LMFDB - Embedded newform 560.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,2,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, 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("560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	560.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:	$4.47162251319$
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:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} +1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} +O(q^{10})$	$q-2.56155 q^{3} +1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} +2.56155 q^{11} -5.68466 q^{13} -2.56155 q^{15} +3.43845 q^{17} -1.12311 q^{19} +2.56155 q^{21} +5.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} +4.56155 q^{29} +10.2462 q^{31} -6.56155 q^{33} -1.00000 q^{35} +8.24621 q^{37} +14.5616 q^{39} +7.12311 q^{41} -1.12311 q^{43} +3.56155 q^{45} -6.56155 q^{47} +1.00000 q^{49} -8.80776 q^{51} -4.87689 q^{53} +2.56155 q^{55} +2.87689 q^{57} +4.00000 q^{59} -15.1231 q^{61} -3.56155 q^{63} -5.68466 q^{65} +14.2462 q^{67} -13.1231 q^{69} +12.2462 q^{73} -2.56155 q^{75} -2.56155 q^{77} +11.6847 q^{79} -7.00000 q^{81} -12.0000 q^{83} +3.43845 q^{85} -11.6847 q^{87} -3.12311 q^{89} +5.68466 q^{91} -26.2462 q^{93} -1.12311 q^{95} +13.6847 q^{97} +9.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9	$ 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + q^{11} + q^{13} - q^{15} + 11 q^{17} + 6 q^{19} + q^{21} + 2 q^{23} + 2 q^{25} - 7 q^{27} + 5 q^{29} + 4 q^{31} - 9 q^{33} - 2 q^{35} + 25 q^{39} + 6 q^{41} + 6 q^{43} + 3 q^{45} - 9 q^{47} + 2 q^{49} + 3 q^{51} - 18 q^{53} + q^{55} + 14 q^{57} + 8 q^{59} - 22 q^{61} - 3 q^{63} + q^{65} + 12 q^{67} - 18 q^{69} + 8 q^{73} - q^{75} - q^{77} + 11 q^{79} - 14 q^{81} - 24 q^{83} + 11 q^{85} - 11 q^{87} + 2 q^{89} - q^{91} - 36 q^{93} + 6 q^{95} + 15 q^{97} + 10 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9 + q^11 + q^13 - q^15 + 11 * q^17 + 6 * q^19 + q^21 + 2 * q^23 + 2 * q^25 - 7 * q^27 + 5 * q^29 + 4 * q^31 - 9 * q^33 - 2 * q^35 + 25 * q^39 + 6 * q^41 + 6 * q^43 + 3 * q^45 - 9 * q^47 + 2 * q^49 + 3 * q^51 - 18 * q^53 + q^55 + 14 * q^57 + 8 * q^59 - 22 * q^61 - 3 * q^63 + q^65 + 12 * q^67 - 18 * q^69 + 8 * q^73 - q^75 - q^77 + 11 * q^79 - 14 * q^81 - 24 * q^83 + 11 * q^85 - 11 * q^87 + 2 * q^89 - q^91 - 36 * q^93 + 6 * q^95 + 15 * q^97 + 10 * 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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	2.56155	0.772337	0.386169	−	0.922428i	$-0.373798\pi$
$11$	2.56155	0.772337	0.386169	+	0.922428i	$0.373798\pi$
$12$	0	0
$13$	−5.68466	−1.57664	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	−5.68466	−1.57664	−0.788320	+	0.615265i	$0.789049\pi$
$14$	0	0
$15$	−2.56155	−0.661390
$16$	0	0
$17$	3.43845	0.833946	0.416973	−	0.908919i	$-0.363091\pi$
$17$	3.43845	0.833946	0.416973	+	0.908919i	$0.363091\pi$
$18$	0	0
$19$	−1.12311	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$20$	0	0
$21$	2.56155	0.558977
$22$	0	0
$23$	5.12311	1.06824	0.534121	−	0.845408i	$-0.320643\pi$
$23$	5.12311	1.06824	0.534121	+	0.845408i	$0.320643\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	4.56155	0.847059	0.423530	−	0.905882i	$-0.360791\pi$
$29$	4.56155	0.847059	0.423530	+	0.905882i	$0.360791\pi$
$30$	0	0
$31$	10.2462	1.84027	0.920137	−	0.391597i	$-0.128077\pi$
$31$	10.2462	1.84027	0.920137	+	0.391597i	$0.128077\pi$
$32$	0	0
$33$	−6.56155	−1.14222
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	8.24621	1.35567	0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621	1.35567	0.677834	+	0.735215i	$0.262919\pi$
$38$	0	0
$39$	14.5616	2.33171
$40$	0	0
$41$	7.12311	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$41$	7.12311	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$42$	0	0
$43$	−1.12311	−0.171272	−0.0856360	−	0.996326i	$-0.527292\pi$
$43$	−1.12311	−0.171272	−0.0856360	+	0.996326i	$0.527292\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	−6.56155	−0.957101	−0.478550	−	0.878060i	$-0.658838\pi$
$47$	−6.56155	−0.957101	−0.478550	+	0.878060i	$0.658838\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−8.80776	−1.23333
$52$	0	0
$53$	−4.87689	−0.669893	−0.334946	−	0.942237i	$-0.608718\pi$
$53$	−4.87689	−0.669893	−0.334946	+	0.942237i	$0.608718\pi$
$54$	0	0
$55$	2.56155	0.345400
$56$	0	0
$57$	2.87689	0.381054
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−15.1231	−1.93632	−0.968158	−	0.250341i	$-0.919457\pi$
$61$	−15.1231	−1.93632	−0.968158	+	0.250341i	$0.919457\pi$
$62$	0	0
$63$	−3.56155	−0.448713
$64$	0	0
$65$	−5.68466	−0.705095
$66$	0	0
$67$	14.2462	1.74045	0.870226	−	0.492653i	$-0.163973\pi$
$67$	14.2462	1.74045	0.870226	+	0.492653i	$0.163973\pi$
$68$	0	0
$69$	−13.1231	−1.57984
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	12.2462	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$73$	12.2462	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$74$	0	0
$75$	−2.56155	−0.295783
$76$	0	0
$77$	−2.56155	−0.291916
$78$	0	0
$79$	11.6847	1.31463	0.657313	−	0.753617i	$-0.271693\pi$
$79$	11.6847	1.31463	0.657313	+	0.753617i	$0.271693\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	3.43845	0.372952
$86$	0	0
$87$	−11.6847	−1.25273
$88$	0	0
$89$	−3.12311	−0.331049	−0.165524	−	0.986206i	$-0.552932\pi$
$89$	−3.12311	−0.331049	−0.165524	+	0.986206i	$0.552932\pi$
$90$	0	0
$91$	5.68466	0.595914
$92$	0	0
$93$	−26.2462	−2.72161
$94$	0	0
$95$	−1.12311	−0.115228
$96$	0	0
$97$	13.6847	1.38947	0.694733	−	0.719267i	$-0.255522\pi$
$97$	13.6847	1.38947	0.694733	+	0.719267i	$0.255522\pi$
$98$	0	0
$99$	9.12311	0.916907
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−8.80776	−0.867855	−0.433927	−	0.900948i	$-0.642873\pi$
$103$	−8.80776	−0.867855	−0.433927	+	0.900948i	$0.642873\pi$
$104$	0	0
$105$	2.56155	0.249982
$106$	0	0
$107$	−6.87689	−0.664814	−0.332407	−	0.943136i	$-0.607861\pi$
$107$	−6.87689	−0.664814	−0.332407	+	0.943136i	$0.607861\pi$
$108$	0	0
$109$	−8.56155	−0.820048	−0.410024	−	0.912075i	$-0.634480\pi$
$109$	−8.56155	−0.820048	−0.410024	+	0.912075i	$0.634480\pi$
$110$	0	0
$111$	−21.1231	−2.00492
$112$	0	0
$113$	−8.24621	−0.775738	−0.387869	−	0.921714i	$-0.626789\pi$
$113$	−8.24621	−0.775738	−0.387869	+	0.921714i	$0.626789\pi$
$114$	0	0
$115$	5.12311	0.477732
$116$	0	0
$117$	−20.2462	−1.87176
$118$	0	0
$119$	−3.43845	−0.315202
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	−18.2462	−1.64521
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.2462	−0.909204	−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462	−0.909204	−0.454602	+	0.890695i	$0.650219\pi$
$128$	0	0
$129$	2.87689	0.253296
$130$	0	0
$131$	−6.87689	−0.600837	−0.300419	−	0.953807i	$-0.597126\pi$
$131$	−6.87689	−0.600837	−0.300419	+	0.953807i	$0.597126\pi$
$132$	0	0
$133$	1.12311	0.0973856
$134$	0	0
$135$	−1.43845	−0.123802
$136$	0	0
$137$	17.3693	1.48396	0.741980	−	0.670422i	$-0.233887\pi$
$137$	17.3693	1.48396	0.741980	+	0.670422i	$0.233887\pi$
$138$	0	0
$139$	−3.36932	−0.285782	−0.142891	−	0.989738i	$-0.545640\pi$
$139$	−3.36932	−0.285782	−0.142891	+	0.989738i	$0.545640\pi$
$140$	0	0
$141$	16.8078	1.41547
$142$	0	0
$143$	−14.5616	−1.21770
$144$	0	0
$145$	4.56155	0.378816
$146$	0	0
$147$	−2.56155	−0.211273
$148$	0	0
$149$	16.2462	1.33094	0.665471	−	0.746424i	$-0.268231\pi$
$149$	16.2462	1.33094	0.665471	+	0.746424i	$0.268231\pi$
$150$	0	0
$151$	−0.807764	−0.0657349	−0.0328675	−	0.999460i	$-0.510464\pi$
$151$	−0.807764	−0.0657349	−0.0328675	+	0.999460i	$0.510464\pi$
$152$	0	0
$153$	12.2462	0.990048
$154$	0	0
$155$	10.2462	0.822995
$156$	0	0
$157$	−12.2462	−0.977354	−0.488677	−	0.872465i	$-0.662520\pi$
$157$	−12.2462	−0.977354	−0.488677	+	0.872465i	$0.662520\pi$
$158$	0	0
$159$	12.4924	0.990714
$160$	0	0
$161$	−5.12311	−0.403757
$162$	0	0
$163$	21.6155	1.69306	0.846529	−	0.532342i	$-0.178688\pi$
$163$	21.6155	1.69306	0.846529	+	0.532342i	$0.178688\pi$
$164$	0	0
$165$	−6.56155	−0.510816
$166$	0	0
$167$	−24.8078	−1.91968	−0.959841	−	0.280544i	$-0.909485\pi$
$167$	−24.8078	−1.91968	−0.959841	+	0.280544i	$0.909485\pi$
$168$	0	0
$169$	19.3153	1.48580
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	22.1771	1.68609	0.843046	−	0.537841i	$-0.180760\pi$
$173$	22.1771	1.68609	0.843046	+	0.537841i	$0.180760\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−10.2462	−0.770152
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	3.12311	0.232139	0.116069	−	0.993241i	$-0.462971\pi$
$181$	3.12311	0.232139	0.116069	+	0.993241i	$0.462971\pi$
$182$	0	0
$183$	38.7386	2.86364
$184$	0	0
$185$	8.24621	0.606274
$186$	0	0
$187$	8.80776	0.644087
$188$	0	0
$189$	1.43845	0.104632
$190$	0	0
$191$	−11.6847	−0.845472	−0.422736	−	0.906253i	$-0.638930\pi$
$191$	−11.6847	−0.845472	−0.422736	+	0.906253i	$0.638930\pi$
$192$	0	0
$193$	4.87689	0.351047	0.175523	−	0.984475i	$-0.443838\pi$
$193$	4.87689	0.351047	0.175523	+	0.984475i	$0.443838\pi$
$194$	0	0
$195$	14.5616	1.04277
$196$	0	0
$197$	−4.87689	−0.347464	−0.173732	−	0.984793i	$-0.555583\pi$
$197$	−4.87689	−0.347464	−0.173732	+	0.984793i	$0.555583\pi$
$198$	0	0
$199$	−2.24621	−0.159230	−0.0796148	−	0.996826i	$-0.525369\pi$
$199$	−2.24621	−0.159230	−0.0796148	+	0.996826i	$0.525369\pi$
$200$	0	0
$201$	−36.4924	−2.57398
$202$	0	0
$203$	−4.56155	−0.320158
$204$	0	0
$205$	7.12311	0.497499
$206$	0	0
$207$	18.2462	1.26820
$208$	0	0
$209$	−2.87689	−0.198999
$210$	0	0
$211$	13.4384	0.925141	0.462570	−	0.886583i	$-0.346927\pi$
$211$	13.4384	0.925141	0.462570	+	0.886583i	$0.346927\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.12311	−0.0765952
$216$	0	0
$217$	−10.2462	−0.695558
$218$	0	0
$219$	−31.3693	−2.11974
$220$	0	0
$221$	−19.5464	−1.31483
$222$	0	0
$223$	−12.3153	−0.824696	−0.412348	−	0.911026i	$-0.635291\pi$
$223$	−12.3153	−0.824696	−0.412348	+	0.911026i	$0.635291\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	23.6847	1.57201	0.786003	−	0.618223i	$-0.212147\pi$
$227$	23.6847	1.57201	0.786003	+	0.618223i	$0.212147\pi$
$228$	0	0
$229$	13.3693	0.883469	0.441735	−	0.897146i	$-0.354363\pi$
$229$	13.3693	0.883469	0.441735	+	0.897146i	$0.354363\pi$
$230$	0	0
$231$	6.56155	0.431718
$232$	0	0
$233$	−19.1231	−1.25280	−0.626398	−	0.779503i	$-0.715472\pi$
$233$	−19.1231	−1.25280	−0.626398	+	0.779503i	$0.715472\pi$
$234$	0	0
$235$	−6.56155	−0.428029
$236$	0	0
$237$	−29.9309	−1.94422
$238$	0	0
$239$	−19.0540	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$239$	−19.0540	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	6.38447	0.406234
$248$	0	0
$249$	30.7386	1.94798
$250$	0	0
$251$	11.3693	0.717625	0.358812	−	0.933410i	$-0.383182\pi$
$251$	11.3693	0.717625	0.358812	+	0.933410i	$0.383182\pi$
$252$	0	0
$253$	13.1231	0.825043
$254$	0	0
$255$	−8.80776	−0.551564
$256$	0	0
$257$	14.4924	0.904012	0.452006	−	0.892015i	$-0.350708\pi$
$257$	14.4924	0.904012	0.452006	+	0.892015i	$0.350708\pi$
$258$	0	0
$259$	−8.24621	−0.512395
$260$	0	0
$261$	16.2462	1.00562
$262$	0	0
$263$	15.3693	0.947713	0.473856	−	0.880602i	$-0.342862\pi$
$263$	15.3693	0.947713	0.473856	+	0.880602i	$0.342862\pi$
$264$	0	0
$265$	−4.87689	−0.299585
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	−12.2462	−0.746665	−0.373332	−	0.927698i	$-0.621785\pi$
$269$	−12.2462	−0.746665	−0.373332	+	0.927698i	$0.621785\pi$
$270$	0	0
$271$	−10.2462	−0.622413	−0.311207	−	0.950342i	$-0.600733\pi$
$271$	−10.2462	−0.622413	−0.311207	+	0.950342i	$0.600733\pi$
$272$	0	0
$273$	−14.5616	−0.881305
$274$	0	0
$275$	2.56155	0.154467
$276$	0	0
$277$	8.24621	0.495467	0.247733	−	0.968828i	$-0.420314\pi$
$277$	8.24621	0.495467	0.247733	+	0.968828i	$0.420314\pi$
$278$	0	0
$279$	36.4924	2.18474
$280$	0	0
$281$	−15.4384	−0.920981	−0.460490	−	0.887665i	$-0.652326\pi$
$281$	−15.4384	−0.920981	−0.460490	+	0.887665i	$0.652326\pi$
$282$	0	0
$283$	−7.68466	−0.456806	−0.228403	−	0.973567i	$-0.573350\pi$
$283$	−7.68466	−0.456806	−0.228403	+	0.973567i	$0.573350\pi$
$284$	0	0
$285$	2.87689	0.170413
$286$	0	0
$287$	−7.12311	−0.420464
$288$	0	0
$289$	−5.17708	−0.304534
$290$	0	0
$291$	−35.0540	−2.05490
$292$	0	0
$293$	−5.05398	−0.295256	−0.147628	−	0.989043i	$-0.547164\pi$
$293$	−5.05398	−0.295256	−0.147628	+	0.989043i	$0.547164\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−3.68466	−0.213806
$298$	0	0
$299$	−29.1231	−1.68423
$300$	0	0
$301$	1.12311	0.0647347
$302$	0	0
$303$	−15.3693	−0.882944
$304$	0	0
$305$	−15.1231	−0.865947
$306$	0	0
$307$	16.3153	0.931166	0.465583	−	0.885004i	$-0.345845\pi$
$307$	16.3153	0.931166	0.465583	+	0.885004i	$0.345845\pi$
$308$	0	0
$309$	22.5616	1.28348
$310$	0	0
$311$	21.1231	1.19778	0.598891	−	0.800831i	$-0.295608\pi$
$311$	21.1231	1.19778	0.598891	+	0.800831i	$0.295608\pi$
$312$	0	0
$313$	14.3153	0.809151	0.404575	−	0.914505i	$-0.367419\pi$
$313$	14.3153	0.809151	0.404575	+	0.914505i	$0.367419\pi$
$314$	0	0
$315$	−3.56155	−0.200671
$316$	0	0
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	11.6847	0.654215
$320$	0	0
$321$	17.6155	0.983203
$322$	0	0
$323$	−3.86174	−0.214873
$324$	0	0
$325$	−5.68466	−0.315328
$326$	0	0
$327$	21.9309	1.21278
$328$	0	0
$329$	6.56155	0.361750
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	29.3693	1.60943
$334$	0	0
$335$	14.2462	0.778354
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	21.1231	1.14725
$340$	0	0
$341$	26.2462	1.42131
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−13.1231	−0.706524
$346$	0	0
$347$	−6.87689	−0.369171	−0.184586	−	0.982816i	$-0.559094\pi$
$347$	−6.87689	−0.369171	−0.184586	+	0.982816i	$0.559094\pi$
$348$	0	0
$349$	−28.2462	−1.51199	−0.755993	−	0.654580i	$-0.772845\pi$
$349$	−28.2462	−1.51199	−0.755993	+	0.654580i	$0.772845\pi$
$350$	0	0
$351$	8.17708	0.436460
$352$	0	0
$353$	−22.8078	−1.21393	−0.606967	−	0.794727i	$-0.707614\pi$
$353$	−22.8078	−1.21393	−0.606967	+	0.794727i	$0.707614\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.80776	0.466156
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	11.3693	0.596734
$364$	0	0
$365$	12.2462	0.640996
$366$	0	0
$367$	13.9309	0.727185	0.363593	−	0.931558i	$-0.381550\pi$
$367$	13.9309	0.727185	0.363593	+	0.931558i	$0.381550\pi$
$368$	0	0
$369$	25.3693	1.32067
$370$	0	0
$371$	4.87689	0.253196
$372$	0	0
$373$	−9.36932	−0.485125	−0.242562	−	0.970136i	$-0.577988\pi$
$373$	−9.36932	−0.485125	−0.242562	+	0.970136i	$0.577988\pi$
$374$	0	0
$375$	−2.56155	−0.132278
$376$	0	0
$377$	−25.9309	−1.33551
$378$	0	0
$379$	0.492423	0.0252940	0.0126470	−	0.999920i	$-0.495974\pi$
$379$	0.492423	0.0252940	0.0126470	+	0.999920i	$0.495974\pi$
$380$	0	0
$381$	26.2462	1.34463
$382$	0	0
$383$	26.2462	1.34112	0.670559	−	0.741856i	$-0.266054\pi$
$383$	26.2462	1.34112	0.670559	+	0.741856i	$0.266054\pi$
$384$	0	0
$385$	−2.56155	−0.130549
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−31.3002	−1.58698	−0.793491	−	0.608582i	$-0.791739\pi$
$389$	−31.3002	−1.58698	−0.793491	+	0.608582i	$0.791739\pi$
$390$	0	0
$391$	17.6155	0.890856
$392$	0	0
$393$	17.6155	0.888586
$394$	0	0
$395$	11.6847	0.587919
$396$	0	0
$397$	22.1771	1.11304	0.556518	−	0.830836i	$-0.312137\pi$
$397$	22.1771	1.11304	0.556518	+	0.830836i	$0.312137\pi$
$398$	0	0
$399$	−2.87689	−0.144025
$400$	0	0
$401$	15.9309	0.795550	0.397775	−	0.917483i	$-0.369783\pi$
$401$	15.9309	0.795550	0.397775	+	0.917483i	$0.369783\pi$
$402$	0	0
$403$	−58.2462	−2.90145
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	21.1231	1.04703
$408$	0	0
$409$	14.4924	0.716604	0.358302	−	0.933606i	$-0.383356\pi$
$409$	14.4924	0.716604	0.358302	+	0.933606i	$0.383356\pi$
$410$	0	0
$411$	−44.4924	−2.19465
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	8.63068	0.422646
$418$	0	0
$419$	1.75379	0.0856782	0.0428391	−	0.999082i	$-0.486360\pi$
$419$	1.75379	0.0856782	0.0428391	+	0.999082i	$0.486360\pi$
$420$	0	0
$421$	−0.561553	−0.0273684	−0.0136842	−	0.999906i	$-0.504356\pi$
$421$	−0.561553	−0.0273684	−0.0136842	+	0.999906i	$0.504356\pi$
$422$	0	0
$423$	−23.3693	−1.13626
$424$	0	0
$425$	3.43845	0.166789
$426$	0	0
$427$	15.1231	0.731858
$428$	0	0
$429$	37.3002	1.80087
$430$	0	0
$431$	−5.93087	−0.285680	−0.142840	−	0.989746i	$-0.545623\pi$
$431$	−5.93087	−0.285680	−0.142840	+	0.989746i	$0.545623\pi$
$432$	0	0
$433$	36.2462	1.74188	0.870941	−	0.491388i	$-0.163510\pi$
$433$	36.2462	1.74188	0.870941	+	0.491388i	$0.163510\pi$
$434$	0	0
$435$	−11.6847	−0.560236
$436$	0	0
$437$	−5.75379	−0.275241
$438$	0	0
$439$	−10.8769	−0.519126	−0.259563	−	0.965726i	$-0.583578\pi$
$439$	−10.8769	−0.519126	−0.259563	+	0.965726i	$0.583578\pi$
$440$	0	0
$441$	3.56155	0.169598
$442$	0	0
$443$	−21.6155	−1.02698	−0.513492	−	0.858094i	$-0.671649\pi$
$443$	−21.6155	−1.02698	−0.513492	+	0.858094i	$0.671649\pi$
$444$	0	0
$445$	−3.12311	−0.148049
$446$	0	0
$447$	−41.6155	−1.96835
$448$	0	0
$449$	−33.6847	−1.58968	−0.794839	−	0.606821i	$-0.792445\pi$
$449$	−33.6847	−1.58968	−0.794839	+	0.606821i	$0.792445\pi$
$450$	0	0
$451$	18.2462	0.859181
$452$	0	0
$453$	2.06913	0.0972162
$454$	0	0
$455$	5.68466	0.266501
$456$	0	0
$457$	33.3693	1.56095	0.780475	−	0.625186i	$-0.214977\pi$
$457$	33.3693	1.56095	0.780475	+	0.625186i	$0.214977\pi$
$458$	0	0
$459$	−4.94602	−0.230861
$460$	0	0
$461$	27.1231	1.26325	0.631624	−	0.775274i	$-0.282388\pi$
$461$	27.1231	1.26325	0.631624	+	0.775274i	$0.282388\pi$
$462$	0	0
$463$	−10.2462	−0.476182	−0.238091	−	0.971243i	$-0.576522\pi$
$463$	−10.2462	−0.476182	−0.238091	+	0.971243i	$0.576522\pi$
$464$	0	0
$465$	−26.2462	−1.21714
$466$	0	0
$467$	−21.4384	−0.992053	−0.496027	−	0.868307i	$-0.665208\pi$
$467$	−21.4384	−0.992053	−0.496027	+	0.868307i	$0.665208\pi$
$468$	0	0
$469$	−14.2462	−0.657829
$470$	0	0
$471$	31.3693	1.44542
$472$	0	0
$473$	−2.87689	−0.132280
$474$	0	0
$475$	−1.12311	−0.0515316
$476$	0	0
$477$	−17.3693	−0.795286
$478$	0	0
$479$	7.36932	0.336713	0.168356	−	0.985726i	$-0.446154\pi$
$479$	7.36932	0.336713	0.168356	+	0.985726i	$0.446154\pi$
$480$	0	0
$481$	−46.8769	−2.13740
$482$	0	0
$483$	13.1231	0.597122
$484$	0	0
$485$	13.6847	0.621388
$486$	0	0
$487$	5.12311	0.232150	0.116075	−	0.993240i	$-0.462969\pi$
$487$	5.12311	0.232150	0.116075	+	0.993240i	$0.462969\pi$
$488$	0	0
$489$	−55.3693	−2.50389
$490$	0	0
$491$	15.6847	0.707839	0.353919	−	0.935276i	$-0.384849\pi$
$491$	15.6847	0.707839	0.353919	+	0.935276i	$0.384849\pi$
$492$	0	0
$493$	15.6847	0.706401
$494$	0	0
$495$	9.12311	0.410053
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.8078	0.931483	0.465742	−	0.884921i	$-0.345788\pi$
$499$	20.8078	0.931483	0.465742	+	0.884921i	$0.345788\pi$
$500$	0	0
$501$	63.5464	2.83904
$502$	0	0
$503$	−16.1771	−0.721300	−0.360650	−	0.932701i	$-0.617445\pi$
$503$	−16.1771	−0.721300	−0.360650	+	0.932701i	$0.617445\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	−49.4773	−2.19736
$508$	0	0
$509$	−16.7386	−0.741927	−0.370963	−	0.928647i	$-0.620972\pi$
$509$	−16.7386	−0.741927	−0.370963	+	0.928647i	$0.620972\pi$
$510$	0	0
$511$	−12.2462	−0.541740
$512$	0	0
$513$	1.61553	0.0713273
$514$	0	0
$515$	−8.80776	−0.388116
$516$	0	0
$517$	−16.8078	−0.739205
$518$	0	0
$519$	−56.8078	−2.49358
$520$	0	0
$521$	4.24621	0.186030	0.0930149	−	0.995665i	$-0.470350\pi$
$521$	4.24621	0.186030	0.0930149	+	0.995665i	$0.470350\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	0	0
$525$	2.56155	0.111795
$526$	0	0
$527$	35.2311	1.53469
$528$	0	0
$529$	3.24621	0.141140
$530$	0	0
$531$	14.2462	0.618233
$532$	0	0
$533$	−40.4924	−1.75392
$534$	0	0
$535$	−6.87689	−0.297314
$536$	0	0
$537$	−10.2462	−0.442157
$538$	0	0
$539$	2.56155	0.110334
$540$	0	0
$541$	−20.4233	−0.878066	−0.439033	−	0.898471i	$-0.644679\pi$
$541$	−20.4233	−0.878066	−0.439033	+	0.898471i	$0.644679\pi$
$542$	0	0
$543$	−8.00000	−0.343313
$544$	0	0
$545$	−8.56155	−0.366737
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	−53.8617	−2.29876
$550$	0	0
$551$	−5.12311	−0.218252
$552$	0	0
$553$	−11.6847	−0.496882
$554$	0	0
$555$	−21.1231	−0.896626
$556$	0	0
$557$	23.6155	1.00062	0.500311	−	0.865846i	$-0.333219\pi$
$557$	23.6155	1.00062	0.500311	+	0.865846i	$0.333219\pi$
$558$	0	0
$559$	6.38447	0.270034
$560$	0	0
$561$	−22.5616	−0.952550
$562$	0	0
$563$	40.4924	1.70655	0.853276	−	0.521459i	$-0.174612\pi$
$563$	40.4924	1.70655	0.853276	+	0.521459i	$0.174612\pi$
$564$	0	0
$565$	−8.24621	−0.346921
$566$	0	0
$567$	7.00000	0.293972
$568$	0	0
$569$	−46.9848	−1.96971	−0.984854	−	0.173388i	$-0.944529\pi$
$569$	−46.9848	−1.96971	−0.984854	+	0.173388i	$0.944529\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	29.9309	1.25038
$574$	0	0
$575$	5.12311	0.213648
$576$	0	0
$577$	0.561553	0.0233777	0.0116889	−	0.999932i	$-0.496279\pi$
$577$	0.561553	0.0233777	0.0116889	+	0.999932i	$0.496279\pi$
$578$	0	0
$579$	−12.4924	−0.519167
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−12.4924	−0.517383
$584$	0	0
$585$	−20.2462	−0.837078
$586$	0	0
$587$	6.24621	0.257809	0.128904	−	0.991657i	$-0.458854\pi$
$587$	6.24621	0.257809	0.128904	+	0.991657i	$0.458854\pi$
$588$	0	0
$589$	−11.5076	−0.474161
$590$	0	0
$591$	12.4924	0.513870
$592$	0	0
$593$	−24.4233	−1.00294	−0.501472	−	0.865174i	$-0.667208\pi$
$593$	−24.4233	−1.00294	−0.501472	+	0.865174i	$0.667208\pi$
$594$	0	0
$595$	−3.43845	−0.140963
$596$	0	0
$597$	5.75379	0.235487
$598$	0	0
$599$	18.0691	0.738285	0.369142	−	0.929373i	$-0.379651\pi$
$599$	18.0691	0.738285	0.369142	+	0.929373i	$0.379651\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	50.7386	2.06624
$604$	0	0
$605$	−4.43845	−0.180449
$606$	0	0
$607$	−3.68466	−0.149556	−0.0747778	−	0.997200i	$-0.523825\pi$
$607$	−3.68466	−0.149556	−0.0747778	+	0.997200i	$0.523825\pi$
$608$	0	0
$609$	11.6847	0.473486
$610$	0	0
$611$	37.3002	1.50900
$612$	0	0
$613$	−32.7386	−1.32230	−0.661150	−	0.750253i	$-0.729932\pi$
$613$	−32.7386	−1.32230	−0.661150	+	0.750253i	$0.729932\pi$
$614$	0	0
$615$	−18.2462	−0.735758
$616$	0	0
$617$	−32.2462	−1.29818	−0.649092	−	0.760710i	$-0.724851\pi$
$617$	−32.2462	−1.29818	−0.649092	+	0.760710i	$0.724851\pi$
$618$	0	0
$619$	−35.3693	−1.42161	−0.710806	−	0.703388i	$-0.751670\pi$
$619$	−35.3693	−1.42161	−0.710806	+	0.703388i	$0.751670\pi$
$620$	0	0
$621$	−7.36932	−0.295720
$622$	0	0
$623$	3.12311	0.125125
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	7.36932	0.294302
$628$	0	0
$629$	28.3542	1.13055
$630$	0	0
$631$	−34.4233	−1.37037	−0.685185	−	0.728369i	$-0.740279\pi$
$631$	−34.4233	−1.37037	−0.685185	+	0.728369i	$0.740279\pi$
$632$	0	0
$633$	−34.4233	−1.36820
$634$	0	0
$635$	−10.2462	−0.406608
$636$	0	0
$637$	−5.68466	−0.225234
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.9848	1.69780	0.848900	−	0.528554i	$-0.177266\pi$
$641$	42.9848	1.69780	0.848900	+	0.528554i	$0.177266\pi$
$642$	0	0
$643$	−7.05398	−0.278182	−0.139091	−	0.990280i	$-0.544418\pi$
$643$	−7.05398	−0.278182	−0.139091	+	0.990280i	$0.544418\pi$
$644$	0	0
$645$	2.87689	0.113278
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	10.2462	0.402199
$650$	0	0
$651$	26.2462	1.02867
$652$	0	0
$653$	−5.50758	−0.215528	−0.107764	−	0.994176i	$-0.534369\pi$
$653$	−5.50758	−0.215528	−0.107764	+	0.994176i	$0.534369\pi$
$654$	0	0
$655$	−6.87689	−0.268702
$656$	0	0
$657$	43.6155	1.70160
$658$	0	0
$659$	20.8078	0.810555	0.405278	−	0.914194i	$-0.367175\pi$
$659$	20.8078	0.810555	0.405278	+	0.914194i	$0.367175\pi$
$660$	0	0
$661$	23.6155	0.918538	0.459269	−	0.888297i	$-0.348111\pi$
$661$	23.6155	0.918538	0.459269	+	0.888297i	$0.348111\pi$
$662$	0	0
$663$	50.0691	1.94452
$664$	0	0
$665$	1.12311	0.0435522
$666$	0	0
$667$	23.3693	0.904864
$668$	0	0
$669$	31.5464	1.21965
$670$	0	0
$671$	−38.7386	−1.49549
$672$	0	0
$673$	0.384472	0.0148203	0.00741015	−	0.999973i	$-0.497641\pi$
$673$	0.384472	0.0148203	0.00741015	+	0.999973i	$0.497641\pi$
$674$	0	0
$675$	−1.43845	−0.0553659
$676$	0	0
$677$	−13.6847	−0.525944	−0.262972	−	0.964803i	$-0.584703\pi$
$677$	−13.6847	−0.525944	−0.262972	+	0.964803i	$0.584703\pi$
$678$	0	0
$679$	−13.6847	−0.525169
$680$	0	0
$681$	−60.6695	−2.32486
$682$	0	0
$683$	20.9848	0.802963	0.401481	−	0.915867i	$-0.368495\pi$
$683$	20.9848	0.802963	0.401481	+	0.915867i	$0.368495\pi$
$684$	0	0
$685$	17.3693	0.663647
$686$	0	0
$687$	−34.2462	−1.30657
$688$	0	0
$689$	27.7235	1.05618
$690$	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
$693$	−9.12311	−0.346558
$694$	0	0
$695$	−3.36932	−0.127805
$696$	0	0
$697$	24.4924	0.927717
$698$	0	0
$699$	48.9848	1.85278
$700$	0	0
$701$	−21.6847	−0.819018	−0.409509	−	0.912306i	$-0.634300\pi$
$701$	−21.6847	−0.819018	−0.409509	+	0.912306i	$0.634300\pi$
$702$	0	0
$703$	−9.26137	−0.349299
$704$	0	0
$705$	16.8078	0.633017
$706$	0	0
$707$	−6.00000	−0.225653
$708$	0	0
$709$	33.0540	1.24137	0.620684	−	0.784061i	$-0.286855\pi$
$709$	33.0540	1.24137	0.620684	+	0.784061i	$0.286855\pi$
$710$	0	0
$711$	41.6155	1.56070
$712$	0	0
$713$	52.4924	1.96586
$714$	0	0
$715$	−14.5616	−0.544571
$716$	0	0
$717$	48.8078	1.82276
$718$	0	0
$719$	−17.6155	−0.656948	−0.328474	−	0.944513i	$-0.606534\pi$
$719$	−17.6155	−0.656948	−0.328474	+	0.944513i	$0.606534\pi$
$720$	0	0
$721$	8.80776	0.328018
$722$	0	0
$723$	−5.12311	−0.190530
$724$	0	0
$725$	4.56155	0.169412
$726$	0	0
$727$	16.9848	0.629933	0.314967	−	0.949103i	$-0.398007\pi$
$727$	16.9848	0.629933	0.314967	+	0.949103i	$0.398007\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−3.86174	−0.142832
$732$	0	0
$733$	20.5616	0.759458	0.379729	−	0.925098i	$-0.376017\pi$
$733$	20.5616	0.759458	0.379729	+	0.925098i	$0.376017\pi$
$734$	0	0
$735$	−2.56155	−0.0944843
$736$	0	0
$737$	36.4924	1.34422
$738$	0	0
$739$	10.5616	0.388513	0.194257	−	0.980951i	$-0.437771\pi$
$739$	10.5616	0.388513	0.194257	+	0.980951i	$0.437771\pi$
$740$	0	0
$741$	−16.3542	−0.600785
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	16.2462	0.595215
$746$	0	0
$747$	−42.7386	−1.56372
$748$	0	0
$749$	6.87689	0.251276
$750$	0	0
$751$	43.6847	1.59408	0.797038	−	0.603929i	$-0.206399\pi$
$751$	43.6847	1.59408	0.797038	+	0.603929i	$0.206399\pi$
$752$	0	0
$753$	−29.1231	−1.06130
$754$	0	0
$755$	−0.807764	−0.0293975
$756$	0	0
$757$	−9.36932	−0.340534	−0.170267	−	0.985398i	$-0.554463\pi$
$757$	−9.36932	−0.340534	−0.170267	+	0.985398i	$0.554463\pi$
$758$	0	0
$759$	−33.6155	−1.22017
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	8.56155	0.309949
$764$	0	0
$765$	12.2462	0.442763
$766$	0	0
$767$	−22.7386	−0.821044
$768$	0	0
$769$	−22.9848	−0.828855	−0.414427	−	0.910082i	$-0.636018\pi$
$769$	−22.9848	−0.828855	−0.414427	+	0.910082i	$0.636018\pi$
$770$	0	0
$771$	−37.1231	−1.33696
$772$	0	0
$773$	−9.19224	−0.330622	−0.165311	−	0.986242i	$-0.552863\pi$
$773$	−9.19224	−0.330622	−0.165311	+	0.986242i	$0.552863\pi$
$774$	0	0
$775$	10.2462	0.368055
$776$	0	0
$777$	21.1231	0.757787
$778$	0	0
$779$	−8.00000	−0.286630
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.56155	−0.234491
$784$	0	0
$785$	−12.2462	−0.437086
$786$	0	0
$787$	12.1771	0.434066	0.217033	−	0.976164i	$-0.430362\pi$
$787$	12.1771	0.434066	0.217033	+	0.976164i	$0.430362\pi$
$788$	0	0
$789$	−39.3693	−1.40158
$790$	0	0
$791$	8.24621	0.293202
$792$	0	0
$793$	85.9697	3.05287
$794$	0	0
$795$	12.4924	0.443061
$796$	0	0
$797$	−2.80776	−0.0994561	−0.0497281	−	0.998763i	$-0.515835\pi$
$797$	−2.80776	−0.0994561	−0.0497281	+	0.998763i	$0.515835\pi$
$798$	0	0
$799$	−22.5616	−0.798170
$800$	0	0
$801$	−11.1231	−0.393016
$802$	0	0
$803$	31.3693	1.10700
$804$	0	0
$805$	−5.12311	−0.180566
$806$	0	0
$807$	31.3693	1.10425
$808$	0	0
$809$	−33.0540	−1.16212	−0.581058	−	0.813862i	$-0.697361\pi$
$809$	−33.0540	−1.16212	−0.581058	+	0.813862i	$0.697361\pi$
$810$	0	0
$811$	−13.6155	−0.478106	−0.239053	−	0.971007i	$-0.576837\pi$
$811$	−13.6155	−0.478106	−0.239053	+	0.971007i	$0.576837\pi$
$812$	0	0
$813$	26.2462	0.920495
$814$	0	0
$815$	21.6155	0.757159
$816$	0	0
$817$	1.26137	0.0441296
$818$	0	0
$819$	20.2462	0.707460
$820$	0	0
$821$	0.699813	0.0244237	0.0122118	−	0.999925i	$-0.496113\pi$
$821$	0.699813	0.0244237	0.0122118	+	0.999925i	$0.496113\pi$
$822$	0	0
$823$	34.2462	1.19375	0.596874	−	0.802335i	$-0.296409\pi$
$823$	34.2462	1.19375	0.596874	+	0.802335i	$0.296409\pi$
$824$	0	0
$825$	−6.56155	−0.228444
$826$	0	0
$827$	−33.1231	−1.15180	−0.575902	−	0.817519i	$-0.695349\pi$
$827$	−33.1231	−1.15180	−0.575902	+	0.817519i	$0.695349\pi$
$828$	0	0
$829$	−35.6155	−1.23698	−0.618489	−	0.785793i	$-0.712255\pi$
$829$	−35.6155	−1.23698	−0.618489	+	0.785793i	$0.712255\pi$
$830$	0	0
$831$	−21.1231	−0.732752
$832$	0	0
$833$	3.43845	0.119135
$834$	0	0
$835$	−24.8078	−0.858508
$836$	0	0
$837$	−14.7386	−0.509442
$838$	0	0
$839$	−10.8769	−0.375512	−0.187756	−	0.982216i	$-0.560121\pi$
$839$	−10.8769	−0.375512	−0.187756	+	0.982216i	$0.560121\pi$
$840$	0	0
$841$	−8.19224	−0.282491
$842$	0	0
$843$	39.5464	1.36205
$844$	0	0
$845$	19.3153	0.664468
$846$	0	0
$847$	4.43845	0.152507
$848$	0	0
$849$	19.6847	0.675576
$850$	0	0
$851$	42.2462	1.44818
$852$	0	0
$853$	32.2462	1.10409	0.552045	−	0.833815i	$-0.313848\pi$
$853$	32.2462	1.10409	0.552045	+	0.833815i	$0.313848\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	3.26137	0.111406	0.0557031	−	0.998447i	$-0.482260\pi$
$857$	3.26137	0.111406	0.0557031	+	0.998447i	$0.482260\pi$
$858$	0	0
$859$	28.9848	0.988950	0.494475	−	0.869192i	$-0.335360\pi$
$859$	28.9848	0.988950	0.494475	+	0.869192i	$0.335360\pi$
$860$	0	0
$861$	18.2462	0.621829
$862$	0	0
$863$	4.49242	0.152924	0.0764619	−	0.997073i	$-0.475638\pi$
$863$	4.49242	0.152924	0.0764619	+	0.997073i	$0.475638\pi$
$864$	0	0
$865$	22.1771	0.754043
$866$	0	0
$867$	13.2614	0.450380
$868$	0	0
$869$	29.9309	1.01534
$870$	0	0
$871$	−80.9848	−2.74407
$872$	0	0
$873$	48.7386	1.64955
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−46.4924	−1.56994	−0.784969	−	0.619535i	$-0.787321\pi$
$877$	−46.4924	−1.56994	−0.784969	+	0.619535i	$0.787321\pi$
$878$	0	0
$879$	12.9460	0.436659
$880$	0	0
$881$	−20.1080	−0.677454	−0.338727	−	0.940885i	$-0.609996\pi$
$881$	−20.1080	−0.677454	−0.338727	+	0.940885i	$0.609996\pi$
$882$	0	0
$883$	−38.2462	−1.28709	−0.643544	−	0.765409i	$-0.722537\pi$
$883$	−38.2462	−1.28709	−0.643544	+	0.765409i	$0.722537\pi$
$884$	0	0
$885$	−10.2462	−0.344423
$886$	0	0
$887$	3.50758	0.117773	0.0588865	−	0.998265i	$-0.481245\pi$
$887$	3.50758	0.117773	0.0588865	+	0.998265i	$0.481245\pi$
$888$	0	0
$889$	10.2462	0.343647
$890$	0	0
$891$	−17.9309	−0.600707
$892$	0	0
$893$	7.36932	0.246605
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	74.6004	2.49083
$898$	0	0
$899$	46.7386	1.55882
$900$	0	0
$901$	−16.7689	−0.558655
$902$	0	0
$903$	−2.87689	−0.0957371
$904$	0	0
$905$	3.12311	0.103816
$906$	0	0
$907$	−26.1080	−0.866900	−0.433450	−	0.901178i	$-0.642704\pi$
$907$	−26.1080	−0.866900	−0.433450	+	0.901178i	$0.642704\pi$
$908$	0	0
$909$	21.3693	0.708776
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	−30.7386	−1.01730
$914$	0	0
$915$	38.7386	1.28066
$916$	0	0
$917$	6.87689	0.227095
$918$	0	0
$919$	−6.56155	−0.216446	−0.108223	−	0.994127i	$-0.534516\pi$
$919$	−6.56155	−0.216446	−0.108223	+	0.994127i	$0.534516\pi$
$920$	0	0
$921$	−41.7926	−1.37711
$922$	0	0
$923$	0	0
$924$	0	0
$925$	8.24621	0.271134
$926$	0	0
$927$	−31.3693	−1.03030
$928$	0	0
$929$	−11.1231	−0.364937	−0.182469	−	0.983212i	$-0.558409\pi$
$929$	−11.1231	−0.364937	−0.182469	+	0.983212i	$0.558409\pi$
$930$	0	0
$931$	−1.12311	−0.0368083
$932$	0	0
$933$	−54.1080	−1.77141
$934$	0	0
$935$	8.80776	0.288045
$936$	0	0
$937$	17.1922	0.561646	0.280823	−	0.959760i	$-0.409393\pi$
$937$	17.1922	0.561646	0.280823	+	0.959760i	$0.409393\pi$
$938$	0	0
$939$	−36.6695	−1.19666
$940$	0	0
$941$	53.3693	1.73979	0.869895	−	0.493237i	$-0.164186\pi$
$941$	53.3693	1.73979	0.869895	+	0.493237i	$0.164186\pi$
$942$	0	0
$943$	36.4924	1.18836
$944$	0	0
$945$	1.43845	0.0467927
$946$	0	0
$947$	−47.2311	−1.53480	−0.767402	−	0.641167i	$-0.778451\pi$
$947$	−47.2311	−1.53480	−0.767402	+	0.641167i	$0.778451\pi$
$948$	0	0
$949$	−69.6155	−2.25982
$950$	0	0
$951$	25.6155	0.830640
$952$	0	0
$953$	5.86174	0.189880	0.0949402	−	0.995483i	$-0.469734\pi$
$953$	5.86174	0.189880	0.0949402	+	0.995483i	$0.469734\pi$
$954$	0	0
$955$	−11.6847	−0.378107
$956$	0	0
$957$	−29.9309	−0.967528
$958$	0	0
$959$	−17.3693	−0.560884
$960$	0	0
$961$	73.9848	2.38661
$962$	0	0
$963$	−24.4924	−0.789257
$964$	0	0
$965$	4.87689	0.156993
$966$	0	0
$967$	30.1080	0.968206	0.484103	−	0.875011i	$-0.339146\pi$
$967$	30.1080	0.968206	0.484103	+	0.875011i	$0.339146\pi$
$968$	0	0
$969$	9.89205	0.317778
$970$	0	0
$971$	−26.7386	−0.858084	−0.429042	−	0.903285i	$-0.641149\pi$
$971$	−26.7386	−0.858084	−0.429042	+	0.903285i	$0.641149\pi$
$972$	0	0
$973$	3.36932	0.108015
$974$	0	0
$975$	14.5616	0.466343
$976$	0	0
$977$	−54.9848	−1.75912	−0.879561	−	0.475787i	$-0.842163\pi$
$977$	−54.9848	−1.75912	−0.879561	+	0.475787i	$0.842163\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	−30.4924	−0.973548
$982$	0	0
$983$	−16.1771	−0.515969	−0.257984	−	0.966149i	$-0.583058\pi$
$983$	−16.1771	−0.515969	−0.257984	+	0.966149i	$0.583058\pi$
$984$	0	0
$985$	−4.87689	−0.155391
$986$	0	0
$987$	−16.8078	−0.534997
$988$	0	0
$989$	−5.75379	−0.182960
$990$	0	0
$991$	−12.4924	−0.396835	−0.198417	−	0.980118i	$-0.563580\pi$
$991$	−12.4924	−0.396835	−0.198417	+	0.980118i	$0.563580\pi$
$992$	0	0
$993$	−30.7386	−0.975461
$994$	0	0
$995$	−2.24621	−0.0712097
$996$	0	0
$997$	−28.0691	−0.888958	−0.444479	−	0.895789i	$-0.646611\pi$
$997$	−28.0691	−0.888958	−0.444479	+	0.895789i	$0.646611\pi$
$998$	0	0
$999$	−11.8617	−0.375289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.a.g.1.1		2
3.2	odd	2		5040.2.a.bq.1.1		2
4.3	odd	2		280.2.a.d.1.2	✓	2
5.2	odd	4		2800.2.g.u.449.4		4
5.3	odd	4		2800.2.g.u.449.1		4
5.4	even	2		2800.2.a.bn.1.2		2
7.6	odd	2		3920.2.a.bu.1.2		2
8.3	odd	2		2240.2.a.be.1.1		2
8.5	even	2		2240.2.a.bi.1.2		2
12.11	even	2		2520.2.a.w.1.2		2
20.3	even	4		1400.2.g.k.449.4		4
20.7	even	4		1400.2.g.k.449.1		4
20.19	odd	2		1400.2.a.p.1.1		2
28.3	even	6		1960.2.q.u.961.2		4
28.11	odd	6		1960.2.q.s.961.1		4
28.19	even	6		1960.2.q.u.361.2		4
28.23	odd	6		1960.2.q.s.361.1		4
28.27	even	2		1960.2.a.r.1.1		2
140.139	even	2		9800.2.a.by.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.d.1.2	✓	2	4.3	odd	2
560.2.a.g.1.1		2	1.1	even	1	trivial
1400.2.a.p.1.1		2	20.19	odd	2
1400.2.g.k.449.1		4	20.7	even	4
1400.2.g.k.449.4		4	20.3	even	4
1960.2.a.r.1.1		2	28.27	even	2
1960.2.q.s.361.1		4	28.23	odd	6
1960.2.q.s.961.1		4	28.11	odd	6
1960.2.q.u.361.2		4	28.19	even	6
1960.2.q.u.961.2		4	28.3	even	6
2240.2.a.be.1.1		2	8.3	odd	2
2240.2.a.bi.1.2		2	8.5	even	2
2520.2.a.w.1.2		2	12.11	even	2
2800.2.a.bn.1.2		2	5.4	even	2
2800.2.g.u.449.1		4	5.3	odd	4
2800.2.g.u.449.4		4	5.2	odd	4
3920.2.a.bu.1.2		2	7.6	odd	2
5040.2.a.bq.1.1		2	3.2	odd	2
9800.2.a.by.1.2		2	140.139	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} +1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q-2.56155 q^{3} +1.00000 q^{5} -1.00000 q^{7} +3.56155 q^{9} +2.56155 q^{11} -5.68466 q^{13} -2.56155 q^{15} +3.43845 q^{17} -1.12311 q^{19} +2.56155 q^{21} +5.12311 q^{23} +1.00000 q^{25} -1.43845 q^{27} +4.56155 q^{29} +10.2462 q^{31} -6.56155 q^{33} -1.00000 q^{35} +8.24621 q^{37} +14.5616 q^{39} +7.12311 q^{41} -1.12311 q^{43} +3.56155 q^{45} -6.56155 q^{47} +1.00000 q^{49} -8.80776 q^{51} -4.87689 q^{53} +2.56155 q^{55} +2.87689 q^{57} +4.00000 q^{59} -15.1231 q^{61} -3.56155 q^{63} -5.68466 q^{65} +14.2462 q^{67} -13.1231 q^{69} +12.2462 q^{73} -2.56155 q^{75} -2.56155 q^{77} +11.6847 q^{79} -7.00000 q^{81} -12.0000 q^{83} +3.43845 q^{85} -11.6847 q^{87} -3.12311 q^{89} +5.68466 q^{91} -26.2462 q^{93} -1.12311 q^{95} +13.6847 q^{97} +9.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9	\( 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} + q^{11} + q^{13} - q^{15} + 11 q^{17} + 6 q^{19} + q^{21} + 2 q^{23} + 2 q^{25} - 7 q^{27} + 5 q^{29} + 4 q^{31} - 9 q^{33} - 2 q^{35} + 25 q^{39} + 6 q^{41} + 6 q^{43} + 3 q^{45} - 9 q^{47} + 2 q^{49} + 3 q^{51} - 18 q^{53} + q^{55} + 14 q^{57} + 8 q^{59} - 22 q^{61} - 3 q^{63} + q^{65} + 12 q^{67} - 18 q^{69} + 8 q^{73} - q^{75} - q^{77} + 11 q^{79} - 14 q^{81} - 24 q^{83} + 11 q^{85} - 11 q^{87} + 2 q^{89} - q^{91} - 36 q^{93} + 6 q^{95} + 15 q^{97} + 10 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9 + q^11 + q^13 - q^15 + 11 * q^17 + 6 * q^19 + q^21 + 2 * q^23 + 2 * q^25 - 7 * q^27 + 5 * q^29 + 4 * q^31 - 9 * q^33 - 2 * q^35 + 25 * q^39 + 6 * q^41 + 6 * q^43 + 3 * q^45 - 9 * q^47 + 2 * q^49 + 3 * q^51 - 18 * q^53 + q^55 + 14 * q^57 + 8 * q^59 - 22 * q^61 - 3 * q^63 + q^65 + 12 * q^67 - 18 * q^69 + 8 * q^73 - q^75 - q^77 + 11 * q^79 - 14 * q^81 - 24 * q^83 + 11 * q^85 - 11 * q^87 + 2 * q^89 - q^91 - 36 * q^93 + 6 * q^95 + 15 * q^97 + 10 * q^99

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