LMFDB - Embedded newform 6008.2.a.b.1.41

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.41
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.62887 q^{3} -0.385949 q^{5} +1.06025 q^{7} +3.91098 q^{9} +O(q^{10})$	$q+2.62887 q^{3} -0.385949 q^{5} +1.06025 q^{7} +3.91098 q^{9} -3.06623 q^{11} -4.43652 q^{13} -1.01461 q^{15} -2.91886 q^{17} -0.422058 q^{19} +2.78728 q^{21} -0.921411 q^{23} -4.85104 q^{25} +2.39486 q^{27} -8.01261 q^{29} +6.77228 q^{31} -8.06074 q^{33} -0.409204 q^{35} +4.94195 q^{37} -11.6631 q^{39} +7.67152 q^{41} +1.55444 q^{43} -1.50944 q^{45} -4.43923 q^{47} -5.87586 q^{49} -7.67332 q^{51} -9.11898 q^{53} +1.18341 q^{55} -1.10954 q^{57} +1.46728 q^{59} +2.75627 q^{61} +4.14664 q^{63} +1.71227 q^{65} -11.8281 q^{67} -2.42228 q^{69} +8.47433 q^{71} -14.0133 q^{73} -12.7528 q^{75} -3.25099 q^{77} -10.1827 q^{79} -5.43717 q^{81} +13.7930 q^{83} +1.12653 q^{85} -21.0642 q^{87} +2.11756 q^{89} -4.70384 q^{91} +17.8035 q^{93} +0.162893 q^{95} +1.01462 q^{97} -11.9920 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.62887	1.51778	0.758891	−	0.651218i	$-0.225742\pi$
$3$	2.62887	1.51778	0.758891	+	0.651218i	$0.225742\pi$
$4$	0	0
$5$	−0.385949	−0.172602	−0.0863008	−	0.996269i	$-0.527505\pi$
$5$	−0.385949	−0.172602	−0.0863008	+	0.996269i	$0.527505\pi$
$6$	0	0
$7$	1.06025	0.400738	0.200369	−	0.979720i	$-0.435786\pi$
$7$	1.06025	0.400738	0.200369	+	0.979720i	$0.435786\pi$
$8$	0	0
$9$	3.91098	1.30366
$10$	0	0
$11$	−3.06623	−0.924504	−0.462252	−	0.886749i	$-0.652959\pi$
$11$	−3.06623	−0.924504	−0.462252	+	0.886749i	$0.652959\pi$
$12$	0	0
$13$	−4.43652	−1.23047	−0.615235	−	0.788344i	$-0.710939\pi$
$13$	−4.43652	−1.23047	−0.615235	+	0.788344i	$0.710939\pi$
$14$	0	0
$15$	−1.01461	−0.261971
$16$	0	0
$17$	−2.91886	−0.707928	−0.353964	−	0.935259i	$-0.615166\pi$
$17$	−2.91886	−0.707928	−0.353964	+	0.935259i	$0.615166\pi$
$18$	0	0
$19$	−0.422058	−0.0968267	−0.0484134	−	0.998827i	$-0.515416\pi$
$19$	−0.422058	−0.0968267	−0.0484134	+	0.998827i	$0.515416\pi$
$20$	0	0
$21$	2.78728	0.608233
$22$	0	0
$23$	−0.921411	−0.192128	−0.0960638	−	0.995375i	$-0.530625\pi$
$23$	−0.921411	−0.192128	−0.0960638	+	0.995375i	$0.530625\pi$
$24$	0	0
$25$	−4.85104	−0.970209
$26$	0	0
$27$	2.39486	0.460890
$28$	0	0
$29$	−8.01261	−1.48791	−0.743953	−	0.668233i	$-0.767051\pi$
$29$	−8.01261	−1.48791	−0.743953	+	0.668233i	$0.767051\pi$
$30$	0	0
$31$	6.77228	1.21634	0.608169	−	0.793808i	$-0.291904\pi$
$31$	6.77228	1.21634	0.608169	+	0.793808i	$0.291904\pi$
$32$	0	0
$33$	−8.06074	−1.40320
$34$	0	0
$35$	−0.409204	−0.0691681
$36$	0	0
$37$	4.94195	0.812452	0.406226	−	0.913773i	$-0.366845\pi$
$37$	4.94195	0.812452	0.406226	+	0.913773i	$0.366845\pi$
$38$	0	0
$39$	−11.6631	−1.86758
$40$	0	0
$41$	7.67152	1.19809	0.599045	−	0.800715i	$-0.295547\pi$
$41$	7.67152	1.19809	0.599045	+	0.800715i	$0.295547\pi$
$42$	0	0
$43$	1.55444	0.237049	0.118525	−	0.992951i	$-0.462184\pi$
$43$	1.55444	0.237049	0.118525	+	0.992951i	$0.462184\pi$
$44$	0	0
$45$	−1.50944	−0.225014
$46$	0	0
$47$	−4.43923	−0.647528	−0.323764	−	0.946138i	$-0.604948\pi$
$47$	−4.43923	−0.647528	−0.323764	+	0.946138i	$0.604948\pi$
$48$	0	0
$49$	−5.87586	−0.839409
$50$	0	0
$51$	−7.67332	−1.07448
$52$	0	0
$53$	−9.11898	−1.25259	−0.626294	−	0.779587i	$-0.715429\pi$
$53$	−9.11898	−1.25259	−0.626294	+	0.779587i	$0.715429\pi$
$54$	0	0
$55$	1.18341	0.159571
$56$	0	0
$57$	−1.10954	−0.146962
$58$	0	0
$59$	1.46728	0.191024	0.0955121	−	0.995428i	$-0.469551\pi$
$59$	1.46728	0.191024	0.0955121	+	0.995428i	$0.469551\pi$
$60$	0	0
$61$	2.75627	0.352904	0.176452	−	0.984309i	$-0.443538\pi$
$61$	2.75627	0.352904	0.176452	+	0.984309i	$0.443538\pi$
$62$	0	0
$63$	4.14664	0.522427
$64$	0	0
$65$	1.71227	0.212381
$66$	0	0
$67$	−11.8281	−1.44503	−0.722516	−	0.691354i	$-0.757014\pi$
$67$	−11.8281	−1.44503	−0.722516	+	0.691354i	$0.757014\pi$
$68$	0	0
$69$	−2.42228	−0.291608
$70$	0	0
$71$	8.47433	1.00572	0.502859	−	0.864368i	$-0.332281\pi$
$71$	8.47433	1.00572	0.502859	+	0.864368i	$0.332281\pi$
$72$	0	0
$73$	−14.0133	−1.64014	−0.820068	−	0.572266i	$-0.806064\pi$
$73$	−14.0133	−1.64014	−0.820068	+	0.572266i	$0.806064\pi$
$74$	0	0
$75$	−12.7528	−1.47256
$76$	0	0
$77$	−3.25099	−0.370484
$78$	0	0
$79$	−10.1827	−1.14564	−0.572822	−	0.819679i	$-0.694152\pi$
$79$	−10.1827	−1.14564	−0.572822	+	0.819679i	$0.694152\pi$
$80$	0	0
$81$	−5.43717	−0.604130
$82$	0	0
$83$	13.7930	1.51398	0.756989	−	0.653427i	$-0.226669\pi$
$83$	13.7930	1.51398	0.756989	+	0.653427i	$0.226669\pi$
$84$	0	0
$85$	1.12653	0.122189
$86$	0	0
$87$	−21.0642	−2.25831
$88$	0	0
$89$	2.11756	0.224461	0.112230	−	0.993682i	$-0.464201\pi$
$89$	2.11756	0.224461	0.112230	+	0.993682i	$0.464201\pi$
$90$	0	0
$91$	−4.70384	−0.493097
$92$	0	0
$93$	17.8035	1.84613
$94$	0	0
$95$	0.162893	0.0167124
$96$	0	0
$97$	1.01462	0.103019	0.0515095	−	0.998673i	$-0.483597\pi$
$97$	1.01462	0.103019	0.0515095	+	0.998673i	$0.483597\pi$
$98$	0	0
$99$	−11.9920	−1.20524
$100$	0	0
$101$	13.3078	1.32417	0.662086	−	0.749428i	$-0.269671\pi$
$101$	13.3078	1.32417	0.662086	+	0.749428i	$0.269671\pi$
$102$	0	0
$103$	−14.9579	−1.47384	−0.736922	−	0.675977i	$-0.763722\pi$
$103$	−14.9579	−1.47384	−0.736922	+	0.675977i	$0.763722\pi$
$104$	0	0
$105$	−1.07575	−0.104982
$106$	0	0
$107$	−6.42206	−0.620844	−0.310422	−	0.950599i	$-0.600470\pi$
$107$	−6.42206	−0.620844	−0.310422	+	0.950599i	$0.600470\pi$
$108$	0	0
$109$	−14.2439	−1.36432	−0.682158	−	0.731204i	$-0.738959\pi$
$109$	−14.2439	−1.36432	−0.682158	+	0.731204i	$0.738959\pi$
$110$	0	0
$111$	12.9918	1.23312
$112$	0	0
$113$	−19.6439	−1.84794	−0.923970	−	0.382464i	$-0.875075\pi$
$113$	−19.6439	−1.84794	−0.923970	+	0.382464i	$0.875075\pi$
$114$	0	0
$115$	0.355618	0.0331615
$116$	0	0
$117$	−17.3512	−1.60412
$118$	0	0
$119$	−3.09473	−0.283694
$120$	0	0
$121$	−1.59821	−0.145292
$122$	0	0
$123$	20.1675	1.81844
$124$	0	0
$125$	3.80200	0.340061
$126$	0	0
$127$	12.9335	1.14766	0.573832	−	0.818973i	$-0.305456\pi$
$127$	12.9335	1.14766	0.573832	+	0.818973i	$0.305456\pi$
$128$	0	0
$129$	4.08642	0.359789
$130$	0	0
$131$	−3.47119	−0.303280	−0.151640	−	0.988436i	$-0.548455\pi$
$131$	−3.47119	−0.303280	−0.151640	+	0.988436i	$0.548455\pi$
$132$	0	0
$133$	−0.447489	−0.0388022
$134$	0	0
$135$	−0.924292	−0.0795504
$136$	0	0
$137$	0.485435	0.0414735	0.0207367	−	0.999785i	$-0.493399\pi$
$137$	0.485435	0.0414735	0.0207367	+	0.999785i	$0.493399\pi$
$138$	0	0
$139$	−3.43389	−0.291259	−0.145629	−	0.989339i	$-0.546521\pi$
$139$	−3.43389	−0.291259	−0.145629	+	0.989339i	$0.546521\pi$
$140$	0	0
$141$	−11.6702	−0.982806
$142$	0	0
$143$	13.6034	1.13757
$144$	0	0
$145$	3.09246	0.256815
$146$	0	0
$147$	−15.4469	−1.27404
$148$	0	0
$149$	12.6436	1.03581	0.517903	−	0.855440i	$-0.326713\pi$
$149$	12.6436	1.03581	0.517903	+	0.855440i	$0.326713\pi$
$150$	0	0
$151$	−6.80235	−0.553568	−0.276784	−	0.960932i	$-0.589269\pi$
$151$	−6.80235	−0.553568	−0.276784	+	0.960932i	$0.589269\pi$
$152$	0	0
$153$	−11.4156	−0.922897
$154$	0	0
$155$	−2.61375	−0.209942
$156$	0	0
$157$	−17.1570	−1.36928	−0.684641	−	0.728881i	$-0.740041\pi$
$157$	−17.1570	−1.36928	−0.684641	+	0.728881i	$0.740041\pi$
$158$	0	0
$159$	−23.9727	−1.90116
$160$	0	0
$161$	−0.976931	−0.0769929
$162$	0	0
$163$	−5.85431	−0.458545	−0.229272	−	0.973362i	$-0.573635\pi$
$163$	−5.85431	−0.458545	−0.229272	+	0.973362i	$0.573635\pi$
$164$	0	0
$165$	3.11103	0.242194
$166$	0	0
$167$	6.83009	0.528528	0.264264	−	0.964450i	$-0.414871\pi$
$167$	6.83009	0.528528	0.264264	+	0.964450i	$0.414871\pi$
$168$	0	0
$169$	6.68273	0.514056
$170$	0	0
$171$	−1.65066	−0.126229
$172$	0	0
$173$	7.75144	0.589331	0.294665	−	0.955600i	$-0.404792\pi$
$173$	7.75144	0.589331	0.294665	+	0.955600i	$0.404792\pi$
$174$	0	0
$175$	−5.14334	−0.388800
$176$	0	0
$177$	3.85731	0.289933
$178$	0	0
$179$	12.8904	0.963471	0.481735	−	0.876317i	$-0.340007\pi$
$179$	12.8904	0.963471	0.481735	+	0.876317i	$0.340007\pi$
$180$	0	0
$181$	19.7202	1.46579	0.732897	−	0.680340i	$-0.238168\pi$
$181$	19.7202	1.46579	0.732897	+	0.680340i	$0.238168\pi$
$182$	0	0
$183$	7.24589	0.535632
$184$	0	0
$185$	−1.90734	−0.140230
$186$	0	0
$187$	8.94991	0.654482
$188$	0	0
$189$	2.53916	0.184696
$190$	0	0
$191$	18.8115	1.36115	0.680576	−	0.732678i	$-0.261730\pi$
$191$	18.8115	1.36115	0.680576	+	0.732678i	$0.261730\pi$
$192$	0	0
$193$	−10.8869	−0.783659	−0.391829	−	0.920038i	$-0.628158\pi$
$193$	−10.8869	−0.783659	−0.391829	+	0.920038i	$0.628158\pi$
$194$	0	0
$195$	4.50134	0.322348
$196$	0	0
$197$	−10.7351	−0.764846	−0.382423	−	0.923987i	$-0.624910\pi$
$197$	−10.7351	−0.764846	−0.382423	+	0.923987i	$0.624910\pi$
$198$	0	0
$199$	2.49440	0.176823	0.0884115	−	0.996084i	$-0.471821\pi$
$199$	2.49440	0.176823	0.0884115	+	0.996084i	$0.471821\pi$
$200$	0	0
$201$	−31.0946	−2.19324
$202$	0	0
$203$	−8.49541	−0.596261
$204$	0	0
$205$	−2.96081	−0.206792
$206$	0	0
$207$	−3.60362	−0.250469
$208$	0	0
$209$	1.29413	0.0895167
$210$	0	0
$211$	14.3338	0.986776	0.493388	−	0.869809i	$-0.335758\pi$
$211$	14.3338	0.986776	0.493388	+	0.869809i	$0.335758\pi$
$212$	0	0
$213$	22.2780	1.52646
$214$	0	0
$215$	−0.599932	−0.0409150
$216$	0	0
$217$	7.18034	0.487433
$218$	0	0
$219$	−36.8393	−2.48937
$220$	0	0
$221$	12.9496	0.871084
$222$	0	0
$223$	11.4907	0.769471	0.384736	−	0.923027i	$-0.374293\pi$
$223$	11.4907	0.769471	0.384736	+	0.923027i	$0.374293\pi$
$224$	0	0
$225$	−18.9723	−1.26482
$226$	0	0
$227$	28.6613	1.90232	0.951159	−	0.308703i	$-0.0998949\pi$
$227$	28.6613	1.90232	0.951159	+	0.308703i	$0.0998949\pi$
$228$	0	0
$229$	25.7970	1.70471	0.852356	−	0.522962i	$-0.175173\pi$
$229$	25.7970	1.70471	0.852356	+	0.522962i	$0.175173\pi$
$230$	0	0
$231$	−8.54644	−0.562314
$232$	0	0
$233$	−12.5583	−0.822719	−0.411360	−	0.911473i	$-0.634946\pi$
$233$	−12.5583	−0.822719	−0.411360	+	0.911473i	$0.634946\pi$
$234$	0	0
$235$	1.71331	0.111764
$236$	0	0
$237$	−26.7691	−1.73884
$238$	0	0
$239$	−28.1629	−1.82171	−0.910854	−	0.412730i	$-0.864575\pi$
$239$	−28.1629	−1.82171	−0.910854	+	0.412730i	$0.864575\pi$
$240$	0	0
$241$	−10.9306	−0.704100	−0.352050	−	0.935981i	$-0.614515\pi$
$241$	−10.9306	−0.704100	−0.352050	+	0.935981i	$0.614515\pi$
$242$	0	0
$243$	−21.4782	−1.37783
$244$	0	0
$245$	2.26778	0.144883
$246$	0	0
$247$	1.87247	0.119142
$248$	0	0
$249$	36.2601	2.29789
$250$	0	0
$251$	21.5941	1.36301	0.681505	−	0.731814i	$-0.261326\pi$
$251$	21.5941	1.36301	0.681505	+	0.731814i	$0.261326\pi$
$252$	0	0
$253$	2.82526	0.177623
$254$	0	0
$255$	2.96151	0.185457
$256$	0	0
$257$	−13.0167	−0.811962	−0.405981	−	0.913882i	$-0.633070\pi$
$257$	−13.0167	−0.811962	−0.405981	+	0.913882i	$0.633070\pi$
$258$	0	0
$259$	5.23973	0.325581
$260$	0	0
$261$	−31.3372	−1.93972
$262$	0	0
$263$	19.6538	1.21190	0.605952	−	0.795501i	$-0.292792\pi$
$263$	19.6538	1.21190	0.605952	+	0.795501i	$0.292792\pi$
$264$	0	0
$265$	3.51946	0.216199
$266$	0	0
$267$	5.56679	0.340682
$268$	0	0
$269$	15.1920	0.926272	0.463136	−	0.886287i	$-0.346724\pi$
$269$	15.1920	0.926272	0.463136	+	0.886287i	$0.346724\pi$
$270$	0	0
$271$	2.16377	0.131440	0.0657199	−	0.997838i	$-0.479066\pi$
$271$	2.16377	0.131440	0.0657199	+	0.997838i	$0.479066\pi$
$272$	0	0
$273$	−12.3658	−0.748413
$274$	0	0
$275$	14.8744	0.896962
$276$	0	0
$277$	−10.4628	−0.628649	−0.314325	−	0.949316i	$-0.601778\pi$
$277$	−10.4628	−0.628649	−0.314325	+	0.949316i	$0.601778\pi$
$278$	0	0
$279$	26.4863	1.58569
$280$	0	0
$281$	−11.7212	−0.699227	−0.349613	−	0.936894i	$-0.613687\pi$
$281$	−11.7212	−0.699227	−0.349613	+	0.936894i	$0.613687\pi$
$282$	0	0
$283$	−14.8020	−0.879887	−0.439944	−	0.898025i	$-0.645002\pi$
$283$	−14.8020	−0.879887	−0.439944	+	0.898025i	$0.645002\pi$
$284$	0	0
$285$	0.428225	0.0253658
$286$	0	0
$287$	8.13376	0.480121
$288$	0	0
$289$	−8.48025	−0.498838
$290$	0	0
$291$	2.66731	0.156360
$292$	0	0
$293$	6.91494	0.403975	0.201987	−	0.979388i	$-0.435260\pi$
$293$	6.91494	0.403975	0.201987	+	0.979388i	$0.435260\pi$
$294$	0	0
$295$	−0.566297	−0.0329711
$296$	0	0
$297$	−7.34319	−0.426095
$298$	0	0
$299$	4.08786	0.236407
$300$	0	0
$301$	1.64810	0.0949947
$302$	0	0
$303$	34.9844	2.00980
$304$	0	0
$305$	−1.06378	−0.0609118
$306$	0	0
$307$	7.00843	0.399992	0.199996	−	0.979797i	$-0.435907\pi$
$307$	7.00843	0.399992	0.199996	+	0.979797i	$0.435907\pi$
$308$	0	0
$309$	−39.3224	−2.23697
$310$	0	0
$311$	−18.9854	−1.07656	−0.538282	−	0.842764i	$-0.680927\pi$
$311$	−18.9854	−1.07656	−0.538282	+	0.842764i	$0.680927\pi$
$312$	0	0
$313$	−4.91551	−0.277841	−0.138921	−	0.990304i	$-0.544363\pi$
$313$	−4.91551	−0.277841	−0.138921	+	0.990304i	$0.544363\pi$
$314$	0	0
$315$	−1.60039	−0.0901717
$316$	0	0
$317$	−27.6581	−1.55343	−0.776716	−	0.629851i	$-0.783116\pi$
$317$	−27.6581	−1.55343	−0.776716	+	0.629851i	$0.783116\pi$
$318$	0	0
$319$	24.5685	1.37557
$320$	0	0
$321$	−16.8828	−0.942305
$322$	0	0
$323$	1.23193	0.0685463
$324$	0	0
$325$	21.5218	1.19381
$326$	0	0
$327$	−37.4454	−2.07073
$328$	0	0
$329$	−4.70671	−0.259489
$330$	0	0
$331$	21.0884	1.15912	0.579562	−	0.814928i	$-0.303224\pi$
$331$	21.0884	1.15912	0.579562	+	0.814928i	$0.303224\pi$
$332$	0	0
$333$	19.3279	1.05916
$334$	0	0
$335$	4.56504	0.249415
$336$	0	0
$337$	27.7334	1.51073	0.755366	−	0.655303i	$-0.227459\pi$
$337$	27.7334	1.51073	0.755366	+	0.655303i	$0.227459\pi$
$338$	0	0
$339$	−51.6413	−2.80477
$340$	0	0
$341$	−20.7654	−1.12451
$342$	0	0
$343$	−13.6517	−0.737122
$344$	0	0
$345$	0.934874	0.0503319
$346$	0	0
$347$	−17.2013	−0.923414	−0.461707	−	0.887032i	$-0.652763\pi$
$347$	−17.2013	−0.923414	−0.461707	+	0.887032i	$0.652763\pi$
$348$	0	0
$349$	16.0824	0.860871	0.430435	−	0.902621i	$-0.358360\pi$
$349$	16.0824	0.860871	0.430435	+	0.902621i	$0.358360\pi$
$350$	0	0
$351$	−10.6248	−0.567112
$352$	0	0
$353$	26.7167	1.42199	0.710993	−	0.703199i	$-0.248246\pi$
$353$	26.7167	1.42199	0.710993	+	0.703199i	$0.248246\pi$
$354$	0	0
$355$	−3.27066	−0.173588
$356$	0	0
$357$	−8.13567	−0.430585
$358$	0	0
$359$	−24.5985	−1.29826	−0.649130	−	0.760678i	$-0.724867\pi$
$359$	−24.5985	−1.29826	−0.649130	+	0.760678i	$0.724867\pi$
$360$	0	0
$361$	−18.8219	−0.990625
$362$	0	0
$363$	−4.20149	−0.220521
$364$	0	0
$365$	5.40843	0.283090
$366$	0	0
$367$	−27.6065	−1.44105	−0.720524	−	0.693430i	$-0.756099\pi$
$367$	−27.6065	−1.44105	−0.720524	+	0.693430i	$0.756099\pi$
$368$	0	0
$369$	30.0032	1.56190
$370$	0	0
$371$	−9.66844	−0.501960
$372$	0	0
$373$	30.8027	1.59490	0.797452	−	0.603382i	$-0.206181\pi$
$373$	30.8027	1.59490	0.797452	+	0.603382i	$0.206181\pi$
$374$	0	0
$375$	9.99497	0.516138
$376$	0	0
$377$	35.5481	1.83082
$378$	0	0
$379$	−12.2838	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$379$	−12.2838	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$380$	0	0
$381$	34.0006	1.74190
$382$	0	0
$383$	0.743755	0.0380041	0.0190020	−	0.999819i	$-0.493951\pi$
$383$	0.743755	0.0380041	0.0190020	+	0.999819i	$0.493951\pi$
$384$	0	0
$385$	1.25471	0.0639462
$386$	0	0
$387$	6.07937	0.309032
$388$	0	0
$389$	−8.43706	−0.427776	−0.213888	−	0.976858i	$-0.568613\pi$
$389$	−8.43706	−0.427776	−0.213888	+	0.976858i	$0.568613\pi$
$390$	0	0
$391$	2.68947	0.136012
$392$	0	0
$393$	−9.12533	−0.460312
$394$	0	0
$395$	3.93001	0.197740
$396$	0	0
$397$	25.4329	1.27644	0.638219	−	0.769855i	$-0.279671\pi$
$397$	25.4329	1.27644	0.638219	+	0.769855i	$0.279671\pi$
$398$	0	0
$399$	−1.17639	−0.0588933
$400$	0	0
$401$	−22.4850	−1.12285	−0.561425	−	0.827528i	$-0.689747\pi$
$401$	−22.4850	−1.12285	−0.561425	+	0.827528i	$0.689747\pi$
$402$	0	0
$403$	−30.0454	−1.49667
$404$	0	0
$405$	2.09847	0.104274
$406$	0	0
$407$	−15.1532	−0.751115
$408$	0	0
$409$	20.5185	1.01457	0.507287	−	0.861777i	$-0.330648\pi$
$409$	20.5185	1.01457	0.507287	+	0.861777i	$0.330648\pi$
$410$	0	0
$411$	1.27615	0.0629477
$412$	0	0
$413$	1.55570	0.0765508
$414$	0	0
$415$	−5.32339	−0.261315
$416$	0	0
$417$	−9.02727	−0.442067
$418$	0	0
$419$	30.2145	1.47607	0.738037	−	0.674761i	$-0.235753\pi$
$419$	30.2145	1.47607	0.738037	+	0.674761i	$0.235753\pi$
$420$	0	0
$421$	31.1839	1.51981	0.759905	−	0.650034i	$-0.225245\pi$
$421$	31.1839	1.51981	0.759905	+	0.650034i	$0.225245\pi$
$422$	0	0
$423$	−17.3617	−0.844156
$424$	0	0
$425$	14.1595	0.686838
$426$	0	0
$427$	2.92235	0.141422
$428$	0	0
$429$	35.7617	1.72659
$430$	0	0
$431$	3.03702	0.146288	0.0731441	−	0.997321i	$-0.476697\pi$
$431$	3.03702	0.146288	0.0731441	+	0.997321i	$0.476697\pi$
$432$	0	0
$433$	−31.7544	−1.52602	−0.763010	−	0.646387i	$-0.776279\pi$
$433$	−31.7544	−1.52602	−0.763010	+	0.646387i	$0.776279\pi$
$434$	0	0
$435$	8.12968	0.389789
$436$	0	0
$437$	0.388889	0.0186031
$438$	0	0
$439$	−20.0020	−0.954643	−0.477321	−	0.878729i	$-0.658392\pi$
$439$	−20.0020	−0.954643	−0.477321	+	0.878729i	$0.658392\pi$
$440$	0	0
$441$	−22.9804	−1.09430
$442$	0	0
$443$	−34.8403	−1.65531	−0.827657	−	0.561234i	$-0.810327\pi$
$443$	−34.8403	−1.65531	−0.827657	+	0.561234i	$0.810327\pi$
$444$	0	0
$445$	−0.817268	−0.0387422
$446$	0	0
$447$	33.2385	1.57213
$448$	0	0
$449$	0.947307	0.0447062	0.0223531	−	0.999750i	$-0.492884\pi$
$449$	0.947307	0.0447062	0.0223531	+	0.999750i	$0.492884\pi$
$450$	0	0
$451$	−23.5227	−1.10764
$452$	0	0
$453$	−17.8825	−0.840195
$454$	0	0
$455$	1.81544	0.0851092
$456$	0	0
$457$	−37.6267	−1.76010	−0.880052	−	0.474878i	$-0.842492\pi$
$457$	−37.6267	−1.76010	−0.880052	+	0.474878i	$0.842492\pi$
$458$	0	0
$459$	−6.99025	−0.326277
$460$	0	0
$461$	0.853835	0.0397671	0.0198835	−	0.999802i	$-0.493670\pi$
$461$	0.853835	0.0397671	0.0198835	+	0.999802i	$0.493670\pi$
$462$	0	0
$463$	11.3267	0.526397	0.263199	−	0.964742i	$-0.415223\pi$
$463$	11.3267	0.526397	0.263199	+	0.964742i	$0.415223\pi$
$464$	0	0
$465$	−6.87123	−0.318646
$466$	0	0
$467$	−41.3143	−1.91180	−0.955899	−	0.293696i	$-0.905115\pi$
$467$	−41.3143	−1.91180	−0.955899	+	0.293696i	$0.905115\pi$
$468$	0	0
$469$	−12.5408	−0.579080
$470$	0	0
$471$	−45.1037	−2.07827
$472$	0	0
$473$	−4.76626	−0.219153
$474$	0	0
$475$	2.04742	0.0939422
$476$	0	0
$477$	−35.6642	−1.63295
$478$	0	0
$479$	24.2435	1.10772	0.553858	−	0.832611i	$-0.313155\pi$
$479$	24.2435	1.10772	0.553858	+	0.832611i	$0.313155\pi$
$480$	0	0
$481$	−21.9251	−0.999698
$482$	0	0
$483$	−2.56823	−0.116858
$484$	0	0
$485$	−0.391591	−0.0177812
$486$	0	0
$487$	−17.2489	−0.781621	−0.390811	−	0.920471i	$-0.627805\pi$
$487$	−17.2489	−0.781621	−0.390811	+	0.920471i	$0.627805\pi$
$488$	0	0
$489$	−15.3902	−0.695971
$490$	0	0
$491$	1.32041	0.0595892	0.0297946	−	0.999556i	$-0.490515\pi$
$491$	1.32041	0.0595892	0.0297946	+	0.999556i	$0.490515\pi$
$492$	0	0
$493$	23.3877	1.05333
$494$	0	0
$495$	4.62829	0.208026
$496$	0	0
$497$	8.98495	0.403030
$498$	0	0
$499$	−4.79336	−0.214580	−0.107290	−	0.994228i	$-0.534217\pi$
$499$	−4.79336	−0.214580	−0.107290	+	0.994228i	$0.534217\pi$
$500$	0	0
$501$	17.9554	0.802190
$502$	0	0
$503$	−12.3724	−0.551657	−0.275829	−	0.961207i	$-0.588952\pi$
$503$	−12.3724	−0.551657	−0.275829	+	0.961207i	$0.588952\pi$
$504$	0	0
$505$	−5.13611	−0.228554
$506$	0	0
$507$	17.5681	0.780225
$508$	0	0
$509$	38.2590	1.69580	0.847901	−	0.530155i	$-0.177866\pi$
$509$	38.2590	1.69580	0.847901	+	0.530155i	$0.177866\pi$
$510$	0	0
$511$	−14.8577	−0.657266
$512$	0	0
$513$	−1.01077	−0.0446265
$514$	0	0
$515$	5.77298	0.254388
$516$	0	0
$517$	13.6117	0.598642
$518$	0	0
$519$	20.3776	0.894475
$520$	0	0
$521$	−15.4874	−0.678514	−0.339257	−	0.940694i	$-0.610176\pi$
$521$	−15.4874	−0.678514	−0.339257	+	0.940694i	$0.610176\pi$
$522$	0	0
$523$	26.0329	1.13834	0.569170	−	0.822220i	$-0.307265\pi$
$523$	26.0329	1.13834	0.569170	+	0.822220i	$0.307265\pi$
$524$	0	0
$525$	−13.5212	−0.590113
$526$	0	0
$527$	−19.7673	−0.861079
$528$	0	0
$529$	−22.1510	−0.963087
$530$	0	0
$531$	5.73852	0.249031
$532$	0	0
$533$	−34.0349	−1.47421
$534$	0	0
$535$	2.47859	0.107159
$536$	0	0
$537$	33.8871	1.46234
$538$	0	0
$539$	18.0168	0.776037
$540$	0	0
$541$	35.8172	1.53990	0.769952	−	0.638102i	$-0.220280\pi$
$541$	35.8172	1.53990	0.769952	+	0.638102i	$0.220280\pi$
$542$	0	0
$543$	51.8420	2.22475
$544$	0	0
$545$	5.49741	0.235483
$546$	0	0
$547$	7.76624	0.332060	0.166030	−	0.986121i	$-0.446905\pi$
$547$	7.76624	0.332060	0.166030	+	0.986121i	$0.446905\pi$
$548$	0	0
$549$	10.7797	0.460067
$550$	0	0
$551$	3.38179	0.144069
$552$	0	0
$553$	−10.7963	−0.459104
$554$	0	0
$555$	−5.01416	−0.212839
$556$	0	0
$557$	38.1611	1.61694	0.808470	−	0.588538i	$-0.200296\pi$
$557$	38.1611	1.61694	0.808470	+	0.588538i	$0.200296\pi$
$558$	0	0
$559$	−6.89629	−0.291682
$560$	0	0
$561$	23.5282	0.993361
$562$	0	0
$563$	−5.95868	−0.251129	−0.125564	−	0.992085i	$-0.540074\pi$
$563$	−5.95868	−0.251129	−0.125564	+	0.992085i	$0.540074\pi$
$564$	0	0
$565$	7.58153	0.318957
$566$	0	0
$567$	−5.76478	−0.242098
$568$	0	0
$569$	40.1828	1.68455	0.842275	−	0.539048i	$-0.181216\pi$
$569$	40.1828	1.68455	0.842275	+	0.539048i	$0.181216\pi$
$570$	0	0
$571$	21.1336	0.884416	0.442208	−	0.896913i	$-0.354195\pi$
$571$	21.1336	0.884416	0.442208	+	0.896913i	$0.354195\pi$
$572$	0	0
$573$	49.4530	2.06593
$574$	0	0
$575$	4.46981	0.186404
$576$	0	0
$577$	19.8944	0.828213	0.414107	−	0.910228i	$-0.364094\pi$
$577$	19.8944	0.828213	0.414107	+	0.910228i	$0.364094\pi$
$578$	0	0
$579$	−28.6204	−1.18942
$580$	0	0
$581$	14.6241	0.606710
$582$	0	0
$583$	27.9609	1.15802
$584$	0	0
$585$	6.69666	0.276873
$586$	0	0
$587$	5.59029	0.230736	0.115368	−	0.993323i	$-0.463195\pi$
$587$	5.59029	0.230736	0.115368	+	0.993323i	$0.463195\pi$
$588$	0	0
$589$	−2.85830	−0.117774
$590$	0	0
$591$	−28.2213	−1.16087
$592$	0	0
$593$	16.9966	0.697967	0.348984	−	0.937129i	$-0.386527\pi$
$593$	16.9966	0.697967	0.348984	+	0.937129i	$0.386527\pi$
$594$	0	0
$595$	1.19441	0.0489660
$596$	0	0
$597$	6.55745	0.268379
$598$	0	0
$599$	37.0485	1.51376	0.756881	−	0.653553i	$-0.226722\pi$
$599$	37.0485	1.51376	0.756881	+	0.653553i	$0.226722\pi$
$600$	0	0
$601$	−6.79139	−0.277027	−0.138513	−	0.990361i	$-0.544232\pi$
$601$	−6.79139	−0.277027	−0.138513	+	0.990361i	$0.544232\pi$
$602$	0	0
$603$	−46.2595	−1.88383
$604$	0	0
$605$	0.616827	0.0250776
$606$	0	0
$607$	23.5147	0.954432	0.477216	−	0.878786i	$-0.341646\pi$
$607$	23.5147	0.954432	0.477216	+	0.878786i	$0.341646\pi$
$608$	0	0
$609$	−22.3334	−0.904994
$610$	0	0
$611$	19.6947	0.796763
$612$	0	0
$613$	−34.4466	−1.39129	−0.695643	−	0.718388i	$-0.744880\pi$
$613$	−34.4466	−1.39129	−0.695643	+	0.718388i	$0.744880\pi$
$614$	0	0
$615$	−7.78361	−0.313865
$616$	0	0
$617$	2.77501	0.111718	0.0558589	−	0.998439i	$-0.482210\pi$
$617$	2.77501	0.111718	0.0558589	+	0.998439i	$0.482210\pi$
$618$	0	0
$619$	−35.8593	−1.44131	−0.720653	−	0.693295i	$-0.756158\pi$
$619$	−35.8593	−1.44131	−0.720653	+	0.693295i	$0.756158\pi$
$620$	0	0
$621$	−2.20665	−0.0885497
$622$	0	0
$623$	2.24515	0.0899500
$624$	0	0
$625$	22.7878	0.911514
$626$	0	0
$627$	3.40210	0.135867
$628$	0	0
$629$	−14.4249	−0.575157
$630$	0	0
$631$	−29.2138	−1.16298	−0.581491	−	0.813553i	$-0.697530\pi$
$631$	−29.2138	−1.16298	−0.581491	+	0.813553i	$0.697530\pi$
$632$	0	0
$633$	37.6816	1.49771
$634$	0	0
$635$	−4.99168	−0.198089
$636$	0	0
$637$	26.0684	1.03287
$638$	0	0
$639$	33.1430	1.31112
$640$	0	0
$641$	−0.769270	−0.0303843	−0.0151922	−	0.999885i	$-0.504836\pi$
$641$	−0.769270	−0.0303843	−0.0151922	+	0.999885i	$0.504836\pi$
$642$	0	0
$643$	−31.0795	−1.22566	−0.612829	−	0.790216i	$-0.709968\pi$
$643$	−31.0795	−1.22566	−0.612829	+	0.790216i	$0.709968\pi$
$644$	0	0
$645$	−1.57715	−0.0621001
$646$	0	0
$647$	24.1262	0.948498	0.474249	−	0.880391i	$-0.342720\pi$
$647$	24.1262	0.948498	0.474249	+	0.880391i	$0.342720\pi$
$648$	0	0
$649$	−4.49904	−0.176603
$650$	0	0
$651$	18.8762	0.739817
$652$	0	0
$653$	−19.6172	−0.767681	−0.383841	−	0.923399i	$-0.625399\pi$
$653$	−19.6172	−0.767681	−0.383841	+	0.923399i	$0.625399\pi$
$654$	0	0
$655$	1.33970	0.0523465
$656$	0	0
$657$	−54.8059	−2.13818
$658$	0	0
$659$	−13.2915	−0.517761	−0.258881	−	0.965909i	$-0.583354\pi$
$659$	−13.2915	−0.517761	−0.258881	+	0.965909i	$0.583354\pi$
$660$	0	0
$661$	39.7632	1.54661	0.773306	−	0.634034i	$-0.218602\pi$
$661$	39.7632	1.54661	0.773306	+	0.634034i	$0.218602\pi$
$662$	0	0
$663$	34.0429	1.32211
$664$	0	0
$665$	0.172708	0.00669732
$666$	0	0
$667$	7.38291	0.285868
$668$	0	0
$669$	30.2075	1.16789
$670$	0	0
$671$	−8.45137	−0.326262
$672$	0	0
$673$	−27.3955	−1.05602	−0.528009	−	0.849238i	$-0.677061\pi$
$673$	−27.3955	−1.05602	−0.528009	+	0.849238i	$0.677061\pi$
$674$	0	0
$675$	−11.6176	−0.447160
$676$	0	0
$677$	29.2926	1.12580	0.562902	−	0.826523i	$-0.309685\pi$
$677$	29.2926	1.12580	0.562902	+	0.826523i	$0.309685\pi$
$678$	0	0
$679$	1.07575	0.0412836
$680$	0	0
$681$	75.3470	2.88730
$682$	0	0
$683$	7.05739	0.270043	0.135022	−	0.990843i	$-0.456890\pi$
$683$	7.05739	0.270043	0.135022	+	0.990843i	$0.456890\pi$
$684$	0	0
$685$	−0.187353	−0.00715839
$686$	0	0
$687$	67.8170	2.58738
$688$	0	0
$689$	40.4566	1.54127
$690$	0	0
$691$	−49.9069	−1.89855	−0.949275	−	0.314448i	$-0.898181\pi$
$691$	−49.9069	−1.89855	−0.949275	+	0.314448i	$0.898181\pi$
$692$	0	0
$693$	−12.7146	−0.482986
$694$	0	0
$695$	1.32531	0.0502717
$696$	0	0
$697$	−22.3921	−0.848161
$698$	0	0
$699$	−33.0141	−1.24871
$700$	0	0
$701$	−4.71795	−0.178195	−0.0890973	−	0.996023i	$-0.528398\pi$
$701$	−4.71795	−0.178195	−0.0890973	+	0.996023i	$0.528398\pi$
$702$	0	0
$703$	−2.08579	−0.0786671
$704$	0	0
$705$	4.50409	0.169634
$706$	0	0
$707$	14.1096	0.530647
$708$	0	0
$709$	13.7207	0.515292	0.257646	−	0.966239i	$-0.417053\pi$
$709$	13.7207	0.515292	0.257646	+	0.966239i	$0.417053\pi$
$710$	0	0
$711$	−39.8244	−1.49353
$712$	0	0
$713$	−6.24006	−0.233692
$714$	0	0
$715$	−5.25022	−0.196347
$716$	0	0
$717$	−74.0368	−2.76495
$718$	0	0
$719$	−1.45561	−0.0542850	−0.0271425	−	0.999632i	$-0.508641\pi$
$719$	−1.45561	−0.0542850	−0.0271425	+	0.999632i	$0.508641\pi$
$720$	0	0
$721$	−15.8592	−0.590626
$722$	0	0
$723$	−28.7351	−1.06867
$724$	0	0
$725$	38.8695	1.44358
$726$	0	0
$727$	36.8028	1.36494	0.682471	−	0.730913i	$-0.260906\pi$
$727$	36.8028	1.36494	0.682471	+	0.730913i	$0.260906\pi$
$728$	0	0
$729$	−40.1520	−1.48711
$730$	0	0
$731$	−4.53718	−0.167814
$732$	0	0
$733$	−44.4744	−1.64270	−0.821350	−	0.570425i	$-0.806778\pi$
$733$	−44.4744	−1.64270	−0.821350	+	0.570425i	$0.806778\pi$
$734$	0	0
$735$	5.96171	0.219901
$736$	0	0
$737$	36.2677	1.33594
$738$	0	0
$739$	−18.5303	−0.681650	−0.340825	−	0.940127i	$-0.610706\pi$
$739$	−18.5303	−0.681650	−0.340825	+	0.940127i	$0.610706\pi$
$740$	0	0
$741$	4.92249	0.180832
$742$	0	0
$743$	−34.1551	−1.25303	−0.626515	−	0.779409i	$-0.715519\pi$
$743$	−34.1551	−1.25303	−0.626515	+	0.779409i	$0.715519\pi$
$744$	0	0
$745$	−4.87979	−0.178782
$746$	0	0
$747$	53.9442	1.97371
$748$	0	0
$749$	−6.80902	−0.248796
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	56.7682	2.06875
$754$	0	0
$755$	2.62536	0.0955466
$756$	0	0
$757$	−23.5644	−0.856463	−0.428231	−	0.903669i	$-0.640863\pi$
$757$	−23.5644	−0.856463	−0.428231	+	0.903669i	$0.640863\pi$
$758$	0	0
$759$	7.42726	0.269593
$760$	0	0
$761$	54.8766	1.98928	0.994638	−	0.103417i	$-0.0329777\pi$
$761$	54.8766	1.98928	0.994638	+	0.103417i	$0.0329777\pi$
$762$	0	0
$763$	−15.1021	−0.546734
$764$	0	0
$765$	4.40584	0.159293
$766$	0	0
$767$	−6.50964	−0.235050
$768$	0	0
$769$	9.01166	0.324969	0.162484	−	0.986711i	$-0.448049\pi$
$769$	9.01166	0.324969	0.162484	+	0.986711i	$0.448049\pi$
$770$	0	0
$771$	−34.2194	−1.23238
$772$	0	0
$773$	−24.4863	−0.880712	−0.440356	−	0.897823i	$-0.645148\pi$
$773$	−24.4863	−0.880712	−0.440356	+	0.897823i	$0.645148\pi$
$774$	0	0
$775$	−32.8526	−1.18010
$776$	0	0
$777$	13.7746	0.494160
$778$	0	0
$779$	−3.23783	−0.116007
$780$	0	0
$781$	−25.9843	−0.929791
$782$	0	0
$783$	−19.1891	−0.685761
$784$	0	0
$785$	6.62174	0.236340
$786$	0	0
$787$	−12.3100	−0.438804	−0.219402	−	0.975635i	$-0.570411\pi$
$787$	−12.3100	−0.438804	−0.219402	+	0.975635i	$0.570411\pi$
$788$	0	0
$789$	51.6674	1.83941
$790$	0	0
$791$	−20.8275	−0.740541
$792$	0	0
$793$	−12.2283	−0.434238
$794$	0	0
$795$	9.25221	0.328142
$796$	0	0
$797$	54.1894	1.91949	0.959744	−	0.280877i	$-0.0906253\pi$
$797$	54.1894	1.91949	0.959744	+	0.280877i	$0.0906253\pi$
$798$	0	0
$799$	12.9575	0.458403
$800$	0	0
$801$	8.28172	0.292620
$802$	0	0
$803$	42.9681	1.51631
$804$	0	0
$805$	0.377045	0.0132891
$806$	0	0
$807$	39.9378	1.40588
$808$	0	0
$809$	−7.73276	−0.271869	−0.135935	−	0.990718i	$-0.543404\pi$
$809$	−7.73276	−0.271869	−0.135935	+	0.990718i	$0.543404\pi$
$810$	0	0
$811$	−14.8591	−0.521774	−0.260887	−	0.965369i	$-0.584015\pi$
$811$	−14.8591	−0.521774	−0.260887	+	0.965369i	$0.584015\pi$
$812$	0	0
$813$	5.68829	0.199497
$814$	0	0
$815$	2.25946	0.0791455
$816$	0	0
$817$	−0.656062	−0.0229527
$818$	0	0
$819$	−18.3966	−0.642831
$820$	0	0
$821$	−9.73848	−0.339875	−0.169938	−	0.985455i	$-0.554357\pi$
$821$	−9.73848	−0.339875	−0.169938	+	0.985455i	$0.554357\pi$
$822$	0	0
$823$	18.3051	0.638075	0.319038	−	0.947742i	$-0.396640\pi$
$823$	18.3051	0.638075	0.319038	+	0.947742i	$0.396640\pi$
$824$	0	0
$825$	39.1030	1.36139
$826$	0	0
$827$	−32.4200	−1.12735	−0.563677	−	0.825995i	$-0.690614\pi$
$827$	−32.4200	−1.12735	−0.563677	+	0.825995i	$0.690614\pi$
$828$	0	0
$829$	−11.7075	−0.406617	−0.203309	−	0.979115i	$-0.565169\pi$
$829$	−11.7075	−0.406617	−0.203309	+	0.979115i	$0.565169\pi$
$830$	0	0
$831$	−27.5054	−0.954152
$832$	0	0
$833$	17.1508	0.594241
$834$	0	0
$835$	−2.63606	−0.0912247
$836$	0	0
$837$	16.2186	0.560598
$838$	0	0
$839$	9.03241	0.311833	0.155917	−	0.987770i	$-0.450167\pi$
$839$	9.03241	0.311833	0.155917	+	0.987770i	$0.450167\pi$
$840$	0	0
$841$	35.2020	1.21386
$842$	0	0
$843$	−30.8135	−1.06127
$844$	0	0
$845$	−2.57919	−0.0887269
$846$	0	0
$847$	−1.69451	−0.0582240
$848$	0	0
$849$	−38.9126	−1.33548
$850$	0	0
$851$	−4.55357	−0.156094
$852$	0	0
$853$	−55.0552	−1.88506	−0.942528	−	0.334128i	$-0.891558\pi$
$853$	−55.0552	−1.88506	−0.942528	+	0.334128i	$0.891558\pi$
$854$	0	0
$855$	0.637070	0.0217874
$856$	0	0
$857$	9.95859	0.340179	0.170089	−	0.985429i	$-0.445594\pi$
$857$	9.95859	0.340179	0.170089	+	0.985429i	$0.445594\pi$
$858$	0	0
$859$	2.52496	0.0861506	0.0430753	−	0.999072i	$-0.486284\pi$
$859$	2.52496	0.0861506	0.0430753	+	0.999072i	$0.486284\pi$
$860$	0	0
$861$	21.3826	0.728718
$862$	0	0
$863$	−6.27195	−0.213500	−0.106750	−	0.994286i	$-0.534044\pi$
$863$	−6.27195	−0.213500	−0.106750	+	0.994286i	$0.534044\pi$
$864$	0	0
$865$	−2.99166	−0.101719
$866$	0	0
$867$	−22.2935	−0.757128
$868$	0	0
$869$	31.2226	1.05915
$870$	0	0
$871$	52.4756	1.77807
$872$	0	0
$873$	3.96815	0.134302
$874$	0	0
$875$	4.03108	0.136276
$876$	0	0
$877$	29.7100	1.00324	0.501618	−	0.865089i	$-0.332738\pi$
$877$	29.7100	1.00324	0.501618	+	0.865089i	$0.332738\pi$
$878$	0	0
$879$	18.1785	0.613146
$880$	0	0
$881$	7.83415	0.263939	0.131970	−	0.991254i	$-0.457870\pi$
$881$	7.83415	0.263939	0.131970	+	0.991254i	$0.457870\pi$
$882$	0	0
$883$	−32.7670	−1.10270	−0.551349	−	0.834275i	$-0.685887\pi$
$883$	−32.7670	−1.10270	−0.551349	+	0.834275i	$0.685887\pi$
$884$	0	0
$885$	−1.48872	−0.0500429
$886$	0	0
$887$	−35.7445	−1.20018	−0.600092	−	0.799931i	$-0.704869\pi$
$887$	−35.7445	−1.20018	−0.600092	+	0.799931i	$0.704869\pi$
$888$	0	0
$889$	13.7128	0.459914
$890$	0	0
$891$	16.6716	0.558521
$892$	0	0
$893$	1.87361	0.0626980
$894$	0	0
$895$	−4.97502	−0.166297
$896$	0	0
$897$	10.7465	0.358814
$898$	0	0
$899$	−54.2637	−1.80980
$900$	0	0
$901$	26.6170	0.886742
$902$	0	0
$903$	4.33264	0.144181
$904$	0	0
$905$	−7.61100	−0.252998
$906$	0	0
$907$	−48.3106	−1.60413	−0.802063	−	0.597239i	$-0.796264\pi$
$907$	−48.3106	−1.60413	−0.802063	+	0.597239i	$0.796264\pi$
$908$	0	0
$909$	52.0464	1.72627
$910$	0	0
$911$	−44.3628	−1.46981	−0.734903	−	0.678172i	$-0.762772\pi$
$911$	−44.3628	−1.46981	−0.734903	+	0.678172i	$0.762772\pi$
$912$	0	0
$913$	−42.2926	−1.39968
$914$	0	0
$915$	−2.79654	−0.0924508
$916$	0	0
$917$	−3.68035	−0.121536
$918$	0	0
$919$	−27.7212	−0.914438	−0.457219	−	0.889354i	$-0.651154\pi$
$919$	−27.7212	−0.914438	−0.457219	+	0.889354i	$0.651154\pi$
$920$	0	0
$921$	18.4243	0.607100
$922$	0	0
$923$	−37.5966	−1.23751
$924$	0	0
$925$	−23.9736	−0.788248
$926$	0	0
$927$	−58.5000	−1.92139
$928$	0	0
$929$	−14.2627	−0.467943	−0.233972	−	0.972243i	$-0.575172\pi$
$929$	−14.2627	−0.467943	−0.233972	+	0.972243i	$0.575172\pi$
$930$	0	0
$931$	2.47995	0.0812772
$932$	0	0
$933$	−49.9103	−1.63399
$934$	0	0
$935$	−3.45421	−0.112965
$936$	0	0
$937$	54.7211	1.78766	0.893831	−	0.448404i	$-0.148007\pi$
$937$	54.7211	1.78766	0.893831	+	0.448404i	$0.148007\pi$
$938$	0	0
$939$	−12.9223	−0.421702
$940$	0	0
$941$	−19.7531	−0.643932	−0.321966	−	0.946751i	$-0.604344\pi$
$941$	−19.7531	−0.643932	−0.321966	+	0.946751i	$0.604344\pi$
$942$	0	0
$943$	−7.06863	−0.230186
$944$	0	0
$945$	−0.979984	−0.0318789
$946$	0	0
$947$	−16.5516	−0.537856	−0.268928	−	0.963160i	$-0.586669\pi$
$947$	−16.5516	−0.537856	−0.268928	+	0.963160i	$0.586669\pi$
$948$	0	0
$949$	62.1704	2.01814
$950$	0	0
$951$	−72.7096	−2.35777
$952$	0	0
$953$	43.4666	1.40802	0.704011	−	0.710189i	$-0.251391\pi$
$953$	43.4666	1.40802	0.704011	+	0.710189i	$0.251391\pi$
$954$	0	0
$955$	−7.26027	−0.234937
$956$	0	0
$957$	64.5876	2.08782
$958$	0	0
$959$	0.514684	0.0166200
$960$	0	0
$961$	14.8638	0.479478
$962$	0	0
$963$	−25.1166	−0.809370
$964$	0	0
$965$	4.20180	0.135261
$966$	0	0
$967$	54.4228	1.75012	0.875059	−	0.484015i	$-0.160822\pi$
$967$	54.4228	1.75012	0.875059	+	0.484015i	$0.160822\pi$
$968$	0	0
$969$	3.23859	0.104038
$970$	0	0
$971$	−15.6010	−0.500660	−0.250330	−	0.968161i	$-0.580539\pi$
$971$	−15.6010	−0.500660	−0.250330	+	0.968161i	$0.580539\pi$
$972$	0	0
$973$	−3.64080	−0.116719
$974$	0	0
$975$	56.5780	1.81195
$976$	0	0
$977$	−32.6341	−1.04406	−0.522029	−	0.852928i	$-0.674825\pi$
$977$	−32.6341	−1.04406	−0.522029	+	0.852928i	$0.674825\pi$
$978$	0	0
$979$	−6.49292	−0.207515
$980$	0	0
$981$	−55.7076	−1.77861
$982$	0	0
$983$	19.6891	0.627984	0.313992	−	0.949426i	$-0.398333\pi$
$983$	19.6891	0.627984	0.313992	+	0.949426i	$0.398333\pi$
$984$	0	0
$985$	4.14321	0.132014
$986$	0	0
$987$	−12.3733	−0.393848
$988$	0	0
$989$	−1.43227	−0.0455437
$990$	0	0
$991$	−33.6329	−1.06838	−0.534192	−	0.845363i	$-0.679384\pi$
$991$	−33.6329	−1.06838	−0.534192	+	0.845363i	$0.679384\pi$
$992$	0	0
$993$	55.4388	1.75930
$994$	0	0
$995$	−0.962709	−0.0305199
$996$	0	0
$997$	13.5851	0.430244	0.215122	−	0.976587i	$-0.430985\pi$
$997$	13.5851	0.430244	0.215122	+	0.976587i	$0.430985\pi$
$998$	0	0
$999$	11.8353	0.374451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.41	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.41	✓	44	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+2.62887 q^{3} -0.385949 q^{5} +1.06025 q^{7} +3.91098 q^{9} +O(q^{10})\)	\(q+2.62887 q^{3} -0.385949 q^{5} +1.06025 q^{7} +3.91098 q^{9} -3.06623 q^{11} -4.43652 q^{13} -1.01461 q^{15} -2.91886 q^{17} -0.422058 q^{19} +2.78728 q^{21} -0.921411 q^{23} -4.85104 q^{25} +2.39486 q^{27} -8.01261 q^{29} +6.77228 q^{31} -8.06074 q^{33} -0.409204 q^{35} +4.94195 q^{37} -11.6631 q^{39} +7.67152 q^{41} +1.55444 q^{43} -1.50944 q^{45} -4.43923 q^{47} -5.87586 q^{49} -7.67332 q^{51} -9.11898 q^{53} +1.18341 q^{55} -1.10954 q^{57} +1.46728 q^{59} +2.75627 q^{61} +4.14664 q^{63} +1.71227 q^{65} -11.8281 q^{67} -2.42228 q^{69} +8.47433 q^{71} -14.0133 q^{73} -12.7528 q^{75} -3.25099 q^{77} -10.1827 q^{79} -5.43717 q^{81} +13.7930 q^{83} +1.12653 q^{85} -21.0642 q^{87} +2.11756 q^{89} -4.70384 q^{91} +17.8035 q^{93} +0.162893 q^{95} +1.01462 q^{97} -11.9920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more