LMFDB - Embedded newform 3381.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 13x^{8} + 41x^{7} + 47x^{6} - 165x^{5} - 45x^{4} + 207x^{3} + 12x^{2} - 76x - 2 $ x^10 - 3x^9 - 13x^8 + 41x^7 + 47x^6 - 165x^5 - 45x^4 + 207x^3 + 12x^2 - 76*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.06506$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.06506 q^{2} +1.00000 q^{3} +2.26446 q^{4} +0.608853 q^{5} -2.06506 q^{6} -0.546135 q^{8} +1.00000 q^{9} -1.25732 q^{10} -1.10243 q^{11} +2.26446 q^{12} -1.84712 q^{13} +0.608853 q^{15} -3.40113 q^{16} -5.88576 q^{17} -2.06506 q^{18} -1.08814 q^{19} +1.37873 q^{20} +2.27659 q^{22} +1.00000 q^{23} -0.546135 q^{24} -4.62930 q^{25} +3.81441 q^{26} +1.00000 q^{27} -0.804169 q^{29} -1.25732 q^{30} -8.77360 q^{31} +8.11580 q^{32} -1.10243 q^{33} +12.1544 q^{34} +2.26446 q^{36} +9.11528 q^{37} +2.24707 q^{38} -1.84712 q^{39} -0.332516 q^{40} +9.79559 q^{41} +5.23469 q^{43} -2.49642 q^{44} +0.608853 q^{45} -2.06506 q^{46} +5.90878 q^{47} -3.40113 q^{48} +9.55977 q^{50} -5.88576 q^{51} -4.18274 q^{52} +4.51013 q^{53} -2.06506 q^{54} -0.671220 q^{55} -1.08814 q^{57} +1.66066 q^{58} +7.40064 q^{59} +1.37873 q^{60} -5.02768 q^{61} +18.1180 q^{62} -9.95734 q^{64} -1.12462 q^{65} +2.27659 q^{66} +2.06305 q^{67} -13.3281 q^{68} +1.00000 q^{69} +4.30092 q^{71} -0.546135 q^{72} +1.15613 q^{73} -18.8236 q^{74} -4.62930 q^{75} -2.46406 q^{76} +3.81441 q^{78} +15.1716 q^{79} -2.07079 q^{80} +1.00000 q^{81} -20.2285 q^{82} +10.7196 q^{83} -3.58356 q^{85} -10.8099 q^{86} -0.804169 q^{87} +0.602078 q^{88} +7.92167 q^{89} -1.25732 q^{90} +2.26446 q^{92} -8.77360 q^{93} -12.2020 q^{94} -0.662518 q^{95} +8.11580 q^{96} +17.0648 q^{97} -1.10243 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 10 q^{3} + 15 q^{4} + 5 q^{5} + 3 q^{6} + 9 q^{8} + 10 q^{9} - 11 q^{10} + 8 q^{11} + 15 q^{12} + 5 q^{15} + 37 q^{16} + 11 q^{17} + 3 q^{18} - q^{19} + 15 q^{20} + 6 q^{22} + 10 q^{23}+ \cdots + 8 q^{99}+O(q^{100}) $ 10 * q + 3 * q^2 + 10 * q^3 + 15 * q^4 + 5 * q^5 + 3 * q^6 + 9 * q^8 + 10 * q^9 - 11 * q^10 + 8 * q^11 + 15 * q^12 + 5 * q^15 + 37 * q^16 + 11 * q^17 + 3 * q^18 - q^19 + 15 * q^20 + 6 * q^22 + 10 * q^23 + 9 * q^24 + 21 * q^25 - q^26 + 10 * q^27 + 22 * q^29 - 11 * q^30 + 3 * q^31 + 11 * q^32 + 8 * q^33 + 3 * q^34 + 15 * q^36 - 3 * q^37 - 16 * q^38 - 39 * q^40 + 26 * q^41 + 27 * q^43 + 16 * q^44 + 5 * q^45 + 3 * q^46 - 11 * q^47 + 37 * q^48 + 2 * q^50 + 11 * q^51 - 29 * q^52 + 5 * q^53 + 3 * q^54 + 18 * q^55 - q^57 + 16 * q^58 + 10 * q^59 + 15 * q^60 - 22 * q^61 + 32 * q^62 + 69 * q^64 - 11 * q^65 + 6 * q^66 - 2 * q^67 + 21 * q^68 + 10 * q^69 + 27 * q^71 + 9 * q^72 + 8 * q^73 + 14 * q^74 + 21 * q^75 + 22 * q^76 - q^78 + 21 * q^79 + 53 * q^80 + 10 * q^81 - 36 * q^82 + 12 * q^83 + 23 * q^85 + 18 * q^86 + 22 * q^87 - 10 * q^88 - 6 * q^89 - 11 * q^90 + 15 * q^92 + 3 * q^93 - 35 * q^94 + 44 * q^95 + 11 * q^96 - 6 * q^97 + 8 * 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 $

$2$	−2.06506	−1.46022	−0.730108	−	0.683331i	$-0.760530\pi$
$2$	−2.06506	−1.46022	−0.730108	+	0.683331i	$0.760530\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.26446	1.13223
$5$	0.608853	0.272287	0.136144	−	0.990689i	$-0.456529\pi$
$5$	0.608853	0.272287	0.136144	+	0.990689i	$0.456529\pi$
$6$	−2.06506	−0.843056
$7$	0	0
$7$	0	0
$8$	−0.546135	−0.193088
$9$	1.00000	0.333333
$10$	−1.25732	−0.397599
$11$	−1.10243	−0.332396	−0.166198	−	0.986092i	$-0.553149\pi$
$11$	−1.10243	−0.332396	−0.166198	+	0.986092i	$0.553149\pi$
$12$	2.26446	0.653695
$13$	−1.84712	−0.512299	−0.256149	−	0.966637i	$-0.582454\pi$
$13$	−1.84712	−0.512299	−0.256149	+	0.966637i	$0.582454\pi$
$14$	0	0
$15$	0.608853	0.157205
$16$	−3.40113	−0.850282
$17$	−5.88576	−1.42751	−0.713753	−	0.700397i	$-0.753006\pi$
$17$	−5.88576	−1.42751	−0.713753	+	0.700397i	$0.753006\pi$
$18$	−2.06506	−0.486739
$19$	−1.08814	−0.249637	−0.124818	−	0.992180i	$-0.539835\pi$
$19$	−1.08814	−0.249637	−0.124818	+	0.992180i	$0.539835\pi$
$20$	1.37873	0.308293
$21$	0	0
$22$	2.27659	0.485370
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−0.546135	−0.111479
$25$	−4.62930	−0.925860
$26$	3.81441	0.748067
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.804169	−0.149330	−0.0746652	−	0.997209i	$-0.523789\pi$
$29$	−0.804169	−0.149330	−0.0746652	+	0.997209i	$0.523789\pi$
$30$	−1.25732	−0.229554
$31$	−8.77360	−1.57579	−0.787893	−	0.615813i	$-0.788828\pi$
$31$	−8.77360	−1.57579	−0.787893	+	0.615813i	$0.788828\pi$
$32$	8.11580	1.43468
$33$	−1.10243	−0.191909
$34$	12.1544	2.08447
$35$	0	0
$36$	2.26446	0.377411
$37$	9.11528	1.49854	0.749271	−	0.662263i	$-0.230404\pi$
$37$	9.11528	1.49854	0.749271	+	0.662263i	$0.230404\pi$
$38$	2.24707	0.364524
$39$	−1.84712	−0.295776
$40$	−0.332516	−0.0525754
$41$	9.79559	1.52981	0.764907	−	0.644140i	$-0.222785\pi$
$41$	9.79559	1.52981	0.764907	+	0.644140i	$0.222785\pi$
$42$	0	0
$43$	5.23469	0.798283	0.399142	−	0.916889i	$-0.369308\pi$
$43$	5.23469	0.798283	0.399142	+	0.916889i	$0.369308\pi$
$44$	−2.49642	−0.376350
$45$	0.608853	0.0907625
$46$	−2.06506	−0.304476
$47$	5.90878	0.861885	0.430942	−	0.902379i	$-0.358181\pi$
$47$	5.90878	0.861885	0.430942	+	0.902379i	$0.358181\pi$
$48$	−3.40113	−0.490911
$49$	0	0
$50$	9.55977	1.35196
$51$	−5.88576	−0.824171
$52$	−4.18274	−0.580041
$53$	4.51013	0.619515	0.309757	−	0.950816i	$-0.399752\pi$
$53$	4.51013	0.619515	0.309757	+	0.950816i	$0.399752\pi$
$54$	−2.06506	−0.281019
$55$	−0.671220	−0.0905072
$56$	0	0
$57$	−1.08814	−0.144128
$58$	1.66066	0.218055
$59$	7.40064	0.963481	0.481741	−	0.876314i	$-0.340005\pi$
$59$	7.40064	0.963481	0.481741	+	0.876314i	$0.340005\pi$
$60$	1.37873	0.177993
$61$	−5.02768	−0.643729	−0.321864	−	0.946786i	$-0.604310\pi$
$61$	−5.02768	−0.643729	−0.321864	+	0.946786i	$0.604310\pi$
$62$	18.1180	2.30099
$63$	0	0
$64$	−9.95734	−1.24467
$65$	−1.12462	−0.139492
$66$	2.27659	0.280229
$67$	2.06305	0.252042	0.126021	−	0.992028i	$-0.459779\pi$
$67$	2.06305	0.252042	0.126021	+	0.992028i	$0.459779\pi$
$68$	−13.3281	−1.61627
$69$	1.00000	0.120386
$70$	0	0
$71$	4.30092	0.510425	0.255213	−	0.966885i	$-0.417855\pi$
$71$	4.30092	0.510425	0.255213	+	0.966885i	$0.417855\pi$
$72$	−0.546135	−0.0643627
$73$	1.15613	0.135315	0.0676577	−	0.997709i	$-0.478447\pi$
$73$	1.15613	0.135315	0.0676577	+	0.997709i	$0.478447\pi$
$74$	−18.8236	−2.18820
$75$	−4.62930	−0.534545
$76$	−2.46406	−0.282647
$77$	0	0
$78$	3.81441	0.431897
$79$	15.1716	1.70694	0.853469	−	0.521144i	$-0.174495\pi$
$79$	15.1716	1.70694	0.853469	+	0.521144i	$0.174495\pi$
$80$	−2.07079	−0.231521
$81$	1.00000	0.111111
$82$	−20.2285	−2.23386
$83$	10.7196	1.17663	0.588315	−	0.808632i	$-0.299791\pi$
$83$	10.7196	1.17663	0.588315	+	0.808632i	$0.299791\pi$
$84$	0	0
$85$	−3.58356	−0.388692
$86$	−10.8099	−1.16567
$87$	−0.804169	−0.0862160
$88$	0.602078	0.0641817
$89$	7.92167	0.839695	0.419847	−	0.907595i	$-0.362084\pi$
$89$	7.92167	0.839695	0.419847	+	0.907595i	$0.362084\pi$
$90$	−1.25732	−0.132533
$91$	0	0
$92$	2.26446	0.236087
$93$	−8.77360	−0.909780
$94$	−12.2020	−1.25854
$95$	−0.662518	−0.0679729
$96$	8.11580	0.828315
$97$	17.0648	1.73267	0.866336	−	0.499461i	$-0.166469\pi$
$97$	17.0648	1.73267	0.866336	+	0.499461i	$0.166469\pi$
$98$	0	0
$99$	−1.10243	−0.110799
$100$	−10.4829	−1.04829
$101$	8.56557	0.852306	0.426153	−	0.904651i	$-0.359869\pi$
$101$	8.56557	0.852306	0.426153	+	0.904651i	$0.359869\pi$
$102$	12.1544	1.20347
$103$	−10.4373	−1.02842	−0.514208	−	0.857666i	$-0.671914\pi$
$103$	−10.4373	−1.02842	−0.514208	+	0.857666i	$0.671914\pi$
$104$	1.00878	0.0989187
$105$	0	0
$106$	−9.31369	−0.904625
$107$	−8.07268	−0.780415	−0.390208	−	0.920727i	$-0.627597\pi$
$107$	−8.07268	−0.780415	−0.390208	+	0.920727i	$0.627597\pi$
$108$	2.26446	0.217898
$109$	3.81163	0.365088	0.182544	−	0.983198i	$-0.441567\pi$
$109$	3.81163	0.365088	0.182544	+	0.983198i	$0.441567\pi$
$110$	1.38611	0.132160
$111$	9.11528	0.865184
$112$	0	0
$113$	8.89428	0.836703	0.418352	−	0.908285i	$-0.362608\pi$
$113$	8.89428	0.836703	0.418352	+	0.908285i	$0.362608\pi$
$114$	2.24707	0.210458
$115$	0.608853	0.0567758
$116$	−1.82101	−0.169077
$117$	−1.84712	−0.170766
$118$	−15.2827	−1.40689
$119$	0	0
$120$	−0.332516	−0.0303544
$121$	−9.78464	−0.889513
$122$	10.3825	0.939984
$123$	9.79559	0.883239
$124$	−19.8675	−1.78416
$125$	−5.86283	−0.524387
$126$	0	0
$127$	7.91043	0.701937	0.350969	−	0.936387i	$-0.385852\pi$
$127$	7.91043	0.701937	0.350969	+	0.936387i	$0.385852\pi$
$128$	4.33089	0.382800
$129$	5.23469	0.460889
$130$	2.32241	0.203689
$131$	−0.475706	−0.0415627	−0.0207813	−	0.999784i	$-0.506615\pi$
$131$	−0.475706	−0.0415627	−0.0207813	+	0.999784i	$0.506615\pi$
$132$	−2.49642	−0.217286
$133$	0	0
$134$	−4.26033	−0.368036
$135$	0.608853	0.0524017
$136$	3.21442	0.275634
$137$	−8.27302	−0.706812	−0.353406	−	0.935470i	$-0.614977\pi$
$137$	−8.27302	−0.706812	−0.353406	+	0.935470i	$0.614977\pi$
$138$	−2.06506	−0.175789
$139$	2.48191	0.210513	0.105257	−	0.994445i	$-0.466434\pi$
$139$	2.48191	0.210513	0.105257	+	0.994445i	$0.466434\pi$
$140$	0	0
$141$	5.90878	0.497609
$142$	−8.88165	−0.745331
$143$	2.03633	0.170286
$144$	−3.40113	−0.283427
$145$	−0.489621	−0.0406608
$146$	−2.38749	−0.197590
$147$	0	0
$148$	20.6412	1.69670
$149$	20.5957	1.68726	0.843631	−	0.536924i	$-0.180414\pi$
$149$	20.5957	1.68726	0.843631	+	0.536924i	$0.180414\pi$
$150$	9.55977	0.780552
$151$	12.7948	1.04123	0.520613	−	0.853793i	$-0.325703\pi$
$151$	12.7948	1.04123	0.520613	+	0.853793i	$0.325703\pi$
$152$	0.594272	0.0482018
$153$	−5.88576	−0.475835
$154$	0	0
$155$	−5.34183	−0.429066
$156$	−4.18274	−0.334887
$157$	4.57005	0.364730	0.182365	−	0.983231i	$-0.441625\pi$
$157$	4.57005	0.364730	0.182365	+	0.983231i	$0.441625\pi$
$158$	−31.3302	−2.49250
$159$	4.51013	0.357677
$160$	4.94133	0.390646
$161$	0	0
$162$	−2.06506	−0.162246
$163$	−21.7544	−1.70393	−0.851967	−	0.523596i	$-0.824590\pi$
$163$	−21.7544	−1.70393	−0.851967	+	0.523596i	$0.824590\pi$
$164$	22.1818	1.73211
$165$	−0.671220	−0.0522544
$166$	−22.1366	−1.71814
$167$	3.92341	0.303602	0.151801	−	0.988411i	$-0.451493\pi$
$167$	3.92341	0.303602	0.151801	+	0.988411i	$0.451493\pi$
$168$	0	0
$169$	−9.58815	−0.737550
$170$	7.40026	0.567574
$171$	−1.08814	−0.0832122
$172$	11.8538	0.903842
$173$	24.0171	1.82599	0.912993	−	0.407976i	$-0.133765\pi$
$173$	24.0171	1.82599	0.912993	+	0.407976i	$0.133765\pi$
$174$	1.66066	0.125894
$175$	0	0
$176$	3.74952	0.282630
$177$	7.40064	0.556266
$178$	−16.3587	−1.22614
$179$	20.1690	1.50750	0.753752	−	0.657159i	$-0.228242\pi$
$179$	20.1690	1.50750	0.753752	+	0.657159i	$0.228242\pi$
$180$	1.37873	0.102764
$181$	−16.7087	−1.24195	−0.620973	−	0.783832i	$-0.713262\pi$
$181$	−16.7087	−1.24195	−0.620973	+	0.783832i	$0.713262\pi$
$182$	0	0
$183$	−5.02768	−0.371657
$184$	−0.546135	−0.0402616
$185$	5.54986	0.408034
$186$	18.1180	1.32848
$187$	6.48865	0.474497
$188$	13.3802	0.975854
$189$	0	0
$190$	1.36814	0.0992551
$191$	3.50958	0.253944	0.126972	−	0.991906i	$-0.459474\pi$
$191$	3.50958	0.253944	0.126972	+	0.991906i	$0.459474\pi$
$192$	−9.95734	−0.718609
$193$	−13.6422	−0.981987	−0.490993	−	0.871163i	$-0.663366\pi$
$193$	−13.6422	−0.981987	−0.490993	+	0.871163i	$0.663366\pi$
$194$	−35.2399	−2.53008
$195$	−1.12462	−0.0805360
$196$	0	0
$197$	−13.4651	−0.959349	−0.479675	−	0.877447i	$-0.659245\pi$
$197$	−13.4651	−0.959349	−0.479675	+	0.877447i	$0.659245\pi$
$198$	2.27659	0.161790
$199$	−14.2974	−1.01352	−0.506760	−	0.862087i	$-0.669157\pi$
$199$	−14.2974	−1.01352	−0.506760	+	0.862087i	$0.669157\pi$
$200$	2.52822	0.178772
$201$	2.06305	0.145517
$202$	−17.6884	−1.24455
$203$	0	0
$204$	−13.3281	−0.933153
$205$	5.96408	0.416549
$206$	21.5536	1.50171
$207$	1.00000	0.0695048
$208$	6.28229	0.435598
$209$	1.19960	0.0829782
$210$	0	0
$211$	2.83108	0.194900	0.0974499	−	0.995240i	$-0.468931\pi$
$211$	2.83108	0.194900	0.0974499	+	0.995240i	$0.468931\pi$
$212$	10.2130	0.701435
$213$	4.30092	0.294694
$214$	16.6706	1.13958
$215$	3.18716	0.217362
$216$	−0.546135	−0.0371598
$217$	0	0
$218$	−7.87123	−0.533107
$219$	1.15613	0.0781243
$220$	−1.51995	−0.102475
$221$	10.8717	0.731310
$222$	−18.8236	−1.26336
$223$	−4.78780	−0.320615	−0.160307	−	0.987067i	$-0.551249\pi$
$223$	−4.78780	−0.320615	−0.160307	+	0.987067i	$0.551249\pi$
$224$	0	0
$225$	−4.62930	−0.308620
$226$	−18.3672	−1.22177
$227$	−19.2951	−1.28066	−0.640331	−	0.768099i	$-0.721203\pi$
$227$	−19.2951	−1.28066	−0.640331	+	0.768099i	$0.721203\pi$
$228$	−2.46406	−0.163186
$229$	−8.12954	−0.537215	−0.268607	−	0.963250i	$-0.586563\pi$
$229$	−8.12954	−0.537215	−0.268607	+	0.963250i	$0.586563\pi$
$230$	−1.25732	−0.0829050
$231$	0	0
$232$	0.439185	0.0288339
$233$	7.94940	0.520783	0.260391	−	0.965503i	$-0.416148\pi$
$233$	7.94940	0.520783	0.260391	+	0.965503i	$0.416148\pi$
$234$	3.81441	0.249356
$235$	3.59758	0.234680
$236$	16.7585	1.09088
$237$	15.1716	0.985501
$238$	0	0
$239$	15.7789	1.02065	0.510326	−	0.859981i	$-0.329525\pi$
$239$	15.7789	1.02065	0.510326	+	0.859981i	$0.329525\pi$
$240$	−2.07079	−0.133669
$241$	11.2656	0.725680	0.362840	−	0.931851i	$-0.381807\pi$
$241$	11.2656	0.725680	0.362840	+	0.931851i	$0.381807\pi$
$242$	20.2059	1.29888
$243$	1.00000	0.0641500
$244$	−11.3850	−0.728851
$245$	0	0
$246$	−20.2285	−1.28972
$247$	2.00993	0.127889
$248$	4.79157	0.304265
$249$	10.7196	0.679328
$250$	12.1071	0.765719
$251$	2.27255	0.143442	0.0717209	−	0.997425i	$-0.477151\pi$
$251$	2.27255	0.143442	0.0717209	+	0.997425i	$0.477151\pi$
$252$	0	0
$253$	−1.10243	−0.0693094
$254$	−16.3355	−1.02498
$255$	−3.58356	−0.224411
$256$	10.9711	0.685697
$257$	0.871285	0.0543492	0.0271746	−	0.999631i	$-0.491349\pi$
$257$	0.871285	0.0543492	0.0271746	+	0.999631i	$0.491349\pi$
$258$	−10.8099	−0.672998
$259$	0	0
$260$	−2.54667	−0.157938
$261$	−0.804169	−0.0497768
$262$	0.982361	0.0606905
$263$	−5.33476	−0.328956	−0.164478	−	0.986381i	$-0.552594\pi$
$263$	−5.33476	−0.328956	−0.164478	+	0.986381i	$0.552594\pi$
$264$	0.602078	0.0370553
$265$	2.74601	0.168686
$266$	0	0
$267$	7.92167	0.484798
$268$	4.67171	0.285370
$269$	−22.7388	−1.38641	−0.693203	−	0.720742i	$-0.743801\pi$
$269$	−22.7388	−1.38641	−0.693203	+	0.720742i	$0.743801\pi$
$270$	−1.25732	−0.0765179
$271$	6.42151	0.390079	0.195039	−	0.980795i	$-0.437517\pi$
$271$	6.42151	0.390079	0.195039	+	0.980795i	$0.437517\pi$
$272$	20.0182	1.21378
$273$	0	0
$274$	17.0843	1.03210
$275$	5.10349	0.307752
$276$	2.26446	0.136305
$277$	14.0353	0.843298	0.421649	−	0.906759i	$-0.361452\pi$
$277$	14.0353	0.843298	0.421649	+	0.906759i	$0.361452\pi$
$278$	−5.12530	−0.307395
$279$	−8.77360	−0.525262
$280$	0	0
$281$	24.5278	1.46321	0.731603	−	0.681731i	$-0.238773\pi$
$281$	24.5278	1.46321	0.731603	+	0.681731i	$0.238773\pi$
$282$	−12.2020	−0.726617
$283$	4.39941	0.261518	0.130759	−	0.991414i	$-0.458259\pi$
$283$	4.39941	0.261518	0.130759	+	0.991414i	$0.458259\pi$
$284$	9.73928	0.577920
$285$	−0.662518	−0.0392442
$286$	−4.20513	−0.248655
$287$	0	0
$288$	8.11580	0.478228
$289$	17.6422	1.03777
$290$	1.01110	0.0593736
$291$	17.0648	1.00036
$292$	2.61803	0.153208
$293$	−13.3413	−0.779407	−0.389704	−	0.920940i	$-0.627423\pi$
$293$	−13.3413	−0.779407	−0.389704	+	0.920940i	$0.627423\pi$
$294$	0	0
$295$	4.50590	0.262344
$296$	−4.97818	−0.289351
$297$	−1.10243	−0.0639696
$298$	−42.5312	−2.46377
$299$	−1.84712	−0.106822
$300$	−10.4829	−0.605230
$301$	0	0
$302$	−26.4220	−1.52042
$303$	8.56557	0.492079
$304$	3.70091	0.212262
$305$	−3.06112	−0.175279
$306$	12.1544	0.694823
$307$	−24.5499	−1.40114	−0.700569	−	0.713585i	$-0.747070\pi$
$307$	−24.5499	−1.40114	−0.700569	+	0.713585i	$0.747070\pi$
$308$	0	0
$309$	−10.4373	−0.593756
$310$	11.0312	0.626530
$311$	−26.0773	−1.47871	−0.739355	−	0.673315i	$-0.764870\pi$
$311$	−26.0773	−1.47871	−0.739355	+	0.673315i	$0.764870\pi$
$312$	1.00878	0.0571108
$313$	−24.1973	−1.36771	−0.683857	−	0.729616i	$-0.739699\pi$
$313$	−24.1973	−1.36771	−0.683857	+	0.729616i	$0.739699\pi$
$314$	−9.43741	−0.532584
$315$	0	0
$316$	34.3555	1.93265
$317$	3.60976	0.202744	0.101372	−	0.994849i	$-0.467677\pi$
$317$	3.60976	0.202744	0.101372	+	0.994849i	$0.467677\pi$
$318$	−9.31369	−0.522286
$319$	0.886542	0.0496368
$320$	−6.06256	−0.338907
$321$	−8.07268	−0.450573
$322$	0	0
$323$	6.40453	0.356358
$324$	2.26446	0.125804
$325$	8.55087	0.474317
$326$	44.9240	2.48811
$327$	3.81163	0.210784
$328$	−5.34972	−0.295389
$329$	0	0
$330$	1.38611	0.0763027
$331$	8.23405	0.452584	0.226292	−	0.974059i	$-0.427340\pi$
$331$	8.23405	0.452584	0.226292	+	0.974059i	$0.427340\pi$
$332$	24.2742	1.33222
$333$	9.11528	0.499514
$334$	−8.10206	−0.443325
$335$	1.25610	0.0686279
$336$	0	0
$337$	26.1865	1.42647	0.713235	−	0.700925i	$-0.247229\pi$
$337$	26.1865	1.42647	0.713235	+	0.700925i	$0.247229\pi$
$338$	19.8001	1.07698
$339$	8.89428	0.483071
$340$	−8.11485	−0.440090
$341$	9.67231	0.523785
$342$	2.24707	0.121508
$343$	0	0
$344$	−2.85885	−0.154139
$345$	0.608853	0.0327795
$346$	−49.5967	−2.66633
$347$	−5.92735	−0.318197	−0.159098	−	0.987263i	$-0.550859\pi$
$347$	−5.92735	−0.318197	−0.159098	+	0.987263i	$0.550859\pi$
$348$	−1.82101	−0.0976165
$349$	−29.6319	−1.58616	−0.793080	−	0.609118i	$-0.791524\pi$
$349$	−29.6319	−1.58616	−0.793080	+	0.609118i	$0.791524\pi$
$350$	0	0
$351$	−1.84712	−0.0985919
$352$	−8.94712	−0.476883
$353$	14.2388	0.757853	0.378926	−	0.925427i	$-0.376293\pi$
$353$	14.2388	0.757853	0.378926	+	0.925427i	$0.376293\pi$
$354$	−15.2827	−0.812269
$355$	2.61863	0.138982
$356$	17.9383	0.950730
$357$	0	0
$358$	−41.6502	−2.20128
$359$	2.78230	0.146844	0.0734220	−	0.997301i	$-0.476608\pi$
$359$	2.78230	0.146844	0.0734220	+	0.997301i	$0.476608\pi$
$360$	−0.332516	−0.0175251
$361$	−17.8159	−0.937682
$362$	34.5044	1.81351
$363$	−9.78464	−0.513561
$364$	0	0
$365$	0.703916	0.0368447
$366$	10.3825	0.542700
$367$	6.66447	0.347883	0.173941	−	0.984756i	$-0.444350\pi$
$367$	6.66447	0.347883	0.173941	+	0.984756i	$0.444350\pi$
$368$	−3.40113	−0.177296
$369$	9.79559	0.509938
$370$	−11.4608	−0.595818
$371$	0	0
$372$	−19.8675	−1.03008
$373$	−19.0824	−0.988048	−0.494024	−	0.869448i	$-0.664475\pi$
$373$	−19.0824	−0.988048	−0.494024	+	0.869448i	$0.664475\pi$
$374$	−13.3994	−0.692869
$375$	−5.86283	−0.302755
$376$	−3.22700	−0.166420
$377$	1.48540	0.0765018
$378$	0	0
$379$	13.2631	0.681282	0.340641	−	0.940193i	$-0.389356\pi$
$379$	13.2631	0.681282	0.340641	+	0.940193i	$0.389356\pi$
$380$	−1.50025	−0.0769611
$381$	7.91043	0.405264
$382$	−7.24748	−0.370813
$383$	3.11602	0.159221	0.0796105	−	0.996826i	$-0.474632\pi$
$383$	3.11602	0.159221	0.0796105	+	0.996826i	$0.474632\pi$
$384$	4.33089	0.221010
$385$	0	0
$386$	28.1719	1.43391
$387$	5.23469	0.266094
$388$	38.6427	1.96179
$389$	−9.98409	−0.506214	−0.253107	−	0.967438i	$-0.581452\pi$
$389$	−9.98409	−0.506214	−0.253107	+	0.967438i	$0.581452\pi$
$390$	2.32241	0.117600
$391$	−5.88576	−0.297656
$392$	0	0
$393$	−0.475706	−0.0239962
$394$	27.8062	1.40086
$395$	9.23727	0.464777
$396$	−2.49642	−0.125450
$397$	−5.64444	−0.283286	−0.141643	−	0.989918i	$-0.545239\pi$
$397$	−5.64444	−0.283286	−0.141643	+	0.989918i	$0.545239\pi$
$398$	29.5251	1.47996
$399$	0	0
$400$	15.7448	0.787242
$401$	38.5622	1.92571	0.962853	−	0.270028i	$-0.0870328\pi$
$401$	38.5622	1.92571	0.962853	+	0.270028i	$0.0870328\pi$
$402$	−4.26033	−0.212486
$403$	16.2059	0.807273
$404$	19.3964	0.965008
$405$	0.608853	0.0302542
$406$	0	0
$407$	−10.0490	−0.498110
$408$	3.21442	0.159138
$409$	36.8071	1.81999	0.909997	−	0.414614i	$-0.136083\pi$
$409$	36.8071	1.81999	0.909997	+	0.414614i	$0.136083\pi$
$410$	−12.3162	−0.608252
$411$	−8.27302	−0.408078
$412$	−23.6349	−1.16441
$413$	0	0
$414$	−2.06506	−0.101492
$415$	6.52667	0.320382
$416$	−14.9908	−0.734987
$417$	2.48191	0.121540
$418$	−2.47725	−0.121166
$419$	6.45390	0.315293	0.157647	−	0.987496i	$-0.449609\pi$
$419$	6.45390	0.315293	0.157647	+	0.987496i	$0.449609\pi$
$420$	0	0
$421$	29.9287	1.45863	0.729317	−	0.684176i	$-0.239838\pi$
$421$	29.9287	1.45863	0.729317	+	0.684176i	$0.239838\pi$
$422$	−5.84635	−0.284596
$423$	5.90878	0.287295
$424$	−2.46314	−0.119621
$425$	27.2469	1.32167
$426$	−8.88165	−0.430317
$427$	0	0
$428$	−18.2803	−0.883612
$429$	2.03633	0.0983147
$430$	−6.58167	−0.317396
$431$	7.74839	0.373227	0.186613	−	0.982433i	$-0.440249\pi$
$431$	7.74839	0.373227	0.186613	+	0.982433i	$0.440249\pi$
$432$	−3.40113	−0.163637
$433$	−5.17696	−0.248789	−0.124394	−	0.992233i	$-0.539699\pi$
$433$	−5.17696	−0.248789	−0.124394	+	0.992233i	$0.539699\pi$
$434$	0	0
$435$	−0.489621	−0.0234755
$436$	8.63130	0.413364
$437$	−1.08814	−0.0520528
$438$	−2.38749	−0.114078
$439$	30.2921	1.44576	0.722881	−	0.690973i	$-0.242818\pi$
$439$	30.2921	1.44576	0.722881	+	0.690973i	$0.242818\pi$
$440$	0.366577	0.0174759
$441$	0	0
$442$	−22.4507	−1.06787
$443$	38.2021	1.81504	0.907519	−	0.420012i	$-0.137974\pi$
$443$	38.2021	1.81504	0.907519	+	0.420012i	$0.137974\pi$
$444$	20.6412	0.979589
$445$	4.82313	0.228638
$446$	9.88708	0.468167
$447$	20.5957	0.974141
$448$	0	0
$449$	2.95568	0.139487	0.0697435	−	0.997565i	$-0.477782\pi$
$449$	2.95568	0.139487	0.0697435	+	0.997565i	$0.477782\pi$
$450$	9.55977	0.450652
$451$	−10.7990	−0.508504
$452$	20.1408	0.947343
$453$	12.7948	0.601152
$454$	39.8455	1.87004
$455$	0	0
$456$	0.594272	0.0278293
$457$	17.6954	0.827758	0.413879	−	0.910332i	$-0.364174\pi$
$457$	17.6954	0.827758	0.413879	+	0.910332i	$0.364174\pi$
$458$	16.7880	0.784450
$459$	−5.88576	−0.274724
$460$	1.37873	0.0642834
$461$	11.7447	0.547005	0.273503	−	0.961871i	$-0.411818\pi$
$461$	11.7447	0.547005	0.273503	+	0.961871i	$0.411818\pi$
$462$	0	0
$463$	27.0872	1.25885	0.629424	−	0.777062i	$-0.283291\pi$
$463$	27.0872	1.25885	0.629424	+	0.777062i	$0.283291\pi$
$464$	2.73508	0.126973
$465$	−5.34183	−0.247722
$466$	−16.4160	−0.760455
$467$	−23.1520	−1.07135	−0.535673	−	0.844426i	$-0.679942\pi$
$467$	−23.1520	−1.07135	−0.535673	+	0.844426i	$0.679942\pi$
$468$	−4.18274	−0.193347
$469$	0	0
$470$	−7.42921	−0.342684
$471$	4.57005	0.210577
$472$	−4.04175	−0.186037
$473$	−5.77090	−0.265346
$474$	−31.3302	−1.43904
$475$	5.03733	0.231128
$476$	0	0
$477$	4.51013	0.206505
$478$	−32.5843	−1.49037
$479$	−22.3675	−1.02200	−0.510999	−	0.859581i	$-0.670725\pi$
$479$	−22.3675	−1.02200	−0.510999	+	0.859581i	$0.670725\pi$
$480$	4.94133	0.225540
$481$	−16.8370	−0.767701
$482$	−23.2641	−1.05965
$483$	0	0
$484$	−22.1570	−1.00714
$485$	10.3900	0.471785
$486$	−2.06506	−0.0936729
$487$	41.8522	1.89650	0.948252	−	0.317518i	$-0.102849\pi$
$487$	41.8522	1.89650	0.948252	+	0.317518i	$0.102849\pi$
$488$	2.74580	0.124296
$489$	−21.7544	−0.983766
$490$	0	0
$491$	16.2658	0.734063	0.367032	−	0.930209i	$-0.380374\pi$
$491$	16.2658	0.734063	0.367032	+	0.930209i	$0.380374\pi$
$492$	22.1818	1.00003
$493$	4.73315	0.213170
$494$	−4.15061	−0.186745
$495$	−0.671220	−0.0301691
$496$	29.8401	1.33986
$497$	0	0
$498$	−22.1366	−0.991966
$499$	−19.8250	−0.887489	−0.443744	−	0.896153i	$-0.646350\pi$
$499$	−19.8250	−0.887489	−0.443744	+	0.896153i	$0.646350\pi$
$500$	−13.2762	−0.593728
$501$	3.92341	0.175285
$502$	−4.69294	−0.209456
$503$	−9.18201	−0.409406	−0.204703	−	0.978824i	$-0.565623\pi$
$503$	−9.18201	−0.409406	−0.204703	+	0.978824i	$0.565623\pi$
$504$	0	0
$505$	5.21517	0.232072
$506$	2.27659	0.101207
$507$	−9.58815	−0.425825
$508$	17.9129	0.794756
$509$	32.8014	1.45390	0.726948	−	0.686692i	$-0.240938\pi$
$509$	32.8014	1.45390	0.726948	+	0.686692i	$0.240938\pi$
$510$	7.40026	0.327689
$511$	0	0
$512$	−31.3178	−1.38407
$513$	−1.08814	−0.0480426
$514$	−1.79925	−0.0793617
$515$	−6.35477	−0.280025
$516$	11.8538	0.521834
$517$	−6.51404	−0.286487
$518$	0	0
$519$	24.0171	1.05423
$520$	0.614197	0.0269343
$521$	−6.86356	−0.300698	−0.150349	−	0.988633i	$-0.548040\pi$
$521$	−6.86356	−0.300698	−0.150349	+	0.988633i	$0.548040\pi$
$522$	1.66066	0.0726849
$523$	−9.76622	−0.427047	−0.213524	−	0.976938i	$-0.568494\pi$
$523$	−9.76622	−0.427047	−0.213524	+	0.976938i	$0.568494\pi$
$524$	−1.07722	−0.0470586
$525$	0	0
$526$	11.0166	0.480347
$527$	51.6393	2.24944
$528$	3.74952	0.163177
$529$	1.00000	0.0434783
$530$	−5.67067	−0.246318
$531$	7.40064	0.321160
$532$	0	0
$533$	−18.0936	−0.783722
$534$	−16.3587	−0.707910
$535$	−4.91508	−0.212497
$536$	−1.12671	−0.0486663
$537$	20.1690	0.870357
$538$	46.9568	2.02445
$539$	0	0
$540$	1.37873	0.0593309
$541$	−4.64207	−0.199578	−0.0997891	−	0.995009i	$-0.531817\pi$
$541$	−4.64207	−0.199578	−0.0997891	+	0.995009i	$0.531817\pi$
$542$	−13.2608	−0.569600
$543$	−16.7087	−0.717038
$544$	−47.7676	−2.04802
$545$	2.32072	0.0994088
$546$	0	0
$547$	21.6976	0.927722	0.463861	−	0.885908i	$-0.346464\pi$
$547$	21.6976	0.927722	0.463861	+	0.885908i	$0.346464\pi$
$548$	−18.7340	−0.800275
$549$	−5.02768	−0.214576
$550$	−10.5390	−0.449385
$551$	0.875049	0.0372783
$552$	−0.546135	−0.0232451
$553$	0	0
$554$	−28.9836	−1.23140
$555$	5.54986	0.235579
$556$	5.62021	0.238350
$557$	−32.1596	−1.36265	−0.681323	−	0.731983i	$-0.738595\pi$
$557$	−32.1596	−1.36265	−0.681323	+	0.731983i	$0.738595\pi$
$558$	18.1180	0.766996
$559$	−9.66910	−0.408960
$560$	0	0
$561$	6.48865	0.273951
$562$	−50.6513	−2.13660
$563$	12.3159	0.519055	0.259528	−	0.965736i	$-0.416433\pi$
$563$	12.3159	0.519055	0.259528	+	0.965736i	$0.416433\pi$
$564$	13.3802	0.563409
$565$	5.41531	0.227824
$566$	−9.08503	−0.381872
$567$	0	0
$568$	−2.34888	−0.0985570
$569$	−26.2350	−1.09983	−0.549914	−	0.835221i	$-0.685340\pi$
$569$	−26.2350	−1.09983	−0.549914	+	0.835221i	$0.685340\pi$
$570$	1.36814	0.0573050
$571$	−5.59829	−0.234281	−0.117141	−	0.993115i	$-0.537373\pi$
$571$	−5.59829	−0.234281	−0.117141	+	0.993115i	$0.537373\pi$
$572$	4.61119	0.192803
$573$	3.50958	0.146615
$574$	0	0
$575$	−4.62930	−0.193055
$576$	−9.95734	−0.414889
$577$	10.1348	0.421915	0.210958	−	0.977495i	$-0.432342\pi$
$577$	10.1348	0.421915	0.210958	+	0.977495i	$0.432342\pi$
$578$	−36.4321	−1.51538
$579$	−13.6422	−0.566950
$580$	−1.10873	−0.0460375
$581$	0	0
$582$	−35.2399	−1.46074
$583$	−4.97212	−0.205924
$584$	−0.631406	−0.0261278
$585$	−1.12462	−0.0464975
$586$	27.5506	1.13810
$587$	−27.2323	−1.12400	−0.561999	−	0.827138i	$-0.689968\pi$
$587$	−27.2323	−1.12400	−0.561999	+	0.827138i	$0.689968\pi$
$588$	0	0
$589$	9.54691	0.393374
$590$	−9.30495	−0.383079
$591$	−13.4651	−0.553880
$592$	−31.0022	−1.27418
$593$	−38.3547	−1.57504	−0.787519	−	0.616290i	$-0.788635\pi$
$593$	−38.3547	−1.57504	−0.787519	+	0.616290i	$0.788635\pi$
$594$	2.27659	0.0934095
$595$	0	0
$596$	46.6381	1.91037
$597$	−14.2974	−0.585155
$598$	3.81441	0.155983
$599$	−6.88378	−0.281264	−0.140632	−	0.990062i	$-0.544913\pi$
$599$	−6.88378	−0.281264	−0.140632	+	0.990062i	$0.544913\pi$
$600$	2.52822	0.103214
$601$	−1.80012	−0.0734283	−0.0367141	−	0.999326i	$-0.511689\pi$
$601$	−1.80012	−0.0734283	−0.0367141	+	0.999326i	$0.511689\pi$
$602$	0	0
$603$	2.06305	0.0840141
$604$	28.9734	1.17891
$605$	−5.95741	−0.242203
$606$	−17.6884	−0.718542
$607$	−2.93517	−0.119135	−0.0595674	−	0.998224i	$-0.518972\pi$
$607$	−2.93517	−0.119135	−0.0595674	+	0.998224i	$0.518972\pi$
$608$	−8.83113	−0.358150
$609$	0	0
$610$	6.32139	0.255946
$611$	−10.9142	−0.441542
$612$	−13.3281	−0.538756
$613$	31.7390	1.28192	0.640962	−	0.767572i	$-0.278535\pi$
$613$	31.7390	1.28192	0.640962	+	0.767572i	$0.278535\pi$
$614$	50.6970	2.04596
$615$	5.96408	0.240495
$616$	0	0
$617$	−41.7807	−1.68203	−0.841013	−	0.541015i	$-0.818040\pi$
$617$	−41.7807	−1.68203	−0.841013	+	0.541015i	$0.818040\pi$
$618$	21.5536	0.867012
$619$	−27.2170	−1.09394	−0.546972	−	0.837151i	$-0.684220\pi$
$619$	−27.2170	−1.09394	−0.546972	+	0.837151i	$0.684220\pi$
$620$	−12.0964	−0.485803
$621$	1.00000	0.0401286
$622$	53.8512	2.15924
$623$	0	0
$624$	6.28229	0.251493
$625$	19.5769	0.783076
$626$	49.9689	1.99716
$627$	1.19960	0.0479075
$628$	10.3487	0.412959
$629$	−53.6503	−2.13918
$630$	0	0
$631$	25.8728	1.02998	0.514990	−	0.857196i	$-0.327795\pi$
$631$	25.8728	1.02998	0.514990	+	0.857196i	$0.327795\pi$
$632$	−8.28574	−0.329589
$633$	2.83108	0.112525
$634$	−7.45437	−0.296051
$635$	4.81629	0.191129
$636$	10.2130	0.404973
$637$	0	0
$638$	−1.83076	−0.0724805
$639$	4.30092	0.170142
$640$	2.63687	0.104232
$641$	−13.3591	−0.527651	−0.263826	−	0.964570i	$-0.584984\pi$
$641$	−13.3591	−0.527651	−0.263826	+	0.964570i	$0.584984\pi$
$642$	16.6706	0.657934
$643$	−43.4719	−1.71436	−0.857182	−	0.515014i	$-0.827787\pi$
$643$	−43.4719	−1.71436	−0.857182	+	0.515014i	$0.827787\pi$
$644$	0	0
$645$	3.18716	0.125494
$646$	−13.2257	−0.520360
$647$	46.1515	1.81440	0.907201	−	0.420698i	$-0.138215\pi$
$647$	46.1515	1.81440	0.907201	+	0.420698i	$0.138215\pi$
$648$	−0.546135	−0.0214542
$649$	−8.15871	−0.320257
$650$	−17.6580	−0.692605
$651$	0	0
$652$	−49.2620	−1.92925
$653$	−33.5965	−1.31473	−0.657367	−	0.753570i	$-0.728330\pi$
$653$	−33.5965	−1.31473	−0.657367	+	0.753570i	$0.728330\pi$
$654$	−7.87123	−0.307790
$655$	−0.289635	−0.0113170
$656$	−33.3161	−1.30077
$657$	1.15613	0.0451051
$658$	0	0
$659$	10.2122	0.397809	0.198905	−	0.980019i	$-0.436262\pi$
$659$	10.2122	0.397809	0.198905	+	0.980019i	$0.436262\pi$
$660$	−1.51995	−0.0591641
$661$	−18.0200	−0.700895	−0.350448	−	0.936582i	$-0.613971\pi$
$661$	−18.0200	−0.700895	−0.350448	+	0.936582i	$0.613971\pi$
$662$	−17.0038	−0.660871
$663$	10.8717	0.422222
$664$	−5.85436	−0.227193
$665$	0	0
$666$	−18.8236	−0.729399
$667$	−0.804169	−0.0311375
$668$	8.88442	0.343748
$669$	−4.78780	−0.185107
$670$	−2.59391	−0.100212
$671$	5.54268	0.213973
$672$	0	0
$673$	−40.5777	−1.56415	−0.782077	−	0.623181i	$-0.785840\pi$
$673$	−40.5777	−1.56415	−0.782077	+	0.623181i	$0.785840\pi$
$674$	−54.0767	−2.08296
$675$	−4.62930	−0.178182
$676$	−21.7120	−0.835078
$677$	−25.9423	−0.997044	−0.498522	−	0.866877i	$-0.666124\pi$
$677$	−25.9423	−0.997044	−0.498522	+	0.866877i	$0.666124\pi$
$678$	−18.3672	−0.705388
$679$	0	0
$680$	1.95711	0.0750518
$681$	−19.2951	−0.739390
$682$	−19.9739	−0.764839
$683$	−25.8167	−0.987848	−0.493924	−	0.869505i	$-0.664438\pi$
$683$	−25.8167	−0.987848	−0.493924	+	0.869505i	$0.664438\pi$
$684$	−2.46406	−0.0942156
$685$	−5.03705	−0.192456
$686$	0	0
$687$	−8.12954	−0.310161
$688$	−17.8039	−0.678766
$689$	−8.33076	−0.317377
$690$	−1.25732	−0.0478652
$691$	15.6020	0.593528	0.296764	−	0.954951i	$-0.404093\pi$
$691$	15.6020	0.593528	0.296764	+	0.954951i	$0.404093\pi$
$692$	54.3859	2.06744
$693$	0	0
$694$	12.2403	0.464636
$695$	1.51112	0.0573201
$696$	0.439185	0.0166473
$697$	−57.6545	−2.18382
$698$	61.1916	2.31614
$699$	7.94940	0.300674
$700$	0	0
$701$	38.6350	1.45922	0.729612	−	0.683861i	$-0.239701\pi$
$701$	38.6350	1.45922	0.729612	+	0.683861i	$0.239701\pi$
$702$	3.81441	0.143966
$703$	−9.91871	−0.374091
$704$	10.9773	0.413722
$705$	3.59758	0.135493
$706$	−29.4039	−1.10663
$707$	0	0
$708$	16.7585	0.629823
$709$	−43.3432	−1.62779	−0.813894	−	0.581013i	$-0.802656\pi$
$709$	−43.3432	−1.62779	−0.813894	+	0.581013i	$0.802656\pi$
$710$	−5.40762	−0.202944
$711$	15.1716	0.568979
$712$	−4.32630	−0.162135
$713$	−8.77360	−0.328574
$714$	0	0
$715$	1.23982	0.0463667
$716$	45.6720	1.70684
$717$	15.7789	0.589274
$718$	−5.74560	−0.214424
$719$	29.4485	1.09824	0.549122	−	0.835742i	$-0.314962\pi$
$719$	29.4485	1.09824	0.549122	+	0.835742i	$0.314962\pi$
$720$	−2.07079	−0.0771737
$721$	0	0
$722$	36.7910	1.36922
$723$	11.2656	0.418971
$724$	−37.8362	−1.40617
$725$	3.72274	0.138259
$726$	20.2059	0.749910
$727$	50.1254	1.85905	0.929524	−	0.368763i	$-0.120218\pi$
$727$	50.1254	1.85905	0.929524	+	0.368763i	$0.120218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−1.45363	−0.0538012
$731$	−30.8101	−1.13955
$732$	−11.3850	−0.420802
$733$	20.9760	0.774766	0.387383	−	0.921919i	$-0.373379\pi$
$733$	20.9760	0.774766	0.387383	+	0.921919i	$0.373379\pi$
$734$	−13.7625	−0.507984
$735$	0	0
$736$	8.11580	0.299152
$737$	−2.27438	−0.0837778
$738$	−20.2285	−0.744620
$739$	11.5852	0.426168	0.213084	−	0.977034i	$-0.431649\pi$
$739$	11.5852	0.426168	0.213084	+	0.977034i	$0.431649\pi$
$740$	12.5675	0.461990
$741$	2.00993	0.0738365
$742$	0	0
$743$	−44.7847	−1.64299	−0.821495	−	0.570216i	$-0.806860\pi$
$743$	−44.7847	−1.64299	−0.821495	+	0.570216i	$0.806860\pi$
$744$	4.79157	0.175668
$745$	12.5397	0.459420
$746$	39.4062	1.44276
$747$	10.7196	0.392210
$748$	14.6933	0.537241
$749$	0	0
$750$	12.1071	0.442088
$751$	−22.2887	−0.813325	−0.406662	−	0.913579i	$-0.633307\pi$
$751$	−22.2887	−0.813325	−0.406662	+	0.913579i	$0.633307\pi$
$752$	−20.0965	−0.732845
$753$	2.27255	0.0828162
$754$	−3.06743	−0.111709
$755$	7.79015	0.283513
$756$	0	0
$757$	−34.3724	−1.24929	−0.624643	−	0.780910i	$-0.714756\pi$
$757$	−34.3724	−1.24929	−0.624643	+	0.780910i	$0.714756\pi$
$758$	−27.3892	−0.994820
$759$	−1.10243	−0.0400158
$760$	0.361824	0.0131248
$761$	−43.7206	−1.58487	−0.792435	−	0.609956i	$-0.791187\pi$
$761$	−43.7206	−1.58487	−0.792435	+	0.609956i	$0.791187\pi$
$762$	−16.3355	−0.591773
$763$	0	0
$764$	7.94731	0.287524
$765$	−3.58356	−0.129564
$766$	−6.43476	−0.232497
$767$	−13.6699	−0.493590
$768$	10.9711	0.395887
$769$	2.96450	0.106903	0.0534513	−	0.998570i	$-0.482978\pi$
$769$	2.96450	0.106903	0.0534513	+	0.998570i	$0.482978\pi$
$770$	0	0
$771$	0.871285	0.0313785
$772$	−30.8923	−1.11184
$773$	−22.2038	−0.798615	−0.399307	−	0.916817i	$-0.630749\pi$
$773$	−22.2038	−0.798615	−0.399307	+	0.916817i	$0.630749\pi$
$774$	−10.8099	−0.388556
$775$	40.6156	1.45896
$776$	−9.31971	−0.334558
$777$	0	0
$778$	20.6177	0.739181
$779$	−10.6590	−0.381898
$780$	−2.54667	−0.0911855
$781$	−4.74147	−0.169663
$782$	12.1544	0.434642
$783$	−0.804169	−0.0287387
$784$	0	0
$785$	2.78249	0.0993112
$786$	0.982361	0.0350397
$787$	34.6515	1.23519	0.617596	−	0.786495i	$-0.288107\pi$
$787$	34.6515	1.23519	0.617596	+	0.786495i	$0.288107\pi$
$788$	−30.4913	−1.08621
$789$	−5.33476	−0.189923
$790$	−19.0755	−0.678676
$791$	0	0
$792$	0.602078	0.0213939
$793$	9.28673	0.329781
$794$	11.6561	0.413660
$795$	2.74601	0.0973909
$796$	−32.3761	−1.14754
$797$	22.4200	0.794158	0.397079	−	0.917784i	$-0.370024\pi$
$797$	22.4200	0.794158	0.397079	+	0.917784i	$0.370024\pi$
$798$	0	0
$799$	−34.7777	−1.23035
$800$	−37.5705	−1.32832
$801$	7.92167	0.279898
$802$	−79.6332	−2.81195
$803$	−1.27456	−0.0449783
$804$	4.67171	0.164759
$805$	0	0
$806$	−33.4661	−1.17879
$807$	−22.7388	−0.800442
$808$	−4.67796	−0.164570
$809$	26.0244	0.914970	0.457485	−	0.889217i	$-0.348750\pi$
$809$	26.0244	0.914970	0.457485	+	0.889217i	$0.348750\pi$
$810$	−1.25732	−0.0441776
$811$	45.2644	1.58945	0.794724	−	0.606970i	$-0.207615\pi$
$811$	45.2644	1.58945	0.794724	+	0.606970i	$0.207615\pi$
$812$	0	0
$813$	6.42151	0.225212
$814$	20.7517	0.727348
$815$	−13.2452	−0.463959
$816$	20.0182	0.700778
$817$	−5.69608	−0.199281
$818$	−76.0088	−2.65759
$819$	0	0
$820$	13.5054	0.471630
$821$	7.28729	0.254328	0.127164	−	0.991882i	$-0.459413\pi$
$821$	7.28729	0.254328	0.127164	+	0.991882i	$0.459413\pi$
$822$	17.0843	0.595882
$823$	3.29869	0.114985	0.0574925	−	0.998346i	$-0.481689\pi$
$823$	3.29869	0.114985	0.0574925	+	0.998346i	$0.481689\pi$
$824$	5.70017	0.198575
$825$	5.10349	0.177681
$826$	0	0
$827$	−48.8022	−1.69702	−0.848510	−	0.529179i	$-0.822500\pi$
$827$	−48.8022	−1.69702	−0.848510	+	0.529179i	$0.822500\pi$
$828$	2.26446	0.0786956
$829$	−24.7978	−0.861263	−0.430631	−	0.902528i	$-0.641709\pi$
$829$	−24.7978	−0.861263	−0.430631	+	0.902528i	$0.641709\pi$
$830$	−13.4780	−0.467827
$831$	14.0353	0.486878
$832$	18.3924	0.637642
$833$	0	0
$834$	−5.12530	−0.177475
$835$	2.38878	0.0826671
$836$	2.71646	0.0939506
$837$	−8.77360	−0.303260
$838$	−13.3277	−0.460397
$839$	6.27230	0.216544	0.108272	−	0.994121i	$-0.465468\pi$
$839$	6.27230	0.216544	0.108272	+	0.994121i	$0.465468\pi$
$840$	0	0
$841$	−28.3533	−0.977700
$842$	−61.8044	−2.12992
$843$	24.5278	0.844782
$844$	6.41089	0.220672
$845$	−5.83777	−0.200826
$846$	−12.2020	−0.419513
$847$	0	0
$848$	−15.3395	−0.526762
$849$	4.39941	0.150987
$850$	−56.2665	−1.92993
$851$	9.11528	0.312468
$852$	9.73928	0.333662
$853$	1.75188	0.0599834	0.0299917	−	0.999550i	$-0.490452\pi$
$853$	1.75188	0.0599834	0.0299917	+	0.999550i	$0.490452\pi$
$854$	0	0
$855$	−0.662518	−0.0226576
$856$	4.40878	0.150689
$857$	15.0955	0.515653	0.257826	−	0.966191i	$-0.416994\pi$
$857$	15.0955	0.515653	0.257826	+	0.966191i	$0.416994\pi$
$858$	−4.20513	−0.143561
$859$	−9.58801	−0.327139	−0.163569	−	0.986532i	$-0.552301\pi$
$859$	−9.58801	−0.327139	−0.163569	+	0.986532i	$0.552301\pi$
$860$	7.21721	0.246105
$861$	0	0
$862$	−16.0009	−0.544992
$863$	34.3335	1.16872	0.584362	−	0.811493i	$-0.301345\pi$
$863$	34.3335	1.16872	0.584362	+	0.811493i	$0.301345\pi$
$864$	8.11580	0.276105
$865$	14.6229	0.497193
$866$	10.6907	0.363285
$867$	17.6422	0.599159
$868$	0	0
$869$	−16.7257	−0.567379
$870$	1.01110	0.0342793
$871$	−3.81071	−0.129121
$872$	−2.08166	−0.0704941
$873$	17.0648	0.577557
$874$	2.24707	0.0760084
$875$	0	0
$876$	2.61803	0.0884549
$877$	17.6058	0.594505	0.297252	−	0.954799i	$-0.403930\pi$
$877$	17.6058	0.594505	0.297252	+	0.954799i	$0.403930\pi$
$878$	−62.5549	−2.11112
$879$	−13.3413	−0.449991
$880$	2.28290	0.0769567
$881$	−39.5494	−1.33245	−0.666227	−	0.745749i	$-0.732092\pi$
$881$	−39.5494	−1.33245	−0.666227	+	0.745749i	$0.732092\pi$
$882$	0	0
$883$	0.340528	0.0114597	0.00572983	−	0.999984i	$-0.498176\pi$
$883$	0.340528	0.0114597	0.00572983	+	0.999984i	$0.498176\pi$
$884$	24.6186	0.828013
$885$	4.50590	0.151464
$886$	−78.8896	−2.65035
$887$	−30.0962	−1.01053	−0.505266	−	0.862964i	$-0.668606\pi$
$887$	−30.0962	−1.01053	−0.505266	+	0.862964i	$0.668606\pi$
$888$	−4.97818	−0.167057
$889$	0	0
$890$	−9.96004	−0.333861
$891$	−1.10243	−0.0369329
$892$	−10.8418	−0.363010
$893$	−6.42959	−0.215158
$894$	−42.5312	−1.42246
$895$	12.2800	0.410474
$896$	0	0
$897$	−1.84712	−0.0616735
$898$	−6.10364	−0.203681
$899$	7.05546	0.235313
$900$	−10.4829	−0.349429
$901$	−26.5456	−0.884361
$902$	22.3005	0.742526
$903$	0	0
$904$	−4.85748	−0.161557
$905$	−10.1731	−0.338166
$906$	−26.4220	−0.877812
$907$	31.8853	1.05873	0.529366	−	0.848393i	$-0.322430\pi$
$907$	31.8853	1.05873	0.529366	+	0.848393i	$0.322430\pi$
$908$	−43.6931	−1.45001
$909$	8.56557	0.284102
$910$	0	0
$911$	36.7297	1.21691	0.608454	−	0.793589i	$-0.291790\pi$
$911$	36.7297	1.21691	0.608454	+	0.793589i	$0.291790\pi$
$912$	3.70091	0.122549
$913$	−11.8177	−0.391107
$914$	−36.5421	−1.20871
$915$	−3.06112	−0.101198
$916$	−18.4090	−0.608252
$917$	0	0
$918$	12.1544	0.401156
$919$	−29.3315	−0.967556	−0.483778	−	0.875191i	$-0.660736\pi$
$919$	−29.3315	−0.967556	−0.483778	+	0.875191i	$0.660736\pi$
$920$	−0.332516	−0.0109627
$921$	−24.5499	−0.808947
$922$	−24.2535	−0.798746
$923$	−7.94431	−0.261490
$924$	0	0
$925$	−42.1973	−1.38744
$926$	−55.9366	−1.83819
$927$	−10.4373	−0.342805
$928$	−6.52647	−0.214242
$929$	1.30704	0.0428826	0.0214413	−	0.999770i	$-0.493174\pi$
$929$	1.30704	0.0428826	0.0214413	+	0.999770i	$0.493174\pi$
$930$	11.0312	0.361727
$931$	0	0
$932$	18.0011	0.589647
$933$	−26.0773	−0.853734
$934$	47.8102	1.56440
$935$	3.95064	0.129200
$936$	1.00878	0.0329729
$937$	4.77786	0.156086	0.0780430	−	0.996950i	$-0.475133\pi$
$937$	4.77786	0.156086	0.0780430	+	0.996950i	$0.475133\pi$
$938$	0	0
$939$	−24.1973	−0.789650
$940$	8.14660	0.265713
$941$	33.4004	1.08882	0.544411	−	0.838819i	$-0.316753\pi$
$941$	33.4004	1.08882	0.544411	+	0.838819i	$0.316753\pi$
$942$	−9.43741	−0.307488
$943$	9.79559	0.318988
$944$	−25.1705	−0.819231
$945$	0	0
$946$	11.9172	0.387463
$947$	35.8911	1.16630	0.583152	−	0.812363i	$-0.301819\pi$
$947$	35.8911	1.16630	0.583152	+	0.812363i	$0.301819\pi$
$948$	34.3555	1.11582
$949$	−2.13552	−0.0693219
$950$	−10.4024	−0.337498
$951$	3.60976	0.117055
$952$	0	0
$953$	−42.4876	−1.37631	−0.688154	−	0.725564i	$-0.741579\pi$
$953$	−42.4876	−1.37631	−0.688154	+	0.725564i	$0.741579\pi$
$954$	−9.31369	−0.301542
$955$	2.13682	0.0691457
$956$	35.7308	1.15562
$957$	0.886542	0.0286578
$958$	46.1903	1.49234
$959$	0	0
$960$	−6.06256	−0.195668
$961$	45.9761	1.48310
$962$	34.7694	1.12101
$963$	−8.07268	−0.260138
$964$	25.5105	0.821638
$965$	−8.30609	−0.267383
$966$	0	0
$967$	−50.9129	−1.63725	−0.818624	−	0.574330i	$-0.805263\pi$
$967$	−50.9129	−1.63725	−0.818624	+	0.574330i	$0.805263\pi$
$968$	5.34374	0.171754
$969$	6.40453	0.205743
$970$	−21.4559	−0.688908
$971$	16.7372	0.537123	0.268561	−	0.963263i	$-0.413452\pi$
$971$	16.7372	0.537123	0.268561	+	0.963263i	$0.413452\pi$
$972$	2.26446	0.0726327
$973$	0	0
$974$	−86.4272	−2.76931
$975$	8.55087	0.273847
$976$	17.0998	0.547351
$977$	47.3729	1.51559	0.757797	−	0.652491i	$-0.226276\pi$
$977$	47.3729	1.51559	0.757797	+	0.652491i	$0.226276\pi$
$978$	44.9240	1.43651
$979$	−8.73310	−0.279111
$980$	0	0
$981$	3.81163	0.121696
$982$	−33.5897	−1.07189
$983$	−35.2609	−1.12465	−0.562323	−	0.826918i	$-0.690092\pi$
$983$	−35.2609	−1.12465	−0.562323	+	0.826918i	$0.690092\pi$
$984$	−5.34972	−0.170543
$985$	−8.19827	−0.261219
$986$	−9.77422	−0.311275
$987$	0	0
$988$	4.55141	0.144800
$989$	5.23469	0.166454
$990$	1.38611	0.0440534
$991$	16.9164	0.537369	0.268684	−	0.963228i	$-0.413411\pi$
$991$	16.9164	0.537369	0.268684	+	0.963228i	$0.413411\pi$
$992$	−71.2048	−2.26075
$993$	8.23405	0.261300
$994$	0	0
$995$	−8.70504	−0.275968
$996$	24.2742	0.769157
$997$	52.3647	1.65841	0.829203	−	0.558947i	$-0.188795\pi$
$997$	52.3647	1.65841	0.829203	+	0.558947i	$0.188795\pi$
$998$	40.9398	1.29593
$999$	9.11528	0.288395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bj.1.2		10
7.3	odd	6		483.2.i.h.415.9	yes	20
7.5	odd	6		483.2.i.h.277.9	✓	20
7.6	odd	2		3381.2.a.bi.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.h.277.9	✓	20	7.5	odd	6
483.2.i.h.415.9	yes	20	7.3	odd	6
3381.2.a.bi.1.2		10	7.6	odd	2
3381.2.a.bj.1.2		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q-2.06506 q^{2} +1.00000 q^{3} +2.26446 q^{4} +0.608853 q^{5} -2.06506 q^{6} -0.546135 q^{8} +1.00000 q^{9} -1.25732 q^{10} -1.10243 q^{11} +2.26446 q^{12} -1.84712 q^{13} +0.608853 q^{15} -3.40113 q^{16} -5.88576 q^{17} -2.06506 q^{18} -1.08814 q^{19} +1.37873 q^{20} +2.27659 q^{22} +1.00000 q^{23} -0.546135 q^{24} -4.62930 q^{25} +3.81441 q^{26} +1.00000 q^{27} -0.804169 q^{29} -1.25732 q^{30} -8.77360 q^{31} +8.11580 q^{32} -1.10243 q^{33} +12.1544 q^{34} +2.26446 q^{36} +9.11528 q^{37} +2.24707 q^{38} -1.84712 q^{39} -0.332516 q^{40} +9.79559 q^{41} +5.23469 q^{43} -2.49642 q^{44} +0.608853 q^{45} -2.06506 q^{46} +5.90878 q^{47} -3.40113 q^{48} +9.55977 q^{50} -5.88576 q^{51} -4.18274 q^{52} +4.51013 q^{53} -2.06506 q^{54} -0.671220 q^{55} -1.08814 q^{57} +1.66066 q^{58} +7.40064 q^{59} +1.37873 q^{60} -5.02768 q^{61} +18.1180 q^{62} -9.95734 q^{64} -1.12462 q^{65} +2.27659 q^{66} +2.06305 q^{67} -13.3281 q^{68} +1.00000 q^{69} +4.30092 q^{71} -0.546135 q^{72} +1.15613 q^{73} -18.8236 q^{74} -4.62930 q^{75} -2.46406 q^{76} +3.81441 q^{78} +15.1716 q^{79} -2.07079 q^{80} +1.00000 q^{81} -20.2285 q^{82} +10.7196 q^{83} -3.58356 q^{85} -10.8099 q^{86} -0.804169 q^{87} +0.602078 q^{88} +7.92167 q^{89} -1.25732 q^{90} +2.26446 q^{92} -8.77360 q^{93} -12.2020 q^{94} -0.662518 q^{95} +8.11580 q^{96} +17.0648 q^{97} -1.10243 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 10 q^{3} + 15 q^{4} + 5 q^{5} + 3 q^{6} + 9 q^{8} + 10 q^{9} - 11 q^{10} + 8 q^{11} + 15 q^{12} + 5 q^{15} + 37 q^{16} + 11 q^{17} + 3 q^{18} - q^{19} + 15 q^{20} + 6 q^{22} + 10 q^{23}+ \cdots + 8 q^{99}+O(q^{100}) \) 10 * q + 3 * q^2 + 10 * q^3 + 15 * q^4 + 5 * q^5 + 3 * q^6 + 9 * q^8 + 10 * q^9 - 11 * q^10 + 8 * q^11 + 15 * q^12 + 5 * q^15 + 37 * q^16 + 11 * q^17 + 3 * q^18 - q^19 + 15 * q^20 + 6 * q^22 + 10 * q^23 + 9 * q^24 + 21 * q^25 - q^26 + 10 * q^27 + 22 * q^29 - 11 * q^30 + 3 * q^31 + 11 * q^32 + 8 * q^33 + 3 * q^34 + 15 * q^36 - 3 * q^37 - 16 * q^38 - 39 * q^40 + 26 * q^41 + 27 * q^43 + 16 * q^44 + 5 * q^45 + 3 * q^46 - 11 * q^47 + 37 * q^48 + 2 * q^50 + 11 * q^51 - 29 * q^52 + 5 * q^53 + 3 * q^54 + 18 * q^55 - q^57 + 16 * q^58 + 10 * q^59 + 15 * q^60 - 22 * q^61 + 32 * q^62 + 69 * q^64 - 11 * q^65 + 6 * q^66 - 2 * q^67 + 21 * q^68 + 10 * q^69 + 27 * q^71 + 9 * q^72 + 8 * q^73 + 14 * q^74 + 21 * q^75 + 22 * q^76 - q^78 + 21 * q^79 + 53 * q^80 + 10 * q^81 - 36 * q^82 + 12 * q^83 + 23 * q^85 + 18 * q^86 + 22 * q^87 - 10 * q^88 - 6 * q^89 - 11 * q^90 + 15 * q^92 + 3 * q^93 - 35 * q^94 + 44 * q^95 + 11 * q^96 - 6 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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