LMFDB - Embedded newform 124.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,4,Mod(1,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("124.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	124.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:	$7.31623684071$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4000044.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 21x^{2} + 16x + 62 $ x^4 - x^3 - 21x^2 + 16x + 62
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.13619$ of defining polynomial
Character	$\chi$	$=$	124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.27238 q^{3} -14.2221 q^{5} -10.5389 q^{7} +41.4323 q^{9} +O(q^{10})$	$q-8.27238 q^{3} -14.2221 q^{5} -10.5389 q^{7} +41.4323 q^{9} +63.0333 q^{11} -7.43887 q^{13} +117.650 q^{15} -92.0710 q^{17} +85.6533 q^{19} +87.1814 q^{21} +28.0075 q^{23} +77.2676 q^{25} -119.390 q^{27} +280.289 q^{29} +31.0000 q^{31} -521.436 q^{33} +149.884 q^{35} -68.5265 q^{37} +61.5372 q^{39} -362.214 q^{41} -253.611 q^{43} -589.254 q^{45} +512.852 q^{47} -231.933 q^{49} +761.647 q^{51} +536.888 q^{53} -896.465 q^{55} -708.557 q^{57} +99.5106 q^{59} +272.454 q^{61} -436.649 q^{63} +105.796 q^{65} +160.654 q^{67} -231.688 q^{69} +811.229 q^{71} +877.634 q^{73} -639.187 q^{75} -664.299 q^{77} -5.11763 q^{79} -131.035 q^{81} +73.4805 q^{83} +1309.44 q^{85} -2318.66 q^{87} -695.497 q^{89} +78.3971 q^{91} -256.444 q^{93} -1218.17 q^{95} +832.569 q^{97} +2611.62 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9	$ 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9} + 96 q^{11} + 42 q^{13} + 182 q^{15} - 2 q^{17} + 180 q^{19} + 150 q^{21} + 142 q^{23} + 150 q^{25} + 20 q^{27} + 262 q^{29} + 124 q^{31} - 444 q^{33} + 224 q^{35} - 284 q^{37} + 60 q^{39} - 526 q^{41} - 326 q^{43} - 870 q^{45} + 468 q^{47} - 978 q^{49} + 356 q^{51} - 252 q^{53} - 876 q^{55} - 1298 q^{57} - 164 q^{59} - 1066 q^{61} - 236 q^{63} - 598 q^{65} - 956 q^{69} + 1504 q^{71} - 732 q^{73} + 188 q^{75} - 288 q^{77} + 822 q^{79} - 536 q^{81} + 1408 q^{83} + 482 q^{85} - 208 q^{87} + 250 q^{89} + 946 q^{91} + 62 q^{93} + 2292 q^{95} + 526 q^{97} + 2952 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9 + 96 * q^11 + 42 * q^13 + 182 * q^15 - 2 * q^17 + 180 * q^19 + 150 * q^21 + 142 * q^23 + 150 * q^25 + 20 * q^27 + 262 * q^29 + 124 * q^31 - 444 * q^33 + 224 * q^35 - 284 * q^37 + 60 * q^39 - 526 * q^41 - 326 * q^43 - 870 * q^45 + 468 * q^47 - 978 * q^49 + 356 * q^51 - 252 * q^53 - 876 * q^55 - 1298 * q^57 - 164 * q^59 - 1066 * q^61 - 236 * q^63 - 598 * q^65 - 956 * q^69 + 1504 * q^71 - 732 * q^73 + 188 * q^75 - 288 * q^77 + 822 * q^79 - 536 * q^81 + 1408 * q^83 + 482 * q^85 - 208 * q^87 + 250 * q^89 + 946 * q^91 + 62 * q^93 + 2292 * q^95 + 526 * q^97 + 2952 * 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
$3$	−8.27238	−1.59202	−0.796010	−	0.605283i	$-0.793060\pi$
$3$	−8.27238	−1.59202	−0.796010	+	0.605283i	$0.793060\pi$
$4$	0	0
$5$	−14.2221	−1.27206	−0.636031	−	0.771664i	$-0.719425\pi$
$5$	−14.2221	−1.27206	−0.636031	+	0.771664i	$0.719425\pi$
$6$	0	0
$7$	−10.5389	−0.569045	−0.284522	−	0.958669i	$-0.591835\pi$
$7$	−10.5389	−0.569045	−0.284522	+	0.958669i	$0.591835\pi$
$8$	0	0
$9$	41.4323	1.53453
$10$	0	0
$11$	63.0333	1.72775	0.863875	−	0.503705i	$-0.168030\pi$
$11$	63.0333	1.72775	0.863875	+	0.503705i	$0.168030\pi$
$12$	0	0
$13$	−7.43887	−0.158705	−0.0793527	−	0.996847i	$-0.525285\pi$
$13$	−7.43887	−0.158705	−0.0793527	+	0.996847i	$0.525285\pi$
$14$	0	0
$15$	117.650	2.02515
$16$	0	0
$17$	−92.0710	−1.31356	−0.656779	−	0.754083i	$-0.728082\pi$
$17$	−92.0710	−1.31356	−0.656779	+	0.754083i	$0.728082\pi$
$18$	0	0
$19$	85.6533	1.03422	0.517111	−	0.855918i	$-0.327007\pi$
$19$	85.6533	1.03422	0.517111	+	0.855918i	$0.327007\pi$
$20$	0	0
$21$	87.1814	0.905931
$22$	0	0
$23$	28.0075	0.253911	0.126956	−	0.991908i	$-0.459479\pi$
$23$	28.0075	0.253911	0.126956	+	0.991908i	$0.459479\pi$
$24$	0	0
$25$	77.2676	0.618140
$26$	0	0
$27$	−119.390	−0.850984
$28$	0	0
$29$	280.289	1.79477	0.897385	−	0.441248i	$-0.145464\pi$
$29$	280.289	1.79477	0.897385	+	0.441248i	$0.145464\pi$
$30$	0	0
$31$	31.0000	0.179605
$31$	31.0000	0.179605
$32$	0	0
$33$	−521.436	−2.75062
$34$	0	0
$35$	149.884	0.723860
$36$	0	0
$37$	−68.5265	−0.304478	−0.152239	−	0.988344i	$-0.548648\pi$
$37$	−68.5265	−0.304478	−0.152239	+	0.988344i	$0.548648\pi$
$38$	0	0
$39$	61.5372	0.252662
$40$	0	0
$41$	−362.214	−1.37971	−0.689857	−	0.723946i	$-0.742327\pi$
$41$	−362.214	−1.37971	−0.689857	+	0.723946i	$0.742327\pi$
$42$	0	0
$43$	−253.611	−0.899426	−0.449713	−	0.893173i	$-0.648474\pi$
$43$	−253.611	−0.899426	−0.449713	+	0.893173i	$0.648474\pi$
$44$	0	0
$45$	−589.254	−1.95202
$46$	0	0
$47$	512.852	1.59164	0.795821	−	0.605531i	$-0.207039\pi$
$47$	512.852	1.59164	0.795821	+	0.605531i	$0.207039\pi$
$48$	0	0
$49$	−231.933	−0.676188
$50$	0	0
$51$	761.647	2.09121
$52$	0	0
$53$	536.888	1.39146	0.695729	−	0.718305i	$-0.255082\pi$
$53$	536.888	1.39146	0.695729	+	0.718305i	$0.255082\pi$
$54$	0	0
$55$	−896.465	−2.19781
$56$	0	0
$57$	−708.557	−1.64650
$58$	0	0
$59$	99.5106	0.219579	0.109790	−	0.993955i	$-0.464982\pi$
$59$	99.5106	0.219579	0.109790	+	0.993955i	$0.464982\pi$
$60$	0	0
$61$	272.454	0.571871	0.285936	−	0.958249i	$-0.407696\pi$
$61$	272.454	0.571871	0.285936	+	0.958249i	$0.407696\pi$
$62$	0	0
$63$	−436.649	−0.873216
$64$	0	0
$65$	105.796	0.201883
$66$	0	0
$67$	160.654	0.292940	0.146470	−	0.989215i	$-0.453209\pi$
$67$	160.654	0.292940	0.146470	+	0.989215i	$0.453209\pi$
$68$	0	0
$69$	−231.688	−0.404232
$70$	0	0
$71$	811.229	1.35599	0.677994	−	0.735068i	$-0.262850\pi$
$71$	811.229	1.35599	0.677994	+	0.735068i	$0.262850\pi$
$72$	0	0
$73$	877.634	1.40711	0.703557	−	0.710639i	$-0.251594\pi$
$73$	877.634	1.40711	0.703557	+	0.710639i	$0.251594\pi$
$74$	0	0
$75$	−639.187	−0.984092
$76$	0	0
$77$	−664.299	−0.983167
$78$	0	0
$79$	−5.11763	−0.00728833	−0.00364417	−	0.999993i	$-0.501160\pi$
$79$	−5.11763	−0.00728833	−0.00364417	+	0.999993i	$0.501160\pi$
$80$	0	0
$81$	−131.035	−0.179746
$82$	0	0
$83$	73.4805	0.0971751	0.0485876	−	0.998819i	$-0.484528\pi$
$83$	73.4805	0.0971751	0.0485876	+	0.998819i	$0.484528\pi$
$84$	0	0
$85$	1309.44	1.67093
$86$	0	0
$87$	−2318.66	−2.85731
$88$	0	0
$89$	−695.497	−0.828343	−0.414172	−	0.910199i	$-0.635929\pi$
$89$	−695.497	−0.828343	−0.414172	+	0.910199i	$0.635929\pi$
$90$	0	0
$91$	78.3971	0.0903105
$92$	0	0
$93$	−256.444	−0.285935
$94$	0	0
$95$	−1218.17	−1.31559
$96$	0	0
$97$	832.569	0.871490	0.435745	−	0.900070i	$-0.356485\pi$
$97$	832.569	0.871490	0.435745	+	0.900070i	$0.356485\pi$
$98$	0	0
$99$	2611.62	2.65129
$100$	0	0
$101$	1687.58	1.66258	0.831288	−	0.555841i	$-0.187604\pi$
$101$	1687.58	1.66258	0.831288	+	0.555841i	$0.187604\pi$
$102$	0	0
$103$	−326.056	−0.311915	−0.155957	−	0.987764i	$-0.549846\pi$
$103$	−326.056	−0.311915	−0.155957	+	0.987764i	$0.549846\pi$
$104$	0	0
$105$	−1239.90	−1.15240
$106$	0	0
$107$	2031.21	1.83518	0.917590	−	0.397528i	$-0.130132\pi$
$107$	2031.21	1.83518	0.917590	+	0.397528i	$0.130132\pi$
$108$	0	0
$109$	−1544.34	−1.35708	−0.678538	−	0.734566i	$-0.737386\pi$
$109$	−1544.34	−1.35708	−0.678538	+	0.734566i	$0.737386\pi$
$110$	0	0
$111$	566.878	0.484736
$112$	0	0
$113$	−20.8013	−0.0173170	−0.00865852	−	0.999963i	$-0.502756\pi$
$113$	−20.8013	−0.0173170	−0.00865852	+	0.999963i	$0.502756\pi$
$114$	0	0
$115$	−398.324	−0.322991
$116$	0	0
$117$	−308.210	−0.243538
$118$	0	0
$119$	970.323	0.747473
$120$	0	0
$121$	2642.20	1.98512
$122$	0	0
$123$	2996.37	2.19653
$124$	0	0
$125$	678.855	0.485749
$126$	0	0
$127$	1354.85	0.946639	0.473319	−	0.880891i	$-0.343056\pi$
$127$	1354.85	0.946639	0.473319	+	0.880891i	$0.343056\pi$
$128$	0	0
$129$	2097.97	1.43191
$130$	0	0
$131$	1103.27	0.735828	0.367914	−	0.929860i	$-0.380072\pi$
$131$	1103.27	0.735828	0.367914	+	0.929860i	$0.380072\pi$
$132$	0	0
$133$	−902.687	−0.588518
$134$	0	0
$135$	1697.97	1.08250
$136$	0	0
$137$	−1087.72	−0.678324	−0.339162	−	0.940728i	$-0.610144\pi$
$137$	−1087.72	−0.678324	−0.339162	+	0.940728i	$0.610144\pi$
$138$	0	0
$139$	−3140.12	−1.91613	−0.958064	−	0.286556i	$-0.907490\pi$
$139$	−3140.12	−1.91613	−0.958064	+	0.286556i	$0.907490\pi$
$140$	0	0
$141$	−4242.51	−2.53393
$142$	0	0
$143$	−468.897	−0.274204
$144$	0	0
$145$	−3986.29	−2.28306
$146$	0	0
$147$	1918.64	1.07651
$148$	0	0
$149$	51.9051	0.0285385	0.0142692	−	0.999898i	$-0.495458\pi$
$149$	51.9051	0.0285385	0.0142692	+	0.999898i	$0.495458\pi$
$150$	0	0
$151$	−351.978	−0.189692	−0.0948461	−	0.995492i	$-0.530236\pi$
$151$	−351.978	−0.189692	−0.0948461	+	0.995492i	$0.530236\pi$
$152$	0	0
$153$	−3814.72	−2.01570
$154$	0	0
$155$	−440.884	−0.228469
$156$	0	0
$157$	−2620.26	−1.33197	−0.665986	−	0.745964i	$-0.731989\pi$
$157$	−2620.26	−1.33197	−0.665986	+	0.745964i	$0.731989\pi$
$158$	0	0
$159$	−4441.34	−2.21523
$160$	0	0
$161$	−295.166	−0.144487
$162$	0	0
$163$	−1387.18	−0.666578	−0.333289	−	0.942825i	$-0.608158\pi$
$163$	−1387.18	−0.666578	−0.333289	+	0.942825i	$0.608158\pi$
$164$	0	0
$165$	7415.90	3.49895
$166$	0	0
$167$	−817.463	−0.378785	−0.189393	−	0.981901i	$-0.560652\pi$
$167$	−817.463	−0.378785	−0.189393	+	0.981901i	$0.560652\pi$
$168$	0	0
$169$	−2141.66	−0.974813
$170$	0	0
$171$	3548.82	1.58705
$172$	0	0
$173$	2934.75	1.28974	0.644869	−	0.764293i	$-0.276912\pi$
$173$	2934.75	1.28974	0.644869	+	0.764293i	$0.276912\pi$
$174$	0	0
$175$	−814.311	−0.351749
$176$	0	0
$177$	−823.190	−0.349575
$178$	0	0
$179$	−4401.96	−1.83809	−0.919044	−	0.394155i	$-0.871037\pi$
$179$	−4401.96	−1.83809	−0.919044	+	0.394155i	$0.871037\pi$
$180$	0	0
$181$	2517.85	1.03398	0.516991	−	0.855991i	$-0.327052\pi$
$181$	2517.85	1.03398	0.516991	+	0.855991i	$0.327052\pi$
$182$	0	0
$183$	−2253.84	−0.910431
$184$	0	0
$185$	974.590	0.387315
$186$	0	0
$187$	−5803.54	−2.26950
$188$	0	0
$189$	1258.23	0.484248
$190$	0	0
$191$	4826.64	1.82850	0.914249	−	0.405152i	$-0.132781\pi$
$191$	4826.64	1.82850	0.914249	+	0.405152i	$0.132781\pi$
$192$	0	0
$193$	1105.03	0.412135	0.206067	−	0.978538i	$-0.433933\pi$
$193$	1105.03	0.412135	0.206067	+	0.978538i	$0.433933\pi$
$194$	0	0
$195$	−875.187	−0.321402
$196$	0	0
$197$	−1474.45	−0.533250	−0.266625	−	0.963800i	$-0.585908\pi$
$197$	−1474.45	−0.533250	−0.266625	+	0.963800i	$0.585908\pi$
$198$	0	0
$199$	1656.51	0.590084	0.295042	−	0.955484i	$-0.404666\pi$
$199$	1656.51	0.590084	0.295042	+	0.955484i	$0.404666\pi$
$200$	0	0
$201$	−1328.99	−0.466367
$202$	0	0
$203$	−2953.92	−1.02130
$204$	0	0
$205$	5151.43	1.75508
$206$	0	0
$207$	1160.41	0.389635
$208$	0	0
$209$	5399.01	1.78688
$210$	0	0
$211$	1368.77	0.446589	0.223295	−	0.974751i	$-0.428319\pi$
$211$	1368.77	0.446589	0.223295	+	0.974751i	$0.428319\pi$
$212$	0	0
$213$	−6710.79	−2.15876
$214$	0	0
$215$	3606.88	1.14413
$216$	0	0
$217$	−326.704	−0.102203
$218$	0	0
$219$	−7260.13	−2.24015
$220$	0	0
$221$	684.904	0.208469
$222$	0	0
$223$	−5727.26	−1.71985	−0.859923	−	0.510424i	$-0.829489\pi$
$223$	−5727.26	−1.71985	−0.859923	+	0.510424i	$0.829489\pi$
$224$	0	0
$225$	3201.37	0.948555
$226$	0	0
$227$	3853.82	1.12681	0.563407	−	0.826179i	$-0.309490\pi$
$227$	3853.82	1.12681	0.563407	+	0.826179i	$0.309490\pi$
$228$	0	0
$229$	2969.95	0.857030	0.428515	−	0.903535i	$-0.359037\pi$
$229$	2969.95	0.857030	0.428515	+	0.903535i	$0.359037\pi$
$230$	0	0
$231$	5495.33	1.56522
$232$	0	0
$233$	−537.311	−0.151075	−0.0755373	−	0.997143i	$-0.524067\pi$
$233$	−537.311	−0.151075	−0.0755373	+	0.997143i	$0.524067\pi$
$234$	0	0
$235$	−7293.83	−2.02467
$236$	0	0
$237$	42.3350	0.0116032
$238$	0	0
$239$	−1199.62	−0.324674	−0.162337	−	0.986735i	$-0.551903\pi$
$239$	−1199.62	−0.324674	−0.162337	+	0.986735i	$0.551903\pi$
$240$	0	0
$241$	1966.80	0.525695	0.262847	−	0.964837i	$-0.415338\pi$
$241$	1966.80	0.525695	0.262847	+	0.964837i	$0.415338\pi$
$242$	0	0
$243$	4307.49	1.13714
$244$	0	0
$245$	3298.56	0.860153
$246$	0	0
$247$	−637.164	−0.164137
$248$	0	0
$249$	−607.859	−0.154705
$250$	0	0
$251$	−5315.68	−1.33675	−0.668373	−	0.743827i	$-0.733009\pi$
$251$	−5315.68	−1.33675	−0.668373	+	0.743827i	$0.733009\pi$
$252$	0	0
$253$	1765.40	0.438695
$254$	0	0
$255$	−10832.2	−2.66015
$256$	0	0
$257$	5212.75	1.26522	0.632612	−	0.774469i	$-0.281983\pi$
$257$	5212.75	1.26522	0.632612	+	0.774469i	$0.281983\pi$
$258$	0	0
$259$	722.191	0.173262
$260$	0	0
$261$	11613.0	2.75413
$262$	0	0
$263$	−742.464	−0.174077	−0.0870386	−	0.996205i	$-0.527740\pi$
$263$	−742.464	−0.174077	−0.0870386	+	0.996205i	$0.527740\pi$
$264$	0	0
$265$	−7635.66	−1.77002
$266$	0	0
$267$	5753.42	1.31874
$268$	0	0
$269$	−3690.08	−0.836386	−0.418193	−	0.908358i	$-0.637337\pi$
$269$	−3690.08	−0.836386	−0.418193	+	0.908358i	$0.637337\pi$
$270$	0	0
$271$	3502.59	0.785120	0.392560	−	0.919726i	$-0.371590\pi$
$271$	3502.59	0.785120	0.392560	+	0.919726i	$0.371590\pi$
$272$	0	0
$273$	−648.531	−0.143776
$274$	0	0
$275$	4870.43	1.06799
$276$	0	0
$277$	−3553.56	−0.770804	−0.385402	−	0.922749i	$-0.625937\pi$
$277$	−3553.56	−0.770804	−0.385402	+	0.922749i	$0.625937\pi$
$278$	0	0
$279$	1284.40	0.275610
$280$	0	0
$281$	3326.30	0.706158	0.353079	−	0.935594i	$-0.385135\pi$
$281$	3326.30	0.706158	0.353079	+	0.935594i	$0.385135\pi$
$282$	0	0
$283$	8340.98	1.75201	0.876007	−	0.482299i	$-0.160198\pi$
$283$	8340.98	1.75201	0.876007	+	0.482299i	$0.160198\pi$
$284$	0	0
$285$	10077.2	2.09445
$286$	0	0
$287$	3817.32	0.785119
$288$	0	0
$289$	3564.07	0.725436
$290$	0	0
$291$	−6887.33	−1.38743
$292$	0	0
$293$	−2426.13	−0.483741	−0.241871	−	0.970309i	$-0.577761\pi$
$293$	−2426.13	−0.483741	−0.241871	+	0.970309i	$0.577761\pi$
$294$	0	0
$295$	−1415.25	−0.279318
$296$	0	0
$297$	−7525.53	−1.47029
$298$	0	0
$299$	−208.344	−0.0402971
$300$	0	0
$301$	2672.77	0.511814
$302$	0	0
$303$	−13960.3	−2.64686
$304$	0	0
$305$	−3874.86	−0.727456
$306$	0	0
$307$	−5206.71	−0.967957	−0.483978	−	0.875080i	$-0.660809\pi$
$307$	−5206.71	−0.967957	−0.483978	+	0.875080i	$0.660809\pi$
$308$	0	0
$309$	2697.26	0.496575
$310$	0	0
$311$	6273.62	1.14387	0.571936	−	0.820298i	$-0.306192\pi$
$311$	6273.62	1.14387	0.571936	+	0.820298i	$0.306192\pi$
$312$	0	0
$313$	4208.93	0.760074	0.380037	−	0.924971i	$-0.375911\pi$
$313$	4208.93	0.760074	0.380037	+	0.924971i	$0.375911\pi$
$314$	0	0
$315$	6210.06	1.11078
$316$	0	0
$317$	−1045.00	−0.185151	−0.0925757	−	0.995706i	$-0.529510\pi$
$317$	−1045.00	−0.185151	−0.0925757	+	0.995706i	$0.529510\pi$
$318$	0	0
$319$	17667.5	3.10092
$320$	0	0
$321$	−16802.9	−2.92164
$322$	0	0
$323$	−7886.18	−1.35851
$324$	0	0
$325$	−574.783	−0.0981023
$326$	0	0
$327$	12775.4	2.16049
$328$	0	0
$329$	−5404.88	−0.905716
$330$	0	0
$331$	4725.18	0.784651	0.392325	−	0.919827i	$-0.371671\pi$
$331$	4725.18	0.784651	0.392325	+	0.919827i	$0.371671\pi$
$332$	0	0
$333$	−2839.21	−0.467231
$334$	0	0
$335$	−2284.83	−0.372638
$336$	0	0
$337$	−8312.98	−1.34373	−0.671864	−	0.740674i	$-0.734506\pi$
$337$	−8312.98	−1.34373	−0.671864	+	0.740674i	$0.734506\pi$
$338$	0	0
$339$	172.077	0.0275691
$340$	0	0
$341$	1954.03	0.310313
$342$	0	0
$343$	6059.13	0.953826
$344$	0	0
$345$	3295.09	0.514208
$346$	0	0
$347$	8018.96	1.24058	0.620289	−	0.784374i	$-0.287016\pi$
$347$	8018.96	1.24058	0.620289	+	0.784374i	$0.287016\pi$
$348$	0	0
$349$	4548.82	0.697687	0.348843	−	0.937181i	$-0.386574\pi$
$349$	4548.82	0.697687	0.348843	+	0.937181i	$0.386574\pi$
$350$	0	0
$351$	888.125	0.135056
$352$	0	0
$353$	−5790.54	−0.873086	−0.436543	−	0.899683i	$-0.643797\pi$
$353$	−5790.54	−0.873086	−0.436543	+	0.899683i	$0.643797\pi$
$354$	0	0
$355$	−11537.4	−1.72490
$356$	0	0
$357$	−8026.88	−1.18999
$358$	0	0
$359$	−1798.04	−0.264337	−0.132168	−	0.991227i	$-0.542194\pi$
$359$	−1798.04	−0.264337	−0.132168	+	0.991227i	$0.542194\pi$
$360$	0	0
$361$	477.486	0.0696146
$362$	0	0
$363$	−21857.3	−3.16036
$364$	0	0
$365$	−12481.8	−1.78994
$366$	0	0
$367$	−1617.59	−0.230075	−0.115038	−	0.993361i	$-0.536699\pi$
$367$	−1617.59	−0.230075	−0.115038	+	0.993361i	$0.536699\pi$
$368$	0	0
$369$	−15007.4	−2.11721
$370$	0	0
$371$	−5658.18	−0.791801
$372$	0	0
$373$	4293.59	0.596015	0.298008	−	0.954564i	$-0.403678\pi$
$373$	4293.59	0.596015	0.298008	+	0.954564i	$0.403678\pi$
$374$	0	0
$375$	−5615.75	−0.773322
$376$	0	0
$377$	−2085.03	−0.284840
$378$	0	0
$379$	4626.81	0.627079	0.313540	−	0.949575i	$-0.398485\pi$
$379$	4626.81	0.627079	0.313540	+	0.949575i	$0.398485\pi$
$380$	0	0
$381$	−11207.8	−1.50707
$382$	0	0
$383$	5725.57	0.763872	0.381936	−	0.924189i	$-0.375257\pi$
$383$	5725.57	0.763872	0.381936	+	0.924189i	$0.375257\pi$
$384$	0	0
$385$	9447.71	1.25065
$386$	0	0
$387$	−10507.7	−1.38020
$388$	0	0
$389$	12753.9	1.66234	0.831170	−	0.556018i	$-0.187671\pi$
$389$	12753.9	1.66234	0.831170	+	0.556018i	$0.187671\pi$
$390$	0	0
$391$	−2578.67	−0.333527
$392$	0	0
$393$	−9126.70	−1.17145
$394$	0	0
$395$	72.7833	0.00927121
$396$	0	0
$397$	11167.2	1.41176	0.705878	−	0.708333i	$-0.250553\pi$
$397$	11167.2	1.41176	0.705878	+	0.708333i	$0.250553\pi$
$398$	0	0
$399$	7467.38	0.936933
$400$	0	0
$401$	5657.21	0.704508	0.352254	−	0.935904i	$-0.385415\pi$
$401$	5657.21	0.704508	0.352254	+	0.935904i	$0.385415\pi$
$402$	0	0
$403$	−230.605	−0.0285043
$404$	0	0
$405$	1863.59	0.228648
$406$	0	0
$407$	−4319.45	−0.526062
$408$	0	0
$409$	−4695.62	−0.567686	−0.283843	−	0.958871i	$-0.591609\pi$
$409$	−4695.62	−0.567686	−0.283843	+	0.958871i	$0.591609\pi$
$410$	0	0
$411$	8998.06	1.07991
$412$	0	0
$413$	−1048.73	−0.124950
$414$	0	0
$415$	−1045.05	−0.123613
$416$	0	0
$417$	25976.3	3.05051
$418$	0	0
$419$	−2436.26	−0.284055	−0.142027	−	0.989863i	$-0.545362\pi$
$419$	−2436.26	−0.284055	−0.142027	+	0.989863i	$0.545362\pi$
$420$	0	0
$421$	4398.72	0.509217	0.254609	−	0.967044i	$-0.418053\pi$
$421$	4398.72	0.509217	0.254609	+	0.967044i	$0.418053\pi$
$422$	0	0
$423$	21248.7	2.44242
$424$	0	0
$425$	−7114.10	−0.811964
$426$	0	0
$427$	−2871.35	−0.325420
$428$	0	0
$429$	3878.89	0.436538
$430$	0	0
$431$	−10812.7	−1.20842	−0.604212	−	0.796824i	$-0.706512\pi$
$431$	−10812.7	−1.20842	−0.604212	+	0.796824i	$0.706512\pi$
$432$	0	0
$433$	−14788.5	−1.64131	−0.820657	−	0.571421i	$-0.806393\pi$
$433$	−14788.5	−1.64131	−0.820657	+	0.571421i	$0.806393\pi$
$434$	0	0
$435$	32976.1	3.63468
$436$	0	0
$437$	2398.93	0.262601
$438$	0	0
$439$	−5645.65	−0.613786	−0.306893	−	0.951744i	$-0.599289\pi$
$439$	−5645.65	−0.613786	−0.306893	+	0.951744i	$0.599289\pi$
$440$	0	0
$441$	−9609.51	−1.03763
$442$	0	0
$443$	5629.49	0.603759	0.301880	−	0.953346i	$-0.402386\pi$
$443$	5629.49	0.603759	0.301880	+	0.953346i	$0.402386\pi$
$444$	0	0
$445$	9891.42	1.05370
$446$	0	0
$447$	−429.379	−0.0454338
$448$	0	0
$449$	5610.99	0.589753	0.294876	−	0.955535i	$-0.404722\pi$
$449$	5610.99	0.589753	0.294876	+	0.955535i	$0.404722\pi$
$450$	0	0
$451$	−22831.5	−2.38380
$452$	0	0
$453$	2911.69	0.301994
$454$	0	0
$455$	−1114.97	−0.114880
$456$	0	0
$457$	3086.92	0.315974	0.157987	−	0.987441i	$-0.449500\pi$
$457$	3086.92	0.315974	0.157987	+	0.987441i	$0.449500\pi$
$458$	0	0
$459$	10992.3	1.11782
$460$	0	0
$461$	−18357.0	−1.85460	−0.927302	−	0.374313i	$-0.877878\pi$
$461$	−18357.0	−1.85460	−0.927302	+	0.374313i	$0.877878\pi$
$462$	0	0
$463$	14184.2	1.42374	0.711872	−	0.702309i	$-0.247847\pi$
$463$	14184.2	1.42374	0.711872	+	0.702309i	$0.247847\pi$
$464$	0	0
$465$	3647.17	0.363727
$466$	0	0
$467$	16382.3	1.62330	0.811651	−	0.584143i	$-0.198569\pi$
$467$	16382.3	1.62330	0.811651	+	0.584143i	$0.198569\pi$
$468$	0	0
$469$	−1693.11	−0.166696
$470$	0	0
$471$	21675.8	2.12053
$472$	0	0
$473$	−15985.9	−1.55398
$474$	0	0
$475$	6618.22	0.639294
$476$	0	0
$477$	22244.5	2.13523
$478$	0	0
$479$	−3654.57	−0.348605	−0.174302	−	0.984692i	$-0.555767\pi$
$479$	−3654.57	−0.348605	−0.174302	+	0.984692i	$0.555767\pi$
$480$	0	0
$481$	509.760	0.0483223
$482$	0	0
$483$	2441.73	0.230026
$484$	0	0
$485$	−11840.9	−1.10859
$486$	0	0
$487$	475.273	0.0442232	0.0221116	−	0.999756i	$-0.492961\pi$
$487$	475.273	0.0442232	0.0221116	+	0.999756i	$0.492961\pi$
$488$	0	0
$489$	11475.3	1.06121
$490$	0	0
$491$	−1025.15	−0.0942252	−0.0471126	−	0.998890i	$-0.515002\pi$
$491$	−1025.15	−0.0942252	−0.0471126	+	0.998890i	$0.515002\pi$
$492$	0	0
$493$	−25806.5	−2.35754
$494$	0	0
$495$	−37142.6	−3.37260
$496$	0	0
$497$	−8549.42	−0.771617
$498$	0	0
$499$	7436.24	0.667118	0.333559	−	0.942729i	$-0.391750\pi$
$499$	7436.24	0.667118	0.333559	+	0.942729i	$0.391750\pi$
$500$	0	0
$501$	6762.37	0.603034
$502$	0	0
$503$	−6206.12	−0.550133	−0.275067	−	0.961425i	$-0.588700\pi$
$503$	−6206.12	−0.550133	−0.275067	+	0.961425i	$0.588700\pi$
$504$	0	0
$505$	−24000.9	−2.11490
$506$	0	0
$507$	17716.7	1.55192
$508$	0	0
$509$	4379.68	0.381387	0.190693	−	0.981650i	$-0.438926\pi$
$509$	4379.68	0.381387	0.190693	+	0.981650i	$0.438926\pi$
$510$	0	0
$511$	−9249.26	−0.800710
$512$	0	0
$513$	−10226.1	−0.880106
$514$	0	0
$515$	4637.19	0.396775
$516$	0	0
$517$	32326.8	2.74996
$518$	0	0
$519$	−24277.3	−2.05329
$520$	0	0
$521$	5992.52	0.503910	0.251955	−	0.967739i	$-0.418926\pi$
$521$	5992.52	0.503910	0.251955	+	0.967739i	$0.418926\pi$
$522$	0	0
$523$	12222.8	1.02193	0.510963	−	0.859602i	$-0.329289\pi$
$523$	12222.8	1.02193	0.510963	+	0.859602i	$0.329289\pi$
$524$	0	0
$525$	6736.30	0.559992
$526$	0	0
$527$	−2854.20	−0.235922
$528$	0	0
$529$	−11382.6	−0.935529
$530$	0	0
$531$	4122.95	0.336951
$532$	0	0
$533$	2694.46	0.218968
$534$	0	0
$535$	−28888.0	−2.33446
$536$	0	0
$537$	36414.7	2.92627
$538$	0	0
$539$	−14619.5	−1.16828
$540$	0	0
$541$	−16378.5	−1.30160	−0.650802	−	0.759247i	$-0.725567\pi$
$541$	−16378.5	−1.30160	−0.650802	+	0.759247i	$0.725567\pi$
$542$	0	0
$543$	−20828.7	−1.64612
$544$	0	0
$545$	21963.8	1.72628
$546$	0	0
$547$	13273.6	1.03755	0.518773	−	0.854912i	$-0.326389\pi$
$547$	13273.6	1.03755	0.518773	+	0.854912i	$0.326389\pi$
$548$	0	0
$549$	11288.4	0.877554
$550$	0	0
$551$	24007.7	1.85619
$552$	0	0
$553$	53.9340	0.00414739
$554$	0	0
$555$	−8062.18	−0.616614
$556$	0	0
$557$	−9434.70	−0.717704	−0.358852	−	0.933394i	$-0.616832\pi$
$557$	−9434.70	−0.717704	−0.358852	+	0.933394i	$0.616832\pi$
$558$	0	0
$559$	1886.58	0.142744
$560$	0	0
$561$	48009.1	3.61310
$562$	0	0
$563$	2245.85	0.168120	0.0840599	−	0.996461i	$-0.473211\pi$
$563$	2245.85	0.168120	0.0840599	+	0.996461i	$0.473211\pi$
$564$	0	0
$565$	295.838	0.0220283
$566$	0	0
$567$	1380.96	0.102284
$568$	0	0
$569$	−22368.4	−1.64803	−0.824017	−	0.566564i	$-0.808272\pi$
$569$	−22368.4	−1.64803	−0.824017	+	0.566564i	$0.808272\pi$
$570$	0	0
$571$	14761.1	1.08184	0.540922	−	0.841073i	$-0.318075\pi$
$571$	14761.1	1.08184	0.540922	+	0.841073i	$0.318075\pi$
$572$	0	0
$573$	−39927.8	−2.91101
$574$	0	0
$575$	2164.07	0.156953
$576$	0	0
$577$	9917.77	0.715567	0.357784	−	0.933805i	$-0.383533\pi$
$577$	9917.77	0.715567	0.357784	+	0.933805i	$0.383533\pi$
$578$	0	0
$579$	−9141.25	−0.656127
$580$	0	0
$581$	−774.400	−0.0552970
$582$	0	0
$583$	33841.8	2.40409
$584$	0	0
$585$	4383.38	0.309796
$586$	0	0
$587$	−5155.53	−0.362507	−0.181253	−	0.983436i	$-0.558015\pi$
$587$	−5155.53	−0.362507	−0.181253	+	0.983436i	$0.558015\pi$
$588$	0	0
$589$	2655.25	0.185752
$590$	0	0
$591$	12197.2	0.848944
$592$	0	0
$593$	−21201.4	−1.46819	−0.734094	−	0.679048i	$-0.762393\pi$
$593$	−21201.4	−1.46819	−0.734094	+	0.679048i	$0.762393\pi$
$594$	0	0
$595$	−13800.0	−0.950832
$596$	0	0
$597$	−13703.3	−0.939427
$598$	0	0
$599$	−20562.4	−1.40260	−0.701299	−	0.712867i	$-0.747396\pi$
$599$	−20562.4	−1.40260	−0.701299	+	0.712867i	$0.747396\pi$
$600$	0	0
$601$	22685.8	1.53972	0.769860	−	0.638212i	$-0.220326\pi$
$601$	22685.8	1.53972	0.769860	+	0.638212i	$0.220326\pi$
$602$	0	0
$603$	6656.27	0.449526
$604$	0	0
$605$	−37577.6	−2.52520
$606$	0	0
$607$	−6234.98	−0.416920	−0.208460	−	0.978031i	$-0.566845\pi$
$607$	−6234.98	−0.416920	−0.208460	+	0.978031i	$0.566845\pi$
$608$	0	0
$609$	24436.0	1.62594
$610$	0	0
$611$	−3815.04	−0.252602
$612$	0	0
$613$	6766.24	0.445817	0.222908	−	0.974839i	$-0.428445\pi$
$613$	6766.24	0.445817	0.222908	+	0.974839i	$0.428445\pi$
$614$	0	0
$615$	−42614.6	−2.79413
$616$	0	0
$617$	−29774.0	−1.94271	−0.971357	−	0.237625i	$-0.923631\pi$
$617$	−29774.0	−1.94271	−0.971357	+	0.237625i	$0.923631\pi$
$618$	0	0
$619$	9546.06	0.619852	0.309926	−	0.950761i	$-0.399696\pi$
$619$	9546.06	0.619852	0.309926	+	0.950761i	$0.399696\pi$
$620$	0	0
$621$	−3343.80	−0.216074
$622$	0	0
$623$	7329.74	0.471364
$624$	0	0
$625$	−19313.2	−1.23604
$626$	0	0
$627$	−44662.7	−2.84475
$628$	0	0
$629$	6309.31	0.399950
$630$	0	0
$631$	6395.10	0.403463	0.201731	−	0.979441i	$-0.435343\pi$
$631$	6395.10	0.403463	0.201731	+	0.979441i	$0.435343\pi$
$632$	0	0
$633$	−11323.0	−0.710979
$634$	0	0
$635$	−19268.7	−1.20418
$636$	0	0
$637$	1725.32	0.107315
$638$	0	0
$639$	33611.1	2.08080
$640$	0	0
$641$	−23877.2	−1.47128	−0.735641	−	0.677372i	$-0.763119\pi$
$641$	−23877.2	−1.47128	−0.735641	+	0.677372i	$0.763119\pi$
$642$	0	0
$643$	12351.7	0.757549	0.378774	−	0.925489i	$-0.376346\pi$
$643$	12351.7	0.757549	0.378774	+	0.925489i	$0.376346\pi$
$644$	0	0
$645$	−29837.5	−1.82147
$646$	0	0
$647$	4898.62	0.297658	0.148829	−	0.988863i	$-0.452450\pi$
$647$	4898.62	0.297658	0.148829	+	0.988863i	$0.452450\pi$
$648$	0	0
$649$	6272.48	0.379378
$650$	0	0
$651$	2702.62	0.162710
$652$	0	0
$653$	−6449.30	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$653$	−6449.30	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$654$	0	0
$655$	−15690.8	−0.936018
$656$	0	0
$657$	36362.4	2.15926
$658$	0	0
$659$	3875.34	0.229077	0.114539	−	0.993419i	$-0.463461\pi$
$659$	3875.34	0.229077	0.114539	+	0.993419i	$0.463461\pi$
$660$	0	0
$661$	−10903.5	−0.641600	−0.320800	−	0.947147i	$-0.603952\pi$
$661$	−10903.5	−0.641600	−0.320800	+	0.947147i	$0.603952\pi$
$662$	0	0
$663$	−5665.79	−0.331887
$664$	0	0
$665$	12838.1	0.748631
$666$	0	0
$667$	7850.18	0.455712
$668$	0	0
$669$	47378.1	2.73803
$670$	0	0
$671$	17173.7	0.988051
$672$	0	0
$673$	−3591.84	−0.205728	−0.102864	−	0.994695i	$-0.532801\pi$
$673$	−3591.84	−0.205728	−0.102864	+	0.994695i	$0.532801\pi$
$674$	0	0
$675$	−9224.96	−0.526028
$676$	0	0
$677$	−11890.0	−0.674992	−0.337496	−	0.941327i	$-0.609580\pi$
$677$	−11890.0	−0.674992	−0.337496	+	0.941327i	$0.609580\pi$
$678$	0	0
$679$	−8774.32	−0.495917
$680$	0	0
$681$	−31880.3	−1.79391
$682$	0	0
$683$	21627.4	1.21164	0.605820	−	0.795602i	$-0.292845\pi$
$683$	21627.4	1.21164	0.605820	+	0.795602i	$0.292845\pi$
$684$	0	0
$685$	15469.7	0.862870
$686$	0	0
$687$	−24568.6	−1.36441
$688$	0	0
$689$	−3993.84	−0.220832
$690$	0	0
$691$	−15212.2	−0.837481	−0.418741	−	0.908106i	$-0.637528\pi$
$691$	−15212.2	−0.837481	−0.418741	+	0.908106i	$0.637528\pi$
$692$	0	0
$693$	−27523.5	−1.50870
$694$	0	0
$695$	44659.1	2.43743
$696$	0	0
$697$	33349.4	1.81234
$698$	0	0
$699$	4444.84	0.240514
$700$	0	0
$701$	−4925.72	−0.265395	−0.132697	−	0.991157i	$-0.542364\pi$
$701$	−4925.72	−0.265395	−0.132697	+	0.991157i	$0.542364\pi$
$702$	0	0
$703$	−5869.52	−0.314898
$704$	0	0
$705$	60337.3	3.22331
$706$	0	0
$707$	−17785.1	−0.946080
$708$	0	0
$709$	15644.7	0.828700	0.414350	−	0.910118i	$-0.364009\pi$
$709$	15644.7	0.828700	0.414350	+	0.910118i	$0.364009\pi$
$710$	0	0
$711$	−212.035	−0.0111842
$712$	0	0
$713$	868.231	0.0456038
$714$	0	0
$715$	6668.68	0.348804
$716$	0	0
$717$	9923.75	0.516889
$718$	0	0
$719$	28366.4	1.47133	0.735665	−	0.677345i	$-0.236870\pi$
$719$	28366.4	1.47133	0.735665	+	0.677345i	$0.236870\pi$
$720$	0	0
$721$	3436.25	0.177493
$722$	0	0
$723$	−16270.1	−0.836917
$724$	0	0
$725$	21657.2	1.10942
$726$	0	0
$727$	−6836.93	−0.348787	−0.174393	−	0.984676i	$-0.555796\pi$
$727$	−6836.93	−0.348787	−0.174393	+	0.984676i	$0.555796\pi$
$728$	0	0
$729$	−32095.3	−1.63061
$730$	0	0
$731$	23350.2	1.18145
$732$	0	0
$733$	−17411.0	−0.877340	−0.438670	−	0.898648i	$-0.644550\pi$
$733$	−17411.0	−0.877340	−0.438670	+	0.898648i	$0.644550\pi$
$734$	0	0
$735$	−27287.0	−1.36938
$736$	0	0
$737$	10126.6	0.506128
$738$	0	0
$739$	7750.13	0.385782	0.192891	−	0.981220i	$-0.438214\pi$
$739$	7750.13	0.385782	0.192891	+	0.981220i	$0.438214\pi$
$740$	0	0
$741$	5270.86	0.261309
$742$	0	0
$743$	13699.1	0.676408	0.338204	−	0.941073i	$-0.390181\pi$
$743$	13699.1	0.676408	0.338204	+	0.941073i	$0.390181\pi$
$744$	0	0
$745$	−738.199	−0.0363027
$746$	0	0
$747$	3044.47	0.149118
$748$	0	0
$749$	−21406.6	−1.04430
$750$	0	0
$751$	11037.2	0.536289	0.268144	−	0.963379i	$-0.413590\pi$
$751$	11037.2	0.536289	0.268144	+	0.963379i	$0.413590\pi$
$752$	0	0
$753$	43973.4	2.12813
$754$	0	0
$755$	5005.85	0.241300
$756$	0	0
$757$	−21064.0	−1.01134	−0.505670	−	0.862727i	$-0.668755\pi$
$757$	−21064.0	−1.01134	−0.505670	+	0.862727i	$0.668755\pi$
$758$	0	0
$759$	−14604.1	−0.698412
$760$	0	0
$761$	26680.1	1.27090	0.635449	−	0.772143i	$-0.280815\pi$
$761$	26680.1	1.27090	0.635449	+	0.772143i	$0.280815\pi$
$762$	0	0
$763$	16275.6	0.772236
$764$	0	0
$765$	54253.2	2.56409
$766$	0	0
$767$	−740.246	−0.0348484
$768$	0	0
$769$	10489.0	0.491862	0.245931	−	0.969287i	$-0.420906\pi$
$769$	10489.0	0.491862	0.245931	+	0.969287i	$0.420906\pi$
$770$	0	0
$771$	−43121.9	−2.01426
$772$	0	0
$773$	5594.28	0.260301	0.130150	−	0.991494i	$-0.458454\pi$
$773$	5594.28	0.260301	0.130150	+	0.991494i	$0.458454\pi$
$774$	0	0
$775$	2395.29	0.111021
$776$	0	0
$777$	−5974.24	−0.275836
$778$	0	0
$779$	−31024.8	−1.42693
$780$	0	0
$781$	51134.4	2.34281
$782$	0	0
$783$	−33463.6	−1.52732
$784$	0	0
$785$	37265.6	1.69435
$786$	0	0
$787$	−40709.4	−1.84388	−0.921940	−	0.387333i	$-0.873396\pi$
$787$	−40709.4	−1.84388	−0.921940	+	0.387333i	$0.873396\pi$
$788$	0	0
$789$	6141.95	0.277135
$790$	0	0
$791$	219.222	0.00985416
$792$	0	0
$793$	−2026.75	−0.0907591
$794$	0	0
$795$	63165.1	2.81791
$796$	0	0
$797$	−28069.4	−1.24751	−0.623757	−	0.781618i	$-0.714394\pi$
$797$	−28069.4	−1.24751	−0.623757	+	0.781618i	$0.714394\pi$
$798$	0	0
$799$	−47218.8	−2.09072
$800$	0	0
$801$	−28816.1	−1.27112
$802$	0	0
$803$	55320.2	2.43114
$804$	0	0
$805$	4197.88	0.183796
$806$	0	0
$807$	30525.7	1.33154
$808$	0	0
$809$	11470.7	0.498502	0.249251	−	0.968439i	$-0.419816\pi$
$809$	11470.7	0.498502	0.249251	+	0.968439i	$0.419816\pi$
$810$	0	0
$811$	−1445.47	−0.0625859	−0.0312929	−	0.999510i	$-0.509962\pi$
$811$	−1445.47	−0.0625859	−0.0312929	+	0.999510i	$0.509962\pi$
$812$	0	0
$813$	−28974.8	−1.24993
$814$	0	0
$815$	19728.6	0.847928
$816$	0	0
$817$	−21722.6	−0.930206
$818$	0	0
$819$	3248.18	0.138584
$820$	0	0
$821$	8498.84	0.361281	0.180641	−	0.983549i	$-0.442183\pi$
$821$	8498.84	0.361281	0.180641	+	0.983549i	$0.442183\pi$
$822$	0	0
$823$	−42728.1	−1.80973	−0.904866	−	0.425697i	$-0.860029\pi$
$823$	−42728.1	−1.80973	−0.904866	+	0.425697i	$0.860029\pi$
$824$	0	0
$825$	−40290.1	−1.70027
$826$	0	0
$827$	−7877.68	−0.331238	−0.165619	−	0.986190i	$-0.552962\pi$
$827$	−7877.68	−0.331238	−0.165619	+	0.986190i	$0.552962\pi$
$828$	0	0
$829$	−14476.2	−0.606491	−0.303245	−	0.952912i	$-0.598070\pi$
$829$	−14476.2	−0.606491	−0.303245	+	0.952912i	$0.598070\pi$
$830$	0	0
$831$	29396.4	1.22714
$832$	0	0
$833$	21354.3	0.888213
$834$	0	0
$835$	11626.0	0.481838
$836$	0	0
$837$	−3701.08	−0.152841
$838$	0	0
$839$	−30240.9	−1.24438	−0.622188	−	0.782868i	$-0.713756\pi$
$839$	−30240.9	−1.24438	−0.622188	+	0.782868i	$0.713756\pi$
$840$	0	0
$841$	54172.9	2.22120
$842$	0	0
$843$	−27516.4	−1.12422
$844$	0	0
$845$	30458.9	1.24002
$846$	0	0
$847$	−27845.7	−1.12962
$848$	0	0
$849$	−68999.8	−2.78924
$850$	0	0
$851$	−1919.25	−0.0773104
$852$	0	0
$853$	28719.4	1.15279	0.576397	−	0.817170i	$-0.304458\pi$
$853$	28719.4	1.15279	0.576397	+	0.817170i	$0.304458\pi$
$854$	0	0
$855$	−50471.5	−2.01882
$856$	0	0
$857$	−13418.8	−0.534861	−0.267431	−	0.963577i	$-0.586175\pi$
$857$	−13418.8	−0.534861	−0.267431	+	0.963577i	$0.586175\pi$
$858$	0	0
$859$	6822.25	0.270980	0.135490	−	0.990779i	$-0.456739\pi$
$859$	6822.25	0.270980	0.135490	+	0.990779i	$0.456739\pi$
$860$	0	0
$861$	−31578.3	−1.24993
$862$	0	0
$863$	114.333	0.00450980	0.00225490	−	0.999997i	$-0.499282\pi$
$863$	114.333	0.00450980	0.00225490	+	0.999997i	$0.499282\pi$
$864$	0	0
$865$	−41738.2	−1.64063
$866$	0	0
$867$	−29483.3	−1.15491
$868$	0	0
$869$	−322.581	−0.0125924
$870$	0	0
$871$	−1195.08	−0.0464912
$872$	0	0
$873$	34495.3	1.33733
$874$	0	0
$875$	−7154.35	−0.276413
$876$	0	0
$877$	−379.341	−0.0146060	−0.00730299	−	0.999973i	$-0.502325\pi$
$877$	−379.341	−0.0146060	−0.00730299	+	0.999973i	$0.502325\pi$
$878$	0	0
$879$	20069.9	0.770126
$880$	0	0
$881$	−26438.0	−1.01103	−0.505517	−	0.862817i	$-0.668698\pi$
$881$	−26438.0	−1.01103	−0.505517	+	0.862817i	$0.668698\pi$
$882$	0	0
$883$	−46370.1	−1.76724	−0.883622	−	0.468201i	$-0.844902\pi$
$883$	−46370.1	−1.76724	−0.883622	+	0.468201i	$0.844902\pi$
$884$	0	0
$885$	11707.5	0.444680
$886$	0	0
$887$	3409.57	0.129067	0.0645333	−	0.997916i	$-0.479444\pi$
$887$	3409.57	0.129067	0.0645333	+	0.997916i	$0.479444\pi$
$888$	0	0
$889$	−14278.5	−0.538680
$890$	0	0
$891$	−8259.56	−0.310556
$892$	0	0
$893$	43927.5	1.64611
$894$	0	0
$895$	62605.0	2.33816
$896$	0	0
$897$	1723.50	0.0641538
$898$	0	0
$899$	8688.96	0.322350
$900$	0	0
$901$	−49431.8	−1.82776
$902$	0	0
$903$	−22110.2	−0.814818
$904$	0	0
$905$	−35809.1	−1.31529
$906$	0	0
$907$	27432.9	1.00429	0.502146	−	0.864783i	$-0.332544\pi$
$907$	27432.9	1.00429	0.502146	+	0.864783i	$0.332544\pi$
$908$	0	0
$909$	69920.3	2.55128
$910$	0	0
$911$	30005.2	1.09124	0.545618	−	0.838034i	$-0.316295\pi$
$911$	30005.2	1.09124	0.545618	+	0.838034i	$0.316295\pi$
$912$	0	0
$913$	4631.72	0.167894
$914$	0	0
$915$	32054.4	1.15812
$916$	0	0
$917$	−11627.2	−0.418719
$918$	0	0
$919$	1728.19	0.0620323	0.0310161	−	0.999519i	$-0.490126\pi$
$919$	1728.19	0.0620323	0.0310161	+	0.999519i	$0.490126\pi$
$920$	0	0
$921$	43071.9	1.54101
$922$	0	0
$923$	−6034.62	−0.215203
$924$	0	0
$925$	−5294.88	−0.188210
$926$	0	0
$927$	−13509.3	−0.478643
$928$	0	0
$929$	−32600.7	−1.15134	−0.575670	−	0.817682i	$-0.695259\pi$
$929$	−32600.7	−1.15134	−0.575670	+	0.817682i	$0.695259\pi$
$930$	0	0
$931$	−19865.8	−0.699329
$932$	0	0
$933$	−51897.8	−1.82107
$934$	0	0
$935$	82538.4	2.88695
$936$	0	0
$937$	49288.1	1.71843	0.859217	−	0.511611i	$-0.170951\pi$
$937$	49288.1	1.71843	0.859217	+	0.511611i	$0.170951\pi$
$938$	0	0
$939$	−34817.9	−1.21005
$940$	0	0
$941$	39333.1	1.36262	0.681309	−	0.731996i	$-0.261411\pi$
$941$	39333.1	1.36262	0.681309	+	0.731996i	$0.261411\pi$
$942$	0	0
$943$	−10144.7	−0.350325
$944$	0	0
$945$	−17894.7	−0.615993
$946$	0	0
$947$	−54186.8	−1.85938	−0.929691	−	0.368340i	$-0.879926\pi$
$947$	−54186.8	−1.85938	−0.929691	+	0.368340i	$0.879926\pi$
$948$	0	0
$949$	−6528.61	−0.223317
$950$	0	0
$951$	8644.64	0.294765
$952$	0	0
$953$	−15282.7	−0.519470	−0.259735	−	0.965680i	$-0.583635\pi$
$953$	−15282.7	−0.519470	−0.259735	+	0.965680i	$0.583635\pi$
$954$	0	0
$955$	−68644.8	−2.32596
$956$	0	0
$957$	−146153.	−4.93672
$958$	0	0
$959$	11463.4	0.385997
$960$	0	0
$961$	961.000	0.0322581
$962$	0	0
$963$	84157.7	2.81614
$964$	0	0
$965$	−15715.9	−0.524260
$966$	0	0
$967$	37500.2	1.24708	0.623539	−	0.781792i	$-0.285694\pi$
$967$	37500.2	1.24708	0.623539	+	0.781792i	$0.285694\pi$
$968$	0	0
$969$	65237.5	2.16278
$970$	0	0
$971$	−1355.18	−0.0447888	−0.0223944	−	0.999749i	$-0.507129\pi$
$971$	−1355.18	−0.0447888	−0.0223944	+	0.999749i	$0.507129\pi$
$972$	0	0
$973$	33093.3	1.09036
$974$	0	0
$975$	4754.83	0.156181
$976$	0	0
$977$	28943.4	0.947782	0.473891	−	0.880584i	$-0.342849\pi$
$977$	28943.4	0.947782	0.473891	+	0.880584i	$0.342849\pi$
$978$	0	0
$979$	−43839.5	−1.43117
$980$	0	0
$981$	−63985.7	−2.08247
$982$	0	0
$983$	−3605.72	−0.116994	−0.0584968	−	0.998288i	$-0.518631\pi$
$983$	−3605.72	−0.116994	−0.0584968	+	0.998288i	$0.518631\pi$
$984$	0	0
$985$	20969.7	0.678326
$986$	0	0
$987$	44711.2	1.44192
$988$	0	0
$989$	−7103.00	−0.228374
$990$	0	0
$991$	29695.8	0.951887	0.475943	−	0.879476i	$-0.342107\pi$
$991$	29695.8	0.951887	0.475943	+	0.879476i	$0.342107\pi$
$992$	0	0
$993$	−39088.5	−1.24918
$994$	0	0
$995$	−23559.0	−0.750624
$996$	0	0
$997$	11764.0	0.373692	0.186846	−	0.982389i	$-0.440173\pi$
$997$	11764.0	0.373692	0.186846	+	0.982389i	$0.440173\pi$
$998$	0	0
$999$	8181.37	0.259106

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.4.a.b.1.1	✓	4
3.2	odd	2		1116.4.a.f.1.4		4
4.3	odd	2		496.4.a.f.1.4		4
8.3	odd	2		1984.4.a.o.1.1		4
8.5	even	2		1984.4.a.m.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.4.a.b.1.1	✓	4	1.1	even	1	trivial
496.4.a.f.1.4		4	4.3	odd	2
1116.4.a.f.1.4		4	3.2	odd	2
1984.4.a.m.1.4		4	8.5	even	2
1984.4.a.o.1.1		4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	124.a (trivial)

\(f(q)\)	\(=\)	\(q-8.27238 q^{3} -14.2221 q^{5} -10.5389 q^{7} +41.4323 q^{9} +O(q^{10})\)	\(q-8.27238 q^{3} -14.2221 q^{5} -10.5389 q^{7} +41.4323 q^{9} +63.0333 q^{11} -7.43887 q^{13} +117.650 q^{15} -92.0710 q^{17} +85.6533 q^{19} +87.1814 q^{21} +28.0075 q^{23} +77.2676 q^{25} -119.390 q^{27} +280.289 q^{29} +31.0000 q^{31} -521.436 q^{33} +149.884 q^{35} -68.5265 q^{37} +61.5372 q^{39} -362.214 q^{41} -253.611 q^{43} -589.254 q^{45} +512.852 q^{47} -231.933 q^{49} +761.647 q^{51} +536.888 q^{53} -896.465 q^{55} -708.557 q^{57} +99.5106 q^{59} +272.454 q^{61} -436.649 q^{63} +105.796 q^{65} +160.654 q^{67} -231.688 q^{69} +811.229 q^{71} +877.634 q^{73} -639.187 q^{75} -664.299 q^{77} -5.11763 q^{79} -131.035 q^{81} +73.4805 q^{83} +1309.44 q^{85} -2318.66 q^{87} -695.497 q^{89} +78.3971 q^{91} -256.444 q^{93} -1218.17 q^{95} +832.569 q^{97} +2611.62 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9	\( 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9} + 96 q^{11} + 42 q^{13} + 182 q^{15} - 2 q^{17} + 180 q^{19} + 150 q^{21} + 142 q^{23} + 150 q^{25} + 20 q^{27} + 262 q^{29} + 124 q^{31} - 444 q^{33} + 224 q^{35} - 284 q^{37} + 60 q^{39} - 526 q^{41} - 326 q^{43} - 870 q^{45} + 468 q^{47} - 978 q^{49} + 356 q^{51} - 252 q^{53} - 876 q^{55} - 1298 q^{57} - 164 q^{59} - 1066 q^{61} - 236 q^{63} - 598 q^{65} - 956 q^{69} + 1504 q^{71} - 732 q^{73} + 188 q^{75} - 288 q^{77} + 822 q^{79} - 536 q^{81} + 1408 q^{83} + 482 q^{85} - 208 q^{87} + 250 q^{89} + 946 q^{91} + 62 q^{93} + 2292 q^{95} + 526 q^{97} + 2952 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^5 + 16 * q^7 + 64 * q^9 + 96 * q^11 + 42 * q^13 + 182 * q^15 - 2 * q^17 + 180 * q^19 + 150 * q^21 + 142 * q^23 + 150 * q^25 + 20 * q^27 + 262 * q^29 + 124 * q^31 - 444 * q^33 + 224 * q^35 - 284 * q^37 + 60 * q^39 - 526 * q^41 - 326 * q^43 - 870 * q^45 + 468 * q^47 - 978 * q^49 + 356 * q^51 - 252 * q^53 - 876 * q^55 - 1298 * q^57 - 164 * q^59 - 1066 * q^61 - 236 * q^63 - 598 * q^65 - 956 * q^69 + 1504 * q^71 - 732 * q^73 + 188 * q^75 - 288 * q^77 + 822 * q^79 - 536 * q^81 + 1408 * q^83 + 482 * q^85 - 208 * q^87 + 250 * q^89 + 946 * q^91 + 62 * q^93 + 2292 * q^95 + 526 * q^97 + 2952 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more