LMFDB - Embedded newform 374.2.a.b.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [374,2,Mod(1,374)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(374, 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("374.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$374 = 2 \cdot 11 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	374.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.98640503560$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.785.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 6x + 5$ x^3 - x^2 - 6*x + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.38849$ of defining polynomial
Character	$\chi$	$=$	374.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.38849 q^{3} +1.00000 q^{4} -1.70488 q^{5} -2.38849 q^{6} +1.68361 q^{7} -1.00000 q^{8} +2.70488 q^{9} +1.70488 q^{10} -1.00000 q^{11} +2.38849 q^{12} +6.38849 q^{13} -1.68361 q^{14} -4.07210 q^{15} +1.00000 q^{16} +1.00000 q^{17} -2.70488 q^{18} +5.09337 q^{19} -1.70488 q^{20} +4.02128 q^{21} +1.00000 q^{22} +2.77698 q^{23} -2.38849 q^{24} -2.09337 q^{25} -6.38849 q^{26} -0.704883 q^{27} +1.68361 q^{28} -2.77698 q^{29} +4.07210 q^{30} -0.776979 q^{31} -1.00000 q^{32} -2.38849 q^{33} -1.00000 q^{34} -2.87035 q^{35} +2.70488 q^{36} -9.87035 q^{37} -5.09337 q^{38} +15.2588 q^{39} +1.70488 q^{40} -5.11465 q^{41} -4.02128 q^{42} +3.70488 q^{43} -1.00000 q^{44} -4.61151 q^{45} -2.77698 q^{46} +13.2588 q^{47} +2.38849 q^{48} -4.16547 q^{49} +2.09337 q^{50} +2.38849 q^{51} +6.38849 q^{52} +6.81953 q^{53} +0.704883 q^{54} +1.70488 q^{55} -1.68361 q^{56} +12.1655 q^{57} +2.77698 q^{58} -9.40977 q^{59} -4.07210 q^{60} +4.77698 q^{61} +0.776979 q^{62} +4.55396 q^{63} +1.00000 q^{64} -10.8916 q^{65} +2.38849 q^{66} -12.7770 q^{67} +1.00000 q^{68} +6.63279 q^{69} +2.87035 q^{70} -11.3672 q^{71} -2.70488 q^{72} +2.97872 q^{73} +9.87035 q^{74} -5.00000 q^{75} +5.09337 q^{76} -1.68361 q^{77} -15.2588 q^{78} -2.38849 q^{79} -1.70488 q^{80} -9.79826 q^{81} +5.11465 q^{82} -9.09337 q^{83} +4.02128 q^{84} -1.70488 q^{85} -3.70488 q^{86} -6.63279 q^{87} +1.00000 q^{88} +8.64733 q^{89} +4.61151 q^{90} +10.7557 q^{91} +2.77698 q^{92} -1.85581 q^{93} -13.2588 q^{94} -8.68361 q^{95} -2.38849 q^{96} -9.55396 q^{97} +4.16547 q^{98} -2.70488 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 3 q^{2} - q^{3} + 3 q^{4} - q^{5} + q^{6} + q^{7} - 3 q^{8} + 4 q^{9} + q^{10} - 3 q^{11} - q^{12} + 11 q^{13} - q^{14} + 3 q^{16} + 3 q^{17} - 4 q^{18} + 3 q^{19} - q^{20} + 12 q^{21} + 3 q^{22}+ \cdots - 4 q^{99}+O(q^{100})$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 - q^5 + q^6 + q^7 - 3 * q^8 + 4 * q^9 + q^10 - 3 * q^11 - q^12 + 11 * q^13 - q^14 + 3 * q^16 + 3 * q^17 - 4 * q^18 + 3 * q^19 - q^20 + 12 * q^21 + 3 * q^22 - 8 * q^23 + q^24 + 6 * q^25 - 11 * q^26 + 2 * q^27 + q^28 + 8 * q^29 + 14 * q^31 - 3 * q^32 + q^33 - 3 * q^34 + 20 * q^35 + 4 * q^36 - q^37 - 3 * q^38 + 9 * q^39 + q^40 - 3 * q^41 - 12 * q^42 + 7 * q^43 - 3 * q^44 - 22 * q^45 + 8 * q^46 + 3 * q^47 - q^48 + 12 * q^49 - 6 * q^50 - q^51 + 11 * q^52 + 4 * q^53 - 2 * q^54 + q^55 - q^56 + 12 * q^57 - 8 * q^58 - 20 * q^59 - 2 * q^61 - 14 * q^62 - 19 * q^63 + 3 * q^64 - 4 * q^65 - q^66 - 22 * q^67 + 3 * q^68 + 28 * q^69 - 20 * q^70 - 26 * q^71 - 4 * q^72 + 9 * q^73 + q^74 - 15 * q^75 + 3 * q^76 - q^77 - 9 * q^78 + q^79 - q^80 - 13 * q^81 + 3 * q^82 - 15 * q^83 + 12 * q^84 - q^85 - 7 * q^86 - 28 * q^87 + 3 * q^88 - 19 * q^89 + 22 * q^90 + 16 * q^91 - 8 * q^92 - 30 * q^93 - 3 * q^94 - 22 * q^95 + q^96 + 4 * q^97 - 12 * q^98 - 4 * 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
$3$	2.38849	1.37900	0.689498	−	0.724288i	$-0.257831\pi$
$3$	2.38849	1.37900	0.689498	+	0.724288i	$0.257831\pi$
$4$	1.00000	0.500000
$5$	−1.70488	−0.762447	−0.381223	−	0.924483i	$-0.624497\pi$
$5$	−1.70488	−0.762447	−0.381223	+	0.924483i	$0.624497\pi$
$6$	−2.38849	−0.975097
$7$	1.68361	0.636343	0.318172	−	0.948033i	$-0.396931\pi$
$7$	1.68361	0.636343	0.318172	+	0.948033i	$0.396931\pi$
$8$	−1.00000	−0.353553
$9$	2.70488	0.901628
$10$	1.70488	0.539131
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.38849	0.689498
$13$	6.38849	1.77185	0.885924	−	0.463830i	$-0.153525\pi$
$13$	6.38849	1.77185	0.885924	+	0.463830i	$0.153525\pi$
$14$	−1.68361	−0.449963
$15$	−4.07210	−1.05141
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.70488	−0.637547
$19$	5.09337	1.16850	0.584250	−	0.811574i	$-0.301389\pi$
$19$	5.09337	1.16850	0.584250	+	0.811574i	$0.301389\pi$
$20$	−1.70488	−0.381223
$21$	4.02128	0.877515
$22$	1.00000	0.213201
$23$	2.77698	0.579040	0.289520	−	0.957172i	$-0.406504\pi$
$23$	2.77698	0.579040	0.289520	+	0.957172i	$0.406504\pi$
$24$	−2.38849	−0.487548
$25$	−2.09337	−0.418675
$26$	−6.38849	−1.25289
$27$	−0.704883	−0.135655
$28$	1.68361	0.318172
$29$	−2.77698	−0.515672	−0.257836	−	0.966189i	$-0.583009\pi$
$29$	−2.77698	−0.515672	−0.257836	+	0.966189i	$0.583009\pi$
$30$	4.07210	0.743460
$31$	−0.776979	−0.139550	−0.0697748	−	0.997563i	$-0.522228\pi$
$31$	−0.776979	−0.139550	−0.0697748	+	0.997563i	$0.522228\pi$
$32$	−1.00000	−0.176777
$33$	−2.38849	−0.415783
$34$	−1.00000	−0.171499
$35$	−2.87035	−0.485178
$36$	2.70488	0.450814
$37$	−9.87035	−1.62268	−0.811338	−	0.584577i	$-0.801260\pi$
$37$	−9.87035	−1.62268	−0.811338	+	0.584577i	$0.801260\pi$
$38$	−5.09337	−0.826254
$39$	15.2588	2.44337
$40$	1.70488	0.269566
$41$	−5.11465	−0.798774	−0.399387	−	0.916782i	$-0.630777\pi$
$41$	−5.11465	−0.798774	−0.399387	+	0.916782i	$0.630777\pi$
$42$	−4.02128	−0.620496
$43$	3.70488	0.564989	0.282495	−	0.959269i	$-0.408838\pi$
$43$	3.70488	0.564989	0.282495	+	0.959269i	$0.408838\pi$
$44$	−1.00000	−0.150756
$45$	−4.61151	−0.687443
$46$	−2.77698	−0.409443
$47$	13.2588	1.93400	0.967000	−	0.254775i	$-0.0820013\pi$
$47$	13.2588	1.93400	0.967000	+	0.254775i	$0.0820013\pi$
$48$	2.38849	0.344749
$49$	−4.16547	−0.595067
$50$	2.09337	0.296048
$51$	2.38849	0.334455
$52$	6.38849	0.885924
$53$	6.81953	0.936735	0.468367	−	0.883534i	$-0.344842\pi$
$53$	6.81953	0.936735	0.468367	+	0.883534i	$0.344842\pi$
$54$	0.704883	0.0959225
$55$	1.70488	0.229886
$56$	−1.68361	−0.224981
$57$	12.1655	1.61136
$58$	2.77698	0.364635
$59$	−9.40977	−1.22505	−0.612524	−	0.790452i	$-0.709846\pi$
$59$	−9.40977	−1.22505	−0.612524	+	0.790452i	$0.709846\pi$
$60$	−4.07210	−0.525705
$61$	4.77698	0.611630	0.305815	−	0.952091i	$-0.401071\pi$
$61$	4.77698	0.611630	0.305815	+	0.952091i	$0.401071\pi$
$62$	0.776979	0.0986765
$63$	4.55396	0.573745
$64$	1.00000	0.125000
$65$	−10.8916	−1.35094
$66$	2.38849	0.294003
$67$	−12.7770	−1.56096	−0.780478	−	0.625183i	$-0.785024\pi$
$67$	−12.7770	−1.56096	−0.780478	+	0.625183i	$0.785024\pi$
$68$	1.00000	0.121268
$69$	6.63279	0.798494
$70$	2.87035	0.343073
$71$	−11.3672	−1.34904	−0.674520	−	0.738257i	$-0.735649\pi$
$71$	−11.3672	−1.34904	−0.674520	+	0.738257i	$0.735649\pi$
$72$	−2.70488	−0.318774
$73$	2.97872	0.348633	0.174317	−	0.984690i	$-0.444228\pi$
$73$	2.97872	0.348633	0.174317	+	0.984690i	$0.444228\pi$
$74$	9.87035	1.14741
$75$	−5.00000	−0.577350
$76$	5.09337	0.584250
$77$	−1.68361	−0.191865
$78$	−15.2588	−1.72772
$79$	−2.38849	−0.268726	−0.134363	−	0.990932i	$-0.542899\pi$
$79$	−2.38849	−0.268726	−0.134363	+	0.990932i	$0.542899\pi$
$80$	−1.70488	−0.190612
$81$	−9.79826	−1.08870
$82$	5.11465	0.564819
$83$	−9.09337	−0.998127	−0.499064	−	0.866565i	$-0.666323\pi$
$83$	−9.09337	−0.998127	−0.499064	+	0.866565i	$0.666323\pi$
$84$	4.02128	0.438757
$85$	−1.70488	−0.184921
$86$	−3.70488	−0.399508
$87$	−6.63279	−0.711109
$88$	1.00000	0.106600
$89$	8.64733	0.916615	0.458308	−	0.888794i	$-0.348456\pi$
$89$	8.64733	0.916615	0.458308	+	0.888794i	$0.348456\pi$
$90$	4.61151	0.486096
$91$	10.7557	1.12750
$92$	2.77698	0.289520
$93$	−1.85581	−0.192438
$94$	−13.2588	−1.36755
$95$	−8.68361	−0.890919
$96$	−2.38849	−0.243774
$97$	−9.55396	−0.970058	−0.485029	−	0.874498i	$-0.661191\pi$
$97$	−9.55396	−0.970058	−0.485029	+	0.874498i	$0.661191\pi$
$98$	4.16547	0.420776
$99$	−2.70488	−0.271851
$100$	−2.09337	−0.209337
$101$	−1.25884	−0.125259	−0.0626297	−	0.998037i	$-0.519949\pi$
$101$	−1.25884	−0.125259	−0.0626297	+	0.998037i	$0.519949\pi$
$102$	−2.38849	−0.236496
$103$	1.02128	0.100629	0.0503147	−	0.998733i	$-0.483978\pi$
$103$	1.02128	0.100629	0.0503147	+	0.998733i	$0.483978\pi$
$104$	−6.38849	−0.626443
$105$	−6.85581	−0.669058
$106$	−6.81953	−0.662372
$107$	−4.73443	−0.457694	−0.228847	−	0.973462i	$-0.573496\pi$
$107$	−4.73443	−0.457694	−0.228847	+	0.973462i	$0.573496\pi$
$108$	−0.704883	−0.0678274
$109$	−3.55396	−0.340408	−0.170204	−	0.985409i	$-0.554443\pi$
$109$	−3.55396	−0.340408	−0.170204	+	0.985409i	$0.554443\pi$
$110$	−1.70488	−0.162554
$111$	−23.5752	−2.23766
$112$	1.68361	0.159086
$113$	−8.59023	−0.808101	−0.404051	−	0.914737i	$-0.632398\pi$
$113$	−8.59023	−0.808101	−0.404051	+	0.914737i	$0.632398\pi$
$114$	−12.1655	−1.13940
$115$	−4.73443	−0.441487
$116$	−2.77698	−0.257836
$117$	17.2801	1.59755
$118$	9.40977	0.866239
$119$	1.68361	0.154336
$120$	4.07210	0.371730
$121$	1.00000	0.0909091
$122$	−4.77698	−0.432487
$123$	−12.2163	−1.10151
$124$	−0.776979	−0.0697748
$125$	12.0934	1.08166
$126$	−4.55396	−0.405699
$127$	−5.95745	−0.528638	−0.264319	−	0.964435i	$-0.585147\pi$
$127$	−5.95745	−0.528638	−0.264319	+	0.964435i	$0.585147\pi$
$128$	−1.00000	−0.0883883
$129$	8.84908	0.779118
$130$	10.8916	0.955259
$131$	−6.81953	−0.595825	−0.297913	−	0.954593i	$-0.596290\pi$
$131$	−6.81953	−0.595825	−0.297913	+	0.954593i	$0.596290\pi$
$132$	−2.38849	−0.207891
$133$	8.57524	0.743567
$134$	12.7770	1.10376
$135$	1.20174	0.103430
$136$	−1.00000	−0.0857493
$137$	1.61151	0.137681	0.0688403	−	0.997628i	$-0.478070\pi$
$137$	1.61151	0.137681	0.0688403	+	0.997628i	$0.478070\pi$
$138$	−6.63279	−0.564620
$139$	22.3309	1.89409	0.947043	−	0.321108i	$-0.104055\pi$
$139$	22.3309	1.89409	0.947043	+	0.321108i	$0.104055\pi$
$140$	−2.87035	−0.242589
$141$	31.6686	2.66698
$142$	11.3672	0.953915
$143$	−6.38849	−0.534232
$144$	2.70488	0.225407
$145$	4.73443	0.393173
$146$	−2.97872	−0.246521
$147$	−9.94918	−0.820595
$148$	−9.87035	−0.811338
$149$	−0.345936	−0.0283402	−0.0141701	−	0.999900i	$-0.504511\pi$
$149$	−0.345936	−0.0283402	−0.0141701	+	0.999900i	$0.504511\pi$
$150$	5.00000	0.408248
$151$	−21.1079	−1.71774	−0.858869	−	0.512195i	$-0.828832\pi$
$151$	−21.1079	−1.71774	−0.858869	+	0.512195i	$0.828832\pi$
$152$	−5.09337	−0.413127
$153$	2.70488	0.218677
$154$	1.68361	0.135669
$155$	1.32466	0.106399
$156$	15.2588	1.22169
$157$	15.5114	1.23794	0.618972	−	0.785413i	$-0.287549\pi$
$157$	15.5114	1.23794	0.618972	+	0.785413i	$0.287549\pi$
$158$	2.38849	0.190018
$159$	16.2884	1.29175
$160$	1.70488	0.134783
$161$	4.67534	0.368468
$162$	9.79826	0.769824
$163$	−12.5244	−0.980988	−0.490494	−	0.871445i	$-0.663184\pi$
$163$	−12.5244	−0.980988	−0.490494	+	0.871445i	$0.663184\pi$
$164$	−5.11465	−0.399387
$165$	4.07210	0.317012
$166$	9.09337	0.705783
$167$	5.11465	0.395783	0.197892	−	0.980224i	$-0.436591\pi$
$167$	5.11465	0.395783	0.197892	+	0.980224i	$0.436591\pi$
$168$	−4.02128	−0.310248
$169$	27.8128	2.13945
$170$	1.70488	0.130759
$171$	13.7770	1.05355
$172$	3.70488	0.282495
$173$	−5.18047	−0.393864	−0.196932	−	0.980417i	$-0.563098\pi$
$173$	−5.18047	−0.393864	−0.196932	+	0.980417i	$0.563098\pi$
$174$	6.63279	0.502830
$175$	−3.52442	−0.266421
$176$	−1.00000	−0.0753778
$177$	−22.4751	−1.68933
$178$	−8.64733	−0.648145
$179$	0.632787	0.0472967	0.0236484	−	0.999720i	$-0.492472\pi$
$179$	0.632787	0.0472967	0.0236484	+	0.999720i	$0.492472\pi$
$180$	−4.61151	−0.343722
$181$	16.6178	1.23519	0.617595	−	0.786496i	$-0.288107\pi$
$181$	16.6178	1.23519	0.617595	+	0.786496i	$0.288107\pi$
$182$	−10.7557	−0.797266
$183$	11.4098	0.843434
$184$	−2.77698	−0.204722
$185$	16.8278	1.23720
$186$	1.85581	0.136074
$187$	−1.00000	−0.0731272
$188$	13.2588	0.967000
$189$	−1.18675	−0.0863231
$190$	8.68361	0.629975
$191$	1.79826	0.130117	0.0650586	−	0.997881i	$-0.479277\pi$
$191$	1.79826	0.130117	0.0650586	+	0.997881i	$0.479277\pi$
$192$	2.38849	0.172374
$193$	−25.3097	−1.82183	−0.910915	−	0.412595i	$-0.864623\pi$
$193$	−25.3097	−1.82183	−0.910915	+	0.412595i	$0.864623\pi$
$194$	9.55396	0.685934
$195$	−26.0145	−1.86294
$196$	−4.16547	−0.297534
$197$	25.0063	1.78162	0.890812	−	0.454372i	$-0.150136\pi$
$197$	25.0063	1.78162	0.890812	+	0.454372i	$0.150136\pi$
$198$	2.70488	0.192228
$199$	14.1867	1.00567	0.502836	−	0.864382i	$-0.332290\pi$
$199$	14.1867	1.00567	0.502836	+	0.864382i	$0.332290\pi$
$200$	2.09337	0.148024
$201$	−30.5177	−2.15255
$202$	1.25884	0.0885718
$203$	−4.67534	−0.328145
$204$	2.38849	0.167228
$205$	8.71988	0.609023
$206$	−1.02128	−0.0711557
$207$	7.51141	0.522079
$208$	6.38849	0.442962
$209$	−5.09337	−0.352316
$210$	6.85581	0.473096
$211$	27.1930	1.87205	0.936023	−	0.351940i	$-0.114478\pi$
$211$	27.1930	1.87205	0.936023	+	0.351940i	$0.114478\pi$
$212$	6.81953	0.468367
$213$	−27.1505	−1.86032
$214$	4.73443	0.323639
$215$	−6.31639	−0.430774
$216$	0.704883	0.0479612
$217$	−1.30813	−0.0888015
$218$	3.55396	0.240704
$219$	7.11465	0.480763
$220$	1.70488	0.114943
$221$	6.38849	0.429736
$222$	23.5752	1.58227
$223$	−7.91291	−0.529887	−0.264944	−	0.964264i	$-0.585353\pi$
$223$	−7.91291	−0.529887	−0.264944	+	0.964264i	$0.585353\pi$
$224$	−1.68361	−0.112491
$225$	−5.66233	−0.377489
$226$	8.59023	0.571414
$227$	−0.186746	−0.0123948	−0.00619739	−	0.999981i	$-0.501973\pi$
$227$	−0.186746	−0.0123948	−0.00619739	+	0.999981i	$0.501973\pi$
$228$	12.1655	0.805678
$229$	−10.6328	−0.702634	−0.351317	−	0.936257i	$-0.614266\pi$
$229$	−10.6328	−0.702634	−0.351317	+	0.936257i	$0.614266\pi$
$230$	4.73443	0.312179
$231$	−4.02128	−0.264581
$232$	2.77698	0.182318
$233$	19.3522	1.26781	0.633903	−	0.773413i	$-0.281452\pi$
$233$	19.3522	1.26781	0.633903	+	0.773413i	$0.281452\pi$
$234$	−17.2801	−1.12964
$235$	−22.6048	−1.47457
$236$	−9.40977	−0.612524
$237$	−5.70488	−0.370572
$238$	−1.68361	−0.109132
$239$	5.18047	0.335097	0.167548	−	0.985864i	$-0.446415\pi$
$239$	5.18047	0.335097	0.167548	+	0.985864i	$0.446415\pi$
$240$	−4.07210	−0.262853
$241$	−15.5540	−1.00192	−0.500959	−	0.865471i	$-0.667019\pi$
$241$	−15.5540	−1.00192	−0.500959	+	0.865471i	$0.667019\pi$
$242$	−1.00000	−0.0642824
$243$	−21.2884	−1.36565
$244$	4.77698	0.305815
$245$	7.10164	0.453707
$246$	12.2163	0.778882
$247$	32.5390	2.07040
$248$	0.776979	0.0493382
$249$	−21.7194	−1.37641
$250$	−12.0934	−0.764852
$251$	−20.7044	−1.30685	−0.653426	−	0.756990i	$-0.726669\pi$
$251$	−20.7044	−1.30685	−0.653426	+	0.756990i	$0.726669\pi$
$252$	4.55396	0.286872
$253$	−2.77698	−0.174587
$254$	5.95745	0.373803
$255$	−4.07210	−0.255005
$256$	1.00000	0.0625000
$257$	10.5457	0.657822	0.328911	−	0.944361i	$-0.393318\pi$
$257$	10.5457	0.657822	0.328911	+	0.944361i	$0.393318\pi$
$258$	−8.84908	−0.550919
$259$	−16.6178	−1.03258
$260$	−10.8916	−0.675470
$261$	−7.51141	−0.464944
$262$	6.81953	0.421312
$263$	−23.3672	−1.44088	−0.720442	−	0.693515i	$-0.756061\pi$
$263$	−23.3672	−1.44088	−0.720442	+	0.693515i	$0.756061\pi$
$264$	2.38849	0.147001
$265$	−11.6265	−0.714211
$266$	−8.57524	−0.525781
$267$	20.6541	1.26401
$268$	−12.7770	−0.780478
$269$	−7.76871	−0.473667	−0.236833	−	0.971550i	$-0.576110\pi$
$269$	−7.76871	−0.473667	−0.236833	+	0.971550i	$0.576110\pi$
$270$	−1.20174	−0.0731358
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	1.00000	0.0606339
$273$	25.6899	1.55482
$274$	−1.61151	−0.0973549
$275$	2.09337	0.126235
$276$	6.63279	0.399247
$277$	21.6982	1.30372	0.651858	−	0.758341i	$-0.273990\pi$
$277$	21.6982	1.30372	0.651858	+	0.758341i	$0.273990\pi$
$278$	−22.3309	−1.33932
$279$	−2.10164	−0.125822
$280$	2.87035	0.171536
$281$	29.5540	1.76304	0.881521	−	0.472145i	$-0.156520\pi$
$281$	29.5540	1.76304	0.881521	+	0.472145i	$0.156520\pi$
$282$	−31.6686	−1.88584
$283$	−14.7344	−0.875871	−0.437935	−	0.899006i	$-0.644290\pi$
$283$	−14.7344	−0.875871	−0.437935	+	0.899006i	$0.644290\pi$
$284$	−11.3672	−0.674520
$285$	−20.7407	−1.22857
$286$	6.38849	0.377759
$287$	−8.61106	−0.508295
$288$	−2.70488	−0.159387
$289$	1.00000	0.0588235
$290$	−4.73443	−0.278015
$291$	−22.8195	−1.33770
$292$	2.97872	0.174317
$293$	3.02954	0.176988	0.0884939	−	0.996077i	$-0.471795\pi$
$293$	3.02954	0.176988	0.0884939	+	0.996077i	$0.471795\pi$
$294$	9.94918	0.580248
$295$	16.0426	0.934034
$296$	9.87035	0.573703
$297$	0.704883	0.0409015
$298$	0.345936	0.0200395
$299$	17.7407	1.02597
$300$	−5.00000	−0.288675
$301$	6.23757	0.359527
$302$	21.1079	1.21462
$303$	−3.00673	−0.172732
$304$	5.09337	0.292125
$305$	−8.14419	−0.466335
$306$	−2.70488	−0.154628
$307$	16.2443	0.927111	0.463556	−	0.886068i	$-0.346573\pi$
$307$	16.2443	0.927111	0.463556	+	0.886068i	$0.346573\pi$
$308$	−1.68361	−0.0959324
$309$	2.43931	0.138767
$310$	−1.32466	−0.0752356
$311$	25.1930	1.42857	0.714283	−	0.699857i	$-0.246753\pi$
$311$	25.1930	1.42857	0.714283	+	0.699857i	$0.246753\pi$
$312$	−15.2588	−0.863862
$313$	4.96373	0.280566	0.140283	−	0.990111i	$-0.455199\pi$
$313$	4.96373	0.280566	0.140283	+	0.990111i	$0.455199\pi$
$314$	−15.5114	−0.875359
$315$	−7.76397	−0.437450
$316$	−2.38849	−0.134363
$317$	−6.37349	−0.357971	−0.178985	−	0.983852i	$-0.557282\pi$
$317$	−6.37349	−0.357971	−0.178985	+	0.983852i	$0.557282\pi$
$318$	−16.2884	−0.913407
$319$	2.77698	0.155481
$320$	−1.70488	−0.0953059
$321$	−11.3081	−0.631158
$322$	−4.67534	−0.260547
$323$	5.09337	0.283403
$324$	−9.79826	−0.544348
$325$	−13.3735	−0.741828
$326$	12.5244	0.693663
$327$	−8.48859	−0.469420
$328$	5.11465	0.282409
$329$	22.3227	1.23069
$330$	−4.07210	−0.224162
$331$	−10.4460	−0.574166	−0.287083	−	0.957906i	$-0.592686\pi$
$331$	−10.4460	−0.574166	−0.287083	+	0.957906i	$0.592686\pi$
$332$	−9.09337	−0.499064
$333$	−26.6982	−1.46305
$334$	−5.11465	−0.279861
$335$	21.7833	1.19015
$336$	4.02128	0.219379
$337$	−6.07838	−0.331110	−0.165555	−	0.986201i	$-0.552942\pi$
$337$	−6.07838	−0.331110	−0.165555	+	0.986201i	$0.552942\pi$
$338$	−27.8128	−1.51282
$339$	−20.5177	−1.11437
$340$	−1.70488	−0.0924603
$341$	0.776979	0.0420758
$342$	−13.7770	−0.744974
$343$	−18.7983	−1.01501
$344$	−3.70488	−0.199754
$345$	−11.3081	−0.608809
$346$	5.18047	0.278504
$347$	−35.9275	−1.92869	−0.964343	−	0.264654i	$-0.914742\pi$
$347$	−35.9275	−1.92869	−0.964343	+	0.264654i	$0.914742\pi$
$348$	−6.63279	−0.355555
$349$	18.2884	0.978955	0.489477	−	0.872016i	$-0.337188\pi$
$349$	18.2884	0.978955	0.489477	+	0.872016i	$0.337188\pi$
$350$	3.52442	0.188388
$351$	−4.50314	−0.240360
$352$	1.00000	0.0533002
$353$	1.25884	0.0670014	0.0335007	−	0.999439i	$-0.489334\pi$
$353$	1.25884	0.0670014	0.0335007	+	0.999439i	$0.489334\pi$
$354$	22.4751	1.19454
$355$	19.3798	1.02857
$356$	8.64733	0.458308
$357$	4.02128	0.212829
$358$	−0.632787	−0.0334438
$359$	26.3309	1.38969	0.694847	−	0.719158i	$-0.255472\pi$
$359$	26.3309	1.38969	0.694847	+	0.719158i	$0.255472\pi$
$360$	4.61151	0.243048
$361$	6.94245	0.365392
$362$	−16.6178	−0.873412
$363$	2.38849	0.125363
$364$	10.7557	0.563752
$365$	−5.07838	−0.265814
$366$	−11.4098	−0.596398
$367$	−23.0654	−1.20400	−0.602001	−	0.798495i	$-0.705630\pi$
$367$	−23.0654	−1.20400	−0.602001	+	0.798495i	$0.705630\pi$
$368$	2.77698	0.144760
$369$	−13.8345	−0.720197
$370$	−16.8278	−0.874836
$371$	11.4814	0.596085
$372$	−1.85581	−0.0962191
$373$	12.7324	0.659261	0.329630	−	0.944110i	$-0.393076\pi$
$373$	12.7324	0.659261	0.329630	+	0.944110i	$0.393076\pi$
$374$	1.00000	0.0517088
$375$	28.8849	1.49161
$376$	−13.2588	−0.683773
$377$	−17.7407	−0.913693
$378$	1.18675	0.0610396
$379$	14.8854	0.764609	0.382305	−	0.924036i	$-0.375131\pi$
$379$	14.8854	0.764609	0.382305	+	0.924036i	$0.375131\pi$
$380$	−8.68361	−0.445460
$381$	−14.2293	−0.728989
$382$	−1.79826	−0.0920068
$383$	3.91489	0.200042	0.100021	−	0.994985i	$-0.468109\pi$
$383$	3.91489	0.200042	0.100021	+	0.994985i	$0.468109\pi$
$384$	−2.38849	−0.121887
$385$	2.87035	0.146287
$386$	25.3097	1.28823
$387$	10.0213	0.509410
$388$	−9.55396	−0.485029
$389$	37.6982	1.91137	0.955686	−	0.294388i	$-0.0951157\pi$
$389$	37.6982	1.91137	0.955686	+	0.294388i	$0.0951157\pi$
$390$	26.0145	1.31730
$391$	2.77698	0.140438
$392$	4.16547	0.210388
$393$	−16.2884	−0.821640
$394$	−25.0063	−1.25980
$395$	4.07210	0.204889
$396$	−2.70488	−0.135925
$397$	19.0508	0.956133	0.478067	−	0.878324i	$-0.341338\pi$
$397$	19.0508	0.956133	0.478067	+	0.878324i	$0.341338\pi$
$398$	−14.1867	−0.711117
$399$	20.4819	1.02538
$400$	−2.09337	−0.104669
$401$	−13.0063	−0.649503	−0.324751	−	0.945799i	$-0.605281\pi$
$401$	−13.0063	−0.649503	−0.324751	+	0.945799i	$0.605281\pi$
$402$	30.5177	1.52208
$403$	−4.96373	−0.247261
$404$	−1.25884	−0.0626297
$405$	16.7049	0.830072
$406$	4.67534	0.232033
$407$	9.87035	0.489255
$408$	−2.38849	−0.118248
$409$	2.81953	0.139417	0.0697085	−	0.997567i	$-0.477793\pi$
$409$	2.81953	0.139417	0.0697085	+	0.997567i	$0.477793\pi$
$410$	−8.71988	−0.430644
$411$	3.84908	0.189861
$412$	1.02128	0.0503147
$413$	−15.8423	−0.779551
$414$	−7.51141	−0.369165
$415$	15.5031	0.761019
$416$	−6.38849	−0.313221
$417$	53.3372	2.61193
$418$	5.09337	0.249125
$419$	−7.32267	−0.357736	−0.178868	−	0.983873i	$-0.557244\pi$
$419$	−7.32267	−0.357736	−0.178868	+	0.983873i	$0.557244\pi$
$420$	−6.85581	−0.334529
$421$	13.5540	0.660580	0.330290	−	0.943880i	$-0.392854\pi$
$421$	13.5540	0.660580	0.330290	+	0.943880i	$0.392854\pi$
$422$	−27.1930	−1.32374
$423$	35.8636	1.74375
$424$	−6.81953	−0.331186
$425$	−2.09337	−0.101544
$426$	27.1505	1.31544
$427$	8.04255	0.389206
$428$	−4.73443	−0.228847
$429$	−15.2588	−0.736704
$430$	6.31639	0.304604
$431$	−2.40150	−0.115676	−0.0578381	−	0.998326i	$-0.518421\pi$
$431$	−2.40150	−0.115676	−0.0578381	+	0.998326i	$0.518421\pi$
$432$	−0.704883	−0.0339137
$433$	25.5472	1.22772	0.613861	−	0.789414i	$-0.289616\pi$
$433$	25.5472	1.22772	0.613861	+	0.789414i	$0.289616\pi$
$434$	1.30813	0.0627921
$435$	11.3081	0.542183
$436$	−3.55396	−0.170204
$437$	14.1442	0.676608
$438$	−7.11465	−0.339951
$439$	12.2226	0.583351	0.291676	−	0.956517i	$-0.405787\pi$
$439$	12.2226	0.583351	0.291676	+	0.956517i	$0.405787\pi$
$440$	−1.70488	−0.0812771
$441$	−11.2671	−0.536529
$442$	−6.38849	−0.303869
$443$	−34.6619	−1.64684	−0.823418	−	0.567436i	$-0.807936\pi$
$443$	−34.6619	−1.64684	−0.823418	+	0.567436i	$0.807936\pi$
$444$	−23.5752	−1.11883
$445$	−14.7427	−0.698871
$446$	7.91291	0.374687
$447$	−0.826265	−0.0390810
$448$	1.68361	0.0795429
$449$	21.9700	1.03683	0.518414	−	0.855130i	$-0.326523\pi$
$449$	21.9700	1.03683	0.518414	+	0.855130i	$0.326523\pi$
$450$	5.66233	0.266925
$451$	5.11465	0.240839
$452$	−8.59023	−0.404051
$453$	−50.4160	−2.36875
$454$	0.186746	0.00876443
$455$	−18.3372	−0.859662
$456$	−12.1655	−0.569700
$457$	−27.7407	−1.29766	−0.648828	−	0.760935i	$-0.724740\pi$
$457$	−27.7407	−1.29766	−0.648828	+	0.760935i	$0.724740\pi$
$458$	10.6328	0.496838
$459$	−0.704883	−0.0329011
$460$	−4.73443	−0.220744
$461$	−27.0504	−1.25986	−0.629931	−	0.776651i	$-0.716917\pi$
$461$	−27.0504	−1.25986	−0.629931	+	0.776651i	$0.716917\pi$
$462$	4.02128	0.187087
$463$	−37.6832	−1.75128	−0.875642	−	0.482960i	$-0.839562\pi$
$463$	−37.6832	−1.75128	−0.875642	+	0.482960i	$0.839562\pi$
$464$	−2.77698	−0.128918
$465$	3.16394	0.146724
$466$	−19.3522	−0.896474
$467$	26.7770	1.23909	0.619545	−	0.784961i	$-0.287317\pi$
$467$	26.7770	1.23909	0.619545	+	0.784961i	$0.287317\pi$
$468$	17.2801	0.798774
$469$	−21.5114	−0.993304
$470$	22.6048	1.04268
$471$	37.0488	1.70712
$472$	9.40977	0.433120
$473$	−3.70488	−0.170351
$474$	5.70488	0.262034
$475$	−10.6623	−0.489221
$476$	1.68361	0.0771680
$477$	18.4460	0.844586
$478$	−5.18047	−0.236949
$479$	−18.2734	−0.834932	−0.417466	−	0.908692i	$-0.637082\pi$
$479$	−18.2734	−0.834932	−0.417466	+	0.908692i	$0.637082\pi$
$480$	4.07210	0.185865
$481$	−63.0566	−2.87514
$482$	15.5540	0.708463
$483$	11.1670	0.508116
$484$	1.00000	0.0454545
$485$	16.2884	0.739617
$486$	21.2884	0.965661
$487$	−42.0716	−1.90645	−0.953224	−	0.302265i	$-0.902257\pi$
$487$	−42.0716	−1.90645	−0.953224	+	0.302265i	$0.902257\pi$
$488$	−4.77698	−0.216244
$489$	−29.9144	−1.35278
$490$	−7.10164	−0.320819
$491$	35.9933	1.62435	0.812177	−	0.583411i	$-0.198282\pi$
$491$	35.9933	1.62435	0.812177	+	0.583411i	$0.198282\pi$
$492$	−12.2163	−0.550753
$493$	−2.77698	−0.125069
$494$	−32.5390	−1.46400
$495$	4.61151	0.207272
$496$	−0.776979	−0.0348874
$497$	−19.1379	−0.858453
$498$	21.7194	0.973271
$499$	−20.3159	−0.909466	−0.454733	−	0.890628i	$-0.650265\pi$
$499$	−20.3159	−0.909466	−0.454733	+	0.890628i	$0.650265\pi$
$500$	12.0934	0.540832
$501$	12.2163	0.545784
$502$	20.7044	0.924084
$503$	−16.7915	−0.748697	−0.374349	−	0.927288i	$-0.622134\pi$
$503$	−16.7915	−0.748697	−0.374349	+	0.927288i	$0.622134\pi$
$504$	−4.55396	−0.202849
$505$	2.14618	0.0955037
$506$	2.77698	0.123452
$507$	66.4306	2.95029
$508$	−5.95745	−0.264319
$509$	12.5602	0.556723	0.278361	−	0.960476i	$-0.410209\pi$
$509$	12.5602	0.556723	0.278361	+	0.960476i	$0.410209\pi$
$510$	4.07210	0.180315
$511$	5.01500	0.221850
$512$	−1.00000	−0.0441942
$513$	−3.59023	−0.158513
$514$	−10.5457	−0.465151
$515$	−1.74116	−0.0767246
$516$	8.84908	0.389559
$517$	−13.2588	−0.583123
$518$	16.6178	0.730144
$519$	−12.3735	−0.543136
$520$	10.8916	0.477630
$521$	31.6982	1.38872	0.694361	−	0.719627i	$-0.255687\pi$
$521$	31.6982	1.38872	0.694361	+	0.719627i	$0.255687\pi$
$522$	7.51141	0.328765
$523$	21.2588	0.929584	0.464792	−	0.885420i	$-0.346129\pi$
$523$	21.2588	0.929584	0.464792	+	0.885420i	$0.346129\pi$
$524$	−6.81953	−0.297913
$525$	−8.41803	−0.367393
$526$	23.3672	1.01886
$527$	−0.776979	−0.0338458
$528$	−2.38849	−0.103946
$529$	−15.2884	−0.664712
$530$	11.6265	0.505023
$531$	−25.4523	−1.10454
$532$	8.57524	0.371784
$533$	−32.6749	−1.41531
$534$	−20.6541	−0.893789
$535$	8.07164	0.348968
$536$	12.7770	0.551881
$537$	1.51141	0.0652220
$538$	7.76871	0.334933
$539$	4.16547	0.179419
$540$	1.20174	0.0517148
$541$	−8.54768	−0.367493	−0.183747	−	0.982974i	$-0.558823\pi$
$541$	−8.54768	−0.367493	−0.183747	+	0.982974i	$0.558823\pi$
$542$	−2.00000	−0.0859074
$543$	39.6914	1.70332
$544$	−1.00000	−0.0428746
$545$	6.05909	0.259543
$546$	−25.6899	−1.09943
$547$	−17.1930	−0.735121	−0.367560	−	0.930000i	$-0.619807\pi$
$547$	−17.1930	−0.735121	−0.367560	+	0.930000i	$0.619807\pi$
$548$	1.61151	0.0688403
$549$	12.9212	0.551462
$550$	−2.09337	−0.0892617
$551$	−14.1442	−0.602563
$552$	−6.63279	−0.282310
$553$	−4.02128	−0.171002
$554$	−21.6982	−0.921866
$555$	40.1930	1.70610
$556$	22.3309	0.947043
$557$	15.0295	0.636822	0.318411	−	0.947953i	$-0.396851\pi$
$557$	15.0295	0.636822	0.318411	+	0.947953i	$0.396851\pi$
$558$	2.10164	0.0889695
$559$	23.6686	1.00108
$560$	−2.87035	−0.121295
$561$	−2.38849	−0.100842
$562$	−29.5540	−1.24666
$563$	15.7900	0.665469	0.332734	−	0.943021i	$-0.392029\pi$
$563$	15.7900	0.665469	0.332734	+	0.943021i	$0.392029\pi$
$564$	31.6686	1.33349
$565$	14.6453	0.616134
$566$	14.7344	0.619334
$567$	−16.4964	−0.692784
$568$	11.3672	0.476958
$569$	27.5965	1.15691	0.578453	−	0.815716i	$-0.303657\pi$
$569$	27.5965	1.15691	0.578453	+	0.815716i	$0.303657\pi$
$570$	20.7407	0.868733
$571$	0.547680	0.0229197	0.0114598	−	0.999934i	$-0.496352\pi$
$571$	0.547680	0.0229197	0.0114598	+	0.999934i	$0.496352\pi$
$572$	−6.38849	−0.267116
$573$	4.29512	0.179431
$574$	8.61106	0.359419
$575$	−5.81325	−0.242429
$576$	2.70488	0.112703
$577$	−40.9357	−1.70418	−0.852088	−	0.523398i	$-0.824664\pi$
$577$	−40.9357	−1.70418	−0.852088	+	0.523398i	$0.824664\pi$
$578$	−1.00000	−0.0415945
$579$	−60.4519	−2.51229
$580$	4.73443	0.196586
$581$	−15.3097	−0.635152
$582$	22.8195	0.945900
$583$	−6.81953	−0.282436
$584$	−2.97872	−0.123260
$585$	−29.4606	−1.21805
$586$	−3.02954	−0.125149
$587$	9.22302	0.380675	0.190337	−	0.981719i	$-0.439042\pi$
$587$	9.22302	0.380675	0.190337	+	0.981719i	$0.439042\pi$
$588$	−9.94918	−0.410297
$589$	−3.95745	−0.163064
$590$	−16.0426	−0.660461
$591$	59.7272	2.45685
$592$	−9.87035	−0.405669
$593$	19.5114	0.801237	0.400619	−	0.916245i	$-0.368795\pi$
$593$	19.5114	0.801237	0.400619	+	0.916245i	$0.368795\pi$
$594$	−0.704883	−0.0289217
$595$	−2.87035	−0.117673
$596$	−0.345936	−0.0141701
$597$	33.8849	1.38682
$598$	−17.7407	−0.725471
$599$	−47.8549	−1.95530	−0.977649	−	0.210243i	$-0.932574\pi$
$599$	−47.8549	−1.95530	−0.977649	+	0.210243i	$0.932574\pi$
$600$	5.00000	0.204124
$601$	−4.94918	−0.201881	−0.100941	−	0.994892i	$-0.532185\pi$
$601$	−4.94918	−0.201881	−0.100941	+	0.994892i	$0.532185\pi$
$602$	−6.23757	−0.254224
$603$	−34.5602	−1.40740
$604$	−21.1079	−0.858869
$605$	−1.70488	−0.0693134
$606$	3.00673	0.122140
$607$	−13.7900	−0.559718	−0.279859	−	0.960041i	$-0.590288\pi$
$607$	−13.7900	−0.559718	−0.279859	+	0.960041i	$0.590288\pi$
$608$	−5.09337	−0.206564
$609$	−11.1670	−0.452510
$610$	8.14419	0.329749
$611$	84.7040	3.42676
$612$	2.70488	0.109338
$613$	46.5748	1.88114	0.940569	−	0.339603i	$-0.110293\pi$
$613$	46.5748	1.88114	0.940569	+	0.339603i	$0.110293\pi$
$614$	−16.2443	−0.655566
$615$	20.8273	0.839840
$616$	1.68361	0.0678344
$617$	−15.2821	−0.615234	−0.307617	−	0.951510i	$-0.599532\pi$
$617$	−15.2821	−0.615234	−0.307617	+	0.951510i	$0.599532\pi$
$618$	−2.43931	−0.0981234
$619$	48.3159	1.94198	0.970991	−	0.239117i	$-0.0768579\pi$
$619$	48.3159	1.94198	0.970991	+	0.239117i	$0.0768579\pi$
$620$	1.32466	0.0531996
$621$	−1.95745	−0.0785496
$622$	−25.1930	−1.01015
$623$	14.5587	0.583282
$624$	15.2588	0.610843
$625$	−10.1509	−0.406037
$626$	−4.96373	−0.198390
$627$	−12.1655	−0.485842
$628$	15.5114	0.618972
$629$	−9.87035	−0.393557
$630$	7.76397	0.309324
$631$	32.9062	1.30997	0.654987	−	0.755640i	$-0.272674\pi$
$631$	32.9062	1.30997	0.654987	+	0.755640i	$0.272674\pi$
$632$	2.38849	0.0950090
$633$	64.9503	2.58154
$634$	6.37349	0.253124
$635$	10.1568	0.403058
$636$	16.2884	0.645876
$637$	−26.6111	−1.05437
$638$	−2.77698	−0.109942
$639$	−30.7470	−1.21633
$640$	1.70488	0.0673914
$641$	−32.3309	−1.27700	−0.638498	−	0.769624i	$-0.720444\pi$
$641$	−32.3309	−1.27700	−0.638498	+	0.769624i	$0.720444\pi$
$642$	11.3081	0.446296
$643$	−5.26557	−0.207654	−0.103827	−	0.994595i	$-0.533109\pi$
$643$	−5.26557	−0.207654	−0.103827	+	0.994595i	$0.533109\pi$
$644$	4.67534	0.184234
$645$	−15.0866	−0.594036
$646$	−5.09337	−0.200396
$647$	−28.8211	−1.13307	−0.566537	−	0.824037i	$-0.691717\pi$
$647$	−28.8211	−1.13307	−0.566537	+	0.824037i	$0.691717\pi$
$648$	9.79826	0.384912
$649$	9.40977	0.369366
$650$	13.3735	0.524552
$651$	−3.12445	−0.122457
$652$	−12.5244	−0.490494
$653$	34.1587	1.33673	0.668367	−	0.743831i	$-0.266993\pi$
$653$	34.1587	1.33673	0.668367	+	0.743831i	$0.266993\pi$
$654$	8.48859	0.331930
$655$	11.6265	0.454285
$656$	−5.11465	−0.199694
$657$	8.05710	0.314337
$658$	−22.3227	−0.870228
$659$	33.2076	1.29358	0.646792	−	0.762667i	$-0.276110\pi$
$659$	33.2076	1.29358	0.646792	+	0.762667i	$0.276110\pi$
$660$	4.07210	0.158506
$661$	−23.0654	−0.897139	−0.448569	−	0.893748i	$-0.648066\pi$
$661$	−23.0654	−0.897139	−0.448569	+	0.893748i	$0.648066\pi$
$662$	10.4460	0.405997
$663$	15.2588	0.592604
$664$	9.09337	0.352891
$665$	−14.6198	−0.566931
$666$	26.6982	1.03453
$667$	−7.71162	−0.298595
$668$	5.11465	0.197892
$669$	−18.8999	−0.730712
$670$	−21.7833	−0.841561
$671$	−4.77698	−0.184413
$672$	−4.02128	−0.155124
$673$	−40.8128	−1.57322	−0.786609	−	0.617452i	$-0.788165\pi$
$673$	−40.8128	−1.57322	−0.786609	+	0.617452i	$0.788165\pi$
$674$	6.07838	0.234130
$675$	1.47558	0.0567952
$676$	27.8128	1.06972
$677$	−27.8423	−1.07007	−0.535034	−	0.844830i	$-0.679701\pi$
$677$	−27.8423	−1.07007	−0.535034	+	0.844830i	$0.679701\pi$
$678$	20.5177	0.787977
$679$	−16.0851	−0.617290
$680$	1.70488	0.0653793
$681$	−0.446041	−0.0170923
$682$	−0.776979	−0.0297521
$683$	4.74116	0.181415	0.0907077	−	0.995878i	$-0.471087\pi$
$683$	4.74116	0.181415	0.0907077	+	0.995878i	$0.471087\pi$
$684$	13.7770	0.526776
$685$	−2.74744	−0.104974
$686$	18.7983	0.717721
$687$	−25.3963	−0.968929
$688$	3.70488	0.141247
$689$	43.5665	1.65975
$690$	11.3081	0.430493
$691$	−28.8766	−1.09852	−0.549259	−	0.835652i	$-0.685090\pi$
$691$	−28.8766	−1.09852	−0.549259	+	0.835652i	$0.685090\pi$
$692$	−5.18047	−0.196932
$693$	−4.55396	−0.172991
$694$	35.9275	1.36379
$695$	−38.0716	−1.44414
$696$	6.63279	0.251415
$697$	−5.11465	−0.193731
$698$	−18.2884	−0.692226
$699$	46.2226	1.74830
$700$	−3.52442	−0.133210
$701$	45.2801	1.71021	0.855103	−	0.518458i	$-0.173494\pi$
$701$	45.2801	1.71021	0.855103	+	0.518458i	$0.173494\pi$
$702$	4.50314	0.169960
$703$	−50.2734	−1.89610
$704$	−1.00000	−0.0376889
$705$	−53.9913	−2.03343
$706$	−1.25884	−0.0473772
$707$	−2.11939	−0.0797081
$708$	−22.4751	−0.844667
$709$	−17.3227	−0.650567	−0.325283	−	0.945617i	$-0.605460\pi$
$709$	−17.3227	−0.650567	−0.325283	+	0.945617i	$0.605460\pi$
$710$	−19.3798	−0.727310
$711$	−6.46059	−0.242291
$712$	−8.64733	−0.324072
$713$	−2.15766	−0.0808049
$714$	−4.02128	−0.150493
$715$	10.8916	0.407324
$716$	0.632787	0.0236484
$717$	12.3735	0.462097
$718$	−26.3309	−0.982662
$719$	36.9337	1.37740	0.688698	−	0.725048i	$-0.258183\pi$
$719$	36.9337	1.37740	0.688698	+	0.725048i	$0.258183\pi$
$720$	−4.61151	−0.171861
$721$	1.71943	0.0640349
$722$	−6.94245	−0.258371
$723$	−37.1505	−1.38164
$724$	16.6178	0.617595
$725$	5.81325	0.215899
$726$	−2.38849	−0.0886452
$727$	18.9487	0.702769	0.351385	−	0.936231i	$-0.385711\pi$
$727$	18.9487	0.702769	0.351385	+	0.936231i	$0.385711\pi$
$728$	−10.7557	−0.398633
$729$	−21.4523	−0.794530
$730$	5.07838	0.187959
$731$	3.70488	0.137030
$732$	11.4098	0.421717
$733$	14.4393	0.533328	0.266664	−	0.963790i	$-0.414079\pi$
$733$	14.4393	0.533328	0.266664	+	0.963790i	$0.414079\pi$
$734$	23.0654	0.851358
$735$	16.9622	0.625660
$736$	−2.77698	−0.102361
$737$	12.7770	0.470646
$738$	13.8345	0.509256
$739$	22.4901	0.827313	0.413656	−	0.910433i	$-0.364251\pi$
$739$	22.4901	0.827313	0.413656	+	0.910433i	$0.364251\pi$
$740$	16.8278	0.618602
$741$	77.7190	2.85508
$742$	−11.4814	−0.421496
$743$	−12.4393	−0.456354	−0.228177	−	0.973620i	$-0.573276\pi$
$743$	−12.4393	−0.456354	−0.228177	+	0.973620i	$0.573276\pi$
$744$	1.85581	0.0680372
$745$	0.589781	0.0216079
$746$	−12.7324	−0.466168
$747$	−24.5965	−0.899939
$748$	−1.00000	−0.0365636
$749$	−7.97091	−0.291251
$750$	−28.8849	−1.05473
$751$	4.40349	0.160686	0.0803428	−	0.996767i	$-0.474399\pi$
$751$	4.40349	0.160686	0.0803428	+	0.996767i	$0.474399\pi$
$752$	13.2588	0.483500
$753$	−49.4523	−1.80214
$754$	17.7407	0.646078
$755$	35.9865	1.30968
$756$	−1.18675	−0.0431615
$757$	−21.8684	−0.794819	−0.397410	−	0.917641i	$-0.630091\pi$
$757$	−21.8684	−0.794819	−0.397410	+	0.917641i	$0.630091\pi$
$758$	−14.8854	−0.540660
$759$	−6.63279	−0.240755
$760$	8.68361	0.314988
$761$	10.4886	0.380211	0.190106	−	0.981764i	$-0.439117\pi$
$761$	10.4886	0.380211	0.190106	+	0.981764i	$0.439117\pi$
$762$	14.2293	0.515473
$763$	−5.98347	−0.216616
$764$	1.79826	0.0650586
$765$	−4.61151	−0.166730
$766$	−3.91489	−0.141451
$767$	−60.1142	−2.17060
$768$	2.38849	0.0861872
$769$	−27.2947	−0.984271	−0.492135	−	0.870519i	$-0.663784\pi$
$769$	−27.2947	−0.984271	−0.492135	+	0.870519i	$0.663784\pi$
$770$	−2.87035	−0.103440
$771$	25.1883	0.907134
$772$	−25.3097	−0.910915
$773$	33.4689	1.20379	0.601895	−	0.798575i	$-0.294412\pi$
$773$	33.4689	1.20379	0.601895	+	0.798575i	$0.294412\pi$
$774$	−10.0213	−0.360207
$775$	1.62651	0.0584259
$776$	9.55396	0.342967
$777$	−39.6914	−1.42392
$778$	−37.6982	−1.35154
$779$	−26.0508	−0.933367
$780$	−26.0145	−0.931470
$781$	11.3672	0.406751
$782$	−2.77698	−0.0993046
$783$	1.95745	0.0699534
$784$	−4.16547	−0.148767
$785$	−26.4451	−0.943867
$786$	16.2884	0.580987
$787$	14.8195	0.528259	0.264130	−	0.964487i	$-0.414915\pi$
$787$	14.8195	0.528259	0.264130	+	0.964487i	$0.414915\pi$
$788$	25.0063	0.890812
$789$	−55.8123	−1.98697
$790$	−4.07210	−0.144879
$791$	−14.4626	−0.514230
$792$	2.70488	0.0961138
$793$	30.5177	1.08371
$794$	−19.0508	−0.676088
$795$	−27.7698	−0.984893
$796$	14.1867	0.502836
$797$	−15.8423	−0.561165	−0.280582	−	0.959830i	$-0.590528\pi$
$797$	−15.8423	−0.561165	−0.280582	+	0.959830i	$0.590528\pi$
$798$	−20.4819	−0.725050
$799$	13.2588	0.469064
$800$	2.09337	0.0740119
$801$	23.3900	0.826446
$802$	13.0063	0.459268
$803$	−2.97872	−0.105117
$804$	−30.5177	−1.07628
$805$	−7.97091	−0.280938
$806$	4.96373	0.174840
$807$	−18.5555	−0.653184
$808$	1.25884	0.0442859
$809$	−12.8195	−0.450711	−0.225355	−	0.974277i	$-0.572354\pi$
$809$	−12.8195	−0.450711	−0.225355	+	0.974277i	$0.572354\pi$
$810$	−16.7049	−0.586950
$811$	15.6391	0.549162	0.274581	−	0.961564i	$-0.411461\pi$
$811$	15.6391	0.549162	0.274581	+	0.961564i	$0.411461\pi$
$812$	−4.67534	−0.164072
$813$	4.77698	0.167536
$814$	−9.87035	−0.345956
$815$	21.3527	0.747951
$816$	2.38849	0.0836139
$817$	18.8704	0.660190
$818$	−2.81953	−0.0985827
$819$	29.0929	1.01659
$820$	8.71988	0.304511
$821$	−24.8195	−0.866208	−0.433104	−	0.901344i	$-0.642582\pi$
$821$	−24.8195	−0.866208	−0.433104	+	0.901344i	$0.642582\pi$
$822$	−3.84908	−0.134252
$823$	−1.66906	−0.0581798	−0.0290899	−	0.999577i	$-0.509261\pi$
$823$	−1.66906	−0.0581798	−0.0290899	+	0.999577i	$0.509261\pi$
$824$	−1.02128	−0.0355779
$825$	5.00000	0.174078
$826$	15.8423	0.551226
$827$	16.8486	0.585884	0.292942	−	0.956130i	$-0.405366\pi$
$827$	16.8486	0.585884	0.292942	+	0.956130i	$0.405366\pi$
$828$	7.51141	0.261039
$829$	−48.0716	−1.66960	−0.834799	−	0.550555i	$-0.814416\pi$
$829$	−48.0716	−1.66960	−0.834799	+	0.550555i	$0.814416\pi$
$830$	−15.5031	−0.538122
$831$	51.8258	1.79782
$832$	6.38849	0.221481
$833$	−4.16547	−0.144325
$834$	−53.3372	−1.84692
$835$	−8.71988	−0.301764
$836$	−5.09337	−0.176158
$837$	0.547680	0.0189306
$838$	7.32267	0.252958
$839$	−50.9637	−1.75946	−0.879732	−	0.475471i	$-0.842278\pi$
$839$	−50.9637	−1.75946	−0.879732	+	0.475471i	$0.842278\pi$
$840$	6.85581	0.236548
$841$	−21.2884	−0.734082
$842$	−13.5540	−0.467100
$843$	70.5893	2.43123
$844$	27.1930	0.936023
$845$	−47.4176	−1.63121
$846$	−35.8636	−1.23302
$847$	1.68361	0.0578494
$848$	6.81953	0.234184
$849$	−35.1930	−1.20782
$850$	2.09337	0.0718021
$851$	−27.4098	−0.939595
$852$	−27.1505	−0.930160
$853$	−37.0228	−1.26764	−0.633818	−	0.773482i	$-0.718513\pi$
$853$	−37.0228	−1.26764	−0.633818	+	0.773482i	$0.718513\pi$
$854$	−8.04255	−0.275211
$855$	−23.4881	−0.803278
$856$	4.73443	0.161819
$857$	−41.2376	−1.40865	−0.704324	−	0.709878i	$-0.748750\pi$
$857$	−41.2376	−1.40865	−0.704324	+	0.709878i	$0.748750\pi$
$858$	15.2588	0.520928
$859$	−30.5177	−1.04125	−0.520625	−	0.853785i	$-0.674301\pi$
$859$	−30.5177	−1.04125	−0.520625	+	0.853785i	$0.674301\pi$
$860$	−6.31639	−0.215387
$861$	−20.5674	−0.700936
$862$	2.40150	0.0817954
$863$	−0.375479	−0.0127814	−0.00639072	−	0.999980i	$-0.502034\pi$
$863$	−0.375479	−0.0127814	−0.00639072	+	0.999980i	$0.502034\pi$
$864$	0.704883	0.0239806
$865$	8.83209	0.300300
$866$	−25.5472	−0.868130
$867$	2.38849	0.0811174
$868$	−1.30813	−0.0444007
$869$	2.38849	0.0810240
$870$	−11.3081	−0.383381
$871$	−81.6256	−2.76578
$872$	3.55396	0.120352
$873$	−25.8423	−0.874631
$874$	−14.1442	−0.478434
$875$	20.3605	0.688310
$876$	7.11465	0.240382
$877$	34.2584	1.15682	0.578412	−	0.815745i	$-0.303673\pi$
$877$	34.2584	1.15682	0.578412	+	0.815745i	$0.303673\pi$
$878$	−12.2226	−0.412492
$879$	7.23603	0.244065
$880$	1.70488	0.0574716
$881$	38.3309	1.29140	0.645701	−	0.763590i	$-0.276565\pi$
$881$	38.3309	1.29140	0.645701	+	0.763590i	$0.276565\pi$
$882$	11.2671	0.379383
$883$	35.7833	1.20420	0.602101	−	0.798420i	$-0.294330\pi$
$883$	35.7833	1.20420	0.602101	+	0.798420i	$0.294330\pi$
$884$	6.38849	0.214868
$885$	38.3175	1.28803
$886$	34.6619	1.16449
$887$	15.4078	0.517343	0.258671	−	0.965965i	$-0.416715\pi$
$887$	15.4078	0.517343	0.258671	+	0.965965i	$0.416715\pi$
$888$	23.5752	0.791133
$889$	−10.0300	−0.336395
$890$	14.7427	0.494176
$891$	9.79826	0.328254
$892$	−7.91291	−0.264944
$893$	67.5322	2.25988
$894$	0.826265	0.0276344
$895$	−1.07883	−0.0360612
$896$	−1.68361	−0.0562453
$897$	42.3735	1.41481
$898$	−21.9700	−0.733149
$899$	2.15766	0.0719619
$900$	−5.66233	−0.188744
$901$	6.81953	0.227192
$902$	−5.11465	−0.170299
$903$	14.8984	0.495786
$904$	8.59023	0.285707
$905$	−28.3314	−0.941767
$906$	50.4160	1.67496
$907$	30.3885	1.00903	0.504517	−	0.863402i	$-0.331671\pi$
$907$	30.3885	1.00903	0.504517	+	0.863402i	$0.331671\pi$
$908$	−0.186746	−0.00619739
$909$	−3.40502	−0.112937
$910$	18.3372	0.607873
$911$	−28.2158	−0.934832	−0.467416	−	0.884037i	$-0.654815\pi$
$911$	−28.2158	−0.934832	−0.467416	+	0.884037i	$0.654815\pi$
$912$	12.1655	0.402839
$913$	9.09337	0.300947
$914$	27.7407	0.917581
$915$	−19.4523	−0.643074
$916$	−10.6328	−0.351317
$917$	−11.4814	−0.379150
$918$	0.704883	0.0232646
$919$	−8.28838	−0.273409	−0.136704	−	0.990612i	$-0.543651\pi$
$919$	−8.28838	−0.273409	−0.136704	+	0.990612i	$0.543651\pi$
$920$	4.73443	0.156089
$921$	38.7993	1.27848
$922$	27.0504	0.890856
$923$	−72.6193	−2.39029
$924$	−4.02128	−0.132290
$925$	20.6623	0.679373
$926$	37.6832	1.23835
$927$	2.76243	0.0907303
$928$	2.77698	0.0911588
$929$	8.01653	0.263014	0.131507	−	0.991315i	$-0.458018\pi$
$929$	8.01653	0.263014	0.131507	+	0.991315i	$0.458018\pi$
$930$	−3.16394	−0.103750
$931$	−21.2163	−0.695336
$932$	19.3522	0.633903
$933$	60.1733	1.96998
$934$	−26.7770	−0.876170
$935$	1.70488	0.0557556
$936$	−17.2801	−0.564818
$937$	20.3735	0.665573	0.332786	−	0.943002i	$-0.392011\pi$
$937$	20.3735	0.665573	0.332786	+	0.943002i	$0.392011\pi$
$938$	21.5114	0.702372
$939$	11.8558	0.386900
$940$	−22.6048	−0.737287
$941$	−12.3144	−0.401438	−0.200719	−	0.979649i	$-0.564328\pi$
$941$	−12.3144	−0.401438	−0.200719	+	0.979649i	$0.564328\pi$
$942$	−37.0488	−1.20712
$943$	−14.2033	−0.462522
$944$	−9.40977	−0.306262
$945$	2.02326	0.0658168
$946$	3.70488	0.120456
$947$	−6.88535	−0.223744	−0.111872	−	0.993723i	$-0.535685\pi$
$947$	−6.88535	−0.223744	−0.111872	+	0.993723i	$0.535685\pi$
$948$	−5.70488	−0.185286
$949$	19.0295	0.617725
$950$	10.6623	0.345932
$951$	−15.2230	−0.493640
$952$	−1.68361	−0.0545660
$953$	−50.1993	−1.62611	−0.813057	−	0.582183i	$-0.802199\pi$
$953$	−50.1993	−1.62611	−0.813057	+	0.582183i	$0.802199\pi$
$954$	−18.4460	−0.597213
$955$	−3.06582	−0.0992075
$956$	5.18047	0.167548
$957$	6.63279	0.214408
$958$	18.2734	0.590386
$959$	2.71315	0.0876122
$960$	−4.07210	−0.131426
$961$	−30.3963	−0.980526
$962$	63.0566	2.03303
$963$	−12.8061	−0.412670
$964$	−15.5540	−0.500959
$965$	43.1500	1.38905
$966$	−11.1670	−0.359292
$967$	8.50513	0.273506	0.136753	−	0.990605i	$-0.456333\pi$
$967$	8.50513	0.273506	0.136753	+	0.990605i	$0.456333\pi$
$968$	−1.00000	−0.0321412
$969$	12.1655	0.390811
$970$	−16.2884	−0.522989
$971$	9.46885	0.303870	0.151935	−	0.988390i	$-0.451450\pi$
$971$	9.46885	0.303870	0.151935	+	0.988390i	$0.451450\pi$
$972$	−21.2884	−0.682825
$973$	37.5965	1.20529
$974$	42.0716	1.34806
$975$	−31.9424	−1.02298
$976$	4.77698	0.152907
$977$	39.7765	1.27256	0.636282	−	0.771457i	$-0.280472\pi$
$977$	39.7765	1.27256	0.636282	+	0.771457i	$0.280472\pi$
$978$	29.9144	0.956558
$979$	−8.64733	−0.276370
$980$	7.10164	0.226854
$981$	−9.61304	−0.306921
$982$	−35.9933	−1.14859
$983$	−14.9637	−0.477269	−0.238634	−	0.971109i	$-0.576700\pi$
$983$	−14.9637	−0.477269	−0.238634	+	0.971109i	$0.576700\pi$
$984$	12.2163	0.389441
$985$	−42.6328	−1.35839
$986$	2.77698	0.0884370
$987$	53.3175	1.69711
$988$	32.5390	1.03520
$989$	10.2884	0.327152
$990$	−4.61151	−0.146563
$991$	−4.63279	−0.147165	−0.0735827	−	0.997289i	$-0.523443\pi$
$991$	−4.63279	−0.147165	−0.0735827	+	0.997289i	$0.523443\pi$
$992$	0.776979	0.0246691
$993$	−24.9503	−0.791773
$994$	19.1379	0.607018
$995$	−24.1867	−0.766771
$996$	−21.7194	−0.688206
$997$	57.8123	1.83094	0.915468	−	0.402391i	$-0.131821\pi$
$997$	57.8123	1.83094	0.915468	+	0.402391i	$0.131821\pi$
$998$	20.3159	0.643090
$999$	6.95745	0.220124

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	374.2.a.b.1.3	✓	3
3.2	odd	2		3366.2.a.bd.1.2		3
4.3	odd	2		2992.2.a.q.1.1		3
5.4	even	2		9350.2.a.cf.1.1		3
11.10	odd	2		4114.2.a.s.1.3		3
17.16	even	2		6358.2.a.o.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.a.b.1.3	✓	3	1.1	even	1	trivial
2992.2.a.q.1.1		3	4.3	odd	2
3366.2.a.bd.1.2		3	3.2	odd	2
4114.2.a.s.1.3		3	11.10	odd	2
6358.2.a.o.1.1		3	17.16	even	2
9350.2.a.cf.1.1		3	5.4	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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	374.2.a.b.1.3

Level	$374$
Weight	$2$
Character	374.1
Self dual	yes
Analytic conductor	$2.986$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$