LMFDB - Embedded newform 8001.2.a.q.1.4

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
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			1.4
Root			$-1.64397$ 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)

$f(q)$	$=$	$q-1.64397 q^{2} +0.702628 q^{4} -1.50234 q^{5} -1.00000 q^{7} +2.13284 q^{8} +O(q^{10})$	\(q-1.64397 q^{2} +0.702628 q^{4} -1.50234 q^{5} -1.00000 q^{7} +2.13284 q^{8} +2.46980 q^{10} -3.11180 q^{11} +0.853816 q^{13} +1.64397 q^{14} -4.91157 q^{16} -2.49595 q^{17} +7.85770 q^{19} -1.05559 q^{20} +5.11569 q^{22} -1.92669 q^{23} -2.74296 q^{25} -1.40365 q^{26} -0.702628 q^{28} -4.22790 q^{29} +10.7031 q^{31} +3.80879 q^{32} +4.10325 q^{34} +1.50234 q^{35} -9.99125 q^{37} -12.9178 q^{38} -3.20425 q^{40} -0.296700 q^{41} +4.14902 q^{43} -2.18644 q^{44} +3.16741 q^{46} +8.57529 q^{47} +1.00000 q^{49} +4.50934 q^{50} +0.599915 q^{52} -5.87916 q^{53} +4.67499 q^{55} -2.13284 q^{56} +6.95053 q^{58} -12.9315 q^{59} +3.16211 q^{61} -17.5956 q^{62} +3.56162 q^{64} -1.28272 q^{65} +7.74810 q^{67} -1.75372 q^{68} -2.46980 q^{70} -0.392291 q^{71} +4.24778 q^{73} +16.4253 q^{74} +5.52104 q^{76} +3.11180 q^{77} -5.04857 q^{79} +7.37887 q^{80} +0.487765 q^{82} +12.1443 q^{83} +3.74977 q^{85} -6.82085 q^{86} -6.63695 q^{88} +5.95600 q^{89} -0.853816 q^{91} -1.35375 q^{92} -14.0975 q^{94} -11.8050 q^{95} -12.7319 q^{97} -1.64397 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

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 $

$2$	−1.64397	−1.16246	−0.581230	−	0.813739i	$-0.697428\pi$
$2$	−1.64397	−1.16246	−0.581230	+	0.813739i	$0.697428\pi$
$3$	0	0
$3$	0	0
$4$	0.702628	0.351314
$5$	−1.50234	−0.671869	−0.335934	−	0.941885i	$-0.609052\pi$
$5$	−1.50234	−0.671869	−0.335934	+	0.941885i	$0.609052\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.13284	0.754072
$9$	0	0
$10$	2.46980	0.781021
$11$	−3.11180	−0.938242	−0.469121	−	0.883134i	$-0.655429\pi$
$11$	−3.11180	−0.938242	−0.469121	+	0.883134i	$0.655429\pi$
$12$	0	0
$13$	0.853816	0.236806	0.118403	−	0.992966i	$-0.462223\pi$
$13$	0.853816	0.236806	0.118403	+	0.992966i	$0.462223\pi$
$14$	1.64397	0.439369
$15$	0	0
$16$	−4.91157	−1.22789
$17$	−2.49595	−0.605356	−0.302678	−	0.953093i	$-0.597881\pi$
$17$	−2.49595	−0.605356	−0.302678	+	0.953093i	$0.597881\pi$
$18$	0	0
$19$	7.85770	1.80268	0.901340	−	0.433113i	$-0.142585\pi$
$19$	7.85770	1.80268	0.901340	+	0.433113i	$0.142585\pi$
$20$	−1.05559	−0.236037
$21$	0	0
$22$	5.11569	1.09067
$23$	−1.92669	−0.401743	−0.200871	−	0.979618i	$-0.564377\pi$
$23$	−1.92669	−0.401743	−0.200871	+	0.979618i	$0.564377\pi$
$24$	0	0
$25$	−2.74296	−0.548593
$26$	−1.40365	−0.275277
$27$	0	0
$28$	−0.702628	−0.132784
$29$	−4.22790	−0.785101	−0.392551	−	0.919730i	$-0.628407\pi$
$29$	−4.22790	−0.785101	−0.392551	+	0.919730i	$0.628407\pi$
$30$	0	0
$31$	10.7031	1.92234	0.961168	−	0.275965i	$-0.0889974\pi$
$31$	10.7031	1.92234	0.961168	+	0.275965i	$0.0889974\pi$
$32$	3.80879	0.673305
$33$	0	0
$34$	4.10325	0.703702
$35$	1.50234	0.253942
$36$	0	0
$37$	−9.99125	−1.64255	−0.821276	−	0.570531i	$-0.806737\pi$
$37$	−9.99125	−1.64255	−0.821276	+	0.570531i	$0.806737\pi$
$38$	−12.9178	−2.09554
$39$	0	0
$40$	−3.20425	−0.506637
$41$	−0.296700	−0.0463368	−0.0231684	−	0.999732i	$-0.507375\pi$
$41$	−0.296700	−0.0463368	−0.0231684	+	0.999732i	$0.507375\pi$
$42$	0	0
$43$	4.14902	0.632719	0.316360	−	0.948639i	$-0.397539\pi$
$43$	4.14902	0.632719	0.316360	+	0.948639i	$0.397539\pi$
$44$	−2.18644	−0.329618
$45$	0	0
$46$	3.16741	0.467010
$47$	8.57529	1.25083	0.625417	−	0.780291i	$-0.284929\pi$
$47$	8.57529	1.25083	0.625417	+	0.780291i	$0.284929\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.50934	0.637717
$51$	0	0
$52$	0.599915	0.0831933
$53$	−5.87916	−0.807565	−0.403783	−	0.914855i	$-0.632305\pi$
$53$	−5.87916	−0.807565	−0.403783	+	0.914855i	$0.632305\pi$
$54$	0	0
$55$	4.67499	0.630375
$56$	−2.13284	−0.285012
$57$	0	0
$58$	6.95053	0.912649
$59$	−12.9315	−1.68354	−0.841772	−	0.539834i	$-0.818487\pi$
$59$	−12.9315	−1.68354	−0.841772	+	0.539834i	$0.818487\pi$
$60$	0	0
$61$	3.16211	0.404866	0.202433	−	0.979296i	$-0.435115\pi$
$61$	3.16211	0.404866	0.202433	+	0.979296i	$0.435115\pi$
$62$	−17.5956	−2.23464
$63$	0	0
$64$	3.56162	0.445202
$65$	−1.28272	−0.159102
$66$	0	0
$67$	7.74810	0.946581	0.473290	−	0.880906i	$-0.343066\pi$
$67$	7.74810	0.946581	0.473290	+	0.880906i	$0.343066\pi$
$68$	−1.75372	−0.212670
$69$	0	0
$70$	−2.46980	−0.295198
$71$	−0.392291	−0.0465564	−0.0232782	−	0.999729i	$-0.507410\pi$
$71$	−0.392291	−0.0465564	−0.0232782	+	0.999729i	$0.507410\pi$
$72$	0	0
$73$	4.24778	0.497165	0.248582	−	0.968611i	$-0.420035\pi$
$73$	4.24778	0.497165	0.248582	+	0.968611i	$0.420035\pi$
$74$	16.4253	1.90940
$75$	0	0
$76$	5.52104	0.633307
$77$	3.11180	0.354622
$78$	0	0
$79$	−5.04857	−0.568009	−0.284004	−	0.958823i	$-0.591663\pi$
$79$	−5.04857	−0.568009	−0.284004	+	0.958823i	$0.591663\pi$
$80$	7.37887	0.824982
$81$	0	0
$82$	0.487765	0.0538647
$83$	12.1443	1.33301	0.666503	−	0.745503i	$-0.267790\pi$
$83$	12.1443	1.33301	0.666503	+	0.745503i	$0.267790\pi$
$84$	0	0
$85$	3.74977	0.406720
$86$	−6.82085	−0.735511
$87$	0	0
$88$	−6.63695	−0.707502
$89$	5.95600	0.631334	0.315667	−	0.948870i	$-0.397772\pi$
$89$	5.95600	0.631334	0.315667	+	0.948870i	$0.397772\pi$
$90$	0	0
$91$	−0.853816	−0.0895042
$92$	−1.35375	−0.141138
$93$	0	0
$94$	−14.0975	−1.45405
$95$	−11.8050	−1.21116
$96$	0	0
$97$	−12.7319	−1.29273	−0.646365	−	0.763029i	$-0.723711\pi$
$97$	−12.7319	−1.29273	−0.646365	+	0.763029i	$0.723711\pi$
$98$	−1.64397	−0.166066
$99$	0	0
$100$	−1.92728	−0.192728
$101$	−5.59746	−0.556968	−0.278484	−	0.960441i	$-0.589832\pi$
$101$	−5.59746	−0.556968	−0.278484	+	0.960441i	$0.589832\pi$
$102$	0	0
$103$	10.1532	1.00043	0.500215	−	0.865902i	$-0.333254\pi$
$103$	10.1532	1.00043	0.500215	+	0.865902i	$0.333254\pi$
$104$	1.82105	0.178569
$105$	0	0
$106$	9.66515	0.938762
$107$	11.6951	1.13061	0.565305	−	0.824882i	$-0.308759\pi$
$107$	11.6951	1.13061	0.565305	+	0.824882i	$0.308759\pi$
$108$	0	0
$109$	−9.63924	−0.923272	−0.461636	−	0.887069i	$-0.652737\pi$
$109$	−9.63924	−0.923272	−0.461636	+	0.887069i	$0.652737\pi$
$110$	−7.68553	−0.732786
$111$	0	0
$112$	4.91157	0.464100
$113$	5.02902	0.473091	0.236545	−	0.971620i	$-0.423985\pi$
$113$	5.02902	0.473091	0.236545	+	0.971620i	$0.423985\pi$
$114$	0	0
$115$	2.89455	0.269918
$116$	−2.97064	−0.275817
$117$	0	0
$118$	21.2590	1.95705
$119$	2.49595	0.228803
$120$	0	0
$121$	−1.31673	−0.119702
$122$	−5.19840	−0.470641
$123$	0	0
$124$	7.52031	0.675344
$125$	11.6326	1.04045
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−13.4728	−1.19083
$129$	0	0
$130$	2.10876	0.184950
$131$	12.3886	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$131$	12.3886	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$132$	0	0
$133$	−7.85770	−0.681349
$134$	−12.7376	−1.10036
$135$	0	0
$136$	−5.32345	−0.456482
$137$	−2.96514	−0.253329	−0.126664	−	0.991946i	$-0.540427\pi$
$137$	−2.96514	−0.253329	−0.126664	+	0.991946i	$0.540427\pi$
$138$	0	0
$139$	22.7593	1.93042	0.965211	−	0.261472i	$-0.0842080\pi$
$139$	22.7593	1.93042	0.965211	+	0.261472i	$0.0842080\pi$
$140$	1.05559	0.0892136
$141$	0	0
$142$	0.644914	0.0541200
$143$	−2.65690	−0.222181
$144$	0	0
$145$	6.35176	0.527485
$146$	−6.98321	−0.577934
$147$	0	0
$148$	−7.02014	−0.577052
$149$	4.40346	0.360746	0.180373	−	0.983598i	$-0.442270\pi$
$149$	4.40346	0.360746	0.180373	+	0.983598i	$0.442270\pi$
$150$	0	0
$151$	−8.05008	−0.655106	−0.327553	−	0.944833i	$-0.606224\pi$
$151$	−8.05008	−0.655106	−0.327553	+	0.944833i	$0.606224\pi$
$152$	16.7592	1.35935
$153$	0	0
$154$	−5.11569	−0.412234
$155$	−16.0798	−1.29156
$156$	0	0
$157$	−7.14755	−0.570437	−0.285218	−	0.958463i	$-0.592066\pi$
$157$	−7.14755	−0.570437	−0.285218	+	0.958463i	$0.592066\pi$
$158$	8.29969	0.660288
$159$	0	0
$160$	−5.72211	−0.452372
$161$	1.92669	0.151844
$162$	0	0
$163$	5.50626	0.431284	0.215642	−	0.976473i	$-0.430816\pi$
$163$	5.50626	0.431284	0.215642	+	0.976473i	$0.430816\pi$
$164$	−0.208470	−0.0162788
$165$	0	0
$166$	−19.9648	−1.54957
$167$	1.48053	0.114567	0.0572835	−	0.998358i	$-0.481756\pi$
$167$	1.48053	0.114567	0.0572835	+	0.998358i	$0.481756\pi$
$168$	0	0
$169$	−12.2710	−0.943923
$170$	−6.16450	−0.472795
$171$	0	0
$172$	2.91522	0.222283
$173$	−17.4367	−1.32569	−0.662844	−	0.748757i	$-0.730651\pi$
$173$	−17.4367	−1.32569	−0.662844	+	0.748757i	$0.730651\pi$
$174$	0	0
$175$	2.74296	0.207349
$176$	15.2838	1.15206
$177$	0	0
$178$	−9.79146	−0.733901
$179$	9.62078	0.719091	0.359545	−	0.933128i	$-0.382932\pi$
$179$	9.62078	0.719091	0.359545	+	0.933128i	$0.382932\pi$
$180$	0	0
$181$	−20.3741	−1.51440	−0.757198	−	0.653186i	$-0.773432\pi$
$181$	−20.3741	−1.51440	−0.757198	+	0.653186i	$0.773432\pi$
$182$	1.40365	0.104045
$183$	0	0
$184$	−4.10931	−0.302943
$185$	15.0103	1.10358
$186$	0	0
$187$	7.76688	0.567970
$188$	6.02524	0.439436
$189$	0	0
$190$	19.4070	1.40793
$191$	−19.6236	−1.41991	−0.709956	−	0.704246i	$-0.751285\pi$
$191$	−19.6236	−1.41991	−0.709956	+	0.704246i	$0.751285\pi$
$192$	0	0
$193$	16.0868	1.15795	0.578977	−	0.815344i	$-0.303452\pi$
$193$	16.0868	1.15795	0.578977	+	0.815344i	$0.303452\pi$
$194$	20.9308	1.50275
$195$	0	0
$196$	0.702628	0.0501877
$197$	−10.0376	−0.715153	−0.357576	−	0.933884i	$-0.616397\pi$
$197$	−10.0376	−0.715153	−0.357576	+	0.933884i	$0.616397\pi$
$198$	0	0
$199$	−11.1859	−0.792947	−0.396473	−	0.918046i	$-0.629766\pi$
$199$	−11.1859	−0.792947	−0.396473	+	0.918046i	$0.629766\pi$
$200$	−5.85029	−0.413678
$201$	0	0
$202$	9.20205	0.647454
$203$	4.22790	0.296740
$204$	0	0
$205$	0.445746	0.0311322
$206$	−16.6916	−1.16296
$207$	0	0
$208$	−4.19358	−0.290772
$209$	−24.4515	−1.69135
$210$	0	0
$211$	−5.18621	−0.357033	−0.178517	−	0.983937i	$-0.557130\pi$
$211$	−5.18621	−0.357033	−0.178517	+	0.983937i	$0.557130\pi$
$212$	−4.13087	−0.283709
$213$	0	0
$214$	−19.2264	−1.31429
$215$	−6.23325	−0.425104
$216$	0	0
$217$	−10.7031	−0.726574
$218$	15.8466	1.07327
$219$	0	0
$220$	3.28478	0.221460
$221$	−2.13108	−0.143352
$222$	0	0
$223$	27.4486	1.83809	0.919046	−	0.394150i	$-0.128961\pi$
$223$	27.4486	1.83809	0.919046	+	0.394150i	$0.128961\pi$
$224$	−3.80879	−0.254485
$225$	0	0
$226$	−8.26755	−0.549949
$227$	28.4170	1.88610	0.943050	−	0.332652i	$-0.107943\pi$
$227$	28.4170	1.88610	0.943050	+	0.332652i	$0.107943\pi$
$228$	0	0
$229$	30.1369	1.99150	0.995750	−	0.0921004i	$-0.0293581\pi$
$229$	30.1369	1.99150	0.995750	+	0.0921004i	$0.0293581\pi$
$230$	−4.75855	−0.313769
$231$	0	0
$232$	−9.01742	−0.592022
$233$	−1.97898	−0.129647	−0.0648235	−	0.997897i	$-0.520648\pi$
$233$	−1.97898	−0.129647	−0.0648235	+	0.997897i	$0.520648\pi$
$234$	0	0
$235$	−12.8830	−0.840396
$236$	−9.08607	−0.591452
$237$	0	0
$238$	−4.10325	−0.265974
$239$	−13.8986	−0.899024	−0.449512	−	0.893274i	$-0.648402\pi$
$239$	−13.8986	−0.899024	−0.449512	+	0.893274i	$0.648402\pi$
$240$	0	0
$241$	29.3019	1.88750	0.943751	−	0.330658i	$-0.107271\pi$
$241$	29.3019	1.88750	0.943751	+	0.330658i	$0.107271\pi$
$242$	2.16465	0.139149
$243$	0	0
$244$	2.22179	0.142235
$245$	−1.50234	−0.0959812
$246$	0	0
$247$	6.70903	0.426885
$248$	22.8280	1.44958
$249$	0	0
$250$	−19.1236	−1.20948
$251$	11.2240	0.708451	0.354226	−	0.935160i	$-0.384745\pi$
$251$	11.2240	0.708451	0.354226	+	0.935160i	$0.384745\pi$
$252$	0	0
$253$	5.99547	0.376932
$254$	−1.64397	−0.103152
$255$	0	0
$256$	15.0255	0.939096
$257$	7.44300	0.464281	0.232141	−	0.972682i	$-0.425427\pi$
$257$	7.44300	0.464281	0.232141	+	0.972682i	$0.425427\pi$
$258$	0	0
$259$	9.99125	0.620826
$260$	−0.901279	−0.0558949
$261$	0	0
$262$	−20.3664	−1.25824
$263$	−17.8004	−1.09762	−0.548811	−	0.835946i	$-0.684919\pi$
$263$	−17.8004	−1.09762	−0.548811	+	0.835946i	$0.684919\pi$
$264$	0	0
$265$	8.83252	0.542578
$266$	12.9178	0.792041
$267$	0	0
$268$	5.44403	0.332547
$269$	−9.38213	−0.572039	−0.286019	−	0.958224i	$-0.592332\pi$
$269$	−9.38213	−0.572039	−0.286019	+	0.958224i	$0.592332\pi$
$270$	0	0
$271$	−1.61168	−0.0979025	−0.0489512	−	0.998801i	$-0.515588\pi$
$271$	−1.61168	−0.0979025	−0.0489512	+	0.998801i	$0.515588\pi$
$272$	12.2590	0.743312
$273$	0	0
$274$	4.87459	0.294485
$275$	8.53554	0.514713
$276$	0	0
$277$	−10.1885	−0.612170	−0.306085	−	0.952004i	$-0.599019\pi$
$277$	−10.1885	−0.612170	−0.306085	+	0.952004i	$0.599019\pi$
$278$	−37.4156	−2.24404
$279$	0	0
$280$	3.20425	0.191491
$281$	9.28607	0.553961	0.276980	−	0.960876i	$-0.410666\pi$
$281$	9.28607	0.553961	0.276980	+	0.960876i	$0.410666\pi$
$282$	0	0
$283$	−18.8789	−1.12224	−0.561118	−	0.827736i	$-0.689629\pi$
$283$	−18.8789	−1.12224	−0.561118	+	0.827736i	$0.689629\pi$
$284$	−0.275635	−0.0163559
$285$	0	0
$286$	4.36786	0.258277
$287$	0.296700	0.0175137
$288$	0	0
$289$	−10.7703	−0.633544
$290$	−10.4421	−0.613180
$291$	0	0
$292$	2.98461	0.174661
$293$	23.2367	1.35750	0.678752	−	0.734368i	$-0.262521\pi$
$293$	23.2367	1.35750	0.678752	+	0.734368i	$0.262521\pi$
$294$	0	0
$295$	19.4276	1.13112
$296$	−21.3097	−1.23860
$297$	0	0
$298$	−7.23915	−0.419353
$299$	−1.64504	−0.0951350
$300$	0	0
$301$	−4.14902	−0.239145
$302$	13.2341	0.761535
$303$	0	0
$304$	−38.5936	−2.21350
$305$	−4.75057	−0.272017
$306$	0	0
$307$	10.7735	0.614874	0.307437	−	0.951568i	$-0.400529\pi$
$307$	10.7735	0.614874	0.307437	+	0.951568i	$0.400529\pi$
$308$	2.18644	0.124584
$309$	0	0
$310$	26.4346	1.50138
$311$	−15.9004	−0.901631	−0.450815	−	0.892617i	$-0.648867\pi$
$311$	−15.9004	−0.901631	−0.450815	+	0.892617i	$0.648867\pi$
$312$	0	0
$313$	2.82592	0.159731	0.0798653	−	0.996806i	$-0.474551\pi$
$313$	2.82592	0.159731	0.0798653	+	0.996806i	$0.474551\pi$
$314$	11.7503	0.663110
$315$	0	0
$316$	−3.54727	−0.199550
$317$	−28.7802	−1.61645	−0.808227	−	0.588871i	$-0.799573\pi$
$317$	−28.7802	−1.61645	−0.808227	+	0.588871i	$0.799573\pi$
$318$	0	0
$319$	13.1564	0.736615
$320$	−5.35078	−0.299118
$321$	0	0
$322$	−3.16741	−0.176513
$323$	−19.6124	−1.09126
$324$	0	0
$325$	−2.34199	−0.129910
$326$	−9.05211	−0.501350
$327$	0	0
$328$	−0.632813	−0.0349413
$329$	−8.57529	−0.472771
$330$	0	0
$331$	−25.1639	−1.38313	−0.691566	−	0.722314i	$-0.743079\pi$
$331$	−25.1639	−1.38313	−0.691566	+	0.722314i	$0.743079\pi$
$332$	8.53290	0.468304
$333$	0	0
$334$	−2.43395	−0.133180
$335$	−11.6403	−0.635978
$336$	0	0
$337$	−10.1621	−0.553563	−0.276782	−	0.960933i	$-0.589268\pi$
$337$	−10.1621	−0.553563	−0.276782	+	0.960933i	$0.589268\pi$
$338$	20.1731	1.09727
$339$	0	0
$340$	2.63469	0.142886
$341$	−33.3059	−1.80362
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	8.84918	0.477116
$345$	0	0
$346$	28.6654	1.54106
$347$	−9.52869	−0.511527	−0.255764	−	0.966739i	$-0.582327\pi$
$347$	−9.52869	−0.511527	−0.255764	+	0.966739i	$0.582327\pi$
$348$	0	0
$349$	11.0117	0.589443	0.294722	−	0.955583i	$-0.404773\pi$
$349$	11.0117	0.589443	0.294722	+	0.955583i	$0.404773\pi$
$350$	−4.50934	−0.241034
$351$	0	0
$352$	−11.8522	−0.631723
$353$	−23.9795	−1.27630	−0.638150	−	0.769912i	$-0.720300\pi$
$353$	−23.9795	−1.27630	−0.638150	+	0.769912i	$0.720300\pi$
$354$	0	0
$355$	0.589356	0.0312798
$356$	4.18485	0.221797
$357$	0	0
$358$	−15.8162	−0.835915
$359$	29.7165	1.56837	0.784187	−	0.620524i	$-0.213080\pi$
$359$	29.7165	1.56837	0.784187	+	0.620524i	$0.213080\pi$
$360$	0	0
$361$	42.7434	2.24965
$362$	33.4944	1.76042
$363$	0	0
$364$	−0.599915	−0.0314441
$365$	−6.38162	−0.334029
$366$	0	0
$367$	−21.7769	−1.13675	−0.568373	−	0.822771i	$-0.692427\pi$
$367$	−21.7769	−1.13675	−0.568373	+	0.822771i	$0.692427\pi$
$368$	9.46307	0.493297
$369$	0	0
$370$	−24.6764	−1.28287
$371$	5.87916	0.305231
$372$	0	0
$373$	−7.66991	−0.397133	−0.198566	−	0.980087i	$-0.563629\pi$
$373$	−7.66991	−0.397133	−0.198566	+	0.980087i	$0.563629\pi$
$374$	−12.7685	−0.660243
$375$	0	0
$376$	18.2897	0.943219
$377$	−3.60985	−0.185917
$378$	0	0
$379$	−11.6717	−0.599538	−0.299769	−	0.954012i	$-0.596910\pi$
$379$	−11.6717	−0.599538	−0.299769	+	0.954012i	$0.596910\pi$
$380$	−8.29450	−0.425499
$381$	0	0
$382$	32.2605	1.65059
$383$	14.8396	0.758269	0.379135	−	0.925342i	$-0.376222\pi$
$383$	14.8396	0.758269	0.379135	+	0.925342i	$0.376222\pi$
$384$	0	0
$385$	−4.67499	−0.238259
$386$	−26.4462	−1.34608
$387$	0	0
$388$	−8.94580	−0.454154
$389$	1.08394	0.0549580	0.0274790	−	0.999622i	$-0.491252\pi$
$389$	1.08394	0.0549580	0.0274790	+	0.999622i	$0.491252\pi$
$390$	0	0
$391$	4.80891	0.243197
$392$	2.13284	0.107725
$393$	0	0
$394$	16.5016	0.831336
$395$	7.58469	0.381627
$396$	0	0
$397$	−10.7362	−0.538836	−0.269418	−	0.963023i	$-0.586831\pi$
$397$	−10.7362	−0.538836	−0.269418	+	0.963023i	$0.586831\pi$
$398$	18.3892	0.921769
$399$	0	0
$400$	13.4723	0.673613
$401$	26.4102	1.31886	0.659431	−	0.751765i	$-0.270797\pi$
$401$	26.4102	1.31886	0.659431	+	0.751765i	$0.270797\pi$
$402$	0	0
$403$	9.13848	0.455220
$404$	−3.93294	−0.195671
$405$	0	0
$406$	−6.95053	−0.344949
$407$	31.0907	1.54111
$408$	0	0
$409$	−31.2144	−1.54345	−0.771727	−	0.635954i	$-0.780607\pi$
$409$	−31.2144	−1.54345	−0.771727	+	0.635954i	$0.780607\pi$
$410$	−0.732791	−0.0361900
$411$	0	0
$412$	7.13396	0.351465
$413$	12.9315	0.636320
$414$	0	0
$415$	−18.2448	−0.895604
$416$	3.25200	0.159443
$417$	0	0
$418$	40.1975	1.96613
$419$	−37.7463	−1.84403	−0.922014	−	0.387157i	$-0.873457\pi$
$419$	−37.7463	−1.84403	−0.922014	+	0.387157i	$0.873457\pi$
$420$	0	0
$421$	1.92008	0.0935791	0.0467895	−	0.998905i	$-0.485101\pi$
$421$	1.92008	0.0935791	0.0467895	+	0.998905i	$0.485101\pi$
$422$	8.52596	0.415037
$423$	0	0
$424$	−12.5393	−0.608962
$425$	6.84629	0.332094
$426$	0	0
$427$	−3.16211	−0.153025
$428$	8.21732	0.397199
$429$	0	0
$430$	10.2473	0.494167
$431$	−32.1264	−1.54747	−0.773737	−	0.633507i	$-0.781615\pi$
$431$	−32.1264	−1.54747	−0.773737	+	0.633507i	$0.781615\pi$
$432$	0	0
$433$	−21.6075	−1.03839	−0.519194	−	0.854656i	$-0.673768\pi$
$433$	−21.6075	−1.03839	−0.519194	+	0.854656i	$0.673768\pi$
$434$	17.5956	0.844614
$435$	0	0
$436$	−6.77281	−0.324359
$437$	−15.1393	−0.724213
$438$	0	0
$439$	−5.86414	−0.279880	−0.139940	−	0.990160i	$-0.544691\pi$
$439$	−5.86414	−0.279880	−0.139940	+	0.990160i	$0.544691\pi$
$440$	9.97098	0.475348
$441$	0	0
$442$	3.50342	0.166641
$443$	20.6185	0.979616	0.489808	−	0.871830i	$-0.337067\pi$
$443$	20.6185	0.979616	0.489808	+	0.871830i	$0.337067\pi$
$444$	0	0
$445$	−8.94795	−0.424174
$446$	−45.1245	−2.13671
$447$	0	0
$448$	−3.56162	−0.168271
$449$	−11.5989	−0.547384	−0.273692	−	0.961817i	$-0.588245\pi$
$449$	−11.5989	−0.547384	−0.273692	+	0.961817i	$0.588245\pi$
$450$	0	0
$451$	0.923270	0.0434751
$452$	3.53353	0.166203
$453$	0	0
$454$	−46.7165	−2.19252
$455$	1.28272	0.0601351
$456$	0	0
$457$	−38.6356	−1.80730	−0.903648	−	0.428276i	$-0.859121\pi$
$457$	−38.6356	−1.80730	−0.903648	+	0.428276i	$0.859121\pi$
$458$	−49.5440	−2.31504
$459$	0	0
$460$	2.03379	0.0948261
$461$	16.8231	0.783529	0.391765	−	0.920065i	$-0.371865\pi$
$461$	16.8231	0.783529	0.391765	+	0.920065i	$0.371865\pi$
$462$	0	0
$463$	2.85654	0.132755	0.0663774	−	0.997795i	$-0.478856\pi$
$463$	2.85654	0.132755	0.0663774	+	0.997795i	$0.478856\pi$
$464$	20.7656	0.964020
$465$	0	0
$466$	3.25337	0.150710
$467$	6.72711	0.311294	0.155647	−	0.987813i	$-0.450254\pi$
$467$	6.72711	0.311294	0.155647	+	0.987813i	$0.450254\pi$
$468$	0	0
$469$	−7.74810	−0.357774
$470$	21.1793	0.976927
$471$	0	0
$472$	−27.5809	−1.26951
$473$	−12.9109	−0.593644
$474$	0	0
$475$	−21.5534	−0.988937
$476$	1.75372	0.0803817
$477$	0	0
$478$	22.8488	1.04508
$479$	−0.115837	−0.00529271	−0.00264635	−	0.999996i	$-0.500842\pi$
$479$	−0.115837	−0.00529271	−0.00264635	+	0.999996i	$0.500842\pi$
$480$	0	0
$481$	−8.53069	−0.388966
$482$	−48.1714	−2.19415
$483$	0	0
$484$	−0.925168	−0.0420531
$485$	19.1277	0.868544
$486$	0	0
$487$	1.35308	0.0613138	0.0306569	−	0.999530i	$-0.490240\pi$
$487$	1.35308	0.0613138	0.0306569	+	0.999530i	$0.490240\pi$
$488$	6.74426	0.305298
$489$	0	0
$490$	2.46980	0.111574
$491$	−13.3445	−0.602230	−0.301115	−	0.953588i	$-0.597359\pi$
$491$	−13.3445	−0.602230	−0.301115	+	0.953588i	$0.597359\pi$
$492$	0	0
$493$	10.5526	0.475266
$494$	−11.0294	−0.496237
$495$	0	0
$496$	−52.5691	−2.36042
$497$	0.392291	0.0175967
$498$	0	0
$499$	0.531521	0.0237942	0.0118971	−	0.999929i	$-0.496213\pi$
$499$	0.531521	0.0237942	0.0118971	+	0.999929i	$0.496213\pi$
$500$	8.17339	0.365525
$501$	0	0
$502$	−18.4518	−0.823546
$503$	31.7938	1.41762	0.708809	−	0.705401i	$-0.249233\pi$
$503$	31.7938	1.41762	0.708809	+	0.705401i	$0.249233\pi$
$504$	0	0
$505$	8.40931	0.374210
$506$	−9.85635	−0.438168
$507$	0	0
$508$	0.702628	0.0311741
$509$	19.4509	0.862147	0.431074	−	0.902317i	$-0.358135\pi$
$509$	19.4509	0.862147	0.431074	+	0.902317i	$0.358135\pi$
$510$	0	0
$511$	−4.24778	−0.187911
$512$	2.24403	0.0991731
$513$	0	0
$514$	−12.2360	−0.539709
$515$	−15.2537	−0.672157
$516$	0	0
$517$	−26.6845	−1.17358
$518$	−16.4253	−0.721686
$519$	0	0
$520$	−2.73584	−0.119975
$521$	−23.6715	−1.03707	−0.518533	−	0.855058i	$-0.673522\pi$
$521$	−23.6715	−1.03707	−0.518533	+	0.855058i	$0.673522\pi$
$522$	0	0
$523$	5.38698	0.235556	0.117778	−	0.993040i	$-0.462423\pi$
$523$	5.38698	0.235556	0.117778	+	0.993040i	$0.462423\pi$
$524$	8.70455	0.380260
$525$	0	0
$526$	29.2633	1.27594
$527$	−26.7144	−1.16370
$528$	0	0
$529$	−19.2879	−0.838603
$530$	−14.5204	−0.630725
$531$	0	0
$532$	−5.52104	−0.239367
$533$	−0.253327	−0.0109728
$534$	0	0
$535$	−17.5701	−0.759621
$536$	16.5254	0.713790
$537$	0	0
$538$	15.4239	0.664972
$539$	−3.11180	−0.134035
$540$	0	0
$541$	20.9826	0.902114	0.451057	−	0.892495i	$-0.351047\pi$
$541$	20.9826	0.902114	0.451057	+	0.892495i	$0.351047\pi$
$542$	2.64955	0.113808
$543$	0	0
$544$	−9.50653	−0.407589
$545$	14.4815	0.620318
$546$	0	0
$547$	−32.8035	−1.40258	−0.701288	−	0.712878i	$-0.747391\pi$
$547$	−32.8035	−1.40258	−0.701288	+	0.712878i	$0.747391\pi$
$548$	−2.08339	−0.0889979
$549$	0	0
$550$	−14.0322	−0.598333
$551$	−33.2215	−1.41529
$552$	0	0
$553$	5.04857	0.214687
$554$	16.7496	0.711623
$555$	0	0
$556$	15.9914	0.678184
$557$	−30.7859	−1.30444	−0.652219	−	0.758030i	$-0.726162\pi$
$557$	−30.7859	−1.30444	−0.652219	+	0.758030i	$0.726162\pi$
$558$	0	0
$559$	3.54250	0.149832
$560$	−7.37887	−0.311814
$561$	0	0
$562$	−15.2660	−0.643957
$563$	7.80410	0.328904	0.164452	−	0.986385i	$-0.447415\pi$
$563$	7.80410	0.328904	0.164452	+	0.986385i	$0.447415\pi$
$564$	0	0
$565$	−7.55532	−0.317855
$566$	31.0364	1.30456
$567$	0	0
$568$	−0.836693	−0.0351069
$569$	17.3419	0.727010	0.363505	−	0.931592i	$-0.381580\pi$
$569$	17.3419	0.727010	0.363505	+	0.931592i	$0.381580\pi$
$570$	0	0
$571$	−31.6340	−1.32384	−0.661920	−	0.749575i	$-0.730258\pi$
$571$	−31.6340	−1.32384	−0.661920	+	0.749575i	$0.730258\pi$
$572$	−1.86681	−0.0780554
$573$	0	0
$574$	−0.487765	−0.0203589
$575$	5.28484	0.220393
$576$	0	0
$577$	−21.4335	−0.892287	−0.446144	−	0.894961i	$-0.647203\pi$
$577$	−21.4335	−0.892287	−0.446144	+	0.894961i	$0.647203\pi$
$578$	17.7059	0.736470
$579$	0	0
$580$	4.46292	0.185313
$581$	−12.1443	−0.503829
$582$	0	0
$583$	18.2948	0.757691
$584$	9.05982	0.374898
$585$	0	0
$586$	−38.2004	−1.57804
$587$	−0.389253	−0.0160662	−0.00803309	−	0.999968i	$-0.502557\pi$
$587$	−0.389253	−0.0160662	−0.00803309	+	0.999968i	$0.502557\pi$
$588$	0	0
$589$	84.1018	3.46535
$590$	−31.9384	−1.31488
$591$	0	0
$592$	49.0727	2.01688
$593$	−6.22335	−0.255562	−0.127781	−	0.991802i	$-0.540786\pi$
$593$	−6.22335	−0.255562	−0.127781	+	0.991802i	$0.540786\pi$
$594$	0	0
$595$	−3.74977	−0.153726
$596$	3.09400	0.126735
$597$	0	0
$598$	2.70439	0.110591
$599$	15.1977	0.620961	0.310481	−	0.950580i	$-0.399510\pi$
$599$	15.1977	0.620961	0.310481	+	0.950580i	$0.399510\pi$
$600$	0	0
$601$	−34.6537	−1.41356	−0.706778	−	0.707436i	$-0.749852\pi$
$601$	−34.6537	−1.41356	−0.706778	+	0.707436i	$0.749852\pi$
$602$	6.82085	0.277997
$603$	0	0
$604$	−5.65621	−0.230148
$605$	1.97817	0.0804242
$606$	0	0
$607$	13.9747	0.567215	0.283607	−	0.958940i	$-0.408469\pi$
$607$	13.9747	0.567215	0.283607	+	0.958940i	$0.408469\pi$
$608$	29.9283	1.21375
$609$	0	0
$610$	7.80979	0.316209
$611$	7.32172	0.296205
$612$	0	0
$613$	1.16254	0.0469547	0.0234774	−	0.999724i	$-0.492526\pi$
$613$	1.16254	0.0469547	0.0234774	+	0.999724i	$0.492526\pi$
$614$	−17.7112	−0.714766
$615$	0	0
$616$	6.63695	0.267410
$617$	−32.9218	−1.32538	−0.662692	−	0.748892i	$-0.730586\pi$
$617$	−32.9218	−1.32538	−0.662692	+	0.748892i	$0.730586\pi$
$618$	0	0
$619$	2.06204	0.0828802	0.0414401	−	0.999141i	$-0.486805\pi$
$619$	2.06204	0.0828802	0.0414401	+	0.999141i	$0.486805\pi$
$620$	−11.2981	−0.453742
$621$	0	0
$622$	26.1398	1.04811
$623$	−5.95600	−0.238622
$624$	0	0
$625$	−3.76133	−0.150453
$626$	−4.64573	−0.185681
$627$	0	0
$628$	−5.02207	−0.200403
$629$	24.9376	0.994329
$630$	0	0
$631$	−10.5678	−0.420697	−0.210348	−	0.977626i	$-0.567460\pi$
$631$	−10.5678	−0.420697	−0.210348	+	0.977626i	$0.567460\pi$
$632$	−10.7678	−0.428319
$633$	0	0
$634$	47.3136	1.87906
$635$	−1.50234	−0.0596187
$636$	0	0
$637$	0.853816	0.0338294
$638$	−21.6286	−0.856285
$639$	0	0
$640$	20.2407	0.800084
$641$	−36.4374	−1.43919	−0.719596	−	0.694393i	$-0.755673\pi$
$641$	−36.4374	−1.43919	−0.719596	+	0.694393i	$0.755673\pi$
$642$	0	0
$643$	−34.0987	−1.34472	−0.672361	−	0.740223i	$-0.734720\pi$
$643$	−34.0987	−1.34472	−0.672361	+	0.740223i	$0.734720\pi$
$644$	1.35375	0.0533451
$645$	0	0
$646$	32.2421	1.26855
$647$	−11.2080	−0.440631	−0.220316	−	0.975429i	$-0.570709\pi$
$647$	−11.2080	−0.440631	−0.220316	+	0.975429i	$0.570709\pi$
$648$	0	0
$649$	40.2403	1.57957
$650$	3.85015	0.151015
$651$	0	0
$652$	3.86885	0.151516
$653$	15.6310	0.611689	0.305844	−	0.952082i	$-0.401061\pi$
$653$	15.6310	0.611689	0.305844	+	0.952082i	$0.401061\pi$
$654$	0	0
$655$	−18.6119	−0.727226
$656$	1.45726	0.0568966
$657$	0	0
$658$	14.0975	0.549577
$659$	7.39070	0.287901	0.143950	−	0.989585i	$-0.454019\pi$
$659$	7.39070	0.287901	0.143950	+	0.989585i	$0.454019\pi$
$660$	0	0
$661$	20.0253	0.778893	0.389447	−	0.921049i	$-0.372666\pi$
$661$	20.0253	0.778893	0.389447	+	0.921049i	$0.372666\pi$
$662$	41.3686	1.60784
$663$	0	0
$664$	25.9017	1.00518
$665$	11.8050	0.457777
$666$	0	0
$667$	8.14585	0.315409
$668$	1.04026	0.0402490
$669$	0	0
$670$	19.1363	0.739299
$671$	−9.83983	−0.379863
$672$	0	0
$673$	39.3004	1.51492	0.757459	−	0.652883i	$-0.226441\pi$
$673$	39.3004	1.51492	0.757459	+	0.652883i	$0.226441\pi$
$674$	16.7061	0.643495
$675$	0	0
$676$	−8.62195	−0.331613
$677$	19.1027	0.734175	0.367088	−	0.930186i	$-0.380355\pi$
$677$	19.1027	0.734175	0.367088	+	0.930186i	$0.380355\pi$
$678$	0	0
$679$	12.7319	0.488606
$680$	7.99765	0.306696
$681$	0	0
$682$	54.7538	2.09663
$683$	−42.8271	−1.63873	−0.819367	−	0.573270i	$-0.805675\pi$
$683$	−42.8271	−1.63873	−0.819367	+	0.573270i	$0.805675\pi$
$684$	0	0
$685$	4.45465	0.170204
$686$	1.64397	0.0627670
$687$	0	0
$688$	−20.3782	−0.776911
$689$	−5.01972	−0.191236
$690$	0	0
$691$	1.27559	0.0485258	0.0242629	−	0.999706i	$-0.492276\pi$
$691$	1.27559	0.0485258	0.0242629	+	0.999706i	$0.492276\pi$
$692$	−12.2515	−0.465733
$693$	0	0
$694$	15.6649	0.594630
$695$	−34.1923	−1.29699
$696$	0	0
$697$	0.740548	0.0280502
$698$	−18.1029	−0.685205
$699$	0	0
$700$	1.92728	0.0728445
$701$	20.0665	0.757901	0.378951	−	0.925417i	$-0.376285\pi$
$701$	20.0665	0.757901	0.378951	+	0.925417i	$0.376285\pi$
$702$	0	0
$703$	−78.5082	−2.96099
$704$	−11.0830	−0.417708
$705$	0	0
$706$	39.4215	1.48365
$707$	5.59746	0.210514
$708$	0	0
$709$	−32.3909	−1.21647	−0.608233	−	0.793759i	$-0.708121\pi$
$709$	−32.3909	−1.21647	−0.608233	+	0.793759i	$0.708121\pi$
$710$	−0.968883	−0.0363615
$711$	0	0
$712$	12.7032	0.476071
$713$	−20.6216	−0.772284
$714$	0	0
$715$	3.99158	0.149277
$716$	6.75983	0.252627
$717$	0	0
$718$	−48.8529	−1.82317
$719$	−21.7675	−0.811791	−0.405896	−	0.913919i	$-0.633040\pi$
$719$	−21.7675	−0.811791	−0.405896	+	0.913919i	$0.633040\pi$
$720$	0	0
$721$	−10.1532	−0.378127
$722$	−70.2687	−2.61513
$723$	0	0
$724$	−14.3154	−0.532028
$725$	11.5970	0.430701
$726$	0	0
$727$	8.46880	0.314090	0.157045	−	0.987591i	$-0.449803\pi$
$727$	8.46880	0.314090	0.157045	+	0.987591i	$0.449803\pi$
$728$	−1.82105	−0.0674926
$729$	0	0
$730$	10.4912	0.388296
$731$	−10.3557	−0.383020
$732$	0	0
$733$	−52.4162	−1.93604	−0.968019	−	0.250877i	$-0.919281\pi$
$733$	−52.4162	−1.93604	−0.968019	+	0.250877i	$0.919281\pi$
$734$	35.8005	1.32142
$735$	0	0
$736$	−7.33835	−0.270495
$737$	−24.1105	−0.888122
$738$	0	0
$739$	−23.4354	−0.862087	−0.431043	−	0.902331i	$-0.641854\pi$
$739$	−23.4354	−0.862087	−0.431043	+	0.902331i	$0.641854\pi$
$740$	10.5467	0.387703
$741$	0	0
$742$	−9.66515	−0.354819
$743$	−2.47308	−0.0907285	−0.0453643	−	0.998971i	$-0.514445\pi$
$743$	−2.47308	−0.0907285	−0.0453643	+	0.998971i	$0.514445\pi$
$744$	0	0
$745$	−6.61551	−0.242374
$746$	12.6091	0.461651
$747$	0	0
$748$	5.45723	0.199536
$749$	−11.6951	−0.427330
$750$	0	0
$751$	−44.0389	−1.60700	−0.803501	−	0.595303i	$-0.797032\pi$
$751$	−44.0389	−1.60700	−0.803501	+	0.595303i	$0.797032\pi$
$752$	−42.1181	−1.53589
$753$	0	0
$754$	5.93447	0.216121
$755$	12.0940	0.440145
$756$	0	0
$757$	−17.5903	−0.639332	−0.319666	−	0.947530i	$-0.603571\pi$
$757$	−17.5903	−0.639332	−0.319666	+	0.947530i	$0.603571\pi$
$758$	19.1880	0.696939
$759$	0	0
$760$	−25.1781	−0.913304
$761$	22.6552	0.821249	0.410625	−	0.911805i	$-0.365311\pi$
$761$	22.6552	0.821249	0.410625	+	0.911805i	$0.365311\pi$
$762$	0	0
$763$	9.63924	0.348964
$764$	−13.7881	−0.498835
$765$	0	0
$766$	−24.3958	−0.881458
$767$	−11.0412	−0.398673
$768$	0	0
$769$	−20.6905	−0.746118	−0.373059	−	0.927808i	$-0.621691\pi$
$769$	−20.6905	−0.746118	−0.373059	+	0.927808i	$0.621691\pi$
$770$	7.68553	0.276967
$771$	0	0
$772$	11.3030	0.406806
$773$	53.9967	1.94213	0.971063	−	0.238825i	$-0.0767622\pi$
$773$	53.9967	1.94213	0.971063	+	0.238825i	$0.0767622\pi$
$774$	0	0
$775$	−29.3582	−1.05458
$776$	−27.1551	−0.974810
$777$	0	0
$778$	−1.78196	−0.0638865
$779$	−2.33138	−0.0835304
$780$	0	0
$781$	1.22073	0.0436812
$782$	−7.90570	−0.282707
$783$	0	0
$784$	−4.91157	−0.175413
$785$	10.7381	0.383259
$786$	0	0
$787$	6.97616	0.248673	0.124337	−	0.992240i	$-0.460320\pi$
$787$	6.97616	0.248673	0.124337	+	0.992240i	$0.460320\pi$
$788$	−7.05273	−0.251243
$789$	0	0
$790$	−12.4690	−0.443627
$791$	−5.02902	−0.178811
$792$	0	0
$793$	2.69986	0.0958748
$794$	17.6500	0.626376
$795$	0	0
$796$	−7.85952	−0.278573
$797$	−3.70408	−0.131205	−0.0656026	−	0.997846i	$-0.520897\pi$
$797$	−3.70408	−0.131205	−0.0656026	+	0.997846i	$0.520897\pi$
$798$	0	0
$799$	−21.4035	−0.757200
$800$	−10.4474	−0.369370
$801$	0	0
$802$	−43.4175	−1.53313
$803$	−13.2182	−0.466461
$804$	0	0
$805$	−2.89455	−0.102019
$806$	−15.0234	−0.529176
$807$	0	0
$808$	−11.9385	−0.419994
$809$	−21.5004	−0.755915	−0.377958	−	0.925823i	$-0.623374\pi$
$809$	−21.5004	−0.755915	−0.377958	+	0.925823i	$0.623374\pi$
$810$	0	0
$811$	−17.0080	−0.597232	−0.298616	−	0.954373i	$-0.596525\pi$
$811$	−17.0080	−0.597232	−0.298616	+	0.954373i	$0.596525\pi$
$812$	2.97064	0.104249
$813$	0	0
$814$	−51.1122	−1.79148
$815$	−8.27229	−0.289766
$816$	0	0
$817$	32.6017	1.14059
$818$	51.3155	1.79420
$819$	0	0
$820$	0.313193	0.0109372
$821$	42.6811	1.48958	0.744790	−	0.667299i	$-0.232550\pi$
$821$	42.6811	1.48958	0.744790	+	0.667299i	$0.232550\pi$
$822$	0	0
$823$	30.3748	1.05880	0.529399	−	0.848373i	$-0.322417\pi$
$823$	30.3748	1.05880	0.529399	+	0.848373i	$0.322417\pi$
$824$	21.6552	0.754395
$825$	0	0
$826$	−21.2590	−0.739696
$827$	41.4154	1.44015	0.720077	−	0.693894i	$-0.244106\pi$
$827$	41.4154	1.44015	0.720077	+	0.693894i	$0.244106\pi$
$828$	0	0
$829$	−24.4111	−0.847832	−0.423916	−	0.905701i	$-0.639345\pi$
$829$	−24.4111	−0.847832	−0.423916	+	0.905701i	$0.639345\pi$
$830$	29.9939	1.04110
$831$	0	0
$832$	3.04097	0.105427
$833$	−2.49595	−0.0864794
$834$	0	0
$835$	−2.22427	−0.0769740
$836$	−17.1803	−0.594195
$837$	0	0
$838$	62.0537	2.14361
$839$	−14.6821	−0.506883	−0.253441	−	0.967351i	$-0.581562\pi$
$839$	−14.6821	−0.506883	−0.253441	+	0.967351i	$0.581562\pi$
$840$	0	0
$841$	−11.1249	−0.383616
$842$	−3.15655	−0.108782
$843$	0	0
$844$	−3.64398	−0.125431
$845$	18.4353	0.634192
$846$	0	0
$847$	1.31673	0.0452432
$848$	28.8759	0.991603
$849$	0	0
$850$	−11.2551	−0.386046
$851$	19.2500	0.659883
$852$	0	0
$853$	24.3034	0.832133	0.416066	−	0.909334i	$-0.363408\pi$
$853$	24.3034	0.832133	0.416066	+	0.909334i	$0.363408\pi$
$854$	5.19840	0.177886
$855$	0	0
$856$	24.9438	0.852560
$857$	−56.6256	−1.93429	−0.967146	−	0.254221i	$-0.918181\pi$
$857$	−56.6256	−1.93429	−0.967146	+	0.254221i	$0.918181\pi$
$858$	0	0
$859$	−19.6310	−0.669802	−0.334901	−	0.942253i	$-0.608703\pi$
$859$	−19.6310	−0.669802	−0.334901	+	0.942253i	$0.608703\pi$
$860$	−4.37966	−0.149345
$861$	0	0
$862$	52.8147	1.79888
$863$	−10.2243	−0.348040	−0.174020	−	0.984742i	$-0.555676\pi$
$863$	−10.2243	−0.348040	−0.174020	+	0.984742i	$0.555676\pi$
$864$	0	0
$865$	26.1959	0.890688
$866$	35.5219	1.20708
$867$	0	0
$868$	−7.52031	−0.255256
$869$	15.7101	0.532930
$870$	0	0
$871$	6.61545	0.224156
$872$	−20.5589	−0.696213
$873$	0	0
$874$	24.8886	0.841869
$875$	−11.6326	−0.393253
$876$	0	0
$877$	−27.8528	−0.940523	−0.470262	−	0.882527i	$-0.655840\pi$
$877$	−27.8528	−0.940523	−0.470262	+	0.882527i	$0.655840\pi$
$878$	9.64045	0.325349
$879$	0	0
$880$	−22.9615	−0.774033
$881$	−42.9697	−1.44769	−0.723843	−	0.689965i	$-0.757626\pi$
$881$	−42.9697	−1.44769	−0.723843	+	0.689965i	$0.757626\pi$
$882$	0	0
$883$	36.0526	1.21327	0.606634	−	0.794982i	$-0.292520\pi$
$883$	36.0526	1.21327	0.606634	+	0.794982i	$0.292520\pi$
$884$	−1.49736	−0.0503615
$885$	0	0
$886$	−33.8962	−1.13877
$887$	−20.7959	−0.698258	−0.349129	−	0.937075i	$-0.613522\pi$
$887$	−20.7959	−0.698258	−0.349129	+	0.937075i	$0.613522\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	14.7101	0.493085
$891$	0	0
$892$	19.2861	0.645748
$893$	67.3820	2.25485
$894$	0	0
$895$	−14.4537	−0.483135
$896$	13.4728	0.450093
$897$	0	0
$898$	19.0681	0.636312
$899$	−45.2517	−1.50923
$900$	0	0
$901$	14.6741	0.488864
$902$	−1.51783	−0.0505381
$903$	0	0
$904$	10.7261	0.356744
$905$	30.6089	1.01747
$906$	0	0
$907$	−43.7901	−1.45403	−0.727014	−	0.686623i	$-0.759092\pi$
$907$	−43.7901	−1.45403	−0.727014	+	0.686623i	$0.759092\pi$
$908$	19.9666	0.662613
$909$	0	0
$910$	−2.10876	−0.0699046
$911$	2.95271	0.0978276	0.0489138	−	0.998803i	$-0.484424\pi$
$911$	2.95271	0.0978276	0.0489138	+	0.998803i	$0.484424\pi$
$912$	0	0
$913$	−37.7905	−1.25068
$914$	63.5156	2.10091
$915$	0	0
$916$	21.1750	0.699642
$917$	−12.3886	−0.409106
$918$	0	0
$919$	−2.93375	−0.0967754	−0.0483877	−	0.998829i	$-0.515408\pi$
$919$	−2.93375	−0.0967754	−0.0483877	+	0.998829i	$0.515408\pi$
$920$	6.17360	0.203538
$921$	0	0
$922$	−27.6566	−0.910822
$923$	−0.334945	−0.0110248
$924$	0	0
$925$	27.4056	0.901092
$926$	−4.69606	−0.154322
$927$	0	0
$928$	−16.1032	−0.528612
$929$	−11.2783	−0.370028	−0.185014	−	0.982736i	$-0.559233\pi$
$929$	−11.2783	−0.370028	−0.185014	+	0.982736i	$0.559233\pi$
$930$	0	0
$931$	7.85770	0.257526
$932$	−1.39048	−0.0455468
$933$	0	0
$934$	−11.0592	−0.361867
$935$	−11.6685	−0.381601
$936$	0	0
$937$	34.1815	1.11666	0.558331	−	0.829619i	$-0.311442\pi$
$937$	34.1815	1.11666	0.558331	+	0.829619i	$0.311442\pi$
$938$	12.7376	0.415898
$939$	0	0
$940$	−9.05198	−0.295243
$941$	−15.0504	−0.490628	−0.245314	−	0.969444i	$-0.578891\pi$
$941$	−15.0504	−0.490628	−0.245314	+	0.969444i	$0.578891\pi$
$942$	0	0
$943$	0.571649	0.0186155
$944$	63.5142	2.06721
$945$	0	0
$946$	21.2251	0.690087
$947$	−22.5752	−0.733595	−0.366798	−	0.930301i	$-0.619546\pi$
$947$	−22.5752	−0.733595	−0.366798	+	0.930301i	$0.619546\pi$
$948$	0	0
$949$	3.62682	0.117732
$950$	35.4330	1.14960
$951$	0	0
$952$	5.32345	0.172534
$953$	9.29671	0.301150	0.150575	−	0.988599i	$-0.451887\pi$
$953$	9.29671	0.301150	0.150575	+	0.988599i	$0.451887\pi$
$954$	0	0
$955$	29.4814	0.953994
$956$	−9.76553	−0.315840
$957$	0	0
$958$	0.190431	0.00615256
$959$	2.96514	0.0957492
$960$	0	0
$961$	83.5566	2.69537
$962$	14.0242	0.452158
$963$	0	0
$964$	20.5883	0.663106
$965$	−24.1679	−0.777993
$966$	0	0
$967$	−21.6311	−0.695608	−0.347804	−	0.937567i	$-0.613073\pi$
$967$	−21.6311	−0.695608	−0.347804	+	0.937567i	$0.613073\pi$
$968$	−2.80836	−0.0902641
$969$	0	0
$970$	−31.4453	−1.00965
$971$	−50.4074	−1.61765	−0.808826	−	0.588048i	$-0.799897\pi$
$971$	−50.4074	−1.61765	−0.808826	+	0.588048i	$0.799897\pi$
$972$	0	0
$973$	−22.7593	−0.729631
$974$	−2.22442	−0.0712749
$975$	0	0
$976$	−15.5309	−0.497132
$977$	49.3226	1.57797	0.788984	−	0.614414i	$-0.210607\pi$
$977$	49.3226	1.57797	0.788984	+	0.614414i	$0.210607\pi$
$978$	0	0
$979$	−18.5338	−0.592344
$980$	−1.05559	−0.0337196
$981$	0	0
$982$	21.9380	0.700069
$983$	−7.54294	−0.240582	−0.120291	−	0.992739i	$-0.538383\pi$
$983$	−7.54294	−0.240582	−0.120291	+	0.992739i	$0.538383\pi$
$984$	0	0
$985$	15.0800	0.480488
$986$	−17.3481	−0.552477
$987$	0	0
$988$	4.71395	0.149971
$989$	−7.99387	−0.254190
$990$	0	0
$991$	−15.5296	−0.493315	−0.246658	−	0.969103i	$-0.579332\pi$
$991$	−15.5296	−0.493315	−0.246658	+	0.969103i	$0.579332\pi$
$992$	40.7659	1.29432
$993$	0	0
$994$	−0.644914	−0.0204554
$995$	16.8051	0.532756
$996$	0	0
$997$	−16.0747	−0.509091	−0.254545	−	0.967061i	$-0.581926\pi$
$997$	−16.0747	−0.509091	−0.254545	+	0.967061i	$0.581926\pi$
$998$	−0.873803	−0.0276598
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.4		15
3.2	odd	2		889.2.a.b.1.12	✓	15
21.20	even	2		6223.2.a.j.1.12		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.12	✓	15	3.2	odd	2
6223.2.a.j.1.12		15	21.20	even	2
8001.2.a.q.1.4		15	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)

\(f(q)\)	\(=\)	\(q-1.64397 q^{2} +0.702628 q^{4} -1.50234 q^{5} -1.00000 q^{7} +2.13284 q^{8} +O(q^{10})\)	\(q-1.64397 q^{2} +0.702628 q^{4} -1.50234 q^{5} -1.00000 q^{7} +2.13284 q^{8} +2.46980 q^{10} -3.11180 q^{11} +0.853816 q^{13} +1.64397 q^{14} -4.91157 q^{16} -2.49595 q^{17} +7.85770 q^{19} -1.05559 q^{20} +5.11569 q^{22} -1.92669 q^{23} -2.74296 q^{25} -1.40365 q^{26} -0.702628 q^{28} -4.22790 q^{29} +10.7031 q^{31} +3.80879 q^{32} +4.10325 q^{34} +1.50234 q^{35} -9.99125 q^{37} -12.9178 q^{38} -3.20425 q^{40} -0.296700 q^{41} +4.14902 q^{43} -2.18644 q^{44} +3.16741 q^{46} +8.57529 q^{47} +1.00000 q^{49} +4.50934 q^{50} +0.599915 q^{52} -5.87916 q^{53} +4.67499 q^{55} -2.13284 q^{56} +6.95053 q^{58} -12.9315 q^{59} +3.16211 q^{61} -17.5956 q^{62} +3.56162 q^{64} -1.28272 q^{65} +7.74810 q^{67} -1.75372 q^{68} -2.46980 q^{70} -0.392291 q^{71} +4.24778 q^{73} +16.4253 q^{74} +5.52104 q^{76} +3.11180 q^{77} -5.04857 q^{79} +7.37887 q^{80} +0.487765 q^{82} +12.1443 q^{83} +3.74977 q^{85} -6.82085 q^{86} -6.63695 q^{88} +5.95600 q^{89} -0.853816 q^{91} -1.35375 q^{92} -14.0975 q^{94} -11.8050 q^{95} -12.7319 q^{97} -1.64397 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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