LMFDB - Embedded newform 43.3.b.b.42.2

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			42.2
Root			$-2.58315i$ of defining polynomial
Character	$\chi$	$=$	43.42
Dual form			43.3.b.b.42.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58315i q^{2} -3.59415i q^{3} -2.67267 q^{4} +7.02284i q^{5} -9.28424 q^{6} -0.845536i q^{7} -3.42869i q^{8} -3.91793 q^{9} +O(q^{10})$	\(q-2.58315i q^{2} -3.59415i q^{3} -2.67267 q^{4} +7.02284i q^{5} -9.28424 q^{6} -0.845536i q^{7} -3.42869i q^{8} -3.91793 q^{9} +18.1411 q^{10} +14.5475 q^{11} +9.60599i q^{12} -15.5505 q^{13} -2.18415 q^{14} +25.2412 q^{15} -19.5475 q^{16} +6.95691 q^{17} +10.1206i q^{18} +30.8865i q^{19} -18.7698i q^{20} -3.03899 q^{21} -37.5784i q^{22} +17.5074 q^{23} -12.3232 q^{24} -24.3203 q^{25} +40.1692i q^{26} -18.2657i q^{27} +2.25984i q^{28} +7.86076i q^{29} -65.2017i q^{30} -57.7456 q^{31} +36.7794i q^{32} -52.2860i q^{33} -17.9708i q^{34} +5.93806 q^{35} +10.4713 q^{36} +32.5310i q^{37} +79.7846 q^{38} +55.8907i q^{39} +24.0791 q^{40} -18.4425 q^{41} +7.85016i q^{42} +(22.2772 - 36.7794i) q^{43} -38.8807 q^{44} -27.5150i q^{45} -45.2242i q^{46} -25.7806 q^{47} +70.2567i q^{48} +48.2851 q^{49} +62.8230i q^{50} -25.0042i q^{51} +41.5613 q^{52} -79.9247 q^{53} -47.1832 q^{54} +102.165i q^{55} -2.89908 q^{56} +111.011 q^{57} +20.3055 q^{58} +18.4243 q^{59} -67.4613 q^{60} -76.1107i q^{61} +149.166i q^{62} +3.31275i q^{63} +16.8168 q^{64} -109.208i q^{65} -135.063 q^{66} +9.00391 q^{67} -18.5935 q^{68} -62.9242i q^{69} -15.3389i q^{70} -51.6630i q^{71} +13.4333i q^{72} -77.4556i q^{73} +84.0326 q^{74} +87.4108i q^{75} -82.5496i q^{76} -12.3004i q^{77} +144.374 q^{78} +7.04014 q^{79} -137.279i q^{80} -100.911 q^{81} +47.6397i q^{82} +83.8385 q^{83} +8.12221 q^{84} +48.8573i q^{85} +(-95.0069 - 57.5454i) q^{86} +28.2528 q^{87} -49.8789i q^{88} +91.8322i q^{89} -71.0753 q^{90} +13.1485i q^{91} -46.7915 q^{92} +207.546i q^{93} +66.5953i q^{94} -216.911 q^{95} +132.191 q^{96} -155.976 q^{97} -124.728i q^{98} -56.9961 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.58315i		−	1.29158i	−0.763517	−	0.645788i	$-0.776529\pi$
$2$		−	2.58315i		−	1.29158i	0.763517	−	0.645788i	$-0.223471\pi$
$3$		−	3.59415i		−	1.19805i	−0.800730	−	0.599025i	$-0.795555\pi$
$3$		−	3.59415i		−	1.19805i	0.800730	−	0.599025i	$-0.204445\pi$
$4$	−2.67267			−0.668168
$5$			7.02284i			1.40457i	0.711897	+	0.702284i	$0.247836\pi$
$5$			7.02284i			1.40457i	−0.711897	+	0.702284i	$0.752164\pi$
$6$	−9.28424			−1.54737
$7$		−	0.845536i		−	0.120791i	−0.998175	−	0.0603954i	$-0.980764\pi$
$7$		−	0.845536i		−	0.120791i	0.998175	−	0.0603954i	$-0.0192362\pi$
$8$		−	3.42869i		−	0.428586i
$9$	−3.91793			−0.435325
$10$	18.1411			1.81411
$11$	14.5475			1.32250			0.661251	−	0.750165i	$-0.270026\pi$
$11$	14.5475			1.32250			0.661251	+	0.750165i	$0.270026\pi$
$12$			9.60599i			0.800499i
$13$	−15.5505			−1.19619			−0.598095	−	0.801425i	$-0.704075\pi$
$13$	−15.5505			−1.19619			−0.598095	+	0.801425i	$0.704075\pi$
$14$	−2.18415			−0.156011
$15$	25.2412			1.68274
$16$	−19.5475			−1.22172
$17$	6.95691			0.409230			0.204615	−	0.978843i	$-0.434406\pi$
$17$	6.95691			0.409230			0.204615	+	0.978843i	$0.434406\pi$
$18$			10.1206i			0.562256i
$19$			30.8865i			1.62561i	0.582539	+	0.812803i	$0.302060\pi$
$19$			30.8865i			1.62561i	−0.582539	+	0.812803i	$0.697940\pi$
$20$		−	18.7698i		−	0.938488i
$21$	−3.03899			−0.144714
$22$		−	37.5784i		−	1.70811i
$23$	17.5074			0.761190			0.380595	−	0.924742i	$-0.375719\pi$
$23$	17.5074			0.761190			0.380595	+	0.924742i	$0.375719\pi$
$24$	−12.3232			−0.513468
$25$	−24.3203			−0.972811
$26$			40.1692i			1.54497i
$27$		−	18.2657i		−	0.676509i
$28$			2.25984i			0.0807086i
$29$			7.86076i			0.271061i	0.990773	+	0.135530i	$0.0432738\pi$
$29$			7.86076i			0.271061i	−0.990773	+	0.135530i	$0.956726\pi$
$30$		−	65.2017i		−	2.17339i
$31$	−57.7456			−1.86276			−0.931380	−	0.364048i	$-0.881394\pi$
$31$	−57.7456			−1.86276			−0.931380	+	0.364048i	$0.881394\pi$
$32$			36.7794i			1.14936i
$33$		−	52.2860i		−	1.58442i
$34$		−	17.9708i		−	0.528552i
$35$	5.93806			0.169659
$36$	10.4713			0.290870
$37$			32.5310i			0.879217i	0.898189	+	0.439609i	$0.144883\pi$
$37$			32.5310i			0.879217i	−0.898189	+	0.439609i	$0.855117\pi$
$38$	79.7846			2.09959
$39$			55.8907i			1.43310i
$40$	24.0791			0.601978
$41$	−18.4425			−0.449817			−0.224908	−	0.974380i	$-0.572208\pi$
$41$	−18.4425			−0.449817			−0.224908	+	0.974380i	$0.572208\pi$
$42$			7.85016i			0.186909i
$43$	22.2772	−	36.7794i	0.518074	−	0.855336i
$43$	22.2772	−	36.7794i	0.518074	−	0.855336i
$44$	−38.8807			−0.883653
$45$		−	27.5150i		−	0.611444i
$46$		−	45.2242i		−	0.983135i
$47$	−25.7806			−0.548524			−0.274262	−	0.961655i	$-0.588434\pi$
$47$	−25.7806			−0.548524			−0.274262	+	0.961655i	$0.588434\pi$
$48$			70.2567i			1.46368i
$49$	48.2851			0.985410
$50$			62.8230i			1.25646i
$51$		−	25.0042i		−	0.490278i
$52$	41.5613			0.799256
$53$	−79.9247			−1.50801			−0.754006	−	0.656867i	$-0.771881\pi$
$53$	−79.9247			−1.50801			−0.754006	+	0.656867i	$0.771881\pi$
$54$	−47.1832			−0.873763
$55$			102.165i			1.85754i
$56$	−2.89908			−0.0517693
$57$	111.011			1.94756
$58$	20.3055			0.350095
$59$	18.4243			0.312277			0.156138	−	0.987735i	$-0.450095\pi$
$59$	18.4243			0.312277			0.156138	+	0.987735i	$0.450095\pi$
$60$	−67.4613			−1.12436
$61$		−	76.1107i		−	1.24772i	−0.781537	−	0.623858i	$-0.785564\pi$
$61$		−	76.1107i		−	1.24772i	0.781537	−	0.623858i	$-0.214436\pi$
$62$			149.166i			2.40590i
$63$			3.31275i			0.0525833i
$64$	16.8168			0.262763
$65$		−	109.208i		−	1.68013i
$66$	−135.063			−2.04640
$67$	9.00391			0.134387			0.0671934	−	0.997740i	$-0.478596\pi$
$67$	9.00391			0.134387			0.0671934	+	0.997740i	$0.478596\pi$
$68$	−18.5935			−0.273435
$69$		−	62.9242i		−	0.911944i
$70$		−	15.3389i		−	0.219127i
$71$		−	51.6630i		−	0.727648i	−0.931468	−	0.363824i	$-0.881471\pi$
$71$		−	51.6630i		−	0.727648i	0.931468	−	0.363824i	$-0.118529\pi$
$72$			13.4333i			0.186574i
$73$		−	77.4556i		−	1.06104i	−0.847674	−	0.530518i	$-0.821997\pi$
$73$		−	77.4556i		−	1.06104i	0.847674	−	0.530518i	$-0.178003\pi$
$74$	84.0326			1.13558
$75$			87.4108i			1.16548i
$76$		−	82.5496i		−	1.08618i
$77$		−	12.3004i		−	0.159746i
$78$	144.374			1.85095
$79$	7.04014			0.0891157			0.0445578	−	0.999007i	$-0.485812\pi$
$79$	7.04014			0.0891157			0.0445578	+	0.999007i	$0.485812\pi$
$80$		−	137.279i		−	1.71599i
$81$	−100.911			−1.24582
$82$			47.6397i			0.580972i
$83$	83.8385			1.01010			0.505051	−	0.863089i	$-0.331474\pi$
$83$	83.8385			1.01010			0.505051	+	0.863089i	$0.331474\pi$
$84$	8.12221			0.0966930
$85$			48.8573i			0.574791i
$86$	−95.0069	−	57.5454i	−1.10473	−	0.669132i
$87$	28.2528			0.324744
$88$		−	49.8789i		−	0.566805i
$89$			91.8322i			1.03182i	0.856642	+	0.515911i	$0.172547\pi$
$89$			91.8322i			1.03182i	−0.856642	+	0.515911i	$0.827453\pi$
$90$	−71.0753			−0.789726
$91$			13.1485i			0.144489i
$92$	−46.7915			−0.508603
$93$			207.546i			2.23168i
$94$			66.5953i			0.708461i
$95$	−216.911			−2.28327
$96$	132.191			1.37699
$97$	−155.976			−1.60800			−0.803998	−	0.594632i	$-0.797298\pi$
$97$	−155.976			−1.60800			−0.803998	+	0.594632i	$0.797298\pi$
$98$		−	124.728i		−	1.27273i
$99$	−56.9961			−0.575718
$100$	65.0001			0.650001
$101$	61.9489			0.613355			0.306678	−	0.951813i	$-0.400783\pi$
$101$	61.9489			0.613355			0.306678	+	0.951813i	$0.400783\pi$
$102$	−64.5896			−0.633232
$103$	−19.2194			−0.186596			−0.0932978	−	0.995638i	$-0.529741\pi$
$103$	−19.2194			−0.186596			−0.0932978	+	0.995638i	$0.529741\pi$
$104$			53.3177i			0.512670i
$105$		−	21.3423i		−	0.203260i
$106$			206.458i			1.94771i
$107$	151.561			1.41646			0.708230	−	0.705981i	$-0.249494\pi$
$107$	151.561			1.41646			0.708230	+	0.705981i	$0.249494\pi$
$108$			48.8184i			0.452022i
$109$	43.6904			0.400829			0.200415	−	0.979711i	$-0.435771\pi$
$109$	43.6904			0.400829			0.200415	+	0.979711i	$0.435771\pi$
$110$	263.907			2.39916
$111$	116.922			1.05335
$112$			16.5281i			0.147573i
$113$		−	30.9047i		−	0.273493i	−0.990606	−	0.136747i	$-0.956335\pi$
$113$		−	30.9047i		−	0.273493i	0.990606	−	0.136747i	$-0.0436646\pi$
$114$		−	286.758i		−	2.51542i
$115$			122.951i			1.06914i
$116$		−	21.0092i		−	0.181114i
$117$	60.9256			0.520731
$118$		−	47.5928i		−	0.403329i
$119$		−	5.88232i		−	0.0494313i
$120$		−	86.5440i		−	0.721200i
$121$	90.6301			0.749009
$122$	−196.606			−1.61152
$123$			66.2851i			0.538903i
$124$	154.335			1.24464
$125$			4.77360i			0.0381888i
$126$	8.55733			0.0679153
$127$	19.3094			0.152043			0.0760214	−	0.997106i	$-0.475778\pi$
$127$	19.3094			0.152043			0.0760214	+	0.997106i	$0.475778\pi$
$128$			103.677i			0.809979i
$129$	−132.191	−	80.0676i	−1.02474	−	0.620679i
$130$	−282.102			−2.17001
$131$		−	19.3546i		−	0.147745i	−0.997268	−	0.0738725i	$-0.976464\pi$
$131$		−	19.3546i		−	0.147745i	0.997268	−	0.0738725i	$-0.0235358\pi$
$132$			139.743i			1.05866i
$133$	26.1157			0.196358
$134$		−	23.2585i		−	0.173571i
$135$	128.277			0.950203
$136$		−	23.8531i		−	0.175390i
$137$		−	74.8826i		−	0.546588i	−0.961931	−	0.273294i	$-0.911887\pi$
$137$		−	74.8826i		−	0.546588i	0.961931	−	0.273294i	$-0.0881132\pi$
$138$	−162.543			−1.17785
$139$	48.3195			0.347622			0.173811	−	0.984779i	$-0.444392\pi$
$139$	48.3195			0.347622			0.173811	+	0.984779i	$0.444392\pi$
$140$	−15.8705			−0.113361
$141$			92.6596i			0.657160i
$142$	−133.453			−0.939813
$143$	−226.221			−1.58196
$144$	76.5857			0.531845
$145$	−55.2048			−0.380723
$146$	−200.080			−1.37041
$147$		−	173.544i		−	1.18057i
$148$		−	86.9448i		−	0.587465i
$149$			56.9508i			0.382220i	0.981569	+	0.191110i	$0.0612087\pi$
$149$			56.9508i			0.382220i	−0.981569	+	0.191110i	$0.938791\pi$
$150$	225.795			1.50530
$151$		−	133.385i		−	0.883346i	−0.897176	−	0.441673i	$-0.854385\pi$
$151$		−	133.385i		−	0.883346i	0.897176	−	0.441673i	$-0.145615\pi$
$152$	105.900			0.696712
$153$	−27.2567			−0.178148
$154$	−31.7739			−0.206324
$155$		−	405.538i		−	2.61637i
$156$		−	149.378i		−	0.957549i
$157$			172.766i			1.10042i	0.835026	+	0.550210i	$0.185452\pi$
$157$			172.766i			1.10042i	−0.835026	+	0.550210i	$0.814548\pi$
$158$		−	18.1857i		−	0.115100i
$159$			287.261i			1.80668i
$160$	−258.296			−1.61435
$161$		−	14.8031i		−	0.0919448i
$162$			260.669i			1.60907i
$163$			243.720i			1.49521i	0.664142	+	0.747607i	$0.268797\pi$
$163$			243.720i			1.49521i	−0.664142	+	0.747607i	$0.731203\pi$
$164$	49.2907			0.300553
$165$	367.196			2.22543
$166$		−	216.568i		−	1.30462i
$167$	246.040			1.47330			0.736648	−	0.676277i	$-0.236408\pi$
$167$	246.040			1.47330			0.736648	+	0.676277i	$0.236408\pi$
$168$			10.4197i			0.0620222i
$169$	72.8168			0.430869
$170$	126.206			0.742387
$171$		−	121.011i		−	0.707667i
$172$	−59.5396	+	98.2994i	−0.346161	+	0.571508i
$173$	−137.397			−0.794201			−0.397100	−	0.917775i	$-0.629984\pi$
$173$	−137.397			−0.794201			−0.397100	+	0.917775i	$0.629984\pi$
$174$		−	72.9812i		−	0.419432i
$175$			20.5637i			0.117507i
$176$	−284.368			−1.61573
$177$		−	66.2199i		−	0.374123i
$178$	237.217			1.33268
$179$		−	131.174i		−	0.732816i	−0.930454	−	0.366408i	$-0.880587\pi$
$179$		−	131.174i		−	0.732816i	0.930454	−	0.366408i	$-0.119413\pi$
$180$			73.5385i			0.408547i
$181$	229.555			1.26826			0.634129	−	0.773227i	$-0.281359\pi$
$181$	229.555			1.26826			0.634129	+	0.773227i	$0.281359\pi$
$182$	33.9645			0.186618
$183$	−273.553			−1.49483
$184$		−	60.0273i		−	0.326235i
$185$	−228.460			−1.23492
$186$	536.124			2.88239
$187$	101.206			0.541207
$188$	68.9032			0.366507
$189$	−15.4443			−0.0817161
$190$			560.314i			2.94902i
$191$		−	209.496i		−	1.09684i	−0.836204	−	0.548419i	$-0.815230\pi$
$191$		−	209.496i		−	1.09684i	0.836204	−	0.548419i	$-0.184770\pi$
$192$		−	60.4422i		−	0.314803i
$193$	−261.300			−1.35389			−0.676943	−	0.736035i	$-0.736696\pi$
$193$	−261.300			−1.35389			−0.676943	+	0.736035i	$0.736696\pi$
$194$			402.909i			2.07685i
$195$	−392.512			−2.01288
$196$	−129.050			−0.658419
$197$	94.0645			0.477485			0.238742	−	0.971083i	$-0.423265\pi$
$197$	94.0645			0.477485			0.238742	+	0.971083i	$0.423265\pi$
$198$			147.230i			0.743584i
$199$			28.5992i			0.143715i	0.997415	+	0.0718573i	$0.0228926\pi$
$199$			28.5992i			0.143715i	−0.997415	+	0.0718573i	$0.977107\pi$
$200$			83.3866i			0.416933i
$201$		−	32.3614i		−	0.161002i
$202$		−	160.023i		−	0.792195i
$203$	6.64655			0.0327416
$204$			66.8280i			0.327588i
$205$		−	129.519i		−	0.631798i
$206$			49.6465i			0.241002i
$207$	−68.5926			−0.331365
$208$	303.973			1.46141
$209$			449.322i			2.14987i
$210$	−55.1304			−0.262526
$211$		−	22.6639i		−	0.107412i	−0.998557	−	0.0537059i	$-0.982897\pi$
$211$		−	22.6639i		−	0.107412i	0.998557	−	0.0537059i	$-0.0171033\pi$
$212$	213.612			1.00761
$213$	−185.685			−0.871760
$214$		−	391.506i		−	1.82947i
$215$	258.296	+	156.449i	1.20138	+	0.727670i
$216$	−62.6275			−0.289942
$217$			48.8260i			0.225004i
$218$		−	112.859i		−	0.517701i
$219$	−278.387			−1.27117
$220$		−	273.053i		−	1.24115i
$221$	−108.183			−0.489517
$222$		−	302.026i		−	1.36048i
$223$		−	263.106i		−	1.17985i	−0.807460	−	0.589923i	$-0.799158\pi$
$223$		−	263.106i		−	1.17985i	0.807460	−	0.589923i	$-0.200842\pi$
$224$	31.0983			0.138832
$225$	95.2851			0.423489
$226$	−79.8316			−0.353237
$227$			54.4887i			0.240038i	0.992772	+	0.120019i	$0.0382956\pi$
$227$			54.4887i			0.240038i	−0.992772	+	0.120019i	$0.961704\pi$
$228$	−296.696			−1.30130
$229$	189.993			0.829664			0.414832	−	0.909898i	$-0.363840\pi$
$229$	189.993			0.829664			0.414832	+	0.909898i	$0.363840\pi$
$230$	317.602			1.38088
$231$	−44.2097			−0.191384
$232$	26.9521			0.116173
$233$		−	209.186i		−	0.897795i	−0.893583	−	0.448898i	$-0.851817\pi$
$233$		−	209.186i		−	0.897795i	0.893583	−	0.448898i	$-0.148183\pi$
$234$		−	157.380i		−	0.672564i
$235$		−	181.053i		−	0.770440i
$236$	−49.2422			−0.208653
$237$		−	25.3033i		−	0.106765i
$238$	−15.1949			−0.0638442
$239$	−68.1912			−0.285319			−0.142659	−	0.989772i	$-0.545565\pi$
$239$	−68.1912			−0.285319			−0.142659	+	0.989772i	$0.545565\pi$
$240$	−493.402			−2.05584
$241$			380.930i			1.58062i	0.612705	+	0.790312i	$0.290082\pi$
$241$			380.930i			1.58062i	−0.612705	+	0.790312i	$0.709918\pi$
$242$		−	234.111i		−	0.967402i
$243$			198.298i			0.816043i
$244$			203.419i			0.833685i
$245$			339.098i			1.38407i
$246$	171.224			0.696034
$247$		−	480.300i		−	1.94453i
$248$			197.992i			0.798353i
$249$		−	301.328i		−	1.21015i
$250$	12.3309			0.0493237
$251$	−175.752			−0.700208			−0.350104	−	0.936711i	$-0.613854\pi$
$251$	−175.752			−0.700208			−0.350104	+	0.936711i	$0.613854\pi$
$252$		−	8.85389i		−	0.0351345i
$253$	254.689			1.00667
$254$		−	49.8792i		−	0.196375i
$255$	175.600			0.688629
$256$	335.082			1.30891
$257$			162.749i			0.633263i	0.948549	+	0.316632i	$0.102552\pi$
$257$			162.749i			0.633263i	−0.948549	+	0.316632i	$0.897448\pi$
$258$	−206.827	+	341.469i	−0.801654	+	1.32352i
$259$	27.5062			0.106201
$260$			291.878i			1.12261i
$261$		−	30.7979i		−	0.118000i
$262$	−49.9959			−0.190824
$263$		−	78.4612i		−	0.298331i	−0.988812	−	0.149166i	$-0.952341\pi$
$263$		−	78.4612i		−	0.298331i	0.988812	−	0.149166i	$-0.0476588\pi$
$264$	−179.272			−0.679062
$265$		−	561.298i		−	2.11811i
$266$		−	67.4607i		−	0.253612i
$267$	330.059			1.23618
$268$	−24.0645			−0.0897930
$269$	−48.1852			−0.179127			−0.0895636	−	0.995981i	$-0.528547\pi$
$269$	−48.1852			−0.179127			−0.0895636	+	0.995981i	$0.528547\pi$
$270$		−	331.360i		−	1.22726i
$271$	81.9474			0.302389			0.151194	−	0.988504i	$-0.451688\pi$
$271$	81.9474			0.302389			0.151194	+	0.988504i	$0.451688\pi$
$272$	−135.990			−0.499964
$273$	47.2576			0.173105
$274$	−193.433			−0.705960
$275$	−353.799			−1.28654
$276$			168.176i			0.609332i
$277$		−	1.40205i		−	0.00506156i	−0.999997	−	0.00253078i	$-0.999194\pi$
$277$		−	1.40205i		−	0.00506156i	0.999997	−	0.00253078i	$-0.000805573\pi$
$278$		−	124.817i		−	0.448981i
$279$	226.243			0.810907
$280$		−	20.3598i		−	0.0727135i
$281$	248.005			0.882580			0.441290	−	0.897364i	$-0.354521\pi$
$281$	248.005			0.882580			0.441290	+	0.897364i	$0.354521\pi$
$282$	239.354			0.848772
$283$	−138.544			−0.489553			−0.244777	−	0.969580i	$-0.578715\pi$
$283$	−138.544			−0.489553			−0.244777	+	0.969580i	$0.578715\pi$
$284$			138.078i			0.486191i
$285$			779.611i			2.73548i
$286$			584.362i			2.04322i
$287$			15.5938i			0.0543338i
$288$		−	144.099i		−	0.500344i
$289$	−240.601			−0.832531
$290$			142.602i			0.491733i
$291$			560.600i			1.92646i
$292$			207.014i			0.708950i
$293$	493.442			1.68410			0.842051	−	0.539398i	$-0.181348\pi$
$293$	493.442			1.68410			0.842051	+	0.539398i	$0.181348\pi$
$294$	−448.290			−1.52480
$295$			129.391i			0.438614i
$296$	111.539			0.376820
$297$		−	265.721i		−	0.894684i
$298$	147.112			0.493666
$299$	−272.248			−0.910527
$300$		−	233.620i		−	0.778735i
$301$	−31.0983	−	18.8362i	−0.103317	−	0.0625786i
$302$	−344.554			−1.14091
$303$		−	222.654i		−	0.734831i
$304$		−	603.755i		−	1.98603i
$305$	534.513			1.75250
$306$			70.4081i			0.230092i
$307$	74.5297			0.242768			0.121384	−	0.992606i	$-0.461267\pi$
$307$	74.5297			0.242768			0.121384	+	0.992606i	$0.461267\pi$
$308$			32.8751i			0.106737i
$309$			69.0773i			0.223551i
$310$	−1047.57			−3.37924
$311$	21.3831			0.0687560			0.0343780	−	0.999409i	$-0.489055\pi$
$311$	21.3831			0.0687560			0.0343780	+	0.999409i	$0.489055\pi$
$312$	191.632			0.614205
$313$			231.173i			0.738571i	0.929316	+	0.369285i	$0.120398\pi$
$313$			231.173i			0.738571i	−0.929316	+	0.369285i	$0.879602\pi$
$314$	446.281			1.42128
$315$	−23.2649			−0.0738568
$316$	−18.8160			−0.0595443
$317$	−160.636			−0.506737			−0.253369	−	0.967370i	$-0.581539\pi$
$317$	−160.636			−0.506737			−0.253369	+	0.967370i	$0.581539\pi$
$318$	742.040			2.33346
$319$			114.354i			0.358478i
$320$			118.102i			0.369068i
$321$		−	544.734i		−	1.69699i
$322$	−38.2387			−0.118754
$323$			214.875i			0.665247i
$324$	269.703			0.832415
$325$	378.191			1.16367
$326$	629.565			1.93118
$327$		−	157.030i		−	0.480213i
$328$			63.2335i			0.192785i
$329$			21.7985i			0.0662567i
$330$		−	948.523i		−	2.87431i
$331$			409.618i			1.23752i	0.785581	+	0.618759i	$0.212364\pi$
$331$			409.618i			1.23752i	−0.785581	+	0.618759i	$0.787636\pi$
$332$	−224.073			−0.674918
$333$		−	127.454i		−	0.382746i
$334$		−	635.559i		−	1.90287i
$335$			63.2330i			0.188755i
$336$	59.4046			0.176799
$337$	−476.795			−1.41482			−0.707411	−	0.706803i	$-0.750137\pi$
$337$	−476.795			−1.41482			−0.707411	+	0.706803i	$0.750137\pi$
$338$		−	188.097i		−	0.556500i
$339$	−111.076			−0.327659
$340$		−	130.580i		−	0.384057i
$341$	−840.054			−2.46350
$342$	−312.590			−0.914006
$343$		−	82.2580i		−	0.239819i
$344$	−126.105	−	76.3815i	−0.366585	−	0.222039i
$345$	441.906			1.28089
$346$			354.917i			1.02577i
$347$		−	180.778i		−	0.520973i	−0.965477	−	0.260487i	$-0.916117\pi$
$347$		−	180.778i		−	0.520973i	0.965477	−	0.260487i	$-0.0838830\pi$
$348$	−75.5104			−0.216984
$349$			280.319i			0.803206i	0.915814	+	0.401603i	$0.131547\pi$
$349$			280.319i			0.803206i	−0.915814	+	0.401603i	$0.868453\pi$
$350$	53.1191			0.151769
$351$			284.041i			0.809233i
$352$			535.049i			1.52003i
$353$	−117.502			−0.332866			−0.166433	−	0.986053i	$-0.553225\pi$
$353$	−117.502			−0.332866			−0.166433	+	0.986053i	$0.553225\pi$
$354$	−171.056			−0.483209
$355$	362.821			1.02203
$356$		−	245.437i		−	0.689431i
$357$	−21.1420			−0.0592212
$358$	−338.843			−0.946488
$359$	−208.289			−0.580193			−0.290096	−	0.956997i	$-0.593687\pi$
$359$	−208.289			−0.580193			−0.290096	+	0.956997i	$0.593687\pi$
$360$	−94.3402			−0.262056
$361$	−592.977			−1.64260
$362$		−	592.975i		−	1.63805i
$363$		−	325.738i		−	0.897351i
$364$		−	35.1416i		−	0.0965428i
$365$	543.958			1.49030
$366$			706.630i			1.93068i
$367$	−444.860			−1.21215			−0.606076	−	0.795407i	$-0.707257\pi$
$367$	−444.860			−1.21215			−0.606076	+	0.795407i	$0.707257\pi$
$368$	−342.226			−0.929961
$369$	72.2563			0.195817
$370$			590.148i			1.59499i
$371$			67.5792i			0.182154i
$372$		−	554.703i		−	1.49114i
$373$			71.8648i			0.192667i	0.995349	+	0.0963335i	$0.0307115\pi$
$373$			71.8648i			0.192667i	−0.995349	+	0.0963335i	$0.969288\pi$
$374$		−	261.430i		−	0.699010i
$375$	17.1570			0.0457521
$376$			88.3938i			0.235090i
$377$		−	122.238i		−	0.324240i
$378$			39.8951i			0.105543i
$379$	146.399			0.386276			0.193138	−	0.981172i	$-0.438133\pi$
$379$	146.399			0.386276			0.193138	+	0.981172i	$0.438133\pi$
$380$	579.732			1.52561
$381$		−	69.4011i		−	0.182155i
$382$	−541.160			−1.41665
$383$		−	698.948i		−	1.82493i	−0.409155	−	0.912465i	$-0.634176\pi$
$383$		−	698.948i		−	1.82493i	0.409155	−	0.912465i	$-0.365824\pi$
$384$	372.632			0.970396
$385$	86.3841			0.224374
$386$			674.978i			1.74865i
$387$	−87.2804	+	144.099i	−0.225531	+	0.372349i
$388$	416.872			1.07441
$389$			500.337i			1.28621i	0.765776	+	0.643107i	$0.222355\pi$
$389$			500.337i			1.28621i	−0.765776	+	0.643107i	$0.777645\pi$
$390$			1013.92i			2.59979i
$391$	121.797			0.311502
$392$		−	165.554i		−	0.422333i
$393$	−69.5633			−0.177006
$394$		−	242.983i		−	0.616708i
$395$			49.4418i			0.125169i
$396$	152.332			0.384676
$397$	−546.245			−1.37593			−0.687966	−	0.725743i	$-0.741496\pi$
$397$	−546.245			−1.37593			−0.687966	+	0.725743i	$0.741496\pi$
$398$	73.8760			0.185618
$399$		−	93.8637i		−	0.235247i
$400$	475.401			1.18850
$401$	393.142			0.980405			0.490203	−	0.871609i	$-0.336923\pi$
$401$	393.142			0.980405			0.490203	+	0.871609i	$0.336923\pi$
$402$	−83.5945			−0.207946
$403$	897.970			2.22821
$404$	−165.569			−0.409825
$405$		−	708.683i		−	1.74983i
$406$		−	17.1691i		−	0.0422883i
$407$			473.246i			1.16277i
$408$	−85.7316			−0.210126
$409$		−	788.894i		−	1.92884i	−0.264382	−	0.964418i	$-0.585168\pi$
$409$		−	788.894i		−	1.92884i	0.264382	−	0.964418i	$-0.414832\pi$
$410$	−334.566			−0.816015
$411$	−269.139			−0.654840
$412$	51.3670			0.124677
$413$		−	15.5784i		−	0.0377202i
$414$			177.185i			0.427983i
$415$			588.784i			1.41876i
$416$		−	571.937i		−	1.37485i
$417$		−	173.668i		−	0.416469i
$418$	1160.67			2.77671
$419$		−	221.714i		−	0.529149i	−0.964365	−	0.264575i	$-0.914768\pi$
$419$		−	221.714i		−	0.529149i	0.964365	−	0.264575i	$-0.0852316\pi$
$420$			57.0410i			0.135812i
$421$		−	253.802i		−	0.602855i	−0.953489	−	0.301427i	$-0.902537\pi$
$421$		−	253.802i		−	0.602855i	0.953489	−	0.301427i	$-0.0974631\pi$
$422$	−58.5442			−0.138730
$423$	101.007			0.238786
$424$			274.037i			0.646313i
$425$	−169.194			−0.398104
$426$			479.652i			1.12594i
$427$	−64.3544			−0.150713
$428$	−405.074			−0.946434
$429$			813.071i			1.89527i
$430$	404.132	−	667.218i	0.939841	−	1.55167i
$431$	455.340			1.05647			0.528237	−	0.849097i	$-0.322853\pi$
$431$	455.340			1.05647			0.528237	+	0.849097i	$0.322853\pi$
$432$			357.050i			0.826504i
$433$		−	611.084i		−	1.41128i	−0.708571	−	0.705640i	$-0.750660\pi$
$433$		−	611.084i		−	1.41128i	0.708571	−	0.705640i	$-0.249340\pi$
$434$	126.125			0.290610
$435$			198.415i			0.456125i
$436$	−116.770			−0.267821
$437$			540.742i			1.23740i
$438$			719.117i			1.64182i
$439$	598.936			1.36432			0.682159	−	0.731204i	$-0.261041\pi$
$439$	598.936			1.36432			0.682159	+	0.731204i	$0.261041\pi$
$440$	350.291			0.796117
$441$	−189.177			−0.428974
$442$			279.454i			0.632248i
$443$	−379.068			−0.855683			−0.427842	−	0.903854i	$-0.640726\pi$
$443$	−379.068			−0.855683			−0.427842	+	0.903854i	$0.640726\pi$
$444$	−312.493			−0.703813
$445$	−644.923			−1.44927
$446$	−679.642			−1.52386
$447$	204.690			0.457919
$448$		−	14.2192i		−	0.0317393i
$449$			260.317i			0.579770i	0.957061	+	0.289885i	$0.0936171\pi$
$449$			260.317i			0.579770i	−0.957061	+	0.289885i	$0.906383\pi$
$450$		−	246.136i		−	0.546968i
$451$	−268.292			−0.594883
$452$			82.5982i			0.182739i
$453$	−479.407			−1.05829
$454$	140.752			0.310027
$455$	−92.3396			−0.202944
$456$		−	380.621i		−	0.834696i
$457$			670.232i			1.46659i	0.679910	+	0.733296i	$0.262019\pi$
$457$			670.232i			1.46659i	−0.679910	+	0.733296i	$0.737981\pi$
$458$		−	490.781i		−	1.07157i
$459$		−	127.073i		−	0.276848i
$460$		−	328.609i		−	0.714367i
$461$	−557.387			−1.20908			−0.604541	−	0.796574i	$-0.706644\pi$
$461$	−557.387			−1.20908			−0.604541	+	0.796574i	$0.706644\pi$
$462$			114.200i			0.247187i
$463$			116.501i			0.251622i	0.992054	+	0.125811i	$0.0401533\pi$
$463$			116.501i			0.251622i	−0.992054	+	0.125811i	$0.959847\pi$
$464$		−	153.658i		−	0.331160i
$465$	−1457.56			−3.13455
$466$	−540.360			−1.15957
$467$		−	465.465i		−	0.996712i	−0.866972	−	0.498356i	$-0.833937\pi$
$467$		−	465.465i		−	0.996712i	0.866972	−	0.498356i	$-0.166063\pi$
$468$	−162.834			−0.347936
$469$		−	7.61313i		−	0.0162327i
$470$	−467.688			−0.995081
$471$	620.947			1.31836
$472$		−	63.1713i		−	0.133837i
$473$	324.078	−	535.049i	0.685154	−	1.13118i
$474$	−65.3623			−0.137895
$475$		−	751.169i		−	1.58141i
$476$			15.7215i			0.0330284i
$477$	313.139			0.656476
$478$			176.148i			0.368511i
$479$	24.7759			0.0517241			0.0258621	−	0.999666i	$-0.491767\pi$
$479$	24.7759			0.0517241			0.0258621	+	0.999666i	$0.491767\pi$
$480$			928.355i			1.93407i
$481$		−	505.873i		−	1.05171i
$482$	984.001			2.04150
$483$	−53.2046			−0.110155
$484$	−242.225			−0.500464
$485$		−	1095.39i		−	2.25854i
$486$	512.235			1.05398
$487$	356.159			0.731333			0.365667	−	0.930746i	$-0.380841\pi$
$487$	356.159			0.731333			0.365667	+	0.930746i	$0.380841\pi$
$488$	−260.960			−0.534754
$489$	875.966			1.79134
$490$	875.942			1.78764
$491$			498.114i			1.01449i	0.861802	+	0.507244i	$0.169336\pi$
$491$			498.114i			1.01449i	−0.861802	+	0.507244i	$0.830664\pi$
$492$		−	177.158i		−	0.360078i
$493$			54.6866i			0.110926i
$494$	−1240.69			−2.51151
$495$		−	400.274i		−	0.808635i
$496$	1128.78			2.27577
$497$	−43.6830			−0.0878933
$498$	−778.376			−1.56300
$499$		−	0.670777i		−	0.00134424i	−1.00000		0.000672122i	$-0.999786\pi$
$499$		−	0.670777i		−	0.00134424i	1.00000		0.000672122i	$-0.000213943\pi$
$500$		−	12.7583i		−	0.0255165i
$501$		−	884.306i		−	1.76508i
$502$			453.995i			0.904372i
$503$			775.171i			1.54109i	0.637383	+	0.770547i	$0.280017\pi$
$503$			775.171i			1.54109i	−0.637383	+	0.770547i	$0.719983\pi$
$504$	11.3584			0.0225365
$505$			435.057i			0.861499i
$506$		−	657.900i		−	1.30020i
$507$		−	261.715i		−	0.516203i
$508$	−51.6078			−0.101590
$509$	825.181			1.62118			0.810591	−	0.585613i	$-0.199146\pi$
$509$	825.181			1.62118			0.810591	+	0.585613i	$0.199146\pi$
$510$		−	453.603i		−	0.889417i
$511$	−65.4915			−0.128163
$512$		−	450.857i		−	0.880580i
$513$	564.165			1.09974
$514$	420.404			0.817908
$515$		−	134.974i		−	0.262086i
$516$	353.303	+	213.994i	0.684696	+	0.414718i
$517$	−375.044			−0.725424
$518$		−	71.0526i		−	0.137167i
$519$			493.825i			0.951493i
$520$	−374.441			−0.720080
$521$			801.382i			1.53816i	0.639151	+	0.769081i	$0.279286\pi$
$521$			801.382i			1.53816i	−0.639151	+	0.769081i	$0.720714\pi$
$522$	−79.5556			−0.152405
$523$		−	274.667i		−	0.525176i	−0.964908	−	0.262588i	$-0.915424\pi$
$523$		−	274.667i		−	0.525176i	0.964908	−	0.262588i	$-0.0845760\pi$
$524$			51.7285i			0.0987185i
$525$	73.9090			0.140779
$526$	−202.677			−0.385318
$527$	−401.731			−0.762298
$528$			1022.06i			1.93572i
$529$	−222.492			−0.420590
$530$	−1449.92			−2.73569
$531$	−72.1852			−0.135942
$532$	−69.7986			−0.131200
$533$	286.789			0.538066
$534$		−	852.592i		−	1.59661i
$535$			1064.39i			1.98952i
$536$		−	30.8716i		−	0.0575963i
$537$	−471.460			−0.877951
$538$			124.470i			0.231356i
$539$	702.428			1.30321
$540$	−342.843			−0.634895
$541$	361.835			0.668826			0.334413	−	0.942427i	$-0.391462\pi$
$541$	361.835			0.668826			0.334413	+	0.942427i	$0.391462\pi$
$542$		−	211.682i		−	0.390558i
$543$		−	825.055i		−	1.51944i
$544$			255.871i			0.470352i
$545$			306.830i			0.562992i
$546$		−	122.074i		−	0.223578i
$547$	−281.855			−0.515273			−0.257637	−	0.966242i	$-0.582944\pi$
$547$	−281.855			−0.515273			−0.257637	+	0.966242i	$0.582944\pi$
$548$			200.137i			0.365213i
$549$			298.196i			0.543163i
$550$			913.918i			1.66167i
$551$	−242.791			−0.440638
$552$	−215.747			−0.390847
$553$		−	5.95269i		−	0.0107644i
$554$	−3.62171			−0.00653739
$555$			821.121i			1.47950i
$556$	−129.142			−0.232270
$557$	227.271			0.408026			0.204013	−	0.978968i	$-0.434601\pi$
$557$	227.271			0.408026			0.204013	+	0.978968i	$0.434601\pi$
$558$		−	584.420i		−	1.04735i
$559$	−346.421	+	571.937i	−0.619715	+	1.02314i
$560$	−116.074			−0.207276
$561$		−	363.749i		−	0.648394i
$562$		−	640.635i		−	1.13992i
$563$	789.251			1.40187			0.700934	−	0.713226i	$-0.252767\pi$
$563$	789.251			1.40187			0.700934	+	0.713226i	$0.252767\pi$
$564$		−	247.649i		−	0.439093i
$565$	217.039			0.384140
$566$			357.879i			0.632295i
$567$			85.3241i			0.150483i
$568$	−177.136			−0.311860
$569$	581.996			1.02284			0.511420	−	0.859331i	$-0.329120\pi$
$569$	581.996			1.02284			0.511420	+	0.859331i	$0.329120\pi$
$570$	2013.85			3.53308
$571$			761.473i			1.33358i	0.745246	+	0.666789i	$0.232332\pi$
$571$			761.473i			1.33358i	−0.745246	+	0.666789i	$0.767668\pi$
$572$	604.613			1.05702
$573$	−752.960			−1.31407
$574$	40.2811			0.0701762
$575$	−425.784			−0.740494
$576$	−65.8871			−0.114387
$577$			570.663i			0.989017i	0.869173	+	0.494509i	$0.164652\pi$
$577$			570.663i			0.989017i	−0.869173	+	0.494509i	$0.835348\pi$
$578$			621.510i			1.07528i
$579$			939.152i			1.62202i
$580$	147.544			0.254387
$581$		−	70.8885i		−	0.122011i
$582$	1448.11			2.48817
$583$	−1162.70			−1.99435
$584$	−265.571			−0.454745
$585$			427.870i			0.731403i
$586$		−	1274.64i		−	2.17515i
$587$		−	188.120i		−	0.320477i	−0.987078	−	0.160238i	$-0.948774\pi$
$587$		−	188.120i		−	0.320477i	0.987078	−	0.160238i	$-0.0512263\pi$
$588$			463.826i			0.788820i
$589$		−	1783.56i		−	3.02811i
$590$	334.237			0.566503
$591$		−	338.082i		−	0.572051i
$592$		−	635.901i		−	1.07416i
$593$		−	527.123i		−	0.888909i	−0.895801	−	0.444455i	$-0.853398\pi$
$593$		−	527.123i		−	0.888909i	0.895801	−	0.444455i	$-0.146602\pi$
$594$	−686.398			−1.15555
$595$	41.3106			0.0694296
$596$		−	152.211i		−	0.255387i
$597$	102.790			0.172177
$598$			703.257i			1.17602i
$599$	−1156.58			−1.93085			−0.965427	−	0.260675i	$-0.916055\pi$
$599$	−1156.58			−1.93085			−0.965427	+	0.260675i	$0.916055\pi$
$600$	299.704			0.499507
$601$		−	119.619i		−	0.199033i	−0.995036	−	0.0995166i	$-0.968270\pi$
$601$		−	119.619i		−	0.199033i	0.995036	−	0.0995166i	$-0.0317297\pi$
$602$	−48.6567	+	80.3317i	−0.0808250	+	0.133441i
$603$	−35.2767			−0.0585020
$604$			356.495i			0.590224i
$605$			636.481i			1.05203i
$606$	−575.148			−0.949090
$607$		−	295.289i		−	0.486472i	−0.969967	−	0.243236i	$-0.921791\pi$
$607$		−	295.289i		−	0.486472i	0.969967	−	0.243236i	$-0.0782090\pi$
$608$	−1135.99			−1.86840
$609$		−	23.8887i		−	0.0392262i
$610$		−	1380.73i		−	2.26349i
$611$	400.901			0.656139
$612$	72.8482			0.119033
$613$	846.563			1.38102			0.690508	−	0.723324i	$-0.257387\pi$
$613$	846.563			1.38102			0.690508	+	0.723324i	$0.257387\pi$
$614$		−	192.522i		−	0.313553i
$615$	−465.510			−0.756926
$616$	−42.1744			−0.0684649
$617$	495.684			0.803377			0.401689	−	0.915776i	$-0.368423\pi$
$617$	495.684			0.803377			0.401689	+	0.915776i	$0.368423\pi$
$618$	178.437			0.288733
$619$	228.258			0.368753			0.184376	−	0.982856i	$-0.440973\pi$
$619$	228.258			0.368753			0.184376	+	0.982856i	$0.440973\pi$
$620$			1083.87i			1.74818i
$621$		−	319.785i		−	0.514952i
$622$		−	55.2358i		−	0.0888035i
$623$	77.6475			0.124635
$624$		−	1092.52i		−	1.75084i
$625$	−641.531			−1.02645
$626$	597.154			0.953920
$627$	1614.93			2.57565
$628$		−	461.747i		−	0.735266i
$629$			226.316i			0.359802i
$630$			60.0968i			0.0953917i
$631$			702.132i			1.11273i	0.830939	+	0.556364i	$0.187804\pi$
$631$			702.132i			1.11273i	−0.830939	+	0.556364i	$0.812196\pi$
$632$		−	24.1384i		−	0.0381937i
$633$	−81.4574			−0.128685
$634$			414.946i			0.654490i
$635$			135.607i			0.213555i
$636$		−	767.756i		−	1.20716i
$637$	−750.855			−1.17874
$638$	295.395			0.463001
$639$			202.412i			0.316764i
$640$	−728.109			−1.13767
$641$			897.002i			1.39938i	0.714447	+	0.699689i	$0.246678\pi$
$641$			897.002i			1.39938i	−0.714447	+	0.699689i	$0.753322\pi$
$642$	−1407.13			−2.19179
$643$	−1054.92			−1.64062			−0.820311	−	0.571918i	$-0.806200\pi$
$643$	−1054.92			−1.64062			−0.820311	+	0.571918i	$0.806200\pi$
$644$			39.5639i			0.0614346i
$645$	562.302	−	928.355i	0.871786	−	1.43931i
$646$	555.054			0.859217
$647$		−	1230.50i		−	1.90185i	−0.309414	−	0.950927i	$-0.600133\pi$
$647$		−	1230.50i		−	1.90185i	0.309414	−	0.950927i	$-0.399867\pi$
$648$			345.993i			0.533940i
$649$	268.028			0.412986
$650$		−	976.926i		−	1.50296i
$651$	175.488			0.269567
$652$		−	651.383i		−	0.999054i
$653$		−	234.774i		−	0.359532i	−0.983709	−	0.179766i	$-0.942466\pi$
$653$		−	234.774i		−	0.359532i	0.983709	−	0.179766i	$-0.0575340\pi$
$654$	−405.632			−0.620232
$655$	135.924			0.207518
$656$	360.505			0.549550
$657$			303.465i			0.461896i
$658$	56.3087			0.0855756
$659$	−493.253			−0.748488			−0.374244	−	0.927330i	$-0.622098\pi$
$659$	−493.253			−0.748488			−0.374244	+	0.927330i	$0.622098\pi$
$660$	−981.395			−1.48696
$661$	429.533			0.649823			0.324911	−	0.945744i	$-0.394666\pi$
$661$	429.533			0.649823			0.324911	+	0.945744i	$0.394666\pi$
$662$	1058.11			1.59835
$663$			388.827i			0.586466i
$664$		−	287.456i		−	0.432916i
$665$			183.406i			0.275799i
$666$	−329.234			−0.494345
$667$			137.621i			0.206329i
$668$	−657.585			−0.984409
$669$	−945.642			−1.41352
$670$	163.341			0.243792
$671$		−	1107.22i		−	1.65011i
$672$		−	111.772i		−	0.166328i
$673$			349.722i			0.519647i	0.965656	+	0.259823i	$0.0836644\pi$
$673$			349.722i			0.519647i	−0.965656	+	0.259823i	$0.916336\pi$
$674$			1231.63i			1.82735i
$675$			444.228i			0.658115i
$676$	−194.616			−0.287893
$677$			788.649i			1.16492i	0.812861	+	0.582458i	$0.197909\pi$
$677$			788.649i			1.16492i	−0.812861	+	0.582458i	$0.802091\pi$
$678$			286.927i			0.423196i
$679$			131.883i			0.194231i
$680$	167.516			0.246348
$681$	195.841			0.287578
$682$			2169.99i			3.18180i
$683$	−94.5265			−0.138399			−0.0691995	−	0.997603i	$-0.522044\pi$
$683$	−94.5265			−0.138399			−0.0691995	+	0.997603i	$0.522044\pi$
$684$			323.423i			0.472841i
$685$	525.888			0.767720
$686$	−212.485			−0.309745
$687$		−	682.864i		−	0.993980i
$688$	−435.464	+	718.946i	−0.632941	+	1.04498i
$689$	1242.87			1.80387
$690$		−	1141.51i		−	1.65436i
$691$		−	942.228i		−	1.36357i	−0.731552	−	0.681786i	$-0.761203\pi$
$691$		−	942.228i		−	1.36357i	0.731552	−	0.681786i	$-0.238797\pi$
$692$	367.217			0.530660
$693$			48.1922i			0.0695415i
$694$	−466.976			−0.672877
$695$			339.340i			0.488259i
$696$		−	96.8699i		−	0.139181i
$697$	−128.303			−0.184079
$698$	724.106			1.03740
$699$	−751.847			−1.07560
$700$		−	54.9600i		−	0.0785142i
$701$	568.235			0.810607			0.405303	−	0.914182i	$-0.367166\pi$
$701$	568.235			0.810607			0.405303	+	0.914182i	$0.367166\pi$
$702$	733.720			1.04519
$703$	−1004.77			−1.42926
$704$	244.643			0.347504
$705$	−650.733			−0.923026
$706$			303.524i			0.429921i
$707$		−	52.3800i		−	0.0740877i
$708$			176.984i			0.249977i
$709$	1211.96			1.70939			0.854696	−	0.519128i	$-0.173743\pi$
$709$	1211.96			1.70939			0.854696	+	0.519128i	$0.173743\pi$
$710$		−	937.222i		−	1.32003i
$711$	−27.5828			−0.0387943
$712$	314.864			0.442225
$713$	−1010.97			−1.41791
$714$			54.6129i			0.0764886i
$715$		−	1588.71i		−	2.22197i
$716$			350.585i			0.489645i
$717$			245.090i			0.341826i
$718$			538.043i			0.749363i
$719$	692.271			0.962825			0.481413	−	0.876494i	$-0.340124\pi$
$719$	692.271			0.962825			0.481413	+	0.876494i	$0.340124\pi$
$720$			537.849i			0.747013i
$721$			16.2507i			0.0225391i
$722$			1531.75i			2.12154i
$723$	1369.12			1.89367
$724$	−613.525			−0.847410
$725$		−	191.176i		−	0.263691i
$726$	−841.431			−1.15900
$727$			441.195i			0.606871i	0.952852	+	0.303435i	$0.0981337\pi$
$727$			441.195i			0.606871i	−0.952852	+	0.303435i	$0.901866\pi$
$728$	45.0820			0.0619258
$729$	−195.486			−0.268156
$730$		−	1405.13i		−	1.92483i
$731$	154.980	−	255.871i	0.212012	−	0.350029i
$732$	731.119			0.998796
$733$			251.039i			0.342481i	0.985229	+	0.171240i	$0.0547775\pi$
$733$			251.039i			0.342481i	−0.985229	+	0.171240i	$0.945222\pi$
$734$			1149.14i			1.56559i
$735$	1218.77			1.65819
$736$			643.911i			0.874879i
$737$	130.985			0.177727
$738$		−	186.649i		−	0.252912i
$739$		−	635.071i		−	0.859365i	−0.902980	−	0.429683i	$-0.858625\pi$
$739$		−	635.071i		−	0.859365i	0.902980	−	0.429683i	$-0.141375\pi$
$740$	610.600			0.825135
$741$	−1726.27			−2.32965
$742$	174.567			0.235266
$743$			899.136i			1.21014i	0.796171	+	0.605071i	$0.206855\pi$
$743$			899.136i			1.21014i	−0.796171	+	0.605071i	$0.793145\pi$
$744$	711.612			0.956467
$745$	−399.956			−0.536854
$746$	185.638			0.248844
$747$	−328.473			−0.439723
$748$	−270.490			−0.361617
$749$		−	128.151i		−	0.171096i
$750$		−	44.3192i		−	0.0590923i
$751$		−	597.422i		−	0.795502i	−0.917493	−	0.397751i	$-0.869791\pi$
$751$		−	597.422i		−	0.795502i	0.917493	−	0.397751i	$-0.130209\pi$
$752$	503.947			0.670143
$753$			631.680i			0.838885i
$754$	−315.760			−0.418780
$755$	936.743			1.24072
$756$	41.2777			0.0546001
$757$			392.757i			0.518834i	0.965765	+	0.259417i	$0.0835304\pi$
$757$			392.757i			0.518834i	−0.965765	+	0.259417i	$0.916470\pi$
$758$		−	378.170i		−	0.498905i
$759$		−	915.390i		−	1.20605i
$760$			743.720i			0.978579i
$761$		−	472.103i		−	0.620371i	−0.950676	−	0.310186i	$-0.899609\pi$
$761$		−	472.103i		−	0.620371i	0.950676	−	0.310186i	$-0.100391\pi$
$762$	−179.274			−0.235267
$763$		−	36.9418i		−	0.0484165i
$764$			559.914i			0.732872i
$765$		−	191.419i		−	0.250221i
$766$	−1805.49			−2.35704
$767$	−286.507			−0.373542
$768$		−	1204.33i		−	1.56814i
$769$	−591.892			−0.769690			−0.384845	−	0.922981i	$-0.625745\pi$
$769$	−591.892			−0.769690			−0.384845	+	0.922981i	$0.625745\pi$
$770$		−	223.143i		−	0.289796i
$771$	584.943			0.758681
$772$	698.369			0.904624
$773$			632.541i			0.818293i	0.912469	+	0.409147i	$0.134174\pi$
$773$			632.541i			0.818293i	−0.912469	+	0.409147i	$0.865826\pi$
$774$	372.230	+	225.458i	0.480917	+	0.291290i
$775$	1404.39			1.81211
$776$			534.792i			0.689164i
$777$		−	98.8614i		−	0.127235i
$778$	1292.45			1.66124
$779$		−	569.624i		−	0.731225i
$780$	1049.05			1.34494
$781$		−	751.569i		−	0.962316i
$782$		−	314.621i		−	0.402328i
$783$	143.583			0.183375
$784$	−943.853			−1.20389
$785$	−1213.31			−1.54562
$786$			179.693i			0.228617i
$787$	26.7235			0.0339561			0.0169781	−	0.999856i	$-0.494595\pi$
$787$	26.7235			0.0339561			0.0169781	+	0.999856i	$0.494595\pi$
$788$	−251.404			−0.319040
$789$	−282.001			−0.357416
$790$	127.716			0.161665
$791$	−26.1311			−0.0330355
$792$			195.422i			0.246745i
$793$			1183.56i			1.49251i
$794$			1411.03i			1.77712i
$795$	−2017.39			−2.53760
$796$		−	76.4363i		−	0.0960255i
$797$	−638.409			−0.801015			−0.400508	−	0.916293i	$-0.631166\pi$
$797$	−638.409			−0.801015			−0.400508	+	0.916293i	$0.631166\pi$
$798$	−242.464			−0.303840
$799$	−179.354			−0.224473
$800$		−	894.486i		−	1.11811i
$801$		−	359.792i		−	0.449178i
$802$		−	1015.55i		−	1.26627i
$803$		−	1126.79i		−	1.40322i
$804$			86.4915i			0.107577i
$805$	103.960			0.129143
$806$		−	2319.59i		−	2.87791i
$807$			173.185i			0.214603i
$808$		−	212.403i		−	0.262876i
$809$	996.366			1.23160			0.615801	−	0.787902i	$-0.288833\pi$
$809$	996.366			1.23160			0.615801	+	0.787902i	$0.288833\pi$
$810$	−1830.64			−2.26004
$811$		−	1348.22i		−	1.66242i	−0.555959	−	0.831209i	$-0.687649\pi$
$811$		−	1348.22i		−	1.66242i	0.555959	−	0.831209i	$-0.312351\pi$
$812$	−17.7641			−0.0218769
$813$		−	294.531i		−	0.362277i
$814$	1222.47			1.50180
$815$	−1711.60			−2.10013
$816$			488.770i			0.598983i
$817$	1135.99	+	688.065i	1.39044	+	0.842184i
$818$	−2037.83			−2.49124
$819$		−	51.5148i		−	0.0628996i
$820$			346.161i			0.422147i
$821$	−1363.54			−1.66083			−0.830417	−	0.557142i	$-0.811898\pi$
$821$	−1363.54			−1.66083			−0.830417	+	0.557142i	$0.811898\pi$
$822$			695.228i			0.845776i
$823$	590.955			0.718050			0.359025	−	0.933328i	$-0.383109\pi$
$823$	590.955			0.718050			0.359025	+	0.933328i	$0.383109\pi$
$824$			65.8972i			0.0799723i
$825$			1271.61i			1.54134i
$826$	−40.2415			−0.0487185
$827$	−1422.73			−1.72035			−0.860175	−	0.509999i	$-0.829646\pi$
$827$	−1422.73			−1.72035			−0.860175	+	0.509999i	$0.829646\pi$
$828$	183.326			0.221408
$829$		−	873.419i		−	1.05358i	−0.849995	−	0.526791i	$-0.823395\pi$
$829$		−	873.419i		−	1.05358i	0.849995	−	0.526791i	$-0.176605\pi$
$830$	1520.92			1.83243
$831$	−5.03919			−0.00606400
$832$	−261.509			−0.314314
$833$	335.915			0.403259
$834$	−448.610			−0.537901
$835$			1727.90i			2.06934i
$836$		−	1200.89i		−	1.43647i
$837$			1054.77i			1.26017i
$838$	−572.720			−0.683437
$839$			1155.58i			1.37732i	0.725082	+	0.688662i	$0.241802\pi$
$839$			1155.58i			1.37732i	−0.725082	+	0.688662i	$0.758198\pi$
$840$	−73.1761			−0.0871144
$841$	779.208			0.926526
$842$	−655.608			−0.778632
$843$		−	891.368i		−	1.05738i
$844$			60.5731i			0.0717691i
$845$			511.381i			0.605184i
$846$		−	260.916i		−	0.308411i
$847$		−	76.6310i		−	0.0904734i
$848$	1562.33			1.84237
$849$			497.947i			0.586509i
$850$			437.054i			0.514181i
$851$			569.533i			0.669252i
$852$	496.275			0.582482
$853$	86.1596			0.101008			0.0505039	−	0.998724i	$-0.483917\pi$
$853$	86.1596			0.101008			0.0505039	+	0.998724i	$0.483917\pi$
$854$			166.237i			0.194657i
$855$	849.842			0.993967
$856$		−	519.656i		−	0.607075i
$857$	−813.017			−0.948678			−0.474339	−	0.880342i	$-0.657313\pi$
$857$	−813.017			−0.948678			−0.474339	+	0.880342i	$0.657313\pi$
$858$	2100.29			2.44788
$859$			282.918i			0.329357i	0.986347	+	0.164678i	$0.0526587\pi$
$859$			282.918i			0.329357i	−0.986347	+	0.164678i	$0.947341\pi$
$860$	−690.341	−	418.137i	−0.802722	−	0.486206i
$861$	56.0464			0.0650946
$862$		−	1176.21i		−	1.36452i
$863$		−	579.879i		−	0.671934i	−0.941874	−	0.335967i	$-0.890937\pi$
$863$		−	579.879i		−	0.671934i	0.941874	−	0.335967i	$-0.109063\pi$
$864$	671.804			0.777551
$865$		−	964.915i		−	1.11551i
$866$	−1578.52			−1.82277
$867$			864.758i			0.997414i
$868$		−	130.496i		−	0.150341i
$869$	102.417			0.117856
$870$	512.535			0.589121
$871$	−140.015			−0.160752
$872$		−	149.801i		−	0.171790i
$873$	611.101			0.700001
$874$	1396.82			1.59819
$875$	4.03625			0.00461286
$876$	744.038			0.849359
$877$	305.763			0.348647			0.174323	−	0.984688i	$-0.444226\pi$
$877$	305.763			0.348647			0.174323	+	0.984688i	$0.444226\pi$
$878$		−	1547.14i		−	1.76212i
$879$		−	1773.51i		−	2.01764i
$880$		−	1997.07i		−	2.26940i
$881$	911.059			1.03412			0.517059	−	0.855950i	$-0.327027\pi$
$881$	911.059			1.03412			0.517059	+	0.855950i	$0.327027\pi$
$882$			488.674i			0.554052i
$883$	396.432			0.448960			0.224480	−	0.974479i	$-0.427932\pi$
$883$	396.432			0.448960			0.224480	+	0.974479i	$0.427932\pi$
$884$	289.138			0.327079
$885$	465.051			0.525482
$886$			979.189i			1.10518i
$887$			266.284i			0.300207i	0.988670	+	0.150104i	$0.0479607\pi$
$887$			266.284i			0.300207i	−0.988670	+	0.150104i	$0.952039\pi$
$888$		−	400.887i		−	0.451450i
$889$		−	16.3268i		−	0.0183654i
$890$			1665.93i			1.87184i
$891$	−1468.01			−1.64759
$892$			703.195i			0.788336i
$893$		−	796.274i		−	0.891685i
$894$		−	528.744i		−	0.591437i
$895$	921.215			1.02929
$896$	87.6629			0.0978381
$897$			978.500i			1.09086i
$898$	672.438			0.748817
$899$		−	453.924i		−	0.504921i
$900$	−254.666			−0.282962
$901$	−556.029			−0.617124
$902$			693.040i			0.768337i
$903$	−67.7000	+	111.772i	−0.0749724	+	0.123779i
$904$	−105.963			−0.117215
$905$			1612.13i			1.78135i
$906$			1238.38i			1.36687i
$907$	−1168.24			−1.28802			−0.644012	−	0.765015i	$-0.722731\pi$
$907$	−1168.24			−1.28802			−0.644012	+	0.765015i	$0.722731\pi$
$908$		−	145.630i		−	0.160386i
$909$	−242.711			−0.267009
$910$			238.527i			0.262118i
$911$			981.585i			1.07748i	0.842472	+	0.538740i	$0.181100\pi$
$911$			981.585i			1.07748i	−0.842472	+	0.538740i	$0.818900\pi$
$912$	−2169.99			−2.37937
$913$	1219.64			1.33586
$914$	1731.31			1.89421
$915$		−	1921.12i		−	2.09959i
$916$	−507.789			−0.554355
$917$	−16.3650			−0.0178462
$918$	−328.249			−0.357570
$919$	−1148.89			−1.25015			−0.625075	−	0.780564i	$-0.714932\pi$
$919$	−1148.89			−1.25015			−0.625075	+	0.780564i	$0.714932\pi$
$920$	421.562			0.458220
$921$		−	267.871i		−	0.290848i
$922$			1439.82i			1.56162i
$923$			803.384i			0.870405i
$924$	118.158			0.127877
$925$		−	791.164i		−	0.855312i
$926$	300.940			0.324989
$927$	75.3000			0.0812298
$928$	−289.114			−0.311546
$929$			870.176i			0.936680i	0.883548	+	0.468340i	$0.155148\pi$
$929$			870.176i			0.936680i	−0.883548	+	0.468340i	$0.844852\pi$
$930$			3765.11i			4.04851i
$931$			1491.36i			1.60189i
$932$			559.086i			0.599878i
$933$		−	76.8541i		−	0.0823731i
$934$	−1202.37			−1.28733
$935$			710.752i			0.760162i
$936$		−	208.895i		−	0.223178i
$937$		−	1301.72i		−	1.38925i	−0.719374	−	0.694623i	$-0.755571\pi$
$937$		−	1301.72i		−	1.38925i	0.719374	−	0.694623i	$-0.244429\pi$
$938$	−19.6659			−0.0209658
$939$	830.870			0.884845
$940$			483.896i			0.514783i
$941$	−539.895			−0.573746			−0.286873	−	0.957969i	$-0.592616\pi$
$941$	−539.895			−0.573746			−0.286873	+	0.957969i	$0.592616\pi$
$942$		−	1604.00i		−	1.70276i
$943$	−322.879			−0.342396
$944$	−360.150			−0.381515
$945$		−	108.463i		−	0.114776i
$946$	−1382.11	−	837.142i	−1.46101	−	0.884928i
$947$	80.2459			0.0847369			0.0423685	−	0.999102i	$-0.486510\pi$
$947$	80.2459			0.0847369			0.0423685	+	0.999102i	$0.486510\pi$
$948$			67.6275i			0.0713370i
$949$			1204.47i			1.26920i
$950$	−1940.38			−2.04251
$951$			577.349i			0.607097i
$952$	−20.1686			−0.0211855
$953$		−	127.129i		−	0.133399i	−0.997773	−	0.0666996i	$-0.978753\pi$
$953$		−	127.129i		−	0.133399i	0.997773	−	0.0666996i	$-0.0212469\pi$
$954$		−	808.885i		−	0.847888i
$955$	1471.26			1.54058
$956$	182.253			0.190641
$957$	411.007			0.429475
$958$		−	63.9998i		−	0.0668057i
$959$	−63.3159			−0.0660229
$960$	424.476			0.442162
$961$	2373.55			2.46988
$962$	−1306.75			−1.35836
$963$	−593.806			−0.616621
$964$		−	1018.10i		−	1.05612i
$965$		−	1835.07i		−	1.90162i
$966$			137.436i			0.142273i
$967$	−487.850			−0.504498			−0.252249	−	0.967662i	$-0.581170\pi$
$967$	−487.850			−0.504498			−0.252249	+	0.967662i	$0.581170\pi$
$968$		−	310.742i		−	0.321015i
$969$	772.293			0.797000
$970$	−2829.56			−2.91707
$971$	706.443			0.727542			0.363771	−	0.931488i	$-0.381489\pi$
$971$	706.443			0.727542			0.363771	+	0.931488i	$0.381489\pi$
$972$		−	529.987i		−	0.545254i
$973$		−	40.8559i		−	0.0419896i
$974$		−	920.013i		−	0.944572i
$975$		−	1359.28i		−	1.39413i
$976$			1487.78i			1.52436i
$977$	1385.87			1.41849			0.709247	−	0.704961i	$-0.249035\pi$
$977$	1385.87			1.41849			0.709247	+	0.704961i	$0.249035\pi$
$978$		−	2262.75i		−	2.31365i
$979$			1335.93i			1.36459i
$980$		−	906.299i		−	0.924795i
$981$	−171.176			−0.174491
$982$	1286.70			1.31029
$983$		−	868.493i		−	0.883513i	−0.897135	−	0.441756i	$-0.854356\pi$
$983$		−	868.493i		−	0.883513i	0.897135	−	0.441756i	$-0.145644\pi$
$984$	227.271			0.230966
$985$			660.600i			0.670660i
$986$	141.264			0.143270
$987$	78.3470			0.0793789
$988$			1283.68i			1.29927i
$989$	390.015	−	643.911i	0.394353	−	0.651073i
$990$	−1033.97			−1.04441
$991$			1678.05i			1.69329i	0.532162	+	0.846643i	$0.321380\pi$
$991$			1678.05i			1.69329i	−0.532162	+	0.846643i	$0.678620\pi$
$992$		−	2123.85i		−	2.14098i
$993$	1472.23			1.48261
$994$			112.840i			0.113521i
$995$	−200.848			−0.201857
$996$			805.352i			0.808586i
$997$			83.7411i			0.0839931i	0.999118	+	0.0419965i	$0.0133718\pi$
$997$			83.7411i			0.0839931i	−0.999118	+	0.0419965i	$0.986628\pi$
$998$	−1.73272			−0.00173619
$999$	594.204			0.594798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.b.b.42.2	✓	6
3.2	odd	2		387.3.b.c.343.5		6
4.3	odd	2		688.3.b.e.257.5		6
43.42	odd	2	inner	43.3.b.b.42.5	yes	6
129.128	even	2		387.3.b.c.343.2		6
172.171	even	2		688.3.b.e.257.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.b.b.42.2	✓	6	1.1	even	1	trivial
43.3.b.b.42.5	yes	6	43.42	odd	2	inner
387.3.b.c.343.2		6	129.128	even	2
387.3.b.c.343.5		6	3.2	odd	2
688.3.b.e.257.2		6	172.171	even	2
688.3.b.e.257.5		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)

\(f(q)\)	\(=\)	\(q-2.58315i q^{2} -3.59415i q^{3} -2.67267 q^{4} +7.02284i q^{5} -9.28424 q^{6} -0.845536i q^{7} -3.42869i q^{8} -3.91793 q^{9} +O(q^{10})\)	\(q-2.58315i q^{2} -3.59415i q^{3} -2.67267 q^{4} +7.02284i q^{5} -9.28424 q^{6} -0.845536i q^{7} -3.42869i q^{8} -3.91793 q^{9} +18.1411 q^{10} +14.5475 q^{11} +9.60599i q^{12} -15.5505 q^{13} -2.18415 q^{14} +25.2412 q^{15} -19.5475 q^{16} +6.95691 q^{17} +10.1206i q^{18} +30.8865i q^{19} -18.7698i q^{20} -3.03899 q^{21} -37.5784i q^{22} +17.5074 q^{23} -12.3232 q^{24} -24.3203 q^{25} +40.1692i q^{26} -18.2657i q^{27} +2.25984i q^{28} +7.86076i q^{29} -65.2017i q^{30} -57.7456 q^{31} +36.7794i q^{32} -52.2860i q^{33} -17.9708i q^{34} +5.93806 q^{35} +10.4713 q^{36} +32.5310i q^{37} +79.7846 q^{38} +55.8907i q^{39} +24.0791 q^{40} -18.4425 q^{41} +7.85016i q^{42} +(22.2772 - 36.7794i) q^{43} -38.8807 q^{44} -27.5150i q^{45} -45.2242i q^{46} -25.7806 q^{47} +70.2567i q^{48} +48.2851 q^{49} +62.8230i q^{50} -25.0042i q^{51} +41.5613 q^{52} -79.9247 q^{53} -47.1832 q^{54} +102.165i q^{55} -2.89908 q^{56} +111.011 q^{57} +20.3055 q^{58} +18.4243 q^{59} -67.4613 q^{60} -76.1107i q^{61} +149.166i q^{62} +3.31275i q^{63} +16.8168 q^{64} -109.208i q^{65} -135.063 q^{66} +9.00391 q^{67} -18.5935 q^{68} -62.9242i q^{69} -15.3389i q^{70} -51.6630i q^{71} +13.4333i q^{72} -77.4556i q^{73} +84.0326 q^{74} +87.4108i q^{75} -82.5496i q^{76} -12.3004i q^{77} +144.374 q^{78} +7.04014 q^{79} -137.279i q^{80} -100.911 q^{81} +47.6397i q^{82} +83.8385 q^{83} +8.12221 q^{84} +48.8573i q^{85} +(-95.0069 - 57.5454i) q^{86} +28.2528 q^{87} -49.8789i q^{88} +91.8322i q^{89} -71.0753 q^{90} +13.1485i q^{91} -46.7915 q^{92} +207.546i q^{93} +66.5953i q^{94} -216.911 q^{95} +132.191 q^{96} -155.976 q^{97} -124.728i q^{98} -56.9961 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10}) \) 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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