LMFDB - Embedded newform 1089.3.c.l.604.4

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:	$16$
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} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641 $ x^16 - 21x^14 + 227x^12 - 1488x^10 + 24225x^8 - 62832x^6 + 64372x^4 + 7986*x^2 + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.4
Root			$-1.83190 + 2.52140i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.l.604.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11662i q^{2} -5.71334 q^{4} +1.42146 q^{5} -3.80859i q^{7} +5.33982i q^{8} +O(q^{10})$	$q-3.11662i q^{2} -5.71334 q^{4} +1.42146 q^{5} -3.80859i q^{7} +5.33982i q^{8} -4.43016i q^{10} +21.6699i q^{13} -11.8699 q^{14} -6.21114 q^{16} +16.9868i q^{17} -19.5394i q^{19} -8.12129 q^{20} -4.82866 q^{23} -22.9794 q^{25} +67.5370 q^{26} +21.7598i q^{28} +45.4827i q^{29} -20.9624 q^{31} +40.7171i q^{32} +52.9416 q^{34} -5.41377i q^{35} -21.5085 q^{37} -60.8969 q^{38} +7.59035i q^{40} +36.7080i q^{41} +10.0872i q^{43} +15.0491i q^{46} -66.4947 q^{47} +34.4946 q^{49} +71.6182i q^{50} -123.808i q^{52} -82.4525 q^{53} +20.3372 q^{56} +141.752 q^{58} +63.9915 q^{59} +10.9083i q^{61} +65.3318i q^{62} +102.055 q^{64} +30.8030i q^{65} +22.6034 q^{67} -97.0515i q^{68} -16.8727 q^{70} +44.4234 q^{71} +70.0750i q^{73} +67.0340i q^{74} +111.635i q^{76} -139.492i q^{79} -8.82891 q^{80} +114.405 q^{82} +112.208i q^{83} +24.1462i q^{85} +31.4380 q^{86} -63.5921 q^{89} +82.5320 q^{91} +27.5878 q^{92} +207.239i q^{94} -27.7745i q^{95} +101.073 q^{97} -107.507i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 44 q^{4}+O(q^{10}) $ 16 * q - 44 * q^4	$ 16 q - 44 q^{4} + 244 q^{16} + 16 q^{25} - 80 q^{31} + 328 q^{34} - 280 q^{37} + 436 q^{49} + 140 q^{58} - 656 q^{64} + 300 q^{67} + 308 q^{70} + 580 q^{82} + 768 q^{91} - 720 q^{97}+O(q^{100}) $ 16 * q - 44 * q^4 + 244 * q^16 + 16 * q^25 - 80 * q^31 + 328 * q^34 - 280 * q^37 + 436 * q^49 + 140 * q^58 - 656 * q^64 + 300 * q^67 + 308 * q^70 + 580 * q^82 + 768 * q^91 - 720 * 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 $

$2$	− 3.11662i	− 1.55831i	−0.626831	−	0.779156i	$-0.715648\pi$
$2$	− 3.11662i	− 1.55831i	0.626831	−	0.779156i	$-0.284352\pi$
$3$	0	0
$3$	0	0
$4$	−5.71334	−1.42833
$5$	1.42146	0.284293	0.142146	−	0.989846i	$-0.454600\pi$
$5$	1.42146	0.284293	0.142146	+	0.989846i	$0.454600\pi$
$6$	0	0
$7$	− 3.80859i	− 0.544084i	−0.962285	−	0.272042i	$-0.912301\pi$
$7$	− 3.80859i	− 0.544084i	0.962285	−	0.272042i	$-0.0876990\pi$
$8$	5.33982i	0.667477i
$9$	0	0
$10$	− 4.43016i	− 0.443016i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	21.6699i	1.66692i	0.552581	+	0.833459i	$0.313643\pi$
$13$	21.6699i	1.66692i	−0.552581	+	0.833459i	$0.686357\pi$
$14$	−11.8699	−0.847853
$15$	0	0
$16$	−6.21114	−0.388196
$17$	16.9868i	0.999226i	0.866249	+	0.499613i	$0.166524\pi$
$17$	16.9868i	0.999226i	−0.866249	+	0.499613i	$0.833476\pi$
$18$	0	0
$19$	− 19.5394i	− 1.02839i	−0.857673	−	0.514195i	$-0.828091\pi$
$19$	− 19.5394i	− 1.02839i	0.857673	−	0.514195i	$-0.171909\pi$
$20$	−8.12129	−0.406065
$21$	0	0
$22$	0	0
$23$	−4.82866	−0.209942	−0.104971	−	0.994475i	$-0.533475\pi$
$23$	−4.82866	−0.209942	−0.104971	+	0.994475i	$0.533475\pi$
$24$	0	0
$25$	−22.9794	−0.919178
$26$	67.5370	2.59758
$27$	0	0
$28$	21.7598i	0.777134i
$29$	45.4827i	1.56837i	0.620528	+	0.784184i	$0.286918\pi$
$29$	45.4827i	1.56837i	−0.620528	+	0.784184i	$0.713082\pi$
$30$	0	0
$31$	−20.9624	−0.676205	−0.338103	−	0.941109i	$-0.609785\pi$
$31$	−20.9624	−0.676205	−0.338103	+	0.941109i	$0.609785\pi$
$32$	40.7171i	1.27241i
$33$	0	0
$34$	52.9416	1.55710
$35$	− 5.41377i	− 0.154679i
$36$	0	0
$37$	−21.5085	−0.581312	−0.290656	−	0.956828i	$-0.593873\pi$
$37$	−21.5085	−0.581312	−0.290656	+	0.956828i	$0.593873\pi$
$38$	−60.8969	−1.60255
$39$	0	0
$40$	7.59035i	0.189759i
$41$	36.7080i	0.895316i	0.894205	+	0.447658i	$0.147742\pi$
$41$	36.7080i	0.895316i	−0.894205	+	0.447658i	$0.852258\pi$
$42$	0	0
$43$	10.0872i	0.234586i	0.993097	+	0.117293i	$0.0374217\pi$
$43$	10.0872i	0.234586i	−0.993097	+	0.117293i	$0.962578\pi$
$44$	0	0
$45$	0	0
$46$	15.0491i	0.327155i
$47$	−66.4947	−1.41478	−0.707391	−	0.706823i	$-0.750128\pi$
$47$	−66.4947	−1.41478	−0.707391	+	0.706823i	$0.750128\pi$
$48$	0	0
$49$	34.4946	0.703972
$50$	71.6182i	1.43236i
$51$	0	0
$52$	− 123.808i	− 2.38092i
$53$	−82.4525	−1.55571	−0.777854	−	0.628445i	$-0.783692\pi$
$53$	−82.4525	−1.55571	−0.777854	+	0.628445i	$0.783692\pi$
$54$	0	0
$55$	0	0
$56$	20.3372	0.363164
$57$	0	0
$58$	141.752	2.44401
$59$	63.9915	1.08460	0.542300	−	0.840185i	$-0.317553\pi$
$59$	63.9915	1.08460	0.542300	+	0.840185i	$0.317553\pi$
$60$	0	0
$61$	10.9083i	0.178824i	0.995995	+	0.0894120i	$0.0284988\pi$
$61$	10.9083i	0.178824i	−0.995995	+	0.0894120i	$0.971501\pi$
$62$	65.3318i	1.05374i
$63$	0	0
$64$	102.055	1.59461
$65$	30.8030i	0.473893i
$66$	0	0
$67$	22.6034	0.337364	0.168682	−	0.985671i	$-0.446049\pi$
$67$	22.6034	0.337364	0.168682	+	0.985671i	$0.446049\pi$
$68$	− 97.0515i	− 1.42723i
$69$	0	0
$70$	−16.8727	−0.241038
$71$	44.4234	0.625682	0.312841	−	0.949806i	$-0.398719\pi$
$71$	44.4234	0.625682	0.312841	+	0.949806i	$0.398719\pi$
$72$	0	0
$73$	70.0750i	0.959931i	0.877287	+	0.479966i	$0.159351\pi$
$73$	70.0750i	0.959931i	−0.877287	+	0.479966i	$0.840649\pi$
$74$	67.0340i	0.905865i
$75$	0	0
$76$	111.635i	1.46888i
$77$	0	0
$78$	0	0
$79$	− 139.492i	− 1.76572i	−0.469638	−	0.882859i	$-0.655615\pi$
$79$	− 139.492i	− 1.76572i	0.469638	−	0.882859i	$-0.344385\pi$
$80$	−8.82891	−0.110361
$81$	0	0
$82$	114.405	1.39518
$83$	112.208i	1.35191i	0.736945	+	0.675953i	$0.236268\pi$
$83$	112.208i	1.35191i	−0.736945	+	0.675953i	$0.763732\pi$
$84$	0	0
$85$	24.1462i	0.284072i
$86$	31.4380	0.365558
$87$	0	0
$88$	0	0
$89$	−63.5921	−0.714518	−0.357259	−	0.934005i	$-0.616289\pi$
$89$	−63.5921	−0.714518	−0.357259	+	0.934005i	$0.616289\pi$
$90$	0	0
$91$	82.5320	0.906945
$92$	27.5878	0.299867
$93$	0	0
$94$	207.239i	2.20467i
$95$	− 27.7745i	− 0.292363i
$96$	0	0
$97$	101.073	1.04199	0.520995	−	0.853559i	$-0.325561\pi$
$97$	101.073	1.04199	0.520995	+	0.853559i	$0.325561\pi$
$98$	− 107.507i	− 1.09701i
$99$	0	0
$100$	131.289	1.31289
$101$	132.184i	1.30876i	0.756167	+	0.654378i	$0.227070\pi$
$101$	132.184i	1.30876i	−0.756167	+	0.654378i	$0.772930\pi$
$102$	0	0
$103$	50.1602	0.486992	0.243496	−	0.969902i	$-0.421706\pi$
$103$	50.1602	0.486992	0.243496	+	0.969902i	$0.421706\pi$
$104$	−115.714	−1.11263
$105$	0	0
$106$	256.973i	2.42428i
$107$	− 159.213i	− 1.48797i	−0.668197	−	0.743984i	$-0.732934\pi$
$107$	− 159.213i	− 1.48797i	0.668197	−	0.743984i	$-0.267066\pi$
$108$	0	0
$109$	93.4090i	0.856963i	0.903551	+	0.428482i	$0.140951\pi$
$109$	93.4090i	0.856963i	−0.903551	+	0.428482i	$0.859049\pi$
$110$	0	0
$111$	0	0
$112$	23.6557i	0.211212i
$113$	38.4909	0.340627	0.170314	−	0.985390i	$-0.445522\pi$
$113$	38.4909	0.340627	0.170314	+	0.985390i	$0.445522\pi$
$114$	0	0
$115$	−6.86376	−0.0596849
$116$	− 259.858i	− 2.24015i
$117$	0	0
$118$	− 199.437i	− 1.69015i
$119$	64.6959	0.543663
$120$	0	0
$121$	0	0
$122$	33.9969	0.278663
$123$	0	0
$124$	119.765	0.965847
$125$	−68.2010	−0.545608
$126$	0	0
$127$	− 81.2914i	− 0.640090i	−0.947402	−	0.320045i	$-0.896302\pi$
$127$	− 81.2914i	− 0.640090i	0.947402	−	0.320045i	$-0.103698\pi$
$128$	− 155.199i	− 1.21249i
$129$	0	0
$130$	96.0014	0.738472
$131$	147.346i	1.12478i	0.826874	+	0.562388i	$0.190117\pi$
$131$	147.346i	1.12478i	−0.826874	+	0.562388i	$0.809883\pi$
$132$	0	0
$133$	−74.4176	−0.559531
$134$	− 70.4463i	− 0.525718i
$135$	0	0
$136$	−90.7066	−0.666961
$137$	212.251	1.54928	0.774638	−	0.632404i	$-0.217932\pi$
$137$	212.251	1.54928	0.774638	+	0.632404i	$0.217932\pi$
$138$	0	0
$139$	− 219.331i	− 1.57792i	−0.614445	−	0.788959i	$-0.710620\pi$
$139$	− 219.331i	− 1.57792i	0.614445	−	0.788959i	$-0.289380\pi$
$140$	30.9307i	0.220933i
$141$	0	0
$142$	− 138.451i	− 0.975007i
$143$	0	0
$144$	0	0
$145$	64.6519i	0.445875i
$146$	218.397	1.49587
$147$	0	0
$148$	122.885	0.830307
$149$	− 46.2081i	− 0.310122i	−0.987905	−	0.155061i	$-0.950443\pi$
$149$	− 46.2081i	− 0.310122i	0.987905	−	0.155061i	$-0.0495573\pi$
$150$	0	0
$151$	− 160.565i	− 1.06334i	−0.846950	−	0.531672i	$-0.821564\pi$
$151$	− 160.565i	− 1.06334i	0.846950	−	0.531672i	$-0.178436\pi$
$152$	104.337	0.686427
$153$	0	0
$154$	0	0
$155$	−29.7972	−0.192240
$156$	0	0
$157$	−87.3010	−0.556057	−0.278029	−	0.960573i	$-0.589681\pi$
$157$	−87.3010	−0.556057	−0.278029	+	0.960573i	$0.589681\pi$
$158$	−434.743	−2.75154
$159$	0	0
$160$	57.8778i	0.361736i
$161$	18.3904i	0.114226i
$162$	0	0
$163$	−154.692	−0.949028	−0.474514	−	0.880248i	$-0.657376\pi$
$163$	−154.692	−0.949028	−0.474514	+	0.880248i	$0.657376\pi$
$164$	− 209.725i	− 1.27881i
$165$	0	0
$166$	349.710	2.10669
$167$	255.452i	1.52965i	0.644236	+	0.764827i	$0.277176\pi$
$167$	255.452i	1.52965i	−0.644236	+	0.764827i	$0.722824\pi$
$168$	0	0
$169$	−300.586	−1.77862
$170$	75.2544	0.442673
$171$	0	0
$172$	− 57.6315i	− 0.335067i
$173$	− 32.8371i	− 0.189810i	−0.995486	−	0.0949048i	$-0.969745\pi$
$173$	− 32.8371i	− 0.189810i	0.995486	−	0.0949048i	$-0.0302547\pi$
$174$	0	0
$175$	87.5193i	0.500110i
$176$	0	0
$177$	0	0
$178$	198.193i	1.11344i
$179$	−166.111	−0.927996	−0.463998	−	0.885836i	$-0.653585\pi$
$179$	−166.111	−0.927996	−0.463998	+	0.885836i	$0.653585\pi$
$180$	0	0
$181$	−68.5237	−0.378584	−0.189292	−	0.981921i	$-0.560619\pi$
$181$	−68.5237	−0.378584	−0.189292	+	0.981921i	$0.560619\pi$
$182$	− 257.221i	− 1.41330i
$183$	0	0
$184$	− 25.7842i	− 0.140131i
$185$	−30.5736	−0.165263
$186$	0	0
$187$	0	0
$188$	379.907	2.02078
$189$	0	0
$190$	−86.5627	−0.455593
$191$	−263.725	−1.38076	−0.690378	−	0.723448i	$-0.742556\pi$
$191$	−263.725	−1.38076	−0.690378	+	0.723448i	$0.742556\pi$
$192$	0	0
$193$	15.0914i	0.0781938i	0.999235	+	0.0390969i	$0.0124481\pi$
$193$	15.0914i	0.0781938i	−0.999235	+	0.0390969i	$0.987552\pi$
$194$	− 315.007i	− 1.62375i
$195$	0	0
$196$	−197.079	−1.00551
$197$	− 205.342i	− 1.04234i	−0.853452	−	0.521172i	$-0.825495\pi$
$197$	− 205.342i	− 1.04234i	0.853452	−	0.521172i	$-0.174505\pi$
$198$	0	0
$199$	−104.559	−0.525420	−0.262710	−	0.964875i	$-0.584616\pi$
$199$	−104.559	−0.525420	−0.262710	+	0.964875i	$0.584616\pi$
$200$	− 122.706i	− 0.613530i
$201$	0	0
$202$	411.969	2.03945
$203$	173.225	0.853325
$204$	0	0
$205$	52.1790i	0.254532i
$206$	− 156.330i	− 0.758886i
$207$	0	0
$208$	− 134.595i	− 0.647092i
$209$	0	0
$210$	0	0
$211$	144.367i	0.684202i	0.939663	+	0.342101i	$0.111138\pi$
$211$	144.367i	0.684202i	−0.939663	+	0.342101i	$0.888862\pi$
$212$	471.079	2.22207
$213$	0	0
$214$	−496.206	−2.31872
$215$	14.3386i	0.0666911i
$216$	0	0
$217$	79.8371i	0.367913i
$218$	291.121	1.33542
$219$	0	0
$220$	0	0
$221$	−368.104	−1.66563
$222$	0	0
$223$	−187.253	−0.839701	−0.419851	−	0.907593i	$-0.637918\pi$
$223$	−187.253	−0.839701	−0.419851	+	0.907593i	$0.637918\pi$
$224$	155.075	0.692297
$225$	0	0
$226$	− 119.962i	− 0.530804i
$227$	48.4710i	0.213529i	0.994284	+	0.106764i	$0.0340490\pi$
$227$	48.4710i	0.213529i	−0.994284	+	0.106764i	$0.965951\pi$
$228$	0	0
$229$	−245.895	−1.07378	−0.536889	−	0.843653i	$-0.680401\pi$
$229$	−245.895	−1.07378	−0.536889	+	0.843653i	$0.680401\pi$
$230$	21.3918i	0.0930077i
$231$	0	0
$232$	−242.869	−1.04685
$233$	− 22.7392i	− 0.0975932i	−0.998809	−	0.0487966i	$-0.984461\pi$
$233$	− 22.7392i	− 0.0975932i	0.998809	−	0.0487966i	$-0.0155386\pi$
$234$	0	0
$235$	−94.5198	−0.402212
$236$	−365.605	−1.54917
$237$	0	0
$238$	− 201.633i	− 0.847196i
$239$	− 61.6198i	− 0.257823i	−0.991656	−	0.128912i	$-0.958852\pi$
$239$	− 61.6198i	− 0.257823i	0.991656	−	0.128912i	$-0.0411484\pi$
$240$	0	0
$241$	165.003i	0.684662i	0.939579	+	0.342331i	$0.111216\pi$
$241$	165.003i	0.684662i	−0.939579	+	0.342331i	$0.888784\pi$
$242$	0	0
$243$	0	0
$244$	− 62.3226i	− 0.255420i
$245$	49.0328	0.200134
$246$	0	0
$247$	423.418	1.71424
$248$	− 111.935i	− 0.451352i
$249$	0	0
$250$	212.557i	0.850227i
$251$	−44.4040	−0.176908	−0.0884542	−	0.996080i	$-0.528193\pi$
$251$	−44.4040	−0.176908	−0.0884542	+	0.996080i	$0.528193\pi$
$252$	0	0
$253$	0	0
$254$	−253.355	−0.997460
$255$	0	0
$256$	−75.4764	−0.294830
$257$	−20.0744	−0.0781106	−0.0390553	−	0.999237i	$-0.512435\pi$
$257$	−20.0744	−0.0781106	−0.0390553	+	0.999237i	$0.512435\pi$
$258$	0	0
$259$	81.9172i	0.316283i
$260$	− 175.988i	− 0.676877i
$261$	0	0
$262$	459.220	1.75275
$263$	212.968i	0.809765i	0.914369	+	0.404883i	$0.132688\pi$
$263$	212.968i	0.809765i	−0.914369	+	0.404883i	$0.867312\pi$
$264$	0	0
$265$	−117.203	−0.442276
$266$	231.932i	0.871923i
$267$	0	0
$268$	−129.141	−0.481869
$269$	408.873	1.51997	0.759987	−	0.649939i	$-0.225205\pi$
$269$	408.873	1.51997	0.759987	+	0.649939i	$0.225205\pi$
$270$	0	0
$271$	− 10.4233i	− 0.0384624i	−0.999815	−	0.0192312i	$-0.993878\pi$
$271$	− 10.4233i	− 0.0384624i	0.999815	−	0.0192312i	$-0.00612187\pi$
$272$	− 105.508i	− 0.387896i
$273$	0	0
$274$	− 661.506i	− 2.41426i
$275$	0	0
$276$	0	0
$277$	− 436.295i	− 1.57507i	−0.616269	−	0.787536i	$-0.711357\pi$
$277$	− 436.295i	− 1.57507i	0.616269	−	0.787536i	$-0.288643\pi$
$278$	−683.571	−2.45889
$279$	0	0
$280$	28.9086	0.103245
$281$	267.368i	0.951487i	0.879584	+	0.475743i	$0.157821\pi$
$281$	267.368i	0.951487i	−0.879584	+	0.475743i	$0.842179\pi$
$282$	0	0
$283$	335.025i	1.18383i	0.805999	+	0.591917i	$0.201629\pi$
$283$	335.025i	1.18383i	−0.805999	+	0.591917i	$0.798371\pi$
$284$	−253.806	−0.893682
$285$	0	0
$286$	0	0
$287$	139.806	0.487128
$288$	0	0
$289$	0.447432	0.00154821
$290$	201.496	0.694813
$291$	0	0
$292$	− 400.362i	− 1.37110i
$293$	269.557i	0.919991i	0.887921	+	0.459995i	$0.152149\pi$
$293$	269.557i	0.919991i	−0.887921	+	0.459995i	$0.847851\pi$
$294$	0	0
$295$	90.9615	0.308344
$296$	− 114.852i	− 0.388013i
$297$	0	0
$298$	−144.013	−0.483266
$299$	− 104.637i	− 0.349956i
$300$	0	0
$301$	38.4180	0.127635
$302$	−500.420	−1.65702
$303$	0	0
$304$	121.362i	0.399217i
$305$	15.5057i	0.0508383i
$306$	0	0
$307$	513.671i	1.67319i	0.547819	+	0.836597i	$0.315458\pi$
$307$	513.671i	1.67319i	−0.547819	+	0.836597i	$0.684542\pi$
$308$	0	0
$309$	0	0
$310$	92.8667i	0.299570i
$311$	309.431	0.994954	0.497477	−	0.867477i	$-0.334260\pi$
$311$	309.431	0.994954	0.497477	+	0.867477i	$0.334260\pi$
$312$	0	0
$313$	−268.586	−0.858102	−0.429051	−	0.903280i	$-0.641152\pi$
$313$	−268.586	−0.858102	−0.429051	+	0.903280i	$0.641152\pi$
$314$	272.084i	0.866510i
$315$	0	0
$316$	796.963i	2.52204i
$317$	−368.997	−1.16403	−0.582015	−	0.813178i	$-0.697735\pi$
$317$	−368.997	−1.16403	−0.582015	+	0.813178i	$0.697735\pi$
$318$	0	0
$319$	0	0
$320$	145.068	0.453336
$321$	0	0
$322$	57.3159	0.178000
$323$	331.913	1.02759
$324$	0	0
$325$	− 497.963i	− 1.53219i
$326$	482.115i	1.47888i
$327$	0	0
$328$	−196.014	−0.597604
$329$	253.251i	0.769761i
$330$	0	0
$331$	−511.298	−1.54471	−0.772354	−	0.635192i	$-0.780921\pi$
$331$	−511.298	−1.54471	−0.772354	+	0.635192i	$0.780921\pi$
$332$	− 641.083i	− 1.93097i
$333$	0	0
$334$	796.148	2.38368
$335$	32.1299	0.0959101
$336$	0	0
$337$	− 38.7519i	− 0.114991i	−0.998346	−	0.0574955i	$-0.981689\pi$
$337$	− 38.7519i	− 0.114991i	0.998346	−	0.0574955i	$-0.0183115\pi$
$338$	936.815i	2.77164i
$339$	0	0
$340$	− 137.955i	− 0.405750i
$341$	0	0
$342$	0	0
$343$	− 317.997i	− 0.927105i
$344$	−53.8638	−0.156581
$345$	0	0
$346$	−102.341	−0.295782
$347$	405.399i	1.16830i	0.811647	+	0.584149i	$0.198572\pi$
$347$	405.399i	1.16830i	−0.811647	+	0.584149i	$0.801428\pi$
$348$	0	0
$349$	− 217.513i	− 0.623247i	−0.950206	−	0.311623i	$-0.899127\pi$
$349$	− 217.513i	− 0.623247i	0.950206	−	0.311623i	$-0.100873\pi$
$350$	272.765	0.779328
$351$	0	0
$352$	0	0
$353$	417.749	1.18343	0.591713	−	0.806149i	$-0.298452\pi$
$353$	417.749	1.18343	0.591713	+	0.806149i	$0.298452\pi$
$354$	0	0
$355$	63.1462	0.177877
$356$	363.323	1.02057
$357$	0	0
$358$	517.706i	1.44611i
$359$	326.871i	0.910504i	0.890363	+	0.455252i	$0.150451\pi$
$359$	326.871i	0.910504i	−0.890363	+	0.455252i	$0.849549\pi$
$360$	0	0
$361$	−20.7881	−0.0575848
$362$	213.562i	0.589952i
$363$	0	0
$364$	−471.533	−1.29542
$365$	99.6090i	0.272901i
$366$	0	0
$367$	−1.56868	−0.00427432	−0.00213716	−	0.999998i	$-0.500680\pi$
$367$	−1.56868	−0.00427432	−0.00213716	+	0.999998i	$0.500680\pi$
$368$	29.9915	0.0814986
$369$	0	0
$370$	95.2863i	0.257531i
$371$	314.028i	0.846436i
$372$	0	0
$373$	294.158i	0.788627i	0.918976	+	0.394314i	$0.129018\pi$
$373$	294.158i	0.788627i	−0.918976	+	0.394314i	$0.870982\pi$
$374$	0	0
$375$	0	0
$376$	− 355.070i	− 0.944335i
$377$	−985.607	−2.61434
$378$	0	0
$379$	−14.3356	−0.0378249	−0.0189124	−	0.999821i	$-0.506020\pi$
$379$	−14.3356	−0.0378249	−0.0189124	+	0.999821i	$0.506020\pi$
$380$	158.685i	0.417593i
$381$	0	0
$382$	821.930i	2.15165i
$383$	−189.073	−0.493662	−0.246831	−	0.969059i	$-0.579389\pi$
$383$	−189.073	−0.493662	−0.246831	+	0.969059i	$0.579389\pi$
$384$	0	0
$385$	0	0
$386$	47.0342	0.121850
$387$	0	0
$388$	−577.465	−1.48831
$389$	535.857	1.37752	0.688762	−	0.724987i	$-0.258154\pi$
$389$	535.857	1.37752	0.688762	+	0.724987i	$0.258154\pi$
$390$	0	0
$391$	− 82.0237i	− 0.209779i
$392$	184.195i	0.469886i
$393$	0	0
$394$	−639.972	−1.62430
$395$	− 198.282i	− 0.501981i
$396$	0	0
$397$	181.111	0.456199	0.228099	−	0.973638i	$-0.426749\pi$
$397$	181.111	0.456199	0.228099	+	0.973638i	$0.426749\pi$
$398$	325.870i	0.818768i
$399$	0	0
$400$	142.729	0.356821
$401$	−620.600	−1.54763	−0.773816	−	0.633411i	$-0.781654\pi$
$401$	−620.600	−1.54763	−0.773816	+	0.633411i	$0.781654\pi$
$402$	0	0
$403$	− 454.253i	− 1.12718i
$404$	− 755.214i	− 1.86934i
$405$	0	0
$406$	− 539.877i	− 1.32975i
$407$	0	0
$408$	0	0
$409$	− 218.490i	− 0.534206i	−0.963668	−	0.267103i	$-0.913934\pi$
$409$	− 218.490i	− 0.534206i	0.963668	−	0.267103i	$-0.0860664\pi$
$410$	162.622	0.396640
$411$	0	0
$412$	−286.582	−0.695588
$413$	− 243.717i	− 0.590114i
$414$	0	0
$415$	159.500i	0.384337i
$416$	−882.336	−2.12100
$417$	0	0
$418$	0	0
$419$	−819.307	−1.95539	−0.977694	−	0.210035i	$-0.932642\pi$
$419$	−819.307	−1.95539	−0.977694	+	0.210035i	$0.932642\pi$
$420$	0	0
$421$	−264.875	−0.629156	−0.314578	−	0.949232i	$-0.601863\pi$
$421$	−264.875	−0.629156	−0.314578	+	0.949232i	$0.601863\pi$
$422$	449.936	1.06620
$423$	0	0
$424$	− 440.282i	− 1.03840i
$425$	− 390.348i	− 0.918466i
$426$	0	0
$427$	41.5451	0.0972953
$428$	909.635i	2.12532i
$429$	0	0
$430$	44.6879	0.103925
$431$	− 85.0354i	− 0.197298i	−0.995122	−	0.0986490i	$-0.968548\pi$
$431$	− 85.0354i	− 0.197298i	0.995122	−	0.0986490i	$-0.0314521\pi$
$432$	0	0
$433$	−394.294	−0.910609	−0.455304	−	0.890336i	$-0.650470\pi$
$433$	−394.294	−0.910609	−0.455304	+	0.890336i	$0.650470\pi$
$434$	248.822	0.573323
$435$	0	0
$436$	− 533.677i	− 1.22403i
$437$	94.3492i	0.215902i
$438$	0	0
$439$	− 609.996i	− 1.38951i	−0.719245	−	0.694756i	$-0.755512\pi$
$439$	− 609.996i	− 1.38951i	0.719245	−	0.694756i	$-0.244488\pi$
$440$	0	0
$441$	0	0
$442$	1147.24i	2.59557i
$443$	168.950	0.381377	0.190689	−	0.981651i	$-0.438928\pi$
$443$	168.950	0.381377	0.190689	+	0.981651i	$0.438928\pi$
$444$	0	0
$445$	−90.3938	−0.203132
$446$	583.598i	1.30852i
$447$	0	0
$448$	− 388.686i	− 0.867603i
$449$	677.380	1.50864	0.754321	−	0.656506i	$-0.227966\pi$
$449$	677.380	1.50864	0.754321	+	0.656506i	$0.227966\pi$
$450$	0	0
$451$	0	0
$452$	−219.911	−0.486530
$453$	0	0
$454$	151.066	0.332744
$455$	117.316	0.257838
$456$	0	0
$457$	− 573.107i	− 1.25406i	−0.778994	−	0.627032i	$-0.784270\pi$
$457$	− 573.107i	− 1.25406i	0.778994	−	0.627032i	$-0.215730\pi$
$458$	766.362i	1.67328i
$459$	0	0
$460$	39.2150	0.0852500
$461$	333.147i	0.722661i	0.932438	+	0.361330i	$0.117677\pi$
$461$	333.147i	0.722661i	−0.932438	+	0.361330i	$0.882323\pi$
$462$	0	0
$463$	35.2648	0.0761659	0.0380829	−	0.999275i	$-0.487875\pi$
$463$	35.2648	0.0761659	0.0380829	+	0.999275i	$0.487875\pi$
$464$	− 282.499i	− 0.608835i
$465$	0	0
$466$	−70.8696	−0.152081
$467$	5.29045	0.0113286	0.00566430	−	0.999984i	$-0.498197\pi$
$467$	5.29045	0.0113286	0.00566430	+	0.999984i	$0.498197\pi$
$468$	0	0
$469$	− 86.0871i	− 0.183555i
$470$	294.582i	0.626771i
$471$	0	0
$472$	341.703i	0.723947i
$473$	0	0
$474$	0	0
$475$	449.005i	0.945273i
$476$	−369.629	−0.776532
$477$	0	0
$478$	−192.046	−0.401769
$479$	− 124.687i	− 0.260307i	−0.991494	−	0.130153i	$-0.958453\pi$
$479$	− 124.687i	− 0.260307i	0.991494	−	0.130153i	$-0.0415470\pi$
$480$	0	0
$481$	− 466.089i	− 0.969000i
$482$	514.254	1.06692
$483$	0	0
$484$	0	0
$485$	143.672	0.296230
$486$	0	0
$487$	−663.951	−1.36335	−0.681675	−	0.731656i	$-0.738748\pi$
$487$	−663.951	−1.36335	−0.681675	+	0.731656i	$0.738748\pi$
$488$	−58.2482	−0.119361
$489$	0	0
$490$	− 152.817i	− 0.311871i
$491$	− 294.176i	− 0.599136i	−0.954075	−	0.299568i	$-0.903157\pi$
$491$	− 294.176i	− 0.599136i	0.954075	−	0.299568i	$-0.0968426\pi$
$492$	0	0
$493$	−772.607	−1.56715
$494$	− 1319.63i	− 2.67132i
$495$	0	0
$496$	130.200	0.262500
$497$	− 169.191i	− 0.340424i
$498$	0	0
$499$	−642.122	−1.28682	−0.643408	−	0.765523i	$-0.722480\pi$
$499$	−642.122	−1.28682	−0.643408	+	0.765523i	$0.722480\pi$
$500$	389.655	0.779310
$501$	0	0
$502$	138.391i	0.275678i
$503$	− 220.597i	− 0.438562i	−0.975662	−	0.219281i	$-0.929629\pi$
$503$	− 220.597i	− 0.438562i	0.975662	−	0.219281i	$-0.0703712\pi$
$504$	0	0
$505$	187.895i	0.372070i
$506$	0	0
$507$	0	0
$508$	464.445i	0.914262i
$509$	683.899	1.34361	0.671807	−	0.740726i	$-0.265518\pi$
$509$	683.899	1.34361	0.671807	+	0.740726i	$0.265518\pi$
$510$	0	0
$511$	266.887	0.522284
$512$	− 385.565i	− 0.753056i
$513$	0	0
$514$	62.5644i	0.121721i
$515$	71.3009	0.138448
$516$	0	0
$517$	0	0
$518$	255.305	0.492867
$519$	0	0
$520$	−164.483	−0.316313
$521$	−291.550	−0.559597	−0.279799	−	0.960059i	$-0.590268\pi$
$521$	−291.550	−0.559597	−0.279799	+	0.960059i	$0.590268\pi$
$522$	0	0
$523$	341.608i	0.653171i	0.945168	+	0.326585i	$0.105898\pi$
$523$	341.608i	0.653171i	−0.945168	+	0.326585i	$0.894102\pi$
$524$	− 841.834i	− 1.60655i
$525$	0	0
$526$	663.742	1.26187
$527$	− 356.084i	− 0.675682i
$528$	0	0
$529$	−505.684	−0.955924
$530$	365.278i	0.689204i
$531$	0	0
$532$	425.173	0.799197
$533$	−795.460	−1.49242
$534$	0	0
$535$	− 226.315i	− 0.423018i
$536$	120.698i	0.225183i
$537$	0	0
$538$	− 1274.30i	− 2.36859i
$539$	0	0
$540$	0	0
$541$	− 384.646i	− 0.710991i	−0.934678	−	0.355495i	$-0.884312\pi$
$541$	− 384.646i	− 0.710991i	0.934678	−	0.355495i	$-0.115688\pi$
$542$	−32.4856	−0.0599365
$543$	0	0
$544$	−691.654	−1.27142
$545$	132.777i	0.243628i
$546$	0	0
$547$	540.598i	0.988296i	0.869378	+	0.494148i	$0.164520\pi$
$547$	540.598i	0.988296i	−0.869378	+	0.494148i	$0.835480\pi$
$548$	−1212.66	−2.21288
$549$	0	0
$550$	0	0
$551$	888.704	1.61289
$552$	0	0
$553$	−531.267	−0.960700
$554$	−1359.77	−2.45445
$555$	0	0
$556$	1253.11i	2.25380i
$557$	− 21.0178i	− 0.0377340i	−0.999822	−	0.0188670i	$-0.993994\pi$
$557$	− 21.0178i	− 0.0377340i	0.999822	−	0.0188670i	$-0.00600590\pi$
$558$	0	0
$559$	−218.589	−0.391036
$560$	33.6257i	0.0600459i
$561$	0	0
$562$	833.284	1.48271
$563$	− 2.79423i	− 0.00496311i	−0.999997	−	0.00248155i	$-0.999210\pi$
$563$	− 2.79423i	− 0.00496311i	0.999997	−	0.00248155i	$-0.000789904\pi$
$564$	0	0
$565$	54.7134	0.0968379
$566$	1044.15	1.84478
$567$	0	0
$568$	237.213i	0.417628i
$569$	− 317.741i	− 0.558420i	−0.960230	−	0.279210i	$-0.909927\pi$
$569$	− 317.741i	− 0.558420i	0.960230	−	0.279210i	$-0.0900725\pi$
$570$	0	0
$571$	684.355i	1.19852i	0.800554	+	0.599260i	$0.204538\pi$
$571$	684.355i	1.19852i	−0.800554	+	0.599260i	$0.795462\pi$
$572$	0	0
$573$	0	0
$574$	− 435.721i	− 0.759097i
$575$	110.960	0.192974
$576$	0	0
$577$	655.275	1.13566	0.567829	−	0.823147i	$-0.307784\pi$
$577$	655.275	1.13566	0.567829	+	0.823147i	$0.307784\pi$
$578$	− 1.39448i	− 0.00241259i
$579$	0	0
$580$	− 369.378i	− 0.636859i
$581$	427.355	0.735551
$582$	0	0
$583$	0	0
$584$	−374.188	−0.640733
$585$	0	0
$586$	840.108	1.43363
$587$	−1023.97	−1.74442	−0.872208	−	0.489135i	$-0.837313\pi$
$587$	−1023.97	−1.74442	−0.872208	+	0.489135i	$0.837313\pi$
$588$	0	0
$589$	409.592i	0.695402i
$590$	− 283.493i	− 0.480496i
$591$	0	0
$592$	133.593	0.225663
$593$	− 168.317i	− 0.283839i	−0.989878	−	0.141920i	$-0.954673\pi$
$593$	− 168.317i	− 0.283839i	0.989878	−	0.141920i	$-0.0453275\pi$
$594$	0	0
$595$	91.9628	0.154559
$596$	264.002i	0.442957i
$597$	0	0
$598$	−326.114	−0.545340
$599$	−72.5800	−0.121169	−0.0605843	−	0.998163i	$-0.519296\pi$
$599$	−72.5800	−0.121169	−0.0605843	+	0.998163i	$0.519296\pi$
$600$	0	0
$601$	356.395i	0.593004i	0.955032	+	0.296502i	$0.0958201\pi$
$601$	356.395i	0.593004i	−0.955032	+	0.296502i	$0.904180\pi$
$602$	− 119.734i	− 0.198894i
$603$	0	0
$604$	917.361i	1.51881i
$605$	0	0
$606$	0	0
$607$	199.569i	0.328779i	0.986395	+	0.164390i	$0.0525654\pi$
$607$	199.569i	0.328779i	−0.986395	+	0.164390i	$0.947435\pi$
$608$	795.587	1.30853
$609$	0	0
$610$	48.3254	0.0792219
$611$	− 1440.94i	− 2.35833i
$612$	0	0
$613$	− 383.528i	− 0.625657i	−0.949810	−	0.312828i	$-0.898724\pi$
$613$	− 383.528i	− 0.625657i	0.949810	−	0.312828i	$-0.101276\pi$
$614$	1600.92	2.60736
$615$	0	0
$616$	0	0
$617$	196.438	0.318376	0.159188	−	0.987248i	$-0.449112\pi$
$617$	196.438	0.318376	0.159188	+	0.987248i	$0.449112\pi$
$618$	0	0
$619$	1092.78	1.76539	0.882694	−	0.469948i	$-0.155727\pi$
$619$	1092.78	1.76539	0.882694	+	0.469948i	$0.155727\pi$
$620$	170.242	0.274583
$621$	0	0
$622$	− 964.379i	− 1.55045i
$623$	242.196i	0.388758i
$624$	0	0
$625$	477.541	0.764065
$626$	837.081i	1.33719i
$627$	0	0
$628$	498.780	0.794235
$629$	− 365.362i	− 0.580862i
$630$	0	0
$631$	800.410	1.26848	0.634239	−	0.773137i	$-0.281313\pi$
$631$	800.410	1.26848	0.634239	+	0.773137i	$0.281313\pi$
$632$	744.861	1.17858
$633$	0	0
$634$	1150.03i	1.81392i
$635$	− 115.553i	− 0.181973i
$636$	0	0
$637$	747.497i	1.17346i
$638$	0	0
$639$	0	0
$640$	− 220.610i	− 0.344703i
$641$	311.711	0.486288	0.243144	−	0.969990i	$-0.421821\pi$
$641$	311.711	0.486288	0.243144	+	0.969990i	$0.421821\pi$
$642$	0	0
$643$	−10.0999	−0.0157075	−0.00785373	−	0.999969i	$-0.502500\pi$
$643$	−10.0999	−0.0157075	−0.00785373	+	0.999969i	$0.502500\pi$
$644$	− 105.071i	− 0.163153i
$645$	0	0
$646$	− 1034.45i	− 1.60131i
$647$	−347.835	−0.537612	−0.268806	−	0.963194i	$-0.586629\pi$
$647$	−347.835	−0.537612	−0.268806	+	0.963194i	$0.586629\pi$
$648$	0	0
$649$	0	0
$650$	−1551.96	−2.38764
$651$	0	0
$652$	883.804	1.35553
$653$	−477.710	−0.731561	−0.365781	−	0.930701i	$-0.619198\pi$
$653$	−477.710	−0.731561	−0.365781	+	0.930701i	$0.619198\pi$
$654$	0	0
$655$	209.446i	0.319765i
$656$	− 227.998i	− 0.347559i
$657$	0	0
$658$	789.289	1.19953
$659$	− 813.433i	− 1.23434i	−0.786828	−	0.617172i	$-0.788278\pi$
$659$	− 813.433i	− 1.23434i	0.786828	−	0.617172i	$-0.211722\pi$
$660$	0	0
$661$	−1010.49	−1.52873	−0.764364	−	0.644784i	$-0.776947\pi$
$661$	−1010.49	−1.52873	−0.764364	+	0.644784i	$0.776947\pi$
$662$	1593.52i	2.40714i
$663$	0	0
$664$	−599.171	−0.902366
$665$	−105.782	−0.159070
$666$	0	0
$667$	− 219.620i	− 0.329266i
$668$	− 1459.48i	− 2.18486i
$669$	0	0
$670$	− 100.137i	− 0.149458i
$671$	0	0
$672$	0	0
$673$	− 270.805i	− 0.402385i	−0.979552	−	0.201192i	$-0.935518\pi$
$673$	− 270.805i	− 0.402385i	0.979552	−	0.201192i	$-0.0644817\pi$
$674$	−120.775	−0.179192
$675$	0	0
$676$	1717.35	2.54046
$677$	− 820.858i	− 1.21249i	−0.795277	−	0.606247i	$-0.792674\pi$
$677$	− 820.858i	− 1.21249i	0.795277	−	0.606247i	$-0.207326\pi$
$678$	0	0
$679$	− 384.946i	− 0.566931i
$680$	−128.936	−0.189612
$681$	0	0
$682$	0	0
$683$	810.930	1.18731	0.593653	−	0.804721i	$-0.297685\pi$
$683$	810.930	1.18731	0.593653	+	0.804721i	$0.297685\pi$
$684$	0	0
$685$	301.707	0.440448
$686$	−991.076	−1.44472
$687$	0	0
$688$	− 62.6530i	− 0.0910654i
$689$	− 1786.74i	− 2.59324i
$690$	0	0
$691$	69.3580	0.100373	0.0501867	−	0.998740i	$-0.484018\pi$
$691$	69.3580	0.100373	0.0501867	+	0.998740i	$0.484018\pi$
$692$	187.609i	0.271112i
$693$	0	0
$694$	1263.48	1.82057
$695$	− 311.770i	− 0.448591i
$696$	0	0
$697$	−623.552	−0.894623
$698$	−677.906	−0.971213
$699$	0	0
$700$	− 500.027i	− 0.714324i
$701$	128.976i	0.183988i	0.995760	+	0.0919942i	$0.0293241\pi$
$701$	128.976i	0.183988i	−0.995760	+	0.0919942i	$0.970676\pi$
$702$	0	0
$703$	420.264i	0.597815i
$704$	0	0
$705$	0	0
$706$	− 1301.97i	− 1.84415i
$707$	503.436	0.712074
$708$	0	0
$709$	−431.847	−0.609093	−0.304547	−	0.952497i	$-0.598505\pi$
$709$	−431.847	−0.609093	−0.304547	+	0.952497i	$0.598505\pi$
$710$	− 196.803i	− 0.277187i
$711$	0	0
$712$	− 339.570i	− 0.476924i
$713$	101.220	0.141964
$714$	0	0
$715$	0	0
$716$	949.049	1.32549
$717$	0	0
$718$	1018.73	1.41885
$719$	545.503	0.758697	0.379348	−	0.925254i	$-0.376148\pi$
$719$	545.503	0.758697	0.379348	+	0.925254i	$0.376148\pi$
$720$	0	0
$721$	− 191.040i	− 0.264965i
$722$	64.7887i	0.0897350i
$723$	0	0
$724$	391.499	0.540744
$725$	− 1045.17i	− 1.44161i
$726$	0	0
$727$	1092.98	1.50341	0.751705	−	0.659500i	$-0.229232\pi$
$727$	1092.98	1.50341	0.751705	+	0.659500i	$0.229232\pi$
$728$	440.706i	0.605365i
$729$	0	0
$730$	310.444	0.425265
$731$	−171.350	−0.234404
$732$	0	0
$733$	1378.08i	1.88005i	0.341102	+	0.940026i	$0.389200\pi$
$733$	1378.08i	1.88005i	−0.341102	+	0.940026i	$0.610800\pi$
$734$	4.88897i	0.00666072i
$735$	0	0
$736$	− 196.609i	− 0.267132i
$737$	0	0
$738$	0	0
$739$	− 467.632i	− 0.632790i	−0.948628	−	0.316395i	$-0.897528\pi$
$739$	− 467.632i	− 0.632790i	0.948628	−	0.316395i	$-0.102472\pi$
$740$	174.677	0.236050
$741$	0	0
$742$	978.706	1.31901
$743$	− 331.864i	− 0.446654i	−0.974744	−	0.223327i	$-0.928308\pi$
$743$	− 331.864i	− 0.446654i	0.974744	−	0.223327i	$-0.0716918\pi$
$744$	0	0
$745$	− 65.6831i	− 0.0881653i
$746$	916.779	1.22893
$747$	0	0
$748$	0	0
$749$	−606.376	−0.809580
$750$	0	0
$751$	243.546	0.324296	0.162148	−	0.986766i	$-0.448158\pi$
$751$	243.546	0.324296	0.162148	+	0.986766i	$0.448158\pi$
$752$	413.008	0.549213
$753$	0	0
$754$	3071.76i	4.07396i
$755$	− 228.237i	− 0.302301i
$756$	0	0
$757$	1407.93	1.85988	0.929941	−	0.367709i	$-0.119858\pi$
$757$	1407.93	1.85988	0.929941	+	0.367709i	$0.119858\pi$
$758$	44.6787i	0.0589429i
$759$	0	0
$760$	148.311	0.195146
$761$	− 736.232i	− 0.967453i	−0.875219	−	0.483727i	$-0.839283\pi$
$761$	− 736.232i	− 0.967453i	0.875219	−	0.483727i	$-0.160717\pi$
$762$	0	0
$763$	355.757	0.466260
$764$	1506.75	1.97218
$765$	0	0
$766$	589.268i	0.769279i
$767$	1386.69i	1.80794i
$768$	0	0
$769$	− 676.556i	− 0.879787i	−0.898050	−	0.439894i	$-0.855016\pi$
$769$	− 676.556i	− 0.879787i	0.898050	−	0.439894i	$-0.144984\pi$
$770$	0	0
$771$	0	0
$772$	− 86.2222i	− 0.111687i
$773$	590.422	0.763806	0.381903	−	0.924203i	$-0.375269\pi$
$773$	590.422	0.763806	0.381903	+	0.924203i	$0.375269\pi$
$774$	0	0
$775$	481.704	0.621553
$776$	539.712i	0.695505i
$777$	0	0
$778$	− 1670.06i	− 2.14661i
$779$	717.252	0.920734
$780$	0	0
$781$	0	0
$782$	−255.637	−0.326901
$783$	0	0
$784$	−214.251	−0.273279
$785$	−124.095	−0.158083
$786$	0	0
$787$	− 918.908i	− 1.16761i	−0.811895	−	0.583804i	$-0.801564\pi$
$787$	− 918.908i	− 1.16761i	0.811895	−	0.583804i	$-0.198436\pi$
$788$	1173.19i	1.48881i
$789$	0	0
$790$	−617.971	−0.782242
$791$	− 146.596i	− 0.185330i
$792$	0	0
$793$	−236.381	−0.298085
$794$	− 564.455i	− 0.710900i
$795$	0	0
$796$	597.378	0.750475
$797$	116.989	0.146787	0.0733936	−	0.997303i	$-0.476617\pi$
$797$	116.989	0.146787	0.0733936	+	0.997303i	$0.476617\pi$
$798$	0	0
$799$	− 1129.54i	− 1.41369i
$800$	− 935.655i	− 1.16957i
$801$	0	0
$802$	1934.18i	2.41169i
$803$	0	0
$804$	0	0
$805$	26.1413i	0.0324736i
$806$	−1415.74	−1.75650
$807$	0	0
$808$	−705.841	−0.873566
$809$	769.316i	0.950947i	0.879730	+	0.475473i	$0.157723\pi$
$809$	769.316i	0.950947i	−0.879730	+	0.475473i	$0.842277\pi$
$810$	0	0
$811$	854.982i	1.05423i	0.849793	+	0.527116i	$0.176727\pi$
$811$	854.982i	1.05423i	−0.849793	+	0.527116i	$0.823273\pi$
$812$	−989.692	−1.21883
$813$	0	0
$814$	0	0
$815$	−219.888	−0.269802
$816$	0	0
$817$	197.098	0.241246
$818$	−680.951	−0.832459
$819$	0	0
$820$	− 298.116i	− 0.363556i
$821$	750.650i	0.914312i	0.889387	+	0.457156i	$0.151132\pi$
$821$	750.650i	0.914312i	−0.889387	+	0.457156i	$0.848868\pi$
$822$	0	0
$823$	−1292.59	−1.57058	−0.785292	−	0.619125i	$-0.787487\pi$
$823$	−1292.59	−1.57058	−0.785292	+	0.619125i	$0.787487\pi$
$824$	267.846i	0.325056i
$825$	0	0
$826$	−759.575	−0.919582
$827$	− 599.131i	− 0.724463i	−0.932088	−	0.362232i	$-0.882015\pi$
$827$	− 599.131i	− 0.724463i	0.932088	−	0.362232i	$-0.117985\pi$
$828$	0	0
$829$	−976.845	−1.17834	−0.589171	−	0.808009i	$-0.700545\pi$
$829$	−976.845	−1.17834	−0.589171	+	0.808009i	$0.700545\pi$
$830$	497.100	0.598916
$831$	0	0
$832$	2211.53i	2.65809i
$833$	585.955i	0.703427i
$834$	0	0
$835$	363.116i	0.434869i
$836$	0	0
$837$	0	0
$838$	2553.47i	3.04710i
$839$	−403.747	−0.481224	−0.240612	−	0.970621i	$-0.577348\pi$
$839$	−403.747	−0.481224	−0.240612	+	0.970621i	$0.577348\pi$
$840$	0	0
$841$	−1227.67	−1.45978
$842$	825.514i	0.980421i
$843$	0	0
$844$	− 824.815i	− 0.977269i
$845$	−427.272	−0.505648
$846$	0	0
$847$	0	0
$848$	512.124	0.603920
$849$	0	0
$850$	−1216.57	−1.43126
$851$	103.857	0.122042
$852$	0	0
$853$	729.021i	0.854656i	0.904097	+	0.427328i	$0.140545\pi$
$853$	729.021i	0.854656i	−0.904097	+	0.427328i	$0.859455\pi$
$854$	− 129.480i	− 0.151616i
$855$	0	0
$856$	850.167	0.993185
$857$	999.714i	1.16653i	0.812283	+	0.583264i	$0.198225\pi$
$857$	999.714i	1.16653i	−0.812283	+	0.583264i	$0.801775\pi$
$858$	0	0
$859$	1116.47	1.29973	0.649864	−	0.760050i	$-0.274826\pi$
$859$	1116.47	1.29973	0.649864	+	0.760050i	$0.274826\pi$
$860$	− 81.9211i	− 0.0952571i
$861$	0	0
$862$	−265.023	−0.307452
$863$	−1464.26	−1.69671	−0.848354	−	0.529430i	$-0.822406\pi$
$863$	−1464.26	−1.69671	−0.848354	+	0.529430i	$0.822406\pi$
$864$	0	0
$865$	− 46.6767i	− 0.0539615i
$866$	1228.86i	1.41901i
$867$	0	0
$868$	− 456.136i	− 0.525502i
$869$	0	0
$870$	0	0
$871$	489.814i	0.562359i
$872$	−498.787	−0.572004
$873$	0	0
$874$	294.051	0.336442
$875$	259.750i	0.296857i
$876$	0	0
$877$	− 244.706i	− 0.279026i	−0.990220	−	0.139513i	$-0.955446\pi$
$877$	− 244.706i	− 0.279026i	0.990220	−	0.139513i	$-0.0445538\pi$
$878$	−1901.13	−2.16529
$879$	0	0
$880$	0	0
$881$	1021.52	1.15950	0.579748	−	0.814796i	$-0.303151\pi$
$881$	1021.52	1.15950	0.579748	+	0.814796i	$0.303151\pi$
$882$	0	0
$883$	360.612	0.408394	0.204197	−	0.978930i	$-0.434542\pi$
$883$	360.612	0.408394	0.204197	+	0.978930i	$0.434542\pi$
$884$	2103.10	2.37907
$885$	0	0
$886$	− 526.553i	− 0.594304i
$887$	− 136.415i	− 0.153793i	−0.997039	−	0.0768966i	$-0.975499\pi$
$887$	− 136.415i	− 0.153793i	0.997039	−	0.0768966i	$-0.0245011\pi$
$888$	0	0
$889$	−309.606	−0.348263
$890$	281.723i	0.316543i
$891$	0	0
$892$	1069.84	1.19937
$893$	1299.27i	1.45495i
$894$	0	0
$895$	−236.121	−0.263822
$896$	−591.090	−0.659699
$897$	0	0
$898$	− 2111.14i	− 2.35093i
$899$	− 953.424i	− 1.06054i
$900$	0	0
$901$	− 1400.61i	− 1.55450i
$902$	0	0
$903$	0	0
$904$	205.534i	0.227361i
$905$	−97.4039	−0.107629
$906$	0	0
$907$	704.903	0.777181	0.388590	−	0.921411i	$-0.372962\pi$
$907$	704.903	0.777181	0.388590	+	0.921411i	$0.372962\pi$
$908$	− 276.931i	− 0.304990i
$909$	0	0
$910$	− 365.630i	− 0.401791i
$911$	191.940	0.210692	0.105346	−	0.994436i	$-0.466405\pi$
$911$	191.940	0.210692	0.105346	+	0.994436i	$0.466405\pi$
$912$	0	0
$913$	0	0
$914$	−1786.16	−1.95422
$915$	0	0
$916$	1404.88	1.53371
$917$	561.179	0.611973
$918$	0	0
$919$	366.933i	0.399275i	0.979870	+	0.199637i	$0.0639764\pi$
$919$	366.933i	0.399275i	−0.979870	+	0.199637i	$0.936024\pi$
$920$	− 36.6513i	− 0.0398383i
$921$	0	0
$922$	1038.29	1.12613
$923$	962.653i	1.04296i
$924$	0	0
$925$	494.254	0.534329
$926$	− 109.907i	− 0.118690i
$927$	0	0
$928$	−1851.92	−1.99560
$929$	−653.282	−0.703210	−0.351605	−	0.936149i	$-0.614364\pi$
$929$	−653.282	−0.703210	−0.351605	+	0.936149i	$0.614364\pi$
$930$	0	0
$931$	− 674.004i	− 0.723957i
$932$	129.917i	0.139396i
$933$	0	0
$934$	− 16.4883i	− 0.0176535i
$935$	0	0
$936$	0	0
$937$	355.518i	0.379421i	0.981840	+	0.189711i	$0.0607550\pi$
$937$	355.518i	0.379421i	−0.981840	+	0.189711i	$0.939245\pi$
$938$	−268.301	−0.286035
$939$	0	0
$940$	540.023	0.574493
$941$	− 315.563i	− 0.335349i	−0.985842	−	0.167675i	$-0.946374\pi$
$941$	− 315.563i	− 0.335349i	0.985842	−	0.167675i	$-0.0536258\pi$
$942$	0	0
$943$	− 177.250i	− 0.187964i
$944$	−397.460	−0.421038
$945$	0	0
$946$	0	0
$947$	548.676	0.579383	0.289691	−	0.957120i	$-0.406447\pi$
$947$	548.676	0.579383	0.289691	+	0.957120i	$0.406447\pi$
$948$	0	0
$949$	−1518.52	−1.60013
$950$	1399.38	1.47303
$951$	0	0
$952$	345.464i	0.362883i
$953$	1671.54i	1.75397i	0.480513	+	0.876987i	$0.340450\pi$
$953$	1671.54i	1.75397i	−0.480513	+	0.876987i	$0.659550\pi$
$954$	0	0
$955$	−374.875	−0.392539
$956$	352.054i	0.368258i
$957$	0	0
$958$	−388.602	−0.405639
$959$	− 808.377i	− 0.842937i
$960$	0	0
$961$	−521.579	−0.542746
$962$	−1452.62	−1.51000
$963$	0	0
$964$	− 942.720i	− 0.977926i
$965$	21.4519i	0.0222299i
$966$	0	0
$967$	− 562.236i	− 0.581423i	−0.956811	−	0.290711i	$-0.906108\pi$
$967$	− 562.236i	− 0.581423i	0.956811	−	0.290711i	$-0.0938919\pi$
$968$	0	0
$969$	0	0
$970$	− 447.770i	− 0.461619i
$971$	−557.371	−0.574018	−0.287009	−	0.957928i	$-0.592661\pi$
$971$	−557.371	−0.574018	−0.287009	+	0.957928i	$0.592661\pi$
$972$	0	0
$973$	−835.341	−0.858521
$974$	2069.28i	2.12452i
$975$	0	0
$976$	− 67.7527i	− 0.0694188i
$977$	429.256	0.439361	0.219681	−	0.975572i	$-0.429498\pi$
$977$	429.256	0.439361	0.219681	+	0.975572i	$0.429498\pi$
$978$	0	0
$979$	0	0
$980$	−280.141	−0.285858
$981$	0	0
$982$	−916.835	−0.933641
$983$	1310.57	1.33324	0.666618	−	0.745399i	$-0.267741\pi$
$983$	1310.57	1.33324	0.666618	+	0.745399i	$0.267741\pi$
$984$	0	0
$985$	− 291.886i	− 0.296331i
$986$	2407.92i	2.44211i
$987$	0	0
$988$	−2419.13	−2.44851
$989$	− 48.7077i	− 0.0492494i
$990$	0	0
$991$	33.4939	0.0337981	0.0168990	−	0.999857i	$-0.494621\pi$
$991$	33.4939	0.0337981	0.0168990	+	0.999857i	$0.494621\pi$
$992$	− 853.526i	− 0.860409i
$993$	0	0
$994$	−527.303	−0.530486
$995$	−148.626	−0.149373
$996$	0	0
$997$	925.721i	0.928506i	0.885703	+	0.464253i	$0.153677\pi$
$997$	925.721i	0.928506i	−0.885703	+	0.464253i	$0.846323\pi$
$998$	2001.25i	2.00526i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.l.604.4		16
3.2	odd	2	inner	1089.3.c.l.604.13		16
11.4	even	5		99.3.k.b.28.4	yes	16
11.8	odd	10		99.3.k.b.46.4	yes	16
11.10	odd	2	inner	1089.3.c.l.604.14		16
33.8	even	10		99.3.k.b.46.1	yes	16
33.26	odd	10		99.3.k.b.28.1	✓	16
33.32	even	2	inner	1089.3.c.l.604.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.k.b.28.1	✓	16	33.26	odd	10
99.3.k.b.28.4	yes	16	11.4	even	5
99.3.k.b.46.1	yes	16	33.8	even	10
99.3.k.b.46.4	yes	16	11.8	odd	10
1089.3.c.l.604.3		16	33.32	even	2	inner
1089.3.c.l.604.4		16	1.1	even	1	trivial
1089.3.c.l.604.13		16	3.2	odd	2	inner
1089.3.c.l.604.14		16	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)

\(f(q)\)	\(=\)	\(q-3.11662i q^{2} -5.71334 q^{4} +1.42146 q^{5} -3.80859i q^{7} +5.33982i q^{8} +O(q^{10})\)	\(q-3.11662i q^{2} -5.71334 q^{4} +1.42146 q^{5} -3.80859i q^{7} +5.33982i q^{8} -4.43016i q^{10} +21.6699i q^{13} -11.8699 q^{14} -6.21114 q^{16} +16.9868i q^{17} -19.5394i q^{19} -8.12129 q^{20} -4.82866 q^{23} -22.9794 q^{25} +67.5370 q^{26} +21.7598i q^{28} +45.4827i q^{29} -20.9624 q^{31} +40.7171i q^{32} +52.9416 q^{34} -5.41377i q^{35} -21.5085 q^{37} -60.8969 q^{38} +7.59035i q^{40} +36.7080i q^{41} +10.0872i q^{43} +15.0491i q^{46} -66.4947 q^{47} +34.4946 q^{49} +71.6182i q^{50} -123.808i q^{52} -82.4525 q^{53} +20.3372 q^{56} +141.752 q^{58} +63.9915 q^{59} +10.9083i q^{61} +65.3318i q^{62} +102.055 q^{64} +30.8030i q^{65} +22.6034 q^{67} -97.0515i q^{68} -16.8727 q^{70} +44.4234 q^{71} +70.0750i q^{73} +67.0340i q^{74} +111.635i q^{76} -139.492i q^{79} -8.82891 q^{80} +114.405 q^{82} +112.208i q^{83} +24.1462i q^{85} +31.4380 q^{86} -63.5921 q^{89} +82.5320 q^{91} +27.5878 q^{92} +207.239i q^{94} -27.7745i q^{95} +101.073 q^{97} -107.507i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 44 q^{4}+O(q^{10}) \) 16 * q - 44 * q^4	\( 16 q - 44 q^{4} + 244 q^{16} + 16 q^{25} - 80 q^{31} + 328 q^{34} - 280 q^{37} + 436 q^{49} + 140 q^{58} - 656 q^{64} + 300 q^{67} + 308 q^{70} + 580 q^{82} + 768 q^{91} - 720 q^{97}+O(q^{100}) \) 16 * q - 44 * q^4 + 244 * q^16 + 16 * q^25 - 80 * q^31 + 328 * q^34 - 280 * q^37 + 436 * q^49 + 140 * q^58 - 656 * q^64 + 300 * q^67 + 308 * q^70 + 580 * q^82 + 768 * q^91 - 720 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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