LMFDB - Embedded newform 4719.2.a.bc.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4719,2,Mod(1,4719)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4719, 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("4719.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4719 = 3 \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4719.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-1,5,3,-2,-1,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$37.6814047138$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.303952.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 2x^{2} + 5x + 1 $ x^5 - x^4 - 6x^3 + 2x^2 + 5*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.97156$ of defining polynomial
Character	$\chi$	$=$	4719.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.97156 q^{2} +1.00000 q^{3} +1.88703 q^{4} -3.56577 q^{5} +1.97156 q^{6} +1.46434 q^{7} -0.222723 q^{8} +1.00000 q^{9} -7.03011 q^{10} +1.88703 q^{12} +1.00000 q^{13} +2.88703 q^{14} -3.56577 q^{15} -4.21317 q^{16} +0.794179 q^{17} +1.97156 q^{18} -2.39136 q^{19} -6.72872 q^{20} +1.46434 q^{21} -6.10021 q^{23} -0.222723 q^{24} +7.71470 q^{25} +1.97156 q^{26} +1.00000 q^{27} +2.76326 q^{28} +1.87549 q^{29} -7.03011 q^{30} +1.17721 q^{31} -7.86106 q^{32} +1.56577 q^{34} -5.22150 q^{35} +1.88703 q^{36} -8.90898 q^{37} -4.71470 q^{38} +1.00000 q^{39} +0.794179 q^{40} +9.06607 q^{41} +2.88703 q^{42} -10.3374 q^{43} -3.56577 q^{45} -12.0269 q^{46} -6.54846 q^{47} -4.21317 q^{48} -4.85570 q^{49} +15.2100 q^{50} +0.794179 q^{51} +1.88703 q^{52} +2.47020 q^{53} +1.97156 q^{54} -0.326143 q^{56} -2.39136 q^{57} +3.69763 q^{58} +5.15957 q^{59} -6.72872 q^{60} -11.3955 q^{61} +2.32094 q^{62} +1.46434 q^{63} -7.07217 q^{64} -3.56577 q^{65} -3.56577 q^{67} +1.49864 q^{68} -6.10021 q^{69} -10.2945 q^{70} -0.837051 q^{71} -0.222723 q^{72} -9.22175 q^{73} -17.5645 q^{74} +7.71470 q^{75} -4.51257 q^{76} +1.97156 q^{78} +2.57995 q^{79} +15.0232 q^{80} +1.00000 q^{81} +17.8743 q^{82} +5.68748 q^{83} +2.76326 q^{84} -2.83186 q^{85} -20.3807 q^{86} +1.87549 q^{87} -14.2488 q^{89} -7.03011 q^{90} +1.46434 q^{91} -11.5113 q^{92} +1.17721 q^{93} -12.9106 q^{94} +8.52704 q^{95} -7.86106 q^{96} -15.9779 q^{97} -9.57329 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 5 q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 6 q^{7} - 9 q^{8} + 5 q^{9} - 6 q^{10} + 3 q^{12} + 5 q^{13} + 8 q^{14} - 2 q^{15} + 11 q^{16} - 2 q^{17} - q^{18} - 12 q^{19} - 8 q^{20} - 6 q^{21}+ \cdots + q^{98}+O(q^{100}) $ 5 * q - q^2 + 5 * q^3 + 3 * q^4 - 2 * q^5 - q^6 - 6 * q^7 - 9 * q^8 + 5 * q^9 - 6 * q^10 + 3 * q^12 + 5 * q^13 + 8 * q^14 - 2 * q^15 + 11 * q^16 - 2 * q^17 - q^18 - 12 * q^19 - 8 * q^20 - 6 * q^21 + 8 * q^23 - 9 * q^24 + 7 * q^25 - q^26 + 5 * q^27 - 6 * q^29 - 6 * q^30 - 4 * q^31 - 19 * q^32 - 8 * q^34 + 2 * q^35 + 3 * q^36 - 10 * q^37 + 8 * q^38 + 5 * q^39 - 2 * q^40 - 6 * q^41 + 8 * q^42 - 26 * q^43 - 2 * q^45 - 26 * q^46 - 16 * q^47 + 11 * q^48 - 11 * q^49 + 39 * q^50 - 2 * q^51 + 3 * q^52 + 6 * q^53 - q^54 + 24 * q^56 - 12 * q^57 - 20 * q^58 + 6 * q^59 - 8 * q^60 - 14 * q^61 + 12 * q^62 - 6 * q^63 + 15 * q^64 - 2 * q^65 - 2 * q^67 + 12 * q^68 + 8 * q^69 - 10 * q^70 - 14 * q^71 - 9 * q^72 - 12 * q^73 - 36 * q^74 + 7 * q^75 - 18 * q^76 - q^78 + 8 * q^79 + 8 * q^80 + 5 * q^81 + 38 * q^82 + 22 * q^83 - 36 * q^85 - 16 * q^86 - 6 * q^87 - 8 * q^89 - 6 * q^90 - 6 * q^91 + 12 * q^92 - 4 * q^93 + 10 * q^94 - 32 * q^95 - 19 * q^96 + q^98

Coefficient data

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

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

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

$2$	1.97156	1.39410	0.697050	−	0.717022i	$-0.254495\pi$
$2$	1.97156	1.39410	0.697050	+	0.717022i	$0.254495\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.88703	0.943516
$5$	−3.56577	−1.59466	−0.797330	−	0.603544i	$-0.793755\pi$
$5$	−3.56577	−1.59466	−0.797330	+	0.603544i	$0.793755\pi$
$6$	1.97156	0.804884
$7$	1.46434	0.553469	0.276735	−	0.960946i	$-0.410748\pi$
$7$	1.46434	0.553469	0.276735	+	0.960946i	$0.410748\pi$
$8$	−0.222723	−0.0787445
$9$	1.00000	0.333333
$10$	−7.03011	−2.22312
$11$	0	0
$11$	0	0
$12$	1.88703	0.544739
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.88703	0.771592
$15$	−3.56577	−0.920677
$16$	−4.21317	−1.05329
$17$	0.794179	0.192617	0.0963083	−	0.995352i	$-0.469297\pi$
$17$	0.794179	0.192617	0.0963083	+	0.995352i	$0.469297\pi$
$18$	1.97156	0.464700
$19$	−2.39136	−0.548616	−0.274308	−	0.961642i	$-0.588449\pi$
$19$	−2.39136	−0.548616	−0.274308	+	0.961642i	$0.588449\pi$
$20$	−6.72872	−1.50459
$21$	1.46434	0.319546
$22$	0	0
$23$	−6.10021	−1.27198	−0.635990	−	0.771697i	$-0.719408\pi$
$23$	−6.10021	−1.27198	−0.635990	+	0.771697i	$0.719408\pi$
$24$	−0.222723	−0.0454632
$25$	7.71470	1.54294
$26$	1.97156	0.386654
$27$	1.00000	0.192450
$28$	2.76326	0.522207
$29$	1.87549	0.348270	0.174135	−	0.984722i	$-0.444287\pi$
$29$	1.87549	0.348270	0.174135	+	0.984722i	$0.444287\pi$
$30$	−7.03011	−1.28352
$31$	1.17721	0.211433	0.105717	−	0.994396i	$-0.466286\pi$
$31$	1.17721	0.211433	0.105717	+	0.994396i	$0.466286\pi$
$32$	−7.86106	−1.38965
$33$	0	0
$34$	1.56577	0.268527
$35$	−5.22150	−0.882595
$36$	1.88703	0.314505
$37$	−8.90898	−1.46463	−0.732314	−	0.680968i	$-0.761559\pi$
$37$	−8.90898	−1.46463	−0.732314	+	0.680968i	$0.761559\pi$
$38$	−4.71470	−0.764825
$39$	1.00000	0.160128
$40$	0.794179	0.125571
$41$	9.06607	1.41588	0.707941	−	0.706271i	$-0.249624\pi$
$41$	9.06607	1.41588	0.707941	+	0.706271i	$0.249624\pi$
$42$	2.88703	0.445479
$43$	−10.3374	−1.57643	−0.788216	−	0.615399i	$-0.788995\pi$
$43$	−10.3374	−1.57643	−0.788216	+	0.615399i	$0.788995\pi$
$44$	0	0
$45$	−3.56577	−0.531553
$46$	−12.0269	−1.77327
$47$	−6.54846	−0.955190	−0.477595	−	0.878580i	$-0.658491\pi$
$47$	−6.54846	−0.955190	−0.477595	+	0.878580i	$0.658491\pi$
$48$	−4.21317	−0.608119
$49$	−4.85570	−0.693672
$50$	15.2100	2.15101
$51$	0.794179	0.111207
$52$	1.88703	0.261684
$53$	2.47020	0.339308	0.169654	−	0.985504i	$-0.445735\pi$
$53$	2.47020	0.339308	0.169654	+	0.985504i	$0.445735\pi$
$54$	1.97156	0.268295
$55$	0	0
$56$	−0.326143	−0.0435827
$57$	−2.39136	−0.316743
$58$	3.69763	0.485523
$59$	5.15957	0.671719	0.335859	−	0.941912i	$-0.390973\pi$
$59$	5.15957	0.671719	0.335859	+	0.941912i	$0.390973\pi$
$60$	−6.72872	−0.868674
$61$	−11.3955	−1.45904	−0.729522	−	0.683957i	$-0.760258\pi$
$61$	−11.3955	−1.45904	−0.729522	+	0.683957i	$0.760258\pi$
$62$	2.32094	0.294759
$63$	1.46434	0.184490
$64$	−7.07217	−0.884022
$65$	−3.56577	−0.442279
$66$	0	0
$67$	−3.56577	−0.435628	−0.217814	−	0.975990i	$-0.569893\pi$
$67$	−3.56577	−0.435628	−0.217814	+	0.975990i	$0.569893\pi$
$68$	1.49864	0.181737
$69$	−6.10021	−0.734379
$70$	−10.2945	−1.23043
$71$	−0.837051	−0.0993396	−0.0496698	−	0.998766i	$-0.515817\pi$
$71$	−0.837051	−0.0993396	−0.0496698	+	0.998766i	$0.515817\pi$
$72$	−0.222723	−0.0262482
$73$	−9.22175	−1.07932	−0.539662	−	0.841882i	$-0.681448\pi$
$73$	−9.22175	−1.07932	−0.539662	+	0.841882i	$0.681448\pi$
$74$	−17.5645	−2.04184
$75$	7.71470	0.890817
$76$	−4.51257	−0.517628
$77$	0	0
$78$	1.97156	0.223235
$79$	2.57995	0.290267	0.145134	−	0.989412i	$-0.453639\pi$
$79$	2.57995	0.290267	0.145134	+	0.989412i	$0.453639\pi$
$80$	15.0232	1.67965
$81$	1.00000	0.111111
$82$	17.8743	1.97388
$83$	5.68748	0.624282	0.312141	−	0.950036i	$-0.398954\pi$
$83$	5.68748	0.624282	0.312141	+	0.950036i	$0.398954\pi$
$84$	2.76326	0.301496
$85$	−2.83186	−0.307158
$86$	−20.3807	−2.19770
$87$	1.87549	0.201074
$88$	0	0
$89$	−14.2488	−1.51037	−0.755185	−	0.655511i	$-0.772453\pi$
$89$	−14.2488	−1.51037	−0.755185	+	0.655511i	$0.772453\pi$
$90$	−7.03011	−0.741039
$91$	1.46434	0.153505
$92$	−11.5113	−1.20013
$93$	1.17721	0.122071
$94$	−12.9106	−1.33163
$95$	8.52704	0.874855
$96$	−7.86106	−0.802316
$97$	−15.9779	−1.62231	−0.811153	−	0.584834i	$-0.801160\pi$
$97$	−15.9779	−1.62231	−0.811153	+	0.584834i	$0.801160\pi$
$98$	−9.57329	−0.967048
$99$	0	0
$100$	14.5579	1.45579
$101$	7.53716	0.749975	0.374988	−	0.927030i	$-0.377647\pi$
$101$	7.53716	0.749975	0.374988	+	0.927030i	$0.377647\pi$
$102$	1.56577	0.155034
$103$	−10.7613	−1.06034	−0.530171	−	0.847891i	$-0.677872\pi$
$103$	−10.7613	−1.06034	−0.530171	+	0.847891i	$0.677872\pi$
$104$	−0.222723	−0.0218398
$105$	−5.22150	−0.509567
$106$	4.87013	0.473029
$107$	−11.1266	−1.07565	−0.537824	−	0.843057i	$-0.680753\pi$
$107$	−11.1266	−1.07565	−0.537824	+	0.843057i	$0.680753\pi$
$108$	1.88703	0.181580
$109$	7.57007	0.725082	0.362541	−	0.931968i	$-0.381909\pi$
$109$	7.57007	0.725082	0.362541	+	0.931968i	$0.381909\pi$
$110$	0	0
$111$	−8.90898	−0.845603
$112$	−6.16953	−0.582966
$113$	−1.91836	−0.180464	−0.0902321	−	0.995921i	$-0.528761\pi$
$113$	−1.91836	−0.180464	−0.0902321	+	0.995921i	$0.528761\pi$
$114$	−4.71470	−0.441572
$115$	21.7519	2.02838
$116$	3.53911	0.328598
$117$	1.00000	0.0924500
$118$	10.1724	0.936443
$119$	1.16295	0.106607
$120$	0.794179	0.0724983
$121$	0	0
$122$	−22.4669	−2.03405
$123$	9.06607	0.817460
$124$	2.22143	0.199491
$125$	−9.67999	−0.865805
$126$	2.88703	0.257197
$127$	15.3105	1.35859	0.679293	−	0.733867i	$-0.262286\pi$
$127$	15.3105	1.35859	0.679293	+	0.733867i	$0.262286\pi$
$128$	1.77894	0.157238
$129$	−10.3374	−0.910154
$130$	−7.03011	−0.616581
$131$	0.371899	0.0324930	0.0162465	−	0.999868i	$-0.494828\pi$
$131$	0.371899	0.0324930	0.0162465	+	0.999868i	$0.494828\pi$
$132$	0	0
$133$	−3.50177	−0.303642
$134$	−7.03011	−0.607309
$135$	−3.56577	−0.306892
$136$	−0.176882	−0.0151675
$137$	−0.528305	−0.0451361	−0.0225681	−	0.999745i	$-0.507184\pi$
$137$	−0.528305	−0.0451361	−0.0225681	+	0.999745i	$0.507184\pi$
$138$	−12.0269	−1.02380
$139$	−22.9271	−1.94465	−0.972324	−	0.233638i	$-0.924937\pi$
$139$	−22.9271	−1.94465	−0.972324	+	0.233638i	$0.924937\pi$
$140$	−9.85314	−0.832743
$141$	−6.54846	−0.551479
$142$	−1.65029	−0.138489
$143$	0	0
$144$	−4.21317	−0.351098
$145$	−6.68756	−0.555372
$146$	−18.1812	−1.50469
$147$	−4.85570	−0.400492
$148$	−16.8115	−1.38190
$149$	16.9332	1.38722	0.693609	−	0.720352i	$-0.256020\pi$
$149$	16.9332	1.38722	0.693609	+	0.720352i	$0.256020\pi$
$150$	15.2100	1.24189
$151$	−15.3083	−1.24577	−0.622887	−	0.782312i	$-0.714041\pi$
$151$	−15.3083	−1.24577	−0.622887	+	0.782312i	$0.714041\pi$
$152$	0.532611	0.0432005
$153$	0.794179	0.0642056
$154$	0	0
$155$	−4.19766	−0.337164
$156$	1.88703	0.151083
$157$	13.8182	1.10281	0.551406	−	0.834237i	$-0.314091\pi$
$157$	13.8182	1.10281	0.551406	+	0.834237i	$0.314091\pi$
$158$	5.08652	0.404662
$159$	2.47020	0.195899
$160$	28.0307	2.21602
$161$	−8.93279	−0.704002
$162$	1.97156	0.154900
$163$	12.2768	0.961597	0.480798	−	0.876831i	$-0.340347\pi$
$163$	12.2768	0.961597	0.480798	+	0.876831i	$0.340347\pi$
$164$	17.1080	1.33591
$165$	0	0
$166$	11.2132	0.870311
$167$	24.5032	1.89612	0.948058	−	0.318097i	$-0.103044\pi$
$167$	24.5032	1.89612	0.948058	+	0.318097i	$0.103044\pi$
$168$	−0.326143	−0.0251625
$169$	1.00000	0.0769231
$170$	−5.58316	−0.428209
$171$	−2.39136	−0.182872
$172$	−19.5069	−1.48739
$173$	1.80861	0.137506	0.0687529	−	0.997634i	$-0.478098\pi$
$173$	1.80861	0.137506	0.0687529	+	0.997634i	$0.478098\pi$
$174$	3.69763	0.280317
$175$	11.2970	0.853970
$176$	0	0
$177$	5.15957	0.387817
$178$	−28.0923	−2.10561
$179$	24.7330	1.84863	0.924317	−	0.381627i	$-0.124636\pi$
$179$	24.7330	1.84863	0.924317	+	0.381627i	$0.124636\pi$
$180$	−6.72872	−0.501529
$181$	12.0722	0.897318	0.448659	−	0.893703i	$-0.351902\pi$
$181$	12.0722	0.897318	0.448659	+	0.893703i	$0.351902\pi$
$182$	2.88703	0.214001
$183$	−11.3955	−0.842380
$184$	1.35866	0.100162
$185$	31.7674	2.33558
$186$	2.32094	0.170179
$187$	0	0
$188$	−12.3571	−0.901237
$189$	1.46434	0.106515
$190$	16.8115	1.21964
$191$	−0.858754	−0.0621373	−0.0310686	−	0.999517i	$-0.509891\pi$
$191$	−0.858754	−0.0621373	−0.0310686	+	0.999517i	$0.509891\pi$
$192$	−7.07217	−0.510390
$193$	−14.8820	−1.07123	−0.535616	−	0.844462i	$-0.679921\pi$
$193$	−14.8820	−1.07123	−0.535616	+	0.844462i	$0.679921\pi$
$194$	−31.5012	−2.26166
$195$	−3.56577	−0.255350
$196$	−9.16287	−0.654490
$197$	−12.1366	−0.864695	−0.432347	−	0.901707i	$-0.642315\pi$
$197$	−12.1366	−0.864695	−0.432347	+	0.901707i	$0.642315\pi$
$198$	0	0
$199$	−2.01898	−0.143122	−0.0715609	−	0.997436i	$-0.522798\pi$
$199$	−2.01898	−0.143122	−0.0715609	+	0.997436i	$0.522798\pi$
$200$	−1.71824	−0.121498
$201$	−3.56577	−0.251510
$202$	14.8599	1.04554
$203$	2.74636	0.192757
$204$	1.49864	0.104926
$205$	−32.3275	−2.25785
$206$	−21.2165	−1.47822
$207$	−6.10021	−0.423994
$208$	−4.21317	−0.292131
$209$	0	0
$210$	−10.2945	−0.710387
$211$	−11.9257	−0.821000	−0.410500	−	0.911861i	$-0.634646\pi$
$211$	−11.9257	−0.821000	−0.410500	+	0.911861i	$0.634646\pi$
$212$	4.66134	0.320142
$213$	−0.837051	−0.0573538
$214$	−21.9367	−1.49956
$215$	36.8606	2.51387
$216$	−0.222723	−0.0151544
$217$	1.72384	0.117022
$218$	14.9248	1.01084
$219$	−9.22175	−0.623148
$220$	0	0
$221$	0.794179	0.0534223
$222$	−17.5645	−1.17886
$223$	23.6029	1.58057	0.790284	−	0.612741i	$-0.209933\pi$
$223$	23.6029	1.58057	0.790284	+	0.612741i	$0.209933\pi$
$224$	−11.5113	−0.769130
$225$	7.71470	0.514313
$226$	−3.78216	−0.251585
$227$	0.578732	0.0384118	0.0192059	−	0.999816i	$-0.493886\pi$
$227$	0.578732	0.0384118	0.0192059	+	0.999816i	$0.493886\pi$
$228$	−4.51257	−0.298852
$229$	−28.3316	−1.87221	−0.936103	−	0.351726i	$-0.885595\pi$
$229$	−28.3316	−1.87221	−0.936103	+	0.351726i	$0.885595\pi$
$230$	42.8851	2.82776
$231$	0	0
$232$	−0.417715	−0.0274243
$233$	12.0612	0.790155	0.395078	−	0.918648i	$-0.370718\pi$
$233$	12.0612	0.790155	0.395078	+	0.918648i	$0.370718\pi$
$234$	1.97156	0.128885
$235$	23.3503	1.52320
$236$	9.73627	0.633777
$237$	2.57995	0.167586
$238$	2.29282	0.148621
$239$	0.895361	0.0579161	0.0289580	−	0.999581i	$-0.490781\pi$
$239$	0.895361	0.0579161	0.0289580	+	0.999581i	$0.490781\pi$
$240$	15.0232	0.969744
$241$	−25.1729	−1.62153	−0.810766	−	0.585370i	$-0.800949\pi$
$241$	−25.1729	−1.62153	−0.810766	+	0.585370i	$0.800949\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−21.5037	−1.37663
$245$	17.3143	1.10617
$246$	17.8743	1.13962
$247$	−2.39136	−0.152159
$248$	−0.262192	−0.0166492
$249$	5.68748	0.360429
$250$	−19.0846	−1.20702
$251$	14.5080	0.915736	0.457868	−	0.889020i	$-0.348613\pi$
$251$	14.5080	0.915736	0.457868	+	0.889020i	$0.348613\pi$
$252$	2.76326	0.174069
$253$	0	0
$254$	30.1855	1.89401
$255$	−2.83186	−0.177338
$256$	17.6516	1.10323
$257$	5.57370	0.347678	0.173839	−	0.984774i	$-0.444383\pi$
$257$	5.57370	0.347678	0.173839	+	0.984774i	$0.444383\pi$
$258$	−20.3807	−1.26885
$259$	−13.0458	−0.810626
$260$	−6.72872	−0.417297
$261$	1.87549	0.116090
$262$	0.733220	0.0452985
$263$	8.97386	0.553352	0.276676	−	0.960963i	$-0.410767\pi$
$263$	8.97386	0.553352	0.276676	+	0.960963i	$0.410767\pi$
$264$	0	0
$265$	−8.80815	−0.541080
$266$	−6.90393	−0.423307
$267$	−14.2488	−0.872013
$268$	−6.72872	−0.411022
$269$	5.52009	0.336566	0.168283	−	0.985739i	$-0.446178\pi$
$269$	5.52009	0.336566	0.168283	+	0.985739i	$0.446178\pi$
$270$	−7.03011	−0.427839
$271$	−2.34995	−0.142749	−0.0713747	−	0.997450i	$-0.522739\pi$
$271$	−2.34995	−0.142749	−0.0713747	+	0.997450i	$0.522739\pi$
$272$	−3.34601	−0.202882
$273$	1.46434	0.0886260
$274$	−1.04158	−0.0629243
$275$	0	0
$276$	−11.5113	−0.692898
$277$	15.9112	0.956009	0.478005	−	0.878357i	$-0.341360\pi$
$277$	15.9112	0.956009	0.478005	+	0.878357i	$0.341360\pi$
$278$	−45.2020	−2.71103
$279$	1.17721	0.0704777
$280$	1.16295	0.0694995
$281$	−32.9366	−1.96484	−0.982418	−	0.186695i	$-0.940222\pi$
$281$	−32.9366	−1.96484	−0.982418	+	0.186695i	$0.940222\pi$
$282$	−12.9106	−0.768818
$283$	−12.1758	−0.723776	−0.361888	−	0.932222i	$-0.617868\pi$
$283$	−12.1758	−0.723776	−0.361888	+	0.932222i	$0.617868\pi$
$284$	−1.57954	−0.0937285
$285$	8.52704	0.505098
$286$	0	0
$287$	13.2758	0.783648
$288$	−7.86106	−0.463218
$289$	−16.3693	−0.962899
$290$	−13.1849	−0.774244
$291$	−15.9779	−0.936639
$292$	−17.4017	−1.01836
$293$	−16.6437	−0.972336	−0.486168	−	0.873865i	$-0.661606\pi$
$293$	−16.6437	−0.972336	−0.486168	+	0.873865i	$0.661606\pi$
$294$	−9.57329	−0.558325
$295$	−18.3978	−1.07116
$296$	1.98424	0.115331
$297$	0	0
$298$	33.3847	1.93392
$299$	−6.10021	−0.352784
$300$	14.5579	0.840500
$301$	−15.1374	−0.872507
$302$	−30.1812	−1.73673
$303$	7.53716	0.432998
$304$	10.0752	0.577853
$305$	40.6337	2.32668
$306$	1.56577	0.0895090
$307$	25.1690	1.43647	0.718236	−	0.695799i	$-0.244950\pi$
$307$	25.1690	1.43647	0.718236	+	0.695799i	$0.244950\pi$
$308$	0	0
$309$	−10.7613	−0.612189
$310$	−8.27592	−0.470041
$311$	8.35466	0.473750	0.236875	−	0.971540i	$-0.423877\pi$
$311$	8.35466	0.473750	0.236875	+	0.971540i	$0.423877\pi$
$312$	−0.222723	−0.0126092
$313$	11.5387	0.652204	0.326102	−	0.945335i	$-0.394265\pi$
$313$	11.5387	0.652204	0.326102	+	0.945335i	$0.394265\pi$
$314$	27.2434	1.53743
$315$	−5.22150	−0.294198
$316$	4.86845	0.273872
$317$	18.4187	1.03450	0.517248	−	0.855836i	$-0.326956\pi$
$317$	18.4187	1.03450	0.517248	+	0.855836i	$0.326956\pi$
$318$	4.87013	0.273103
$319$	0	0
$320$	25.2177	1.40971
$321$	−11.1266	−0.621025
$322$	−17.6115	−0.981450
$323$	−1.89917	−0.105673
$324$	1.88703	0.104835
$325$	7.71470	0.427935
$326$	24.2045	1.34056
$327$	7.57007	0.418626
$328$	−2.01922	−0.111493
$329$	−9.58918	−0.528669
$330$	0	0
$331$	−15.2177	−0.836442	−0.418221	−	0.908345i	$-0.637346\pi$
$331$	−15.2177	−0.836442	−0.418221	+	0.908345i	$0.637346\pi$
$332$	10.7324	0.589020
$333$	−8.90898	−0.488209
$334$	48.3095	2.64338
$335$	12.7147	0.694678
$336$	−6.16953	−0.336575
$337$	−0.248364	−0.0135293	−0.00676464	−	0.999977i	$-0.502153\pi$
$337$	−0.248364	−0.0135293	−0.00676464	+	0.999977i	$0.502153\pi$
$338$	1.97156	0.107238
$339$	−1.91836	−0.104191
$340$	−5.34381	−0.289809
$341$	0	0
$342$	−4.71470	−0.254942
$343$	−17.3608	−0.937395
$344$	2.30237	0.124135
$345$	21.7519	1.17108
$346$	3.56577	0.191697
$347$	23.7166	1.27317	0.636587	−	0.771205i	$-0.280345\pi$
$347$	23.7166	1.27317	0.636587	+	0.771205i	$0.280345\pi$
$348$	3.53911	0.189716
$349$	−10.7275	−0.574230	−0.287115	−	0.957896i	$-0.592696\pi$
$349$	−10.7275	−0.574230	−0.287115	+	0.957896i	$0.592696\pi$
$350$	22.2726	1.19052
$351$	1.00000	0.0533761
$352$	0	0
$353$	−10.0176	−0.533185	−0.266593	−	0.963809i	$-0.585898\pi$
$353$	−10.0176	−0.533185	−0.266593	+	0.963809i	$0.585898\pi$
$354$	10.1724	0.540656
$355$	2.98473	0.158413
$356$	−26.8880	−1.42506
$357$	1.16295	0.0615498
$358$	48.7625	2.57718
$359$	−30.6675	−1.61857	−0.809284	−	0.587418i	$-0.800144\pi$
$359$	−30.6675	−1.61857	−0.809284	+	0.587418i	$0.800144\pi$
$360$	0.794179	0.0418569
$361$	−13.2814	−0.699021
$362$	23.8010	1.25095
$363$	0	0
$364$	2.76326	0.144834
$365$	32.8826	1.72115
$366$	−22.4669	−1.17436
$367$	23.2498	1.21363	0.606815	−	0.794843i	$-0.292447\pi$
$367$	23.2498	1.21363	0.606815	+	0.794843i	$0.292447\pi$
$368$	25.7012	1.33977
$369$	9.06607	0.471961
$370$	62.6311	3.25604
$371$	3.61721	0.187796
$372$	2.22143	0.115176
$373$	1.88704	0.0977074	0.0488537	−	0.998806i	$-0.484443\pi$
$373$	1.88704	0.0977074	0.0488537	+	0.998806i	$0.484443\pi$
$374$	0	0
$375$	−9.67999	−0.499873
$376$	1.45849	0.0752160
$377$	1.87549	0.0965926
$378$	2.88703	0.148493
$379$	18.2495	0.937412	0.468706	−	0.883354i	$-0.344720\pi$
$379$	18.2495	0.937412	0.468706	+	0.883354i	$0.344720\pi$
$380$	16.0908	0.825440
$381$	15.3105	0.784380
$382$	−1.69308	−0.0866256
$383$	6.87757	0.351427	0.175714	−	0.984441i	$-0.443777\pi$
$383$	6.87757	0.351427	0.175714	+	0.984441i	$0.443777\pi$
$384$	1.77894	0.0907813
$385$	0	0
$386$	−29.3408	−1.49341
$387$	−10.3374	−0.525477
$388$	−30.1507	−1.53067
$389$	27.2413	1.38119	0.690593	−	0.723243i	$-0.257350\pi$
$389$	27.2413	1.38119	0.690593	+	0.723243i	$0.257350\pi$
$390$	−7.03011	−0.355983
$391$	−4.84465	−0.245005
$392$	1.08148	0.0546228
$393$	0.371899	0.0187598
$394$	−23.9279	−1.20547
$395$	−9.19951	−0.462877
$396$	0	0
$397$	28.2690	1.41878	0.709389	−	0.704817i	$-0.248971\pi$
$397$	28.2690	1.41878	0.709389	+	0.704817i	$0.248971\pi$
$398$	−3.98053	−0.199526
$399$	−3.50177	−0.175308
$400$	−32.5034	−1.62517
$401$	−13.1139	−0.654877	−0.327438	−	0.944873i	$-0.606185\pi$
$401$	−13.1139	−0.654877	−0.327438	+	0.944873i	$0.606185\pi$
$402$	−7.03011	−0.350630
$403$	1.17721	0.0586410
$404$	14.2229	0.707613
$405$	−3.56577	−0.177184
$406$	5.41460	0.268722
$407$	0	0
$408$	−0.176882	−0.00875696
$409$	22.1307	1.09429	0.547146	−	0.837037i	$-0.315714\pi$
$409$	22.1307	1.09429	0.547146	+	0.837037i	$0.315714\pi$
$410$	−63.7355	−3.14767
$411$	−0.528305	−0.0260593
$412$	−20.3069	−1.00045
$413$	7.55537	0.371776
$414$	−12.0269	−0.591090
$415$	−20.2802	−0.995517
$416$	−7.86106	−0.385420
$417$	−22.9271	−1.12274
$418$	0	0
$419$	−5.05014	−0.246716	−0.123358	−	0.992362i	$-0.539366\pi$
$419$	−5.05014	−0.246716	−0.123358	+	0.992362i	$0.539366\pi$
$420$	−9.85314	−0.480784
$421$	23.4319	1.14200	0.571001	−	0.820950i	$-0.306555\pi$
$421$	23.4319	1.14200	0.571001	+	0.820950i	$0.306555\pi$
$422$	−23.5122	−1.14456
$423$	−6.54846	−0.318397
$424$	−0.550170	−0.0267186
$425$	6.12685	0.297196
$426$	−1.65029	−0.0799569
$427$	−16.6869	−0.807536
$428$	−20.9962	−1.01489
$429$	0	0
$430$	72.6728	3.50459
$431$	−12.3399	−0.594394	−0.297197	−	0.954816i	$-0.596052\pi$
$431$	−12.3399	−0.594394	−0.297197	+	0.954816i	$0.596052\pi$
$432$	−4.21317	−0.202706
$433$	39.1779	1.88277	0.941386	−	0.337332i	$-0.109524\pi$
$433$	39.1779	1.88277	0.941386	+	0.337332i	$0.109524\pi$
$434$	3.39864	0.163140
$435$	−6.68756	−0.320644
$436$	14.2850	0.684126
$437$	14.5878	0.697829
$438$	−18.1812	−0.868731
$439$	−16.3918	−0.782338	−0.391169	−	0.920319i	$-0.627929\pi$
$439$	−16.3918	−0.782338	−0.391169	+	0.920319i	$0.627929\pi$
$440$	0	0
$441$	−4.85570	−0.231224
$442$	1.56577	0.0744760
$443$	−25.2127	−1.19789	−0.598947	−	0.800789i	$-0.704414\pi$
$443$	−25.2127	−1.19789	−0.598947	+	0.800789i	$0.704414\pi$
$444$	−16.8115	−0.797840
$445$	50.8080	2.40853
$446$	46.5344	2.20347
$447$	16.9332	0.800911
$448$	−10.3561	−0.489279
$449$	−20.2181	−0.954149	−0.477074	−	0.878863i	$-0.658303\pi$
$449$	−20.2181	−0.954149	−0.477074	+	0.878863i	$0.658303\pi$
$450$	15.2100	0.717004
$451$	0	0
$452$	−3.62001	−0.170271
$453$	−15.3083	−0.719248
$454$	1.14100	0.0535499
$455$	−5.22150	−0.244788
$456$	0.532611	0.0249418
$457$	−5.00882	−0.234303	−0.117151	−	0.993114i	$-0.537376\pi$
$457$	−5.00882	−0.234303	−0.117151	+	0.993114i	$0.537376\pi$
$458$	−55.8574	−2.61004
$459$	0.794179	0.0370691
$460$	41.0466	1.91381
$461$	−23.5062	−1.09479	−0.547397	−	0.836873i	$-0.684381\pi$
$461$	−23.5062	−1.09479	−0.547397	+	0.836873i	$0.684381\pi$
$462$	0	0
$463$	22.6379	1.05207	0.526035	−	0.850463i	$-0.323678\pi$
$463$	22.6379	1.05207	0.526035	+	0.850463i	$0.323678\pi$
$464$	−7.90176	−0.366830
$465$	−4.19766	−0.194662
$466$	23.7793	1.10156
$467$	−3.16908	−0.146648	−0.0733238	−	0.997308i	$-0.523361\pi$
$467$	−3.16908	−0.146648	−0.0733238	+	0.997308i	$0.523361\pi$
$468$	1.88703	0.0872281
$469$	−5.22150	−0.241107
$470$	46.0364	2.12350
$471$	13.8182	0.636709
$472$	−1.14916	−0.0528942
$473$	0	0
$474$	5.08652	0.233631
$475$	−18.4486	−0.846481
$476$	2.19452	0.100586
$477$	2.47020	0.113103
$478$	1.76525	0.0807408
$479$	−9.67248	−0.441947	−0.220973	−	0.975280i	$-0.570923\pi$
$479$	−9.67248	−0.441947	−0.220973	+	0.975280i	$0.570923\pi$
$480$	28.0307	1.27942
$481$	−8.90898	−0.406214
$482$	−49.6299	−2.26058
$483$	−8.93279	−0.406456
$484$	0	0
$485$	56.9733	2.58703
$486$	1.97156	0.0894316
$487$	−26.7944	−1.21417	−0.607084	−	0.794637i	$-0.707661\pi$
$487$	−26.7944	−1.21417	−0.607084	+	0.794637i	$0.707661\pi$
$488$	2.53804	0.114892
$489$	12.2768	0.555178
$490$	34.1361	1.54211
$491$	−21.7650	−0.982239	−0.491119	−	0.871092i	$-0.663412\pi$
$491$	−21.7650	−0.982239	−0.491119	+	0.871092i	$0.663412\pi$
$492$	17.1080	0.771287
$493$	1.48947	0.0670825
$494$	−4.71470	−0.212124
$495$	0	0
$496$	−4.95979	−0.222701
$497$	−1.22573	−0.0549814
$498$	11.2132	0.502474
$499$	−26.9793	−1.20776	−0.603880	−	0.797075i	$-0.706379\pi$
$499$	−26.9793	−1.20776	−0.603880	+	0.797075i	$0.706379\pi$
$500$	−18.2665	−0.816901
$501$	24.5032	1.09472
$502$	28.6033	1.27663
$503$	−35.6860	−1.59116	−0.795581	−	0.605847i	$-0.792834\pi$
$503$	−35.6860	−1.59116	−0.795581	+	0.605847i	$0.792834\pi$
$504$	−0.326143	−0.0145276
$505$	−26.8758	−1.19596
$506$	0	0
$507$	1.00000	0.0444116
$508$	28.8914	1.28185
$509$	−9.63181	−0.426922	−0.213461	−	0.976952i	$-0.568474\pi$
$509$	−9.63181	−0.426922	−0.213461	+	0.976952i	$0.568474\pi$
$510$	−5.58316	−0.247227
$511$	−13.5038	−0.597373
$512$	31.2433	1.38077
$513$	−2.39136	−0.105581
$514$	10.9889	0.484698
$515$	38.3723	1.69089
$516$	−19.5069	−0.858744
$517$	0	0
$518$	−25.7205	−1.13009
$519$	1.80861	0.0793890
$520$	0.794179	0.0348270
$521$	20.6888	0.906391	0.453196	−	0.891411i	$-0.350284\pi$
$521$	20.6888	0.906391	0.453196	+	0.891411i	$0.350284\pi$
$522$	3.69763	0.161841
$523$	−28.4093	−1.24225	−0.621126	−	0.783711i	$-0.713324\pi$
$523$	−28.4093	−1.24225	−0.621126	+	0.783711i	$0.713324\pi$
$524$	0.701786	0.0306577
$525$	11.2970	0.493040
$526$	17.6925	0.771428
$527$	0.934916	0.0407256
$528$	0	0
$529$	14.2125	0.617936
$530$	−17.3658	−0.754320
$531$	5.15957	0.223906
$532$	−6.60795	−0.286491
$533$	9.06607	0.392695
$534$	−28.0923	−1.21567
$535$	39.6748	1.71529
$536$	0.794179	0.0343033
$537$	24.7330	1.06731
$538$	10.8832	0.469207
$539$	0	0
$540$	−6.72872	−0.289558
$541$	16.3418	0.702587	0.351294	−	0.936265i	$-0.385742\pi$
$541$	16.3418	0.702587	0.351294	+	0.936265i	$0.385742\pi$
$542$	−4.63306	−0.199007
$543$	12.0722	0.518067
$544$	−6.24309	−0.267670
$545$	−26.9931	−1.15626
$546$	2.88703	0.123554
$547$	29.6110	1.26608	0.633038	−	0.774121i	$-0.281808\pi$
$547$	29.6110	1.26608	0.633038	+	0.774121i	$0.281808\pi$
$548$	−0.996927	−0.0425866
$549$	−11.3955	−0.486348
$550$	0	0
$551$	−4.48497	−0.191066
$552$	1.35866	0.0578283
$553$	3.77793	0.160654
$554$	31.3697	1.33277
$555$	31.7674	1.34845
$556$	−43.2641	−1.83481
$557$	45.1188	1.91174	0.955872	−	0.293785i	$-0.0949149\pi$
$557$	45.1188	1.91174	0.955872	+	0.293785i	$0.0949149\pi$
$558$	2.32094	0.0982530
$559$	−10.3374	−0.437224
$560$	21.9991	0.929632
$561$	0	0
$562$	−64.9364	−2.73918
$563$	3.81739	0.160884	0.0804419	−	0.996759i	$-0.474367\pi$
$563$	3.81739	0.160884	0.0804419	+	0.996759i	$0.474367\pi$
$564$	−12.3571	−0.520330
$565$	6.84043	0.287779
$566$	−24.0053	−1.00902
$567$	1.46434	0.0614966
$568$	0.186430	0.00782245
$569$	−18.3747	−0.770306	−0.385153	−	0.922853i	$-0.625851\pi$
$569$	−18.3747	−0.770306	−0.385153	+	0.922853i	$0.625851\pi$
$570$	16.8115	0.704157
$571$	24.0809	1.00776	0.503878	−	0.863775i	$-0.331906\pi$
$571$	24.0809	1.00776	0.503878	+	0.863775i	$0.331906\pi$
$572$	0	0
$573$	−0.858754	−0.0358750
$574$	26.1740	1.09248
$575$	−47.0613	−1.96259
$576$	−7.07217	−0.294674
$577$	−7.25685	−0.302107	−0.151053	−	0.988526i	$-0.548267\pi$
$577$	−7.25685	−0.302107	−0.151053	+	0.988526i	$0.548267\pi$
$578$	−32.2729	−1.34238
$579$	−14.8820	−0.618476
$580$	−12.6196	−0.524002
$581$	8.32841	0.345521
$582$	−31.5012	−1.30577
$583$	0	0
$584$	2.05390	0.0849908
$585$	−3.56577	−0.147426
$586$	−32.8140	−1.35553
$587$	25.0761	1.03500	0.517501	−	0.855682i	$-0.326862\pi$
$587$	25.0761	1.03500	0.517501	+	0.855682i	$0.326862\pi$
$588$	−9.16287	−0.377870
$589$	−2.81513	−0.115996
$590$	−36.2723	−1.49331
$591$	−12.1366	−0.499232
$592$	37.5351	1.54268
$593$	−35.7124	−1.46653	−0.733267	−	0.679941i	$-0.762005\pi$
$593$	−35.7124	−1.46653	−0.733267	+	0.679941i	$0.762005\pi$
$594$	0	0
$595$	−4.14681	−0.170003
$596$	31.9534	1.30886
$597$	−2.01898	−0.0826314
$598$	−12.0269	−0.491816
$599$	16.3472	0.667929	0.333965	−	0.942586i	$-0.391613\pi$
$599$	16.3472	0.667929	0.333965	+	0.942586i	$0.391613\pi$
$600$	−1.71824	−0.0701469
$601$	31.2413	1.27436	0.637179	−	0.770716i	$-0.280101\pi$
$601$	31.2413	1.27436	0.637179	+	0.770716i	$0.280101\pi$
$602$	−29.8443	−1.21636
$603$	−3.56577	−0.145209
$604$	−28.8873	−1.17541
$605$	0	0
$606$	14.8599	0.603643
$607$	−19.2140	−0.779873	−0.389937	−	0.920842i	$-0.627503\pi$
$607$	−19.2140	−0.779873	−0.389937	+	0.920842i	$0.627503\pi$
$608$	18.7986	0.762385
$609$	2.74636	0.111288
$610$	80.1116	3.24362
$611$	−6.54846	−0.264922
$612$	1.49864	0.0605790
$613$	39.8258	1.60855	0.804274	−	0.594259i	$-0.202555\pi$
$613$	39.8258	1.60855	0.804274	+	0.594259i	$0.202555\pi$
$614$	49.6222	2.00259
$615$	−32.3275	−1.30357
$616$	0	0
$617$	−2.59800	−0.104591	−0.0522957	−	0.998632i	$-0.516654\pi$
$617$	−2.59800	−0.104591	−0.0522957	+	0.998632i	$0.516654\pi$
$618$	−21.2165	−0.853453
$619$	46.2702	1.85976	0.929878	−	0.367869i	$-0.119912\pi$
$619$	46.2702	1.85976	0.929878	+	0.367869i	$0.119912\pi$
$620$	−7.92112	−0.318120
$621$	−6.10021	−0.244793
$622$	16.4717	0.660454
$623$	−20.8651	−0.835944
$624$	−4.21317	−0.168662
$625$	−4.05690	−0.162276
$626$	22.7491	0.909237
$627$	0	0
$628$	26.0754	1.04052
$629$	−7.07532	−0.282112
$630$	−10.2945	−0.410142
$631$	−2.92662	−0.116507	−0.0582534	−	0.998302i	$-0.518553\pi$
$631$	−2.92662	−0.116507	−0.0582534	+	0.998302i	$0.518553\pi$
$632$	−0.574615	−0.0228569
$633$	−11.9257	−0.474005
$634$	36.3135	1.44219
$635$	−54.5937	−2.16648
$636$	4.66134	0.184834
$637$	−4.85570	−0.192390
$638$	0	0
$639$	−0.837051	−0.0331132
$640$	−6.34330	−0.250741
$641$	37.5952	1.48492	0.742460	−	0.669890i	$-0.233659\pi$
$641$	37.5952	1.48492	0.742460	+	0.669890i	$0.233659\pi$
$642$	−21.9367	−0.865772
$643$	−46.5117	−1.83424	−0.917121	−	0.398610i	$-0.869493\pi$
$643$	−46.5117	−1.83424	−0.917121	+	0.398610i	$0.869493\pi$
$644$	−16.8565	−0.664237
$645$	36.8606	1.45139
$646$	−3.74432	−0.147318
$647$	−28.5385	−1.12197	−0.560983	−	0.827827i	$-0.689577\pi$
$647$	−28.5385	−1.12197	−0.560983	+	0.827827i	$0.689577\pi$
$648$	−0.222723	−0.00874939
$649$	0	0
$650$	15.2100	0.596584
$651$	1.72384	0.0675626
$652$	23.1668	0.907282
$653$	39.6486	1.55157	0.775785	−	0.630997i	$-0.217354\pi$
$653$	39.6486	1.55157	0.775785	+	0.630997i	$0.217354\pi$
$654$	14.9248	0.583607
$655$	−1.32611	−0.0518153
$656$	−38.1970	−1.49134
$657$	−9.22175	−0.359775
$658$	−18.9056	−0.737017
$659$	21.9068	0.853367	0.426683	−	0.904401i	$-0.359682\pi$
$659$	21.9068	0.853367	0.426683	+	0.904401i	$0.359682\pi$
$660$	0	0
$661$	−18.5956	−0.723283	−0.361642	−	0.932317i	$-0.617784\pi$
$661$	−18.5956	−0.723283	−0.361642	+	0.932317i	$0.617784\pi$
$662$	−30.0026	−1.16608
$663$	0.794179	0.0308434
$664$	−1.26673	−0.0491587
$665$	12.4865	0.484206
$666$	−17.5645	−0.680612
$667$	−11.4409	−0.442992
$668$	46.2384	1.78902
$669$	23.6029	0.912541
$670$	25.0677	0.968451
$671$	0	0
$672$	−11.5113	−0.444057
$673$	12.1523	0.468436	0.234218	−	0.972184i	$-0.424747\pi$
$673$	12.1523	0.468436	0.234218	+	0.972184i	$0.424747\pi$
$674$	−0.489664	−0.0188612
$675$	7.71470	0.296939
$676$	1.88703	0.0725781
$677$	−29.5968	−1.13750	−0.568749	−	0.822511i	$-0.692572\pi$
$677$	−29.5968	−1.13750	−0.568749	+	0.822511i	$0.692572\pi$
$678$	−3.78216	−0.145253
$679$	−23.3970	−0.897896
$680$	0.630720	0.0241870
$681$	0.578732	0.0221771
$682$	0	0
$683$	6.76714	0.258938	0.129469	−	0.991583i	$-0.458673\pi$
$683$	6.76714	0.258938	0.129469	+	0.991583i	$0.458673\pi$
$684$	−4.51257	−0.172543
$685$	1.88381	0.0719767
$686$	−34.2278	−1.30682
$687$	−28.3316	−1.08092
$688$	43.5531	1.66045
$689$	2.47020	0.0941070
$690$	42.8851	1.63261
$691$	6.24930	0.237735	0.118867	−	0.992910i	$-0.462074\pi$
$691$	6.24930	0.237735	0.118867	+	0.992910i	$0.462074\pi$
$692$	3.41290	0.129739
$693$	0	0
$694$	46.7586	1.77493
$695$	81.7525	3.10105
$696$	−0.417715	−0.0158334
$697$	7.20008	0.272723
$698$	−21.1499	−0.800534
$699$	12.0612	0.456196
$700$	21.3177	0.805734
$701$	41.9595	1.58479	0.792395	−	0.610009i	$-0.208834\pi$
$701$	41.9595	1.58479	0.792395	+	0.610009i	$0.208834\pi$
$702$	1.97156	0.0744116
$703$	21.3046	0.803517
$704$	0	0
$705$	23.3503	0.879422
$706$	−19.7503	−0.743314
$707$	11.0370	0.415088
$708$	9.73627	0.365912
$709$	−15.5081	−0.582419	−0.291209	−	0.956659i	$-0.594058\pi$
$709$	−15.5081	−0.582419	−0.291209	+	0.956659i	$0.594058\pi$
$710$	5.88456	0.220843
$711$	2.57995	0.0967557
$712$	3.17354	0.118933
$713$	−7.18123	−0.268939
$714$	2.29282	0.0858066
$715$	0	0
$716$	46.6720	1.74421
$717$	0.895361	0.0334379
$718$	−60.4626	−2.25645
$719$	51.3406	1.91468	0.957340	−	0.288964i	$-0.0933108\pi$
$719$	51.3406	1.91468	0.957340	+	0.288964i	$0.0933108\pi$
$720$	15.0232	0.559882
$721$	−15.7582	−0.586867
$722$	−26.1850	−0.974505
$723$	−25.1729	−0.936192
$724$	22.7806	0.846633
$725$	14.4688	0.537359
$726$	0	0
$727$	−26.3703	−0.978020	−0.489010	−	0.872278i	$-0.662642\pi$
$727$	−26.3703	−0.978020	−0.489010	+	0.872278i	$0.662642\pi$
$728$	−0.326143	−0.0120877
$729$	1.00000	0.0370370
$730$	64.8299	2.39946
$731$	−8.20971	−0.303647
$732$	−21.5037	−0.794799
$733$	46.0603	1.70128	0.850638	−	0.525752i	$-0.176216\pi$
$733$	46.0603	1.70128	0.850638	+	0.525752i	$0.176216\pi$
$734$	45.8383	1.69192
$735$	17.3143	0.638648
$736$	47.9541	1.76761
$737$	0	0
$738$	17.8743	0.657961
$739$	−27.4321	−1.00910	−0.504552	−	0.863381i	$-0.668342\pi$
$739$	−27.4321	−1.00910	−0.504552	+	0.863381i	$0.668342\pi$
$740$	59.9460	2.20366
$741$	−2.39136	−0.0878488
$742$	7.13154	0.261807
$743$	−48.0019	−1.76102	−0.880509	−	0.474029i	$-0.842799\pi$
$743$	−48.0019	−1.76102	−0.880509	+	0.474029i	$0.842799\pi$
$744$	−0.262192	−0.00961242
$745$	−60.3797	−2.21214
$746$	3.72041	0.136214
$747$	5.68748	0.208094
$748$	0	0
$749$	−16.2931	−0.595338
$750$	−19.0846	−0.696873
$751$	22.2285	0.811129	0.405565	−	0.914066i	$-0.367075\pi$
$751$	22.2285	0.811129	0.405565	+	0.914066i	$0.367075\pi$
$752$	27.5898	1.00610
$753$	14.5080	0.528700
$754$	3.69763	0.134660
$755$	54.5860	1.98659
$756$	2.76326	0.100499
$757$	21.0624	0.765525	0.382762	−	0.923847i	$-0.374973\pi$
$757$	21.0624	0.765525	0.382762	+	0.923847i	$0.374973\pi$
$758$	35.9798	1.30685
$759$	0	0
$760$	−1.89917	−0.0688901
$761$	−38.6383	−1.40064	−0.700319	−	0.713830i	$-0.746959\pi$
$761$	−38.6383	−1.40064	−0.700319	+	0.713830i	$0.746959\pi$
$762$	30.1855	1.09351
$763$	11.0852	0.401310
$764$	−1.62050	−0.0586275
$765$	−2.83186	−0.102386
$766$	13.5595	0.489925
$767$	5.15957	0.186301
$768$	17.6516	0.636948
$769$	32.8225	1.18361	0.591805	−	0.806081i	$-0.298415\pi$
$769$	32.8225	1.18361	0.591805	+	0.806081i	$0.298415\pi$
$770$	0	0
$771$	5.57370	0.200732
$772$	−28.0829	−1.01072
$773$	−18.2932	−0.657960	−0.328980	−	0.944337i	$-0.606705\pi$
$773$	−18.2932	−0.657960	−0.328980	+	0.944337i	$0.606705\pi$
$774$	−20.3807	−0.732568
$775$	9.08182	0.326229
$776$	3.55864	0.127748
$777$	−13.0458	−0.468015
$778$	53.7077	1.92551
$779$	−21.6803	−0.776776
$780$	−6.72872	−0.240927
$781$	0	0
$782$	−9.55151	−0.341561
$783$	1.87549	0.0670245
$784$	20.4579	0.730640
$785$	−49.2725	−1.75861
$786$	0.733220	0.0261531
$787$	−39.2422	−1.39883	−0.699417	−	0.714714i	$-0.746557\pi$
$787$	−39.2422	−1.39883	−0.699417	+	0.714714i	$0.746557\pi$
$788$	−22.9021	−0.815853
$789$	8.97386	0.319478
$790$	−18.1373	−0.645298
$791$	−2.80914	−0.0998814
$792$	0	0
$793$	−11.3955	−0.404666
$794$	55.7338	1.97792
$795$	−8.80815	−0.312393
$796$	−3.80988	−0.135038
$797$	−6.89589	−0.244265	−0.122132	−	0.992514i	$-0.538973\pi$
$797$	−6.89589	−0.244265	−0.122132	+	0.992514i	$0.538973\pi$
$798$	−6.90393	−0.244397
$799$	−5.20064	−0.183986
$800$	−60.6457	−2.14415
$801$	−14.2488	−0.503457
$802$	−25.8548	−0.912964
$803$	0	0
$804$	−6.72872	−0.237304
$805$	31.8522	1.12264
$806$	2.32094	0.0817515
$807$	5.52009	0.194316
$808$	−1.67870	−0.0590564
$809$	−31.0251	−1.09078	−0.545392	−	0.838181i	$-0.683619\pi$
$809$	−31.0251	−1.09078	−0.545392	+	0.838181i	$0.683619\pi$
$810$	−7.03011	−0.247013
$811$	−25.4035	−0.892038	−0.446019	−	0.895023i	$-0.647159\pi$
$811$	−25.4035	−0.892038	−0.446019	+	0.895023i	$0.647159\pi$
$812$	5.18246	0.181869
$813$	−2.34995	−0.0824164
$814$	0	0
$815$	−43.7764	−1.53342
$816$	−3.34601	−0.117134
$817$	24.7203	0.864855
$818$	43.6319	1.52555
$819$	1.46434	0.0511683
$820$	−61.0031	−2.13032
$821$	9.46516	0.330336	0.165168	−	0.986265i	$-0.447183\pi$
$821$	9.46516	0.330336	0.165168	+	0.986265i	$0.447183\pi$
$822$	−1.04158	−0.0363293
$823$	24.4605	0.852639	0.426319	−	0.904573i	$-0.359810\pi$
$823$	24.4605	0.852639	0.426319	+	0.904573i	$0.359810\pi$
$824$	2.39679	0.0834962
$825$	0	0
$826$	14.8958	0.518293
$827$	−14.1762	−0.492954	−0.246477	−	0.969149i	$-0.579273\pi$
$827$	−14.1762	−0.492954	−0.246477	+	0.969149i	$0.579273\pi$
$828$	−11.5113	−0.400045
$829$	−3.12178	−0.108424	−0.0542119	−	0.998529i	$-0.517265\pi$
$829$	−3.12178	−0.108424	−0.0542119	+	0.998529i	$0.517265\pi$
$830$	−39.9836	−1.38785
$831$	15.9112	0.551952
$832$	−7.07217	−0.245183
$833$	−3.85630	−0.133613
$834$	−45.2020	−1.56522
$835$	−87.3728	−3.02366
$836$	0	0
$837$	1.17721	0.0406903
$838$	−9.95663	−0.343946
$839$	9.00239	0.310797	0.155399	−	0.987852i	$-0.450334\pi$
$839$	9.00239	0.310797	0.155399	+	0.987852i	$0.450334\pi$
$840$	1.16295	0.0401256
$841$	−25.4825	−0.878708
$842$	46.1973	1.59206
$843$	−32.9366	−1.13440
$844$	−22.5042	−0.774627
$845$	−3.56577	−0.122666
$846$	−12.9106	−0.443877
$847$	0	0
$848$	−10.4074	−0.357391
$849$	−12.1758	−0.417872
$850$	12.0794	0.414321
$851$	54.3466	1.86298
$852$	−1.57954	−0.0541142
$853$	−27.4270	−0.939081	−0.469541	−	0.882911i	$-0.655580\pi$
$853$	−27.4270	−0.939081	−0.469541	+	0.882911i	$0.655580\pi$
$854$	−32.8992	−1.12579
$855$	8.52704	0.291618
$856$	2.47815	0.0847013
$857$	−39.9386	−1.36427	−0.682137	−	0.731224i	$-0.738949\pi$
$857$	−39.9386	−1.36427	−0.682137	+	0.731224i	$0.738949\pi$
$858$	0	0
$859$	4.76174	0.162469	0.0812343	−	0.996695i	$-0.474114\pi$
$859$	4.76174	0.162469	0.0812343	+	0.996695i	$0.474114\pi$
$860$	69.5572	2.37188
$861$	13.2758	0.452439
$862$	−24.3289	−0.828646
$863$	5.20346	0.177128	0.0885640	−	0.996070i	$-0.471772\pi$
$863$	5.20346	0.177128	0.0885640	+	0.996070i	$0.471772\pi$
$864$	−7.86106	−0.267439
$865$	−6.44907	−0.219275
$866$	77.2415	2.62477
$867$	−16.3693	−0.555930
$868$	3.25294	0.110412
$869$	0	0
$870$	−13.1849	−0.447010
$871$	−3.56577	−0.120821
$872$	−1.68603	−0.0570962
$873$	−15.9779	−0.540769
$874$	28.7606	0.972843
$875$	−14.1748	−0.479196
$876$	−17.4017	−0.587950
$877$	36.7369	1.24052	0.620259	−	0.784397i	$-0.287027\pi$
$877$	36.7369	1.24052	0.620259	+	0.784397i	$0.287027\pi$
$878$	−32.3174	−1.09066
$879$	−16.6437	−0.561378
$880$	0	0
$881$	−45.1493	−1.52112	−0.760558	−	0.649269i	$-0.775075\pi$
$881$	−45.1493	−1.52112	−0.760558	+	0.649269i	$0.775075\pi$
$882$	−9.57329	−0.322349
$883$	4.44216	0.149491	0.0747453	−	0.997203i	$-0.476186\pi$
$883$	4.44216	0.149491	0.0747453	+	0.997203i	$0.476186\pi$
$884$	1.49864	0.0504047
$885$	−18.3978	−0.618436
$886$	−49.7083	−1.66998
$887$	6.44690	0.216466	0.108233	−	0.994126i	$-0.465481\pi$
$887$	6.44690	0.216466	0.108233	+	0.994126i	$0.465481\pi$
$888$	1.98424	0.0665866
$889$	22.4198	0.751936
$890$	100.171	3.35773
$891$	0	0
$892$	44.5394	1.49129
$893$	15.6597	0.524032
$894$	33.3847	1.11655
$895$	−88.1922	−2.94794
$896$	2.60498	0.0870263
$897$	−6.10021	−0.203680
$898$	−39.8610	−1.33018
$899$	2.20785	0.0736358
$900$	14.5579	0.485263
$901$	1.96178	0.0653563
$902$	0	0
$903$	−15.1374	−0.503742
$904$	0.427263	0.0142106
$905$	−43.0466	−1.43092
$906$	−30.1812	−1.00270
$907$	41.4762	1.37720	0.688598	−	0.725143i	$-0.258226\pi$
$907$	41.4762	1.37720	0.688598	+	0.725143i	$0.258226\pi$
$908$	1.09209	0.0362421
$909$	7.53716	0.249992
$910$	−10.2945	−0.341259
$911$	22.6767	0.751313	0.375656	−	0.926759i	$-0.377417\pi$
$911$	22.6767	0.751313	0.375656	+	0.926759i	$0.377417\pi$
$912$	10.0752	0.333624
$913$	0	0
$914$	−9.87517	−0.326642
$915$	40.6337	1.34331
$916$	−53.4627	−1.76646
$917$	0.544588	0.0179839
$918$	1.56577	0.0516780
$919$	46.2181	1.52460	0.762298	−	0.647226i	$-0.224071\pi$
$919$	46.2181	1.52460	0.762298	+	0.647226i	$0.224071\pi$
$920$	−4.84465	−0.159724
$921$	25.1690	0.829348
$922$	−46.3439	−1.52625
$923$	−0.837051	−0.0275519
$924$	0	0
$925$	−68.7301	−2.25983
$926$	44.6318	1.46669
$927$	−10.7613	−0.353448
$928$	−14.7433	−0.483974
$929$	19.7434	0.647759	0.323880	−	0.946098i	$-0.395013\pi$
$929$	19.7434	0.647759	0.323880	+	0.946098i	$0.395013\pi$
$930$	−8.27592	−0.271378
$931$	11.6117	0.380559
$932$	22.7599	0.745524
$933$	8.35466	0.273519
$934$	−6.24802	−0.204442
$935$	0	0
$936$	−0.222723	−0.00727993
$937$	19.0667	0.622882	0.311441	−	0.950266i	$-0.399188\pi$
$937$	19.0667	0.622882	0.311441	+	0.950266i	$0.399188\pi$
$938$	−10.2945	−0.336127
$939$	11.5387	0.376550
$940$	44.0627	1.43717
$941$	−45.8478	−1.49460	−0.747298	−	0.664489i	$-0.768649\pi$
$941$	−45.8478	−1.49460	−0.747298	+	0.664489i	$0.768649\pi$
$942$	27.2434	0.887636
$943$	−55.3049	−1.80098
$944$	−21.7382	−0.707517
$945$	−5.22150	−0.169856
$946$	0	0
$947$	0.652498	0.0212034	0.0106017	−	0.999944i	$-0.496625\pi$
$947$	0.652498	0.0212034	0.0106017	+	0.999944i	$0.496625\pi$
$948$	4.86845	0.158120
$949$	−9.22175	−0.299351
$950$	−36.3725	−1.18008
$951$	18.4187	0.597267
$952$	−0.259016	−0.00839475
$953$	−37.0781	−1.20108	−0.600539	−	0.799595i	$-0.705047\pi$
$953$	−37.0781	−1.20108	−0.600539	+	0.799595i	$0.705047\pi$
$954$	4.87013	0.157676
$955$	3.06212	0.0990878
$956$	1.68957	0.0546447
$957$	0	0
$958$	−19.0698	−0.616118
$959$	−0.773619	−0.0249814
$960$	25.2177	0.813899
$961$	−29.6142	−0.955296
$962$	−17.5645	−0.566304
$963$	−11.1266	−0.358549
$964$	−47.5021	−1.52994
$965$	53.0659	1.70825
$966$	−17.6115	−0.566640
$967$	−34.1386	−1.09782	−0.548912	−	0.835880i	$-0.684958\pi$
$967$	−34.1386	−1.09782	−0.548912	+	0.835880i	$0.684958\pi$
$968$	0	0
$969$	−1.89917	−0.0610101
$970$	112.326	3.60657
$971$	−55.4998	−1.78107	−0.890537	−	0.454910i	$-0.849671\pi$
$971$	−55.4998	−1.78107	−0.890537	+	0.454910i	$0.849671\pi$
$972$	1.88703	0.0605266
$973$	−33.5730	−1.07630
$974$	−52.8266	−1.69267
$975$	7.71470	0.247068
$976$	48.0112	1.53680
$977$	−35.4894	−1.13541	−0.567703	−	0.823233i	$-0.692168\pi$
$977$	−35.4894	−1.13541	−0.567703	+	0.823233i	$0.692168\pi$
$978$	24.2045	0.773974
$979$	0	0
$980$	32.6726	1.04369
$981$	7.57007	0.241694
$982$	−42.9108	−1.36934
$983$	−51.9490	−1.65691	−0.828457	−	0.560052i	$-0.810781\pi$
$983$	−51.9490	−1.65691	−0.828457	+	0.560052i	$0.810781\pi$
$984$	−2.01922	−0.0643705
$985$	43.2762	1.37889
$986$	2.93658	0.0935198
$987$	−9.58918	−0.305227
$988$	−4.51257	−0.143564
$989$	63.0600	2.00519
$990$	0	0
$991$	1.40629	0.0446724	0.0223362	−	0.999751i	$-0.492890\pi$
$991$	1.40629	0.0446724	0.0223362	+	0.999751i	$0.492890\pi$
$992$	−9.25412	−0.293819
$993$	−15.2177	−0.482920
$994$	−2.41659	−0.0766496
$995$	7.19921	0.228230
$996$	10.7324	0.340071
$997$	−17.6234	−0.558138	−0.279069	−	0.960271i	$-0.590026\pi$
$997$	−17.6234	−0.558138	−0.279069	+	0.960271i	$0.590026\pi$
$998$	−53.1912	−1.68374
$999$	−8.90898	−0.281868

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4719.2.a.bc.1.5	✓	5
11.10	odd	2		4719.2.a.be.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4719.2.a.bc.1.5	✓	5	1.1	even	1	trivial
4719.2.a.be.1.1	yes	5	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4719 = 3 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4719.a (trivial)

\(f(q)\)	\(=\)	\(q+1.97156 q^{2} +1.00000 q^{3} +1.88703 q^{4} -3.56577 q^{5} +1.97156 q^{6} +1.46434 q^{7} -0.222723 q^{8} +1.00000 q^{9} -7.03011 q^{10} +1.88703 q^{12} +1.00000 q^{13} +2.88703 q^{14} -3.56577 q^{15} -4.21317 q^{16} +0.794179 q^{17} +1.97156 q^{18} -2.39136 q^{19} -6.72872 q^{20} +1.46434 q^{21} -6.10021 q^{23} -0.222723 q^{24} +7.71470 q^{25} +1.97156 q^{26} +1.00000 q^{27} +2.76326 q^{28} +1.87549 q^{29} -7.03011 q^{30} +1.17721 q^{31} -7.86106 q^{32} +1.56577 q^{34} -5.22150 q^{35} +1.88703 q^{36} -8.90898 q^{37} -4.71470 q^{38} +1.00000 q^{39} +0.794179 q^{40} +9.06607 q^{41} +2.88703 q^{42} -10.3374 q^{43} -3.56577 q^{45} -12.0269 q^{46} -6.54846 q^{47} -4.21317 q^{48} -4.85570 q^{49} +15.2100 q^{50} +0.794179 q^{51} +1.88703 q^{52} +2.47020 q^{53} +1.97156 q^{54} -0.326143 q^{56} -2.39136 q^{57} +3.69763 q^{58} +5.15957 q^{59} -6.72872 q^{60} -11.3955 q^{61} +2.32094 q^{62} +1.46434 q^{63} -7.07217 q^{64} -3.56577 q^{65} -3.56577 q^{67} +1.49864 q^{68} -6.10021 q^{69} -10.2945 q^{70} -0.837051 q^{71} -0.222723 q^{72} -9.22175 q^{73} -17.5645 q^{74} +7.71470 q^{75} -4.51257 q^{76} +1.97156 q^{78} +2.57995 q^{79} +15.0232 q^{80} +1.00000 q^{81} +17.8743 q^{82} +5.68748 q^{83} +2.76326 q^{84} -2.83186 q^{85} -20.3807 q^{86} +1.87549 q^{87} -14.2488 q^{89} -7.03011 q^{90} +1.46434 q^{91} -11.5113 q^{92} +1.17721 q^{93} -12.9106 q^{94} +8.52704 q^{95} -7.86106 q^{96} -15.9779 q^{97} -9.57329 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 5 q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 6 q^{7} - 9 q^{8} + 5 q^{9} - 6 q^{10} + 3 q^{12} + 5 q^{13} + 8 q^{14} - 2 q^{15} + 11 q^{16} - 2 q^{17} - q^{18} - 12 q^{19} - 8 q^{20} - 6 q^{21}+ \cdots + q^{98}+O(q^{100}) \) 5 * q - q^2 + 5 * q^3 + 3 * q^4 - 2 * q^5 - q^6 - 6 * q^7 - 9 * q^8 + 5 * q^9 - 6 * q^10 + 3 * q^12 + 5 * q^13 + 8 * q^14 - 2 * q^15 + 11 * q^16 - 2 * q^17 - q^18 - 12 * q^19 - 8 * q^20 - 6 * q^21 + 8 * q^23 - 9 * q^24 + 7 * q^25 - q^26 + 5 * q^27 - 6 * q^29 - 6 * q^30 - 4 * q^31 - 19 * q^32 - 8 * q^34 + 2 * q^35 + 3 * q^36 - 10 * q^37 + 8 * q^38 + 5 * q^39 - 2 * q^40 - 6 * q^41 + 8 * q^42 - 26 * q^43 - 2 * q^45 - 26 * q^46 - 16 * q^47 + 11 * q^48 - 11 * q^49 + 39 * q^50 - 2 * q^51 + 3 * q^52 + 6 * q^53 - q^54 + 24 * q^56 - 12 * q^57 - 20 * q^58 + 6 * q^59 - 8 * q^60 - 14 * q^61 + 12 * q^62 - 6 * q^63 + 15 * q^64 - 2 * q^65 - 2 * q^67 + 12 * q^68 + 8 * q^69 - 10 * q^70 - 14 * q^71 - 9 * q^72 - 12 * q^73 - 36 * q^74 + 7 * q^75 - 18 * q^76 - q^78 + 8 * q^79 + 8 * q^80 + 5 * q^81 + 38 * q^82 + 22 * q^83 - 36 * q^85 - 16 * q^86 - 6 * q^87 - 8 * q^89 - 6 * q^90 - 6 * q^91 + 12 * q^92 - 4 * q^93 + 10 * q^94 - 32 * q^95 - 19 * q^96 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more