LMFDB - Embedded newform 1089.3.c.e.604.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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([0, 1]))

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

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (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:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.1
Root			$-0.309017 + 0.951057i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.e.604.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.07768i q^{2} -5.47214 q^{4} -4.00000 q^{5} -0.898056i q^{7} +4.53077i q^{8} +O(q^{10})$	$q-3.07768i q^{2} -5.47214 q^{4} -4.00000 q^{5} -0.898056i q^{7} +4.53077i q^{8} +12.3107i q^{10} +8.50651i q^{13} -2.76393 q^{14} -7.94427 q^{16} +24.7275i q^{17} -11.8617i q^{19} +21.8885 q^{20} +7.23607 q^{23} -9.00000 q^{25} +26.1803 q^{26} +4.91428i q^{28} +3.46120i q^{29} +33.1246 q^{31} +42.5730i q^{32} +76.1033 q^{34} +3.59222i q^{35} +40.2492 q^{37} -36.5066 q^{38} -18.1231i q^{40} +1.30660i q^{41} +33.0625i q^{43} -22.2703i q^{46} -22.7639 q^{47} +48.1935 q^{49} +27.6992i q^{50} -46.5488i q^{52} +78.5410 q^{53} +4.06888 q^{56} +10.6525 q^{58} -31.2705 q^{59} -28.0827i q^{61} -101.947i q^{62} +99.2492 q^{64} -34.0260i q^{65} -76.5066 q^{67} -135.312i q^{68} +11.0557 q^{70} +62.3181 q^{71} +94.1421i q^{73} -123.874i q^{74} +64.9089i q^{76} -65.9129i q^{79} +31.7771 q^{80} +4.02129 q^{82} -56.7049i q^{83} -98.9099i q^{85} +101.756 q^{86} +62.2968 q^{89} +7.63932 q^{91} -39.5967 q^{92} +70.0602i q^{94} +47.4468i q^{95} +72.4083 q^{97} -148.324i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 16 q^{5}+O(q^{10}) $ 4 * q - 4 * q^4 - 16 * q^5	$ 4 q - 4 q^{4} - 16 q^{5} - 20 q^{14} + 4 q^{16} + 16 q^{20} + 20 q^{23} - 36 q^{25} + 60 q^{26} + 52 q^{31} + 130 q^{34} - 70 q^{38} - 100 q^{47} - 4 q^{49} + 180 q^{53} - 100 q^{56} - 20 q^{58} - 58 q^{59} + 236 q^{64} - 230 q^{67} + 80 q^{70} - 28 q^{71} - 16 q^{80} + 110 q^{82} + 170 q^{86} - 122 q^{89} + 120 q^{91} - 60 q^{92} - 10 q^{97}+O(q^{100}) $ 4 * q - 4 * q^4 - 16 * q^5 - 20 * q^14 + 4 * q^16 + 16 * q^20 + 20 * q^23 - 36 * q^25 + 60 * q^26 + 52 * q^31 + 130 * q^34 - 70 * q^38 - 100 * q^47 - 4 * q^49 + 180 * q^53 - 100 * q^56 - 20 * q^58 - 58 * q^59 + 236 * q^64 - 230 * q^67 + 80 * q^70 - 28 * q^71 - 16 * q^80 + 110 * q^82 + 170 * q^86 - 122 * q^89 + 120 * q^91 - 60 * q^92 - 10 * 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 $

$2$	− 3.07768i	− 1.53884i	−0.638742	−	0.769421i	$-0.720545\pi$
$2$	− 3.07768i	− 1.53884i	0.638742	−	0.769421i	$-0.279455\pi$
$3$	0	0
$3$	0	0
$4$	−5.47214	−1.36803
$5$	−4.00000	−0.800000	−0.400000	−	0.916515i	$-0.630990\pi$
$5$	−4.00000	−0.800000	−0.400000	+	0.916515i	$0.630990\pi$
$6$	0	0
$7$	− 0.898056i	− 0.128294i	−0.997940	−	0.0641469i	$-0.979567\pi$
$7$	− 0.898056i	− 0.128294i	0.997940	−	0.0641469i	$-0.0204326\pi$
$8$	4.53077i	0.566346i
$9$	0	0
$10$	12.3107i	1.23107i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	8.50651i	0.654347i	0.944964	+	0.327173i	$0.106096\pi$
$13$	8.50651i	0.654347i	−0.944964	+	0.327173i	$0.893904\pi$
$14$	−2.76393	−0.197424
$15$	0	0
$16$	−7.94427	−0.496517
$17$	24.7275i	1.45456i	0.686342	+	0.727279i	$0.259215\pi$
$17$	24.7275i	1.45456i	−0.686342	+	0.727279i	$0.740785\pi$
$18$	0	0
$19$	− 11.8617i	− 0.624300i	−0.950033	−	0.312150i	$-0.898951\pi$
$19$	− 11.8617i	− 0.624300i	0.950033	−	0.312150i	$-0.101049\pi$
$20$	21.8885	1.09443
$21$	0	0
$22$	0	0
$23$	7.23607	0.314612	0.157306	−	0.987550i	$-0.449719\pi$
$23$	7.23607	0.314612	0.157306	+	0.987550i	$0.449719\pi$
$24$	0	0
$25$	−9.00000	−0.360000
$26$	26.1803	1.00694
$27$	0	0
$28$	4.91428i	0.175510i
$29$	3.46120i	0.119352i	0.998218	+	0.0596758i	$0.0190067\pi$
$29$	3.46120i	0.119352i	−0.998218	+	0.0596758i	$0.980993\pi$
$30$	0	0
$31$	33.1246	1.06854	0.534268	−	0.845315i	$-0.320587\pi$
$31$	33.1246	1.06854	0.534268	+	0.845315i	$0.320587\pi$
$32$	42.5730i	1.33041i
$33$	0	0
$34$	76.1033	2.23833
$35$	3.59222i	0.102635i
$36$	0	0
$37$	40.2492	1.08782	0.543908	−	0.839145i	$-0.316944\pi$
$37$	40.2492	1.08782	0.543908	+	0.839145i	$0.316944\pi$
$38$	−36.5066	−0.960699
$39$	0	0
$40$	− 18.1231i	− 0.453077i
$41$	1.30660i	0.0318682i	0.999873	+	0.0159341i	$0.00507219\pi$
$41$	1.30660i	0.0318682i	−0.999873	+	0.0159341i	$0.994928\pi$
$42$	0	0
$43$	33.0625i	0.768895i	0.923147	+	0.384447i	$0.125608\pi$
$43$	33.0625i	0.768895i	−0.923147	+	0.384447i	$0.874392\pi$
$44$	0	0
$45$	0	0
$46$	− 22.2703i	− 0.484138i
$47$	−22.7639	−0.484339	−0.242169	−	0.970234i	$-0.577859\pi$
$47$	−22.7639	−0.484339	−0.242169	+	0.970234i	$0.577859\pi$
$48$	0	0
$49$	48.1935	0.983541
$50$	27.6992i	0.553983i
$51$	0	0
$52$	− 46.5488i	− 0.895169i
$53$	78.5410	1.48191	0.740953	−	0.671557i	$-0.234374\pi$
$53$	78.5410	1.48191	0.740953	+	0.671557i	$0.234374\pi$
$54$	0	0
$55$	0	0
$56$	4.06888	0.0726586
$57$	0	0
$58$	10.6525	0.183663
$59$	−31.2705	−0.530009	−0.265004	−	0.964247i	$-0.585373\pi$
$59$	−31.2705	−0.530009	−0.265004	+	0.964247i	$0.585373\pi$
$60$	0	0
$61$	− 28.0827i	− 0.460372i	−0.973147	−	0.230186i	$-0.926067\pi$
$61$	− 28.0827i	− 0.460372i	0.973147	−	0.230186i	$-0.0739334\pi$
$62$	− 101.947i	− 1.64431i
$63$	0	0
$64$	99.2492	1.55077
$65$	− 34.0260i	− 0.523477i
$66$	0	0
$67$	−76.5066	−1.14189	−0.570945	−	0.820989i	$-0.693423\pi$
$67$	−76.5066	−1.14189	−0.570945	+	0.820989i	$0.693423\pi$
$68$	− 135.312i	− 1.98988i
$69$	0	0
$70$	11.0557	0.157939
$71$	62.3181	0.877720	0.438860	−	0.898555i	$-0.355382\pi$
$71$	62.3181	0.877720	0.438860	+	0.898555i	$0.355382\pi$
$72$	0	0
$73$	94.1421i	1.28962i	0.764344	+	0.644809i	$0.223063\pi$
$73$	94.1421i	1.28962i	−0.764344	+	0.644809i	$0.776937\pi$
$74$	− 123.874i	− 1.67398i
$75$	0	0
$76$	64.9089i	0.854064i
$77$	0	0
$78$	0	0
$79$	− 65.9129i	− 0.834341i	−0.908828	−	0.417170i	$-0.863022\pi$
$79$	− 65.9129i	− 0.834341i	0.908828	−	0.417170i	$-0.136978\pi$
$80$	31.7771	0.397214
$81$	0	0
$82$	4.02129	0.0490401
$83$	− 56.7049i	− 0.683192i	−0.939847	−	0.341596i	$-0.889033\pi$
$83$	− 56.7049i	− 0.683192i	0.939847	−	0.341596i	$-0.110967\pi$
$84$	0	0
$85$	− 98.9099i	− 1.16365i
$86$	101.756	1.18321
$87$	0	0
$88$	0	0
$89$	62.2968	0.699964	0.349982	−	0.936756i	$-0.386188\pi$
$89$	62.2968	0.699964	0.349982	+	0.936756i	$0.386188\pi$
$90$	0	0
$91$	7.63932	0.0839486
$92$	−39.5967	−0.430399
$93$	0	0
$94$	70.0602i	0.745321i
$95$	47.4468i	0.499440i
$96$	0	0
$97$	72.4083	0.746477	0.373239	−	0.927735i	$-0.378247\pi$
$97$	72.4083	0.746477	0.373239	+	0.927735i	$0.378247\pi$
$98$	− 148.324i	− 1.51351i
$99$	0	0
$100$	49.2492	0.492492
$101$	− 64.3789i	− 0.637414i	−0.947853	−	0.318707i	$-0.896751\pi$
$101$	− 64.3789i	− 0.637414i	0.947853	−	0.318707i	$-0.103249\pi$
$102$	0	0
$103$	−32.6099	−0.316601	−0.158301	−	0.987391i	$-0.550601\pi$
$103$	−32.6099	−0.316601	−0.158301	+	0.987391i	$0.550601\pi$
$104$	−38.5410	−0.370587
$105$	0	0
$106$	− 241.724i	− 2.28042i
$107$	− 5.40380i	− 0.0505028i	−0.999681	−	0.0252514i	$-0.991961\pi$
$107$	− 5.40380i	− 0.0505028i	0.999681	−	0.0252514i	$-0.00803863\pi$
$108$	0	0
$109$	− 137.002i	− 1.25690i	−0.777850	−	0.628450i	$-0.783690\pi$
$109$	− 137.002i	− 1.25690i	0.777850	−	0.628450i	$-0.216310\pi$
$110$	0	0
$111$	0	0
$112$	7.13440i	0.0637000i
$113$	46.5066	0.411563	0.205781	−	0.978598i	$-0.434026\pi$
$113$	46.5066	0.411563	0.205781	+	0.978598i	$0.434026\pi$
$114$	0	0
$115$	−28.9443	−0.251689
$116$	− 18.9402i	− 0.163277i
$117$	0	0
$118$	96.2407i	0.815599i
$119$	22.2067	0.186611
$120$	0	0
$121$	0	0
$122$	−86.4296	−0.708439
$123$	0	0
$124$	−181.262	−1.46179
$125$	136.000	1.08800
$126$	0	0
$127$	83.4309i	0.656936i	0.944515	+	0.328468i	$0.106532\pi$
$127$	83.4309i	0.656936i	−0.944515	+	0.328468i	$0.893468\pi$
$128$	− 135.166i	− 1.05598i
$129$	0	0
$130$	−104.721	−0.805549
$131$	− 85.0901i	− 0.649543i	−0.945793	−	0.324771i	$-0.894713\pi$
$131$	− 85.0901i	− 0.649543i	0.945793	−	0.324771i	$-0.105287\pi$
$132$	0	0
$133$	−10.6525	−0.0800938
$134$	235.463i	1.75719i
$135$	0	0
$136$	−112.034	−0.823783
$137$	−58.7426	−0.428778	−0.214389	−	0.976748i	$-0.568776\pi$
$137$	−58.7426	−0.428778	−0.214389	+	0.976748i	$0.568776\pi$
$138$	0	0
$139$	160.988i	1.15818i	0.815262	+	0.579092i	$0.196593\pi$
$139$	160.988i	1.15818i	−0.815262	+	0.579092i	$0.803407\pi$
$140$	− 19.6571i	− 0.140408i
$141$	0	0
$142$	− 191.795i	− 1.35067i
$143$	0	0
$144$	0	0
$145$	− 13.8448i	− 0.0954814i
$146$	289.740	1.98452
$147$	0	0
$148$	−220.249	−1.48817
$149$	257.072i	1.72532i	0.505786	+	0.862659i	$0.331202\pi$
$149$	257.072i	1.72532i	−0.505786	+	0.862659i	$0.668798\pi$
$150$	0	0
$151$	288.192i	1.90856i	0.298918	+	0.954279i	$0.403374\pi$
$151$	288.192i	1.90856i	−0.298918	+	0.954279i	$0.596626\pi$
$152$	53.7426	0.353570
$153$	0	0
$154$	0	0
$155$	−132.498	−0.854829
$156$	0	0
$157$	276.620	1.76191	0.880955	−	0.473200i	$-0.156901\pi$
$157$	276.620	1.76191	0.880955	+	0.473200i	$0.156901\pi$
$158$	−202.859	−1.28392
$159$	0	0
$160$	− 170.292i	− 1.06433i
$161$	− 6.49839i	− 0.0403627i
$162$	0	0
$163$	−69.3657	−0.425556	−0.212778	−	0.977101i	$-0.568251\pi$
$163$	−69.3657	−0.425556	−0.212778	+	0.977101i	$0.568251\pi$
$164$	− 7.14987i	− 0.0435967i
$165$	0	0
$166$	−174.520	−1.05132
$167$	− 288.990i	− 1.73048i	−0.501358	−	0.865240i	$-0.667166\pi$
$167$	− 288.990i	− 1.73048i	0.501358	−	0.865240i	$-0.332834\pi$
$168$	0	0
$169$	96.6393	0.571830
$170$	−304.413	−1.79067
$171$	0	0
$172$	− 180.922i	− 1.05187i
$173$	183.208i	1.05901i	0.848308	+	0.529503i	$0.177621\pi$
$173$	183.208i	1.05901i	−0.848308	+	0.529503i	$0.822379\pi$
$174$	0	0
$175$	8.08250i	0.0461857i
$176$	0	0
$177$	0	0
$178$	− 191.730i	− 1.07713i
$179$	−2.05070	−0.0114564	−0.00572822	−	0.999984i	$-0.501823\pi$
$179$	−2.05070	−0.0114564	−0.00572822	+	0.999984i	$0.501823\pi$
$180$	0	0
$181$	−72.9017	−0.402772	−0.201386	−	0.979512i	$-0.564545\pi$
$181$	−72.9017	−0.402772	−0.201386	+	0.979512i	$0.564545\pi$
$182$	− 23.5114i	− 0.129184i
$183$	0	0
$184$	32.7849i	0.178179i
$185$	−160.997	−0.870253
$186$	0	0
$187$	0	0
$188$	124.567	0.662592
$189$	0	0
$190$	146.026	0.768560
$191$	103.803	0.543473	0.271737	−	0.962372i	$-0.412402\pi$
$191$	103.803	0.543473	0.271737	+	0.962372i	$0.412402\pi$
$192$	0	0
$193$	344.015i	1.78246i	0.453553	+	0.891229i	$0.350156\pi$
$193$	344.015i	1.78246i	−0.453553	+	0.891229i	$0.649844\pi$
$194$	− 222.850i	− 1.14871i
$195$	0	0
$196$	−263.721	−1.34552
$197$	99.0718i	0.502903i	0.967870	+	0.251451i	$0.0809078\pi$
$197$	99.0718i	0.502903i	−0.967870	+	0.251451i	$0.919092\pi$
$198$	0	0
$199$	−153.469	−0.771201	−0.385601	−	0.922666i	$-0.626006\pi$
$199$	−153.469	−0.771201	−0.385601	+	0.922666i	$0.626006\pi$
$200$	− 40.7769i	− 0.203885i
$201$	0	0
$202$	−198.138	−0.980880
$203$	3.10835	0.0153121
$204$	0	0
$205$	− 5.22638i	− 0.0254945i
$206$	100.363i	0.487199i
$207$	0	0
$208$	− 67.5780i	− 0.324894i
$209$	0	0
$210$	0	0
$211$	333.515i	1.58064i	0.612693	+	0.790321i	$0.290086\pi$
$211$	333.515i	1.58064i	−0.612693	+	0.790321i	$0.709914\pi$
$212$	−429.787	−2.02730
$213$	0	0
$214$	−16.6312	−0.0777158
$215$	− 132.250i	− 0.615116i
$216$	0	0
$217$	− 29.7478i	− 0.137086i
$218$	−421.649	−1.93417
$219$	0	0
$220$	0	0
$221$	−210.344	−0.951785
$222$	0	0
$223$	113.974	0.511093	0.255546	−	0.966797i	$-0.417745\pi$
$223$	113.974	0.511093	0.255546	+	0.966797i	$0.417745\pi$
$224$	38.2330	0.170683
$225$	0	0
$226$	− 143.133i	− 0.633330i
$227$	39.0154i	0.171874i	0.996301	+	0.0859370i	$0.0273884\pi$
$227$	39.0154i	0.171874i	−0.996301	+	0.0859370i	$0.972612\pi$
$228$	0	0
$229$	134.207	0.586055	0.293028	−	0.956104i	$-0.405337\pi$
$229$	134.207	0.586055	0.293028	+	0.956104i	$0.405337\pi$
$230$	89.0813i	0.387310i
$231$	0	0
$232$	−15.6819	−0.0675944
$233$	267.340i	1.14738i	0.819071	+	0.573692i	$0.194489\pi$
$233$	267.340i	1.14738i	−0.819071	+	0.573692i	$0.805511\pi$
$234$	0	0
$235$	91.0557	0.387471
$236$	171.116	0.725070
$237$	0	0
$238$	− 68.3450i	− 0.287164i
$239$	110.141i	0.460843i	0.973091	+	0.230422i	$0.0740105\pi$
$239$	110.141i	0.460843i	−0.973091	+	0.230422i	$0.925990\pi$
$240$	0	0
$241$	− 191.103i	− 0.792960i	−0.918043	−	0.396480i	$-0.870232\pi$
$241$	− 191.103i	− 0.792960i	0.918043	−	0.396480i	$-0.129768\pi$
$242$	0	0
$243$	0	0
$244$	153.672i	0.629804i
$245$	−192.774	−0.786833
$246$	0	0
$247$	100.902	0.408509
$248$	150.080i	0.605161i
$249$	0	0
$250$	− 418.565i	− 1.67426i
$251$	427.135	1.70173	0.850866	−	0.525383i	$-0.176078\pi$
$251$	427.135	1.70173	0.850866	+	0.525383i	$0.176078\pi$
$252$	0	0
$253$	0	0
$254$	256.774	1.01092
$255$	0	0
$256$	−19.0000	−0.0742188
$257$	−28.2806	−0.110041	−0.0550205	−	0.998485i	$-0.517522\pi$
$257$	−28.2806	−0.110041	−0.0550205	+	0.998485i	$0.517522\pi$
$258$	0	0
$259$	− 36.1461i	− 0.139560i
$260$	186.195i	0.716135i
$261$	0	0
$262$	−261.880	−0.999544
$263$	− 94.5506i	− 0.359508i	−0.983712	−	0.179754i	$-0.942470\pi$
$263$	− 94.5506i	− 0.359508i	0.983712	−	0.179754i	$-0.0575302\pi$
$264$	0	0
$265$	−314.164	−1.18552
$266$	32.7849i	0.123252i
$267$	0	0
$268$	418.654	1.56214
$269$	−233.994	−0.869865	−0.434933	−	0.900463i	$-0.643228\pi$
$269$	−233.994	−0.869865	−0.434933	+	0.900463i	$0.643228\pi$
$270$	0	0
$271$	143.420i	0.529224i	0.964355	+	0.264612i	$0.0852438\pi$
$271$	143.420i	0.529224i	−0.964355	+	0.264612i	$0.914756\pi$
$272$	− 196.442i	− 0.722212i
$273$	0	0
$274$	180.791i	0.659822i
$275$	0	0
$276$	0	0
$277$	− 389.079i	− 1.40462i	−0.711872	−	0.702309i	$-0.752152\pi$
$277$	− 389.079i	− 1.40462i	0.711872	−	0.702309i	$-0.247848\pi$
$278$	495.469	1.78226
$279$	0	0
$280$	−16.2755	−0.0581269
$281$	101.034i	0.359550i	0.983708	+	0.179775i	$0.0575370\pi$
$281$	101.034i	0.359550i	−0.983708	+	0.179775i	$0.942463\pi$
$282$	0	0
$283$	2.12002i	0.00749125i	0.999993	+	0.00374562i	$0.00119227\pi$
$283$	2.12002i	0.00749125i	−0.999993	+	0.00374562i	$0.998808\pi$
$284$	−341.013	−1.20075
$285$	0	0
$286$	0	0
$287$	1.17340	0.00408849
$288$	0	0
$289$	−322.448	−1.11574
$290$	−42.6099	−0.146931
$291$	0	0
$292$	− 515.158i	− 1.76424i
$293$	470.109i	1.60447i	0.597010	+	0.802234i	$0.296355\pi$
$293$	470.109i	1.60447i	−0.597010	+	0.802234i	$0.703645\pi$
$294$	0	0
$295$	125.082	0.424007
$296$	182.360i	0.616081i
$297$	0	0
$298$	791.187	2.65499
$299$	61.5537i	0.205865i
$300$	0	0
$301$	29.6919	0.0986443
$302$	886.964	2.93697
$303$	0	0
$304$	94.2326i	0.309976i
$305$	112.331i	0.368297i
$306$	0	0
$307$	99.4185i	0.323839i	0.986804	+	0.161919i	$0.0517685\pi$
$307$	99.4185i	0.323839i	−0.986804	+	0.161919i	$0.948232\pi$
$308$	0	0
$309$	0	0
$310$	407.788i	1.31545i
$311$	−339.214	−1.09072	−0.545359	−	0.838202i	$-0.683607\pi$
$311$	−339.214	−1.09072	−0.545359	+	0.838202i	$0.683607\pi$
$312$	0	0
$313$	386.389	1.23447	0.617235	−	0.786779i	$-0.288253\pi$
$313$	386.389	1.23447	0.617235	+	0.786779i	$0.288253\pi$
$314$	− 851.349i	− 2.71130i
$315$	0	0
$316$	360.684i	1.14141i
$317$	253.072	0.798334	0.399167	−	0.916878i	$-0.369299\pi$
$317$	253.072	0.798334	0.399167	+	0.916878i	$0.369299\pi$
$318$	0	0
$319$	0	0
$320$	−396.997	−1.24062
$321$	0	0
$322$	−20.0000	−0.0621118
$323$	293.310	0.908080
$324$	0	0
$325$	− 76.5586i	− 0.235565i
$326$	213.486i	0.654864i
$327$	0	0
$328$	−5.91988	−0.0180484
$329$	20.4433i	0.0621376i
$330$	0	0
$331$	372.116	1.12422	0.562109	−	0.827063i	$-0.309990\pi$
$331$	372.116	1.12422	0.562109	+	0.827063i	$0.309990\pi$
$332$	310.297i	0.934629i
$333$	0	0
$334$	−889.420	−2.66293
$335$	306.026	0.913511
$336$	0	0
$337$	− 82.2149i	− 0.243961i	−0.992532	−	0.121981i	$-0.961075\pi$
$337$	− 82.2149i	− 0.243961i	0.992532	−	0.121981i	$-0.0389245\pi$
$338$	− 297.425i	− 0.879956i
$339$	0	0
$340$	541.248i	1.59191i
$341$	0	0
$342$	0	0
$343$	− 87.2852i	− 0.254476i
$344$	−149.798	−0.435460
$345$	0	0
$346$	563.856	1.62964
$347$	229.439i	0.661207i	0.943770	+	0.330603i	$0.107252\pi$
$347$	229.439i	0.661207i	−0.943770	+	0.330603i	$0.892748\pi$
$348$	0	0
$349$	− 232.370i	− 0.665816i	−0.942959	−	0.332908i	$-0.891970\pi$
$349$	− 232.370i	− 0.665816i	0.942959	−	0.332908i	$-0.108030\pi$
$350$	24.8754	0.0710725
$351$	0	0
$352$	0	0
$353$	−34.6443	−0.0981426	−0.0490713	−	0.998795i	$-0.515626\pi$
$353$	−34.6443	−0.0981426	−0.0490713	+	0.998795i	$0.515626\pi$
$354$	0	0
$355$	−249.272	−0.702176
$356$	−340.897	−0.957575
$357$	0	0
$358$	6.31142i	0.0176297i
$359$	451.793i	1.25848i	0.777213	+	0.629238i	$0.216633\pi$
$359$	451.793i	1.25848i	−0.777213	+	0.629238i	$0.783367\pi$
$360$	0	0
$361$	220.300	0.610249
$362$	224.368i	0.619802i
$363$	0	0
$364$	−41.8034	−0.114845
$365$	− 376.568i	− 1.03169i
$366$	0	0
$367$	368.827	1.00498	0.502489	−	0.864584i	$-0.332418\pi$
$367$	368.827	1.00498	0.502489	+	0.864584i	$0.332418\pi$
$368$	−57.4853	−0.156210
$369$	0	0
$370$	495.497i	1.33918i
$371$	− 70.5342i	− 0.190119i
$372$	0	0
$373$	593.166i	1.59026i	0.606440	+	0.795129i	$0.292597\pi$
$373$	593.166i	1.59026i	−0.606440	+	0.795129i	$0.707403\pi$
$374$	0	0
$375$	0	0
$376$	− 103.138i	− 0.274303i
$377$	−29.4427	−0.0780974
$378$	0	0
$379$	−281.251	−0.742087	−0.371044	−	0.928615i	$-0.621000\pi$
$379$	−281.251	−0.742087	−0.371044	+	0.928615i	$0.621000\pi$
$380$	− 259.635i	− 0.683251i
$381$	0	0
$382$	− 319.474i	− 0.836319i
$383$	−637.558	−1.66464	−0.832321	−	0.554294i	$-0.812988\pi$
$383$	−637.558	−1.66464	−0.832321	+	0.554294i	$0.812988\pi$
$384$	0	0
$385$	0	0
$386$	1058.77	2.74292
$387$	0	0
$388$	−396.228	−1.02121
$389$	−667.469	−1.71586	−0.857929	−	0.513768i	$-0.828249\pi$
$389$	−667.469	−1.71586	−0.857929	+	0.513768i	$0.828249\pi$
$390$	0	0
$391$	178.930i	0.457621i
$392$	218.354i	0.557024i
$393$	0	0
$394$	304.912	0.773888
$395$	263.652i	0.667473i
$396$	0	0
$397$	−5.37384	−0.0135361	−0.00676805	−	0.999977i	$-0.502154\pi$
$397$	−5.37384	−0.0135361	−0.00676805	+	0.999977i	$0.502154\pi$
$398$	472.329i	1.18676i
$399$	0	0
$400$	71.4984	0.178746
$401$	282.795	0.705225	0.352613	−	0.935769i	$-0.385293\pi$
$401$	282.795	0.705225	0.352613	+	0.935769i	$0.385293\pi$
$402$	0	0
$403$	281.775i	0.699193i
$404$	352.290i	0.872005i
$405$	0	0
$406$	− 9.56652i	− 0.0235629i
$407$	0	0
$408$	0	0
$409$	31.0580i	0.0759364i	0.999279	+	0.0379682i	$0.0120886\pi$
$409$	31.0580i	0.0759364i	−0.999279	+	0.0379682i	$0.987911\pi$
$410$	−16.0851	−0.0392321
$411$	0	0
$412$	178.446	0.433121
$413$	28.0827i	0.0679968i
$414$	0	0
$415$	226.820i	0.546553i
$416$	−362.148	−0.870548
$417$	0	0
$418$	0	0
$419$	245.156	0.585098	0.292549	−	0.956251i	$-0.405497\pi$
$419$	245.156	0.585098	0.292549	+	0.956251i	$0.405497\pi$
$420$	0	0
$421$	−77.4102	−0.183872	−0.0919361	−	0.995765i	$-0.529306\pi$
$421$	−77.4102	−0.183872	−0.0919361	+	0.995765i	$0.529306\pi$
$422$	1026.45	2.43236
$423$	0	0
$424$	355.851i	0.839272i
$425$	− 222.547i	− 0.523641i
$426$	0	0
$427$	−25.2198	−0.0590628
$428$	29.5703i	0.0690896i
$429$	0	0
$430$	−407.023	−0.946566
$431$	449.473i	1.04286i	0.853294	+	0.521430i	$0.174601\pi$
$431$	449.473i	1.04286i	−0.853294	+	0.521430i	$0.825399\pi$
$432$	0	0
$433$	144.359	0.333392	0.166696	−	0.986008i	$-0.446690\pi$
$433$	144.359	0.333392	0.166696	+	0.986008i	$0.446690\pi$
$434$	−91.5542	−0.210954
$435$	0	0
$436$	749.695i	1.71948i
$437$	− 85.8321i	− 0.196412i
$438$	0	0
$439$	− 155.406i	− 0.354001i	−0.984211	−	0.177000i	$-0.943361\pi$
$439$	− 155.406i	− 0.354001i	0.984211	−	0.177000i	$-0.0566394\pi$
$440$	0	0
$441$	0	0
$442$	647.374i	1.46465i
$443$	−324.307	−0.732070	−0.366035	−	0.930601i	$-0.619285\pi$
$443$	−324.307	−0.732070	−0.366035	+	0.930601i	$0.619285\pi$
$444$	0	0
$445$	−249.187	−0.559971
$446$	− 350.775i	− 0.786491i
$447$	0	0
$448$	− 89.1314i	− 0.198954i
$449$	391.402	0.871720	0.435860	−	0.900015i	$-0.356444\pi$
$449$	391.402	0.871720	0.435860	+	0.900015i	$0.356444\pi$
$450$	0	0
$451$	0	0
$452$	−254.490	−0.563032
$453$	0	0
$454$	120.077	0.264487
$455$	−30.5573	−0.0671589
$456$	0	0
$457$	− 52.2455i	− 0.114323i	−0.998365	−	0.0571614i	$-0.981795\pi$
$457$	− 52.2455i	− 0.114323i	0.998365	−	0.0571614i	$-0.0182050\pi$
$458$	− 413.046i	− 0.901846i
$459$	0	0
$460$	158.387	0.344320
$461$	339.293i	0.735994i	0.929827	+	0.367997i	$0.119956\pi$
$461$	339.293i	0.735994i	−0.929827	+	0.367997i	$0.880044\pi$
$462$	0	0
$463$	676.869	1.46192	0.730960	−	0.682420i	$-0.239072\pi$
$463$	676.869	1.46192	0.730960	+	0.682420i	$0.239072\pi$
$464$	− 27.4967i	− 0.0592601i
$465$	0	0
$466$	822.789	1.76564
$467$	−368.885	−0.789905	−0.394952	−	0.918702i	$-0.629239\pi$
$467$	−368.885	−0.789905	−0.394952	+	0.918702i	$0.629239\pi$
$468$	0	0
$469$	68.7072i	0.146497i
$470$	− 280.241i	− 0.596257i
$471$	0	0
$472$	− 141.679i	− 0.300168i
$473$	0	0
$474$	0	0
$475$	106.755i	0.224748i
$476$	−121.518	−0.255290
$477$	0	0
$478$	338.981	0.709165
$479$	475.435i	0.992558i	0.868163	+	0.496279i	$0.165301\pi$
$479$	475.435i	0.992558i	−0.868163	+	0.496279i	$0.834699\pi$
$480$	0	0
$481$	342.380i	0.711809i
$482$	−588.156	−1.22024
$483$	0	0
$484$	0	0
$485$	−289.633	−0.597182
$486$	0	0
$487$	−179.157	−0.367879	−0.183940	−	0.982938i	$-0.558885\pi$
$487$	−179.157	−0.367879	−0.183940	+	0.982938i	$0.558885\pi$
$488$	127.236	0.260730
$489$	0	0
$490$	593.297i	1.21081i
$491$	− 642.756i	− 1.30908i	−0.756029	−	0.654538i	$-0.772863\pi$
$491$	− 642.756i	− 1.30908i	0.756029	−	0.654538i	$-0.227137\pi$
$492$	0	0
$493$	−85.5867	−0.173604
$494$	− 310.543i	− 0.628631i
$495$	0	0
$496$	−263.151	−0.530546
$497$	− 55.9651i	− 0.112606i
$498$	0	0
$499$	−525.687	−1.05348	−0.526740	−	0.850026i	$-0.676586\pi$
$499$	−525.687	−1.05348	−0.526740	+	0.850026i	$0.676586\pi$
$500$	−744.210	−1.48842
$501$	0	0
$502$	− 1314.59i	− 2.61870i
$503$	641.330i	1.27501i	0.770446	+	0.637505i	$0.220033\pi$
$503$	641.330i	1.27501i	−0.770446	+	0.637505i	$0.779967\pi$
$504$	0	0
$505$	257.515i	0.509932i
$506$	0	0
$507$	0	0
$508$	− 456.545i	− 0.898711i
$509$	455.331	0.894560	0.447280	−	0.894394i	$-0.352393\pi$
$509$	455.331	0.894560	0.447280	+	0.894394i	$0.352393\pi$
$510$	0	0
$511$	84.5449	0.165450
$512$	− 482.186i	− 0.941770i
$513$	0	0
$514$	87.0386i	0.169336i
$515$	130.440	0.253281
$516$	0	0
$517$	0	0
$518$	−111.246	−0.214761
$519$	0	0
$520$	154.164	0.296469
$521$	181.234	0.347858	0.173929	−	0.984758i	$-0.444354\pi$
$521$	181.234	0.347858	0.173929	+	0.984758i	$0.444354\pi$
$522$	0	0
$523$	− 342.768i	− 0.655387i	−0.944784	−	0.327694i	$-0.893729\pi$
$523$	− 342.768i	− 0.655387i	0.944784	−	0.327694i	$-0.106271\pi$
$524$	465.625i	0.888597i
$525$	0	0
$526$	−290.997	−0.553226
$527$	819.088i	1.55425i
$528$	0	0
$529$	−476.639	−0.901020
$530$	966.898i	1.82434i
$531$	0	0
$532$	58.2918	0.109571
$533$	−11.1146	−0.0208528
$534$	0	0
$535$	21.6152i	0.0404023i
$536$	− 346.634i	− 0.646704i
$537$	0	0
$538$	720.159i	1.33859i
$539$	0	0
$540$	0	0
$541$	− 305.155i	− 0.564058i	−0.959406	−	0.282029i	$-0.908993\pi$
$541$	− 305.155i	− 0.564058i	0.959406	−	0.282029i	$-0.0910074\pi$
$542$	441.400	0.814391
$543$	0	0
$544$	−1052.72	−1.93515
$545$	548.009i	1.00552i
$546$	0	0
$547$	− 151.284i	− 0.276571i	−0.990392	−	0.138285i	$-0.955841\pi$
$547$	− 151.284i	− 0.276571i	0.990392	−	0.138285i	$-0.0441591\pi$
$548$	321.448	0.586583
$549$	0	0
$550$	0	0
$551$	41.0557	0.0745113
$552$	0	0
$553$	−59.1935	−0.107041
$554$	−1197.46	−2.16149
$555$	0	0
$556$	− 880.946i	− 1.58444i
$557$	109.012i	0.195713i	0.995201	+	0.0978567i	$0.0311987\pi$
$557$	109.012i	0.195713i	−0.995201	+	0.0978567i	$0.968801\pi$
$558$	0	0
$559$	−281.246	−0.503124
$560$	− 28.5376i	− 0.0509600i
$561$	0	0
$562$	310.949	0.553291
$563$	− 996.668i	− 1.77028i	−0.465324	−	0.885140i	$-0.654062\pi$
$563$	− 996.668i	− 1.77028i	0.465324	−	0.885140i	$-0.345938\pi$
$564$	0	0
$565$	−186.026	−0.329250
$566$	6.52476	0.0115278
$567$	0	0
$568$	282.349i	0.497093i
$569$	99.3280i	0.174566i	0.996184	+	0.0872829i	$0.0278184\pi$
$569$	99.3280i	0.174566i	−0.996184	+	0.0872829i	$0.972182\pi$
$570$	0	0
$571$	− 930.127i	− 1.62894i	−0.580202	−	0.814472i	$-0.697027\pi$
$571$	− 930.127i	− 1.62894i	0.580202	−	0.814472i	$-0.302973\pi$
$572$	0	0
$573$	0	0
$574$	− 3.61134i	− 0.00629153i
$575$	−65.1246	−0.113260
$576$	0	0
$577$	−347.349	−0.601992	−0.300996	−	0.953625i	$-0.597319\pi$
$577$	−347.349	−0.601992	−0.300996	+	0.953625i	$0.597319\pi$
$578$	992.392i	1.71694i
$579$	0	0
$580$	75.7606i	0.130622i
$581$	−50.9242	−0.0876492
$582$	0	0
$583$	0	0
$584$	−426.536	−0.730370
$585$	0	0
$586$	1446.85	2.46902
$587$	635.392	1.08244	0.541220	−	0.840881i	$-0.317963\pi$
$587$	635.392	1.08244	0.541220	+	0.840881i	$0.317963\pi$
$588$	0	0
$589$	− 392.914i	− 0.667087i
$590$	− 384.963i	− 0.652480i
$591$	0	0
$592$	−319.751	−0.540120
$593$	− 341.152i	− 0.575299i	−0.957736	−	0.287650i	$-0.907126\pi$
$593$	− 341.152i	− 0.575299i	0.957736	−	0.287650i	$-0.0928739\pi$
$594$	0	0
$595$	−88.8266	−0.149288
$596$	− 1406.73i	− 2.36029i
$597$	0	0
$598$	189.443	0.316794
$599$	447.659	0.747345	0.373672	−	0.927561i	$-0.378098\pi$
$599$	447.659	0.747345	0.373672	+	0.927561i	$0.378098\pi$
$600$	0	0
$601$	− 801.828i	− 1.33416i	−0.744988	−	0.667078i	$-0.767545\pi$
$601$	− 801.828i	− 1.33416i	0.744988	−	0.667078i	$-0.232455\pi$
$602$	− 91.3824i	− 0.151798i
$603$	0	0
$604$	− 1577.03i	− 2.61097i
$605$	0	0
$606$	0	0
$607$	− 1019.48i	− 1.67954i	−0.542940	−	0.839771i	$-0.682689\pi$
$607$	− 1019.48i	− 1.67954i	0.542940	−	0.839771i	$-0.317311\pi$
$608$	504.989	0.830574
$609$	0	0
$610$	345.718	0.566751
$611$	− 193.642i	− 0.316926i
$612$	0	0
$613$	748.218i	1.22058i	0.792177	+	0.610292i	$0.208948\pi$
$613$	748.218i	1.22058i	−0.792177	+	0.610292i	$0.791052\pi$
$614$	305.979	0.498337
$615$	0	0
$616$	0	0
$617$	−517.100	−0.838088	−0.419044	−	0.907966i	$-0.637635\pi$
$617$	−517.100	−0.838088	−0.419044	+	0.907966i	$0.637635\pi$
$618$	0	0
$619$	−693.228	−1.11992	−0.559958	−	0.828521i	$-0.689183\pi$
$619$	−693.228	−1.11992	−0.559958	+	0.828521i	$0.689183\pi$
$620$	725.050	1.16943
$621$	0	0
$622$	1043.99i	1.67844i
$623$	− 55.9460i	− 0.0898010i
$624$	0	0
$625$	−319.000	−0.510400
$626$	− 1189.18i	− 1.89965i
$627$	0	0
$628$	−1513.70	−2.41035
$629$	995.261i	1.58229i
$630$	0	0
$631$	−896.804	−1.42124	−0.710621	−	0.703575i	$-0.751586\pi$
$631$	−896.804	−1.42124	−0.710621	+	0.703575i	$0.751586\pi$
$632$	298.636	0.472526
$633$	0	0
$634$	− 778.875i	− 1.22851i
$635$	− 333.724i	− 0.525549i
$636$	0	0
$637$	409.958i	0.643577i
$638$	0	0
$639$	0	0
$640$	540.662i	0.844785i
$641$	−568.759	−0.887299	−0.443650	−	0.896200i	$-0.646317\pi$
$641$	−568.759	−0.887299	−0.443650	+	0.896200i	$0.646317\pi$
$642$	0	0
$643$	660.510	1.02723	0.513616	−	0.858020i	$-0.328306\pi$
$643$	660.510	1.02723	0.513616	+	0.858020i	$0.328306\pi$
$644$	35.5601i	0.0552175i
$645$	0	0
$646$	− 902.715i	− 1.39739i
$647$	−35.3514	−0.0546389	−0.0273194	−	0.999627i	$-0.508697\pi$
$647$	−35.3514	−0.0546389	−0.0273194	+	0.999627i	$0.508697\pi$
$648$	0	0
$649$	0	0
$650$	−235.623	−0.362497
$651$	0	0
$652$	379.579	0.582176
$653$	−419.443	−0.642332	−0.321166	−	0.947023i	$-0.604075\pi$
$653$	−419.443	−0.642332	−0.321166	+	0.947023i	$0.604075\pi$
$654$	0	0
$655$	340.360i	0.519634i
$656$	− 10.3799i	− 0.0158231i
$657$	0	0
$658$	62.9180	0.0956200
$659$	− 1281.39i	− 1.94445i	−0.234058	−	0.972223i	$-0.575200\pi$
$659$	− 1281.39i	− 1.94445i	0.234058	−	0.972223i	$-0.424800\pi$
$660$	0	0
$661$	70.6950	0.106952	0.0534758	−	0.998569i	$-0.482970\pi$
$661$	70.6950	0.106952	0.0534758	+	0.998569i	$0.482970\pi$
$662$	− 1145.26i	− 1.73000i
$663$	0	0
$664$	256.917	0.386923
$665$	42.6099	0.0640750
$666$	0	0
$667$	25.0455i	0.0375494i
$668$	1581.39i	2.36736i
$669$	0	0
$670$	− 941.852i	− 1.40575i
$671$	0	0
$672$	0	0
$673$	800.003i	1.18871i	0.804202	+	0.594356i	$0.202593\pi$
$673$	800.003i	1.18871i	−0.804202	+	0.594356i	$0.797407\pi$
$674$	−253.031	−0.375417
$675$	0	0
$676$	−528.823	−0.782283
$677$	− 936.059i	− 1.38266i	−0.722541	−	0.691329i	$-0.757026\pi$
$677$	− 936.059i	− 1.38266i	0.722541	−	0.691329i	$-0.242974\pi$
$678$	0	0
$679$	− 65.0267i	− 0.0957683i
$680$	448.138	0.659026
$681$	0	0
$682$	0	0
$683$	241.319	0.353322	0.176661	−	0.984272i	$-0.443470\pi$
$683$	241.319	0.353322	0.176661	+	0.984272i	$0.443470\pi$
$684$	0	0
$685$	234.971	0.343023
$686$	−268.636	−0.391598
$687$	0	0
$688$	− 262.657i	− 0.381769i
$689$	668.110i	0.969680i
$690$	0	0
$691$	−910.877	−1.31820	−0.659101	−	0.752055i	$-0.729063\pi$
$691$	−910.877	−1.31820	−0.659101	+	0.752055i	$0.729063\pi$
$692$	− 1002.54i	− 1.44876i
$693$	0	0
$694$	706.140	1.01749
$695$	− 643.951i	− 0.926548i
$696$	0	0
$697$	−32.3088	−0.0463541
$698$	−715.161	−1.02459
$699$	0	0
$700$	− 44.2286i	− 0.0631837i
$701$	− 432.048i	− 0.616330i	−0.951333	−	0.308165i	$-0.900285\pi$
$701$	− 432.048i	− 0.616330i	0.951333	−	0.308165i	$-0.0997150\pi$
$702$	0	0
$703$	− 477.424i	− 0.679124i
$704$	0	0
$705$	0	0
$706$	106.624i	0.151026i
$707$	−57.8158	−0.0817763
$708$	0	0
$709$	−876.318	−1.23599	−0.617996	−	0.786181i	$-0.712055\pi$
$709$	−876.318	−1.23599	−0.617996	+	0.786181i	$0.712055\pi$
$710$	767.182i	1.08054i
$711$	0	0
$712$	282.252i	0.396422i
$713$	239.692	0.336174
$714$	0	0
$715$	0	0
$716$	11.2217	0.0156728
$717$	0	0
$718$	1390.48	1.93660
$719$	−502.296	−0.698603	−0.349302	−	0.937010i	$-0.613581\pi$
$719$	−502.296	−0.698603	−0.349302	+	0.937010i	$0.613581\pi$
$720$	0	0
$721$	29.2855i	0.0406179i
$722$	− 678.013i	− 0.939077i
$723$	0	0
$724$	398.928	0.551006
$725$	− 31.1508i	− 0.0429666i
$726$	0	0
$727$	−393.878	−0.541786	−0.270893	−	0.962609i	$-0.587319\pi$
$727$	−393.878	−0.541786	−0.270893	+	0.962609i	$0.587319\pi$
$728$	34.6120i	0.0475439i
$729$	0	0
$730$	−1158.96	−1.58761
$731$	−817.551	−1.11840
$732$	0	0
$733$	− 620.763i	− 0.846880i	−0.905924	−	0.423440i	$-0.860822\pi$
$733$	− 620.763i	− 0.846880i	0.905924	−	0.423440i	$-0.139178\pi$
$734$	− 1135.13i	− 1.54650i
$735$	0	0
$736$	308.061i	0.418562i
$737$	0	0
$738$	0	0
$739$	850.002i	1.15021i	0.818081	+	0.575103i	$0.195038\pi$
$739$	850.002i	1.15021i	−0.818081	+	0.575103i	$0.804962\pi$
$740$	880.997	1.19054
$741$	0	0
$742$	−217.082	−0.292563
$743$	− 405.618i	− 0.545920i	−0.962025	−	0.272960i	$-0.911997\pi$
$743$	− 405.618i	− 0.545920i	0.962025	−	0.272960i	$-0.0880026\pi$
$744$	0	0
$745$	− 1028.29i	− 1.38025i
$746$	1825.58	2.44716
$747$	0	0
$748$	0	0
$749$	−4.85292	−0.00647919
$750$	0	0
$751$	−849.423	−1.13106	−0.565528	−	0.824729i	$-0.691327\pi$
$751$	−849.423	−1.13106	−0.565528	+	0.824729i	$0.691327\pi$
$752$	180.843	0.240483
$753$	0	0
$754$	90.6154i	0.120180i
$755$	− 1152.77i	− 1.52685i
$756$	0	0
$757$	600.521	0.793291	0.396645	−	0.917972i	$-0.370174\pi$
$757$	600.521	0.793291	0.396645	+	0.917972i	$0.370174\pi$
$758$	865.602i	1.14196i
$759$	0	0
$760$	−214.971	−0.282856
$761$	349.235i	0.458916i	0.973319	+	0.229458i	$0.0736953\pi$
$761$	349.235i	0.458916i	−0.973319	+	0.229458i	$0.926305\pi$
$762$	0	0
$763$	−123.036	−0.161252
$764$	−568.026	−0.743490
$765$	0	0
$766$	1962.20i	2.56162i
$767$	− 266.003i	− 0.346809i
$768$	0	0
$769$	− 768.616i	− 0.999500i	−0.866170	−	0.499750i	$-0.833425\pi$
$769$	− 768.616i	− 0.999500i	0.866170	−	0.499750i	$-0.166575\pi$
$770$	0	0
$771$	0	0
$772$	− 1882.49i	− 2.43846i
$773$	−6.98684	−0.00903861	−0.00451930	−	0.999990i	$-0.501439\pi$
$773$	−6.98684	−0.00903861	−0.00451930	+	0.999990i	$0.501439\pi$
$774$	0	0
$775$	−298.122	−0.384673
$776$	328.065i	0.422764i
$777$	0	0
$778$	2054.26i	2.64044i
$779$	15.4984	0.0198953
$780$	0	0
$781$	0	0
$782$	550.689	0.704206
$783$	0	0
$784$	−382.862	−0.488345
$785$	−1106.48	−1.40953
$786$	0	0
$787$	− 1102.71i	− 1.40116i	−0.713573	−	0.700581i	$-0.752924\pi$
$787$	− 1102.71i	− 1.40116i	0.713573	−	0.700581i	$-0.247076\pi$
$788$	− 542.135i	− 0.687988i
$789$	0	0
$790$	811.437	1.02713
$791$	− 41.7655i	− 0.0528009i
$792$	0	0
$793$	238.885	0.301243
$794$	16.5390i	0.0208299i
$795$	0	0
$796$	839.803	1.05503
$797$	721.364	0.905099	0.452549	−	0.891739i	$-0.350515\pi$
$797$	721.364	0.905099	0.452549	+	0.891739i	$0.350515\pi$
$798$	0	0
$799$	− 562.894i	− 0.704499i
$800$	− 383.157i	− 0.478947i
$801$	0	0
$802$	− 870.354i	− 1.08523i
$803$	0	0
$804$	0	0
$805$	25.9936i	0.0322902i
$806$	867.214	1.07595
$807$	0	0
$808$	291.686	0.360997
$809$	802.474i	0.991933i	0.868342	+	0.495966i	$0.165186\pi$
$809$	802.474i	0.991933i	−0.868342	+	0.495966i	$0.834814\pi$
$810$	0	0
$811$	495.868i	0.611428i	0.952123	+	0.305714i	$0.0988951\pi$
$811$	495.868i	0.611428i	−0.952123	+	0.305714i	$0.901105\pi$
$812$	−17.0093	−0.0209474
$813$	0	0
$814$	0	0
$815$	277.463	0.340445
$816$	0	0
$817$	392.177	0.480021
$818$	95.5867	0.116854
$819$	0	0
$820$	28.5995i	0.0348774i
$821$	291.172i	0.354655i	0.984152	+	0.177328i	$0.0567452\pi$
$821$	291.172i	0.354655i	−0.984152	+	0.177328i	$0.943255\pi$
$822$	0	0
$823$	1074.98	1.30618	0.653089	−	0.757281i	$-0.273473\pi$
$823$	1074.98	1.30618	0.653089	+	0.757281i	$0.273473\pi$
$824$	− 147.748i	− 0.179306i
$825$	0	0
$826$	86.4296	0.104636
$827$	− 456.794i	− 0.552351i	−0.961107	−	0.276175i	$-0.910933\pi$
$827$	− 456.794i	− 0.552351i	0.961107	−	0.276175i	$-0.0890671\pi$
$828$	0	0
$829$	2.20044	0.00265433	0.00132717	−	0.999999i	$-0.499578\pi$
$829$	2.20044	0.00265433	0.00132717	+	0.999999i	$0.499578\pi$
$830$	698.079	0.841059
$831$	0	0
$832$	844.264i	1.01474i
$833$	1191.70i	1.43062i
$834$	0	0
$835$	1155.96i	1.38438i
$836$	0	0
$837$	0	0
$838$	− 754.512i	− 0.900373i
$839$	−1452.42	−1.73113	−0.865564	−	0.500798i	$-0.833040\pi$
$839$	−1452.42	−1.73113	−0.865564	+	0.500798i	$0.833040\pi$
$840$	0	0
$841$	829.020	0.985755
$842$	238.244i	0.282950i
$843$	0	0
$844$	− 1825.04i	− 2.16237i
$845$	−386.557	−0.457464
$846$	0	0
$847$	0	0
$848$	−623.951	−0.735792
$849$	0	0
$850$	−684.930	−0.805800
$851$	291.246	0.342240
$852$	0	0
$853$	1059.50i	1.24208i	0.783777	+	0.621042i	$0.213290\pi$
$853$	1059.50i	1.24208i	−0.783777	+	0.621042i	$0.786710\pi$
$854$	77.6186i	0.0908883i
$855$	0	0
$856$	24.4834	0.0286021
$857$	951.067i	1.10976i	0.831929	+	0.554882i	$0.187237\pi$
$857$	951.067i	1.10976i	−0.831929	+	0.554882i	$0.812763\pi$
$858$	0	0
$859$	1146.23	1.33438	0.667188	−	0.744890i	$-0.267498\pi$
$859$	1146.23	1.33438	0.667188	+	0.744890i	$0.267498\pi$
$860$	723.689i	0.841499i
$861$	0	0
$862$	1383.34	1.60480
$863$	−945.293	−1.09536	−0.547678	−	0.836689i	$-0.684488\pi$
$863$	−945.293	−1.09536	−0.547678	+	0.836689i	$0.684488\pi$
$864$	0	0
$865$	− 732.832i	− 0.847204i
$866$	− 444.291i	− 0.513038i
$867$	0	0
$868$	162.784i	0.187539i
$869$	0	0
$870$	0	0
$871$	− 650.804i	− 0.747192i
$872$	620.725	0.711841
$873$	0	0
$874$	−264.164	−0.302247
$875$	− 122.136i	− 0.139584i
$876$	0	0
$877$	− 158.698i	− 0.180956i	−0.995898	−	0.0904780i	$-0.971161\pi$
$877$	− 158.698i	− 0.180956i	0.995898	−	0.0904780i	$-0.0288395\pi$
$878$	−478.292	−0.544751
$879$	0	0
$880$	0	0
$881$	−402.370	−0.456719	−0.228360	−	0.973577i	$-0.573336\pi$
$881$	−402.370	−0.456719	−0.228360	+	0.973577i	$0.573336\pi$
$882$	0	0
$883$	791.257	0.896101	0.448051	−	0.894008i	$-0.352118\pi$
$883$	791.257	0.896101	0.448051	+	0.894008i	$0.352118\pi$
$884$	1151.03	1.30207
$885$	0	0
$886$	998.114i	1.12654i
$887$	− 847.120i	− 0.955040i	−0.878621	−	0.477520i	$-0.841536\pi$
$887$	− 847.120i	− 0.955040i	0.878621	−	0.477520i	$-0.158464\pi$
$888$	0	0
$889$	74.9256	0.0842808
$890$	766.920i	0.861707i
$891$	0	0
$892$	−623.680	−0.699192
$893$	270.019i	0.302373i
$894$	0	0
$895$	8.20281	0.00916515
$896$	−121.386	−0.135476
$897$	0	0
$898$	− 1204.61i	− 1.34144i
$899$	114.651i	0.127532i
$900$	0	0
$901$	1942.12i	2.15552i
$902$	0	0
$903$	0	0
$904$	210.711i	0.233087i
$905$	291.607	0.322217
$906$	0	0
$907$	426.711	0.470464	0.235232	−	0.971939i	$-0.424415\pi$
$907$	426.711	0.470464	0.235232	+	0.971939i	$0.424415\pi$
$908$	− 213.498i	− 0.235129i
$909$	0	0
$910$	94.0456i	0.103347i
$911$	−140.111	−0.153800	−0.0768998	−	0.997039i	$-0.524502\pi$
$911$	−140.111	−0.153800	−0.0768998	+	0.997039i	$0.524502\pi$
$912$	0	0
$913$	0	0
$914$	−160.795	−0.175925
$915$	0	0
$916$	−734.397	−0.801743
$917$	−76.4157	−0.0833323
$918$	0	0
$919$	1039.71i	1.13135i	0.824630	+	0.565673i	$0.191383\pi$
$919$	1039.71i	1.13135i	−0.824630	+	0.565673i	$0.808617\pi$
$920$	− 131.140i	− 0.142543i
$921$	0	0
$922$	1044.24	1.13258
$923$	530.109i	0.574333i
$924$	0	0
$925$	−362.243	−0.391614
$926$	− 2083.19i	− 2.24966i
$927$	0	0
$928$	−147.354	−0.158786
$929$	−1643.92	−1.76956	−0.884778	−	0.466014i	$-0.845690\pi$
$929$	−1643.92	−1.76956	−0.884778	+	0.466014i	$0.845690\pi$
$930$	0	0
$931$	− 571.657i	− 0.614025i
$932$	− 1462.92i	− 1.56966i
$933$	0	0
$934$	1135.31i	1.21554i
$935$	0	0
$936$	0	0
$937$	725.529i	0.774311i	0.922014	+	0.387156i	$0.126542\pi$
$937$	725.529i	0.774311i	−0.922014	+	0.387156i	$0.873458\pi$
$938$	211.459	0.225436
$939$	0	0
$940$	−498.269	−0.530074
$941$	183.401i	0.194900i	0.995240	+	0.0974500i	$0.0310686\pi$
$941$	183.401i	0.194900i	−0.995240	+	0.0974500i	$0.968931\pi$
$942$	0	0
$943$	9.45461i	0.0100261i
$944$	248.421	0.263158
$945$	0	0
$946$	0	0
$947$	926.439	0.978288	0.489144	−	0.872203i	$-0.337309\pi$
$947$	926.439	0.978288	0.489144	+	0.872203i	$0.337309\pi$
$948$	0	0
$949$	−800.820	−0.843857
$950$	328.559	0.345852
$951$	0	0
$952$	100.613i	0.105686i
$953$	− 249.834i	− 0.262155i	−0.991372	−	0.131078i	$-0.958156\pi$
$953$	− 249.834i	− 0.262155i	0.991372	−	0.131078i	$-0.0418438\pi$
$954$	0	0
$955$	−415.214	−0.434779
$956$	− 602.709i	− 0.630449i
$957$	0	0
$958$	1463.24	1.52739
$959$	52.7542i	0.0550096i
$960$	0	0
$961$	136.240	0.141769
$962$	1053.74	1.09536
$963$	0	0
$964$	1045.74i	1.08480i
$965$	− 1376.06i	− 1.42597i
$966$	0	0
$967$	554.026i	0.572933i	0.958090	+	0.286466i	$0.0924806\pi$
$967$	554.026i	0.572933i	−0.958090	+	0.286466i	$0.907519\pi$
$968$	0	0
$969$	0	0
$970$	891.399i	0.918968i
$971$	1141.15	1.17523	0.587615	−	0.809141i	$-0.300067\pi$
$971$	1141.15	1.17523	0.587615	+	0.809141i	$0.300067\pi$
$972$	0	0
$973$	144.576	0.148588
$974$	551.389i	0.566108i
$975$	0	0
$976$	223.096i	0.228582i
$977$	−128.695	−0.131725	−0.0658624	−	0.997829i	$-0.520980\pi$
$977$	−128.695	−0.131725	−0.0658624	+	0.997829i	$0.520980\pi$
$978$	0	0
$979$	0	0
$980$	1054.89	1.07641
$981$	0	0
$982$	−1978.20	−2.01446
$983$	1334.87	1.35795	0.678976	−	0.734160i	$-0.262424\pi$
$983$	1334.87	1.35795	0.678976	+	0.734160i	$0.262424\pi$
$984$	0	0
$985$	− 396.287i	− 0.402322i
$986$	263.409i	0.267149i
$987$	0	0
$988$	−552.148	−0.558854
$989$	239.242i	0.241903i
$990$	0	0
$991$	762.024	0.768944	0.384472	−	0.923137i	$-0.374383\pi$
$991$	762.024	0.768944	0.384472	+	0.923137i	$0.374383\pi$
$992$	1410.22i	1.42159i
$993$	0	0
$994$	−172.243	−0.173283
$995$	613.876	0.616961
$996$	0	0
$997$	1205.57i	1.20919i	0.796532	+	0.604597i	$0.206666\pi$
$997$	1205.57i	1.20919i	−0.796532	+	0.604597i	$0.793334\pi$
$998$	1617.90i	1.62114i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.e.604.1		4
3.2	odd	2		121.3.b.b.120.4		4
11.2	odd	10		99.3.k.a.73.1		4
11.5	even	5		99.3.k.a.19.1		4
11.10	odd	2	inner	1089.3.c.e.604.4		4
33.2	even	10		11.3.d.a.7.1	✓	4
33.5	odd	10		11.3.d.a.8.1	yes	4
33.8	even	10		121.3.d.a.112.1		4
33.14	odd	10		121.3.d.c.112.1		4
33.17	even	10		121.3.d.d.118.1		4
33.20	odd	10		121.3.d.d.40.1		4
33.26	odd	10		121.3.d.a.94.1		4
33.29	even	10		121.3.d.c.94.1		4
33.32	even	2		121.3.b.b.120.1		4
132.35	odd	10		176.3.n.a.161.1		4
132.71	even	10		176.3.n.a.129.1		4
165.2	odd	20		275.3.q.d.249.1		8
165.38	even	20		275.3.q.d.74.1		8
165.68	odd	20		275.3.q.d.249.2		8
165.104	odd	10		275.3.x.e.151.1		4
165.134	even	10		275.3.x.e.51.1		4
165.137	even	20		275.3.q.d.74.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.3.d.a.7.1	✓	4	33.2	even	10
11.3.d.a.8.1	yes	4	33.5	odd	10
99.3.k.a.19.1		4	11.5	even	5
99.3.k.a.73.1		4	11.2	odd	10
121.3.b.b.120.1		4	33.32	even	2
121.3.b.b.120.4		4	3.2	odd	2
121.3.d.a.94.1		4	33.26	odd	10
121.3.d.a.112.1		4	33.8	even	10
121.3.d.c.94.1		4	33.29	even	10
121.3.d.c.112.1		4	33.14	odd	10
121.3.d.d.40.1		4	33.20	odd	10
121.3.d.d.118.1		4	33.17	even	10
176.3.n.a.129.1		4	132.71	even	10
176.3.n.a.161.1		4	132.35	odd	10
275.3.q.d.74.1		8	165.38	even	20
275.3.q.d.74.2		8	165.137	even	20
275.3.q.d.249.1		8	165.2	odd	20
275.3.q.d.249.2		8	165.68	odd	20
275.3.x.e.51.1		4	165.134	even	10
275.3.x.e.151.1		4	165.104	odd	10
1089.3.c.e.604.1		4	1.1	even	1	trivial
1089.3.c.e.604.4		4	11.10	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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.07768i q^{2} -5.47214 q^{4} -4.00000 q^{5} -0.898056i q^{7} +4.53077i q^{8} +O(q^{10})\)	\(q-3.07768i q^{2} -5.47214 q^{4} -4.00000 q^{5} -0.898056i q^{7} +4.53077i q^{8} +12.3107i q^{10} +8.50651i q^{13} -2.76393 q^{14} -7.94427 q^{16} +24.7275i q^{17} -11.8617i q^{19} +21.8885 q^{20} +7.23607 q^{23} -9.00000 q^{25} +26.1803 q^{26} +4.91428i q^{28} +3.46120i q^{29} +33.1246 q^{31} +42.5730i q^{32} +76.1033 q^{34} +3.59222i q^{35} +40.2492 q^{37} -36.5066 q^{38} -18.1231i q^{40} +1.30660i q^{41} +33.0625i q^{43} -22.2703i q^{46} -22.7639 q^{47} +48.1935 q^{49} +27.6992i q^{50} -46.5488i q^{52} +78.5410 q^{53} +4.06888 q^{56} +10.6525 q^{58} -31.2705 q^{59} -28.0827i q^{61} -101.947i q^{62} +99.2492 q^{64} -34.0260i q^{65} -76.5066 q^{67} -135.312i q^{68} +11.0557 q^{70} +62.3181 q^{71} +94.1421i q^{73} -123.874i q^{74} +64.9089i q^{76} -65.9129i q^{79} +31.7771 q^{80} +4.02129 q^{82} -56.7049i q^{83} -98.9099i q^{85} +101.756 q^{86} +62.2968 q^{89} +7.63932 q^{91} -39.5967 q^{92} +70.0602i q^{94} +47.4468i q^{95} +72.4083 q^{97} -148.324i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 16 q^{5}+O(q^{10}) \) 4 * q - 4 * q^4 - 16 * q^5	\( 4 q - 4 q^{4} - 16 q^{5} - 20 q^{14} + 4 q^{16} + 16 q^{20} + 20 q^{23} - 36 q^{25} + 60 q^{26} + 52 q^{31} + 130 q^{34} - 70 q^{38} - 100 q^{47} - 4 q^{49} + 180 q^{53} - 100 q^{56} - 20 q^{58} - 58 q^{59} + 236 q^{64} - 230 q^{67} + 80 q^{70} - 28 q^{71} - 16 q^{80} + 110 q^{82} + 170 q^{86} - 122 q^{89} + 120 q^{91} - 60 q^{92} - 10 q^{97}+O(q^{100}) \) 4 * q - 4 * q^4 - 16 * q^5 - 20 * q^14 + 4 * q^16 + 16 * q^20 + 20 * q^23 - 36 * q^25 + 60 * q^26 + 52 * q^31 + 130 * q^34 - 70 * q^38 - 100 * q^47 - 4 * q^49 + 180 * q^53 - 100 * q^56 - 20 * q^58 - 58 * q^59 + 236 * q^64 - 230 * q^67 + 80 * q^70 - 28 * q^71 - 16 * q^80 + 110 * q^82 + 170 * q^86 - 122 * q^89 + 120 * q^91 - 60 * q^92 - 10 * q^97

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