LMFDB - Embedded newform 8001.2.a.o.1.6

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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.18273$ 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)

$f(q)$	$=$	$q-1.18273 q^{2} -0.601156 q^{4} -4.39766 q^{5} +1.00000 q^{7} +3.07646 q^{8} +O(q^{10})$	\(q-1.18273 q^{2} -0.601156 q^{4} -4.39766 q^{5} +1.00000 q^{7} +3.07646 q^{8} +5.20123 q^{10} +1.66461 q^{11} +1.04052 q^{13} -1.18273 q^{14} -2.43630 q^{16} -2.31427 q^{17} -1.96198 q^{19} +2.64368 q^{20} -1.96878 q^{22} -2.53372 q^{23} +14.3394 q^{25} -1.23065 q^{26} -0.601156 q^{28} -3.94733 q^{29} +2.35543 q^{31} -3.27144 q^{32} +2.73716 q^{34} -4.39766 q^{35} -3.81874 q^{37} +2.32049 q^{38} -13.5292 q^{40} -5.25377 q^{41} +1.58885 q^{43} -1.00069 q^{44} +2.99670 q^{46} +5.23626 q^{47} +1.00000 q^{49} -16.9596 q^{50} -0.625517 q^{52} -6.13449 q^{53} -7.32038 q^{55} +3.07646 q^{56} +4.66862 q^{58} +1.48950 q^{59} +0.959646 q^{61} -2.78583 q^{62} +8.74182 q^{64} -4.57586 q^{65} +4.40683 q^{67} +1.39124 q^{68} +5.20123 q^{70} -4.45038 q^{71} +1.66690 q^{73} +4.51652 q^{74} +1.17946 q^{76} +1.66461 q^{77} +0.363151 q^{79} +10.7140 q^{80} +6.21377 q^{82} +9.34690 q^{83} +10.1774 q^{85} -1.87918 q^{86} +5.12110 q^{88} +3.44222 q^{89} +1.04052 q^{91} +1.52316 q^{92} -6.19307 q^{94} +8.62813 q^{95} +13.6324 q^{97} -1.18273 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * 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 $

$2$	−1.18273	−0.836314	−0.418157	−	0.908375i	$-0.637324\pi$
$2$	−1.18273	−0.836314	−0.418157	+	0.908375i	$0.637324\pi$
$3$	0	0
$3$	0	0
$4$	−0.601156	−0.300578
$5$	−4.39766	−1.96669	−0.983347	−	0.181739i	$-0.941827\pi$
$5$	−4.39766	−1.96669	−0.983347	+	0.181739i	$0.941827\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.07646	1.08769
$9$	0	0
$10$	5.20123	1.64477
$11$	1.66461	0.501898	0.250949	−	0.968000i	$-0.419257\pi$
$11$	1.66461	0.501898	0.250949	+	0.968000i	$0.419257\pi$
$12$	0	0
$13$	1.04052	0.288589	0.144294	−	0.989535i	$-0.453909\pi$
$13$	1.04052	0.288589	0.144294	+	0.989535i	$0.453909\pi$
$14$	−1.18273	−0.316097
$15$	0	0
$16$	−2.43630	−0.609075
$17$	−2.31427	−0.561294	−0.280647	−	0.959811i	$-0.590549\pi$
$17$	−2.31427	−0.561294	−0.280647	+	0.959811i	$0.590549\pi$
$18$	0	0
$19$	−1.96198	−0.450109	−0.225055	−	0.974346i	$-0.572256\pi$
$19$	−1.96198	−0.450109	−0.225055	+	0.974346i	$0.572256\pi$
$20$	2.64368	0.591145
$21$	0	0
$22$	−1.96878	−0.419745
$23$	−2.53372	−0.528318	−0.264159	−	0.964479i	$-0.585094\pi$
$23$	−2.53372	−0.528318	−0.264159	+	0.964479i	$0.585094\pi$
$24$	0	0
$25$	14.3394	2.86788
$26$	−1.23065	−0.241351
$27$	0	0
$28$	−0.601156	−0.113608
$29$	−3.94733	−0.733001	−0.366501	−	0.930418i	$-0.619444\pi$
$29$	−3.94733	−0.733001	−0.366501	+	0.930418i	$0.619444\pi$
$30$	0	0
$31$	2.35543	0.423048	0.211524	−	0.977373i	$-0.432157\pi$
$31$	2.35543	0.423048	0.211524	+	0.977373i	$0.432157\pi$
$32$	−3.27144	−0.578314
$33$	0	0
$34$	2.73716	0.469418
$35$	−4.39766	−0.743340
$36$	0	0
$37$	−3.81874	−0.627797	−0.313898	−	0.949457i	$-0.601635\pi$
$37$	−3.81874	−0.627797	−0.313898	+	0.949457i	$0.601635\pi$
$38$	2.32049	0.376433
$39$	0	0
$40$	−13.5292	−2.13916
$41$	−5.25377	−0.820500	−0.410250	−	0.911973i	$-0.634559\pi$
$41$	−5.25377	−0.820500	−0.410250	+	0.911973i	$0.634559\pi$
$42$	0	0
$43$	1.58885	0.242298	0.121149	−	0.992634i	$-0.461342\pi$
$43$	1.58885	0.242298	0.121149	+	0.992634i	$0.461342\pi$
$44$	−1.00069	−0.150860
$45$	0	0
$46$	2.99670	0.441840
$47$	5.23626	0.763787	0.381893	−	0.924206i	$-0.375272\pi$
$47$	5.23626	0.763787	0.381893	+	0.924206i	$0.375272\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−16.9596	−2.39845
$51$	0	0
$52$	−0.625517	−0.0867435
$53$	−6.13449	−0.842637	−0.421318	−	0.906913i	$-0.638432\pi$
$53$	−6.13449	−0.842637	−0.421318	+	0.906913i	$0.638432\pi$
$54$	0	0
$55$	−7.32038	−0.987080
$56$	3.07646	0.411109
$57$	0	0
$58$	4.66862	0.613020
$59$	1.48950	0.193916	0.0969580	−	0.995288i	$-0.469089\pi$
$59$	1.48950	0.193916	0.0969580	+	0.995288i	$0.469089\pi$
$60$	0	0
$61$	0.959646	0.122870	0.0614350	−	0.998111i	$-0.480432\pi$
$61$	0.959646	0.122870	0.0614350	+	0.998111i	$0.480432\pi$
$62$	−2.78583	−0.353801
$63$	0	0
$64$	8.74182	1.09273
$65$	−4.57586	−0.567566
$66$	0	0
$67$	4.40683	0.538380	0.269190	−	0.963087i	$-0.413244\pi$
$67$	4.40683	0.538380	0.269190	+	0.963087i	$0.413244\pi$
$68$	1.39124	0.168713
$69$	0	0
$70$	5.20123	0.621666
$71$	−4.45038	−0.528163	−0.264082	−	0.964500i	$-0.585069\pi$
$71$	−4.45038	−0.528163	−0.264082	+	0.964500i	$0.585069\pi$
$72$	0	0
$73$	1.66690	0.195096	0.0975479	−	0.995231i	$-0.468900\pi$
$73$	1.66690	0.195096	0.0975479	+	0.995231i	$0.468900\pi$
$74$	4.51652	0.525035
$75$	0	0
$76$	1.17946	0.135293
$77$	1.66461	0.189700
$78$	0	0
$79$	0.363151	0.0408577	0.0204288	−	0.999791i	$-0.493497\pi$
$79$	0.363151	0.0408577	0.0204288	+	0.999791i	$0.493497\pi$
$80$	10.7140	1.19786
$81$	0	0
$82$	6.21377	0.686196
$83$	9.34690	1.02596	0.512978	−	0.858402i	$-0.328542\pi$
$83$	9.34690	1.02596	0.512978	+	0.858402i	$0.328542\pi$
$84$	0	0
$85$	10.1774	1.10389
$86$	−1.87918	−0.202637
$87$	0	0
$88$	5.12110	0.545911
$89$	3.44222	0.364874	0.182437	−	0.983218i	$-0.441601\pi$
$89$	3.44222	0.364874	0.182437	+	0.983218i	$0.441601\pi$
$90$	0	0
$91$	1.04052	0.109076
$92$	1.52316	0.158801
$93$	0	0
$94$	−6.19307	−0.638766
$95$	8.62813	0.885227
$96$	0	0
$97$	13.6324	1.38416	0.692082	−	0.721819i	$-0.256694\pi$
$97$	13.6324	1.38416	0.692082	+	0.721819i	$0.256694\pi$
$98$	−1.18273	−0.119473
$99$	0	0
$100$	−8.62023	−0.862023
$101$	11.5755	1.15181	0.575903	−	0.817518i	$-0.304651\pi$
$101$	11.5755	1.15181	0.575903	+	0.817518i	$0.304651\pi$
$102$	0	0
$103$	−3.97805	−0.391969	−0.195984	−	0.980607i	$-0.562790\pi$
$103$	−3.97805	−0.391969	−0.195984	+	0.980607i	$0.562790\pi$
$104$	3.20112	0.313896
$105$	0	0
$106$	7.25543	0.704709
$107$	−7.19510	−0.695576	−0.347788	−	0.937573i	$-0.613067\pi$
$107$	−7.19510	−0.695576	−0.347788	+	0.937573i	$0.613067\pi$
$108$	0	0
$109$	5.38609	0.515894	0.257947	−	0.966159i	$-0.416954\pi$
$109$	5.38609	0.515894	0.257947	+	0.966159i	$0.416954\pi$
$110$	8.65802	0.825510
$111$	0	0
$112$	−2.43630	−0.230209
$113$	3.43506	0.323144	0.161572	−	0.986861i	$-0.448344\pi$
$113$	3.43506	0.323144	0.161572	+	0.986861i	$0.448344\pi$
$114$	0	0
$115$	11.1425	1.03904
$116$	2.37296	0.220324
$117$	0	0
$118$	−1.76167	−0.162175
$119$	−2.31427	−0.212149
$120$	0	0
$121$	−8.22908	−0.748098
$122$	−1.13500	−0.102758
$123$	0	0
$124$	−1.41598	−0.127159
$125$	−41.0716	−3.67355
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−3.79631	−0.335549
$129$	0	0
$130$	5.41200	0.474664
$131$	6.97230	0.609173	0.304586	−	0.952485i	$-0.401482\pi$
$131$	6.97230	0.609173	0.304586	+	0.952485i	$0.401482\pi$
$132$	0	0
$133$	−1.96198	−0.170125
$134$	−5.21208	−0.450255
$135$	0	0
$136$	−7.11977	−0.610515
$137$	1.28547	0.109825	0.0549127	−	0.998491i	$-0.482512\pi$
$137$	1.28547	0.109825	0.0549127	+	0.998491i	$0.482512\pi$
$138$	0	0
$139$	19.7131	1.67204	0.836020	−	0.548699i	$-0.184877\pi$
$139$	19.7131	1.67204	0.836020	+	0.548699i	$0.184877\pi$
$140$	2.64368	0.223432
$141$	0	0
$142$	5.26359	0.441711
$143$	1.73206	0.144842
$144$	0	0
$145$	17.3590	1.44159
$146$	−1.97149	−0.163161
$147$	0	0
$148$	2.29566	0.188702
$149$	−8.22055	−0.673454	−0.336727	−	0.941602i	$-0.609320\pi$
$149$	−8.22055	−0.673454	−0.336727	+	0.941602i	$0.609320\pi$
$150$	0	0
$151$	−0.422522	−0.0343843	−0.0171922	−	0.999852i	$-0.505473\pi$
$151$	−0.422522	−0.0343843	−0.0171922	+	0.999852i	$0.505473\pi$
$152$	−6.03595	−0.489580
$153$	0	0
$154$	−1.96878	−0.158649
$155$	−10.3584	−0.832006
$156$	0	0
$157$	13.6917	1.09272	0.546359	−	0.837551i	$-0.316013\pi$
$157$	13.6917	1.09272	0.546359	+	0.837551i	$0.316013\pi$
$158$	−0.429508	−0.0341698
$159$	0	0
$160$	14.3867	1.13737
$161$	−2.53372	−0.199685
$162$	0	0
$163$	−2.08608	−0.163394	−0.0816971	−	0.996657i	$-0.526034\pi$
$163$	−2.08608	−0.163394	−0.0816971	+	0.996657i	$0.526034\pi$
$164$	3.15833	0.246624
$165$	0	0
$166$	−11.0548	−0.858022
$167$	−10.5152	−0.813688	−0.406844	−	0.913498i	$-0.633371\pi$
$167$	−10.5152	−0.813688	−0.406844	+	0.913498i	$0.633371\pi$
$168$	0	0
$169$	−11.9173	−0.916716
$170$	−12.0371	−0.923202
$171$	0	0
$172$	−0.955148	−0.0728294
$173$	−12.2940	−0.934692	−0.467346	−	0.884075i	$-0.654790\pi$
$173$	−12.2940	−0.934692	−0.467346	+	0.884075i	$0.654790\pi$
$174$	0	0
$175$	14.3394	1.08396
$176$	−4.05548	−0.305694
$177$	0	0
$178$	−4.07121	−0.305150
$179$	5.87375	0.439025	0.219512	−	0.975610i	$-0.429553\pi$
$179$	5.87375	0.439025	0.219512	+	0.975610i	$0.429553\pi$
$180$	0	0
$181$	4.74037	0.352349	0.176175	−	0.984359i	$-0.443628\pi$
$181$	4.74037	0.352349	0.176175	+	0.984359i	$0.443628\pi$
$182$	−1.23065	−0.0912222
$183$	0	0
$184$	−7.79490	−0.574647
$185$	16.7935	1.23468
$186$	0	0
$187$	−3.85236	−0.281713
$188$	−3.14781	−0.229578
$189$	0	0
$190$	−10.2047	−0.740328
$191$	5.44920	0.394290	0.197145	−	0.980374i	$-0.436833\pi$
$191$	5.44920	0.394290	0.197145	+	0.980374i	$0.436833\pi$
$192$	0	0
$193$	−1.41857	−0.102111	−0.0510556	−	0.998696i	$-0.516259\pi$
$193$	−1.41857	−0.102111	−0.0510556	+	0.998696i	$0.516259\pi$
$194$	−16.1234	−1.15760
$195$	0	0
$196$	−0.601156	−0.0429397
$197$	15.5285	1.10636	0.553179	−	0.833063i	$-0.313415\pi$
$197$	15.5285	1.10636	0.553179	+	0.833063i	$0.313415\pi$
$198$	0	0
$199$	26.0728	1.84825	0.924127	−	0.382085i	$-0.124794\pi$
$199$	26.0728	1.84825	0.924127	+	0.382085i	$0.124794\pi$
$200$	44.1146	3.11937
$201$	0	0
$202$	−13.6907	−0.963271
$203$	−3.94733	−0.277049
$204$	0	0
$205$	23.1043	1.61367
$206$	4.70495	0.327809
$207$	0	0
$208$	−2.53502	−0.175772
$209$	−3.26593	−0.225909
$210$	0	0
$211$	−2.16643	−0.149143	−0.0745716	−	0.997216i	$-0.523759\pi$
$211$	−2.16643	−0.149143	−0.0745716	+	0.997216i	$0.523759\pi$
$212$	3.68779	0.253278
$213$	0	0
$214$	8.50984	0.581721
$215$	−6.98723	−0.476525
$216$	0	0
$217$	2.35543	0.159897
$218$	−6.37028	−0.431450
$219$	0	0
$220$	4.40069	0.296695
$221$	−2.40805	−0.161983
$222$	0	0
$223$	−18.5131	−1.23973	−0.619863	−	0.784710i	$-0.712812\pi$
$223$	−18.5131	−1.23973	−0.619863	+	0.784710i	$0.712812\pi$
$224$	−3.27144	−0.218582
$225$	0	0
$226$	−4.06274	−0.270250
$227$	27.7262	1.84025	0.920125	−	0.391625i	$-0.128087\pi$
$227$	27.7262	1.84025	0.920125	+	0.391625i	$0.128087\pi$
$228$	0	0
$229$	13.4725	0.890285	0.445142	−	0.895460i	$-0.353153\pi$
$229$	13.4725	0.890285	0.445142	+	0.895460i	$0.353153\pi$
$230$	−13.1785	−0.868964
$231$	0	0
$232$	−12.1438	−0.797280
$233$	−0.714943	−0.0468375	−0.0234187	−	0.999726i	$-0.507455\pi$
$233$	−0.714943	−0.0468375	−0.0234187	+	0.999726i	$0.507455\pi$
$234$	0	0
$235$	−23.0273	−1.50213
$236$	−0.895420	−0.0582869
$237$	0	0
$238$	2.73716	0.177423
$239$	4.14047	0.267825	0.133912	−	0.990993i	$-0.457246\pi$
$239$	4.14047	0.267825	0.133912	+	0.990993i	$0.457246\pi$
$240$	0	0
$241$	−20.3753	−1.31249	−0.656244	−	0.754549i	$-0.727856\pi$
$241$	−20.3753	−1.31249	−0.656244	+	0.754549i	$0.727856\pi$
$242$	9.73276	0.625645
$243$	0	0
$244$	−0.576897	−0.0369320
$245$	−4.39766	−0.280956
$246$	0	0
$247$	−2.04149	−0.129897
$248$	7.24639	0.460146
$249$	0	0
$250$	48.5765	3.07225
$251$	−4.17045	−0.263237	−0.131618	−	0.991300i	$-0.542017\pi$
$251$	−4.17045	−0.263237	−0.131618	+	0.991300i	$0.542017\pi$
$252$	0	0
$253$	−4.21766	−0.265162
$254$	1.18273	0.0742109
$255$	0	0
$256$	−12.9936	−0.812103
$257$	23.0198	1.43593	0.717967	−	0.696077i	$-0.245073\pi$
$257$	23.0198	1.43593	0.717967	+	0.696077i	$0.245073\pi$
$258$	0	0
$259$	−3.81874	−0.237285
$260$	2.75081	0.170598
$261$	0	0
$262$	−8.24633	−0.509460
$263$	−21.0044	−1.29519	−0.647595	−	0.761985i	$-0.724225\pi$
$263$	−21.0044	−1.29519	−0.647595	+	0.761985i	$0.724225\pi$
$264$	0	0
$265$	26.9774	1.65721
$266$	2.32049	0.142278
$267$	0	0
$268$	−2.64919	−0.161825
$269$	1.62594	0.0991355	0.0495678	−	0.998771i	$-0.484216\pi$
$269$	1.62594	0.0991355	0.0495678	+	0.998771i	$0.484216\pi$
$270$	0	0
$271$	−25.5156	−1.54996	−0.774981	−	0.631985i	$-0.782241\pi$
$271$	−25.5156	−1.54996	−0.774981	+	0.631985i	$0.782241\pi$
$272$	5.63826	0.341870
$273$	0	0
$274$	−1.52036	−0.0918485
$275$	23.8695	1.43939
$276$	0	0
$277$	8.26564	0.496634	0.248317	−	0.968679i	$-0.420122\pi$
$277$	8.26564	0.496634	0.248317	+	0.968679i	$0.420122\pi$
$278$	−23.3152	−1.39835
$279$	0	0
$280$	−13.5292	−0.808525
$281$	−20.3442	−1.21363	−0.606817	−	0.794842i	$-0.707554\pi$
$281$	−20.3442	−1.21363	−0.606817	+	0.794842i	$0.707554\pi$
$282$	0	0
$283$	22.0601	1.31134	0.655670	−	0.755048i	$-0.272386\pi$
$283$	22.0601	1.31134	0.655670	+	0.755048i	$0.272386\pi$
$284$	2.67538	0.158754
$285$	0	0
$286$	−2.04856	−0.121134
$287$	−5.25377	−0.310120
$288$	0	0
$289$	−11.6441	−0.684949
$290$	−20.5310	−1.20562
$291$	0	0
$292$	−1.00207	−0.0586415
$293$	−18.5933	−1.08624	−0.543118	−	0.839657i	$-0.682756\pi$
$293$	−18.5933	−1.08624	−0.543118	+	0.839657i	$0.682756\pi$
$294$	0	0
$295$	−6.55030	−0.381373
$296$	−11.7482	−0.682849
$297$	0	0
$298$	9.72267	0.563219
$299$	−2.63640	−0.152467
$300$	0	0
$301$	1.58885	0.0915799
$302$	0.499728	0.0287561
$303$	0	0
$304$	4.77997	0.274150
$305$	−4.22020	−0.241648
$306$	0	0
$307$	−12.6306	−0.720865	−0.360432	−	0.932785i	$-0.617371\pi$
$307$	−12.6306	−0.720865	−0.360432	+	0.932785i	$0.617371\pi$
$308$	−1.00069	−0.0570196
$309$	0	0
$310$	12.2511	0.695818
$311$	27.8668	1.58018	0.790090	−	0.612991i	$-0.210034\pi$
$311$	27.8668	1.58018	0.790090	+	0.612991i	$0.210034\pi$
$312$	0	0
$313$	−5.08549	−0.287449	−0.143724	−	0.989618i	$-0.545908\pi$
$313$	−5.08549	−0.287449	−0.143724	+	0.989618i	$0.545908\pi$
$314$	−16.1936	−0.913856
$315$	0	0
$316$	−0.218310	−0.0122809
$317$	24.7396	1.38951	0.694756	−	0.719246i	$-0.255512\pi$
$317$	24.7396	1.38951	0.694756	+	0.719246i	$0.255512\pi$
$318$	0	0
$319$	−6.57077	−0.367892
$320$	−38.4435	−2.14906
$321$	0	0
$322$	2.99670	0.167000
$323$	4.54056	0.252644
$324$	0	0
$325$	14.9205	0.827640
$326$	2.46726	0.136649
$327$	0	0
$328$	−16.1630	−0.892452
$329$	5.23626	0.288684
$330$	0	0
$331$	−6.06469	−0.333345	−0.166673	−	0.986012i	$-0.553302\pi$
$331$	−6.06469	−0.333345	−0.166673	+	0.986012i	$0.553302\pi$
$332$	−5.61895	−0.308380
$333$	0	0
$334$	12.4366	0.680499
$335$	−19.3797	−1.05883
$336$	0	0
$337$	−34.7463	−1.89275	−0.946377	−	0.323065i	$-0.895287\pi$
$337$	−34.7463	−1.89275	−0.946377	+	0.323065i	$0.895287\pi$
$338$	14.0949	0.766663
$339$	0	0
$340$	−6.11820	−0.331806
$341$	3.92087	0.212327
$342$	0	0
$343$	1.00000	0.0539949
$344$	4.88804	0.263545
$345$	0	0
$346$	14.5404	0.781696
$347$	−8.74382	−0.469393	−0.234697	−	0.972069i	$-0.575410\pi$
$347$	−8.74382	−0.469393	−0.234697	+	0.972069i	$0.575410\pi$
$348$	0	0
$349$	−1.46070	−0.0781897	−0.0390948	−	0.999236i	$-0.512447\pi$
$349$	−1.46070	−0.0781897	−0.0390948	+	0.999236i	$0.512447\pi$
$350$	−16.9596	−0.906530
$351$	0	0
$352$	−5.44567	−0.290255
$353$	13.6042	0.724078	0.362039	−	0.932163i	$-0.382081\pi$
$353$	13.6042	0.724078	0.362039	+	0.932163i	$0.382081\pi$
$354$	0	0
$355$	19.5713	1.03874
$356$	−2.06931	−0.109673
$357$	0	0
$358$	−6.94704	−0.367163
$359$	24.1334	1.27371	0.636856	−	0.770983i	$-0.280235\pi$
$359$	24.1334	1.27371	0.636856	+	0.770983i	$0.280235\pi$
$360$	0	0
$361$	−15.1506	−0.797402
$362$	−5.60657	−0.294675
$363$	0	0
$364$	−0.625517	−0.0327860
$365$	−7.33046	−0.383694
$366$	0	0
$367$	−37.0828	−1.93571	−0.967853	−	0.251518i	$-0.919070\pi$
$367$	−37.0828	−1.93571	−0.967853	+	0.251518i	$0.919070\pi$
$368$	6.17291	0.321785
$369$	0	0
$370$	−19.8621	−1.03258
$371$	−6.13449	−0.318487
$372$	0	0
$373$	−28.9626	−1.49963	−0.749814	−	0.661649i	$-0.769857\pi$
$373$	−28.9626	−1.49963	−0.749814	+	0.661649i	$0.769857\pi$
$374$	4.55629	0.235600
$375$	0	0
$376$	16.1091	0.830765
$377$	−4.10729	−0.211536
$378$	0	0
$379$	23.9193	1.22865	0.614325	−	0.789053i	$-0.289428\pi$
$379$	23.9193	1.22865	0.614325	+	0.789053i	$0.289428\pi$
$380$	−5.18685	−0.266080
$381$	0	0
$382$	−6.44492	−0.329751
$383$	−1.39678	−0.0713721	−0.0356860	−	0.999363i	$-0.511362\pi$
$383$	−1.39678	−0.0713721	−0.0356860	+	0.999363i	$0.511362\pi$
$384$	0	0
$385$	−7.32038	−0.373081
$386$	1.67779	0.0853971
$387$	0	0
$388$	−8.19522	−0.416049
$389$	−16.0305	−0.812776	−0.406388	−	0.913701i	$-0.633212\pi$
$389$	−16.0305	−0.812776	−0.406388	+	0.913701i	$0.633212\pi$
$390$	0	0
$391$	5.86373	0.296542
$392$	3.07646	0.155385
$393$	0	0
$394$	−18.3659	−0.925263
$395$	−1.59701	−0.0803545
$396$	0	0
$397$	−11.9774	−0.601129	−0.300565	−	0.953761i	$-0.597175\pi$
$397$	−11.9774	−0.601129	−0.300565	+	0.953761i	$0.597175\pi$
$398$	−30.8371	−1.54572
$399$	0	0
$400$	−34.9351	−1.74676
$401$	−28.8147	−1.43894	−0.719470	−	0.694524i	$-0.755615\pi$
$401$	−28.8147	−1.43894	−0.719470	+	0.694524i	$0.755615\pi$
$402$	0	0
$403$	2.45088	0.122087
$404$	−6.95868	−0.346207
$405$	0	0
$406$	4.66862	0.231700
$407$	−6.35670	−0.315090
$408$	0	0
$409$	25.7429	1.27290	0.636451	−	0.771317i	$-0.280402\pi$
$409$	25.7429	1.27290	0.636451	+	0.771317i	$0.280402\pi$
$410$	−27.3261	−1.34954
$411$	0	0
$412$	2.39143	0.117817
$413$	1.48950	0.0732934
$414$	0	0
$415$	−41.1045	−2.01774
$416$	−3.40401	−0.166895
$417$	0	0
$418$	3.86271	0.188931
$419$	−34.0800	−1.66491	−0.832457	−	0.554089i	$-0.813067\pi$
$419$	−34.0800	−1.66491	−0.832457	+	0.554089i	$0.813067\pi$
$420$	0	0
$421$	−37.0186	−1.80418	−0.902089	−	0.431551i	$-0.857967\pi$
$421$	−37.0186	−1.80418	−0.902089	+	0.431551i	$0.857967\pi$
$422$	2.56230	0.124731
$423$	0	0
$424$	−18.8725	−0.916529
$425$	−33.1853	−1.60973
$426$	0	0
$427$	0.959646	0.0464405
$428$	4.32538	0.209075
$429$	0	0
$430$	8.26399	0.398525
$431$	−9.99515	−0.481449	−0.240725	−	0.970593i	$-0.577385\pi$
$431$	−9.99515	−0.481449	−0.240725	+	0.970593i	$0.577385\pi$
$432$	0	0
$433$	6.52632	0.313635	0.156817	−	0.987628i	$-0.449877\pi$
$433$	6.52632	0.313635	0.156817	+	0.987628i	$0.449877\pi$
$434$	−2.78583	−0.133724
$435$	0	0
$436$	−3.23788	−0.155066
$437$	4.97112	0.237801
$438$	0	0
$439$	−33.3741	−1.59286	−0.796429	−	0.604733i	$-0.793280\pi$
$439$	−33.3741	−1.59286	−0.796429	+	0.604733i	$0.793280\pi$
$440$	−22.5209	−1.07364
$441$	0	0
$442$	2.84807	0.135469
$443$	33.4161	1.58765	0.793824	−	0.608148i	$-0.208087\pi$
$443$	33.4161	1.58765	0.793824	+	0.608148i	$0.208087\pi$
$444$	0	0
$445$	−15.1377	−0.717596
$446$	21.8959	1.03680
$447$	0	0
$448$	8.74182	0.413012
$449$	12.9184	0.609657	0.304829	−	0.952407i	$-0.401401\pi$
$449$	12.9184	0.609657	0.304829	+	0.952407i	$0.401401\pi$
$450$	0	0
$451$	−8.74546	−0.411808
$452$	−2.06501	−0.0971299
$453$	0	0
$454$	−32.7925	−1.53903
$455$	−4.57586	−0.214520
$456$	0	0
$457$	−8.04193	−0.376186	−0.188093	−	0.982151i	$-0.560231\pi$
$457$	−8.04193	−0.376186	−0.188093	+	0.982151i	$0.560231\pi$
$458$	−15.9342	−0.744558
$459$	0	0
$460$	−6.69836	−0.312313
$461$	8.93259	0.416032	0.208016	−	0.978125i	$-0.433299\pi$
$461$	8.93259	0.416032	0.208016	+	0.978125i	$0.433299\pi$
$462$	0	0
$463$	−3.88415	−0.180512	−0.0902559	−	0.995919i	$-0.528769\pi$
$463$	−3.88415	−0.180512	−0.0902559	+	0.995919i	$0.528769\pi$
$464$	9.61688	0.446453
$465$	0	0
$466$	0.845583	0.0391709
$467$	8.51733	0.394135	0.197068	−	0.980390i	$-0.436858\pi$
$467$	8.51733	0.394135	0.197068	+	0.980390i	$0.436858\pi$
$468$	0	0
$469$	4.40683	0.203489
$470$	27.2350	1.25626
$471$	0	0
$472$	4.58238	0.210921
$473$	2.64482	0.121609
$474$	0	0
$475$	−28.1337	−1.29086
$476$	1.39124	0.0637674
$477$	0	0
$478$	−4.89705	−0.223986
$479$	2.35800	0.107740	0.0538699	−	0.998548i	$-0.482844\pi$
$479$	2.35800	0.107740	0.0538699	+	0.998548i	$0.482844\pi$
$480$	0	0
$481$	−3.97348	−0.181175
$482$	24.0984	1.09765
$483$	0	0
$484$	4.94696	0.224862
$485$	−59.9508	−2.72223
$486$	0	0
$487$	−30.0551	−1.36193	−0.680964	−	0.732317i	$-0.738439\pi$
$487$	−30.0551	−1.36193	−0.680964	+	0.732317i	$0.738439\pi$
$488$	2.95231	0.133645
$489$	0	0
$490$	5.20123	0.234968
$491$	28.8366	1.30138	0.650689	−	0.759345i	$-0.274480\pi$
$491$	28.8366	1.30138	0.650689	+	0.759345i	$0.274480\pi$
$492$	0	0
$493$	9.13521	0.411429
$494$	2.41452	0.108634
$495$	0	0
$496$	−5.73853	−0.257668
$497$	−4.45038	−0.199627
$498$	0	0
$499$	−16.6515	−0.745425	−0.372712	−	0.927947i	$-0.621572\pi$
$499$	−16.6515	−0.745425	−0.372712	+	0.927947i	$0.621572\pi$
$500$	24.6904	1.10419
$501$	0	0
$502$	4.93251	0.220149
$503$	−22.9160	−1.02177	−0.510887	−	0.859648i	$-0.670683\pi$
$503$	−22.9160	−1.02177	−0.510887	+	0.859648i	$0.670683\pi$
$504$	0	0
$505$	−50.9051	−2.26525
$506$	4.98834	0.221759
$507$	0	0
$508$	0.601156	0.0266720
$509$	14.8044	0.656195	0.328098	−	0.944644i	$-0.393593\pi$
$509$	14.8044	0.656195	0.328098	+	0.944644i	$0.393593\pi$
$510$	0	0
$511$	1.66690	0.0737393
$512$	22.9605	1.01472
$513$	0	0
$514$	−27.2261	−1.20089
$515$	17.4941	0.770883
$516$	0	0
$517$	8.71632	0.383343
$518$	4.51652	0.198445
$519$	0	0
$520$	−14.0775	−0.617337
$521$	−34.1375	−1.49559	−0.747796	−	0.663929i	$-0.768888\pi$
$521$	−34.1375	−1.49559	−0.747796	+	0.663929i	$0.768888\pi$
$522$	0	0
$523$	20.4142	0.892651	0.446326	−	0.894871i	$-0.352732\pi$
$523$	20.4142	0.892651	0.446326	+	0.894871i	$0.352732\pi$
$524$	−4.19144	−0.183104
$525$	0	0
$526$	24.8425	1.08319
$527$	−5.45111	−0.237454
$528$	0	0
$529$	−16.5802	−0.720880
$530$	−31.9069	−1.38595
$531$	0	0
$532$	1.17946	0.0511360
$533$	−5.46666	−0.236787
$534$	0	0
$535$	31.6416	1.36799
$536$	13.5574	0.585592
$537$	0	0
$538$	−1.92305	−0.0829085
$539$	1.66461	0.0716998
$540$	0	0
$541$	−34.3273	−1.47585	−0.737924	−	0.674884i	$-0.764194\pi$
$541$	−34.3273	−1.47585	−0.737924	+	0.674884i	$0.764194\pi$
$542$	30.1780	1.29626
$543$	0	0
$544$	7.57101	0.324604
$545$	−23.6862	−1.01461
$546$	0	0
$547$	−26.5778	−1.13638	−0.568192	−	0.822896i	$-0.692357\pi$
$547$	−26.5778	−1.13638	−0.568192	+	0.822896i	$0.692357\pi$
$548$	−0.772770	−0.0330111
$549$	0	0
$550$	−28.2311	−1.20378
$551$	7.74459	0.329931
$552$	0	0
$553$	0.363151	0.0154427
$554$	−9.77600	−0.415342
$555$	0	0
$556$	−11.8506	−0.502579
$557$	−37.6319	−1.59452	−0.797258	−	0.603639i	$-0.793717\pi$
$557$	−37.6319	−1.59452	−0.797258	+	0.603639i	$0.793717\pi$
$558$	0	0
$559$	1.65324	0.0699244
$560$	10.7140	0.452750
$561$	0	0
$562$	24.0617	1.01498
$563$	−41.5353	−1.75050	−0.875251	−	0.483668i	$-0.839304\pi$
$563$	−41.5353	−1.75050	−0.875251	+	0.483668i	$0.839304\pi$
$564$	0	0
$565$	−15.1062	−0.635525
$566$	−26.0911	−1.09669
$567$	0	0
$568$	−13.6914	−0.574479
$569$	−7.29064	−0.305639	−0.152820	−	0.988254i	$-0.548835\pi$
$569$	−7.29064	−0.305639	−0.152820	+	0.988254i	$0.548835\pi$
$570$	0	0
$571$	−2.75961	−0.115486	−0.0577431	−	0.998331i	$-0.518390\pi$
$571$	−2.75961	−0.115486	−0.0577431	+	0.998331i	$0.518390\pi$
$572$	−1.04124	−0.0435364
$573$	0	0
$574$	6.21377	0.259358
$575$	−36.3321	−1.51515
$576$	0	0
$577$	−17.5687	−0.731393	−0.365697	−	0.930734i	$-0.619169\pi$
$577$	−17.5687	−0.731393	−0.365697	+	0.930734i	$0.619169\pi$
$578$	13.7718	0.572833
$579$	0	0
$580$	−10.4355	−0.433310
$581$	9.34690	0.387775
$582$	0	0
$583$	−10.2115	−0.422918
$584$	5.12815	0.212204
$585$	0	0
$586$	21.9909	0.908434
$587$	12.6926	0.523878	0.261939	−	0.965084i	$-0.415638\pi$
$587$	12.6926	0.523878	0.261939	+	0.965084i	$0.415638\pi$
$588$	0	0
$589$	−4.62131	−0.190418
$590$	7.74722	0.318948
$591$	0	0
$592$	9.30358	0.382375
$593$	−8.14472	−0.334464	−0.167232	−	0.985918i	$-0.553483\pi$
$593$	−8.14472	−0.334464	−0.167232	+	0.985918i	$0.553483\pi$
$594$	0	0
$595$	10.1774	0.417232
$596$	4.94184	0.202425
$597$	0	0
$598$	3.11814	0.127510
$599$	13.8428	0.565603	0.282802	−	0.959178i	$-0.408736\pi$
$599$	13.8428	0.565603	0.282802	+	0.959178i	$0.408736\pi$
$600$	0	0
$601$	−11.5041	−0.469262	−0.234631	−	0.972084i	$-0.575388\pi$
$601$	−11.5041	−0.469262	−0.234631	+	0.972084i	$0.575388\pi$
$602$	−1.87918	−0.0765896
$603$	0	0
$604$	0.254002	0.0103352
$605$	36.1887	1.47128
$606$	0	0
$607$	−1.32544	−0.0537980	−0.0268990	−	0.999638i	$-0.508563\pi$
$607$	−1.32544	−0.0537980	−0.0268990	+	0.999638i	$0.508563\pi$
$608$	6.41850	0.260305
$609$	0	0
$610$	4.99134	0.202093
$611$	5.44844	0.220420
$612$	0	0
$613$	−38.1974	−1.54278	−0.771389	−	0.636364i	$-0.780438\pi$
$613$	−38.1974	−1.54278	−0.771389	+	0.636364i	$0.780438\pi$
$614$	14.9385	0.602870
$615$	0	0
$616$	5.12110	0.206335
$617$	39.5442	1.59199	0.795994	−	0.605304i	$-0.206949\pi$
$617$	39.5442	1.59199	0.795994	+	0.605304i	$0.206949\pi$
$618$	0	0
$619$	13.9857	0.562131	0.281065	−	0.959689i	$-0.409312\pi$
$619$	13.9857	0.562131	0.281065	+	0.959689i	$0.409312\pi$
$620$	6.22701	0.250083
$621$	0	0
$622$	−32.9588	−1.32153
$623$	3.44222	0.137910
$624$	0	0
$625$	108.922	4.35687
$626$	6.01475	0.240398
$627$	0	0
$628$	−8.23087	−0.328447
$629$	8.83760	0.352378
$630$	0	0
$631$	18.8959	0.752234	0.376117	−	0.926572i	$-0.377259\pi$
$631$	18.8959	0.752234	0.376117	+	0.926572i	$0.377259\pi$
$632$	1.11722	0.0444406
$633$	0	0
$634$	−29.2601	−1.16207
$635$	4.39766	0.174516
$636$	0	0
$637$	1.04052	0.0412270
$638$	7.77142	0.307674
$639$	0	0
$640$	16.6949	0.659923
$641$	35.1899	1.38992	0.694959	−	0.719049i	$-0.255422\pi$
$641$	35.1899	1.38992	0.694959	+	0.719049i	$0.255422\pi$
$642$	0	0
$643$	3.65365	0.144086	0.0720429	−	0.997402i	$-0.477048\pi$
$643$	3.65365	0.144086	0.0720429	+	0.997402i	$0.477048\pi$
$644$	1.52316	0.0600211
$645$	0	0
$646$	−5.37025	−0.211290
$647$	22.0251	0.865897	0.432949	−	0.901419i	$-0.357473\pi$
$647$	22.0251	0.865897	0.432949	+	0.901419i	$0.357473\pi$
$648$	0	0
$649$	2.47943	0.0973261
$650$	−17.6469	−0.692167
$651$	0	0
$652$	1.25406	0.0491127
$653$	−3.18838	−0.124771	−0.0623855	−	0.998052i	$-0.519871\pi$
$653$	−3.18838	−0.124771	−0.0623855	+	0.998052i	$0.519871\pi$
$654$	0	0
$655$	−30.6618	−1.19806
$656$	12.7997	0.499746
$657$	0	0
$658$	−6.19307	−0.241431
$659$	13.0590	0.508708	0.254354	−	0.967111i	$-0.418137\pi$
$659$	13.0590	0.508708	0.254354	+	0.967111i	$0.418137\pi$
$660$	0	0
$661$	16.6245	0.646620	0.323310	−	0.946293i	$-0.395204\pi$
$661$	16.6245	0.646620	0.323310	+	0.946293i	$0.395204\pi$
$662$	7.17287	0.278782
$663$	0	0
$664$	28.7554	1.11592
$665$	8.62813	0.334584
$666$	0	0
$667$	10.0015	0.387258
$668$	6.32126	0.244577
$669$	0	0
$670$	22.9210	0.885514
$671$	1.59743	0.0616683
$672$	0	0
$673$	−5.00147	−0.192792	−0.0963962	−	0.995343i	$-0.530732\pi$
$673$	−5.00147	−0.192792	−0.0963962	+	0.995343i	$0.530732\pi$
$674$	41.0954	1.58294
$675$	0	0
$676$	7.16417	0.275545
$677$	18.0201	0.692568	0.346284	−	0.938130i	$-0.387443\pi$
$677$	18.0201	0.692568	0.346284	+	0.938130i	$0.387443\pi$
$678$	0	0
$679$	13.6324	0.523165
$680$	31.3103	1.20070
$681$	0	0
$682$	−4.63732	−0.177572
$683$	−10.1203	−0.387241	−0.193621	−	0.981076i	$-0.562023\pi$
$683$	−10.1203	−0.387241	−0.193621	+	0.981076i	$0.562023\pi$
$684$	0	0
$685$	−5.65307	−0.215993
$686$	−1.18273	−0.0451567
$687$	0	0
$688$	−3.87092	−0.147577
$689$	−6.38307	−0.243176
$690$	0	0
$691$	−16.4471	−0.625677	−0.312838	−	0.949806i	$-0.601280\pi$
$691$	−16.4471	−0.625677	−0.312838	+	0.949806i	$0.601280\pi$
$692$	7.39059	0.280948
$693$	0	0
$694$	10.3416	0.392560
$695$	−86.6914	−3.28839
$696$	0	0
$697$	12.1587	0.460542
$698$	1.72761	0.0653911
$699$	0	0
$700$	−8.62023	−0.325814
$701$	−24.1464	−0.911998	−0.455999	−	0.889980i	$-0.650718\pi$
$701$	−24.1464	−0.911998	−0.455999	+	0.889980i	$0.650718\pi$
$702$	0	0
$703$	7.49229	0.282577
$704$	14.5517	0.548438
$705$	0	0
$706$	−16.0900	−0.605557
$707$	11.5755	0.435341
$708$	0	0
$709$	47.5814	1.78696	0.893479	−	0.449105i	$-0.148257\pi$
$709$	47.5814	1.78696	0.893479	+	0.449105i	$0.148257\pi$
$710$	−23.1475	−0.868710
$711$	0	0
$712$	10.5898	0.396871
$713$	−5.96801	−0.223504
$714$	0	0
$715$	−7.61702	−0.284861
$716$	−3.53104	−0.131961
$717$	0	0
$718$	−28.5432	−1.06522
$719$	−12.1620	−0.453568	−0.226784	−	0.973945i	$-0.572821\pi$
$719$	−12.1620	−0.453568	−0.226784	+	0.973945i	$0.572821\pi$
$720$	0	0
$721$	−3.97805	−0.148150
$722$	17.9191	0.666878
$723$	0	0
$724$	−2.84971	−0.105908
$725$	−56.6025	−2.10216
$726$	0	0
$727$	34.5875	1.28278	0.641390	−	0.767215i	$-0.278358\pi$
$727$	34.5875	1.28278	0.641390	+	0.767215i	$0.278358\pi$
$728$	3.20112	0.118642
$729$	0	0
$730$	8.66993	0.320889
$731$	−3.67704	−0.136000
$732$	0	0
$733$	−26.1614	−0.966293	−0.483146	−	0.875540i	$-0.660506\pi$
$733$	−26.1614	−0.966293	−0.483146	+	0.875540i	$0.660506\pi$
$734$	43.8588	1.61886
$735$	0	0
$736$	8.28893	0.305534
$737$	7.33565	0.270212
$738$	0	0
$739$	24.5920	0.904632	0.452316	−	0.891858i	$-0.350598\pi$
$739$	24.5920	0.904632	0.452316	+	0.891858i	$0.350598\pi$
$740$	−10.0955	−0.371119
$741$	0	0
$742$	7.25543	0.266355
$743$	−7.02907	−0.257871	−0.128936	−	0.991653i	$-0.541156\pi$
$743$	−7.02907	−0.257871	−0.128936	+	0.991653i	$0.541156\pi$
$744$	0	0
$745$	36.1512	1.32448
$746$	34.2549	1.25416
$747$	0	0
$748$	2.31587	0.0846766
$749$	−7.19510	−0.262903
$750$	0	0
$751$	39.5507	1.44323	0.721614	−	0.692296i	$-0.243401\pi$
$751$	39.5507	1.44323	0.721614	+	0.692296i	$0.243401\pi$
$752$	−12.7571	−0.465203
$753$	0	0
$754$	4.85780	0.176911
$755$	1.85811	0.0676235
$756$	0	0
$757$	19.8991	0.723247	0.361623	−	0.932324i	$-0.382223\pi$
$757$	19.8991	0.723247	0.361623	+	0.932324i	$0.382223\pi$
$758$	−28.2900	−1.02754
$759$	0	0
$760$	26.5441	0.962855
$761$	−21.1800	−0.767774	−0.383887	−	0.923380i	$-0.625415\pi$
$761$	−21.1800	−0.767774	−0.383887	+	0.923380i	$0.625415\pi$
$762$	0	0
$763$	5.38609	0.194990
$764$	−3.27582	−0.118515
$765$	0	0
$766$	1.65201	0.0596895
$767$	1.54985	0.0559620
$768$	0	0
$769$	−16.1861	−0.583684	−0.291842	−	0.956466i	$-0.594268\pi$
$769$	−16.1861	−0.583684	−0.291842	+	0.956466i	$0.594268\pi$
$770$	8.65802	0.312013
$771$	0	0
$772$	0.852785	0.0306924
$773$	26.5228	0.953959	0.476980	−	0.878914i	$-0.341732\pi$
$773$	26.5228	0.953959	0.476980	+	0.878914i	$0.341732\pi$
$774$	0	0
$775$	33.7755	1.21325
$776$	41.9396	1.50554
$777$	0	0
$778$	18.9597	0.679737
$779$	10.3078	0.369315
$780$	0	0
$781$	−7.40815	−0.265084
$782$	−6.93520	−0.248002
$783$	0	0
$784$	−2.43630	−0.0870107
$785$	−60.2116	−2.14904
$786$	0	0
$787$	49.9403	1.78018	0.890090	−	0.455784i	$-0.150641\pi$
$787$	49.9403	1.78018	0.890090	+	0.455784i	$0.150641\pi$
$788$	−9.33503	−0.332547
$789$	0	0
$790$	1.88883	0.0672016
$791$	3.43506	0.122137
$792$	0	0
$793$	0.998533	0.0354589
$794$	14.1660	0.502733
$795$	0	0
$796$	−15.6738	−0.555545
$797$	−23.1748	−0.820895	−0.410447	−	0.911884i	$-0.634627\pi$
$797$	−23.1748	−0.820895	−0.410447	+	0.911884i	$0.634627\pi$
$798$	0	0
$799$	−12.1181	−0.428709
$800$	−46.9105	−1.65854
$801$	0	0
$802$	34.0800	1.20341
$803$	2.77474	0.0979183
$804$	0	0
$805$	11.1425	0.392720
$806$	−2.89872	−0.102103
$807$	0	0
$808$	35.6115	1.25281
$809$	−44.9574	−1.58062	−0.790308	−	0.612710i	$-0.790080\pi$
$809$	−44.9574	−1.58062	−0.790308	+	0.612710i	$0.790080\pi$
$810$	0	0
$811$	−28.9877	−1.01790	−0.508949	−	0.860797i	$-0.669966\pi$
$811$	−28.9877	−1.01790	−0.508949	+	0.860797i	$0.669966\pi$
$812$	2.37296	0.0832747
$813$	0	0
$814$	7.51825	0.263514
$815$	9.17386	0.321346
$816$	0	0
$817$	−3.11730	−0.109060
$818$	−30.4468	−1.06455
$819$	0	0
$820$	−13.8893	−0.485035
$821$	−40.5069	−1.41370	−0.706850	−	0.707363i	$-0.749885\pi$
$821$	−40.5069	−1.41370	−0.706850	+	0.707363i	$0.749885\pi$
$822$	0	0
$823$	19.0163	0.662866	0.331433	−	0.943479i	$-0.392468\pi$
$823$	19.0163	0.662866	0.331433	+	0.943479i	$0.392468\pi$
$824$	−12.2383	−0.426342
$825$	0	0
$826$	−1.76167	−0.0612963
$827$	−9.67417	−0.336404	−0.168202	−	0.985753i	$-0.553796\pi$
$827$	−9.67417	−0.336404	−0.168202	+	0.985753i	$0.553796\pi$
$828$	0	0
$829$	19.7818	0.687049	0.343525	−	0.939144i	$-0.388379\pi$
$829$	19.7818	0.687049	0.343525	+	0.939144i	$0.388379\pi$
$830$	48.6154	1.68747
$831$	0	0
$832$	9.09606	0.315349
$833$	−2.31427	−0.0801848
$834$	0	0
$835$	46.2421	1.60028
$836$	1.96333	0.0679033
$837$	0	0
$838$	40.3073	1.39239
$839$	−20.1087	−0.694229	−0.347114	−	0.937823i	$-0.612838\pi$
$839$	−20.1087	−0.694229	−0.347114	+	0.937823i	$0.612838\pi$
$840$	0	0
$841$	−13.4186	−0.462709
$842$	43.7829	1.50886
$843$	0	0
$844$	1.30236	0.0448292
$845$	52.4083	1.80290
$846$	0	0
$847$	−8.22908	−0.282754
$848$	14.9454	0.513229
$849$	0	0
$850$	39.2492	1.34624
$851$	9.67563	0.331676
$852$	0	0
$853$	48.0343	1.64466	0.822332	−	0.569008i	$-0.192672\pi$
$853$	48.0343	1.64466	0.822332	+	0.569008i	$0.192672\pi$
$854$	−1.13500	−0.0388389
$855$	0	0
$856$	−22.1354	−0.756573
$857$	−21.9389	−0.749417	−0.374709	−	0.927143i	$-0.622257\pi$
$857$	−21.9389	−0.749417	−0.374709	+	0.927143i	$0.622257\pi$
$858$	0	0
$859$	−36.0049	−1.22847	−0.614236	−	0.789122i	$-0.710536\pi$
$859$	−36.0049	−1.22847	−0.614236	+	0.789122i	$0.710536\pi$
$860$	4.20042	0.143233
$861$	0	0
$862$	11.8215	0.402643
$863$	−37.6822	−1.28272	−0.641358	−	0.767242i	$-0.721629\pi$
$863$	−37.6822	−1.28272	−0.641358	+	0.767242i	$0.721629\pi$
$864$	0	0
$865$	54.0646	1.83825
$866$	−7.71885	−0.262297
$867$	0	0
$868$	−1.41598	−0.0480616
$869$	0.604504	0.0205064
$870$	0	0
$871$	4.58541	0.155371
$872$	16.5701	0.561134
$873$	0	0
$874$	−5.87948	−0.198876
$875$	−41.0716	−1.38847
$876$	0	0
$877$	−5.87617	−0.198424	−0.0992122	−	0.995066i	$-0.531632\pi$
$877$	−5.87617	−0.198424	−0.0992122	+	0.995066i	$0.531632\pi$
$878$	39.4724	1.33213
$879$	0	0
$880$	17.8346	0.601206
$881$	−12.8017	−0.431299	−0.215649	−	0.976471i	$-0.569187\pi$
$881$	−12.8017	−0.431299	−0.215649	+	0.976471i	$0.569187\pi$
$882$	0	0
$883$	−40.7111	−1.37004	−0.685018	−	0.728526i	$-0.740206\pi$
$883$	−40.7111	−1.37004	−0.685018	+	0.728526i	$0.740206\pi$
$884$	1.44762	0.0486886
$885$	0	0
$886$	−39.5222	−1.32777
$887$	−8.81316	−0.295917	−0.147958	−	0.988994i	$-0.547270\pi$
$887$	−8.81316	−0.295917	−0.147958	+	0.988994i	$0.547270\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	17.9038	0.600136
$891$	0	0
$892$	11.1292	0.372634
$893$	−10.2734	−0.343788
$894$	0	0
$895$	−25.8308	−0.863427
$896$	−3.79631	−0.126826
$897$	0	0
$898$	−15.2789	−0.509865
$899$	−9.29767	−0.310095
$900$	0	0
$901$	14.1969	0.472967
$902$	10.3435	0.344401
$903$	0	0
$904$	10.5678	0.351481
$905$	−20.8466	−0.692963
$906$	0	0
$907$	11.6102	0.385511	0.192756	−	0.981247i	$-0.438258\pi$
$907$	11.6102	0.385511	0.192756	+	0.981247i	$0.438258\pi$
$908$	−16.6678	−0.553139
$909$	0	0
$910$	5.41200	0.179406
$911$	36.4476	1.20756	0.603782	−	0.797150i	$-0.293660\pi$
$911$	36.4476	1.20756	0.603782	+	0.797150i	$0.293660\pi$
$912$	0	0
$913$	15.5589	0.514926
$914$	9.51141	0.314610
$915$	0	0
$916$	−8.09905	−0.267600
$917$	6.97230	0.230246
$918$	0	0
$919$	−40.0025	−1.31956	−0.659780	−	0.751459i	$-0.729351\pi$
$919$	−40.0025	−1.31956	−0.659780	+	0.751459i	$0.729351\pi$
$920$	34.2793	1.13016
$921$	0	0
$922$	−10.5648	−0.347934
$923$	−4.63072	−0.152422
$924$	0	0
$925$	−54.7585	−1.80045
$926$	4.59389	0.150965
$927$	0	0
$928$	12.9135	0.423905
$929$	28.4033	0.931881	0.465940	−	0.884816i	$-0.345716\pi$
$929$	28.4033	0.931881	0.465940	+	0.884816i	$0.345716\pi$
$930$	0	0
$931$	−1.96198	−0.0643013
$932$	0.429793	0.0140783
$933$	0	0
$934$	−10.0737	−0.329621
$935$	16.9414	0.554042
$936$	0	0
$937$	19.9227	0.650848	0.325424	−	0.945568i	$-0.394493\pi$
$937$	19.9227	0.650848	0.325424	+	0.945568i	$0.394493\pi$
$938$	−5.21208	−0.170180
$939$	0	0
$940$	13.8430	0.451509
$941$	21.0072	0.684815	0.342408	−	0.939551i	$-0.388758\pi$
$941$	21.0072	0.684815	0.342408	+	0.939551i	$0.388758\pi$
$942$	0	0
$943$	13.3116	0.433485
$944$	−3.62886	−0.118109
$945$	0	0
$946$	−3.12810	−0.101703
$947$	−16.8945	−0.548998	−0.274499	−	0.961587i	$-0.588512\pi$
$947$	−16.8945	−0.548998	−0.274499	+	0.961587i	$0.588512\pi$
$948$	0	0
$949$	1.73445	0.0563025
$950$	33.2745	1.07957
$951$	0	0
$952$	−7.11977	−0.230753
$953$	−18.3733	−0.595169	−0.297584	−	0.954696i	$-0.596181\pi$
$953$	−18.3733	−0.595169	−0.297584	+	0.954696i	$0.596181\pi$
$954$	0	0
$955$	−23.9637	−0.775448
$956$	−2.48907	−0.0805023
$957$	0	0
$958$	−2.78887	−0.0901044
$959$	1.28547	0.0415101
$960$	0	0
$961$	−25.4519	−0.821030
$962$	4.69954	0.151519
$963$	0	0
$964$	12.2487	0.394505
$965$	6.23841	0.200821
$966$	0	0
$967$	−40.9671	−1.31741	−0.658707	−	0.752399i	$-0.728896\pi$
$967$	−40.9671	−1.31741	−0.658707	+	0.752399i	$0.728896\pi$
$968$	−25.3164	−0.813700
$969$	0	0
$970$	70.9055	2.27664
$971$	41.1426	1.32033	0.660164	−	0.751121i	$-0.270487\pi$
$971$	41.1426	1.32033	0.660164	+	0.751121i	$0.270487\pi$
$972$	0	0
$973$	19.7131	0.631972
$974$	35.5470	1.13900
$975$	0	0
$976$	−2.33798	−0.0748370
$977$	−16.2215	−0.518972	−0.259486	−	0.965747i	$-0.583553\pi$
$977$	−16.2215	−0.518972	−0.259486	+	0.965747i	$0.583553\pi$
$978$	0	0
$979$	5.72995	0.183130
$980$	2.64368	0.0844493
$981$	0	0
$982$	−34.1058	−1.08836
$983$	32.5695	1.03881	0.519403	−	0.854529i	$-0.326154\pi$
$983$	32.5695	1.03881	0.519403	+	0.854529i	$0.326154\pi$
$984$	0	0
$985$	−68.2889	−2.17587
$986$	−10.8045	−0.344084
$987$	0	0
$988$	1.22725	0.0390441
$989$	−4.02571	−0.128010
$990$	0	0
$991$	−20.4145	−0.648487	−0.324244	−	0.945974i	$-0.605110\pi$
$991$	−20.4145	−0.648487	−0.324244	+	0.945974i	$0.605110\pi$
$992$	−7.70565	−0.244655
$993$	0	0
$994$	5.26359	0.166951
$995$	−114.659	−3.63495
$996$	0	0
$997$	−9.35391	−0.296241	−0.148121	−	0.988969i	$-0.547322\pi$
$997$	−9.35391	−0.296241	−0.148121	+	0.988969i	$0.547322\pi$
$998$	19.6942	0.623410
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.6		13
3.2	odd	2		2667.2.a.l.1.8	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.8	✓	13	3.2	odd	2
8001.2.a.o.1.6		13	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)

\(f(q)\)	\(=\)	\(q-1.18273 q^{2} -0.601156 q^{4} -4.39766 q^{5} +1.00000 q^{7} +3.07646 q^{8} +O(q^{10})\)	\(q-1.18273 q^{2} -0.601156 q^{4} -4.39766 q^{5} +1.00000 q^{7} +3.07646 q^{8} +5.20123 q^{10} +1.66461 q^{11} +1.04052 q^{13} -1.18273 q^{14} -2.43630 q^{16} -2.31427 q^{17} -1.96198 q^{19} +2.64368 q^{20} -1.96878 q^{22} -2.53372 q^{23} +14.3394 q^{25} -1.23065 q^{26} -0.601156 q^{28} -3.94733 q^{29} +2.35543 q^{31} -3.27144 q^{32} +2.73716 q^{34} -4.39766 q^{35} -3.81874 q^{37} +2.32049 q^{38} -13.5292 q^{40} -5.25377 q^{41} +1.58885 q^{43} -1.00069 q^{44} +2.99670 q^{46} +5.23626 q^{47} +1.00000 q^{49} -16.9596 q^{50} -0.625517 q^{52} -6.13449 q^{53} -7.32038 q^{55} +3.07646 q^{56} +4.66862 q^{58} +1.48950 q^{59} +0.959646 q^{61} -2.78583 q^{62} +8.74182 q^{64} -4.57586 q^{65} +4.40683 q^{67} +1.39124 q^{68} +5.20123 q^{70} -4.45038 q^{71} +1.66690 q^{73} +4.51652 q^{74} +1.17946 q^{76} +1.66461 q^{77} +0.363151 q^{79} +10.7140 q^{80} +6.21377 q^{82} +9.34690 q^{83} +10.1774 q^{85} -1.87918 q^{86} +5.12110 q^{88} +3.44222 q^{89} +1.04052 q^{91} +1.52316 q^{92} -6.19307 q^{94} +8.62813 q^{95} +13.6324 q^{97} -1.18273 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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