LMFDB - Embedded newform 560.4.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,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, 4, names="a")

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

chi := DirichletCharacter("560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
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:	$33.0410696032$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+4.65685 q^{3} -5.00000 q^{5} +7.00000 q^{7} -5.31371 q^{9} +52.2548 q^{11} +30.6569 q^{13} -23.2843 q^{15} +37.2254 q^{17} -80.2254 q^{19} +32.5980 q^{21} -25.8335 q^{23} +25.0000 q^{25} -150.480 q^{27} +20.9411 q^{29} +314.558 q^{31} +243.343 q^{33} -35.0000 q^{35} +197.147 q^{37} +142.765 q^{39} +11.3625 q^{41} +33.8335 q^{43} +26.5685 q^{45} +361.676 q^{47} +49.0000 q^{49} +173.353 q^{51} +153.019 q^{53} -261.274 q^{55} -373.598 q^{57} +616.000 q^{59} +15.2649 q^{61} -37.1960 q^{63} -153.284 q^{65} +166.510 q^{67} -120.303 q^{69} +952.000 q^{71} -148.489 q^{73} +116.421 q^{75} +365.784 q^{77} -857.725 q^{79} -557.294 q^{81} -660.528 q^{83} -186.127 q^{85} +97.5198 q^{87} -45.7746 q^{89} +214.598 q^{91} +1464.85 q^{93} +401.127 q^{95} +1682.13 q^{97} -277.667 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 10 q^{5} + 14 q^{7} + 12 q^{9} + 14 q^{11} + 50 q^{13} + 10 q^{15} - 50 q^{17} - 36 q^{19} - 14 q^{21} - 244 q^{23} + 50 q^{25} - 86 q^{27} - 26 q^{29} + 120 q^{31} + 498 q^{33} - 70 q^{35}+ \cdots - 940 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 10 * q^5 + 14 * q^7 + 12 * q^9 + 14 * q^11 + 50 * q^13 + 10 * q^15 - 50 * q^17 - 36 * q^19 - 14 * q^21 - 244 * q^23 + 50 * q^25 - 86 * q^27 - 26 * q^29 + 120 * q^31 + 498 * q^33 - 70 * q^35 + 564 * q^37 + 14 * q^39 - 328 * q^41 + 260 * q^43 - 60 * q^45 + 350 * q^47 + 98 * q^49 + 754 * q^51 - 56 * q^53 - 70 * q^55 - 668 * q^57 + 1232 * q^59 + 336 * q^61 + 84 * q^63 - 250 * q^65 + 152 * q^67 + 1332 * q^69 + 1904 * q^71 + 676 * q^73 - 50 * q^75 + 98 * q^77 - 1014 * q^79 - 1454 * q^81 + 376 * q^83 + 250 * q^85 + 410 * q^87 - 216 * q^89 + 350 * q^91 + 2760 * q^93 + 180 * q^95 + 2742 * q^97 - 940 * q^99

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
$3$	4.65685	0.896212	0.448106	−	0.893980i	$-0.352099\pi$
$3$	4.65685	0.896212	0.448106	+	0.893980i	$0.352099\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−5.31371	−0.196804
$10$	0	0
$11$	52.2548	1.43231	0.716156	−	0.697941i	$-0.245900\pi$
$11$	52.2548	1.43231	0.716156	+	0.697941i	$0.245900\pi$
$12$	0	0
$13$	30.6569	0.654052	0.327026	−	0.945015i	$-0.393953\pi$
$13$	30.6569	0.654052	0.327026	+	0.945015i	$0.393953\pi$
$14$	0	0
$15$	−23.2843	−0.400798
$16$	0	0
$17$	37.2254	0.531087	0.265544	−	0.964099i	$-0.414449\pi$
$17$	37.2254	0.531087	0.265544	+	0.964099i	$0.414449\pi$
$18$	0	0
$19$	−80.2254	−0.968683	−0.484341	−	0.874879i	$-0.660941\pi$
$19$	−80.2254	−0.968683	−0.484341	+	0.874879i	$0.660941\pi$
$20$	0	0
$21$	32.5980	0.338736
$22$	0	0
$23$	−25.8335	−0.234202	−0.117101	−	0.993120i	$-0.537360\pi$
$23$	−25.8335	−0.234202	−0.117101	+	0.993120i	$0.537360\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−150.480	−1.07259
$28$	0	0
$29$	20.9411	0.134092	0.0670460	−	0.997750i	$-0.478643\pi$
$29$	20.9411	0.134092	0.0670460	+	0.997750i	$0.478643\pi$
$30$	0	0
$31$	314.558	1.82246	0.911232	−	0.411894i	$-0.135133\pi$
$31$	314.558	1.82246	0.911232	+	0.411894i	$0.135133\pi$
$32$	0	0
$33$	243.343	1.28365
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	197.147	0.875968	0.437984	−	0.898983i	$-0.355693\pi$
$37$	197.147	0.875968	0.437984	+	0.898983i	$0.355693\pi$
$38$	0	0
$39$	142.765	0.586170
$40$	0	0
$41$	11.3625	0.0432810	0.0216405	−	0.999766i	$-0.493111\pi$
$41$	11.3625	0.0432810	0.0216405	+	0.999766i	$0.493111\pi$
$42$	0	0
$43$	33.8335	0.119990	0.0599948	−	0.998199i	$-0.480892\pi$
$43$	33.8335	0.119990	0.0599948	+	0.998199i	$0.480892\pi$
$44$	0	0
$45$	26.5685	0.0880134
$46$	0	0
$47$	361.676	1.12247	0.561233	−	0.827658i	$-0.310327\pi$
$47$	361.676	1.12247	0.561233	+	0.827658i	$0.310327\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	173.353	0.475967
$52$	0	0
$53$	153.019	0.396582	0.198291	−	0.980143i	$-0.436461\pi$
$53$	153.019	0.396582	0.198291	+	0.980143i	$0.436461\pi$
$54$	0	0
$55$	−261.274	−0.640549
$56$	0	0
$57$	−373.598	−0.868145
$58$	0	0
$59$	616.000	1.35926	0.679630	−	0.733555i	$-0.262140\pi$
$59$	616.000	1.35926	0.679630	+	0.733555i	$0.262140\pi$
$60$	0	0
$61$	15.2649	0.0320406	0.0160203	−	0.999872i	$-0.494900\pi$
$61$	15.2649	0.0320406	0.0160203	+	0.999872i	$0.494900\pi$
$62$	0	0
$63$	−37.1960	−0.0743849
$64$	0	0
$65$	−153.284	−0.292501
$66$	0	0
$67$	166.510	0.303618	0.151809	−	0.988410i	$-0.451490\pi$
$67$	166.510	0.303618	0.151809	+	0.988410i	$0.451490\pi$
$68$	0	0
$69$	−120.303	−0.209895
$70$	0	0
$71$	952.000	1.59129	0.795645	−	0.605763i	$-0.207132\pi$
$71$	952.000	1.59129	0.795645	+	0.605763i	$0.207132\pi$
$72$	0	0
$73$	−148.489	−0.238074	−0.119037	−	0.992890i	$-0.537981\pi$
$73$	−148.489	−0.238074	−0.119037	+	0.992890i	$0.537981\pi$
$74$	0	0
$75$	116.421	0.179242
$76$	0	0
$77$	365.784	0.541363
$78$	0	0
$79$	−857.725	−1.22154	−0.610770	−	0.791808i	$-0.709140\pi$
$79$	−857.725	−1.22154	−0.610770	+	0.791808i	$0.709140\pi$
$80$	0	0
$81$	−557.294	−0.764464
$82$	0	0
$83$	−660.528	−0.873523	−0.436761	−	0.899577i	$-0.643875\pi$
$83$	−660.528	−0.873523	−0.436761	+	0.899577i	$0.643875\pi$
$84$	0	0
$85$	−186.127	−0.237509
$86$	0	0
$87$	97.5198	0.120175
$88$	0	0
$89$	−45.7746	−0.0545180	−0.0272590	−	0.999628i	$-0.508678\pi$
$89$	−45.7746	−0.0545180	−0.0272590	+	0.999628i	$0.508678\pi$
$90$	0	0
$91$	214.598	0.247209
$92$	0	0
$93$	1464.85	1.63331
$94$	0	0
$95$	401.127	0.433208
$96$	0	0
$97$	1682.13	1.76076	0.880382	−	0.474265i	$-0.157286\pi$
$97$	1682.13	1.76076	0.880382	+	0.474265i	$0.157286\pi$
$98$	0	0
$99$	−277.667	−0.281885
$100$	0	0
$101$	−434.167	−0.427734	−0.213867	−	0.976863i	$-0.568606\pi$
$101$	−434.167	−0.427734	−0.213867	+	0.976863i	$0.568606\pi$
$102$	0	0
$103$	−345.577	−0.330589	−0.165295	−	0.986244i	$-0.552858\pi$
$103$	−345.577	−0.330589	−0.165295	+	0.986244i	$0.552858\pi$
$104$	0	0
$105$	−162.990	−0.151487
$106$	0	0
$107$	−217.119	−0.196165	−0.0980825	−	0.995178i	$-0.531271\pi$
$107$	−217.119	−0.196165	−0.0980825	+	0.995178i	$0.531271\pi$
$108$	0	0
$109$	1734.41	1.52409	0.762047	−	0.647521i	$-0.224194\pi$
$109$	1734.41	1.52409	0.762047	+	0.647521i	$0.224194\pi$
$110$	0	0
$111$	918.086	0.785053
$112$	0	0
$113$	−1854.20	−1.54362	−0.771809	−	0.635855i	$-0.780648\pi$
$113$	−1854.20	−1.54362	−0.771809	+	0.635855i	$0.780648\pi$
$114$	0	0
$115$	129.167	0.104738
$116$	0	0
$117$	−162.902	−0.128720
$118$	0	0
$119$	260.578	0.200732
$120$	0	0
$121$	1399.57	1.05152
$122$	0	0
$123$	52.9134	0.0387890
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1394.51	−0.974352	−0.487176	−	0.873304i	$-0.661973\pi$
$127$	−1394.51	−0.974352	−0.487176	+	0.873304i	$0.661973\pi$
$128$	0	0
$129$	157.558	0.107536
$130$	0	0
$131$	−1762.42	−1.17544	−0.587722	−	0.809063i	$-0.699975\pi$
$131$	−1762.42	−1.17544	−0.587722	+	0.809063i	$0.699975\pi$
$132$	0	0
$133$	−561.578	−0.366128
$134$	0	0
$135$	752.401	0.479677
$136$	0	0
$137$	−922.949	−0.575568	−0.287784	−	0.957695i	$-0.592919\pi$
$137$	−922.949	−0.575568	−0.287784	+	0.957695i	$0.592919\pi$
$138$	0	0
$139$	196.039	0.119624	0.0598122	−	0.998210i	$-0.480950\pi$
$139$	196.039	0.119624	0.0598122	+	0.998210i	$0.480950\pi$
$140$	0	0
$141$	1684.27	1.00597
$142$	0	0
$143$	1601.97	0.936807
$144$	0	0
$145$	−104.706	−0.0599678
$146$	0	0
$147$	228.186	0.128030
$148$	0	0
$149$	780.372	0.429064	0.214532	−	0.976717i	$-0.431177\pi$
$149$	780.372	0.429064	0.214532	+	0.976717i	$0.431177\pi$
$150$	0	0
$151$	2319.43	1.25002	0.625008	−	0.780618i	$-0.285096\pi$
$151$	2319.43	1.25002	0.625008	+	0.780618i	$0.285096\pi$
$152$	0	0
$153$	−197.805	−0.104520
$154$	0	0
$155$	−1572.79	−0.815030
$156$	0	0
$157$	1022.90	0.519977	0.259989	−	0.965612i	$-0.416281\pi$
$157$	1022.90	0.519977	0.259989	+	0.965612i	$0.416281\pi$
$158$	0	0
$159$	712.589	0.355421
$160$	0	0
$161$	−180.834	−0.0885201
$162$	0	0
$163$	1350.63	0.649013	0.324507	−	0.945883i	$-0.394802\pi$
$163$	1350.63	0.649013	0.324507	+	0.945883i	$0.394802\pi$
$164$	0	0
$165$	−1216.72	−0.574068
$166$	0	0
$167$	1230.58	0.570209	0.285105	−	0.958496i	$-0.407972\pi$
$167$	1230.58	0.570209	0.285105	+	0.958496i	$0.407972\pi$
$168$	0	0
$169$	−1257.16	−0.572215
$170$	0	0
$171$	426.294	0.190641
$172$	0	0
$173$	−2487.65	−1.09325	−0.546626	−	0.837377i	$-0.684088\pi$
$173$	−2487.65	−1.09325	−0.546626	+	0.837377i	$0.684088\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	2868.62	1.21819
$178$	0	0
$179$	−1621.18	−0.676941	−0.338471	−	0.940977i	$-0.609910\pi$
$179$	−1621.18	−0.676941	−0.338471	+	0.940977i	$0.609910\pi$
$180$	0	0
$181$	2593.69	1.06512	0.532561	−	0.846392i	$-0.321230\pi$
$181$	2593.69	1.06512	0.532561	+	0.846392i	$0.321230\pi$
$182$	0	0
$183$	71.0866	0.0287151
$184$	0	0
$185$	−985.736	−0.391745
$186$	0	0
$187$	1945.21	0.760682
$188$	0	0
$189$	−1053.36	−0.405401
$190$	0	0
$191$	1823.08	0.690645	0.345323	−	0.938484i	$-0.387769\pi$
$191$	1823.08	0.690645	0.345323	+	0.938484i	$0.387769\pi$
$192$	0	0
$193$	−1541.03	−0.574744	−0.287372	−	0.957819i	$-0.592782\pi$
$193$	−1541.03	−0.574744	−0.287372	+	0.957819i	$0.592782\pi$
$194$	0	0
$195$	−713.823	−0.262143
$196$	0	0
$197$	701.243	0.253612	0.126806	−	0.991928i	$-0.459527\pi$
$197$	701.243	0.253612	0.126806	+	0.991928i	$0.459527\pi$
$198$	0	0
$199$	−3294.96	−1.17374	−0.586868	−	0.809682i	$-0.699639\pi$
$199$	−3294.96	−1.17374	−0.586868	+	0.809682i	$0.699639\pi$
$200$	0	0
$201$	775.411	0.272106
$202$	0	0
$203$	146.588	0.0506820
$204$	0	0
$205$	−56.8124	−0.0193559
$206$	0	0
$207$	137.272	0.0460920
$208$	0	0
$209$	−4192.16	−1.38746
$210$	0	0
$211$	−4082.35	−1.33195	−0.665974	−	0.745975i	$-0.731984\pi$
$211$	−4082.35	−1.33195	−0.665974	+	0.745975i	$0.731984\pi$
$212$	0	0
$213$	4433.33	1.42613
$214$	0	0
$215$	−169.167	−0.0536610
$216$	0	0
$217$	2201.91	0.688826
$218$	0	0
$219$	−691.494	−0.213364
$220$	0	0
$221$	1141.21	0.347359
$222$	0	0
$223$	−747.161	−0.224366	−0.112183	−	0.993688i	$-0.535784\pi$
$223$	−747.161	−0.224366	−0.112183	+	0.993688i	$0.535784\pi$
$224$	0	0
$225$	−132.843	−0.0393608
$226$	0	0
$227$	−1665.67	−0.487025	−0.243513	−	0.969898i	$-0.578300\pi$
$227$	−1665.67	−0.487025	−0.243513	+	0.969898i	$0.578300\pi$
$228$	0	0
$229$	−6628.35	−1.91272	−0.956362	−	0.292183i	$-0.905618\pi$
$229$	−6628.35	−1.91272	−0.956362	+	0.292183i	$0.905618\pi$
$230$	0	0
$231$	1703.40	0.485176
$232$	0	0
$233$	−432.431	−0.121586	−0.0607929	−	0.998150i	$-0.519363\pi$
$233$	−432.431	−0.121586	−0.0607929	+	0.998150i	$0.519363\pi$
$234$	0	0
$235$	−1808.38	−0.501982
$236$	0	0
$237$	−3994.30	−1.09476
$238$	0	0
$239$	−5580.44	−1.51033	−0.755165	−	0.655535i	$-0.772443\pi$
$239$	−5580.44	−1.51033	−0.755165	+	0.655535i	$0.772443\pi$
$240$	0	0
$241$	−6296.87	−1.68306	−0.841529	−	0.540212i	$-0.818344\pi$
$241$	−6296.87	−1.68306	−0.841529	+	0.540212i	$0.818344\pi$
$242$	0	0
$243$	1467.73	0.387468
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−2459.46	−0.633569
$248$	0	0
$249$	−3075.98	−0.782862
$250$	0	0
$251$	−311.921	−0.0784393	−0.0392197	−	0.999231i	$-0.512487\pi$
$251$	−311.921	−0.0784393	−0.0392197	+	0.999231i	$0.512487\pi$
$252$	0	0
$253$	−1349.92	−0.335451
$254$	0	0
$255$	−866.766	−0.212859
$256$	0	0
$257$	−7861.39	−1.90809	−0.954046	−	0.299659i	$-0.903127\pi$
$257$	−7861.39	−1.90809	−0.954046	+	0.299659i	$0.903127\pi$
$258$	0	0
$259$	1380.03	0.331085
$260$	0	0
$261$	−111.275	−0.0263899
$262$	0	0
$263$	−5227.09	−1.22554	−0.612769	−	0.790262i	$-0.709944\pi$
$263$	−5227.09	−1.22554	−0.612769	+	0.790262i	$0.709944\pi$
$264$	0	0
$265$	−765.097	−0.177357
$266$	0	0
$267$	−213.166	−0.0488596
$268$	0	0
$269$	1281.71	0.290510	0.145255	−	0.989394i	$-0.453600\pi$
$269$	1281.71	0.290510	0.145255	+	0.989394i	$0.453600\pi$
$270$	0	0
$271$	−4704.14	−1.05445	−0.527226	−	0.849725i	$-0.676768\pi$
$271$	−4704.14	−1.05445	−0.527226	+	0.849725i	$0.676768\pi$
$272$	0	0
$273$	999.352	0.221551
$274$	0	0
$275$	1306.37	0.286462
$276$	0	0
$277$	8958.56	1.94321	0.971603	−	0.236619i	$-0.0760393\pi$
$277$	8958.56	1.94321	0.971603	+	0.236619i	$0.0760393\pi$
$278$	0	0
$279$	−1671.47	−0.358668
$280$	0	0
$281$	−370.904	−0.0787412	−0.0393706	−	0.999225i	$-0.512535\pi$
$281$	−370.904	−0.0787412	−0.0393706	+	0.999225i	$0.512535\pi$
$282$	0	0
$283$	5822.26	1.22296	0.611479	−	0.791261i	$-0.290575\pi$
$283$	5822.26	1.22296	0.611479	+	0.791261i	$0.290575\pi$
$284$	0	0
$285$	1867.99	0.388246
$286$	0	0
$287$	79.5374	0.0163587
$288$	0	0
$289$	−3527.27	−0.717946
$290$	0	0
$291$	7833.42	1.57802
$292$	0	0
$293$	7443.79	1.48420	0.742100	−	0.670289i	$-0.233830\pi$
$293$	7443.79	1.48420	0.742100	+	0.670289i	$0.233830\pi$
$294$	0	0
$295$	−3080.00	−0.607880
$296$	0	0
$297$	−7863.32	−1.53628
$298$	0	0
$299$	−791.973	−0.153181
$300$	0	0
$301$	236.834	0.0453518
$302$	0	0
$303$	−2021.85	−0.383341
$304$	0	0
$305$	−76.3247	−0.0143290
$306$	0	0
$307$	761.674	0.141600	0.0707998	−	0.997491i	$-0.477445\pi$
$307$	761.674	0.141600	0.0707998	+	0.997491i	$0.477445\pi$
$308$	0	0
$309$	−1609.30	−0.296278
$310$	0	0
$311$	−7718.69	−1.40735	−0.703677	−	0.710520i	$-0.748460\pi$
$311$	−7718.69	−1.40735	−0.703677	+	0.710520i	$0.748460\pi$
$312$	0	0
$313$	8556.00	1.54509	0.772546	−	0.634959i	$-0.218983\pi$
$313$	8556.00	1.54509	0.772546	+	0.634959i	$0.218983\pi$
$314$	0	0
$315$	185.980	0.0332660
$316$	0	0
$317$	−7780.95	−1.37862	−0.689309	−	0.724468i	$-0.742086\pi$
$317$	−7780.95	−1.37862	−0.689309	+	0.724468i	$0.742086\pi$
$318$	0	0
$319$	1094.28	0.192062
$320$	0	0
$321$	−1011.09	−0.175805
$322$	0	0
$323$	−2986.42	−0.514455
$324$	0	0
$325$	766.421	0.130810
$326$	0	0
$327$	8076.89	1.36591
$328$	0	0
$329$	2531.73	0.424252
$330$	0	0
$331$	4932.12	0.819015	0.409507	−	0.912307i	$-0.365701\pi$
$331$	4932.12	0.819015	0.409507	+	0.912307i	$0.365701\pi$
$332$	0	0
$333$	−1047.58	−0.172394
$334$	0	0
$335$	−832.548	−0.135782
$336$	0	0
$337$	−7121.13	−1.15108	−0.575538	−	0.817775i	$-0.695207\pi$
$337$	−7121.13	−1.15108	−0.575538	+	0.817775i	$0.695207\pi$
$338$	0	0
$339$	−8634.76	−1.38341
$340$	0	0
$341$	16437.2	2.61034
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	601.514	0.0938679
$346$	0	0
$347$	−9540.58	−1.47598	−0.737991	−	0.674811i	$-0.764225\pi$
$347$	−9540.58	−1.47598	−0.737991	+	0.674811i	$0.764225\pi$
$348$	0	0
$349$	1281.65	0.196576	0.0982880	−	0.995158i	$-0.468663\pi$
$349$	1281.65	0.196576	0.0982880	+	0.995158i	$0.468663\pi$
$350$	0	0
$351$	−4613.25	−0.701530
$352$	0	0
$353$	5798.07	0.874221	0.437110	−	0.899408i	$-0.356002\pi$
$353$	5798.07	0.874221	0.437110	+	0.899408i	$0.356002\pi$
$354$	0	0
$355$	−4760.00	−0.711647
$356$	0	0
$357$	1213.47	0.179899
$358$	0	0
$359$	−2267.29	−0.333323	−0.166662	−	0.986014i	$-0.553299\pi$
$359$	−2267.29	−0.333323	−0.166662	+	0.986014i	$0.553299\pi$
$360$	0	0
$361$	−422.886	−0.0616541
$362$	0	0
$363$	6517.58	0.942381
$364$	0	0
$365$	742.447	0.106470
$366$	0	0
$367$	7372.85	1.04866	0.524332	−	0.851514i	$-0.324315\pi$
$367$	7372.85	1.04866	0.524332	+	0.851514i	$0.324315\pi$
$368$	0	0
$369$	−60.3769	−0.00851788
$370$	0	0
$371$	1071.14	0.149894
$372$	0	0
$373$	6447.14	0.894961	0.447480	−	0.894294i	$-0.352321\pi$
$373$	6447.14	0.894961	0.447480	+	0.894294i	$0.352321\pi$
$374$	0	0
$375$	−582.107	−0.0801596
$376$	0	0
$377$	641.989	0.0877032
$378$	0	0
$379$	4247.57	0.575680	0.287840	−	0.957678i	$-0.407063\pi$
$379$	4247.57	0.575680	0.287840	+	0.957678i	$0.407063\pi$
$380$	0	0
$381$	−6494.03	−0.873226
$382$	0	0
$383$	6681.86	0.891454	0.445727	−	0.895169i	$-0.352945\pi$
$383$	6681.86	0.891454	0.445727	+	0.895169i	$0.352945\pi$
$384$	0	0
$385$	−1828.92	−0.242105
$386$	0	0
$387$	−179.781	−0.0236145
$388$	0	0
$389$	−6371.78	−0.830494	−0.415247	−	0.909709i	$-0.636305\pi$
$389$	−6371.78	−0.830494	−0.415247	+	0.909709i	$0.636305\pi$
$390$	0	0
$391$	−961.661	−0.124382
$392$	0	0
$393$	−8207.33	−1.05345
$394$	0	0
$395$	4288.62	0.546289
$396$	0	0
$397$	4247.93	0.537021	0.268510	−	0.963277i	$-0.413469\pi$
$397$	4247.93	0.537021	0.268510	+	0.963277i	$0.413469\pi$
$398$	0	0
$399$	−2615.19	−0.328128
$400$	0	0
$401$	−8833.62	−1.10008	−0.550038	−	0.835140i	$-0.685387\pi$
$401$	−8833.62	−1.10008	−0.550038	+	0.835140i	$0.685387\pi$
$402$	0	0
$403$	9643.37	1.19199
$404$	0	0
$405$	2786.47	0.341879
$406$	0	0
$407$	10301.9	1.25466
$408$	0	0
$409$	−319.205	−0.0385908	−0.0192954	−	0.999814i	$-0.506142\pi$
$409$	−319.205	−0.0385908	−0.0192954	+	0.999814i	$0.506142\pi$
$410$	0	0
$411$	−4298.04	−0.515831
$412$	0	0
$413$	4312.00	0.513752
$414$	0	0
$415$	3302.64	0.390651
$416$	0	0
$417$	912.924	0.107209
$418$	0	0
$419$	12789.2	1.49115	0.745577	−	0.666420i	$-0.232174\pi$
$419$	12789.2	1.49115	0.745577	+	0.666420i	$0.232174\pi$
$420$	0	0
$421$	−6747.40	−0.781112	−0.390556	−	0.920579i	$-0.627717\pi$
$421$	−6747.40	−0.781112	−0.390556	+	0.920579i	$0.627717\pi$
$422$	0	0
$423$	−1921.84	−0.220906
$424$	0	0
$425$	930.635	0.106217
$426$	0	0
$427$	106.855	0.0121102
$428$	0	0
$429$	7460.14	0.839577
$430$	0	0
$431$	5184.75	0.579444	0.289722	−	0.957111i	$-0.406437\pi$
$431$	5184.75	0.579444	0.289722	+	0.957111i	$0.406437\pi$
$432$	0	0
$433$	−4242.03	−0.470806	−0.235403	−	0.971898i	$-0.575641\pi$
$433$	−4242.03	−0.470806	−0.235403	+	0.971898i	$0.575641\pi$
$434$	0	0
$435$	−487.599	−0.0537439
$436$	0	0
$437$	2072.50	0.226868
$438$	0	0
$439$	5434.12	0.590789	0.295394	−	0.955375i	$-0.404549\pi$
$439$	5434.12	0.590789	0.295394	+	0.955375i	$0.404549\pi$
$440$	0	0
$441$	−260.372	−0.0281149
$442$	0	0
$443$	11493.8	1.23270	0.616350	−	0.787472i	$-0.288611\pi$
$443$	11493.8	1.23270	0.616350	+	0.787472i	$0.288611\pi$
$444$	0	0
$445$	228.873	0.0243812
$446$	0	0
$447$	3634.08	0.384532
$448$	0	0
$449$	−16849.3	−1.77098	−0.885489	−	0.464661i	$-0.846176\pi$
$449$	−16849.3	−1.77098	−0.885489	+	0.464661i	$0.846176\pi$
$450$	0	0
$451$	593.745	0.0619919
$452$	0	0
$453$	10801.2	1.12028
$454$	0	0
$455$	−1072.99	−0.110555
$456$	0	0
$457$	15348.5	1.57106	0.785528	−	0.618826i	$-0.212391\pi$
$457$	15348.5	1.57106	0.785528	+	0.618826i	$0.212391\pi$
$458$	0	0
$459$	−5601.69	−0.569639
$460$	0	0
$461$	14038.4	1.41830	0.709148	−	0.705059i	$-0.249080\pi$
$461$	14038.4	1.41830	0.709148	+	0.705059i	$0.249080\pi$
$462$	0	0
$463$	8661.23	0.869377	0.434689	−	0.900581i	$-0.356858\pi$
$463$	8661.23	0.869377	0.434689	+	0.900581i	$0.356858\pi$
$464$	0	0
$465$	−7324.26	−0.730440
$466$	0	0
$467$	−7014.71	−0.695079	−0.347539	−	0.937665i	$-0.612983\pi$
$467$	−7014.71	−0.695079	−0.347539	+	0.937665i	$0.612983\pi$
$468$	0	0
$469$	1165.57	0.114757
$470$	0	0
$471$	4763.50	0.466010
$472$	0	0
$473$	1767.96	0.171863
$474$	0	0
$475$	−2005.63	−0.193737
$476$	0	0
$477$	−813.100	−0.0780488
$478$	0	0
$479$	−18134.7	−1.72984	−0.864922	−	0.501907i	$-0.832632\pi$
$479$	−18134.7	−1.72984	−0.864922	+	0.501907i	$0.832632\pi$
$480$	0	0
$481$	6043.91	0.572929
$482$	0	0
$483$	−842.119	−0.0793328
$484$	0	0
$485$	−8410.63	−0.787438
$486$	0	0
$487$	−16537.8	−1.53881	−0.769405	−	0.638761i	$-0.779447\pi$
$487$	−16537.8	−1.53881	−0.769405	+	0.638761i	$0.779447\pi$
$488$	0	0
$489$	6289.67	0.581654
$490$	0	0
$491$	−220.608	−0.0202768	−0.0101384	−	0.999949i	$-0.503227\pi$
$491$	−220.608	−0.0202768	−0.0101384	+	0.999949i	$0.503227\pi$
$492$	0	0
$493$	779.542	0.0712146
$494$	0	0
$495$	1388.33	0.126063
$496$	0	0
$497$	6664.00	0.601451
$498$	0	0
$499$	−5939.04	−0.532801	−0.266401	−	0.963862i	$-0.585834\pi$
$499$	−5939.04	−0.532801	−0.266401	+	0.963862i	$0.585834\pi$
$500$	0	0
$501$	5730.62	0.511029
$502$	0	0
$503$	11604.8	1.02869	0.514345	−	0.857584i	$-0.328035\pi$
$503$	11604.8	1.02869	0.514345	+	0.857584i	$0.328035\pi$
$504$	0	0
$505$	2170.83	0.191289
$506$	0	0
$507$	−5854.40	−0.512826
$508$	0	0
$509$	−1867.67	−0.162639	−0.0813193	−	0.996688i	$-0.525913\pi$
$509$	−1867.67	−0.162639	−0.0813193	+	0.996688i	$0.525913\pi$
$510$	0	0
$511$	−1039.43	−0.0899834
$512$	0	0
$513$	12072.3	1.03900
$514$	0	0
$515$	1727.88	0.147844
$516$	0	0
$517$	18899.3	1.60772
$518$	0	0
$519$	−11584.6	−0.979786
$520$	0	0
$521$	6117.21	0.514395	0.257197	−	0.966359i	$-0.417201\pi$
$521$	6117.21	0.514395	0.257197	+	0.966359i	$0.417201\pi$
$522$	0	0
$523$	16685.6	1.39505	0.697524	−	0.716561i	$-0.254285\pi$
$523$	16685.6	1.39505	0.697524	+	0.716561i	$0.254285\pi$
$524$	0	0
$525$	814.949	0.0677473
$526$	0	0
$527$	11709.6	0.967887
$528$	0	0
$529$	−11499.6	−0.945149
$530$	0	0
$531$	−3273.24	−0.267508
$532$	0	0
$533$	348.338	0.0283081
$534$	0	0
$535$	1085.59	0.0877276
$536$	0	0
$537$	−7549.59	−0.606683
$538$	0	0
$539$	2560.49	0.204616
$540$	0	0
$541$	9309.03	0.739790	0.369895	−	0.929074i	$-0.379394\pi$
$541$	9309.03	0.739790	0.369895	+	0.929074i	$0.379394\pi$
$542$	0	0
$543$	12078.4	0.954575
$544$	0	0
$545$	−8672.05	−0.681596
$546$	0	0
$547$	−10894.7	−0.851598	−0.425799	−	0.904818i	$-0.640007\pi$
$547$	−10894.7	−0.851598	−0.425799	+	0.904818i	$0.640007\pi$
$548$	0	0
$549$	−81.1134	−0.00630571
$550$	0	0
$551$	−1680.01	−0.129893
$552$	0	0
$553$	−6004.07	−0.461698
$554$	0	0
$555$	−4590.43	−0.351086
$556$	0	0
$557$	−7873.90	−0.598973	−0.299486	−	0.954101i	$-0.596815\pi$
$557$	−7873.90	−0.598973	−0.299486	+	0.954101i	$0.596815\pi$
$558$	0	0
$559$	1037.23	0.0784796
$560$	0	0
$561$	9058.55	0.681733
$562$	0	0
$563$	−21770.7	−1.62971	−0.814854	−	0.579666i	$-0.803183\pi$
$563$	−21770.7	−1.62971	−0.814854	+	0.579666i	$0.803183\pi$
$564$	0	0
$565$	9271.02	0.690327
$566$	0	0
$567$	−3901.06	−0.288940
$568$	0	0
$569$	−12381.3	−0.912213	−0.456106	−	0.889925i	$-0.650756\pi$
$569$	−12381.3	−0.912213	−0.456106	+	0.889925i	$0.650756\pi$
$570$	0	0
$571$	5768.38	0.422765	0.211383	−	0.977403i	$-0.432203\pi$
$571$	5768.38	0.422765	0.211383	+	0.977403i	$0.432203\pi$
$572$	0	0
$573$	8489.81	0.618965
$574$	0	0
$575$	−645.837	−0.0468405
$576$	0	0
$577$	4733.38	0.341513	0.170757	−	0.985313i	$-0.445379\pi$
$577$	4733.38	0.341513	0.170757	+	0.985313i	$0.445379\pi$
$578$	0	0
$579$	−7176.34	−0.515093
$580$	0	0
$581$	−4623.70	−0.330161
$582$	0	0
$583$	7996.00	0.568028
$584$	0	0
$585$	814.508	0.0575654
$586$	0	0
$587$	8441.67	0.593569	0.296785	−	0.954944i	$-0.404086\pi$
$587$	8441.67	0.593569	0.296785	+	0.954944i	$0.404086\pi$
$588$	0	0
$589$	−25235.6	−1.76539
$590$	0	0
$591$	3265.59	0.227290
$592$	0	0
$593$	18939.9	1.31158	0.655791	−	0.754943i	$-0.272335\pi$
$593$	18939.9	1.31158	0.655791	+	0.754943i	$0.272335\pi$
$594$	0	0
$595$	−1302.89	−0.0897701
$596$	0	0
$597$	−15344.2	−1.05192
$598$	0	0
$599$	−22655.3	−1.54536	−0.772681	−	0.634794i	$-0.781085\pi$
$599$	−22655.3	−1.54536	−0.772681	+	0.634794i	$0.781085\pi$
$600$	0	0
$601$	−15947.4	−1.08237	−0.541187	−	0.840902i	$-0.682025\pi$
$601$	−15947.4	−1.08237	−0.541187	+	0.840902i	$0.682025\pi$
$602$	0	0
$603$	−884.784	−0.0597532
$604$	0	0
$605$	−6997.84	−0.470252
$606$	0	0
$607$	25993.2	1.73811	0.869053	−	0.494719i	$-0.164729\pi$
$607$	25993.2	1.73811	0.869053	+	0.494719i	$0.164729\pi$
$608$	0	0
$609$	682.638	0.0454218
$610$	0	0
$611$	11087.9	0.734152
$612$	0	0
$613$	665.408	0.0438427	0.0219213	−	0.999760i	$-0.493022\pi$
$613$	665.408	0.0438427	0.0219213	+	0.999760i	$0.493022\pi$
$614$	0	0
$615$	−264.567	−0.0173470
$616$	0	0
$617$	18401.3	1.20066	0.600330	−	0.799752i	$-0.295036\pi$
$617$	18401.3	1.20066	0.600330	+	0.799752i	$0.295036\pi$
$618$	0	0
$619$	11150.6	0.724040	0.362020	−	0.932170i	$-0.382087\pi$
$619$	11150.6	0.724040	0.362020	+	0.932170i	$0.382087\pi$
$620$	0	0
$621$	3887.43	0.251203
$622$	0	0
$623$	−320.422	−0.0206059
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−19522.3	−1.24345
$628$	0	0
$629$	7338.88	0.465215
$630$	0	0
$631$	−5381.79	−0.339534	−0.169767	−	0.985484i	$-0.554301\pi$
$631$	−5381.79	−0.339534	−0.169767	+	0.985484i	$0.554301\pi$
$632$	0	0
$633$	−19010.9	−1.19371
$634$	0	0
$635$	6972.55	0.435744
$636$	0	0
$637$	1502.19	0.0934361
$638$	0	0
$639$	−5058.65	−0.313172
$640$	0	0
$641$	−19455.1	−1.19880	−0.599398	−	0.800451i	$-0.704593\pi$
$641$	−19455.1	−1.19880	−0.599398	+	0.800451i	$0.704593\pi$
$642$	0	0
$643$	14695.8	0.901317	0.450658	−	0.892696i	$-0.351189\pi$
$643$	14695.8	0.901317	0.450658	+	0.892696i	$0.351189\pi$
$644$	0	0
$645$	−787.788	−0.0480917
$646$	0	0
$647$	12694.8	0.771383	0.385691	−	0.922628i	$-0.373963\pi$
$647$	12694.8	0.771383	0.385691	+	0.922628i	$0.373963\pi$
$648$	0	0
$649$	32189.0	1.94688
$650$	0	0
$651$	10254.0	0.617334
$652$	0	0
$653$	−12385.6	−0.742247	−0.371124	−	0.928583i	$-0.621027\pi$
$653$	−12385.6	−0.742247	−0.371124	+	0.928583i	$0.621027\pi$
$654$	0	0
$655$	8812.09	0.525675
$656$	0	0
$657$	789.030	0.0468539
$658$	0	0
$659$	2072.18	0.122489	0.0612447	−	0.998123i	$-0.480493\pi$
$659$	2072.18	0.122489	0.0612447	+	0.998123i	$0.480493\pi$
$660$	0	0
$661$	1074.36	0.0632193	0.0316096	−	0.999500i	$-0.489937\pi$
$661$	1074.36	0.0632193	0.0316096	+	0.999500i	$0.489937\pi$
$662$	0	0
$663$	5314.47	0.311307
$664$	0	0
$665$	2807.89	0.163737
$666$	0	0
$667$	−540.982	−0.0314047
$668$	0	0
$669$	−3479.42	−0.201080
$670$	0	0
$671$	797.667	0.0458921
$672$	0	0
$673$	26195.2	1.50037	0.750186	−	0.661226i	$-0.229964\pi$
$673$	26195.2	1.50037	0.750186	+	0.661226i	$0.229964\pi$
$674$	0	0
$675$	−3762.01	−0.214518
$676$	0	0
$677$	−4228.44	−0.240047	−0.120024	−	0.992771i	$-0.538297\pi$
$677$	−4228.44	−0.240047	−0.120024	+	0.992771i	$0.538297\pi$
$678$	0	0
$679$	11774.9	0.665506
$680$	0	0
$681$	−7756.80	−0.436478
$682$	0	0
$683$	−27525.5	−1.54207	−0.771036	−	0.636792i	$-0.780261\pi$
$683$	−27525.5	−1.54207	−0.771036	+	0.636792i	$0.780261\pi$
$684$	0	0
$685$	4614.74	0.257402
$686$	0	0
$687$	−30867.3	−1.71421
$688$	0	0
$689$	4691.09	0.259385
$690$	0	0
$691$	33324.4	1.83462	0.917309	−	0.398177i	$-0.130357\pi$
$691$	33324.4	1.83462	0.917309	+	0.398177i	$0.130357\pi$
$692$	0	0
$693$	−1943.67	−0.106542
$694$	0	0
$695$	−980.193	−0.0534976
$696$	0	0
$697$	422.973	0.0229860
$698$	0	0
$699$	−2013.77	−0.108967
$700$	0	0
$701$	−33262.9	−1.79219	−0.896094	−	0.443864i	$-0.853607\pi$
$701$	−33262.9	−1.79219	−0.896094	+	0.443864i	$0.853607\pi$
$702$	0	0
$703$	−15816.2	−0.848534
$704$	0	0
$705$	−8421.37	−0.449882
$706$	0	0
$707$	−3039.17	−0.161668
$708$	0	0
$709$	13703.0	0.725851	0.362926	−	0.931818i	$-0.381778\pi$
$709$	13703.0	0.725851	0.362926	+	0.931818i	$0.381778\pi$
$710$	0	0
$711$	4557.70	0.240404
$712$	0	0
$713$	−8126.14	−0.426825
$714$	0	0
$715$	−8009.84	−0.418953
$716$	0	0
$717$	−25987.3	−1.35358
$718$	0	0
$719$	8074.93	0.418838	0.209419	−	0.977826i	$-0.432843\pi$
$719$	8074.93	0.418838	0.209419	+	0.977826i	$0.432843\pi$
$720$	0	0
$721$	−2419.04	−0.124951
$722$	0	0
$723$	−29323.6	−1.50838
$724$	0	0
$725$	523.528	0.0268184
$726$	0	0
$727$	3668.70	0.187159	0.0935794	−	0.995612i	$-0.470169\pi$
$727$	3668.70	0.187159	0.0935794	+	0.995612i	$0.470169\pi$
$728$	0	0
$729$	21881.9	1.11172
$730$	0	0
$731$	1259.46	0.0637250
$732$	0	0
$733$	−14980.3	−0.754857	−0.377428	−	0.926039i	$-0.623192\pi$
$733$	−14980.3	−0.754857	−0.377428	+	0.926039i	$0.623192\pi$
$734$	0	0
$735$	−1140.93	−0.0572569
$736$	0	0
$737$	8700.94	0.434875
$738$	0	0
$739$	−6530.59	−0.325077	−0.162538	−	0.986702i	$-0.551968\pi$
$739$	−6530.59	−0.325077	−0.162538	+	0.986702i	$0.551968\pi$
$740$	0	0
$741$	−11453.3	−0.567812
$742$	0	0
$743$	−25952.0	−1.28141	−0.640704	−	0.767788i	$-0.721357\pi$
$743$	−25952.0	−1.28141	−0.640704	+	0.767788i	$0.721357\pi$
$744$	0	0
$745$	−3901.86	−0.191883
$746$	0	0
$747$	3509.85	0.171913
$748$	0	0
$749$	−1519.83	−0.0741434
$750$	0	0
$751$	14093.9	0.684813	0.342407	−	0.939552i	$-0.388758\pi$
$751$	14093.9	0.684813	0.342407	+	0.939552i	$0.388758\pi$
$752$	0	0
$753$	−1452.57	−0.0702983
$754$	0	0
$755$	−11597.1	−0.559024
$756$	0	0
$757$	−2554.41	−0.122644	−0.0613220	−	0.998118i	$-0.519532\pi$
$757$	−2554.41	−0.122644	−0.0613220	+	0.998118i	$0.519532\pi$
$758$	0	0
$759$	−6286.40	−0.300635
$760$	0	0
$761$	2219.08	0.105705	0.0528527	−	0.998602i	$-0.483169\pi$
$761$	2219.08	0.105705	0.0528527	+	0.998602i	$0.483169\pi$
$762$	0	0
$763$	12140.9	0.576054
$764$	0	0
$765$	989.025	0.0467428
$766$	0	0
$767$	18884.6	0.889028
$768$	0	0
$769$	−22466.2	−1.05352	−0.526758	−	0.850015i	$-0.676592\pi$
$769$	−22466.2	−1.05352	−0.526758	+	0.850015i	$0.676592\pi$
$770$	0	0
$771$	−36609.3	−1.71006
$772$	0	0
$773$	9674.79	0.450165	0.225083	−	0.974340i	$-0.427735\pi$
$773$	9674.79	0.450165	0.225083	+	0.974340i	$0.427735\pi$
$774$	0	0
$775$	7863.96	0.364493
$776$	0	0
$777$	6426.60	0.296722
$778$	0	0
$779$	−911.560	−0.0419256
$780$	0	0
$781$	49746.6	2.27922
$782$	0	0
$783$	−3151.23	−0.143826
$784$	0	0
$785$	−5114.51	−0.232541
$786$	0	0
$787$	20942.8	0.948577	0.474288	−	0.880370i	$-0.342705\pi$
$787$	20942.8	0.948577	0.474288	+	0.880370i	$0.342705\pi$
$788$	0	0
$789$	−24341.8	−1.09834
$790$	0	0
$791$	−12979.4	−0.583433
$792$	0	0
$793$	467.975	0.0209562
$794$	0	0
$795$	−3562.94	−0.158949
$796$	0	0
$797$	23526.6	1.04561	0.522807	−	0.852451i	$-0.324885\pi$
$797$	23526.6	1.04561	0.522807	+	0.852451i	$0.324885\pi$
$798$	0	0
$799$	13463.5	0.596127
$800$	0	0
$801$	243.233	0.0107294
$802$	0	0
$803$	−7759.29	−0.340996
$804$	0	0
$805$	904.172	0.0395874
$806$	0	0
$807$	5968.72	0.260358
$808$	0	0
$809$	−18202.2	−0.791047	−0.395523	−	0.918456i	$-0.629437\pi$
$809$	−18202.2	−0.791047	−0.395523	+	0.918456i	$0.629437\pi$
$810$	0	0
$811$	2510.24	0.108689	0.0543443	−	0.998522i	$-0.482693\pi$
$811$	2510.24	0.108689	0.0543443	+	0.998522i	$0.482693\pi$
$812$	0	0
$813$	−21906.5	−0.945012
$814$	0	0
$815$	−6753.13	−0.290248
$816$	0	0
$817$	−2714.30	−0.116232
$818$	0	0
$819$	−1140.31	−0.0486516
$820$	0	0
$821$	17899.6	0.760903	0.380451	−	0.924801i	$-0.375769\pi$
$821$	17899.6	0.760903	0.380451	+	0.924801i	$0.375769\pi$
$822$	0	0
$823$	−14039.5	−0.594637	−0.297318	−	0.954778i	$-0.596092\pi$
$823$	−14039.5	−0.594637	−0.297318	+	0.954778i	$0.596092\pi$
$824$	0	0
$825$	6083.58	0.256731
$826$	0	0
$827$	−15127.4	−0.636073	−0.318036	−	0.948079i	$-0.603023\pi$
$827$	−15127.4	−0.636073	−0.318036	+	0.948079i	$0.603023\pi$
$828$	0	0
$829$	21986.5	0.921136	0.460568	−	0.887624i	$-0.347646\pi$
$829$	21986.5	0.921136	0.460568	+	0.887624i	$0.347646\pi$
$830$	0	0
$831$	41718.7	1.74152
$832$	0	0
$833$	1824.04	0.0758696
$834$	0	0
$835$	−6152.89	−0.255005
$836$	0	0
$837$	−47334.8	−1.95476
$838$	0	0
$839$	2276.89	0.0936914	0.0468457	−	0.998902i	$-0.485083\pi$
$839$	2276.89	0.0936914	0.0468457	+	0.998902i	$0.485083\pi$
$840$	0	0
$841$	−23950.5	−0.982019
$842$	0	0
$843$	−1727.25	−0.0705688
$844$	0	0
$845$	6285.79	0.255903
$846$	0	0
$847$	9796.97	0.397436
$848$	0	0
$849$	27113.4	1.09603
$850$	0	0
$851$	−5093.00	−0.205154
$852$	0	0
$853$	13342.6	0.535570	0.267785	−	0.963479i	$-0.413708\pi$
$853$	13342.6	0.535570	0.267785	+	0.963479i	$0.413708\pi$
$854$	0	0
$855$	−2131.47	−0.0852571
$856$	0	0
$857$	18690.9	0.745003	0.372502	−	0.928032i	$-0.378500\pi$
$857$	18690.9	0.745003	0.372502	+	0.928032i	$0.378500\pi$
$858$	0	0
$859$	−18318.9	−0.727628	−0.363814	−	0.931472i	$-0.618526\pi$
$859$	−18318.9	−0.727628	−0.363814	+	0.931472i	$0.618526\pi$
$860$	0	0
$861$	370.394	0.0146609
$862$	0	0
$863$	−38133.1	−1.50413	−0.752067	−	0.659087i	$-0.770943\pi$
$863$	−38133.1	−1.50413	−0.752067	+	0.659087i	$0.770943\pi$
$864$	0	0
$865$	12438.3	0.488917
$866$	0	0
$867$	−16426.0	−0.643432
$868$	0	0
$869$	−44820.3	−1.74962
$870$	0	0
$871$	5104.66	0.198582
$872$	0	0
$873$	−8938.33	−0.346525
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−19707.5	−0.758807	−0.379404	−	0.925231i	$-0.623871\pi$
$877$	−19707.5	−0.758807	−0.379404	+	0.925231i	$0.623871\pi$
$878$	0	0
$879$	34664.6	1.33016
$880$	0	0
$881$	−14091.5	−0.538883	−0.269441	−	0.963017i	$-0.586839\pi$
$881$	−14091.5	−0.538883	−0.269441	+	0.963017i	$0.586839\pi$
$882$	0	0
$883$	−3115.87	−0.118751	−0.0593757	−	0.998236i	$-0.518911\pi$
$883$	−3115.87	−0.118751	−0.0593757	+	0.998236i	$0.518911\pi$
$884$	0	0
$885$	−14343.1	−0.544789
$886$	0	0
$887$	−38734.6	−1.46627	−0.733134	−	0.680084i	$-0.761943\pi$
$887$	−38734.6	−1.46627	−0.733134	+	0.680084i	$0.761943\pi$
$888$	0	0
$889$	−9761.57	−0.368270
$890$	0	0
$891$	−29121.3	−1.09495
$892$	0	0
$893$	−29015.6	−1.08731
$894$	0	0
$895$	8105.89	0.302737
$896$	0	0
$897$	−3688.10	−0.137282
$898$	0	0
$899$	6587.21	0.244378
$900$	0	0
$901$	5696.21	0.210619
$902$	0	0
$903$	1102.90	0.0406449
$904$	0	0
$905$	−12968.4	−0.476337
$906$	0	0
$907$	−19242.9	−0.704464	−0.352232	−	0.935913i	$-0.614577\pi$
$907$	−19242.9	−0.704464	−0.352232	+	0.935913i	$0.614577\pi$
$908$	0	0
$909$	2307.03	0.0841799
$910$	0	0
$911$	−34613.3	−1.25882	−0.629412	−	0.777072i	$-0.716704\pi$
$911$	−34613.3	−1.25882	−0.629412	+	0.777072i	$0.716704\pi$
$912$	0	0
$913$	−34515.8	−1.25116
$914$	0	0
$915$	−355.433	−0.0128418
$916$	0	0
$917$	−12336.9	−0.444276
$918$	0	0
$919$	−25826.4	−0.927022	−0.463511	−	0.886091i	$-0.653411\pi$
$919$	−25826.4	−0.927022	−0.463511	+	0.886091i	$0.653411\pi$
$920$	0	0
$921$	3547.01	0.126903
$922$	0	0
$923$	29185.3	1.04079
$924$	0	0
$925$	4928.68	0.175194
$926$	0	0
$927$	1836.29	0.0650613
$928$	0	0
$929$	19451.6	0.686960	0.343480	−	0.939160i	$-0.388394\pi$
$929$	19451.6	0.686960	0.343480	+	0.939160i	$0.388394\pi$
$930$	0	0
$931$	−3931.04	−0.138383
$932$	0	0
$933$	−35944.8	−1.26129
$934$	0	0
$935$	−9726.03	−0.340188
$936$	0	0
$937$	34469.1	1.20177	0.600884	−	0.799336i	$-0.294815\pi$
$937$	34469.1	1.20177	0.600884	+	0.799336i	$0.294815\pi$
$938$	0	0
$939$	39844.1	1.38473
$940$	0	0
$941$	14156.4	0.490419	0.245209	−	0.969470i	$-0.421143\pi$
$941$	14156.4	0.490419	0.245209	+	0.969470i	$0.421143\pi$
$942$	0	0
$943$	−293.532	−0.0101365
$944$	0	0
$945$	5266.81	0.181301
$946$	0	0
$947$	38092.4	1.30711	0.653557	−	0.756877i	$-0.273276\pi$
$947$	38092.4	1.30711	0.653557	+	0.756877i	$0.273276\pi$
$948$	0	0
$949$	−4552.22	−0.155713
$950$	0	0
$951$	−36234.8	−1.23553
$952$	0	0
$953$	5037.40	0.171225	0.0856126	−	0.996329i	$-0.472715\pi$
$953$	5037.40	0.171225	0.0856126	+	0.996329i	$0.472715\pi$
$954$	0	0
$955$	−9115.39	−0.308866
$956$	0	0
$957$	5095.88	0.172128
$958$	0	0
$959$	−6460.64	−0.217544
$960$	0	0
$961$	69156.0	2.32137
$962$	0	0
$963$	1153.71	0.0386060
$964$	0	0
$965$	7705.14	0.257033
$966$	0	0
$967$	−11495.3	−0.382278	−0.191139	−	0.981563i	$-0.561218\pi$
$967$	−11495.3	−0.382278	−0.191139	+	0.981563i	$0.561218\pi$
$968$	0	0
$969$	−13907.3	−0.461061
$970$	0	0
$971$	−22352.7	−0.738757	−0.369379	−	0.929279i	$-0.620429\pi$
$971$	−22352.7	−0.738757	−0.369379	+	0.929279i	$0.620429\pi$
$972$	0	0
$973$	1372.27	0.0452138
$974$	0	0
$975$	3569.11	0.117234
$976$	0	0
$977$	14345.7	0.469765	0.234882	−	0.972024i	$-0.424530\pi$
$977$	14345.7	0.469765	0.234882	+	0.972024i	$0.424530\pi$
$978$	0	0
$979$	−2391.94	−0.0780867
$980$	0	0
$981$	−9216.15	−0.299948
$982$	0	0
$983$	−34460.9	−1.11814	−0.559070	−	0.829120i	$-0.688842\pi$
$983$	−34460.9	−1.11814	−0.559070	+	0.829120i	$0.688842\pi$
$984$	0	0
$985$	−3506.21	−0.113419
$986$	0	0
$987$	11789.9	0.380220
$988$	0	0
$989$	−874.036	−0.0281019
$990$	0	0
$991$	35189.6	1.12799	0.563993	−	0.825780i	$-0.309265\pi$
$991$	35189.6	1.12799	0.563993	+	0.825780i	$0.309265\pi$
$992$	0	0
$993$	22968.2	0.734011
$994$	0	0
$995$	16474.8	0.524911
$996$	0	0
$997$	−50730.0	−1.61147	−0.805734	−	0.592277i	$-0.798229\pi$
$997$	−50730.0	−1.61147	−0.805734	+	0.592277i	$0.798229\pi$
$998$	0	0
$999$	−29666.8	−0.939554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.4.a.r.1.2		2
4.3	odd	2		35.4.a.b.1.2	✓	2
8.3	odd	2		2240.4.a.bn.1.2		2
8.5	even	2		2240.4.a.bo.1.1		2
12.11	even	2		315.4.a.f.1.1		2
20.3	even	4		175.4.b.c.99.1		4
20.7	even	4		175.4.b.c.99.4		4
20.19	odd	2		175.4.a.c.1.1		2
28.3	even	6		245.4.e.i.226.1		4
28.11	odd	6		245.4.e.h.226.1		4
28.19	even	6		245.4.e.i.116.1		4
28.23	odd	6		245.4.e.h.116.1		4
28.27	even	2		245.4.a.k.1.2		2
60.59	even	2		1575.4.a.z.1.2		2
84.83	odd	2		2205.4.a.u.1.1		2
140.139	even	2		1225.4.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.b.1.2	✓	2	4.3	odd	2
175.4.a.c.1.1		2	20.19	odd	2
175.4.b.c.99.1		4	20.3	even	4
175.4.b.c.99.4		4	20.7	even	4
245.4.a.k.1.2		2	28.27	even	2
245.4.e.h.116.1		4	28.23	odd	6
245.4.e.h.226.1		4	28.11	odd	6
245.4.e.i.116.1		4	28.19	even	6
245.4.e.i.226.1		4	28.3	even	6
315.4.a.f.1.1		2	12.11	even	2
560.4.a.r.1.2		2	1.1	even	1	trivial
1225.4.a.m.1.1		2	140.139	even	2
1575.4.a.z.1.2		2	60.59	even	2
2205.4.a.u.1.1		2	84.83	odd	2
2240.4.a.bn.1.2		2	8.3	odd	2
2240.4.a.bo.1.1		2	8.5	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q+4.65685 q^{3} -5.00000 q^{5} +7.00000 q^{7} -5.31371 q^{9} +52.2548 q^{11} +30.6569 q^{13} -23.2843 q^{15} +37.2254 q^{17} -80.2254 q^{19} +32.5980 q^{21} -25.8335 q^{23} +25.0000 q^{25} -150.480 q^{27} +20.9411 q^{29} +314.558 q^{31} +243.343 q^{33} -35.0000 q^{35} +197.147 q^{37} +142.765 q^{39} +11.3625 q^{41} +33.8335 q^{43} +26.5685 q^{45} +361.676 q^{47} +49.0000 q^{49} +173.353 q^{51} +153.019 q^{53} -261.274 q^{55} -373.598 q^{57} +616.000 q^{59} +15.2649 q^{61} -37.1960 q^{63} -153.284 q^{65} +166.510 q^{67} -120.303 q^{69} +952.000 q^{71} -148.489 q^{73} +116.421 q^{75} +365.784 q^{77} -857.725 q^{79} -557.294 q^{81} -660.528 q^{83} -186.127 q^{85} +97.5198 q^{87} -45.7746 q^{89} +214.598 q^{91} +1464.85 q^{93} +401.127 q^{95} +1682.13 q^{97} -277.667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 10 q^{5} + 14 q^{7} + 12 q^{9} + 14 q^{11} + 50 q^{13} + 10 q^{15} - 50 q^{17} - 36 q^{19} - 14 q^{21} - 244 q^{23} + 50 q^{25} - 86 q^{27} - 26 q^{29} + 120 q^{31} + 498 q^{33} - 70 q^{35}+ \cdots - 940 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 10 * q^5 + 14 * q^7 + 12 * q^9 + 14 * q^11 + 50 * q^13 + 10 * q^15 - 50 * q^17 - 36 * q^19 - 14 * q^21 - 244 * q^23 + 50 * q^25 - 86 * q^27 - 26 * q^29 + 120 * q^31 + 498 * q^33 - 70 * q^35 + 564 * q^37 + 14 * q^39 - 328 * q^41 + 260 * q^43 - 60 * q^45 + 350 * q^47 + 98 * q^49 + 754 * q^51 - 56 * q^53 - 70 * q^55 - 668 * q^57 + 1232 * q^59 + 336 * q^61 + 84 * q^63 - 250 * q^65 + 152 * q^67 + 1332 * q^69 + 1904 * q^71 + 676 * q^73 - 50 * q^75 + 98 * q^77 - 1014 * q^79 - 1454 * q^81 + 376 * q^83 + 250 * q^85 + 410 * q^87 - 216 * q^89 + 350 * q^91 + 2760 * q^93 + 180 * q^95 + 2742 * q^97 - 940 * q^99

Introduction

L-functions

Modular forms

Varieties

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