LMFDB - Embedded newform 1520.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68133 q^{3} +1.00000 q^{5} +3.18953 q^{7} +4.18953 q^{9} +O(q^{10})$	$q+2.68133 q^{3} +1.00000 q^{5} +3.18953 q^{7} +4.18953 q^{9} -0.681331 q^{13} +2.68133 q^{15} -1.18953 q^{17} -1.00000 q^{19} +8.55220 q^{21} +2.17313 q^{23} +1.00000 q^{25} +3.18953 q^{27} +2.81047 q^{29} -6.37907 q^{31} +3.18953 q^{35} +7.87086 q^{37} -1.82687 q^{39} +0.983593 q^{41} -1.36266 q^{43} +4.18953 q^{45} -11.7417 q^{47} +3.17313 q^{49} -3.18953 q^{51} -1.69774 q^{53} -2.68133 q^{57} -11.5358 q^{59} +7.36266 q^{61} +13.3627 q^{63} -0.681331 q^{65} -7.02759 q^{67} +5.82687 q^{69} +12.7581 q^{71} -5.53579 q^{73} +2.68133 q^{75} -5.36266 q^{79} -4.01641 q^{81} -2.37907 q^{83} -1.18953 q^{85} +7.53579 q^{87} +3.01641 q^{89} -2.17313 q^{91} -17.1044 q^{93} -1.00000 q^{95} +4.88727 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9	$ 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9} + 5 q^{13} + q^{15} + 5 q^{17} - 3 q^{19} + 3 q^{21} + q^{23} + 3 q^{25} + q^{27} + 17 q^{29} - 2 q^{31} + q^{35} + 8 q^{37} - 11 q^{39} + 6 q^{41} + 10 q^{43} + 4 q^{45} - 4 q^{47} + 4 q^{49} - q^{51} + 5 q^{53} - q^{57} - 15 q^{59} + 8 q^{61} + 26 q^{63} + 5 q^{65} - 3 q^{67} + 23 q^{69} + 4 q^{71} + 3 q^{73} + q^{75} - 2 q^{79} - 9 q^{81} + 10 q^{83} + 5 q^{85} + 3 q^{87} + 6 q^{89} - q^{91} - 6 q^{93} - 3 q^{95} - 4 q^{97}+O(q^{100}) $ 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9 + 5 * q^13 + q^15 + 5 * q^17 - 3 * q^19 + 3 * q^21 + q^23 + 3 * q^25 + q^27 + 17 * q^29 - 2 * q^31 + q^35 + 8 * q^37 - 11 * q^39 + 6 * q^41 + 10 * q^43 + 4 * q^45 - 4 * q^47 + 4 * q^49 - q^51 + 5 * q^53 - q^57 - 15 * q^59 + 8 * q^61 + 26 * q^63 + 5 * q^65 - 3 * q^67 + 23 * q^69 + 4 * q^71 + 3 * q^73 + q^75 - 2 * q^79 - 9 * q^81 + 10 * q^83 + 5 * q^85 + 3 * q^87 + 6 * q^89 - q^91 - 6 * q^93 - 3 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.68133	1.54807	0.774033	−	0.633145i	$-0.218236\pi$
$3$	2.68133	1.54807	0.774033	+	0.633145i	$0.218236\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.18953	1.20553	0.602765	−	0.797919i	$-0.294066\pi$
$7$	3.18953	1.20553	0.602765	+	0.797919i	$0.294066\pi$
$8$	0	0
$9$	4.18953	1.39651
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−0.681331	−0.188967	−0.0944836	−	0.995526i	$-0.530120\pi$
$13$	−0.681331	−0.188967	−0.0944836	+	0.995526i	$0.530120\pi$
$14$	0	0
$15$	2.68133	0.692317
$16$	0	0
$17$	−1.18953	−0.288504	−0.144252	−	0.989541i	$-0.546078\pi$
$17$	−1.18953	−0.288504	−0.144252	+	0.989541i	$0.546078\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	8.55220	1.86624
$22$	0	0
$23$	2.17313	0.453128	0.226564	−	0.973996i	$-0.427251\pi$
$23$	2.17313	0.453128	0.226564	+	0.973996i	$0.427251\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.18953	0.613826
$28$	0	0
$29$	2.81047	0.521890	0.260945	−	0.965354i	$-0.415966\pi$
$29$	2.81047	0.521890	0.260945	+	0.965354i	$0.415966\pi$
$30$	0	0
$31$	−6.37907	−1.14571	−0.572857	−	0.819655i	$-0.694165\pi$
$31$	−6.37907	−1.14571	−0.572857	+	0.819655i	$0.694165\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.18953	0.539130
$36$	0	0
$37$	7.87086	1.29396	0.646981	−	0.762506i	$-0.276031\pi$
$37$	7.87086	1.29396	0.646981	+	0.762506i	$0.276031\pi$
$38$	0	0
$39$	−1.82687	−0.292534
$40$	0	0
$41$	0.983593	0.153611	0.0768057	−	0.997046i	$-0.475528\pi$
$41$	0.983593	0.153611	0.0768057	+	0.997046i	$0.475528\pi$
$42$	0	0
$43$	−1.36266	−0.207804	−0.103902	−	0.994588i	$-0.533133\pi$
$43$	−1.36266	−0.207804	−0.103902	+	0.994588i	$0.533133\pi$
$44$	0	0
$45$	4.18953	0.624539
$46$	0	0
$47$	−11.7417	−1.71271	−0.856354	−	0.516390i	$-0.827276\pi$
$47$	−11.7417	−1.71271	−0.856354	+	0.516390i	$0.827276\pi$
$48$	0	0
$49$	3.17313	0.453304
$50$	0	0
$51$	−3.18953	−0.446624
$52$	0	0
$53$	−1.69774	−0.233202	−0.116601	−	0.993179i	$-0.537200\pi$
$53$	−1.69774	−0.233202	−0.116601	+	0.993179i	$0.537200\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.68133	−0.355151
$58$	0	0
$59$	−11.5358	−1.50183	−0.750916	−	0.660398i	$-0.770388\pi$
$59$	−11.5358	−1.50183	−0.750916	+	0.660398i	$0.770388\pi$
$60$	0	0
$61$	7.36266	0.942692	0.471346	−	0.881948i	$-0.343768\pi$
$61$	7.36266	0.942692	0.471346	+	0.881948i	$0.343768\pi$
$62$	0	0
$63$	13.3627	1.68354
$64$	0	0
$65$	−0.681331	−0.0845087
$66$	0	0
$67$	−7.02759	−0.858556	−0.429278	−	0.903172i	$-0.641232\pi$
$67$	−7.02759	−0.858556	−0.429278	+	0.903172i	$0.641232\pi$
$68$	0	0
$69$	5.82687	0.701473
$70$	0	0
$71$	12.7581	1.51411	0.757056	−	0.653350i	$-0.226637\pi$
$71$	12.7581	1.51411	0.757056	+	0.653350i	$0.226637\pi$
$72$	0	0
$73$	−5.53579	−0.647915	−0.323958	−	0.946072i	$-0.605013\pi$
$73$	−5.53579	−0.647915	−0.323958	+	0.946072i	$0.605013\pi$
$74$	0	0
$75$	2.68133	0.309613
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.36266	−0.603347	−0.301673	−	0.953411i	$-0.597545\pi$
$79$	−5.36266	−0.603347	−0.301673	+	0.953411i	$0.597545\pi$
$80$	0	0
$81$	−4.01641	−0.446267
$82$	0	0
$83$	−2.37907	−0.261137	−0.130568	−	0.991439i	$-0.541680\pi$
$83$	−2.37907	−0.261137	−0.130568	+	0.991439i	$0.541680\pi$
$84$	0	0
$85$	−1.18953	−0.129023
$86$	0	0
$87$	7.53579	0.807921
$88$	0	0
$89$	3.01641	0.319738	0.159869	−	0.987138i	$-0.448893\pi$
$89$	3.01641	0.319738	0.159869	+	0.987138i	$0.448893\pi$
$90$	0	0
$91$	−2.17313	−0.227806
$92$	0	0
$93$	−17.1044	−1.77364
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	4.88727	0.496227	0.248114	−	0.968731i	$-0.420189\pi$
$97$	4.88727	0.496227	0.248114	+	0.968731i	$0.420189\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.1208	1.20606	0.603032	−	0.797717i	$-0.293959\pi$
$101$	12.1208	1.20606	0.603032	+	0.797717i	$0.293959\pi$
$102$	0	0
$103$	17.8709	1.76087	0.880434	−	0.474168i	$-0.157251\pi$
$103$	17.8709	1.76087	0.880434	+	0.474168i	$0.157251\pi$
$104$	0	0
$105$	8.55220	0.834609
$106$	0	0
$107$	−14.4231	−1.39433	−0.697165	−	0.716911i	$-0.745555\pi$
$107$	−14.4231	−1.39433	−0.697165	+	0.716911i	$0.745555\pi$
$108$	0	0
$109$	10.2059	0.977552	0.488776	−	0.872409i	$-0.337444\pi$
$109$	10.2059	0.977552	0.488776	+	0.872409i	$0.337444\pi$
$110$	0	0
$111$	21.1044	2.00314
$112$	0	0
$113$	−1.49180	−0.140336	−0.0701682	−	0.997535i	$-0.522354\pi$
$113$	−1.49180	−0.140336	−0.0701682	+	0.997535i	$0.522354\pi$
$114$	0	0
$115$	2.17313	0.202645
$116$	0	0
$117$	−2.85446	−0.263895
$118$	0	0
$119$	−3.79406	−0.347801
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	2.63734	0.237801
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.24993	−0.732063	−0.366032	−	0.930602i	$-0.619284\pi$
$127$	−8.24993	−0.732063	−0.366032	+	0.930602i	$0.619284\pi$
$128$	0	0
$129$	−3.65375	−0.321694
$130$	0	0
$131$	−8.75814	−0.765202	−0.382601	−	0.923914i	$-0.624972\pi$
$131$	−8.75814	−0.765202	−0.382601	+	0.923914i	$0.624972\pi$
$132$	0	0
$133$	−3.18953	−0.276568
$134$	0	0
$135$	3.18953	0.274511
$136$	0	0
$137$	−11.9149	−1.01795	−0.508977	−	0.860780i	$-0.669976\pi$
$137$	−11.9149	−1.01795	−0.508977	+	0.860780i	$0.669976\pi$
$138$	0	0
$139$	23.4835	1.99184	0.995920	−	0.0902352i	$-0.0287619\pi$
$139$	23.4835	1.99184	0.995920	+	0.0902352i	$0.0287619\pi$
$140$	0	0
$141$	−31.4835	−2.65139
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.81047	0.233396
$146$	0	0
$147$	8.50820	0.701745
$148$	0	0
$149$	−18.8461	−1.54393	−0.771967	−	0.635662i	$-0.780727\pi$
$149$	−18.8461	−1.54393	−0.771967	+	0.635662i	$0.780727\pi$
$150$	0	0
$151$	16.0880	1.30922	0.654611	−	0.755966i	$-0.272833\pi$
$151$	16.0880	1.30922	0.654611	+	0.755966i	$0.272833\pi$
$152$	0	0
$153$	−4.98359	−0.402900
$154$	0	0
$155$	−6.37907	−0.512379
$156$	0	0
$157$	−13.7417	−1.09671	−0.548355	−	0.836246i	$-0.684746\pi$
$157$	−13.7417	−1.09671	−0.548355	+	0.836246i	$0.684746\pi$
$158$	0	0
$159$	−4.55220	−0.361013
$160$	0	0
$161$	6.93126	0.546260
$162$	0	0
$163$	14.6373	1.14648	0.573242	−	0.819386i	$-0.305685\pi$
$163$	14.6373	1.14648	0.573242	+	0.819386i	$0.305685\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.9588	−1.38970	−0.694849	−	0.719156i	$-0.744529\pi$
$167$	−17.9588	−1.38970	−0.694849	+	0.719156i	$0.744529\pi$
$168$	0	0
$169$	−12.5358	−0.964291
$170$	0	0
$171$	−4.18953	−0.320382
$172$	0	0
$173$	6.85446	0.521135	0.260567	−	0.965456i	$-0.416090\pi$
$173$	6.85446	0.521135	0.260567	+	0.965456i	$0.416090\pi$
$174$	0	0
$175$	3.18953	0.241106
$176$	0	0
$177$	−30.9313	−2.32494
$178$	0	0
$179$	−5.01641	−0.374944	−0.187472	−	0.982270i	$-0.560029\pi$
$179$	−5.01641	−0.374944	−0.187472	+	0.982270i	$0.560029\pi$
$180$	0	0
$181$	16.7253	1.24318	0.621592	−	0.783341i	$-0.286486\pi$
$181$	16.7253	1.24318	0.621592	+	0.783341i	$0.286486\pi$
$182$	0	0
$183$	19.7417	1.45935
$184$	0	0
$185$	7.87086	0.578677
$186$	0	0
$187$	0	0
$188$	0	0
$189$	10.1731	0.739986
$190$	0	0
$191$	11.2775	0.816013	0.408006	−	0.912979i	$-0.366224\pi$
$191$	11.2775	0.816013	0.408006	+	0.912979i	$0.366224\pi$
$192$	0	0
$193$	13.5798	0.977494	0.488747	−	0.872426i	$-0.337454\pi$
$193$	13.5798	0.977494	0.488747	+	0.872426i	$0.337454\pi$
$194$	0	0
$195$	−1.82687	−0.130825
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−25.6566	−1.81875	−0.909374	−	0.415980i	$-0.863438\pi$
$199$	−25.6566	−1.81875	−0.909374	+	0.415980i	$0.863438\pi$
$200$	0	0
$201$	−18.8433	−1.32910
$202$	0	0
$203$	8.96408	0.629155
$204$	0	0
$205$	0.983593	0.0686971
$206$	0	0
$207$	9.10439	0.632799
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.2939	−1.12172	−0.560860	−	0.827911i	$-0.689529\pi$
$211$	−16.2939	−1.12172	−0.560860	+	0.827911i	$0.689529\pi$
$212$	0	0
$213$	34.2088	2.34395
$214$	0	0
$215$	−1.36266	−0.0929327
$216$	0	0
$217$	−20.3463	−1.38119
$218$	0	0
$219$	−14.8433	−1.00302
$220$	0	0
$221$	0.810466	0.0545178
$222$	0	0
$223$	24.2499	1.62390	0.811948	−	0.583730i	$-0.198407\pi$
$223$	24.2499	1.62390	0.811948	+	0.583730i	$0.198407\pi$
$224$	0	0
$225$	4.18953	0.279302
$226$	0	0
$227$	−5.31867	−0.353012	−0.176506	−	0.984300i	$-0.556480\pi$
$227$	−5.31867	−0.353012	−0.176506	+	0.984300i	$0.556480\pi$
$228$	0	0
$229$	11.7089	0.773747	0.386873	−	0.922133i	$-0.373555\pi$
$229$	11.7089	0.773747	0.386873	+	0.922133i	$0.373555\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.5163	1.80265	0.901325	−	0.433142i	$-0.142595\pi$
$233$	27.5163	1.80265	0.901325	+	0.433142i	$0.142595\pi$
$234$	0	0
$235$	−11.7417	−0.765946
$236$	0	0
$237$	−14.3791	−0.934021
$238$	0	0
$239$	−14.9313	−0.965823	−0.482912	−	0.875669i	$-0.660421\pi$
$239$	−14.9313	−0.965823	−0.482912	+	0.875669i	$0.660421\pi$
$240$	0	0
$241$	−9.32985	−0.600988	−0.300494	−	0.953784i	$-0.597152\pi$
$241$	−9.32985	−0.600988	−0.300494	+	0.953784i	$0.597152\pi$
$242$	0	0
$243$	−20.3379	−1.30468
$244$	0	0
$245$	3.17313	0.202724
$246$	0	0
$247$	0.681331	0.0433520
$248$	0	0
$249$	−6.37907	−0.404257
$250$	0	0
$251$	−1.27468	−0.0804569	−0.0402285	−	0.999191i	$-0.512809\pi$
$251$	−1.27468	−0.0804569	−0.0402285	+	0.999191i	$0.512809\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.18953	−0.199736
$256$	0	0
$257$	31.2887	1.95174	0.975868	−	0.218363i	$-0.0700717\pi$
$257$	31.2887	1.95174	0.975868	+	0.218363i	$0.0700717\pi$
$258$	0	0
$259$	25.1044	1.55991
$260$	0	0
$261$	11.7745	0.728826
$262$	0	0
$263$	−25.5163	−1.57340	−0.786700	−	0.617335i	$-0.788212\pi$
$263$	−25.5163	−1.57340	−0.786700	+	0.617335i	$0.788212\pi$
$264$	0	0
$265$	−1.69774	−0.104291
$266$	0	0
$267$	8.08798	0.494977
$268$	0	0
$269$	26.7581	1.63147	0.815736	−	0.578424i	$-0.196332\pi$
$269$	26.7581	1.63147	0.815736	+	0.578424i	$0.196332\pi$
$270$	0	0
$271$	−14.1731	−0.860956	−0.430478	−	0.902601i	$-0.641655\pi$
$271$	−14.1731	−0.860956	−0.430478	+	0.902601i	$0.641655\pi$
$272$	0	0
$273$	−5.82687	−0.352658
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.08798	0.365791	0.182896	−	0.983132i	$-0.441453\pi$
$277$	6.08798	0.365791	0.182896	+	0.983132i	$0.441453\pi$
$278$	0	0
$279$	−26.7253	−1.60000
$280$	0	0
$281$	−15.7089	−0.937115	−0.468558	−	0.883433i	$-0.655226\pi$
$281$	−15.7089	−0.937115	−0.468558	+	0.883433i	$0.655226\pi$
$282$	0	0
$283$	−0.346255	−0.0205827	−0.0102913	−	0.999947i	$-0.503276\pi$
$283$	−0.346255	−0.0205827	−0.0102913	+	0.999947i	$0.503276\pi$
$284$	0	0
$285$	−2.68133	−0.158828
$286$	0	0
$287$	3.13720	0.185183
$288$	0	0
$289$	−15.5850	−0.916765
$290$	0	0
$291$	13.1044	0.768193
$292$	0	0
$293$	6.71414	0.392244	0.196122	−	0.980579i	$-0.437165\pi$
$293$	6.71414	0.392244	0.196122	+	0.980579i	$0.437165\pi$
$294$	0	0
$295$	−11.5358	−0.671640
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.48062	−0.0856264
$300$	0	0
$301$	−4.34625	−0.250514
$302$	0	0
$303$	32.4999	1.86707
$304$	0	0
$305$	7.36266	0.421585
$306$	0	0
$307$	17.6126	1.00520	0.502602	−	0.864518i	$-0.332376\pi$
$307$	17.6126	1.00520	0.502602	+	0.864518i	$0.332376\pi$
$308$	0	0
$309$	47.9177	2.72594
$310$	0	0
$311$	6.58501	0.373402	0.186701	−	0.982417i	$-0.440221\pi$
$311$	6.58501	0.373402	0.186701	+	0.982417i	$0.440221\pi$
$312$	0	0
$313$	−18.6178	−1.05234	−0.526171	−	0.850379i	$-0.676373\pi$
$313$	−18.6178	−1.05234	−0.526171	+	0.850379i	$0.676373\pi$
$314$	0	0
$315$	13.3627	0.752901
$316$	0	0
$317$	−17.7857	−0.998946	−0.499473	−	0.866330i	$-0.666473\pi$
$317$	−17.7857	−0.998946	−0.499473	+	0.866330i	$0.666473\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−38.6730	−2.15852
$322$	0	0
$323$	1.18953	0.0661874
$324$	0	0
$325$	−0.681331	−0.0377934
$326$	0	0
$327$	27.3655	1.51332
$328$	0	0
$329$	−37.4506	−2.06472
$330$	0	0
$331$	3.53579	0.194345	0.0971723	−	0.995268i	$-0.469020\pi$
$331$	3.53579	0.194345	0.0971723	+	0.995268i	$0.469020\pi$
$332$	0	0
$333$	32.9753	1.80703
$334$	0	0
$335$	−7.02759	−0.383958
$336$	0	0
$337$	−7.87086	−0.428753	−0.214377	−	0.976751i	$-0.568772\pi$
$337$	−7.87086	−0.428753	−0.214377	+	0.976751i	$0.568772\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−12.2059	−0.659059
$344$	0	0
$345$	5.82687	0.313708
$346$	0	0
$347$	−15.6537	−0.840337	−0.420169	−	0.907446i	$-0.638029\pi$
$347$	−15.6537	−0.840337	−0.420169	+	0.907446i	$0.638029\pi$
$348$	0	0
$349$	−21.4178	−1.14647	−0.573235	−	0.819391i	$-0.694312\pi$
$349$	−21.4178	−1.14647	−0.573235	+	0.819391i	$0.694312\pi$
$350$	0	0
$351$	−2.17313	−0.115993
$352$	0	0
$353$	3.15672	0.168015	0.0840076	−	0.996465i	$-0.473228\pi$
$353$	3.15672	0.168015	0.0840076	+	0.996465i	$0.473228\pi$
$354$	0	0
$355$	12.7581	0.677132
$356$	0	0
$357$	−10.1731	−0.538419
$358$	0	0
$359$	−17.8269	−0.940866	−0.470433	−	0.882436i	$-0.655902\pi$
$359$	−17.8269	−0.940866	−0.470433	+	0.882436i	$0.655902\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−29.4946	−1.54807
$364$	0	0
$365$	−5.53579	−0.289756
$366$	0	0
$367$	−19.2252	−1.00355	−0.501773	−	0.864999i	$-0.667319\pi$
$367$	−19.2252	−1.00355	−0.501773	+	0.864999i	$0.667319\pi$
$368$	0	0
$369$	4.12080	0.214520
$370$	0	0
$371$	−5.41499	−0.281132
$372$	0	0
$373$	9.95601	0.515503	0.257751	−	0.966211i	$-0.417018\pi$
$373$	9.95601	0.515503	0.257751	+	0.966211i	$0.417018\pi$
$374$	0	0
$375$	2.68133	0.138463
$376$	0	0
$377$	−1.91486	−0.0986201
$378$	0	0
$379$	19.6238	1.00801	0.504003	−	0.863702i	$-0.331860\pi$
$379$	19.6238	1.00801	0.504003	+	0.863702i	$0.331860\pi$
$380$	0	0
$381$	−22.1208	−1.13328
$382$	0	0
$383$	−11.9037	−0.608250	−0.304125	−	0.952632i	$-0.598364\pi$
$383$	−11.9037	−0.608250	−0.304125	+	0.952632i	$0.598364\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.70892	−0.290201
$388$	0	0
$389$	10.3463	0.524576	0.262288	−	0.964990i	$-0.415523\pi$
$389$	10.3463	0.524576	0.262288	+	0.964990i	$0.415523\pi$
$390$	0	0
$391$	−2.58501	−0.130730
$392$	0	0
$393$	−23.4835	−1.18458
$394$	0	0
$395$	−5.36266	−0.269825
$396$	0	0
$397$	26.1536	1.31261	0.656306	−	0.754495i	$-0.272118\pi$
$397$	26.1536	1.31261	0.656306	+	0.754495i	$0.272118\pi$
$398$	0	0
$399$	−8.55220	−0.428145
$400$	0	0
$401$	−32.7253	−1.63422	−0.817112	−	0.576479i	$-0.804426\pi$
$401$	−32.7253	−1.63422	−0.817112	+	0.576479i	$0.804426\pi$
$402$	0	0
$403$	4.34625	0.216502
$404$	0	0
$405$	−4.01641	−0.199577
$406$	0	0
$407$	0	0
$408$	0	0
$409$	4.03281	0.199410	0.0997049	−	0.995017i	$-0.468210\pi$
$409$	4.03281	0.199410	0.0997049	+	0.995017i	$0.468210\pi$
$410$	0	0
$411$	−31.9477	−1.57586
$412$	0	0
$413$	−36.7938	−1.81050
$414$	0	0
$415$	−2.37907	−0.116784
$416$	0	0
$417$	62.9669	3.08350
$418$	0	0
$419$	15.8297	0.773332	0.386666	−	0.922220i	$-0.373627\pi$
$419$	15.8297	0.773332	0.386666	+	0.922220i	$0.373627\pi$
$420$	0	0
$421$	6.05233	0.294973	0.147486	−	0.989064i	$-0.452882\pi$
$421$	6.05233	0.294973	0.147486	+	0.989064i	$0.452882\pi$
$422$	0	0
$423$	−49.1924	−2.39182
$424$	0	0
$425$	−1.18953	−0.0577009
$426$	0	0
$427$	23.4835	1.13644
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.7089	−1.23835	−0.619177	−	0.785251i	$-0.712534\pi$
$431$	−25.7089	−1.23835	−0.619177	+	0.785251i	$0.712534\pi$
$432$	0	0
$433$	2.57383	0.123690	0.0618452	−	0.998086i	$-0.480301\pi$
$433$	2.57383	0.123690	0.0618452	+	0.998086i	$0.480301\pi$
$434$	0	0
$435$	7.53579	0.361313
$436$	0	0
$437$	−2.17313	−0.103955
$438$	0	0
$439$	−26.8133	−1.27973	−0.639865	−	0.768488i	$-0.721010\pi$
$439$	−26.8133	−1.27973	−0.639865	+	0.768488i	$0.721010\pi$
$440$	0	0
$441$	13.2939	0.633044
$442$	0	0
$443$	−10.7909	−0.512693	−0.256347	−	0.966585i	$-0.582519\pi$
$443$	−10.7909	−0.512693	−0.256347	+	0.966585i	$0.582519\pi$
$444$	0	0
$445$	3.01641	0.142991
$446$	0	0
$447$	−50.5327	−2.39011
$448$	0	0
$449$	−29.4835	−1.39141	−0.695705	−	0.718327i	$-0.744908\pi$
$449$	−29.4835	−1.39141	−0.695705	+	0.718327i	$0.744908\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	43.1372	2.02676
$454$	0	0
$455$	−2.17313	−0.101878
$456$	0	0
$457$	−11.8269	−0.553238	−0.276619	−	0.960980i	$-0.589214\pi$
$457$	−11.8269	−0.553238	−0.276619	+	0.960980i	$0.589214\pi$
$458$	0	0
$459$	−3.79406	−0.177092
$460$	0	0
$461$	−7.45065	−0.347011	−0.173506	−	0.984833i	$-0.555509\pi$
$461$	−7.45065	−0.347011	−0.173506	+	0.984833i	$0.555509\pi$
$462$	0	0
$463$	37.4506	1.74048	0.870240	−	0.492629i	$-0.163964\pi$
$463$	37.4506	1.74048	0.870240	+	0.492629i	$0.163964\pi$
$464$	0	0
$465$	−17.1044	−0.793197
$466$	0	0
$467$	27.4835	1.27178	0.635891	−	0.771779i	$-0.280633\pi$
$467$	27.4835	1.27178	0.635891	+	0.771779i	$0.280633\pi$
$468$	0	0
$469$	−22.4147	−1.03502
$470$	0	0
$471$	−36.8461	−1.69778
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−7.11273	−0.325669
$478$	0	0
$479$	−10.6597	−0.487054	−0.243527	−	0.969894i	$-0.578304\pi$
$479$	−10.6597	−0.487054	−0.243527	+	0.969894i	$0.578304\pi$
$480$	0	0
$481$	−5.36266	−0.244516
$482$	0	0
$483$	18.5850	0.845647
$484$	0	0
$485$	4.88727	0.221920
$486$	0	0
$487$	−7.92604	−0.359163	−0.179581	−	0.983743i	$-0.557474\pi$
$487$	−7.92604	−0.359163	−0.179581	+	0.983743i	$0.557474\pi$
$488$	0	0
$489$	39.2475	1.77484
$490$	0	0
$491$	1.68656	0.0761133	0.0380567	−	0.999276i	$-0.487883\pi$
$491$	1.68656	0.0761133	0.0380567	+	0.999276i	$0.487883\pi$
$492$	0	0
$493$	−3.34314	−0.150568
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.6925	1.82531
$498$	0	0
$499$	20.3463	0.910823	0.455412	−	0.890281i	$-0.349492\pi$
$499$	20.3463	0.910823	0.455412	+	0.890281i	$0.349492\pi$
$500$	0	0
$501$	−48.1536	−2.15134
$502$	0	0
$503$	−1.06874	−0.0476526	−0.0238263	−	0.999716i	$-0.507585\pi$
$503$	−1.06874	−0.0476526	−0.0238263	+	0.999716i	$0.507585\pi$
$504$	0	0
$505$	12.1208	0.539368
$506$	0	0
$507$	−33.6126	−1.49279
$508$	0	0
$509$	−19.0164	−0.842887	−0.421444	−	0.906855i	$-0.638476\pi$
$509$	−19.0164	−0.842887	−0.421444	+	0.906855i	$0.638476\pi$
$510$	0	0
$511$	−17.6566	−0.781081
$512$	0	0
$513$	−3.18953	−0.140821
$514$	0	0
$515$	17.8709	0.787484
$516$	0	0
$517$	0	0
$518$	0	0
$519$	18.3791	0.806752
$520$	0	0
$521$	9.07158	0.397433	0.198717	−	0.980057i	$-0.436323\pi$
$521$	9.07158	0.397433	0.198717	+	0.980057i	$0.436323\pi$
$522$	0	0
$523$	13.4946	0.590079	0.295040	−	0.955485i	$-0.404667\pi$
$523$	13.4946	0.590079	0.295040	+	0.955485i	$0.404667\pi$
$524$	0	0
$525$	8.55220	0.373248
$526$	0	0
$527$	7.58812	0.330544
$528$	0	0
$529$	−18.2775	−0.794675
$530$	0	0
$531$	−48.3296	−2.09733
$532$	0	0
$533$	−0.670152	−0.0290275
$534$	0	0
$535$	−14.4231	−0.623563
$536$	0	0
$537$	−13.4506	−0.580438
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.63734	0.199375	0.0996874	−	0.995019i	$-0.468216\pi$
$541$	4.63734	0.199375	0.0996874	+	0.995019i	$0.468216\pi$
$542$	0	0
$543$	44.8461	1.92453
$544$	0	0
$545$	10.2059	0.437174
$546$	0	0
$547$	18.1291	0.775146	0.387573	−	0.921839i	$-0.373314\pi$
$547$	18.1291	0.775146	0.387573	+	0.921839i	$0.373314\pi$
$548$	0	0
$549$	30.8461	1.31648
$550$	0	0
$551$	−2.81047	−0.119730
$552$	0	0
$553$	−17.1044	−0.727353
$554$	0	0
$555$	21.1044	0.895831
$556$	0	0
$557$	32.2968	1.36846	0.684229	−	0.729267i	$-0.260139\pi$
$557$	32.2968	1.36846	0.684229	+	0.729267i	$0.260139\pi$
$558$	0	0
$559$	0.928423	0.0392681
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.3871	1.32281	0.661405	−	0.750029i	$-0.269960\pi$
$563$	31.3871	1.32281	0.661405	+	0.750029i	$0.269960\pi$
$564$	0	0
$565$	−1.49180	−0.0627604
$566$	0	0
$567$	−12.8105	−0.537989
$568$	0	0
$569$	−31.7969	−1.33300	−0.666498	−	0.745507i	$-0.732207\pi$
$569$	−31.7969	−1.33300	−0.666498	+	0.745507i	$0.732207\pi$
$570$	0	0
$571$	−18.3134	−0.766394	−0.383197	−	0.923667i	$-0.625177\pi$
$571$	−18.3134	−0.766394	−0.383197	+	0.923667i	$0.625177\pi$
$572$	0	0
$573$	30.2388	1.26324
$574$	0	0
$575$	2.17313	0.0906257
$576$	0	0
$577$	26.7282	1.11271	0.556354	−	0.830945i	$-0.312200\pi$
$577$	26.7282	1.11271	0.556354	+	0.830945i	$0.312200\pi$
$578$	0	0
$579$	36.4119	1.51323
$580$	0	0
$581$	−7.58812	−0.314808
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.85446	−0.118017
$586$	0	0
$587$	−24.3463	−1.00488	−0.502439	−	0.864613i	$-0.667564\pi$
$587$	−24.3463	−1.00488	−0.502439	+	0.864613i	$0.667564\pi$
$588$	0	0
$589$	6.37907	0.262845
$590$	0	0
$591$	−5.36266	−0.220590
$592$	0	0
$593$	12.7253	0.522566	0.261283	−	0.965262i	$-0.415854\pi$
$593$	12.7253	0.522566	0.261283	+	0.965262i	$0.415854\pi$
$594$	0	0
$595$	−3.79406	−0.155541
$596$	0	0
$597$	−68.7938	−2.81554
$598$	0	0
$599$	−10.6373	−0.434630	−0.217315	−	0.976102i	$-0.569730\pi$
$599$	−10.6373	−0.434630	−0.217315	+	0.976102i	$0.569730\pi$
$600$	0	0
$601$	11.5329	0.470439	0.235219	−	0.971942i	$-0.424419\pi$
$601$	11.5329	0.470439	0.235219	+	0.971942i	$0.424419\pi$
$602$	0	0
$603$	−29.4423	−1.19898
$604$	0	0
$605$	−11.0000	−0.447214
$606$	0	0
$607$	18.2827	0.742074	0.371037	−	0.928618i	$-0.379002\pi$
$607$	18.2827	0.742074	0.371037	+	0.928618i	$0.379002\pi$
$608$	0	0
$609$	24.0357	0.973974
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−6.84612	−0.276512	−0.138256	−	0.990397i	$-0.544150\pi$
$613$	−6.84612	−0.276512	−0.138256	+	0.990397i	$0.544150\pi$
$614$	0	0
$615$	2.63734	0.106348
$616$	0	0
$617$	15.0820	0.607180	0.303590	−	0.952803i	$-0.401815\pi$
$617$	15.0820	0.607180	0.303590	+	0.952803i	$0.401815\pi$
$618$	0	0
$619$	42.7253	1.71728	0.858638	−	0.512583i	$-0.171311\pi$
$619$	42.7253	1.71728	0.858638	+	0.512583i	$0.171311\pi$
$620$	0	0
$621$	6.93126	0.278142
$622$	0	0
$623$	9.62093	0.385454
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.36266	−0.373314
$630$	0	0
$631$	−26.7909	−1.06653	−0.533265	−	0.845948i	$-0.679035\pi$
$631$	−26.7909	−1.06653	−0.533265	+	0.845948i	$0.679035\pi$
$632$	0	0
$633$	−43.6894	−1.73650
$634$	0	0
$635$	−8.24993	−0.327389
$636$	0	0
$637$	−2.16195	−0.0856595
$638$	0	0
$639$	53.4506	2.11447
$640$	0	0
$641$	28.1208	1.11070	0.555352	−	0.831615i	$-0.312583\pi$
$641$	28.1208	1.11070	0.555352	+	0.831615i	$0.312583\pi$
$642$	0	0
$643$	−15.1372	−0.596953	−0.298477	−	0.954417i	$-0.596478\pi$
$643$	−15.1372	−0.596953	−0.298477	+	0.954417i	$0.596478\pi$
$644$	0	0
$645$	−3.65375	−0.143866
$646$	0	0
$647$	−1.06874	−0.0420164	−0.0210082	−	0.999779i	$-0.506688\pi$
$647$	−1.06874	−0.0420164	−0.0210082	+	0.999779i	$0.506688\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−54.5550	−2.13818
$652$	0	0
$653$	−21.3298	−0.834701	−0.417351	−	0.908745i	$-0.637041\pi$
$653$	−21.3298	−0.834701	−0.417351	+	0.908745i	$0.637041\pi$
$654$	0	0
$655$	−8.75814	−0.342209
$656$	0	0
$657$	−23.1924	−0.904821
$658$	0	0
$659$	−45.4639	−1.77102	−0.885512	−	0.464617i	$-0.846192\pi$
$659$	−45.4639	−1.77102	−0.885512	+	0.464617i	$0.846192\pi$
$660$	0	0
$661$	28.6074	1.11270	0.556349	−	0.830949i	$-0.312202\pi$
$661$	28.6074	1.11270	0.556349	+	0.830949i	$0.312202\pi$
$662$	0	0
$663$	2.17313	0.0843973
$664$	0	0
$665$	−3.18953	−0.123685
$666$	0	0
$667$	6.10750	0.236483
$668$	0	0
$669$	65.0221	2.51390
$670$	0	0
$671$	0	0
$672$	0	0
$673$	47.1840	1.81881	0.909405	−	0.415911i	$-0.136537\pi$
$673$	47.1840	1.81881	0.909405	+	0.415911i	$0.136537\pi$
$674$	0	0
$675$	3.18953	0.122765
$676$	0	0
$677$	20.8573	0.801611	0.400806	−	0.916163i	$-0.368730\pi$
$677$	20.8573	0.801611	0.400806	+	0.916163i	$0.368730\pi$
$678$	0	0
$679$	15.5881	0.598217
$680$	0	0
$681$	−14.2611	−0.546487
$682$	0	0
$683$	17.2887	0.661534	0.330767	−	0.943713i	$-0.392693\pi$
$683$	17.2887	0.661534	0.330767	+	0.943713i	$0.392693\pi$
$684$	0	0
$685$	−11.9149	−0.455243
$686$	0	0
$687$	31.3955	1.19781
$688$	0	0
$689$	1.15672	0.0440675
$690$	0	0
$691$	35.8297	1.36303	0.681513	−	0.731806i	$-0.261322\pi$
$691$	35.8297	1.36303	0.681513	+	0.731806i	$0.261322\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.4835	0.890778
$696$	0	0
$697$	−1.17002	−0.0443176
$698$	0	0
$699$	73.7802	2.79062
$700$	0	0
$701$	−38.0880	−1.43856	−0.719282	−	0.694719i	$-0.755529\pi$
$701$	−38.0880	−1.43856	−0.719282	+	0.694719i	$0.755529\pi$
$702$	0	0
$703$	−7.87086	−0.296855
$704$	0	0
$705$	−31.4835	−1.18574
$706$	0	0
$707$	38.6597	1.45395
$708$	0	0
$709$	−15.2747	−0.573653	−0.286826	−	0.957983i	$-0.592600\pi$
$709$	−15.2747	−0.573653	−0.286826	+	0.957983i	$0.592600\pi$
$710$	0	0
$711$	−22.4671	−0.842580
$712$	0	0
$713$	−13.8625	−0.519156
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−40.0357	−1.49516
$718$	0	0
$719$	27.8597	1.03899	0.519495	−	0.854473i	$-0.326120\pi$
$719$	27.8597	1.03899	0.519495	+	0.854473i	$0.326120\pi$
$720$	0	0
$721$	56.9997	2.12278
$722$	0	0
$723$	−25.0164	−0.930370
$724$	0	0
$725$	2.81047	0.104378
$726$	0	0
$727$	39.4311	1.46242	0.731210	−	0.682153i	$-0.238956\pi$
$727$	39.4311	1.46242	0.731210	+	0.682153i	$0.238956\pi$
$728$	0	0
$729$	−42.4835	−1.57346
$730$	0	0
$731$	1.62093	0.0599523
$732$	0	0
$733$	33.1372	1.22395	0.611975	−	0.790877i	$-0.290375\pi$
$733$	33.1372	1.22395	0.611975	+	0.790877i	$0.290375\pi$
$734$	0	0
$735$	8.50820	0.313830
$736$	0	0
$737$	0	0
$738$	0	0
$739$	25.8625	0.951368	0.475684	−	0.879616i	$-0.342201\pi$
$739$	25.8625	0.951368	0.475684	+	0.879616i	$0.342201\pi$
$740$	0	0
$741$	1.82687	0.0671118
$742$	0	0
$743$	28.6842	1.05232	0.526160	−	0.850386i	$-0.323631\pi$
$743$	28.6842	1.05232	0.526160	+	0.850386i	$0.323631\pi$
$744$	0	0
$745$	−18.8461	−0.690468
$746$	0	0
$747$	−9.96719	−0.364680
$748$	0	0
$749$	−46.0028	−1.68091
$750$	0	0
$751$	34.2968	1.25151	0.625753	−	0.780021i	$-0.284792\pi$
$751$	34.2968	1.25151	0.625753	+	0.780021i	$0.284792\pi$
$752$	0	0
$753$	−3.41783	−0.124553
$754$	0	0
$755$	16.0880	0.585502
$756$	0	0
$757$	20.6597	0.750889	0.375445	−	0.926845i	$-0.377490\pi$
$757$	20.6597	0.750889	0.375445	+	0.926845i	$0.377490\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.1236	1.38198	0.690990	−	0.722864i	$-0.257175\pi$
$761$	38.1236	1.38198	0.690990	+	0.722864i	$0.257175\pi$
$762$	0	0
$763$	32.5522	1.17847
$764$	0	0
$765$	−4.98359	−0.180182
$766$	0	0
$767$	7.85969	0.283797
$768$	0	0
$769$	20.3267	0.733001	0.366500	−	0.930418i	$-0.380556\pi$
$769$	20.3267	0.733001	0.366500	+	0.930418i	$0.380556\pi$
$770$	0	0
$771$	83.8953	3.02142
$772$	0	0
$773$	26.4559	0.951552	0.475776	−	0.879567i	$-0.342167\pi$
$773$	26.4559	0.951552	0.475776	+	0.879567i	$0.342167\pi$
$774$	0	0
$775$	−6.37907	−0.229143
$776$	0	0
$777$	67.3132	2.41485
$778$	0	0
$779$	−0.983593	−0.0352409
$780$	0	0
$781$	0	0
$782$	0	0
$783$	8.96408	0.320350
$784$	0	0
$785$	−13.7417	−0.490463
$786$	0	0
$787$	27.3738	0.975772	0.487886	−	0.872907i	$-0.337768\pi$
$787$	27.3738	0.975772	0.487886	+	0.872907i	$0.337768\pi$
$788$	0	0
$789$	−68.4176	−2.43573
$790$	0	0
$791$	−4.75814	−0.169180
$792$	0	0
$793$	−5.01641	−0.178138
$794$	0	0
$795$	−4.55220	−0.161450
$796$	0	0
$797$	36.4454	1.29096	0.645481	−	0.763776i	$-0.276657\pi$
$797$	36.4454	1.29096	0.645481	+	0.763776i	$0.276657\pi$
$798$	0	0
$799$	13.9672	0.494124
$800$	0	0
$801$	12.6373	0.446518
$802$	0	0
$803$	0	0
$804$	0	0
$805$	6.93126	0.244295
$806$	0	0
$807$	71.7474	2.52563
$808$	0	0
$809$	52.2444	1.83682	0.918408	−	0.395634i	$-0.129475\pi$
$809$	52.2444	1.83682	0.918408	+	0.395634i	$0.129475\pi$
$810$	0	0
$811$	−32.8984	−1.15522	−0.577610	−	0.816313i	$-0.696015\pi$
$811$	−32.8984	−1.15522	−0.577610	+	0.816313i	$0.696015\pi$
$812$	0	0
$813$	−38.0028	−1.33282
$814$	0	0
$815$	14.6373	0.512724
$816$	0	0
$817$	1.36266	0.0476735
$818$	0	0
$819$	−9.10439	−0.318133
$820$	0	0
$821$	−4.44470	−0.155121	−0.0775605	−	0.996988i	$-0.524713\pi$
$821$	−4.44470	−0.155121	−0.0775605	+	0.996988i	$0.524713\pi$
$822$	0	0
$823$	27.7774	0.968259	0.484129	−	0.874996i	$-0.339136\pi$
$823$	27.7774	0.968259	0.484129	+	0.874996i	$0.339136\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.5110	−1.61735	−0.808674	−	0.588257i	$-0.799814\pi$
$827$	−46.5110	−1.61735	−0.808674	+	0.588257i	$0.799814\pi$
$828$	0	0
$829$	28.2388	0.980772	0.490386	−	0.871505i	$-0.336856\pi$
$829$	28.2388	0.980772	0.490386	+	0.871505i	$0.336856\pi$
$830$	0	0
$831$	16.3239	0.566270
$832$	0	0
$833$	−3.77454	−0.130780
$834$	0	0
$835$	−17.9588	−0.621492
$836$	0	0
$837$	−20.3463	−0.703269
$838$	0	0
$839$	−9.01641	−0.311281	−0.155640	−	0.987814i	$-0.549744\pi$
$839$	−9.01641	−0.311281	−0.155640	+	0.987814i	$0.549744\pi$
$840$	0	0
$841$	−21.1013	−0.727630
$842$	0	0
$843$	−42.1208	−1.45072
$844$	0	0
$845$	−12.5358	−0.431244
$846$	0	0
$847$	−35.0849	−1.20553
$848$	0	0
$849$	−0.928423	−0.0318634
$850$	0	0
$851$	17.1044	0.586331
$852$	0	0
$853$	5.30749	0.181725	0.0908625	−	0.995863i	$-0.471038\pi$
$853$	5.30749	0.181725	0.0908625	+	0.995863i	$0.471038\pi$
$854$	0	0
$855$	−4.18953	−0.143279
$856$	0	0
$857$	−38.9258	−1.32968	−0.664839	−	0.746986i	$-0.731500\pi$
$857$	−38.9258	−1.32968	−0.664839	+	0.746986i	$0.731500\pi$
$858$	0	0
$859$	−20.1760	−0.688395	−0.344198	−	0.938897i	$-0.611849\pi$
$859$	−20.1760	−0.688395	−0.344198	+	0.938897i	$0.611849\pi$
$860$	0	0
$861$	8.41188	0.286676
$862$	0	0
$863$	30.3051	1.03160	0.515799	−	0.856710i	$-0.327495\pi$
$863$	30.3051	1.03160	0.515799	+	0.856710i	$0.327495\pi$
$864$	0	0
$865$	6.85446	0.233059
$866$	0	0
$867$	−41.7886	−1.41921
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.78811	0.162239
$872$	0	0
$873$	20.4754	0.692987
$874$	0	0
$875$	3.18953	0.107826
$876$	0	0
$877$	9.43947	0.318748	0.159374	−	0.987218i	$-0.449052\pi$
$877$	9.43947	0.318748	0.159374	+	0.987218i	$0.449052\pi$
$878$	0	0
$879$	18.0028	0.607221
$880$	0	0
$881$	−24.4671	−0.824316	−0.412158	−	0.911112i	$-0.635225\pi$
$881$	−24.4671	−0.824316	−0.412158	+	0.911112i	$0.635225\pi$
$882$	0	0
$883$	−6.12080	−0.205981	−0.102991	−	0.994682i	$-0.532841\pi$
$883$	−6.12080	−0.205981	−0.102991	+	0.994682i	$0.532841\pi$
$884$	0	0
$885$	−30.9313	−1.03974
$886$	0	0
$887$	−45.4200	−1.52505	−0.762526	−	0.646957i	$-0.776041\pi$
$887$	−45.4200	−1.52505	−0.762526	+	0.646957i	$0.776041\pi$
$888$	0	0
$889$	−26.3134	−0.882524
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.7417	0.392922
$894$	0	0
$895$	−5.01641	−0.167680
$896$	0	0
$897$	−3.97003	−0.132555
$898$	0	0
$899$	−17.9282	−0.597937
$900$	0	0
$901$	2.01952	0.0672798
$902$	0	0
$903$	−11.6537	−0.387812
$904$	0	0
$905$	16.7253	0.555969
$906$	0	0
$907$	10.2694	0.340991	0.170496	−	0.985358i	$-0.445463\pi$
$907$	10.2694	0.340991	0.170496	+	0.985358i	$0.445463\pi$
$908$	0	0
$909$	50.7805	1.68428
$910$	0	0
$911$	10.9180	0.361728	0.180864	−	0.983508i	$-0.442111\pi$
$911$	10.9180	0.361728	0.180864	+	0.983508i	$0.442111\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	19.7417	0.652641
$916$	0	0
$917$	−27.9344	−0.922474
$918$	0	0
$919$	38.9313	1.28422	0.642112	−	0.766611i	$-0.278058\pi$
$919$	38.9313	1.28422	0.642112	+	0.766611i	$0.278058\pi$
$920$	0	0
$921$	47.2252	1.55612
$922$	0	0
$923$	−8.69251	−0.286117
$924$	0	0
$925$	7.87086	0.258792
$926$	0	0
$927$	74.8706	2.45907
$928$	0	0
$929$	21.9700	0.720813	0.360407	−	0.932795i	$-0.382638\pi$
$929$	21.9700	0.720813	0.360407	+	0.932795i	$0.382638\pi$
$930$	0	0
$931$	−3.17313	−0.103995
$932$	0	0
$933$	17.6566	0.578051
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.6402	0.347600	0.173800	−	0.984781i	$-0.444395\pi$
$937$	10.6402	0.347600	0.173800	+	0.984781i	$0.444395\pi$
$938$	0	0
$939$	−49.9205	−1.62910
$940$	0	0
$941$	−20.8656	−0.680200	−0.340100	−	0.940389i	$-0.610461\pi$
$941$	−20.8656	−0.680200	−0.340100	+	0.940389i	$0.610461\pi$
$942$	0	0
$943$	2.13747	0.0696057
$944$	0	0
$945$	10.1731	0.330932
$946$	0	0
$947$	−12.6925	−0.412451	−0.206226	−	0.978504i	$-0.566118\pi$
$947$	−12.6925	−0.412451	−0.206226	+	0.978504i	$0.566118\pi$
$948$	0	0
$949$	3.77170	0.122435
$950$	0	0
$951$	−47.6894	−1.54643
$952$	0	0
$953$	16.7337	0.542056	0.271028	−	0.962571i	$-0.412636\pi$
$953$	16.7337	0.542056	0.271028	+	0.962571i	$0.412636\pi$
$954$	0	0
$955$	11.2775	0.364932
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−38.0028	−1.22718
$960$	0	0
$961$	9.69251	0.312662
$962$	0	0
$963$	−60.4259	−1.94720
$964$	0	0
$965$	13.5798	0.437149
$966$	0	0
$967$	−4.62688	−0.148790	−0.0743952	−	0.997229i	$-0.523703\pi$
$967$	−4.62688	−0.148790	−0.0743952	+	0.997229i	$0.523703\pi$
$968$	0	0
$969$	3.18953	0.102463
$970$	0	0
$971$	49.4506	1.58695	0.793473	−	0.608605i	$-0.208271\pi$
$971$	49.4506	1.58695	0.793473	+	0.608605i	$0.208271\pi$
$972$	0	0
$973$	74.9013	2.40123
$974$	0	0
$975$	−1.82687	−0.0585068
$976$	0	0
$977$	22.5962	0.722916	0.361458	−	0.932388i	$-0.382279\pi$
$977$	22.5962	0.722916	0.361458	+	0.932388i	$0.382279\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	42.7581	1.36516
$982$	0	0
$983$	−32.9424	−1.05070	−0.525350	−	0.850886i	$-0.676066\pi$
$983$	−32.9424	−1.05070	−0.525350	+	0.850886i	$0.676066\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	−100.418	−3.19633
$988$	0	0
$989$	−2.96124	−0.0941618
$990$	0	0
$991$	49.7802	1.58132	0.790660	−	0.612255i	$-0.209737\pi$
$991$	49.7802	1.58132	0.790660	+	0.612255i	$0.209737\pi$
$992$	0	0
$993$	9.48062	0.300858
$994$	0	0
$995$	−25.6566	−0.813368
$996$	0	0
$997$	−36.4014	−1.15284	−0.576422	−	0.817152i	$-0.695552\pi$
$997$	−36.4014	−1.15284	−0.576422	+	0.817152i	$0.695552\pi$
$998$	0	0
$999$	25.1044	0.794268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.s.1.3		3
4.3	odd	2		760.2.a.j.1.1	✓	3
5.4	even	2		7600.2.a.bq.1.1		3
8.3	odd	2		6080.2.a.bv.1.3		3
8.5	even	2		6080.2.a.bq.1.1		3
12.11	even	2		6840.2.a.bg.1.1		3
20.3	even	4		3800.2.d.l.3649.1		6
20.7	even	4		3800.2.d.l.3649.6		6
20.19	odd	2		3800.2.a.x.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.j.1.1	✓	3	4.3	odd	2
1520.2.a.s.1.3		3	1.1	even	1	trivial
3800.2.a.x.1.3		3	20.19	odd	2
3800.2.d.l.3649.1		6	20.3	even	4
3800.2.d.l.3649.6		6	20.7	even	4
6080.2.a.bq.1.1		3	8.5	even	2
6080.2.a.bv.1.3		3	8.3	odd	2
6840.2.a.bg.1.1		3	12.11	even	2
7600.2.a.bq.1.1		3	5.4	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68133 q^{3} +1.00000 q^{5} +3.18953 q^{7} +4.18953 q^{9} +O(q^{10})\)	\(q+2.68133 q^{3} +1.00000 q^{5} +3.18953 q^{7} +4.18953 q^{9} -0.681331 q^{13} +2.68133 q^{15} -1.18953 q^{17} -1.00000 q^{19} +8.55220 q^{21} +2.17313 q^{23} +1.00000 q^{25} +3.18953 q^{27} +2.81047 q^{29} -6.37907 q^{31} +3.18953 q^{35} +7.87086 q^{37} -1.82687 q^{39} +0.983593 q^{41} -1.36266 q^{43} +4.18953 q^{45} -11.7417 q^{47} +3.17313 q^{49} -3.18953 q^{51} -1.69774 q^{53} -2.68133 q^{57} -11.5358 q^{59} +7.36266 q^{61} +13.3627 q^{63} -0.681331 q^{65} -7.02759 q^{67} +5.82687 q^{69} +12.7581 q^{71} -5.53579 q^{73} +2.68133 q^{75} -5.36266 q^{79} -4.01641 q^{81} -2.37907 q^{83} -1.18953 q^{85} +7.53579 q^{87} +3.01641 q^{89} -2.17313 q^{91} -17.1044 q^{93} -1.00000 q^{95} +4.88727 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9	\( 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9} + 5 q^{13} + q^{15} + 5 q^{17} - 3 q^{19} + 3 q^{21} + q^{23} + 3 q^{25} + q^{27} + 17 q^{29} - 2 q^{31} + q^{35} + 8 q^{37} - 11 q^{39} + 6 q^{41} + 10 q^{43} + 4 q^{45} - 4 q^{47} + 4 q^{49} - q^{51} + 5 q^{53} - q^{57} - 15 q^{59} + 8 q^{61} + 26 q^{63} + 5 q^{65} - 3 q^{67} + 23 q^{69} + 4 q^{71} + 3 q^{73} + q^{75} - 2 q^{79} - 9 q^{81} + 10 q^{83} + 5 q^{85} + 3 q^{87} + 6 q^{89} - q^{91} - 6 q^{93} - 3 q^{95} - 4 q^{97}+O(q^{100}) \) 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9 + 5 * q^13 + q^15 + 5 * q^17 - 3 * q^19 + 3 * q^21 + q^23 + 3 * q^25 + q^27 + 17 * q^29 - 2 * q^31 + q^35 + 8 * q^37 - 11 * q^39 + 6 * q^41 + 10 * q^43 + 4 * q^45 - 4 * q^47 + 4 * q^49 - q^51 + 5 * q^53 - q^57 - 15 * q^59 + 8 * q^61 + 26 * q^63 + 5 * q^65 - 3 * q^67 + 23 * q^69 + 4 * q^71 + 3 * q^73 + q^75 - 2 * q^79 - 9 * q^81 + 10 * q^83 + 5 * q^85 + 3 * q^87 + 6 * q^89 - q^91 - 6 * q^93 - 3 * q^95 - 4 * q^97

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