LMFDB - Embedded newform 531.3.c.d.235.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,3,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, 3, names="a")

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

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
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:	$14.4687020375$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 264 x^{18} + 33518 x^{16} - 2330346 x^{14} + 94572949 x^{12} - 2154660388 x^{10} + \cdots + 545424869874889 $ x^20 - 264x^18 + 33518x^16 - 2330346x^14 + 94572949x^12 - 2154660388x^10 + 19607696037x^8 + 202461271054x^6 - 5241140987242x^4 - 4683513358960*x^2 + 545424869874889
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.2
Root			$8.96460 - 3.66991i$ of defining polynomial
Character	$\chi$	$=$	531.235
Dual form			531.3.c.d.235.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.66991i q^{2} -9.46822 q^{4} +8.96460 q^{5} +4.32189 q^{7} +20.0678i q^{8} +O(q^{10})$	\(q-3.66991i q^{2} -9.46822 q^{4} +8.96460 q^{5} +4.32189 q^{7} +20.0678i q^{8} -32.8992i q^{10} +13.6358i q^{11} +9.53274i q^{13} -15.8609i q^{14} +35.7743 q^{16} +14.2419 q^{17} +19.5898 q^{19} -84.8788 q^{20} +50.0422 q^{22} -12.0912i q^{23} +55.3640 q^{25} +34.9843 q^{26} -40.9206 q^{28} +28.4114 q^{29} -4.47637i q^{31} -51.0168i q^{32} -52.2666i q^{34} +38.7440 q^{35} -42.7697i q^{37} -71.8927i q^{38} +179.900i q^{40} -69.3413 q^{41} +67.1934i q^{43} -129.107i q^{44} -44.3736 q^{46} +48.9942i q^{47} -30.3213 q^{49} -203.181i q^{50} -90.2580i q^{52} -65.4320 q^{53} +122.240i q^{55} +86.7310i q^{56} -104.267i q^{58} +(17.7234 - 56.2751i) q^{59} -71.1567i q^{61} -16.4278 q^{62} -44.1300 q^{64} +85.4572i q^{65} -66.2757i q^{67} -134.846 q^{68} -142.187i q^{70} -83.7826 q^{71} -22.7269i q^{73} -156.961 q^{74} -185.480 q^{76} +58.9325i q^{77} +107.019 q^{79} +320.702 q^{80} +254.476i q^{82} +31.9307i q^{83} +127.673 q^{85} +246.593 q^{86} -273.641 q^{88} -2.39998i q^{89} +41.1994i q^{91} +114.482i q^{92} +179.804 q^{94} +175.614 q^{95} -46.2558i q^{97} +111.276i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) $ 20 * q - 40 * q^4 - 8 * q^7	$ 20 q - 40 q^{4} - 8 q^{7} + 88 q^{16} + 60 q^{19} + 136 q^{22} + 148 q^{25} - 136 q^{28} - 84 q^{46} - 100 q^{49} + 36 q^{64} - 552 q^{76} + 252 q^{79} + 180 q^{85} - 788 q^{88} + 584 q^{94}+O(q^{100}) $ 20 * q - 40 * q^4 - 8 * q^7 + 88 * q^16 + 60 * q^19 + 136 * q^22 + 148 * q^25 - 136 * q^28 - 84 * q^46 - 100 * q^49 + 36 * q^64 - 552 * q^76 + 252 * q^79 + 180 * q^85 - 788 * q^88 + 584 * q^94

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 $

$2$		−	3.66991i		−	1.83495i	−0.397789	−	0.917477i	$-0.630223\pi$
$2$		−	3.66991i		−	1.83495i	0.397789	−	0.917477i	$-0.369777\pi$
$3$		0			0
$3$		0			0
$4$	−9.46822			−2.36705
$5$	8.96460			1.79292			0.896460	−	0.443125i	$-0.146130\pi$
$5$	8.96460			1.79292			0.896460	+	0.443125i	$0.146130\pi$
$6$		0			0
$7$	4.32189			0.617413			0.308706	−	0.951157i	$-0.400104\pi$
$7$	4.32189			0.617413			0.308706	+	0.951157i	$0.400104\pi$
$8$			20.0678i			2.50848i
$9$		0			0
$10$		−	32.8992i		−	3.28992i
$11$			13.6358i			1.23962i	0.784752	+	0.619810i	$0.212790\pi$
$11$			13.6358i			1.23962i	−0.784752	+	0.619810i	$0.787210\pi$
$12$		0			0
$13$			9.53274i			0.733287i	0.930361	+	0.366644i	$0.119493\pi$
$13$			9.53274i			0.733287i	−0.930361	+	0.366644i	$0.880507\pi$
$14$		−	15.8609i		−	1.13292i
$15$		0			0
$16$	35.7743			2.23589
$17$	14.2419			0.837761			0.418881	−	0.908041i	$-0.362423\pi$
$17$	14.2419			0.837761			0.418881	+	0.908041i	$0.362423\pi$
$18$		0			0
$19$	19.5898			1.03104			0.515520	−	0.856877i	$-0.327599\pi$
$19$	19.5898			1.03104			0.515520	+	0.856877i	$0.327599\pi$
$20$	−84.8788			−4.24394
$21$		0			0
$22$	50.0422			2.27464
$23$		−	12.0912i		−	0.525705i	−0.964836	−	0.262852i	$-0.915337\pi$
$23$		−	12.0912i		−	0.525705i	0.964836	−	0.262852i	$-0.0846632\pi$
$24$		0			0
$25$	55.3640			2.21456
$26$	34.9843			1.34555
$27$		0			0
$28$	−40.9206			−1.46145
$29$	28.4114			0.979703			0.489851	−	0.871806i	$-0.337051\pi$
$29$	28.4114			0.979703			0.489851	+	0.871806i	$0.337051\pi$
$30$		0			0
$31$		−	4.47637i		−	0.144399i	−0.997390	−	0.0721995i	$-0.976998\pi$
$31$		−	4.47637i		−	0.144399i	0.997390	−	0.0721995i	$-0.0230018\pi$
$32$		−	51.0168i		−	1.59428i
$33$		0			0
$34$		−	52.2666i		−	1.53725i
$35$	38.7440			1.10697
$36$		0			0
$37$		−	42.7697i		−	1.15594i	−0.816059	−	0.577968i	$-0.803846\pi$
$37$		−	42.7697i		−	1.15594i	0.816059	−	0.577968i	$-0.196154\pi$
$38$		−	71.8927i		−	1.89191i
$39$		0			0
$40$			179.900i			4.49751i
$41$	−69.3413			−1.69125			−0.845626	−	0.533776i	$-0.820772\pi$
$41$	−69.3413			−1.69125			−0.845626	+	0.533776i	$0.820772\pi$
$42$		0			0
$43$			67.1934i			1.56264i	0.624133	+	0.781318i	$0.285453\pi$
$43$			67.1934i			1.56264i	−0.624133	+	0.781318i	$0.714547\pi$
$44$		−	129.107i		−	2.93425i
$45$		0			0
$46$	−44.3736			−0.964644
$47$			48.9942i			1.04243i	0.853426	+	0.521215i	$0.174521\pi$
$47$			48.9942i			1.04243i	−0.853426	+	0.521215i	$0.825479\pi$
$48$		0			0
$49$	−30.3213			−0.618801
$50$		−	203.181i		−	4.06362i
$51$		0			0
$52$		−	90.2580i		−	1.73573i
$53$	−65.4320			−1.23457			−0.617283	−	0.786741i	$-0.711767\pi$
$53$	−65.4320			−1.23457			−0.617283	+	0.786741i	$0.711767\pi$
$54$		0			0
$55$			122.240i			2.22254i
$56$			86.7310i			1.54877i
$57$		0			0
$58$		−	104.267i		−	1.79771i
$59$	17.7234	−	56.2751i	0.300396	−	0.953814i
$59$	17.7234	−	56.2751i	0.300396	−	0.953814i
$60$		0			0
$61$		−	71.1567i		−	1.16650i	−0.812291	−	0.583252i	$-0.801780\pi$
$61$		−	71.1567i		−	1.16650i	0.812291	−	0.583252i	$-0.198220\pi$
$62$	−16.4278			−0.264965
$63$		0			0
$64$	−44.1300			−0.689531
$65$			85.4572i			1.31473i
$66$		0			0
$67$		−	66.2757i		−	0.989190i	−0.869124	−	0.494595i	$-0.835316\pi$
$67$		−	66.2757i		−	0.989190i	0.869124	−	0.494595i	$-0.164684\pi$
$68$	−134.846			−1.98303
$69$		0			0
$70$		−	142.187i		−	2.03124i
$71$	−83.7826			−1.18004			−0.590018	−	0.807390i	$-0.700879\pi$
$71$	−83.7826			−1.18004			−0.590018	+	0.807390i	$0.700879\pi$
$72$		0			0
$73$		−	22.7269i		−	0.311328i	−0.987810	−	0.155664i	$-0.950248\pi$
$73$		−	22.7269i		−	0.311328i	0.987810	−	0.155664i	$-0.0497517\pi$
$74$	−156.961			−2.12109
$75$		0			0
$76$	−185.480			−2.44053
$77$			58.9325i			0.765357i
$78$		0			0
$79$	107.019			1.35468			0.677338	−	0.735672i	$-0.263133\pi$
$79$	107.019			1.35468			0.677338	+	0.735672i	$0.263133\pi$
$80$	320.702			4.00878
$81$		0			0
$82$			254.476i			3.10337i
$83$			31.9307i			0.384707i	0.981326	+	0.192353i	$0.0616120\pi$
$83$			31.9307i			0.384707i	−0.981326	+	0.192353i	$0.938388\pi$
$84$		0			0
$85$	127.673			1.50204
$86$	246.593			2.86737
$87$		0			0
$88$	−273.641			−3.10956
$89$		−	2.39998i		−	0.0269660i	−0.999909	−	0.0134830i	$-0.995708\pi$
$89$		−	2.39998i		−	0.0269660i	0.999909	−	0.0134830i	$-0.00429190\pi$
$90$		0			0
$91$			41.1994i			0.452741i
$92$			114.482i			1.24437i
$93$		0			0
$94$	179.804			1.91281
$95$	175.614			1.84857
$96$		0			0
$97$		−	46.2558i		−	0.476864i	−0.971159	−	0.238432i	$-0.923367\pi$
$97$		−	46.2558i		−	0.476864i	0.971159	−	0.238432i	$-0.0766334\pi$
$98$			111.276i			1.13547i
$99$		0			0
$100$	−524.199			−5.24199
$101$			190.622i			1.88735i	0.330874	+	0.943675i	$0.392656\pi$
$101$			190.622i			1.88735i	−0.330874	+	0.943675i	$0.607344\pi$
$102$		0			0
$103$			93.8837i			0.911492i	0.890110	+	0.455746i	$0.150627\pi$
$103$			93.8837i			0.911492i	−0.890110	+	0.455746i	$0.849373\pi$
$104$	−191.302			−1.83944
$105$		0			0
$106$			240.129i			2.26537i
$107$	−166.847			−1.55932			−0.779659	−	0.626204i	$-0.784608\pi$
$107$	−166.847			−1.55932			−0.779659	+	0.626204i	$0.784608\pi$
$108$		0			0
$109$		−	146.663i		−	1.34553i	−0.739854	−	0.672767i	$-0.765105\pi$
$109$		−	146.663i		−	1.34553i	0.739854	−	0.672767i	$-0.234895\pi$
$110$	448.608			4.07825
$111$		0			0
$112$	154.612			1.38047
$113$		−	211.179i		−	1.86884i	−0.356177	−	0.934419i	$-0.615920\pi$
$113$		−	211.179i		−	1.86884i	0.356177	−	0.934419i	$-0.384080\pi$
$114$		0			0
$115$		−	108.393i		−	0.942546i
$116$	−269.005			−2.31901
$117$		0			0
$118$	−206.524	−	65.0432i	−1.75021	−	0.551213i
$119$	61.5521			0.517245
$120$		0			0
$121$	−64.9353			−0.536656
$122$	−261.139			−2.14048
$123$		0			0
$124$			42.3832i			0.341800i
$125$	272.202			2.17761
$126$		0			0
$127$	15.5695			0.122594			0.0612972	−	0.998120i	$-0.480476\pi$
$127$	15.5695			0.122594			0.0612972	+	0.998120i	$0.480476\pi$
$128$		−	42.1144i		−	0.329018i
$129$		0			0
$130$	313.620			2.41246
$131$			14.5772i			0.111276i	0.998451	+	0.0556382i	$0.0177193\pi$
$131$			14.5772i			0.111276i	−0.998451	+	0.0556382i	$0.982281\pi$
$132$		0			0
$133$	84.6649			0.636578
$134$	−243.226			−1.81512
$135$		0			0
$136$			285.805i			2.10151i
$137$	98.5758			0.719531			0.359766	−	0.933043i	$-0.382857\pi$
$137$	98.5758			0.719531			0.359766	+	0.933043i	$0.382857\pi$
$138$		0			0
$139$	−2.47568			−0.0178106			−0.00890532	−	0.999960i	$-0.502835\pi$
$139$	−2.47568			−0.0178106			−0.00890532	+	0.999960i	$0.502835\pi$
$140$	−366.837			−2.62026
$141$		0			0
$142$			307.474i			2.16531i
$143$	−129.987			−0.908997
$144$		0			0
$145$	254.697			1.75653
$146$	−83.4058			−0.571272
$147$		0			0
$148$			404.952i			2.73617i
$149$			225.682i			1.51464i	0.653041	+	0.757322i	$0.273493\pi$
$149$			225.682i			1.51464i	−0.653041	+	0.757322i	$0.726507\pi$
$150$		0			0
$151$		−	257.836i		−	1.70752i	−0.520666	−	0.853761i	$-0.674316\pi$
$151$		−	257.836i		−	1.70752i	0.520666	−	0.853761i	$-0.325684\pi$
$152$			393.125i			2.58635i
$153$		0			0
$154$	216.277			1.40439
$155$		−	40.1288i		−	0.258896i
$156$		0			0
$157$		−	214.262i		−	1.36473i	−0.731014	−	0.682363i	$-0.760952\pi$
$157$		−	214.262i		−	1.36473i	0.731014	−	0.682363i	$-0.239048\pi$
$158$		−	392.751i		−	2.48577i
$159$		0			0
$160$		−	457.346i		−	2.85841i
$161$		−	52.2569i		−	0.324577i
$162$		0			0
$163$	−197.164			−1.20960			−0.604799	−	0.796378i	$-0.706746\pi$
$163$	−197.164			−1.20960			−0.604799	+	0.796378i	$0.706746\pi$
$164$	656.539			4.00328
$165$		0			0
$166$	117.183			0.705919
$167$	220.901			1.32276			0.661381	−	0.750051i	$-0.269971\pi$
$167$	220.901			1.32276			0.661381	+	0.750051i	$0.269971\pi$
$168$		0			0
$169$	78.1269			0.462290
$170$		−	468.549i		−	2.75617i
$171$		0			0
$172$		−	636.202i		−	3.69885i
$173$		−	198.023i		−	1.14464i	−0.820030	−	0.572321i	$-0.806043\pi$
$173$		−	198.023i		−	1.14464i	0.820030	−	0.572321i	$-0.193957\pi$
$174$		0			0
$175$	239.277			1.36730
$176$			487.811i			2.77165i
$177$		0			0
$178$	−8.80769			−0.0494814
$179$			246.980i			1.37978i	0.723915	+	0.689889i	$0.242341\pi$
$179$			246.980i			1.37978i	−0.723915	+	0.689889i	$0.757659\pi$
$180$		0			0
$181$	−125.340			−0.692485			−0.346243	−	0.938145i	$-0.612543\pi$
$181$	−125.340			−0.692485			−0.346243	+	0.938145i	$0.612543\pi$
$182$	151.198			0.830759
$183$		0			0
$184$	242.645			1.31872
$185$		−	383.413i		−	2.07250i
$186$		0			0
$187$			194.200i			1.03850i
$188$		−	463.888i		−	2.46749i
$189$		0			0
$190$		−	644.489i		−	3.39205i
$191$			40.1016i			0.209956i	0.994475	+	0.104978i	$0.0334772\pi$
$191$			40.1016i			0.209956i	−0.994475	+	0.104978i	$0.966523\pi$
$192$		0			0
$193$	−94.7477			−0.490921			−0.245460	−	0.969407i	$-0.578939\pi$
$193$	−94.7477			−0.490921			−0.245460	+	0.969407i	$0.578939\pi$
$194$	−169.755			−0.875023
$195$		0			0
$196$	287.088			1.46474
$197$	−332.185			−1.68622			−0.843110	−	0.537742i	$-0.819278\pi$
$197$	−332.185			−1.68622			−0.843110	+	0.537742i	$0.819278\pi$
$198$		0			0
$199$	330.004			1.65831			0.829156	−	0.559017i	$-0.188821\pi$
$199$	330.004			1.65831			0.829156	+	0.559017i	$0.188821\pi$
$200$			1111.04i			5.55519i
$201$		0			0
$202$	699.566			3.46320
$203$	122.791			0.604881
$204$		0			0
$205$	−621.617			−3.03228
$206$	344.544			1.67255
$207$		0			0
$208$			341.027i			1.63955i
$209$			267.122i			1.27810i
$210$		0			0
$211$			298.987i			1.41700i	0.705711	+	0.708499i	$0.250628\pi$
$211$			298.987i			1.41700i	−0.705711	+	0.708499i	$0.749372\pi$
$212$	619.524			2.92228
$213$		0			0
$214$			612.313i			2.86128i
$215$			602.362i			2.80168i
$216$		0			0
$217$		−	19.3464i		−	0.0891538i
$218$	−538.241			−2.46899
$219$		0			0
$220$		−	1157.39i		−	5.26087i
$221$			135.765i			0.614320i
$222$		0			0
$223$	36.5690			0.163987			0.0819934	−	0.996633i	$-0.473871\pi$
$223$	36.5690			0.163987			0.0819934	+	0.996633i	$0.473871\pi$
$224$		−	220.489i		−	0.984327i
$225$		0			0
$226$	−775.006			−3.42923
$227$		−	102.568i		−	0.451841i	−0.974146	−	0.225921i	$-0.927461\pi$
$227$		−	102.568i		−	0.451841i	0.974146	−	0.225921i	$-0.0725390\pi$
$228$		0			0
$229$		−	123.970i		−	0.541355i	−0.962670	−	0.270678i	$-0.912752\pi$
$229$		−	123.970i		−	0.541355i	0.962670	−	0.270678i	$-0.0872477\pi$
$230$	−397.792			−1.72953
$231$		0			0
$232$			570.155i			2.45757i
$233$			228.102i			0.978976i	0.872010	+	0.489488i	$0.162816\pi$
$233$			228.102i			0.978976i	−0.872010	+	0.489488i	$0.837184\pi$
$234$		0			0
$235$			439.213i			1.86899i
$236$	−167.809	+	532.824i	−0.711054	+	2.25773i
$237$		0			0
$238$		−	225.891i		−	0.949120i
$239$	22.7258			0.0950872			0.0475436	−	0.998869i	$-0.484861\pi$
$239$	22.7258			0.0950872			0.0475436	+	0.998869i	$0.484861\pi$
$240$		0			0
$241$	108.364			0.449644			0.224822	−	0.974400i	$-0.427820\pi$
$241$	108.364			0.449644			0.224822	+	0.974400i	$0.427820\pi$
$242$			238.307i			0.984738i
$243$		0			0
$244$			673.728i			2.76118i
$245$	−271.818			−1.10946
$246$		0			0
$247$			186.744i			0.756049i
$248$	89.8311			0.362222
$249$		0			0
$250$		−	998.954i		−	3.99582i
$251$	−27.1592			−0.108204			−0.0541019	−	0.998535i	$-0.517230\pi$
$251$	−27.1592			−0.108204			−0.0541019	+	0.998535i	$0.517230\pi$
$252$		0			0
$253$	164.873			0.651674
$254$		−	57.1385i		−	0.224955i
$255$		0			0
$256$	−331.076			−1.29326
$257$	144.817			0.563489			0.281745	−	0.959489i	$-0.409087\pi$
$257$	144.817			0.563489			0.281745	+	0.959489i	$0.409087\pi$
$258$		0			0
$259$		−	184.846i		−	0.713690i
$260$		−	809.127i		−	3.11203i
$261$		0			0
$262$	53.4970			0.204187
$263$	−196.961			−0.748903			−0.374451	−	0.927247i	$-0.622169\pi$
$263$	−196.961			−0.748903			−0.374451	+	0.927247i	$0.622169\pi$
$264$		0			0
$265$	−586.572			−2.21348
$266$		−	310.712i		−	1.16809i
$267$		0			0
$268$			627.513i			2.34147i
$269$		−	169.628i		−	0.630586i	−0.948994	−	0.315293i	$-0.897897\pi$
$269$		−	169.628i		−	0.630586i	0.948994	−	0.315293i	$-0.102103\pi$
$270$		0			0
$271$	−132.729			−0.489776			−0.244888	−	0.969551i	$-0.578751\pi$
$271$	−132.729			−0.489776			−0.244888	+	0.969551i	$0.578751\pi$
$272$	509.495			1.87314
$273$		0			0
$274$		−	361.764i		−	1.32031i
$275$			754.934i			2.74521i
$276$		0			0
$277$	−62.9826			−0.227374			−0.113687	−	0.993517i	$-0.536266\pi$
$277$	−62.9826			−0.227374			−0.113687	+	0.993517i	$0.536266\pi$
$278$			9.08551i			0.0326817i
$279$		0			0
$280$			777.509i			2.77682i
$281$	183.858			0.654297			0.327149	−	0.944973i	$-0.393912\pi$
$281$	183.858			0.654297			0.327149	+	0.944973i	$0.393912\pi$
$282$		0			0
$283$			29.6551i			0.104788i	0.998626	+	0.0523941i	$0.0166852\pi$
$283$			29.6551i			0.104788i	−0.998626	+	0.0523941i	$0.983315\pi$
$284$	793.272			2.79321
$285$		0			0
$286$			477.039i			1.66797i
$287$	−299.686			−1.04420
$288$		0			0
$289$	−86.1671			−0.298156
$290$		−	934.713i		−	3.22315i
$291$		0			0
$292$			215.184i			0.736930i
$293$	46.9157			0.160122			0.0800608	−	0.996790i	$-0.474489\pi$
$293$	46.9157			0.160122			0.0800608	+	0.996790i	$0.474489\pi$
$294$		0			0
$295$	158.883	−	504.483i	0.538587	−	1.71011i
$296$	858.295			2.89965
$297$		0			0
$298$	828.232			2.77930
$299$	115.262			0.385493
$300$		0			0
$301$			290.402i			0.964792i
$302$	−946.233			−3.13322
$303$		0			0
$304$	700.810			2.30530
$305$		−	637.892i		−	2.09145i
$306$		0			0
$307$	−128.974			−0.420112			−0.210056	−	0.977689i	$-0.567365\pi$
$307$	−128.974			−0.420112			−0.210056	+	0.977689i	$0.567365\pi$
$308$		−	557.986i		−	1.81164i
$309$		0			0
$310$	−147.269			−0.475062
$311$	138.827			0.446388			0.223194	−	0.974774i	$-0.428352\pi$
$311$	138.827			0.446388			0.223194	+	0.974774i	$0.428352\pi$
$312$		0			0
$313$			189.766i			0.606280i	0.952946	+	0.303140i	$0.0980350\pi$
$313$			189.766i			0.606280i	−0.952946	+	0.303140i	$0.901965\pi$
$314$	−786.321			−2.50421
$315$		0			0
$316$	−1013.28			−3.20659
$317$	−12.7187			−0.0401222			−0.0200611	−	0.999799i	$-0.506386\pi$
$317$	−12.7187			−0.0401222			−0.0200611	+	0.999799i	$0.506386\pi$
$318$		0			0
$319$			387.412i			1.21446i
$320$	−395.608			−1.23627
$321$		0			0
$322$	−191.778			−0.595583
$323$	278.996			0.863766
$324$		0			0
$325$			527.771i			1.62391i
$326$			723.575i			2.21956i
$327$		0			0
$328$		−	1391.53i		−	4.24247i
$329$			211.747i			0.643609i
$330$		0			0
$331$	245.719			0.742353			0.371176	−	0.928562i	$-0.378955\pi$
$331$	245.719			0.742353			0.371176	+	0.928562i	$0.378955\pi$
$332$		−	302.326i		−	0.910622i
$333$		0			0
$334$		−	810.686i		−	2.42720i
$335$		−	594.135i		−	1.77354i
$336$		0			0
$337$		−	12.5592i		−	0.0372678i	−0.999826	−	0.0186339i	$-0.994068\pi$
$337$		−	12.5592i		−	0.0372678i	0.999826	−	0.0186339i	$-0.00593169\pi$
$338$		−	286.719i		−	0.848280i
$339$		0			0
$340$	−1208.84			−3.55541
$341$	61.0389			0.179000
$342$		0			0
$343$	−342.818			−0.999469
$344$	−1348.43			−3.91985
$345$		0			0
$346$	−726.726			−2.10036
$347$		−	154.507i		−	0.445265i	−0.974902	−	0.222633i	$-0.928535\pi$
$347$		−	154.507i		−	0.445265i	0.974902	−	0.222633i	$-0.0714650\pi$
$348$		0			0
$349$		−	478.604i		−	1.37136i	−0.727904	−	0.685679i	$-0.759505\pi$
$349$		−	478.604i		−	1.37136i	0.727904	−	0.685679i	$-0.240495\pi$
$350$		−	878.126i		−	2.50893i
$351$		0			0
$352$	695.656			1.97630
$353$		−	324.397i		−	0.918971i	−0.888185	−	0.459485i	$-0.848034\pi$
$353$		−	324.397i		−	0.918971i	0.888185	−	0.459485i	$-0.151966\pi$
$354$		0			0
$355$	−751.077			−2.11571
$356$			22.7235i			0.0638300i
$357$		0			0
$358$	906.394			2.53183
$359$	116.884			0.325583			0.162792	−	0.986660i	$-0.447950\pi$
$359$	116.884			0.325583			0.162792	+	0.986660i	$0.447950\pi$
$360$		0			0
$361$	22.7593			0.0630451
$362$			459.985i			1.27068i
$363$		0			0
$364$		−	390.085i		−	1.07166i
$365$		−	203.738i		−	0.558186i
$366$		0			0
$367$			314.105i			0.855873i	0.903809	+	0.427937i	$0.140759\pi$
$367$			314.105i			0.855873i	−0.903809	+	0.427937i	$0.859241\pi$
$368$		−	432.554i		−	1.17542i
$369$		0			0
$370$	−1407.09			−3.80295
$371$	−282.790			−0.762237
$372$		0			0
$373$	81.4817			0.218450			0.109225	−	0.994017i	$-0.465163\pi$
$373$	81.4817			0.218450			0.109225	+	0.994017i	$0.465163\pi$
$374$	712.698			1.90561
$375$		0			0
$376$	−983.208			−2.61491
$377$			270.838i			0.718404i
$378$		0			0
$379$	−454.159			−1.19831			−0.599154	−	0.800633i	$-0.704496\pi$
$379$	−454.159			−1.19831			−0.599154	+	0.800633i	$0.704496\pi$
$380$	−1662.76			−4.37567
$381$		0			0
$382$	147.169			0.385259
$383$	−454.875			−1.18766			−0.593832	−	0.804589i	$-0.702386\pi$
$383$	−454.875			−1.18766			−0.593832	+	0.804589i	$0.702386\pi$
$384$		0			0
$385$			528.306i			1.37222i
$386$			347.715i			0.900817i
$387$		0			0
$388$			437.960i			1.12876i
$389$	−324.780			−0.834910			−0.417455	−	0.908698i	$-0.637078\pi$
$389$	−324.780			−0.834910			−0.417455	+	0.908698i	$0.637078\pi$
$390$		0			0
$391$		−	172.202i		−	0.440415i
$392$		−	608.483i		−	1.55225i
$393$		0			0
$394$			1219.09i			3.09413i
$395$	959.386			2.42883
$396$		0			0
$397$			50.8140i			0.127995i	0.997950	+	0.0639974i	$0.0203850\pi$
$397$			50.8140i			0.127995i	−0.997950	+	0.0639974i	$0.979615\pi$
$398$		−	1211.08i		−	3.04293i
$399$		0			0
$400$	1980.61			4.95152
$401$			406.223i			1.01302i	0.862233	+	0.506512i	$0.169065\pi$
$401$			406.223i			1.01302i	−0.862233	+	0.506512i	$0.830935\pi$
$402$		0			0
$403$	42.6720			0.105886
$404$		−	1804.85i		−	4.46746i
$405$		0			0
$406$		−	450.631i		−	1.10993i
$407$	583.199			1.43292
$408$		0			0
$409$			462.100i			1.12983i	0.825149	+	0.564915i	$0.191091\pi$
$409$			462.100i			1.12983i	−0.825149	+	0.564915i	$0.808909\pi$
$410$			2281.28i			5.56409i
$411$		0			0
$412$		−	888.911i		−	2.15755i
$413$	76.5985	−	243.215i	0.185469	−	0.588897i
$414$		0			0
$415$			286.246i			0.689748i
$416$	486.330			1.16906
$417$		0			0
$418$	980.315			2.34525
$419$			449.125i			1.07190i	0.844250	+	0.535949i	$0.180046\pi$
$419$			449.125i			1.07190i	−0.844250	+	0.535949i	$0.819954\pi$
$420$		0			0
$421$		−	423.462i		−	1.00585i	−0.864331	−	0.502924i	$-0.832258\pi$
$421$		−	423.462i		−	1.00585i	0.864331	−	0.502924i	$-0.167742\pi$
$422$	1097.25			2.60013
$423$		0			0
$424$		−	1313.08i		−	3.09689i
$425$	788.492			1.85527
$426$		0			0
$427$		−	307.532i		−	0.720215i
$428$	1579.74			3.69099
$429$		0			0
$430$	2210.61			5.14096
$431$			374.380i			0.868632i	0.900761	+	0.434316i	$0.143010\pi$
$431$			374.380i			0.868632i	−0.900761	+	0.434316i	$0.856990\pi$
$432$		0			0
$433$	−372.144			−0.859455			−0.429728	−	0.902959i	$-0.641390\pi$
$433$	−372.144			−0.859455			−0.429728	+	0.902959i	$0.641390\pi$
$434$	−70.9994			−0.163593
$435$		0			0
$436$			1388.64i			3.18495i
$437$		−	236.864i		−	0.542023i
$438$		0			0
$439$	−208.178			−0.474209			−0.237104	−	0.971484i	$-0.576198\pi$
$439$	−208.178			−0.474209			−0.237104	+	0.971484i	$0.576198\pi$
$440$	−2453.09			−5.57519
$441$		0			0
$442$	498.244			1.12725
$443$			211.679i			0.477832i	0.971040	+	0.238916i	$0.0767920\pi$
$443$			211.679i			0.477832i	−0.971040	+	0.238916i	$0.923208\pi$
$444$		0			0
$445$		−	21.5148i		−	0.0483479i
$446$		−	134.205i		−	0.300908i
$447$		0			0
$448$	−190.725			−0.425726
$449$	−336.684			−0.749853			−0.374926	−	0.927055i	$-0.622332\pi$
$449$	−336.684			−0.749853			−0.374926	+	0.927055i	$0.622332\pi$
$450$		0			0
$451$		−	945.525i		−	2.09651i
$452$			1999.48i			4.42364i
$453$		0			0
$454$	−376.415			−0.829108
$455$			369.336i			0.811729i
$456$		0			0
$457$		−	430.659i		−	0.942360i	−0.882037	−	0.471180i	$-0.843828\pi$
$457$		−	430.659i		−	0.942360i	0.882037	−	0.471180i	$-0.156172\pi$
$458$	−454.960			−0.993362
$459$		0			0
$460$			1026.29i			2.23106i
$461$	43.1204			0.0935366			0.0467683	−	0.998906i	$-0.485108\pi$
$461$	43.1204			0.0935366			0.0467683	+	0.998906i	$0.485108\pi$
$462$		0			0
$463$		−	767.797i		−	1.65831i	−0.559019	−	0.829155i	$-0.688822\pi$
$463$		−	767.797i		−	1.65831i	0.559019	−	0.829155i	$-0.311178\pi$
$464$	1016.40			2.19051
$465$		0			0
$466$	837.111			1.79638
$467$		−	146.035i		−	0.312709i	−0.987701	−	0.156354i	$-0.950026\pi$
$467$		−	146.035i		−	0.312709i	0.987701	−	0.156354i	$-0.0499742\pi$
$468$		0			0
$469$		−	286.436i		−	0.610738i
$470$	1611.87			3.42951
$471$		0			0
$472$	1129.32	+	355.670i	2.39263	+	0.753538i
$473$	−916.236			−1.93707
$474$		0			0
$475$	1084.57			2.28330
$476$	−582.789			−1.22435
$477$		0			0
$478$		−	83.4017i		−	0.174481i
$479$	−498.711			−1.04115			−0.520576	−	0.853816i	$-0.674283\pi$
$479$	−498.711			−1.04115			−0.520576	+	0.853816i	$0.674283\pi$
$480$		0			0
$481$	407.712			0.847634
$482$		−	397.687i		−	0.825076i
$483$		0			0
$484$	614.822			1.27029
$485$		−	414.665i		−	0.854979i
$486$		0			0
$487$	50.3547			0.103398			0.0516989	−	0.998663i	$-0.483536\pi$
$487$	50.3547			0.103398			0.0516989	+	0.998663i	$0.483536\pi$
$488$	1427.96			2.92615
$489$		0			0
$490$			997.547i			2.03581i
$491$	−189.197			−0.385330			−0.192665	−	0.981265i	$-0.561713\pi$
$491$	−189.197			−0.385330			−0.192665	+	0.981265i	$0.561713\pi$
$492$		0			0
$493$	404.633			0.820757
$494$	685.334			1.38732
$495$		0			0
$496$		−	160.139i		−	0.322860i
$497$	−362.099			−0.728570
$498$		0			0
$499$	−506.086			−1.01420			−0.507100	−	0.861887i	$-0.669283\pi$
$499$	−506.086			−1.01420			−0.507100	+	0.861887i	$0.669283\pi$
$500$	−2577.26			−5.15453
$501$		0			0
$502$			99.6716i			0.198549i
$503$			254.679i			0.506321i	0.967424	+	0.253160i	$0.0814701\pi$
$503$			254.679i			0.506321i	−0.967424	+	0.253160i	$0.918530\pi$
$504$		0			0
$505$			1708.85i			3.38387i
$506$		−	605.070i		−	1.19579i
$507$		0			0
$508$	−147.415			−0.290187
$509$		−	433.410i		−	0.851493i	−0.904842	−	0.425747i	$-0.860012\pi$
$509$		−	433.410i		−	0.851493i	0.904842	−	0.425747i	$-0.139988\pi$
$510$		0			0
$511$		−	98.2234i		−	0.192218i
$512$			1046.56i			2.04406i
$513$		0			0
$514$		−	531.464i		−	1.03398i
$515$			841.630i			1.63423i
$516$		0			0
$517$	−668.075			−1.29222
$518$	−678.367			−1.30959
$519$		0			0
$520$	−1714.94			−3.29796
$521$	−369.864			−0.709912			−0.354956	−	0.934883i	$-0.615504\pi$
$521$	−369.864			−0.709912			−0.354956	+	0.934883i	$0.615504\pi$
$522$		0			0
$523$	−36.8743			−0.0705054			−0.0352527	−	0.999378i	$-0.511224\pi$
$523$	−36.8743			−0.0705054			−0.0352527	+	0.999378i	$0.511224\pi$
$524$		−	138.020i		−	0.263397i
$525$		0			0
$526$			722.830i			1.37420i
$527$		−	63.7522i		−	0.120972i
$528$		0			0
$529$	382.803			0.723635
$530$			2152.66i			4.06163i
$531$		0			0
$532$	−801.625			−1.50681
$533$		−	661.012i		−	1.24017i
$534$		0			0
$535$	−1495.72			−2.79573
$536$	1330.01			2.48136
$537$		0			0
$538$	−622.518			−1.15710
$539$		−	413.455i		−	0.767078i
$540$		0			0
$541$		−	56.6164i		−	0.104651i	−0.998630	−	0.0523257i	$-0.983337\pi$
$541$		−	56.6164i		−	0.104651i	0.998630	−	0.0523257i	$-0.0166634\pi$
$542$			487.105i			0.898717i
$543$		0			0
$544$		−	726.579i		−	1.33562i
$545$		−	1314.78i		−	2.41244i
$546$		0			0
$547$	−135.503			−0.247720			−0.123860	−	0.992300i	$-0.539527\pi$
$547$	−135.503			−0.247720			−0.123860	+	0.992300i	$0.539527\pi$
$548$	−933.337			−1.70317
$549$		0			0
$550$	2770.54			5.03734
$551$	556.573			1.01011
$552$		0			0
$553$	462.526			0.836395
$554$			231.140i			0.417221i
$555$		0			0
$556$	23.4403			0.0421587
$557$	−486.948			−0.874233			−0.437117	−	0.899405i	$-0.644000\pi$
$557$	−486.948			−0.874233			−0.437117	+	0.899405i	$0.644000\pi$
$558$		0			0
$559$	−640.537			−1.14586
$560$	1386.04			2.47507
$561$		0			0
$562$		−	674.740i		−	1.20061i
$563$			424.331i			0.753697i	0.926275	+	0.376848i	$0.122992\pi$
$563$			424.331i			0.753697i	−0.926275	+	0.376848i	$0.877008\pi$
$564$		0			0
$565$		−	1893.13i		−	3.35068i
$566$	108.831			0.192281
$567$		0			0
$568$		−	1681.34i		−	2.96010i
$569$		−	535.710i		−	0.941495i	−0.882268	−	0.470747i	$-0.843984\pi$
$569$		−	535.710i		−	0.941495i	0.882268	−	0.470747i	$-0.156016\pi$
$570$		0			0
$571$			139.630i			0.244536i	0.992497	+	0.122268i	$0.0390167\pi$
$571$			139.630i			0.244536i	−0.992497	+	0.122268i	$0.960983\pi$
$572$	1230.74			2.15165
$573$		0			0
$574$			1099.82i			1.91606i
$575$		−	669.418i		−	1.16421i
$576$		0			0
$577$	915.488			1.58663			0.793317	−	0.608809i	$-0.208352\pi$
$577$	915.488			1.58663			0.793317	+	0.608809i	$0.208352\pi$
$578$			316.225i			0.547102i
$579$		0			0
$580$	−2411.52			−4.15780
$581$			138.001i			0.237523i
$582$		0			0
$583$		−	892.218i		−	1.53039i
$584$	456.081			0.780960
$585$		0			0
$586$		−	172.176i		−	0.293816i
$587$			255.575i			0.435391i	0.976017	+	0.217696i	$0.0698540\pi$
$587$			255.575i			0.435391i	−0.976017	+	0.217696i	$0.930146\pi$
$588$		0			0
$589$		−	87.6910i		−	0.148881i
$590$	−1851.41	−	583.086i	−3.13798	−	0.988281i
$591$		0			0
$592$		−	1530.05i		−	2.58455i
$593$	514.899			0.868295			0.434148	−	0.900842i	$-0.357050\pi$
$593$	514.899			0.868295			0.434148	+	0.900842i	$0.357050\pi$
$594$		0			0
$595$	551.790			0.927378
$596$		−	2136.81i		−	3.58525i
$597$		0			0
$598$		−	423.002i		−	0.707361i
$599$	187.458			0.312951			0.156475	−	0.987682i	$-0.449987\pi$
$599$	187.458			0.312951			0.156475	+	0.987682i	$0.449987\pi$
$600$		0			0
$601$			1103.75i			1.83653i	0.395967	+	0.918265i	$0.370410\pi$
$601$			1103.75i			1.83653i	−0.395967	+	0.918265i	$0.629590\pi$
$602$	1065.75			1.77035
$603$		0			0
$604$			2441.24i			4.04180i
$605$	−582.119			−0.962180
$606$		0			0
$607$	916.930			1.51059			0.755296	−	0.655384i	$-0.227493\pi$
$607$	916.930			1.51059			0.755296	+	0.655384i	$0.227493\pi$
$608$		−	999.409i		−	1.64376i
$609$		0			0
$610$	−2341.00			−3.83771
$611$	−467.049			−0.764400
$612$		0			0
$613$			575.358i			0.938594i	0.883040	+	0.469297i	$0.155493\pi$
$613$			575.358i			0.938594i	−0.883040	+	0.469297i	$0.844507\pi$
$614$			473.324i			0.770887i
$615$		0			0
$616$	−1182.65			−1.91988
$617$	−490.156			−0.794419			−0.397209	−	0.917728i	$-0.630021\pi$
$617$	−490.156			−0.794419			−0.397209	+	0.917728i	$0.630021\pi$
$618$		0			0
$619$	−989.393			−1.59837			−0.799187	−	0.601083i	$-0.794736\pi$
$619$	−989.393			−1.59837			−0.799187	+	0.601083i	$0.794736\pi$
$620$			379.949i			0.612820i
$621$		0			0
$622$		−	509.480i		−	0.819100i
$623$		−	10.3724i		−	0.0166492i
$624$		0			0
$625$	1056.08			1.68972
$626$	696.422			1.11250
$627$		0			0
$628$			2028.68i			3.23038i
$629$		−	609.123i		−	0.968399i
$630$		0			0
$631$	791.969			1.25510			0.627550	−	0.778576i	$-0.284058\pi$
$631$	791.969			1.25510			0.627550	+	0.778576i	$0.284058\pi$
$632$			2147.65i			3.39818i
$633$		0			0
$634$			46.6766i			0.0736224i
$635$	139.574			0.219802
$636$		0			0
$637$		−	289.045i		−	0.453759i
$638$	1421.77			2.22847
$639$		0			0
$640$		−	377.538i		−	0.589904i
$641$	−1130.03			−1.76292			−0.881460	−	0.472259i	$-0.843439\pi$
$641$	−1130.03			−1.76292			−0.881460	+	0.472259i	$0.843439\pi$
$642$		0			0
$643$	1087.76			1.69169			0.845847	−	0.533426i	$-0.179096\pi$
$643$	1087.76			1.69169			0.845847	+	0.533426i	$0.179096\pi$
$644$			494.779i			0.768291i
$645$		0			0
$646$		−	1023.89i		−	1.58497i
$647$	−331.288			−0.512037			−0.256018	−	0.966672i	$-0.582411\pi$
$647$	−331.288			−0.512037			−0.256018	+	0.966672i	$0.582411\pi$
$648$		0			0
$649$	767.356	+	241.673i	1.18237	+	0.372377i
$650$	1936.87			2.97980
$651$		0			0
$652$	1866.80			2.86318
$653$	−108.450			−0.166079			−0.0830397	−	0.996546i	$-0.526463\pi$
$653$	−108.450			−0.166079			−0.0830397	+	0.996546i	$0.526463\pi$
$654$		0			0
$655$			130.679i			0.199510i
$656$	−2480.63			−3.78146
$657$		0			0
$658$	777.094			1.18099
$659$			760.081i			1.15339i	0.816961	+	0.576693i	$0.195657\pi$
$659$			760.081i			1.15339i	−0.816961	+	0.576693i	$0.804343\pi$
$660$		0			0
$661$	−811.117			−1.22711			−0.613553	−	0.789654i	$-0.710260\pi$
$661$	−811.117			−1.22711			−0.613553	+	0.789654i	$0.710260\pi$
$662$		−	901.765i		−	1.36218i
$663$		0			0
$664$	−640.780			−0.965030
$665$	758.987			1.14133
$666$		0			0
$667$		−	343.528i		−	0.515034i
$668$	−2091.54			−3.13105
$669$		0			0
$670$	−2180.42			−3.25436
$671$	970.280			1.44602
$672$		0			0
$673$			254.007i			0.377424i	0.982032	+	0.188712i	$0.0604313\pi$
$673$			254.007i			0.377424i	−0.982032	+	0.188712i	$0.939569\pi$
$674$	−46.0912			−0.0683846
$675$		0			0
$676$	−739.723			−1.09426
$677$	1111.68			1.64207			0.821035	−	0.570878i	$-0.193397\pi$
$677$	1111.68			1.64207			0.821035	+	0.570878i	$0.193397\pi$
$678$		0			0
$679$		−	199.913i		−	0.294422i
$680$			2562.13i			3.76784i
$681$		0			0
$682$		−	224.007i		−	0.328456i
$683$			446.263i			0.653387i	0.945130	+	0.326693i	$0.105934\pi$
$683$			446.263i			0.653387i	−0.945130	+	0.326693i	$0.894066\pi$
$684$		0			0
$685$	883.692			1.29006
$686$			1258.11i			1.83398i
$687$		0			0
$688$			2403.79i			3.49389i
$689$		−	623.746i		−	0.905292i
$690$		0			0
$691$			997.782i			1.44397i	0.691910	+	0.721984i	$0.256770\pi$
$691$			997.782i			1.44397i	−0.691910	+	0.721984i	$0.743230\pi$
$692$			1874.93i			2.70943i
$693$		0			0
$694$	−567.026			−0.817041
$695$	−22.1935			−0.0319330
$696$		0			0
$697$	−987.555			−1.41686
$698$	−1756.43			−2.51638
$699$		0			0
$700$	−2265.53			−3.23647
$701$		−	483.823i		−	0.690190i	−0.938568	−	0.345095i	$-0.887847\pi$
$701$		−	483.823i		−	0.690190i	0.938568	−	0.345095i	$-0.112153\pi$
$702$		0			0
$703$		−	837.848i		−	1.19182i
$704$		−	601.748i		−	0.854756i
$705$		0			0
$706$	−1190.51			−1.68627
$707$			823.849i			1.16527i
$708$		0			0
$709$	−585.052			−0.825179			−0.412590	−	0.910917i	$-0.635376\pi$
$709$	−585.052			−0.825179			−0.412590	+	0.910917i	$0.635376\pi$
$710$			2756.38i			3.88223i
$711$		0			0
$712$	48.1623			0.0676437
$713$	−54.1247			−0.0759112
$714$		0			0
$715$	−1165.28			−1.62976
$716$		−	2338.46i		−	3.26601i
$717$		0			0
$718$		−	428.955i		−	0.597430i
$719$		−	928.729i		−	1.29170i	−0.763466	−	0.645848i	$-0.776504\pi$
$719$		−	928.729i		−	1.29170i	0.763466	−	0.645848i	$-0.223496\pi$
$720$		0			0
$721$			405.755i			0.562767i
$722$		−	83.5245i		−	0.115685i
$723$		0			0
$724$	1186.74			1.63915
$725$	1572.97			2.16961
$726$		0			0
$727$	983.354			1.35262			0.676309	−	0.736618i	$-0.263578\pi$
$727$	983.354			1.35262			0.676309	+	0.736618i	$0.263578\pi$
$728$	−826.784			−1.13569
$729$		0			0
$730$	−747.699			−1.02425
$731$			956.964i			1.30912i
$732$		0			0
$733$	−439.097			−0.599041			−0.299520	−	0.954090i	$-0.596827\pi$
$733$	−439.097			−0.599041			−0.299520	+	0.954090i	$0.596827\pi$
$734$	1152.74			1.57049
$735$		0			0
$736$	−616.855			−0.838119
$737$	903.723			1.22622
$738$		0			0
$739$		−	1203.22i		−	1.62817i	−0.580744	−	0.814086i	$-0.697238\pi$
$739$		−	1203.22i		−	1.62817i	0.580744	−	0.814086i	$-0.302762\pi$
$740$			3630.24i			4.90573i
$741$		0			0
$742$			1037.81i			1.39867i
$743$	316.428			0.425878			0.212939	−	0.977065i	$-0.431696\pi$
$743$	316.428			0.425878			0.212939	+	0.977065i	$0.431696\pi$
$744$		0			0
$745$			2023.15i			2.71564i
$746$		−	299.030i		−	0.400845i
$747$		0			0
$748$		−	1838.73i		−	2.45820i
$749$	−721.095			−0.962743
$750$		0			0
$751$			1010.01i			1.34488i	0.740151	+	0.672441i	$0.234754\pi$
$751$			1010.01i			1.34488i	−0.740151	+	0.672441i	$0.765246\pi$
$752$			1752.73i			2.33076i
$753$		0			0
$754$	993.951			1.31824
$755$		−	2311.39i		−	3.06145i
$756$		0			0
$757$	892.928			1.17956			0.589781	−	0.807563i	$-0.299214\pi$
$757$	892.928			1.17956			0.589781	+	0.807563i	$0.299214\pi$
$758$			1666.72i			2.19884i
$759$		0			0
$760$			3524.21i			4.63711i
$761$	−799.760			−1.05093			−0.525467	−	0.850814i	$-0.676109\pi$
$761$	−799.760			−1.05093			−0.525467	+	0.850814i	$0.676109\pi$
$762$		0			0
$763$		−	633.863i		−	0.830751i
$764$		−	379.690i		−	0.496977i
$765$		0			0
$766$			1669.35i			2.17931i
$767$	536.455	+	168.952i	0.699420	+	0.220277i
$768$		0			0
$769$			707.413i			0.919913i	0.887941	+	0.459957i	$0.152135\pi$
$769$			707.413i			0.919913i	−0.887941	+	0.459957i	$0.847865\pi$
$770$	1938.83			2.51797
$771$		0			0
$772$	897.092			1.16204
$773$		−	321.206i		−	0.415532i	−0.978179	−	0.207766i	$-0.933381\pi$
$773$		−	321.206i		−	0.415532i	0.978179	−	0.207766i	$-0.0666192\pi$
$774$		0			0
$775$		−	247.830i		−	0.319780i
$776$	928.255			1.19620
$777$		0			0
$778$			1191.91i			1.53202i
$779$	−1358.38			−1.74375
$780$		0			0
$781$		−	1142.44i		−	1.46280i
$782$	−631.966			−0.808141
$783$		0			0
$784$	−1084.72			−1.38357
$785$		−	1920.77i		−	2.44684i
$786$		0			0
$787$	1213.75			1.54225			0.771124	−	0.636685i	$-0.219695\pi$
$787$	1213.75			1.54225			0.771124	+	0.636685i	$0.219695\pi$
$788$	3145.20			3.99137
$789$		0			0
$790$		−	3520.86i		−	4.45678i
$791$		−	912.691i		−	1.15384i
$792$		0			0
$793$	678.318			0.855383
$794$	186.483			0.234865
$795$		0			0
$796$	−3124.55			−3.92532
$797$			914.338i			1.14722i	0.819127	+	0.573612i	$0.194458\pi$
$797$			914.338i			1.14722i	−0.819127	+	0.573612i	$0.805542\pi$
$798$		0			0
$799$			697.772i			0.873307i
$800$		−	2824.50i		−	3.53062i
$801$		0			0
$802$	1490.80			1.85885
$803$	309.900			0.385928
$804$		0			0
$805$		−	468.462i		−	0.581940i
$806$		−	156.602i		−	0.194296i
$807$		0			0
$808$	−3825.38			−4.73438
$809$		−	384.689i		−	0.475512i	−0.971325	−	0.237756i	$-0.923588\pi$
$809$		−	384.689i		−	0.475512i	0.971325	−	0.237756i	$-0.0764119\pi$
$810$		0			0
$811$		−	1032.29i		−	1.27286i	−0.771332	−	0.636432i	$-0.780409\pi$
$811$		−	1032.29i		−	1.27286i	0.771332	−	0.636432i	$-0.219591\pi$
$812$	−1162.61			−1.43179
$813$		0			0
$814$		−	2140.29i		−	2.62934i
$815$	−1767.50			−2.16871
$816$		0			0
$817$			1316.30i			1.61114i
$818$	1695.86			2.07318
$819$		0			0
$820$	5885.61			7.17757
$821$			80.1892i			0.0976726i	0.998807	+	0.0488363i	$0.0155513\pi$
$821$			80.1892i			0.0976726i	−0.998807	+	0.0488363i	$0.984449\pi$
$822$		0			0
$823$			22.5732i			0.0274280i	0.999906	+	0.0137140i	$0.00436543\pi$
$823$			22.5732i			0.0274280i	−0.999906	+	0.0137140i	$0.995635\pi$
$824$	−1884.04			−2.28646
$825$		0			0
$826$	−892.575	−	281.109i	−1.08060	−	0.340326i
$827$	−264.781			−0.320171			−0.160086	−	0.987103i	$-0.551177\pi$
$827$	−264.781			−0.320171			−0.160086	+	0.987103i	$0.551177\pi$
$828$		0			0
$829$	409.691			0.494199			0.247100	−	0.968990i	$-0.420523\pi$
$829$	409.691			0.494199			0.247100	+	0.968990i	$0.420523\pi$
$830$	1050.49			1.26566
$831$		0			0
$832$		−	420.680i		−	0.505625i
$833$	−431.834			−0.518408
$834$		0			0
$835$	1980.29			2.37160
$836$		−	2529.17i		−	3.02533i
$837$		0			0
$838$	1648.25			1.96688
$839$		−	270.040i		−	0.321860i	−0.986966	−	0.160930i	$-0.948551\pi$
$839$		−	270.040i		−	0.321860i	0.986966	−	0.160930i	$-0.0514493\pi$
$840$		0			0
$841$	−33.7936			−0.0401826
$842$	−1554.07			−1.84568
$843$		0			0
$844$		−	2830.87i		−	3.35411i
$845$	700.377			0.828848
$846$		0			0
$847$	−280.643			−0.331338
$848$	−2340.78			−2.76036
$849$		0			0
$850$		−	2893.69i		−	3.40434i
$851$	−517.137			−0.607681
$852$		0			0
$853$	536.801			0.629309			0.314655	−	0.949206i	$-0.398111\pi$
$853$	536.801			0.629309			0.314655	+	0.949206i	$0.398111\pi$
$854$	−1128.61			−1.32156
$855$		0			0
$856$		−	3348.26i		−	3.91152i
$857$		−	858.124i		−	1.00131i	−0.865646	−	0.500656i	$-0.833092\pi$
$857$		−	858.124i		−	1.00131i	0.865646	−	0.500656i	$-0.166908\pi$
$858$		0			0
$859$			528.552i			0.615310i	0.951498	+	0.307655i	$0.0995443\pi$
$859$			528.552i			0.615310i	−0.951498	+	0.307655i	$0.900456\pi$
$860$		−	5703.29i		−	6.63174i
$861$		0			0
$862$	1373.94			1.59390
$863$			1292.84i			1.49808i	0.662524	+	0.749041i	$0.269485\pi$
$863$			1292.84i			1.49808i	−0.662524	+	0.749041i	$0.730515\pi$
$864$		0			0
$865$		−	1775.20i		−	2.05225i
$866$			1365.73i			1.57706i
$867$		0			0
$868$			183.176i			0.211032i
$869$			1459.30i			1.67928i
$870$		0			0
$871$	631.789			0.725360
$872$	2943.22			3.37525
$873$		0			0
$874$	−869.269			−0.994587
$875$	1176.43			1.34449
$876$		0			0
$877$	97.8137			0.111532			0.0557661	−	0.998444i	$-0.482240\pi$
$877$	97.8137			0.111532			0.0557661	+	0.998444i	$0.482240\pi$
$878$			763.993i			0.870151i
$879$		0			0
$880$			4373.03i			4.96935i
$881$			298.429i			0.338739i	0.985553	+	0.169369i	$0.0541731\pi$
$881$			298.429i			0.338739i	−0.985553	+	0.169369i	$0.945827\pi$
$882$		0			0
$883$	−201.494			−0.228193			−0.114096	−	0.993470i	$-0.536397\pi$
$883$	−201.494			−0.228193			−0.114096	+	0.993470i	$0.536397\pi$
$884$		−	1285.45i		−	1.45413i
$885$		0			0
$886$	776.844			0.876799
$887$		−	1024.69i		−	1.15523i	−0.816309	−	0.577615i	$-0.803984\pi$
$887$		−	1024.69i		−	1.15523i	0.816309	−	0.577615i	$-0.196016\pi$
$888$		0			0
$889$	67.2896			0.0756913
$890$	−78.9574			−0.0887162
$891$		0			0
$892$	−346.244			−0.388166
$893$			959.785i			1.07479i
$894$		0			0
$895$			2214.08i			2.47383i
$896$		−	182.014i		−	0.203140i
$897$		0			0
$898$			1235.60i			1.37594i
$899$		−	127.180i		−	0.141468i
$900$		0			0
$901$	−931.879			−1.03427
$902$	−3469.99			−3.84699
$903$		0			0
$904$	4237.90			4.68794
$905$	−1123.62			−1.24157
$906$		0			0
$907$	6.21913			0.00685681			0.00342840	−	0.999994i	$-0.498909\pi$
$907$	6.21913			0.00685681			0.00342840	+	0.999994i	$0.498909\pi$
$908$			971.136i			1.06953i
$909$		0			0
$910$	1355.43			1.48948
$911$	−553.376			−0.607438			−0.303719	−	0.952762i	$-0.598228\pi$
$911$	−553.376			−0.607438			−0.303719	+	0.952762i	$0.598228\pi$
$912$		0			0
$913$	−435.400			−0.476890
$914$	−1580.48			−1.72919
$915$		0			0
$916$			1173.78i			1.28142i
$917$			63.0011i			0.0687035i
$918$		0			0
$919$			197.684i			0.215107i	0.994199	+	0.107554i	$0.0343017\pi$
$919$			197.684i			0.215107i	−0.994199	+	0.107554i	$0.965698\pi$
$920$	2175.21			2.36436
$921$		0			0
$922$		−	158.248i		−	0.171635i
$923$		−	798.677i		−	0.865306i
$924$		0			0
$925$		−	2367.90i		−	2.55989i
$926$	−2817.74			−3.04292
$927$		0			0
$928$		−	1449.46i		−	1.56192i
$929$		−	647.999i		−	0.697523i	−0.937212	−	0.348761i	$-0.886602\pi$
$929$		−	647.999i		−	0.697523i	0.937212	−	0.348761i	$-0.113398\pi$
$930$		0			0
$931$	−593.987			−0.638009
$932$		−	2159.71i		−	2.31729i
$933$		0			0
$934$	−535.935			−0.573806
$935$			1740.93i			1.86196i
$936$		0			0
$937$		−	841.351i		−	0.897920i	−0.893552	−	0.448960i	$-0.851795\pi$
$937$		−	841.351i		−	0.897920i	0.893552	−	0.448960i	$-0.148205\pi$
$938$	−1051.19			−1.12068
$939$		0			0
$940$		−	4158.57i		−	4.42401i
$941$			696.791i			0.740479i	0.928936	+	0.370239i	$0.120724\pi$
$941$			696.791i			0.740479i	−0.928936	+	0.370239i	$0.879276\pi$
$942$		0			0
$943$			838.420i			0.889099i
$944$	634.041	−	2013.20i	0.671654	−	2.13263i
$945$		0			0
$946$			3362.50i			3.55444i
$947$	−1839.80			−1.94276			−0.971382	−	0.237524i	$-0.923664\pi$
$947$	−1839.80			−1.94276			−0.971382	+	0.237524i	$0.923664\pi$
$948$		0			0
$949$	216.650			0.228293
$950$		−	3980.27i		−	4.18976i
$951$		0			0
$952$			1235.22i			1.29750i
$953$	719.684			0.755178			0.377589	−	0.925973i	$-0.376753\pi$
$953$	719.684			0.755178			0.377589	+	0.925973i	$0.376753\pi$
$954$		0			0
$955$			359.494i			0.376434i
$956$	−215.173			−0.225077
$957$		0			0
$958$			1830.22i			1.91046i
$959$	426.034			0.444248
$960$		0			0
$961$	940.962			0.979149
$962$		−	1496.26i		−	1.55537i
$963$		0			0
$964$	−1026.02			−1.06433
$965$	−849.375			−0.880181
$966$		0			0
$967$		−	639.556i		−	0.661381i	−0.943739	−	0.330691i	$-0.892718\pi$
$967$		−	639.556i		−	0.661381i	0.943739	−	0.330691i	$-0.107282\pi$
$968$		−	1303.11i		−	1.34619i
$969$		0			0
$970$	−1521.78			−1.56885
$971$	1762.15			1.81478			0.907389	−	0.420291i	$-0.138072\pi$
$971$	1762.15			1.81478			0.907389	+	0.420291i	$0.138072\pi$
$972$		0			0
$973$	−10.6996			−0.0109965
$974$		−	184.797i		−	0.189730i
$975$		0			0
$976$		−	2545.58i		−	2.60818i
$977$		−	472.107i		−	0.483221i	−0.970373	−	0.241610i	$-0.922324\pi$
$977$		−	472.107i		−	0.483221i	0.970373	−	0.241610i	$-0.0776756\pi$
$978$		0			0
$979$	32.7256			0.0334276
$980$	2573.63			2.62615
$981$		0			0
$982$			694.336i			0.707063i
$983$			50.9730i			0.0518545i	0.999664	+	0.0259273i	$0.00825383\pi$
$983$			50.9730i			0.0518545i	−0.999664	+	0.0259273i	$0.991746\pi$
$984$		0			0
$985$	−2977.91			−3.02326
$986$		−	1484.97i		−	1.50605i
$987$		0			0
$988$		−	1768.13i		−	1.78961i
$989$	812.449			0.821486
$990$		0			0
$991$		−	762.578i		−	0.769503i	−0.923020	−	0.384752i	$-0.874287\pi$
$991$		−	762.578i		−	0.769503i	0.923020	−	0.384752i	$-0.125713\pi$
$992$	−228.370			−0.230212
$993$		0			0
$994$			1328.87i			1.33689i
$995$	2958.36			2.97322
$996$		0			0
$997$	−1475.40			−1.47984			−0.739918	−	0.672698i	$-0.765136\pi$
$997$	−1475.40			−1.47984			−0.739918	+	0.672698i	$0.765136\pi$
$998$			1857.29i			1.86101i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.3.c.d.235.2	yes	20
3.2	odd	2	inner	531.3.c.d.235.19	yes	20
59.58	odd	2	inner	531.3.c.d.235.20	yes	20
177.176	even	2	inner	531.3.c.d.235.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.3.c.d.235.1	✓	20	177.176	even	2	inner
531.3.c.d.235.2	yes	20	1.1	even	1	trivial
531.3.c.d.235.19	yes	20	3.2	odd	2	inner
531.3.c.d.235.20	yes	20	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.66991i q^{2} -9.46822 q^{4} +8.96460 q^{5} +4.32189 q^{7} +20.0678i q^{8} +O(q^{10})\)	\(q-3.66991i q^{2} -9.46822 q^{4} +8.96460 q^{5} +4.32189 q^{7} +20.0678i q^{8} -32.8992i q^{10} +13.6358i q^{11} +9.53274i q^{13} -15.8609i q^{14} +35.7743 q^{16} +14.2419 q^{17} +19.5898 q^{19} -84.8788 q^{20} +50.0422 q^{22} -12.0912i q^{23} +55.3640 q^{25} +34.9843 q^{26} -40.9206 q^{28} +28.4114 q^{29} -4.47637i q^{31} -51.0168i q^{32} -52.2666i q^{34} +38.7440 q^{35} -42.7697i q^{37} -71.8927i q^{38} +179.900i q^{40} -69.3413 q^{41} +67.1934i q^{43} -129.107i q^{44} -44.3736 q^{46} +48.9942i q^{47} -30.3213 q^{49} -203.181i q^{50} -90.2580i q^{52} -65.4320 q^{53} +122.240i q^{55} +86.7310i q^{56} -104.267i q^{58} +(17.7234 - 56.2751i) q^{59} -71.1567i q^{61} -16.4278 q^{62} -44.1300 q^{64} +85.4572i q^{65} -66.2757i q^{67} -134.846 q^{68} -142.187i q^{70} -83.7826 q^{71} -22.7269i q^{73} -156.961 q^{74} -185.480 q^{76} +58.9325i q^{77} +107.019 q^{79} +320.702 q^{80} +254.476i q^{82} +31.9307i q^{83} +127.673 q^{85} +246.593 q^{86} -273.641 q^{88} -2.39998i q^{89} +41.1994i q^{91} +114.482i q^{92} +179.804 q^{94} +175.614 q^{95} -46.2558i q^{97} +111.276i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) \) 20 * q - 40 * q^4 - 8 * q^7	\( 20 q - 40 q^{4} - 8 q^{7} + 88 q^{16} + 60 q^{19} + 136 q^{22} + 148 q^{25} - 136 q^{28} - 84 q^{46} - 100 q^{49} + 36 q^{64} - 552 q^{76} + 252 q^{79} + 180 q^{85} - 788 q^{88} + 584 q^{94}+O(q^{100}) \) 20 * q - 40 * q^4 - 8 * q^7 + 88 * q^16 + 60 * q^19 + 136 * q^22 + 148 * q^25 - 136 * q^28 - 84 * q^46 - 100 * q^49 + 36 * q^64 - 552 * q^76 + 252 * q^79 + 180 * q^85 - 788 * q^88 + 584 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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