LMFDB - Embedded newform 3381.2.a.s.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3381,2,Mod(1,3381)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,-1,2,5,-1,-1,0,-9,2,9,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.9974209234$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
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			$-1.56155$ 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)

$f(q)$	$=$	\(q+1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} +1.56155 q^{5} +1.56155 q^{6} -2.43845 q^{8} +1.00000 q^{9} +2.43845 q^{10} -2.00000 q^{11} +0.438447 q^{12} -6.12311 q^{13} +1.56155 q^{15} -4.68466 q^{16} -7.56155 q^{17} +1.56155 q^{18} +1.43845 q^{19} +0.684658 q^{20} -3.12311 q^{22} +1.00000 q^{23} -2.43845 q^{24} -2.56155 q^{25} -9.56155 q^{26} +1.00000 q^{27} -9.12311 q^{29} +2.43845 q^{30} +5.68466 q^{31} -2.43845 q^{32} -2.00000 q^{33} -11.8078 q^{34} +0.438447 q^{36} +3.43845 q^{37} +2.24621 q^{38} -6.12311 q^{39} -3.80776 q^{40} -10.2462 q^{41} -0.315342 q^{43} -0.876894 q^{44} +1.56155 q^{45} +1.56155 q^{46} +6.68466 q^{47} -4.68466 q^{48} -4.00000 q^{50} -7.56155 q^{51} -2.68466 q^{52} -7.80776 q^{53} +1.56155 q^{54} -3.12311 q^{55} +1.43845 q^{57} -14.2462 q^{58} +9.12311 q^{59} +0.684658 q^{60} +6.00000 q^{61} +8.87689 q^{62} +5.56155 q^{64} -9.56155 q^{65} -3.12311 q^{66} -14.1231 q^{67} -3.31534 q^{68} +1.00000 q^{69} +13.8078 q^{71} -2.43845 q^{72} +5.87689 q^{73} +5.36932 q^{74} -2.56155 q^{75} +0.630683 q^{76} -9.56155 q^{78} -5.43845 q^{79} -7.31534 q^{80} +1.00000 q^{81} -16.0000 q^{82} +4.87689 q^{83} -11.8078 q^{85} -0.492423 q^{86} -9.12311 q^{87} +4.87689 q^{88} -10.0000 q^{89} +2.43845 q^{90} +0.438447 q^{92} +5.68466 q^{93} +10.4384 q^{94} +2.24621 q^{95} -2.43845 q^{96} +15.3693 q^{97} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} + 5 q^{4} - q^{5} - q^{6} - 9 q^{8} + 2 q^{9} + 9 q^{10} - 4 q^{11} + 5 q^{12} - 4 q^{13} - q^{15} + 3 q^{16} - 11 q^{17} - q^{18} + 7 q^{19} - 11 q^{20} + 2 q^{22} + 2 q^{23}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 + 5 * q^4 - q^5 - q^6 - 9 * q^8 + 2 * q^9 + 9 * q^10 - 4 * q^11 + 5 * q^12 - 4 * q^13 - q^15 + 3 * q^16 - 11 * q^17 - q^18 + 7 * q^19 - 11 * q^20 + 2 * q^22 + 2 * q^23 - 9 * q^24 - q^25 - 15 * q^26 + 2 * q^27 - 10 * q^29 + 9 * q^30 - q^31 - 9 * q^32 - 4 * q^33 - 3 * q^34 + 5 * q^36 + 11 * q^37 - 12 * q^38 - 4 * q^39 + 13 * q^40 - 4 * q^41 - 13 * q^43 - 10 * q^44 - q^45 - q^46 + q^47 + 3 * q^48 - 8 * q^50 - 11 * q^51 + 7 * q^52 + 5 * q^53 - q^54 + 2 * q^55 + 7 * q^57 - 12 * q^58 + 10 * q^59 - 11 * q^60 + 12 * q^61 + 26 * q^62 + 7 * q^64 - 15 * q^65 + 2 * q^66 - 20 * q^67 - 19 * q^68 + 2 * q^69 + 7 * q^71 - 9 * q^72 + 20 * q^73 - 14 * q^74 - q^75 + 26 * q^76 - 15 * q^78 - 15 * q^79 - 27 * q^80 + 2 * q^81 - 32 * q^82 + 18 * q^83 - 3 * q^85 + 32 * q^86 - 10 * q^87 + 18 * q^88 - 20 * q^89 + 9 * q^90 + 5 * q^92 - q^93 + 25 * q^94 - 12 * q^95 - 9 * q^96 + 6 * q^97 - 4 * q^99

Coefficient data

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

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

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

$2$	1.56155	1.10418	0.552092	−	0.833783i	$-0.313830\pi$
$2$	1.56155	1.10418	0.552092	+	0.833783i	$0.313830\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.438447	0.219224
$5$	1.56155	0.698348	0.349174	−	0.937058i	$-0.386462\pi$
$5$	1.56155	0.698348	0.349174	+	0.937058i	$0.386462\pi$
$6$	1.56155	0.637501
$7$	0	0
$7$	0	0
$8$	−2.43845	−0.862121
$9$	1.00000	0.333333
$10$	2.43845	0.771105
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0.438447	0.126569
$13$	−6.12311	−1.69824	−0.849122	−	0.528197i	$-0.822868\pi$
$13$	−6.12311	−1.69824	−0.849122	+	0.528197i	$0.822868\pi$
$14$	0	0
$15$	1.56155	0.403191
$16$	−4.68466	−1.17116
$17$	−7.56155	−1.83395	−0.916973	−	0.398949i	$-0.869375\pi$
$17$	−7.56155	−1.83395	−0.916973	+	0.398949i	$0.869375\pi$
$18$	1.56155	0.368062
$19$	1.43845	0.330002	0.165001	−	0.986293i	$-0.447237\pi$
$19$	1.43845	0.330002	0.165001	+	0.986293i	$0.447237\pi$
$20$	0.684658	0.153094
$21$	0	0
$22$	−3.12311	−0.665848
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−2.43845	−0.497746
$25$	−2.56155	−0.512311
$26$	−9.56155	−1.87517
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.12311	−1.69412	−0.847059	−	0.531499i	$-0.821629\pi$
$29$	−9.12311	−1.69412	−0.847059	+	0.531499i	$0.821629\pi$
$30$	2.43845	0.445198
$31$	5.68466	1.02099	0.510497	−	0.859879i	$-0.329461\pi$
$31$	5.68466	1.02099	0.510497	+	0.859879i	$0.329461\pi$
$32$	−2.43845	−0.431061
$33$	−2.00000	−0.348155
$34$	−11.8078	−2.02501
$35$	0	0
$36$	0.438447	0.0730745
$37$	3.43845	0.565277	0.282639	−	0.959226i	$-0.408790\pi$
$37$	3.43845	0.565277	0.282639	+	0.959226i	$0.408790\pi$
$38$	2.24621	0.364384
$39$	−6.12311	−0.980482
$40$	−3.80776	−0.602060
$41$	−10.2462	−1.60019	−0.800095	−	0.599874i	$-0.795217\pi$
$41$	−10.2462	−1.60019	−0.800095	+	0.599874i	$0.795217\pi$
$42$	0	0
$43$	−0.315342	−0.0480891	−0.0240446	−	0.999711i	$-0.507654\pi$
$43$	−0.315342	−0.0480891	−0.0240446	+	0.999711i	$0.507654\pi$
$44$	−0.876894	−0.132197
$45$	1.56155	0.232783
$46$	1.56155	0.230238
$47$	6.68466	0.975058	0.487529	−	0.873107i	$-0.337898\pi$
$47$	6.68466	0.975058	0.487529	+	0.873107i	$0.337898\pi$
$48$	−4.68466	−0.676172
$49$	0	0
$50$	−4.00000	−0.565685
$51$	−7.56155	−1.05883
$52$	−2.68466	−0.372295
$53$	−7.80776	−1.07248	−0.536239	−	0.844066i	$-0.680156\pi$
$53$	−7.80776	−1.07248	−0.536239	+	0.844066i	$0.680156\pi$
$54$	1.56155	0.212500
$55$	−3.12311	−0.421119
$56$	0	0
$57$	1.43845	0.190527
$58$	−14.2462	−1.87062
$59$	9.12311	1.18773	0.593864	−	0.804566i	$-0.297602\pi$
$59$	9.12311	1.18773	0.593864	+	0.804566i	$0.297602\pi$
$60$	0.684658	0.0883890
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	8.87689	1.12737
$63$	0	0
$64$	5.56155	0.695194
$65$	−9.56155	−1.18596
$66$	−3.12311	−0.384428
$67$	−14.1231	−1.72541	−0.862706	−	0.505706i	$-0.831232\pi$
$67$	−14.1231	−1.72541	−0.862706	+	0.505706i	$0.831232\pi$
$68$	−3.31534	−0.402044
$69$	1.00000	0.120386
$70$	0	0
$71$	13.8078	1.63868	0.819340	−	0.573308i	$-0.194340\pi$
$71$	13.8078	1.63868	0.819340	+	0.573308i	$0.194340\pi$
$72$	−2.43845	−0.287374
$73$	5.87689	0.687838	0.343919	−	0.938999i	$-0.388245\pi$
$73$	5.87689	0.687838	0.343919	+	0.938999i	$0.388245\pi$
$74$	5.36932	0.624170
$75$	−2.56155	−0.295783
$76$	0.630683	0.0723443
$77$	0	0
$78$	−9.56155	−1.08263
$79$	−5.43845	−0.611873	−0.305937	−	0.952052i	$-0.598970\pi$
$79$	−5.43845	−0.611873	−0.305937	+	0.952052i	$0.598970\pi$
$80$	−7.31534	−0.817880
$81$	1.00000	0.111111
$82$	−16.0000	−1.76690
$83$	4.87689	0.535309	0.267654	−	0.963515i	$-0.413751\pi$
$83$	4.87689	0.535309	0.267654	+	0.963515i	$0.413751\pi$
$84$	0	0
$85$	−11.8078	−1.28073
$86$	−0.492423	−0.0530993
$87$	−9.12311	−0.978100
$88$	4.87689	0.519879
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	2.43845	0.257035
$91$	0	0
$92$	0.438447	0.0457113
$93$	5.68466	0.589472
$94$	10.4384	1.07664
$95$	2.24621	0.230456
$96$	−2.43845	−0.248873
$97$	15.3693	1.56052	0.780259	−	0.625457i	$-0.215087\pi$
$97$	15.3693	1.56052	0.780259	+	0.625457i	$0.215087\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	−1.12311	−0.112311
$101$	−0.876894	−0.0872543	−0.0436271	−	0.999048i	$-0.513891\pi$
$101$	−0.876894	−0.0872543	−0.0436271	+	0.999048i	$0.513891\pi$
$102$	−11.8078	−1.16914
$103$	14.1231	1.39159	0.695795	−	0.718240i	$-0.255052\pi$
$103$	14.1231	1.39159	0.695795	+	0.718240i	$0.255052\pi$
$104$	14.9309	1.46409
$105$	0	0
$106$	−12.1922	−1.18421
$107$	−6.24621	−0.603844	−0.301922	−	0.953333i	$-0.597628\pi$
$107$	−6.24621	−0.603844	−0.301922	+	0.953333i	$0.597628\pi$
$108$	0.438447	0.0421896
$109$	−10.5616	−1.01161	−0.505807	−	0.862647i	$-0.668805\pi$
$109$	−10.5616	−1.01161	−0.505807	+	0.862647i	$0.668805\pi$
$110$	−4.87689	−0.464994
$111$	3.43845	0.326363
$112$	0	0
$113$	9.80776	0.922637	0.461318	−	0.887235i	$-0.347377\pi$
$113$	9.80776	0.922637	0.461318	+	0.887235i	$0.347377\pi$
$114$	2.24621	0.210377
$115$	1.56155	0.145616
$116$	−4.00000	−0.371391
$117$	−6.12311	−0.566081
$118$	14.2462	1.31147
$119$	0	0
$120$	−3.80776	−0.347600
$121$	−7.00000	−0.636364
$122$	9.36932	0.848258
$123$	−10.2462	−0.923870
$124$	2.49242	0.223826
$125$	−11.8078	−1.05612
$126$	0	0
$127$	1.68466	0.149489	0.0747446	−	0.997203i	$-0.476186\pi$
$127$	1.68466	0.149489	0.0747446	+	0.997203i	$0.476186\pi$
$128$	13.5616	1.19868
$129$	−0.315342	−0.0277643
$130$	−14.9309	−1.30952
$131$	12.6847	1.10826	0.554132	−	0.832429i	$-0.313050\pi$
$131$	12.6847	1.10826	0.554132	+	0.832429i	$0.313050\pi$
$132$	−0.876894	−0.0763239
$133$	0	0
$134$	−22.0540	−1.90517
$135$	1.56155	0.134397
$136$	18.4384	1.58108
$137$	−7.80776	−0.667062	−0.333531	−	0.942739i	$-0.608240\pi$
$137$	−7.80776	−0.667062	−0.333531	+	0.942739i	$0.608240\pi$
$138$	1.56155	0.132928
$139$	11.9309	1.01196	0.505982	−	0.862544i	$-0.331130\pi$
$139$	11.9309	1.01196	0.505982	+	0.862544i	$0.331130\pi$
$140$	0	0
$141$	6.68466	0.562950
$142$	21.5616	1.80941
$143$	12.2462	1.02408
$144$	−4.68466	−0.390388
$145$	−14.2462	−1.18308
$146$	9.17708	0.759501
$147$	0	0
$148$	1.50758	0.123922
$149$	−1.31534	−0.107757	−0.0538785	−	0.998547i	$-0.517158\pi$
$149$	−1.31534	−0.107757	−0.0538785	+	0.998547i	$0.517158\pi$
$150$	−4.00000	−0.326599
$151$	−18.2462	−1.48486	−0.742428	−	0.669926i	$-0.766326\pi$
$151$	−18.2462	−1.48486	−0.742428	+	0.669926i	$0.766326\pi$
$152$	−3.50758	−0.284502
$153$	−7.56155	−0.611315
$154$	0	0
$155$	8.87689	0.713009
$156$	−2.68466	−0.214945
$157$	4.24621	0.338885	0.169442	−	0.985540i	$-0.445803\pi$
$157$	4.24621	0.338885	0.169442	+	0.985540i	$0.445803\pi$
$158$	−8.49242	−0.675621
$159$	−7.80776	−0.619196
$160$	−3.80776	−0.301030
$161$	0	0
$162$	1.56155	0.122687
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−4.49242	−0.350799
$165$	−3.12311	−0.243133
$166$	7.61553	0.591080
$167$	−3.31534	−0.256549	−0.128274	−	0.991739i	$-0.540944\pi$
$167$	−3.31534	−0.256549	−0.128274	+	0.991739i	$0.540944\pi$
$168$	0	0
$169$	24.4924	1.88403
$170$	−18.4384	−1.41416
$171$	1.43845	0.110001
$172$	−0.138261	−0.0105423
$173$	−23.3693	−1.77674	−0.888368	−	0.459132i	$-0.848161\pi$
$173$	−23.3693	−1.77674	−0.888368	+	0.459132i	$0.848161\pi$
$174$	−14.2462	−1.08000
$175$	0	0
$176$	9.36932	0.706239
$177$	9.12311	0.685735
$178$	−15.6155	−1.17043
$179$	6.93087	0.518038	0.259019	−	0.965872i	$-0.416601\pi$
$179$	6.93087	0.518038	0.259019	+	0.965872i	$0.416601\pi$
$180$	0.684658	0.0510314
$181$	−7.93087	−0.589497	−0.294748	−	0.955575i	$-0.595236\pi$
$181$	−7.93087	−0.589497	−0.294748	+	0.955575i	$0.595236\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	−2.43845	−0.179765
$185$	5.36932	0.394760
$186$	8.87689	0.650885
$187$	15.1231	1.10591
$188$	2.93087	0.213756
$189$	0	0
$190$	3.50758	0.254466
$191$	−1.12311	−0.0812651	−0.0406325	−	0.999174i	$-0.512937\pi$
$191$	−1.12311	−0.0812651	−0.0406325	+	0.999174i	$0.512937\pi$
$192$	5.56155	0.401371
$193$	7.87689	0.566991	0.283496	−	0.958974i	$-0.408506\pi$
$193$	7.87689	0.566991	0.283496	+	0.958974i	$0.408506\pi$
$194$	24.0000	1.72310
$195$	−9.56155	−0.684717
$196$	0	0
$197$	−21.3693	−1.52250	−0.761250	−	0.648458i	$-0.775414\pi$
$197$	−21.3693	−1.52250	−0.761250	+	0.648458i	$0.775414\pi$
$198$	−3.12311	−0.221949
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	6.24621	0.441674
$201$	−14.1231	−0.996167
$202$	−1.36932	−0.0963448
$203$	0	0
$204$	−3.31534	−0.232120
$205$	−16.0000	−1.11749
$206$	22.0540	1.53657
$207$	1.00000	0.0695048
$208$	28.6847	1.98892
$209$	−2.87689	−0.198999
$210$	0	0
$211$	−7.12311	−0.490375	−0.245187	−	0.969476i	$-0.578849\pi$
$211$	−7.12311	−0.490375	−0.245187	+	0.969476i	$0.578849\pi$
$212$	−3.42329	−0.235113
$213$	13.8078	0.946092
$214$	−9.75379	−0.666755
$215$	−0.492423	−0.0335829
$216$	−2.43845	−0.165915
$217$	0	0
$218$	−16.4924	−1.11701
$219$	5.87689	0.397124
$220$	−1.36932	−0.0923193
$221$	46.3002	3.11449
$222$	5.36932	0.360365
$223$	−20.4924	−1.37227	−0.686137	−	0.727472i	$-0.740695\pi$
$223$	−20.4924	−1.37227	−0.686137	+	0.727472i	$0.740695\pi$
$224$	0	0
$225$	−2.56155	−0.170770
$226$	15.3153	1.01876
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0.630683	0.0417680
$229$	−29.3002	−1.93621	−0.968105	−	0.250543i	$-0.919391\pi$
$229$	−29.3002	−1.93621	−0.968105	+	0.250543i	$0.919391\pi$
$230$	2.43845	0.160786
$231$	0	0
$232$	22.2462	1.46054
$233$	−10.8769	−0.712569	−0.356285	−	0.934378i	$-0.615957\pi$
$233$	−10.8769	−0.712569	−0.356285	+	0.934378i	$0.615957\pi$
$234$	−9.56155	−0.625058
$235$	10.4384	0.680929
$236$	4.00000	0.260378
$237$	−5.43845	−0.353265
$238$	0	0
$239$	−18.2462	−1.18025	−0.590125	−	0.807312i	$-0.700921\pi$
$239$	−18.2462	−1.18025	−0.590125	+	0.807312i	$0.700921\pi$
$240$	−7.31534	−0.472203
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−10.9309	−0.702663
$243$	1.00000	0.0641500
$244$	2.63068	0.168412
$245$	0	0
$246$	−16.0000	−1.02012
$247$	−8.80776	−0.560425
$248$	−13.8617	−0.880221
$249$	4.87689	0.309061
$250$	−18.4384	−1.16615
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	2.63068	0.165064
$255$	−11.8078	−0.739431
$256$	10.0540	0.628373
$257$	−6.63068	−0.413611	−0.206805	−	0.978382i	$-0.566307\pi$
$257$	−6.63068	−0.413611	−0.206805	+	0.978382i	$0.566307\pi$
$258$	−0.492423	−0.0306569
$259$	0	0
$260$	−4.19224	−0.259991
$261$	−9.12311	−0.564706
$262$	19.8078	1.22373
$263$	4.49242	0.277015	0.138507	−	0.990361i	$-0.455770\pi$
$263$	4.49242	0.277015	0.138507	+	0.990361i	$0.455770\pi$
$264$	4.87689	0.300152
$265$	−12.1922	−0.748963
$266$	0	0
$267$	−10.0000	−0.611990
$268$	−6.19224	−0.378251
$269$	−8.49242	−0.517792	−0.258896	−	0.965905i	$-0.583359\pi$
$269$	−8.49242	−0.517792	−0.258896	+	0.965905i	$0.583359\pi$
$270$	2.43845	0.148399
$271$	−19.6155	−1.19156	−0.595779	−	0.803148i	$-0.703157\pi$
$271$	−19.6155	−1.19156	−0.595779	+	0.803148i	$0.703157\pi$
$272$	35.4233	2.14785
$273$	0	0
$274$	−12.1922	−0.736560
$275$	5.12311	0.308935
$276$	0.438447	0.0263914
$277$	−29.2462	−1.75723	−0.878617	−	0.477527i	$-0.841533\pi$
$277$	−29.2462	−1.75723	−0.878617	+	0.477527i	$0.841533\pi$
$278$	18.6307	1.11739
$279$	5.68466	0.340332
$280$	0	0
$281$	19.5616	1.16694	0.583472	−	0.812133i	$-0.301694\pi$
$281$	19.5616	1.16694	0.583472	+	0.812133i	$0.301694\pi$
$282$	10.4384	0.621600
$283$	−24.1231	−1.43397	−0.716985	−	0.697089i	$-0.754478\pi$
$283$	−24.1231	−1.43397	−0.716985	+	0.697089i	$0.754478\pi$
$284$	6.05398	0.359237
$285$	2.24621	0.133054
$286$	19.1231	1.13077
$287$	0	0
$288$	−2.43845	−0.143687
$289$	40.1771	2.36336
$290$	−22.2462	−1.30634
$291$	15.3693	0.900965
$292$	2.57671	0.150790
$293$	24.0540	1.40525	0.702624	−	0.711561i	$-0.252012\pi$
$293$	24.0540	1.40525	0.702624	+	0.711561i	$0.252012\pi$
$294$	0	0
$295$	14.2462	0.829446
$296$	−8.38447	−0.487338
$297$	−2.00000	−0.116052
$298$	−2.05398	−0.118984
$299$	−6.12311	−0.354108
$300$	−1.12311	−0.0648425
$301$	0	0
$302$	−28.4924	−1.63955
$303$	−0.876894	−0.0503763
$304$	−6.73863	−0.386487
$305$	9.36932	0.536486
$306$	−11.8078	−0.675005
$307$	11.6847	0.666879	0.333439	−	0.942772i	$-0.391791\pi$
$307$	11.6847	0.666879	0.333439	+	0.942772i	$0.391791\pi$
$308$	0	0
$309$	14.1231	0.803435
$310$	13.8617	0.787294
$311$	7.56155	0.428776	0.214388	−	0.976749i	$-0.431224\pi$
$311$	7.56155	0.428776	0.214388	+	0.976749i	$0.431224\pi$
$312$	14.9309	0.845294
$313$	29.3002	1.65614	0.828072	−	0.560621i	$-0.189438\pi$
$313$	29.3002	1.65614	0.828072	+	0.560621i	$0.189438\pi$
$314$	6.63068	0.374191
$315$	0	0
$316$	−2.38447	−0.134137
$317$	−11.7538	−0.660159	−0.330079	−	0.943953i	$-0.607075\pi$
$317$	−11.7538	−0.660159	−0.330079	+	0.943953i	$0.607075\pi$
$318$	−12.1922	−0.683707
$319$	18.2462	1.02159
$320$	8.68466	0.485487
$321$	−6.24621	−0.348630
$322$	0	0
$323$	−10.8769	−0.605207
$324$	0.438447	0.0243582
$325$	15.6847	0.870028
$326$	6.24621	0.345946
$327$	−10.5616	−0.584055
$328$	24.9848	1.37956
$329$	0	0
$330$	−4.87689	−0.268464
$331$	−15.4384	−0.848574	−0.424287	−	0.905528i	$-0.639475\pi$
$331$	−15.4384	−0.848574	−0.424287	+	0.905528i	$0.639475\pi$
$332$	2.13826	0.117352
$333$	3.43845	0.188426
$334$	−5.17708	−0.283277
$335$	−22.0540	−1.20494
$336$	0	0
$337$	−20.8078	−1.13347	−0.566736	−	0.823900i	$-0.691794\pi$
$337$	−20.8078	−1.13347	−0.566736	+	0.823900i	$0.691794\pi$
$338$	38.2462	2.08032
$339$	9.80776	0.532685
$340$	−5.17708	−0.280767
$341$	−11.3693	−0.615683
$342$	2.24621	0.121461
$343$	0	0
$344$	0.768944	0.0414587
$345$	1.56155	0.0840712
$346$	−36.4924	−1.96184
$347$	−12.6847	−0.680948	−0.340474	−	0.940254i	$-0.610588\pi$
$347$	−12.6847	−0.680948	−0.340474	+	0.940254i	$0.610588\pi$
$348$	−4.00000	−0.214423
$349$	0.930870	0.0498283	0.0249142	−	0.999690i	$-0.492069\pi$
$349$	0.930870	0.0498283	0.0249142	+	0.999690i	$0.492069\pi$
$350$	0	0
$351$	−6.12311	−0.326827
$352$	4.87689	0.259939
$353$	−7.36932	−0.392229	−0.196115	−	0.980581i	$-0.562832\pi$
$353$	−7.36932	−0.392229	−0.196115	+	0.980581i	$0.562832\pi$
$354$	14.2462	0.757178
$355$	21.5616	1.14437
$356$	−4.38447	−0.232377
$357$	0	0
$358$	10.8229	0.572009
$359$	12.7386	0.672319	0.336160	−	0.941805i	$-0.390872\pi$
$359$	12.7386	0.672319	0.336160	+	0.941805i	$0.390872\pi$
$360$	−3.80776	−0.200687
$361$	−16.9309	−0.891098
$362$	−12.3845	−0.650913
$363$	−7.00000	−0.367405
$364$	0	0
$365$	9.17708	0.480350
$366$	9.36932	0.489742
$367$	−8.36932	−0.436875	−0.218437	−	0.975851i	$-0.570096\pi$
$367$	−8.36932	−0.436875	−0.218437	+	0.975851i	$0.570096\pi$
$368$	−4.68466	−0.244205
$369$	−10.2462	−0.533396
$370$	8.38447	0.435888
$371$	0	0
$372$	2.49242	0.129226
$373$	9.68466	0.501453	0.250726	−	0.968058i	$-0.419331\pi$
$373$	9.68466	0.501453	0.250726	+	0.968058i	$0.419331\pi$
$374$	23.6155	1.22113
$375$	−11.8078	−0.609750
$376$	−16.3002	−0.840618
$377$	55.8617	2.87703
$378$	0	0
$379$	8.61553	0.442550	0.221275	−	0.975211i	$-0.428978\pi$
$379$	8.61553	0.442550	0.221275	+	0.975211i	$0.428978\pi$
$380$	0.984845	0.0505215
$381$	1.68466	0.0863077
$382$	−1.75379	−0.0897316
$383$	4.24621	0.216971	0.108486	−	0.994098i	$-0.465400\pi$
$383$	4.24621	0.216971	0.108486	+	0.994098i	$0.465400\pi$
$384$	13.5616	0.692060
$385$	0	0
$386$	12.3002	0.626063
$387$	−0.315342	−0.0160297
$388$	6.73863	0.342102
$389$	15.6155	0.791739	0.395869	−	0.918307i	$-0.370443\pi$
$389$	15.6155	0.791739	0.395869	+	0.918307i	$0.370443\pi$
$390$	−14.9309	−0.756054
$391$	−7.56155	−0.382404
$392$	0	0
$393$	12.6847	0.639856
$394$	−33.3693	−1.68112
$395$	−8.49242	−0.427300
$396$	−0.876894	−0.0440656
$397$	22.6155	1.13504	0.567520	−	0.823359i	$-0.307903\pi$
$397$	22.6155	1.13504	0.567520	+	0.823359i	$0.307903\pi$
$398$	−12.4924	−0.626189
$399$	0	0
$400$	12.0000	0.600000
$401$	−0.684658	−0.0341902	−0.0170951	−	0.999854i	$-0.505442\pi$
$401$	−0.684658	−0.0341902	−0.0170951	+	0.999854i	$0.505442\pi$
$402$	−22.0540	−1.09995
$403$	−34.8078	−1.73390
$404$	−0.384472	−0.0191282
$405$	1.56155	0.0775942
$406$	0	0
$407$	−6.87689	−0.340875
$408$	18.4384	0.912839
$409$	18.8617	0.932653	0.466326	−	0.884613i	$-0.345577\pi$
$409$	18.8617	0.932653	0.466326	+	0.884613i	$0.345577\pi$
$410$	−24.9848	−1.23391
$411$	−7.80776	−0.385129
$412$	6.19224	0.305070
$413$	0	0
$414$	1.56155	0.0767461
$415$	7.61553	0.373832
$416$	14.9309	0.732046
$417$	11.9309	0.584257
$418$	−4.49242	−0.219732
$419$	−28.4924	−1.39195	−0.695973	−	0.718068i	$-0.745027\pi$
$419$	−28.4924	−1.39195	−0.695973	+	0.718068i	$0.745027\pi$
$420$	0	0
$421$	−4.31534	−0.210317	−0.105158	−	0.994455i	$-0.533535\pi$
$421$	−4.31534	−0.210317	−0.105158	+	0.994455i	$0.533535\pi$
$422$	−11.1231	−0.541464
$423$	6.68466	0.325019
$424$	19.0388	0.924607
$425$	19.3693	0.939550
$426$	21.5616	1.04466
$427$	0	0
$428$	−2.73863	−0.132377
$429$	12.2462	0.591253
$430$	−0.768944	−0.0370818
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	−4.68466	−0.225391
$433$	−6.80776	−0.327160	−0.163580	−	0.986530i	$-0.552304\pi$
$433$	−6.80776	−0.327160	−0.163580	+	0.986530i	$0.552304\pi$
$434$	0	0
$435$	−14.2462	−0.683054
$436$	−4.63068	−0.221770
$437$	1.43845	0.0688103
$438$	9.17708	0.438498
$439$	15.6155	0.745288	0.372644	−	0.927974i	$-0.378451\pi$
$439$	15.6155	0.745288	0.372644	+	0.927974i	$0.378451\pi$
$440$	7.61553	0.363056
$441$	0	0
$442$	72.3002	3.43897
$443$	13.3153	0.632631	0.316315	−	0.948654i	$-0.397554\pi$
$443$	13.3153	0.632631	0.316315	+	0.948654i	$0.397554\pi$
$444$	1.50758	0.0715465
$445$	−15.6155	−0.740247
$446$	−32.0000	−1.51524
$447$	−1.31534	−0.0622135
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	−4.00000	−0.188562
$451$	20.4924	0.964950
$452$	4.30019	0.202264
$453$	−18.2462	−0.857282
$454$	3.12311	0.146575
$455$	0	0
$456$	−3.50758	−0.164257
$457$	1.43845	0.0672877	0.0336439	−	0.999434i	$-0.489289\pi$
$457$	1.43845	0.0672877	0.0336439	+	0.999434i	$0.489289\pi$
$458$	−45.7538	−2.13793
$459$	−7.56155	−0.352943
$460$	0.684658	0.0319224
$461$	−4.24621	−0.197766	−0.0988829	−	0.995099i	$-0.531527\pi$
$461$	−4.24621	−0.197766	−0.0988829	+	0.995099i	$0.531527\pi$
$462$	0	0
$463$	19.6847	0.914824	0.457412	−	0.889255i	$-0.348777\pi$
$463$	19.6847	0.914824	0.457412	+	0.889255i	$0.348777\pi$
$464$	42.7386	1.98409
$465$	8.87689	0.411656
$466$	−16.9848	−0.786808
$467$	1.12311	0.0519711	0.0259856	−	0.999662i	$-0.491728\pi$
$467$	1.12311	0.0519711	0.0259856	+	0.999662i	$0.491728\pi$
$468$	−2.68466	−0.124098
$469$	0	0
$470$	16.3002	0.751872
$471$	4.24621	0.195655
$472$	−22.2462	−1.02396
$473$	0.630683	0.0289988
$474$	−8.49242	−0.390070
$475$	−3.68466	−0.169064
$476$	0	0
$477$	−7.80776	−0.357493
$478$	−28.4924	−1.30321
$479$	−19.3693	−0.885007	−0.442503	−	0.896767i	$-0.645910\pi$
$479$	−19.3693	−0.885007	−0.442503	+	0.896767i	$0.645910\pi$
$480$	−3.80776	−0.173800
$481$	−21.0540	−0.959979
$482$	3.12311	0.142254
$483$	0	0
$484$	−3.06913	−0.139506
$485$	24.0000	1.08978
$486$	1.56155	0.0708335
$487$	22.4233	1.01610	0.508048	−	0.861329i	$-0.330367\pi$
$487$	22.4233	1.01610	0.508048	+	0.861329i	$0.330367\pi$
$488$	−14.6307	−0.662300
$489$	4.00000	0.180886
$490$	0	0
$491$	−30.6847	−1.38478	−0.692390	−	0.721524i	$-0.743442\pi$
$491$	−30.6847	−1.38478	−0.692390	+	0.721524i	$0.743442\pi$
$492$	−4.49242	−0.202534
$493$	68.9848	3.10692
$494$	−13.7538	−0.618812
$495$	−3.12311	−0.140373
$496$	−26.6307	−1.19575
$497$	0	0
$498$	7.61553	0.341260
$499$	−35.6847	−1.59746	−0.798732	−	0.601686i	$-0.794496\pi$
$499$	−35.6847	−1.59746	−0.798732	+	0.601686i	$0.794496\pi$
$500$	−5.17708	−0.231526
$501$	−3.31534	−0.148119
$502$	3.12311	0.139391
$503$	9.36932	0.417757	0.208879	−	0.977942i	$-0.433019\pi$
$503$	9.36932	0.417757	0.208879	+	0.977942i	$0.433019\pi$
$504$	0	0
$505$	−1.36932	−0.0609338
$506$	−3.12311	−0.138839
$507$	24.4924	1.08775
$508$	0.738634	0.0327716
$509$	4.87689	0.216165	0.108082	−	0.994142i	$-0.465529\pi$
$509$	4.87689	0.216165	0.108082	+	0.994142i	$0.465529\pi$
$510$	−18.4384	−0.816468
$511$	0	0
$512$	−11.4233	−0.504843
$513$	1.43845	0.0635090
$514$	−10.3542	−0.456703
$515$	22.0540	0.971814
$516$	−0.138261	−0.00608658
$517$	−13.3693	−0.587982
$518$	0	0
$519$	−23.3693	−1.02580
$520$	23.3153	1.02245
$521$	−30.6847	−1.34432	−0.672160	−	0.740406i	$-0.734633\pi$
$521$	−30.6847	−1.34432	−0.672160	+	0.740406i	$0.734633\pi$
$522$	−14.2462	−0.623540
$523$	18.3693	0.803234	0.401617	−	0.915808i	$-0.368448\pi$
$523$	18.3693	0.803234	0.401617	+	0.915808i	$0.368448\pi$
$524$	5.56155	0.242958
$525$	0	0
$526$	7.01515	0.305875
$527$	−42.9848	−1.87245
$528$	9.36932	0.407747
$529$	1.00000	0.0434783
$530$	−19.0388	−0.826994
$531$	9.12311	0.395909
$532$	0	0
$533$	62.7386	2.71751
$534$	−15.6155	−0.675750
$535$	−9.75379	−0.421693
$536$	34.4384	1.48751
$537$	6.93087	0.299089
$538$	−13.2614	−0.571738
$539$	0	0
$540$	0.684658	0.0294630
$541$	3.73863	0.160736	0.0803682	−	0.996765i	$-0.474390\pi$
$541$	3.73863	0.160736	0.0803682	+	0.996765i	$0.474390\pi$
$542$	−30.6307	−1.31570
$543$	−7.93087	−0.340346
$544$	18.4384	0.790542
$545$	−16.4924	−0.706458
$546$	0	0
$547$	29.3693	1.25574	0.627871	−	0.778318i	$-0.283927\pi$
$547$	29.3693	1.25574	0.627871	+	0.778318i	$0.283927\pi$
$548$	−3.42329	−0.146236
$549$	6.00000	0.256074
$550$	8.00000	0.341121
$551$	−13.1231	−0.559063
$552$	−2.43845	−0.103787
$553$	0	0
$554$	−45.6695	−1.94031
$555$	5.36932	0.227915
$556$	5.23106	0.221846
$557$	32.2462	1.36632	0.683158	−	0.730271i	$-0.260606\pi$
$557$	32.2462	1.36632	0.683158	+	0.730271i	$0.260606\pi$
$558$	8.87689	0.375789
$559$	1.93087	0.0816671
$560$	0	0
$561$	15.1231	0.638498
$562$	30.5464	1.28852
$563$	−14.2462	−0.600406	−0.300203	−	0.953875i	$-0.597054\pi$
$563$	−14.2462	−0.600406	−0.300203	+	0.953875i	$0.597054\pi$
$564$	2.93087	0.123412
$565$	15.3153	0.644321
$566$	−37.6695	−1.58337
$567$	0	0
$568$	−33.6695	−1.41274
$569$	−25.1771	−1.05548	−0.527739	−	0.849407i	$-0.676960\pi$
$569$	−25.1771	−1.05548	−0.527739	+	0.849407i	$0.676960\pi$
$570$	3.50758	0.146916
$571$	40.8617	1.71001	0.855005	−	0.518619i	$-0.173554\pi$
$571$	40.8617	1.71001	0.855005	+	0.518619i	$0.173554\pi$
$572$	5.36932	0.224502
$573$	−1.12311	−0.0469184
$574$	0	0
$575$	−2.56155	−0.106824
$576$	5.56155	0.231731
$577$	10.3153	0.429433	0.214717	−	0.976676i	$-0.431117\pi$
$577$	10.3153	0.429433	0.214717	+	0.976676i	$0.431117\pi$
$578$	62.7386	2.60958
$579$	7.87689	0.327353
$580$	−6.24621	−0.259360
$581$	0	0
$582$	24.0000	0.994832
$583$	15.6155	0.646729
$584$	−14.3305	−0.593000
$585$	−9.56155	−0.395322
$586$	37.5616	1.55165
$587$	0.438447	0.0180967	0.00904833	−	0.999959i	$-0.497120\pi$
$587$	0.438447	0.0180967	0.00904833	+	0.999959i	$0.497120\pi$
$588$	0	0
$589$	8.17708	0.336931
$590$	22.2462	0.915862
$591$	−21.3693	−0.879016
$592$	−16.1080	−0.662033
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	−3.12311	−0.128143
$595$	0	0
$596$	−0.576708	−0.0236229
$597$	−8.00000	−0.327418
$598$	−9.56155	−0.391001
$599$	−37.1771	−1.51901	−0.759507	−	0.650499i	$-0.774560\pi$
$599$	−37.1771	−1.51901	−0.759507	+	0.650499i	$0.774560\pi$
$600$	6.24621	0.255001
$601$	5.05398	0.206156	0.103078	−	0.994673i	$-0.467131\pi$
$601$	5.05398	0.206156	0.103078	+	0.994673i	$0.467131\pi$
$602$	0	0
$603$	−14.1231	−0.575137
$604$	−8.00000	−0.325515
$605$	−10.9309	−0.444403
$606$	−1.36932	−0.0556247
$607$	42.8078	1.73751	0.868757	−	0.495239i	$-0.164920\pi$
$607$	42.8078	1.73751	0.868757	+	0.495239i	$0.164920\pi$
$608$	−3.50758	−0.142251
$609$	0	0
$610$	14.6307	0.592379
$611$	−40.9309	−1.65589
$612$	−3.31534	−0.134015
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	18.2462	0.736357
$615$	−16.0000	−0.645182
$616$	0	0
$617$	−0.930870	−0.0374754	−0.0187377	−	0.999824i	$-0.505965\pi$
$617$	−0.930870	−0.0374754	−0.0187377	+	0.999824i	$0.505965\pi$
$618$	22.0540	0.887141
$619$	−22.1231	−0.889203	−0.444601	−	0.895729i	$-0.646655\pi$
$619$	−22.1231	−0.889203	−0.444601	+	0.895729i	$0.646655\pi$
$620$	3.89205	0.156308
$621$	1.00000	0.0401286
$622$	11.8078	0.473448
$623$	0	0
$624$	28.6847	1.14831
$625$	−5.63068	−0.225227
$626$	45.7538	1.82869
$627$	−2.87689	−0.114892
$628$	1.86174	0.0742915
$629$	−26.0000	−1.03669
$630$	0	0
$631$	5.94602	0.236708	0.118354	−	0.992971i	$-0.462238\pi$
$631$	5.94602	0.236708	0.118354	+	0.992971i	$0.462238\pi$
$632$	13.2614	0.527509
$633$	−7.12311	−0.283118
$634$	−18.3542	−0.728937
$635$	2.63068	0.104395
$636$	−3.42329	−0.135742
$637$	0	0
$638$	28.4924	1.12803
$639$	13.8078	0.546227
$640$	21.1771	0.837098
$641$	−34.4384	−1.36024	−0.680118	−	0.733102i	$-0.738072\pi$
$641$	−34.4384	−1.36024	−0.680118	+	0.733102i	$0.738072\pi$
$642$	−9.75379	−0.384951
$643$	37.4384	1.47643	0.738214	−	0.674566i	$-0.235669\pi$
$643$	37.4384	1.47643	0.738214	+	0.674566i	$0.235669\pi$
$644$	0	0
$645$	−0.492423	−0.0193891
$646$	−16.9848	−0.668260
$647$	0.630683	0.0247947	0.0123974	−	0.999923i	$-0.496054\pi$
$647$	0.630683	0.0247947	0.0123974	+	0.999923i	$0.496054\pi$
$648$	−2.43845	−0.0957913
$649$	−18.2462	−0.716226
$650$	24.4924	0.960672
$651$	0	0
$652$	1.75379	0.0686837
$653$	10.4924	0.410600	0.205300	−	0.978699i	$-0.434183\pi$
$653$	10.4924	0.410600	0.205300	+	0.978699i	$0.434183\pi$
$654$	−16.4924	−0.644905
$655$	19.8078	0.773953
$656$	48.0000	1.87409
$657$	5.87689	0.229279
$658$	0	0
$659$	−0.384472	−0.0149769	−0.00748845	−	0.999972i	$-0.502384\pi$
$659$	−0.384472	−0.0149769	−0.00748845	+	0.999972i	$0.502384\pi$
$660$	−1.36932	−0.0533006
$661$	28.8078	1.12049	0.560246	−	0.828326i	$-0.310707\pi$
$661$	28.8078	1.12049	0.560246	+	0.828326i	$0.310707\pi$
$662$	−24.1080	−0.936982
$663$	46.3002	1.79815
$664$	−11.8920	−0.461501
$665$	0	0
$666$	5.36932	0.208057
$667$	−9.12311	−0.353248
$668$	−1.45360	−0.0562416
$669$	−20.4924	−0.792283
$670$	−34.4384	−1.33047
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−1.68466	−0.0649388	−0.0324694	−	0.999473i	$-0.510337\pi$
$673$	−1.68466	−0.0649388	−0.0324694	+	0.999473i	$0.510337\pi$
$674$	−32.4924	−1.25156
$675$	−2.56155	−0.0985942
$676$	10.7386	0.413024
$677$	40.9309	1.57310	0.786551	−	0.617526i	$-0.211865\pi$
$677$	40.9309	1.57310	0.786551	+	0.617526i	$0.211865\pi$
$678$	15.3153	0.588182
$679$	0	0
$680$	28.7926	1.10415
$681$	2.00000	0.0766402
$682$	−17.7538	−0.679828
$683$	−14.0540	−0.537760	−0.268880	−	0.963174i	$-0.586654\pi$
$683$	−14.0540	−0.537760	−0.268880	+	0.963174i	$0.586654\pi$
$684$	0.630683	0.0241148
$685$	−12.1922	−0.465841
$686$	0	0
$687$	−29.3002	−1.11787
$688$	1.47727	0.0563203
$689$	47.8078	1.82133
$690$	2.43845	0.0928301
$691$	14.8078	0.563314	0.281657	−	0.959515i	$-0.409116\pi$
$691$	14.8078	0.563314	0.281657	+	0.959515i	$0.409116\pi$
$692$	−10.2462	−0.389503
$693$	0	0
$694$	−19.8078	−0.751892
$695$	18.6307	0.706702
$696$	22.2462	0.843240
$697$	77.4773	2.93466
$698$	1.45360	0.0550197
$699$	−10.8769	−0.411402
$700$	0	0
$701$	7.17708	0.271075	0.135537	−	0.990772i	$-0.456724\pi$
$701$	7.17708	0.271075	0.135537	+	0.990772i	$0.456724\pi$
$702$	−9.56155	−0.360878
$703$	4.94602	0.186543
$704$	−11.1231	−0.419218
$705$	10.4384	0.393135
$706$	−11.5076	−0.433093
$707$	0	0
$708$	4.00000	0.150329
$709$	20.6307	0.774802	0.387401	−	0.921911i	$-0.373373\pi$
$709$	20.6307	0.774802	0.387401	+	0.921911i	$0.373373\pi$
$710$	33.6695	1.26359
$711$	−5.43845	−0.203958
$712$	24.3845	0.913847
$713$	5.68466	0.212892
$714$	0	0
$715$	19.1231	0.715164
$716$	3.03882	0.113566
$717$	−18.2462	−0.681417
$718$	19.8920	0.742365
$719$	−36.5464	−1.36295	−0.681475	−	0.731841i	$-0.738661\pi$
$719$	−36.5464	−1.36295	−0.681475	+	0.731841i	$0.738661\pi$
$720$	−7.31534	−0.272627
$721$	0	0
$722$	−26.4384	−0.983937
$723$	2.00000	0.0743808
$724$	−3.47727	−0.129232
$725$	23.3693	0.867915
$726$	−10.9309	−0.405683
$727$	28.3153	1.05016	0.525079	−	0.851054i	$-0.324036\pi$
$727$	28.3153	1.05016	0.525079	+	0.851054i	$0.324036\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	14.3305	0.530395
$731$	2.38447	0.0881929
$732$	2.63068	0.0972328
$733$	−1.30019	−0.0480235	−0.0240117	−	0.999712i	$-0.507644\pi$
$733$	−1.30019	−0.0480235	−0.0240117	+	0.999712i	$0.507644\pi$
$734$	−13.0691	−0.482390
$735$	0	0
$736$	−2.43845	−0.0898824
$737$	28.2462	1.04046
$738$	−16.0000	−0.588968
$739$	−28.6695	−1.05462	−0.527312	−	0.849672i	$-0.676800\pi$
$739$	−28.6695	−1.05462	−0.527312	+	0.849672i	$0.676800\pi$
$740$	2.35416	0.0865407
$741$	−8.80776	−0.323561
$742$	0	0
$743$	27.6155	1.01312	0.506558	−	0.862206i	$-0.330918\pi$
$743$	27.6155	1.01312	0.506558	+	0.862206i	$0.330918\pi$
$744$	−13.8617	−0.508196
$745$	−2.05398	−0.0752518
$746$	15.1231	0.553696
$747$	4.87689	0.178436
$748$	6.63068	0.242442
$749$	0	0
$750$	−18.4384	−0.673277
$751$	29.9309	1.09219	0.546096	−	0.837722i	$-0.316113\pi$
$751$	29.9309	1.09219	0.546096	+	0.837722i	$0.316113\pi$
$752$	−31.3153	−1.14195
$753$	2.00000	0.0728841
$754$	87.2311	3.17677
$755$	−28.4924	−1.03695
$756$	0	0
$757$	−21.6155	−0.785630	−0.392815	−	0.919618i	$-0.628499\pi$
$757$	−21.6155	−0.785630	−0.392815	+	0.919618i	$0.628499\pi$
$758$	13.4536	0.488657
$759$	−2.00000	−0.0725954
$760$	−5.47727	−0.198681
$761$	−6.38447	−0.231437	−0.115718	−	0.993282i	$-0.536917\pi$
$761$	−6.38447	−0.231437	−0.115718	+	0.993282i	$0.536917\pi$
$762$	2.63068	0.0952996
$763$	0	0
$764$	−0.492423	−0.0178152
$765$	−11.8078	−0.426911
$766$	6.63068	0.239576
$767$	−55.8617	−2.01705
$768$	10.0540	0.362792
$769$	−18.8078	−0.678225	−0.339113	−	0.940746i	$-0.610127\pi$
$769$	−18.8078	−0.678225	−0.339113	+	0.940746i	$0.610127\pi$
$770$	0	0
$771$	−6.63068	−0.238798
$772$	3.45360	0.124298
$773$	−30.3002	−1.08982	−0.544911	−	0.838494i	$-0.683437\pi$
$773$	−30.3002	−1.08982	−0.544911	+	0.838494i	$0.683437\pi$
$774$	−0.492423	−0.0176998
$775$	−14.5616	−0.523066
$776$	−37.4773	−1.34536
$777$	0	0
$778$	24.3845	0.874226
$779$	−14.7386	−0.528066
$780$	−4.19224	−0.150106
$781$	−27.6155	−0.988161
$782$	−11.8078	−0.422245
$783$	−9.12311	−0.326033
$784$	0	0
$785$	6.63068	0.236659
$786$	19.8078	0.706520
$787$	−45.5616	−1.62409	−0.812047	−	0.583592i	$-0.801647\pi$
$787$	−45.5616	−1.62409	−0.812047	+	0.583592i	$0.801647\pi$
$788$	−9.36932	−0.333768
$789$	4.49242	0.159934
$790$	−13.2614	−0.471818
$791$	0	0
$792$	4.87689	0.173293
$793$	−36.7386	−1.30463
$794$	35.3153	1.25329
$795$	−12.1922	−0.432414
$796$	−3.50758	−0.124323
$797$	−30.0540	−1.06457	−0.532283	−	0.846566i	$-0.678666\pi$
$797$	−30.0540	−1.06457	−0.532283	+	0.846566i	$0.678666\pi$
$798$	0	0
$799$	−50.5464	−1.78820
$800$	6.24621	0.220837
$801$	−10.0000	−0.353333
$802$	−1.06913	−0.0377523
$803$	−11.7538	−0.414782
$804$	−6.19224	−0.218383
$805$	0	0
$806$	−54.3542	−1.91454
$807$	−8.49242	−0.298947
$808$	2.13826	0.0752237
$809$	−18.6307	−0.655020	−0.327510	−	0.944848i	$-0.606210\pi$
$809$	−18.6307	−0.655020	−0.327510	+	0.944848i	$0.606210\pi$
$810$	2.43845	0.0856783
$811$	−19.6155	−0.688794	−0.344397	−	0.938824i	$-0.611917\pi$
$811$	−19.6155	−0.688794	−0.344397	+	0.938824i	$0.611917\pi$
$812$	0	0
$813$	−19.6155	−0.687947
$814$	−10.7386	−0.376389
$815$	6.24621	0.218795
$816$	35.4233	1.24006
$817$	−0.453602	−0.0158695
$818$	29.4536	1.02982
$819$	0	0
$820$	−7.01515	−0.244980
$821$	−12.7386	−0.444581	−0.222291	−	0.974980i	$-0.571353\pi$
$821$	−12.7386	−0.444581	−0.222291	+	0.974980i	$0.571353\pi$
$822$	−12.1922	−0.425253
$823$	29.8617	1.04091	0.520457	−	0.853888i	$-0.325761\pi$
$823$	29.8617	1.04091	0.520457	+	0.853888i	$0.325761\pi$
$824$	−34.4384	−1.19972
$825$	5.12311	0.178364
$826$	0	0
$827$	22.7386	0.790700	0.395350	−	0.918531i	$-0.370623\pi$
$827$	22.7386	0.790700	0.395350	+	0.918531i	$0.370623\pi$
$828$	0.438447	0.0152371
$829$	1.49242	0.0518340	0.0259170	−	0.999664i	$-0.491749\pi$
$829$	1.49242	0.0518340	0.0259170	+	0.999664i	$0.491749\pi$
$830$	11.8920	0.412779
$831$	−29.2462	−1.01454
$832$	−34.0540	−1.18061
$833$	0	0
$834$	18.6307	0.645128
$835$	−5.17708	−0.179160
$836$	−1.26137	−0.0436253
$837$	5.68466	0.196491
$838$	−44.4924	−1.53697
$839$	−20.6307	−0.712250	−0.356125	−	0.934438i	$-0.615902\pi$
$839$	−20.6307	−0.712250	−0.356125	+	0.934438i	$0.615902\pi$
$840$	0	0
$841$	54.2311	1.87004
$842$	−6.73863	−0.232229
$843$	19.5616	0.673736
$844$	−3.12311	−0.107502
$845$	38.2462	1.31571
$846$	10.4384	0.358881
$847$	0	0
$848$	36.5767	1.25605
$849$	−24.1231	−0.827903
$850$	30.2462	1.03744
$851$	3.43845	0.117868
$852$	6.05398	0.207406
$853$	−31.9309	−1.09329	−0.546646	−	0.837364i	$-0.684096\pi$
$853$	−31.9309	−1.09329	−0.546646	+	0.837364i	$0.684096\pi$
$854$	0	0
$855$	2.24621	0.0768188
$856$	15.2311	0.520587
$857$	17.3693	0.593325	0.296662	−	0.954982i	$-0.404126\pi$
$857$	17.3693	0.593325	0.296662	+	0.954982i	$0.404126\pi$
$858$	19.1231	0.652852
$859$	−26.2462	−0.895509	−0.447755	−	0.894156i	$-0.647776\pi$
$859$	−26.2462	−0.895509	−0.447755	+	0.894156i	$0.647776\pi$
$860$	−0.215901	−0.00736217
$861$	0	0
$862$	12.4924	0.425494
$863$	−6.30019	−0.214461	−0.107230	−	0.994234i	$-0.534198\pi$
$863$	−6.30019	−0.214461	−0.107230	+	0.994234i	$0.534198\pi$
$864$	−2.43845	−0.0829577
$865$	−36.4924	−1.24078
$866$	−10.6307	−0.361245
$867$	40.1771	1.36449
$868$	0	0
$869$	10.8769	0.368973
$870$	−22.2462	−0.754217
$871$	86.4773	2.93017
$872$	25.7538	0.872133
$873$	15.3693	0.520173
$874$	2.24621	0.0759792
$875$	0	0
$876$	2.57671	0.0870589
$877$	−24.0540	−0.812245	−0.406123	−	0.913819i	$-0.633119\pi$
$877$	−24.0540	−0.812245	−0.406123	+	0.913819i	$0.633119\pi$
$878$	24.3845	0.822936
$879$	24.0540	0.811320
$880$	14.6307	0.493200
$881$	5.31534	0.179078	0.0895392	−	0.995983i	$-0.471461\pi$
$881$	5.31534	0.179078	0.0895392	+	0.995983i	$0.471461\pi$
$882$	0	0
$883$	−2.06913	−0.0696318	−0.0348159	−	0.999394i	$-0.511084\pi$
$883$	−2.06913	−0.0696318	−0.0348159	+	0.999394i	$0.511084\pi$
$884$	20.3002	0.682769
$885$	14.2462	0.478881
$886$	20.7926	0.698541
$887$	0.630683	0.0211763	0.0105881	−	0.999944i	$-0.496630\pi$
$887$	0.630683	0.0211763	0.0105881	+	0.999944i	$0.496630\pi$
$888$	−8.38447	−0.281364
$889$	0	0
$890$	−24.3845	−0.817369
$891$	−2.00000	−0.0670025
$892$	−8.98485	−0.300835
$893$	9.61553	0.321771
$894$	−2.05398	−0.0686952
$895$	10.8229	0.361770
$896$	0	0
$897$	−6.12311	−0.204445
$898$	−21.8617	−0.729536
$899$	−51.8617	−1.72969
$900$	−1.12311	−0.0374369
$901$	59.0388	1.96687
$902$	32.0000	1.06548
$903$	0	0
$904$	−23.9157	−0.795425
$905$	−12.3845	−0.411674
$906$	−28.4924	−0.946597
$907$	7.73863	0.256957	0.128479	−	0.991712i	$-0.458991\pi$
$907$	7.73863	0.256957	0.128479	+	0.991712i	$0.458991\pi$
$908$	0.876894	0.0291008
$909$	−0.876894	−0.0290848
$910$	0	0
$911$	40.4924	1.34157	0.670787	−	0.741650i	$-0.265957\pi$
$911$	40.4924	1.34157	0.670787	+	0.741650i	$0.265957\pi$
$912$	−6.73863	−0.223138
$913$	−9.75379	−0.322803
$914$	2.24621	0.0742981
$915$	9.36932	0.309740
$916$	−12.8466	−0.424463
$917$	0	0
$918$	−11.8078	−0.389714
$919$	−39.4924	−1.30273	−0.651367	−	0.758762i	$-0.725804\pi$
$919$	−39.4924	−1.30273	−0.651367	+	0.758762i	$0.725804\pi$
$920$	−3.80776	−0.125538
$921$	11.6847	0.385023
$922$	−6.63068	−0.218370
$923$	−84.5464	−2.78288
$924$	0	0
$925$	−8.80776	−0.289597
$926$	30.7386	1.01013
$927$	14.1231	0.463864
$928$	22.2462	0.730268
$929$	3.50758	0.115080	0.0575399	−	0.998343i	$-0.481674\pi$
$929$	3.50758	0.115080	0.0575399	+	0.998343i	$0.481674\pi$
$930$	13.8617	0.454544
$931$	0	0
$932$	−4.76894	−0.156212
$933$	7.56155	0.247554
$934$	1.75379	0.0573857
$935$	23.6155	0.772310
$936$	14.9309	0.488031
$937$	−1.93087	−0.0630788	−0.0315394	−	0.999503i	$-0.510041\pi$
$937$	−1.93087	−0.0630788	−0.0315394	+	0.999503i	$0.510041\pi$
$938$	0	0
$939$	29.3002	0.956175
$940$	4.57671	0.149276
$941$	−15.1231	−0.492999	−0.246500	−	0.969143i	$-0.579280\pi$
$941$	−15.1231	−0.492999	−0.246500	+	0.969143i	$0.579280\pi$
$942$	6.63068	0.216039
$943$	−10.2462	−0.333663
$944$	−42.7386	−1.39102
$945$	0	0
$946$	0.984845	0.0320201
$947$	−17.3153	−0.562673	−0.281336	−	0.959609i	$-0.590778\pi$
$947$	−17.3153	−0.562673	−0.281336	+	0.959609i	$0.590778\pi$
$948$	−2.38447	−0.0774440
$949$	−35.9848	−1.16812
$950$	−5.75379	−0.186678
$951$	−11.7538	−0.381143
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	−12.1922	−0.394738
$955$	−1.75379	−0.0567513
$956$	−8.00000	−0.258738
$957$	18.2462	0.589816
$958$	−30.2462	−0.977211
$959$	0	0
$960$	8.68466	0.280296
$961$	1.31534	0.0424304
$962$	−32.8769	−1.05999
$963$	−6.24621	−0.201281
$964$	0.876894	0.0282429
$965$	12.3002	0.395957
$966$	0	0
$967$	−40.8078	−1.31229	−0.656145	−	0.754635i	$-0.727814\pi$
$967$	−40.8078	−1.31229	−0.656145	+	0.754635i	$0.727814\pi$
$968$	17.0691	0.548623
$969$	−10.8769	−0.349416
$970$	37.4773	1.20332
$971$	−9.12311	−0.292774	−0.146387	−	0.989227i	$-0.546765\pi$
$971$	−9.12311	−0.292774	−0.146387	+	0.989227i	$0.546765\pi$
$972$	0.438447	0.0140632
$973$	0	0
$974$	35.0152	1.12196
$975$	15.6847	0.502311
$976$	−28.1080	−0.899714
$977$	−8.43845	−0.269970	−0.134985	−	0.990848i	$-0.543099\pi$
$977$	−8.43845	−0.269970	−0.134985	+	0.990848i	$0.543099\pi$
$978$	6.24621	0.199732
$979$	20.0000	0.639203
$980$	0	0
$981$	−10.5616	−0.337204
$982$	−47.9157	−1.52905
$983$	5.36932	0.171255	0.0856273	−	0.996327i	$-0.472711\pi$
$983$	5.36932	0.171255	0.0856273	+	0.996327i	$0.472711\pi$
$984$	24.9848	0.796488
$985$	−33.3693	−1.06323
$986$	107.723	3.43061
$987$	0	0
$988$	−3.86174	−0.122858
$989$	−0.315342	−0.0100273
$990$	−4.87689	−0.154998
$991$	−7.30019	−0.231898	−0.115949	−	0.993255i	$-0.536991\pi$
$991$	−7.30019	−0.231898	−0.115949	+	0.993255i	$0.536991\pi$
$992$	−13.8617	−0.440111
$993$	−15.4384	−0.489924
$994$	0	0
$995$	−12.4924	−0.396036
$996$	2.13826	0.0677534
$997$	31.3002	0.991287	0.495643	−	0.868526i	$-0.334932\pi$
$997$	31.3002	0.991287	0.495643	+	0.868526i	$0.334932\pi$
$998$	−55.7235	−1.76390
$999$	3.43845	0.108788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.s.1.2		2
7.2	even	3		483.2.i.e.277.1	✓	4
7.4	even	3		483.2.i.e.415.1	yes	4
7.6	odd	2		3381.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.e.277.1	✓	4	7.2	even	3
483.2.i.e.415.1	yes	4	7.4	even	3
3381.2.a.q.1.2		2	7.6	odd	2
3381.2.a.s.1.2		2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.56155 q^{2} +1.00000 q^{3} +0.438447 q^{4} +1.56155 q^{5} +1.56155 q^{6} -2.43845 q^{8} +1.00000 q^{9} +2.43845 q^{10} -2.00000 q^{11} +0.438447 q^{12} -6.12311 q^{13} +1.56155 q^{15} -4.68466 q^{16} -7.56155 q^{17} +1.56155 q^{18} +1.43845 q^{19} +0.684658 q^{20} -3.12311 q^{22} +1.00000 q^{23} -2.43845 q^{24} -2.56155 q^{25} -9.56155 q^{26} +1.00000 q^{27} -9.12311 q^{29} +2.43845 q^{30} +5.68466 q^{31} -2.43845 q^{32} -2.00000 q^{33} -11.8078 q^{34} +0.438447 q^{36} +3.43845 q^{37} +2.24621 q^{38} -6.12311 q^{39} -3.80776 q^{40} -10.2462 q^{41} -0.315342 q^{43} -0.876894 q^{44} +1.56155 q^{45} +1.56155 q^{46} +6.68466 q^{47} -4.68466 q^{48} -4.00000 q^{50} -7.56155 q^{51} -2.68466 q^{52} -7.80776 q^{53} +1.56155 q^{54} -3.12311 q^{55} +1.43845 q^{57} -14.2462 q^{58} +9.12311 q^{59} +0.684658 q^{60} +6.00000 q^{61} +8.87689 q^{62} +5.56155 q^{64} -9.56155 q^{65} -3.12311 q^{66} -14.1231 q^{67} -3.31534 q^{68} +1.00000 q^{69} +13.8078 q^{71} -2.43845 q^{72} +5.87689 q^{73} +5.36932 q^{74} -2.56155 q^{75} +0.630683 q^{76} -9.56155 q^{78} -5.43845 q^{79} -7.31534 q^{80} +1.00000 q^{81} -16.0000 q^{82} +4.87689 q^{83} -11.8078 q^{85} -0.492423 q^{86} -9.12311 q^{87} +4.87689 q^{88} -10.0000 q^{89} +2.43845 q^{90} +0.438447 q^{92} +5.68466 q^{93} +10.4384 q^{94} +2.24621 q^{95} -2.43845 q^{96} +15.3693 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} + 5 q^{4} - q^{5} - q^{6} - 9 q^{8} + 2 q^{9} + 9 q^{10} - 4 q^{11} + 5 q^{12} - 4 q^{13} - q^{15} + 3 q^{16} - 11 q^{17} - q^{18} + 7 q^{19} - 11 q^{20} + 2 q^{22} + 2 q^{23}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 + 5 * q^4 - q^5 - q^6 - 9 * q^8 + 2 * q^9 + 9 * q^10 - 4 * q^11 + 5 * q^12 - 4 * q^13 - q^15 + 3 * q^16 - 11 * q^17 - q^18 + 7 * q^19 - 11 * q^20 + 2 * q^22 + 2 * q^23 - 9 * q^24 - q^25 - 15 * q^26 + 2 * q^27 - 10 * q^29 + 9 * q^30 - q^31 - 9 * q^32 - 4 * q^33 - 3 * q^34 + 5 * q^36 + 11 * q^37 - 12 * q^38 - 4 * q^39 + 13 * q^40 - 4 * q^41 - 13 * q^43 - 10 * q^44 - q^45 - q^46 + q^47 + 3 * q^48 - 8 * q^50 - 11 * q^51 + 7 * q^52 + 5 * q^53 - q^54 + 2 * q^55 + 7 * q^57 - 12 * q^58 + 10 * q^59 - 11 * q^60 + 12 * q^61 + 26 * q^62 + 7 * q^64 - 15 * q^65 + 2 * q^66 - 20 * q^67 - 19 * q^68 + 2 * q^69 + 7 * q^71 - 9 * q^72 + 20 * q^73 - 14 * q^74 - q^75 + 26 * q^76 - 15 * q^78 - 15 * q^79 - 27 * q^80 + 2 * q^81 - 32 * q^82 + 18 * q^83 - 3 * q^85 + 32 * q^86 - 10 * q^87 + 18 * q^88 - 20 * q^89 + 9 * q^90 + 5 * q^92 - q^93 + 25 * q^94 - 12 * q^95 - 9 * q^96 + 6 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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