LMFDB - Embedded newform 8001.2.a.l.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:	$7$
Coefficient field:	7.7.118870813.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 9x^{5} - 3x^{4} + 20x^{3} + 7x^{2} - 13x - 4 $ x^7 - 9x^5 - 3x^4 + 20x^3 + 7x^2 - 13*x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.14753$ 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)

$f(q)$	$=$	$q+2.15625 q^{2} +2.64943 q^{4} +0.239094 q^{5} +1.00000 q^{7} +1.40032 q^{8} +O(q^{10})$	\(q+2.15625 q^{2} +2.64943 q^{4} +0.239094 q^{5} +1.00000 q^{7} +1.40032 q^{8} +0.515547 q^{10} +3.99784 q^{11} -1.61682 q^{13} +2.15625 q^{14} -2.27940 q^{16} -4.61421 q^{17} -4.17289 q^{19} +0.633462 q^{20} +8.62035 q^{22} -5.22408 q^{23} -4.94283 q^{25} -3.48627 q^{26} +2.64943 q^{28} -9.10130 q^{29} -6.57815 q^{31} -7.71560 q^{32} -9.94939 q^{34} +0.239094 q^{35} -0.353708 q^{37} -8.99781 q^{38} +0.334809 q^{40} +7.72038 q^{41} -7.33166 q^{43} +10.5920 q^{44} -11.2644 q^{46} -3.52396 q^{47} +1.00000 q^{49} -10.6580 q^{50} -4.28365 q^{52} +0.655025 q^{53} +0.955860 q^{55} +1.40032 q^{56} -19.6247 q^{58} +11.8415 q^{59} -7.05891 q^{61} -14.1842 q^{62} -12.0780 q^{64} -0.386572 q^{65} -0.167504 q^{67} -12.2250 q^{68} +0.515547 q^{70} -6.98566 q^{71} -2.79662 q^{73} -0.762683 q^{74} -11.0558 q^{76} +3.99784 q^{77} +14.2526 q^{79} -0.544990 q^{80} +16.6471 q^{82} +6.75526 q^{83} -1.10323 q^{85} -15.8089 q^{86} +5.59827 q^{88} +11.9943 q^{89} -1.61682 q^{91} -13.8408 q^{92} -7.59856 q^{94} -0.997714 q^{95} -1.79091 q^{97} +2.15625 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * 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 $

$2$	2.15625	1.52470	0.762350	−	0.647164i	$-0.224045\pi$
$2$	2.15625	1.52470	0.762350	+	0.647164i	$0.224045\pi$
$3$	0	0
$3$	0	0
$4$	2.64943	1.32471
$5$	0.239094	0.106926	0.0534631	−	0.998570i	$-0.482974\pi$
$5$	0.239094	0.106926	0.0534631	+	0.998570i	$0.482974\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.40032	0.495090
$9$	0	0
$10$	0.515547	0.163030
$11$	3.99784	1.20539	0.602697	−	0.797970i	$-0.294093\pi$
$11$	3.99784	1.20539	0.602697	+	0.797970i	$0.294093\pi$
$12$	0	0
$13$	−1.61682	−0.448425	−0.224213	−	0.974540i	$-0.571981\pi$
$13$	−1.61682	−0.448425	−0.224213	+	0.974540i	$0.571981\pi$
$14$	2.15625	0.576283
$15$	0	0
$16$	−2.27940	−0.569849
$17$	−4.61421	−1.11911	−0.559555	−	0.828793i	$-0.689028\pi$
$17$	−4.61421	−1.11911	−0.559555	+	0.828793i	$0.689028\pi$
$18$	0	0
$19$	−4.17289	−0.957328	−0.478664	−	0.877998i	$-0.658879\pi$
$19$	−4.17289	−0.957328	−0.478664	+	0.877998i	$0.658879\pi$
$20$	0.633462	0.141646
$21$	0	0
$22$	8.62035	1.83786
$23$	−5.22408	−1.08930	−0.544648	−	0.838665i	$-0.683337\pi$
$23$	−5.22408	−1.08930	−0.544648	+	0.838665i	$0.683337\pi$
$24$	0	0
$25$	−4.94283	−0.988567
$26$	−3.48627	−0.683714
$27$	0	0
$28$	2.64943	0.500694
$29$	−9.10130	−1.69007	−0.845035	−	0.534712i	$-0.820420\pi$
$29$	−9.10130	−1.69007	−0.845035	+	0.534712i	$0.820420\pi$
$30$	0	0
$31$	−6.57815	−1.18147	−0.590736	−	0.806865i	$-0.701162\pi$
$31$	−6.57815	−1.18147	−0.590736	+	0.806865i	$0.701162\pi$
$32$	−7.71560	−1.36394
$33$	0	0
$34$	−9.94939	−1.70631
$35$	0.239094	0.0404143
$36$	0	0
$37$	−0.353708	−0.0581492	−0.0290746	−	0.999577i	$-0.509256\pi$
$37$	−0.353708	−0.0581492	−0.0290746	+	0.999577i	$0.509256\pi$
$38$	−8.99781	−1.45964
$39$	0	0
$40$	0.334809	0.0529380
$41$	7.72038	1.20572	0.602860	−	0.797847i	$-0.294028\pi$
$41$	7.72038	1.20572	0.602860	+	0.797847i	$0.294028\pi$
$42$	0	0
$43$	−7.33166	−1.11807	−0.559034	−	0.829145i	$-0.688828\pi$
$43$	−7.33166	−1.11807	−0.559034	+	0.829145i	$0.688828\pi$
$44$	10.5920	1.59680
$45$	0	0
$46$	−11.2644	−1.66085
$47$	−3.52396	−0.514023	−0.257011	−	0.966408i	$-0.582738\pi$
$47$	−3.52396	−0.514023	−0.257011	+	0.966408i	$0.582738\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.6580	−1.50727
$51$	0	0
$52$	−4.28365	−0.594035
$53$	0.655025	0.0899745	0.0449873	−	0.998988i	$-0.485675\pi$
$53$	0.655025	0.0899745	0.0449873	+	0.998988i	$0.485675\pi$
$54$	0	0
$55$	0.955860	0.128888
$56$	1.40032	0.187126
$57$	0	0
$58$	−19.6247	−2.57685
$59$	11.8415	1.54163	0.770814	−	0.637060i	$-0.219850\pi$
$59$	11.8415	1.54163	0.770814	+	0.637060i	$0.219850\pi$
$60$	0	0
$61$	−7.05891	−0.903801	−0.451901	−	0.892068i	$-0.649254\pi$
$61$	−7.05891	−0.903801	−0.451901	+	0.892068i	$0.649254\pi$
$62$	−14.1842	−1.80139
$63$	0	0
$64$	−12.0780	−1.50975
$65$	−0.386572	−0.0479484
$66$	0	0
$67$	−0.167504	−0.0204638	−0.0102319	−	0.999948i	$-0.503257\pi$
$67$	−0.167504	−0.0204638	−0.0102319	+	0.999948i	$0.503257\pi$
$68$	−12.2250	−1.48250
$69$	0	0
$70$	0.515547	0.0616197
$71$	−6.98566	−0.829045	−0.414523	−	0.910039i	$-0.636051\pi$
$71$	−6.98566	−0.829045	−0.414523	+	0.910039i	$0.636051\pi$
$72$	0	0
$73$	−2.79662	−0.327320	−0.163660	−	0.986517i	$-0.552330\pi$
$73$	−2.79662	−0.327320	−0.163660	+	0.986517i	$0.552330\pi$
$74$	−0.762683	−0.0886601
$75$	0	0
$76$	−11.0558	−1.26818
$77$	3.99784	0.455596
$78$	0	0
$79$	14.2526	1.60354	0.801770	−	0.597633i	$-0.203892\pi$
$79$	14.2526	1.60354	0.801770	+	0.597633i	$0.203892\pi$
$80$	−0.544990	−0.0609317
$81$	0	0
$82$	16.6471	1.83836
$83$	6.75526	0.741486	0.370743	−	0.928735i	$-0.379103\pi$
$83$	6.75526	0.741486	0.370743	+	0.928735i	$0.379103\pi$
$84$	0	0
$85$	−1.10323	−0.119662
$86$	−15.8089	−1.70472
$87$	0	0
$88$	5.59827	0.596778
$89$	11.9943	1.27140	0.635698	−	0.771938i	$-0.280712\pi$
$89$	11.9943	1.27140	0.635698	+	0.771938i	$0.280712\pi$
$90$	0	0
$91$	−1.61682	−0.169489
$92$	−13.8408	−1.44300
$93$	0	0
$94$	−7.59856	−0.783731
$95$	−0.997714	−0.102363
$96$	0	0
$97$	−1.79091	−0.181840	−0.0909199	−	0.995858i	$-0.528981\pi$
$97$	−1.79091	−0.181840	−0.0909199	+	0.995858i	$0.528981\pi$
$98$	2.15625	0.217814
$99$	0	0
$100$	−13.0957	−1.30957
$101$	11.0925	1.10375	0.551873	−	0.833928i	$-0.313913\pi$
$101$	11.0925	1.10375	0.551873	+	0.833928i	$0.313913\pi$
$102$	0	0
$103$	14.5981	1.43839	0.719197	−	0.694806i	$-0.244510\pi$
$103$	14.5981	1.43839	0.719197	+	0.694806i	$0.244510\pi$
$104$	−2.26407	−0.222011
$105$	0	0
$106$	1.41240	0.137184
$107$	−10.6926	−1.03369	−0.516845	−	0.856079i	$-0.672894\pi$
$107$	−10.6926	−1.03369	−0.516845	+	0.856079i	$0.672894\pi$
$108$	0	0
$109$	−19.1750	−1.83664	−0.918318	−	0.395844i	$-0.870452\pi$
$109$	−19.1750	−1.83664	−0.918318	+	0.395844i	$0.870452\pi$
$110$	2.06107	0.196516
$111$	0	0
$112$	−2.27940	−0.215383
$113$	13.9603	1.31327	0.656636	−	0.754208i	$-0.271979\pi$
$113$	13.9603	1.31327	0.656636	+	0.754208i	$0.271979\pi$
$114$	0	0
$115$	−1.24905	−0.116474
$116$	−24.1132	−2.23886
$117$	0	0
$118$	25.5332	2.35052
$119$	−4.61421	−0.422984
$120$	0	0
$121$	4.98272	0.452974
$122$	−15.2208	−1.37803
$123$	0	0
$124$	−17.4283	−1.56511
$125$	−2.37727	−0.212630
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−10.6120	−0.937978
$129$	0	0
$130$	−0.833547	−0.0731069
$131$	1.23738	0.108111	0.0540553	−	0.998538i	$-0.482785\pi$
$131$	1.23738	0.108111	0.0540553	+	0.998538i	$0.482785\pi$
$132$	0	0
$133$	−4.17289	−0.361836
$134$	−0.361180	−0.0312012
$135$	0	0
$136$	−6.46139	−0.554059
$137$	5.10054	0.435769	0.217884	−	0.975975i	$-0.430084\pi$
$137$	5.10054	0.435769	0.217884	+	0.975975i	$0.430084\pi$
$138$	0	0
$139$	12.5215	1.06206	0.531032	−	0.847352i	$-0.321805\pi$
$139$	12.5215	1.06206	0.531032	+	0.847352i	$0.321805\pi$
$140$	0.633462	0.0535373
$141$	0	0
$142$	−15.0628	−1.26405
$143$	−6.46379	−0.540529
$144$	0	0
$145$	−2.17607	−0.180713
$146$	−6.03022	−0.499065
$147$	0	0
$148$	−0.937122	−0.0770310
$149$	−9.58767	−0.785452	−0.392726	−	0.919655i	$-0.628468\pi$
$149$	−9.58767	−0.785452	−0.392726	+	0.919655i	$0.628468\pi$
$150$	0	0
$151$	−17.1326	−1.39423	−0.697117	−	0.716957i	$-0.745534\pi$
$151$	−17.1326	−1.39423	−0.697117	+	0.716957i	$0.745534\pi$
$152$	−5.84341	−0.473963
$153$	0	0
$154$	8.62035	0.694648
$155$	−1.57280	−0.126330
$156$	0	0
$157$	7.98480	0.637257	0.318628	−	0.947880i	$-0.396778\pi$
$157$	7.98480	0.637257	0.318628	+	0.947880i	$0.396778\pi$
$158$	30.7322	2.44492
$159$	0	0
$160$	−1.84476	−0.145841
$161$	−5.22408	−0.411715
$162$	0	0
$163$	5.01915	0.393130	0.196565	−	0.980491i	$-0.437021\pi$
$163$	5.01915	0.393130	0.196565	+	0.980491i	$0.437021\pi$
$164$	20.4546	1.59723
$165$	0	0
$166$	14.5661	1.13055
$167$	−7.58516	−0.586958	−0.293479	−	0.955966i	$-0.594813\pi$
$167$	−7.58516	−0.586958	−0.293479	+	0.955966i	$0.594813\pi$
$168$	0	0
$169$	−10.3859	−0.798915
$170$	−2.37884	−0.182449
$171$	0	0
$172$	−19.4247	−1.48112
$173$	4.01343	0.305135	0.152568	−	0.988293i	$-0.451246\pi$
$173$	4.01343	0.305135	0.152568	+	0.988293i	$0.451246\pi$
$174$	0	0
$175$	−4.94283	−0.373643
$176$	−9.11266	−0.686893
$177$	0	0
$178$	25.8628	1.93850
$179$	24.8302	1.85590	0.927948	−	0.372711i	$-0.121572\pi$
$179$	24.8302	1.85590	0.927948	+	0.372711i	$0.121572\pi$
$180$	0	0
$181$	7.31346	0.543606	0.271803	−	0.962353i	$-0.412380\pi$
$181$	7.31346	0.543606	0.271803	+	0.962353i	$0.412380\pi$
$182$	−3.48627	−0.258420
$183$	0	0
$184$	−7.31540	−0.539299
$185$	−0.0845694	−0.00621767
$186$	0	0
$187$	−18.4469	−1.34897
$188$	−9.33648	−0.680933
$189$	0	0
$190$	−2.15132	−0.156073
$191$	12.4008	0.897293	0.448646	−	0.893709i	$-0.351906\pi$
$191$	12.4008	0.897293	0.448646	+	0.893709i	$0.351906\pi$
$192$	0	0
$193$	16.0448	1.15493	0.577465	−	0.816415i	$-0.304042\pi$
$193$	16.0448	1.15493	0.577465	+	0.816415i	$0.304042\pi$
$194$	−3.86166	−0.277251
$195$	0	0
$196$	2.64943	0.189245
$197$	−12.1872	−0.868304	−0.434152	−	0.900840i	$-0.642952\pi$
$197$	−12.1872	−0.868304	−0.434152	+	0.900840i	$0.642952\pi$
$198$	0	0
$199$	13.7816	0.976950	0.488475	−	0.872578i	$-0.337553\pi$
$199$	13.7816	0.976950	0.488475	+	0.872578i	$0.337553\pi$
$200$	−6.92157	−0.489429
$201$	0	0
$202$	23.9183	1.68288
$203$	−9.10130	−0.638786
$204$	0	0
$205$	1.84590	0.128923
$206$	31.4772	2.19312
$207$	0	0
$208$	3.68537	0.255535
$209$	−16.6826	−1.15396
$210$	0	0
$211$	−12.3776	−0.852111	−0.426056	−	0.904697i	$-0.640097\pi$
$211$	−12.3776	−0.852111	−0.426056	+	0.904697i	$0.640097\pi$
$212$	1.73544	0.119190
$213$	0	0
$214$	−23.0559	−1.57607
$215$	−1.75296	−0.119551
$216$	0	0
$217$	−6.57815	−0.446554
$218$	−41.3462	−2.80032
$219$	0	0
$220$	2.53248	0.170740
$221$	7.46034	0.501837
$222$	0	0
$223$	−16.9298	−1.13370	−0.566852	−	0.823819i	$-0.691839\pi$
$223$	−16.9298	−1.13370	−0.566852	+	0.823819i	$0.691839\pi$
$224$	−7.71560	−0.515520
$225$	0	0
$226$	30.1019	2.00235
$227$	14.4225	0.957256	0.478628	−	0.878018i	$-0.341134\pi$
$227$	14.4225	0.957256	0.478628	+	0.878018i	$0.341134\pi$
$228$	0	0
$229$	−16.4164	−1.08483	−0.542414	−	0.840111i	$-0.682490\pi$
$229$	−16.4164	−1.08483	−0.542414	+	0.840111i	$0.682490\pi$
$230$	−2.69326	−0.177588
$231$	0	0
$232$	−12.7448	−0.836736
$233$	7.64997	0.501167	0.250583	−	0.968095i	$-0.419378\pi$
$233$	7.64997	0.501167	0.250583	+	0.968095i	$0.419378\pi$
$234$	0	0
$235$	−0.842559	−0.0549625
$236$	31.3731	2.04221
$237$	0	0
$238$	−9.94939	−0.644923
$239$	−8.03695	−0.519867	−0.259933	−	0.965627i	$-0.583701\pi$
$239$	−8.03695	−0.519867	−0.259933	+	0.965627i	$0.583701\pi$
$240$	0	0
$241$	−22.8036	−1.46891	−0.734456	−	0.678657i	$-0.762562\pi$
$241$	−22.8036	−1.46891	−0.734456	+	0.678657i	$0.762562\pi$
$242$	10.7440	0.690650
$243$	0	0
$244$	−18.7021	−1.19728
$245$	0.239094	0.0152752
$246$	0	0
$247$	6.74682	0.429290
$248$	−9.21155	−0.584934
$249$	0	0
$250$	−5.12600	−0.324197
$251$	2.22670	0.140548	0.0702739	−	0.997528i	$-0.477613\pi$
$251$	2.22670	0.140548	0.0702739	+	0.997528i	$0.477613\pi$
$252$	0	0
$253$	−20.8850	−1.31303
$254$	−2.15625	−0.135295
$255$	0	0
$256$	1.27383	0.0796142
$257$	−12.7786	−0.797110	−0.398555	−	0.917144i	$-0.630488\pi$
$257$	−12.7786	−0.797110	−0.398555	+	0.917144i	$0.630488\pi$
$258$	0	0
$259$	−0.353708	−0.0219783
$260$	−1.02419	−0.0635178
$261$	0	0
$262$	2.66811	0.164836
$263$	−1.91222	−0.117912	−0.0589562	−	0.998261i	$-0.518777\pi$
$263$	−1.91222	−0.117912	−0.0589562	+	0.998261i	$0.518777\pi$
$264$	0	0
$265$	0.156612	0.00962063
$266$	−8.99781	−0.551691
$267$	0	0
$268$	−0.443788	−0.0271087
$269$	−7.78586	−0.474712	−0.237356	−	0.971423i	$-0.576281\pi$
$269$	−7.78586	−0.474712	−0.237356	+	0.971423i	$0.576281\pi$
$270$	0	0
$271$	−10.3228	−0.627064	−0.313532	−	0.949578i	$-0.601512\pi$
$271$	−10.3228	−0.627064	−0.313532	+	0.949578i	$0.601512\pi$
$272$	10.5176	0.637723
$273$	0	0
$274$	10.9981	0.664417
$275$	−19.7607	−1.19161
$276$	0	0
$277$	−13.1278	−0.788776	−0.394388	−	0.918944i	$-0.629043\pi$
$277$	−13.1278	−0.788776	−0.394388	+	0.918944i	$0.629043\pi$
$278$	26.9996	1.61933
$279$	0	0
$280$	0.334809	0.0200087
$281$	−18.1404	−1.08216	−0.541081	−	0.840970i	$-0.681985\pi$
$281$	−18.1404	−1.08216	−0.541081	+	0.840970i	$0.681985\pi$
$282$	0	0
$283$	−19.1287	−1.13708	−0.568542	−	0.822655i	$-0.692492\pi$
$283$	−19.1287	−1.13708	−0.568542	+	0.822655i	$0.692492\pi$
$284$	−18.5080	−1.09825
$285$	0	0
$286$	−13.9376	−0.824145
$287$	7.72038	0.455720
$288$	0	0
$289$	4.29089	0.252406
$290$	−4.69215	−0.275533
$291$	0	0
$292$	−7.40944	−0.433604
$293$	26.6893	1.55921	0.779603	−	0.626274i	$-0.215421\pi$
$293$	26.6893	1.55921	0.779603	+	0.626274i	$0.215421\pi$
$294$	0	0
$295$	2.83122	0.164840
$296$	−0.495306	−0.0287891
$297$	0	0
$298$	−20.6734	−1.19758
$299$	8.44639	0.488468
$300$	0	0
$301$	−7.33166	−0.422590
$302$	−36.9423	−2.12579
$303$	0	0
$304$	9.51168	0.545532
$305$	−1.68774	−0.0966400
$306$	0	0
$307$	9.45867	0.539835	0.269917	−	0.962883i	$-0.413004\pi$
$307$	9.45867	0.539835	0.269917	+	0.962883i	$0.413004\pi$
$308$	10.5920	0.603534
$309$	0	0
$310$	−3.39135	−0.192616
$311$	−1.98631	−0.112633	−0.0563167	−	0.998413i	$-0.517936\pi$
$311$	−1.98631	−0.112633	−0.0563167	+	0.998413i	$0.517936\pi$
$312$	0	0
$313$	−20.0441	−1.13296	−0.566480	−	0.824075i	$-0.691695\pi$
$313$	−20.0441	−1.13296	−0.566480	+	0.824075i	$0.691695\pi$
$314$	17.2173	0.971626
$315$	0	0
$316$	37.7611	2.12423
$317$	27.2490	1.53046	0.765228	−	0.643760i	$-0.222626\pi$
$317$	27.2490	1.53046	0.765228	+	0.643760i	$0.222626\pi$
$318$	0	0
$319$	−36.3855	−2.03720
$320$	−2.88778	−0.161432
$321$	0	0
$322$	−11.2644	−0.627742
$323$	19.2546	1.07135
$324$	0	0
$325$	7.99168	0.443298
$326$	10.8226	0.599406
$327$	0	0
$328$	10.8110	0.596940
$329$	−3.52396	−0.194282
$330$	0	0
$331$	34.0423	1.87113	0.935567	−	0.353150i	$-0.114889\pi$
$331$	34.0423	1.87113	0.935567	+	0.353150i	$0.114889\pi$
$332$	17.8976	0.982256
$333$	0	0
$334$	−16.3555	−0.894935
$335$	−0.0400491	−0.00218812
$336$	0	0
$337$	−13.1131	−0.714318	−0.357159	−	0.934044i	$-0.616255\pi$
$337$	−13.1131	−0.714318	−0.357159	+	0.934044i	$0.616255\pi$
$338$	−22.3946	−1.21811
$339$	0	0
$340$	−2.92292	−0.158518
$341$	−26.2984	−1.42414
$342$	0	0
$343$	1.00000	0.0539949
$344$	−10.2667	−0.553543
$345$	0	0
$346$	8.65396	0.465240
$347$	8.22380	0.441477	0.220738	−	0.975333i	$-0.429153\pi$
$347$	8.22380	0.441477	0.220738	+	0.975333i	$0.429153\pi$
$348$	0	0
$349$	−22.6433	−1.21207	−0.606035	−	0.795438i	$-0.707241\pi$
$349$	−22.6433	−1.21207	−0.606035	+	0.795438i	$0.707241\pi$
$350$	−10.6580	−0.569694
$351$	0	0
$352$	−30.8457	−1.64408
$353$	22.9217	1.22000	0.610000	−	0.792401i	$-0.291169\pi$
$353$	22.9217	1.22000	0.610000	+	0.792401i	$0.291169\pi$
$354$	0	0
$355$	−1.67023	−0.0886466
$356$	31.7781	1.68423
$357$	0	0
$358$	53.5402	2.82969
$359$	−22.9728	−1.21246	−0.606230	−	0.795289i	$-0.707319\pi$
$359$	−22.9728	−1.21246	−0.606230	+	0.795289i	$0.707319\pi$
$360$	0	0
$361$	−1.58696	−0.0835240
$362$	15.7697	0.828836
$363$	0	0
$364$	−4.28365	−0.224524
$365$	−0.668655	−0.0349990
$366$	0	0
$367$	−2.80411	−0.146373	−0.0731866	−	0.997318i	$-0.523317\pi$
$367$	−2.80411	−0.146373	−0.0731866	+	0.997318i	$0.523317\pi$
$368$	11.9077	0.620734
$369$	0	0
$370$	−0.182353	−0.00948008
$371$	0.655025	0.0340072
$372$	0	0
$373$	−35.2417	−1.82475	−0.912374	−	0.409358i	$-0.865753\pi$
$373$	−35.2417	−1.82475	−0.912374	+	0.409358i	$0.865753\pi$
$374$	−39.7761	−2.05677
$375$	0	0
$376$	−4.93469	−0.254487
$377$	14.7152	0.757870
$378$	0	0
$379$	−8.26039	−0.424308	−0.212154	−	0.977236i	$-0.568048\pi$
$379$	−8.26039	−0.424308	−0.212154	+	0.977236i	$0.568048\pi$
$380$	−2.64337	−0.135602
$381$	0	0
$382$	26.7393	1.36810
$383$	6.69505	0.342101	0.171050	−	0.985262i	$-0.445284\pi$
$383$	6.69505	0.342101	0.171050	+	0.985262i	$0.445284\pi$
$384$	0	0
$385$	0.955860	0.0487151
$386$	34.5966	1.76092
$387$	0	0
$388$	−4.74489	−0.240885
$389$	13.4050	0.679658	0.339829	−	0.940487i	$-0.389631\pi$
$389$	13.4050	0.679658	0.339829	+	0.940487i	$0.389631\pi$
$390$	0	0
$391$	24.1050	1.21904
$392$	1.40032	0.0707271
$393$	0	0
$394$	−26.2787	−1.32390
$395$	3.40771	0.171460
$396$	0	0
$397$	1.27937	0.0642097	0.0321048	−	0.999485i	$-0.489779\pi$
$397$	1.27937	0.0642097	0.0321048	+	0.999485i	$0.489779\pi$
$398$	29.7165	1.48956
$399$	0	0
$400$	11.2667	0.563334
$401$	25.8637	1.29157	0.645785	−	0.763519i	$-0.276530\pi$
$401$	25.8637	1.29157	0.645785	+	0.763519i	$0.276530\pi$
$402$	0	0
$403$	10.6357	0.529802
$404$	29.3888	1.46215
$405$	0	0
$406$	−19.6247	−0.973958
$407$	−1.41407	−0.0700927
$408$	0	0
$409$	−8.40634	−0.415667	−0.207833	−	0.978164i	$-0.566641\pi$
$409$	−8.40634	−0.415667	−0.207833	+	0.978164i	$0.566641\pi$
$410$	3.98022	0.196569
$411$	0	0
$412$	38.6766	1.90546
$413$	11.8415	0.582681
$414$	0	0
$415$	1.61514	0.0792843
$416$	12.4747	0.611625
$417$	0	0
$418$	−35.9718	−1.75944
$419$	−12.0193	−0.587183	−0.293592	−	0.955931i	$-0.594851\pi$
$419$	−12.0193	−0.587183	−0.293592	+	0.955931i	$0.594851\pi$
$420$	0	0
$421$	27.5197	1.34123	0.670614	−	0.741807i	$-0.266031\pi$
$421$	27.5197	1.34123	0.670614	+	0.741807i	$0.266031\pi$
$422$	−26.6893	−1.29922
$423$	0	0
$424$	0.917247	0.0445455
$425$	22.8073	1.10631
$426$	0	0
$427$	−7.05891	−0.341605
$428$	−28.3292	−1.36934
$429$	0	0
$430$	−3.77981	−0.182279
$431$	−35.4733	−1.70869	−0.854345	−	0.519707i	$-0.826041\pi$
$431$	−35.4733	−1.70869	−0.854345	+	0.519707i	$0.826041\pi$
$432$	0	0
$433$	26.5237	1.27465	0.637323	−	0.770597i	$-0.280042\pi$
$433$	26.5237	1.27465	0.637323	+	0.770597i	$0.280042\pi$
$434$	−14.1842	−0.680862
$435$	0	0
$436$	−50.8028	−2.43301
$437$	21.7995	1.04281
$438$	0	0
$439$	−32.6301	−1.55735	−0.778674	−	0.627428i	$-0.784108\pi$
$439$	−32.6301	−1.55735	−0.778674	+	0.627428i	$0.784108\pi$
$440$	1.33851	0.0638111
$441$	0	0
$442$	16.0864	0.765151
$443$	−16.0947	−0.764681	−0.382340	−	0.924022i	$-0.624882\pi$
$443$	−16.0947	−0.764681	−0.382340	+	0.924022i	$0.624882\pi$
$444$	0	0
$445$	2.86777	0.135945
$446$	−36.5050	−1.72856
$447$	0	0
$448$	−12.0780	−0.570632
$449$	−33.1478	−1.56434	−0.782171	−	0.623064i	$-0.785888\pi$
$449$	−33.1478	−1.56434	−0.782171	+	0.623064i	$0.785888\pi$
$450$	0	0
$451$	30.8648	1.45337
$452$	36.9867	1.73971
$453$	0	0
$454$	31.0986	1.45953
$455$	−0.386572	−0.0181228
$456$	0	0
$457$	−20.7795	−0.972026	−0.486013	−	0.873952i	$-0.661549\pi$
$457$	−20.7795	−0.972026	−0.486013	+	0.873952i	$0.661549\pi$
$458$	−35.3980	−1.65404
$459$	0	0
$460$	−3.30925	−0.154295
$461$	−26.0027	−1.21106	−0.605532	−	0.795821i	$-0.707040\pi$
$461$	−26.0027	−1.21106	−0.605532	+	0.795821i	$0.707040\pi$
$462$	0	0
$463$	1.40695	0.0653867	0.0326934	−	0.999465i	$-0.489592\pi$
$463$	1.40695	0.0653867	0.0326934	+	0.999465i	$0.489592\pi$
$464$	20.7455	0.963084
$465$	0	0
$466$	16.4953	0.764129
$467$	4.52880	0.209568	0.104784	−	0.994495i	$-0.466585\pi$
$467$	4.52880	0.209568	0.104784	+	0.994495i	$0.466585\pi$
$468$	0	0
$469$	−0.167504	−0.00773460
$470$	−1.81677	−0.0838013
$471$	0	0
$472$	16.5819	0.763244
$473$	−29.3108	−1.34771
$474$	0	0
$475$	20.6259	0.946382
$476$	−12.2250	−0.560332
$477$	0	0
$478$	−17.3297	−0.792641
$479$	−1.74794	−0.0798653	−0.0399327	−	0.999202i	$-0.512714\pi$
$479$	−1.74794	−0.0798653	−0.0399327	+	0.999202i	$0.512714\pi$
$480$	0	0
$481$	0.571882	0.0260756
$482$	−49.1704	−2.23965
$483$	0	0
$484$	13.2013	0.600061
$485$	−0.428197	−0.0194434
$486$	0	0
$487$	−25.2382	−1.14365	−0.571827	−	0.820375i	$-0.693765\pi$
$487$	−25.2382	−1.14365	−0.571827	+	0.820375i	$0.693765\pi$
$488$	−9.88477	−0.447463
$489$	0	0
$490$	0.515547	0.0232900
$491$	27.1705	1.22619	0.613093	−	0.790010i	$-0.289925\pi$
$491$	27.1705	1.22619	0.613093	+	0.790010i	$0.289925\pi$
$492$	0	0
$493$	41.9953	1.89137
$494$	14.5478	0.654539
$495$	0	0
$496$	14.9942	0.673260
$497$	−6.98566	−0.313350
$498$	0	0
$499$	−18.9409	−0.847909	−0.423955	−	0.905683i	$-0.639358\pi$
$499$	−18.9409	−0.847909	−0.423955	+	0.905683i	$0.639358\pi$
$500$	−6.29841	−0.281673
$501$	0	0
$502$	4.80132	0.214293
$503$	2.52810	0.112722	0.0563611	−	0.998410i	$-0.482050\pi$
$503$	2.52810	0.112722	0.0563611	+	0.998410i	$0.482050\pi$
$504$	0	0
$505$	2.65216	0.118019
$506$	−45.0334	−2.00198
$507$	0	0
$508$	−2.64943	−0.117549
$509$	−34.9093	−1.54733	−0.773663	−	0.633598i	$-0.781578\pi$
$509$	−34.9093	−1.54733	−0.773663	+	0.633598i	$0.781578\pi$
$510$	0	0
$511$	−2.79662	−0.123715
$512$	23.9707	1.05937
$513$	0	0
$514$	−27.5540	−1.21535
$515$	3.49032	0.153802
$516$	0	0
$517$	−14.0882	−0.619600
$518$	−0.762683	−0.0335104
$519$	0	0
$520$	−0.541327	−0.0237387
$521$	−8.56576	−0.375273	−0.187636	−	0.982239i	$-0.560083\pi$
$521$	−8.56576	−0.375273	−0.187636	+	0.982239i	$0.560083\pi$
$522$	0	0
$523$	2.00260	0.0875675	0.0437838	−	0.999041i	$-0.486059\pi$
$523$	2.00260	0.0875675	0.0437838	+	0.999041i	$0.486059\pi$
$524$	3.27835	0.143215
$525$	0	0
$526$	−4.12322	−0.179781
$527$	30.3530	1.32220
$528$	0	0
$529$	4.29098	0.186564
$530$	0.337696	0.0146686
$531$	0	0
$532$	−11.0558	−0.479328
$533$	−12.4825	−0.540676
$534$	0	0
$535$	−2.55653	−0.110528
$536$	−0.234559	−0.0101314
$537$	0	0
$538$	−16.7883	−0.723794
$539$	3.99784	0.172199
$540$	0	0
$541$	−14.8542	−0.638632	−0.319316	−	0.947648i	$-0.603453\pi$
$541$	−14.8542	−0.638632	−0.319316	+	0.947648i	$0.603453\pi$
$542$	−22.2585	−0.956085
$543$	0	0
$544$	35.6014	1.52640
$545$	−4.58464	−0.196384
$546$	0	0
$547$	9.86356	0.421735	0.210868	−	0.977515i	$-0.432371\pi$
$547$	9.86356	0.421735	0.210868	+	0.977515i	$0.432371\pi$
$548$	13.5135	0.577268
$549$	0	0
$550$	−42.6090	−1.81685
$551$	37.9788	1.61795
$552$	0	0
$553$	14.2526	0.606081
$554$	−28.3069	−1.20265
$555$	0	0
$556$	33.1749	1.40693
$557$	1.50087	0.0635939	0.0317970	−	0.999494i	$-0.489877\pi$
$557$	1.50087	0.0635939	0.0317970	+	0.999494i	$0.489877\pi$
$558$	0	0
$559$	11.8540	0.501370
$560$	−0.544990	−0.0230300
$561$	0	0
$562$	−39.1152	−1.64997
$563$	−40.5430	−1.70868	−0.854341	−	0.519713i	$-0.826039\pi$
$563$	−40.5430	−1.70868	−0.854341	+	0.519713i	$0.826039\pi$
$564$	0	0
$565$	3.33782	0.140423
$566$	−41.2463	−1.73371
$567$	0	0
$568$	−9.78219	−0.410452
$569$	−3.51593	−0.147396	−0.0736978	−	0.997281i	$-0.523480\pi$
$569$	−3.51593	−0.147396	−0.0736978	+	0.997281i	$0.523480\pi$
$570$	0	0
$571$	−39.7252	−1.66245	−0.831224	−	0.555937i	$-0.812359\pi$
$571$	−39.7252	−1.66245	−0.831224	+	0.555937i	$0.812359\pi$
$572$	−17.1253	−0.716046
$573$	0	0
$574$	16.6471	0.694836
$575$	25.8217	1.07684
$576$	0	0
$577$	34.1969	1.42364	0.711818	−	0.702364i	$-0.247872\pi$
$577$	34.1969	1.42364	0.711818	+	0.702364i	$0.247872\pi$
$578$	9.25225	0.384843
$579$	0	0
$580$	−5.76533	−0.239392
$581$	6.75526	0.280256
$582$	0	0
$583$	2.61868	0.108455
$584$	−3.91618	−0.162053
$585$	0	0
$586$	57.5489	2.37732
$587$	14.5520	0.600625	0.300313	−	0.953841i	$-0.402909\pi$
$587$	14.5520	0.600625	0.300313	+	0.953841i	$0.402909\pi$
$588$	0	0
$589$	27.4499	1.13105
$590$	6.10483	0.251332
$591$	0	0
$592$	0.806240	0.0331363
$593$	22.6700	0.930947	0.465473	−	0.885062i	$-0.345884\pi$
$593$	22.6700	0.930947	0.465473	+	0.885062i	$0.345884\pi$
$594$	0	0
$595$	−1.10323	−0.0452280
$596$	−25.4018	−1.04050
$597$	0	0
$598$	18.2126	0.744767
$599$	−19.5665	−0.799466	−0.399733	−	0.916632i	$-0.630897\pi$
$599$	−19.5665	−0.799466	−0.399733	+	0.916632i	$0.630897\pi$
$600$	0	0
$601$	7.31719	0.298475	0.149237	−	0.988801i	$-0.452318\pi$
$601$	7.31719	0.298475	0.149237	+	0.988801i	$0.452318\pi$
$602$	−15.8089	−0.644323
$603$	0	0
$604$	−45.3917	−1.84696
$605$	1.19134	0.0484348
$606$	0	0
$607$	25.1482	1.02074	0.510368	−	0.859956i	$-0.329509\pi$
$607$	25.1482	1.02074	0.510368	+	0.859956i	$0.329509\pi$
$608$	32.1964	1.30574
$609$	0	0
$610$	−3.63920	−0.147347
$611$	5.69762	0.230501
$612$	0	0
$613$	40.8861	1.65137	0.825686	−	0.564129i	$-0.190788\pi$
$613$	40.8861	1.65137	0.825686	+	0.564129i	$0.190788\pi$
$614$	20.3953	0.823087
$615$	0	0
$616$	5.59827	0.225561
$617$	−23.3040	−0.938185	−0.469092	−	0.883149i	$-0.655419\pi$
$617$	−23.3040	−0.938185	−0.469092	+	0.883149i	$0.655419\pi$
$618$	0	0
$619$	36.2218	1.45588	0.727939	−	0.685642i	$-0.240478\pi$
$619$	36.2218	1.45588	0.727939	+	0.685642i	$0.240478\pi$
$620$	−4.16701	−0.167351
$621$	0	0
$622$	−4.28299	−0.171732
$623$	11.9943	0.480543
$624$	0	0
$625$	24.1458	0.965831
$626$	−43.2202	−1.72743
$627$	0	0
$628$	21.1551	0.844182
$629$	1.63208	0.0650753
$630$	0	0
$631$	44.1474	1.75748	0.878741	−	0.477299i	$-0.158384\pi$
$631$	44.1474	1.75748	0.878741	+	0.477299i	$0.158384\pi$
$632$	19.9582	0.793896
$633$	0	0
$634$	58.7557	2.33349
$635$	−0.239094	−0.00948816
$636$	0	0
$637$	−1.61682	−0.0640608
$638$	−78.4564	−3.10612
$639$	0	0
$640$	−2.53727	−0.100294
$641$	−20.5976	−0.813557	−0.406779	−	0.913527i	$-0.633348\pi$
$641$	−20.5976	−0.813557	−0.406779	+	0.913527i	$0.633348\pi$
$642$	0	0
$643$	−3.13591	−0.123668	−0.0618342	−	0.998086i	$-0.519695\pi$
$643$	−3.13591	−0.123668	−0.0618342	+	0.998086i	$0.519695\pi$
$644$	−13.8408	−0.545404
$645$	0	0
$646$	41.5178	1.63349
$647$	−6.98764	−0.274712	−0.137356	−	0.990522i	$-0.543861\pi$
$647$	−6.98764	−0.274712	−0.137356	+	0.990522i	$0.543861\pi$
$648$	0	0
$649$	47.3403	1.85827
$650$	17.2321	0.675897
$651$	0	0
$652$	13.2979	0.520784
$653$	−18.7511	−0.733786	−0.366893	−	0.930263i	$-0.619578\pi$
$653$	−18.7511	−0.733786	−0.366893	+	0.930263i	$0.619578\pi$
$654$	0	0
$655$	0.295851	0.0115598
$656$	−17.5978	−0.687079
$657$	0	0
$658$	−7.59856	−0.296223
$659$	−28.0273	−1.09179	−0.545895	−	0.837854i	$-0.683810\pi$
$659$	−28.0273	−1.09179	−0.545895	+	0.837854i	$0.683810\pi$
$660$	0	0
$661$	−46.8900	−1.82381	−0.911905	−	0.410401i	$-0.865389\pi$
$661$	−46.8900	−1.82381	−0.911905	+	0.410401i	$0.865389\pi$
$662$	73.4038	2.85292
$663$	0	0
$664$	9.45956	0.367102
$665$	−0.997714	−0.0386897
$666$	0	0
$667$	47.5459	1.84098
$668$	−20.0963	−0.777550
$669$	0	0
$670$	−0.0863560	−0.00333622
$671$	−28.2204	−1.08944
$672$	0	0
$673$	30.2264	1.16514	0.582571	−	0.812780i	$-0.302047\pi$
$673$	30.2264	1.16514	0.582571	+	0.812780i	$0.302047\pi$
$674$	−28.2752	−1.08912
$675$	0	0
$676$	−27.5166	−1.05833
$677$	4.66520	0.179298	0.0896491	−	0.995973i	$-0.471425\pi$
$677$	4.66520	0.179298	0.0896491	+	0.995973i	$0.471425\pi$
$678$	0	0
$679$	−1.79091	−0.0687290
$680$	−1.54488	−0.0592434
$681$	0	0
$682$	−56.7060	−2.17138
$683$	34.8003	1.33160	0.665798	−	0.746132i	$-0.268091\pi$
$683$	34.8003	1.33160	0.665798	+	0.746132i	$0.268091\pi$
$684$	0	0
$685$	1.21951	0.0465951
$686$	2.15625	0.0823261
$687$	0	0
$688$	16.7118	0.637130
$689$	−1.05906	−0.0403469
$690$	0	0
$691$	36.6726	1.39509	0.697546	−	0.716540i	$-0.254275\pi$
$691$	36.6726	1.39509	0.697546	+	0.716540i	$0.254275\pi$
$692$	10.6333	0.404216
$693$	0	0
$694$	17.7326	0.673120
$695$	2.99383	0.113562
$696$	0	0
$697$	−35.6234	−1.34933
$698$	−48.8247	−1.84804
$699$	0	0
$700$	−13.0957	−0.494970
$701$	11.0457	0.417192	0.208596	−	0.978002i	$-0.433111\pi$
$701$	11.0457	0.417192	0.208596	+	0.978002i	$0.433111\pi$
$702$	0	0
$703$	1.47598	0.0556678
$704$	−48.2859	−1.81984
$705$	0	0
$706$	49.4250	1.86014
$707$	11.0925	0.417177
$708$	0	0
$709$	−7.73369	−0.290445	−0.145222	−	0.989399i	$-0.546390\pi$
$709$	−7.73369	−0.290445	−0.145222	+	0.989399i	$0.546390\pi$
$710$	−3.60144	−0.135160
$711$	0	0
$712$	16.7960	0.629455
$713$	34.3648	1.28697
$714$	0	0
$715$	−1.54545	−0.0577967
$716$	65.7857	2.45853
$717$	0	0
$718$	−49.5353	−1.84864
$719$	50.9775	1.90114	0.950571	−	0.310509i	$-0.100499\pi$
$719$	50.9775	1.90114	0.950571	+	0.310509i	$0.100499\pi$
$720$	0	0
$721$	14.5981	0.543662
$722$	−3.42188	−0.127349
$723$	0	0
$724$	19.3765	0.720121
$725$	44.9862	1.67075
$726$	0	0
$727$	−6.89848	−0.255850	−0.127925	−	0.991784i	$-0.540832\pi$
$727$	−6.89848	−0.255850	−0.127925	+	0.991784i	$0.540832\pi$
$728$	−2.26407	−0.0839122
$729$	0	0
$730$	−1.44179	−0.0533630
$731$	33.8298	1.25124
$732$	0	0
$733$	6.77021	0.250063	0.125032	−	0.992153i	$-0.460097\pi$
$733$	6.77021	0.250063	0.125032	+	0.992153i	$0.460097\pi$
$734$	−6.04636	−0.223175
$735$	0	0
$736$	40.3069	1.48573
$737$	−0.669652	−0.0246670
$738$	0	0
$739$	10.2163	0.375814	0.187907	−	0.982187i	$-0.439830\pi$
$739$	10.2163	0.375814	0.187907	+	0.982187i	$0.439830\pi$
$740$	−0.224060	−0.00823662
$741$	0	0
$742$	1.41240	0.0518508
$743$	−3.12087	−0.114493	−0.0572467	−	0.998360i	$-0.518232\pi$
$743$	−3.12087	−0.114493	−0.0572467	+	0.998360i	$0.518232\pi$
$744$	0	0
$745$	−2.29235	−0.0839854
$746$	−75.9901	−2.78219
$747$	0	0
$748$	−48.8736	−1.78699
$749$	−10.6926	−0.390698
$750$	0	0
$751$	17.9651	0.655557	0.327778	−	0.944755i	$-0.393700\pi$
$751$	17.9651	0.655557	0.327778	+	0.944755i	$0.393700\pi$
$752$	8.03251	0.292916
$753$	0	0
$754$	31.7296	1.15552
$755$	−4.09631	−0.149080
$756$	0	0
$757$	−41.8801	−1.52216	−0.761080	−	0.648658i	$-0.775330\pi$
$757$	−41.8801	−1.52216	−0.761080	+	0.648658i	$0.775330\pi$
$758$	−17.8115	−0.646942
$759$	0	0
$760$	−1.39712	−0.0506790
$761$	−47.4903	−1.72152	−0.860761	−	0.509010i	$-0.830012\pi$
$761$	−47.4903	−1.72152	−0.860761	+	0.509010i	$0.830012\pi$
$762$	0	0
$763$	−19.1750	−0.694183
$764$	32.8551	1.18866
$765$	0	0
$766$	14.4362	0.521602
$767$	−19.1455	−0.691305
$768$	0	0
$769$	44.5562	1.60674	0.803369	−	0.595481i	$-0.203039\pi$
$769$	44.5562	1.60674	0.803369	+	0.595481i	$0.203039\pi$
$770$	2.06107	0.0742760
$771$	0	0
$772$	42.5095	1.52995
$773$	−28.7248	−1.03316	−0.516580	−	0.856239i	$-0.672795\pi$
$773$	−28.7248	−1.03316	−0.516580	+	0.856239i	$0.672795\pi$
$774$	0	0
$775$	32.5147	1.16796
$776$	−2.50786	−0.0900270
$777$	0	0
$778$	28.9045	1.03628
$779$	−32.2163	−1.15427
$780$	0	0
$781$	−27.9275	−0.999326
$782$	51.9764	1.85867
$783$	0	0
$784$	−2.27940	−0.0814070
$785$	1.90912	0.0681394
$786$	0	0
$787$	−6.88490	−0.245420	−0.122710	−	0.992443i	$-0.539159\pi$
$787$	−6.88490	−0.245420	−0.122710	+	0.992443i	$0.539159\pi$
$788$	−32.2891	−1.15025
$789$	0	0
$790$	7.34788	0.261426
$791$	13.9603	0.496370
$792$	0	0
$793$	11.4130	0.405287
$794$	2.75864	0.0979006
$795$	0	0
$796$	36.5132	1.29418
$797$	−19.8500	−0.703123	−0.351562	−	0.936165i	$-0.614349\pi$
$797$	−19.8500	−0.703123	−0.351562	+	0.936165i	$0.614349\pi$
$798$	0	0
$799$	16.2603	0.575248
$800$	38.1370	1.34834
$801$	0	0
$802$	55.7686	1.96926
$803$	−11.1804	−0.394549
$804$	0	0
$805$	−1.24905	−0.0440231
$806$	22.9332	0.807789
$807$	0	0
$808$	15.5331	0.546454
$809$	18.8305	0.662046	0.331023	−	0.943623i	$-0.392606\pi$
$809$	18.8305	0.662046	0.331023	+	0.943623i	$0.392606\pi$
$810$	0	0
$811$	−47.7100	−1.67533	−0.837663	−	0.546188i	$-0.816078\pi$
$811$	−47.7100	−1.67533	−0.837663	+	0.546188i	$0.816078\pi$
$812$	−24.1132	−0.846208
$813$	0	0
$814$	−3.04908	−0.106870
$815$	1.20005	0.0420359
$816$	0	0
$817$	30.5942	1.07036
$818$	−18.1262	−0.633768
$819$	0	0
$820$	4.89057	0.170786
$821$	−25.1137	−0.876475	−0.438238	−	0.898859i	$-0.644397\pi$
$821$	−25.1137	−0.876475	−0.438238	+	0.898859i	$0.644397\pi$
$822$	0	0
$823$	43.0088	1.49919	0.749597	−	0.661895i	$-0.230247\pi$
$823$	43.0088	1.49919	0.749597	+	0.661895i	$0.230247\pi$
$824$	20.4421	0.712134
$825$	0	0
$826$	25.5332	0.888413
$827$	−13.8440	−0.481403	−0.240701	−	0.970599i	$-0.577377\pi$
$827$	−13.8440	−0.481403	−0.240701	+	0.970599i	$0.577377\pi$
$828$	0	0
$829$	12.5853	0.437105	0.218553	−	0.975825i	$-0.429867\pi$
$829$	12.5853	0.437105	0.218553	+	0.975825i	$0.429867\pi$
$830$	3.48266	0.120885
$831$	0	0
$832$	19.5280	0.677010
$833$	−4.61421	−0.159873
$834$	0	0
$835$	−1.81357	−0.0627611
$836$	−44.1992	−1.52866
$837$	0	0
$838$	−25.9167	−0.895279
$839$	4.46813	0.154257	0.0771284	−	0.997021i	$-0.475425\pi$
$839$	4.46813	0.154257	0.0771284	+	0.997021i	$0.475425\pi$
$840$	0	0
$841$	53.8337	1.85633
$842$	59.3394	2.04497
$843$	0	0
$844$	−32.7936	−1.12880
$845$	−2.48320	−0.0854248
$846$	0	0
$847$	4.98272	0.171208
$848$	−1.49306	−0.0512719
$849$	0	0
$850$	49.1782	1.68680
$851$	1.84780	0.0633416
$852$	0	0
$853$	−20.9580	−0.717589	−0.358795	−	0.933416i	$-0.616812\pi$
$853$	−20.9580	−0.717589	−0.358795	+	0.933416i	$0.616812\pi$
$854$	−15.2208	−0.520845
$855$	0	0
$856$	−14.9731	−0.511769
$857$	3.36198	0.114843	0.0574216	−	0.998350i	$-0.481712\pi$
$857$	3.36198	0.114843	0.0574216	+	0.998350i	$0.481712\pi$
$858$	0	0
$859$	31.0025	1.05779	0.528895	−	0.848687i	$-0.322607\pi$
$859$	31.0025	1.05779	0.528895	+	0.848687i	$0.322607\pi$
$860$	−4.64432	−0.158370
$861$	0	0
$862$	−76.4894	−2.60524
$863$	−8.87723	−0.302185	−0.151092	−	0.988520i	$-0.548279\pi$
$863$	−8.87723	−0.302185	−0.151092	+	0.988520i	$0.548279\pi$
$864$	0	0
$865$	0.959586	0.0326269
$866$	57.1917	1.94345
$867$	0	0
$868$	−17.4283	−0.591556
$869$	56.9795	1.93290
$870$	0	0
$871$	0.270823	0.00917650
$872$	−26.8513	−0.909299
$873$	0	0
$874$	47.0053	1.58998
$875$	−2.37727	−0.0803665
$876$	0	0
$877$	−19.3103	−0.652061	−0.326030	−	0.945359i	$-0.605711\pi$
$877$	−19.3103	−0.652061	−0.326030	+	0.945359i	$0.605711\pi$
$878$	−70.3587	−2.37449
$879$	0	0
$880$	−2.17878	−0.0734468
$881$	−3.30078	−0.111206	−0.0556030	−	0.998453i	$-0.517708\pi$
$881$	−3.30078	−0.111206	−0.0556030	+	0.998453i	$0.517708\pi$
$882$	0	0
$883$	51.9315	1.74764	0.873818	−	0.486253i	$-0.161637\pi$
$883$	51.9315	1.74764	0.873818	+	0.486253i	$0.161637\pi$
$884$	19.7656	0.664790
$885$	0	0
$886$	−34.7042	−1.16591
$887$	−9.29948	−0.312246	−0.156123	−	0.987738i	$-0.549900\pi$
$887$	−9.29948	−0.312246	−0.156123	+	0.987738i	$0.549900\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	6.18364	0.207276
$891$	0	0
$892$	−44.8543	−1.50183
$893$	14.7051	0.492088
$894$	0	0
$895$	5.93675	0.198444
$896$	−10.6120	−0.354522
$897$	0	0
$898$	−71.4750	−2.38515
$899$	59.8698	1.99677
$900$	0	0
$901$	−3.02242	−0.100691
$902$	66.5524	2.21595
$903$	0	0
$904$	19.5489	0.650187
$905$	1.74861	0.0581256
$906$	0	0
$907$	−31.2600	−1.03797	−0.518985	−	0.854783i	$-0.673690\pi$
$907$	−31.2600	−1.03797	−0.518985	+	0.854783i	$0.673690\pi$
$908$	38.2114	1.26809
$909$	0	0
$910$	−0.833547	−0.0276318
$911$	30.7338	1.01826	0.509128	−	0.860691i	$-0.329968\pi$
$911$	30.7338	1.01826	0.509128	+	0.860691i	$0.329968\pi$
$912$	0	0
$913$	27.0065	0.893783
$914$	−44.8059	−1.48205
$915$	0	0
$916$	−43.4941	−1.43709
$917$	1.23738	0.0408619
$918$	0	0
$919$	7.17530	0.236691	0.118346	−	0.992972i	$-0.462241\pi$
$919$	7.17530	0.236691	0.118346	+	0.992972i	$0.462241\pi$
$920$	−1.74907	−0.0576651
$921$	0	0
$922$	−56.0683	−1.84651
$923$	11.2946	0.371765
$924$	0	0
$925$	1.74832	0.0574844
$926$	3.03375	0.0996952
$927$	0	0
$928$	70.2220	2.30515
$929$	−23.3488	−0.766051	−0.383025	−	0.923738i	$-0.625118\pi$
$929$	−23.3488	−0.766051	−0.383025	+	0.923738i	$0.625118\pi$
$930$	0	0
$931$	−4.17289	−0.136761
$932$	20.2680	0.663902
$933$	0	0
$934$	9.76524	0.319529
$935$	−4.41053	−0.144240
$936$	0	0
$937$	−54.8510	−1.79191	−0.895953	−	0.444150i	$-0.853506\pi$
$937$	−54.8510	−1.79191	−0.895953	+	0.444150i	$0.853506\pi$
$938$	−0.361180	−0.0117929
$939$	0	0
$940$	−2.23230	−0.0728095
$941$	2.79391	0.0910788	0.0455394	−	0.998963i	$-0.485499\pi$
$941$	2.79391	0.0910788	0.0455394	+	0.998963i	$0.485499\pi$
$942$	0	0
$943$	−40.3319	−1.31339
$944$	−26.9914	−0.878495
$945$	0	0
$946$	−63.2014	−2.05486
$947$	−6.53039	−0.212209	−0.106105	−	0.994355i	$-0.533838\pi$
$947$	−6.53039	−0.212209	−0.106105	+	0.994355i	$0.533838\pi$
$948$	0	0
$949$	4.52163	0.146778
$950$	44.4747	1.44295
$951$	0	0
$952$	−6.46139	−0.209415
$953$	−19.4446	−0.629873	−0.314937	−	0.949113i	$-0.601983\pi$
$953$	−19.4446	−0.629873	−0.314937	+	0.949113i	$0.601983\pi$
$954$	0	0
$955$	2.96497	0.0959440
$956$	−21.2933	−0.688674
$957$	0	0
$958$	−3.76899	−0.121771
$959$	5.10054	0.164705
$960$	0	0
$961$	12.2721	0.395874
$962$	1.23312	0.0397574
$963$	0	0
$964$	−60.4165	−1.94589
$965$	3.83622	0.123492
$966$	0	0
$967$	−4.67554	−0.150355	−0.0751777	−	0.997170i	$-0.523952\pi$
$967$	−4.67554	−0.150355	−0.0751777	+	0.997170i	$0.523952\pi$
$968$	6.97742	0.224263
$969$	0	0
$970$	−0.923301	−0.0296454
$971$	−4.75185	−0.152494	−0.0762470	−	0.997089i	$-0.524294\pi$
$971$	−4.75185	−0.152494	−0.0762470	+	0.997089i	$0.524294\pi$
$972$	0	0
$973$	12.5215	0.401422
$974$	−54.4200	−1.74373
$975$	0	0
$976$	16.0901	0.515030
$977$	37.3562	1.19513	0.597565	−	0.801821i	$-0.296135\pi$
$977$	37.3562	1.19513	0.597565	+	0.801821i	$0.296135\pi$
$978$	0	0
$979$	47.9514	1.53253
$980$	0.633462	0.0202352
$981$	0	0
$982$	58.5864	1.86957
$983$	−45.4192	−1.44865	−0.724324	−	0.689459i	$-0.757848\pi$
$983$	−45.4192	−1.44865	−0.724324	+	0.689459i	$0.757848\pi$
$984$	0	0
$985$	−2.91389	−0.0928444
$986$	90.5524	2.88378
$987$	0	0
$988$	17.8752	0.568686
$989$	38.3011	1.21791
$990$	0	0
$991$	−8.11368	−0.257739	−0.128870	−	0.991662i	$-0.541135\pi$
$991$	−8.11368	−0.257739	−0.128870	+	0.991662i	$0.541135\pi$
$992$	50.7544	1.61145
$993$	0	0
$994$	−15.0628	−0.477764
$995$	3.29509	0.104461
$996$	0	0
$997$	−33.0431	−1.04648	−0.523242	−	0.852184i	$-0.675278\pi$
$997$	−33.0431	−1.04648	−0.523242	+	0.852184i	$0.675278\pi$
$998$	−40.8413	−1.29281
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.l.1.6		7
3.2	odd	2		2667.2.a.j.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.j.1.2	✓	7	3.2	odd	2
8001.2.a.l.1.6		7	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)

\(f(q)\)	\(=\)	\(q+2.15625 q^{2} +2.64943 q^{4} +0.239094 q^{5} +1.00000 q^{7} +1.40032 q^{8} +O(q^{10})\)	\(q+2.15625 q^{2} +2.64943 q^{4} +0.239094 q^{5} +1.00000 q^{7} +1.40032 q^{8} +0.515547 q^{10} +3.99784 q^{11} -1.61682 q^{13} +2.15625 q^{14} -2.27940 q^{16} -4.61421 q^{17} -4.17289 q^{19} +0.633462 q^{20} +8.62035 q^{22} -5.22408 q^{23} -4.94283 q^{25} -3.48627 q^{26} +2.64943 q^{28} -9.10130 q^{29} -6.57815 q^{31} -7.71560 q^{32} -9.94939 q^{34} +0.239094 q^{35} -0.353708 q^{37} -8.99781 q^{38} +0.334809 q^{40} +7.72038 q^{41} -7.33166 q^{43} +10.5920 q^{44} -11.2644 q^{46} -3.52396 q^{47} +1.00000 q^{49} -10.6580 q^{50} -4.28365 q^{52} +0.655025 q^{53} +0.955860 q^{55} +1.40032 q^{56} -19.6247 q^{58} +11.8415 q^{59} -7.05891 q^{61} -14.1842 q^{62} -12.0780 q^{64} -0.386572 q^{65} -0.167504 q^{67} -12.2250 q^{68} +0.515547 q^{70} -6.98566 q^{71} -2.79662 q^{73} -0.762683 q^{74} -11.0558 q^{76} +3.99784 q^{77} +14.2526 q^{79} -0.544990 q^{80} +16.6471 q^{82} +6.75526 q^{83} -1.10323 q^{85} -15.8089 q^{86} +5.59827 q^{88} +11.9943 q^{89} -1.61682 q^{91} -13.8408 q^{92} -7.59856 q^{94} -0.997714 q^{95} -1.79091 q^{97} +2.15625 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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