LMFDB - Embedded newform 460.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [460,2,Mod(1,460)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(460, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("460.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	460.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:	$3.67311849298$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	460.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} -0.561553 q^{9} +O(q^{10})$	$q-1.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} -0.561553 q^{9} +2.00000 q^{11} -3.56155 q^{13} -1.56155 q^{15} +2.56155 q^{17} +6.00000 q^{19} -4.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.56155 q^{27} +6.12311 q^{29} +7.24621 q^{31} -3.12311 q^{33} +2.56155 q^{35} -4.56155 q^{37} +5.56155 q^{39} +4.12311 q^{41} -0.561553 q^{45} +4.68466 q^{47} -0.438447 q^{49} -4.00000 q^{51} -4.56155 q^{53} +2.00000 q^{55} -9.36932 q^{57} -3.68466 q^{59} -7.12311 q^{61} -1.43845 q^{63} -3.56155 q^{65} -8.56155 q^{67} -1.56155 q^{69} +10.1231 q^{71} -4.43845 q^{73} -1.56155 q^{75} +5.12311 q^{77} +4.87689 q^{79} -7.00000 q^{81} -13.9309 q^{83} +2.56155 q^{85} -9.56155 q^{87} +14.2462 q^{89} -9.12311 q^{91} -11.3153 q^{93} +6.00000 q^{95} -13.1231 q^{97} -1.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9	$ 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} + 4 q^{11} - 3 q^{13} + q^{15} + q^{17} + 12 q^{19} - 8 q^{21} + 2 q^{23} + 2 q^{25} + 7 q^{27} + 4 q^{29} - 2 q^{31} + 2 q^{33} + q^{35} - 5 q^{37} + 7 q^{39} + 3 q^{45} - 3 q^{47} - 5 q^{49} - 8 q^{51} - 5 q^{53} + 4 q^{55} + 6 q^{57} + 5 q^{59} - 6 q^{61} - 7 q^{63} - 3 q^{65} - 13 q^{67} + q^{69} + 12 q^{71} - 13 q^{73} + q^{75} + 2 q^{77} + 18 q^{79} - 14 q^{81} + q^{83} + q^{85} - 15 q^{87} + 12 q^{89} - 10 q^{91} - 35 q^{93} + 12 q^{95} - 18 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 + 4 * q^11 - 3 * q^13 + q^15 + q^17 + 12 * q^19 - 8 * q^21 + 2 * q^23 + 2 * q^25 + 7 * q^27 + 4 * q^29 - 2 * q^31 + 2 * q^33 + q^35 - 5 * q^37 + 7 * q^39 + 3 * q^45 - 3 * q^47 - 5 * q^49 - 8 * q^51 - 5 * q^53 + 4 * q^55 + 6 * q^57 + 5 * q^59 - 6 * q^61 - 7 * q^63 - 3 * q^65 - 13 * q^67 + q^69 + 12 * q^71 - 13 * q^73 + q^75 + 2 * q^77 + 18 * q^79 - 14 * q^81 + q^83 + q^85 - 15 * q^87 + 12 * q^89 - 10 * q^91 - 35 * q^93 + 12 * q^95 - 18 * q^97 + 6 * q^99

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
$3$	−1.56155	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−3.56155	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$13$	−3.56155	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$14$	0	0
$15$	−1.56155	−0.403191
$16$	0	0
$17$	2.56155	0.621268	0.310634	−	0.950530i	$-0.399459\pi$
$17$	2.56155	0.621268	0.310634	+	0.950530i	$0.399459\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.56155	1.07032
$28$	0	0
$29$	6.12311	1.13703	0.568516	−	0.822672i	$-0.307518\pi$
$29$	6.12311	1.13703	0.568516	+	0.822672i	$0.307518\pi$
$30$	0	0
$31$	7.24621	1.30146	0.650729	−	0.759310i	$-0.274463\pi$
$31$	7.24621	1.30146	0.650729	+	0.759310i	$0.274463\pi$
$32$	0	0
$33$	−3.12311	−0.543663
$34$	0	0
$35$	2.56155	0.432981
$36$	0	0
$37$	−4.56155	−0.749915	−0.374957	−	0.927042i	$-0.622343\pi$
$37$	−4.56155	−0.749915	−0.374957	+	0.927042i	$0.622343\pi$
$38$	0	0
$39$	5.56155	0.890561
$40$	0	0
$41$	4.12311	0.643921	0.321960	−	0.946753i	$-0.395658\pi$
$41$	4.12311	0.643921	0.321960	+	0.946753i	$0.395658\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−0.561553	−0.0837114
$46$	0	0
$47$	4.68466	0.683328	0.341664	−	0.939822i	$-0.389010\pi$
$47$	4.68466	0.683328	0.341664	+	0.939822i	$0.389010\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−4.56155	−0.626577	−0.313289	−	0.949658i	$-0.601431\pi$
$53$	−4.56155	−0.626577	−0.313289	+	0.949658i	$0.601431\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−9.36932	−1.24100
$58$	0	0
$59$	−3.68466	−0.479702	−0.239851	−	0.970810i	$-0.577099\pi$
$59$	−3.68466	−0.479702	−0.239851	+	0.970810i	$0.577099\pi$
$60$	0	0
$61$	−7.12311	−0.912020	−0.456010	−	0.889975i	$-0.650722\pi$
$61$	−7.12311	−0.912020	−0.456010	+	0.889975i	$0.650722\pi$
$62$	0	0
$63$	−1.43845	−0.181227
$64$	0	0
$65$	−3.56155	−0.441756
$66$	0	0
$67$	−8.56155	−1.04596	−0.522980	−	0.852345i	$-0.675180\pi$
$67$	−8.56155	−1.04596	−0.522980	+	0.852345i	$0.675180\pi$
$68$	0	0
$69$	−1.56155	−0.187989
$70$	0	0
$71$	10.1231	1.20139	0.600696	−	0.799478i	$-0.294890\pi$
$71$	10.1231	1.20139	0.600696	+	0.799478i	$0.294890\pi$
$72$	0	0
$73$	−4.43845	−0.519481	−0.259740	−	0.965678i	$-0.583637\pi$
$73$	−4.43845	−0.519481	−0.259740	+	0.965678i	$0.583637\pi$
$74$	0	0
$75$	−1.56155	−0.180313
$76$	0	0
$77$	5.12311	0.583832
$78$	0	0
$79$	4.87689	0.548693	0.274347	−	0.961631i	$-0.411538\pi$
$79$	4.87689	0.548693	0.274347	+	0.961631i	$0.411538\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−13.9309	−1.52911	−0.764556	−	0.644558i	$-0.777042\pi$
$83$	−13.9309	−1.52911	−0.764556	+	0.644558i	$0.777042\pi$
$84$	0	0
$85$	2.56155	0.277839
$86$	0	0
$87$	−9.56155	−1.02511
$88$	0	0
$89$	14.2462	1.51010	0.755048	−	0.655670i	$-0.227614\pi$
$89$	14.2462	1.51010	0.755048	+	0.655670i	$0.227614\pi$
$90$	0	0
$91$	−9.12311	−0.956361
$92$	0	0
$93$	−11.3153	−1.17335
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	−13.1231	−1.33245	−0.666225	−	0.745751i	$-0.732091\pi$
$97$	−13.1231	−1.33245	−0.666225	+	0.745751i	$0.732091\pi$
$98$	0	0
$99$	−1.12311	−0.112876
$100$	0	0
$101$	−17.6847	−1.75969	−0.879845	−	0.475261i	$-0.842353\pi$
$101$	−17.6847	−1.75969	−0.879845	+	0.475261i	$0.842353\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	3.43845	0.332407	0.166204	−	0.986091i	$-0.446849\pi$
$107$	3.43845	0.332407	0.166204	+	0.986091i	$0.446849\pi$
$108$	0	0
$109$	15.3693	1.47211	0.736057	−	0.676920i	$-0.236686\pi$
$109$	15.3693	1.47211	0.736057	+	0.676920i	$0.236686\pi$
$110$	0	0
$111$	7.12311	0.676095
$112$	0	0
$113$	12.8078	1.20485	0.602427	−	0.798174i	$-0.294201\pi$
$113$	12.8078	1.20485	0.602427	+	0.798174i	$0.294201\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	6.56155	0.601497
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−6.43845	−0.580535
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.80776	0.692827	0.346414	−	0.938082i	$-0.387399\pi$
$127$	7.80776	0.692827	0.346414	+	0.938082i	$0.387399\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.192236	0.0167957	0.00839787	−	0.999965i	$-0.497327\pi$
$131$	0.192236	0.0167957	0.00839787	+	0.999965i	$0.497327\pi$
$132$	0	0
$133$	15.3693	1.33269
$134$	0	0
$135$	5.56155	0.478662
$136$	0	0
$137$	6.87689	0.587533	0.293766	−	0.955877i	$-0.405091\pi$
$137$	6.87689	0.587533	0.293766	+	0.955877i	$0.405091\pi$
$138$	0	0
$139$	−11.2462	−0.953891	−0.476946	−	0.878933i	$-0.658256\pi$
$139$	−11.2462	−0.953891	−0.476946	+	0.878933i	$0.658256\pi$
$140$	0	0
$141$	−7.31534	−0.616063
$142$	0	0
$143$	−7.12311	−0.595664
$144$	0	0
$145$	6.12311	0.508496
$146$	0	0
$147$	0.684658	0.0564697
$148$	0	0
$149$	−9.36932	−0.767564	−0.383782	−	0.923424i	$-0.625379\pi$
$149$	−9.36932	−0.767564	−0.383782	+	0.923424i	$0.625379\pi$
$150$	0	0
$151$	−14.0540	−1.14370	−0.571848	−	0.820359i	$-0.693773\pi$
$151$	−14.0540	−1.14370	−0.571848	+	0.820359i	$0.693773\pi$
$152$	0	0
$153$	−1.43845	−0.116292
$154$	0	0
$155$	7.24621	0.582030
$156$	0	0
$157$	−4.31534	−0.344402	−0.172201	−	0.985062i	$-0.555088\pi$
$157$	−4.31534	−0.344402	−0.172201	+	0.985062i	$0.555088\pi$
$158$	0	0
$159$	7.12311	0.564899
$160$	0	0
$161$	2.56155	0.201879
$162$	0	0
$163$	16.9309	1.32613	0.663064	−	0.748563i	$-0.269256\pi$
$163$	16.9309	1.32613	0.663064	+	0.748563i	$0.269256\pi$
$164$	0	0
$165$	−3.12311	−0.243133
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	−3.36932	−0.257658
$172$	0	0
$173$	5.36932	0.408222	0.204111	−	0.978948i	$-0.434570\pi$
$173$	5.36932	0.408222	0.204111	+	0.978948i	$0.434570\pi$
$174$	0	0
$175$	2.56155	0.193635
$176$	0	0
$177$	5.75379	0.432481
$178$	0	0
$179$	−6.93087	−0.518038	−0.259019	−	0.965872i	$-0.583399\pi$
$179$	−6.93087	−0.518038	−0.259019	+	0.965872i	$0.583399\pi$
$180$	0	0
$181$	−5.36932	−0.399098	−0.199549	−	0.979888i	$-0.563948\pi$
$181$	−5.36932	−0.399098	−0.199549	+	0.979888i	$0.563948\pi$
$182$	0	0
$183$	11.1231	0.822244
$184$	0	0
$185$	−4.56155	−0.335372
$186$	0	0
$187$	5.12311	0.374639
$188$	0	0
$189$	14.2462	1.03626
$190$	0	0
$191$	17.3693	1.25680	0.628400	−	0.777891i	$-0.283710\pi$
$191$	17.3693	1.25680	0.628400	+	0.777891i	$0.283710\pi$
$192$	0	0
$193$	2.43845	0.175523	0.0877616	−	0.996142i	$-0.472029\pi$
$193$	2.43845	0.175523	0.0877616	+	0.996142i	$0.472029\pi$
$194$	0	0
$195$	5.56155	0.398271
$196$	0	0
$197$	−18.6847	−1.33123	−0.665613	−	0.746297i	$-0.731830\pi$
$197$	−18.6847	−1.33123	−0.665613	+	0.746297i	$0.731830\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	13.3693	0.942999
$202$	0	0
$203$	15.6847	1.10085
$204$	0	0
$205$	4.12311	0.287970
$206$	0	0
$207$	−0.561553	−0.0390306
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	0.315342	0.0217090	0.0108545	−	0.999941i	$-0.496545\pi$
$211$	0.315342	0.0217090	0.0108545	+	0.999941i	$0.496545\pi$
$212$	0	0
$213$	−15.8078	−1.08313
$214$	0	0
$215$	0	0
$216$	0	0
$217$	18.5616	1.26004
$218$	0	0
$219$	6.93087	0.468345
$220$	0	0
$221$	−9.12311	−0.613686
$222$	0	0
$223$	−17.6155	−1.17962	−0.589812	−	0.807541i	$-0.700798\pi$
$223$	−17.6155	−1.17962	−0.589812	+	0.807541i	$0.700798\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	26.2462	1.74202	0.871011	−	0.491263i	$-0.163465\pi$
$227$	26.2462	1.74202	0.871011	+	0.491263i	$0.163465\pi$
$228$	0	0
$229$	−26.7386	−1.76694	−0.883469	−	0.468489i	$-0.844799\pi$
$229$	−26.7386	−1.76694	−0.883469	+	0.468489i	$0.844799\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−0.684658	−0.0448535	−0.0224267	−	0.999748i	$-0.507139\pi$
$233$	−0.684658	−0.0448535	−0.0224267	+	0.999748i	$0.507139\pi$
$234$	0	0
$235$	4.68466	0.305593
$236$	0	0
$237$	−7.61553	−0.494682
$238$	0	0
$239$	2.75379	0.178128	0.0890639	−	0.996026i	$-0.471612\pi$
$239$	2.75379	0.178128	0.0890639	+	0.996026i	$0.471612\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	−5.75379	−0.369106
$244$	0	0
$245$	−0.438447	−0.0280114
$246$	0	0
$247$	−21.3693	−1.35970
$248$	0	0
$249$	21.7538	1.37859
$250$	0	0
$251$	−7.36932	−0.465147	−0.232574	−	0.972579i	$-0.574715\pi$
$251$	−7.36932	−0.465147	−0.232574	+	0.972579i	$0.574715\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	31.8078	1.98411	0.992057	−	0.125790i	$-0.0401465\pi$
$257$	31.8078	1.98411	0.992057	+	0.125790i	$0.0401465\pi$
$258$	0	0
$259$	−11.6847	−0.726049
$260$	0	0
$261$	−3.43845	−0.212835
$262$	0	0
$263$	0.0691303	0.00426276	0.00213138	−	0.999998i	$-0.499322\pi$
$263$	0.0691303	0.00426276	0.00213138	+	0.999998i	$0.499322\pi$
$264$	0	0
$265$	−4.56155	−0.280214
$266$	0	0
$267$	−22.2462	−1.36145
$268$	0	0
$269$	23.2462	1.41735	0.708673	−	0.705537i	$-0.249294\pi$
$269$	23.2462	1.41735	0.708673	+	0.705537i	$0.249294\pi$
$270$	0	0
$271$	21.9309	1.33221	0.666103	−	0.745860i	$-0.267961\pi$
$271$	21.9309	1.33221	0.666103	+	0.745860i	$0.267961\pi$
$272$	0	0
$273$	14.2462	0.862220
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	−6.68466	−0.401642	−0.200821	−	0.979628i	$-0.564361\pi$
$277$	−6.68466	−0.401642	−0.200821	+	0.979628i	$0.564361\pi$
$278$	0	0
$279$	−4.06913	−0.243612
$280$	0	0
$281$	−6.87689	−0.410241	−0.205121	−	0.978737i	$-0.565759\pi$
$281$	−6.87689	−0.410241	−0.205121	+	0.978737i	$0.565759\pi$
$282$	0	0
$283$	14.8078	0.880230	0.440115	−	0.897941i	$-0.354938\pi$
$283$	14.8078	0.880230	0.440115	+	0.897941i	$0.354938\pi$
$284$	0	0
$285$	−9.36932	−0.554990
$286$	0	0
$287$	10.5616	0.623429
$288$	0	0
$289$	−10.4384	−0.614026
$290$	0	0
$291$	20.4924	1.20129
$292$	0	0
$293$	−16.8078	−0.981920	−0.490960	−	0.871182i	$-0.663354\pi$
$293$	−16.8078	−0.981920	−0.490960	+	0.871182i	$0.663354\pi$
$294$	0	0
$295$	−3.68466	−0.214529
$296$	0	0
$297$	11.1231	0.645428
$298$	0	0
$299$	−3.56155	−0.205970
$300$	0	0
$301$	0	0
$302$	0	0
$303$	27.6155	1.58647
$304$	0	0
$305$	−7.12311	−0.407868
$306$	0	0
$307$	−21.1231	−1.20556	−0.602780	−	0.797908i	$-0.705940\pi$
$307$	−21.1231	−1.20556	−0.602780	+	0.797908i	$0.705940\pi$
$308$	0	0
$309$	24.9848	1.42134
$310$	0	0
$311$	8.68466	0.492462	0.246231	−	0.969211i	$-0.420808\pi$
$311$	8.68466	0.492462	0.246231	+	0.969211i	$0.420808\pi$
$312$	0	0
$313$	−0.807764	−0.0456575	−0.0228288	−	0.999739i	$-0.507267\pi$
$313$	−0.807764	−0.0456575	−0.0228288	+	0.999739i	$0.507267\pi$
$314$	0	0
$315$	−1.43845	−0.0810473
$316$	0	0
$317$	15.1231	0.849398	0.424699	−	0.905335i	$-0.360380\pi$
$317$	15.1231	0.849398	0.424699	+	0.905335i	$0.360380\pi$
$318$	0	0
$319$	12.2462	0.685656
$320$	0	0
$321$	−5.36932	−0.299686
$322$	0	0
$323$	15.3693	0.855172
$324$	0	0
$325$	−3.56155	−0.197559
$326$	0	0
$327$	−24.0000	−1.32720
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	28.6155	1.57285	0.786426	−	0.617685i	$-0.211929\pi$
$331$	28.6155	1.57285	0.786426	+	0.617685i	$0.211929\pi$
$332$	0	0
$333$	2.56155	0.140372
$334$	0	0
$335$	−8.56155	−0.467768
$336$	0	0
$337$	13.6155	0.741685	0.370843	−	0.928696i	$-0.379069\pi$
$337$	13.6155	0.741685	0.370843	+	0.928696i	$0.379069\pi$
$338$	0	0
$339$	−20.0000	−1.08625
$340$	0	0
$341$	14.4924	0.784809
$342$	0	0
$343$	−19.0540	−1.02882
$344$	0	0
$345$	−1.56155	−0.0840712
$346$	0	0
$347$	16.4924	0.885360	0.442680	−	0.896680i	$-0.354028\pi$
$347$	16.4924	0.885360	0.442680	+	0.896680i	$0.354028\pi$
$348$	0	0
$349$	−26.3693	−1.41152	−0.705759	−	0.708452i	$-0.749394\pi$
$349$	−26.3693	−1.41152	−0.705759	+	0.708452i	$0.749394\pi$
$350$	0	0
$351$	−19.8078	−1.05726
$352$	0	0
$353$	−2.05398	−0.109322	−0.0546610	−	0.998505i	$-0.517408\pi$
$353$	−2.05398	−0.109322	−0.0546610	+	0.998505i	$0.517408\pi$
$354$	0	0
$355$	10.1231	0.537279
$356$	0	0
$357$	−10.2462	−0.542287
$358$	0	0
$359$	15.6155	0.824156	0.412078	−	0.911149i	$-0.364803\pi$
$359$	15.6155	0.824156	0.412078	+	0.911149i	$0.364803\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	10.9309	0.573722
$364$	0	0
$365$	−4.43845	−0.232319
$366$	0	0
$367$	−22.5616	−1.17770	−0.588852	−	0.808241i	$-0.700420\pi$
$367$	−22.5616	−1.17770	−0.588852	+	0.808241i	$0.700420\pi$
$368$	0	0
$369$	−2.31534	−0.120532
$370$	0	0
$371$	−11.6847	−0.606637
$372$	0	0
$373$	−20.2462	−1.04831	−0.524155	−	0.851623i	$-0.675619\pi$
$373$	−20.2462	−1.04831	−0.524155	+	0.851623i	$0.675619\pi$
$374$	0	0
$375$	−1.56155	−0.0806382
$376$	0	0
$377$	−21.8078	−1.12316
$378$	0	0
$379$	−36.4924	−1.87449	−0.937245	−	0.348672i	$-0.886633\pi$
$379$	−36.4924	−1.87449	−0.937245	+	0.348672i	$0.886633\pi$
$380$	0	0
$381$	−12.1922	−0.624627
$382$	0	0
$383$	3.19224	0.163116	0.0815578	−	0.996669i	$-0.474010\pi$
$383$	3.19224	0.163116	0.0815578	+	0.996669i	$0.474010\pi$
$384$	0	0
$385$	5.12311	0.261098
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.8769	−0.551480	−0.275740	−	0.961232i	$-0.588923\pi$
$389$	−10.8769	−0.551480	−0.275740	+	0.961232i	$0.588923\pi$
$390$	0	0
$391$	2.56155	0.129543
$392$	0	0
$393$	−0.300187	−0.0151424
$394$	0	0
$395$	4.87689	0.245383
$396$	0	0
$397$	34.5464	1.73383	0.866917	−	0.498453i	$-0.166098\pi$
$397$	34.5464	1.73383	0.866917	+	0.498453i	$0.166098\pi$
$398$	0	0
$399$	−24.0000	−1.20150
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	−25.8078	−1.28558
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	−9.12311	−0.452216
$408$	0	0
$409$	−35.9848	−1.77934	−0.889668	−	0.456608i	$-0.849064\pi$
$409$	−35.9848	−1.77934	−0.889668	+	0.456608i	$0.849064\pi$
$410$	0	0
$411$	−10.7386	−0.529698
$412$	0	0
$413$	−9.43845	−0.464436
$414$	0	0
$415$	−13.9309	−0.683839
$416$	0	0
$417$	17.5616	0.859993
$418$	0	0
$419$	−18.8769	−0.922197	−0.461098	−	0.887349i	$-0.652544\pi$
$419$	−18.8769	−0.922197	−0.461098	+	0.887349i	$0.652544\pi$
$420$	0	0
$421$	25.1231	1.22443	0.612213	−	0.790693i	$-0.290280\pi$
$421$	25.1231	1.22443	0.612213	+	0.790693i	$0.290280\pi$
$422$	0	0
$423$	−2.63068	−0.127908
$424$	0	0
$425$	2.56155	0.124254
$426$	0	0
$427$	−18.2462	−0.882996
$428$	0	0
$429$	11.1231	0.537029
$430$	0	0
$431$	−10.2462	−0.493543	−0.246771	−	0.969074i	$-0.579370\pi$
$431$	−10.2462	−0.493543	−0.246771	+	0.969074i	$0.579370\pi$
$432$	0	0
$433$	39.9309	1.91896	0.959478	−	0.281785i	$-0.0909265\pi$
$433$	39.9309	1.91896	0.959478	+	0.281785i	$0.0909265\pi$
$434$	0	0
$435$	−9.56155	−0.458441
$436$	0	0
$437$	6.00000	0.287019
$438$	0	0
$439$	19.8078	0.945373	0.472686	−	0.881231i	$-0.343284\pi$
$439$	19.8078	0.945373	0.472686	+	0.881231i	$0.343284\pi$
$440$	0	0
$441$	0.246211	0.0117243
$442$	0	0
$443$	−11.8078	−0.561004	−0.280502	−	0.959853i	$-0.590501\pi$
$443$	−11.8078	−0.561004	−0.280502	+	0.959853i	$0.590501\pi$
$444$	0	0
$445$	14.2462	0.675335
$446$	0	0
$447$	14.6307	0.692008
$448$	0	0
$449$	−21.6847	−1.02336	−0.511681	−	0.859175i	$-0.670977\pi$
$449$	−21.6847	−1.02336	−0.511681	+	0.859175i	$0.670977\pi$
$450$	0	0
$451$	8.24621	0.388299
$452$	0	0
$453$	21.9460	1.03111
$454$	0	0
$455$	−9.12311	−0.427698
$456$	0	0
$457$	9.43845	0.441512	0.220756	−	0.975329i	$-0.429148\pi$
$457$	9.43845	0.441512	0.220756	+	0.975329i	$0.429148\pi$
$458$	0	0
$459$	14.2462	0.664956
$460$	0	0
$461$	−40.9309	−1.90634	−0.953170	−	0.302434i	$-0.902201\pi$
$461$	−40.9309	−1.90634	−0.953170	+	0.302434i	$0.902201\pi$
$462$	0	0
$463$	−31.3693	−1.45786	−0.728928	−	0.684590i	$-0.759981\pi$
$463$	−31.3693	−1.45786	−0.728928	+	0.684590i	$0.759981\pi$
$464$	0	0
$465$	−11.3153	−0.524736
$466$	0	0
$467$	−22.3153	−1.03263	−0.516315	−	0.856398i	$-0.672697\pi$
$467$	−22.3153	−1.03263	−0.516315	+	0.856398i	$0.672697\pi$
$468$	0	0
$469$	−21.9309	−1.01267
$470$	0	0
$471$	6.73863	0.310500
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	2.56155	0.117285
$478$	0	0
$479$	−43.2311	−1.97528	−0.987639	−	0.156748i	$-0.949899\pi$
$479$	−43.2311	−1.97528	−0.987639	+	0.156748i	$0.949899\pi$
$480$	0	0
$481$	16.2462	0.740763
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	−13.1231	−0.595890
$486$	0	0
$487$	7.80776	0.353804	0.176902	−	0.984229i	$-0.443393\pi$
$487$	7.80776	0.353804	0.176902	+	0.984229i	$0.443393\pi$
$488$	0	0
$489$	−26.4384	−1.19559
$490$	0	0
$491$	11.4924	0.518646	0.259323	−	0.965791i	$-0.416501\pi$
$491$	11.4924	0.518646	0.259323	+	0.965791i	$0.416501\pi$
$492$	0	0
$493$	15.6847	0.706401
$494$	0	0
$495$	−1.12311	−0.0504798
$496$	0	0
$497$	25.9309	1.16316
$498$	0	0
$499$	22.6155	1.01241	0.506205	−	0.862413i	$-0.331048\pi$
$499$	22.6155	1.01241	0.506205	+	0.862413i	$0.331048\pi$
$500$	0	0
$501$	12.4924	0.558120
$502$	0	0
$503$	−3.93087	−0.175269	−0.0876344	−	0.996153i	$-0.527931\pi$
$503$	−3.93087	−0.175269	−0.0876344	+	0.996153i	$0.527931\pi$
$504$	0	0
$505$	−17.6847	−0.786957
$506$	0	0
$507$	0.492423	0.0218693
$508$	0	0
$509$	−27.1771	−1.20460	−0.602301	−	0.798269i	$-0.705749\pi$
$509$	−27.1771	−1.20460	−0.602301	+	0.798269i	$0.705749\pi$
$510$	0	0
$511$	−11.3693	−0.502949
$512$	0	0
$513$	33.3693	1.47329
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	9.36932	0.412062
$518$	0	0
$519$	−8.38447	−0.368037
$520$	0	0
$521$	−13.6155	−0.596507	−0.298254	−	0.954487i	$-0.596404\pi$
$521$	−13.6155	−0.596507	−0.298254	+	0.954487i	$0.596404\pi$
$522$	0	0
$523$	−14.7386	−0.644475	−0.322238	−	0.946659i	$-0.604435\pi$
$523$	−14.7386	−0.644475	−0.322238	+	0.946659i	$0.604435\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	18.5616	0.808554
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.06913	0.0897926
$532$	0	0
$533$	−14.6847	−0.636063
$534$	0	0
$535$	3.43845	0.148657
$536$	0	0
$537$	10.8229	0.467043
$538$	0	0
$539$	−0.876894	−0.0377705
$540$	0	0
$541$	−2.68466	−0.115422	−0.0577112	−	0.998333i	$-0.518380\pi$
$541$	−2.68466	−0.115422	−0.0577112	+	0.998333i	$0.518380\pi$
$542$	0	0
$543$	8.38447	0.359812
$544$	0	0
$545$	15.3693	0.658349
$546$	0	0
$547$	−21.1771	−0.905467	−0.452733	−	0.891646i	$-0.649551\pi$
$547$	−21.1771	−0.905467	−0.452733	+	0.891646i	$0.649551\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	36.7386	1.56512
$552$	0	0
$553$	12.4924	0.531232
$554$	0	0
$555$	7.12311	0.302359
$556$	0	0
$557$	−15.6847	−0.664580	−0.332290	−	0.943177i	$-0.607821\pi$
$557$	−15.6847	−0.664580	−0.332290	+	0.943177i	$0.607821\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	32.6695	1.37686	0.688428	−	0.725305i	$-0.258301\pi$
$563$	32.6695	1.37686	0.688428	+	0.725305i	$0.258301\pi$
$564$	0	0
$565$	12.8078	0.538827
$566$	0	0
$567$	−17.9309	−0.753026
$568$	0	0
$569$	16.7386	0.701720	0.350860	−	0.936428i	$-0.385889\pi$
$569$	16.7386	0.701720	0.350860	+	0.936428i	$0.385889\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	−27.1231	−1.13308
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−29.5616	−1.23066	−0.615332	−	0.788268i	$-0.710978\pi$
$577$	−29.5616	−1.23066	−0.615332	+	0.788268i	$0.710978\pi$
$578$	0	0
$579$	−3.80776	−0.158245
$580$	0	0
$581$	−35.6847	−1.48045
$582$	0	0
$583$	−9.12311	−0.377840
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	7.06913	0.291774	0.145887	−	0.989301i	$-0.453396\pi$
$587$	7.06913	0.291774	0.145887	+	0.989301i	$0.453396\pi$
$588$	0	0
$589$	43.4773	1.79145
$590$	0	0
$591$	29.1771	1.20018
$592$	0	0
$593$	15.6155	0.641253	0.320626	−	0.947206i	$-0.396107\pi$
$593$	15.6155	0.641253	0.320626	+	0.947206i	$0.396107\pi$
$594$	0	0
$595$	6.56155	0.268997
$596$	0	0
$597$	15.6155	0.639101
$598$	0	0
$599$	−32.9848	−1.34772	−0.673862	−	0.738857i	$-0.735366\pi$
$599$	−32.9848	−1.34772	−0.673862	+	0.738857i	$0.735366\pi$
$600$	0	0
$601$	36.3693	1.48354	0.741768	−	0.670657i	$-0.233988\pi$
$601$	36.3693	1.48354	0.741768	+	0.670657i	$0.233988\pi$
$602$	0	0
$603$	4.80776	0.195787
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	8.49242	0.344697	0.172348	−	0.985036i	$-0.444865\pi$
$607$	8.49242	0.344697	0.172348	+	0.985036i	$0.444865\pi$
$608$	0	0
$609$	−24.4924	−0.992483
$610$	0	0
$611$	−16.6847	−0.674989
$612$	0	0
$613$	−5.61553	−0.226809	−0.113405	−	0.993549i	$-0.536176\pi$
$613$	−5.61553	−0.226809	−0.113405	+	0.993549i	$0.536176\pi$
$614$	0	0
$615$	−6.43845	−0.259623
$616$	0	0
$617$	0.561553	0.0226073	0.0113036	−	0.999936i	$-0.496402\pi$
$617$	0.561553	0.0226073	0.0113036	+	0.999936i	$0.496402\pi$
$618$	0	0
$619$	12.4924	0.502113	0.251056	−	0.967972i	$-0.419222\pi$
$619$	12.4924	0.502113	0.251056	+	0.967972i	$0.419222\pi$
$620$	0	0
$621$	5.56155	0.223177
$622$	0	0
$623$	36.4924	1.46204
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−18.7386	−0.748349
$628$	0	0
$629$	−11.6847	−0.465898
$630$	0	0
$631$	30.2462	1.20408	0.602041	−	0.798465i	$-0.294354\pi$
$631$	30.2462	1.20408	0.602041	+	0.798465i	$0.294354\pi$
$632$	0	0
$633$	−0.492423	−0.0195720
$634$	0	0
$635$	7.80776	0.309842
$636$	0	0
$637$	1.56155	0.0618710
$638$	0	0
$639$	−5.68466	−0.224882
$640$	0	0
$641$	6.87689	0.271621	0.135810	−	0.990735i	$-0.456636\pi$
$641$	6.87689	0.271621	0.135810	+	0.990735i	$0.456636\pi$
$642$	0	0
$643$	20.4233	0.805416	0.402708	−	0.915328i	$-0.368069\pi$
$643$	20.4233	0.805416	0.402708	+	0.915328i	$0.368069\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.31534	0.208968	0.104484	−	0.994527i	$-0.466681\pi$
$647$	5.31534	0.208968	0.104484	+	0.994527i	$0.466681\pi$
$648$	0	0
$649$	−7.36932	−0.289271
$650$	0	0
$651$	−28.9848	−1.13601
$652$	0	0
$653$	39.6695	1.55239	0.776194	−	0.630494i	$-0.217148\pi$
$653$	39.6695	1.55239	0.776194	+	0.630494i	$0.217148\pi$
$654$	0	0
$655$	0.192236	0.00751128
$656$	0	0
$657$	2.49242	0.0972387
$658$	0	0
$659$	15.3693	0.598704	0.299352	−	0.954143i	$-0.403230\pi$
$659$	15.3693	0.598704	0.299352	+	0.954143i	$0.403230\pi$
$660$	0	0
$661$	−6.49242	−0.252526	−0.126263	−	0.991997i	$-0.540298\pi$
$661$	−6.49242	−0.252526	−0.126263	+	0.991997i	$0.540298\pi$
$662$	0	0
$663$	14.2462	0.553277
$664$	0	0
$665$	15.3693	0.595997
$666$	0	0
$667$	6.12311	0.237088
$668$	0	0
$669$	27.5076	1.06350
$670$	0	0
$671$	−14.2462	−0.549969
$672$	0	0
$673$	34.5464	1.33167	0.665833	−	0.746101i	$-0.268076\pi$
$673$	34.5464	1.33167	0.665833	+	0.746101i	$0.268076\pi$
$674$	0	0
$675$	5.56155	0.214064
$676$	0	0
$677$	26.8078	1.03031	0.515153	−	0.857098i	$-0.327735\pi$
$677$	26.8078	1.03031	0.515153	+	0.857098i	$0.327735\pi$
$678$	0	0
$679$	−33.6155	−1.29005
$680$	0	0
$681$	−40.9848	−1.57054
$682$	0	0
$683$	42.0540	1.60915	0.804575	−	0.593851i	$-0.202393\pi$
$683$	42.0540	1.60915	0.804575	+	0.593851i	$0.202393\pi$
$684$	0	0
$685$	6.87689	0.262753
$686$	0	0
$687$	41.7538	1.59301
$688$	0	0
$689$	16.2462	0.618931
$690$	0	0
$691$	16.4924	0.627401	0.313701	−	0.949522i	$-0.398431\pi$
$691$	16.4924	0.627401	0.313701	+	0.949522i	$0.398431\pi$
$692$	0	0
$693$	−2.87689	−0.109284
$694$	0	0
$695$	−11.2462	−0.426593
$696$	0	0
$697$	10.5616	0.400047
$698$	0	0
$699$	1.06913	0.0404382
$700$	0	0
$701$	−1.75379	−0.0662397	−0.0331198	−	0.999451i	$-0.510544\pi$
$701$	−1.75379	−0.0662397	−0.0331198	+	0.999451i	$0.510544\pi$
$702$	0	0
$703$	−27.3693	−1.03225
$704$	0	0
$705$	−7.31534	−0.275512
$706$	0	0
$707$	−45.3002	−1.70369
$708$	0	0
$709$	29.7538	1.11743	0.558713	−	0.829361i	$-0.311295\pi$
$709$	29.7538	1.11743	0.558713	+	0.829361i	$0.311295\pi$
$710$	0	0
$711$	−2.73863	−0.102707
$712$	0	0
$713$	7.24621	0.271373
$714$	0	0
$715$	−7.12311	−0.266389
$716$	0	0
$717$	−4.30019	−0.160593
$718$	0	0
$719$	19.0540	0.710593	0.355297	−	0.934754i	$-0.384380\pi$
$719$	19.0540	0.710593	0.355297	+	0.934754i	$0.384380\pi$
$720$	0	0
$721$	−40.9848	−1.52636
$722$	0	0
$723$	−9.36932	−0.348449
$724$	0	0
$725$	6.12311	0.227406
$726$	0	0
$727$	−21.1922	−0.785977	−0.392988	−	0.919543i	$-0.628559\pi$
$727$	−21.1922	−0.785977	−0.392988	+	0.919543i	$0.628559\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0.315342	0.0116474	0.00582370	−	0.999983i	$-0.498146\pi$
$733$	0.315342	0.0116474	0.00582370	+	0.999983i	$0.498146\pi$
$734$	0	0
$735$	0.684658	0.0252540
$736$	0	0
$737$	−17.1231	−0.630738
$738$	0	0
$739$	0.615528	0.0226426	0.0113213	−	0.999936i	$-0.496396\pi$
$739$	0.615528	0.0226426	0.0113213	+	0.999936i	$0.496396\pi$
$740$	0	0
$741$	33.3693	1.22585
$742$	0	0
$743$	3.50758	0.128681	0.0643403	−	0.997928i	$-0.479506\pi$
$743$	3.50758	0.128681	0.0643403	+	0.997928i	$0.479506\pi$
$744$	0	0
$745$	−9.36932	−0.343265
$746$	0	0
$747$	7.82292	0.286226
$748$	0	0
$749$	8.80776	0.321829
$750$	0	0
$751$	−13.1231	−0.478869	−0.239434	−	0.970913i	$-0.576962\pi$
$751$	−13.1231	−0.478869	−0.239434	+	0.970913i	$0.576962\pi$
$752$	0	0
$753$	11.5076	0.419359
$754$	0	0
$755$	−14.0540	−0.511477
$756$	0	0
$757$	12.5616	0.456557	0.228279	−	0.973596i	$-0.426690\pi$
$757$	12.5616	0.456557	0.228279	+	0.973596i	$0.426690\pi$
$758$	0	0
$759$	−3.12311	−0.113362
$760$	0	0
$761$	−43.9848	−1.59445	−0.797225	−	0.603683i	$-0.793699\pi$
$761$	−43.9848	−1.59445	−0.797225	+	0.603683i	$0.793699\pi$
$762$	0	0
$763$	39.3693	1.42526
$764$	0	0
$765$	−1.43845	−0.0520072
$766$	0	0
$767$	13.1231	0.473848
$768$	0	0
$769$	−10.6307	−0.383352	−0.191676	−	0.981458i	$-0.561392\pi$
$769$	−10.6307	−0.383352	−0.191676	+	0.981458i	$0.561392\pi$
$770$	0	0
$771$	−49.6695	−1.78880
$772$	0	0
$773$	−41.1231	−1.47910	−0.739548	−	0.673104i	$-0.764961\pi$
$773$	−41.1231	−1.47910	−0.739548	+	0.673104i	$0.764961\pi$
$774$	0	0
$775$	7.24621	0.260292
$776$	0	0
$777$	18.2462	0.654579
$778$	0	0
$779$	24.7386	0.886354
$780$	0	0
$781$	20.2462	0.724466
$782$	0	0
$783$	34.0540	1.21699
$784$	0	0
$785$	−4.31534	−0.154021
$786$	0	0
$787$	−39.6847	−1.41461	−0.707303	−	0.706911i	$-0.750088\pi$
$787$	−39.6847	−1.41461	−0.707303	+	0.706911i	$0.750088\pi$
$788$	0	0
$789$	−0.107951	−0.00384314
$790$	0	0
$791$	32.8078	1.16651
$792$	0	0
$793$	25.3693	0.900891
$794$	0	0
$795$	7.12311	0.252631
$796$	0	0
$797$	−52.4233	−1.85693	−0.928464	−	0.371422i	$-0.878870\pi$
$797$	−52.4233	−1.85693	−0.928464	+	0.371422i	$0.878870\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	−8.00000	−0.282666
$802$	0	0
$803$	−8.87689	−0.313259
$804$	0	0
$805$	2.56155	0.0902829
$806$	0	0
$807$	−36.3002	−1.27783
$808$	0	0
$809$	−54.1771	−1.90476	−0.952382	−	0.304906i	$-0.901375\pi$
$809$	−54.1771	−1.90476	−0.952382	+	0.304906i	$0.901375\pi$
$810$	0	0
$811$	47.2462	1.65904	0.829519	−	0.558478i	$-0.188614\pi$
$811$	47.2462	1.65904	0.829519	+	0.558478i	$0.188614\pi$
$812$	0	0
$813$	−34.2462	−1.20107
$814$	0	0
$815$	16.9309	0.593062
$816$	0	0
$817$	0	0
$818$	0	0
$819$	5.12311	0.179016
$820$	0	0
$821$	−34.4924	−1.20379	−0.601897	−	0.798574i	$-0.705588\pi$
$821$	−34.4924	−1.20379	−0.601897	+	0.798574i	$0.705588\pi$
$822$	0	0
$823$	−38.0540	−1.32648	−0.663239	−	0.748408i	$-0.730819\pi$
$823$	−38.0540	−1.32648	−0.663239	+	0.748408i	$0.730819\pi$
$824$	0	0
$825$	−3.12311	−0.108733
$826$	0	0
$827$	19.6847	0.684503	0.342251	−	0.939608i	$-0.388811\pi$
$827$	19.6847	0.684503	0.342251	+	0.939608i	$0.388811\pi$
$828$	0	0
$829$	−29.5464	−1.02619	−0.513094	−	0.858332i	$-0.671501\pi$
$829$	−29.5464	−1.02619	−0.513094	+	0.858332i	$0.671501\pi$
$830$	0	0
$831$	10.4384	0.362106
$832$	0	0
$833$	−1.12311	−0.0389133
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	40.3002	1.39298
$838$	0	0
$839$	−42.8769	−1.48027	−0.740137	−	0.672456i	$-0.765240\pi$
$839$	−42.8769	−1.48027	−0.740137	+	0.672456i	$0.765240\pi$
$840$	0	0
$841$	8.49242	0.292842
$842$	0	0
$843$	10.7386	0.369858
$844$	0	0
$845$	−0.315342	−0.0108481
$846$	0	0
$847$	−17.9309	−0.616112
$848$	0	0
$849$	−23.1231	−0.793583
$850$	0	0
$851$	−4.56155	−0.156368
$852$	0	0
$853$	−20.2462	−0.693217	−0.346609	−	0.938010i	$-0.612667\pi$
$853$	−20.2462	−0.693217	−0.346609	+	0.938010i	$0.612667\pi$
$854$	0	0
$855$	−3.36932	−0.115228
$856$	0	0
$857$	−33.3153	−1.13803	−0.569015	−	0.822327i	$-0.692675\pi$
$857$	−33.3153	−1.13803	−0.569015	+	0.822327i	$0.692675\pi$
$858$	0	0
$859$	−14.5076	−0.494992	−0.247496	−	0.968889i	$-0.579608\pi$
$859$	−14.5076	−0.494992	−0.247496	+	0.968889i	$0.579608\pi$
$860$	0	0
$861$	−16.4924	−0.562060
$862$	0	0
$863$	−7.56155	−0.257398	−0.128699	−	0.991684i	$-0.541080\pi$
$863$	−7.56155	−0.257398	−0.128699	+	0.991684i	$0.541080\pi$
$864$	0	0
$865$	5.36932	0.182562
$866$	0	0
$867$	16.3002	0.553583
$868$	0	0
$869$	9.75379	0.330875
$870$	0	0
$871$	30.4924	1.03320
$872$	0	0
$873$	7.36932	0.249414
$874$	0	0
$875$	2.56155	0.0865963
$876$	0	0
$877$	−18.9848	−0.641073	−0.320536	−	0.947236i	$-0.603863\pi$
$877$	−18.9848	−0.641073	−0.320536	+	0.947236i	$0.603863\pi$
$878$	0	0
$879$	26.2462	0.885263
$880$	0	0
$881$	41.4773	1.39740	0.698702	−	0.715413i	$-0.253761\pi$
$881$	41.4773	1.39740	0.698702	+	0.715413i	$0.253761\pi$
$882$	0	0
$883$	10.7386	0.361384	0.180692	−	0.983540i	$-0.442166\pi$
$883$	10.7386	0.361384	0.180692	+	0.983540i	$0.442166\pi$
$884$	0	0
$885$	5.75379	0.193411
$886$	0	0
$887$	42.7926	1.43684	0.718418	−	0.695612i	$-0.244867\pi$
$887$	42.7926	1.43684	0.718418	+	0.695612i	$0.244867\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	−14.0000	−0.469018
$892$	0	0
$893$	28.1080	0.940597
$894$	0	0
$895$	−6.93087	−0.231673
$896$	0	0
$897$	5.56155	0.185695
$898$	0	0
$899$	44.3693	1.47980
$900$	0	0
$901$	−11.6847	−0.389272
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.36932	−0.178482
$906$	0	0
$907$	21.6847	0.720027	0.360014	−	0.932947i	$-0.382772\pi$
$907$	21.6847	0.720027	0.360014	+	0.932947i	$0.382772\pi$
$908$	0	0
$909$	9.93087	0.329386
$910$	0	0
$911$	−3.12311	−0.103473	−0.0517366	−	0.998661i	$-0.516476\pi$
$911$	−3.12311	−0.103473	−0.0517366	+	0.998661i	$0.516476\pi$
$912$	0	0
$913$	−27.8617	−0.922089
$914$	0	0
$915$	11.1231	0.367719
$916$	0	0
$917$	0.492423	0.0162612
$918$	0	0
$919$	46.7386	1.54177	0.770883	−	0.636977i	$-0.219815\pi$
$919$	46.7386	1.54177	0.770883	+	0.636977i	$0.219815\pi$
$920$	0	0
$921$	32.9848	1.08689
$922$	0	0
$923$	−36.0540	−1.18673
$924$	0	0
$925$	−4.56155	−0.149983
$926$	0	0
$927$	8.98485	0.295101
$928$	0	0
$929$	−56.7235	−1.86104	−0.930518	−	0.366245i	$-0.880643\pi$
$929$	−56.7235	−1.86104	−0.930518	+	0.366245i	$0.880643\pi$
$930$	0	0
$931$	−2.63068	−0.0862172
$932$	0	0
$933$	−13.5616	−0.443985
$934$	0	0
$935$	5.12311	0.167543
$936$	0	0
$937$	−16.2462	−0.530741	−0.265370	−	0.964147i	$-0.585494\pi$
$937$	−16.2462	−0.530741	−0.265370	+	0.964147i	$0.585494\pi$
$938$	0	0
$939$	1.26137	0.0411631
$940$	0	0
$941$	−37.7538	−1.23074	−0.615369	−	0.788239i	$-0.710993\pi$
$941$	−37.7538	−1.23074	−0.615369	+	0.788239i	$0.710993\pi$
$942$	0	0
$943$	4.12311	0.134267
$944$	0	0
$945$	14.2462	0.463429
$946$	0	0
$947$	56.6847	1.84200	0.921002	−	0.389558i	$-0.127372\pi$
$947$	56.6847	1.84200	0.921002	+	0.389558i	$0.127372\pi$
$948$	0	0
$949$	15.8078	0.513142
$950$	0	0
$951$	−23.6155	−0.765786
$952$	0	0
$953$	−38.4924	−1.24689	−0.623446	−	0.781866i	$-0.714268\pi$
$953$	−38.4924	−1.24689	−0.623446	+	0.781866i	$0.714268\pi$
$954$	0	0
$955$	17.3693	0.562058
$956$	0	0
$957$	−19.1231	−0.618162
$958$	0	0
$959$	17.6155	0.568835
$960$	0	0
$961$	21.5076	0.693793
$962$	0	0
$963$	−1.93087	−0.0622214
$964$	0	0
$965$	2.43845	0.0784964
$966$	0	0
$967$	−56.6847	−1.82286	−0.911428	−	0.411460i	$-0.865019\pi$
$967$	−56.6847	−1.82286	−0.911428	+	0.411460i	$0.865019\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−22.6307	−0.726253	−0.363127	−	0.931740i	$-0.618291\pi$
$971$	−22.6307	−0.726253	−0.363127	+	0.931740i	$0.618291\pi$
$972$	0	0
$973$	−28.8078	−0.923535
$974$	0	0
$975$	5.56155	0.178112
$976$	0	0
$977$	59.7926	1.91294	0.956468	−	0.291839i	$-0.0942671\pi$
$977$	59.7926	1.91294	0.956468	+	0.291839i	$0.0942671\pi$
$978$	0	0
$979$	28.4924	0.910622
$980$	0	0
$981$	−8.63068	−0.275557
$982$	0	0
$983$	−37.0540	−1.18184	−0.590919	−	0.806731i	$-0.701235\pi$
$983$	−37.0540	−1.18184	−0.590919	+	0.806731i	$0.701235\pi$
$984$	0	0
$985$	−18.6847	−0.595343
$986$	0	0
$987$	−18.7386	−0.596457
$988$	0	0
$989$	0	0
$990$	0	0
$991$	53.3002	1.69314	0.846568	−	0.532280i	$-0.178665\pi$
$991$	53.3002	1.69314	0.846568	+	0.532280i	$0.178665\pi$
$992$	0	0
$993$	−44.6847	−1.41802
$994$	0	0
$995$	−10.0000	−0.317021
$996$	0	0
$997$	55.6155	1.76136	0.880681	−	0.473710i	$-0.157086\pi$
$997$	55.6155	1.76136	0.880681	+	0.473710i	$0.157086\pi$
$998$	0	0
$999$	−25.3693	−0.802650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.a.e.1.1	✓	2
3.2	odd	2		4140.2.a.m.1.2		2
4.3	odd	2		1840.2.a.m.1.2		2
5.2	odd	4		2300.2.c.h.1749.3		4
5.3	odd	4		2300.2.c.h.1749.2		4
5.4	even	2		2300.2.a.i.1.2		2
8.3	odd	2		7360.2.a.bo.1.1		2
8.5	even	2		7360.2.a.bi.1.2		2
20.19	odd	2		9200.2.a.bv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.a.e.1.1	✓	2	1.1	even	1	trivial
1840.2.a.m.1.2		2	4.3	odd	2
2300.2.a.i.1.2		2	5.4	even	2
2300.2.c.h.1749.2		4	5.3	odd	4
2300.2.c.h.1749.3		4	5.2	odd	4
4140.2.a.m.1.2		2	3.2	odd	2
7360.2.a.bi.1.2		2	8.5	even	2
7360.2.a.bo.1.1		2	8.3	odd	2
9200.2.a.bv.1.1		2	20.19	odd	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:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q-1.56155 q^{3} +1.00000 q^{5} +2.56155 q^{7} -0.561553 q^{9} +2.00000 q^{11} -3.56155 q^{13} -1.56155 q^{15} +2.56155 q^{17} +6.00000 q^{19} -4.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.56155 q^{27} +6.12311 q^{29} +7.24621 q^{31} -3.12311 q^{33} +2.56155 q^{35} -4.56155 q^{37} +5.56155 q^{39} +4.12311 q^{41} -0.561553 q^{45} +4.68466 q^{47} -0.438447 q^{49} -4.00000 q^{51} -4.56155 q^{53} +2.00000 q^{55} -9.36932 q^{57} -3.68466 q^{59} -7.12311 q^{61} -1.43845 q^{63} -3.56155 q^{65} -8.56155 q^{67} -1.56155 q^{69} +10.1231 q^{71} -4.43845 q^{73} -1.56155 q^{75} +5.12311 q^{77} +4.87689 q^{79} -7.00000 q^{81} -13.9309 q^{83} +2.56155 q^{85} -9.56155 q^{87} +14.2462 q^{89} -9.12311 q^{91} -11.3153 q^{93} +6.00000 q^{95} -13.1231 q^{97} -1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9	\( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} + 4 q^{11} - 3 q^{13} + q^{15} + q^{17} + 12 q^{19} - 8 q^{21} + 2 q^{23} + 2 q^{25} + 7 q^{27} + 4 q^{29} - 2 q^{31} + 2 q^{33} + q^{35} - 5 q^{37} + 7 q^{39} + 3 q^{45} - 3 q^{47} - 5 q^{49} - 8 q^{51} - 5 q^{53} + 4 q^{55} + 6 q^{57} + 5 q^{59} - 6 q^{61} - 7 q^{63} - 3 q^{65} - 13 q^{67} + q^{69} + 12 q^{71} - 13 q^{73} + q^{75} + 2 q^{77} + 18 q^{79} - 14 q^{81} + q^{83} + q^{85} - 15 q^{87} + 12 q^{89} - 10 q^{91} - 35 q^{93} + 12 q^{95} - 18 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 + 4 * q^11 - 3 * q^13 + q^15 + q^17 + 12 * q^19 - 8 * q^21 + 2 * q^23 + 2 * q^25 + 7 * q^27 + 4 * q^29 - 2 * q^31 + 2 * q^33 + q^35 - 5 * q^37 + 7 * q^39 + 3 * q^45 - 3 * q^47 - 5 * q^49 - 8 * q^51 - 5 * q^53 + 4 * q^55 + 6 * q^57 + 5 * q^59 - 6 * q^61 - 7 * q^63 - 3 * q^65 - 13 * q^67 + q^69 + 12 * q^71 - 13 * q^73 + q^75 + 2 * q^77 + 18 * q^79 - 14 * q^81 + q^83 + q^85 - 15 * q^87 + 12 * q^89 - 10 * q^91 - 35 * q^93 + 12 * q^95 - 18 * q^97 + 6 * q^99

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