LMFDB - Embedded newform 6579.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6579,2,Mod(1,6579)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6579, 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("6579.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6579 = 3^{2} \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6579.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:	$52.5335794898$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.2460365.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6x^{4} + 6x^{3} + 7x^{2} - 7x + 1 $ x^6 - x^5 - 6x^4 + 6x^3 + 7x^2 - 7x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 731)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.48737$ of defining polynomial
Character	$\chi$	$=$	6579.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.212277 q^{2} -1.95494 q^{4} +1.65901 q^{5} +2.53919 q^{7} -0.839542 q^{8} +O(q^{10})$	\(q+0.212277 q^{2} -1.95494 q^{4} +1.65901 q^{5} +2.53919 q^{7} -0.839542 q^{8} +0.352169 q^{10} -1.11540 q^{11} +2.39509 q^{13} +0.539012 q^{14} +3.73166 q^{16} +1.00000 q^{17} -5.01350 q^{19} -3.24326 q^{20} -0.236773 q^{22} -3.24795 q^{23} -2.24769 q^{25} +0.508423 q^{26} -4.96396 q^{28} +1.28054 q^{29} +0.498801 q^{31} +2.47123 q^{32} +0.212277 q^{34} +4.21254 q^{35} -11.9871 q^{37} -1.06425 q^{38} -1.39281 q^{40} -10.5784 q^{41} -1.00000 q^{43} +2.18053 q^{44} -0.689465 q^{46} -0.400325 q^{47} -0.552508 q^{49} -0.477132 q^{50} -4.68226 q^{52} +5.37685 q^{53} -1.85045 q^{55} -2.13176 q^{56} +0.271830 q^{58} -5.21308 q^{59} +0.0130677 q^{61} +0.105884 q^{62} -6.93874 q^{64} +3.97349 q^{65} -8.24406 q^{67} -1.95494 q^{68} +0.894226 q^{70} +8.15605 q^{71} +5.99199 q^{73} -2.54459 q^{74} +9.80108 q^{76} -2.83221 q^{77} -1.43914 q^{79} +6.19086 q^{80} -2.24555 q^{82} +1.09306 q^{83} +1.65901 q^{85} -0.212277 q^{86} +0.936422 q^{88} +10.7179 q^{89} +6.08160 q^{91} +6.34955 q^{92} -0.0849797 q^{94} -8.31744 q^{95} -6.18621 q^{97} -0.117285 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 5 q^{4} - 3 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) $ 6 * q + q^2 + 5 * q^4 - 3 * q^5 - 7 * q^7 + 9 * q^8	$ 6 q + q^{2} + 5 q^{4} - 3 q^{5} - 7 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 10 q^{13} + 7 q^{14} - q^{16} + 6 q^{17} - 20 q^{19} - q^{20} + 2 q^{22} + 3 q^{23} - 7 q^{25} + 3 q^{26} - 11 q^{28} + 15 q^{29} + 12 q^{31} - q^{32} + q^{34} + 9 q^{35} - 14 q^{37} - 27 q^{38} - 7 q^{40} + 2 q^{41} - 6 q^{43} + 12 q^{44} + 14 q^{46} + 11 q^{47} + 3 q^{49} - 7 q^{50} - 5 q^{52} - 3 q^{53} + 6 q^{55} - 22 q^{56} - 21 q^{58} - 2 q^{59} - 20 q^{61} - 3 q^{62} - 39 q^{64} + 34 q^{65} - 2 q^{67} + 5 q^{68} - q^{70} - q^{71} + 13 q^{73} - 28 q^{74} - 29 q^{76} + 11 q^{77} - 26 q^{79} - 12 q^{80} - 9 q^{82} - 10 q^{83} - 3 q^{85} - q^{86} - 6 q^{88} + 15 q^{89} + 8 q^{91} + 9 q^{92} - 33 q^{94} + 21 q^{95} - 22 q^{97} + 3 q^{98}+O(q^{100}) $ 6 * q + q^2 + 5 * q^4 - 3 * q^5 - 7 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 10 * q^13 + 7 * q^14 - q^16 + 6 * q^17 - 20 * q^19 - q^20 + 2 * q^22 + 3 * q^23 - 7 * q^25 + 3 * q^26 - 11 * q^28 + 15 * q^29 + 12 * q^31 - q^32 + q^34 + 9 * q^35 - 14 * q^37 - 27 * q^38 - 7 * q^40 + 2 * q^41 - 6 * q^43 + 12 * q^44 + 14 * q^46 + 11 * q^47 + 3 * q^49 - 7 * q^50 - 5 * q^52 - 3 * q^53 + 6 * q^55 - 22 * q^56 - 21 * q^58 - 2 * q^59 - 20 * q^61 - 3 * q^62 - 39 * q^64 + 34 * q^65 - 2 * q^67 + 5 * q^68 - q^70 - q^71 + 13 * q^73 - 28 * q^74 - 29 * q^76 + 11 * q^77 - 26 * q^79 - 12 * q^80 - 9 * q^82 - 10 * q^83 - 3 * q^85 - q^86 - 6 * q^88 + 15 * q^89 + 8 * q^91 + 9 * q^92 - 33 * q^94 + 21 * q^95 - 22 * q^97 + 3 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.212277	0.150102	0.0750512	−	0.997180i	$-0.476088\pi$
$2$	0.212277	0.150102	0.0750512	+	0.997180i	$0.476088\pi$
$3$	0	0
$3$	0	0
$4$	−1.95494	−0.977469
$5$	1.65901	0.741932	0.370966	−	0.928646i	$-0.379027\pi$
$5$	1.65901	0.741932	0.370966	+	0.928646i	$0.379027\pi$
$6$	0	0
$7$	2.53919	0.959724	0.479862	−	0.877344i	$-0.340687\pi$
$7$	2.53919	0.959724	0.479862	+	0.877344i	$0.340687\pi$
$8$	−0.839542	−0.296823
$9$	0	0
$10$	0.352169	0.111366
$11$	−1.11540	−0.336305	−0.168152	−	0.985761i	$-0.553780\pi$
$11$	−1.11540	−0.336305	−0.168152	+	0.985761i	$0.553780\pi$
$12$	0	0
$13$	2.39509	0.664280	0.332140	−	0.943230i	$-0.392229\pi$
$13$	2.39509	0.664280	0.332140	+	0.943230i	$0.392229\pi$
$14$	0.539012	0.144057
$15$	0	0
$16$	3.73166	0.932915
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−5.01350	−1.15017	−0.575087	−	0.818092i	$-0.695032\pi$
$19$	−5.01350	−1.15017	−0.575087	+	0.818092i	$0.695032\pi$
$20$	−3.24326	−0.725216
$21$	0	0
$22$	−0.236773	−0.0504801
$23$	−3.24795	−0.677245	−0.338622	−	0.940922i	$-0.609961\pi$
$23$	−3.24795	−0.677245	−0.338622	+	0.940922i	$0.609961\pi$
$24$	0	0
$25$	−2.24769	−0.449537
$26$	0.508423	0.0997100
$27$	0	0
$28$	−4.96396	−0.938101
$29$	1.28054	0.237791	0.118896	−	0.992907i	$-0.462065\pi$
$29$	1.28054	0.237791	0.118896	+	0.992907i	$0.462065\pi$
$30$	0	0
$31$	0.498801	0.0895873	0.0447936	−	0.998996i	$-0.485737\pi$
$31$	0.498801	0.0895873	0.0447936	+	0.998996i	$0.485737\pi$
$32$	2.47123	0.436856
$33$	0	0
$34$	0.212277	0.0364052
$35$	4.21254	0.712050
$36$	0	0
$37$	−11.9871	−1.97067	−0.985336	−	0.170625i	$-0.945421\pi$
$37$	−11.9871	−1.97067	−0.985336	+	0.170625i	$0.945421\pi$
$38$	−1.06425	−0.172644
$39$	0	0
$40$	−1.39281	−0.220222
$41$	−10.5784	−1.65207	−0.826036	−	0.563617i	$-0.809409\pi$
$41$	−10.5784	−1.65207	−0.826036	+	0.563617i	$0.809409\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	2.18053	0.328728
$45$	0	0
$46$	−0.689465	−0.101656
$47$	−0.400325	−0.0583934	−0.0291967	−	0.999574i	$-0.509295\pi$
$47$	−0.400325	−0.0583934	−0.0291967	+	0.999574i	$0.509295\pi$
$48$	0	0
$49$	−0.552508	−0.0789297
$50$	−0.477132	−0.0674766
$51$	0	0
$52$	−4.68226	−0.649313
$53$	5.37685	0.738566	0.369283	−	0.929317i	$-0.379603\pi$
$53$	5.37685	0.738566	0.369283	+	0.929317i	$0.379603\pi$
$54$	0	0
$55$	−1.85045	−0.249515
$56$	−2.13176	−0.284868
$57$	0	0
$58$	0.271830	0.0356930
$59$	−5.21308	−0.678685	−0.339342	−	0.940663i	$-0.610204\pi$
$59$	−5.21308	−0.678685	−0.339342	+	0.940663i	$0.610204\pi$
$60$	0	0
$61$	0.0130677	0.00167314	0.000836571	−	1.00000i	$-0.499734\pi$
$61$	0.0130677	0.00167314	0.000836571		1.00000i	$0.499734\pi$
$62$	0.105884	0.0134473
$63$	0	0
$64$	−6.93874	−0.867342
$65$	3.97349	0.492850
$66$	0	0
$67$	−8.24406	−1.00717	−0.503586	−	0.863945i	$-0.667986\pi$
$67$	−8.24406	−1.00717	−0.503586	+	0.863945i	$0.667986\pi$
$68$	−1.95494	−0.237071
$69$	0	0
$70$	0.894226	0.106880
$71$	8.15605	0.967946	0.483973	−	0.875083i	$-0.339193\pi$
$71$	8.15605	0.967946	0.483973	+	0.875083i	$0.339193\pi$
$72$	0	0
$73$	5.99199	0.701310	0.350655	−	0.936505i	$-0.385959\pi$
$73$	5.99199	0.701310	0.350655	+	0.936505i	$0.385959\pi$
$74$	−2.54459	−0.295803
$75$	0	0
$76$	9.80108	1.12426
$77$	−2.83221	−0.322760
$78$	0	0
$79$	−1.43914	−0.161916	−0.0809579	−	0.996718i	$-0.525798\pi$
$79$	−1.43914	−0.161916	−0.0809579	+	0.996718i	$0.525798\pi$
$80$	6.19086	0.692160
$81$	0	0
$82$	−2.24555	−0.247980
$83$	1.09306	0.119978	0.0599892	−	0.998199i	$-0.480893\pi$
$83$	1.09306	0.119978	0.0599892	+	0.998199i	$0.480893\pi$
$84$	0	0
$85$	1.65901	0.179945
$86$	−0.212277	−0.0228904
$87$	0	0
$88$	0.936422	0.0998229
$89$	10.7179	1.13610	0.568048	−	0.822995i	$-0.307699\pi$
$89$	10.7179	1.13610	0.568048	+	0.822995i	$0.307699\pi$
$90$	0	0
$91$	6.08160	0.637525
$92$	6.34955	0.661986
$93$	0	0
$94$	−0.0849797	−0.00876499
$95$	−8.31744	−0.853351
$96$	0	0
$97$	−6.18621	−0.628115	−0.314057	−	0.949404i	$-0.601688\pi$
$97$	−6.18621	−0.628115	−0.314057	+	0.949404i	$0.601688\pi$
$98$	−0.117285	−0.0118475
$99$	0	0
$100$	4.39409	0.439409
$101$	1.95718	0.194747	0.0973736	−	0.995248i	$-0.468956\pi$
$101$	1.95718	0.194747	0.0973736	+	0.995248i	$0.468956\pi$
$102$	0	0
$103$	7.26547	0.715888	0.357944	−	0.933743i	$-0.383478\pi$
$103$	7.26547	0.715888	0.357944	+	0.933743i	$0.383478\pi$
$104$	−2.01078	−0.197173
$105$	0	0
$106$	1.14138	0.110861
$107$	−8.14989	−0.787880	−0.393940	−	0.919136i	$-0.628888\pi$
$107$	−8.14989	−0.787880	−0.393940	+	0.919136i	$0.628888\pi$
$108$	0	0
$109$	−16.1863	−1.55036	−0.775181	−	0.631739i	$-0.782342\pi$
$109$	−16.1863	−1.55036	−0.775181	+	0.631739i	$0.782342\pi$
$110$	−0.392809	−0.0374528
$111$	0	0
$112$	9.47540	0.895341
$113$	−4.46484	−0.420017	−0.210008	−	0.977700i	$-0.567349\pi$
$113$	−4.46484	−0.420017	−0.210008	+	0.977700i	$0.567349\pi$
$114$	0	0
$115$	−5.38839	−0.502470
$116$	−2.50339	−0.232433
$117$	0	0
$118$	−1.10662	−0.101872
$119$	2.53919	0.232767
$120$	0	0
$121$	−9.75589	−0.886899
$122$	0.00277396	0.000251143 0
$123$	0	0
$124$	−0.975125	−0.0875688
$125$	−12.0240	−1.07546
$126$	0	0
$127$	−0.291837	−0.0258963	−0.0129482	−	0.999916i	$-0.504122\pi$
$127$	−0.291837	−0.0258963	−0.0129482	+	0.999916i	$0.504122\pi$
$128$	−6.41539	−0.567046
$129$	0	0
$130$	0.843479	0.0739780
$131$	−0.474123	−0.0414243	−0.0207122	−	0.999785i	$-0.506593\pi$
$131$	−0.474123	−0.0414243	−0.0207122	+	0.999785i	$0.506593\pi$
$132$	0	0
$133$	−12.7302	−1.10385
$134$	−1.75002	−0.151179
$135$	0	0
$136$	−0.839542	−0.0719901
$137$	19.1328	1.63463	0.817314	−	0.576192i	$-0.195462\pi$
$137$	19.1328	1.63463	0.817314	+	0.576192i	$0.195462\pi$
$138$	0	0
$139$	−9.70162	−0.822881	−0.411440	−	0.911437i	$-0.634974\pi$
$139$	−9.70162	−0.822881	−0.411440	+	0.911437i	$0.634974\pi$
$140$	−8.23526	−0.696007
$141$	0	0
$142$	1.73134	0.145291
$143$	−2.67148	−0.223400
$144$	0	0
$145$	2.12444	0.176425
$146$	1.27196	0.105268
$147$	0	0
$148$	23.4341	1.92627
$149$	20.2460	1.65861	0.829307	−	0.558793i	$-0.188735\pi$
$149$	20.2460	1.65861	0.829307	+	0.558793i	$0.188735\pi$
$150$	0	0
$151$	−5.52591	−0.449692	−0.224846	−	0.974394i	$-0.572188\pi$
$151$	−5.52591	−0.449692	−0.224846	+	0.974394i	$0.572188\pi$
$152$	4.20904	0.341398
$153$	0	0
$154$	−0.601212	−0.0484470
$155$	0.827516	0.0664677
$156$	0	0
$157$	−8.65527	−0.690766	−0.345383	−	0.938462i	$-0.612251\pi$
$157$	−8.65527	−0.690766	−0.345383	+	0.938462i	$0.612251\pi$
$158$	−0.305496	−0.0243040
$159$	0	0
$160$	4.09979	0.324117
$161$	−8.24717	−0.649968
$162$	0	0
$163$	19.2922	1.51108	0.755541	−	0.655102i	$-0.227374\pi$
$163$	19.2922	1.51108	0.755541	+	0.655102i	$0.227374\pi$
$164$	20.6802	1.61485
$165$	0	0
$166$	0.232030	0.0180091
$167$	−2.26648	−0.175385	−0.0876926	−	0.996148i	$-0.527949\pi$
$167$	−2.26648	−0.175385	−0.0876926	+	0.996148i	$0.527949\pi$
$168$	0	0
$169$	−7.26352	−0.558733
$170$	0.352169	0.0270102
$171$	0	0
$172$	1.95494	0.149063
$173$	2.78795	0.211964	0.105982	−	0.994368i	$-0.466201\pi$
$173$	2.78795	0.211964	0.105982	+	0.994368i	$0.466201\pi$
$174$	0	0
$175$	−5.70730	−0.431432
$176$	−4.16228	−0.313744
$177$	0	0
$178$	2.27517	0.170531
$179$	24.8463	1.85710	0.928549	−	0.371209i	$-0.121057\pi$
$179$	24.8463	1.85710	0.928549	+	0.371209i	$0.121057\pi$
$180$	0	0
$181$	−1.71039	−0.127132	−0.0635662	−	0.997978i	$-0.520247\pi$
$181$	−1.71039	−0.127132	−0.0635662	+	0.997978i	$0.520247\pi$
$182$	1.29098	0.0956941
$183$	0	0
$184$	2.72679	0.201022
$185$	−19.8868	−1.46210
$186$	0	0
$187$	−1.11540	−0.0815659
$188$	0.782611	0.0570778
$189$	0	0
$190$	−1.76560	−0.128090
$191$	−20.5373	−1.48602	−0.743012	−	0.669278i	$-0.766603\pi$
$191$	−20.5373	−1.48602	−0.743012	+	0.669278i	$0.766603\pi$
$192$	0	0
$193$	−13.1576	−0.947105	−0.473553	−	0.880766i	$-0.657029\pi$
$193$	−13.1576	−0.947105	−0.473553	+	0.880766i	$0.657029\pi$
$194$	−1.31319	−0.0942815
$195$	0	0
$196$	1.08012	0.0771513
$197$	−15.4265	−1.09909	−0.549547	−	0.835463i	$-0.685200\pi$
$197$	−15.4265	−1.09909	−0.549547	+	0.835463i	$0.685200\pi$
$198$	0	0
$199$	9.69335	0.687144	0.343572	−	0.939126i	$-0.388363\pi$
$199$	9.69335	0.687144	0.343572	+	0.939126i	$0.388363\pi$
$200$	1.88703	0.133433
$201$	0	0
$202$	0.415465	0.0292320
$203$	3.25155	0.228214
$204$	0	0
$205$	−17.5497	−1.22572
$206$	1.54229	0.107457
$207$	0	0
$208$	8.93768	0.619717
$209$	5.59204	0.386809
$210$	0	0
$211$	−20.9824	−1.44449	−0.722244	−	0.691639i	$-0.756889\pi$
$211$	−20.9824	−1.44449	−0.722244	+	0.691639i	$0.756889\pi$
$212$	−10.5114	−0.721926
$213$	0	0
$214$	−1.73003	−0.118263
$215$	−1.65901	−0.113144
$216$	0	0
$217$	1.26655	0.0859791
$218$	−3.43597	−0.232713
$219$	0	0
$220$	3.61752	0.243893
$221$	2.39509	0.161111
$222$	0	0
$223$	14.4719	0.969109	0.484555	−	0.874761i	$-0.338982\pi$
$223$	14.4719	0.969109	0.484555	+	0.874761i	$0.338982\pi$
$224$	6.27492	0.419261
$225$	0	0
$226$	−0.947783	−0.0630456
$227$	−19.4270	−1.28941	−0.644706	−	0.764430i	$-0.723020\pi$
$227$	−19.4270	−1.28941	−0.644706	+	0.764430i	$0.723020\pi$
$228$	0	0
$229$	−17.1473	−1.13312	−0.566562	−	0.824019i	$-0.691727\pi$
$229$	−17.1473	−1.13312	−0.566562	+	0.824019i	$0.691727\pi$
$230$	−1.14383	−0.0754219
$231$	0	0
$232$	−1.07507	−0.0705818
$233$	1.27543	0.0835559	0.0417780	−	0.999127i	$-0.486698\pi$
$233$	1.27543	0.0835559	0.0417780	+	0.999127i	$0.486698\pi$
$234$	0	0
$235$	−0.664143	−0.0433239
$236$	10.1912	0.663393
$237$	0	0
$238$	0.539012	0.0349389
$239$	9.91245	0.641183	0.320592	−	0.947218i	$-0.396118\pi$
$239$	9.91245	0.641183	0.320592	+	0.947218i	$0.396118\pi$
$240$	0	0
$241$	−29.4709	−1.89839	−0.949193	−	0.314694i	$-0.898098\pi$
$241$	−29.4709	−1.89839	−0.949193	+	0.314694i	$0.898098\pi$
$242$	−2.07095	−0.133126
$243$	0	0
$244$	−0.0255465	−0.00163544
$245$	−0.916616	−0.0585604
$246$	0	0
$247$	−12.0078	−0.764038
$248$	−0.418764	−0.0265916
$249$	0	0
$250$	−2.55241	−0.161429
$251$	−16.6596	−1.05155	−0.525774	−	0.850624i	$-0.676224\pi$
$251$	−16.6596	−1.05155	−0.525774	+	0.850624i	$0.676224\pi$
$252$	0	0
$253$	3.62276	0.227761
$254$	−0.0619502	−0.00388710
$255$	0	0
$256$	12.5156	0.782227
$257$	13.6655	0.852434	0.426217	−	0.904621i	$-0.359846\pi$
$257$	13.6655	0.852434	0.426217	+	0.904621i	$0.359846\pi$
$258$	0	0
$259$	−30.4376	−1.89130
$260$	−7.76792	−0.481746
$261$	0	0
$262$	−0.100645	−0.00621789
$263$	13.8793	0.855835	0.427917	−	0.903818i	$-0.359247\pi$
$263$	13.8793	0.855835	0.427917	+	0.903818i	$0.359247\pi$
$264$	0	0
$265$	8.92024	0.547966
$266$	−2.70233	−0.165691
$267$	0	0
$268$	16.1166	0.984480
$269$	1.13342	0.0691057	0.0345528	−	0.999403i	$-0.488999\pi$
$269$	1.13342	0.0691057	0.0345528	+	0.999403i	$0.488999\pi$
$270$	0	0
$271$	15.3569	0.932868	0.466434	−	0.884556i	$-0.345538\pi$
$271$	15.3569	0.932868	0.466434	+	0.884556i	$0.345538\pi$
$272$	3.73166	0.226265
$273$	0	0
$274$	4.06146	0.245362
$275$	2.50706	0.151181
$276$	0	0
$277$	24.7714	1.48837	0.744183	−	0.667975i	$-0.232839\pi$
$277$	24.7714	1.48837	0.744183	+	0.667975i	$0.232839\pi$
$278$	−2.05943	−0.123516
$279$	0	0
$280$	−3.53661	−0.211353
$281$	0.0469321	0.00279973	0.00139987	−	0.999999i	$-0.499554\pi$
$281$	0.0469321	0.00279973	0.00139987	+	0.999999i	$0.499554\pi$
$282$	0	0
$283$	−16.9012	−1.00467	−0.502335	−	0.864673i	$-0.667526\pi$
$283$	−16.9012	−1.00467	−0.502335	+	0.864673i	$0.667526\pi$
$284$	−15.9446	−0.946137
$285$	0	0
$286$	−0.567093	−0.0335329
$287$	−26.8606	−1.58553
$288$	0	0
$289$	1.00000	0.0588235
$290$	0.450969	0.0264818
$291$	0	0
$292$	−11.7140	−0.685509
$293$	−30.8356	−1.80143	−0.900717	−	0.434407i	$-0.856958\pi$
$293$	−30.8356	−1.80143	−0.900717	+	0.434407i	$0.856958\pi$
$294$	0	0
$295$	−8.64854	−0.503538
$296$	10.0637	0.584941
$297$	0	0
$298$	4.29775	0.248962
$299$	−7.77915	−0.449880
$300$	0	0
$301$	−2.53919	−0.146357
$302$	−1.17302	−0.0674999
$303$	0	0
$304$	−18.7087	−1.07302
$305$	0.0216794	0.00124136
$306$	0	0
$307$	3.07155	0.175303	0.0876513	−	0.996151i	$-0.472064\pi$
$307$	3.07155	0.175303	0.0876513	+	0.996151i	$0.472064\pi$
$308$	5.53679	0.315488
$309$	0	0
$310$	0.175662	0.00997696
$311$	32.2382	1.82806	0.914030	−	0.405646i	$-0.132953\pi$
$311$	32.2382	1.82806	0.914030	+	0.405646i	$0.132953\pi$
$312$	0	0
$313$	0.706328	0.0399240	0.0199620	−	0.999801i	$-0.493645\pi$
$313$	0.706328	0.0399240	0.0199620	+	0.999801i	$0.493645\pi$
$314$	−1.83731	−0.103686
$315$	0	0
$316$	2.81343	0.158268
$317$	−11.8604	−0.666148	−0.333074	−	0.942901i	$-0.608086\pi$
$317$	−11.8604	−0.666148	−0.333074	+	0.942901i	$0.608086\pi$
$318$	0	0
$319$	−1.42831	−0.0799703
$320$	−11.5114	−0.643509
$321$	0	0
$322$	−1.75068	−0.0975618
$323$	−5.01350	−0.278958
$324$	0	0
$325$	−5.38342	−0.298618
$326$	4.09529	0.226817
$327$	0	0
$328$	8.88103	0.490373
$329$	−1.01650	−0.0560416
$330$	0	0
$331$	−24.7566	−1.36075	−0.680373	−	0.732866i	$-0.738182\pi$
$331$	−24.7566	−1.36075	−0.680373	+	0.732866i	$0.738182\pi$
$332$	−2.13686	−0.117275
$333$	0	0
$334$	−0.481120	−0.0263257
$335$	−13.6770	−0.747253
$336$	0	0
$337$	13.2780	0.723296	0.361648	−	0.932315i	$-0.382214\pi$
$337$	13.2780	0.723296	0.361648	+	0.932315i	$0.382214\pi$
$338$	−1.54188	−0.0838671
$339$	0	0
$340$	−3.24326	−0.175891
$341$	−0.556361	−0.0301286
$342$	0	0
$343$	−19.1773	−1.03547
$344$	0.839542	0.0452651
$345$	0	0
$346$	0.591818	0.0318163
$347$	−0.888881	−0.0477177	−0.0238588	−	0.999715i	$-0.507595\pi$
$347$	−0.888881	−0.0477177	−0.0238588	+	0.999715i	$0.507595\pi$
$348$	0	0
$349$	26.9773	1.44406	0.722030	−	0.691862i	$-0.243209\pi$
$349$	26.9773	1.44406	0.722030	+	0.691862i	$0.243209\pi$
$350$	−1.21153	−0.0647589
$351$	0	0
$352$	−2.75640	−0.146917
$353$	−22.5168	−1.19845	−0.599224	−	0.800581i	$-0.704524\pi$
$353$	−22.5168	−1.19845	−0.599224	+	0.800581i	$0.704524\pi$
$354$	0	0
$355$	13.5310	0.718150
$356$	−20.9529	−1.11050
$357$	0	0
$358$	5.27429	0.278755
$359$	−19.5922	−1.03404	−0.517018	−	0.855975i	$-0.672958\pi$
$359$	−19.5922	−1.03404	−0.517018	+	0.855975i	$0.672958\pi$
$360$	0	0
$361$	6.13514	0.322902
$362$	−0.363077	−0.0190829
$363$	0	0
$364$	−11.8892	−0.623161
$365$	9.94078	0.520324
$366$	0	0
$367$	−8.26173	−0.431259	−0.215630	−	0.976475i	$-0.569180\pi$
$367$	−8.26173	−0.431259	−0.215630	+	0.976475i	$0.569180\pi$
$368$	−12.1203	−0.631812
$369$	0	0
$370$	−4.22150	−0.219465
$371$	13.6528	0.708820
$372$	0	0
$373$	−18.9938	−0.983462	−0.491731	−	0.870747i	$-0.663636\pi$
$373$	−18.9938	−0.983462	−0.491731	+	0.870747i	$0.663636\pi$
$374$	−0.236773	−0.0122432
$375$	0	0
$376$	0.336090	0.0173325
$377$	3.06702	0.157960
$378$	0	0
$379$	22.6083	1.16131	0.580654	−	0.814150i	$-0.302797\pi$
$379$	22.6083	1.16131	0.580654	+	0.814150i	$0.302797\pi$
$380$	16.2601	0.834125
$381$	0	0
$382$	−4.35958	−0.223056
$383$	16.3572	0.835814	0.417907	−	0.908490i	$-0.362764\pi$
$383$	16.3572	0.835814	0.417907	+	0.908490i	$0.362764\pi$
$384$	0	0
$385$	−4.69866	−0.239466
$386$	−2.79306	−0.142163
$387$	0	0
$388$	12.0937	0.613963
$389$	31.5393	1.59911	0.799554	−	0.600594i	$-0.205069\pi$
$389$	31.5393	1.59911	0.799554	+	0.600594i	$0.205069\pi$
$390$	0	0
$391$	−3.24795	−0.164256
$392$	0.463853	0.0234281
$393$	0	0
$394$	−3.27469	−0.164977
$395$	−2.38755	−0.120131
$396$	0	0
$397$	−22.6234	−1.13543	−0.567717	−	0.823224i	$-0.692173\pi$
$397$	−22.6234	−1.13543	−0.567717	+	0.823224i	$0.692173\pi$
$398$	2.05767	0.103142
$399$	0	0
$400$	−8.38760	−0.419380
$401$	−17.2056	−0.859205	−0.429603	−	0.903018i	$-0.641346\pi$
$401$	−17.2056	−0.859205	−0.429603	+	0.903018i	$0.641346\pi$
$402$	0	0
$403$	1.19468	0.0595110
$404$	−3.82618	−0.190359
$405$	0	0
$406$	0.690228	0.0342554
$407$	13.3704	0.662746
$408$	0	0
$409$	−0.928413	−0.0459071	−0.0229535	−	0.999737i	$-0.507307\pi$
$409$	−0.928413	−0.0459071	−0.0229535	+	0.999737i	$0.507307\pi$
$410$	−3.72540	−0.183984
$411$	0	0
$412$	−14.2036	−0.699759
$413$	−13.2370	−0.651350
$414$	0	0
$415$	1.81339	0.0890159
$416$	5.91883	0.290194
$417$	0	0
$418$	1.18706	0.0580610
$419$	20.6047	1.00660	0.503302	−	0.864111i	$-0.332118\pi$
$419$	20.6047	1.00660	0.503302	+	0.864111i	$0.332118\pi$
$420$	0	0
$421$	−25.6177	−1.24853	−0.624265	−	0.781213i	$-0.714601\pi$
$421$	−25.6177	−1.24853	−0.624265	+	0.781213i	$0.714601\pi$
$422$	−4.45408	−0.216821
$423$	0	0
$424$	−4.51409	−0.219223
$425$	−2.24769	−0.109029
$426$	0	0
$427$	0.0331813	0.00160575
$428$	15.9325	0.770129
$429$	0	0
$430$	−0.352169	−0.0169831
$431$	−31.2758	−1.50650	−0.753250	−	0.657734i	$-0.771515\pi$
$431$	−31.2758	−1.50650	−0.753250	+	0.657734i	$0.771515\pi$
$432$	0	0
$433$	−30.1102	−1.44700	−0.723502	−	0.690322i	$-0.757469\pi$
$433$	−30.1102	−1.44700	−0.723502	+	0.690322i	$0.757469\pi$
$434$	0.268859	0.0129057
$435$	0	0
$436$	31.6431	1.51543
$437$	16.2836	0.778950
$438$	0	0
$439$	7.51969	0.358895	0.179448	−	0.983768i	$-0.442569\pi$
$439$	7.51969	0.358895	0.179448	+	0.983768i	$0.442569\pi$
$440$	1.55353	0.0740618
$441$	0	0
$442$	0.508423	0.0241832
$443$	4.83603	0.229767	0.114883	−	0.993379i	$-0.463351\pi$
$443$	4.83603	0.229767	0.114883	+	0.993379i	$0.463351\pi$
$444$	0	0
$445$	17.7811	0.842907
$446$	3.07205	0.145466
$447$	0	0
$448$	−17.6188	−0.832409
$449$	−24.1377	−1.13913	−0.569565	−	0.821947i	$-0.692888\pi$
$449$	−24.1377	−1.13913	−0.569565	+	0.821947i	$0.692888\pi$
$450$	0	0
$451$	11.7991	0.555600
$452$	8.72849	0.410554
$453$	0	0
$454$	−4.12389	−0.193544
$455$	10.0894	0.473000
$456$	0	0
$457$	−0.253686	−0.0118669	−0.00593347	−	0.999982i	$-0.501889\pi$
$457$	−0.253686	−0.0118669	−0.00593347	+	0.999982i	$0.501889\pi$
$458$	−3.63997	−0.170085
$459$	0	0
$460$	10.5340	0.491149
$461$	24.3643	1.13476	0.567379	−	0.823457i	$-0.307958\pi$
$461$	24.3643	1.13476	0.567379	+	0.823457i	$0.307958\pi$
$462$	0	0
$463$	−3.20234	−0.148825	−0.0744127	−	0.997228i	$-0.523708\pi$
$463$	−3.20234	−0.148825	−0.0744127	+	0.997228i	$0.523708\pi$
$464$	4.77856	0.221839
$465$	0	0
$466$	0.270743	0.0125419
$467$	−3.56529	−0.164982	−0.0824909	−	0.996592i	$-0.526288\pi$
$467$	−3.56529	−0.164982	−0.0824909	+	0.996592i	$0.526288\pi$
$468$	0	0
$469$	−20.9332	−0.966608
$470$	−0.140982	−0.00650303
$471$	0	0
$472$	4.37660	0.201449
$473$	1.11540	0.0512860
$474$	0	0
$475$	11.2688	0.517046
$476$	−4.96396	−0.227523
$477$	0	0
$478$	2.10418	0.0962431
$479$	6.80763	0.311048	0.155524	−	0.987832i	$-0.450293\pi$
$479$	6.80763	0.311048	0.155524	+	0.987832i	$0.450293\pi$
$480$	0	0
$481$	−28.7103	−1.30908
$482$	−6.25599	−0.284952
$483$	0	0
$484$	19.0722	0.866917
$485$	−10.2630	−0.466018
$486$	0	0
$487$	2.20169	0.0997679	0.0498840	−	0.998755i	$-0.484115\pi$
$487$	2.20169	0.0997679	0.0498840	+	0.998755i	$0.484115\pi$
$488$	−0.0109708	−0.000496627 0
$489$	0	0
$490$	−0.194576	−0.00879006
$491$	−24.9880	−1.12769	−0.563847	−	0.825879i	$-0.690679\pi$
$491$	−24.9880	−1.12769	−0.563847	+	0.825879i	$0.690679\pi$
$492$	0	0
$493$	1.28054	0.0576728
$494$	−2.54898	−0.114684
$495$	0	0
$496$	1.86136	0.0835774
$497$	20.7098	0.928961
$498$	0	0
$499$	5.04332	0.225770	0.112885	−	0.993608i	$-0.463991\pi$
$499$	5.04332	0.225770	0.112885	+	0.993608i	$0.463991\pi$
$500$	23.5061	1.05123
$501$	0	0
$502$	−3.53646	−0.157840
$503$	20.4975	0.913938	0.456969	−	0.889483i	$-0.348935\pi$
$503$	20.4975	0.913938	0.456969	+	0.889483i	$0.348935\pi$
$504$	0	0
$505$	3.24699	0.144489
$506$	0.769027	0.0341874
$507$	0	0
$508$	0.570523	0.0253129
$509$	27.7373	1.22944	0.614718	−	0.788747i	$-0.289270\pi$
$509$	27.7373	1.22944	0.614718	+	0.788747i	$0.289270\pi$
$510$	0	0
$511$	15.2148	0.673064
$512$	15.4876	0.684460
$513$	0	0
$514$	2.90088	0.127952
$515$	12.0535	0.531140
$516$	0	0
$517$	0.446521	0.0196380
$518$	−6.46120	−0.283889
$519$	0	0
$520$	−3.33591	−0.146289
$521$	9.37969	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$521$	9.37969	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$522$	0	0
$523$	−36.4986	−1.59597	−0.797986	−	0.602675i	$-0.794101\pi$
$523$	−36.4986	−1.59597	−0.797986	+	0.602675i	$0.794101\pi$
$524$	0.926881	0.0404910
$525$	0	0
$526$	2.94626	0.128463
$527$	0.498801	0.0217281
$528$	0	0
$529$	−12.4508	−0.541339
$530$	1.89356	0.0822510
$531$	0	0
$532$	24.8868	1.07898
$533$	−25.3363	−1.09744
$534$	0	0
$535$	−13.5208	−0.584553
$536$	6.92124	0.298952
$537$	0	0
$538$	0.240598	0.0103729
$539$	0.616265	0.0265444
$540$	0	0
$541$	−21.2609	−0.914075	−0.457038	−	0.889447i	$-0.651090\pi$
$541$	−21.2609	−0.914075	−0.457038	+	0.889447i	$0.651090\pi$
$542$	3.25992	0.140026
$543$	0	0
$544$	2.47123	0.105953
$545$	−26.8532	−1.15026
$546$	0	0
$547$	14.5000	0.619976	0.309988	−	0.950740i	$-0.399675\pi$
$547$	14.5000	0.619976	0.309988	+	0.950740i	$0.399675\pi$
$548$	−37.4035	−1.59780
$549$	0	0
$550$	0.532191	0.0226927
$551$	−6.42000	−0.273501
$552$	0	0
$553$	−3.65425	−0.155395
$554$	5.25839	0.223407
$555$	0	0
$556$	18.9661	0.804341
$557$	−41.8703	−1.77410	−0.887052	−	0.461670i	$-0.847250\pi$
$557$	−41.8703	−1.77410	−0.887052	+	0.461670i	$0.847250\pi$
$558$	0	0
$559$	−2.39509	−0.101302
$560$	15.7198	0.664282
$561$	0	0
$562$	0.00996259	0.000420247 0
$563$	−30.3605	−1.27954	−0.639772	−	0.768565i	$-0.720971\pi$
$563$	−30.3605	−1.27954	−0.639772	+	0.768565i	$0.720971\pi$
$564$	0	0
$565$	−7.40722	−0.311624
$566$	−3.58773	−0.150803
$567$	0	0
$568$	−6.84735	−0.287308
$569$	1.79244	0.0751431	0.0375716	−	0.999294i	$-0.488038\pi$
$569$	1.79244	0.0751431	0.0375716	+	0.999294i	$0.488038\pi$
$570$	0	0
$571$	18.7085	0.782926	0.391463	−	0.920194i	$-0.371969\pi$
$571$	18.7085	0.782926	0.391463	+	0.920194i	$0.371969\pi$
$572$	5.22258	0.218367
$573$	0	0
$574$	−5.70189	−0.237992
$575$	7.30037	0.304447
$576$	0	0
$577$	0.286316	0.0119195	0.00595975	−	0.999982i	$-0.498103\pi$
$577$	0.286316	0.0119195	0.00595975	+	0.999982i	$0.498103\pi$
$578$	0.212277	0.00882955
$579$	0	0
$580$	−4.15314	−0.172450
$581$	2.77548	0.115146
$582$	0	0
$583$	−5.99731	−0.248383
$584$	−5.03053	−0.208165
$585$	0	0
$586$	−6.54568	−0.270399
$587$	23.7797	0.981495	0.490748	−	0.871302i	$-0.336724\pi$
$587$	23.7797	0.981495	0.490748	+	0.871302i	$0.336724\pi$
$588$	0	0
$589$	−2.50074	−0.103041
$590$	−1.83589	−0.0755822
$591$	0	0
$592$	−44.7319	−1.83847
$593$	−16.1975	−0.665152	−0.332576	−	0.943076i	$-0.607918\pi$
$593$	−16.1975	−0.665152	−0.332576	+	0.943076i	$0.607918\pi$
$594$	0	0
$595$	4.21254	0.172697
$596$	−39.5796	−1.62124
$597$	0	0
$598$	−1.65133	−0.0675281
$599$	−8.90490	−0.363844	−0.181922	−	0.983313i	$-0.558232\pi$
$599$	−8.90490	−0.363844	−0.181922	+	0.983313i	$0.558232\pi$
$600$	0	0
$601$	−5.00807	−0.204283	−0.102142	−	0.994770i	$-0.532569\pi$
$601$	−5.00807	−0.204283	−0.102142	+	0.994770i	$0.532569\pi$
$602$	−0.539012	−0.0219685
$603$	0	0
$604$	10.8028	0.439561
$605$	−16.1851	−0.658019
$606$	0	0
$607$	35.2057	1.42895	0.714477	−	0.699659i	$-0.246665\pi$
$607$	35.2057	1.42895	0.714477	+	0.699659i	$0.246665\pi$
$608$	−12.3895	−0.502461
$609$	0	0
$610$	0.00460203	0.000186331 0
$611$	−0.958816	−0.0387895
$612$	0	0
$613$	23.4816	0.948411	0.474205	−	0.880414i	$-0.342735\pi$
$613$	23.4816	0.948411	0.474205	+	0.880414i	$0.342735\pi$
$614$	0.652019	0.0263134
$615$	0	0
$616$	2.37776	0.0958025
$617$	−8.17066	−0.328939	−0.164469	−	0.986382i	$-0.552591\pi$
$617$	−8.17066	−0.328939	−0.164469	+	0.986382i	$0.552591\pi$
$618$	0	0
$619$	19.2859	0.775166	0.387583	−	0.921835i	$-0.373310\pi$
$619$	19.2859	0.775166	0.387583	+	0.921835i	$0.373310\pi$
$620$	−1.61774	−0.0649701
$621$	0	0
$622$	6.84342	0.274396
$623$	27.2148	1.09034
$624$	0	0
$625$	−8.70948	−0.348379
$626$	0.149937	0.00599269
$627$	0	0
$628$	16.9205	0.675203
$629$	−11.9871	−0.477958
$630$	0	0
$631$	18.3501	0.730508	0.365254	−	0.930908i	$-0.380982\pi$
$631$	18.3501	0.730508	0.365254	+	0.930908i	$0.380982\pi$
$632$	1.20822	0.0480603
$633$	0	0
$634$	−2.51770	−0.0999905
$635$	−0.484160	−0.0192133
$636$	0	0
$637$	−1.32331	−0.0524314
$638$	−0.303198	−0.0120037
$639$	0	0
$640$	−10.6432	−0.420709
$641$	−40.9509	−1.61746	−0.808732	−	0.588177i	$-0.799846\pi$
$641$	−40.9509	−1.61746	−0.808732	+	0.588177i	$0.799846\pi$
$642$	0	0
$643$	−29.1707	−1.15038	−0.575191	−	0.818019i	$-0.695072\pi$
$643$	−29.1707	−1.15038	−0.575191	+	0.818019i	$0.695072\pi$
$644$	16.1227	0.635324
$645$	0	0
$646$	−1.06425	−0.0418723
$647$	−33.5423	−1.31868	−0.659341	−	0.751844i	$-0.729165\pi$
$647$	−33.5423	−1.31868	−0.659341	+	0.751844i	$0.729165\pi$
$648$	0	0
$649$	5.81465	0.228245
$650$	−1.14278	−0.0448233
$651$	0	0
$652$	−37.7151	−1.47704
$653$	1.36258	0.0533218	0.0266609	−	0.999645i	$-0.491513\pi$
$653$	1.36258	0.0533218	0.0266609	+	0.999645i	$0.491513\pi$
$654$	0	0
$655$	−0.786575	−0.0307340
$656$	−39.4751	−1.54124
$657$	0	0
$658$	−0.215780	−0.00841197
$659$	3.00400	0.117019	0.0585097	−	0.998287i	$-0.481365\pi$
$659$	3.00400	0.117019	0.0585097	+	0.998287i	$0.481365\pi$
$660$	0	0
$661$	−45.7661	−1.78010	−0.890048	−	0.455868i	$-0.849329\pi$
$661$	−45.7661	−1.78010	−0.890048	+	0.455868i	$0.849329\pi$
$662$	−5.25525	−0.204251
$663$	0	0
$664$	−0.917666	−0.0356124
$665$	−21.1196	−0.818982
$666$	0	0
$667$	−4.15915	−0.161043
$668$	4.43082	0.171434
$669$	0	0
$670$	−2.90331	−0.112165
$671$	−0.0145756	−0.000562686 0
$672$	0	0
$673$	−18.0592	−0.696130	−0.348065	−	0.937470i	$-0.613161\pi$
$673$	−18.0592	−0.696130	−0.348065	+	0.937470i	$0.613161\pi$
$674$	2.81860	0.108568
$675$	0	0
$676$	14.1997	0.546144
$677$	−21.1629	−0.813355	−0.406677	−	0.913572i	$-0.633313\pi$
$677$	−21.1629	−0.813355	−0.406677	+	0.913572i	$0.633313\pi$
$678$	0	0
$679$	−15.7080	−0.602817
$680$	−1.39281	−0.0534118
$681$	0	0
$682$	−0.118103	−0.00452238
$683$	−5.23314	−0.200241	−0.100120	−	0.994975i	$-0.531923\pi$
$683$	−5.23314	−0.200241	−0.100120	+	0.994975i	$0.531923\pi$
$684$	0	0
$685$	31.7416	1.21278
$686$	−4.07089	−0.155427
$687$	0	0
$688$	−3.73166	−0.142268
$689$	12.8781	0.490615
$690$	0	0
$691$	−18.4677	−0.702543	−0.351272	−	0.936274i	$-0.614251\pi$
$691$	−18.4677	−0.702543	−0.351272	+	0.936274i	$0.614251\pi$
$692$	−5.45028	−0.207188
$693$	0	0
$694$	−0.188689	−0.00716253
$695$	−16.0951	−0.610522
$696$	0	0
$697$	−10.5784	−0.400686
$698$	5.72665	0.216757
$699$	0	0
$700$	11.1574	0.421711
$701$	31.0116	1.17129	0.585645	−	0.810567i	$-0.300841\pi$
$701$	31.0116	1.17129	0.585645	+	0.810567i	$0.300841\pi$
$702$	0	0
$703$	60.0974	2.26662
$704$	7.73945	0.291691
$705$	0	0
$706$	−4.77980	−0.179890
$707$	4.96967	0.186904
$708$	0	0
$709$	39.7743	1.49376	0.746878	−	0.664961i	$-0.231552\pi$
$709$	39.7743	1.49376	0.746878	+	0.664961i	$0.231552\pi$
$710$	2.87231	0.107796
$711$	0	0
$712$	−8.99814	−0.337220
$713$	−1.62008	−0.0606725
$714$	0	0
$715$	−4.43201	−0.165748
$716$	−48.5730	−1.81526
$717$	0	0
$718$	−4.15897	−0.155211
$719$	−27.8343	−1.03804	−0.519022	−	0.854761i	$-0.673704\pi$
$719$	−27.8343	−1.03804	−0.519022	+	0.854761i	$0.673704\pi$
$720$	0	0
$721$	18.4484	0.687055
$722$	1.30235	0.0484684
$723$	0	0
$724$	3.34371	0.124268
$725$	−2.87826	−0.106896
$726$	0	0
$727$	20.7165	0.768332	0.384166	−	0.923264i	$-0.374489\pi$
$727$	20.7165	0.768332	0.384166	+	0.923264i	$0.374489\pi$
$728$	−5.10576	−0.189232
$729$	0	0
$730$	2.11020	0.0781019
$731$	−1.00000	−0.0369863
$732$	0	0
$733$	33.5896	1.24066	0.620330	−	0.784341i	$-0.286999\pi$
$733$	33.5896	1.24066	0.620330	+	0.784341i	$0.286999\pi$
$734$	−1.75378	−0.0647330
$735$	0	0
$736$	−8.02644	−0.295858
$737$	9.19540	0.338717
$738$	0	0
$739$	24.8390	0.913716	0.456858	−	0.889540i	$-0.348975\pi$
$739$	24.8390	0.913716	0.456858	+	0.889540i	$0.348975\pi$
$740$	38.8774	1.42916
$741$	0	0
$742$	2.89818	0.106396
$743$	−3.14892	−0.115523	−0.0577614	−	0.998330i	$-0.518396\pi$
$743$	−3.14892	−0.115523	−0.0577614	+	0.998330i	$0.518396\pi$
$744$	0	0
$745$	33.5883	1.23058
$746$	−4.03194	−0.147620
$747$	0	0
$748$	2.18053	0.0797281
$749$	−20.6941	−0.756147
$750$	0	0
$751$	50.6694	1.84895	0.924476	−	0.381240i	$-0.124503\pi$
$751$	50.6694	1.84895	0.924476	+	0.381240i	$0.124503\pi$
$752$	−1.49388	−0.0544761
$753$	0	0
$754$	0.651058	0.0237101
$755$	−9.16755	−0.333641
$756$	0	0
$757$	−33.9812	−1.23507	−0.617534	−	0.786544i	$-0.711868\pi$
$757$	−33.9812	−1.23507	−0.617534	+	0.786544i	$0.711868\pi$
$758$	4.79921	0.174315
$759$	0	0
$760$	6.98284	0.253294
$761$	−20.2111	−0.732652	−0.366326	−	0.930487i	$-0.619384\pi$
$761$	−20.2111	−0.732652	−0.366326	+	0.930487i	$0.619384\pi$
$762$	0	0
$763$	−41.1000	−1.48792
$764$	40.1491	1.45254
$765$	0	0
$766$	3.47225	0.125458
$767$	−12.4858	−0.450836
$768$	0	0
$769$	54.2117	1.95492	0.977461	−	0.211114i	$-0.0677091\pi$
$769$	54.2117	1.95492	0.977461	+	0.211114i	$0.0677091\pi$
$770$	−0.997416	−0.0359444
$771$	0	0
$772$	25.7223	0.925766
$773$	6.03236	0.216969	0.108485	−	0.994098i	$-0.465400\pi$
$773$	6.03236	0.216969	0.108485	+	0.994098i	$0.465400\pi$
$774$	0	0
$775$	−1.12115	−0.0402728
$776$	5.19359	0.186439
$777$	0	0
$778$	6.69507	0.240030
$779$	53.0349	1.90017
$780$	0	0
$781$	−9.09724	−0.325525
$782$	−0.689465	−0.0246552
$783$	0	0
$784$	−2.06177	−0.0736347
$785$	−14.3592	−0.512501
$786$	0	0
$787$	−42.1924	−1.50400	−0.751998	−	0.659165i	$-0.770910\pi$
$787$	−42.1924	−1.50400	−0.751998	+	0.659165i	$0.770910\pi$
$788$	30.1579	1.07433
$789$	0	0
$790$	−0.506821	−0.0180319
$791$	−11.3371	−0.403100
$792$	0	0
$793$	0.0312983	0.00111143
$794$	−4.80242	−0.170431
$795$	0	0
$796$	−18.9499	−0.671662
$797$	−7.85342	−0.278182	−0.139091	−	0.990280i	$-0.544418\pi$
$797$	−7.85342	−0.278182	−0.139091	+	0.990280i	$0.544418\pi$
$798$	0	0
$799$	−0.400325	−0.0141625
$800$	−5.55455	−0.196383
$801$	0	0
$802$	−3.65234	−0.128969
$803$	−6.68345	−0.235854
$804$	0	0
$805$	−13.6821	−0.482232
$806$	0.253602	0.00893274
$807$	0	0
$808$	−1.64314	−0.0578054
$809$	−33.1598	−1.16584	−0.582919	−	0.812531i	$-0.698089\pi$
$809$	−33.1598	−1.16584	−0.582919	+	0.812531i	$0.698089\pi$
$810$	0	0
$811$	−17.7207	−0.622259	−0.311130	−	0.950367i	$-0.600707\pi$
$811$	−17.7207	−0.622259	−0.311130	+	0.950367i	$0.600707\pi$
$812$	−6.35657	−0.223072
$813$	0	0
$814$	2.83823	0.0994798
$815$	32.0059	1.12112
$816$	0	0
$817$	5.01350	0.175400
$818$	−0.197081	−0.00689076
$819$	0	0
$820$	34.3086	1.19811
$821$	46.1710	1.61138	0.805689	−	0.592339i	$-0.201795\pi$
$821$	46.1710	1.61138	0.805689	+	0.592339i	$0.201795\pi$
$822$	0	0
$823$	29.8326	1.03990	0.519949	−	0.854197i	$-0.325951\pi$
$823$	29.8326	1.03990	0.519949	+	0.854197i	$0.325951\pi$
$824$	−6.09967	−0.212492
$825$	0	0
$826$	−2.80991	−0.0977692
$827$	32.2143	1.12020	0.560101	−	0.828424i	$-0.310762\pi$
$827$	32.2143	1.12020	0.560101	+	0.828424i	$0.310762\pi$
$828$	0	0
$829$	29.4265	1.02203	0.511013	−	0.859573i	$-0.329270\pi$
$829$	29.4265	1.02203	0.511013	+	0.859573i	$0.329270\pi$
$830$	0.384941	0.0133615
$831$	0	0
$832$	−16.6189	−0.576158
$833$	−0.552508	−0.0191433
$834$	0	0
$835$	−3.76011	−0.130124
$836$	−10.9321	−0.378094
$837$	0	0
$838$	4.37389	0.151094
$839$	35.6961	1.23237	0.616183	−	0.787603i	$-0.288678\pi$
$839$	35.6961	1.23237	0.616183	+	0.787603i	$0.288678\pi$
$840$	0	0
$841$	−27.3602	−0.943455
$842$	−5.43804	−0.187407
$843$	0	0
$844$	41.0193	1.41194
$845$	−12.0503	−0.414542
$846$	0	0
$847$	−24.7721	−0.851178
$848$	20.0646	0.689020
$849$	0	0
$850$	−0.477132	−0.0163655
$851$	38.9336	1.33463
$852$	0	0
$853$	44.7203	1.53120	0.765598	−	0.643320i	$-0.222443\pi$
$853$	44.7203	1.53120	0.765598	+	0.643320i	$0.222443\pi$
$854$	0.00704362	0.000241028 0
$855$	0	0
$856$	6.84218	0.233861
$857$	45.1768	1.54321	0.771605	−	0.636102i	$-0.219454\pi$
$857$	45.1768	1.54321	0.771605	+	0.636102i	$0.219454\pi$
$858$	0	0
$859$	39.5488	1.34939	0.674694	−	0.738098i	$-0.264276\pi$
$859$	39.5488	1.34939	0.674694	+	0.738098i	$0.264276\pi$
$860$	3.24326	0.110594
$861$	0	0
$862$	−6.63912	−0.226129
$863$	47.9021	1.63061	0.815303	−	0.579034i	$-0.196570\pi$
$863$	47.9021	1.63061	0.815303	+	0.579034i	$0.196570\pi$
$864$	0	0
$865$	4.62524	0.157263
$866$	−6.39170	−0.217199
$867$	0	0
$868$	−2.47603	−0.0840419
$869$	1.60521	0.0544531
$870$	0	0
$871$	−19.7453	−0.669044
$872$	13.5890	0.460183
$873$	0	0
$874$	3.45663	0.116922
$875$	−30.5312	−1.03214
$876$	0	0
$877$	33.5981	1.13453	0.567264	−	0.823536i	$-0.308002\pi$
$877$	33.5981	1.13453	0.567264	+	0.823536i	$0.308002\pi$
$878$	1.59626	0.0538710
$879$	0	0
$880$	−6.90527	−0.232777
$881$	−5.00316	−0.168561	−0.0842804	−	0.996442i	$-0.526859\pi$
$881$	−5.00316	−0.168561	−0.0842804	+	0.996442i	$0.526859\pi$
$882$	0	0
$883$	−18.9288	−0.637005	−0.318503	−	0.947922i	$-0.603180\pi$
$883$	−18.9288	−0.637005	−0.318503	+	0.947922i	$0.603180\pi$
$884$	−4.68226	−0.157482
$885$	0	0
$886$	1.02658	0.0344885
$887$	−22.5167	−0.756037	−0.378019	−	0.925798i	$-0.623394\pi$
$887$	−22.5167	−0.756037	−0.378019	+	0.925798i	$0.623394\pi$
$888$	0	0
$889$	−0.741029	−0.0248533
$890$	3.77452	0.126522
$891$	0	0
$892$	−28.2916	−0.947274
$893$	2.00703	0.0671626
$894$	0	0
$895$	41.2202	1.37784
$896$	−16.2899	−0.544208
$897$	0	0
$898$	−5.12388	−0.170986
$899$	0.638737	0.0213031
$900$	0	0
$901$	5.37685	0.179129
$902$	2.50468	0.0833968
$903$	0	0
$904$	3.74842	0.124671
$905$	−2.83756	−0.0943236
$906$	0	0
$907$	−19.7476	−0.655709	−0.327855	−	0.944728i	$-0.606326\pi$
$907$	−19.7476	−0.655709	−0.327855	+	0.944728i	$0.606326\pi$
$908$	37.9785	1.26036
$909$	0	0
$910$	2.14175	0.0709985
$911$	45.3377	1.50210	0.751052	−	0.660243i	$-0.229547\pi$
$911$	45.3377	1.50210	0.751052	+	0.660243i	$0.229547\pi$
$912$	0	0
$913$	−1.21919	−0.0403493
$914$	−0.0538517	−0.00178126
$915$	0	0
$916$	33.5219	1.10759
$917$	−1.20389	−0.0397559
$918$	0	0
$919$	−51.7641	−1.70754	−0.853771	−	0.520649i	$-0.825690\pi$
$919$	−51.7641	−1.70754	−0.853771	+	0.520649i	$0.825690\pi$
$920$	4.52378	0.149144
$921$	0	0
$922$	5.17197	0.170330
$923$	19.5345	0.642987
$924$	0	0
$925$	26.9433	0.885890
$926$	−0.679783	−0.0223391
$927$	0	0
$928$	3.16452	0.103880
$929$	55.7602	1.82943	0.914716	−	0.404098i	$-0.132415\pi$
$929$	55.7602	1.82943	0.914716	+	0.404098i	$0.132415\pi$
$930$	0	0
$931$	2.76999	0.0907829
$932$	−2.49338	−0.0816734
$933$	0	0
$934$	−0.756828	−0.0247642
$935$	−1.85045	−0.0605163
$936$	0	0
$937$	−3.30658	−0.108021	−0.0540106	−	0.998540i	$-0.517200\pi$
$937$	−3.30658	−0.108021	−0.0540106	+	0.998540i	$0.517200\pi$
$938$	−4.44364	−0.145090
$939$	0	0
$940$	1.29836	0.0423478
$941$	13.2953	0.433415	0.216708	−	0.976237i	$-0.430468\pi$
$941$	13.2953	0.433415	0.216708	+	0.976237i	$0.430468\pi$
$942$	0	0
$943$	34.3582	1.11886
$944$	−19.4534	−0.633155
$945$	0	0
$946$	0.236773	0.00769815
$947$	7.42234	0.241194	0.120597	−	0.992702i	$-0.461519\pi$
$947$	7.42234	0.241194	0.120597	+	0.992702i	$0.461519\pi$
$948$	0	0
$949$	14.3514	0.465866
$950$	2.39210	0.0776099
$951$	0	0
$952$	−2.13176	−0.0690907
$953$	13.2058	0.427777	0.213888	−	0.976858i	$-0.431387\pi$
$953$	13.2058	0.427777	0.213888	+	0.976858i	$0.431387\pi$
$954$	0	0
$955$	−34.0715	−1.10253
$956$	−19.3782	−0.626737
$957$	0	0
$958$	1.44510	0.0466891
$959$	48.5819	1.56879
$960$	0	0
$961$	−30.7512	−0.991974
$962$	−6.09453	−0.196496
$963$	0	0
$964$	57.6138	1.85561
$965$	−21.8286	−0.702688
$966$	0	0
$967$	−16.1744	−0.520135	−0.260067	−	0.965591i	$-0.583745\pi$
$967$	−16.1744	−0.520135	−0.260067	+	0.965591i	$0.583745\pi$
$968$	8.19048	0.263252
$969$	0	0
$970$	−2.17860	−0.0699505
$971$	−51.3760	−1.64873	−0.824367	−	0.566056i	$-0.808469\pi$
$971$	−51.3760	−1.64873	−0.824367	+	0.566056i	$0.808469\pi$
$972$	0	0
$973$	−24.6343	−0.789739
$974$	0.467367	0.0149754
$975$	0	0
$976$	0.0487641	0.00156090
$977$	−9.39621	−0.300611	−0.150306	−	0.988640i	$-0.548026\pi$
$977$	−9.39621	−0.300611	−0.150306	+	0.988640i	$0.548026\pi$
$978$	0	0
$979$	−11.9547	−0.382075
$980$	1.79193	0.0572410
$981$	0	0
$982$	−5.30438	−0.169270
$983$	17.8674	0.569883	0.284942	−	0.958545i	$-0.408026\pi$
$983$	17.8674	0.569883	0.284942	+	0.958545i	$0.408026\pi$
$984$	0	0
$985$	−25.5928	−0.815453
$986$	0.271830	0.00865683
$987$	0	0
$988$	23.4745	0.746823
$989$	3.24795	0.103279
$990$	0	0
$991$	−29.1968	−0.927467	−0.463733	−	0.885975i	$-0.653490\pi$
$991$	−29.1968	−0.927467	−0.463733	+	0.885975i	$0.653490\pi$
$992$	1.23265	0.0391367
$993$	0	0
$994$	4.39621	0.139439
$995$	16.0814	0.509814
$996$	0	0
$997$	−40.5691	−1.28484	−0.642419	−	0.766354i	$-0.722069\pi$
$997$	−40.5691	−1.28484	−0.642419	+	0.766354i	$0.722069\pi$
$998$	1.07058	0.0338886
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6579.2.a.j.1.4		6
3.2	odd	2		731.2.a.c.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.a.c.1.3	✓	6	3.2	odd	2
6579.2.a.j.1.4		6	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:	\( N \)	\(=\)	\( 6579 = 3^{2} \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6579.a (trivial)

\(f(q)\)	\(=\)	\(q+0.212277 q^{2} -1.95494 q^{4} +1.65901 q^{5} +2.53919 q^{7} -0.839542 q^{8} +O(q^{10})\)	\(q+0.212277 q^{2} -1.95494 q^{4} +1.65901 q^{5} +2.53919 q^{7} -0.839542 q^{8} +0.352169 q^{10} -1.11540 q^{11} +2.39509 q^{13} +0.539012 q^{14} +3.73166 q^{16} +1.00000 q^{17} -5.01350 q^{19} -3.24326 q^{20} -0.236773 q^{22} -3.24795 q^{23} -2.24769 q^{25} +0.508423 q^{26} -4.96396 q^{28} +1.28054 q^{29} +0.498801 q^{31} +2.47123 q^{32} +0.212277 q^{34} +4.21254 q^{35} -11.9871 q^{37} -1.06425 q^{38} -1.39281 q^{40} -10.5784 q^{41} -1.00000 q^{43} +2.18053 q^{44} -0.689465 q^{46} -0.400325 q^{47} -0.552508 q^{49} -0.477132 q^{50} -4.68226 q^{52} +5.37685 q^{53} -1.85045 q^{55} -2.13176 q^{56} +0.271830 q^{58} -5.21308 q^{59} +0.0130677 q^{61} +0.105884 q^{62} -6.93874 q^{64} +3.97349 q^{65} -8.24406 q^{67} -1.95494 q^{68} +0.894226 q^{70} +8.15605 q^{71} +5.99199 q^{73} -2.54459 q^{74} +9.80108 q^{76} -2.83221 q^{77} -1.43914 q^{79} +6.19086 q^{80} -2.24555 q^{82} +1.09306 q^{83} +1.65901 q^{85} -0.212277 q^{86} +0.936422 q^{88} +10.7179 q^{89} +6.08160 q^{91} +6.34955 q^{92} -0.0849797 q^{94} -8.31744 q^{95} -6.18621 q^{97} -0.117285 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 5 q^{4} - 3 q^{5} - 7 q^{7} + 9 q^{8}+O(q^{10}) \) 6 * q + q^2 + 5 * q^4 - 3 * q^5 - 7 * q^7 + 9 * q^8	\( 6 q + q^{2} + 5 q^{4} - 3 q^{5} - 7 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 10 q^{13} + 7 q^{14} - q^{16} + 6 q^{17} - 20 q^{19} - q^{20} + 2 q^{22} + 3 q^{23} - 7 q^{25} + 3 q^{26} - 11 q^{28} + 15 q^{29} + 12 q^{31} - q^{32} + q^{34} + 9 q^{35} - 14 q^{37} - 27 q^{38} - 7 q^{40} + 2 q^{41} - 6 q^{43} + 12 q^{44} + 14 q^{46} + 11 q^{47} + 3 q^{49} - 7 q^{50} - 5 q^{52} - 3 q^{53} + 6 q^{55} - 22 q^{56} - 21 q^{58} - 2 q^{59} - 20 q^{61} - 3 q^{62} - 39 q^{64} + 34 q^{65} - 2 q^{67} + 5 q^{68} - q^{70} - q^{71} + 13 q^{73} - 28 q^{74} - 29 q^{76} + 11 q^{77} - 26 q^{79} - 12 q^{80} - 9 q^{82} - 10 q^{83} - 3 q^{85} - q^{86} - 6 q^{88} + 15 q^{89} + 8 q^{91} + 9 q^{92} - 33 q^{94} + 21 q^{95} - 22 q^{97} + 3 q^{98}+O(q^{100}) \) 6 * q + q^2 + 5 * q^4 - 3 * q^5 - 7 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 10 * q^13 + 7 * q^14 - q^16 + 6 * q^17 - 20 * q^19 - q^20 + 2 * q^22 + 3 * q^23 - 7 * q^25 + 3 * q^26 - 11 * q^28 + 15 * q^29 + 12 * q^31 - q^32 + q^34 + 9 * q^35 - 14 * q^37 - 27 * q^38 - 7 * q^40 + 2 * q^41 - 6 * q^43 + 12 * q^44 + 14 * q^46 + 11 * q^47 + 3 * q^49 - 7 * q^50 - 5 * q^52 - 3 * q^53 + 6 * q^55 - 22 * q^56 - 21 * q^58 - 2 * q^59 - 20 * q^61 - 3 * q^62 - 39 * q^64 + 34 * q^65 - 2 * q^67 + 5 * q^68 - q^70 - q^71 + 13 * q^73 - 28 * q^74 - 29 * q^76 + 11 * q^77 - 26 * q^79 - 12 * q^80 - 9 * q^82 - 10 * q^83 - 3 * q^85 - q^86 - 6 * q^88 + 15 * q^89 + 8 * q^91 + 9 * q^92 - 33 * q^94 + 21 * q^95 - 22 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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