LMFDB - Embedded newform 8001.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.45188$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.45188 q^{2} +0.107956 q^{4} -0.584615 q^{5} -1.00000 q^{7} +2.74702 q^{8} +O(q^{10})$	\(q-1.45188 q^{2} +0.107956 q^{4} -0.584615 q^{5} -1.00000 q^{7} +2.74702 q^{8} +0.848791 q^{10} +5.41545 q^{11} -3.18339 q^{13} +1.45188 q^{14} -4.20426 q^{16} -5.06086 q^{17} -5.11413 q^{19} -0.0631129 q^{20} -7.86259 q^{22} -3.75579 q^{23} -4.65823 q^{25} +4.62190 q^{26} -0.107956 q^{28} -8.71861 q^{29} +3.51720 q^{31} +0.610037 q^{32} +7.34776 q^{34} +0.584615 q^{35} -2.23456 q^{37} +7.42511 q^{38} -1.60595 q^{40} +6.65414 q^{41} -4.51015 q^{43} +0.584632 q^{44} +5.45296 q^{46} -9.14467 q^{47} +1.00000 q^{49} +6.76319 q^{50} -0.343667 q^{52} -2.10337 q^{53} -3.16596 q^{55} -2.74702 q^{56} +12.6584 q^{58} +5.83902 q^{59} +14.0585 q^{61} -5.10655 q^{62} +7.52281 q^{64} +1.86106 q^{65} -3.86784 q^{67} -0.546352 q^{68} -0.848791 q^{70} +12.1700 q^{71} -11.2754 q^{73} +3.24431 q^{74} -0.552103 q^{76} -5.41545 q^{77} -3.62482 q^{79} +2.45787 q^{80} -9.66101 q^{82} +14.1521 q^{83} +2.95866 q^{85} +6.54820 q^{86} +14.8764 q^{88} +6.96813 q^{89} +3.18339 q^{91} -0.405461 q^{92} +13.2770 q^{94} +2.98980 q^{95} -15.9899 q^{97} -1.45188 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * 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 $

$2$	−1.45188	−1.02663	−0.513317	−	0.858199i	$-0.671584\pi$
$2$	−1.45188	−1.02663	−0.513317	+	0.858199i	$0.671584\pi$
$3$	0	0
$3$	0	0
$4$	0.107956	0.0539781
$5$	−0.584615	−0.261448	−0.130724	−	0.991419i	$-0.541730\pi$
$5$	−0.584615	−0.261448	−0.130724	+	0.991419i	$0.541730\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.74702	0.971219
$9$	0	0
$10$	0.848791	0.268411
$11$	5.41545	1.63282	0.816410	−	0.577472i	$-0.195961\pi$
$11$	5.41545	1.63282	0.816410	+	0.577472i	$0.195961\pi$
$12$	0	0
$13$	−3.18339	−0.882913	−0.441457	−	0.897283i	$-0.645538\pi$
$13$	−3.18339	−0.882913	−0.441457	+	0.897283i	$0.645538\pi$
$14$	1.45188	0.388031
$15$	0	0
$16$	−4.20426	−1.05106
$17$	−5.06086	−1.22744	−0.613720	−	0.789524i	$-0.710327\pi$
$17$	−5.06086	−1.22744	−0.613720	+	0.789524i	$0.710327\pi$
$18$	0	0
$19$	−5.11413	−1.17326	−0.586631	−	0.809854i	$-0.699546\pi$
$19$	−5.11413	−1.17326	−0.586631	+	0.809854i	$0.699546\pi$
$20$	−0.0631129	−0.0141125
$21$	0	0
$22$	−7.86259	−1.67631
$23$	−3.75579	−0.783137	−0.391568	−	0.920149i	$-0.628067\pi$
$23$	−3.75579	−0.783137	−0.391568	+	0.920149i	$0.628067\pi$
$24$	0	0
$25$	−4.65823	−0.931645
$26$	4.62190	0.906429
$27$	0	0
$28$	−0.107956	−0.0204018
$29$	−8.71861	−1.61901	−0.809503	−	0.587116i	$-0.800263\pi$
$29$	−8.71861	−1.61901	−0.809503	+	0.587116i	$0.800263\pi$
$30$	0	0
$31$	3.51720	0.631707	0.315854	−	0.948808i	$-0.397709\pi$
$31$	3.51720	0.631707	0.315854	+	0.948808i	$0.397709\pi$
$32$	0.610037	0.107840
$33$	0	0
$34$	7.34776	1.26013
$35$	0.584615	0.0988180
$36$	0	0
$37$	−2.23456	−0.367359	−0.183680	−	0.982986i	$-0.558801\pi$
$37$	−2.23456	−0.367359	−0.183680	+	0.982986i	$0.558801\pi$
$38$	7.42511	1.20451
$39$	0	0
$40$	−1.60595	−0.253923
$41$	6.65414	1.03920	0.519601	−	0.854409i	$-0.326081\pi$
$41$	6.65414	1.03920	0.519601	+	0.854409i	$0.326081\pi$
$42$	0	0
$43$	−4.51015	−0.687791	−0.343896	−	0.939008i	$-0.611747\pi$
$43$	−4.51015	−0.687791	−0.343896	+	0.939008i	$0.611747\pi$
$44$	0.584632	0.0881366
$45$	0	0
$46$	5.45296	0.803995
$47$	−9.14467	−1.33389	−0.666944	−	0.745108i	$-0.732398\pi$
$47$	−9.14467	−1.33389	−0.666944	+	0.745108i	$0.732398\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.76319	0.956459
$51$	0	0
$52$	−0.343667	−0.0476580
$53$	−2.10337	−0.288920	−0.144460	−	0.989511i	$-0.546145\pi$
$53$	−2.10337	−0.288920	−0.144460	+	0.989511i	$0.546145\pi$
$54$	0	0
$55$	−3.16596	−0.426897
$56$	−2.74702	−0.367086
$57$	0	0
$58$	12.6584	1.66213
$59$	5.83902	0.760175	0.380088	−	0.924950i	$-0.375894\pi$
$59$	5.83902	0.760175	0.380088	+	0.924950i	$0.375894\pi$
$60$	0	0
$61$	14.0585	1.80001	0.900005	−	0.435879i	$-0.143562\pi$
$61$	14.0585	1.80001	0.900005	+	0.435879i	$0.143562\pi$
$62$	−5.10655	−0.648533
$63$	0	0
$64$	7.52281	0.940352
$65$	1.86106	0.230836
$66$	0	0
$67$	−3.86784	−0.472532	−0.236266	−	0.971688i	$-0.575924\pi$
$67$	−3.86784	−0.472532	−0.236266	+	0.971688i	$0.575924\pi$
$68$	−0.546352	−0.0662549
$69$	0	0
$70$	−0.848791	−0.101450
$71$	12.1700	1.44431	0.722156	−	0.691731i	$-0.243151\pi$
$71$	12.1700	1.44431	0.722156	+	0.691731i	$0.243151\pi$
$72$	0	0
$73$	−11.2754	−1.31969	−0.659844	−	0.751402i	$-0.729378\pi$
$73$	−11.2754	−1.31969	−0.659844	+	0.751402i	$0.729378\pi$
$74$	3.24431	0.377143
$75$	0	0
$76$	−0.552103	−0.0633305
$77$	−5.41545	−0.617148
$78$	0	0
$79$	−3.62482	−0.407824	−0.203912	−	0.978989i	$-0.565366\pi$
$79$	−3.62482	−0.407824	−0.203912	+	0.978989i	$0.565366\pi$
$80$	2.45787	0.274799
$81$	0	0
$82$	−9.66101	−1.06688
$83$	14.1521	1.55339	0.776695	−	0.629877i	$-0.216895\pi$
$83$	14.1521	1.55339	0.776695	+	0.629877i	$0.216895\pi$
$84$	0	0
$85$	2.95866	0.320911
$86$	6.54820	0.706110
$87$	0	0
$88$	14.8764	1.58583
$89$	6.96813	0.738621	0.369310	−	0.929306i	$-0.379594\pi$
$89$	6.96813	0.738621	0.369310	+	0.929306i	$0.379594\pi$
$90$	0	0
$91$	3.18339	0.333710
$92$	−0.405461	−0.0422723
$93$	0	0
$94$	13.2770	1.36941
$95$	2.98980	0.306747
$96$	0	0
$97$	−15.9899	−1.62353	−0.811764	−	0.583985i	$-0.801493\pi$
$97$	−15.9899	−1.62353	−0.811764	+	0.583985i	$0.801493\pi$
$98$	−1.45188	−0.146662
$99$	0	0
$100$	−0.502885	−0.0502885
$101$	10.4766	1.04246	0.521231	−	0.853416i	$-0.325473\pi$
$101$	10.4766	1.04246	0.521231	+	0.853416i	$0.325473\pi$
$102$	0	0
$103$	−16.7816	−1.65354	−0.826772	−	0.562537i	$-0.809826\pi$
$103$	−16.7816	−1.65354	−0.826772	+	0.562537i	$0.809826\pi$
$104$	−8.74483	−0.857502
$105$	0	0
$106$	3.05384	0.296616
$107$	−5.72707	−0.553657	−0.276828	−	0.960919i	$-0.589283\pi$
$107$	−5.72707	−0.553657	−0.276828	+	0.960919i	$0.589283\pi$
$108$	0	0
$109$	17.7933	1.70429	0.852143	−	0.523309i	$-0.175303\pi$
$109$	17.7933	1.70429	0.852143	+	0.523309i	$0.175303\pi$
$110$	4.59659	0.438267
$111$	0	0
$112$	4.20426	0.397265
$113$	−8.78023	−0.825974	−0.412987	−	0.910737i	$-0.635515\pi$
$113$	−8.78023	−0.825974	−0.412987	+	0.910737i	$0.635515\pi$
$114$	0	0
$115$	2.19569	0.204749
$116$	−0.941229	−0.0873909
$117$	0	0
$118$	−8.47755	−0.780422
$119$	5.06086	0.463928
$120$	0	0
$121$	18.3271	1.66610
$122$	−20.4113	−1.84795
$123$	0	0
$124$	0.379704	0.0340984
$125$	5.64634	0.505024
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−12.1423	−1.07324
$129$	0	0
$130$	−2.70203	−0.236984
$131$	−1.64272	−0.143525	−0.0717625	−	0.997422i	$-0.522862\pi$
$131$	−1.64272	−0.143525	−0.0717625	+	0.997422i	$0.522862\pi$
$132$	0	0
$133$	5.11413	0.443451
$134$	5.61565	0.485118
$135$	0	0
$136$	−13.9023	−1.19211
$137$	19.4601	1.66259	0.831294	−	0.555833i	$-0.187601\pi$
$137$	19.4601	1.66259	0.831294	+	0.555833i	$0.187601\pi$
$138$	0	0
$139$	−1.07654	−0.0913113	−0.0456556	−	0.998957i	$-0.514538\pi$
$139$	−1.07654	−0.0913113	−0.0456556	+	0.998957i	$0.514538\pi$
$140$	0.0631129	0.00533401
$141$	0	0
$142$	−17.6694	−1.48278
$143$	−17.2395	−1.44164
$144$	0	0
$145$	5.09703	0.423285
$146$	16.3706	1.35484
$147$	0	0
$148$	−0.241235	−0.0198294
$149$	−16.0590	−1.31561	−0.657804	−	0.753189i	$-0.728514\pi$
$149$	−16.0590	−1.31561	−0.657804	+	0.753189i	$0.728514\pi$
$150$	0	0
$151$	−3.56737	−0.290308	−0.145154	−	0.989409i	$-0.546368\pi$
$151$	−3.56737	−0.290308	−0.145154	+	0.989409i	$0.546368\pi$
$152$	−14.0486	−1.13949
$153$	0	0
$154$	7.86259	0.633585
$155$	−2.05621	−0.165159
$156$	0	0
$157$	0.684661	0.0546419	0.0273210	−	0.999627i	$-0.491302\pi$
$157$	0.684661	0.0546419	0.0273210	+	0.999627i	$0.491302\pi$
$158$	5.26281	0.418686
$159$	0	0
$160$	−0.356637	−0.0281946
$161$	3.75579	0.295998
$162$	0	0
$163$	20.5205	1.60729	0.803644	−	0.595110i	$-0.202892\pi$
$163$	20.5205	1.60729	0.803644	+	0.595110i	$0.202892\pi$
$164$	0.718356	0.0560941
$165$	0	0
$166$	−20.5471	−1.59476
$167$	3.47768	0.269111	0.134555	−	0.990906i	$-0.457039\pi$
$167$	3.47768	0.269111	0.134555	+	0.990906i	$0.457039\pi$
$168$	0	0
$169$	−2.86604	−0.220465
$170$	−4.29561	−0.329459
$171$	0	0
$172$	−0.486899	−0.0371257
$173$	−10.7751	−0.819217	−0.409609	−	0.912261i	$-0.634335\pi$
$173$	−10.7751	−0.819217	−0.409609	+	0.912261i	$0.634335\pi$
$174$	0	0
$175$	4.65823	0.352129
$176$	−22.7680	−1.71620
$177$	0	0
$178$	−10.1169	−0.758293
$179$	−19.4287	−1.45217	−0.726085	−	0.687605i	$-0.758662\pi$
$179$	−19.4287	−1.45217	−0.726085	+	0.687605i	$0.758662\pi$
$180$	0	0
$181$	18.3686	1.36533	0.682664	−	0.730732i	$-0.260821\pi$
$181$	18.3686	1.36533	0.682664	+	0.730732i	$0.260821\pi$
$182$	−4.62190	−0.342598
$183$	0	0
$184$	−10.3172	−0.760597
$185$	1.30636	0.0960452
$186$	0	0
$187$	−27.4069	−2.00419
$188$	−0.987225	−0.0720008
$189$	0	0
$190$	−4.34083	−0.314917
$191$	−3.24268	−0.234632	−0.117316	−	0.993095i	$-0.537429\pi$
$191$	−3.24268	−0.234632	−0.117316	+	0.993095i	$0.537429\pi$
$192$	0	0
$193$	−19.0343	−1.37012	−0.685059	−	0.728487i	$-0.740224\pi$
$193$	−19.0343	−1.37012	−0.685059	+	0.728487i	$0.740224\pi$
$194$	23.2154	1.66677
$195$	0	0
$196$	0.107956	0.00771116
$197$	9.92678	0.707254	0.353627	−	0.935387i	$-0.384948\pi$
$197$	9.92678	0.707254	0.353627	+	0.935387i	$0.384948\pi$
$198$	0	0
$199$	8.65223	0.613340	0.306670	−	0.951816i	$-0.400785\pi$
$199$	8.65223	0.613340	0.306670	+	0.951816i	$0.400785\pi$
$200$	−12.7962	−0.904831
$201$	0	0
$202$	−15.2108	−1.07023
$203$	8.71861	0.611927
$204$	0	0
$205$	−3.89011	−0.271697
$206$	24.3649	1.69759
$207$	0	0
$208$	13.3838	0.927999
$209$	−27.6953	−1.91573
$210$	0	0
$211$	−28.4387	−1.95780	−0.978901	−	0.204336i	$-0.934496\pi$
$211$	−28.4387	−1.95780	−0.978901	+	0.204336i	$0.934496\pi$
$212$	−0.227072	−0.0155954
$213$	0	0
$214$	8.31502	0.568403
$215$	2.63670	0.179822
$216$	0	0
$217$	−3.51720	−0.238763
$218$	−25.8337	−1.74968
$219$	0	0
$220$	−0.341785	−0.0230431
$221$	16.1107	1.08372
$222$	0	0
$223$	1.32183	0.0885163	0.0442581	−	0.999020i	$-0.485908\pi$
$223$	1.32183	0.0885163	0.0442581	+	0.999020i	$0.485908\pi$
$224$	−0.610037	−0.0407598
$225$	0	0
$226$	12.7478	0.847974
$227$	−3.52975	−0.234277	−0.117139	−	0.993116i	$-0.537372\pi$
$227$	−3.52975	−0.234277	−0.117139	+	0.993116i	$0.537372\pi$
$228$	0	0
$229$	−13.5473	−0.895230	−0.447615	−	0.894226i	$-0.647726\pi$
$229$	−13.5473	−0.895230	−0.447615	+	0.894226i	$0.647726\pi$
$230$	−3.18788	−0.210203
$231$	0	0
$232$	−23.9502	−1.57241
$233$	9.26564	0.607012	0.303506	−	0.952829i	$-0.401843\pi$
$233$	9.26564	0.607012	0.303506	+	0.952829i	$0.401843\pi$
$234$	0	0
$235$	5.34611	0.348742
$236$	0.630358	0.0410328
$237$	0	0
$238$	−7.34776	−0.476285
$239$	−5.54274	−0.358530	−0.179265	−	0.983801i	$-0.557372\pi$
$239$	−5.54274	−0.358530	−0.179265	+	0.983801i	$0.557372\pi$
$240$	0	0
$241$	18.1165	1.16699	0.583494	−	0.812117i	$-0.301685\pi$
$241$	18.1165	1.16699	0.583494	+	0.812117i	$0.301685\pi$
$242$	−26.6088	−1.71048
$243$	0	0
$244$	1.51771	0.0971612
$245$	−0.584615	−0.0373497
$246$	0	0
$247$	16.2803	1.03589
$248$	9.66182	0.613526
$249$	0	0
$250$	−8.19782	−0.518475
$251$	16.8288	1.06223	0.531113	−	0.847301i	$-0.321774\pi$
$251$	16.8288	1.06223	0.531113	+	0.847301i	$0.321774\pi$
$252$	0	0
$253$	−20.3393	−1.27872
$254$	1.45188	0.0910991
$255$	0	0
$256$	2.58354	0.161471
$257$	0.164440	0.0102575	0.00512876	−	0.999987i	$-0.498367\pi$
$257$	0.164440	0.0102575	0.00512876	+	0.999987i	$0.498367\pi$
$258$	0	0
$259$	2.23456	0.138849
$260$	0.200913	0.0124601
$261$	0	0
$262$	2.38503	0.147348
$263$	12.6341	0.779053	0.389527	−	0.921015i	$-0.372639\pi$
$263$	12.6341	0.779053	0.389527	+	0.921015i	$0.372639\pi$
$264$	0	0
$265$	1.22966	0.0755376
$266$	−7.42511	−0.455263
$267$	0	0
$268$	−0.417558	−0.0255064
$269$	5.50729	0.335785	0.167893	−	0.985805i	$-0.446304\pi$
$269$	5.50729	0.335785	0.167893	+	0.985805i	$0.446304\pi$
$270$	0	0
$271$	−28.0965	−1.70674	−0.853370	−	0.521305i	$-0.825445\pi$
$271$	−28.0965	−1.70674	−0.853370	+	0.521305i	$0.825445\pi$
$272$	21.2772	1.29012
$273$	0	0
$274$	−28.2537	−1.70687
$275$	−25.2264	−1.52121
$276$	0	0
$277$	19.2463	1.15640	0.578198	−	0.815897i	$-0.303756\pi$
$277$	19.2463	1.15640	0.578198	+	0.815897i	$0.303756\pi$
$278$	1.56301	0.0937433
$279$	0	0
$280$	1.60595	0.0959739
$281$	10.0850	0.601623	0.300812	−	0.953684i	$-0.402742\pi$
$281$	10.0850	0.601623	0.300812	+	0.953684i	$0.402742\pi$
$282$	0	0
$283$	28.0175	1.66546	0.832732	−	0.553676i	$-0.186775\pi$
$283$	28.0175	1.66546	0.832732	+	0.553676i	$0.186775\pi$
$284$	1.31383	0.0779612
$285$	0	0
$286$	25.0297	1.48004
$287$	−6.65414	−0.392781
$288$	0	0
$289$	8.61231	0.506607
$290$	−7.40028	−0.434559
$291$	0	0
$292$	−1.21725	−0.0712343
$293$	24.0283	1.40375	0.701875	−	0.712300i	$-0.252346\pi$
$293$	24.0283	1.40375	0.701875	+	0.712300i	$0.252346\pi$
$294$	0	0
$295$	−3.41358	−0.198746
$296$	−6.13838	−0.356786
$297$	0	0
$298$	23.3158	1.35065
$299$	11.9561	0.691442
$300$	0	0
$301$	4.51015	0.259961
$302$	5.17939	0.298041
$303$	0	0
$304$	21.5011	1.23317
$305$	−8.21883	−0.470609
$306$	0	0
$307$	−16.4884	−0.941045	−0.470523	−	0.882388i	$-0.655935\pi$
$307$	−16.4884	−0.941045	−0.470523	+	0.882388i	$0.655935\pi$
$308$	−0.584632	−0.0333125
$309$	0	0
$310$	2.98537	0.169557
$311$	30.3835	1.72289	0.861446	−	0.507849i	$-0.169559\pi$
$311$	30.3835	1.72289	0.861446	+	0.507849i	$0.169559\pi$
$312$	0	0
$313$	19.7235	1.11484	0.557420	−	0.830230i	$-0.311791\pi$
$313$	19.7235	1.11484	0.557420	+	0.830230i	$0.311791\pi$
$314$	−0.994046	−0.0560973
$315$	0	0
$316$	−0.391322	−0.0220136
$317$	−22.1712	−1.24526	−0.622630	−	0.782516i	$-0.713936\pi$
$317$	−22.1712	−1.24526	−0.622630	+	0.782516i	$0.713936\pi$
$318$	0	0
$319$	−47.2152	−2.64355
$320$	−4.39795	−0.245853
$321$	0	0
$322$	−5.45296	−0.303882
$323$	25.8819	1.44011
$324$	0	0
$325$	14.8289	0.822562
$326$	−29.7933	−1.65010
$327$	0	0
$328$	18.2790	1.00929
$329$	9.14467	0.504162
$330$	0	0
$331$	−5.62100	−0.308958	−0.154479	−	0.987996i	$-0.549370\pi$
$331$	−5.62100	−0.308958	−0.154479	+	0.987996i	$0.549370\pi$
$332$	1.52780	0.0838491
$333$	0	0
$334$	−5.04917	−0.276278
$335$	2.26120	0.123543
$336$	0	0
$337$	8.56317	0.466465	0.233233	−	0.972421i	$-0.425070\pi$
$337$	8.56317	0.466465	0.233233	+	0.972421i	$0.425070\pi$
$338$	4.16115	0.226337
$339$	0	0
$340$	0.319405	0.0173222
$341$	19.0472	1.03146
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−12.3895	−0.667996
$345$	0	0
$346$	15.6442	0.841037
$347$	0.292076	0.0156795	0.00783973	−	0.999969i	$-0.497505\pi$
$347$	0.292076	0.0156795	0.00783973	+	0.999969i	$0.497505\pi$
$348$	0	0
$349$	1.71700	0.0919090	0.0459545	−	0.998944i	$-0.485367\pi$
$349$	1.71700	0.0919090	0.0459545	+	0.998944i	$0.485367\pi$
$350$	−6.76319	−0.361507
$351$	0	0
$352$	3.30363	0.176084
$353$	−2.50642	−0.133403	−0.0667016	−	0.997773i	$-0.521248\pi$
$353$	−2.50642	−0.133403	−0.0667016	+	0.997773i	$0.521248\pi$
$354$	0	0
$355$	−7.11476	−0.377612
$356$	0.752254	0.0398694
$357$	0	0
$358$	28.2081	1.49085
$359$	30.7410	1.62245	0.811224	−	0.584736i	$-0.198802\pi$
$359$	30.7410	1.62245	0.811224	+	0.584736i	$0.198802\pi$
$360$	0	0
$361$	7.15435	0.376545
$362$	−26.6690	−1.40169
$363$	0	0
$364$	0.343667	0.0180130
$365$	6.59178	0.345030
$366$	0	0
$367$	0.365574	0.0190828	0.00954140	−	0.999954i	$-0.496963\pi$
$367$	0.365574	0.0190828	0.00954140	+	0.999954i	$0.496963\pi$
$368$	15.7903	0.823127
$369$	0	0
$370$	−1.89667	−0.0986033
$371$	2.10337	0.109202
$372$	0	0
$373$	28.3498	1.46790	0.733950	−	0.679204i	$-0.237675\pi$
$373$	28.3498	1.46790	0.733950	+	0.679204i	$0.237675\pi$
$374$	39.7915	2.05757
$375$	0	0
$376$	−25.1206	−1.29550
$377$	27.7547	1.42944
$378$	0	0
$379$	25.0643	1.28747	0.643734	−	0.765250i	$-0.277384\pi$
$379$	25.0643	1.28747	0.643734	+	0.765250i	$0.277384\pi$
$380$	0.322767	0.0165576
$381$	0	0
$382$	4.70799	0.240882
$383$	11.0735	0.565828	0.282914	−	0.959145i	$-0.408699\pi$
$383$	11.0735	0.565828	0.282914	+	0.959145i	$0.408699\pi$
$384$	0	0
$385$	3.16596	0.161352
$386$	27.6355	1.40661
$387$	0	0
$388$	−1.72621	−0.0876350
$389$	6.67744	0.338560	0.169280	−	0.985568i	$-0.445856\pi$
$389$	6.67744	0.338560	0.169280	+	0.985568i	$0.445856\pi$
$390$	0	0
$391$	19.0075	0.961253
$392$	2.74702	0.138746
$393$	0	0
$394$	−14.4125	−0.726091
$395$	2.11913	0.106625
$396$	0	0
$397$	23.2489	1.16683	0.583414	−	0.812175i	$-0.301717\pi$
$397$	23.2489	1.16683	0.583414	+	0.812175i	$0.301717\pi$
$398$	−12.5620	−0.629676
$399$	0	0
$400$	19.5844	0.979219
$401$	12.7702	0.637712	0.318856	−	0.947803i	$-0.396701\pi$
$401$	12.7702	0.637712	0.318856	+	0.947803i	$0.396701\pi$
$402$	0	0
$403$	−11.1966	−0.557743
$404$	1.13102	0.0562701
$405$	0	0
$406$	−12.6584	−0.628225
$407$	−12.1011	−0.599831
$408$	0	0
$409$	2.99975	0.148328	0.0741641	−	0.997246i	$-0.476371\pi$
$409$	2.99975	0.148328	0.0741641	+	0.997246i	$0.476371\pi$
$410$	5.64797	0.278933
$411$	0	0
$412$	−1.81168	−0.0892553
$413$	−5.83902	−0.287319
$414$	0	0
$415$	−8.27351	−0.406131
$416$	−1.94199	−0.0952137
$417$	0	0
$418$	40.2103	1.96675
$419$	16.0099	0.782134	0.391067	−	0.920362i	$-0.372106\pi$
$419$	16.0099	0.782134	0.391067	+	0.920362i	$0.372106\pi$
$420$	0	0
$421$	−17.0046	−0.828751	−0.414376	−	0.910106i	$-0.636000\pi$
$421$	−17.0046	−0.828751	−0.414376	+	0.910106i	$0.636000\pi$
$422$	41.2896	2.00995
$423$	0	0
$424$	−5.77801	−0.280605
$425$	23.5746	1.14354
$426$	0	0
$427$	−14.0585	−0.680340
$428$	−0.618273	−0.0298854
$429$	0	0
$430$	−3.82817	−0.184611
$431$	−36.2193	−1.74462	−0.872311	−	0.488951i	$-0.837380\pi$
$431$	−36.2193	−1.74462	−0.872311	+	0.488951i	$0.837380\pi$
$432$	0	0
$433$	−23.3247	−1.12091	−0.560457	−	0.828184i	$-0.689374\pi$
$433$	−23.3247	−1.12091	−0.560457	+	0.828184i	$0.689374\pi$
$434$	5.10655	0.245122
$435$	0	0
$436$	1.92089	0.0919942
$437$	19.2076	0.918825
$438$	0	0
$439$	31.0060	1.47984	0.739919	−	0.672696i	$-0.234864\pi$
$439$	31.0060	1.47984	0.739919	+	0.672696i	$0.234864\pi$
$440$	−8.69695	−0.414611
$441$	0	0
$442$	−23.3908	−1.11259
$443$	1.04527	0.0496622	0.0248311	−	0.999692i	$-0.492095\pi$
$443$	1.04527	0.0496622	0.0248311	+	0.999692i	$0.492095\pi$
$444$	0	0
$445$	−4.07368	−0.193111
$446$	−1.91914	−0.0908739
$447$	0	0
$448$	−7.52281	−0.355420
$449$	−30.8628	−1.45651	−0.728253	−	0.685309i	$-0.759667\pi$
$449$	−30.8628	−1.45651	−0.728253	+	0.685309i	$0.759667\pi$
$450$	0	0
$451$	36.0352	1.69683
$452$	−0.947880	−0.0445845
$453$	0	0
$454$	5.12477	0.240517
$455$	−1.86106	−0.0872477
$456$	0	0
$457$	−3.39633	−0.158874	−0.0794369	−	0.996840i	$-0.525312\pi$
$457$	−3.39633	−0.158874	−0.0794369	+	0.996840i	$0.525312\pi$
$458$	19.6690	0.919074
$459$	0	0
$460$	0.237039	0.0110520
$461$	10.3660	0.482792	0.241396	−	0.970427i	$-0.422395\pi$
$461$	10.3660	0.482792	0.241396	+	0.970427i	$0.422395\pi$
$462$	0	0
$463$	−23.6359	−1.09846	−0.549228	−	0.835673i	$-0.685078\pi$
$463$	−23.6359	−1.09846	−0.549228	+	0.835673i	$0.685078\pi$
$464$	36.6553	1.70168
$465$	0	0
$466$	−13.4526	−0.623180
$467$	19.0928	0.883510	0.441755	−	0.897136i	$-0.354356\pi$
$467$	19.0928	0.883510	0.441755	+	0.897136i	$0.354356\pi$
$468$	0	0
$469$	3.86784	0.178600
$470$	−7.76192	−0.358031
$471$	0	0
$472$	16.0399	0.738296
$473$	−24.4245	−1.12304
$474$	0	0
$475$	23.8228	1.09306
$476$	0.546352	0.0250420
$477$	0	0
$478$	8.04739	0.368079
$479$	13.9575	0.637733	0.318867	−	0.947800i	$-0.396698\pi$
$479$	13.9575	0.637733	0.318867	+	0.947800i	$0.396698\pi$
$480$	0	0
$481$	7.11347	0.324346
$482$	−26.3030	−1.19807
$483$	0	0
$484$	1.97853	0.0899331
$485$	9.34794	0.424468
$486$	0	0
$487$	17.8690	0.809722	0.404861	−	0.914378i	$-0.367320\pi$
$487$	17.8690	0.809722	0.404861	+	0.914378i	$0.367320\pi$
$488$	38.6191	1.74820
$489$	0	0
$490$	0.848791	0.0383445
$491$	−1.53715	−0.0693706	−0.0346853	−	0.999398i	$-0.511043\pi$
$491$	−1.53715	−0.0693706	−0.0346853	+	0.999398i	$0.511043\pi$
$492$	0	0
$493$	44.1237	1.98723
$494$	−23.6370	−1.06348
$495$	0	0
$496$	−14.7872	−0.663965
$497$	−12.1700	−0.545898
$498$	0	0
$499$	−35.1272	−1.57251	−0.786255	−	0.617902i	$-0.787983\pi$
$499$	−35.1272	−1.57251	−0.786255	+	0.617902i	$0.787983\pi$
$500$	0.609558	0.0272603
$501$	0	0
$502$	−24.4334	−1.09052
$503$	−23.3017	−1.03897	−0.519486	−	0.854479i	$-0.673877\pi$
$503$	−23.3017	−1.03897	−0.519486	+	0.854479i	$0.673877\pi$
$504$	0	0
$505$	−6.12479	−0.272549
$506$	29.5303	1.31278
$507$	0	0
$508$	−0.107956	−0.00478978
$509$	−40.7781	−1.80746	−0.903730	−	0.428104i	$-0.859182\pi$
$509$	−40.7781	−1.80746	−0.903730	+	0.428104i	$0.859182\pi$
$510$	0	0
$511$	11.2754	0.498796
$512$	20.5336	0.907466
$513$	0	0
$514$	−0.238748	−0.0105307
$515$	9.81080	0.432316
$516$	0	0
$517$	−49.5225	−2.17800
$518$	−3.24431	−0.142547
$519$	0	0
$520$	5.11236	0.224192
$521$	−6.66265	−0.291896	−0.145948	−	0.989292i	$-0.546623\pi$
$521$	−6.66265	−0.291896	−0.145948	+	0.989292i	$0.546623\pi$
$522$	0	0
$523$	−24.2779	−1.06160	−0.530799	−	0.847497i	$-0.678108\pi$
$523$	−24.2779	−1.06160	−0.530799	+	0.847497i	$0.678108\pi$
$524$	−0.177342	−0.00774721
$525$	0	0
$526$	−18.3432	−0.799803
$527$	−17.8001	−0.775382
$528$	0	0
$529$	−8.89402	−0.386696
$530$	−1.78532	−0.0775495
$531$	0	0
$532$	0.552103	0.0239367
$533$	−21.1827	−0.917524
$534$	0	0
$535$	3.34813	0.144752
$536$	−10.6250	−0.458932
$537$	0	0
$538$	−7.99592	−0.344729
$539$	5.41545	0.233260
$540$	0	0
$541$	28.1614	1.21075	0.605377	−	0.795939i	$-0.293023\pi$
$541$	28.1614	1.21075	0.605377	+	0.795939i	$0.293023\pi$
$542$	40.7927	1.75220
$543$	0	0
$544$	−3.08731	−0.132367
$545$	−10.4022	−0.445582
$546$	0	0
$547$	−31.8313	−1.36101	−0.680503	−	0.732745i	$-0.738239\pi$
$547$	−31.8313	−1.36101	−0.680503	+	0.732745i	$0.738239\pi$
$548$	2.10084	0.0897434
$549$	0	0
$550$	36.6257	1.56173
$551$	44.5881	1.89952
$552$	0	0
$553$	3.62482	0.154143
$554$	−27.9433	−1.18720
$555$	0	0
$556$	−0.116220	−0.00492881
$557$	23.7141	1.00480	0.502399	−	0.864636i	$-0.332451\pi$
$557$	23.7141	1.00480	0.502399	+	0.864636i	$0.332451\pi$
$558$	0	0
$559$	14.3576	0.607260
$560$	−2.45787	−0.103864
$561$	0	0
$562$	−14.6423	−0.617647
$563$	−6.95606	−0.293163	−0.146582	−	0.989199i	$-0.546827\pi$
$563$	−6.95606	−0.293163	−0.146582	+	0.989199i	$0.546827\pi$
$564$	0	0
$565$	5.13305	0.215949
$566$	−40.6780	−1.70982
$567$	0	0
$568$	33.4312	1.40274
$569$	14.2834	0.598790	0.299395	−	0.954129i	$-0.403215\pi$
$569$	14.2834	0.598790	0.299395	+	0.954129i	$0.403215\pi$
$570$	0	0
$571$	−4.94964	−0.207136	−0.103568	−	0.994622i	$-0.533026\pi$
$571$	−4.94964	−0.207136	−0.103568	+	0.994622i	$0.533026\pi$
$572$	−1.86111	−0.0778169
$573$	0	0
$574$	9.66101	0.403243
$575$	17.4953	0.729606
$576$	0	0
$577$	4.53775	0.188909	0.0944546	−	0.995529i	$-0.469889\pi$
$577$	4.53775	0.188909	0.0944546	+	0.995529i	$0.469889\pi$
$578$	−12.5040	−0.520100
$579$	0	0
$580$	0.550256	0.0228482
$581$	−14.1521	−0.587126
$582$	0	0
$583$	−11.3907	−0.471755
$584$	−30.9738	−1.28171
$585$	0	0
$586$	−34.8863	−1.44114
$587$	−0.284682	−0.0117501	−0.00587505	−	0.999983i	$-0.501870\pi$
$587$	−0.284682	−0.0117501	−0.00587505	+	0.999983i	$0.501870\pi$
$588$	0	0
$589$	−17.9874	−0.741159
$590$	4.95611	0.204040
$591$	0	0
$592$	9.39466	0.386118
$593$	11.4352	0.469589	0.234795	−	0.972045i	$-0.424558\pi$
$593$	11.4352	0.469589	0.234795	+	0.972045i	$0.424558\pi$
$594$	0	0
$595$	−2.95866	−0.121293
$596$	−1.73367	−0.0710140
$597$	0	0
$598$	−17.3589	−0.709858
$599$	−11.9250	−0.487241	−0.243620	−	0.969871i	$-0.578335\pi$
$599$	−11.9250	−0.487241	−0.243620	+	0.969871i	$0.578335\pi$
$600$	0	0
$601$	29.4693	1.20208	0.601039	−	0.799220i	$-0.294754\pi$
$601$	29.4693	1.20208	0.601039	+	0.799220i	$0.294754\pi$
$602$	−6.54820	−0.266885
$603$	0	0
$604$	−0.385120	−0.0156703
$605$	−10.7143	−0.435599
$606$	0	0
$607$	−7.54814	−0.306369	−0.153185	−	0.988198i	$-0.548953\pi$
$607$	−7.54814	−0.306369	−0.153185	+	0.988198i	$0.548953\pi$
$608$	−3.11981	−0.126525
$609$	0	0
$610$	11.9328	0.483143
$611$	29.1110	1.17771
$612$	0	0
$613$	42.7869	1.72815	0.864074	−	0.503365i	$-0.167905\pi$
$613$	42.7869	1.72815	0.864074	+	0.503365i	$0.167905\pi$
$614$	23.9393	0.966110
$615$	0	0
$616$	−14.8764	−0.599386
$617$	−22.1822	−0.893020	−0.446510	−	0.894779i	$-0.647333\pi$
$617$	−22.1822	−0.893020	−0.446510	+	0.894779i	$0.647333\pi$
$618$	0	0
$619$	8.41815	0.338354	0.169177	−	0.985586i	$-0.445889\pi$
$619$	8.41815	0.338354	0.169177	+	0.985586i	$0.445889\pi$
$620$	−0.221980	−0.00891495
$621$	0	0
$622$	−44.1133	−1.76878
$623$	−6.96813	−0.279172
$624$	0	0
$625$	19.9902	0.799608
$626$	−28.6362	−1.14453
$627$	0	0
$628$	0.0739135	0.00294947
$629$	11.3088	0.450911
$630$	0	0
$631$	−42.9906	−1.71143	−0.855715	−	0.517448i	$-0.826882\pi$
$631$	−42.9906	−1.71143	−0.855715	+	0.517448i	$0.826882\pi$
$632$	−9.95746	−0.396087
$633$	0	0
$634$	32.1900	1.27843
$635$	0.584615	0.0231997
$636$	0	0
$637$	−3.18339	−0.126130
$638$	68.5509	2.71395
$639$	0	0
$640$	7.09857	0.280596
$641$	21.6825	0.856407	0.428203	−	0.903682i	$-0.359147\pi$
$641$	21.6825	0.856407	0.428203	+	0.903682i	$0.359147\pi$
$642$	0	0
$643$	7.73727	0.305128	0.152564	−	0.988294i	$-0.451247\pi$
$643$	7.73727	0.305128	0.152564	+	0.988294i	$0.451247\pi$
$644$	0.405461	0.0159774
$645$	0	0
$646$	−37.5774	−1.47846
$647$	20.9296	0.822827	0.411414	−	0.911449i	$-0.365035\pi$
$647$	20.9296	0.822827	0.411414	+	0.911449i	$0.365035\pi$
$648$	0	0
$649$	31.6209	1.24123
$650$	−21.5298	−0.844470
$651$	0	0
$652$	2.21531	0.0867584
$653$	−4.52343	−0.177015	−0.0885077	−	0.996075i	$-0.528210\pi$
$653$	−4.52343	−0.177015	−0.0885077	+	0.996075i	$0.528210\pi$
$654$	0	0
$655$	0.960358	0.0375243
$656$	−27.9757	−1.09227
$657$	0	0
$658$	−13.2770	−0.517590
$659$	11.5665	0.450567	0.225283	−	0.974293i	$-0.427669\pi$
$659$	11.5665	0.450567	0.225283	+	0.974293i	$0.427669\pi$
$660$	0	0
$661$	−0.556631	−0.0216504	−0.0108252	−	0.999941i	$-0.503446\pi$
$661$	−0.556631	−0.0216504	−0.0108252	+	0.999941i	$0.503446\pi$
$662$	8.16101	0.317187
$663$	0	0
$664$	38.8760	1.50868
$665$	−2.98980	−0.115939
$666$	0	0
$667$	32.7453	1.26790
$668$	0.375437	0.0145261
$669$	0	0
$670$	−3.28299	−0.126833
$671$	76.1333	2.93909
$672$	0	0
$673$	−23.9745	−0.924150	−0.462075	−	0.886841i	$-0.652895\pi$
$673$	−23.9745	−0.924150	−0.462075	+	0.886841i	$0.652895\pi$
$674$	−12.4327	−0.478889
$675$	0	0
$676$	−0.309407	−0.0119003
$677$	51.7020	1.98707	0.993535	−	0.113524i	$-0.0362138\pi$
$677$	51.7020	1.98707	0.993535	+	0.113524i	$0.0362138\pi$
$678$	0	0
$679$	15.9899	0.613636
$680$	8.12749	0.311675
$681$	0	0
$682$	−27.6543	−1.05894
$683$	−29.9749	−1.14696	−0.573479	−	0.819220i	$-0.694406\pi$
$683$	−29.9749	−1.14696	−0.573479	+	0.819220i	$0.694406\pi$
$684$	0	0
$685$	−11.3767	−0.434680
$686$	1.45188	0.0554330
$687$	0	0
$688$	18.9618	0.722913
$689$	6.69585	0.255091
$690$	0	0
$691$	−39.1395	−1.48894	−0.744468	−	0.667658i	$-0.767297\pi$
$691$	−39.1395	−1.48894	−0.744468	+	0.667658i	$0.767297\pi$
$692$	−1.16324	−0.0442198
$693$	0	0
$694$	−0.424059	−0.0160971
$695$	0.629364	0.0238731
$696$	0	0
$697$	−33.6757	−1.27556
$698$	−2.49288	−0.0943570
$699$	0	0
$700$	0.502885	0.0190073
$701$	−29.4425	−1.11203	−0.556013	−	0.831173i	$-0.687670\pi$
$701$	−29.4425	−1.11203	−0.556013	+	0.831173i	$0.687670\pi$
$702$	0	0
$703$	11.4278	0.431009
$704$	40.7394	1.53543
$705$	0	0
$706$	3.63902	0.136956
$707$	−10.4766	−0.394014
$708$	0	0
$709$	19.0173	0.714210	0.357105	−	0.934064i	$-0.383764\pi$
$709$	19.0173	0.714210	0.357105	+	0.934064i	$0.383764\pi$
$710$	10.3298	0.387669
$711$	0	0
$712$	19.1416	0.717362
$713$	−13.2099	−0.494713
$714$	0	0
$715$	10.0785	0.376913
$716$	−2.09745	−0.0783854
$717$	0	0
$718$	−44.6322	−1.66566
$719$	−0.197837	−0.00737807	−0.00368903	−	0.999993i	$-0.501174\pi$
$719$	−0.197837	−0.00737807	−0.00368903	+	0.999993i	$0.501174\pi$
$720$	0	0
$721$	16.7816	0.624981
$722$	−10.3873	−0.386574
$723$	0	0
$724$	1.98301	0.0736978
$725$	40.6133	1.50834
$726$	0	0
$727$	34.0943	1.26449	0.632244	−	0.774769i	$-0.282134\pi$
$727$	34.0943	1.26449	0.632244	+	0.774769i	$0.282134\pi$
$728$	8.74483	0.324105
$729$	0	0
$730$	−9.57048	−0.354219
$731$	22.8252	0.844222
$732$	0	0
$733$	−2.86047	−0.105654	−0.0528270	−	0.998604i	$-0.516823\pi$
$733$	−2.86047	−0.105654	−0.0528270	+	0.998604i	$0.516823\pi$
$734$	−0.530769	−0.0195911
$735$	0	0
$736$	−2.29117	−0.0844538
$737$	−20.9461	−0.771560
$738$	0	0
$739$	20.3860	0.749912	0.374956	−	0.927043i	$-0.377658\pi$
$739$	20.3860	0.749912	0.374956	+	0.927043i	$0.377658\pi$
$740$	0.141029	0.00518434
$741$	0	0
$742$	−3.05384	−0.112110
$743$	20.0394	0.735174	0.367587	−	0.929989i	$-0.380184\pi$
$743$	20.0394	0.735174	0.367587	+	0.929989i	$0.380184\pi$
$744$	0	0
$745$	9.38835	0.343963
$746$	−41.1606	−1.50700
$747$	0	0
$748$	−2.95874	−0.108182
$749$	5.72707	0.209263
$750$	0	0
$751$	−1.06573	−0.0388890	−0.0194445	−	0.999811i	$-0.506190\pi$
$751$	−1.06573	−0.0388890	−0.0194445	+	0.999811i	$0.506190\pi$
$752$	38.4466	1.40200
$753$	0	0
$754$	−40.2965	−1.46751
$755$	2.08554	0.0759005
$756$	0	0
$757$	37.0228	1.34562	0.672808	−	0.739817i	$-0.265088\pi$
$757$	37.0228	1.34562	0.672808	+	0.739817i	$0.265088\pi$
$758$	−36.3904	−1.32176
$759$	0	0
$760$	8.21304	0.297918
$761$	−3.94528	−0.143016	−0.0715081	−	0.997440i	$-0.522781\pi$
$761$	−3.94528	−0.143016	−0.0715081	+	0.997440i	$0.522781\pi$
$762$	0	0
$763$	−17.7933	−0.644159
$764$	−0.350068	−0.0126650
$765$	0	0
$766$	−16.0773	−0.580898
$767$	−18.5879	−0.671169
$768$	0	0
$769$	−22.9790	−0.828644	−0.414322	−	0.910130i	$-0.635981\pi$
$769$	−22.9790	−0.828644	−0.414322	+	0.910130i	$0.635981\pi$
$770$	−4.59659	−0.165650
$771$	0	0
$772$	−2.05487	−0.0739565
$773$	15.6983	0.564627	0.282314	−	0.959322i	$-0.408898\pi$
$773$	15.6983	0.564627	0.282314	+	0.959322i	$0.408898\pi$
$774$	0	0
$775$	−16.3839	−0.588527
$776$	−43.9246	−1.57680
$777$	0	0
$778$	−9.69485	−0.347577
$779$	−34.0301	−1.21926
$780$	0	0
$781$	65.9060	2.35830
$782$	−27.5967	−0.986855
$783$	0	0
$784$	−4.20426	−0.150152
$785$	−0.400263	−0.0142860
$786$	0	0
$787$	−19.5950	−0.698485	−0.349242	−	0.937032i	$-0.613561\pi$
$787$	−19.5950	−0.698485	−0.349242	+	0.937032i	$0.613561\pi$
$788$	1.07166	0.0381762
$789$	0	0
$790$	−3.07672	−0.109465
$791$	8.78023	0.312189
$792$	0	0
$793$	−44.7538	−1.58925
$794$	−33.7546	−1.19791
$795$	0	0
$796$	0.934063	0.0331070
$797$	−32.2337	−1.14178	−0.570888	−	0.821028i	$-0.693401\pi$
$797$	−32.2337	−1.14178	−0.570888	+	0.821028i	$0.693401\pi$
$798$	0	0
$799$	46.2799	1.63727
$800$	−2.84169	−0.100469
$801$	0	0
$802$	−18.5407	−0.654697
$803$	−61.0615	−2.15482
$804$	0	0
$805$	−2.19569	−0.0773880
$806$	16.2561	0.572598
$807$	0	0
$808$	28.7795	1.01246
$809$	−50.1803	−1.76424	−0.882122	−	0.471021i	$-0.843886\pi$
$809$	−50.1803	−1.76424	−0.882122	+	0.471021i	$0.843886\pi$
$810$	0	0
$811$	−32.2851	−1.13368	−0.566841	−	0.823827i	$-0.691835\pi$
$811$	−32.2851	−1.13368	−0.566841	+	0.823827i	$0.691835\pi$
$812$	0.941229	0.0330306
$813$	0	0
$814$	17.5694	0.615808
$815$	−11.9966	−0.420222
$816$	0	0
$817$	23.0655	0.806960
$818$	−4.35528	−0.152279
$819$	0	0
$820$	−0.419961	−0.0146657
$821$	2.84821	0.0994032	0.0497016	−	0.998764i	$-0.484173\pi$
$821$	2.84821	0.0994032	0.0497016	+	0.998764i	$0.484173\pi$
$822$	0	0
$823$	−20.4935	−0.714358	−0.357179	−	0.934036i	$-0.616261\pi$
$823$	−20.4935	−0.714358	−0.357179	+	0.934036i	$0.616261\pi$
$824$	−46.0995	−1.60595
$825$	0	0
$826$	8.47755	0.294972
$827$	−12.8868	−0.448117	−0.224059	−	0.974576i	$-0.571931\pi$
$827$	−12.8868	−0.448117	−0.224059	+	0.974576i	$0.571931\pi$
$828$	0	0
$829$	9.04317	0.314082	0.157041	−	0.987592i	$-0.449804\pi$
$829$	9.04317	0.314082	0.157041	+	0.987592i	$0.449804\pi$
$830$	12.0121	0.416948
$831$	0	0
$832$	−23.9480	−0.830249
$833$	−5.06086	−0.175348
$834$	0	0
$835$	−2.03310	−0.0703584
$836$	−2.98989	−0.103407
$837$	0	0
$838$	−23.2444	−0.802965
$839$	−14.0717	−0.485807	−0.242904	−	0.970050i	$-0.578100\pi$
$839$	−14.0717	−0.485807	−0.242904	+	0.970050i	$0.578100\pi$
$840$	0	0
$841$	47.0142	1.62118
$842$	24.6886	0.850825
$843$	0	0
$844$	−3.07014	−0.105678
$845$	1.67553	0.0576400
$846$	0	0
$847$	−18.3271	−0.629728
$848$	8.84312	0.303674
$849$	0	0
$850$	−34.2275	−1.17399
$851$	8.39254	0.287693
$852$	0	0
$853$	−36.9212	−1.26416	−0.632080	−	0.774903i	$-0.717799\pi$
$853$	−36.9212	−1.26416	−0.632080	+	0.774903i	$0.717799\pi$
$854$	20.4113	0.698461
$855$	0	0
$856$	−15.7324	−0.537722
$857$	26.3947	0.901626	0.450813	−	0.892619i	$-0.351134\pi$
$857$	26.3947	0.901626	0.450813	+	0.892619i	$0.351134\pi$
$858$	0	0
$859$	53.7029	1.83232	0.916160	−	0.400813i	$-0.131272\pi$
$859$	53.7029	1.83232	0.916160	+	0.400813i	$0.131272\pi$
$860$	0.284648	0.00970643
$861$	0	0
$862$	52.5861	1.79109
$863$	−4.39848	−0.149726	−0.0748630	−	0.997194i	$-0.523852\pi$
$863$	−4.39848	−0.149726	−0.0748630	+	0.997194i	$0.523852\pi$
$864$	0	0
$865$	6.29930	0.214183
$866$	33.8647	1.15077
$867$	0	0
$868$	−0.379704	−0.0128880
$869$	−19.6301	−0.665904
$870$	0	0
$871$	12.3128	0.417205
$872$	48.8785	1.65523
$873$	0	0
$874$	−27.8872	−0.943297
$875$	−5.64634	−0.190881
$876$	0	0
$877$	27.9653	0.944321	0.472161	−	0.881513i	$-0.343474\pi$
$877$	27.9653	0.944321	0.472161	+	0.881513i	$0.343474\pi$
$878$	−45.0171	−1.51925
$879$	0	0
$880$	13.3105	0.448697
$881$	31.8646	1.07355	0.536773	−	0.843727i	$-0.319643\pi$
$881$	31.8646	1.07355	0.536773	+	0.843727i	$0.319643\pi$
$882$	0	0
$883$	22.9214	0.771367	0.385684	−	0.922631i	$-0.373966\pi$
$883$	22.9214	0.771367	0.385684	+	0.922631i	$0.373966\pi$
$884$	1.73925	0.0584973
$885$	0	0
$886$	−1.51760	−0.0509849
$887$	−52.3948	−1.75924	−0.879622	−	0.475673i	$-0.842204\pi$
$887$	−52.3948	−1.75924	−0.879622	+	0.475673i	$0.842204\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	5.91449	0.198254
$891$	0	0
$892$	0.142700	0.00477794
$893$	46.7671	1.56500
$894$	0	0
$895$	11.3583	0.379666
$896$	12.1423	0.405646
$897$	0	0
$898$	44.8091	1.49530
$899$	−30.6651	−1.02274
$900$	0	0
$901$	10.6449	0.354632
$902$	−52.3187	−1.74202
$903$	0	0
$904$	−24.1195	−0.802201
$905$	−10.7386	−0.356962
$906$	0	0
$907$	8.25274	0.274028	0.137014	−	0.990569i	$-0.456250\pi$
$907$	8.25274	0.274028	0.137014	+	0.990569i	$0.456250\pi$
$908$	−0.381058	−0.0126459
$909$	0	0
$910$	2.70203	0.0895715
$911$	41.0478	1.35998	0.679988	−	0.733223i	$-0.261985\pi$
$911$	41.0478	1.35998	0.679988	+	0.733223i	$0.261985\pi$
$912$	0	0
$913$	76.6398	2.53641
$914$	4.93107	0.163105
$915$	0	0
$916$	−1.46251	−0.0483228
$917$	1.64272	0.0542473
$918$	0	0
$919$	18.6013	0.613601	0.306801	−	0.951774i	$-0.400741\pi$
$919$	18.6013	0.613601	0.306801	+	0.951774i	$0.400741\pi$
$920$	6.03162	0.198856
$921$	0	0
$922$	−15.0502	−0.495651
$923$	−38.7418	−1.27520
$924$	0	0
$925$	10.4091	0.342248
$926$	34.3166	1.12771
$927$	0	0
$928$	−5.31868	−0.174594
$929$	−24.4625	−0.802588	−0.401294	−	0.915949i	$-0.631439\pi$
$929$	−24.4625	−0.802588	−0.401294	+	0.915949i	$0.631439\pi$
$930$	0	0
$931$	−5.11413	−0.167609
$932$	1.00028	0.0327654
$933$	0	0
$934$	−27.7205	−0.907042
$935$	16.0225	0.523990
$936$	0	0
$937$	52.8188	1.72552	0.862758	−	0.505617i	$-0.168735\pi$
$937$	52.8188	1.72552	0.862758	+	0.505617i	$0.168735\pi$
$938$	−5.61565	−0.183357
$939$	0	0
$940$	0.577146	0.0188244
$941$	7.92118	0.258223	0.129111	−	0.991630i	$-0.458787\pi$
$941$	7.92118	0.258223	0.129111	+	0.991630i	$0.458787\pi$
$942$	0	0
$943$	−24.9916	−0.813837
$944$	−24.5487	−0.798993
$945$	0	0
$946$	35.4614	1.15295
$947$	56.1375	1.82422	0.912111	−	0.409943i	$-0.134451\pi$
$947$	56.1375	1.82422	0.912111	+	0.409943i	$0.134451\pi$
$948$	0	0
$949$	35.8941	1.16517
$950$	−34.5878	−1.12218
$951$	0	0
$952$	13.9023	0.450576
$953$	1.73578	0.0562273	0.0281137	−	0.999605i	$-0.491050\pi$
$953$	1.73578	0.0562273	0.0281137	+	0.999605i	$0.491050\pi$
$954$	0	0
$955$	1.89572	0.0613441
$956$	−0.598373	−0.0193528
$957$	0	0
$958$	−20.2646	−0.654719
$959$	−19.4601	−0.628399
$960$	0	0
$961$	−18.6293	−0.600946
$962$	−10.3279	−0.332985
$963$	0	0
$964$	1.95579	0.0629918
$965$	11.1277	0.358215
$966$	0	0
$967$	−26.7443	−0.860038	−0.430019	−	0.902820i	$-0.641493\pi$
$967$	−26.7443	−0.860038	−0.430019	+	0.902820i	$0.641493\pi$
$968$	50.3450	1.61815
$969$	0	0
$970$	−13.5721	−0.435773
$971$	33.1012	1.06227	0.531134	−	0.847288i	$-0.321766\pi$
$971$	33.1012	1.06227	0.531134	+	0.847288i	$0.321766\pi$
$972$	0	0
$973$	1.07654	0.0345124
$974$	−25.9437	−0.831288
$975$	0	0
$976$	−59.1057	−1.89193
$977$	−7.19435	−0.230168	−0.115084	−	0.993356i	$-0.536714\pi$
$977$	−7.19435	−0.230168	−0.115084	+	0.993356i	$0.536714\pi$
$978$	0	0
$979$	37.7356	1.20603
$980$	−0.0631129	−0.00201607
$981$	0	0
$982$	2.23176	0.0712182
$983$	22.7413	0.725334	0.362667	−	0.931919i	$-0.381866\pi$
$983$	22.7413	0.725334	0.362667	+	0.931919i	$0.381866\pi$
$984$	0	0
$985$	−5.80335	−0.184910
$986$	−64.0623	−2.04016
$987$	0	0
$988$	1.75756	0.0559153
$989$	16.9392	0.538635
$990$	0	0
$991$	50.6822	1.60997	0.804987	−	0.593292i	$-0.202172\pi$
$991$	50.6822	1.60997	0.804987	+	0.593292i	$0.202172\pi$
$992$	2.14562	0.0681236
$993$	0	0
$994$	17.6694	0.560438
$995$	−5.05823	−0.160357
$996$	0	0
$997$	−14.1678	−0.448700	−0.224350	−	0.974509i	$-0.572026\pi$
$997$	−14.1678	−0.448700	−0.224350	+	0.974509i	$0.572026\pi$
$998$	51.0005	1.61439
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.5		16
3.2	odd	2		889.2.a.c.1.12	✓	16
21.20	even	2		6223.2.a.k.1.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.12	✓	16	3.2	odd	2
6223.2.a.k.1.12		16	21.20	even	2
8001.2.a.t.1.5		16	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.45188 q^{2} +0.107956 q^{4} -0.584615 q^{5} -1.00000 q^{7} +2.74702 q^{8} +O(q^{10})\)	\(q-1.45188 q^{2} +0.107956 q^{4} -0.584615 q^{5} -1.00000 q^{7} +2.74702 q^{8} +0.848791 q^{10} +5.41545 q^{11} -3.18339 q^{13} +1.45188 q^{14} -4.20426 q^{16} -5.06086 q^{17} -5.11413 q^{19} -0.0631129 q^{20} -7.86259 q^{22} -3.75579 q^{23} -4.65823 q^{25} +4.62190 q^{26} -0.107956 q^{28} -8.71861 q^{29} +3.51720 q^{31} +0.610037 q^{32} +7.34776 q^{34} +0.584615 q^{35} -2.23456 q^{37} +7.42511 q^{38} -1.60595 q^{40} +6.65414 q^{41} -4.51015 q^{43} +0.584632 q^{44} +5.45296 q^{46} -9.14467 q^{47} +1.00000 q^{49} +6.76319 q^{50} -0.343667 q^{52} -2.10337 q^{53} -3.16596 q^{55} -2.74702 q^{56} +12.6584 q^{58} +5.83902 q^{59} +14.0585 q^{61} -5.10655 q^{62} +7.52281 q^{64} +1.86106 q^{65} -3.86784 q^{67} -0.546352 q^{68} -0.848791 q^{70} +12.1700 q^{71} -11.2754 q^{73} +3.24431 q^{74} -0.552103 q^{76} -5.41545 q^{77} -3.62482 q^{79} +2.45787 q^{80} -9.66101 q^{82} +14.1521 q^{83} +2.95866 q^{85} +6.54820 q^{86} +14.8764 q^{88} +6.96813 q^{89} +3.18339 q^{91} -0.405461 q^{92} +13.2770 q^{94} +2.98980 q^{95} -15.9899 q^{97} -1.45188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

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