LMFDB - Embedded newform 420.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,4,Mod(1,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, 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("420.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	420.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:	$24.7808022024$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{130}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 130 $ x^2 - 130
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-11.4018$ of defining polynomial
Character	$\chi$	$=$	420.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -8.00000 q^{11} +47.6070 q^{13} -15.0000 q^{15} -3.60702 q^{17} -107.214 q^{19} +21.0000 q^{21} -29.6070 q^{23} +25.0000 q^{25} -27.0000 q^{27} +250.035 q^{29} +299.249 q^{31} +24.0000 q^{33} -35.0000 q^{35} -14.0351 q^{37} -142.821 q^{39} -234.856 q^{41} -120.035 q^{43} +45.0000 q^{45} +503.249 q^{47} +49.0000 q^{49} +10.8211 q^{51} +715.642 q^{53} -40.0000 q^{55} +321.642 q^{57} -148.070 q^{59} +137.965 q^{61} -63.0000 q^{63} +238.035 q^{65} +820.821 q^{67} +88.8211 q^{69} -726.463 q^{71} +968.393 q^{73} -75.0000 q^{75} +56.0000 q^{77} +1182.57 q^{79} +81.0000 q^{81} +724.070 q^{83} -18.0351 q^{85} -750.105 q^{87} +325.144 q^{89} -333.249 q^{91} -897.747 q^{93} -536.070 q^{95} +976.463 q^{97} -72.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9} - 16 q^{11} + 4 q^{13} - 30 q^{15} + 84 q^{17} - 32 q^{19} + 42 q^{21} + 32 q^{23} + 50 q^{25} - 54 q^{27} + 44 q^{29} - 40 q^{31} + 48 q^{33} - 70 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 - 16 * q^11 + 4 * q^13 - 30 * q^15 + 84 * q^17 - 32 * q^19 + 42 * q^21 + 32 * q^23 + 50 * q^25 - 54 * q^27 + 44 * q^29 - 40 * q^31 + 48 * q^33 - 70 * q^35 + 428 * q^37 - 12 * q^39 + 260 * q^41 + 216 * q^43 + 90 * q^45 + 368 * q^47 + 98 * q^49 - 252 * q^51 + 884 * q^53 - 80 * q^55 + 96 * q^57 + 616 * q^59 + 732 * q^61 - 126 * q^63 + 20 * q^65 + 1368 * q^67 - 96 * q^69 - 632 * q^71 + 2028 * q^73 - 150 * q^75 + 112 * q^77 + 176 * q^79 + 162 * q^81 + 536 * q^83 + 420 * q^85 - 132 * q^87 + 1380 * q^89 - 28 * q^91 + 120 * q^93 - 160 * q^95 + 1132 * q^97 - 144 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	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
$9$	9.00000	0.333333
$10$	0	0
$11$	−8.00000	−0.219281	−0.109640	−	0.993971i	$-0.534970\pi$
$11$	−8.00000	−0.219281	−0.109640	+	0.993971i	$0.534970\pi$
$12$	0	0
$13$	47.6070	1.01568	0.507839	−	0.861452i	$-0.330445\pi$
$13$	47.6070	1.01568	0.507839	+	0.861452i	$0.330445\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−3.60702	−0.0514606	−0.0257303	−	0.999669i	$-0.508191\pi$
$17$	−3.60702	−0.0514606	−0.0257303	+	0.999669i	$0.508191\pi$
$18$	0	0
$19$	−107.214	−1.29456	−0.647279	−	0.762254i	$-0.724093\pi$
$19$	−107.214	−1.29456	−0.647279	+	0.762254i	$0.724093\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	−29.6070	−0.268413	−0.134206	−	0.990953i	$-0.542848\pi$
$23$	−29.6070	−0.268413	−0.134206	+	0.990953i	$0.542848\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	250.035	1.60105	0.800523	−	0.599302i	$-0.204555\pi$
$29$	250.035	1.60105	0.800523	+	0.599302i	$0.204555\pi$
$30$	0	0
$31$	299.249	1.73377	0.866883	−	0.498512i	$-0.166120\pi$
$31$	299.249	1.73377	0.866883	+	0.498512i	$0.166120\pi$
$32$	0	0
$33$	24.0000	0.126602
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−14.0351	−0.0623609	−0.0311805	−	0.999514i	$-0.509927\pi$
$37$	−14.0351	−0.0623609	−0.0311805	+	0.999514i	$0.509927\pi$
$38$	0	0
$39$	−142.821	−0.586402
$40$	0	0
$41$	−234.856	−0.894594	−0.447297	−	0.894385i	$-0.647613\pi$
$41$	−234.856	−0.894594	−0.447297	+	0.894385i	$0.647613\pi$
$42$	0	0
$43$	−120.035	−0.425702	−0.212851	−	0.977085i	$-0.568275\pi$
$43$	−120.035	−0.425702	−0.212851	+	0.977085i	$0.568275\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	503.249	1.56184	0.780919	−	0.624632i	$-0.214751\pi$
$47$	503.249	1.56184	0.780919	+	0.624632i	$0.214751\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	10.8211	0.0297108
$52$	0	0
$53$	715.642	1.85474	0.927368	−	0.374151i	$-0.122066\pi$
$53$	715.642	1.85474	0.927368	+	0.374151i	$0.122066\pi$
$54$	0	0
$55$	−40.0000	−0.0980654
$56$	0	0
$57$	321.642	0.747413
$58$	0	0
$59$	−148.070	−0.326730	−0.163365	−	0.986566i	$-0.552235\pi$
$59$	−148.070	−0.326730	−0.163365	+	0.986566i	$0.552235\pi$
$60$	0	0
$61$	137.965	0.289584	0.144792	−	0.989462i	$-0.453749\pi$
$61$	137.965	0.289584	0.144792	+	0.989462i	$0.453749\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	238.035	0.454225
$66$	0	0
$67$	820.821	1.49671	0.748353	−	0.663301i	$-0.230845\pi$
$67$	820.821	1.49671	0.748353	+	0.663301i	$0.230845\pi$
$68$	0	0
$69$	88.8211	0.154968
$70$	0	0
$71$	−726.463	−1.21430	−0.607150	−	0.794587i	$-0.707687\pi$
$71$	−726.463	−1.21430	−0.607150	+	0.794587i	$0.707687\pi$
$72$	0	0
$73$	968.393	1.55263	0.776314	−	0.630347i	$-0.217087\pi$
$73$	968.393	1.55263	0.776314	+	0.630347i	$0.217087\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	56.0000	0.0828804
$78$	0	0
$79$	1182.57	1.68417	0.842084	−	0.539346i	$-0.181328\pi$
$79$	1182.57	1.68417	0.842084	+	0.539346i	$0.181328\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	724.070	0.957555	0.478777	−	0.877936i	$-0.341080\pi$
$83$	724.070	0.957555	0.478777	+	0.877936i	$0.341080\pi$
$84$	0	0
$85$	−18.0351	−0.0230139
$86$	0	0
$87$	−750.105	−0.924365
$88$	0	0
$89$	325.144	0.387249	0.193625	−	0.981076i	$-0.437976\pi$
$89$	325.144	0.387249	0.193625	+	0.981076i	$0.437976\pi$
$90$	0	0
$91$	−333.249	−0.383890
$92$	0	0
$93$	−897.747	−1.00099
$94$	0	0
$95$	−536.070	−0.578944
$96$	0	0
$97$	976.463	1.02211	0.511056	−	0.859548i	$-0.329255\pi$
$97$	976.463	1.02211	0.511056	+	0.859548i	$0.329255\pi$
$98$	0	0
$99$	−72.0000	−0.0730937
$100$	0	0
$101$	−13.0737	−0.0128800	−0.00644001	−	0.999979i	$-0.502050\pi$
$101$	−13.0737	−0.0128800	−0.00644001	+	0.999979i	$0.502050\pi$
$102$	0	0
$103$	712.000	0.681121	0.340560	−	0.940223i	$-0.389383\pi$
$103$	712.000	0.681121	0.340560	+	0.940223i	$0.389383\pi$
$104$	0	0
$105$	105.000	0.0975900
$106$	0	0
$107$	−1814.46	−1.63935	−0.819677	−	0.572827i	$-0.805847\pi$
$107$	−1814.46	−1.63935	−0.819677	+	0.572827i	$0.805847\pi$
$108$	0	0
$109$	1039.57	0.913513	0.456757	−	0.889592i	$-0.349011\pi$
$109$	1039.57	0.913513	0.456757	+	0.889592i	$0.349011\pi$
$110$	0	0
$111$	42.1053	0.0360041
$112$	0	0
$113$	−587.572	−0.489151	−0.244576	−	0.969630i	$-0.578649\pi$
$113$	−587.572	−0.489151	−0.244576	+	0.969630i	$0.578649\pi$
$114$	0	0
$115$	−148.035	−0.120038
$116$	0	0
$117$	428.463	0.338559
$118$	0	0
$119$	25.2491	0.0194503
$120$	0	0
$121$	−1267.00	−0.951916
$122$	0	0
$123$	704.568	0.516494
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−580.856	−0.405848	−0.202924	−	0.979195i	$-0.565044\pi$
$127$	−580.856	−0.405848	−0.202924	+	0.979195i	$0.565044\pi$
$128$	0	0
$129$	360.105	0.245779
$130$	0	0
$131$	−923.860	−0.616168	−0.308084	−	0.951359i	$-0.599688\pi$
$131$	−923.860	−0.616168	−0.308084	+	0.951359i	$0.599688\pi$
$132$	0	0
$133$	750.498	0.489297
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−1744.64	−1.08799	−0.543995	−	0.839089i	$-0.683089\pi$
$137$	−1744.64	−1.08799	−0.543995	+	0.839089i	$0.683089\pi$
$138$	0	0
$139$	−2851.64	−1.74009	−0.870045	−	0.492972i	$-0.835910\pi$
$139$	−2851.64	−1.74009	−0.870045	+	0.492972i	$0.835910\pi$
$140$	0	0
$141$	−1509.75	−0.901728
$142$	0	0
$143$	−380.856	−0.222719
$144$	0	0
$145$	1250.18	0.716010
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−1342.04	−0.737878	−0.368939	−	0.929454i	$-0.620279\pi$
$149$	−1342.04	−0.737878	−0.368939	+	0.929454i	$0.620279\pi$
$150$	0	0
$151$	−897.712	−0.483807	−0.241903	−	0.970300i	$-0.577772\pi$
$151$	−897.712	−0.483807	−0.241903	+	0.970300i	$0.577772\pi$
$152$	0	0
$153$	−32.4632	−0.0171535
$154$	0	0
$155$	1496.25	0.775363
$156$	0	0
$157$	1266.96	0.644042	0.322021	−	0.946733i	$-0.395638\pi$
$157$	1266.96	0.644042	0.322021	+	0.946733i	$0.395638\pi$
$158$	0	0
$159$	−2146.93	−1.07083
$160$	0	0
$161$	207.249	0.101450
$162$	0	0
$163$	555.179	0.266779	0.133389	−	0.991064i	$-0.457414\pi$
$163$	555.179	0.266779	0.133389	+	0.991064i	$0.457414\pi$
$164$	0	0
$165$	120.000	0.0566181
$166$	0	0
$167$	656.961	0.304414	0.152207	−	0.988349i	$-0.451362\pi$
$167$	656.961	0.304414	0.152207	+	0.988349i	$0.451362\pi$
$168$	0	0
$169$	69.4281	0.0316013
$170$	0	0
$171$	−964.926	−0.431519
$172$	0	0
$173$	−1115.99	−0.490447	−0.245224	−	0.969467i	$-0.578861\pi$
$173$	−1115.99	−0.490447	−0.245224	+	0.969467i	$0.578861\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	444.211	0.188638
$178$	0	0
$179$	1801.99	0.752443	0.376221	−	0.926530i	$-0.377223\pi$
$179$	1801.99	0.752443	0.376221	+	0.926530i	$0.377223\pi$
$180$	0	0
$181$	2570.25	1.05550	0.527748	−	0.849401i	$-0.323036\pi$
$181$	2570.25	1.05550	0.527748	+	0.849401i	$0.323036\pi$
$182$	0	0
$183$	−413.895	−0.167191
$184$	0	0
$185$	−70.1754	−0.0278886
$186$	0	0
$187$	28.8561	0.0112843
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−5224.25	−1.97913	−0.989563	−	0.144099i	$-0.953971\pi$
$191$	−5224.25	−1.97913	−0.989563	+	0.144099i	$0.953971\pi$
$192$	0	0
$193$	2109.07	0.786603	0.393302	−	0.919409i	$-0.371333\pi$
$193$	2109.07	0.786603	0.393302	+	0.919409i	$0.371333\pi$
$194$	0	0
$195$	−714.105	−0.262247
$196$	0	0
$197$	−1338.86	−0.484211	−0.242105	−	0.970250i	$-0.577838\pi$
$197$	−1338.86	−0.484211	−0.242105	+	0.970250i	$0.577838\pi$
$198$	0	0
$199$	2637.10	0.939393	0.469696	−	0.882828i	$-0.344363\pi$
$199$	2637.10	0.939393	0.469696	+	0.882828i	$0.344363\pi$
$200$	0	0
$201$	−2462.46	−0.864123
$202$	0	0
$203$	−1750.25	−0.605139
$204$	0	0
$205$	−1174.28	−0.400075
$206$	0	0
$207$	−266.463	−0.0894709
$208$	0	0
$209$	857.712	0.283872
$210$	0	0
$211$	−3.64915	−0.00119061	−0.000595303	−	1.00000i	$-0.500189\pi$
$211$	−3.64915	−0.00119061	−0.000595303		1.00000i	$0.500189\pi$
$212$	0	0
$213$	2179.39	0.701077
$214$	0	0
$215$	−600.175	−0.190380
$216$	0	0
$217$	−2094.74	−0.655302
$218$	0	0
$219$	−2905.18	−0.896410
$220$	0	0
$221$	−171.719	−0.0522674
$222$	0	0
$223$	5381.00	1.61587	0.807933	−	0.589274i	$-0.200586\pi$
$223$	5381.00	1.61587	0.807933	+	0.589274i	$0.200586\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−2159.07	−0.631290	−0.315645	−	0.948877i	$-0.602221\pi$
$227$	−2159.07	−0.631290	−0.315645	+	0.948877i	$0.602221\pi$
$228$	0	0
$229$	−423.467	−0.122199	−0.0610993	−	0.998132i	$-0.519461\pi$
$229$	−423.467	−0.122199	−0.0610993	+	0.998132i	$0.519461\pi$
$230$	0	0
$231$	−168.000	−0.0478510
$232$	0	0
$233$	3336.34	0.938074	0.469037	−	0.883179i	$-0.344601\pi$
$233$	3336.34	0.938074	0.469037	+	0.883179i	$0.344601\pi$
$234$	0	0
$235$	2516.25	0.698476
$236$	0	0
$237$	−3547.71	−0.972355
$238$	0	0
$239$	−5800.60	−1.56991	−0.784957	−	0.619550i	$-0.787315\pi$
$239$	−5800.60	−1.56991	−0.784957	+	0.619550i	$0.787315\pi$
$240$	0	0
$241$	−4531.56	−1.21122	−0.605609	−	0.795763i	$-0.707070\pi$
$241$	−4531.56	−1.21122	−0.605609	+	0.795763i	$0.707070\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	−5104.14	−1.31485
$248$	0	0
$249$	−2172.21	−0.552844
$250$	0	0
$251$	−2527.71	−0.635647	−0.317823	−	0.948150i	$-0.602952\pi$
$251$	−2527.71	−0.635647	−0.317823	+	0.948150i	$0.602952\pi$
$252$	0	0
$253$	236.856	0.0588578
$254$	0	0
$255$	54.1053	0.0132871
$256$	0	0
$257$	508.884	0.123515	0.0617574	−	0.998091i	$-0.480329\pi$
$257$	508.884	0.123515	0.0617574	+	0.998091i	$0.480329\pi$
$258$	0	0
$259$	98.2456	0.0235702
$260$	0	0
$261$	2250.32	0.533682
$262$	0	0
$263$	−1048.89	−0.245922	−0.122961	−	0.992412i	$-0.539239\pi$
$263$	−1048.89	−0.245922	−0.122961	+	0.992412i	$0.539239\pi$
$264$	0	0
$265$	3578.21	0.829463
$266$	0	0
$267$	−975.432	−0.223578
$268$	0	0
$269$	4642.72	1.05231	0.526155	−	0.850389i	$-0.323633\pi$
$269$	4642.72	1.05231	0.526155	+	0.850389i	$0.323633\pi$
$270$	0	0
$271$	−3659.53	−0.820297	−0.410149	−	0.912019i	$-0.634523\pi$
$271$	−3659.53	−0.820297	−0.410149	+	0.912019i	$0.634523\pi$
$272$	0	0
$273$	999.747	0.221639
$274$	0	0
$275$	−200.000	−0.0438562
$276$	0	0
$277$	5066.18	1.09891	0.549453	−	0.835525i	$-0.314836\pi$
$277$	5066.18	1.09891	0.549453	+	0.835525i	$0.314836\pi$
$278$	0	0
$279$	2693.24	0.577922
$280$	0	0
$281$	−1405.72	−0.298428	−0.149214	−	0.988805i	$-0.547674\pi$
$281$	−1405.72	−0.298428	−0.149214	+	0.988805i	$0.547674\pi$
$282$	0	0
$283$	5147.72	1.08127	0.540636	−	0.841256i	$-0.318184\pi$
$283$	5147.72	1.08127	0.540636	+	0.841256i	$0.318184\pi$
$284$	0	0
$285$	1608.21	0.334253
$286$	0	0
$287$	1643.99	0.338125
$288$	0	0
$289$	−4899.99	−0.997352
$290$	0	0
$291$	−2929.39	−0.590116
$292$	0	0
$293$	6426.65	1.28139	0.640697	−	0.767794i	$-0.278645\pi$
$293$	6426.65	1.28139	0.640697	+	0.767794i	$0.278645\pi$
$294$	0	0
$295$	−740.351	−0.146118
$296$	0	0
$297$	216.000	0.0422006
$298$	0	0
$299$	−1409.50	−0.272621
$300$	0	0
$301$	840.246	0.160900
$302$	0	0
$303$	39.2211	0.00743628
$304$	0	0
$305$	689.825	0.129506
$306$	0	0
$307$	3511.35	0.652780	0.326390	−	0.945235i	$-0.394168\pi$
$307$	3511.35	0.652780	0.326390	+	0.945235i	$0.394168\pi$
$308$	0	0
$309$	−2136.00	−0.393245
$310$	0	0
$311$	3487.13	0.635810	0.317905	−	0.948123i	$-0.397021\pi$
$311$	3487.13	0.635810	0.317905	+	0.948123i	$0.397021\pi$
$312$	0	0
$313$	3936.53	0.710882	0.355441	−	0.934699i	$-0.384331\pi$
$313$	3936.53	0.710882	0.355441	+	0.934699i	$0.384331\pi$
$314$	0	0
$315$	−315.000	−0.0563436
$316$	0	0
$317$	−5641.64	−0.999577	−0.499788	−	0.866148i	$-0.666589\pi$
$317$	−5641.64	−0.999577	−0.499788	+	0.866148i	$0.666589\pi$
$318$	0	0
$319$	−2000.28	−0.351079
$320$	0	0
$321$	5443.39	0.946481
$322$	0	0
$323$	386.723	0.0666187
$324$	0	0
$325$	1190.18	0.203136
$326$	0	0
$327$	−3118.72	−0.527417
$328$	0	0
$329$	−3522.74	−0.590320
$330$	0	0
$331$	7988.77	1.32659	0.663297	−	0.748356i	$-0.269157\pi$
$331$	7988.77	1.32659	0.663297	+	0.748356i	$0.269157\pi$
$332$	0	0
$333$	−126.316	−0.0207870
$334$	0	0
$335$	4104.11	0.669347
$336$	0	0
$337$	−1701.65	−0.275059	−0.137529	−	0.990498i	$-0.543916\pi$
$337$	−1701.65	−0.275059	−0.137529	+	0.990498i	$0.543916\pi$
$338$	0	0
$339$	1762.72	0.282412
$340$	0	0
$341$	−2393.99	−0.380182
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	444.105	0.0693038
$346$	0	0
$347$	−7223.09	−1.11745	−0.558726	−	0.829352i	$-0.688710\pi$
$347$	−7223.09	−1.11745	−0.558726	+	0.829352i	$0.688710\pi$
$348$	0	0
$349$	9167.31	1.40606	0.703030	−	0.711160i	$-0.251830\pi$
$349$	9167.31	1.40606	0.703030	+	0.711160i	$0.251830\pi$
$350$	0	0
$351$	−1285.39	−0.195467
$352$	0	0
$353$	−895.102	−0.134962	−0.0674808	−	0.997721i	$-0.521496\pi$
$353$	−895.102	−0.134962	−0.0674808	+	0.997721i	$0.521496\pi$
$354$	0	0
$355$	−3632.32	−0.543052
$356$	0	0
$357$	−75.7474	−0.0112296
$358$	0	0
$359$	2200.39	0.323487	0.161744	−	0.986833i	$-0.448288\pi$
$359$	2200.39	0.323487	0.161744	+	0.986833i	$0.448288\pi$
$360$	0	0
$361$	4635.85	0.675878
$362$	0	0
$363$	3801.00	0.549589
$364$	0	0
$365$	4841.96	0.694356
$366$	0	0
$367$	−8403.92	−1.19532	−0.597658	−	0.801751i	$-0.703902\pi$
$367$	−8403.92	−1.19532	−0.597658	+	0.801751i	$0.703902\pi$
$368$	0	0
$369$	−2113.71	−0.298198
$370$	0	0
$371$	−5009.49	−0.701024
$372$	0	0
$373$	5820.17	0.807927	0.403964	−	0.914775i	$-0.367632\pi$
$373$	5820.17	0.807927	0.403964	+	0.914775i	$0.367632\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	11903.4	1.62615
$378$	0	0
$379$	154.849	0.0209870	0.0104935	−	0.999945i	$-0.496660\pi$
$379$	154.849	0.0209870	0.0104935	+	0.999945i	$0.496660\pi$
$380$	0	0
$381$	1742.57	0.234316
$382$	0	0
$383$	8189.51	1.09260	0.546298	−	0.837591i	$-0.316037\pi$
$383$	8189.51	1.09260	0.546298	+	0.837591i	$0.316037\pi$
$384$	0	0
$385$	280.000	0.0370653
$386$	0	0
$387$	−1080.32	−0.141901
$388$	0	0
$389$	11037.6	1.43863	0.719317	−	0.694682i	$-0.244455\pi$
$389$	11037.6	1.43863	0.719317	+	0.694682i	$0.244455\pi$
$390$	0	0
$391$	106.793	0.0138127
$392$	0	0
$393$	2771.58	0.355745
$394$	0	0
$395$	5912.84	0.753183
$396$	0	0
$397$	−4297.16	−0.543246	−0.271623	−	0.962404i	$-0.587560\pi$
$397$	−4297.16	−0.543246	−0.271623	+	0.962404i	$0.587560\pi$
$398$	0	0
$399$	−2251.49	−0.282496
$400$	0	0
$401$	−6429.34	−0.800663	−0.400332	−	0.916370i	$-0.631105\pi$
$401$	−6429.34	−0.800663	−0.400332	+	0.916370i	$0.631105\pi$
$402$	0	0
$403$	14246.4	1.76095
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	112.281	0.0136746
$408$	0	0
$409$	−2714.65	−0.328192	−0.164096	−	0.986444i	$-0.552471\pi$
$409$	−2714.65	−0.328192	−0.164096	+	0.986444i	$0.552471\pi$
$410$	0	0
$411$	5233.92	0.628151
$412$	0	0
$413$	1036.49	0.123492
$414$	0	0
$415$	3620.35	0.428231
$416$	0	0
$417$	8554.91	1.00464
$418$	0	0
$419$	12058.8	1.40600	0.702999	−	0.711191i	$-0.251844\pi$
$419$	12058.8	1.40600	0.702999	+	0.711191i	$0.251844\pi$
$420$	0	0
$421$	−15177.7	−1.75704	−0.878522	−	0.477702i	$-0.841470\pi$
$421$	−15177.7	−1.75704	−0.878522	+	0.477702i	$0.841470\pi$
$422$	0	0
$423$	4529.24	0.520613
$424$	0	0
$425$	−90.1754	−0.0102921
$426$	0	0
$427$	−965.754	−0.109452
$428$	0	0
$429$	1142.57	0.128587
$430$	0	0
$431$	−6139.60	−0.686158	−0.343079	−	0.939307i	$-0.611470\pi$
$431$	−6139.60	−0.686158	−0.343079	+	0.939307i	$0.611470\pi$
$432$	0	0
$433$	−7673.29	−0.851628	−0.425814	−	0.904811i	$-0.640012\pi$
$433$	−7673.29	−0.851628	−0.425814	+	0.904811i	$0.640012\pi$
$434$	0	0
$435$	−3750.53	−0.413388
$436$	0	0
$437$	3174.29	0.347475
$438$	0	0
$439$	−6200.11	−0.674066	−0.337033	−	0.941493i	$-0.609423\pi$
$439$	−6200.11	−0.674066	−0.337033	+	0.941493i	$0.609423\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−7082.66	−0.759610	−0.379805	−	0.925067i	$-0.624009\pi$
$443$	−7082.66	−0.759610	−0.379805	+	0.925067i	$0.624009\pi$
$444$	0	0
$445$	1625.72	0.173183
$446$	0	0
$447$	4026.11	0.426014
$448$	0	0
$449$	1677.35	0.176301	0.0881506	−	0.996107i	$-0.471904\pi$
$449$	1677.35	0.176301	0.0881506	+	0.996107i	$0.471904\pi$
$450$	0	0
$451$	1878.85	0.196168
$452$	0	0
$453$	2693.14	0.279326
$454$	0	0
$455$	−1666.25	−0.171681
$456$	0	0
$457$	−4232.65	−0.433250	−0.216625	−	0.976255i	$-0.569505\pi$
$457$	−4232.65	−0.433250	−0.216625	+	0.976255i	$0.569505\pi$
$458$	0	0
$459$	97.3895	0.00990360
$460$	0	0
$461$	7543.64	0.762131	0.381066	−	0.924548i	$-0.375557\pi$
$461$	7543.64	0.762131	0.381066	+	0.924548i	$0.375557\pi$
$462$	0	0
$463$	1533.63	0.153939	0.0769695	−	0.997033i	$-0.475476\pi$
$463$	1533.63	0.153939	0.0769695	+	0.997033i	$0.475476\pi$
$464$	0	0
$465$	−4488.74	−0.447656
$466$	0	0
$467$	−16601.1	−1.64498	−0.822492	−	0.568776i	$-0.807417\pi$
$467$	−16601.1	−1.64498	−0.822492	+	0.568776i	$0.807417\pi$
$468$	0	0
$469$	−5745.75	−0.565701
$470$	0	0
$471$	−3800.88	−0.371838
$472$	0	0
$473$	960.281	0.0933483
$474$	0	0
$475$	−2680.35	−0.258911
$476$	0	0
$477$	6440.78	0.618245
$478$	0	0
$479$	−6104.39	−0.582290	−0.291145	−	0.956679i	$-0.594036\pi$
$479$	−6104.39	−0.582290	−0.291145	+	0.956679i	$0.594036\pi$
$480$	0	0
$481$	−668.169	−0.0633386
$482$	0	0
$483$	−621.747	−0.0585724
$484$	0	0
$485$	4882.32	0.457102
$486$	0	0
$487$	15817.6	1.47180	0.735898	−	0.677093i	$-0.236760\pi$
$487$	15817.6	1.47180	0.735898	+	0.677093i	$0.236760\pi$
$488$	0	0
$489$	−1665.54	−0.154025
$490$	0	0
$491$	7648.25	0.702975	0.351488	−	0.936193i	$-0.385676\pi$
$491$	7648.25	0.702975	0.351488	+	0.936193i	$0.385676\pi$
$492$	0	0
$493$	−901.881	−0.0823908
$494$	0	0
$495$	−360.000	−0.0326885
$496$	0	0
$497$	5085.24	0.458962
$498$	0	0
$499$	−5096.55	−0.457220	−0.228610	−	0.973518i	$-0.573418\pi$
$499$	−5096.55	−0.457220	−0.228610	+	0.973518i	$0.573418\pi$
$500$	0	0
$501$	−1970.88	−0.175754
$502$	0	0
$503$	2139.61	0.189663	0.0948317	−	0.995493i	$-0.469769\pi$
$503$	2139.61	0.189663	0.0948317	+	0.995493i	$0.469769\pi$
$504$	0	0
$505$	−65.3685	−0.00576012
$506$	0	0
$507$	−208.284	−0.0182450
$508$	0	0
$509$	16551.0	1.44128	0.720638	−	0.693312i	$-0.243849\pi$
$509$	16551.0	1.44128	0.720638	+	0.693312i	$0.243849\pi$
$510$	0	0
$511$	−6778.75	−0.586838
$512$	0	0
$513$	2894.78	0.249138
$514$	0	0
$515$	3560.00	0.304606
$516$	0	0
$517$	−4025.99	−0.342482
$518$	0	0
$519$	3347.98	0.283160
$520$	0	0
$521$	−10252.9	−0.862166	−0.431083	−	0.902312i	$-0.641868\pi$
$521$	−10252.9	−0.862166	−0.431083	+	0.902312i	$0.641868\pi$
$522$	0	0
$523$	−4139.62	−0.346105	−0.173053	−	0.984913i	$-0.555363\pi$
$523$	−4139.62	−0.346105	−0.173053	+	0.984913i	$0.555363\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	−1079.40	−0.0892206
$528$	0	0
$529$	−11290.4	−0.927955
$530$	0	0
$531$	−1332.63	−0.108910
$532$	0	0
$533$	−11180.8	−0.908620
$534$	0	0
$535$	−9072.32	−0.733141
$536$	0	0
$537$	−5405.98	−0.434423
$538$	0	0
$539$	−392.000	−0.0313259
$540$	0	0
$541$	−17964.3	−1.42763	−0.713815	−	0.700335i	$-0.753034\pi$
$541$	−17964.3	−1.42763	−0.713815	+	0.700335i	$0.753034\pi$
$542$	0	0
$543$	−7710.74	−0.609391
$544$	0	0
$545$	5197.86	0.408535
$546$	0	0
$547$	−9606.29	−0.750888	−0.375444	−	0.926845i	$-0.622510\pi$
$547$	−9606.29	−0.750888	−0.375444	+	0.926845i	$0.622510\pi$
$548$	0	0
$549$	1241.68	0.0965279
$550$	0	0
$551$	−26807.3	−2.07265
$552$	0	0
$553$	−8277.98	−0.636556
$554$	0	0
$555$	210.526	0.0161015
$556$	0	0
$557$	14761.5	1.12292	0.561459	−	0.827505i	$-0.310240\pi$
$557$	14761.5	1.12292	0.561459	+	0.827505i	$0.310240\pi$
$558$	0	0
$559$	−5714.51	−0.432376
$560$	0	0
$561$	−86.5684	−0.00651501
$562$	0	0
$563$	14119.8	1.05698	0.528491	−	0.848939i	$-0.322758\pi$
$563$	14119.8	1.05698	0.528491	+	0.848939i	$0.322758\pi$
$564$	0	0
$565$	−2937.86	−0.218755
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	3622.62	0.266903	0.133452	−	0.991055i	$-0.457394\pi$
$569$	3622.62	0.266903	0.133452	+	0.991055i	$0.457394\pi$
$570$	0	0
$571$	1247.31	0.0914158	0.0457079	−	0.998955i	$-0.485446\pi$
$571$	1247.31	0.0914158	0.0457079	+	0.998955i	$0.485446\pi$
$572$	0	0
$573$	15672.7	1.14265
$574$	0	0
$575$	−740.175	−0.0536825
$576$	0	0
$577$	−8345.35	−0.602117	−0.301058	−	0.953606i	$-0.597340\pi$
$577$	−8345.35	−0.602117	−0.301058	+	0.953606i	$0.597340\pi$
$578$	0	0
$579$	−6327.22	−0.454146
$580$	0	0
$581$	−5068.49	−0.361922
$582$	0	0
$583$	−5725.14	−0.406708
$584$	0	0
$585$	2142.32	0.151408
$586$	0	0
$587$	6191.99	0.435384	0.217692	−	0.976017i	$-0.430147\pi$
$587$	6191.99	0.435384	0.217692	+	0.976017i	$0.430147\pi$
$588$	0	0
$589$	−32083.7	−2.24446
$590$	0	0
$591$	4016.57	0.279559
$592$	0	0
$593$	−17840.9	−1.23548	−0.617738	−	0.786384i	$-0.711951\pi$
$593$	−17840.9	−1.23548	−0.617738	+	0.786384i	$0.711951\pi$
$594$	0	0
$595$	126.246	0.00869843
$596$	0	0
$597$	−7911.31	−0.542359
$598$	0	0
$599$	−8996.65	−0.613678	−0.306839	−	0.951761i	$-0.599271\pi$
$599$	−8996.65	−0.613678	−0.306839	+	0.951761i	$0.599271\pi$
$600$	0	0
$601$	11346.7	0.770122	0.385061	−	0.922891i	$-0.374180\pi$
$601$	11346.7	0.770122	0.385061	+	0.922891i	$0.374180\pi$
$602$	0	0
$603$	7387.39	0.498902
$604$	0	0
$605$	−6335.00	−0.425710
$606$	0	0
$607$	1461.98	0.0977593	0.0488796	−	0.998805i	$-0.484435\pi$
$607$	1461.98	0.0977593	0.0488796	+	0.998805i	$0.484435\pi$
$608$	0	0
$609$	5250.74	0.349377
$610$	0	0
$611$	23958.2	1.58633
$612$	0	0
$613$	−8018.02	−0.528295	−0.264147	−	0.964482i	$-0.585091\pi$
$613$	−8018.02	−0.528295	−0.264147	+	0.964482i	$0.585091\pi$
$614$	0	0
$615$	3522.84	0.230983
$616$	0	0
$617$	−21890.6	−1.42833	−0.714166	−	0.699976i	$-0.753194\pi$
$617$	−21890.6	−1.42833	−0.714166	+	0.699976i	$0.753194\pi$
$618$	0	0
$619$	17702.0	1.14944	0.574719	−	0.818351i	$-0.305111\pi$
$619$	17702.0	1.14944	0.574719	+	0.818351i	$0.305111\pi$
$620$	0	0
$621$	799.389	0.0516560
$622$	0	0
$623$	−2276.01	−0.146366
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−2573.14	−0.163893
$628$	0	0
$629$	50.6248	0.00320913
$630$	0	0
$631$	−12482.9	−0.787538	−0.393769	−	0.919209i	$-0.628829\pi$
$631$	−12482.9	−0.787538	−0.393769	+	0.919209i	$0.628829\pi$
$632$	0	0
$633$	10.9474	0.000687397 0
$634$	0	0
$635$	−2904.28	−0.181501
$636$	0	0
$637$	2332.74	0.145097
$638$	0	0
$639$	−6538.17	−0.404767
$640$	0	0
$641$	2433.03	0.149920	0.0749602	−	0.997187i	$-0.476117\pi$
$641$	2433.03	0.149920	0.0749602	+	0.997187i	$0.476117\pi$
$642$	0	0
$643$	25638.6	1.57246	0.786228	−	0.617937i	$-0.212031\pi$
$643$	25638.6	1.57246	0.786228	+	0.617937i	$0.212031\pi$
$644$	0	0
$645$	1800.53	0.109916
$646$	0	0
$647$	−1377.10	−0.0836777	−0.0418388	−	0.999124i	$-0.513322\pi$
$647$	−1377.10	−0.0836777	−0.0418388	+	0.999124i	$0.513322\pi$
$648$	0	0
$649$	1184.56	0.0716458
$650$	0	0
$651$	6284.23	0.378339
$652$	0	0
$653$	−30887.6	−1.85103	−0.925517	−	0.378707i	$-0.876369\pi$
$653$	−30887.6	−1.85103	−0.925517	+	0.378707i	$0.876369\pi$
$654$	0	0
$655$	−4619.30	−0.275559
$656$	0	0
$657$	8715.54	0.517543
$658$	0	0
$659$	8566.50	0.506378	0.253189	−	0.967417i	$-0.418520\pi$
$659$	8566.50	0.506378	0.253189	+	0.967417i	$0.418520\pi$
$660$	0	0
$661$	−31788.0	−1.87052	−0.935258	−	0.353968i	$-0.884832\pi$
$661$	−31788.0	−1.87052	−0.935258	+	0.353968i	$0.884832\pi$
$662$	0	0
$663$	515.158	0.0301766
$664$	0	0
$665$	3752.49	0.218820
$666$	0	0
$667$	−7402.79	−0.429741
$668$	0	0
$669$	−16143.0	−0.932921
$670$	0	0
$671$	−1103.72	−0.0635002
$672$	0	0
$673$	−12617.1	−0.722667	−0.361334	−	0.932437i	$-0.617678\pi$
$673$	−12617.1	−0.722667	−0.361334	+	0.932437i	$0.617678\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	18994.5	1.07831	0.539157	−	0.842205i	$-0.318743\pi$
$677$	18994.5	1.07831	0.539157	+	0.842205i	$0.318743\pi$
$678$	0	0
$679$	−6835.24	−0.386322
$680$	0	0
$681$	6477.22	0.364475
$682$	0	0
$683$	2154.07	0.120678	0.0603391	−	0.998178i	$-0.480782\pi$
$683$	2154.07	0.120678	0.0603391	+	0.998178i	$0.480782\pi$
$684$	0	0
$685$	−8723.19	−0.486564
$686$	0	0
$687$	1270.40	0.0705514
$688$	0	0
$689$	34069.6	1.88381
$690$	0	0
$691$	26566.9	1.46260	0.731298	−	0.682059i	$-0.238915\pi$
$691$	26566.9	1.46260	0.731298	+	0.682059i	$0.238915\pi$
$692$	0	0
$693$	504.000	0.0276268
$694$	0	0
$695$	−14258.2	−0.778192
$696$	0	0
$697$	847.130	0.0460364
$698$	0	0
$699$	−10009.0	−0.541597
$700$	0	0
$701$	12469.2	0.671831	0.335915	−	0.941892i	$-0.390954\pi$
$701$	12469.2	0.671831	0.335915	+	0.941892i	$0.390954\pi$
$702$	0	0
$703$	1504.76	0.0807298
$704$	0	0
$705$	−7548.74	−0.403265
$706$	0	0
$707$	91.5159	0.00486819
$708$	0	0
$709$	−16393.4	−0.868358	−0.434179	−	0.900827i	$-0.642961\pi$
$709$	−16393.4	−0.868358	−0.434179	+	0.900827i	$0.642961\pi$
$710$	0	0
$711$	10643.1	0.561390
$712$	0	0
$713$	−8859.87	−0.465364
$714$	0	0
$715$	−1904.28	−0.0996029
$716$	0	0
$717$	17401.8	0.906390
$718$	0	0
$719$	30451.8	1.57950	0.789750	−	0.613428i	$-0.210210\pi$
$719$	30451.8	1.57950	0.789750	+	0.613428i	$0.210210\pi$
$720$	0	0
$721$	−4984.00	−0.257439
$722$	0	0
$723$	13594.7	0.699296
$724$	0	0
$725$	6250.88	0.320209
$726$	0	0
$727$	−31913.8	−1.62809	−0.814043	−	0.580804i	$-0.802738\pi$
$727$	−31913.8	−1.62809	−0.814043	+	0.580804i	$0.802738\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	432.969	0.0219069
$732$	0	0
$733$	−5065.68	−0.255260	−0.127630	−	0.991822i	$-0.540737\pi$
$733$	−5065.68	−0.255260	−0.127630	+	0.991822i	$0.540737\pi$
$734$	0	0
$735$	−735.000	−0.0368856
$736$	0	0
$737$	−6566.57	−0.328199
$738$	0	0
$739$	30785.2	1.53241	0.766206	−	0.642595i	$-0.222142\pi$
$739$	30785.2	1.53241	0.766206	+	0.642595i	$0.222142\pi$
$740$	0	0
$741$	15312.4	0.759131
$742$	0	0
$743$	−31310.8	−1.54601	−0.773004	−	0.634402i	$-0.781247\pi$
$743$	−31310.8	−1.54601	−0.773004	+	0.634402i	$0.781247\pi$
$744$	0	0
$745$	−6710.18	−0.329989
$746$	0	0
$747$	6516.63	0.319185
$748$	0	0
$749$	12701.2	0.619617
$750$	0	0
$751$	19339.4	0.939688	0.469844	−	0.882749i	$-0.344310\pi$
$751$	19339.4	0.939688	0.469844	+	0.882749i	$0.344310\pi$
$752$	0	0
$753$	7583.12	0.366991
$754$	0	0
$755$	−4488.56	−0.216365
$756$	0	0
$757$	27657.0	1.32789	0.663943	−	0.747783i	$-0.268882\pi$
$757$	27657.0	1.32789	0.663943	+	0.747783i	$0.268882\pi$
$758$	0	0
$759$	−710.568	−0.0339816
$760$	0	0
$761$	−33018.3	−1.57281	−0.786407	−	0.617708i	$-0.788061\pi$
$761$	−33018.3	−1.57281	−0.786407	+	0.617708i	$0.788061\pi$
$762$	0	0
$763$	−7277.00	−0.345276
$764$	0	0
$765$	−162.316	−0.00767129
$766$	0	0
$767$	−7049.18	−0.331853
$768$	0	0
$769$	−32218.5	−1.51083	−0.755416	−	0.655245i	$-0.772565\pi$
$769$	−32218.5	−1.51083	−0.755416	+	0.655245i	$0.772565\pi$
$770$	0	0
$771$	−1526.65	−0.0713113
$772$	0	0
$773$	−17244.8	−0.802395	−0.401197	−	0.915992i	$-0.631406\pi$
$773$	−17244.8	−0.802395	−0.401197	+	0.915992i	$0.631406\pi$
$774$	0	0
$775$	7481.23	0.346753
$776$	0	0
$777$	−294.737	−0.0136083
$778$	0	0
$779$	25179.9	1.15810
$780$	0	0
$781$	5811.71	0.266273
$782$	0	0
$783$	−6750.95	−0.308122
$784$	0	0
$785$	6334.81	0.288024
$786$	0	0
$787$	29840.4	1.35158	0.675791	−	0.737093i	$-0.263802\pi$
$787$	29840.4	1.35158	0.675791	+	0.737093i	$0.263802\pi$
$788$	0	0
$789$	3146.67	0.141983
$790$	0	0
$791$	4113.00	0.184882
$792$	0	0
$793$	6568.10	0.294124
$794$	0	0
$795$	−10734.6	−0.478891
$796$	0	0
$797$	38261.5	1.70049	0.850246	−	0.526386i	$-0.176453\pi$
$797$	38261.5	1.70049	0.850246	+	0.526386i	$0.176453\pi$
$798$	0	0
$799$	−1815.23	−0.0803732
$800$	0	0
$801$	2926.29	0.129083
$802$	0	0
$803$	−7747.14	−0.340462
$804$	0	0
$805$	1036.25	0.0453700
$806$	0	0
$807$	−13928.1	−0.607551
$808$	0	0
$809$	−9614.18	−0.417820	−0.208910	−	0.977935i	$-0.566992\pi$
$809$	−9614.18	−0.417820	−0.208910	+	0.977935i	$0.566992\pi$
$810$	0	0
$811$	6295.26	0.272573	0.136286	−	0.990669i	$-0.456483\pi$
$811$	6295.26	0.272573	0.136286	+	0.990669i	$0.456483\pi$
$812$	0	0
$813$	10978.6	0.473599
$814$	0	0
$815$	2775.89	0.119307
$816$	0	0
$817$	12869.4	0.551095
$818$	0	0
$819$	−2999.24	−0.127963
$820$	0	0
$821$	−1947.06	−0.0827684	−0.0413842	−	0.999143i	$-0.513177\pi$
$821$	−1947.06	−0.0827684	−0.0413842	+	0.999143i	$0.513177\pi$
$822$	0	0
$823$	21340.5	0.903867	0.451933	−	0.892052i	$-0.350734\pi$
$823$	21340.5	0.903867	0.451933	+	0.892052i	$0.350734\pi$
$824$	0	0
$825$	600.000	0.0253204
$826$	0	0
$827$	26708.7	1.12304	0.561519	−	0.827464i	$-0.310217\pi$
$827$	26708.7	1.12304	0.561519	+	0.827464i	$0.310217\pi$
$828$	0	0
$829$	−41080.8	−1.72110	−0.860551	−	0.509364i	$-0.829881\pi$
$829$	−41080.8	−1.72110	−0.860551	+	0.509364i	$0.829881\pi$
$830$	0	0
$831$	−15198.5	−0.634454
$832$	0	0
$833$	−176.744	−0.00735151
$834$	0	0
$835$	3284.81	0.136138
$836$	0	0
$837$	−8079.73	−0.333663
$838$	0	0
$839$	21123.1	0.869189	0.434595	−	0.900626i	$-0.356892\pi$
$839$	21123.1	0.869189	0.434595	+	0.900626i	$0.356892\pi$
$840$	0	0
$841$	38128.5	1.56335
$842$	0	0
$843$	4217.16	0.172297
$844$	0	0
$845$	347.140	0.0141325
$846$	0	0
$847$	8869.00	0.359790
$848$	0	0
$849$	−15443.2	−0.624273
$850$	0	0
$851$	415.537	0.0167385
$852$	0	0
$853$	−32643.6	−1.31031	−0.655156	−	0.755494i	$-0.727397\pi$
$853$	−32643.6	−1.31031	−0.655156	+	0.755494i	$0.727397\pi$
$854$	0	0
$855$	−4824.63	−0.192981
$856$	0	0
$857$	−16653.4	−0.663790	−0.331895	−	0.943316i	$-0.607688\pi$
$857$	−16653.4	−0.663790	−0.331895	+	0.943316i	$0.607688\pi$
$858$	0	0
$859$	29551.0	1.17377	0.586885	−	0.809671i	$-0.300354\pi$
$859$	29551.0	1.17377	0.586885	+	0.809671i	$0.300354\pi$
$860$	0	0
$861$	−4931.98	−0.195216
$862$	0	0
$863$	−15877.7	−0.626284	−0.313142	−	0.949706i	$-0.601382\pi$
$863$	−15877.7	−0.626284	−0.313142	+	0.949706i	$0.601382\pi$
$864$	0	0
$865$	−5579.96	−0.219335
$866$	0	0
$867$	14700.0	0.575821
$868$	0	0
$869$	−9460.55	−0.369306
$870$	0	0
$871$	39076.8	1.52017
$872$	0	0
$873$	8788.17	0.340704
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−15580.5	−0.599905	−0.299952	−	0.953954i	$-0.596971\pi$
$877$	−15580.5	−0.599905	−0.299952	+	0.953954i	$0.596971\pi$
$878$	0	0
$879$	−19279.9	−0.739814
$880$	0	0
$881$	19212.6	0.734720	0.367360	−	0.930079i	$-0.380262\pi$
$881$	19212.6	0.734720	0.367360	+	0.930079i	$0.380262\pi$
$882$	0	0
$883$	−5438.36	−0.207266	−0.103633	−	0.994616i	$-0.533047\pi$
$883$	−5438.36	−0.207266	−0.103633	+	0.994616i	$0.533047\pi$
$884$	0	0
$885$	2221.05	0.0843614
$886$	0	0
$887$	−14961.8	−0.566367	−0.283183	−	0.959066i	$-0.591390\pi$
$887$	−14961.8	−0.566367	−0.283183	+	0.959066i	$0.591390\pi$
$888$	0	0
$889$	4065.99	0.153396
$890$	0	0
$891$	−648.000	−0.0243646
$892$	0	0
$893$	−53955.4	−2.02189
$894$	0	0
$895$	9009.96	0.336503
$896$	0	0
$897$	4228.51	0.157398
$898$	0	0
$899$	74822.8	2.77584
$900$	0	0
$901$	−2581.33	−0.0954458
$902$	0	0
$903$	−2520.74	−0.0928958
$904$	0	0
$905$	12851.2	0.472032
$906$	0	0
$907$	33326.6	1.22006	0.610029	−	0.792379i	$-0.291158\pi$
$907$	33326.6	1.22006	0.610029	+	0.792379i	$0.291158\pi$
$908$	0	0
$909$	−117.663	−0.00429334
$910$	0	0
$911$	−1597.21	−0.0580879	−0.0290439	−	0.999578i	$-0.509246\pi$
$911$	−1597.21	−0.0580879	−0.0290439	+	0.999578i	$0.509246\pi$
$912$	0	0
$913$	−5792.56	−0.209973
$914$	0	0
$915$	−2069.47	−0.0747702
$916$	0	0
$917$	6467.02	0.232890
$918$	0	0
$919$	5165.28	0.185405	0.0927023	−	0.995694i	$-0.470450\pi$
$919$	5165.28	0.185405	0.0927023	+	0.995694i	$0.470450\pi$
$920$	0	0
$921$	−10534.1	−0.376883
$922$	0	0
$923$	−34584.7	−1.23334
$924$	0	0
$925$	−350.877	−0.0124722
$926$	0	0
$927$	6408.00	0.227040
$928$	0	0
$929$	−46519.1	−1.64289	−0.821444	−	0.570290i	$-0.806831\pi$
$929$	−46519.1	−1.64289	−0.821444	+	0.570290i	$0.806831\pi$
$930$	0	0
$931$	−5253.49	−0.184937
$932$	0	0
$933$	−10461.4	−0.367085
$934$	0	0
$935$	144.281	0.00504651
$936$	0	0
$937$	12275.1	0.427973	0.213987	−	0.976837i	$-0.431355\pi$
$937$	12275.1	0.427973	0.213987	+	0.976837i	$0.431355\pi$
$938$	0	0
$939$	−11809.6	−0.410428
$940$	0	0
$941$	31881.3	1.10446	0.552232	−	0.833690i	$-0.313776\pi$
$941$	31881.3	1.10446	0.552232	+	0.833690i	$0.313776\pi$
$942$	0	0
$943$	6953.39	0.240120
$944$	0	0
$945$	945.000	0.0325300
$946$	0	0
$947$	−53055.6	−1.82057	−0.910283	−	0.413987i	$-0.864136\pi$
$947$	−53055.6	−1.82057	−0.910283	+	0.413987i	$0.864136\pi$
$948$	0	0
$949$	46102.3	1.57697
$950$	0	0
$951$	16924.9	0.577106
$952$	0	0
$953$	31377.1	1.06653	0.533265	−	0.845948i	$-0.320965\pi$
$953$	31377.1	1.06653	0.533265	+	0.845948i	$0.320965\pi$
$954$	0	0
$955$	−26121.2	−0.885092
$956$	0	0
$957$	6000.84	0.202696
$958$	0	0
$959$	12212.5	0.411221
$960$	0	0
$961$	59759.0	2.00594
$962$	0	0
$963$	−16330.2	−0.546451
$964$	0	0
$965$	10545.4	0.351780
$966$	0	0
$967$	12719.5	0.422989	0.211494	−	0.977379i	$-0.432167\pi$
$967$	12719.5	0.422989	0.211494	+	0.977379i	$0.432167\pi$
$968$	0	0
$969$	−1160.17	−0.0384623
$970$	0	0
$971$	58288.2	1.92642	0.963212	−	0.268742i	$-0.0866079\pi$
$971$	58288.2	1.92642	0.963212	+	0.268742i	$0.0866079\pi$
$972$	0	0
$973$	19961.4	0.657692
$974$	0	0
$975$	−3570.53	−0.117280
$976$	0	0
$977$	29987.3	0.981965	0.490983	−	0.871169i	$-0.336638\pi$
$977$	29987.3	0.981965	0.490983	+	0.871169i	$0.336638\pi$
$978$	0	0
$979$	−2601.15	−0.0849164
$980$	0	0
$981$	9356.15	0.304504
$982$	0	0
$983$	21551.4	0.699270	0.349635	−	0.936886i	$-0.386306\pi$
$983$	21551.4	0.699270	0.349635	+	0.936886i	$0.386306\pi$
$984$	0	0
$985$	−6694.28	−0.216546
$986$	0	0
$987$	10568.2	0.340821
$988$	0	0
$989$	3553.88	0.114264
$990$	0	0
$991$	31116.1	0.997413	0.498707	−	0.866771i	$-0.333809\pi$
$991$	31116.1	0.997413	0.498707	+	0.866771i	$0.333809\pi$
$992$	0	0
$993$	−23966.3	−0.765909
$994$	0	0
$995$	13185.5	0.420109
$996$	0	0
$997$	−58953.1	−1.87268	−0.936340	−	0.351093i	$-0.885810\pi$
$997$	−58953.1	−1.87268	−0.936340	+	0.351093i	$0.885810\pi$
$998$	0	0
$999$	378.947	0.0120014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.4.a.h.1.2	✓	2
3.2	odd	2		1260.4.a.m.1.2		2
4.3	odd	2		1680.4.a.bp.1.2		2
5.2	odd	4		2100.4.k.k.1849.3		4
5.3	odd	4		2100.4.k.k.1849.2		4
5.4	even	2		2100.4.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.4.a.h.1.2	✓	2	1.1	even	1	trivial
1260.4.a.m.1.2		2	3.2	odd	2
1680.4.a.bp.1.2		2	4.3	odd	2
2100.4.a.u.1.1		2	5.4	even	2
2100.4.k.k.1849.2		4	5.3	odd	4
2100.4.k.k.1849.3		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	420.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -8.00000 q^{11} +47.6070 q^{13} -15.0000 q^{15} -3.60702 q^{17} -107.214 q^{19} +21.0000 q^{21} -29.6070 q^{23} +25.0000 q^{25} -27.0000 q^{27} +250.035 q^{29} +299.249 q^{31} +24.0000 q^{33} -35.0000 q^{35} -14.0351 q^{37} -142.821 q^{39} -234.856 q^{41} -120.035 q^{43} +45.0000 q^{45} +503.249 q^{47} +49.0000 q^{49} +10.8211 q^{51} +715.642 q^{53} -40.0000 q^{55} +321.642 q^{57} -148.070 q^{59} +137.965 q^{61} -63.0000 q^{63} +238.035 q^{65} +820.821 q^{67} +88.8211 q^{69} -726.463 q^{71} +968.393 q^{73} -75.0000 q^{75} +56.0000 q^{77} +1182.57 q^{79} +81.0000 q^{81} +724.070 q^{83} -18.0351 q^{85} -750.105 q^{87} +325.144 q^{89} -333.249 q^{91} -897.747 q^{93} -536.070 q^{95} +976.463 q^{97} -72.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 10 q^{5} - 14 q^{7} + 18 q^{9} - 16 q^{11} + 4 q^{13} - 30 q^{15} + 84 q^{17} - 32 q^{19} + 42 q^{21} + 32 q^{23} + 50 q^{25} - 54 q^{27} + 44 q^{29} - 40 q^{31} + 48 q^{33} - 70 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 10 * q^5 - 14 * q^7 + 18 * q^9 - 16 * q^11 + 4 * q^13 - 30 * q^15 + 84 * q^17 - 32 * q^19 + 42 * q^21 + 32 * q^23 + 50 * q^25 - 54 * q^27 + 44 * q^29 - 40 * q^31 + 48 * q^33 - 70 * q^35 + 428 * q^37 - 12 * q^39 + 260 * q^41 + 216 * q^43 + 90 * q^45 + 368 * q^47 + 98 * q^49 - 252 * q^51 + 884 * q^53 - 80 * q^55 + 96 * q^57 + 616 * q^59 + 732 * q^61 - 126 * q^63 + 20 * q^65 + 1368 * q^67 - 96 * q^69 - 632 * q^71 + 2028 * q^73 - 150 * q^75 + 112 * q^77 + 176 * q^79 + 162 * q^81 + 536 * q^83 + 420 * q^85 - 132 * q^87 + 1380 * q^89 - 28 * q^91 + 120 * q^93 - 160 * q^95 + 1132 * q^97 - 144 * q^99

Introduction

L-functions

Modular forms

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