LMFDB - Embedded newform 1840.4.a.s.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 81x^{4} + 161x^{3} + 1520x^{2} - 3915x + 588 $ x^6 - 2x^5 - 81x^4 + 161x^3 + 1520x^2 - 3915*x + 588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-6.42830$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.42830 q^{3} -5.00000 q^{5} -8.32258 q^{7} +14.3231 q^{9} +O(q^{10})$	$q+6.42830 q^{3} -5.00000 q^{5} -8.32258 q^{7} +14.3231 q^{9} -1.59730 q^{11} +10.1241 q^{13} -32.1415 q^{15} -43.6876 q^{17} +48.9819 q^{19} -53.5001 q^{21} +23.0000 q^{23} +25.0000 q^{25} -81.4910 q^{27} +200.736 q^{29} -34.6948 q^{31} -10.2679 q^{33} +41.6129 q^{35} +92.5057 q^{37} +65.0806 q^{39} +253.390 q^{41} -446.151 q^{43} -71.6154 q^{45} -315.028 q^{47} -273.735 q^{49} -280.837 q^{51} -515.242 q^{53} +7.98651 q^{55} +314.870 q^{57} -98.7342 q^{59} -309.934 q^{61} -119.205 q^{63} -50.6204 q^{65} +395.169 q^{67} +147.851 q^{69} +700.646 q^{71} -531.139 q^{73} +160.708 q^{75} +13.2937 q^{77} -107.360 q^{79} -910.573 q^{81} -531.243 q^{83} +218.438 q^{85} +1290.39 q^{87} +228.493 q^{89} -84.2584 q^{91} -223.028 q^{93} -244.909 q^{95} -618.384 q^{97} -22.8783 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 30 * q^5 - 28 * q^7 + 4 * q^9	$ 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9} + 3 q^{11} - 28 q^{13} + 10 q^{15} + 24 q^{17} + 3 q^{19} + 60 q^{21} + 138 q^{23} + 150 q^{25} + 97 q^{27} + 76 q^{29} + 381 q^{31} - 3 q^{33} + 140 q^{35} + 131 q^{37} - 41 q^{39} - 95 q^{41} + 202 q^{43} - 20 q^{45} + 119 q^{47} - 578 q^{49} + 997 q^{51} + 137 q^{53} - 15 q^{55} - 894 q^{57} + 39 q^{59} - 573 q^{61} + 355 q^{63} + 140 q^{65} + 563 q^{67} - 46 q^{69} + 83 q^{71} - 1799 q^{73} - 50 q^{75} - 1410 q^{77} + 1636 q^{79} - 2006 q^{81} + 1191 q^{83} - 120 q^{85} + 1821 q^{87} - 1370 q^{89} + 1251 q^{91} - 2215 q^{93} - 15 q^{95} - 3021 q^{97} + 525 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 30 * q^5 - 28 * q^7 + 4 * q^9 + 3 * q^11 - 28 * q^13 + 10 * q^15 + 24 * q^17 + 3 * q^19 + 60 * q^21 + 138 * q^23 + 150 * q^25 + 97 * q^27 + 76 * q^29 + 381 * q^31 - 3 * q^33 + 140 * q^35 + 131 * q^37 - 41 * q^39 - 95 * q^41 + 202 * q^43 - 20 * q^45 + 119 * q^47 - 578 * q^49 + 997 * q^51 + 137 * q^53 - 15 * q^55 - 894 * q^57 + 39 * q^59 - 573 * q^61 + 355 * q^63 + 140 * q^65 + 563 * q^67 - 46 * q^69 + 83 * q^71 - 1799 * q^73 - 50 * q^75 - 1410 * q^77 + 1636 * q^79 - 2006 * q^81 + 1191 * q^83 - 120 * q^85 + 1821 * q^87 - 1370 * q^89 + 1251 * q^91 - 2215 * q^93 - 15 * q^95 - 3021 * q^97 + 525 * 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
$3$	6.42830	1.23713	0.618564	−	0.785735i	$-0.287715\pi$
$3$	6.42830	1.23713	0.618564	+	0.785735i	$0.287715\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−8.32258	−0.449377	−0.224689	−	0.974431i	$-0.572136\pi$
$7$	−8.32258	−0.449377	−0.224689	+	0.974431i	$0.572136\pi$
$8$	0	0
$9$	14.3231	0.530485
$10$	0	0
$11$	−1.59730	−0.0437822	−0.0218911	−	0.999760i	$-0.506969\pi$
$11$	−1.59730	−0.0437822	−0.0218911	+	0.999760i	$0.506969\pi$
$12$	0	0
$13$	10.1241	0.215993	0.107997	−	0.994151i	$-0.465556\pi$
$13$	10.1241	0.215993	0.107997	+	0.994151i	$0.465556\pi$
$14$	0	0
$15$	−32.1415	−0.553260
$16$	0	0
$17$	−43.6876	−0.623282	−0.311641	−	0.950200i	$-0.600879\pi$
$17$	−43.6876	−0.623282	−0.311641	+	0.950200i	$0.600879\pi$
$18$	0	0
$19$	48.9819	0.591432	0.295716	−	0.955276i	$-0.404442\pi$
$19$	48.9819	0.591432	0.295716	+	0.955276i	$0.404442\pi$
$20$	0	0
$21$	−53.5001	−0.555937
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−81.4910	−0.580850
$28$	0	0
$29$	200.736	1.28537	0.642685	−	0.766130i	$-0.277820\pi$
$29$	200.736	1.28537	0.642685	+	0.766130i	$0.277820\pi$
$30$	0	0
$31$	−34.6948	−0.201012	−0.100506	−	0.994936i	$-0.532046\pi$
$31$	−34.6948	−0.201012	−0.100506	+	0.994936i	$0.532046\pi$
$32$	0	0
$33$	−10.2679	−0.0541642
$34$	0	0
$35$	41.6129	0.200968
$36$	0	0
$37$	92.5057	0.411023	0.205511	−	0.978655i	$-0.434114\pi$
$37$	92.5057	0.411023	0.205511	+	0.978655i	$0.434114\pi$
$38$	0	0
$39$	65.0806	0.267211
$40$	0	0
$41$	253.390	0.965190	0.482595	−	0.875844i	$-0.339694\pi$
$41$	253.390	0.965190	0.482595	+	0.875844i	$0.339694\pi$
$42$	0	0
$43$	−446.151	−1.58227	−0.791133	−	0.611645i	$-0.790508\pi$
$43$	−446.151	−1.58227	−0.791133	+	0.611645i	$0.790508\pi$
$44$	0	0
$45$	−71.6154	−0.237240
$46$	0	0
$47$	−315.028	−0.977693	−0.488847	−	0.872370i	$-0.662582\pi$
$47$	−315.028	−0.977693	−0.488847	+	0.872370i	$0.662582\pi$
$48$	0	0
$49$	−273.735	−0.798060
$50$	0	0
$51$	−280.837	−0.771079
$52$	0	0
$53$	−515.242	−1.33536	−0.667678	−	0.744450i	$-0.732712\pi$
$53$	−515.242	−1.33536	−0.667678	+	0.744450i	$0.732712\pi$
$54$	0	0
$55$	7.98651	0.0195800
$56$	0	0
$57$	314.870	0.731677
$58$	0	0
$59$	−98.7342	−0.217866	−0.108933	−	0.994049i	$-0.534743\pi$
$59$	−98.7342	−0.217866	−0.108933	+	0.994049i	$0.534743\pi$
$60$	0	0
$61$	−309.934	−0.650541	−0.325271	−	0.945621i	$-0.605455\pi$
$61$	−309.934	−0.650541	−0.325271	+	0.945621i	$0.605455\pi$
$62$	0	0
$63$	−119.205	−0.238388
$64$	0	0
$65$	−50.6204	−0.0965951
$66$	0	0
$67$	395.169	0.720560	0.360280	−	0.932844i	$-0.382681\pi$
$67$	395.169	0.720560	0.360280	+	0.932844i	$0.382681\pi$
$68$	0	0
$69$	147.851	0.257959
$70$	0	0
$71$	700.646	1.17115	0.585573	−	0.810620i	$-0.300869\pi$
$71$	700.646	1.17115	0.585573	+	0.810620i	$0.300869\pi$
$72$	0	0
$73$	−531.139	−0.851578	−0.425789	−	0.904823i	$-0.640003\pi$
$73$	−531.139	−0.851578	−0.425789	+	0.904823i	$0.640003\pi$
$74$	0	0
$75$	160.708	0.247426
$76$	0	0
$77$	13.2937	0.0196747
$78$	0	0
$79$	−107.360	−0.152898	−0.0764492	−	0.997073i	$-0.524358\pi$
$79$	−107.360	−0.152898	−0.0764492	+	0.997073i	$0.524358\pi$
$80$	0	0
$81$	−910.573	−1.24907
$82$	0	0
$83$	−531.243	−0.702549	−0.351274	−	0.936273i	$-0.614252\pi$
$83$	−531.243	−0.702549	−0.351274	+	0.936273i	$0.614252\pi$
$84$	0	0
$85$	218.438	0.278740
$86$	0	0
$87$	1290.39	1.59017
$88$	0	0
$89$	228.493	0.272137	0.136068	−	0.990699i	$-0.456553\pi$
$89$	228.493	0.272137	0.136068	+	0.990699i	$0.456553\pi$
$90$	0	0
$91$	−84.2584	−0.0970625
$92$	0	0
$93$	−223.028	−0.248677
$94$	0	0
$95$	−244.909	−0.264496
$96$	0	0
$97$	−618.384	−0.647292	−0.323646	−	0.946178i	$-0.604909\pi$
$97$	−618.384	−0.647292	−0.323646	+	0.946178i	$0.604909\pi$
$98$	0	0
$99$	−22.8783	−0.0232258
$100$	0	0
$101$	−1075.77	−1.05983	−0.529915	−	0.848051i	$-0.677776\pi$
$101$	−1075.77	−1.05983	−0.529915	+	0.848051i	$0.677776\pi$
$102$	0	0
$103$	−459.840	−0.439897	−0.219948	−	0.975512i	$-0.570589\pi$
$103$	−459.840	−0.439897	−0.219948	+	0.975512i	$0.570589\pi$
$104$	0	0
$105$	267.500	0.248623
$106$	0	0
$107$	−1518.68	−1.37212	−0.686058	−	0.727547i	$-0.740660\pi$
$107$	−1518.68	−1.37212	−0.686058	+	0.727547i	$0.740660\pi$
$108$	0	0
$109$	−1186.57	−1.04268	−0.521342	−	0.853348i	$-0.674568\pi$
$109$	−1186.57	−1.04268	−0.521342	+	0.853348i	$0.674568\pi$
$110$	0	0
$111$	594.655	0.508488
$112$	0	0
$113$	−1678.95	−1.39772	−0.698862	−	0.715257i	$-0.746310\pi$
$113$	−1678.95	−1.39772	−0.698862	+	0.715257i	$0.746310\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	145.008	0.114581
$118$	0	0
$119$	363.594	0.280089
$120$	0	0
$121$	−1328.45	−0.998083
$122$	0	0
$123$	1628.87	1.19406
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	692.369	0.483763	0.241881	−	0.970306i	$-0.422236\pi$
$127$	692.369	0.483763	0.241881	+	0.970306i	$0.422236\pi$
$128$	0	0
$129$	−2868.00	−1.95746
$130$	0	0
$131$	269.315	0.179620	0.0898099	−	0.995959i	$-0.471374\pi$
$131$	269.315	0.179620	0.0898099	+	0.995959i	$0.471374\pi$
$132$	0	0
$133$	−407.656	−0.265776
$134$	0	0
$135$	407.455	0.259764
$136$	0	0
$137$	−995.345	−0.620716	−0.310358	−	0.950620i	$-0.600449\pi$
$137$	−995.345	−0.620716	−0.310358	+	0.950620i	$0.600449\pi$
$138$	0	0
$139$	15.3496	0.00936647	0.00468323	−	0.999989i	$-0.498509\pi$
$139$	15.3496	0.00936647	0.00468323	+	0.999989i	$0.498509\pi$
$140$	0	0
$141$	−2025.10	−1.20953
$142$	0	0
$143$	−16.1712	−0.00945667
$144$	0	0
$145$	−1003.68	−0.574835
$146$	0	0
$147$	−1759.65	−0.987302
$148$	0	0
$149$	1861.91	1.02372	0.511859	−	0.859070i	$-0.328957\pi$
$149$	1861.91	1.02372	0.511859	+	0.859070i	$0.328957\pi$
$150$	0	0
$151$	3009.27	1.62180	0.810898	−	0.585187i	$-0.198979\pi$
$151$	3009.27	1.62180	0.810898	+	0.585187i	$0.198979\pi$
$152$	0	0
$153$	−625.741	−0.330642
$154$	0	0
$155$	173.474	0.0898952
$156$	0	0
$157$	2413.34	1.22679	0.613394	−	0.789777i	$-0.289804\pi$
$157$	2413.34	1.22679	0.613394	+	0.789777i	$0.289804\pi$
$158$	0	0
$159$	−3312.13	−1.65201
$160$	0	0
$161$	−191.419	−0.0937016
$162$	0	0
$163$	−386.040	−0.185503	−0.0927515	−	0.995689i	$-0.529566\pi$
$163$	−386.040	−0.185503	−0.0927515	+	0.995689i	$0.529566\pi$
$164$	0	0
$165$	51.3397	0.0242230
$166$	0	0
$167$	−1032.55	−0.478448	−0.239224	−	0.970964i	$-0.576893\pi$
$167$	−1032.55	−0.478448	−0.239224	+	0.970964i	$0.576893\pi$
$168$	0	0
$169$	−2094.50	−0.953347
$170$	0	0
$171$	701.571	0.313746
$172$	0	0
$173$	−2988.81	−1.31350	−0.656748	−	0.754111i	$-0.728068\pi$
$173$	−2988.81	−1.31350	−0.656748	+	0.754111i	$0.728068\pi$
$174$	0	0
$175$	−208.065	−0.0898754
$176$	0	0
$177$	−634.693	−0.269528
$178$	0	0
$179$	−1517.25	−0.633545	−0.316773	−	0.948502i	$-0.602599\pi$
$179$	−1517.25	−0.633545	−0.316773	+	0.948502i	$0.602599\pi$
$180$	0	0
$181$	−3680.36	−1.51138	−0.755689	−	0.654931i	$-0.772698\pi$
$181$	−3680.36	−1.51138	−0.755689	+	0.654931i	$0.772698\pi$
$182$	0	0
$183$	−1992.35	−0.804803
$184$	0	0
$185$	−462.529	−0.183815
$186$	0	0
$187$	69.7823	0.0272887
$188$	0	0
$189$	678.216	0.261021
$190$	0	0
$191$	568.256	0.215275	0.107638	−	0.994190i	$-0.465671\pi$
$191$	568.256	0.215275	0.107638	+	0.994190i	$0.465671\pi$
$192$	0	0
$193$	1256.14	0.468492	0.234246	−	0.972177i	$-0.424738\pi$
$193$	1256.14	0.468492	0.234246	+	0.972177i	$0.424738\pi$
$194$	0	0
$195$	−325.403	−0.119500
$196$	0	0
$197$	−3655.52	−1.32206	−0.661028	−	0.750361i	$-0.729879\pi$
$197$	−3655.52	−1.32206	−0.661028	+	0.750361i	$0.729879\pi$
$198$	0	0
$199$	3030.78	1.07963	0.539815	−	0.841784i	$-0.318494\pi$
$199$	3030.78	1.07963	0.539815	+	0.841784i	$0.318494\pi$
$200$	0	0
$201$	2540.26	0.891425
$202$	0	0
$203$	−1670.64	−0.577616
$204$	0	0
$205$	−1266.95	−0.431646
$206$	0	0
$207$	329.431	0.110614
$208$	0	0
$209$	−78.2388	−0.0258942
$210$	0	0
$211$	2306.60	0.752574	0.376287	−	0.926503i	$-0.377201\pi$
$211$	2306.60	0.752574	0.376287	+	0.926503i	$0.377201\pi$
$212$	0	0
$213$	4503.97	1.44886
$214$	0	0
$215$	2230.76	0.707611
$216$	0	0
$217$	288.750	0.0903301
$218$	0	0
$219$	−3414.33	−1.05351
$220$	0	0
$221$	−442.296	−0.134625
$222$	0	0
$223$	−984.280	−0.295571	−0.147785	−	0.989019i	$-0.547214\pi$
$223$	−984.280	−0.295571	−0.147785	+	0.989019i	$0.547214\pi$
$224$	0	0
$225$	358.077	0.106097
$226$	0	0
$227$	1638.64	0.479120	0.239560	−	0.970882i	$-0.422997\pi$
$227$	1638.64	0.479120	0.239560	+	0.970882i	$0.422997\pi$
$228$	0	0
$229$	2472.33	0.713432	0.356716	−	0.934213i	$-0.383896\pi$
$229$	2472.33	0.713432	0.356716	+	0.934213i	$0.383896\pi$
$230$	0	0
$231$	85.4558	0.0243402
$232$	0	0
$233$	−304.031	−0.0854837	−0.0427419	−	0.999086i	$-0.513609\pi$
$233$	−304.031	−0.0854837	−0.0427419	+	0.999086i	$0.513609\pi$
$234$	0	0
$235$	1575.14	0.437238
$236$	0	0
$237$	−690.144	−0.189155
$238$	0	0
$239$	−220.665	−0.0597224	−0.0298612	−	0.999554i	$-0.509507\pi$
$239$	−220.665	−0.0597224	−0.0298612	+	0.999554i	$0.509507\pi$
$240$	0	0
$241$	−3743.94	−1.00070	−0.500349	−	0.865824i	$-0.666795\pi$
$241$	−3743.94	−1.00070	−0.500349	+	0.865824i	$0.666795\pi$
$242$	0	0
$243$	−3653.18	−0.964409
$244$	0	0
$245$	1368.67	0.356903
$246$	0	0
$247$	495.896	0.127745
$248$	0	0
$249$	−3414.99	−0.869142
$250$	0	0
$251$	−1936.75	−0.487038	−0.243519	−	0.969896i	$-0.578302\pi$
$251$	−1936.75	−0.487038	−0.243519	+	0.969896i	$0.578302\pi$
$252$	0	0
$253$	−36.7379	−0.00912923
$254$	0	0
$255$	1404.19	0.344837
$256$	0	0
$257$	3292.09	0.799047	0.399523	−	0.916723i	$-0.369176\pi$
$257$	3292.09	0.799047	0.399523	+	0.916723i	$0.369176\pi$
$258$	0	0
$259$	−769.887	−0.184704
$260$	0	0
$261$	2875.16	0.681869
$262$	0	0
$263$	2260.19	0.529920	0.264960	−	0.964259i	$-0.414641\pi$
$263$	2260.19	0.529920	0.264960	+	0.964259i	$0.414641\pi$
$264$	0	0
$265$	2576.21	0.597190
$266$	0	0
$267$	1468.82	0.336668
$268$	0	0
$269$	2151.40	0.487633	0.243817	−	0.969821i	$-0.421600\pi$
$269$	2151.40	0.487633	0.243817	+	0.969821i	$0.421600\pi$
$270$	0	0
$271$	5832.47	1.30737	0.653685	−	0.756767i	$-0.273222\pi$
$271$	5832.47	1.30737	0.653685	+	0.756767i	$0.273222\pi$
$272$	0	0
$273$	−541.639	−0.120079
$274$	0	0
$275$	−39.9325	−0.00875645
$276$	0	0
$277$	4848.03	1.05159	0.525795	−	0.850612i	$-0.323768\pi$
$277$	4848.03	1.05159	0.525795	+	0.850612i	$0.323768\pi$
$278$	0	0
$279$	−496.936	−0.106634
$280$	0	0
$281$	−3055.68	−0.648706	−0.324353	−	0.945936i	$-0.605147\pi$
$281$	−3055.68	−0.648706	−0.324353	+	0.945936i	$0.605147\pi$
$282$	0	0
$283$	−6960.33	−1.46201	−0.731005	−	0.682372i	$-0.760948\pi$
$283$	−6960.33	−1.46201	−0.731005	+	0.682372i	$0.760948\pi$
$284$	0	0
$285$	−1574.35	−0.327216
$286$	0	0
$287$	−2108.86	−0.433735
$288$	0	0
$289$	−3004.39	−0.611519
$290$	0	0
$291$	−3975.16	−0.800783
$292$	0	0
$293$	3248.07	0.647626	0.323813	−	0.946121i	$-0.395035\pi$
$293$	3248.07	0.647626	0.323813	+	0.946121i	$0.395035\pi$
$294$	0	0
$295$	493.671	0.0974326
$296$	0	0
$297$	130.166	0.0254309
$298$	0	0
$299$	232.854	0.0450377
$300$	0	0
$301$	3713.13	0.711034
$302$	0	0
$303$	−6915.36	−1.31115
$304$	0	0
$305$	1549.67	0.290931
$306$	0	0
$307$	1354.90	0.251883	0.125941	−	0.992038i	$-0.459805\pi$
$307$	1354.90	0.251883	0.125941	+	0.992038i	$0.459805\pi$
$308$	0	0
$309$	−2955.99	−0.544208
$310$	0	0
$311$	907.288	0.165426	0.0827132	−	0.996573i	$-0.473641\pi$
$311$	907.288	0.165426	0.0827132	+	0.996573i	$0.473641\pi$
$312$	0	0
$313$	5522.93	0.997363	0.498681	−	0.866785i	$-0.333818\pi$
$313$	5522.93	0.997363	0.498681	+	0.866785i	$0.333818\pi$
$314$	0	0
$315$	596.025	0.106610
$316$	0	0
$317$	5610.21	0.994009	0.497004	−	0.867748i	$-0.334433\pi$
$317$	5610.21	0.994009	0.497004	+	0.867748i	$0.334433\pi$
$318$	0	0
$319$	−320.636	−0.0562764
$320$	0	0
$321$	−9762.54	−1.69748
$322$	0	0
$323$	−2139.90	−0.368629
$324$	0	0
$325$	253.102	0.0431987
$326$	0	0
$327$	−7627.62	−1.28993
$328$	0	0
$329$	2621.85	0.439353
$330$	0	0
$331$	6674.82	1.10840	0.554202	−	0.832383i	$-0.313024\pi$
$331$	6674.82	1.10840	0.554202	+	0.832383i	$0.313024\pi$
$332$	0	0
$333$	1324.97	0.218041
$334$	0	0
$335$	−1975.84	−0.322244
$336$	0	0
$337$	7375.54	1.19220	0.596100	−	0.802910i	$-0.296716\pi$
$337$	7375.54	1.19220	0.596100	+	0.802910i	$0.296716\pi$
$338$	0	0
$339$	−10792.8	−1.72916
$340$	0	0
$341$	55.4180	0.00880074
$342$	0	0
$343$	5132.83	0.808007
$344$	0	0
$345$	−739.255	−0.115363
$346$	0	0
$347$	2053.90	0.317750	0.158875	−	0.987299i	$-0.449213\pi$
$347$	2053.90	0.317750	0.158875	+	0.987299i	$0.449213\pi$
$348$	0	0
$349$	−478.454	−0.0733841	−0.0366920	−	0.999327i	$-0.511682\pi$
$349$	−478.454	−0.0733841	−0.0366920	+	0.999327i	$0.511682\pi$
$350$	0	0
$351$	−825.021	−0.125460
$352$	0	0
$353$	109.570	0.0165207	0.00826034	−	0.999966i	$-0.497371\pi$
$353$	109.570	0.0165207	0.00826034	+	0.999966i	$0.497371\pi$
$354$	0	0
$355$	−3503.23	−0.523753
$356$	0	0
$357$	2337.29	0.346506
$358$	0	0
$359$	5835.58	0.857912	0.428956	−	0.903325i	$-0.358882\pi$
$359$	5835.58	0.857912	0.428956	+	0.903325i	$0.358882\pi$
$360$	0	0
$361$	−4459.78	−0.650208
$362$	0	0
$363$	−8539.67	−1.23476
$364$	0	0
$365$	2655.70	0.380837
$366$	0	0
$367$	7693.84	1.09432	0.547160	−	0.837028i	$-0.315709\pi$
$367$	7693.84	1.09432	0.547160	+	0.837028i	$0.315709\pi$
$368$	0	0
$369$	3629.32	0.512019
$370$	0	0
$371$	4288.14	0.600079
$372$	0	0
$373$	−8726.57	−1.21138	−0.605690	−	0.795701i	$-0.707103\pi$
$373$	−8726.57	−1.21138	−0.605690	+	0.795701i	$0.707103\pi$
$374$	0	0
$375$	−803.538	−0.110652
$376$	0	0
$377$	2032.27	0.277631
$378$	0	0
$379$	−9893.96	−1.34095	−0.670473	−	0.741934i	$-0.733909\pi$
$379$	−9893.96	−1.34095	−0.670473	+	0.741934i	$0.733909\pi$
$380$	0	0
$381$	4450.76	0.598476
$382$	0	0
$383$	−3010.04	−0.401582	−0.200791	−	0.979634i	$-0.564351\pi$
$383$	−3010.04	−0.401582	−0.200791	+	0.979634i	$0.564351\pi$
$384$	0	0
$385$	−66.4684	−0.00879881
$386$	0	0
$387$	−6390.26	−0.839368
$388$	0	0
$389$	−8778.85	−1.14423	−0.572115	−	0.820173i	$-0.693877\pi$
$389$	−8778.85	−1.14423	−0.572115	+	0.820173i	$0.693877\pi$
$390$	0	0
$391$	−1004.81	−0.129963
$392$	0	0
$393$	1731.24	0.222213
$394$	0	0
$395$	536.801	0.0683782
$396$	0	0
$397$	−10697.1	−1.35232	−0.676161	−	0.736753i	$-0.736358\pi$
$397$	−10697.1	−1.35232	−0.676161	+	0.736753i	$0.736358\pi$
$398$	0	0
$399$	−2620.53	−0.328799
$400$	0	0
$401$	−369.210	−0.0459787	−0.0229893	−	0.999736i	$-0.507318\pi$
$401$	−369.210	−0.0459787	−0.0229893	+	0.999736i	$0.507318\pi$
$402$	0	0
$403$	−351.252	−0.0434172
$404$	0	0
$405$	4552.86	0.558601
$406$	0	0
$407$	−147.760	−0.0179955
$408$	0	0
$409$	−6327.37	−0.764959	−0.382480	−	0.923964i	$-0.624930\pi$
$409$	−6327.37	−0.764959	−0.382480	+	0.923964i	$0.624930\pi$
$410$	0	0
$411$	−6398.38	−0.767905
$412$	0	0
$413$	821.723	0.0979040
$414$	0	0
$415$	2656.22	0.314189
$416$	0	0
$417$	98.6721	0.0115875
$418$	0	0
$419$	4130.29	0.481571	0.240785	−	0.970578i	$-0.422595\pi$
$419$	4130.29	0.481571	0.240785	+	0.970578i	$0.422595\pi$
$420$	0	0
$421$	−5578.43	−0.645786	−0.322893	−	0.946435i	$-0.604655\pi$
$421$	−5578.43	−0.645786	−0.322893	+	0.946435i	$0.604655\pi$
$422$	0	0
$423$	−4512.18	−0.518651
$424$	0	0
$425$	−1092.19	−0.124656
$426$	0	0
$427$	2579.45	0.292338
$428$	0	0
$429$	−103.953	−0.0116991
$430$	0	0
$431$	−1572.83	−0.175779	−0.0878894	−	0.996130i	$-0.528012\pi$
$431$	−1572.83	−0.175779	−0.0878894	+	0.996130i	$0.528012\pi$
$432$	0	0
$433$	−723.228	−0.0802681	−0.0401341	−	0.999194i	$-0.512779\pi$
$433$	−723.228	−0.0802681	−0.0401341	+	0.999194i	$0.512779\pi$
$434$	0	0
$435$	−6451.96	−0.711144
$436$	0	0
$437$	1126.58	0.123322
$438$	0	0
$439$	−10176.6	−1.10638	−0.553192	−	0.833054i	$-0.686590\pi$
$439$	−10176.6	−1.10638	−0.553192	+	0.833054i	$0.686590\pi$
$440$	0	0
$441$	−3920.72	−0.423359
$442$	0	0
$443$	−10665.1	−1.14382	−0.571910	−	0.820316i	$-0.693797\pi$
$443$	−10665.1	−1.14382	−0.571910	+	0.820316i	$0.693797\pi$
$444$	0	0
$445$	−1142.46	−0.121703
$446$	0	0
$447$	11968.9	1.26647
$448$	0	0
$449$	−6922.86	−0.727640	−0.363820	−	0.931469i	$-0.618528\pi$
$449$	−6922.86	−0.727640	−0.363820	+	0.931469i	$0.618528\pi$
$450$	0	0
$451$	−404.740	−0.0422582
$452$	0	0
$453$	19344.5	2.00637
$454$	0	0
$455$	421.292	0.0434076
$456$	0	0
$457$	606.753	0.0621065	0.0310533	−	0.999518i	$-0.490114\pi$
$457$	606.753	0.0621065	0.0310533	+	0.999518i	$0.490114\pi$
$458$	0	0
$459$	3560.15	0.362034
$460$	0	0
$461$	14239.6	1.43862	0.719308	−	0.694691i	$-0.244459\pi$
$461$	14239.6	1.43862	0.719308	+	0.694691i	$0.244459\pi$
$462$	0	0
$463$	2927.25	0.293825	0.146913	−	0.989149i	$-0.453066\pi$
$463$	2927.25	0.293825	0.146913	+	0.989149i	$0.453066\pi$
$464$	0	0
$465$	1115.14	0.111212
$466$	0	0
$467$	−2195.75	−0.217575	−0.108787	−	0.994065i	$-0.534697\pi$
$467$	−2195.75	−0.217575	−0.108787	+	0.994065i	$0.534697\pi$
$468$	0	0
$469$	−3288.82	−0.323803
$470$	0	0
$471$	15513.7	1.51769
$472$	0	0
$473$	712.638	0.0692751
$474$	0	0
$475$	1224.55	0.118286
$476$	0	0
$477$	−7379.85	−0.708386
$478$	0	0
$479$	−8272.52	−0.789104	−0.394552	−	0.918874i	$-0.629100\pi$
$479$	−8272.52	−0.789104	−0.394552	+	0.918874i	$0.629100\pi$
$480$	0	0
$481$	936.535	0.0887782
$482$	0	0
$483$	−1230.50	−0.115921
$484$	0	0
$485$	3091.92	0.289478
$486$	0	0
$487$	−7983.87	−0.742883	−0.371441	−	0.928456i	$-0.621136\pi$
$487$	−7983.87	−0.742883	−0.371441	+	0.928456i	$0.621136\pi$
$488$	0	0
$489$	−2481.58	−0.229491
$490$	0	0
$491$	519.669	0.0477645	0.0238822	−	0.999715i	$-0.492397\pi$
$491$	519.669	0.0477645	0.0238822	+	0.999715i	$0.492397\pi$
$492$	0	0
$493$	−8769.67	−0.801148
$494$	0	0
$495$	114.391	0.0103869
$496$	0	0
$497$	−5831.19	−0.526287
$498$	0	0
$499$	−862.162	−0.0773460	−0.0386730	−	0.999252i	$-0.512313\pi$
$499$	−862.162	−0.0773460	−0.0386730	+	0.999252i	$0.512313\pi$
$500$	0	0
$501$	−6637.52	−0.591901
$502$	0	0
$503$	14142.2	1.25362	0.626809	−	0.779173i	$-0.284361\pi$
$503$	14142.2	1.25362	0.626809	+	0.779173i	$0.284361\pi$
$504$	0	0
$505$	5378.84	0.473970
$506$	0	0
$507$	−13464.1	−1.17941
$508$	0	0
$509$	−7163.46	−0.623801	−0.311900	−	0.950115i	$-0.600966\pi$
$509$	−7163.46	−0.623801	−0.311900	+	0.950115i	$0.600966\pi$
$510$	0	0
$511$	4420.45	0.382680
$512$	0	0
$513$	−3991.58	−0.343533
$514$	0	0
$515$	2299.20	0.196728
$516$	0	0
$517$	503.195	0.0428056
$518$	0	0
$519$	−19212.9	−1.62496
$520$	0	0
$521$	5219.95	0.438944	0.219472	−	0.975619i	$-0.429566\pi$
$521$	5219.95	0.438944	0.219472	+	0.975619i	$0.429566\pi$
$522$	0	0
$523$	−13888.9	−1.16123	−0.580613	−	0.814180i	$-0.697187\pi$
$523$	−13888.9	−1.16123	−0.580613	+	0.814180i	$0.697187\pi$
$524$	0	0
$525$	−1337.50	−0.111187
$526$	0	0
$527$	1515.73	0.125287
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−1414.18	−0.115575
$532$	0	0
$533$	2565.33	0.208475
$534$	0	0
$535$	7593.40	0.613629
$536$	0	0
$537$	−9753.34	−0.783776
$538$	0	0
$539$	437.237	0.0349409
$540$	0	0
$541$	−10331.1	−0.821012	−0.410506	−	0.911858i	$-0.634648\pi$
$541$	−10331.1	−0.821012	−0.410506	+	0.911858i	$0.634648\pi$
$542$	0	0
$543$	−23658.5	−1.86977
$544$	0	0
$545$	5932.84	0.466303
$546$	0	0
$547$	19121.6	1.49466	0.747332	−	0.664451i	$-0.231335\pi$
$547$	19121.6	1.49466	0.747332	+	0.664451i	$0.231335\pi$
$548$	0	0
$549$	−4439.22	−0.345102
$550$	0	0
$551$	9832.42	0.760209
$552$	0	0
$553$	893.514	0.0687090
$554$	0	0
$555$	−2973.27	−0.227403
$556$	0	0
$557$	−10080.2	−0.766808	−0.383404	−	0.923581i	$-0.625248\pi$
$557$	−10080.2	−0.766808	−0.383404	+	0.923581i	$0.625248\pi$
$558$	0	0
$559$	−4516.87	−0.341759
$560$	0	0
$561$	448.582	0.0337596
$562$	0	0
$563$	−11517.1	−0.862144	−0.431072	−	0.902318i	$-0.641864\pi$
$563$	−11517.1	−0.862144	−0.431072	+	0.902318i	$0.641864\pi$
$564$	0	0
$565$	8394.77	0.625081
$566$	0	0
$567$	7578.32	0.561304
$568$	0	0
$569$	−5771.86	−0.425253	−0.212627	−	0.977134i	$-0.568202\pi$
$569$	−5771.86	−0.425253	−0.212627	+	0.977134i	$0.568202\pi$
$570$	0	0
$571$	12608.7	0.924092	0.462046	−	0.886856i	$-0.347116\pi$
$571$	12608.7	0.924092	0.462046	+	0.886856i	$0.347116\pi$
$572$	0	0
$573$	3652.92	0.266323
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	9818.70	0.708419	0.354209	−	0.935166i	$-0.384750\pi$
$577$	9818.70	0.708419	0.354209	+	0.935166i	$0.384750\pi$
$578$	0	0
$579$	8074.86	0.579585
$580$	0	0
$581$	4421.32	0.315709
$582$	0	0
$583$	822.997	0.0584649
$584$	0	0
$585$	−725.040	−0.0512422
$586$	0	0
$587$	−7452.02	−0.523983	−0.261991	−	0.965070i	$-0.584379\pi$
$587$	−7452.02	−0.523983	−0.261991	+	0.965070i	$0.584379\pi$
$588$	0	0
$589$	−1699.41	−0.118885
$590$	0	0
$591$	−23498.8	−1.63555
$592$	0	0
$593$	5386.06	0.372983	0.186492	−	0.982457i	$-0.440288\pi$
$593$	5386.06	0.372983	0.186492	+	0.982457i	$0.440288\pi$
$594$	0	0
$595$	−1817.97	−0.125260
$596$	0	0
$597$	19482.8	1.33564
$598$	0	0
$599$	−5750.26	−0.392236	−0.196118	−	0.980580i	$-0.562834\pi$
$599$	−5750.26	−0.392236	−0.196118	+	0.980580i	$0.562834\pi$
$600$	0	0
$601$	−894.738	−0.0607273	−0.0303637	−	0.999539i	$-0.509667\pi$
$601$	−894.738	−0.0607273	−0.0303637	+	0.999539i	$0.509667\pi$
$602$	0	0
$603$	5660.03	0.382246
$604$	0	0
$605$	6642.24	0.446356
$606$	0	0
$607$	6015.58	0.402248	0.201124	−	0.979566i	$-0.435541\pi$
$607$	6015.58	0.402248	0.201124	+	0.979566i	$0.435541\pi$
$608$	0	0
$609$	−10739.4	−0.714585
$610$	0	0
$611$	−3189.37	−0.211175
$612$	0	0
$613$	4533.12	0.298680	0.149340	−	0.988786i	$-0.452285\pi$
$613$	4533.12	0.298680	0.149340	+	0.988786i	$0.452285\pi$
$614$	0	0
$615$	−8144.33	−0.534002
$616$	0	0
$617$	−11378.8	−0.742455	−0.371227	−	0.928542i	$-0.621063\pi$
$617$	−11378.8	−0.742455	−0.371227	+	0.928542i	$0.621063\pi$
$618$	0	0
$619$	17167.2	1.11471	0.557357	−	0.830273i	$-0.311816\pi$
$619$	17167.2	1.11471	0.557357	+	0.830273i	$0.311816\pi$
$620$	0	0
$621$	−1874.29	−0.121116
$622$	0	0
$623$	−1901.65	−0.122292
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−502.943	−0.0320344
$628$	0	0
$629$	−4041.35	−0.256183
$630$	0	0
$631$	−24289.5	−1.53241	−0.766206	−	0.642595i	$-0.777858\pi$
$631$	−24289.5	−1.53241	−0.766206	+	0.642595i	$0.777858\pi$
$632$	0	0
$633$	14827.5	0.931030
$634$	0	0
$635$	−3461.85	−0.216345
$636$	0	0
$637$	−2771.31	−0.172376
$638$	0	0
$639$	10035.4	0.621275
$640$	0	0
$641$	−22941.6	−1.41363	−0.706816	−	0.707397i	$-0.749869\pi$
$641$	−22941.6	−1.41363	−0.706816	+	0.707397i	$0.749869\pi$
$642$	0	0
$643$	8623.66	0.528902	0.264451	−	0.964399i	$-0.414809\pi$
$643$	8623.66	0.528902	0.264451	+	0.964399i	$0.414809\pi$
$644$	0	0
$645$	14340.0	0.875405
$646$	0	0
$647$	30451.5	1.85034	0.925172	−	0.379547i	$-0.123920\pi$
$647$	30451.5	1.85034	0.925172	+	0.379547i	$0.123920\pi$
$648$	0	0
$649$	157.708	0.00953866
$650$	0	0
$651$	1856.17	0.111750
$652$	0	0
$653$	20330.4	1.21836	0.609182	−	0.793031i	$-0.291498\pi$
$653$	20330.4	1.21836	0.609182	+	0.793031i	$0.291498\pi$
$654$	0	0
$655$	−1346.58	−0.0803284
$656$	0	0
$657$	−7607.56	−0.451749
$658$	0	0
$659$	−29029.8	−1.71599	−0.857996	−	0.513656i	$-0.828291\pi$
$659$	−29029.8	−1.71599	−0.857996	+	0.513656i	$0.828291\pi$
$660$	0	0
$661$	31721.6	1.86661	0.933304	−	0.359088i	$-0.116912\pi$
$661$	31721.6	1.86661	0.933304	+	0.359088i	$0.116912\pi$
$662$	0	0
$663$	−2843.21	−0.166548
$664$	0	0
$665$	2038.28	0.118859
$666$	0	0
$667$	4616.93	0.268018
$668$	0	0
$669$	−6327.25	−0.365658
$670$	0	0
$671$	495.059	0.0284822
$672$	0	0
$673$	−266.739	−0.0152779	−0.00763895	−	0.999971i	$-0.502432\pi$
$673$	−266.739	−0.0152779	−0.00763895	+	0.999971i	$0.502432\pi$
$674$	0	0
$675$	−2037.28	−0.116170
$676$	0	0
$677$	−5439.22	−0.308783	−0.154391	−	0.988010i	$-0.549342\pi$
$677$	−5439.22	−0.308783	−0.154391	+	0.988010i	$0.549342\pi$
$678$	0	0
$679$	5146.55	0.290878
$680$	0	0
$681$	10533.7	0.592733
$682$	0	0
$683$	2914.19	0.163263	0.0816314	−	0.996663i	$-0.473987\pi$
$683$	2914.19	0.163263	0.0816314	+	0.996663i	$0.473987\pi$
$684$	0	0
$685$	4976.73	0.277593
$686$	0	0
$687$	15892.9	0.882606
$688$	0	0
$689$	−5216.35	−0.288428
$690$	0	0
$691$	29045.8	1.59907	0.799533	−	0.600622i	$-0.205080\pi$
$691$	29045.8	1.59907	0.799533	+	0.600622i	$0.205080\pi$
$692$	0	0
$693$	190.406	0.0104371
$694$	0	0
$695$	−76.7482	−0.00418881
$696$	0	0
$697$	−11070.0	−0.601586
$698$	0	0
$699$	−1954.40	−0.105754
$700$	0	0
$701$	19177.9	1.03329	0.516646	−	0.856199i	$-0.327180\pi$
$701$	19177.9	1.03329	0.516646	+	0.856199i	$0.327180\pi$
$702$	0	0
$703$	4531.10	0.243092
$704$	0	0
$705$	10125.5	0.540919
$706$	0	0
$707$	8953.16	0.476264
$708$	0	0
$709$	20890.1	1.10655	0.553276	−	0.832998i	$-0.313377\pi$
$709$	20890.1	1.10655	0.553276	+	0.832998i	$0.313377\pi$
$710$	0	0
$711$	−1537.73	−0.0811102
$712$	0	0
$713$	−797.979	−0.0419138
$714$	0	0
$715$	80.8560	0.00422915
$716$	0	0
$717$	−1418.50	−0.0738842
$718$	0	0
$719$	−6739.88	−0.349590	−0.174795	−	0.984605i	$-0.555926\pi$
$719$	−6739.88	−0.349590	−0.174795	+	0.984605i	$0.555926\pi$
$720$	0	0
$721$	3827.06	0.197680
$722$	0	0
$723$	−24067.2	−1.23799
$724$	0	0
$725$	5018.40	0.257074
$726$	0	0
$727$	−9983.95	−0.509332	−0.254666	−	0.967029i	$-0.581966\pi$
$727$	−9983.95	−0.509332	−0.254666	+	0.967029i	$0.581966\pi$
$728$	0	0
$729$	1101.72	0.0559731
$730$	0	0
$731$	19491.3	0.986198
$732$	0	0
$733$	13884.2	0.699622	0.349811	−	0.936820i	$-0.386246\pi$
$733$	13884.2	0.699622	0.349811	+	0.936820i	$0.386246\pi$
$734$	0	0
$735$	8798.25	0.441535
$736$	0	0
$737$	−631.203	−0.0315477
$738$	0	0
$739$	27623.4	1.37502	0.687512	−	0.726173i	$-0.258703\pi$
$739$	27623.4	1.37502	0.687512	+	0.726173i	$0.258703\pi$
$740$	0	0
$741$	3187.77	0.158037
$742$	0	0
$743$	35.0253	0.00172941	0.000864706	−	1.00000i	$-0.499725\pi$
$743$	35.0253	0.00172941	0.000864706		1.00000i	$0.499725\pi$
$744$	0	0
$745$	−9309.57	−0.457820
$746$	0	0
$747$	−7609.04	−0.372691
$748$	0	0
$749$	12639.3	0.616598
$750$	0	0
$751$	25497.7	1.23891	0.619456	−	0.785031i	$-0.287353\pi$
$751$	25497.7	1.23891	0.619456	+	0.785031i	$0.287353\pi$
$752$	0	0
$753$	−12450.0	−0.602528
$754$	0	0
$755$	−15046.4	−0.725289
$756$	0	0
$757$	−16363.0	−0.785630	−0.392815	−	0.919617i	$-0.628499\pi$
$757$	−16363.0	−0.785630	−0.392815	+	0.919617i	$0.628499\pi$
$758$	0	0
$759$	−236.163	−0.0112940
$760$	0	0
$761$	−19026.0	−0.906299	−0.453150	−	0.891435i	$-0.649700\pi$
$761$	−19026.0	−0.906299	−0.453150	+	0.891435i	$0.649700\pi$
$762$	0	0
$763$	9875.31	0.468559
$764$	0	0
$765$	3128.71	0.147867
$766$	0	0
$767$	−999.592	−0.0470576
$768$	0	0
$769$	−16422.8	−0.770121	−0.385060	−	0.922891i	$-0.625819\pi$
$769$	−16422.8	−0.770121	−0.385060	+	0.922891i	$0.625819\pi$
$770$	0	0
$771$	21162.6	0.988523
$772$	0	0
$773$	−30887.7	−1.43720	−0.718599	−	0.695425i	$-0.755216\pi$
$773$	−30887.7	−1.43720	−0.718599	+	0.695425i	$0.755216\pi$
$774$	0	0
$775$	−867.369	−0.0402023
$776$	0	0
$777$	−4949.06	−0.228503
$778$	0	0
$779$	12411.5	0.570844
$780$	0	0
$781$	−1119.14	−0.0512754
$782$	0	0
$783$	−16358.2	−0.746608
$784$	0	0
$785$	−12066.7	−0.548636
$786$	0	0
$787$	18026.5	0.816486	0.408243	−	0.912873i	$-0.366142\pi$
$787$	18026.5	0.816486	0.408243	+	0.912873i	$0.366142\pi$
$788$	0	0
$789$	14529.2	0.655579
$790$	0	0
$791$	13973.2	0.628105
$792$	0	0
$793$	−3137.80	−0.140513
$794$	0	0
$795$	16560.7	0.738800
$796$	0	0
$797$	−5024.03	−0.223288	−0.111644	−	0.993748i	$-0.535612\pi$
$797$	−5024.03	−0.223288	−0.111644	+	0.993748i	$0.535612\pi$
$798$	0	0
$799$	13762.8	0.609379
$800$	0	0
$801$	3272.72	0.144364
$802$	0	0
$803$	848.390	0.0372840
$804$	0	0
$805$	957.097	0.0419046
$806$	0	0
$807$	13829.9	0.603265
$808$	0	0
$809$	13539.5	0.588408	0.294204	−	0.955743i	$-0.404946\pi$
$809$	13539.5	0.588408	0.294204	+	0.955743i	$0.404946\pi$
$810$	0	0
$811$	8701.18	0.376745	0.188372	−	0.982098i	$-0.439679\pi$
$811$	8701.18	0.376745	0.188372	+	0.982098i	$0.439679\pi$
$812$	0	0
$813$	37492.9	1.61738
$814$	0	0
$815$	1930.20	0.0829595
$816$	0	0
$817$	−21853.3	−0.935802
$818$	0	0
$819$	−1206.84	−0.0514901
$820$	0	0
$821$	24130.0	1.02575	0.512876	−	0.858463i	$-0.328580\pi$
$821$	24130.0	1.02575	0.512876	+	0.858463i	$0.328580\pi$
$822$	0	0
$823$	−740.011	−0.0313428	−0.0156714	−	0.999877i	$-0.504989\pi$
$823$	−740.011	−0.0313428	−0.0156714	+	0.999877i	$0.504989\pi$
$824$	0	0
$825$	−256.699	−0.0108328
$826$	0	0
$827$	26735.3	1.12416	0.562078	−	0.827084i	$-0.310002\pi$
$827$	26735.3	1.12416	0.562078	+	0.827084i	$0.310002\pi$
$828$	0	0
$829$	−16817.0	−0.704558	−0.352279	−	0.935895i	$-0.614593\pi$
$829$	−16817.0	−0.704558	−0.352279	+	0.935895i	$0.614593\pi$
$830$	0	0
$831$	31164.6	1.30095
$832$	0	0
$833$	11958.8	0.497417
$834$	0	0
$835$	5162.73	0.213968
$836$	0	0
$837$	2827.31	0.116758
$838$	0	0
$839$	−4657.60	−0.191655	−0.0958273	−	0.995398i	$-0.530550\pi$
$839$	−4657.60	−0.191655	−0.0958273	+	0.995398i	$0.530550\pi$
$840$	0	0
$841$	15905.9	0.652177
$842$	0	0
$843$	−19642.8	−0.802532
$844$	0	0
$845$	10472.5	0.426350
$846$	0	0
$847$	11056.1	0.448516
$848$	0	0
$849$	−44743.1	−1.80869
$850$	0	0
$851$	2127.63	0.0857042
$852$	0	0
$853$	28392.0	1.13965	0.569826	−	0.821765i	$-0.307011\pi$
$853$	28392.0	1.13965	0.569826	+	0.821765i	$0.307011\pi$
$854$	0	0
$855$	−3507.86	−0.140311
$856$	0	0
$857$	−9369.46	−0.373459	−0.186730	−	0.982411i	$-0.559789\pi$
$857$	−9369.46	−0.373459	−0.186730	+	0.982411i	$0.559789\pi$
$858$	0	0
$859$	1622.41	0.0644423	0.0322211	−	0.999481i	$-0.489742\pi$
$859$	1622.41	0.0644423	0.0322211	+	0.999481i	$0.489742\pi$
$860$	0	0
$861$	−13556.4	−0.536585
$862$	0	0
$863$	29507.8	1.16391	0.581956	−	0.813220i	$-0.302288\pi$
$863$	29507.8	1.16391	0.581956	+	0.813220i	$0.302288\pi$
$864$	0	0
$865$	14944.0	0.587413
$866$	0	0
$867$	−19313.2	−0.756528
$868$	0	0
$869$	171.487	0.00669423
$870$	0	0
$871$	4000.72	0.155636
$872$	0	0
$873$	−8857.16	−0.343379
$874$	0	0
$875$	1040.32	0.0401935
$876$	0	0
$877$	−50395.7	−1.94041	−0.970207	−	0.242279i	$-0.922105\pi$
$877$	−50395.7	−1.94041	−0.970207	+	0.242279i	$0.922105\pi$
$878$	0	0
$879$	20879.6	0.801196
$880$	0	0
$881$	2847.84	0.108906	0.0544529	−	0.998516i	$-0.482659\pi$
$881$	2847.84	0.108906	0.0544529	+	0.998516i	$0.482659\pi$
$882$	0	0
$883$	47218.4	1.79958	0.899788	−	0.436328i	$-0.143721\pi$
$883$	47218.4	1.79958	0.899788	+	0.436328i	$0.143721\pi$
$884$	0	0
$885$	3173.47	0.120537
$886$	0	0
$887$	22706.5	0.859536	0.429768	−	0.902939i	$-0.358595\pi$
$887$	22706.5	0.859536	0.429768	+	0.902939i	$0.358595\pi$
$888$	0	0
$889$	−5762.30	−0.217392
$890$	0	0
$891$	1454.46	0.0546871
$892$	0	0
$893$	−15430.7	−0.578239
$894$	0	0
$895$	7586.25	0.283330
$896$	0	0
$897$	1496.85	0.0557174
$898$	0	0
$899$	−6964.49	−0.258374
$900$	0	0
$901$	22509.7	0.832304
$902$	0	0
$903$	23869.1	0.879640
$904$	0	0
$905$	18401.8	0.675908
$906$	0	0
$907$	32574.6	1.19253	0.596264	−	0.802788i	$-0.296651\pi$
$907$	32574.6	1.19253	0.596264	+	0.802788i	$0.296651\pi$
$908$	0	0
$909$	−15408.3	−0.562224
$910$	0	0
$911$	−21063.3	−0.766037	−0.383018	−	0.923741i	$-0.625115\pi$
$911$	−21063.3	−0.766037	−0.383018	+	0.923741i	$0.625115\pi$
$912$	0	0
$913$	848.556	0.0307591
$914$	0	0
$915$	9961.76	0.359919
$916$	0	0
$917$	−2241.40	−0.0807170
$918$	0	0
$919$	40325.8	1.44747	0.723736	−	0.690077i	$-0.242423\pi$
$919$	40325.8	1.44747	0.723736	+	0.690077i	$0.242423\pi$
$920$	0	0
$921$	8709.69	0.311611
$922$	0	0
$923$	7093.39	0.252960
$924$	0	0
$925$	2312.64	0.0822046
$926$	0	0
$927$	−6586.33	−0.233359
$928$	0	0
$929$	30365.3	1.07239	0.536196	−	0.844093i	$-0.319861\pi$
$929$	30365.3	1.07239	0.536196	+	0.844093i	$0.319861\pi$
$930$	0	0
$931$	−13408.0	−0.471998
$932$	0	0
$933$	5832.32	0.204653
$934$	0	0
$935$	−348.911	−0.0122039
$936$	0	0
$937$	−5133.76	−0.178989	−0.0894945	−	0.995987i	$-0.528525\pi$
$937$	−5133.76	−0.178989	−0.0894945	+	0.995987i	$0.528525\pi$
$938$	0	0
$939$	35503.1	1.23387
$940$	0	0
$941$	−14510.2	−0.502677	−0.251339	−	0.967899i	$-0.580871\pi$
$941$	−14510.2	−0.502677	−0.251339	+	0.967899i	$0.580871\pi$
$942$	0	0
$943$	5827.96	0.201256
$944$	0	0
$945$	−3391.08	−0.116732
$946$	0	0
$947$	−28776.6	−0.987448	−0.493724	−	0.869619i	$-0.664365\pi$
$947$	−28776.6	−0.987448	−0.493724	+	0.869619i	$0.664365\pi$
$948$	0	0
$949$	−5377.29	−0.183935
$950$	0	0
$951$	36064.1	1.22972
$952$	0	0
$953$	−658.129	−0.0223703	−0.0111851	−	0.999937i	$-0.503560\pi$
$953$	−658.129	−0.0223703	−0.0111851	+	0.999937i	$0.503560\pi$
$954$	0	0
$955$	−2841.28	−0.0962740
$956$	0	0
$957$	−2061.15	−0.0696211
$958$	0	0
$959$	8283.84	0.278936
$960$	0	0
$961$	−28587.3	−0.959594
$962$	0	0
$963$	−21752.2	−0.727886
$964$	0	0
$965$	−6280.71	−0.209516
$966$	0	0
$967$	−26064.0	−0.866765	−0.433382	−	0.901210i	$-0.642680\pi$
$967$	−26064.0	−0.866765	−0.433382	+	0.901210i	$0.642680\pi$
$968$	0	0
$969$	−13755.9	−0.456041
$970$	0	0
$971$	25739.5	0.850691	0.425345	−	0.905031i	$-0.360153\pi$
$971$	25739.5	0.850691	0.425345	+	0.905031i	$0.360153\pi$
$972$	0	0
$973$	−127.749	−0.00420908
$974$	0	0
$975$	1627.02	0.0534422
$976$	0	0
$977$	−7083.18	−0.231946	−0.115973	−	0.993252i	$-0.536999\pi$
$977$	−7083.18	−0.231946	−0.115973	+	0.993252i	$0.536999\pi$
$978$	0	0
$979$	−364.972	−0.0119148
$980$	0	0
$981$	−16995.3	−0.553128
$982$	0	0
$983$	−42357.2	−1.37435	−0.687175	−	0.726492i	$-0.741149\pi$
$983$	−42357.2	−1.37435	−0.687175	+	0.726492i	$0.741149\pi$
$984$	0	0
$985$	18277.6	0.591241
$986$	0	0
$987$	16854.0	0.543536
$988$	0	0
$989$	−10261.5	−0.329925
$990$	0	0
$991$	23366.7	0.749007	0.374504	−	0.927225i	$-0.377813\pi$
$991$	23366.7	0.749007	0.374504	+	0.927225i	$0.377813\pi$
$992$	0	0
$993$	42907.8	1.37124
$994$	0	0
$995$	−15153.9	−0.482825
$996$	0	0
$997$	48861.4	1.55211	0.776056	−	0.630664i	$-0.217218\pi$
$997$	48861.4	1.55211	0.776056	+	0.630664i	$0.217218\pi$
$998$	0	0
$999$	−7538.39	−0.238743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.s.1.6		6
4.3	odd	2		920.4.a.b.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.b.1.1	✓	6	4.3	odd	2
1840.4.a.s.1.6		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+6.42830 q^{3} -5.00000 q^{5} -8.32258 q^{7} +14.3231 q^{9} +O(q^{10})\)	\(q+6.42830 q^{3} -5.00000 q^{5} -8.32258 q^{7} +14.3231 q^{9} -1.59730 q^{11} +10.1241 q^{13} -32.1415 q^{15} -43.6876 q^{17} +48.9819 q^{19} -53.5001 q^{21} +23.0000 q^{23} +25.0000 q^{25} -81.4910 q^{27} +200.736 q^{29} -34.6948 q^{31} -10.2679 q^{33} +41.6129 q^{35} +92.5057 q^{37} +65.0806 q^{39} +253.390 q^{41} -446.151 q^{43} -71.6154 q^{45} -315.028 q^{47} -273.735 q^{49} -280.837 q^{51} -515.242 q^{53} +7.98651 q^{55} +314.870 q^{57} -98.7342 q^{59} -309.934 q^{61} -119.205 q^{63} -50.6204 q^{65} +395.169 q^{67} +147.851 q^{69} +700.646 q^{71} -531.139 q^{73} +160.708 q^{75} +13.2937 q^{77} -107.360 q^{79} -910.573 q^{81} -531.243 q^{83} +218.438 q^{85} +1290.39 q^{87} +228.493 q^{89} -84.2584 q^{91} -223.028 q^{93} -244.909 q^{95} -618.384 q^{97} -22.8783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 30 * q^5 - 28 * q^7 + 4 * q^9	\( 6 q - 2 q^{3} - 30 q^{5} - 28 q^{7} + 4 q^{9} + 3 q^{11} - 28 q^{13} + 10 q^{15} + 24 q^{17} + 3 q^{19} + 60 q^{21} + 138 q^{23} + 150 q^{25} + 97 q^{27} + 76 q^{29} + 381 q^{31} - 3 q^{33} + 140 q^{35} + 131 q^{37} - 41 q^{39} - 95 q^{41} + 202 q^{43} - 20 q^{45} + 119 q^{47} - 578 q^{49} + 997 q^{51} + 137 q^{53} - 15 q^{55} - 894 q^{57} + 39 q^{59} - 573 q^{61} + 355 q^{63} + 140 q^{65} + 563 q^{67} - 46 q^{69} + 83 q^{71} - 1799 q^{73} - 50 q^{75} - 1410 q^{77} + 1636 q^{79} - 2006 q^{81} + 1191 q^{83} - 120 q^{85} + 1821 q^{87} - 1370 q^{89} + 1251 q^{91} - 2215 q^{93} - 15 q^{95} - 3021 q^{97} + 525 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 30 * q^5 - 28 * q^7 + 4 * q^9 + 3 * q^11 - 28 * q^13 + 10 * q^15 + 24 * q^17 + 3 * q^19 + 60 * q^21 + 138 * q^23 + 150 * q^25 + 97 * q^27 + 76 * q^29 + 381 * q^31 - 3 * q^33 + 140 * q^35 + 131 * q^37 - 41 * q^39 - 95 * q^41 + 202 * q^43 - 20 * q^45 + 119 * q^47 - 578 * q^49 + 997 * q^51 + 137 * q^53 - 15 * q^55 - 894 * q^57 + 39 * q^59 - 573 * q^61 + 355 * q^63 + 140 * q^65 + 563 * q^67 - 46 * q^69 + 83 * q^71 - 1799 * q^73 - 50 * q^75 - 1410 * q^77 + 1636 * q^79 - 2006 * q^81 + 1191 * q^83 - 120 * q^85 + 1821 * q^87 - 1370 * q^89 + 1251 * q^91 - 2215 * q^93 - 15 * q^95 - 3021 * q^97 + 525 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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