LMFDB - Embedded newform 43.3.b.b.42.6

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			42.6
Root			$3.18991i$ of defining polynomial
Character	$\chi$	$=$	43.42
Dual form			43.3.b.b.42.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.18991i q^{2} -0.724539i q^{3} -6.17554 q^{4} +7.66434i q^{5} +2.31122 q^{6} -10.1297i q^{7} -6.93980i q^{8} +8.47504 q^{9} +O(q^{10})$	\(q+3.18991i q^{2} -0.724539i q^{3} -6.17554 q^{4} +7.66434i q^{5} +2.31122 q^{6} -10.1297i q^{7} -6.93980i q^{8} +8.47504 q^{9} -24.4486 q^{10} -2.43517 q^{11} +4.47442i q^{12} +18.1491 q^{13} +32.3129 q^{14} +5.55311 q^{15} -2.56483 q^{16} -1.13567 q^{17} +27.0346i q^{18} -12.3269i q^{19} -47.3314i q^{20} -7.33937 q^{21} -7.76798i q^{22} -24.2847 q^{23} -5.02815 q^{24} -33.7420 q^{25} +57.8939i q^{26} -12.6614i q^{27} +62.5565i q^{28} -35.5218i q^{29} +17.7139i q^{30} -12.9823 q^{31} -35.9408i q^{32} +1.76438i q^{33} -3.62270i q^{34} +77.6375 q^{35} -52.3380 q^{36} +41.5116i q^{37} +39.3216 q^{38} -13.1497i q^{39} +53.1889 q^{40} -65.0667 q^{41} -23.4120i q^{42} +(23.6064 + 35.9408i) q^{43} +15.0385 q^{44} +64.9556i q^{45} -77.4662i q^{46} +51.0402 q^{47} +1.85832i q^{48} -53.6110 q^{49} -107.634i q^{50} +0.822839i q^{51} -112.080 q^{52} +56.2026 q^{53} +40.3886 q^{54} -18.6640i q^{55} -70.2981 q^{56} -8.93130 q^{57} +113.312 q^{58} -88.8355 q^{59} -34.2935 q^{60} -65.1393i q^{61} -41.4123i q^{62} -85.8497i q^{63} +104.389 q^{64} +139.101i q^{65} -5.62821 q^{66} +45.3618 q^{67} +7.01339 q^{68} +17.5952i q^{69} +247.657i q^{70} +63.7983i q^{71} -58.8151i q^{72} +8.80247i q^{73} -132.418 q^{74} +24.4474i q^{75} +76.1251i q^{76} +24.6676i q^{77} +41.9464 q^{78} +31.8496 q^{79} -19.6577i q^{80} +67.1017 q^{81} -207.557i q^{82} -68.4740 q^{83} +45.3246 q^{84} -8.70417i q^{85} +(-114.648 + 75.3023i) q^{86} -25.7370 q^{87} +16.8996i q^{88} -5.90432i q^{89} -207.203 q^{90} -183.845i q^{91} +149.971 q^{92} +9.40616i q^{93} +162.814i q^{94} +94.4773 q^{95} -26.0405 q^{96} -100.657 q^{97} -171.015i q^{98} -20.6382 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			3.18991i			1.59496i	0.603348	+	0.797478i	$0.293833\pi$
$2$			3.18991i			1.59496i	−0.603348	+	0.797478i	$0.706167\pi$
$3$		−	0.724539i		−	0.241513i	−0.992682	−	0.120757i	$-0.961468\pi$
$3$		−	0.724539i		−	0.241513i	0.992682	−	0.120757i	$-0.0385320\pi$
$4$	−6.17554			−1.54389
$5$			7.66434i			1.53287i	0.642324	+	0.766434i	$0.277971\pi$
$5$			7.66434i			1.53287i	−0.642324	+	0.766434i	$0.722029\pi$
$6$	2.31122			0.385203
$7$		−	10.1297i		−	1.44710i	−0.690271	−	0.723551i	$-0.742509\pi$
$7$		−	10.1297i		−	1.44710i	0.690271	−	0.723551i	$-0.257491\pi$
$8$		−	6.93980i		−	0.867475i
$9$	8.47504			0.941671
$10$	−24.4486			−2.44486
$11$	−2.43517			−0.221379			−0.110690	−	0.993855i	$-0.535306\pi$
$11$	−2.43517			−0.221379			−0.110690	+	0.993855i	$0.535306\pi$
$12$			4.47442i			0.372869i
$13$	18.1491			1.39608			0.698041	−	0.716058i	$-0.254055\pi$
$13$	18.1491			1.39608			0.698041	+	0.716058i	$0.254055\pi$
$14$	32.3129			2.30806
$15$	5.55311			0.370207
$16$	−2.56483			−0.160302
$17$	−1.13567			−0.0668042			−0.0334021	−	0.999442i	$-0.510634\pi$
$17$	−1.13567			−0.0668042			−0.0334021	+	0.999442i	$0.510634\pi$
$18$			27.0346i			1.50192i
$19$		−	12.3269i		−	0.648783i	−0.945923	−	0.324391i	$-0.894841\pi$
$19$		−	12.3269i		−	0.648783i	0.945923	−	0.324391i	$-0.105159\pi$
$20$		−	47.3314i		−	2.36657i
$21$	−7.33937			−0.349494
$22$		−	7.76798i		−	0.353090i
$23$	−24.2847			−1.05586			−0.527929	−	0.849288i	$-0.677031\pi$
$23$	−24.2847			−1.05586			−0.527929	+	0.849288i	$0.677031\pi$
$24$	−5.02815			−0.209506
$25$	−33.7420			−1.34968
$26$			57.8939i			2.22669i
$27$		−	12.6614i		−	0.468939i
$28$			62.5565i			2.23416i
$29$		−	35.5218i		−	1.22489i	−0.790513	−	0.612445i	$-0.790186\pi$
$29$		−	35.5218i		−	1.22489i	0.790513	−	0.612445i	$-0.209814\pi$
$30$			17.7139i			0.590465i
$31$	−12.9823			−0.418783			−0.209391	−	0.977832i	$-0.567148\pi$
$31$	−12.9823			−0.418783			−0.209391	+	0.977832i	$0.567148\pi$
$32$		−	35.9408i		−	1.12315i
$33$			1.76438i			0.0534660i
$34$		−	3.62270i		−	0.106550i
$35$	77.6375			2.21821
$36$	−52.3380			−1.45383
$37$			41.5116i			1.12193i	0.827838	+	0.560967i	$0.189571\pi$
$37$			41.5116i			1.12193i	−0.827838	+	0.560967i	$0.810429\pi$
$38$	39.3216			1.03478
$39$		−	13.1497i		−	0.337172i
$40$	53.1889			1.32972
$41$	−65.0667			−1.58699			−0.793496	−	0.608576i	$-0.791741\pi$
$41$	−65.0667			−1.58699			−0.793496	+	0.608576i	$0.791741\pi$
$42$		−	23.4120i		−	0.557427i
$43$	23.6064	+	35.9408i	0.548986	+	0.835832i
$43$	23.6064	+	35.9408i	0.548986	+	0.835832i
$44$	15.0385			0.341784
$45$			64.9556i			1.44346i
$46$		−	77.4662i		−	1.68405i
$47$	51.0402			1.08596			0.542981	−	0.839745i	$-0.317296\pi$
$47$	51.0402			1.08596			0.542981	+	0.839745i	$0.317296\pi$
$48$			1.85832i			0.0387150i
$49$	−53.6110			−1.09410
$50$		−	107.634i		−	2.15268i
$51$			0.822839i			0.0161341i
$52$	−112.080			−2.15539
$53$	56.2026			1.06043			0.530214	−	0.847864i	$-0.322112\pi$
$53$	56.2026			1.06043			0.530214	+	0.847864i	$0.322112\pi$
$54$	40.3886			0.747937
$55$		−	18.6640i		−	0.339345i
$56$	−70.2981			−1.25532
$57$	−8.93130			−0.156689
$58$	113.312			1.95365
$59$	−88.8355			−1.50569			−0.752843	−	0.658200i	$-0.771318\pi$
$59$	−88.8355			−1.50569			−0.752843	+	0.658200i	$0.771318\pi$
$60$	−34.2935			−0.571558
$61$		−	65.1393i		−	1.06786i	−0.845529	−	0.533929i	$-0.820715\pi$
$61$		−	65.1393i		−	1.06786i	0.845529	−	0.533929i	$-0.179285\pi$
$62$		−	41.4123i		−	0.667940i
$63$		−	85.8497i		−	1.36269i
$64$	104.389			1.63107
$65$			139.101i			2.14001i
$66$	−5.62821			−0.0852759
$67$	45.3618			0.677042			0.338521	−	0.940959i	$-0.390073\pi$
$67$	45.3618			0.677042			0.338521	+	0.940959i	$0.390073\pi$
$68$	7.01339			0.103138
$69$			17.5952i			0.255004i
$70$			247.657i			3.53796i
$71$			63.7983i			0.898567i	0.893389	+	0.449284i	$0.148321\pi$
$71$			63.7983i			0.898567i	−0.893389	+	0.449284i	$0.851679\pi$
$72$		−	58.8151i		−	0.816876i
$73$			8.80247i			0.120582i	0.998181	+	0.0602909i	$0.0192028\pi$
$73$			8.80247i			0.120582i	−0.998181	+	0.0602909i	$0.980797\pi$
$74$	−132.418			−1.78944
$75$			24.4474i			0.325966i
$76$			76.1251i			1.00165i
$77$			24.6676i			0.320358i
$78$	41.9464			0.537775
$79$	31.8496			0.403159			0.201580	−	0.979472i	$-0.435393\pi$
$79$	31.8496			0.403159			0.201580	+	0.979472i	$0.435393\pi$
$80$		−	19.6577i		−	0.245721i
$81$	67.1017			0.828417
$82$		−	207.557i		−	2.53118i
$83$	−68.4740			−0.824988			−0.412494	−	0.910960i	$-0.635342\pi$
$83$	−68.4740			−0.824988			−0.412494	+	0.910960i	$0.635342\pi$
$84$	45.3246			0.539579
$85$		−	8.70417i		−	0.102402i
$86$	−114.648	+	75.3023i	−1.33312	+	0.875608i
$87$	−25.7370			−0.295827
$88$			16.8996i			0.192041i
$89$		−	5.90432i		−	0.0663406i	−0.999450	−	0.0331703i	$-0.989440\pi$
$89$		−	5.90432i		−	0.0663406i	0.999450	−	0.0331703i	$-0.0105604\pi$
$90$	−207.203			−2.30225
$91$		−	183.845i		−	2.02027i
$92$	149.971			1.63012
$93$			9.40616i			0.101142i
$94$			162.814i			1.73206i
$95$	94.4773			0.994498
$96$	−26.0405			−0.271255
$97$	−100.657			−1.03770			−0.518849	−	0.854866i	$-0.673639\pi$
$97$	−100.657			−1.03770			−0.518849	+	0.854866i	$0.673639\pi$
$98$		−	171.015i		−	1.74505i
$99$	−20.6382			−0.208466
$100$	208.376			2.08376
$101$	13.7062			0.135705			0.0678525	−	0.997695i	$-0.478385\pi$
$101$	13.7062			0.135705			0.0678525	+	0.997695i	$0.478385\pi$
$102$	−2.62478			−0.0257332
$103$	−96.0402			−0.932429			−0.466214	−	0.884672i	$-0.654382\pi$
$103$	−96.0402			−0.932429			−0.466214	+	0.884672i	$0.654382\pi$
$104$		−	125.951i		−	1.21107i
$105$		−	56.2514i		−	0.535728i
$106$			179.282i			1.69134i
$107$	−2.08036			−0.0194427			−0.00972133	−	0.999953i	$-0.503094\pi$
$107$	−2.08036			−0.0194427			−0.00972133	+	0.999953i	$0.503094\pi$
$108$			78.1907i			0.723988i
$109$	−53.1078			−0.487228			−0.243614	−	0.969872i	$-0.578333\pi$
$109$	−53.1078			−0.487228			−0.243614	+	0.969872i	$0.578333\pi$
$110$	59.5364			0.541240
$111$	30.0768			0.270962
$112$			25.9810i			0.231973i
$113$		−	28.8791i		−	0.255568i	−0.991802	−	0.127784i	$-0.959214\pi$
$113$		−	28.8791i		−	0.255568i	0.991802	−	0.127784i	$-0.0407864\pi$
$114$		−	28.4901i		−	0.249913i
$115$		−	186.126i		−	1.61849i
$116$			219.367i			1.89109i
$117$	153.814			1.31465
$118$		−	283.378i		−	2.40150i
$119$			11.5040i			0.0966725i
$120$		−	38.5375i		−	0.321146i
$121$	−115.070			−0.950991
$122$	207.789			1.70319
$123$			47.1433i			0.383279i
$124$	80.1726			0.646553
$125$		−	67.0020i		−	0.536016i
$126$	273.853			2.17344
$127$	148.673			1.17066			0.585328	−	0.810797i	$-0.300966\pi$
$127$	148.673			1.17066			0.585328	+	0.810797i	$0.300966\pi$
$128$			189.227i			1.47834i
$129$	26.0405	−	17.1037i	0.201864	−	0.132587i
$130$	−443.719			−3.41322
$131$			157.214i			1.20011i	0.799960	+	0.600054i	$0.204854\pi$
$131$			157.214i			1.20011i	−0.799960	+	0.600054i	$0.795146\pi$
$132$		−	10.8960i		−	0.0825453i
$133$	−124.868			−0.938854
$134$			144.700i			1.07985i
$135$	97.0409			0.718821
$136$			7.88133i			0.0579510i
$137$			121.604i			0.887618i	0.896122	+	0.443809i	$0.146373\pi$
$137$			121.604i			0.887618i	−0.896122	+	0.443809i	$0.853627\pi$
$138$	−56.1273			−0.406719
$139$	−27.8992			−0.200714			−0.100357	−	0.994952i	$-0.531998\pi$
$139$	−27.8992			−0.200714			−0.100357	+	0.994952i	$0.531998\pi$
$140$	−479.454			−3.42467
$141$		−	36.9806i		−	0.262274i
$142$	−203.511			−1.43318
$143$	−44.1961			−0.309063
$144$	−21.7370			−0.150952
$145$	272.251			1.87759
$146$	−28.0791			−0.192323
$147$			38.8433i			0.264240i
$148$		−	256.357i		−	1.73214i
$149$		−	212.121i		−	1.42363i	−0.702366	−	0.711816i	$-0.747873\pi$
$149$		−	212.121i		−	1.42363i	0.702366	−	0.711816i	$-0.252127\pi$
$150$	−77.9852			−0.519901
$151$			108.847i			0.720842i	0.932790	+	0.360421i	$0.117367\pi$
$151$			108.847i			0.720842i	−0.932790	+	0.360421i	$0.882633\pi$
$152$	−85.5460			−0.562802
$153$	−9.62487			−0.0629077
$154$	−78.6874			−0.510957
$155$		−	99.5005i		−	0.641938i
$156$			81.2066i			0.520555i
$157$		−	106.772i		−	0.680075i	−0.940412	−	0.340038i	$-0.889560\pi$
$157$		−	106.772i		−	0.680075i	0.940412	−	0.340038i	$-0.110440\pi$
$158$			101.597i			0.643021i
$159$		−	40.7210i		−	0.256107i
$160$	275.462			1.72164
$161$			245.997i			1.52793i
$162$			214.049i			1.32129i
$163$			202.231i			1.24068i	0.784334	+	0.620339i	$0.213005\pi$
$163$			202.231i			1.24068i	−0.784334	+	0.620339i	$0.786995\pi$
$164$	401.822			2.45013
$165$	−13.5228			−0.0819562
$166$		−	218.426i		−	1.31582i
$167$	−41.0703			−0.245930			−0.122965	−	0.992411i	$-0.539240\pi$
$167$	−41.0703			−0.245930			−0.122965	+	0.992411i	$0.539240\pi$
$168$			50.9337i			0.303177i
$169$	160.389			0.949045
$170$	27.7656			0.163327
$171$		−	104.471i		−	0.610940i
$172$	−145.782	−	221.954i	−0.847571	−	1.29043i
$173$	−24.1006			−0.139310			−0.0696549	−	0.997571i	$-0.522190\pi$
$173$	−24.1006			−0.139310			−0.0696549	+	0.997571i	$0.522190\pi$
$174$		−	82.0986i		−	0.471831i
$175$			341.797i			1.95313i
$176$	6.24580			0.0354875
$177$			64.3648i			0.363643i
$178$	18.8343			0.105810
$179$		−	83.0659i		−	0.464055i	−0.972709	−	0.232028i	$-0.925464\pi$
$179$		−	83.0659i		−	0.464055i	0.972709	−	0.232028i	$-0.0745360\pi$
$180$		−	401.136i		−	2.22853i
$181$	−112.273			−0.620291			−0.310145	−	0.950689i	$-0.600378\pi$
$181$	−112.273			−0.620291			−0.310145	+	0.950689i	$0.600378\pi$
$182$	586.449			3.22225
$183$	−47.1960			−0.257902
$184$			168.531i			0.915930i
$185$	−318.159			−1.71978
$186$	−30.0048			−0.161316
$187$	2.76556			0.0147891
$188$	−315.201			−1.67660
$189$	−128.256			−0.678602
$190$			301.374i			1.58618i
$191$			43.7078i			0.228836i	0.993433	+	0.114418i	$0.0365004\pi$
$191$			43.7078i			0.228836i	−0.993433	+	0.114418i	$0.963500\pi$
$192$		−	75.6336i		−	0.393925i
$193$	236.100			1.22332			0.611659	−	0.791122i	$-0.290503\pi$
$193$	236.100			1.22332			0.611659	+	0.791122i	$0.290503\pi$
$194$		−	321.086i		−	1.65508i
$195$	100.784			0.516840
$196$	331.077			1.68917
$197$	350.134			1.77733			0.888665	−	0.458557i	$-0.151634\pi$
$197$	350.134			1.77733			0.888665	+	0.458557i	$0.151634\pi$
$198$		−	65.8340i		−	0.332495i
$199$		−	353.573i		−	1.77675i	−0.459122	−	0.888373i	$-0.651836\pi$
$199$		−	353.573i		−	1.77675i	0.459122	−	0.888373i	$-0.348164\pi$
$200$			234.163i			1.17081i
$201$		−	32.8664i		−	0.163514i
$202$			43.7216i			0.216444i
$203$	−359.826			−1.77254
$204$		−	5.08148i		−	0.0249092i
$205$		−	498.693i		−	2.43265i
$206$		−	306.360i		−	1.48718i
$207$	−205.814			−0.994272
$208$	−46.5493			−0.223794
$209$			30.0180i			0.143627i
$210$	179.437			0.854462
$211$			281.519i			1.33421i	0.744962	+	0.667107i	$0.232468\pi$
$211$			281.519i			1.33421i	−0.744962	+	0.667107i	$0.767532\pi$
$212$	−347.082			−1.63718
$213$	46.2243			0.217016
$214$		−	6.63618i		−	0.0310102i
$215$	−275.462	+	180.927i	−1.28122	+	0.841522i
$216$	−87.8672			−0.406793
$217$			131.507i			0.606021i
$218$		−	169.409i		−	0.777107i
$219$	6.37774			0.0291221
$220$			115.260i			0.523910i
$221$	−20.6114			−0.0932642
$222$			95.9423i			0.432172i
$223$			95.1942i			0.426880i	0.976956	+	0.213440i	$0.0684668\pi$
$223$			95.1942i			0.426880i	−0.976956	+	0.213440i	$0.931533\pi$
$224$	−364.070			−1.62531
$225$	−285.965			−1.27096
$226$	92.1219			0.407619
$227$			106.481i			0.469079i	0.972107	+	0.234540i	$0.0753583\pi$
$227$			106.481i			0.469079i	−0.972107	+	0.234540i	$0.924642\pi$
$228$	55.1556			0.241911
$229$	170.352			0.743896			0.371948	−	0.928254i	$-0.378690\pi$
$229$	170.352			0.743896			0.371948	+	0.928254i	$0.378690\pi$
$230$	593.727			2.58142
$231$	17.8726			0.0773707
$232$	−246.514			−1.06256
$233$			55.4970i			0.238184i	0.992883	+	0.119092i	$0.0379984\pi$
$233$			55.4970i			0.238184i	−0.992883	+	0.119092i	$0.962002\pi$
$234$			490.654i			2.09681i
$235$			391.189i			1.66463i
$236$	548.608			2.32461
$237$		−	23.0763i		−	0.0973682i
$238$	−36.6969			−0.154188
$239$	−20.7695			−0.0869017			−0.0434509	−	0.999056i	$-0.513835\pi$
$239$	−20.7695			−0.0869017			−0.0434509	+	0.999056i	$0.513835\pi$
$240$	−14.2428			−0.0593449
$241$			303.282i			1.25843i	0.777231	+	0.629216i	$0.216624\pi$
$241$			303.282i			1.25843i	−0.777231	+	0.629216i	$0.783376\pi$
$242$		−	367.063i		−	1.51679i
$243$		−	162.570i		−	0.669012i
$244$			402.271i			1.64865i
$245$		−	410.893i		−	1.67711i
$246$	−150.383			−0.611314
$247$		−	223.721i		−	0.905754i
$248$			90.0943i			0.363283i
$249$			49.6121i			0.199245i
$250$	213.731			0.854922
$251$	−7.25467			−0.0289031			−0.0144515	−	0.999896i	$-0.504600\pi$
$251$	−7.25467			−0.0289031			−0.0144515	+	0.999896i	$0.504600\pi$
$252$			530.169i			2.10384i
$253$	59.1375			0.233745
$254$			474.255i			1.86715i
$255$	−6.30651			−0.0247314
$256$	−186.065			−0.726816
$257$		−	316.956i		−	1.23329i	−0.787241	−	0.616645i	$-0.788491\pi$
$257$		−	316.956i		−	1.23329i	0.787241	−	0.616645i	$-0.211509\pi$
$258$	54.5595	+	83.0669i	0.211471	+	0.321965i
$259$	420.500			1.62355
$260$		−	859.022i		−	3.30393i
$261$		−	301.049i		−	1.15344i
$262$	−501.499			−1.91412
$263$		−	229.845i		−	0.873934i	−0.899478	−	0.436967i	$-0.856053\pi$
$263$		−	229.845i		−	0.873934i	0.899478	−	0.436967i	$-0.143947\pi$
$264$	12.2444			0.0463804
$265$			430.756i			1.62549i
$266$		−	398.317i		−	1.49743i
$267$	−4.27791			−0.0160221
$268$	−280.134			−1.04528
$269$	285.119			1.05992			0.529961	−	0.848022i	$-0.322206\pi$
$269$	285.119			1.05992			0.529961	+	0.848022i	$0.322206\pi$
$270$			309.552i			1.14649i
$271$	78.3270			0.289030			0.144515	−	0.989503i	$-0.453838\pi$
$271$	78.3270			0.289030			0.144515	+	0.989503i	$0.453838\pi$
$272$	2.91281			0.0107088
$273$	−133.203			−0.487922
$274$	−387.905			−1.41571
$275$	82.1677			0.298791
$276$		−	108.660i		−	0.393696i
$277$			119.631i			0.431880i	0.976407	+	0.215940i	$0.0692815\pi$
$277$			119.631i			0.431880i	−0.976407	+	0.215940i	$0.930718\pi$
$278$		−	88.9960i		−	0.320129i
$279$	−110.025			−0.394356
$280$		−	538.789i		−	1.92424i
$281$	−150.423			−0.535314			−0.267657	−	0.963514i	$-0.586249\pi$
$281$	−150.423			−0.535314			−0.267657	+	0.963514i	$0.586249\pi$
$282$	117.965			0.418315
$283$	−261.144			−0.922771			−0.461385	−	0.887200i	$-0.652647\pi$
$283$	−261.144			−0.922771			−0.461385	+	0.887200i	$0.652647\pi$
$284$		−	393.989i		−	1.38729i
$285$		−	68.4525i		−	0.240184i
$286$		−	140.982i		−	0.492943i
$287$			659.106i			2.29654i
$288$		−	304.600i		−	1.05764i
$289$	−287.710			−0.995537
$290$			868.458i			2.99468i
$291$			72.9297i			0.250618i
$292$		−	54.3601i		−	0.186165i
$293$	358.409			1.22324			0.611619	−	0.791152i	$-0.290518\pi$
$293$	358.409			1.22324			0.611619	+	0.791152i	$0.290518\pi$
$294$	−123.907			−0.421451
$295$		−	680.865i		−	2.30802i
$296$	288.082			0.973250
$297$			30.8326i			0.103813i
$298$	676.648			2.27063
$299$	−440.745			−1.47406
$300$		−	150.976i		−	0.503254i
$301$	364.070	−	239.126i	1.20953	−	0.794438i
$302$	−347.213			−1.14971
$303$		−	9.93068i		−	0.0327745i
$304$			31.6163i			0.104001i
$305$	499.250			1.63688
$306$		−	30.7025i		−	0.100335i
$307$	509.730			1.66036			0.830180	−	0.557496i	$-0.188238\pi$
$307$	509.730			1.66036			0.830180	+	0.557496i	$0.188238\pi$
$308$		−	152.336i		−	0.494596i
$309$			69.5849i			0.225194i
$310$	317.398			1.02386
$311$	−27.7820			−0.0893312			−0.0446656	−	0.999002i	$-0.514222\pi$
$311$	−27.7820			−0.0893312			−0.0446656	+	0.999002i	$0.514222\pi$
$312$	−91.2563			−0.292488
$313$			38.5686i			0.123222i	0.998100	+	0.0616111i	$0.0196239\pi$
$313$			38.5686i			0.123222i	−0.998100	+	0.0616111i	$0.980376\pi$
$314$	340.593			1.08469
$315$	657.981			2.08883
$316$	−196.688			−0.622432
$317$	85.8945			0.270961			0.135480	−	0.990780i	$-0.456742\pi$
$317$	85.8945			0.270961			0.135480	+	0.990780i	$0.456742\pi$
$318$	129.896			0.408480
$319$			86.5017i			0.271165i
$320$			800.069i			2.50022i
$321$			1.50731i			0.00469566i
$322$	−784.710			−2.43699
$323$			13.9993i			0.0433414i
$324$	−414.390			−1.27898
$325$	−612.387			−1.88427
$326$	−645.098			−1.97883
$327$			38.4787i			0.117672i
$328$			451.549i			1.37668i
$329$		−	517.022i		−	1.57150i
$330$		−	43.1365i		−	0.130717i
$331$		−	363.666i		−	1.09869i	−0.835596	−	0.549345i	$-0.814877\pi$
$331$		−	363.666i		−	1.09869i	0.835596	−	0.549345i	$-0.185123\pi$
$332$	422.864			1.27369
$333$			351.813i			1.05649i
$334$		−	131.011i		−	0.392247i
$335$			347.668i			1.03782i
$336$	18.8242			0.0560245
$337$	8.64293			0.0256467			0.0128233	−	0.999918i	$-0.495918\pi$
$337$	8.64293			0.0256467			0.0128233	+	0.999918i	$0.495918\pi$
$338$			511.626i			1.51369i
$339$	−20.9241			−0.0617229
$340$			53.7530i			0.158097i
$341$	31.6140			0.0927098
$342$	333.253			0.974423
$343$			46.7084i			0.136176i
$344$	249.422	−	163.823i	0.725063	−	0.476231i
$345$	−134.856			−0.390887
$346$		−	76.8788i		−	0.222193i
$347$			278.736i			0.803273i	0.915799	+	0.401636i	$0.131558\pi$
$347$			278.736i			0.803273i	−0.915799	+	0.401636i	$0.868442\pi$
$348$	158.940			0.456723
$349$			169.755i			0.486403i	0.969976	+	0.243201i	$0.0781976\pi$
$349$			169.755i			0.486403i	−0.969976	+	0.243201i	$0.921802\pi$
$350$	−1090.30			−3.11515
$351$		−	229.792i		−	0.654677i
$352$			87.5219i			0.248642i
$353$	26.8357			0.0760219			0.0380109	−	0.999277i	$-0.487898\pi$
$353$	26.8357			0.0760219			0.0380109	+	0.999277i	$0.487898\pi$
$354$	−205.318			−0.579995
$355$	−488.971			−1.37738
$356$			36.4624i			0.102422i
$357$	8.33512			0.0233477
$358$	264.973			0.740148
$359$	169.169			0.471223			0.235611	−	0.971847i	$-0.424291\pi$
$359$	169.169			0.471223			0.235611	+	0.971847i	$0.424291\pi$
$360$	450.779			1.25216
$361$	209.048			0.579081
$362$		−	358.140i		−	0.989336i
$363$			83.3727i			0.229677i
$364$			1135.34i			3.11907i
$365$	−67.4651			−0.184836
$366$		−	150.551i		−	0.411342i
$367$	−540.524			−1.47282			−0.736409	−	0.676536i	$-0.763480\pi$
$367$	−540.524			−1.47282			−0.736409	+	0.676536i	$0.763480\pi$
$368$	62.2862			0.169256
$369$	−551.443			−1.49442
$370$		−	1014.90i		−	2.74297i
$371$		−	569.316i		−	1.53455i
$372$		−	58.0882i		−	0.156151i
$373$		−	26.7152i		−	0.0716226i	−0.999359	−	0.0358113i	$-0.988598\pi$
$373$		−	26.7152i		−	0.0716226i	0.999359	−	0.0358113i	$-0.0114015\pi$
$374$			8.82188i			0.0235879i
$375$	−48.5456			−0.129455
$376$		−	354.209i		−	0.942044i
$377$		−	644.688i		−	1.71005i
$378$		−	409.125i		−	1.08234i
$379$	315.193			0.831644			0.415822	−	0.909446i	$-0.363494\pi$
$379$	315.193			0.831644			0.415822	+	0.909446i	$0.363494\pi$
$380$	−583.449			−1.53539
$381$		−	107.720i		−	0.282729i
$382$	−139.424			−0.364984
$383$		−	131.099i		−	0.342295i	−0.985245	−	0.171148i	$-0.945252\pi$
$383$		−	131.099i		−	0.342295i	0.985245	−	0.171148i	$-0.0547475\pi$
$384$	137.103			0.357038
$385$	−189.061			−0.491066
$386$			753.139i			1.95114i
$387$	200.065	+	304.600i	0.516964	+	0.787079i
$388$	621.610			1.60209
$389$		−	110.050i		−	0.282904i	−0.989945	−	0.141452i	$-0.954823\pi$
$389$		−	110.050i		−	0.282904i	0.989945	−	0.141452i	$-0.0451771\pi$
$390$			321.491i			0.824337i
$391$	27.5795			0.0705358
$392$			372.050i			0.949106i
$393$	113.908			0.289841
$394$			1116.90i			2.83476i
$395$			244.106i			0.617989i
$396$	127.452			0.321848
$397$	68.3215			0.172094			0.0860472	−	0.996291i	$-0.472576\pi$
$397$	68.3215			0.172094			0.0860472	+	0.996291i	$0.472576\pi$
$398$	1127.87			2.83383
$399$			90.4715i			0.226746i
$400$	86.5426			0.216356
$401$	−675.508			−1.68456			−0.842279	−	0.539042i	$-0.818787\pi$
$401$	−675.508			−1.68456			−0.842279	+	0.539042i	$0.818787\pi$
$402$	104.841			0.260798
$403$	−235.616			−0.584655
$404$	−84.6433			−0.209513
$405$			514.290i			1.26985i
$406$		−	1147.81i		−	2.82713i
$407$		−	101.088i		−	0.248373i
$408$	5.71034			0.0139959
$409$		−	1.76834i		−	0.00432356i	−0.999998	−	0.00216178i	$-0.999312\pi$
$409$		−	1.76834i		−	0.00432356i	0.999998	−	0.00216178i	$-0.000688116\pi$
$410$	1590.79			3.87997
$411$	88.1066			0.214371
$412$	593.100			1.43956
$413$			899.878i			2.17888i
$414$		−	656.529i		−	1.58582i
$415$		−	524.808i		−	1.26460i
$416$		−	652.291i		−	1.56801i
$417$			20.2141i			0.0484749i
$418$	−95.7549			−0.229079
$419$		−	444.028i		−	1.05973i	−0.848081	−	0.529867i	$-0.822242\pi$
$419$		−	444.028i		−	1.05973i	0.848081	−	0.529867i	$-0.177758\pi$
$420$			347.383i			0.827103i
$421$			141.195i			0.335381i	0.985840	+	0.167690i	$0.0536309\pi$
$421$			141.195i			0.335381i	−0.985840	+	0.167690i	$0.946369\pi$
$422$	−898.022			−2.12801
$423$	432.568			1.02262
$424$		−	390.035i		−	0.919894i
$425$	38.3199			0.0901645
$426$			147.452i			0.346131i
$427$	−659.843			−1.54530
$428$	12.8474			0.0300173
$429$			32.0218i			0.0746429i
$430$	−577.142	−	878.700i	−1.34219	−	2.04349i
$431$	302.066			0.700849			0.350424	−	0.936591i	$-0.386037\pi$
$431$	302.066			0.700849			0.350424	+	0.936591i	$0.386037\pi$
$432$			32.4742i			0.0751718i
$433$			380.497i			0.878747i	0.898304	+	0.439374i	$0.144800\pi$
$433$			380.497i			0.878747i	−0.898304	+	0.439374i	$0.855200\pi$
$434$	−419.495			−0.966578
$435$		−	197.257i		−	0.453464i
$436$	327.970			0.752224
$437$			299.355i			0.685022i
$438$			20.3444i			0.0464485i
$439$	558.769			1.27282			0.636412	−	0.771350i	$-0.280418\pi$
$439$	558.769			1.27282			0.636412	+	0.771350i	$0.280418\pi$
$440$	−129.524			−0.294373
$441$	−454.356			−1.03029
$442$		−	65.7485i		−	0.148752i
$443$	−346.462			−0.782082			−0.391041	−	0.920373i	$-0.627885\pi$
$443$	−346.462			−0.782082			−0.391041	+	0.920373i	$0.627885\pi$
$444$	−185.740			−0.418334
$445$	45.2527			0.101691
$446$	−303.661			−0.680855
$447$	−153.690			−0.343826
$448$		−	1057.43i		−	2.36033i
$449$			250.699i			0.558350i	0.960240	+	0.279175i	$0.0900609\pi$
$449$			250.699i			0.558350i	−0.960240	+	0.279175i	$0.909939\pi$
$450$		−	912.204i		−	2.02712i
$451$	158.448			0.351327
$452$			178.344i			0.394567i
$453$	78.8640			0.174093
$454$	−339.665			−0.748161
$455$	1409.05			3.09681
$456$			61.9814i			0.135924i
$457$			745.356i			1.63098i	0.578773	+	0.815488i	$0.303532\pi$
$457$			745.356i			1.63098i	−0.578773	+	0.815488i	$0.696468\pi$
$458$			543.408i			1.18648i
$459$			14.3791i			0.0313271i
$460$			1149.43i			2.49876i
$461$	6.73204			0.0146031			0.00730156	−	0.999973i	$-0.497676\pi$
$461$	6.73204			0.0146031			0.00730156	+	0.999973i	$0.497676\pi$
$462$			57.0121i			0.123403i
$463$			448.198i			0.968031i	0.875059	+	0.484016i	$0.160822\pi$
$463$			448.198i			0.968031i	−0.875059	+	0.484016i	$0.839178\pi$
$464$			91.1074i			0.196352i
$465$	−72.0920			−0.155037
$466$	−177.031			−0.379894
$467$		−	506.048i		−	1.08361i	−0.840503	−	0.541807i	$-0.817740\pi$
$467$		−	506.048i		−	1.08361i	0.840503	−	0.541807i	$-0.182260\pi$
$468$	−949.886			−2.02967
$469$		−	459.502i		−	0.979749i
$470$	−1247.86			−2.65502
$471$	−77.3603			−0.164247
$472$			616.500i			1.30614i
$473$	−57.4856	−	87.5219i	−0.121534	−	0.185036i
$474$	73.6112			0.155298
$475$			415.934i			0.875650i
$476$		−	71.0436i		−	0.149251i
$477$	476.320			0.998574
$478$		−	66.2529i		−	0.138604i
$479$	−772.714			−1.61318			−0.806591	−	0.591110i	$-0.798690\pi$
$479$	−772.714			−1.61318			−0.806591	+	0.591110i	$0.798690\pi$
$480$		−	199.583i		−	0.415798i
$481$			753.397i			1.56631i
$482$	−967.443			−2.00714
$483$	178.235			0.369016
$484$	710.620			1.46822
$485$		−	771.467i		−	1.59065i
$486$	518.584			1.06705
$487$	−762.454			−1.56561			−0.782807	−	0.622265i	$-0.786213\pi$
$487$	−762.454			−1.56561			−0.782807	+	0.622265i	$0.786213\pi$
$488$	−452.054			−0.926340
$489$	146.524			0.299640
$490$	1310.71			2.67492
$491$			928.398i			1.89083i	0.325868	+	0.945415i	$0.394343\pi$
$491$			928.398i			1.89083i	−0.325868	+	0.945415i	$0.605657\pi$
$492$		−	291.136i		−	0.591739i
$493$			40.3411i			0.0818279i
$494$	713.651			1.44464
$495$		−	158.178i		−	0.319551i
$496$	33.2973			0.0671317
$497$	646.258			1.30032
$498$	−158.258			−0.317788
$499$		−	669.978i		−	1.34264i	−0.741167	−	0.671320i	$-0.765728\pi$
$499$		−	669.978i		−	1.34264i	0.741167	−	0.671320i	$-0.234272\pi$
$500$			413.774i			0.827548i
$501$			29.7570i			0.0593952i
$502$		−	23.1418i		−	0.0460991i
$503$			274.168i			0.545065i	0.962147	+	0.272532i	$0.0878612\pi$
$503$			274.168i			0.545065i	−0.962147	+	0.272532i	$0.912139\pi$
$504$	−595.780			−1.18210
$505$			105.049i			0.208018i
$506$			188.643i			0.372813i
$507$		−	116.208i		−	0.229207i
$508$	−918.139			−1.80736
$509$	703.988			1.38308			0.691540	−	0.722338i	$-0.256933\pi$
$509$	703.988			1.38308			0.691540	+	0.722338i	$0.256933\pi$
$510$		−	20.1172i		−	0.0394456i
$511$	89.1665			0.174494
$512$			163.380i			0.319101i
$513$	−156.075			−0.304239
$514$	1011.06			1.96704
$515$		−	736.084i		−	1.42929i
$516$	−160.814	+	105.625i	−0.311655	+	0.204699i
$517$	−124.292			−0.240409
$518$			1341.36i			2.58950i
$519$			17.4618i			0.0336451i
$520$	965.330			1.85640
$521$			588.189i			1.12896i	0.825446	+	0.564480i	$0.190923\pi$
$521$			588.189i			1.12896i	−0.825446	+	0.564480i	$0.809077\pi$
$522$	960.320			1.83969
$523$		−	743.907i		−	1.42238i	−0.702998	−	0.711192i	$-0.748156\pi$
$523$		−	743.907i		−	1.42238i	0.702998	−	0.711192i	$-0.251844\pi$
$524$		−	970.882i		−	1.85283i
$525$	247.645			0.471706
$526$	733.184			1.39389
$527$	14.7436			0.0279765
$528$		−	4.52532i		−	0.00857069i
$529$	60.7485			0.114837
$530$	−1374.07			−2.59259
$531$	−752.885			−1.41786
$532$	771.125			1.44948
$533$	−1180.90			−2.21557
$534$		−	13.6462i		−	0.0255546i
$535$		−	15.9446i		−	0.0298030i
$536$		−	314.802i		−	0.587317i
$537$	−60.1845			−0.112075
$538$			909.506i			1.69053i
$539$	130.552			0.242212
$540$	−599.280			−1.10978
$541$	−2.77685			−0.00513282			−0.00256641	−	0.999997i	$-0.500817\pi$
$541$	−2.77685			−0.00513282			−0.00256641	+	0.999997i	$0.500817\pi$
$542$			249.856i			0.460990i
$543$			81.3459i			0.149808i
$544$			40.8169i			0.0750311i
$545$		−	407.036i		−	0.746856i
$546$		−	424.905i		−	0.778214i
$547$	700.748			1.28108			0.640538	−	0.767927i	$-0.278711\pi$
$547$	700.748			1.28108			0.640538	+	0.767927i	$0.278711\pi$
$548$		−	750.968i		−	1.37038i
$549$		−	552.059i		−	1.00557i
$550$			262.108i			0.476559i
$551$	−437.873			−0.794688
$552$	122.107			0.221209
$553$		−	322.627i		−	0.583412i
$554$	−381.612			−0.688830
$555$			230.518i			0.415349i
$556$	172.293			0.309879
$557$	−590.822			−1.06072			−0.530361	−	0.847772i	$-0.677944\pi$
$557$	−590.822			−1.06072			−0.530361	+	0.847772i	$0.677944\pi$
$558$		−	350.971i		−	0.628980i
$559$	428.434	+	652.291i	0.766429	+	1.16689i
$560$	−199.127			−0.355584
$561$		−	2.00375i		−	0.00357175i
$562$		−	479.837i		−	0.853803i
$563$	−306.648			−0.544668			−0.272334	−	0.962203i	$-0.587795\pi$
$563$	−306.648			−0.544668			−0.272334	+	0.962203i	$0.587795\pi$
$564$			228.375i			0.404921i
$565$	221.339			0.391751
$566$		−	833.027i		−	1.47178i
$567$		−	679.721i		−	1.19880i
$568$	442.747			0.779484
$569$	−5.67367			−0.00997130			−0.00498565	−	0.999988i	$-0.501587\pi$
$569$	−5.67367			−0.00997130			−0.00498565	+	0.999988i	$0.501587\pi$
$570$	218.357			0.383083
$571$		−	811.264i		−	1.42078i	−0.703810	−	0.710389i	$-0.748519\pi$
$571$		−	811.264i		−	1.42078i	0.703810	−	0.710389i	$-0.251481\pi$
$572$	272.935			0.477159
$573$	31.6680			0.0552670
$574$	−2102.49			−3.66288
$575$	819.417			1.42507
$576$	884.698			1.53593
$577$		−	976.820i		−	1.69293i	−0.532445	−	0.846464i	$-0.678727\pi$
$577$		−	976.820i		−	1.69293i	0.532445	−	0.846464i	$-0.321273\pi$
$578$		−	917.771i		−	1.58784i
$579$		−	171.064i		−	0.295447i
$580$	−1681.30			−2.89879
$581$			693.622i			1.19384i
$582$	−232.639			−0.399724
$583$	−136.863			−0.234757
$584$	61.0874			0.104602
$585$			1178.88i			2.01518i
$586$			1143.29i			1.95101i
$587$			681.557i			1.16109i	0.814230	+	0.580543i	$0.197160\pi$
$587$			681.557i			1.16109i	−0.814230	+	0.580543i	$0.802840\pi$
$588$		−	239.878i		−	0.407957i
$589$			160.031i			0.271699i
$590$	2171.90			3.68119
$591$		−	253.686i		−	0.429248i
$592$		−	106.470i		−	0.179848i
$593$		−	1030.13i		−	1.73715i	−0.495561	−	0.868573i	$-0.665038\pi$
$593$		−	1030.13i		−	1.73715i	0.495561	−	0.868573i	$-0.334962\pi$
$594$	−98.3532			−0.165578
$595$	−88.1708			−0.148186
$596$			1309.96i			2.19793i
$597$	−256.177			−0.429107
$598$		−	1405.94i		−	2.35107i
$599$	1028.30			1.71669			0.858345	−	0.513073i	$-0.171493\pi$
$599$	1028.30			1.71669			0.858345	+	0.513073i	$0.171493\pi$
$600$	169.660			0.282767
$601$			655.529i			1.09073i	0.838198	+	0.545365i	$0.183609\pi$
$601$			655.529i			1.09073i	−0.838198	+	0.545365i	$0.816391\pi$
$602$	762.790	+	1161.35i	1.26709	+	1.92915i
$603$	384.443			0.637551
$604$		−	672.190i		−	1.11290i
$605$		−	881.935i		−	1.45774i
$606$	31.6780			0.0522739
$607$		−	1016.42i		−	1.67450i	−0.546822	−	0.837249i	$-0.684163\pi$
$607$		−	1016.42i		−	1.67450i	0.546822	−	0.837249i	$-0.315837\pi$
$608$	−443.037			−0.728680
$609$			260.708i			0.428092i
$610$			1592.56i			2.61076i
$611$	926.332			1.51609
$612$	59.4388			0.0971223
$613$	−1029.51			−1.67946			−0.839728	−	0.543007i	$-0.817286\pi$
$613$	−1029.51			−1.67946			−0.839728	+	0.543007i	$0.817286\pi$
$614$			1626.00i			2.64820i
$615$	−361.322			−0.587516
$616$	171.188			0.277903
$617$	426.603			0.691415			0.345708	−	0.938342i	$-0.387639\pi$
$617$	426.603			0.691415			0.345708	+	0.938342i	$0.387639\pi$
$618$	−221.970			−0.359174
$619$	−503.515			−0.813432			−0.406716	−	0.913555i	$-0.633326\pi$
$619$	−503.515			−0.813432			−0.406716	+	0.913555i	$0.633326\pi$
$620$			614.470i			0.991080i
$621$			307.478i			0.495133i
$622$		−	88.6222i		−	0.142479i
$623$	−59.8090			−0.0960016
$624$			33.7268i			0.0540493i
$625$	−330.025			−0.528041
$626$	−123.030			−0.196534
$627$	21.7492			0.0346878
$628$			659.374i			1.04996i
$629$		−	47.1436i		−	0.0749500i
$630$			2098.90i			3.33159i
$631$		−	299.884i		−	0.475252i	−0.971357	−	0.237626i	$-0.923631\pi$
$631$		−	299.884i		−	0.475252i	0.971357	−	0.237626i	$-0.0763692\pi$
$632$		−	221.030i		−	0.349730i
$633$	203.972			0.322230
$634$			273.996i			0.432170i
$635$			1139.48i			1.79446i
$636$			251.474i			0.395400i
$637$	−972.990			−1.52746
$638$	−275.933			−0.432497
$639$			540.693i			0.846155i
$640$	−1450.30			−2.26610
$641$		−	724.346i		−	1.13002i	−0.825082	−	0.565012i	$-0.808871\pi$
$641$		−	724.346i		−	1.13002i	0.825082	−	0.565012i	$-0.191129\pi$
$642$	−4.80817			−0.00748937
$643$	743.353			1.15607			0.578035	−	0.816012i	$-0.303820\pi$
$643$	743.353			1.15607			0.578035	+	0.816012i	$0.303820\pi$
$644$		−	1519.17i		−	2.35896i
$645$	131.089	+	199.583i	0.203239	+	0.309431i
$646$	−44.6565			−0.0691277
$647$		−	61.7177i		−	0.0953905i	−0.998862	−	0.0476953i	$-0.984812\pi$
$647$		−	61.7177i		−	0.0953905i	0.998862	−	0.0476953i	$-0.0151876\pi$
$648$		−	465.672i		−	0.718630i
$649$	216.330			0.333328
$650$		−	1953.46i		−	3.00532i
$651$	95.2817			0.146362
$652$		−	1248.88i		−	1.91547i
$653$		−	422.488i		−	0.646996i	−0.946229	−	0.323498i	$-0.895141\pi$
$653$		−	422.488i		−	0.646996i	0.946229	−	0.323498i	$-0.104859\pi$
$654$	−122.744			−0.187681
$655$	−1204.94			−1.83960
$656$	166.885			0.254398
$657$			74.6013i			0.113548i
$658$	1649.26			2.50647
$659$	−620.921			−0.942217			−0.471108	−	0.882075i	$-0.656146\pi$
$659$	−620.921			−0.942217			−0.471108	+	0.882075i	$0.656146\pi$
$660$	83.5105			0.126531
$661$	496.134			0.750581			0.375291	−	0.926907i	$-0.377543\pi$
$661$	496.134			0.750581			0.375291	+	0.926907i	$0.377543\pi$
$662$	1160.06			1.75236
$663$			14.9338i			0.0225245i
$664$			475.196i			0.715656i
$665$		−	957.027i		−	1.43914i
$666$	−1122.25			−1.68506
$667$			862.638i			1.29331i
$668$	253.631			0.379687
$669$	68.9719			0.103097
$670$	−1109.03			−1.65527
$671$			158.625i			0.236402i
$672$			263.783i			0.392534i
$673$		−	453.069i		−	0.673209i	−0.941646	−	0.336604i	$-0.890722\pi$
$673$		−	453.069i		−	0.673209i	0.941646	−	0.336604i	$-0.109278\pi$
$674$			27.5702i			0.0409053i
$675$			427.220i			0.632918i
$676$	−990.487			−1.46522
$677$		−	233.751i		−	0.345275i	−0.984985	−	0.172638i	$-0.944771\pi$
$677$		−	233.751i		−	0.345275i	0.984985	−	0.172638i	$-0.0552289\pi$
$678$		−	66.7459i		−	0.0984453i
$679$			1019.62i			1.50165i
$680$	−60.4052			−0.0888312
$681$	77.1497			0.113289
$682$			100.846i			0.147868i
$683$	578.370			0.846808			0.423404	−	0.905941i	$-0.360835\pi$
$683$	578.370			0.846808			0.423404	+	0.905941i	$0.360835\pi$
$684$			645.164i			0.943222i
$685$	−932.011			−1.36060
$686$	−148.996			−0.217195
$687$		−	123.427i		−	0.179660i
$688$	−60.5463	−	92.1819i	−0.0880034	−	0.133985i
$689$	1020.03			1.48044
$690$		−	430.178i		−	0.623447i
$691$			127.474i			0.184477i	0.995737	+	0.0922387i	$0.0294023\pi$
$691$			127.474i			0.184477i	−0.995737	+	0.0922387i	$0.970598\pi$
$692$	148.834			0.215078
$693$			209.059i			0.301672i
$694$	−889.142			−1.28118
$695$		−	213.829i		−	0.307667i
$696$			178.609i			0.256622i
$697$	73.8944			0.106018
$698$	−541.502			−0.775791
$699$	40.2097			0.0575247
$700$		−	2110.78i		−	3.01541i
$701$	126.686			0.180721			0.0903606	−	0.995909i	$-0.471198\pi$
$701$	126.686			0.180721			0.0903606	+	0.995909i	$0.471198\pi$
$702$	733.016			1.04418
$703$	511.708			0.727892
$704$	−254.204			−0.361085
$705$	283.432			0.402031
$706$			85.6036i			0.121252i
$707$		−	138.840i		−	0.196379i
$708$		−	397.488i		−	0.561423i
$709$	733.535			1.03461			0.517303	−	0.855803i	$-0.326936\pi$
$709$	733.535			1.03461			0.517303	+	0.855803i	$0.326936\pi$
$710$		−	1559.78i		−	2.19687i
$711$	269.926			0.379643
$712$	−40.9748			−0.0575488
$713$	315.271			0.442175
$714$			26.5883i			0.0372385i
$715$		−	338.734i		−	0.473753i
$716$			512.977i			0.716449i
$717$			15.0483i			0.0209879i
$718$			539.634i			0.751580i
$719$	−885.641			−1.23177			−0.615884	−	0.787837i	$-0.711201\pi$
$719$	−885.641			−1.23177			−0.615884	+	0.787837i	$0.711201\pi$
$720$		−	166.600i		−	0.231389i
$721$			972.859i			1.34932i
$722$			666.846i			0.923609i
$723$	219.740			0.303928
$724$	693.344			0.957658
$725$			1198.58i			1.65321i
$726$	−265.952			−0.366324
$727$		−	244.282i		−	0.336014i	−0.985786	−	0.168007i	$-0.946267\pi$
$727$		−	244.282i		−	0.336014i	0.985786	−	0.168007i	$-0.0537331\pi$
$728$	−1275.85			−1.75253
$729$	486.127			0.666841
$730$		−	215.208i		−	0.294805i
$731$	−26.8091	−	40.8169i	−0.0366746	−	0.0558371i
$732$	291.461			0.398171
$733$			1317.81i			1.79783i	0.438120	+	0.898917i	$0.355645\pi$
$733$			1317.81i			1.79783i	−0.438120	+	0.898917i	$0.644355\pi$
$734$		−	1724.23i		−	2.34908i
$735$	−297.708			−0.405045
$736$			872.812i			1.18589i
$737$	−110.464			−0.149883
$738$		−	1759.05i		−	2.38354i
$739$			126.907i			0.171728i	0.996307	+	0.0858641i	$0.0273651\pi$
$739$			126.907i			0.171728i	−0.996307	+	0.0858641i	$0.972635\pi$
$740$	1964.80			2.65514
$741$	−162.095			−0.218751
$742$	1816.07			2.44753
$743$			483.885i			0.651259i	0.945497	+	0.325629i	$0.105576\pi$
$743$			483.885i			0.651259i	−0.945497	+	0.325629i	$0.894424\pi$
$744$	65.2768			0.0877377
$745$	1625.77			2.18224
$746$	85.2192			0.114235
$747$	−580.320			−0.776867
$748$	−17.0788			−0.0228326
$749$			21.0735i			0.0281355i
$750$		−	154.856i		−	0.206475i
$751$			11.4506i			0.0152471i	0.999971	+	0.00762354i	$0.00242667\pi$
$751$			11.4506i			0.0152471i	−0.999971	+	0.00762354i	$0.997573\pi$
$752$	−130.909			−0.174082
$753$			5.25629i			0.00698047i
$754$	2056.50			2.72745
$755$	−834.241			−1.10495
$756$	792.050			1.04768
$757$		−	995.145i		−	1.31459i	−0.753633	−	0.657295i	$-0.771701\pi$
$757$		−	995.145i		−	1.31459i	0.753633	−	0.657295i	$-0.228299\pi$
$758$			1005.44i			1.32644i
$759$		−	42.8474i		−	0.0564525i
$760$		−	655.653i		−	0.862701i
$761$		−	1067.64i		−	1.40295i	−0.712695	−	0.701474i	$-0.752526\pi$
$761$		−	1067.64i		−	1.40295i	0.712695	−	0.701474i	$-0.247474\pi$
$762$	343.616			0.450940
$763$			537.967i			0.705068i
$764$		−	269.919i		−	0.353297i
$765$		−	73.7682i		−	0.0964291i
$766$	418.195			0.545946
$767$	−1612.28			−2.10206
$768$			134.811i			0.175535i
$769$	−361.976			−0.470710			−0.235355	−	0.971909i	$-0.575625\pi$
$769$	−361.976			−0.470710			−0.235355	+	0.971909i	$0.575625\pi$
$770$		−	603.087i		−	0.783230i
$771$	−229.647			−0.297856
$772$	−1458.05			−1.88866
$773$			579.373i			0.749513i	0.927123	+	0.374756i	$0.122274\pi$
$773$			579.373i			0.749513i	−0.927123	+	0.374756i	$0.877726\pi$
$774$	−971.646	+	638.190i	−1.25536	+	0.824535i
$775$	438.048			0.565224
$776$			698.537i			0.900176i
$777$		−	304.669i		−	0.392109i
$778$	351.049			0.451220
$779$			802.068i			1.02961i
$780$	−622.395			−0.797942
$781$		−	155.360i		−	0.198924i
$782$			87.9762i			0.112502i
$783$	−449.754			−0.574399
$784$	137.503			0.175387
$785$	818.335			1.04246
$786$			363.356i			0.462285i
$787$	−1046.34			−1.32954			−0.664768	−	0.747050i	$-0.731470\pi$
$787$	−1046.34			−1.32954			−0.664768	+	0.747050i	$0.731470\pi$
$788$	−2162.27			−2.74399
$789$	−166.531			−0.211066
$790$	−778.676			−0.985666
$791$	−292.537			−0.369832
$792$			143.225i			0.180839i
$793$		−	1182.22i		−	1.49082i
$794$			217.940i			0.274483i
$795$	312.100			0.392578
$796$			2183.50i			2.74309i
$797$	−959.219			−1.20354			−0.601769	−	0.798670i	$-0.705537\pi$
$797$	−959.219			−1.20354			−0.601769	+	0.798670i	$0.705537\pi$
$798$	−288.596			−0.361649
$799$	−57.9649			−0.0725468
$800$			1212.72i			1.51589i
$801$		−	50.0393i		−	0.0624711i
$802$		−	2154.81i		−	2.68680i
$803$		−	21.4355i		−	0.0266943i
$804$			202.968i			0.252448i
$805$	−1885.41			−2.34212
$806$		−	751.595i		−	0.932500i
$807$		−	206.580i		−	0.255985i
$808$		−	95.1183i		−	0.117721i
$809$	639.565			0.790562			0.395281	−	0.918560i	$-0.370647\pi$
$809$	639.565			0.790562			0.395281	+	0.918560i	$0.370647\pi$
$810$	−1640.54			−2.02536
$811$		−	515.867i		−	0.636087i	−0.948076	−	0.318044i	$-0.896974\pi$
$811$		−	515.867i		−	0.636087i	0.948076	−	0.318044i	$-0.103026\pi$
$812$	2222.12			2.73660
$813$		−	56.7510i		−	0.0698044i
$814$	322.461			0.396144
$815$	−1549.96			−1.90180
$816$		−	2.11044i		−	0.00258632i
$817$	443.037	−	290.993i	0.542273	−	0.356172i
$818$	5.64084			0.00689589
$819$		−	1558.09i		−	1.90243i
$820$			3079.70i			3.75573i
$821$	1288.57			1.56951			0.784756	−	0.619805i	$-0.212788\pi$
$821$	1288.57			1.56951			0.784756	+	0.619805i	$0.212788\pi$
$822$			281.052i			0.341913i
$823$	516.675			0.627795			0.313897	−	0.949457i	$-0.398365\pi$
$823$	516.675			0.627795			0.313897	+	0.949457i	$0.398365\pi$
$824$			666.499i			0.808858i
$825$		−	59.5337i		−	0.0721620i
$826$	−2870.53			−3.47522
$827$	35.6545			0.0431131			0.0215565	−	0.999768i	$-0.493138\pi$
$827$	35.6545			0.0431131			0.0215565	+	0.999768i	$0.493138\pi$
$828$	1271.01			1.53504
$829$			617.780i			0.745212i	0.927990	+	0.372606i	$0.121536\pi$
$829$			617.780i			0.745212i	−0.927990	+	0.372606i	$0.878464\pi$
$830$	1674.09			2.01698
$831$	86.6771			0.104305
$832$	1894.56			2.27711
$833$	60.8846			0.0730907
$834$	−64.4811			−0.0773154
$835$		−	314.776i		−	0.376978i
$836$		−	185.378i		−	0.221744i
$837$			164.373i			0.196384i
$838$	1416.41			1.69023
$839$		−	98.1185i		−	0.116947i	−0.998289	−	0.0584735i	$-0.981377\pi$
$839$		−	98.1185i		−	0.116947i	0.998289	−	0.0584735i	$-0.0186233\pi$
$840$	−390.373			−0.464730
$841$	−420.800			−0.500357
$842$	−450.401			−0.534918
$843$			108.988i			0.129285i
$844$		−	1738.53i		−	2.05987i
$845$			1229.27i			1.45476i
$846$			1379.85i			1.63103i
$847$			1165.63i			1.37618i
$848$	−144.150			−0.169988
$849$			189.209i			0.222861i
$850$			122.237i			0.143808i
$851$		−	1008.10i		−	1.18460i
$852$	−285.460			−0.335047
$853$	564.129			0.661347			0.330674	−	0.943745i	$-0.392724\pi$
$853$	564.129			0.661347			0.330674	+	0.943745i	$0.392724\pi$
$854$		−	2104.84i		−	2.46468i
$855$	800.699			0.936490
$856$			14.4373i			0.0168660i
$857$	−699.603			−0.816340			−0.408170	−	0.912906i	$-0.633833\pi$
$857$	−699.603			−0.816340			−0.408170	+	0.912906i	$0.633833\pi$
$858$	−102.147			−0.119052
$859$			1395.67i			1.62476i	0.583128	+	0.812380i	$0.301829\pi$
$859$			1395.67i			1.62476i	−0.583128	+	0.812380i	$0.698171\pi$
$860$	1701.13	−	1117.32i	1.97806	−	1.29921i
$861$	477.548			0.554644
$862$			963.564i			1.11782i
$863$			768.514i			0.890515i	0.895403	+	0.445257i	$0.146888\pi$
$863$			768.514i			0.890515i	−0.895403	+	0.445257i	$0.853112\pi$
$864$	−455.059			−0.526688
$865$		−	184.715i		−	0.213543i
$866$	−1213.75			−1.40156
$867$			208.457i			0.240435i
$868$		−	812.125i		−	0.935628i
$869$	−77.5591			−0.0892510
$870$	629.232			0.723255
$871$	823.275			0.945206
$872$			368.558i			0.422658i
$873$	−853.070			−0.977170
$874$	−954.916			−1.09258
$875$	−678.711			−0.775670
$876$	−39.3860			−0.0449612
$877$	1318.08			1.50294			0.751470	−	0.659767i	$-0.229345\pi$
$877$	1318.08			1.50294			0.751470	+	0.659767i	$0.229345\pi$
$878$			1782.43i			2.03010i
$879$		−	259.681i		−	0.295428i
$880$			47.8699i			0.0543976i
$881$	−691.467			−0.784867			−0.392433	−	0.919780i	$-0.628367\pi$
$881$	−691.467			−0.784867			−0.392433	+	0.919780i	$0.628367\pi$
$882$		−	1449.36i		−	1.64326i
$883$	−173.332			−0.196299			−0.0981493	−	0.995172i	$-0.531292\pi$
$883$	−173.332			−0.196299			−0.0981493	+	0.995172i	$0.531292\pi$
$884$	127.287			0.143989
$885$	−493.313			−0.557416
$886$		−	1105.19i		−	1.24739i
$887$			142.316i			0.160446i	0.996777	+	0.0802232i	$0.0255633\pi$
$887$			142.316i			0.160446i	−0.996777	+	0.0802232i	$0.974437\pi$
$888$		−	208.727i		−	0.235053i
$889$		−	1506.02i		−	1.69406i
$890$			144.352i			0.162193i
$891$	−163.404			−0.183394
$892$		−	587.876i		−	0.659054i
$893$		−	629.166i		−	0.704553i
$894$		−	490.258i		−	0.548387i
$895$	636.645			0.711335
$896$	1916.82			2.13931
$897$			319.337i			0.356006i
$898$	−799.708			−0.890543
$899$			461.154i			0.512963i
$900$	1765.99			1.96221
$901$	−63.8278			−0.0708410
$902$			505.437i			0.560351i
$903$	−173.256	−	263.783i	−0.191867	−	0.292118i
$904$	−200.415			−0.221698
$905$		−	860.495i		−	0.950823i
$906$			251.569i			0.277670i
$907$	1795.52			1.97963			0.989814	−	0.142370i	$-0.0454721\pi$
$907$	1795.52			1.97963			0.989814	+	0.142370i	$0.0454721\pi$
$908$		−	657.578i		−	0.724205i
$909$	116.161			0.127790
$910$			4494.74i			4.93928i
$911$		−	365.632i		−	0.401353i	−0.979658	−	0.200676i	$-0.935686\pi$
$911$		−	365.632i		−	0.401353i	0.979658	−	0.200676i	$-0.0643140\pi$
$912$	22.9073			0.0251176
$913$	166.746			0.182635
$914$	−2377.62			−2.60134
$915$		−	361.726i		−	0.395329i
$916$	−1052.02			−1.14849
$917$	1592.53			1.73668
$918$	−45.8682			−0.0499654
$919$	1083.62			1.17913			0.589566	−	0.807720i	$-0.299299\pi$
$919$	1083.62			1.17913			0.589566	+	0.807720i	$0.299299\pi$
$920$	−1291.68			−1.40400
$921$		−	369.320i		−	0.400998i
$922$			21.4746i			0.0232913i
$923$			1157.88i			1.25447i
$924$	−110.373			−0.119451
$925$		−	1400.69i		−	1.51426i
$926$	−1429.71			−1.54397
$927$	−813.945			−0.878042
$928$	−1276.68			−1.37573
$929$		−	1400.56i		−	1.50760i	−0.657105	−	0.753799i	$-0.728219\pi$
$929$		−	1400.56i		−	1.50760i	0.657105	−	0.753799i	$-0.271781\pi$
$930$		−	229.967i		−	0.247276i
$931$			660.856i			0.709835i
$932$		−	342.724i		−	0.367730i
$933$			20.1292i			0.0215747i
$934$	1614.25			1.72832
$935$			21.1962i			0.0226697i
$936$		−	1067.44i		−	1.14043i
$937$		−	846.473i		−	0.903387i	−0.892173	−	0.451693i	$-0.850820\pi$
$937$		−	846.473i		−	0.903387i	0.892173	−	0.451693i	$-0.149180\pi$
$938$	1465.77			1.56266
$939$	27.9444			0.0297598
$940$		−	2415.81i		−	2.57001i
$941$	−1354.63			−1.43956			−0.719780	−	0.694203i	$-0.755757\pi$
$941$	−1354.63			−1.43956			−0.719780	+	0.694203i	$0.755757\pi$
$942$		−	246.773i		−	0.261967i
$943$	1580.13			1.67564
$944$	227.848			0.241364
$945$		−	982.996i		−	1.04021i
$946$	279.187	−	183.374i	0.295124	−	0.193841i
$947$	−209.288			−0.221001			−0.110501	−	0.993876i	$-0.535245\pi$
$947$	−209.288			−0.221001			−0.110501	+	0.993876i	$0.535245\pi$
$948$			142.508i			0.150325i
$949$			159.757i			0.168342i
$950$	−1326.79			−1.39662
$951$		−	62.2339i		−	0.0654405i
$952$	79.8356			0.0838610
$953$		−	293.406i		−	0.307876i	−0.988080	−	0.153938i	$-0.950804\pi$
$953$		−	293.406i		−	0.307876i	0.988080	−	0.153938i	$-0.0491957\pi$
$954$			1519.42i			1.59268i
$955$	−334.991			−0.350776
$956$	128.263			0.134166
$957$	62.6739			0.0654899
$958$		−	2464.89i		−	2.57295i
$959$	1231.81			1.28447
$960$	579.682			0.603835
$961$	−792.461			−0.824621
$962$	−2403.27			−2.49820
$963$	−17.6312			−0.0183086
$964$		−	1872.93i		−	1.94288i
$965$			1809.55i			1.87518i
$966$			568.553i			0.588564i
$967$	−1028.49			−1.06359			−0.531794	−	0.846874i	$-0.678482\pi$
$967$	−1028.49			−1.06359			−0.531794	+	0.846874i	$0.678482\pi$
$968$			798.562i			0.824961i
$969$	10.1430			0.0104675
$970$	2460.91			2.53702
$971$	−438.558			−0.451656			−0.225828	−	0.974167i	$-0.572509\pi$
$971$	−438.558			−0.451656			−0.225828	+	0.974167i	$0.572509\pi$
$972$			1003.96i			1.03288i
$973$			282.611i			0.290453i
$974$		−	2432.16i		−	2.49709i
$975$			443.698i			0.455075i
$976$			167.071i			0.171180i
$977$	−626.212			−0.640954			−0.320477	−	0.947256i	$-0.603843\pi$
$977$	−626.212			−0.640954			−0.320477	+	0.947256i	$0.603843\pi$
$978$			467.399i			0.477913i
$979$			14.3780i			0.0146864i
$980$			2537.49i			2.58927i
$981$	−450.091			−0.458808
$982$	−2961.51			−3.01579
$983$			1720.98i			1.75074i	0.483454	+	0.875370i	$0.339382\pi$
$983$			1720.98i			1.75074i	−0.483454	+	0.875370i	$0.660618\pi$
$984$	327.165			0.332485
$985$			2683.54i			2.72441i
$986$	−128.685			−0.130512
$987$	−374.603			−0.379537
$988$			1381.60i			1.39838i
$989$	−573.275	−	872.812i	−0.579651	−	0.882520i
$990$	504.574			0.509671
$991$			753.187i			0.760027i	0.924981	+	0.380014i	$0.124081\pi$
$991$			753.187i			0.760027i	−0.924981	+	0.380014i	$0.875919\pi$
$992$			466.593i			0.470356i
$993$	−263.490			−0.265348
$994$			2061.51i			2.07395i
$995$	2709.90			2.72352
$996$		−	306.382i		−	0.307612i
$997$		−	533.031i		−	0.534635i	−0.963609	−	0.267317i	$-0.913863\pi$
$997$		−	533.031i		−	0.534635i	0.963609	−	0.267317i	$-0.0861372\pi$
$998$	2137.17			2.14145
$999$	525.593			0.526119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.b.b.42.6	yes	6
3.2	odd	2		387.3.b.c.343.1		6
4.3	odd	2		688.3.b.e.257.4		6
43.42	odd	2	inner	43.3.b.b.42.1	✓	6
129.128	even	2		387.3.b.c.343.6		6
172.171	even	2		688.3.b.e.257.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.b.b.42.1	✓	6	43.42	odd	2	inner
43.3.b.b.42.6	yes	6	1.1	even	1	trivial
387.3.b.c.343.1		6	3.2	odd	2
387.3.b.c.343.6		6	129.128	even	2
688.3.b.e.257.3		6	172.171	even	2
688.3.b.e.257.4		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.18991i q^{2} -0.724539i q^{3} -6.17554 q^{4} +7.66434i q^{5} +2.31122 q^{6} -10.1297i q^{7} -6.93980i q^{8} +8.47504 q^{9} +O(q^{10})\)	\(q+3.18991i q^{2} -0.724539i q^{3} -6.17554 q^{4} +7.66434i q^{5} +2.31122 q^{6} -10.1297i q^{7} -6.93980i q^{8} +8.47504 q^{9} -24.4486 q^{10} -2.43517 q^{11} +4.47442i q^{12} +18.1491 q^{13} +32.3129 q^{14} +5.55311 q^{15} -2.56483 q^{16} -1.13567 q^{17} +27.0346i q^{18} -12.3269i q^{19} -47.3314i q^{20} -7.33937 q^{21} -7.76798i q^{22} -24.2847 q^{23} -5.02815 q^{24} -33.7420 q^{25} +57.8939i q^{26} -12.6614i q^{27} +62.5565i q^{28} -35.5218i q^{29} +17.7139i q^{30} -12.9823 q^{31} -35.9408i q^{32} +1.76438i q^{33} -3.62270i q^{34} +77.6375 q^{35} -52.3380 q^{36} +41.5116i q^{37} +39.3216 q^{38} -13.1497i q^{39} +53.1889 q^{40} -65.0667 q^{41} -23.4120i q^{42} +(23.6064 + 35.9408i) q^{43} +15.0385 q^{44} +64.9556i q^{45} -77.4662i q^{46} +51.0402 q^{47} +1.85832i q^{48} -53.6110 q^{49} -107.634i q^{50} +0.822839i q^{51} -112.080 q^{52} +56.2026 q^{53} +40.3886 q^{54} -18.6640i q^{55} -70.2981 q^{56} -8.93130 q^{57} +113.312 q^{58} -88.8355 q^{59} -34.2935 q^{60} -65.1393i q^{61} -41.4123i q^{62} -85.8497i q^{63} +104.389 q^{64} +139.101i q^{65} -5.62821 q^{66} +45.3618 q^{67} +7.01339 q^{68} +17.5952i q^{69} +247.657i q^{70} +63.7983i q^{71} -58.8151i q^{72} +8.80247i q^{73} -132.418 q^{74} +24.4474i q^{75} +76.1251i q^{76} +24.6676i q^{77} +41.9464 q^{78} +31.8496 q^{79} -19.6577i q^{80} +67.1017 q^{81} -207.557i q^{82} -68.4740 q^{83} +45.3246 q^{84} -8.70417i q^{85} +(-114.648 + 75.3023i) q^{86} -25.7370 q^{87} +16.8996i q^{88} -5.90432i q^{89} -207.203 q^{90} -183.845i q^{91} +149.971 q^{92} +9.40616i q^{93} +162.814i q^{94} +94.4773 q^{95} -26.0405 q^{96} -100.657 q^{97} -171.015i q^{98} -20.6382 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10}) \) 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

Modular forms

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