LMFDB - Embedded newform 8001.2.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 $ x^11 - 2x^10 - 15x^9 + 25x^8 + 88x^7 - 112x^6 - 247x^5 + 215x^4 + 313x^3 - 170x^2 - 121x + 57
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.5
Root			$-0.910036$ 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)

$f(q)$	$=$	$q-0.910036 q^{2} -1.17183 q^{4} -2.84869 q^{5} -1.00000 q^{7} +2.88648 q^{8} +O(q^{10})$	\(q-0.910036 q^{2} -1.17183 q^{4} -2.84869 q^{5} -1.00000 q^{7} +2.88648 q^{8} +2.59241 q^{10} +1.86466 q^{11} -6.19675 q^{13} +0.910036 q^{14} -0.283136 q^{16} -2.49732 q^{17} -6.22887 q^{19} +3.33819 q^{20} -1.69691 q^{22} +5.12980 q^{23} +3.11503 q^{25} +5.63926 q^{26} +1.17183 q^{28} +7.68531 q^{29} +4.32504 q^{31} -5.51530 q^{32} +2.27265 q^{34} +2.84869 q^{35} +2.90361 q^{37} +5.66849 q^{38} -8.22269 q^{40} -4.60795 q^{41} -9.98321 q^{43} -2.18507 q^{44} -4.66830 q^{46} +10.9278 q^{47} +1.00000 q^{49} -2.83479 q^{50} +7.26156 q^{52} +10.7010 q^{53} -5.31184 q^{55} -2.88648 q^{56} -6.99391 q^{58} -11.4224 q^{59} +1.39989 q^{61} -3.93594 q^{62} +5.58540 q^{64} +17.6526 q^{65} +14.3600 q^{67} +2.92644 q^{68} -2.59241 q^{70} -9.49609 q^{71} -16.1291 q^{73} -2.64239 q^{74} +7.29920 q^{76} -1.86466 q^{77} +12.5570 q^{79} +0.806567 q^{80} +4.19340 q^{82} +6.86150 q^{83} +7.11407 q^{85} +9.08509 q^{86} +5.38232 q^{88} +8.88528 q^{89} +6.19675 q^{91} -6.01128 q^{92} -9.94466 q^{94} +17.7441 q^{95} +1.23342 q^{97} -0.910036 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * 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 $

$2$	−0.910036	−0.643493	−0.321746	−	0.946826i	$-0.604270\pi$
$2$	−0.910036	−0.643493	−0.321746	+	0.946826i	$0.604270\pi$
$3$	0	0
$3$	0	0
$4$	−1.17183	−0.585917
$5$	−2.84869	−1.27397	−0.636986	−	0.770875i	$-0.719819\pi$
$5$	−2.84869	−1.27397	−0.636986	+	0.770875i	$0.719819\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.88648	1.02053
$9$	0	0
$10$	2.59241	0.819792
$11$	1.86466	0.562217	0.281108	−	0.959676i	$-0.409298\pi$
$11$	1.86466	0.562217	0.281108	+	0.959676i	$0.409298\pi$
$12$	0	0
$13$	−6.19675	−1.71867	−0.859334	−	0.511415i	$-0.829122\pi$
$13$	−6.19675	−1.71867	−0.859334	+	0.511415i	$0.829122\pi$
$14$	0.910036	0.243217
$15$	0	0
$16$	−0.283136	−0.0707841
$17$	−2.49732	−0.605688	−0.302844	−	0.953040i	$-0.597936\pi$
$17$	−2.49732	−0.605688	−0.302844	+	0.953040i	$0.597936\pi$
$18$	0	0
$19$	−6.22887	−1.42900	−0.714500	−	0.699635i	$-0.753346\pi$
$19$	−6.22887	−1.42900	−0.714500	+	0.699635i	$0.753346\pi$
$20$	3.33819	0.746442
$21$	0	0
$22$	−1.69691	−0.361782
$23$	5.12980	1.06964	0.534819	−	0.844967i	$-0.320380\pi$
$23$	5.12980	1.06964	0.534819	+	0.844967i	$0.320380\pi$
$24$	0	0
$25$	3.11503	0.623006
$26$	5.63926	1.10595
$27$	0	0
$28$	1.17183	0.221456
$29$	7.68531	1.42713	0.713563	−	0.700591i	$-0.247080\pi$
$29$	7.68531	1.42713	0.713563	+	0.700591i	$0.247080\pi$
$30$	0	0
$31$	4.32504	0.776800	0.388400	−	0.921491i	$-0.373028\pi$
$31$	4.32504	0.776800	0.388400	+	0.921491i	$0.373028\pi$
$32$	−5.51530	−0.974977
$33$	0	0
$34$	2.27265	0.389756
$35$	2.84869	0.481516
$36$	0	0
$37$	2.90361	0.477350	0.238675	−	0.971100i	$-0.423287\pi$
$37$	2.90361	0.477350	0.238675	+	0.971100i	$0.423287\pi$
$38$	5.66849	0.919551
$39$	0	0
$40$	−8.22269	−1.30012
$41$	−4.60795	−0.719640	−0.359820	−	0.933022i	$-0.617162\pi$
$41$	−4.60795	−0.719640	−0.359820	+	0.933022i	$0.617162\pi$
$42$	0	0
$43$	−9.98321	−1.52243	−0.761213	−	0.648502i	$-0.775396\pi$
$43$	−9.98321	−1.52243	−0.761213	+	0.648502i	$0.775396\pi$
$44$	−2.18507	−0.329412
$45$	0	0
$46$	−4.66830	−0.688304
$47$	10.9278	1.59398	0.796989	−	0.603994i	$-0.206425\pi$
$47$	10.9278	1.59398	0.796989	+	0.603994i	$0.206425\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.83479	−0.400900
$51$	0	0
$52$	7.26156	1.00700
$53$	10.7010	1.46990	0.734948	−	0.678123i	$-0.237207\pi$
$53$	10.7010	1.46990	0.734948	+	0.678123i	$0.237207\pi$
$54$	0	0
$55$	−5.31184	−0.716249
$56$	−2.88648	−0.385723
$57$	0	0
$58$	−6.99391	−0.918345
$59$	−11.4224	−1.48707	−0.743533	−	0.668699i	$-0.766851\pi$
$59$	−11.4224	−1.48707	−0.743533	+	0.668699i	$0.766851\pi$
$60$	0	0
$61$	1.39989	0.179237	0.0896185	−	0.995976i	$-0.471435\pi$
$61$	1.39989	0.179237	0.0896185	+	0.995976i	$0.471435\pi$
$62$	−3.93594	−0.499865
$63$	0	0
$64$	5.58540	0.698175
$65$	17.6526	2.18954
$66$	0	0
$67$	14.3600	1.75436	0.877178	−	0.480164i	$-0.159423\pi$
$67$	14.3600	1.75436	0.877178	+	0.480164i	$0.159423\pi$
$68$	2.92644	0.354883
$69$	0	0
$70$	−2.59241	−0.309852
$71$	−9.49609	−1.12698	−0.563489	−	0.826123i	$-0.690541\pi$
$71$	−9.49609	−1.12698	−0.563489	+	0.826123i	$0.690541\pi$
$72$	0	0
$73$	−16.1291	−1.88777	−0.943887	−	0.330269i	$-0.892861\pi$
$73$	−16.1291	−1.88777	−0.943887	+	0.330269i	$0.892861\pi$
$74$	−2.64239	−0.307171
$75$	0	0
$76$	7.29920	0.837276
$77$	−1.86466	−0.212498
$78$	0	0
$79$	12.5570	1.41277	0.706384	−	0.707829i	$-0.250325\pi$
$79$	12.5570	1.41277	0.706384	+	0.707829i	$0.250325\pi$
$80$	0.806567	0.0901770
$81$	0	0
$82$	4.19340	0.463083
$83$	6.86150	0.753148	0.376574	−	0.926387i	$-0.377102\pi$
$83$	6.86150	0.753148	0.376574	+	0.926387i	$0.377102\pi$
$84$	0	0
$85$	7.11407	0.771630
$86$	9.08509	0.979670
$87$	0	0
$88$	5.38232	0.573757
$89$	8.88528	0.941838	0.470919	−	0.882176i	$-0.343922\pi$
$89$	8.88528	0.941838	0.470919	+	0.882176i	$0.343922\pi$
$90$	0	0
$91$	6.19675	0.649596
$92$	−6.01128	−0.626719
$93$	0	0
$94$	−9.94466	−1.02571
$95$	17.7441	1.82051
$96$	0	0
$97$	1.23342	0.125235	0.0626176	−	0.998038i	$-0.480055\pi$
$97$	1.23342	0.125235	0.0626176	+	0.998038i	$0.480055\pi$
$98$	−0.910036	−0.0919275
$99$	0	0
$100$	−3.65030	−0.365030
$101$	4.39135	0.436956	0.218478	−	0.975842i	$-0.429891\pi$
$101$	4.39135	0.436956	0.218478	+	0.975842i	$0.429891\pi$
$102$	0	0
$103$	4.10635	0.404610	0.202305	−	0.979323i	$-0.435157\pi$
$103$	4.10635	0.404610	0.202305	+	0.979323i	$0.435157\pi$
$104$	−17.8868	−1.75395
$105$	0	0
$106$	−9.73830	−0.945867
$107$	−16.0178	−1.54850	−0.774250	−	0.632880i	$-0.781873\pi$
$107$	−16.0178	−1.54850	−0.774250	+	0.632880i	$0.781873\pi$
$108$	0	0
$109$	4.60049	0.440647	0.220324	−	0.975427i	$-0.429289\pi$
$109$	4.60049	0.440647	0.220324	+	0.975427i	$0.429289\pi$
$110$	4.83397	0.460901
$111$	0	0
$112$	0.283136	0.0267539
$113$	10.3473	0.973393	0.486697	−	0.873571i	$-0.338202\pi$
$113$	10.3473	0.973393	0.486697	+	0.873571i	$0.338202\pi$
$114$	0	0
$115$	−14.6132	−1.36269
$116$	−9.00591	−0.836178
$117$	0	0
$118$	10.3948	0.956916
$119$	2.49732	0.228929
$120$	0	0
$121$	−7.52304	−0.683912
$122$	−1.27395	−0.115338
$123$	0	0
$124$	−5.06823	−0.455141
$125$	5.36970	0.480281
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	5.94769	0.525707
$129$	0	0
$130$	−16.0645	−1.40895
$131$	−1.45092	−0.126768	−0.0633838	−	0.997989i	$-0.520189\pi$
$131$	−1.45092	−0.126768	−0.0633838	+	0.997989i	$0.520189\pi$
$132$	0	0
$133$	6.22887	0.540111
$134$	−13.0681	−1.12892
$135$	0	0
$136$	−7.20846	−0.618120
$137$	−8.15630	−0.696840	−0.348420	−	0.937339i	$-0.613282\pi$
$137$	−8.15630	−0.696840	−0.348420	+	0.937339i	$0.613282\pi$
$138$	0	0
$139$	17.9807	1.52510	0.762550	−	0.646929i	$-0.223947\pi$
$139$	17.9807	1.52510	0.762550	+	0.646929i	$0.223947\pi$
$140$	−3.33819	−0.282129
$141$	0	0
$142$	8.64179	0.725203
$143$	−11.5548	−0.966264
$144$	0	0
$145$	−21.8931	−1.81812
$146$	14.6781	1.21477
$147$	0	0
$148$	−3.40255	−0.279688
$149$	7.83149	0.641580	0.320790	−	0.947150i	$-0.396052\pi$
$149$	7.83149	0.641580	0.320790	+	0.947150i	$0.396052\pi$
$150$	0	0
$151$	−2.72817	−0.222015	−0.111008	−	0.993820i	$-0.535408\pi$
$151$	−2.72817	−0.222015	−0.111008	+	0.993820i	$0.535408\pi$
$152$	−17.9795	−1.45833
$153$	0	0
$154$	1.69691	0.136741
$155$	−12.3207	−0.989622
$156$	0	0
$157$	−19.3454	−1.54393	−0.771964	−	0.635666i	$-0.780726\pi$
$157$	−19.3454	−1.54393	−0.771964	+	0.635666i	$0.780726\pi$
$158$	−11.4273	−0.909106
$159$	0	0
$160$	15.7114	1.24209
$161$	−5.12980	−0.404285
$162$	0	0
$163$	−4.44858	−0.348440	−0.174220	−	0.984707i	$-0.555740\pi$
$163$	−4.44858	−0.348440	−0.174220	+	0.984707i	$0.555740\pi$
$164$	5.39975	0.421649
$165$	0	0
$166$	−6.24422	−0.484645
$167$	11.3915	0.881499	0.440750	−	0.897630i	$-0.354713\pi$
$167$	11.3915	0.881499	0.440750	+	0.897630i	$0.354713\pi$
$168$	0	0
$169$	25.3997	1.95382
$170$	−6.47406	−0.496538
$171$	0	0
$172$	11.6987	0.892015
$173$	1.47427	0.112086	0.0560432	−	0.998428i	$-0.482152\pi$
$173$	1.47427	0.112086	0.0560432	+	0.998428i	$0.482152\pi$
$174$	0	0
$175$	−3.11503	−0.235474
$176$	−0.527954	−0.0397960
$177$	0	0
$178$	−8.08593	−0.606066
$179$	1.38218	0.103309	0.0516545	−	0.998665i	$-0.483551\pi$
$179$	1.38218	0.103309	0.0516545	+	0.998665i	$0.483551\pi$
$180$	0	0
$181$	12.5132	0.930098	0.465049	−	0.885285i	$-0.346037\pi$
$181$	12.5132	0.930098	0.465049	+	0.885285i	$0.346037\pi$
$182$	−5.63926	−0.418010
$183$	0	0
$184$	14.8071	1.09159
$185$	−8.27147	−0.608131
$186$	0	0
$187$	−4.65665	−0.340528
$188$	−12.8055	−0.933939
$189$	0	0
$190$	−16.1478	−1.17148
$191$	−7.25363	−0.524855	−0.262427	−	0.964952i	$-0.584523\pi$
$191$	−7.25363	−0.524855	−0.262427	+	0.964952i	$0.584523\pi$
$192$	0	0
$193$	−3.03198	−0.218247	−0.109123	−	0.994028i	$-0.534804\pi$
$193$	−3.03198	−0.218247	−0.109123	+	0.994028i	$0.534804\pi$
$194$	−1.12246	−0.0805879
$195$	0	0
$196$	−1.17183	−0.0837024
$197$	25.6531	1.82771	0.913853	−	0.406045i	$-0.133092\pi$
$197$	25.6531	1.82771	0.913853	+	0.406045i	$0.133092\pi$
$198$	0	0
$199$	−10.3539	−0.733971	−0.366985	−	0.930227i	$-0.619610\pi$
$199$	−10.3539	−0.733971	−0.366985	+	0.930227i	$0.619610\pi$
$200$	8.99148	0.635793
$201$	0	0
$202$	−3.99629	−0.281178
$203$	−7.68531	−0.539403
$204$	0	0
$205$	13.1266	0.916802
$206$	−3.73692	−0.260364
$207$	0	0
$208$	1.75452	0.121654
$209$	−11.6147	−0.803408
$210$	0	0
$211$	18.9676	1.30578	0.652891	−	0.757452i	$-0.273556\pi$
$211$	18.9676	1.30578	0.652891	+	0.757452i	$0.273556\pi$
$212$	−12.5398	−0.861237
$213$	0	0
$214$	14.5768	0.996448
$215$	28.4391	1.93953
$216$	0	0
$217$	−4.32504	−0.293603
$218$	−4.18661	−0.283553
$219$	0	0
$220$	6.22460	0.419662
$221$	15.4752	1.04098
$222$	0	0
$223$	9.10091	0.609442	0.304721	−	0.952442i	$-0.401437\pi$
$223$	9.10091	0.609442	0.304721	+	0.952442i	$0.401437\pi$
$224$	5.51530	0.368507
$225$	0	0
$226$	−9.41643	−0.626372
$227$	4.71243	0.312775	0.156388	−	0.987696i	$-0.450015\pi$
$227$	4.71243	0.312775	0.156388	+	0.987696i	$0.450015\pi$
$228$	0	0
$229$	−11.1018	−0.733629	−0.366815	−	0.930294i	$-0.619552\pi$
$229$	−11.1018	−0.733629	−0.366815	+	0.930294i	$0.619552\pi$
$230$	13.2985	0.876880
$231$	0	0
$232$	22.1835	1.45642
$233$	−9.44652	−0.618862	−0.309431	−	0.950922i	$-0.600139\pi$
$233$	−9.44652	−0.618862	−0.309431	+	0.950922i	$0.600139\pi$
$234$	0	0
$235$	−31.1298	−2.03068
$236$	13.3851	0.871297
$237$	0	0
$238$	−2.27265	−0.147314
$239$	−0.884259	−0.0571979	−0.0285990	−	0.999591i	$-0.509105\pi$
$239$	−0.884259	−0.0571979	−0.0285990	+	0.999591i	$0.509105\pi$
$240$	0	0
$241$	−16.4107	−1.05711	−0.528554	−	0.848900i	$-0.677266\pi$
$241$	−16.4107	−1.05711	−0.528554	+	0.848900i	$0.677266\pi$
$242$	6.84623	0.440093
$243$	0	0
$244$	−1.64043	−0.105018
$245$	−2.84869	−0.181996
$246$	0	0
$247$	38.5987	2.45598
$248$	12.4842	0.792745
$249$	0	0
$250$	−4.88662	−0.309057
$251$	31.1797	1.96804	0.984022	−	0.178045i	$-0.0569772\pi$
$251$	31.1797	1.96804	0.984022	+	0.178045i	$0.0569772\pi$
$252$	0	0
$253$	9.56534	0.601368
$254$	−0.910036	−0.0571007
$255$	0	0
$256$	−16.5834	−1.03646
$257$	8.99673	0.561201	0.280600	−	0.959825i	$-0.409466\pi$
$257$	8.99673	0.561201	0.280600	+	0.959825i	$0.409466\pi$
$258$	0	0
$259$	−2.90361	−0.180421
$260$	−20.6859	−1.28289
$261$	0	0
$262$	1.32039	0.0815740
$263$	10.2900	0.634511	0.317256	−	0.948340i	$-0.397239\pi$
$263$	10.2900	0.634511	0.317256	+	0.948340i	$0.397239\pi$
$264$	0	0
$265$	−30.4838	−1.87261
$266$	−5.66849	−0.347558
$267$	0	0
$268$	−16.8276	−1.02791
$269$	−21.1306	−1.28835	−0.644177	−	0.764877i	$-0.722800\pi$
$269$	−21.1306	−1.28835	−0.644177	+	0.764877i	$0.722800\pi$
$270$	0	0
$271$	−19.6434	−1.19325	−0.596625	−	0.802520i	$-0.703492\pi$
$271$	−19.6434	−1.19325	−0.596625	+	0.802520i	$0.703492\pi$
$272$	0.707081	0.0428731
$273$	0	0
$274$	7.42253	0.448411
$275$	5.80847	0.350264
$276$	0	0
$277$	−17.9075	−1.07596	−0.537979	−	0.842959i	$-0.680812\pi$
$277$	−17.9075	−1.07596	−0.537979	+	0.842959i	$0.680812\pi$
$278$	−16.3631	−0.981391
$279$	0	0
$280$	8.22269	0.491400
$281$	−15.3397	−0.915089	−0.457545	−	0.889187i	$-0.651271\pi$
$281$	−15.3397	−0.915089	−0.457545	+	0.889187i	$0.651271\pi$
$282$	0	0
$283$	3.98371	0.236807	0.118404	−	0.992966i	$-0.462222\pi$
$283$	3.98371	0.236807	0.118404	+	0.992966i	$0.462222\pi$
$284$	11.1278	0.660316
$285$	0	0
$286$	10.5153	0.621784
$287$	4.60795	0.271998
$288$	0	0
$289$	−10.7634	−0.633142
$290$	19.9235	1.16995
$291$	0	0
$292$	18.9007	1.10608
$293$	−14.2180	−0.830627	−0.415313	−	0.909678i	$-0.636328\pi$
$293$	−14.2180	−0.830627	−0.415313	+	0.909678i	$0.636328\pi$
$294$	0	0
$295$	32.5388	1.89448
$296$	8.38121	0.487148
$297$	0	0
$298$	−7.12694	−0.412852
$299$	−31.7881	−1.83835
$300$	0	0
$301$	9.98321	0.575423
$302$	2.48273	0.142865
$303$	0	0
$304$	1.76362	0.101150
$305$	−3.98784	−0.228343
$306$	0	0
$307$	25.3729	1.44811	0.724054	−	0.689744i	$-0.242277\pi$
$307$	25.3729	1.44811	0.724054	+	0.689744i	$0.242277\pi$
$308$	2.18507	0.124506
$309$	0	0
$310$	11.2123	0.636815
$311$	−22.4753	−1.27446	−0.637228	−	0.770675i	$-0.719919\pi$
$311$	−22.4753	−1.27446	−0.637228	+	0.770675i	$0.719919\pi$
$312$	0	0
$313$	28.4319	1.60707	0.803533	−	0.595261i	$-0.202951\pi$
$313$	28.4319	1.60707	0.803533	+	0.595261i	$0.202951\pi$
$314$	17.6050	0.993507
$315$	0	0
$316$	−14.7147	−0.827765
$317$	6.99427	0.392838	0.196419	−	0.980520i	$-0.437069\pi$
$317$	6.99427	0.392838	0.196419	+	0.980520i	$0.437069\pi$
$318$	0	0
$319$	14.3305	0.802354
$320$	−15.9111	−0.889455
$321$	0	0
$322$	4.66830	0.260154
$323$	15.5554	0.865528
$324$	0	0
$325$	−19.3030	−1.07074
$326$	4.04837	0.224218
$327$	0	0
$328$	−13.3008	−0.734412
$329$	−10.9278	−0.602467
$330$	0	0
$331$	2.79465	0.153608	0.0768040	−	0.997046i	$-0.475528\pi$
$331$	2.79465	0.153608	0.0768040	+	0.997046i	$0.475528\pi$
$332$	−8.04054	−0.441282
$333$	0	0
$334$	−10.3667	−0.567238
$335$	−40.9073	−2.23500
$336$	0	0
$337$	−9.11926	−0.496758	−0.248379	−	0.968663i	$-0.579898\pi$
$337$	−9.11926	−0.496758	−0.248379	+	0.968663i	$0.579898\pi$
$338$	−23.1146	−1.25727
$339$	0	0
$340$	−8.33652	−0.452111
$341$	8.06474	0.436730
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−28.8164	−1.55368
$345$	0	0
$346$	−1.34164	−0.0721268
$347$	32.7437	1.75778	0.878888	−	0.477029i	$-0.158286\pi$
$347$	32.7437	1.75778	0.878888	+	0.477029i	$0.158286\pi$
$348$	0	0
$349$	−20.9539	−1.12164	−0.560818	−	0.827939i	$-0.689513\pi$
$349$	−20.9539	−1.12164	−0.560818	+	0.827939i	$0.689513\pi$
$350$	2.83479	0.151526
$351$	0	0
$352$	−10.2842	−0.548148
$353$	−25.5672	−1.36080	−0.680401	−	0.732840i	$-0.738194\pi$
$353$	−25.5672	−1.36080	−0.680401	+	0.732840i	$0.738194\pi$
$354$	0	0
$355$	27.0514	1.43574
$356$	−10.4121	−0.551839
$357$	0	0
$358$	−1.25783	−0.0664786
$359$	−0.235612	−0.0124351	−0.00621757	−	0.999981i	$-0.501979\pi$
$359$	−0.235612	−0.0124351	−0.00621757	+	0.999981i	$0.501979\pi$
$360$	0	0
$361$	19.7988	1.04204
$362$	−11.3874	−0.598511
$363$	0	0
$364$	−7.26156	−0.380609
$365$	45.9469	2.40497
$366$	0	0
$367$	−14.6165	−0.762977	−0.381488	−	0.924374i	$-0.624588\pi$
$367$	−14.6165	−0.762977	−0.381488	+	0.924374i	$0.624588\pi$
$368$	−1.45243	−0.0757133
$369$	0	0
$370$	7.52734	0.391328
$371$	−10.7010	−0.555568
$372$	0	0
$373$	8.31248	0.430404	0.215202	−	0.976570i	$-0.430959\pi$
$373$	8.31248	0.430404	0.215202	+	0.976570i	$0.430959\pi$
$374$	4.23772	0.219127
$375$	0	0
$376$	31.5428	1.62670
$377$	−47.6239	−2.45276
$378$	0	0
$379$	−29.3794	−1.50912	−0.754560	−	0.656231i	$-0.772150\pi$
$379$	−29.3794	−1.50912	−0.754560	+	0.656231i	$0.772150\pi$
$380$	−20.7931	−1.06667
$381$	0	0
$382$	6.60107	0.337740
$383$	−32.6225	−1.66693	−0.833466	−	0.552570i	$-0.813647\pi$
$383$	−32.6225	−1.66693	−0.833466	+	0.552570i	$0.813647\pi$
$384$	0	0
$385$	5.31184	0.270717
$386$	2.75921	0.140440
$387$	0	0
$388$	−1.44537	−0.0733774
$389$	−15.2301	−0.772198	−0.386099	−	0.922457i	$-0.626178\pi$
$389$	−15.2301	−0.772198	−0.386099	+	0.922457i	$0.626178\pi$
$390$	0	0
$391$	−12.8107	−0.647866
$392$	2.88648	0.145789
$393$	0	0
$394$	−23.3452	−1.17612
$395$	−35.7708	−1.79983
$396$	0	0
$397$	−18.7221	−0.939634	−0.469817	−	0.882764i	$-0.655680\pi$
$397$	−18.7221	−0.939634	−0.469817	+	0.882764i	$0.655680\pi$
$398$	9.42245	0.472305
$399$	0	0
$400$	−0.881977	−0.0440989
$401$	5.46933	0.273125	0.136563	−	0.990631i	$-0.456395\pi$
$401$	5.46933	0.273125	0.136563	+	0.990631i	$0.456395\pi$
$402$	0	0
$403$	−26.8012	−1.33506
$404$	−5.14594	−0.256020
$405$	0	0
$406$	6.99391	0.347102
$407$	5.41425	0.268374
$408$	0	0
$409$	10.4460	0.516522	0.258261	−	0.966075i	$-0.416851\pi$
$409$	10.4460	0.516522	0.258261	+	0.966075i	$0.416851\pi$
$410$	−11.9457	−0.589955
$411$	0	0
$412$	−4.81196	−0.237068
$413$	11.4224	0.562058
$414$	0	0
$415$	−19.5463	−0.959490
$416$	34.1769	1.67566
$417$	0	0
$418$	10.5698	0.516987
$419$	8.46705	0.413642	0.206821	−	0.978379i	$-0.433688\pi$
$419$	8.46705	0.413642	0.206821	+	0.978379i	$0.433688\pi$
$420$	0	0
$421$	5.21912	0.254365	0.127182	−	0.991879i	$-0.459407\pi$
$421$	5.21912	0.254365	0.127182	+	0.991879i	$0.459407\pi$
$422$	−17.2612	−0.840261
$423$	0	0
$424$	30.8883	1.50007
$425$	−7.77921	−0.377347
$426$	0	0
$427$	−1.39989	−0.0677452
$428$	18.7702	0.907292
$429$	0	0
$430$	−25.8806	−1.24807
$431$	−12.8810	−0.620458	−0.310229	−	0.950662i	$-0.600406\pi$
$431$	−12.8810	−0.620458	−0.310229	+	0.950662i	$0.600406\pi$
$432$	0	0
$433$	−28.9525	−1.39137	−0.695684	−	0.718348i	$-0.744899\pi$
$433$	−28.9525	−1.39137	−0.695684	+	0.718348i	$0.744899\pi$
$434$	3.93594	0.188931
$435$	0	0
$436$	−5.39101	−0.258183
$437$	−31.9528	−1.52851
$438$	0	0
$439$	−18.4236	−0.879311	−0.439656	−	0.898166i	$-0.644900\pi$
$439$	−18.4236	−0.879311	−0.439656	+	0.898166i	$0.644900\pi$
$440$	−15.3325	−0.730950
$441$	0	0
$442$	−14.0830	−0.669861
$443$	−17.7779	−0.844655	−0.422328	−	0.906443i	$-0.638787\pi$
$443$	−17.7779	−0.844655	−0.422328	+	0.906443i	$0.638787\pi$
$444$	0	0
$445$	−25.3114	−1.19988
$446$	−8.28216	−0.392171
$447$	0	0
$448$	−5.58540	−0.263885
$449$	−8.68627	−0.409930	−0.204965	−	0.978769i	$-0.565708\pi$
$449$	−8.68627	−0.409930	−0.204965	+	0.978769i	$0.565708\pi$
$450$	0	0
$451$	−8.59226	−0.404594
$452$	−12.1253	−0.570328
$453$	0	0
$454$	−4.28848	−0.201269
$455$	−17.6526	−0.827567
$456$	0	0
$457$	5.23318	0.244798	0.122399	−	0.992481i	$-0.460941\pi$
$457$	5.23318	0.244798	0.122399	+	0.992481i	$0.460941\pi$
$458$	10.1031	0.472085
$459$	0	0
$460$	17.1243	0.798422
$461$	−9.90168	−0.461167	−0.230584	−	0.973052i	$-0.574064\pi$
$461$	−9.90168	−0.461167	−0.230584	+	0.973052i	$0.574064\pi$
$462$	0	0
$463$	−25.6117	−1.19028	−0.595139	−	0.803623i	$-0.702903\pi$
$463$	−25.6117	−1.19028	−0.595139	+	0.803623i	$0.702903\pi$
$464$	−2.17599	−0.101018
$465$	0	0
$466$	8.59667	0.398233
$467$	−8.39998	−0.388705	−0.194352	−	0.980932i	$-0.562261\pi$
$467$	−8.39998	−0.388705	−0.194352	+	0.980932i	$0.562261\pi$
$468$	0	0
$469$	−14.3600	−0.663085
$470$	28.3292	1.30673
$471$	0	0
$472$	−32.9705	−1.51759
$473$	−18.6153	−0.855933
$474$	0	0
$475$	−19.4031	−0.890275
$476$	−2.92644	−0.134133
$477$	0	0
$478$	0.804707	0.0368065
$479$	−43.2028	−1.97399	−0.986993	−	0.160763i	$-0.948605\pi$
$479$	−43.2028	−1.97399	−0.986993	+	0.160763i	$0.948605\pi$
$480$	0	0
$481$	−17.9929	−0.820406
$482$	14.9344	0.680241
$483$	0	0
$484$	8.81575	0.400716
$485$	−3.51364	−0.159546
$486$	0	0
$487$	−6.80293	−0.308270	−0.154135	−	0.988050i	$-0.549259\pi$
$487$	−6.80293	−0.308270	−0.154135	+	0.988050i	$0.549259\pi$
$488$	4.04075	0.182916
$489$	0	0
$490$	2.59241	0.117113
$491$	−29.7692	−1.34347	−0.671733	−	0.740794i	$-0.734450\pi$
$491$	−29.7692	−1.34347	−0.671733	+	0.740794i	$0.734450\pi$
$492$	0	0
$493$	−19.1926	−0.864393
$494$	−35.1262	−1.58040
$495$	0	0
$496$	−1.22458	−0.0549851
$497$	9.49609	0.425958
$498$	0	0
$499$	30.0896	1.34700	0.673498	−	0.739189i	$-0.264791\pi$
$499$	30.0896	1.34700	0.673498	+	0.739189i	$0.264791\pi$
$500$	−6.29240	−0.281405
$501$	0	0
$502$	−28.3746	−1.26642
$503$	−24.1598	−1.07723	−0.538616	−	0.842551i	$-0.681053\pi$
$503$	−24.1598	−1.07723	−0.538616	+	0.842551i	$0.681053\pi$
$504$	0	0
$505$	−12.5096	−0.556670
$506$	−8.70481	−0.386976
$507$	0	0
$508$	−1.17183	−0.0519917
$509$	−1.01922	−0.0451761	−0.0225881	−	0.999745i	$-0.507191\pi$
$509$	−1.01922	−0.0451761	−0.0225881	+	0.999745i	$0.507191\pi$
$510$	0	0
$511$	16.1291	0.713511
$512$	3.19612	0.141250
$513$	0	0
$514$	−8.18735	−0.361129
$515$	−11.6977	−0.515462
$516$	0	0
$517$	20.3766	0.896161
$518$	2.64239	0.116100
$519$	0	0
$520$	50.9539	2.23448
$521$	−23.8097	−1.04312	−0.521561	−	0.853214i	$-0.674650\pi$
$521$	−23.8097	−1.04312	−0.521561	+	0.853214i	$0.674650\pi$
$522$	0	0
$523$	5.03806	0.220299	0.110149	−	0.993915i	$-0.464867\pi$
$523$	5.03806	0.220299	0.110149	+	0.993915i	$0.464867\pi$
$524$	1.70024	0.0742752
$525$	0	0
$526$	−9.36431	−0.408304
$527$	−10.8010	−0.470499
$528$	0	0
$529$	3.31485	0.144124
$530$	27.7414	1.20501
$531$	0	0
$532$	−7.29920	−0.316460
$533$	28.5543	1.23682
$534$	0	0
$535$	45.6297	1.97275
$536$	41.4500	1.79037
$537$	0	0
$538$	19.2296	0.829046
$539$	1.86466	0.0803167
$540$	0	0
$541$	−5.22853	−0.224792	−0.112396	−	0.993663i	$-0.535853\pi$
$541$	−5.22853	−0.224792	−0.112396	+	0.993663i	$0.535853\pi$
$542$	17.8762	0.767848
$543$	0	0
$544$	13.7735	0.590532
$545$	−13.1054	−0.561372
$546$	0	0
$547$	−6.31818	−0.270146	−0.135073	−	0.990836i	$-0.543127\pi$
$547$	−6.31818	−0.270146	−0.135073	+	0.990836i	$0.543127\pi$
$548$	9.55783	0.408290
$549$	0	0
$550$	−5.28592	−0.225392
$551$	−47.8708	−2.03936
$552$	0	0
$553$	−12.5570	−0.533976
$554$	16.2965	0.692371
$555$	0	0
$556$	−21.0704	−0.893582
$557$	13.0444	0.552711	0.276356	−	0.961055i	$-0.410873\pi$
$557$	13.0444	0.552711	0.276356	+	0.961055i	$0.410873\pi$
$558$	0	0
$559$	61.8634	2.61654
$560$	−0.806567	−0.0340837
$561$	0	0
$562$	13.9597	0.588853
$563$	45.5329	1.91898	0.959491	−	0.281738i	$-0.0909109\pi$
$563$	45.5329	1.91898	0.959491	+	0.281738i	$0.0909109\pi$
$564$	0	0
$565$	−29.4763	−1.24008
$566$	−3.62532	−0.152384
$567$	0	0
$568$	−27.4103	−1.15011
$569$	−2.80352	−0.117530	−0.0587649	−	0.998272i	$-0.518716\pi$
$569$	−2.80352	−0.117530	−0.0587649	+	0.998272i	$0.518716\pi$
$570$	0	0
$571$	−35.8981	−1.50229	−0.751144	−	0.660138i	$-0.770498\pi$
$571$	−35.8981	−1.50229	−0.751144	+	0.660138i	$0.770498\pi$
$572$	13.5404	0.566151
$573$	0	0
$574$	−4.19340	−0.175029
$575$	15.9795	0.666390
$576$	0	0
$577$	32.3851	1.34821	0.674105	−	0.738636i	$-0.264530\pi$
$577$	32.3851	1.34821	0.674105	+	0.738636i	$0.264530\pi$
$578$	9.79510	0.407422
$579$	0	0
$580$	25.6550	1.06527
$581$	−6.86150	−0.284663
$582$	0	0
$583$	19.9538	0.826400
$584$	−46.5565	−1.92652
$585$	0	0
$586$	12.9389	0.534502
$587$	12.0855	0.498824	0.249412	−	0.968397i	$-0.419763\pi$
$587$	12.0855	0.498824	0.249412	+	0.968397i	$0.419763\pi$
$588$	0	0
$589$	−26.9401	−1.11005
$590$	−29.6115	−1.21908
$591$	0	0
$592$	−0.822116	−0.0337888
$593$	−15.7649	−0.647385	−0.323693	−	0.946162i	$-0.604924\pi$
$593$	−15.7649	−0.647385	−0.323693	+	0.946162i	$0.604924\pi$
$594$	0	0
$595$	−7.11407	−0.291649
$596$	−9.17720	−0.375913
$597$	0	0
$598$	28.9283	1.18297
$599$	28.8610	1.17923	0.589614	−	0.807685i	$-0.299280\pi$
$599$	28.8610	1.17923	0.589614	+	0.807685i	$0.299280\pi$
$600$	0	0
$601$	23.7006	0.966766	0.483383	−	0.875409i	$-0.339408\pi$
$601$	23.7006	0.966766	0.483383	+	0.875409i	$0.339408\pi$
$602$	−9.08509	−0.370280
$603$	0	0
$604$	3.19696	0.130083
$605$	21.4308	0.871285
$606$	0	0
$607$	−5.11336	−0.207545	−0.103772	−	0.994601i	$-0.533091\pi$
$607$	−5.11336	−0.207545	−0.103772	+	0.994601i	$0.533091\pi$
$608$	34.3541	1.39324
$609$	0	0
$610$	3.62908	0.146937
$611$	−67.7166	−2.73952
$612$	0	0
$613$	−25.7355	−1.03945	−0.519724	−	0.854334i	$-0.673965\pi$
$613$	−25.7355	−1.03945	−0.519724	+	0.854334i	$0.673965\pi$
$614$	−23.0902	−0.931846
$615$	0	0
$616$	−5.38232	−0.216860
$617$	8.33851	0.335696	0.167848	−	0.985813i	$-0.446318\pi$
$617$	8.33851	0.335696	0.167848	+	0.985813i	$0.446318\pi$
$618$	0	0
$619$	34.3577	1.38095	0.690476	−	0.723355i	$-0.257401\pi$
$619$	34.3577	1.38095	0.690476	+	0.723355i	$0.257401\pi$
$620$	14.4378	0.579837
$621$	0	0
$622$	20.4533	0.820103
$623$	−8.88528	−0.355981
$624$	0	0
$625$	−30.8717	−1.23487
$626$	−25.8740	−1.03413
$627$	0	0
$628$	22.6696	0.904614
$629$	−7.25122	−0.289125
$630$	0	0
$631$	12.8597	0.511935	0.255968	−	0.966685i	$-0.417606\pi$
$631$	12.8597	0.511935	0.255968	+	0.966685i	$0.417606\pi$
$632$	36.2454	1.44177
$633$	0	0
$634$	−6.36504	−0.252788
$635$	−2.84869	−0.113047
$636$	0	0
$637$	−6.19675	−0.245524
$638$	−13.0413	−0.516309
$639$	0	0
$640$	−16.9431	−0.669736
$641$	−35.1723	−1.38922	−0.694611	−	0.719386i	$-0.744423\pi$
$641$	−35.1723	−1.38922	−0.694611	+	0.719386i	$0.744423\pi$
$642$	0	0
$643$	40.3387	1.59080	0.795401	−	0.606084i	$-0.207260\pi$
$643$	40.3387	1.59080	0.795401	+	0.606084i	$0.207260\pi$
$644$	6.01128	0.236877
$645$	0	0
$646$	−14.1560	−0.556961
$647$	38.7448	1.52321	0.761607	−	0.648039i	$-0.224411\pi$
$647$	38.7448	1.52321	0.761607	+	0.648039i	$0.224411\pi$
$648$	0	0
$649$	−21.2989	−0.836053
$650$	17.5665	0.689013
$651$	0	0
$652$	5.21300	0.204157
$653$	−6.11456	−0.239281	−0.119641	−	0.992817i	$-0.538174\pi$
$653$	−6.11456	−0.239281	−0.119641	+	0.992817i	$0.538174\pi$
$654$	0	0
$655$	4.13322	0.161498
$656$	1.30468	0.0509391
$657$	0	0
$658$	9.94466	0.387683
$659$	40.4854	1.57709	0.788543	−	0.614979i	$-0.210836\pi$
$659$	40.4854	1.57709	0.788543	+	0.614979i	$0.210836\pi$
$660$	0	0
$661$	−47.4100	−1.84404	−0.922018	−	0.387146i	$-0.873461\pi$
$661$	−47.4100	−1.84404	−0.922018	+	0.387146i	$0.873461\pi$
$662$	−2.54323	−0.0988456
$663$	0	0
$664$	19.8056	0.768607
$665$	−17.7441	−0.688087
$666$	0	0
$667$	39.4241	1.52651
$668$	−13.3489	−0.516485
$669$	0	0
$670$	37.2271	1.43821
$671$	2.61031	0.100770
$672$	0	0
$673$	−45.1817	−1.74163	−0.870813	−	0.491615i	$-0.836407\pi$
$673$	−45.1817	−1.74163	−0.870813	+	0.491615i	$0.836407\pi$
$674$	8.29886	0.319660
$675$	0	0
$676$	−29.7642	−1.14478
$677$	39.3478	1.51226	0.756130	−	0.654422i	$-0.227088\pi$
$677$	39.3478	1.51226	0.756130	+	0.654422i	$0.227088\pi$
$678$	0	0
$679$	−1.23342	−0.0473344
$680$	20.5347	0.787468
$681$	0	0
$682$	−7.33921	−0.281033
$683$	−8.53221	−0.326476	−0.163238	−	0.986587i	$-0.552194\pi$
$683$	−8.53221	−0.326476	−0.163238	+	0.986587i	$0.552194\pi$
$684$	0	0
$685$	23.2348	0.887755
$686$	0.910036	0.0347453
$687$	0	0
$688$	2.82661	0.107764
$689$	−66.3114	−2.52626
$690$	0	0
$691$	51.8607	1.97287	0.986437	−	0.164143i	$-0.0524857\pi$
$691$	51.8607	1.97287	0.986437	+	0.164143i	$0.0524857\pi$
$692$	−1.72760	−0.0656734
$693$	0	0
$694$	−29.7980	−1.13112
$695$	−51.2213	−1.94294
$696$	0	0
$697$	11.5075	0.435877
$698$	19.0688	0.721765
$699$	0	0
$700$	3.65030	0.137968
$701$	−30.3697	−1.14705	−0.573524	−	0.819189i	$-0.694424\pi$
$701$	−30.3697	−1.14705	−0.573524	+	0.819189i	$0.694424\pi$
$702$	0	0
$703$	−18.0862	−0.682133
$704$	10.4149	0.392526
$705$	0	0
$706$	23.2670	0.875666
$707$	−4.39135	−0.165154
$708$	0	0
$709$	7.22623	0.271387	0.135693	−	0.990751i	$-0.456674\pi$
$709$	7.22623	0.271387	0.135693	+	0.990751i	$0.456674\pi$
$710$	−24.6178	−0.923888
$711$	0	0
$712$	25.6472	0.961170
$713$	22.1866	0.830895
$714$	0	0
$715$	32.9161	1.23099
$716$	−1.61969	−0.0605305
$717$	0	0
$718$	0.214416	0.00800192
$719$	5.93481	0.221331	0.110666	−	0.993858i	$-0.464702\pi$
$719$	5.93481	0.221331	0.110666	+	0.993858i	$0.464702\pi$
$720$	0	0
$721$	−4.10635	−0.152928
$722$	−18.0176	−0.670546
$723$	0	0
$724$	−14.6634	−0.544960
$725$	23.9400	0.889107
$726$	0	0
$727$	11.2737	0.418120	0.209060	−	0.977903i	$-0.432960\pi$
$727$	11.2737	0.418120	0.209060	+	0.977903i	$0.432960\pi$
$728$	17.8868	0.662929
$729$	0	0
$730$	−41.8134	−1.54758
$731$	24.9312	0.922115
$732$	0	0
$733$	−15.2465	−0.563144	−0.281572	−	0.959540i	$-0.590856\pi$
$733$	−15.2465	−0.563144	−0.281572	+	0.959540i	$0.590856\pi$
$734$	13.3016	0.490970
$735$	0	0
$736$	−28.2924	−1.04287
$737$	26.7766	0.986329
$738$	0	0
$739$	−44.7526	−1.64625	−0.823126	−	0.567859i	$-0.807772\pi$
$739$	−44.7526	−1.64625	−0.823126	+	0.567859i	$0.807772\pi$
$740$	9.69279	0.356314
$741$	0	0
$742$	9.73830	0.357504
$743$	−34.8351	−1.27798	−0.638988	−	0.769217i	$-0.720647\pi$
$743$	−34.8351	−1.27798	−0.638988	+	0.769217i	$0.720647\pi$
$744$	0	0
$745$	−22.3095	−0.817356
$746$	−7.56466	−0.276962
$747$	0	0
$748$	5.45682	0.199521
$749$	16.0178	0.585278
$750$	0	0
$751$	−51.3734	−1.87464	−0.937321	−	0.348467i	$-0.886702\pi$
$751$	−51.3734	−1.87464	−0.937321	+	0.348467i	$0.886702\pi$
$752$	−3.09405	−0.112828
$753$	0	0
$754$	43.3395	1.57833
$755$	7.77171	0.282841
$756$	0	0
$757$	−0.972360	−0.0353410	−0.0176705	−	0.999844i	$-0.505625\pi$
$757$	−0.972360	−0.0353410	−0.0176705	+	0.999844i	$0.505625\pi$
$758$	26.7363	0.971107
$759$	0	0
$760$	51.2181	1.85787
$761$	14.6105	0.529631	0.264815	−	0.964299i	$-0.414689\pi$
$761$	14.6105	0.529631	0.264815	+	0.964299i	$0.414689\pi$
$762$	0	0
$763$	−4.60049	−0.166549
$764$	8.50006	0.307521
$765$	0	0
$766$	29.6877	1.07266
$767$	70.7815	2.55577
$768$	0	0
$769$	−34.5936	−1.24748	−0.623738	−	0.781633i	$-0.714387\pi$
$769$	−34.5936	−1.24748	−0.623738	+	0.781633i	$0.714387\pi$
$770$	−4.83397	−0.174204
$771$	0	0
$772$	3.55298	0.127875
$773$	12.4898	0.449226	0.224613	−	0.974448i	$-0.427888\pi$
$773$	12.4898	0.449226	0.224613	+	0.974448i	$0.427888\pi$
$774$	0	0
$775$	13.4726	0.483951
$776$	3.56026	0.127806
$777$	0	0
$778$	13.8600	0.496904
$779$	28.7023	1.02837
$780$	0	0
$781$	−17.7070	−0.633606
$782$	11.6582	0.416897
$783$	0	0
$784$	−0.283136	−0.0101120
$785$	55.1089	1.96692
$786$	0	0
$787$	2.75373	0.0981597	0.0490798	−	0.998795i	$-0.484371\pi$
$787$	2.75373	0.0981597	0.0490798	+	0.998795i	$0.484371\pi$
$788$	−30.0612	−1.07088
$789$	0	0
$790$	32.5528	1.15818
$791$	−10.3473	−0.367908
$792$	0	0
$793$	−8.67473	−0.308049
$794$	17.0378	0.604648
$795$	0	0
$796$	12.1331	0.430046
$797$	−21.4910	−0.761252	−0.380626	−	0.924729i	$-0.624291\pi$
$797$	−21.4910	−0.761252	−0.380626	+	0.924729i	$0.624291\pi$
$798$	0	0
$799$	−27.2901	−0.965453
$800$	−17.1803	−0.607416
$801$	0	0
$802$	−4.97729	−0.175754
$803$	−30.0754	−1.06134
$804$	0	0
$805$	14.6132	0.515048
$806$	24.3900	0.859103
$807$	0	0
$808$	12.6756	0.445925
$809$	7.69502	0.270543	0.135271	−	0.990809i	$-0.456809\pi$
$809$	7.69502	0.270543	0.135271	+	0.990809i	$0.456809\pi$
$810$	0	0
$811$	−39.7158	−1.39461	−0.697306	−	0.716774i	$-0.745618\pi$
$811$	−39.7158	−1.39461	−0.697306	+	0.716774i	$0.745618\pi$
$812$	9.00591	0.316045
$813$	0	0
$814$	−4.92716	−0.172697
$815$	12.6726	0.443902
$816$	0	0
$817$	62.1841	2.17555
$818$	−9.50624	−0.332378
$819$	0	0
$820$	−15.3822	−0.537170
$821$	31.6422	1.10432	0.552160	−	0.833738i	$-0.313804\pi$
$821$	31.6422	1.10432	0.552160	+	0.833738i	$0.313804\pi$
$822$	0	0
$823$	50.2891	1.75297	0.876484	−	0.481431i	$-0.159883\pi$
$823$	50.2891	1.75297	0.876484	+	0.481431i	$0.159883\pi$
$824$	11.8529	0.412915
$825$	0	0
$826$	−10.3948	−0.361680
$827$	27.3110	0.949698	0.474849	−	0.880067i	$-0.342503\pi$
$827$	27.3110	0.949698	0.474849	+	0.880067i	$0.342503\pi$
$828$	0	0
$829$	−5.34901	−0.185779	−0.0928894	−	0.995676i	$-0.529610\pi$
$829$	−5.34901	−0.185779	−0.0928894	+	0.995676i	$0.529610\pi$
$830$	17.7878	0.617425
$831$	0	0
$832$	−34.6113	−1.19993
$833$	−2.49732	−0.0865269
$834$	0	0
$835$	−32.4508	−1.12301
$836$	13.6105	0.470730
$837$	0	0
$838$	−7.70532	−0.266176
$839$	6.12466	0.211447	0.105723	−	0.994396i	$-0.466284\pi$
$839$	6.12466	0.211447	0.105723	+	0.994396i	$0.466284\pi$
$840$	0	0
$841$	30.0640	1.03669
$842$	−4.74959	−0.163682
$843$	0	0
$844$	−22.2269	−0.765080
$845$	−72.3557	−2.48911
$846$	0	0
$847$	7.52304	0.258495
$848$	−3.02984	−0.104045
$849$	0	0
$850$	7.07936	0.242820
$851$	14.8949	0.510591
$852$	0	0
$853$	−12.7046	−0.434996	−0.217498	−	0.976061i	$-0.569789\pi$
$853$	−12.7046	−0.434996	−0.217498	+	0.976061i	$0.569789\pi$
$854$	1.27395	0.0435935
$855$	0	0
$856$	−46.2351	−1.58028
$857$	−11.0623	−0.377880	−0.188940	−	0.981989i	$-0.560505\pi$
$857$	−11.0623	−0.377880	−0.188940	+	0.981989i	$0.560505\pi$
$858$	0	0
$859$	−5.41515	−0.184763	−0.0923813	−	0.995724i	$-0.529448\pi$
$859$	−5.41515	−0.184763	−0.0923813	+	0.995724i	$0.529448\pi$
$860$	−33.3259	−1.13640
$861$	0	0
$862$	11.7222	0.399260
$863$	−21.3557	−0.726956	−0.363478	−	0.931603i	$-0.618411\pi$
$863$	−21.3557	−0.726956	−0.363478	+	0.931603i	$0.618411\pi$
$864$	0	0
$865$	−4.19973	−0.142795
$866$	26.3478	0.895335
$867$	0	0
$868$	5.06823	0.172027
$869$	23.4145	0.794282
$870$	0	0
$871$	−88.9855	−3.01516
$872$	13.2792	0.449692
$873$	0	0
$874$	29.0782	0.983586
$875$	−5.36970	−0.181529
$876$	0	0
$877$	−9.00207	−0.303978	−0.151989	−	0.988382i	$-0.548568\pi$
$877$	−9.00207	−0.303978	−0.151989	+	0.988382i	$0.548568\pi$
$878$	16.7662	0.565830
$879$	0	0
$880$	1.50398	0.0506990
$881$	38.3952	1.29357	0.646783	−	0.762674i	$-0.276114\pi$
$881$	38.3952	1.29357	0.646783	+	0.762674i	$0.276114\pi$
$882$	0	0
$883$	−52.5943	−1.76994	−0.884969	−	0.465650i	$-0.845821\pi$
$883$	−52.5943	−1.76994	−0.884969	+	0.465650i	$0.845821\pi$
$884$	−18.1344	−0.609926
$885$	0	0
$886$	16.1786	0.543529
$887$	−0.462980	−0.0155453	−0.00777267	−	0.999970i	$-0.502474\pi$
$887$	−0.462980	−0.0155453	−0.00777267	+	0.999970i	$0.502474\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	23.0343	0.772111
$891$	0	0
$892$	−10.6648	−0.357082
$893$	−68.0676	−2.27779
$894$	0	0
$895$	−3.93740	−0.131613
$896$	−5.94769	−0.198698
$897$	0	0
$898$	7.90482	0.263787
$899$	33.2393	1.10859
$900$	0	0
$901$	−26.7238	−0.890298
$902$	7.81927	0.260353
$903$	0	0
$904$	29.8674	0.993373
$905$	−35.6462	−1.18492
$906$	0	0
$907$	9.91810	0.329325	0.164663	−	0.986350i	$-0.447346\pi$
$907$	9.91810	0.329325	0.164663	+	0.986350i	$0.447346\pi$
$908$	−5.52219	−0.183260
$909$	0	0
$910$	16.0645	0.532533
$911$	8.34507	0.276485	0.138242	−	0.990398i	$-0.455855\pi$
$911$	8.34507	0.276485	0.138242	+	0.990398i	$0.455855\pi$
$912$	0	0
$913$	12.7944	0.423432
$914$	−4.76238	−0.157526
$915$	0	0
$916$	13.0095	0.429846
$917$	1.45092	0.0479136
$918$	0	0
$919$	27.2474	0.898807	0.449404	−	0.893329i	$-0.351637\pi$
$919$	27.2474	0.898807	0.449404	+	0.893329i	$0.351637\pi$
$920$	−42.1808	−1.39066
$921$	0	0
$922$	9.01089	0.296758
$923$	58.8449	1.93690
$924$	0	0
$925$	9.04481	0.297392
$926$	23.3076	0.765936
$927$	0	0
$928$	−42.3868	−1.39142
$929$	22.7028	0.744854	0.372427	−	0.928062i	$-0.378526\pi$
$929$	22.7028	0.744854	0.372427	+	0.928062i	$0.378526\pi$
$930$	0	0
$931$	−6.22887	−0.204143
$932$	11.0698	0.362602
$933$	0	0
$934$	7.64429	0.250129
$935$	13.2653	0.433823
$936$	0	0
$937$	28.1759	0.920468	0.460234	−	0.887798i	$-0.347765\pi$
$937$	28.1759	0.920468	0.460234	+	0.887798i	$0.347765\pi$
$938$	13.0681	0.426690
$939$	0	0
$940$	36.4789	1.18981
$941$	−22.7468	−0.741523	−0.370762	−	0.928728i	$-0.620903\pi$
$941$	−22.7468	−0.741523	−0.370762	+	0.928728i	$0.620903\pi$
$942$	0	0
$943$	−23.6378	−0.769754
$944$	3.23409	0.105261
$945$	0	0
$946$	16.9406	0.550787
$947$	18.0753	0.587367	0.293683	−	0.955903i	$-0.405119\pi$
$947$	18.0753	0.587367	0.293683	+	0.955903i	$0.405119\pi$
$948$	0	0
$949$	99.9482	3.24446
$950$	17.6575	0.572885
$951$	0	0
$952$	7.20846	0.233628
$953$	−37.0138	−1.19899	−0.599497	−	0.800377i	$-0.704633\pi$
$953$	−37.0138	−1.19899	−0.599497	+	0.800377i	$0.704633\pi$
$954$	0	0
$955$	20.6633	0.668650
$956$	1.03620	0.0335132
$957$	0	0
$958$	39.3161	1.27025
$959$	8.15630	0.263381
$960$	0	0
$961$	−12.2940	−0.396581
$962$	16.3742	0.527925
$963$	0	0
$964$	19.2307	0.619378
$965$	8.63717	0.278040
$966$	0	0
$967$	−1.69090	−0.0543755	−0.0271878	−	0.999630i	$-0.508655\pi$
$967$	−1.69090	−0.0543755	−0.0271878	+	0.999630i	$0.508655\pi$
$968$	−21.7151	−0.697950
$969$	0	0
$970$	3.19754	0.102667
$971$	−4.21235	−0.135181	−0.0675904	−	0.997713i	$-0.521531\pi$
$971$	−4.21235	−0.135181	−0.0675904	+	0.997713i	$0.521531\pi$
$972$	0	0
$973$	−17.9807	−0.576434
$974$	6.19091	0.198370
$975$	0	0
$976$	−0.396358	−0.0126871
$977$	−6.21411	−0.198807	−0.0994036	−	0.995047i	$-0.531693\pi$
$977$	−6.21411	−0.198807	−0.0994036	+	0.995047i	$0.531693\pi$
$978$	0	0
$979$	16.5680	0.529517
$980$	3.33819	0.106635
$981$	0	0
$982$	27.0911	0.864510
$983$	−25.0905	−0.800263	−0.400131	−	0.916458i	$-0.631036\pi$
$983$	−25.0905	−0.800263	−0.400131	+	0.916458i	$0.631036\pi$
$984$	0	0
$985$	−73.0776	−2.32845
$986$	17.4660	0.556231
$987$	0	0
$988$	−45.2313	−1.43900
$989$	−51.2119	−1.62844
$990$	0	0
$991$	−46.2979	−1.47070	−0.735351	−	0.677687i	$-0.762983\pi$
$991$	−46.2979	−1.47070	−0.735351	+	0.677687i	$0.762983\pi$
$992$	−23.8539	−0.757363
$993$	0	0
$994$	−8.64179	−0.274101
$995$	29.4951	0.935059
$996$	0	0
$997$	12.5863	0.398614	0.199307	−	0.979937i	$-0.436131\pi$
$997$	12.5863	0.398614	0.199307	+	0.979937i	$0.436131\pi$
$998$	−27.3826	−0.866782
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.m.1.5		11
3.2	odd	2		2667.2.a.k.1.7	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.k.1.7	✓	11	3.2	odd	2
8001.2.a.m.1.5		11	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)

\(f(q)\)	\(=\)	\(q-0.910036 q^{2} -1.17183 q^{4} -2.84869 q^{5} -1.00000 q^{7} +2.88648 q^{8} +O(q^{10})\)	\(q-0.910036 q^{2} -1.17183 q^{4} -2.84869 q^{5} -1.00000 q^{7} +2.88648 q^{8} +2.59241 q^{10} +1.86466 q^{11} -6.19675 q^{13} +0.910036 q^{14} -0.283136 q^{16} -2.49732 q^{17} -6.22887 q^{19} +3.33819 q^{20} -1.69691 q^{22} +5.12980 q^{23} +3.11503 q^{25} +5.63926 q^{26} +1.17183 q^{28} +7.68531 q^{29} +4.32504 q^{31} -5.51530 q^{32} +2.27265 q^{34} +2.84869 q^{35} +2.90361 q^{37} +5.66849 q^{38} -8.22269 q^{40} -4.60795 q^{41} -9.98321 q^{43} -2.18507 q^{44} -4.66830 q^{46} +10.9278 q^{47} +1.00000 q^{49} -2.83479 q^{50} +7.26156 q^{52} +10.7010 q^{53} -5.31184 q^{55} -2.88648 q^{56} -6.99391 q^{58} -11.4224 q^{59} +1.39989 q^{61} -3.93594 q^{62} +5.58540 q^{64} +17.6526 q^{65} +14.3600 q^{67} +2.92644 q^{68} -2.59241 q^{70} -9.49609 q^{71} -16.1291 q^{73} -2.64239 q^{74} +7.29920 q^{76} -1.86466 q^{77} +12.5570 q^{79} +0.806567 q^{80} +4.19340 q^{82} +6.86150 q^{83} +7.11407 q^{85} +9.08509 q^{86} +5.38232 q^{88} +8.88528 q^{89} +6.19675 q^{91} -6.01128 q^{92} -9.94466 q^{94} +17.7441 q^{95} +1.23342 q^{97} -0.910036 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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