LMFDB - Embedded newform 5950.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5950,2,Mod(1,5950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5950, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5950.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5950.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$47.5109892027$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1190)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} -0.854102 q^{11} +0.618034 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -2.61803 q^{18} -3.61803 q^{19} +0.618034 q^{21} -0.854102 q^{22} +5.09017 q^{23} +0.618034 q^{24} -4.00000 q^{26} -3.47214 q^{27} +1.00000 q^{28} -8.61803 q^{29} -0.381966 q^{31} +1.00000 q^{32} -0.527864 q^{33} +1.00000 q^{34} -2.61803 q^{36} -1.52786 q^{37} -3.61803 q^{38} -2.47214 q^{39} +10.0902 q^{41} +0.618034 q^{42} +5.38197 q^{43} -0.854102 q^{44} +5.09017 q^{46} -12.6180 q^{47} +0.618034 q^{48} +1.00000 q^{49} +0.618034 q^{51} -4.00000 q^{52} -0.618034 q^{53} -3.47214 q^{54} +1.00000 q^{56} -2.23607 q^{57} -8.61803 q^{58} -13.5623 q^{59} -10.4721 q^{61} -0.381966 q^{62} -2.61803 q^{63} +1.00000 q^{64} -0.527864 q^{66} +3.32624 q^{67} +1.00000 q^{68} +3.14590 q^{69} +5.52786 q^{71} -2.61803 q^{72} -4.00000 q^{73} -1.52786 q^{74} -3.61803 q^{76} -0.854102 q^{77} -2.47214 q^{78} -16.1803 q^{79} +5.70820 q^{81} +10.0902 q^{82} -10.1803 q^{83} +0.618034 q^{84} +5.38197 q^{86} -5.32624 q^{87} -0.854102 q^{88} +18.1803 q^{89} -4.00000 q^{91} +5.09017 q^{92} -0.236068 q^{93} -12.6180 q^{94} +0.618034 q^{96} -8.76393 q^{97} +1.00000 q^{98} +2.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^6 + 2 * q^7 + 2 * q^8 - 3 * q^9	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} + 5 q^{11} - q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 3 q^{18} - 5 q^{19} - q^{21} + 5 q^{22} - q^{23} - q^{24} - 8 q^{26} + 2 q^{27} + 2 q^{28} - 15 q^{29} - 3 q^{31} + 2 q^{32} - 10 q^{33} + 2 q^{34} - 3 q^{36} - 12 q^{37} - 5 q^{38} + 4 q^{39} + 9 q^{41} - q^{42} + 13 q^{43} + 5 q^{44} - q^{46} - 23 q^{47} - q^{48} + 2 q^{49} - q^{51} - 8 q^{52} + q^{53} + 2 q^{54} + 2 q^{56} - 15 q^{58} - 7 q^{59} - 12 q^{61} - 3 q^{62} - 3 q^{63} + 2 q^{64} - 10 q^{66} - 9 q^{67} + 2 q^{68} + 13 q^{69} + 20 q^{71} - 3 q^{72} - 8 q^{73} - 12 q^{74} - 5 q^{76} + 5 q^{77} + 4 q^{78} - 10 q^{79} - 2 q^{81} + 9 q^{82} + 2 q^{83} - q^{84} + 13 q^{86} + 5 q^{87} + 5 q^{88} + 14 q^{89} - 8 q^{91} - q^{92} + 4 q^{93} - 23 q^{94} - q^{96} - 22 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^6 + 2 * q^7 + 2 * q^8 - 3 * q^9 + 5 * q^11 - q^12 - 8 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 3 * q^18 - 5 * q^19 - q^21 + 5 * q^22 - q^23 - q^24 - 8 * q^26 + 2 * q^27 + 2 * q^28 - 15 * q^29 - 3 * q^31 + 2 * q^32 - 10 * q^33 + 2 * q^34 - 3 * q^36 - 12 * q^37 - 5 * q^38 + 4 * q^39 + 9 * q^41 - q^42 + 13 * q^43 + 5 * q^44 - q^46 - 23 * q^47 - q^48 + 2 * q^49 - q^51 - 8 * q^52 + q^53 + 2 * q^54 + 2 * q^56 - 15 * q^58 - 7 * q^59 - 12 * q^61 - 3 * q^62 - 3 * q^63 + 2 * q^64 - 10 * q^66 - 9 * q^67 + 2 * q^68 + 13 * q^69 + 20 * q^71 - 3 * q^72 - 8 * q^73 - 12 * q^74 - 5 * q^76 + 5 * q^77 + 4 * q^78 - 10 * q^79 - 2 * q^81 + 9 * q^82 + 2 * q^83 - q^84 + 13 * q^86 + 5 * q^87 + 5 * q^88 + 14 * q^89 - 8 * q^91 - q^92 + 4 * q^93 - 23 * q^94 - q^96 - 22 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−0.854102	−0.257521	−0.128761	−	0.991676i	$-0.541100\pi$
$11$	−0.854102	−0.257521	−0.128761	+	0.991676i	$0.541100\pi$
$12$	0.618034	0.178411
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.61803	−0.617077
$19$	−3.61803	−0.830034	−0.415017	−	0.909814i	$-0.636224\pi$
$19$	−3.61803	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$20$	0	0
$21$	0.618034	0.134866
$22$	−0.854102	−0.182095
$23$	5.09017	1.06137	0.530687	−	0.847568i	$-0.321934\pi$
$23$	5.09017	1.06137	0.530687	+	0.847568i	$0.321934\pi$
$24$	0.618034	0.126156
$25$	0	0
$26$	−4.00000	−0.784465
$27$	−3.47214	−0.668213
$28$	1.00000	0.188982
$29$	−8.61803	−1.60033	−0.800164	−	0.599781i	$-0.795254\pi$
$29$	−8.61803	−1.60033	−0.800164	+	0.599781i	$0.795254\pi$
$30$	0	0
$31$	−0.381966	−0.0686031	−0.0343016	−	0.999412i	$-0.510921\pi$
$31$	−0.381966	−0.0686031	−0.0343016	+	0.999412i	$0.510921\pi$
$32$	1.00000	0.176777
$33$	−0.527864	−0.0918893
$34$	1.00000	0.171499
$35$	0	0
$36$	−2.61803	−0.436339
$37$	−1.52786	−0.251179	−0.125590	−	0.992082i	$-0.540082\pi$
$37$	−1.52786	−0.251179	−0.125590	+	0.992082i	$0.540082\pi$
$38$	−3.61803	−0.586923
$39$	−2.47214	−0.395859
$40$	0	0
$41$	10.0902	1.57582	0.787910	−	0.615791i	$-0.211163\pi$
$41$	10.0902	1.57582	0.787910	+	0.615791i	$0.211163\pi$
$42$	0.618034	0.0953647
$43$	5.38197	0.820742	0.410371	−	0.911919i	$-0.365399\pi$
$43$	5.38197	0.820742	0.410371	+	0.911919i	$0.365399\pi$
$44$	−0.854102	−0.128761
$45$	0	0
$46$	5.09017	0.750505
$47$	−12.6180	−1.84053	−0.920265	−	0.391296i	$-0.872027\pi$
$47$	−12.6180	−1.84053	−0.920265	+	0.391296i	$0.872027\pi$
$48$	0.618034	0.0892055
$49$	1.00000	0.142857
$50$	0	0
$51$	0.618034	0.0865421
$52$	−4.00000	−0.554700
$53$	−0.618034	−0.0848935	−0.0424467	−	0.999099i	$-0.513515\pi$
$53$	−0.618034	−0.0848935	−0.0424467	+	0.999099i	$0.513515\pi$
$54$	−3.47214	−0.472498
$55$	0	0
$56$	1.00000	0.133631
$57$	−2.23607	−0.296174
$58$	−8.61803	−1.13160
$59$	−13.5623	−1.76566	−0.882831	−	0.469691i	$-0.844365\pi$
$59$	−13.5623	−1.76566	−0.882831	+	0.469691i	$0.844365\pi$
$60$	0	0
$61$	−10.4721	−1.34082	−0.670410	−	0.741991i	$-0.733882\pi$
$61$	−10.4721	−1.34082	−0.670410	+	0.741991i	$0.733882\pi$
$62$	−0.381966	−0.0485097
$63$	−2.61803	−0.329841
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.527864	−0.0649756
$67$	3.32624	0.406365	0.203182	−	0.979141i	$-0.434872\pi$
$67$	3.32624	0.406365	0.203182	+	0.979141i	$0.434872\pi$
$68$	1.00000	0.121268
$69$	3.14590	0.378722
$70$	0	0
$71$	5.52786	0.656037	0.328018	−	0.944671i	$-0.393619\pi$
$71$	5.52786	0.656037	0.328018	+	0.944671i	$0.393619\pi$
$72$	−2.61803	−0.308538
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−1.52786	−0.177611
$75$	0	0
$76$	−3.61803	−0.415017
$77$	−0.854102	−0.0973340
$78$	−2.47214	−0.279914
$79$	−16.1803	−1.82043	−0.910215	−	0.414136i	$-0.864084\pi$
$79$	−16.1803	−1.82043	−0.910215	+	0.414136i	$0.864084\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	10.0902	1.11427
$83$	−10.1803	−1.11744	−0.558719	−	0.829357i	$-0.688707\pi$
$83$	−10.1803	−1.11744	−0.558719	+	0.829357i	$0.688707\pi$
$84$	0.618034	0.0674330
$85$	0	0
$86$	5.38197	0.580352
$87$	−5.32624	−0.571033
$88$	−0.854102	−0.0910476
$89$	18.1803	1.92711	0.963556	−	0.267506i	$-0.0861996\pi$
$89$	18.1803	1.92711	0.963556	+	0.267506i	$0.0861996\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	5.09017	0.530687
$93$	−0.236068	−0.0244791
$94$	−12.6180	−1.30145
$95$	0	0
$96$	0.618034	0.0630778
$97$	−8.76393	−0.889842	−0.444921	−	0.895570i	$-0.646768\pi$
$97$	−8.76393	−0.889842	−0.444921	+	0.895570i	$0.646768\pi$
$98$	1.00000	0.101015
$99$	2.23607	0.224733
$100$	0	0
$101$	−11.6180	−1.15604	−0.578019	−	0.816023i	$-0.696174\pi$
$101$	−11.6180	−1.15604	−0.578019	+	0.816023i	$0.696174\pi$
$102$	0.618034	0.0611945
$103$	12.0344	1.18579	0.592894	−	0.805280i	$-0.297985\pi$
$103$	12.0344	1.18579	0.592894	+	0.805280i	$0.297985\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−0.618034	−0.0600288
$107$	−12.6525	−1.22316	−0.611581	−	0.791182i	$-0.709466\pi$
$107$	−12.6525	−1.22316	−0.611581	+	0.791182i	$0.709466\pi$
$108$	−3.47214	−0.334106
$109$	−16.0902	−1.54116	−0.770579	−	0.637344i	$-0.780033\pi$
$109$	−16.0902	−1.54116	−0.770579	+	0.637344i	$0.780033\pi$
$110$	0	0
$111$	−0.944272	−0.0896263
$112$	1.00000	0.0944911
$113$	−18.5623	−1.74619	−0.873097	−	0.487546i	$-0.837892\pi$
$113$	−18.5623	−1.74619	−0.873097	+	0.487546i	$0.837892\pi$
$114$	−2.23607	−0.209427
$115$	0	0
$116$	−8.61803	−0.800164
$117$	10.4721	0.968149
$118$	−13.5623	−1.24851
$119$	1.00000	0.0916698
$120$	0	0
$121$	−10.2705	−0.933683
$122$	−10.4721	−0.948103
$123$	6.23607	0.562287
$124$	−0.381966	−0.0343016
$125$	0	0
$126$	−2.61803	−0.233233
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000	0.0883883
$129$	3.32624	0.292859
$130$	0	0
$131$	5.41641	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$131$	5.41641	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$132$	−0.527864	−0.0459447
$133$	−3.61803	−0.313723
$134$	3.32624	0.287343
$135$	0	0
$136$	1.00000	0.0857493
$137$	9.41641	0.804498	0.402249	−	0.915530i	$-0.368229\pi$
$137$	9.41641	0.804498	0.402249	+	0.915530i	$0.368229\pi$
$138$	3.14590	0.267797
$139$	18.6525	1.58208	0.791041	−	0.611763i	$-0.209539\pi$
$139$	18.6525	1.58208	0.791041	+	0.611763i	$0.209539\pi$
$140$	0	0
$141$	−7.79837	−0.656742
$142$	5.52786	0.463888
$143$	3.41641	0.285694
$144$	−2.61803	−0.218169
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0.618034	0.0509746
$148$	−1.52786	−0.125590
$149$	0.763932	0.0625837	0.0312919	−	0.999510i	$-0.490038\pi$
$149$	0.763932	0.0625837	0.0312919	+	0.999510i	$0.490038\pi$
$150$	0	0
$151$	8.09017	0.658369	0.329184	−	0.944266i	$-0.393226\pi$
$151$	8.09017	0.658369	0.329184	+	0.944266i	$0.393226\pi$
$152$	−3.61803	−0.293461
$153$	−2.61803	−0.211656
$154$	−0.854102	−0.0688255
$155$	0	0
$156$	−2.47214	−0.197929
$157$	−12.6525	−1.00978	−0.504889	−	0.863184i	$-0.668466\pi$
$157$	−12.6525	−1.00978	−0.504889	+	0.863184i	$0.668466\pi$
$158$	−16.1803	−1.28724
$159$	−0.381966	−0.0302919
$160$	0	0
$161$	5.09017	0.401162
$162$	5.70820	0.448479
$163$	7.41641	0.580898	0.290449	−	0.956890i	$-0.406195\pi$
$163$	7.41641	0.580898	0.290449	+	0.956890i	$0.406195\pi$
$164$	10.0902	0.787910
$165$	0	0
$166$	−10.1803	−0.790148
$167$	5.41641	0.419134	0.209567	−	0.977794i	$-0.432795\pi$
$167$	5.41641	0.419134	0.209567	+	0.977794i	$0.432795\pi$
$168$	0.618034	0.0476824
$169$	3.00000	0.230769
$170$	0	0
$171$	9.47214	0.724352
$172$	5.38197	0.410371
$173$	−16.5623	−1.25921	−0.629604	−	0.776916i	$-0.716783\pi$
$173$	−16.5623	−1.25921	−0.629604	+	0.776916i	$0.716783\pi$
$174$	−5.32624	−0.403781
$175$	0	0
$176$	−0.854102	−0.0643804
$177$	−8.38197	−0.630027
$178$	18.1803	1.36267
$179$	1.52786	0.114198	0.0570990	−	0.998369i	$-0.481815\pi$
$179$	1.52786	0.114198	0.0570990	+	0.998369i	$0.481815\pi$
$180$	0	0
$181$	−8.76393	−0.651418	−0.325709	−	0.945470i	$-0.605603\pi$
$181$	−8.76393	−0.651418	−0.325709	+	0.945470i	$0.605603\pi$
$182$	−4.00000	−0.296500
$183$	−6.47214	−0.478434
$184$	5.09017	0.375252
$185$	0	0
$186$	−0.236068	−0.0173093
$187$	−0.854102	−0.0624581
$188$	−12.6180	−0.920265
$189$	−3.47214	−0.252561
$190$	0	0
$191$	10.5623	0.764262	0.382131	−	0.924108i	$-0.375190\pi$
$191$	10.5623	0.764262	0.382131	+	0.924108i	$0.375190\pi$
$192$	0.618034	0.0446028
$193$	−22.5066	−1.62006	−0.810029	−	0.586389i	$-0.800549\pi$
$193$	−22.5066	−1.62006	−0.810029	+	0.586389i	$0.800549\pi$
$194$	−8.76393	−0.629214
$195$	0	0
$196$	1.00000	0.0714286
$197$	−8.29180	−0.590766	−0.295383	−	0.955379i	$-0.595447\pi$
$197$	−8.29180	−0.590766	−0.295383	+	0.955379i	$0.595447\pi$
$198$	2.23607	0.158910
$199$	7.09017	0.502609	0.251304	−	0.967908i	$-0.419140\pi$
$199$	7.09017	0.502609	0.251304	+	0.967908i	$0.419140\pi$
$200$	0	0
$201$	2.05573	0.145000
$202$	−11.6180	−0.817442
$203$	−8.61803	−0.604867
$204$	0.618034	0.0432710
$205$	0	0
$206$	12.0344	0.838479
$207$	−13.3262	−0.926238
$208$	−4.00000	−0.277350
$209$	3.09017	0.213752
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−0.618034	−0.0424467
$213$	3.41641	0.234088
$214$	−12.6525	−0.864905
$215$	0	0
$216$	−3.47214	−0.236249
$217$	−0.381966	−0.0259295
$218$	−16.0902	−1.08976
$219$	−2.47214	−0.167051
$220$	0	0
$221$	−4.00000	−0.269069
$222$	−0.944272	−0.0633754
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−18.5623	−1.23475
$227$	−4.38197	−0.290841	−0.145421	−	0.989370i	$-0.546454\pi$
$227$	−4.38197	−0.290841	−0.145421	+	0.989370i	$0.546454\pi$
$228$	−2.23607	−0.148087
$229$	6.03444	0.398767	0.199384	−	0.979922i	$-0.436106\pi$
$229$	6.03444	0.398767	0.199384	+	0.979922i	$0.436106\pi$
$230$	0	0
$231$	−0.527864	−0.0347309
$232$	−8.61803	−0.565802
$233$	12.4721	0.817077	0.408538	−	0.912741i	$-0.366039\pi$
$233$	12.4721	0.817077	0.408538	+	0.912741i	$0.366039\pi$
$234$	10.4721	0.684585
$235$	0	0
$236$	−13.5623	−0.882831
$237$	−10.0000	−0.649570
$238$	1.00000	0.0648204
$239$	14.3820	0.930292	0.465146	−	0.885234i	$-0.346002\pi$
$239$	14.3820	0.930292	0.465146	+	0.885234i	$0.346002\pi$
$240$	0	0
$241$	9.67376	0.623142	0.311571	−	0.950223i	$-0.399145\pi$
$241$	9.67376	0.623142	0.311571	+	0.950223i	$0.399145\pi$
$242$	−10.2705	−0.660213
$243$	13.9443	0.894525
$244$	−10.4721	−0.670410
$245$	0	0
$246$	6.23607	0.397597
$247$	14.4721	0.920840
$248$	−0.381966	−0.0242549
$249$	−6.29180	−0.398726
$250$	0	0
$251$	−27.9787	−1.76600	−0.883000	−	0.469372i	$-0.844480\pi$
$251$	−27.9787	−1.76600	−0.883000	+	0.469372i	$0.844480\pi$
$252$	−2.61803	−0.164921
$253$	−4.34752	−0.273327
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.9787	1.68289	0.841443	−	0.540346i	$-0.181707\pi$
$257$	26.9787	1.68289	0.841443	+	0.540346i	$0.181707\pi$
$258$	3.32624	0.207083
$259$	−1.52786	−0.0949369
$260$	0	0
$261$	22.5623	1.39657
$262$	5.41641	0.334627
$263$	13.1246	0.809298	0.404649	−	0.914472i	$-0.367394\pi$
$263$	13.1246	0.809298	0.404649	+	0.914472i	$0.367394\pi$
$264$	−0.527864	−0.0324878
$265$	0	0
$266$	−3.61803	−0.221836
$267$	11.2361	0.687636
$268$	3.32624	0.203182
$269$	9.23607	0.563133	0.281567	−	0.959542i	$-0.409146\pi$
$269$	9.23607	0.563133	0.281567	+	0.959542i	$0.409146\pi$
$270$	0	0
$271$	−18.0000	−1.09342	−0.546711	−	0.837321i	$-0.684120\pi$
$271$	−18.0000	−1.09342	−0.546711	+	0.837321i	$0.684120\pi$
$272$	1.00000	0.0606339
$273$	−2.47214	−0.149620
$274$	9.41641	0.568866
$275$	0	0
$276$	3.14590	0.189361
$277$	−9.23607	−0.554942	−0.277471	−	0.960734i	$-0.589496\pi$
$277$	−9.23607	−0.554942	−0.277471	+	0.960734i	$0.589496\pi$
$278$	18.6525	1.11870
$279$	1.00000	0.0598684
$280$	0	0
$281$	25.8541	1.54233	0.771163	−	0.636637i	$-0.219675\pi$
$281$	25.8541	1.54233	0.771163	+	0.636637i	$0.219675\pi$
$282$	−7.79837	−0.464386
$283$	15.4164	0.916410	0.458205	−	0.888846i	$-0.348492\pi$
$283$	15.4164	0.916410	0.458205	+	0.888846i	$0.348492\pi$
$284$	5.52786	0.328018
$285$	0	0
$286$	3.41641	0.202016
$287$	10.0902	0.595604
$288$	−2.61803	−0.154269
$289$	1.00000	0.0588235
$290$	0	0
$291$	−5.41641	−0.317515
$292$	−4.00000	−0.234082
$293$	20.1803	1.17895	0.589474	−	0.807787i	$-0.299335\pi$
$293$	20.1803	1.17895	0.589474	+	0.807787i	$0.299335\pi$
$294$	0.618034	0.0360445
$295$	0	0
$296$	−1.52786	−0.0888053
$297$	2.96556	0.172079
$298$	0.763932	0.0442534
$299$	−20.3607	−1.17749
$300$	0	0
$301$	5.38197	0.310211
$302$	8.09017	0.465537
$303$	−7.18034	−0.412500
$304$	−3.61803	−0.207508
$305$	0	0
$306$	−2.61803	−0.149663
$307$	−7.23607	−0.412984	−0.206492	−	0.978448i	$-0.566205\pi$
$307$	−7.23607	−0.412984	−0.206492	+	0.978448i	$0.566205\pi$
$308$	−0.854102	−0.0486670
$309$	7.43769	0.423116
$310$	0	0
$311$	14.1459	0.802140	0.401070	−	0.916047i	$-0.368638\pi$
$311$	14.1459	0.802140	0.401070	+	0.916047i	$0.368638\pi$
$312$	−2.47214	−0.139957
$313$	−5.41641	−0.306153	−0.153077	−	0.988214i	$-0.548918\pi$
$313$	−5.41641	−0.306153	−0.153077	+	0.988214i	$0.548918\pi$
$314$	−12.6525	−0.714021
$315$	0	0
$316$	−16.1803	−0.910215
$317$	−16.6525	−0.935296	−0.467648	−	0.883915i	$-0.654899\pi$
$317$	−16.6525	−0.935296	−0.467648	+	0.883915i	$0.654899\pi$
$318$	−0.381966	−0.0214196
$319$	7.36068	0.412119
$320$	0	0
$321$	−7.81966	−0.436451
$322$	5.09017	0.283664
$323$	−3.61803	−0.201313
$324$	5.70820	0.317122
$325$	0	0
$326$	7.41641	0.410757
$327$	−9.94427	−0.549919
$328$	10.0902	0.557136
$329$	−12.6180	−0.695655
$330$	0	0
$331$	−34.6525	−1.90467	−0.952336	−	0.305051i	$-0.901327\pi$
$331$	−34.6525	−1.90467	−0.952336	+	0.305051i	$0.901327\pi$
$332$	−10.1803	−0.558719
$333$	4.00000	0.219199
$334$	5.41641	0.296373
$335$	0	0
$336$	0.618034	0.0337165
$337$	31.4508	1.71324	0.856618	−	0.515950i	$-0.172561\pi$
$337$	31.4508	1.71324	0.856618	+	0.515950i	$0.172561\pi$
$338$	3.00000	0.163178
$339$	−11.4721	−0.623081
$340$	0	0
$341$	0.326238	0.0176668
$342$	9.47214	0.512194
$343$	1.00000	0.0539949
$344$	5.38197	0.290176
$345$	0	0
$346$	−16.5623	−0.890395
$347$	24.1803	1.29807	0.649034	−	0.760759i	$-0.275173\pi$
$347$	24.1803	1.29807	0.649034	+	0.760759i	$0.275173\pi$
$348$	−5.32624	−0.285516
$349$	31.3050	1.67572	0.837858	−	0.545889i	$-0.183808\pi$
$349$	31.3050	1.67572	0.837858	+	0.545889i	$0.183808\pi$
$350$	0	0
$351$	13.8885	0.741316
$352$	−0.854102	−0.0455238
$353$	−21.3262	−1.13508	−0.567541	−	0.823345i	$-0.692105\pi$
$353$	−21.3262	−1.13508	−0.567541	+	0.823345i	$0.692105\pi$
$354$	−8.38197	−0.445496
$355$	0	0
$356$	18.1803	0.963556
$357$	0.618034	0.0327098
$358$	1.52786	0.0807501
$359$	28.5066	1.50452	0.752260	−	0.658867i	$-0.228964\pi$
$359$	28.5066	1.50452	0.752260	+	0.658867i	$0.228964\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	−8.76393	−0.460622
$363$	−6.34752	−0.333159
$364$	−4.00000	−0.209657
$365$	0	0
$366$	−6.47214	−0.338304
$367$	−25.1246	−1.31149	−0.655747	−	0.754981i	$-0.727646\pi$
$367$	−25.1246	−1.31149	−0.655747	+	0.754981i	$0.727646\pi$
$368$	5.09017	0.265343
$369$	−26.4164	−1.37518
$370$	0	0
$371$	−0.618034	−0.0320867
$372$	−0.236068	−0.0122396
$373$	−24.3262	−1.25956	−0.629782	−	0.776772i	$-0.716856\pi$
$373$	−24.3262	−1.25956	−0.629782	+	0.776772i	$0.716856\pi$
$374$	−0.854102	−0.0441646
$375$	0	0
$376$	−12.6180	−0.650725
$377$	34.4721	1.77541
$378$	−3.47214	−0.178587
$379$	27.2705	1.40079	0.700396	−	0.713754i	$-0.253007\pi$
$379$	27.2705	1.40079	0.700396	+	0.713754i	$0.253007\pi$
$380$	0	0
$381$	8.65248	0.443280
$382$	10.5623	0.540415
$383$	−27.6180	−1.41122	−0.705608	−	0.708603i	$-0.749326\pi$
$383$	−27.6180	−1.41122	−0.705608	+	0.708603i	$0.749326\pi$
$384$	0.618034	0.0315389
$385$	0	0
$386$	−22.5066	−1.14555
$387$	−14.0902	−0.716244
$388$	−8.76393	−0.444921
$389$	−1.81966	−0.0922604	−0.0461302	−	0.998935i	$-0.514689\pi$
$389$	−1.81966	−0.0922604	−0.0461302	+	0.998935i	$0.514689\pi$
$390$	0	0
$391$	5.09017	0.257421
$392$	1.00000	0.0505076
$393$	3.34752	0.168860
$394$	−8.29180	−0.417735
$395$	0	0
$396$	2.23607	0.112367
$397$	−14.3262	−0.719013	−0.359507	−	0.933143i	$-0.617055\pi$
$397$	−14.3262	−0.719013	−0.359507	+	0.933143i	$0.617055\pi$
$398$	7.09017	0.355398
$399$	−2.23607	−0.111943
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	2.05573	0.102530
$403$	1.52786	0.0761083
$404$	−11.6180	−0.578019
$405$	0	0
$406$	−8.61803	−0.427706
$407$	1.30495	0.0646841
$408$	0.618034	0.0305972
$409$	3.52786	0.174442	0.0872208	−	0.996189i	$-0.472201\pi$
$409$	3.52786	0.174442	0.0872208	+	0.996189i	$0.472201\pi$
$410$	0	0
$411$	5.81966	0.287063
$412$	12.0344	0.592894
$413$	−13.5623	−0.667357
$414$	−13.3262	−0.654949
$415$	0	0
$416$	−4.00000	−0.196116
$417$	11.5279	0.564522
$418$	3.09017	0.151145
$419$	2.00000	0.0977064	0.0488532	−	0.998806i	$-0.484443\pi$
$419$	2.00000	0.0977064	0.0488532	+	0.998806i	$0.484443\pi$
$420$	0	0
$421$	14.4721	0.705329	0.352664	−	0.935750i	$-0.385276\pi$
$421$	14.4721	0.705329	0.352664	+	0.935750i	$0.385276\pi$
$422$	12.0000	0.584151
$423$	33.0344	1.60619
$424$	−0.618034	−0.0300144
$425$	0	0
$426$	3.41641	0.165526
$427$	−10.4721	−0.506782
$428$	−12.6525	−0.611581
$429$	2.11146	0.101942
$430$	0	0
$431$	1.70820	0.0822813	0.0411406	−	0.999153i	$-0.486901\pi$
$431$	1.70820	0.0822813	0.0411406	+	0.999153i	$0.486901\pi$
$432$	−3.47214	−0.167053
$433$	−27.8885	−1.34024	−0.670119	−	0.742254i	$-0.733757\pi$
$433$	−27.8885	−1.34024	−0.670119	+	0.742254i	$0.733757\pi$
$434$	−0.381966	−0.0183350
$435$	0	0
$436$	−16.0902	−0.770579
$437$	−18.4164	−0.880976
$438$	−2.47214	−0.118123
$439$	11.0902	0.529305	0.264652	−	0.964344i	$-0.414743\pi$
$439$	11.0902	0.529305	0.264652	+	0.964344i	$0.414743\pi$
$440$	0	0
$441$	−2.61803	−0.124668
$442$	−4.00000	−0.190261
$443$	−26.9787	−1.28180	−0.640899	−	0.767625i	$-0.721438\pi$
$443$	−26.9787	−1.28180	−0.640899	+	0.767625i	$0.721438\pi$
$444$	−0.944272	−0.0448132
$445$	0	0
$446$	−20.0000	−0.947027
$447$	0.472136	0.0223313
$448$	1.00000	0.0472456
$449$	16.7639	0.791139	0.395569	−	0.918436i	$-0.370547\pi$
$449$	16.7639	0.791139	0.395569	+	0.918436i	$0.370547\pi$
$450$	0	0
$451$	−8.61803	−0.405807
$452$	−18.5623	−0.873097
$453$	5.00000	0.234920
$454$	−4.38197	−0.205656
$455$	0	0
$456$	−2.23607	−0.104713
$457$	−15.5279	−0.726363	−0.363181	−	0.931718i	$-0.618309\pi$
$457$	−15.5279	−0.726363	−0.363181	+	0.931718i	$0.618309\pi$
$458$	6.03444	0.281971
$459$	−3.47214	−0.162065
$460$	0	0
$461$	−15.7984	−0.735804	−0.367902	−	0.929865i	$-0.619924\pi$
$461$	−15.7984	−0.735804	−0.367902	+	0.929865i	$0.619924\pi$
$462$	−0.527864	−0.0245585
$463$	20.1803	0.937860	0.468930	−	0.883235i	$-0.344640\pi$
$463$	20.1803	0.937860	0.468930	+	0.883235i	$0.344640\pi$
$464$	−8.61803	−0.400082
$465$	0	0
$466$	12.4721	0.577761
$467$	−2.58359	−0.119554	−0.0597772	−	0.998212i	$-0.519039\pi$
$467$	−2.58359	−0.119554	−0.0597772	+	0.998212i	$0.519039\pi$
$468$	10.4721	0.484075
$469$	3.32624	0.153591
$470$	0	0
$471$	−7.81966	−0.360311
$472$	−13.5623	−0.624256
$473$	−4.59675	−0.211359
$474$	−10.0000	−0.459315
$475$	0	0
$476$	1.00000	0.0458349
$477$	1.61803	0.0740847
$478$	14.3820	0.657816
$479$	−36.9443	−1.68803	−0.844013	−	0.536322i	$-0.819813\pi$
$479$	−36.9443	−1.68803	−0.844013	+	0.536322i	$0.819813\pi$
$480$	0	0
$481$	6.11146	0.278658
$482$	9.67376	0.440628
$483$	3.14590	0.143143
$484$	−10.2705	−0.466841
$485$	0	0
$486$	13.9443	0.632525
$487$	−5.03444	−0.228132	−0.114066	−	0.993473i	$-0.536388\pi$
$487$	−5.03444	−0.228132	−0.114066	+	0.993473i	$0.536388\pi$
$488$	−10.4721	−0.474051
$489$	4.58359	0.207277
$490$	0	0
$491$	1.70820	0.0770902	0.0385451	−	0.999257i	$-0.487728\pi$
$491$	1.70820	0.0770902	0.0385451	+	0.999257i	$0.487728\pi$
$492$	6.23607	0.281144
$493$	−8.61803	−0.388137
$494$	14.4721	0.651132
$495$	0	0
$496$	−0.381966	−0.0171508
$497$	5.52786	0.247959
$498$	−6.29180	−0.281942
$499$	10.2016	0.456688	0.228344	−	0.973581i	$-0.426669\pi$
$499$	10.2016	0.456688	0.228344	+	0.973581i	$0.426669\pi$
$500$	0	0
$501$	3.34752	0.149556
$502$	−27.9787	−1.24875
$503$	−23.8885	−1.06514	−0.532569	−	0.846387i	$-0.678773\pi$
$503$	−23.8885	−1.06514	−0.532569	+	0.846387i	$0.678773\pi$
$504$	−2.61803	−0.116617
$505$	0	0
$506$	−4.34752	−0.193271
$507$	1.85410	0.0823436
$508$	14.0000	0.621150
$509$	8.14590	0.361061	0.180530	−	0.983569i	$-0.442219\pi$
$509$	8.14590	0.361061	0.180530	+	0.983569i	$0.442219\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	1.00000	0.0441942
$513$	12.5623	0.554639
$514$	26.9787	1.18998
$515$	0	0
$516$	3.32624	0.146429
$517$	10.7771	0.473976
$518$	−1.52786	−0.0671305
$519$	−10.2361	−0.449314
$520$	0	0
$521$	38.9787	1.70769	0.853844	−	0.520529i	$-0.174265\pi$
$521$	38.9787	1.70769	0.853844	+	0.520529i	$0.174265\pi$
$522$	22.5623	0.987525
$523$	24.8328	1.08586	0.542932	−	0.839777i	$-0.317314\pi$
$523$	24.8328	1.08586	0.542932	+	0.839777i	$0.317314\pi$
$524$	5.41641	0.236617
$525$	0	0
$526$	13.1246	0.572260
$527$	−0.381966	−0.0166387
$528$	−0.527864	−0.0229723
$529$	2.90983	0.126514
$530$	0	0
$531$	35.5066	1.54085
$532$	−3.61803	−0.156862
$533$	−40.3607	−1.74822
$534$	11.2361	0.486232
$535$	0	0
$536$	3.32624	0.143672
$537$	0.944272	0.0407483
$538$	9.23607	0.398195
$539$	−0.854102	−0.0367888
$540$	0	0
$541$	5.79837	0.249292	0.124646	−	0.992201i	$-0.460221\pi$
$541$	5.79837	0.249292	0.124646	+	0.992201i	$0.460221\pi$
$542$	−18.0000	−0.773166
$543$	−5.41641	−0.232440
$544$	1.00000	0.0428746
$545$	0	0
$546$	−2.47214	−0.105798
$547$	13.7082	0.586120	0.293060	−	0.956094i	$-0.405326\pi$
$547$	13.7082	0.586120	0.293060	+	0.956094i	$0.405326\pi$
$548$	9.41641	0.402249
$549$	27.4164	1.17010
$550$	0	0
$551$	31.1803	1.32833
$552$	3.14590	0.133898
$553$	−16.1803	−0.688058
$554$	−9.23607	−0.392403
$555$	0	0
$556$	18.6525	0.791041
$557$	10.3820	0.439898	0.219949	−	0.975511i	$-0.429411\pi$
$557$	10.3820	0.439898	0.219949	+	0.975511i	$0.429411\pi$
$558$	1.00000	0.0423334
$559$	−21.5279	−0.910532
$560$	0	0
$561$	−0.527864	−0.0222864
$562$	25.8541	1.09059
$563$	−21.2361	−0.894994	−0.447497	−	0.894286i	$-0.647684\pi$
$563$	−21.2361	−0.894994	−0.447497	+	0.894286i	$0.647684\pi$
$564$	−7.79837	−0.328371
$565$	0	0
$566$	15.4164	0.648000
$567$	5.70820	0.239722
$568$	5.52786	0.231944
$569$	−38.1591	−1.59971	−0.799855	−	0.600193i	$-0.795091\pi$
$569$	−38.1591	−1.59971	−0.799855	+	0.600193i	$0.795091\pi$
$570$	0	0
$571$	5.79837	0.242654	0.121327	−	0.992613i	$-0.461285\pi$
$571$	5.79837	0.242654	0.121327	+	0.992613i	$0.461285\pi$
$572$	3.41641	0.142847
$573$	6.52786	0.272705
$574$	10.0902	0.421156
$575$	0	0
$576$	−2.61803	−0.109085
$577$	28.5623	1.18906	0.594532	−	0.804072i	$-0.297337\pi$
$577$	28.5623	1.18906	0.594532	+	0.804072i	$0.297337\pi$
$578$	1.00000	0.0415945
$579$	−13.9098	−0.578073
$580$	0	0
$581$	−10.1803	−0.422352
$582$	−5.41641	−0.224517
$583$	0.527864	0.0218619
$584$	−4.00000	−0.165521
$585$	0	0
$586$	20.1803	0.833642
$587$	−43.3050	−1.78739	−0.893693	−	0.448678i	$-0.851895\pi$
$587$	−43.3050	−1.78739	−0.893693	+	0.448678i	$0.851895\pi$
$588$	0.618034	0.0254873
$589$	1.38197	0.0569429
$590$	0	0
$591$	−5.12461	−0.210798
$592$	−1.52786	−0.0627948
$593$	−12.7984	−0.525566	−0.262783	−	0.964855i	$-0.584640\pi$
$593$	−12.7984	−0.525566	−0.262783	+	0.964855i	$0.584640\pi$
$594$	2.96556	0.121678
$595$	0	0
$596$	0.763932	0.0312919
$597$	4.38197	0.179342
$598$	−20.3607	−0.832610
$599$	28.7984	1.17667	0.588335	−	0.808617i	$-0.299784\pi$
$599$	28.7984	1.17667	0.588335	+	0.808617i	$0.299784\pi$
$600$	0	0
$601$	−2.27051	−0.0926160	−0.0463080	−	0.998927i	$-0.514746\pi$
$601$	−2.27051	−0.0926160	−0.0463080	+	0.998927i	$0.514746\pi$
$602$	5.38197	0.219353
$603$	−8.70820	−0.354625
$604$	8.09017	0.329184
$605$	0	0
$606$	−7.18034	−0.291681
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−3.61803	−0.146731
$609$	−5.32624	−0.215830
$610$	0	0
$611$	50.4721	2.04188
$612$	−2.61803	−0.105828
$613$	0.472136	0.0190694	0.00953470	−	0.999955i	$-0.496965\pi$
$613$	0.472136	0.0190694	0.00953470	+	0.999955i	$0.496965\pi$
$614$	−7.23607	−0.292024
$615$	0	0
$616$	−0.854102	−0.0344127
$617$	−21.5623	−0.868066	−0.434033	−	0.900897i	$-0.642910\pi$
$617$	−21.5623	−0.868066	−0.434033	+	0.900897i	$0.642910\pi$
$618$	7.43769	0.299188
$619$	22.7639	0.914960	0.457480	−	0.889220i	$-0.348752\pi$
$619$	22.7639	0.914960	0.457480	+	0.889220i	$0.348752\pi$
$620$	0	0
$621$	−17.6738	−0.709224
$622$	14.1459	0.567199
$623$	18.1803	0.728380
$624$	−2.47214	−0.0989646
$625$	0	0
$626$	−5.41641	−0.216483
$627$	1.90983	0.0762713
$628$	−12.6525	−0.504889
$629$	−1.52786	−0.0609199
$630$	0	0
$631$	−10.5623	−0.420479	−0.210239	−	0.977650i	$-0.567424\pi$
$631$	−10.5623	−0.420479	−0.210239	+	0.977650i	$0.567424\pi$
$632$	−16.1803	−0.643619
$633$	7.41641	0.294776
$634$	−16.6525	−0.661354
$635$	0	0
$636$	−0.381966	−0.0151459
$637$	−4.00000	−0.158486
$638$	7.36068	0.291412
$639$	−14.4721	−0.572509
$640$	0	0
$641$	−8.11146	−0.320383	−0.160192	−	0.987086i	$-0.551211\pi$
$641$	−8.11146	−0.320383	−0.160192	+	0.987086i	$0.551211\pi$
$642$	−7.81966	−0.308617
$643$	−20.3607	−0.802947	−0.401473	−	0.915871i	$-0.631502\pi$
$643$	−20.3607	−0.802947	−0.401473	+	0.915871i	$0.631502\pi$
$644$	5.09017	0.200581
$645$	0	0
$646$	−3.61803	−0.142350
$647$	−20.9443	−0.823404	−0.411702	−	0.911318i	$-0.635066\pi$
$647$	−20.9443	−0.823404	−0.411702	+	0.911318i	$0.635066\pi$
$648$	5.70820	0.224239
$649$	11.5836	0.454696
$650$	0	0
$651$	−0.236068	−0.00925223
$652$	7.41641	0.290449
$653$	−42.6525	−1.66912	−0.834560	−	0.550917i	$-0.814278\pi$
$653$	−42.6525	−1.66912	−0.834560	+	0.550917i	$0.814278\pi$
$654$	−9.94427	−0.388852
$655$	0	0
$656$	10.0902	0.393955
$657$	10.4721	0.408557
$658$	−12.6180	−0.491902
$659$	−27.1246	−1.05662	−0.528312	−	0.849050i	$-0.677175\pi$
$659$	−27.1246	−1.05662	−0.528312	+	0.849050i	$0.677175\pi$
$660$	0	0
$661$	14.0902	0.548044	0.274022	−	0.961723i	$-0.411646\pi$
$661$	14.0902	0.548044	0.274022	+	0.961723i	$0.411646\pi$
$662$	−34.6525	−1.34681
$663$	−2.47214	−0.0960098
$664$	−10.1803	−0.395074
$665$	0	0
$666$	4.00000	0.154997
$667$	−43.8673	−1.69855
$668$	5.41641	0.209567
$669$	−12.3607	−0.477891
$670$	0	0
$671$	8.94427	0.345290
$672$	0.618034	0.0238412
$673$	16.2705	0.627182	0.313591	−	0.949558i	$-0.398468\pi$
$673$	16.2705	0.627182	0.313591	+	0.949558i	$0.398468\pi$
$674$	31.4508	1.21144
$675$	0	0
$676$	3.00000	0.115385
$677$	11.8541	0.455590	0.227795	−	0.973709i	$-0.426848\pi$
$677$	11.8541	0.455590	0.227795	+	0.973709i	$0.426848\pi$
$678$	−11.4721	−0.440585
$679$	−8.76393	−0.336329
$680$	0	0
$681$	−2.70820	−0.103779
$682$	0.326238	0.0124923
$683$	−9.59675	−0.367209	−0.183605	−	0.983000i	$-0.558777\pi$
$683$	−9.59675	−0.367209	−0.183605	+	0.983000i	$0.558777\pi$
$684$	9.47214	0.362176
$685$	0	0
$686$	1.00000	0.0381802
$687$	3.72949	0.142289
$688$	5.38197	0.205186
$689$	2.47214	0.0941809
$690$	0	0
$691$	5.12461	0.194949	0.0974747	−	0.995238i	$-0.468923\pi$
$691$	5.12461	0.194949	0.0974747	+	0.995238i	$0.468923\pi$
$692$	−16.5623	−0.629604
$693$	2.23607	0.0849412
$694$	24.1803	0.917873
$695$	0	0
$696$	−5.32624	−0.201891
$697$	10.0902	0.382192
$698$	31.3050	1.18491
$699$	7.70820	0.291551
$700$	0	0
$701$	37.1246	1.40218	0.701089	−	0.713074i	$-0.252698\pi$
$701$	37.1246	1.40218	0.701089	+	0.713074i	$0.252698\pi$
$702$	13.8885	0.524189
$703$	5.52786	0.208487
$704$	−0.854102	−0.0321902
$705$	0	0
$706$	−21.3262	−0.802624
$707$	−11.6180	−0.436941
$708$	−8.38197	−0.315014
$709$	44.6869	1.67825	0.839126	−	0.543937i	$-0.183067\pi$
$709$	44.6869	1.67825	0.839126	+	0.543937i	$0.183067\pi$
$710$	0	0
$711$	42.3607	1.58865
$712$	18.1803	0.681337
$713$	−1.94427	−0.0728136
$714$	0.618034	0.0231293
$715$	0	0
$716$	1.52786	0.0570990
$717$	8.88854	0.331949
$718$	28.5066	1.06386
$719$	−27.9787	−1.04343	−0.521715	−	0.853120i	$-0.674708\pi$
$719$	−27.9787	−1.04343	−0.521715	+	0.853120i	$0.674708\pi$
$720$	0	0
$721$	12.0344	0.448186
$722$	−5.90983	−0.219941
$723$	5.97871	0.222351
$724$	−8.76393	−0.325709
$725$	0	0
$726$	−6.34752	−0.235579
$727$	11.9656	0.443778	0.221889	−	0.975072i	$-0.428778\pi$
$727$	11.9656	0.443778	0.221889	+	0.975072i	$0.428778\pi$
$728$	−4.00000	−0.148250
$729$	−8.50658	−0.315058
$730$	0	0
$731$	5.38197	0.199059
$732$	−6.47214	−0.239217
$733$	13.5279	0.499663	0.249832	−	0.968289i	$-0.419625\pi$
$733$	13.5279	0.499663	0.249832	+	0.968289i	$0.419625\pi$
$734$	−25.1246	−0.927366
$735$	0	0
$736$	5.09017	0.187626
$737$	−2.84095	−0.104648
$738$	−26.4164	−0.972401
$739$	−4.76393	−0.175244	−0.0876220	−	0.996154i	$-0.527927\pi$
$739$	−4.76393	−0.175244	−0.0876220	+	0.996154i	$0.527927\pi$
$740$	0	0
$741$	8.94427	0.328576
$742$	−0.618034	−0.0226887
$743$	−28.9443	−1.06186	−0.530931	−	0.847415i	$-0.678158\pi$
$743$	−28.9443	−1.06186	−0.530931	+	0.847415i	$0.678158\pi$
$744$	−0.236068	−0.00865467
$745$	0	0
$746$	−24.3262	−0.890647
$747$	26.6525	0.975163
$748$	−0.854102	−0.0312291
$749$	−12.6525	−0.462311
$750$	0	0
$751$	26.3607	0.961915	0.480957	−	0.876744i	$-0.340289\pi$
$751$	26.3607	0.961915	0.480957	+	0.876744i	$0.340289\pi$
$752$	−12.6180	−0.460132
$753$	−17.2918	−0.630148
$754$	34.4721	1.25540
$755$	0	0
$756$	−3.47214	−0.126280
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	27.2705	0.990510
$759$	−2.68692	−0.0975289
$760$	0	0
$761$	−3.41641	−0.123845	−0.0619223	−	0.998081i	$-0.519723\pi$
$761$	−3.41641	−0.123845	−0.0619223	+	0.998081i	$0.519723\pi$
$762$	8.65248	0.313446
$763$	−16.0902	−0.582503
$764$	10.5623	0.382131
$765$	0	0
$766$	−27.6180	−0.997880
$767$	54.2492	1.95883
$768$	0.618034	0.0223014
$769$	27.5967	0.995164	0.497582	−	0.867417i	$-0.334221\pi$
$769$	27.5967	0.995164	0.497582	+	0.867417i	$0.334221\pi$
$770$	0	0
$771$	16.6738	0.600491
$772$	−22.5066	−0.810029
$773$	34.7639	1.25037	0.625186	−	0.780476i	$-0.285023\pi$
$773$	34.7639	1.25037	0.625186	+	0.780476i	$0.285023\pi$
$774$	−14.0902	−0.506461
$775$	0	0
$776$	−8.76393	−0.314607
$777$	−0.944272	−0.0338756
$778$	−1.81966	−0.0652380
$779$	−36.5066	−1.30798
$780$	0	0
$781$	−4.72136	−0.168944
$782$	5.09017	0.182024
$783$	29.9230	1.06936
$784$	1.00000	0.0357143
$785$	0	0
$786$	3.34752	0.119402
$787$	−4.74265	−0.169057	−0.0845285	−	0.996421i	$-0.526938\pi$
$787$	−4.74265	−0.169057	−0.0845285	+	0.996421i	$0.526938\pi$
$788$	−8.29180	−0.295383
$789$	8.11146	0.288775
$790$	0	0
$791$	−18.5623	−0.660000
$792$	2.23607	0.0794552
$793$	41.8885	1.48751
$794$	−14.3262	−0.508419
$795$	0	0
$796$	7.09017	0.251304
$797$	−37.1246	−1.31502	−0.657511	−	0.753445i	$-0.728391\pi$
$797$	−37.1246	−1.31502	−0.657511	+	0.753445i	$0.728391\pi$
$798$	−2.23607	−0.0791559
$799$	−12.6180	−0.446394
$800$	0	0
$801$	−47.5967	−1.68175
$802$	18.0000	0.635602
$803$	3.41641	0.120562
$804$	2.05573	0.0724999
$805$	0	0
$806$	1.52786	0.0538167
$807$	5.70820	0.200938
$808$	−11.6180	−0.408721
$809$	52.8328	1.85750	0.928751	−	0.370703i	$-0.120883\pi$
$809$	52.8328	1.85750	0.928751	+	0.370703i	$0.120883\pi$
$810$	0	0
$811$	−31.1246	−1.09293	−0.546466	−	0.837481i	$-0.684027\pi$
$811$	−31.1246	−1.09293	−0.546466	+	0.837481i	$0.684027\pi$
$812$	−8.61803	−0.302434
$813$	−11.1246	−0.390157
$814$	1.30495	0.0457385
$815$	0	0
$816$	0.618034	0.0216355
$817$	−19.4721	−0.681244
$818$	3.52786	0.123349
$819$	10.4721	0.365926
$820$	0	0
$821$	−6.58359	−0.229769	−0.114884	−	0.993379i	$-0.536650\pi$
$821$	−6.58359	−0.229769	−0.114884	+	0.993379i	$0.536650\pi$
$822$	5.81966	0.202984
$823$	42.3820	1.47734	0.738671	−	0.674066i	$-0.235454\pi$
$823$	42.3820	1.47734	0.738671	+	0.674066i	$0.235454\pi$
$824$	12.0344	0.419240
$825$	0	0
$826$	−13.5623	−0.471893
$827$	−13.8197	−0.480557	−0.240278	−	0.970704i	$-0.577239\pi$
$827$	−13.8197	−0.480557	−0.240278	+	0.970704i	$0.577239\pi$
$828$	−13.3262	−0.463119
$829$	39.0344	1.35572	0.677861	−	0.735190i	$-0.262907\pi$
$829$	39.0344	1.35572	0.677861	+	0.735190i	$0.262907\pi$
$830$	0	0
$831$	−5.70820	−0.198015
$832$	−4.00000	−0.138675
$833$	1.00000	0.0346479
$834$	11.5279	0.399177
$835$	0	0
$836$	3.09017	0.106876
$837$	1.32624	0.0458415
$838$	2.00000	0.0690889
$839$	1.85410	0.0640107	0.0320054	−	0.999488i	$-0.489811\pi$
$839$	1.85410	0.0640107	0.0320054	+	0.999488i	$0.489811\pi$
$840$	0	0
$841$	45.2705	1.56105
$842$	14.4721	0.498743
$843$	15.9787	0.550336
$844$	12.0000	0.413057
$845$	0	0
$846$	33.0344	1.13575
$847$	−10.2705	−0.352899
$848$	−0.618034	−0.0212234
$849$	9.52786	0.326995
$850$	0	0
$851$	−7.77709	−0.266595
$852$	3.41641	0.117044
$853$	0.437694	0.0149864	0.00749318	−	0.999972i	$-0.497615\pi$
$853$	0.437694	0.0149864	0.00749318	+	0.999972i	$0.497615\pi$
$854$	−10.4721	−0.358349
$855$	0	0
$856$	−12.6525	−0.432453
$857$	−11.5967	−0.396137	−0.198069	−	0.980188i	$-0.563467\pi$
$857$	−11.5967	−0.396137	−0.198069	+	0.980188i	$0.563467\pi$
$858$	2.11146	0.0720839
$859$	−7.05573	−0.240738	−0.120369	−	0.992729i	$-0.538408\pi$
$859$	−7.05573	−0.240738	−0.120369	+	0.992729i	$0.538408\pi$
$860$	0	0
$861$	6.23607	0.212525
$862$	1.70820	0.0581817
$863$	48.5410	1.65236	0.826178	−	0.563410i	$-0.190511\pi$
$863$	48.5410	1.65236	0.826178	+	0.563410i	$0.190511\pi$
$864$	−3.47214	−0.118124
$865$	0	0
$866$	−27.8885	−0.947691
$867$	0.618034	0.0209895
$868$	−0.381966	−0.0129648
$869$	13.8197	0.468800
$870$	0	0
$871$	−13.3050	−0.450821
$872$	−16.0902	−0.544882
$873$	22.9443	0.776546
$874$	−18.4164	−0.622944
$875$	0	0
$876$	−2.47214	−0.0835257
$877$	−11.4164	−0.385505	−0.192752	−	0.981247i	$-0.561741\pi$
$877$	−11.4164	−0.385505	−0.192752	+	0.981247i	$0.561741\pi$
$878$	11.0902	0.374275
$879$	12.4721	0.420675
$880$	0	0
$881$	−15.2705	−0.514477	−0.257238	−	0.966348i	$-0.582813\pi$
$881$	−15.2705	−0.514477	−0.257238	+	0.966348i	$0.582813\pi$
$882$	−2.61803	−0.0881538
$883$	−10.1115	−0.340278	−0.170139	−	0.985420i	$-0.554422\pi$
$883$	−10.1115	−0.340278	−0.170139	+	0.985420i	$0.554422\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−26.9787	−0.906368
$887$	1.88854	0.0634111	0.0317055	−	0.999497i	$-0.489906\pi$
$887$	1.88854	0.0634111	0.0317055	+	0.999497i	$0.489906\pi$
$888$	−0.944272	−0.0316877
$889$	14.0000	0.469545
$890$	0	0
$891$	−4.87539	−0.163332
$892$	−20.0000	−0.669650
$893$	45.6525	1.52770
$894$	0.472136	0.0157906
$895$	0	0
$896$	1.00000	0.0334077
$897$	−12.5836	−0.420154
$898$	16.7639	0.559420
$899$	3.29180	0.109788
$900$	0	0
$901$	−0.618034	−0.0205897
$902$	−8.61803	−0.286949
$903$	3.32624	0.110690
$904$	−18.5623	−0.617373
$905$	0	0
$906$	5.00000	0.166114
$907$	22.0000	0.730498	0.365249	−	0.930910i	$-0.380984\pi$
$907$	22.0000	0.730498	0.365249	+	0.930910i	$0.380984\pi$
$908$	−4.38197	−0.145421
$909$	30.4164	1.00885
$910$	0	0
$911$	40.1803	1.33123	0.665617	−	0.746293i	$-0.268168\pi$
$911$	40.1803	1.33123	0.665617	+	0.746293i	$0.268168\pi$
$912$	−2.23607	−0.0740436
$913$	8.69505	0.287764
$914$	−15.5279	−0.513616
$915$	0	0
$916$	6.03444	0.199384
$917$	5.41641	0.178866
$918$	−3.47214	−0.114598
$919$	−24.4377	−0.806125	−0.403063	−	0.915172i	$-0.632054\pi$
$919$	−24.4377	−0.806125	−0.403063	+	0.915172i	$0.632054\pi$
$920$	0	0
$921$	−4.47214	−0.147362
$922$	−15.7984	−0.520292
$923$	−22.1115	−0.727807
$924$	−0.527864	−0.0173655
$925$	0	0
$926$	20.1803	0.663167
$927$	−31.5066	−1.03481
$928$	−8.61803	−0.282901
$929$	−45.7771	−1.50190	−0.750949	−	0.660360i	$-0.770403\pi$
$929$	−45.7771	−1.50190	−0.750949	+	0.660360i	$0.770403\pi$
$930$	0	0
$931$	−3.61803	−0.118576
$932$	12.4721	0.408538
$933$	8.74265	0.286221
$934$	−2.58359	−0.0845377
$935$	0	0
$936$	10.4721	0.342292
$937$	50.6869	1.65587	0.827935	−	0.560824i	$-0.189516\pi$
$937$	50.6869	1.65587	0.827935	+	0.560824i	$0.189516\pi$
$938$	3.32624	0.108606
$939$	−3.34752	−0.109242
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−7.81966	−0.254778
$943$	51.3607	1.67253
$944$	−13.5623	−0.441415
$945$	0	0
$946$	−4.59675	−0.149453
$947$	−4.83282	−0.157045	−0.0785227	−	0.996912i	$-0.525020\pi$
$947$	−4.83282	−0.157045	−0.0785227	+	0.996912i	$0.525020\pi$
$948$	−10.0000	−0.324785
$949$	16.0000	0.519382
$950$	0	0
$951$	−10.2918	−0.333734
$952$	1.00000	0.0324102
$953$	18.5410	0.600603	0.300301	−	0.953844i	$-0.402913\pi$
$953$	18.5410	0.600603	0.300301	+	0.953844i	$0.402913\pi$
$954$	1.61803	0.0523858
$955$	0	0
$956$	14.3820	0.465146
$957$	4.54915	0.147053
$958$	−36.9443	−1.19362
$959$	9.41641	0.304072
$960$	0	0
$961$	−30.8541	−0.995294
$962$	6.11146	0.197041
$963$	33.1246	1.06743
$964$	9.67376	0.311571
$965$	0	0
$966$	3.14590	0.101218
$967$	−24.6525	−0.792770	−0.396385	−	0.918084i	$-0.629735\pi$
$967$	−24.6525	−0.792770	−0.396385	+	0.918084i	$0.629735\pi$
$968$	−10.2705	−0.330107
$969$	−2.23607	−0.0718329
$970$	0	0
$971$	−47.6180	−1.52814	−0.764068	−	0.645136i	$-0.776801\pi$
$971$	−47.6180	−1.52814	−0.764068	+	0.645136i	$0.776801\pi$
$972$	13.9443	0.447263
$973$	18.6525	0.597971
$974$	−5.03444	−0.161314
$975$	0	0
$976$	−10.4721	−0.335205
$977$	−46.4296	−1.48541	−0.742707	−	0.669617i	$-0.766458\pi$
$977$	−46.4296	−1.48541	−0.742707	+	0.669617i	$0.766458\pi$
$978$	4.58359	0.146567
$979$	−15.5279	−0.496273
$980$	0	0
$981$	42.1246	1.34494
$982$	1.70820	0.0545110
$983$	−0.832816	−0.0265627	−0.0132813	−	0.999912i	$-0.504228\pi$
$983$	−0.832816	−0.0265627	−0.0132813	+	0.999912i	$0.504228\pi$
$984$	6.23607	0.198799
$985$	0	0
$986$	−8.61803	−0.274454
$987$	−7.79837	−0.248225
$988$	14.4721	0.460420
$989$	27.3951	0.871114
$990$	0	0
$991$	−29.3050	−0.930902	−0.465451	−	0.885074i	$-0.654108\pi$
$991$	−29.3050	−0.930902	−0.465451	+	0.885074i	$0.654108\pi$
$992$	−0.381966	−0.0121274
$993$	−21.4164	−0.679629
$994$	5.52786	0.175333
$995$	0	0
$996$	−6.29180	−0.199363
$997$	34.0000	1.07679	0.538395	−	0.842692i	$-0.319031\pi$
$997$	34.0000	1.07679	0.538395	+	0.842692i	$0.319031\pi$
$998$	10.2016	0.322927
$999$	5.30495	0.167841

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5950.2.a.z.1.2		2
5.2	odd	4		1190.2.e.d.239.3	yes	4
5.3	odd	4		1190.2.e.d.239.2	✓	4
5.4	even	2		5950.2.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.e.d.239.2	✓	4	5.3	odd	4
1190.2.e.d.239.3	yes	4	5.2	odd	4
5950.2.a.u.1.1		2	5.4	even	2
5950.2.a.z.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5950.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.61803 q^{9} -0.854102 q^{11} +0.618034 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -2.61803 q^{18} -3.61803 q^{19} +0.618034 q^{21} -0.854102 q^{22} +5.09017 q^{23} +0.618034 q^{24} -4.00000 q^{26} -3.47214 q^{27} +1.00000 q^{28} -8.61803 q^{29} -0.381966 q^{31} +1.00000 q^{32} -0.527864 q^{33} +1.00000 q^{34} -2.61803 q^{36} -1.52786 q^{37} -3.61803 q^{38} -2.47214 q^{39} +10.0902 q^{41} +0.618034 q^{42} +5.38197 q^{43} -0.854102 q^{44} +5.09017 q^{46} -12.6180 q^{47} +0.618034 q^{48} +1.00000 q^{49} +0.618034 q^{51} -4.00000 q^{52} -0.618034 q^{53} -3.47214 q^{54} +1.00000 q^{56} -2.23607 q^{57} -8.61803 q^{58} -13.5623 q^{59} -10.4721 q^{61} -0.381966 q^{62} -2.61803 q^{63} +1.00000 q^{64} -0.527864 q^{66} +3.32624 q^{67} +1.00000 q^{68} +3.14590 q^{69} +5.52786 q^{71} -2.61803 q^{72} -4.00000 q^{73} -1.52786 q^{74} -3.61803 q^{76} -0.854102 q^{77} -2.47214 q^{78} -16.1803 q^{79} +5.70820 q^{81} +10.0902 q^{82} -10.1803 q^{83} +0.618034 q^{84} +5.38197 q^{86} -5.32624 q^{87} -0.854102 q^{88} +18.1803 q^{89} -4.00000 q^{91} +5.09017 q^{92} -0.236068 q^{93} -12.6180 q^{94} +0.618034 q^{96} -8.76393 q^{97} +1.00000 q^{98} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^6 + 2 * q^7 + 2 * q^8 - 3 * q^9	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} + 5 q^{11} - q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 3 q^{18} - 5 q^{19} - q^{21} + 5 q^{22} - q^{23} - q^{24} - 8 q^{26} + 2 q^{27} + 2 q^{28} - 15 q^{29} - 3 q^{31} + 2 q^{32} - 10 q^{33} + 2 q^{34} - 3 q^{36} - 12 q^{37} - 5 q^{38} + 4 q^{39} + 9 q^{41} - q^{42} + 13 q^{43} + 5 q^{44} - q^{46} - 23 q^{47} - q^{48} + 2 q^{49} - q^{51} - 8 q^{52} + q^{53} + 2 q^{54} + 2 q^{56} - 15 q^{58} - 7 q^{59} - 12 q^{61} - 3 q^{62} - 3 q^{63} + 2 q^{64} - 10 q^{66} - 9 q^{67} + 2 q^{68} + 13 q^{69} + 20 q^{71} - 3 q^{72} - 8 q^{73} - 12 q^{74} - 5 q^{76} + 5 q^{77} + 4 q^{78} - 10 q^{79} - 2 q^{81} + 9 q^{82} + 2 q^{83} - q^{84} + 13 q^{86} + 5 q^{87} + 5 q^{88} + 14 q^{89} - 8 q^{91} - q^{92} + 4 q^{93} - 23 q^{94} - q^{96} - 22 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - q^6 + 2 * q^7 + 2 * q^8 - 3 * q^9 + 5 * q^11 - q^12 - 8 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 3 * q^18 - 5 * q^19 - q^21 + 5 * q^22 - q^23 - q^24 - 8 * q^26 + 2 * q^27 + 2 * q^28 - 15 * q^29 - 3 * q^31 + 2 * q^32 - 10 * q^33 + 2 * q^34 - 3 * q^36 - 12 * q^37 - 5 * q^38 + 4 * q^39 + 9 * q^41 - q^42 + 13 * q^43 + 5 * q^44 - q^46 - 23 * q^47 - q^48 + 2 * q^49 - q^51 - 8 * q^52 + q^53 + 2 * q^54 + 2 * q^56 - 15 * q^58 - 7 * q^59 - 12 * q^61 - 3 * q^62 - 3 * q^63 + 2 * q^64 - 10 * q^66 - 9 * q^67 + 2 * q^68 + 13 * q^69 + 20 * q^71 - 3 * q^72 - 8 * q^73 - 12 * q^74 - 5 * q^76 + 5 * q^77 + 4 * q^78 - 10 * q^79 - 2 * q^81 + 9 * q^82 + 2 * q^83 - q^84 + 13 * q^86 + 5 * q^87 + 5 * q^88 + 14 * q^89 - 8 * q^91 - q^92 + 4 * q^93 - 23 * q^94 - q^96 - 22 * q^97 + 2 * q^98

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