LMFDB - Embedded newform 1089.3.b.j.485.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(485,1089)]

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, 0]))

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

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

chi := DirichletCharacter("1089.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.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:	$29.6731007888$
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} + 48x^{14} + 921x^{12} + 8986x^{10} + 46812x^{8} + 125072x^{6} + 152129x^{4} + 65614x^{2} + 5041 $ x^16 + 48x^14 + 921x^12 + 8986x^10 + 46812x^8 + 125072x^6 + 152129x^4 + 65614*x^2 + 5041
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.6
Root			$-1.35141i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.j.485.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.35141i q^{2} +2.17369 q^{4} -5.10469i q^{5} +13.1679 q^{7} -8.34318i q^{8} +O(q^{10})$	$q-1.35141i q^{2} +2.17369 q^{4} -5.10469i q^{5} +13.1679 q^{7} -8.34318i q^{8} -6.89852 q^{10} +2.67027 q^{13} -17.7952i q^{14} -2.58027 q^{16} -17.2161i q^{17} -30.6226 q^{19} -11.0960i q^{20} -15.5309i q^{23} -1.05786 q^{25} -3.60862i q^{26} +28.6230 q^{28} +10.3828i q^{29} +16.8678 q^{31} -29.8857i q^{32} -23.2660 q^{34} -67.2181i q^{35} +18.7510 q^{37} +41.3837i q^{38} -42.5894 q^{40} +13.3582i q^{41} -10.5356 q^{43} -20.9886 q^{46} +82.2065i q^{47} +124.394 q^{49} +1.42960i q^{50} +5.80435 q^{52} +62.9849i q^{53} -109.862i q^{56} +14.0314 q^{58} -65.9058i q^{59} -36.5562 q^{61} -22.7954i q^{62} -50.7089 q^{64} -13.6309i q^{65} +60.5815 q^{67} -37.4225i q^{68} -90.8392 q^{70} +49.1893i q^{71} -6.37308 q^{73} -25.3402i q^{74} -66.5643 q^{76} -115.053 q^{79} +13.1715i q^{80} +18.0523 q^{82} +40.4929i q^{83} -87.8828 q^{85} +14.2379i q^{86} -71.1308i q^{89} +35.1619 q^{91} -33.7594i q^{92} +111.095 q^{94} +156.319i q^{95} +18.6682 q^{97} -168.107i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 8 q^{7}+O(q^{10}) $ 16 * q - 32 * q^4 + 8 * q^7	$ 16 q - 32 q^{4} + 8 q^{7} - 24 q^{10} - 4 q^{13} + 28 q^{16} + 20 q^{19} - 44 q^{25} - 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} - 224 q^{40} - 272 q^{43} - 208 q^{46} + 348 q^{49} - 520 q^{52} - 44 q^{58} - 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} + 4 q^{73} - 1052 q^{76} - 216 q^{79} + 348 q^{82} - 416 q^{85} - 168 q^{91} - 1140 q^{94} - 44 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 8 * q^7 - 24 * q^10 - 4 * q^13 + 28 * q^16 + 20 * q^19 - 44 * q^25 - 16 * q^28 + 28 * q^31 + 148 * q^34 - 148 * q^37 - 224 * q^40 - 272 * q^43 - 208 * q^46 + 348 * q^49 - 520 * q^52 - 44 * q^58 - 224 * q^61 + 436 * q^64 + 24 * q^67 - 664 * q^70 + 4 * q^73 - 1052 * q^76 - 216 * q^79 + 348 * q^82 - 416 * q^85 - 168 * q^91 - 1140 * q^94 - 44 * q^97

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 $

$2$	− 1.35141i	− 0.675704i	−0.941199	−	0.337852i	$-0.890300\pi$
$2$	− 1.35141i	− 0.675704i	0.941199	−	0.337852i	$-0.109700\pi$
$3$	0	0
$3$	0	0
$4$	2.17369	0.543424
$5$	− 5.10469i	− 1.02094i	−0.859896	−	0.510469i	$-0.829472\pi$
$5$	− 5.10469i	− 1.02094i	0.859896	−	0.510469i	$-0.170528\pi$
$6$	0	0
$7$	13.1679	1.88113	0.940565	−	0.339613i	$-0.110296\pi$
$7$	13.1679	1.88113	0.940565	+	0.339613i	$0.110296\pi$
$8$	− 8.34318i	− 1.04290i
$9$	0	0
$10$	−6.89852	−0.689852
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.67027	0.205405	0.102703	−	0.994712i	$-0.467251\pi$
$13$	2.67027	0.205405	0.102703	+	0.994712i	$0.467251\pi$
$14$	− 17.7952i	− 1.27109i
$15$	0	0
$16$	−2.58027	−0.161267
$17$	− 17.2161i	− 1.01271i	−0.862325	−	0.506356i	$-0.830992\pi$
$17$	− 17.2161i	− 1.01271i	0.862325	−	0.506356i	$-0.169008\pi$
$18$	0	0
$19$	−30.6226	−1.61172	−0.805859	−	0.592107i	$-0.798296\pi$
$19$	−30.6226	−1.61172	−0.805859	+	0.592107i	$0.798296\pi$
$20$	− 11.0960i	− 0.554802i
$21$	0	0
$22$	0	0
$23$	− 15.5309i	− 0.675257i	−0.941279	−	0.337628i	$-0.890375\pi$
$23$	− 15.5309i	− 0.675257i	0.941279	−	0.337628i	$-0.109625\pi$
$24$	0	0
$25$	−1.05786	−0.0423145
$26$	− 3.60862i	− 0.138793i
$27$	0	0
$28$	28.6230	1.02225
$29$	10.3828i	0.358028i	0.983847	+	0.179014i	$0.0572908\pi$
$29$	10.3828i	0.358028i	−0.983847	+	0.179014i	$0.942709\pi$
$30$	0	0
$31$	16.8678	0.544124	0.272062	−	0.962280i	$-0.412294\pi$
$31$	16.8678	0.544124	0.272062	+	0.962280i	$0.412294\pi$
$32$	− 29.8857i	− 0.933929i
$33$	0	0
$34$	−23.2660	−0.684294
$35$	− 67.2181i	− 1.92052i
$36$	0	0
$37$	18.7510	0.506783	0.253391	−	0.967364i	$-0.418454\pi$
$37$	18.7510	0.506783	0.253391	+	0.967364i	$0.418454\pi$
$38$	41.3837i	1.08904i
$39$	0	0
$40$	−42.5894	−1.06473
$41$	13.3582i	0.325809i	0.986642	+	0.162904i	$0.0520862\pi$
$41$	13.3582i	0.325809i	−0.986642	+	0.162904i	$0.947914\pi$
$42$	0	0
$43$	−10.5356	−0.245015	−0.122507	−	0.992468i	$-0.539093\pi$
$43$	−10.5356	−0.245015	−0.122507	+	0.992468i	$0.539093\pi$
$44$	0	0
$45$	0	0
$46$	−20.9886	−0.456274
$47$	82.2065i	1.74907i	0.484958	+	0.874537i	$0.338835\pi$
$47$	82.2065i	1.74907i	−0.484958	+	0.874537i	$0.661165\pi$
$48$	0	0
$49$	124.394	2.53865
$50$	1.42960i	0.0285921i
$51$	0	0
$52$	5.80435	0.111622
$53$	62.9849i	1.18839i	0.804320	+	0.594197i	$0.202530\pi$
$53$	62.9849i	1.18839i	−0.804320	+	0.594197i	$0.797470\pi$
$54$	0	0
$55$	0	0
$56$	− 109.862i	− 1.96183i
$57$	0	0
$58$	14.0314	0.241921
$59$	− 65.9058i	− 1.11705i	−0.829488	−	0.558524i	$-0.811368\pi$
$59$	− 65.9058i	− 1.11705i	0.829488	−	0.558524i	$-0.188632\pi$
$60$	0	0
$61$	−36.5562	−0.599282	−0.299641	−	0.954052i	$-0.596867\pi$
$61$	−36.5562	−0.599282	−0.299641	+	0.954052i	$0.596867\pi$
$62$	− 22.7954i	− 0.367667i
$63$	0	0
$64$	−50.7089	−0.792327
$65$	− 13.6309i	− 0.209706i
$66$	0	0
$67$	60.5815	0.904202	0.452101	−	0.891967i	$-0.350675\pi$
$67$	60.5815	0.904202	0.452101	+	0.891967i	$0.350675\pi$
$68$	− 37.4225i	− 0.550331i
$69$	0	0
$70$	−90.8392	−1.29770
$71$	49.1893i	0.692807i	0.938086	+	0.346404i	$0.112597\pi$
$71$	49.1893i	0.692807i	−0.938086	+	0.346404i	$0.887403\pi$
$72$	0	0
$73$	−6.37308	−0.0873024	−0.0436512	−	0.999047i	$-0.513899\pi$
$73$	−6.37308	−0.0873024	−0.0436512	+	0.999047i	$0.513899\pi$
$74$	− 25.3402i	− 0.342435i
$75$	0	0
$76$	−66.5643	−0.875846
$77$	0	0
$78$	0	0
$79$	−115.053	−1.45637	−0.728183	−	0.685383i	$-0.759635\pi$
$79$	−115.053	−1.45637	−0.728183	+	0.685383i	$0.759635\pi$
$80$	13.1715i	0.164644i
$81$	0	0
$82$	18.0523	0.220150
$83$	40.4929i	0.487867i	0.969792	+	0.243933i	$0.0784378\pi$
$83$	40.4929i	0.487867i	−0.969792	+	0.243933i	$0.921562\pi$
$84$	0	0
$85$	−87.8828	−1.03392
$86$	14.2379i	0.165557i
$87$	0	0
$88$	0	0
$89$	− 71.1308i	− 0.799222i	−0.916685	−	0.399611i	$-0.869145\pi$
$89$	− 71.1308i	− 0.799222i	0.916685	−	0.399611i	$-0.130855\pi$
$90$	0	0
$91$	35.1619	0.386394
$92$	− 33.7594i	− 0.366950i
$93$	0	0
$94$	111.095	1.18186
$95$	156.319i	1.64546i
$96$	0	0
$97$	18.6682	0.192456	0.0962279	−	0.995359i	$-0.469322\pi$
$97$	18.6682	0.192456	0.0962279	+	0.995359i	$0.469322\pi$
$98$	− 168.107i	− 1.71538i
$99$	0	0
$100$	−2.29947	−0.0229947
$101$	− 47.0627i	− 0.465967i	−0.972481	−	0.232983i	$-0.925151\pi$
$101$	− 47.0627i	− 0.465967i	0.972481	−	0.232983i	$-0.0748488\pi$
$102$	0	0
$103$	111.438	1.08192	0.540960	−	0.841048i	$-0.318061\pi$
$103$	111.438	1.08192	0.540960	+	0.841048i	$0.318061\pi$
$104$	− 22.2785i	− 0.214217i
$105$	0	0
$106$	85.1183	0.803003
$107$	− 110.880i	− 1.03626i	−0.855301	−	0.518131i	$-0.826628\pi$
$107$	− 110.880i	− 1.03626i	0.855301	−	0.518131i	$-0.173372\pi$
$108$	0	0
$109$	−118.337	−1.08566	−0.542832	−	0.839841i	$-0.682648\pi$
$109$	−118.337	−1.08566	−0.542832	+	0.839841i	$0.682648\pi$
$110$	0	0
$111$	0	0
$112$	−33.9768	−0.303364
$113$	− 69.3440i	− 0.613664i	−0.951764	−	0.306832i	$-0.900731\pi$
$113$	− 69.3440i	− 0.613664i	0.951764	−	0.306832i	$-0.0992689\pi$
$114$	0	0
$115$	−79.2805	−0.689395
$116$	22.5691i	0.194561i
$117$	0	0
$118$	−89.0657	−0.754794
$119$	− 226.700i	− 1.90504i
$120$	0	0
$121$	0	0
$122$	49.4023i	0.404937i
$123$	0	0
$124$	36.6655	0.295690
$125$	− 122.217i	− 0.977738i
$126$	0	0
$127$	−10.7748	−0.0848412	−0.0424206	−	0.999100i	$-0.513507\pi$
$127$	−10.7748	−0.0848412	−0.0424206	+	0.999100i	$0.513507\pi$
$128$	− 51.0145i	− 0.398550i
$129$	0	0
$130$	−18.4209	−0.141699
$131$	− 92.0724i	− 0.702843i	−0.936217	−	0.351421i	$-0.885698\pi$
$131$	− 92.0724i	− 0.702843i	0.936217	−	0.351421i	$-0.114302\pi$
$132$	0	0
$133$	−403.236	−3.03185
$134$	− 81.8704i	− 0.610973i
$135$	0	0
$136$	−143.637	−1.05615
$137$	198.651i	1.45001i	0.688746	+	0.725003i	$0.258162\pi$
$137$	198.651i	1.45001i	−0.688746	+	0.725003i	$0.741838\pi$
$138$	0	0
$139$	59.5350	0.428310	0.214155	−	0.976800i	$-0.431300\pi$
$139$	59.5350	0.428310	0.214155	+	0.976800i	$0.431300\pi$
$140$	− 146.112i	− 1.04365i
$141$	0	0
$142$	66.4748	0.468133
$143$	0	0
$144$	0	0
$145$	53.0011	0.365525
$146$	8.61263i	0.0589906i
$147$	0	0
$148$	40.7588	0.275398
$149$	14.5644i	0.0977477i	0.998805	+	0.0488738i	$0.0155632\pi$
$149$	14.5644i	0.0977477i	−0.998805	+	0.0488738i	$0.984437\pi$
$150$	0	0
$151$	243.598	1.61323	0.806616	−	0.591076i	$-0.201297\pi$
$151$	243.598	1.61323	0.806616	+	0.591076i	$0.201297\pi$
$152$	255.490i	1.68086i
$153$	0	0
$154$	0	0
$155$	− 86.1051i	− 0.555517i
$156$	0	0
$157$	−111.033	−0.707218	−0.353609	−	0.935393i	$-0.615046\pi$
$157$	−111.033	−0.707218	−0.353609	+	0.935393i	$0.615046\pi$
$158$	155.483i	0.984072i
$159$	0	0
$160$	−152.557	−0.953484
$161$	− 204.510i	− 1.27025i
$162$	0	0
$163$	123.154	0.755546	0.377773	−	0.925898i	$-0.376690\pi$
$163$	123.154	0.755546	0.377773	+	0.925898i	$0.376690\pi$
$164$	29.0365i	0.177052i
$165$	0	0
$166$	54.7225	0.329654
$167$	− 229.010i	− 1.37132i	−0.727924	−	0.685658i	$-0.759515\pi$
$167$	− 229.010i	− 1.37132i	0.727924	−	0.685658i	$-0.240485\pi$
$168$	0	0
$169$	−161.870	−0.957809
$170$	118.766i	0.698621i
$171$	0	0
$172$	−22.9012	−0.133147
$173$	172.678i	0.998138i	0.866562	+	0.499069i	$0.166325\pi$
$173$	172.678i	0.998138i	−0.866562	+	0.499069i	$0.833675\pi$
$174$	0	0
$175$	−13.9298	−0.0795991
$176$	0	0
$177$	0	0
$178$	−96.1267	−0.540038
$179$	182.492i	1.01951i	0.860320	+	0.509755i	$0.170264\pi$
$179$	182.492i	1.01951i	−0.860320	+	0.509755i	$0.829736\pi$
$180$	0	0
$181$	121.741	0.672600	0.336300	−	0.941755i	$-0.390824\pi$
$181$	121.741	0.672600	0.336300	+	0.941755i	$0.390824\pi$
$182$	− 47.5181i	− 0.261088i
$183$	0	0
$184$	−129.577	−0.704224
$185$	− 95.7178i	− 0.517394i
$186$	0	0
$187$	0	0
$188$	178.692i	0.950488i
$189$	0	0
$190$	211.251	1.11185
$191$	314.416i	1.64616i	0.567926	+	0.823080i	$0.307746\pi$
$191$	314.416i	1.64616i	−0.567926	+	0.823080i	$0.692254\pi$
$192$	0	0
$193$	−124.391	−0.644511	−0.322255	−	0.946653i	$-0.604441\pi$
$193$	−124.391	−0.644511	−0.322255	+	0.946653i	$0.604441\pi$
$194$	− 25.2284i	− 0.130043i
$195$	0	0
$196$	270.395	1.37956
$197$	193.993i	0.984738i	0.870387	+	0.492369i	$0.163869\pi$
$197$	193.993i	0.984738i	−0.870387	+	0.492369i	$0.836131\pi$
$198$	0	0
$199$	−67.6934	−0.340168	−0.170084	−	0.985430i	$-0.554404\pi$
$199$	−67.6934	−0.340168	−0.170084	+	0.985430i	$0.554404\pi$
$200$	8.82594i	0.0441297i
$201$	0	0
$202$	−63.6009	−0.314856
$203$	136.720i	0.673498i
$204$	0	0
$205$	68.1892	0.332630
$206$	− 150.598i	− 0.731058i
$207$	0	0
$208$	−6.89002	−0.0331251
$209$	0	0
$210$	0	0
$211$	−75.0484	−0.355679	−0.177840	−	0.984059i	$-0.556911\pi$
$211$	−75.0484	−0.355679	−0.177840	+	0.984059i	$0.556911\pi$
$212$	136.910i	0.645801i
$213$	0	0
$214$	−149.844	−0.700207
$215$	53.7811i	0.250145i
$216$	0	0
$217$	222.114	1.02357
$218$	159.922i	0.733588i
$219$	0	0
$220$	0	0
$221$	− 45.9716i	− 0.208016i
$222$	0	0
$223$	−5.97825	−0.0268083	−0.0134042	−	0.999910i	$-0.504267\pi$
$223$	−5.97825	−0.0268083	−0.0134042	+	0.999910i	$0.504267\pi$
$224$	− 393.533i	− 1.75684i
$225$	0	0
$226$	−93.7121	−0.414655
$227$	168.380i	0.741762i	0.928680	+	0.370881i	$0.120944\pi$
$227$	168.380i	0.741762i	−0.928680	+	0.370881i	$0.879056\pi$
$228$	0	0
$229$	−352.188	−1.53794	−0.768969	−	0.639287i	$-0.779230\pi$
$229$	−352.188	−1.53794	−0.768969	+	0.639287i	$0.779230\pi$
$230$	107.140i	0.465827i
$231$	0	0
$232$	86.6257	0.373387
$233$	356.092i	1.52829i	0.645043	+	0.764146i	$0.276840\pi$
$233$	356.092i	1.52829i	−0.645043	+	0.764146i	$0.723160\pi$
$234$	0	0
$235$	419.639	1.78570
$236$	− 143.259i	− 0.607030i
$237$	0	0
$238$	−306.364	−1.28725
$239$	17.4384i	0.0729639i	0.999334	+	0.0364820i	$0.0116151\pi$
$239$	17.4384i	0.0729639i	−0.999334	+	0.0364820i	$0.988385\pi$
$240$	0	0
$241$	98.6722	0.409428	0.204714	−	0.978822i	$-0.434374\pi$
$241$	98.6722	0.409428	0.204714	+	0.978822i	$0.434374\pi$
$242$	0	0
$243$	0	0
$244$	−79.4620	−0.325664
$245$	− 634.993i	− 2.59181i
$246$	0	0
$247$	−81.7707	−0.331055
$248$	− 140.732i	− 0.567466i
$249$	0	0
$250$	−165.165	−0.660661
$251$	− 177.305i	− 0.706394i	−0.935549	−	0.353197i	$-0.885094\pi$
$251$	− 177.305i	− 0.706394i	0.935549	−	0.353197i	$-0.114906\pi$
$252$	0	0
$253$	0	0
$254$	14.5612i	0.0573275i
$255$	0	0
$256$	−271.777	−1.06163
$257$	115.325i	0.448735i	0.974505	+	0.224367i	$0.0720315\pi$
$257$	115.325i	0.448735i	−0.974505	+	0.224367i	$0.927968\pi$
$258$	0	0
$259$	246.911	0.953324
$260$	− 29.6294i	− 0.113959i
$261$	0	0
$262$	−124.427	−0.474914
$263$	− 236.953i	− 0.900964i	−0.892786	−	0.450482i	$-0.851252\pi$
$263$	− 236.953i	− 0.900964i	0.892786	−	0.450482i	$-0.148748\pi$
$264$	0	0
$265$	321.518	1.21328
$266$	544.937i	2.04864i
$267$	0	0
$268$	131.686	0.491365
$269$	− 528.479i	− 1.96461i	−0.187297	−	0.982303i	$-0.559973\pi$
$269$	− 528.479i	− 1.96461i	0.187297	−	0.982303i	$-0.440027\pi$
$270$	0	0
$271$	178.936	0.660280	0.330140	−	0.943932i	$-0.392904\pi$
$271$	178.936	0.660280	0.330140	+	0.943932i	$0.392904\pi$
$272$	44.4222i	0.163317i
$273$	0	0
$274$	268.458	0.979775
$275$	0	0
$276$	0	0
$277$	286.215	1.03327	0.516634	−	0.856206i	$-0.327185\pi$
$277$	286.215	1.03327	0.516634	+	0.856206i	$0.327185\pi$
$278$	− 80.4562i	− 0.289411i
$279$	0	0
$280$	−560.813	−2.00290
$281$	253.434i	0.901902i	0.892549	+	0.450951i	$0.148915\pi$
$281$	253.434i	0.901902i	−0.892549	+	0.450951i	$0.851085\pi$
$282$	0	0
$283$	283.270	1.00095	0.500477	−	0.865750i	$-0.333158\pi$
$283$	283.270	1.00095	0.500477	+	0.865750i	$0.333158\pi$
$284$	106.923i	0.376488i
$285$	0	0
$286$	0	0
$287$	175.899i	0.612889i
$288$	0	0
$289$	−7.39394	−0.0255846
$290$	− 71.6261i	− 0.246986i
$291$	0	0
$292$	−13.8531	−0.0474422
$293$	299.293i	1.02148i	0.859736	+	0.510739i	$0.170628\pi$
$293$	299.293i	1.02148i	−0.859736	+	0.510739i	$0.829372\pi$
$294$	0	0
$295$	−336.429	−1.14044
$296$	− 156.443i	− 0.528523i
$297$	0	0
$298$	19.6825	0.0660485
$299$	− 41.4717i	− 0.138701i
$300$	0	0
$301$	−138.732	−0.460905
$302$	− 329.200i	− 1.09007i
$303$	0	0
$304$	79.0148	0.259917
$305$	186.608i	0.611829i
$306$	0	0
$307$	−111.839	−0.364296	−0.182148	−	0.983271i	$-0.558305\pi$
$307$	−111.839	−0.364296	−0.182148	+	0.983271i	$0.558305\pi$
$308$	0	0
$309$	0	0
$310$	−116.363	−0.375365
$311$	356.284i	1.14561i	0.819692	+	0.572805i	$0.194145\pi$
$311$	356.284i	1.14561i	−0.819692	+	0.572805i	$0.805855\pi$
$312$	0	0
$313$	338.742	1.08224	0.541121	−	0.840945i	$-0.318000\pi$
$313$	338.742	1.08224	0.541121	+	0.840945i	$0.318000\pi$
$314$	150.051i	0.477870i
$315$	0	0
$316$	−250.090	−0.791423
$317$	− 404.642i	− 1.27647i	−0.769840	−	0.638237i	$-0.779664\pi$
$317$	− 404.642i	− 1.27647i	0.769840	−	0.638237i	$-0.220336\pi$
$318$	0	0
$319$	0	0
$320$	258.853i	0.808917i
$321$	0	0
$322$	−276.376	−0.858311
$323$	527.202i	1.63221i
$324$	0	0
$325$	−2.82478	−0.00869162
$326$	− 166.431i	− 0.510526i
$327$	0	0
$328$	111.450	0.339785
$329$	1082.49i	3.29024i
$330$	0	0
$331$	634.065	1.91561	0.957803	−	0.287426i	$-0.0927995\pi$
$331$	634.065	1.91561	0.957803	+	0.287426i	$0.0927995\pi$
$332$	88.0193i	0.265118i
$333$	0	0
$334$	−309.486	−0.926604
$335$	− 309.250i	− 0.923134i
$336$	0	0
$337$	−237.412	−0.704488	−0.352244	−	0.935908i	$-0.614581\pi$
$337$	−237.412	−0.704488	−0.352244	+	0.935908i	$0.614581\pi$
$338$	218.752i	0.647195i
$339$	0	0
$340$	−191.030	−0.561854
$341$	0	0
$342$	0	0
$343$	992.782	2.89441
$344$	87.9007i	0.255525i
$345$	0	0
$346$	233.358	0.674446
$347$	− 487.484i	− 1.40485i	−0.711757	−	0.702426i	$-0.752100\pi$
$347$	− 487.484i	− 1.40485i	0.711757	−	0.702426i	$-0.247900\pi$
$348$	0	0
$349$	−38.0498	−0.109025	−0.0545126	−	0.998513i	$-0.517361\pi$
$349$	−38.0498	−0.109025	−0.0545126	+	0.998513i	$0.517361\pi$
$350$	18.8249i	0.0537855i
$351$	0	0
$352$	0	0
$353$	90.5379i	0.256481i	0.991743	+	0.128241i	$0.0409330\pi$
$353$	90.5379i	0.256481i	−0.991743	+	0.128241i	$0.959067\pi$
$354$	0	0
$355$	251.096	0.707313
$356$	− 154.617i	− 0.434316i
$357$	0	0
$358$	246.621	0.688887
$359$	111.987i	0.311941i	0.987762	+	0.155971i	$0.0498505\pi$
$359$	111.987i	0.311941i	−0.987762	+	0.155971i	$0.950150\pi$
$360$	0	0
$361$	576.747	1.59764
$362$	− 164.521i	− 0.454478i
$363$	0	0
$364$	76.4312	0.209976
$365$	32.5326i	0.0891304i
$366$	0	0
$367$	338.791	0.923137	0.461568	−	0.887105i	$-0.347287\pi$
$367$	338.791	0.923137	0.461568	+	0.887105i	$0.347287\pi$
$368$	40.0740i	0.108897i
$369$	0	0
$370$	−129.354	−0.349605
$371$	829.380i	2.23552i
$372$	0	0
$373$	554.596	1.48685	0.743427	−	0.668818i	$-0.233199\pi$
$373$	554.596	1.48685	0.743427	+	0.668818i	$0.233199\pi$
$374$	0	0
$375$	0	0
$376$	685.864	1.82411
$377$	27.7249i	0.0735409i
$378$	0	0
$379$	−251.309	−0.663084	−0.331542	−	0.943440i	$-0.607569\pi$
$379$	−251.309	−0.663084	−0.331542	+	0.943440i	$0.607569\pi$
$380$	339.790i	0.894184i
$381$	0	0
$382$	424.905	1.11232
$383$	514.214i	1.34259i	0.741188	+	0.671297i	$0.234263\pi$
$383$	514.214i	1.34259i	−0.741188	+	0.671297i	$0.765737\pi$
$384$	0	0
$385$	0	0
$386$	168.102i	0.435499i
$387$	0	0
$388$	40.5790	0.104585
$389$	− 194.151i	− 0.499102i	−0.968362	−	0.249551i	$-0.919717\pi$
$389$	− 194.151i	− 0.499102i	0.968362	−	0.249551i	$-0.0802830\pi$
$390$	0	0
$391$	−267.382	−0.683840
$392$	− 1037.84i	− 2.64756i
$393$	0	0
$394$	262.164	0.665392
$395$	587.309i	1.48686i
$396$	0	0
$397$	−260.248	−0.655537	−0.327769	−	0.944758i	$-0.606297\pi$
$397$	−260.248	−0.655537	−0.327769	+	0.944758i	$0.606297\pi$
$398$	91.4815i	0.229853i
$399$	0	0
$400$	2.72957	0.00682393
$401$	379.370i	0.946059i	0.881047	+	0.473029i	$0.156840\pi$
$401$	379.370i	0.946059i	−0.881047	+	0.473029i	$0.843160\pi$
$402$	0	0
$403$	45.0417	0.111766
$404$	− 102.300i	− 0.253217i
$405$	0	0
$406$	184.765	0.455085
$407$	0	0
$408$	0	0
$409$	−682.061	−1.66763	−0.833815	−	0.552044i	$-0.813848\pi$
$409$	−682.061	−1.66763	−0.833815	+	0.552044i	$0.813848\pi$
$410$	− 92.1515i	− 0.224760i
$411$	0	0
$412$	242.232	0.587941
$413$	− 867.842i	− 2.10131i
$414$	0	0
$415$	206.704	0.498082
$416$	− 79.8030i	− 0.191834i
$417$	0	0
$418$	0	0
$419$	275.319i	0.657085i	0.944489	+	0.328543i	$0.106557\pi$
$419$	275.319i	0.657085i	−0.944489	+	0.328543i	$0.893443\pi$
$420$	0	0
$421$	121.470	0.288527	0.144263	−	0.989539i	$-0.453919\pi$
$421$	121.470	0.288527	0.144263	+	0.989539i	$0.453919\pi$
$422$	101.421i	0.240334i
$423$	0	0
$424$	525.494	1.23937
$425$	18.2123i	0.0428524i
$426$	0	0
$427$	−481.369	−1.12733
$428$	− 241.019i	− 0.563130i
$429$	0	0
$430$	72.6803	0.169024
$431$	98.7461i	0.229109i	0.993417	+	0.114555i	$0.0365441\pi$
$431$	98.7461i	0.229109i	−0.993417	+	0.114555i	$0.963456\pi$
$432$	0	0
$433$	559.992	1.29328	0.646642	−	0.762794i	$-0.276173\pi$
$433$	559.992	1.29328	0.646642	+	0.762794i	$0.276173\pi$
$434$	− 300.167i	− 0.691630i
$435$	0	0
$436$	−257.230	−0.589976
$437$	475.597i	1.08832i
$438$	0	0
$439$	577.439	1.31535	0.657676	−	0.753301i	$-0.271540\pi$
$439$	577.439	1.31535	0.657676	+	0.753301i	$0.271540\pi$
$440$	0	0
$441$	0	0
$442$	−62.1264	−0.140558
$443$	− 87.3800i	− 0.197246i	−0.995125	−	0.0986230i	$-0.968556\pi$
$443$	− 87.3800i	− 0.197246i	0.995125	−	0.0986230i	$-0.0314438\pi$
$444$	0	0
$445$	−363.101	−0.815956
$446$	8.07906i	0.0181145i
$447$	0	0
$448$	−667.731	−1.49047
$449$	− 96.2092i	− 0.214274i	−0.994244	−	0.107137i	$-0.965832\pi$
$449$	− 96.2092i	− 0.214274i	0.994244	−	0.107137i	$-0.0341684\pi$
$450$	0	0
$451$	0	0
$452$	− 150.733i	− 0.333479i
$453$	0	0
$454$	227.550	0.501212
$455$	− 179.490i	− 0.394485i
$456$	0	0
$457$	691.788	1.51376	0.756879	−	0.653555i	$-0.226723\pi$
$457$	691.788	1.51376	0.756879	+	0.653555i	$0.226723\pi$
$458$	475.949i	1.03919i
$459$	0	0
$460$	−172.332	−0.374634
$461$	− 70.5491i	− 0.153035i	−0.997068	−	0.0765175i	$-0.975620\pi$
$461$	− 70.5491i	− 0.153035i	0.997068	−	0.0765175i	$-0.0243801\pi$
$462$	0	0
$463$	248.292	0.536267	0.268134	−	0.963382i	$-0.413593\pi$
$463$	248.292	0.536267	0.268134	+	0.963382i	$0.413593\pi$
$464$	− 26.7905i	− 0.0577381i
$465$	0	0
$466$	481.226	1.03267
$467$	− 254.158i	− 0.544235i	−0.962264	−	0.272118i	$-0.912276\pi$
$467$	− 254.158i	− 0.544235i	0.962264	−	0.272118i	$-0.0877240\pi$
$468$	0	0
$469$	797.732	1.70092
$470$	− 567.103i	− 1.20660i
$471$	0	0
$472$	−549.864	−1.16497
$473$	0	0
$474$	0	0
$475$	32.3946	0.0681991
$476$	− 492.777i	− 1.03525i
$477$	0	0
$478$	23.5664	0.0493020
$479$	18.2786i	0.0381599i	0.999818	+	0.0190800i	$0.00607371\pi$
$479$	18.2786i	0.0381599i	−0.999818	+	0.0190800i	$0.993926\pi$
$480$	0	0
$481$	50.0701	0.104096
$482$	− 133.346i	− 0.276652i
$483$	0	0
$484$	0	0
$485$	− 95.2954i	− 0.196485i
$486$	0	0
$487$	−243.375	−0.499744	−0.249872	−	0.968279i	$-0.580389\pi$
$487$	−243.375	−0.499744	−0.249872	+	0.968279i	$0.580389\pi$
$488$	304.995i	0.624990i
$489$	0	0
$490$	−858.135	−1.75130
$491$	316.374i	0.644347i	0.946681	+	0.322173i	$0.104413\pi$
$491$	316.374i	0.644347i	−0.946681	+	0.322173i	$0.895587\pi$
$492$	0	0
$493$	178.752	0.362579
$494$	110.506i	0.223696i
$495$	0	0
$496$	−43.5236	−0.0877493
$497$	647.721i	1.30326i
$498$	0	0
$499$	26.8038	0.0537149	0.0268575	−	0.999639i	$-0.491450\pi$
$499$	26.8038	0.0537149	0.0268575	+	0.999639i	$0.491450\pi$
$500$	− 265.663i	− 0.531326i
$501$	0	0
$502$	−239.611	−0.477314
$503$	− 182.625i	− 0.363071i	−0.983384	−	0.181535i	$-0.941893\pi$
$503$	− 182.625i	− 0.363071i	0.983384	−	0.181535i	$-0.0581067\pi$
$504$	0	0
$505$	−240.240	−0.475723
$506$	0	0
$507$	0	0
$508$	−23.4212	−0.0461047
$509$	− 216.345i	− 0.425040i	−0.977157	−	0.212520i	$-0.931833\pi$
$509$	− 216.345i	− 0.425040i	0.977157	−	0.212520i	$-0.0681670\pi$
$510$	0	0
$511$	−83.9201	−0.164227
$512$	163.224i	0.318797i
$513$	0	0
$514$	155.851	0.303212
$515$	− 568.855i	− 1.10457i
$516$	0	0
$517$	0	0
$518$	− 333.678i	− 0.644165i
$519$	0	0
$520$	−113.725	−0.218702
$521$	571.189i	1.09633i	0.836370	+	0.548166i	$0.184674\pi$
$521$	571.189i	1.09633i	−0.836370	+	0.548166i	$0.815326\pi$
$522$	0	0
$523$	−767.452	−1.46740	−0.733702	−	0.679472i	$-0.762209\pi$
$523$	−767.452	−1.46740	−0.733702	+	0.679472i	$0.762209\pi$
$524$	− 200.137i	− 0.381941i
$525$	0	0
$526$	−320.221	−0.608785
$527$	− 290.398i	− 0.551041i
$528$	0	0
$529$	287.791	0.544028
$530$	− 434.503i	− 0.819816i
$531$	0	0
$532$	−876.513	−1.64758
$533$	35.6699i	0.0669228i
$534$	0	0
$535$	−566.008	−1.05796
$536$	− 505.443i	− 0.942990i
$537$	0	0
$538$	−714.191	−1.32749
$539$	0	0
$540$	0	0
$541$	−205.807	−0.380420	−0.190210	−	0.981743i	$-0.560917\pi$
$541$	−205.807	−0.380420	−0.190210	+	0.981743i	$0.560917\pi$
$542$	− 241.815i	− 0.446154i
$543$	0	0
$544$	−514.516	−0.945801
$545$	604.076i	1.10840i
$546$	0	0
$547$	−435.348	−0.795883	−0.397942	−	0.917411i	$-0.630275\pi$
$547$	−435.348	−0.795883	−0.397942	+	0.917411i	$0.630275\pi$
$548$	431.806i	0.787968i
$549$	0	0
$550$	0	0
$551$	− 317.949i	− 0.577040i
$552$	0	0
$553$	−1515.01	−2.73961
$554$	− 386.794i	− 0.698184i
$555$	0	0
$556$	129.411	0.232754
$557$	738.780i	1.32636i	0.748462	+	0.663178i	$0.230793\pi$
$557$	738.780i	1.32636i	−0.748462	+	0.663178i	$0.769207\pi$
$558$	0	0
$559$	−28.1330	−0.0503273
$560$	173.441i	0.309716i
$561$	0	0
$562$	342.493	0.609419
$563$	− 65.8590i	− 0.116979i	−0.998288	−	0.0584893i	$-0.981372\pi$
$563$	− 65.8590i	− 0.116979i	0.998288	−	0.0584893i	$-0.0186284\pi$
$564$	0	0
$565$	−353.980	−0.626513
$566$	− 382.813i	− 0.676348i
$567$	0	0
$568$	410.395	0.722527
$569$	− 697.641i	− 1.22608i	−0.790051	−	0.613041i	$-0.789946\pi$
$569$	− 697.641i	− 1.22608i	0.790051	−	0.613041i	$-0.210054\pi$
$570$	0	0
$571$	256.388	0.449017	0.224508	−	0.974472i	$-0.427922\pi$
$571$	256.388	0.449017	0.224508	+	0.974472i	$0.427922\pi$
$572$	0	0
$573$	0	0
$574$	237.711	0.414132
$575$	16.4296i	0.0285732i
$576$	0	0
$577$	454.911	0.788407	0.394203	−	0.919023i	$-0.371021\pi$
$577$	454.911	0.788407	0.394203	+	0.919023i	$0.371021\pi$
$578$	9.99223i	0.0172876i
$579$	0	0
$580$	115.208	0.198635
$581$	533.207i	0.917741i
$582$	0	0
$583$	0	0
$584$	53.1718i	0.0910475i
$585$	0	0
$586$	404.467	0.690217
$587$	110.294i	0.187894i	0.995577	+	0.0939470i	$0.0299484\pi$
$587$	110.294i	0.187894i	−0.995577	+	0.0939470i	$0.970052\pi$
$588$	0	0
$589$	−516.538	−0.876975
$590$	454.653i	0.770598i
$591$	0	0
$592$	−48.3826	−0.0817273
$593$	− 250.416i	− 0.422287i	−0.977455	−	0.211144i	$-0.932281\pi$
$593$	− 250.416i	− 0.422287i	0.977455	−	0.211144i	$-0.0677188\pi$
$594$	0	0
$595$	−1157.23	−1.94493
$596$	31.6586i	0.0531184i
$597$	0	0
$598$	−56.0452	−0.0937211
$599$	285.443i	0.476533i	0.971200	+	0.238267i	$0.0765792\pi$
$599$	285.443i	0.476533i	−0.971200	+	0.238267i	$0.923421\pi$
$600$	0	0
$601$	223.484	0.371854	0.185927	−	0.982564i	$-0.440471\pi$
$601$	223.484	0.371854	0.185927	+	0.982564i	$0.440471\pi$
$602$	187.484i	0.311435i
$603$	0	0
$604$	529.508	0.876668
$605$	0	0
$606$	0	0
$607$	−954.863	−1.57309	−0.786543	−	0.617536i	$-0.788131\pi$
$607$	−954.863	−1.57309	−0.786543	+	0.617536i	$0.788131\pi$
$608$	915.180i	1.50523i
$609$	0	0
$610$	252.184	0.413416
$611$	219.513i	0.359269i
$612$	0	0
$613$	340.137	0.554872	0.277436	−	0.960744i	$-0.410515\pi$
$613$	340.137	0.554872	0.277436	+	0.960744i	$0.410515\pi$
$614$	151.140i	0.246156i
$615$	0	0
$616$	0	0
$617$	812.456i	1.31678i	0.752675	+	0.658392i	$0.228763\pi$
$617$	812.456i	1.31678i	−0.752675	+	0.658392i	$0.771237\pi$
$618$	0	0
$619$	913.830	1.47630	0.738150	−	0.674636i	$-0.235699\pi$
$619$	913.830	1.47630	0.738150	+	0.674636i	$0.235699\pi$
$620$	− 187.166i	− 0.301881i
$621$	0	0
$622$	481.486	0.774093
$623$	− 936.644i	− 1.50344i
$624$	0	0
$625$	−650.327	−1.04052
$626$	− 457.779i	− 0.731276i
$627$	0	0
$628$	−241.352	−0.384319
$629$	− 322.818i	− 0.513224i
$630$	0	0
$631$	−603.870	−0.957005	−0.478502	−	0.878086i	$-0.658820\pi$
$631$	−603.870	−0.957005	−0.478502	+	0.878086i	$0.658820\pi$
$632$	959.907i	1.51884i
$633$	0	0
$634$	−546.837	−0.862518
$635$	55.0021i	0.0866176i
$636$	0	0
$637$	332.165	0.521453
$638$	0	0
$639$	0	0
$640$	−260.413	−0.406895
$641$	1077.99i	1.68174i	0.541241	+	0.840868i	$0.317955\pi$
$641$	1077.99i	1.68174i	−0.541241	+	0.840868i	$0.682045\pi$
$642$	0	0
$643$	384.547	0.598052	0.299026	−	0.954245i	$-0.403338\pi$
$643$	384.547	0.598052	0.299026	+	0.954245i	$0.403338\pi$
$644$	− 444.542i	− 0.690282i
$645$	0	0
$646$	712.466	1.10289
$647$	− 595.205i	− 0.919946i	−0.887933	−	0.459973i	$-0.847859\pi$
$647$	− 595.205i	− 0.919946i	0.887933	−	0.459973i	$-0.152141\pi$
$648$	0	0
$649$	0	0
$650$	3.81743i	0.00587297i
$651$	0	0
$652$	267.699	0.410582
$653$	418.619i	0.641071i	0.947237	+	0.320535i	$0.103863\pi$
$653$	418.619i	0.641071i	−0.947237	+	0.320535i	$0.896137\pi$
$654$	0	0
$655$	−470.001	−0.717559
$656$	− 34.4677i	− 0.0525422i
$657$	0	0
$658$	1462.88	2.22323
$659$	− 1112.39i	− 1.68800i	−0.536347	−	0.843998i	$-0.680196\pi$
$659$	− 1112.39i	− 1.68800i	0.536347	−	0.843998i	$-0.319804\pi$
$660$	0	0
$661$	−736.025	−1.11350	−0.556751	−	0.830679i	$-0.687952\pi$
$661$	−736.025	−1.11350	−0.556751	+	0.830679i	$0.687952\pi$
$662$	− 856.882i	− 1.29438i
$663$	0	0
$664$	337.840	0.508795
$665$	2058.40i	3.09533i
$666$	0	0
$667$	161.255	0.241761
$668$	− 497.797i	− 0.745206i
$669$	0	0
$670$	−417.923	−0.623765
$671$	0	0
$672$	0	0
$673$	−545.629	−0.810742	−0.405371	−	0.914152i	$-0.632858\pi$
$673$	−545.629	−0.810742	−0.405371	+	0.914152i	$0.632858\pi$
$674$	320.841i	0.476025i
$675$	0	0
$676$	−351.855	−0.520496
$677$	984.875i	1.45476i	0.686233	+	0.727382i	$0.259263\pi$
$677$	984.875i	1.45476i	−0.686233	+	0.727382i	$0.740737\pi$
$678$	0	0
$679$	245.821	0.362034
$680$	733.223i	1.07827i
$681$	0	0
$682$	0	0
$683$	− 924.817i	− 1.35405i	−0.735959	−	0.677026i	$-0.763269\pi$
$683$	− 924.817i	− 1.35405i	0.735959	−	0.677026i	$-0.236731\pi$
$684$	0	0
$685$	1014.05	1.48037
$686$	− 1341.65i	− 1.95576i
$687$	0	0
$688$	27.1848	0.0395128
$689$	168.187i	0.244102i
$690$	0	0
$691$	−79.1299	−0.114515	−0.0572575	−	0.998359i	$-0.518236\pi$
$691$	−79.1299	−0.114515	−0.0572575	+	0.998359i	$0.518236\pi$
$692$	375.349i	0.542412i
$693$	0	0
$694$	−658.790	−0.949265
$695$	− 303.908i	− 0.437278i
$696$	0	0
$697$	229.975	0.329950
$698$	51.4208i	0.0736688i
$699$	0	0
$700$	−30.2792	−0.0432560
$701$	1232.29i	1.75790i	0.476912	+	0.878951i	$0.341756\pi$
$701$	1232.29i	1.75790i	−0.476912	+	0.878951i	$0.658244\pi$
$702$	0	0
$703$	−574.204	−0.816791
$704$	0	0
$705$	0	0
$706$	122.354	0.173306
$707$	− 619.717i	− 0.876545i
$708$	0	0
$709$	−267.568	−0.377388	−0.188694	−	0.982036i	$-0.560425\pi$
$709$	−267.568	−0.377388	−0.188694	+	0.982036i	$0.560425\pi$
$710$	− 339.333i	− 0.477934i
$711$	0	0
$712$	−593.457	−0.833507
$713$	− 261.973i	− 0.367423i
$714$	0	0
$715$	0	0
$716$	396.682i	0.554025i
$717$	0	0
$718$	151.340	0.210780
$719$	− 686.620i	− 0.954965i	−0.878641	−	0.477482i	$-0.841549\pi$
$719$	− 686.620i	− 0.954965i	0.878641	−	0.477482i	$-0.158451\pi$
$720$	0	0
$721$	1467.40	2.03523
$722$	− 779.420i	− 1.07953i
$723$	0	0
$724$	264.627	0.365507
$725$	− 10.9836i	− 0.0151498i
$726$	0	0
$727$	−1265.58	−1.74082	−0.870410	−	0.492327i	$-0.836146\pi$
$727$	−1265.58	−1.74082	−0.870410	+	0.492327i	$0.836146\pi$
$728$	− 293.362i	− 0.402970i
$729$	0	0
$730$	43.9648	0.0602258
$731$	181.382i	0.248129i
$732$	0	0
$733$	−99.5964	−0.135875	−0.0679375	−	0.997690i	$-0.521642\pi$
$733$	−99.5964	−0.135875	−0.0679375	+	0.997690i	$0.521642\pi$
$734$	− 457.845i	− 0.623767i
$735$	0	0
$736$	−464.152	−0.630642
$737$	0	0
$738$	0	0
$739$	320.536	0.433743	0.216872	−	0.976200i	$-0.430415\pi$
$739$	320.536	0.433743	0.216872	+	0.976200i	$0.430415\pi$
$740$	− 208.061i	− 0.281164i
$741$	0	0
$742$	1120.83	1.51055
$743$	54.6611i	0.0735681i	0.999323	+	0.0367840i	$0.0117114\pi$
$743$	54.6611i	0.0735681i	−0.999323	+	0.0367840i	$0.988289\pi$
$744$	0	0
$745$	74.3468	0.0997943
$746$	− 749.486i	− 1.00467i
$747$	0	0
$748$	0	0
$749$	− 1460.06i	− 1.94935i
$750$	0	0
$751$	145.039	0.193128	0.0965641	−	0.995327i	$-0.469215\pi$
$751$	145.039	0.193128	0.0965641	+	0.995327i	$0.469215\pi$
$752$	− 212.115i	− 0.282068i
$753$	0	0
$754$	37.4677	0.0496919
$755$	− 1243.49i	− 1.64701i
$756$	0	0
$757$	261.648	0.345638	0.172819	−	0.984954i	$-0.444712\pi$
$757$	261.648	0.345638	0.172819	+	0.984954i	$0.444712\pi$
$758$	339.621i	0.448049i
$759$	0	0
$760$	1304.20	1.71605
$761$	− 1004.63i	− 1.32014i	−0.751204	−	0.660070i	$-0.770527\pi$
$761$	− 1004.63i	− 1.32014i	0.751204	−	0.660070i	$-0.229473\pi$
$762$	0	0
$763$	−1558.26	−2.04228
$764$	683.445i	0.894562i
$765$	0	0
$766$	694.913	0.907197
$767$	− 175.986i	− 0.229448i
$768$	0	0
$769$	−1067.91	−1.38869	−0.694347	−	0.719641i	$-0.744307\pi$
$769$	−1067.91	−1.38869	−0.694347	+	0.719641i	$0.744307\pi$
$770$	0	0
$771$	0	0
$772$	−270.387	−0.350242
$773$	− 177.483i	− 0.229603i	−0.993388	−	0.114801i	$-0.963377\pi$
$773$	− 177.483i	− 0.229603i	0.993388	−	0.114801i	$-0.0366232\pi$
$774$	0	0
$775$	−17.8439	−0.0230243
$776$	− 155.752i	− 0.200712i
$777$	0	0
$778$	−262.377	−0.337245
$779$	− 409.062i	− 0.525112i
$780$	0	0
$781$	0	0
$782$	361.342i	0.462074i
$783$	0	0
$784$	−320.970	−0.409401
$785$	566.790i	0.722026i
$786$	0	0
$787$	−660.507	−0.839271	−0.419636	−	0.907693i	$-0.637842\pi$
$787$	−660.507	−0.839271	−0.419636	+	0.907693i	$0.637842\pi$
$788$	421.682i	0.535130i
$789$	0	0
$790$	793.695	1.00468
$791$	− 913.116i	− 1.15438i
$792$	0	0
$793$	−97.6148	−0.123096
$794$	351.702i	0.442949i
$795$	0	0
$796$	−147.145	−0.184855
$797$	− 398.568i	− 0.500086i	−0.968235	−	0.250043i	$-0.919555\pi$
$797$	− 398.568i	− 0.500086i	0.968235	−	0.250043i	$-0.0804447\pi$
$798$	0	0
$799$	1415.27	1.77131
$800$	31.6150i	0.0395188i
$801$	0	0
$802$	512.683	0.639256
$803$	0	0
$804$	0	0
$805$	−1043.96	−1.29684
$806$	− 60.8697i	− 0.0755207i
$807$	0	0
$808$	−392.652	−0.485956
$809$	752.975i	0.930748i	0.885114	+	0.465374i	$0.154080\pi$
$809$	752.975i	0.930748i	−0.885114	+	0.465374i	$0.845920\pi$
$810$	0	0
$811$	−1002.83	−1.23654	−0.618270	−	0.785966i	$-0.712166\pi$
$811$	−1002.83	−1.23654	−0.618270	+	0.785966i	$0.712166\pi$
$812$	297.188i	0.365995i
$813$	0	0
$814$	0	0
$815$	− 628.663i	− 0.771366i
$816$	0	0
$817$	322.629	0.394895
$818$	921.743i	1.12683i
$819$	0	0
$820$	148.223	0.180759
$821$	1368.74i	1.66717i	0.552394	+	0.833583i	$0.313714\pi$
$821$	1368.74i	1.66717i	−0.552394	+	0.833583i	$0.686286\pi$
$822$	0	0
$823$	−1395.78	−1.69597	−0.847985	−	0.530019i	$-0.822185\pi$
$823$	−1395.78	−1.69597	−0.847985	+	0.530019i	$0.822185\pi$
$824$	− 929.746i	− 1.12833i
$825$	0	0
$826$	−1172.81	−1.41987
$827$	202.834i	0.245265i	0.992452	+	0.122632i	$0.0391336\pi$
$827$	202.834i	0.245265i	−0.992452	+	0.122632i	$0.960866\pi$
$828$	0	0
$829$	569.961	0.687528	0.343764	−	0.939056i	$-0.388298\pi$
$829$	569.961	0.687528	0.343764	+	0.939056i	$0.388298\pi$
$830$	− 279.341i	− 0.336556i
$831$	0	0
$832$	−135.406	−0.162748
$833$	− 2141.58i	− 2.57092i
$834$	0	0
$835$	−1169.02	−1.40003
$836$	0	0
$837$	0	0
$838$	372.068	0.443995
$839$	− 129.965i	− 0.154904i	−0.996996	−	0.0774522i	$-0.975321\pi$
$839$	− 129.965i	− 0.154904i	0.996996	−	0.0774522i	$-0.0246785\pi$
$840$	0	0
$841$	733.197	0.871816
$842$	− 164.155i	− 0.194959i
$843$	0	0
$844$	−163.132	−0.193285
$845$	826.294i	0.977863i
$846$	0	0
$847$	0	0
$848$	− 162.518i	− 0.191649i
$849$	0	0
$850$	24.6122	0.0289555
$851$	− 291.219i	− 0.342208i
$852$	0	0
$853$	−322.301	−0.377844	−0.188922	−	0.981992i	$-0.560499\pi$
$853$	−322.301	−0.377844	−0.188922	+	0.981992i	$0.560499\pi$
$854$	650.526i	0.761740i
$855$	0	0
$856$	−925.093	−1.08072
$857$	− 634.845i	− 0.740776i	−0.928877	−	0.370388i	$-0.879225\pi$
$857$	− 634.845i	− 0.740776i	0.928877	−	0.370388i	$-0.120775\pi$
$858$	0	0
$859$	1658.25	1.93044	0.965219	−	0.261443i	$-0.0841984\pi$
$859$	1658.25	1.93044	0.965219	+	0.261443i	$0.0841984\pi$
$860$	116.904i	0.135935i
$861$	0	0
$862$	133.446	0.154810
$863$	− 862.333i	− 0.999227i	−0.866249	−	0.499613i	$-0.833475\pi$
$863$	− 862.333i	− 0.999227i	0.866249	−	0.499613i	$-0.166525\pi$
$864$	0	0
$865$	881.467	1.01904
$866$	− 756.778i	− 0.873877i
$867$	0	0
$868$	482.809	0.556231
$869$	0	0
$870$	0	0
$871$	161.769	0.185728
$872$	987.311i	1.13224i
$873$	0	0
$874$	642.726	0.735385
$875$	− 1609.35i	− 1.83925i
$876$	0	0
$877$	−685.376	−0.781500	−0.390750	−	0.920497i	$-0.627784\pi$
$877$	−685.376	−0.781500	−0.390750	+	0.920497i	$0.627784\pi$
$878$	− 780.357i	− 0.888789i
$879$	0	0
$880$	0	0
$881$	551.003i	0.625429i	0.949847	+	0.312715i	$0.101238\pi$
$881$	551.003i	0.625429i	−0.949847	+	0.312715i	$0.898762\pi$
$882$	0	0
$883$	−1592.69	−1.80373	−0.901863	−	0.432022i	$-0.857800\pi$
$883$	−1592.69	−1.80373	−0.901863	+	0.432022i	$0.857800\pi$
$884$	− 99.9282i	− 0.113041i
$885$	0	0
$886$	−118.086	−0.133280
$887$	384.029i	0.432952i	0.976288	+	0.216476i	$0.0694564\pi$
$887$	384.029i	0.432952i	−0.976288	+	0.216476i	$0.930544\pi$
$888$	0	0
$889$	−141.882	−0.159597
$890$	490.697i	0.551345i
$891$	0	0
$892$	−12.9949	−0.0145683
$893$	− 2517.38i	− 2.81902i
$894$	0	0
$895$	931.566	1.04086
$896$	− 671.754i	− 0.749725i
$897$	0	0
$898$	−130.018	−0.144786
$899$	175.136i	0.194812i
$900$	0	0
$901$	1084.35	1.20350
$902$	0	0
$903$	0	0
$904$	−578.550	−0.639989
$905$	− 621.448i	− 0.686682i
$906$	0	0
$907$	−1417.57	−1.56292	−0.781462	−	0.623953i	$-0.785526\pi$
$907$	−1417.57	−1.56292	−0.781462	+	0.623953i	$0.785526\pi$
$908$	366.007i	0.403091i
$909$	0	0
$910$	−242.565	−0.266555
$911$	874.814i	0.960278i	0.877192	+	0.480139i	$0.159414\pi$
$911$	874.814i	0.960278i	−0.877192	+	0.480139i	$0.840586\pi$
$912$	0	0
$913$	0	0
$914$	− 934.888i	− 1.02285i
$915$	0	0
$916$	−765.548	−0.835751
$917$	− 1212.40i	− 1.32214i
$918$	0	0
$919$	1162.91	1.26541	0.632705	−	0.774393i	$-0.281945\pi$
$919$	1162.91	1.26541	0.632705	+	0.774393i	$0.281945\pi$
$920$	661.451i	0.718969i
$921$	0	0
$922$	−95.3407	−0.103406
$923$	131.349i	0.142306i
$924$	0	0
$925$	−19.8359	−0.0214443
$926$	− 335.544i	− 0.362358i
$927$	0	0
$928$	310.298	0.334373
$929$	554.284i	0.596646i	0.954465	+	0.298323i	$0.0964273\pi$
$929$	554.284i	0.596646i	−0.954465	+	0.298323i	$0.903573\pi$
$930$	0	0
$931$	−3809.27	−4.09159
$932$	774.036i	0.830511i
$933$	0	0
$934$	−343.471	−0.367742
$935$	0	0
$936$	0	0
$937$	857.873	0.915553	0.457776	−	0.889067i	$-0.348646\pi$
$937$	857.873	0.915553	0.457776	+	0.889067i	$0.348646\pi$
$938$	− 1078.06i	− 1.14932i
$939$	0	0
$940$	912.166	0.970390
$941$	337.080i	0.358214i	0.983830	+	0.179107i	$0.0573209\pi$
$941$	337.080i	0.358214i	−0.983830	+	0.179107i	$0.942679\pi$
$942$	0	0
$943$	207.464	0.220004
$944$	170.055i	0.180143i
$945$	0	0
$946$	0	0
$947$	405.388i	0.428076i	0.976825	+	0.214038i	$0.0686616\pi$
$947$	405.388i	0.428076i	−0.976825	+	0.214038i	$0.931338\pi$
$948$	0	0
$949$	−17.0178	−0.0179324
$950$	− 43.7783i	− 0.0460824i
$951$	0	0
$952$	−1891.40	−1.98677
$953$	747.947i	0.784834i	0.919787	+	0.392417i	$0.128361\pi$
$953$	747.947i	0.784834i	−0.919787	+	0.392417i	$0.871639\pi$
$954$	0	0
$955$	1605.00	1.68063
$956$	37.9057i	0.0396503i
$957$	0	0
$958$	24.7019	0.0257848
$959$	2615.82i	2.72765i
$960$	0	0
$961$	−676.476	−0.703929
$962$	− 67.6652i	− 0.0703380i
$963$	0	0
$964$	214.483	0.222493
$965$	634.975i	0.658005i
$966$	0	0
$967$	284.479	0.294187	0.147094	−	0.989123i	$-0.453008\pi$
$967$	284.479	0.294187	0.147094	+	0.989123i	$0.453008\pi$
$968$	0	0
$969$	0	0
$970$	−128.783	−0.132766
$971$	1179.42i	1.21464i	0.794456	+	0.607322i	$0.207756\pi$
$971$	1179.42i	1.21464i	−0.794456	+	0.607322i	$0.792244\pi$
$972$	0	0
$973$	783.952	0.805707
$974$	328.900i	0.337679i
$975$	0	0
$976$	94.3249	0.0966443
$977$	− 13.6463i	− 0.0139676i	−0.999976	−	0.00698379i	$-0.997777\pi$
$977$	− 13.6463i	− 0.0139676i	0.999976	−	0.00698379i	$-0.00222303\pi$
$978$	0	0
$979$	0	0
$980$	− 1380.28i	− 1.40845i
$981$	0	0
$982$	427.551	0.435388
$983$	501.274i	0.509943i	0.966949	+	0.254971i	$0.0820660\pi$
$983$	501.274i	0.509943i	−0.966949	+	0.254971i	$0.917934\pi$
$984$	0	0
$985$	990.276	1.00536
$986$	− 241.566i	− 0.244996i
$987$	0	0
$988$	−177.745	−0.179903
$989$	163.628i	0.165448i
$990$	0	0
$991$	118.253	0.119327	0.0596635	−	0.998219i	$-0.480997\pi$
$991$	118.253	0.119327	0.0596635	+	0.998219i	$0.480997\pi$
$992$	− 504.108i	− 0.508173i
$993$	0	0
$994$	875.335	0.880619
$995$	345.554i	0.347290i
$996$	0	0
$997$	−495.885	−0.497377	−0.248689	−	0.968583i	$-0.580000\pi$
$997$	−495.885	−0.497377	−0.248689	+	0.968583i	$0.580000\pi$
$998$	− 36.2228i	− 0.0362954i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.j.485.6		16
3.2	odd	2	inner	1089.3.b.j.485.11		16
11.7	odd	10		99.3.l.a.71.3	yes	32
11.8	odd	10		99.3.l.a.53.6	yes	32
11.10	odd	2		1089.3.b.i.485.11		16
33.8	even	10		99.3.l.a.53.3	✓	32
33.29	even	10		99.3.l.a.71.6	yes	32
33.32	even	2		1089.3.b.i.485.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.l.a.53.3	✓	32	33.8	even	10
99.3.l.a.53.6	yes	32	11.8	odd	10
99.3.l.a.71.3	yes	32	11.7	odd	10
99.3.l.a.71.6	yes	32	33.29	even	10
1089.3.b.i.485.6		16	33.32	even	2
1089.3.b.i.485.11		16	11.10	odd	2
1089.3.b.j.485.6		16	1.1	even	1	trivial
1089.3.b.j.485.11		16	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.35141i q^{2} +2.17369 q^{4} -5.10469i q^{5} +13.1679 q^{7} -8.34318i q^{8} +O(q^{10})\)	\(q-1.35141i q^{2} +2.17369 q^{4} -5.10469i q^{5} +13.1679 q^{7} -8.34318i q^{8} -6.89852 q^{10} +2.67027 q^{13} -17.7952i q^{14} -2.58027 q^{16} -17.2161i q^{17} -30.6226 q^{19} -11.0960i q^{20} -15.5309i q^{23} -1.05786 q^{25} -3.60862i q^{26} +28.6230 q^{28} +10.3828i q^{29} +16.8678 q^{31} -29.8857i q^{32} -23.2660 q^{34} -67.2181i q^{35} +18.7510 q^{37} +41.3837i q^{38} -42.5894 q^{40} +13.3582i q^{41} -10.5356 q^{43} -20.9886 q^{46} +82.2065i q^{47} +124.394 q^{49} +1.42960i q^{50} +5.80435 q^{52} +62.9849i q^{53} -109.862i q^{56} +14.0314 q^{58} -65.9058i q^{59} -36.5562 q^{61} -22.7954i q^{62} -50.7089 q^{64} -13.6309i q^{65} +60.5815 q^{67} -37.4225i q^{68} -90.8392 q^{70} +49.1893i q^{71} -6.37308 q^{73} -25.3402i q^{74} -66.5643 q^{76} -115.053 q^{79} +13.1715i q^{80} +18.0523 q^{82} +40.4929i q^{83} -87.8828 q^{85} +14.2379i q^{86} -71.1308i q^{89} +35.1619 q^{91} -33.7594i q^{92} +111.095 q^{94} +156.319i q^{95} +18.6682 q^{97} -168.107i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 8 q^{7}+O(q^{10}) \) 16 * q - 32 * q^4 + 8 * q^7	\( 16 q - 32 q^{4} + 8 q^{7} - 24 q^{10} - 4 q^{13} + 28 q^{16} + 20 q^{19} - 44 q^{25} - 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} - 224 q^{40} - 272 q^{43} - 208 q^{46} + 348 q^{49} - 520 q^{52} - 44 q^{58} - 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} + 4 q^{73} - 1052 q^{76} - 216 q^{79} + 348 q^{82} - 416 q^{85} - 168 q^{91} - 1140 q^{94} - 44 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 8 * q^7 - 24 * q^10 - 4 * q^13 + 28 * q^16 + 20 * q^19 - 44 * q^25 - 16 * q^28 + 28 * q^31 + 148 * q^34 - 148 * q^37 - 224 * q^40 - 272 * q^43 - 208 * q^46 + 348 * q^49 - 520 * q^52 - 44 * q^58 - 224 * q^61 + 436 * q^64 + 24 * q^67 - 664 * q^70 + 4 * q^73 - 1052 * q^76 - 216 * q^79 + 348 * q^82 - 416 * q^85 - 168 * q^91 - 1140 * q^94 - 44 * q^97

Introduction

L-functions

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