LMFDB - Embedded newform 1449.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$11.5703232529$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2147108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 $ x^5 - 2x^4 - 9x^3 + 17x^2 + 16x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.50216$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50216 q^{2} +0.256481 q^{4} +3.82405 q^{5} +1.00000 q^{7} -2.61904 q^{8} +O(q^{10})$	\(q+1.50216 q^{2} +0.256481 q^{4} +3.82405 q^{5} +1.00000 q^{7} -2.61904 q^{8} +5.74433 q^{10} +0.542019 q^{11} -1.21662 q^{13} +1.50216 q^{14} -4.44718 q^{16} +3.66299 q^{17} +5.00432 q^{19} +0.980794 q^{20} +0.814198 q^{22} +1.00000 q^{23} +9.62336 q^{25} -1.82756 q^{26} +0.256481 q^{28} -3.72840 q^{29} +9.04067 q^{31} -1.44228 q^{32} +5.50239 q^{34} +3.82405 q^{35} -9.08566 q^{37} +7.51728 q^{38} -10.0154 q^{40} +4.86472 q^{41} +3.64810 q^{43} +0.139017 q^{44} +1.50216 q^{46} -10.6393 q^{47} +1.00000 q^{49} +14.4558 q^{50} -0.312039 q^{52} -7.48704 q^{53} +2.07271 q^{55} -2.61904 q^{56} -5.60065 q^{58} -5.18027 q^{59} +3.26161 q^{61} +13.5805 q^{62} +6.72782 q^{64} -4.65242 q^{65} -12.0377 q^{67} +0.939485 q^{68} +5.74433 q^{70} +5.29796 q^{71} +8.19070 q^{73} -13.6481 q^{74} +1.28351 q^{76} +0.542019 q^{77} +10.0291 q^{79} -17.0062 q^{80} +7.30758 q^{82} -7.64810 q^{83} +14.0075 q^{85} +5.48003 q^{86} -1.41957 q^{88} +4.74703 q^{89} -1.21662 q^{91} +0.256481 q^{92} -15.9819 q^{94} +19.1368 q^{95} -10.9894 q^{97} +1.50216 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8	$ 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8} - 8 q^{10} + 4 q^{11} - 6 q^{13} - 2 q^{14} + 10 q^{16} + 12 q^{17} + 6 q^{19} + 14 q^{22} + 5 q^{23} + 19 q^{25} - q^{26} + 12 q^{28} + 4 q^{29} + 30 q^{31} - 8 q^{32} + 6 q^{34} + 4 q^{35} + 4 q^{37} + 40 q^{38} - 50 q^{40} - 6 q^{41} - 12 q^{43} + 26 q^{44} - 2 q^{46} - 10 q^{47} + 5 q^{49} + 2 q^{50} - 21 q^{52} - 16 q^{53} + 18 q^{55} - 3 q^{56} + 13 q^{58} - 22 q^{59} - 18 q^{61} - 15 q^{62} + 25 q^{64} + 26 q^{65} - 2 q^{67} - 12 q^{68} - 8 q^{70} - 4 q^{71} - 2 q^{73} - 38 q^{74} + 10 q^{76} + 4 q^{77} + 30 q^{79} + 10 q^{80} - 7 q^{82} - 8 q^{83} - 12 q^{85} - 8 q^{86} + 4 q^{88} + 20 q^{89} - 6 q^{91} + 12 q^{92} - 25 q^{94} - 8 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8 - 8 * q^10 + 4 * q^11 - 6 * q^13 - 2 * q^14 + 10 * q^16 + 12 * q^17 + 6 * q^19 + 14 * q^22 + 5 * q^23 + 19 * q^25 - q^26 + 12 * q^28 + 4 * q^29 + 30 * q^31 - 8 * q^32 + 6 * q^34 + 4 * q^35 + 4 * q^37 + 40 * q^38 - 50 * q^40 - 6 * q^41 - 12 * q^43 + 26 * q^44 - 2 * q^46 - 10 * q^47 + 5 * q^49 + 2 * q^50 - 21 * q^52 - 16 * q^53 + 18 * q^55 - 3 * q^56 + 13 * q^58 - 22 * q^59 - 18 * q^61 - 15 * q^62 + 25 * q^64 + 26 * q^65 - 2 * q^67 - 12 * q^68 - 8 * q^70 - 4 * q^71 - 2 * q^73 - 38 * q^74 + 10 * q^76 + 4 * q^77 + 30 * q^79 + 10 * q^80 - 7 * q^82 - 8 * q^83 - 12 * q^85 - 8 * q^86 + 4 * q^88 + 20 * q^89 - 6 * q^91 + 12 * q^92 - 25 * q^94 - 8 * q^95 - 12 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.50216	1.06219	0.531093	−	0.847313i	$-0.321781\pi$
$2$	1.50216	1.06219	0.531093	+	0.847313i	$0.321781\pi$
$3$	0	0
$3$	0	0
$4$	0.256481	0.128240
$5$	3.82405	1.71017	0.855084	−	0.518490i	$-0.173506\pi$
$5$	3.82405	1.71017	0.855084	+	0.518490i	$0.173506\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.61904	−0.925971
$9$	0	0
$10$	5.74433	1.81652
$11$	0.542019	0.163425	0.0817124	−	0.996656i	$-0.473961\pi$
$11$	0.542019	0.163425	0.0817124	+	0.996656i	$0.473961\pi$
$12$	0	0
$13$	−1.21662	−0.337430	−0.168715	−	0.985665i	$-0.553962\pi$
$13$	−1.21662	−0.337430	−0.168715	+	0.985665i	$0.553962\pi$
$14$	1.50216	0.401469
$15$	0	0
$16$	−4.44718	−1.11179
$17$	3.66299	0.888405	0.444203	−	0.895926i	$-0.353487\pi$
$17$	3.66299	0.888405	0.444203	+	0.895926i	$0.353487\pi$
$18$	0	0
$19$	5.00432	1.14807	0.574035	−	0.818831i	$-0.305378\pi$
$19$	5.00432	1.14807	0.574035	+	0.818831i	$0.305378\pi$
$20$	0.980794	0.219312
$21$	0	0
$22$	0.814198	0.173588
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	9.62336	1.92467
$26$	−1.82756	−0.358413
$27$	0	0
$28$	0.256481	0.0484703
$29$	−3.72840	−0.692346	−0.346173	−	0.938171i	$-0.612519\pi$
$29$	−3.72840	−0.692346	−0.346173	+	0.938171i	$0.612519\pi$
$30$	0	0
$31$	9.04067	1.62375	0.811876	−	0.583830i	$-0.198446\pi$
$31$	9.04067	1.62375	0.811876	+	0.583830i	$0.198446\pi$
$32$	−1.44228	−0.254962
$33$	0	0
$34$	5.50239	0.943652
$35$	3.82405	0.646382
$36$	0	0
$37$	−9.08566	−1.49367	−0.746837	−	0.665008i	$-0.768428\pi$
$37$	−9.08566	−1.49367	−0.746837	+	0.665008i	$0.768428\pi$
$38$	7.51728	1.21946
$39$	0	0
$40$	−10.0154	−1.58357
$41$	4.86472	0.759742	0.379871	−	0.925040i	$-0.375968\pi$
$41$	4.86472	0.759742	0.379871	+	0.925040i	$0.375968\pi$
$42$	0	0
$43$	3.64810	0.556330	0.278165	−	0.960533i	$-0.410274\pi$
$43$	3.64810	0.556330	0.278165	+	0.960533i	$0.410274\pi$
$44$	0.139017	0.0209576
$45$	0	0
$46$	1.50216	0.221481
$47$	−10.6393	−1.55190	−0.775950	−	0.630794i	$-0.782729\pi$
$47$	−10.6393	−1.55190	−0.775950	+	0.630794i	$0.782729\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	14.4558	2.04436
$51$	0	0
$52$	−0.312039	−0.0432721
$53$	−7.48704	−1.02842	−0.514212	−	0.857663i	$-0.671916\pi$
$53$	−7.48704	−1.02842	−0.514212	+	0.857663i	$0.671916\pi$
$54$	0	0
$55$	2.07271	0.279484
$56$	−2.61904	−0.349984
$57$	0	0
$58$	−5.60065	−0.735401
$59$	−5.18027	−0.674413	−0.337207	−	0.941431i	$-0.609482\pi$
$59$	−5.18027	−0.674413	−0.337207	+	0.941431i	$0.609482\pi$
$60$	0	0
$61$	3.26161	0.417606	0.208803	−	0.977958i	$-0.433043\pi$
$61$	3.26161	0.417606	0.208803	+	0.977958i	$0.433043\pi$
$62$	13.5805	1.72473
$63$	0	0
$64$	6.72782	0.840978
$65$	−4.65242	−0.577061
$66$	0	0
$67$	−12.0377	−1.47064	−0.735319	−	0.677721i	$-0.762968\pi$
$67$	−12.0377	−1.47064	−0.735319	+	0.677721i	$0.762968\pi$
$68$	0.939485	0.113929
$69$	0	0
$70$	5.74433	0.686579
$71$	5.29796	0.628752	0.314376	−	0.949298i	$-0.398205\pi$
$71$	5.29796	0.628752	0.314376	+	0.949298i	$0.398205\pi$
$72$	0	0
$73$	8.19070	0.958649	0.479324	−	0.877638i	$-0.340882\pi$
$73$	8.19070	0.958649	0.479324	+	0.877638i	$0.340882\pi$
$74$	−13.6481	−1.58656
$75$	0	0
$76$	1.28351	0.147229
$77$	0.542019	0.0617688
$78$	0	0
$79$	10.0291	1.12836	0.564179	−	0.825653i	$-0.309193\pi$
$79$	10.0291	1.12836	0.564179	+	0.825653i	$0.309193\pi$
$80$	−17.0062	−1.90135
$81$	0	0
$82$	7.30758	0.806987
$83$	−7.64810	−0.839488	−0.419744	−	0.907643i	$-0.637880\pi$
$83$	−7.64810	−0.839488	−0.419744	+	0.907643i	$0.637880\pi$
$84$	0	0
$85$	14.0075	1.51932
$86$	5.48003	0.590926
$87$	0	0
$88$	−1.41957	−0.151327
$89$	4.74703	0.503184	0.251592	−	0.967833i	$-0.419046\pi$
$89$	4.74703	0.503184	0.251592	+	0.967833i	$0.419046\pi$
$90$	0	0
$91$	−1.21662	−0.127536
$92$	0.256481	0.0267399
$93$	0	0
$94$	−15.9819	−1.64841
$95$	19.1368	1.96339
$96$	0	0
$97$	−10.9894	−1.11581	−0.557904	−	0.829906i	$-0.688394\pi$
$97$	−10.9894	−1.11581	−0.557904	+	0.829906i	$0.688394\pi$
$98$	1.50216	0.151741
$99$	0	0
$100$	2.46820	0.246820
$101$	−3.59862	−0.358076	−0.179038	−	0.983842i	$-0.557299\pi$
$101$	−3.59862	−0.358076	−0.179038	+	0.983842i	$0.557299\pi$
$102$	0	0
$103$	9.48704	0.934786	0.467393	−	0.884050i	$-0.345193\pi$
$103$	9.48704	0.934786	0.467393	+	0.884050i	$0.345193\pi$
$104$	3.18638	0.312450
$105$	0	0
$106$	−11.2467	−1.09238
$107$	−6.60294	−0.638330	−0.319165	−	0.947699i	$-0.603402\pi$
$107$	−6.60294	−0.638330	−0.319165	+	0.947699i	$0.603402\pi$
$108$	0	0
$109$	1.10726	0.106057	0.0530283	−	0.998593i	$-0.483113\pi$
$109$	1.10726	0.106057	0.0530283	+	0.998593i	$0.483113\pi$
$110$	3.11353	0.296864
$111$	0	0
$112$	−4.44718	−0.420219
$113$	−0.433241	−0.0407559	−0.0203779	−	0.999792i	$-0.506487\pi$
$113$	−0.433241	−0.0407559	−0.0203779	+	0.999792i	$0.506487\pi$
$114$	0	0
$115$	3.82405	0.356595
$116$	−0.956262	−0.0887867
$117$	0	0
$118$	−7.78158	−0.716353
$119$	3.66299	0.335786
$120$	0	0
$121$	−10.7062	−0.973292
$122$	4.89945	0.443576
$123$	0	0
$124$	2.31876	0.208230
$125$	17.6800	1.58134
$126$	0	0
$127$	8.84149	0.784556	0.392278	−	0.919847i	$-0.371687\pi$
$127$	8.84149	0.784556	0.392278	+	0.919847i	$0.371687\pi$
$128$	12.9908	1.14824
$129$	0	0
$130$	−6.98867	−0.612947
$131$	2.54931	0.222735	0.111367	−	0.993779i	$-0.464477\pi$
$131$	2.54931	0.222735	0.111367	+	0.993779i	$0.464477\pi$
$132$	0	0
$133$	5.00432	0.433929
$134$	−18.0825	−1.56209
$135$	0	0
$136$	−9.59352	−0.822638
$137$	−13.6265	−1.16419	−0.582095	−	0.813121i	$-0.697767\pi$
$137$	−13.6265	−1.16419	−0.582095	+	0.813121i	$0.697767\pi$
$138$	0	0
$139$	−18.0466	−1.53069	−0.765347	−	0.643618i	$-0.777432\pi$
$139$	−18.0466	−1.53069	−0.765347	+	0.643618i	$0.777432\pi$
$140$	0.980794	0.0828923
$141$	0	0
$142$	7.95838	0.667852
$143$	−0.659431	−0.0551444
$144$	0	0
$145$	−14.2576	−1.18403
$146$	12.3037	1.01826
$147$	0	0
$148$	−2.33029	−0.191549
$149$	−13.8178	−1.13200	−0.565999	−	0.824406i	$-0.691509\pi$
$149$	−13.8178	−1.13200	−0.565999	+	0.824406i	$0.691509\pi$
$150$	0	0
$151$	16.1412	1.31355	0.656777	−	0.754085i	$-0.271919\pi$
$151$	16.1412	1.31355	0.656777	+	0.754085i	$0.271919\pi$
$152$	−13.1065	−1.06308
$153$	0	0
$154$	0.814198	0.0656100
$155$	34.5720	2.77689
$156$	0	0
$157$	−15.3111	−1.22196	−0.610979	−	0.791647i	$-0.709224\pi$
$157$	−15.3111	−1.22196	−0.610979	+	0.791647i	$0.709224\pi$
$158$	15.0652	1.19853
$159$	0	0
$160$	−5.51536	−0.436028
$161$	1.00000	0.0788110
$162$	0	0
$163$	−3.94876	−0.309291	−0.154645	−	0.987970i	$-0.549423\pi$
$163$	−3.94876	−0.309291	−0.154645	+	0.987970i	$0.549423\pi$
$164$	1.24771	0.0974295
$165$	0	0
$166$	−11.4887	−0.891693
$167$	−1.87752	−0.145287	−0.0726433	−	0.997358i	$-0.523143\pi$
$167$	−1.87752	−0.145287	−0.0726433	+	0.997358i	$0.523143\pi$
$168$	0	0
$169$	−11.5198	−0.886141
$170$	21.0414	1.61380
$171$	0	0
$172$	0.935667	0.0713439
$173$	−10.9246	−0.830582	−0.415291	−	0.909689i	$-0.636320\pi$
$173$	−10.9246	−0.830582	−0.415291	+	0.909689i	$0.636320\pi$
$174$	0	0
$175$	9.62336	0.727458
$176$	−2.41045	−0.181695
$177$	0	0
$178$	7.13079	0.534475
$179$	−4.74253	−0.354474	−0.177237	−	0.984168i	$-0.556716\pi$
$179$	−4.74253	−0.354474	−0.177237	+	0.984168i	$0.556716\pi$
$180$	0	0
$181$	2.82837	0.210231	0.105115	−	0.994460i	$-0.466479\pi$
$181$	2.82837	0.210231	0.105115	+	0.994460i	$0.466479\pi$
$182$	−1.82756	−0.135468
$183$	0	0
$184$	−2.61904	−0.193078
$185$	−34.7440	−2.55443
$186$	0	0
$187$	1.98541	0.145187
$188$	−2.72877	−0.199016
$189$	0	0
$190$	28.7464	2.08549
$191$	3.56838	0.258199	0.129099	−	0.991632i	$-0.458791\pi$
$191$	3.56838	0.258199	0.129099	+	0.991632i	$0.458791\pi$
$192$	0	0
$193$	20.9158	1.50555	0.752777	−	0.658276i	$-0.228714\pi$
$193$	20.9158	1.50555	0.752777	+	0.658276i	$0.228714\pi$
$194$	−16.5079	−1.18520
$195$	0	0
$196$	0.256481	0.0183200
$197$	−5.54576	−0.395119	−0.197559	−	0.980291i	$-0.563302\pi$
$197$	−5.54576	−0.395119	−0.197559	+	0.980291i	$0.563302\pi$
$198$	0	0
$199$	−7.73808	−0.548538	−0.274269	−	0.961653i	$-0.588436\pi$
$199$	−7.73808	−0.548538	−0.274269	+	0.961653i	$0.588436\pi$
$200$	−25.2040	−1.78219
$201$	0	0
$202$	−5.40570	−0.380344
$203$	−3.72840	−0.261682
$204$	0	0
$205$	18.6029	1.29929
$206$	14.2510	0.992917
$207$	0	0
$208$	5.41053	0.375153
$209$	2.71243	0.187623
$210$	0	0
$211$	9.43918	0.649820	0.324910	−	0.945745i	$-0.394666\pi$
$211$	9.43918	0.649820	0.324910	+	0.945745i	$0.394666\pi$
$212$	−1.92028	−0.131885
$213$	0	0
$214$	−9.91866	−0.678026
$215$	13.9505	0.951418
$216$	0	0
$217$	9.04067	0.613721
$218$	1.66329	0.112652
$219$	0	0
$220$	0.531609	0.0358411
$221$	−4.45647	−0.299774
$222$	0	0
$223$	11.1010	0.743379	0.371689	−	0.928357i	$-0.378779\pi$
$223$	11.1010	0.743379	0.371689	+	0.928357i	$0.378779\pi$
$224$	−1.44228	−0.0963665
$225$	0	0
$226$	−0.650797	−0.0432903
$227$	−20.9827	−1.39267	−0.696336	−	0.717716i	$-0.745188\pi$
$227$	−20.9827	−1.39267	−0.696336	+	0.717716i	$0.745188\pi$
$228$	0	0
$229$	4.63545	0.306319	0.153159	−	0.988201i	$-0.451055\pi$
$229$	4.63545	0.306319	0.153159	+	0.988201i	$0.451055\pi$
$230$	5.74433	0.378770
$231$	0	0
$232$	9.76484	0.641093
$233$	−16.1703	−1.05935	−0.529675	−	0.848201i	$-0.677686\pi$
$233$	−16.1703	−1.05935	−0.529675	+	0.848201i	$0.677686\pi$
$234$	0	0
$235$	−40.6852	−2.65401
$236$	−1.32864	−0.0864869
$237$	0	0
$238$	5.50239	0.356667
$239$	−1.26610	−0.0818973	−0.0409486	−	0.999161i	$-0.513038\pi$
$239$	−1.26610	−0.0818973	−0.0409486	+	0.999161i	$0.513038\pi$
$240$	0	0
$241$	−2.09623	−0.135030	−0.0675150	−	0.997718i	$-0.521507\pi$
$241$	−2.09623	−0.135030	−0.0675150	+	0.997718i	$0.521507\pi$
$242$	−16.0824	−1.03382
$243$	0	0
$244$	0.836539	0.0535539
$245$	3.82405	0.244310
$246$	0	0
$247$	−6.08835	−0.387393
$248$	−23.6779	−1.50355
$249$	0	0
$250$	26.5581	1.67968
$251$	−11.5365	−0.728179	−0.364089	−	0.931364i	$-0.618620\pi$
$251$	−11.5365	−0.728179	−0.364089	+	0.931364i	$0.618620\pi$
$252$	0	0
$253$	0.542019	0.0340764
$254$	13.2813	0.833345
$255$	0	0
$256$	6.05863	0.378664
$257$	−14.6546	−0.914131	−0.457066	−	0.889433i	$-0.651100\pi$
$257$	−14.6546	−0.914131	−0.457066	+	0.889433i	$0.651100\pi$
$258$	0	0
$259$	−9.08566	−0.564555
$260$	−1.19325	−0.0740025
$261$	0	0
$262$	3.82947	0.236586
$263$	22.7133	1.40056	0.700282	−	0.713866i	$-0.253058\pi$
$263$	22.7133	1.40056	0.700282	+	0.713866i	$0.253058\pi$
$264$	0	0
$265$	−28.6308	−1.75878
$266$	7.51728	0.460914
$267$	0	0
$268$	−3.08743	−0.188595
$269$	−25.0855	−1.52949	−0.764746	−	0.644332i	$-0.777135\pi$
$269$	−25.0855	−1.52949	−0.764746	+	0.644332i	$0.777135\pi$
$270$	0	0
$271$	−28.0232	−1.70229	−0.851145	−	0.524930i	$-0.824091\pi$
$271$	−28.0232	−1.70229	−0.851145	+	0.524930i	$0.824091\pi$
$272$	−16.2900	−0.987724
$273$	0	0
$274$	−20.4692	−1.23659
$275$	5.21604	0.314539
$276$	0	0
$277$	−16.2692	−0.977524	−0.488762	−	0.872417i	$-0.662551\pi$
$277$	−16.2692	−0.977524	−0.488762	+	0.872417i	$0.662551\pi$
$278$	−27.1089	−1.62588
$279$	0	0
$280$	−10.0154	−0.598532
$281$	13.4694	0.803518	0.401759	−	0.915745i	$-0.368399\pi$
$281$	13.4694	0.803518	0.401759	+	0.915745i	$0.368399\pi$
$282$	0	0
$283$	18.0516	1.07305	0.536527	−	0.843883i	$-0.319736\pi$
$283$	18.0516	1.07305	0.536527	+	0.843883i	$0.319736\pi$
$284$	1.35882	0.0806314
$285$	0	0
$286$	−0.990570	−0.0585736
$287$	4.86472	0.287155
$288$	0	0
$289$	−3.58251	−0.210736
$290$	−21.4172	−1.25766
$291$	0	0
$292$	2.10075	0.122937
$293$	−23.1521	−1.35256	−0.676280	−	0.736644i	$-0.736409\pi$
$293$	−23.1521	−1.35256	−0.676280	+	0.736644i	$0.736409\pi$
$294$	0	0
$295$	−19.8096	−1.15336
$296$	23.7957	1.38310
$297$	0	0
$298$	−20.7565	−1.20239
$299$	−1.21662	−0.0703590
$300$	0	0
$301$	3.64810	0.210273
$302$	24.2467	1.39524
$303$	0	0
$304$	−22.2551	−1.27642
$305$	12.4726	0.714176
$306$	0	0
$307$	−1.14301	−0.0652353	−0.0326176	−	0.999468i	$-0.510384\pi$
$307$	−1.14301	−0.0652353	−0.0326176	+	0.999468i	$0.510384\pi$
$308$	0.139017	0.00792124
$309$	0	0
$310$	51.9326	2.94957
$311$	19.3196	1.09551	0.547757	−	0.836638i	$-0.315482\pi$
$311$	19.3196	1.09551	0.547757	+	0.836638i	$0.315482\pi$
$312$	0	0
$313$	−17.4140	−0.984299	−0.492150	−	0.870511i	$-0.663789\pi$
$313$	−17.4140	−0.984299	−0.492150	+	0.870511i	$0.663789\pi$
$314$	−22.9997	−1.29795
$315$	0	0
$316$	2.57226	0.144701
$317$	−1.42612	−0.0800989	−0.0400495	−	0.999198i	$-0.512752\pi$
$317$	−1.42612	−0.0800989	−0.0400495	+	0.999198i	$0.512752\pi$
$318$	0	0
$319$	−2.02086	−0.113147
$320$	25.7275	1.43821
$321$	0	0
$322$	1.50216	0.0837120
$323$	18.3308	1.01995
$324$	0	0
$325$	−11.7080	−0.649442
$326$	−5.93166	−0.328524
$327$	0	0
$328$	−12.7409	−0.703499
$329$	−10.6393	−0.586563
$330$	0	0
$331$	3.91690	0.215292	0.107646	−	0.994189i	$-0.465669\pi$
$331$	3.91690	0.215292	0.107646	+	0.994189i	$0.465669\pi$
$332$	−1.96159	−0.107656
$333$	0	0
$334$	−2.82033	−0.154321
$335$	−46.0327	−2.51504
$336$	0	0
$337$	7.73646	0.421432	0.210716	−	0.977547i	$-0.432421\pi$
$337$	7.73646	0.421432	0.210716	+	0.977547i	$0.432421\pi$
$338$	−17.3046	−0.941247
$339$	0	0
$340$	3.59264	0.194838
$341$	4.90021	0.265361
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.55453	−0.515146
$345$	0	0
$346$	−16.4105	−0.882233
$347$	32.0316	1.71955	0.859773	−	0.510677i	$-0.170605\pi$
$347$	32.0316	1.71955	0.859773	+	0.510677i	$0.170605\pi$
$348$	0	0
$349$	29.0447	1.55473	0.777363	−	0.629052i	$-0.216557\pi$
$349$	29.0447	1.55473	0.777363	+	0.629052i	$0.216557\pi$
$350$	14.4558	0.772696
$351$	0	0
$352$	−0.781744	−0.0416671
$353$	−14.1412	−0.752661	−0.376331	−	0.926485i	$-0.622814\pi$
$353$	−14.1412	−0.752661	−0.376331	+	0.926485i	$0.622814\pi$
$354$	0	0
$355$	20.2597	1.07527
$356$	1.21752	0.0645284
$357$	0	0
$358$	−7.12404	−0.376517
$359$	−7.25851	−0.383089	−0.191545	−	0.981484i	$-0.561350\pi$
$359$	−7.25851	−0.383089	−0.191545	+	0.981484i	$0.561350\pi$
$360$	0	0
$361$	6.04319	0.318063
$362$	4.24866	0.223304
$363$	0	0
$364$	−0.312039	−0.0163553
$365$	31.3216	1.63945
$366$	0	0
$367$	−15.7667	−0.823015	−0.411507	−	0.911406i	$-0.634998\pi$
$367$	−15.7667	−0.823015	−0.411507	+	0.911406i	$0.634998\pi$
$368$	−4.44718	−0.231825
$369$	0	0
$370$	−52.1910	−2.71328
$371$	−7.48704	−0.388708
$372$	0	0
$373$	4.16807	0.215815	0.107907	−	0.994161i	$-0.465585\pi$
$373$	4.16807	0.215815	0.107907	+	0.994161i	$0.465585\pi$
$374$	2.98240	0.154216
$375$	0	0
$376$	27.8648	1.43702
$377$	4.53605	0.233618
$378$	0	0
$379$	18.6572	0.958356	0.479178	−	0.877718i	$-0.340935\pi$
$379$	18.6572	0.958356	0.479178	+	0.877718i	$0.340935\pi$
$380$	4.90821	0.251786
$381$	0	0
$382$	5.36027	0.274255
$383$	30.2829	1.54738	0.773692	−	0.633562i	$-0.218408\pi$
$383$	30.2829	1.54738	0.773692	+	0.633562i	$0.218408\pi$
$384$	0	0
$385$	2.07271	0.105635
$386$	31.4189	1.59918
$387$	0	0
$388$	−2.81857	−0.143091
$389$	14.0727	0.713515	0.356757	−	0.934197i	$-0.383882\pi$
$389$	14.0727	0.713515	0.356757	+	0.934197i	$0.383882\pi$
$390$	0	0
$391$	3.66299	0.185245
$392$	−2.61904	−0.132282
$393$	0	0
$394$	−8.33061	−0.419690
$395$	38.3516	1.92968
$396$	0	0
$397$	30.2423	1.51782	0.758908	−	0.651198i	$-0.225733\pi$
$397$	30.2423	1.51782	0.758908	+	0.651198i	$0.225733\pi$
$398$	−11.6238	−0.582649
$399$	0	0
$400$	−42.7968	−2.13984
$401$	0.668630	0.0333898	0.0166949	−	0.999861i	$-0.494686\pi$
$401$	0.668630	0.0333898	0.0166949	+	0.999861i	$0.494686\pi$
$402$	0	0
$403$	−10.9991	−0.547902
$404$	−0.922976	−0.0459198
$405$	0	0
$406$	−5.60065	−0.277955
$407$	−4.92460	−0.244103
$408$	0	0
$409$	−9.05602	−0.447791	−0.223896	−	0.974613i	$-0.571878\pi$
$409$	−9.05602	−0.447791	−0.223896	+	0.974613i	$0.571878\pi$
$410$	27.9446	1.38008
$411$	0	0
$412$	2.43324	0.119877
$413$	−5.18027	−0.254904
$414$	0	0
$415$	−29.2467	−1.43567
$416$	1.75471	0.0860318
$417$	0	0
$418$	4.07451	0.199291
$419$	19.9592	0.975068	0.487534	−	0.873104i	$-0.337896\pi$
$419$	19.9592	0.975068	0.487534	+	0.873104i	$0.337896\pi$
$420$	0	0
$421$	−6.86296	−0.334480	−0.167240	−	0.985916i	$-0.553486\pi$
$421$	−6.86296	−0.334480	−0.167240	+	0.985916i	$0.553486\pi$
$422$	14.1791	0.690230
$423$	0	0
$424$	19.6089	0.952291
$425$	35.2503	1.70989
$426$	0	0
$427$	3.26161	0.157840
$428$	−1.69352	−0.0818596
$429$	0	0
$430$	20.9559	1.01058
$431$	−4.52160	−0.217798	−0.108899	−	0.994053i	$-0.534732\pi$
$431$	−4.52160	−0.217798	−0.108899	+	0.994053i	$0.534732\pi$
$432$	0	0
$433$	14.5170	0.697641	0.348820	−	0.937190i	$-0.386582\pi$
$433$	14.5170	0.697641	0.348820	+	0.937190i	$0.386582\pi$
$434$	13.5805	0.651886
$435$	0	0
$436$	0.283992	0.0136007
$437$	5.00432	0.239389
$438$	0	0
$439$	25.5058	1.21732	0.608662	−	0.793430i	$-0.291707\pi$
$439$	25.5058	1.21732	0.608662	+	0.793430i	$0.291707\pi$
$440$	−5.42851	−0.258794
$441$	0	0
$442$	−6.69432	−0.318416
$443$	−13.9434	−0.662469	−0.331235	−	0.943548i	$-0.607465\pi$
$443$	−13.9434	−0.662469	−0.331235	+	0.943548i	$0.607465\pi$
$444$	0	0
$445$	18.1529	0.860528
$446$	16.6755	0.789607
$447$	0	0
$448$	6.72782	0.317860
$449$	22.5307	1.06329	0.531644	−	0.846968i	$-0.321574\pi$
$449$	22.5307	1.06329	0.531644	+	0.846968i	$0.321574\pi$
$450$	0	0
$451$	2.63677	0.124161
$452$	−0.111118	−0.00522654
$453$	0	0
$454$	−31.5194	−1.47928
$455$	−4.65242	−0.218109
$456$	0	0
$457$	−12.9589	−0.606191	−0.303096	−	0.952960i	$-0.598020\pi$
$457$	−12.9589	−0.606191	−0.303096	+	0.952960i	$0.598020\pi$
$458$	6.96317	0.325368
$459$	0	0
$460$	0.980794	0.0457298
$461$	−14.9315	−0.695428	−0.347714	−	0.937601i	$-0.613042\pi$
$461$	−14.9315	−0.695428	−0.347714	+	0.937601i	$0.613042\pi$
$462$	0	0
$463$	17.4941	0.813018	0.406509	−	0.913647i	$-0.366746\pi$
$463$	17.4941	0.813018	0.406509	+	0.913647i	$0.366746\pi$
$464$	16.5809	0.769747
$465$	0	0
$466$	−24.2903	−1.12523
$467$	17.6281	0.815732	0.407866	−	0.913042i	$-0.366273\pi$
$467$	17.6281	0.815732	0.407866	+	0.913042i	$0.366273\pi$
$468$	0	0
$469$	−12.0377	−0.555849
$470$	−61.1156	−2.81905
$471$	0	0
$472$	13.5673	0.624487
$473$	1.97734	0.0909181
$474$	0	0
$475$	48.1583	2.20966
$476$	0.939485	0.0430612
$477$	0	0
$478$	−1.90188	−0.0869902
$479$	11.7457	0.536673	0.268336	−	0.963325i	$-0.413526\pi$
$479$	11.7457	0.536673	0.268336	+	0.963325i	$0.413526\pi$
$480$	0	0
$481$	11.0538	0.504010
$482$	−3.14887	−0.143427
$483$	0	0
$484$	−2.74594	−0.124815
$485$	−42.0241	−1.90822
$486$	0	0
$487$	22.9380	1.03942	0.519710	−	0.854342i	$-0.326040\pi$
$487$	22.9380	1.03942	0.519710	+	0.854342i	$0.326040\pi$
$488$	−8.54229	−0.386691
$489$	0	0
$490$	5.74433	0.259502
$491$	−11.5682	−0.522067	−0.261034	−	0.965330i	$-0.584063\pi$
$491$	−11.5682	−0.522067	−0.261034	+	0.965330i	$0.584063\pi$
$492$	0	0
$493$	−13.6571	−0.615084
$494$	−9.14567	−0.411483
$495$	0	0
$496$	−40.2055	−1.80528
$497$	5.29796	0.237646
$498$	0	0
$499$	25.0063	1.11943	0.559717	−	0.828684i	$-0.310910\pi$
$499$	25.0063	1.11943	0.559717	+	0.828684i	$0.310910\pi$
$500$	4.53457	0.202792
$501$	0	0
$502$	−17.3297	−0.773462
$503$	−16.1238	−0.718925	−0.359463	−	0.933160i	$-0.617040\pi$
$503$	−16.1238	−0.718925	−0.359463	+	0.933160i	$0.617040\pi$
$504$	0	0
$505$	−13.7613	−0.612370
$506$	0.814198	0.0361955
$507$	0	0
$508$	2.26767	0.100612
$509$	1.46604	0.0649809	0.0324905	−	0.999472i	$-0.489656\pi$
$509$	1.46604	0.0649809	0.0324905	+	0.999472i	$0.489656\pi$
$510$	0	0
$511$	8.19070	0.362335
$512$	−16.8806	−0.746025
$513$	0	0
$514$	−22.0136	−0.970978
$515$	36.2789	1.59864
$516$	0	0
$517$	−5.76669	−0.253619
$518$	−13.6481	−0.599663
$519$	0	0
$520$	12.1849	0.534342
$521$	31.1921	1.36655	0.683276	−	0.730160i	$-0.260554\pi$
$521$	31.1921	1.36655	0.683276	+	0.730160i	$0.260554\pi$
$522$	0	0
$523$	−25.0064	−1.09345	−0.546727	−	0.837311i	$-0.684126\pi$
$523$	−25.0064	−1.09345	−0.546727	+	0.837311i	$0.684126\pi$
$524$	0.653850	0.0285636
$525$	0	0
$526$	34.1190	1.48766
$527$	33.1159	1.44255
$528$	0	0
$529$	1.00000	0.0434783
$530$	−43.0080	−1.86815
$531$	0	0
$532$	1.28351	0.0556472
$533$	−5.91852	−0.256360
$534$	0	0
$535$	−25.2500	−1.09165
$536$	31.5272	1.36177
$537$	0	0
$538$	−37.6824	−1.62460
$539$	0.542019	0.0233464
$540$	0	0
$541$	−29.8577	−1.28368	−0.641842	−	0.766837i	$-0.721829\pi$
$541$	−29.8577	−1.28368	−0.641842	+	0.766837i	$0.721829\pi$
$542$	−42.0954	−1.80815
$543$	0	0
$544$	−5.28306	−0.226510
$545$	4.23423	0.181375
$546$	0	0
$547$	29.1669	1.24708	0.623542	−	0.781789i	$-0.285693\pi$
$547$	29.1669	1.24708	0.623542	+	0.781789i	$0.285693\pi$
$548$	−3.49493	−0.149296
$549$	0	0
$550$	7.83532	0.334099
$551$	−18.6581	−0.794862
$552$	0	0
$553$	10.0291	0.426479
$554$	−24.4390	−1.03831
$555$	0	0
$556$	−4.62860	−0.196297
$557$	−40.2543	−1.70563	−0.852814	−	0.522214i	$-0.825106\pi$
$557$	−40.2543	−1.70563	−0.852814	+	0.522214i	$0.825106\pi$
$558$	0	0
$559$	−4.43835	−0.187722
$560$	−17.0062	−0.718645
$561$	0	0
$562$	20.2332	0.853486
$563$	22.5130	0.948808	0.474404	−	0.880307i	$-0.342664\pi$
$563$	22.5130	0.948808	0.474404	+	0.880307i	$0.342664\pi$
$564$	0	0
$565$	−1.65673	−0.0696994
$566$	27.1163	1.13978
$567$	0	0
$568$	−13.8756	−0.582207
$569$	−6.81862	−0.285852	−0.142926	−	0.989733i	$-0.545651\pi$
$569$	−6.81862	−0.285852	−0.142926	+	0.989733i	$0.545651\pi$
$570$	0	0
$571$	−0.873495	−0.0365546	−0.0182773	−	0.999833i	$-0.505818\pi$
$571$	−0.873495	−0.0365546	−0.0182773	+	0.999833i	$0.505818\pi$
$572$	−0.169131	−0.00707173
$573$	0	0
$574$	7.30758	0.305013
$575$	9.62336	0.401322
$576$	0	0
$577$	−16.4150	−0.683366	−0.341683	−	0.939815i	$-0.610997\pi$
$577$	−16.4150	−0.683366	−0.341683	+	0.939815i	$0.610997\pi$
$578$	−5.38150	−0.223841
$579$	0	0
$580$	−3.65679	−0.151840
$581$	−7.64810	−0.317297
$582$	0	0
$583$	−4.05812	−0.168070
$584$	−21.4518	−0.887681
$585$	0	0
$586$	−34.7781	−1.43667
$587$	9.97331	0.411643	0.205821	−	0.978590i	$-0.434013\pi$
$587$	9.97331	0.411643	0.205821	+	0.978590i	$0.434013\pi$
$588$	0	0
$589$	45.2424	1.86418
$590$	−29.7572	−1.22508
$591$	0	0
$592$	40.4055	1.66066
$593$	20.5516	0.843951	0.421976	−	0.906607i	$-0.361337\pi$
$593$	20.5516	0.843951	0.421976	+	0.906607i	$0.361337\pi$
$594$	0	0
$595$	14.0075	0.574250
$596$	−3.54400	−0.145168
$597$	0	0
$598$	−1.82756	−0.0747344
$599$	−15.5468	−0.635224	−0.317612	−	0.948221i	$-0.602881\pi$
$599$	−15.5468	−0.635224	−0.317612	+	0.948221i	$0.602881\pi$
$600$	0	0
$601$	−29.3814	−1.19849	−0.599246	−	0.800565i	$-0.704533\pi$
$601$	−29.3814	−1.19849	−0.599246	+	0.800565i	$0.704533\pi$
$602$	5.48003	0.223349
$603$	0	0
$604$	4.13991	0.168450
$605$	−40.9411	−1.66449
$606$	0	0
$607$	31.1108	1.26275	0.631374	−	0.775478i	$-0.282491\pi$
$607$	31.1108	1.26275	0.631374	+	0.775478i	$0.282491\pi$
$608$	−7.21764	−0.292714
$609$	0	0
$610$	18.7358	0.758589
$611$	12.9440	0.523657
$612$	0	0
$613$	−27.7365	−1.12026	−0.560132	−	0.828403i	$-0.689250\pi$
$613$	−27.7365	−1.12026	−0.560132	+	0.828403i	$0.689250\pi$
$614$	−1.71699	−0.0692920
$615$	0	0
$616$	−1.41957	−0.0571961
$617$	20.6041	0.829490	0.414745	−	0.909938i	$-0.363871\pi$
$617$	20.6041	0.829490	0.414745	+	0.909938i	$0.363871\pi$
$618$	0	0
$619$	−9.83894	−0.395460	−0.197730	−	0.980256i	$-0.563357\pi$
$619$	−9.83894	−0.395460	−0.197730	+	0.980256i	$0.563357\pi$
$620$	8.86704	0.356109
$621$	0	0
$622$	29.0211	1.16364
$623$	4.74703	0.190186
$624$	0	0
$625$	19.4923	0.779690
$626$	−26.1586	−1.04551
$627$	0	0
$628$	−3.92700	−0.156704
$629$	−33.2807	−1.32699
$630$	0	0
$631$	2.65314	0.105620	0.0528098	−	0.998605i	$-0.483182\pi$
$631$	2.65314	0.105620	0.0528098	+	0.998605i	$0.483182\pi$
$632$	−26.2665	−1.04483
$633$	0	0
$634$	−2.14226	−0.0850800
$635$	33.8103	1.34172
$636$	0	0
$637$	−1.21662	−0.0482043
$638$	−3.03566	−0.120183
$639$	0	0
$640$	49.6775	1.96368
$641$	46.9550	1.85461	0.927306	−	0.374304i	$-0.122118\pi$
$641$	46.9550	1.85461	0.927306	+	0.374304i	$0.122118\pi$
$642$	0	0
$643$	−8.01924	−0.316248	−0.158124	−	0.987419i	$-0.550545\pi$
$643$	−8.01924	−0.316248	−0.158124	+	0.987419i	$0.550545\pi$
$644$	0.256481	0.0101067
$645$	0	0
$646$	27.5357	1.08338
$647$	−20.3950	−0.801810	−0.400905	−	0.916120i	$-0.631304\pi$
$647$	−20.3950	−0.801810	−0.400905	+	0.916120i	$0.631304\pi$
$648$	0	0
$649$	−2.80780	−0.110216
$650$	−17.5872	−0.689828
$651$	0	0
$652$	−1.01278	−0.0396635
$653$	30.9134	1.20973	0.604867	−	0.796326i	$-0.293226\pi$
$653$	30.9134	1.20973	0.604867	+	0.796326i	$0.293226\pi$
$654$	0	0
$655$	9.74871	0.380913
$656$	−21.6343	−0.844677
$657$	0	0
$658$	−15.9819	−0.623039
$659$	−8.74476	−0.340647	−0.170324	−	0.985388i	$-0.554481\pi$
$659$	−8.74476	−0.340647	−0.170324	+	0.985388i	$0.554481\pi$
$660$	0	0
$661$	36.1030	1.40424	0.702121	−	0.712057i	$-0.252236\pi$
$661$	36.1030	1.40424	0.702121	+	0.712057i	$0.252236\pi$
$662$	5.88380	0.228680
$663$	0	0
$664$	20.0307	0.777342
$665$	19.1368	0.742092
$666$	0	0
$667$	−3.72840	−0.144364
$668$	−0.481546	−0.0186316
$669$	0	0
$670$	−69.1485	−2.67144
$671$	1.76785	0.0682472
$672$	0	0
$673$	−37.5762	−1.44845	−0.724227	−	0.689561i	$-0.757803\pi$
$673$	−37.5762	−1.44845	−0.724227	+	0.689561i	$0.757803\pi$
$674$	11.6214	0.447639
$675$	0	0
$676$	−2.95461	−0.113639
$677$	21.5389	0.827807	0.413904	−	0.910321i	$-0.364165\pi$
$677$	21.5389	0.827807	0.413904	+	0.910321i	$0.364165\pi$
$678$	0	0
$679$	−10.9894	−0.421736
$680$	−36.6861	−1.40685
$681$	0	0
$682$	7.36090	0.281863
$683$	−20.0545	−0.767364	−0.383682	−	0.923465i	$-0.625344\pi$
$683$	−20.0545	−0.767364	−0.383682	+	0.923465i	$0.625344\pi$
$684$	0	0
$685$	−52.1084	−1.99096
$686$	1.50216	0.0573527
$687$	0	0
$688$	−16.2238	−0.618525
$689$	9.10888	0.347021
$690$	0	0
$691$	36.3245	1.38185	0.690925	−	0.722927i	$-0.257204\pi$
$691$	36.3245	1.38185	0.690925	+	0.722927i	$0.257204\pi$
$692$	−2.80195	−0.106514
$693$	0	0
$694$	48.1165	1.82648
$695$	−69.0111	−2.61774
$696$	0	0
$697$	17.8194	0.674959
$698$	43.6297	1.65141
$699$	0	0
$700$	2.46820	0.0932894
$701$	−28.3217	−1.06969	−0.534847	−	0.844949i	$-0.679631\pi$
$701$	−28.3217	−1.06969	−0.534847	+	0.844949i	$0.679631\pi$
$702$	0	0
$703$	−45.4675	−1.71484
$704$	3.64660	0.137437
$705$	0	0
$706$	−21.2424	−0.799466
$707$	−3.59862	−0.135340
$708$	0	0
$709$	21.4326	0.804919	0.402460	−	0.915438i	$-0.368155\pi$
$709$	21.4326	0.804919	0.402460	+	0.915438i	$0.368155\pi$
$710$	30.4332	1.14214
$711$	0	0
$712$	−12.4327	−0.465934
$713$	9.04067	0.338576
$714$	0	0
$715$	−2.52170	−0.0943061
$716$	−1.21637	−0.0454578
$717$	0	0
$718$	−10.9034	−0.406912
$719$	−46.7125	−1.74208	−0.871040	−	0.491212i	$-0.836554\pi$
$719$	−46.7125	−1.74208	−0.871040	+	0.491212i	$0.836554\pi$
$720$	0	0
$721$	9.48704	0.353316
$722$	9.07783	0.337842
$723$	0	0
$724$	0.725421	0.0269601
$725$	−35.8797	−1.33254
$726$	0	0
$727$	17.3705	0.644237	0.322119	−	0.946699i	$-0.395605\pi$
$727$	17.3705	0.644237	0.322119	+	0.946699i	$0.395605\pi$
$728$	3.18638	0.118095
$729$	0	0
$730$	47.0501	1.74140
$731$	13.3630	0.494247
$732$	0	0
$733$	15.4702	0.571404	0.285702	−	0.958318i	$-0.407773\pi$
$733$	15.4702	0.571404	0.285702	+	0.958318i	$0.407773\pi$
$734$	−23.6841	−0.874195
$735$	0	0
$736$	−1.44228	−0.0531632
$737$	−6.52465	−0.240339
$738$	0	0
$739$	−17.5944	−0.647221	−0.323611	−	0.946190i	$-0.604897\pi$
$739$	−17.5944	−0.647221	−0.323611	+	0.946190i	$0.604897\pi$
$740$	−8.91116	−0.327581
$741$	0	0
$742$	−11.2467	−0.412880
$743$	32.0161	1.17456	0.587278	−	0.809385i	$-0.300199\pi$
$743$	32.0161	1.17456	0.587278	+	0.809385i	$0.300199\pi$
$744$	0	0
$745$	−52.8399	−1.93591
$746$	6.26111	0.229236
$747$	0	0
$748$	0.509219	0.0186189
$749$	−6.60294	−0.241266
$750$	0	0
$751$	38.1097	1.39064	0.695321	−	0.718700i	$-0.255262\pi$
$751$	38.1097	1.39064	0.695321	+	0.718700i	$0.255262\pi$
$752$	47.3148	1.72539
$753$	0	0
$754$	6.81386	0.248146
$755$	61.7248	2.24640
$756$	0	0
$757$	19.5846	0.711814	0.355907	−	0.934521i	$-0.384172\pi$
$757$	19.5846	0.711814	0.355907	+	0.934521i	$0.384172\pi$
$758$	28.0261	1.01795
$759$	0	0
$760$	−50.1200	−1.81804
$761$	1.99465	0.0723058	0.0361529	−	0.999346i	$-0.488490\pi$
$761$	1.99465	0.0723058	0.0361529	+	0.999346i	$0.488490\pi$
$762$	0	0
$763$	1.10726	0.0400856
$764$	0.915220	0.0331115
$765$	0	0
$766$	45.4897	1.64361
$767$	6.30242	0.227567
$768$	0	0
$769$	11.5731	0.417338	0.208669	−	0.977986i	$-0.433087\pi$
$769$	11.5731	0.417338	0.208669	+	0.977986i	$0.433087\pi$
$770$	3.11353	0.112204
$771$	0	0
$772$	5.36450	0.193073
$773$	47.3961	1.70472	0.852360	−	0.522955i	$-0.175170\pi$
$773$	47.3961	1.70472	0.852360	+	0.522955i	$0.175170\pi$
$774$	0	0
$775$	87.0016	3.12519
$776$	28.7818	1.03321
$777$	0	0
$778$	21.1394	0.757886
$779$	24.3446	0.872236
$780$	0	0
$781$	2.87159	0.102754
$782$	5.50239	0.196765
$783$	0	0
$784$	−4.44718	−0.158828
$785$	−58.5504	−2.08975
$786$	0	0
$787$	−30.1964	−1.07639	−0.538193	−	0.842822i	$-0.680893\pi$
$787$	−30.1964	−1.07639	−0.538193	+	0.842822i	$0.680893\pi$
$788$	−1.42238	−0.0506701
$789$	0	0
$790$	57.6102	2.04968
$791$	−0.433241	−0.0154043
$792$	0	0
$793$	−3.96814	−0.140913
$794$	45.4287	1.61220
$795$	0	0
$796$	−1.98467	−0.0703446
$797$	41.0916	1.45554	0.727770	−	0.685821i	$-0.240557\pi$
$797$	41.0916	1.45554	0.727770	+	0.685821i	$0.240557\pi$
$798$	0	0
$799$	−38.9716	−1.37872
$800$	−13.8796	−0.490718
$801$	0	0
$802$	1.00439	0.0354662
$803$	4.43951	0.156667
$804$	0	0
$805$	3.82405	0.134780
$806$	−16.5223	−0.581975
$807$	0	0
$808$	9.42494	0.331568
$809$	−17.9154	−0.629870	−0.314935	−	0.949113i	$-0.601983\pi$
$809$	−17.9154	−0.629870	−0.314935	+	0.949113i	$0.601983\pi$
$810$	0	0
$811$	10.1577	0.356686	0.178343	−	0.983968i	$-0.442926\pi$
$811$	10.1577	0.356686	0.178343	+	0.983968i	$0.442926\pi$
$812$	−0.956262	−0.0335582
$813$	0	0
$814$	−7.39753	−0.259283
$815$	−15.1003	−0.528939
$816$	0	0
$817$	18.2563	0.638705
$818$	−13.6036	−0.475638
$819$	0	0
$820$	4.77129	0.166621
$821$	15.2289	0.531491	0.265745	−	0.964043i	$-0.414382\pi$
$821$	15.2289	0.531491	0.265745	+	0.964043i	$0.414382\pi$
$822$	0	0
$823$	−20.0090	−0.697468	−0.348734	−	0.937222i	$-0.613388\pi$
$823$	−20.0090	−0.697468	−0.348734	+	0.937222i	$0.613388\pi$
$824$	−24.8470	−0.865585
$825$	0	0
$826$	−7.78158	−0.270756
$827$	−18.0259	−0.626822	−0.313411	−	0.949618i	$-0.601472\pi$
$827$	−18.0259	−0.626822	−0.313411	+	0.949618i	$0.601472\pi$
$828$	0	0
$829$	14.2616	0.495327	0.247664	−	0.968846i	$-0.420337\pi$
$829$	14.2616	0.495327	0.247664	+	0.968846i	$0.420337\pi$
$830$	−43.9332	−1.52494
$831$	0	0
$832$	−8.18520	−0.283771
$833$	3.66299	0.126915
$834$	0	0
$835$	−7.17971	−0.248464
$836$	0.695686	0.0240608
$837$	0	0
$838$	29.9818	1.03570
$839$	−51.5877	−1.78101	−0.890503	−	0.454977i	$-0.849647\pi$
$839$	−51.5877	−1.78101	−0.890503	+	0.454977i	$0.849647\pi$
$840$	0	0
$841$	−15.0990	−0.520656
$842$	−10.3093	−0.355280
$843$	0	0
$844$	2.42097	0.0833331
$845$	−44.0524	−1.51545
$846$	0	0
$847$	−10.7062	−0.367870
$848$	33.2962	1.14340
$849$	0	0
$850$	52.9515	1.81622
$851$	−9.08566	−0.311452
$852$	0	0
$853$	48.2610	1.65243	0.826213	−	0.563358i	$-0.190491\pi$
$853$	48.2610	1.65243	0.826213	+	0.563358i	$0.190491\pi$
$854$	4.89945	0.167656
$855$	0	0
$856$	17.2934	0.591075
$857$	1.37665	0.0470256	0.0235128	−	0.999724i	$-0.492515\pi$
$857$	1.37665	0.0470256	0.0235128	+	0.999724i	$0.492515\pi$
$858$	0	0
$859$	−36.3622	−1.24066	−0.620331	−	0.784340i	$-0.713002\pi$
$859$	−36.3622	−1.24066	−0.620331	+	0.784340i	$0.713002\pi$
$860$	3.57804	0.122010
$861$	0	0
$862$	−6.79215	−0.231342
$863$	−44.8731	−1.52750	−0.763749	−	0.645514i	$-0.776643\pi$
$863$	−44.8731	−1.52750	−0.763749	+	0.645514i	$0.776643\pi$
$864$	0	0
$865$	−41.7762	−1.42043
$866$	21.8068	0.741025
$867$	0	0
$868$	2.31876	0.0787037
$869$	5.43594	0.184402
$870$	0	0
$871$	14.6453	0.496237
$872$	−2.89997	−0.0982054
$873$	0	0
$874$	7.51728	0.254276
$875$	17.6800	0.597692
$876$	0	0
$877$	42.2559	1.42688	0.713441	−	0.700716i	$-0.247136\pi$
$877$	42.2559	1.42688	0.713441	+	0.700716i	$0.247136\pi$
$878$	38.3137	1.29303
$879$	0	0
$880$	−9.21770	−0.310729
$881$	−51.1432	−1.72306	−0.861529	−	0.507708i	$-0.830493\pi$
$881$	−51.1432	−1.72306	−0.861529	+	0.507708i	$0.830493\pi$
$882$	0	0
$883$	52.6889	1.77312	0.886562	−	0.462610i	$-0.153087\pi$
$883$	52.6889	1.77312	0.886562	+	0.462610i	$0.153087\pi$
$884$	−1.14300	−0.0384432
$885$	0	0
$886$	−20.9451	−0.703666
$887$	42.9102	1.44078	0.720391	−	0.693568i	$-0.243962\pi$
$887$	42.9102	1.44078	0.720391	+	0.693568i	$0.243962\pi$
$888$	0	0
$889$	8.84149	0.296534
$890$	27.2685	0.914042
$891$	0	0
$892$	2.84719	0.0953311
$893$	−53.2424	−1.78169
$894$	0	0
$895$	−18.1357	−0.606209
$896$	12.9908	0.433993
$897$	0	0
$898$	33.8446	1.12941
$899$	−33.7072	−1.12420
$900$	0	0
$901$	−27.4249	−0.913657
$902$	3.96085	0.131882
$903$	0	0
$904$	1.13468	0.0377388
$905$	10.8158	0.359530
$906$	0	0
$907$	−24.5213	−0.814217	−0.407108	−	0.913380i	$-0.633463\pi$
$907$	−24.5213	−0.814217	−0.407108	+	0.913380i	$0.633463\pi$
$908$	−5.38166	−0.178597
$909$	0	0
$910$	−6.98867	−0.231672
$911$	31.6347	1.04810	0.524052	−	0.851686i	$-0.324420\pi$
$911$	31.6347	1.04810	0.524052	+	0.851686i	$0.324420\pi$
$912$	0	0
$913$	−4.14541	−0.137193
$914$	−19.4663	−0.643888
$915$	0	0
$916$	1.18890	0.0392824
$917$	2.54931	0.0841858
$918$	0	0
$919$	−43.7293	−1.44250	−0.721249	−	0.692676i	$-0.756431\pi$
$919$	−43.7293	−1.44250	−0.721249	+	0.692676i	$0.756431\pi$
$920$	−10.0154	−0.330196
$921$	0	0
$922$	−22.4294	−0.738674
$923$	−6.44561	−0.212160
$924$	0	0
$925$	−87.4346	−2.87483
$926$	26.2788	0.863576
$927$	0	0
$928$	5.37741	0.176522
$929$	−3.63411	−0.119231	−0.0596157	−	0.998221i	$-0.518988\pi$
$929$	−3.63411	−0.119231	−0.0596157	+	0.998221i	$0.518988\pi$
$930$	0	0
$931$	5.00432	0.164010
$932$	−4.14736	−0.135851
$933$	0	0
$934$	26.4802	0.866460
$935$	7.59230	0.248295
$936$	0	0
$937$	−10.3649	−0.338606	−0.169303	−	0.985564i	$-0.554152\pi$
$937$	−10.3649	−0.338606	−0.169303	+	0.985564i	$0.554152\pi$
$938$	−18.0825	−0.590415
$939$	0	0
$940$	−10.4350	−0.340351
$941$	29.8720	0.973800	0.486900	−	0.873458i	$-0.338128\pi$
$941$	29.8720	0.973800	0.486900	+	0.873458i	$0.338128\pi$
$942$	0	0
$943$	4.86472	0.158417
$944$	23.0376	0.749809
$945$	0	0
$946$	2.97028	0.0965720
$947$	−7.17017	−0.232999	−0.116500	−	0.993191i	$-0.537167\pi$
$947$	−7.17017	−0.232999	−0.116500	+	0.993191i	$0.537167\pi$
$948$	0	0
$949$	−9.96497	−0.323477
$950$	72.3415	2.34707
$951$	0	0
$952$	−9.59352	−0.310928
$953$	−16.0495	−0.519894	−0.259947	−	0.965623i	$-0.583705\pi$
$953$	−16.0495	−0.519894	−0.259947	+	0.965623i	$0.583705\pi$
$954$	0	0
$955$	13.6457	0.441563
$956$	−0.324730	−0.0105025
$957$	0	0
$958$	17.6438	0.570046
$959$	−13.6265	−0.440022
$960$	0	0
$961$	50.7337	1.63657
$962$	16.6046	0.535352
$963$	0	0
$964$	−0.537642	−0.0173163
$965$	79.9832	2.57475
$966$	0	0
$967$	33.5360	1.07845	0.539223	−	0.842163i	$-0.318718\pi$
$967$	33.5360	1.07845	0.539223	+	0.842163i	$0.318718\pi$
$968$	28.0400	0.901241
$969$	0	0
$970$	−63.1269	−2.02688
$971$	17.5189	0.562208	0.281104	−	0.959677i	$-0.409299\pi$
$971$	17.5189	0.562208	0.281104	+	0.959677i	$0.409299\pi$
$972$	0	0
$973$	−18.0466	−0.578548
$974$	34.4565	1.10406
$975$	0	0
$976$	−14.5050	−0.464292
$977$	−50.1902	−1.60573	−0.802863	−	0.596163i	$-0.796691\pi$
$977$	−50.1902	−1.60573	−0.802863	+	0.596163i	$0.796691\pi$
$978$	0	0
$979$	2.57298	0.0822327
$980$	0.980794	0.0313303
$981$	0	0
$982$	−17.3773	−0.554533
$983$	22.7924	0.726966	0.363483	−	0.931601i	$-0.381587\pi$
$983$	22.7924	0.726966	0.363483	+	0.931601i	$0.381587\pi$
$984$	0	0
$985$	−21.2073	−0.675719
$986$	−20.5151	−0.653334
$987$	0	0
$988$	−1.56154	−0.0496793
$989$	3.64810	0.116003
$990$	0	0
$991$	−0.965443	−0.0306683	−0.0153342	−	0.999882i	$-0.504881\pi$
$991$	−0.965443	−0.0306683	−0.0153342	+	0.999882i	$0.504881\pi$
$992$	−13.0392	−0.413995
$993$	0	0
$994$	7.95838	0.252424
$995$	−29.5908	−0.938091
$996$	0	0
$997$	−4.22619	−0.133845	−0.0669224	−	0.997758i	$-0.521318\pi$
$997$	−4.22619	−0.133845	−0.0669224	+	0.997758i	$0.521318\pi$
$998$	37.5634	1.18905
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.r.1.4		5
3.2	odd	2		161.2.a.d.1.2	✓	5
12.11	even	2		2576.2.a.bd.1.5		5
15.14	odd	2		4025.2.a.p.1.4		5
21.20	even	2		1127.2.a.h.1.2		5
69.68	even	2		3703.2.a.j.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.d.1.2	✓	5	3.2	odd	2
1127.2.a.h.1.2		5	21.20	even	2
1449.2.a.r.1.4		5	1.1	even	1	trivial
2576.2.a.bd.1.5		5	12.11	even	2
3703.2.a.j.1.2		5	69.68	even	2
4025.2.a.p.1.4		5	15.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q+1.50216 q^{2} +0.256481 q^{4} +3.82405 q^{5} +1.00000 q^{7} -2.61904 q^{8} +O(q^{10})\)	\(q+1.50216 q^{2} +0.256481 q^{4} +3.82405 q^{5} +1.00000 q^{7} -2.61904 q^{8} +5.74433 q^{10} +0.542019 q^{11} -1.21662 q^{13} +1.50216 q^{14} -4.44718 q^{16} +3.66299 q^{17} +5.00432 q^{19} +0.980794 q^{20} +0.814198 q^{22} +1.00000 q^{23} +9.62336 q^{25} -1.82756 q^{26} +0.256481 q^{28} -3.72840 q^{29} +9.04067 q^{31} -1.44228 q^{32} +5.50239 q^{34} +3.82405 q^{35} -9.08566 q^{37} +7.51728 q^{38} -10.0154 q^{40} +4.86472 q^{41} +3.64810 q^{43} +0.139017 q^{44} +1.50216 q^{46} -10.6393 q^{47} +1.00000 q^{49} +14.4558 q^{50} -0.312039 q^{52} -7.48704 q^{53} +2.07271 q^{55} -2.61904 q^{56} -5.60065 q^{58} -5.18027 q^{59} +3.26161 q^{61} +13.5805 q^{62} +6.72782 q^{64} -4.65242 q^{65} -12.0377 q^{67} +0.939485 q^{68} +5.74433 q^{70} +5.29796 q^{71} +8.19070 q^{73} -13.6481 q^{74} +1.28351 q^{76} +0.542019 q^{77} +10.0291 q^{79} -17.0062 q^{80} +7.30758 q^{82} -7.64810 q^{83} +14.0075 q^{85} +5.48003 q^{86} -1.41957 q^{88} +4.74703 q^{89} -1.21662 q^{91} +0.256481 q^{92} -15.9819 q^{94} +19.1368 q^{95} -10.9894 q^{97} +1.50216 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8	\( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8} - 8 q^{10} + 4 q^{11} - 6 q^{13} - 2 q^{14} + 10 q^{16} + 12 q^{17} + 6 q^{19} + 14 q^{22} + 5 q^{23} + 19 q^{25} - q^{26} + 12 q^{28} + 4 q^{29} + 30 q^{31} - 8 q^{32} + 6 q^{34} + 4 q^{35} + 4 q^{37} + 40 q^{38} - 50 q^{40} - 6 q^{41} - 12 q^{43} + 26 q^{44} - 2 q^{46} - 10 q^{47} + 5 q^{49} + 2 q^{50} - 21 q^{52} - 16 q^{53} + 18 q^{55} - 3 q^{56} + 13 q^{58} - 22 q^{59} - 18 q^{61} - 15 q^{62} + 25 q^{64} + 26 q^{65} - 2 q^{67} - 12 q^{68} - 8 q^{70} - 4 q^{71} - 2 q^{73} - 38 q^{74} + 10 q^{76} + 4 q^{77} + 30 q^{79} + 10 q^{80} - 7 q^{82} - 8 q^{83} - 12 q^{85} - 8 q^{86} + 4 q^{88} + 20 q^{89} - 6 q^{91} + 12 q^{92} - 25 q^{94} - 8 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8 - 8 * q^10 + 4 * q^11 - 6 * q^13 - 2 * q^14 + 10 * q^16 + 12 * q^17 + 6 * q^19 + 14 * q^22 + 5 * q^23 + 19 * q^25 - q^26 + 12 * q^28 + 4 * q^29 + 30 * q^31 - 8 * q^32 + 6 * q^34 + 4 * q^35 + 4 * q^37 + 40 * q^38 - 50 * q^40 - 6 * q^41 - 12 * q^43 + 26 * q^44 - 2 * q^46 - 10 * q^47 + 5 * q^49 + 2 * q^50 - 21 * q^52 - 16 * q^53 + 18 * q^55 - 3 * q^56 + 13 * q^58 - 22 * q^59 - 18 * q^61 - 15 * q^62 + 25 * q^64 + 26 * q^65 - 2 * q^67 - 12 * q^68 - 8 * q^70 - 4 * q^71 - 2 * q^73 - 38 * q^74 + 10 * q^76 + 4 * q^77 + 30 * q^79 + 10 * q^80 - 7 * q^82 - 8 * q^83 - 12 * q^85 - 8 * q^86 + 4 * q^88 + 20 * q^89 - 6 * q^91 + 12 * q^92 - 25 * q^94 - 8 * q^95 - 12 * q^97 - 2 * q^98

Introduction

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