LMFDB - Embedded newform 8001.2.a.u.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.665348$ 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)

$f(q)$	$=$	$q-0.665348 q^{2} -1.55731 q^{4} +3.70415 q^{5} +1.00000 q^{7} +2.36685 q^{8} +O(q^{10})$	\(q-0.665348 q^{2} -1.55731 q^{4} +3.70415 q^{5} +1.00000 q^{7} +2.36685 q^{8} -2.46455 q^{10} -1.00383 q^{11} +3.77620 q^{13} -0.665348 q^{14} +1.53985 q^{16} -4.06728 q^{17} -8.44025 q^{19} -5.76853 q^{20} +0.667895 q^{22} -0.442232 q^{23} +8.72076 q^{25} -2.51249 q^{26} -1.55731 q^{28} -5.65212 q^{29} +8.26348 q^{31} -5.75823 q^{32} +2.70616 q^{34} +3.70415 q^{35} -10.2212 q^{37} +5.61570 q^{38} +8.76718 q^{40} +5.95725 q^{41} +2.00895 q^{43} +1.56327 q^{44} +0.294238 q^{46} -3.17856 q^{47} +1.00000 q^{49} -5.80234 q^{50} -5.88073 q^{52} +0.363462 q^{53} -3.71833 q^{55} +2.36685 q^{56} +3.76063 q^{58} +10.0577 q^{59} +3.57488 q^{61} -5.49809 q^{62} +0.751538 q^{64} +13.9876 q^{65} +3.95922 q^{67} +6.33403 q^{68} -2.46455 q^{70} +10.6852 q^{71} -13.5615 q^{73} +6.80067 q^{74} +13.1441 q^{76} -1.00383 q^{77} +14.7485 q^{79} +5.70383 q^{80} -3.96364 q^{82} +3.52844 q^{83} -15.0658 q^{85} -1.33665 q^{86} -2.37591 q^{88} +5.89993 q^{89} +3.77620 q^{91} +0.688694 q^{92} +2.11485 q^{94} -31.2640 q^{95} -6.24359 q^{97} -0.665348 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	−0.665348	−0.470472	−0.235236	−	0.971938i	$-0.575586\pi$
$2$	−0.665348	−0.470472	−0.235236	+	0.971938i	$0.575586\pi$
$3$	0	0
$3$	0	0
$4$	−1.55731	−0.778656
$5$	3.70415	1.65655	0.828274	−	0.560323i	$-0.189323\pi$
$5$	3.70415	1.65655	0.828274	+	0.560323i	$0.189323\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.36685	0.836808
$9$	0	0
$10$	−2.46455	−0.779360
$11$	−1.00383	−0.302665	−0.151333	−	0.988483i	$-0.548356\pi$
$11$	−1.00383	−0.302665	−0.151333	+	0.988483i	$0.548356\pi$
$12$	0	0
$13$	3.77620	1.04733	0.523665	−	0.851924i	$-0.324564\pi$
$13$	3.77620	1.04733	0.523665	+	0.851924i	$0.324564\pi$
$14$	−0.665348	−0.177822
$15$	0	0
$16$	1.53985	0.384961
$17$	−4.06728	−0.986461	−0.493231	−	0.869899i	$-0.664184\pi$
$17$	−4.06728	−0.986461	−0.493231	+	0.869899i	$0.664184\pi$
$18$	0	0
$19$	−8.44025	−1.93633	−0.968163	−	0.250321i	$-0.919464\pi$
$19$	−8.44025	−1.93633	−0.968163	+	0.250321i	$0.919464\pi$
$20$	−5.76853	−1.28988
$21$	0	0
$22$	0.667895	0.142396
$23$	−0.442232	−0.0922118	−0.0461059	−	0.998937i	$-0.514681\pi$
$23$	−0.442232	−0.0922118	−0.0461059	+	0.998937i	$0.514681\pi$
$24$	0	0
$25$	8.72076	1.74415
$26$	−2.51249	−0.492740
$27$	0	0
$28$	−1.55731	−0.294304
$29$	−5.65212	−1.04957	−0.524786	−	0.851234i	$-0.675855\pi$
$29$	−5.65212	−1.04957	−0.524786	+	0.851234i	$0.675855\pi$
$30$	0	0
$31$	8.26348	1.48417	0.742083	−	0.670308i	$-0.233838\pi$
$31$	8.26348	1.48417	0.742083	+	0.670308i	$0.233838\pi$
$32$	−5.75823	−1.01792
$33$	0	0
$34$	2.70616	0.464102
$35$	3.70415	0.626116
$36$	0	0
$37$	−10.2212	−1.68036	−0.840179	−	0.542309i	$-0.817550\pi$
$37$	−10.2212	−1.68036	−0.840179	+	0.542309i	$0.817550\pi$
$38$	5.61570	0.910987
$39$	0	0
$40$	8.76718	1.38621
$41$	5.95725	0.930366	0.465183	−	0.885215i	$-0.345989\pi$
$41$	5.95725	0.930366	0.465183	+	0.885215i	$0.345989\pi$
$42$	0	0
$43$	2.00895	0.306362	0.153181	−	0.988198i	$-0.451048\pi$
$43$	2.00895	0.306362	0.153181	+	0.988198i	$0.451048\pi$
$44$	1.56327	0.235672
$45$	0	0
$46$	0.294238	0.0433831
$47$	−3.17856	−0.463640	−0.231820	−	0.972759i	$-0.574468\pi$
$47$	−3.17856	−0.463640	−0.231820	+	0.972759i	$0.574468\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.80234	−0.820575
$51$	0	0
$52$	−5.88073	−0.815510
$53$	0.363462	0.0499253	0.0249627	−	0.999688i	$-0.492053\pi$
$53$	0.363462	0.0499253	0.0249627	+	0.999688i	$0.492053\pi$
$54$	0	0
$55$	−3.71833	−0.501380
$56$	2.36685	0.316284
$57$	0	0
$58$	3.76063	0.493795
$59$	10.0577	1.30940	0.654700	−	0.755889i	$-0.272795\pi$
$59$	10.0577	1.30940	0.654700	+	0.755889i	$0.272795\pi$
$60$	0	0
$61$	3.57488	0.457717	0.228858	−	0.973460i	$-0.426501\pi$
$61$	3.57488	0.457717	0.228858	+	0.973460i	$0.426501\pi$
$62$	−5.49809	−0.698258
$63$	0	0
$64$	0.751538	0.0939423
$65$	13.9876	1.73495
$66$	0	0
$67$	3.95922	0.483695	0.241848	−	0.970314i	$-0.422247\pi$
$67$	3.95922	0.483695	0.241848	+	0.970314i	$0.422247\pi$
$68$	6.33403	0.768114
$69$	0	0
$70$	−2.46455	−0.294570
$71$	10.6852	1.26810	0.634052	−	0.773291i	$-0.281391\pi$
$71$	10.6852	1.26810	0.634052	+	0.773291i	$0.281391\pi$
$72$	0	0
$73$	−13.5615	−1.58725	−0.793625	−	0.608407i	$-0.791809\pi$
$73$	−13.5615	−1.58725	−0.793625	+	0.608407i	$0.791809\pi$
$74$	6.80067	0.790562
$75$	0	0
$76$	13.1441	1.50773
$77$	−1.00383	−0.114397
$78$	0	0
$79$	14.7485	1.65934	0.829669	−	0.558255i	$-0.188529\pi$
$79$	14.7485	1.65934	0.829669	+	0.558255i	$0.188529\pi$
$80$	5.70383	0.637707
$81$	0	0
$82$	−3.96364	−0.437711
$83$	3.52844	0.387297	0.193648	−	0.981071i	$-0.437968\pi$
$83$	3.52844	0.387297	0.193648	+	0.981071i	$0.437968\pi$
$84$	0	0
$85$	−15.0658	−1.63412
$86$	−1.33665	−0.144135
$87$	0	0
$88$	−2.37591	−0.253273
$89$	5.89993	0.625392	0.312696	−	0.949853i	$-0.398768\pi$
$89$	5.89993	0.625392	0.312696	+	0.949853i	$0.398768\pi$
$90$	0	0
$91$	3.77620	0.395854
$92$	0.688694	0.0718013
$93$	0	0
$94$	2.11485	0.218130
$95$	−31.2640	−3.20762
$96$	0	0
$97$	−6.24359	−0.633941	−0.316970	−	0.948435i	$-0.602666\pi$
$97$	−6.24359	−0.633941	−0.316970	+	0.948435i	$0.602666\pi$
$98$	−0.665348	−0.0672103
$99$	0	0
$100$	−13.5810	−1.35810
$101$	18.8071	1.87138	0.935688	−	0.352828i	$-0.114780\pi$
$101$	18.8071	1.87138	0.935688	+	0.352828i	$0.114780\pi$
$102$	0	0
$103$	11.1061	1.09431	0.547157	−	0.837030i	$-0.315710\pi$
$103$	11.1061	1.09431	0.547157	+	0.837030i	$0.315710\pi$
$104$	8.93771	0.876415
$105$	0	0
$106$	−0.241829	−0.0234885
$107$	0.580119	0.0560822	0.0280411	−	0.999607i	$-0.491073\pi$
$107$	0.580119	0.0560822	0.0280411	+	0.999607i	$0.491073\pi$
$108$	0	0
$109$	17.5353	1.67958	0.839788	−	0.542914i	$-0.182679\pi$
$109$	17.5353	1.67958	0.839788	+	0.542914i	$0.182679\pi$
$110$	2.47399	0.235885
$111$	0	0
$112$	1.53985	0.145502
$113$	19.4000	1.82499	0.912497	−	0.409083i	$-0.134151\pi$
$113$	19.4000	1.82499	0.912497	+	0.409083i	$0.134151\pi$
$114$	0	0
$115$	−1.63810	−0.152753
$116$	8.80212	0.817256
$117$	0	0
$118$	−6.69186	−0.616036
$119$	−4.06728	−0.372847
$120$	0	0
$121$	−9.99233	−0.908394
$122$	−2.37854	−0.215343
$123$	0	0
$124$	−12.8688	−1.15565
$125$	13.7823	1.23272
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	11.0164	0.973724
$129$	0	0
$130$	−9.30665	−0.816247
$131$	19.5438	1.70755	0.853775	−	0.520642i	$-0.174307\pi$
$131$	19.5438	1.70755	0.853775	+	0.520642i	$0.174307\pi$
$132$	0	0
$133$	−8.44025	−0.731862
$134$	−2.63426	−0.227565
$135$	0	0
$136$	−9.62665	−0.825478
$137$	6.60487	0.564292	0.282146	−	0.959371i	$-0.408954\pi$
$137$	6.60487	0.564292	0.282146	+	0.959371i	$0.408954\pi$
$138$	0	0
$139$	−5.63615	−0.478052	−0.239026	−	0.971013i	$-0.576828\pi$
$139$	−5.63615	−0.478052	−0.239026	+	0.971013i	$0.576828\pi$
$140$	−5.76853	−0.487529
$141$	0	0
$142$	−7.10939	−0.596607
$143$	−3.79066	−0.316991
$144$	0	0
$145$	−20.9363	−1.73867
$146$	9.02310	0.746757
$147$	0	0
$148$	15.9176	1.30842
$149$	−3.12232	−0.255790	−0.127895	−	0.991788i	$-0.540822\pi$
$149$	−3.12232	−0.255790	−0.127895	+	0.991788i	$0.540822\pi$
$150$	0	0
$151$	13.7417	1.11828	0.559140	−	0.829073i	$-0.311131\pi$
$151$	13.7417	1.11828	0.559140	+	0.829073i	$0.311131\pi$
$152$	−19.9768	−1.62033
$153$	0	0
$154$	0.667895	0.0538205
$155$	30.6092	2.45859
$156$	0	0
$157$	18.9828	1.51499	0.757497	−	0.652838i	$-0.226422\pi$
$157$	18.9828	1.51499	0.757497	+	0.652838i	$0.226422\pi$
$158$	−9.81290	−0.780672
$159$	0	0
$160$	−21.3294	−1.68624
$161$	−0.442232	−0.0348528
$162$	0	0
$163$	−22.2933	−1.74615	−0.873075	−	0.487586i	$-0.837877\pi$
$163$	−22.2933	−1.74615	−0.873075	+	0.487586i	$0.837877\pi$
$164$	−9.27729	−0.724435
$165$	0	0
$166$	−2.34764	−0.182212
$167$	1.57767	0.122084	0.0610421	−	0.998135i	$-0.480558\pi$
$167$	1.57767	0.122084	0.0610421	+	0.998135i	$0.480558\pi$
$168$	0	0
$169$	1.25972	0.0969017
$170$	10.0240	0.768808
$171$	0	0
$172$	−3.12856	−0.238551
$173$	4.11390	0.312774	0.156387	−	0.987696i	$-0.450015\pi$
$173$	4.11390	0.312774	0.156387	+	0.987696i	$0.450015\pi$
$174$	0	0
$175$	8.72076	0.659228
$176$	−1.54574	−0.116514
$177$	0	0
$178$	−3.92551	−0.294229
$179$	−5.78579	−0.432450	−0.216225	−	0.976344i	$-0.569374\pi$
$179$	−5.78579	−0.432450	−0.216225	+	0.976344i	$0.569374\pi$
$180$	0	0
$181$	−6.54333	−0.486362	−0.243181	−	0.969981i	$-0.578191\pi$
$181$	−6.54333	−0.486362	−0.243181	+	0.969981i	$0.578191\pi$
$182$	−2.51249	−0.186238
$183$	0	0
$184$	−1.04670	−0.0771636
$185$	−37.8610	−2.78360
$186$	0	0
$187$	4.08285	0.298568
$188$	4.95000	0.361016
$189$	0	0
$190$	20.8014	1.50909
$191$	7.70840	0.557760	0.278880	−	0.960326i	$-0.410037\pi$
$191$	7.70840	0.557760	0.278880	+	0.960326i	$0.410037\pi$
$192$	0	0
$193$	26.8743	1.93445	0.967227	−	0.253914i	$-0.0817179\pi$
$193$	26.8743	1.93445	0.967227	+	0.253914i	$0.0817179\pi$
$194$	4.15416	0.298251
$195$	0	0
$196$	−1.55731	−0.111237
$197$	−9.18051	−0.654084	−0.327042	−	0.945010i	$-0.606052\pi$
$197$	−9.18051	−0.654084	−0.327042	+	0.945010i	$0.606052\pi$
$198$	0	0
$199$	−15.6252	−1.10764	−0.553821	−	0.832636i	$-0.686831\pi$
$199$	−15.6252	−1.10764	−0.553821	+	0.832636i	$0.686831\pi$
$200$	20.6407	1.45952
$201$	0	0
$202$	−12.5133	−0.880430
$203$	−5.65212	−0.396701
$204$	0	0
$205$	22.0666	1.54120
$206$	−7.38941	−0.514845
$207$	0	0
$208$	5.81477	0.403182
$209$	8.47256	0.586059
$210$	0	0
$211$	0.566441	0.0389954	0.0194977	−	0.999810i	$-0.493793\pi$
$211$	0.566441	0.0389954	0.0194977	+	0.999810i	$0.493793\pi$
$212$	−0.566023	−0.0388746
$213$	0	0
$214$	−0.385981	−0.0263851
$215$	7.44147	0.507504
$216$	0	0
$217$	8.26348	0.560962
$218$	−11.6671	−0.790194
$219$	0	0
$220$	5.79061	0.390403
$221$	−15.3589	−1.03315
$222$	0	0
$223$	−5.32002	−0.356255	−0.178128	−	0.984007i	$-0.557004\pi$
$223$	−5.32002	−0.356255	−0.178128	+	0.984007i	$0.557004\pi$
$224$	−5.75823	−0.384738
$225$	0	0
$226$	−12.9077	−0.858609
$227$	−25.5949	−1.69879	−0.849397	−	0.527754i	$-0.823034\pi$
$227$	−25.5949	−1.69879	−0.849397	+	0.527754i	$0.823034\pi$
$228$	0	0
$229$	−16.0579	−1.06114	−0.530568	−	0.847642i	$-0.678021\pi$
$229$	−16.0579	−1.06114	−0.530568	+	0.847642i	$0.678021\pi$
$230$	1.08990	0.0718661
$231$	0	0
$232$	−13.3777	−0.878291
$233$	−23.0614	−1.51080	−0.755401	−	0.655262i	$-0.772558\pi$
$233$	−23.0614	−1.51080	−0.755401	+	0.655262i	$0.772558\pi$
$234$	0	0
$235$	−11.7739	−0.768042
$236$	−15.6630	−1.01957
$237$	0	0
$238$	2.70616	0.175414
$239$	21.7293	1.40555	0.702777	−	0.711410i	$-0.251943\pi$
$239$	21.7293	1.40555	0.702777	+	0.711410i	$0.251943\pi$
$240$	0	0
$241$	13.4723	0.867830	0.433915	−	0.900954i	$-0.357132\pi$
$241$	13.4723	0.867830	0.433915	+	0.900954i	$0.357132\pi$
$242$	6.64838	0.427374
$243$	0	0
$244$	−5.56721	−0.356404
$245$	3.70415	0.236650
$246$	0	0
$247$	−31.8721	−2.02797
$248$	19.5584	1.24196
$249$	0	0
$250$	−9.17001	−0.579963
$251$	21.8629	1.37997	0.689987	−	0.723822i	$-0.257616\pi$
$251$	21.8629	1.37997	0.689987	+	0.723822i	$0.257616\pi$
$252$	0	0
$253$	0.443925	0.0279093
$254$	−0.665348	−0.0417476
$255$	0	0
$256$	−8.83284	−0.552052
$257$	−18.8653	−1.17679	−0.588393	−	0.808575i	$-0.700239\pi$
$257$	−18.8653	−1.17679	−0.588393	+	0.808575i	$0.700239\pi$
$258$	0	0
$259$	−10.2212	−0.635116
$260$	−21.7831	−1.35093
$261$	0	0
$262$	−13.0034	−0.803354
$263$	5.02576	0.309901	0.154951	−	0.987922i	$-0.450478\pi$
$263$	5.02576	0.309901	0.154951	+	0.987922i	$0.450478\pi$
$264$	0	0
$265$	1.34632	0.0827037
$266$	5.61570	0.344321
$267$	0	0
$268$	−6.16574	−0.376632
$269$	14.3454	0.874652	0.437326	−	0.899303i	$-0.355926\pi$
$269$	14.3454	0.874652	0.437326	+	0.899303i	$0.355926\pi$
$270$	0	0
$271$	11.0365	0.670418	0.335209	−	0.942144i	$-0.391193\pi$
$271$	11.0365	0.670418	0.335209	+	0.942144i	$0.391193\pi$
$272$	−6.26299	−0.379749
$273$	0	0
$274$	−4.39454	−0.265484
$275$	−8.75414	−0.527895
$276$	0	0
$277$	−17.6419	−1.06000	−0.530000	−	0.847998i	$-0.677808\pi$
$277$	−17.6419	−1.06000	−0.530000	+	0.847998i	$0.677808\pi$
$278$	3.75000	0.224910
$279$	0	0
$280$	8.76718	0.523939
$281$	−7.54428	−0.450054	−0.225027	−	0.974353i	$-0.572247\pi$
$281$	−7.54428	−0.450054	−0.225027	+	0.974353i	$0.572247\pi$
$282$	0	0
$283$	−10.6025	−0.630253	−0.315127	−	0.949050i	$-0.602047\pi$
$283$	−10.6025	−0.630253	−0.315127	+	0.949050i	$0.602047\pi$
$284$	−16.6402	−0.987416
$285$	0	0
$286$	2.52211	0.149135
$287$	5.95725	0.351645
$288$	0	0
$289$	−0.457208	−0.0268946
$290$	13.9299	0.817995
$291$	0	0
$292$	21.1194	1.23592
$293$	31.2181	1.82378	0.911890	−	0.410435i	$-0.134623\pi$
$293$	31.2181	1.82378	0.911890	+	0.410435i	$0.134623\pi$
$294$	0	0
$295$	37.2552	2.16908
$296$	−24.1921	−1.40614
$297$	0	0
$298$	2.07743	0.120342
$299$	−1.66996	−0.0965762
$300$	0	0
$301$	2.00895	0.115794
$302$	−9.14299	−0.526120
$303$	0	0
$304$	−12.9967	−0.745411
$305$	13.2419	0.758230
$306$	0	0
$307$	26.3560	1.50422	0.752109	−	0.659038i	$-0.229036\pi$
$307$	26.3560	1.50422	0.752109	+	0.659038i	$0.229036\pi$
$308$	1.56327	0.0890757
$309$	0	0
$310$	−20.3658	−1.15670
$311$	19.7761	1.12140	0.560699	−	0.828020i	$-0.310533\pi$
$311$	19.7761	1.12140	0.560699	+	0.828020i	$0.310533\pi$
$312$	0	0
$313$	−11.3455	−0.641288	−0.320644	−	0.947200i	$-0.603899\pi$
$313$	−11.3455	−0.641288	−0.320644	+	0.947200i	$0.603899\pi$
$314$	−12.6302	−0.712763
$315$	0	0
$316$	−22.9681	−1.29205
$317$	−15.5335	−0.872448	−0.436224	−	0.899838i	$-0.643684\pi$
$317$	−15.5335	−0.872448	−0.436224	+	0.899838i	$0.643684\pi$
$318$	0	0
$319$	5.67376	0.317669
$320$	2.78381	0.155620
$321$	0	0
$322$	0.294238	0.0163973
$323$	34.3289	1.91011
$324$	0	0
$325$	32.9314	1.82670
$326$	14.8328	0.821515
$327$	0	0
$328$	14.0999	0.778537
$329$	−3.17856	−0.175239
$330$	0	0
$331$	2.58933	0.142322	0.0711612	−	0.997465i	$-0.477330\pi$
$331$	2.58933	0.142322	0.0711612	+	0.997465i	$0.477330\pi$
$332$	−5.49488	−0.301571
$333$	0	0
$334$	−1.04970	−0.0574372
$335$	14.6656	0.801265
$336$	0	0
$337$	29.4115	1.60214	0.801072	−	0.598568i	$-0.204263\pi$
$337$	29.4115	1.60214	0.801072	+	0.598568i	$0.204263\pi$
$338$	−0.838154	−0.0455896
$339$	0	0
$340$	23.4622	1.27242
$341$	−8.29511	−0.449205
$342$	0	0
$343$	1.00000	0.0539949
$344$	4.75489	0.256366
$345$	0	0
$346$	−2.73717	−0.147151
$347$	4.05583	0.217728	0.108864	−	0.994057i	$-0.465279\pi$
$347$	4.05583	0.217728	0.108864	+	0.994057i	$0.465279\pi$
$348$	0	0
$349$	−23.9595	−1.28252	−0.641261	−	0.767323i	$-0.721588\pi$
$349$	−23.9595	−1.28252	−0.641261	+	0.767323i	$0.721588\pi$
$350$	−5.80234	−0.310148
$351$	0	0
$352$	5.78027	0.308090
$353$	20.2726	1.07900	0.539501	−	0.841985i	$-0.318613\pi$
$353$	20.2726	1.07900	0.539501	+	0.841985i	$0.318613\pi$
$354$	0	0
$355$	39.5797	2.10067
$356$	−9.18804	−0.486965
$357$	0	0
$358$	3.84956	0.203456
$359$	−29.0431	−1.53284	−0.766418	−	0.642342i	$-0.777963\pi$
$359$	−29.0431	−1.53284	−0.766418	+	0.642342i	$0.777963\pi$
$360$	0	0
$361$	52.2378	2.74936
$362$	4.35359	0.228820
$363$	0	0
$364$	−5.88073	−0.308234
$365$	−50.2338	−2.62936
$366$	0	0
$367$	6.09658	0.318239	0.159120	−	0.987259i	$-0.449134\pi$
$367$	6.09658	0.318239	0.159120	+	0.987259i	$0.449134\pi$
$368$	−0.680969	−0.0354980
$369$	0	0
$370$	25.1907	1.30960
$371$	0.363462	0.0188700
$372$	0	0
$373$	−27.3423	−1.41573	−0.707865	−	0.706348i	$-0.750342\pi$
$373$	−27.3423	−1.41573	−0.707865	+	0.706348i	$0.750342\pi$
$374$	−2.71652	−0.140468
$375$	0	0
$376$	−7.52316	−0.387978
$377$	−21.3436	−1.09925
$378$	0	0
$379$	10.1244	0.520054	0.260027	−	0.965601i	$-0.416268\pi$
$379$	10.1244	0.520054	0.260027	+	0.965601i	$0.416268\pi$
$380$	48.6878	2.49763
$381$	0	0
$382$	−5.12877	−0.262410
$383$	−3.42254	−0.174883	−0.0874417	−	0.996170i	$-0.527869\pi$
$383$	−3.42254	−0.174883	−0.0874417	+	0.996170i	$0.527869\pi$
$384$	0	0
$385$	−3.71833	−0.189504
$386$	−17.8808	−0.910106
$387$	0	0
$388$	9.72322	0.493622
$389$	17.2100	0.872582	0.436291	−	0.899806i	$-0.356292\pi$
$389$	17.2100	0.872582	0.436291	+	0.899806i	$0.356292\pi$
$390$	0	0
$391$	1.79868	0.0909633
$392$	2.36685	0.119544
$393$	0	0
$394$	6.10823	0.307728
$395$	54.6308	2.74877
$396$	0	0
$397$	−21.9471	−1.10149	−0.550747	−	0.834672i	$-0.685657\pi$
$397$	−21.9471	−1.10149	−0.550747	+	0.834672i	$0.685657\pi$
$398$	10.3962	0.521114
$399$	0	0
$400$	13.4286	0.671431
$401$	18.4565	0.921673	0.460837	−	0.887485i	$-0.347549\pi$
$401$	18.4565	0.921673	0.460837	+	0.887485i	$0.347549\pi$
$402$	0	0
$403$	31.2046	1.55441
$404$	−29.2885	−1.45716
$405$	0	0
$406$	3.76063	0.186637
$407$	10.2603	0.508586
$408$	0	0
$409$	−11.7867	−0.582817	−0.291408	−	0.956599i	$-0.594124\pi$
$409$	−11.7867	−0.582817	−0.291408	+	0.956599i	$0.594124\pi$
$410$	−14.6819	−0.725090
$411$	0	0
$412$	−17.2956	−0.852095
$413$	10.0577	0.494907
$414$	0	0
$415$	13.0699	0.641576
$416$	−21.7443	−1.06610
$417$	0	0
$418$	−5.63720	−0.275724
$419$	−14.7894	−0.722511	−0.361255	−	0.932467i	$-0.617652\pi$
$419$	−14.7894	−0.722511	−0.361255	+	0.932467i	$0.617652\pi$
$420$	0	0
$421$	−14.4728	−0.705360	−0.352680	−	0.935744i	$-0.614730\pi$
$421$	−14.4728	−0.705360	−0.352680	+	0.935744i	$0.614730\pi$
$422$	−0.376880	−0.0183462
$423$	0	0
$424$	0.860260	0.0417779
$425$	−35.4698	−1.72054
$426$	0	0
$427$	3.57488	0.173001
$428$	−0.903427	−0.0436688
$429$	0	0
$430$	−4.95117	−0.238766
$431$	17.0346	0.820529	0.410264	−	0.911967i	$-0.365436\pi$
$431$	17.0346	0.820529	0.410264	+	0.911967i	$0.365436\pi$
$432$	0	0
$433$	−6.44778	−0.309860	−0.154930	−	0.987925i	$-0.549515\pi$
$433$	−6.44778	−0.309860	−0.154930	+	0.987925i	$0.549515\pi$
$434$	−5.49809	−0.263917
$435$	0	0
$436$	−27.3079	−1.30781
$437$	3.73255	0.178552
$438$	0	0
$439$	−2.94169	−0.140399	−0.0701995	−	0.997533i	$-0.522364\pi$
$439$	−2.94169	−0.140399	−0.0701995	+	0.997533i	$0.522364\pi$
$440$	−8.80074	−0.419559
$441$	0	0
$442$	10.2190	0.486069
$443$	4.26199	0.202493	0.101247	−	0.994861i	$-0.467717\pi$
$443$	4.26199	0.202493	0.101247	+	0.994861i	$0.467717\pi$
$444$	0	0
$445$	21.8543	1.03599
$446$	3.53967	0.167608
$447$	0	0
$448$	0.751538	0.0355068
$449$	13.2152	0.623666	0.311833	−	0.950137i	$-0.399057\pi$
$449$	13.2152	0.623666	0.311833	+	0.950137i	$0.399057\pi$
$450$	0	0
$451$	−5.98005	−0.281590
$452$	−30.2118	−1.42104
$453$	0	0
$454$	17.0295	0.799235
$455$	13.9876	0.655751
$456$	0	0
$457$	−35.9385	−1.68113	−0.840566	−	0.541709i	$-0.817778\pi$
$457$	−35.9385	−1.68113	−0.840566	+	0.541709i	$0.817778\pi$
$458$	10.6841	0.499235
$459$	0	0
$460$	2.55103	0.118942
$461$	0.433465	0.0201885	0.0100942	−	0.999949i	$-0.496787\pi$
$461$	0.433465	0.0201885	0.0100942	+	0.999949i	$0.496787\pi$
$462$	0	0
$463$	−3.27537	−0.152219	−0.0761096	−	0.997099i	$-0.524250\pi$
$463$	−3.27537	−0.152219	−0.0761096	+	0.997099i	$0.524250\pi$
$464$	−8.70339	−0.404045
$465$	0	0
$466$	15.3439	0.710790
$467$	22.0551	1.02059	0.510293	−	0.860000i	$-0.329537\pi$
$467$	22.0551	1.02059	0.510293	+	0.860000i	$0.329537\pi$
$468$	0	0
$469$	3.95922	0.182820
$470$	7.83371	0.361342
$471$	0	0
$472$	23.8050	1.09572
$473$	−2.01664	−0.0927253
$474$	0	0
$475$	−73.6054	−3.37725
$476$	6.33403	0.290320
$477$	0	0
$478$	−14.4576	−0.661274
$479$	2.91609	0.133240	0.0666198	−	0.997778i	$-0.478779\pi$
$479$	2.91609	0.133240	0.0666198	+	0.997778i	$0.478779\pi$
$480$	0	0
$481$	−38.5974	−1.75989
$482$	−8.96379	−0.408290
$483$	0	0
$484$	15.5612	0.707326
$485$	−23.1272	−1.05015
$486$	0	0
$487$	23.2024	1.05140	0.525702	−	0.850669i	$-0.323803\pi$
$487$	23.2024	1.05140	0.525702	+	0.850669i	$0.323803\pi$
$488$	8.46121	0.383021
$489$	0	0
$490$	−2.46455	−0.111337
$491$	−34.2673	−1.54646	−0.773231	−	0.634125i	$-0.781361\pi$
$491$	−34.2673	−1.54646	−0.773231	+	0.634125i	$0.781361\pi$
$492$	0	0
$493$	22.9888	1.03536
$494$	21.2060	0.954105
$495$	0	0
$496$	12.7245	0.571346
$497$	10.6852	0.479298
$498$	0	0
$499$	−28.5434	−1.27778	−0.638890	−	0.769298i	$-0.720606\pi$
$499$	−28.5434	−1.27778	−0.638890	+	0.769298i	$0.720606\pi$
$500$	−21.4633	−0.959869
$501$	0	0
$502$	−14.5464	−0.649239
$503$	34.3100	1.52981	0.764904	−	0.644145i	$-0.222787\pi$
$503$	34.3100	1.52981	0.764904	+	0.644145i	$0.222787\pi$
$504$	0	0
$505$	69.6644	3.10003
$506$	−0.295365	−0.0131306
$507$	0	0
$508$	−1.55731	−0.0690946
$509$	23.2579	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$509$	23.2579	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$510$	0	0
$511$	−13.5615	−0.599924
$512$	−16.1560	−0.713999
$513$	0	0
$514$	12.5520	0.553645
$515$	41.1387	1.81279
$516$	0	0
$517$	3.19072	0.140328
$518$	6.80067	0.298804
$519$	0	0
$520$	33.1067	1.45182
$521$	−24.7606	−1.08478	−0.542391	−	0.840126i	$-0.682481\pi$
$521$	−24.7606	−1.08478	−0.542391	+	0.840126i	$0.682481\pi$
$522$	0	0
$523$	2.59245	0.113360	0.0566799	−	0.998392i	$-0.481949\pi$
$523$	2.59245	0.113360	0.0566799	+	0.998392i	$0.481949\pi$
$524$	−30.4358	−1.32959
$525$	0	0
$526$	−3.34388	−0.145800
$527$	−33.6099	−1.46407
$528$	0	0
$529$	−22.8044	−0.991497
$530$	−0.895770	−0.0389098
$531$	0	0
$532$	13.1441	0.569869
$533$	22.4958	0.974401
$534$	0	0
$535$	2.14885	0.0929030
$536$	9.37087	0.404760
$537$	0	0
$538$	−9.54465	−0.411499
$539$	−1.00383	−0.0432379
$540$	0	0
$541$	−17.7457	−0.762946	−0.381473	−	0.924380i	$-0.624583\pi$
$541$	−17.7457	−0.762946	−0.381473	+	0.924380i	$0.624583\pi$
$542$	−7.34310	−0.315413
$543$	0	0
$544$	23.4204	1.00414
$545$	64.9535	2.78230
$546$	0	0
$547$	17.1810	0.734609	0.367304	−	0.930101i	$-0.380281\pi$
$547$	17.1810	0.734609	0.367304	+	0.930101i	$0.380281\pi$
$548$	−10.2858	−0.439389
$549$	0	0
$550$	5.82455	0.248360
$551$	47.7053	2.03232
$552$	0	0
$553$	14.7485	0.627171
$554$	11.7380	0.498700
$555$	0	0
$556$	8.77725	0.372238
$557$	24.3474	1.03163	0.515815	−	0.856700i	$-0.327489\pi$
$557$	24.3474	1.03163	0.515815	+	0.856700i	$0.327489\pi$
$558$	0	0
$559$	7.58621	0.320863
$560$	5.70383	0.241031
$561$	0	0
$562$	5.01957	0.211738
$563$	15.3841	0.648363	0.324181	−	0.945995i	$-0.394911\pi$
$563$	15.3841	0.648363	0.324181	+	0.945995i	$0.394911\pi$
$564$	0	0
$565$	71.8605	3.02319
$566$	7.05435	0.296517
$567$	0	0
$568$	25.2903	1.06116
$569$	−15.5356	−0.651288	−0.325644	−	0.945492i	$-0.605581\pi$
$569$	−15.5356	−0.651288	−0.325644	+	0.945492i	$0.605581\pi$
$570$	0	0
$571$	1.68607	0.0705597	0.0352799	−	0.999377i	$-0.488768\pi$
$571$	1.68607	0.0705597	0.0352799	+	0.999377i	$0.488768\pi$
$572$	5.90324	0.246827
$573$	0	0
$574$	−3.96364	−0.165439
$575$	−3.85660	−0.160831
$576$	0	0
$577$	−19.6600	−0.818458	−0.409229	−	0.912432i	$-0.634202\pi$
$577$	−19.6600	−0.818458	−0.409229	+	0.912432i	$0.634202\pi$
$578$	0.304203	0.0126532
$579$	0	0
$580$	32.6044	1.35382
$581$	3.52844	0.146384
$582$	0	0
$583$	−0.364853	−0.0151107
$584$	−32.0980	−1.32822
$585$	0	0
$586$	−20.7709	−0.858038
$587$	1.52868	0.0630953	0.0315476	−	0.999502i	$-0.489956\pi$
$587$	1.52868	0.0630953	0.0315476	+	0.999502i	$0.489956\pi$
$588$	0	0
$589$	−69.7459	−2.87383
$590$	−24.7877	−1.02049
$591$	0	0
$592$	−15.7391	−0.646873
$593$	22.5830	0.927373	0.463686	−	0.885999i	$-0.346526\pi$
$593$	22.5830	0.927373	0.463686	+	0.885999i	$0.346526\pi$
$594$	0	0
$595$	−15.0658	−0.617639
$596$	4.86242	0.199173
$597$	0	0
$598$	1.11110	0.0454364
$599$	39.1873	1.60115	0.800575	−	0.599232i	$-0.204527\pi$
$599$	39.1873	1.60115	0.800575	+	0.599232i	$0.204527\pi$
$600$	0	0
$601$	33.3105	1.35877	0.679383	−	0.733784i	$-0.262248\pi$
$601$	33.3105	1.35877	0.679383	+	0.733784i	$0.262248\pi$
$602$	−1.33665	−0.0544779
$603$	0	0
$604$	−21.4001	−0.870756
$605$	−37.0131	−1.50480
$606$	0	0
$607$	25.4329	1.03229	0.516145	−	0.856501i	$-0.327367\pi$
$607$	25.4329	1.03229	0.516145	+	0.856501i	$0.327367\pi$
$608$	48.6009	1.97103
$609$	0	0
$610$	−8.81048	−0.356726
$611$	−12.0029	−0.485584
$612$	0	0
$613$	−47.2216	−1.90726	−0.953631	−	0.300978i	$-0.902687\pi$
$613$	−47.2216	−1.90726	−0.953631	+	0.300978i	$0.902687\pi$
$614$	−17.5359	−0.707693
$615$	0	0
$616$	−2.37591	−0.0957281
$617$	−6.12983	−0.246778	−0.123389	−	0.992358i	$-0.539376\pi$
$617$	−6.12983	−0.246778	−0.123389	+	0.992358i	$0.539376\pi$
$618$	0	0
$619$	−24.5315	−0.986003	−0.493002	−	0.870028i	$-0.664100\pi$
$619$	−24.5315	−0.986003	−0.493002	+	0.870028i	$0.664100\pi$
$620$	−47.6681	−1.91440
$621$	0	0
$622$	−13.1580	−0.527586
$623$	5.89993	0.236376
$624$	0	0
$625$	7.44790	0.297916
$626$	7.54873	0.301708
$627$	0	0
$628$	−29.5622	−1.17966
$629$	41.5726	1.65761
$630$	0	0
$631$	−3.62627	−0.144360	−0.0721798	−	0.997392i	$-0.522996\pi$
$631$	−3.62627	−0.144360	−0.0721798	+	0.997392i	$0.522996\pi$
$632$	34.9075	1.38855
$633$	0	0
$634$	10.3352	0.410462
$635$	3.70415	0.146995
$636$	0	0
$637$	3.77620	0.149619
$638$	−3.77502	−0.149455
$639$	0	0
$640$	40.8066	1.61302
$641$	−39.9869	−1.57939	−0.789693	−	0.613502i	$-0.789760\pi$
$641$	−39.9869	−1.57939	−0.789693	+	0.613502i	$0.789760\pi$
$642$	0	0
$643$	−28.0038	−1.10436	−0.552181	−	0.833724i	$-0.686204\pi$
$643$	−28.0038	−1.10436	−0.552181	+	0.833724i	$0.686204\pi$
$644$	0.688694	0.0271383
$645$	0	0
$646$	−22.8407	−0.898653
$647$	−7.74784	−0.304599	−0.152299	−	0.988334i	$-0.548668\pi$
$647$	−7.74784	−0.304599	−0.152299	+	0.988334i	$0.548668\pi$
$648$	0	0
$649$	−10.0962	−0.396310
$650$	−21.9108	−0.859414
$651$	0	0
$652$	34.7177	1.35965
$653$	−4.00319	−0.156657	−0.0783284	−	0.996928i	$-0.524958\pi$
$653$	−4.00319	−0.156657	−0.0783284	+	0.996928i	$0.524958\pi$
$654$	0	0
$655$	72.3933	2.82864
$656$	9.17324	0.358155
$657$	0	0
$658$	2.11485	0.0824453
$659$	25.1729	0.980598	0.490299	−	0.871554i	$-0.336887\pi$
$659$	25.1729	0.980598	0.490299	+	0.871554i	$0.336887\pi$
$660$	0	0
$661$	−16.8774	−0.656453	−0.328227	−	0.944599i	$-0.606451\pi$
$661$	−16.8774	−0.656453	−0.328227	+	0.944599i	$0.606451\pi$
$662$	−1.72281	−0.0669587
$663$	0	0
$664$	8.35129	0.324093
$665$	−31.2640	−1.21237
$666$	0	0
$667$	2.49955	0.0967830
$668$	−2.45693	−0.0950615
$669$	0	0
$670$	−9.75770	−0.376973
$671$	−3.58857	−0.138535
$672$	0	0
$673$	−13.8992	−0.535776	−0.267888	−	0.963450i	$-0.586326\pi$
$673$	−13.8992	−0.535776	−0.267888	+	0.963450i	$0.586326\pi$
$674$	−19.5689	−0.753764
$675$	0	0
$676$	−1.96178	−0.0754531
$677$	17.9526	0.689973	0.344986	−	0.938608i	$-0.387884\pi$
$677$	17.9526	0.689973	0.344986	+	0.938608i	$0.387884\pi$
$678$	0	0
$679$	−6.24359	−0.239607
$680$	−35.6586	−1.36745
$681$	0	0
$682$	5.51914	0.211339
$683$	5.72385	0.219017	0.109508	−	0.993986i	$-0.465072\pi$
$683$	5.72385	0.219017	0.109508	+	0.993986i	$0.465072\pi$
$684$	0	0
$685$	24.4655	0.934777
$686$	−0.665348	−0.0254031
$687$	0	0
$688$	3.09347	0.117938
$689$	1.37251	0.0522883
$690$	0	0
$691$	12.8512	0.488884	0.244442	−	0.969664i	$-0.421395\pi$
$691$	12.8512	0.488884	0.244442	+	0.969664i	$0.421395\pi$
$692$	−6.40662	−0.243543
$693$	0	0
$694$	−2.69854	−0.102435
$695$	−20.8772	−0.791917
$696$	0	0
$697$	−24.2298	−0.917770
$698$	15.9414	0.603390
$699$	0	0
$700$	−13.5810	−0.513312
$701$	−44.1253	−1.66659	−0.833295	−	0.552829i	$-0.813548\pi$
$701$	−44.1253	−1.66659	−0.833295	+	0.552829i	$0.813548\pi$
$702$	0	0
$703$	86.2697	3.25372
$704$	−0.754415	−0.0284331
$705$	0	0
$706$	−13.4883	−0.507640
$707$	18.8071	0.707314
$708$	0	0
$709$	−5.03792	−0.189203	−0.0946016	−	0.995515i	$-0.530158\pi$
$709$	−5.03792	−0.189203	−0.0946016	+	0.995515i	$0.530158\pi$
$710$	−26.3343	−0.988309
$711$	0	0
$712$	13.9643	0.523333
$713$	−3.65438	−0.136858
$714$	0	0
$715$	−14.0412	−0.525111
$716$	9.01028	0.336730
$717$	0	0
$718$	19.3238	0.721156
$719$	23.6695	0.882724	0.441362	−	0.897329i	$-0.354495\pi$
$719$	23.6695	0.882724	0.441362	+	0.897329i	$0.354495\pi$
$720$	0	0
$721$	11.1061	0.413612
$722$	−34.7563	−1.29350
$723$	0	0
$724$	10.1900	0.378708
$725$	−49.2908	−1.83062
$726$	0	0
$727$	43.0170	1.59541	0.797707	−	0.603045i	$-0.206046\pi$
$727$	43.0170	1.59541	0.797707	+	0.603045i	$0.206046\pi$
$728$	8.93771	0.331254
$729$	0	0
$730$	33.4229	1.23704
$731$	−8.17097	−0.302214
$732$	0	0
$733$	19.0267	0.702768	0.351384	−	0.936231i	$-0.385711\pi$
$733$	19.0267	0.702768	0.351384	+	0.936231i	$0.385711\pi$
$734$	−4.05635	−0.149723
$735$	0	0
$736$	2.54648	0.0938644
$737$	−3.97437	−0.146398
$738$	0	0
$739$	24.5099	0.901609	0.450805	−	0.892623i	$-0.351137\pi$
$739$	24.5099	0.901609	0.450805	+	0.892623i	$0.351137\pi$
$740$	58.9614	2.16746
$741$	0	0
$742$	−0.241829	−0.00887781
$743$	46.4668	1.70470	0.852350	−	0.522971i	$-0.175177\pi$
$743$	46.4668	1.70470	0.852350	+	0.522971i	$0.175177\pi$
$744$	0	0
$745$	−11.5655	−0.423729
$746$	18.1921	0.666062
$747$	0	0
$748$	−6.35827	−0.232482
$749$	0.580119	0.0211971
$750$	0	0
$751$	−52.6553	−1.92142	−0.960709	−	0.277556i	$-0.910476\pi$
$751$	−52.6553	−1.92142	−0.960709	+	0.277556i	$0.910476\pi$
$752$	−4.89448	−0.178483
$753$	0	0
$754$	14.2009	0.517166
$755$	50.9012	1.85249
$756$	0	0
$757$	−22.1215	−0.804019	−0.402009	−	0.915636i	$-0.631688\pi$
$757$	−22.1215	−0.804019	−0.402009	+	0.915636i	$0.631688\pi$
$758$	−6.73623	−0.244671
$759$	0	0
$760$	−73.9972	−2.68416
$761$	−31.7423	−1.15066	−0.575329	−	0.817922i	$-0.695126\pi$
$761$	−31.7423	−1.15066	−0.575329	+	0.817922i	$0.695126\pi$
$762$	0	0
$763$	17.5353	0.634820
$764$	−12.0044	−0.434303
$765$	0	0
$766$	2.27718	0.0822778
$767$	37.9799	1.37137
$768$	0	0
$769$	22.2299	0.801630	0.400815	−	0.916159i	$-0.368727\pi$
$769$	22.2299	0.801630	0.400815	+	0.916159i	$0.368727\pi$
$770$	2.47399	0.0891562
$771$	0	0
$772$	−41.8517	−1.50627
$773$	5.30786	0.190910	0.0954552	−	0.995434i	$-0.469569\pi$
$773$	5.30786	0.190910	0.0954552	+	0.995434i	$0.469569\pi$
$774$	0	0
$775$	72.0639	2.58861
$776$	−14.7776	−0.530487
$777$	0	0
$778$	−11.4506	−0.410525
$779$	−50.2807	−1.80149
$780$	0	0
$781$	−10.7261	−0.383811
$782$	−1.19675	−0.0427957
$783$	0	0
$784$	1.53985	0.0549945
$785$	70.3154	2.50966
$786$	0	0
$787$	25.0906	0.894383	0.447191	−	0.894438i	$-0.352424\pi$
$787$	25.0906	0.894383	0.447191	+	0.894438i	$0.352424\pi$
$788$	14.2969	0.509306
$789$	0	0
$790$	−36.3485	−1.29322
$791$	19.4000	0.689783
$792$	0	0
$793$	13.4995	0.479381
$794$	14.6025	0.518222
$795$	0	0
$796$	24.3333	0.862471
$797$	−19.1556	−0.678528	−0.339264	−	0.940691i	$-0.610178\pi$
$797$	−19.1556	−0.678528	−0.339264	+	0.940691i	$0.610178\pi$
$798$	0	0
$799$	12.9281	0.457363
$800$	−50.2162	−1.77541
$801$	0	0
$802$	−12.2800	−0.433621
$803$	13.6134	0.480406
$804$	0	0
$805$	−1.63810	−0.0577353
$806$	−20.7619	−0.731307
$807$	0	0
$808$	44.5136	1.56598
$809$	−3.57506	−0.125692	−0.0628462	−	0.998023i	$-0.520018\pi$
$809$	−3.57506	−0.125692	−0.0628462	+	0.998023i	$0.520018\pi$
$810$	0	0
$811$	7.26875	0.255240	0.127620	−	0.991823i	$-0.459266\pi$
$811$	7.26875	0.255240	0.127620	+	0.991823i	$0.459266\pi$
$812$	8.80212	0.308894
$813$	0	0
$814$	−6.82670	−0.239276
$815$	−82.5780	−2.89258
$816$	0	0
$817$	−16.9561	−0.593217
$818$	7.84228	0.274199
$819$	0	0
$820$	−34.3645	−1.20006
$821$	50.6189	1.76661	0.883306	−	0.468797i	$-0.155312\pi$
$821$	50.6189	1.76661	0.883306	+	0.468797i	$0.155312\pi$
$822$	0	0
$823$	−23.9420	−0.834565	−0.417282	−	0.908777i	$-0.637017\pi$
$823$	−23.9420	−0.834565	−0.417282	+	0.908777i	$0.637017\pi$
$824$	26.2864	0.915731
$825$	0	0
$826$	−6.69186	−0.232840
$827$	−6.33527	−0.220299	−0.110150	−	0.993915i	$-0.535133\pi$
$827$	−6.33527	−0.220299	−0.110150	+	0.993915i	$0.535133\pi$
$828$	0	0
$829$	−15.8957	−0.552080	−0.276040	−	0.961146i	$-0.589022\pi$
$829$	−15.8957	−0.552080	−0.276040	+	0.961146i	$0.589022\pi$
$830$	−8.69602	−0.301843
$831$	0	0
$832$	2.83796	0.0983887
$833$	−4.06728	−0.140923
$834$	0	0
$835$	5.84395	0.202238
$836$	−13.1944	−0.456338
$837$	0	0
$838$	9.84012	0.339921
$839$	−10.1898	−0.351790	−0.175895	−	0.984409i	$-0.556282\pi$
$839$	−10.1898	−0.351790	−0.175895	+	0.984409i	$0.556282\pi$
$840$	0	0
$841$	2.94649	0.101603
$842$	9.62944	0.331852
$843$	0	0
$844$	−0.882125	−0.0303640
$845$	4.66621	0.160522
$846$	0	0
$847$	−9.99233	−0.343341
$848$	0.559675	0.0192193
$849$	0	0
$850$	23.5998	0.809465
$851$	4.52015	0.154949
$852$	0	0
$853$	33.6392	1.15178	0.575892	−	0.817526i	$-0.304655\pi$
$853$	33.6392	1.15178	0.575892	+	0.817526i	$0.304655\pi$
$854$	−2.37854	−0.0813920
$855$	0	0
$856$	1.37306	0.0469301
$857$	37.5136	1.28144	0.640721	−	0.767774i	$-0.278636\pi$
$857$	37.5136	1.28144	0.640721	+	0.767774i	$0.278636\pi$
$858$	0	0
$859$	17.2642	0.589048	0.294524	−	0.955644i	$-0.404839\pi$
$859$	17.2642	0.589048	0.294524	+	0.955644i	$0.404839\pi$
$860$	−11.5887	−0.395171
$861$	0	0
$862$	−11.3339	−0.386036
$863$	−46.7804	−1.59242	−0.796212	−	0.605017i	$-0.793166\pi$
$863$	−46.7804	−1.59242	−0.796212	+	0.605017i	$0.793166\pi$
$864$	0	0
$865$	15.2385	0.518125
$866$	4.29002	0.145781
$867$	0	0
$868$	−12.8688	−0.436796
$869$	−14.8050	−0.502224
$870$	0	0
$871$	14.9508	0.506589
$872$	41.5034	1.40548
$873$	0	0
$874$	−2.48344	−0.0840038
$875$	13.7823	0.465926
$876$	0	0
$877$	7.19645	0.243007	0.121503	−	0.992591i	$-0.461228\pi$
$877$	7.19645	0.243007	0.121503	+	0.992591i	$0.461228\pi$
$878$	1.95725	0.0660539
$879$	0	0
$880$	−5.72566	−0.193012
$881$	−20.8602	−0.702797	−0.351399	−	0.936226i	$-0.614294\pi$
$881$	−20.8602	−0.702797	−0.351399	+	0.936226i	$0.614294\pi$
$882$	0	0
$883$	26.7712	0.900922	0.450461	−	0.892796i	$-0.351260\pi$
$883$	26.7712	0.900922	0.450461	+	0.892796i	$0.351260\pi$
$884$	23.9186	0.804469
$885$	0	0
$886$	−2.83571	−0.0952675
$887$	25.5549	0.858050	0.429025	−	0.903292i	$-0.358857\pi$
$887$	25.5549	0.858050	0.429025	+	0.903292i	$0.358857\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−14.5407	−0.487405
$891$	0	0
$892$	8.28494	0.277400
$893$	26.8278	0.897758
$894$	0	0
$895$	−21.4315	−0.716374
$896$	11.0164	0.368033
$897$	0	0
$898$	−8.79274	−0.293417
$899$	−46.7062	−1.55774
$900$	0	0
$901$	−1.47830	−0.0492494
$902$	3.97881	0.132480
$903$	0	0
$904$	45.9168	1.52717
$905$	−24.2375	−0.805682
$906$	0	0
$907$	−6.56205	−0.217889	−0.108945	−	0.994048i	$-0.534747\pi$
$907$	−6.56205	−0.217889	−0.108945	+	0.994048i	$0.534747\pi$
$908$	39.8593	1.32278
$909$	0	0
$910$	−9.30665	−0.308513
$911$	−34.3146	−1.13689	−0.568447	−	0.822720i	$-0.692456\pi$
$911$	−34.3146	−1.13689	−0.568447	+	0.822720i	$0.692456\pi$
$912$	0	0
$913$	−3.54195	−0.117221
$914$	23.9116	0.790926
$915$	0	0
$916$	25.0072	0.826260
$917$	19.5438	0.645393
$918$	0	0
$919$	33.0506	1.09024	0.545119	−	0.838359i	$-0.316485\pi$
$919$	33.0506	1.09024	0.545119	+	0.838359i	$0.316485\pi$
$920$	−3.87713	−0.127825
$921$	0	0
$922$	−0.288405	−0.00949811
$923$	40.3496	1.32812
$924$	0	0
$925$	−89.1369	−2.93080
$926$	2.17926	0.0716149
$927$	0	0
$928$	32.5462	1.06838
$929$	53.9455	1.76989	0.884947	−	0.465691i	$-0.154194\pi$
$929$	53.9455	1.76989	0.884947	+	0.465691i	$0.154194\pi$
$930$	0	0
$931$	−8.44025	−0.276618
$932$	35.9138	1.17640
$933$	0	0
$934$	−14.6743	−0.480157
$935$	15.1235	0.494592
$936$	0	0
$937$	13.0572	0.426559	0.213279	−	0.976991i	$-0.431586\pi$
$937$	13.0572	0.426559	0.213279	+	0.976991i	$0.431586\pi$
$938$	−2.63426	−0.0860115
$939$	0	0
$940$	18.3356	0.598041
$941$	3.15078	0.102713	0.0513563	−	0.998680i	$-0.483646\pi$
$941$	3.15078	0.102713	0.0513563	+	0.998680i	$0.483646\pi$
$942$	0	0
$943$	−2.63449	−0.0857907
$944$	15.4873	0.504068
$945$	0	0
$946$	1.34177	0.0436246
$947$	−46.2541	−1.50306	−0.751528	−	0.659702i	$-0.770683\pi$
$947$	−46.2541	−1.50306	−0.751528	+	0.659702i	$0.770683\pi$
$948$	0	0
$949$	−51.2109	−1.66238
$950$	48.9732	1.58890
$951$	0	0
$952$	−9.62665	−0.312002
$953$	13.1147	0.424826	0.212413	−	0.977180i	$-0.431868\pi$
$953$	13.1147	0.424826	0.212413	+	0.977180i	$0.431868\pi$
$954$	0	0
$955$	28.5531	0.923956
$956$	−33.8393	−1.09444
$957$	0	0
$958$	−1.94022	−0.0626855
$959$	6.60487	0.213282
$960$	0	0
$961$	37.2851	1.20275
$962$	25.6807	0.827980
$963$	0	0
$964$	−20.9806	−0.675741
$965$	99.5465	3.20452
$966$	0	0
$967$	31.4093	1.01005	0.505027	−	0.863103i	$-0.331482\pi$
$967$	31.4093	1.01005	0.505027	+	0.863103i	$0.331482\pi$
$968$	−23.6503	−0.760151
$969$	0	0
$970$	15.3877	0.494068
$971$	−37.6772	−1.20912	−0.604560	−	0.796560i	$-0.706651\pi$
$971$	−37.6772	−1.20912	−0.604560	+	0.796560i	$0.706651\pi$
$972$	0	0
$973$	−5.63615	−0.180687
$974$	−15.4377	−0.494656
$975$	0	0
$976$	5.50477	0.176203
$977$	−27.2703	−0.872454	−0.436227	−	0.899837i	$-0.643686\pi$
$977$	−27.2703	−0.872454	−0.436227	+	0.899837i	$0.643686\pi$
$978$	0	0
$979$	−5.92251	−0.189284
$980$	−5.76853	−0.184269
$981$	0	0
$982$	22.7997	0.727567
$983$	−39.9820	−1.27523	−0.637614	−	0.770356i	$-0.720079\pi$
$983$	−39.9820	−1.27523	−0.637614	+	0.770356i	$0.720079\pi$
$984$	0	0
$985$	−34.0060	−1.08352
$986$	−15.2955	−0.487109
$987$	0	0
$988$	49.6348	1.57909
$989$	−0.888423	−0.0282502
$990$	0	0
$991$	34.6355	1.10023	0.550116	−	0.835088i	$-0.314584\pi$
$991$	34.6355	1.10023	0.550116	+	0.835088i	$0.314584\pi$
$992$	−47.5831	−1.51076
$993$	0	0
$994$	−7.10939	−0.225496
$995$	−57.8782	−1.83486
$996$	0	0
$997$	−15.7082	−0.497483	−0.248742	−	0.968570i	$-0.580017\pi$
$997$	−15.7082	−0.497483	−0.248742	+	0.968570i	$0.580017\pi$
$998$	18.9913	0.601160
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.6		18
3.2	odd	2		2667.2.a.p.1.13	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.13	✓	18	3.2	odd	2
8001.2.a.u.1.6		18	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)

\(f(q)\)	\(=\)	\(q-0.665348 q^{2} -1.55731 q^{4} +3.70415 q^{5} +1.00000 q^{7} +2.36685 q^{8} +O(q^{10})\)	\(q-0.665348 q^{2} -1.55731 q^{4} +3.70415 q^{5} +1.00000 q^{7} +2.36685 q^{8} -2.46455 q^{10} -1.00383 q^{11} +3.77620 q^{13} -0.665348 q^{14} +1.53985 q^{16} -4.06728 q^{17} -8.44025 q^{19} -5.76853 q^{20} +0.667895 q^{22} -0.442232 q^{23} +8.72076 q^{25} -2.51249 q^{26} -1.55731 q^{28} -5.65212 q^{29} +8.26348 q^{31} -5.75823 q^{32} +2.70616 q^{34} +3.70415 q^{35} -10.2212 q^{37} +5.61570 q^{38} +8.76718 q^{40} +5.95725 q^{41} +2.00895 q^{43} +1.56327 q^{44} +0.294238 q^{46} -3.17856 q^{47} +1.00000 q^{49} -5.80234 q^{50} -5.88073 q^{52} +0.363462 q^{53} -3.71833 q^{55} +2.36685 q^{56} +3.76063 q^{58} +10.0577 q^{59} +3.57488 q^{61} -5.49809 q^{62} +0.751538 q^{64} +13.9876 q^{65} +3.95922 q^{67} +6.33403 q^{68} -2.46455 q^{70} +10.6852 q^{71} -13.5615 q^{73} +6.80067 q^{74} +13.1441 q^{76} -1.00383 q^{77} +14.7485 q^{79} +5.70383 q^{80} -3.96364 q^{82} +3.52844 q^{83} -15.0658 q^{85} -1.33665 q^{86} -2.37591 q^{88} +5.89993 q^{89} +3.77620 q^{91} +0.688694 q^{92} +2.11485 q^{94} -31.2640 q^{95} -6.24359 q^{97} -0.665348 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

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Database

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