LMFDB - Embedded newform 415.4.b.a.84.122

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [415,4,Mod(84,415)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("415.84"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 415 = 5 \cdot 83 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	415.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4857926524$
Analytic rank:	$0$
Dimension:	$124$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			84.122
Character	$\chi$	$=$	415.84
Dual form			415.4.b.a.84.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.43165i q^{2} +5.71216i q^{3} -21.5028 q^{4} +(10.3260 - 4.28658i) q^{5} -31.0264 q^{6} -16.9337i q^{7} -73.3426i q^{8} -5.62872 q^{9} +(23.2832 + 56.0869i) q^{10} +50.7444 q^{11} -122.827i q^{12} -11.1969i q^{13} +91.9778 q^{14} +(24.4856 + 58.9834i) q^{15} +226.349 q^{16} -90.3524i q^{17} -30.5732i q^{18} -16.3166 q^{19} +(-222.037 + 92.1735i) q^{20} +96.7278 q^{21} +275.626i q^{22} -142.132i q^{23} +418.944 q^{24} +(88.2505 - 88.5260i) q^{25} +60.8176 q^{26} +122.076i q^{27} +364.122i q^{28} +160.663 q^{29} +(-320.377 + 132.997i) q^{30} -70.0129 q^{31} +642.705i q^{32} +289.860i q^{33} +490.763 q^{34} +(-72.5875 - 174.856i) q^{35} +121.033 q^{36} +226.468i q^{37} -88.6260i q^{38} +63.9584 q^{39} +(-314.389 - 757.332i) q^{40} -219.178 q^{41} +525.391i q^{42} -525.750i q^{43} -1091.15 q^{44} +(-58.1219 + 24.1279i) q^{45} +772.013 q^{46} -256.333i q^{47} +1292.94i q^{48} +56.2505 q^{49} +(480.842 + 479.346i) q^{50} +516.107 q^{51} +240.765i q^{52} -462.654i q^{53} -663.074 q^{54} +(523.985 - 217.520i) q^{55} -1241.96 q^{56} -93.2029i q^{57} +872.663i q^{58} -161.025 q^{59} +(-526.509 - 1268.31i) q^{60} -219.277 q^{61} -380.286i q^{62} +95.3149i q^{63} -1680.16 q^{64} +(-47.9964 - 115.619i) q^{65} -1574.42 q^{66} +806.850i q^{67} +1942.83i q^{68} +811.882 q^{69} +(949.758 - 394.270i) q^{70} +643.214 q^{71} +412.824i q^{72} -718.854i q^{73} -1230.09 q^{74} +(505.674 + 504.101i) q^{75} +350.852 q^{76} -859.290i q^{77} +347.400i q^{78} +952.969 q^{79} +(2337.26 - 970.260i) q^{80} -849.293 q^{81} -1190.50i q^{82} -83.0000i q^{83} -2079.92 q^{84} +(-387.303 - 932.975i) q^{85} +2855.69 q^{86} +917.730i q^{87} -3721.73i q^{88} -1243.99 q^{89} +(-131.054 - 315.698i) q^{90} -189.605 q^{91} +3056.25i q^{92} -399.925i q^{93} +1392.31 q^{94} +(-168.484 + 69.9423i) q^{95} -3671.23 q^{96} +1251.39i q^{97} +305.533i q^{98} -285.626 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 124 q - 504 q^{4} + 6 q^{5} - 1088 q^{9} + 62 q^{10} + 60 q^{11} + 124 q^{14} - 120 q^{15} + 2080 q^{16} + 124 q^{19} - 206 q^{20} + 24 q^{21} + 368 q^{24} - 430 q^{25} - 176 q^{26} + 208 q^{29} - 52 q^{30}+ \cdots + 2668 q^{99}+O(q^{100}) $ 124 * q - 504 * q^4 + 6 * q^5 - 1088 * q^9 + 62 * q^10 + 60 * q^11 + 124 * q^14 - 120 * q^15 + 2080 * q^16 + 124 * q^19 - 206 * q^20 + 24 * q^21 + 368 * q^24 - 430 * q^25 - 176 * q^26 + 208 * q^29 - 52 * q^30 + 372 * q^31 + 496 * q^34 + 320 * q^35 + 4592 * q^36 - 592 * q^39 - 430 * q^40 - 384 * q^41 - 636 * q^44 - 118 * q^45 - 1216 * q^46 - 6504 * q^49 + 440 * q^50 - 696 * q^51 + 8 * q^54 - 684 * q^55 - 1328 * q^56 - 1068 * q^59 + 4698 * q^60 + 952 * q^61 - 9180 * q^64 - 16 * q^65 - 284 * q^66 + 3496 * q^69 - 1296 * q^70 - 2976 * q^71 - 1732 * q^74 + 1796 * q^75 + 804 * q^76 + 1080 * q^79 - 1758 * q^80 + 8700 * q^81 + 6752 * q^84 + 1036 * q^85 - 2692 * q^86 + 832 * q^89 - 5586 * q^90 + 1464 * q^91 + 2620 * q^94 + 1240 * q^95 + 5892 * q^96 + 2668 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/415\mathbb{Z}\right)^\times$.

$n$	$167$	$251$
$\chi(n)$	$-1$	$1$

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 $

$2$			5.43165i			1.92038i	0.279351	+	0.960189i	$0.409881\pi$
$2$			5.43165i			1.92038i	−0.279351	+	0.960189i	$0.590119\pi$
$3$			5.71216i			1.09930i	0.835393	+	0.549652i	$0.185240\pi$
$3$			5.71216i			1.09930i	−0.835393	+	0.549652i	$0.814760\pi$
$4$	−21.5028			−2.68785
$5$	10.3260	−	4.28658i	0.923581	−	0.383403i
$5$	10.3260	−	4.28658i	0.923581	−	0.383403i
$6$	−31.0264			−2.11108
$7$		−	16.9337i		−	0.914333i	−0.889381	−	0.457166i	$-0.848864\pi$
$7$		−	16.9337i		−	0.914333i	0.889381	−	0.457166i	$-0.151136\pi$
$8$		−	73.3426i		−	3.24131i
$9$	−5.62872			−0.208471
$10$	23.2832	+	56.0869i	0.736279	+	1.77362i
$11$	50.7444			1.39091			0.695456	−	0.718569i	$-0.255203\pi$
$11$	50.7444			1.39091			0.695456	+	0.718569i	$0.255203\pi$
$12$		−	122.827i		−	2.95477i
$13$		−	11.1969i		−	0.238882i	−0.992841	−	0.119441i	$-0.961890\pi$
$13$		−	11.1969i		−	0.238882i	0.992841	−	0.119441i	$-0.0381102\pi$
$14$	91.9778			1.75586
$15$	24.4856	+	58.9834i	0.421477	+	1.01530i
$16$	226.349			3.53670
$17$		−	90.3524i		−	1.28904i	−0.764587	−	0.644520i	$-0.777057\pi$
$17$		−	90.3524i		−	1.28904i	0.764587	−	0.644520i	$-0.222943\pi$
$18$		−	30.5732i		−	0.400343i
$19$	−16.3166			−0.197015			−0.0985074	−	0.995136i	$-0.531407\pi$
$19$	−16.3166			−0.197015			−0.0985074	+	0.995136i	$0.531407\pi$
$20$	−222.037	+	92.1735i	−2.48245	+	1.03053i
$21$	96.7278			1.00513
$22$			275.626i			2.67108i
$23$		−	142.132i		−	1.28855i	−0.764794	−	0.644275i	$-0.777159\pi$
$23$		−	142.132i		−	1.28855i	0.764794	−	0.644275i	$-0.222841\pi$
$24$	418.944			3.56319
$25$	88.2505	−	88.5260i	0.706004	−	0.708208i
$26$	60.8176			0.458743
$27$			122.076i			0.870132i
$28$			364.122i			2.45759i
$29$	160.663			1.02877			0.514384	−	0.857560i	$-0.328020\pi$
$29$	160.663			1.02877			0.514384	+	0.857560i	$0.328020\pi$
$30$	−320.377	+	132.997i	−1.94975	+	0.809395i
$31$	−70.0129			−0.405635			−0.202818	−	0.979217i	$-0.565010\pi$
$31$	−70.0129			−0.405635			−0.202818	+	0.979217i	$0.565010\pi$
$32$			642.705i			3.55048i
$33$			289.860i			1.52904i
$34$	490.763			2.47544
$35$	−72.5875	−	174.856i	−0.350558	−	0.844460i
$36$	121.033			0.560339
$37$			226.468i			1.00624i	0.864215	+	0.503122i	$0.167815\pi$
$37$			226.468i			1.00624i	−0.864215	+	0.503122i	$0.832185\pi$
$38$		−	88.6260i		−	0.378343i
$39$	63.9584			0.262604
$40$	−314.389	−	757.332i	−1.24273	−	2.99362i
$41$	−219.178			−0.834873			−0.417437	−	0.908706i	$-0.637071\pi$
$41$	−219.178			−0.834873			−0.417437	+	0.908706i	$0.637071\pi$
$42$			525.391i			1.93023i
$43$		−	525.750i		−	1.86456i	−0.361734	−	0.932281i	$-0.617815\pi$
$43$		−	525.750i		−	1.86456i	0.361734	−	0.932281i	$-0.382185\pi$
$44$	−1091.15			−3.73856
$45$	−58.1219	+	24.1279i	−0.192540	+	0.0799284i
$46$	772.013			2.47450
$47$		−	256.333i		−	0.795533i	−0.917487	−	0.397767i	$-0.869785\pi$
$47$		−	256.333i		−	0.795533i	0.917487	−	0.397767i	$-0.130215\pi$
$48$			1292.94i			3.88791i
$49$	56.2505			0.163996
$50$	480.842	+	479.346i	1.36003	+	1.35579i
$51$	516.107			1.41705
$52$			240.765i			0.642078i
$53$		−	462.654i		−	1.19906i	−0.800351	−	0.599532i	$-0.795354\pi$
$53$		−	462.654i		−	1.19906i	0.800351	−	0.599532i	$-0.204646\pi$
$54$	−663.074			−1.67098
$55$	523.985	−	217.520i	1.28462	−	0.533280i
$56$	−1241.96			−2.96364
$57$		−	93.2029i		−	0.216579i
$58$			872.663i			1.97562i
$59$	−161.025			−0.355317			−0.177658	−	0.984092i	$-0.556852\pi$
$59$	−161.025			−0.355317			−0.177658	+	0.984092i	$0.556852\pi$
$60$	−526.509	−	1268.31i	−1.13287	−	2.72897i
$61$	−219.277			−0.460255			−0.230128	−	0.973160i	$-0.573914\pi$
$61$	−219.277			−0.460255			−0.230128	+	0.973160i	$0.573914\pi$
$62$		−	380.286i		−	0.778973i
$63$			95.3149i			0.190612i
$64$	−1680.16			−3.28157
$65$	−47.9964	−	115.619i	−0.0915879	−	0.220626i
$66$	−1574.42			−2.93633
$67$			806.850i			1.47123i	0.677400	+	0.735615i	$0.263107\pi$
$67$			806.850i			1.47123i	−0.677400	+	0.735615i	$0.736893\pi$
$68$			1942.83i			3.46475i
$69$	811.882			1.41651
$70$	949.758	−	394.270i	1.62168	−	0.673204i
$71$	643.214			1.07515			0.537574	−	0.843217i	$-0.319341\pi$
$71$	643.214			1.07515			0.537574	+	0.843217i	$0.319341\pi$
$72$			412.824i			0.675720i
$73$		−	718.854i		−	1.15254i	−0.817259	−	0.576270i	$-0.804508\pi$
$73$		−	718.854i		−	1.15254i	0.817259	−	0.576270i	$-0.195492\pi$
$74$	−1230.09			−1.93237
$75$	505.674	+	504.101i	0.778536	+	0.776114i
$76$	350.852			0.529547
$77$		−	859.290i		−	1.27176i
$78$			347.400i			0.504298i
$79$	952.969			1.35718			0.678591	−	0.734516i	$-0.262591\pi$
$79$	952.969			1.35718			0.678591	+	0.734516i	$0.262591\pi$
$80$	2337.26	−	970.260i	3.26643	−	1.35598i
$81$	−849.293			−1.16501
$82$		−	1190.50i		−	1.60327i
$83$		−	83.0000i		−	0.109764i
$83$		−	83.0000i		−	0.109764i
$84$	−2079.92			−2.70164
$85$	−387.303	−	932.975i	−0.494222	−	1.19053i
$86$	2855.69			3.58066
$87$			917.730i			1.13093i
$88$		−	3721.73i		−	4.50838i
$89$	−1243.99			−1.48160			−0.740802	−	0.671723i	$-0.765554\pi$
$89$	−1243.99			−1.48160			−0.740802	+	0.671723i	$0.765554\pi$
$90$	−131.054	−	315.698i	−0.153493	−	0.369749i
$91$	−189.605			−0.218417
$92$			3056.25i			3.46343i
$93$		−	399.925i		−	0.445917i
$94$	1392.31			1.52772
$95$	−168.484	+	69.9423i	−0.181959	+	0.0755361i
$96$	−3671.23			−3.90306
$97$			1251.39i			1.30990i	0.755674	+	0.654948i	$0.227309\pi$
$97$			1251.39i			1.30990i	−0.755674	+	0.654948i	$0.772691\pi$
$98$			305.533i			0.314934i
$99$	−285.626			−0.289965
$100$	−1897.63	+	1903.56i	−1.89763	+	1.90356i
$101$	−1022.30			−1.00716			−0.503580	−	0.863949i	$-0.667984\pi$
$101$	−1022.30			−1.00716			−0.503580	+	0.863949i	$0.667984\pi$
$102$			2803.31i			2.72127i
$103$		−	421.633i		−	0.403347i	−0.979453	−	0.201674i	$-0.935362\pi$
$103$		−	421.633i		−	0.403347i	0.979453	−	0.201674i	$-0.0646380\pi$
$104$	−821.209			−0.774290
$105$	998.806	−	414.631i	0.928319	−	0.385370i
$106$	2512.97			2.30265
$107$			1123.54i			1.01511i	0.861620	+	0.507555i	$0.169450\pi$
$107$			1123.54i			1.01511i	−0.861620	+	0.507555i	$0.830550\pi$
$108$		−	2624.98i		−	2.33878i
$109$	−1076.48			−0.945948			−0.472974	−	0.881076i	$-0.656820\pi$
$109$	−1076.48			−0.945948			−0.472974	+	0.881076i	$0.656820\pi$
$110$	1181.49	+	2846.10i	1.02410	+	2.46696i
$111$	−1293.62			−1.10617
$112$		−	3832.91i		−	3.23372i
$113$			797.445i			0.663870i	0.943302	+	0.331935i	$0.107701\pi$
$113$			797.445i			0.663870i	−0.943302	+	0.331935i	$0.892299\pi$
$114$	506.245			0.415914
$115$	−609.262	−	1467.65i	−0.494034	−	1.19008i
$116$	−3454.70			−2.76518
$117$			63.0242i			0.0497999i
$118$		−	874.632i		−	0.682342i
$119$	−1530.00			−1.17861
$120$	4326.00	−	1795.84i	3.29090	−	1.36614i
$121$	1244.00			0.934635
$122$		−	1191.04i		−	0.883864i
$123$		−	1251.98i		−	0.917780i
$124$	1505.47			1.09029
$125$	531.797	−	1292.41i	0.380523	−	0.924771i
$126$	−517.717			−0.366047
$127$			1776.02i			1.24091i	0.784240	+	0.620457i	$0.213053\pi$
$127$			1776.02i			1.24091i	−0.784240	+	0.620457i	$0.786947\pi$
$128$		−	3984.41i		−	2.75137i
$129$	3003.17			2.04972
$130$	628.000	−	260.699i	0.423686	−	0.175883i
$131$	541.816			0.361364			0.180682	−	0.983542i	$-0.442169\pi$
$131$	541.816			0.361364			0.180682	+	0.983542i	$0.442169\pi$
$132$		−	6232.81i		−	4.10982i
$133$			276.300i			0.180137i
$134$	−4382.53			−2.82532
$135$	523.289	+	1260.55i	0.333611	+	0.803637i
$136$	−6626.68			−4.17818
$137$			2314.73i			1.44351i	0.692150	+	0.721753i	$0.256663\pi$
$137$			2314.73i			1.44351i	−0.692150	+	0.721753i	$0.743337\pi$
$138$			4409.86i			2.72023i
$139$	−628.096			−0.383269			−0.191634	−	0.981466i	$-0.561379\pi$
$139$	−628.096			−0.383269			−0.191634	+	0.981466i	$0.561379\pi$
$140$	1560.84	+	3759.90i	0.942248	+	2.26978i
$141$	1464.22			0.874534
$142$			3493.71i			2.06469i
$143$		−	568.180i		−	0.332263i
$144$	−1274.05			−0.737298
$145$	1658.99	−	688.693i	0.950151	−	0.394433i
$146$	3904.56			2.21331
$147$			321.312i			0.180281i
$148$		−	4869.69i		−	2.70464i
$149$	1379.27			0.758348			0.379174	−	0.925325i	$-0.376208\pi$
$149$	1379.27			0.758348			0.379174	+	0.925325i	$0.376208\pi$
$150$	−2738.10	+	2746.64i	−1.49043	+	1.49508i
$151$	352.495			0.189971			0.0949856	−	0.995479i	$-0.469719\pi$
$151$	352.495			0.189971			0.0949856	+	0.995479i	$0.469719\pi$
$152$			1196.70i			0.638587i
$153$			508.568i			0.268728i
$154$	4667.36			2.44225
$155$	−722.950	+	300.116i	−0.374637	+	0.155522i
$156$	−1375.29			−0.705840
$157$			265.419i			0.134922i	0.997722	+	0.0674610i	$0.0214898\pi$
$157$			265.419i			0.134922i	−0.997722	+	0.0674610i	$0.978510\pi$
$158$			5176.19i			2.60630i
$159$	2642.75			1.31814
$160$	2755.01	+	6636.54i	1.36126	+	3.27916i
$161$	−2406.82			−1.17816
$162$		−	4613.06i		−	2.23726i
$163$			2727.24i			1.31052i	0.755406	+	0.655258i	$0.227440\pi$
$163$			2727.24i			1.31052i	−0.755406	+	0.655258i	$0.772560\pi$
$164$	4712.94			2.24402
$165$	1242.51	+	2993.08i	0.586237	+	1.41219i
$166$	450.827			0.210789
$167$			975.207i			0.451879i	0.974141	+	0.225940i	$0.0725452\pi$
$167$			975.207i			0.451879i	−0.974141	+	0.225940i	$0.927455\pi$
$168$		−	7094.26i		−	3.25794i
$169$	2071.63			0.942936
$170$	5067.59	−	2103.69i	2.28627	−	0.949093i
$171$	91.8414			0.0410719
$172$			11305.1i			5.01167i
$173$			918.398i			0.403610i	0.979426	+	0.201805i	$0.0646807\pi$
$173$			918.398i			0.403610i	−0.979426	+	0.201805i	$0.935319\pi$
$174$	−4984.79			−2.17181
$175$	−1499.07	−	1494.41i	−0.647537	−	0.645523i
$176$	11485.9			4.91923
$177$		−	919.800i		−	0.390601i
$178$		−	6756.92i		−	2.84524i
$179$	2559.18			1.06862			0.534308	−	0.845290i	$-0.320572\pi$
$179$	2559.18			1.06862			0.534308	+	0.845290i	$0.320572\pi$
$180$	1249.78	−	518.818i	0.517519	−	0.214836i
$181$	−801.727			−0.329237			−0.164619	−	0.986357i	$-0.552639\pi$
$181$	−801.727			−0.329237			−0.164619	+	0.986357i	$0.552639\pi$
$182$		−	1029.87i		−	0.419444i
$183$		−	1252.55i		−	0.505961i
$184$	−10424.4			−4.17660
$185$	970.770	+	2338.49i	0.385797	+	0.929348i
$186$	2172.25			0.856329
$187$		−	4584.88i		−	1.79294i
$188$			5511.89i			2.13828i
$189$	2067.20			0.795590
$190$	−379.902	−	915.147i	−0.145058	−	0.349430i
$191$	4365.70			1.65388			0.826940	−	0.562290i	$-0.190079\pi$
$191$	4365.70			1.65388			0.826940	+	0.562290i	$0.190079\pi$
$192$		−	9597.35i		−	3.60744i
$193$		−	360.181i		−	0.134334i	−0.997742	−	0.0671669i	$-0.978604\pi$
$193$		−	360.181i		−	0.134334i	0.997742	−	0.0671669i	$-0.0213960\pi$
$194$	−6797.14			−2.51549
$195$	660.431	−	274.163i	0.242536	−	0.100683i
$196$	−1209.54			−0.440796
$197$		−	3730.39i		−	1.34913i	−0.738214	−	0.674567i	$-0.764330\pi$
$197$		−	3730.39i		−	1.34913i	0.738214	−	0.674567i	$-0.235670\pi$
$198$		−	1551.42i		−	0.556842i
$199$	−3088.29			−1.10011			−0.550057	−	0.835127i	$-0.685394\pi$
$199$	−3088.29			−1.10011			−0.550057	+	0.835127i	$0.685394\pi$
$200$	−6492.72	−	6472.52i	−2.29552	−	2.28838i
$201$	−4608.85			−1.61733
$202$		−	5552.80i		−	1.93413i
$203$		−	2720.61i		−	0.940637i
$204$	−11097.8			−3.80882
$205$	−2263.22	+	939.522i	−0.771073	+	0.320093i
$206$	2290.16			0.774579
$207$			800.023i			0.268625i
$208$		−	2534.40i		−	0.844851i
$209$	−827.976			−0.274030
$210$	2252.13	+	5425.17i	0.740056	+	1.78272i
$211$	4260.36			1.39002			0.695012	−	0.718998i	$-0.255399\pi$
$211$	4260.36			1.39002			0.695012	+	0.718998i	$0.255399\pi$
$212$			9948.35i			3.22290i
$213$			3674.14i			1.18191i
$214$	−6102.67			−1.94939
$215$	−2253.67	−	5428.87i	−0.714879	−	1.72207i
$216$	8953.37			2.82037
$217$			1185.58i			0.370885i
$218$		−	5847.08i		−	1.81658i
$219$	4106.20			1.26699
$220$	−11267.1	+	4677.29i	−3.45287	+	1.43338i
$221$	−1011.67			−0.307928
$222$		−	7026.48i		−	2.12426i
$223$		−	1561.00i		−	0.468755i	−0.972146	−	0.234378i	$-0.924695\pi$
$223$		−	1561.00i		−	0.468755i	0.972146	−	0.234378i	$-0.0753052\pi$
$224$	10883.4			3.24632
$225$	−496.737	+	498.288i	−0.147181	+	0.147641i
$226$	−4331.44			−1.27488
$227$		−	2438.36i		−	0.712950i	−0.934305	−	0.356475i	$-0.883979\pi$
$227$		−	2438.36i		−	0.712950i	0.934305	−	0.356475i	$-0.116021\pi$
$228$			2004.12i			0.582133i
$229$	−6771.10			−1.95392			−0.976958	−	0.213429i	$-0.931537\pi$
$229$	−6771.10			−1.95392			−0.976958	+	0.213429i	$0.931537\pi$
$230$	7971.77	−	3309.30i	2.28540	−	0.948732i
$231$	4908.40			1.39805
$232$		−	11783.4i		−	3.33456i
$233$			2764.75i			0.777359i	0.921373	+	0.388679i	$0.127069\pi$
$233$			2764.75i			0.777359i	−0.921373	+	0.388679i	$0.872931\pi$
$234$	−342.325			−0.0956346
$235$	−1098.79	−	2646.89i	−0.305010	−	0.734740i
$236$	3462.49			0.955038
$237$			5443.51i			1.49196i
$238$		−	8310.42i		−	2.26338i
$239$	158.861			0.0429952			0.0214976	−	0.999769i	$-0.493157\pi$
$239$	158.861			0.0429952			0.0214976	+	0.999769i	$0.493157\pi$
$240$	5542.28	+	13350.8i	1.49064	+	3.59080i
$241$	6522.30			1.74331			0.871656	−	0.490119i	$-0.163047\pi$
$241$	6522.30			1.74331			0.871656	+	0.490119i	$0.163047\pi$
$242$			6756.96i			1.79485i
$243$		−	1555.24i		−	0.410570i
$244$	4715.08			1.23710
$245$	580.840	−	241.122i	0.151463	−	0.0628765i
$246$	6800.30			1.76248
$247$			182.695i			0.0470632i
$248$			5134.93i			1.31479i
$249$	474.109			0.120664
$250$	7019.90	+	2888.54i	1.77591	+	0.730748i
$251$	−3908.68			−0.982924			−0.491462	−	0.870899i	$-0.663537\pi$
$251$	−3908.68			−0.982924			−0.491462	+	0.870899i	$0.663537\pi$
$252$		−	2049.54i		−	0.512336i
$253$		−	7212.43i		−	1.79226i
$254$	−9646.71			−2.38303
$255$	5329.30	−	2212.33i	1.30876	−	0.543301i
$256$	8200.61			2.00210
$257$		−	2309.60i		−	0.560579i	−0.959915	−	0.280290i	$-0.909570\pi$
$257$		−	2309.60i		−	0.560579i	0.959915	−	0.280290i	$-0.0904305\pi$
$258$			16312.2i			3.93624i
$259$	3834.93			0.920042
$260$	1032.06	+	2486.13i	0.246175	+	0.593011i
$261$	−904.324			−0.214468
$262$			2942.96i			0.693956i
$263$			5385.33i			1.26264i	0.775523	+	0.631319i	$0.217486\pi$
$263$			5385.33i			1.26264i	−0.775523	+	0.631319i	$0.782514\pi$
$264$	21259.1			4.95608
$265$	−1983.20	−	4777.34i	−0.459725	−	1.10743i
$266$	−1500.76			−0.345931
$267$		−	7105.87i		−	1.62873i
$268$		−	17349.5i		−	3.95445i
$269$	5596.72			1.26854			0.634271	−	0.773111i	$-0.281300\pi$
$269$	5596.72			1.26854			0.634271	+	0.773111i	$0.281300\pi$
$270$	−6846.87	+	2842.32i	−1.54329	+	0.640660i
$271$	2451.43			0.549497			0.274749	−	0.961516i	$-0.411405\pi$
$271$	2451.43			0.549497			0.274749	+	0.961516i	$0.411405\pi$
$272$		−	20451.1i		−	4.55894i
$273$		−	1083.05i		−	0.240107i
$274$	−12572.8			−2.77208
$275$	4478.22	−	4492.20i	0.981989	−	0.985054i
$276$	−17457.8			−3.80737
$277$			4657.27i			1.01021i	0.863058	+	0.505105i	$0.168546\pi$
$277$			4657.27i			1.01021i	−0.863058	+	0.505105i	$0.831454\pi$
$278$		−	3411.59i		−	0.736021i
$279$	394.083			0.0845632
$280$	−12824.4	+	5323.75i	−2.73716	+	1.13627i
$281$	−5615.34			−1.19211			−0.596056	−	0.802943i	$-0.703266\pi$
$281$	−5615.34			−1.19211			−0.596056	+	0.802943i	$0.703266\pi$
$282$			7953.11i			1.67944i
$283$			3941.89i			0.827990i	0.910279	+	0.413995i	$0.135867\pi$
$283$			3941.89i			0.827990i	−0.910279	+	0.413995i	$0.864133\pi$
$284$	−13830.9			−2.88984
$285$	−399.521	−	962.408i	−0.0830372	−	0.200029i
$286$	3086.16			0.638071
$287$			3711.48i			0.763352i
$288$		−	3617.61i		−	0.740172i
$289$	−3250.56			−0.661625
$290$	3740.74	+	9011.07i	0.757461	+	1.82465i
$291$	−7148.16			−1.43997
$292$			15457.4i			3.09786i
$293$		−	9749.32i		−	1.94390i	−0.235197	−	0.971948i	$-0.575574\pi$
$293$		−	9749.32i		−	1.94390i	0.235197	−	0.971948i	$-0.424426\pi$
$294$	−1745.25			−0.346208
$295$	−1662.74	+	690.246i	−0.328164	+	0.136229i
$296$	16609.7			3.26155
$297$			6194.68i			1.21028i
$298$			7491.69i			1.45632i
$299$	−1591.44			−0.307811
$300$	−10873.4	−	10839.6i	−2.09259	−	2.08608i
$301$	−8902.89			−1.70483
$302$			1914.63i			0.364817i
$303$		−	5839.56i		−	1.10718i
$304$	−3693.23			−0.696781
$305$	−2264.25	+	939.948i	−0.425083	+	0.176463i
$306$	−2762.36			−0.516058
$307$		−	143.211i		−	0.0266236i	−0.999911	−	0.0133118i	$-0.995763\pi$
$307$		−	143.211i		−	0.0266236i	0.999911	−	0.0133118i	$-0.00423741\pi$
$308$			18477.2i			3.41829i
$309$	2408.43			0.443401
$310$	−1630.12	−	3926.81i	−0.298661	−	0.719445i
$311$	6811.30			1.24191			0.620954	−	0.783847i	$-0.286745\pi$
$311$	6811.30			1.24191			0.620954	+	0.783847i	$0.286745\pi$
$312$		−	4690.87i		−	0.851181i
$313$			4897.45i			0.884410i	0.896914	+	0.442205i	$0.145804\pi$
$313$			4897.45i			0.884410i	−0.896914	+	0.442205i	$0.854196\pi$
$314$	−1441.66			−0.259101
$315$	408.575	+	984.217i	0.0730812	+	0.176046i
$316$	−20491.5			−3.64790
$317$		−	9377.66i		−	1.66152i	−0.556631	−	0.830760i	$-0.687906\pi$
$317$		−	9377.66i		−	1.66152i	0.556631	−	0.830760i	$-0.312094\pi$
$318$			14354.5i			2.53132i
$319$	8152.73			1.43093
$320$	−17349.3	+	7202.14i	−3.03079	+	1.25816i
$321$	−6417.83			−1.11591
$322$		−	13073.0i		−	2.26252i
$323$			1474.24i			0.253960i
$324$	18262.2			3.13138
$325$	−991.216	−	988.132i	−0.169178	−	0.168651i
$326$	−14813.4			−2.51668
$327$		−	6149.03i		−	1.03989i
$328$			16075.0i			2.70609i
$329$	−4340.67			−0.727382
$330$	−16257.4	+	6748.87i	−2.71194	+	1.12580i
$331$	−3987.87			−0.662215			−0.331107	−	0.943593i	$-0.607422\pi$
$331$	−3987.87			−0.662215			−0.331107	+	0.943593i	$0.607422\pi$
$332$			1784.73i			0.295030i
$333$		−	1274.72i		−	0.209773i
$334$	−5296.98			−0.867779
$335$	3458.63	+	8331.50i	0.564074	+	1.35880i
$336$	21894.2			3.55484
$337$		−	966.436i		−	0.156217i	−0.996945	−	0.0781085i	$-0.975112\pi$
$337$		−	966.436i		−	0.156217i	0.996945	−	0.0781085i	$-0.0248881\pi$
$338$			11252.4i			1.81079i
$339$	−4555.13			−0.729795
$340$	8328.10	+	20061.6i	1.32840	+	3.19998i
$341$	−3552.77			−0.564203
$342$			498.850i			0.0788735i
$343$		−	6760.78i		−	1.06428i
$344$	−38559.9			−6.04363
$345$	8383.46	−	3480.20i	1.30826	−	0.543094i
$346$	−4988.42			−0.775084
$347$		−	8497.96i		−	1.31468i	−0.753593	−	0.657341i	$-0.771681\pi$
$347$		−	8497.96i		−	1.31468i	0.753593	−	0.657341i	$-0.228319\pi$
$348$		−	19733.8i		−	3.03977i
$349$	11168.6			1.71301			0.856504	−	0.516141i	$-0.172632\pi$
$349$	11168.6			1.71301			0.856504	+	0.516141i	$0.172632\pi$
$350$	8117.09	−	8142.42i	1.23965	−	1.24352i
$351$	1366.87			0.207858
$352$			32613.7i			4.93840i
$353$			12010.4i			1.81090i	0.424451	+	0.905451i	$0.360467\pi$
$353$			12010.4i			1.81090i	−0.424451	+	0.905451i	$0.639533\pi$
$354$	4996.03			0.750102
$355$	6641.80	−	2757.19i	0.992986	−	0.412215i
$356$	26749.3			3.98233
$357$		−	8739.59i		−	1.29565i
$358$			13900.6i			2.05215i
$359$	555.854			0.0817182			0.0408591	−	0.999165i	$-0.486991\pi$
$359$	555.854			0.0817182			0.0408591	+	0.999165i	$0.486991\pi$
$360$	1769.60	+	4262.81i	0.259073	+	0.624082i
$361$	−6592.77			−0.961185
$362$		−	4354.70i		−	0.632260i
$363$			7105.91i			1.02745i
$364$	4077.03			0.587073
$365$	−3081.42	−	7422.85i	−0.441888	−	1.06446i
$366$	6803.39			0.971636
$367$		−	3299.31i		−	0.469271i	−0.972083	−	0.234636i	$-0.924610\pi$
$367$		−	3299.31i		−	0.469271i	0.972083	−	0.234636i	$-0.0753897\pi$
$368$		−	32171.5i		−	4.55721i
$369$	1233.69			0.174047
$370$	−12701.9	+	5272.88i	−1.78470	+	0.740876i
$371$	−7834.43			−1.09634
$372$			8599.50i			1.19856i
$373$			4776.45i			0.663043i	0.943448	+	0.331522i	$0.107562\pi$
$373$			4776.45i			0.663043i	−0.943448	+	0.331522i	$0.892438\pi$
$374$	24903.5			3.44312
$375$	7382.43	+	3037.71i	1.01661	+	0.418311i
$376$	−18800.1			−2.57857
$377$		−	1798.92i		−	0.245754i
$378$			11228.3i			1.52783i
$379$	5771.41			0.782210			0.391105	−	0.920346i	$-0.372093\pi$
$379$	5771.41			0.782210			0.391105	+	0.920346i	$0.372093\pi$
$380$	3622.89	−	1503.96i	0.489079	−	0.203030i
$381$	−10144.9			−1.36414
$382$			23713.0i			3.17608i
$383$		−	5086.95i		−	0.678671i	−0.940665	−	0.339336i	$-0.889798\pi$
$383$		−	5086.95i		−	0.678671i	0.940665	−	0.339336i	$-0.110202\pi$
$384$	22759.6			3.02459
$385$	−3683.41	−	8872.99i	−0.487595	−	1.17457i
$386$	1956.38			0.257971
$387$			2959.30i			0.388707i
$388$		−	26908.5i		−	3.52081i
$389$	−6396.80			−0.833755			−0.416877	−	0.908963i	$-0.636876\pi$
$389$	−6396.80			−0.833755			−0.416877	+	0.908963i	$0.636876\pi$
$390$	1489.16	+	3587.23i	0.193350	+	0.465760i
$391$	−12842.0			−1.66099
$392$		−	4125.56i		−	0.531562i
$393$			3094.94i			0.397249i
$394$	20262.2			2.59085
$395$	9840.31	−	4084.97i	1.25347	−	0.520348i
$396$	6141.77			0.779382
$397$			10982.2i			1.38836i	0.719802	+	0.694179i	$0.244233\pi$
$397$			10982.2i			1.38836i	−0.719802	+	0.694179i	$0.755767\pi$
$398$		−	16774.5i		−	2.11264i
$399$	−1578.27			−0.198026
$400$	19975.4	−	20037.7i	2.49692	−	2.50472i
$401$	10978.7			1.36721			0.683607	−	0.729851i	$-0.260410\pi$
$401$	10978.7			1.36721			0.683607	+	0.729851i	$0.260410\pi$
$402$		−	25033.7i		−	3.10589i
$403$			783.927i			0.0968987i
$404$	21982.4			2.70710
$405$	−8769.76	+	3640.56i	−1.07598	+	0.446669i
$406$	14777.4			1.80638
$407$			11492.0i			1.39960i
$408$		−	37852.6i		−	4.59310i
$409$	−7617.67			−0.920953			−0.460476	−	0.887672i	$-0.652321\pi$
$409$	−7617.67			−0.920953			−0.460476	+	0.887672i	$0.652321\pi$
$410$	−5103.15	−	12293.0i	−0.614700	−	1.48075i
$411$	−13222.1			−1.58685
$412$			9066.30i			1.08414i
$413$			2726.75i			0.324878i
$414$	−4345.45			−0.515862
$415$	−355.786	−	857.054i	−0.0420840	−	0.101376i
$416$	7196.30			0.848144
$417$		−	3587.78i		−	0.421329i
$418$		−	4497.28i		−	0.526241i
$419$	3269.97			0.381261			0.190631	−	0.981662i	$-0.438947\pi$
$419$	3269.97			0.381261			0.190631	+	0.981662i	$0.438947\pi$
$420$	−21477.2	+	8915.74i	−2.49518	+	1.03582i
$421$	−12491.7			−1.44610			−0.723049	−	0.690797i	$-0.757260\pi$
$421$	−12491.7			−1.44610			−0.723049	+	0.690797i	$0.757260\pi$
$422$			23140.8i			2.66937i
$423$			1442.83i			0.165846i
$424$	−33932.2			−3.88654
$425$	−7998.54	−	7973.65i	−0.912908	−	0.910068i
$426$	−19956.6			−2.26972
$427$			3713.17i			0.420826i
$428$		−	24159.3i		−	2.72846i
$429$	3245.53			0.365258
$430$	29487.7	−	12241.1i	3.30703	−	1.37284i
$431$	−5673.97			−0.634120			−0.317060	−	0.948406i	$-0.602696\pi$
$431$	−5673.97			−0.634120			−0.317060	+	0.948406i	$0.602696\pi$
$432$			27631.7i			3.07739i
$433$			10027.0i			1.11286i	0.830896	+	0.556428i	$0.187828\pi$
$433$			10027.0i			1.11286i	−0.830896	+	0.556428i	$0.812172\pi$
$434$	−6439.63			−0.712240
$435$	3933.92	+	9476.43i	0.433602	+	1.04451i
$436$	23147.4			2.54257
$437$			2319.12i			0.253863i
$438$			22303.5i			2.43311i
$439$	6417.87			0.697741			0.348870	−	0.937171i	$-0.386565\pi$
$439$	6417.87			0.697741			0.348870	+	0.937171i	$0.386565\pi$
$440$	−15953.5	−	38430.4i	−1.72853	−	4.16385i
$441$	−316.618			−0.0341883
$442$		−	5495.02i		−	0.591338i
$443$		−	17033.0i		−	1.82677i	−0.407094	−	0.913386i	$-0.633458\pi$
$443$		−	17033.0i		−	1.82677i	0.407094	−	0.913386i	$-0.366542\pi$
$444$	27816.4			2.97322
$445$	−12845.4	+	5332.46i	−1.36838	+	0.568052i
$446$	8478.82			0.900188
$447$			7878.58i			0.833656i
$448$			28451.3i			3.00044i
$449$	−5362.91			−0.563677			−0.281839	−	0.959462i	$-0.590944\pi$
$449$	−5362.91			−0.563677			−0.281839	+	0.959462i	$0.590944\pi$
$450$	−2706.52	−	2698.10i	−0.283526	−	0.282644i
$451$	−11122.0			−1.16123
$452$		−	17147.3i		−	1.78438i
$453$			2013.51i			0.208836i
$454$	13244.3			1.36913
$455$	−1957.85	+	812.755i	−0.201726	+	0.0837418i
$456$	−6835.74			−0.702001
$457$			505.071i			0.0516985i	0.999666	+	0.0258492i	$0.00822899\pi$
$457$			505.071i			0.0516985i	−0.999666	+	0.0258492i	$0.991771\pi$
$458$		−	36778.2i		−	3.75226i
$459$	11029.9			1.12163
$460$	13100.8	+	31558.7i	1.32789	+	3.19876i
$461$	8884.40			0.897587			0.448794	−	0.893635i	$-0.351854\pi$
$461$	8884.40			0.897587			0.448794	+	0.893635i	$0.351854\pi$
$462$			26660.7i			2.68478i
$463$			13283.3i			1.33332i	0.745362	+	0.666660i	$0.232277\pi$
$463$			13283.3i			1.33332i	−0.745362	+	0.666660i	$0.767723\pi$
$464$	36365.7			3.63844
$465$	−1714.31	−	4129.60i	−0.170966	−	0.411840i
$466$	−15017.1			−1.49282
$467$			6150.88i			0.609484i	0.952435	+	0.304742i	$0.0985702\pi$
$467$			6150.88i			0.609484i	−0.952435	+	0.304742i	$0.901430\pi$
$468$		−	1355.20i		−	0.133855i
$469$	13662.9			1.34519
$470$	14377.0	−	5968.26i	1.41098	−	0.585734i
$471$	−1516.12			−0.148320
$472$			11810.0i			1.15169i
$473$		−	26678.9i		−	2.59344i
$474$	−29567.2			−2.86512
$475$	−1439.95	+	1444.44i	−0.139093	+	0.139527i
$476$	32899.3			3.16793
$477$			2604.15i			0.249970i
$478$			862.876i			0.0825670i
$479$	954.857			0.0910825			0.0455413	−	0.998962i	$-0.485499\pi$
$479$	954.857			0.0910825			0.0455413	+	0.998962i	$0.485499\pi$
$480$	−37909.0	+	15737.0i	−3.60479	+	1.49644i
$481$	2535.73			0.240373
$482$			35426.8i			3.34782i
$483$		−	13748.2i		−	1.29516i
$484$	−26749.5			−2.51216
$485$	5364.20	+	12921.8i	0.502218	+	1.20979i
$486$	8447.51			0.788450
$487$			116.784i			0.0108665i	0.999985	+	0.00543323i	$0.00172946\pi$
$487$			116.784i			0.0108665i	−0.999985	+	0.00543323i	$0.998271\pi$
$488$			16082.3i			1.49183i
$489$	−15578.4			−1.44066
$490$	1309.69	+	3154.92i	0.120747	+	0.290867i
$491$	7087.04			0.651393			0.325696	−	0.945474i	$-0.394401\pi$
$491$	7087.04			0.651393			0.325696	+	0.945474i	$0.394401\pi$
$492$			26921.0i			2.46686i
$493$		−	14516.3i		−	1.32612i
$494$	−992.336			−0.0903791
$495$	−2949.36	+	1224.36i	−0.267806	+	0.111173i
$496$	−15847.3			−1.43461
$497$		−	10892.0i		−	0.983042i
$498$			2575.19i			0.231721i
$499$	15393.0			1.38094			0.690468	−	0.723363i	$-0.257405\pi$
$499$	15393.0			1.38094			0.690468	+	0.723363i	$0.257405\pi$
$500$	−11435.1	+	27790.4i	−1.02279	+	2.48565i
$501$	−5570.54			−0.496753
$502$		−	21230.6i		−	1.88759i
$503$			3902.29i			0.345914i	0.984929	+	0.172957i	$0.0553321\pi$
$503$			3902.29i			0.345914i	−0.984929	+	0.172957i	$0.944668\pi$
$504$	6990.64			0.617833
$505$	−10556.3	+	4382.19i	−0.930193	+	0.386148i
$506$	39175.4			3.44182
$507$			11833.5i			1.03657i
$508$		−	38189.4i		−	3.33540i
$509$	−10935.8			−0.952303			−0.476152	−	0.879363i	$-0.657969\pi$
$509$	−10935.8			−0.952303			−0.476152	+	0.879363i	$0.657969\pi$
$510$	12016.6	+	28946.9i	1.04334	+	2.51331i
$511$	−12172.8			−1.05381
$512$			12667.6i			1.09342i
$513$		−	1991.86i		−	0.171429i
$514$	12544.9			1.07652
$515$	−1807.36	−	4353.76i	−0.154645	−	0.372524i
$516$	−64576.6			−5.50935
$517$		−	13007.5i		−	1.10652i
$518$			20830.0i			1.76683i
$519$	−5246.03			−0.443690
$520$	−8479.76	+	3520.18i	−0.715120	+	0.296865i
$521$	−863.633			−0.0726228			−0.0363114	−	0.999341i	$-0.511561\pi$
$521$	−863.633			−0.0726228			−0.0363114	+	0.999341i	$0.511561\pi$
$522$		−	4911.97i		−	0.411860i
$523$		−	3289.97i		−	0.275068i	−0.990497	−	0.137534i	$-0.956082\pi$
$523$		−	3289.97i		−	0.275068i	0.990497	−	0.137534i	$-0.0439176\pi$
$524$	−11650.6			−0.971294
$525$	8536.28	−	8562.92i	0.709626	−	0.711841i
$526$	−29251.2			−2.42474
$527$			6325.84i			0.522880i
$528$			65609.4i			5.40773i
$529$	−8034.62			−0.660362
$530$	25948.8	−	10772.0i	2.12669	−	0.882845i
$531$	906.364			0.0740732
$532$		−	5941.22i		−	0.484182i
$533$			2454.11i			0.199436i
$534$	38596.6			3.12779
$535$	4816.14	+	11601.6i	0.389196	+	0.937536i
$536$	59176.5			4.76872
$537$			14618.4i			1.17473i
$538$			30399.4i			2.43608i
$539$	2854.40			0.228104
$540$	−11252.2	−	27105.4i	−0.896697	−	2.16006i
$541$	2487.39			0.197673			0.0988366	−	0.995104i	$-0.468488\pi$
$541$	2487.39			0.197673			0.0988366	+	0.995104i	$0.468488\pi$
$542$			13315.3i			1.05524i
$543$		−	4579.59i		−	0.361932i
$544$	58070.0			4.57671
$545$	−11115.7	+	4614.42i	−0.873660	+	0.362679i
$546$	5882.75			0.461096
$547$		−	18868.9i		−	1.47491i	−0.675394	−	0.737457i	$-0.736027\pi$
$547$		−	18868.9i		−	1.47491i	0.675394	−	0.737457i	$-0.263973\pi$
$548$		−	49773.1i		−	3.87993i
$549$	1234.25			0.0959498
$550$	24400.1	+	24324.1i	1.89168	+	1.88579i
$551$	−2621.46			−0.202683
$552$		−	59545.5i		−	4.59135i
$553$		−	16137.3i		−	1.24092i
$554$	−25296.7			−1.93999
$555$	−13357.8	+	5545.19i	−1.02164	+	0.424109i
$556$	13505.8			1.03017
$557$			7279.63i			0.553766i	0.960904	+	0.276883i	$0.0893015\pi$
$557$			7279.63i			0.553766i	−0.960904	+	0.276883i	$0.910699\pi$
$558$			2140.52i			0.162393i
$559$	−5886.77			−0.445410
$560$	−16430.1	−	39578.5i	−1.23982	−	2.98660i
$561$	26189.6			1.97099
$562$		−	30500.6i		−	2.28930i
$563$			4082.14i			0.305580i	0.988259	+	0.152790i	$0.0488258\pi$
$563$			4082.14i			0.305580i	−0.988259	+	0.152790i	$0.951174\pi$
$564$	−31484.8			−2.35062
$565$	3418.31	+	8234.38i	0.254530	+	0.613138i
$566$	−21411.0			−1.59005
$567$			14381.7i			1.06521i
$568$		−	47175.0i		−	3.48489i
$569$	−11688.2			−0.861154			−0.430577	−	0.902554i	$-0.641690\pi$
$569$	−11688.2			−0.861154			−0.430577	+	0.902554i	$0.641690\pi$
$570$	5227.46	−	2170.06i	0.384130	−	0.159463i
$571$	−1359.11			−0.0996096			−0.0498048	−	0.998759i	$-0.515860\pi$
$571$	−1359.11			−0.0996096			−0.0498048	+	0.998759i	$0.515860\pi$
$572$			12217.5i			0.893074i
$573$			24937.6i			1.81812i
$574$	−20159.5			−1.46592
$575$	−12582.4	−	12543.3i	−0.912561	−	0.909722i
$576$	9457.16			0.684111
$577$			669.912i			0.0483342i	0.999708	+	0.0241671i	$0.00769337\pi$
$577$			669.912i			0.0483342i	−0.999708	+	0.0241671i	$0.992307\pi$
$578$		−	17655.9i		−	1.27057i
$579$	2057.41			0.147674
$580$	−35673.0	+	14808.8i	−2.55387	+	1.06018i
$581$	−1405.50			−0.100361
$582$		−	38826.3i		−	2.76530i
$583$		−	23477.1i		−	1.66779i
$584$	−52722.6			−3.73575
$585$	270.158	+	650.784i	0.0190934	+	0.0459942i
$586$	52954.9			3.73301
$587$		−	9749.87i		−	0.685554i	−0.939417	−	0.342777i	$-0.888632\pi$
$587$		−	9749.87i		−	0.685554i	0.939417	−	0.342777i	$-0.111368\pi$
$588$		−	6909.11i		−	0.484569i
$589$	1142.37			0.0799161
$590$	−3749.18	−	9031.40i	−0.261612	−	0.630198i
$591$	21308.6			1.48311
$592$			51260.6i			3.55878i
$593$			25333.1i			1.75431i	0.480205	+	0.877156i	$0.340562\pi$
$593$			25333.1i			1.75431i	−0.480205	+	0.877156i	$0.659438\pi$
$594$	−33647.3			−2.32419
$595$	−15798.7	+	6558.46i	−1.08854	+	0.451883i
$596$	−29658.1			−2.03833
$597$		−	17640.8i		−	1.20936i
$598$		−	8644.15i		−	0.591113i
$599$	9336.51			0.636861			0.318430	−	0.947946i	$-0.396844\pi$
$599$	9336.51			0.636861			0.318430	+	0.947946i	$0.396844\pi$
$600$	36972.0	−	37087.4i	2.51563	−	2.52348i
$601$	5696.77			0.386649			0.193325	−	0.981135i	$-0.438073\pi$
$601$	5696.77			0.386649			0.193325	+	0.981135i	$0.438073\pi$
$602$		−	48357.4i		−	3.27392i
$603$		−	4541.53i		−	0.306709i
$604$	−7579.64			−0.510615
$605$	12845.5	−	5332.50i	0.863211	−	0.358342i
$606$	31718.5			2.12619
$607$		−	8881.61i		−	0.593894i	−0.954894	−	0.296947i	$-0.904032\pi$
$607$		−	8881.61i		−	0.593894i	0.954894	−	0.296947i	$-0.0959684\pi$
$608$		−	10486.8i		−	0.699497i
$609$	15540.5			1.03405
$610$	−5105.47	−	12298.6i	−0.338876	−	0.816320i
$611$	−2870.14			−0.190038
$612$		−	10935.6i		−	0.722300i
$613$		−	8383.75i		−	0.552393i	−0.961101	−	0.276196i	$-0.910926\pi$
$613$		−	8383.75i		−	0.552393i	0.961101	−	0.276196i	$-0.0890740\pi$
$614$	777.869			0.0511274
$615$	−5366.70	−	12927.9i	−0.351880	−	0.847644i
$616$	−63022.5			−4.12216
$617$			4630.92i			0.302162i	0.988521	+	0.151081i	$0.0482754\pi$
$617$			4630.92i			0.302162i	−0.988521	+	0.151081i	$0.951725\pi$
$618$			13081.8i			0.851498i
$619$	9349.71			0.607102			0.303551	−	0.952815i	$-0.401828\pi$
$619$	9349.71			0.607102			0.303551	+	0.952815i	$0.401828\pi$
$620$	15545.5	−	6453.33i	1.00697	−	0.418019i
$621$	17351.0			1.12121
$622$			36996.6i			2.38493i
$623$			21065.3i			1.35468i
$624$	14476.9			0.928749
$625$	−48.6933	−	15624.9i	−0.00311637	−	0.999995i
$626$	−26601.2			−1.69840
$627$		−	4729.53i		−	0.301243i
$628$		−	5707.26i		−	0.362650i
$629$	20461.9			1.29709
$630$	−5345.92	+	2219.23i	−0.338074	+	0.140343i
$631$	5242.29			0.330733			0.165366	−	0.986232i	$-0.447119\pi$
$631$	5242.29			0.330733			0.165366	+	0.986232i	$0.447119\pi$
$632$		−	69893.2i		−	4.39905i
$633$			24335.8i			1.52806i
$634$	50936.2			3.19075
$635$	7613.04	+	18339.1i	0.475771	+	1.14609i
$636$	−56826.5			−3.54295
$637$		−	629.831i		−	0.0391755i
$638$			44282.8i			2.74792i
$639$	−3620.47			−0.224137
$640$	−17079.5	−	41142.8i	−1.05488	−	2.54111i
$641$	−795.339			−0.0490078			−0.0245039	−	0.999700i	$-0.507801\pi$
$641$	−795.339			−0.0490078			−0.0245039	+	0.999700i	$0.507801\pi$
$642$		−	34859.4i		−	2.14298i
$643$		−	25157.0i		−	1.54292i	−0.636280	−	0.771458i	$-0.719528\pi$
$643$		−	25157.0i		−	1.54292i	0.636280	−	0.771458i	$-0.280472\pi$
$644$	51753.5			3.16673
$645$	31010.6	−	12873.3i	1.89308	−	0.785870i
$646$	−8007.57			−0.487699
$647$			21290.9i			1.29371i	0.762612	+	0.646856i	$0.223917\pi$
$647$			21290.9i			1.29371i	−0.762612	+	0.646856i	$0.776083\pi$
$648$			62289.3i			3.77617i
$649$	−8171.13			−0.494214
$650$	5367.19	−	5383.94i	0.323874	−	0.324885i
$651$	−6772.19			−0.407716
$652$		−	58643.3i		−	3.52247i
$653$			12072.4i			0.723473i	0.932280	+	0.361736i	$0.117816\pi$
$653$			12072.4i			0.723473i	−0.932280	+	0.361736i	$0.882184\pi$
$654$	33399.4			1.99697
$655$	5594.77	−	2322.54i	0.333749	−	0.138548i
$656$	−49610.5			−2.95269
$657$			4046.22i			0.240271i
$658$		−	23577.0i		−	1.39685i
$659$	10344.4			0.611471			0.305736	−	0.952116i	$-0.401098\pi$
$659$	10344.4			0.611471			0.305736	+	0.952116i	$0.401098\pi$
$660$	−26717.4	−	64359.7i	−1.57572	−	3.79575i
$661$	−27742.2			−1.63245			−0.816223	−	0.577737i	$-0.803936\pi$
$661$	−27742.2			−1.63245			−0.816223	+	0.577737i	$0.803936\pi$
$662$		−	21660.7i		−	1.27170i
$663$		−	5778.80i		−	0.338507i
$664$	−6087.43			−0.355780
$665$	1184.38	+	2853.06i	0.0690651	+	0.166371i
$666$	6923.84			0.402843
$667$		−	22835.4i		−	1.32562i
$668$		−	20969.7i		−	1.21458i
$669$	8916.69			0.515305
$670$	−45253.8	+	18786.0i	−2.60941	+	1.08324i
$671$	−11127.1			−0.640174
$672$			62167.5i			3.56869i
$673$			19752.9i			1.13138i	0.824618	+	0.565689i	$0.191390\pi$
$673$			19752.9i			1.13138i	−0.824618	+	0.565689i	$0.808610\pi$
$674$	5249.34			0.299996
$675$	10806.9	+	10773.3i	0.616234	+	0.614317i
$676$	−44545.9			−2.53447
$677$		−	4090.68i		−	0.232227i	−0.993236	−	0.116113i	$-0.962956\pi$
$677$		−	4090.68i		−	0.232227i	0.993236	−	0.116113i	$-0.0370436\pi$
$678$		−	24741.9i		−	1.40148i
$679$	21190.7			1.19768
$680$	−68426.8	+	28405.8i	−3.85889	+	1.60193i
$681$	13928.3			0.783749
$682$		−	19297.4i		−	1.08348i
$683$			10438.9i			0.584822i	0.956293	+	0.292411i	$0.0944576\pi$
$683$			10438.9i			0.584822i	−0.956293	+	0.292411i	$0.905542\pi$
$684$	−1974.85			−0.110395
$685$	9922.25	+	23901.7i	0.553445	+	1.33320i
$686$	36722.2			2.04382
$687$		−	38677.6i		−	2.14795i
$688$		−	119003.i		−	6.59439i
$689$	−5180.28			−0.286434
$690$	18903.2	+	45536.0i	1.04295	+	2.51236i
$691$	−32006.9			−1.76208			−0.881042	−	0.473038i	$-0.843157\pi$
$691$	−32006.9			−1.76208			−0.881042	+	0.473038i	$0.843157\pi$
$692$		−	19748.2i		−	1.08484i
$693$			4836.70i			0.265124i
$694$	46158.0			2.52469
$695$	−6485.68	+	2692.38i	−0.353980	+	0.146946i
$696$	67308.6			3.66570
$697$			19803.2i			1.07619i
$698$			60663.7i			3.28962i
$699$	−15792.7			−0.854554
$700$	32234.2	+	32133.9i	1.74048	+	1.73507i
$701$	27100.4			1.46015			0.730077	−	0.683364i	$-0.239484\pi$
$701$	27100.4			1.46015			0.730077	+	0.683364i	$0.239484\pi$
$702$			7424.38i			0.399167i
$703$		−	3695.18i		−	0.198245i
$704$	−85258.9			−4.56437
$705$	15119.4	−	6276.48i	0.807703	−	0.335299i
$706$	−65236.2			−3.47762
$707$			17311.4i			0.920879i
$708$			19778.3i			1.04988i
$709$	−1290.48			−0.0683569			−0.0341785	−	0.999416i	$-0.510881\pi$
$709$	−1290.48			−0.0683569			−0.0341785	+	0.999416i	$0.510881\pi$
$710$	14976.1	+	36075.9i	0.791608	+	1.90691i
$711$	−5363.99			−0.282933
$712$			91237.5i			4.80234i
$713$			9951.10i			0.522681i
$714$	47470.4			2.48814
$715$	−2435.55	−	5867.00i	−0.127391	−	0.306872i
$716$	−55029.6			−2.87228
$717$			907.438i			0.0472648i
$718$			3019.20i			0.156930i
$719$	35368.8			1.83454			0.917270	−	0.398267i	$-0.130388\pi$
$719$	35368.8			1.83454			0.917270	+	0.398267i	$0.130388\pi$
$720$	−13155.8	+	5461.32i	−0.680955	+	0.282683i
$721$	−7139.80			−0.368793
$722$		−	35809.6i		−	1.84584i
$723$			37256.4i			1.91643i
$724$	17239.4			0.884941
$725$	14178.6	−	14222.8i	0.726315	−	0.728582i
$726$	−38596.8			−1.97309
$727$			21339.0i			1.08861i	0.838887	+	0.544305i	$0.183207\pi$
$727$			21339.0i			1.08861i	−0.838887	+	0.544305i	$0.816793\pi$
$728$			13906.1i			0.707959i
$729$	−14047.1			−0.713669
$730$	40318.3	−	16737.2i	2.04417	−	0.848591i
$731$	−47502.8			−2.40350
$732$			26933.2i			1.35995i
$733$			12359.9i			0.622815i	0.950277	+	0.311407i	$0.100800\pi$
$733$			12359.9i			0.622815i	−0.950277	+	0.311407i	$0.899200\pi$
$734$	17920.7			0.901178
$735$	1377.33	+	3317.85i	0.0691204	+	0.166504i
$736$	91349.3			4.57497
$737$			40943.2i			2.04635i
$738$			6700.97i			0.334236i
$739$	−35695.7			−1.77684			−0.888421	−	0.459029i	$-0.848197\pi$
$739$	−35695.7			−1.77684			−0.888421	+	0.459029i	$0.848197\pi$
$740$	−20874.3	−	50284.2i	−1.03697	−	2.49795i
$741$	−1043.58			−0.0517368
$742$		−	42553.9i		−	2.10539i
$743$		−	3339.05i		−	0.164869i	−0.996596	−	0.0824346i	$-0.973730\pi$
$743$		−	3339.05i		−	0.164869i	0.996596	−	0.0824346i	$-0.0262695\pi$
$744$	−29331.5			−1.44536
$745$	14242.2	−	5912.33i	0.700396	−	0.290753i
$746$	−25944.0			−1.27329
$747$			467.183i			0.0228827i
$748$			98587.9i			4.81916i
$749$	19025.7			0.928148
$750$	−16499.8	+	40098.8i	−0.803315	+	1.95227i
$751$	−9278.11			−0.450816			−0.225408	−	0.974264i	$-0.572372\pi$
$751$	−9278.11			−0.450816			−0.225408	+	0.974264i	$0.572372\pi$
$752$		−	58020.7i		−	2.81356i
$753$		−	22327.0i		−	1.08053i
$754$	9771.11			0.471940
$755$	3639.85	−	1511.00i	0.175454	−	0.0728356i
$756$	−44450.6			−2.13843
$757$		−	1472.08i		−	0.0706787i	−0.999375	−	0.0353393i	$-0.988749\pi$
$757$		−	1472.08i		−	0.0706787i	0.999375	−	0.0353393i	$-0.0112512\pi$
$758$			31348.3i			1.50214i
$759$	41198.5			1.97024
$760$	5129.75	+	12357.1i	0.244836	+	0.589787i
$761$	−8823.27			−0.420293			−0.210147	−	0.977670i	$-0.567394\pi$
$761$	−8823.27			−0.420293			−0.210147	+	0.977670i	$0.567394\pi$
$762$		−	55103.5i		−	2.61967i
$763$			18228.8i			0.864911i
$764$	−93874.9			−4.44539
$765$	2180.02	+	5251.45i	0.103031	+	0.248192i
$766$	27630.5			1.30331
$767$			1802.98i			0.0848786i
$768$			46843.2i			2.20092i
$769$	−27106.6			−1.27112			−0.635558	−	0.772053i	$-0.719230\pi$
$769$	−27106.6			−1.27112			−0.635558	+	0.772053i	$0.719230\pi$
$770$	48195.0	−	20007.0i	2.25562	−	0.936367i
$771$	13192.8			0.616247
$772$			7744.91i			0.361069i
$773$			18170.6i			0.845473i	0.906253	+	0.422736i	$0.138930\pi$
$773$			18170.6i			0.845473i	−0.906253	+	0.422736i	$0.861070\pi$
$774$	−16073.9			−0.746465
$775$	−6178.67	+	6197.96i	−0.286380	+	0.287274i
$776$	91780.5			4.24578
$777$			21905.7i			1.01141i
$778$		−	34745.2i		−	1.60112i
$779$	3576.23			0.164482
$780$	−14201.1	+	5895.27i	−0.651900	+	0.270621i
$781$	32639.5			1.49543
$782$		−	69753.3i		−	3.18974i
$783$			19613.1i			0.895164i
$784$	12732.2			0.580003
$785$	1137.74	+	2740.70i	0.0517295	+	0.124611i
$786$	−16810.6			−0.762869
$787$		−	39458.2i		−	1.78721i	−0.448854	−	0.893605i	$-0.648168\pi$
$787$		−	39458.2i		−	1.78721i	0.448854	−	0.893605i	$-0.351832\pi$
$788$			80213.9i			3.62627i
$789$	−30761.9			−1.38802
$790$	22188.1	+	53449.1i	0.999264	+	2.40713i
$791$	13503.7			0.606998
$792$			20948.5i			0.939866i
$793$			2455.22i			0.109946i
$794$	−59651.2			−2.66617
$795$	27288.9	−	11328.3i	1.21741	−	0.505377i
$796$	66406.8			2.95694
$797$			39744.6i			1.76640i	0.468992	+	0.883202i	$0.344617\pi$
$797$			39744.6i			1.76640i	−0.468992	+	0.883202i	$0.655383\pi$
$798$		−	8572.59i		−	0.380284i
$799$	−23160.3			−1.02547
$800$	56896.1	+	56719.1i	2.51448	+	2.50665i
$801$	7002.07			0.308872
$802$			59632.7i			2.62557i
$803$		−	36477.8i		−	1.60308i
$804$	99103.3			4.34715
$805$	−24852.8	+	10317.0i	−1.08813	+	0.451712i
$806$	−4258.02			−0.186082
$807$			31969.3i			1.39451i
$808$			74978.4i			3.26452i
$809$	−34428.6			−1.49622			−0.748112	−	0.663573i	$-0.769039\pi$
$809$	−34428.6			−1.49622			−0.748112	+	0.663573i	$0.769039\pi$
$810$	−19774.2	−	47634.2i	−0.857773	−	2.06629i
$811$	−8239.84			−0.356769			−0.178385	−	0.983961i	$-0.557087\pi$
$811$	−8239.84			−0.356769			−0.178385	+	0.983961i	$0.557087\pi$
$812$			58500.7i			2.52829i
$813$			14003.0i			0.604065i
$814$	−62420.3			−2.68775
$815$	11690.5	+	28161.3i	0.502456	+	1.21037i
$816$	116820.			5.01167
$817$			8578.45i			0.367346i
$818$		−	41376.5i		−	1.76858i
$819$	1067.23			0.0455337
$820$	48665.6	−	20202.4i	2.07253	−	0.860362i
$821$	−24987.6			−1.06221			−0.531104	−	0.847307i	$-0.678223\pi$
$821$	−24987.6			−1.06221			−0.531104	+	0.847307i	$0.678223\pi$
$822$		−	71817.7i		−	3.04736i
$823$		−	18313.9i		−	0.775678i	−0.921727	−	0.387839i	$-0.873222\pi$
$823$		−	18313.9i		−	0.775678i	0.921727	−	0.387839i	$-0.126778\pi$
$824$	−30923.7			−1.30737
$825$	25660.1	+	25580.3i	1.08287	+	1.07951i
$826$	−14810.7			−0.623888
$827$			1622.47i			0.0682209i	0.999418	+	0.0341104i	$0.0108598\pi$
$827$			1622.47i			0.0682209i	−0.999418	+	0.0341104i	$0.989140\pi$
$828$		−	17202.7i		−	0.722025i
$829$	6758.88			0.283167			0.141584	−	0.989926i	$-0.454781\pi$
$829$	6758.88			0.283167			0.141584	+	0.989926i	$0.454781\pi$
$830$	4655.22	−	1932.50i	0.194681	−	0.0808171i
$831$	−26603.1			−1.11053
$832$			18812.6i			0.783906i
$833$		−	5082.37i		−	0.211397i
$834$	19487.6			0.809111
$835$	4180.30	+	10069.9i	0.173252	+	0.417347i
$836$	17803.8			0.736552
$837$		−	8546.90i		−	0.352956i
$838$			17761.3i			0.732166i
$839$	28954.2			1.19143			0.595715	−	0.803196i	$-0.296869\pi$
$839$	28954.2			1.19143			0.595715	+	0.803196i	$0.296869\pi$
$840$	−30410.1	−	73255.0i	−1.24911	−	3.00897i
$841$	1423.47			0.0583650
$842$		−	67850.4i		−	2.77705i
$843$		−	32075.7i		−	1.31049i
$844$	−91609.7			−3.73618
$845$	21391.5	−	8880.20i	0.870878	−	0.361524i
$846$	−7836.94			−0.318486
$847$		−	21065.5i		−	0.854567i
$848$		−	104721.i		−	4.24072i
$849$	−22516.7			−0.910213
$850$	43310.1	−	43445.2i	1.74767	−	1.75313i
$851$	32188.4			1.29660
$852$		−	79004.3i		−	3.17681i
$853$		−	32830.1i		−	1.31780i	−0.752231	−	0.658899i	$-0.771022\pi$
$853$		−	32830.1i		−	1.31780i	0.752231	−	0.658899i	$-0.228978\pi$
$854$	−20168.6			−0.808146
$855$	948.350	−	393.685i	0.0379332	−	0.0157471i
$856$	82403.3			3.29029
$857$			28174.0i			1.12299i	0.827479	+	0.561497i	$0.189774\pi$
$857$			28174.0i			1.12299i	−0.827479	+	0.561497i	$0.810226\pi$
$858$			17628.6i			0.701434i
$859$	−788.825			−0.0313322			−0.0156661	−	0.999877i	$-0.504987\pi$
$859$	−788.825			−0.0313322			−0.0156661	+	0.999877i	$0.504987\pi$
$860$	48460.2	+	116736.i	1.92149	+	4.62868i
$861$	−21200.6			−0.839156
$862$		−	30819.0i		−	1.21775i
$863$		−	3457.73i		−	0.136388i	−0.997672	−	0.0681938i	$-0.978276\pi$
$863$		−	3457.73i		−	0.136388i	0.997672	−	0.0681938i	$-0.0217236\pi$
$864$	−78459.0			−3.08938
$865$	3936.79	+	9483.34i	0.154745	+	0.372767i
$866$	−54463.1			−2.13710
$867$		−	18567.7i		−	0.727327i
$868$		−	25493.2i		−	0.996885i
$869$	48357.9			1.88772
$870$	−51472.6	+	21367.7i	−2.00585	+	0.832680i
$871$	9034.22			0.351450
$872$			78952.0i			3.06611i
$873$		−	7043.75i		−	0.273075i
$874$	−12596.6			−0.487514
$875$	−21885.2	−	9005.28i	−0.845549	−	0.347925i
$876$	−88294.9			−3.40549
$877$			23671.4i			0.911432i	0.890125	+	0.455716i	$0.150617\pi$
$877$			23671.4i			0.911432i	−0.890125	+	0.455716i	$0.849383\pi$
$878$			34859.6i			1.33993i
$879$	55689.6			2.13693
$880$	118603.	−	49235.3i	4.54331	−	1.88605i
$881$	14351.7			0.548832			0.274416	−	0.961611i	$-0.411515\pi$
$881$	14351.7			0.548832			0.274416	+	0.961611i	$0.411515\pi$
$882$		−	1719.76i		−	0.0656546i
$883$			17104.0i			0.651864i	0.945393	+	0.325932i	$0.105678\pi$
$883$			17104.0i			0.651864i	−0.945393	+	0.325932i	$0.894322\pi$
$884$	21753.7			0.827665
$885$	−3942.79	−	9497.81i	−0.149758	−	0.360752i
$886$	92517.0			3.50809
$887$			680.407i			0.0257563i	0.999917	+	0.0128781i	$0.00409935\pi$
$887$			680.407i			0.0257563i	−0.999917	+	0.0128781i	$0.995901\pi$
$888$			94877.2i			3.58544i
$889$	30074.5			1.13461
$890$	−28964.1	−	69771.7i	−1.09087	−	2.62781i
$891$	−43096.9			−1.62043
$892$			33565.9i			1.25994i
$893$			4182.49i			0.156732i
$894$	−42793.7			−1.60093
$895$	26426.0	−	10970.1i	0.986953	−	0.409711i
$896$	−67470.7			−2.51567
$897$		−	9090.56i		−	0.338378i
$898$		−	29129.4i		−	1.08247i
$899$	−11248.5			−0.417305
$900$	10681.2	−	10714.6i	0.395602	−	0.396837i
$901$	−41801.9			−1.54564
$902$		−	60411.1i		−	2.23001i
$903$		−	50854.7i		−	1.87413i
$904$	58486.6			2.15181
$905$	−8278.60	+	3436.67i	−0.304077	+	0.126231i
$906$	−10936.7			−0.401045
$907$		−	21786.8i		−	0.797596i	−0.917039	−	0.398798i	$-0.869427\pi$
$907$		−	21786.8i		−	0.797596i	0.917039	−	0.398798i	$-0.130573\pi$
$908$			52431.6i			1.91630i
$909$	5754.26			0.209964
$910$	−4414.60	−	10634.3i	−0.160816	−	0.387390i
$911$	36241.5			1.31804			0.659020	−	0.752125i	$-0.270971\pi$
$911$	36241.5			1.31804			0.659020	+	0.752125i	$0.270971\pi$
$912$		−	21096.3i		−	0.765975i
$913$		−	4211.79i		−	0.152672i
$914$	−2743.37			−0.0992806
$915$	−5369.13	−	12933.7i	−0.193987	−	0.467296i
$916$	145598.			5.25184
$917$		−	9174.94i		−	0.330407i
$918$			59910.4i			2.15396i
$919$	38708.3			1.38941			0.694706	−	0.719293i	$-0.255534\pi$
$919$	38708.3			1.38941			0.694706	+	0.719293i	$0.255534\pi$
$920$	−107641.	+	44684.8i	−3.85742	+	1.60132i
$921$	818.041			0.0292675
$922$			48256.9i			1.72371i
$923$		−	7202.00i		−	0.256833i
$924$	−105544.			−3.75774
$925$	20048.3	+	19985.9i	0.712630	+	0.710412i
$926$	−72150.2			−2.56048
$927$			2373.25i			0.0840862i
$928$			103259.i			3.65262i
$929$	32082.9			1.13305			0.566526	−	0.824044i	$-0.308287\pi$
$929$	32082.9			1.13305			0.566526	+	0.824044i	$0.308287\pi$
$930$	22430.5	−	9311.52i	0.790889	−	0.328319i
$931$	−917.816			−0.0323096
$932$		−	59449.8i		−	2.08943i
$933$			38907.2i			1.36523i
$934$	−33409.4			−1.17044
$935$	−19653.5	−	47343.3i	−0.687419	−	1.65593i
$936$	4622.35			0.161417
$937$			218.425i			0.00761539i	0.999993	+	0.00380769i	$0.00121203\pi$
$937$			218.425i			0.00761539i	−0.999993	+	0.00380769i	$0.998788\pi$
$938$			74212.3i			2.58328i
$939$	−27975.0			−0.972236
$940$	23627.1	+	56915.5i	0.819822	+	1.97487i
$941$	−10322.1			−0.357589			−0.178795	−	0.983886i	$-0.557220\pi$
$941$	−10322.1			−0.357589			−0.178795	+	0.983886i	$0.557220\pi$
$942$		−	8235.01i		−	0.284831i
$943$			31152.3i			1.07578i
$944$	−36447.8			−1.25665
$945$	21345.8	−	8861.20i	0.734792	−	0.305032i
$946$	144910.			4.98039
$947$			27221.7i			0.934093i	0.884233	+	0.467047i	$0.154682\pi$
$947$			27221.7i			0.934093i	−0.884233	+	0.467047i	$0.845318\pi$
$948$		−	117051.i		−	4.01016i
$949$	−8048.93			−0.275321
$950$	−7845.70	−	7821.29i	−0.267945	−	0.267112i
$951$	53566.7			1.82652
$952$			112214.i			3.82025i
$953$			40514.2i			1.37711i	0.725186	+	0.688553i	$0.241754\pi$
$953$			40514.2i			1.37711i	−0.725186	+	0.688553i	$0.758246\pi$
$954$	−14144.8			−0.480037
$955$	45080.0	−	18713.9i	1.52749	−	0.634103i
$956$	−3415.95			−0.115565
$957$			46569.7i			1.57302i
$958$			5186.45i			0.174913i
$959$	39196.8			1.31985
$960$	−41139.8	−	99101.7i	−1.38310	−	3.33176i
$961$	−24889.2			−0.835460
$962$			13773.2i			0.461607i
$963$		−	6324.09i		−	0.211621i
$964$	−140248.			−4.68576
$965$	−1543.94	−	3719.21i	−0.0515040	−	0.124068i
$966$	74675.2			2.48720
$967$			41649.8i			1.38508i	0.721382	+	0.692538i	$0.243507\pi$
$967$			41649.8i			1.38508i	−0.721382	+	0.692538i	$0.756493\pi$
$968$		−	91238.0i		−	3.02944i
$969$	−8421.11			−0.279179
$970$	−70186.9	+	29136.4i	−2.32326	+	0.964448i
$971$	−39344.6			−1.30034			−0.650170	−	0.759789i	$-0.725302\pi$
$971$	−39344.6			−1.30034			−0.650170	+	0.759789i	$0.725302\pi$
$972$			33442.0i			1.10355i
$973$			10636.0i			0.350435i
$974$	−634.327			−0.0208677
$975$	5644.36	−	5661.98i	0.185399	−	0.185978i
$976$	−49633.1			−1.62778
$977$		−	8834.33i		−	0.289289i	−0.989484	−	0.144644i	$-0.953796\pi$
$977$		−	8834.33i		−	0.289289i	0.989484	−	0.144644i	$-0.0462038\pi$
$978$		−	84616.5i		−	2.76660i
$979$	−63125.6			−2.06078
$980$	−12489.7	+	5184.81i	−0.407111	+	0.169003i
$981$	6059.21			0.197203
$982$			38494.3i			1.25092i
$983$			31253.8i			1.01408i	0.861922	+	0.507041i	$0.169261\pi$
$983$			31253.8i			1.01408i	−0.861922	+	0.507041i	$0.830739\pi$
$984$	−91823.2			−2.97481
$985$	−15990.6	−	38519.8i	−0.517262	−	1.24603i
$986$	78847.2			2.54666
$987$		−	24794.6i		−	0.799615i
$988$		−	3928.46i		−	0.126499i
$989$	−74726.2			−2.40258
$990$	−6650.28	−	16019.9i	−0.213495	−	0.514289i
$991$	−28789.8			−0.922844			−0.461422	−	0.887181i	$-0.652661\pi$
$991$	−28789.8			−0.922844			−0.461422	+	0.887181i	$0.652661\pi$
$992$		−	44997.7i		−	1.44020i
$993$		−	22779.3i		−	0.727976i
$994$	59161.4			1.88781
$995$	−31889.5	+	13238.2i	−1.01604	+	0.421787i
$996$	−10194.7			−0.324328
$997$		−	29380.4i		−	0.933285i	−0.884446	−	0.466643i	$-0.845463\pi$
$997$		−	29380.4i		−	0.933285i	0.884446	−	0.466643i	$-0.154537\pi$
$998$			83609.6i			2.65192i
$999$	−27646.3			−0.875565

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	415.4.b.a.84.122	yes	124
5.2	odd	4		2075.4.a.m.1.3		62
5.3	odd	4		2075.4.a.n.1.60		62
5.4	even	2	inner	415.4.b.a.84.3	✓	124

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.4.b.a.84.3	✓	124	5.4	even	2	inner
415.4.b.a.84.122	yes	124	1.1	even	1	trivial
2075.4.a.m.1.3		62	5.2	odd	4
2075.4.a.n.1.60		62	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 415 = 5 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	415.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.43165i q^{2} +5.71216i q^{3} -21.5028 q^{4} +(10.3260 - 4.28658i) q^{5} -31.0264 q^{6} -16.9337i q^{7} -73.3426i q^{8} -5.62872 q^{9} +(23.2832 + 56.0869i) q^{10} +50.7444 q^{11} -122.827i q^{12} -11.1969i q^{13} +91.9778 q^{14} +(24.4856 + 58.9834i) q^{15} +226.349 q^{16} -90.3524i q^{17} -30.5732i q^{18} -16.3166 q^{19} +(-222.037 + 92.1735i) q^{20} +96.7278 q^{21} +275.626i q^{22} -142.132i q^{23} +418.944 q^{24} +(88.2505 - 88.5260i) q^{25} +60.8176 q^{26} +122.076i q^{27} +364.122i q^{28} +160.663 q^{29} +(-320.377 + 132.997i) q^{30} -70.0129 q^{31} +642.705i q^{32} +289.860i q^{33} +490.763 q^{34} +(-72.5875 - 174.856i) q^{35} +121.033 q^{36} +226.468i q^{37} -88.6260i q^{38} +63.9584 q^{39} +(-314.389 - 757.332i) q^{40} -219.178 q^{41} +525.391i q^{42} -525.750i q^{43} -1091.15 q^{44} +(-58.1219 + 24.1279i) q^{45} +772.013 q^{46} -256.333i q^{47} +1292.94i q^{48} +56.2505 q^{49} +(480.842 + 479.346i) q^{50} +516.107 q^{51} +240.765i q^{52} -462.654i q^{53} -663.074 q^{54} +(523.985 - 217.520i) q^{55} -1241.96 q^{56} -93.2029i q^{57} +872.663i q^{58} -161.025 q^{59} +(-526.509 - 1268.31i) q^{60} -219.277 q^{61} -380.286i q^{62} +95.3149i q^{63} -1680.16 q^{64} +(-47.9964 - 115.619i) q^{65} -1574.42 q^{66} +806.850i q^{67} +1942.83i q^{68} +811.882 q^{69} +(949.758 - 394.270i) q^{70} +643.214 q^{71} +412.824i q^{72} -718.854i q^{73} -1230.09 q^{74} +(505.674 + 504.101i) q^{75} +350.852 q^{76} -859.290i q^{77} +347.400i q^{78} +952.969 q^{79} +(2337.26 - 970.260i) q^{80} -849.293 q^{81} -1190.50i q^{82} -83.0000i q^{83} -2079.92 q^{84} +(-387.303 - 932.975i) q^{85} +2855.69 q^{86} +917.730i q^{87} -3721.73i q^{88} -1243.99 q^{89} +(-131.054 - 315.698i) q^{90} -189.605 q^{91} +3056.25i q^{92} -399.925i q^{93} +1392.31 q^{94} +(-168.484 + 69.9423i) q^{95} -3671.23 q^{96} +1251.39i q^{97} +305.533i q^{98} -285.626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 504 q^{4} + 6 q^{5} - 1088 q^{9} + 62 q^{10} + 60 q^{11} + 124 q^{14} - 120 q^{15} + 2080 q^{16} + 124 q^{19} - 206 q^{20} + 24 q^{21} + 368 q^{24} - 430 q^{25} - 176 q^{26} + 208 q^{29} - 52 q^{30}+ \cdots + 2668 q^{99}+O(q^{100}) \) 124 * q - 504 * q^4 + 6 * q^5 - 1088 * q^9 + 62 * q^10 + 60 * q^11 + 124 * q^14 - 120 * q^15 + 2080 * q^16 + 124 * q^19 - 206 * q^20 + 24 * q^21 + 368 * q^24 - 430 * q^25 - 176 * q^26 + 208 * q^29 - 52 * q^30 + 372 * q^31 + 496 * q^34 + 320 * q^35 + 4592 * q^36 - 592 * q^39 - 430 * q^40 - 384 * q^41 - 636 * q^44 - 118 * q^45 - 1216 * q^46 - 6504 * q^49 + 440 * q^50 - 696 * q^51 + 8 * q^54 - 684 * q^55 - 1328 * q^56 - 1068 * q^59 + 4698 * q^60 + 952 * q^61 - 9180 * q^64 - 16 * q^65 - 284 * q^66 + 3496 * q^69 - 1296 * q^70 - 2976 * q^71 - 1732 * q^74 + 1796 * q^75 + 804 * q^76 + 1080 * q^79 - 1758 * q^80 + 8700 * q^81 + 6752 * q^84 + 1036 * q^85 - 2692 * q^86 + 832 * q^89 - 5586 * q^90 + 1464 * q^91 + 2620 * q^94 + 1240 * q^95 + 5892 * q^96 + 2668 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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