
$700$	− 18.6122i	− 0.703474i
$701$	−14.7015	−0.555268	−0.277634	−	0.960687i	$-0.589550\pi$
$701$	−14.7015	−0.555268	−0.277634	+	0.960687i	$0.589550\pi$
$702$	0	0
$703$	− 11.3642i	− 0.428611i
$704$	0	0
$705$	0	0
$706$	4.13372i	0.155575i
$707$	24.6000i	0.925179i
$708$	0	0
$709$	10.7824	0.404940	0.202470	−	0.979288i	$-0.435103\pi$
$709$	10.7824	0.404940	0.202470	+	0.979288i	$0.435103\pi$
$710$	−2.67500	−0.100391
$711$	0	0
$712$	− 2.06495i	− 0.0773871i
$713$	4.88683i	0.183013i
$714$	0	0
$715$	0	0
$716$	− 10.0448i	− 0.375391i
$717$	0	0
$718$	6.28715	0.234634
$719$	− 31.7500i	− 1.18407i	−0.805911	−	0.592037i	$-0.798324\pi$
$719$	− 31.7500i	− 1.18407i	0.805911	−	0.592037i	$-0.201676\pi$
$720$	0	0
$721$	45.0607i	1.67815i
$722$	3.89328	0.144893
$723$	0	0
$724$	−10.2827	−0.382154
$725$	−33.7614	−1.25387
$726$	0	0
$727$	23.6418	0.876828	0.438414	−	0.898773i	$-0.355540\pi$
$727$	23.6418	0.876828	0.438414	+	0.898773i	$0.355540\pi$
$728$	7.75257	0.287329
$729$	0	0
$730$	−1.30409	−0.0482664
$731$	52.9944i	1.96007i
$732$	0	0
$733$	− 16.4244i	− 0.606651i	−0.952887	−	0.303325i	$-0.901903\pi$
$733$	− 16.4244i	− 0.606651i	0.952887	−	0.303325i	$-0.0980969\pi$
$734$	2.27132	0.0838361
$735$	0	0
$736$	− 2.37936i	− 0.0877043i
$737$	0	0
$738$	0	0
$739$	34.8449i	1.28179i	0.767629	+	0.640895i	$0.221437\pi$
$739$	34.8449i	1.28179i	−0.767629	+	0.640895i	$0.778563\pi$
$740$	− 33.6288i	− 1.23622i
$741$	0	0
$742$	−7.21149	−0.264742
$743$	52.7926	1.93677	0.968386	−	0.249458i	$-0.0802525\pi$
$743$	52.7926	1.93677	0.968386	+	0.249458i	$0.0802525\pi$
$744$	0	0
$745$	16.5269i	0.605497i
$746$	4.28730i	0.156969i
$747$	0	0
$748$	0	0
$749$	19.3397i	0.706657i
$750$	0	0
$751$	51.3777	1.87480	0.937400	−	0.348255i	$-0.113226\pi$
$751$	51.3777	1.87480	0.937400	+	0.348255i	$0.113226\pi$
$752$	− 27.9246i	− 1.01830i
$753$	0	0
$754$	− 6.91639i	− 0.251880i
$755$	−26.8083	−0.975652
$756$	0	0
$757$	−7.06878	−0.256919	−0.128460	−	0.991715i	$-0.541003\pi$
$757$	−7.06878	−0.256919	−0.128460	+	0.991715i	$0.541003\pi$
$758$	5.00521	0.181797
$759$	0	0
$760$	−5.57481	−0.202219
$761$	−29.1327	−1.05606	−0.528030	−	0.849226i	$-0.677069\pi$
$761$	−29.1327	−1.05606	−0.528030	+	0.849226i	$0.677069\pi$
$762$	0	0
$763$	23.4384	0.848528
$764$	21.0979i	0.763294i
$765$	0	0
$766$	− 1.26836i	− 0.0458277i
$767$	−0.266460	−0.00962132
$768$	0	0
$769$	44.2408i	1.59536i	0.603079	+	0.797682i	$0.293940\pi$
$769$	44.2408i	1.59536i	−0.603079	+	0.797682i	$0.706060\pi$
$770$	0	0
$771$	0	0
$772$	29.7440i	1.07051i
$773$	40.0382i	1.44007i	0.693936	+	0.720037i	$0.255875\pi$
$773$	40.0382i	1.44007i	−0.693936	+	0.720037i	$0.744125\pi$
$774$	0	0
$775$	21.1696	0.760434
$776$	−1.66773	−0.0598679
$777$	0	0
$778$	− 2.49957i	− 0.0896139i
$779$	− 16.1174i	− 0.577466i
$780$	0	0
$781$	0	0
$782$	0.924086i	0.0330453i
$783$	0	0
$784$	−1.42042	−0.0507292
$785$	− 8.92215i	− 0.318445i
$786$	0	0
$787$	28.3641i	1.01107i	0.862806	+	0.505535i	$0.168705\pi$
$787$	28.3641i	1.01107i	−0.862806	+	0.505535i	$0.831295\pi$
$788$	14.0170	0.499334
$789$	0	0
$790$	5.28668	0.188092
$791$	−31.0173	−1.10285
$792$	0	0
$793$	−6.12332	−0.217446
$794$	−4.82605	−0.171270
$795$	0	0
$796$	22.0133	0.780241
$797$	− 14.9437i	− 0.529334i	−0.964340	−	0.264667i	$-0.914738\pi$
$797$	− 14.9437i	− 0.529334i	0.964340	−	0.264667i	$-0.0852621\pi$
$798$	0	0
$799$	34.4398i	1.21839i
$800$	−10.3073	−0.364418
$801$	0	0
$802$	7.78923i	0.275047i
$803$	0	0
$804$	0	0
$805$	− 6.48522i	− 0.228574i
$806$	4.33682i	0.152758i
$807$	0	0
$808$	9.03299	0.317779
$809$	−29.7759	−1.04687	−0.523433	−	0.852067i	$-0.675349\pi$
$809$	−29.7759	−1.04687	−0.523433	+	0.852067i	$0.675349\pi$
$810$	0	0
$811$	44.1463i	1.55019i	0.631846	+	0.775094i	$0.282297\pi$
$811$	44.1463i	1.55019i	−0.631846	+	0.775094i	$0.717703\pi$
$812$	50.2428i	1.76318i
$813$	0	0
$814$	0	0
$815$	23.0948i	0.808975i
$816$	0	0
$817$	−22.6975	−0.794085
$818$	− 3.01200i	− 0.105312i
$819$	0	0
$820$	− 47.6942i	− 1.66555i
$821$	−4.00495	−0.139774	−0.0698868	−	0.997555i	$-0.522264\pi$
$821$	−4.00495	−0.139774	−0.0698868	+	0.997555i	$0.522264\pi$
$822$	0	0
$823$	42.1823	1.47038	0.735191	−	0.677860i	$-0.237093\pi$
$823$	42.1823	1.47038	0.735191	+	0.677860i	$0.237093\pi$
$824$	16.5460	0.576409
$825$	0	0
$826$	−0.0643512	−0.00223906
$827$	26.5981	0.924907	0.462454	−	0.886643i	$-0.346969\pi$
$827$	26.5981	0.924907	0.462454	+	0.886643i	$0.346969\pi$
$828$	0	0
$829$	12.2316	0.424819	0.212410	−	0.977181i	$-0.431869\pi$
$829$	12.2316	0.424819	0.212410	+	0.977181i	$0.431869\pi$
$830$	5.80655i	0.201548i
$831$	0	0
$832$	18.5544i	0.643257i
$833$	1.75182	0.0606971
$834$	0	0
$835$	− 48.0560i	− 1.66305i
$836$	0	0
$837$	0	0
$838$	2.55271i	0.0881820i
$839$	− 9.64647i	− 0.333033i	−0.986039	−	0.166517i	$-0.946748\pi$
$839$	− 9.64647i	− 0.333033i	0.986039	−	0.166517i	$-0.0532519\pi$
$840$	0	0
$841$	62.1376	2.14268
$842$	−1.58712	−0.0546958
$843$	0	0
$844$	7.15846i	0.246405i
$845$	14.1513i	0.486820i
$846$	0	0
$847$	0	0
$848$	− 37.8287i	− 1.29904i
$849$	0	0
$850$	4.00311	0.137306
$851$	− 4.85436i	− 0.166405i
$852$	0	0
$853$	11.7386i	0.401921i	0.979599	+	0.200960i	$0.0644063\pi$
$853$	11.7386i	0.401921i	−0.979599	+	0.200960i	$0.935594\pi$
$854$	−1.47881	−0.0506037
$855$	0	0
$856$	7.10143	0.242722
$857$	−48.7738	−1.66608	−0.833041	−	0.553212i	$-0.813402\pi$
$857$	−48.7738	−1.66608	−0.833041	+	0.553212i	$0.813402\pi$
$858$	0	0
$859$	−31.0880	−1.06071	−0.530355	−	0.847776i	$-0.677941\pi$
$859$	−31.0880	−1.06071	−0.530355	+	0.847776i	$0.677941\pi$
$860$	−67.1659	−2.29034
$861$	0	0
$862$	4.98199	0.169687
$863$	− 52.1380i	− 1.77480i	−0.461002	−	0.887399i	$-0.652510\pi$
$863$	− 52.1380i	− 1.77480i	0.461002	−	0.887399i	$-0.347490\pi$
$864$	0	0
$865$	45.3522i	1.54202i
$866$	7.93723	0.269718
$867$	0	0
$868$	− 31.5040i	− 1.06931i
$869$	0	0
$870$	0	0
$871$	4.02058i	0.136232i
$872$	− 8.60646i	− 0.291452i
$873$	0	0
$874$	−0.395786	−0.0133877
$875$	11.6261	0.393034
$876$	0	0
$877$	16.9878i	0.573637i	0.957985	+	0.286819i	$0.0925977\pi$
$877$	16.9878i	0.573637i	−0.957985	+	0.286819i	$0.907402\pi$
$878$	− 5.12259i	− 0.172879i
$879$	0	0
$880$	0	0
$881$	50.3115i	1.69504i	0.530765	+	0.847519i	$0.321905\pi$
$881$	50.3115i	1.69504i	−0.530765	+	0.847519i	$0.678095\pi$
$882$	0	0
$883$	14.3693	0.483567	0.241783	−	0.970330i	$-0.422268\pi$
$883$	14.3693	0.483567	0.241783	+	0.970330i	$0.422268\pi$
$884$	− 24.6676i	− 0.829660i
$885$	0	0
$886$	6.18153i	0.207673i
$887$	41.3003	1.38673	0.693365	−	0.720587i	$-0.256127\pi$
$887$	41.3003	1.38673	0.693365	+	0.720587i	$0.256127\pi$
$888$	0	0
$889$	7.83891	0.262909
$890$	−1.53296	−0.0513851
$891$	0	0
$892$	0.548221	0.0183558
$893$	−14.7506	−0.493610
$894$	0	0
$895$	−15.1619	−0.506807
$896$	20.3299i	0.679176i
$897$	0	0
$898$	0.837616i	0.0279516i
$899$	−57.1464	−1.90594
$900$	0	0
$901$	46.6547i	1.55429i
$902$	0	0
$903$	0	0
$904$	11.3894i	0.378805i
$905$	15.5210i	0.515937i
$906$	0	0
$907$	−26.1167	−0.867190	−0.433595	−	0.901108i	$-0.642755\pi$
$907$	−26.1167	−0.867190	−0.433595	+	0.901108i	$0.642755\pi$
$908$	35.3371	1.17270
$909$	0	0
$910$	− 5.75531i	− 0.190787i
$911$	45.4127i	1.50459i	0.658826	+	0.752296i	$0.271054\pi$
$911$	45.4127i	1.50459i	−0.658826	+	0.752296i	$0.728946\pi$
$912$	0	0
$913$	0	0
$914$	3.96175i	0.131043i
$915$	0	0
$916$	−51.0543	−1.68688
$917$	5.71398i	0.188692i
$918$	0	0
$919$	27.8566i	0.918905i	0.888202	+	0.459453i	$0.151954\pi$
$919$	27.8566i	0.918905i	−0.888202	+	0.459453i	$0.848046\pi$
$920$	−2.38134	−0.0785104
$921$	0	0
$922$	−9.27652	−0.305506
$923$	10.3076	0.339280
$924$	0	0
$925$	−21.0289	−0.691427
$926$	1.13020	0.0371408
$927$	0	0
$928$	27.8241	0.913372
$929$	− 49.7545i	− 1.63239i	−0.577774	−	0.816197i	$-0.696079\pi$
$929$	− 49.7545i	− 1.63239i	0.577774	−	0.816197i	$-0.303921\pi$
$930$	0	0
$931$	0.750306i	0.0245903i
$932$	20.9093	0.684905
$933$	0	0
$934$	10.5597i	0.345524i
$935$	0	0
$936$	0	0
$937$	20.4059i	0.666633i	0.942815	+	0.333316i	$0.108168\pi$
$937$	20.4059i	0.666633i	−0.942815	+	0.333316i	$0.891832\pi$
$938$	0.970986i	0.0317038i
$939$	0	0
$940$	−43.6496	−1.42369
$941$	−17.5434	−0.571898	−0.285949	−	0.958245i	$-0.592309\pi$
$941$	−17.5434	−0.571898	−0.285949	+	0.958245i	$0.592309\pi$
$942$	0	0
$943$	− 6.88472i	− 0.224197i
$944$	− 0.337561i	− 0.0109867i
$945$	0	0
$946$	0	0
$947$	− 47.0672i	− 1.52948i	−0.644340	−	0.764739i	$-0.722868\pi$
$947$	− 47.0672i	− 1.52948i	0.644340	−	0.764739i	$-0.277132\pi$
$948$	0	0
$949$	5.02505	0.163120
$950$	1.71453i	0.0556268i
$951$	0	0
$952$	− 12.1127i	− 0.392574i
$953$	−51.6298	−1.67245	−0.836227	−	0.548384i	$-0.815243\pi$
$953$	−51.6298	−1.67245	−0.836227	+	0.548384i	$0.815243\pi$
$954$	0	0
$955$	31.8457	1.03050
$956$	−14.3689	−0.464725
$957$	0	0
$958$	−0.893318	−0.0288618
$959$	13.7960	0.445495
$960$	0	0
$961$	4.83277	0.155896
$962$	− 4.30800i	− 0.138896i
$963$	0	0
$964$	− 5.16572i	− 0.166376i
$965$	44.8965	1.44527
$966$	0	0
$967$	33.6964i	1.08360i	0.840506	+	0.541802i	$0.182258\pi$
$967$	33.6964i	1.08360i	−0.840506	+	0.541802i	$0.817742\pi$
$968$	0	0
$969$	0	0
$970$	1.23808i	0.0397523i
$971$	55.8818i	1.79333i	0.442708	+	0.896666i	$0.354018\pi$
$971$	55.8818i	1.79333i	−0.442708	+	0.896666i	$0.645982\pi$
$972$	0	0
$973$	−47.5376	−1.52399
$974$	8.09233	0.259295
$975$	0	0
$976$	− 7.75724i	− 0.248303i
$977$	20.6593i	0.660949i	0.943815	+	0.330475i	$0.107209\pi$
$977$	20.6593i	0.660949i	−0.943815	+	0.330475i	$0.892791\pi$
$978$	0	0
$979$	0	0
$980$	2.22029i	0.0709245i
$981$	0	0
$982$	6.19455	0.197676
$983$	50.1274i	1.59882i	0.600789	+	0.799408i	$0.294853\pi$
$983$	50.1274i	1.59882i	−0.600789	+	0.799408i	$0.705147\pi$
$984$	0	0
$985$	− 21.1577i	− 0.674139i
$986$	−10.8062	−0.344141
$987$	0	0
$988$	10.5651	0.336121
$989$	−9.69548	−0.308298
$990$	0	0
$991$	45.3068	1.43922	0.719609	−	0.694379i	$-0.244321\pi$
$991$	45.3068	1.43922	0.719609	+	0.694379i	$0.244321\pi$
$992$	−17.4467	−0.553933
$993$	0	0
$994$	2.48933	0.0789568
$995$	− 33.2276i	− 1.05338i
$996$	0	0
$997$	− 10.2799i	− 0.325567i	−0.986662	−	0.162784i	$-0.947953\pi$
$997$	− 10.2799i	− 0.325567i	0.986662	−	0.162784i	$-0.0520473\pi$
$998$	−5.90282	−0.186850
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.d.g.1088.10		16
3.2	odd	2	inner	1089.2.d.g.1088.7		16
11.6	odd	10		99.2.j.a.8.3	yes	16
11.9	even	5		99.2.j.a.62.2	yes	16
11.10	odd	2	inner	1089.2.d.g.1088.8		16
33.17	even	10		99.2.j.a.8.2	✓	16
33.20	odd	10		99.2.j.a.62.3	yes	16
33.32	even	2	inner	1089.2.d.g.1088.9		16
44.31	odd	10		1584.2.cd.c.161.2		16
44.39	even	10		1584.2.cd.c.305.3		16
99.20	odd	30		891.2.u.c.458.3		32
99.31	even	15		891.2.u.c.755.3		32
99.50	even	30		891.2.u.c.107.2		32
99.61	odd	30		891.2.u.c.701.2		32
99.83	even	30		891.2.u.c.701.3		32
99.86	odd	30		891.2.u.c.755.2		32
99.94	odd	30		891.2.u.c.107.3		32
99.97	even	15		891.2.u.c.458.2		32
132.83	odd	10		1584.2.cd.c.305.2		16
132.119	even	10		1584.2.cd.c.161.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.j.a.8.2	✓	16	33.17	even	10
99.2.j.a.8.3	yes	16	11.6	odd	10
99.2.j.a.62.2	yes	16	11.9	even	5
99.2.j.a.62.3	yes	16	33.20	odd	10
891.2.u.c.107.2		32	99.50	even	30
891.2.u.c.107.3		32	99.94	odd	30
891.2.u.c.458.2		32	99.97	even	15
891.2.u.c.458.3		32	99.20	odd	30
891.2.u.c.701.2		32	99.61	odd	30
891.2.u.c.701.3		32	99.83	even	30
891.2.u.c.755.2		32	99.86	odd	30
891.2.u.c.755.3		32	99.31	even	15
1089.2.d.g.1088.7		16	3.2	odd	2	inner
1089.2.d.g.1088.8		16	11.10	odd	2	inner
1089.2.d.g.1088.9		16	33.32	even	2	inner
1089.2.d.g.1088.10		16	1.1	even	1	trivial
1584.2.cd.c.161.2		16	44.31	odd	10
1584.2.cd.c.161.3		16	132.119	even	10
1584.2.cd.c.305.2		16	132.83	odd	10
1584.2.cd.c.305.3		16	44.39	even	10

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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.253675 q^{2} -1.93565 q^{4} +2.92173i q^{5} -2.71893i q^{7} -0.998377 q^{8} +O(q^{10})\)	\(q+0.253675 q^{2} -1.93565 q^{4} +2.92173i q^{5} -2.71893i q^{7} -0.998377 q^{8} +0.741170i q^{10} -2.85596i q^{13} -0.689726i q^{14} +3.61803 q^{16} -4.46218 q^{17} -1.91115i q^{19} -5.65544i q^{20} -0.816370i q^{23} -3.53649 q^{25} -0.724487i q^{26} +5.26290i q^{28} +9.54660 q^{29} -5.98605 q^{31} +2.91456 q^{32} -1.13195 q^{34} +7.94398 q^{35} +5.94628 q^{37} -0.484812i q^{38} -2.91698i q^{40} +8.43334 q^{41} -11.8763i q^{43} -0.207093i q^{46} -7.71816i q^{47} -0.392594 q^{49} -0.897120 q^{50} +5.52814i q^{52} -10.4556i q^{53} +2.71452i q^{56} +2.42174 q^{58} -0.0932996i q^{59} -2.14405i q^{61} -1.51851 q^{62} -6.49672 q^{64} +8.34434 q^{65} -1.40778 q^{67} +8.63722 q^{68} +2.01519 q^{70} +3.60916i q^{71} +1.75950i q^{73} +1.50842 q^{74} +3.69932i q^{76} -7.13289i q^{79} +10.5709i q^{80} +2.13933 q^{82} +7.83430 q^{83} -13.0373i q^{85} -3.01273i q^{86} +2.06830i q^{89} -7.76517 q^{91} +1.58021i q^{92} -1.95791i q^{94} +5.58387 q^{95} +1.67044 q^{97} -0.0995913 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{4}+O(q^{10}) \) 16 * q + 16 * q^4	\( 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91}+O(q^{100}) \) 16 * q + 16 * q^4 + 40 * q^16 - 32 * q^25 + 16 * q^31 + 40 * q^34 + 8 * q^37 + 16 * q^49 + 32 * q^58 - 104 * q^64 + 96 * q^67 - 64 * q^70 + 88 * q^82 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more