LMFDB - Embedded newform 531.5.c.d.235.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(235,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.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:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.4
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.37

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.76718i q^{2} -29.7948 q^{4} -6.77685 q^{5} +45.8846 q^{7} +93.3518i q^{8} +O(q^{10})$	\(q-6.76718i q^{2} -29.7948 q^{4} -6.77685 q^{5} +45.8846 q^{7} +93.3518i q^{8} +45.8602i q^{10} -152.372i q^{11} +138.574i q^{13} -310.510i q^{14} +155.012 q^{16} -61.1978 q^{17} -184.090 q^{19} +201.915 q^{20} -1031.13 q^{22} -55.4703i q^{23} -579.074 q^{25} +937.758 q^{26} -1367.12 q^{28} -290.483 q^{29} +681.108i q^{31} +444.632i q^{32} +414.137i q^{34} -310.953 q^{35} +1422.17i q^{37} +1245.77i q^{38} -632.631i q^{40} -1532.90 q^{41} -404.708i q^{43} +4539.89i q^{44} -375.377 q^{46} +1601.56i q^{47} -295.602 q^{49} +3918.70i q^{50} -4128.79i q^{52} +1325.56 q^{53} +1032.60i q^{55} +4283.41i q^{56} +1965.75i q^{58} +(-3430.65 + 589.891i) q^{59} -2732.09i q^{61} +4609.19 q^{62} +5489.10 q^{64} -939.097i q^{65} +2225.52i q^{67} +1823.37 q^{68} +2104.28i q^{70} +6157.04 q^{71} +2816.89i q^{73} +9624.09 q^{74} +5484.93 q^{76} -6991.53i q^{77} +7747.95 q^{79} -1050.49 q^{80} +10373.4i q^{82} +648.806i q^{83} +414.728 q^{85} -2738.73 q^{86} +14224.2 q^{88} +5751.21i q^{89} +6358.43i q^{91} +1652.72i q^{92} +10838.1 q^{94} +1247.55 q^{95} -1510.67i q^{97} +2000.39i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7	$ 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	6.76718i		−	1.69180i	−0.533345	−	0.845898i	$-0.679065\pi$
$2$		−	6.76718i		−	1.69180i	0.533345	−	0.845898i	$-0.320935\pi$
$3$		0			0
$3$		0			0
$4$	−29.7948			−1.86217
$5$	−6.77685			−0.271074			−0.135537	−	0.990772i	$-0.543276\pi$
$5$	−6.77685			−0.271074			−0.135537	+	0.990772i	$0.543276\pi$
$6$		0			0
$7$	45.8846			0.936421			0.468210	−	0.883617i	$-0.344899\pi$
$7$	45.8846			0.936421			0.468210	+	0.883617i	$0.344899\pi$
$8$			93.3518i			1.45862i
$9$		0			0
$10$			45.8602i			0.458602i
$11$		−	152.372i		−	1.25927i	−0.776890	−	0.629636i	$-0.783204\pi$
$11$		−	152.372i		−	1.25927i	0.776890	−	0.629636i	$-0.216796\pi$
$12$		0			0
$13$			138.574i			0.819966i	0.912093	+	0.409983i	$0.134465\pi$
$13$			138.574i			0.819966i	−0.912093	+	0.409983i	$0.865535\pi$
$14$		−	310.510i		−	1.58423i
$15$		0			0
$16$	155.012			0.605517
$17$	−61.1978			−0.211757			−0.105878	−	0.994379i	$-0.533765\pi$
$17$	−61.1978			−0.211757			−0.105878	+	0.994379i	$0.533765\pi$
$18$		0			0
$19$	−184.090			−0.509946			−0.254973	−	0.966948i	$-0.582067\pi$
$19$	−184.090			−0.509946			−0.254973	+	0.966948i	$0.582067\pi$
$20$	201.915			0.504787
$21$		0			0
$22$	−1031.13			−2.13043
$23$		−	55.4703i		−	0.104859i	−0.998625	−	0.0524294i	$-0.983304\pi$
$23$		−	55.4703i		−	0.104859i	0.998625	−	0.0524294i	$-0.0166964\pi$
$24$		0			0
$25$	−579.074			−0.926519
$26$	937.758			1.38722
$27$		0			0
$28$	−1367.12			−1.74378
$29$	−290.483			−0.345402			−0.172701	−	0.984974i	$-0.555250\pi$
$29$	−290.483			−0.345402			−0.172701	+	0.984974i	$0.555250\pi$
$30$		0			0
$31$			681.108i			0.708750i	0.935103	+	0.354375i	$0.115306\pi$
$31$			681.108i			0.708750i	−0.935103	+	0.354375i	$0.884694\pi$
$32$			444.632i			0.434211i
$33$		0			0
$34$			414.137i			0.358250i
$35$	−310.953			−0.253839
$36$		0			0
$37$			1422.17i			1.03884i	0.854519	+	0.519420i	$0.173852\pi$
$37$			1422.17i			1.03884i	−0.854519	+	0.519420i	$0.826148\pi$
$38$			1245.77i			0.862725i
$39$		0			0
$40$		−	632.631i		−	0.395394i
$41$	−1532.90			−0.911895			−0.455948	−	0.890007i	$-0.650700\pi$
$41$	−1532.90			−0.911895			−0.455948	+	0.890007i	$0.650700\pi$
$42$		0			0
$43$		−	404.708i		−	0.218879i	−0.993993	−	0.109440i	$-0.965094\pi$
$43$		−	404.708i		−	0.218879i	0.993993	−	0.109440i	$-0.0349056\pi$
$44$			4539.89i			2.34498i
$45$		0			0
$46$	−375.377			−0.177400
$47$			1601.56i			0.725016i	0.931981	+	0.362508i	$0.118079\pi$
$47$			1601.56i			0.725016i	−0.931981	+	0.362508i	$0.881921\pi$
$48$		0			0
$49$	−295.602			−0.123116
$50$			3918.70i			1.56748i
$51$		0			0
$52$		−	4128.79i		−	1.52692i
$53$	1325.56			0.471897			0.235949	−	0.971766i	$-0.424180\pi$
$53$	1325.56			0.471897			0.235949	+	0.971766i	$0.424180\pi$
$54$		0			0
$55$			1032.60i			0.341356i
$56$			4283.41i			1.36588i
$57$		0			0
$58$			1965.75i			0.584350i
$59$	−3430.65	+	589.891i	−0.985537	+	0.169460i
$59$	−3430.65	+	589.891i	−0.985537	+	0.169460i
$60$		0			0
$61$		−	2732.09i		−	0.734236i	−0.930174	−	0.367118i	$-0.880344\pi$
$61$		−	2732.09i		−	0.734236i	0.930174	−	0.367118i	$-0.119656\pi$
$62$	4609.19			1.19906
$63$		0			0
$64$	5489.10			1.34011
$65$		−	939.097i		−	0.222271i
$66$		0			0
$67$			2225.52i			0.495771i	0.968789	+	0.247885i	$0.0797357\pi$
$67$			2225.52i			0.495771i	−0.968789	+	0.247885i	$0.920264\pi$
$68$	1823.37			0.394328
$69$		0			0
$70$			2104.28i			0.429444i
$71$	6157.04			1.22139			0.610696	−	0.791865i	$-0.290890\pi$
$71$	6157.04			1.22139			0.610696	+	0.791865i	$0.290890\pi$
$72$		0			0
$73$			2816.89i			0.528597i	0.964441	+	0.264298i	$0.0851404\pi$
$73$			2816.89i			0.528597i	−0.964441	+	0.264298i	$0.914860\pi$
$74$	9624.09			1.75750
$75$		0			0
$76$	5484.93			0.949608
$77$		−	6991.53i		−	1.17921i
$78$		0			0
$79$	7747.95			1.24146			0.620730	−	0.784025i	$-0.286836\pi$
$79$	7747.95			1.24146			0.620730	+	0.784025i	$0.286836\pi$
$80$	−1050.49			−0.164140
$81$		0			0
$82$			10373.4i			1.54274i
$83$			648.806i			0.0941800i	0.998891	+	0.0470900i	$0.0149948\pi$
$83$			648.806i			0.0941800i	−0.998891	+	0.0470900i	$0.985005\pi$
$84$		0			0
$85$	414.728			0.0574018
$86$	−2738.73			−0.370299
$87$		0			0
$88$	14224.2			1.83680
$89$			5751.21i			0.726071i	0.931775	+	0.363036i	$0.118260\pi$
$89$			5751.21i			0.726071i	−0.931775	+	0.363036i	$0.881740\pi$
$90$		0			0
$91$			6358.43i			0.767833i
$92$			1652.72i			0.195265i
$93$		0			0
$94$	10838.1			1.22658
$95$	1247.55			0.138233
$96$		0			0
$97$		−	1510.67i		−	0.160556i	−0.996773	−	0.0802779i	$-0.974419\pi$
$97$		−	1510.67i		−	0.160556i	0.996773	−	0.0802779i	$-0.0255808\pi$
$98$			2000.39i			0.208287i
$99$		0			0
$100$	17253.4			1.72534
$101$			2457.45i			0.240903i	0.992719	+	0.120451i	$0.0384342\pi$
$101$			2457.45i			0.240903i	−0.992719	+	0.120451i	$0.961566\pi$
$102$		0			0
$103$			8751.92i			0.824952i	0.910968	+	0.412476i	$0.135336\pi$
$103$			8751.92i			0.824952i	−0.910968	+	0.412476i	$0.864664\pi$
$104$	−12936.2			−1.19602
$105$		0			0
$106$		−	8970.31i		−	0.798354i
$107$	−21034.2			−1.83721			−0.918603	−	0.395182i	$-0.870681\pi$
$107$	−21034.2			−1.83721			−0.918603	+	0.395182i	$0.870681\pi$
$108$		0			0
$109$			9627.62i			0.810338i	0.914242	+	0.405169i	$0.132787\pi$
$109$			9627.62i			0.810338i	−0.914242	+	0.405169i	$0.867213\pi$
$110$	6987.81			0.577505
$111$		0			0
$112$	7112.68			0.567018
$113$		−	23222.8i		−	1.81869i	−0.416044	−	0.909344i	$-0.636584\pi$
$113$		−	23222.8i		−	1.81869i	0.416044	−	0.909344i	$-0.363416\pi$
$114$		0			0
$115$			375.914i			0.0284245i
$116$	8654.89			0.643199
$117$		0			0
$118$	3991.90	+	23215.9i	0.286692	+	1.66733i
$119$	−2808.04			−0.198294
$120$		0			0
$121$	−8576.22			−0.585768
$122$	−18488.6			−1.24218
$123$		0			0
$124$		−	20293.5i		−	1.31981i
$125$	8159.83			0.522229
$126$		0			0
$127$	3258.32			0.202016			0.101008	−	0.994886i	$-0.467793\pi$
$127$	3258.32			0.202016			0.101008	+	0.994886i	$0.467793\pi$
$128$		−	30031.7i		−	1.83299i
$129$		0			0
$130$	−6355.04			−0.376038
$131$			23879.6i			1.39151i	0.718281	+	0.695753i	$0.244929\pi$
$131$			23879.6i			1.39151i	−0.718281	+	0.695753i	$0.755071\pi$
$132$		0			0
$133$	−8446.92			−0.477524
$134$	15060.5			0.838743
$135$		0			0
$136$		−	5712.92i		−	0.308873i
$137$	−25992.0			−1.38483			−0.692417	−	0.721497i	$-0.743454\pi$
$137$	−25992.0			−1.38483			−0.692417	+	0.721497i	$0.743454\pi$
$138$		0			0
$139$	10296.8			0.532931			0.266466	−	0.963844i	$-0.414144\pi$
$139$	10296.8			0.532931			0.266466	+	0.963844i	$0.414144\pi$
$140$	9264.78			0.472693
$141$		0			0
$142$		−	41665.8i		−	2.06635i
$143$	21114.8			1.03256
$144$		0			0
$145$	1968.56			0.0936296
$146$	19062.4			0.894278
$147$		0			0
$148$		−	42373.3i		−	1.93450i
$149$			34672.4i			1.56175i	0.624687	+	0.780875i	$0.285227\pi$
$149$			34672.4i			1.56175i	−0.624687	+	0.780875i	$0.714773\pi$
$150$		0			0
$151$			8100.65i			0.355276i	0.984096	+	0.177638i	$0.0568456\pi$
$151$			8100.65i			0.355276i	−0.984096	+	0.177638i	$0.943154\pi$
$152$		−	17185.2i		−	0.743818i
$153$		0			0
$154$	−47313.0			−1.99498
$155$		−	4615.77i		−	0.192124i
$156$		0			0
$157$		−	35250.4i		−	1.43009i	−0.699077	−	0.715046i	$-0.746406\pi$
$157$		−	35250.4i		−	1.43009i	0.699077	−	0.715046i	$-0.253594\pi$
$158$		−	52431.8i		−	2.10030i
$159$		0			0
$160$		−	3013.21i		−	0.117703i
$161$		−	2545.23i		−	0.0981919i
$162$		0			0
$163$	−12118.8			−0.456126			−0.228063	−	0.973646i	$-0.573239\pi$
$163$	−12118.8			−0.456126			−0.228063	+	0.973646i	$0.573239\pi$
$164$	45672.3			1.69811
$165$		0			0
$166$	4390.59			0.159333
$167$	−24887.2			−0.892365			−0.446182	−	0.894942i	$-0.647217\pi$
$167$	−24887.2			−0.892365			−0.446182	+	0.894942i	$0.647217\pi$
$168$		0			0
$169$	9358.17			0.327656
$170$		−	2806.54i		−	0.0971121i
$171$		0			0
$172$			12058.2i			0.407591i
$173$			6149.40i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$173$			6149.40i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$174$		0			0
$175$	−26570.6			−0.867612
$176$		−	23619.5i		−	0.762510i
$177$		0			0
$178$	38919.5			1.22836
$179$			17529.1i			0.547082i	0.961860	+	0.273541i	$0.0881949\pi$
$179$			17529.1i			0.547082i	−0.961860	+	0.273541i	$0.911805\pi$
$180$		0			0
$181$	20780.3			0.634299			0.317150	−	0.948376i	$-0.397274\pi$
$181$	20780.3			0.634299			0.317150	+	0.948376i	$0.397274\pi$
$182$	43028.6			1.29902
$183$		0			0
$184$	5178.25			0.152949
$185$		−	9637.84i		−	0.281602i
$186$		0			0
$187$			9324.83i			0.266660i
$188$		−	47718.1i		−	1.35011i
$189$		0			0
$190$		−	8442.42i		−	0.233862i
$191$			13668.6i			0.374676i	0.982296	+	0.187338i	$0.0599860\pi$
$191$			13668.6i			0.374676i	−0.982296	+	0.187338i	$0.940014\pi$
$192$		0			0
$193$	24606.0			0.660582			0.330291	−	0.943879i	$-0.392853\pi$
$193$	24606.0			0.660582			0.330291	+	0.943879i	$0.392853\pi$
$194$	−10223.0			−0.271628
$195$		0			0
$196$	8807.38			0.229263
$197$	−7657.45			−0.197311			−0.0986556	−	0.995122i	$-0.531454\pi$
$197$	−7657.45			−0.197311			−0.0986556	+	0.995122i	$0.531454\pi$
$198$		0			0
$199$	50335.0			1.27105			0.635527	−	0.772079i	$-0.280783\pi$
$199$	50335.0			1.27105			0.635527	+	0.772079i	$0.280783\pi$
$200$		−	54057.6i		−	1.35144i
$201$		0			0
$202$	16630.0			0.407558
$203$	−13328.7			−0.323442
$204$		0			0
$205$	10388.2			0.247191
$206$	59225.8			1.39565
$207$		0			0
$208$			21480.7i			0.496503i
$209$			28050.2i			0.642161i
$210$		0			0
$211$			34304.2i			0.770518i	0.922809	+	0.385259i	$0.125888\pi$
$211$			34304.2i			0.770518i	−0.922809	+	0.385259i	$0.874112\pi$
$212$	−39494.7			−0.878755
$213$		0			0
$214$			142342.i			3.10818i
$215$			2742.64i			0.0593324i
$216$		0			0
$217$			31252.4i			0.663688i
$218$	65151.9			1.37093
$219$		0			0
$220$		−	30766.1i		−	0.635664i
$221$		−	8480.44i		−	0.173634i
$222$		0			0
$223$	12252.7			0.246390			0.123195	−	0.992383i	$-0.460686\pi$
$223$	12252.7			0.246390			0.123195	+	0.992383i	$0.460686\pi$
$224$			20401.8i			0.406604i
$225$		0			0
$226$	−157153.			−3.07685
$227$			83145.6i			1.61357i	0.590846	+	0.806784i	$0.298794\pi$
$227$			83145.6i			1.61357i	−0.590846	+	0.806784i	$0.701206\pi$
$228$		0			0
$229$		−	84147.1i		−	1.60460i	−0.596918	−	0.802302i	$-0.703608\pi$
$229$		−	84147.1i		−	1.60460i	0.596918	−	0.802302i	$-0.296392\pi$
$230$	2543.88			0.0480884
$231$		0			0
$232$		−	27117.1i		−	0.503811i
$233$		−	10889.2i		−	0.200578i	−0.994958	−	0.100289i	$-0.968023\pi$
$233$		−	10889.2i		−	0.200578i	0.994958	−	0.100289i	$-0.0319767\pi$
$234$		0			0
$235$		−	10853.5i		−	0.196533i
$236$	102216.	−	17575.7i	1.83524	−	0.315564i
$237$		0			0
$238$			19002.5i			0.335472i
$239$	−62954.4			−1.10212			−0.551061	−	0.834465i	$-0.685777\pi$
$239$	−62954.4			−1.10212			−0.551061	+	0.834465i	$0.685777\pi$
$240$		0			0
$241$	−63118.5			−1.08673			−0.543366	−	0.839496i	$-0.682851\pi$
$241$	−63118.5			−1.08673			−0.543366	+	0.839496i	$0.682851\pi$
$242$			58036.9i			0.990999i
$243$		0			0
$244$			81402.1i			1.36728i
$245$	2003.25			0.0333736
$246$		0			0
$247$		−	25510.2i		−	0.418138i
$248$	−63582.7			−1.03380
$249$		0			0
$250$		−	55219.1i		−	0.883505i
$251$	66048.7			1.04838			0.524188	−	0.851603i	$-0.324369\pi$
$251$	66048.7			1.04838			0.524188	+	0.851603i	$0.324369\pi$
$252$		0			0
$253$	−8452.12			−0.132046
$254$		−	22049.6i		−	0.341770i
$255$		0			0
$256$	−115404.			−1.76093
$257$	−91076.3			−1.37892			−0.689460	−	0.724323i	$-0.742152\pi$
$257$	−91076.3			−1.37892			−0.689460	+	0.724323i	$0.742152\pi$
$258$		0			0
$259$			65255.8i			0.972791i
$260$			27980.2i			0.413908i
$261$		0			0
$262$	161598.			2.35414
$263$	83369.8			1.20531			0.602653	−	0.798003i	$-0.294110\pi$
$263$	83369.8			1.20531			0.602653	+	0.798003i	$0.294110\pi$
$264$		0			0
$265$	−8983.12			−0.127919
$266$			57161.9i			0.807873i
$267$		0			0
$268$		−	66308.7i		−	0.923211i
$269$			13600.6i			0.187955i	0.995574	+	0.0939775i	$0.0299582\pi$
$269$			13600.6i			0.187955i	−0.995574	+	0.0939775i	$0.970042\pi$
$270$		0			0
$271$	−374.789			−0.00510327			−0.00255164	−	0.999997i	$-0.500812\pi$
$271$	−374.789			−0.00510327			−0.00255164	+	0.999997i	$0.500812\pi$
$272$	−9486.40			−0.128222
$273$		0			0
$274$			175892.i			2.34286i
$275$			88234.7i			1.16674i
$276$		0			0
$277$	−34517.8			−0.449866			−0.224933	−	0.974374i	$-0.572216\pi$
$277$	−34517.8			−0.449866			−0.224933	+	0.974374i	$0.572216\pi$
$278$		−	69680.1i		−	0.901611i
$279$		0			0
$280$		−	29028.0i		−	0.370255i
$281$	−100630.			−1.27442			−0.637210	−	0.770690i	$-0.719912\pi$
$281$	−100630.			−1.27442			−0.637210	+	0.770690i	$0.719912\pi$
$282$		0			0
$283$		−	85403.6i		−	1.06636i	−0.846002	−	0.533179i	$-0.820997\pi$
$283$		−	85403.6i		−	1.06636i	0.846002	−	0.533179i	$-0.179003\pi$
$284$	−183448.			−2.27444
$285$		0			0
$286$		−	142888.i		−	1.74688i
$287$	−70336.4			−0.853918
$288$		0			0
$289$	−79775.8			−0.955159
$290$		−	13321.6i		−	0.158402i
$291$		0			0
$292$		−	83928.7i		−	0.984339i
$293$	−133357.			−1.55339			−0.776696	−	0.629876i	$-0.783106\pi$
$293$	−133357.			−1.55339			−0.776696	+	0.629876i	$0.783106\pi$
$294$		0			0
$295$	23249.0	−	3997.60i	0.267153	−	0.0459362i
$296$	−132762.			−1.51527
$297$		0			0
$298$	234635.			2.64216
$299$	7686.75			0.0859806
$300$		0			0
$301$		−	18569.9i		−	0.204963i
$302$	54818.6			0.601055
$303$		0			0
$304$	−28536.3			−0.308781
$305$			18515.0i			0.199032i
$306$		0			0
$307$	20011.7			0.212328			0.106164	−	0.994349i	$-0.466143\pi$
$307$	20011.7			0.212328			0.106164	+	0.994349i	$0.466143\pi$
$308$			208311.i			2.19589i
$309$		0			0
$310$	−31235.8			−0.325034
$311$	−32402.1			−0.335006			−0.167503	−	0.985872i	$-0.553570\pi$
$311$	−32402.1			−0.335006			−0.167503	+	0.985872i	$0.553570\pi$
$312$		0			0
$313$			191288.i			1.95254i	0.216568	+	0.976268i	$0.430514\pi$
$313$			191288.i			1.95254i	−0.216568	+	0.976268i	$0.569486\pi$
$314$	−238546.			−2.41943
$315$		0			0
$316$	−230848.			−2.31181
$317$	127089.			1.26470			0.632351	−	0.774682i	$-0.282090\pi$
$317$	127089.			1.26470			0.632351	+	0.774682i	$0.282090\pi$
$318$		0			0
$319$			44261.5i			0.434956i
$320$	−37198.8			−0.363270
$321$		0			0
$322$	−17224.1			−0.166121
$323$	11265.9			0.107985
$324$		0			0
$325$		−	80244.8i		−	0.759714i
$326$			82010.3i			0.771673i
$327$		0			0
$328$		−	143099.i		−	1.33011i
$329$			73487.0i			0.678920i
$330$		0			0
$331$	−190707.			−1.74064			−0.870321	−	0.492484i	$-0.836089\pi$
$331$	−190707.			−1.74064			−0.870321	+	0.492484i	$0.836089\pi$
$332$		−	19331.0i		−	0.175379i
$333$		0			0
$334$			168416.i			1.50970i
$335$		−	15082.0i		−	0.134391i
$336$		0			0
$337$			121592.i			1.07065i	0.844648	+	0.535323i	$0.179810\pi$
$337$			121592.i			1.07065i	−0.844648	+	0.535323i	$0.820190\pi$
$338$		−	63328.5i		−	0.554326i
$339$		0			0
$340$	−12356.7			−0.106892
$341$	103782.			0.892509
$342$		0			0
$343$	−123733.			−1.05171
$344$	37780.2			0.319262
$345$		0			0
$346$	41614.1			0.347607
$347$			11383.8i			0.0945426i	0.998882	+	0.0472713i	$0.0150525\pi$
$347$			11383.8i			0.0945426i	−0.998882	+	0.0472713i	$0.984947\pi$
$348$		0			0
$349$			133595.i			1.09683i	0.836207	+	0.548414i	$0.184768\pi$
$349$			133595.i			1.09683i	−0.836207	+	0.548414i	$0.815232\pi$
$350$			179808.i			1.46782i
$351$		0			0
$352$	67749.5			0.546790
$353$			152806.i			1.22628i	0.789973	+	0.613141i	$0.210094\pi$
$353$			152806.i			1.22628i	−0.789973	+	0.613141i	$0.789906\pi$
$354$		0			0
$355$	−41725.3			−0.331088
$356$		−	171356.i		−	1.35207i
$357$		0			0
$358$	118622.			0.925551
$359$	84352.3			0.654497			0.327249	−	0.944938i	$-0.393879\pi$
$359$	84352.3			0.654497			0.327249	+	0.944938i	$0.393879\pi$
$360$		0			0
$361$	−96431.7			−0.739955
$362$		−	140624.i		−	1.07310i
$363$		0			0
$364$		−	189448.i		−	1.42984i
$365$		−	19089.6i		−	0.143289i
$366$		0			0
$367$		−	22612.6i		−	0.167888i	−0.996470	−	0.0839439i	$-0.973248\pi$
$367$		−	22612.6i		−	0.167888i	0.996470	−	0.0839439i	$-0.0267517\pi$
$368$		−	8598.57i		−	0.0634937i
$369$		0			0
$370$	−65221.0			−0.476414
$371$	60822.8			0.441894
$372$		0			0
$373$	−142008.			−1.02069			−0.510347	−	0.859968i	$-0.670483\pi$
$373$	−142008.			−1.02069			−0.510347	+	0.859968i	$0.670483\pi$
$374$	63102.8			0.451134
$375$		0			0
$376$	−149509.			−1.05752
$377$		−	40253.5i		−	0.283218i
$378$		0			0
$379$	−71134.7			−0.495226			−0.247613	−	0.968859i	$-0.579646\pi$
$379$	−71134.7			−0.495226			−0.247613	+	0.968859i	$0.579646\pi$
$380$	−37170.6			−0.257414
$381$		0			0
$382$	92497.6			0.633875
$383$	−5389.15			−0.0367386			−0.0183693	−	0.999831i	$-0.505847\pi$
$383$	−5389.15			−0.0367386			−0.0183693	+	0.999831i	$0.505847\pi$
$384$		0			0
$385$			47380.5i			0.319653i
$386$		−	166514.i		−	1.11757i
$387$		0			0
$388$			45010.1i			0.298983i
$389$	−227686.			−1.50466			−0.752329	−	0.658787i	$-0.771070\pi$
$389$	−227686.			−1.50466			−0.752329	+	0.658787i	$0.771070\pi$
$390$		0			0
$391$			3394.66i			0.0222046i
$392$		−	27594.9i		−	0.179580i
$393$		0			0
$394$			51819.4i			0.333810i
$395$	−52506.7			−0.336527
$396$		0			0
$397$			66414.0i			0.421385i	0.977552	+	0.210692i	$0.0675718\pi$
$397$			66414.0i			0.421385i	−0.977552	+	0.210692i	$0.932428\pi$
$398$		−	340626.i		−	2.15036i
$399$		0			0
$400$	−89763.6			−0.561022
$401$		−	185618.i		−	1.15434i	−0.816625	−	0.577168i	$-0.804158\pi$
$401$		−	185618.i		−	1.15434i	0.816625	−	0.577168i	$-0.195842\pi$
$402$		0			0
$403$	−94384.1			−0.581151
$404$		−	73219.1i		−	0.448602i
$405$		0			0
$406$			90197.9i			0.547198i
$407$	216699.			1.30818
$408$		0			0
$409$		−	69799.6i		−	0.417260i	−0.977995	−	0.208630i	$-0.933100\pi$
$409$		−	69799.6i		−	0.417260i	0.977995	−	0.208630i	$-0.0669004\pi$
$410$		−	70298.9i		−	0.418197i
$411$		0			0
$412$		−	260761.i		−	1.53620i
$413$	−157414.	+	27066.9i	−0.922877	+	0.158686i
$414$		0			0
$415$		−	4396.86i		−	0.0255297i
$416$	−61614.6			−0.356038
$417$		0			0
$418$	189821.			1.08641
$419$		−	68851.9i		−	0.392182i	−0.980586	−	0.196091i	$-0.937175\pi$
$419$		−	68851.9i		−	0.392182i	0.980586	−	0.196091i	$-0.0628248\pi$
$420$		0			0
$421$			52611.8i			0.296837i	0.988925	+	0.148419i	$0.0474183\pi$
$421$			52611.8i			0.296837i	−0.988925	+	0.148419i	$0.952582\pi$
$422$	232143.			1.30356
$423$		0			0
$424$			123743.i			0.688320i
$425$	35438.1			0.196197
$426$		0			0
$427$		−	125361.i		−	0.687554i
$428$	626708.			3.42120
$429$		0			0
$430$	18560.0			0.100378
$431$		−	127282.i		−	0.685193i	−0.939483	−	0.342596i	$-0.888694\pi$
$431$		−	127282.i		−	0.685193i	0.939483	−	0.342596i	$-0.111306\pi$
$432$		0			0
$433$	67016.8			0.357444			0.178722	−	0.983900i	$-0.442804\pi$
$433$	67016.8			0.357444			0.178722	+	0.983900i	$0.442804\pi$
$434$	211491.			1.12282
$435$		0			0
$436$		−	286853.i		−	1.50899i
$437$			10211.5i			0.0534723i
$438$		0			0
$439$	131134.			0.680434			0.340217	−	0.940347i	$-0.389499\pi$
$439$	131134.			0.680434			0.340217	+	0.940347i	$0.389499\pi$
$440$	−96395.2			−0.497909
$441$		0			0
$442$	−57388.7			−0.293753
$443$		−	238014.i		−	1.21282i	−0.795154	−	0.606408i	$-0.792610\pi$
$443$		−	238014.i		−	1.21282i	0.795154	−	0.606408i	$-0.207390\pi$
$444$		0			0
$445$		−	38975.1i		−	0.196819i
$446$		−	82916.3i		−	0.416841i
$447$		0			0
$448$	251865.			1.25491
$449$	140599.			0.697411			0.348706	−	0.937232i	$-0.386621\pi$
$449$	140599.			0.697411			0.348706	+	0.937232i	$0.386621\pi$
$450$		0			0
$451$			233570.i			1.14832i
$452$			691919.i			3.38671i
$453$		0			0
$454$	562661.			2.72983
$455$		−	43090.1i		−	0.208140i
$456$		0			0
$457$		−	381250.i		−	1.82548i	−0.408539	−	0.912741i	$-0.633962\pi$
$457$		−	381250.i		−	1.82548i	0.408539	−	0.912741i	$-0.366038\pi$
$458$	−569439.			−2.71466
$459$		0			0
$460$		−	11200.3i		−	0.0529313i
$461$	66518.7			0.312998			0.156499	−	0.987678i	$-0.449979\pi$
$461$	66518.7			0.312998			0.156499	+	0.987678i	$0.449979\pi$
$462$		0			0
$463$		−	362433.i		−	1.69070i	−0.534216	−	0.845348i	$-0.679393\pi$
$463$		−	362433.i		−	1.69070i	0.534216	−	0.845348i	$-0.320607\pi$
$464$	−45028.5			−0.209147
$465$		0			0
$466$	−73689.1			−0.339337
$467$		−	184210.i		−	0.844656i	−0.906443	−	0.422328i	$-0.861213\pi$
$467$		−	184210.i		−	0.844656i	0.906443	−	0.422328i	$-0.138787\pi$
$468$		0			0
$469$			102117.i			0.464250i
$470$	−73447.9			−0.332494
$471$		0			0
$472$	−55067.4	−	320258.i	−0.247178	−	1.43753i
$473$	−61666.1			−0.275629
$474$		0			0
$475$	106602.			0.472475
$476$	83664.8			0.369257
$477$		0			0
$478$			426024.i			1.86457i
$479$	348835.			1.52037			0.760185	−	0.649707i	$-0.225108\pi$
$479$	348835.			1.52037			0.760185	+	0.649707i	$0.225108\pi$
$480$		0			0
$481$	−197076.			−0.851813
$482$			427135.i			1.83853i
$483$		0			0
$484$	255527.			1.09080
$485$			10237.6i			0.0435225i
$486$		0			0
$487$	−451419.			−1.90336			−0.951681	−	0.307087i	$-0.900646\pi$
$487$	−451419.			−1.90336			−0.951681	+	0.307087i	$0.900646\pi$
$488$	255046.			1.07097
$489$		0			0
$490$		−	13556.3i		−	0.0564612i
$491$	64824.7			0.268892			0.134446	−	0.990921i	$-0.457075\pi$
$491$	64824.7			0.268892			0.134446	+	0.990921i	$0.457075\pi$
$492$		0			0
$493$	17776.9			0.0731414
$494$	−172632.			−0.707405
$495$		0			0
$496$			105580.i			0.429160i
$497$	282513.			1.14374
$498$		0			0
$499$	−151617.			−0.608900			−0.304450	−	0.952528i	$-0.598473\pi$
$499$	−151617.			−0.608900			−0.304450	+	0.952528i	$0.598473\pi$
$500$	−243120.			−0.972481
$501$		0			0
$502$		−	446964.i		−	1.77364i
$503$		−	154559.i		−	0.610883i	−0.952211	−	0.305442i	$-0.901196\pi$
$503$		−	154559.i		−	0.610883i	0.952211	−	0.305442i	$-0.0988040\pi$
$504$		0			0
$505$		−	16653.8i		−	0.0653024i
$506$			57197.0i			0.223394i
$507$		0			0
$508$	−97080.8			−0.376189
$509$			6251.16i			0.0241282i	0.999927	+	0.0120641i	$0.00384022\pi$
$509$			6251.16i			0.0241282i	−0.999927	+	0.0120641i	$0.996160\pi$
$510$		0			0
$511$			129252.i			0.494989i
$512$			300454.i			1.14614i
$513$		0			0
$514$			616330.i			2.33285i
$515$		−	59310.4i		−	0.223623i
$516$		0			0
$517$	244033.			0.912993
$518$	441598.			1.64576
$519$		0			0
$520$	87666.4			0.324210
$521$	−351023.			−1.29318			−0.646591	−	0.762836i	$-0.723806\pi$
$521$	−351023.			−1.29318			−0.646591	+	0.762836i	$0.723806\pi$
$522$		0			0
$523$	−58841.2			−0.215119			−0.107559	−	0.994199i	$-0.534304\pi$
$523$	−58841.2			−0.215119			−0.107559	+	0.994199i	$0.534304\pi$
$524$		−	711489.i		−	2.59123i
$525$		0			0
$526$		−	564179.i		−	2.03913i
$527$		−	41682.3i		−	0.150083i
$528$		0			0
$529$	276764.			0.989005
$530$			60790.4i			0.216413i
$531$		0			0
$532$	251674.			0.889233
$533$		−	212420.i		−	0.747723i
$534$		0			0
$535$	142545.			0.498019
$536$	−207756.			−0.723142
$537$		0			0
$538$	92037.8			0.317981
$539$			45041.4i			0.155037i
$540$		0			0
$541$		−	451192.i		−	1.54158i	−0.637088	−	0.770791i	$-0.719861\pi$
$541$		−	451192.i		−	1.54158i	0.637088	−	0.770791i	$-0.280139\pi$
$542$			2536.27i			0.00863369i
$543$		0			0
$544$		−	27210.5i		−	0.0919472i
$545$		−	65244.9i		−	0.219661i
$546$		0			0
$547$	−341064.			−1.13988			−0.569942	−	0.821685i	$-0.693034\pi$
$547$	−341064.			−1.13988			−0.569942	+	0.821685i	$0.693034\pi$
$548$	774425.			2.57880
$549$		0			0
$550$	597100.			1.97389
$551$	53475.2			0.176137
$552$		0			0
$553$	355512.			1.16253
$554$			233588.i			0.761082i
$555$		0			0
$556$	−306790.			−0.992410
$557$	−333403.			−1.07463			−0.537315	−	0.843382i	$-0.680561\pi$
$557$	−333403.			−1.07463			−0.537315	+	0.843382i	$0.680561\pi$
$558$		0			0
$559$	56082.1			0.179474
$560$	−48201.5			−0.153704
$561$		0			0
$562$			680978.i			2.15606i
$563$			48980.2i			0.154527i	0.997011	+	0.0772634i	$0.0246182\pi$
$563$			48980.2i			0.154527i	−0.997011	+	0.0772634i	$0.975382\pi$
$564$		0			0
$565$			157378.i			0.492999i
$566$	−577942.			−1.80406
$567$		0			0
$568$			574771.i			1.78155i
$569$			47923.5i			0.148021i	0.997257	+	0.0740106i	$0.0235799\pi$
$569$			47923.5i			0.148021i	−0.997257	+	0.0740106i	$0.976420\pi$
$570$		0			0
$571$		−	518303.i		−	1.58969i	−0.606815	−	0.794843i	$-0.707553\pi$
$571$		−	518303.i		−	1.58969i	0.606815	−	0.794843i	$-0.292447\pi$
$572$	−629112.			−1.92281
$573$		0			0
$574$			475979.i			1.44465i
$575$			32121.4i			0.0971536i
$576$		0			0
$577$	92997.1			0.279330			0.139665	−	0.990199i	$-0.455397\pi$
$577$	92997.1			0.279330			0.139665	+	0.990199i	$0.455397\pi$
$578$			539858.i			1.61593i
$579$		0			0
$580$	−58652.9			−0.174355
$581$			29770.2i			0.0881921i
$582$		0			0
$583$		−	201978.i		−	0.594247i
$584$	−262962.			−0.771022
$585$		0			0
$586$			902452.i			2.62802i
$587$			333846.i			0.968881i	0.874824	+	0.484440i	$0.160977\pi$
$587$			333846.i			0.968881i	−0.874824	+	0.484440i	$0.839023\pi$
$588$		0			0
$589$		−	125386.i		−	0.361424i
$590$	−27052.5	−	157330.i	−0.0777147	−	0.451969i
$591$		0			0
$592$			220454.i			0.629034i
$593$	−122051.			−0.347081			−0.173541	−	0.984827i	$-0.555521\pi$
$593$	−122051.			−0.347081			−0.173541	+	0.984827i	$0.555521\pi$
$594$		0			0
$595$	19029.6			0.0537522
$596$		−	1.03306e6i		−	2.90825i
$597$		0			0
$598$		−	52017.7i		−	0.145462i
$599$	118410.			0.330015			0.165008	−	0.986292i	$-0.447235\pi$
$599$	118410.			0.330015			0.165008	+	0.986292i	$0.447235\pi$
$600$		0			0
$601$		−	316761.i		−	0.876966i	−0.898740	−	0.438483i	$-0.855516\pi$
$601$		−	316761.i		−	0.876966i	0.898740	−	0.438483i	$-0.144484\pi$
$602$	−125666.			−0.346756
$603$		0			0
$604$		−	241357.i		−	0.661586i
$605$	58119.8			0.158786
$606$		0			0
$607$	−635965.			−1.72606			−0.863030	−	0.505153i	$-0.831436\pi$
$607$	−635965.			−1.72606			−0.863030	+	0.505153i	$0.831436\pi$
$608$		−	81852.6i		−	0.221424i
$609$		0			0
$610$	125294.			0.336722
$611$	−221935.			−0.594489
$612$		0			0
$613$			664506.i			1.76839i	0.467119	+	0.884195i	$0.345292\pi$
$613$			664506.i			1.76839i	−0.467119	+	0.884195i	$0.654708\pi$
$614$		−	135423.i		−	0.359215i
$615$		0			0
$616$	652672.			1.72002
$617$	−214690.			−0.563952			−0.281976	−	0.959422i	$-0.590990\pi$
$617$	−214690.			−0.563952			−0.281976	+	0.959422i	$0.590990\pi$
$618$		0			0
$619$	−638321.			−1.66594			−0.832968	−	0.553322i	$-0.813360\pi$
$619$	−638321.			−1.66594			−0.832968	+	0.553322i	$0.813360\pi$
$620$			137526.i			0.357767i
$621$		0			0
$622$			219271.i			0.566761i
$623$			263892.i			0.679908i
$624$		0			0
$625$	306624.			0.784956
$626$	1.29448e6			3.30329
$627$		0			0
$628$			1.05028e6i			2.66308i
$629$		−	87033.7i		−	0.219981i
$630$		0			0
$631$	−375421.			−0.942888			−0.471444	−	0.881896i	$-0.656267\pi$
$631$	−375421.			−0.942888			−0.471444	+	0.881896i	$0.656267\pi$
$632$			723285.i			1.81082i
$633$		0			0
$634$		−	860032.i		−	2.13962i
$635$	−22081.1			−0.0547613
$636$		0			0
$637$		−	40962.8i		−	0.100951i
$638$	299526.			0.735857
$639$		0			0
$640$			203520.i			0.496875i
$641$	−704608.			−1.71487			−0.857435	−	0.514592i	$-0.827943\pi$
$641$	−704608.			−1.71487			−0.857435	+	0.514592i	$0.827943\pi$
$642$		0			0
$643$	−448900.			−1.08574			−0.542872	−	0.839816i	$-0.682663\pi$
$643$	−448900.			−1.08574			−0.542872	+	0.839816i	$0.682663\pi$
$644$			75834.6i			0.182850i
$645$		0			0
$646$		−	76238.6i		−	0.182688i
$647$	211597.			0.505477			0.252738	−	0.967535i	$-0.418669\pi$
$647$	211597.			0.505477			0.252738	+	0.967535i	$0.418669\pi$
$648$		0			0
$649$	89882.8	+	522736.i	0.213397	+	1.24106i
$650$	−543031.			−1.28528
$651$		0			0
$652$	361078.			0.849387
$653$	171313.			0.401757			0.200878	−	0.979616i	$-0.435620\pi$
$653$	171313.			0.401757			0.200878	+	0.979616i	$0.435620\pi$
$654$		0			0
$655$		−	161829.i		−	0.377201i
$656$	−237618.			−0.552168
$657$		0			0
$658$	497300.			1.14859
$659$		−	123614.i		−	0.284641i	−0.989821	−	0.142321i	$-0.954544\pi$
$659$		−	123614.i		−	0.284641i	0.989821	−	0.142321i	$-0.0454564\pi$
$660$		0			0
$661$	759419.			1.73811			0.869057	−	0.494712i	$-0.164726\pi$
$661$	759419.			1.73811			0.869057	+	0.494712i	$0.164726\pi$
$662$			1.29055e6i			2.94481i
$663$		0			0
$664$	−60567.2			−0.137373
$665$	57243.5			0.129444
$666$		0			0
$667$			16113.2i			0.0362185i
$668$	741507.			1.66174
$669$		0			0
$670$	−102063.			−0.227361
$671$	−416295.			−0.924604
$672$		0			0
$673$		−	366990.i		−	0.810260i	−0.914259	−	0.405130i	$-0.867226\pi$
$673$		−	366990.i		−	0.810260i	0.914259	−	0.405130i	$-0.132774\pi$
$674$	822836.			1.81131
$675$		0			0
$676$	−278825.			−0.610152
$677$	−411628.			−0.898105			−0.449053	−	0.893505i	$-0.648238\pi$
$677$	−411628.			−0.898105			−0.449053	+	0.893505i	$0.648238\pi$
$678$		0			0
$679$		−	69316.5i		−	0.150348i
$680$			38715.6i			0.0837275i
$681$		0			0
$682$		−	702311.i		−	1.50994i
$683$			654877.i			1.40384i	0.712254	+	0.701921i	$0.247674\pi$
$683$			654877.i			1.40384i	−0.712254	+	0.701921i	$0.752326\pi$
$684$		0			0
$685$	176144.			0.375393
$686$			837321.i			1.77928i
$687$		0			0
$688$		−	62734.6i		−	0.132535i
$689$			183688.i			0.386940i
$690$		0			0
$691$		−	628200.i		−	1.31565i	−0.753169	−	0.657827i	$-0.771476\pi$
$691$		−	628200.i		−	1.31565i	0.753169	−	0.657827i	$-0.228524\pi$
$692$		−	183220.i		−	0.382614i
$693$		0			0
$694$	77036.1			0.159947
$695$	−69779.6			−0.144464
$696$		0			0
$697$	93809.8			0.193100
$698$	904060.			1.85561
$699$		0			0
$700$	791665.			1.61564
$701$		−	517988.i		−	1.05410i	−0.849833	−	0.527052i	$-0.823297\pi$
$701$		−	517988.i		−	1.05410i	0.849833	−	0.527052i	$-0.176703\pi$
$702$		0			0
$703$		−	261808.i		−	0.529752i
$704$		−	836386.i		−	1.68757i
$705$		0			0
$706$	1.03407e6			2.07462
$707$			112759.i			0.225586i
$708$		0			0
$709$	417989.			0.831519			0.415759	−	0.909475i	$-0.363516\pi$
$709$	417989.			0.831519			0.415759	+	0.909475i	$0.363516\pi$
$710$			282363.i			0.560133i
$711$		0			0
$712$	−536886.			−1.05906
$713$	37781.3			0.0743186
$714$		0			0
$715$	−143092.			−0.279900
$716$		−	522274.i		−	1.01876i
$717$		0			0
$718$		−	570827.i		−	1.10728i
$719$			88171.5i			0.170557i	0.996357	+	0.0852787i	$0.0271781\pi$
$719$			88171.5i			0.170557i	−0.996357	+	0.0852787i	$0.972822\pi$
$720$		0			0
$721$			401578.i			0.772502i
$722$			652571.i			1.25185i
$723$		0			0
$724$	−619144.			−1.18117
$725$	168212.			0.320022
$726$		0			0
$727$	765157.			1.44771			0.723855	−	0.689952i	$-0.242368\pi$
$727$	765157.			1.44771			0.723855	+	0.689952i	$0.242368\pi$
$728$	−593571.			−1.11998
$729$		0			0
$730$	−129183.			−0.242415
$731$			24767.2i			0.0463492i
$732$		0			0
$733$	45126.1			0.0839885			0.0419943	−	0.999118i	$-0.486629\pi$
$733$	45126.1			0.0839885			0.0419943	+	0.999118i	$0.486629\pi$
$734$	−153024.			−0.284032
$735$		0			0
$736$	24663.9			0.0455308
$737$	339106.			0.624311
$738$		0			0
$739$			1.03826e6i			1.90116i	0.310476	+	0.950581i	$0.399512\pi$
$739$			1.03826e6i			1.90116i	−0.310476	+	0.950581i	$0.600488\pi$
$740$			287157.i			0.524392i
$741$		0			0
$742$		−	411599.i		−	0.747595i
$743$	432123.			0.782763			0.391381	−	0.920229i	$-0.371997\pi$
$743$	432123.			0.782763			0.391381	+	0.920229i	$0.371997\pi$
$744$		0			0
$745$		−	234970.i		−	0.423350i
$746$			960995.i			1.72681i
$747$		0			0
$748$		−	277831.i		−	0.496567i
$749$	−965145.			−1.72040
$750$		0			0
$751$			586786.i			1.04040i	0.854045	+	0.520200i	$0.174142\pi$
$751$			586786.i			1.04040i	−0.854045	+	0.520200i	$0.825858\pi$
$752$			248262.i			0.439009i
$753$		0			0
$754$	−272403.			−0.479148
$755$		−	54896.9i		−	0.0963061i
$756$		0			0
$757$	−69647.3			−0.121538			−0.0607690	−	0.998152i	$-0.519355\pi$
$757$	−69647.3			−0.121538			−0.0607690	+	0.998152i	$0.519355\pi$
$758$			481382.i			0.837821i
$759$		0			0
$760$			116461.i			0.201630i
$761$	533970.			0.922035			0.461017	−	0.887391i	$-0.347485\pi$
$761$	533970.			0.922035			0.461017	+	0.887391i	$0.347485\pi$
$762$		0			0
$763$			441760.i			0.758817i
$764$		−	407252.i		−	0.697712i
$765$		0			0
$766$			36469.4i			0.0621542i
$767$	−81743.7	−	475400.i	−0.138952	−	0.808107i
$768$		0			0
$769$			316009.i			0.534376i	0.963644	+	0.267188i	$0.0860945\pi$
$769$			316009.i			0.534376i	−0.963644	+	0.267188i	$0.913905\pi$
$770$	320633.			0.540787
$771$		0			0
$772$	−733131.			−1.23012
$773$		−	42404.6i		−	0.0709666i	−0.999370	−	0.0354833i	$-0.988703\pi$
$773$		−	42404.6i		−	0.0709666i	0.999370	−	0.0354833i	$-0.0112971\pi$
$774$		0			0
$775$		−	394412.i		−	0.656670i
$776$	141024.			0.234190
$777$		0			0
$778$			1.54080e6i			2.54558i
$779$	282192.			0.465017
$780$		0			0
$781$		−	938160.i		−	1.53807i
$782$	22972.3			0.0375656
$783$		0			0
$784$	−45821.9			−0.0745488
$785$			238886.i			0.387661i
$786$		0			0
$787$	−952711.			−1.53820			−0.769098	−	0.639131i	$-0.779294\pi$
$787$	−952711.			−1.53820			−0.769098	+	0.639131i	$0.779294\pi$
$788$	228152.			0.367428
$789$		0			0
$790$			355322.i			0.569335i
$791$		−	1.06557e6i		−	1.70306i
$792$		0			0
$793$	378598.			0.602049
$794$	449436.			0.712897
$795$		0			0
$796$	−1.49972e6			−2.36692
$797$		−	821853.i		−	1.29383i	−0.762562	−	0.646915i	$-0.776059\pi$
$797$		−	821853.i		−	1.29383i	0.762562	−	0.646915i	$-0.223941\pi$
$798$		0			0
$799$		−	98012.0i		−	0.153527i
$800$		−	257475.i		−	0.402305i
$801$		0			0
$802$	−1.25611e6			−1.95290
$803$	429215.			0.665647
$804$		0			0
$805$			17248.7i			0.0266173i
$806$			638715.i			0.983188i
$807$		0			0
$808$	−229407.			−0.351386
$809$			460917.i			0.704249i	0.935953	+	0.352124i	$0.114541\pi$
$809$			460917.i			0.704249i	−0.935953	+	0.352124i	$0.885459\pi$
$810$		0			0
$811$		−	690476.i		−	1.04980i	−0.851164	−	0.524901i	$-0.824102\pi$
$811$		−	690476.i		−	1.04980i	0.851164	−	0.524901i	$-0.175898\pi$
$812$	397126.			0.602305
$813$		0			0
$814$		−	1.46644e6i		−	2.21318i
$815$	82127.4			0.123644
$816$		0			0
$817$			74502.8i			0.111617i
$818$	−472347.			−0.705918
$819$		0			0
$820$	−309514.			−0.460313
$821$		−	936452.i		−	1.38931i	−0.719343	−	0.694655i	$-0.755557\pi$
$821$		−	936452.i		−	1.38931i	0.719343	−	0.694655i	$-0.244443\pi$
$822$		0			0
$823$			348374.i			0.514336i	0.966367	+	0.257168i	$0.0827893\pi$
$823$			348374.i			0.514336i	−0.966367	+	0.257168i	$0.917211\pi$
$824$	−817007.			−1.20329
$825$		0			0
$826$	183167.	+	1.06525e6i	0.268464	+	1.56132i
$827$	−441747.			−0.645896			−0.322948	−	0.946417i	$-0.604674\pi$
$827$	−441747.			−0.645896			−0.322948	+	0.946417i	$0.604674\pi$
$828$		0			0
$829$	104727.			0.152387			0.0761936	−	0.997093i	$-0.475723\pi$
$829$	104727.			0.152387			0.0761936	+	0.997093i	$0.475723\pi$
$830$	−29754.4			−0.0431911
$831$		0			0
$832$			760649.i			1.09885i
$833$	18090.2			0.0260707
$834$		0			0
$835$	168657.			0.241897
$836$		−	835750.i		−	1.19582i
$837$		0			0
$838$	−465933.			−0.663492
$839$		−	750204.i		−	1.06575i	−0.846194	−	0.532875i	$-0.821112\pi$
$839$		−	750204.i		−	1.06575i	0.846194	−	0.532875i	$-0.178888\pi$
$840$		0			0
$841$	−622900.			−0.880697
$842$	356033.			0.502188
$843$		0			0
$844$		−	1.02209e6i		−	1.43484i
$845$	−63418.9			−0.0888189
$846$		0			0
$847$	−393517.			−0.548525
$848$	205478.			0.285742
$849$		0			0
$850$		−	239816.i		−	0.331925i
$851$	78888.2			0.108931
$852$		0			0
$853$	389367.			0.535132			0.267566	−	0.963540i	$-0.413781\pi$
$853$	389367.			0.535132			0.267566	+	0.963540i	$0.413781\pi$
$854$	−848341.			−1.16320
$855$		0			0
$856$		−	1.96358e6i		−	2.67979i
$857$		−	608191.i		−	0.828091i	−0.910256	−	0.414046i	$-0.864115\pi$
$857$		−	608191.i		−	0.828091i	0.910256	−	0.414046i	$-0.135885\pi$
$858$		0			0
$859$			667593.i			0.904743i	0.891830	+	0.452371i	$0.149422\pi$
$859$			667593.i			0.904743i	−0.891830	+	0.452371i	$0.850578\pi$
$860$		−	81716.4i		−	0.110487i
$861$		0			0
$862$	−861341.			−1.15921
$863$			507853.i			0.681894i	0.940083	+	0.340947i	$0.110748\pi$
$863$			507853.i			0.681894i	−0.940083	+	0.340947i	$0.889252\pi$
$864$		0			0
$865$		−	41673.6i		−	0.0556966i
$866$		−	453515.i		−	0.604722i
$867$		0			0
$868$		−	931158.i		−	1.23590i
$869$		−	1.18057e6i		−	1.56334i
$870$		0			0
$871$	−308399.			−0.406515
$872$	−898756.			−1.18198
$873$		0			0
$874$	69103.4			0.0904642
$875$	374411.			0.489026
$876$		0			0
$877$	501363.			0.651858			0.325929	−	0.945394i	$-0.394323\pi$
$877$	501363.			0.651858			0.325929	+	0.945394i	$0.394323\pi$
$878$		−	887408.i		−	1.15116i
$879$		0			0
$880$			160066.i			0.206697i
$881$		−	1.42120e6i		−	1.83106i	−0.402248	−	0.915531i	$-0.631771\pi$
$881$		−	1.42120e6i		−	1.83106i	0.402248	−	0.915531i	$-0.368229\pi$
$882$		0			0
$883$	352600.			0.452231			0.226116	−	0.974100i	$-0.427397\pi$
$883$	352600.			0.452231			0.226116	+	0.974100i	$0.427397\pi$
$884$			252673.i			0.323336i
$885$		0			0
$886$	−1.61068e6			−2.05184
$887$		−	1.03662e6i		−	1.31756i	−0.752336	−	0.658780i	$-0.771073\pi$
$887$		−	1.03662e6i		−	1.31756i	0.752336	−	0.658780i	$-0.228927\pi$
$888$		0			0
$889$	149507.			0.189172
$890$	−263752.			−0.332978
$891$		0			0
$892$	−365067.			−0.458820
$893$		−	294832.i		−	0.369719i
$894$		0			0
$895$		−	118792.i		−	0.148300i
$896$		−	1.37799e6i		−	1.71645i
$897$		0			0
$898$		−	951458.i		−	1.17988i
$899$		−	197851.i		−	0.244804i
$900$		0			0
$901$	−81121.3			−0.0999275
$902$	1.58061e6			1.94273
$903$		0			0
$904$	2.16789e6			2.65278
$905$	−140825.			−0.171942
$906$		0			0
$907$	−1.36132e6			−1.65480			−0.827402	−	0.561610i	$-0.810182\pi$
$907$	−1.36132e6			−1.65480			−0.827402	+	0.561610i	$0.810182\pi$
$908$		−	2.47730e6i		−	3.00474i
$909$		0			0
$910$	−291599.			−0.352130
$911$	−863937.			−1.04099			−0.520493	−	0.853866i	$-0.674252\pi$
$911$	−863937.			−1.04099			−0.520493	+	0.853866i	$0.674252\pi$
$912$		0			0
$913$	98859.8			0.118598
$914$	−2.57999e6			−3.08834
$915$		0			0
$916$			2.50714e6i			2.98805i
$917$			1.09571e6i			1.30304i
$918$		0			0
$919$		−	354665.i		−	0.419940i	−0.977708	−	0.209970i	$-0.932663\pi$
$919$		−	354665.i		−	0.419940i	0.977708	−	0.209970i	$-0.0673367\pi$
$920$	−35092.2			−0.0414605
$921$		0			0
$922$		−	450144.i		−	0.529529i
$923$			853207.i			1.00150i
$924$		0			0
$925$		−	823543.i		−	0.962504i
$926$	−2.45265e6			−2.86031
$927$		0			0
$928$		−	129158.i		−	0.149978i
$929$			163853.i			0.189855i	0.995484	+	0.0949275i	$0.0302619\pi$
$929$			163853.i			0.189855i	−0.995484	+	0.0949275i	$0.969738\pi$
$930$		0			0
$931$	54417.5			0.0627825
$932$			324441.i			0.373511i
$933$		0			0
$934$	−1.24658e6			−1.42899
$935$		−	63192.9i		−	0.0722845i
$936$		0			0
$937$			89081.6i			0.101463i	0.998712	+	0.0507316i	$0.0161553\pi$
$937$			89081.6i			0.101463i	−0.998712	+	0.0507316i	$0.983845\pi$
$938$	691044.			0.785417
$939$		0			0
$940$			323379.i			0.365979i
$941$			1.32463e6i			1.49594i	0.663730	+	0.747972i	$0.268972\pi$
$941$			1.32463e6i			1.49594i	−0.663730	+	0.747972i	$0.731028\pi$
$942$		0			0
$943$			85030.2i			0.0956202i
$944$	−531793.	+	91440.3i	−0.596759	+	0.102611i
$945$		0			0
$946$			417306.i			0.466307i
$947$	−406812.			−0.453622			−0.226811	−	0.973939i	$-0.572830\pi$
$947$	−406812.			−0.453622			−0.226811	+	0.973939i	$0.572830\pi$
$948$		0			0
$949$	−390349.			−0.433431
$950$		−	721396.i		−	0.799331i
$951$		0			0
$952$		−	262135.i		−	0.289235i
$953$	618044.			0.680508			0.340254	−	0.940334i	$-0.389487\pi$
$953$	618044.			0.680508			0.340254	+	0.940334i	$0.389487\pi$
$954$		0			0
$955$		−	92629.7i		−	0.101565i
$956$	1.87571e6			2.05234
$957$		0			0
$958$		−	2.36063e6i		−	2.57216i
$959$	−1.19263e6			−1.29679
$960$		0			0
$961$	459612.			0.497674
$962$			1.33365e6i			1.44109i
$963$		0			0
$964$	1.88060e6			2.02368
$965$	−166751.			−0.179067
$966$		0			0
$967$			1.04471e6i			1.11723i	0.829426	+	0.558617i	$0.188668\pi$
$967$			1.04471e6i			1.11723i	−0.829426	+	0.558617i	$0.811332\pi$
$968$		−	800606.i		−	0.854413i
$969$		0			0
$970$	69279.6			0.0736312
$971$	−1.24646e6			−1.32203			−0.661013	−	0.750375i	$-0.729873\pi$
$971$	−1.24646e6			−1.32203			−0.661013	+	0.750375i	$0.729873\pi$
$972$		0			0
$973$	472463.			0.499048
$974$			3.05483e6i			3.22010i
$975$		0			0
$976$		−	423508.i		−	0.444592i
$977$			1.30541e6i			1.36759i	0.729673	+	0.683796i	$0.239672\pi$
$977$			1.30541e6i			1.36759i	−0.729673	+	0.683796i	$0.760328\pi$
$978$		0			0
$979$	876324.			0.914322
$980$	−59686.3			−0.0621473
$981$		0			0
$982$		−	438681.i		−	0.454910i
$983$			1.73019e6i			1.79055i	0.445510	+	0.895277i	$0.353022\pi$
$983$			1.73019e6i			1.79055i	−0.445510	+	0.895277i	$0.646978\pi$
$984$		0			0
$985$	51893.4			0.0534859
$986$		−	120300.i		−	0.123740i
$987$		0			0
$988$			760071.i			0.778646i
$989$	−22449.2			−0.0229514
$990$		0			0
$991$			88862.9i			0.0904843i	0.998976	+	0.0452421i	$0.0144059\pi$
$991$			88862.9i			0.0904843i	−0.998976	+	0.0452421i	$0.985594\pi$
$992$	−302843.			−0.307747
$993$		0			0
$994$		−	1.91182e6i		−	1.93497i
$995$	−341113.			−0.344549
$996$		0			0
$997$	1.35339e6			1.36155			0.680773	−	0.732494i	$-0.261644\pi$
$997$	1.35339e6			1.36155			0.680773	+	0.732494i	$0.261644\pi$
$998$			1.02602e6i			1.03013i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.4		40
3.2	odd	2		177.5.c.a.58.37	yes	40
59.58	odd	2	inner	531.5.c.d.235.37		40
177.176	even	2		177.5.c.a.58.4	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.4	✓	40	177.176	even	2
177.5.c.a.58.37	yes	40	3.2	odd	2
531.5.c.d.235.4		40	1.1	even	1	trivial
531.5.c.d.235.37		40	59.58	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.76718i q^{2} -29.7948 q^{4} -6.77685 q^{5} +45.8846 q^{7} +93.3518i q^{8} +O(q^{10})\)	\(q-6.76718i q^{2} -29.7948 q^{4} -6.77685 q^{5} +45.8846 q^{7} +93.3518i q^{8} +45.8602i q^{10} -152.372i q^{11} +138.574i q^{13} -310.510i q^{14} +155.012 q^{16} -61.1978 q^{17} -184.090 q^{19} +201.915 q^{20} -1031.13 q^{22} -55.4703i q^{23} -579.074 q^{25} +937.758 q^{26} -1367.12 q^{28} -290.483 q^{29} +681.108i q^{31} +444.632i q^{32} +414.137i q^{34} -310.953 q^{35} +1422.17i q^{37} +1245.77i q^{38} -632.631i q^{40} -1532.90 q^{41} -404.708i q^{43} +4539.89i q^{44} -375.377 q^{46} +1601.56i q^{47} -295.602 q^{49} +3918.70i q^{50} -4128.79i q^{52} +1325.56 q^{53} +1032.60i q^{55} +4283.41i q^{56} +1965.75i q^{58} +(-3430.65 + 589.891i) q^{59} -2732.09i q^{61} +4609.19 q^{62} +5489.10 q^{64} -939.097i q^{65} +2225.52i q^{67} +1823.37 q^{68} +2104.28i q^{70} +6157.04 q^{71} +2816.89i q^{73} +9624.09 q^{74} +5484.93 q^{76} -6991.53i q^{77} +7747.95 q^{79} -1050.49 q^{80} +10373.4i q^{82} +648.806i q^{83} +414.728 q^{85} -2738.73 q^{86} +14224.2 q^{88} +5751.21i q^{89} +6358.43i q^{91} +1652.72i q^{92} +10838.1 q^{94} +1247.55 q^{95} -1510.67i q^{97} +2000.39i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7	\( 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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