LMFDB - Embedded newform 531.5.c.d.235.10

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			235.10
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.64719i q^{2} -5.59638 q^{4} +0.691812 q^{5} -76.1022 q^{7} -48.3476i q^{8} +O(q^{10})$	\(q-4.64719i q^{2} -5.59638 q^{4} +0.691812 q^{5} -76.1022 q^{7} -48.3476i q^{8} -3.21498i q^{10} -74.8765i q^{11} -105.330i q^{13} +353.662i q^{14} -314.223 q^{16} +141.930 q^{17} -170.820 q^{19} -3.87164 q^{20} -347.966 q^{22} -226.464i q^{23} -624.521 q^{25} -489.487 q^{26} +425.897 q^{28} +677.129 q^{29} -114.554i q^{31} +686.691i q^{32} -659.575i q^{34} -52.6484 q^{35} +488.383i q^{37} +793.831i q^{38} -33.4475i q^{40} +1825.63 q^{41} +527.862i q^{43} +419.037i q^{44} -1052.42 q^{46} +1903.37i q^{47} +3390.55 q^{49} +2902.27i q^{50} +589.465i q^{52} -4683.03 q^{53} -51.8005i q^{55} +3679.36i q^{56} -3146.75i q^{58} +(-2331.20 + 2585.12i) q^{59} +6414.67i q^{61} -532.355 q^{62} -1836.38 q^{64} -72.8684i q^{65} -1772.67i q^{67} -794.294 q^{68} +244.667i q^{70} -170.581 q^{71} +8999.83i q^{73} +2269.61 q^{74} +955.971 q^{76} +5698.27i q^{77} -204.796 q^{79} -217.383 q^{80} -8484.05i q^{82} -1391.38i q^{83} +98.1888 q^{85} +2453.07 q^{86} -3620.10 q^{88} -3001.60i q^{89} +8015.83i q^{91} +1267.38i q^{92} +8845.32 q^{94} -118.175 q^{95} +9224.90i q^{97} -15756.5i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	4.64719i		−	1.16180i	−0.813976	−	0.580899i	$-0.802701\pi$
$2$		−	4.64719i		−	1.16180i	0.813976	−	0.580899i	$-0.197299\pi$
$3$		0			0
$3$		0			0
$4$	−5.59638			−0.349774
$5$	0.691812			0.0276725			0.0138362	−	0.999904i	$-0.495596\pi$
$5$	0.691812			0.0276725			0.0138362	+	0.999904i	$0.495596\pi$
$6$		0			0
$7$	−76.1022			−1.55311			−0.776553	−	0.630051i	$-0.783034\pi$
$7$	−76.1022			−1.55311			−0.776553	+	0.630051i	$0.783034\pi$
$8$		−	48.3476i		−	0.755431i
$9$		0			0
$10$		−	3.21498i		−	0.0321498i
$11$		−	74.8765i		−	0.618814i	−0.950930	−	0.309407i	$-0.899869\pi$
$11$		−	74.8765i		−	0.618814i	0.950930	−	0.309407i	$-0.100131\pi$
$12$		0			0
$13$		−	105.330i		−	0.623253i	−0.950205	−	0.311626i	$-0.899126\pi$
$13$		−	105.330i		−	0.623253i	0.950205	−	0.311626i	$-0.100874\pi$
$14$			353.662i			1.80440i
$15$		0			0
$16$	−314.223			−1.22743
$17$	141.930			0.491107			0.245553	−	0.969383i	$-0.421030\pi$
$17$	141.930			0.491107			0.245553	+	0.969383i	$0.421030\pi$
$18$		0			0
$19$	−170.820			−0.473184			−0.236592	−	0.971609i	$-0.576031\pi$
$19$	−170.820			−0.473184			−0.236592	+	0.971609i	$0.576031\pi$
$20$	−3.87164			−0.00967911
$21$		0			0
$22$	−347.966			−0.718937
$23$		−	226.464i		−	0.428098i	−0.976823	−	0.214049i	$-0.931335\pi$
$23$		−	226.464i		−	0.428098i	0.976823	−	0.214049i	$-0.0686653\pi$
$24$		0			0
$25$	−624.521			−0.999234
$26$	−489.487			−0.724093
$27$		0			0
$28$	425.897			0.543236
$29$	677.129			0.805148			0.402574	−	0.915388i	$-0.368116\pi$
$29$	677.129			0.805148			0.402574	+	0.915388i	$0.368116\pi$
$30$		0			0
$31$		−	114.554i		−	0.119203i	−0.998222	−	0.0596016i	$-0.981017\pi$
$31$		−	114.554i		−	0.119203i	0.998222	−	0.0596016i	$-0.0189830\pi$
$32$			686.691i			0.670596i
$33$		0			0
$34$		−	659.575i		−	0.570567i
$35$	−52.6484			−0.0429783
$36$		0			0
$37$			488.383i			0.356745i	0.983963	+	0.178372i	$0.0570831\pi$
$37$			488.383i			0.356745i	−0.983963	+	0.178372i	$0.942917\pi$
$38$			793.831i			0.549744i
$39$		0			0
$40$		−	33.4475i		−	0.0209047i
$41$	1825.63			1.08604			0.543019	−	0.839720i	$-0.317281\pi$
$41$	1825.63			1.08604			0.543019	+	0.839720i	$0.317281\pi$
$42$		0			0
$43$			527.862i			0.285485i	0.989760	+	0.142742i	$0.0455921\pi$
$43$			527.862i			0.285485i	−0.989760	+	0.142742i	$0.954408\pi$
$44$			419.037i			0.216445i
$45$		0			0
$46$	−1052.42			−0.497364
$47$			1903.37i			0.861643i	0.902437	+	0.430822i	$0.141776\pi$
$47$			1903.37i			0.861643i	−0.902437	+	0.430822i	$0.858224\pi$
$48$		0			0
$49$	3390.55			1.41214
$50$			2902.27i			1.16091i
$51$		0			0
$52$			589.465i			0.217997i
$53$	−4683.03			−1.66715			−0.833576	−	0.552404i	$-0.813710\pi$
$53$	−4683.03			−1.66715			−0.833576	+	0.552404i	$0.813710\pi$
$54$		0			0
$55$		−	51.8005i		−	0.0171241i
$56$			3679.36i			1.17327i
$57$		0			0
$58$		−	3146.75i		−	0.935419i
$59$	−2331.20	+	2585.12i	−0.669693	+	0.742638i
$59$	−2331.20	+	2585.12i	−0.669693	+	0.742638i
$60$		0			0
$61$			6414.67i			1.72391i	0.506985	+	0.861955i	$0.330760\pi$
$61$			6414.67i			1.72391i	−0.506985	+	0.861955i	$0.669240\pi$
$62$	−532.355			−0.138490
$63$		0			0
$64$	−1836.38			−0.448335
$65$		−	72.8684i		−	0.0172469i
$66$		0			0
$67$		−	1772.67i		−	0.394892i	−0.980314	−	0.197446i	$-0.936735\pi$
$67$		−	1772.67i		−	0.394892i	0.980314	−	0.197446i	$-0.0632647\pi$
$68$	−794.294			−0.171776
$69$		0			0
$70$			244.667i			0.0499321i
$71$	−170.581			−0.0338387			−0.0169193	−	0.999857i	$-0.505386\pi$
$71$	−170.581			−0.0338387			−0.0169193	+	0.999857i	$0.505386\pi$
$72$		0			0
$73$			8999.83i			1.68884i	0.535682	+	0.844420i	$0.320055\pi$
$73$			8999.83i			1.68884i	−0.535682	+	0.844420i	$0.679945\pi$
$74$	2269.61			0.414465
$75$		0			0
$76$	955.971			0.165507
$77$			5698.27i			0.961085i
$78$		0			0
$79$	−204.796			−0.0328146			−0.0164073	−	0.999865i	$-0.505223\pi$
$79$	−204.796			−0.0328146			−0.0164073	+	0.999865i	$0.505223\pi$
$80$	−217.383			−0.0339661
$81$		0			0
$82$		−	8484.05i		−	1.26176i
$83$		−	1391.38i		−	0.201971i	−0.994888	−	0.100986i	$-0.967800\pi$
$83$		−	1391.38i		−	0.201971i	0.994888	−	0.100986i	$-0.0321996\pi$
$84$		0			0
$85$	98.1888			0.0135901
$86$	2453.07			0.331676
$87$		0			0
$88$	−3620.10			−0.467472
$89$		−	3001.60i		−	0.378942i	−0.981886	−	0.189471i	$-0.939323\pi$
$89$		−	3001.60i		−	0.378942i	0.981886	−	0.189471i	$-0.0606773\pi$
$90$		0			0
$91$			8015.83i			0.967978i
$92$			1267.38i			0.149738i
$93$		0			0
$94$	8845.32			1.00105
$95$	−118.175			−0.0130942
$96$		0			0
$97$			9224.90i			0.980434i	0.871600	+	0.490217i	$0.163082\pi$
$97$			9224.90i			0.980434i	−0.871600	+	0.490217i	$0.836918\pi$
$98$		−	15756.5i		−	1.64062i
$99$		0			0
$100$	3495.06			0.349506
$101$		−	15773.4i		−	1.54626i	−0.634250	−	0.773128i	$-0.718691\pi$
$101$		−	15773.4i		−	1.54626i	0.634250	−	0.773128i	$-0.281309\pi$
$102$		0			0
$103$			12464.1i			1.17486i	0.809273	+	0.587432i	$0.199861\pi$
$103$			12464.1i			1.17486i	−0.809273	+	0.587432i	$0.800139\pi$
$104$	−5092.44			−0.470825
$105$		0			0
$106$			21762.9i			1.93689i
$107$	5179.20			0.452372			0.226186	−	0.974084i	$-0.427374\pi$
$107$	5179.20			0.452372			0.226186	+	0.974084i	$0.427374\pi$
$108$		0			0
$109$			653.130i			0.0549727i	0.999622	+	0.0274863i	$0.00875028\pi$
$109$			653.130i			0.0549727i	−0.999622	+	0.0274863i	$0.991250\pi$
$110$	−240.727			−0.0198948
$111$		0			0
$112$	23913.0			1.90633
$113$			12708.3i			0.995247i	0.867393	+	0.497623i	$0.165794\pi$
$113$			12708.3i			0.995247i	−0.867393	+	0.497623i	$0.834206\pi$
$114$		0			0
$115$		−	156.671i		−	0.0118465i
$116$	−3789.47			−0.281619
$117$		0			0
$118$	12013.6	+	10833.5i	0.862795	+	0.778048i
$119$	−10801.2			−0.762742
$120$		0			0
$121$	9034.50			0.617069
$122$	29810.2			2.00283
$123$		0			0
$124$			641.089i			0.0416941i
$125$	−864.434			−0.0553238
$126$		0			0
$127$	−3763.18			−0.233318			−0.116659	−	0.993172i	$-0.537218\pi$
$127$	−3763.18			−0.233318			−0.116659	+	0.993172i	$0.537218\pi$
$128$			19521.1i			1.19147i
$129$		0			0
$130$	−338.633			−0.0200375
$131$			1872.87i			0.109135i	0.998510	+	0.0545677i	$0.0173781\pi$
$131$			1872.87i			0.109135i	−0.998510	+	0.0545677i	$0.982622\pi$
$132$		0			0
$133$	12999.7			0.734906
$134$	−8237.93			−0.458784
$135$		0			0
$136$		−	6861.97i		−	0.370998i
$137$	7217.96			0.384568			0.192284	−	0.981339i	$-0.438410\pi$
$137$	7217.96			0.384568			0.192284	+	0.981339i	$0.438410\pi$
$138$		0			0
$139$	−4580.22			−0.237059			−0.118530	−	0.992951i	$-0.537818\pi$
$139$	−4580.22			−0.237059			−0.118530	+	0.992951i	$0.537818\pi$
$140$	294.641			0.0150327
$141$		0			0
$142$			792.721i			0.0393137i
$143$	−7886.72			−0.385678
$144$		0			0
$145$	468.446			0.0222804
$146$	41823.9			1.96209
$147$		0			0
$148$		−	2733.18i		−	0.124780i
$149$		−	26501.4i		−	1.19370i	−0.802352	−	0.596851i	$-0.796418\pi$
$149$		−	26501.4i		−	1.19370i	0.802352	−	0.596851i	$-0.203582\pi$
$150$		0			0
$151$		−	28179.8i		−	1.23590i	−0.786217	−	0.617950i	$-0.787963\pi$
$151$		−	28179.8i		−	1.23590i	0.786217	−	0.617950i	$-0.212037\pi$
$152$			8258.72i			0.357458i
$153$		0			0
$154$	26481.0			1.11659
$155$		−	79.2500i		−	0.00329865i
$156$		0			0
$157$		−	12508.2i		−	0.507452i	−0.967276	−	0.253726i	$-0.918344\pi$
$157$		−	12508.2i		−	0.507452i	0.967276	−	0.253726i	$-0.0816562\pi$
$158$			951.725i			0.0381239i
$159$		0			0
$160$			475.061i			0.0185571i
$161$			17234.4i			0.664882i
$162$		0			0
$163$	−17112.4			−0.644075			−0.322038	−	0.946727i	$-0.604368\pi$
$163$	−17112.4			−0.644075			−0.322038	+	0.946727i	$0.604368\pi$
$164$	−10216.9			−0.379867
$165$		0			0
$166$	−6466.00			−0.234650
$167$	5275.71			0.189168			0.0945841	−	0.995517i	$-0.469848\pi$
$167$	5275.71			0.189168			0.0945841	+	0.995517i	$0.469848\pi$
$168$		0			0
$169$	17466.7			0.611556
$170$		−	456.302i		−	0.0157890i
$171$		0			0
$172$		−	2954.11i		−	0.0998551i
$173$			19148.2i			0.639786i	0.947454	+	0.319893i	$0.103647\pi$
$173$			19148.2i			0.639786i	−0.947454	+	0.319893i	$0.896353\pi$
$174$		0			0
$175$	47527.5			1.55192
$176$			23527.9i			0.759553i
$177$		0			0
$178$	−13949.0			−0.440254
$179$		−	59089.1i		−	1.84417i	−0.386987	−	0.922085i	$-0.626484\pi$
$179$		−	59089.1i		−	1.84417i	0.386987	−	0.922085i	$-0.373516\pi$
$180$		0			0
$181$	21278.1			0.649494			0.324747	−	0.945801i	$-0.394721\pi$
$181$	21278.1			0.649494			0.324747	+	0.945801i	$0.394721\pi$
$182$	37251.1			1.12459
$183$		0			0
$184$	−10949.0			−0.323399
$185$			337.869i			0.00987201i
$186$		0			0
$187$		−	10627.2i		−	0.303904i
$188$		−	10652.0i		−	0.301380i
$189$		0			0
$190$			549.182i			0.0152128i
$191$		−	12072.7i		−	0.330932i	−0.986215	−	0.165466i	$-0.947087\pi$
$191$		−	12072.7i		−	0.330932i	0.986215	−	0.165466i	$-0.0529129\pi$
$192$		0			0
$193$	−43480.2			−1.16728			−0.583642	−	0.812011i	$-0.698373\pi$
$193$	−43480.2			−1.16728			−0.583642	+	0.812011i	$0.698373\pi$
$194$	42869.9			1.13907
$195$		0			0
$196$	−18974.8			−0.493930
$197$	−28225.1			−0.727283			−0.363642	−	0.931539i	$-0.618467\pi$
$197$	−28225.1			−0.727283			−0.363642	+	0.931539i	$0.618467\pi$
$198$		0			0
$199$	−46347.4			−1.17036			−0.585180	−	0.810903i	$-0.698976\pi$
$199$	−46347.4			−1.17036			−0.585180	+	0.810903i	$0.698976\pi$
$200$			30194.1i			0.754853i
$201$		0			0
$202$	−73301.8			−1.79644
$203$	−51531.0			−1.25048
$204$		0			0
$205$	1262.99			0.0300534
$206$	57923.2			1.36495
$207$		0			0
$208$			33097.0i			0.765000i
$209$			12790.4i			0.292813i
$210$		0			0
$211$		−	49595.9i		−	1.11399i	−0.830516	−	0.556994i	$-0.811955\pi$
$211$		−	49595.9i		−	1.11399i	0.830516	−	0.556994i	$-0.188045\pi$
$212$	26208.0			0.583126
$213$		0			0
$214$		−	24068.7i		−	0.525564i
$215$			365.181i			0.00790008i
$216$		0			0
$217$			8717.83i			0.185135i
$218$	3035.22			0.0638671
$219$		0			0
$220$			289.895i			0.00598957i
$221$		−	14949.4i		−	0.306084i
$222$		0			0
$223$	27897.9			0.560998			0.280499	−	0.959854i	$-0.409500\pi$
$223$	27897.9			0.560998			0.280499	+	0.959854i	$0.409500\pi$
$224$		−	52258.7i		−	1.04151i
$225$		0			0
$226$	59057.9			1.15628
$227$			34726.4i			0.673919i	0.941519	+	0.336960i	$0.109399\pi$
$227$			34726.4i			0.673919i	−0.941519	+	0.336960i	$0.890601\pi$
$228$		0			0
$229$			57357.6i			1.09376i	0.837212	+	0.546878i	$0.184184\pi$
$229$			57357.6i			1.09376i	−0.837212	+	0.546878i	$0.815816\pi$
$230$	−728.078			−0.0137633
$231$		0			0
$232$		−	32737.6i		−	0.608234i
$233$			64148.1i			1.18160i	0.806816	+	0.590802i	$0.201189\pi$
$233$			64148.1i			1.18160i	−0.806816	+	0.590802i	$0.798811\pi$
$234$		0			0
$235$			1316.77i			0.0238438i
$236$	13046.3	−	14467.3i	0.234241	−	0.259755i
$237$		0			0
$238$			50195.2i			0.886151i
$239$	33929.5			0.593994			0.296997	−	0.954878i	$-0.404015\pi$
$239$	33929.5			0.593994			0.296997	+	0.954878i	$0.404015\pi$
$240$		0			0
$241$	−35255.6			−0.607007			−0.303503	−	0.952830i	$-0.598156\pi$
$241$	−35255.6			−0.607007			−0.303503	+	0.952830i	$0.598156\pi$
$242$		−	41985.1i		−	0.716909i
$243$		0			0
$244$		−	35898.9i		−	0.602978i
$245$	2345.62			0.0390775
$246$		0			0
$247$			17992.4i			0.294913i
$248$	−5538.42			−0.0900498
$249$		0			0
$250$			4017.19i			0.0642750i
$251$	−57278.6			−0.909170			−0.454585	−	0.890703i	$-0.650212\pi$
$251$	−57278.6			−0.909170			−0.454585	+	0.890703i	$0.650212\pi$
$252$		0			0
$253$	−16956.8			−0.264913
$254$			17488.2i			0.271068i
$255$		0			0
$256$	61336.0			0.935913
$257$	15290.7			0.231506			0.115753	−	0.993278i	$-0.463072\pi$
$257$	15290.7			0.231506			0.115753	+	0.993278i	$0.463072\pi$
$258$		0			0
$259$		−	37167.1i		−	0.554062i
$260$			407.799i			0.00603253i
$261$		0			0
$262$	8703.60			0.126793
$263$	−90982.9			−1.31537			−0.657685	−	0.753293i	$-0.728464\pi$
$263$	−90982.9			−1.31537			−0.657685	+	0.753293i	$0.728464\pi$
$264$		0			0
$265$	−3239.78			−0.0461342
$266$		−	60412.3i		−	0.853812i
$267$		0			0
$268$			9920.53i			0.138123i
$269$			68865.4i			0.951692i	0.879529	+	0.475846i	$0.157858\pi$
$269$			68865.4i			0.951692i	−0.879529	+	0.475846i	$0.842142\pi$
$270$		0			0
$271$	−120932.			−1.64666			−0.823330	−	0.567563i	$-0.807886\pi$
$271$	−120932.			−1.64666			−0.823330	+	0.567563i	$0.807886\pi$
$272$	−44597.6			−0.602800
$273$		0			0
$274$		−	33543.2i		−	0.446790i
$275$			46762.0i			0.618340i
$276$		0			0
$277$	81260.6			1.05906			0.529530	−	0.848291i	$-0.322368\pi$
$277$	81260.6			1.05906			0.529530	+	0.848291i	$0.322368\pi$
$278$			21285.1i			0.275415i
$279$		0			0
$280$			2545.43i			0.0324672i
$281$	58788.8			0.744530			0.372265	−	0.928127i	$-0.378581\pi$
$281$	58788.8			0.744530			0.372265	+	0.928127i	$0.378581\pi$
$282$		0			0
$283$			74223.1i			0.926757i	0.886160	+	0.463379i	$0.153363\pi$
$283$			74223.1i			0.926757i	−0.886160	+	0.463379i	$0.846637\pi$
$284$	954.634			0.0118359
$285$		0			0
$286$			36651.1i			0.448079i
$287$	−138934.			−1.68673
$288$		0			0
$289$	−63376.9			−0.758814
$290$		−	2176.96i		−	0.0258854i
$291$		0			0
$292$		−	50366.4i		−	0.590712i
$293$	−105265.			−1.22617			−0.613083	−	0.790019i	$-0.710071\pi$
$293$	−105265.			−1.22617			−0.613083	+	0.790019i	$0.710071\pi$
$294$		0			0
$295$	−1612.75	+	1788.42i	−0.0185321	+	0.0205506i
$296$	23612.2			0.269496
$297$		0			0
$298$	−123157.			−1.38684
$299$	−23853.4			−0.266813
$300$		0			0
$301$		−	40171.5i		−	0.443389i
$302$	−130957.			−1.43587
$303$		0			0
$304$	53675.4			0.580802
$305$			4437.75i			0.0477049i
$306$		0			0
$307$	3547.26			0.0376371			0.0188185	−	0.999823i	$-0.494010\pi$
$307$	3547.26			0.0376371			0.0188185	+	0.999823i	$0.494010\pi$
$308$		−	31889.7i		−	0.336162i
$309$		0			0
$310$	−368.290			−0.00383236
$311$	−151291.			−1.56420			−0.782102	−	0.623150i	$-0.785853\pi$
$311$	−151291.			−1.56420			−0.782102	+	0.623150i	$0.785853\pi$
$312$		0			0
$313$		−	25650.3i		−	0.261821i	−0.991394	−	0.130911i	$-0.958210\pi$
$313$		−	25650.3i		−	0.261821i	0.991394	−	0.130911i	$-0.0417901\pi$
$314$	−58128.0			−0.589557
$315$		0			0
$316$	1146.11			0.0114777
$317$	−129307.			−1.28677			−0.643387	−	0.765541i	$-0.722471\pi$
$317$	−129307.			−1.28677			−0.643387	+	0.765541i	$0.722471\pi$
$318$		0			0
$319$		−	50701.1i		−	0.498237i
$320$	−1270.43			−0.0124065
$321$		0			0
$322$	80091.6			0.772459
$323$	−24244.4			−0.232384
$324$		0			0
$325$			65780.6i			0.622775i
$326$			79524.7i			0.748285i
$327$		0			0
$328$		−	88264.8i		−	0.820427i
$329$		−	144851.i		−	1.33822i
$330$		0			0
$331$	−132570.			−1.21001			−0.605004	−	0.796223i	$-0.706828\pi$
$331$	−132570.			−1.21001			−0.605004	+	0.796223i	$0.706828\pi$
$332$			7786.68i			0.0706442i
$333$		0			0
$334$		−	24517.2i		−	0.219775i
$335$		−	1226.35i		−	0.0109276i
$336$		0			0
$337$		−	182026.i		−	1.60277i	−0.598146	−	0.801387i	$-0.704096\pi$
$337$		−	182026.i		−	1.60277i	0.598146	−	0.801387i	$-0.295904\pi$
$338$		−	81170.9i		−	0.710505i
$339$		0			0
$340$	−549.502			−0.00475348
$341$	−8577.42			−0.0737646
$342$		0			0
$343$	−75307.0			−0.640099
$344$	25520.8			0.215664
$345$		0			0
$346$	88985.1			0.743302
$347$			36084.6i			0.299684i	0.988710	+	0.149842i	$0.0478765\pi$
$347$			36084.6i			0.299684i	−0.988710	+	0.149842i	$0.952124\pi$
$348$		0			0
$349$			155958.i			1.28043i	0.768194	+	0.640217i	$0.221156\pi$
$349$			155958.i			1.28043i	−0.768194	+	0.640217i	$0.778844\pi$
$350$		−	220869.i		−	1.80301i
$351$		0			0
$352$	51417.0			0.414975
$353$		−	106349.i		−	0.853464i	−0.904378	−	0.426732i	$-0.859665\pi$
$353$		−	106349.i		−	0.853464i	0.904378	−	0.426732i	$-0.140335\pi$
$354$		0			0
$355$	−118.010			−0.000936400
$356$			16798.1i			0.132544i
$357$		0			0
$358$	−274598.			−2.14255
$359$	221323.			1.71726			0.858632	−	0.512592i	$-0.171315\pi$
$359$	221323.			1.71726			0.858632	+	0.512592i	$0.171315\pi$
$360$		0			0
$361$	−101142.			−0.776097
$362$		−	98883.2i		−	0.754580i
$363$		0			0
$364$		−	44859.6i		−	0.338573i
$365$			6226.19i			0.0467344i
$366$		0			0
$367$		−	231374.i		−	1.71784i	−0.512108	−	0.858921i	$-0.671135\pi$
$367$		−	231374.i		−	1.71784i	0.512108	−	0.858921i	$-0.328865\pi$
$368$			71160.1i			0.525462i
$369$		0			0
$370$	1570.14			0.0114693
$371$	356389.			2.58927
$372$		0			0
$373$	129853.			0.933327			0.466663	−	0.884435i	$-0.345456\pi$
$373$	129853.			0.933327			0.466663	+	0.884435i	$0.345456\pi$
$374$	−49386.7			−0.353075
$375$		0			0
$376$	92023.4			0.650912
$377$		−	71321.8i		−	0.501810i
$378$		0			0
$379$	−271372.			−1.88924			−0.944618	−	0.328173i	$-0.893567\pi$
$379$	−271372.			−1.88924			−0.944618	+	0.328173i	$0.893567\pi$
$380$	661.352			0.00458000
$381$		0			0
$382$	−56104.4			−0.384477
$383$	228592.			1.55834			0.779171	−	0.626811i	$-0.215640\pi$
$383$	228592.			1.55834			0.779171	+	0.626811i	$0.215640\pi$
$384$		0			0
$385$			3942.13i			0.0265956i
$386$			202061.i			1.35615i
$387$		0			0
$388$		−	51626.1i		−	0.342930i
$389$	−19842.8			−0.131131			−0.0655654	−	0.997848i	$-0.520885\pi$
$389$	−19842.8			−0.131131			−0.0655654	+	0.997848i	$0.520885\pi$
$390$		0			0
$391$		−	32142.0i		−	0.210242i
$392$		−	163925.i		−	1.06678i
$393$		0			0
$394$			131168.i			0.844956i
$395$	−141.680			−0.000908061
$396$		0			0
$397$			286794.i			1.81966i	0.414986	+	0.909828i	$0.363787\pi$
$397$			286794.i			1.81966i	−0.414986	+	0.909828i	$0.636213\pi$
$398$			215385.i			1.35972i
$399$		0			0
$400$	196239.			1.22649
$401$			227378.i			1.41403i	0.707198	+	0.707016i	$0.249959\pi$
$401$			227378.i			1.41403i	−0.707198	+	0.707016i	$0.750041\pi$
$402$		0			0
$403$	−12066.0			−0.0742937
$404$			88273.7i			0.540839i
$405$		0			0
$406$			239475.i			1.45281i
$407$	36568.4			0.220759
$408$		0			0
$409$		−	158393.i		−	0.946870i	−0.880829	−	0.473435i	$-0.843014\pi$
$409$		−	158393.i		−	0.946870i	0.880829	−	0.473435i	$-0.156986\pi$
$410$		−	5869.37i		−	0.0349159i
$411$		0			0
$412$		−	69754.0i		−	0.410937i
$413$	177410.	−	196734.i	1.04011	−	1.15340i
$414$		0			0
$415$		−	962.573i		−	0.00558904i
$416$	72328.9			0.417951
$417$		0			0
$418$	59439.3			0.340190
$419$		−	122537.i		−	0.697976i	−0.937127	−	0.348988i	$-0.886525\pi$
$419$		−	122537.i		−	0.697976i	0.937127	−	0.348988i	$-0.113475\pi$
$420$		0			0
$421$		−	46500.7i		−	0.262359i	−0.991359	−	0.131179i	$-0.958124\pi$
$421$		−	46500.7i		−	0.262359i	0.991359	−	0.131179i	$-0.0418764\pi$
$422$	−230481.			−1.29423
$423$		0			0
$424$			226413.i			1.25942i
$425$	−88638.3			−0.490731
$426$		0			0
$427$		−	488171.i		−	2.67742i
$428$	−28984.8			−0.158228
$429$		0			0
$430$	1697.07			0.00917829
$431$			17223.7i			0.0927198i	0.998925	+	0.0463599i	$0.0147621\pi$
$431$			17223.7i			0.0927198i	−0.998925	+	0.0463599i	$0.985238\pi$
$432$		0			0
$433$	28363.6			0.151281			0.0756407	−	0.997135i	$-0.475900\pi$
$433$	28363.6			0.151281			0.0756407	+	0.997135i	$0.475900\pi$
$434$	40513.4			0.215090
$435$		0			0
$436$		−	3655.17i		−	0.0192280i
$437$			38684.5i			0.202569i
$438$		0			0
$439$	−222696.			−1.15554			−0.577769	−	0.816200i	$-0.696077\pi$
$439$	−222696.			−1.15554			−0.577769	+	0.816200i	$0.696077\pi$
$440$	−2504.43			−0.0129361
$441$		0			0
$442$	−69472.9			−0.355607
$443$			174192.i			0.887609i	0.896124	+	0.443804i	$0.146371\pi$
$443$			174192.i			0.887609i	−0.896124	+	0.443804i	$0.853629\pi$
$444$		0			0
$445$		−	2076.54i		−	0.0104863i
$446$		−	129647.i		−	0.651766i
$447$		0			0
$448$	139753.			0.696312
$449$	24381.9			0.120941			0.0604706	−	0.998170i	$-0.480740\pi$
$449$	24381.9			0.120941			0.0604706	+	0.998170i	$0.480740\pi$
$450$		0			0
$451$		−	136697.i		−	0.672056i
$452$		−	71120.5i		−	0.348111i
$453$		0			0
$454$	161380.			0.782958
$455$			5545.44i			0.0267864i
$456$		0			0
$457$		−	278285.i		−	1.33247i	−0.745741	−	0.666236i	$-0.767904\pi$
$457$		−	278285.i		−	1.33247i	0.745741	−	0.666236i	$-0.232096\pi$
$458$	266552.			1.27072
$459$		0			0
$460$			876.788i			0.00414361i
$461$	−13805.0			−0.0649581			−0.0324791	−	0.999472i	$-0.510340\pi$
$461$	−13805.0			−0.0649581			−0.0324791	+	0.999472i	$0.510340\pi$
$462$		0			0
$463$			129315.i			0.603234i	0.953429	+	0.301617i	$0.0975264\pi$
$463$			129315.i			0.603234i	−0.953429	+	0.301617i	$0.902474\pi$
$464$	−212769.			−0.988264
$465$		0			0
$466$	298109.			1.37279
$467$			294700.i			1.35128i	0.737230	+	0.675641i	$0.236133\pi$
$467$			294700.i			1.35128i	−0.737230	+	0.675641i	$0.763867\pi$
$468$		0			0
$469$			134904.i			0.613309i
$470$	6119.30			0.0277017
$471$		0			0
$472$	124984.	+	112708.i	0.561012	+	0.505907i
$473$	39524.5			0.176662
$474$		0			0
$475$	106680.			0.472822
$476$	60447.5			0.266787
$477$		0			0
$478$		−	157677.i		−	0.690100i
$479$	144629.			0.630355			0.315177	−	0.949033i	$-0.397936\pi$
$479$	144629.			0.630355			0.315177	+	0.949033i	$0.397936\pi$
$480$		0			0
$481$	51441.3			0.222342
$482$			163839.i			0.705219i
$483$		0			0
$484$	−50560.5			−0.215834
$485$			6381.90i			0.0271310i
$486$		0			0
$487$	−275005.			−1.15953			−0.579766	−	0.814783i	$-0.696856\pi$
$487$	−275005.			−1.15953			−0.579766	+	0.814783i	$0.696856\pi$
$488$	310134.			1.30230
$489$		0			0
$490$		−	10900.6i		−	0.0454001i
$491$	−32951.9			−0.136684			−0.0683419	−	0.997662i	$-0.521771\pi$
$491$	−32951.9			−0.136684			−0.0683419	+	0.997662i	$0.521771\pi$
$492$		0			0
$493$	96104.9			0.395414
$494$	83614.0			0.342630
$495$		0			0
$496$			35995.5i			0.146314i
$497$	12981.6			0.0525551
$498$		0			0
$499$	−17305.7			−0.0695005			−0.0347503	−	0.999396i	$-0.511064\pi$
$499$	−17305.7			−0.0695005			−0.0347503	+	0.999396i	$0.511064\pi$
$500$	4837.70			0.0193508
$501$		0			0
$502$			266185.i			1.05627i
$503$		−	264972.i		−	1.04728i	−0.851938	−	0.523642i	$-0.824573\pi$
$503$		−	264972.i		−	1.04728i	0.851938	−	0.523642i	$-0.175427\pi$
$504$		0			0
$505$		−	10912.2i		−	0.0427887i
$506$			78801.7i			0.307776i
$507$		0			0
$508$	21060.2			0.0816084
$509$		−	446664.i		−	1.72403i	−0.506882	−	0.862015i	$-0.669202\pi$
$509$		−	446664.i		−	1.72403i	0.506882	−	0.862015i	$-0.330798\pi$
$510$		0			0
$511$		−	684907.i		−	2.62295i
$512$			27296.9i			0.104129i
$513$		0			0
$514$		−	71059.0i		−	0.268963i
$515$			8622.84i			0.0325114i
$516$		0			0
$517$	142518.			0.533197
$518$	−172722.			−0.643708
$519$		0			0
$520$	−3523.01			−0.0130289
$521$	−14595.5			−0.0537703			−0.0268852	−	0.999639i	$-0.508559\pi$
$521$	−14595.5			−0.0537703			−0.0268852	+	0.999639i	$0.508559\pi$
$522$		0			0
$523$	−40396.8			−0.147687			−0.0738437	−	0.997270i	$-0.523527\pi$
$523$	−40396.8			−0.147687			−0.0738437	+	0.997270i	$0.523527\pi$
$524$		−	10481.3i		−	0.0381727i
$525$		0			0
$526$			422815.i			1.52819i
$527$		−	16258.7i		−	0.0585415i
$528$		0			0
$529$	228555.			0.816732
$530$			15055.9i			0.0535987i
$531$		0			0
$532$	−72751.5			−0.257051
$533$		−	192293.i		−	0.676876i
$534$		0			0
$535$	3583.03			0.0125182
$536$	−85704.3			−0.298314
$537$		0			0
$538$	320030.			1.10567
$539$		−	253873.i		−	0.873853i
$540$		0			0
$541$			528895.i			1.80707i	0.428513	+	0.903536i	$0.359038\pi$
$541$			528895.i			1.80707i	−0.428513	+	0.903536i	$0.640962\pi$
$542$			561996.i			1.91309i
$543$		0			0
$544$			97461.9i			0.329334i
$545$			451.843i			0.00152123i
$546$		0			0
$547$	−223190.			−0.745935			−0.372967	−	0.927844i	$-0.621660\pi$
$547$	−223190.			−0.745935			−0.372967	+	0.927844i	$0.621660\pi$
$548$	−40394.4			−0.134512
$549$		0			0
$550$	217312.			0.718386
$551$	−115667.			−0.380983
$552$		0			0
$553$	15585.4			0.0509645
$554$		−	377634.i		−	1.23041i
$555$		0			0
$556$	25632.6			0.0829170
$557$	−497882.			−1.60478			−0.802392	−	0.596798i	$-0.796439\pi$
$557$	−497882.			−1.60478			−0.802392	+	0.596798i	$0.796439\pi$
$558$		0			0
$559$	55599.5			0.177929
$560$	16543.3			0.0527530
$561$		0			0
$562$		−	273203.i		−	0.864993i
$563$			268686.i			0.847671i	0.905739	+	0.423836i	$0.139317\pi$
$563$			268686.i			0.847671i	−0.905739	+	0.423836i	$0.860683\pi$
$564$		0			0
$565$			8791.76i			0.0275409i
$566$	344929.			1.07670
$567$		0			0
$568$			8247.17i			0.0255628i
$569$		−	122187.i		−	0.377398i	−0.982035	−	0.188699i	$-0.939573\pi$
$569$		−	122187.i		−	0.377398i	0.982035	−	0.188699i	$-0.0604271\pi$
$570$		0			0
$571$			20881.3i			0.0640449i	0.999487	+	0.0320224i	$0.0101948\pi$
$571$			20881.3i			0.0640449i	−0.999487	+	0.0320224i	$0.989805\pi$
$572$	44137.1			0.134900
$573$		0			0
$574$			645655.i			1.95964i
$575$			141432.i			0.427770i
$576$		0			0
$577$	−309645.			−0.930063			−0.465032	−	0.885294i	$-0.653957\pi$
$577$	−309645.			−0.930063			−0.465032	+	0.885294i	$0.653957\pi$
$578$			294525.i			0.881588i
$579$		0			0
$580$	−2621.60			−0.00779311
$581$			105887.i			0.313683i
$582$		0			0
$583$			350649.i			1.03166i
$584$	435120.			1.27580
$585$		0			0
$586$			489187.i			1.42456i
$587$		−	245888.i		−	0.713611i	−0.934179	−	0.356806i	$-0.883866\pi$
$587$		−	245888.i		−	0.713611i	0.934179	−	0.356806i	$-0.116134\pi$
$588$		0			0
$589$			19568.1i			0.0564050i
$590$	8311.12	+	7494.77i	0.0238757	+	0.0215305i
$591$		0			0
$592$		−	153461.i		−	0.437880i
$593$	−396294.			−1.12696			−0.563480	−	0.826130i	$-0.690538\pi$
$593$	−396294.			−1.12696			−0.563480	+	0.826130i	$0.690538\pi$
$594$		0			0
$595$	−7472.39			−0.0211070
$596$			148312.i			0.417526i
$597$		0			0
$598$			110851.i			0.309983i
$599$	107130.			0.298579			0.149289	−	0.988794i	$-0.452301\pi$
$599$	107130.			0.298579			0.149289	+	0.988794i	$0.452301\pi$
$600$		0			0
$601$			479681.i			1.32802i	0.747725	+	0.664008i	$0.231146\pi$
$601$			479681.i			1.32802i	−0.747725	+	0.664008i	$0.768854\pi$
$602$	−186684.			−0.515128
$603$		0			0
$604$			157705.i			0.432285i
$605$	6250.18			0.0170758
$606$		0			0
$607$	−182930.			−0.496486			−0.248243	−	0.968698i	$-0.579853\pi$
$607$	−182930.			−0.496486			−0.248243	+	0.968698i	$0.579853\pi$
$608$		−	117300.i		−	0.317316i
$609$		0			0
$610$	20623.1			0.0554234
$611$	200481.			0.537021
$612$		0			0
$613$		−	474523.i		−	1.26281i	−0.775455	−	0.631403i	$-0.782479\pi$
$613$		−	474523.i		−	1.26281i	0.775455	−	0.631403i	$-0.217521\pi$
$614$		−	16484.8i		−	0.0437267i
$615$		0			0
$616$	275498.			0.726034
$617$	339241.			0.891123			0.445562	−	0.895251i	$-0.353004\pi$
$617$	339241.			0.891123			0.445562	+	0.895251i	$0.353004\pi$
$618$		0			0
$619$	307348.			0.802137			0.401069	−	0.916048i	$-0.368639\pi$
$619$	307348.			0.802137			0.401069	+	0.916048i	$0.368639\pi$
$620$			443.513i			0.00115378i
$621$		0			0
$622$			703080.i			1.81729i
$623$			228429.i			0.588538i
$624$		0			0
$625$	389728.			0.997703
$626$	−119202.			−0.304183
$627$		0			0
$628$			70000.6i			0.177493i
$629$			69316.2i			0.175200i
$630$		0			0
$631$	478230.			1.20110			0.600548	−	0.799589i	$-0.294949\pi$
$631$	478230.			1.20110			0.600548	+	0.799589i	$0.294949\pi$
$632$			9901.38i			0.0247892i
$633$		0			0
$634$			600913.i			1.49497i
$635$	−2603.41			−0.00645648
$636$		0			0
$637$		−	357126.i		−	0.880121i
$638$	−235618.			−0.578850
$639$		0			0
$640$			13504.9i			0.0329710i
$641$	−84112.5			−0.204713			−0.102356	−	0.994748i	$-0.532638\pi$
$641$	−84112.5			−0.204713			−0.102356	+	0.994748i	$0.532638\pi$
$642$		0			0
$643$	202387.			0.489508			0.244754	−	0.969585i	$-0.421293\pi$
$643$	202387.			0.489508			0.244754	+	0.969585i	$0.421293\pi$
$644$		−	96450.3i		−	0.232558i
$645$		0			0
$646$			112668.i			0.269983i
$647$	−729383.			−1.74240			−0.871199	−	0.490930i	$-0.836657\pi$
$647$	−729383.			−1.74240			−0.871199	+	0.490930i	$0.836657\pi$
$648$		0			0
$649$	193565.	+	174552.i	0.459555	+	0.414416i
$650$	305695.			0.723539
$651$		0			0
$652$	95767.6			0.225280
$653$	33872.3			0.0794362			0.0397181	−	0.999211i	$-0.487354\pi$
$653$	33872.3			0.0794362			0.0397181	+	0.999211i	$0.487354\pi$
$654$		0			0
$655$			1295.68i			0.00302005i
$656$	−573654.			−1.33304
$657$		0			0
$658$	−673149.			−1.55475
$659$			12252.6i			0.0282134i	0.999900	+	0.0141067i	$0.00449046\pi$
$659$			12252.6i			0.0282134i	−0.999900	+	0.0141067i	$0.995510\pi$
$660$		0			0
$661$	−193919.			−0.443832			−0.221916	−	0.975066i	$-0.571231\pi$
$661$	−193919.			−0.443832			−0.221916	+	0.975066i	$0.571231\pi$
$662$			616076.i			1.40578i
$663$		0			0
$664$	−67269.8			−0.152575
$665$	8993.38			0.0203367
$666$		0			0
$667$		−	153345.i		−	0.344682i
$668$	−29524.9			−0.0661660
$669$		0			0
$670$	−5699.10			−0.0126957
$671$	480308.			1.06678
$672$		0			0
$673$			527119.i			1.16380i	0.813260	+	0.581900i	$0.197690\pi$
$673$			527119.i			1.16380i	−0.813260	+	0.581900i	$0.802310\pi$
$674$	−845907.			−1.86210
$675$		0			0
$676$	−97750.0			−0.213906
$677$	−160122.			−0.349359			−0.174680	−	0.984625i	$-0.555889\pi$
$677$	−160122.			−0.349359			−0.174680	+	0.984625i	$0.555889\pi$
$678$		0			0
$679$		−	702036.i		−	1.52272i
$680$		−	4747.19i		−	0.0102664i
$681$		0			0
$682$			39860.9i			0.0856995i
$683$		−	96946.8i		−	0.207822i	−0.994587	−	0.103911i	$-0.966864\pi$
$683$		−	96946.8i		−	0.207822i	0.994587	−	0.103911i	$-0.0331358\pi$
$684$		0			0
$685$	4993.47			0.0106420
$686$			349966.i			0.743666i
$687$		0			0
$688$		−	165866.i		−	0.350413i
$689$			493262.i			1.03906i
$690$		0			0
$691$			292122.i			0.611798i	0.952064	+	0.305899i	$0.0989570\pi$
$691$			292122.i			0.611798i	−0.952064	+	0.305899i	$0.901043\pi$
$692$		−	107160.i		−	0.223780i
$693$		0			0
$694$	167692.			0.348172
$695$	−3168.65			−0.00656001
$696$		0			0
$697$	259111.			0.533361
$698$	724767.			1.48761
$699$		0			0
$700$	−265982.			−0.542820
$701$		−	761446.i		−	1.54954i	−0.632243	−	0.774770i	$-0.717866\pi$
$701$		−	761446.i		−	1.54954i	0.632243	−	0.774770i	$-0.282134\pi$
$702$		0			0
$703$		−	83425.4i		−	0.168806i
$704$			137502.i			0.277436i
$705$		0			0
$706$	−494225.			−0.991552
$707$			1.20039e6i			2.40150i
$708$		0			0
$709$	9379.41			0.0186588			0.00932938	−	0.999956i	$-0.497030\pi$
$709$	9379.41			0.0186588			0.00932938	+	0.999956i	$0.497030\pi$
$710$			548.414i			0.00108791i
$711$		0			0
$712$	−145120.			−0.286265
$713$	−25942.4			−0.0510307
$714$		0			0
$715$	−5456.13			−0.0106727
$716$			330685.i			0.645042i
$717$		0			0
$718$		−	1.02853e6i		−	1.99511i
$719$		−	474152.i		−	0.917192i	−0.888645	−	0.458596i	$-0.848353\pi$
$719$		−	474152.i		−	0.917192i	0.888645	−	0.458596i	$-0.151647\pi$
$720$		0			0
$721$		−	948549.i		−	1.82469i
$722$			470025.i			0.901667i
$723$		0			0
$724$	−119080.			−0.227176
$725$	−422882.			−0.804531
$726$		0			0
$727$	−172753.			−0.326857			−0.163429	−	0.986555i	$-0.552255\pi$
$727$	−172753.			−0.326857			−0.163429	+	0.986555i	$0.552255\pi$
$728$	387546.			0.731241
$729$		0			0
$730$	28934.3			0.0542959
$731$			74919.4i			0.140204i
$732$		0			0
$733$	192952.			0.359122			0.179561	−	0.983747i	$-0.442532\pi$
$733$	192952.			0.359122			0.179561	+	0.983747i	$0.442532\pi$
$734$	−1.07524e6			−1.99578
$735$		0			0
$736$	155511.			0.287081
$737$	−132731.			−0.244365
$738$		0			0
$739$			894229.i			1.63742i	0.574208	+	0.818709i	$0.305310\pi$
$739$			894229.i			1.63742i	−0.574208	+	0.818709i	$0.694690\pi$
$740$		−	1890.85i		−	0.00345297i
$741$		0			0
$742$		−	1.65621e6i		−	3.00820i
$743$	814173.			1.47482			0.737410	−	0.675445i	$-0.236048\pi$
$743$	814173.			1.47482			0.737410	+	0.675445i	$0.236048\pi$
$744$		0			0
$745$		−	18334.0i		−	0.0330327i
$746$		−	603451.i		−	1.08434i
$747$		0			0
$748$			59473.9i			0.106298i
$749$	−394149.			−0.702582
$750$		0			0
$751$		−	417689.i		−	0.740581i	−0.928916	−	0.370291i	$-0.879258\pi$
$751$		−	417689.i		−	0.740581i	0.928916	−	0.370291i	$-0.120742\pi$
$752$		−	598082.i		−	1.05761i
$753$		0			0
$754$	−331446.			−0.583002
$755$		−	19495.1i		−	0.0342004i
$756$		0			0
$757$	−999639.			−1.74442			−0.872210	−	0.489131i	$-0.837314\pi$
$757$	−999639.			−1.74442			−0.872210	+	0.489131i	$0.837314\pi$
$758$			1.26112e6i			2.19491i
$759$		0			0
$760$			5713.48i			0.00989176i
$761$	698337.			1.20586			0.602928	−	0.797795i	$-0.294001\pi$
$761$	698337.			1.20586			0.602928	+	0.797795i	$0.294001\pi$
$762$		0			0
$763$		−	49704.7i		−	0.0853784i
$764$			67563.7i			0.115751i
$765$		0			0
$766$		−	1.06231e6i		−	1.81048i
$767$	272290.	+	245545.i	0.462851	+	0.417388i
$768$		0			0
$769$			1.02416e6i			1.73187i	0.500155	+	0.865936i	$0.333276\pi$
$769$			1.02416e6i			1.73187i	−0.500155	+	0.865936i	$0.666724\pi$
$770$	18319.8			0.0308987
$771$		0			0
$772$	243332.			0.408286
$773$		−	907633.i		−	1.51898i	−0.650521	−	0.759488i	$-0.725449\pi$
$773$		−	907633.i		−	1.51898i	0.650521	−	0.759488i	$-0.274551\pi$
$774$		0			0
$775$			71541.5i			0.119112i
$776$	446002.			0.740651
$777$		0			0
$778$			92213.4i			0.152347i
$779$	−311853.			−0.513896
$780$		0			0
$781$			12772.5i			0.0209399i
$782$	−149370.			−0.244259
$783$		0			0
$784$	−1.06539e6			−1.73331
$785$		−	8653.32i		−	0.0140425i
$786$		0			0
$787$	418956.			0.676423			0.338212	−	0.941070i	$-0.390178\pi$
$787$	418956.			0.676423			0.338212	+	0.941070i	$0.390178\pi$
$788$	157959.			0.254385
$789$		0			0
$790$			658.415i			0.00105498i
$791$		−	967131.i		−	1.54572i
$792$		0			0
$793$	675655.			1.07443
$794$	1.33279e6			2.11407
$795$		0			0
$796$	259378.			0.409361
$797$			429165.i			0.675628i	0.941213	+	0.337814i	$0.109688\pi$
$797$			429165.i			0.675628i	−0.941213	+	0.337814i	$0.890312\pi$
$798$		0			0
$799$			270145.i			0.423159i
$800$		−	428853.i		−	0.670083i
$801$		0			0
$802$	1.05667e6			1.64282
$803$	673876.			1.04508
$804$		0			0
$805$			11923.0i			0.0183989i
$806$			56072.8i			0.0863142i
$807$		0			0
$808$	−762604.			−1.16809
$809$		−	94992.3i		−	0.145141i	−0.997363	−	0.0725707i	$-0.976880\pi$
$809$		−	94992.3i		−	0.145141i	0.997363	−	0.0725707i	$-0.0231203\pi$
$810$		0			0
$811$			738552.i			1.12290i	0.827512	+	0.561448i	$0.189756\pi$
$811$			738552.i			1.12290i	−0.827512	+	0.561448i	$0.810244\pi$
$812$	288387.			0.437385
$813$		0			0
$814$		−	169941.i		−	0.256477i
$815$	−11838.6			−0.0178232
$816$		0			0
$817$		−	90169.1i		−	0.135087i
$818$	−736084.			−1.10007
$819$		0			0
$820$	−7068.18			−0.0105119
$821$		−	29461.7i		−	0.0437091i	−0.999761	−	0.0218545i	$-0.993043\pi$
$821$		−	29461.7i		−	0.0437091i	0.999761	−	0.0218545i	$-0.00695707\pi$
$822$		0			0
$823$		−	130911.i		−	0.193275i	−0.995320	−	0.0966375i	$-0.969191\pi$
$823$		−	130911.i		−	0.193275i	0.995320	−	0.0966375i	$-0.0308087\pi$
$824$	602611.			0.887529
$825$		0			0
$826$	−914258.	−	824457.i	−1.34001	−	1.20839i
$827$	−238564.			−0.348813			−0.174407	−	0.984674i	$-0.555801\pi$
$827$	−238564.			−0.348813			−0.174407	+	0.984674i	$0.555801\pi$
$828$		0			0
$829$	1.03043e6			1.49937			0.749685	−	0.661795i	$-0.230205\pi$
$829$	1.03043e6			1.49937			0.749685	+	0.661795i	$0.230205\pi$
$830$	−4473.26			−0.00649333
$831$		0			0
$832$			193425.i			0.279426i
$833$	481221.			0.693512
$834$		0			0
$835$	3649.80			0.00523475
$836$		−	71579.8i		−	0.102418i
$837$		0			0
$838$	−569454.			−0.810906
$839$		−	199610.i		−	0.283568i	−0.989898	−	0.141784i	$-0.954716\pi$
$839$		−	199610.i		−	0.283568i	0.989898	−	0.141784i	$-0.0452839\pi$
$840$		0			0
$841$	−248777.			−0.351737
$842$	−216098.			−0.304808
$843$		0			0
$844$			277557.i			0.389644i
$845$	12083.6			0.0169233
$846$		0			0
$847$	−687546.			−0.958374
$848$	1.47151e6			2.04632
$849$		0			0
$850$			411919.i			0.570130i
$851$	110601.			0.152722
$852$		0			0
$853$	−667639.			−0.917580			−0.458790	−	0.888545i	$-0.651717\pi$
$853$	−667639.			−0.917580			−0.458790	+	0.888545i	$0.651717\pi$
$854$	−2.26862e6			−3.11062
$855$		0			0
$856$		−	250402.i		−	0.341736i
$857$		−	470276.i		−	0.640312i	−0.947365	−	0.320156i	$-0.896265\pi$
$857$		−	470276.i		−	0.640312i	0.947365	−	0.320156i	$-0.103735\pi$
$858$		0			0
$859$		−	919460.i		−	1.24608i	−0.782189	−	0.623041i	$-0.785897\pi$
$859$		−	919460.i		−	1.24608i	0.782189	−	0.623041i	$-0.214103\pi$
$860$		−	2043.69i		−	0.00276324i
$861$		0			0
$862$	80041.9			0.107722
$863$			344188.i			0.462141i	0.972937	+	0.231071i	$0.0742228\pi$
$863$			344188.i			0.462141i	−0.972937	+	0.231071i	$0.925777\pi$
$864$		0			0
$865$			13246.9i			0.0177045i
$866$		−	131811.i		−	0.175758i
$867$		0			0
$868$		−	48788.3i		−	0.0647554i
$869$			15334.4i			0.0203061i
$870$		0			0
$871$	−186715.			−0.246117
$872$	31577.3			0.0415281
$873$		0			0
$874$	179774.			0.235345
$875$	65785.4			0.0859237
$876$		0			0
$877$	−826042.			−1.07400			−0.536998	−	0.843583i	$-0.680442\pi$
$877$	−826042.			−1.07400			−0.536998	+	0.843583i	$0.680442\pi$
$878$			1.03491e6i			1.34250i
$879$		0			0
$880$			16276.9i			0.0210187i
$881$		−	737152.i		−	0.949742i	−0.880056	−	0.474871i	$-0.842495\pi$
$881$		−	737152.i		−	0.949742i	0.880056	−	0.474871i	$-0.157505\pi$
$882$		0			0
$883$	−310160.			−0.397800			−0.198900	−	0.980020i	$-0.563737\pi$
$883$	−310160.			−0.397800			−0.198900	+	0.980020i	$0.563737\pi$
$884$			83662.7i			0.107060i
$885$		0			0
$886$	809505.			1.03122
$887$		−	959122.i		−	1.21906i	−0.792762	−	0.609532i	$-0.791357\pi$
$887$		−	959122.i		−	1.21906i	0.792762	−	0.609532i	$-0.208643\pi$
$888$		0			0
$889$	286386.			0.362367
$890$	−9650.09			−0.0121829
$891$		0			0
$892$	−156127.			−0.196222
$893$		−	325133.i		−	0.407716i
$894$		0			0
$895$		−	40878.5i		−	0.0510328i
$896$		−	1.48560e6i		−	1.85048i
$897$		0			0
$898$		−	113307.i		−	0.140509i
$899$		−	77568.0i		−	0.0959761i
$900$		0			0
$901$	−664662.			−0.818750
$902$	−635256.			−0.780793
$903$		0			0
$904$	614416.			0.751841
$905$	14720.4			0.0179731
$906$		0			0
$907$	11680.6			0.0141988			0.00709939	−	0.999975i	$-0.497740\pi$
$907$	11680.6			0.0141988			0.00709939	+	0.999975i	$0.497740\pi$
$908$		−	194342.i		−	0.235719i
$909$		0			0
$910$	25770.7			0.0311203
$911$	−637090.			−0.767651			−0.383826	−	0.923406i	$-0.625394\pi$
$911$	−637090.			−0.767651			−0.383826	+	0.923406i	$0.625394\pi$
$912$		0			0
$913$	−104182.			−0.124983
$914$	−1.29325e6			−1.54806
$915$		0			0
$916$		−	320995.i		−	0.382567i
$917$		−	142530.i		−	0.169499i
$918$		0			0
$919$		−	702495.i		−	0.831787i	−0.909413	−	0.415893i	$-0.863469\pi$
$919$		−	702495.i		−	0.831787i	0.909413	−	0.415893i	$-0.136531\pi$
$920$	−7574.64			−0.00894925
$921$		0			0
$922$			64154.3i			0.0754682i
$923$			17967.2i			0.0210900i
$924$		0			0
$925$		−	305006.i		−	0.356471i
$926$	600950.			0.700836
$927$		0			0
$928$			464978.i			0.539929i
$929$		−	631639.i		−	0.731875i	−0.930639	−	0.365938i	$-0.880748\pi$
$929$		−	631639.i		−	0.731875i	0.930639	−	0.365938i	$-0.119252\pi$
$930$		0			0
$931$	−579172.			−0.668203
$932$		−	358997.i		−	0.413294i
$933$		0			0
$934$	1.36953e6			1.56992
$935$		−	7352.04i		−	0.00840978i
$936$		0			0
$937$			550717.i			0.627263i	0.949545	+	0.313631i	$0.101546\pi$
$937$			550717.i			0.627263i	−0.949545	+	0.313631i	$0.898454\pi$
$938$	626925.			0.712541
$939$		0			0
$940$		−	7369.17i		−	0.00833994i
$941$		−	625271.i		−	0.706137i	−0.935597	−	0.353069i	$-0.885138\pi$
$941$		−	625271.i		−	0.706137i	0.935597	−	0.353069i	$-0.114862\pi$
$942$		0			0
$943$		−	413439.i		−	0.464931i
$944$	732516.	−	812304.i	0.822003	−	0.911537i
$945$		0			0
$946$		−	183678.i		−	0.205246i
$947$	−62423.0			−0.0696057			−0.0348028	−	0.999394i	$-0.511080\pi$
$947$	−62423.0			−0.0696057			−0.0348028	+	0.999394i	$0.511080\pi$
$948$		0			0
$949$	947949.			1.05257
$950$		−	495764.i		−	0.549323i
$951$		0			0
$952$			522211.i			0.576199i
$953$	−817236.			−0.899832			−0.449916	−	0.893071i	$-0.648546\pi$
$953$	−817236.			−0.899832			−0.449916	+	0.893071i	$0.648546\pi$
$954$		0			0
$955$		−	8352.07i		−	0.00915772i
$956$	−189882.			−0.207763
$957$		0			0
$958$		−	672119.i		−	0.732344i
$959$	−549303.			−0.597276
$960$		0			0
$961$	910398.			0.985791
$962$		−	239057.i		−	0.258316i
$963$		0			0
$964$	197303.			0.212315
$965$	−30080.1			−0.0323017
$966$		0			0
$967$			991071.i			1.05987i	0.848039	+	0.529934i	$0.177783\pi$
$967$			991071.i			1.05987i	−0.848039	+	0.529934i	$0.822217\pi$
$968$		−	436797.i		−	0.466153i
$969$		0			0
$970$	29657.9			0.0315208
$971$	1.08960e6			1.15566			0.577829	−	0.816158i	$-0.303900\pi$
$971$	1.08960e6			1.15566			0.577829	+	0.816158i	$0.303900\pi$
$972$		0			0
$973$	348565.			0.368178
$974$			1.27800e6i			1.34714i
$975$		0			0
$976$		−	2.01563e6i		−	2.11598i
$977$			10538.2i			0.0110402i	0.999985	+	0.00552010i	$0.00175711\pi$
$977$			10538.2i			0.0110402i	−0.999985	+	0.00552010i	$0.998243\pi$
$978$		0			0
$979$	−224749.			−0.234495
$980$	−13127.0			−0.0136683
$981$		0			0
$982$			153134.i			0.158799i
$983$			825144.i			0.853931i	0.904268	+	0.426965i	$0.140417\pi$
$983$			825144.i			0.853931i	−0.904268	+	0.426965i	$0.859583\pi$
$984$		0			0
$985$	−19526.5			−0.0201257
$986$		−	446618.i		−	0.459391i
$987$		0			0
$988$		−	100692.i		−	0.103153i
$989$	119542.			0.122216
$990$		0			0
$991$		−	1.64789e6i		−	1.67795i	−0.544167	−	0.838977i	$-0.683154\pi$
$991$		−	1.64789e6i		−	1.67795i	0.544167	−	0.838977i	$-0.316846\pi$
$992$	78663.3			0.0799372
$993$		0			0
$994$		−	60327.8i		−	0.0610584i
$995$	−32063.7			−0.0323868
$996$		0			0
$997$	596888.			0.600486			0.300243	−	0.953863i	$-0.402932\pi$
$997$	596888.			0.600486			0.300243	+	0.953863i	$0.402932\pi$
$998$			80422.9i			0.0807456i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.10		40
3.2	odd	2		177.5.c.a.58.31	yes	40
59.58	odd	2	inner	531.5.c.d.235.31		40
177.176	even	2		177.5.c.a.58.10	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.10	✓	40	177.176	even	2
177.5.c.a.58.31	yes	40	3.2	odd	2
531.5.c.d.235.10		40	1.1	even	1	trivial
531.5.c.d.235.31		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)

\(f(q)\)	\(=\)	\(q-4.64719i q^{2} -5.59638 q^{4} +0.691812 q^{5} -76.1022 q^{7} -48.3476i q^{8} +O(q^{10})\)	\(q-4.64719i q^{2} -5.59638 q^{4} +0.691812 q^{5} -76.1022 q^{7} -48.3476i q^{8} -3.21498i q^{10} -74.8765i q^{11} -105.330i q^{13} +353.662i q^{14} -314.223 q^{16} +141.930 q^{17} -170.820 q^{19} -3.87164 q^{20} -347.966 q^{22} -226.464i q^{23} -624.521 q^{25} -489.487 q^{26} +425.897 q^{28} +677.129 q^{29} -114.554i q^{31} +686.691i q^{32} -659.575i q^{34} -52.6484 q^{35} +488.383i q^{37} +793.831i q^{38} -33.4475i q^{40} +1825.63 q^{41} +527.862i q^{43} +419.037i q^{44} -1052.42 q^{46} +1903.37i q^{47} +3390.55 q^{49} +2902.27i q^{50} +589.465i q^{52} -4683.03 q^{53} -51.8005i q^{55} +3679.36i q^{56} -3146.75i q^{58} +(-2331.20 + 2585.12i) q^{59} +6414.67i q^{61} -532.355 q^{62} -1836.38 q^{64} -72.8684i q^{65} -1772.67i q^{67} -794.294 q^{68} +244.667i q^{70} -170.581 q^{71} +8999.83i q^{73} +2269.61 q^{74} +955.971 q^{76} +5698.27i q^{77} -204.796 q^{79} -217.383 q^{80} -8484.05i q^{82} -1391.38i q^{83} +98.1888 q^{85} +2453.07 q^{86} -3620.10 q^{88} -3001.60i q^{89} +8015.83i q^{91} +1267.38i q^{92} +8845.32 q^{94} -118.175 q^{95} +9224.90i q^{97} -15756.5i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 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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