LMFDB - Embedded newform 666.2.a.h.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +2.00000 q^{10} +1.56155 q^{11} +6.68466 q^{13} +1.56155 q^{14} +1.00000 q^{16} -3.56155 q^{17} -3.56155 q^{19} -2.00000 q^{20} -1.56155 q^{22} -8.68466 q^{23} -1.00000 q^{25} -6.68466 q^{26} -1.56155 q^{28} +1.12311 q^{29} -9.12311 q^{31} -1.00000 q^{32} +3.56155 q^{34} +3.12311 q^{35} -1.00000 q^{37} +3.56155 q^{38} +2.00000 q^{40} -11.1231 q^{41} +1.12311 q^{43} +1.56155 q^{44} +8.68466 q^{46} +10.2462 q^{47} -4.56155 q^{49} +1.00000 q^{50} +6.68466 q^{52} +1.56155 q^{53} -3.12311 q^{55} +1.56155 q^{56} -1.12311 q^{58} -0.876894 q^{59} -12.2462 q^{61} +9.12311 q^{62} +1.00000 q^{64} -13.3693 q^{65} +2.24621 q^{67} -3.56155 q^{68} -3.12311 q^{70} -2.24621 q^{71} +3.56155 q^{73} +1.00000 q^{74} -3.56155 q^{76} -2.43845 q^{77} -6.00000 q^{79} -2.00000 q^{80} +11.1231 q^{82} -14.9309 q^{83} +7.12311 q^{85} -1.12311 q^{86} -1.56155 q^{88} +12.9309 q^{89} -10.4384 q^{91} -8.68466 q^{92} -10.2462 q^{94} +7.12311 q^{95} +2.87689 q^{97} +4.56155 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8} + 4 q^{10} - q^{11} + q^{13} - q^{14} + 2 q^{16} - 3 q^{17} - 3 q^{19} - 4 q^{20} + q^{22} - 5 q^{23} - 2 q^{25} - q^{26} + q^{28} - 6 q^{29} - 10 q^{31} - 2 q^{32} + 3 q^{34} - 2 q^{35} - 2 q^{37} + 3 q^{38} + 4 q^{40} - 14 q^{41} - 6 q^{43} - q^{44} + 5 q^{46} + 4 q^{47} - 5 q^{49} + 2 q^{50} + q^{52} - q^{53} + 2 q^{55} - q^{56} + 6 q^{58} - 10 q^{59} - 8 q^{61} + 10 q^{62} + 2 q^{64} - 2 q^{65} - 12 q^{67} - 3 q^{68} + 2 q^{70} + 12 q^{71} + 3 q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} - 12 q^{79} - 4 q^{80} + 14 q^{82} - q^{83} + 6 q^{85} + 6 q^{86} + q^{88} - 3 q^{89} - 25 q^{91} - 5 q^{92} - 4 q^{94} + 6 q^{95} + 14 q^{97} + 5 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + q^7 - 2 * q^8 + 4 * q^10 - q^11 + q^13 - q^14 + 2 * q^16 - 3 * q^17 - 3 * q^19 - 4 * q^20 + q^22 - 5 * q^23 - 2 * q^25 - q^26 + q^28 - 6 * q^29 - 10 * q^31 - 2 * q^32 + 3 * q^34 - 2 * q^35 - 2 * q^37 + 3 * q^38 + 4 * q^40 - 14 * q^41 - 6 * q^43 - q^44 + 5 * q^46 + 4 * q^47 - 5 * q^49 + 2 * q^50 + q^52 - q^53 + 2 * q^55 - q^56 + 6 * q^58 - 10 * q^59 - 8 * q^61 + 10 * q^62 + 2 * q^64 - 2 * q^65 - 12 * q^67 - 3 * q^68 + 2 * q^70 + 12 * q^71 + 3 * q^73 + 2 * q^74 - 3 * q^76 - 9 * q^77 - 12 * q^79 - 4 * q^80 + 14 * q^82 - q^83 + 6 * q^85 + 6 * q^86 + q^88 - 3 * q^89 - 25 * q^91 - 5 * q^92 - 4 * q^94 + 6 * q^95 + 14 * q^97 + 5 * 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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	6.68466	1.85399	0.926995	−	0.375073i	$-0.122382\pi$
$13$	6.68466	1.85399	0.926995	+	0.375073i	$0.122382\pi$
$14$	1.56155	0.417343
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.56155	−0.863803	−0.431902	−	0.901921i	$-0.642157\pi$
$17$	−3.56155	−0.863803	−0.431902	+	0.901921i	$0.642157\pi$
$18$	0	0
$19$	−3.56155	−0.817076	−0.408538	−	0.912741i	$-0.633961\pi$
$19$	−3.56155	−0.817076	−0.408538	+	0.912741i	$0.633961\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.56155	−0.332924
$23$	−8.68466	−1.81088	−0.905438	−	0.424478i	$-0.860458\pi$
$23$	−8.68466	−1.81088	−0.905438	+	0.424478i	$0.860458\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−6.68466	−1.31097
$27$	0	0
$28$	−1.56155	−0.295106
$29$	1.12311	0.208555	0.104278	−	0.994548i	$-0.466747\pi$
$29$	1.12311	0.208555	0.104278	+	0.994548i	$0.466747\pi$
$30$	0	0
$31$	−9.12311	−1.63856	−0.819279	−	0.573395i	$-0.805626\pi$
$31$	−9.12311	−1.63856	−0.819279	+	0.573395i	$0.805626\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.56155	0.610801
$35$	3.12311	0.527901
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	3.56155	0.577760
$39$	0	0
$40$	2.00000	0.316228
$41$	−11.1231	−1.73714	−0.868569	−	0.495569i	$-0.834960\pi$
$41$	−11.1231	−1.73714	−0.868569	+	0.495569i	$0.834960\pi$
$42$	0	0
$43$	1.12311	0.171272	0.0856360	−	0.996326i	$-0.472708\pi$
$43$	1.12311	0.171272	0.0856360	+	0.996326i	$0.472708\pi$
$44$	1.56155	0.235413
$45$	0	0
$46$	8.68466	1.28048
$47$	10.2462	1.49456	0.747282	−	0.664507i	$-0.231359\pi$
$47$	10.2462	1.49456	0.747282	+	0.664507i	$0.231359\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	1.00000	0.141421
$51$	0	0
$52$	6.68466	0.926995
$53$	1.56155	0.214496	0.107248	−	0.994232i	$-0.465796\pi$
$53$	1.56155	0.214496	0.107248	+	0.994232i	$0.465796\pi$
$54$	0	0
$55$	−3.12311	−0.421119
$56$	1.56155	0.208671
$57$	0	0
$58$	−1.12311	−0.147471
$59$	−0.876894	−0.114162	−0.0570810	−	0.998370i	$-0.518179\pi$
$59$	−0.876894	−0.114162	−0.0570810	+	0.998370i	$0.518179\pi$
$60$	0	0
$61$	−12.2462	−1.56797	−0.783983	−	0.620782i	$-0.786815\pi$
$61$	−12.2462	−1.56797	−0.783983	+	0.620782i	$0.786815\pi$
$62$	9.12311	1.15864
$63$	0	0
$64$	1.00000	0.125000
$65$	−13.3693	−1.65826
$66$	0	0
$67$	2.24621	0.274418	0.137209	−	0.990542i	$-0.456187\pi$
$67$	2.24621	0.274418	0.137209	+	0.990542i	$0.456187\pi$
$68$	−3.56155	−0.431902
$69$	0	0
$70$	−3.12311	−0.373283
$71$	−2.24621	−0.266576	−0.133288	−	0.991077i	$-0.542554\pi$
$71$	−2.24621	−0.266576	−0.133288	+	0.991077i	$0.542554\pi$
$72$	0	0
$73$	3.56155	0.416848	0.208424	−	0.978039i	$-0.433167\pi$
$73$	3.56155	0.416848	0.208424	+	0.978039i	$0.433167\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	−3.56155	−0.408538
$77$	−2.43845	−0.277887
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	11.1231	1.22834
$83$	−14.9309	−1.63888	−0.819438	−	0.573168i	$-0.805714\pi$
$83$	−14.9309	−1.63888	−0.819438	+	0.573168i	$0.805714\pi$
$84$	0	0
$85$	7.12311	0.772609
$86$	−1.12311	−0.121108
$87$	0	0
$88$	−1.56155	−0.166462
$89$	12.9309	1.37067	0.685335	−	0.728228i	$-0.259656\pi$
$89$	12.9309	1.37067	0.685335	+	0.728228i	$0.259656\pi$
$90$	0	0
$91$	−10.4384	−1.09425
$92$	−8.68466	−0.905438
$93$	0	0
$94$	−10.2462	−1.05682
$95$	7.12311	0.730815
$96$	0	0
$97$	2.87689	0.292104	0.146052	−	0.989277i	$-0.453343\pi$
$97$	2.87689	0.292104	0.146052	+	0.989277i	$0.453343\pi$
$98$	4.56155	0.460786
$99$	0	0
$100$	−1.00000	−0.100000
$101$	11.1231	1.10679	0.553395	−	0.832919i	$-0.313332\pi$
$101$	11.1231	1.10679	0.553395	+	0.832919i	$0.313332\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	−6.68466	−0.655485
$105$	0	0
$106$	−1.56155	−0.151671
$107$	6.43845	0.622428	0.311214	−	0.950340i	$-0.399264\pi$
$107$	6.43845	0.622428	0.311214	+	0.950340i	$0.399264\pi$
$108$	0	0
$109$	3.56155	0.341135	0.170567	−	0.985346i	$-0.445440\pi$
$109$	3.56155	0.341135	0.170567	+	0.985346i	$0.445440\pi$
$110$	3.12311	0.297776
$111$	0	0
$112$	−1.56155	−0.147553
$113$	12.2462	1.15203	0.576013	−	0.817440i	$-0.304608\pi$
$113$	12.2462	1.15203	0.576013	+	0.817440i	$0.304608\pi$
$114$	0	0
$115$	17.3693	1.61970
$116$	1.12311	0.104278
$117$	0	0
$118$	0.876894	0.0807247
$119$	5.56155	0.509827
$120$	0	0
$121$	−8.56155	−0.778323
$122$	12.2462	1.10872
$123$	0	0
$124$	−9.12311	−0.819279
$125$	12.0000	1.07331
$126$	0	0
$127$	5.56155	0.493508	0.246754	−	0.969078i	$-0.420636\pi$
$127$	5.56155	0.493508	0.246754	+	0.969078i	$0.420636\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	13.3693	1.17257
$131$	−15.1231	−1.32131	−0.660656	−	0.750689i	$-0.729722\pi$
$131$	−15.1231	−1.32131	−0.660656	+	0.750689i	$0.729722\pi$
$132$	0	0
$133$	5.56155	0.482248
$134$	−2.24621	−0.194043
$135$	0	0
$136$	3.56155	0.305401
$137$	5.75379	0.491579	0.245790	−	0.969323i	$-0.420953\pi$
$137$	5.75379	0.491579	0.245790	+	0.969323i	$0.420953\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	3.12311	0.263951
$141$	0	0
$142$	2.24621	0.188498
$143$	10.4384	0.872907
$144$	0	0
$145$	−2.24621	−0.186538
$146$	−3.56155	−0.294756
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−3.12311	−0.255855	−0.127927	−	0.991784i	$-0.540832\pi$
$149$	−3.12311	−0.255855	−0.127927	+	0.991784i	$0.540832\pi$
$150$	0	0
$151$	3.31534	0.269799	0.134899	−	0.990859i	$-0.456929\pi$
$151$	3.31534	0.269799	0.134899	+	0.990859i	$0.456929\pi$
$152$	3.56155	0.288880
$153$	0	0
$154$	2.43845	0.196496
$155$	18.2462	1.46557
$156$	0	0
$157$	−10.4924	−0.837386	−0.418693	−	0.908128i	$-0.637512\pi$
$157$	−10.4924	−0.837386	−0.418693	+	0.908128i	$0.637512\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	2.00000	0.158114
$161$	13.5616	1.06880
$162$	0	0
$163$	−20.9309	−1.63943	−0.819716	−	0.572770i	$-0.805869\pi$
$163$	−20.9309	−1.63943	−0.819716	+	0.572770i	$0.805869\pi$
$164$	−11.1231	−0.868569
$165$	0	0
$166$	14.9309	1.15886
$167$	16.6847	1.29110	0.645549	−	0.763719i	$-0.276629\pi$
$167$	16.6847	1.29110	0.645549	+	0.763719i	$0.276629\pi$
$168$	0	0
$169$	31.6847	2.43728
$170$	−7.12311	−0.546317
$171$	0	0
$172$	1.12311	0.0856360
$173$	−1.56155	−0.118723	−0.0593613	−	0.998237i	$-0.518906\pi$
$173$	−1.56155	−0.118723	−0.0593613	+	0.998237i	$0.518906\pi$
$174$	0	0
$175$	1.56155	0.118042
$176$	1.56155	0.117706
$177$	0	0
$178$	−12.9309	−0.969210
$179$	−7.12311	−0.532406	−0.266203	−	0.963917i	$-0.585769\pi$
$179$	−7.12311	−0.532406	−0.266203	+	0.963917i	$0.585769\pi$
$180$	0	0
$181$	14.4924	1.07721	0.538607	−	0.842557i	$-0.318951\pi$
$181$	14.4924	1.07721	0.538607	+	0.842557i	$0.318951\pi$
$182$	10.4384	0.773749
$183$	0	0
$184$	8.68466	0.640242
$185$	2.00000	0.147043
$186$	0	0
$187$	−5.56155	−0.406701
$188$	10.2462	0.747282
$189$	0	0
$190$	−7.12311	−0.516764
$191$	−2.43845	−0.176440	−0.0882199	−	0.996101i	$-0.528118\pi$
$191$	−2.43845	−0.176440	−0.0882199	+	0.996101i	$0.528118\pi$
$192$	0	0
$193$	0.630683	0.0453976	0.0226988	−	0.999742i	$-0.492774\pi$
$193$	0.630683	0.0453976	0.0226988	+	0.999742i	$0.492774\pi$
$194$	−2.87689	−0.206549
$195$	0	0
$196$	−4.56155	−0.325825
$197$	−11.8078	−0.841268	−0.420634	−	0.907230i	$-0.638192\pi$
$197$	−11.8078	−0.841268	−0.420634	+	0.907230i	$0.638192\pi$
$198$	0	0
$199$	13.1231	0.930272	0.465136	−	0.885239i	$-0.346005\pi$
$199$	13.1231	0.930272	0.465136	+	0.885239i	$0.346005\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−11.1231	−0.782619
$203$	−1.75379	−0.123092
$204$	0	0
$205$	22.2462	1.55374
$206$	10.0000	0.696733
$207$	0	0
$208$	6.68466	0.463498
$209$	−5.56155	−0.384701
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	1.56155	0.107248
$213$	0	0
$214$	−6.43845	−0.440123
$215$	−2.24621	−0.153190
$216$	0	0
$217$	14.2462	0.967096
$218$	−3.56155	−0.241219
$219$	0	0
$220$	−3.12311	−0.210560
$221$	−23.8078	−1.60148
$222$	0	0
$223$	28.4924	1.90799	0.953997	−	0.299817i	$-0.0969255\pi$
$223$	28.4924	1.90799	0.953997	+	0.299817i	$0.0969255\pi$
$224$	1.56155	0.104336
$225$	0	0
$226$	−12.2462	−0.814606
$227$	5.36932	0.356374	0.178187	−	0.983997i	$-0.442977\pi$
$227$	5.36932	0.356374	0.178187	+	0.983997i	$0.442977\pi$
$228$	0	0
$229$	27.3693	1.80862	0.904308	−	0.426881i	$-0.140388\pi$
$229$	27.3693	1.80862	0.904308	+	0.426881i	$0.140388\pi$
$230$	−17.3693	−1.14530
$231$	0	0
$232$	−1.12311	−0.0737355
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	0	0
$235$	−20.4924	−1.33678
$236$	−0.876894	−0.0570810
$237$	0	0
$238$	−5.56155	−0.360502
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−8.24621	−0.531185	−0.265593	−	0.964085i	$-0.585568\pi$
$241$	−8.24621	−0.531185	−0.265593	+	0.964085i	$0.585568\pi$
$242$	8.56155	0.550357
$243$	0	0
$244$	−12.2462	−0.783983
$245$	9.12311	0.582854
$246$	0	0
$247$	−23.8078	−1.51485
$248$	9.12311	0.579318
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−7.12311	−0.449606	−0.224803	−	0.974404i	$-0.572174\pi$
$251$	−7.12311	−0.449606	−0.224803	+	0.974404i	$0.572174\pi$
$252$	0	0
$253$	−13.5616	−0.852608
$254$	−5.56155	−0.348963
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.5616	−1.46973	−0.734865	−	0.678214i	$-0.762754\pi$
$257$	−23.5616	−1.46973	−0.734865	+	0.678214i	$0.762754\pi$
$258$	0	0
$259$	1.56155	0.0970302
$260$	−13.3693	−0.829130
$261$	0	0
$262$	15.1231	0.934309
$263$	−13.7538	−0.848095	−0.424047	−	0.905640i	$-0.639391\pi$
$263$	−13.7538	−0.848095	−0.424047	+	0.905640i	$0.639391\pi$
$264$	0	0
$265$	−3.12311	−0.191851
$266$	−5.56155	−0.341001
$267$	0	0
$268$	2.24621	0.137209
$269$	−18.0540	−1.10077	−0.550385	−	0.834911i	$-0.685519\pi$
$269$	−18.0540	−1.10077	−0.550385	+	0.834911i	$0.685519\pi$
$270$	0	0
$271$	−5.75379	−0.349518	−0.174759	−	0.984611i	$-0.555915\pi$
$271$	−5.75379	−0.349518	−0.174759	+	0.984611i	$0.555915\pi$
$272$	−3.56155	−0.215951
$273$	0	0
$274$	−5.75379	−0.347599
$275$	−1.56155	−0.0941652
$276$	0	0
$277$	27.5616	1.65601	0.828007	−	0.560718i	$-0.189475\pi$
$277$	27.5616	1.65601	0.828007	+	0.560718i	$0.189475\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	−3.12311	−0.186641
$281$	12.9309	0.771391	0.385696	−	0.922626i	$-0.373962\pi$
$281$	12.9309	0.771391	0.385696	+	0.922626i	$0.373962\pi$
$282$	0	0
$283$	−6.19224	−0.368090	−0.184045	−	0.982918i	$-0.558919\pi$
$283$	−6.19224	−0.368090	−0.184045	+	0.982918i	$0.558919\pi$
$284$	−2.24621	−0.133288
$285$	0	0
$286$	−10.4384	−0.617238
$287$	17.3693	1.02528
$288$	0	0
$289$	−4.31534	−0.253844
$290$	2.24621	0.131902
$291$	0	0
$292$	3.56155	0.208424
$293$	28.3002	1.65331	0.826657	−	0.562706i	$-0.190240\pi$
$293$	28.3002	1.65331	0.826657	+	0.562706i	$0.190240\pi$
$294$	0	0
$295$	1.75379	0.102110
$296$	1.00000	0.0581238
$297$	0	0
$298$	3.12311	0.180917
$299$	−58.0540	−3.35735
$300$	0	0
$301$	−1.75379	−0.101087
$302$	−3.31534	−0.190776
$303$	0	0
$304$	−3.56155	−0.204269
$305$	24.4924	1.40243
$306$	0	0
$307$	−17.3693	−0.991319	−0.495660	−	0.868517i	$-0.665074\pi$
$307$	−17.3693	−0.991319	−0.495660	+	0.868517i	$0.665074\pi$
$308$	−2.43845	−0.138943
$309$	0	0
$310$	−18.2462	−1.03632
$311$	−30.2462	−1.71511	−0.857553	−	0.514396i	$-0.828016\pi$
$311$	−30.2462	−1.71511	−0.857553	+	0.514396i	$0.828016\pi$
$312$	0	0
$313$	−2.87689	−0.162612	−0.0813058	−	0.996689i	$-0.525909\pi$
$313$	−2.87689	−0.162612	−0.0813058	+	0.996689i	$0.525909\pi$
$314$	10.4924	0.592122
$315$	0	0
$316$	−6.00000	−0.337526
$317$	7.12311	0.400073	0.200037	−	0.979788i	$-0.435894\pi$
$317$	7.12311	0.400073	0.200037	+	0.979788i	$0.435894\pi$
$318$	0	0
$319$	1.75379	0.0981933
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−13.5616	−0.755756
$323$	12.6847	0.705793
$324$	0	0
$325$	−6.68466	−0.370798
$326$	20.9309	1.15925
$327$	0	0
$328$	11.1231	0.614171
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−23.3693	−1.28449	−0.642247	−	0.766498i	$-0.721998\pi$
$331$	−23.3693	−1.28449	−0.642247	+	0.766498i	$0.721998\pi$
$332$	−14.9309	−0.819438
$333$	0	0
$334$	−16.6847	−0.912944
$335$	−4.49242	−0.245447
$336$	0	0
$337$	−4.93087	−0.268602	−0.134301	−	0.990941i	$-0.542879\pi$
$337$	−4.93087	−0.268602	−0.134301	+	0.990941i	$0.542879\pi$
$338$	−31.6847	−1.72342
$339$	0	0
$340$	7.12311	0.386305
$341$	−14.2462	−0.771476
$342$	0	0
$343$	18.0540	0.974823
$344$	−1.12311	−0.0605538
$345$	0	0
$346$	1.56155	0.0839496
$347$	−11.6155	−0.623554	−0.311777	−	0.950155i	$-0.600924\pi$
$347$	−11.6155	−0.623554	−0.311777	+	0.950155i	$0.600924\pi$
$348$	0	0
$349$	−6.49242	−0.347531	−0.173766	−	0.984787i	$-0.555594\pi$
$349$	−6.49242	−0.347531	−0.173766	+	0.984787i	$0.555594\pi$
$350$	−1.56155	−0.0834685
$351$	0	0
$352$	−1.56155	−0.0832310
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	4.49242	0.238433
$356$	12.9309	0.685335
$357$	0	0
$358$	7.12311	0.376468
$359$	28.4924	1.50377	0.751886	−	0.659293i	$-0.229144\pi$
$359$	28.4924	1.50377	0.751886	+	0.659293i	$0.229144\pi$
$360$	0	0
$361$	−6.31534	−0.332386
$362$	−14.4924	−0.761705
$363$	0	0
$364$	−10.4384	−0.547123
$365$	−7.12311	−0.372840
$366$	0	0
$367$	24.6847	1.28853	0.644264	−	0.764803i	$-0.277164\pi$
$367$	24.6847	1.28853	0.644264	+	0.764803i	$0.277164\pi$
$368$	−8.68466	−0.452719
$369$	0	0
$370$	−2.00000	−0.103975
$371$	−2.43845	−0.126598
$372$	0	0
$373$	27.3693	1.41713	0.708565	−	0.705646i	$-0.249343\pi$
$373$	27.3693	1.41713	0.708565	+	0.705646i	$0.249343\pi$
$374$	5.56155	0.287581
$375$	0	0
$376$	−10.2462	−0.528408
$377$	7.50758	0.386660
$378$	0	0
$379$	14.2462	0.731779	0.365889	−	0.930658i	$-0.380765\pi$
$379$	14.2462	0.731779	0.365889	+	0.930658i	$0.380765\pi$
$380$	7.12311	0.365408
$381$	0	0
$382$	2.43845	0.124762
$383$	−11.8078	−0.603349	−0.301674	−	0.953411i	$-0.597546\pi$
$383$	−11.8078	−0.603349	−0.301674	+	0.953411i	$0.597546\pi$
$384$	0	0
$385$	4.87689	0.248550
$386$	−0.630683	−0.0321009
$387$	0	0
$388$	2.87689	0.146052
$389$	−16.7386	−0.848682	−0.424341	−	0.905502i	$-0.639494\pi$
$389$	−16.7386	−0.848682	−0.424341	+	0.905502i	$0.639494\pi$
$390$	0	0
$391$	30.9309	1.56424
$392$	4.56155	0.230393
$393$	0	0
$394$	11.8078	0.594867
$395$	12.0000	0.603786
$396$	0	0
$397$	−14.4924	−0.727354	−0.363677	−	0.931525i	$-0.618479\pi$
$397$	−14.4924	−0.727354	−0.363677	+	0.931525i	$0.618479\pi$
$398$	−13.1231	−0.657802
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−11.5616	−0.577356	−0.288678	−	0.957426i	$-0.593216\pi$
$401$	−11.5616	−0.577356	−0.288678	+	0.957426i	$0.593216\pi$
$402$	0	0
$403$	−60.9848	−3.03787
$404$	11.1231	0.553395
$405$	0	0
$406$	1.75379	0.0870391
$407$	−1.56155	−0.0774033
$408$	0	0
$409$	20.2462	1.00111	0.500555	−	0.865705i	$-0.333129\pi$
$409$	20.2462	1.00111	0.500555	+	0.865705i	$0.333129\pi$
$410$	−22.2462	−1.09866
$411$	0	0
$412$	−10.0000	−0.492665
$413$	1.36932	0.0673797
$414$	0	0
$415$	29.8617	1.46586
$416$	−6.68466	−0.327742
$417$	0	0
$418$	5.56155	0.272024
$419$	−23.3153	−1.13903	−0.569514	−	0.821981i	$-0.692869\pi$
$419$	−23.3153	−1.13903	−0.569514	+	0.821981i	$0.692869\pi$
$420$	0	0
$421$	20.2462	0.986740	0.493370	−	0.869820i	$-0.335765\pi$
$421$	20.2462	0.986740	0.493370	+	0.869820i	$0.335765\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	−1.56155	−0.0758357
$425$	3.56155	0.172761
$426$	0	0
$427$	19.1231	0.925432
$428$	6.43845	0.311214
$429$	0	0
$430$	2.24621	0.108322
$431$	37.1771	1.79076	0.895378	−	0.445306i	$-0.146905\pi$
$431$	37.1771	1.79076	0.895378	+	0.445306i	$0.146905\pi$
$432$	0	0
$433$	20.4384	0.982209	0.491105	−	0.871101i	$-0.336593\pi$
$433$	20.4384	0.982209	0.491105	+	0.871101i	$0.336593\pi$
$434$	−14.2462	−0.683840
$435$	0	0
$436$	3.56155	0.170567
$437$	30.9309	1.47962
$438$	0	0
$439$	−10.4924	−0.500776	−0.250388	−	0.968146i	$-0.580558\pi$
$439$	−10.4924	−0.500776	−0.250388	+	0.968146i	$0.580558\pi$
$440$	3.12311	0.148888
$441$	0	0
$442$	23.8078	1.13242
$443$	18.7386	0.890299	0.445150	−	0.895456i	$-0.353150\pi$
$443$	18.7386	0.890299	0.445150	+	0.895456i	$0.353150\pi$
$444$	0	0
$445$	−25.8617	−1.22596
$446$	−28.4924	−1.34916
$447$	0	0
$448$	−1.56155	−0.0737764
$449$	−32.2462	−1.52179	−0.760896	−	0.648873i	$-0.775240\pi$
$449$	−32.2462	−1.52179	−0.760896	+	0.648873i	$0.775240\pi$
$450$	0	0
$451$	−17.3693	−0.817889
$452$	12.2462	0.576013
$453$	0	0
$454$	−5.36932	−0.251995
$455$	20.8769	0.978724
$456$	0	0
$457$	5.61553	0.262683	0.131342	−	0.991337i	$-0.458072\pi$
$457$	5.61553	0.262683	0.131342	+	0.991337i	$0.458072\pi$
$458$	−27.3693	−1.27888
$459$	0	0
$460$	17.3693	0.809849
$461$	26.9848	1.25681	0.628405	−	0.777887i	$-0.283708\pi$
$461$	26.9848	1.25681	0.628405	+	0.777887i	$0.283708\pi$
$462$	0	0
$463$	−11.3693	−0.528377	−0.264188	−	0.964471i	$-0.585104\pi$
$463$	−11.3693	−0.528377	−0.264188	+	0.964471i	$0.585104\pi$
$464$	1.12311	0.0521389
$465$	0	0
$466$	−16.0000	−0.741186
$467$	19.6155	0.907698	0.453849	−	0.891079i	$-0.350050\pi$
$467$	19.6155	0.907698	0.453849	+	0.891079i	$0.350050\pi$
$468$	0	0
$469$	−3.50758	−0.161965
$470$	20.4924	0.945245
$471$	0	0
$472$	0.876894	0.0403623
$473$	1.75379	0.0806393
$474$	0	0
$475$	3.56155	0.163415
$476$	5.56155	0.254913
$477$	0	0
$478$	8.00000	0.365911
$479$	−42.0540	−1.92150	−0.960748	−	0.277424i	$-0.910519\pi$
$479$	−42.0540	−1.92150	−0.960748	+	0.277424i	$0.910519\pi$
$480$	0	0
$481$	−6.68466	−0.304794
$482$	8.24621	0.375605
$483$	0	0
$484$	−8.56155	−0.389161
$485$	−5.75379	−0.261266
$486$	0	0
$487$	13.1231	0.594665	0.297332	−	0.954774i	$-0.403903\pi$
$487$	13.1231	0.594665	0.297332	+	0.954774i	$0.403903\pi$
$488$	12.2462	0.554360
$489$	0	0
$490$	−9.12311	−0.412140
$491$	1.06913	0.0482492	0.0241246	−	0.999709i	$-0.492320\pi$
$491$	1.06913	0.0482492	0.0241246	+	0.999709i	$0.492320\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	23.8078	1.07116
$495$	0	0
$496$	−9.12311	−0.409640
$497$	3.50758	0.157336
$498$	0	0
$499$	−33.4233	−1.49623	−0.748116	−	0.663568i	$-0.769041\pi$
$499$	−33.4233	−1.49623	−0.748116	+	0.663568i	$0.769041\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	7.12311	0.317920
$503$	−32.9848	−1.47072	−0.735361	−	0.677676i	$-0.762987\pi$
$503$	−32.9848	−1.47072	−0.735361	+	0.677676i	$0.762987\pi$
$504$	0	0
$505$	−22.2462	−0.989943
$506$	13.5616	0.602885
$507$	0	0
$508$	5.56155	0.246754
$509$	−1.94602	−0.0862560	−0.0431280	−	0.999070i	$-0.513732\pi$
$509$	−1.94602	−0.0862560	−0.0431280	+	0.999070i	$0.513732\pi$
$510$	0	0
$511$	−5.56155	−0.246029
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	23.5616	1.03926
$515$	20.0000	0.881305
$516$	0	0
$517$	16.0000	0.703679
$518$	−1.56155	−0.0686107
$519$	0	0
$520$	13.3693	0.586283
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	−29.1231	−1.27346	−0.636732	−	0.771085i	$-0.719714\pi$
$523$	−29.1231	−1.27346	−0.636732	+	0.771085i	$0.719714\pi$
$524$	−15.1231	−0.660656
$525$	0	0
$526$	13.7538	0.599694
$527$	32.4924	1.41539
$528$	0	0
$529$	52.4233	2.27927
$530$	3.12311	0.135659
$531$	0	0
$532$	5.56155	0.241124
$533$	−74.3542	−3.22064
$534$	0	0
$535$	−12.8769	−0.556717
$536$	−2.24621	−0.0970215
$537$	0	0
$538$	18.0540	0.778362
$539$	−7.12311	−0.306814
$540$	0	0
$541$	−3.06913	−0.131952	−0.0659761	−	0.997821i	$-0.521016\pi$
$541$	−3.06913	−0.131952	−0.0659761	+	0.997821i	$0.521016\pi$
$542$	5.75379	0.247146
$543$	0	0
$544$	3.56155	0.152700
$545$	−7.12311	−0.305120
$546$	0	0
$547$	−2.19224	−0.0937332	−0.0468666	−	0.998901i	$-0.514924\pi$
$547$	−2.19224	−0.0937332	−0.0468666	+	0.998901i	$0.514924\pi$
$548$	5.75379	0.245790
$549$	0	0
$550$	1.56155	0.0665848
$551$	−4.00000	−0.170406
$552$	0	0
$553$	9.36932	0.398424
$554$	−27.5616	−1.17098
$555$	0	0
$556$	−12.0000	−0.508913
$557$	34.0000	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$557$	34.0000	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$558$	0	0
$559$	7.50758	0.317537
$560$	3.12311	0.131975
$561$	0	0
$562$	−12.9309	−0.545456
$563$	8.87689	0.374116	0.187058	−	0.982349i	$-0.440105\pi$
$563$	8.87689	0.374116	0.187058	+	0.982349i	$0.440105\pi$
$564$	0	0
$565$	−24.4924	−1.03040
$566$	6.19224	0.260279
$567$	0	0
$568$	2.24621	0.0942489
$569$	38.3002	1.60563	0.802814	−	0.596230i	$-0.203335\pi$
$569$	38.3002	1.60563	0.802814	+	0.596230i	$0.203335\pi$
$570$	0	0
$571$	10.7386	0.449398	0.224699	−	0.974428i	$-0.427860\pi$
$571$	10.7386	0.449398	0.224699	+	0.974428i	$0.427860\pi$
$572$	10.4384	0.436453
$573$	0	0
$574$	−17.3693	−0.724981
$575$	8.68466	0.362175
$576$	0	0
$577$	−9.12311	−0.379800	−0.189900	−	0.981803i	$-0.560816\pi$
$577$	−9.12311	−0.379800	−0.189900	+	0.981803i	$0.560816\pi$
$578$	4.31534	0.179495
$579$	0	0
$580$	−2.24621	−0.0932688
$581$	23.3153	0.967283
$582$	0	0
$583$	2.43845	0.100990
$584$	−3.56155	−0.147378
$585$	0	0
$586$	−28.3002	−1.16907
$587$	13.7538	0.567680	0.283840	−	0.958872i	$-0.408392\pi$
$587$	13.7538	0.567680	0.283840	+	0.958872i	$0.408392\pi$
$588$	0	0
$589$	32.4924	1.33883
$590$	−1.75379	−0.0722023
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−33.3693	−1.37031	−0.685157	−	0.728396i	$-0.740266\pi$
$593$	−33.3693	−1.37031	−0.685157	+	0.728396i	$0.740266\pi$
$594$	0	0
$595$	−11.1231	−0.456003
$596$	−3.12311	−0.127927
$597$	0	0
$598$	58.0540	2.37400
$599$	34.2462	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$599$	34.2462	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$600$	0	0
$601$	−9.31534	−0.379981	−0.189990	−	0.981786i	$-0.560846\pi$
$601$	−9.31534	−0.379981	−0.189990	+	0.981786i	$0.560846\pi$
$602$	1.75379	0.0714791
$603$	0	0
$604$	3.31534	0.134899
$605$	17.1231	0.696153
$606$	0	0
$607$	35.8617	1.45558	0.727792	−	0.685798i	$-0.240547\pi$
$607$	35.8617	1.45558	0.727792	+	0.685798i	$0.240547\pi$
$608$	3.56155	0.144440
$609$	0	0
$610$	−24.4924	−0.991669
$611$	68.4924	2.77091
$612$	0	0
$613$	−30.9848	−1.25147	−0.625733	−	0.780037i	$-0.715200\pi$
$613$	−30.9848	−1.25147	−0.625733	+	0.780037i	$0.715200\pi$
$614$	17.3693	0.700969
$615$	0	0
$616$	2.43845	0.0982478
$617$	23.1231	0.930901	0.465451	−	0.885074i	$-0.345892\pi$
$617$	23.1231	0.930901	0.465451	+	0.885074i	$0.345892\pi$
$618$	0	0
$619$	18.7386	0.753169	0.376585	−	0.926382i	$-0.377098\pi$
$619$	18.7386	0.753169	0.376585	+	0.926382i	$0.377098\pi$
$620$	18.2462	0.732785
$621$	0	0
$622$	30.2462	1.21276
$623$	−20.1922	−0.808985
$624$	0	0
$625$	−19.0000	−0.760000
$626$	2.87689	0.114984
$627$	0	0
$628$	−10.4924	−0.418693
$629$	3.56155	0.142008
$630$	0	0
$631$	−12.7386	−0.507117	−0.253559	−	0.967320i	$-0.581601\pi$
$631$	−12.7386	−0.507117	−0.253559	+	0.967320i	$0.581601\pi$
$632$	6.00000	0.238667
$633$	0	0
$634$	−7.12311	−0.282895
$635$	−11.1231	−0.441407
$636$	0	0
$637$	−30.4924	−1.20815
$638$	−1.75379	−0.0694332
$639$	0	0
$640$	2.00000	0.0790569
$641$	21.8617	0.863487	0.431743	−	0.901996i	$-0.357899\pi$
$641$	21.8617	0.863487	0.431743	+	0.901996i	$0.357899\pi$
$642$	0	0
$643$	2.19224	0.0864533	0.0432267	−	0.999065i	$-0.486236\pi$
$643$	2.19224	0.0864533	0.0432267	+	0.999065i	$0.486236\pi$
$644$	13.5616	0.534400
$645$	0	0
$646$	−12.6847	−0.499071
$647$	−14.9309	−0.586993	−0.293497	−	0.955960i	$-0.594819\pi$
$647$	−14.9309	−0.586993	−0.293497	+	0.955960i	$0.594819\pi$
$648$	0	0
$649$	−1.36932	−0.0537504
$650$	6.68466	0.262194
$651$	0	0
$652$	−20.9309	−0.819716
$653$	−16.7386	−0.655033	−0.327517	−	0.944845i	$-0.606212\pi$
$653$	−16.7386	−0.655033	−0.327517	+	0.944845i	$0.606212\pi$
$654$	0	0
$655$	30.2462	1.18182
$656$	−11.1231	−0.434284
$657$	0	0
$658$	16.0000	0.623745
$659$	18.2462	0.710771	0.355386	−	0.934720i	$-0.384350\pi$
$659$	18.2462	0.710771	0.355386	+	0.934720i	$0.384350\pi$
$660$	0	0
$661$	−19.5616	−0.760856	−0.380428	−	0.924810i	$-0.624223\pi$
$661$	−19.5616	−0.760856	−0.380428	+	0.924810i	$0.624223\pi$
$662$	23.3693	0.908274
$663$	0	0
$664$	14.9309	0.579430
$665$	−11.1231	−0.431336
$666$	0	0
$667$	−9.75379	−0.377668
$668$	16.6847	0.645549
$669$	0	0
$670$	4.49242	0.173557
$671$	−19.1231	−0.738239
$672$	0	0
$673$	0.0539753	0.00208060	0.00104030	−	0.999999i	$-0.499669\pi$
$673$	0.0539753	0.00208060	0.00104030	+	0.999999i	$0.499669\pi$
$674$	4.93087	0.189930
$675$	0	0
$676$	31.6847	1.21864
$677$	22.4384	0.862380	0.431190	−	0.902261i	$-0.358094\pi$
$677$	22.4384	0.862380	0.431190	+	0.902261i	$0.358094\pi$
$678$	0	0
$679$	−4.49242	−0.172403
$680$	−7.12311	−0.273159
$681$	0	0
$682$	14.2462	0.545516
$683$	−24.1080	−0.922465	−0.461233	−	0.887279i	$-0.652593\pi$
$683$	−24.1080	−0.922465	−0.461233	+	0.887279i	$0.652593\pi$
$684$	0	0
$685$	−11.5076	−0.439682
$686$	−18.0540	−0.689304
$687$	0	0
$688$	1.12311	0.0428180
$689$	10.4384	0.397673
$690$	0	0
$691$	−31.1231	−1.18398	−0.591989	−	0.805946i	$-0.701657\pi$
$691$	−31.1231	−1.18398	−0.591989	+	0.805946i	$0.701657\pi$
$692$	−1.56155	−0.0593613
$693$	0	0
$694$	11.6155	0.440919
$695$	24.0000	0.910372
$696$	0	0
$697$	39.6155	1.50055
$698$	6.49242	0.245742
$699$	0	0
$700$	1.56155	0.0590211
$701$	−20.6307	−0.779210	−0.389605	−	0.920982i	$-0.627388\pi$
$701$	−20.6307	−0.779210	−0.389605	+	0.920982i	$0.627388\pi$
$702$	0	0
$703$	3.56155	0.134327
$704$	1.56155	0.0588532
$705$	0	0
$706$	26.0000	0.978523
$707$	−17.3693	−0.653240
$708$	0	0
$709$	−42.3002	−1.58862	−0.794308	−	0.607515i	$-0.792167\pi$
$709$	−42.3002	−1.58862	−0.794308	+	0.607515i	$0.792167\pi$
$710$	−4.49242	−0.168598
$711$	0	0
$712$	−12.9309	−0.484605
$713$	79.2311	2.96723
$714$	0	0
$715$	−20.8769	−0.780752
$716$	−7.12311	−0.266203
$717$	0	0
$718$	−28.4924	−1.06333
$719$	−22.6307	−0.843982	−0.421991	−	0.906600i	$-0.638669\pi$
$719$	−22.6307	−0.843982	−0.421991	+	0.906600i	$0.638669\pi$
$720$	0	0
$721$	15.6155	0.581553
$722$	6.31534	0.235033
$723$	0	0
$724$	14.4924	0.538607
$725$	−1.12311	−0.0417111
$726$	0	0
$727$	−31.3693	−1.16342	−0.581712	−	0.813395i	$-0.697617\pi$
$727$	−31.3693	−1.16342	−0.581712	+	0.813395i	$0.697617\pi$
$728$	10.4384	0.386875
$729$	0	0
$730$	7.12311	0.263638
$731$	−4.00000	−0.147945
$732$	0	0
$733$	−47.8617	−1.76781	−0.883907	−	0.467663i	$-0.845096\pi$
$733$	−47.8617	−1.76781	−0.883907	+	0.467663i	$0.845096\pi$
$734$	−24.6847	−0.911127
$735$	0	0
$736$	8.68466	0.320121
$737$	3.50758	0.129203
$738$	0	0
$739$	47.6155	1.75157	0.875783	−	0.482705i	$-0.160346\pi$
$739$	47.6155	1.75157	0.875783	+	0.482705i	$0.160346\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	2.43845	0.0895182
$743$	12.9848	0.476368	0.238184	−	0.971220i	$-0.423448\pi$
$743$	12.9848	0.476368	0.238184	+	0.971220i	$0.423448\pi$
$744$	0	0
$745$	6.24621	0.228843
$746$	−27.3693	−1.00206
$747$	0	0
$748$	−5.56155	−0.203351
$749$	−10.0540	−0.367364
$750$	0	0
$751$	−39.2311	−1.43156	−0.715781	−	0.698325i	$-0.753929\pi$
$751$	−39.2311	−1.43156	−0.715781	+	0.698325i	$0.753929\pi$
$752$	10.2462	0.373641
$753$	0	0
$754$	−7.50758	−0.273410
$755$	−6.63068	−0.241315
$756$	0	0
$757$	−10.6847	−0.388341	−0.194170	−	0.980968i	$-0.562201\pi$
$757$	−10.6847	−0.388341	−0.194170	+	0.980968i	$0.562201\pi$
$758$	−14.2462	−0.517446
$759$	0	0
$760$	−7.12311	−0.258382
$761$	−40.1080	−1.45391	−0.726956	−	0.686684i	$-0.759066\pi$
$761$	−40.1080	−1.45391	−0.726956	+	0.686684i	$0.759066\pi$
$762$	0	0
$763$	−5.56155	−0.201342
$764$	−2.43845	−0.0882199
$765$	0	0
$766$	11.8078	0.426632
$767$	−5.86174	−0.211655
$768$	0	0
$769$	−5.61553	−0.202501	−0.101251	−	0.994861i	$-0.532284\pi$
$769$	−5.61553	−0.202501	−0.101251	+	0.994861i	$0.532284\pi$
$770$	−4.87689	−0.175751
$771$	0	0
$772$	0.630683	0.0226988
$773$	16.6847	0.600105	0.300053	−	0.953923i	$-0.402996\pi$
$773$	16.6847	0.600105	0.300053	+	0.953923i	$0.402996\pi$
$774$	0	0
$775$	9.12311	0.327712
$776$	−2.87689	−0.103274
$777$	0	0
$778$	16.7386	0.600109
$779$	39.6155	1.41937
$780$	0	0
$781$	−3.50758	−0.125511
$782$	−30.9309	−1.10609
$783$	0	0
$784$	−4.56155	−0.162913
$785$	20.9848	0.748981
$786$	0	0
$787$	35.2311	1.25585	0.627926	−	0.778273i	$-0.283904\pi$
$787$	35.2311	1.25585	0.627926	+	0.778273i	$0.283904\pi$
$788$	−11.8078	−0.420634
$789$	0	0
$790$	−12.0000	−0.426941
$791$	−19.1231	−0.679939
$792$	0	0
$793$	−81.8617	−2.90700
$794$	14.4924	0.514317
$795$	0	0
$796$	13.1231	0.465136
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	−36.4924	−1.29101
$800$	1.00000	0.0353553
$801$	0	0
$802$	11.5616	0.408253
$803$	5.56155	0.196263
$804$	0	0
$805$	−27.1231	−0.955964
$806$	60.9848	2.14810
$807$	0	0
$808$	−11.1231	−0.391309
$809$	37.4233	1.31573	0.657866	−	0.753135i	$-0.271459\pi$
$809$	37.4233	1.31573	0.657866	+	0.753135i	$0.271459\pi$
$810$	0	0
$811$	17.3693	0.609919	0.304960	−	0.952365i	$-0.401357\pi$
$811$	17.3693	0.609919	0.304960	+	0.952365i	$0.401357\pi$
$812$	−1.75379	−0.0615459
$813$	0	0
$814$	1.56155	0.0547324
$815$	41.8617	1.46635
$816$	0	0
$817$	−4.00000	−0.139942
$818$	−20.2462	−0.707892
$819$	0	0
$820$	22.2462	0.776871
$821$	35.3153	1.23251	0.616257	−	0.787545i	$-0.288648\pi$
$821$	35.3153	1.23251	0.616257	+	0.787545i	$0.288648\pi$
$822$	0	0
$823$	19.8078	0.690455	0.345228	−	0.938519i	$-0.387802\pi$
$823$	19.8078	0.690455	0.345228	+	0.938519i	$0.387802\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	−1.36932	−0.0476446
$827$	50.2462	1.74723	0.873616	−	0.486616i	$-0.161769\pi$
$827$	50.2462	1.74723	0.873616	+	0.486616i	$0.161769\pi$
$828$	0	0
$829$	3.56155	0.123698	0.0618489	−	0.998086i	$-0.480300\pi$
$829$	3.56155	0.123698	0.0618489	+	0.998086i	$0.480300\pi$
$830$	−29.8617	−1.03652
$831$	0	0
$832$	6.68466	0.231749
$833$	16.2462	0.562898
$834$	0	0
$835$	−33.3693	−1.15479
$836$	−5.56155	−0.192350
$837$	0	0
$838$	23.3153	0.805415
$839$	−31.1231	−1.07449	−0.537244	−	0.843427i	$-0.680535\pi$
$839$	−31.1231	−1.07449	−0.537244	+	0.843427i	$0.680535\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	−20.2462	−0.697731
$843$	0	0
$844$	−4.00000	−0.137686
$845$	−63.3693	−2.17997
$846$	0	0
$847$	13.3693	0.459375
$848$	1.56155	0.0536239
$849$	0	0
$850$	−3.56155	−0.122160
$851$	8.68466	0.297706
$852$	0	0
$853$	36.4384	1.24763	0.623814	−	0.781573i	$-0.285582\pi$
$853$	36.4384	1.24763	0.623814	+	0.781573i	$0.285582\pi$
$854$	−19.1231	−0.654379
$855$	0	0
$856$	−6.43845	−0.220062
$857$	−43.6695	−1.49172	−0.745861	−	0.666102i	$-0.767962\pi$
$857$	−43.6695	−1.49172	−0.745861	+	0.666102i	$0.767962\pi$
$858$	0	0
$859$	0.0539753	0.00184161	0.000920807	−	1.00000i	$-0.499707\pi$
$859$	0.0539753	0.00184161	0.000920807		1.00000i	$0.499707\pi$
$860$	−2.24621	−0.0765952
$861$	0	0
$862$	−37.1771	−1.26626
$863$	−57.3693	−1.95287	−0.976437	−	0.215802i	$-0.930763\pi$
$863$	−57.3693	−1.95287	−0.976437	+	0.215802i	$0.930763\pi$
$864$	0	0
$865$	3.12311	0.106189
$866$	−20.4384	−0.694527
$867$	0	0
$868$	14.2462	0.483548
$869$	−9.36932	−0.317832
$870$	0	0
$871$	15.0152	0.508769
$872$	−3.56155	−0.120609
$873$	0	0
$874$	−30.9309	−1.04625
$875$	−18.7386	−0.633481
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	10.4924	0.354102
$879$	0	0
$880$	−3.12311	−0.105280
$881$	−16.4924	−0.555644	−0.277822	−	0.960633i	$-0.589613\pi$
$881$	−16.4924	−0.555644	−0.277822	+	0.960633i	$0.589613\pi$
$882$	0	0
$883$	−28.4384	−0.957030	−0.478515	−	0.878079i	$-0.658825\pi$
$883$	−28.4384	−0.957030	−0.478515	+	0.878079i	$0.658825\pi$
$884$	−23.8078	−0.800742
$885$	0	0
$886$	−18.7386	−0.629537
$887$	−23.1231	−0.776398	−0.388199	−	0.921576i	$-0.626903\pi$
$887$	−23.1231	−0.776398	−0.388199	+	0.921576i	$0.626903\pi$
$888$	0	0
$889$	−8.68466	−0.291274
$890$	25.8617	0.866887
$891$	0	0
$892$	28.4924	0.953997
$893$	−36.4924	−1.22117
$894$	0	0
$895$	14.2462	0.476198
$896$	1.56155	0.0521678
$897$	0	0
$898$	32.2462	1.07607
$899$	−10.2462	−0.341730
$900$	0	0
$901$	−5.56155	−0.185282
$902$	17.3693	0.578335
$903$	0	0
$904$	−12.2462	−0.407303
$905$	−28.9848	−0.963489
$906$	0	0
$907$	38.3002	1.27174	0.635868	−	0.771797i	$-0.280642\pi$
$907$	38.3002	1.27174	0.635868	+	0.771797i	$0.280642\pi$
$908$	5.36932	0.178187
$909$	0	0
$910$	−20.8769	−0.692062
$911$	−36.4924	−1.20905	−0.604524	−	0.796587i	$-0.706637\pi$
$911$	−36.4924	−1.20905	−0.604524	+	0.796587i	$0.706637\pi$
$912$	0	0
$913$	−23.3153	−0.771625
$914$	−5.61553	−0.185745
$915$	0	0
$916$	27.3693	0.904308
$917$	23.6155	0.779853
$918$	0	0
$919$	44.2462	1.45955	0.729774	−	0.683689i	$-0.239625\pi$
$919$	44.2462	1.45955	0.729774	+	0.683689i	$0.239625\pi$
$920$	−17.3693	−0.572649
$921$	0	0
$922$	−26.9848	−0.888699
$923$	−15.0152	−0.494230
$924$	0	0
$925$	1.00000	0.0328798
$926$	11.3693	0.373619
$927$	0	0
$928$	−1.12311	−0.0368677
$929$	−46.2462	−1.51729	−0.758644	−	0.651505i	$-0.774138\pi$
$929$	−46.2462	−1.51729	−0.758644	+	0.651505i	$0.774138\pi$
$930$	0	0
$931$	16.2462	0.532448
$932$	16.0000	0.524097
$933$	0	0
$934$	−19.6155	−0.641840
$935$	11.1231	0.363764
$936$	0	0
$937$	−32.2462	−1.05344	−0.526719	−	0.850040i	$-0.676578\pi$
$937$	−32.2462	−1.05344	−0.526719	+	0.850040i	$0.676578\pi$
$938$	3.50758	0.114526
$939$	0	0
$940$	−20.4924	−0.668389
$941$	49.8617	1.62545	0.812723	−	0.582650i	$-0.197984\pi$
$941$	49.8617	1.62545	0.812723	+	0.582650i	$0.197984\pi$
$942$	0	0
$943$	96.6004	3.14574
$944$	−0.876894	−0.0285405
$945$	0	0
$946$	−1.75379	−0.0570206
$947$	5.36932	0.174479	0.0872397	−	0.996187i	$-0.472195\pi$
$947$	5.36932	0.174479	0.0872397	+	0.996187i	$0.472195\pi$
$948$	0	0
$949$	23.8078	0.772833
$950$	−3.56155	−0.115552
$951$	0	0
$952$	−5.56155	−0.180251
$953$	−49.8617	−1.61518	−0.807590	−	0.589744i	$-0.799229\pi$
$953$	−49.8617	−1.61518	−0.807590	+	0.589744i	$0.799229\pi$
$954$	0	0
$955$	4.87689	0.157813
$956$	−8.00000	−0.258738
$957$	0	0
$958$	42.0540	1.35870
$959$	−8.98485	−0.290136
$960$	0	0
$961$	52.2311	1.68487
$962$	6.68466	0.215522
$963$	0	0
$964$	−8.24621	−0.265593
$965$	−1.26137	−0.0406048
$966$	0	0
$967$	−3.36932	−0.108350	−0.0541750	−	0.998531i	$-0.517253\pi$
$967$	−3.36932	−0.108350	−0.0541750	+	0.998531i	$0.517253\pi$
$968$	8.56155	0.275179
$969$	0	0
$970$	5.75379	0.184743
$971$	−9.75379	−0.313014	−0.156507	−	0.987677i	$-0.550023\pi$
$971$	−9.75379	−0.313014	−0.156507	+	0.987677i	$0.550023\pi$
$972$	0	0
$973$	18.7386	0.600733
$974$	−13.1231	−0.420491
$975$	0	0
$976$	−12.2462	−0.391992
$977$	33.9157	1.08506	0.542530	−	0.840036i	$-0.317467\pi$
$977$	33.9157	1.08506	0.542530	+	0.840036i	$0.317467\pi$
$978$	0	0
$979$	20.1922	0.645347
$980$	9.12311	0.291427
$981$	0	0
$982$	−1.06913	−0.0341173
$983$	19.6155	0.625638	0.312819	−	0.949813i	$-0.398727\pi$
$983$	19.6155	0.625638	0.312819	+	0.949813i	$0.398727\pi$
$984$	0	0
$985$	23.6155	0.752453
$986$	4.00000	0.127386
$987$	0	0
$988$	−23.8078	−0.757426
$989$	−9.75379	−0.310152
$990$	0	0
$991$	60.7386	1.92943	0.964713	−	0.263303i	$-0.0848119\pi$
$991$	60.7386	1.92943	0.964713	+	0.263303i	$0.0848119\pi$
$992$	9.12311	0.289659
$993$	0	0
$994$	−3.50758	−0.111254
$995$	−26.2462	−0.832061
$996$	0	0
$997$	−36.9309	−1.16961	−0.584806	−	0.811173i	$-0.698829\pi$
$997$	−36.9309	−1.16961	−0.584806	+	0.811173i	$0.698829\pi$
$998$	33.4233	1.05800
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	666.2.a.h.1.1	✓	2
3.2	odd	2		666.2.a.k.1.1	yes	2
4.3	odd	2		5328.2.a.y.1.2		2
12.11	even	2		5328.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
666.2.a.h.1.1	✓	2	1.1	even	1	trivial
666.2.a.k.1.1	yes	2	3.2	odd	2
5328.2.a.y.1.2		2	4.3	odd	2
5328.2.a.bh.1.2		2	12.11	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +2.00000 q^{10} +1.56155 q^{11} +6.68466 q^{13} +1.56155 q^{14} +1.00000 q^{16} -3.56155 q^{17} -3.56155 q^{19} -2.00000 q^{20} -1.56155 q^{22} -8.68466 q^{23} -1.00000 q^{25} -6.68466 q^{26} -1.56155 q^{28} +1.12311 q^{29} -9.12311 q^{31} -1.00000 q^{32} +3.56155 q^{34} +3.12311 q^{35} -1.00000 q^{37} +3.56155 q^{38} +2.00000 q^{40} -11.1231 q^{41} +1.12311 q^{43} +1.56155 q^{44} +8.68466 q^{46} +10.2462 q^{47} -4.56155 q^{49} +1.00000 q^{50} +6.68466 q^{52} +1.56155 q^{53} -3.12311 q^{55} +1.56155 q^{56} -1.12311 q^{58} -0.876894 q^{59} -12.2462 q^{61} +9.12311 q^{62} +1.00000 q^{64} -13.3693 q^{65} +2.24621 q^{67} -3.56155 q^{68} -3.12311 q^{70} -2.24621 q^{71} +3.56155 q^{73} +1.00000 q^{74} -3.56155 q^{76} -2.43845 q^{77} -6.00000 q^{79} -2.00000 q^{80} +11.1231 q^{82} -14.9309 q^{83} +7.12311 q^{85} -1.12311 q^{86} -1.56155 q^{88} +12.9309 q^{89} -10.4384 q^{91} -8.68466 q^{92} -10.2462 q^{94} +7.12311 q^{95} +2.87689 q^{97} +4.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 2 q^{8} + 4 q^{10} - q^{11} + q^{13} - q^{14} + 2 q^{16} - 3 q^{17} - 3 q^{19} - 4 q^{20} + q^{22} - 5 q^{23} - 2 q^{25} - q^{26} + q^{28} - 6 q^{29} - 10 q^{31} - 2 q^{32} + 3 q^{34} - 2 q^{35} - 2 q^{37} + 3 q^{38} + 4 q^{40} - 14 q^{41} - 6 q^{43} - q^{44} + 5 q^{46} + 4 q^{47} - 5 q^{49} + 2 q^{50} + q^{52} - q^{53} + 2 q^{55} - q^{56} + 6 q^{58} - 10 q^{59} - 8 q^{61} + 10 q^{62} + 2 q^{64} - 2 q^{65} - 12 q^{67} - 3 q^{68} + 2 q^{70} + 12 q^{71} + 3 q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} - 12 q^{79} - 4 q^{80} + 14 q^{82} - q^{83} + 6 q^{85} + 6 q^{86} + q^{88} - 3 q^{89} - 25 q^{91} - 5 q^{92} - 4 q^{94} + 6 q^{95} + 14 q^{97} + 5 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + q^7 - 2 * q^8 + 4 * q^10 - q^11 + q^13 - q^14 + 2 * q^16 - 3 * q^17 - 3 * q^19 - 4 * q^20 + q^22 - 5 * q^23 - 2 * q^25 - q^26 + q^28 - 6 * q^29 - 10 * q^31 - 2 * q^32 + 3 * q^34 - 2 * q^35 - 2 * q^37 + 3 * q^38 + 4 * q^40 - 14 * q^41 - 6 * q^43 - q^44 + 5 * q^46 + 4 * q^47 - 5 * q^49 + 2 * q^50 + q^52 - q^53 + 2 * q^55 - q^56 + 6 * q^58 - 10 * q^59 - 8 * q^61 + 10 * q^62 + 2 * q^64 - 2 * q^65 - 12 * q^67 - 3 * q^68 + 2 * q^70 + 12 * q^71 + 3 * q^73 + 2 * q^74 - 3 * q^76 - 9 * q^77 - 12 * q^79 - 4 * q^80 + 14 * q^82 - q^83 + 6 * q^85 + 6 * q^86 + q^88 - 3 * q^89 - 25 * q^91 - 5 * q^92 - 4 * q^94 + 6 * q^95 + 14 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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