LMFDB - Embedded newform 1089.3.c.m.604.12

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} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 77x^12 + 88x^11 - 577x^10 + 578x^9 + 1520x^8 + 1868x^7 - 1619x^6 - 16804x^5 + 32427x^4 + 43316x^3 - 71672x^2 + 83521
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.12
Root			$1.60675 + 1.36085i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.m.604.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21517i q^{2} -0.906963 q^{4} -8.69502 q^{5} +6.67189i q^{7} +6.85159i q^{8} +O(q^{10})$	$q+2.21517i q^{2} -0.906963 q^{4} -8.69502 q^{5} +6.67189i q^{7} +6.85159i q^{8} -19.2609i q^{10} +11.1547i q^{13} -14.7794 q^{14} -18.8053 q^{16} +7.82687i q^{17} +8.44217i q^{19} +7.88606 q^{20} -9.30611 q^{23} +50.6034 q^{25} -24.7094 q^{26} -6.05116i q^{28} -7.17852i q^{29} -27.4243 q^{31} -14.2504i q^{32} -17.3378 q^{34} -58.0122i q^{35} -52.9416 q^{37} -18.7008 q^{38} -59.5747i q^{40} +47.8535i q^{41} -45.8381i q^{43} -20.6146i q^{46} +16.0778 q^{47} +4.48587 q^{49} +112.095i q^{50} -10.1169i q^{52} +54.9322 q^{53} -45.7131 q^{56} +15.9016 q^{58} +71.2113 q^{59} -1.30131i q^{61} -60.7494i q^{62} -43.6540 q^{64} -96.9901i q^{65} -28.9406 q^{67} -7.09868i q^{68} +128.507 q^{70} -22.0521 q^{71} -125.227i q^{73} -117.274i q^{74} -7.65674i q^{76} -42.8764i q^{79} +163.512 q^{80} -106.004 q^{82} -133.214i q^{83} -68.0548i q^{85} +101.539 q^{86} +9.48441 q^{89} -74.4227 q^{91} +8.44030 q^{92} +35.6151i q^{94} -73.4049i q^{95} +156.020 q^{97} +9.93695i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) $ 16 * q - 20 * q^4 + 4 * q^5	$ 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) $ 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * 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 $

$2$	2.21517i	1.10758i	0.832655	+	0.553792i	$0.186820\pi$
$2$	2.21517i	1.10758i	−0.832655	+	0.553792i	$0.813180\pi$
$3$	0	0
$3$	0	0
$4$	−0.906963	−0.226741
$5$	−8.69502	−1.73900	−0.869502	−	0.493929i	$-0.835560\pi$
$5$	−8.69502	−1.73900	−0.869502	+	0.493929i	$0.835560\pi$
$6$	0	0
$7$	6.67189i	0.953127i	0.879140	+	0.476564i	$0.158118\pi$
$7$	6.67189i	0.953127i	−0.879140	+	0.476564i	$0.841882\pi$
$8$	6.85159i	0.856449i
$9$	0	0
$10$	− 19.2609i	− 1.92609i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	11.1547i	0.858051i	0.903292	+	0.429026i	$0.141143\pi$
$13$	11.1547i	0.858051i	−0.903292	+	0.429026i	$0.858857\pi$
$14$	−14.7794	−1.05567
$15$	0	0
$16$	−18.8053	−1.17533
$17$	7.82687i	0.460404i	0.973143	+	0.230202i	$0.0739387\pi$
$17$	7.82687i	0.460404i	−0.973143	+	0.230202i	$0.926061\pi$
$18$	0	0
$19$	8.44217i	0.444325i	0.975010	+	0.222162i	$0.0713115\pi$
$19$	8.44217i	0.444325i	−0.975010	+	0.222162i	$0.928688\pi$
$20$	7.88606	0.394303
$21$	0	0
$22$	0	0
$23$	−9.30611	−0.404613	−0.202307	−	0.979322i	$-0.564844\pi$
$23$	−9.30611	−0.404613	−0.202307	+	0.979322i	$0.564844\pi$
$24$	0	0
$25$	50.6034	2.02414
$26$	−24.7094	−0.950363
$27$	0	0
$28$	− 6.05116i	− 0.216113i
$29$	− 7.17852i	− 0.247535i	−0.992311	−	0.123768i	$-0.960502\pi$
$29$	− 7.17852i	− 0.247535i	0.992311	−	0.123768i	$-0.0394978\pi$
$30$	0	0
$31$	−27.4243	−0.884655	−0.442328	−	0.896854i	$-0.645847\pi$
$31$	−27.4243	−0.884655	−0.442328	+	0.896854i	$0.645847\pi$
$32$	− 14.2504i	− 0.445326i
$33$	0	0
$34$	−17.3378	−0.509936
$35$	− 58.0122i	− 1.65749i
$36$	0	0
$37$	−52.9416	−1.43085	−0.715427	−	0.698688i	$-0.753768\pi$
$37$	−52.9416	−1.43085	−0.715427	+	0.698688i	$0.753768\pi$
$38$	−18.7008	−0.492127
$39$	0	0
$40$	− 59.5747i	− 1.48937i
$41$	47.8535i	1.16716i	0.812056	+	0.583580i	$0.198348\pi$
$41$	47.8535i	1.16716i	−0.812056	+	0.583580i	$0.801652\pi$
$42$	0	0
$43$	− 45.8381i	− 1.06600i	−0.846114	−	0.533002i	$-0.821064\pi$
$43$	− 45.8381i	− 1.06600i	0.846114	−	0.533002i	$-0.178936\pi$
$44$	0	0
$45$	0	0
$46$	− 20.6146i	− 0.448143i
$47$	16.0778	0.342082	0.171041	−	0.985264i	$-0.445287\pi$
$47$	16.0778	0.342082	0.171041	+	0.985264i	$0.445287\pi$
$48$	0	0
$49$	4.48587	0.0915484
$50$	112.095i	2.24190i
$51$	0	0
$52$	− 10.1169i	− 0.194555i
$53$	54.9322	1.03646	0.518228	−	0.855243i	$-0.326592\pi$
$53$	54.9322	1.03646	0.518228	+	0.855243i	$0.326592\pi$
$54$	0	0
$55$	0	0
$56$	−45.7131	−0.816305
$57$	0	0
$58$	15.9016	0.274166
$59$	71.2113	1.20697	0.603486	−	0.797374i	$-0.293778\pi$
$59$	71.2113	1.20697	0.603486	+	0.797374i	$0.293778\pi$
$60$	0	0
$61$	− 1.30131i	− 0.0213330i	−0.999943	−	0.0106665i	$-0.996605\pi$
$61$	− 1.30131i	− 0.0213330i	0.999943	−	0.0106665i	$-0.00339531\pi$
$62$	− 60.7494i	− 0.979830i
$63$	0	0
$64$	−43.6540	−0.682094
$65$	− 96.9901i	− 1.49216i
$66$	0	0
$67$	−28.9406	−0.431949	−0.215975	−	0.976399i	$-0.569293\pi$
$67$	−28.9406	−0.431949	−0.215975	+	0.976399i	$0.569293\pi$
$68$	− 7.09868i	− 0.104392i
$69$	0	0
$70$	128.507	1.83581
$71$	−22.0521	−0.310593	−0.155296	−	0.987868i	$-0.549633\pi$
$71$	−22.0521	−0.310593	−0.155296	+	0.987868i	$0.549633\pi$
$72$	0	0
$73$	− 125.227i	− 1.71544i	−0.514120	−	0.857718i	$-0.671881\pi$
$73$	− 125.227i	− 1.71544i	0.514120	−	0.857718i	$-0.328119\pi$
$74$	− 117.274i	− 1.58479i
$75$	0	0
$76$	− 7.65674i	− 0.100747i
$77$	0	0
$78$	0	0
$79$	− 42.8764i	− 0.542739i	−0.962475	−	0.271370i	$-0.912523\pi$
$79$	− 42.8764i	− 0.542739i	0.962475	−	0.271370i	$-0.0874765\pi$
$80$	163.512	2.04390
$81$	0	0
$82$	−106.004	−1.29273
$83$	− 133.214i	− 1.60499i	−0.596661	−	0.802493i	$-0.703506\pi$
$83$	− 133.214i	− 1.60499i	0.596661	−	0.802493i	$-0.296494\pi$
$84$	0	0
$85$	− 68.0548i	− 0.800645i
$86$	101.539	1.18069
$87$	0	0
$88$	0	0
$89$	9.48441	0.106566	0.0532832	−	0.998579i	$-0.483031\pi$
$89$	9.48441	0.106566	0.0532832	+	0.998579i	$0.483031\pi$
$90$	0	0
$91$	−74.4227	−0.817832
$92$	8.44030	0.0917423
$93$	0	0
$94$	35.6151i	0.378884i
$95$	− 73.4049i	− 0.772683i
$96$	0	0
$97$	156.020	1.60845	0.804226	−	0.594324i	$-0.202580\pi$
$97$	156.020	1.60845	0.804226	+	0.594324i	$0.202580\pi$
$98$	9.93695i	0.101397i
$99$	0	0
$100$	−45.8954	−0.458954
$101$	− 55.0762i	− 0.545309i	−0.962112	−	0.272655i	$-0.912098\pi$
$101$	− 55.0762i	− 0.545309i	0.962112	−	0.272655i	$-0.0879016\pi$
$102$	0	0
$103$	−81.0177	−0.786579	−0.393290	−	0.919415i	$-0.628663\pi$
$103$	−81.0177	−0.786579	−0.393290	+	0.919415i	$0.628663\pi$
$104$	−76.4272	−0.734877
$105$	0	0
$106$	121.684i	1.14796i
$107$	− 78.1987i	− 0.730829i	−0.930845	−	0.365415i	$-0.880927\pi$
$107$	− 78.1987i	− 0.730829i	0.930845	−	0.365415i	$-0.119073\pi$
$108$	0	0
$109$	2.67841i	0.0245725i	0.999925	+	0.0122863i	$0.00391094\pi$
$109$	2.67841i	0.0245725i	−0.999925	+	0.0122863i	$0.996089\pi$
$110$	0	0
$111$	0	0
$112$	− 125.467i	− 1.12024i
$113$	−184.206	−1.63014	−0.815069	−	0.579364i	$-0.803301\pi$
$113$	−184.206	−1.63014	−0.815069	+	0.579364i	$0.803301\pi$
$114$	0	0
$115$	80.9168	0.703624
$116$	6.51066i	0.0561263i
$117$	0	0
$118$	157.745i	1.33682i
$119$	−52.2200	−0.438824
$120$	0	0
$121$	0	0
$122$	2.88262	0.0236280
$123$	0	0
$124$	24.8728	0.200587
$125$	−222.622	−1.78098
$126$	0	0
$127$	119.107i	0.937849i	0.883238	+	0.468924i	$0.155358\pi$
$127$	119.107i	0.937849i	−0.883238	+	0.468924i	$0.844642\pi$
$128$	− 153.703i	− 1.20080i
$129$	0	0
$130$	214.849	1.65269
$131$	− 17.9999i	− 0.137403i	−0.997637	−	0.0687017i	$-0.978114\pi$
$131$	− 17.9999i	− 0.137403i	0.997637	−	0.0687017i	$-0.0218857\pi$
$132$	0	0
$133$	−56.3253	−0.423498
$134$	− 64.1083i	− 0.478420i
$135$	0	0
$136$	−53.6265	−0.394313
$137$	−92.5606	−0.675625	−0.337812	−	0.941213i	$-0.609687\pi$
$137$	−92.5606	−0.675625	−0.337812	+	0.941213i	$0.609687\pi$
$138$	0	0
$139$	216.817i	1.55983i	0.625883	+	0.779917i	$0.284739\pi$
$139$	216.817i	1.55983i	−0.625883	+	0.779917i	$0.715261\pi$
$140$	52.6150i	0.375821i
$141$	0	0
$142$	− 48.8490i	− 0.344007i
$143$	0	0
$144$	0	0
$145$	62.4174i	0.430465i
$146$	277.398	1.89999
$147$	0	0
$148$	48.0161	0.324433
$149$	140.721i	0.944438i	0.881481	+	0.472219i	$0.156547\pi$
$149$	140.721i	0.944438i	−0.881481	+	0.472219i	$0.843453\pi$
$150$	0	0
$151$	11.5088i	0.0762172i	0.999274	+	0.0381086i	$0.0121333\pi$
$151$	11.5088i	0.0762172i	−0.999274	+	0.0381086i	$0.987867\pi$
$152$	−57.8423	−0.380542
$153$	0	0
$154$	0	0
$155$	238.455	1.53842
$156$	0	0
$157$	−149.729	−0.953690	−0.476845	−	0.878987i	$-0.658220\pi$
$157$	−149.729	−0.953690	−0.476845	+	0.878987i	$0.658220\pi$
$158$	94.9783	0.601129
$159$	0	0
$160$	123.908i	0.774424i
$161$	− 62.0893i	− 0.385648i
$162$	0	0
$163$	−54.8173	−0.336302	−0.168151	−	0.985761i	$-0.553780\pi$
$163$	−54.8173	−0.336302	−0.168151	+	0.985761i	$0.553780\pi$
$164$	− 43.4014i	− 0.264643i
$165$	0	0
$166$	295.091	1.77766
$167$	− 42.9417i	− 0.257136i	−0.991701	−	0.128568i	$-0.958962\pi$
$167$	− 42.9417i	− 0.257136i	0.991701	−	0.128568i	$-0.0410381\pi$
$168$	0	0
$169$	44.5734	0.263748
$170$	150.753	0.886781
$171$	0	0
$172$	41.5735i	0.241706i
$173$	− 16.2431i	− 0.0938907i	−0.998897	−	0.0469453i	$-0.985051\pi$
$173$	− 16.2431i	− 0.0938907i	0.998897	−	0.0469453i	$-0.0149487\pi$
$174$	0	0
$175$	337.620i	1.92926i
$176$	0	0
$177$	0	0
$178$	21.0095i	0.118031i
$179$	−130.353	−0.728228	−0.364114	−	0.931354i	$-0.618628\pi$
$179$	−130.353	−0.728228	−0.364114	+	0.931354i	$0.618628\pi$
$180$	0	0
$181$	−101.541	−0.560998	−0.280499	−	0.959854i	$-0.590500\pi$
$181$	−101.541	−0.560998	−0.280499	+	0.959854i	$0.590500\pi$
$182$	− 164.859i	− 0.905817i
$183$	0	0
$184$	− 63.7617i	− 0.346531i
$185$	460.328	2.48826
$186$	0	0
$187$	0	0
$188$	−14.5820	−0.0775639
$189$	0	0
$190$	162.604	0.855811
$191$	161.775	0.846987	0.423494	−	0.905899i	$-0.360804\pi$
$191$	161.775	0.846987	0.423494	+	0.905899i	$0.360804\pi$
$192$	0	0
$193$	− 34.1916i	− 0.177159i	−0.996069	−	0.0885793i	$-0.971767\pi$
$193$	− 34.1916i	− 0.177159i	0.996069	−	0.0885793i	$-0.0282327\pi$
$194$	345.610i	1.78149i
$195$	0	0
$196$	−4.06852	−0.0207577
$197$	61.8792i	0.314107i	0.987590	+	0.157054i	$0.0501996\pi$
$197$	61.8792i	0.314107i	−0.987590	+	0.157054i	$0.949800\pi$
$198$	0	0
$199$	238.301	1.19749	0.598747	−	0.800938i	$-0.295666\pi$
$199$	238.301	1.19749	0.598747	+	0.800938i	$0.295666\pi$
$200$	346.714i	1.73357i
$201$	0	0
$202$	122.003	0.603975
$203$	47.8943	0.235933
$204$	0	0
$205$	− 416.087i	− 2.02969i
$206$	− 179.468i	− 0.871202i
$207$	0	0
$208$	− 209.767i	− 1.00849i
$209$	0	0
$210$	0	0
$211$	188.443i	0.893096i	0.894760	+	0.446548i	$0.147347\pi$
$211$	188.443i	0.893096i	−0.894760	+	0.446548i	$0.852653\pi$
$212$	−49.8214	−0.235007
$213$	0	0
$214$	173.223	0.809454
$215$	398.564i	1.85378i
$216$	0	0
$217$	− 182.972i	− 0.843189i
$218$	−5.93311	−0.0272161
$219$	0	0
$220$	0	0
$221$	−87.3062	−0.395050
$222$	0	0
$223$	−258.263	−1.15813	−0.579065	−	0.815281i	$-0.696582\pi$
$223$	−258.263	−1.15813	−0.579065	+	0.815281i	$0.696582\pi$
$224$	95.0774	0.424453
$225$	0	0
$226$	− 408.046i	− 1.80551i
$227$	44.3088i	0.195193i	0.995226	+	0.0975965i	$0.0311155\pi$
$227$	44.3088i	0.195193i	−0.995226	+	0.0975965i	$0.968885\pi$
$228$	0	0
$229$	−3.83877	−0.0167632	−0.00838160	−	0.999965i	$-0.502668\pi$
$229$	−3.83877	−0.0167632	−0.00838160	+	0.999965i	$0.502668\pi$
$230$	179.244i	0.779323i
$231$	0	0
$232$	49.1843	0.212001
$233$	− 241.342i	− 1.03580i	−0.855440	−	0.517902i	$-0.826713\pi$
$233$	− 241.342i	− 1.03580i	0.855440	−	0.517902i	$-0.173287\pi$
$234$	0	0
$235$	−139.797	−0.594882
$236$	−64.5860	−0.273670
$237$	0	0
$238$	− 115.676i	− 0.486034i
$239$	463.393i	1.93888i	0.245323	+	0.969441i	$0.421106\pi$
$239$	463.393i	1.93888i	−0.245323	+	0.969441i	$0.578894\pi$
$240$	0	0
$241$	447.213i	1.85565i	0.373011	+	0.927827i	$0.378325\pi$
$241$	447.213i	1.85565i	−0.373011	+	0.927827i	$0.621675\pi$
$242$	0	0
$243$	0	0
$244$	1.18024i	0.00483705i
$245$	−39.0047	−0.159203
$246$	0	0
$247$	−94.1696	−0.381254
$248$	− 187.900i	− 0.757662i
$249$	0	0
$250$	− 493.145i	− 1.97258i
$251$	−380.631	−1.51646	−0.758230	−	0.651987i	$-0.773936\pi$
$251$	−380.631	−1.51646	−0.758230	+	0.651987i	$0.773936\pi$
$252$	0	0
$253$	0	0
$254$	−263.841	−1.03875
$255$	0	0
$256$	165.861	0.647894
$257$	−3.33118	−0.0129618	−0.00648089	−	0.999979i	$-0.502063\pi$
$257$	−3.33118	−0.0129618	−0.00648089	+	0.999979i	$0.502063\pi$
$258$	0	0
$259$	− 353.220i	− 1.36379i
$260$	87.9664i	0.338332i
$261$	0	0
$262$	39.8727	0.152186
$263$	379.793i	1.44408i	0.691850	+	0.722041i	$0.256796\pi$
$263$	379.793i	1.44408i	−0.691850	+	0.722041i	$0.743204\pi$
$264$	0	0
$265$	−477.636	−1.80240
$266$	− 124.770i	− 0.469060i
$267$	0	0
$268$	26.2481	0.0979405
$269$	187.216	0.695972	0.347986	−	0.937500i	$-0.386866\pi$
$269$	187.216	0.695972	0.347986	+	0.937500i	$0.386866\pi$
$270$	0	0
$271$	305.158i	1.12604i	0.826442	+	0.563022i	$0.190361\pi$
$271$	305.158i	1.12604i	−0.826442	+	0.563022i	$0.809639\pi$
$272$	− 147.186i	− 0.541127i
$273$	0	0
$274$	− 205.037i	− 0.748311i
$275$	0	0
$276$	0	0
$277$	− 480.172i	− 1.73347i	−0.498765	−	0.866737i	$-0.666213\pi$
$277$	− 480.172i	− 1.73347i	0.498765	−	0.866737i	$-0.333787\pi$
$278$	−480.285	−1.72765
$279$	0	0
$280$	397.476	1.41956
$281$	9.05822i	0.0322357i	0.999870	+	0.0161178i	$0.00513069\pi$
$281$	9.05822i	0.0322357i	−0.999870	+	0.0161178i	$0.994869\pi$
$282$	0	0
$283$	266.704i	0.942416i	0.882022	+	0.471208i	$0.156182\pi$
$283$	266.704i	0.942416i	−0.882022	+	0.471208i	$0.843818\pi$
$284$	20.0004	0.0704240
$285$	0	0
$286$	0	0
$287$	−319.274	−1.11245
$288$	0	0
$289$	227.740	0.788028
$290$	−138.265	−0.476776
$291$	0	0
$292$	113.576i	0.388959i
$293$	170.234i	0.581004i	0.956874	+	0.290502i	$0.0938223\pi$
$293$	170.234i	0.581004i	−0.956874	+	0.290502i	$0.906178\pi$
$294$	0	0
$295$	−619.184	−2.09893
$296$	− 362.734i	− 1.22545i
$297$	0	0
$298$	−311.721	−1.04604
$299$	− 103.807i	− 0.347179i
$300$	0	0
$301$	305.827	1.01604
$302$	−25.4939	−0.0844169
$303$	0	0
$304$	− 158.757i	− 0.522228i
$305$	11.3149i	0.0370981i
$306$	0	0
$307$	− 228.869i	− 0.745501i	−0.927932	−	0.372750i	$-0.878415\pi$
$307$	− 228.869i	− 0.745501i	0.927932	−	0.372750i	$-0.121585\pi$
$308$	0	0
$309$	0	0
$310$	528.218i	1.70393i
$311$	−282.560	−0.908555	−0.454277	−	0.890860i	$-0.650102\pi$
$311$	−282.560	−0.908555	−0.454277	+	0.890860i	$0.650102\pi$
$312$	0	0
$313$	−350.176	−1.11877	−0.559386	−	0.828907i	$-0.688963\pi$
$313$	−350.176	−1.11877	−0.559386	+	0.828907i	$0.688963\pi$
$314$	− 331.675i	− 1.05629i
$315$	0	0
$316$	38.8873i	0.123061i
$317$	−374.415	−1.18112	−0.590559	−	0.806994i	$-0.701093\pi$
$317$	−374.415	−1.18112	−0.590559	+	0.806994i	$0.701093\pi$
$318$	0	0
$319$	0	0
$320$	379.572	1.18616
$321$	0	0
$322$	137.538	0.427137
$323$	−66.0758	−0.204569
$324$	0	0
$325$	564.464i	1.73681i
$326$	− 121.429i	− 0.372483i
$327$	0	0
$328$	−327.873	−0.999612
$329$	107.270i	0.326048i
$330$	0	0
$331$	262.925	0.794334	0.397167	−	0.917746i	$-0.369993\pi$
$331$	262.925	0.794334	0.397167	+	0.917746i	$0.369993\pi$
$332$	120.820i	0.363916i
$333$	0	0
$334$	95.1230	0.284799
$335$	251.639	0.751162
$336$	0	0
$337$	31.6019i	0.0937743i	0.998900	+	0.0468872i	$0.0149301\pi$
$337$	31.6019i	0.0937743i	−0.998900	+	0.0468872i	$0.985070\pi$
$338$	98.7375i	0.292123i
$339$	0	0
$340$	61.7232i	0.181539i
$341$	0	0
$342$	0	0
$343$	356.852i	1.04038i
$344$	314.064	0.912977
$345$	0	0
$346$	35.9811	0.103992
$347$	− 413.522i	− 1.19170i	−0.803094	−	0.595852i	$-0.796814\pi$
$347$	− 413.522i	− 1.19170i	0.803094	−	0.595852i	$-0.203186\pi$
$348$	0	0
$349$	367.213i	1.05219i	0.850427	+	0.526094i	$0.176344\pi$
$349$	367.213i	1.05219i	−0.850427	+	0.526094i	$0.823656\pi$
$350$	−747.885	−2.13682
$351$	0	0
$352$	0	0
$353$	415.577	1.17727	0.588636	−	0.808398i	$-0.299665\pi$
$353$	415.577	1.17727	0.588636	+	0.808398i	$0.299665\pi$
$354$	0	0
$355$	191.743	0.540122
$356$	−8.60201	−0.0241629
$357$	0	0
$358$	− 288.753i	− 0.806574i
$359$	154.752i	0.431065i	0.976497	+	0.215532i	$0.0691487\pi$
$359$	154.752i	0.431065i	−0.976497	+	0.215532i	$0.930851\pi$
$360$	0	0
$361$	289.730	0.802575
$362$	− 224.929i	− 0.621352i
$363$	0	0
$364$	67.4987	0.185436
$365$	1088.85i	2.98315i
$366$	0	0
$367$	−108.936	−0.296829	−0.148414	−	0.988925i	$-0.547417\pi$
$367$	−108.936	−0.296829	−0.148414	+	0.988925i	$0.547417\pi$
$368$	175.004	0.475554
$369$	0	0
$370$	1019.70i	2.75596i
$371$	366.501i	0.987875i
$372$	0	0
$373$	− 721.229i	− 1.93359i	−0.255555	−	0.966795i	$-0.582258\pi$
$373$	− 721.229i	− 1.93359i	0.255555	−	0.966795i	$-0.417742\pi$
$374$	0	0
$375$	0	0
$376$	110.159i	0.292976i
$377$	80.0741	0.212398
$378$	0	0
$379$	−12.9375	−0.0341358	−0.0170679	−	0.999854i	$-0.505433\pi$
$379$	−12.9375	−0.0341358	−0.0170679	+	0.999854i	$0.505433\pi$
$380$	66.5755i	0.175199i
$381$	0	0
$382$	358.358i	0.938109i
$383$	99.6126	0.260085	0.130043	−	0.991508i	$-0.458489\pi$
$383$	99.6126	0.260085	0.130043	+	0.991508i	$0.458489\pi$
$384$	0	0
$385$	0	0
$386$	75.7401	0.196218
$387$	0	0
$388$	−141.504	−0.364701
$389$	−202.122	−0.519595	−0.259797	−	0.965663i	$-0.583656\pi$
$389$	−202.122	−0.519595	−0.259797	+	0.965663i	$0.583656\pi$
$390$	0	0
$391$	− 72.8377i	− 0.186286i
$392$	30.7353i	0.0784065i
$393$	0	0
$394$	−137.073	−0.347900
$395$	372.811i	0.943826i
$396$	0	0
$397$	332.729	0.838108	0.419054	−	0.907961i	$-0.362362\pi$
$397$	332.729	0.838108	0.419054	+	0.907961i	$0.362362\pi$
$398$	527.877i	1.32632i
$399$	0	0
$400$	−951.611	−2.37903
$401$	−656.880	−1.63811	−0.819053	−	0.573718i	$-0.805500\pi$
$401$	−656.880	−1.63811	−0.819053	+	0.573718i	$0.805500\pi$
$402$	0	0
$403$	− 305.909i	− 0.759080i
$404$	49.9521i	0.123644i
$405$	0	0
$406$	106.094i	0.261315i
$407$	0	0
$408$	0	0
$409$	− 469.056i	− 1.14684i	−0.819263	−	0.573418i	$-0.805617\pi$
$409$	− 469.056i	− 1.14684i	0.819263	−	0.573418i	$-0.194383\pi$
$410$	921.703	2.24806
$411$	0	0
$412$	73.4800	0.178350
$413$	475.114i	1.15040i
$414$	0	0
$415$	1158.30i	2.79108i
$416$	158.959	0.382113
$417$	0	0
$418$	0	0
$419$	−242.229	−0.578112	−0.289056	−	0.957312i	$-0.593341\pi$
$419$	−242.229	−0.578112	−0.289056	+	0.957312i	$0.593341\pi$
$420$	0	0
$421$	−737.569	−1.75195	−0.875973	−	0.482360i	$-0.839780\pi$
$421$	−737.569	−1.75195	−0.875973	+	0.482360i	$0.839780\pi$
$422$	−417.433	−0.989178
$423$	0	0
$424$	376.373i	0.887672i
$425$	396.066i	0.931921i
$426$	0	0
$427$	8.68220	0.0203330
$428$	70.9233i	0.165709i
$429$	0	0
$430$	−882.885	−2.05322
$431$	− 491.440i	− 1.14023i	−0.821564	−	0.570116i	$-0.806898\pi$
$431$	− 491.440i	− 1.14023i	0.821564	−	0.570116i	$-0.193102\pi$
$432$	0	0
$433$	−402.664	−0.929939	−0.464970	−	0.885327i	$-0.653935\pi$
$433$	−402.664	−0.929939	−0.464970	+	0.885327i	$0.653935\pi$
$434$	405.314	0.933902
$435$	0	0
$436$	− 2.42921i	− 0.00557159i
$437$	− 78.5638i	− 0.179780i
$438$	0	0
$439$	− 68.4030i	− 0.155815i	−0.996961	−	0.0779077i	$-0.975176\pi$
$439$	− 68.4030i	− 0.155815i	0.996961	−	0.0779077i	$-0.0248240\pi$
$440$	0	0
$441$	0	0
$442$	− 193.398i	− 0.437551i
$443$	−313.441	−0.707542	−0.353771	−	0.935332i	$-0.615101\pi$
$443$	−313.441	−0.707542	−0.353771	+	0.935332i	$0.615101\pi$
$444$	0	0
$445$	−82.4671	−0.185319
$446$	− 572.096i	− 1.28273i
$447$	0	0
$448$	− 291.255i	− 0.650122i
$449$	−329.111	−0.732987	−0.366494	−	0.930421i	$-0.619442\pi$
$449$	−329.111	−0.732987	−0.366494	+	0.930421i	$0.619442\pi$
$450$	0	0
$451$	0	0
$452$	167.068	0.369619
$453$	0	0
$454$	−98.1514	−0.216193
$455$	647.107	1.42221
$456$	0	0
$457$	88.5487i	0.193761i	0.995296	+	0.0968804i	$0.0308864\pi$
$457$	88.5487i	0.193761i	−0.995296	+	0.0968804i	$0.969114\pi$
$458$	− 8.50352i	− 0.0185666i
$459$	0	0
$460$	−73.3886	−0.159540
$461$	− 607.310i	− 1.31737i	−0.752417	−	0.658687i	$-0.771112\pi$
$461$	− 607.310i	− 1.31737i	0.752417	−	0.658687i	$-0.228888\pi$
$462$	0	0
$463$	−40.7126	−0.0879321	−0.0439660	−	0.999033i	$-0.513999\pi$
$463$	−40.7126	−0.0879321	−0.0439660	+	0.999033i	$0.513999\pi$
$464$	134.994i	0.290936i
$465$	0	0
$466$	534.614	1.14724
$467$	447.800	0.958887	0.479443	−	0.877573i	$-0.340839\pi$
$467$	447.800	0.958887	0.479443	+	0.877573i	$0.340839\pi$
$468$	0	0
$469$	− 193.089i	− 0.411703i
$470$	− 309.674i	− 0.658881i
$471$	0	0
$472$	487.911i	1.03371i
$473$	0	0
$474$	0	0
$475$	427.203i	0.899374i
$476$	47.3616	0.0994992
$477$	0	0
$478$	−1026.49	−2.14747
$479$	− 58.4924i	− 0.122114i	−0.998134	−	0.0610568i	$-0.980553\pi$
$479$	− 58.4924i	− 0.122114i	0.998134	−	0.0610568i	$-0.0194471\pi$
$480$	0	0
$481$	− 590.546i	− 1.22775i
$482$	−990.650	−2.05529
$483$	0	0
$484$	0	0
$485$	−1356.60	−2.79710
$486$	0	0
$487$	−653.283	−1.34144	−0.670722	−	0.741709i	$-0.734015\pi$
$487$	−653.283	−1.34144	−0.670722	+	0.741709i	$0.734015\pi$
$488$	8.91605	0.0182706
$489$	0	0
$490$	− 86.4020i	− 0.176331i
$491$	− 294.767i	− 0.600339i	−0.953886	−	0.300170i	$-0.902957\pi$
$491$	− 294.767i	− 0.600339i	0.953886	−	0.300170i	$-0.0970433\pi$
$492$	0	0
$493$	56.1854	0.113966
$494$	− 208.601i	− 0.422270i
$495$	0	0
$496$	515.722	1.03976
$497$	− 147.129i	− 0.296034i
$498$	0	0
$499$	41.3213	0.0828083	0.0414042	−	0.999142i	$-0.486817\pi$
$499$	41.3213	0.0828083	0.0414042	+	0.999142i	$0.486817\pi$
$500$	201.910	0.403820
$501$	0	0
$502$	− 843.162i	− 1.67961i
$503$	− 612.314i	− 1.21733i	−0.793429	−	0.608663i	$-0.791706\pi$
$503$	− 612.314i	− 1.21733i	0.793429	−	0.608663i	$-0.208294\pi$
$504$	0	0
$505$	478.889i	0.948295i
$506$	0	0
$507$	0	0
$508$	− 108.025i	− 0.212648i
$509$	299.983	0.589358	0.294679	−	0.955596i	$-0.404787\pi$
$509$	299.983	0.589358	0.294679	+	0.955596i	$0.404787\pi$
$510$	0	0
$511$	835.500	1.63503
$512$	− 247.401i	− 0.483205i
$513$	0	0
$514$	− 7.37911i	− 0.0143562i
$515$	704.450	1.36786
$516$	0	0
$517$	0	0
$518$	782.442	1.51051
$519$	0	0
$520$	664.537	1.27795
$521$	−635.801	−1.22035	−0.610174	−	0.792268i	$-0.708900\pi$
$521$	−635.801	−1.22035	−0.610174	+	0.792268i	$0.708900\pi$
$522$	0	0
$523$	8.23200i	0.0157400i	0.999969	+	0.00786998i	$0.00250512\pi$
$523$	8.23200i	0.0157400i	−0.999969	+	0.00786998i	$0.997495\pi$
$524$	16.3252i	0.0311550i
$525$	0	0
$526$	−841.306	−1.59944
$527$	− 214.647i	− 0.407299i
$528$	0	0
$529$	−442.396	−0.836288
$530$	− 1058.04i	− 1.99631i
$531$	0	0
$532$	51.0849	0.0960243
$533$	−533.790	−1.00148
$534$	0	0
$535$	679.939i	1.27091i
$536$	− 198.289i	− 0.369943i
$537$	0	0
$538$	414.716i	0.770847i
$539$	0	0
$540$	0	0
$541$	205.864i	0.380526i	0.981733	+	0.190263i	$0.0609340\pi$
$541$	205.864i	0.380526i	−0.981733	+	0.190263i	$0.939066\pi$
$542$	−675.975	−1.24719
$543$	0	0
$544$	111.536	0.205030
$545$	− 23.2888i	− 0.0427317i
$546$	0	0
$547$	− 932.049i	− 1.70393i	−0.523600	−	0.851964i	$-0.675411\pi$
$547$	− 932.049i	− 1.70393i	0.523600	−	0.851964i	$-0.324589\pi$
$548$	83.9491	0.153192
$549$	0	0
$550$	0	0
$551$	60.6023	0.109986
$552$	0	0
$553$	286.067	0.517299
$554$	1063.66	1.91997
$555$	0	0
$556$	− 196.645i	− 0.353678i
$557$	745.673i	1.33873i	0.742933	+	0.669366i	$0.233434\pi$
$557$	745.673i	1.33873i	−0.742933	+	0.669366i	$0.766566\pi$
$558$	0	0
$559$	511.309	0.914686
$560$	1090.94i	1.94810i
$561$	0	0
$562$	−20.0655	−0.0357037
$563$	766.304i	1.36111i	0.732698	+	0.680554i	$0.238261\pi$
$563$	766.304i	1.36111i	−0.732698	+	0.680554i	$0.761739\pi$
$564$	0	0
$565$	1601.67	2.83482
$566$	−590.793	−1.04380
$567$	0	0
$568$	− 151.092i	− 0.266007i
$569$	886.251i	1.55756i	0.627297	+	0.778780i	$0.284161\pi$
$569$	886.251i	1.55756i	−0.627297	+	0.778780i	$0.715839\pi$
$570$	0	0
$571$	− 504.852i	− 0.884154i	−0.896977	−	0.442077i	$-0.854242\pi$
$571$	− 504.852i	− 0.884154i	0.896977	−	0.442077i	$-0.145758\pi$
$572$	0	0
$573$	0	0
$574$	− 707.244i	− 1.23213i
$575$	−470.921	−0.818993
$576$	0	0
$577$	865.114	1.49933	0.749666	−	0.661817i	$-0.230214\pi$
$577$	865.114	1.49933	0.749666	+	0.661817i	$0.230214\pi$
$578$	504.482i	0.872807i
$579$	0	0
$580$	− 56.6103i	− 0.0976040i
$581$	888.788	1.52976
$582$	0	0
$583$	0	0
$584$	858.004	1.46918
$585$	0	0
$586$	−377.097	−0.643511
$587$	733.629	1.24979	0.624897	−	0.780708i	$-0.285141\pi$
$587$	733.629	1.24979	0.624897	+	0.780708i	$0.285141\pi$
$588$	0	0
$589$	− 231.521i	− 0.393074i
$590$	− 1371.60i	− 2.32474i
$591$	0	0
$592$	995.581	1.68172
$593$	1126.19i	1.89915i	0.313544	+	0.949574i	$0.398484\pi$
$593$	1126.19i	1.89915i	−0.313544	+	0.949574i	$0.601516\pi$
$594$	0	0
$595$	454.054	0.763117
$596$	− 127.629i	− 0.214143i
$597$	0	0
$598$	229.949	0.384530
$599$	36.7240	0.0613089	0.0306545	−	0.999530i	$-0.490241\pi$
$599$	36.7240	0.0613089	0.0306545	+	0.999530i	$0.490241\pi$
$600$	0	0
$601$	642.204i	1.06856i	0.845308	+	0.534279i	$0.179417\pi$
$601$	642.204i	1.06856i	−0.845308	+	0.534279i	$0.820583\pi$
$602$	677.458i	1.12535i
$603$	0	0
$604$	− 10.4381i	− 0.0172815i
$605$	0	0
$606$	0	0
$607$	− 660.993i	− 1.08895i	−0.838777	−	0.544475i	$-0.816729\pi$
$607$	− 660.993i	− 1.08895i	0.838777	−	0.544475i	$-0.183271\pi$
$608$	120.305	0.197869
$609$	0	0
$610$	−25.0644	−0.0410892
$611$	179.343i	0.293524i
$612$	0	0
$613$	− 729.719i	− 1.19041i	−0.803575	−	0.595203i	$-0.797072\pi$
$613$	− 729.719i	− 1.19041i	0.803575	−	0.595203i	$-0.202928\pi$
$614$	506.982	0.825704
$615$	0	0
$616$	0	0
$617$	329.848	0.534600	0.267300	−	0.963613i	$-0.413868\pi$
$617$	329.848	0.534600	0.267300	+	0.963613i	$0.413868\pi$
$618$	0	0
$619$	1033.26	1.66924	0.834620	−	0.550826i	$-0.185687\pi$
$619$	1033.26	1.66924	0.834620	+	0.550826i	$0.185687\pi$
$620$	−216.270	−0.348822
$621$	0	0
$622$	− 625.919i	− 1.00630i
$623$	63.2789i	0.101571i
$624$	0	0
$625$	670.620	1.07299
$626$	− 775.697i	− 1.23913i
$627$	0	0
$628$	135.799	0.216240
$629$	− 414.367i	− 0.658771i
$630$	0	0
$631$	−854.749	−1.35459	−0.677297	−	0.735709i	$-0.736849\pi$
$631$	−854.749	−1.35459	−0.677297	+	0.735709i	$0.736849\pi$
$632$	293.771	0.464828
$633$	0	0
$634$	− 829.391i	− 1.30819i
$635$	− 1035.64i	− 1.63092i
$636$	0	0
$637$	50.0384i	0.0785532i
$638$	0	0
$639$	0	0
$640$	1336.45i	2.08820i
$641$	−452.979	−0.706676	−0.353338	−	0.935496i	$-0.614953\pi$
$641$	−452.979	−0.706676	−0.353338	+	0.935496i	$0.614953\pi$
$642$	0	0
$643$	−265.255	−0.412527	−0.206264	−	0.978496i	$-0.566130\pi$
$643$	−265.255	−0.412527	−0.206264	+	0.978496i	$0.566130\pi$
$644$	56.3127i	0.0874421i
$645$	0	0
$646$	− 146.369i	− 0.226577i
$647$	342.818	0.529859	0.264929	−	0.964268i	$-0.414651\pi$
$647$	342.818	0.529859	0.264929	+	0.964268i	$0.414651\pi$
$648$	0	0
$649$	0	0
$650$	−1250.38	−1.92366
$651$	0	0
$652$	49.7172	0.0762534
$653$	−543.167	−0.831802	−0.415901	−	0.909410i	$-0.636534\pi$
$653$	−543.167	−0.831802	−0.415901	+	0.909410i	$0.636534\pi$
$654$	0	0
$655$	156.509i	0.238945i
$656$	− 899.899i	− 1.37180i
$657$	0	0
$658$	−237.620	−0.361125
$659$	− 309.878i	− 0.470224i	−0.971968	−	0.235112i	$-0.924454\pi$
$659$	− 309.878i	− 0.470224i	0.971968	−	0.235112i	$-0.0755457\pi$
$660$	0	0
$661$	389.483	0.589234	0.294617	−	0.955615i	$-0.404808\pi$
$661$	389.483	0.589234	0.294617	+	0.955615i	$0.404808\pi$
$662$	582.422i	0.879791i
$663$	0	0
$664$	912.727	1.37459
$665$	489.749	0.736465
$666$	0	0
$667$	66.8041i	0.100156i
$668$	38.9465i	0.0583032i
$669$	0	0
$670$	557.423i	0.831974i
$671$	0	0
$672$	0	0
$673$	− 721.948i	− 1.07273i	−0.843986	−	0.536365i	$-0.819797\pi$
$673$	− 721.948i	− 1.07273i	0.843986	−	0.536365i	$-0.180203\pi$
$674$	−70.0036	−0.103863
$675$	0	0
$676$	−40.4264	−0.0598024
$677$	846.044i	1.24970i	0.780746	+	0.624848i	$0.214839\pi$
$677$	846.044i	1.24970i	−0.780746	+	0.624848i	$0.785161\pi$
$678$	0	0
$679$	1040.95i	1.53306i
$680$	466.284	0.685712
$681$	0	0
$682$	0	0
$683$	−49.2192	−0.0720632	−0.0360316	−	0.999351i	$-0.511472\pi$
$683$	−49.2192	−0.0720632	−0.0360316	+	0.999351i	$0.511472\pi$
$684$	0	0
$685$	804.817	1.17491
$686$	−790.486	−1.15231
$687$	0	0
$688$	861.999i	1.25290i
$689$	612.750i	0.889333i
$690$	0	0
$691$	705.693	1.02126	0.510632	−	0.859799i	$-0.329411\pi$
$691$	705.693	1.02126	0.510632	+	0.859799i	$0.329411\pi$
$692$	14.7319i	0.0212888i
$693$	0	0
$694$	916.019	1.31991
$695$	− 1885.23i	− 2.71256i
$696$	0	0
$697$	−374.543	−0.537365
$698$	−813.439	−1.16538
$699$	0	0
$700$	− 306.209i	− 0.437442i
$701$	− 56.9789i	− 0.0812823i	−0.999174	−	0.0406412i	$-0.987060\pi$
$701$	− 56.9789i	− 0.0812823i	0.999174	−	0.0406412i	$-0.0129400\pi$
$702$	0	0
$703$	− 446.942i	− 0.635764i
$704$	0	0
$705$	0	0
$706$	920.573i	1.30393i
$707$	367.463	0.519749
$708$	0	0
$709$	481.533	0.679172	0.339586	−	0.940575i	$-0.389713\pi$
$709$	481.533	0.679172	0.339586	+	0.940575i	$0.389713\pi$
$710$	424.743i	0.598230i
$711$	0	0
$712$	64.9833i	0.0912687i
$713$	255.214	0.357943
$714$	0	0
$715$	0	0
$716$	118.225	0.165119
$717$	0	0
$718$	−342.802	−0.477440
$719$	−987.683	−1.37369	−0.686845	−	0.726804i	$-0.741005\pi$
$719$	−987.683	−1.37369	−0.686845	+	0.726804i	$0.741005\pi$
$720$	0	0
$721$	− 540.541i	− 0.749710i
$722$	641.800i	0.888919i
$723$	0	0
$724$	92.0936	0.127201
$725$	− 363.258i	− 0.501045i
$726$	0	0
$727$	−146.472	−0.201474	−0.100737	−	0.994913i	$-0.532120\pi$
$727$	−146.472	−0.201474	−0.100737	+	0.994913i	$0.532120\pi$
$728$	− 509.914i	− 0.700432i
$729$	0	0
$730$	−2411.99	−3.30409
$731$	358.769	0.490792
$732$	0	0
$733$	140.454i	0.191615i	0.995400	+	0.0958076i	$0.0305433\pi$
$733$	140.454i	0.191615i	−0.995400	+	0.0958076i	$0.969457\pi$
$734$	− 241.312i	− 0.328763i
$735$	0	0
$736$	132.616i	0.180185i
$737$	0	0
$738$	0	0
$739$	− 662.223i	− 0.896107i	−0.894007	−	0.448053i	$-0.852117\pi$
$739$	− 662.223i	− 0.896107i	0.894007	−	0.448053i	$-0.147883\pi$
$740$	−417.501	−0.564190
$741$	0	0
$742$	−811.862	−1.09415
$743$	− 191.960i	− 0.258358i	−0.991621	−	0.129179i	$-0.958766\pi$
$743$	− 191.960i	− 0.258358i	0.991621	−	0.129179i	$-0.0412341\pi$
$744$	0	0
$745$	− 1223.57i	− 1.64238i
$746$	1597.64	2.14161
$747$	0	0
$748$	0	0
$749$	521.733	0.696573
$750$	0	0
$751$	−1199.85	−1.59767	−0.798833	−	0.601552i	$-0.794549\pi$
$751$	−1199.85	−1.59767	−0.798833	+	0.601552i	$0.794549\pi$
$752$	−302.348	−0.402059
$753$	0	0
$754$	177.377i	0.235249i
$755$	− 100.069i	− 0.132542i
$756$	0	0
$757$	−572.112	−0.755762	−0.377881	−	0.925854i	$-0.623347\pi$
$757$	−572.112	−0.755762	−0.377881	+	0.925854i	$0.623347\pi$
$758$	− 28.6586i	− 0.0378082i
$759$	0	0
$760$	502.940	0.661764
$761$	− 71.9549i	− 0.0945531i	−0.998882	−	0.0472765i	$-0.984946\pi$
$761$	− 71.9549i	− 0.0945531i	0.998882	−	0.0472765i	$-0.0150542\pi$
$762$	0	0
$763$	−17.8700	−0.0234207
$764$	−146.724	−0.192047
$765$	0	0
$766$	220.658i	0.288066i
$767$	794.338i	1.03564i
$768$	0	0
$769$	− 87.6070i	− 0.113923i	−0.998376	−	0.0569616i	$-0.981859\pi$
$769$	− 87.6070i	− 0.113923i	0.998376	−	0.0569616i	$-0.0181413\pi$
$770$	0	0
$771$	0	0
$772$	31.0105i	0.0401691i
$773$	767.814	0.993291	0.496646	−	0.867953i	$-0.334565\pi$
$773$	767.814	0.993291	0.496646	+	0.867953i	$0.334565\pi$
$774$	0	0
$775$	−1387.76	−1.79066
$776$	1068.98i	1.37756i
$777$	0	0
$778$	− 447.735i	− 0.575495i
$779$	−403.988	−0.518598
$780$	0	0
$781$	0	0
$782$	161.348	0.206327
$783$	0	0
$784$	−84.3580	−0.107599
$785$	1301.90	1.65847
$786$	0	0
$787$	313.510i	0.398361i	0.979963	+	0.199181i	$0.0638280\pi$
$787$	313.510i	0.398361i	−0.979963	+	0.199181i	$0.936172\pi$
$788$	− 56.1221i	− 0.0712210i
$789$	0	0
$790$	−825.839	−1.04537
$791$	− 1229.00i	− 1.55373i
$792$	0	0
$793$	14.5157	0.0183048
$794$	737.050i	0.928274i
$795$	0	0
$796$	−216.130	−0.271521
$797$	671.426	0.842442	0.421221	−	0.906958i	$-0.361602\pi$
$797$	671.426	0.842442	0.421221	+	0.906958i	$0.361602\pi$
$798$	0	0
$799$	125.839i	0.157496i
$800$	− 721.121i	− 0.901401i
$801$	0	0
$802$	− 1455.10i	− 1.81434i
$803$	0	0
$804$	0	0
$805$	539.868i	0.670644i
$806$	677.640	0.840744
$807$	0	0
$808$	377.360	0.467029
$809$	1059.92i	1.31016i	0.755562	+	0.655078i	$0.227364\pi$
$809$	1059.92i	1.31016i	−0.755562	+	0.655078i	$0.772636\pi$
$810$	0	0
$811$	868.931i	1.07143i	0.844398	+	0.535716i	$0.179958\pi$
$811$	868.931i	1.07143i	−0.844398	+	0.535716i	$0.820042\pi$
$812$	−43.4384	−0.0534956
$813$	0	0
$814$	0	0
$815$	476.637	0.584831
$816$	0	0
$817$	386.974	0.473652
$818$	1039.04	1.27022
$819$	0	0
$820$	377.376i	0.460215i
$821$	− 749.652i	− 0.913096i	−0.889699	−	0.456548i	$-0.849086\pi$
$821$	− 749.652i	− 0.913096i	0.889699	−	0.456548i	$-0.150914\pi$
$822$	0	0
$823$	1184.79	1.43960	0.719800	−	0.694182i	$-0.244234\pi$
$823$	1184.79	1.43960	0.719800	+	0.694182i	$0.244234\pi$
$824$	− 555.100i	− 0.673665i
$825$	0	0
$826$	−1052.46	−1.27416
$827$	100.344i	0.121335i	0.998158	+	0.0606674i	$0.0193229\pi$
$827$	100.344i	0.121335i	−0.998158	+	0.0606674i	$0.980677\pi$
$828$	0	0
$829$	249.143	0.300535	0.150267	−	0.988645i	$-0.451987\pi$
$829$	249.143	0.300535	0.150267	+	0.988645i	$0.451987\pi$
$830$	−2565.82	−3.09135
$831$	0	0
$832$	− 486.946i	− 0.585271i
$833$	35.1103i	0.0421492i
$834$	0	0
$835$	373.379i	0.447161i
$836$	0	0
$837$	0	0
$838$	− 536.577i	− 0.640307i
$839$	121.681	0.145030	0.0725152	−	0.997367i	$-0.476897\pi$
$839$	121.681	0.145030	0.0725152	+	0.997367i	$0.476897\pi$
$840$	0	0
$841$	789.469	0.938726
$842$	− 1633.84i	− 1.94043i
$843$	0	0
$844$	− 170.911i	− 0.202501i
$845$	−387.567	−0.458659
$846$	0	0
$847$	0	0
$848$	−1033.01	−1.21818
$849$	0	0
$850$	−877.353	−1.03218
$851$	492.680	0.578942
$852$	0	0
$853$	144.122i	0.168959i	0.996425	+	0.0844795i	$0.0269228\pi$
$853$	144.122i	0.168959i	−0.996425	+	0.0844795i	$0.973077\pi$
$854$	19.2325i	0.0225205i
$855$	0	0
$856$	535.786	0.625918
$857$	527.163i	0.615126i	0.951528	+	0.307563i	$0.0995134\pi$
$857$	527.163i	0.615126i	−0.951528	+	0.307563i	$0.900487\pi$
$858$	0	0
$859$	122.027	0.142057	0.0710284	−	0.997474i	$-0.477372\pi$
$859$	122.027	0.142057	0.0710284	+	0.997474i	$0.477372\pi$
$860$	− 361.482i	− 0.420328i
$861$	0	0
$862$	1088.62	1.26290
$863$	−845.250	−0.979432	−0.489716	−	0.871882i	$-0.662900\pi$
$863$	−845.250	−0.979432	−0.489716	+	0.871882i	$0.662900\pi$
$864$	0	0
$865$	141.234i	0.163276i
$866$	− 891.967i	− 1.02998i
$867$	0	0
$868$	165.949i	0.191185i
$869$	0	0
$870$	0	0
$871$	− 322.823i	− 0.370635i
$872$	−18.3513	−0.0210451
$873$	0	0
$874$	174.032	0.199121
$875$	− 1485.31i	− 1.69750i
$876$	0	0
$877$	914.615i	1.04289i	0.853285	+	0.521445i	$0.174607\pi$
$877$	914.615i	1.04289i	−0.853285	+	0.521445i	$0.825393\pi$
$878$	151.524	0.172579
$879$	0	0
$880$	0	0
$881$	−618.978	−0.702586	−0.351293	−	0.936266i	$-0.614258\pi$
$881$	−618.978	−0.702586	−0.351293	+	0.936266i	$0.614258\pi$
$882$	0	0
$883$	−58.5805	−0.0663425	−0.0331713	−	0.999450i	$-0.510561\pi$
$883$	−58.5805	−0.0663425	−0.0331713	+	0.999450i	$0.510561\pi$
$884$	79.1835	0.0895740
$885$	0	0
$886$	− 694.324i	− 0.783661i
$887$	− 1060.58i	− 1.19569i	−0.801612	−	0.597845i	$-0.796024\pi$
$887$	− 1060.58i	− 1.19569i	0.801612	−	0.597845i	$-0.203976\pi$
$888$	0	0
$889$	−794.667	−0.893889
$890$	− 182.678i	− 0.205257i
$891$	0	0
$892$	234.235	0.262595
$893$	135.732i	0.151995i
$894$	0	0
$895$	1133.42	1.26639
$896$	1025.49	1.14452
$897$	0	0
$898$	− 729.036i	− 0.811845i
$899$	196.866i	0.218983i
$900$	0	0
$901$	429.947i	0.477189i
$902$	0	0
$903$	0	0
$904$	− 1262.10i	− 1.39613i
$905$	882.898	0.975578
$906$	0	0
$907$	−1248.22	−1.37621	−0.688105	−	0.725611i	$-0.741557\pi$
$907$	−1248.22	−1.37621	−0.688105	+	0.725611i	$0.741557\pi$
$908$	− 40.1865i	− 0.0442582i
$909$	0	0
$910$	1433.45i	1.57522i
$911$	619.331	0.679837	0.339918	−	0.940455i	$-0.389601\pi$
$911$	619.331	0.679837	0.339918	+	0.940455i	$0.389601\pi$
$912$	0	0
$913$	0	0
$914$	−196.150	−0.214606
$915$	0	0
$916$	3.48162	0.00380090
$917$	120.093	0.130963
$918$	0	0
$919$	1704.20i	1.85441i	0.374558	+	0.927204i	$0.377795\pi$
$919$	1704.20i	1.85441i	−0.374558	+	0.927204i	$0.622205\pi$
$920$	554.409i	0.602618i
$921$	0	0
$922$	1345.29	1.45910
$923$	− 245.984i	− 0.266504i
$924$	0	0
$925$	−2679.02	−2.89624
$926$	− 90.1851i	− 0.0973921i
$927$	0	0
$928$	−102.297	−0.110234
$929$	505.092	0.543694	0.271847	−	0.962340i	$-0.412365\pi$
$929$	505.092	0.543694	0.271847	+	0.962340i	$0.412365\pi$
$930$	0	0
$931$	37.8705i	0.0406772i
$932$	218.889i	0.234859i
$933$	0	0
$934$	991.952i	1.06205i
$935$	0	0
$936$	0	0
$937$	1305.95i	1.39376i	0.717189	+	0.696878i	$0.245428\pi$
$937$	1305.95i	1.39376i	−0.717189	+	0.696878i	$0.754572\pi$
$938$	427.723	0.455995
$939$	0	0
$940$	126.791	0.134884
$941$	31.8347i	0.0338307i	0.999857	+	0.0169154i	$0.00538459\pi$
$941$	31.8347i	0.0338307i	−0.999857	+	0.0169154i	$0.994615\pi$
$942$	0	0
$943$	− 445.330i	− 0.472248i
$944$	−1339.15	−1.41859
$945$	0	0
$946$	0	0
$947$	749.175	0.791103	0.395552	−	0.918444i	$-0.370553\pi$
$947$	749.175	0.791103	0.395552	+	0.918444i	$0.370553\pi$
$948$	0	0
$949$	1396.86	1.47193
$950$	−946.325	−0.996132
$951$	0	0
$952$	− 357.790i	− 0.375830i
$953$	− 1452.72i	− 1.52437i	−0.647361	−	0.762183i	$-0.724127\pi$
$953$	− 1452.72i	− 1.52437i	0.647361	−	0.762183i	$-0.275873\pi$
$954$	0	0
$955$	−1406.63	−1.47291
$956$	− 420.280i	− 0.439624i
$957$	0	0
$958$	129.571	0.135251
$959$	− 617.554i	− 0.643957i
$960$	0	0
$961$	−208.907	−0.217385
$962$	1308.16	1.35983
$963$	0	0
$964$	− 405.605i	− 0.420752i
$965$	297.297i	0.308079i
$966$	0	0
$967$	− 950.193i	− 0.982619i	−0.870985	−	0.491310i	$-0.836518\pi$
$967$	− 950.193i	− 0.982619i	0.870985	−	0.491310i	$-0.163482\pi$
$968$	0	0
$969$	0	0
$970$	− 3005.09i	− 3.09803i
$971$	659.055	0.678738	0.339369	−	0.940653i	$-0.389786\pi$
$971$	659.055	0.678738	0.339369	+	0.940653i	$0.389786\pi$
$972$	0	0
$973$	−1446.58	−1.48672
$974$	− 1447.13i	− 1.48576i
$975$	0	0
$976$	24.4715i	0.0250733i
$977$	−206.627	−0.211491	−0.105746	−	0.994393i	$-0.533723\pi$
$977$	−206.627	−0.211491	−0.105746	+	0.994393i	$0.533723\pi$
$978$	0	0
$979$	0	0
$980$	35.3758	0.0360978
$981$	0	0
$982$	652.957	0.664926
$983$	−228.441	−0.232391	−0.116196	−	0.993226i	$-0.537070\pi$
$983$	−228.441	−0.232391	−0.116196	+	0.993226i	$0.537070\pi$
$984$	0	0
$985$	− 538.041i	− 0.546234i
$986$	124.460i	0.126227i
$987$	0	0
$988$	85.4084	0.0864457
$989$	426.575i	0.431319i
$990$	0	0
$991$	−1872.78	−1.88979	−0.944895	−	0.327373i	$-0.893837\pi$
$991$	−1872.78	−1.88979	−0.944895	+	0.327373i	$0.893837\pi$
$992$	390.809i	0.393960i
$993$	0	0
$994$	325.915	0.327883
$995$	−2072.03	−2.08245
$996$	0	0
$997$	− 1536.35i	− 1.54097i	−0.637459	−	0.770484i	$-0.720015\pi$
$997$	− 1536.35i	− 1.54097i	0.637459	−	0.770484i	$-0.279985\pi$
$998$	91.5337i	0.0917171i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.m.604.12		16
3.2	odd	2		363.3.c.e.241.5		16
11.2	odd	10		99.3.k.c.73.2		16
11.5	even	5		99.3.k.c.19.2		16
11.10	odd	2	inner	1089.3.c.m.604.5		16
33.2	even	10		33.3.g.a.7.3	✓	16
33.5	odd	10		33.3.g.a.19.3	yes	16
33.8	even	10		363.3.g.g.112.3		16
33.14	odd	10		363.3.g.a.112.2		16
33.17	even	10		363.3.g.f.118.2		16
33.20	odd	10		363.3.g.f.40.2		16
33.26	odd	10		363.3.g.g.94.3		16
33.29	even	10		363.3.g.a.94.2		16
33.32	even	2		363.3.c.e.241.12		16
132.35	odd	10		528.3.bf.b.337.1		16
132.71	even	10		528.3.bf.b.481.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.7.3	✓	16	33.2	even	10
33.3.g.a.19.3	yes	16	33.5	odd	10
99.3.k.c.19.2		16	11.5	even	5
99.3.k.c.73.2		16	11.2	odd	10
363.3.c.e.241.5		16	3.2	odd	2
363.3.c.e.241.12		16	33.32	even	2
363.3.g.a.94.2		16	33.29	even	10
363.3.g.a.112.2		16	33.14	odd	10
363.3.g.f.40.2		16	33.20	odd	10
363.3.g.f.118.2		16	33.17	even	10
363.3.g.g.94.3		16	33.26	odd	10
363.3.g.g.112.3		16	33.8	even	10
528.3.bf.b.337.1		16	132.35	odd	10
528.3.bf.b.481.1		16	132.71	even	10
1089.3.c.m.604.5		16	11.10	odd	2	inner
1089.3.c.m.604.12		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.21517i q^{2} -0.906963 q^{4} -8.69502 q^{5} +6.67189i q^{7} +6.85159i q^{8} +O(q^{10})\)	\(q+2.21517i q^{2} -0.906963 q^{4} -8.69502 q^{5} +6.67189i q^{7} +6.85159i q^{8} -19.2609i q^{10} +11.1547i q^{13} -14.7794 q^{14} -18.8053 q^{16} +7.82687i q^{17} +8.44217i q^{19} +7.88606 q^{20} -9.30611 q^{23} +50.6034 q^{25} -24.7094 q^{26} -6.05116i q^{28} -7.17852i q^{29} -27.4243 q^{31} -14.2504i q^{32} -17.3378 q^{34} -58.0122i q^{35} -52.9416 q^{37} -18.7008 q^{38} -59.5747i q^{40} +47.8535i q^{41} -45.8381i q^{43} -20.6146i q^{46} +16.0778 q^{47} +4.48587 q^{49} +112.095i q^{50} -10.1169i q^{52} +54.9322 q^{53} -45.7131 q^{56} +15.9016 q^{58} +71.2113 q^{59} -1.30131i q^{61} -60.7494i q^{62} -43.6540 q^{64} -96.9901i q^{65} -28.9406 q^{67} -7.09868i q^{68} +128.507 q^{70} -22.0521 q^{71} -125.227i q^{73} -117.274i q^{74} -7.65674i q^{76} -42.8764i q^{79} +163.512 q^{80} -106.004 q^{82} -133.214i q^{83} -68.0548i q^{85} +101.539 q^{86} +9.48441 q^{89} -74.4227 q^{91} +8.44030 q^{92} +35.6151i q^{94} -73.4049i q^{95} +156.020 q^{97} +9.93695i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) \) 16 * q - 20 * q^4 + 4 * q^5	\( 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) \) 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Introduction

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