LMFDB - Embedded newform 1008.4.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.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:	$59.4739252858$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.69042$ of defining polynomial
Character	$\chi$	$=$	1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.3808 q^{5} +7.00000 q^{7} +O(q^{10})$	$q+15.3808 q^{5} +7.00000 q^{7} +7.38083 q^{11} -67.5233 q^{13} -138.427 q^{17} +116.570 q^{19} -133.474 q^{23} +111.570 q^{25} -120.000 q^{29} -295.617 q^{31} +107.666 q^{35} -337.617 q^{37} -409.189 q^{41} -346.093 q^{43} +345.808 q^{47} +49.0000 q^{49} +381.995 q^{53} +113.523 q^{55} +438.192 q^{59} +707.897 q^{61} -1038.56 q^{65} -264.953 q^{67} +391.951 q^{71} -237.047 q^{73} +51.6658 q^{77} +1143.61 q^{79} -49.1400 q^{83} -2129.13 q^{85} +333.847 q^{89} -472.663 q^{91} +1792.94 q^{95} -397.047 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{5} + 14 q^{7}+O(q^{10}) $ 2 * q + 12 * q^5 + 14 * q^7	$ 2 q + 12 q^{5} + 14 q^{7} - 4 q^{11} - 60 q^{13} - 108 q^{17} + 8 q^{19} + 52 q^{23} - 2 q^{25} - 240 q^{29} - 216 q^{31} + 84 q^{35} - 300 q^{37} - 612 q^{41} - 392 q^{43} + 504 q^{47} + 98 q^{49} - 24 q^{53} + 152 q^{55} + 1064 q^{59} + 140 q^{61} - 1064 q^{65} - 680 q^{67} + 540 q^{71} - 324 q^{73} - 28 q^{77} + 336 q^{79} + 352 q^{83} - 2232 q^{85} - 852 q^{89} - 420 q^{91} + 2160 q^{95} - 644 q^{97}+O(q^{100}) $ 2 * q + 12 * q^5 + 14 * q^7 - 4 * q^11 - 60 * q^13 - 108 * q^17 + 8 * q^19 + 52 * q^23 - 2 * q^25 - 240 * q^29 - 216 * q^31 + 84 * q^35 - 300 * q^37 - 612 * q^41 - 392 * q^43 + 504 * q^47 + 98 * q^49 - 24 * q^53 + 152 * q^55 + 1064 * q^59 + 140 * q^61 - 1064 * q^65 - 680 * q^67 + 540 * q^71 - 324 * q^73 - 28 * q^77 + 336 * q^79 + 352 * q^83 - 2232 * q^85 - 852 * q^89 - 420 * q^91 + 2160 * q^95 - 644 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	15.3808	1.37570	0.687852	−	0.725851i	$-0.258554\pi$
$5$	15.3808	1.37570	0.687852	+	0.725851i	$0.258554\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	7.38083	0.202309	0.101155	−	0.994871i	$-0.467746\pi$
$11$	7.38083	0.202309	0.101155	+	0.994871i	$0.467746\pi$
$12$	0	0
$13$	−67.5233	−1.44058	−0.720292	−	0.693671i	$-0.755992\pi$
$13$	−67.5233	−1.44058	−0.720292	+	0.693671i	$0.755992\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−138.427	−1.97492	−0.987459	−	0.157877i	$-0.949535\pi$
$17$	−138.427	−1.97492	−0.987459	+	0.157877i	$0.949535\pi$
$18$	0	0
$19$	116.570	1.40753	0.703763	−	0.710435i	$-0.251502\pi$
$19$	116.570	1.40753	0.703763	+	0.710435i	$0.251502\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−133.474	−1.21006	−0.605028	−	0.796204i	$-0.706838\pi$
$23$	−133.474	−1.21006	−0.605028	+	0.796204i	$0.706838\pi$
$24$	0	0
$25$	111.570	0.892560
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−120.000	−0.768395	−0.384197	−	0.923251i	$-0.625522\pi$
$29$	−120.000	−0.768395	−0.384197	+	0.923251i	$0.625522\pi$
$30$	0	0
$31$	−295.617	−1.71272	−0.856360	−	0.516379i	$-0.827279\pi$
$31$	−295.617	−1.71272	−0.856360	+	0.516379i	$0.827279\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	107.666	0.519967
$36$	0	0
$37$	−337.617	−1.50010	−0.750052	−	0.661379i	$-0.769971\pi$
$37$	−337.617	−1.50010	−0.750052	+	0.661379i	$0.769971\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−409.189	−1.55865	−0.779324	−	0.626621i	$-0.784438\pi$
$41$	−409.189	−1.55865	−0.779324	+	0.626621i	$0.784438\pi$
$42$	0	0
$43$	−346.093	−1.22741	−0.613706	−	0.789534i	$-0.710322\pi$
$43$	−346.093	−1.22741	−0.613706	+	0.789534i	$0.710322\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	345.808	1.07322	0.536610	−	0.843830i	$-0.319705\pi$
$47$	345.808	1.07322	0.536610	+	0.843830i	$0.319705\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	381.995	0.990020	0.495010	−	0.868887i	$-0.335164\pi$
$53$	381.995	0.990020	0.495010	+	0.868887i	$0.335164\pi$
$54$	0	0
$55$	113.523	0.278318
$56$	0	0
$57$	0	0
$58$	0	0
$59$	438.192	0.966910	0.483455	−	0.875369i	$-0.339382\pi$
$59$	438.192	0.966910	0.483455	+	0.875369i	$0.339382\pi$
$60$	0	0
$61$	707.897	1.48585	0.742925	−	0.669375i	$-0.233438\pi$
$61$	707.897	1.48585	0.742925	+	0.669375i	$0.233438\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1038.56	−1.98182
$66$	0	0
$67$	−264.953	−0.483122	−0.241561	−	0.970386i	$-0.577659\pi$
$67$	−264.953	−0.483122	−0.241561	+	0.970386i	$0.577659\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	391.951	0.655155	0.327577	−	0.944824i	$-0.393768\pi$
$71$	391.951	0.655155	0.327577	+	0.944824i	$0.393768\pi$
$72$	0	0
$73$	−237.047	−0.380058	−0.190029	−	0.981779i	$-0.560858\pi$
$73$	−237.047	−0.380058	−0.190029	+	0.981779i	$0.560858\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	51.6658	0.0764658
$78$	0	0
$79$	1143.61	1.62868	0.814340	−	0.580388i	$-0.197099\pi$
$79$	1143.61	1.62868	0.814340	+	0.580388i	$0.197099\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−49.1400	−0.0649857	−0.0324928	−	0.999472i	$-0.510345\pi$
$83$	−49.1400	−0.0649857	−0.0324928	+	0.999472i	$0.510345\pi$
$84$	0	0
$85$	−2129.13	−2.71690
$86$	0	0
$87$	0	0
$88$	0	0
$89$	333.847	0.397615	0.198808	−	0.980039i	$-0.436293\pi$
$89$	333.847	0.397615	0.198808	+	0.980039i	$0.436293\pi$
$90$	0	0
$91$	−472.663	−0.544490
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1792.94	1.93634
$96$	0	0
$97$	−397.047	−0.415608	−0.207804	−	0.978170i	$-0.566632\pi$
$97$	−397.047	−0.415608	−0.207804	+	0.978170i	$0.566632\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−440.148	−0.433627	−0.216813	−	0.976213i	$-0.569566\pi$
$101$	−440.148	−0.433627	−0.216813	+	0.976213i	$0.569566\pi$
$102$	0	0
$103$	−1700.94	−1.62717	−0.813587	−	0.581443i	$-0.802488\pi$
$103$	−1700.94	−1.62717	−0.813587	+	0.581443i	$0.802488\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	966.044	0.872813	0.436407	−	0.899750i	$-0.356251\pi$
$107$	966.044	0.872813	0.436407	+	0.899750i	$0.356251\pi$
$108$	0	0
$109$	−468.860	−0.412006	−0.206003	−	0.978551i	$-0.566046\pi$
$109$	−468.860	−0.412006	−0.206003	+	0.978551i	$0.566046\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−497.140	−0.413867	−0.206934	−	0.978355i	$-0.566348\pi$
$113$	−497.140	−0.413867	−0.206934	+	0.978355i	$0.566348\pi$
$114$	0	0
$115$	−2052.94	−1.66468
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−968.992	−0.746449
$120$	0	0
$121$	−1276.52	−0.959071
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−206.565	−0.147806
$126$	0	0
$127$	−539.813	−0.377171	−0.188585	−	0.982057i	$-0.560390\pi$
$127$	−539.813	−0.377171	−0.188585	+	0.982057i	$0.560390\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2340.36	1.56090	0.780452	−	0.625215i	$-0.214989\pi$
$131$	2340.36	1.56090	0.780452	+	0.625215i	$0.214989\pi$
$132$	0	0
$133$	815.990	0.531995
$134$	0	0
$135$	0	0
$136$	0	0
$137$	278.388	0.173608	0.0868041	−	0.996225i	$-0.472335\pi$
$137$	278.388	0.173608	0.0868041	+	0.996225i	$0.472335\pi$
$138$	0	0
$139$	945.886	0.577187	0.288594	−	0.957452i	$-0.406812\pi$
$139$	945.886	0.577187	0.288594	+	0.957452i	$0.406812\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−498.378	−0.291444
$144$	0	0
$145$	−1845.70	−1.05708
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2511.68	1.38097	0.690487	−	0.723345i	$-0.257396\pi$
$149$	2511.68	1.38097	0.690487	+	0.723345i	$0.257396\pi$
$150$	0	0
$151$	−3126.07	−1.68474	−0.842372	−	0.538897i	$-0.818841\pi$
$151$	−3126.07	−1.68474	−0.842372	+	0.538897i	$0.818841\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4546.83	−2.35619
$156$	0	0
$157$	825.596	0.419680	0.209840	−	0.977736i	$-0.432706\pi$
$157$	825.596	0.419680	0.209840	+	0.977736i	$0.432706\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−934.319	−0.457358
$162$	0	0
$163$	−967.047	−0.464693	−0.232346	−	0.972633i	$-0.574640\pi$
$163$	−967.047	−0.464693	−0.232346	+	0.972633i	$0.574640\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3567.31	−1.65298	−0.826488	−	0.562955i	$-0.809664\pi$
$167$	−3567.31	−1.65298	−0.826488	+	0.562955i	$0.809664\pi$
$168$	0	0
$169$	2362.40	1.07528
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−637.651	−0.280229	−0.140115	−	0.990135i	$-0.544747\pi$
$173$	−637.651	−0.280229	−0.140115	+	0.990135i	$0.544747\pi$
$174$	0	0
$175$	780.990	0.337356
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1329.48	−0.555141	−0.277571	−	0.960705i	$-0.589529\pi$
$179$	−1329.48	−0.555141	−0.277571	+	0.960705i	$0.589529\pi$
$180$	0	0
$181$	−2102.94	−0.863594	−0.431797	−	0.901971i	$-0.642120\pi$
$181$	−2102.94	−0.863594	−0.431797	+	0.901971i	$0.642120\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5192.82	−2.06370
$186$	0	0
$187$	−1021.71	−0.399545
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2605.45	0.987037	0.493518	−	0.869735i	$-0.335711\pi$
$191$	2605.45	0.987037	0.493518	+	0.869735i	$0.335711\pi$
$192$	0	0
$193$	−750.363	−0.279857	−0.139928	−	0.990162i	$-0.544687\pi$
$193$	−750.363	−0.279857	−0.139928	+	0.990162i	$0.544687\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1738.97	0.628916	0.314458	−	0.949271i	$-0.398177\pi$
$197$	1738.97	0.628916	0.314458	+	0.949271i	$0.398177\pi$
$198$	0	0
$199$	1897.12	0.675795	0.337898	−	0.941183i	$-0.390284\pi$
$199$	1897.12	0.675795	0.337898	+	0.941183i	$0.390284\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−840.000	−0.290426
$204$	0	0
$205$	−6293.67	−2.14424
$206$	0	0
$207$	0	0
$208$	0	0
$209$	860.383	0.284756
$210$	0	0
$211$	−1533.14	−0.500217	−0.250108	−	0.968218i	$-0.580466\pi$
$211$	−1533.14	−0.500217	−0.250108	+	0.968218i	$0.580466\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5323.20	−1.68856
$216$	0	0
$217$	−2069.32	−0.647347
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9347.08	2.84504
$222$	0	0
$223$	−851.793	−0.255786	−0.127893	−	0.991788i	$-0.540821\pi$
$223$	−851.793	−0.255786	−0.127893	+	0.991788i	$0.540821\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1896.63	−0.554553	−0.277277	−	0.960790i	$-0.589432\pi$
$227$	−1896.63	−0.554553	−0.277277	+	0.960790i	$0.589432\pi$
$228$	0	0
$229$	−1924.68	−0.555400	−0.277700	−	0.960668i	$-0.589572\pi$
$229$	−1924.68	−0.555400	−0.277700	+	0.960668i	$0.589572\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1923.91	0.540943	0.270471	−	0.962728i	$-0.412820\pi$
$233$	1923.91	0.540943	0.270471	+	0.962728i	$0.412820\pi$
$234$	0	0
$235$	5318.82	1.47643
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3625.89	0.981337	0.490669	−	0.871346i	$-0.336753\pi$
$239$	3625.89	0.981337	0.490669	+	0.871346i	$0.336753\pi$
$240$	0	0
$241$	274.974	0.0734963	0.0367482	−	0.999325i	$-0.488300\pi$
$241$	274.974	0.0734963	0.0367482	+	0.999325i	$0.488300\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	753.661	0.196529
$246$	0	0
$247$	−7871.19	−2.02766
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1362.22	−0.342561	−0.171280	−	0.985222i	$-0.554790\pi$
$251$	−1362.22	−0.342561	−0.171280	+	0.985222i	$0.554790\pi$
$252$	0	0
$253$	−985.150	−0.244806
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4928.48	1.19623	0.598113	−	0.801412i	$-0.295917\pi$
$257$	4928.48	1.19623	0.598113	+	0.801412i	$0.295917\pi$
$258$	0	0
$259$	−2363.32	−0.566986
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2060.55	−0.483114	−0.241557	−	0.970387i	$-0.577658\pi$
$263$	−2060.55	−0.483114	−0.241557	+	0.970387i	$0.577658\pi$
$264$	0	0
$265$	5875.40	1.36197
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5309.36	−1.20341	−0.601705	−	0.798719i	$-0.705512\pi$
$269$	−5309.36	−1.20341	−0.601705	+	0.798719i	$0.705512\pi$
$270$	0	0
$271$	−80.4037	−0.0180228	−0.00901139	−	0.999959i	$-0.502868\pi$
$271$	−80.4037	−0.0180228	−0.00901139	+	0.999959i	$0.502868\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	823.479	0.180573
$276$	0	0
$277$	−213.637	−0.0463401	−0.0231700	−	0.999732i	$-0.507376\pi$
$277$	−213.637	−0.0463401	−0.0231700	+	0.999732i	$0.507376\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5748.26	1.22033	0.610165	−	0.792274i	$-0.291103\pi$
$281$	5748.26	1.22033	0.610165	+	0.792274i	$0.291103\pi$
$282$	0	0
$283$	4158.08	0.873401	0.436700	−	0.899607i	$-0.356147\pi$
$283$	4158.08	0.873401	0.436700	+	0.899607i	$0.356147\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2864.32	−0.589114
$288$	0	0
$289$	14249.2	2.90030
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2748.47	−0.548011	−0.274006	−	0.961728i	$-0.588349\pi$
$293$	−2748.47	−0.548011	−0.274006	+	0.961728i	$0.588349\pi$
$294$	0	0
$295$	6739.75	1.33018
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9012.62	1.74319
$300$	0	0
$301$	−2422.65	−0.463918
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10888.0	2.04409
$306$	0	0
$307$	−7532.86	−1.40040	−0.700201	−	0.713946i	$-0.746906\pi$
$307$	−7532.86	−1.40040	−0.700201	+	0.713946i	$0.746906\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8038.39	1.46564	0.732822	−	0.680421i	$-0.238203\pi$
$311$	8038.39	1.46564	0.732822	+	0.680421i	$0.238203\pi$
$312$	0	0
$313$	−804.053	−0.145200	−0.0726002	−	0.997361i	$-0.523130\pi$
$313$	−804.053	−0.145200	−0.0726002	+	0.997361i	$0.523130\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8673.02	−1.53667	−0.768337	−	0.640046i	$-0.778915\pi$
$317$	−8673.02	−1.53667	−0.768337	+	0.640046i	$0.778915\pi$
$318$	0	0
$319$	−885.700	−0.155454
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−16136.5	−2.77975
$324$	0	0
$325$	−7533.58	−1.28581
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2420.66	0.405639
$330$	0	0
$331$	−105.327	−0.0174902	−0.00874512	−	0.999962i	$-0.502784\pi$
$331$	−105.327	−0.0174902	−0.00874512	+	0.999962i	$0.502784\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4075.20	−0.664633
$336$	0	0
$337$	−1360.43	−0.219902	−0.109951	−	0.993937i	$-0.535069\pi$
$337$	−1360.43	−0.219902	−0.109951	+	0.993937i	$0.535069\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2181.90	−0.346499
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10512.0	1.62626	0.813130	−	0.582083i	$-0.197762\pi$
$347$	10512.0	1.62626	0.813130	+	0.582083i	$0.197762\pi$
$348$	0	0
$349$	2560.50	0.392723	0.196361	−	0.980532i	$-0.437087\pi$
$349$	2560.50	0.392723	0.196361	+	0.980532i	$0.437087\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3139.63	−0.473387	−0.236693	−	0.971584i	$-0.576064\pi$
$353$	−3139.63	−0.473387	−0.236693	+	0.971584i	$0.576064\pi$
$354$	0	0
$355$	6028.53	0.901299
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1693.89	0.249025	0.124513	−	0.992218i	$-0.460263\pi$
$359$	1693.89	0.249025	0.124513	+	0.992218i	$0.460263\pi$
$360$	0	0
$361$	6729.56	0.981128
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3645.97	−0.522847
$366$	0	0
$367$	3239.63	0.460782	0.230391	−	0.973098i	$-0.425999\pi$
$367$	3239.63	0.460782	0.230391	+	0.973098i	$0.425999\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2673.96	0.374192
$372$	0	0
$373$	443.679	0.0615894	0.0307947	−	0.999526i	$-0.490196\pi$
$373$	443.679	0.0615894	0.0307947	+	0.999526i	$0.490196\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8102.80	1.10694
$378$	0	0
$379$	−10317.1	−1.39829	−0.699145	−	0.714980i	$-0.746436\pi$
$379$	−10317.1	−1.39829	−0.699145	+	0.714980i	$0.746436\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6015.80	0.802593	0.401297	−	0.915948i	$-0.368560\pi$
$383$	6015.80	0.802593	0.401297	+	0.915948i	$0.368560\pi$
$384$	0	0
$385$	794.663	0.105194
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11903.3	−1.55147	−0.775735	−	0.631059i	$-0.782621\pi$
$389$	−11903.3	−1.55147	−0.775735	+	0.631059i	$0.782621\pi$
$390$	0	0
$391$	18476.5	2.38976
$392$	0	0
$393$	0	0
$394$	0	0
$395$	17589.6	2.24058
$396$	0	0
$397$	−8891.67	−1.12408	−0.562040	−	0.827110i	$-0.689983\pi$
$397$	−8891.67	−1.12408	−0.562040	+	0.827110i	$0.689983\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11175.9	−1.39177	−0.695884	−	0.718154i	$-0.744987\pi$
$401$	−11175.9	−1.39177	−0.695884	+	0.718154i	$0.744987\pi$
$402$	0	0
$403$	19961.0	2.46732
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2491.89	−0.303485
$408$	0	0
$409$	3961.48	0.478931	0.239465	−	0.970905i	$-0.423028\pi$
$409$	3961.48	0.478931	0.239465	+	0.970905i	$0.423028\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3067.34	0.365458
$414$	0	0
$415$	−755.813	−0.0894010
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12490.0	−1.45627	−0.728136	−	0.685432i	$-0.759613\pi$
$419$	−12490.0	−1.45627	−0.728136	+	0.685432i	$0.759613\pi$
$420$	0	0
$421$	4345.54	0.503060	0.251530	−	0.967849i	$-0.419066\pi$
$421$	4345.54	0.503060	0.251530	+	0.967849i	$0.419066\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15444.4	−1.76273
$426$	0	0
$427$	4955.28	0.561599
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12328.2	−1.37779	−0.688894	−	0.724862i	$-0.741904\pi$
$431$	−12328.2	−1.37779	−0.688894	+	0.724862i	$0.741904\pi$
$432$	0	0
$433$	17230.6	1.91235	0.956175	−	0.292795i	$-0.0945852\pi$
$433$	17230.6	1.91235	0.956175	+	0.292795i	$0.0945852\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15559.1	−1.70318
$438$	0	0
$439$	1530.34	0.166376	0.0831881	−	0.996534i	$-0.473490\pi$
$439$	1530.34	0.166376	0.0831881	+	0.996534i	$0.473490\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3104.81	−0.332989	−0.166494	−	0.986042i	$-0.553245\pi$
$443$	−3104.81	−0.332989	−0.166494	+	0.986042i	$0.553245\pi$
$444$	0	0
$445$	5134.85	0.547001
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2226.56	0.234026	0.117013	−	0.993130i	$-0.462668\pi$
$449$	2226.56	0.234026	0.117013	+	0.993130i	$0.462668\pi$
$450$	0	0
$451$	−3020.16	−0.315329
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7269.95	−0.749057
$456$	0	0
$457$	−6793.88	−0.695414	−0.347707	−	0.937603i	$-0.613040\pi$
$457$	−6793.88	−0.695414	−0.347707	+	0.937603i	$0.613040\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−429.847	−0.0434273	−0.0217137	−	0.999764i	$-0.506912\pi$
$461$	−429.847	−0.0434273	−0.0217137	+	0.999764i	$0.506912\pi$
$462$	0	0
$463$	−14571.3	−1.46260	−0.731301	−	0.682055i	$-0.761087\pi$
$463$	−14571.3	−1.46260	−0.731301	+	0.682055i	$0.761087\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5586.53	0.553563	0.276782	−	0.960933i	$-0.410732\pi$
$467$	5586.53	0.553563	0.276782	+	0.960933i	$0.410732\pi$
$468$	0	0
$469$	−1854.67	−0.182603
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2554.46	−0.248317
$474$	0	0
$475$	13005.7	1.25630
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15703.7	1.49796	0.748979	−	0.662594i	$-0.230545\pi$
$479$	15703.7	1.49796	0.748979	+	0.662594i	$0.230545\pi$
$480$	0	0
$481$	22797.0	2.16103
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6106.91	−0.571753
$486$	0	0
$487$	−6752.64	−0.628319	−0.314160	−	0.949370i	$-0.601723\pi$
$487$	−6752.64	−0.628319	−0.314160	+	0.949370i	$0.601723\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10092.9	0.927674	0.463837	−	0.885920i	$-0.346472\pi$
$491$	10092.9	0.927674	0.463837	+	0.885920i	$0.346472\pi$
$492$	0	0
$493$	16611.3	1.51752
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2743.66	0.247625
$498$	0	0
$499$	925.493	0.0830276	0.0415138	−	0.999138i	$-0.486782\pi$
$499$	925.493	0.0830276	0.0415138	+	0.999138i	$0.486782\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16024.5	1.42047	0.710237	−	0.703963i	$-0.248588\pi$
$503$	16024.5	1.42047	0.710237	+	0.703963i	$0.248588\pi$
$504$	0	0
$505$	−6769.84	−0.596542
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1484.94	0.129310	0.0646550	−	0.997908i	$-0.479405\pi$
$509$	1484.94	0.129310	0.0646550	+	0.997908i	$0.479405\pi$
$510$	0	0
$511$	−1659.33	−0.143648
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−26161.9	−2.23851
$516$	0	0
$517$	2552.35	0.217123
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5627.65	−0.473228	−0.236614	−	0.971604i	$-0.576038\pi$
$521$	−5627.65	−0.473228	−0.236614	+	0.971604i	$0.576038\pi$
$522$	0	0
$523$	305.493	0.0255416	0.0127708	−	0.999918i	$-0.495935\pi$
$523$	305.493	0.0255416	0.0127708	+	0.999918i	$0.495935\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	40921.5	3.38248
$528$	0	0
$529$	5648.34	0.464235
$530$	0	0
$531$	0	0
$532$	0	0
$533$	27629.8	2.24537
$534$	0	0
$535$	14858.6	1.20073
$536$	0	0
$537$	0	0
$538$	0	0
$539$	361.661	0.0289014
$540$	0	0
$541$	−12789.3	−1.01636	−0.508182	−	0.861249i	$-0.669682\pi$
$541$	−12789.3	−1.01636	−0.508182	+	0.861249i	$0.669682\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7211.46	−0.566798
$546$	0	0
$547$	9612.58	0.751379	0.375690	−	0.926746i	$-0.377406\pi$
$547$	9612.58	0.751379	0.375690	+	0.926746i	$0.377406\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−13988.4	−1.08153
$552$	0	0
$553$	8005.25	0.615583
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8702.82	0.662029	0.331015	−	0.943626i	$-0.392609\pi$
$557$	8702.82	0.662029	0.331015	+	0.943626i	$0.392609\pi$
$558$	0	0
$559$	23369.4	1.76819
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25350.3	1.89767	0.948836	−	0.315768i	$-0.102262\pi$
$563$	25350.3	1.89767	0.948836	+	0.315768i	$0.102262\pi$
$564$	0	0
$565$	−7646.43	−0.569359
$566$	0	0
$567$	0	0
$568$	0	0
$569$	102.151	0.00752618	0.00376309	−	0.999993i	$-0.498802\pi$
$569$	102.151	0.00752618	0.00376309	+	0.999993i	$0.498802\pi$
$570$	0	0
$571$	−24948.8	−1.82850	−0.914252	−	0.405145i	$-0.867221\pi$
$571$	−24948.8	−1.82850	−0.914252	+	0.405145i	$0.867221\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−14891.7	−1.08005
$576$	0	0
$577$	16478.1	1.18889	0.594446	−	0.804135i	$-0.297371\pi$
$577$	16478.1	1.18889	0.594446	+	0.804135i	$0.297371\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−343.980	−0.0245623
$582$	0	0
$583$	2819.44	0.200290
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11868.7	0.834536	0.417268	−	0.908784i	$-0.362988\pi$
$587$	11868.7	0.834536	0.417268	+	0.908784i	$0.362988\pi$
$588$	0	0
$589$	−34460.0	−2.41070
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2932.17	−0.203052	−0.101526	−	0.994833i	$-0.532373\pi$
$593$	−2932.17	−0.203052	−0.101526	+	0.994833i	$0.532373\pi$
$594$	0	0
$595$	−14903.9	−1.02689
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−19700.9	−1.34383	−0.671917	−	0.740626i	$-0.734529\pi$
$599$	−19700.9	−1.34383	−0.671917	+	0.740626i	$0.734529\pi$
$600$	0	0
$601$	24055.7	1.63270	0.816350	−	0.577557i	$-0.195994\pi$
$601$	24055.7	1.63270	0.816350	+	0.577557i	$0.195994\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−19634.0	−1.31940
$606$	0	0
$607$	−17230.4	−1.15216	−0.576081	−	0.817393i	$-0.695419\pi$
$607$	−17230.4	−1.15216	−0.576081	+	0.817393i	$0.695419\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−23350.1	−1.54606
$612$	0	0
$613$	−20182.8	−1.32982	−0.664908	−	0.746925i	$-0.731529\pi$
$613$	−20182.8	−1.32982	−0.664908	+	0.746925i	$0.731529\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6973.78	0.455030	0.227515	−	0.973775i	$-0.426940\pi$
$617$	6973.78	0.455030	0.227515	+	0.973775i	$0.426940\pi$
$618$	0	0
$619$	−2633.76	−0.171018	−0.0855088	−	0.996337i	$-0.527252\pi$
$619$	−2633.76	−0.171018	−0.0855088	+	0.996337i	$0.527252\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2336.93	0.150284
$624$	0	0
$625$	−17123.4	−1.09590
$626$	0	0
$627$	0	0
$628$	0	0
$629$	46735.4	2.96258
$630$	0	0
$631$	16214.3	1.02295	0.511474	−	0.859299i	$-0.329100\pi$
$631$	16214.3	1.02295	0.511474	+	0.859299i	$0.329100\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8302.78	−0.518875
$636$	0	0
$637$	−3308.64	−0.205798
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8501.01	0.523822	0.261911	−	0.965092i	$-0.415647\pi$
$641$	8501.01	0.523822	0.261911	+	0.965092i	$0.415647\pi$
$642$	0	0
$643$	10440.0	0.640301	0.320151	−	0.947367i	$-0.396266\pi$
$643$	10440.0	0.640301	0.320151	+	0.947367i	$0.396266\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24887.7	−1.51227	−0.756133	−	0.654418i	$-0.772914\pi$
$647$	−24887.7	−1.51227	−0.756133	+	0.654418i	$0.772914\pi$
$648$	0	0
$649$	3234.22	0.195615
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7687.17	0.460677	0.230339	−	0.973111i	$-0.426017\pi$
$653$	7687.17	0.460677	0.230339	+	0.973111i	$0.426017\pi$
$654$	0	0
$655$	35996.7	2.14734
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13833.6	0.817723	0.408862	−	0.912596i	$-0.365926\pi$
$659$	13833.6	0.817723	0.408862	+	0.912596i	$0.365926\pi$
$660$	0	0
$661$	−28253.7	−1.66255	−0.831273	−	0.555864i	$-0.812388\pi$
$661$	−28253.7	−1.66255	−0.831273	+	0.555864i	$0.812388\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12550.6	0.731867
$666$	0	0
$667$	16016.9	0.929800
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5224.87	0.300602
$672$	0	0
$673$	22016.0	1.26100	0.630500	−	0.776189i	$-0.282850\pi$
$673$	22016.0	1.26100	0.630500	+	0.776189i	$0.282850\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10816.8	−0.614069	−0.307034	−	0.951698i	$-0.599337\pi$
$677$	−10816.8	−0.614069	−0.307034	+	0.951698i	$0.599337\pi$
$678$	0	0
$679$	−2779.33	−0.157085
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16676.4	−0.934270	−0.467135	−	0.884186i	$-0.654714\pi$
$683$	−16676.4	−0.934270	−0.467135	+	0.884186i	$0.654714\pi$
$684$	0	0
$685$	4281.85	0.238833
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−25793.6	−1.42621
$690$	0	0
$691$	−12840.7	−0.706920	−0.353460	−	0.935450i	$-0.614995\pi$
$691$	−12840.7	−0.706920	−0.353460	+	0.935450i	$0.614995\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14548.5	0.794039
$696$	0	0
$697$	56643.0	3.07820
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17635.0	0.950166	0.475083	−	0.879941i	$-0.342418\pi$
$701$	17635.0	0.950166	0.475083	+	0.879941i	$0.342418\pi$
$702$	0	0
$703$	−39356.0	−2.11143
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3081.03	−0.163896
$708$	0	0
$709$	−8031.26	−0.425417	−0.212708	−	0.977116i	$-0.568228\pi$
$709$	−8031.26	−0.425417	−0.212708	+	0.977116i	$0.568228\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39457.2	2.07249
$714$	0	0
$715$	−7665.47	−0.400940
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−929.655	−0.0482202	−0.0241101	−	0.999709i	$-0.507675\pi$
$719$	−929.655	−0.0482202	−0.0241101	+	0.999709i	$0.507675\pi$
$720$	0	0
$721$	−11906.6	−0.615014
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−13388.4	−0.685838
$726$	0	0
$727$	−5520.10	−0.281608	−0.140804	−	0.990037i	$-0.544969\pi$
$727$	−5520.10	−0.281608	−0.140804	+	0.990037i	$0.544969\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	47908.8	2.42404
$732$	0	0
$733$	−8128.99	−0.409620	−0.204810	−	0.978802i	$-0.565658\pi$
$733$	−8128.99	−0.409620	−0.204810	+	0.978802i	$0.565658\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1955.58	−0.0977403
$738$	0	0
$739$	−4945.82	−0.246191	−0.123095	−	0.992395i	$-0.539282\pi$
$739$	−4945.82	−0.246191	−0.123095	+	0.992395i	$0.539282\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31743.1	−1.56735	−0.783675	−	0.621171i	$-0.786657\pi$
$743$	−31743.1	−1.56735	−0.783675	+	0.621171i	$0.786657\pi$
$744$	0	0
$745$	38631.8	1.89981
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6762.31	0.329892
$750$	0	0
$751$	−18434.0	−0.895695	−0.447848	−	0.894110i	$-0.647809\pi$
$751$	−18434.0	−0.895695	−0.447848	+	0.894110i	$0.647809\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−48081.6	−2.31771
$756$	0	0
$757$	−3179.53	−0.152658	−0.0763289	−	0.997083i	$-0.524320\pi$
$757$	−3179.53	−0.152658	−0.0763289	+	0.997083i	$0.524320\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30346.5	−1.44555	−0.722773	−	0.691086i	$-0.757133\pi$
$761$	−30346.5	−1.44555	−0.722773	+	0.691086i	$0.757133\pi$
$762$	0	0
$763$	−3282.02	−0.155724
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−29588.2	−1.39292
$768$	0	0
$769$	−27931.7	−1.30981	−0.654904	−	0.755712i	$-0.727291\pi$
$769$	−27931.7	−1.30981	−0.654904	+	0.755712i	$0.727291\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38460.7	1.78957	0.894784	−	0.446499i	$-0.147330\pi$
$773$	38460.7	1.78957	0.894784	+	0.446499i	$0.147330\pi$
$774$	0	0
$775$	−32981.9	−1.52870
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−47699.2	−2.19384
$780$	0	0
$781$	2892.92	0.132544
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12698.4	0.577355
$786$	0	0
$787$	23991.8	1.08668	0.543339	−	0.839513i	$-0.317160\pi$
$787$	23991.8	1.08668	0.543339	+	0.839513i	$0.317160\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3479.98	−0.156427
$792$	0	0
$793$	−47799.5	−2.14049
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25623.9	−1.13883	−0.569414	−	0.822051i	$-0.692830\pi$
$797$	−25623.9	−1.13883	−0.569414	+	0.822051i	$0.692830\pi$
$798$	0	0
$799$	−47869.4	−2.11952
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1749.60	−0.0768893
$804$	0	0
$805$	−14370.6	−0.629189
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−32864.3	−1.42824	−0.714122	−	0.700022i	$-0.753174\pi$
$809$	−32864.3	−1.42824	−0.714122	+	0.700022i	$0.753174\pi$
$810$	0	0
$811$	34023.4	1.47315	0.736575	−	0.676356i	$-0.236442\pi$
$811$	34023.4	1.47315	0.736575	+	0.676356i	$0.236442\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14874.0	−0.639280
$816$	0	0
$817$	−40344.1	−1.72761
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38417.6	−1.63311	−0.816555	−	0.577267i	$-0.804119\pi$
$821$	−38417.6	−1.63311	−0.816555	+	0.577267i	$0.804119\pi$
$822$	0	0
$823$	26077.9	1.10452	0.552259	−	0.833672i	$-0.313766\pi$
$823$	26077.9	1.10452	0.552259	+	0.833672i	$0.313766\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35834.5	−1.50675	−0.753377	−	0.657588i	$-0.771577\pi$
$827$	−35834.5	−1.50675	−0.753377	+	0.657588i	$0.771577\pi$
$828$	0	0
$829$	−14282.0	−0.598353	−0.299177	−	0.954198i	$-0.596712\pi$
$829$	−14282.0	−0.598353	−0.299177	+	0.954198i	$0.596712\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6782.95	−0.282131
$834$	0	0
$835$	−54868.2	−2.27400
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15296.6	0.629437	0.314718	−	0.949185i	$-0.398090\pi$
$839$	15296.6	0.629437	0.314718	+	0.949185i	$0.398090\pi$
$840$	0	0
$841$	−9989.00	−0.409570
$842$	0	0
$843$	0	0
$844$	0	0
$845$	36335.7	1.47927
$846$	0	0
$847$	−8935.66	−0.362495
$848$	0	0
$849$	0	0
$850$	0	0
$851$	45063.1	1.81521
$852$	0	0
$853$	36704.7	1.47333	0.736663	−	0.676260i	$-0.236401\pi$
$853$	36704.7	1.47333	0.736663	+	0.676260i	$0.236401\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27077.7	1.07930	0.539649	−	0.841890i	$-0.318557\pi$
$857$	27077.7	1.07930	0.539649	+	0.841890i	$0.318557\pi$
$858$	0	0
$859$	−2090.02	−0.0830157	−0.0415079	−	0.999138i	$-0.513216\pi$
$859$	−2090.02	−0.0830157	−0.0415079	+	0.999138i	$0.513216\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26267.8	1.03611	0.518056	−	0.855346i	$-0.326656\pi$
$863$	26267.8	1.03611	0.518056	+	0.855346i	$0.326656\pi$
$864$	0	0
$865$	−9807.60	−0.385512
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8440.77	0.329498
$870$	0	0
$871$	17890.5	0.695979
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1445.95	−0.0558653
$876$	0	0
$877$	−21598.4	−0.831614	−0.415807	−	0.909453i	$-0.636501\pi$
$877$	−21598.4	−0.831614	−0.415807	+	0.909453i	$0.636501\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34607.8	−1.32346	−0.661730	−	0.749742i	$-0.730177\pi$
$881$	−34607.8	−1.32346	−0.661730	+	0.749742i	$0.730177\pi$
$882$	0	0
$883$	−29039.5	−1.10675	−0.553374	−	0.832933i	$-0.686660\pi$
$883$	−29039.5	−1.10675	−0.553374	+	0.832933i	$0.686660\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13238.4	0.501131	0.250565	−	0.968100i	$-0.419384\pi$
$887$	13238.4	0.501131	0.250565	+	0.968100i	$0.419384\pi$
$888$	0	0
$889$	−3778.69	−0.142557
$890$	0	0
$891$	0	0
$892$	0	0
$893$	40310.9	1.51058
$894$	0	0
$895$	−20448.6	−0.763710
$896$	0	0
$897$	0	0
$898$	0	0
$899$	35474.0	1.31604
$900$	0	0
$901$	−52878.6	−1.95521
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−32345.0	−1.18805
$906$	0	0
$907$	18301.1	0.669986	0.334993	−	0.942221i	$-0.391266\pi$
$907$	18301.1	0.669986	0.334993	+	0.942221i	$0.391266\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	32499.5	1.18195	0.590975	−	0.806690i	$-0.298743\pi$
$911$	32499.5	1.18195	0.590975	+	0.806690i	$0.298743\pi$
$912$	0	0
$913$	−362.694	−0.0131472
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16382.5	0.589966
$918$	0	0
$919$	−30535.9	−1.09607	−0.548035	−	0.836456i	$-0.684624\pi$
$919$	−30535.9	−1.09607	−0.548035	+	0.836456i	$0.684624\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−26465.8	−0.943806
$924$	0	0
$925$	−37667.9	−1.33893
$926$	0	0
$927$	0	0
$928$	0	0
$929$	31312.6	1.10585	0.552925	−	0.833231i	$-0.313512\pi$
$929$	31312.6	1.10585	0.552925	+	0.833231i	$0.313512\pi$
$930$	0	0
$931$	5711.93	0.201075
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15714.7	−0.549655
$936$	0	0
$937$	20032.8	0.698444	0.349222	−	0.937040i	$-0.386446\pi$
$937$	20032.8	0.698444	0.349222	+	0.937040i	$0.386446\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8355.34	−0.289454	−0.144727	−	0.989472i	$-0.546230\pi$
$941$	−8355.34	−0.289454	−0.144727	+	0.989472i	$0.546230\pi$
$942$	0	0
$943$	54616.2	1.88605
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6758.68	0.231919	0.115960	−	0.993254i	$-0.463006\pi$
$947$	6758.68	0.231919	0.115960	+	0.993254i	$0.463006\pi$
$948$	0	0
$949$	16006.2	0.547505
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21459.4	−0.729421	−0.364711	−	0.931121i	$-0.618832\pi$
$953$	−21459.4	−0.729421	−0.364711	+	0.931121i	$0.618832\pi$
$954$	0	0
$955$	40074.0	1.35787
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1948.72	0.0656178
$960$	0	0
$961$	57598.2	1.93341
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−11541.2	−0.385000
$966$	0	0
$967$	19262.1	0.640566	0.320283	−	0.947322i	$-0.396222\pi$
$967$	19262.1	0.640566	0.320283	+	0.947322i	$0.396222\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30865.8	1.02011	0.510056	−	0.860141i	$-0.329625\pi$
$971$	30865.8	1.02011	0.510056	+	0.860141i	$0.329625\pi$
$972$	0	0
$973$	6621.20	0.218156
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42508.2	1.39197	0.695986	−	0.718055i	$-0.254967\pi$
$977$	42508.2	1.39197	0.695986	+	0.718055i	$0.254967\pi$
$978$	0	0
$979$	2464.07	0.0804413
$980$	0	0
$981$	0	0
$982$	0	0
$983$	875.203	0.0283974	0.0141987	−	0.999899i	$-0.495480\pi$
$983$	875.203	0.0283974	0.0141987	+	0.999899i	$0.495480\pi$
$984$	0	0
$985$	26746.8	0.865201
$986$	0	0
$987$	0	0
$988$	0	0
$989$	46194.5	1.48524
$990$	0	0
$991$	−16092.7	−0.515844	−0.257922	−	0.966166i	$-0.583038\pi$
$991$	−16092.7	−0.515844	−0.257922	+	0.966166i	$0.583038\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	29179.3	0.929694
$996$	0	0
$997$	−20580.7	−0.653758	−0.326879	−	0.945066i	$-0.605997\pi$
$997$	−20580.7	−0.653758	−0.326879	+	0.945066i	$0.605997\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.a.bh.1.2		2
3.2	odd	2		1008.4.a.z.1.1		2
4.3	odd	2		504.4.a.o.1.2	yes	2
12.11	even	2		504.4.a.k.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.a.k.1.1	✓	2	12.11	even	2
504.4.a.o.1.2	yes	2	4.3	odd	2
1008.4.a.z.1.1		2	3.2	odd	2
1008.4.a.bh.1.2		2	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q+15.3808 q^{5} +7.00000 q^{7} +O(q^{10})\)	\(q+15.3808 q^{5} +7.00000 q^{7} +7.38083 q^{11} -67.5233 q^{13} -138.427 q^{17} +116.570 q^{19} -133.474 q^{23} +111.570 q^{25} -120.000 q^{29} -295.617 q^{31} +107.666 q^{35} -337.617 q^{37} -409.189 q^{41} -346.093 q^{43} +345.808 q^{47} +49.0000 q^{49} +381.995 q^{53} +113.523 q^{55} +438.192 q^{59} +707.897 q^{61} -1038.56 q^{65} -264.953 q^{67} +391.951 q^{71} -237.047 q^{73} +51.6658 q^{77} +1143.61 q^{79} -49.1400 q^{83} -2129.13 q^{85} +333.847 q^{89} -472.663 q^{91} +1792.94 q^{95} -397.047 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{5} + 14 q^{7}+O(q^{10}) \) 2 * q + 12 * q^5 + 14 * q^7	\( 2 q + 12 q^{5} + 14 q^{7} - 4 q^{11} - 60 q^{13} - 108 q^{17} + 8 q^{19} + 52 q^{23} - 2 q^{25} - 240 q^{29} - 216 q^{31} + 84 q^{35} - 300 q^{37} - 612 q^{41} - 392 q^{43} + 504 q^{47} + 98 q^{49} - 24 q^{53} + 152 q^{55} + 1064 q^{59} + 140 q^{61} - 1064 q^{65} - 680 q^{67} + 540 q^{71} - 324 q^{73} - 28 q^{77} + 336 q^{79} + 352 q^{83} - 2232 q^{85} - 852 q^{89} - 420 q^{91} + 2160 q^{95} - 644 q^{97}+O(q^{100}) \) 2 * q + 12 * q^5 + 14 * q^7 - 4 * q^11 - 60 * q^13 - 108 * q^17 + 8 * q^19 + 52 * q^23 - 2 * q^25 - 240 * q^29 - 216 * q^31 + 84 * q^35 - 300 * q^37 - 612 * q^41 - 392 * q^43 + 504 * q^47 + 98 * q^49 - 24 * q^53 + 152 * q^55 + 1064 * q^59 + 140 * q^61 - 1064 * q^65 - 680 * q^67 + 540 * q^71 - 324 * q^73 - 28 * q^77 + 336 * q^79 + 352 * q^83 - 2232 * q^85 - 852 * q^89 - 420 * q^91 + 2160 * q^95 - 644 * q^97

Introduction

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