LMFDB - Embedded newform 415.4.b.a.84.3

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

$q$-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	$-0.590119\pi$
$2$		−	5.43165i		−	1.92038i	0.279351	−	0.960189i	$-0.409881\pi$
$3$		−	5.71216i		−	1.09930i	−0.835393	−	0.549652i	$-0.814760\pi$
$3$		−	5.71216i		−	1.09930i	0.835393	−	0.549652i	$-0.185240\pi$
$4$	−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	$0.151136\pi$
$7$			16.9337i			0.914333i	−0.889381	+	0.457166i	$0.848864\pi$
$8$			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	$0.0381102\pi$
$13$			11.1969i			0.238882i	−0.992841	+	0.119441i	$0.961890\pi$
$14$	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	$0.222943\pi$
$17$			90.3524i			1.28904i	−0.764587	+	0.644520i	$0.777057\pi$
$18$			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	$0.222841\pi$
$23$			142.132i			1.28855i	−0.764794	+	0.644275i	$0.777159\pi$
$24$	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	$-0.832185\pi$
$37$		−	226.468i		−	1.00624i	0.864215	−	0.503122i	$-0.167815\pi$
$38$			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	$0.382185\pi$
$43$			525.750i			1.86456i	−0.361734	+	0.932281i	$0.617815\pi$
$44$	−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	$0.130215\pi$
$47$			256.333i			0.795533i	−0.917487	+	0.397767i	$0.869785\pi$
$48$		−	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	$0.204646\pi$
$53$			462.654i			1.19906i	−0.800351	+	0.599532i	$0.795354\pi$
$54$	−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	$-0.736893\pi$
$67$		−	806.850i		−	1.47123i	0.677400	−	0.735615i	$-0.263107\pi$
$68$		−	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	$0.195492\pi$
$73$			718.854i			1.15254i	−0.817259	+	0.576270i	$0.804508\pi$
$74$	−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	$-0.772691\pi$
$97$		−	1251.39i		−	1.30990i	0.755674	−	0.654948i	$-0.227309\pi$
$98$		−	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	$0.0646380\pi$
$103$			421.633i			0.403347i	−0.979453	+	0.201674i	$0.935362\pi$
$104$	−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	$-0.830550\pi$
$107$		−	1123.54i		−	1.01511i	0.861620	−	0.507555i	$-0.169450\pi$
$108$			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	$-0.892299\pi$
$113$		−	797.445i		−	0.663870i	0.943302	−	0.331935i	$-0.107701\pi$
$114$	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	$-0.786947\pi$
$127$		−	1776.02i		−	1.24091i	0.784240	−	0.620457i	$-0.213053\pi$
$128$			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	$-0.743337\pi$
$137$		−	2314.73i		−	1.44351i	0.692150	−	0.721753i	$-0.256663\pi$
$138$		−	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	$-0.978510\pi$
$157$		−	265.419i		−	0.134922i	0.997722	−	0.0674610i	$-0.0214898\pi$
$158$		−	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	$-0.772560\pi$
$163$		−	2727.24i		−	1.31052i	0.755406	−	0.655258i	$-0.227440\pi$
$164$	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	$-0.927455\pi$
$167$		−	975.207i		−	0.451879i	0.974141	−	0.225940i	$-0.0725452\pi$
$168$			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	$-0.935319\pi$
$173$		−	918.398i		−	0.403610i	0.979426	−	0.201805i	$-0.0646807\pi$
$174$	−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	$0.0213960\pi$
$193$			360.181i			0.134334i	−0.997742	+	0.0671669i	$0.978604\pi$
$194$	−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	$0.235670\pi$
$197$			3730.39i			1.34913i	−0.738214	+	0.674567i	$0.764330\pi$
$198$			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	$0.0753052\pi$
$223$			1561.00i			0.468755i	−0.972146	+	0.234378i	$0.924695\pi$
$224$	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	$0.116021\pi$
$227$			2438.36i			0.712950i	−0.934305	+	0.356475i	$0.883979\pi$
$228$		−	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	$-0.872931\pi$
$233$		−	2764.75i		−	0.777359i	0.921373	−	0.388679i	$-0.127069\pi$
$234$	−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	$0.0904305\pi$
$257$			2309.60i			0.560579i	−0.959915	+	0.280290i	$0.909570\pi$
$258$		−	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	$-0.782514\pi$
$263$		−	5385.33i		−	1.26264i	0.775523	−	0.631319i	$-0.217486\pi$
$264$	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	$-0.831454\pi$
$277$		−	4657.27i		−	1.01021i	0.863058	−	0.505105i	$-0.168546\pi$
$278$			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	$-0.864133\pi$
$283$		−	3941.89i		−	0.827990i	0.910279	−	0.413995i	$-0.135867\pi$
$284$	−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	$0.424426\pi$
$293$			9749.32i			1.94390i	−0.235197	+	0.971948i	$0.575574\pi$
$294$	−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	$0.00423741\pi$
$307$			143.211i			0.0266236i	−0.999911	+	0.0133118i	$0.995763\pi$
$308$		−	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	$-0.854196\pi$
$313$		−	4897.45i		−	0.884410i	0.896914	−	0.442205i	$-0.145804\pi$
$314$	−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	$0.312094\pi$
$317$			9377.66i			1.66152i	−0.556631	+	0.830760i	$0.687906\pi$
$318$		−	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	$0.0248881\pi$
$337$			966.436i			0.156217i	−0.996945	+	0.0781085i	$0.975112\pi$
$338$		−	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	$0.228319\pi$
$347$			8497.96i			1.31468i	−0.753593	+	0.657341i	$0.771681\pi$
$348$			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	$-0.639533\pi$
$353$		−	12010.4i		−	1.81090i	0.424451	−	0.905451i	$-0.360467\pi$
$354$	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	$0.0753897\pi$
$367$			3299.31i			0.469271i	−0.972083	+	0.234636i	$0.924610\pi$
$368$			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	$-0.892438\pi$
$373$		−	4776.45i		−	0.663043i	0.943448	−	0.331522i	$-0.107562\pi$
$374$	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	$0.110202\pi$
$383$			5086.95i			0.678671i	−0.940665	+	0.339336i	$0.889798\pi$
$384$	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	$-0.755767\pi$
$397$		−	10982.2i		−	1.38836i	0.719802	−	0.694179i	$-0.244233\pi$
$398$			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	$-0.812172\pi$
$433$		−	10027.0i		−	1.11286i	0.830896	−	0.556428i	$-0.187828\pi$
$434$	−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	$0.366542\pi$
$443$			17033.0i			1.82677i	−0.407094	+	0.913386i	$0.633458\pi$
$444$	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	$-0.991771\pi$
$457$		−	505.071i		−	0.0516985i	0.999666	−	0.0258492i	$-0.00822899\pi$
$458$			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	$-0.767723\pi$
$463$		−	13283.3i		−	1.33332i	0.745362	−	0.666660i	$-0.232277\pi$
$464$	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	$-0.901430\pi$
$467$		−	6150.88i		−	0.609484i	0.952435	−	0.304742i	$-0.0985702\pi$
$468$			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	$-0.998271\pi$
$487$		−	116.784i		−	0.0108665i	0.999985	−	0.00543323i	$-0.00172946\pi$
$488$		−	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	$-0.944668\pi$
$503$		−	3902.29i		−	0.345914i	0.984929	−	0.172957i	$-0.0553321\pi$
$504$	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	$0.0439176\pi$
$523$			3289.97i			0.275068i	−0.990497	+	0.137534i	$0.956082\pi$
$524$	−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	$0.263973\pi$
$547$			18868.9i			1.47491i	−0.675394	+	0.737457i	$0.736027\pi$
$548$			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	$-0.910699\pi$
$557$		−	7279.63i		−	0.553766i	0.960904	−	0.276883i	$-0.0893015\pi$
$558$		−	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	$-0.951174\pi$
$563$		−	4082.14i		−	0.305580i	0.988259	−	0.152790i	$-0.0488258\pi$
$564$	−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	$-0.992307\pi$
$577$		−	669.912i		−	0.0483342i	0.999708	−	0.0241671i	$-0.00769337\pi$
$578$			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	$0.111368\pi$
$587$			9749.87i			0.685554i	−0.939417	+	0.342777i	$0.888632\pi$
$588$			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	$-0.659438\pi$
$593$		−	25333.1i		−	1.75431i	0.480205	−	0.877156i	$-0.340562\pi$
$594$	−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	$0.0959684\pi$
$607$			8881.61i			0.593894i	−0.954894	+	0.296947i	$0.904032\pi$
$608$			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	$0.0890740\pi$
$613$			8383.75i			0.552393i	−0.961101	+	0.276196i	$0.910926\pi$
$614$	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	$-0.951725\pi$
$617$		−	4630.92i		−	0.302162i	0.988521	−	0.151081i	$-0.0482754\pi$
$618$		−	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	$0.280472\pi$
$643$			25157.0i			1.54292i	−0.636280	+	0.771458i	$0.719528\pi$
$644$	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	$-0.776083\pi$
$647$		−	21290.9i		−	1.29371i	0.762612	−	0.646856i	$-0.223917\pi$
$648$		−	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	$-0.882184\pi$
$653$		−	12072.4i		−	0.723473i	0.932280	−	0.361736i	$-0.117816\pi$
$654$	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	$-0.808610\pi$
$673$		−	19752.9i		−	1.13138i	0.824618	−	0.565689i	$-0.191390\pi$
$674$	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	$0.0370436\pi$
$677$			4090.68i			0.232227i	−0.993236	+	0.116113i	$0.962956\pi$
$678$			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	$-0.905542\pi$
$683$		−	10438.9i		−	0.584822i	0.956293	−	0.292411i	$-0.0944576\pi$
$684$	−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	$-0.816793\pi$
$727$		−	21339.0i		−	1.08861i	0.838887	−	0.544305i	$-0.183207\pi$
$728$		−	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	$-0.899200\pi$
$733$		−	12359.9i		−	0.622815i	0.950277	−	0.311407i	$-0.100800\pi$
$734$	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	$0.0262695\pi$
$743$			3339.05i			0.164869i	−0.996596	+	0.0824346i	$0.973730\pi$
$744$	−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	$0.0112512\pi$
$757$			1472.08i			0.0706787i	−0.999375	+	0.0353393i	$0.988749\pi$
$758$		−	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	$-0.861070\pi$
$773$		−	18170.6i		−	0.845473i	0.906253	−	0.422736i	$-0.138930\pi$
$774$	−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	$0.351832\pi$
$787$			39458.2i			1.78721i	−0.448854	+	0.893605i	$0.648168\pi$
$788$		−	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	$-0.655383\pi$
$797$		−	39744.6i		−	1.76640i	0.468992	−	0.883202i	$-0.344617\pi$
$798$			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	$0.126778\pi$
$823$			18313.9i			0.775678i	−0.921727	+	0.387839i	$0.873222\pi$
$824$	−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	$-0.989140\pi$
$827$		−	1622.47i		−	0.0682209i	0.999418	−	0.0341104i	$-0.0108598\pi$
$828$			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	$0.228978\pi$
$853$			32830.1i			1.31780i	−0.752231	+	0.658899i	$0.771022\pi$
$854$	−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	$-0.810226\pi$
$857$		−	28174.0i		−	1.12299i	0.827479	−	0.561497i	$-0.189774\pi$
$858$		−	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	$0.0217236\pi$
$863$			3457.73i			0.136388i	−0.997672	+	0.0681938i	$0.978276\pi$
$864$	−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	$-0.849383\pi$
$877$		−	23671.4i		−	0.911432i	0.890125	−	0.455716i	$-0.150617\pi$
$878$		−	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	$-0.894322\pi$
$883$		−	17104.0i		−	0.651864i	0.945393	−	0.325932i	$-0.105678\pi$
$884$	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	$-0.995901\pi$
$887$		−	680.407i		−	0.0257563i	0.999917	−	0.0128781i	$-0.00409935\pi$
$888$		−	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	$0.130573\pi$
$907$			21786.8i			0.797596i	−0.917039	+	0.398798i	$0.869427\pi$
$908$		−	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	$-0.998788\pi$
$937$		−	218.425i		−	0.00761539i	0.999993	−	0.00380769i	$-0.00121203\pi$
$938$		−	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	$-0.845318\pi$
$947$		−	27221.7i		−	0.934093i	0.884233	−	0.467047i	$-0.154682\pi$
$948$			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	$-0.758246\pi$
$953$		−	40514.2i		−	1.37711i	0.725186	−	0.688553i	$-0.241754\pi$
$954$	−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	$-0.756493\pi$
$967$		−	41649.8i		−	1.38508i	0.721382	−	0.692538i	$-0.243507\pi$
$968$			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	$0.0462038\pi$
$977$			8834.33i			0.289289i	−0.989484	+	0.144644i	$0.953796\pi$
$978$			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	$-0.830739\pi$
$983$		−	31253.8i		−	1.01408i	0.861922	−	0.507041i	$-0.169261\pi$
$984$	−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	$0.154537\pi$
$997$			29380.4i			0.933285i	−0.884446	+	0.466643i	$0.845463\pi$
$998$		−	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.84.3	✓	124
5.2	odd	4		2075.4.a.n.1.60		62
5.3	odd	4		2075.4.a.m.1.3		62
5.4	even	2	inner	415.4.b.a.84.122	yes	124

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