LMFDB - Embedded newform 8001.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} + \cdots - 8 $ x^13 - 4x^12 - 10x^11 + 53x^10 + 19x^9 - 242x^8 + 61x^7 + 467x^6 - 211x^5 - 372x^4 + 146x^3 + 116x^2 - 12x - 8
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			1.2
Root			$2.29288$ 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)

$f(q)$	$=$	$q-2.29288 q^{2} +3.25728 q^{4} +0.132682 q^{5} +1.00000 q^{7} -2.88278 q^{8} +O(q^{10})$	\(q-2.29288 q^{2} +3.25728 q^{4} +0.132682 q^{5} +1.00000 q^{7} -2.88278 q^{8} -0.304223 q^{10} +1.38028 q^{11} +3.00012 q^{13} -2.29288 q^{14} +0.0952967 q^{16} -4.60865 q^{17} -1.78380 q^{19} +0.432182 q^{20} -3.16481 q^{22} -6.02678 q^{23} -4.98240 q^{25} -6.87890 q^{26} +3.25728 q^{28} -2.00225 q^{29} +9.54114 q^{31} +5.54705 q^{32} +10.5670 q^{34} +0.132682 q^{35} +2.84160 q^{37} +4.09003 q^{38} -0.382493 q^{40} -2.18268 q^{41} +9.51578 q^{43} +4.49595 q^{44} +13.8187 q^{46} -2.52668 q^{47} +1.00000 q^{49} +11.4240 q^{50} +9.77222 q^{52} +8.38962 q^{53} +0.183138 q^{55} -2.88278 q^{56} +4.59090 q^{58} -7.78977 q^{59} +3.77711 q^{61} -21.8766 q^{62} -12.9093 q^{64} +0.398062 q^{65} -10.1772 q^{67} -15.0116 q^{68} -0.304223 q^{70} +2.19559 q^{71} +1.19059 q^{73} -6.51543 q^{74} -5.81033 q^{76} +1.38028 q^{77} -10.3605 q^{79} +0.0126442 q^{80} +5.00462 q^{82} -1.81059 q^{83} -0.611485 q^{85} -21.8185 q^{86} -3.97904 q^{88} +7.25065 q^{89} +3.00012 q^{91} -19.6309 q^{92} +5.79337 q^{94} -0.236678 q^{95} +2.14183 q^{97} -2.29288 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * 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 $

$2$	−2.29288	−1.62131	−0.810654	−	0.585526i	$-0.800888\pi$
$2$	−2.29288	−1.62131	−0.810654	+	0.585526i	$0.800888\pi$
$3$	0	0
$3$	0	0
$4$	3.25728	1.62864
$5$	0.132682	0.0593372	0.0296686	−	0.999560i	$-0.490555\pi$
$5$	0.132682	0.0593372	0.0296686	+	0.999560i	$0.490555\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.88278	−1.01922
$9$	0	0
$10$	−0.304223	−0.0962039
$11$	1.38028	0.416170	0.208085	−	0.978111i	$-0.433277\pi$
$11$	1.38028	0.416170	0.208085	+	0.978111i	$0.433277\pi$
$12$	0	0
$13$	3.00012	0.832083	0.416042	−	0.909346i	$-0.363417\pi$
$13$	3.00012	0.832083	0.416042	+	0.909346i	$0.363417\pi$
$14$	−2.29288	−0.612797
$15$	0	0
$16$	0.0952967	0.0238242
$17$	−4.60865	−1.11776	−0.558880	−	0.829248i	$-0.688769\pi$
$17$	−4.60865	−1.11776	−0.558880	+	0.829248i	$0.688769\pi$
$18$	0	0
$19$	−1.78380	−0.409232	−0.204616	−	0.978842i	$-0.565595\pi$
$19$	−1.78380	−0.409232	−0.204616	+	0.978842i	$0.565595\pi$
$20$	0.432182	0.0966388
$21$	0	0
$22$	−3.16481	−0.674739
$23$	−6.02678	−1.25667	−0.628336	−	0.777942i	$-0.716264\pi$
$23$	−6.02678	−1.25667	−0.628336	+	0.777942i	$0.716264\pi$
$24$	0	0
$25$	−4.98240	−0.996479
$26$	−6.87890	−1.34906
$27$	0	0
$28$	3.25728	0.615567
$29$	−2.00225	−0.371808	−0.185904	−	0.982568i	$-0.559521\pi$
$29$	−2.00225	−0.371808	−0.185904	+	0.982568i	$0.559521\pi$
$30$	0	0
$31$	9.54114	1.71364	0.856820	−	0.515616i	$-0.172437\pi$
$31$	9.54114	1.71364	0.856820	+	0.515616i	$0.172437\pi$
$32$	5.54705	0.980590
$33$	0	0
$34$	10.5670	1.81223
$35$	0.132682	0.0224274
$36$	0	0
$37$	2.84160	0.467156	0.233578	−	0.972338i	$-0.424957\pi$
$37$	2.84160	0.467156	0.233578	+	0.972338i	$0.424957\pi$
$38$	4.09003	0.663491
$39$	0	0
$40$	−0.382493	−0.0604774
$41$	−2.18268	−0.340878	−0.170439	−	0.985368i	$-0.554519\pi$
$41$	−2.18268	−0.340878	−0.170439	+	0.985368i	$0.554519\pi$
$42$	0	0
$43$	9.51578	1.45114	0.725571	−	0.688147i	$-0.241576\pi$
$43$	9.51578	1.45114	0.725571	+	0.688147i	$0.241576\pi$
$44$	4.49595	0.677790
$45$	0	0
$46$	13.8187	2.03745
$47$	−2.52668	−0.368554	−0.184277	−	0.982874i	$-0.558994\pi$
$47$	−2.52668	−0.368554	−0.184277	+	0.982874i	$0.558994\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	11.4240	1.61560
$51$	0	0
$52$	9.77222	1.35516
$53$	8.38962	1.15240	0.576202	−	0.817308i	$-0.304534\pi$
$53$	8.38962	1.15240	0.576202	+	0.817308i	$0.304534\pi$
$54$	0	0
$55$	0.183138	0.0246944
$56$	−2.88278	−0.385227
$57$	0	0
$58$	4.59090	0.602815
$59$	−7.78977	−1.01414	−0.507071	−	0.861904i	$-0.669272\pi$
$59$	−7.78977	−1.01414	−0.507071	+	0.861904i	$0.669272\pi$
$60$	0	0
$61$	3.77711	0.483609	0.241804	−	0.970325i	$-0.422261\pi$
$61$	3.77711	0.483609	0.241804	+	0.970325i	$0.422261\pi$
$62$	−21.8766	−2.77834
$63$	0	0
$64$	−12.9093	−1.61366
$65$	0.398062	0.0493735
$66$	0	0
$67$	−10.1772	−1.24334	−0.621670	−	0.783280i	$-0.713545\pi$
$67$	−10.1772	−1.24334	−0.621670	+	0.783280i	$0.713545\pi$
$68$	−15.0116	−1.82043
$69$	0	0
$70$	−0.304223	−0.0363616
$71$	2.19559	0.260569	0.130284	−	0.991477i	$-0.458411\pi$
$71$	2.19559	0.260569	0.130284	+	0.991477i	$0.458411\pi$
$72$	0	0
$73$	1.19059	0.139348	0.0696738	−	0.997570i	$-0.477804\pi$
$73$	1.19059	0.139348	0.0696738	+	0.997570i	$0.477804\pi$
$74$	−6.51543	−0.757403
$75$	0	0
$76$	−5.81033	−0.666491
$77$	1.38028	0.157297
$78$	0	0
$79$	−10.3605	−1.16565	−0.582826	−	0.812597i	$-0.698053\pi$
$79$	−10.3605	−1.16565	−0.582826	+	0.812597i	$0.698053\pi$
$80$	0.0126442	0.00141366
$81$	0	0
$82$	5.00462	0.552668
$83$	−1.81059	−0.198738	−0.0993688	−	0.995051i	$-0.531682\pi$
$83$	−1.81059	−0.198738	−0.0993688	+	0.995051i	$0.531682\pi$
$84$	0	0
$85$	−0.611485	−0.0663248
$86$	−21.8185	−2.35275
$87$	0	0
$88$	−3.97904	−0.424167
$89$	7.25065	0.768567	0.384284	−	0.923215i	$-0.374448\pi$
$89$	7.25065	0.768567	0.384284	+	0.923215i	$0.374448\pi$
$90$	0	0
$91$	3.00012	0.314498
$92$	−19.6309	−2.04666
$93$	0	0
$94$	5.79337	0.597540
$95$	−0.236678	−0.0242827
$96$	0	0
$97$	2.14183	0.217469	0.108735	−	0.994071i	$-0.465320\pi$
$97$	2.14183	0.217469	0.108735	+	0.994071i	$0.465320\pi$
$98$	−2.29288	−0.231615
$99$	0	0
$100$	−16.2290	−1.62290
$101$	−16.2730	−1.61922	−0.809610	−	0.586969i	$-0.800321\pi$
$101$	−16.2730	−1.61922	−0.809610	+	0.586969i	$0.800321\pi$
$102$	0	0
$103$	−12.7663	−1.25790	−0.628952	−	0.777444i	$-0.716516\pi$
$103$	−12.7663	−1.25790	−0.628952	+	0.777444i	$0.716516\pi$
$104$	−8.64868	−0.848073
$105$	0	0
$106$	−19.2364	−1.86840
$107$	−1.40448	−0.135776	−0.0678880	−	0.997693i	$-0.521626\pi$
$107$	−1.40448	−0.135776	−0.0678880	+	0.997693i	$0.521626\pi$
$108$	0	0
$109$	11.1108	1.06422	0.532111	−	0.846674i	$-0.321399\pi$
$109$	11.1108	1.06422	0.532111	+	0.846674i	$0.321399\pi$
$110$	−0.419913	−0.0400371
$111$	0	0
$112$	0.0952967	0.00900469
$113$	−17.6560	−1.66094	−0.830469	−	0.557065i	$-0.811927\pi$
$113$	−17.6560	−1.66094	−0.830469	+	0.557065i	$0.811927\pi$
$114$	0	0
$115$	−0.799646	−0.0745674
$116$	−6.52187	−0.605540
$117$	0	0
$118$	17.8610	1.64424
$119$	−4.60865	−0.422474
$120$	0	0
$121$	−9.09483	−0.826803
$122$	−8.66043	−0.784079
$123$	0	0
$124$	31.0781	2.79090
$125$	−1.32448	−0.118466
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	18.5053	1.63565
$129$	0	0
$130$	−0.912706	−0.0800497
$131$	−5.73840	−0.501367	−0.250683	−	0.968069i	$-0.580655\pi$
$131$	−5.73840	−0.501367	−0.250683	+	0.968069i	$0.580655\pi$
$132$	0	0
$133$	−1.78380	−0.154675
$134$	23.3350	2.01584
$135$	0	0
$136$	13.2857	1.13924
$137$	9.98542	0.853112	0.426556	−	0.904461i	$-0.359727\pi$
$137$	9.98542	0.853112	0.426556	+	0.904461i	$0.359727\pi$
$138$	0	0
$139$	7.44343	0.631344	0.315672	−	0.948868i	$-0.397770\pi$
$139$	7.44343	0.631344	0.315672	+	0.948868i	$0.397770\pi$
$140$	0.432182	0.0365261
$141$	0	0
$142$	−5.03422	−0.422462
$143$	4.14100	0.346288
$144$	0	0
$145$	−0.265662	−0.0220620
$146$	−2.72987	−0.225925
$147$	0	0
$148$	9.25587	0.760828
$149$	−8.64989	−0.708626	−0.354313	−	0.935127i	$-0.615285\pi$
$149$	−8.64989	−0.708626	−0.354313	+	0.935127i	$0.615285\pi$
$150$	0	0
$151$	0.112505	0.00915555	0.00457778	−	0.999990i	$-0.498543\pi$
$151$	0.112505	0.00915555	0.00457778	+	0.999990i	$0.498543\pi$
$152$	5.14230	0.417096
$153$	0	0
$154$	−3.16481	−0.255027
$155$	1.26594	0.101683
$156$	0	0
$157$	10.2079	0.814679	0.407340	−	0.913277i	$-0.366457\pi$
$157$	10.2079	0.814679	0.407340	+	0.913277i	$0.366457\pi$
$158$	23.7554	1.88988
$159$	0	0
$160$	0.735994	0.0581854
$161$	−6.02678	−0.474977
$162$	0	0
$163$	−12.2432	−0.958964	−0.479482	−	0.877552i	$-0.659175\pi$
$163$	−12.2432	−0.958964	−0.479482	+	0.877552i	$0.659175\pi$
$164$	−7.10961	−0.555167
$165$	0	0
$166$	4.15145	0.322215
$167$	21.7101	1.67998	0.839990	−	0.542602i	$-0.182561\pi$
$167$	21.7101	1.67998	0.839990	+	0.542602i	$0.182561\pi$
$168$	0	0
$169$	−3.99928	−0.307637
$170$	1.40206	0.107533
$171$	0	0
$172$	30.9955	2.36339
$173$	2.24784	0.170900	0.0854501	−	0.996342i	$-0.472767\pi$
$173$	2.24784	0.170900	0.0854501	+	0.996342i	$0.472767\pi$
$174$	0	0
$175$	−4.98240	−0.376634
$176$	0.131536	0.00991490
$177$	0	0
$178$	−16.6248	−1.24608
$179$	2.39212	0.178796	0.0893979	−	0.995996i	$-0.471506\pi$
$179$	2.39212	0.178796	0.0893979	+	0.995996i	$0.471506\pi$
$180$	0	0
$181$	−20.6405	−1.53420	−0.767098	−	0.641530i	$-0.778300\pi$
$181$	−20.6405	−1.53420	−0.767098	+	0.641530i	$0.778300\pi$
$182$	−6.87890	−0.509898
$183$	0	0
$184$	17.3739	1.28082
$185$	0.377029	0.0277197
$186$	0	0
$187$	−6.36122	−0.465178
$188$	−8.23010	−0.600242
$189$	0	0
$190$	0.542674	0.0393697
$191$	16.1222	1.16656	0.583279	−	0.812272i	$-0.301769\pi$
$191$	16.1222	1.16656	0.583279	+	0.812272i	$0.301769\pi$
$192$	0	0
$193$	8.63114	0.621283	0.310642	−	0.950527i	$-0.399456\pi$
$193$	8.63114	0.621283	0.310642	+	0.950527i	$0.399456\pi$
$194$	−4.91094	−0.352585
$195$	0	0
$196$	3.25728	0.232663
$197$	−0.747074	−0.0532268	−0.0266134	−	0.999646i	$-0.508472\pi$
$197$	−0.747074	−0.0532268	−0.0266134	+	0.999646i	$0.508472\pi$
$198$	0	0
$199$	−1.78469	−0.126513	−0.0632565	−	0.997997i	$-0.520149\pi$
$199$	−1.78469	−0.126513	−0.0632565	+	0.997997i	$0.520149\pi$
$200$	14.3631	1.01563
$201$	0	0
$202$	37.3118	2.62525
$203$	−2.00225	−0.140530
$204$	0	0
$205$	−0.289603	−0.0202268
$206$	29.2716	2.03945
$207$	0	0
$208$	0.285902	0.0198237
$209$	−2.46214	−0.170310
$210$	0	0
$211$	−13.0567	−0.898862	−0.449431	−	0.893315i	$-0.648373\pi$
$211$	−13.0567	−0.898862	−0.449431	+	0.893315i	$0.648373\pi$
$212$	27.3273	1.87685
$213$	0	0
$214$	3.22029	0.220135
$215$	1.26257	0.0861067
$216$	0	0
$217$	9.54114	0.647695
$218$	−25.4757	−1.72543
$219$	0	0
$220$	0.596532	0.0402182
$221$	−13.8265	−0.930070
$222$	0	0
$223$	−9.47976	−0.634812	−0.317406	−	0.948290i	$-0.602812\pi$
$223$	−9.47976	−0.634812	−0.317406	+	0.948290i	$0.602812\pi$
$224$	5.54705	0.370628
$225$	0	0
$226$	40.4830	2.69289
$227$	−3.99593	−0.265219	−0.132610	−	0.991168i	$-0.542336\pi$
$227$	−3.99593	−0.265219	−0.132610	+	0.991168i	$0.542336\pi$
$228$	0	0
$229$	5.69325	0.376220	0.188110	−	0.982148i	$-0.439764\pi$
$229$	5.69325	0.376220	0.188110	+	0.982148i	$0.439764\pi$
$230$	1.83349	0.120897
$231$	0	0
$232$	5.77203	0.378952
$233$	11.1279	0.729014	0.364507	−	0.931201i	$-0.381238\pi$
$233$	11.1279	0.729014	0.364507	+	0.931201i	$0.381238\pi$
$234$	0	0
$235$	−0.335245	−0.0218690
$236$	−25.3734	−1.65167
$237$	0	0
$238$	10.5670	0.684960
$239$	8.65428	0.559799	0.279900	−	0.960029i	$-0.409699\pi$
$239$	8.65428	0.559799	0.279900	+	0.960029i	$0.409699\pi$
$240$	0	0
$241$	−26.5941	−1.71308	−0.856538	−	0.516083i	$-0.827390\pi$
$241$	−26.5941	−1.71308	−0.856538	+	0.516083i	$0.827390\pi$
$242$	20.8533	1.34050
$243$	0	0
$244$	12.3031	0.787624
$245$	0.132682	0.00847674
$246$	0	0
$247$	−5.35162	−0.340515
$248$	−27.5050	−1.74657
$249$	0	0
$250$	3.03688	0.192069
$251$	6.80462	0.429504	0.214752	−	0.976669i	$-0.431106\pi$
$251$	6.80462	0.429504	0.214752	+	0.976669i	$0.431106\pi$
$252$	0	0
$253$	−8.31864	−0.522989
$254$	2.29288	0.143868
$255$	0	0
$256$	−16.6117	−1.03823
$257$	−25.0375	−1.56180	−0.780898	−	0.624658i	$-0.785238\pi$
$257$	−25.0375	−1.56180	−0.780898	+	0.624658i	$0.785238\pi$
$258$	0	0
$259$	2.84160	0.176568
$260$	1.29660	0.0804116
$261$	0	0
$262$	13.1574	0.812870
$263$	18.7441	1.15581	0.577905	−	0.816104i	$-0.303870\pi$
$263$	18.7441	1.15581	0.577905	+	0.816104i	$0.303870\pi$
$264$	0	0
$265$	1.11315	0.0683804
$266$	4.09003	0.250776
$267$	0	0
$268$	−33.1499	−2.02495
$269$	−11.3046	−0.689253	−0.344627	−	0.938740i	$-0.611995\pi$
$269$	−11.3046	−0.689253	−0.344627	+	0.938740i	$0.611995\pi$
$270$	0	0
$271$	−19.6892	−1.19604	−0.598018	−	0.801483i	$-0.704045\pi$
$271$	−19.6892	−1.19604	−0.598018	+	0.801483i	$0.704045\pi$
$272$	−0.439189	−0.0266297
$273$	0	0
$274$	−22.8953	−1.38316
$275$	−6.87710	−0.414704
$276$	0	0
$277$	−19.1796	−1.15239	−0.576196	−	0.817311i	$-0.695464\pi$
$277$	−19.1796	−1.15239	−0.576196	+	0.817311i	$0.695464\pi$
$278$	−17.0669	−1.02360
$279$	0	0
$280$	−0.382493	−0.0228583
$281$	5.36195	0.319867	0.159934	−	0.987128i	$-0.448872\pi$
$281$	5.36195	0.319867	0.159934	+	0.987128i	$0.448872\pi$
$282$	0	0
$283$	−25.2685	−1.50206	−0.751028	−	0.660271i	$-0.770442\pi$
$283$	−25.2685	−1.50206	−0.751028	+	0.660271i	$0.770442\pi$
$284$	7.15165	0.424372
$285$	0	0
$286$	−9.49480	−0.561439
$287$	−2.18268	−0.128840
$288$	0	0
$289$	4.23962	0.249389
$290$	0.609130	0.0357694
$291$	0	0
$292$	3.87807	0.226947
$293$	−18.0751	−1.05596	−0.527980	−	0.849257i	$-0.677050\pi$
$293$	−18.0751	−1.05596	−0.527980	+	0.849257i	$0.677050\pi$
$294$	0	0
$295$	−1.03356	−0.0601763
$296$	−8.19169	−0.476132
$297$	0	0
$298$	19.8331	1.14890
$299$	−18.0811	−1.04566
$300$	0	0
$301$	9.51578	0.548480
$302$	−0.257961	−0.0148440
$303$	0	0
$304$	−0.169990	−0.00974962
$305$	0.501154	0.0286960
$306$	0	0
$307$	19.0494	1.08721	0.543604	−	0.839342i	$-0.317059\pi$
$307$	19.0494	1.08721	0.543604	+	0.839342i	$0.317059\pi$
$308$	4.49595	0.256181
$309$	0	0
$310$	−2.90264	−0.164859
$311$	−4.75698	−0.269744	−0.134872	−	0.990863i	$-0.543062\pi$
$311$	−4.75698	−0.269744	−0.134872	+	0.990863i	$0.543062\pi$
$312$	0	0
$313$	1.35571	0.0766292	0.0383146	−	0.999266i	$-0.487801\pi$
$313$	1.35571	0.0766292	0.0383146	+	0.999266i	$0.487801\pi$
$314$	−23.4054	−1.32085
$315$	0	0
$316$	−33.7472	−1.89843
$317$	13.3742	0.751173	0.375586	−	0.926787i	$-0.377441\pi$
$317$	13.3742	0.751173	0.375586	+	0.926787i	$0.377441\pi$
$318$	0	0
$319$	−2.76366	−0.154735
$320$	−1.71283	−0.0957502
$321$	0	0
$322$	13.8187	0.770084
$323$	8.22091	0.457424
$324$	0	0
$325$	−14.9478	−0.829154
$326$	28.0722	1.55477
$327$	0	0
$328$	6.29219	0.347428
$329$	−2.52668	−0.139300
$330$	0	0
$331$	−11.6504	−0.640367	−0.320183	−	0.947356i	$-0.603744\pi$
$331$	−11.6504	−0.640367	−0.320183	+	0.947356i	$0.603744\pi$
$332$	−5.89758	−0.323672
$333$	0	0
$334$	−49.7786	−2.72376
$335$	−1.35033	−0.0737763
$336$	0	0
$337$	16.5945	0.903962	0.451981	−	0.892028i	$-0.350718\pi$
$337$	16.5945	0.903962	0.451981	+	0.892028i	$0.350718\pi$
$338$	9.16985	0.498774
$339$	0	0
$340$	−1.99177	−0.108019
$341$	13.1694	0.713165
$342$	0	0
$343$	1.00000	0.0539949
$344$	−27.4319	−1.47903
$345$	0	0
$346$	−5.15402	−0.277082
$347$	−1.74448	−0.0936485	−0.0468242	−	0.998903i	$-0.514910\pi$
$347$	−1.74448	−0.0936485	−0.0468242	+	0.998903i	$0.514910\pi$
$348$	0	0
$349$	32.0356	1.71483	0.857413	−	0.514629i	$-0.172070\pi$
$349$	32.0356	1.71483	0.857413	+	0.514629i	$0.172070\pi$
$350$	11.4240	0.610639
$351$	0	0
$352$	7.65648	0.408092
$353$	15.9793	0.850494	0.425247	−	0.905077i	$-0.360187\pi$
$353$	15.9793	0.850494	0.425247	+	0.905077i	$0.360187\pi$
$354$	0	0
$355$	0.291316	0.0154614
$356$	23.6174	1.25172
$357$	0	0
$358$	−5.48484	−0.289883
$359$	22.8017	1.20343	0.601713	−	0.798712i	$-0.294485\pi$
$359$	22.8017	1.20343	0.601713	+	0.798712i	$0.294485\pi$
$360$	0	0
$361$	−15.8181	−0.832529
$362$	47.3261	2.48740
$363$	0	0
$364$	9.77222	0.512203
$365$	0.157969	0.00826850
$366$	0	0
$367$	−2.46175	−0.128502	−0.0642511	−	0.997934i	$-0.520466\pi$
$367$	−2.46175	−0.128502	−0.0642511	+	0.997934i	$0.520466\pi$
$368$	−0.574333	−0.0299392
$369$	0	0
$370$	−0.864480	−0.0449422
$371$	8.38962	0.435567
$372$	0	0
$373$	−25.6753	−1.32942	−0.664708	−	0.747103i	$-0.731444\pi$
$373$	−25.6753	−1.32942	−0.664708	+	0.747103i	$0.731444\pi$
$374$	14.5855	0.754197
$375$	0	0
$376$	7.28386	0.375637
$377$	−6.00698	−0.309375
$378$	0	0
$379$	28.6117	1.46968	0.734842	−	0.678239i	$-0.237256\pi$
$379$	28.6117	1.46968	0.734842	+	0.678239i	$0.237256\pi$
$380$	−0.770927	−0.0395477
$381$	0	0
$382$	−36.9661	−1.89135
$383$	28.5878	1.46077	0.730385	−	0.683036i	$-0.239341\pi$
$383$	28.5878	1.46077	0.730385	+	0.683036i	$0.239341\pi$
$384$	0	0
$385$	0.183138	0.00933359
$386$	−19.7901	−1.00729
$387$	0	0
$388$	6.97652	0.354179
$389$	−17.3188	−0.878096	−0.439048	−	0.898464i	$-0.644684\pi$
$389$	−17.3188	−0.878096	−0.439048	+	0.898464i	$0.644684\pi$
$390$	0	0
$391$	27.7753	1.40466
$392$	−2.88278	−0.145602
$393$	0	0
$394$	1.71295	0.0862970
$395$	−1.37466	−0.0691666
$396$	0	0
$397$	−15.8409	−0.795031	−0.397515	−	0.917595i	$-0.630127\pi$
$397$	−15.8409	−0.795031	−0.397515	+	0.917595i	$0.630127\pi$
$398$	4.09206	0.205116
$399$	0	0
$400$	−0.474806	−0.0237403
$401$	−25.1098	−1.25392	−0.626962	−	0.779050i	$-0.715702\pi$
$401$	−25.1098	−1.25392	−0.626962	+	0.779050i	$0.715702\pi$
$402$	0	0
$403$	28.6246	1.42589
$404$	−53.0055	−2.63712
$405$	0	0
$406$	4.59090	0.227843
$407$	3.92220	0.194416
$408$	0	0
$409$	8.42996	0.416835	0.208417	−	0.978040i	$-0.433169\pi$
$409$	8.42996	0.416835	0.208417	+	0.978040i	$0.433169\pi$
$410$	0.664024	0.0327938
$411$	0	0
$412$	−41.5835	−2.04867
$413$	−7.78977	−0.383310
$414$	0	0
$415$	−0.240232	−0.0117925
$416$	16.6418	0.815932
$417$	0	0
$418$	5.64539	0.276125
$419$	17.2527	0.842851	0.421425	−	0.906863i	$-0.361530\pi$
$419$	17.2527	0.842851	0.421425	+	0.906863i	$0.361530\pi$
$420$	0	0
$421$	27.9955	1.36442	0.682209	−	0.731157i	$-0.261019\pi$
$421$	27.9955	1.36442	0.682209	+	0.731157i	$0.261019\pi$
$422$	29.9375	1.45733
$423$	0	0
$424$	−24.1854	−1.17455
$425$	22.9621	1.11383
$426$	0	0
$427$	3.77711	0.182787
$428$	−4.57477	−0.221130
$429$	0	0
$430$	−2.89492	−0.139605
$431$	12.7499	0.614142	0.307071	−	0.951687i	$-0.400651\pi$
$431$	12.7499	0.614142	0.307071	+	0.951687i	$0.400651\pi$
$432$	0	0
$433$	5.67855	0.272894	0.136447	−	0.990647i	$-0.456432\pi$
$433$	5.67855	0.272894	0.136447	+	0.990647i	$0.456432\pi$
$434$	−21.8766	−1.05011
$435$	0	0
$436$	36.1910	1.73323
$437$	10.7506	0.514270
$438$	0	0
$439$	−37.3733	−1.78373	−0.891865	−	0.452302i	$-0.850603\pi$
$439$	−37.3733	−1.78373	−0.891865	+	0.452302i	$0.850603\pi$
$440$	−0.527947	−0.0251689
$441$	0	0
$442$	31.7024	1.50793
$443$	−22.9515	−1.09046	−0.545230	−	0.838287i	$-0.683558\pi$
$443$	−22.9515	−1.09046	−0.545230	+	0.838287i	$0.683558\pi$
$444$	0	0
$445$	0.962031	0.0456046
$446$	21.7359	1.02922
$447$	0	0
$448$	−12.9093	−0.609907
$449$	−30.4308	−1.43612	−0.718059	−	0.695982i	$-0.754969\pi$
$449$	−30.4308	−1.43612	−0.718059	+	0.695982i	$0.754969\pi$
$450$	0	0
$451$	−3.01271	−0.141863
$452$	−57.5105	−2.70507
$453$	0	0
$454$	9.16216	0.430002
$455$	0.398062	0.0186614
$456$	0	0
$457$	−32.1203	−1.50252	−0.751261	−	0.660005i	$-0.770554\pi$
$457$	−32.1203	−1.50252	−0.751261	+	0.660005i	$0.770554\pi$
$458$	−13.0539	−0.609969
$459$	0	0
$460$	−2.60467	−0.121443
$461$	−11.1704	−0.520259	−0.260129	−	0.965574i	$-0.583765\pi$
$461$	−11.1704	−0.520259	−0.260129	+	0.965574i	$0.583765\pi$
$462$	0	0
$463$	25.3971	1.18030	0.590152	−	0.807292i	$-0.299068\pi$
$463$	25.3971	1.18030	0.590152	+	0.807292i	$0.299068\pi$
$464$	−0.190807	−0.00885802
$465$	0	0
$466$	−25.5149	−1.18196
$467$	−37.9241	−1.75492	−0.877459	−	0.479652i	$-0.840763\pi$
$467$	−37.9241	−1.75492	−0.877459	+	0.479652i	$0.840763\pi$
$468$	0	0
$469$	−10.1772	−0.469938
$470$	0.768676	0.0354564
$471$	0	0
$472$	22.4562	1.03363
$473$	13.1344	0.603921
$474$	0	0
$475$	8.88760	0.407791
$476$	−15.0116	−0.688057
$477$	0	0
$478$	−19.8432	−0.907606
$479$	9.94380	0.454344	0.227172	−	0.973855i	$-0.427052\pi$
$479$	9.94380	0.454344	0.227172	+	0.973855i	$0.427052\pi$
$480$	0	0
$481$	8.52513	0.388713
$482$	60.9770	2.77742
$483$	0	0
$484$	−29.6244	−1.34656
$485$	0.284182	0.0129040
$486$	0	0
$487$	−3.73337	−0.169175	−0.0845876	−	0.996416i	$-0.526957\pi$
$487$	−3.73337	−0.169175	−0.0845876	+	0.996416i	$0.526957\pi$
$488$	−10.8886	−0.492902
$489$	0	0
$490$	−0.304223	−0.0137434
$491$	−37.0833	−1.67355	−0.836773	−	0.547550i	$-0.815561\pi$
$491$	−37.0833	−1.67355	−0.836773	+	0.547550i	$0.815561\pi$
$492$	0	0
$493$	9.22765	0.415592
$494$	12.2706	0.552080
$495$	0	0
$496$	0.909239	0.0408260
$497$	2.19559	0.0984858
$498$	0	0
$499$	24.7993	1.11017	0.555084	−	0.831795i	$-0.312686\pi$
$499$	24.7993	1.11017	0.555084	+	0.831795i	$0.312686\pi$
$500$	−4.31421	−0.192937
$501$	0	0
$502$	−15.6021	−0.696357
$503$	27.4762	1.22510	0.612551	−	0.790431i	$-0.290143\pi$
$503$	27.4762	1.22510	0.612551	+	0.790431i	$0.290143\pi$
$504$	0	0
$505$	−2.15913	−0.0960800
$506$	19.0736	0.847925
$507$	0	0
$508$	−3.25728	−0.144518
$509$	−2.58766	−0.114696	−0.0573481	−	0.998354i	$-0.518264\pi$
$509$	−2.58766	−0.114696	−0.0573481	+	0.998354i	$0.518264\pi$
$510$	0	0
$511$	1.19059	0.0526684
$512$	1.07805	0.0476437
$513$	0	0
$514$	57.4079	2.53215
$515$	−1.69386	−0.0746405
$516$	0	0
$517$	−3.48753	−0.153381
$518$	−6.51543	−0.286271
$519$	0	0
$520$	−1.14752	−0.0503223
$521$	0.605284	0.0265180	0.0132590	−	0.999912i	$-0.495779\pi$
$521$	0.605284	0.0265180	0.0132590	+	0.999912i	$0.495779\pi$
$522$	0	0
$523$	−17.0952	−0.747522	−0.373761	−	0.927525i	$-0.621932\pi$
$523$	−17.0952	−0.747522	−0.373761	+	0.927525i	$0.621932\pi$
$524$	−18.6916	−0.816545
$525$	0	0
$526$	−42.9779	−1.87392
$527$	−43.9717	−1.91544
$528$	0	0
$529$	13.3221	0.579223
$530$	−2.55232	−0.110866
$531$	0	0
$532$	−5.81033	−0.251910
$533$	−6.54831	−0.283639
$534$	0	0
$535$	−0.186349	−0.00805656
$536$	29.3385	1.26723
$537$	0	0
$538$	25.9200	1.11749
$539$	1.38028	0.0594528
$540$	0	0
$541$	3.19188	0.137230	0.0686149	−	0.997643i	$-0.478142\pi$
$541$	3.19188	0.137230	0.0686149	+	0.997643i	$0.478142\pi$
$542$	45.1449	1.93914
$543$	0	0
$544$	−25.5644	−1.09606
$545$	1.47421	0.0631480
$546$	0	0
$547$	−44.2108	−1.89032	−0.945159	−	0.326610i	$-0.894094\pi$
$547$	−44.2108	−1.89032	−0.945159	+	0.326610i	$0.894094\pi$
$548$	32.5253	1.38941
$549$	0	0
$550$	15.7683	0.672363
$551$	3.57161	0.152156
$552$	0	0
$553$	−10.3605	−0.440575
$554$	43.9765	1.86838
$555$	0	0
$556$	24.2453	1.02823
$557$	35.3458	1.49765	0.748826	−	0.662767i	$-0.230618\pi$
$557$	35.3458	1.49765	0.748826	+	0.662767i	$0.230618\pi$
$558$	0	0
$559$	28.5485	1.20747
$560$	0.0126442	0.000534313 0
$561$	0	0
$562$	−12.2943	−0.518603
$563$	18.2180	0.767798	0.383899	−	0.923375i	$-0.374581\pi$
$563$	18.2180	0.767798	0.383899	+	0.923375i	$0.374581\pi$
$564$	0	0
$565$	−2.34264	−0.0985554
$566$	57.9375	2.43529
$567$	0	0
$568$	−6.32941	−0.265576
$569$	−30.8334	−1.29260	−0.646301	−	0.763083i	$-0.723685\pi$
$569$	−30.8334	−1.29260	−0.646301	+	0.763083i	$0.723685\pi$
$570$	0	0
$571$	−19.1022	−0.799401	−0.399701	−	0.916646i	$-0.630886\pi$
$571$	−19.1022	−0.799401	−0.399701	+	0.916646i	$0.630886\pi$
$572$	13.4884	0.563978
$573$	0	0
$574$	5.00462	0.208889
$575$	30.0278	1.25225
$576$	0	0
$577$	19.0295	0.792207	0.396103	−	0.918206i	$-0.370362\pi$
$577$	19.0295	0.792207	0.396103	+	0.918206i	$0.370362\pi$
$578$	−9.72091	−0.404337
$579$	0	0
$580$	−0.865335	−0.0359311
$581$	−1.81059	−0.0751158
$582$	0	0
$583$	11.5800	0.479595
$584$	−3.43220	−0.142025
$585$	0	0
$586$	41.4440	1.71203
$587$	−13.9522	−0.575868	−0.287934	−	0.957650i	$-0.592968\pi$
$587$	−13.9522	−0.575868	−0.287934	+	0.957650i	$0.592968\pi$
$588$	0	0
$589$	−17.0195	−0.701276
$590$	2.36983	0.0975644
$591$	0	0
$592$	0.270795	0.0111296
$593$	11.3635	0.466642	0.233321	−	0.972400i	$-0.425041\pi$
$593$	11.3635	0.466642	0.233321	+	0.972400i	$0.425041\pi$
$594$	0	0
$595$	−0.611485	−0.0250684
$596$	−28.1751	−1.15410
$597$	0	0
$598$	41.4576	1.69533
$599$	7.97960	0.326037	0.163019	−	0.986623i	$-0.447877\pi$
$599$	7.97960	0.326037	0.163019	+	0.986623i	$0.447877\pi$
$600$	0	0
$601$	−27.4921	−1.12143	−0.560713	−	0.828010i	$-0.689473\pi$
$601$	−27.4921	−1.12143	−0.560713	+	0.828010i	$0.689473\pi$
$602$	−21.8185	−0.889255
$603$	0	0
$604$	0.366461	0.0149111
$605$	−1.20672	−0.0490602
$606$	0	0
$607$	−1.64188	−0.0666420	−0.0333210	−	0.999445i	$-0.510608\pi$
$607$	−1.64188	−0.0666420	−0.0333210	+	0.999445i	$0.510608\pi$
$608$	−9.89484	−0.401289
$609$	0	0
$610$	−1.14908	−0.0465250
$611$	−7.58035	−0.306668
$612$	0	0
$613$	−21.9275	−0.885642	−0.442821	−	0.896610i	$-0.646022\pi$
$613$	−21.9275	−0.885642	−0.442821	+	0.896610i	$0.646022\pi$
$614$	−43.6780	−1.76270
$615$	0	0
$616$	−3.97904	−0.160320
$617$	28.4427	1.14506	0.572530	−	0.819883i	$-0.305962\pi$
$617$	28.4427	1.14506	0.572530	+	0.819883i	$0.305962\pi$
$618$	0	0
$619$	28.9635	1.16414	0.582071	−	0.813138i	$-0.302243\pi$
$619$	28.9635	1.16414	0.582071	+	0.813138i	$0.302243\pi$
$620$	4.12351	0.165604
$621$	0	0
$622$	10.9072	0.437338
$623$	7.25065	0.290491
$624$	0	0
$625$	24.7362	0.989450
$626$	−3.10847	−0.124240
$627$	0	0
$628$	33.2499	1.32682
$629$	−13.0959	−0.522168
$630$	0	0
$631$	−26.5333	−1.05627	−0.528136	−	0.849160i	$-0.677109\pi$
$631$	−26.5333	−1.05627	−0.528136	+	0.849160i	$0.677109\pi$
$632$	29.8672	1.18805
$633$	0	0
$634$	−30.6655	−1.21788
$635$	−0.132682	−0.00526533
$636$	0	0
$637$	3.00012	0.118869
$638$	6.33672	0.250873
$639$	0	0
$640$	2.45532	0.0970550
$641$	−25.6409	−1.01276	−0.506378	−	0.862312i	$-0.669016\pi$
$641$	−25.6409	−1.01276	−0.506378	+	0.862312i	$0.669016\pi$
$642$	0	0
$643$	−43.4896	−1.71506	−0.857532	−	0.514431i	$-0.828003\pi$
$643$	−43.4896	−1.71506	−0.857532	+	0.514431i	$0.828003\pi$
$644$	−19.6309	−0.773566
$645$	0	0
$646$	−18.8495	−0.741624
$647$	−7.81527	−0.307250	−0.153625	−	0.988129i	$-0.549095\pi$
$647$	−7.81527	−0.307250	−0.153625	+	0.988129i	$0.549095\pi$
$648$	0	0
$649$	−10.7521	−0.422055
$650$	34.2734	1.34431
$651$	0	0
$652$	−39.8796	−1.56180
$653$	20.2875	0.793912	0.396956	−	0.917838i	$-0.370067\pi$
$653$	20.2875	0.793912	0.396956	+	0.917838i	$0.370067\pi$
$654$	0	0
$655$	−0.761383	−0.0297497
$656$	−0.208003	−0.00812114
$657$	0	0
$658$	5.79337	0.225849
$659$	−44.0567	−1.71620	−0.858102	−	0.513479i	$-0.828356\pi$
$659$	−44.0567	−1.71620	−0.858102	+	0.513479i	$0.828356\pi$
$660$	0	0
$661$	−7.35149	−0.285940	−0.142970	−	0.989727i	$-0.545665\pi$
$661$	−7.35149	−0.285940	−0.142970	+	0.989727i	$0.545665\pi$
$662$	26.7130	1.03823
$663$	0	0
$664$	5.21952	0.202557
$665$	−0.236678	−0.00917799
$666$	0	0
$667$	12.0671	0.467240
$668$	70.7159	2.73608
$669$	0	0
$670$	3.09613	0.119614
$671$	5.21346	0.201263
$672$	0	0
$673$	19.8435	0.764912	0.382456	−	0.923974i	$-0.375078\pi$
$673$	19.8435	0.764912	0.382456	+	0.923974i	$0.375078\pi$
$674$	−38.0492	−1.46560
$675$	0	0
$676$	−13.0268	−0.501030
$677$	13.3359	0.512539	0.256269	−	0.966605i	$-0.417507\pi$
$677$	13.3359	0.512539	0.256269	+	0.966605i	$0.417507\pi$
$678$	0	0
$679$	2.14183	0.0821957
$680$	1.76277	0.0675993
$681$	0	0
$682$	−30.1959	−1.15626
$683$	15.1569	0.579962	0.289981	−	0.957032i	$-0.406351\pi$
$683$	15.1569	0.579962	0.289981	+	0.957032i	$0.406351\pi$
$684$	0	0
$685$	1.32489	0.0506213
$686$	−2.29288	−0.0875424
$687$	0	0
$688$	0.906822	0.0345723
$689$	25.1699	0.958895
$690$	0	0
$691$	−43.0546	−1.63788	−0.818938	−	0.573882i	$-0.805437\pi$
$691$	−43.0546	−1.63788	−0.818938	+	0.573882i	$0.805437\pi$
$692$	7.32184	0.278334
$693$	0	0
$694$	3.99987	0.151833
$695$	0.987610	0.0374622
$696$	0	0
$697$	10.0592	0.381020
$698$	−73.4536	−2.78026
$699$	0	0
$700$	−16.2290	−0.613400
$701$	−30.1134	−1.13737	−0.568685	−	0.822556i	$-0.692547\pi$
$701$	−30.1134	−1.13737	−0.568685	+	0.822556i	$0.692547\pi$
$702$	0	0
$703$	−5.06884	−0.191175
$704$	−17.8184	−0.671557
$705$	0	0
$706$	−36.6386	−1.37891
$707$	−16.2730	−0.612007
$708$	0	0
$709$	−23.1425	−0.869136	−0.434568	−	0.900639i	$-0.643099\pi$
$709$	−23.1425	−0.869136	−0.434568	+	0.900639i	$0.643099\pi$
$710$	−0.667951	−0.0250677
$711$	0	0
$712$	−20.9020	−0.783336
$713$	−57.5024	−2.15348
$714$	0	0
$715$	0.549437	0.0205478
$716$	7.79181	0.291194
$717$	0	0
$718$	−52.2814	−1.95112
$719$	−6.88533	−0.256780	−0.128390	−	0.991724i	$-0.540981\pi$
$719$	−6.88533	−0.256780	−0.128390	+	0.991724i	$0.540981\pi$
$720$	0	0
$721$	−12.7663	−0.475443
$722$	36.2688	1.34979
$723$	0	0
$724$	−67.2318	−2.49865
$725$	9.97598	0.370499
$726$	0	0
$727$	−3.64321	−0.135119	−0.0675596	−	0.997715i	$-0.521521\pi$
$727$	−3.64321	−0.135119	−0.0675596	+	0.997715i	$0.521521\pi$
$728$	−8.64868	−0.320541
$729$	0	0
$730$	−0.362204	−0.0134058
$731$	−43.8548	−1.62203
$732$	0	0
$733$	27.4346	1.01332	0.506660	−	0.862146i	$-0.330880\pi$
$733$	27.4346	1.01332	0.506660	+	0.862146i	$0.330880\pi$
$734$	5.64448	0.208342
$735$	0	0
$736$	−33.4309	−1.23228
$737$	−14.0473	−0.517440
$738$	0	0
$739$	−3.92099	−0.144236	−0.0721179	−	0.997396i	$-0.522976\pi$
$739$	−3.92099	−0.144236	−0.0721179	+	0.997396i	$0.522976\pi$
$740$	1.22809	0.0451454
$741$	0	0
$742$	−19.2364	−0.706189
$743$	−50.9410	−1.86885	−0.934423	−	0.356165i	$-0.884084\pi$
$743$	−50.9410	−1.86885	−0.934423	+	0.356165i	$0.884084\pi$
$744$	0	0
$745$	−1.14768	−0.0420479
$746$	58.8703	2.15539
$747$	0	0
$748$	−20.7202	−0.757607
$749$	−1.40448	−0.0513185
$750$	0	0
$751$	3.73352	0.136238	0.0681191	−	0.997677i	$-0.478300\pi$
$751$	3.73352	0.136238	0.0681191	+	0.997677i	$0.478300\pi$
$752$	−0.240784	−0.00878051
$753$	0	0
$754$	13.7733	0.501592
$755$	0.0149274	0.000543265 0
$756$	0	0
$757$	−0.802306	−0.0291603	−0.0145802	−	0.999894i	$-0.504641\pi$
$757$	−0.802306	−0.0291603	−0.0145802	+	0.999894i	$0.504641\pi$
$758$	−65.6030	−2.38281
$759$	0	0
$760$	0.682291	0.0247493
$761$	28.3858	1.02899	0.514493	−	0.857495i	$-0.327980\pi$
$761$	28.3858	1.02899	0.514493	+	0.857495i	$0.327980\pi$
$762$	0	0
$763$	11.1108	0.402238
$764$	52.5143	1.89990
$765$	0	0
$766$	−65.5483	−2.36836
$767$	−23.3702	−0.843851
$768$	0	0
$769$	37.0161	1.33483	0.667417	−	0.744684i	$-0.267400\pi$
$769$	37.0161	1.33483	0.667417	+	0.744684i	$0.267400\pi$
$770$	−0.419913	−0.0151326
$771$	0	0
$772$	28.1140	1.01185
$773$	−13.8879	−0.499514	−0.249757	−	0.968309i	$-0.580351\pi$
$773$	−13.8879	−0.499514	−0.249757	+	0.968309i	$0.580351\pi$
$774$	0	0
$775$	−47.5377	−1.70761
$776$	−6.17441	−0.221648
$777$	0	0
$778$	39.7098	1.42366
$779$	3.89347	0.139498
$780$	0	0
$781$	3.03053	0.108441
$782$	−63.6853	−2.27738
$783$	0	0
$784$	0.0952967	0.00340345
$785$	1.35440	0.0483408
$786$	0	0
$787$	−4.72348	−0.168374	−0.0841869	−	0.996450i	$-0.526829\pi$
$787$	−4.72348	−0.168374	−0.0841869	+	0.996450i	$0.526829\pi$
$788$	−2.43343	−0.0866872
$789$	0	0
$790$	3.15192	0.112140
$791$	−17.6560	−0.627775
$792$	0	0
$793$	11.3318	0.402403
$794$	36.3212	1.28899
$795$	0	0
$796$	−5.81321	−0.206044
$797$	−7.11438	−0.252004	−0.126002	−	0.992030i	$-0.540215\pi$
$797$	−7.11438	−0.252004	−0.126002	+	0.992030i	$0.540215\pi$
$798$	0	0
$799$	11.6446	0.411956
$800$	−27.6376	−0.977137
$801$	0	0
$802$	57.5736	2.03300
$803$	1.64334	0.0579923
$804$	0	0
$805$	−0.799646	−0.0281838
$806$	−65.6325	−2.31181
$807$	0	0
$808$	46.9113	1.65033
$809$	49.9190	1.75506	0.877529	−	0.479523i	$-0.159190\pi$
$809$	49.9190	1.75506	0.877529	+	0.479523i	$0.159190\pi$
$810$	0	0
$811$	24.6467	0.865464	0.432732	−	0.901523i	$-0.357550\pi$
$811$	24.6467	0.865464	0.432732	+	0.901523i	$0.357550\pi$
$812$	−6.52187	−0.228873
$813$	0	0
$814$	−8.99311	−0.315208
$815$	−1.62446	−0.0569022
$816$	0	0
$817$	−16.9743	−0.593854
$818$	−19.3288	−0.675817
$819$	0	0
$820$	−0.943317	−0.0329421
$821$	−19.0309	−0.664183	−0.332092	−	0.943247i	$-0.607754\pi$
$821$	−19.0309	−0.664183	−0.332092	+	0.943247i	$0.607754\pi$
$822$	0	0
$823$	20.8785	0.727780	0.363890	−	0.931442i	$-0.381448\pi$
$823$	20.8785	0.727780	0.363890	+	0.931442i	$0.381448\pi$
$824$	36.8025	1.28208
$825$	0	0
$826$	17.8610	0.621463
$827$	−30.7068	−1.06778	−0.533890	−	0.845554i	$-0.679270\pi$
$827$	−30.7068	−1.06778	−0.533890	+	0.845554i	$0.679270\pi$
$828$	0	0
$829$	9.14687	0.317684	0.158842	−	0.987304i	$-0.449224\pi$
$829$	9.14687	0.317684	0.158842	+	0.987304i	$0.449224\pi$
$830$	0.550822	0.0191193
$831$	0	0
$832$	−38.7294	−1.34270
$833$	−4.60865	−0.159680
$834$	0	0
$835$	2.88054	0.0996853
$836$	−8.01988	−0.277373
$837$	0	0
$838$	−39.5583	−1.36652
$839$	−48.3089	−1.66781	−0.833904	−	0.551909i	$-0.813899\pi$
$839$	−48.3089	−1.66781	−0.833904	+	0.551909i	$0.813899\pi$
$840$	0	0
$841$	−24.9910	−0.861759
$842$	−64.1902	−2.21214
$843$	0	0
$844$	−42.5294	−1.46392
$845$	−0.530633	−0.0182543
$846$	0	0
$847$	−9.09483	−0.312502
$848$	0.799503	0.0274551
$849$	0	0
$850$	−52.6492	−1.80585
$851$	−17.1257	−0.587061
$852$	0	0
$853$	1.91745	0.0656523	0.0328261	−	0.999461i	$-0.489549\pi$
$853$	1.91745	0.0656523	0.0328261	+	0.999461i	$0.489549\pi$
$854$	−8.66043	−0.296354
$855$	0	0
$856$	4.04880	0.138385
$857$	−42.3948	−1.44818	−0.724090	−	0.689705i	$-0.757740\pi$
$857$	−42.3948	−1.44818	−0.724090	+	0.689705i	$0.757740\pi$
$858$	0	0
$859$	−12.3574	−0.421629	−0.210815	−	0.977526i	$-0.567612\pi$
$859$	−12.3574	−0.421629	−0.210815	+	0.977526i	$0.567612\pi$
$860$	4.11255	0.140237
$861$	0	0
$862$	−29.2340	−0.995714
$863$	6.84551	0.233024	0.116512	−	0.993189i	$-0.462829\pi$
$863$	6.84551	0.233024	0.116512	+	0.993189i	$0.462829\pi$
$864$	0	0
$865$	0.298248	0.0101407
$866$	−13.0202	−0.442445
$867$	0	0
$868$	31.0781	1.05486
$869$	−14.3004	−0.485109
$870$	0	0
$871$	−30.5327	−1.03456
$872$	−32.0300	−1.08467
$873$	0	0
$874$	−24.6497	−0.833790
$875$	−1.32448	−0.0447758
$876$	0	0
$877$	24.0490	0.812077	0.406038	−	0.913856i	$-0.366910\pi$
$877$	24.0490	0.812077	0.406038	+	0.913856i	$0.366910\pi$
$878$	85.6923	2.89197
$879$	0	0
$880$	0.0174525	0.000588323 0
$881$	13.2032	0.444826	0.222413	−	0.974953i	$-0.428607\pi$
$881$	13.2032	0.444826	0.222413	+	0.974953i	$0.428607\pi$
$882$	0	0
$883$	13.8993	0.467749	0.233874	−	0.972267i	$-0.424860\pi$
$883$	13.8993	0.467749	0.233874	+	0.972267i	$0.424860\pi$
$884$	−45.0367	−1.51475
$885$	0	0
$886$	52.6250	1.76797
$887$	−19.1216	−0.642041	−0.321020	−	0.947072i	$-0.604026\pi$
$887$	−19.1216	−0.642041	−0.321020	+	0.947072i	$0.604026\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−2.20582	−0.0739391
$891$	0	0
$892$	−30.8782	−1.03388
$893$	4.50710	0.150824
$894$	0	0
$895$	0.317392	0.0106092
$896$	18.5053	0.618218
$897$	0	0
$898$	69.7741	2.32839
$899$	−19.1037	−0.637145
$900$	0	0
$901$	−38.6648	−1.28811
$902$	6.90778	0.230004
$903$	0	0
$904$	50.8983	1.69285
$905$	−2.73862	−0.0910349
$906$	0	0
$907$	−4.48188	−0.148818	−0.0744091	−	0.997228i	$-0.523707\pi$
$907$	−4.48188	−0.148818	−0.0744091	+	0.997228i	$0.523707\pi$
$908$	−13.0158	−0.431946
$909$	0	0
$910$	−0.912706	−0.0302559
$911$	−10.0337	−0.332430	−0.166215	−	0.986090i	$-0.553155\pi$
$911$	−10.0337	−0.332430	−0.166215	+	0.986090i	$0.553155\pi$
$912$	0	0
$913$	−2.49911	−0.0827086
$914$	73.6477	2.43605
$915$	0	0
$916$	18.5445	0.612727
$917$	−5.73840	−0.189499
$918$	0	0
$919$	14.4554	0.476839	0.238419	−	0.971162i	$-0.423371\pi$
$919$	14.4554	0.476839	0.238419	+	0.971162i	$0.423371\pi$
$920$	2.30520	0.0760003
$921$	0	0
$922$	25.6124	0.843500
$923$	6.58704	0.216815
$924$	0	0
$925$	−14.1580	−0.465511
$926$	−58.2324	−1.91364
$927$	0	0
$928$	−11.1066	−0.364591
$929$	−53.1010	−1.74219	−0.871093	−	0.491117i	$-0.836589\pi$
$929$	−53.1010	−1.74219	−0.871093	+	0.491117i	$0.836589\pi$
$930$	0	0
$931$	−1.78380	−0.0584617
$932$	36.2467	1.18730
$933$	0	0
$934$	86.9552	2.84526
$935$	−0.844019	−0.0276024
$936$	0	0
$937$	−0.659156	−0.0215337	−0.0107668	−	0.999942i	$-0.503427\pi$
$937$	−0.659156	−0.0215337	−0.0107668	+	0.999942i	$0.503427\pi$
$938$	23.3350	0.761914
$939$	0	0
$940$	−1.09199	−0.0356167
$941$	59.8621	1.95145	0.975724	−	0.219006i	$-0.0702814\pi$
$941$	59.8621	1.95145	0.975724	+	0.219006i	$0.0702814\pi$
$942$	0	0
$943$	13.1546	0.428372
$944$	−0.742339	−0.0241611
$945$	0	0
$946$	−30.1156	−0.979142
$947$	−40.1186	−1.30368	−0.651840	−	0.758357i	$-0.726002\pi$
$947$	−40.1186	−1.30368	−0.651840	+	0.758357i	$0.726002\pi$
$948$	0	0
$949$	3.57190	0.115949
$950$	−20.3782	−0.661155
$951$	0	0
$952$	13.2857	0.430592
$953$	−24.9277	−0.807487	−0.403744	−	0.914872i	$-0.632291\pi$
$953$	−24.9277	−0.807487	−0.403744	+	0.914872i	$0.632291\pi$
$954$	0	0
$955$	2.13912	0.0692203
$956$	28.1894	0.911710
$957$	0	0
$958$	−22.7999	−0.736631
$959$	9.98542	0.322446
$960$	0	0
$961$	60.0334	1.93656
$962$	−19.5471	−0.630223
$963$	0	0
$964$	−86.6244	−2.78998
$965$	1.14520	0.0368652
$966$	0	0
$967$	26.4870	0.851764	0.425882	−	0.904779i	$-0.359964\pi$
$967$	26.4870	0.851764	0.425882	+	0.904779i	$0.359964\pi$
$968$	26.2184	0.842690
$969$	0	0
$970$	−0.651594	−0.0209214
$971$	−5.35158	−0.171740	−0.0858702	−	0.996306i	$-0.527367\pi$
$971$	−5.35158	−0.171740	−0.0858702	+	0.996306i	$0.527367\pi$
$972$	0	0
$973$	7.44343	0.238625
$974$	8.56016	0.274285
$975$	0	0
$976$	0.359946	0.0115216
$977$	34.3412	1.09867	0.549336	−	0.835602i	$-0.314881\pi$
$977$	34.3412	1.09867	0.549336	+	0.835602i	$0.314881\pi$
$978$	0	0
$979$	10.0079	0.319854
$980$	0.432182	0.0138055
$981$	0	0
$982$	85.0274	2.71333
$983$	29.9723	0.955968	0.477984	−	0.878369i	$-0.341368\pi$
$983$	29.9723	0.955968	0.477984	+	0.878369i	$0.341368\pi$
$984$	0	0
$985$	−0.0991233	−0.00315833
$986$	−21.1578	−0.673803
$987$	0	0
$988$	−17.4317	−0.554576
$989$	−57.3495	−1.82361
$990$	0	0
$991$	27.5987	0.876703	0.438351	−	0.898804i	$-0.355563\pi$
$991$	27.5987	0.876703	0.438351	+	0.898804i	$0.355563\pi$
$992$	52.9252	1.68038
$993$	0	0
$994$	−5.03422	−0.159676
$995$	−0.236796	−0.00750693
$996$	0	0
$997$	53.8873	1.70663	0.853314	−	0.521397i	$-0.174589\pi$
$997$	53.8873	1.70663	0.853314	+	0.521397i	$0.174589\pi$
$998$	−56.8616	−1.79992
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.o.1.2		13
3.2	odd	2		2667.2.a.l.1.12	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.l.1.12	✓	13	3.2	odd	2
8001.2.a.o.1.2		13	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)

\(f(q)\)	\(=\)	\(q-2.29288 q^{2} +3.25728 q^{4} +0.132682 q^{5} +1.00000 q^{7} -2.88278 q^{8} +O(q^{10})\)	\(q-2.29288 q^{2} +3.25728 q^{4} +0.132682 q^{5} +1.00000 q^{7} -2.88278 q^{8} -0.304223 q^{10} +1.38028 q^{11} +3.00012 q^{13} -2.29288 q^{14} +0.0952967 q^{16} -4.60865 q^{17} -1.78380 q^{19} +0.432182 q^{20} -3.16481 q^{22} -6.02678 q^{23} -4.98240 q^{25} -6.87890 q^{26} +3.25728 q^{28} -2.00225 q^{29} +9.54114 q^{31} +5.54705 q^{32} +10.5670 q^{34} +0.132682 q^{35} +2.84160 q^{37} +4.09003 q^{38} -0.382493 q^{40} -2.18268 q^{41} +9.51578 q^{43} +4.49595 q^{44} +13.8187 q^{46} -2.52668 q^{47} +1.00000 q^{49} +11.4240 q^{50} +9.77222 q^{52} +8.38962 q^{53} +0.183138 q^{55} -2.88278 q^{56} +4.59090 q^{58} -7.78977 q^{59} +3.77711 q^{61} -21.8766 q^{62} -12.9093 q^{64} +0.398062 q^{65} -10.1772 q^{67} -15.0116 q^{68} -0.304223 q^{70} +2.19559 q^{71} +1.19059 q^{73} -6.51543 q^{74} -5.81033 q^{76} +1.38028 q^{77} -10.3605 q^{79} +0.0126442 q^{80} +5.00462 q^{82} -1.81059 q^{83} -0.611485 q^{85} -21.8185 q^{86} -3.97904 q^{88} +7.25065 q^{89} +3.00012 q^{91} -19.6309 q^{92} +5.79337 q^{94} -0.236678 q^{95} +2.14183 q^{97} -2.29288 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 10 q^{4} - 12 q^{5} + 13 q^{7} - 9 q^{8} + 6 q^{10} - 3 q^{11} + 21 q^{13} - 4 q^{14} + 8 q^{16} - 17 q^{17} + 5 q^{19} - 29 q^{20} + q^{22} - 4 q^{23} + q^{25} - 22 q^{26} + 10 q^{28} - 21 q^{29} - 7 q^{31} - 12 q^{32} + 2 q^{34} - 12 q^{35} + 7 q^{37} + 9 q^{38} + 29 q^{40} - 21 q^{41} - 9 q^{43} + 2 q^{44} - 28 q^{46} - 23 q^{47} + 13 q^{49} - 15 q^{50} + 15 q^{52} - 31 q^{53} - 8 q^{55} - 9 q^{56} - 25 q^{58} - 28 q^{59} + 29 q^{61} + 3 q^{62} + 9 q^{64} - 30 q^{65} - 18 q^{67} - 34 q^{68} + 6 q^{70} - 10 q^{71} + 24 q^{73} + 19 q^{74} - 3 q^{77} - 28 q^{79} - 26 q^{80} + 18 q^{82} - 26 q^{83} + 20 q^{85} + 2 q^{86} - 17 q^{88} - 44 q^{89} + 21 q^{91} - 6 q^{92} - 9 q^{94} + 2 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 10 * q^4 - 12 * q^5 + 13 * q^7 - 9 * q^8 + 6 * q^10 - 3 * q^11 + 21 * q^13 - 4 * q^14 + 8 * q^16 - 17 * q^17 + 5 * q^19 - 29 * q^20 + q^22 - 4 * q^23 + q^25 - 22 * q^26 + 10 * q^28 - 21 * q^29 - 7 * q^31 - 12 * q^32 + 2 * q^34 - 12 * q^35 + 7 * q^37 + 9 * q^38 + 29 * q^40 - 21 * q^41 - 9 * q^43 + 2 * q^44 - 28 * q^46 - 23 * q^47 + 13 * q^49 - 15 * q^50 + 15 * q^52 - 31 * q^53 - 8 * q^55 - 9 * q^56 - 25 * q^58 - 28 * q^59 + 29 * q^61 + 3 * q^62 + 9 * q^64 - 30 * q^65 - 18 * q^67 - 34 * q^68 + 6 * q^70 - 10 * q^71 + 24 * q^73 + 19 * q^74 - 3 * q^77 - 28 * q^79 - 26 * q^80 + 18 * q^82 - 26 * q^83 + 20 * q^85 + 2 * q^86 - 17 * q^88 - 44 * q^89 + 21 * q^91 - 6 * q^92 - 9 * q^94 + 2 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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