LMFDB - Embedded newform 8001.2.a.u.1.9

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			1.9
Root			$0.0366427$ 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)

$f(q)$	$=$	$q+0.0366427 q^{2} -1.99866 q^{4} -1.02187 q^{5} +1.00000 q^{7} -0.146522 q^{8} +O(q^{10})$	\(q+0.0366427 q^{2} -1.99866 q^{4} -1.02187 q^{5} +1.00000 q^{7} -0.146522 q^{8} -0.0374440 q^{10} +4.36561 q^{11} -3.87219 q^{13} +0.0366427 q^{14} +3.99195 q^{16} -6.07872 q^{17} +6.90344 q^{19} +2.04236 q^{20} +0.159968 q^{22} +5.17354 q^{23} -3.95579 q^{25} -0.141888 q^{26} -1.99866 q^{28} +1.01222 q^{29} +8.54462 q^{31} +0.439319 q^{32} -0.222741 q^{34} -1.02187 q^{35} -8.38368 q^{37} +0.252961 q^{38} +0.149726 q^{40} -1.56657 q^{41} -6.37089 q^{43} -8.72536 q^{44} +0.189573 q^{46} -4.23682 q^{47} +1.00000 q^{49} -0.144951 q^{50} +7.73918 q^{52} +5.22125 q^{53} -4.46107 q^{55} -0.146522 q^{56} +0.0370905 q^{58} +7.49969 q^{59} +14.3692 q^{61} +0.313098 q^{62} -7.96779 q^{64} +3.95686 q^{65} +7.55792 q^{67} +12.1493 q^{68} -0.0374440 q^{70} -15.2047 q^{71} -1.99435 q^{73} -0.307201 q^{74} -13.7976 q^{76} +4.36561 q^{77} -14.9551 q^{79} -4.07924 q^{80} -0.0574035 q^{82} +15.0270 q^{83} +6.21164 q^{85} -0.233447 q^{86} -0.639657 q^{88} +3.14237 q^{89} -3.87219 q^{91} -10.3401 q^{92} -0.155249 q^{94} -7.05440 q^{95} -16.4469 q^{97} +0.0366427 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.0366427	0.0259103	0.0129552	−	0.999916i	$-0.495876\pi$
$2$	0.0366427	0.0259103	0.0129552	+	0.999916i	$0.495876\pi$
$3$	0	0
$3$	0	0
$4$	−1.99866	−0.999329
$5$	−1.02187	−0.456993	−0.228497	−	0.973545i	$-0.573381\pi$
$5$	−1.02187	−0.456993	−0.228497	+	0.973545i	$0.573381\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.146522	−0.0518033
$9$	0	0
$10$	−0.0374440	−0.0118408
$11$	4.36561	1.31628	0.658140	−	0.752895i	$-0.271343\pi$
$11$	4.36561	1.31628	0.658140	+	0.752895i	$0.271343\pi$
$12$	0	0
$13$	−3.87219	−1.07395	−0.536976	−	0.843598i	$-0.680433\pi$
$13$	−3.87219	−1.07395	−0.536976	+	0.843598i	$0.680433\pi$
$14$	0.0366427	0.00979319
$15$	0	0
$16$	3.99195	0.997986
$17$	−6.07872	−1.47431	−0.737153	−	0.675726i	$-0.763830\pi$
$17$	−6.07872	−1.47431	−0.737153	+	0.675726i	$0.763830\pi$
$18$	0	0
$19$	6.90344	1.58376	0.791879	−	0.610678i	$-0.209103\pi$
$19$	6.90344	1.58376	0.791879	+	0.610678i	$0.209103\pi$
$20$	2.04236	0.456686
$21$	0	0
$22$	0.159968	0.0341053
$23$	5.17354	1.07876	0.539379	−	0.842063i	$-0.318659\pi$
$23$	5.17354	1.07876	0.539379	+	0.842063i	$0.318659\pi$
$24$	0	0
$25$	−3.95579	−0.791157
$26$	−0.141888	−0.0278265
$27$	0	0
$28$	−1.99866	−0.377711
$29$	1.01222	0.187965	0.0939823	−	0.995574i	$-0.470040\pi$
$29$	1.01222	0.187965	0.0939823	+	0.995574i	$0.470040\pi$
$30$	0	0
$31$	8.54462	1.53466	0.767329	−	0.641253i	$-0.221585\pi$
$31$	8.54462	1.53466	0.767329	+	0.641253i	$0.221585\pi$
$32$	0.439319	0.0776614
$33$	0	0
$34$	−0.222741	−0.0381997
$35$	−1.02187	−0.172727
$36$	0	0
$37$	−8.38368	−1.37827	−0.689134	−	0.724634i	$-0.742009\pi$
$37$	−8.38368	−1.37827	−0.689134	+	0.724634i	$0.742009\pi$
$38$	0.252961	0.0410357
$39$	0	0
$40$	0.149726	0.0236737
$41$	−1.56657	−0.244657	−0.122329	−	0.992490i	$-0.539036\pi$
$41$	−1.56657	−0.244657	−0.122329	+	0.992490i	$0.539036\pi$
$42$	0	0
$43$	−6.37089	−0.971552	−0.485776	−	0.874083i	$-0.661463\pi$
$43$	−6.37089	−0.971552	−0.485776	+	0.874083i	$0.661463\pi$
$44$	−8.72536	−1.31540
$45$	0	0
$46$	0.189573	0.0279510
$47$	−4.23682	−0.618003	−0.309002	−	0.951062i	$-0.599995\pi$
$47$	−4.23682	−0.618003	−0.309002	+	0.951062i	$0.599995\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.144951	−0.0204992
$51$	0	0
$52$	7.73918	1.07323
$53$	5.22125	0.717194	0.358597	−	0.933492i	$-0.383255\pi$
$53$	5.22125	0.717194	0.358597	+	0.933492i	$0.383255\pi$
$54$	0	0
$55$	−4.46107	−0.601531
$56$	−0.146522	−0.0195798
$57$	0	0
$58$	0.0370905	0.00487022
$59$	7.49969	0.976377	0.488188	−	0.872738i	$-0.337658\pi$
$59$	7.49969	0.976377	0.488188	+	0.872738i	$0.337658\pi$
$60$	0	0
$61$	14.3692	1.83979	0.919895	−	0.392165i	$-0.128274\pi$
$61$	14.3692	1.83979	0.919895	+	0.392165i	$0.128274\pi$
$62$	0.313098	0.0397635
$63$	0	0
$64$	−7.96779	−0.995974
$65$	3.95686	0.490789
$66$	0	0
$67$	7.55792	0.923346	0.461673	−	0.887050i	$-0.347249\pi$
$67$	7.55792	0.923346	0.461673	+	0.887050i	$0.347249\pi$
$68$	12.1493	1.47332
$69$	0	0
$70$	−0.0374440	−0.00447542
$71$	−15.2047	−1.80447	−0.902233	−	0.431248i	$-0.858073\pi$
$71$	−15.2047	−1.80447	−0.902233	+	0.431248i	$0.858073\pi$
$72$	0	0
$73$	−1.99435	−0.233421	−0.116711	−	0.993166i	$-0.537235\pi$
$73$	−1.99435	−0.233421	−0.116711	+	0.993166i	$0.537235\pi$
$74$	−0.307201	−0.0357114
$75$	0	0
$76$	−13.7976	−1.58269
$77$	4.36561	0.497507
$78$	0	0
$79$	−14.9551	−1.68258	−0.841288	−	0.540587i	$-0.818202\pi$
$79$	−14.9551	−1.68258	−0.841288	+	0.540587i	$0.818202\pi$
$80$	−4.07924	−0.456073
$81$	0	0
$82$	−0.0574035	−0.00633916
$83$	15.0270	1.64943	0.824717	−	0.565546i	$-0.191335\pi$
$83$	15.0270	1.64943	0.824717	+	0.565546i	$0.191335\pi$
$84$	0	0
$85$	6.21164	0.673747
$86$	−0.233447	−0.0251732
$87$	0	0
$88$	−0.639657	−0.0681876
$89$	3.14237	0.333090	0.166545	−	0.986034i	$-0.446739\pi$
$89$	3.14237	0.333090	0.166545	+	0.986034i	$0.446739\pi$
$90$	0	0
$91$	−3.87219	−0.405916
$92$	−10.3401	−1.07803
$93$	0	0
$94$	−0.155249	−0.0160127
$95$	−7.05440	−0.723767
$96$	0	0
$97$	−16.4469	−1.66993	−0.834966	−	0.550301i	$-0.814513\pi$
$97$	−16.4469	−1.66993	−0.834966	+	0.550301i	$0.814513\pi$
$98$	0.0366427	0.00370148
$99$	0	0
$100$	7.90626	0.790626
$101$	−2.20148	−0.219055	−0.109528	−	0.993984i	$-0.534934\pi$
$101$	−2.20148	−0.219055	−0.109528	+	0.993984i	$0.534934\pi$
$102$	0	0
$103$	−12.2962	−1.21158	−0.605789	−	0.795625i	$-0.707142\pi$
$103$	−12.2962	−1.21158	−0.605789	+	0.795625i	$0.707142\pi$
$104$	0.567360	0.0556342
$105$	0	0
$106$	0.191321	0.0185827
$107$	−2.63179	−0.254424	−0.127212	−	0.991876i	$-0.540603\pi$
$107$	−2.63179	−0.254424	−0.127212	+	0.991876i	$0.540603\pi$
$108$	0	0
$109$	0.425217	0.0407284	0.0203642	−	0.999793i	$-0.493517\pi$
$109$	0.425217	0.0407284	0.0203642	+	0.999793i	$0.493517\pi$
$110$	−0.163466	−0.0155859
$111$	0	0
$112$	3.99195	0.377203
$113$	17.2777	1.62535	0.812674	−	0.582719i	$-0.198011\pi$
$113$	17.2777	1.62535	0.812674	+	0.582719i	$0.198011\pi$
$114$	0	0
$115$	−5.28667	−0.492985
$116$	−2.02308	−0.187838
$117$	0	0
$118$	0.274809	0.0252983
$119$	−6.07872	−0.557235
$120$	0	0
$121$	8.05854	0.732595
$122$	0.526528	0.0476696
$123$	0	0
$124$	−17.0778	−1.53363
$125$	9.15163	0.818547
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−1.17060	−0.103467
$129$	0	0
$130$	0.144990	0.0127165
$131$	−2.45915	−0.214857	−0.107428	−	0.994213i	$-0.534262\pi$
$131$	−2.45915	−0.214857	−0.107428	+	0.994213i	$0.534262\pi$
$132$	0	0
$133$	6.90344	0.598604
$134$	0.276943	0.0239242
$135$	0	0
$136$	0.890665	0.0763739
$137$	−3.53859	−0.302322	−0.151161	−	0.988509i	$-0.548301\pi$
$137$	−3.53859	−0.302322	−0.151161	+	0.988509i	$0.548301\pi$
$138$	0	0
$139$	−7.30754	−0.619817	−0.309909	−	0.950766i	$-0.600298\pi$
$139$	−7.30754	−0.619817	−0.309909	+	0.950766i	$0.600298\pi$
$140$	2.04236	0.172611
$141$	0	0
$142$	−0.557142	−0.0467543
$143$	−16.9045	−1.41362
$144$	0	0
$145$	−1.03435	−0.0858985
$146$	−0.0730785	−0.00604802
$147$	0	0
$148$	16.7561	1.37734
$149$	14.4377	1.18278	0.591390	−	0.806386i	$-0.298579\pi$
$149$	14.4377	1.18278	0.591390	+	0.806386i	$0.298579\pi$
$150$	0	0
$151$	1.71468	0.139539	0.0697693	−	0.997563i	$-0.477774\pi$
$151$	1.71468	0.139539	0.0697693	+	0.997563i	$0.477774\pi$
$152$	−1.01150	−0.0820439
$153$	0	0
$154$	0.159968	0.0128906
$155$	−8.73147	−0.701329
$156$	0	0
$157$	−13.0418	−1.04085	−0.520426	−	0.853907i	$-0.674227\pi$
$157$	−13.0418	−1.04085	−0.520426	+	0.853907i	$0.674227\pi$
$158$	−0.547995	−0.0435961
$159$	0	0
$160$	−0.448926	−0.0354907
$161$	5.17354	0.407732
$162$	0	0
$163$	−2.54240	−0.199136	−0.0995679	−	0.995031i	$-0.531746\pi$
$163$	−2.54240	−0.199136	−0.0995679	+	0.995031i	$0.531746\pi$
$164$	3.13104	0.244493
$165$	0	0
$166$	0.550632	0.0427374
$167$	−2.11145	−0.163389	−0.0816943	−	0.996657i	$-0.526033\pi$
$167$	−2.11145	−0.163389	−0.0816943	+	0.996657i	$0.526033\pi$
$168$	0	0
$169$	1.99384	0.153372
$170$	0.227612	0.0174570
$171$	0	0
$172$	12.7332	0.970900
$173$	1.36849	0.104045	0.0520224	−	0.998646i	$-0.483433\pi$
$173$	1.36849	0.104045	0.0520224	+	0.998646i	$0.483433\pi$
$174$	0	0
$175$	−3.95579	−0.299029
$176$	17.4273	1.31363
$177$	0	0
$178$	0.115145	0.00863048
$179$	22.1352	1.65446	0.827231	−	0.561862i	$-0.189915\pi$
$179$	22.1352	1.65446	0.827231	+	0.561862i	$0.189915\pi$
$180$	0	0
$181$	9.74075	0.724024	0.362012	−	0.932173i	$-0.382090\pi$
$181$	9.74075	0.724024	0.362012	+	0.932173i	$0.382090\pi$
$182$	−0.141888	−0.0105174
$183$	0	0
$184$	−0.758036	−0.0558832
$185$	8.56701	0.629859
$186$	0	0
$187$	−26.5373	−1.94060
$188$	8.46794	0.617588
$189$	0	0
$190$	−0.258493	−0.0187530
$191$	14.6153	1.05752	0.528762	−	0.848770i	$-0.322657\pi$
$191$	14.6153	1.05752	0.528762	+	0.848770i	$0.322657\pi$
$192$	0	0
$193$	13.9523	1.00431	0.502154	−	0.864778i	$-0.332541\pi$
$193$	13.9523	1.00431	0.502154	+	0.864778i	$0.332541\pi$
$194$	−0.602661	−0.0432685
$195$	0	0
$196$	−1.99866	−0.142761
$197$	24.5804	1.75128	0.875639	−	0.482966i	$-0.160440\pi$
$197$	24.5804	1.75128	0.875639	+	0.482966i	$0.160440\pi$
$198$	0	0
$199$	19.8486	1.40703	0.703515	−	0.710681i	$-0.251613\pi$
$199$	19.8486	1.40703	0.703515	+	0.710681i	$0.251613\pi$
$200$	0.579609	0.0409845
$201$	0	0
$202$	−0.0806681	−0.00567579
$203$	1.01222	0.0710439
$204$	0	0
$205$	1.60083	0.111807
$206$	−0.450566	−0.0313924
$207$	0	0
$208$	−15.4576	−1.07179
$209$	30.1377	2.08467
$210$	0	0
$211$	−15.9995	−1.10145	−0.550725	−	0.834687i	$-0.685649\pi$
$211$	−15.9995	−1.10145	−0.550725	+	0.834687i	$0.685649\pi$
$212$	−10.4355	−0.716713
$213$	0	0
$214$	−0.0964359	−0.00659222
$215$	6.51021	0.443993
$216$	0	0
$217$	8.54462	0.580047
$218$	0.0155811	0.00105529
$219$	0	0
$220$	8.91616	0.601127
$221$	23.5379	1.58333
$222$	0	0
$223$	13.5951	0.910394	0.455197	−	0.890391i	$-0.349569\pi$
$223$	13.5951	0.910394	0.455197	+	0.890391i	$0.349569\pi$
$224$	0.439319	0.0293533
$225$	0	0
$226$	0.633102	0.0421133
$227$	4.86970	0.323214	0.161607	−	0.986855i	$-0.448332\pi$
$227$	4.86970	0.323214	0.161607	+	0.986855i	$0.448332\pi$
$228$	0	0
$229$	−23.0172	−1.52102	−0.760509	−	0.649328i	$-0.775050\pi$
$229$	−23.0172	−1.52102	−0.760509	+	0.649328i	$0.775050\pi$
$230$	−0.193718	−0.0127734
$231$	0	0
$232$	−0.148312	−0.00973718
$233$	18.3244	1.20047	0.600237	−	0.799822i	$-0.295073\pi$
$233$	18.3244	1.20047	0.600237	+	0.799822i	$0.295073\pi$
$234$	0	0
$235$	4.32947	0.282423
$236$	−14.9893	−0.975721
$237$	0	0
$238$	−0.222741	−0.0144381
$239$	16.8820	1.09201	0.546003	−	0.837783i	$-0.316149\pi$
$239$	16.8820	1.09201	0.546003	+	0.837783i	$0.316149\pi$
$240$	0	0
$241$	−19.2430	−1.23955	−0.619776	−	0.784779i	$-0.712777\pi$
$241$	−19.2430	−1.23955	−0.619776	+	0.784779i	$0.712777\pi$
$242$	0.295287	0.0189818
$243$	0	0
$244$	−28.7191	−1.83855
$245$	−1.02187	−0.0652847
$246$	0	0
$247$	−26.7314	−1.70088
$248$	−1.25197	−0.0795004
$249$	0	0
$250$	0.335341	0.0212088
$251$	−14.0644	−0.887736	−0.443868	−	0.896092i	$-0.646394\pi$
$251$	−14.0644	−0.887736	−0.443868	+	0.896092i	$0.646394\pi$
$252$	0	0
$253$	22.5856	1.41995
$254$	0.0366427	0.00229917
$255$	0	0
$256$	15.8927	0.993293
$257$	−3.70073	−0.230845	−0.115422	−	0.993316i	$-0.536822\pi$
$257$	−3.70073	−0.230845	−0.115422	+	0.993316i	$0.536822\pi$
$258$	0	0
$259$	−8.38368	−0.520936
$260$	−7.90841	−0.490459
$261$	0	0
$262$	−0.0901100	−0.00556701
$263$	0.927045	0.0571641	0.0285820	−	0.999591i	$-0.490901\pi$
$263$	0.927045	0.0571641	0.0285820	+	0.999591i	$0.490901\pi$
$264$	0	0
$265$	−5.33543	−0.327753
$266$	0.252961	0.0155100
$267$	0	0
$268$	−15.1057	−0.922726
$269$	22.9177	1.39732	0.698659	−	0.715455i	$-0.253781\pi$
$269$	22.9177	1.39732	0.698659	+	0.715455i	$0.253781\pi$
$270$	0	0
$271$	4.10911	0.249610	0.124805	−	0.992181i	$-0.460169\pi$
$271$	4.10911	0.249610	0.124805	+	0.992181i	$0.460169\pi$
$272$	−24.2659	−1.47134
$273$	0	0
$274$	−0.129664	−0.00783327
$275$	−17.2694	−1.04138
$276$	0	0
$277$	−2.61597	−0.157179	−0.0785893	−	0.996907i	$-0.525042\pi$
$277$	−2.61597	−0.157179	−0.0785893	+	0.996907i	$0.525042\pi$
$278$	−0.267768	−0.0160597
$279$	0	0
$280$	0.149726	0.00894783
$281$	2.34509	0.139896	0.0699482	−	0.997551i	$-0.477717\pi$
$281$	2.34509	0.139896	0.0699482	+	0.997551i	$0.477717\pi$
$282$	0	0
$283$	−8.87213	−0.527393	−0.263697	−	0.964606i	$-0.584942\pi$
$283$	−8.87213	−0.527393	−0.263697	+	0.964606i	$0.584942\pi$
$284$	30.3890	1.80325
$285$	0	0
$286$	−0.619426	−0.0366274
$287$	−1.56657	−0.0924718
$288$	0	0
$289$	19.9508	1.17358
$290$	−0.0379016	−0.00222566
$291$	0	0
$292$	3.98602	0.233264
$293$	12.1220	0.708176	0.354088	−	0.935212i	$-0.384791\pi$
$293$	12.1220	0.708176	0.354088	+	0.935212i	$0.384791\pi$
$294$	0	0
$295$	−7.66369	−0.446198
$296$	1.22839	0.0713988
$297$	0	0
$298$	0.529036	0.0306462
$299$	−20.0329	−1.15853
$300$	0	0
$301$	−6.37089	−0.367212
$302$	0.0628306	0.00361549
$303$	0	0
$304$	27.5582	1.58057
$305$	−14.6834	−0.840771
$306$	0	0
$307$	24.8259	1.41689	0.708443	−	0.705768i	$-0.249398\pi$
$307$	24.8259	1.41689	0.708443	+	0.705768i	$0.249398\pi$
$308$	−8.72536	−0.497173
$309$	0	0
$310$	−0.319945	−0.0181717
$311$	−28.3900	−1.60985	−0.804926	−	0.593375i	$-0.797795\pi$
$311$	−28.3900	−1.60985	−0.804926	+	0.593375i	$0.797795\pi$
$312$	0	0
$313$	−24.0947	−1.36191	−0.680957	−	0.732323i	$-0.738436\pi$
$313$	−24.0947	−1.36191	−0.680957	+	0.732323i	$0.738436\pi$
$314$	−0.477889	−0.0269688
$315$	0	0
$316$	29.8901	1.68145
$317$	26.5259	1.48984	0.744921	−	0.667153i	$-0.232487\pi$
$317$	26.5259	1.48984	0.744921	+	0.667153i	$0.232487\pi$
$318$	0	0
$319$	4.41896	0.247414
$320$	8.14203	0.455153
$321$	0	0
$322$	0.189573	0.0105645
$323$	−41.9641	−2.33494
$324$	0	0
$325$	15.3175	0.849665
$326$	−0.0931604	−0.00515968
$327$	0	0
$328$	0.229537	0.0126741
$329$	−4.23682	−0.233583
$330$	0	0
$331$	18.8041	1.03357	0.516783	−	0.856116i	$-0.327129\pi$
$331$	18.8041	1.03357	0.516783	+	0.856116i	$0.327129\pi$
$332$	−30.0339	−1.64833
$333$	0	0
$334$	−0.0773692	−0.00423345
$335$	−7.72319	−0.421963
$336$	0	0
$337$	22.4878	1.22499	0.612494	−	0.790475i	$-0.290166\pi$
$337$	22.4878	1.22499	0.612494	+	0.790475i	$0.290166\pi$
$338$	0.0730598	0.00397393
$339$	0	0
$340$	−12.4149	−0.673295
$341$	37.3025	2.02004
$342$	0	0
$343$	1.00000	0.0539949
$344$	0.933475	0.0503296
$345$	0	0
$346$	0.0501454	0.00269583
$347$	−17.8599	−0.958772	−0.479386	−	0.877604i	$-0.659141\pi$
$347$	−17.8599	−0.958772	−0.479386	+	0.877604i	$0.659141\pi$
$348$	0	0
$349$	30.3258	1.62330	0.811652	−	0.584141i	$-0.198569\pi$
$349$	30.3258	1.62330	0.811652	+	0.584141i	$0.198569\pi$
$350$	−0.144951	−0.00774795
$351$	0	0
$352$	1.91790	0.102224
$353$	−23.4387	−1.24752	−0.623759	−	0.781617i	$-0.714395\pi$
$353$	−23.4387	−1.24752	−0.623759	+	0.781617i	$0.714395\pi$
$354$	0	0
$355$	15.5372	0.824629
$356$	−6.28051	−0.332867
$357$	0	0
$358$	0.811094	0.0428676
$359$	−33.4031	−1.76295	−0.881473	−	0.472234i	$-0.843448\pi$
$359$	−33.4031	−1.76295	−0.881473	+	0.472234i	$0.843448\pi$
$360$	0	0
$361$	28.6575	1.50829
$362$	0.356928	0.0187597
$363$	0	0
$364$	7.73918	0.405643
$365$	2.03796	0.106672
$366$	0	0
$367$	5.05854	0.264054	0.132027	−	0.991246i	$-0.457852\pi$
$367$	5.05854	0.264054	0.132027	+	0.991246i	$0.457852\pi$
$368$	20.6525	1.07659
$369$	0	0
$370$	0.313919	0.0163199
$371$	5.22125	0.271074
$372$	0	0
$373$	19.5903	1.01435	0.507173	−	0.861844i	$-0.330690\pi$
$373$	19.5903	1.01435	0.507173	+	0.861844i	$0.330690\pi$
$374$	−0.972400	−0.0502816
$375$	0	0
$376$	0.620786	0.0320146
$377$	−3.91951	−0.201865
$378$	0	0
$379$	33.1031	1.70039	0.850195	−	0.526467i	$-0.176484\pi$
$379$	33.1031	1.70039	0.850195	+	0.526467i	$0.176484\pi$
$380$	14.0993	0.723281
$381$	0	0
$382$	0.535543	0.0274008
$383$	29.4067	1.50261	0.751307	−	0.659953i	$-0.229424\pi$
$383$	29.4067	1.50261	0.751307	+	0.659953i	$0.229424\pi$
$384$	0	0
$385$	−4.46107	−0.227357
$386$	0.511251	0.0260220
$387$	0	0
$388$	32.8718	1.66881
$389$	22.2547	1.12836	0.564178	−	0.825653i	$-0.309193\pi$
$389$	22.2547	1.12836	0.564178	+	0.825653i	$0.309193\pi$
$390$	0	0
$391$	−31.4485	−1.59042
$392$	−0.146522	−0.00740047
$393$	0	0
$394$	0.900692	0.0453762
$395$	15.2821	0.768926
$396$	0	0
$397$	15.2378	0.764765	0.382383	−	0.924004i	$-0.375104\pi$
$397$	15.2378	0.764765	0.382383	+	0.924004i	$0.375104\pi$
$398$	0.727307	0.0364566
$399$	0	0
$400$	−15.7913	−0.789564
$401$	7.18602	0.358853	0.179426	−	0.983771i	$-0.442576\pi$
$401$	7.18602	0.358853	0.179426	+	0.983771i	$0.442576\pi$
$402$	0	0
$403$	−33.0864	−1.64815
$404$	4.40000	0.218908
$405$	0	0
$406$	0.0370905	0.00184077
$407$	−36.5999	−1.81419
$408$	0	0
$409$	13.1266	0.649067	0.324534	−	0.945874i	$-0.394793\pi$
$409$	13.1266	0.649067	0.324534	+	0.945874i	$0.394793\pi$
$410$	0.0586588	0.00289695
$411$	0	0
$412$	24.5758	1.21076
$413$	7.49969	0.369036
$414$	0	0
$415$	−15.3557	−0.753780
$416$	−1.70113	−0.0834046
$417$	0	0
$418$	1.10433	0.0540145
$419$	23.9348	1.16929	0.584645	−	0.811289i	$-0.301234\pi$
$419$	23.9348	1.16929	0.584645	+	0.811289i	$0.301234\pi$
$420$	0	0
$421$	−3.33899	−0.162732	−0.0813662	−	0.996684i	$-0.525928\pi$
$421$	−3.33899	−0.162732	−0.0813662	+	0.996684i	$0.525928\pi$
$422$	−0.586265	−0.0285389
$423$	0	0
$424$	−0.765027	−0.0371530
$425$	24.0461	1.16641
$426$	0	0
$427$	14.3692	0.695375
$428$	5.26004	0.254254
$429$	0	0
$430$	0.238552	0.0115040
$431$	18.6000	0.895932	0.447966	−	0.894051i	$-0.352149\pi$
$431$	18.6000	0.895932	0.447966	+	0.894051i	$0.352149\pi$
$432$	0	0
$433$	−10.1711	−0.488792	−0.244396	−	0.969676i	$-0.578590\pi$
$433$	−10.1711	−0.488792	−0.244396	+	0.969676i	$0.578590\pi$
$434$	0.313098	0.0150292
$435$	0	0
$436$	−0.849862	−0.0407010
$437$	35.7152	1.70849
$438$	0	0
$439$	5.54172	0.264492	0.132246	−	0.991217i	$-0.457781\pi$
$439$	5.54172	0.264492	0.132246	+	0.991217i	$0.457781\pi$
$440$	0.653645	0.0311613
$441$	0	0
$442$	0.862495	0.0410247
$443$	11.7233	0.556989	0.278495	−	0.960438i	$-0.410165\pi$
$443$	11.7233	0.556989	0.278495	+	0.960438i	$0.410165\pi$
$444$	0	0
$445$	−3.21108	−0.152220
$446$	0.498161	0.0235886
$447$	0	0
$448$	−7.96779	−0.376443
$449$	26.5397	1.25249	0.626243	−	0.779628i	$-0.284592\pi$
$449$	26.5397	1.25249	0.626243	+	0.779628i	$0.284592\pi$
$450$	0	0
$451$	−6.83904	−0.322038
$452$	−34.5322	−1.62426
$453$	0	0
$454$	0.178439	0.00837457
$455$	3.95686	0.185501
$456$	0	0
$457$	9.91495	0.463802	0.231901	−	0.972739i	$-0.425505\pi$
$457$	9.91495	0.463802	0.231901	+	0.972739i	$0.425505\pi$
$458$	−0.843412	−0.0394101
$459$	0	0
$460$	10.5662	0.492654
$461$	1.17683	0.0548107	0.0274053	−	0.999624i	$-0.491276\pi$
$461$	1.17683	0.0548107	0.0274053	+	0.999624i	$0.491276\pi$
$462$	0	0
$463$	19.2312	0.893751	0.446875	−	0.894596i	$-0.352537\pi$
$463$	19.2312	0.893751	0.446875	+	0.894596i	$0.352537\pi$
$464$	4.04073	0.187586
$465$	0	0
$466$	0.671458	0.0311047
$467$	−22.2523	−1.02972	−0.514858	−	0.857276i	$-0.672155\pi$
$467$	−22.2523	−1.02972	−0.514858	+	0.857276i	$0.672155\pi$
$468$	0	0
$469$	7.55792	0.348992
$470$	0.158644	0.00731768
$471$	0	0
$472$	−1.09887	−0.0505795
$473$	−27.8128	−1.27883
$474$	0	0
$475$	−27.3085	−1.25300
$476$	12.1493	0.556861
$477$	0	0
$478$	0.618603	0.0282943
$479$	−5.63308	−0.257382	−0.128691	−	0.991685i	$-0.541078\pi$
$479$	−5.63308	−0.257382	−0.128691	+	0.991685i	$0.541078\pi$
$480$	0	0
$481$	32.4632	1.48019
$482$	−0.705117	−0.0321172
$483$	0	0
$484$	−16.1063	−0.732103
$485$	16.8066	0.763148
$486$	0	0
$487$	25.9817	1.17734	0.588672	−	0.808372i	$-0.299651\pi$
$487$	25.9817	1.17734	0.588672	+	0.808372i	$0.299651\pi$
$488$	−2.10540	−0.0953071
$489$	0	0
$490$	−0.0374440	−0.00169155
$491$	25.7518	1.16216	0.581081	−	0.813846i	$-0.302630\pi$
$491$	25.7518	1.16216	0.581081	+	0.813846i	$0.302630\pi$
$492$	0	0
$493$	−6.15300	−0.277117
$494$	−0.979513	−0.0440704
$495$	0	0
$496$	34.1097	1.53157
$497$	−15.2047	−0.682024
$498$	0	0
$499$	29.1717	1.30591	0.652953	−	0.757398i	$-0.273530\pi$
$499$	29.1717	1.30591	0.652953	+	0.757398i	$0.273530\pi$
$500$	−18.2910	−0.817997
$501$	0	0
$502$	−0.515358	−0.0230015
$503$	13.8331	0.616787	0.308393	−	0.951259i	$-0.400209\pi$
$503$	13.8331	0.616787	0.308393	+	0.951259i	$0.400209\pi$
$504$	0	0
$505$	2.24962	0.100107
$506$	0.827600	0.0367913
$507$	0	0
$508$	−1.99866	−0.0886761
$509$	−6.56616	−0.291040	−0.145520	−	0.989355i	$-0.546485\pi$
$509$	−6.56616	−0.291040	−0.145520	+	0.989355i	$0.546485\pi$
$510$	0	0
$511$	−1.99435	−0.0882249
$512$	2.92355	0.129204
$513$	0	0
$514$	−0.135605	−0.00598127
$515$	12.5651	0.553683
$516$	0	0
$517$	−18.4963	−0.813466
$518$	−0.307201	−0.0134976
$519$	0	0
$520$	−0.579767	−0.0254245
$521$	5.05524	0.221474	0.110737	−	0.993850i	$-0.464679\pi$
$521$	5.05524	0.221474	0.110737	+	0.993850i	$0.464679\pi$
$522$	0	0
$523$	11.1440	0.487292	0.243646	−	0.969864i	$-0.421656\pi$
$523$	11.1440	0.487292	0.243646	+	0.969864i	$0.421656\pi$
$524$	4.91500	0.214713
$525$	0	0
$526$	0.0339695	0.00148114
$527$	−51.9403	−2.26256
$528$	0	0
$529$	3.76551	0.163718
$530$	−0.195505	−0.00849219
$531$	0	0
$532$	−13.7976	−0.598202
$533$	6.06606	0.262750
$534$	0	0
$535$	2.68934	0.116270
$536$	−1.10740	−0.0478324
$537$	0	0
$538$	0.839768	0.0362050
$539$	4.36561	0.188040
$540$	0	0
$541$	−10.6947	−0.459801	−0.229901	−	0.973214i	$-0.573840\pi$
$541$	−10.6947	−0.459801	−0.229901	+	0.973214i	$0.573840\pi$
$542$	0.150569	0.00646749
$543$	0	0
$544$	−2.67050	−0.114497
$545$	−0.434515	−0.0186126
$546$	0	0
$547$	−34.1696	−1.46099	−0.730494	−	0.682919i	$-0.760710\pi$
$547$	−34.1696	−1.46099	−0.730494	+	0.682919i	$0.760710\pi$
$548$	7.07243	0.302119
$549$	0	0
$550$	−0.632799	−0.0269826
$551$	6.98780	0.297690
$552$	0	0
$553$	−14.9551	−0.635954
$554$	−0.0958564	−0.00407255
$555$	0	0
$556$	14.6053	0.619401
$557$	−41.7086	−1.76725	−0.883625	−	0.468195i	$-0.844904\pi$
$557$	−41.7086	−1.76725	−0.883625	+	0.468195i	$0.844904\pi$
$558$	0	0
$559$	24.6693	1.04340
$560$	−4.07924	−0.172379
$561$	0	0
$562$	0.0859306	0.00362476
$563$	−21.3267	−0.898814	−0.449407	−	0.893327i	$-0.648365\pi$
$563$	−21.3267	−0.898814	−0.449407	+	0.893327i	$0.648365\pi$
$564$	0	0
$565$	−17.6555	−0.742773
$566$	−0.325099	−0.0136649
$567$	0	0
$568$	2.22782	0.0934773
$569$	−28.0181	−1.17458	−0.587290	−	0.809377i	$-0.699805\pi$
$569$	−28.0181	−1.17458	−0.587290	+	0.809377i	$0.699805\pi$
$570$	0	0
$571$	40.4176	1.69143	0.845713	−	0.533638i	$-0.179176\pi$
$571$	40.4176	1.69143	0.845713	+	0.533638i	$0.179176\pi$
$572$	33.7862	1.41267
$573$	0	0
$574$	−0.0574035	−0.00239598
$575$	−20.4654	−0.853467
$576$	0	0
$577$	−19.8471	−0.826244	−0.413122	−	0.910676i	$-0.635562\pi$
$577$	−19.8471	−0.826244	−0.413122	+	0.910676i	$0.635562\pi$
$578$	0.731052	0.0304078
$579$	0	0
$580$	2.06732	0.0858408
$581$	15.0270	0.623427
$582$	0	0
$583$	22.7940	0.944029
$584$	0.292216	0.0120920
$585$	0	0
$586$	0.444184	0.0183491
$587$	19.8820	0.820616	0.410308	−	0.911947i	$-0.365421\pi$
$587$	19.8820	0.820616	0.410308	+	0.911947i	$0.365421\pi$
$588$	0	0
$589$	58.9873	2.43053
$590$	−0.280819	−0.0115611
$591$	0	0
$592$	−33.4672	−1.37549
$593$	−29.7636	−1.22225	−0.611123	−	0.791536i	$-0.709282\pi$
$593$	−29.7636	−1.22225	−0.611123	+	0.791536i	$0.709282\pi$
$594$	0	0
$595$	6.21164	0.254653
$596$	−28.8560	−1.18199
$597$	0	0
$598$	−0.734061	−0.0300180
$599$	39.5210	1.61478	0.807392	−	0.590016i	$-0.200878\pi$
$599$	39.5210	1.61478	0.807392	+	0.590016i	$0.200878\pi$
$600$	0	0
$601$	−8.39771	−0.342550	−0.171275	−	0.985223i	$-0.554789\pi$
$601$	−8.39771	−0.342550	−0.171275	+	0.985223i	$0.554789\pi$
$602$	−0.233447	−0.00951459
$603$	0	0
$604$	−3.42706	−0.139445
$605$	−8.23476	−0.334791
$606$	0	0
$607$	−8.95531	−0.363485	−0.181742	−	0.983346i	$-0.558174\pi$
$607$	−8.95531	−0.363485	−0.181742	+	0.983346i	$0.558174\pi$
$608$	3.03282	0.122997
$609$	0	0
$610$	−0.538041	−0.0217847
$611$	16.4058	0.663706
$612$	0	0
$613$	−20.6982	−0.835991	−0.417995	−	0.908449i	$-0.637267\pi$
$613$	−20.6982	−0.835991	−0.417995	+	0.908449i	$0.637267\pi$
$614$	0.909688	0.0367120
$615$	0	0
$616$	−0.639657	−0.0257725
$617$	−5.71815	−0.230204	−0.115102	−	0.993354i	$-0.536720\pi$
$617$	−5.71815	−0.230204	−0.115102	+	0.993354i	$0.536720\pi$
$618$	0	0
$619$	−41.8747	−1.68309	−0.841544	−	0.540189i	$-0.818353\pi$
$619$	−41.8747	−1.68309	−0.841544	+	0.540189i	$0.818353\pi$
$620$	17.4512	0.700858
$621$	0	0
$622$	−1.04029	−0.0417118
$623$	3.14237	0.125896
$624$	0	0
$625$	10.4272	0.417087
$626$	−0.882897	−0.0352876
$627$	0	0
$628$	26.0662	1.04015
$629$	50.9620	2.03199
$630$	0	0
$631$	−0.116518	−0.00463852	−0.00231926	−	0.999997i	$-0.500738\pi$
$631$	−0.116518	−0.00463852	−0.00231926	+	0.999997i	$0.500738\pi$
$632$	2.19124	0.0871630
$633$	0	0
$634$	0.971981	0.0386023
$635$	−1.02187	−0.0405516
$636$	0	0
$637$	−3.87219	−0.153422
$638$	0.161923	0.00641058
$639$	0	0
$640$	1.19620	0.0472839
$641$	12.7160	0.502250	0.251125	−	0.967955i	$-0.419199\pi$
$641$	12.7160	0.502250	0.251125	+	0.967955i	$0.419199\pi$
$642$	0	0
$643$	5.87970	0.231873	0.115936	−	0.993257i	$-0.463013\pi$
$643$	5.87970	0.231873	0.115936	+	0.993257i	$0.463013\pi$
$644$	−10.3401	−0.407458
$645$	0	0
$646$	−1.53768	−0.0604992
$647$	−24.5196	−0.963966	−0.481983	−	0.876181i	$-0.660083\pi$
$647$	−24.5196	−0.963966	−0.481983	+	0.876181i	$0.660083\pi$
$648$	0	0
$649$	32.7407	1.28519
$650$	0.561277	0.0220151
$651$	0	0
$652$	5.08138	0.199002
$653$	26.9393	1.05421	0.527107	−	0.849799i	$-0.323277\pi$
$653$	26.9393	1.05421	0.527107	+	0.849799i	$0.323277\pi$
$654$	0	0
$655$	2.51292	0.0981881
$656$	−6.25367	−0.244165
$657$	0	0
$658$	−0.155249	−0.00605222
$659$	−25.2200	−0.982432	−0.491216	−	0.871038i	$-0.663447\pi$
$659$	−25.2200	−0.982432	−0.491216	+	0.871038i	$0.663447\pi$
$660$	0	0
$661$	−18.6507	−0.725429	−0.362714	−	0.931900i	$-0.618150\pi$
$661$	−18.6507	−0.725429	−0.362714	+	0.931900i	$0.618150\pi$
$662$	0.689034	0.0267801
$663$	0	0
$664$	−2.20179	−0.0854460
$665$	−7.05440	−0.273558
$666$	0	0
$667$	5.23676	0.202768
$668$	4.22006	0.163279
$669$	0	0
$670$	−0.282999	−0.0109332
$671$	62.7304	2.42168
$672$	0	0
$673$	17.2225	0.663879	0.331939	−	0.943301i	$-0.392297\pi$
$673$	17.2225	0.663879	0.331939	+	0.943301i	$0.392297\pi$
$674$	0.824015	0.0317399
$675$	0	0
$676$	−3.98501	−0.153269
$677$	14.8363	0.570207	0.285104	−	0.958497i	$-0.407972\pi$
$677$	14.8363	0.570207	0.285104	+	0.958497i	$0.407972\pi$
$678$	0	0
$679$	−16.4469	−0.631175
$680$	−0.910141	−0.0349023
$681$	0	0
$682$	1.36686	0.0523400
$683$	6.22360	0.238139	0.119070	−	0.992886i	$-0.462009\pi$
$683$	6.22360	0.238139	0.119070	+	0.992886i	$0.462009\pi$
$684$	0	0
$685$	3.61597	0.138159
$686$	0.0366427	0.00139903
$687$	0	0
$688$	−25.4323	−0.969596
$689$	−20.2177	−0.770232
$690$	0	0
$691$	−16.4440	−0.625561	−0.312780	−	0.949825i	$-0.601260\pi$
$691$	−16.4440	−0.625561	−0.312780	+	0.949825i	$0.601260\pi$
$692$	−2.73515	−0.103975
$693$	0	0
$694$	−0.654437	−0.0248421
$695$	7.46733	0.283252
$696$	0	0
$697$	9.52275	0.360700
$698$	1.11122	0.0420604
$699$	0	0
$700$	7.90626	0.298829
$701$	33.7827	1.27595	0.637977	−	0.770055i	$-0.279771\pi$
$701$	33.7827	1.27595	0.637977	+	0.770055i	$0.279771\pi$
$702$	0	0
$703$	−57.8762	−2.18284
$704$	−34.7843	−1.31098
$705$	0	0
$706$	−0.858859	−0.0323236
$707$	−2.20148	−0.0827950
$708$	0	0
$709$	49.4454	1.85696	0.928481	−	0.371380i	$-0.121115\pi$
$709$	49.4454	1.85696	0.928481	+	0.371380i	$0.121115\pi$
$710$	0.569326	0.0213664
$711$	0	0
$712$	−0.460425	−0.0172552
$713$	44.2059	1.65552
$714$	0	0
$715$	17.2741	0.646015
$716$	−44.2406	−1.65335
$717$	0	0
$718$	−1.22398	−0.0456785
$719$	39.8435	1.48591	0.742955	−	0.669341i	$-0.233424\pi$
$719$	39.8435	1.48591	0.742955	+	0.669341i	$0.233424\pi$
$720$	0	0
$721$	−12.2962	−0.457933
$722$	1.05009	0.0390803
$723$	0	0
$724$	−19.4684	−0.723538
$725$	−4.00413	−0.148709
$726$	0	0
$727$	−15.6229	−0.579421	−0.289710	−	0.957114i	$-0.593559\pi$
$727$	−15.6229	−0.579421	−0.289710	+	0.957114i	$0.593559\pi$
$728$	0.567360	0.0210278
$729$	0	0
$730$	0.0746766	0.00276390
$731$	38.7269	1.43236
$732$	0	0
$733$	37.9306	1.40100	0.700499	−	0.713653i	$-0.252961\pi$
$733$	37.9306	1.40100	0.700499	+	0.713653i	$0.252961\pi$
$734$	0.185359	0.00684171
$735$	0	0
$736$	2.27284	0.0837779
$737$	32.9949	1.21538
$738$	0	0
$739$	24.3400	0.895362	0.447681	−	0.894193i	$-0.352250\pi$
$739$	24.3400	0.895362	0.447681	+	0.894193i	$0.352250\pi$
$740$	−17.1225	−0.629436
$741$	0	0
$742$	0.191321	0.00702362
$743$	−29.3253	−1.07584	−0.537920	−	0.842996i	$-0.680790\pi$
$743$	−29.3253	−1.07584	−0.537920	+	0.842996i	$0.680790\pi$
$744$	0	0
$745$	−14.7534	−0.540522
$746$	0.717842	0.0262821
$747$	0	0
$748$	53.0390	1.93930
$749$	−2.63179	−0.0961634
$750$	0	0
$751$	−1.30594	−0.0476543	−0.0238272	−	0.999716i	$-0.507585\pi$
$751$	−1.30594	−0.0476543	−0.0238272	+	0.999716i	$0.507585\pi$
$752$	−16.9131	−0.616759
$753$	0	0
$754$	−0.143621	−0.00523039
$755$	−1.75218	−0.0637682
$756$	0	0
$757$	−16.6342	−0.604580	−0.302290	−	0.953216i	$-0.597751\pi$
$757$	−16.6342	−0.604580	−0.302290	+	0.953216i	$0.597751\pi$
$758$	1.21299	0.0440577
$759$	0	0
$760$	1.03362	0.0374935
$761$	−20.4601	−0.741677	−0.370838	−	0.928697i	$-0.620930\pi$
$761$	−20.4601	−0.741677	−0.370838	+	0.928697i	$0.620930\pi$
$762$	0	0
$763$	0.425217	0.0153939
$764$	−29.2109	−1.05681
$765$	0	0
$766$	1.07754	0.0389332
$767$	−29.0402	−1.04858
$768$	0	0
$769$	37.8309	1.36422	0.682108	−	0.731251i	$-0.261063\pi$
$769$	37.8309	1.36422	0.682108	+	0.731251i	$0.261063\pi$
$770$	−0.163466	−0.00589091
$771$	0	0
$772$	−27.8859	−1.00363
$773$	−23.3436	−0.839613	−0.419806	−	0.907614i	$-0.637902\pi$
$773$	−23.3436	−0.839613	−0.419806	+	0.907614i	$0.637902\pi$
$774$	0	0
$775$	−33.8007	−1.21416
$776$	2.40983	0.0865080
$777$	0	0
$778$	0.815472	0.0292361
$779$	−10.8147	−0.387478
$780$	0	0
$781$	−66.3778	−2.37518
$782$	−1.15236	−0.0412083
$783$	0	0
$784$	3.99195	0.142569
$785$	13.3270	0.475662
$786$	0	0
$787$	11.4350	0.407613	0.203806	−	0.979011i	$-0.434669\pi$
$787$	11.4350	0.407613	0.203806	+	0.979011i	$0.434669\pi$
$788$	−49.1277	−1.75010
$789$	0	0
$790$	0.559978	0.0199231
$791$	17.2777	0.614324
$792$	0	0
$793$	−55.6403	−1.97585
$794$	0.558356	0.0198153
$795$	0	0
$796$	−39.6705	−1.40609
$797$	46.3966	1.64345	0.821726	−	0.569883i	$-0.193012\pi$
$797$	46.3966	1.64345	0.821726	+	0.569883i	$0.193012\pi$
$798$	0	0
$799$	25.7544	0.911126
$800$	−1.73785	−0.0614424
$801$	0	0
$802$	0.263316	0.00929800
$803$	−8.70656	−0.307248
$804$	0	0
$805$	−5.28667	−0.186331
$806$	−1.21238	−0.0427041
$807$	0	0
$808$	0.322564	0.0113478
$809$	38.6606	1.35923	0.679617	−	0.733567i	$-0.262146\pi$
$809$	38.6606	1.35923	0.679617	+	0.733567i	$0.262146\pi$
$810$	0	0
$811$	18.9107	0.664043	0.332021	−	0.943272i	$-0.392269\pi$
$811$	18.9107	0.664043	0.332021	+	0.943272i	$0.392269\pi$
$812$	−2.02308	−0.0709962
$813$	0	0
$814$	−1.34112	−0.0470062
$815$	2.59799	0.0910037
$816$	0	0
$817$	−43.9811	−1.53870
$818$	0.480994	0.0168175
$819$	0	0
$820$	−3.19951	−0.111732
$821$	41.0661	1.43322	0.716608	−	0.697477i	$-0.245694\pi$
$821$	41.0661	1.43322	0.716608	+	0.697477i	$0.245694\pi$
$822$	0	0
$823$	−10.9130	−0.380402	−0.190201	−	0.981745i	$-0.560914\pi$
$823$	−10.9130	−0.380402	−0.190201	+	0.981745i	$0.560914\pi$
$824$	1.80166	0.0627637
$825$	0	0
$826$	0.274809	0.00956184
$827$	16.0479	0.558041	0.279021	−	0.960285i	$-0.409990\pi$
$827$	16.0479	0.558041	0.279021	+	0.960285i	$0.409990\pi$
$828$	0	0
$829$	−48.1452	−1.67215	−0.836076	−	0.548614i	$-0.815156\pi$
$829$	−48.1452	−1.67215	−0.836076	+	0.548614i	$0.815156\pi$
$830$	−0.562673	−0.0195307
$831$	0	0
$832$	30.8528	1.06963
$833$	−6.07872	−0.210615
$834$	0	0
$835$	2.15762	0.0746675
$836$	−60.2350	−2.08327
$837$	0	0
$838$	0.877036	0.0302967
$839$	28.2575	0.975558	0.487779	−	0.872967i	$-0.337807\pi$
$839$	28.2575	0.975558	0.487779	+	0.872967i	$0.337807\pi$
$840$	0	0
$841$	−27.9754	−0.964669
$842$	−0.122350	−0.00421645
$843$	0	0
$844$	31.9775	1.10071
$845$	−2.03744	−0.0700901
$846$	0	0
$847$	8.05854	0.276895
$848$	20.8430	0.715750
$849$	0	0
$850$	0.881115	0.0302220
$851$	−43.3733	−1.48682
$852$	0	0
$853$	−8.89198	−0.304456	−0.152228	−	0.988345i	$-0.548645\pi$
$853$	−8.89198	−0.304456	−0.152228	+	0.988345i	$0.548645\pi$
$854$	0.526528	0.0180174
$855$	0	0
$856$	0.385614	0.0131800
$857$	−6.16814	−0.210700	−0.105350	−	0.994435i	$-0.533596\pi$
$857$	−6.16814	−0.210700	−0.105350	+	0.994435i	$0.533596\pi$
$858$	0	0
$859$	−13.4714	−0.459640	−0.229820	−	0.973233i	$-0.573814\pi$
$859$	−13.4714	−0.459640	−0.229820	+	0.973233i	$0.573814\pi$
$860$	−13.0117	−0.443694
$861$	0	0
$862$	0.681556	0.0232139
$863$	−8.88588	−0.302479	−0.151239	−	0.988497i	$-0.548326\pi$
$863$	−8.88588	−0.302479	−0.151239	+	0.988497i	$0.548326\pi$
$864$	0	0
$865$	−1.39842	−0.0475477
$866$	−0.372697	−0.0126648
$867$	0	0
$868$	−17.0778	−0.579657
$869$	−65.2880	−2.21474
$870$	0	0
$871$	−29.2657	−0.991629
$872$	−0.0623035	−0.00210986
$873$	0	0
$874$	1.30870	0.0442676
$875$	9.15163	0.309382
$876$	0	0
$877$	−41.0939	−1.38764	−0.693821	−	0.720148i	$-0.744074\pi$
$877$	−41.0939	−1.38764	−0.693821	+	0.720148i	$0.744074\pi$
$878$	0.203064	0.00685307
$879$	0	0
$880$	−17.8084	−0.600320
$881$	−48.4814	−1.63338	−0.816690	−	0.577076i	$-0.804194\pi$
$881$	−48.4814	−1.63338	−0.816690	+	0.577076i	$0.804194\pi$
$882$	0	0
$883$	−14.3235	−0.482025	−0.241012	−	0.970522i	$-0.577479\pi$
$883$	−14.3235	−0.482025	−0.241012	+	0.970522i	$0.577479\pi$
$884$	−47.0443	−1.58227
$885$	0	0
$886$	0.429573	0.0144318
$887$	−16.4012	−0.550700	−0.275350	−	0.961344i	$-0.588794\pi$
$887$	−16.4012	−0.550700	−0.275350	+	0.961344i	$0.588794\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−0.117663	−0.00394407
$891$	0	0
$892$	−27.1719	−0.909783
$893$	−29.2486	−0.978768
$894$	0	0
$895$	−22.6192	−0.756077
$896$	−1.17060	−0.0391070
$897$	0	0
$898$	0.972488	0.0324523
$899$	8.64903	0.288461
$900$	0	0
$901$	−31.7385	−1.05736
$902$	−0.250601	−0.00834411
$903$	0	0
$904$	−2.53156	−0.0841983
$905$	−9.95376	−0.330874
$906$	0	0
$907$	3.75725	0.124757	0.0623787	−	0.998053i	$-0.480131\pi$
$907$	3.75725	0.124757	0.0623787	+	0.998053i	$0.480131\pi$
$908$	−9.73287	−0.322997
$909$	0	0
$910$	0.144990	0.00480638
$911$	36.8735	1.22167	0.610837	−	0.791756i	$-0.290833\pi$
$911$	36.8735	1.22167	0.610837	+	0.791756i	$0.290833\pi$
$912$	0	0
$913$	65.6022	2.17112
$914$	0.363311	0.0120173
$915$	0	0
$916$	46.0034	1.52000
$917$	−2.45915	−0.0812083
$918$	0	0
$919$	23.2231	0.766060	0.383030	−	0.923736i	$-0.374881\pi$
$919$	23.2231	0.766060	0.383030	+	0.923736i	$0.374881\pi$
$920$	0.774613	0.0255382
$921$	0	0
$922$	0.0431225	0.00142016
$923$	58.8755	1.93791
$924$	0	0
$925$	33.1640	1.09043
$926$	0.704685	0.0231574
$927$	0	0
$928$	0.444688	0.0145976
$929$	−46.8936	−1.53853	−0.769265	−	0.638930i	$-0.779377\pi$
$929$	−46.8936	−1.53853	−0.769265	+	0.638930i	$0.779377\pi$
$930$	0	0
$931$	6.90344	0.226251
$932$	−36.6243	−1.19967
$933$	0	0
$934$	−0.815387	−0.0266803
$935$	27.1176	0.886841
$936$	0	0
$937$	−6.16009	−0.201241	−0.100621	−	0.994925i	$-0.532083\pi$
$937$	−6.16009	−0.201241	−0.100621	+	0.994925i	$0.532083\pi$
$938$	0.276943	0.00904250
$939$	0	0
$940$	−8.65312	−0.282234
$941$	−40.2409	−1.31182	−0.655908	−	0.754841i	$-0.727714\pi$
$941$	−40.2409	−1.31182	−0.655908	+	0.754841i	$0.727714\pi$
$942$	0	0
$943$	−8.10472	−0.263926
$944$	29.9384	0.974411
$945$	0	0
$946$	−1.01914	−0.0331350
$947$	−0.731230	−0.0237618	−0.0118809	−	0.999929i	$-0.503782\pi$
$947$	−0.731230	−0.0237618	−0.0118809	+	0.999929i	$0.503782\pi$
$948$	0	0
$949$	7.72250	0.250683
$950$	−1.00066	−0.0324657
$951$	0	0
$952$	0.890665	0.0288666
$953$	−2.38255	−0.0771785	−0.0385892	−	0.999255i	$-0.512286\pi$
$953$	−2.38255	−0.0771785	−0.0385892	+	0.999255i	$0.512286\pi$
$954$	0	0
$955$	−14.9349	−0.483281
$956$	−33.7414	−1.09127
$957$	0	0
$958$	−0.206412	−0.00666885
$959$	−3.53859	−0.114267
$960$	0	0
$961$	42.0105	1.35518
$962$	1.18954	0.0383523
$963$	0	0
$964$	38.4602	1.23872
$965$	−14.2574	−0.458962
$966$	0	0
$967$	−53.9701	−1.73556	−0.867781	−	0.496948i	$-0.834454\pi$
$967$	−53.9701	−1.73556	−0.867781	+	0.496948i	$0.834454\pi$
$968$	−1.18075	−0.0379508
$969$	0	0
$970$	0.615839	0.0197734
$971$	31.0823	0.997477	0.498739	−	0.866752i	$-0.333797\pi$
$971$	31.0823	0.997477	0.498739	+	0.866752i	$0.333797\pi$
$972$	0	0
$973$	−7.30754	−0.234269
$974$	0.952041	0.0305054
$975$	0	0
$976$	57.3611	1.83608
$977$	12.7232	0.407051	0.203526	−	0.979070i	$-0.434760\pi$
$977$	12.7232	0.407051	0.203526	+	0.979070i	$0.434760\pi$
$978$	0	0
$979$	13.7183	0.438440
$980$	2.04236	0.0652409
$981$	0	0
$982$	0.943617	0.0301120
$983$	−49.5307	−1.57978	−0.789892	−	0.613246i	$-0.789863\pi$
$983$	−49.5307	−1.57978	−0.789892	+	0.613246i	$0.789863\pi$
$984$	0	0
$985$	−25.1179	−0.800322
$986$	−0.225463	−0.00718020
$987$	0	0
$988$	53.4270	1.69974
$989$	−32.9601	−1.04807
$990$	0	0
$991$	38.0811	1.20969	0.604843	−	0.796345i	$-0.293236\pi$
$991$	38.0811	1.20969	0.604843	+	0.796345i	$0.293236\pi$
$992$	3.75382	0.119184
$993$	0	0
$994$	−0.557142	−0.0176715
$995$	−20.2826	−0.643003
$996$	0	0
$997$	11.9772	0.379320	0.189660	−	0.981850i	$-0.439261\pi$
$997$	11.9772	0.379320	0.189660	+	0.981850i	$0.439261\pi$
$998$	1.06893	0.0338365
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.9		18
3.2	odd	2		2667.2.a.p.1.10	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.10	✓	18	3.2	odd	2
8001.2.a.u.1.9		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)

\(f(q)\)	\(=\)	\(q+0.0366427 q^{2} -1.99866 q^{4} -1.02187 q^{5} +1.00000 q^{7} -0.146522 q^{8} +O(q^{10})\)	\(q+0.0366427 q^{2} -1.99866 q^{4} -1.02187 q^{5} +1.00000 q^{7} -0.146522 q^{8} -0.0374440 q^{10} +4.36561 q^{11} -3.87219 q^{13} +0.0366427 q^{14} +3.99195 q^{16} -6.07872 q^{17} +6.90344 q^{19} +2.04236 q^{20} +0.159968 q^{22} +5.17354 q^{23} -3.95579 q^{25} -0.141888 q^{26} -1.99866 q^{28} +1.01222 q^{29} +8.54462 q^{31} +0.439319 q^{32} -0.222741 q^{34} -1.02187 q^{35} -8.38368 q^{37} +0.252961 q^{38} +0.149726 q^{40} -1.56657 q^{41} -6.37089 q^{43} -8.72536 q^{44} +0.189573 q^{46} -4.23682 q^{47} +1.00000 q^{49} -0.144951 q^{50} +7.73918 q^{52} +5.22125 q^{53} -4.46107 q^{55} -0.146522 q^{56} +0.0370905 q^{58} +7.49969 q^{59} +14.3692 q^{61} +0.313098 q^{62} -7.96779 q^{64} +3.95686 q^{65} +7.55792 q^{67} +12.1493 q^{68} -0.0374440 q^{70} -15.2047 q^{71} -1.99435 q^{73} -0.307201 q^{74} -13.7976 q^{76} +4.36561 q^{77} -14.9551 q^{79} -4.07924 q^{80} -0.0574035 q^{82} +15.0270 q^{83} +6.21164 q^{85} -0.233447 q^{86} -0.639657 q^{88} +3.14237 q^{89} -3.87219 q^{91} -10.3401 q^{92} -0.155249 q^{94} -7.05440 q^{95} -16.4469 q^{97} +0.0366427 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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