LMFDB - Embedded newform 1089.3.b.i.485.8

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			485.8
Root			$-0.311356i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.i.485.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.311356i q^{2} +3.90306 q^{4} -5.94641i q^{5} +8.67101 q^{7} -2.46067i q^{8} +O(q^{10})$	$q-0.311356i q^{2} +3.90306 q^{4} -5.94641i q^{5} +8.67101 q^{7} -2.46067i q^{8} -1.85145 q^{10} +18.6982 q^{13} -2.69977i q^{14} +14.8461 q^{16} -9.59948i q^{17} +5.76332 q^{19} -23.2092i q^{20} +41.5830i q^{23} -10.3598 q^{25} -5.82181i q^{26} +33.8434 q^{28} +17.3733i q^{29} -13.6771 q^{31} -14.4651i q^{32} -2.98886 q^{34} -51.5613i q^{35} -7.21391 q^{37} -1.79445i q^{38} -14.6321 q^{40} -53.1361i q^{41} +43.3682 q^{43} +12.9471 q^{46} +18.8439i q^{47} +26.1864 q^{49} +3.22558i q^{50} +72.9802 q^{52} +54.5516i q^{53} -21.3365i q^{56} +5.40928 q^{58} +35.3312i q^{59} -117.852 q^{61} +4.25847i q^{62} +54.8805 q^{64} -111.187i q^{65} -91.5111 q^{67} -37.4673i q^{68} -16.0540 q^{70} -115.816i q^{71} -52.9453 q^{73} +2.24610i q^{74} +22.4946 q^{76} +2.04912 q^{79} -88.2809i q^{80} -16.5443 q^{82} +28.8194i q^{83} -57.0824 q^{85} -13.5030i q^{86} +134.980i q^{89} +162.132 q^{91} +162.301i q^{92} +5.86717 q^{94} -34.2710i q^{95} -9.97627 q^{97} -8.15330i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 0.311356i	− 0.155678i	−0.996966	−	0.0778391i	$-0.975198\pi$
$2$	− 0.311356i	− 0.155678i	0.996966	−	0.0778391i	$-0.0248020\pi$
$3$	0	0
$3$	0	0
$4$	3.90306	0.975764
$5$	− 5.94641i	− 1.18928i	−0.803992	−	0.594641i	$-0.797294\pi$
$5$	− 5.94641i	− 1.18928i	0.803992	−	0.594641i	$-0.202706\pi$
$6$	0	0
$7$	8.67101	1.23872	0.619358	−	0.785109i	$-0.287393\pi$
$7$	8.67101	1.23872	0.619358	+	0.785109i	$0.287393\pi$
$8$	− 2.46067i	− 0.307583i
$9$	0	0
$10$	−1.85145	−0.185145
$11$	0	0
$11$	0	0
$12$	0	0
$13$	18.6982	1.43832	0.719162	−	0.694842i	$-0.244526\pi$
$13$	18.6982	1.43832	0.719162	+	0.694842i	$0.244526\pi$
$14$	− 2.69977i	− 0.192841i
$15$	0	0
$16$	14.8461	0.927880
$17$	− 9.59948i	− 0.564675i	−0.959315	−	0.282338i	$-0.908890\pi$
$17$	− 9.59948i	− 0.564675i	0.959315	−	0.282338i	$-0.0911098\pi$
$18$	0	0
$19$	5.76332	0.303333	0.151666	−	0.988432i	$-0.451536\pi$
$19$	5.76332	0.303333	0.151666	+	0.988432i	$0.451536\pi$
$20$	− 23.2092i	− 1.16046i
$21$	0	0
$22$	0	0
$23$	41.5830i	1.80796i	0.427578	+	0.903979i	$0.359367\pi$
$23$	41.5830i	1.80796i	−0.427578	+	0.903979i	$0.640633\pi$
$24$	0	0
$25$	−10.3598	−0.414390
$26$	− 5.82181i	− 0.223916i
$27$	0	0
$28$	33.8434	1.20869
$29$	17.3733i	0.599078i	0.954084	+	0.299539i	$0.0968329\pi$
$29$	17.3733i	0.599078i	−0.954084	+	0.299539i	$0.903167\pi$
$30$	0	0
$31$	−13.6771	−0.441198	−0.220599	−	0.975365i	$-0.570801\pi$
$31$	−13.6771	−0.441198	−0.220599	+	0.975365i	$0.570801\pi$
$32$	− 14.4651i	− 0.452034i
$33$	0	0
$34$	−2.98886	−0.0879076
$35$	− 51.5613i	− 1.47318i
$36$	0	0
$37$	−7.21391	−0.194970	−0.0974852	−	0.995237i	$-0.531080\pi$
$37$	−7.21391	−0.194970	−0.0974852	+	0.995237i	$0.531080\pi$
$38$	− 1.79445i	− 0.0472223i
$39$	0	0
$40$	−14.6321	−0.365803
$41$	− 53.1361i	− 1.29600i	−0.761640	−	0.648001i	$-0.775605\pi$
$41$	− 53.1361i	− 1.29600i	0.761640	−	0.648001i	$-0.224395\pi$
$42$	0	0
$43$	43.3682	1.00856	0.504282	−	0.863539i	$-0.331757\pi$
$43$	43.3682	1.00856	0.504282	+	0.863539i	$0.331757\pi$
$44$	0	0
$45$	0	0
$46$	12.9471	0.281459
$47$	18.8439i	0.400934i	0.979700	+	0.200467i	$0.0642460\pi$
$47$	18.8439i	0.400934i	−0.979700	+	0.200467i	$0.935754\pi$
$48$	0	0
$49$	26.1864	0.534416
$50$	3.22558i	0.0645115i
$51$	0	0
$52$	72.9802	1.40347
$53$	54.5516i	1.02928i	0.857408	+	0.514638i	$0.172074\pi$
$53$	54.5516i	1.02928i	−0.857408	+	0.514638i	$0.827926\pi$
$54$	0	0
$55$	0	0
$56$	− 21.3365i	− 0.381008i
$57$	0	0
$58$	5.40928	0.0932634
$59$	35.3312i	0.598833i	0.954122	+	0.299417i	$0.0967920\pi$
$59$	35.3312i	0.598833i	−0.954122	+	0.299417i	$0.903208\pi$
$60$	0	0
$61$	−117.852	−1.93201	−0.966004	−	0.258528i	$-0.916762\pi$
$61$	−117.852	−1.93201	−0.966004	+	0.258528i	$0.916762\pi$
$62$	4.25847i	0.0686849i
$63$	0	0
$64$	54.8805	0.857508
$65$	− 111.187i	− 1.71057i
$66$	0	0
$67$	−91.5111	−1.36584	−0.682919	−	0.730494i	$-0.739290\pi$
$67$	−91.5111	−1.36584	−0.682919	+	0.730494i	$0.739290\pi$
$68$	− 37.4673i	− 0.550990i
$69$	0	0
$70$	−16.0540	−0.229342
$71$	− 115.816i	− 1.63120i	−0.578613	−	0.815602i	$-0.696406\pi$
$71$	− 115.816i	− 1.63120i	0.578613	−	0.815602i	$-0.303594\pi$
$72$	0	0
$73$	−52.9453	−0.725278	−0.362639	−	0.931930i	$-0.618124\pi$
$73$	−52.9453	−0.725278	−0.362639	+	0.931930i	$0.618124\pi$
$74$	2.24610i	0.0303526i
$75$	0	0
$76$	22.4946	0.295981
$77$	0	0
$78$	0	0
$79$	2.04912	0.0259383	0.0129691	−	0.999916i	$-0.495872\pi$
$79$	2.04912	0.0259383	0.0129691	+	0.999916i	$0.495872\pi$
$80$	− 88.2809i	− 1.10351i
$81$	0	0
$82$	−16.5443	−0.201759
$83$	28.8194i	0.347222i	0.984814	+	0.173611i	$0.0555435\pi$
$83$	28.8194i	0.347222i	−0.984814	+	0.173611i	$0.944457\pi$
$84$	0	0
$85$	−57.0824	−0.671558
$86$	− 13.5030i	− 0.157011i
$87$	0	0
$88$	0	0
$89$	134.980i	1.51663i	0.651891	+	0.758313i	$0.273976\pi$
$89$	134.980i	1.51663i	−0.651891	+	0.758313i	$0.726024\pi$
$90$	0	0
$91$	162.132	1.78168
$92$	162.301i	1.76414i
$93$	0	0
$94$	5.86717	0.0624168
$95$	− 34.2710i	− 0.360748i
$96$	0	0
$97$	−9.97627	−0.102848	−0.0514241	−	0.998677i	$-0.516376\pi$
$97$	−9.97627	−0.102848	−0.0514241	+	0.998677i	$0.516376\pi$
$98$	− 8.15330i	− 0.0831970i
$99$	0	0
$100$	−40.4347	−0.404347
$101$	− 172.426i	− 1.70719i	−0.520939	−	0.853594i	$-0.674418\pi$
$101$	− 172.426i	− 1.70719i	0.520939	−	0.853594i	$-0.325582\pi$
$102$	0	0
$103$	−113.639	−1.10329	−0.551643	−	0.834080i	$-0.685999\pi$
$103$	−113.639	−1.10329	−0.551643	+	0.834080i	$0.685999\pi$
$104$	− 46.0101i	− 0.442405i
$105$	0	0
$106$	16.9850	0.160236
$107$	83.1724i	0.777312i	0.921383	+	0.388656i	$0.127061\pi$
$107$	83.1724i	0.777312i	−0.921383	+	0.388656i	$0.872939\pi$
$108$	0	0
$109$	125.432	1.15075	0.575377	−	0.817888i	$-0.304855\pi$
$109$	125.432	1.15075	0.575377	+	0.817888i	$0.304855\pi$
$110$	0	0
$111$	0	0
$112$	128.731	1.14938
$113$	− 101.186i	− 0.895451i	−0.894171	−	0.447726i	$-0.852234\pi$
$113$	− 101.186i	− 0.895451i	0.894171	−	0.447726i	$-0.147766\pi$
$114$	0	0
$115$	247.270	2.15017
$116$	67.8089i	0.584559i
$117$	0	0
$118$	11.0006	0.0932252
$119$	− 83.2372i	− 0.699472i
$120$	0	0
$121$	0	0
$122$	36.6941i	0.300771i
$123$	0	0
$124$	−53.3827	−0.430505
$125$	− 87.0569i	− 0.696455i
$126$	0	0
$127$	123.996	0.976346	0.488173	−	0.872747i	$-0.337663\pi$
$127$	123.996	0.976346	0.488173	+	0.872747i	$0.337663\pi$
$128$	− 74.9478i	− 0.585529i
$129$	0	0
$130$	−34.6189	−0.266299
$131$	− 52.5607i	− 0.401227i	−0.979670	−	0.200614i	$-0.935706\pi$
$131$	− 52.5607i	− 0.401227i	0.979670	−	0.200614i	$-0.0642935\pi$
$132$	0	0
$133$	49.9738	0.375743
$134$	28.4926i	0.212631i
$135$	0	0
$136$	−23.6211	−0.173685
$137$	− 125.783i	− 0.918121i	−0.888405	−	0.459060i	$-0.848186\pi$
$137$	− 125.783i	− 0.918121i	0.888405	−	0.459060i	$-0.151814\pi$
$138$	0	0
$139$	53.6460	0.385943	0.192971	−	0.981204i	$-0.438188\pi$
$139$	53.6460	0.385943	0.192971	+	0.981204i	$0.438188\pi$
$140$	− 201.247i	− 1.43748i
$141$	0	0
$142$	−36.0599	−0.253943
$143$	0	0
$144$	0	0
$145$	103.309	0.712473
$146$	16.4848i	0.112910i
$147$	0	0
$148$	−28.1563	−0.190245
$149$	4.23732i	0.0284384i	0.999899	+	0.0142192i	$0.00452626\pi$
$149$	4.23732i	0.0284384i	−0.999899	+	0.0142192i	$0.995474\pi$
$150$	0	0
$151$	94.3874	0.625082	0.312541	−	0.949904i	$-0.398820\pi$
$151$	94.3874	0.625082	0.312541	+	0.949904i	$0.398820\pi$
$152$	− 14.1816i	− 0.0933001i
$153$	0	0
$154$	0	0
$155$	81.3299i	0.524709i
$156$	0	0
$157$	−148.683	−0.947028	−0.473514	−	0.880786i	$-0.657015\pi$
$157$	−148.683	−0.947028	−0.473514	+	0.880786i	$0.657015\pi$
$158$	− 0.638008i	− 0.00403802i
$159$	0	0
$160$	−86.0153	−0.537596
$161$	360.567i	2.23954i
$162$	0	0
$163$	−91.7512	−0.562891	−0.281445	−	0.959577i	$-0.590814\pi$
$163$	−91.7512	−0.562891	−0.281445	+	0.959577i	$0.590814\pi$
$164$	− 207.393i	− 1.26459i
$165$	0	0
$166$	8.97310	0.0540548
$167$	− 14.2677i	− 0.0854351i	−0.999087	−	0.0427175i	$-0.986398\pi$
$167$	− 14.2677i	− 0.0854351i	0.999087	−	0.0427175i	$-0.0136016\pi$
$168$	0	0
$169$	180.623	1.06878
$170$	17.7730i	0.104547i
$171$	0	0
$172$	169.269	0.984120
$173$	− 232.280i	− 1.34266i	−0.741158	−	0.671331i	$-0.765723\pi$
$173$	− 232.280i	− 1.34266i	0.741158	−	0.671331i	$-0.234277\pi$
$174$	0	0
$175$	−89.8295	−0.513312
$176$	0	0
$177$	0	0
$178$	42.0268	0.236105
$179$	143.658i	0.802560i	0.915956	+	0.401280i	$0.131434\pi$
$179$	143.658i	0.802560i	−0.915956	+	0.401280i	$0.868566\pi$
$180$	0	0
$181$	−239.706	−1.32434	−0.662171	−	0.749353i	$-0.730365\pi$
$181$	−239.706	−1.32434	−0.662171	+	0.749353i	$0.730365\pi$
$182$	− 50.4810i	− 0.277368i
$183$	0	0
$184$	102.322	0.556098
$185$	42.8968i	0.231875i
$186$	0	0
$187$	0	0
$188$	73.5489i	0.391218i
$189$	0	0
$190$	−10.6705	−0.0561606
$191$	− 37.5317i	− 0.196501i	−0.995162	−	0.0982506i	$-0.968675\pi$
$191$	− 37.5317i	− 0.196501i	0.995162	−	0.0982506i	$-0.0313247\pi$
$192$	0	0
$193$	203.572	1.05478	0.527389	−	0.849624i	$-0.323171\pi$
$193$	203.572	1.05478	0.527389	+	0.849624i	$0.323171\pi$
$194$	3.10617i	0.0160112i
$195$	0	0
$196$	102.207	0.521464
$197$	208.477i	1.05826i	0.848541	+	0.529129i	$0.177481\pi$
$197$	208.477i	1.05826i	−0.848541	+	0.529129i	$0.822519\pi$
$198$	0	0
$199$	−249.874	−1.25565	−0.627825	−	0.778355i	$-0.716055\pi$
$199$	−249.874	−1.25565	−0.627825	+	0.778355i	$0.716055\pi$
$200$	25.4919i	0.127460i
$201$	0	0
$202$	−53.6859	−0.265772
$203$	150.644i	0.742088i
$204$	0	0
$205$	−315.969	−1.54131
$206$	35.3821i	0.171758i
$207$	0	0
$208$	277.595	1.33459
$209$	0	0
$210$	0	0
$211$	121.553	0.576081	0.288040	−	0.957618i	$-0.406996\pi$
$211$	121.553	0.576081	0.288040	+	0.957618i	$0.406996\pi$
$212$	212.918i	1.00433i
$213$	0	0
$214$	25.8963	0.121011
$215$	− 257.885i	− 1.19947i
$216$	0	0
$217$	−118.595	−0.546519
$218$	− 39.0541i	− 0.179147i
$219$	0	0
$220$	0	0
$221$	− 179.493i	− 0.812186i
$222$	0	0
$223$	−71.8121	−0.322027	−0.161014	−	0.986952i	$-0.551476\pi$
$223$	−71.8121	−0.322027	−0.161014	+	0.986952i	$0.551476\pi$
$224$	− 125.427i	− 0.559942i
$225$	0	0
$226$	−31.5049	−0.139402
$227$	− 234.045i	− 1.03104i	−0.856879	−	0.515518i	$-0.827600\pi$
$227$	− 234.045i	− 1.03104i	0.856879	−	0.515518i	$-0.172400\pi$
$228$	0	0
$229$	68.1310	0.297515	0.148758	−	0.988874i	$-0.452473\pi$
$229$	68.1310	0.297515	0.148758	+	0.988874i	$0.452473\pi$
$230$	− 76.9889i	− 0.334735i
$231$	0	0
$232$	42.7498	0.184267
$233$	− 106.735i	− 0.458090i	−0.973416	−	0.229045i	$-0.926440\pi$
$233$	− 106.735i	− 0.458090i	0.973416	−	0.229045i	$-0.0735603\pi$
$234$	0	0
$235$	112.054	0.476824
$236$	137.900i	0.584320i
$237$	0	0
$238$	−25.9164	−0.108893
$239$	348.588i	1.45853i	0.684232	+	0.729265i	$0.260138\pi$
$239$	348.588i	1.45853i	−0.684232	+	0.729265i	$0.739862\pi$
$240$	0	0
$241$	−377.429	−1.56609	−0.783047	−	0.621962i	$-0.786336\pi$
$241$	−377.429	−1.56609	−0.783047	+	0.621962i	$0.786336\pi$
$242$	0	0
$243$	0	0
$244$	−459.985	−1.88518
$245$	− 155.715i	− 0.635571i
$246$	0	0
$247$	107.764	0.436291
$248$	33.6549i	0.135705i
$249$	0	0
$250$	−27.1057	−0.108423
$251$	301.660i	1.20183i	0.799311	+	0.600917i	$0.205198\pi$
$251$	301.660i	1.20183i	−0.799311	+	0.600917i	$0.794802\pi$
$252$	0	0
$253$	0	0
$254$	− 38.6069i	− 0.151996i
$255$	0	0
$256$	196.187	0.766354
$257$	− 14.9765i	− 0.0582743i	−0.999575	−	0.0291372i	$-0.990724\pi$
$257$	− 14.9765i	− 0.0582743i	0.999575	−	0.0291372i	$-0.00927596\pi$
$258$	0	0
$259$	−62.5519	−0.241513
$260$	− 433.970i	− 1.66912i
$261$	0	0
$262$	−16.3651	−0.0624623
$263$	470.671i	1.78962i	0.446443	+	0.894812i	$0.352691\pi$
$263$	470.671i	1.78962i	−0.446443	+	0.894812i	$0.647309\pi$
$264$	0	0
$265$	324.386	1.22410
$266$	− 15.5597i	− 0.0584950i
$267$	0	0
$268$	−357.173	−1.33274
$269$	− 199.826i	− 0.742848i	−0.928463	−	0.371424i	$-0.878870\pi$
$269$	− 199.826i	− 0.742848i	0.928463	−	0.371424i	$-0.121130\pi$
$270$	0	0
$271$	−87.2697	−0.322028	−0.161014	−	0.986952i	$-0.551476\pi$
$271$	−87.2697	−0.322028	−0.161014	+	0.986952i	$0.551476\pi$
$272$	− 142.515i	− 0.523951i
$273$	0	0
$274$	−39.1632	−0.142931
$275$	0	0
$276$	0	0
$277$	−299.219	−1.08021	−0.540106	−	0.841597i	$-0.681616\pi$
$277$	−299.219	−1.08021	−0.540106	+	0.841597i	$0.681616\pi$
$278$	− 16.7030i	− 0.0600828i
$279$	0	0
$280$	−126.875	−0.453126
$281$	298.705i	1.06301i	0.847056	+	0.531504i	$0.178373\pi$
$281$	298.705i	1.06301i	−0.847056	+	0.531504i	$0.821627\pi$
$282$	0	0
$283$	−204.517	−0.722674	−0.361337	−	0.932435i	$-0.617680\pi$
$283$	−204.517	−0.722674	−0.361337	+	0.932435i	$0.617680\pi$
$284$	− 452.035i	− 1.59167i
$285$	0	0
$286$	0	0
$287$	− 460.743i	− 1.60538i
$288$	0	0
$289$	196.850	0.681142
$290$	− 32.1658i	− 0.110916i
$291$	0	0
$292$	−206.648	−0.707700
$293$	102.843i	0.351001i	0.984479	+	0.175500i	$0.0561543\pi$
$293$	102.843i	0.351001i	−0.984479	+	0.175500i	$0.943846\pi$
$294$	0	0
$295$	210.093	0.712181
$296$	17.7510i	0.0599697i
$297$	0	0
$298$	1.31932	0.00442724
$299$	777.528i	2.60043i
$300$	0	0
$301$	376.046	1.24932
$302$	− 29.3881i	− 0.0973116i
$303$	0	0
$304$	85.5627	0.281456
$305$	700.799i	2.29770i
$306$	0	0
$307$	39.5643	0.128874	0.0644369	−	0.997922i	$-0.479475\pi$
$307$	39.5643	0.128874	0.0644369	+	0.997922i	$0.479475\pi$
$308$	0	0
$309$	0	0
$310$	25.3226	0.0816857
$311$	− 139.455i	− 0.448407i	−0.974542	−	0.224203i	$-0.928022\pi$
$311$	− 139.455i	− 0.448407i	0.974542	−	0.224203i	$-0.0719780\pi$
$312$	0	0
$313$	−226.078	−0.722295	−0.361148	−	0.932509i	$-0.617615\pi$
$313$	−226.078	−0.722295	−0.361148	+	0.932509i	$0.617615\pi$
$314$	46.2935i	0.147432i
$315$	0	0
$316$	7.99784	0.0253096
$317$	352.182i	1.11098i	0.831522	+	0.555492i	$0.187470\pi$
$317$	352.182i	1.11098i	−0.831522	+	0.555492i	$0.812530\pi$
$318$	0	0
$319$	0	0
$320$	− 326.342i	− 1.01982i
$321$	0	0
$322$	112.265	0.348648
$323$	− 55.3249i	− 0.171284i
$324$	0	0
$325$	−193.709	−0.596028
$326$	28.5673i	0.0876298i
$327$	0	0
$328$	−130.750	−0.398629
$329$	163.396i	0.496644i
$330$	0	0
$331$	−84.8580	−0.256369	−0.128184	−	0.991750i	$-0.540915\pi$
$331$	−84.8580	−0.256369	−0.128184	+	0.991750i	$0.540915\pi$
$332$	112.484i	0.338806i
$333$	0	0
$334$	−4.44233	−0.0133004
$335$	544.162i	1.62437i
$336$	0	0
$337$	527.174	1.56432	0.782158	−	0.623080i	$-0.214119\pi$
$337$	527.174	1.56432	0.782158	+	0.623080i	$0.214119\pi$
$338$	− 56.2383i	− 0.166385i
$339$	0	0
$340$	−222.796	−0.655282
$341$	0	0
$342$	0	0
$343$	−197.817	−0.576726
$344$	− 106.715i	− 0.310217i
$345$	0	0
$346$	−72.3220	−0.209023
$347$	− 126.318i	− 0.364029i	−0.983296	−	0.182015i	$-0.941738\pi$
$347$	− 126.318i	− 0.364029i	0.983296	−	0.182015i	$-0.0582618\pi$
$348$	0	0
$349$	−451.455	−1.29357	−0.646784	−	0.762673i	$-0.723887\pi$
$349$	−451.455	−1.29357	−0.646784	+	0.762673i	$0.723887\pi$
$350$	27.9690i	0.0799114i
$351$	0	0
$352$	0	0
$353$	145.710i	0.412775i	0.978470	+	0.206388i	$0.0661708\pi$
$353$	145.710i	0.412775i	−0.978470	+	0.206388i	$0.933829\pi$
$354$	0	0
$355$	−688.686	−1.93996
$356$	526.833i	1.47987i
$357$	0	0
$358$	44.7289	0.124941
$359$	429.208i	1.19557i	0.801658	+	0.597783i	$0.203952\pi$
$359$	429.208i	1.19557i	−0.801658	+	0.597783i	$0.796048\pi$
$360$	0	0
$361$	−327.784	−0.907989
$362$	74.6339i	0.206171i
$363$	0	0
$364$	632.812	1.73850
$365$	314.834i	0.862559i
$366$	0	0
$367$	132.361	0.360658	0.180329	−	0.983606i	$-0.442284\pi$
$367$	132.361	0.360658	0.180329	+	0.983606i	$0.442284\pi$
$368$	617.345i	1.67757i
$369$	0	0
$370$	13.3562	0.0360978
$371$	473.017i	1.27498i
$372$	0	0
$373$	−478.180	−1.28198	−0.640992	−	0.767547i	$-0.721477\pi$
$373$	−478.180	−1.28198	−0.640992	+	0.767547i	$0.721477\pi$
$374$	0	0
$375$	0	0
$376$	46.3686	0.123321
$377$	324.849i	0.861669i
$378$	0	0
$379$	−264.206	−0.697113	−0.348557	−	0.937288i	$-0.613328\pi$
$379$	−264.206	−0.697113	−0.348557	+	0.937288i	$0.613328\pi$
$380$	− 133.762i	− 0.352005i
$381$	0	0
$382$	−11.6857	−0.0305909
$383$	236.431i	0.617313i	0.951174	+	0.308657i	$0.0998794\pi$
$383$	236.431i	0.617313i	−0.951174	+	0.308657i	$0.900121\pi$
$384$	0	0
$385$	0	0
$386$	− 63.3834i	− 0.164206i
$387$	0	0
$388$	−38.9379	−0.100356
$389$	− 511.217i	− 1.31418i	−0.753811	−	0.657091i	$-0.771787\pi$
$389$	− 511.217i	− 1.31418i	0.753811	−	0.657091i	$-0.228213\pi$
$390$	0	0
$391$	399.175	1.02091
$392$	− 64.4360i	− 0.164378i
$393$	0	0
$394$	64.9106	0.164748
$395$	− 12.1849i	− 0.0308479i
$396$	0	0
$397$	389.224	0.980413	0.490207	−	0.871606i	$-0.336921\pi$
$397$	389.224	0.980413	0.490207	+	0.871606i	$0.336921\pi$
$398$	77.7999i	0.195477i
$399$	0	0
$400$	−153.802	−0.384504
$401$	226.050i	0.563715i	0.959456	+	0.281858i	$0.0909506\pi$
$401$	226.050i	0.563715i	−0.959456	+	0.281858i	$0.909049\pi$
$402$	0	0
$403$	−255.738	−0.634586
$404$	− 672.988i	− 1.66581i
$405$	0	0
$406$	46.9039	0.115527
$407$	0	0
$408$	0	0
$409$	−181.766	−0.444415	−0.222208	−	0.974999i	$-0.571326\pi$
$409$	−181.766	−0.444415	−0.222208	+	0.974999i	$0.571326\pi$
$410$	98.3789i	0.239948i
$411$	0	0
$412$	−443.538	−1.07655
$413$	306.357i	0.741784i
$414$	0	0
$415$	171.372	0.412944
$416$	− 270.471i	− 0.650172i
$417$	0	0
$418$	0	0
$419$	171.909i	0.410284i	0.978732	+	0.205142i	$0.0657656\pi$
$419$	171.909i	0.410284i	−0.978732	+	0.205142i	$0.934234\pi$
$420$	0	0
$421$	−120.846	−0.287046	−0.143523	−	0.989647i	$-0.545843\pi$
$421$	−120.846	−0.287046	−0.143523	+	0.989647i	$0.545843\pi$
$422$	− 37.8463i	− 0.0896832i
$423$	0	0
$424$	134.233	0.316588
$425$	99.4482i	0.233996i
$426$	0	0
$427$	−1021.90	−2.39321
$428$	324.627i	0.758473i
$429$	0	0
$430$	−80.2942	−0.186731
$431$	474.296i	1.10045i	0.835015	+	0.550227i	$0.185459\pi$
$431$	474.296i	1.10045i	−0.835015	+	0.550227i	$0.814541\pi$
$432$	0	0
$433$	254.226	0.587127	0.293564	−	0.955940i	$-0.405159\pi$
$433$	254.226	0.587127	0.293564	+	0.955940i	$0.405159\pi$
$434$	36.9252i	0.0850811i
$435$	0	0
$436$	489.569	1.12286
$437$	239.656i	0.548412i
$438$	0	0
$439$	515.246	1.17368	0.586841	−	0.809703i	$-0.300372\pi$
$439$	515.246	1.17368	0.586841	+	0.809703i	$0.300372\pi$
$440$	0	0
$441$	0	0
$442$	−55.8863	−0.126440
$443$	526.041i	1.18745i	0.804668	+	0.593726i	$0.202344\pi$
$443$	526.041i	1.18745i	−0.804668	+	0.593726i	$0.797656\pi$
$444$	0	0
$445$	802.644	1.80369
$446$	22.3592i	0.0501326i
$447$	0	0
$448$	475.870	1.06221
$449$	− 541.858i	− 1.20681i	−0.797435	−	0.603405i	$-0.793810\pi$
$449$	− 541.858i	− 1.20681i	0.797435	−	0.603405i	$-0.206190\pi$
$450$	0	0
$451$	0	0
$452$	− 394.935i	− 0.873749i
$453$	0	0
$454$	−72.8714	−0.160510
$455$	− 964.106i	− 2.11891i
$456$	0	0
$457$	122.430	0.267899	0.133950	−	0.990988i	$-0.457234\pi$
$457$	122.430	0.267899	0.133950	+	0.990988i	$0.457234\pi$
$458$	− 21.2130i	− 0.0463166i
$459$	0	0
$460$	965.107	2.09806
$461$	308.300i	0.668764i	0.942438	+	0.334382i	$0.108528\pi$
$461$	308.300i	0.668764i	−0.942438	+	0.334382i	$0.891472\pi$
$462$	0	0
$463$	369.654	0.798389	0.399195	−	0.916866i	$-0.369290\pi$
$463$	369.654	0.798389	0.399195	+	0.916866i	$0.369290\pi$
$464$	257.925i	0.555873i
$465$	0	0
$466$	−33.2326	−0.0713146
$467$	108.850i	0.233084i	0.993186	+	0.116542i	$0.0371809\pi$
$467$	108.850i	0.233084i	−0.993186	+	0.116542i	$0.962819\pi$
$468$	0	0
$469$	−793.494	−1.69188
$470$	− 34.8886i	− 0.0742311i
$471$	0	0
$472$	86.9382	0.184191
$473$	0	0
$474$	0	0
$475$	−59.7066	−0.125698
$476$	− 324.879i	− 0.682520i
$477$	0	0
$478$	108.535	0.227061
$479$	− 302.678i	− 0.631897i	−0.948776	−	0.315948i	$-0.897677\pi$
$479$	− 302.678i	− 0.631897i	0.948776	−	0.315948i	$-0.102323\pi$
$480$	0	0
$481$	−134.887	−0.280431
$482$	117.515i	0.243807i
$483$	0	0
$484$	0	0
$485$	59.3229i	0.122315i
$486$	0	0
$487$	464.193	0.953169	0.476584	−	0.879129i	$-0.341875\pi$
$487$	464.193	0.953169	0.476584	+	0.879129i	$0.341875\pi$
$488$	289.996i	0.594253i
$489$	0	0
$490$	−48.4829	−0.0989446
$491$	− 347.094i	− 0.706913i	−0.935451	−	0.353456i	$-0.885006\pi$
$491$	− 347.094i	− 0.706913i	0.935451	−	0.353456i	$-0.114994\pi$
$492$	0	0
$493$	166.774	0.338285
$494$	− 33.5530i	− 0.0679210i
$495$	0	0
$496$	−203.052	−0.409379
$497$	− 1004.24i	− 2.02060i
$498$	0	0
$499$	454.952	0.911728	0.455864	−	0.890049i	$-0.349330\pi$
$499$	454.952	0.911728	0.455864	+	0.890049i	$0.349330\pi$
$500$	− 339.788i	− 0.679576i
$501$	0	0
$502$	93.9239	0.187099
$503$	− 414.477i	− 0.824010i	−0.911182	−	0.412005i	$-0.864829\pi$
$503$	− 414.477i	− 0.824010i	0.911182	−	0.412005i	$-0.135171\pi$
$504$	0	0
$505$	−1025.31	−2.03033
$506$	0	0
$507$	0	0
$508$	483.963	0.952684
$509$	150.010i	0.294714i	0.989083	+	0.147357i	$0.0470767\pi$
$509$	150.010i	0.294714i	−0.989083	+	0.147357i	$0.952923\pi$
$510$	0	0
$511$	−459.089	−0.898413
$512$	− 360.875i	− 0.704834i
$513$	0	0
$514$	−4.66303	−0.00907204
$515$	675.741i	1.31212i
$516$	0	0
$517$	0	0
$518$	19.4759i	0.0375983i
$519$	0	0
$520$	−273.595	−0.526144
$521$	437.080i	0.838924i	0.907773	+	0.419462i	$0.137781\pi$
$521$	437.080i	0.838924i	−0.907773	+	0.419462i	$0.862219\pi$
$522$	0	0
$523$	−410.273	−0.784461	−0.392231	−	0.919867i	$-0.628297\pi$
$523$	−410.273	−0.784461	−0.392231	+	0.919867i	$0.628297\pi$
$524$	− 205.148i	− 0.391503i
$525$	0	0
$526$	146.546	0.278605
$527$	131.293i	0.249134i
$528$	0	0
$529$	−1200.15	−2.26871
$530$	− 101.000i	− 0.190565i
$531$	0	0
$532$	195.051	0.366636
$533$	− 993.550i	− 1.86407i
$534$	0	0
$535$	494.577	0.924443
$536$	225.178i	0.420109i
$537$	0	0
$538$	−62.2171	−0.115645
$539$	0	0
$540$	0	0
$541$	−499.682	−0.923626	−0.461813	−	0.886977i	$-0.652801\pi$
$541$	−499.682	−0.923626	−0.461813	+	0.886977i	$0.652801\pi$
$542$	27.1720i	0.0501328i
$543$	0	0
$544$	−138.857	−0.255252
$545$	− 745.871i	− 1.36857i
$546$	0	0
$547$	−695.564	−1.27160	−0.635798	−	0.771855i	$-0.719329\pi$
$547$	−695.564	−1.27160	−0.635798	+	0.771855i	$0.719329\pi$
$548$	− 490.936i	− 0.895869i
$549$	0	0
$550$	0	0
$551$	100.128i	0.181720i
$552$	0	0
$553$	17.7680	0.0321301
$554$	93.1637i	0.168165i
$555$	0	0
$556$	209.383	0.376589
$557$	− 433.734i	− 0.778697i	−0.921091	−	0.389348i	$-0.872700\pi$
$557$	− 433.734i	− 0.778697i	0.921091	−	0.389348i	$-0.127300\pi$
$558$	0	0
$559$	810.909	1.45064
$560$	− 765.484i	− 1.36694i
$561$	0	0
$562$	93.0037	0.165487
$563$	− 672.668i	− 1.19479i	−0.801946	−	0.597396i	$-0.796202\pi$
$563$	− 672.668i	− 1.19479i	0.801946	−	0.597396i	$-0.203798\pi$
$564$	0	0
$565$	−601.693	−1.06494
$566$	63.6776i	0.112505i
$567$	0	0
$568$	−284.984	−0.501732
$569$	543.176i	0.954614i	0.878737	+	0.477307i	$0.158387\pi$
$569$	543.176i	0.954614i	−0.878737	+	0.477307i	$0.841613\pi$
$570$	0	0
$571$	607.861	1.06456	0.532278	−	0.846570i	$-0.321336\pi$
$571$	607.861	1.06456	0.532278	+	0.846570i	$0.321336\pi$
$572$	0	0
$573$	0	0
$574$	−143.455	−0.249922
$575$	− 430.790i	− 0.749200i
$576$	0	0
$577$	131.051	0.227125	0.113562	−	0.993531i	$-0.463774\pi$
$577$	131.051	0.227125	0.113562	+	0.993531i	$0.463774\pi$
$578$	− 61.2905i	− 0.106039i
$579$	0	0
$580$	403.219	0.695205
$581$	249.893i	0.430109i
$582$	0	0
$583$	0	0
$584$	130.281i	0.223083i
$585$	0	0
$586$	32.0209	0.0546431
$587$	− 133.668i	− 0.227713i	−0.993497	−	0.113857i	$-0.963680\pi$
$587$	− 133.668i	− 0.227713i	0.993497	−	0.113857i	$-0.0363204\pi$
$588$	0	0
$589$	−78.8257	−0.133830
$590$	− 65.4139i	− 0.110871i
$591$	0	0
$592$	−107.098	−0.180909
$593$	164.004i	0.276567i	0.990393	+	0.138283i	$0.0441585\pi$
$593$	164.004i	0.276567i	−0.990393	+	0.138283i	$0.955841\pi$
$594$	0	0
$595$	−494.962	−0.831869
$596$	16.5385i	0.0277492i
$597$	0	0
$598$	242.088	0.404830
$599$	42.8102i	0.0714694i	0.999361	+	0.0357347i	$0.0113771\pi$
$599$	42.8102i	0.0714694i	−0.999361	+	0.0357347i	$0.988623\pi$
$600$	0	0
$601$	−311.489	−0.518284	−0.259142	−	0.965839i	$-0.583440\pi$
$601$	−311.489	−0.518284	−0.259142	+	0.965839i	$0.583440\pi$
$602$	− 117.084i	− 0.194492i
$603$	0	0
$604$	368.399	0.609933
$605$	0	0
$606$	0	0
$607$	1126.14	1.85526	0.927631	−	0.373498i	$-0.121842\pi$
$607$	1126.14	1.85526	0.927631	+	0.373498i	$0.121842\pi$
$608$	− 83.3669i	− 0.137117i
$609$	0	0
$610$	218.198	0.357702
$611$	352.348i	0.576674i
$612$	0	0
$613$	466.854	0.761589	0.380795	−	0.924660i	$-0.375650\pi$
$613$	466.854	0.761589	0.380795	+	0.924660i	$0.375650\pi$
$614$	− 12.3186i	− 0.0200628i
$615$	0	0
$616$	0	0
$617$	130.650i	0.211750i	0.994379	+	0.105875i	$0.0337644\pi$
$617$	130.650i	0.211750i	−0.994379	+	0.105875i	$0.966236\pi$
$618$	0	0
$619$	1096.95	1.77214	0.886068	−	0.463555i	$-0.153426\pi$
$619$	1096.95	1.77214	0.886068	+	0.463555i	$0.153426\pi$
$620$	317.435i	0.511992i
$621$	0	0
$622$	−43.4200	−0.0698072
$623$	1170.41i	1.87867i
$624$	0	0
$625$	−776.669	−1.24267
$626$	70.3909i	0.112446i
$627$	0	0
$628$	−580.320	−0.924076
$629$	69.2497i	0.110095i
$630$	0	0
$631$	−1161.69	−1.84102	−0.920511	−	0.390716i	$-0.872227\pi$
$631$	−1161.69	−1.84102	−0.920511	+	0.390716i	$0.872227\pi$
$632$	− 5.04221i	− 0.00797818i
$633$	0	0
$634$	109.654	0.172956
$635$	− 737.331i	− 1.16115i
$636$	0	0
$637$	489.639	0.768664
$638$	0	0
$639$	0	0
$640$	−445.670	−0.696359
$641$	387.958i	0.605238i	0.953112	+	0.302619i	$0.0978610\pi$
$641$	387.958i	0.605238i	−0.953112	+	0.302619i	$0.902139\pi$
$642$	0	0
$643$	−115.275	−0.179277	−0.0896383	−	0.995974i	$-0.528571\pi$
$643$	−115.275	−0.179277	−0.0896383	+	0.995974i	$0.528571\pi$
$644$	1407.31i	2.18527i
$645$	0	0
$646$	−17.2257	−0.0266652
$647$	182.585i	0.282203i	0.989995	+	0.141101i	$0.0450643\pi$
$647$	182.585i	0.282203i	−0.989995	+	0.141101i	$0.954936\pi$
$648$	0	0
$649$	0	0
$650$	60.3125i	0.0927885i
$651$	0	0
$652$	−358.110	−0.549249
$653$	− 10.8119i	− 0.0165573i	−0.999966	−	0.00827865i	$-0.997365\pi$
$653$	− 10.8119i	− 0.0165573i	0.999966	−	0.00827865i	$-0.00263521\pi$
$654$	0	0
$655$	−312.548	−0.477172
$656$	− 788.863i	− 1.20253i
$657$	0	0
$658$	50.8743	0.0773166
$659$	− 875.394i	− 1.32837i	−0.747569	−	0.664184i	$-0.768779\pi$
$659$	− 875.394i	− 1.32837i	0.747569	−	0.664184i	$-0.231221\pi$
$660$	0	0
$661$	1015.92	1.53695	0.768475	−	0.639880i	$-0.221016\pi$
$661$	1015.92	1.53695	0.768475	+	0.639880i	$0.221016\pi$
$662$	26.4211i	0.0399110i
$663$	0	0
$664$	70.9149	0.106800
$665$	− 297.165i	− 0.446864i
$666$	0	0
$667$	−722.433	−1.08311
$668$	− 55.6875i	− 0.0833645i
$669$	0	0
$670$	169.428	0.252878
$671$	0	0
$672$	0	0
$673$	−113.798	−0.169091	−0.0845454	−	0.996420i	$-0.526944\pi$
$673$	−113.798	−0.169091	−0.0845454	+	0.996420i	$0.526944\pi$
$674$	− 164.139i	− 0.243530i
$675$	0	0
$676$	704.984	1.04288
$677$	853.302i	1.26042i	0.776426	+	0.630208i	$0.217031\pi$
$677$	853.302i	1.26042i	−0.776426	+	0.630208i	$0.782969\pi$
$678$	0	0
$679$	−86.5043	−0.127400
$680$	140.461i	0.206560i
$681$	0	0
$682$	0	0
$683$	193.740i	0.283661i	0.989891	+	0.141830i	$0.0452987\pi$
$683$	193.740i	0.283661i	−0.989891	+	0.141830i	$0.954701\pi$
$684$	0	0
$685$	−747.954	−1.09190
$686$	61.5916i	0.0897836i
$687$	0	0
$688$	643.848	0.935826
$689$	1020.02i	1.48043i
$690$	0	0
$691$	−253.670	−0.367106	−0.183553	−	0.983010i	$-0.558760\pi$
$691$	−253.670	−0.367106	−0.183553	+	0.983010i	$0.558760\pi$
$692$	− 906.604i	− 1.31012i
$693$	0	0
$694$	−39.3300	−0.0566714
$695$	− 319.001i	− 0.458994i
$696$	0	0
$697$	−510.078	−0.731820
$698$	140.563i	0.201380i
$699$	0	0
$700$	−350.610	−0.500871
$701$	− 1139.33i	− 1.62529i	−0.582760	−	0.812644i	$-0.698027\pi$
$701$	− 1139.33i	− 1.62529i	0.582760	−	0.812644i	$-0.301973\pi$
$702$	0	0
$703$	−41.5761	−0.0591409
$704$	0	0
$705$	0	0
$706$	45.3676	0.0642601
$707$	− 1495.11i	− 2.11472i
$708$	0	0
$709$	374.159	0.527728	0.263864	−	0.964560i	$-0.415003\pi$
$709$	374.159	0.527728	0.263864	+	0.964560i	$0.415003\pi$
$710$	214.427i	0.302010i
$711$	0	0
$712$	332.140	0.466489
$713$	− 568.737i	− 0.797667i
$714$	0	0
$715$	0	0
$716$	560.706i	0.783109i
$717$	0	0
$718$	133.637	0.186124
$719$	757.515i	1.05357i	0.849999	+	0.526784i	$0.176602\pi$
$719$	757.515i	1.05357i	−0.849999	+	0.526784i	$0.823398\pi$
$720$	0	0
$721$	−985.361	−1.36666
$722$	102.058i	0.141354i
$723$	0	0
$724$	−935.585	−1.29224
$725$	− 179.983i	− 0.248252i
$726$	0	0
$727$	845.080	1.16242	0.581211	−	0.813753i	$-0.302579\pi$
$727$	845.080	1.16242	0.581211	+	0.813753i	$0.302579\pi$
$728$	− 398.954i	− 0.548014i
$729$	0	0
$730$	98.0256	0.134282
$731$	− 416.312i	− 0.569511i
$732$	0	0
$733$	1358.08	1.85278	0.926388	−	0.376570i	$-0.122897\pi$
$733$	1358.08	1.85278	0.926388	+	0.376570i	$0.122897\pi$
$734$	− 41.2115i	− 0.0561465i
$735$	0	0
$736$	601.502	0.817258
$737$	0	0
$738$	0	0
$739$	801.192	1.08416	0.542078	−	0.840328i	$-0.317638\pi$
$739$	801.192	1.08416	0.542078	+	0.840328i	$0.317638\pi$
$740$	167.429i	0.226255i
$741$	0	0
$742$	147.277	0.198487
$743$	− 893.596i	− 1.20269i	−0.798991	−	0.601343i	$-0.794632\pi$
$743$	− 893.596i	− 1.20269i	0.798991	−	0.601343i	$-0.205368\pi$
$744$	0	0
$745$	25.1968	0.0338213
$746$	148.884i	0.199577i
$747$	0	0
$748$	0	0
$749$	721.189i	0.962868i
$750$	0	0
$751$	397.035	0.528676	0.264338	−	0.964430i	$-0.414847\pi$
$751$	397.035	0.528676	0.264338	+	0.964430i	$0.414847\pi$
$752$	279.758i	0.372019i
$753$	0	0
$754$	101.144	0.134143
$755$	− 561.266i	− 0.743398i
$756$	0	0
$757$	593.110	0.783500	0.391750	−	0.920072i	$-0.371870\pi$
$757$	593.110	0.783500	0.391750	+	0.920072i	$0.371870\pi$
$758$	82.2622i	0.108525i
$759$	0	0
$760$	−84.3296	−0.110960
$761$	409.540i	0.538160i	0.963118	+	0.269080i	$0.0867196\pi$
$761$	409.540i	0.538160i	−0.963118	+	0.269080i	$0.913280\pi$
$762$	0	0
$763$	1087.62	1.42546
$764$	− 146.488i	− 0.191739i
$765$	0	0
$766$	73.6143	0.0961022
$767$	660.630i	0.861316i
$768$	0	0
$769$	119.029	0.154784	0.0773921	−	0.997001i	$-0.475341\pi$
$769$	119.029	0.154784	0.0773921	+	0.997001i	$0.475341\pi$
$770$	0	0
$771$	0	0
$772$	794.553	1.02921
$773$	701.535i	0.907548i	0.891117	+	0.453774i	$0.149923\pi$
$773$	701.535i	0.907548i	−0.891117	+	0.453774i	$0.850077\pi$
$774$	0	0
$775$	141.692	0.182828
$776$	24.5483i	0.0316344i
$777$	0	0
$778$	−159.171	−0.204590
$779$	− 306.240i	− 0.393120i
$780$	0	0
$781$	0	0
$782$	− 124.286i	− 0.158933i
$783$	0	0
$784$	388.766	0.495874
$785$	884.132i	1.12628i
$786$	0	0
$787$	1539.13	1.95569	0.977844	−	0.209336i	$-0.0671303\pi$
$787$	1539.13	1.95569	0.977844	+	0.209336i	$0.0671303\pi$
$788$	813.697i	1.03261i
$789$	0	0
$790$	−3.79385	−0.00480234
$791$	− 877.384i	− 1.10921i
$792$	0	0
$793$	−2203.63	−2.77885
$794$	− 121.187i	− 0.152629i
$795$	0	0
$796$	−975.273	−1.22522
$797$	− 989.760i	− 1.24186i	−0.783867	−	0.620928i	$-0.786756\pi$
$797$	− 989.760i	− 1.24186i	0.783867	−	0.620928i	$-0.213244\pi$
$798$	0	0
$799$	180.892	0.226398
$800$	149.855i	0.187319i
$801$	0	0
$802$	70.3821	0.0877582
$803$	0	0
$804$	0	0
$805$	2144.08	2.66345
$806$	79.6257i	0.0987912i
$807$	0	0
$808$	−424.283	−0.525103
$809$	− 94.1076i	− 0.116326i	−0.998307	−	0.0581629i	$-0.981476\pi$
$809$	− 94.1076i	− 0.116326i	0.998307	−	0.0581629i	$-0.0185243\pi$
$810$	0	0
$811$	−370.312	−0.456612	−0.228306	−	0.973589i	$-0.573319\pi$
$811$	−370.312	−0.456612	−0.228306	+	0.973589i	$0.573319\pi$
$812$	587.971i	0.724103i
$813$	0	0
$814$	0	0
$815$	545.590i	0.669436i
$816$	0	0
$817$	249.945	0.305930
$818$	56.5940i	0.0691858i
$819$	0	0
$820$	−1233.24	−1.50396
$821$	213.696i	0.260288i	0.991495	+	0.130144i	$0.0415439\pi$
$821$	213.696i	0.260288i	−0.991495	+	0.130144i	$0.958456\pi$
$822$	0	0
$823$	104.216	0.126630	0.0633148	−	0.997994i	$-0.479833\pi$
$823$	104.216	0.126630	0.0633148	+	0.997994i	$0.479833\pi$
$824$	279.627i	0.339353i
$825$	0	0
$826$	95.3861	0.115480
$827$	− 1329.26i	− 1.60732i	−0.595086	−	0.803662i	$-0.702882\pi$
$827$	− 1329.26i	− 1.60732i	0.595086	−	0.803662i	$-0.297118\pi$
$828$	0	0
$829$	−350.092	−0.422306	−0.211153	−	0.977453i	$-0.567722\pi$
$829$	−350.092	−0.422306	−0.211153	+	0.977453i	$0.567722\pi$
$830$	− 53.3577i	− 0.0642864i
$831$	0	0
$832$	1026.17	1.23338
$833$	− 251.376i	− 0.301772i
$834$	0	0
$835$	−84.8413	−0.101606
$836$	0	0
$837$	0	0
$838$	53.5250	0.0638723
$839$	− 1542.17i	− 1.83811i	−0.394132	−	0.919054i	$-0.628955\pi$
$839$	− 1542.17i	− 1.83811i	0.394132	−	0.919054i	$-0.371045\pi$
$840$	0	0
$841$	539.169	0.641105
$842$	37.6263i	0.0446868i
$843$	0	0
$844$	474.428	0.562119
$845$	− 1074.06i	− 1.27108i
$846$	0	0
$847$	0	0
$848$	809.878i	0.955044i
$849$	0	0
$850$	30.9638	0.0364280
$851$	− 299.976i	− 0.352498i
$852$	0	0
$853$	53.1329	0.0622895	0.0311448	−	0.999515i	$-0.490085\pi$
$853$	53.1329	0.0622895	0.0311448	+	0.999515i	$0.490085\pi$
$854$	318.175i	0.372570i
$855$	0	0
$856$	204.660	0.239088
$857$	668.807i	0.780405i	0.920729	+	0.390202i	$0.127595\pi$
$857$	668.807i	0.780405i	−0.920729	+	0.390202i	$0.872405\pi$
$858$	0	0
$859$	192.579	0.224189	0.112095	−	0.993698i	$-0.464244\pi$
$859$	192.579	0.224189	0.112095	+	0.993698i	$0.464244\pi$
$860$	− 1006.54i	− 1.17040i
$861$	0	0
$862$	147.675	0.171317
$863$	− 851.072i	− 0.986179i	−0.869979	−	0.493089i	$-0.835868\pi$
$863$	− 851.072i	− 0.986179i	0.869979	−	0.493089i	$-0.164132\pi$
$864$	0	0
$865$	−1381.23	−1.59680
$866$	− 79.1549i	− 0.0914029i
$867$	0	0
$868$	−462.882	−0.533274
$869$	0	0
$870$	0	0
$871$	−1711.10	−1.96452
$872$	− 308.647i	− 0.353953i
$873$	0	0
$874$	74.6185	0.0853758
$875$	− 754.871i	− 0.862709i
$876$	0	0
$877$	440.628	0.502427	0.251213	−	0.967932i	$-0.419170\pi$
$877$	440.628	0.502427	0.251213	+	0.967932i	$0.419170\pi$
$878$	− 160.425i	− 0.182717i
$879$	0	0
$880$	0	0
$881$	448.427i	0.508997i	0.967073	+	0.254499i	$0.0819104\pi$
$881$	448.427i	0.508997i	−0.967073	+	0.254499i	$0.918090\pi$
$882$	0	0
$883$	−310.831	−0.352017	−0.176009	−	0.984389i	$-0.556319\pi$
$883$	−310.831	−0.352017	−0.176009	+	0.984389i	$0.556319\pi$
$884$	− 700.572i	− 0.792502i
$885$	0	0
$886$	163.786	0.184860
$887$	− 1292.83i	− 1.45754i	−0.684761	−	0.728768i	$-0.740093\pi$
$887$	− 1292.83i	− 1.45754i	0.684761	−	0.728768i	$-0.259907\pi$
$888$	0	0
$889$	1075.17	1.20942
$890$	− 249.908i	− 0.280796i
$891$	0	0
$892$	−280.287	−0.314223
$893$	108.604i	0.121617i
$894$	0	0
$895$	854.250	0.954469
$896$	− 649.873i	− 0.725304i
$897$	0	0
$898$	−168.711	−0.187874
$899$	− 237.617i	− 0.264312i
$900$	0	0
$901$	523.667	0.581206
$902$	0	0
$903$	0	0
$904$	−248.985	−0.275426
$905$	1425.39i	1.57501i
$906$	0	0
$907$	815.519	0.899139	0.449570	−	0.893245i	$-0.351577\pi$
$907$	815.519	0.899139	0.449570	+	0.893245i	$0.351577\pi$
$908$	− 913.491i	− 1.00605i
$909$	0	0
$910$	−300.180	−0.329869
$911$	− 243.043i	− 0.266787i	−0.991063	−	0.133393i	$-0.957413\pi$
$911$	− 243.043i	− 0.266787i	0.991063	−	0.133393i	$-0.0425874\pi$
$912$	0	0
$913$	0	0
$914$	− 38.1193i	− 0.0417061i
$915$	0	0
$916$	265.919	0.290305
$917$	− 455.755i	− 0.497006i
$918$	0	0
$919$	−290.283	−0.315868	−0.157934	−	0.987450i	$-0.550483\pi$
$919$	−290.283	−0.315868	−0.157934	+	0.987450i	$0.550483\pi$
$920$	− 608.448i	− 0.661357i
$921$	0	0
$922$	95.9913	0.104112
$923$	− 2165.54i	− 2.34620i
$924$	0	0
$925$	74.7343	0.0807938
$926$	− 115.094i	− 0.124292i
$927$	0	0
$928$	251.306	0.270804
$929$	295.075i	0.317627i	0.987309	+	0.158813i	$0.0507668\pi$
$929$	295.075i	0.317627i	−0.987309	+	0.158813i	$0.949233\pi$
$930$	0	0
$931$	150.921	0.162106
$932$	− 416.593i	− 0.446988i
$933$	0	0
$934$	33.8912	0.0362861
$935$	0	0
$936$	0	0
$937$	−918.143	−0.979875	−0.489938	−	0.871758i	$-0.662980\pi$
$937$	−918.143	−0.979875	−0.489938	+	0.871758i	$0.662980\pi$
$938$	247.059i	0.263389i
$939$	0	0
$940$	437.352	0.465268
$941$	998.146i	1.06073i	0.847770	+	0.530365i	$0.177945\pi$
$941$	998.146i	1.06073i	−0.847770	+	0.530365i	$0.822055\pi$
$942$	0	0
$943$	2209.56	2.34312
$944$	524.529i	0.555645i
$945$	0	0
$946$	0	0
$947$	− 780.779i	− 0.824476i	−0.911076	−	0.412238i	$-0.864747\pi$
$947$	− 780.779i	− 0.824476i	0.911076	−	0.412238i	$-0.135253\pi$
$948$	0	0
$949$	−989.982	−1.04318
$950$	18.5900i	0.0195684i
$951$	0	0
$952$	−204.819	−0.215146
$953$	− 93.2267i	− 0.0978245i	−0.998803	−	0.0489122i	$-0.984425\pi$
$953$	− 93.2267i	− 0.0978245i	0.998803	−	0.0489122i	$-0.0155755\pi$
$954$	0	0
$955$	−223.179	−0.233695
$956$	1360.56i	1.42318i
$957$	0	0
$958$	−94.2409	−0.0983725
$959$	− 1090.66i	− 1.13729i
$960$	0	0
$961$	−773.936	−0.805344
$962$	41.9980i	0.0436570i
$963$	0	0
$964$	−1473.13	−1.52814
$965$	− 1210.52i	− 1.25443i
$966$	0	0
$967$	365.949	0.378437	0.189219	−	0.981935i	$-0.439405\pi$
$967$	365.949	0.378437	0.189219	+	0.981935i	$0.439405\pi$
$968$	0	0
$969$	0	0
$970$	18.4706	0.0190418
$971$	− 284.163i	− 0.292650i	−0.989237	−	0.146325i	$-0.953256\pi$
$971$	− 284.163i	− 0.292650i	0.989237	−	0.146325i	$-0.0467445\pi$
$972$	0	0
$973$	465.165	0.478073
$974$	− 144.529i	− 0.148388i
$975$	0	0
$976$	−1749.65	−1.79267
$977$	882.999i	0.903786i	0.892072	+	0.451893i	$0.149251\pi$
$977$	882.999i	0.903786i	−0.892072	+	0.451893i	$0.850749\pi$
$978$	0	0
$979$	0	0
$980$	− 607.765i	− 0.620168i
$981$	0	0
$982$	−108.070	−0.110051
$983$	− 1364.76i	− 1.38836i	−0.719799	−	0.694182i	$-0.755766\pi$
$983$	− 1364.76i	− 1.38836i	0.719799	−	0.694182i	$-0.244234\pi$
$984$	0	0
$985$	1239.69	1.25857
$986$	− 51.9262i	− 0.0526635i
$987$	0	0
$988$	420.608	0.425717
$989$	1803.38i	1.82344i
$990$	0	0
$991$	−842.288	−0.849938	−0.424969	−	0.905208i	$-0.639715\pi$
$991$	−842.288	−0.849938	−0.424969	+	0.905208i	$0.639715\pi$
$992$	197.841i	0.199437i
$993$	0	0
$994$	−312.676	−0.314563
$995$	1485.85i	1.49332i
$996$	0	0
$997$	−366.417	−0.367520	−0.183760	−	0.982971i	$-0.558827\pi$
$997$	−366.417	−0.367520	−0.183760	+	0.982971i	$0.558827\pi$
$998$	− 141.652i	− 0.141936i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.i.485.8		16
3.2	odd	2	inner	1089.3.b.i.485.9		16
11.3	even	5		99.3.l.a.53.4	✓	32
11.4	even	5		99.3.l.a.71.5	yes	32
11.10	odd	2		1089.3.b.j.485.9		16
33.14	odd	10		99.3.l.a.53.5	yes	32
33.26	odd	10		99.3.l.a.71.4	yes	32
33.32	even	2		1089.3.b.j.485.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.l.a.53.4	✓	32	11.3	even	5
99.3.l.a.53.5	yes	32	33.14	odd	10
99.3.l.a.71.4	yes	32	33.26	odd	10
99.3.l.a.71.5	yes	32	11.4	even	5
1089.3.b.i.485.8		16	1.1	even	1	trivial
1089.3.b.i.485.9		16	3.2	odd	2	inner
1089.3.b.j.485.8		16	33.32	even	2
1089.3.b.j.485.9		16	11.10	odd	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.311356i q^{2} +3.90306 q^{4} -5.94641i q^{5} +8.67101 q^{7} -2.46067i q^{8} +O(q^{10})\)	\(q-0.311356i q^{2} +3.90306 q^{4} -5.94641i q^{5} +8.67101 q^{7} -2.46067i q^{8} -1.85145 q^{10} +18.6982 q^{13} -2.69977i q^{14} +14.8461 q^{16} -9.59948i q^{17} +5.76332 q^{19} -23.2092i q^{20} +41.5830i q^{23} -10.3598 q^{25} -5.82181i q^{26} +33.8434 q^{28} +17.3733i q^{29} -13.6771 q^{31} -14.4651i q^{32} -2.98886 q^{34} -51.5613i q^{35} -7.21391 q^{37} -1.79445i q^{38} -14.6321 q^{40} -53.1361i q^{41} +43.3682 q^{43} +12.9471 q^{46} +18.8439i q^{47} +26.1864 q^{49} +3.22558i q^{50} +72.9802 q^{52} +54.5516i q^{53} -21.3365i q^{56} +5.40928 q^{58} +35.3312i q^{59} -117.852 q^{61} +4.25847i q^{62} +54.8805 q^{64} -111.187i q^{65} -91.5111 q^{67} -37.4673i q^{68} -16.0540 q^{70} -115.816i q^{71} -52.9453 q^{73} +2.24610i q^{74} +22.4946 q^{76} +2.04912 q^{79} -88.2809i q^{80} -16.5443 q^{82} +28.8194i q^{83} -57.0824 q^{85} -13.5030i q^{86} +134.980i q^{89} +162.132 q^{91} +162.301i q^{92} +5.86717 q^{94} -34.2710i q^{95} -9.97627 q^{97} -8.15330i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 8 q^{7}+O(q^{10}) \) 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

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