LMFDB - Embedded newform 43.3.b.b.42.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [43,3,Mod(42,43)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(43, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("43.42");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	43.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.17166513675$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 20x^{4} + 121x^{2} + 214 $ x^6 + 20x^4 + 121x^2 + 214
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			42.3
Root			$-1.77533i$ of defining polynomial
Character	$\chi$	$=$	43.42
Dual form			43.3.b.b.42.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77533i q^{2} +5.61757i q^{3} +0.848217 q^{4} +2.98959i q^{5} +9.97302 q^{6} -6.83184i q^{7} -8.60717i q^{8} -22.5571 q^{9} +O(q^{10})$	\(q-1.77533i q^{2} +5.61757i q^{3} +0.848217 q^{4} +2.98959i q^{5} +9.97302 q^{6} -6.83184i q^{7} -8.60717i q^{8} -22.5571 q^{9} +5.30751 q^{10} +6.88766 q^{11} +4.76492i q^{12} +12.4014 q^{13} -12.1287 q^{14} -16.7943 q^{15} -11.8877 q^{16} -15.8212 q^{17} +40.0462i q^{18} -19.2112i q^{19} +2.53583i q^{20} +38.3784 q^{21} -12.2278i q^{22} -33.2226 q^{23} +48.3514 q^{24} +16.0623 q^{25} -22.0165i q^{26} -76.1581i q^{27} -5.79488i q^{28} +45.8411i q^{29} +29.8153i q^{30} +14.7278 q^{31} -13.3242i q^{32} +38.6919i q^{33} +28.0879i q^{34} +20.4244 q^{35} -19.1333 q^{36} +13.1726i q^{37} -34.1062 q^{38} +69.6657i q^{39} +25.7319 q^{40} -2.49085 q^{41} -68.1341i q^{42} +(-40.8836 + 13.3242i) q^{43} +5.84223 q^{44} -67.4366i q^{45} +58.9810i q^{46} -10.2595 q^{47} -66.7798i q^{48} +2.32596 q^{49} -28.5159i q^{50} -88.8770i q^{51} +10.5191 q^{52} -31.2780 q^{53} -135.205 q^{54} +20.5913i q^{55} -58.8028 q^{56} +107.920 q^{57} +81.3829 q^{58} +64.4112 q^{59} -14.2452 q^{60} +78.1922i q^{61} -26.1467i q^{62} +154.107i q^{63} -71.2054 q^{64} +37.0751i q^{65} +68.6908 q^{66} -89.3657 q^{67} -13.4198 q^{68} -186.631i q^{69} -36.2600i q^{70} -35.5065i q^{71} +194.153i q^{72} -35.9603i q^{73} +23.3857 q^{74} +90.2313i q^{75} -16.2953i q^{76} -47.0554i q^{77} +123.679 q^{78} +50.1103 q^{79} -35.5393i q^{80} +224.809 q^{81} +4.42207i q^{82} -10.3645 q^{83} +32.5532 q^{84} -47.2991i q^{85} +(23.6548 + 72.5817i) q^{86} -257.516 q^{87} -59.2832i q^{88} +13.4900i q^{89} -119.722 q^{90} -84.7243i q^{91} -28.1800 q^{92} +82.7347i q^{93} +18.2140i q^{94} +57.4338 q^{95} +74.8496 q^{96} +66.6322 q^{97} -4.12933i q^{98} -155.366 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10}) $ 6 * q - 16 * q^4 + 6 * q^6 - 36 * q^9	$ 6 q - 16 q^{4} + 6 q^{6} - 36 q^{9} - 2 q^{10} + 38 q^{11} + 30 q^{13} + 36 q^{14} + 28 q^{15} - 68 q^{16} - 20 q^{17} + 56 q^{21} - 80 q^{23} + 62 q^{24} - 84 q^{25} - 112 q^{31} + 208 q^{35} - 122 q^{36} + 170 q^{38} + 206 q^{40} - 172 q^{41} + 10 q^{43} - 36 q^{44} + 30 q^{47} - 6 q^{49} - 120 q^{52} - 110 q^{53} - 284 q^{54} - 264 q^{56} + 420 q^{57} + 430 q^{58} - 12 q^{59} - 232 q^{60} + 100 q^{64} - 144 q^{66} - 70 q^{67} - 50 q^{68} - 50 q^{74} + 620 q^{78} + 178 q^{79} + 382 q^{81} + 10 q^{83} + 172 q^{84} - 372 q^{86} - 510 q^{87} - 796 q^{90} + 150 q^{92} - 130 q^{95} + 362 q^{96} - 380 q^{97} - 466 q^{99}+O(q^{100}) $ 6 * q - 16 * q^4 + 6 * q^6 - 36 * q^9 - 2 * q^10 + 38 * q^11 + 30 * q^13 + 36 * q^14 + 28 * q^15 - 68 * q^16 - 20 * q^17 + 56 * q^21 - 80 * q^23 + 62 * q^24 - 84 * q^25 - 112 * q^31 + 208 * q^35 - 122 * q^36 + 170 * q^38 + 206 * q^40 - 172 * q^41 + 10 * q^43 - 36 * q^44 + 30 * q^47 - 6 * q^49 - 120 * q^52 - 110 * q^53 - 284 * q^54 - 264 * q^56 + 420 * q^57 + 430 * q^58 - 12 * q^59 - 232 * q^60 + 100 * q^64 - 144 * q^66 - 70 * q^67 - 50 * q^68 - 50 * q^74 + 620 * q^78 + 178 * q^79 + 382 * q^81 + 10 * q^83 + 172 * q^84 - 372 * q^86 - 510 * q^87 - 796 * q^90 + 150 * q^92 - 130 * q^95 + 362 * q^96 - 380 * q^97 - 466 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/43\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$		−	1.77533i		−	0.887663i	−0.896110	−	0.443832i	$-0.853619\pi$
$2$		−	1.77533i		−	0.887663i	0.896110	−	0.443832i	$-0.146381\pi$
$3$			5.61757i			1.87252i	0.351302	+	0.936262i	$0.385739\pi$
$3$			5.61757i			1.87252i	−0.351302	+	0.936262i	$0.614261\pi$
$4$	0.848217			0.212054
$5$			2.98959i			0.597919i	0.954266	+	0.298959i	$0.0966395\pi$
$5$			2.98959i			0.597919i	−0.954266	+	0.298959i	$0.903360\pi$
$6$	9.97302			1.66217
$7$		−	6.83184i		−	0.975977i	−0.872850	−	0.487989i	$-0.837731\pi$
$7$		−	6.83184i		−	0.975977i	0.872850	−	0.487989i	$-0.162269\pi$
$8$		−	8.60717i		−	1.07590i
$9$	−22.5571			−2.50635
$10$	5.30751			0.530751
$11$	6.88766			0.626151			0.313075	−	0.949728i	$-0.398641\pi$
$11$	6.88766			0.626151			0.313075	+	0.949728i	$0.398641\pi$
$12$			4.76492i			0.397077i
$13$	12.4014			0.953953			0.476977	−	0.878916i	$-0.341733\pi$
$13$	12.4014			0.953953			0.476977	+	0.878916i	$0.341733\pi$
$14$	−12.1287			−0.866339
$15$	−16.7943			−1.11962
$16$	−11.8877			−0.742979
$17$	−15.8212			−0.930661			−0.465331	−	0.885137i	$-0.654065\pi$
$17$	−15.8212			−0.930661			−0.465331	+	0.885137i	$0.654065\pi$
$18$			40.0462i			2.22479i
$19$		−	19.2112i		−	1.01112i	−0.862792	−	0.505559i	$-0.831286\pi$
$19$		−	19.2112i		−	1.01112i	0.862792	−	0.505559i	$-0.168714\pi$
$20$			2.53583i			0.126791i
$21$	38.3784			1.82754
$22$		−	12.2278i		−	0.555811i
$23$	−33.2226			−1.44446			−0.722231	−	0.691652i	$-0.756883\pi$
$23$	−33.2226			−1.44446			−0.722231	+	0.691652i	$0.756883\pi$
$24$	48.3514			2.01464
$25$	16.0623			0.642493
$26$		−	22.0165i		−	0.846789i
$27$		−	76.1581i		−	2.82067i
$28$		−	5.79488i		−	0.206960i
$29$			45.8411i			1.58073i	0.612637	+	0.790364i	$0.290109\pi$
$29$			45.8411i			1.58073i	−0.612637	+	0.790364i	$0.709891\pi$
$30$			29.8153i			0.993843i
$31$	14.7278			0.475092			0.237546	−	0.971376i	$-0.423657\pi$
$31$	14.7278			0.475092			0.237546	+	0.971376i	$0.423657\pi$
$32$		−	13.3242i		−	0.416381i
$33$			38.6919i			1.17248i
$34$			28.0879i			0.826114i
$35$	20.4244			0.583555
$36$	−19.1333			−0.531481
$37$			13.1726i			0.356017i	0.984029	+	0.178009i	$0.0569655\pi$
$37$			13.1726i			0.356017i	−0.984029	+	0.178009i	$0.943034\pi$
$38$	−34.1062			−0.897532
$39$			69.6657i			1.78630i
$40$	25.7319			0.643298
$41$	−2.49085			−0.0607525			−0.0303762	−	0.999539i	$-0.509671\pi$
$41$	−2.49085			−0.0607525			−0.0303762	+	0.999539i	$0.509671\pi$
$42$		−	68.1341i		−	1.62224i
$43$	−40.8836	+	13.3242i	−0.950781	+	0.309865i
$43$	−40.8836	+	13.3242i	−0.950781	+	0.309865i
$44$	5.84223			0.132778
$45$		−	67.4366i		−	1.49859i
$46$			58.9810i			1.28220i
$47$	−10.2595			−0.218288			−0.109144	−	0.994026i	$-0.534811\pi$
$47$	−10.2595			−0.218288			−0.109144	+	0.994026i	$0.534811\pi$
$48$		−	66.7798i		−	1.39125i
$49$	2.32596			0.0474685
$50$		−	28.5159i		−	0.570317i
$51$		−	88.8770i		−	1.74269i
$52$	10.5191			0.202290
$53$	−31.2780			−0.590151			−0.295075	−	0.955474i	$-0.595345\pi$
$53$	−31.2780			−0.590151			−0.295075	+	0.955474i	$0.595345\pi$
$54$	−135.205			−2.50380
$55$			20.5913i			0.374387i
$56$	−58.8028			−1.05005
$57$	107.920			1.89334
$58$	81.3829			1.40315
$59$	64.4112			1.09171			0.545857	−	0.837878i	$-0.316204\pi$
$59$	64.4112			1.09171			0.545857	+	0.837878i	$0.316204\pi$
$60$	−14.2452			−0.237420
$61$			78.1922i			1.28184i	0.767608	+	0.640920i	$0.221447\pi$
$61$			78.1922i			1.28184i	−0.767608	+	0.640920i	$0.778553\pi$
$62$		−	26.1467i		−	0.421721i
$63$			154.107i			2.44614i
$64$	−71.2054			−1.11258
$65$			37.0751i			0.570387i
$66$	68.6908			1.04077
$67$	−89.3657			−1.33382			−0.666908	−	0.745140i	$-0.732383\pi$
$67$	−89.3657			−1.33382			−0.666908	+	0.745140i	$0.732383\pi$
$68$	−13.4198			−0.197351
$69$		−	186.631i		−	2.70479i
$70$		−	36.2600i		−	0.518000i
$71$		−	35.5065i		−	0.500092i	−0.968234	−	0.250046i	$-0.919554\pi$
$71$		−	35.5065i		−	0.500092i	0.968234	−	0.250046i	$-0.0804457\pi$
$72$			194.153i			2.69657i
$73$		−	35.9603i		−	0.492607i	−0.969193	−	0.246303i	$-0.920784\pi$
$73$		−	35.9603i		−	0.492607i	0.969193	−	0.246303i	$-0.0792160\pi$
$74$	23.3857			0.316024
$75$			90.2313i			1.20308i
$76$		−	16.2953i		−	0.214412i
$77$		−	47.0554i		−	0.611109i
$78$	123.679			1.58563
$79$	50.1103			0.634308			0.317154	−	0.948374i	$-0.397273\pi$
$79$	50.1103			0.634308			0.317154	+	0.948374i	$0.397273\pi$
$80$		−	35.5393i		−	0.444241i
$81$	224.809			2.77543
$82$			4.42207i			0.0539277i
$83$	−10.3645			−0.124873			−0.0624367	−	0.998049i	$-0.519887\pi$
$83$	−10.3645			−0.124873			−0.0624367	+	0.998049i	$0.519887\pi$
$84$	32.5532			0.387538
$85$		−	47.2991i		−	0.556460i
$86$	23.6548	+	72.5817i	0.275056	+	0.843973i
$87$	−257.516			−2.95995
$88$		−	59.2832i		−	0.673673i
$89$			13.4900i			0.151573i	0.997124	+	0.0757865i	$0.0241468\pi$
$89$			13.4900i			0.151573i	−0.997124	+	0.0757865i	$0.975853\pi$
$90$	−119.722			−1.33024
$91$		−	84.7243i		−	0.931037i
$92$	−28.1800			−0.306304
$93$			82.7347i			0.889621i
$94$			18.2140i			0.193766i
$95$	57.4338			0.604566
$96$	74.8496			0.779684
$97$	66.6322			0.686930			0.343465	−	0.939165i	$-0.388399\pi$
$97$	66.6322			0.686930			0.343465	+	0.939165i	$0.388399\pi$
$98$		−	4.12933i		−	0.0421360i
$99$	−155.366			−1.56935
$100$	13.6243			0.136243
$101$	73.3449			0.726187			0.363094	−	0.931753i	$-0.381721\pi$
$101$	73.3449			0.726187			0.363094	+	0.931753i	$0.381721\pi$
$102$	−157.786			−1.54692
$103$	−34.7405			−0.337286			−0.168643	−	0.985677i	$-0.553939\pi$
$103$	−34.7405			−0.337286			−0.168643	+	0.985677i	$0.553939\pi$
$104$		−	106.741i		−	1.02635i
$105$			114.736i			1.09272i
$106$			55.5286i			0.523855i
$107$	120.519			1.12635			0.563173	−	0.826339i	$-0.309580\pi$
$107$	120.519			1.12635			0.563173	+	0.826339i	$0.309580\pi$
$108$		−	64.5986i		−	0.598135i
$109$	−81.5825			−0.748464			−0.374232	−	0.927335i	$-0.622094\pi$
$109$	−81.5825			−0.748464			−0.374232	+	0.927335i	$0.622094\pi$
$110$	36.5563			0.332330
$111$	−73.9983			−0.666651
$112$			81.2146i			0.725130i
$113$		−	58.7443i		−	0.519861i	−0.965627	−	0.259931i	$-0.916300\pi$
$113$		−	58.7443i		−	0.519861i	0.965627	−	0.259931i	$-0.0836997\pi$
$114$		−	191.594i		−	1.68065i
$115$		−	99.3222i		−	0.863671i
$116$			38.8832i			0.335200i
$117$	−279.740			−2.39094
$118$		−	114.351i		−	0.969075i
$119$			108.088i			0.908304i
$120$			144.551i			1.20459i
$121$	−73.5601			−0.607935
$122$	138.817			1.13784
$123$		−	13.9925i		−	0.113760i
$124$	12.4924			0.100745
$125$			122.760i			0.982078i
$126$	273.590			2.17135
$127$	−17.9828			−0.141597			−0.0707984	−	0.997491i	$-0.522555\pi$
$127$	−17.9828			−0.141597			−0.0707984	+	0.997491i	$0.522555\pi$
$128$			73.1161i			0.571219i
$129$	−74.8496	−	229.666i	−0.580230	−	1.78036i
$130$	65.8205			0.506311
$131$		−	103.364i		−	0.789039i	−0.918887	−	0.394520i	$-0.870911\pi$
$131$		−	103.364i		−	0.789039i	0.918887	−	0.394520i	$-0.129089\pi$
$132$			32.8192i			0.248630i
$133$	−131.248			−0.986827
$134$			158.653i			1.18398i
$135$	227.682			1.68653
$136$			136.176i			1.00129i
$137$			140.446i			1.02516i	0.858641	+	0.512578i	$0.171309\pi$
$137$			140.446i			1.02516i	−0.858641	+	0.512578i	$0.828691\pi$
$138$	−331.330			−2.40094
$139$	248.580			1.78834			0.894172	−	0.447725i	$-0.147765\pi$
$139$	248.580			1.78834			0.894172	+	0.447725i	$0.147765\pi$
$140$	17.3244			0.123745
$141$		−	57.6337i		−	0.408750i
$142$	−63.0357			−0.443913
$143$	85.4166			0.597319
$144$	268.151			1.86216
$145$	−137.046			−0.945147
$146$	−63.8412			−0.437269
$147$			13.0662i			0.0888859i
$148$			11.1733i			0.0754950i
$149$		−	95.0589i		−	0.637979i	−0.947758	−	0.318990i	$-0.896656\pi$
$149$		−	95.0589i		−	0.637979i	0.947758	−	0.318990i	$-0.103344\pi$
$150$	160.190			1.06793
$151$		−	122.926i		−	0.814076i	−0.913411	−	0.407038i	$-0.866562\pi$
$151$		−	122.926i		−	0.814076i	0.913411	−	0.407038i	$-0.133438\pi$
$152$	−165.354			−1.08786
$153$	356.882			2.33256
$154$	−83.5387			−0.542459
$155$			44.0303i			0.284066i
$156$			59.0917i			0.378793i
$157$			189.332i			1.20594i	0.797765	+	0.602969i	$0.206016\pi$
$157$			189.332i			1.20594i	−0.797765	+	0.602969i	$0.793984\pi$
$158$		−	88.9621i		−	0.563051i
$159$		−	175.706i		−	1.10507i
$160$	39.8339			0.248962
$161$			226.972i			1.40976i
$162$		−	399.110i		−	2.46364i
$163$		−	264.440i		−	1.62233i	−0.584817	−	0.811165i	$-0.698834\pi$
$163$		−	264.440i		−	1.62233i	0.584817	−	0.811165i	$-0.301166\pi$
$164$	−2.11278			−0.0128828
$165$	−115.673			−0.701049
$166$			18.4004i			0.110845i
$167$	40.0299			0.239700			0.119850	−	0.992792i	$-0.461759\pi$
$167$	40.0299			0.239700			0.119850	+	0.992792i	$0.461759\pi$
$168$		−	330.329i		−	1.96624i
$169$	−15.2054			−0.0899729
$170$	−83.9713			−0.493949
$171$			433.350i			2.53421i
$172$	−34.6781	+	11.3018i	−0.201617	+	0.0657082i
$173$	181.497			1.04912			0.524559	−	0.851374i	$-0.324230\pi$
$173$	181.497			1.04912			0.524559	+	0.851374i	$0.324230\pi$
$174$			457.175i			2.62744i
$175$		−	109.735i		−	0.627059i
$176$	−81.8782			−0.465217
$177$			361.834i			2.04426i
$178$	23.9492			0.134546
$179$		−	62.5972i		−	0.349705i	−0.984595	−	0.174852i	$-0.944055\pi$
$179$		−	62.5972i		−	0.349705i	0.984595	−	0.174852i	$-0.0559448\pi$
$180$		−	57.2009i		−	0.317783i
$181$	16.7178			0.0923638			0.0461819	−	0.998933i	$-0.485295\pi$
$181$	16.7178			0.0923638			0.0461819	+	0.998933i	$0.485295\pi$
$182$	−150.413			−0.826447
$183$	−439.251			−2.40028
$184$			285.953i			1.55409i
$185$	−39.3809			−0.212870
$186$	146.881			0.789683
$187$	−108.971			−0.582734
$188$	−8.70231			−0.0462889
$189$	−520.300			−2.75291
$190$		−	101.964i		−	0.536651i
$191$		−	44.7333i		−	0.234206i	−0.993120	−	0.117103i	$-0.962639\pi$
$191$		−	44.7333i		−	0.234206i	0.993120	−	0.117103i	$-0.0373607\pi$
$192$		−	400.002i		−	2.08334i
$193$	135.200			0.700516			0.350258	−	0.936653i	$-0.386094\pi$
$193$	135.200			0.700516			0.350258	+	0.936653i	$0.386094\pi$
$194$		−	118.294i		−	0.609763i
$195$	−208.272			−1.06806
$196$	1.97292			0.0100659
$197$	145.802			0.740109			0.370055	−	0.929010i	$-0.379339\pi$
$197$	145.802			0.740109			0.370055	+	0.929010i	$0.379339\pi$
$198$			275.825i			1.39305i
$199$			280.657i			1.41034i	0.709039	+	0.705169i	$0.249129\pi$
$199$			280.657i			1.41034i	−0.709039	+	0.705169i	$0.750871\pi$
$200$		−	138.251i		−	0.691256i
$201$		−	502.018i		−	2.49760i
$202$		−	130.211i		−	0.644609i
$203$	313.179			1.54275
$204$		−	75.3870i		−	0.369544i
$205$		−	7.44664i		−	0.0363251i
$206$			61.6757i			0.299396i
$207$	749.407			3.62032
$208$	−147.424			−0.708767
$209$		−	132.320i		−	0.633112i
$210$	203.693			0.969968
$211$			181.131i			0.858439i	0.903200	+	0.429219i	$0.141211\pi$
$211$			181.131i			0.858439i	−0.903200	+	0.429219i	$0.858789\pi$
$212$	−26.5305			−0.125144
$213$	199.460			0.936434
$214$		−	213.961i		−	0.999816i
$215$	−39.8339	−	122.225i	−0.185274	−	0.568490i
$216$	−655.505			−3.03475
$217$		−	100.618i		−	0.463679i
$218$			144.836i			0.664384i
$219$	202.010			0.922418
$220$			17.4659i			0.0793905i
$221$	−196.205			−0.887807
$222$			131.371i			0.591762i
$223$			107.757i			0.483213i	0.970374	+	0.241607i	$0.0776744\pi$
$223$			107.757i			0.483213i	−0.970374	+	0.241607i	$0.922326\pi$
$224$	−91.0288			−0.406378
$225$	−362.320			−1.61031
$226$	−104.290			−0.461461
$227$		−	139.742i		−	0.615603i	−0.951451	−	0.307801i	$-0.900407\pi$
$227$		−	139.742i		−	0.615603i	0.951451	−	0.307801i	$-0.0995932\pi$
$228$	91.5400			0.401491
$229$	308.655			1.34784			0.673919	−	0.738805i	$-0.264610\pi$
$229$	308.655			1.34784			0.673919	+	0.738805i	$0.264610\pi$
$230$	−176.329			−0.766649
$231$	264.337			1.14432
$232$	394.562			1.70070
$233$		−	301.133i		−	1.29242i	−0.763161	−	0.646209i	$-0.776354\pi$
$233$		−	301.133i		−	1.29242i	0.763161	−	0.646209i	$-0.223646\pi$
$234$			496.629i			2.12235i
$235$		−	30.6719i		−	0.130519i
$236$	54.6346			0.231503
$237$			281.498i			1.18776i
$238$	191.892			0.806268
$239$	−122.039			−0.510625			−0.255312	−	0.966859i	$-0.582178\pi$
$239$	−122.039			−0.510625			−0.255312	+	0.966859i	$0.582178\pi$
$240$	199.644			0.831852
$241$			475.092i			1.97133i	0.168701	+	0.985667i	$0.446043\pi$
$241$			475.092i			1.97133i	−0.168701	+	0.985667i	$0.553957\pi$
$242$			130.593i			0.539642i
$243$			577.461i			2.37638i
$244$			66.3240i			0.271820i
$245$			6.95367i			0.0283823i
$246$	−24.8413			−0.100981
$247$		−	238.246i		−	0.964559i
$248$		−	126.765i		−	0.511149i
$249$		−	58.2233i		−	0.233828i
$250$	217.939			0.871754
$251$	−35.9930			−0.143399			−0.0716993	−	0.997426i	$-0.522842\pi$
$251$	−35.9930			−0.143399			−0.0716993	+	0.997426i	$0.522842\pi$
$252$			130.716i			0.518714i
$253$	−228.826			−0.904451
$254$			31.9253i			0.125690i
$255$	265.706			1.04198
$256$	−155.017			−0.605534
$257$		−	62.7857i		−	0.244302i	−0.992512	−	0.122151i	$-0.961021\pi$
$257$		−	62.7857i		−	0.244302i	0.992512	−	0.122151i	$-0.0389793\pi$
$258$	−407.733	+	132.882i	−1.58036	+	0.515048i
$259$	89.9934			0.347465
$260$			31.4478i			0.120953i
$261$		−	1034.04i		−	3.96185i
$262$	−183.505			−0.700401
$263$		−	17.1841i		−	0.0653386i	−0.999466	−	0.0326693i	$-0.989599\pi$
$263$		−	17.1841i		−	0.0653386i	0.999466	−	0.0326693i	$-0.0104008\pi$
$264$	333.028			1.26147
$265$		−	93.5085i		−	0.352862i
$266$			233.008i			0.875970i
$267$	−75.7811			−0.283824
$268$	−75.8015			−0.282842
$269$	−407.934			−1.51648			−0.758242	−	0.651974i	$-0.773941\pi$
$269$	−407.934			−1.51648			−0.758242	+	0.651974i	$0.773941\pi$
$270$		−	404.209i		−	1.49707i
$271$	−316.274			−1.16706			−0.583532	−	0.812090i	$-0.698330\pi$
$271$	−316.274			−1.16706			−0.583532	+	0.812090i	$0.698330\pi$
$272$	188.078			0.691461
$273$	475.945			1.74339
$274$	249.338			0.909993
$275$	110.632			0.402298
$276$		−	158.303i		−	0.573562i
$277$		−	209.408i		−	0.755984i	−0.925809	−	0.377992i	$-0.876615\pi$
$277$		−	209.408i		−	0.755984i	0.925809	−	0.377992i	$-0.123385\pi$
$278$		−	441.310i		−	1.58745i
$279$	−332.218			−1.19074
$280$		−	175.796i		−	0.627845i
$281$	36.4182			0.129602			0.0648011	−	0.997898i	$-0.479359\pi$
$281$	36.4182			0.129602			0.0648011	+	0.997898i	$0.479359\pi$
$282$	−102.319			−0.362832
$283$	−15.3124			−0.0541074			−0.0270537	−	0.999634i	$-0.508613\pi$
$283$	−15.3124			−0.0541074			−0.0270537	+	0.999634i	$0.508613\pi$
$284$		−	30.1172i		−	0.106047i
$285$			322.638i			1.13206i
$286$		−	151.642i		−	0.530218i
$287$			17.0171i			0.0592930i
$288$			300.555i			1.04360i
$289$	−38.6884			−0.133870
$290$			243.302i			0.838972i
$291$			374.311i			1.28629i
$292$		−	30.5021i		−	0.104459i
$293$	−421.851			−1.43976			−0.719882	−	0.694097i	$-0.755804\pi$
$293$	−421.851			−1.43976			−0.719882	+	0.694097i	$0.755804\pi$
$294$	23.1968			0.0789008
$295$			192.563i			0.652757i
$296$	113.379			0.383038
$297$		−	524.551i		−	1.76616i
$298$	−168.761			−0.566311
$299$	−412.007			−1.37795
$300$			76.5357i			0.255119i
$301$	91.0288	+	279.310i	0.302421	+	0.927940i
$302$	−218.233			−0.722625
$303$			412.020i			1.35980i
$304$			228.377i			0.751239i
$305$	−233.763			−0.766436
$306$		−	633.581i		−	2.07053i
$307$	−239.260			−0.779349			−0.389674	−	0.920953i	$-0.627412\pi$
$307$	−239.260			−0.779349			−0.389674	+	0.920953i	$0.627412\pi$
$308$		−	39.9132i		−	0.129588i
$309$		−	195.157i		−	0.631576i
$310$	78.1681			0.252155
$311$	565.399			1.81800			0.909001	−	0.416793i	$-0.136846\pi$
$311$	565.399			1.81800			0.909001	+	0.416793i	$0.136846\pi$
$312$	599.624			1.92187
$313$			340.149i			1.08674i	0.839493	+	0.543370i	$0.182852\pi$
$313$			340.149i			1.08674i	−0.839493	+	0.543370i	$0.817148\pi$
$314$	336.126			1.07047
$315$	−460.716			−1.46259
$316$	42.5044			0.134508
$317$	−470.259			−1.48347			−0.741733	−	0.670695i	$-0.765996\pi$
$317$	−470.259			−1.48347			−0.741733	+	0.670695i	$0.765996\pi$
$318$	−311.936			−0.980931
$319$			315.738i			0.989774i
$320$		−	212.875i		−	0.665235i
$321$			677.025i			2.10911i
$322$	402.949			1.25139
$323$			303.945i			0.941008i
$324$	190.687			0.588541
$325$	199.195			0.612908
$326$	−469.467			−1.44008
$327$		−	458.296i		−	1.40152i
$328$			21.4392i			0.0653633i
$329$			70.0915i			0.213044i
$330$			205.358i			0.622296i
$331$		−	297.506i		−	0.898809i	−0.893328	−	0.449405i	$-0.851636\pi$
$331$		−	297.506i		−	0.898809i	0.893328	−	0.449405i	$-0.148364\pi$
$332$	−8.79134			−0.0264799
$333$		−	297.137i		−	0.892303i
$334$		−	71.0662i		−	0.212773i
$335$		−	267.167i		−	0.797514i
$336$	−456.229			−1.35782
$337$	488.152			1.44852			0.724261	−	0.689526i	$-0.242181\pi$
$337$	488.152			1.44852			0.724261	+	0.689526i	$0.242181\pi$
$338$			26.9946i			0.0798657i
$339$	330.000			0.973452
$340$		−	40.1199i		−	0.118000i
$341$	101.440			0.297479
$342$	769.337			2.24952
$343$		−	350.651i		−	1.02231i
$344$	114.684	+	351.892i	0.333382	+	1.02294i
$345$	557.950			1.61725
$346$		−	322.217i		−	0.931263i
$347$			620.235i			1.78742i	0.448646	+	0.893710i	$0.351907\pi$
$347$			620.235i			1.78742i	−0.448646	+	0.893710i	$0.648093\pi$
$348$	−218.429			−0.627670
$349$			97.1065i			0.278242i	0.990275	+	0.139121i	$0.0444277\pi$
$349$			97.1065i			0.278242i	−0.990275	+	0.139121i	$0.955572\pi$
$350$	−194.816			−0.556617
$351$		−	944.466i		−	2.69079i
$352$		−	91.7725i		−	0.260717i
$353$	355.666			1.00755			0.503776	−	0.863834i	$-0.331944\pi$
$353$	355.666			1.00755			0.503776	+	0.863834i	$0.331944\pi$
$354$	642.374			1.81462
$355$	106.150			0.299014
$356$			11.4425i			0.0321417i
$357$	−607.193			−1.70082
$358$	−111.130			−0.310420
$359$	185.120			0.515655			0.257827	−	0.966191i	$-0.416993\pi$
$359$	185.120			0.515655			0.257827	+	0.966191i	$0.416993\pi$
$360$	−580.438			−1.61233
$361$	−8.07135			−0.0223583
$362$		−	29.6796i		−	0.0819879i
$363$		−	413.229i		−	1.13837i
$364$		−	71.8646i		−	0.197430i
$365$	107.507			0.294539
$366$			779.813i			2.13064i
$367$	−379.616			−1.03438			−0.517188	−	0.855872i	$-0.673021\pi$
$367$	−379.616			−1.03438			−0.517188	+	0.855872i	$0.673021\pi$
$368$	394.939			1.07320
$369$	56.1864			0.152267
$370$			69.9139i			0.188956i
$371$			213.686i			0.575974i
$372$			70.1770i			0.188648i
$373$		−	190.307i		−	0.510206i	−0.966914	−	0.255103i	$-0.917891\pi$
$373$		−	190.307i		−	0.510206i	0.966914	−	0.255103i	$-0.0821094\pi$
$374$			193.460i			0.517272i
$375$	−689.611			−1.83896
$376$			88.3055i			0.234855i
$377$			568.494i			1.50794i
$378$			923.702i			2.44366i
$379$	−542.592			−1.43164			−0.715820	−	0.698285i	$-0.753947\pi$
$379$	−542.592			−1.43164			−0.715820	+	0.698285i	$0.753947\pi$
$380$	48.7163			0.128201
$381$		−	101.020i		−	0.265143i
$382$	−79.4161			−0.207896
$383$			126.903i			0.331340i	0.986181	+	0.165670i	$0.0529786\pi$
$383$			126.903i			0.331340i	−0.986181	+	0.165670i	$0.947021\pi$
$384$	−410.735			−1.06962
$385$	140.677			0.365394
$386$		−	240.024i		−	0.621823i
$387$	922.215	−	300.555i	2.38299	−	0.776629i
$388$	56.5186			0.145667
$389$		−	689.167i		−	1.77164i	−0.464031	−	0.885819i	$-0.653597\pi$
$389$		−	689.167i		−	1.77164i	0.464031	−	0.885819i	$-0.346403\pi$
$390$			369.751i			0.948080i
$391$	525.623			1.34430
$392$		−	20.0199i		−	0.0510712i
$393$	580.656			1.47750
$394$		−	258.845i		−	0.656968i
$395$			149.809i			0.379264i
$396$	−131.784			−0.332788
$397$	−422.076			−1.06316			−0.531582	−	0.847007i	$-0.678402\pi$
$397$	−422.076			−1.06316			−0.531582	+	0.847007i	$0.678402\pi$
$398$	498.258			1.25191
$399$		−	737.295i		−	1.84786i
$400$	−190.943			−0.477359
$401$	−393.635			−0.981632			−0.490816	−	0.871263i	$-0.663301\pi$
$401$	−393.635			−0.981632			−0.490816	+	0.871263i	$0.663301\pi$
$402$	−891.247			−2.21703
$403$	182.646			0.453215
$404$	62.2124			0.153991
$405$			672.089i			1.65948i
$406$		−	555.995i		−	1.36945i
$407$			90.7287i			0.222921i
$408$	−764.979			−1.87495
$409$		−	319.833i		−	0.781988i	−0.920393	−	0.390994i	$-0.872131\pi$
$409$		−	319.833i		−	0.781988i	0.920393	−	0.390994i	$-0.127869\pi$
$410$	−13.2202			−0.0322444
$411$	−788.967			−1.91963
$412$	−29.4675			−0.0715229
$413$		−	440.047i		−	1.06549i
$414$		−	1330.44i		−	3.21363i
$415$		−	30.9856i		−	0.0746642i
$416$		−	165.239i		−	0.397208i
$417$			1396.41i			3.34872i
$418$	−234.912			−0.561990
$419$		−	334.413i		−	0.798121i	−0.916925	−	0.399061i	$-0.869336\pi$
$419$		−	334.413i		−	0.798121i	0.916925	−	0.399061i	$-0.130664\pi$
$420$			97.3208i			0.231716i
$421$			332.902i			0.790740i	0.918522	+	0.395370i	$0.129384\pi$
$421$			332.902i			0.790740i	−0.918522	+	0.395370i	$0.870616\pi$
$422$	321.566			0.762005
$423$	231.426			0.547105
$424$			269.215i			0.634941i
$425$	−254.126			−0.597943
$426$		−	354.107i		−	0.831238i
$427$	534.197			1.25105
$428$	102.226			0.238847
$429$			479.834i			1.11849i
$430$	−216.990	+	70.7182i	−0.504627	+	0.164461i
$431$	−163.406			−0.379132			−0.189566	−	0.981868i	$-0.560708\pi$
$431$	−163.406			−0.379132			−0.189566	+	0.981868i	$0.560708\pi$
$432$			905.341i			2.09570i
$433$		−	190.232i		−	0.439335i	−0.975575	−	0.219668i	$-0.929503\pi$
$433$		−	190.232i		−	0.439335i	0.975575	−	0.219668i	$-0.0704973\pi$
$434$	−178.630			−0.411590
$435$		−	769.868i		−	1.76981i
$436$	−69.1997			−0.158715
$437$			638.248i			1.46052i
$438$		−	358.633i		−	0.818796i
$439$	−343.705			−0.782927			−0.391463	−	0.920194i	$-0.628031\pi$
$439$	−343.705			−0.782927			−0.391463	+	0.920194i	$0.628031\pi$
$440$	177.233			0.402802
$441$	−52.4669			−0.118973
$442$			348.329i			0.788074i
$443$	515.530			1.16373			0.581863	−	0.813287i	$-0.302324\pi$
$443$	515.530			1.16373			0.581863	+	0.813287i	$0.302324\pi$
$444$	−62.7666			−0.141366
$445$	−40.3296			−0.0906284
$446$	191.303			0.428931
$447$	534.000			1.19463
$448$			486.464i			1.08586i
$449$		−	598.611i		−	1.33321i	−0.745411	−	0.666605i	$-0.767747\pi$
$449$		−	598.611i		−	1.33321i	0.745411	−	0.666605i	$-0.232253\pi$
$450$			643.236i			1.42941i
$451$	−17.1561			−0.0380402
$452$		−	49.8279i		−	0.110239i
$453$	690.543			1.52438
$454$	−248.087			−0.546448
$455$	253.291			0.556684
$456$		−	928.889i		−	2.03704i
$457$		−	459.459i		−	1.00538i	−0.864467	−	0.502690i	$-0.832344\pi$
$457$		−	459.459i		−	1.00538i	0.864467	−	0.502690i	$-0.167656\pi$
$458$		−	547.963i		−	1.19643i
$459$			1204.92i			2.62509i
$460$		−	84.2468i		−	0.183145i
$461$	334.655			0.725933			0.362967	−	0.931802i	$-0.381764\pi$
$461$	334.655			0.725933			0.362967	+	0.931802i	$0.381764\pi$
$462$		−	469.284i		−	1.01577i
$463$			696.646i			1.50464i	0.658801	+	0.752318i	$0.271064\pi$
$463$			696.646i			1.50464i	−0.658801	+	0.752318i	$0.728936\pi$
$464$		−	544.944i		−	1.17445i
$465$	−247.343			−0.531921
$466$	−534.610			−1.14723
$467$		−	123.292i		−	0.264009i	−0.991249	−	0.132004i	$-0.957859\pi$
$467$		−	123.292i		−	0.264009i	0.991249	−	0.132004i	$-0.0421413\pi$
$468$	−237.280			−0.507009
$469$			610.532i			1.30177i
$470$	−54.4525			−0.115856
$471$	−1063.59			−2.25815
$472$		−	554.398i		−	1.17457i
$473$	−281.592	+	91.7725i	−0.595332	+	0.194022i
$474$	499.751			1.05433
$475$		−	308.577i		−	0.649636i
$476$			91.6822i			0.192610i
$477$	705.541			1.47912
$478$			216.660i			0.453263i
$479$	201.938			0.421583			0.210791	−	0.977531i	$-0.432396\pi$
$479$	201.938			0.421583			0.210791	+	0.977531i	$0.432396\pi$
$480$			223.770i			0.466188i
$481$			163.359i			0.339624i
$482$	843.443			1.74988
$483$	−1275.03			−2.63981
$484$	−62.3950			−0.128915
$485$			199.203i			0.410729i
$486$	1025.18			2.10943
$487$	151.294			0.310666			0.155333	−	0.987862i	$-0.450355\pi$
$487$	151.294			0.310666			0.155333	+	0.987862i	$0.450355\pi$
$488$	673.014			1.37913
$489$	1485.51			3.03785
$490$	12.3450			0.0251939
$491$		−	56.4377i		−	0.114944i	−0.998347	−	0.0574722i	$-0.981696\pi$
$491$		−	56.4377i		−	0.114944i	0.998347	−	0.0574722i	$-0.0183041\pi$
$492$		−	11.8687i		−	0.0241234i
$493$		−	725.263i		−	1.47112i
$494$	−422.964			−0.856203
$495$		−	464.481i		−	0.938345i
$496$	−175.080			−0.352983
$497$	−242.575			−0.488078
$498$	−103.365			−0.207561
$499$			337.720i			0.676793i	0.941004	+	0.338396i	$0.109884\pi$
$499$			337.720i			0.676793i	−0.941004	+	0.338396i	$0.890116\pi$
$500$			104.127i			0.208254i
$501$			224.871i			0.448844i
$502$			63.8994i			0.127290i
$503$			124.026i			0.246573i	0.992371	+	0.123287i	$0.0393434\pi$
$503$			124.026i			0.246573i	−0.992371	+	0.123287i	$0.960657\pi$
$504$	1326.42			2.63179
$505$			219.272i			0.434201i
$506$			406.241i			0.802848i
$507$		−	85.4176i		−	0.168476i
$508$	−15.2533			−0.0300262
$509$	229.831			0.451535			0.225767	−	0.974181i	$-0.427511\pi$
$509$	229.831			0.451535			0.225767	+	0.974181i	$0.427511\pi$
$510$		−	471.715i		−	0.924931i
$511$	−245.675			−0.480773
$512$			567.670i			1.10873i
$513$	−1463.09			−2.85203
$514$	−111.465			−0.216858
$515$		−	103.860i		−	0.201670i
$516$	−63.4887	−	194.807i	−0.123040	−	0.377533i
$517$	−70.6642			−0.136681
$518$		−	159.768i		−	0.308432i
$519$			1019.57i			1.96450i
$520$	319.112			0.613677
$521$			335.813i			0.644555i	0.946645	+	0.322278i	$0.104448\pi$
$521$			335.813i			0.644555i	−0.946645	+	0.322278i	$0.895552\pi$
$522$	−1835.76			−3.51679
$523$		−	630.020i		−	1.20463i	−0.798259	−	0.602314i	$-0.794246\pi$
$523$		−	630.020i		−	1.20463i	0.798259	−	0.602314i	$-0.205754\pi$
$524$		−	87.6752i		−	0.167319i
$525$	616.446			1.17418
$526$	−30.5073			−0.0579987
$527$	−233.013			−0.442149
$528$		−	459.956i		−	0.871130i
$529$	574.743			1.08647
$530$	−166.008			−0.313223
$531$	−1452.93			−2.73621
$532$	−111.327			−0.209261
$533$	−30.8900			−0.0579550
$534$			134.536i			0.251940i
$535$			360.303i			0.673464i
$536$			769.186i			1.43505i
$537$	351.644			0.654831
$538$			724.216i			1.34613i
$539$	16.0204			0.0297224
$540$	193.124			0.357636
$541$	579.942			1.07198			0.535991	−	0.844224i	$-0.319938\pi$
$541$	579.942			1.07198			0.535991	+	0.844224i	$0.319938\pi$
$542$			561.490i			1.03596i
$543$			93.9137i			0.172953i
$544$			210.805i			0.387510i
$545$		−	243.899i		−	0.447521i
$546$		−	844.958i		−	1.54754i
$547$	276.106			0.504764			0.252382	−	0.967628i	$-0.418786\pi$
$547$	276.106			0.504764			0.252382	+	0.967628i	$0.418786\pi$
$548$			119.129i			0.217389i
$549$		−	1763.79i		−	3.21273i
$550$		−	196.408i		−	0.357105i
$551$	880.664			1.59830
$552$	−1606.36			−2.91007
$553$		−	342.346i		−	0.619070i
$554$	−371.767			−0.671059
$555$		−	221.225i		−	0.398603i
$556$	210.850			0.379226
$557$	−1.44850			−0.00260054			−0.00130027	−	0.999999i	$-0.500414\pi$
$557$	−1.44850			−0.00260054			−0.00130027	+	0.999999i	$0.500414\pi$
$558$			589.795i			1.05698i
$559$	−507.013	+	165.239i	−0.907000	+	0.295597i
$560$	−242.799			−0.433569
$561$		−	612.154i		−	1.09118i
$562$		−	64.6542i		−	0.115043i
$563$	−87.6035			−0.155601			−0.0778006	−	0.996969i	$-0.524790\pi$
$563$	−87.6035			−0.155601			−0.0778006	+	0.996969i	$0.524790\pi$
$564$		−	48.8859i		−	0.0866771i
$565$	175.622			0.310835
$566$			27.1845i			0.0480292i
$567$		−	1535.86i		−	2.70875i
$568$	−305.611			−0.538047
$569$	−657.323			−1.15522			−0.577612	−	0.816311i	$-0.696015\pi$
$569$	−657.323			−1.15522			−0.577612	+	0.816311i	$0.696015\pi$
$570$	572.788			1.00489
$571$			321.145i			0.562426i	0.959645	+	0.281213i	$0.0907368\pi$
$571$			321.145i			0.562426i	−0.959645	+	0.281213i	$0.909263\pi$
$572$	72.4518			0.126664
$573$	251.292			0.438556
$574$	30.2109			0.0526322
$575$	−533.633			−0.928057
$576$	1606.19			2.78852
$577$		−	440.527i		−	0.763479i	−0.924270	−	0.381740i	$-0.875325\pi$
$577$		−	440.527i		−	0.763479i	0.924270	−	0.381740i	$-0.124675\pi$
$578$			68.6845i			0.118831i
$579$			759.494i			1.31173i
$580$	−116.245			−0.200423
$581$			70.8085i			0.121874i
$582$	664.525			1.14180
$583$	−215.432			−0.369523
$584$	−309.516			−0.529994
$585$		−	836.308i		−	1.42959i
$586$			748.923i			1.27803i
$587$		−	975.567i		−	1.66195i	−0.556307	−	0.830977i	$-0.687782\pi$
$587$		−	975.567i		−	1.66195i	0.556307	−	0.830977i	$-0.312218\pi$
$588$			11.0830i			0.0188486i
$589$		−	282.940i		−	0.480373i
$590$	341.863			0.579428
$591$			819.051i			1.38587i
$592$		−	156.592i		−	0.264513i
$593$			14.4319i			0.0243370i	0.999926	+	0.0121685i	$0.00387345\pi$
$593$			14.4319i			0.0243370i	−0.999926	+	0.0121685i	$0.996127\pi$
$594$	−931.249			−1.56776
$595$	−323.140			−0.543092
$596$		−	80.6306i		−	0.135286i
$597$	−1576.61			−2.64089
$598$			731.447i			1.22316i
$599$	−947.716			−1.58216			−0.791082	−	0.611710i	$-0.790482\pi$
$599$	−947.716			−1.58216			−0.791082	+	0.611710i	$0.790482\pi$
$600$	776.636			1.29439
$601$		−	1115.25i		−	1.85565i	−0.373016	−	0.927825i	$-0.621676\pi$
$601$		−	1115.25i		−	1.85565i	0.373016	−	0.927825i	$-0.378324\pi$
$602$	495.866	−	161.606i	0.823698	−	0.268448i
$603$	2015.83			3.34301
$604$		−	104.268i		−	0.172628i
$605$		−	219.915i		−	0.363496i
$606$	731.470			1.20705
$607$			297.682i			0.490416i	0.969471	+	0.245208i	$0.0788562\pi$
$607$			297.682i			0.490416i	−0.969471	+	0.245208i	$0.921144\pi$
$608$	−255.974			−0.421010
$609$			1759.31i			2.88885i
$610$			415.006i			0.680337i
$611$	−127.233			−0.208237
$612$	302.713			0.494629
$613$	382.944			0.624705			0.312352	−	0.949966i	$-0.398883\pi$
$613$	382.944			0.624705			0.312352	+	0.949966i	$0.398883\pi$
$614$			424.765i			0.691799i
$615$	41.8320			0.0680195
$616$	−405.014			−0.657490
$617$	−702.287			−1.13823			−0.569114	−	0.822258i	$-0.692714\pi$
$617$	−702.287			−1.13823			−0.569114	+	0.822258i	$0.692714\pi$
$618$	−346.467			−0.560627
$619$	−680.743			−1.09975			−0.549873	−	0.835248i	$-0.685324\pi$
$619$	−680.743			−1.09975			−0.549873	+	0.835248i	$0.685324\pi$
$620$			37.3472i			0.0602375i
$621$			2530.17i			4.07435i
$622$		−	1003.77i		−	1.61377i
$623$	92.1616			0.147932
$624$		−	828.162i		−	1.32718i
$625$	34.5564			0.0552903
$626$	603.876			0.964659
$627$	743.319			1.18552
$628$			160.595i			0.255724i
$629$		−	208.408i		−	0.331332i
$630$			817.922i			1.29829i
$631$			1019.35i			1.61546i	0.589555	+	0.807728i	$0.299303\pi$
$631$			1019.35i			1.61546i	−0.589555	+	0.807728i	$0.700697\pi$
$632$		−	431.308i		−	0.682449i
$633$	−1017.51			−1.60745
$634$			834.863i			1.31682i
$635$		−	53.7613i		−	0.0846634i
$636$		−	149.037i		−	0.234335i
$637$	28.8451			0.0452827
$638$	560.538			0.878586
$639$			800.925i			1.25340i
$640$	−218.587			−0.341543
$641$		−	815.451i		−	1.27215i	−0.771625	−	0.636077i	$-0.780556\pi$
$641$		−	815.451i		−	1.27215i	0.771625	−	0.636077i	$-0.219444\pi$
$642$	1201.94			1.87218
$643$	661.567			1.02888			0.514438	−	0.857528i	$-0.328001\pi$
$643$	661.567			1.02888			0.514438	+	0.857528i	$0.328001\pi$
$644$			192.521i			0.298946i
$645$	686.609	−	223.770i	1.06451	−	0.346930i
$646$	539.602			0.835298
$647$			1013.73i			1.56681i	0.621512	+	0.783405i	$0.286519\pi$
$647$			1013.73i			1.56681i	−0.621512	+	0.783405i	$0.713481\pi$
$648$		−	1934.97i		−	2.98607i
$649$	443.642			0.683578
$650$		−	353.636i		−	0.544056i
$651$	565.230			0.868249
$652$		−	224.302i		−	0.344022i
$653$			812.108i			1.24366i	0.783153	+	0.621829i	$0.213610\pi$
$653$			812.108i			1.24366i	−0.783153	+	0.621829i	$0.786390\pi$
$654$	−813.624			−1.24407
$655$	309.017			0.471782
$656$	29.6104			0.0451378
$657$			811.160i			1.23464i
$658$	124.435			0.189111
$659$	−1106.83			−1.67955			−0.839777	−	0.542932i	$-0.817314\pi$
$659$	−1106.83			−1.67955			−0.839777	+	0.542932i	$0.817314\pi$
$660$	−98.1159			−0.148661
$661$	523.333			0.791729			0.395865	−	0.918309i	$-0.370445\pi$
$661$	523.333			0.791729			0.395865	+	0.918309i	$0.370445\pi$
$662$	−528.170			−0.797840
$663$		−	1102.20i		−	1.66244i
$664$			89.2089i			0.134351i
$665$		−	392.378i		−	0.590043i
$666$	−527.515			−0.792064
$667$		−	1522.96i		−	2.28330i
$668$	33.9541			0.0508295
$669$	−605.330			−0.904828
$670$	−474.309			−0.707924
$671$			538.562i			0.802625i
$672$		−	511.361i		−	0.760953i
$673$		−	320.300i		−	0.475929i	−0.971274	−	0.237965i	$-0.923520\pi$
$673$		−	320.300i		−	0.475929i	0.971274	−	0.237965i	$-0.0764803\pi$
$674$		−	866.629i		−	1.28580i
$675$		−	1223.28i		−	1.81226i
$676$	−12.8975			−0.0190791
$677$		−	854.967i		−	1.26288i	−0.775426	−	0.631438i	$-0.782465\pi$
$677$		−	854.967i		−	1.26288i	0.775426	−	0.631438i	$-0.217535\pi$
$678$		−	585.858i		−	0.864098i
$679$		−	455.221i		−	0.670428i
$680$	−407.111			−0.598693
$681$	785.010			1.15273
$682$		−	180.090i		−	0.264061i
$683$	1331.16			1.94898			0.974492	−	0.224420i	$-0.0720488\pi$
$683$	1331.16			1.94898			0.974492	+	0.224420i	$0.0720488\pi$
$684$			367.575i			0.537390i
$685$	−419.877			−0.612960
$686$	−622.519			−0.907463
$687$			1733.89i			2.52386i
$688$	486.010	−	158.393i	0.706410	−	0.230223i
$689$	−387.891			−0.562976
$690$		−	990.543i		−	1.43557i
$691$		−	568.600i		−	0.822866i	−0.911440	−	0.411433i	$-0.865028\pi$
$691$		−	568.600i		−	0.822866i	0.911440	−	0.411433i	$-0.134972\pi$
$692$	153.949			0.222470
$693$			1061.43i			1.53165i
$694$	1101.12			1.58663
$695$			743.152i			1.06928i
$696$			2216.48i			3.18460i
$697$	39.4084			0.0565400
$698$	172.396			0.246985
$699$	1691.64			2.42008
$700$		−	93.0793i		−	0.132970i
$701$	369.079			0.526504			0.263252	−	0.964727i	$-0.415205\pi$
$701$	369.079			0.526504			0.263252	+	0.964727i	$0.415205\pi$
$702$	−1676.74			−2.38851
$703$	253.063			0.359975
$704$	−490.439			−0.696646
$705$	172.301			0.244399
$706$		−	631.423i		−	0.894367i
$707$		−	501.081i		−	0.708742i
$708$			306.914i			0.433494i
$709$	−656.495			−0.925945			−0.462972	−	0.886373i	$-0.653217\pi$
$709$	−656.495			−0.925945			−0.462972	+	0.886373i	$0.653217\pi$
$710$		−	188.451i		−	0.265424i
$711$	−1130.34			−1.58979
$712$	116.111			0.163077
$713$	−489.298			−0.686252
$714$			1077.97i			1.50976i
$715$			255.361i			0.357148i
$716$		−	53.0960i		−	0.0741564i
$717$		−	685.564i		−	0.956157i
$718$		−	328.649i		−	0.457728i
$719$	−497.631			−0.692115			−0.346057	−	0.938213i	$-0.612480\pi$
$719$	−497.631			−0.692115			−0.346057	+	0.938213i	$0.612480\pi$
$720$			801.664i			1.11342i
$721$			237.341i			0.329183i
$722$			14.3293i			0.0198466i
$723$	−2668.86			−3.69137
$724$	14.1804			0.0195861
$725$			736.315i			1.01561i
$726$	−733.617			−1.01049
$727$			1020.14i			1.40322i	0.712559	+	0.701612i	$0.247536\pi$
$727$			1020.14i			1.40322i	−0.712559	+	0.701612i	$0.752464\pi$
$728$	−729.237			−1.00170
$729$	−1220.64			−1.67441
$730$		−	190.859i		−	0.261451i
$731$	646.829	−	210.805i	0.884855	−	0.288379i
$732$	−372.580			−0.508989
$733$		−	310.797i		−	0.424007i	−0.977269	−	0.212003i	$-0.932001\pi$
$733$		−	310.797i		−	0.424007i	0.977269	−	0.212003i	$-0.0679987\pi$
$734$			673.942i			0.918177i
$735$	−39.0627			−0.0531466
$736$			442.665i			0.601447i
$737$	−615.521			−0.835171
$738$		−	99.7492i		−	0.135162i
$739$		−	1188.34i		−	1.60804i	−0.594603	−	0.804020i	$-0.702691\pi$
$739$		−	1188.34i		−	1.60804i	0.594603	−	0.804020i	$-0.297309\pi$
$740$	−33.4035			−0.0451399
$741$	1338.36			1.80616
$742$	379.363			0.511271
$743$			258.515i			0.347934i	0.984752	+	0.173967i	$0.0556585\pi$
$743$			258.515i			0.347934i	−0.984752	+	0.173967i	$0.944341\pi$
$744$	712.111			0.957139
$745$	284.188			0.381460
$746$	−337.857			−0.452891
$747$	233.793			0.312976
$748$	−92.4313			−0.123571
$749$		−	823.367i		−	1.09929i
$750$			1224.29i			1.63238i
$751$		−	816.888i		−	1.08773i	−0.839171	−	0.543867i	$-0.816960\pi$
$751$		−	816.888i		−	1.08773i	0.839171	−	0.543867i	$-0.183040\pi$
$752$	121.962			0.162183
$753$		−	202.193i		−	0.268517i
$754$	1009.26			1.33854
$755$	367.497			0.486752
$756$	−441.327			−0.583766
$757$			807.752i			1.06704i	0.845786	+	0.533522i	$0.179132\pi$
$757$			807.752i			1.06704i	−0.845786	+	0.533522i	$0.820868\pi$
$758$			963.277i			1.27081i
$759$		−	1285.45i		−	1.69361i
$760$		−	494.342i		−	0.650450i
$761$		−	589.286i		−	0.774358i	−0.922005	−	0.387179i	$-0.873450\pi$
$761$		−	589.286i		−	0.774358i	0.922005	−	0.387179i	$-0.126550\pi$
$762$	−179.343			−0.235358
$763$			557.359i			0.730483i
$764$		−	37.9435i		−	0.0496643i
$765$			1066.93i			1.39468i
$766$	225.294			0.294118
$767$	798.788			1.04144
$768$		−	870.818i		−	1.13388i
$769$	1272.87			1.65522			0.827612	−	0.561300i	$-0.189699\pi$
$769$	1272.87			1.65522			0.827612	+	0.561300i	$0.189699\pi$
$770$		−	249.747i		−	0.324346i
$771$	352.703			0.457462
$772$	114.679			0.148547
$773$		−	176.192i		−	0.227933i	−0.993485	−	0.113967i	$-0.963644\pi$
$773$		−	176.192i		−	0.227933i	0.993485	−	0.113967i	$-0.0363557\pi$
$774$	−533.584	−	1637.23i	−0.689385	−	2.11529i
$775$	236.563			0.305243
$776$		−	573.515i		−	0.739065i
$777$			505.544i			0.650636i
$778$	−1223.50			−1.57262
$779$			47.8523i			0.0614279i
$780$	−176.660			−0.226487
$781$		−	244.557i		−	0.313133i
$782$		−	933.153i		−	1.19329i
$783$	3491.17			4.45871
$784$	−27.6502			−0.0352681
$785$	−566.026			−0.721053
$786$		−	1030.85i		−	1.31152i
$787$	−1120.38			−1.42361			−0.711804	−	0.702379i	$-0.752121\pi$
$787$	−1120.38			−1.42361			−0.711804	+	0.702379i	$0.752121\pi$
$788$	123.671			0.156943
$789$	96.5327			0.122348
$790$	265.961			0.336659
$791$	−401.332			−0.507373
$792$			1337.26i			1.68846i
$793$			969.693i			1.22282i
$794$			749.323i			0.943732i
$795$	525.291			0.660743
$796$			238.058i			0.299068i
$797$	−872.371			−1.09457			−0.547284	−	0.836947i	$-0.684338\pi$
$797$	−872.371			−1.09457			−0.547284	+	0.836947i	$0.684338\pi$
$798$	−1308.94			−1.64028
$799$	162.319			0.203152
$800$		−	214.018i		−	0.267522i
$801$		−	304.296i		−	0.379895i
$802$			698.830i			0.871359i
$803$		−	247.682i		−	0.308446i
$804$		−	425.821i		−	0.529628i
$805$	−678.553			−0.842924
$806$		−	324.256i		−	0.402303i
$807$		−	2291.60i		−	2.83965i
$808$		−	631.292i		−	0.781302i
$809$	273.069			0.337539			0.168770	−	0.985656i	$-0.446021\pi$
$809$	273.069			0.337539			0.168770	+	0.985656i	$0.446021\pi$
$810$	1193.18			1.47306
$811$			1420.64i			1.75171i	0.482576	+	0.875854i	$0.339701\pi$
$811$			1420.64i			1.75171i	−0.482576	+	0.875854i	$0.660299\pi$
$812$	265.644			0.327148
$813$		−	1776.69i		−	2.18536i
$814$	161.073			0.197878
$815$	790.568			0.970022
$816$			1056.54i			1.29478i
$817$	255.974	+	785.424i	0.313310	+	0.961351i
$818$	−567.808			−0.694142
$819$			1911.14i			2.33350i
$820$		−	6.31636i		−	0.00770288i
$821$	1163.98			1.41775			0.708877	−	0.705332i	$-0.249202\pi$
$821$	1163.98			1.41775			0.708877	+	0.705332i	$0.249202\pi$
$822$			1400.67i			1.70398i
$823$	47.3698			0.0575574			0.0287787	−	0.999586i	$-0.490838\pi$
$823$	47.3698			0.0575574			0.0287787	+	0.999586i	$0.490838\pi$
$824$			299.017i			0.362885i
$825$			621.482i			0.753312i
$826$	−781.227			−0.945795
$827$	−662.925			−0.801602			−0.400801	−	0.916165i	$-0.631268\pi$
$827$	−662.925			−0.801602			−0.400801	+	0.916165i	$0.631268\pi$
$828$	635.660			0.767705
$829$			952.975i			1.14955i	0.818312	+	0.574774i	$0.194910\pi$
$829$			952.975i			1.14955i	−0.818312	+	0.574774i	$0.805090\pi$
$830$	−55.0096			−0.0662766
$831$	1176.36			1.41560
$832$	−883.047			−1.06135
$833$	−36.7995			−0.0441771
$834$	2479.09			2.97253
$835$			119.673i			0.143321i
$836$		−	112.236i		−	0.134254i
$837$		−	1121.64i		−	1.34008i
$838$	−593.692			−0.708463
$839$		−	218.560i		−	0.260501i	−0.991481	−	0.130250i	$-0.958422\pi$
$839$		−	218.560i		−	0.260501i	0.991481	−	0.130250i	$-0.0415781\pi$
$840$	987.549			1.17565
$841$	−1260.41			−1.49870
$842$	591.009			0.701911
$843$			204.582i			0.242683i
$844$			153.638i			0.182036i
$845$		−	45.4581i		−	0.0537965i
$846$		−	410.856i		−	0.485645i
$847$			502.551i			0.593331i
$848$	371.822			0.438469
$849$		−	86.0185i		−	0.101317i
$850$			451.156i			0.530772i
$851$		−	437.630i		−	0.514254i
$852$	169.186			0.198575
$853$	−1445.29			−1.69436			−0.847180	−	0.531306i	$-0.821701\pi$
$853$	−1445.29			−1.69436			−0.847180	+	0.531306i	$0.821701\pi$
$854$		−	948.374i		−	1.11051i
$855$	−1295.54			−1.51525
$856$		−	1037.33i		−	1.21183i
$857$	−92.3800			−0.107795			−0.0538973	−	0.998546i	$-0.517164\pi$
$857$	−92.3800			−0.107795			−0.0538973	+	0.998546i	$0.517164\pi$
$858$	851.861			0.992846
$859$			786.473i			0.915568i	0.889064	+	0.457784i	$0.151357\pi$
$859$			786.473i			0.915568i	−0.889064	+	0.457784i	$0.848643\pi$
$860$	−33.7878	−	103.674i	−0.0392882	−	0.120551i
$861$	−95.5948			−0.111028
$862$			290.099i			0.336542i
$863$			1277.18i			1.47993i	0.672646	+	0.739965i	$0.265158\pi$
$863$			1277.18i			1.47993i	−0.672646	+	0.739965i	$0.734842\pi$
$864$	−1014.74			−1.17447
$865$			542.603i			0.627287i
$866$	−337.724			−0.389982
$867$		−	217.335i		−	0.250674i
$868$		−	85.3461i		−	0.0983250i
$869$	345.143			0.397172
$870$	−1366.77			−1.57100
$871$	−1108.26			−1.27240
$872$			702.194i			0.805269i
$873$	−1503.03			−1.72169
$874$	1133.10			1.29645
$875$	838.675			0.958485
$876$	171.348			0.195603
$877$	1081.16			1.23279			0.616396	−	0.787436i	$-0.288592\pi$
$877$	1081.16			1.23279			0.616396	+	0.787436i	$0.288592\pi$
$878$			610.188i			0.694975i
$879$		−	2369.78i		−	2.69599i
$880$		−	244.782i		−	0.278162i
$881$	−575.591			−0.653338			−0.326669	−	0.945139i	$-0.605926\pi$
$881$	−575.591			−0.653338			−0.326669	+	0.945139i	$0.605926\pi$
$882$			93.1458i			0.105608i
$883$	1401.90			1.58766			0.793828	−	0.608143i	$-0.208085\pi$
$883$	1401.90			1.58766			0.793828	+	0.608143i	$0.208085\pi$
$884$	−166.425			−0.188263
$885$	−1081.74			−1.22230
$886$		−	915.234i		−	1.03300i
$887$			205.103i			0.231232i	0.993294	+	0.115616i	$0.0368842\pi$
$887$			205.103i			0.231232i	−0.993294	+	0.115616i	$0.963116\pi$
$888$			636.916i			0.717247i
$889$			122.856i			0.138195i
$890$			71.5983i			0.0804475i
$891$	1548.41			1.73783
$892$			91.4010i			0.102467i
$893$			197.098i			0.220715i
$894$		−	948.025i		−	1.06043i
$895$	187.140			0.209095
$896$	499.517			0.557497
$897$		−	2314.48i		−	2.58024i
$898$	−1062.73			−1.18344
$899$			675.141i			0.750991i
$900$	−307.326			−0.341473
$901$	494.857			0.549230
$902$			30.4577i			0.0337669i
$903$	−1569.04	+	511.361i	−1.73759	+	0.566291i
$904$	−505.622			−0.559316
$905$			49.9796i			0.0552260i
$906$		−	1225.94i		−	1.35313i
$907$	−252.284			−0.278152			−0.139076	−	0.990282i	$-0.544413\pi$
$907$	−252.284			−0.278152			−0.139076	+	0.990282i	$0.544413\pi$
$908$		−	118.531i		−	0.130541i
$909$	−1654.45			−1.82008
$910$		−	449.675i		−	0.494148i
$911$			1367.68i			1.50130i	0.660702	+	0.750649i	$0.270259\pi$
$911$			1367.68i			1.50130i	−0.660702	+	0.750649i	$0.729741\pi$
$912$	−1282.92			−1.40671
$913$	−71.3871			−0.0781896
$914$	−815.689			−0.892439
$915$		−	1313.18i		−	1.43517i
$916$	261.806			0.285815
$917$	−706.167			−0.770084
$918$	2139.12			2.33019
$919$	−270.734			−0.294597			−0.147298	−	0.989092i	$-0.547058\pi$
$919$	−270.734			−0.294597			−0.147298	+	0.989092i	$0.547058\pi$
$920$	−854.883			−0.929220
$921$		−	1344.06i		−	1.45935i
$922$		−	594.122i		−	0.644384i
$923$		−	440.330i		−	0.477064i
$924$	224.215			0.242657
$925$			211.583i			0.228739i
$926$	1236.77			1.33561
$927$	783.645			0.845356
$928$	610.796			0.658185
$929$		−	791.662i		−	0.852165i	−0.904684	−	0.426083i	$-0.859893\pi$
$929$		−	791.662i		−	0.852165i	0.904684	−	0.426083i	$-0.140107\pi$
$930$			439.115i			0.472167i
$931$		−	44.6845i		−	0.0479962i
$932$		−	255.426i		−	0.274063i
$933$			3176.17i			3.40425i
$934$	−218.884			−0.234351
$935$		−	325.780i		−	0.348428i
$936$			2407.77i			2.57240i
$937$			1025.37i			1.09431i	0.837030	+	0.547157i	$0.184290\pi$
$937$			1025.37i			1.09431i	−0.837030	+	0.547157i	$0.815710\pi$
$938$	1083.89			1.15554
$939$	−1910.81			−2.03495
$940$		−	26.0164i		−	0.0276770i
$941$	1515.52			1.61054			0.805271	−	0.592907i	$-0.202020\pi$
$941$	1515.52			1.61054			0.805271	+	0.592907i	$0.202020\pi$
$942$			1888.21i			2.00447i
$943$	82.7527			0.0877547
$944$	−765.698			−0.811121
$945$		−	1555.49i		−	1.64602i
$946$	162.926	+	499.918i	0.172226	+	0.528454i
$947$	804.042			0.849042			0.424521	−	0.905418i	$-0.360443\pi$
$947$	804.042			0.849042			0.424521	+	0.905418i	$0.360443\pi$
$948$			238.772i			0.251869i
$949$		−	445.958i		−	0.469924i
$950$	−547.825			−0.576658
$951$		−	2641.71i		−	2.77783i
$952$	930.333			0.977241
$953$			1307.06i			1.37152i	0.727828	+	0.685760i	$0.240530\pi$
$953$			1307.06i			1.37152i	−0.727828	+	0.685760i	$0.759470\pi$
$954$		−	1252.57i		−	1.31296i
$955$	133.734			0.140036
$956$	−103.516			−0.108280
$957$	−1773.68			−1.85338
$958$		−	358.506i		−	0.374224i
$959$	959.507			1.00053
$960$	1195.84			1.24567
$961$	−744.091			−0.774288
$962$	290.016			0.301472
$963$	−2718.56			−2.82301
$964$			402.981i			0.418030i
$965$			404.192i			0.418852i
$966$			2263.59i			2.34327i
$967$	−1273.66			−1.31713			−0.658563	−	0.752525i	$-0.728836\pi$
$967$	−1273.66			−1.31713			−0.658563	+	0.752525i	$0.728836\pi$
$968$			633.144i			0.654075i
$969$	−1707.44			−1.76206
$970$	353.651			0.364589
$971$	1511.12			1.55625			0.778123	−	0.628112i	$-0.216172\pi$
$971$	1511.12			1.55625			0.778123	+	0.628112i	$0.216172\pi$
$972$			489.812i			0.503922i
$973$		−	1698.26i		−	1.74538i
$974$		−	268.597i		−	0.275767i
$975$			1118.99i			1.14769i
$976$		−	929.523i		−	0.952380i
$977$	1275.34			1.30537			0.652684	−	0.757630i	$-0.273643\pi$
$977$	1275.34			1.30537			0.652684	+	0.757630i	$0.273643\pi$
$978$		−	2637.27i		−	2.69659i
$979$			92.9146i			0.0949076i
$980$			5.89822i			0.00601859i
$981$	1840.27			1.87591
$982$	−100.195			−0.102032
$983$			643.947i			0.655084i	0.944837	+	0.327542i	$0.106220\pi$
$983$			643.947i			0.655084i	−0.944837	+	0.327542i	$0.893780\pi$
$984$	−120.436			−0.122394
$985$			435.887i			0.442525i
$986$	−1287.58			−1.30586
$987$	−393.744			−0.398930
$988$		−	202.084i		−	0.204539i
$989$	1358.26	−	442.665i	1.37337	−	0.447588i
$990$	−824.604			−0.832934
$991$		−	778.474i		−	0.785544i	−0.919636	−	0.392772i	$-0.871516\pi$
$991$		−	778.474i		−	0.785544i	0.919636	−	0.392772i	$-0.128484\pi$
$992$		−	196.237i		−	0.197819i
$993$	1671.26			1.68304
$994$			430.650i			0.433249i
$995$	−839.052			−0.843268
$996$		−	49.3860i		−	0.0495843i
$997$		−	457.177i		−	0.458552i	−0.973361	−	0.229276i	$-0.926364\pi$
$997$		−	457.177i		−	0.458552i	0.973361	−	0.229276i	$-0.0736359\pi$
$998$	599.562			0.600764
$999$	1003.20			1.00421

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.b.b.42.3	✓	6
3.2	odd	2		387.3.b.c.343.4		6
4.3	odd	2		688.3.b.e.257.1		6
43.42	odd	2	inner	43.3.b.b.42.4	yes	6
129.128	even	2		387.3.b.c.343.3		6
172.171	even	2		688.3.b.e.257.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.b.b.42.3	✓	6	1.1	even	1	trivial
43.3.b.b.42.4	yes	6	43.42	odd	2	inner
387.3.b.c.343.3		6	129.128	even	2
387.3.b.c.343.4		6	3.2	odd	2
688.3.b.e.257.1		6	4.3	odd	2
688.3.b.e.257.6		6	172.171	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.77533i q^{2} +5.61757i q^{3} +0.848217 q^{4} +2.98959i q^{5} +9.97302 q^{6} -6.83184i q^{7} -8.60717i q^{8} -22.5571 q^{9} +O(q^{10})\)	\(q-1.77533i q^{2} +5.61757i q^{3} +0.848217 q^{4} +2.98959i q^{5} +9.97302 q^{6} -6.83184i q^{7} -8.60717i q^{8} -22.5571 q^{9} +5.30751 q^{10} +6.88766 q^{11} +4.76492i q^{12} +12.4014 q^{13} -12.1287 q^{14} -16.7943 q^{15} -11.8877 q^{16} -15.8212 q^{17} +40.0462i q^{18} -19.2112i q^{19} +2.53583i q^{20} +38.3784 q^{21} -12.2278i q^{22} -33.2226 q^{23} +48.3514 q^{24} +16.0623 q^{25} -22.0165i q^{26} -76.1581i q^{27} -5.79488i q^{28} +45.8411i q^{29} +29.8153i q^{30} +14.7278 q^{31} -13.3242i q^{32} +38.6919i q^{33} +28.0879i q^{34} +20.4244 q^{35} -19.1333 q^{36} +13.1726i q^{37} -34.1062 q^{38} +69.6657i q^{39} +25.7319 q^{40} -2.49085 q^{41} -68.1341i q^{42} +(-40.8836 + 13.3242i) q^{43} +5.84223 q^{44} -67.4366i q^{45} +58.9810i q^{46} -10.2595 q^{47} -66.7798i q^{48} +2.32596 q^{49} -28.5159i q^{50} -88.8770i q^{51} +10.5191 q^{52} -31.2780 q^{53} -135.205 q^{54} +20.5913i q^{55} -58.8028 q^{56} +107.920 q^{57} +81.3829 q^{58} +64.4112 q^{59} -14.2452 q^{60} +78.1922i q^{61} -26.1467i q^{62} +154.107i q^{63} -71.2054 q^{64} +37.0751i q^{65} +68.6908 q^{66} -89.3657 q^{67} -13.4198 q^{68} -186.631i q^{69} -36.2600i q^{70} -35.5065i q^{71} +194.153i q^{72} -35.9603i q^{73} +23.3857 q^{74} +90.2313i q^{75} -16.2953i q^{76} -47.0554i q^{77} +123.679 q^{78} +50.1103 q^{79} -35.5393i q^{80} +224.809 q^{81} +4.42207i q^{82} -10.3645 q^{83} +32.5532 q^{84} -47.2991i q^{85} +(23.6548 + 72.5817i) q^{86} -257.516 q^{87} -59.2832i q^{88} +13.4900i q^{89} -119.722 q^{90} -84.7243i q^{91} -28.1800 q^{92} +82.7347i q^{93} +18.2140i q^{94} +57.4338 q^{95} +74.8496 q^{96} +66.6322 q^{97} -4.12933i q^{98} -155.366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10}) \) 6 * q - 16 * q^4 + 6 * q^6 - 36 * q^9	\( 6 q - 16 q^{4} + 6 q^{6} - 36 q^{9} - 2 q^{10} + 38 q^{11} + 30 q^{13} + 36 q^{14} + 28 q^{15} - 68 q^{16} - 20 q^{17} + 56 q^{21} - 80 q^{23} + 62 q^{24} - 84 q^{25} - 112 q^{31} + 208 q^{35} - 122 q^{36} + 170 q^{38} + 206 q^{40} - 172 q^{41} + 10 q^{43} - 36 q^{44} + 30 q^{47} - 6 q^{49} - 120 q^{52} - 110 q^{53} - 284 q^{54} - 264 q^{56} + 420 q^{57} + 430 q^{58} - 12 q^{59} - 232 q^{60} + 100 q^{64} - 144 q^{66} - 70 q^{67} - 50 q^{68} - 50 q^{74} + 620 q^{78} + 178 q^{79} + 382 q^{81} + 10 q^{83} + 172 q^{84} - 372 q^{86} - 510 q^{87} - 796 q^{90} + 150 q^{92} - 130 q^{95} + 362 q^{96} - 380 q^{97} - 466 q^{99}+O(q^{100}) \) 6 * q - 16 * q^4 + 6 * q^6 - 36 * q^9 - 2 * q^10 + 38 * q^11 + 30 * q^13 + 36 * q^14 + 28 * q^15 - 68 * q^16 - 20 * q^17 + 56 * q^21 - 80 * q^23 + 62 * q^24 - 84 * q^25 - 112 * q^31 + 208 * q^35 - 122 * q^36 + 170 * q^38 + 206 * q^40 - 172 * q^41 + 10 * q^43 - 36 * q^44 + 30 * q^47 - 6 * q^49 - 120 * q^52 - 110 * q^53 - 284 * q^54 - 264 * q^56 + 420 * q^57 + 430 * q^58 - 12 * q^59 - 232 * q^60 + 100 * q^64 - 144 * q^66 - 70 * q^67 - 50 * q^68 - 50 * q^74 + 620 * q^78 + 178 * q^79 + 382 * q^81 + 10 * q^83 + 172 * q^84 - 372 * q^86 - 510 * q^87 - 796 * q^90 + 150 * q^92 - 130 * q^95 + 362 * q^96 - 380 * q^97 - 466 * q^99

Introduction

L-functions

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