LMFDB - Embedded newform 8450.2.a.cx.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, 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("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,9,-7,9,0,-7,1,9,8,0,-4,-7,0,1,0,9,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 14x^{7} + 17x^{6} + 53x^{5} - 69x^{4} - 33x^{3} + 26x^{2} + 8x - 1 $ x^9 - x^8 - 14x^7 + 17x^6 + 53x^5 - 69x^4 - 33x^3 + 26x^2 + 8*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1690)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.430845$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -2.55203 q^{3} +1.00000 q^{4} -2.55203 q^{6} +1.86919 q^{7} +1.00000 q^{8} +3.51288 q^{9} +4.42801 q^{11} -2.55203 q^{12} +1.86919 q^{14} +1.00000 q^{16} -4.16688 q^{17} +3.51288 q^{18} +4.52055 q^{19} -4.77024 q^{21} +4.42801 q^{22} -7.56163 q^{23} -2.55203 q^{24} -1.30889 q^{27} +1.86919 q^{28} +3.22160 q^{29} -8.51124 q^{31} +1.00000 q^{32} -11.3004 q^{33} -4.16688 q^{34} +3.51288 q^{36} -0.488194 q^{37} +4.52055 q^{38} -10.9095 q^{41} -4.77024 q^{42} -4.38872 q^{43} +4.42801 q^{44} -7.56163 q^{46} -12.5176 q^{47} -2.55203 q^{48} -3.50612 q^{49} +10.6340 q^{51} +0.191394 q^{53} -1.30889 q^{54} +1.86919 q^{56} -11.5366 q^{57} +3.22160 q^{58} -11.2785 q^{59} +6.28685 q^{61} -8.51124 q^{62} +6.56625 q^{63} +1.00000 q^{64} -11.3004 q^{66} +5.69785 q^{67} -4.16688 q^{68} +19.2975 q^{69} +10.1438 q^{71} +3.51288 q^{72} -4.10663 q^{73} -0.488194 q^{74} +4.52055 q^{76} +8.27681 q^{77} -2.33064 q^{79} -7.19832 q^{81} -10.9095 q^{82} +2.49654 q^{83} -4.77024 q^{84} -4.38872 q^{86} -8.22163 q^{87} +4.42801 q^{88} +11.9415 q^{89} -7.56163 q^{92} +21.7210 q^{93} -12.5176 q^{94} -2.55203 q^{96} -1.91384 q^{97} -3.50612 q^{98} +15.5551 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 7 q^{3} + 9 q^{4} - 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} - 4 q^{11} - 7 q^{12} + q^{14} + 9 q^{16} - 12 q^{17} + 8 q^{18} - 6 q^{19} - 8 q^{21} - 4 q^{22} - 11 q^{23} - 7 q^{24} - 34 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 - 7 * q^3 + 9 * q^4 - 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 - 4 * q^11 - 7 * q^12 + q^14 + 9 * q^16 - 12 * q^17 + 8 * q^18 - 6 * q^19 - 8 * q^21 - 4 * q^22 - 11 * q^23 - 7 * q^24 - 34 * q^27 + q^28 + 15 * q^29 + 6 * q^31 + 9 * q^32 - 14 * q^33 - 12 * q^34 + 8 * q^36 - 10 * q^37 - 6 * q^38 - 27 * q^41 - 8 * q^42 - 41 * q^43 - 4 * q^44 - 11 * q^46 + 5 * q^47 - 7 * q^48 - 10 * q^49 + 12 * q^51 - 18 * q^53 - 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 - 20 * q^59 - 19 * q^61 + 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 - 12 * q^68 - 14 * q^69 + 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 - 6 * q^76 - 30 * q^77 - 14 * q^79 + 53 * q^81 - 27 * q^82 + q^83 - 8 * q^84 - 41 * q^86 - 33 * q^87 - 4 * q^88 - 43 * q^89 - 11 * q^92 - 28 * q^93 + 5 * q^94 - 7 * q^96 + 6 * q^97 - 10 * q^98 - 16 * q^99

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
$3$	−2.55203	−1.47342	−0.736709	−	0.676210i	$-0.763621\pi$
$3$	−2.55203	−1.47342	−0.736709	+	0.676210i	$0.763621\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.55203	−1.04186
$7$	1.86919	0.706488	0.353244	−	0.935531i	$-0.385078\pi$
$7$	1.86919	0.706488	0.353244	+	0.935531i	$0.385078\pi$
$8$	1.00000	0.353553
$9$	3.51288	1.17096
$10$	0	0
$11$	4.42801	1.33510	0.667548	−	0.744567i	$-0.267344\pi$
$11$	4.42801	1.33510	0.667548	+	0.744567i	$0.267344\pi$
$12$	−2.55203	−0.736709
$13$	0	0
$13$	0	0
$14$	1.86919	0.499563
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.16688	−1.01062	−0.505308	−	0.862939i	$-0.668621\pi$
$17$	−4.16688	−1.01062	−0.505308	+	0.862939i	$0.668621\pi$
$18$	3.51288	0.827994
$19$	4.52055	1.03709	0.518543	−	0.855052i	$-0.326475\pi$
$19$	4.52055	1.03709	0.518543	+	0.855052i	$0.326475\pi$
$20$	0	0
$21$	−4.77024	−1.04095
$22$	4.42801	0.944056
$23$	−7.56163	−1.57671	−0.788354	−	0.615221i	$-0.789067\pi$
$23$	−7.56163	−1.57671	−0.788354	+	0.615221i	$0.789067\pi$
$24$	−2.55203	−0.520932
$25$	0	0
$26$	0	0
$27$	−1.30889	−0.251895
$28$	1.86919	0.353244
$29$	3.22160	0.598236	0.299118	−	0.954216i	$-0.403308\pi$
$29$	3.22160	0.598236	0.299118	+	0.954216i	$0.403308\pi$
$30$	0	0
$31$	−8.51124	−1.52866	−0.764332	−	0.644823i	$-0.776931\pi$
$31$	−8.51124	−1.52866	−0.764332	+	0.644823i	$0.776931\pi$
$32$	1.00000	0.176777
$33$	−11.3004	−1.96715
$34$	−4.16688	−0.714614
$35$	0	0
$36$	3.51288	0.585480
$37$	−0.488194	−0.0802585	−0.0401293	−	0.999194i	$-0.512777\pi$
$37$	−0.488194	−0.0802585	−0.0401293	+	0.999194i	$0.512777\pi$
$38$	4.52055	0.733330
$39$	0	0
$40$	0	0
$41$	−10.9095	−1.70378	−0.851888	−	0.523724i	$-0.824542\pi$
$41$	−10.9095	−1.70378	−0.851888	+	0.523724i	$0.824542\pi$
$42$	−4.77024	−0.736065
$43$	−4.38872	−0.669274	−0.334637	−	0.942347i	$-0.608614\pi$
$43$	−4.38872	−0.669274	−0.334637	+	0.942347i	$0.608614\pi$
$44$	4.42801	0.667548
$45$	0	0
$46$	−7.56163	−1.11490
$47$	−12.5176	−1.82587	−0.912937	−	0.408101i	$-0.866191\pi$
$47$	−12.5176	−1.82587	−0.912937	+	0.408101i	$0.866191\pi$
$48$	−2.55203	−0.368354
$49$	−3.50612	−0.500874
$50$	0	0
$51$	10.6340	1.48906
$52$	0	0
$53$	0.191394	0.0262900	0.0131450	−	0.999914i	$-0.495816\pi$
$53$	0.191394	0.0262900	0.0131450	+	0.999914i	$0.495816\pi$
$54$	−1.30889	−0.178117
$55$	0	0
$56$	1.86919	0.249781
$57$	−11.5366	−1.52806
$58$	3.22160	0.423016
$59$	−11.2785	−1.46833	−0.734166	−	0.678970i	$-0.762427\pi$
$59$	−11.2785	−1.46833	−0.734166	+	0.678970i	$0.762427\pi$
$60$	0	0
$61$	6.28685	0.804949	0.402474	−	0.915431i	$-0.368150\pi$
$61$	6.28685	0.804949	0.402474	+	0.915431i	$0.368150\pi$
$62$	−8.51124	−1.08093
$63$	6.56625	0.827270
$64$	1.00000	0.125000
$65$	0	0
$66$	−11.3004	−1.39099
$67$	5.69785	0.696103	0.348051	−	0.937475i	$-0.386843\pi$
$67$	5.69785	0.696103	0.348051	+	0.937475i	$0.386843\pi$
$68$	−4.16688	−0.505308
$69$	19.2975	2.32315
$70$	0	0
$71$	10.1438	1.20384	0.601922	−	0.798555i	$-0.294402\pi$
$71$	10.1438	1.20384	0.601922	+	0.798555i	$0.294402\pi$
$72$	3.51288	0.413997
$73$	−4.10663	−0.480645	−0.240323	−	0.970693i	$-0.577253\pi$
$73$	−4.10663	−0.480645	−0.240323	+	0.970693i	$0.577253\pi$
$74$	−0.488194	−0.0567514
$75$	0	0
$76$	4.52055	0.518543
$77$	8.27681	0.943230
$78$	0	0
$79$	−2.33064	−0.262217	−0.131108	−	0.991368i	$-0.541854\pi$
$79$	−2.33064	−0.262217	−0.131108	+	0.991368i	$0.541854\pi$
$80$	0	0
$81$	−7.19832	−0.799813
$82$	−10.9095	−1.20475
$83$	2.49654	0.274031	0.137016	−	0.990569i	$-0.456249\pi$
$83$	2.49654	0.274031	0.137016	+	0.990569i	$0.456249\pi$
$84$	−4.77024	−0.520476
$85$	0	0
$86$	−4.38872	−0.473248
$87$	−8.22163	−0.881451
$88$	4.42801	0.472028
$89$	11.9415	1.26579	0.632896	−	0.774237i	$-0.281866\pi$
$89$	11.9415	1.26579	0.632896	+	0.774237i	$0.281866\pi$
$90$	0	0
$91$	0	0
$92$	−7.56163	−0.788354
$93$	21.7210	2.25236
$94$	−12.5176	−1.29109
$95$	0	0
$96$	−2.55203	−0.260466
$97$	−1.91384	−0.194322	−0.0971608	−	0.995269i	$-0.530976\pi$
$97$	−1.91384	−0.194322	−0.0971608	+	0.995269i	$0.530976\pi$
$98$	−3.50612	−0.354171
$99$	15.5551	1.56334
$100$	0	0
$101$	16.3092	1.62282	0.811411	−	0.584476i	$-0.198700\pi$
$101$	16.3092	1.62282	0.811411	+	0.584476i	$0.198700\pi$
$102$	10.6340	1.05292
$103$	−14.5800	−1.43661	−0.718305	−	0.695728i	$-0.755082\pi$
$103$	−14.5800	−1.43661	−0.718305	+	0.695728i	$0.755082\pi$
$104$	0	0
$105$	0	0
$106$	0.191394	0.0185898
$107$	11.4218	1.10418	0.552091	−	0.833784i	$-0.313830\pi$
$107$	11.4218	1.10418	0.552091	+	0.833784i	$0.313830\pi$
$108$	−1.30889	−0.125948
$109$	−10.3606	−0.992363	−0.496181	−	0.868219i	$-0.665265\pi$
$109$	−10.3606	−0.992363	−0.496181	+	0.868219i	$0.665265\pi$
$110$	0	0
$111$	1.24589	0.118254
$112$	1.86919	0.176622
$113$	0.559115	0.0525971	0.0262985	−	0.999654i	$-0.491628\pi$
$113$	0.559115	0.0525971	0.0262985	+	0.999654i	$0.491628\pi$
$114$	−11.5366	−1.08050
$115$	0	0
$116$	3.22160	0.299118
$117$	0	0
$118$	−11.2785	−1.03827
$119$	−7.78870	−0.713989
$120$	0	0
$121$	8.60730	0.782482
$122$	6.28685	0.569185
$123$	27.8414	2.51037
$124$	−8.51124	−0.764332
$125$	0	0
$126$	6.56625	0.584968
$127$	1.52200	0.135055	0.0675277	−	0.997717i	$-0.478489\pi$
$127$	1.52200	0.135055	0.0675277	+	0.997717i	$0.478489\pi$
$128$	1.00000	0.0883883
$129$	11.2002	0.986120
$130$	0	0
$131$	7.16465	0.625979	0.312989	−	0.949757i	$-0.398670\pi$
$131$	7.16465	0.625979	0.312989	+	0.949757i	$0.398670\pi$
$132$	−11.3004	−0.983577
$133$	8.44978	0.732689
$134$	5.69785	0.492219
$135$	0	0
$136$	−4.16688	−0.357307
$137$	−16.0494	−1.37119	−0.685596	−	0.727983i	$-0.740458\pi$
$137$	−16.0494	−1.37119	−0.685596	+	0.727983i	$0.740458\pi$
$138$	19.2975	1.64272
$139$	−17.7302	−1.50386	−0.751930	−	0.659243i	$-0.770877\pi$
$139$	−17.7302	−1.50386	−0.751930	+	0.659243i	$0.770877\pi$
$140$	0	0
$141$	31.9452	2.69027
$142$	10.1438	0.851246
$143$	0	0
$144$	3.51288	0.292740
$145$	0	0
$146$	−4.10663	−0.339867
$147$	8.94773	0.737997
$148$	−0.488194	−0.0401293
$149$	−9.94873	−0.815032	−0.407516	−	0.913198i	$-0.633605\pi$
$149$	−9.94873	−0.815032	−0.407516	+	0.913198i	$0.633605\pi$
$150$	0	0
$151$	−11.9580	−0.973130	−0.486565	−	0.873644i	$-0.661750\pi$
$151$	−11.9580	−0.973130	−0.486565	+	0.873644i	$0.661750\pi$
$152$	4.52055	0.366665
$153$	−14.6377	−1.18339
$154$	8.27681	0.666964
$155$	0	0
$156$	0	0
$157$	−6.72590	−0.536785	−0.268392	−	0.963310i	$-0.586492\pi$
$157$	−6.72590	−0.536785	−0.268392	+	0.963310i	$0.586492\pi$
$158$	−2.33064	−0.185415
$159$	−0.488443	−0.0387361
$160$	0	0
$161$	−14.1341	−1.11393
$162$	−7.19832	−0.565553
$163$	−0.00838925	−0.000657097 0	−0.000328549	−	1.00000i	$-0.500105\pi$
$163$	−0.00838925	−0.000657097 0	−0.000328549		1.00000i	$0.500105\pi$
$164$	−10.9095	−0.851888
$165$	0	0
$166$	2.49654	0.193769
$167$	−8.77138	−0.678749	−0.339375	−	0.940651i	$-0.610215\pi$
$167$	−8.77138	−0.678749	−0.339375	+	0.940651i	$0.610215\pi$
$168$	−4.77024	−0.368032
$169$	0	0
$170$	0	0
$171$	15.8801	1.21439
$172$	−4.38872	−0.334637
$173$	5.17428	0.393393	0.196697	−	0.980464i	$-0.436979\pi$
$173$	5.17428	0.393393	0.196697	+	0.980464i	$0.436979\pi$
$174$	−8.22163	−0.623280
$175$	0	0
$176$	4.42801	0.333774
$177$	28.7831	2.16347
$178$	11.9415	0.895050
$179$	−7.82042	−0.584526	−0.292263	−	0.956338i	$-0.594408\pi$
$179$	−7.82042	−0.584526	−0.292263	+	0.956338i	$0.594408\pi$
$180$	0	0
$181$	−12.2270	−0.908823	−0.454411	−	0.890792i	$-0.650150\pi$
$181$	−12.2270	−0.908823	−0.454411	+	0.890792i	$0.650150\pi$
$182$	0	0
$183$	−16.0443	−1.18603
$184$	−7.56163	−0.557451
$185$	0	0
$186$	21.7210	1.59266
$187$	−18.4510	−1.34927
$188$	−12.5176	−0.912937
$189$	−2.44656	−0.177961
$190$	0	0
$191$	18.1313	1.31194	0.655968	−	0.754789i	$-0.272261\pi$
$191$	18.1313	1.31194	0.655968	+	0.754789i	$0.272261\pi$
$192$	−2.55203	−0.184177
$193$	6.34605	0.456799	0.228399	−	0.973568i	$-0.426651\pi$
$193$	6.34605	0.456799	0.228399	+	0.973568i	$0.426651\pi$
$194$	−1.91384	−0.137406
$195$	0	0
$196$	−3.50612	−0.250437
$197$	2.19339	0.156273	0.0781364	−	0.996943i	$-0.475103\pi$
$197$	2.19339	0.156273	0.0781364	+	0.996943i	$0.475103\pi$
$198$	15.5551	1.10545
$199$	−4.80668	−0.340737	−0.170368	−	0.985380i	$-0.554496\pi$
$199$	−4.80668	−0.340737	−0.170368	+	0.985380i	$0.554496\pi$
$200$	0	0
$201$	−14.5411	−1.02565
$202$	16.3092	1.14751
$203$	6.02179	0.422646
$204$	10.6340	0.744530
$205$	0	0
$206$	−14.5800	−1.01584
$207$	−26.5631	−1.84626
$208$	0	0
$209$	20.0171	1.38461
$210$	0	0
$211$	−1.98463	−0.136628	−0.0683139	−	0.997664i	$-0.521762\pi$
$211$	−1.98463	−0.136628	−0.0683139	+	0.997664i	$0.521762\pi$
$212$	0.191394	0.0131450
$213$	−25.8872	−1.77376
$214$	11.4218	0.780775
$215$	0	0
$216$	−1.30889	−0.0890584
$217$	−15.9092	−1.07998
$218$	−10.3606	−0.701706
$219$	10.4803	0.708191
$220$	0	0
$221$	0	0
$222$	1.24589	0.0836185
$223$	−6.23337	−0.417417	−0.208709	−	0.977978i	$-0.566926\pi$
$223$	−6.23337	−0.417417	−0.208709	+	0.977978i	$0.566926\pi$
$224$	1.86919	0.124891
$225$	0	0
$226$	0.559115	0.0371917
$227$	13.3667	0.887180	0.443590	−	0.896230i	$-0.353705\pi$
$227$	13.3667	0.887180	0.443590	+	0.896230i	$0.353705\pi$
$228$	−11.5366	−0.764030
$229$	−5.10916	−0.337623	−0.168811	−	0.985648i	$-0.553993\pi$
$229$	−5.10916	−0.337623	−0.168811	+	0.985648i	$0.553993\pi$
$230$	0	0
$231$	−21.1227	−1.38977
$232$	3.22160	0.211508
$233$	−25.7364	−1.68605	−0.843025	−	0.537874i	$-0.819228\pi$
$233$	−25.7364	−1.68605	−0.843025	+	0.537874i	$0.819228\pi$
$234$	0	0
$235$	0	0
$236$	−11.2785	−0.734166
$237$	5.94786	0.386355
$238$	−7.78870	−0.504866
$239$	9.42688	0.609774	0.304887	−	0.952389i	$-0.401381\pi$
$239$	9.42688	0.609774	0.304887	+	0.952389i	$0.401381\pi$
$240$	0	0
$241$	−8.55884	−0.551324	−0.275662	−	0.961255i	$-0.588897\pi$
$241$	−8.55884	−0.551324	−0.275662	+	0.961255i	$0.588897\pi$
$242$	8.60730	0.553298
$243$	22.2970	1.43035
$244$	6.28685	0.402474
$245$	0	0
$246$	27.8414	1.77510
$247$	0	0
$248$	−8.51124	−0.540464
$249$	−6.37126	−0.403762
$250$	0	0
$251$	−2.91331	−0.183887	−0.0919433	−	0.995764i	$-0.529308\pi$
$251$	−2.91331	−0.183887	−0.0919433	+	0.995764i	$0.529308\pi$
$252$	6.56625	0.413635
$253$	−33.4830	−2.10506
$254$	1.52200	0.0954986
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.3335	1.76740	0.883698	−	0.468057i	$-0.155046\pi$
$257$	28.3335	1.76740	0.883698	+	0.468057i	$0.155046\pi$
$258$	11.2002	0.697292
$259$	−0.912528	−0.0567017
$260$	0	0
$261$	11.3171	0.700510
$262$	7.16465	0.442634
$263$	6.57342	0.405335	0.202667	−	0.979248i	$-0.435039\pi$
$263$	6.57342	0.405335	0.202667	+	0.979248i	$0.435039\pi$
$264$	−11.3004	−0.695494
$265$	0	0
$266$	8.44978	0.518089
$267$	−30.4750	−1.86504
$268$	5.69785	0.348051
$269$	13.1504	0.801797	0.400898	−	0.916123i	$-0.368698\pi$
$269$	13.1504	0.801797	0.400898	+	0.916123i	$0.368698\pi$
$270$	0	0
$271$	22.3779	1.35936	0.679682	−	0.733507i	$-0.262118\pi$
$271$	22.3779	1.35936	0.679682	+	0.733507i	$0.262118\pi$
$272$	−4.16688	−0.252654
$273$	0	0
$274$	−16.0494	−0.969579
$275$	0	0
$276$	19.2975	1.16158
$277$	−11.0451	−0.663634	−0.331817	−	0.943344i	$-0.607662\pi$
$277$	−11.0451	−0.663634	−0.331817	+	0.943344i	$0.607662\pi$
$278$	−17.7302	−1.06339
$279$	−29.8990	−1.79000
$280$	0	0
$281$	20.1064	1.19945	0.599724	−	0.800207i	$-0.295277\pi$
$281$	20.1064	1.19945	0.599724	+	0.800207i	$0.295277\pi$
$282$	31.9452	1.90231
$283$	20.2670	1.20475	0.602373	−	0.798215i	$-0.294222\pi$
$283$	20.2670	1.20475	0.602373	+	0.798215i	$0.294222\pi$
$284$	10.1438	0.601922
$285$	0	0
$286$	0	0
$287$	−20.3919	−1.20370
$288$	3.51288	0.206998
$289$	0.362874	0.0213455
$290$	0	0
$291$	4.88420	0.286317
$292$	−4.10663	−0.240323
$293$	−29.8007	−1.74097	−0.870487	−	0.492191i	$-0.836196\pi$
$293$	−29.8007	−1.74097	−0.870487	+	0.492191i	$0.836196\pi$
$294$	8.94773	0.521842
$295$	0	0
$296$	−0.488194	−0.0283757
$297$	−5.79577	−0.336304
$298$	−9.94873	−0.576315
$299$	0	0
$300$	0	0
$301$	−8.20337	−0.472834
$302$	−11.9580	−0.688107
$303$	−41.6215	−2.39109
$304$	4.52055	0.259271
$305$	0	0
$306$	−14.6377	−0.836784
$307$	−9.13426	−0.521320	−0.260660	−	0.965431i	$-0.583940\pi$
$307$	−9.13426	−0.521320	−0.260660	+	0.965431i	$0.583940\pi$
$308$	8.27681	0.471615
$309$	37.2087	2.11673
$310$	0	0
$311$	−5.34622	−0.303157	−0.151578	−	0.988445i	$-0.548436\pi$
$311$	−5.34622	−0.303157	−0.151578	+	0.988445i	$0.548436\pi$
$312$	0	0
$313$	−1.32226	−0.0747385	−0.0373692	−	0.999302i	$-0.511898\pi$
$313$	−1.32226	−0.0747385	−0.0373692	+	0.999302i	$0.511898\pi$
$314$	−6.72590	−0.379564
$315$	0	0
$316$	−2.33064	−0.131108
$317$	3.84953	0.216211	0.108106	−	0.994139i	$-0.465522\pi$
$317$	3.84953	0.216211	0.108106	+	0.994139i	$0.465522\pi$
$318$	−0.488443	−0.0273905
$319$	14.2653	0.798702
$320$	0	0
$321$	−29.1487	−1.62692
$322$	−14.1341	−0.787665
$323$	−18.8366	−1.04810
$324$	−7.19832	−0.399906
$325$	0	0
$326$	−0.00838925	−0.000464638 0
$327$	26.4405	1.46216
$328$	−10.9095	−0.602376
$329$	−23.3977	−1.28996
$330$	0	0
$331$	4.96398	0.272845	0.136423	−	0.990651i	$-0.456439\pi$
$331$	4.96398	0.272845	0.136423	+	0.990651i	$0.456439\pi$
$332$	2.49654	0.137016
$333$	−1.71497	−0.0939795
$334$	−8.77138	−0.479948
$335$	0	0
$336$	−4.77024	−0.260238
$337$	−11.2699	−0.613911	−0.306956	−	0.951724i	$-0.599310\pi$
$337$	−11.2699	−0.613911	−0.306956	+	0.951724i	$0.599310\pi$
$338$	0	0
$339$	−1.42688	−0.0774975
$340$	0	0
$341$	−37.6879	−2.04091
$342$	15.8801	0.858700
$343$	−19.6380	−1.06035
$344$	−4.38872	−0.236624
$345$	0	0
$346$	5.17428	0.278171
$347$	15.6806	0.841781	0.420890	−	0.907112i	$-0.361718\pi$
$347$	15.6806	0.841781	0.420890	+	0.907112i	$0.361718\pi$
$348$	−8.22163	−0.440725
$349$	−0.557452	−0.0298397	−0.0149198	−	0.999889i	$-0.504749\pi$
$349$	−0.557452	−0.0298397	−0.0149198	+	0.999889i	$0.504749\pi$
$350$	0	0
$351$	0	0
$352$	4.42801	0.236014
$353$	−24.1725	−1.28657	−0.643285	−	0.765627i	$-0.722429\pi$
$353$	−24.1725	−1.28657	−0.643285	+	0.765627i	$0.722429\pi$
$354$	28.7831	1.52980
$355$	0	0
$356$	11.9415	0.632896
$357$	19.8770	1.05200
$358$	−7.82042	−0.413322
$359$	0.910208	0.0480389	0.0240195	−	0.999711i	$-0.492354\pi$
$359$	0.910208	0.0480389	0.0240195	+	0.999711i	$0.492354\pi$
$360$	0	0
$361$	1.43537	0.0755457
$362$	−12.2270	−0.642635
$363$	−21.9661	−1.15292
$364$	0	0
$365$	0	0
$366$	−16.0443	−0.838647
$367$	−4.41148	−0.230277	−0.115139	−	0.993349i	$-0.536731\pi$
$367$	−4.41148	−0.230277	−0.115139	+	0.993349i	$0.536731\pi$
$368$	−7.56163	−0.394177
$369$	−38.3237	−1.99505
$370$	0	0
$371$	0.357752	0.0185735
$372$	21.7210	1.12618
$373$	−7.59431	−0.393219	−0.196609	−	0.980482i	$-0.562993\pi$
$373$	−7.59431	−0.393219	−0.196609	+	0.980482i	$0.562993\pi$
$374$	−18.4510	−0.954078
$375$	0	0
$376$	−12.5176	−0.645544
$377$	0	0
$378$	−2.44656	−0.125838
$379$	30.3412	1.55852	0.779262	−	0.626698i	$-0.215594\pi$
$379$	30.3412	1.55852	0.779262	+	0.626698i	$0.215594\pi$
$380$	0	0
$381$	−3.88419	−0.198993
$382$	18.1313	0.927679
$383$	7.99851	0.408705	0.204352	−	0.978897i	$-0.434491\pi$
$383$	7.99851	0.408705	0.204352	+	0.978897i	$0.434491\pi$
$384$	−2.55203	−0.130233
$385$	0	0
$386$	6.34605	0.323006
$387$	−15.4170	−0.783693
$388$	−1.91384	−0.0971608
$389$	22.4951	1.14055	0.570273	−	0.821455i	$-0.306837\pi$
$389$	22.4951	1.14055	0.570273	+	0.821455i	$0.306837\pi$
$390$	0	0
$391$	31.5084	1.59345
$392$	−3.50612	−0.177086
$393$	−18.2844	−0.922328
$394$	2.19339	0.110502
$395$	0	0
$396$	15.5551	0.781672
$397$	−29.6953	−1.49036	−0.745181	−	0.666862i	$-0.767637\pi$
$397$	−29.6953	−1.49036	−0.745181	+	0.666862i	$0.767637\pi$
$398$	−4.80668	−0.240937
$399$	−21.5641	−1.07956
$400$	0	0
$401$	−26.3944	−1.31807	−0.659037	−	0.752110i	$-0.729036\pi$
$401$	−26.3944	−1.31807	−0.659037	+	0.752110i	$0.729036\pi$
$402$	−14.5411	−0.725244
$403$	0	0
$404$	16.3092	0.811411
$405$	0	0
$406$	6.02179	0.298856
$407$	−2.16173	−0.107153
$408$	10.6340	0.526462
$409$	9.89433	0.489243	0.244622	−	0.969619i	$-0.421336\pi$
$409$	9.89433	0.489243	0.244622	+	0.969619i	$0.421336\pi$
$410$	0	0
$411$	40.9586	2.02034
$412$	−14.5800	−0.718305
$413$	−21.0817	−1.03736
$414$	−26.5631	−1.30550
$415$	0	0
$416$	0	0
$417$	45.2482	2.21581
$418$	20.0171	0.979066
$419$	−17.9386	−0.876359	−0.438179	−	0.898888i	$-0.644377\pi$
$419$	−17.9386	−0.876359	−0.438179	+	0.898888i	$0.644377\pi$
$420$	0	0
$421$	15.5977	0.760184	0.380092	−	0.924949i	$-0.375892\pi$
$421$	15.5977	0.760184	0.380092	+	0.924949i	$0.375892\pi$
$422$	−1.98463	−0.0966105
$423$	−43.9727	−2.13802
$424$	0.191394	0.00929490
$425$	0	0
$426$	−25.8872	−1.25424
$427$	11.7513	0.568687
$428$	11.4218	0.552091
$429$	0	0
$430$	0	0
$431$	30.0631	1.44809	0.724045	−	0.689753i	$-0.242281\pi$
$431$	30.0631	1.44809	0.724045	+	0.689753i	$0.242281\pi$
$432$	−1.30889	−0.0629738
$433$	−22.5878	−1.08550	−0.542750	−	0.839894i	$-0.682617\pi$
$433$	−22.5878	−1.08550	−0.542750	+	0.839894i	$0.682617\pi$
$434$	−15.9092	−0.763664
$435$	0	0
$436$	−10.3606	−0.496181
$437$	−34.1827	−1.63518
$438$	10.4803	0.500767
$439$	−34.8184	−1.66179	−0.830895	−	0.556429i	$-0.812171\pi$
$439$	−34.8184	−1.66179	−0.830895	+	0.556429i	$0.812171\pi$
$440$	0	0
$441$	−12.3166	−0.586503
$442$	0	0
$443$	−15.5022	−0.736533	−0.368267	−	0.929720i	$-0.620049\pi$
$443$	−15.5022	−0.736533	−0.368267	+	0.929720i	$0.620049\pi$
$444$	1.24589	0.0591272
$445$	0	0
$446$	−6.23337	−0.295159
$447$	25.3895	1.20088
$448$	1.86919	0.0883111
$449$	−0.894606	−0.0422191	−0.0211095	−	0.999777i	$-0.506720\pi$
$449$	−0.894606	−0.0422191	−0.0211095	+	0.999777i	$0.506720\pi$
$450$	0	0
$451$	−48.3074	−2.27471
$452$	0.559115	0.0262985
$453$	30.5173	1.43383
$454$	13.3667	0.627331
$455$	0	0
$456$	−11.5366	−0.540251
$457$	−10.7958	−0.505007	−0.252504	−	0.967596i	$-0.581254\pi$
$457$	−10.7958	−0.505007	−0.252504	+	0.967596i	$0.581254\pi$
$458$	−5.10916	−0.238735
$459$	5.45397	0.254570
$460$	0	0
$461$	−10.8827	−0.506859	−0.253429	−	0.967354i	$-0.581559\pi$
$461$	−10.8827	−0.506859	−0.253429	+	0.967354i	$0.581559\pi$
$462$	−21.1227	−0.982717
$463$	−3.15999	−0.146857	−0.0734287	−	0.997300i	$-0.523394\pi$
$463$	−3.15999	−0.146857	−0.0734287	+	0.997300i	$0.523394\pi$
$464$	3.22160	0.149559
$465$	0	0
$466$	−25.7364	−1.19222
$467$	−23.0475	−1.06651	−0.533256	−	0.845954i	$-0.679032\pi$
$467$	−23.0475	−1.06651	−0.533256	+	0.845954i	$0.679032\pi$
$468$	0	0
$469$	10.6504	0.491789
$470$	0	0
$471$	17.1647	0.790908
$472$	−11.2785	−0.519134
$473$	−19.4333	−0.893545
$474$	5.94786	0.273194
$475$	0	0
$476$	−7.78870	−0.356994
$477$	0.672343	0.0307845
$478$	9.42688	0.431175
$479$	−9.16688	−0.418845	−0.209423	−	0.977825i	$-0.567158\pi$
$479$	−9.16688	−0.418845	−0.209423	+	0.977825i	$0.567158\pi$
$480$	0	0
$481$	0	0
$482$	−8.55884	−0.389845
$483$	36.0708	1.64128
$484$	8.60730	0.391241
$485$	0	0
$486$	22.2970	1.01141
$487$	−24.1155	−1.09278	−0.546389	−	0.837531i	$-0.683998\pi$
$487$	−24.1155	−1.09278	−0.546389	+	0.837531i	$0.683998\pi$
$488$	6.28685	0.284592
$489$	0.0214097	0.000968178 0
$490$	0	0
$491$	5.68528	0.256573	0.128286	−	0.991737i	$-0.459052\pi$
$491$	5.68528	0.256573	0.128286	+	0.991737i	$0.459052\pi$
$492$	27.8414	1.25519
$493$	−13.4240	−0.604587
$494$	0	0
$495$	0	0
$496$	−8.51124	−0.382166
$497$	18.9607	0.850501
$498$	−6.37126	−0.285503
$499$	6.68270	0.299159	0.149579	−	0.988750i	$-0.452208\pi$
$499$	6.68270	0.299159	0.149579	+	0.988750i	$0.452208\pi$
$500$	0	0
$501$	22.3849	1.00008
$502$	−2.91331	−0.130027
$503$	17.4365	0.777455	0.388727	−	0.921353i	$-0.372915\pi$
$503$	17.4365	0.777455	0.388727	+	0.921353i	$0.372915\pi$
$504$	6.56625	0.292484
$505$	0	0
$506$	−33.4830	−1.48850
$507$	0	0
$508$	1.52200	0.0675277
$509$	−28.6424	−1.26955	−0.634776	−	0.772696i	$-0.718908\pi$
$509$	−28.6424	−1.26955	−0.634776	+	0.772696i	$0.718908\pi$
$510$	0	0
$511$	−7.67609	−0.339570
$512$	1.00000	0.0441942
$513$	−5.91689	−0.261237
$514$	28.3335	1.24974
$515$	0	0
$516$	11.2002	0.493060
$517$	−55.4279	−2.43772
$518$	−0.912528	−0.0400942
$519$	−13.2049	−0.579632
$520$	0	0
$521$	−24.2436	−1.06213	−0.531066	−	0.847331i	$-0.678208\pi$
$521$	−24.2436	−1.06213	−0.531066	+	0.847331i	$0.678208\pi$
$522$	11.3171	0.495335
$523$	7.25966	0.317443	0.158721	−	0.987323i	$-0.449263\pi$
$523$	7.25966	0.317443	0.158721	+	0.987323i	$0.449263\pi$
$524$	7.16465	0.312989
$525$	0	0
$526$	6.57342	0.286615
$527$	35.4653	1.54489
$528$	−11.3004	−0.491789
$529$	34.1782	1.48601
$530$	0	0
$531$	−39.6199	−1.71936
$532$	8.44978	0.366344
$533$	0	0
$534$	−30.4750	−1.31878
$535$	0	0
$536$	5.69785	0.246109
$537$	19.9580	0.861251
$538$	13.1504	0.566956
$539$	−15.5251	−0.668715
$540$	0	0
$541$	−9.90908	−0.426025	−0.213012	−	0.977049i	$-0.568327\pi$
$541$	−9.90908	−0.426025	−0.213012	+	0.977049i	$0.568327\pi$
$542$	22.3779	0.961215
$543$	31.2036	1.33908
$544$	−4.16688	−0.178653
$545$	0	0
$546$	0	0
$547$	15.7348	0.672769	0.336385	−	0.941725i	$-0.390796\pi$
$547$	15.7348	0.672769	0.336385	+	0.941725i	$0.390796\pi$
$548$	−16.0494	−0.685596
$549$	22.0850	0.942563
$550$	0	0
$551$	14.5634	0.620421
$552$	19.2975	0.821358
$553$	−4.35641	−0.185253
$554$	−11.0451	−0.469260
$555$	0	0
$556$	−17.7302	−0.751930
$557$	11.6684	0.494408	0.247204	−	0.968963i	$-0.420488\pi$
$557$	11.6684	0.494408	0.247204	+	0.968963i	$0.420488\pi$
$558$	−29.8990	−1.26572
$559$	0	0
$560$	0	0
$561$	47.0876	1.98804
$562$	20.1064	0.848138
$563$	−34.0262	−1.43403	−0.717017	−	0.697056i	$-0.754493\pi$
$563$	−34.0262	−1.43403	−0.717017	+	0.697056i	$0.754493\pi$
$564$	31.9452	1.34514
$565$	0	0
$566$	20.2670	0.851884
$567$	−13.4550	−0.565059
$568$	10.1438	0.425623
$569$	−44.3266	−1.85827	−0.929135	−	0.369741i	$-0.879446\pi$
$569$	−44.3266	−1.85827	−0.929135	+	0.369741i	$0.879446\pi$
$570$	0	0
$571$	15.3587	0.642740	0.321370	−	0.946954i	$-0.395857\pi$
$571$	15.3587	0.642740	0.321370	+	0.946954i	$0.395857\pi$
$572$	0	0
$573$	−46.2717	−1.93303
$574$	−20.3919	−0.851143
$575$	0	0
$576$	3.51288	0.146370
$577$	−31.5537	−1.31360	−0.656799	−	0.754065i	$-0.728090\pi$
$577$	−31.5537	−1.31360	−0.656799	+	0.754065i	$0.728090\pi$
$578$	0.362874	0.0150936
$579$	−16.1953	−0.673056
$580$	0	0
$581$	4.66652	0.193600
$582$	4.88420	0.202457
$583$	0.847494	0.0350996
$584$	−4.10663	−0.169934
$585$	0	0
$586$	−29.8007	−1.23106
$587$	−22.9587	−0.947607	−0.473803	−	0.880631i	$-0.657119\pi$
$587$	−22.9587	−0.947607	−0.473803	+	0.880631i	$0.657119\pi$
$588$	8.94773	0.368998
$589$	−38.4755	−1.58535
$590$	0	0
$591$	−5.59761	−0.230255
$592$	−0.488194	−0.0200646
$593$	27.6241	1.13439	0.567194	−	0.823584i	$-0.308029\pi$
$593$	27.6241	1.13439	0.567194	+	0.823584i	$0.308029\pi$
$594$	−5.79577	−0.237803
$595$	0	0
$596$	−9.94873	−0.407516
$597$	12.2668	0.502048
$598$	0	0
$599$	−1.54774	−0.0632389	−0.0316194	−	0.999500i	$-0.510066\pi$
$599$	−1.54774	−0.0632389	−0.0316194	+	0.999500i	$0.510066\pi$
$600$	0	0
$601$	25.3606	1.03448	0.517241	−	0.855840i	$-0.326959\pi$
$601$	25.3606	1.03448	0.517241	+	0.855840i	$0.326959\pi$
$602$	−8.20337	−0.334344
$603$	20.0158	0.815108
$604$	−11.9580	−0.486565
$605$	0	0
$606$	−41.6215	−1.69076
$607$	−19.1839	−0.778652	−0.389326	−	0.921100i	$-0.627292\pi$
$607$	−19.1839	−0.778652	−0.389326	+	0.921100i	$0.627292\pi$
$608$	4.52055	0.183332
$609$	−15.3678	−0.622735
$610$	0	0
$611$	0	0
$612$	−14.6377	−0.591696
$613$	−3.32275	−0.134205	−0.0671024	−	0.997746i	$-0.521375\pi$
$613$	−3.32275	−0.134205	−0.0671024	+	0.997746i	$0.521375\pi$
$614$	−9.13426	−0.368629
$615$	0	0
$616$	8.27681	0.333482
$617$	13.7220	0.552425	0.276213	−	0.961097i	$-0.410921\pi$
$617$	13.7220	0.552425	0.276213	+	0.961097i	$0.410921\pi$
$618$	37.2087	1.49675
$619$	−40.9597	−1.64631	−0.823155	−	0.567817i	$-0.807788\pi$
$619$	−40.9597	−1.64631	−0.823155	+	0.567817i	$0.807788\pi$
$620$	0	0
$621$	9.89731	0.397166
$622$	−5.34622	−0.214364
$623$	22.3209	0.894267
$624$	0	0
$625$	0	0
$626$	−1.32226	−0.0528481
$627$	−51.0842	−2.04011
$628$	−6.72590	−0.268392
$629$	2.03424	0.0811106
$630$	0	0
$631$	14.9553	0.595361	0.297680	−	0.954666i	$-0.403787\pi$
$631$	14.9553	0.595361	0.297680	+	0.954666i	$0.403787\pi$
$632$	−2.33064	−0.0927077
$633$	5.06485	0.201310
$634$	3.84953	0.152885
$635$	0	0
$636$	−0.488443	−0.0193680
$637$	0	0
$638$	14.2653	0.564768
$639$	35.6338	1.40965
$640$	0	0
$641$	−36.5876	−1.44512	−0.722562	−	0.691306i	$-0.757036\pi$
$641$	−36.5876	−1.44512	−0.722562	+	0.691306i	$0.757036\pi$
$642$	−29.1487	−1.15041
$643$	43.3819	1.71082	0.855408	−	0.517954i	$-0.173306\pi$
$643$	43.3819	1.71082	0.855408	+	0.517954i	$0.173306\pi$
$644$	−14.1341	−0.556963
$645$	0	0
$646$	−18.8366	−0.741115
$647$	42.6622	1.67722	0.838611	−	0.544730i	$-0.183368\pi$
$647$	42.6622	1.67722	0.838611	+	0.544730i	$0.183368\pi$
$648$	−7.19832	−0.282777
$649$	−49.9413	−1.96037
$650$	0	0
$651$	40.6007	1.59127
$652$	−0.00838925	−0.000328549 0
$653$	44.8543	1.75528	0.877642	−	0.479316i	$-0.159115\pi$
$653$	44.8543	1.75528	0.877642	+	0.479316i	$0.159115\pi$
$654$	26.4405	1.03391
$655$	0	0
$656$	−10.9095	−0.425944
$657$	−14.4261	−0.562816
$658$	−23.3977	−0.912138
$659$	−28.7835	−1.12125	−0.560624	−	0.828071i	$-0.689439\pi$
$659$	−28.7835	−1.12125	−0.560624	+	0.828071i	$0.689439\pi$
$660$	0	0
$661$	−9.90351	−0.385202	−0.192601	−	0.981277i	$-0.561692\pi$
$661$	−9.90351	−0.385202	−0.192601	+	0.981277i	$0.561692\pi$
$662$	4.96398	0.192931
$663$	0	0
$664$	2.49654	0.0968846
$665$	0	0
$666$	−1.71497	−0.0664536
$667$	−24.3605	−0.943243
$668$	−8.77138	−0.339375
$669$	15.9078	0.615030
$670$	0	0
$671$	27.8383	1.07468
$672$	−4.77024	−0.184016
$673$	−46.6987	−1.80010	−0.900051	−	0.435784i	$-0.856471\pi$
$673$	−46.6987	−1.80010	−0.900051	+	0.435784i	$0.856471\pi$
$674$	−11.2699	−0.434101
$675$	0	0
$676$	0	0
$677$	−15.1897	−0.583787	−0.291893	−	0.956451i	$-0.594285\pi$
$677$	−15.1897	−0.583787	−0.291893	+	0.956451i	$0.594285\pi$
$678$	−1.42688	−0.0547990
$679$	−3.57735	−0.137286
$680$	0	0
$681$	−34.1123	−1.30719
$682$	−37.6879	−1.44314
$683$	−5.91381	−0.226286	−0.113143	−	0.993579i	$-0.536092\pi$
$683$	−5.91381	−0.226286	−0.113143	+	0.993579i	$0.536092\pi$
$684$	15.8801	0.607193
$685$	0	0
$686$	−19.6380	−0.749781
$687$	13.0387	0.497459
$688$	−4.38872	−0.167318
$689$	0	0
$690$	0	0
$691$	−23.5531	−0.896004	−0.448002	−	0.894033i	$-0.647864\pi$
$691$	−23.5531	−0.896004	−0.448002	+	0.894033i	$0.647864\pi$
$692$	5.17428	0.196697
$693$	29.0754	1.10448
$694$	15.6806	0.595229
$695$	0	0
$696$	−8.22163	−0.311640
$697$	45.4585	1.72186
$698$	−0.557452	−0.0210999
$699$	65.6803	2.48426
$700$	0	0
$701$	−39.9205	−1.50778	−0.753888	−	0.657003i	$-0.771824\pi$
$701$	−39.9205	−1.50778	−0.753888	+	0.657003i	$0.771824\pi$
$702$	0	0
$703$	−2.20690	−0.0832349
$704$	4.42801	0.166887
$705$	0	0
$706$	−24.1725	−0.909742
$707$	30.4850	1.14650
$708$	28.7831	1.08173
$709$	8.96987	0.336870	0.168435	−	0.985713i	$-0.446129\pi$
$709$	8.96987	0.336870	0.168435	+	0.985713i	$0.446129\pi$
$710$	0	0
$711$	−8.18724	−0.307046
$712$	11.9415	0.447525
$713$	64.3589	2.41026
$714$	19.8770	0.743879
$715$	0	0
$716$	−7.82042	−0.292263
$717$	−24.0577	−0.898452
$718$	0.910208	0.0339687
$719$	19.5515	0.729149	0.364575	−	0.931174i	$-0.381214\pi$
$719$	19.5515	0.729149	0.364575	+	0.931174i	$0.381214\pi$
$720$	0	0
$721$	−27.2528	−1.01495
$722$	1.43537	0.0534189
$723$	21.8425	0.812330
$724$	−12.2270	−0.454411
$725$	0	0
$726$	−21.9661	−0.815240
$727$	−21.2591	−0.788458	−0.394229	−	0.919012i	$-0.628988\pi$
$727$	−21.2591	−0.788458	−0.394229	+	0.919012i	$0.628988\pi$
$728$	0	0
$729$	−35.3078	−1.30770
$730$	0	0
$731$	18.2873	0.676379
$732$	−16.0443	−0.593013
$733$	51.7515	1.91149	0.955743	−	0.294203i	$-0.0950542\pi$
$733$	51.7515	1.91149	0.955743	+	0.294203i	$0.0950542\pi$
$734$	−4.41148	−0.162831
$735$	0	0
$736$	−7.56163	−0.278725
$737$	25.2301	0.929364
$738$	−38.3237	−1.41072
$739$	−2.48566	−0.0914364	−0.0457182	−	0.998954i	$-0.514558\pi$
$739$	−2.48566	−0.0914364	−0.0457182	+	0.998954i	$0.514558\pi$
$740$	0	0
$741$	0	0
$742$	0.357752	0.0131335
$743$	41.9532	1.53911	0.769557	−	0.638578i	$-0.220477\pi$
$743$	41.9532	1.53911	0.769557	+	0.638578i	$0.220477\pi$
$744$	21.7210	0.796330
$745$	0	0
$746$	−7.59431	−0.278048
$747$	8.77005	0.320879
$748$	−18.4510	−0.674635
$749$	21.3495	0.780092
$750$	0	0
$751$	4.26126	0.155496	0.0777478	−	0.996973i	$-0.475227\pi$
$751$	4.26126	0.155496	0.0777478	+	0.996973i	$0.475227\pi$
$752$	−12.5176	−0.456468
$753$	7.43487	0.270942
$754$	0	0
$755$	0	0
$756$	−2.44656	−0.0889806
$757$	32.1223	1.16750	0.583752	−	0.811932i	$-0.301584\pi$
$757$	32.1223	1.16750	0.583752	+	0.811932i	$0.301584\pi$
$758$	30.3412	1.10204
$759$	85.4498	3.10163
$760$	0	0
$761$	50.8769	1.84429	0.922143	−	0.386850i	$-0.126437\pi$
$761$	50.8769	1.84429	0.922143	+	0.386850i	$0.126437\pi$
$762$	−3.88419	−0.140709
$763$	−19.3659	−0.701093
$764$	18.1313	0.655968
$765$	0	0
$766$	7.99851	0.288998
$767$	0	0
$768$	−2.55203	−0.0920886
$769$	10.8452	0.391089	0.195545	−	0.980695i	$-0.437353\pi$
$769$	10.8452	0.391089	0.195545	+	0.980695i	$0.437353\pi$
$770$	0	0
$771$	−72.3081	−2.60411
$772$	6.34605	0.228399
$773$	−17.0142	−0.611959	−0.305979	−	0.952038i	$-0.598984\pi$
$773$	−17.0142	−0.611959	−0.305979	+	0.952038i	$0.598984\pi$
$774$	−15.4170	−0.554154
$775$	0	0
$776$	−1.91384	−0.0687030
$777$	2.32880	0.0835453
$778$	22.4951	0.806488
$779$	−49.3169	−1.76696
$780$	0	0
$781$	44.9167	1.60725
$782$	31.5084	1.12674
$783$	−4.21670	−0.150693
$784$	−3.50612	−0.125219
$785$	0	0
$786$	−18.2844	−0.652184
$787$	27.0297	0.963505	0.481752	−	0.876307i	$-0.340000\pi$
$787$	27.0297	0.963505	0.481752	+	0.876307i	$0.340000\pi$
$788$	2.19339	0.0781364
$789$	−16.7756	−0.597227
$790$	0	0
$791$	1.04509	0.0371592
$792$	15.5551	0.552726
$793$	0	0
$794$	−29.6953	−1.05385
$795$	0	0
$796$	−4.80668	−0.170368
$797$	−15.1585	−0.536942	−0.268471	−	0.963288i	$-0.586518\pi$
$797$	−15.1585	−0.536942	−0.268471	+	0.963288i	$0.586518\pi$
$798$	−21.5641	−0.763362
$799$	52.1591	1.84526
$800$	0	0
$801$	41.9489	1.48219
$802$	−26.3944	−0.932019
$803$	−18.1842	−0.641708
$804$	−14.5411	−0.512825
$805$	0	0
$806$	0	0
$807$	−33.5604	−1.18138
$808$	16.3092	0.573754
$809$	−39.3369	−1.38301	−0.691506	−	0.722371i	$-0.743052\pi$
$809$	−39.3369	−1.38301	−0.691506	+	0.722371i	$0.743052\pi$
$810$	0	0
$811$	9.57933	0.336376	0.168188	−	0.985755i	$-0.446208\pi$
$811$	9.57933	0.336376	0.168188	+	0.985755i	$0.446208\pi$
$812$	6.02179	0.211323
$813$	−57.1093	−2.00291
$814$	−2.16173	−0.0757685
$815$	0	0
$816$	10.6340	0.372265
$817$	−19.8394	−0.694094
$818$	9.89433	0.345947
$819$	0	0
$820$	0	0
$821$	−7.82037	−0.272933	−0.136466	−	0.990645i	$-0.543575\pi$
$821$	−7.82037	−0.272933	−0.136466	+	0.990645i	$0.543575\pi$
$822$	40.9586	1.42859
$823$	34.1839	1.19158	0.595788	−	0.803142i	$-0.296840\pi$
$823$	34.1839	1.19158	0.595788	+	0.803142i	$0.296840\pi$
$824$	−14.5800	−0.507919
$825$	0	0
$826$	−21.0817	−0.733524
$827$	3.47145	0.120714	0.0603570	−	0.998177i	$-0.480776\pi$
$827$	3.47145	0.120714	0.0603570	+	0.998177i	$0.480776\pi$
$828$	−26.5631	−0.923131
$829$	13.3557	0.463864	0.231932	−	0.972732i	$-0.425495\pi$
$829$	13.3557	0.463864	0.231932	+	0.972732i	$0.425495\pi$
$830$	0	0
$831$	28.1874	0.977811
$832$	0	0
$833$	14.6096	0.506192
$834$	45.2482	1.56682
$835$	0	0
$836$	20.0171	0.692304
$837$	11.1402	0.385063
$838$	−17.9386	−0.619679
$839$	47.9304	1.65474	0.827370	−	0.561657i	$-0.189836\pi$
$839$	47.9304	1.65474	0.827370	+	0.561657i	$0.189836\pi$
$840$	0	0
$841$	−18.6213	−0.642114
$842$	15.5977	0.537532
$843$	−51.3123	−1.76729
$844$	−1.98463	−0.0683139
$845$	0	0
$846$	−43.9727	−1.51181
$847$	16.0887	0.552814
$848$	0.191394	0.00657249
$849$	−51.7220	−1.77509
$850$	0	0
$851$	3.69154	0.126544
$852$	−25.8872	−0.886882
$853$	36.4405	1.24770	0.623850	−	0.781544i	$-0.285568\pi$
$853$	36.4405	1.24770	0.623850	+	0.781544i	$0.285568\pi$
$854$	11.7513	0.402123
$855$	0	0
$856$	11.4218	0.390388
$857$	29.9745	1.02391	0.511954	−	0.859013i	$-0.328922\pi$
$857$	29.9745	1.02391	0.511954	+	0.859013i	$0.328922\pi$
$858$	0	0
$859$	34.8453	1.18891	0.594454	−	0.804130i	$-0.297368\pi$
$859$	34.8453	1.18891	0.594454	+	0.804130i	$0.297368\pi$
$860$	0	0
$861$	52.0409	1.77355
$862$	30.0631	1.02395
$863$	−25.2753	−0.860381	−0.430190	−	0.902738i	$-0.641554\pi$
$863$	−25.2753	−0.860381	−0.430190	+	0.902738i	$0.641554\pi$
$864$	−1.30889	−0.0445292
$865$	0	0
$866$	−22.5878	−0.767565
$867$	−0.926067	−0.0314509
$868$	−15.9092	−0.539992
$869$	−10.3201	−0.350085
$870$	0	0
$871$	0	0
$872$	−10.3606	−0.350853
$873$	−6.72311	−0.227543
$874$	−34.1827	−1.15625
$875$	0	0
$876$	10.4803	0.354096
$877$	−6.44566	−0.217654	−0.108827	−	0.994061i	$-0.534710\pi$
$877$	−6.44566	−0.217654	−0.108827	+	0.994061i	$0.534710\pi$
$878$	−34.8184	−1.17506
$879$	76.0524	2.56518
$880$	0	0
$881$	−12.5890	−0.424133	−0.212066	−	0.977255i	$-0.568019\pi$
$881$	−12.5890	−0.424133	−0.212066	+	0.977255i	$0.568019\pi$
$882$	−12.3166	−0.414721
$883$	−17.7501	−0.597339	−0.298669	−	0.954357i	$-0.596543\pi$
$883$	−17.7501	−0.597339	−0.298669	+	0.954357i	$0.596543\pi$
$884$	0	0
$885$	0	0
$886$	−15.5022	−0.520808
$887$	−6.79925	−0.228296	−0.114148	−	0.993464i	$-0.536414\pi$
$887$	−6.79925	−0.228296	−0.114148	+	0.993464i	$0.536414\pi$
$888$	1.24589	0.0418092
$889$	2.84491	0.0954151
$890$	0	0
$891$	−31.8742	−1.06783
$892$	−6.23337	−0.208709
$893$	−56.5862	−1.89359
$894$	25.3895	0.849152
$895$	0	0
$896$	1.86919	0.0624453
$897$	0	0
$898$	−0.894606	−0.0298534
$899$	−27.4198	−0.914501
$900$	0	0
$901$	−0.797514	−0.0265691
$902$	−48.3074	−1.60846
$903$	20.9353	0.696682
$904$	0.559115	0.0185959
$905$	0	0
$906$	30.5173	1.01387
$907$	10.4184	0.345937	0.172969	−	0.984927i	$-0.444664\pi$
$907$	10.4184	0.345937	0.172969	+	0.984927i	$0.444664\pi$
$908$	13.3667	0.443590
$909$	57.2921	1.90026
$910$	0	0
$911$	−9.38030	−0.310783	−0.155392	−	0.987853i	$-0.549664\pi$
$911$	−9.38030	−0.310783	−0.155392	+	0.987853i	$0.549664\pi$
$912$	−11.5366	−0.382015
$913$	11.0547	0.365858
$914$	−10.7958	−0.357094
$915$	0	0
$916$	−5.10916	−0.168811
$917$	13.3921	0.442247
$918$	5.45397	0.180008
$919$	39.0419	1.28787	0.643937	−	0.765078i	$-0.277300\pi$
$919$	39.0419	1.28787	0.643937	+	0.765078i	$0.277300\pi$
$920$	0	0
$921$	23.3109	0.768121
$922$	−10.8827	−0.358403
$923$	0	0
$924$	−21.1227	−0.694886
$925$	0	0
$926$	−3.15999	−0.103844
$927$	−51.2178	−1.68221
$928$	3.22160	0.105754
$929$	29.3209	0.961988	0.480994	−	0.876724i	$-0.340276\pi$
$929$	29.3209	0.961988	0.480994	+	0.876724i	$0.340276\pi$
$930$	0	0
$931$	−15.8496	−0.519449
$932$	−25.7364	−0.843025
$933$	13.6437	0.446676
$934$	−23.0475	−0.754138
$935$	0	0
$936$	0	0
$937$	−55.0322	−1.79782	−0.898911	−	0.438130i	$-0.855641\pi$
$937$	−55.0322	−1.79782	−0.898911	+	0.438130i	$0.855641\pi$
$938$	10.6504	0.347747
$939$	3.37445	0.110121
$940$	0	0
$941$	−12.9833	−0.423243	−0.211621	−	0.977352i	$-0.567874\pi$
$941$	−12.9833	−0.423243	−0.211621	+	0.977352i	$0.567874\pi$
$942$	17.1647	0.559257
$943$	82.4935	2.68636
$944$	−11.2785	−0.367083
$945$	0	0
$946$	−19.4333	−0.631832
$947$	43.0090	1.39760	0.698802	−	0.715316i	$-0.253717\pi$
$947$	43.0090	1.39760	0.698802	+	0.715316i	$0.253717\pi$
$948$	5.94786	0.193178
$949$	0	0
$950$	0	0
$951$	−9.82414	−0.318570
$952$	−7.78870	−0.252433
$953$	26.9615	0.873369	0.436684	−	0.899615i	$-0.356153\pi$
$953$	26.9615	0.873369	0.436684	+	0.899615i	$0.356153\pi$
$954$	0.672343	0.0217679
$955$	0	0
$956$	9.42688	0.304887
$957$	−36.4055	−1.17682
$958$	−9.16688	−0.296168
$959$	−29.9994	−0.968731
$960$	0	0
$961$	41.4412	1.33681
$962$	0	0
$963$	40.1232	1.29295
$964$	−8.55884	−0.275662
$965$	0	0
$966$	36.0708	1.16056
$967$	−3.03699	−0.0976630	−0.0488315	−	0.998807i	$-0.515550\pi$
$967$	−3.03699	−0.0976630	−0.0488315	+	0.998807i	$0.515550\pi$
$968$	8.60730	0.276649
$969$	48.0716	1.54428
$970$	0	0
$971$	29.5149	0.947180	0.473590	−	0.880746i	$-0.342958\pi$
$971$	29.5149	0.947180	0.473590	+	0.880746i	$0.342958\pi$
$972$	22.2970	0.715177
$973$	−33.1412	−1.06246
$974$	−24.1155	−0.772711
$975$	0	0
$976$	6.28685	0.201237
$977$	38.3621	1.22731	0.613656	−	0.789574i	$-0.289698\pi$
$977$	38.3621	1.22731	0.613656	+	0.789574i	$0.289698\pi$
$978$	0.0214097	0.000684606 0
$979$	52.8769	1.68995
$980$	0	0
$981$	−36.3954	−1.16202
$982$	5.68528	0.181424
$983$	−30.0081	−0.957109	−0.478554	−	0.878058i	$-0.658839\pi$
$983$	−30.0081	−0.957109	−0.478554	+	0.878058i	$0.658839\pi$
$984$	27.8414	0.887551
$985$	0	0
$986$	−13.4240	−0.427507
$987$	59.7118	1.90065
$988$	0	0
$989$	33.1859	1.05525
$990$	0	0
$991$	−24.3851	−0.774618	−0.387309	−	0.921950i	$-0.626595\pi$
$991$	−24.3851	−0.774618	−0.387309	+	0.921950i	$0.626595\pi$
$992$	−8.51124	−0.270232
$993$	−12.6683	−0.402015
$994$	18.9607	0.601395
$995$	0	0
$996$	−6.37126	−0.201881
$997$	29.2603	0.926681	0.463341	−	0.886180i	$-0.346651\pi$
$997$	29.2603	0.926681	0.463341	+	0.886180i	$0.346651\pi$
$998$	6.68270	0.211537
$999$	0.638990	0.0202167

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cx.1.3		9
5.2	odd	4		1690.2.b.f.339.16	yes	18
5.3	odd	4		1690.2.b.f.339.3	✓	18
5.4	even	2		8450.2.a.cw.1.7		9
13.12	even	2		8450.2.a.ct.1.3		9
65.8	even	4		1690.2.c.h.1689.3		18
65.12	odd	4		1690.2.b.g.339.7	yes	18
65.18	even	4		1690.2.c.g.1689.3		18
65.38	odd	4		1690.2.b.g.339.12	yes	18
65.47	even	4		1690.2.c.g.1689.16		18
65.57	even	4		1690.2.c.h.1689.16		18
65.64	even	2		8450.2.a.da.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.b.f.339.3	✓	18	5.3	odd	4
1690.2.b.f.339.16	yes	18	5.2	odd	4
1690.2.b.g.339.7	yes	18	65.12	odd	4
1690.2.b.g.339.12	yes	18	65.38	odd	4
1690.2.c.g.1689.3		18	65.18	even	4
1690.2.c.g.1689.16		18	65.47	even	4
1690.2.c.h.1689.3		18	65.8	even	4
1690.2.c.h.1689.16		18	65.57	even	4
8450.2.a.ct.1.3		9	13.12	even	2
8450.2.a.cw.1.7		9	5.4	even	2
8450.2.a.cx.1.3		9	1.1	even	1	trivial
8450.2.a.da.1.7		9	65.64	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.55203 q^{3} +1.00000 q^{4} -2.55203 q^{6} +1.86919 q^{7} +1.00000 q^{8} +3.51288 q^{9} +4.42801 q^{11} -2.55203 q^{12} +1.86919 q^{14} +1.00000 q^{16} -4.16688 q^{17} +3.51288 q^{18} +4.52055 q^{19} -4.77024 q^{21} +4.42801 q^{22} -7.56163 q^{23} -2.55203 q^{24} -1.30889 q^{27} +1.86919 q^{28} +3.22160 q^{29} -8.51124 q^{31} +1.00000 q^{32} -11.3004 q^{33} -4.16688 q^{34} +3.51288 q^{36} -0.488194 q^{37} +4.52055 q^{38} -10.9095 q^{41} -4.77024 q^{42} -4.38872 q^{43} +4.42801 q^{44} -7.56163 q^{46} -12.5176 q^{47} -2.55203 q^{48} -3.50612 q^{49} +10.6340 q^{51} +0.191394 q^{53} -1.30889 q^{54} +1.86919 q^{56} -11.5366 q^{57} +3.22160 q^{58} -11.2785 q^{59} +6.28685 q^{61} -8.51124 q^{62} +6.56625 q^{63} +1.00000 q^{64} -11.3004 q^{66} +5.69785 q^{67} -4.16688 q^{68} +19.2975 q^{69} +10.1438 q^{71} +3.51288 q^{72} -4.10663 q^{73} -0.488194 q^{74} +4.52055 q^{76} +8.27681 q^{77} -2.33064 q^{79} -7.19832 q^{81} -10.9095 q^{82} +2.49654 q^{83} -4.77024 q^{84} -4.38872 q^{86} -8.22163 q^{87} +4.42801 q^{88} +11.9415 q^{89} -7.56163 q^{92} +21.7210 q^{93} -12.5176 q^{94} -2.55203 q^{96} -1.91384 q^{97} -3.50612 q^{98} +15.5551 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 7 q^{3} + 9 q^{4} - 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} - 4 q^{11} - 7 q^{12} + q^{14} + 9 q^{16} - 12 q^{17} + 8 q^{18} - 6 q^{19} - 8 q^{21} - 4 q^{22} - 11 q^{23} - 7 q^{24} - 34 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 7 * q^3 + 9 * q^4 - 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 - 4 * q^11 - 7 * q^12 + q^14 + 9 * q^16 - 12 * q^17 + 8 * q^18 - 6 * q^19 - 8 * q^21 - 4 * q^22 - 11 * q^23 - 7 * q^24 - 34 * q^27 + q^28 + 15 * q^29 + 6 * q^31 + 9 * q^32 - 14 * q^33 - 12 * q^34 + 8 * q^36 - 10 * q^37 - 6 * q^38 - 27 * q^41 - 8 * q^42 - 41 * q^43 - 4 * q^44 - 11 * q^46 + 5 * q^47 - 7 * q^48 - 10 * q^49 + 12 * q^51 - 18 * q^53 - 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 - 20 * q^59 - 19 * q^61 + 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 - 12 * q^68 - 14 * q^69 + 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 - 6 * q^76 - 30 * q^77 - 14 * q^79 + 53 * q^81 - 27 * q^82 + q^83 - 8 * q^84 - 41 * q^86 - 33 * q^87 - 4 * q^88 - 43 * q^89 - 11 * q^92 - 28 * q^93 + 5 * q^94 - 7 * q^96 + 6 * q^97 - 10 * q^98 - 16 * q^99

Introduction

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