LMFDB - Embedded newform 1053.2.b.i.649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1053,2,Mod(649,1053)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1053, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1053.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1053 = 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1053.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.40824733284$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 16x^{8} + 91x^{6} + 222x^{4} + 228x^{2} + 81 $ x^10 + 16x^8 + 91x^6 + 222x^4 + 228x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$-1.07384i$ of defining polynomial
Character	$\chi$	$=$	1053.649
Dual form			1053.2.b.i.649.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.07384i q^{2} +0.846877 q^{4} +1.27644i q^{5} +1.02875i q^{7} -3.05708i q^{8} +1.37069 q^{10} -4.66263i q^{11} +(1.25783 + 3.37903i) q^{13} +1.10471 q^{14} -1.58904 q^{16} +0.476187 q^{17} -6.69096i q^{19} +1.08099i q^{20} -5.00690 q^{22} -0.959735 q^{23} +3.37069 q^{25} +(3.62852 - 1.35071i) q^{26} +0.871227i q^{28} +9.37759 q^{29} +1.92751i q^{31} -4.40778i q^{32} -0.511346i q^{34} -1.31314 q^{35} +4.94666i q^{37} -7.18499 q^{38} +3.90219 q^{40} -1.52414i q^{41} +2.62852 q^{43} -3.94868i q^{44} +1.03060i q^{46} -6.83948i q^{47} +5.94167 q^{49} -3.61957i q^{50} +(1.06523 + 2.86163i) q^{52} -0.582145 q^{53} +5.95159 q^{55} +3.14498 q^{56} -10.0700i q^{58} -4.21233i q^{59} +9.43290 q^{61} +2.06983 q^{62} -7.91132 q^{64} +(-4.31314 + 1.60555i) q^{65} +2.32275i q^{67} +0.403272 q^{68} +1.41010i q^{70} -1.35071i q^{71} +12.8687i q^{73} +5.31190 q^{74} -5.66642i q^{76} +4.79669 q^{77} -12.9083 q^{79} -2.02833i q^{80} -1.63667 q^{82} +10.2328i q^{83} +0.607825i q^{85} -2.82260i q^{86} -14.2540 q^{88} +6.85985i q^{89} +(-3.47619 + 1.29400i) q^{91} -0.812778 q^{92} -7.34448 q^{94} +8.54063 q^{95} -17.2784i q^{97} -6.38037i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} - 18 q^{14} - 4 q^{16} + 6 q^{17} + 10 q^{22} + 24 q^{23} + 12 q^{25} + 6 q^{26} + 12 q^{29} + 6 q^{35} + 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} - 54 q^{53} + 10 q^{55}+ \cdots + 24 q^{95}+O(q^{100}) $ 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 - 18 * q^14 - 4 * q^16 + 6 * q^17 + 10 * q^22 + 24 * q^23 + 12 * q^25 + 6 * q^26 + 12 * q^29 + 6 * q^35 + 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 - 54 * q^53 + 10 * q^55 + 36 * q^56 + 2 * q^61 + 36 * q^62 + 4 * q^64 - 24 * q^65 + 24 * q^68 - 42 * q^74 - 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 - 84 * q^92 - 20 * q^94 + 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times$.

$n$	$326$	$730$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.07384i		−	0.759316i	−0.925127	−	0.379658i	$-0.876042\pi$
$2$		−	1.07384i		−	0.759316i	0.925127	−	0.379658i	$-0.123958\pi$
$3$		0			0
$3$		0			0
$4$	0.846877			0.423439
$5$			1.27644i			0.570843i	0.958402	+	0.285422i	$0.0921336\pi$
$5$			1.27644i			0.570843i	−0.958402	+	0.285422i	$0.907866\pi$
$6$		0			0
$7$			1.02875i			0.388832i	0.980919	+	0.194416i	$0.0622811\pi$
$7$			1.02875i			0.388832i	−0.980919	+	0.194416i	$0.937719\pi$
$8$		−	3.05708i		−	1.08084i
$9$		0			0
$10$	1.37069			0.433450
$11$		−	4.66263i		−	1.40584i	−0.711271	−	0.702918i	$-0.751880\pi$
$11$		−	4.66263i		−	1.40584i	0.711271	−	0.702918i	$-0.248120\pi$
$12$		0			0
$13$	1.25783	+	3.37903i	0.348860	+	0.937175i
$13$	1.25783	+	3.37903i	0.348860	+	0.937175i
$14$	1.10471			0.295246
$15$		0			0
$16$	−1.58904			−0.397261
$17$	0.476187			0.115492			0.0577461	−	0.998331i	$-0.481609\pi$
$17$	0.476187			0.115492			0.0577461	+	0.998331i	$0.481609\pi$
$18$		0			0
$19$		−	6.69096i		−	1.53501i	−0.641042	−	0.767505i	$-0.721498\pi$
$19$		−	6.69096i		−	1.53501i	0.641042	−	0.767505i	$-0.278502\pi$
$20$			1.08099i			0.241717i
$21$		0			0
$22$	−5.00690			−1.06747
$23$	−0.959735			−0.200119			−0.100059	−	0.994981i	$-0.531903\pi$
$23$	−0.959735			−0.200119			−0.100059	+	0.994981i	$0.531903\pi$
$24$		0			0
$25$	3.37069			0.674138
$26$	3.62852	−	1.35071i	0.711612	−	0.264895i
$27$		0			0
$28$			0.871227i			0.164646i
$29$	9.37759			1.74137			0.870687	−	0.491837i	$-0.163674\pi$
$29$	9.37759			1.74137			0.870687	+	0.491837i	$0.163674\pi$
$30$		0			0
$31$			1.92751i			0.346191i	0.984905	+	0.173095i	$0.0553769\pi$
$31$			1.92751i			0.346191i	−0.984905	+	0.173095i	$0.944623\pi$
$32$		−	4.40778i		−	0.779193i
$33$		0			0
$34$		−	0.511346i		−	0.0876951i
$35$	−1.31314			−0.221962
$36$		0			0
$37$			4.94666i			0.813226i	0.913601	+	0.406613i	$0.133290\pi$
$37$			4.94666i			0.813226i	−0.913601	+	0.406613i	$0.866710\pi$
$38$	−7.18499			−1.16556
$39$		0			0
$40$	3.90219			0.616990
$41$		−	1.52414i		−	0.238030i	−0.992892	−	0.119015i	$-0.962026\pi$
$41$		−	1.52414i		−	0.238030i	0.992892	−	0.119015i	$-0.0379737\pi$
$42$		0			0
$43$	2.62852			0.400846			0.200423	−	0.979709i	$-0.435768\pi$
$43$	2.62852			0.400846			0.200423	+	0.979709i	$0.435768\pi$
$44$		−	3.94868i		−	0.595285i
$45$		0			0
$46$			1.03060i			0.151953i
$47$		−	6.83948i		−	0.997641i	−0.866705	−	0.498820i	$-0.833767\pi$
$47$		−	6.83948i		−	0.997641i	0.866705	−	0.498820i	$-0.166233\pi$
$48$		0			0
$49$	5.94167			0.848810
$50$		−	3.61957i		−	0.511884i
$51$		0			0
$52$	1.06523	+	2.86163i	0.147721	+	0.396836i
$53$	−0.582145			−0.0799637			−0.0399819	−	0.999200i	$-0.512730\pi$
$53$	−0.582145			−0.0799637			−0.0399819	+	0.999200i	$0.512730\pi$
$54$		0			0
$55$	5.95159			0.802512
$56$	3.14498			0.420265
$57$		0			0
$58$		−	10.0700i		−	1.32225i
$59$		−	4.21233i		−	0.548399i	−0.961673	−	0.274199i	$-0.911587\pi$
$59$		−	4.21233i		−	0.548399i	0.961673	−	0.274199i	$-0.0884128\pi$
$60$		0			0
$61$	9.43290			1.20776			0.603880	−	0.797076i	$-0.293621\pi$
$61$	9.43290			1.20776			0.603880	+	0.797076i	$0.293621\pi$
$62$	2.06983			0.262868
$63$		0			0
$64$	−7.91132			−0.988915
$65$	−4.31314	+	1.60555i	−0.534980	+	0.199144i
$66$		0			0
$67$			2.32275i			0.283769i	0.989883	+	0.141885i	$0.0453162\pi$
$67$			2.32275i			0.283769i	−0.989883	+	0.141885i	$0.954684\pi$
$68$	0.403272			0.0489039
$69$		0			0
$70$			1.41010i			0.168539i
$71$		−	1.35071i		−	0.160299i	−0.996783	−	0.0801497i	$-0.974460\pi$
$71$		−	1.35071i		−	0.160299i	0.996783	−	0.0801497i	$-0.0255398\pi$
$72$		0			0
$73$			12.8687i			1.50617i	0.657923	+	0.753085i	$0.271435\pi$
$73$			12.8687i			1.50617i	−0.657923	+	0.753085i	$0.728565\pi$
$74$	5.31190			0.617496
$75$		0			0
$76$		−	5.66642i		−	0.649983i
$77$	4.79669			0.546634
$78$		0			0
$79$	−12.9083			−1.45230			−0.726149	−	0.687538i	$-0.758692\pi$
$79$	−12.9083			−1.45230			−0.726149	+	0.687538i	$0.758692\pi$
$80$		−	2.02833i		−	0.226774i
$81$		0			0
$82$	−1.63667			−0.180740
$83$			10.2328i			1.12320i	0.827409	+	0.561599i	$0.189814\pi$
$83$			10.2328i			1.12320i	−0.827409	+	0.561599i	$0.810186\pi$
$84$		0			0
$85$			0.607825i			0.0659279i
$86$		−	2.82260i		−	0.304369i
$87$		0			0
$88$	−14.2540			−1.51948
$89$			6.85985i			0.727143i	0.931566	+	0.363572i	$0.118443\pi$
$89$			6.85985i			0.727143i	−0.931566	+	0.363572i	$0.881557\pi$
$90$		0			0
$91$	−3.47619	+	1.29400i	−0.364403	+	0.135648i
$92$	−0.812778			−0.0847379
$93$		0			0
$94$	−7.34448			−0.757525
$95$	8.54063			0.876250
$96$		0			0
$97$		−	17.2784i		−	1.75435i	−0.480170	−	0.877175i	$-0.659425\pi$
$97$		−	17.2784i		−	1.75435i	0.480170	−	0.877175i	$-0.340575\pi$
$98$		−	6.38037i		−	0.644515i
$99$		0			0
$100$	2.85456			0.285456
$101$	−13.5287			−1.34615			−0.673077	−	0.739572i	$-0.735028\pi$
$101$	−13.5287			−1.34615			−0.673077	+	0.739572i	$0.735028\pi$
$102$		0			0
$103$	0.104711			0.0103175			0.00515873	−	0.999987i	$-0.498358\pi$
$103$	0.104711			0.0103175			0.00515873	+	0.999987i	$0.498358\pi$
$104$	10.3300	−	3.84529i	1.01294	−	0.377062i
$105$		0			0
$106$			0.625128i			0.0607178i
$107$	−12.4240			−1.20107			−0.600535	−	0.799598i	$-0.705046\pi$
$107$	−12.4240			−1.20107			−0.600535	+	0.799598i	$0.705046\pi$
$108$		0			0
$109$		−	14.1160i		−	1.35207i	−0.736869	−	0.676036i	$-0.763696\pi$
$109$		−	14.1160i		−	1.35207i	0.736869	−	0.676036i	$-0.236304\pi$
$110$		−	6.39103i		−	0.609360i
$111$		0			0
$112$		−	1.63473i		−	0.154468i
$113$	−11.6856			−1.09929			−0.549645	−	0.835398i	$-0.685237\pi$
$113$	−11.6856			−1.09929			−0.549645	+	0.835398i	$0.685237\pi$
$114$		0			0
$115$		−	1.22505i		−	0.114236i
$116$	7.94167			0.737365
$117$		0			0
$118$	−4.52335			−0.416408
$119$			0.489878i			0.0449070i
$120$		0			0
$121$	−10.7401			−0.976376
$122$		−	10.1294i		−	0.917071i
$123$		0			0
$124$			1.63236i			0.146590i
$125$			10.6847i			0.955670i
$126$		0			0
$127$	−8.11161			−0.719789			−0.359894	−	0.932993i	$-0.617187\pi$
$127$	−8.11161			−0.719789			−0.359894	+	0.932993i	$0.617187\pi$
$128$		−	0.320109i		−	0.0282939i
$129$		0			0
$130$	1.72410	+	4.63161i	0.151214	+	0.406219i
$131$	−9.36077			−0.817854			−0.408927	−	0.912567i	$-0.634097\pi$
$131$	−9.36077			−0.817854			−0.408927	+	0.912567i	$0.634097\pi$
$132$		0			0
$133$	6.88334			0.596861
$134$	2.49425			0.215471
$135$		0			0
$136$		−	1.45574i		−	0.124829i
$137$			0.591668i			0.0505496i	0.999681	+	0.0252748i	$0.00804608\pi$
$137$			0.591668i			0.0505496i	−0.999681	+	0.0252748i	$0.991954\pi$
$138$		0			0
$139$	−12.6727			−1.07488			−0.537441	−	0.843301i	$-0.680609\pi$
$139$	−12.6727			−1.07488			−0.537441	+	0.843301i	$0.680609\pi$
$140$	−1.11207			−0.0939873
$141$		0			0
$142$	−1.45044			−0.121718
$143$	15.7552	−	5.86481i	1.31751	−	0.490440i
$144$		0			0
$145$			11.9700i			0.994052i
$146$	13.8189			1.14366
$147$		0			0
$148$			4.18921i			0.344351i
$149$			11.6904i			0.957716i	0.877892	+	0.478858i	$0.158949\pi$
$149$			11.6904i			0.957716i	−0.877892	+	0.478858i	$0.841051\pi$
$150$		0			0
$151$			10.2052i			0.830491i	0.909709	+	0.415245i	$0.136304\pi$
$151$			10.2052i			0.830491i	−0.909709	+	0.415245i	$0.863696\pi$
$152$	−20.4548			−1.65910
$153$		0			0
$154$		−	5.15086i		−	0.415068i
$155$	−2.46036			−0.197620
$156$		0			0
$157$	10.9404			0.873141			0.436570	−	0.899670i	$-0.356193\pi$
$157$	10.9404			0.873141			0.436570	+	0.899670i	$0.356193\pi$
$158$			13.8614i			1.10275i
$159$		0			0
$160$	5.62629			0.444797
$161$		−	0.987329i		−	0.0778125i
$162$		0			0
$163$		−	16.9633i		−	1.32867i	−0.747436	−	0.664334i	$-0.768715\pi$
$163$		−	16.9633i		−	1.32867i	0.747436	−	0.664334i	$-0.231285\pi$
$164$		−	1.29076i		−	0.100791i
$165$		0			0
$166$	10.9884			0.852863
$167$			16.8077i			1.30062i	0.759670	+	0.650309i	$0.225360\pi$
$167$			16.8077i			1.30062i	−0.759670	+	0.650309i	$0.774640\pi$
$168$		0			0
$169$	−9.83571	+	8.50052i	−0.756593	+	0.653886i
$170$	0.652705			0.0500602
$171$		0			0
$172$	2.22604			0.169734
$173$	19.0864			1.45111			0.725556	−	0.688163i	$-0.241583\pi$
$173$	19.0864			1.45111			0.725556	+	0.688163i	$0.241583\pi$
$174$		0			0
$175$			3.46761i			0.262126i
$176$			7.40913i			0.558484i
$177$		0			0
$178$	7.36635			0.552132
$179$	8.33634			0.623087			0.311544	−	0.950232i	$-0.399154\pi$
$179$	8.33634			0.623087			0.311544	+	0.950232i	$0.399154\pi$
$180$		0			0
$181$	9.93629			0.738558			0.369279	−	0.929318i	$-0.379605\pi$
$181$	9.93629			0.738558			0.369279	+	0.929318i	$0.379605\pi$
$182$	1.38954	+	3.73285i	0.103000	+	0.276697i
$183$		0			0
$184$			2.93398i			0.216296i
$185$	−6.31413			−0.464224
$186$		0			0
$187$		−	2.22028i		−	0.162363i
$188$		−	5.79220i		−	0.422440i
$189$		0			0
$190$		−	9.17123i		−	0.665351i
$191$	6.07594			0.439640			0.219820	−	0.975540i	$-0.429453\pi$
$191$	6.07594			0.439640			0.219820	+	0.975540i	$0.429453\pi$
$192$		0			0
$193$		−	16.5130i		−	1.18863i	−0.804232	−	0.594316i	$-0.797423\pi$
$193$		−	16.5130i		−	1.18863i	0.804232	−	0.594316i	$-0.202577\pi$
$194$	−18.5541			−1.33211
$195$		0			0
$196$	5.03186			0.359419
$197$			19.1696i			1.36578i	0.730523	+	0.682888i	$0.239276\pi$
$197$			19.1696i			1.36578i	−0.730523	+	0.682888i	$0.760724\pi$
$198$		0			0
$199$	9.06267			0.642436			0.321218	−	0.947005i	$-0.395908\pi$
$199$	9.06267			0.642436			0.321218	+	0.947005i	$0.395908\pi$
$200$		−	10.3045i		−	0.728636i
$201$		0			0
$202$			14.5276i			1.02216i
$203$			9.64722i			0.677102i
$204$		0			0
$205$	1.94547			0.135878
$206$		−	0.112442i		−	0.00783421i
$207$		0			0
$208$	−1.99875	−	5.36943i	−0.138589	−	0.372303i
$209$	−31.1975			−2.15797
$210$		0			0
$211$	−14.6976			−1.01183			−0.505913	−	0.862584i	$-0.668844\pi$
$211$	−14.6976			−1.01183			−0.505913	+	0.862584i	$0.668844\pi$
$212$	−0.493005			−0.0338597
$213$		0			0
$214$			13.3413i			0.911992i
$215$			3.35516i			0.228820i
$216$		0			0
$217$	−1.98293			−0.134610
$218$	−15.1583			−1.02665
$219$		0			0
$220$	5.04027			0.339815
$221$	0.598963	+	1.60905i	0.0402906	+	0.108236i
$222$		0			0
$223$			20.2052i			1.35304i	0.736423	+	0.676521i	$0.236513\pi$
$223$			20.2052i			1.35304i	−0.736423	+	0.676521i	$0.763487\pi$
$224$	4.53452			0.302975
$225$		0			0
$226$			12.5484i			0.834709i
$227$			20.5540i			1.36422i	0.731251	+	0.682109i	$0.238937\pi$
$227$			20.5540i			1.36422i	−0.731251	+	0.682109i	$0.761063\pi$
$228$		0			0
$229$			19.8920i			1.31450i	0.753672	+	0.657251i	$0.228281\pi$
$229$			19.8920i			1.31450i	−0.753672	+	0.657251i	$0.771719\pi$
$230$	−1.31550			−0.0867415
$231$		0			0
$232$		−	28.6680i		−	1.88215i
$233$	−28.3932			−1.86010			−0.930049	−	0.367436i	$-0.880236\pi$
$233$	−28.3932			−1.86010			−0.930049	+	0.367436i	$0.880236\pi$
$234$		0			0
$235$	8.73021			0.569496
$236$		−	3.56733i		−	0.232213i
$237$		0			0
$238$	0.526048			0.0340987
$239$			14.7148i			0.951822i	0.879493	+	0.475911i	$0.157882\pi$
$239$			14.7148i			0.951822i	−0.879493	+	0.475911i	$0.842118\pi$
$240$		0			0
$241$		−	13.2364i		−	0.852634i	−0.904574	−	0.426317i	$-0.859811\pi$
$241$		−	13.2364i		−	0.852634i	0.904574	−	0.426317i	$-0.140189\pi$
$242$			11.5331i			0.741378i
$243$		0			0
$244$	7.98851			0.511412
$245$			7.58421i			0.484537i
$246$		0			0
$247$	22.6090	−	8.41611i	1.43857	−	0.535504i
$248$	5.89254			0.374177
$249$		0			0
$250$	11.4736			0.725656
$251$	11.9439			0.753893			0.376946	−	0.926235i	$-0.376974\pi$
$251$	11.9439			0.753893			0.376946	+	0.926235i	$0.376974\pi$
$252$		0			0
$253$			4.47489i			0.281334i
$254$			8.71053i			0.546548i
$255$		0			0
$256$	−16.1664			−1.01040
$257$	14.9649			0.933482			0.466741	−	0.884394i	$-0.345428\pi$
$257$	14.9649			0.933482			0.466741	+	0.884394i	$0.345428\pi$
$258$		0			0
$259$	−5.08889			−0.316208
$260$	−3.65270	+	1.35971i	−0.226531	+	0.0843255i
$261$		0			0
$262$			10.0519i			0.621010i
$263$	−1.54924			−0.0955303			−0.0477652	−	0.998859i	$-0.515210\pi$
$263$	−1.54924			−0.0955303			−0.0477652	+	0.998859i	$0.515210\pi$
$264$		0			0
$265$		−	0.743075i		−	0.0456467i
$266$		−	7.39157i		−	0.453206i
$267$		0			0
$268$			1.96709i			0.120159i
$269$	−21.0293			−1.28218			−0.641090	−	0.767466i	$-0.721517\pi$
$269$	−21.0293			−1.28218			−0.641090	+	0.767466i	$0.721517\pi$
$270$		0			0
$271$		−	12.7508i		−	0.774554i	−0.921963	−	0.387277i	$-0.873416\pi$
$271$		−	12.7508i		−	0.774554i	0.921963	−	0.387277i	$-0.126584\pi$
$272$	−0.756681			−0.0458805
$273$		0			0
$274$	0.635354			0.0383832
$275$		−	15.7163i		−	0.947728i
$276$		0			0
$277$	−11.6273			−0.698615			−0.349308	−	0.937008i	$-0.613583\pi$
$277$	−11.6273			−0.698615			−0.349308	+	0.937008i	$0.613583\pi$
$278$			13.6084i			0.816175i
$279$		0			0
$280$			4.01439i			0.239905i
$281$			17.2837i			1.03106i	0.856872	+	0.515528i	$0.172404\pi$
$281$			17.2837i			1.03106i	−0.856872	+	0.515528i	$0.827596\pi$
$282$		0			0
$283$	21.6094			1.28455			0.642273	−	0.766476i	$-0.277991\pi$
$283$	21.6094			1.28455			0.642273	+	0.766476i	$0.277991\pi$
$284$		−	1.14388i		−	0.0678769i
$285$		0			0
$286$	−6.29784	−	16.9185i	−0.372399	−	1.00041i
$287$	1.56796			0.0925536
$288$		0			0
$289$	−16.7732			−0.986662
$290$	12.8538			0.754800
$291$		0			0
$292$			10.8982i			0.637771i
$293$			0.0714448i			0.00417385i	0.999998	+	0.00208692i	$0.000664289\pi$
$293$			0.0714448i			0.00417385i	−0.999998	+	0.00208692i	$0.999336\pi$
$294$		0			0
$295$	5.37681			0.313050
$296$	15.1223			0.878967
$297$		0			0
$298$	12.5536			0.727209
$299$	−1.20719	−	3.24297i	−0.0698134	−	0.187546i
$300$		0			0
$301$			2.70410i			0.155862i
$302$	10.9587			0.630605
$303$		0			0
$304$			10.6322i			0.609800i
$305$			12.0406i			0.689441i
$306$		0			0
$307$			19.9335i			1.13766i	0.822454	+	0.568831i	$0.192604\pi$
$307$			19.9335i			1.13766i	−0.822454	+	0.568831i	$0.807396\pi$
$308$	4.06221			0.231466
$309$		0			0
$310$			2.64202i			0.150056i
$311$	−6.08028			−0.344781			−0.172390	−	0.985029i	$-0.555149\pi$
$311$	−6.08028			−0.344781			−0.172390	+	0.985029i	$0.555149\pi$
$312$		0			0
$313$	6.09243			0.344365			0.172182	−	0.985065i	$-0.444918\pi$
$313$	6.09243			0.344365			0.172182	+	0.985065i	$0.444918\pi$
$314$		−	11.7482i		−	0.662990i
$315$		0			0
$316$	−10.9317			−0.614959
$317$			9.52665i			0.535070i	0.963548	+	0.267535i	$0.0862091\pi$
$317$			9.52665i			0.535070i	−0.963548	+	0.267535i	$0.913791\pi$
$318$		0			0
$319$		−	43.7242i		−	2.44809i
$320$		−	10.0984i		−	0.564516i
$321$		0			0
$322$	−1.06023			−0.0590843
$323$		−	3.18614i		−	0.177282i
$324$		0			0
$325$	4.23977	+	11.3897i	0.235180	+	0.631785i
$326$	−18.2158			−1.00888
$327$		0			0
$328$	−4.65940			−0.257272
$329$	7.03613			0.387915
$330$		0			0
$331$			18.1454i			0.997359i	0.866786	+	0.498680i	$0.166182\pi$
$331$			18.1454i			0.997359i	−0.866786	+	0.498680i	$0.833818\pi$
$332$			8.66595i			0.475606i
$333$		0			0
$334$	18.0487			0.987580
$335$	−2.96486			−0.161988
$336$		0			0
$337$	−8.01859			−0.436800			−0.218400	−	0.975859i	$-0.570084\pi$
$337$	−8.01859			−0.436800			−0.218400	+	0.975859i	$0.570084\pi$
$338$	9.12816	+	10.5619i	0.496506	+	0.574494i
$339$		0			0
$340$			0.514754i			0.0279164i
$341$	8.98726			0.486687
$342$		0			0
$343$			13.3138i			0.718876i
$344$		−	8.03560i		−	0.433251i
$345$		0			0
$346$		−	20.4956i		−	1.10185i
$347$	−16.6273			−0.892599			−0.446299	−	0.894884i	$-0.647258\pi$
$347$	−16.6273			−0.892599			−0.446299	+	0.894884i	$0.647258\pi$
$348$		0			0
$349$			0.178993i			0.00958129i	0.999989	+	0.00479065i	$0.00152492\pi$
$349$			0.178993i			0.00958129i	−0.999989	+	0.00479065i	$0.998475\pi$
$350$	3.72364			0.199037
$351$		0			0
$352$	−20.5519			−1.09542
$353$			16.9210i			0.900616i	0.892873	+	0.450308i	$0.148686\pi$
$353$			16.9210i			0.900616i	−0.892873	+	0.450308i	$0.851314\pi$
$354$		0			0
$355$	1.72410			0.0915058
$356$			5.80945i			0.307900i
$357$		0			0
$358$		−	8.95186i		−	0.473120i
$359$		−	34.4410i		−	1.81773i	−0.417093	−	0.908864i	$-0.636951\pi$
$359$		−	34.4410i		−	1.81773i	0.417093	−	0.908864i	$-0.363049\pi$
$360$		0			0
$361$	−25.7689			−1.35626
$362$		−	10.6699i		−	0.560800i
$363$		0			0
$364$	−2.94390	+	1.09586i	−0.154302	+	0.0574386i
$365$	−16.4262			−0.859787
$366$		0			0
$367$	25.0851			1.30943			0.654717	−	0.755874i	$-0.272788\pi$
$367$	25.0851			1.30943			0.654717	+	0.755874i	$0.272788\pi$
$368$	1.52506			0.0794993
$369$		0			0
$370$			6.78034i			0.352493i
$371$		−	0.598883i		−	0.0310924i
$372$		0			0
$373$	10.9506			0.567001			0.283500	−	0.958972i	$-0.408504\pi$
$373$	10.9506			0.567001			0.283500	+	0.958972i	$0.408504\pi$
$374$	−2.38422			−0.123285
$375$		0			0
$376$	−20.9088			−1.07829
$377$	11.7954	+	31.6872i	0.607496	+	1.63197i
$378$		0			0
$379$		−	11.1048i		−	0.570415i	−0.958466	−	0.285208i	$-0.907937\pi$
$379$		−	11.1048i		−	0.570415i	0.958466	−	0.285208i	$-0.0920626\pi$
$380$	7.23287			0.371038
$381$		0			0
$382$		−	6.52456i		−	0.333826i
$383$		−	5.87871i		−	0.300388i	−0.988657	−	0.150194i	$-0.952010\pi$
$383$		−	5.87871i		−	0.300388i	0.988657	−	0.150194i	$-0.0479898\pi$
$384$		0			0
$385$			6.12271i			0.312042i
$386$	−17.7322			−0.902548
$387$		0			0
$388$		−	14.6326i		−	0.742860i
$389$	0.0802767			0.00407019			0.00203510	−	0.999998i	$-0.499352\pi$
$389$	0.0802767			0.00407019			0.00203510	+	0.999998i	$0.499352\pi$
$390$		0			0
$391$	−0.457013			−0.0231121
$392$		−	18.1641i		−	0.917428i
$393$		0			0
$394$	20.5850			1.03706
$395$		−	16.4767i		−	0.829034i
$396$		0			0
$397$		−	1.33964i		−	0.0672345i	−0.999435	−	0.0336172i	$-0.989297\pi$
$397$		−	1.33964i		−	0.0672345i	0.999435	−	0.0336172i	$-0.0107027\pi$
$398$		−	9.73182i		−	0.487812i
$399$		0			0
$400$	−5.35618			−0.267809
$401$		−	31.6772i		−	1.58188i	−0.611891	−	0.790942i	$-0.709591\pi$
$401$		−	31.6772i		−	1.58188i	0.611891	−	0.790942i	$-0.290409\pi$
$402$		0			0
$403$	−6.51311	+	2.42448i	−0.324441	+	0.120772i
$404$	−11.4571			−0.570014
$405$		0			0
$406$	10.3595			0.514135
$407$	23.0644			1.14326
$408$		0			0
$409$		−	3.37903i		−	0.167082i	−0.996504	−	0.0835412i	$-0.973377\pi$
$409$		−	3.37903i		−	0.167082i	0.996504	−	0.0835412i	$-0.0266230\pi$
$410$		−	2.08912i		−	0.103174i
$411$		0			0
$412$	0.0886771			0.00436881
$413$	4.33345			0.213235
$414$		0			0
$415$	−13.0616			−0.641170
$416$	14.8940	−	5.54426i	0.730240	−	0.271830i
$417$		0			0
$418$			33.5009i			1.63859i
$419$	21.6660			1.05846			0.529228	−	0.848480i	$-0.322482\pi$
$419$	21.6660			1.05846			0.529228	+	0.848480i	$0.322482\pi$
$420$		0			0
$421$		−	14.0513i		−	0.684819i	−0.939551	−	0.342409i	$-0.888757\pi$
$421$		−	14.0513i		−	0.684819i	0.939551	−	0.342409i	$-0.111243\pi$
$422$			15.7828i			0.768297i
$423$		0			0
$424$			1.77966i			0.0864280i
$425$	1.60508			0.0778577
$426$		0			0
$427$			9.70412i			0.469615i
$428$	−10.5216			−0.508580
$429$		0			0
$430$	3.60289			0.173747
$431$		−	26.5547i		−	1.27909i	−0.768752	−	0.639547i	$-0.779122\pi$
$431$		−	26.5547i		−	1.27909i	0.768752	−	0.639547i	$-0.220878\pi$
$432$		0			0
$433$	21.7861			1.04697			0.523485	−	0.852035i	$-0.324631\pi$
$433$	21.7861			1.04697			0.523485	+	0.852035i	$0.324631\pi$
$434$			2.12934i			0.102212i
$435$		0			0
$436$		−	11.9546i		−	0.572520i
$437$			6.42154i			0.307184i
$438$		0			0
$439$	−7.79380			−0.371978			−0.185989	−	0.982552i	$-0.559549\pi$
$439$	−7.79380			−0.371978			−0.185989	+	0.982552i	$0.559549\pi$
$440$		−	18.1945i		−	0.867387i
$441$		0			0
$442$	1.72785	−	0.643188i	0.0821857	−	0.0305933i
$443$	−16.0031			−0.760330			−0.380165	−	0.924919i	$-0.624133\pi$
$443$	−16.0031			−0.760330			−0.380165	+	0.924919i	$0.624133\pi$
$444$		0			0
$445$	−8.75622			−0.415085
$446$	21.6971			1.02739
$447$		0			0
$448$		−	8.13879i		−	0.384522i
$449$			8.58501i			0.405151i	0.979267	+	0.202576i	$0.0649312\pi$
$449$			8.58501i			0.405151i	−0.979267	+	0.202576i	$0.935069\pi$
$450$		0			0
$451$	−7.10648			−0.334631
$452$	−9.89628			−0.465482
$453$		0			0
$454$	22.0716			1.03587
$455$	−1.65172	−	4.43716i	−0.0774337	−	0.208017i
$456$		0			0
$457$		−	5.50364i		−	0.257450i	−0.991680	−	0.128725i	$-0.958912\pi$
$457$		−	5.50364i		−	0.257450i	0.991680	−	0.128725i	$-0.0410884\pi$
$458$	21.3608			0.998123
$459$		0			0
$460$		−	1.03747i		−	0.0483721i
$461$		−	22.9875i		−	1.07063i	−0.844651	−	0.535317i	$-0.820192\pi$
$461$		−	22.9875i		−	1.07063i	0.844651	−	0.535317i	$-0.179808\pi$
$462$		0			0
$463$			25.2200i			1.17207i	0.810284	+	0.586037i	$0.199313\pi$
$463$			25.2200i			1.17207i	−0.810284	+	0.586037i	$0.800687\pi$
$464$	−14.9014			−0.691780
$465$		0			0
$466$			30.4896i			1.41240i
$467$	−18.1098			−0.838023			−0.419012	−	0.907981i	$-0.637623\pi$
$467$	−18.1098			−0.838023			−0.419012	+	0.907981i	$0.637623\pi$
$468$		0			0
$469$	−2.38954			−0.110338
$470$		−	9.37481i		−	0.432428i
$471$		0			0
$472$	−12.8774			−0.592732
$473$		−	12.2558i		−	0.563524i
$474$		0			0
$475$		−	22.5531i		−	1.03481i
$476$			0.414867i			0.0190154i
$477$		0			0
$478$	15.8013			0.722734
$479$			16.5774i			0.757439i	0.925511	+	0.378720i	$0.123636\pi$
$479$			16.5774i			0.757439i	−0.925511	+	0.378720i	$0.876364\pi$
$480$		0			0
$481$	−16.7149	+	6.22207i	−0.762135	+	0.283702i
$482$	−14.2138			−0.647419
$483$		0			0
$484$	−9.09558			−0.413435
$485$	22.0548			1.00146
$486$		0			0
$487$		−	5.78932i		−	0.262339i	−0.991360	−	0.131170i	$-0.958127\pi$
$487$		−	5.78932i		−	0.262339i	0.991360	−	0.131170i	$-0.0418732\pi$
$488$		−	28.8371i		−	1.30539i
$489$		0			0
$490$	8.14419			0.367917
$491$	10.4706			0.472532			0.236266	−	0.971688i	$-0.424076\pi$
$491$	10.4706			0.472532			0.236266	+	0.971688i	$0.424076\pi$
$492$		0			0
$493$	4.46548			0.201115
$494$	−9.03752	−	24.2783i	−0.406617	−	1.09233i
$495$		0			0
$496$		−	3.06289i		−	0.137528i
$497$	1.38954			0.0623295
$498$		0			0
$499$			9.03702i			0.404553i	0.979328	+	0.202276i	$0.0648339\pi$
$499$			9.03702i			0.404553i	−0.979328	+	0.202276i	$0.935166\pi$
$500$			9.04865i			0.404668i
$501$		0			0
$502$		−	12.8258i		−	0.572443i
$503$	9.96486			0.444311			0.222155	−	0.975011i	$-0.428691\pi$
$503$	9.96486			0.444311			0.222155	+	0.975011i	$0.428691\pi$
$504$		0			0
$505$		−	17.2686i		−	0.768443i
$506$	4.80530			0.213621
$507$		0			0
$508$	−6.86954			−0.304787
$509$			35.2868i			1.56406i	0.623240	+	0.782030i	$0.285816\pi$
$509$			35.2868i			1.56406i	−0.623240	+	0.782030i	$0.714184\pi$
$510$		0			0
$511$	−13.2387			−0.585647
$512$			16.7198i			0.738919i
$513$		0			0
$514$		−	16.0698i		−	0.708808i
$515$			0.133657i			0.00588965i
$516$		0			0
$517$	−31.8900			−1.40252
$518$			5.46463i			0.240102i
$519$		0			0
$520$	4.90830	+	13.1856i	0.215243	+	0.578228i
$521$	−12.9544			−0.567541			−0.283770	−	0.958892i	$-0.591585\pi$
$521$	−12.9544			−0.567541			−0.283770	+	0.958892i	$0.591585\pi$
$522$		0			0
$523$	−0.367139			−0.0160539			−0.00802694	−	0.999968i	$-0.502555\pi$
$523$	−0.367139			−0.0160539			−0.00802694	+	0.999968i	$0.502555\pi$
$524$	−7.92743			−0.346311
$525$		0			0
$526$			1.66363i			0.0725377i
$527$			0.917853i			0.0399823i
$528$		0			0
$529$	−22.0789			−0.959953
$530$	−0.797940			−0.0346603
$531$		0			0
$532$	5.82934			0.252734
$533$	5.15010	−	1.91711i	0.223076	−	0.0830392i
$534$		0			0
$535$		−	15.8585i		−	0.685623i
$536$	7.10083			0.306709
$537$		0			0
$538$			22.5820i			0.973580i
$539$		−	27.7038i		−	1.19329i
$540$		0			0
$541$		−	42.0316i		−	1.80708i	−0.428504	−	0.903540i	$-0.640959\pi$
$541$		−	42.0316i		−	1.80708i	0.428504	−	0.903540i	$-0.359041\pi$
$542$	−13.6922			−0.588132
$543$		0			0
$544$		−	2.09893i		−	0.0899908i
$545$	18.0183			0.771821
$546$		0			0
$547$	−28.5784			−1.22192			−0.610962	−	0.791660i	$-0.709217\pi$
$547$	−28.5784			−1.22192			−0.610962	+	0.791660i	$0.709217\pi$
$548$			0.501071i			0.0214047i
$549$		0			0
$550$	−16.8767			−0.719625
$551$		−	62.7451i		−	2.67303i
$552$		0			0
$553$		−	13.2794i		−	0.564699i
$554$			12.4858i			0.530470i
$555$		0			0
$556$	−10.7322			−0.455146
$557$		−	3.59187i		−	0.152192i	−0.997100	−	0.0760961i	$-0.975754\pi$
$557$		−	3.59187i		−	0.152192i	0.997100	−	0.0760961i	$-0.0242456\pi$
$558$		0			0
$559$	3.30625	+	8.88187i	0.139839	+	0.375663i
$560$	2.08664			0.0881768
$561$		0			0
$562$	18.5598			0.782898
$563$	30.6453			1.29155			0.645774	−	0.763529i	$-0.276535\pi$
$563$	30.6453			1.29155			0.645774	+	0.763529i	$0.276535\pi$
$564$		0			0
$565$		−	14.9160i		−	0.627522i
$566$		−	23.2050i		−	0.975377i
$567$		0			0
$568$	−4.12921			−0.173258
$569$	34.6648			1.45322			0.726612	−	0.687048i	$-0.241094\pi$
$569$	34.6648			1.45322			0.726612	+	0.687048i	$0.241094\pi$
$570$		0			0
$571$	−15.5210			−0.649533			−0.324767	−	0.945794i	$-0.605286\pi$
$571$	−15.5210			−0.649533			−0.324767	+	0.945794i	$0.605286\pi$
$572$	13.3427	−	4.96678i	0.557887	−	0.207671i
$573$		0			0
$574$		−	1.68373i		−	0.0702775i
$575$	−3.23497			−0.134908
$576$		0			0
$577$			18.6264i			0.775426i	0.921780	+	0.387713i	$0.126735\pi$
$577$			18.6264i			0.775426i	−0.921780	+	0.387713i	$0.873265\pi$
$578$			18.0117i			0.749188i
$579$		0			0
$580$			10.1371i			0.420920i
$581$	−10.5270			−0.436735
$582$		0			0
$583$			2.71433i			0.112416i
$584$	39.3407			1.62793
$585$		0			0
$586$	0.0767199			0.00316927
$587$		−	35.3932i		−	1.46084i	−0.683001	−	0.730418i	$-0.739325\pi$
$587$		−	35.3932i		−	1.46084i	0.683001	−	0.730418i	$-0.260675\pi$
$588$		0			0
$589$	12.8969			0.531406
$590$		−	5.77380i		−	0.237704i
$591$		0			0
$592$		−	7.86046i		−	0.323063i
$593$		−	2.41535i		−	0.0991864i	−0.998770	−	0.0495932i	$-0.984208\pi$
$593$		−	2.41535i		−	0.0991864i	0.998770	−	0.0495932i	$-0.0157925\pi$
$594$		0			0
$595$	−0.625302			−0.0256349
$596$			9.90034i			0.405534i
$597$		0			0
$598$	−3.48242	+	1.29632i	−0.142407	+	0.0530105i
$599$	−40.9367			−1.67263			−0.836314	−	0.548251i	$-0.815294\pi$
$599$	−40.9367			−1.67263			−0.836314	+	0.548251i	$0.815294\pi$
$600$		0			0
$601$	−17.6415			−0.719613			−0.359807	−	0.933027i	$-0.617157\pi$
$601$	−17.6415			−0.719613			−0.359807	+	0.933027i	$0.617157\pi$
$602$	2.90376			0.118348
$603$		0			0
$604$			8.64259i			0.351662i
$605$		−	13.7092i		−	0.557357i
$606$		0			0
$607$	−18.4591			−0.749232			−0.374616	−	0.927180i	$-0.622226\pi$
$607$	−18.4591			−0.749232			−0.374616	+	0.927180i	$0.622226\pi$
$608$	−29.4923			−1.19607
$609$		0			0
$610$	12.9296			0.523504
$611$	23.1108	−	8.60293i	0.934964	−	0.348037i
$612$		0			0
$613$		−	7.01548i		−	0.283352i	−0.989913	−	0.141676i	$-0.954751\pi$
$613$		−	7.01548i		−	0.283352i	0.989913	−	0.141676i	$-0.0452492\pi$
$614$	21.4053			0.863846
$615$		0			0
$616$		−	14.6639i		−	0.590824i
$617$			10.1422i			0.408311i	0.978938	+	0.204156i	$0.0654449\pi$
$617$			10.1422i			0.408311i	−0.978938	+	0.204156i	$0.934555\pi$
$618$		0			0
$619$			9.36568i			0.376439i	0.982127	+	0.188219i	$0.0602716\pi$
$619$			9.36568i			0.376439i	−0.982127	+	0.188219i	$0.939728\pi$
$620$	−2.08362			−0.0836802
$621$		0			0
$622$			6.52922i			0.261798i
$623$	−7.05709			−0.282736
$624$		0			0
$625$	3.21501			0.128600
$626$		−	6.54227i		−	0.261482i
$627$		0			0
$628$	9.26520			0.369722
$629$			2.35553i			0.0939212i
$630$		0			0
$631$			0.999379i			0.0397846i	0.999802	+	0.0198923i	$0.00633234\pi$
$631$			0.999379i			0.0397846i	−0.999802	+	0.0198923i	$0.993668\pi$
$632$			39.4617i			1.56970i
$633$		0			0
$634$	10.2301			0.406287
$635$		−	10.3540i		−	0.410887i
$636$		0			0
$637$	7.47363	+	20.0771i	0.296116	+	0.795483i
$638$	−46.9526			−1.85887
$639$		0			0
$640$	0.408601			0.0161514
$641$	8.54213			0.337394			0.168697	−	0.985668i	$-0.446044\pi$
$641$	8.54213			0.337394			0.168697	+	0.985668i	$0.446044\pi$
$642$		0			0
$643$		−	0.904290i		−	0.0356617i	−0.999841	−	0.0178309i	$-0.994324\pi$
$643$		−	0.904290i		−	0.0356617i	0.999841	−	0.0178309i	$-0.00567604\pi$
$644$		−	0.836147i		−	0.0329488i
$645$		0			0
$646$	−3.42139			−0.134613
$647$	−20.5148			−0.806518			−0.403259	−	0.915086i	$-0.632123\pi$
$647$	−20.5148			−0.806518			−0.403259	+	0.915086i	$0.632123\pi$
$648$		0			0
$649$	−19.6406			−0.770959
$650$	12.2306	−	4.55281i	0.479725	−	0.178576i
$651$		0			0
$652$		−	14.3658i		−	0.562610i
$653$	−9.36899			−0.366637			−0.183318	−	0.983054i	$-0.558684\pi$
$653$	−9.36899			−0.366637			−0.183318	+	0.983054i	$0.558684\pi$
$654$		0			0
$655$		−	11.9485i		−	0.466867i
$656$			2.42192i			0.0945600i
$657$		0			0
$658$		−	7.55565i		−	0.294550i
$659$	−35.3042			−1.37525			−0.687627	−	0.726064i	$-0.741348\pi$
$659$	−35.3042			−1.37525			−0.687627	+	0.726064i	$0.741348\pi$
$660$		0			0
$661$			23.3442i			0.907983i	0.891006	+	0.453992i	$0.150000\pi$
$661$			23.3442i			0.907983i	−0.891006	+	0.453992i	$0.850000\pi$
$662$	19.4851			0.757311
$663$		0			0
$664$	31.2825			1.21400
$665$			8.78619i			0.340714i
$666$		0			0
$667$	−9.00000			−0.348481
$668$			14.2340i			0.550732i
$669$		0			0
$670$			3.18377i			0.123000i
$671$		−	43.9821i		−	1.69791i
$672$		0			0
$673$	49.8528			1.92168			0.960842	−	0.277096i	$-0.0893720\pi$
$673$	49.8528			1.92168			0.960842	+	0.277096i	$0.0893720\pi$
$674$			8.61065i			0.331670i
$675$		0			0
$676$	−8.32964	+	7.19890i	−0.320371	+	0.276881i
$677$	−10.9118			−0.419376			−0.209688	−	0.977768i	$-0.567245\pi$
$677$	−10.9118			−0.419376			−0.209688	+	0.977768i	$0.567245\pi$
$678$		0			0
$679$	17.7751			0.682147
$680$	1.85817			0.0712576
$681$		0			0
$682$		−	9.65084i		−	0.369550i
$683$			24.0595i			0.920610i	0.887761	+	0.460305i	$0.152260\pi$
$683$			24.0595i			0.920610i	−0.887761	+	0.460305i	$0.847740\pi$
$684$		0			0
$685$	−0.755232			−0.0288559
$686$	14.2968			0.545854
$687$		0			0
$688$	−4.17684			−0.159241
$689$	−0.732241	−	1.96709i	−0.0278962	−	0.0749400i
$690$		0			0
$691$		−	29.4307i		−	1.11960i	−0.828629	−	0.559799i	$-0.810878\pi$
$691$		−	29.4307i		−	1.11960i	0.828629	−	0.559799i	$-0.189122\pi$
$692$	16.1638			0.614457
$693$		0			0
$694$			17.8550i			0.677765i
$695$		−	16.1760i		−	0.613589i
$696$		0			0
$697$		−	0.725773i		−	0.0274906i
$698$	0.192209			0.00727523
$699$		0			0
$700$			2.93664i			0.110994i
$701$	−16.6961			−0.630604			−0.315302	−	0.948991i	$-0.602106\pi$
$701$	−16.6961			−0.630604			−0.315302	+	0.948991i	$0.602106\pi$
$702$		0			0
$703$	33.0979			1.24831
$704$			36.8876i			1.39025i
$705$		0			0
$706$	18.1704			0.683852
$707$		−	13.9177i		−	0.523427i
$708$		0			0
$709$			33.4537i			1.25638i	0.778059	+	0.628191i	$0.216204\pi$
$709$			33.4537i			1.25638i	−0.778059	+	0.628191i	$0.783796\pi$
$710$		−	1.85140i		−	0.0694818i
$711$		0			0
$712$	20.9711			0.785925
$713$		−	1.84990i		−	0.0692792i
$714$		0			0
$715$	7.48611	+	20.1106i	0.279964	+	0.752094i
$716$	7.05986			0.263839
$717$		0			0
$718$	−36.9840			−1.38023
$719$	31.3183			1.16797			0.583987	−	0.811763i	$-0.301492\pi$
$719$	31.3183			1.16797			0.583987	+	0.811763i	$0.301492\pi$
$720$		0			0
$721$			0.107721i			0.00401175i
$722$			27.6716i			1.02983i
$723$		0			0
$724$	8.41482			0.312734
$725$	31.6090			1.17393
$726$		0			0
$727$	37.7830			1.40129			0.700646	−	0.713509i	$-0.252895\pi$
$727$	37.7830			1.40129			0.700646	+	0.713509i	$0.252895\pi$
$728$	3.95586	+	10.6270i	0.146614	+	0.393862i
$729$		0			0
$730$			17.6390i			0.652850i
$731$	1.25167			0.0462946
$732$		0			0
$733$		−	6.40942i		−	0.236737i	−0.992970	−	0.118369i	$-0.962234\pi$
$733$		−	6.40942i		−	0.236737i	0.992970	−	0.118369i	$-0.0377665\pi$
$734$		−	26.9373i		−	0.994275i
$735$		0			0
$736$			4.23030i			0.155931i
$737$	10.8301			0.398933
$738$		0			0
$739$			26.2292i			0.964856i	0.875936	+	0.482428i	$0.160245\pi$
$739$			26.2292i			0.964856i	−0.875936	+	0.482428i	$0.839755\pi$
$740$	−5.34730			−0.196570
$741$		0			0
$742$	−0.643101			−0.0236090
$743$		−	27.4289i		−	1.00627i	−0.864208	−	0.503134i	$-0.832180\pi$
$743$		−	27.4289i		−	1.00627i	0.864208	−	0.503134i	$-0.167820\pi$
$744$		0			0
$745$	−14.9222			−0.546705
$746$		−	11.7591i		−	0.430533i
$747$		0			0
$748$		−	1.88031i		−	0.0687508i
$749$		−	12.7812i		−	0.467014i
$750$		0			0
$751$	−14.6367			−0.534102			−0.267051	−	0.963682i	$-0.586049\pi$
$751$	−14.6367			−0.534102			−0.267051	+	0.963682i	$0.586049\pi$
$752$			10.8682i			0.396324i
$753$		0			0
$754$	34.0268	−	12.6664i	1.23918	−	0.461282i
$755$	−13.0264			−0.474080
$756$		0			0
$757$	4.79203			0.174169			0.0870846	−	0.996201i	$-0.472245\pi$
$757$	4.79203			0.174169			0.0870846	+	0.996201i	$0.472245\pi$
$758$	−11.9247			−0.433126
$759$		0			0
$760$		−	26.1094i		−	0.947087i
$761$		−	19.3385i		−	0.701022i	−0.936559	−	0.350511i	$-0.886008\pi$
$761$		−	19.3385i		−	0.701022i	0.936559	−	0.350511i	$-0.113992\pi$
$762$		0			0
$763$	14.5219			0.525729
$764$	5.14558			0.186160
$765$		0			0
$766$	−6.31277			−0.228089
$767$	14.2336	−	5.29841i	0.513946	−	0.191315i
$768$		0			0
$769$		−	10.0925i		−	0.363946i	−0.983303	−	0.181973i	$-0.941752\pi$
$769$		−	10.0925i		−	0.363946i	0.983303	−	0.181973i	$-0.0582484\pi$
$770$	6.57478			0.236939
$771$		0			0
$772$		−	13.9845i		−	0.503313i
$773$		−	8.43470i		−	0.303375i	−0.988428	−	0.151688i	$-0.951529\pi$
$773$		−	8.43470i		−	0.303375i	0.988428	−	0.151688i	$-0.0484708\pi$
$774$		0			0
$775$			6.49703i			0.233380i
$776$	−52.8213			−1.89617
$777$		0			0
$778$		−	0.0862040i		−	0.00309056i
$779$	−10.1979			−0.365379
$780$		0			0
$781$	−6.29784			−0.225355
$782$			0.490757i			0.0175494i
$783$		0			0
$784$	−9.44157			−0.337199
$785$			13.9648i			0.498426i
$786$		0			0
$787$			12.8145i			0.456787i	0.973569	+	0.228393i	$0.0733472\pi$
$787$			12.8145i			0.456787i	−0.973569	+	0.228393i	$0.926653\pi$
$788$			16.2343i			0.578323i
$789$		0			0
$790$	−17.6933			−0.629499
$791$		−	12.0216i		−	0.427439i
$792$		0			0
$793$	11.8650	+	31.8741i	0.421339	+	1.13188i
$794$	−1.43855			−0.0510522
$795$		0			0
$796$	7.67497			0.272032
$797$	−29.2148			−1.03484			−0.517421	−	0.855731i	$-0.673108\pi$
$797$	−29.2148			−1.03484			−0.517421	+	0.855731i	$0.673108\pi$
$798$		0			0
$799$		−	3.25687i		−	0.115220i
$800$		−	14.8573i		−	0.525284i
$801$		0			0
$802$	−34.0161			−1.20115
$803$	60.0021			2.11743
$804$		0			0
$805$	1.26027			0.0444187
$806$	2.60350	+	6.99401i	0.0917042	+	0.246353i
$807$		0			0
$808$			41.3582i			1.45498i
$809$	11.3570			0.399292			0.199646	−	0.979868i	$-0.436021\pi$
$809$	11.3570			0.399292			0.199646	+	0.979868i	$0.436021\pi$
$810$		0			0
$811$		−	26.8826i		−	0.943974i	−0.881605	−	0.471987i	$-0.843537\pi$
$811$		−	26.8826i		−	0.943974i	0.881605	−	0.471987i	$-0.156463\pi$
$812$			8.17001i			0.286711i
$813$		0			0
$814$		−	24.7674i		−	0.868098i
$815$	21.6527			0.758461
$816$		0			0
$817$		−	17.5873i		−	0.615303i
$818$	−3.62852			−0.126868
$819$		0			0
$820$	1.64758			0.0575359
$821$			16.4776i			0.575072i	0.957770	+	0.287536i	$0.0928361\pi$
$821$			16.4776i			0.575072i	−0.957770	+	0.287536i	$0.907164\pi$
$822$		0			0
$823$	8.20410			0.285977			0.142988	−	0.989724i	$-0.454329\pi$
$823$	8.20410			0.285977			0.142988	+	0.989724i	$0.454329\pi$
$824$		−	0.320109i		−	0.0111515i
$825$		0			0
$826$		−	4.65341i		−	0.161913i
$827$			43.6569i			1.51810i	0.651033	+	0.759049i	$0.274336\pi$
$827$			43.6569i			1.51810i	−0.651033	+	0.759049i	$0.725664\pi$
$828$		0			0
$829$	23.3338			0.810415			0.405208	−	0.914225i	$-0.367199\pi$
$829$	23.3338			0.810415			0.405208	+	0.914225i	$0.367199\pi$
$830$			14.0260i			0.486851i
$831$		0			0
$832$	−9.95113	−	26.7326i	−0.344993	−	0.926787i
$833$	2.82934			0.0980309
$834$		0			0
$835$	−21.4541			−0.742448
$836$	−26.4204			−0.913770
$837$		0			0
$838$		−	23.2658i		−	0.803702i
$839$		−	13.7220i		−	0.473735i	−0.971542	−	0.236868i	$-0.923879\pi$
$839$		−	13.7220i		−	0.473735i	0.971542	−	0.236868i	$-0.0761208\pi$
$840$		0			0
$841$	58.9392			2.03239
$842$	−15.0888			−0.519994
$843$		0			0
$844$	−12.4471			−0.428447
$845$	−10.8504	−	12.5547i	−0.373266	−	0.431896i
$846$		0			0
$847$		−	11.0489i		−	0.379646i
$848$	0.925054			0.0317665
$849$		0			0
$850$		−	1.72359i		−	0.0591186i
$851$		−	4.74748i		−	0.162742i
$852$		0			0
$853$		−	15.3309i		−	0.524920i	−0.964943	−	0.262460i	$-0.915466\pi$
$853$		−	15.3309i		−	0.524920i	0.964943	−	0.262460i	$-0.0845337\pi$
$854$	10.4206			0.356587
$855$		0			0
$856$			37.9810i			1.29817i
$857$	−42.7921			−1.46175			−0.730875	−	0.682511i	$-0.760888\pi$
$857$	−42.7921			−1.46175			−0.730875	+	0.682511i	$0.760888\pi$
$858$		0			0
$859$	−26.3071			−0.897587			−0.448793	−	0.893636i	$-0.648146\pi$
$859$	−26.3071			−0.897587			−0.448793	+	0.893636i	$0.648146\pi$
$860$			2.84141i			0.0968914i
$861$		0			0
$862$	−28.5154			−0.971237
$863$			8.03444i			0.273495i	0.990606	+	0.136748i	$0.0436650\pi$
$863$			8.03444i			0.273495i	−0.990606	+	0.136748i	$0.956335\pi$
$864$		0			0
$865$			24.3627i			0.828357i
$866$		−	23.3946i		−	0.794982i
$867$		0			0
$868$	−1.67930			−0.0569990
$869$			60.1867i			2.04169i
$870$		0			0
$871$	−7.84865	+	2.92163i	−0.265941	+	0.0989958i
$872$	−43.1539			−1.46137
$873$		0			0
$874$	6.89568			0.233250
$875$	−10.9919			−0.371595
$876$		0			0
$877$		−	6.97323i		−	0.235469i	−0.993045	−	0.117735i	$-0.962437\pi$
$877$		−	6.97323i		−	0.235469i	0.993045	−	0.117735i	$-0.0375632\pi$
$878$			8.36926i			0.282449i
$879$		0			0
$880$	−9.45734			−0.318807
$881$	−30.4317			−1.02527			−0.512635	−	0.858606i	$-0.671331\pi$
$881$	−30.4317			−1.02527			−0.512635	+	0.858606i	$0.671331\pi$
$882$		0			0
$883$	−1.51291			−0.0509134			−0.0254567	−	0.999676i	$-0.508104\pi$
$883$	−1.51291			−0.0509134			−0.0254567	+	0.999676i	$0.508104\pi$
$884$	0.507248	+	1.36267i	0.0170606	+	0.0458315i
$885$		0			0
$886$			17.1847i			0.577331i
$887$	36.8943			1.23879			0.619395	−	0.785080i	$-0.287378\pi$
$887$	36.8943			1.23879			0.619395	+	0.785080i	$0.287378\pi$
$888$		0			0
$889$		−	8.34484i		−	0.279877i
$890$			9.40274i			0.315181i
$891$		0			0
$892$			17.1114i			0.572931i
$893$	−45.7627			−1.53139
$894$		0			0
$895$			10.6409i			0.355685i
$896$	0.329313			0.0110016
$897$		0			0
$898$	9.21888			0.307638
$899$			18.0754i			0.602848i
$900$		0			0
$901$	−0.277209			−0.00923519
$902$			7.63119i			0.254091i
$903$		0			0
$904$			35.7238i			1.18816i
$905$			12.6831i			0.421601i
$906$		0			0
$907$	28.7881			0.955893			0.477946	−	0.878389i	$-0.341381\pi$
$907$	28.7881			0.955893			0.477946	+	0.878389i	$0.341381\pi$
$908$			17.4067i			0.577662i
$909$		0			0
$910$	−4.76478	+	1.77367i	−0.157951	+	0.0587967i
$911$	39.7863			1.31818			0.659089	−	0.752065i	$-0.270942\pi$
$911$	39.7863			1.31818			0.659089	+	0.752065i	$0.270942\pi$
$912$		0			0
$913$	47.7119			1.57903
$914$	−5.91001			−0.195486
$915$		0			0
$916$			16.8461i			0.556611i
$917$		−	9.62991i		−	0.318008i
$918$		0			0
$919$	28.4990			0.940095			0.470047	−	0.882641i	$-0.344237\pi$
$919$	28.4990			0.940095			0.470047	+	0.882641i	$0.344237\pi$
$920$	−3.74507			−0.123471
$921$		0			0
$922$	−24.6848			−0.812950
$923$	4.56408	−	1.69896i	0.150228	−	0.0559221i
$924$		0			0
$925$			16.6737i			0.548226i
$926$	27.0822			0.889975
$927$		0			0
$928$		−	41.3344i		−	1.35687i
$929$			30.2119i			0.991219i	0.868545	+	0.495610i	$0.165055\pi$
$929$			30.2119i			0.991219i	−0.868545	+	0.495610i	$0.834945\pi$
$930$		0			0
$931$		−	39.7555i		−	1.30293i
$932$	−24.0455			−0.787637
$933$		0			0
$934$			19.4470i			0.636325i
$935$	2.83407			0.0926839
$936$		0			0
$937$	6.31683			0.206362			0.103181	−	0.994663i	$-0.467098\pi$
$937$	6.31683			0.206362			0.103181	+	0.994663i	$0.467098\pi$
$938$			2.56597i			0.0837818i
$939$		0			0
$940$	7.39342			0.241147
$941$			42.1966i			1.37557i	0.725915	+	0.687784i	$0.241416\pi$
$941$			42.1966i			1.37557i	−0.725915	+	0.687784i	$0.758584\pi$
$942$		0			0
$943$			1.46277i			0.0476342i
$944$			6.69358i			0.217857i
$945$		0			0
$946$	−13.1608			−0.427893
$947$		−	13.6229i		−	0.442684i	−0.975196	−	0.221342i	$-0.928956\pi$
$947$		−	13.6229i		−	0.442684i	0.975196	−	0.221342i	$-0.0710437\pi$
$948$		0			0
$949$	−43.4838	+	16.1867i	−1.41154	+	0.525443i
$950$	−24.2184			−0.785748
$951$		0			0
$952$	1.49760			0.0485373
$953$	43.5443			1.41054			0.705268	−	0.708940i	$-0.250826\pi$
$953$	43.5443			1.41054			0.705268	+	0.708940i	$0.250826\pi$
$954$		0			0
$955$			7.75560i			0.250965i
$956$			12.4616i			0.403038i
$957$		0			0
$958$	17.8014			0.575136
$959$	−0.608680			−0.0196553
$960$		0			0
$961$	27.2847			0.880152
$962$	6.68148	+	17.9491i	0.215420	+	0.578701i
$963$		0			0
$964$		−	11.2096i		−	0.361038i
$965$	21.0779			0.678522
$966$		0			0
$967$			16.1442i			0.519162i	0.965721	+	0.259581i	$0.0835845\pi$
$967$			16.1442i			0.519162i	−0.965721	+	0.259581i	$0.916415\pi$
$968$			32.8334i			1.05531i
$969$		0			0
$970$		−	23.6833i		−	0.760424i
$971$	0.206193			0.00661704			0.00330852	−	0.999995i	$-0.498947\pi$
$971$	0.206193			0.00661704			0.00330852	+	0.999995i	$0.498947\pi$
$972$		0			0
$973$		−	13.0370i		−	0.417948i
$974$	−6.21678			−0.199198
$975$		0			0
$976$	−14.9893			−0.479796
$977$			7.98554i			0.255480i	0.991808	+	0.127740i	$0.0407723\pi$
$977$			7.98554i			0.255480i	−0.991808	+	0.127740i	$0.959228\pi$
$978$		0			0
$979$	31.9850			1.02224
$980$			6.42289i			0.205172i
$981$		0			0
$982$		−	11.2437i		−	0.358801i
$983$		−	27.8874i		−	0.889470i	−0.895662	−	0.444735i	$-0.853298\pi$
$983$		−	27.8874i		−	0.889470i	0.895662	−	0.444735i	$-0.146702\pi$
$984$		0			0
$985$	−24.4689			−0.779644
$986$		−	4.79519i		−	0.152710i
$987$		0			0
$988$	19.1470	−	7.12741i	0.609148	−	0.226753i
$989$	−2.52269			−0.0802168
$990$		0			0
$991$	−8.84780			−0.281059			−0.140530	−	0.990076i	$-0.544881\pi$
$991$	−8.84780			−0.281059			−0.140530	+	0.990076i	$0.544881\pi$
$992$	8.49604			0.269749
$993$		0			0
$994$		−	1.49214i		−	0.0473278i
$995$			11.5680i			0.366730i
$996$		0			0
$997$	−25.4452			−0.805857			−0.402929	−	0.915231i	$-0.632008\pi$
$997$	−25.4452			−0.805857			−0.402929	+	0.915231i	$0.632008\pi$
$998$	9.70427			0.307183
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1053.2.b.i.649.4		10
3.2	odd	2		1053.2.b.j.649.7		10
9.2	odd	6		117.2.t.c.103.7	yes	20
9.4	even	3		351.2.t.c.181.7		20
9.5	odd	6		117.2.t.c.25.4	✓	20
9.7	even	3		351.2.t.c.64.4		20
13.12	even	2	inner	1053.2.b.i.649.7		10
39.38	odd	2		1053.2.b.j.649.4		10
117.25	even	6		351.2.t.c.64.7		20
117.38	odd	6		117.2.t.c.103.4	yes	20
117.77	odd	6		117.2.t.c.25.7	yes	20
117.103	even	6		351.2.t.c.181.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.t.c.25.4	✓	20	9.5	odd	6
117.2.t.c.25.7	yes	20	117.77	odd	6
117.2.t.c.103.4	yes	20	117.38	odd	6
117.2.t.c.103.7	yes	20	9.2	odd	6
351.2.t.c.64.4		20	9.7	even	3
351.2.t.c.64.7		20	117.25	even	6
351.2.t.c.181.4		20	117.103	even	6
351.2.t.c.181.7		20	9.4	even	3
1053.2.b.i.649.4		10	1.1	even	1	trivial
1053.2.b.i.649.7		10	13.12	even	2	inner
1053.2.b.j.649.4		10	39.38	odd	2
1053.2.b.j.649.7		10	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.07384i q^{2} +0.846877 q^{4} +1.27644i q^{5} +1.02875i q^{7} -3.05708i q^{8} +1.37069 q^{10} -4.66263i q^{11} +(1.25783 + 3.37903i) q^{13} +1.10471 q^{14} -1.58904 q^{16} +0.476187 q^{17} -6.69096i q^{19} +1.08099i q^{20} -5.00690 q^{22} -0.959735 q^{23} +3.37069 q^{25} +(3.62852 - 1.35071i) q^{26} +0.871227i q^{28} +9.37759 q^{29} +1.92751i q^{31} -4.40778i q^{32} -0.511346i q^{34} -1.31314 q^{35} +4.94666i q^{37} -7.18499 q^{38} +3.90219 q^{40} -1.52414i q^{41} +2.62852 q^{43} -3.94868i q^{44} +1.03060i q^{46} -6.83948i q^{47} +5.94167 q^{49} -3.61957i q^{50} +(1.06523 + 2.86163i) q^{52} -0.582145 q^{53} +5.95159 q^{55} +3.14498 q^{56} -10.0700i q^{58} -4.21233i q^{59} +9.43290 q^{61} +2.06983 q^{62} -7.91132 q^{64} +(-4.31314 + 1.60555i) q^{65} +2.32275i q^{67} +0.403272 q^{68} +1.41010i q^{70} -1.35071i q^{71} +12.8687i q^{73} +5.31190 q^{74} -5.66642i q^{76} +4.79669 q^{77} -12.9083 q^{79} -2.02833i q^{80} -1.63667 q^{82} +10.2328i q^{83} +0.607825i q^{85} -2.82260i q^{86} -14.2540 q^{88} +6.85985i q^{89} +(-3.47619 + 1.29400i) q^{91} -0.812778 q^{92} -7.34448 q^{94} +8.54063 q^{95} -17.2784i q^{97} -6.38037i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} - 18 q^{14} - 4 q^{16} + 6 q^{17} + 10 q^{22} + 24 q^{23} + 12 q^{25} + 6 q^{26} + 12 q^{29} + 6 q^{35} + 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} - 54 q^{53} + 10 q^{55}+ \cdots + 24 q^{95}+O(q^{100}) \) 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 - 18 * q^14 - 4 * q^16 + 6 * q^17 + 10 * q^22 + 24 * q^23 + 12 * q^25 + 6 * q^26 + 12 * q^29 + 6 * q^35 + 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 - 54 * q^53 + 10 * q^55 + 36 * q^56 + 2 * q^61 + 36 * q^62 + 4 * q^64 - 24 * q^65 + 24 * q^68 - 42 * q^74 - 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 - 84 * q^92 - 20 * q^94 + 24 * q^95

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