LMFDB - Embedded newform 138.3.b.a.91.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,3,Mod(91,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("138.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	138.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:	$3.76022764817$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1358954496.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 $ x^8 + 8x^6 + 20x^4 + 16*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.3
Root			$-0.261052i$ of defining polynomial
Character	$\chi$	$=$	138.91
Dual form			138.3.b.a.91.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -0.918288i q^{5} -2.44949 q^{6} -10.4925i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -0.918288i q^{5} -2.44949 q^{6} -10.4925i q^{7} -2.82843 q^{8} +3.00000 q^{9} +1.29866i q^{10} +1.55678i q^{11} +3.46410 q^{12} +9.98414 q^{13} +14.8387i q^{14} -1.59052i q^{15} +4.00000 q^{16} -6.91346i q^{17} -4.24264 q^{18} -33.2762i q^{19} -1.83658i q^{20} -18.1736i q^{21} -2.20162i q^{22} +(13.6729 + 18.4947i) q^{23} -4.89898 q^{24} +24.1567 q^{25} -14.1197 q^{26} +5.19615 q^{27} -20.9851i q^{28} -7.68857 q^{29} +2.24934i q^{30} -3.45627 q^{31} -5.65685 q^{32} +2.69642i q^{33} +9.77711i q^{34} -9.63516 q^{35} +6.00000 q^{36} +24.0224i q^{37} +47.0597i q^{38} +17.2930 q^{39} +2.59731i q^{40} -46.4390 q^{41} +25.7013i q^{42} +60.6740i q^{43} +3.11356i q^{44} -2.75486i q^{45} +(-19.3364 - 26.1554i) q^{46} -45.1976 q^{47} +6.92820 q^{48} -61.0931 q^{49} -34.1628 q^{50} -11.9745i q^{51} +19.9683 q^{52} +73.3590i q^{53} -7.34847 q^{54} +1.42957 q^{55} +29.6773i q^{56} -57.6361i q^{57} +10.8733 q^{58} +40.1329 q^{59} -3.18104i q^{60} +69.4778i q^{61} +4.88791 q^{62} -31.4776i q^{63} +8.00000 q^{64} -9.16832i q^{65} -3.81332i q^{66} -33.2319i q^{67} -13.8269i q^{68} +(23.6821 + 32.0337i) q^{69} +13.6262 q^{70} +94.1677 q^{71} -8.48528 q^{72} -82.1035 q^{73} -33.9728i q^{74} +41.8407 q^{75} -66.5525i q^{76} +16.3346 q^{77} -24.4561 q^{78} -32.5334i q^{79} -3.67315i q^{80} +9.00000 q^{81} +65.6746 q^{82} -25.3364i q^{83} -36.3472i q^{84} -6.34855 q^{85} -85.8060i q^{86} -13.3170 q^{87} -4.40324i q^{88} -78.8166i q^{89} +3.89597i q^{90} -104.759i q^{91} +(27.3457 + 36.9893i) q^{92} -5.98644 q^{93} +63.9190 q^{94} -30.5572 q^{95} -9.79796 q^{96} +114.288i q^{97} +86.3987 q^{98} +4.67034i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) $ 8 * q + 16 * q^4 + 24 * q^9	$ 8 q + 16 q^{4} + 24 q^{9} + 16 q^{13} + 32 q^{16} + 16 q^{23} + 72 q^{25} + 32 q^{26} - 144 q^{29} - 128 q^{31} - 112 q^{35} + 48 q^{36} + 48 q^{39} - 16 q^{41} - 80 q^{46} - 112 q^{47} + 40 q^{49} - 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} + 80 q^{59} - 96 q^{62} + 64 q^{64} - 72 q^{69} - 144 q^{70} + 32 q^{71} + 64 q^{73} + 48 q^{75} + 224 q^{77} - 144 q^{78} + 72 q^{81} + 48 q^{85} + 96 q^{87} + 32 q^{92} + 192 q^{93} - 16 q^{94} + 112 q^{95} + 224 q^{98}+O(q^{100}) $ 8 * q + 16 * q^4 + 24 * q^9 + 16 * q^13 + 32 * q^16 + 16 * q^23 + 72 * q^25 + 32 * q^26 - 144 * q^29 - 128 * q^31 - 112 * q^35 + 48 * q^36 + 48 * q^39 - 16 * q^41 - 80 * q^46 - 112 * q^47 + 40 * q^49 - 160 * q^50 + 32 * q^52 - 64 * q^55 + 128 * q^58 + 80 * q^59 - 96 * q^62 + 64 * q^64 - 72 * q^69 - 144 * q^70 + 32 * q^71 + 64 * q^73 + 48 * q^75 + 224 * q^77 - 144 * q^78 + 72 * q^81 + 48 * q^85 + 96 * q^87 + 32 * q^92 + 192 * q^93 - 16 * q^94 + 112 * q^95 + 224 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	2.00000			0.500000
$5$		−	0.918288i		−	0.183658i	−0.995775	−	0.0918288i	$-0.970729\pi$
$5$		−	0.918288i		−	0.183658i	0.995775	−	0.0918288i	$-0.0292712\pi$
$6$	−2.44949			−0.408248
$7$		−	10.4925i		−	1.49893i	−0.662043	−	0.749466i	$-0.730310\pi$
$7$		−	10.4925i		−	1.49893i	0.662043	−	0.749466i	$-0.269690\pi$
$8$	−2.82843			−0.353553
$9$	3.00000			0.333333
$10$			1.29866i			0.129866i
$11$			1.55678i			0.141526i	0.997493	+	0.0707628i	$0.0225433\pi$
$11$			1.55678i			0.141526i	−0.997493	+	0.0707628i	$0.977457\pi$
$12$	3.46410			0.288675
$13$	9.98414			0.768011			0.384006	−	0.923331i	$-0.374544\pi$
$13$	9.98414			0.768011			0.384006	+	0.923331i	$0.374544\pi$
$14$			14.8387i			1.05991i
$15$		−	1.59052i		−	0.106035i
$16$	4.00000			0.250000
$17$		−	6.91346i		−	0.406674i	−0.979109	−	0.203337i	$-0.934821\pi$
$17$		−	6.91346i		−	0.406674i	0.979109	−	0.203337i	$-0.0651787\pi$
$18$	−4.24264			−0.235702
$19$		−	33.2762i		−	1.75138i	−0.482873	−	0.875691i	$-0.660407\pi$
$19$		−	33.2762i		−	1.75138i	0.482873	−	0.875691i	$-0.339593\pi$
$20$		−	1.83658i		−	0.0918288i
$21$		−	18.1736i		−	0.865409i
$22$		−	2.20162i		−	0.100074i
$23$	13.6729	+	18.4947i	0.594473	+	0.804116i
$23$	13.6729	+	18.4947i	0.594473	+	0.804116i
$24$	−4.89898			−0.204124
$25$	24.1567			0.966270
$26$	−14.1197			−0.543066
$27$	5.19615			0.192450
$28$		−	20.9851i		−	0.749466i
$29$	−7.68857			−0.265123			−0.132561	−	0.991175i	$-0.542320\pi$
$29$	−7.68857			−0.265123			−0.132561	+	0.991175i	$0.542320\pi$
$30$			2.24934i			0.0749779i
$31$	−3.45627			−0.111493			−0.0557463	−	0.998445i	$-0.517754\pi$
$31$	−3.45627			−0.111493			−0.0557463	+	0.998445i	$0.517754\pi$
$32$	−5.65685			−0.176777
$33$			2.69642i			0.0817098i
$34$			9.77711i			0.287562i
$35$	−9.63516			−0.275290
$36$	6.00000			0.166667
$37$			24.0224i			0.649253i	0.945842	+	0.324627i	$0.105239\pi$
$37$			24.0224i			0.649253i	−0.945842	+	0.324627i	$0.894761\pi$
$38$			47.0597i			1.23841i
$39$	17.2930			0.443411
$40$			2.59731i			0.0649328i
$41$	−46.4390			−1.13266			−0.566329	−	0.824179i	$-0.691637\pi$
$41$	−46.4390			−1.13266			−0.566329	+	0.824179i	$0.691637\pi$
$42$			25.7013i			0.611937i
$43$			60.6740i			1.41102i	0.708699	+	0.705511i	$0.249283\pi$
$43$			60.6740i			1.41102i	−0.708699	+	0.705511i	$0.750717\pi$
$44$			3.11356i			0.0707628i
$45$		−	2.75486i		−	0.0612192i
$46$	−19.3364	−	26.1554i	−0.420356	−	0.568596i
$47$	−45.1976			−0.961651			−0.480825	−	0.876816i	$-0.659663\pi$
$47$	−45.1976			−0.961651			−0.480825	+	0.876816i	$0.659663\pi$
$48$	6.92820			0.144338
$49$	−61.0931			−1.24680
$50$	−34.1628			−0.683256
$51$		−	11.9745i		−	0.234793i
$52$	19.9683			0.384006
$53$			73.3590i			1.38413i	0.721834	+	0.692066i	$0.243299\pi$
$53$			73.3590i			1.38413i	−0.721834	+	0.692066i	$0.756701\pi$
$54$	−7.34847			−0.136083
$55$	1.42957			0.0259922
$56$			29.6773i			0.529953i
$57$		−	57.6361i		−	1.01116i
$58$	10.8733			0.187470
$59$	40.1329			0.680218			0.340109	−	0.940386i	$-0.389536\pi$
$59$	40.1329			0.680218			0.340109	+	0.940386i	$0.389536\pi$
$60$		−	3.18104i		−	0.0530174i
$61$			69.4778i			1.13898i	0.821998	+	0.569490i	$0.192859\pi$
$61$			69.4778i			1.13898i	−0.821998	+	0.569490i	$0.807141\pi$
$62$	4.88791			0.0788372
$63$		−	31.4776i		−	0.499644i
$64$	8.00000			0.125000
$65$		−	9.16832i		−	0.141051i
$66$		−	3.81332i		−	0.0577775i
$67$		−	33.2319i		−	0.495999i	−0.968760	−	0.247999i	$-0.920227\pi$
$67$		−	33.2319i		−	0.495999i	0.968760	−	0.247999i	$-0.0797731\pi$
$68$		−	13.8269i		−	0.203337i
$69$	23.6821	+	32.0337i	0.343219	+	0.464256i
$70$	13.6262			0.194660
$71$	94.1677			1.32631			0.663153	−	0.748484i	$-0.269218\pi$
$71$	94.1677			1.32631			0.663153	+	0.748484i	$0.269218\pi$
$72$	−8.48528			−0.117851
$73$	−82.1035			−1.12471			−0.562353	−	0.826897i	$-0.690104\pi$
$73$	−82.1035			−1.12471			−0.562353	+	0.826897i	$0.690104\pi$
$74$		−	33.9728i		−	0.459092i
$75$	41.8407			0.557876
$76$		−	66.5525i		−	0.875691i
$77$	16.3346			0.212137
$78$	−24.4561			−0.313539
$79$		−	32.5334i		−	0.411815i	−0.978571	−	0.205908i	$-0.933985\pi$
$79$		−	32.5334i		−	0.411815i	0.978571	−	0.205908i	$-0.0660146\pi$
$80$		−	3.67315i		−	0.0459144i
$81$	9.00000			0.111111
$82$	65.6746			0.800910
$83$		−	25.3364i		−	0.305258i	−0.988284	−	0.152629i	$-0.951226\pi$
$83$		−	25.3364i		−	0.305258i	0.988284	−	0.152629i	$-0.0487739\pi$
$84$		−	36.3472i		−	0.432704i
$85$	−6.34855			−0.0746888
$86$		−	85.8060i		−	0.997744i
$87$	−13.3170			−0.153069
$88$		−	4.40324i		−	0.0500368i
$89$		−	78.8166i		−	0.885580i	−0.896625	−	0.442790i	$-0.853989\pi$
$89$		−	78.8166i		−	0.885580i	0.896625	−	0.442790i	$-0.146011\pi$
$90$			3.89597i			0.0432885i
$91$		−	104.759i		−	1.15120i
$92$	27.3457	+	36.9893i	0.297236	+	0.402058i
$93$	−5.98644			−0.0643703
$94$	63.9190			0.679990
$95$	−30.5572			−0.321654
$96$	−9.79796			−0.102062
$97$			114.288i			1.17823i	0.808050	+	0.589114i	$0.200523\pi$
$97$			114.288i			1.17823i	−0.808050	+	0.589114i	$0.799477\pi$
$98$	86.3987			0.881619
$99$			4.67034i			0.0471752i
$100$	48.3135			0.483135
$101$	23.5665			0.233332			0.116666	−	0.993171i	$-0.462779\pi$
$101$	23.5665			0.233332			0.116666	+	0.993171i	$0.462779\pi$
$102$			16.9345i			0.166024i
$103$			56.2801i			0.546409i	0.961956	+	0.273204i	$0.0880835\pi$
$103$			56.2801i			0.546409i	−0.961956	+	0.273204i	$0.911917\pi$
$104$	−28.2394			−0.271533
$105$	−16.6886			−0.158939
$106$		−	103.745i		−	0.978729i
$107$		−	208.674i		−	1.95022i	−0.221713	−	0.975112i	$-0.571165\pi$
$107$		−	208.674i		−	1.95022i	0.221713	−	0.975112i	$-0.428835\pi$
$108$	10.3923			0.0962250
$109$			181.768i			1.66760i	0.552070	+	0.833798i	$0.313838\pi$
$109$			181.768i			1.66760i	−0.552070	+	0.833798i	$0.686162\pi$
$110$	−2.02172			−0.0183793
$111$			41.6080i			0.374847i
$112$		−	41.9701i		−	0.374733i
$113$			51.9427i			0.459670i	0.973230	+	0.229835i	$0.0738187\pi$
$113$			51.9427i			0.459670i	−0.973230	+	0.229835i	$0.926181\pi$
$114$			81.5098i			0.714998i
$115$	16.9834	−	12.5556i	0.147682	−	0.109179i
$116$	−15.3771			−0.132561
$117$	29.9524			0.256004
$118$	−56.7565			−0.480987
$119$	−72.5397			−0.609577
$120$			4.49867i			0.0374889i
$121$	118.576			0.979971
$122$		−	98.2565i		−	0.805381i
$123$	−80.4347			−0.653940
$124$	−6.91254			−0.0557463
$125$		−	45.1400i		−	0.361120i
$126$			44.5160i			0.353302i
$127$	−61.0805			−0.480949			−0.240474	−	0.970656i	$-0.577303\pi$
$127$	−61.0805			−0.480949			−0.240474	+	0.970656i	$0.577303\pi$
$128$	−11.3137			−0.0883883
$129$			105.090i			0.814654i
$130$			12.9660i			0.0997381i
$131$	166.079			1.26778			0.633888	−	0.773425i	$-0.281458\pi$
$131$	166.079			1.26778			0.633888	+	0.773425i	$0.281458\pi$
$132$			5.39285i			0.0408549i
$133$	−349.152			−2.62520
$134$			46.9970i			0.350724i
$135$		−	4.77156i		−	0.0353449i
$136$			19.5542i			0.143781i
$137$			147.259i			1.07489i	0.843300	+	0.537443i	$0.180610\pi$
$137$			147.259i			1.07489i	−0.843300	+	0.537443i	$0.819390\pi$
$138$	−33.4916	−	45.3025i	−0.242693	−	0.328279i
$139$	117.615			0.846153			0.423076	−	0.906094i	$-0.360950\pi$
$139$	117.615			0.846153			0.423076	+	0.906094i	$0.360950\pi$
$140$	−19.2703			−0.137645
$141$	−78.2845			−0.555209
$142$	−133.173			−0.937840
$143$			15.5431i			0.108693i
$144$	12.0000			0.0833333
$145$			7.06032i			0.0486918i
$146$	116.112			0.795287
$147$	−105.816			−0.719839
$148$			48.0448i			0.324627i
$149$		−	47.3519i		−	0.317798i	−0.987295	−	0.158899i	$-0.949206\pi$
$149$		−	47.3519i		−	0.317798i	0.987295	−	0.158899i	$-0.0507944\pi$
$150$	−59.1717			−0.394478
$151$	41.4878			0.274754			0.137377	−	0.990519i	$-0.456133\pi$
$151$	41.4878			0.274754			0.137377	+	0.990519i	$0.456133\pi$
$152$			94.1194i			0.619207i
$153$		−	20.7404i		−	0.135558i
$154$	−23.1006			−0.150004
$155$			3.17385i			0.0204765i
$156$	34.5861			0.221706
$157$			4.69030i			0.0298745i	0.999888	+	0.0149373i	$0.00475485\pi$
$157$			4.69030i			0.0298745i	−0.999888	+	0.0149373i	$0.995245\pi$
$158$			46.0092i			0.291197i
$159$			127.061i			0.799129i
$160$			5.19462i			0.0324664i
$161$	194.056	−	143.463i	1.20532	−	0.891074i
$162$	−12.7279			−0.0785674
$163$	291.064			1.78567			0.892834	−	0.450386i	$-0.148713\pi$
$163$	291.064			1.78567			0.892834	+	0.450386i	$0.148713\pi$
$164$	−92.8780			−0.566329
$165$	2.47609			0.0150066
$166$			35.8311i			0.215850i
$167$	169.155			1.01290			0.506452	−	0.862268i	$-0.330957\pi$
$167$	169.155			1.01290			0.506452	+	0.862268i	$0.330957\pi$
$168$			51.4027i			0.305968i
$169$	−69.3169			−0.410159
$170$	8.97820			0.0528129
$171$		−	99.8287i		−	0.583794i
$172$			121.348i			0.705511i
$173$	−244.417			−1.41281			−0.706407	−	0.707806i	$-0.749685\pi$
$173$	−244.417			−1.41281			−0.706407	+	0.707806i	$0.749685\pi$
$174$	18.8331			0.108236
$175$		−	253.465i		−	1.44837i
$176$			6.22712i			0.0353814i
$177$	69.5122			0.392724
$178$			111.463i			0.626199i
$179$	−320.939			−1.79295			−0.896477	−	0.443090i	$-0.853882\pi$
$179$	−320.939			−1.79295			−0.896477	+	0.443090i	$0.853882\pi$
$180$		−	5.50973i		−	0.0306096i
$181$		−	100.890i		−	0.557401i	−0.960378	−	0.278700i	$-0.910096\pi$
$181$		−	100.890i		−	0.557401i	0.960378	−	0.278700i	$-0.0899037\pi$
$182$			148.151i			0.814019i
$183$			120.339i			0.657591i
$184$	−38.6727	−	52.3108i	−0.210178	−	0.284298i
$185$	22.0595			0.119240
$186$	8.46610			0.0455167
$187$	10.7627			0.0575548
$188$	−90.3952			−0.480825
$189$		−	54.5208i		−	0.288470i
$190$	43.2144			0.227444
$191$		−	40.7431i		−	0.213315i	−0.994296	−	0.106657i	$-0.965985\pi$
$191$		−	40.7431i		−	0.213315i	0.994296	−	0.106657i	$-0.0340148\pi$
$192$	13.8564			0.0721688
$193$	−147.447			−0.763976			−0.381988	−	0.924167i	$-0.624761\pi$
$193$	−147.447			−0.763976			−0.381988	+	0.924167i	$0.624761\pi$
$194$		−	161.628i		−	0.833133i
$195$		−	15.8800i		−	0.0814359i
$196$	−122.186			−0.623399
$197$	108.177			0.549123			0.274562	−	0.961570i	$-0.411467\pi$
$197$	108.177			0.549123			0.274562	+	0.961570i	$0.411467\pi$
$198$		−	6.60486i		−	0.0333579i
$199$			93.6416i			0.470561i	0.971928	+	0.235280i	$0.0756008\pi$
$199$			93.6416i			0.470561i	−0.971928	+	0.235280i	$0.924399\pi$
$200$	−68.3256			−0.341628
$201$		−	57.5594i		−	0.286365i
$202$	−33.3281			−0.164990
$203$			80.6725i			0.397401i
$204$		−	23.9489i		−	0.117397i
$205$			42.6443i			0.208021i
$206$		−	79.5920i		−	0.386369i
$207$	41.0186	+	55.4840i	0.198158	+	0.268039i
$208$	39.9366			0.192003
$209$	51.8038			0.247865
$210$	23.6012			0.112387
$211$	343.986			1.63027			0.815133	−	0.579274i	$-0.196664\pi$
$211$	343.986			1.63027			0.815133	+	0.579274i	$0.196664\pi$
$212$			146.718i			0.692066i
$213$	163.103			0.765743
$214$			295.110i			1.37902i
$215$	55.7162			0.259145
$216$	−14.6969			−0.0680414
$217$			36.2650i			0.167120i
$218$		−	257.059i		−	1.17917i
$219$	−142.207			−0.649349
$220$	2.85915			0.0129961
$221$		−	69.0250i		−	0.312330i
$222$		−	58.8426i		−	0.265057i
$223$	−203.749			−0.913671			−0.456835	−	0.889551i	$-0.651017\pi$
$223$	−203.749			−0.913671			−0.456835	+	0.889551i	$0.651017\pi$
$224$			59.3547i			0.264976i
$225$	72.4702			0.322090
$226$		−	73.4581i		−	0.325036i
$227$			236.903i			1.04362i	0.853060	+	0.521812i	$0.174744\pi$
$227$			236.903i			1.04362i	−0.853060	+	0.521812i	$0.825256\pi$
$228$		−	115.272i		−	0.505580i
$229$		−	176.802i		−	0.772060i	−0.922486	−	0.386030i	$-0.873846\pi$
$229$		−	176.802i		−	0.772060i	0.922486	−	0.386030i	$-0.126154\pi$
$230$	−24.0182	+	17.7563i	−0.104427	+	0.0772015i
$231$	28.2923			0.122477
$232$	21.7466			0.0937351
$233$	336.747			1.44527			0.722634	−	0.691231i	$-0.242931\pi$
$233$	336.747			1.44527			0.722634	+	0.691231i	$0.242931\pi$
$234$	−42.3591			−0.181022
$235$			41.5044i			0.176614i
$236$	80.2657			0.340109
$237$		−	56.3495i		−	0.237762i
$238$	102.587			0.431036
$239$	−314.540			−1.31607			−0.658034	−	0.752988i	$-0.728612\pi$
$239$	−314.540			−1.31607			−0.658034	+	0.752988i	$0.728612\pi$
$240$		−	6.36208i		−	0.0265087i
$241$		−	53.5196i		−	0.222073i	−0.993816	−	0.111037i	$-0.964583\pi$
$241$		−	53.5196i		−	0.222073i	0.993816	−	0.111037i	$-0.0354171\pi$
$242$	−167.692			−0.692944
$243$	15.5885			0.0641500
$244$			138.956i			0.569490i
$245$			56.1011i			0.228984i
$246$	113.752			0.462406
$247$		−	332.235i		−	1.34508i
$248$	9.77581			0.0394186
$249$		−	43.8839i		−	0.176241i
$250$			63.8377i			0.255351i
$251$		−	422.499i		−	1.68326i	−0.540054	−	0.841630i	$-0.681596\pi$
$251$		−	422.499i		−	1.68326i	0.540054	−	0.841630i	$-0.318404\pi$
$252$		−	62.9552i		−	0.249822i
$253$	−28.7921	+	21.2857i	−0.113803	+	0.0841331i
$254$	86.3808			0.340082
$255$	−10.9960			−0.0431216
$256$	16.0000			0.0625000
$257$	−226.197			−0.880145			−0.440072	−	0.897962i	$-0.645047\pi$
$257$	−226.197			−0.880145			−0.440072	+	0.897962i	$0.645047\pi$
$258$		−	148.620i		−	0.576048i
$259$	252.055			0.973187
$260$		−	18.3366i		−	0.0705255i
$261$	−23.0657			−0.0883743
$262$	−234.871			−0.896453
$263$			513.378i			1.95201i	0.217757	+	0.976003i	$0.430126\pi$
$263$			513.378i			1.95201i	−0.217757	+	0.976003i	$0.569874\pi$
$264$		−	7.62664i		−	0.0288888i
$265$	67.3647			0.254206
$266$	493.775			1.85630
$267$		−	136.514i		−	0.511290i
$268$		−	66.4638i		−	0.247999i
$269$	100.582			0.373911			0.186956	−	0.982368i	$-0.440138\pi$
$269$	100.582			0.373911			0.186956	+	0.982368i	$0.440138\pi$
$270$			6.74801i			0.0249926i
$271$	−355.369			−1.31132			−0.655662	−	0.755055i	$-0.727610\pi$
$271$	−355.369			−1.31132			−0.655662	+	0.755055i	$0.727610\pi$
$272$		−	27.6538i		−	0.101669i
$273$		−	181.448i		−	0.664644i
$274$		−	208.256i		−	0.760060i
$275$			37.6068i			0.136752i
$276$	47.3642	+	64.0674i	0.171610	+	0.232128i
$277$	67.3195			0.243031			0.121515	−	0.992590i	$-0.461225\pi$
$277$	67.3195			0.243031			0.121515	+	0.992590i	$0.461225\pi$
$278$	−166.333			−0.598320
$279$	−10.3688			−0.0371642
$280$	27.2523			0.0973298
$281$		−	398.789i		−	1.41918i	−0.704616	−	0.709588i	$-0.748881\pi$
$281$		−	398.789i		−	1.41918i	0.704616	−	0.709588i	$-0.251119\pi$
$282$	110.711			0.392592
$283$		−	410.969i		−	1.45219i	−0.687596	−	0.726094i	$-0.741334\pi$
$283$		−	410.969i		−	1.45219i	0.687596	−	0.726094i	$-0.258666\pi$
$284$	188.335			0.663153
$285$	−52.9266			−0.185707
$286$		−	21.9813i		−	0.0768577i
$287$			487.262i			1.69778i
$288$	−16.9706			−0.0589256
$289$	241.204			0.834616
$290$		−	9.98480i		−	0.0344303i
$291$			197.953i			0.680250i
$292$	−164.207			−0.562353
$293$			15.6841i			0.0535293i	0.999642	+	0.0267647i	$0.00852048\pi$
$293$			15.6841i			0.0535293i	−0.999642	+	0.0267647i	$0.991480\pi$
$294$	149.647			0.509003
$295$		−	36.8535i		−	0.124927i
$296$		−	67.9455i		−	0.229546i
$297$			8.08927i			0.0272366i
$298$			66.9657i			0.224717i
$299$	136.512	+	184.653i	0.456562	+	0.617570i
$300$	83.6814			0.278938
$301$	636.623			2.11503
$302$	−58.6726			−0.194280
$303$	40.8184			0.134714
$304$		−	133.105i		−	0.437845i
$305$	63.8006			0.209182
$306$			29.3313i			0.0958540i
$307$	−33.6253			−0.109529			−0.0547643	−	0.998499i	$-0.517441\pi$
$307$	−33.6253			−0.109529			−0.0547643	+	0.998499i	$0.517441\pi$
$308$	32.6691			0.106069
$309$			97.4800i			0.315469i
$310$		−	4.48850i		−	0.0144790i
$311$	−573.735			−1.84481			−0.922403	−	0.386229i	$-0.873777\pi$
$311$	−573.735			−1.84481			−0.922403	+	0.386229i	$0.873777\pi$
$312$	−48.9121			−0.156770
$313$			524.876i			1.67692i	0.544963	+	0.838460i	$0.316544\pi$
$313$			524.876i			1.67692i	−0.544963	+	0.838460i	$0.683456\pi$
$314$		−	6.63309i		−	0.0211245i
$315$	−28.9055			−0.0917634
$316$		−	65.0668i		−	0.205908i
$317$	−456.839			−1.44113			−0.720567	−	0.693386i	$-0.756118\pi$
$317$	−456.839			−1.44113			−0.720567	+	0.693386i	$0.756118\pi$
$318$		−	179.692i		−	0.565069i
$319$		−	11.9694i		−	0.0375217i
$320$		−	7.34630i		−	0.0229572i
$321$		−	361.434i		−	1.12596i
$322$	−274.436	+	202.887i	−0.852286	+	0.630085i
$323$	−230.054			−0.712241
$324$	18.0000			0.0555556
$325$	241.184			0.742106
$326$	−411.627			−1.26266
$327$			314.831i			0.962787i
$328$	131.349			0.400455
$329$			474.237i			1.44145i
$330$	−3.50172			−0.0106113
$331$	−388.620			−1.17408			−0.587039	−	0.809559i	$-0.699707\pi$
$331$	−388.620			−1.17408			−0.587039	+	0.809559i	$0.699707\pi$
$332$		−	50.6728i		−	0.152629i
$333$			72.0671i			0.216418i
$334$	−239.221			−0.716231
$335$	−30.5165			−0.0910939
$336$		−	72.6944i		−	0.216352i
$337$			3.14404i			0.00932951i	0.999989	+	0.00466475i	$0.00148484\pi$
$337$			3.14404i			0.00932951i	−0.999989	+	0.00466475i	$0.998515\pi$
$338$	98.0289			0.290026
$339$			89.9674i			0.265391i
$340$	−12.6971			−0.0373444
$341$		−	5.38066i		−	0.0157790i
$342$			141.179i			0.412804i
$343$			126.887i			0.369934i
$344$		−	171.612i		−	0.498872i
$345$	29.4162	−	21.7470i	0.0852642	−	0.0630348i
$346$	345.657			0.999010
$347$	−426.628			−1.22948			−0.614738	−	0.788731i	$-0.710738\pi$
$347$	−426.628			−1.22948			−0.614738	+	0.788731i	$0.710738\pi$
$348$	−26.6340			−0.0765344
$349$	155.575			0.445773			0.222887	−	0.974844i	$-0.428452\pi$
$349$	155.575			0.445773			0.222887	+	0.974844i	$0.428452\pi$
$350$			358.454i			1.02415i
$351$	51.8791			0.147804
$352$		−	8.80648i		−	0.0250184i
$353$	−138.608			−0.392658			−0.196329	−	0.980538i	$-0.562902\pi$
$353$	−138.608			−0.392658			−0.196329	+	0.980538i	$0.562902\pi$
$354$	−98.3051			−0.277698
$355$		−	86.4731i		−	0.243586i
$356$		−	157.633i		−	0.442790i
$357$	−125.642			−0.351939
$358$	453.876			1.26781
$359$			399.688i			1.11334i	0.830735	+	0.556668i	$0.187921\pi$
$359$			399.688i			1.11334i	−0.830735	+	0.556668i	$0.812079\pi$
$360$			7.79193i			0.0216443i
$361$	−746.308			−2.06734
$362$			142.679i			0.394142i
$363$	205.380			0.565786
$364$		−	209.518i		−	0.575598i
$365$			75.3947i			0.206561i
$366$		−	170.185i		−	0.464987i
$367$			679.558i			1.85166i	0.377944	+	0.925828i	$0.376631\pi$
$367$			679.558i			1.85166i	−0.377944	+	0.925828i	$0.623369\pi$
$368$	54.6915	+	73.9786i	0.148618	+	0.201029i
$369$	−139.317			−0.377553
$370$	−31.1968			−0.0843156
$371$	769.721			2.07472
$372$	−11.9729			−0.0321851
$373$		−	683.028i		−	1.83118i	−0.402119	−	0.915588i	$-0.631726\pi$
$373$		−	683.028i		−	1.83118i	0.402119	−	0.915588i	$-0.368274\pi$
$374$	−15.2208			−0.0406974
$375$		−	78.1848i		−	0.208493i
$376$	127.838			0.339995
$377$	−76.7638			−0.203617
$378$			77.1040i			0.203979i
$379$		−	409.564i		−	1.08064i	−0.841459	−	0.540322i	$-0.818303\pi$
$379$		−	409.564i		−	1.08064i	0.841459	−	0.540322i	$-0.181697\pi$
$380$	−61.1143			−0.160827
$381$	−105.794			−0.277676
$382$			57.6194i			0.150836i
$383$			579.190i			1.51224i	0.654430	+	0.756122i	$0.272908\pi$
$383$			579.190i			1.51224i	−0.654430	+	0.756122i	$0.727092\pi$
$384$	−19.5959			−0.0510310
$385$		−	14.9998i		−	0.0389606i
$386$	208.522			0.540213
$387$			182.022i			0.470341i
$388$			228.576i			0.589114i
$389$			208.569i			0.536168i	0.963396	+	0.268084i	$0.0863905\pi$
$389$			208.569i			0.536168i	−0.963396	+	0.268084i	$0.913610\pi$
$390$			22.4577i			0.0575838i
$391$	127.862	−	94.5269i	0.327013	−	0.241757i
$392$	172.797			0.440810
$393$	287.657			0.731951
$394$	−152.986			−0.388289
$395$	−29.8750			−0.0756330
$396$			9.34068i			0.0235876i
$397$	−213.354			−0.537416			−0.268708	−	0.963222i	$-0.586597\pi$
$397$	−213.354			−0.537416			−0.268708	+	0.963222i	$0.586597\pi$
$398$		−	132.429i		−	0.332737i
$399$	−604.749			−1.51566
$400$	96.6270			0.241567
$401$		−	420.020i		−	1.04743i	−0.851893	−	0.523715i	$-0.824546\pi$
$401$		−	420.020i		−	1.04743i	0.851893	−	0.523715i	$-0.175454\pi$
$402$			81.4012i			0.202491i
$403$	−34.5079			−0.0856276
$404$	47.1330			0.116666
$405$		−	8.26459i		−	0.0204064i
$406$		−	114.088i		−	0.281005i
$407$	−37.3976			−0.0918859
$408$			33.8689i			0.0830120i
$409$	−298.404			−0.729595			−0.364797	−	0.931087i	$-0.618862\pi$
$409$	−298.404			−0.729595			−0.364797	+	0.931087i	$0.618862\pi$
$410$		−	60.3082i		−	0.147093i
$411$			255.061i			0.620586i
$412$			112.560i			0.273204i
$413$		−	421.095i		−	1.01960i
$414$	−58.0091	−	78.4662i	−0.140119	−	0.189532i
$415$	−23.2661			−0.0560629
$416$	−56.4788			−0.135766
$417$	203.716			0.488527
$418$	−73.2616			−0.175267
$419$		−	32.5936i		−	0.0777891i	−0.999243	−	0.0388945i	$-0.987616\pi$
$419$		−	32.5936i		−	0.0777891i	0.999243	−	0.0388945i	$-0.0123836\pi$
$420$	−33.3772			−0.0794695
$421$			215.582i			0.512071i	0.966667	+	0.256036i	$0.0824164\pi$
$421$			215.582i			0.512071i	−0.966667	+	0.256036i	$0.917584\pi$
$422$	−486.470			−1.15277
$423$	−135.593			−0.320550
$424$		−	207.491i		−	0.489364i
$425$		−	167.007i		−	0.392957i
$426$	−230.663			−0.541462
$427$	728.998			1.70725
$428$		−	417.348i		−	0.975112i
$429$			26.9215i			0.0627540i
$430$	−78.7946			−0.183243
$431$		−	204.018i		−	0.473359i	−0.971588	−	0.236679i	$-0.923941\pi$
$431$		−	204.018i		−	0.473359i	0.971588	−	0.236679i	$-0.0760591\pi$
$432$	20.7846			0.0481125
$433$			391.100i			0.903232i	0.892212	+	0.451616i	$0.149152\pi$
$433$			391.100i			0.903232i	−0.892212	+	0.451616i	$0.850848\pi$
$434$		−	51.2865i		−	0.118172i
$435$			12.2288i			0.0281122i
$436$			363.536i			0.833798i
$437$	615.433	−	454.982i	1.40831	−	1.04115i
$438$	201.112			0.459159
$439$	−13.0370			−0.0296970			−0.0148485	−	0.999890i	$-0.504727\pi$
$439$	−13.0370			−0.0296970			−0.0148485	+	0.999890i	$0.504727\pi$
$440$	−4.04344			−0.00918964
$441$	−183.279			−0.415599
$442$			97.6161i			0.220851i
$443$	318.201			0.718286			0.359143	−	0.933283i	$-0.383069\pi$
$443$	318.201			0.718286			0.359143	+	0.933283i	$0.383069\pi$
$444$			83.2160i			0.187423i
$445$	−72.3763			−0.162643
$446$	288.144			0.646063
$447$		−	82.0159i		−	0.183481i
$448$		−	83.9402i		−	0.187367i
$449$	423.915			0.944132			0.472066	−	0.881563i	$-0.343508\pi$
$449$	423.915			0.944132			0.472066	+	0.881563i	$0.343508\pi$
$450$	−102.488			−0.227752
$451$		−	72.2953i		−	0.160300i
$452$			103.885i			0.229835i
$453$	71.8590			0.158629
$454$		−	335.031i		−	0.737954i
$455$	−96.1988			−0.211426
$456$			163.020i			0.357499i
$457$		−	0.903833i		−	0.00197775i	−1.00000		0.000988876i	$-0.999685\pi$
$457$		−	0.903833i		−	0.00197775i	1.00000		0.000988876i	$-0.000314769\pi$
$458$			250.035i			0.545929i
$459$		−	35.9234i		−	0.0782645i
$460$	33.9668	−	25.1113i	0.0738410	−	0.0545897i
$461$	537.745			1.16648			0.583238	−	0.812301i	$-0.301786\pi$
$461$	537.745			1.16648			0.583238	+	0.812301i	$0.301786\pi$
$462$	−40.0113			−0.0866046
$463$	273.023			0.589683			0.294841	−	0.955546i	$-0.404733\pi$
$463$	273.023			0.589683			0.294841	+	0.955546i	$0.404733\pi$
$464$	−30.7543			−0.0662807
$465$			5.49727i			0.0118221i
$466$	−476.232			−1.02196
$467$		−	595.600i		−	1.27538i	−0.770295	−	0.637688i	$-0.779891\pi$
$467$		−	595.600i		−	1.27538i	0.770295	−	0.637688i	$-0.220109\pi$
$468$	59.9049			0.128002
$469$	−348.687			−0.743469
$470$		−	58.6961i		−	0.124885i
$471$			8.12384i			0.0172481i
$472$	−113.513			−0.240493
$473$	−94.4561			−0.199696
$474$			79.6902i			0.168123i
$475$		−	803.846i		−	1.69231i
$476$	−145.079			−0.304789
$477$			220.077i			0.461377i
$478$	444.827			0.930600
$479$		−	802.514i		−	1.67540i	−0.546134	−	0.837698i	$-0.683901\pi$
$479$		−	802.514i		−	1.67540i	0.546134	−	0.837698i	$-0.316099\pi$
$480$			8.99735i			0.0187445i
$481$			239.843i			0.498634i
$482$			75.6882i			0.157029i
$483$	336.114	−	248.485i	0.695889	−	0.514462i
$484$	237.153			0.489985
$485$	104.949			0.216390
$486$	−22.0454			−0.0453609
$487$	421.109			0.864701			0.432351	−	0.901706i	$-0.357684\pi$
$487$	421.109			0.864701			0.432351	+	0.901706i	$0.357684\pi$
$488$		−	196.513i		−	0.402690i
$489$	504.138			1.03096
$490$		−	79.3389i		−	0.161916i
$491$	162.568			0.331095			0.165548	−	0.986202i	$-0.447061\pi$
$491$	162.568			0.331095			0.165548	+	0.986202i	$0.447061\pi$
$492$	−160.869			−0.326970
$493$			53.1546i			0.107819i
$494$			469.851i			0.951115i
$495$	4.28872			0.00866408
$496$	−13.8251			−0.0278732
$497$		−	988.057i		−	1.98804i
$498$			62.0612i			0.124621i
$499$	−683.170			−1.36908			−0.684539	−	0.728976i	$-0.739997\pi$
$499$	−683.170			−1.36908			−0.684539	+	0.728976i	$0.739997\pi$
$500$		−	90.2801i		−	0.180560i
$501$	292.985			0.584800
$502$			597.503i			1.19025i
$503$		−	710.649i		−	1.41282i	−0.707803	−	0.706410i	$-0.750313\pi$
$503$		−	710.649i		−	1.41282i	0.707803	−	0.706410i	$-0.249687\pi$
$504$			89.0320i			0.176651i
$505$		−	21.6408i		−	0.0428531i
$506$	40.7182	−	30.1025i	0.0804708	−	0.0594911i
$507$	−120.060			−0.236805
$508$	−122.161			−0.240474
$509$	571.826			1.12343			0.561715	−	0.827330i	$-0.310142\pi$
$509$	571.826			1.12343			0.561715	+	0.827330i	$0.310142\pi$
$510$	15.5507			0.0304916
$511$			861.473i			1.68586i
$512$	−22.6274			−0.0441942
$513$		−	172.908i		−	0.337053i
$514$	319.891			0.622356
$515$	51.6813			0.100352
$516$			210.181i			0.407327i
$517$		−	70.3627i		−	0.136098i
$518$	−356.460			−0.688147
$519$	−423.342			−0.815688
$520$			25.9319i			0.0498691i
$521$		−	701.698i		−	1.34683i	−0.739265	−	0.673415i	$-0.764827\pi$
$521$		−	701.698i		−	1.34683i	0.739265	−	0.673415i	$-0.235173\pi$
$522$	32.6198			0.0624901
$523$		−	3.66182i		−	0.00700157i	−0.999994	−	0.00350078i	$-0.998886\pi$
$523$		−	3.66182i		−	0.00700157i	0.999994	−	0.00350078i	$-0.00111434\pi$
$524$	332.157			0.633888
$525$		−	439.015i		−	0.836219i
$526$		−	726.026i		−	1.38028i
$527$			23.8948i			0.0453412i
$528$			10.7857i			0.0204274i
$529$	−155.105	+	505.750i	−0.293204	+	0.956050i
$530$	−95.2680			−0.179751
$531$	120.399			0.226739
$532$	−698.304			−1.31260
$533$	−463.653			−0.869894
$534$			193.060i			0.361536i
$535$	−191.623			−0.358173
$536$			93.9941i			0.175362i
$537$	−555.882			−1.03516
$538$	−142.245			−0.264395
$539$		−	95.1086i		−	0.176454i
$540$		−	9.54313i		−	0.0176725i
$541$	720.807			1.33236			0.666181	−	0.745790i	$-0.267928\pi$
$541$	720.807			1.33236			0.666181	+	0.745790i	$0.267928\pi$
$542$	502.567			0.927246
$543$		−	174.746i		−	0.321816i
$544$			39.1084i			0.0718905i
$545$	166.915			0.306266
$546$			256.606i			0.469974i
$547$	−279.379			−0.510748			−0.255374	−	0.966842i	$-0.582199\pi$
$547$	−279.379			−0.510748			−0.255374	+	0.966842i	$0.582199\pi$
$548$			294.519i			0.537443i
$549$			208.433i			0.379660i
$550$		−	53.1840i		−	0.0966981i
$551$			255.847i			0.464331i
$552$	−66.9831	−	90.6050i	−0.121346	−	0.164139i
$553$	−341.358			−0.617283
$554$	−95.2041			−0.171849
$555$	38.2081			0.0688434
$556$	235.231			0.423076
$557$			946.855i			1.69992i	0.526848	+	0.849960i	$0.323374\pi$
$557$			946.855i			1.69992i	−0.526848	+	0.849960i	$0.676626\pi$
$558$	14.6637			0.0262791
$559$			605.778i			1.08368i
$560$	−38.5406			−0.0688226
$561$	18.6416			0.0332293
$562$			563.972i			1.00351i
$563$			71.3191i			0.126677i	0.997992	+	0.0633385i	$0.0201748\pi$
$563$			71.3191i			0.126677i	−0.997992	+	0.0633385i	$0.979825\pi$
$564$	−156.569			−0.277605
$565$	47.6984			0.0844219
$566$			581.198i			1.02685i
$567$		−	94.4327i		−	0.166548i
$568$	−266.347			−0.468920
$569$			776.277i			1.36428i	0.731220	+	0.682142i	$0.238951\pi$
$569$			776.277i			1.36428i	−0.731220	+	0.682142i	$0.761049\pi$
$570$	74.8495			0.131315
$571$			871.414i			1.52612i	0.646328	+	0.763059i	$0.276304\pi$
$571$			871.414i			1.52612i	−0.646328	+	0.763059i	$0.723696\pi$
$572$			31.0862i			0.0543466i
$573$		−	70.5691i		−	0.123157i
$574$		−	689.093i		−	1.20051i
$575$	330.292	+	446.771i	0.574421	+	0.776993i
$576$	24.0000			0.0416667
$577$	−30.1717			−0.0522907			−0.0261453	−	0.999658i	$-0.508323\pi$
$577$	−30.1717			−0.0522907			−0.0261453	+	0.999658i	$0.508323\pi$
$578$	−341.114			−0.590163
$579$	−255.386			−0.441082
$580$			14.1206i			0.0243459i
$581$	−265.843			−0.457561
$582$		−	279.947i		−	0.481009i
$583$	−114.204			−0.195890
$584$	232.224			0.397644
$585$		−	27.5050i		−	0.0470170i
$586$		−	22.1807i		−	0.0378510i
$587$	691.140			1.17741			0.588706	−	0.808347i	$-0.299638\pi$
$587$	691.140			1.17741			0.588706	+	0.808347i	$0.299638\pi$
$588$	−211.633			−0.359920
$589$			115.012i			0.195266i
$590$			52.1188i			0.0883369i
$591$	187.369			0.317036
$592$			96.0895i			0.162313i
$593$	−220.592			−0.371993			−0.185996	−	0.982550i	$-0.559551\pi$
$593$	−220.592			−0.371993			−0.185996	+	0.982550i	$0.559551\pi$
$594$		−	11.4400i		−	0.0192592i
$595$			66.6123i			0.111953i
$596$		−	94.7037i		−	0.158899i
$597$			162.192i			0.271678i
$598$	−193.057	−	261.139i	−0.322838	−	0.436688i
$599$	−743.322			−1.24094			−0.620469	−	0.784231i	$-0.713058\pi$
$599$	−743.322			−1.24094			−0.620469	+	0.784231i	$0.713058\pi$
$600$	−118.343			−0.197239
$601$	−970.289			−1.61446			−0.807229	−	0.590238i	$-0.799034\pi$
$601$	−970.289			−1.61446			−0.807229	+	0.590238i	$0.799034\pi$
$602$	−900.321			−1.49555
$603$		−	99.6957i		−	0.165333i
$604$	82.9756			0.137377
$605$		−	108.887i		−	0.179979i
$606$	−57.7259			−0.0952573
$607$	−530.990			−0.874777			−0.437388	−	0.899273i	$-0.644097\pi$
$607$	−530.990			−0.874777			−0.437388	+	0.899273i	$0.644097\pi$
$608$			188.239i			0.309603i
$609$			139.729i			0.229440i
$610$	−90.2277			−0.147914
$611$	−451.259			−0.738558
$612$		−	41.4808i		−	0.0677790i
$613$			99.5936i			0.162469i	0.996695	+	0.0812346i	$0.0258863\pi$
$613$			99.5936i			0.162469i	−0.996695	+	0.0812346i	$0.974114\pi$
$614$	47.5533			0.0774484
$615$			73.8622i			0.120101i
$616$	−46.2011			−0.0750018
$617$			520.814i			0.844107i	0.906571	+	0.422053i	$0.138691\pi$
$617$			520.814i			0.844107i	−0.906571	+	0.422053i	$0.861309\pi$
$618$		−	137.857i		−	0.223070i
$619$		−	195.756i		−	0.316245i	−0.987419	−	0.158123i	$-0.949456\pi$
$619$		−	195.756i		−	0.316245i	0.987419	−	0.158123i	$-0.0505441\pi$
$620$			6.34770i			0.0102382i
$621$	71.0463	+	96.1011i	0.114406	+	0.154752i
$622$	811.383			1.30447
$623$	−826.985			−1.32742
$624$	69.1722			0.110853
$625$	562.467			0.899947
$626$		−	742.287i		−	1.18576i
$627$	89.7268			0.143105
$628$			9.38060i			0.0149373i
$629$	166.078			0.264035
$630$	40.8785			0.0648865
$631$			90.4647i			0.143367i	0.997427	+	0.0716836i	$0.0228372\pi$
$631$			90.4647i			0.143367i	−0.997427	+	0.0716836i	$0.977163\pi$
$632$			92.0184i			0.145599i
$633$	595.802			0.941235
$634$	646.068			1.01904
$635$			56.0895i			0.0883298i
$636$			254.123i			0.399564i
$637$	−609.962			−0.957555
$638$			16.9273i			0.0265318i
$639$	282.503			0.442102
$640$			10.3892i			0.0162332i
$641$			100.463i			0.156728i	0.996925	+	0.0783640i	$0.0249696\pi$
$641$			100.463i			0.156728i	−0.996925	+	0.0783640i	$0.975030\pi$
$642$			511.145i			0.796176i
$643$		−	502.742i		−	0.781870i	−0.920418	−	0.390935i	$-0.872152\pi$
$643$		−	502.742i		−	0.781870i	0.920418	−	0.390935i	$-0.127848\pi$
$644$	388.111	−	286.926i	0.602658	−	0.445537i
$645$	96.5033			0.149617
$646$	325.345			0.503631
$647$	806.509			1.24654			0.623268	−	0.782008i	$-0.285805\pi$
$647$	806.509			1.24654			0.623268	+	0.782008i	$0.285805\pi$
$648$	−25.4558			−0.0392837
$649$			62.4781i			0.0962682i
$650$	−341.086			−0.524748
$651$			62.8129i			0.0964867i
$652$	582.128			0.892834
$653$	−696.968			−1.06733			−0.533666	−	0.845695i	$-0.679186\pi$
$653$	−696.968			−1.06733			−0.533666	+	0.845695i	$0.679186\pi$
$654$		−	445.239i		−	0.680793i
$655$		−	152.508i		−	0.232837i
$656$	−185.756			−0.283164
$657$	−246.311			−0.374902
$658$		−	670.672i		−	1.01926i
$659$			11.1238i			0.0168798i	0.999964	+	0.00843991i	$0.00268654\pi$
$659$			11.1238i			0.0168798i	−0.999964	+	0.00843991i	$0.997313\pi$
$660$	4.95218			0.00750331
$661$		−	48.1913i		−	0.0729067i	−0.999335	−	0.0364533i	$-0.988394\pi$
$661$		−	48.1913i		−	0.0729067i	0.999335	−	0.0364533i	$-0.0116060\pi$
$662$	549.592			0.830199
$663$		−	119.555i		−	0.180324i
$664$			71.6621i			0.107925i
$665$			320.622i			0.482138i
$666$		−	101.918i		−	0.153031i
$667$	−105.125	−	142.197i	−0.157608	−	0.213190i
$668$	338.310			0.506452
$669$	−352.903			−0.527508
$670$	43.1568			0.0644131
$671$	−108.162			−0.161195
$672$			102.805i			0.152984i
$673$	−430.700			−0.639971			−0.319985	−	0.947422i	$-0.603678\pi$
$673$	−430.700			−0.639971			−0.319985	+	0.947422i	$0.603678\pi$
$674$		−	4.44635i		−	0.00659696i
$675$	125.522			0.185959
$676$	−138.634			−0.205080
$677$		−	481.634i		−	0.711424i	−0.934596	−	0.355712i	$-0.884238\pi$
$677$		−	481.634i		−	0.711424i	0.934596	−	0.355712i	$-0.115762\pi$
$678$		−	127.233i		−	0.187660i
$679$	1199.17			1.76608
$680$	17.9564			0.0264065
$681$			410.328i			0.602537i
$682$			7.60940i			0.0111575i
$683$	140.267			0.205370			0.102685	−	0.994714i	$-0.467257\pi$
$683$	140.267			0.205370			0.102685	+	0.994714i	$0.467257\pi$
$684$		−	199.657i		−	0.291897i
$685$	135.227			0.197411
$686$		−	179.446i		−	0.261583i
$687$		−	306.230i		−	0.445749i
$688$			242.696i			0.352756i
$689$			732.427i			1.06303i
$690$	−41.6007	+	30.7549i	−0.0602909	+	0.0445723i
$691$	−681.676			−0.986507			−0.493253	−	0.869886i	$-0.664192\pi$
$691$	−681.676			−0.986507			−0.493253	+	0.869886i	$0.664192\pi$
$692$	−488.833			−0.706407
$693$	49.0037			0.0707124
$694$	603.344			0.869371
$695$		−	108.005i		−	0.155402i
$696$	37.6661			0.0541180
$697$			321.054i			0.460623i
$698$	−220.016			−0.315209
$699$	583.263			0.834425
$700$		−	506.931i		−	0.724187i
$701$			38.3055i			0.0546441i	0.999627	+	0.0273221i	$0.00869797\pi$
$701$			38.3055i			0.0546441i	−0.999627	+	0.0273221i	$0.991302\pi$
$702$	−73.3682			−0.104513
$703$	799.374			1.13709
$704$			12.4542i			0.0176907i
$705$			71.8877i			0.101968i
$706$	196.022			0.277651
$707$		−	247.272i		−	0.349748i
$708$	139.024			0.196362
$709$		−	1370.51i		−	1.93301i	−0.256645	−	0.966506i	$-0.582617\pi$
$709$		−	1370.51i		−	1.93301i	0.256645	−	0.966506i	$-0.417383\pi$
$710$			122.291i			0.172241i
$711$		−	97.6002i		−	0.137272i
$712$			222.927i			0.313100i
$713$	−47.2572	−	63.9226i	−0.0662793	−	0.0896530i
$714$	177.685			0.248859
$715$	14.2731			0.0199623
$716$	−641.878			−0.896477
$717$	−544.800			−0.759832
$718$		−	565.244i		−	0.787248i
$719$	−214.779			−0.298718			−0.149359	−	0.988783i	$-0.547721\pi$
$719$	−214.779			−0.298718			−0.149359	+	0.988783i	$0.547721\pi$
$720$		−	11.0195i		−	0.0153048i
$721$	590.520			0.819029
$722$	1055.44			1.46183
$723$		−	92.6987i		−	0.128214i
$724$		−	201.779i		−	0.278700i
$725$	−185.731			−0.256180
$726$	−290.452			−0.400071
$727$		−	277.261i		−	0.381377i	−0.981651	−	0.190689i	$-0.938928\pi$
$727$		−	277.261i		−	0.381377i	0.981651	−	0.190689i	$-0.0610721\pi$
$728$			296.303i			0.407009i
$729$	27.0000			0.0370370
$730$		−	106.624i		−	0.146060i
$731$	419.467			0.573827
$732$			240.678i			0.328795i
$733$			377.997i			0.515684i	0.966187	+	0.257842i	$0.0830115\pi$
$733$			377.997i			0.515684i	−0.966187	+	0.257842i	$0.916989\pi$
$734$		−	961.040i		−	1.30932i
$735$			97.1699i			0.132204i
$736$	−77.3455	−	104.622i	−0.105089	−	0.142149i
$737$	51.7348			0.0701965
$738$	197.024			0.266970
$739$	−536.592			−0.726106			−0.363053	−	0.931769i	$-0.618266\pi$
$739$	−536.592			−0.726106			−0.363053	+	0.931769i	$0.618266\pi$
$740$	44.1189			0.0596202
$741$		−	575.448i		−	0.776582i
$742$	−1088.55			−1.46705
$743$			366.532i			0.493313i	0.969103	+	0.246657i	$0.0793319\pi$
$743$			366.532i			0.493313i	−0.969103	+	0.246657i	$0.920668\pi$
$744$	16.9322			0.0227583
$745$	−43.4826			−0.0583660
$746$			965.948i			1.29484i
$747$		−	76.0091i		−	0.101753i
$748$	21.5255			0.0287774
$749$	−2189.52			−2.92325
$750$			110.570i			0.147427i
$751$			627.256i			0.835227i	0.908625	+	0.417614i	$0.137133\pi$
$751$			627.256i			0.835227i	−0.908625	+	0.417614i	$0.862867\pi$
$752$	−180.790			−0.240413
$753$		−	731.789i		−	0.971831i
$754$	108.560			0.143979
$755$		−	38.0977i		−	0.0504606i
$756$		−	109.042i		−	0.144235i
$757$		−	904.004i		−	1.19419i	−0.802169	−	0.597097i	$-0.796321\pi$
$757$		−	904.004i		−	1.19419i	0.802169	−	0.597097i	$-0.203679\pi$
$758$			579.211i			0.764130i
$759$	−49.8694	+	36.8679i	−0.0657041	+	0.0485742i
$760$	86.4287			0.113722
$761$	617.464			0.811385			0.405692	−	0.914010i	$-0.367030\pi$
$761$	617.464			0.811385			0.405692	+	0.914010i	$0.367030\pi$
$762$	149.616			0.196346
$763$	1907.20			2.49961
$764$		−	81.4861i		−	0.106657i
$765$	−19.0456			−0.0248963
$766$		−	819.098i		−	1.06932i
$767$	400.692			0.522415
$768$	27.7128			0.0360844
$769$		−	113.096i		−	0.147070i	−0.997293	−	0.0735348i	$-0.976572\pi$
$769$		−	113.096i		−	0.147070i	0.997293	−	0.0735348i	$-0.0234280\pi$
$770$			21.2130i			0.0275493i
$771$	−391.785			−0.508152
$772$	−294.895			−0.381988
$773$			1182.99i			1.53039i	0.643799	+	0.765195i	$0.277357\pi$
$773$			1182.99i			1.53039i	−0.643799	+	0.765195i	$0.722643\pi$
$774$		−	257.418i		−	0.332581i
$775$	−83.4923			−0.107732
$776$		−	323.255i		−	0.416566i
$777$	436.573			0.561870
$778$		−	294.962i		−	0.379128i
$779$			1545.31i			1.98372i
$780$		−	31.7600i		−	0.0407179i
$781$			146.599i			0.187706i
$782$	−180.824	+	133.681i	−0.231233	+	0.170948i
$783$	−39.9510			−0.0510229
$784$	−244.372			−0.311700
$785$	4.30705			0.00548668
$786$	−406.808			−0.517567
$787$		−	1258.66i		−	1.59932i	−0.600455	−	0.799658i	$-0.705014\pi$
$787$		−	1258.66i		−	1.59932i	0.600455	−	0.799658i	$-0.294986\pi$
$788$	216.355			0.274562
$789$			889.196i			1.12699i
$790$	42.2497			0.0534806
$791$	545.010			0.689014
$792$		−	13.2097i		−	0.0166789i
$793$			693.677i			0.874750i
$794$	301.728			0.380011
$795$	116.679			0.146766
$796$			187.283i			0.235280i
$797$			466.365i			0.585151i	0.956242	+	0.292575i	$0.0945122\pi$
$797$			466.365i			0.585151i	−0.956242	+	0.292575i	$0.905488\pi$
$798$	855.244			1.07173
$799$			312.472i			0.391078i
$800$	−136.651			−0.170814
$801$		−	236.450i		−	0.295193i
$802$			593.998i			0.740645i
$803$		−	127.817i		−	0.159175i
$804$		−	115.119i		−	0.143183i
$805$	−131.740	−	178.199i	−0.163653	−	0.221365i
$806$	48.8016			0.0605478
$807$	174.213			0.215878
$808$	−66.6561			−0.0824952
$809$	−632.706			−0.782084			−0.391042	−	0.920373i	$-0.627885\pi$
$809$	−632.706			−0.782084			−0.391042	+	0.920373i	$0.627885\pi$
$810$			11.6879i			0.0144295i
$811$	−1044.17			−1.28751			−0.643757	−	0.765230i	$-0.722625\pi$
$811$	−1044.17			−1.28751			−0.643757	+	0.765230i	$0.722625\pi$
$812$			161.345i			0.198701i
$813$	−615.517			−0.757093
$814$	52.8882			0.0649732
$815$		−	267.280i		−	0.327951i
$816$		−	47.8979i		−	0.0586984i
$817$	2019.00			2.47124
$818$	422.007			0.515902
$819$		−	314.277i		−	0.383732i
$820$			85.2887i			0.104011i
$821$	935.808			1.13984			0.569920	−	0.821700i	$-0.306974\pi$
$821$	935.808			1.13984			0.569920	+	0.821700i	$0.306974\pi$
$822$		−	360.711i		−	0.438821i
$823$	−948.181			−1.15210			−0.576051	−	0.817413i	$-0.695407\pi$
$823$	−948.181			−1.15210			−0.576051	+	0.817413i	$0.695407\pi$
$824$		−	159.184i		−	0.193185i
$825$			65.1368i			0.0789537i
$826$			595.519i			0.720967i
$827$		−	1547.20i		−	1.87086i	−0.353517	−	0.935428i	$-0.615014\pi$
$827$		−	1547.20i		−	1.87086i	0.353517	−	0.935428i	$-0.384986\pi$
$828$	82.0372	+	110.968i	0.0990788	+	0.134019i
$829$	−219.577			−0.264869			−0.132435	−	0.991192i	$-0.542279\pi$
$829$	−219.577			−0.264869			−0.132435	+	0.991192i	$0.542279\pi$
$830$	32.9032			0.0396424
$831$	116.601			0.140314
$832$	79.8731			0.0960014
$833$			422.365i			0.507041i
$834$	−288.097			−0.345440
$835$		−	155.333i		−	0.186027i
$836$	103.608			0.123933
$837$	−17.9593			−0.0214568
$838$			46.0943i			0.0550052i
$839$		−	1289.59i		−	1.53705i	−0.639818	−	0.768526i	$-0.720990\pi$
$839$		−	1289.59i		−	1.53705i	0.639818	−	0.768526i	$-0.279010\pi$
$840$	47.2024			0.0561934
$841$	−781.886			−0.929710
$842$		−	304.879i		−	0.362089i
$843$		−	690.722i		−	0.819362i
$844$	687.972			0.815133
$845$			63.6528i			0.0753288i
$846$	191.757			0.226663
$847$		−	1244.17i		−	1.46891i
$848$			293.436i			0.346033i
$849$		−	711.819i		−	0.838421i
$850$			236.183i			0.277863i
$851$	−444.286	+	328.455i	−0.522075	+	0.385964i
$852$	326.207			0.382872
$853$	1129.14			1.32373			0.661864	−	0.749624i	$-0.269766\pi$
$853$	1129.14			1.32373			0.661864	+	0.749624i	$0.269766\pi$
$854$	−1030.96			−1.20721
$855$	−91.6715			−0.107218
$856$			590.219i			0.689508i
$857$	−577.192			−0.673503			−0.336752	−	0.941593i	$-0.609328\pi$
$857$	−577.192			−0.673503			−0.336752	+	0.941593i	$0.609328\pi$
$858$		−	38.0727i		−	0.0443738i
$859$	157.765			0.183661			0.0918304	−	0.995775i	$-0.470728\pi$
$859$	157.765			0.183661			0.0918304	+	0.995775i	$0.470728\pi$
$860$	111.432			0.129573
$861$			843.963i			0.980212i
$862$			288.524i			0.334715i
$863$	−547.299			−0.634182			−0.317091	−	0.948395i	$-0.602706\pi$
$863$	−547.299			−0.634182			−0.317091	+	0.948395i	$0.602706\pi$
$864$	−29.3939			−0.0340207
$865$			224.445i			0.259474i
$866$		−	553.098i		−	0.638682i
$867$	417.778			0.481866
$868$			72.5300i			0.0835599i
$869$	50.6474			0.0582824
$870$		−	17.2942i		−	0.0198784i
$871$		−	331.792i		−	0.380933i
$872$		−	514.117i		−	0.589584i
$873$			342.864i			0.392742i
$874$	−870.353	+	643.442i	−0.995828	+	0.736203i
$875$	−473.633			−0.541295
$876$	−284.415			−0.324675
$877$	−175.201			−0.199773			−0.0998866	−	0.994999i	$-0.531848\pi$
$877$	−175.201			−0.199773			−0.0998866	+	0.994999i	$0.531848\pi$
$878$	18.4371			0.0209990
$879$			27.1657i			0.0309052i
$880$	5.71829			0.00649806
$881$		−	1105.27i		−	1.25456i	−0.778793	−	0.627281i	$-0.784168\pi$
$881$		−	1105.27i		−	1.25456i	0.778793	−	0.627281i	$-0.215832\pi$
$882$	259.196			0.293873
$883$	604.458			0.684550			0.342275	−	0.939600i	$-0.388803\pi$
$883$	604.458			0.684550			0.342275	+	0.939600i	$0.388803\pi$
$884$		−	138.050i		−	0.156165i
$885$		−	63.8322i		−	0.0721268i
$886$	−450.004			−0.507905
$887$	−938.809			−1.05841			−0.529205	−	0.848494i	$-0.677510\pi$
$887$	−938.809			−1.05841			−0.529205	+	0.848494i	$0.677510\pi$
$888$		−	117.685i		−	0.132528i
$889$			640.888i			0.720909i
$890$	102.356			0.115006
$891$			14.0110i			0.0157251i
$892$	−407.497			−0.456835
$893$			1504.01i			1.68422i
$894$			115.988i			0.129740i
$895$			294.714i			0.329290i
$896$			118.709i			0.132488i
$897$	236.446	+	319.829i	0.263596	+	0.356554i
$898$	−599.507			−0.667602
$899$	26.5738			0.0295593
$900$	144.940			0.161045
$901$	507.164			0.562891
$902$			102.241i			0.113349i
$903$	1102.66			1.22111
$904$		−	146.916i		−	0.162518i
$905$	−92.6457			−0.102371
$906$	−101.624			−0.112168
$907$			140.388i			0.154783i	0.997001	+	0.0773914i	$0.0246591\pi$
$907$			140.388i			0.154783i	−0.997001	+	0.0773914i	$0.975341\pi$
$908$			473.806i			0.521812i
$909$	70.6995			0.0777772
$910$	136.046			0.149501
$911$			794.086i			0.871665i	0.900028	+	0.435832i	$0.143546\pi$
$911$			794.086i			0.871665i	−0.900028	+	0.435832i	$0.856454\pi$
$912$		−	230.545i		−	0.252790i
$913$	39.4432			0.0432017
$914$			1.27821i			0.00139848i
$915$	110.506			0.120772
$916$		−	353.603i		−	0.386030i
$917$		−	1742.58i		−	1.90031i
$918$			50.8034i			0.0553413i
$919$		−	298.074i		−	0.324346i	−0.986762	−	0.162173i	$-0.948150\pi$
$919$		−	298.074i		−	0.324346i	0.986762	−	0.162173i	$-0.0518503\pi$
$920$	−48.0364	+	35.5127i	−0.0522134	+	0.0386008i
$921$	−58.2407			−0.0632364
$922$	−760.487			−0.824823
$923$	940.184			1.01862
$924$	56.5846			0.0612387
$925$			580.303i			0.627354i
$926$	−386.113			−0.416969
$927$			168.840i			0.182136i
$928$	43.4931			0.0468676
$929$	950.600			1.02325			0.511626	−	0.859208i	$-0.329043\pi$
$929$	950.600			1.02325			0.511626	+	0.859208i	$0.329043\pi$
$930$		−	7.77432i		−	0.00835948i
$931$			2032.95i			2.18362i
$932$	673.494			0.722634
$933$	−993.737			−1.06510
$934$			842.306i			0.901827i
$935$		−	9.88329i		−	0.0105704i
$936$	−84.7183			−0.0905110
$937$			150.371i			0.160481i	0.996776	+	0.0802407i	$0.0255689\pi$
$937$			150.371i			0.160481i	−0.996776	+	0.0802407i	$0.974431\pi$
$938$	493.118			0.525712
$939$			909.112i			0.968170i
$940$			83.0088i			0.0883072i
$941$			578.230i			0.614484i	0.951631	+	0.307242i	$0.0994061\pi$
$941$			578.230i			0.614484i	−0.951631	+	0.307242i	$0.900594\pi$
$942$		−	11.4888i		−	0.0121962i
$943$	−634.954	−	858.873i	−0.673334	−	0.910788i
$944$	160.531			0.170055
$945$	−50.0658			−0.0529796
$946$	133.581			0.141206
$947$	717.050			0.757180			0.378590	−	0.925564i	$-0.376409\pi$
$947$	717.050			0.757180			0.378590	+	0.925564i	$0.376409\pi$
$948$		−	112.699i		−	0.118881i
$949$	−819.733			−0.863786
$950$			1136.81i			1.19664i
$951$	−791.269			−0.832039
$952$	205.173			0.215518
$953$			1507.38i			1.58172i	0.611998	+	0.790859i	$0.290366\pi$
$953$			1507.38i			1.58172i	−0.611998	+	0.790859i	$0.709634\pi$
$954$		−	311.236i		−	0.326243i
$955$	−37.4139			−0.0391768
$956$	−629.080			−0.658034
$957$		−	20.7316i		−	0.0216631i
$958$			1134.93i			1.18468i
$959$	1545.12			1.61118
$960$		−	12.7242i		−	0.0132543i
$961$	−949.054			−0.987569
$962$		−	339.189i		−	0.352587i
$963$		−	626.022i		−	0.650075i
$964$		−	107.039i		−	0.111037i
$965$			135.399i			0.140310i
$966$	−475.338	+	351.411i	−0.492068	+	0.363780i
$967$	−1661.59			−1.71829			−0.859147	−	0.511728i	$-0.829005\pi$
$967$	−1661.59			−1.71829			−0.859147	+	0.511728i	$0.829005\pi$
$968$	−335.385			−0.346472
$969$	−398.465			−0.411213
$970$	−148.421			−0.153011
$971$		−	1839.27i		−	1.89420i	−0.320935	−	0.947101i	$-0.603997\pi$
$971$		−	1839.27i		−	1.89420i	0.320935	−	0.947101i	$-0.396003\pi$
$972$	31.1769			0.0320750
$973$		−	1234.08i		−	1.26833i
$974$	−595.539			−0.611436
$975$	417.744			0.428455
$976$			277.911i			0.284745i
$977$		−	1162.46i		−	1.18982i	−0.803791	−	0.594912i	$-0.797187\pi$
$977$		−	1162.46i		−	1.18982i	0.803791	−	0.594912i	$-0.202813\pi$
$978$	−712.958			−0.728996
$979$	122.700			0.125332
$980$			112.202i			0.114492i
$981$			545.304i			0.555865i
$982$	−229.906			−0.234120
$983$		−	989.130i		−	1.00624i	−0.864218	−	0.503118i	$-0.832186\pi$
$983$		−	989.130i		−	1.00624i	0.864218	−	0.503118i	$-0.167814\pi$
$984$	227.504			0.231203
$985$		−	99.3379i		−	0.100851i
$986$		−	75.1720i		−	0.0762393i
$987$			821.402i			0.832221i
$988$		−	664.470i		−	0.672540i
$989$	−1122.14	+	829.588i	−1.13463	+	0.838815i
$990$	−6.06516			−0.00612643
$991$	−1308.63			−1.32052			−0.660259	−	0.751038i	$-0.729554\pi$
$991$	−1308.63			−1.32052			−0.660259	+	0.751038i	$0.729554\pi$
$992$	19.5516			0.0197093
$993$	−673.109			−0.677854
$994$			1397.32i			1.40576i
$995$	85.9899			0.0864220
$996$		−	87.7678i		−	0.0881203i
$997$	1788.55			1.79393			0.896964	−	0.442103i	$-0.145768\pi$
$997$	1788.55			1.79393			0.896964	+	0.442103i	$0.145768\pi$
$998$	966.149			0.968085
$999$			124.824i			0.124949i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.b.a.91.3	✓	8
3.2	odd	2		414.3.b.c.91.7		8
4.3	odd	2		1104.3.c.c.1057.2		8
23.22	odd	2	inner	138.3.b.a.91.4	yes	8
69.68	even	2		414.3.b.c.91.6		8
92.91	even	2		1104.3.c.c.1057.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.b.a.91.3	✓	8	1.1	even	1	trivial
138.3.b.a.91.4	yes	8	23.22	odd	2	inner
414.3.b.c.91.6		8	69.68	even	2
414.3.b.c.91.7		8	3.2	odd	2
1104.3.c.c.1057.2		8	4.3	odd	2
1104.3.c.c.1057.3		8	92.91	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -0.918288i q^{5} -2.44949 q^{6} -10.4925i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -0.918288i q^{5} -2.44949 q^{6} -10.4925i q^{7} -2.82843 q^{8} +3.00000 q^{9} +1.29866i q^{10} +1.55678i q^{11} +3.46410 q^{12} +9.98414 q^{13} +14.8387i q^{14} -1.59052i q^{15} +4.00000 q^{16} -6.91346i q^{17} -4.24264 q^{18} -33.2762i q^{19} -1.83658i q^{20} -18.1736i q^{21} -2.20162i q^{22} +(13.6729 + 18.4947i) q^{23} -4.89898 q^{24} +24.1567 q^{25} -14.1197 q^{26} +5.19615 q^{27} -20.9851i q^{28} -7.68857 q^{29} +2.24934i q^{30} -3.45627 q^{31} -5.65685 q^{32} +2.69642i q^{33} +9.77711i q^{34} -9.63516 q^{35} +6.00000 q^{36} +24.0224i q^{37} +47.0597i q^{38} +17.2930 q^{39} +2.59731i q^{40} -46.4390 q^{41} +25.7013i q^{42} +60.6740i q^{43} +3.11356i q^{44} -2.75486i q^{45} +(-19.3364 - 26.1554i) q^{46} -45.1976 q^{47} +6.92820 q^{48} -61.0931 q^{49} -34.1628 q^{50} -11.9745i q^{51} +19.9683 q^{52} +73.3590i q^{53} -7.34847 q^{54} +1.42957 q^{55} +29.6773i q^{56} -57.6361i q^{57} +10.8733 q^{58} +40.1329 q^{59} -3.18104i q^{60} +69.4778i q^{61} +4.88791 q^{62} -31.4776i q^{63} +8.00000 q^{64} -9.16832i q^{65} -3.81332i q^{66} -33.2319i q^{67} -13.8269i q^{68} +(23.6821 + 32.0337i) q^{69} +13.6262 q^{70} +94.1677 q^{71} -8.48528 q^{72} -82.1035 q^{73} -33.9728i q^{74} +41.8407 q^{75} -66.5525i q^{76} +16.3346 q^{77} -24.4561 q^{78} -32.5334i q^{79} -3.67315i q^{80} +9.00000 q^{81} +65.6746 q^{82} -25.3364i q^{83} -36.3472i q^{84} -6.34855 q^{85} -85.8060i q^{86} -13.3170 q^{87} -4.40324i q^{88} -78.8166i q^{89} +3.89597i q^{90} -104.759i q^{91} +(27.3457 + 36.9893i) q^{92} -5.98644 q^{93} +63.9190 q^{94} -30.5572 q^{95} -9.79796 q^{96} +114.288i q^{97} +86.3987 q^{98} +4.67034i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) \) 8 * q + 16 * q^4 + 24 * q^9	\( 8 q + 16 q^{4} + 24 q^{9} + 16 q^{13} + 32 q^{16} + 16 q^{23} + 72 q^{25} + 32 q^{26} - 144 q^{29} - 128 q^{31} - 112 q^{35} + 48 q^{36} + 48 q^{39} - 16 q^{41} - 80 q^{46} - 112 q^{47} + 40 q^{49} - 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} + 80 q^{59} - 96 q^{62} + 64 q^{64} - 72 q^{69} - 144 q^{70} + 32 q^{71} + 64 q^{73} + 48 q^{75} + 224 q^{77} - 144 q^{78} + 72 q^{81} + 48 q^{85} + 96 q^{87} + 32 q^{92} + 192 q^{93} - 16 q^{94} + 112 q^{95} + 224 q^{98}+O(q^{100}) \) 8 * q + 16 * q^4 + 24 * q^9 + 16 * q^13 + 32 * q^16 + 16 * q^23 + 72 * q^25 + 32 * q^26 - 144 * q^29 - 128 * q^31 - 112 * q^35 + 48 * q^36 + 48 * q^39 - 16 * q^41 - 80 * q^46 - 112 * q^47 + 40 * q^49 - 160 * q^50 + 32 * q^52 - 64 * q^55 + 128 * q^58 + 80 * q^59 - 96 * q^62 + 64 * q^64 - 72 * q^69 - 144 * q^70 + 32 * q^71 + 64 * q^73 + 48 * q^75 + 224 * q^77 - 144 * q^78 + 72 * q^81 + 48 * q^85 + 96 * q^87 + 32 * q^92 + 192 * q^93 - 16 * q^94 + 112 * q^95 + 224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more