LMFDB - Embedded newform 6050.2.a.ce.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6050,2,Mod(1,6050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6050.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6050 = 2 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6050.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,2,2,0,-2,-1,-2,6,0,0,2,7,1,0,2,4,-6,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.3094932229$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6050.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} +1.23607 q^{6} +0.618034 q^{7} -1.00000 q^{8} -1.47214 q^{9} -1.23607 q^{12} +2.38197 q^{13} -0.618034 q^{14} +1.00000 q^{16} +6.47214 q^{17} +1.47214 q^{18} +3.61803 q^{19} -0.763932 q^{21} +4.61803 q^{23} +1.23607 q^{24} -2.38197 q^{26} +5.52786 q^{27} +0.618034 q^{28} +4.47214 q^{29} +2.00000 q^{31} -1.00000 q^{32} -6.47214 q^{34} -1.47214 q^{36} +5.61803 q^{37} -3.61803 q^{38} -2.94427 q^{39} +0.618034 q^{41} +0.763932 q^{42} -8.47214 q^{43} -4.61803 q^{46} +10.0902 q^{47} -1.23607 q^{48} -6.61803 q^{49} -8.00000 q^{51} +2.38197 q^{52} -7.09017 q^{53} -5.52786 q^{54} -0.618034 q^{56} -4.47214 q^{57} -4.47214 q^{58} -10.8541 q^{59} -6.94427 q^{61} -2.00000 q^{62} -0.909830 q^{63} +1.00000 q^{64} +9.23607 q^{67} +6.47214 q^{68} -5.70820 q^{69} +2.00000 q^{71} +1.47214 q^{72} +1.52786 q^{73} -5.61803 q^{74} +3.61803 q^{76} +2.94427 q^{78} -6.18034 q^{79} -2.41641 q^{81} -0.618034 q^{82} -10.1803 q^{83} -0.763932 q^{84} +8.47214 q^{86} -5.52786 q^{87} +11.3820 q^{89} +1.47214 q^{91} +4.61803 q^{92} -2.47214 q^{93} -10.0902 q^{94} +1.23607 q^{96} +9.23607 q^{97} +6.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{12} + 7 q^{13} + q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 5 q^{19} - 6 q^{21} + 7 q^{23} - 2 q^{24} - 7 q^{26} + 20 q^{27}+ \cdots + 11 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 6 * q^9 + 2 * q^12 + 7 * q^13 + q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 + 5 * q^19 - 6 * q^21 + 7 * q^23 - 2 * q^24 - 7 * q^26 + 20 * q^27 - q^28 + 4 * q^31 - 2 * q^32 - 4 * q^34 + 6 * q^36 + 9 * q^37 - 5 * q^38 + 12 * q^39 - q^41 + 6 * q^42 - 8 * q^43 - 7 * q^46 + 9 * q^47 + 2 * q^48 - 11 * q^49 - 16 * q^51 + 7 * q^52 - 3 * q^53 - 20 * q^54 + q^56 - 15 * q^59 + 4 * q^61 - 4 * q^62 - 13 * q^63 + 2 * q^64 + 14 * q^67 + 4 * q^68 + 2 * q^69 + 4 * q^71 - 6 * q^72 + 12 * q^73 - 9 * q^74 + 5 * q^76 - 12 * q^78 + 10 * q^79 + 22 * q^81 + q^82 + 2 * q^83 - 6 * q^84 + 8 * q^86 - 20 * q^87 + 25 * q^89 - 6 * q^91 + 7 * q^92 + 4 * q^93 - 9 * q^94 - 2 * q^96 + 14 * q^97 + 11 * q^98

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
$3$	−1.23607	−0.713644	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	−1.23607	−0.713644	−0.356822	+	0.934172i	$0.616140\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.23607	0.504623
$7$	0.618034	0.233595	0.116797	−	0.993156i	$-0.462737\pi$
$7$	0.618034	0.233595	0.116797	+	0.993156i	$0.462737\pi$
$8$	−1.00000	−0.353553
$9$	−1.47214	−0.490712
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−1.23607	−0.356822
$13$	2.38197	0.660639	0.330319	−	0.943869i	$-0.392844\pi$
$13$	2.38197	0.660639	0.330319	+	0.943869i	$0.392844\pi$
$14$	−0.618034	−0.165177
$15$	0	0
$16$	1.00000	0.250000
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	1.47214	0.346986
$19$	3.61803	0.830034	0.415017	−	0.909814i	$-0.363776\pi$
$19$	3.61803	0.830034	0.415017	+	0.909814i	$0.363776\pi$
$20$	0	0
$21$	−0.763932	−0.166704
$22$	0	0
$23$	4.61803	0.962927	0.481463	−	0.876466i	$-0.340105\pi$
$23$	4.61803	0.962927	0.481463	+	0.876466i	$0.340105\pi$
$24$	1.23607	0.252311
$25$	0	0
$26$	−2.38197	−0.467142
$27$	5.52786	1.06384
$28$	0.618034	0.116797
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.47214	−1.10996
$35$	0	0
$36$	−1.47214	−0.245356
$37$	5.61803	0.923599	0.461800	−	0.886984i	$-0.347204\pi$
$37$	5.61803	0.923599	0.461800	+	0.886984i	$0.347204\pi$
$38$	−3.61803	−0.586923
$39$	−2.94427	−0.471461
$40$	0	0
$41$	0.618034	0.0965207	0.0482603	−	0.998835i	$-0.484632\pi$
$41$	0.618034	0.0965207	0.0482603	+	0.998835i	$0.484632\pi$
$42$	0.763932	0.117877
$43$	−8.47214	−1.29199	−0.645994	−	0.763342i	$-0.723557\pi$
$43$	−8.47214	−1.29199	−0.645994	+	0.763342i	$0.723557\pi$
$44$	0	0
$45$	0	0
$46$	−4.61803	−0.680892
$47$	10.0902	1.47180	0.735901	−	0.677089i	$-0.236759\pi$
$47$	10.0902	1.47180	0.735901	+	0.677089i	$0.236759\pi$
$48$	−1.23607	−0.178411
$49$	−6.61803	−0.945433
$50$	0	0
$51$	−8.00000	−1.12022
$52$	2.38197	0.330319
$53$	−7.09017	−0.973910	−0.486955	−	0.873427i	$-0.661892\pi$
$53$	−7.09017	−0.973910	−0.486955	+	0.873427i	$0.661892\pi$
$54$	−5.52786	−0.752247
$55$	0	0
$56$	−0.618034	−0.0825883
$57$	−4.47214	−0.592349
$58$	−4.47214	−0.587220
$59$	−10.8541	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$59$	−10.8541	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$60$	0	0
$61$	−6.94427	−0.889123	−0.444561	−	0.895748i	$-0.646640\pi$
$61$	−6.94427	−0.889123	−0.444561	+	0.895748i	$0.646640\pi$
$62$	−2.00000	−0.254000
$63$	−0.909830	−0.114628
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	9.23607	1.12837	0.564183	−	0.825650i	$-0.309191\pi$
$67$	9.23607	1.12837	0.564183	+	0.825650i	$0.309191\pi$
$68$	6.47214	0.784862
$69$	−5.70820	−0.687187
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	1.47214	0.173493
$73$	1.52786	0.178823	0.0894115	−	0.995995i	$-0.471501\pi$
$73$	1.52786	0.178823	0.0894115	+	0.995995i	$0.471501\pi$
$74$	−5.61803	−0.653083
$75$	0	0
$76$	3.61803	0.415017
$77$	0	0
$78$	2.94427	0.333373
$79$	−6.18034	−0.695343	−0.347671	−	0.937616i	$-0.613027\pi$
$79$	−6.18034	−0.695343	−0.347671	+	0.937616i	$0.613027\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	−0.618034	−0.0682504
$83$	−10.1803	−1.11744	−0.558719	−	0.829357i	$-0.688707\pi$
$83$	−10.1803	−1.11744	−0.558719	+	0.829357i	$0.688707\pi$
$84$	−0.763932	−0.0833518
$85$	0	0
$86$	8.47214	0.913574
$87$	−5.52786	−0.592649
$88$	0	0
$89$	11.3820	1.20649	0.603243	−	0.797557i	$-0.293875\pi$
$89$	11.3820	1.20649	0.603243	+	0.797557i	$0.293875\pi$
$90$	0	0
$91$	1.47214	0.154322
$92$	4.61803	0.481463
$93$	−2.47214	−0.256349
$94$	−10.0902	−1.04072
$95$	0	0
$96$	1.23607	0.126156
$97$	9.23607	0.937781	0.468890	−	0.883256i	$-0.344654\pi$
$97$	9.23607	0.937781	0.468890	+	0.883256i	$0.344654\pi$
$98$	6.61803	0.668522
$99$	0	0
$100$	0	0
$101$	−19.7082	−1.96104	−0.980520	−	0.196420i	$-0.937068\pi$
$101$	−19.7082	−1.96104	−0.980520	+	0.196420i	$0.937068\pi$
$102$	8.00000	0.792118
$103$	6.85410	0.675355	0.337677	−	0.941262i	$-0.390359\pi$
$103$	6.85410	0.675355	0.337677	+	0.941262i	$0.390359\pi$
$104$	−2.38197	−0.233571
$105$	0	0
$106$	7.09017	0.688658
$107$	15.4164	1.49036	0.745180	−	0.666863i	$-0.232363\pi$
$107$	15.4164	1.49036	0.745180	+	0.666863i	$0.232363\pi$
$108$	5.52786	0.531919
$109$	−5.52786	−0.529473	−0.264737	−	0.964321i	$-0.585285\pi$
$109$	−5.52786	−0.529473	−0.264737	+	0.964321i	$0.585285\pi$
$110$	0	0
$111$	−6.94427	−0.659121
$112$	0.618034	0.0583987
$113$	9.41641	0.885821	0.442911	−	0.896566i	$-0.353946\pi$
$113$	9.41641	0.885821	0.442911	+	0.896566i	$0.353946\pi$
$114$	4.47214	0.418854
$115$	0	0
$116$	4.47214	0.415227
$117$	−3.50658	−0.324183
$118$	10.8541	0.999201
$119$	4.00000	0.366679
$120$	0	0
$121$	0	0
$122$	6.94427	0.628705
$123$	−0.763932	−0.0688814
$124$	2.00000	0.179605
$125$	0	0
$126$	0.909830	0.0810541
$127$	2.85410	0.253261	0.126630	−	0.991950i	$-0.459584\pi$
$127$	2.85410	0.253261	0.126630	+	0.991950i	$0.459584\pi$
$128$	−1.00000	−0.0883883
$129$	10.4721	0.922020
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	2.23607	0.193892
$134$	−9.23607	−0.797875
$135$	0	0
$136$	−6.47214	−0.554981
$137$	−21.4164	−1.82973	−0.914864	−	0.403763i	$-0.867702\pi$
$137$	−21.4164	−1.82973	−0.914864	+	0.403763i	$0.867702\pi$
$138$	5.70820	0.485915
$139$	0.326238	0.0276711	0.0138356	−	0.999904i	$-0.495596\pi$
$139$	0.326238	0.0276711	0.0138356	+	0.999904i	$0.495596\pi$
$140$	0	0
$141$	−12.4721	−1.05034
$142$	−2.00000	−0.167836
$143$	0	0
$144$	−1.47214	−0.122678
$145$	0	0
$146$	−1.52786	−0.126447
$147$	8.18034	0.674703
$148$	5.61803	0.461800
$149$	21.7082	1.77841	0.889203	−	0.457513i	$-0.151260\pi$
$149$	21.7082	1.77841	0.889203	+	0.457513i	$0.151260\pi$
$150$	0	0
$151$	13.7082	1.11556	0.557779	−	0.829990i	$-0.311654\pi$
$151$	13.7082	1.11556	0.557779	+	0.829990i	$0.311654\pi$
$152$	−3.61803	−0.293461
$153$	−9.52786	−0.770282
$154$	0	0
$155$	0	0
$156$	−2.94427	−0.235730
$157$	6.14590	0.490496	0.245248	−	0.969460i	$-0.421131\pi$
$157$	6.14590	0.490496	0.245248	+	0.969460i	$0.421131\pi$
$158$	6.18034	0.491681
$159$	8.76393	0.695025
$160$	0	0
$161$	2.85410	0.224935
$162$	2.41641	0.189851
$163$	9.41641	0.737550	0.368775	−	0.929519i	$-0.379777\pi$
$163$	9.41641	0.737550	0.368775	+	0.929519i	$0.379777\pi$
$164$	0.618034	0.0482603
$165$	0	0
$166$	10.1803	0.790148
$167$	−13.3262	−1.03122	−0.515608	−	0.856825i	$-0.672434\pi$
$167$	−13.3262	−1.03122	−0.515608	+	0.856825i	$0.672434\pi$
$168$	0.763932	0.0589386
$169$	−7.32624	−0.563557
$170$	0	0
$171$	−5.32624	−0.407308
$172$	−8.47214	−0.645994
$173$	1.85410	0.140965	0.0704824	−	0.997513i	$-0.477546\pi$
$173$	1.85410	0.140965	0.0704824	+	0.997513i	$0.477546\pi$
$174$	5.52786	0.419066
$175$	0	0
$176$	0	0
$177$	13.4164	1.00844
$178$	−11.3820	−0.853114
$179$	−9.27051	−0.692910	−0.346455	−	0.938067i	$-0.612615\pi$
$179$	−9.27051	−0.692910	−0.346455	+	0.938067i	$0.612615\pi$
$180$	0	0
$181$	19.8885	1.47830	0.739152	−	0.673539i	$-0.235227\pi$
$181$	19.8885	1.47830	0.739152	+	0.673539i	$0.235227\pi$
$182$	−1.47214	−0.109122
$183$	8.58359	0.634517
$184$	−4.61803	−0.340446
$185$	0	0
$186$	2.47214	0.181266
$187$	0	0
$188$	10.0902	0.735901
$189$	3.41641	0.248507
$190$	0	0
$191$	−9.05573	−0.655249	−0.327625	−	0.944808i	$-0.606248\pi$
$191$	−9.05573	−0.655249	−0.327625	+	0.944808i	$0.606248\pi$
$192$	−1.23607	−0.0892055
$193$	13.2361	0.952753	0.476377	−	0.879241i	$-0.341950\pi$
$193$	13.2361	0.952753	0.476377	+	0.879241i	$0.341950\pi$
$194$	−9.23607	−0.663111
$195$	0	0
$196$	−6.61803	−0.472717
$197$	−26.2148	−1.86773	−0.933863	−	0.357631i	$-0.883585\pi$
$197$	−26.2148	−1.86773	−0.933863	+	0.357631i	$0.883585\pi$
$198$	0	0
$199$	−5.52786	−0.391860	−0.195930	−	0.980618i	$-0.562773\pi$
$199$	−5.52786	−0.391860	−0.195930	+	0.980618i	$0.562773\pi$
$200$	0	0
$201$	−11.4164	−0.805251
$202$	19.7082	1.38666
$203$	2.76393	0.193990
$204$	−8.00000	−0.560112
$205$	0	0
$206$	−6.85410	−0.477548
$207$	−6.79837	−0.472520
$208$	2.38197	0.165160
$209$	0	0
$210$	0	0
$211$	26.4721	1.82242	0.911208	−	0.411945i	$-0.135151\pi$
$211$	26.4721	1.82242	0.911208	+	0.411945i	$0.135151\pi$
$212$	−7.09017	−0.486955
$213$	−2.47214	−0.169388
$214$	−15.4164	−1.05384
$215$	0	0
$216$	−5.52786	−0.376124
$217$	1.23607	0.0839098
$218$	5.52786	0.374394
$219$	−1.88854	−0.127616
$220$	0	0
$221$	15.4164	1.03702
$222$	6.94427	0.466069
$223$	19.0902	1.27837	0.639186	−	0.769052i	$-0.279271\pi$
$223$	19.0902	1.27837	0.639186	+	0.769052i	$0.279271\pi$
$224$	−0.618034	−0.0412941
$225$	0	0
$226$	−9.41641	−0.626370
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−4.47214	−0.296174
$229$	−7.23607	−0.478173	−0.239086	−	0.970998i	$-0.576848\pi$
$229$	−7.23607	−0.478173	−0.239086	+	0.970998i	$0.576848\pi$
$230$	0	0
$231$	0	0
$232$	−4.47214	−0.293610
$233$	−19.1246	−1.25289	−0.626447	−	0.779464i	$-0.715492\pi$
$233$	−19.1246	−1.25289	−0.626447	+	0.779464i	$0.715492\pi$
$234$	3.50658	0.229232
$235$	0	0
$236$	−10.8541	−0.706542
$237$	7.63932	0.496227
$238$	−4.00000	−0.259281
$239$	7.23607	0.468062	0.234031	−	0.972229i	$-0.424808\pi$
$239$	7.23607	0.468062	0.234031	+	0.972229i	$0.424808\pi$
$240$	0	0
$241$	−4.90983	−0.316270	−0.158135	−	0.987418i	$-0.550548\pi$
$241$	−4.90983	−0.316270	−0.158135	+	0.987418i	$0.550548\pi$
$242$	0	0
$243$	−13.5967	−0.872232
$244$	−6.94427	−0.444561
$245$	0	0
$246$	0.763932	0.0487065
$247$	8.61803	0.548352
$248$	−2.00000	−0.127000
$249$	12.5836	0.797453
$250$	0	0
$251$	6.79837	0.429110	0.214555	−	0.976712i	$-0.431170\pi$
$251$	6.79837	0.429110	0.214555	+	0.976712i	$0.431170\pi$
$252$	−0.909830	−0.0573139
$253$	0	0
$254$	−2.85410	−0.179082
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.9443	1.30647	0.653234	−	0.757156i	$-0.273412\pi$
$257$	20.9443	1.30647	0.653234	+	0.757156i	$0.273412\pi$
$258$	−10.4721	−0.651967
$259$	3.47214	0.215748
$260$	0	0
$261$	−6.58359	−0.407514
$262$	8.00000	0.494242
$263$	5.14590	0.317310	0.158655	−	0.987334i	$-0.449284\pi$
$263$	5.14590	0.317310	0.158655	+	0.987334i	$0.449284\pi$
$264$	0	0
$265$	0	0
$266$	−2.23607	−0.137102
$267$	−14.0689	−0.861002
$268$	9.23607	0.564183
$269$	22.3607	1.36335	0.681677	−	0.731653i	$-0.261251\pi$
$269$	22.3607	1.36335	0.681677	+	0.731653i	$0.261251\pi$
$270$	0	0
$271$	−1.41641	−0.0860407	−0.0430203	−	0.999074i	$-0.513698\pi$
$271$	−1.41641	−0.0860407	−0.0430203	+	0.999074i	$0.513698\pi$
$272$	6.47214	0.392431
$273$	−1.81966	−0.110131
$274$	21.4164	1.29381
$275$	0	0
$276$	−5.70820	−0.343594
$277$	−0.0344419	−0.00206941	−0.00103471	−	0.999999i	$-0.500329\pi$
$277$	−0.0344419	−0.00206941	−0.00103471	+	0.999999i	$0.500329\pi$
$278$	−0.326238	−0.0195665
$279$	−2.94427	−0.176269
$280$	0	0
$281$	−9.05573	−0.540219	−0.270110	−	0.962830i	$-0.587060\pi$
$281$	−9.05573	−0.540219	−0.270110	+	0.962830i	$0.587060\pi$
$282$	12.4721	0.742705
$283$	−0.180340	−0.0107201	−0.00536005	−	0.999986i	$-0.501706\pi$
$283$	−0.180340	−0.0107201	−0.00536005	+	0.999986i	$0.501706\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	0	0
$287$	0.381966	0.0225467
$288$	1.47214	0.0867464
$289$	24.8885	1.46403
$290$	0	0
$291$	−11.4164	−0.669242
$292$	1.52786	0.0894115
$293$	−29.8541	−1.74410	−0.872048	−	0.489421i	$-0.837208\pi$
$293$	−29.8541	−1.74410	−0.872048	+	0.489421i	$0.837208\pi$
$294$	−8.18034	−0.477087
$295$	0	0
$296$	−5.61803	−0.326542
$297$	0	0
$298$	−21.7082	−1.25752
$299$	11.0000	0.636146
$300$	0	0
$301$	−5.23607	−0.301802
$302$	−13.7082	−0.788818
$303$	24.3607	1.39948
$304$	3.61803	0.207508
$305$	0	0
$306$	9.52786	0.544672
$307$	−15.2361	−0.869568	−0.434784	−	0.900535i	$-0.643175\pi$
$307$	−15.2361	−0.869568	−0.434784	+	0.900535i	$0.643175\pi$
$308$	0	0
$309$	−8.47214	−0.481963
$310$	0	0
$311$	−4.18034	−0.237045	−0.118523	−	0.992951i	$-0.537816\pi$
$311$	−4.18034	−0.237045	−0.118523	+	0.992951i	$0.537816\pi$
$312$	2.94427	0.166687
$313$	11.5279	0.651593	0.325797	−	0.945440i	$-0.394368\pi$
$313$	11.5279	0.651593	0.325797	+	0.945440i	$0.394368\pi$
$314$	−6.14590	−0.346833
$315$	0	0
$316$	−6.18034	−0.347671
$317$	−14.3820	−0.807772	−0.403886	−	0.914809i	$-0.632341\pi$
$317$	−14.3820	−0.807772	−0.403886	+	0.914809i	$0.632341\pi$
$318$	−8.76393	−0.491457
$319$	0	0
$320$	0	0
$321$	−19.0557	−1.06359
$322$	−2.85410	−0.159053
$323$	23.4164	1.30292
$324$	−2.41641	−0.134245
$325$	0	0
$326$	−9.41641	−0.521527
$327$	6.83282	0.377856
$328$	−0.618034	−0.0341252
$329$	6.23607	0.343806
$330$	0	0
$331$	−6.09017	−0.334746	−0.167373	−	0.985894i	$-0.553528\pi$
$331$	−6.09017	−0.334746	−0.167373	+	0.985894i	$0.553528\pi$
$332$	−10.1803	−0.558719
$333$	−8.27051	−0.453221
$334$	13.3262	0.729179
$335$	0	0
$336$	−0.763932	−0.0416759
$337$	−1.81966	−0.0991232	−0.0495616	−	0.998771i	$-0.515782\pi$
$337$	−1.81966	−0.0991232	−0.0495616	+	0.998771i	$0.515782\pi$
$338$	7.32624	0.398495
$339$	−11.6393	−0.632161
$340$	0	0
$341$	0	0
$342$	5.32624	0.288010
$343$	−8.41641	−0.454443
$344$	8.47214	0.456787
$345$	0	0
$346$	−1.85410	−0.0996771
$347$	2.65248	0.142392	0.0711962	−	0.997462i	$-0.477318\pi$
$347$	2.65248	0.142392	0.0711962	+	0.997462i	$0.477318\pi$
$348$	−5.52786	−0.296325
$349$	−7.23607	−0.387338	−0.193669	−	0.981067i	$-0.562039\pi$
$349$	−7.23607	−0.387338	−0.193669	+	0.981067i	$0.562039\pi$
$350$	0	0
$351$	13.1672	0.702812
$352$	0	0
$353$	16.0000	0.851594	0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000	0.851594	0.425797	+	0.904819i	$0.359994\pi$
$354$	−13.4164	−0.713074
$355$	0	0
$356$	11.3820	0.603243
$357$	−4.94427	−0.261679
$358$	9.27051	0.489962
$359$	−32.3607	−1.70793	−0.853966	−	0.520329i	$-0.825809\pi$
$359$	−32.3607	−1.70793	−0.853966	+	0.520329i	$0.825809\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	−19.8885	−1.04532
$363$	0	0
$364$	1.47214	0.0771609
$365$	0	0
$366$	−8.58359	−0.448672
$367$	29.8885	1.56017	0.780085	−	0.625674i	$-0.215176\pi$
$367$	29.8885	1.56017	0.780085	+	0.625674i	$0.215176\pi$
$368$	4.61803	0.240732
$369$	−0.909830	−0.0473639
$370$	0	0
$371$	−4.38197	−0.227500
$372$	−2.47214	−0.128174
$373$	−6.56231	−0.339783	−0.169892	−	0.985463i	$-0.554342\pi$
$373$	−6.56231	−0.339783	−0.169892	+	0.985463i	$0.554342\pi$
$374$	0	0
$375$	0	0
$376$	−10.0902	−0.520361
$377$	10.6525	0.548630
$378$	−3.41641	−0.175721
$379$	32.0344	1.64550	0.822749	−	0.568404i	$-0.192439\pi$
$379$	32.0344	1.64550	0.822749	+	0.568404i	$0.192439\pi$
$380$	0	0
$381$	−3.52786	−0.180738
$382$	9.05573	0.463331
$383$	−22.2148	−1.13512	−0.567561	−	0.823331i	$-0.692113\pi$
$383$	−22.2148	−1.13512	−0.567561	+	0.823331i	$0.692113\pi$
$384$	1.23607	0.0630778
$385$	0	0
$386$	−13.2361	−0.673698
$387$	12.4721	0.633994
$388$	9.23607	0.468890
$389$	25.1246	1.27387	0.636934	−	0.770918i	$-0.280202\pi$
$389$	25.1246	1.27387	0.636934	+	0.770918i	$0.280202\pi$
$390$	0	0
$391$	29.8885	1.51153
$392$	6.61803	0.334261
$393$	9.88854	0.498811
$394$	26.2148	1.32068
$395$	0	0
$396$	0	0
$397$	12.3262	0.618636	0.309318	−	0.950959i	$-0.399899\pi$
$397$	12.3262	0.618636	0.309318	+	0.950959i	$0.399899\pi$
$398$	5.52786	0.277087
$399$	−2.76393	−0.138370
$400$	0	0
$401$	0.0901699	0.00450287	0.00225144	−	0.999997i	$-0.499283\pi$
$401$	0.0901699	0.00450287	0.00225144	+	0.999997i	$0.499283\pi$
$402$	11.4164	0.569399
$403$	4.76393	0.237308
$404$	−19.7082	−0.980520
$405$	0	0
$406$	−2.76393	−0.137172
$407$	0	0
$408$	8.00000	0.396059
$409$	25.9787	1.28456	0.642282	−	0.766468i	$-0.277988\pi$
$409$	25.9787	1.28456	0.642282	+	0.766468i	$0.277988\pi$
$410$	0	0
$411$	26.4721	1.30577
$412$	6.85410	0.337677
$413$	−6.70820	−0.330089
$414$	6.79837	0.334122
$415$	0	0
$416$	−2.38197	−0.116785
$417$	−0.403252	−0.0197473
$418$	0	0
$419$	35.9787	1.75768	0.878838	−	0.477121i	$-0.158320\pi$
$419$	35.9787	1.75768	0.878838	+	0.477121i	$0.158320\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−26.4721	−1.28864
$423$	−14.8541	−0.722231
$424$	7.09017	0.344329
$425$	0	0
$426$	2.47214	0.119775
$427$	−4.29180	−0.207695
$428$	15.4164	0.745180
$429$	0	0
$430$	0	0
$431$	28.1803	1.35740	0.678700	−	0.734416i	$-0.262544\pi$
$431$	28.1803	1.35740	0.678700	+	0.734416i	$0.262544\pi$
$432$	5.52786	0.265959
$433$	−7.41641	−0.356410	−0.178205	−	0.983993i	$-0.557029\pi$
$433$	−7.41641	−0.356410	−0.178205	+	0.983993i	$0.557029\pi$
$434$	−1.23607	−0.0593332
$435$	0	0
$436$	−5.52786	−0.264737
$437$	16.7082	0.799262
$438$	1.88854	0.0902381
$439$	−6.18034	−0.294972	−0.147486	−	0.989064i	$-0.547118\pi$
$439$	−6.18034	−0.294972	−0.147486	+	0.989064i	$0.547118\pi$
$440$	0	0
$441$	9.74265	0.463936
$442$	−15.4164	−0.733284
$443$	−0.180340	−0.00856821	−0.00428410	−	0.999991i	$-0.501364\pi$
$443$	−0.180340	−0.00856821	−0.00428410	+	0.999991i	$0.501364\pi$
$444$	−6.94427	−0.329561
$445$	0	0
$446$	−19.0902	−0.903946
$447$	−26.8328	−1.26915
$448$	0.618034	0.0291994
$449$	−20.8541	−0.984166	−0.492083	−	0.870548i	$-0.663764\pi$
$449$	−20.8541	−0.984166	−0.492083	+	0.870548i	$0.663764\pi$
$450$	0	0
$451$	0	0
$452$	9.41641	0.442911
$453$	−16.9443	−0.796111
$454$	−12.0000	−0.563188
$455$	0	0
$456$	4.47214	0.209427
$457$	8.18034	0.382660	0.191330	−	0.981526i	$-0.438720\pi$
$457$	8.18034	0.382660	0.191330	+	0.981526i	$0.438720\pi$
$458$	7.23607	0.338119
$459$	35.7771	1.66993
$460$	0	0
$461$	−9.70820	−0.452156	−0.226078	−	0.974109i	$-0.572590\pi$
$461$	−9.70820	−0.452156	−0.226078	+	0.974109i	$0.572590\pi$
$462$	0	0
$463$	1.32624	0.0616355	0.0308178	−	0.999525i	$-0.490189\pi$
$463$	1.32624	0.0616355	0.0308178	+	0.999525i	$0.490189\pi$
$464$	4.47214	0.207614
$465$	0	0
$466$	19.1246	0.885931
$467$	−18.6525	−0.863134	−0.431567	−	0.902081i	$-0.642039\pi$
$467$	−18.6525	−0.863134	−0.431567	+	0.902081i	$0.642039\pi$
$468$	−3.50658	−0.162092
$469$	5.70820	0.263580
$470$	0	0
$471$	−7.59675	−0.350040
$472$	10.8541	0.499601
$473$	0	0
$474$	−7.63932	−0.350886
$475$	0	0
$476$	4.00000	0.183340
$477$	10.4377	0.477909
$478$	−7.23607	−0.330970
$479$	−27.2361	−1.24445	−0.622224	−	0.782839i	$-0.713771\pi$
$479$	−27.2361	−1.24445	−0.622224	+	0.782839i	$0.713771\pi$
$480$	0	0
$481$	13.3820	0.610165
$482$	4.90983	0.223637
$483$	−3.52786	−0.160523
$484$	0	0
$485$	0	0
$486$	13.5967	0.616761
$487$	40.9443	1.85536	0.927681	−	0.373374i	$-0.121799\pi$
$487$	40.9443	1.85536	0.927681	+	0.373374i	$0.121799\pi$
$488$	6.94427	0.314352
$489$	−11.6393	−0.526348
$490$	0	0
$491$	26.2705	1.18557	0.592786	−	0.805360i	$-0.298028\pi$
$491$	26.2705	1.18557	0.592786	+	0.805360i	$0.298028\pi$
$492$	−0.763932	−0.0344407
$493$	28.9443	1.30358
$494$	−8.61803	−0.387744
$495$	0	0
$496$	2.00000	0.0898027
$497$	1.23607	0.0554452
$498$	−12.5836	−0.563884
$499$	−0.326238	−0.0146044	−0.00730221	−	0.999973i	$-0.502324\pi$
$499$	−0.326238	−0.0146044	−0.00730221	+	0.999973i	$0.502324\pi$
$500$	0	0
$501$	16.4721	0.735921
$502$	−6.79837	−0.303426
$503$	12.3820	0.552085	0.276042	−	0.961145i	$-0.410977\pi$
$503$	12.3820	0.552085	0.276042	+	0.961145i	$0.410977\pi$
$504$	0.909830	0.0405271
$505$	0	0
$506$	0	0
$507$	9.05573	0.402179
$508$	2.85410	0.126630
$509$	−11.7082	−0.518957	−0.259479	−	0.965749i	$-0.583551\pi$
$509$	−11.7082	−0.518957	−0.259479	+	0.965749i	$0.583551\pi$
$510$	0	0
$511$	0.944272	0.0417721
$512$	−1.00000	−0.0441942
$513$	20.0000	0.883022
$514$	−20.9443	−0.923812
$515$	0	0
$516$	10.4721	0.461010
$517$	0	0
$518$	−3.47214	−0.152557
$519$	−2.29180	−0.100599
$520$	0	0
$521$	−31.0902	−1.36209	−0.681043	−	0.732244i	$-0.738473\pi$
$521$	−31.0902	−1.36209	−0.681043	+	0.732244i	$0.738473\pi$
$522$	6.58359	0.288156
$523$	1.12461	0.0491758	0.0245879	−	0.999698i	$-0.492173\pi$
$523$	1.12461	0.0491758	0.0245879	+	0.999698i	$0.492173\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	−5.14590	−0.224372
$527$	12.9443	0.563861
$528$	0	0
$529$	−1.67376	−0.0727723
$530$	0	0
$531$	15.9787	0.693417
$532$	2.23607	0.0969458
$533$	1.47214	0.0637653
$534$	14.0689	0.608820
$535$	0	0
$536$	−9.23607	−0.398937
$537$	11.4590	0.494492
$538$	−22.3607	−0.964037
$539$	0	0
$540$	0	0
$541$	15.8197	0.680140	0.340070	−	0.940400i	$-0.389549\pi$
$541$	15.8197	0.680140	0.340070	+	0.940400i	$0.389549\pi$
$542$	1.41641	0.0608399
$543$	−24.5836	−1.05498
$544$	−6.47214	−0.277491
$545$	0	0
$546$	1.81966	0.0778743
$547$	19.8885	0.850373	0.425186	−	0.905106i	$-0.360209\pi$
$547$	19.8885	0.850373	0.425186	+	0.905106i	$0.360209\pi$
$548$	−21.4164	−0.914864
$549$	10.2229	0.436303
$550$	0	0
$551$	16.1803	0.689306
$552$	5.70820	0.242957
$553$	−3.81966	−0.162428
$554$	0.0344419	0.00146329
$555$	0	0
$556$	0.326238	0.0138356
$557$	−10.5623	−0.447539	−0.223770	−	0.974642i	$-0.571836\pi$
$557$	−10.5623	−0.447539	−0.223770	+	0.974642i	$0.571836\pi$
$558$	2.94427	0.124641
$559$	−20.1803	−0.853537
$560$	0	0
$561$	0	0
$562$	9.05573	0.381993
$563$	45.5967	1.92167	0.960837	−	0.277115i	$-0.0893781\pi$
$563$	45.5967	1.92167	0.960837	+	0.277115i	$0.0893781\pi$
$564$	−12.4721	−0.525172
$565$	0	0
$566$	0.180340	0.00758025
$567$	−1.49342	−0.0627178
$568$	−2.00000	−0.0839181
$569$	5.85410	0.245417	0.122708	−	0.992443i	$-0.460842\pi$
$569$	5.85410	0.245417	0.122708	+	0.992443i	$0.460842\pi$
$570$	0	0
$571$	44.6869	1.87009	0.935045	−	0.354530i	$-0.115359\pi$
$571$	44.6869	1.87009	0.935045	+	0.354530i	$0.115359\pi$
$572$	0	0
$573$	11.1935	0.467615
$574$	−0.381966	−0.0159430
$575$	0	0
$576$	−1.47214	−0.0613390
$577$	17.5279	0.729695	0.364847	−	0.931067i	$-0.381121\pi$
$577$	17.5279	0.729695	0.364847	+	0.931067i	$0.381121\pi$
$578$	−24.8885	−1.03523
$579$	−16.3607	−0.679927
$580$	0	0
$581$	−6.29180	−0.261028
$582$	11.4164	0.473225
$583$	0	0
$584$	−1.52786	−0.0632235
$585$	0	0
$586$	29.8541	1.23326
$587$	−24.1803	−0.998029	−0.499015	−	0.866594i	$-0.666305\pi$
$587$	−24.1803	−0.998029	−0.499015	+	0.866594i	$0.666305\pi$
$588$	8.18034	0.337352
$589$	7.23607	0.298157
$590$	0	0
$591$	32.4033	1.33289
$592$	5.61803	0.230900
$593$	−6.11146	−0.250967	−0.125484	−	0.992096i	$-0.540048\pi$
$593$	−6.11146	−0.250967	−0.125484	+	0.992096i	$0.540048\pi$
$594$	0	0
$595$	0	0
$596$	21.7082	0.889203
$597$	6.83282	0.279649
$598$	−11.0000	−0.449823
$599$	−6.18034	−0.252522	−0.126261	−	0.991997i	$-0.540298\pi$
$599$	−6.18034	−0.252522	−0.126261	+	0.991997i	$0.540298\pi$
$600$	0	0
$601$	0.0901699	0.00367811	0.00183905	−	0.999998i	$-0.499415\pi$
$601$	0.0901699	0.00367811	0.00183905	+	0.999998i	$0.499415\pi$
$602$	5.23607	0.213406
$603$	−13.5967	−0.553702
$604$	13.7082	0.557779
$605$	0	0
$606$	−24.3607	−0.989585
$607$	−33.5279	−1.36085	−0.680427	−	0.732816i	$-0.738206\pi$
$607$	−33.5279	−1.36085	−0.680427	+	0.732816i	$0.738206\pi$
$608$	−3.61803	−0.146731
$609$	−3.41641	−0.138440
$610$	0	0
$611$	24.0344	0.972329
$612$	−9.52786	−0.385141
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	15.2361	0.614878
$615$	0	0
$616$	0	0
$617$	−9.05573	−0.364570	−0.182285	−	0.983246i	$-0.558349\pi$
$617$	−9.05573	−0.364570	−0.182285	+	0.983246i	$0.558349\pi$
$618$	8.47214	0.340799
$619$	−7.43769	−0.298946	−0.149473	−	0.988766i	$-0.547758\pi$
$619$	−7.43769	−0.298946	−0.149473	+	0.988766i	$0.547758\pi$
$620$	0	0
$621$	25.5279	1.02440
$622$	4.18034	0.167616
$623$	7.03444	0.281829
$624$	−2.94427	−0.117865
$625$	0	0
$626$	−11.5279	−0.460746
$627$	0	0
$628$	6.14590	0.245248
$629$	36.3607	1.44980
$630$	0	0
$631$	27.1246	1.07981	0.539907	−	0.841725i	$-0.318459\pi$
$631$	27.1246	1.07981	0.539907	+	0.841725i	$0.318459\pi$
$632$	6.18034	0.245841
$633$	−32.7214	−1.30056
$634$	14.3820	0.571181
$635$	0	0
$636$	8.76393	0.347513
$637$	−15.7639	−0.624590
$638$	0	0
$639$	−2.94427	−0.116474
$640$	0	0
$641$	30.7426	1.21426	0.607131	−	0.794602i	$-0.292320\pi$
$641$	30.7426	1.21426	0.607131	+	0.794602i	$0.292320\pi$
$642$	19.0557	0.752070
$643$	−20.8328	−0.821566	−0.410783	−	0.911733i	$-0.634745\pi$
$643$	−20.8328	−0.821566	−0.410783	+	0.911733i	$0.634745\pi$
$644$	2.85410	0.112467
$645$	0	0
$646$	−23.4164	−0.921306
$647$	27.7771	1.09203	0.546015	−	0.837775i	$-0.316144\pi$
$647$	27.7771	1.09203	0.546015	+	0.837775i	$0.316144\pi$
$648$	2.41641	0.0949255
$649$	0	0
$650$	0	0
$651$	−1.52786	−0.0598817
$652$	9.41641	0.368775
$653$	29.2148	1.14326	0.571631	−	0.820511i	$-0.306311\pi$
$653$	29.2148	1.14326	0.571631	+	0.820511i	$0.306311\pi$
$654$	−6.83282	−0.267184
$655$	0	0
$656$	0.618034	0.0241302
$657$	−2.24922	−0.0877506
$658$	−6.23607	−0.243107
$659$	−11.3820	−0.443378	−0.221689	−	0.975117i	$-0.571157\pi$
$659$	−11.3820	−0.443378	−0.221689	+	0.975117i	$0.571157\pi$
$660$	0	0
$661$	−5.88854	−0.229038	−0.114519	−	0.993421i	$-0.536533\pi$
$661$	−5.88854	−0.229038	−0.114519	+	0.993421i	$0.536533\pi$
$662$	6.09017	0.236701
$663$	−19.0557	−0.740063
$664$	10.1803	0.395074
$665$	0	0
$666$	8.27051	0.320476
$667$	20.6525	0.799667
$668$	−13.3262	−0.515608
$669$	−23.5967	−0.912303
$670$	0	0
$671$	0	0
$672$	0.763932	0.0294693
$673$	2.18034	0.0840459	0.0420230	−	0.999117i	$-0.486620\pi$
$673$	2.18034	0.0840459	0.0420230	+	0.999117i	$0.486620\pi$
$674$	1.81966	0.0700907
$675$	0	0
$676$	−7.32624	−0.281778
$677$	16.4721	0.633076	0.316538	−	0.948580i	$-0.397480\pi$
$677$	16.4721	0.633076	0.316538	+	0.948580i	$0.397480\pi$
$678$	11.6393	0.447005
$679$	5.70820	0.219061
$680$	0	0
$681$	−14.8328	−0.568395
$682$	0	0
$683$	15.5967	0.596793	0.298396	−	0.954442i	$-0.403548\pi$
$683$	15.5967	0.596793	0.298396	+	0.954442i	$0.403548\pi$
$684$	−5.32624	−0.203654
$685$	0	0
$686$	8.41641	0.321340
$687$	8.94427	0.341245
$688$	−8.47214	−0.322997
$689$	−16.8885	−0.643402
$690$	0	0
$691$	39.1591	1.48968	0.744840	−	0.667243i	$-0.232526\pi$
$691$	39.1591	1.48968	0.744840	+	0.667243i	$0.232526\pi$
$692$	1.85410	0.0704824
$693$	0	0
$694$	−2.65248	−0.100687
$695$	0	0
$696$	5.52786	0.209533
$697$	4.00000	0.151511
$698$	7.23607	0.273889
$699$	23.6393	0.894121
$700$	0	0
$701$	−26.5410	−1.00244	−0.501220	−	0.865320i	$-0.667115\pi$
$701$	−26.5410	−1.00244	−0.501220	+	0.865320i	$0.667115\pi$
$702$	−13.1672	−0.496963
$703$	20.3262	0.766619
$704$	0	0
$705$	0	0
$706$	−16.0000	−0.602168
$707$	−12.1803	−0.458089
$708$	13.4164	0.504219
$709$	−4.47214	−0.167955	−0.0839773	−	0.996468i	$-0.526762\pi$
$709$	−4.47214	−0.167955	−0.0839773	+	0.996468i	$0.526762\pi$
$710$	0	0
$711$	9.09830	0.341213
$712$	−11.3820	−0.426557
$713$	9.23607	0.345893
$714$	4.94427	0.185035
$715$	0	0
$716$	−9.27051	−0.346455
$717$	−8.94427	−0.334030
$718$	32.3607	1.20769
$719$	19.5967	0.730835	0.365418	−	0.930844i	$-0.380926\pi$
$719$	19.5967	0.730835	0.365418	+	0.930844i	$0.380926\pi$
$720$	0	0
$721$	4.23607	0.157759
$722$	5.90983	0.219941
$723$	6.06888	0.225704
$724$	19.8885	0.739152
$725$	0	0
$726$	0	0
$727$	−31.7426	−1.17727	−0.588635	−	0.808399i	$-0.700334\pi$
$727$	−31.7426	−1.17727	−0.588635	+	0.808399i	$0.700334\pi$
$728$	−1.47214	−0.0545610
$729$	24.0557	0.890953
$730$	0	0
$731$	−54.8328	−2.02806
$732$	8.58359	0.317259
$733$	9.41641	0.347803	0.173901	−	0.984763i	$-0.444363\pi$
$733$	9.41641	0.347803	0.173901	+	0.984763i	$0.444363\pi$
$734$	−29.8885	−1.10321
$735$	0	0
$736$	−4.61803	−0.170223
$737$	0	0
$738$	0.909830	0.0334913
$739$	8.61803	0.317020	0.158510	−	0.987357i	$-0.449331\pi$
$739$	8.61803	0.317020	0.158510	+	0.987357i	$0.449331\pi$
$740$	0	0
$741$	−10.6525	−0.391328
$742$	4.38197	0.160867
$743$	15.6738	0.575015	0.287507	−	0.957778i	$-0.407173\pi$
$743$	15.6738	0.575015	0.287507	+	0.957778i	$0.407173\pi$
$744$	2.47214	0.0906329
$745$	0	0
$746$	6.56231	0.240263
$747$	14.9868	0.548340
$748$	0	0
$749$	9.52786	0.348141
$750$	0	0
$751$	−23.1246	−0.843829	−0.421915	−	0.906636i	$-0.638642\pi$
$751$	−23.1246	−0.843829	−0.421915	+	0.906636i	$0.638642\pi$
$752$	10.0902	0.367951
$753$	−8.40325	−0.306232
$754$	−10.6525	−0.387940
$755$	0	0
$756$	3.41641	0.124254
$757$	7.72949	0.280933	0.140467	−	0.990085i	$-0.455140\pi$
$757$	7.72949	0.280933	0.140467	+	0.990085i	$0.455140\pi$
$758$	−32.0344	−1.16354
$759$	0	0
$760$	0	0
$761$	−10.3607	−0.375574	−0.187787	−	0.982210i	$-0.560132\pi$
$761$	−10.3607	−0.375574	−0.187787	+	0.982210i	$0.560132\pi$
$762$	3.52786	0.127801
$763$	−3.41641	−0.123682
$764$	−9.05573	−0.327625
$765$	0	0
$766$	22.2148	0.802653
$767$	−25.8541	−0.933538
$768$	−1.23607	−0.0446028
$769$	−37.0344	−1.33550	−0.667748	−	0.744387i	$-0.732742\pi$
$769$	−37.0344	−1.33550	−0.667748	+	0.744387i	$0.732742\pi$
$770$	0	0
$771$	−25.8885	−0.932353
$772$	13.2361	0.476377
$773$	−39.4508	−1.41895	−0.709474	−	0.704731i	$-0.751068\pi$
$773$	−39.4508	−1.41895	−0.709474	+	0.704731i	$0.751068\pi$
$774$	−12.4721	−0.448302
$775$	0	0
$776$	−9.23607	−0.331556
$777$	−4.29180	−0.153967
$778$	−25.1246	−0.900761
$779$	2.23607	0.0801154
$780$	0	0
$781$	0	0
$782$	−29.8885	−1.06881
$783$	24.7214	0.883469
$784$	−6.61803	−0.236358
$785$	0	0
$786$	−9.88854	−0.352713
$787$	−18.6525	−0.664889	−0.332444	−	0.943123i	$-0.607873\pi$
$787$	−18.6525	−0.664889	−0.332444	+	0.943123i	$0.607873\pi$
$788$	−26.2148	−0.933863
$789$	−6.36068	−0.226446
$790$	0	0
$791$	5.81966	0.206923
$792$	0	0
$793$	−16.5410	−0.587389
$794$	−12.3262	−0.437442
$795$	0	0
$796$	−5.52786	−0.195930
$797$	5.61803	0.199001	0.0995005	−	0.995038i	$-0.468276\pi$
$797$	5.61803	0.199001	0.0995005	+	0.995038i	$0.468276\pi$
$798$	2.76393	0.0978421
$799$	65.3050	2.31032
$800$	0	0
$801$	−16.7558	−0.592037
$802$	−0.0901699	−0.00318401
$803$	0	0
$804$	−11.4164	−0.402626
$805$	0	0
$806$	−4.76393	−0.167802
$807$	−27.6393	−0.972950
$808$	19.7082	0.693332
$809$	−2.56231	−0.0900859	−0.0450429	−	0.998985i	$-0.514342\pi$
$809$	−2.56231	−0.0900859	−0.0450429	+	0.998985i	$0.514342\pi$
$810$	0	0
$811$	−42.1459	−1.47994	−0.739971	−	0.672638i	$-0.765161\pi$
$811$	−42.1459	−1.47994	−0.739971	+	0.672638i	$0.765161\pi$
$812$	2.76393	0.0969950
$813$	1.75078	0.0614024
$814$	0	0
$815$	0	0
$816$	−8.00000	−0.280056
$817$	−30.6525	−1.07239
$818$	−25.9787	−0.908324
$819$	−2.16718	−0.0757275
$820$	0	0
$821$	2.65248	0.0925720	0.0462860	−	0.998928i	$-0.485261\pi$
$821$	2.65248	0.0925720	0.0462860	+	0.998928i	$0.485261\pi$
$822$	−26.4721	−0.923322
$823$	6.45085	0.224862	0.112431	−	0.993660i	$-0.464136\pi$
$823$	6.45085	0.224862	0.112431	+	0.993660i	$0.464136\pi$
$824$	−6.85410	−0.238774
$825$	0	0
$826$	6.70820	0.233408
$827$	8.83282	0.307147	0.153574	−	0.988137i	$-0.450922\pi$
$827$	8.83282	0.307147	0.153574	+	0.988137i	$0.450922\pi$
$828$	−6.79837	−0.236260
$829$	−25.5279	−0.886619	−0.443310	−	0.896369i	$-0.646196\pi$
$829$	−25.5279	−0.886619	−0.443310	+	0.896369i	$0.646196\pi$
$830$	0	0
$831$	0.0425725	0.00147682
$832$	2.38197	0.0825798
$833$	−42.8328	−1.48407
$834$	0.403252	0.0139635
$835$	0	0
$836$	0	0
$837$	11.0557	0.382142
$838$	−35.9787	−1.24286
$839$	−7.23607	−0.249817	−0.124908	−	0.992168i	$-0.539864\pi$
$839$	−7.23607	−0.249817	−0.124908	+	0.992168i	$0.539864\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−2.00000	−0.0689246
$843$	11.1935	0.385524
$844$	26.4721	0.911208
$845$	0	0
$846$	14.8541	0.510695
$847$	0	0
$848$	−7.09017	−0.243477
$849$	0.222912	0.00765033
$850$	0	0
$851$	25.9443	0.889358
$852$	−2.47214	−0.0846940
$853$	5.27051	0.180459	0.0902294	−	0.995921i	$-0.471240\pi$
$853$	5.27051	0.180459	0.0902294	+	0.995921i	$0.471240\pi$
$854$	4.29180	0.146862
$855$	0	0
$856$	−15.4164	−0.526922
$857$	8.18034	0.279435	0.139718	−	0.990191i	$-0.455381\pi$
$857$	8.18034	0.279435	0.139718	+	0.990191i	$0.455381\pi$
$858$	0	0
$859$	8.61803	0.294044	0.147022	−	0.989133i	$-0.453031\pi$
$859$	8.61803	0.294044	0.147022	+	0.989133i	$0.453031\pi$
$860$	0	0
$861$	−0.472136	−0.0160904
$862$	−28.1803	−0.959826
$863$	23.0344	0.784102	0.392051	−	0.919944i	$-0.371766\pi$
$863$	23.0344	0.784102	0.392051	+	0.919944i	$0.371766\pi$
$864$	−5.52786	−0.188062
$865$	0	0
$866$	7.41641	0.252020
$867$	−30.7639	−1.04480
$868$	1.23607	0.0419549
$869$	0	0
$870$	0	0
$871$	22.0000	0.745442
$872$	5.52786	0.187197
$873$	−13.5967	−0.460180
$874$	−16.7082	−0.565163
$875$	0	0
$876$	−1.88854	−0.0638080
$877$	−51.8673	−1.75143	−0.875716	−	0.482826i	$-0.839610\pi$
$877$	−51.8673	−1.75143	−0.875716	+	0.482826i	$0.839610\pi$
$878$	6.18034	0.208576
$879$	36.9017	1.24466
$880$	0	0
$881$	6.79837	0.229043	0.114522	−	0.993421i	$-0.463467\pi$
$881$	6.79837	0.229043	0.114522	+	0.993421i	$0.463467\pi$
$882$	−9.74265	−0.328052
$883$	14.2918	0.480957	0.240479	−	0.970654i	$-0.422696\pi$
$883$	14.2918	0.480957	0.240479	+	0.970654i	$0.422696\pi$
$884$	15.4164	0.518510
$885$	0	0
$886$	0.180340	0.00605864
$887$	39.1591	1.31483	0.657416	−	0.753528i	$-0.271649\pi$
$887$	39.1591	1.31483	0.657416	+	0.753528i	$0.271649\pi$
$888$	6.94427	0.233035
$889$	1.76393	0.0591604
$890$	0	0
$891$	0	0
$892$	19.0902	0.639186
$893$	36.5066	1.22165
$894$	26.8328	0.897424
$895$	0	0
$896$	−0.618034	−0.0206471
$897$	−13.5967	−0.453982
$898$	20.8541	0.695910
$899$	8.94427	0.298308
$900$	0	0
$901$	−45.8885	−1.52877
$902$	0	0
$903$	6.47214	0.215379
$904$	−9.41641	−0.313185
$905$	0	0
$906$	16.9443	0.562936
$907$	29.4853	0.979043	0.489522	−	0.871991i	$-0.337171\pi$
$907$	29.4853	0.979043	0.489522	+	0.871991i	$0.337171\pi$
$908$	12.0000	0.398234
$909$	29.0132	0.962306
$910$	0	0
$911$	−24.1803	−0.801130	−0.400565	−	0.916268i	$-0.631186\pi$
$911$	−24.1803	−0.801130	−0.400565	+	0.916268i	$0.631186\pi$
$912$	−4.47214	−0.148087
$913$	0	0
$914$	−8.18034	−0.270582
$915$	0	0
$916$	−7.23607	−0.239086
$917$	−4.94427	−0.163274
$918$	−35.7771	−1.18082
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	18.8328	0.620562
$922$	9.70820	0.319723
$923$	4.76393	0.156807
$924$	0	0
$925$	0	0
$926$	−1.32624	−0.0435829
$927$	−10.0902	−0.331405
$928$	−4.47214	−0.146805
$929$	−6.50658	−0.213474	−0.106737	−	0.994287i	$-0.534040\pi$
$929$	−6.50658	−0.213474	−0.106737	+	0.994287i	$0.534040\pi$
$930$	0	0
$931$	−23.9443	−0.784742
$932$	−19.1246	−0.626447
$933$	5.16718	0.169166
$934$	18.6525	0.610328
$935$	0	0
$936$	3.50658	0.114616
$937$	53.3050	1.74140	0.870698	−	0.491817i	$-0.163667\pi$
$937$	53.3050	1.74140	0.870698	+	0.491817i	$0.163667\pi$
$938$	−5.70820	−0.186379
$939$	−14.2492	−0.465006
$940$	0	0
$941$	19.8885	0.648348	0.324174	−	0.945997i	$-0.394914\pi$
$941$	19.8885	0.648348	0.324174	+	0.945997i	$0.394914\pi$
$942$	7.59675	0.247515
$943$	2.85410	0.0929423
$944$	−10.8541	−0.353271
$945$	0	0
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	7.63932	0.248114
$949$	3.63932	0.118137
$950$	0	0
$951$	17.7771	0.576462
$952$	−4.00000	−0.129641
$953$	−6.76393	−0.219105	−0.109553	−	0.993981i	$-0.534942\pi$
$953$	−6.76393	−0.219105	−0.109553	+	0.993981i	$0.534942\pi$
$954$	−10.4377	−0.337933
$955$	0	0
$956$	7.23607	0.234031
$957$	0	0
$958$	27.2361	0.879957
$959$	−13.2361	−0.427415
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−13.3820	−0.431452
$963$	−22.6950	−0.731338
$964$	−4.90983	−0.158135
$965$	0	0
$966$	3.52786	0.113507
$967$	44.5623	1.43303	0.716514	−	0.697573i	$-0.245737\pi$
$967$	44.5623	1.43303	0.716514	+	0.697573i	$0.245737\pi$
$968$	0	0
$969$	−28.9443	−0.929824
$970$	0	0
$971$	−35.1591	−1.12831	−0.564154	−	0.825670i	$-0.690798\pi$
$971$	−35.1591	−1.12831	−0.564154	+	0.825670i	$0.690798\pi$
$972$	−13.5967	−0.436116
$973$	0.201626	0.00646384
$974$	−40.9443	−1.31194
$975$	0	0
$976$	−6.94427	−0.222281
$977$	8.83282	0.282587	0.141293	−	0.989968i	$-0.454874\pi$
$977$	8.83282	0.282587	0.141293	+	0.989968i	$0.454874\pi$
$978$	11.6393	0.372184
$979$	0	0
$980$	0	0
$981$	8.13777	0.259819
$982$	−26.2705	−0.838326
$983$	−22.2148	−0.708541	−0.354271	−	0.935143i	$-0.615271\pi$
$983$	−22.2148	−0.708541	−0.354271	+	0.935143i	$0.615271\pi$
$984$	0.763932	0.0243533
$985$	0	0
$986$	−28.9443	−0.921773
$987$	−7.70820	−0.245355
$988$	8.61803	0.274176
$989$	−39.1246	−1.24409
$990$	0	0
$991$	35.8197	1.13785	0.568925	−	0.822390i	$-0.307360\pi$
$991$	35.8197	1.13785	0.568925	+	0.822390i	$0.307360\pi$
$992$	−2.00000	−0.0635001
$993$	7.52786	0.238890
$994$	−1.23607	−0.0392057
$995$	0	0
$996$	12.5836	0.398726
$997$	−26.9443	−0.853334	−0.426667	−	0.904409i	$-0.640312\pi$
$997$	−26.9443	−0.853334	−0.426667	+	0.904409i	$0.640312\pi$
$998$	0.326238	0.0103269
$999$	31.0557	0.982560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6050.2.a.ce.1.1		2
5.2	odd	4		1210.2.b.f.969.2		4
5.3	odd	4		1210.2.b.f.969.3		4
5.4	even	2		6050.2.a.cl.1.2		2
11.5	even	5		550.2.h.e.201.1		4
11.9	even	5		550.2.h.e.301.1		4
11.10	odd	2		6050.2.a.ct.1.1		2
55.9	even	10		550.2.h.d.301.1		4
55.27	odd	20		110.2.j.a.69.2	yes	8
55.32	even	4		1210.2.b.g.969.4		4
55.38	odd	20		110.2.j.a.69.1	yes	8
55.42	odd	20		110.2.j.a.59.1	✓	8
55.43	even	4		1210.2.b.g.969.1		4
55.49	even	10		550.2.h.d.201.1		4
55.53	odd	20		110.2.j.a.59.2	yes	8
55.54	odd	2		6050.2.a.bv.1.2		2
165.38	even	20		990.2.ba.b.289.2		8
165.53	even	20		990.2.ba.b.829.1		8
165.137	even	20		990.2.ba.b.289.1		8
165.152	even	20		990.2.ba.b.829.2		8
220.27	even	20		880.2.cd.a.289.2		8
220.163	even	20		880.2.cd.a.609.2		8
220.203	even	20		880.2.cd.a.289.1		8
220.207	even	20		880.2.cd.a.609.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.j.a.59.1	✓	8	55.42	odd	20
110.2.j.a.59.2	yes	8	55.53	odd	20
110.2.j.a.69.1	yes	8	55.38	odd	20
110.2.j.a.69.2	yes	8	55.27	odd	20
550.2.h.d.201.1		4	55.49	even	10
550.2.h.d.301.1		4	55.9	even	10
550.2.h.e.201.1		4	11.5	even	5
550.2.h.e.301.1		4	11.9	even	5
880.2.cd.a.289.1		8	220.203	even	20
880.2.cd.a.289.2		8	220.27	even	20
880.2.cd.a.609.1		8	220.207	even	20
880.2.cd.a.609.2		8	220.163	even	20
990.2.ba.b.289.1		8	165.137	even	20
990.2.ba.b.289.2		8	165.38	even	20
990.2.ba.b.829.1		8	165.53	even	20
990.2.ba.b.829.2		8	165.152	even	20
1210.2.b.f.969.2		4	5.2	odd	4
1210.2.b.f.969.3		4	5.3	odd	4
1210.2.b.g.969.1		4	55.43	even	4
1210.2.b.g.969.4		4	55.32	even	4
6050.2.a.bv.1.2		2	55.54	odd	2
6050.2.a.ce.1.1		2	1.1	even	1	trivial
6050.2.a.cl.1.2		2	5.4	even	2
6050.2.a.ct.1.1		2	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6050 = 2 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6050.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} +1.23607 q^{6} +0.618034 q^{7} -1.00000 q^{8} -1.47214 q^{9} -1.23607 q^{12} +2.38197 q^{13} -0.618034 q^{14} +1.00000 q^{16} +6.47214 q^{17} +1.47214 q^{18} +3.61803 q^{19} -0.763932 q^{21} +4.61803 q^{23} +1.23607 q^{24} -2.38197 q^{26} +5.52786 q^{27} +0.618034 q^{28} +4.47214 q^{29} +2.00000 q^{31} -1.00000 q^{32} -6.47214 q^{34} -1.47214 q^{36} +5.61803 q^{37} -3.61803 q^{38} -2.94427 q^{39} +0.618034 q^{41} +0.763932 q^{42} -8.47214 q^{43} -4.61803 q^{46} +10.0902 q^{47} -1.23607 q^{48} -6.61803 q^{49} -8.00000 q^{51} +2.38197 q^{52} -7.09017 q^{53} -5.52786 q^{54} -0.618034 q^{56} -4.47214 q^{57} -4.47214 q^{58} -10.8541 q^{59} -6.94427 q^{61} -2.00000 q^{62} -0.909830 q^{63} +1.00000 q^{64} +9.23607 q^{67} +6.47214 q^{68} -5.70820 q^{69} +2.00000 q^{71} +1.47214 q^{72} +1.52786 q^{73} -5.61803 q^{74} +3.61803 q^{76} +2.94427 q^{78} -6.18034 q^{79} -2.41641 q^{81} -0.618034 q^{82} -10.1803 q^{83} -0.763932 q^{84} +8.47214 q^{86} -5.52786 q^{87} +11.3820 q^{89} +1.47214 q^{91} +4.61803 q^{92} -2.47214 q^{93} -10.0902 q^{94} +1.23607 q^{96} +9.23607 q^{97} +6.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{12} + 7 q^{13} + q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 5 q^{19} - 6 q^{21} + 7 q^{23} - 2 q^{24} - 7 q^{26} + 20 q^{27}+ \cdots + 11 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 6 * q^9 + 2 * q^12 + 7 * q^13 + q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 + 5 * q^19 - 6 * q^21 + 7 * q^23 - 2 * q^24 - 7 * q^26 + 20 * q^27 - q^28 + 4 * q^31 - 2 * q^32 - 4 * q^34 + 6 * q^36 + 9 * q^37 - 5 * q^38 + 12 * q^39 - q^41 + 6 * q^42 - 8 * q^43 - 7 * q^46 + 9 * q^47 + 2 * q^48 - 11 * q^49 - 16 * q^51 + 7 * q^52 - 3 * q^53 - 20 * q^54 + q^56 - 15 * q^59 + 4 * q^61 - 4 * q^62 - 13 * q^63 + 2 * q^64 + 14 * q^67 + 4 * q^68 + 2 * q^69 + 4 * q^71 - 6 * q^72 + 12 * q^73 - 9 * q^74 + 5 * q^76 - 12 * q^78 + 10 * q^79 + 22 * q^81 + q^82 + 2 * q^83 - 6 * q^84 + 8 * q^86 - 20 * q^87 + 25 * q^89 - 6 * q^91 + 7 * q^92 + 4 * q^93 - 9 * q^94 - 2 * q^96 + 14 * q^97 + 11 * q^98

Introduction

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