LMFDB - Embedded newform 666.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [666,2,Mod(1,666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(666, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 666 = 2 \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	666.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:	$5.31803677462$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	666.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} -3.23607 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} -3.23607 q^{7} -1.00000 q^{8} +3.85410 q^{10} +1.38197 q^{11} -2.85410 q^{13} +3.23607 q^{14} +1.00000 q^{16} +4.47214 q^{17} +4.47214 q^{19} -3.85410 q^{20} -1.38197 q^{22} -2.85410 q^{23} +9.85410 q^{25} +2.85410 q^{26} -3.23607 q^{28} +9.32624 q^{29} +7.38197 q^{31} -1.00000 q^{32} -4.47214 q^{34} +12.4721 q^{35} -1.00000 q^{37} -4.47214 q^{38} +3.85410 q^{40} -9.61803 q^{41} -5.23607 q^{43} +1.38197 q^{44} +2.85410 q^{46} +1.23607 q^{47} +3.47214 q^{49} -9.85410 q^{50} -2.85410 q^{52} -0.472136 q^{53} -5.32624 q^{55} +3.23607 q^{56} -9.32624 q^{58} +4.76393 q^{59} +10.6180 q^{61} -7.38197 q^{62} +1.00000 q^{64} +11.0000 q^{65} +1.09017 q^{67} +4.47214 q^{68} -12.4721 q^{70} -2.94427 q^{71} +7.09017 q^{73} +1.00000 q^{74} +4.47214 q^{76} -4.47214 q^{77} -8.56231 q^{79} -3.85410 q^{80} +9.61803 q^{82} +14.4721 q^{83} -17.2361 q^{85} +5.23607 q^{86} -1.38197 q^{88} +1.52786 q^{89} +9.23607 q^{91} -2.85410 q^{92} -1.23607 q^{94} -17.2361 q^{95} -0.472136 q^{97} -3.47214 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + 5 q^{11} + q^{13} + 2 q^{14} + 2 q^{16} - q^{20} - 5 q^{22} + q^{23} + 13 q^{25} - q^{26} - 2 q^{28} + 3 q^{29} + 17 q^{31} - 2 q^{32} + 16 q^{35} - 2 q^{37} + q^{40} - 17 q^{41} - 6 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} - 2 q^{49} - 13 q^{50} + q^{52} + 8 q^{53} + 5 q^{55} + 2 q^{56} - 3 q^{58} + 14 q^{59} + 19 q^{61} - 17 q^{62} + 2 q^{64} + 22 q^{65} - 9 q^{67} - 16 q^{70} + 12 q^{71} + 3 q^{73} + 2 q^{74} + 3 q^{79} - q^{80} + 17 q^{82} + 20 q^{83} - 30 q^{85} + 6 q^{86} - 5 q^{88} + 12 q^{89} + 14 q^{91} + q^{92} + 2 q^{94} - 30 q^{95} + 8 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8 + q^10 + 5 * q^11 + q^13 + 2 * q^14 + 2 * q^16 - q^20 - 5 * q^22 + q^23 + 13 * q^25 - q^26 - 2 * q^28 + 3 * q^29 + 17 * q^31 - 2 * q^32 + 16 * q^35 - 2 * q^37 + q^40 - 17 * q^41 - 6 * q^43 + 5 * q^44 - q^46 - 2 * q^47 - 2 * q^49 - 13 * q^50 + q^52 + 8 * q^53 + 5 * q^55 + 2 * q^56 - 3 * q^58 + 14 * q^59 + 19 * q^61 - 17 * q^62 + 2 * q^64 + 22 * q^65 - 9 * q^67 - 16 * q^70 + 12 * q^71 + 3 * q^73 + 2 * q^74 + 3 * q^79 - q^80 + 17 * q^82 + 20 * q^83 - 30 * q^85 + 6 * q^86 - 5 * q^88 + 12 * q^89 + 14 * q^91 + q^92 + 2 * q^94 - 30 * q^95 + 8 * q^97 + 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.85410	−1.72361	−0.861803	−	0.507242i	$-0.830665\pi$
$5$	−3.85410	−1.72361	−0.861803	+	0.507242i	$0.830665\pi$
$6$	0	0
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.85410	1.21877
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	0	0
$13$	−2.85410	−0.791585	−0.395793	−	0.918340i	$-0.629530\pi$
$13$	−2.85410	−0.791585	−0.395793	+	0.918340i	$0.629530\pi$
$14$	3.23607	0.864876
$15$	0	0
$16$	1.00000	0.250000
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	−3.85410	−0.861803
$21$	0	0
$22$	−1.38197	−0.294636
$23$	−2.85410	−0.595121	−0.297561	−	0.954703i	$-0.596173\pi$
$23$	−2.85410	−0.595121	−0.297561	+	0.954703i	$0.596173\pi$
$24$	0	0
$25$	9.85410	1.97082
$26$	2.85410	0.559735
$27$	0	0
$28$	−3.23607	−0.611559
$29$	9.32624	1.73184	0.865919	−	0.500183i	$-0.166734\pi$
$29$	9.32624	1.73184	0.865919	+	0.500183i	$0.166734\pi$
$30$	0	0
$31$	7.38197	1.32584	0.662920	−	0.748690i	$-0.269317\pi$
$31$	7.38197	1.32584	0.662920	+	0.748690i	$0.269317\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.47214	−0.766965
$35$	12.4721	2.10818
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−4.47214	−0.725476
$39$	0	0
$40$	3.85410	0.609387
$41$	−9.61803	−1.50208	−0.751042	−	0.660254i	$-0.770449\pi$
$41$	−9.61803	−1.50208	−0.751042	+	0.660254i	$0.770449\pi$
$42$	0	0
$43$	−5.23607	−0.798493	−0.399246	−	0.916844i	$-0.630728\pi$
$43$	−5.23607	−0.798493	−0.399246	+	0.916844i	$0.630728\pi$
$44$	1.38197	0.208339
$45$	0	0
$46$	2.85410	0.420814
$47$	1.23607	0.180299	0.0901495	−	0.995928i	$-0.471266\pi$
$47$	1.23607	0.180299	0.0901495	+	0.995928i	$0.471266\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	−9.85410	−1.39358
$51$	0	0
$52$	−2.85410	−0.395793
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	−5.32624	−0.718190
$56$	3.23607	0.432438
$57$	0	0
$58$	−9.32624	−1.22460
$59$	4.76393	0.620211	0.310106	−	0.950702i	$-0.399636\pi$
$59$	4.76393	0.620211	0.310106	+	0.950702i	$0.399636\pi$
$60$	0	0
$61$	10.6180	1.35950	0.679750	−	0.733444i	$-0.262088\pi$
$61$	10.6180	1.35950	0.679750	+	0.733444i	$0.262088\pi$
$62$	−7.38197	−0.937511
$63$	0	0
$64$	1.00000	0.125000
$65$	11.0000	1.36438
$66$	0	0
$67$	1.09017	0.133185	0.0665927	−	0.997780i	$-0.478787\pi$
$67$	1.09017	0.133185	0.0665927	+	0.997780i	$0.478787\pi$
$68$	4.47214	0.542326
$69$	0	0
$70$	−12.4721	−1.49071
$71$	−2.94427	−0.349421	−0.174710	−	0.984620i	$-0.555899\pi$
$71$	−2.94427	−0.349421	−0.174710	+	0.984620i	$0.555899\pi$
$72$	0	0
$73$	7.09017	0.829842	0.414921	−	0.909858i	$-0.363809\pi$
$73$	7.09017	0.829842	0.414921	+	0.909858i	$0.363809\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	4.47214	0.512989
$77$	−4.47214	−0.509647
$78$	0	0
$79$	−8.56231	−0.963335	−0.481667	−	0.876354i	$-0.659969\pi$
$79$	−8.56231	−0.963335	−0.481667	+	0.876354i	$0.659969\pi$
$80$	−3.85410	−0.430902
$81$	0	0
$82$	9.61803	1.06213
$83$	14.4721	1.58852	0.794262	−	0.607576i	$-0.207858\pi$
$83$	14.4721	1.58852	0.794262	+	0.607576i	$0.207858\pi$
$84$	0	0
$85$	−17.2361	−1.86951
$86$	5.23607	0.564620
$87$	0	0
$88$	−1.38197	−0.147318
$89$	1.52786	0.161953	0.0809766	−	0.996716i	$-0.474196\pi$
$89$	1.52786	0.161953	0.0809766	+	0.996716i	$0.474196\pi$
$90$	0	0
$91$	9.23607	0.968203
$92$	−2.85410	−0.297561
$93$	0	0
$94$	−1.23607	−0.127491
$95$	−17.2361	−1.76838
$96$	0	0
$97$	−0.472136	−0.0479381	−0.0239691	−	0.999713i	$-0.507630\pi$
$97$	−0.472136	−0.0479381	−0.0239691	+	0.999713i	$0.507630\pi$
$98$	−3.47214	−0.350739
$99$	0	0
$100$	9.85410	0.985410
$101$	−3.52786	−0.351036	−0.175518	−	0.984476i	$-0.556160\pi$
$101$	−3.52786	−0.351036	−0.175518	+	0.984476i	$0.556160\pi$
$102$	0	0
$103$	17.2705	1.70171	0.850857	−	0.525397i	$-0.176083\pi$
$103$	17.2705	1.70171	0.850857	+	0.525397i	$0.176083\pi$
$104$	2.85410	0.279868
$105$	0	0
$106$	0.472136	0.0458579
$107$	7.32624	0.708254	0.354127	−	0.935197i	$-0.384778\pi$
$107$	7.32624	0.708254	0.354127	+	0.935197i	$0.384778\pi$
$108$	0	0
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	5.32624	0.507837
$111$	0	0
$112$	−3.23607	−0.305780
$113$	6.94427	0.653262	0.326631	−	0.945152i	$-0.394087\pi$
$113$	6.94427	0.653262	0.326631	+	0.945152i	$0.394087\pi$
$114$	0	0
$115$	11.0000	1.02576
$116$	9.32624	0.865919
$117$	0	0
$118$	−4.76393	−0.438555
$119$	−14.4721	−1.32666
$120$	0	0
$121$	−9.09017	−0.826379
$122$	−10.6180	−0.961312
$123$	0	0
$124$	7.38197	0.662920
$125$	−18.7082	−1.67331
$126$	0	0
$127$	−8.47214	−0.751780	−0.375890	−	0.926664i	$-0.622663\pi$
$127$	−8.47214	−0.751780	−0.375890	+	0.926664i	$0.622663\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−11.0000	−0.964764
$131$	22.6525	1.97916	0.989578	−	0.143998i	$-0.0459958\pi$
$131$	22.6525	1.97916	0.989578	+	0.143998i	$0.0459958\pi$
$132$	0	0
$133$	−14.4721	−1.25489
$134$	−1.09017	−0.0941763
$135$	0	0
$136$	−4.47214	−0.383482
$137$	−3.67376	−0.313871	−0.156935	−	0.987609i	$-0.550161\pi$
$137$	−3.67376	−0.313871	−0.156935	+	0.987609i	$0.550161\pi$
$138$	0	0
$139$	4.85410	0.411720	0.205860	−	0.978581i	$-0.434001\pi$
$139$	4.85410	0.411720	0.205860	+	0.978581i	$0.434001\pi$
$140$	12.4721	1.05409
$141$	0	0
$142$	2.94427	0.247078
$143$	−3.94427	−0.329837
$144$	0	0
$145$	−35.9443	−2.98501
$146$	−7.09017	−0.586787
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−16.1803	−1.32555	−0.662773	−	0.748821i	$-0.730620\pi$
$149$	−16.1803	−1.32555	−0.662773	+	0.748821i	$0.730620\pi$
$150$	0	0
$151$	4.29180	0.349261	0.174631	−	0.984634i	$-0.444127\pi$
$151$	4.29180	0.349261	0.174631	+	0.984634i	$0.444127\pi$
$152$	−4.47214	−0.362738
$153$	0	0
$154$	4.47214	0.360375
$155$	−28.4508	−2.28523
$156$	0	0
$157$	−16.4721	−1.31462	−0.657310	−	0.753620i	$-0.728306\pi$
$157$	−16.4721	−1.31462	−0.657310	+	0.753620i	$0.728306\pi$
$158$	8.56231	0.681180
$159$	0	0
$160$	3.85410	0.304694
$161$	9.23607	0.727904
$162$	0	0
$163$	−3.52786	−0.276324	−0.138162	−	0.990410i	$-0.544119\pi$
$163$	−3.52786	−0.276324	−0.138162	+	0.990410i	$0.544119\pi$
$164$	−9.61803	−0.751042
$165$	0	0
$166$	−14.4721	−1.12326
$167$	−13.8541	−1.07206	−0.536031	−	0.844198i	$-0.680077\pi$
$167$	−13.8541	−1.07206	−0.536031	+	0.844198i	$0.680077\pi$
$168$	0	0
$169$	−4.85410	−0.373392
$170$	17.2361	1.32195
$171$	0	0
$172$	−5.23607	−0.399246
$173$	0.472136	0.0358958	0.0179479	−	0.999839i	$-0.494287\pi$
$173$	0.472136	0.0358958	0.0179479	+	0.999839i	$0.494287\pi$
$174$	0	0
$175$	−31.8885	−2.41055
$176$	1.38197	0.104170
$177$	0	0
$178$	−1.52786	−0.114518
$179$	12.6525	0.945690	0.472845	−	0.881146i	$-0.343227\pi$
$179$	12.6525	0.945690	0.472845	+	0.881146i	$0.343227\pi$
$180$	0	0
$181$	14.4721	1.07571	0.537853	−	0.843039i	$-0.319236\pi$
$181$	14.4721	1.07571	0.537853	+	0.843039i	$0.319236\pi$
$182$	−9.23607	−0.684623
$183$	0	0
$184$	2.85410	0.210407
$185$	3.85410	0.283359
$186$	0	0
$187$	6.18034	0.451951
$188$	1.23607	0.0901495
$189$	0	0
$190$	17.2361	1.25044
$191$	7.09017	0.513027	0.256513	−	0.966541i	$-0.417426\pi$
$191$	7.09017	0.513027	0.256513	+	0.966541i	$0.417426\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0.472136	0.0338974
$195$	0	0
$196$	3.47214	0.248010
$197$	7.52786	0.536338	0.268169	−	0.963372i	$-0.413581\pi$
$197$	7.52786	0.536338	0.268169	+	0.963372i	$0.413581\pi$
$198$	0	0
$199$	3.05573	0.216615	0.108307	−	0.994117i	$-0.465457\pi$
$199$	3.05573	0.216615	0.108307	+	0.994117i	$0.465457\pi$
$200$	−9.85410	−0.696790
$201$	0	0
$202$	3.52786	0.248220
$203$	−30.1803	−2.11824
$204$	0	0
$205$	37.0689	2.58900
$206$	−17.2705	−1.20329
$207$	0	0
$208$	−2.85410	−0.197896
$209$	6.18034	0.427503
$210$	0	0
$211$	11.2705	0.775894	0.387947	−	0.921682i	$-0.373184\pi$
$211$	11.2705	0.775894	0.387947	+	0.921682i	$0.373184\pi$
$212$	−0.472136	−0.0324264
$213$	0	0
$214$	−7.32624	−0.500811
$215$	20.1803	1.37629
$216$	0	0
$217$	−23.8885	−1.62166
$218$	−2.94427	−0.199411
$219$	0	0
$220$	−5.32624	−0.359095
$221$	−12.7639	−0.858595
$222$	0	0
$223$	14.1803	0.949586	0.474793	−	0.880098i	$-0.342523\pi$
$223$	14.1803	0.949586	0.474793	+	0.880098i	$0.342523\pi$
$224$	3.23607	0.216219
$225$	0	0
$226$	−6.94427	−0.461926
$227$	4.29180	0.284857	0.142428	−	0.989805i	$-0.454509\pi$
$227$	4.29180	0.284857	0.142428	+	0.989805i	$0.454509\pi$
$228$	0	0
$229$	−23.1246	−1.52812	−0.764059	−	0.645147i	$-0.776796\pi$
$229$	−23.1246	−1.52812	−0.764059	+	0.645147i	$0.776796\pi$
$230$	−11.0000	−0.725319
$231$	0	0
$232$	−9.32624	−0.612298
$233$	6.56231	0.429911	0.214955	−	0.976624i	$-0.431039\pi$
$233$	6.56231	0.429911	0.214955	+	0.976624i	$0.431039\pi$
$234$	0	0
$235$	−4.76393	−0.310765
$236$	4.76393	0.310106
$237$	0	0
$238$	14.4721	0.938089
$239$	−9.85410	−0.637409	−0.318704	−	0.947854i	$-0.603248\pi$
$239$	−9.85410	−0.637409	−0.318704	+	0.947854i	$0.603248\pi$
$240$	0	0
$241$	−1.52786	−0.0984184	−0.0492092	−	0.998788i	$-0.515670\pi$
$241$	−1.52786	−0.0984184	−0.0492092	+	0.998788i	$0.515670\pi$
$242$	9.09017	0.584338
$243$	0	0
$244$	10.6180	0.679750
$245$	−13.3820	−0.854942
$246$	0	0
$247$	−12.7639	−0.812150
$248$	−7.38197	−0.468755
$249$	0	0
$250$	18.7082	1.18321
$251$	20.9443	1.32199	0.660995	−	0.750390i	$-0.270134\pi$
$251$	20.9443	1.32199	0.660995	+	0.750390i	$0.270134\pi$
$252$	0	0
$253$	−3.94427	−0.247974
$254$	8.47214	0.531589
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.05573	0.0658545	0.0329273	−	0.999458i	$-0.489517\pi$
$257$	1.05573	0.0658545	0.0329273	+	0.999458i	$0.489517\pi$
$258$	0	0
$259$	3.23607	0.201079
$260$	11.0000	0.682191
$261$	0	0
$262$	−22.6525	−1.39947
$263$	13.2361	0.816171	0.408085	−	0.912944i	$-0.366197\pi$
$263$	13.2361	0.816171	0.408085	+	0.912944i	$0.366197\pi$
$264$	0	0
$265$	1.81966	0.111781
$266$	14.4721	0.887344
$267$	0	0
$268$	1.09017	0.0665927
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	12.9443	0.786309	0.393154	−	0.919473i	$-0.371384\pi$
$271$	12.9443	0.786309	0.393154	+	0.919473i	$0.371384\pi$
$272$	4.47214	0.271163
$273$	0	0
$274$	3.67376	0.221940
$275$	13.6180	0.821198
$276$	0	0
$277$	16.7984	1.00932	0.504658	−	0.863319i	$-0.331619\pi$
$277$	16.7984	1.00932	0.504658	+	0.863319i	$0.331619\pi$
$278$	−4.85410	−0.291130
$279$	0	0
$280$	−12.4721	−0.745353
$281$	−29.8885	−1.78300	−0.891501	−	0.453020i	$-0.850347\pi$
$281$	−29.8885	−1.78300	−0.891501	+	0.453020i	$0.850347\pi$
$282$	0	0
$283$	6.76393	0.402074	0.201037	−	0.979584i	$-0.435569\pi$
$283$	6.76393	0.402074	0.201037	+	0.979584i	$0.435569\pi$
$284$	−2.94427	−0.174710
$285$	0	0
$286$	3.94427	0.233230
$287$	31.1246	1.83723
$288$	0	0
$289$	3.00000	0.176471
$290$	35.9443	2.11072
$291$	0	0
$292$	7.09017	0.414921
$293$	12.6525	0.739166	0.369583	−	0.929198i	$-0.379501\pi$
$293$	12.6525	0.739166	0.369583	+	0.929198i	$0.379501\pi$
$294$	0	0
$295$	−18.3607	−1.06900
$296$	1.00000	0.0581238
$297$	0	0
$298$	16.1803	0.937302
$299$	8.14590	0.471089
$300$	0	0
$301$	16.9443	0.976652
$302$	−4.29180	−0.246965
$303$	0	0
$304$	4.47214	0.256495
$305$	−40.9230	−2.34324
$306$	0	0
$307$	−12.8541	−0.733622	−0.366811	−	0.930295i	$-0.619550\pi$
$307$	−12.8541	−0.733622	−0.366811	+	0.930295i	$0.619550\pi$
$308$	−4.47214	−0.254824
$309$	0	0
$310$	28.4508	1.61590
$311$	27.0344	1.53298	0.766491	−	0.642255i	$-0.222001\pi$
$311$	27.0344	1.53298	0.766491	+	0.642255i	$0.222001\pi$
$312$	0	0
$313$	28.1803	1.59285	0.796423	−	0.604739i	$-0.206723\pi$
$313$	28.1803	1.59285	0.796423	+	0.604739i	$0.206723\pi$
$314$	16.4721	0.929576
$315$	0	0
$316$	−8.56231	−0.481667
$317$	−20.9443	−1.17635	−0.588174	−	0.808735i	$-0.700153\pi$
$317$	−20.9443	−1.17635	−0.588174	+	0.808735i	$0.700153\pi$
$318$	0	0
$319$	12.8885	0.721620
$320$	−3.85410	−0.215451
$321$	0	0
$322$	−9.23607	−0.514706
$323$	20.0000	1.11283
$324$	0	0
$325$	−28.1246	−1.56007
$326$	3.52786	0.195390
$327$	0	0
$328$	9.61803	0.531067
$329$	−4.00000	−0.220527
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	14.4721	0.794262
$333$	0	0
$334$	13.8541	0.758063
$335$	−4.20163	−0.229559
$336$	0	0
$337$	12.0344	0.655558	0.327779	−	0.944754i	$-0.393700\pi$
$337$	12.0344	0.655558	0.327779	+	0.944754i	$0.393700\pi$
$338$	4.85410	0.264028
$339$	0	0
$340$	−17.2361	−0.934757
$341$	10.2016	0.552449
$342$	0	0
$343$	11.4164	0.616428
$344$	5.23607	0.282310
$345$	0	0
$346$	−0.472136	−0.0253822
$347$	−17.2361	−0.925281	−0.462640	−	0.886546i	$-0.653098\pi$
$347$	−17.2361	−0.925281	−0.462640	+	0.886546i	$0.653098\pi$
$348$	0	0
$349$	−10.1803	−0.544941	−0.272471	−	0.962164i	$-0.587841\pi$
$349$	−10.1803	−0.544941	−0.272471	+	0.962164i	$0.587841\pi$
$350$	31.8885	1.70451
$351$	0	0
$352$	−1.38197	−0.0736590
$353$	−16.2918	−0.867125	−0.433562	−	0.901124i	$-0.642744\pi$
$353$	−16.2918	−0.867125	−0.433562	+	0.901124i	$0.642744\pi$
$354$	0	0
$355$	11.3475	0.602264
$356$	1.52786	0.0809766
$357$	0	0
$358$	−12.6525	−0.668704
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−14.4721	−0.760639
$363$	0	0
$364$	9.23607	0.484102
$365$	−27.3262	−1.43032
$366$	0	0
$367$	13.1246	0.685099	0.342550	−	0.939500i	$-0.388710\pi$
$367$	13.1246	0.685099	0.342550	+	0.939500i	$0.388710\pi$
$368$	−2.85410	−0.148780
$369$	0	0
$370$	−3.85410	−0.200365
$371$	1.52786	0.0793227
$372$	0	0
$373$	27.7082	1.43468	0.717338	−	0.696725i	$-0.245360\pi$
$373$	27.7082	1.43468	0.717338	+	0.696725i	$0.245360\pi$
$374$	−6.18034	−0.319578
$375$	0	0
$376$	−1.23607	−0.0637453
$377$	−26.6180	−1.37090
$378$	0	0
$379$	28.0902	1.44290	0.721448	−	0.692469i	$-0.243477\pi$
$379$	28.0902	1.44290	0.721448	+	0.692469i	$0.243477\pi$
$380$	−17.2361	−0.884192
$381$	0	0
$382$	−7.09017	−0.362765
$383$	−17.8885	−0.914062	−0.457031	−	0.889451i	$-0.651087\pi$
$383$	−17.8885	−0.914062	−0.457031	+	0.889451i	$0.651087\pi$
$384$	0	0
$385$	17.2361	0.878431
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−0.472136	−0.0239691
$389$	6.85410	0.347517	0.173758	−	0.984788i	$-0.444409\pi$
$389$	6.85410	0.347517	0.173758	+	0.984788i	$0.444409\pi$
$390$	0	0
$391$	−12.7639	−0.645500
$392$	−3.47214	−0.175369
$393$	0	0
$394$	−7.52786	−0.379248
$395$	33.0000	1.66041
$396$	0	0
$397$	20.6525	1.03652	0.518259	−	0.855224i	$-0.326580\pi$
$397$	20.6525	1.03652	0.518259	+	0.855224i	$0.326580\pi$
$398$	−3.05573	−0.153170
$399$	0	0
$400$	9.85410	0.492705
$401$	4.76393	0.237899	0.118950	−	0.992900i	$-0.462047\pi$
$401$	4.76393	0.237899	0.118950	+	0.992900i	$0.462047\pi$
$402$	0	0
$403$	−21.0689	−1.04952
$404$	−3.52786	−0.175518
$405$	0	0
$406$	30.1803	1.49783
$407$	−1.38197	−0.0685015
$408$	0	0
$409$	−26.1803	−1.29453	−0.647267	−	0.762263i	$-0.724088\pi$
$409$	−26.1803	−1.29453	−0.647267	+	0.762263i	$0.724088\pi$
$410$	−37.0689	−1.83070
$411$	0	0
$412$	17.2705	0.850857
$413$	−15.4164	−0.758592
$414$	0	0
$415$	−55.7771	−2.73799
$416$	2.85410	0.139934
$417$	0	0
$418$	−6.18034	−0.302290
$419$	10.5623	0.516002	0.258001	−	0.966145i	$-0.416936\pi$
$419$	10.5623	0.516002	0.258001	+	0.966145i	$0.416936\pi$
$420$	0	0
$421$	−31.0344	−1.51253	−0.756263	−	0.654268i	$-0.772977\pi$
$421$	−31.0344	−1.51253	−0.756263	+	0.654268i	$0.772977\pi$
$422$	−11.2705	−0.548640
$423$	0	0
$424$	0.472136	0.0229289
$425$	44.0689	2.13765
$426$	0	0
$427$	−34.3607	−1.66283
$428$	7.32624	0.354127
$429$	0	0
$430$	−20.1803	−0.973182
$431$	−12.3607	−0.595393	−0.297696	−	0.954661i	$-0.596218\pi$
$431$	−12.3607	−0.595393	−0.297696	+	0.954661i	$0.596218\pi$
$432$	0	0
$433$	−20.6738	−0.993518	−0.496759	−	0.867889i	$-0.665477\pi$
$433$	−20.6738	−0.993518	−0.496759	+	0.867889i	$0.665477\pi$
$434$	23.8885	1.14669
$435$	0	0
$436$	2.94427	0.141005
$437$	−12.7639	−0.610582
$438$	0	0
$439$	7.79837	0.372196	0.186098	−	0.982531i	$-0.440416\pi$
$439$	7.79837	0.372196	0.186098	+	0.982531i	$0.440416\pi$
$440$	5.32624	0.253918
$441$	0	0
$442$	12.7639	0.607118
$443$	−15.2705	−0.725524	−0.362762	−	0.931882i	$-0.618166\pi$
$443$	−15.2705	−0.725524	−0.362762	+	0.931882i	$0.618166\pi$
$444$	0	0
$445$	−5.88854	−0.279144
$446$	−14.1803	−0.671459
$447$	0	0
$448$	−3.23607	−0.152890
$449$	28.4721	1.34368	0.671842	−	0.740695i	$-0.265504\pi$
$449$	28.4721	1.34368	0.671842	+	0.740695i	$0.265504\pi$
$450$	0	0
$451$	−13.2918	−0.625886
$452$	6.94427	0.326631
$453$	0	0
$454$	−4.29180	−0.201424
$455$	−35.5967	−1.66880
$456$	0	0
$457$	−24.7639	−1.15841	−0.579204	−	0.815183i	$-0.696637\pi$
$457$	−24.7639	−1.15841	−0.579204	+	0.815183i	$0.696637\pi$
$458$	23.1246	1.08054
$459$	0	0
$460$	11.0000	0.512878
$461$	38.9443	1.81382	0.906908	−	0.421329i	$-0.138436\pi$
$461$	38.9443	1.81382	0.906908	+	0.421329i	$0.138436\pi$
$462$	0	0
$463$	−4.56231	−0.212028	−0.106014	−	0.994365i	$-0.533809\pi$
$463$	−4.56231	−0.212028	−0.106014	+	0.994365i	$0.533809\pi$
$464$	9.32624	0.432960
$465$	0	0
$466$	−6.56231	−0.303993
$467$	24.3607	1.12728	0.563639	−	0.826021i	$-0.309401\pi$
$467$	24.3607	1.12728	0.563639	+	0.826021i	$0.309401\pi$
$468$	0	0
$469$	−3.52786	−0.162902
$470$	4.76393	0.219744
$471$	0	0
$472$	−4.76393	−0.219278
$473$	−7.23607	−0.332715
$474$	0	0
$475$	44.0689	2.02202
$476$	−14.4721	−0.663329
$477$	0	0
$478$	9.85410	0.450716
$479$	−36.5623	−1.67057	−0.835287	−	0.549814i	$-0.814699\pi$
$479$	−36.5623	−1.67057	−0.835287	+	0.549814i	$0.814699\pi$
$480$	0	0
$481$	2.85410	0.130136
$482$	1.52786	0.0695923
$483$	0	0
$484$	−9.09017	−0.413190
$485$	1.81966	0.0826265
$486$	0	0
$487$	37.3050	1.69045	0.845224	−	0.534412i	$-0.179467\pi$
$487$	37.3050	1.69045	0.845224	+	0.534412i	$0.179467\pi$
$488$	−10.6180	−0.480656
$489$	0	0
$490$	13.3820	0.604536
$491$	28.4508	1.28397	0.641984	−	0.766718i	$-0.278111\pi$
$491$	28.4508	1.28397	0.641984	+	0.766718i	$0.278111\pi$
$492$	0	0
$493$	41.7082	1.87844
$494$	12.7639	0.574276
$495$	0	0
$496$	7.38197	0.331460
$497$	9.52786	0.427383
$498$	0	0
$499$	10.2918	0.460724	0.230362	−	0.973105i	$-0.426009\pi$
$499$	10.2918	0.460724	0.230362	+	0.973105i	$0.426009\pi$
$500$	−18.7082	−0.836656
$501$	0	0
$502$	−20.9443	−0.934789
$503$	−19.0902	−0.851189	−0.425594	−	0.904914i	$-0.639935\pi$
$503$	−19.0902	−0.851189	−0.425594	+	0.904914i	$0.639935\pi$
$504$	0	0
$505$	13.5967	0.605047
$506$	3.94427	0.175344
$507$	0	0
$508$	−8.47214	−0.375890
$509$	17.7082	0.784902	0.392451	−	0.919773i	$-0.371627\pi$
$509$	17.7082	0.784902	0.392451	+	0.919773i	$0.371627\pi$
$510$	0	0
$511$	−22.9443	−1.01499
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−1.05573	−0.0465662
$515$	−66.5623	−2.93309
$516$	0	0
$517$	1.70820	0.0751267
$518$	−3.23607	−0.142185
$519$	0	0
$520$	−11.0000	−0.482382
$521$	1.41641	0.0620540	0.0310270	−	0.999519i	$-0.490122\pi$
$521$	1.41641	0.0620540	0.0310270	+	0.999519i	$0.490122\pi$
$522$	0	0
$523$	11.8197	0.516838	0.258419	−	0.966033i	$-0.416799\pi$
$523$	11.8197	0.516838	0.258419	+	0.966033i	$0.416799\pi$
$524$	22.6525	0.989578
$525$	0	0
$526$	−13.2361	−0.577120
$527$	33.0132	1.43808
$528$	0	0
$529$	−14.8541	−0.645831
$530$	−1.81966	−0.0790410
$531$	0	0
$532$	−14.4721	−0.627447
$533$	27.4508	1.18903
$534$	0	0
$535$	−28.2361	−1.22075
$536$	−1.09017	−0.0470882
$537$	0	0
$538$	4.00000	0.172452
$539$	4.79837	0.206681
$540$	0	0
$541$	−10.3262	−0.443960	−0.221980	−	0.975051i	$-0.571252\pi$
$541$	−10.3262	−0.443960	−0.221980	+	0.975051i	$0.571252\pi$
$542$	−12.9443	−0.556004
$543$	0	0
$544$	−4.47214	−0.191741
$545$	−11.3475	−0.486075
$546$	0	0
$547$	−16.0689	−0.687056	−0.343528	−	0.939142i	$-0.611622\pi$
$547$	−16.0689	−0.687056	−0.343528	+	0.939142i	$0.611622\pi$
$548$	−3.67376	−0.156935
$549$	0	0
$550$	−13.6180	−0.580675
$551$	41.7082	1.77683
$552$	0	0
$553$	27.7082	1.17827
$554$	−16.7984	−0.713695
$555$	0	0
$556$	4.85410	0.205860
$557$	−19.5623	−0.828882	−0.414441	−	0.910076i	$-0.636023\pi$
$557$	−19.5623	−0.828882	−0.414441	+	0.910076i	$0.636023\pi$
$558$	0	0
$559$	14.9443	0.632075
$560$	12.4721	0.527044
$561$	0	0
$562$	29.8885	1.26077
$563$	−7.88854	−0.332462	−0.166231	−	0.986087i	$-0.553160\pi$
$563$	−7.88854	−0.332462	−0.166231	+	0.986087i	$0.553160\pi$
$564$	0	0
$565$	−26.7639	−1.12597
$566$	−6.76393	−0.284309
$567$	0	0
$568$	2.94427	0.123539
$569$	−13.8885	−0.582238	−0.291119	−	0.956687i	$-0.594028\pi$
$569$	−13.8885	−0.582238	−0.291119	+	0.956687i	$0.594028\pi$
$570$	0	0
$571$	−10.5623	−0.442019	−0.221009	−	0.975272i	$-0.570935\pi$
$571$	−10.5623	−0.442019	−0.221009	+	0.975272i	$0.570935\pi$
$572$	−3.94427	−0.164918
$573$	0	0
$574$	−31.1246	−1.29912
$575$	−28.1246	−1.17288
$576$	0	0
$577$	10.6525	0.443468	0.221734	−	0.975107i	$-0.428828\pi$
$577$	10.6525	0.443468	0.221734	+	0.975107i	$0.428828\pi$
$578$	−3.00000	−0.124784
$579$	0	0
$580$	−35.9443	−1.49250
$581$	−46.8328	−1.94295
$582$	0	0
$583$	−0.652476	−0.0270228
$584$	−7.09017	−0.293393
$585$	0	0
$586$	−12.6525	−0.522669
$587$	13.0557	0.538868	0.269434	−	0.963019i	$-0.413163\pi$
$587$	13.0557	0.538868	0.269434	+	0.963019i	$0.413163\pi$
$588$	0	0
$589$	33.0132	1.36028
$590$	18.3607	0.755897
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−17.5623	−0.721197	−0.360599	−	0.932721i	$-0.617428\pi$
$593$	−17.5623	−0.721197	−0.360599	+	0.932721i	$0.617428\pi$
$594$	0	0
$595$	55.7771	2.28664
$596$	−16.1803	−0.662773
$597$	0	0
$598$	−8.14590	−0.333111
$599$	6.36068	0.259890	0.129945	−	0.991521i	$-0.458520\pi$
$599$	6.36068	0.259890	0.129945	+	0.991521i	$0.458520\pi$
$600$	0	0
$601$	−24.6869	−1.00700	−0.503500	−	0.863995i	$-0.667955\pi$
$601$	−24.6869	−1.00700	−0.503500	+	0.863995i	$0.667955\pi$
$602$	−16.9443	−0.690597
$603$	0	0
$604$	4.29180	0.174631
$605$	35.0344	1.42435
$606$	0	0
$607$	35.0344	1.42200	0.711002	−	0.703190i	$-0.248242\pi$
$607$	35.0344	1.42200	0.711002	+	0.703190i	$0.248242\pi$
$608$	−4.47214	−0.181369
$609$	0	0
$610$	40.9230	1.65692
$611$	−3.52786	−0.142722
$612$	0	0
$613$	−13.8197	−0.558171	−0.279085	−	0.960266i	$-0.590031\pi$
$613$	−13.8197	−0.558171	−0.279085	+	0.960266i	$0.590031\pi$
$614$	12.8541	0.518749
$615$	0	0
$616$	4.47214	0.180187
$617$	−0.0901699	−0.00363011	−0.00181505	−	0.999998i	$-0.500578\pi$
$617$	−0.0901699	−0.00363011	−0.00181505	+	0.999998i	$0.500578\pi$
$618$	0	0
$619$	−15.2705	−0.613774	−0.306887	−	0.951746i	$-0.599287\pi$
$619$	−15.2705	−0.613774	−0.306887	+	0.951746i	$0.599287\pi$
$620$	−28.4508	−1.14261
$621$	0	0
$622$	−27.0344	−1.08398
$623$	−4.94427	−0.198088
$624$	0	0
$625$	22.8328	0.913313
$626$	−28.1803	−1.12631
$627$	0	0
$628$	−16.4721	−0.657310
$629$	−4.47214	−0.178316
$630$	0	0
$631$	−47.3951	−1.88677	−0.943385	−	0.331700i	$-0.892378\pi$
$631$	−47.3951	−1.88677	−0.943385	+	0.331700i	$0.892378\pi$
$632$	8.56231	0.340590
$633$	0	0
$634$	20.9443	0.831803
$635$	32.6525	1.29577
$636$	0	0
$637$	−9.90983	−0.392642
$638$	−12.8885	−0.510262
$639$	0	0
$640$	3.85410	0.152347
$641$	−15.5066	−0.612473	−0.306236	−	0.951955i	$-0.599070\pi$
$641$	−15.5066	−0.612473	−0.306236	+	0.951955i	$0.599070\pi$
$642$	0	0
$643$	−28.7639	−1.13434	−0.567169	−	0.823601i	$-0.691962\pi$
$643$	−28.7639	−1.13434	−0.567169	+	0.823601i	$0.691962\pi$
$644$	9.23607	0.363952
$645$	0	0
$646$	−20.0000	−0.786889
$647$	−30.0902	−1.18297	−0.591483	−	0.806317i	$-0.701457\pi$
$647$	−30.0902	−1.18297	−0.591483	+	0.806317i	$0.701457\pi$
$648$	0	0
$649$	6.58359	0.258429
$650$	28.1246	1.10314
$651$	0	0
$652$	−3.52786	−0.138162
$653$	38.2705	1.49764	0.748820	−	0.662773i	$-0.230621\pi$
$653$	38.2705	1.49764	0.748820	+	0.662773i	$0.230621\pi$
$654$	0	0
$655$	−87.3050	−3.41129
$656$	−9.61803	−0.375521
$657$	0	0
$658$	4.00000	0.155936
$659$	−40.4508	−1.57574	−0.787871	−	0.615841i	$-0.788816\pi$
$659$	−40.4508	−1.57574	−0.787871	+	0.615841i	$0.788816\pi$
$660$	0	0
$661$	17.3262	0.673913	0.336956	−	0.941520i	$-0.390603\pi$
$661$	17.3262	0.673913	0.336956	+	0.941520i	$0.390603\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	−14.4721	−0.561628
$665$	55.7771	2.16294
$666$	0	0
$667$	−26.6180	−1.03065
$668$	−13.8541	−0.536031
$669$	0	0
$670$	4.20163	0.162323
$671$	14.6738	0.566474
$672$	0	0
$673$	11.1459	0.429643	0.214821	−	0.976653i	$-0.431083\pi$
$673$	11.1459	0.429643	0.214821	+	0.976653i	$0.431083\pi$
$674$	−12.0344	−0.463549
$675$	0	0
$676$	−4.85410	−0.186696
$677$	−34.6525	−1.33180	−0.665901	−	0.746040i	$-0.731953\pi$
$677$	−34.6525	−1.33180	−0.665901	+	0.746040i	$0.731953\pi$
$678$	0	0
$679$	1.52786	0.0586340
$680$	17.2361	0.660973
$681$	0	0
$682$	−10.2016	−0.390640
$683$	8.58359	0.328442	0.164221	−	0.986424i	$-0.447489\pi$
$683$	8.58359	0.328442	0.164221	+	0.986424i	$0.447489\pi$
$684$	0	0
$685$	14.1591	0.540990
$686$	−11.4164	−0.435880
$687$	0	0
$688$	−5.23607	−0.199623
$689$	1.34752	0.0513366
$690$	0	0
$691$	35.7771	1.36102	0.680512	−	0.732737i	$-0.261757\pi$
$691$	35.7771	1.36102	0.680512	+	0.732737i	$0.261757\pi$
$692$	0.472136	0.0179479
$693$	0	0
$694$	17.2361	0.654272
$695$	−18.7082	−0.709643
$696$	0	0
$697$	−43.0132	−1.62924
$698$	10.1803	0.385332
$699$	0	0
$700$	−31.8885	−1.20527
$701$	42.9787	1.62328	0.811642	−	0.584155i	$-0.198574\pi$
$701$	42.9787	1.62328	0.811642	+	0.584155i	$0.198574\pi$
$702$	0	0
$703$	−4.47214	−0.168670
$704$	1.38197	0.0520848
$705$	0	0
$706$	16.2918	0.613150
$707$	11.4164	0.429358
$708$	0	0
$709$	−8.21478	−0.308513	−0.154256	−	0.988031i	$-0.549298\pi$
$709$	−8.21478	−0.308513	−0.154256	+	0.988031i	$0.549298\pi$
$710$	−11.3475	−0.425865
$711$	0	0
$712$	−1.52786	−0.0572591
$713$	−21.0689	−0.789036
$714$	0	0
$715$	15.2016	0.568509
$716$	12.6525	0.472845
$717$	0	0
$718$	−4.47214	−0.166899
$719$	−27.4164	−1.02246	−0.511230	−	0.859444i	$-0.670810\pi$
$719$	−27.4164	−1.02246	−0.511230	+	0.859444i	$0.670810\pi$
$720$	0	0
$721$	−55.8885	−2.08140
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	14.4721	0.537853
$725$	91.9017	3.41314
$726$	0	0
$727$	−29.8541	−1.10723	−0.553614	−	0.832774i	$-0.686752\pi$
$727$	−29.8541	−1.10723	−0.553614	+	0.832774i	$0.686752\pi$
$728$	−9.23607	−0.342311
$729$	0	0
$730$	27.3262	1.01139
$731$	−23.4164	−0.866087
$732$	0	0
$733$	27.5279	1.01676	0.508382	−	0.861131i	$-0.330244\pi$
$733$	27.5279	1.01676	0.508382	+	0.861131i	$0.330244\pi$
$734$	−13.1246	−0.484438
$735$	0	0
$736$	2.85410	0.105204
$737$	1.50658	0.0554955
$738$	0	0
$739$	−10.9098	−0.401325	−0.200662	−	0.979660i	$-0.564309\pi$
$739$	−10.9098	−0.401325	−0.200662	+	0.979660i	$0.564309\pi$
$740$	3.85410	0.141680
$741$	0	0
$742$	−1.52786	−0.0560897
$743$	48.0689	1.76348	0.881738	−	0.471739i	$-0.156374\pi$
$743$	48.0689	1.76348	0.881738	+	0.471739i	$0.156374\pi$
$744$	0	0
$745$	62.3607	2.28472
$746$	−27.7082	−1.01447
$747$	0	0
$748$	6.18034	0.225976
$749$	−23.7082	−0.866279
$750$	0	0
$751$	−1.05573	−0.0385241	−0.0192620	−	0.999814i	$-0.506132\pi$
$751$	−1.05573	−0.0385241	−0.0192620	+	0.999814i	$0.506132\pi$
$752$	1.23607	0.0450748
$753$	0	0
$754$	26.6180	0.969372
$755$	−16.5410	−0.601989
$756$	0	0
$757$	−4.14590	−0.150685	−0.0753426	−	0.997158i	$-0.524005\pi$
$757$	−4.14590	−0.150685	−0.0753426	+	0.997158i	$0.524005\pi$
$758$	−28.0902	−1.02028
$759$	0	0
$760$	17.2361	0.625218
$761$	19.1459	0.694038	0.347019	−	0.937858i	$-0.387194\pi$
$761$	19.1459	0.694038	0.347019	+	0.937858i	$0.387194\pi$
$762$	0	0
$763$	−9.52786	−0.344932
$764$	7.09017	0.256513
$765$	0	0
$766$	17.8885	0.646339
$767$	−13.5967	−0.490950
$768$	0	0
$769$	23.8885	0.861443	0.430721	−	0.902485i	$-0.358259\pi$
$769$	23.8885	0.861443	0.430721	+	0.902485i	$0.358259\pi$
$770$	−17.2361	−0.621145
$771$	0	0
$772$	4.00000	0.143963
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	72.7426	2.61299
$776$	0.472136	0.0169487
$777$	0	0
$778$	−6.85410	−0.245731
$779$	−43.0132	−1.54111
$780$	0	0
$781$	−4.06888	−0.145596
$782$	12.7639	0.456437
$783$	0	0
$784$	3.47214	0.124005
$785$	63.4853	2.26589
$786$	0	0
$787$	−25.5279	−0.909970	−0.454985	−	0.890499i	$-0.650355\pi$
$787$	−25.5279	−0.909970	−0.454985	+	0.890499i	$0.650355\pi$
$788$	7.52786	0.268169
$789$	0	0
$790$	−33.0000	−1.17409
$791$	−22.4721	−0.799017
$792$	0	0
$793$	−30.3050	−1.07616
$794$	−20.6525	−0.732929
$795$	0	0
$796$	3.05573	0.108307
$797$	−46.2705	−1.63899	−0.819493	−	0.573090i	$-0.805745\pi$
$797$	−46.2705	−1.63899	−0.819493	+	0.573090i	$0.805745\pi$
$798$	0	0
$799$	5.52786	0.195562
$800$	−9.85410	−0.348395
$801$	0	0
$802$	−4.76393	−0.168220
$803$	9.79837	0.345777
$804$	0	0
$805$	−35.5967	−1.25462
$806$	21.0689	0.742120
$807$	0	0
$808$	3.52786	0.124110
$809$	13.1246	0.461437	0.230718	−	0.973021i	$-0.425892\pi$
$809$	13.1246	0.461437	0.230718	+	0.973021i	$0.425892\pi$
$810$	0	0
$811$	−34.1033	−1.19753	−0.598765	−	0.800925i	$-0.704342\pi$
$811$	−34.1033	−1.19753	−0.598765	+	0.800925i	$0.704342\pi$
$812$	−30.1803	−1.05912
$813$	0	0
$814$	1.38197	0.0484379
$815$	13.5967	0.476273
$816$	0	0
$817$	−23.4164	−0.819236
$818$	26.1803	0.915374
$819$	0	0
$820$	37.0689	1.29450
$821$	5.41641	0.189034	0.0945170	−	0.995523i	$-0.469869\pi$
$821$	5.41641	0.189034	0.0945170	+	0.995523i	$0.469869\pi$
$822$	0	0
$823$	1.88854	0.0658305	0.0329152	−	0.999458i	$-0.489521\pi$
$823$	1.88854	0.0658305	0.0329152	+	0.999458i	$0.489521\pi$
$824$	−17.2705	−0.601647
$825$	0	0
$826$	15.4164	0.536405
$827$	2.06888	0.0719421	0.0359711	−	0.999353i	$-0.488548\pi$
$827$	2.06888	0.0719421	0.0359711	+	0.999353i	$0.488548\pi$
$828$	0	0
$829$	55.7984	1.93796	0.968979	−	0.247144i	$-0.0794920\pi$
$829$	55.7984	1.93796	0.968979	+	0.247144i	$0.0794920\pi$
$830$	55.7771	1.93605
$831$	0	0
$832$	−2.85410	−0.0989482
$833$	15.5279	0.538009
$834$	0	0
$835$	53.3951	1.84781
$836$	6.18034	0.213752
$837$	0	0
$838$	−10.5623	−0.364869
$839$	36.6525	1.26538	0.632692	−	0.774404i	$-0.281950\pi$
$839$	36.6525	1.26538	0.632692	+	0.774404i	$0.281950\pi$
$840$	0	0
$841$	57.9787	1.99927
$842$	31.0344	1.06952
$843$	0	0
$844$	11.2705	0.387947
$845$	18.7082	0.643582
$846$	0	0
$847$	29.4164	1.01076
$848$	−0.472136	−0.0162132
$849$	0	0
$850$	−44.0689	−1.51155
$851$	2.85410	0.0978374
$852$	0	0
$853$	29.7426	1.01837	0.509184	−	0.860657i	$-0.329947\pi$
$853$	29.7426	1.01837	0.509184	+	0.860657i	$0.329947\pi$
$854$	34.3607	1.17580
$855$	0	0
$856$	−7.32624	−0.250406
$857$	9.05573	0.309338	0.154669	−	0.987966i	$-0.450569\pi$
$857$	9.05573	0.309338	0.154669	+	0.987966i	$0.450569\pi$
$858$	0	0
$859$	53.4164	1.82254	0.911272	−	0.411805i	$-0.135101\pi$
$859$	53.4164	1.82254	0.911272	+	0.411805i	$0.135101\pi$
$860$	20.1803	0.688144
$861$	0	0
$862$	12.3607	0.421006
$863$	−19.4164	−0.660942	−0.330471	−	0.943816i	$-0.607208\pi$
$863$	−19.4164	−0.660942	−0.330471	+	0.943816i	$0.607208\pi$
$864$	0	0
$865$	−1.81966	−0.0618703
$866$	20.6738	0.702523
$867$	0	0
$868$	−23.8885	−0.810830
$869$	−11.8328	−0.401401
$870$	0	0
$871$	−3.11146	−0.105428
$872$	−2.94427	−0.0997056
$873$	0	0
$874$	12.7639	0.431746
$875$	60.5410	2.04666
$876$	0	0
$877$	−16.8328	−0.568404	−0.284202	−	0.958764i	$-0.591729\pi$
$877$	−16.8328	−0.568404	−0.284202	+	0.958764i	$0.591729\pi$
$878$	−7.79837	−0.263182
$879$	0	0
$880$	−5.32624	−0.179547
$881$	−19.7426	−0.665147	−0.332573	−	0.943077i	$-0.607917\pi$
$881$	−19.7426	−0.665147	−0.332573	+	0.943077i	$0.607917\pi$
$882$	0	0
$883$	−33.3050	−1.12080	−0.560400	−	0.828222i	$-0.689353\pi$
$883$	−33.3050	−1.12080	−0.560400	+	0.828222i	$0.689353\pi$
$884$	−12.7639	−0.429297
$885$	0	0
$886$	15.2705	0.513023
$887$	−27.8885	−0.936406	−0.468203	−	0.883621i	$-0.655098\pi$
$887$	−27.8885	−0.936406	−0.468203	+	0.883621i	$0.655098\pi$
$888$	0	0
$889$	27.4164	0.919517
$890$	5.88854	0.197384
$891$	0	0
$892$	14.1803	0.474793
$893$	5.52786	0.184983
$894$	0	0
$895$	−48.7639	−1.63000
$896$	3.23607	0.108109
$897$	0	0
$898$	−28.4721	−0.950127
$899$	68.8460	2.29614
$900$	0	0
$901$	−2.11146	−0.0703428
$902$	13.2918	0.442568
$903$	0	0
$904$	−6.94427	−0.230963
$905$	−55.7771	−1.85409
$906$	0	0
$907$	−55.8885	−1.85575	−0.927874	−	0.372893i	$-0.878366\pi$
$907$	−55.8885	−1.85575	−0.927874	+	0.372893i	$0.878366\pi$
$908$	4.29180	0.142428
$909$	0	0
$910$	35.5967	1.18002
$911$	−13.8885	−0.460148	−0.230074	−	0.973173i	$-0.573897\pi$
$911$	−13.8885	−0.460148	−0.230074	+	0.973173i	$0.573897\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	24.7639	0.819118
$915$	0	0
$916$	−23.1246	−0.764059
$917$	−73.3050	−2.42074
$918$	0	0
$919$	−5.88854	−0.194245	−0.0971226	−	0.995272i	$-0.530964\pi$
$919$	−5.88854	−0.194245	−0.0971226	+	0.995272i	$0.530964\pi$
$920$	−11.0000	−0.362659
$921$	0	0
$922$	−38.9443	−1.28256
$923$	8.40325	0.276596
$924$	0	0
$925$	−9.85410	−0.324001
$926$	4.56231	0.149927
$927$	0	0
$928$	−9.32624	−0.306149
$929$	27.4508	0.900633	0.450317	−	0.892869i	$-0.351311\pi$
$929$	27.4508	0.900633	0.450317	+	0.892869i	$0.351311\pi$
$930$	0	0
$931$	15.5279	0.508905
$932$	6.56231	0.214955
$933$	0	0
$934$	−24.3607	−0.797106
$935$	−23.8197	−0.778986
$936$	0	0
$937$	−48.0476	−1.56965	−0.784823	−	0.619720i	$-0.787246\pi$
$937$	−48.0476	−1.56965	−0.784823	+	0.619720i	$0.787246\pi$
$938$	3.52786	0.115189
$939$	0	0
$940$	−4.76393	−0.155382
$941$	26.1803	0.853455	0.426727	−	0.904380i	$-0.359666\pi$
$941$	26.1803	0.853455	0.426727	+	0.904380i	$0.359666\pi$
$942$	0	0
$943$	27.4508	0.893923
$944$	4.76393	0.155053
$945$	0	0
$946$	7.23607	0.235265
$947$	−18.8328	−0.611984	−0.305992	−	0.952034i	$-0.598988\pi$
$947$	−18.8328	−0.611984	−0.305992	+	0.952034i	$0.598988\pi$
$948$	0	0
$949$	−20.2361	−0.656891
$950$	−44.0689	−1.42978
$951$	0	0
$952$	14.4721	0.469045
$953$	−11.4508	−0.370929	−0.185465	−	0.982651i	$-0.559379\pi$
$953$	−11.4508	−0.370929	−0.185465	+	0.982651i	$0.559379\pi$
$954$	0	0
$955$	−27.3262	−0.884256
$956$	−9.85410	−0.318704
$957$	0	0
$958$	36.5623	1.18127
$959$	11.8885	0.383901
$960$	0	0
$961$	23.4934	0.757852
$962$	−2.85410	−0.0920199
$963$	0	0
$964$	−1.52786	−0.0492092
$965$	−15.4164	−0.496272
$966$	0	0
$967$	45.2705	1.45580	0.727901	−	0.685683i	$-0.240496\pi$
$967$	45.2705	1.45580	0.727901	+	0.685683i	$0.240496\pi$
$968$	9.09017	0.292169
$969$	0	0
$970$	−1.81966	−0.0584258
$971$	−42.3262	−1.35831	−0.679157	−	0.733993i	$-0.737654\pi$
$971$	−42.3262	−1.35831	−0.679157	+	0.733993i	$0.737654\pi$
$972$	0	0
$973$	−15.7082	−0.503582
$974$	−37.3050	−1.19533
$975$	0	0
$976$	10.6180	0.339875
$977$	−43.5279	−1.39258	−0.696290	−	0.717761i	$-0.745167\pi$
$977$	−43.5279	−1.39258	−0.696290	+	0.717761i	$0.745167\pi$
$978$	0	0
$979$	2.11146	0.0674824
$980$	−13.3820	−0.427471
$981$	0	0
$982$	−28.4508	−0.907903
$983$	31.7771	1.01353	0.506766	−	0.862084i	$-0.330841\pi$
$983$	31.7771	1.01353	0.506766	+	0.862084i	$0.330841\pi$
$984$	0	0
$985$	−29.0132	−0.924436
$986$	−41.7082	−1.32826
$987$	0	0
$988$	−12.7639	−0.406075
$989$	14.9443	0.475200
$990$	0	0
$991$	33.1033	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$991$	33.1033	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$992$	−7.38197	−0.234378
$993$	0	0
$994$	−9.52786	−0.302205
$995$	−11.7771	−0.373359
$996$	0	0
$997$	−17.7771	−0.563006	−0.281503	−	0.959560i	$-0.590833\pi$
$997$	−17.7771	−0.563006	−0.281503	+	0.959560i	$0.590833\pi$
$998$	−10.2918	−0.325781
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	666.2.a.i.1.1		2
3.2	odd	2		74.2.a.b.1.1	✓	2
4.3	odd	2		5328.2.a.bc.1.1		2
12.11	even	2		592.2.a.g.1.2		2
15.2	even	4		1850.2.b.j.149.4		4
15.8	even	4		1850.2.b.j.149.1		4
15.14	odd	2		1850.2.a.t.1.2		2
21.20	even	2		3626.2.a.s.1.2		2
24.5	odd	2		2368.2.a.y.1.2		2
24.11	even	2		2368.2.a.u.1.1		2
33.32	even	2		8954.2.a.j.1.1		2
111.110	odd	2		2738.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.b.1.1	✓	2	3.2	odd	2
592.2.a.g.1.2		2	12.11	even	2
666.2.a.i.1.1		2	1.1	even	1	trivial
1850.2.a.t.1.2		2	15.14	odd	2
1850.2.b.j.149.1		4	15.8	even	4
1850.2.b.j.149.4		4	15.2	even	4
2368.2.a.u.1.1		2	24.11	even	2
2368.2.a.y.1.2		2	24.5	odd	2
2738.2.a.g.1.1		2	111.110	odd	2
3626.2.a.s.1.2		2	21.20	even	2
5328.2.a.bc.1.1		2	4.3	odd	2
8954.2.a.j.1.1		2	33.32	even	2

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:	\( N \)	\(=\)	\( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	666.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} -3.23607 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.85410 q^{5} -3.23607 q^{7} -1.00000 q^{8} +3.85410 q^{10} +1.38197 q^{11} -2.85410 q^{13} +3.23607 q^{14} +1.00000 q^{16} +4.47214 q^{17} +4.47214 q^{19} -3.85410 q^{20} -1.38197 q^{22} -2.85410 q^{23} +9.85410 q^{25} +2.85410 q^{26} -3.23607 q^{28} +9.32624 q^{29} +7.38197 q^{31} -1.00000 q^{32} -4.47214 q^{34} +12.4721 q^{35} -1.00000 q^{37} -4.47214 q^{38} +3.85410 q^{40} -9.61803 q^{41} -5.23607 q^{43} +1.38197 q^{44} +2.85410 q^{46} +1.23607 q^{47} +3.47214 q^{49} -9.85410 q^{50} -2.85410 q^{52} -0.472136 q^{53} -5.32624 q^{55} +3.23607 q^{56} -9.32624 q^{58} +4.76393 q^{59} +10.6180 q^{61} -7.38197 q^{62} +1.00000 q^{64} +11.0000 q^{65} +1.09017 q^{67} +4.47214 q^{68} -12.4721 q^{70} -2.94427 q^{71} +7.09017 q^{73} +1.00000 q^{74} +4.47214 q^{76} -4.47214 q^{77} -8.56231 q^{79} -3.85410 q^{80} +9.61803 q^{82} +14.4721 q^{83} -17.2361 q^{85} +5.23607 q^{86} -1.38197 q^{88} +1.52786 q^{89} +9.23607 q^{91} -2.85410 q^{92} -1.23607 q^{94} -17.2361 q^{95} -0.472136 q^{97} -3.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + 5 q^{11} + q^{13} + 2 q^{14} + 2 q^{16} - q^{20} - 5 q^{22} + q^{23} + 13 q^{25} - q^{26} - 2 q^{28} + 3 q^{29} + 17 q^{31} - 2 q^{32} + 16 q^{35} - 2 q^{37} + q^{40} - 17 q^{41} - 6 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} - 2 q^{49} - 13 q^{50} + q^{52} + 8 q^{53} + 5 q^{55} + 2 q^{56} - 3 q^{58} + 14 q^{59} + 19 q^{61} - 17 q^{62} + 2 q^{64} + 22 q^{65} - 9 q^{67} - 16 q^{70} + 12 q^{71} + 3 q^{73} + 2 q^{74} + 3 q^{79} - q^{80} + 17 q^{82} + 20 q^{83} - 30 q^{85} + 6 q^{86} - 5 q^{88} + 12 q^{89} + 14 q^{91} + q^{92} + 2 q^{94} - 30 q^{95} + 8 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8 + q^10 + 5 * q^11 + q^13 + 2 * q^14 + 2 * q^16 - q^20 - 5 * q^22 + q^23 + 13 * q^25 - q^26 - 2 * q^28 + 3 * q^29 + 17 * q^31 - 2 * q^32 + 16 * q^35 - 2 * q^37 + q^40 - 17 * q^41 - 6 * q^43 + 5 * q^44 - q^46 - 2 * q^47 - 2 * q^49 - 13 * q^50 + q^52 + 8 * q^53 + 5 * q^55 + 2 * q^56 - 3 * q^58 + 14 * q^59 + 19 * q^61 - 17 * q^62 + 2 * q^64 + 22 * q^65 - 9 * q^67 - 16 * q^70 + 12 * q^71 + 3 * q^73 + 2 * q^74 + 3 * q^79 - q^80 + 17 * q^82 + 20 * q^83 - 30 * q^85 + 6 * q^86 - 5 * q^88 + 12 * q^89 + 14 * q^91 + q^92 + 2 * q^94 - 30 * q^95 + 8 * q^97 + 2 * q^98

Introduction

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