LMFDB - Embedded newform 8001.2.a.r.1.15

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-2.36717$ 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)

$f(q)$	$=$	$q+2.36717 q^{2} +3.60352 q^{4} -4.44142 q^{5} -1.00000 q^{7} +3.79580 q^{8} +O(q^{10})$	\(q+2.36717 q^{2} +3.60352 q^{4} -4.44142 q^{5} -1.00000 q^{7} +3.79580 q^{8} -10.5136 q^{10} -1.84770 q^{11} +6.85313 q^{13} -2.36717 q^{14} +1.77830 q^{16} +3.10566 q^{17} -0.484625 q^{19} -16.0047 q^{20} -4.37383 q^{22} +2.45447 q^{23} +14.7262 q^{25} +16.2226 q^{26} -3.60352 q^{28} -5.48819 q^{29} -8.84399 q^{31} -3.38206 q^{32} +7.35164 q^{34} +4.44142 q^{35} -4.16374 q^{37} -1.14719 q^{38} -16.8588 q^{40} +1.10507 q^{41} +12.4806 q^{43} -6.65821 q^{44} +5.81016 q^{46} -7.32054 q^{47} +1.00000 q^{49} +34.8595 q^{50} +24.6954 q^{52} -10.6136 q^{53} +8.20641 q^{55} -3.79580 q^{56} -12.9915 q^{58} -7.61137 q^{59} -13.3762 q^{61} -20.9353 q^{62} -11.5625 q^{64} -30.4377 q^{65} +7.56439 q^{67} +11.1913 q^{68} +10.5136 q^{70} +3.49842 q^{71} -1.95384 q^{73} -9.85629 q^{74} -1.74636 q^{76} +1.84770 q^{77} +2.84735 q^{79} -7.89818 q^{80} +2.61590 q^{82} -0.436650 q^{83} -13.7935 q^{85} +29.5437 q^{86} -7.01350 q^{88} -10.1276 q^{89} -6.85313 q^{91} +8.84472 q^{92} -17.3290 q^{94} +2.15242 q^{95} -9.94808 q^{97} +2.36717 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * 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 $

$2$	2.36717	1.67385	0.836923	−	0.547321i	$-0.184352\pi$
$2$	2.36717	1.67385	0.836923	+	0.547321i	$0.184352\pi$
$3$	0	0
$3$	0	0
$4$	3.60352	1.80176
$5$	−4.44142	−1.98626	−0.993132	−	0.117000i	$-0.962672\pi$
$5$	−4.44142	−1.98626	−0.993132	+	0.117000i	$0.962672\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.79580	1.34202
$9$	0	0
$10$	−10.5136	−3.32470
$11$	−1.84770	−0.557102	−0.278551	−	0.960421i	$-0.589854\pi$
$11$	−1.84770	−0.557102	−0.278551	+	0.960421i	$0.589854\pi$
$12$	0	0
$13$	6.85313	1.90072	0.950359	−	0.311156i	$-0.100716\pi$
$13$	6.85313	1.90072	0.950359	+	0.311156i	$0.100716\pi$
$14$	−2.36717	−0.632654
$15$	0	0
$16$	1.77830	0.444575
$17$	3.10566	0.753233	0.376617	−	0.926369i	$-0.377087\pi$
$17$	3.10566	0.753233	0.376617	+	0.926369i	$0.377087\pi$
$18$	0	0
$19$	−0.484625	−0.111181	−0.0555903	−	0.998454i	$-0.517704\pi$
$19$	−0.484625	−0.111181	−0.0555903	+	0.998454i	$0.517704\pi$
$20$	−16.0047	−3.57877
$21$	0	0
$22$	−4.37383	−0.932503
$23$	2.45447	0.511792	0.255896	−	0.966704i	$-0.417630\pi$
$23$	2.45447	0.511792	0.255896	+	0.966704i	$0.417630\pi$
$24$	0	0
$25$	14.7262	2.94524
$26$	16.2226	3.18151
$27$	0	0
$28$	−3.60352	−0.681001
$29$	−5.48819	−1.01913	−0.509565	−	0.860432i	$-0.670194\pi$
$29$	−5.48819	−1.01913	−0.509565	+	0.860432i	$0.670194\pi$
$30$	0	0
$31$	−8.84399	−1.58843	−0.794214	−	0.607639i	$-0.792117\pi$
$31$	−8.84399	−1.58843	−0.794214	+	0.607639i	$0.792117\pi$
$32$	−3.38206	−0.597870
$33$	0	0
$34$	7.35164	1.26080
$35$	4.44142	0.750737
$36$	0	0
$37$	−4.16374	−0.684514	−0.342257	−	0.939606i	$-0.611191\pi$
$37$	−4.16374	−0.684514	−0.342257	+	0.939606i	$0.611191\pi$
$38$	−1.14719	−0.186099
$39$	0	0
$40$	−16.8588	−2.66560
$41$	1.10507	0.172583	0.0862916	−	0.996270i	$-0.472498\pi$
$41$	1.10507	0.172583	0.0862916	+	0.996270i	$0.472498\pi$
$42$	0	0
$43$	12.4806	1.90327	0.951635	−	0.307230i	$-0.0994020\pi$
$43$	12.4806	1.90327	0.951635	+	0.307230i	$0.0994020\pi$
$44$	−6.65821	−1.00376
$45$	0	0
$46$	5.81016	0.856661
$47$	−7.32054	−1.06781	−0.533905	−	0.845544i	$-0.679276\pi$
$47$	−7.32054	−1.06781	−0.533905	+	0.845544i	$0.679276\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	34.8595	4.92988
$51$	0	0
$52$	24.6954	3.42463
$53$	−10.6136	−1.45789	−0.728946	−	0.684571i	$-0.759990\pi$
$53$	−10.6136	−1.45789	−0.728946	+	0.684571i	$0.759990\pi$
$54$	0	0
$55$	8.20641	1.10655
$56$	−3.79580	−0.507236
$57$	0	0
$58$	−12.9915	−1.70587
$59$	−7.61137	−0.990916	−0.495458	−	0.868632i	$-0.665000\pi$
$59$	−7.61137	−0.990916	−0.495458	+	0.868632i	$0.665000\pi$
$60$	0	0
$61$	−13.3762	−1.71264	−0.856321	−	0.516444i	$-0.827255\pi$
$61$	−13.3762	−1.71264	−0.856321	+	0.516444i	$0.827255\pi$
$62$	−20.9353	−2.65878
$63$	0	0
$64$	−11.5625	−1.44532
$65$	−30.4377	−3.77533
$66$	0	0
$67$	7.56439	0.924137	0.462069	−	0.886844i	$-0.347107\pi$
$67$	7.56439	0.924137	0.462069	+	0.886844i	$0.347107\pi$
$68$	11.1913	1.35714
$69$	0	0
$70$	10.5136	1.25662
$71$	3.49842	0.415186	0.207593	−	0.978215i	$-0.433437\pi$
$71$	3.49842	0.415186	0.207593	+	0.978215i	$0.433437\pi$
$72$	0	0
$73$	−1.95384	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$73$	−1.95384	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$74$	−9.85629	−1.14577
$75$	0	0
$76$	−1.74636	−0.200321
$77$	1.84770	0.210565
$78$	0	0
$79$	2.84735	0.320352	0.160176	−	0.987088i	$-0.448794\pi$
$79$	2.84735	0.320352	0.160176	+	0.987088i	$0.448794\pi$
$80$	−7.89818	−0.883043
$81$	0	0
$82$	2.61590	0.288878
$83$	−0.436650	−0.0479285	−0.0239643	−	0.999713i	$-0.507629\pi$
$83$	−0.436650	−0.0479285	−0.0239643	+	0.999713i	$0.507629\pi$
$84$	0	0
$85$	−13.7935	−1.49612
$86$	29.5437	3.18578
$87$	0	0
$88$	−7.01350	−0.747642
$89$	−10.1276	−1.07352	−0.536761	−	0.843734i	$-0.680352\pi$
$89$	−10.1276	−1.07352	−0.536761	+	0.843734i	$0.680352\pi$
$90$	0	0
$91$	−6.85313	−0.718404
$92$	8.84472	0.922126
$93$	0	0
$94$	−17.3290	−1.78735
$95$	2.15242	0.220834
$96$	0	0
$97$	−9.94808	−1.01007	−0.505037	−	0.863097i	$-0.668521\pi$
$97$	−9.94808	−1.01007	−0.505037	+	0.863097i	$0.668521\pi$
$98$	2.36717	0.239121
$99$	0	0
$100$	53.0662	5.30662
$101$	−6.45088	−0.641887	−0.320943	−	0.947098i	$-0.604000\pi$
$101$	−6.45088	−0.641887	−0.320943	+	0.947098i	$0.604000\pi$
$102$	0	0
$103$	−6.10263	−0.601310	−0.300655	−	0.953733i	$-0.597205\pi$
$103$	−6.10263	−0.601310	−0.300655	+	0.953733i	$0.597205\pi$
$104$	26.0132	2.55080
$105$	0	0
$106$	−25.1243	−2.44029
$107$	7.58115	0.732897	0.366449	−	0.930438i	$-0.380574\pi$
$107$	7.58115	0.732897	0.366449	+	0.930438i	$0.380574\pi$
$108$	0	0
$109$	−0.833514	−0.0798362	−0.0399181	−	0.999203i	$-0.512710\pi$
$109$	−0.833514	−0.0798362	−0.0399181	+	0.999203i	$0.512710\pi$
$110$	19.4260	1.85220
$111$	0	0
$112$	−1.77830	−0.168033
$113$	2.85163	0.268259	0.134129	−	0.990964i	$-0.457176\pi$
$113$	2.85163	0.268259	0.134129	+	0.990964i	$0.457176\pi$
$114$	0	0
$115$	−10.9013	−1.01655
$116$	−19.7768	−1.83623
$117$	0	0
$118$	−18.0174	−1.65864
$119$	−3.10566	−0.284695
$120$	0	0
$121$	−7.58601	−0.689637
$122$	−31.6637	−2.86670
$123$	0	0
$124$	−31.8695	−2.86196
$125$	−43.1982	−3.86377
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−20.6064	−1.82137
$129$	0	0
$130$	−72.0513	−6.31931
$131$	9.74655	0.851560	0.425780	−	0.904827i	$-0.360000\pi$
$131$	9.74655	0.851560	0.425780	+	0.904827i	$0.360000\pi$
$132$	0	0
$133$	0.484625	0.0420223
$134$	17.9062	1.54686
$135$	0	0
$136$	11.7885	1.01085
$137$	6.55908	0.560380	0.280190	−	0.959945i	$-0.409602\pi$
$137$	6.55908	0.560380	0.280190	+	0.959945i	$0.409602\pi$
$138$	0	0
$139$	−20.9922	−1.78053	−0.890266	−	0.455440i	$-0.849482\pi$
$139$	−20.9922	−1.78053	−0.890266	+	0.455440i	$0.849482\pi$
$140$	16.0047	1.35265
$141$	0	0
$142$	8.28137	0.694957
$143$	−12.6625	−1.05889
$144$	0	0
$145$	24.3753	2.02426
$146$	−4.62509	−0.382775
$147$	0	0
$148$	−15.0041	−1.23333
$149$	−3.11905	−0.255523	−0.127761	−	0.991805i	$-0.540779\pi$
$149$	−3.11905	−0.255523	−0.127761	+	0.991805i	$0.540779\pi$
$150$	0	0
$151$	−14.9881	−1.21972	−0.609858	−	0.792510i	$-0.708774\pi$
$151$	−14.9881	−1.21972	−0.609858	+	0.792510i	$0.708774\pi$
$152$	−1.83954	−0.149207
$153$	0	0
$154$	4.37383	0.352453
$155$	39.2799	3.15504
$156$	0	0
$157$	20.3733	1.62596	0.812981	−	0.582290i	$-0.197843\pi$
$157$	20.3733	1.62596	0.812981	+	0.582290i	$0.197843\pi$
$158$	6.74018	0.536220
$159$	0	0
$160$	15.0212	1.18753
$161$	−2.45447	−0.193439
$162$	0	0
$163$	−21.1466	−1.65633	−0.828163	−	0.560487i	$-0.810614\pi$
$163$	−21.1466	−1.65633	−0.828163	+	0.560487i	$0.810614\pi$
$164$	3.98214	0.310953
$165$	0	0
$166$	−1.03363	−0.0802249
$167$	−22.2366	−1.72072	−0.860360	−	0.509687i	$-0.829761\pi$
$167$	−22.2366	−1.72072	−0.860360	+	0.509687i	$0.829761\pi$
$168$	0	0
$169$	33.9655	2.61273
$170$	−32.6517	−2.50427
$171$	0	0
$172$	44.9740	3.42923
$173$	−8.69447	−0.661028	−0.330514	−	0.943801i	$-0.607222\pi$
$173$	−8.69447	−0.661028	−0.330514	+	0.943801i	$0.607222\pi$
$174$	0	0
$175$	−14.7262	−1.11320
$176$	−3.28576	−0.247674
$177$	0	0
$178$	−23.9738	−1.79691
$179$	7.84200	0.586138	0.293069	−	0.956091i	$-0.405323\pi$
$179$	7.84200	0.586138	0.293069	+	0.956091i	$0.405323\pi$
$180$	0	0
$181$	−5.78184	−0.429761	−0.214880	−	0.976640i	$-0.568936\pi$
$181$	−5.78184	−0.429761	−0.214880	+	0.976640i	$0.568936\pi$
$182$	−16.2226	−1.20250
$183$	0	0
$184$	9.31668	0.686835
$185$	18.4929	1.35963
$186$	0	0
$187$	−5.73833	−0.419628
$188$	−26.3797	−1.92394
$189$	0	0
$190$	5.09517	0.369642
$191$	17.4716	1.26420	0.632100	−	0.774887i	$-0.282193\pi$
$191$	17.4716	1.26420	0.632100	+	0.774887i	$0.282193\pi$
$192$	0	0
$193$	−7.17520	−0.516482	−0.258241	−	0.966081i	$-0.583143\pi$
$193$	−7.17520	−0.516482	−0.258241	+	0.966081i	$0.583143\pi$
$194$	−23.5489	−1.69071
$195$	0	0
$196$	3.60352	0.257394
$197$	−12.1243	−0.863822	−0.431911	−	0.901916i	$-0.642161\pi$
$197$	−12.1243	−0.863822	−0.431911	+	0.901916i	$0.642161\pi$
$198$	0	0
$199$	−11.9318	−0.845826	−0.422913	−	0.906170i	$-0.638992\pi$
$199$	−11.9318	−0.845826	−0.422913	+	0.906170i	$0.638992\pi$
$200$	55.8979	3.95257
$201$	0	0
$202$	−15.2704	−1.07442
$203$	5.48819	0.385195
$204$	0	0
$205$	−4.90809	−0.342796
$206$	−14.4460	−1.00650
$207$	0	0
$208$	12.1869	0.845011
$209$	0.895442	0.0619390
$210$	0	0
$211$	23.1739	1.59536	0.797680	−	0.603081i	$-0.206060\pi$
$211$	23.1739	1.59536	0.797680	+	0.603081i	$0.206060\pi$
$212$	−38.2463	−2.62677
$213$	0	0
$214$	17.9459	1.22676
$215$	−55.4315	−3.78040
$216$	0	0
$217$	8.84399	0.600369
$218$	−1.97307	−0.133633
$219$	0	0
$220$	29.5719	1.99374
$221$	21.2835	1.43168
$222$	0	0
$223$	1.07211	0.0717940	0.0358970	−	0.999355i	$-0.488571\pi$
$223$	1.07211	0.0717940	0.0358970	+	0.999355i	$0.488571\pi$
$224$	3.38206	0.225974
$225$	0	0
$226$	6.75031	0.449024
$227$	5.69631	0.378077	0.189039	−	0.981970i	$-0.439463\pi$
$227$	5.69631	0.378077	0.189039	+	0.981970i	$0.439463\pi$
$228$	0	0
$229$	−3.45297	−0.228179	−0.114089	−	0.993470i	$-0.536395\pi$
$229$	−3.45297	−0.228179	−0.114089	+	0.993470i	$0.536395\pi$
$230$	−25.8054	−1.70155
$231$	0	0
$232$	−20.8321	−1.36769
$233$	−17.3845	−1.13890	−0.569450	−	0.822026i	$-0.692844\pi$
$233$	−17.3845	−1.13890	−0.569450	+	0.822026i	$0.692844\pi$
$234$	0	0
$235$	32.5136	2.12095
$236$	−27.4277	−1.78539
$237$	0	0
$238$	−7.35164	−0.476536
$239$	−5.12430	−0.331463	−0.165732	−	0.986171i	$-0.552999\pi$
$239$	−5.12430	−0.331463	−0.165732	+	0.986171i	$0.552999\pi$
$240$	0	0
$241$	26.2742	1.69247	0.846235	−	0.532810i	$-0.178864\pi$
$241$	26.2742	1.69247	0.846235	+	0.532810i	$0.178864\pi$
$242$	−17.9574	−1.15435
$243$	0	0
$244$	−48.2012	−3.08577
$245$	−4.44142	−0.283752
$246$	0	0
$247$	−3.32120	−0.211323
$248$	−33.5701	−2.13170
$249$	0	0
$250$	−102.258	−6.46735
$251$	18.9132	1.19379	0.596894	−	0.802320i	$-0.296401\pi$
$251$	18.9132	1.19379	0.596894	+	0.802320i	$0.296401\pi$
$252$	0	0
$253$	−4.53512	−0.285121
$254$	2.36717	0.148530
$255$	0	0
$256$	−25.6539	−1.60337
$257$	−9.82622	−0.612943	−0.306471	−	0.951880i	$-0.599148\pi$
$257$	−9.82622	−0.612943	−0.306471	+	0.951880i	$0.599148\pi$
$258$	0	0
$259$	4.16374	0.258722
$260$	−109.683	−6.80223
$261$	0	0
$262$	23.0718	1.42538
$263$	−26.6641	−1.64418	−0.822090	−	0.569357i	$-0.807192\pi$
$263$	−26.6641	−1.64418	−0.822090	+	0.569357i	$0.807192\pi$
$264$	0	0
$265$	47.1395	2.89576
$266$	1.14719	0.0703389
$267$	0	0
$268$	27.2584	1.66507
$269$	4.36031	0.265853	0.132926	−	0.991126i	$-0.457563\pi$
$269$	4.36031	0.265853	0.132926	+	0.991126i	$0.457563\pi$
$270$	0	0
$271$	3.10531	0.188634	0.0943169	−	0.995542i	$-0.469933\pi$
$271$	3.10531	0.188634	0.0943169	+	0.995542i	$0.469933\pi$
$272$	5.52279	0.334869
$273$	0	0
$274$	15.5265	0.937990
$275$	−27.2096	−1.64080
$276$	0	0
$277$	−3.75493	−0.225612	−0.112806	−	0.993617i	$-0.535984\pi$
$277$	−3.75493	−0.225612	−0.112806	+	0.993617i	$0.535984\pi$
$278$	−49.6921	−2.98034
$279$	0	0
$280$	16.8588	1.00750
$281$	−16.3541	−0.975606	−0.487803	−	0.872954i	$-0.662202\pi$
$281$	−16.3541	−0.975606	−0.487803	+	0.872954i	$0.662202\pi$
$282$	0	0
$283$	6.42572	0.381969	0.190985	−	0.981593i	$-0.438832\pi$
$283$	6.42572	0.381969	0.190985	+	0.981593i	$0.438832\pi$
$284$	12.6066	0.748065
$285$	0	0
$286$	−29.9744	−1.77242
$287$	−1.10507	−0.0652303
$288$	0	0
$289$	−7.35487	−0.432640
$290$	57.7007	3.38830
$291$	0	0
$292$	−7.04071	−0.412026
$293$	−4.90401	−0.286495	−0.143248	−	0.989687i	$-0.545755\pi$
$293$	−4.90401	−0.286495	−0.143248	+	0.989687i	$0.545755\pi$
$294$	0	0
$295$	33.8053	1.96822
$296$	−15.8047	−0.918631
$297$	0	0
$298$	−7.38334	−0.427705
$299$	16.8208	0.972772
$300$	0	0
$301$	−12.4806	−0.719369
$302$	−35.4795	−2.04162
$303$	0	0
$304$	−0.861809	−0.0494281
$305$	59.4092	3.40176
$306$	0	0
$307$	−13.2374	−0.755496	−0.377748	−	0.925909i	$-0.623301\pi$
$307$	−13.2374	−0.755496	−0.377748	+	0.925909i	$0.623301\pi$
$308$	6.65821	0.379387
$309$	0	0
$310$	92.9823	5.28104
$311$	−23.5718	−1.33664	−0.668318	−	0.743876i	$-0.732985\pi$
$311$	−23.5718	−1.33664	−0.668318	+	0.743876i	$0.732985\pi$
$312$	0	0
$313$	−7.11014	−0.401889	−0.200944	−	0.979603i	$-0.564401\pi$
$313$	−7.11014	−0.401889	−0.200944	+	0.979603i	$0.564401\pi$
$314$	48.2271	2.72161
$315$	0	0
$316$	10.2605	0.577197
$317$	2.92096	0.164058	0.0820289	−	0.996630i	$-0.473860\pi$
$317$	2.92096	0.164058	0.0820289	+	0.996630i	$0.473860\pi$
$318$	0	0
$319$	10.1405	0.567760
$320$	51.3541	2.87078
$321$	0	0
$322$	−5.81016	−0.323787
$323$	−1.50508	−0.0837450
$324$	0	0
$325$	100.921	5.59808
$326$	−50.0576	−2.77244
$327$	0	0
$328$	4.19464	0.231610
$329$	7.32054	0.403594
$330$	0	0
$331$	16.1362	0.886926	0.443463	−	0.896293i	$-0.353750\pi$
$331$	16.1362	0.886926	0.443463	+	0.896293i	$0.353750\pi$
$332$	−1.57347	−0.0863556
$333$	0	0
$334$	−52.6379	−2.88022
$335$	−33.5966	−1.83558
$336$	0	0
$337$	33.0104	1.79819	0.899096	−	0.437751i	$-0.144225\pi$
$337$	33.0104	1.79819	0.899096	+	0.437751i	$0.144225\pi$
$338$	80.4022	4.37330
$339$	0	0
$340$	−49.7053	−2.69565
$341$	16.3410	0.884916
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	47.3738	2.55423
$345$	0	0
$346$	−20.5813	−1.10646
$347$	1.10656	0.0594034	0.0297017	−	0.999559i	$-0.490544\pi$
$347$	1.10656	0.0594034	0.0297017	+	0.999559i	$0.490544\pi$
$348$	0	0
$349$	24.6199	1.31787	0.658936	−	0.752199i	$-0.271007\pi$
$349$	24.6199	1.31787	0.658936	+	0.752199i	$0.271007\pi$
$350$	−34.8595	−1.86332
$351$	0	0
$352$	6.24904	0.333075
$353$	27.9422	1.48722	0.743608	−	0.668616i	$-0.233113\pi$
$353$	27.9422	1.48722	0.743608	+	0.668616i	$0.233113\pi$
$354$	0	0
$355$	−15.5380	−0.824669
$356$	−36.4949	−1.93423
$357$	0	0
$358$	18.5634	0.981105
$359$	−2.61657	−0.138098	−0.0690488	−	0.997613i	$-0.521996\pi$
$359$	−2.61657	−0.138098	−0.0690488	+	0.997613i	$0.521996\pi$
$360$	0	0
$361$	−18.7651	−0.987639
$362$	−13.6866	−0.719353
$363$	0	0
$364$	−24.6954	−1.29439
$365$	8.67784	0.454219
$366$	0	0
$367$	−20.6547	−1.07816	−0.539082	−	0.842253i	$-0.681229\pi$
$367$	−20.6547	−1.07816	−0.539082	+	0.842253i	$0.681229\pi$
$368$	4.36478	0.227530
$369$	0	0
$370$	43.7760	2.27580
$371$	10.6136	0.551031
$372$	0	0
$373$	−19.3343	−1.00109	−0.500546	−	0.865710i	$-0.666867\pi$
$373$	−19.3343	−1.00109	−0.500546	+	0.865710i	$0.666867\pi$
$374$	−13.5836	−0.702392
$375$	0	0
$376$	−27.7873	−1.43302
$377$	−37.6113	−1.93708
$378$	0	0
$379$	15.4094	0.791526	0.395763	−	0.918353i	$-0.370480\pi$
$379$	15.4094	0.791526	0.395763	+	0.918353i	$0.370480\pi$
$380$	7.75630	0.397890
$381$	0	0
$382$	41.3583	2.11608
$383$	−20.5369	−1.04939	−0.524693	−	0.851292i	$-0.675820\pi$
$383$	−20.5369	−1.04939	−0.524693	+	0.851292i	$0.675820\pi$
$384$	0	0
$385$	−8.20641	−0.418237
$386$	−16.9849	−0.864511
$387$	0	0
$388$	−35.8481	−1.81991
$389$	29.4969	1.49555	0.747777	−	0.663950i	$-0.231121\pi$
$389$	29.4969	1.49555	0.747777	+	0.663950i	$0.231121\pi$
$390$	0	0
$391$	7.62275	0.385499
$392$	3.79580	0.191717
$393$	0	0
$394$	−28.7004	−1.44590
$395$	−12.6463	−0.636303
$396$	0	0
$397$	11.6410	0.584246	0.292123	−	0.956381i	$-0.405638\pi$
$397$	11.6410	0.584246	0.292123	+	0.956381i	$0.405638\pi$
$398$	−28.2448	−1.41578
$399$	0	0
$400$	26.1876	1.30938
$401$	8.20039	0.409508	0.204754	−	0.978813i	$-0.434361\pi$
$401$	8.20039	0.409508	0.204754	+	0.978813i	$0.434361\pi$
$402$	0	0
$403$	−60.6090	−3.01915
$404$	−23.2459	−1.15652
$405$	0	0
$406$	12.9915	0.644757
$407$	7.69333	0.381344
$408$	0	0
$409$	−27.2288	−1.34638	−0.673189	−	0.739470i	$-0.735076\pi$
$409$	−27.2288	−1.34638	−0.673189	+	0.739470i	$0.735076\pi$
$410$	−11.6183	−0.573787
$411$	0	0
$412$	−21.9909	−1.08342
$413$	7.61137	0.374531
$414$	0	0
$415$	1.93934	0.0951987
$416$	−23.1777	−1.13638
$417$	0	0
$418$	2.11967	0.103676
$419$	−25.0184	−1.22223	−0.611114	−	0.791543i	$-0.709278\pi$
$419$	−25.0184	−1.22223	−0.611114	+	0.791543i	$0.709278\pi$
$420$	0	0
$421$	3.36212	0.163860	0.0819298	−	0.996638i	$-0.473892\pi$
$421$	3.36212	0.163860	0.0819298	+	0.996638i	$0.473892\pi$
$422$	54.8568	2.67039
$423$	0	0
$424$	−40.2872	−1.95652
$425$	45.7346	2.21846
$426$	0	0
$427$	13.3762	0.647318
$428$	27.3188	1.32050
$429$	0	0
$430$	−131.216	−6.32780
$431$	12.4497	0.599683	0.299841	−	0.953989i	$-0.403066\pi$
$431$	12.4497	0.599683	0.299841	+	0.953989i	$0.403066\pi$
$432$	0	0
$433$	12.5325	0.602272	0.301136	−	0.953581i	$-0.402634\pi$
$433$	12.5325	0.602272	0.301136	+	0.953581i	$0.402634\pi$
$434$	20.9353	1.00492
$435$	0	0
$436$	−3.00358	−0.143846
$437$	−1.18950	−0.0569014
$438$	0	0
$439$	−11.8612	−0.566107	−0.283053	−	0.959104i	$-0.591347\pi$
$439$	−11.8612	−0.566107	−0.283053	+	0.959104i	$0.591347\pi$
$440$	31.1499	1.48501
$441$	0	0
$442$	50.3818	2.39642
$443$	3.31365	0.157436	0.0787181	−	0.996897i	$-0.474917\pi$
$443$	3.31365	0.157436	0.0787181	+	0.996897i	$0.474917\pi$
$444$	0	0
$445$	44.9809	2.13230
$446$	2.53788	0.120172
$447$	0	0
$448$	11.5625	0.546278
$449$	8.23924	0.388834	0.194417	−	0.980919i	$-0.437719\pi$
$449$	8.23924	0.388834	0.194417	+	0.980919i	$0.437719\pi$
$450$	0	0
$451$	−2.04184	−0.0961465
$452$	10.2759	0.483338
$453$	0	0
$454$	13.4842	0.632843
$455$	30.4377	1.42694
$456$	0	0
$457$	−15.1854	−0.710343	−0.355171	−	0.934801i	$-0.615577\pi$
$457$	−15.1854	−0.710343	−0.355171	+	0.934801i	$0.615577\pi$
$458$	−8.17379	−0.381936
$459$	0	0
$460$	−39.2831	−1.83159
$461$	26.7978	1.24810	0.624050	−	0.781385i	$-0.285486\pi$
$461$	26.7978	1.24810	0.624050	+	0.781385i	$0.285486\pi$
$462$	0	0
$463$	−4.34158	−0.201770	−0.100885	−	0.994898i	$-0.532167\pi$
$463$	−4.34158	−0.201770	−0.100885	+	0.994898i	$0.532167\pi$
$464$	−9.75964	−0.453080
$465$	0	0
$466$	−41.1522	−1.90634
$467$	30.2887	1.40160	0.700798	−	0.713360i	$-0.252828\pi$
$467$	30.2887	1.40160	0.700798	+	0.713360i	$0.252828\pi$
$468$	0	0
$469$	−7.56439	−0.349291
$470$	76.9653	3.55015
$471$	0	0
$472$	−28.8913	−1.32983
$473$	−23.0604	−1.06032
$474$	0	0
$475$	−7.13670	−0.327454
$476$	−11.1913	−0.512952
$477$	0	0
$478$	−12.1301	−0.554818
$479$	29.7493	1.35928	0.679639	−	0.733547i	$-0.262136\pi$
$479$	29.7493	1.35928	0.679639	+	0.733547i	$0.262136\pi$
$480$	0	0
$481$	−28.5347	−1.30107
$482$	62.1956	2.83293
$483$	0	0
$484$	−27.3363	−1.24256
$485$	44.1836	2.00627
$486$	0	0
$487$	13.7263	0.622000	0.311000	−	0.950410i	$-0.399336\pi$
$487$	13.7263	0.622000	0.311000	+	0.950410i	$0.399336\pi$
$488$	−50.7733	−2.29840
$489$	0	0
$490$	−10.5136	−0.474957
$491$	−18.7897	−0.847969	−0.423984	−	0.905670i	$-0.639369\pi$
$491$	−18.7897	−0.847969	−0.423984	+	0.905670i	$0.639369\pi$
$492$	0	0
$493$	−17.0444	−0.767643
$494$	−7.86187	−0.353722
$495$	0	0
$496$	−15.7273	−0.706175
$497$	−3.49842	−0.156926
$498$	0	0
$499$	20.1796	0.903361	0.451681	−	0.892180i	$-0.350825\pi$
$499$	20.1796	0.903361	0.451681	+	0.892180i	$0.350825\pi$
$500$	−155.666	−6.96158
$501$	0	0
$502$	44.7708	1.99822
$503$	−22.5712	−1.00640	−0.503199	−	0.864171i	$-0.667844\pi$
$503$	−22.5712	−1.00640	−0.503199	+	0.864171i	$0.667844\pi$
$504$	0	0
$505$	28.6511	1.27496
$506$	−10.7354	−0.477248
$507$	0	0
$508$	3.60352	0.159880
$509$	40.5012	1.79518	0.897591	−	0.440829i	$-0.145316\pi$
$509$	40.5012	1.79518	0.897591	+	0.440829i	$0.145316\pi$
$510$	0	0
$511$	1.95384	0.0864329
$512$	−19.5145	−0.862426
$513$	0	0
$514$	−23.2604	−1.02597
$515$	27.1044	1.19436
$516$	0	0
$517$	13.5261	0.594879
$518$	9.85629	0.433061
$519$	0	0
$520$	−115.535	−5.06656
$521$	30.8517	1.35164	0.675820	−	0.737067i	$-0.263790\pi$
$521$	30.8517	1.35164	0.675820	+	0.737067i	$0.263790\pi$
$522$	0	0
$523$	30.2690	1.32357	0.661786	−	0.749693i	$-0.269799\pi$
$523$	30.2690	1.32357	0.661786	+	0.749693i	$0.269799\pi$
$524$	35.1218	1.53431
$525$	0	0
$526$	−63.1187	−2.75210
$527$	−27.4664	−1.19646
$528$	0	0
$529$	−16.9756	−0.738069
$530$	111.588	4.84705
$531$	0	0
$532$	1.74636	0.0757141
$533$	7.57320	0.328032
$534$	0	0
$535$	−33.6711	−1.45573
$536$	28.7129	1.24021
$537$	0	0
$538$	10.3216	0.444997
$539$	−1.84770	−0.0795860
$540$	0	0
$541$	25.6889	1.10445	0.552227	−	0.833694i	$-0.313778\pi$
$541$	25.6889	1.10445	0.552227	+	0.833694i	$0.313778\pi$
$542$	7.35080	0.315744
$543$	0	0
$544$	−10.5035	−0.450336
$545$	3.70199	0.158576
$546$	0	0
$547$	−15.0932	−0.645339	−0.322670	−	0.946512i	$-0.604580\pi$
$547$	−15.0932	−0.645339	−0.322670	+	0.946512i	$0.604580\pi$
$548$	23.6358	1.00967
$549$	0	0
$550$	−64.4099	−2.74645
$551$	2.65971	0.113308
$552$	0	0
$553$	−2.84735	−0.121082
$554$	−8.88859	−0.377640
$555$	0	0
$556$	−75.6456	−3.20809
$557$	8.54632	0.362119	0.181060	−	0.983472i	$-0.442047\pi$
$557$	8.54632	0.362119	0.181060	+	0.983472i	$0.442047\pi$
$558$	0	0
$559$	85.5311	3.61758
$560$	7.89818	0.333759
$561$	0	0
$562$	−38.7131	−1.63301
$563$	−28.6914	−1.20920	−0.604600	−	0.796529i	$-0.706667\pi$
$563$	−28.6914	−1.20920	−0.604600	+	0.796529i	$0.706667\pi$
$564$	0	0
$565$	−12.6653	−0.532833
$566$	15.2108	0.639358
$567$	0	0
$568$	13.2793	0.557188
$569$	−41.3273	−1.73253	−0.866265	−	0.499584i	$-0.833486\pi$
$569$	−41.3273	−1.73253	−0.866265	+	0.499584i	$0.833486\pi$
$570$	0	0
$571$	−9.68792	−0.405427	−0.202713	−	0.979238i	$-0.564976\pi$
$571$	−9.68792	−0.405427	−0.202713	+	0.979238i	$0.564976\pi$
$572$	−45.6296	−1.90787
$573$	0	0
$574$	−2.61590	−0.109185
$575$	36.1450	1.50735
$576$	0	0
$577$	24.3971	1.01566	0.507832	−	0.861456i	$-0.330447\pi$
$577$	24.3971	1.01566	0.507832	+	0.861456i	$0.330447\pi$
$578$	−17.4103	−0.724172
$579$	0	0
$580$	87.8370	3.64723
$581$	0.436650	0.0181153
$582$	0	0
$583$	19.6108	0.812195
$584$	−7.41641	−0.306893
$585$	0	0
$586$	−11.6086	−0.479549
$587$	−3.72657	−0.153812	−0.0769060	−	0.997038i	$-0.524504\pi$
$587$	−3.72657	−0.153812	−0.0769060	+	0.997038i	$0.524504\pi$
$588$	0	0
$589$	4.28602	0.176602
$590$	80.0230	3.29450
$591$	0	0
$592$	−7.40437	−0.304318
$593$	−35.8037	−1.47028	−0.735141	−	0.677914i	$-0.762884\pi$
$593$	−35.8037	−1.47028	−0.735141	+	0.677914i	$0.762884\pi$
$594$	0	0
$595$	13.7935	0.565480
$596$	−11.2396	−0.460390
$597$	0	0
$598$	39.8178	1.62827
$599$	−26.6137	−1.08741	−0.543703	−	0.839278i	$-0.682978\pi$
$599$	−26.6137	−1.08741	−0.543703	+	0.839278i	$0.682978\pi$
$600$	0	0
$601$	36.5912	1.49258	0.746292	−	0.665618i	$-0.231832\pi$
$601$	36.5912	1.49258	0.746292	+	0.665618i	$0.231832\pi$
$602$	−29.5437	−1.20411
$603$	0	0
$604$	−54.0100	−2.19763
$605$	33.6927	1.36980
$606$	0	0
$607$	−20.9111	−0.848754	−0.424377	−	0.905486i	$-0.639507\pi$
$607$	−20.9111	−0.848754	−0.424377	+	0.905486i	$0.639507\pi$
$608$	1.63903	0.0664716
$609$	0	0
$610$	140.632	5.69402
$611$	−50.1686	−2.02961
$612$	0	0
$613$	8.37205	0.338144	0.169072	−	0.985604i	$-0.445923\pi$
$613$	8.37205	0.338144	0.169072	+	0.985604i	$0.445923\pi$
$614$	−31.3351	−1.26458
$615$	0	0
$616$	7.01350	0.282582
$617$	40.9757	1.64962	0.824811	−	0.565409i	$-0.191282\pi$
$617$	40.9757	1.64962	0.824811	+	0.565409i	$0.191282\pi$
$618$	0	0
$619$	15.7042	0.631203	0.315602	−	0.948892i	$-0.397794\pi$
$619$	15.7042	0.631203	0.315602	+	0.948892i	$0.397794\pi$
$620$	141.546	5.68461
$621$	0	0
$622$	−55.7986	−2.23732
$623$	10.1276	0.405753
$624$	0	0
$625$	118.230	4.72922
$626$	−16.8309	−0.672700
$627$	0	0
$628$	73.4154	2.92959
$629$	−12.9312	−0.515599
$630$	0	0
$631$	44.5714	1.77436	0.887179	−	0.461425i	$-0.152662\pi$
$631$	44.5714	1.77436	0.887179	+	0.461425i	$0.152662\pi$
$632$	10.8080	0.429919
$633$	0	0
$634$	6.91443	0.274607
$635$	−4.44142	−0.176252
$636$	0	0
$637$	6.85313	0.271531
$638$	24.0044	0.950342
$639$	0	0
$640$	91.5218	3.61771
$641$	−33.7741	−1.33400	−0.666998	−	0.745060i	$-0.732421\pi$
$641$	−33.7741	−1.33400	−0.666998	+	0.745060i	$0.732421\pi$
$642$	0	0
$643$	−30.2210	−1.19180	−0.595900	−	0.803059i	$-0.703205\pi$
$643$	−30.2210	−1.19180	−0.595900	+	0.803059i	$0.703205\pi$
$644$	−8.84472	−0.348531
$645$	0	0
$646$	−3.56279	−0.140176
$647$	−25.2719	−0.993541	−0.496771	−	0.867882i	$-0.665481\pi$
$647$	−25.2719	−0.993541	−0.496771	+	0.867882i	$0.665481\pi$
$648$	0	0
$649$	14.0635	0.552041
$650$	238.897	9.37032
$651$	0	0
$652$	−76.2020	−2.98430
$653$	33.9769	1.32962	0.664810	−	0.747013i	$-0.268513\pi$
$653$	33.9769	1.32962	0.664810	+	0.747013i	$0.268513\pi$
$654$	0	0
$655$	−43.2885	−1.69142
$656$	1.96515	0.0767261
$657$	0	0
$658$	17.3290	0.675554
$659$	−21.9867	−0.856482	−0.428241	−	0.903665i	$-0.640867\pi$
$659$	−21.9867	−0.856482	−0.428241	+	0.903665i	$0.640867\pi$
$660$	0	0
$661$	−18.2024	−0.707993	−0.353996	−	0.935247i	$-0.615178\pi$
$661$	−18.2024	−0.707993	−0.353996	+	0.935247i	$0.615178\pi$
$662$	38.1972	1.48458
$663$	0	0
$664$	−1.65744	−0.0643210
$665$	−2.15242	−0.0834675
$666$	0	0
$667$	−13.4706	−0.521583
$668$	−80.1300	−3.10032
$669$	0	0
$670$	−79.5291	−3.07248
$671$	24.7151	0.954117
$672$	0	0
$673$	41.4992	1.59968	0.799839	−	0.600215i	$-0.204918\pi$
$673$	41.4992	1.59968	0.799839	+	0.600215i	$0.204918\pi$
$674$	78.1415	3.00990
$675$	0	0
$676$	122.395	4.70750
$677$	0.104008	0.00399734	0.00199867	−	0.999998i	$-0.499364\pi$
$677$	0.104008	0.00399734	0.00199867	+	0.999998i	$0.499364\pi$
$678$	0	0
$679$	9.94808	0.381772
$680$	−52.3576	−2.00782
$681$	0	0
$682$	38.6821	1.48121
$683$	−13.7612	−0.526556	−0.263278	−	0.964720i	$-0.584804\pi$
$683$	−13.7612	−0.526556	−0.263278	+	0.964720i	$0.584804\pi$
$684$	0	0
$685$	−29.1317	−1.11306
$686$	−2.36717	−0.0903792
$687$	0	0
$688$	22.1942	0.846146
$689$	−72.7365	−2.77104
$690$	0	0
$691$	18.9767	0.721909	0.360954	−	0.932583i	$-0.382451\pi$
$691$	18.9767	0.721909	0.360954	+	0.932583i	$0.382451\pi$
$692$	−31.3307	−1.19101
$693$	0	0
$694$	2.61943	0.0994322
$695$	93.2351	3.53661
$696$	0	0
$697$	3.43198	0.129995
$698$	58.2796	2.20591
$699$	0	0
$700$	−53.0662	−2.00571
$701$	36.3475	1.37283	0.686413	−	0.727212i	$-0.259184\pi$
$701$	36.3475	1.37283	0.686413	+	0.727212i	$0.259184\pi$
$702$	0	0
$703$	2.01785	0.0761047
$704$	21.3641	0.805189
$705$	0	0
$706$	66.1442	2.48937
$707$	6.45088	0.242610
$708$	0	0
$709$	−11.8677	−0.445702	−0.222851	−	0.974852i	$-0.571536\pi$
$709$	−11.8677	−0.445702	−0.222851	+	0.974852i	$0.571536\pi$
$710$	−36.7811	−1.38037
$711$	0	0
$712$	−38.4423	−1.44069
$713$	−21.7073	−0.812945
$714$	0	0
$715$	56.2396	2.10324
$716$	28.2588	1.05608
$717$	0	0
$718$	−6.19389	−0.231154
$719$	3.13658	0.116975	0.0584874	−	0.998288i	$-0.481372\pi$
$719$	3.13658	0.116975	0.0584874	+	0.998288i	$0.481372\pi$
$720$	0	0
$721$	6.10263	0.227274
$722$	−44.4204	−1.65315
$723$	0	0
$724$	−20.8349	−0.774325
$725$	−80.8202	−3.00159
$726$	0	0
$727$	25.8875	0.960113	0.480056	−	0.877238i	$-0.340616\pi$
$727$	25.8875	0.960113	0.480056	+	0.877238i	$0.340616\pi$
$728$	−26.0132	−0.964112
$729$	0	0
$730$	20.5420	0.760292
$731$	38.7604	1.43361
$732$	0	0
$733$	−34.6744	−1.28073	−0.640364	−	0.768072i	$-0.721216\pi$
$733$	−34.6744	−1.28073	−0.640364	+	0.768072i	$0.721216\pi$
$734$	−48.8932	−1.80468
$735$	0	0
$736$	−8.30117	−0.305985
$737$	−13.9767	−0.514839
$738$	0	0
$739$	4.87609	0.179370	0.0896850	−	0.995970i	$-0.471414\pi$
$739$	4.87609	0.179370	0.0896850	+	0.995970i	$0.471414\pi$
$740$	66.6395	2.44972
$741$	0	0
$742$	25.1243	0.922341
$743$	26.0846	0.956950	0.478475	−	0.878101i	$-0.341190\pi$
$743$	26.0846	0.956950	0.478475	+	0.878101i	$0.341190\pi$
$744$	0	0
$745$	13.8530	0.507535
$746$	−45.7677	−1.67567
$747$	0	0
$748$	−20.6782	−0.756068
$749$	−7.58115	−0.277009
$750$	0	0
$751$	−31.3869	−1.14533	−0.572663	−	0.819791i	$-0.694090\pi$
$751$	−31.3869	−1.14533	−0.572663	+	0.819791i	$0.694090\pi$
$752$	−13.0181	−0.474721
$753$	0	0
$754$	−89.0325	−3.24237
$755$	66.5686	2.42268
$756$	0	0
$757$	−12.9321	−0.470027	−0.235013	−	0.971992i	$-0.575513\pi$
$757$	−12.9321	−0.470027	−0.235013	+	0.971992i	$0.575513\pi$
$758$	36.4767	1.32489
$759$	0	0
$760$	8.17018	0.296364
$761$	−27.1971	−0.985893	−0.492947	−	0.870059i	$-0.664080\pi$
$761$	−27.1971	−0.985893	−0.492947	+	0.870059i	$0.664080\pi$
$762$	0	0
$763$	0.833514	0.0301752
$764$	62.9592	2.27778
$765$	0	0
$766$	−48.6144	−1.75651
$767$	−52.1617	−1.88345
$768$	0	0
$769$	−51.5663	−1.85953	−0.929765	−	0.368154i	$-0.879990\pi$
$769$	−51.5663	−1.85953	−0.929765	+	0.368154i	$0.879990\pi$
$770$	−19.4260	−0.700065
$771$	0	0
$772$	−25.8559	−0.930576
$773$	−7.46723	−0.268578	−0.134289	−	0.990942i	$-0.542875\pi$
$773$	−7.46723	−0.268578	−0.134289	+	0.990942i	$0.542875\pi$
$774$	0	0
$775$	−130.239	−4.67831
$776$	−37.7610	−1.35554
$777$	0	0
$778$	69.8244	2.50333
$779$	−0.535546	−0.0191879
$780$	0	0
$781$	−6.46403	−0.231301
$782$	18.0444	0.645266
$783$	0	0
$784$	1.77830	0.0635107
$785$	−90.4862	−3.22959
$786$	0	0
$787$	−49.1743	−1.75287	−0.876437	−	0.481516i	$-0.840086\pi$
$787$	−49.1743	−1.75287	−0.876437	+	0.481516i	$0.840086\pi$
$788$	−43.6902	−1.55640
$789$	0	0
$790$	−29.9360	−1.06507
$791$	−2.85163	−0.101392
$792$	0	0
$793$	−91.6686	−3.25525
$794$	27.5563	0.977937
$795$	0	0
$796$	−42.9966	−1.52397
$797$	0.389128	0.0137836	0.00689182	−	0.999976i	$-0.497806\pi$
$797$	0.389128	0.0137836	0.00689182	+	0.999976i	$0.497806\pi$
$798$	0	0
$799$	−22.7351	−0.804310
$800$	−49.8050	−1.76087
$801$	0	0
$802$	19.4118	0.685453
$803$	3.61011	0.127398
$804$	0	0
$805$	10.9013	0.384221
$806$	−143.472	−5.05359
$807$	0	0
$808$	−24.4863	−0.861424
$809$	−5.22432	−0.183677	−0.0918386	−	0.995774i	$-0.529274\pi$
$809$	−5.22432	−0.183677	−0.0918386	+	0.995774i	$0.529274\pi$
$810$	0	0
$811$	−5.84462	−0.205232	−0.102616	−	0.994721i	$-0.532721\pi$
$811$	−5.84462	−0.205232	−0.102616	+	0.994721i	$0.532721\pi$
$812$	19.7768	0.694029
$813$	0	0
$814$	18.2115	0.638312
$815$	93.9208	3.28990
$816$	0	0
$817$	−6.04840	−0.211607
$818$	−64.4554	−2.25363
$819$	0	0
$820$	−17.6864	−0.617635
$821$	−6.28267	−0.219267	−0.109633	−	0.993972i	$-0.534968\pi$
$821$	−6.28267	−0.219267	−0.109633	+	0.993972i	$0.534968\pi$
$822$	0	0
$823$	15.6962	0.547135	0.273567	−	0.961853i	$-0.411796\pi$
$823$	15.6962	0.547135	0.273567	+	0.961853i	$0.411796\pi$
$824$	−23.1644	−0.806970
$825$	0	0
$826$	18.0174	0.626907
$827$	27.2279	0.946808	0.473404	−	0.880846i	$-0.343025\pi$
$827$	27.2279	0.946808	0.473404	+	0.880846i	$0.343025\pi$
$828$	0	0
$829$	10.0133	0.347777	0.173888	−	0.984765i	$-0.444367\pi$
$829$	10.0133	0.347777	0.173888	+	0.984765i	$0.444367\pi$
$830$	4.59077	0.159348
$831$	0	0
$832$	−79.2396	−2.74714
$833$	3.10566	0.107605
$834$	0	0
$835$	98.7621	3.41780
$836$	3.22674	0.111599
$837$	0	0
$838$	−59.2229	−2.04582
$839$	−1.38422	−0.0477888	−0.0238944	−	0.999714i	$-0.507607\pi$
$839$	−1.38422	−0.0477888	−0.0238944	+	0.999714i	$0.507607\pi$
$840$	0	0
$841$	1.12018	0.0386269
$842$	7.95872	0.274276
$843$	0	0
$844$	83.5077	2.87445
$845$	−150.855	−5.18957
$846$	0	0
$847$	7.58601	0.260658
$848$	−18.8742	−0.648142
$849$	0	0
$850$	108.262	3.71335
$851$	−10.2198	−0.350329
$852$	0	0
$853$	−46.9422	−1.60727	−0.803635	−	0.595122i	$-0.797104\pi$
$853$	−46.9422	−1.60727	−0.803635	+	0.595122i	$0.797104\pi$
$854$	31.6637	1.08351
$855$	0	0
$856$	28.7766	0.983562
$857$	−9.82991	−0.335783	−0.167892	−	0.985805i	$-0.553696\pi$
$857$	−9.82991	−0.335783	−0.167892	+	0.985805i	$0.553696\pi$
$858$	0	0
$859$	−15.1727	−0.517687	−0.258843	−	0.965919i	$-0.583341\pi$
$859$	−15.1727	−0.517687	−0.258843	+	0.965919i	$0.583341\pi$
$860$	−199.748	−6.81136
$861$	0	0
$862$	29.4707	1.00378
$863$	−26.4532	−0.900476	−0.450238	−	0.892908i	$-0.648661\pi$
$863$	−26.4532	−0.900476	−0.450238	+	0.892908i	$0.648661\pi$
$864$	0	0
$865$	38.6158	1.31298
$866$	29.6665	1.00811
$867$	0	0
$868$	31.8695	1.08172
$869$	−5.26105	−0.178469
$870$	0	0
$871$	51.8398	1.75652
$872$	−3.16386	−0.107142
$873$	0	0
$874$	−2.81575	−0.0952441
$875$	43.1982	1.46037
$876$	0	0
$877$	42.1630	1.42374	0.711871	−	0.702310i	$-0.247848\pi$
$877$	42.1630	1.42374	0.711871	+	0.702310i	$0.247848\pi$
$878$	−28.0776	−0.947575
$879$	0	0
$880$	14.5935	0.491945
$881$	47.4179	1.59755	0.798775	−	0.601630i	$-0.205482\pi$
$881$	47.4179	1.59755	0.798775	+	0.601630i	$0.205482\pi$
$882$	0	0
$883$	15.1860	0.511051	0.255525	−	0.966802i	$-0.417752\pi$
$883$	15.1860	0.511051	0.255525	+	0.966802i	$0.417752\pi$
$884$	76.6955	2.57955
$885$	0	0
$886$	7.84399	0.263524
$887$	11.8833	0.399004	0.199502	−	0.979897i	$-0.436068\pi$
$887$	11.8833	0.399004	0.199502	+	0.979897i	$0.436068\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	106.478	3.56914
$891$	0	0
$892$	3.86338	0.129356
$893$	3.54772	0.118720
$894$	0	0
$895$	−34.8296	−1.16423
$896$	20.6064	0.688412
$897$	0	0
$898$	19.5037	0.650848
$899$	48.5375	1.61881
$900$	0	0
$901$	−32.9623	−1.09813
$902$	−4.83339	−0.160934
$903$	0	0
$904$	10.8242	0.360009
$905$	25.6796	0.853618
$906$	0	0
$907$	−48.7557	−1.61891	−0.809453	−	0.587184i	$-0.800236\pi$
$907$	−48.7557	−1.61891	−0.809453	+	0.587184i	$0.800236\pi$
$908$	20.5267	0.681204
$909$	0	0
$910$	72.0513	2.38848
$911$	24.6116	0.815420	0.407710	−	0.913112i	$-0.366328\pi$
$911$	24.6116	0.815420	0.407710	+	0.913112i	$0.366328\pi$
$912$	0	0
$913$	0.806797	0.0267011
$914$	−35.9465	−1.18900
$915$	0	0
$916$	−12.4428	−0.411123
$917$	−9.74655	−0.321859
$918$	0	0
$919$	16.3444	0.539150	0.269575	−	0.962979i	$-0.413117\pi$
$919$	16.3444	0.539150	0.269575	+	0.962979i	$0.413117\pi$
$920$	−41.3793	−1.36424
$921$	0	0
$922$	63.4352	2.08913
$923$	23.9751	0.789151
$924$	0	0
$925$	−61.3161	−2.01606
$926$	−10.2773	−0.337732
$927$	0	0
$928$	18.5614	0.609308
$929$	4.38031	0.143713	0.0718566	−	0.997415i	$-0.477108\pi$
$929$	4.38031	0.143713	0.0718566	+	0.997415i	$0.477108\pi$
$930$	0	0
$931$	−0.484625	−0.0158830
$932$	−62.6455	−2.05202
$933$	0	0
$934$	71.6987	2.34605
$935$	25.4863	0.833492
$936$	0	0
$937$	3.77424	0.123299	0.0616495	−	0.998098i	$-0.480364\pi$
$937$	3.77424	0.123299	0.0616495	+	0.998098i	$0.480364\pi$
$938$	−17.9062	−0.584659
$939$	0	0
$940$	117.163	3.82144
$941$	2.27918	0.0742992	0.0371496	−	0.999310i	$-0.488172\pi$
$941$	2.27918	0.0742992	0.0371496	+	0.999310i	$0.488172\pi$
$942$	0	0
$943$	2.71236	0.0883267
$944$	−13.5353	−0.440536
$945$	0	0
$946$	−54.5879	−1.77481
$947$	−1.40505	−0.0456580	−0.0228290	−	0.999739i	$-0.507267\pi$
$947$	−1.40505	−0.0456580	−0.0228290	+	0.999739i	$0.507267\pi$
$948$	0	0
$949$	−13.3899	−0.434656
$950$	−16.8938	−0.548108
$951$	0	0
$952$	−11.7885	−0.382067
$953$	−4.50211	−0.145838	−0.0729188	−	0.997338i	$-0.523231\pi$
$953$	−4.50211	−0.145838	−0.0729188	+	0.997338i	$0.523231\pi$
$954$	0	0
$955$	−77.5987	−2.51103
$956$	−18.4655	−0.597217
$957$	0	0
$958$	70.4217	2.27522
$959$	−6.55908	−0.211804
$960$	0	0
$961$	47.2161	1.52310
$962$	−67.5465	−2.17779
$963$	0	0
$964$	94.6795	3.04942
$965$	31.8681	1.02587
$966$	0	0
$967$	−38.6707	−1.24357	−0.621784	−	0.783189i	$-0.713592\pi$
$967$	−38.6707	−1.24357	−0.621784	+	0.783189i	$0.713592\pi$
$968$	−28.7950	−0.925506
$969$	0	0
$970$	104.590	3.35819
$971$	−28.0570	−0.900393	−0.450196	−	0.892930i	$-0.648646\pi$
$971$	−28.0570	−0.900393	−0.450196	+	0.892930i	$0.648646\pi$
$972$	0	0
$973$	20.9922	0.672978
$974$	32.4927	1.04113
$975$	0	0
$976$	−23.7868	−0.761397
$977$	27.5009	0.879831	0.439916	−	0.898039i	$-0.355008\pi$
$977$	27.5009	0.879831	0.439916	+	0.898039i	$0.355008\pi$
$978$	0	0
$979$	18.7127	0.598062
$980$	−16.0047	−0.511252
$981$	0	0
$982$	−44.4786	−1.41937
$983$	16.8113	0.536198	0.268099	−	0.963391i	$-0.413605\pi$
$983$	16.8113	0.536198	0.268099	+	0.963391i	$0.413605\pi$
$984$	0	0
$985$	53.8492	1.71578
$986$	−40.3472	−1.28492
$987$	0	0
$988$	−11.9680	−0.380753
$989$	30.6332	0.974079
$990$	0	0
$991$	−46.4589	−1.47582	−0.737908	−	0.674901i	$-0.764186\pi$
$991$	−46.4589	−1.47582	−0.737908	+	0.674901i	$0.764186\pi$
$992$	29.9109	0.949673
$993$	0	0
$994$	−8.28137	−0.262669
$995$	52.9943	1.68003
$996$	0	0
$997$	10.3466	0.327680	0.163840	−	0.986487i	$-0.447612\pi$
$997$	10.3466	0.327680	0.163840	+	0.986487i	$0.447612\pi$
$998$	47.7686	1.51209
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.r.1.15		16
3.2	odd	2		2667.2.a.o.1.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.2	✓	16	3.2	odd	2
8001.2.a.r.1.15		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.36717 q^{2} +3.60352 q^{4} -4.44142 q^{5} -1.00000 q^{7} +3.79580 q^{8} +O(q^{10})\)	\(q+2.36717 q^{2} +3.60352 q^{4} -4.44142 q^{5} -1.00000 q^{7} +3.79580 q^{8} -10.5136 q^{10} -1.84770 q^{11} +6.85313 q^{13} -2.36717 q^{14} +1.77830 q^{16} +3.10566 q^{17} -0.484625 q^{19} -16.0047 q^{20} -4.37383 q^{22} +2.45447 q^{23} +14.7262 q^{25} +16.2226 q^{26} -3.60352 q^{28} -5.48819 q^{29} -8.84399 q^{31} -3.38206 q^{32} +7.35164 q^{34} +4.44142 q^{35} -4.16374 q^{37} -1.14719 q^{38} -16.8588 q^{40} +1.10507 q^{41} +12.4806 q^{43} -6.65821 q^{44} +5.81016 q^{46} -7.32054 q^{47} +1.00000 q^{49} +34.8595 q^{50} +24.6954 q^{52} -10.6136 q^{53} +8.20641 q^{55} -3.79580 q^{56} -12.9915 q^{58} -7.61137 q^{59} -13.3762 q^{61} -20.9353 q^{62} -11.5625 q^{64} -30.4377 q^{65} +7.56439 q^{67} +11.1913 q^{68} +10.5136 q^{70} +3.49842 q^{71} -1.95384 q^{73} -9.85629 q^{74} -1.74636 q^{76} +1.84770 q^{77} +2.84735 q^{79} -7.89818 q^{80} +2.61590 q^{82} -0.436650 q^{83} -13.7935 q^{85} +29.5437 q^{86} -7.01350 q^{88} -10.1276 q^{89} -6.85313 q^{91} +8.84472 q^{92} -17.3290 q^{94} +2.15242 q^{95} -9.94808 q^{97} +2.36717 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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