LMFDB - Embedded newform 8001.2.a.s.1.3

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:	$16$
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} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.45059$ 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)

$f(q)$	$=$	$q-2.45059 q^{2} +4.00539 q^{4} +2.70231 q^{5} -1.00000 q^{7} -4.91439 q^{8} +O(q^{10})$	\(q-2.45059 q^{2} +4.00539 q^{4} +2.70231 q^{5} -1.00000 q^{7} -4.91439 q^{8} -6.62224 q^{10} -3.42638 q^{11} +0.0836936 q^{13} +2.45059 q^{14} +4.03237 q^{16} -2.76183 q^{17} +2.02842 q^{19} +10.8238 q^{20} +8.39666 q^{22} -8.87114 q^{23} +2.30245 q^{25} -0.205099 q^{26} -4.00539 q^{28} -6.99655 q^{29} +3.05039 q^{31} -0.0529059 q^{32} +6.76810 q^{34} -2.70231 q^{35} -2.92280 q^{37} -4.97083 q^{38} -13.2802 q^{40} +2.94777 q^{41} +4.21313 q^{43} -13.7240 q^{44} +21.7395 q^{46} -4.16631 q^{47} +1.00000 q^{49} -5.64237 q^{50} +0.335225 q^{52} +4.44373 q^{53} -9.25913 q^{55} +4.91439 q^{56} +17.1457 q^{58} +6.96022 q^{59} -1.97387 q^{61} -7.47525 q^{62} -7.93509 q^{64} +0.226166 q^{65} +2.82114 q^{67} -11.0622 q^{68} +6.62224 q^{70} -4.92396 q^{71} -10.4342 q^{73} +7.16258 q^{74} +8.12462 q^{76} +3.42638 q^{77} -2.96655 q^{79} +10.8967 q^{80} -7.22379 q^{82} +0.158273 q^{83} -7.46330 q^{85} -10.3246 q^{86} +16.8386 q^{88} +13.7012 q^{89} -0.0836936 q^{91} -35.5324 q^{92} +10.2099 q^{94} +5.48141 q^{95} -5.43065 q^{97} -2.45059 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * 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 $

$2$	−2.45059	−1.73283	−0.866414	−	0.499326i	$-0.833581\pi$
$2$	−2.45059	−1.73283	−0.866414	+	0.499326i	$0.833581\pi$
$3$	0	0
$3$	0	0
$4$	4.00539	2.00270
$5$	2.70231	1.20851	0.604254	−	0.796792i	$-0.293471\pi$
$5$	2.70231	1.20851	0.604254	+	0.796792i	$0.293471\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−4.91439	−1.73750
$9$	0	0
$10$	−6.62224	−2.09414
$11$	−3.42638	−1.03309	−0.516547	−	0.856259i	$-0.672783\pi$
$11$	−3.42638	−1.03309	−0.516547	+	0.856259i	$0.672783\pi$
$12$	0	0
$13$	0.0836936	0.0232124	0.0116062	−	0.999933i	$-0.496306\pi$
$13$	0.0836936	0.0232124	0.0116062	+	0.999933i	$0.496306\pi$
$14$	2.45059	0.654948
$15$	0	0
$16$	4.03237	1.00809
$17$	−2.76183	−0.669841	−0.334921	−	0.942246i	$-0.608709\pi$
$17$	−2.76183	−0.669841	−0.334921	+	0.942246i	$0.608709\pi$
$18$	0	0
$19$	2.02842	0.465352	0.232676	−	0.972554i	$-0.425252\pi$
$19$	2.02842	0.465352	0.232676	+	0.972554i	$0.425252\pi$
$20$	10.8238	2.42027
$21$	0	0
$22$	8.39666	1.79017
$23$	−8.87114	−1.84976	−0.924881	−	0.380257i	$-0.875835\pi$
$23$	−8.87114	−1.84976	−0.924881	+	0.380257i	$0.875835\pi$
$24$	0	0
$25$	2.30245	0.460491
$26$	−0.205099	−0.0402232
$27$	0	0
$28$	−4.00539	−0.756948
$29$	−6.99655	−1.29923	−0.649613	−	0.760265i	$-0.725069\pi$
$29$	−6.99655	−1.29923	−0.649613	+	0.760265i	$0.725069\pi$
$30$	0	0
$31$	3.05039	0.547866	0.273933	−	0.961749i	$-0.411675\pi$
$31$	3.05039	0.547866	0.273933	+	0.961749i	$0.411675\pi$
$32$	−0.0529059	−0.00935253
$33$	0	0
$34$	6.76810	1.16072
$35$	−2.70231	−0.456773
$36$	0	0
$37$	−2.92280	−0.480505	−0.240252	−	0.970710i	$-0.577230\pi$
$37$	−2.92280	−0.480505	−0.240252	+	0.970710i	$0.577230\pi$
$38$	−4.97083	−0.806375
$39$	0	0
$40$	−13.2802	−2.09978
$41$	2.94777	0.460365	0.230183	−	0.973147i	$-0.426068\pi$
$41$	2.94777	0.460365	0.230183	+	0.973147i	$0.426068\pi$
$42$	0	0
$43$	4.21313	0.642496	0.321248	−	0.946995i	$-0.395898\pi$
$43$	4.21313	0.642496	0.321248	+	0.946995i	$0.395898\pi$
$44$	−13.7240	−2.06897
$45$	0	0
$46$	21.7395	3.20532
$47$	−4.16631	−0.607719	−0.303859	−	0.952717i	$-0.598275\pi$
$47$	−4.16631	−0.607719	−0.303859	+	0.952717i	$0.598275\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.64237	−0.797952
$51$	0	0
$52$	0.335225	0.0464874
$53$	4.44373	0.610393	0.305197	−	0.952289i	$-0.401278\pi$
$53$	4.44373	0.610393	0.305197	+	0.952289i	$0.401278\pi$
$54$	0	0
$55$	−9.25913	−1.24850
$56$	4.91439	0.656713
$57$	0	0
$58$	17.1457	2.25134
$59$	6.96022	0.906144	0.453072	−	0.891474i	$-0.350328\pi$
$59$	6.96022	0.906144	0.453072	+	0.891474i	$0.350328\pi$
$60$	0	0
$61$	−1.97387	−0.252729	−0.126364	−	0.991984i	$-0.540331\pi$
$61$	−1.97387	−0.252729	−0.126364	+	0.991984i	$0.540331\pi$
$62$	−7.47525	−0.949358
$63$	0	0
$64$	−7.93509	−0.991886
$65$	0.226166	0.0280524
$66$	0	0
$67$	2.82114	0.344657	0.172328	−	0.985040i	$-0.444871\pi$
$67$	2.82114	0.344657	0.172328	+	0.985040i	$0.444871\pi$
$68$	−11.0622	−1.34149
$69$	0	0
$70$	6.62224	0.791509
$71$	−4.92396	−0.584367	−0.292183	−	0.956362i	$-0.594382\pi$
$71$	−4.92396	−0.584367	−0.292183	+	0.956362i	$0.594382\pi$
$72$	0	0
$73$	−10.4342	−1.22123	−0.610616	−	0.791927i	$-0.709078\pi$
$73$	−10.4342	−1.22123	−0.610616	+	0.791927i	$0.709078\pi$
$74$	7.16258	0.832633
$75$	0	0
$76$	8.12462	0.931958
$77$	3.42638	0.390473
$78$	0	0
$79$	−2.96655	−0.333763	−0.166881	−	0.985977i	$-0.553370\pi$
$79$	−2.96655	−0.333763	−0.166881	+	0.985977i	$0.553370\pi$
$80$	10.8967	1.21829
$81$	0	0
$82$	−7.22379	−0.797734
$83$	0.158273	0.0173727	0.00868635	−	0.999962i	$-0.497235\pi$
$83$	0.158273	0.0173727	0.00868635	+	0.999962i	$0.497235\pi$
$84$	0	0
$85$	−7.46330	−0.809508
$86$	−10.3246	−1.11334
$87$	0	0
$88$	16.8386	1.79500
$89$	13.7012	1.45233	0.726163	−	0.687523i	$-0.241302\pi$
$89$	13.7012	1.45233	0.726163	+	0.687523i	$0.241302\pi$
$90$	0	0
$91$	−0.0836936	−0.00877347
$92$	−35.5324	−3.70451
$93$	0	0
$94$	10.2099	1.05307
$95$	5.48141	0.562381
$96$	0	0
$97$	−5.43065	−0.551399	−0.275700	−	0.961244i	$-0.588910\pi$
$97$	−5.43065	−0.551399	−0.275700	+	0.961244i	$0.588910\pi$
$98$	−2.45059	−0.247547
$99$	0	0
$100$	9.22223	0.922223
$101$	4.30013	0.427879	0.213939	−	0.976847i	$-0.431370\pi$
$101$	4.30013	0.427879	0.213939	+	0.976847i	$0.431370\pi$
$102$	0	0
$103$	−2.58455	−0.254664	−0.127332	−	0.991860i	$-0.540641\pi$
$103$	−2.58455	−0.254664	−0.127332	+	0.991860i	$0.540641\pi$
$104$	−0.411303	−0.0403316
$105$	0	0
$106$	−10.8898	−1.05771
$107$	7.75671	0.749870	0.374935	−	0.927051i	$-0.377665\pi$
$107$	7.75671	0.749870	0.374935	+	0.927051i	$0.377665\pi$
$108$	0	0
$109$	−0.0610158	−0.00584425	−0.00292213	−	0.999996i	$-0.500930\pi$
$109$	−0.0610158	−0.00584425	−0.00292213	+	0.999996i	$0.500930\pi$
$110$	22.6903	2.16344
$111$	0	0
$112$	−4.03237	−0.381023
$113$	12.2332	1.15080	0.575399	−	0.817873i	$-0.304847\pi$
$113$	12.2332	1.15080	0.575399	+	0.817873i	$0.304847\pi$
$114$	0	0
$115$	−23.9725	−2.23545
$116$	−28.0239	−2.60196
$117$	0	0
$118$	−17.0566	−1.57019
$119$	2.76183	0.253176
$120$	0	0
$121$	0.740100	0.0672818
$122$	4.83716	0.437936
$123$	0	0
$124$	12.2180	1.09721
$125$	−7.28959	−0.652001
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	19.5515	1.72812
$129$	0	0
$130$	−0.554239	−0.0486100
$131$	22.1146	1.93216	0.966081	−	0.258238i	$-0.0831419\pi$
$131$	22.1146	1.93216	0.966081	+	0.258238i	$0.0831419\pi$
$132$	0	0
$133$	−2.02842	−0.175886
$134$	−6.91345	−0.597231
$135$	0	0
$136$	13.5727	1.16385
$137$	−14.0408	−1.19959	−0.599794	−	0.800154i	$-0.704751\pi$
$137$	−14.0408	−1.19959	−0.599794	+	0.800154i	$0.704751\pi$
$138$	0	0
$139$	19.6595	1.66750	0.833749	−	0.552143i	$-0.186190\pi$
$139$	19.6595	1.66750	0.833749	+	0.552143i	$0.186190\pi$
$140$	−10.8238	−0.914777
$141$	0	0
$142$	12.0666	1.01261
$143$	−0.286766	−0.0239806
$144$	0	0
$145$	−18.9068	−1.57013
$146$	25.5700	2.11618
$147$	0	0
$148$	−11.7069	−0.962305
$149$	−2.11694	−0.173427	−0.0867133	−	0.996233i	$-0.527636\pi$
$149$	−2.11694	−0.173427	−0.0867133	+	0.996233i	$0.527636\pi$
$150$	0	0
$151$	22.7239	1.84925	0.924624	−	0.380881i	$-0.124379\pi$
$151$	22.7239	1.84925	0.924624	+	0.380881i	$0.124379\pi$
$152$	−9.96845	−0.808548
$153$	0	0
$154$	−8.39666	−0.676622
$155$	8.24309	0.662101
$156$	0	0
$157$	0.885324	0.0706565	0.0353283	−	0.999376i	$-0.488752\pi$
$157$	0.885324	0.0706565	0.0353283	+	0.999376i	$0.488752\pi$
$158$	7.26979	0.578354
$159$	0	0
$160$	−0.142968	−0.0113026
$161$	8.87114	0.699144
$162$	0	0
$163$	12.6258	0.988932	0.494466	−	0.869197i	$-0.335364\pi$
$163$	12.6258	0.988932	0.494466	+	0.869197i	$0.335364\pi$
$164$	11.8070	0.921971
$165$	0	0
$166$	−0.387862	−0.0301039
$167$	2.83261	0.219194	0.109597	−	0.993976i	$-0.465044\pi$
$167$	2.83261	0.219194	0.109597	+	0.993976i	$0.465044\pi$
$168$	0	0
$169$	−12.9930	−0.999461
$170$	18.2895	1.40274
$171$	0	0
$172$	16.8752	1.28672
$173$	0.671811	0.0510768	0.0255384	−	0.999674i	$-0.491870\pi$
$173$	0.671811	0.0510768	0.0255384	+	0.999674i	$0.491870\pi$
$174$	0	0
$175$	−2.30245	−0.174049
$176$	−13.8164	−1.04145
$177$	0	0
$178$	−33.5760	−2.51663
$179$	14.4098	1.07704	0.538520	−	0.842613i	$-0.318984\pi$
$179$	14.4098	1.07704	0.538520	+	0.842613i	$0.318984\pi$
$180$	0	0
$181$	9.68040	0.719538	0.359769	−	0.933041i	$-0.382855\pi$
$181$	9.68040	0.719538	0.359769	+	0.933041i	$0.382855\pi$
$182$	0.205099	0.0152029
$183$	0	0
$184$	43.5962	3.21396
$185$	−7.89829	−0.580694
$186$	0	0
$187$	9.46307	0.692008
$188$	−16.6877	−1.21708
$189$	0	0
$190$	−13.4327	−0.974510
$191$	14.5340	1.05165	0.525823	−	0.850594i	$-0.323757\pi$
$191$	14.5340	1.05165	0.525823	+	0.850594i	$0.323757\pi$
$192$	0	0
$193$	16.7131	1.20304	0.601519	−	0.798859i	$-0.294563\pi$
$193$	16.7131	1.20304	0.601519	+	0.798859i	$0.294563\pi$
$194$	13.3083	0.955480
$195$	0	0
$196$	4.00539	0.286099
$197$	16.3336	1.16372	0.581860	−	0.813289i	$-0.302325\pi$
$197$	16.3336	1.16372	0.581860	+	0.813289i	$0.302325\pi$
$198$	0	0
$199$	9.49240	0.672898	0.336449	−	0.941702i	$-0.390774\pi$
$199$	9.49240	0.672898	0.336449	+	0.941702i	$0.390774\pi$
$200$	−11.3152	−0.800102
$201$	0	0
$202$	−10.5379	−0.741441
$203$	6.99655	0.491062
$204$	0	0
$205$	7.96579	0.556355
$206$	6.33368	0.441288
$207$	0	0
$208$	0.337483	0.0234003
$209$	−6.95015	−0.480752
$210$	0	0
$211$	15.0877	1.03868	0.519340	−	0.854568i	$-0.326178\pi$
$211$	15.0877	1.03868	0.519340	+	0.854568i	$0.326178\pi$
$212$	17.7989	1.22243
$213$	0	0
$214$	−19.0085	−1.29940
$215$	11.3852	0.776462
$216$	0	0
$217$	−3.05039	−0.207074
$218$	0.149525	0.0101271
$219$	0	0
$220$	−37.0864	−2.50037
$221$	−0.231147	−0.0155486
$222$	0	0
$223$	27.4243	1.83647	0.918234	−	0.396038i	$-0.129615\pi$
$223$	27.4243	1.83647	0.918234	+	0.396038i	$0.129615\pi$
$224$	0.0529059	0.00353493
$225$	0	0
$226$	−29.9784	−1.99414
$227$	0.168916	0.0112114	0.00560568	−	0.999984i	$-0.498216\pi$
$227$	0.168916	0.0112114	0.00560568	+	0.999984i	$0.498216\pi$
$228$	0	0
$229$	−11.6864	−0.772258	−0.386129	−	0.922445i	$-0.626188\pi$
$229$	−11.6864	−0.772258	−0.386129	+	0.922445i	$0.626188\pi$
$230$	58.7469	3.87365
$231$	0	0
$232$	34.3838	2.25740
$233$	0.787277	0.0515762	0.0257881	−	0.999667i	$-0.491790\pi$
$233$	0.787277	0.0515762	0.0257881	+	0.999667i	$0.491790\pi$
$234$	0	0
$235$	−11.2586	−0.734433
$236$	27.8784	1.81473
$237$	0	0
$238$	−6.76810	−0.438711
$239$	7.85461	0.508072	0.254036	−	0.967195i	$-0.418242\pi$
$239$	7.85461	0.508072	0.254036	+	0.967195i	$0.418242\pi$
$240$	0	0
$241$	−14.1811	−0.913484	−0.456742	−	0.889599i	$-0.650984\pi$
$241$	−14.1811	−0.913484	−0.456742	+	0.889599i	$0.650984\pi$
$242$	−1.81368	−0.116588
$243$	0	0
$244$	−7.90614	−0.506139
$245$	2.70231	0.172644
$246$	0	0
$247$	0.169766	0.0108019
$248$	−14.9908	−0.951917
$249$	0	0
$250$	17.8638	1.12981
$251$	−23.1623	−1.46199	−0.730997	−	0.682381i	$-0.760945\pi$
$251$	−23.1623	−1.46199	−0.730997	+	0.682381i	$0.760945\pi$
$252$	0	0
$253$	30.3959	1.91098
$254$	2.45059	0.153764
$255$	0	0
$256$	−32.0424	−2.00265
$257$	18.0285	1.12459	0.562295	−	0.826937i	$-0.309919\pi$
$257$	18.0285	1.12459	0.562295	+	0.826937i	$0.309919\pi$
$258$	0	0
$259$	2.92280	0.181614
$260$	0.905882	0.0561804
$261$	0	0
$262$	−54.1938	−3.34811
$263$	21.4369	1.32186	0.660929	−	0.750448i	$-0.270162\pi$
$263$	21.4369	1.32186	0.660929	+	0.750448i	$0.270162\pi$
$264$	0	0
$265$	12.0083	0.737665
$266$	4.97083	0.304781
$267$	0	0
$268$	11.2998	0.690243
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	−14.7956	−0.898766	−0.449383	−	0.893339i	$-0.648356\pi$
$271$	−14.7956	−0.898766	−0.449383	+	0.893339i	$0.648356\pi$
$272$	−11.1367	−0.675262
$273$	0	0
$274$	34.4083	2.07868
$275$	−7.88909	−0.475730
$276$	0	0
$277$	29.5965	1.77828	0.889140	−	0.457636i	$-0.151304\pi$
$277$	29.5965	1.77828	0.889140	+	0.457636i	$0.151304\pi$
$278$	−48.1774	−2.88949
$279$	0	0
$280$	13.2802	0.793642
$281$	−5.69309	−0.339621	−0.169811	−	0.985477i	$-0.554316\pi$
$281$	−5.69309	−0.339621	−0.169811	+	0.985477i	$0.554316\pi$
$282$	0	0
$283$	−11.9040	−0.707621	−0.353810	−	0.935317i	$-0.615114\pi$
$283$	−11.9040	−0.707621	−0.353810	+	0.935317i	$0.615114\pi$
$284$	−19.7224	−1.17031
$285$	0	0
$286$	0.702746	0.0415543
$287$	−2.94777	−0.174002
$288$	0	0
$289$	−9.37232	−0.551313
$290$	46.3329	2.72076
$291$	0	0
$292$	−41.7931	−2.44575
$293$	1.48566	0.0867931	0.0433966	−	0.999058i	$-0.486182\pi$
$293$	1.48566	0.0867931	0.0433966	+	0.999058i	$0.486182\pi$
$294$	0	0
$295$	18.8086	1.09508
$296$	14.3638	0.834876
$297$	0	0
$298$	5.18775	0.300519
$299$	−0.742458	−0.0429374
$300$	0	0
$301$	−4.21313	−0.242841
$302$	−55.6870	−3.20443
$303$	0	0
$304$	8.17934	0.469118
$305$	−5.33401	−0.305425
$306$	0	0
$307$	−14.7553	−0.842129	−0.421065	−	0.907031i	$-0.638343\pi$
$307$	−14.7553	−0.842129	−0.421065	+	0.907031i	$0.638343\pi$
$308$	13.7240	0.781997
$309$	0	0
$310$	−20.2004	−1.14731
$311$	−12.3354	−0.699479	−0.349740	−	0.936847i	$-0.613730\pi$
$311$	−12.3354	−0.699479	−0.349740	+	0.936847i	$0.613730\pi$
$312$	0	0
$313$	−28.0987	−1.58823	−0.794117	−	0.607765i	$-0.792066\pi$
$313$	−28.0987	−1.58823	−0.794117	+	0.607765i	$0.792066\pi$
$314$	−2.16957	−0.122436
$315$	0	0
$316$	−11.8822	−0.668425
$317$	20.7025	1.16277	0.581383	−	0.813630i	$-0.302512\pi$
$317$	20.7025	1.16277	0.581383	+	0.813630i	$0.302512\pi$
$318$	0	0
$319$	23.9729	1.34222
$320$	−21.4430	−1.19870
$321$	0	0
$322$	−21.7395	−1.21150
$323$	−5.60215	−0.311712
$324$	0	0
$325$	0.192701	0.0106891
$326$	−30.9408	−1.71365
$327$	0	0
$328$	−14.4865	−0.799884
$329$	4.16631	0.229696
$330$	0	0
$331$	13.1837	0.724644	0.362322	−	0.932053i	$-0.381984\pi$
$331$	13.1837	0.724644	0.362322	+	0.932053i	$0.381984\pi$
$332$	0.633944	0.0347922
$333$	0	0
$334$	−6.94157	−0.379826
$335$	7.62358	0.416521
$336$	0	0
$337$	−23.5606	−1.28343	−0.641714	−	0.766944i	$-0.721776\pi$
$337$	−23.5606	−1.28343	−0.641714	+	0.766944i	$0.721776\pi$
$338$	31.8405	1.73189
$339$	0	0
$340$	−29.8934	−1.62120
$341$	−10.4518	−0.565997
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−20.7049	−1.11634
$345$	0	0
$346$	−1.64633	−0.0885074
$347$	11.8299	0.635062	0.317531	−	0.948248i	$-0.397146\pi$
$347$	11.8299	0.635062	0.317531	+	0.948248i	$0.397146\pi$
$348$	0	0
$349$	−1.12684	−0.0603183	−0.0301592	−	0.999545i	$-0.509601\pi$
$349$	−1.12684	−0.0603183	−0.0301592	+	0.999545i	$0.509601\pi$
$350$	5.64237	0.301597
$351$	0	0
$352$	0.181276	0.00966204
$353$	−23.7282	−1.26292	−0.631461	−	0.775407i	$-0.717545\pi$
$353$	−23.7282	−1.26292	−0.631461	+	0.775407i	$0.717545\pi$
$354$	0	0
$355$	−13.3061	−0.706212
$356$	54.8787	2.90856
$357$	0	0
$358$	−35.3126	−1.86633
$359$	−17.8988	−0.944663	−0.472331	−	0.881421i	$-0.656587\pi$
$359$	−17.8988	−0.944663	−0.472331	+	0.881421i	$0.656587\pi$
$360$	0	0
$361$	−14.8855	−0.783448
$362$	−23.7227	−1.24684
$363$	0	0
$364$	−0.335225	−0.0175706
$365$	−28.1964	−1.47587
$366$	0	0
$367$	2.49468	0.130221	0.0651107	−	0.997878i	$-0.479260\pi$
$367$	2.49468	0.130221	0.0651107	+	0.997878i	$0.479260\pi$
$368$	−35.7717	−1.86473
$369$	0	0
$370$	19.3555	1.00624
$371$	−4.44373	−0.230707
$372$	0	0
$373$	−5.50561	−0.285070	−0.142535	−	0.989790i	$-0.545525\pi$
$373$	−5.50561	−0.285070	−0.142535	+	0.989790i	$0.545525\pi$
$374$	−23.1901	−1.19913
$375$	0	0
$376$	20.4749	1.05591
$377$	−0.585566	−0.0301582
$378$	0	0
$379$	−13.7392	−0.705734	−0.352867	−	0.935674i	$-0.614793\pi$
$379$	−13.7392	−0.705734	−0.352867	+	0.935674i	$0.614793\pi$
$380$	21.9552	1.12628
$381$	0	0
$382$	−35.6170	−1.82232
$383$	−19.1520	−0.978620	−0.489310	−	0.872110i	$-0.662751\pi$
$383$	−19.1520	−0.978620	−0.489310	+	0.872110i	$0.662751\pi$
$384$	0	0
$385$	9.25913	0.471889
$386$	−40.9570	−2.08466
$387$	0	0
$388$	−21.7519	−1.10428
$389$	16.5935	0.841325	0.420662	−	0.907217i	$-0.361798\pi$
$389$	16.5935	0.841325	0.420662	+	0.907217i	$0.361798\pi$
$390$	0	0
$391$	24.5006	1.23905
$392$	−4.91439	−0.248214
$393$	0	0
$394$	−40.0269	−2.01653
$395$	−8.01652	−0.403355
$396$	0	0
$397$	24.1336	1.21123	0.605616	−	0.795757i	$-0.292927\pi$
$397$	24.1336	1.21123	0.605616	+	0.795757i	$0.292927\pi$
$398$	−23.2620	−1.16602
$399$	0	0
$400$	9.28435	0.464217
$401$	0.596275	0.0297765	0.0148883	−	0.999889i	$-0.495261\pi$
$401$	0.596275	0.0297765	0.0148883	+	0.999889i	$0.495261\pi$
$402$	0	0
$403$	0.255298	0.0127173
$404$	17.2237	0.856911
$405$	0	0
$406$	−17.1457	−0.850926
$407$	10.0146	0.496406
$408$	0	0
$409$	34.1836	1.69027	0.845136	−	0.534551i	$-0.179519\pi$
$409$	34.1836	1.69027	0.845136	+	0.534551i	$0.179519\pi$
$410$	−19.5209	−0.964067
$411$	0	0
$412$	−10.3521	−0.510014
$413$	−6.96022	−0.342490
$414$	0	0
$415$	0.427702	0.0209950
$416$	−0.00442789	−0.000217095 0
$417$	0	0
$418$	17.0320	0.833061
$419$	14.9093	0.728367	0.364183	−	0.931327i	$-0.381348\pi$
$419$	14.9093	0.728367	0.364183	+	0.931327i	$0.381348\pi$
$420$	0	0
$421$	16.2341	0.791200	0.395600	−	0.918423i	$-0.370537\pi$
$421$	16.2341	0.791200	0.395600	+	0.918423i	$0.370537\pi$
$422$	−36.9738	−1.79986
$423$	0	0
$424$	−21.8382	−1.06056
$425$	−6.35898	−0.308456
$426$	0	0
$427$	1.97387	0.0955225
$428$	31.0687	1.50176
$429$	0	0
$430$	−27.9004	−1.34547
$431$	−2.63668	−0.127005	−0.0635023	−	0.997982i	$-0.520227\pi$
$431$	−2.63668	−0.127005	−0.0635023	+	0.997982i	$0.520227\pi$
$432$	0	0
$433$	−38.6820	−1.85894	−0.929468	−	0.368902i	$-0.879734\pi$
$433$	−38.6820	−1.85894	−0.929468	+	0.368902i	$0.879734\pi$
$434$	7.47525	0.358824
$435$	0	0
$436$	−0.244392	−0.0117043
$437$	−17.9944	−0.860790
$438$	0	0
$439$	24.3517	1.16224	0.581122	−	0.813816i	$-0.302614\pi$
$439$	24.3517	1.16224	0.581122	+	0.813816i	$0.302614\pi$
$440$	45.5030	2.16927
$441$	0	0
$442$	0.566447	0.0269431
$443$	−17.9259	−0.851686	−0.425843	−	0.904797i	$-0.640022\pi$
$443$	−17.9259	−0.851686	−0.425843	+	0.904797i	$0.640022\pi$
$444$	0	0
$445$	37.0249	1.75515
$446$	−67.2058	−3.18229
$447$	0	0
$448$	7.93509	0.374898
$449$	13.2455	0.625092	0.312546	−	0.949903i	$-0.398818\pi$
$449$	13.2455	0.625092	0.312546	+	0.949903i	$0.398818\pi$
$450$	0	0
$451$	−10.1002	−0.475600
$452$	48.9985	2.30470
$453$	0	0
$454$	−0.413944	−0.0194274
$455$	−0.226166	−0.0106028
$456$	0	0
$457$	−18.4683	−0.863912	−0.431956	−	0.901895i	$-0.642176\pi$
$457$	−18.4683	−0.863912	−0.431956	+	0.901895i	$0.642176\pi$
$458$	28.6385	1.33819
$459$	0	0
$460$	−96.0194	−4.47693
$461$	15.2020	0.708027	0.354013	−	0.935240i	$-0.384817\pi$
$461$	15.2020	0.708027	0.354013	+	0.935240i	$0.384817\pi$
$462$	0	0
$463$	−7.06945	−0.328545	−0.164273	−	0.986415i	$-0.552528\pi$
$463$	−7.06945	−0.328545	−0.164273	+	0.986415i	$0.552528\pi$
$464$	−28.2127	−1.30974
$465$	0	0
$466$	−1.92929	−0.0893728
$467$	−13.5348	−0.626315	−0.313158	−	0.949701i	$-0.601387\pi$
$467$	−13.5348	−0.626315	−0.313158	+	0.949701i	$0.601387\pi$
$468$	0	0
$469$	−2.82114	−0.130268
$470$	27.5903	1.27265
$471$	0	0
$472$	−34.2052	−1.57442
$473$	−14.4358	−0.663758
$474$	0	0
$475$	4.67035	0.214290
$476$	11.0622	0.507035
$477$	0	0
$478$	−19.2484	−0.880402
$479$	9.01528	0.411918	0.205959	−	0.978561i	$-0.433969\pi$
$479$	9.01528	0.411918	0.205959	+	0.978561i	$0.433969\pi$
$480$	0	0
$481$	−0.244619	−0.0111537
$482$	34.7520	1.58291
$483$	0	0
$484$	2.96439	0.134745
$485$	−14.6753	−0.666370
$486$	0	0
$487$	4.57661	0.207386	0.103693	−	0.994609i	$-0.466934\pi$
$487$	4.57661	0.207386	0.103693	+	0.994609i	$0.466934\pi$
$488$	9.70039	0.439116
$489$	0	0
$490$	−6.62224	−0.299162
$491$	−21.7974	−0.983705	−0.491852	−	0.870679i	$-0.663680\pi$
$491$	−21.7974	−0.983705	−0.491852	+	0.870679i	$0.663680\pi$
$492$	0	0
$493$	19.3233	0.870276
$494$	−0.416026	−0.0187179
$495$	0	0
$496$	12.3003	0.552300
$497$	4.92396	0.220870
$498$	0	0
$499$	1.14000	0.0510335	0.0255167	−	0.999674i	$-0.491877\pi$
$499$	1.14000	0.0510335	0.0255167	+	0.999674i	$0.491877\pi$
$500$	−29.1977	−1.30576
$501$	0	0
$502$	56.7614	2.53338
$503$	−1.32624	−0.0591341	−0.0295671	−	0.999563i	$-0.509413\pi$
$503$	−1.32624	−0.0591341	−0.0295671	+	0.999563i	$0.509413\pi$
$504$	0	0
$505$	11.6203	0.517095
$506$	−74.4880	−3.31139
$507$	0	0
$508$	−4.00539	−0.177710
$509$	−21.6355	−0.958975	−0.479488	−	0.877549i	$-0.659177\pi$
$509$	−21.6355	−0.958975	−0.479488	+	0.877549i	$0.659177\pi$
$510$	0	0
$511$	10.4342	0.461582
$512$	39.4199	1.74213
$513$	0	0
$514$	−44.1806	−1.94872
$515$	−6.98425	−0.307763
$516$	0	0
$517$	14.2754	0.627830
$518$	−7.16258	−0.314706
$519$	0	0
$520$	−1.11147	−0.0487410
$521$	−3.88126	−0.170041	−0.0850205	−	0.996379i	$-0.527096\pi$
$521$	−3.88126	−0.170041	−0.0850205	+	0.996379i	$0.527096\pi$
$522$	0	0
$523$	11.8877	0.519811	0.259906	−	0.965634i	$-0.416309\pi$
$523$	11.8877	0.519811	0.259906	+	0.965634i	$0.416309\pi$
$524$	88.5776	3.86953
$525$	0	0
$526$	−52.5332	−2.29055
$527$	−8.42465	−0.366983
$528$	0	0
$529$	55.6972	2.42162
$530$	−29.4275	−1.27825
$531$	0	0
$532$	−8.12462	−0.352247
$533$	0.246710	0.0106862
$534$	0	0
$535$	20.9610	0.906224
$536$	−13.8642	−0.598841
$537$	0	0
$538$	−36.7588	−1.58479
$539$	−3.42638	−0.147585
$540$	0	0
$541$	−12.4372	−0.534719	−0.267360	−	0.963597i	$-0.586151\pi$
$541$	−12.4372	−0.534719	−0.267360	+	0.963597i	$0.586151\pi$
$542$	36.2578	1.55741
$543$	0	0
$544$	0.146117	0.00626471
$545$	−0.164883	−0.00706283
$546$	0	0
$547$	−2.09590	−0.0896140	−0.0448070	−	0.998996i	$-0.514267\pi$
$547$	−2.09590	−0.0896140	−0.0448070	+	0.998996i	$0.514267\pi$
$548$	−56.2390	−2.40241
$549$	0	0
$550$	19.3329	0.824359
$551$	−14.1920	−0.604598
$552$	0	0
$553$	2.96655	0.126150
$554$	−72.5288	−3.08145
$555$	0	0
$556$	78.7440	3.33949
$557$	−21.9349	−0.929409	−0.464705	−	0.885466i	$-0.653840\pi$
$557$	−21.9349	−0.929409	−0.464705	+	0.885466i	$0.653840\pi$
$558$	0	0
$559$	0.352612	0.0149139
$560$	−10.8967	−0.460469
$561$	0	0
$562$	13.9514	0.588505
$563$	29.4859	1.24268	0.621342	−	0.783540i	$-0.286588\pi$
$563$	29.4859	1.24268	0.621342	+	0.783540i	$0.286588\pi$
$564$	0	0
$565$	33.0577	1.39075
$566$	29.1719	1.22619
$567$	0	0
$568$	24.1983	1.01534
$569$	15.2138	0.637796	0.318898	−	0.947789i	$-0.396687\pi$
$569$	15.2138	0.637796	0.318898	+	0.947789i	$0.396687\pi$
$570$	0	0
$571$	1.52209	0.0636973	0.0318487	−	0.999493i	$-0.489861\pi$
$571$	1.52209	0.0636973	0.0318487	+	0.999493i	$0.489861\pi$
$572$	−1.14861	−0.0480258
$573$	0	0
$574$	7.22379	0.301515
$575$	−20.4254	−0.851798
$576$	0	0
$577$	6.63152	0.276074	0.138037	−	0.990427i	$-0.455921\pi$
$577$	6.63152	0.276074	0.138037	+	0.990427i	$0.455921\pi$
$578$	22.9677	0.955331
$579$	0	0
$580$	−75.7292	−3.14448
$581$	−0.158273	−0.00656626
$582$	0	0
$583$	−15.2259	−0.630593
$584$	51.2777	2.12189
$585$	0	0
$586$	−3.64074	−0.150398
$587$	14.4120	0.594848	0.297424	−	0.954746i	$-0.403873\pi$
$587$	14.4120	0.594848	0.297424	+	0.954746i	$0.403873\pi$
$588$	0	0
$589$	6.18748	0.254951
$590$	−46.0923	−1.89759
$591$	0	0
$592$	−11.7858	−0.484393
$593$	−34.4555	−1.41492	−0.707459	−	0.706755i	$-0.750158\pi$
$593$	−34.4555	−1.41492	−0.707459	+	0.706755i	$0.750158\pi$
$594$	0	0
$595$	7.46330	0.305965
$596$	−8.47917	−0.347321
$597$	0	0
$598$	1.81946	0.0744032
$599$	19.8531	0.811175	0.405588	−	0.914056i	$-0.367067\pi$
$599$	19.8531	0.811175	0.405588	+	0.914056i	$0.367067\pi$
$600$	0	0
$601$	−8.70757	−0.355189	−0.177595	−	0.984104i	$-0.556832\pi$
$601$	−8.70757	−0.355189	−0.177595	+	0.984104i	$0.556832\pi$
$602$	10.3246	0.420801
$603$	0	0
$604$	91.0182	3.70348
$605$	1.99998	0.0813106
$606$	0	0
$607$	−8.62465	−0.350064	−0.175032	−	0.984563i	$-0.556003\pi$
$607$	−8.62465	−0.350064	−0.175032	+	0.984563i	$0.556003\pi$
$608$	−0.107316	−0.00435222
$609$	0	0
$610$	13.0715	0.529249
$611$	−0.348693	−0.0141066
$612$	0	0
$613$	10.1332	0.409276	0.204638	−	0.978838i	$-0.434398\pi$
$613$	10.1332	0.409276	0.204638	+	0.978838i	$0.434398\pi$
$614$	36.1592	1.45927
$615$	0	0
$616$	−16.8386	−0.678446
$617$	−17.6913	−0.712225	−0.356113	−	0.934443i	$-0.615898\pi$
$617$	−17.6913	−0.712225	−0.356113	+	0.934443i	$0.615898\pi$
$618$	0	0
$619$	−4.55734	−0.183175	−0.0915875	−	0.995797i	$-0.529194\pi$
$619$	−4.55734	−0.183175	−0.0915875	+	0.995797i	$0.529194\pi$
$620$	33.0168	1.32599
$621$	0	0
$622$	30.2291	1.21208
$623$	−13.7012	−0.548927
$624$	0	0
$625$	−31.2110	−1.24844
$626$	68.8585	2.75214
$627$	0	0
$628$	3.54607	0.141503
$629$	8.07226	0.321862
$630$	0	0
$631$	−4.36700	−0.173848	−0.0869238	−	0.996215i	$-0.527704\pi$
$631$	−4.36700	−0.173848	−0.0869238	+	0.996215i	$0.527704\pi$
$632$	14.5788	0.579912
$633$	0	0
$634$	−50.7333	−2.01488
$635$	−2.70231	−0.107238
$636$	0	0
$637$	0.0836936	0.00331606
$638$	−58.7476	−2.32584
$639$	0	0
$640$	52.8340	2.08845
$641$	18.6507	0.736660	0.368330	−	0.929695i	$-0.379930\pi$
$641$	18.6507	0.736660	0.368330	+	0.929695i	$0.379930\pi$
$642$	0	0
$643$	30.8696	1.21738	0.608690	−	0.793408i	$-0.291695\pi$
$643$	30.8696	1.21738	0.608690	+	0.793408i	$0.291695\pi$
$644$	35.5324	1.40017
$645$	0	0
$646$	13.7286	0.540143
$647$	18.2783	0.718592	0.359296	−	0.933224i	$-0.383017\pi$
$647$	18.2783	0.718592	0.359296	+	0.933224i	$0.383017\pi$
$648$	0	0
$649$	−23.8484	−0.936131
$650$	−0.472230	−0.0185224
$651$	0	0
$652$	50.5714	1.98053
$653$	20.6449	0.807898	0.403949	−	0.914781i	$-0.367637\pi$
$653$	20.6449	0.807898	0.403949	+	0.914781i	$0.367637\pi$
$654$	0	0
$655$	59.7604	2.33503
$656$	11.8865	0.464090
$657$	0	0
$658$	−10.2099	−0.398024
$659$	9.69654	0.377723	0.188862	−	0.982004i	$-0.439520\pi$
$659$	9.69654	0.377723	0.188862	+	0.982004i	$0.439520\pi$
$660$	0	0
$661$	−27.7254	−1.07839	−0.539197	−	0.842180i	$-0.681272\pi$
$661$	−27.7254	−1.07839	−0.539197	+	0.842180i	$0.681272\pi$
$662$	−32.3079	−1.25568
$663$	0	0
$664$	−0.777814	−0.0301850
$665$	−5.48141	−0.212560
$666$	0	0
$667$	62.0674	2.40326
$668$	11.3457	0.438979
$669$	0	0
$670$	−18.6823	−0.721759
$671$	6.76325	0.261092
$672$	0	0
$673$	−41.0167	−1.58108	−0.790538	−	0.612413i	$-0.790199\pi$
$673$	−41.0167	−1.58108	−0.790538	+	0.612413i	$0.790199\pi$
$674$	57.7374	2.22396
$675$	0	0
$676$	−52.0420	−2.00162
$677$	−10.1404	−0.389727	−0.194863	−	0.980830i	$-0.562426\pi$
$677$	−10.1404	−0.389727	−0.194863	+	0.980830i	$0.562426\pi$
$678$	0	0
$679$	5.43065	0.208409
$680$	36.6775	1.40652
$681$	0	0
$682$	25.6131	0.980776
$683$	−20.5856	−0.787686	−0.393843	−	0.919178i	$-0.628855\pi$
$683$	−20.5856	−0.787686	−0.393843	+	0.919178i	$0.628855\pi$
$684$	0	0
$685$	−37.9426	−1.44971
$686$	2.45059	0.0935640
$687$	0	0
$688$	16.9889	0.647695
$689$	0.371912	0.0141687
$690$	0	0
$691$	−34.3178	−1.30551	−0.652756	−	0.757568i	$-0.726387\pi$
$691$	−34.3178	−1.30551	−0.652756	+	0.757568i	$0.726387\pi$
$692$	2.69086	0.102291
$693$	0	0
$694$	−28.9902	−1.10045
$695$	53.1260	2.01519
$696$	0	0
$697$	−8.14124	−0.308371
$698$	2.76142	0.104521
$699$	0	0
$700$	−9.22223	−0.348567
$701$	24.9235	0.941347	0.470673	−	0.882307i	$-0.344011\pi$
$701$	24.9235	0.941347	0.470673	+	0.882307i	$0.344011\pi$
$702$	0	0
$703$	−5.92866	−0.223604
$704$	27.1886	1.02471
$705$	0	0
$706$	58.1480	2.18843
$707$	−4.30013	−0.161723
$708$	0	0
$709$	−9.94960	−0.373665	−0.186832	−	0.982392i	$-0.559822\pi$
$709$	−9.94960	−0.373665	−0.186832	+	0.982392i	$0.559822\pi$
$710$	32.6077	1.22374
$711$	0	0
$712$	−67.3331	−2.52341
$713$	−27.0605	−1.01342
$714$	0	0
$715$	−0.774930	−0.0289807
$716$	57.7170	2.15698
$717$	0	0
$718$	43.8626	1.63694
$719$	−0.464282	−0.0173148	−0.00865740	−	0.999963i	$-0.502756\pi$
$719$	−0.464282	−0.0173148	−0.00865740	+	0.999963i	$0.502756\pi$
$720$	0	0
$721$	2.58455	0.0962538
$722$	36.4783	1.35758
$723$	0	0
$724$	38.7738	1.44102
$725$	−16.1092	−0.598282
$726$	0	0
$727$	−7.22529	−0.267971	−0.133986	−	0.990983i	$-0.542778\pi$
$727$	−7.22529	−0.267971	−0.133986	+	0.990983i	$0.542778\pi$
$728$	0.411303	0.0152439
$729$	0	0
$730$	69.0978	2.55743
$731$	−11.6359	−0.430370
$732$	0	0
$733$	17.9405	0.662648	0.331324	−	0.943517i	$-0.392505\pi$
$733$	17.9405	0.662648	0.331324	+	0.943517i	$0.392505\pi$
$734$	−6.11344	−0.225651
$735$	0	0
$736$	0.469336	0.0173000
$737$	−9.66630	−0.356063
$738$	0	0
$739$	52.8059	1.94250	0.971248	−	0.238069i	$-0.0765146\pi$
$739$	52.8059	1.94250	0.971248	+	0.238069i	$0.0765146\pi$
$740$	−31.6357	−1.16295
$741$	0	0
$742$	10.8898	0.399776
$743$	23.1811	0.850432	0.425216	−	0.905092i	$-0.360198\pi$
$743$	23.1811	0.850432	0.425216	+	0.905092i	$0.360198\pi$
$744$	0	0
$745$	−5.72062	−0.209587
$746$	13.4920	0.493977
$747$	0	0
$748$	37.9033	1.38588
$749$	−7.75671	−0.283424
$750$	0	0
$751$	20.4907	0.747718	0.373859	−	0.927486i	$-0.378034\pi$
$751$	20.4907	0.747718	0.373859	+	0.927486i	$0.378034\pi$
$752$	−16.8001	−0.612637
$753$	0	0
$754$	1.43498	0.0522590
$755$	61.4070	2.23483
$756$	0	0
$757$	−52.2660	−1.89964	−0.949819	−	0.312799i	$-0.898733\pi$
$757$	−52.2660	−1.89964	−0.949819	+	0.312799i	$0.898733\pi$
$758$	33.6691	1.22292
$759$	0	0
$760$	−26.9378	−0.977137
$761$	31.5922	1.14522	0.572609	−	0.819829i	$-0.305931\pi$
$761$	31.5922	1.14522	0.572609	+	0.819829i	$0.305931\pi$
$762$	0	0
$763$	0.0610158	0.00220892
$764$	58.2145	2.10613
$765$	0	0
$766$	46.9336	1.69578
$767$	0.582526	0.0210338
$768$	0	0
$769$	18.8353	0.679217	0.339609	−	0.940567i	$-0.389705\pi$
$769$	18.8353	0.679217	0.339609	+	0.940567i	$0.389705\pi$
$770$	−22.6903	−0.817703
$771$	0	0
$772$	66.9426	2.40932
$773$	−43.9559	−1.58099	−0.790493	−	0.612471i	$-0.790175\pi$
$773$	−43.9559	−1.58099	−0.790493	+	0.612471i	$0.790175\pi$
$774$	0	0
$775$	7.02339	0.252287
$776$	26.6883	0.958055
$777$	0	0
$778$	−40.6639	−1.45787
$779$	5.97933	0.214232
$780$	0	0
$781$	16.8714	0.603706
$782$	−60.0408	−2.14705
$783$	0	0
$784$	4.03237	0.144013
$785$	2.39242	0.0853890
$786$	0	0
$787$	27.1558	0.967998	0.483999	−	0.875068i	$-0.339184\pi$
$787$	27.1558	0.967998	0.483999	+	0.875068i	$0.339184\pi$
$788$	65.4224	2.33058
$789$	0	0
$790$	19.6452	0.698945
$791$	−12.2332	−0.434961
$792$	0	0
$793$	−0.165201	−0.00586645
$794$	−59.1416	−2.09886
$795$	0	0
$796$	38.0208	1.34761
$797$	−16.7805	−0.594397	−0.297198	−	0.954816i	$-0.596052\pi$
$797$	−16.7805	−0.594397	−0.297198	+	0.954816i	$0.596052\pi$
$798$	0	0
$799$	11.5066	0.407075
$800$	−0.121813	−0.00430676
$801$	0	0
$802$	−1.46122	−0.0515976
$803$	35.7516	1.26165
$804$	0	0
$805$	23.9725	0.844921
$806$	−0.625631	−0.0220369
$807$	0	0
$808$	−21.1325	−0.743439
$809$	18.4092	0.647234	0.323617	−	0.946188i	$-0.395101\pi$
$809$	18.4092	0.647234	0.323617	+	0.946188i	$0.395101\pi$
$810$	0	0
$811$	5.51082	0.193511	0.0967555	−	0.995308i	$-0.469154\pi$
$811$	5.51082	0.193511	0.0967555	+	0.995308i	$0.469154\pi$
$812$	28.0239	0.983447
$813$	0	0
$814$	−24.5417	−0.860187
$815$	34.1189	1.19513
$816$	0	0
$817$	8.54600	0.298987
$818$	−83.7701	−2.92895
$819$	0	0
$820$	31.9061	1.11421
$821$	20.5500	0.717200	0.358600	−	0.933491i	$-0.383254\pi$
$821$	20.5500	0.717200	0.358600	+	0.933491i	$0.383254\pi$
$822$	0	0
$823$	14.9187	0.520035	0.260017	−	0.965604i	$-0.416272\pi$
$823$	14.9187	0.520035	0.260017	+	0.965604i	$0.416272\pi$
$824$	12.7015	0.442478
$825$	0	0
$826$	17.0566	0.593477
$827$	22.3698	0.777873	0.388936	−	0.921265i	$-0.372843\pi$
$827$	22.3698	0.777873	0.388936	+	0.921265i	$0.372843\pi$
$828$	0	0
$829$	45.2096	1.57019	0.785097	−	0.619372i	$-0.212613\pi$
$829$	45.2096	1.57019	0.785097	+	0.619372i	$0.212613\pi$
$830$	−1.04812	−0.0363808
$831$	0	0
$832$	−0.664116	−0.0230241
$833$	−2.76183	−0.0956916
$834$	0	0
$835$	7.65459	0.264898
$836$	−27.8381	−0.962799
$837$	0	0
$838$	−36.5366	−1.26213
$839$	6.00052	0.207161	0.103580	−	0.994621i	$-0.466970\pi$
$839$	6.00052	0.207161	0.103580	+	0.994621i	$0.466970\pi$
$840$	0	0
$841$	19.9517	0.687990
$842$	−39.7830	−1.37101
$843$	0	0
$844$	60.4321	2.08016
$845$	−35.1110	−1.20786
$846$	0	0
$847$	−0.740100	−0.0254301
$848$	17.9188	0.615333
$849$	0	0
$850$	15.5832	0.534501
$851$	25.9286	0.888819
$852$	0	0
$853$	−40.5650	−1.38892	−0.694460	−	0.719531i	$-0.744357\pi$
$853$	−40.5650	−1.38892	−0.694460	+	0.719531i	$0.744357\pi$
$854$	−4.83716	−0.165524
$855$	0	0
$856$	−38.1195	−1.30290
$857$	38.0835	1.30091	0.650454	−	0.759546i	$-0.274579\pi$
$857$	38.0835	1.30091	0.650454	+	0.759546i	$0.274579\pi$
$858$	0	0
$859$	−42.0689	−1.43537	−0.717687	−	0.696366i	$-0.754799\pi$
$859$	−42.0689	−1.43537	−0.717687	+	0.696366i	$0.754799\pi$
$860$	45.6020	1.55502
$861$	0	0
$862$	6.46143	0.220077
$863$	−20.3607	−0.693086	−0.346543	−	0.938034i	$-0.612645\pi$
$863$	−20.3607	−0.693086	−0.346543	+	0.938034i	$0.612645\pi$
$864$	0	0
$865$	1.81544	0.0617267
$866$	94.7936	3.22122
$867$	0	0
$868$	−12.2180	−0.414706
$869$	10.1645	0.344808
$870$	0	0
$871$	0.236111	0.00800032
$872$	0.299855	0.0101544
$873$	0	0
$874$	44.0969	1.49160
$875$	7.28959	0.246433
$876$	0	0
$877$	−39.9863	−1.35024	−0.675121	−	0.737707i	$-0.735908\pi$
$877$	−39.9863	−1.35024	−0.675121	+	0.737707i	$0.735908\pi$
$878$	−59.6761	−2.01397
$879$	0	0
$880$	−37.3362	−1.25860
$881$	53.9457	1.81748	0.908738	−	0.417367i	$-0.137047\pi$
$881$	53.9457	1.81748	0.908738	+	0.417367i	$0.137047\pi$
$882$	0	0
$883$	36.0654	1.21370	0.606848	−	0.794818i	$-0.292433\pi$
$883$	36.0654	1.21370	0.606848	+	0.794818i	$0.292433\pi$
$884$	−0.925834	−0.0311392
$885$	0	0
$886$	43.9291	1.47583
$887$	−51.7244	−1.73674	−0.868368	−	0.495921i	$-0.834831\pi$
$887$	−51.7244	−1.73674	−0.868368	+	0.495921i	$0.834831\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−90.7327	−3.04137
$891$	0	0
$892$	109.845	3.67789
$893$	−8.45103	−0.282803
$894$	0	0
$895$	38.9397	1.30161
$896$	−19.5515	−0.653168
$897$	0	0
$898$	−32.4592	−1.08318
$899$	−21.3422	−0.711803
$900$	0	0
$901$	−12.2728	−0.408867
$902$	24.7515	0.824133
$903$	0	0
$904$	−60.1185	−1.99951
$905$	26.1594	0.869568
$906$	0	0
$907$	−32.6700	−1.08479	−0.542395	−	0.840124i	$-0.682482\pi$
$907$	−32.6700	−1.08479	−0.542395	+	0.840124i	$0.682482\pi$
$908$	0.676575	0.0224529
$909$	0	0
$910$	0.554239	0.0183728
$911$	30.6978	1.01706	0.508532	−	0.861043i	$-0.330188\pi$
$911$	30.6978	1.01706	0.508532	+	0.861043i	$0.330188\pi$
$912$	0	0
$913$	−0.542303	−0.0179476
$914$	45.2583	1.49701
$915$	0	0
$916$	−46.8085	−1.54660
$917$	−22.1146	−0.730289
$918$	0	0
$919$	0.338191	0.0111559	0.00557794	−	0.999984i	$-0.498224\pi$
$919$	0.338191	0.0111559	0.00557794	+	0.999984i	$0.498224\pi$
$920$	117.810	3.88409
$921$	0	0
$922$	−37.2538	−1.22689
$923$	−0.412104	−0.0135646
$924$	0	0
$925$	−6.72961	−0.221268
$926$	17.3243	0.569312
$927$	0	0
$928$	0.370159	0.0121511
$929$	−11.3223	−0.371472	−0.185736	−	0.982600i	$-0.559467\pi$
$929$	−11.3223	−0.371472	−0.185736	+	0.982600i	$0.559467\pi$
$930$	0	0
$931$	2.02842	0.0664788
$932$	3.15335	0.103291
$933$	0	0
$934$	33.1682	1.08530
$935$	25.5721	0.836297
$936$	0	0
$937$	−14.3039	−0.467287	−0.233643	−	0.972322i	$-0.575065\pi$
$937$	−14.3039	−0.467287	−0.233643	+	0.972322i	$0.575065\pi$
$938$	6.91345	0.225732
$939$	0	0
$940$	−45.0952	−1.47084
$941$	40.5443	1.32171	0.660853	−	0.750515i	$-0.270195\pi$
$941$	40.5443	1.32171	0.660853	+	0.750515i	$0.270195\pi$
$942$	0	0
$943$	−26.1501	−0.851566
$944$	28.0662	0.913476
$945$	0	0
$946$	35.3762	1.15018
$947$	14.4457	0.469423	0.234711	−	0.972065i	$-0.424585\pi$
$947$	14.4457	0.469423	0.234711	+	0.972065i	$0.424585\pi$
$948$	0	0
$949$	−0.873276	−0.0283477
$950$	−11.4451	−0.371328
$951$	0	0
$952$	−13.5727	−0.439893
$953$	−36.9336	−1.19640	−0.598199	−	0.801348i	$-0.704117\pi$
$953$	−36.9336	−1.19640	−0.598199	+	0.801348i	$0.704117\pi$
$954$	0	0
$955$	39.2754	1.27092
$956$	31.4608	1.01751
$957$	0	0
$958$	−22.0927	−0.713784
$959$	14.0408	0.453402
$960$	0	0
$961$	−21.6951	−0.699843
$962$	0.599462	0.0193274
$963$	0	0
$964$	−56.8007	−1.82943
$965$	45.1640	1.45388
$966$	0	0
$967$	−52.5490	−1.68986	−0.844930	−	0.534876i	$-0.820358\pi$
$967$	−52.5490	−1.68986	−0.844930	+	0.534876i	$0.820358\pi$
$968$	−3.63714	−0.116902
$969$	0	0
$970$	35.9631	1.15471
$971$	−28.0397	−0.899836	−0.449918	−	0.893070i	$-0.648547\pi$
$971$	−28.0397	−0.899836	−0.449918	+	0.893070i	$0.648547\pi$
$972$	0	0
$973$	−19.6595	−0.630255
$974$	−11.2154	−0.359365
$975$	0	0
$976$	−7.95939	−0.254774
$977$	−49.0013	−1.56769	−0.783845	−	0.620957i	$-0.786744\pi$
$977$	−49.0013	−1.56769	−0.783845	+	0.620957i	$0.786744\pi$
$978$	0	0
$979$	−46.9456	−1.50039
$980$	10.8238	0.345753
$981$	0	0
$982$	53.4166	1.70459
$983$	−13.8095	−0.440453	−0.220227	−	0.975449i	$-0.570680\pi$
$983$	−13.8095	−0.440453	−0.220227	+	0.975449i	$0.570680\pi$
$984$	0	0
$985$	44.1383	1.40636
$986$	−47.3534	−1.50804
$987$	0	0
$988$	0.679979	0.0216330
$989$	−37.3753	−1.18846
$990$	0	0
$991$	5.70845	0.181335	0.0906674	−	0.995881i	$-0.471100\pi$
$991$	5.70845	0.181335	0.0906674	+	0.995881i	$0.471100\pi$
$992$	−0.161384	−0.00512394
$993$	0	0
$994$	−12.0666	−0.382730
$995$	25.6514	0.813203
$996$	0	0
$997$	56.3977	1.78613	0.893066	−	0.449926i	$-0.148550\pi$
$997$	56.3977	1.78613	0.893066	+	0.449926i	$0.148550\pi$
$998$	−2.79368	−0.0884323
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.3		16
3.2	odd	2		2667.2.a.n.1.14	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.14	✓	16	3.2	odd	2
8001.2.a.s.1.3		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)

\(f(q)\)	\(=\)	\(q-2.45059 q^{2} +4.00539 q^{4} +2.70231 q^{5} -1.00000 q^{7} -4.91439 q^{8} +O(q^{10})\)	\(q-2.45059 q^{2} +4.00539 q^{4} +2.70231 q^{5} -1.00000 q^{7} -4.91439 q^{8} -6.62224 q^{10} -3.42638 q^{11} +0.0836936 q^{13} +2.45059 q^{14} +4.03237 q^{16} -2.76183 q^{17} +2.02842 q^{19} +10.8238 q^{20} +8.39666 q^{22} -8.87114 q^{23} +2.30245 q^{25} -0.205099 q^{26} -4.00539 q^{28} -6.99655 q^{29} +3.05039 q^{31} -0.0529059 q^{32} +6.76810 q^{34} -2.70231 q^{35} -2.92280 q^{37} -4.97083 q^{38} -13.2802 q^{40} +2.94777 q^{41} +4.21313 q^{43} -13.7240 q^{44} +21.7395 q^{46} -4.16631 q^{47} +1.00000 q^{49} -5.64237 q^{50} +0.335225 q^{52} +4.44373 q^{53} -9.25913 q^{55} +4.91439 q^{56} +17.1457 q^{58} +6.96022 q^{59} -1.97387 q^{61} -7.47525 q^{62} -7.93509 q^{64} +0.226166 q^{65} +2.82114 q^{67} -11.0622 q^{68} +6.62224 q^{70} -4.92396 q^{71} -10.4342 q^{73} +7.16258 q^{74} +8.12462 q^{76} +3.42638 q^{77} -2.96655 q^{79} +10.8967 q^{80} -7.22379 q^{82} +0.158273 q^{83} -7.46330 q^{85} -10.3246 q^{86} +16.8386 q^{88} +13.7012 q^{89} -0.0836936 q^{91} -35.5324 q^{92} +10.2099 q^{94} +5.48141 q^{95} -5.43065 q^{97} -2.45059 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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