LMFDB - Embedded newform 2277.2.a.n.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2277,2,Mod(1,2277)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2277, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("2277.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2277 = 3^{2} \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2277.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-2,0,4,-6,0,0,-6,0,8,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.1819365402$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4222000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 8x^{2} - 8x + 1 $ x^6 - 2x^5 - 6x^4 + 10x^3 + 8x^2 - 8*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 759)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.537960$ of defining polynomial
Character	$\chi$	$=$	2277.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.537960 q^{2} -1.71060 q^{4} -3.67524 q^{5} -2.55698 q^{7} +1.99615 q^{8} +1.97713 q^{10} +1.00000 q^{11} +4.47214 q^{13} +1.37555 q^{14} +2.34735 q^{16} -2.66496 q^{17} +2.95381 q^{19} +6.28686 q^{20} -0.537960 q^{22} -1.00000 q^{23} +8.50740 q^{25} -2.40583 q^{26} +4.37397 q^{28} +2.49712 q^{29} -3.20786 q^{31} -5.25509 q^{32} +1.43364 q^{34} +9.39752 q^{35} +9.75632 q^{37} -1.58903 q^{38} -7.33635 q^{40} -5.69445 q^{41} +7.13878 q^{43} -1.71060 q^{44} +0.537960 q^{46} -12.6064 q^{47} -0.461849 q^{49} -4.57665 q^{50} -7.65003 q^{52} +11.5228 q^{53} -3.67524 q^{55} -5.10413 q^{56} -1.34335 q^{58} +1.28418 q^{59} +8.34528 q^{61} +1.72570 q^{62} -1.86766 q^{64} -16.4362 q^{65} -6.33697 q^{67} +4.55867 q^{68} -5.05549 q^{70} +10.6270 q^{71} -3.90102 q^{73} -5.24851 q^{74} -5.05279 q^{76} -2.55698 q^{77} +5.41035 q^{79} -8.62706 q^{80} +3.06339 q^{82} -12.5491 q^{83} +9.79436 q^{85} -3.84038 q^{86} +1.99615 q^{88} -14.7009 q^{89} -11.4352 q^{91} +1.71060 q^{92} +6.78174 q^{94} -10.8560 q^{95} -10.6347 q^{97} +0.248457 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 4 q^{4} - 6 q^{5} - 6 q^{8} + 8 q^{10} + 6 q^{11} - 12 q^{14} - 10 q^{17} - 2 q^{19} - 12 q^{20} - 2 q^{22} - 6 q^{23} + 10 q^{25} + 18 q^{28} - 16 q^{29} - 12 q^{31} - 4 q^{32} - 2 q^{34}+ \cdots - 16 q^{98}+O(q^{100}) $ 6 * q - 2 * q^2 + 4 * q^4 - 6 * q^5 - 6 * q^8 + 8 * q^10 + 6 * q^11 - 12 * q^14 - 10 * q^17 - 2 * q^19 - 12 * q^20 - 2 * q^22 - 6 * q^23 + 10 * q^25 + 18 * q^28 - 16 * q^29 - 12 * q^31 - 4 * q^32 - 2 * q^34 - 14 * q^35 + 12 * q^37 - 6 * q^38 - 4 * q^40 - 22 * q^41 + 4 * q^44 + 2 * q^46 - 8 * q^47 + 14 * q^49 - 28 * q^50 + 20 * q^52 - 10 * q^53 - 6 * q^55 - 22 * q^56 - 8 * q^58 - 12 * q^59 + 24 * q^61 + 34 * q^62 - 26 * q^64 - 20 * q^65 - 18 * q^67 + 2 * q^68 + 46 * q^70 - 24 * q^71 - 4 * q^73 - 36 * q^74 - 30 * q^76 + 16 * q^79 + 20 * q^80 - 18 * q^82 - 16 * q^83 - 16 * q^85 + 24 * q^86 - 6 * q^88 - 2 * q^89 - 40 * q^91 - 4 * q^92 + 12 * q^94 + 28 * q^95 - 12 * q^97 - 16 * 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 $

$2$	−0.537960	−0.380395	−0.190198	−	0.981746i	$-0.560913\pi$
$2$	−0.537960	−0.380395	−0.190198	+	0.981746i	$0.560913\pi$
$3$	0	0
$3$	0	0
$4$	−1.71060	−0.855299
$5$	−3.67524	−1.64362	−0.821809	−	0.569763i	$-0.807035\pi$
$5$	−3.67524	−1.64362	−0.821809	+	0.569763i	$0.807035\pi$
$6$	0	0
$7$	−2.55698	−0.966448	−0.483224	−	0.875497i	$-0.660534\pi$
$7$	−2.55698	−0.966448	−0.483224	+	0.875497i	$0.660534\pi$
$8$	1.99615	0.705747
$9$	0	0
$10$	1.97713	0.625225
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	1.37555	0.367632
$15$	0	0
$16$	2.34735	0.586836
$17$	−2.66496	−0.646347	−0.323173	−	0.946340i	$-0.604750\pi$
$17$	−2.66496	−0.646347	−0.323173	+	0.946340i	$0.604750\pi$
$18$	0	0
$19$	2.95381	0.677651	0.338826	−	0.940849i	$-0.389970\pi$
$19$	2.95381	0.677651	0.338826	+	0.940849i	$0.389970\pi$
$20$	6.28686	1.40579
$21$	0	0
$22$	−0.537960	−0.114694
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	8.50740	1.70148
$26$	−2.40583	−0.471822
$27$	0	0
$28$	4.37397	0.826602
$29$	2.49712	0.463703	0.231852	−	0.972751i	$-0.425522\pi$
$29$	2.49712	0.463703	0.231852	+	0.972751i	$0.425522\pi$
$30$	0	0
$31$	−3.20786	−0.576148	−0.288074	−	0.957608i	$-0.593015\pi$
$31$	−3.20786	−0.576148	−0.288074	+	0.957608i	$0.593015\pi$
$32$	−5.25509	−0.928977
$33$	0	0
$34$	1.43364	0.245867
$35$	9.39752	1.58847
$36$	0	0
$37$	9.75632	1.60393	0.801964	−	0.597372i	$-0.203788\pi$
$37$	9.75632	1.60393	0.801964	+	0.597372i	$0.203788\pi$
$38$	−1.58903	−0.257775
$39$	0	0
$40$	−7.33635	−1.15998
$41$	−5.69445	−0.889324	−0.444662	−	0.895698i	$-0.646676\pi$
$41$	−5.69445	−0.889324	−0.444662	+	0.895698i	$0.646676\pi$
$42$	0	0
$43$	7.13878	1.08865	0.544327	−	0.838873i	$-0.316785\pi$
$43$	7.13878	1.08865	0.544327	+	0.838873i	$0.316785\pi$
$44$	−1.71060	−0.257882
$45$	0	0
$46$	0.537960	0.0793179
$47$	−12.6064	−1.83883	−0.919416	−	0.393286i	$-0.871338\pi$
$47$	−12.6064	−1.83883	−0.919416	+	0.393286i	$0.871338\pi$
$48$	0	0
$49$	−0.461849	−0.0659785
$50$	−4.57665	−0.647236
$51$	0	0
$52$	−7.65003	−1.06087
$53$	11.5228	1.58278	0.791392	−	0.611309i	$-0.209357\pi$
$53$	11.5228	1.58278	0.791392	+	0.611309i	$0.209357\pi$
$54$	0	0
$55$	−3.67524	−0.495570
$56$	−5.10413	−0.682068
$57$	0	0
$58$	−1.34335	−0.176391
$59$	1.28418	0.167186	0.0835930	−	0.996500i	$-0.473360\pi$
$59$	1.28418	0.167186	0.0835930	+	0.996500i	$0.473360\pi$
$60$	0	0
$61$	8.34528	1.06850	0.534252	−	0.845326i	$-0.320593\pi$
$61$	8.34528	1.06850	0.534252	+	0.845326i	$0.320593\pi$
$62$	1.72570	0.219164
$63$	0	0
$64$	−1.86766	−0.233458
$65$	−16.4362	−2.03866
$66$	0	0
$67$	−6.33697	−0.774184	−0.387092	−	0.922041i	$-0.626520\pi$
$67$	−6.33697	−0.774184	−0.387092	+	0.922041i	$0.626520\pi$
$68$	4.55867	0.552820
$69$	0	0
$70$	−5.05549	−0.604247
$71$	10.6270	1.26119	0.630595	−	0.776112i	$-0.282811\pi$
$71$	10.6270	1.26119	0.630595	+	0.776112i	$0.282811\pi$
$72$	0	0
$73$	−3.90102	−0.456580	−0.228290	−	0.973593i	$-0.573313\pi$
$73$	−3.90102	−0.456580	−0.228290	+	0.973593i	$0.573313\pi$
$74$	−5.24851	−0.610127
$75$	0	0
$76$	−5.05279	−0.579595
$77$	−2.55698	−0.291395
$78$	0	0
$79$	5.41035	0.608712	0.304356	−	0.952558i	$-0.401559\pi$
$79$	5.41035	0.608712	0.304356	+	0.952558i	$0.401559\pi$
$80$	−8.62706	−0.964535
$81$	0	0
$82$	3.06339	0.338295
$83$	−12.5491	−1.37745	−0.688723	−	0.725025i	$-0.741828\pi$
$83$	−12.5491	−1.37745	−0.688723	+	0.725025i	$0.741828\pi$
$84$	0	0
$85$	9.79436	1.06235
$86$	−3.84038	−0.414119
$87$	0	0
$88$	1.99615	0.212791
$89$	−14.7009	−1.55829	−0.779147	−	0.626841i	$-0.784347\pi$
$89$	−14.7009	−1.55829	−0.779147	+	0.626841i	$0.784347\pi$
$90$	0	0
$91$	−11.4352	−1.19873
$92$	1.71060	0.178342
$93$	0	0
$94$	6.78174	0.699483
$95$	−10.8560	−1.11380
$96$	0	0
$97$	−10.6347	−1.07979	−0.539893	−	0.841733i	$-0.681535\pi$
$97$	−10.6347	−1.07979	−0.539893	+	0.841733i	$0.681535\pi$
$98$	0.248457	0.0250979
$99$	0	0
$100$	−14.5528	−1.45528
$101$	−11.6062	−1.15486	−0.577428	−	0.816441i	$-0.695944\pi$
$101$	−11.6062	−1.15486	−0.577428	+	0.816441i	$0.695944\pi$
$102$	0	0
$103$	−11.5074	−1.13385	−0.566926	−	0.823768i	$-0.691868\pi$
$103$	−11.5074	−1.13385	−0.566926	+	0.823768i	$0.691868\pi$
$104$	8.92708	0.875372
$105$	0	0
$106$	−6.19883	−0.602084
$107$	1.98570	0.191965	0.0959827	−	0.995383i	$-0.469401\pi$
$107$	1.98570	0.191965	0.0959827	+	0.995383i	$0.469401\pi$
$108$	0	0
$109$	−16.7273	−1.60218	−0.801090	−	0.598543i	$-0.795746\pi$
$109$	−16.7273	−1.60218	−0.801090	+	0.598543i	$0.795746\pi$
$110$	1.97713	0.188512
$111$	0	0
$112$	−6.00212	−0.567147
$113$	14.8002	1.39228	0.696142	−	0.717904i	$-0.254898\pi$
$113$	14.8002	1.39228	0.696142	+	0.717904i	$0.254898\pi$
$114$	0	0
$115$	3.67524	0.342718
$116$	−4.27157	−0.396605
$117$	0	0
$118$	−0.690838	−0.0635968
$119$	6.81424	0.624660
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.48943	−0.406454
$123$	0	0
$124$	5.48736	0.492779
$125$	−12.8906	−1.15297
$126$	0	0
$127$	−11.5297	−1.02309	−0.511547	−	0.859255i	$-0.670927\pi$
$127$	−11.5297	−1.02309	−0.511547	+	0.859255i	$0.670927\pi$
$128$	11.5149	1.01778
$129$	0	0
$130$	8.84201	0.775496
$131$	−14.5897	−1.27471	−0.637355	−	0.770571i	$-0.719971\pi$
$131$	−14.5897	−1.27471	−0.637355	+	0.770571i	$0.719971\pi$
$132$	0	0
$133$	−7.55285	−0.654915
$134$	3.40904	0.294496
$135$	0	0
$136$	−5.31966	−0.456157
$137$	14.3153	1.22304	0.611518	−	0.791230i	$-0.290559\pi$
$137$	14.3153	1.22304	0.611518	+	0.791230i	$0.290559\pi$
$138$	0	0
$139$	−2.42588	−0.205760	−0.102880	−	0.994694i	$-0.532806\pi$
$139$	−2.42588	−0.205760	−0.102880	+	0.994694i	$0.532806\pi$
$140$	−16.0754	−1.35862
$141$	0	0
$142$	−5.71689	−0.479751
$143$	4.47214	0.373979
$144$	0	0
$145$	−9.17751	−0.762151
$146$	2.09860	0.173681
$147$	0	0
$148$	−16.6891	−1.37184
$149$	15.5525	1.27411	0.637056	−	0.770817i	$-0.280152\pi$
$149$	15.5525	1.27411	0.637056	+	0.770817i	$0.280152\pi$
$150$	0	0
$151$	−12.6020	−1.02554	−0.512768	−	0.858527i	$-0.671380\pi$
$151$	−12.6020	−1.02554	−0.512768	+	0.858527i	$0.671380\pi$
$152$	5.89627	0.478251
$153$	0	0
$154$	1.37555	0.110845
$155$	11.7897	0.946968
$156$	0	0
$157$	−12.2083	−0.974325	−0.487163	−	0.873311i	$-0.661968\pi$
$157$	−12.2083	−0.974325	−0.487163	+	0.873311i	$0.661968\pi$
$158$	−2.91055	−0.231551
$159$	0	0
$160$	19.3137	1.52688
$161$	2.55698	0.201518
$162$	0	0
$163$	11.4322	0.895437	0.447718	−	0.894175i	$-0.352237\pi$
$163$	11.4322	0.895437	0.447718	+	0.894175i	$0.352237\pi$
$164$	9.74092	0.760638
$165$	0	0
$166$	6.75093	0.523974
$167$	0.298238	0.0230783	0.0115392	−	0.999933i	$-0.496327\pi$
$167$	0.298238	0.0230783	0.0115392	+	0.999933i	$0.496327\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	−5.26898	−0.404112
$171$	0	0
$172$	−12.2116	−0.931125
$173$	−2.18141	−0.165850	−0.0829249	−	0.996556i	$-0.526426\pi$
$173$	−2.18141	−0.165850	−0.0829249	+	0.996556i	$0.526426\pi$
$174$	0	0
$175$	−21.7533	−1.64439
$176$	2.34735	0.176938
$177$	0	0
$178$	7.90851	0.592768
$179$	7.20633	0.538627	0.269313	−	0.963053i	$-0.413203\pi$
$179$	7.20633	0.538627	0.269313	+	0.963053i	$0.413203\pi$
$180$	0	0
$181$	24.3892	1.81284	0.906418	−	0.422382i	$-0.138806\pi$
$181$	24.3892	1.81284	0.906418	+	0.422382i	$0.138806\pi$
$182$	6.15167	0.455992
$183$	0	0
$184$	−1.99615	−0.147158
$185$	−35.8568	−2.63625
$186$	0	0
$187$	−2.66496	−0.194881
$188$	21.5645	1.57275
$189$	0	0
$190$	5.84009	0.423685
$191$	4.85569	0.351346	0.175673	−	0.984449i	$-0.443790\pi$
$191$	4.85569	0.351346	0.175673	+	0.984449i	$0.443790\pi$
$192$	0	0
$193$	−26.0684	−1.87644	−0.938221	−	0.346037i	$-0.887527\pi$
$193$	−26.0684	−1.87644	−0.938221	+	0.346037i	$0.887527\pi$
$194$	5.72103	0.410746
$195$	0	0
$196$	0.790039	0.0564314
$197$	−24.6143	−1.75370	−0.876849	−	0.480767i	$-0.840358\pi$
$197$	−24.6143	−1.75370	−0.876849	+	0.480767i	$0.840358\pi$
$198$	0	0
$199$	−9.24320	−0.655233	−0.327617	−	0.944811i	$-0.606245\pi$
$199$	−9.24320	−0.655233	−0.327617	+	0.944811i	$0.606245\pi$
$200$	16.9821	1.20082
$201$	0	0
$202$	6.24366	0.439302
$203$	−6.38508	−0.448145
$204$	0	0
$205$	20.9285	1.46171
$206$	6.19050	0.431312
$207$	0	0
$208$	10.4976	0.727881
$209$	2.95381	0.204320
$210$	0	0
$211$	21.6182	1.48826	0.744130	−	0.668035i	$-0.232864\pi$
$211$	21.6182	1.48826	0.744130	+	0.668035i	$0.232864\pi$
$212$	−19.7110	−1.35375
$213$	0	0
$214$	−1.06823	−0.0730227
$215$	−26.2367	−1.78933
$216$	0	0
$217$	8.20243	0.556817
$218$	8.99860	0.609462
$219$	0	0
$220$	6.28686	0.423860
$221$	−11.9180	−0.801694
$222$	0	0
$223$	6.53721	0.437764	0.218882	−	0.975751i	$-0.429759\pi$
$223$	6.53721	0.437764	0.218882	+	0.975751i	$0.429759\pi$
$224$	13.4372	0.897808
$225$	0	0
$226$	−7.96191	−0.529618
$227$	10.4617	0.694367	0.347183	−	0.937797i	$-0.387138\pi$
$227$	10.4617	0.694367	0.347183	+	0.937797i	$0.387138\pi$
$228$	0	0
$229$	5.83401	0.385523	0.192761	−	0.981246i	$-0.438256\pi$
$229$	5.83401	0.385523	0.192761	+	0.981246i	$0.438256\pi$
$230$	−1.97713	−0.130368
$231$	0	0
$232$	4.98463	0.327257
$233$	12.0725	0.790894	0.395447	−	0.918489i	$-0.370590\pi$
$233$	12.0725	0.790894	0.395447	+	0.918489i	$0.370590\pi$
$234$	0	0
$235$	46.3316	3.02234
$236$	−2.19672	−0.142994
$237$	0	0
$238$	−3.66579	−0.237618
$239$	2.88457	0.186587	0.0932937	−	0.995639i	$-0.470260\pi$
$239$	2.88457	0.186587	0.0932937	+	0.995639i	$0.470260\pi$
$240$	0	0
$241$	23.3698	1.50538	0.752690	−	0.658375i	$-0.228756\pi$
$241$	23.3698	1.50538	0.752690	+	0.658375i	$0.228756\pi$
$242$	−0.537960	−0.0345814
$243$	0	0
$244$	−14.2754	−0.913890
$245$	1.69741	0.108443
$246$	0	0
$247$	13.2099	0.840523
$248$	−6.40338	−0.406615
$249$	0	0
$250$	6.93461	0.438583
$251$	−4.65750	−0.293979	−0.146989	−	0.989138i	$-0.546958\pi$
$251$	−4.65750	−0.293979	−0.146989	+	0.989138i	$0.546958\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	6.20251	0.389180
$255$	0	0
$256$	−2.45924	−0.153702
$257$	−8.80355	−0.549151	−0.274575	−	0.961566i	$-0.588537\pi$
$257$	−8.80355	−0.549151	−0.274575	+	0.961566i	$0.588537\pi$
$258$	0	0
$259$	−24.9467	−1.55011
$260$	28.1157	1.74366
$261$	0	0
$262$	7.84869	0.484893
$263$	−10.8144	−0.666845	−0.333423	−	0.942777i	$-0.608204\pi$
$263$	−10.8144	−0.666845	−0.333423	+	0.942777i	$0.608204\pi$
$264$	0	0
$265$	−42.3492	−2.60149
$266$	4.06313	0.249127
$267$	0	0
$268$	10.8400	0.662159
$269$	3.30384	0.201439	0.100719	−	0.994915i	$-0.467886\pi$
$269$	3.30384	0.201439	0.100719	+	0.994915i	$0.467886\pi$
$270$	0	0
$271$	−3.77454	−0.229287	−0.114644	−	0.993407i	$-0.536573\pi$
$271$	−3.77454	−0.229287	−0.114644	+	0.993407i	$0.536573\pi$
$272$	−6.25557	−0.379300
$273$	0	0
$274$	−7.70105	−0.465237
$275$	8.50740	0.513016
$276$	0	0
$277$	9.59858	0.576723	0.288361	−	0.957522i	$-0.406890\pi$
$277$	9.59858	0.576723	0.288361	+	0.957522i	$0.406890\pi$
$278$	1.30503	0.0782702
$279$	0	0
$280$	18.7589	1.12106
$281$	−14.7114	−0.877607	−0.438803	−	0.898583i	$-0.644598\pi$
$281$	−14.7114	−0.877607	−0.438803	+	0.898583i	$0.644598\pi$
$282$	0	0
$283$	4.25274	0.252799	0.126400	−	0.991979i	$-0.459658\pi$
$283$	4.25274	0.252799	0.126400	+	0.991979i	$0.459658\pi$
$284$	−18.1785	−1.07869
$285$	0	0
$286$	−2.40583	−0.142260
$287$	14.5606	0.859485
$288$	0	0
$289$	−9.89801	−0.582236
$290$	4.93714	0.289919
$291$	0	0
$292$	6.67309	0.390513
$293$	13.1306	0.767100	0.383550	−	0.923520i	$-0.374701\pi$
$293$	13.1306	0.767100	0.383550	+	0.923520i	$0.374701\pi$
$294$	0	0
$295$	−4.71967	−0.274790
$296$	19.4751	1.13197
$297$	0	0
$298$	−8.36664	−0.484667
$299$	−4.47214	−0.258630
$300$	0	0
$301$	−18.2537	−1.05213
$302$	6.77937	0.390109
$303$	0	0
$304$	6.93362	0.397670
$305$	−30.6709	−1.75621
$306$	0	0
$307$	−9.17543	−0.523669	−0.261835	−	0.965113i	$-0.584327\pi$
$307$	−9.17543	−0.523669	−0.261835	+	0.965113i	$0.584327\pi$
$308$	4.37397	0.249230
$309$	0	0
$310$	−6.34237	−0.360222
$311$	−25.5144	−1.44679	−0.723396	−	0.690434i	$-0.757420\pi$
$311$	−25.5144	−1.44679	−0.723396	+	0.690434i	$0.757420\pi$
$312$	0	0
$313$	−11.1238	−0.628754	−0.314377	−	0.949298i	$-0.601796\pi$
$313$	−11.1238	−0.628754	−0.314377	+	0.949298i	$0.601796\pi$
$314$	6.56756	0.370629
$315$	0	0
$316$	−9.25493	−0.520631
$317$	3.31191	0.186016	0.0930078	−	0.995665i	$-0.470352\pi$
$317$	3.31191	0.186016	0.0930078	+	0.995665i	$0.470352\pi$
$318$	0	0
$319$	2.49712	0.139812
$320$	6.86411	0.383715
$321$	0	0
$322$	−1.37555	−0.0766566
$323$	−7.87178	−0.437998
$324$	0	0
$325$	38.0463	2.11043
$326$	−6.15006	−0.340620
$327$	0	0
$328$	−11.3670	−0.627638
$329$	32.2343	1.77714
$330$	0	0
$331$	−20.0495	−1.10202	−0.551009	−	0.834499i	$-0.685757\pi$
$331$	−20.0495	−1.10202	−0.551009	+	0.834499i	$0.685757\pi$
$332$	21.4665	1.17813
$333$	0	0
$334$	−0.160440	−0.00877888
$335$	23.2899	1.27246
$336$	0	0
$337$	−9.89437	−0.538981	−0.269490	−	0.963003i	$-0.586855\pi$
$337$	−9.89437	−0.538981	−0.269490	+	0.963003i	$0.586855\pi$
$338$	−3.76572	−0.204828
$339$	0	0
$340$	−16.7542	−0.908625
$341$	−3.20786	−0.173715
$342$	0	0
$343$	19.0798	1.03021
$344$	14.2501	0.768314
$345$	0	0
$346$	1.17351	0.0630885
$347$	−20.5929	−1.10548	−0.552742	−	0.833352i	$-0.686418\pi$
$347$	−20.5929	−1.10548	−0.552742	+	0.833352i	$0.686418\pi$
$348$	0	0
$349$	25.9496	1.38905	0.694524	−	0.719469i	$-0.255615\pi$
$349$	25.9496	1.38905	0.694524	+	0.719469i	$0.255615\pi$
$350$	11.7024	0.625519
$351$	0	0
$352$	−5.25509	−0.280097
$353$	−5.31928	−0.283117	−0.141558	−	0.989930i	$-0.545211\pi$
$353$	−5.31928	−0.283117	−0.141558	+	0.989930i	$0.545211\pi$
$354$	0	0
$355$	−39.0567	−2.07291
$356$	25.1474	1.33281
$357$	0	0
$358$	−3.87672	−0.204891
$359$	−7.40335	−0.390734	−0.195367	−	0.980730i	$-0.562590\pi$
$359$	−7.40335	−0.390734	−0.195367	+	0.980730i	$0.562590\pi$
$360$	0	0
$361$	−10.2750	−0.540789
$362$	−13.1204	−0.689595
$363$	0	0
$364$	19.5610	1.02527
$365$	14.3372	0.750444
$366$	0	0
$367$	22.4832	1.17361	0.586807	−	0.809727i	$-0.300385\pi$
$367$	22.4832	1.17361	0.586807	+	0.809727i	$0.300385\pi$
$368$	−2.34735	−0.122364
$369$	0	0
$370$	19.2895	1.00282
$371$	−29.4637	−1.52968
$372$	0	0
$373$	−28.1533	−1.45772	−0.728862	−	0.684661i	$-0.759950\pi$
$373$	−28.1533	−1.45772	−0.728862	+	0.684661i	$0.759950\pi$
$374$	1.43364	0.0741318
$375$	0	0
$376$	−25.1643	−1.29775
$377$	11.1675	0.575153
$378$	0	0
$379$	25.1251	1.29059	0.645296	−	0.763933i	$-0.276734\pi$
$379$	25.1251	1.29059	0.645296	+	0.763933i	$0.276734\pi$
$380$	18.5702	0.952633
$381$	0	0
$382$	−2.61217	−0.133650
$383$	−12.3110	−0.629061	−0.314530	−	0.949247i	$-0.601847\pi$
$383$	−12.3110	−0.629061	−0.314530	+	0.949247i	$0.601847\pi$
$384$	0	0
$385$	9.39752	0.478942
$386$	14.0237	0.713790
$387$	0	0
$388$	18.1916	0.923541
$389$	15.4219	0.781920	0.390960	−	0.920408i	$-0.372143\pi$
$389$	15.4219	0.781920	0.390960	+	0.920408i	$0.372143\pi$
$390$	0	0
$391$	2.66496	0.134773
$392$	−0.921923	−0.0465641
$393$	0	0
$394$	13.2415	0.667098
$395$	−19.8843	−1.00049
$396$	0	0
$397$	−17.8183	−0.894277	−0.447138	−	0.894465i	$-0.647557\pi$
$397$	−17.8183	−0.894277	−0.447138	+	0.894465i	$0.647557\pi$
$398$	4.97248	0.249248
$399$	0	0
$400$	19.9698	0.998491
$401$	−30.0385	−1.50005	−0.750025	−	0.661409i	$-0.769959\pi$
$401$	−30.0385	−1.50005	−0.750025	+	0.661409i	$0.769959\pi$
$402$	0	0
$403$	−14.3460	−0.714624
$404$	19.8535	0.987748
$405$	0	0
$406$	3.43492	0.170472
$407$	9.75632	0.483603
$408$	0	0
$409$	−24.9081	−1.23163	−0.615814	−	0.787891i	$-0.711173\pi$
$409$	−24.9081	−1.23163	−0.615814	+	0.787891i	$0.711173\pi$
$410$	−11.2587	−0.556028
$411$	0	0
$412$	19.6845	0.969784
$413$	−3.28362	−0.161577
$414$	0	0
$415$	46.1211	2.26399
$416$	−23.5015	−1.15225
$417$	0	0
$418$	−1.58903	−0.0777222
$419$	18.0499	0.881797	0.440898	−	0.897557i	$-0.354660\pi$
$419$	18.0499	0.881797	0.440898	+	0.897557i	$0.354660\pi$
$420$	0	0
$421$	−1.42818	−0.0696054	−0.0348027	−	0.999394i	$-0.511080\pi$
$421$	−1.42818	−0.0696054	−0.0348027	+	0.999394i	$0.511080\pi$
$422$	−11.6297	−0.566127
$423$	0	0
$424$	23.0014	1.11705
$425$	−22.6719	−1.09975
$426$	0	0
$427$	−21.3387	−1.03265
$428$	−3.39674	−0.164188
$429$	0	0
$430$	14.1143	0.680653
$431$	−26.1677	−1.26045	−0.630226	−	0.776412i	$-0.717038\pi$
$431$	−26.1677	−1.26045	−0.630226	+	0.776412i	$0.717038\pi$
$432$	0	0
$433$	−29.2047	−1.40349	−0.701744	−	0.712429i	$-0.747595\pi$
$433$	−29.2047	−1.40349	−0.701744	+	0.712429i	$0.747595\pi$
$434$	−4.41258	−0.211811
$435$	0	0
$436$	28.6136	1.37034
$437$	−2.95381	−0.141300
$438$	0	0
$439$	−7.30712	−0.348750	−0.174375	−	0.984679i	$-0.555791\pi$
$439$	−7.30712	−0.348750	−0.174375	+	0.984679i	$0.555791\pi$
$440$	−7.33635	−0.349747
$441$	0	0
$442$	6.41143	0.304961
$443$	0.632904	0.0300702	0.0150351	−	0.999887i	$-0.495214\pi$
$443$	0.632904	0.0300702	0.0150351	+	0.999887i	$0.495214\pi$
$444$	0	0
$445$	54.0294	2.56124
$446$	−3.51676	−0.166523
$447$	0	0
$448$	4.77557	0.225625
$449$	−16.3423	−0.771239	−0.385619	−	0.922658i	$-0.626012\pi$
$449$	−16.3423	−0.771239	−0.385619	+	0.922658i	$0.626012\pi$
$450$	0	0
$451$	−5.69445	−0.268141
$452$	−25.3172	−1.19082
$453$	0	0
$454$	−5.62798	−0.264134
$455$	42.0270	1.97026
$456$	0	0
$457$	−18.7587	−0.877495	−0.438747	−	0.898610i	$-0.644578\pi$
$457$	−18.7587	−0.877495	−0.438747	+	0.898610i	$0.644578\pi$
$458$	−3.13847	−0.146651
$459$	0	0
$460$	−6.28686	−0.293127
$461$	2.75084	0.128119	0.0640596	−	0.997946i	$-0.479595\pi$
$461$	2.75084	0.128119	0.0640596	+	0.997946i	$0.479595\pi$
$462$	0	0
$463$	−41.0048	−1.90566	−0.952828	−	0.303511i	$-0.901841\pi$
$463$	−41.0048	−1.90566	−0.952828	+	0.303511i	$0.901841\pi$
$464$	5.86160	0.272118
$465$	0	0
$466$	−6.49451	−0.300852
$467$	7.83216	0.362429	0.181215	−	0.983444i	$-0.441997\pi$
$467$	7.83216	0.362429	0.181215	+	0.983444i	$0.441997\pi$
$468$	0	0
$469$	16.2035	0.748209
$470$	−24.9245	−1.14968
$471$	0	0
$472$	2.56342	0.117991
$473$	7.13878	0.328241
$474$	0	0
$475$	25.1293	1.15301
$476$	−11.6564	−0.534272
$477$	0	0
$478$	−1.55179	−0.0709770
$479$	−9.41104	−0.430001	−0.215001	−	0.976614i	$-0.568975\pi$
$479$	−9.41104	−0.430001	−0.215001	+	0.976614i	$0.568975\pi$
$480$	0	0
$481$	43.6316	1.98943
$482$	−12.5720	−0.572640
$483$	0	0
$484$	−1.71060	−0.0777545
$485$	39.0850	1.77476
$486$	0	0
$487$	−27.3406	−1.23892	−0.619460	−	0.785028i	$-0.712649\pi$
$487$	−27.3406	−1.23892	−0.619460	+	0.785028i	$0.712649\pi$
$488$	16.6585	0.754093
$489$	0	0
$490$	−0.913138	−0.0412514
$491$	−9.18674	−0.414592	−0.207296	−	0.978278i	$-0.566466\pi$
$491$	−9.18674	−0.414592	−0.207296	+	0.978278i	$0.566466\pi$
$492$	0	0
$493$	−6.65471	−0.299713
$494$	−7.10638	−0.319731
$495$	0	0
$496$	−7.52995	−0.338105
$497$	−27.1730	−1.21887
$498$	0	0
$499$	−31.4155	−1.40635	−0.703174	−	0.711017i	$-0.748235\pi$
$499$	−31.4155	−1.40635	−0.703174	+	0.711017i	$0.748235\pi$
$500$	22.0506	0.986132
$501$	0	0
$502$	2.50555	0.111828
$503$	35.7776	1.59524	0.797621	−	0.603158i	$-0.206091\pi$
$503$	35.7776	1.59524	0.797621	+	0.603158i	$0.206091\pi$
$504$	0	0
$505$	42.6555	1.89814
$506$	0.537960	0.0239153
$507$	0	0
$508$	19.7227	0.875051
$509$	19.3038	0.855628	0.427814	−	0.903867i	$-0.359284\pi$
$509$	19.3038	0.855628	0.427814	+	0.903867i	$0.359284\pi$
$510$	0	0
$511$	9.97484	0.441261
$512$	−21.7068	−0.959316
$513$	0	0
$514$	4.73596	0.208894
$515$	42.2923	1.86362
$516$	0	0
$517$	−12.6064	−0.554429
$518$	13.4203	0.589656
$519$	0	0
$520$	−32.8092	−1.43878
$521$	−35.5002	−1.55529	−0.777647	−	0.628701i	$-0.783587\pi$
$521$	−35.5002	−1.55529	−0.777647	+	0.628701i	$0.783587\pi$
$522$	0	0
$523$	−33.9882	−1.48620	−0.743100	−	0.669180i	$-0.766645\pi$
$523$	−33.9882	−1.48620	−0.743100	+	0.669180i	$0.766645\pi$
$524$	24.9571	1.09026
$525$	0	0
$526$	5.81773	0.253665
$527$	8.54880	0.372392
$528$	0	0
$529$	1.00000	0.0434783
$530$	22.7822	0.989596
$531$	0	0
$532$	12.9199	0.560148
$533$	−25.4664	−1.10307
$534$	0	0
$535$	−7.29795	−0.315518
$536$	−12.6496	−0.546378
$537$	0	0
$538$	−1.77734	−0.0766264
$539$	−0.461849	−0.0198933
$540$	0	0
$541$	−5.08429	−0.218591	−0.109295	−	0.994009i	$-0.534859\pi$
$541$	−5.08429	−0.218591	−0.109295	+	0.994009i	$0.534859\pi$
$542$	2.03056	0.0872198
$543$	0	0
$544$	14.0046	0.600441
$545$	61.4767	2.63337
$546$	0	0
$547$	10.5365	0.450508	0.225254	−	0.974300i	$-0.427679\pi$
$547$	10.5365	0.450508	0.225254	+	0.974300i	$0.427679\pi$
$548$	−24.4877	−1.04606
$549$	0	0
$550$	−4.57665	−0.195149
$551$	7.37602	0.314229
$552$	0	0
$553$	−13.8342	−0.588288
$554$	−5.16365	−0.219383
$555$	0	0
$556$	4.14970	0.175986
$557$	−4.63405	−0.196351	−0.0981755	−	0.995169i	$-0.531301\pi$
$557$	−4.63405	−0.196351	−0.0981755	+	0.995169i	$0.531301\pi$
$558$	0	0
$559$	31.9256	1.35031
$560$	22.0592	0.932173
$561$	0	0
$562$	7.91413	0.333837
$563$	−23.7172	−0.999560	−0.499780	−	0.866152i	$-0.666586\pi$
$563$	−23.7172	−0.999560	−0.499780	+	0.866152i	$0.666586\pi$
$564$	0	0
$565$	−54.3942	−2.28838
$566$	−2.28781	−0.0961637
$567$	0	0
$568$	21.2131	0.890081
$569$	24.5801	1.03045	0.515226	−	0.857054i	$-0.327708\pi$
$569$	24.5801	1.03045	0.515226	+	0.857054i	$0.327708\pi$
$570$	0	0
$571$	−39.4422	−1.65061	−0.825303	−	0.564690i	$-0.808996\pi$
$571$	−39.4422	−1.65061	−0.825303	+	0.564690i	$0.808996\pi$
$572$	−7.65003	−0.319864
$573$	0	0
$574$	−7.83303	−0.326944
$575$	−8.50740	−0.354783
$576$	0	0
$577$	−13.9585	−0.581101	−0.290550	−	0.956860i	$-0.593838\pi$
$577$	−13.9585	−0.581101	−0.290550	+	0.956860i	$0.593838\pi$
$578$	5.32474	0.221480
$579$	0	0
$580$	15.6990	0.651867
$581$	32.0879	1.33123
$582$	0	0
$583$	11.5228	0.477227
$584$	−7.78705	−0.322230
$585$	0	0
$586$	−7.06376	−0.291801
$587$	21.4624	0.885848	0.442924	−	0.896559i	$-0.353941\pi$
$587$	21.4624	0.885848	0.442924	+	0.896559i	$0.353941\pi$
$588$	0	0
$589$	−9.47542	−0.390428
$590$	2.53900	0.104529
$591$	0	0
$592$	22.9014	0.941243
$593$	44.3750	1.82226	0.911132	−	0.412114i	$-0.135209\pi$
$593$	44.3750	1.82226	0.911132	+	0.412114i	$0.135209\pi$
$594$	0	0
$595$	−25.0440	−1.02670
$596$	−26.6041	−1.08975
$597$	0	0
$598$	2.40583	0.0983818
$599$	37.1513	1.51796	0.758980	−	0.651114i	$-0.225698\pi$
$599$	37.1513	1.51796	0.758980	+	0.651114i	$0.225698\pi$
$600$	0	0
$601$	2.23068	0.0909911	0.0454956	−	0.998965i	$-0.485513\pi$
$601$	2.23068	0.0909911	0.0454956	+	0.998965i	$0.485513\pi$
$602$	9.81978	0.400224
$603$	0	0
$604$	21.5570	0.877140
$605$	−3.67524	−0.149420
$606$	0	0
$607$	37.9370	1.53981	0.769907	−	0.638156i	$-0.220303\pi$
$607$	37.9370	1.53981	0.769907	+	0.638156i	$0.220303\pi$
$608$	−15.5226	−0.629523
$609$	0	0
$610$	16.4997	0.668055
$611$	−56.3775	−2.28079
$612$	0	0
$613$	35.9975	1.45392	0.726962	−	0.686678i	$-0.240932\pi$
$613$	35.9975	1.45392	0.726962	+	0.686678i	$0.240932\pi$
$614$	4.93602	0.199201
$615$	0	0
$616$	−5.10413	−0.205651
$617$	−7.61360	−0.306512	−0.153256	−	0.988187i	$-0.548976\pi$
$617$	−7.61360	−0.306512	−0.153256	+	0.988187i	$0.548976\pi$
$618$	0	0
$619$	−11.3517	−0.456262	−0.228131	−	0.973630i	$-0.573261\pi$
$619$	−11.3517	−0.456262	−0.228131	+	0.973630i	$0.573261\pi$
$620$	−20.1674	−0.809941
$621$	0	0
$622$	13.7258	0.550353
$623$	37.5900	1.50601
$624$	0	0
$625$	4.83891	0.193556
$626$	5.98416	0.239175
$627$	0	0
$628$	20.8834	0.833340
$629$	−26.0001	−1.03669
$630$	0	0
$631$	−0.156214	−0.00621876	−0.00310938	−	0.999995i	$-0.500990\pi$
$631$	−0.156214	−0.00621876	−0.00310938	+	0.999995i	$0.500990\pi$
$632$	10.7999	0.429597
$633$	0	0
$634$	−1.78168	−0.0707595
$635$	42.3744	1.68158
$636$	0	0
$637$	−2.06545	−0.0818362
$638$	−1.34335	−0.0531838
$639$	0	0
$640$	−42.3201	−1.67285
$641$	0.747839	0.0295379	0.0147689	−	0.999891i	$-0.495299\pi$
$641$	0.747839	0.0295379	0.0147689	+	0.999891i	$0.495299\pi$
$642$	0	0
$643$	6.55554	0.258525	0.129263	−	0.991610i	$-0.458739\pi$
$643$	6.55554	0.258525	0.129263	+	0.991610i	$0.458739\pi$
$644$	−4.37397	−0.172358
$645$	0	0
$646$	4.23471	0.166612
$647$	−11.5431	−0.453806	−0.226903	−	0.973917i	$-0.572860\pi$
$647$	−11.5431	−0.453806	−0.226903	+	0.973917i	$0.572860\pi$
$648$	0	0
$649$	1.28418	0.0504085
$650$	−20.4674	−0.802797
$651$	0	0
$652$	−19.5559	−0.765867
$653$	−30.3196	−1.18650	−0.593249	−	0.805019i	$-0.702155\pi$
$653$	−30.3196	−1.18650	−0.593249	+	0.805019i	$0.702155\pi$
$654$	0	0
$655$	53.6207	2.09514
$656$	−13.3668	−0.521888
$657$	0	0
$658$	−17.3408	−0.676014
$659$	−32.7960	−1.27755	−0.638775	−	0.769394i	$-0.720558\pi$
$659$	−32.7960	−1.27755	−0.638775	+	0.769394i	$0.720558\pi$
$660$	0	0
$661$	13.9008	0.540678	0.270339	−	0.962765i	$-0.412864\pi$
$661$	13.9008	0.540678	0.270339	+	0.962765i	$0.412864\pi$
$662$	10.7858	0.419202
$663$	0	0
$664$	−25.0500	−0.972129
$665$	27.7585	1.07643
$666$	0	0
$667$	−2.49712	−0.0966888
$668$	−0.510165	−0.0197389
$669$	0	0
$670$	−12.5290	−0.484039
$671$	8.34528	0.322166
$672$	0	0
$673$	23.9001	0.921281	0.460640	−	0.887587i	$-0.347620\pi$
$673$	23.9001	0.921281	0.460640	+	0.887587i	$0.347620\pi$
$674$	5.32278	0.205026
$675$	0	0
$676$	−11.9742	−0.460546
$677$	−15.0523	−0.578508	−0.289254	−	0.957252i	$-0.593407\pi$
$677$	−15.0523	−0.578508	−0.289254	+	0.957252i	$0.593407\pi$
$678$	0	0
$679$	27.1926	1.04356
$680$	19.5511	0.749749
$681$	0	0
$682$	1.72570	0.0660805
$683$	−10.7552	−0.411536	−0.205768	−	0.978601i	$-0.565969\pi$
$683$	−10.7552	−0.411536	−0.205768	+	0.978601i	$0.565969\pi$
$684$	0	0
$685$	−52.6121	−2.01021
$686$	−10.2642	−0.391888
$687$	0	0
$688$	16.7572	0.638862
$689$	51.5317	1.96320
$690$	0	0
$691$	−2.78853	−0.106081	−0.0530404	−	0.998592i	$-0.516891\pi$
$691$	−2.78853	−0.106081	−0.0530404	+	0.998592i	$0.516891\pi$
$692$	3.73152	0.141851
$693$	0	0
$694$	11.0782	0.420521
$695$	8.91568	0.338191
$696$	0	0
$697$	15.1755	0.574812
$698$	−13.9598	−0.528388
$699$	0	0
$700$	37.2111	1.40645
$701$	40.2559	1.52045	0.760223	−	0.649663i	$-0.225090\pi$
$701$	40.2559	1.52045	0.760223	+	0.649663i	$0.225090\pi$
$702$	0	0
$703$	28.8183	1.08690
$704$	−1.86766	−0.0703901
$705$	0	0
$706$	2.86156	0.107696
$707$	29.6767	1.11611
$708$	0	0
$709$	−34.1628	−1.28301	−0.641505	−	0.767119i	$-0.721690\pi$
$709$	−34.1628	−1.28301	−0.641505	+	0.767119i	$0.721690\pi$
$710$	21.0110	0.788527
$711$	0	0
$712$	−29.3453	−1.09976
$713$	3.20786	0.120135
$714$	0	0
$715$	−16.4362	−0.614678
$716$	−12.3271	−0.460687
$717$	0	0
$718$	3.98271	0.148633
$719$	11.3401	0.422915	0.211458	−	0.977387i	$-0.432179\pi$
$719$	11.3401	0.422915	0.211458	+	0.977387i	$0.432179\pi$
$720$	0	0
$721$	29.4241	1.09581
$722$	5.52753	0.205713
$723$	0	0
$724$	−41.7202	−1.55052
$725$	21.2440	0.788982
$726$	0	0
$727$	12.5470	0.465342	0.232671	−	0.972555i	$-0.425253\pi$
$727$	12.5470	0.465342	0.232671	+	0.972555i	$0.425253\pi$
$728$	−22.8264	−0.846001
$729$	0	0
$730$	−7.71285	−0.285465
$731$	−19.0245	−0.703648
$732$	0	0
$733$	14.2724	0.527164	0.263582	−	0.964637i	$-0.415096\pi$
$733$	14.2724	0.527164	0.263582	+	0.964637i	$0.415096\pi$
$734$	−12.0951	−0.446437
$735$	0	0
$736$	5.25509	0.193705
$737$	−6.33697	−0.233425
$738$	0	0
$739$	−22.1646	−0.815339	−0.407669	−	0.913130i	$-0.633658\pi$
$739$	−22.1646	−0.815339	−0.407669	+	0.913130i	$0.633658\pi$
$740$	61.3366	2.25478
$741$	0	0
$742$	15.8503	0.581883
$743$	−10.9678	−0.402369	−0.201185	−	0.979553i	$-0.564479\pi$
$743$	−10.9678	−0.402369	−0.201185	+	0.979553i	$0.564479\pi$
$744$	0	0
$745$	−57.1593	−2.09415
$746$	15.1454	0.554511
$747$	0	0
$748$	4.55867	0.166681
$749$	−5.07741	−0.185524
$750$	0	0
$751$	7.44519	0.271679	0.135839	−	0.990731i	$-0.456627\pi$
$751$	7.44519	0.271679	0.135839	+	0.990731i	$0.456627\pi$
$752$	−29.5916	−1.07909
$753$	0	0
$754$	−6.00765	−0.218786
$755$	46.3154	1.68559
$756$	0	0
$757$	10.5354	0.382915	0.191458	−	0.981501i	$-0.438679\pi$
$757$	10.5354	0.382915	0.191458	+	0.981501i	$0.438679\pi$
$758$	−13.5163	−0.490935
$759$	0	0
$760$	−21.6702	−0.786062
$761$	28.4307	1.03061	0.515306	−	0.857006i	$-0.327678\pi$
$761$	28.4307	1.03061	0.515306	+	0.857006i	$0.327678\pi$
$762$	0	0
$763$	42.7713	1.54842
$764$	−8.30614	−0.300506
$765$	0	0
$766$	6.62281	0.239292
$767$	5.74303	0.207369
$768$	0	0
$769$	28.7684	1.03742	0.518708	−	0.854952i	$-0.326413\pi$
$769$	28.7684	1.03742	0.518708	+	0.854952i	$0.326413\pi$
$770$	−5.05549	−0.182187
$771$	0	0
$772$	44.5925	1.60492
$773$	−1.94737	−0.0700420	−0.0350210	−	0.999387i	$-0.511150\pi$
$773$	−1.94737	−0.0700420	−0.0350210	+	0.999387i	$0.511150\pi$
$774$	0	0
$775$	−27.2906	−0.980306
$776$	−21.2284	−0.762056
$777$	0	0
$778$	−8.29635	−0.297439
$779$	−16.8204	−0.602652
$780$	0	0
$781$	10.6270	0.380263
$782$	−1.43364	−0.0512669
$783$	0	0
$784$	−1.08412	−0.0387186
$785$	44.8683	1.60142
$786$	0	0
$787$	−34.7354	−1.23818	−0.619091	−	0.785319i	$-0.712499\pi$
$787$	−34.7354	−1.23818	−0.619091	+	0.785319i	$0.712499\pi$
$788$	42.1052	1.49994
$789$	0	0
$790$	10.6970	0.380582
$791$	−37.8438	−1.34557
$792$	0	0
$793$	37.3212	1.32532
$794$	9.58556	0.340179
$795$	0	0
$796$	15.8114	0.560421
$797$	24.5987	0.871330	0.435665	−	0.900109i	$-0.356513\pi$
$797$	24.5987	0.871330	0.435665	+	0.900109i	$0.356513\pi$
$798$	0	0
$799$	33.5955	1.18852
$800$	−44.7072	−1.58064
$801$	0	0
$802$	16.1595	0.570612
$803$	−3.90102	−0.137664
$804$	0	0
$805$	−9.39752	−0.331219
$806$	7.71757	0.271840
$807$	0	0
$808$	−23.1677	−0.815037
$809$	23.0270	0.809586	0.404793	−	0.914408i	$-0.367344\pi$
$809$	23.0270	0.809586	0.404793	+	0.914408i	$0.367344\pi$
$810$	0	0
$811$	7.14992	0.251067	0.125534	−	0.992089i	$-0.459936\pi$
$811$	7.14992	0.251067	0.125534	+	0.992089i	$0.459936\pi$
$812$	10.9223	0.383298
$813$	0	0
$814$	−5.24851	−0.183960
$815$	−42.0160	−1.47176
$816$	0	0
$817$	21.0866	0.737728
$818$	13.3996	0.468506
$819$	0	0
$820$	−35.8003	−1.25020
$821$	−0.461118	−0.0160931	−0.00804656	−	0.999968i	$-0.502561\pi$
$821$	−0.461118	−0.0160931	−0.00804656	+	0.999968i	$0.502561\pi$
$822$	0	0
$823$	−41.5177	−1.44722	−0.723608	−	0.690211i	$-0.757518\pi$
$823$	−41.5177	−1.44722	−0.723608	+	0.690211i	$0.757518\pi$
$824$	−22.9705	−0.800214
$825$	0	0
$826$	1.76646	0.0614630
$827$	−30.3002	−1.05364	−0.526821	−	0.849976i	$-0.676616\pi$
$827$	−30.3002	−1.05364	−0.526821	+	0.849976i	$0.676616\pi$
$828$	0	0
$829$	6.24929	0.217047	0.108523	−	0.994094i	$-0.465388\pi$
$829$	6.24929	0.217047	0.108523	+	0.994094i	$0.465388\pi$
$830$	−24.8113	−0.861213
$831$	0	0
$832$	−8.35244	−0.289569
$833$	1.23081	0.0426450
$834$	0	0
$835$	−1.09610	−0.0379319
$836$	−5.05279	−0.174754
$837$	0	0
$838$	−9.71014	−0.335431
$839$	6.24066	0.215452	0.107726	−	0.994181i	$-0.465643\pi$
$839$	6.24066	0.215452	0.107726	+	0.994181i	$0.465643\pi$
$840$	0	0
$841$	−22.7644	−0.784979
$842$	0.768306	0.0264776
$843$	0	0
$844$	−36.9801	−1.27291
$845$	−25.7267	−0.885025
$846$	0	0
$847$	−2.55698	−0.0878589
$848$	27.0481	0.928835
$849$	0	0
$850$	12.1966	0.418339
$851$	−9.75632	−0.334442
$852$	0	0
$853$	−52.8477	−1.80947	−0.904735	−	0.425974i	$-0.859931\pi$
$853$	−52.8477	−1.80947	−0.904735	+	0.425974i	$0.859931\pi$
$854$	11.4794	0.392816
$855$	0	0
$856$	3.96377	0.135479
$857$	−39.4372	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$857$	−39.4372	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$858$	0	0
$859$	−31.4955	−1.07461	−0.537307	−	0.843387i	$-0.680558\pi$
$859$	−31.4955	−1.07461	−0.537307	+	0.843387i	$0.680558\pi$
$860$	44.8805	1.53041
$861$	0	0
$862$	14.0772	0.479470
$863$	−28.0451	−0.954666	−0.477333	−	0.878722i	$-0.658397\pi$
$863$	−28.0451	−0.954666	−0.477333	+	0.878722i	$0.658397\pi$
$864$	0	0
$865$	8.01722	0.272594
$866$	15.7110	0.533880
$867$	0	0
$868$	−14.0311	−0.476246
$869$	5.41035	0.183533
$870$	0	0
$871$	−28.3398	−0.960257
$872$	−33.3902	−1.13073
$873$	0	0
$874$	1.58903	0.0537499
$875$	32.9609	1.11428
$876$	0	0
$877$	−55.3045	−1.86750	−0.933750	−	0.357926i	$-0.883484\pi$
$877$	−55.3045	−1.86750	−0.933750	+	0.357926i	$0.883484\pi$
$878$	3.93094	0.132663
$879$	0	0
$880$	−8.62706	−0.290818
$881$	13.8151	0.465443	0.232721	−	0.972543i	$-0.425237\pi$
$881$	13.8151	0.465443	0.232721	+	0.972543i	$0.425237\pi$
$882$	0	0
$883$	−24.5296	−0.825487	−0.412743	−	0.910847i	$-0.635429\pi$
$883$	−24.5296	−0.825487	−0.412743	+	0.910847i	$0.635429\pi$
$884$	20.3870	0.685689
$885$	0	0
$886$	−0.340477	−0.0114386
$887$	45.2629	1.51978	0.759890	−	0.650052i	$-0.225253\pi$
$887$	45.2629	1.51978	0.759890	+	0.650052i	$0.225253\pi$
$888$	0	0
$889$	29.4812	0.988767
$890$	−29.0657	−0.974284
$891$	0	0
$892$	−11.1825	−0.374419
$893$	−37.2370	−1.24609
$894$	0	0
$895$	−26.4850	−0.885297
$896$	−29.4434	−0.983635
$897$	0	0
$898$	8.79148	0.293376
$899$	−8.01040	−0.267162
$900$	0	0
$901$	−30.7079	−1.02303
$902$	3.06339	0.102000
$903$	0	0
$904$	29.5435	0.982601
$905$	−89.6363	−2.97961
$906$	0	0
$907$	−45.6061	−1.51432	−0.757162	−	0.653227i	$-0.773415\pi$
$907$	−45.6061	−1.51432	−0.757162	+	0.653227i	$0.773415\pi$
$908$	−17.8958	−0.593892
$909$	0	0
$910$	−22.6089	−0.749476
$911$	26.1972	0.867950	0.433975	−	0.900925i	$-0.357111\pi$
$911$	26.1972	0.867950	0.433975	+	0.900925i	$0.357111\pi$
$912$	0	0
$913$	−12.5491	−0.415315
$914$	10.0914	0.333795
$915$	0	0
$916$	−9.97966	−0.329737
$917$	37.3056	1.23194
$918$	0	0
$919$	45.3241	1.49510	0.747552	−	0.664203i	$-0.231229\pi$
$919$	45.3241	1.49510	0.747552	+	0.664203i	$0.231229\pi$
$920$	7.33635	0.241872
$921$	0	0
$922$	−1.47984	−0.0487360
$923$	47.5253	1.56431
$924$	0	0
$925$	83.0009	2.72905
$926$	22.0590	0.724903
$927$	0	0
$928$	−13.1226	−0.430770
$929$	−53.7418	−1.76321	−0.881606	−	0.471985i	$-0.843537\pi$
$929$	−53.7418	−1.76321	−0.881606	+	0.471985i	$0.843537\pi$
$930$	0	0
$931$	−1.36422	−0.0447104
$932$	−20.6512	−0.676451
$933$	0	0
$934$	−4.21339	−0.137866
$935$	9.79436	0.320310
$936$	0	0
$937$	−0.902045	−0.0294685	−0.0147343	−	0.999891i	$-0.504690\pi$
$937$	−0.902045	−0.0294685	−0.0147343	+	0.999891i	$0.504690\pi$
$938$	−8.71685	−0.284615
$939$	0	0
$940$	−79.2547	−2.58500
$941$	25.5981	0.834474	0.417237	−	0.908798i	$-0.362999\pi$
$941$	25.5981	0.834474	0.417237	+	0.908798i	$0.362999\pi$
$942$	0	0
$943$	5.69445	0.185437
$944$	3.01441	0.0981108
$945$	0	0
$946$	−3.84038	−0.124862
$947$	37.7889	1.22797	0.613987	−	0.789316i	$-0.289565\pi$
$947$	37.7889	1.22797	0.613987	+	0.789316i	$0.289565\pi$
$948$	0	0
$949$	−17.4459	−0.566318
$950$	−13.5186	−0.438600
$951$	0	0
$952$	13.6023	0.440852
$953$	−61.0266	−1.97684	−0.988422	−	0.151730i	$-0.951516\pi$
$953$	−61.0266	−1.97684	−0.988422	+	0.151730i	$0.951516\pi$
$954$	0	0
$955$	−17.8458	−0.577478
$956$	−4.93435	−0.159588
$957$	0	0
$958$	5.06277	0.163571
$959$	−36.6039	−1.18200
$960$	0	0
$961$	−20.7096	−0.668053
$962$	−23.4721	−0.756769
$963$	0	0
$964$	−39.9763	−1.28755
$965$	95.8075	3.08415
$966$	0	0
$967$	30.0467	0.966238	0.483119	−	0.875555i	$-0.339504\pi$
$967$	30.0467	0.966238	0.483119	+	0.875555i	$0.339504\pi$
$968$	1.99615	0.0641588
$969$	0	0
$970$	−21.0262	−0.675109
$971$	−30.6106	−0.982340	−0.491170	−	0.871064i	$-0.663431\pi$
$971$	−30.6106	−0.982340	−0.491170	+	0.871064i	$0.663431\pi$
$972$	0	0
$973$	6.20292	0.198856
$974$	14.7082	0.471280
$975$	0	0
$976$	19.5892	0.627036
$977$	30.5807	0.978364	0.489182	−	0.872182i	$-0.337295\pi$
$977$	30.5807	0.978364	0.489182	+	0.872182i	$0.337295\pi$
$978$	0	0
$979$	−14.7009	−0.469843
$980$	−2.90358	−0.0927516
$981$	0	0
$982$	4.94210	0.157709
$983$	30.8962	0.985435	0.492717	−	0.870189i	$-0.336004\pi$
$983$	30.8962	0.985435	0.492717	+	0.870189i	$0.336004\pi$
$984$	0	0
$985$	90.4636	2.88241
$986$	3.57997	0.114009
$987$	0	0
$988$	−22.5968	−0.718899
$989$	−7.13878	−0.227000
$990$	0	0
$991$	8.82232	0.280250	0.140125	−	0.990134i	$-0.455250\pi$
$991$	8.82232	0.280250	0.140125	+	0.990134i	$0.455250\pi$
$992$	16.8576	0.535229
$993$	0	0
$994$	14.6180	0.463654
$995$	33.9710	1.07695
$996$	0	0
$997$	52.2841	1.65585	0.827927	−	0.560836i	$-0.189520\pi$
$997$	52.2841	1.65585	0.827927	+	0.560836i	$0.189520\pi$
$998$	16.9003	0.534969
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2277.2.a.n.1.3		6
3.2	odd	2		759.2.a.h.1.4	✓	6
33.32	even	2		8349.2.a.n.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
759.2.a.h.1.4	✓	6	3.2	odd	2
2277.2.a.n.1.3		6	1.1	even	1	trivial
8349.2.a.n.1.3		6	33.32	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:	\( N \)	\(=\)	\( 2277 = 3^{2} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2277.a (trivial)

\(f(q)\)	\(=\)	\(q-0.537960 q^{2} -1.71060 q^{4} -3.67524 q^{5} -2.55698 q^{7} +1.99615 q^{8} +1.97713 q^{10} +1.00000 q^{11} +4.47214 q^{13} +1.37555 q^{14} +2.34735 q^{16} -2.66496 q^{17} +2.95381 q^{19} +6.28686 q^{20} -0.537960 q^{22} -1.00000 q^{23} +8.50740 q^{25} -2.40583 q^{26} +4.37397 q^{28} +2.49712 q^{29} -3.20786 q^{31} -5.25509 q^{32} +1.43364 q^{34} +9.39752 q^{35} +9.75632 q^{37} -1.58903 q^{38} -7.33635 q^{40} -5.69445 q^{41} +7.13878 q^{43} -1.71060 q^{44} +0.537960 q^{46} -12.6064 q^{47} -0.461849 q^{49} -4.57665 q^{50} -7.65003 q^{52} +11.5228 q^{53} -3.67524 q^{55} -5.10413 q^{56} -1.34335 q^{58} +1.28418 q^{59} +8.34528 q^{61} +1.72570 q^{62} -1.86766 q^{64} -16.4362 q^{65} -6.33697 q^{67} +4.55867 q^{68} -5.05549 q^{70} +10.6270 q^{71} -3.90102 q^{73} -5.24851 q^{74} -5.05279 q^{76} -2.55698 q^{77} +5.41035 q^{79} -8.62706 q^{80} +3.06339 q^{82} -12.5491 q^{83} +9.79436 q^{85} -3.84038 q^{86} +1.99615 q^{88} -14.7009 q^{89} -11.4352 q^{91} +1.71060 q^{92} +6.78174 q^{94} -10.8560 q^{95} -10.6347 q^{97} +0.248457 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 4 q^{4} - 6 q^{5} - 6 q^{8} + 8 q^{10} + 6 q^{11} - 12 q^{14} - 10 q^{17} - 2 q^{19} - 12 q^{20} - 2 q^{22} - 6 q^{23} + 10 q^{25} + 18 q^{28} - 16 q^{29} - 12 q^{31} - 4 q^{32} - 2 q^{34}+ \cdots - 16 q^{98}+O(q^{100}) \) 6 * q - 2 * q^2 + 4 * q^4 - 6 * q^5 - 6 * q^8 + 8 * q^10 + 6 * q^11 - 12 * q^14 - 10 * q^17 - 2 * q^19 - 12 * q^20 - 2 * q^22 - 6 * q^23 + 10 * q^25 + 18 * q^28 - 16 * q^29 - 12 * q^31 - 4 * q^32 - 2 * q^34 - 14 * q^35 + 12 * q^37 - 6 * q^38 - 4 * q^40 - 22 * q^41 + 4 * q^44 + 2 * q^46 - 8 * q^47 + 14 * q^49 - 28 * q^50 + 20 * q^52 - 10 * q^53 - 6 * q^55 - 22 * q^56 - 8 * q^58 - 12 * q^59 + 24 * q^61 + 34 * q^62 - 26 * q^64 - 20 * q^65 - 18 * q^67 + 2 * q^68 + 46 * q^70 - 24 * q^71 - 4 * q^73 - 36 * q^74 - 30 * q^76 + 16 * q^79 + 20 * q^80 - 18 * q^82 - 16 * q^83 - 16 * q^85 + 24 * q^86 - 6 * q^88 - 2 * q^89 - 40 * q^91 - 4 * q^92 + 12 * q^94 + 28 * q^95 - 12 * q^97 - 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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