LMFDB - Embedded newform 387.4.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,4,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, 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("387.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	387.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:	$22.8337391722$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 $ x^6 - 32x^4 - 16x^3 + 251x^2 + 276x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.299707$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29971 q^{2} -6.31076 q^{4} -20.4116 q^{5} +29.9522 q^{7} +18.5998 q^{8} +O(q^{10})$	\(q-1.29971 q^{2} -6.31076 q^{4} -20.4116 q^{5} +29.9522 q^{7} +18.5998 q^{8} +26.5291 q^{10} +22.8719 q^{11} -44.4397 q^{13} -38.9292 q^{14} +26.3118 q^{16} +13.0970 q^{17} +5.41527 q^{19} +128.813 q^{20} -29.7268 q^{22} +175.226 q^{23} +291.634 q^{25} +57.7586 q^{26} -189.021 q^{28} -165.972 q^{29} -155.581 q^{31} -182.996 q^{32} -17.0222 q^{34} -611.374 q^{35} -95.3991 q^{37} -7.03827 q^{38} -379.652 q^{40} -189.928 q^{41} -43.0000 q^{43} -144.339 q^{44} -227.743 q^{46} +37.2349 q^{47} +554.137 q^{49} -379.039 q^{50} +280.448 q^{52} -559.862 q^{53} -466.852 q^{55} +557.106 q^{56} +215.715 q^{58} +82.3042 q^{59} -640.304 q^{61} +202.210 q^{62} +27.3469 q^{64} +907.085 q^{65} -509.592 q^{67} -82.6519 q^{68} +794.607 q^{70} -792.932 q^{71} +612.727 q^{73} +123.991 q^{74} -34.1745 q^{76} +685.065 q^{77} +237.047 q^{79} -537.066 q^{80} +246.850 q^{82} -418.683 q^{83} -267.330 q^{85} +55.8874 q^{86} +425.413 q^{88} -113.788 q^{89} -1331.07 q^{91} -1105.81 q^{92} -48.3944 q^{94} -110.534 q^{95} +1649.00 q^{97} -720.216 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8	$ 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8} + 57 q^{10} + 28 q^{11} + 56 q^{13} + 184 q^{14} - 54 q^{16} - 19 q^{17} - 75 q^{19} - 135 q^{20} - 504 q^{22} - 131 q^{23} + 105 q^{25} - 44 q^{26} - 404 q^{28} - 515 q^{29} + 237 q^{31} - 558 q^{32} - 107 q^{34} - 198 q^{35} + 269 q^{37} - 527 q^{38} + 613 q^{40} - 471 q^{41} - 258 q^{43} + 428 q^{44} - 67 q^{46} - 415 q^{47} + 350 q^{49} - 1335 q^{50} - 8 q^{52} - 450 q^{53} - 1732 q^{55} + 780 q^{56} - 1055 q^{58} - 356 q^{59} - 1328 q^{61} - 1603 q^{62} + 466 q^{64} + 62 q^{65} - 632 q^{67} - 571 q^{68} - 1902 q^{70} + 144 q^{71} + 864 q^{73} - 1207 q^{74} + 1005 q^{76} - 2660 q^{77} - 1613 q^{79} - 2399 q^{80} + 1673 q^{82} + 682 q^{83} + 84 q^{85} + 258 q^{86} - 608 q^{88} - 3378 q^{89} - 3900 q^{91} - 3491 q^{92} + 3197 q^{94} + 79 q^{95} - 55 q^{97} - 2398 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8 + 57 * q^10 + 28 * q^11 + 56 * q^13 + 184 * q^14 - 54 * q^16 - 19 * q^17 - 75 * q^19 - 135 * q^20 - 504 * q^22 - 131 * q^23 + 105 * q^25 - 44 * q^26 - 404 * q^28 - 515 * q^29 + 237 * q^31 - 558 * q^32 - 107 * q^34 - 198 * q^35 + 269 * q^37 - 527 * q^38 + 613 * q^40 - 471 * q^41 - 258 * q^43 + 428 * q^44 - 67 * q^46 - 415 * q^47 + 350 * q^49 - 1335 * q^50 - 8 * q^52 - 450 * q^53 - 1732 * q^55 + 780 * q^56 - 1055 * q^58 - 356 * q^59 - 1328 * q^61 - 1603 * q^62 + 466 * q^64 + 62 * q^65 - 632 * q^67 - 571 * q^68 - 1902 * q^70 + 144 * q^71 + 864 * q^73 - 1207 * q^74 + 1005 * q^76 - 2660 * q^77 - 1613 * q^79 - 2399 * q^80 + 1673 * q^82 + 682 * q^83 + 84 * q^85 + 258 * q^86 - 608 * q^88 - 3378 * q^89 - 3900 * q^91 - 3491 * q^92 + 3197 * q^94 + 79 * q^95 - 55 * q^97 - 2398 * q^98

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 $

$2$	−1.29971	−0.459516	−0.229758	−	0.973248i	$-0.573793\pi$
$2$	−1.29971	−0.459516	−0.229758	+	0.973248i	$0.573793\pi$
$3$	0	0
$3$	0	0
$4$	−6.31076	−0.788845
$5$	−20.4116	−1.82567	−0.912835	−	0.408329i	$-0.866112\pi$
$5$	−20.4116	−1.82567	−0.912835	+	0.408329i	$0.866112\pi$
$6$	0	0
$7$	29.9522	1.61727	0.808635	−	0.588311i	$-0.200207\pi$
$7$	29.9522	1.61727	0.808635	+	0.588311i	$0.200207\pi$
$8$	18.5998	0.822003
$9$	0	0
$10$	26.5291	0.838924
$11$	22.8719	0.626922	0.313461	−	0.949601i	$-0.398512\pi$
$11$	22.8719	0.626922	0.313461	+	0.949601i	$0.398512\pi$
$12$	0	0
$13$	−44.4397	−0.948104	−0.474052	−	0.880497i	$-0.657209\pi$
$13$	−44.4397	−0.948104	−0.474052	+	0.880497i	$0.657209\pi$
$14$	−38.9292	−0.743161
$15$	0	0
$16$	26.3118	0.411122
$17$	13.0970	0.186852	0.0934260	−	0.995626i	$-0.470218\pi$
$17$	13.0970	0.186852	0.0934260	+	0.995626i	$0.470218\pi$
$18$	0	0
$19$	5.41527	0.0653868	0.0326934	−	0.999465i	$-0.489592\pi$
$19$	5.41527	0.0653868	0.0326934	+	0.999465i	$0.489592\pi$
$20$	128.813	1.44017
$21$	0	0
$22$	−29.7268	−0.288081
$23$	175.226	1.58857	0.794287	−	0.607542i	$-0.207845\pi$
$23$	175.226	1.58857	0.794287	+	0.607542i	$0.207845\pi$
$24$	0	0
$25$	291.634	2.33307
$26$	57.7586	0.435669
$27$	0	0
$28$	−189.021	−1.27578
$29$	−165.972	−1.06277	−0.531383	−	0.847132i	$-0.678328\pi$
$29$	−165.972	−1.06277	−0.531383	+	0.847132i	$0.678328\pi$
$30$	0	0
$31$	−155.581	−0.901395	−0.450697	−	0.892677i	$-0.648825\pi$
$31$	−155.581	−0.901395	−0.450697	+	0.892677i	$0.648825\pi$
$32$	−182.996	−1.01092
$33$	0	0
$34$	−17.0222	−0.0858615
$35$	−611.374	−2.95260
$36$	0	0
$37$	−95.3991	−0.423879	−0.211939	−	0.977283i	$-0.567978\pi$
$37$	−95.3991	−0.423879	−0.211939	+	0.977283i	$0.567978\pi$
$38$	−7.03827	−0.0300463
$39$	0	0
$40$	−379.652	−1.50071
$41$	−189.928	−0.723457	−0.361728	−	0.932284i	$-0.617813\pi$
$41$	−189.928	−0.723457	−0.361728	+	0.932284i	$0.617813\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	−144.339	−0.494544
$45$	0	0
$46$	−227.743	−0.729975
$47$	37.2349	0.115559	0.0577794	−	0.998329i	$-0.481598\pi$
$47$	37.2349	0.115559	0.0577794	+	0.998329i	$0.481598\pi$
$48$	0	0
$49$	554.137	1.61556
$50$	−379.039	−1.07208
$51$	0	0
$52$	280.448	0.747907
$53$	−559.862	−1.45100	−0.725500	−	0.688223i	$-0.758391\pi$
$53$	−559.862	−1.45100	−0.725500	+	0.688223i	$0.758391\pi$
$54$	0	0
$55$	−466.852	−1.14455
$56$	557.106	1.32940
$57$	0	0
$58$	215.715	0.488358
$59$	82.3042	0.181612	0.0908059	−	0.995869i	$-0.471056\pi$
$59$	82.3042	0.181612	0.0908059	+	0.995869i	$0.471056\pi$
$60$	0	0
$61$	−640.304	−1.34398	−0.671988	−	0.740562i	$-0.734559\pi$
$61$	−640.304	−1.34398	−0.671988	+	0.740562i	$0.734559\pi$
$62$	202.210	0.414205
$63$	0	0
$64$	27.3469	0.0534120
$65$	907.085	1.73092
$66$	0	0
$67$	−509.592	−0.929203	−0.464602	−	0.885520i	$-0.653802\pi$
$67$	−509.592	−0.929203	−0.464602	+	0.885520i	$0.653802\pi$
$68$	−82.6519	−0.147397
$69$	0	0
$70$	794.607	1.35677
$71$	−792.932	−1.32540	−0.662702	−	0.748883i	$-0.730590\pi$
$71$	−792.932	−1.32540	−0.662702	+	0.748883i	$0.730590\pi$
$72$	0	0
$73$	612.727	0.982387	0.491194	−	0.871050i	$-0.336561\pi$
$73$	612.727	0.982387	0.491194	+	0.871050i	$0.336561\pi$
$74$	123.991	0.194779
$75$	0	0
$76$	−34.1745	−0.0515800
$77$	685.065	1.01390
$78$	0	0
$79$	237.047	0.337594	0.168797	−	0.985651i	$-0.446012\pi$
$79$	237.047	0.337594	0.168797	+	0.985651i	$0.446012\pi$
$80$	−537.066	−0.750573
$81$	0	0
$82$	246.850	0.332440
$83$	−418.683	−0.553691	−0.276846	−	0.960914i	$-0.589289\pi$
$83$	−418.683	−0.553691	−0.276846	+	0.960914i	$0.589289\pi$
$84$	0	0
$85$	−267.330	−0.341130
$86$	55.8874	0.0700755
$87$	0	0
$88$	425.413	0.515331
$89$	−113.788	−0.135523	−0.0677615	−	0.997702i	$-0.521586\pi$
$89$	−113.788	−0.135523	−0.0677615	+	0.997702i	$0.521586\pi$
$90$	0	0
$91$	−1331.07	−1.53334
$92$	−1105.81	−1.25314
$93$	0	0
$94$	−48.3944	−0.0531011
$95$	−110.534	−0.119375
$96$	0	0
$97$	1649.00	1.72609	0.863045	−	0.505127i	$-0.168554\pi$
$97$	1649.00	1.72609	0.863045	+	0.505127i	$0.168554\pi$
$98$	−720.216	−0.742376
$99$	0	0
$100$	−1840.43	−1.84043
$101$	165.065	0.162619	0.0813097	−	0.996689i	$-0.474090\pi$
$101$	165.065	0.162619	0.0813097	+	0.996689i	$0.474090\pi$
$102$	0	0
$103$	−1665.32	−1.59309	−0.796547	−	0.604576i	$-0.793342\pi$
$103$	−1665.32	−1.59309	−0.796547	+	0.604576i	$0.793342\pi$
$104$	−826.569	−0.779344
$105$	0	0
$106$	727.657	0.666757
$107$	631.683	0.570721	0.285360	−	0.958420i	$-0.407887\pi$
$107$	631.683	0.570721	0.285360	+	0.958420i	$0.407887\pi$
$108$	0	0
$109$	839.368	0.737586	0.368793	−	0.929512i	$-0.379771\pi$
$109$	839.368	0.737586	0.368793	+	0.929512i	$0.379771\pi$
$110$	606.771	0.525940
$111$	0	0
$112$	788.097	0.664895
$113$	1080.84	0.899796	0.449898	−	0.893080i	$-0.351460\pi$
$113$	1080.84	0.899796	0.449898	+	0.893080i	$0.351460\pi$
$114$	0	0
$115$	−3576.65	−2.90021
$116$	1047.41	0.838358
$117$	0	0
$118$	−106.971	−0.0834535
$119$	392.284	0.302190
$120$	0	0
$121$	−807.876	−0.606969
$122$	832.207	0.617578
$123$	0	0
$124$	981.837	0.711061
$125$	−3401.26	−2.43375
$126$	0	0
$127$	594.837	0.415616	0.207808	−	0.978170i	$-0.433367\pi$
$127$	594.837	0.415616	0.207808	+	0.978170i	$0.433367\pi$
$128$	1428.43	0.986376
$129$	0	0
$130$	−1178.95	−0.795387
$131$	−1987.90	−1.32583	−0.662916	−	0.748694i	$-0.730681\pi$
$131$	−1987.90	−1.32583	−0.662916	+	0.748694i	$0.730681\pi$
$132$	0	0
$133$	162.200	0.105748
$134$	662.321	0.426984
$135$	0	0
$136$	243.601	0.153593
$137$	−1717.01	−1.07076	−0.535379	−	0.844612i	$-0.679831\pi$
$137$	−1717.01	−1.07076	−0.535379	+	0.844612i	$0.679831\pi$
$138$	0	0
$139$	−1735.09	−1.05876	−0.529381	−	0.848384i	$-0.677576\pi$
$139$	−1735.09	−1.05876	−0.529381	+	0.848384i	$0.677576\pi$
$140$	3858.23	2.32914
$141$	0	0
$142$	1030.58	0.609044
$143$	−1016.42	−0.594387
$144$	0	0
$145$	3387.76	1.94026
$146$	−796.366	−0.451423
$147$	0	0
$148$	602.041	0.334375
$149$	−456.600	−0.251048	−0.125524	−	0.992091i	$-0.540061\pi$
$149$	−456.600	−0.251048	−0.125524	+	0.992091i	$0.540061\pi$
$150$	0	0
$151$	−483.618	−0.260638	−0.130319	−	0.991472i	$-0.541600\pi$
$151$	−483.618	−0.260638	−0.130319	+	0.991472i	$0.541600\pi$
$152$	100.723	0.0537481
$153$	0	0
$154$	−890.384	−0.465904
$155$	3175.67	1.64565
$156$	0	0
$157$	2300.64	1.16950	0.584749	−	0.811214i	$-0.301193\pi$
$157$	2300.64	1.16950	0.584749	+	0.811214i	$0.301193\pi$
$158$	−308.092	−0.155130
$159$	0	0
$160$	3735.24	1.84561
$161$	5248.42	2.56915
$162$	0	0
$163$	314.362	0.151060	0.0755299	−	0.997144i	$-0.475935\pi$
$163$	314.362	0.151060	0.0755299	+	0.997144i	$0.475935\pi$
$164$	1198.59	0.570695
$165$	0	0
$166$	544.165	0.254430
$167$	282.441	0.130874	0.0654369	−	0.997857i	$-0.479156\pi$
$167$	282.441	0.130874	0.0654369	+	0.997857i	$0.479156\pi$
$168$	0	0
$169$	−222.114	−0.101099
$170$	347.451	0.156755
$171$	0	0
$172$	271.363	0.120298
$173$	−1774.10	−0.779667	−0.389834	−	0.920885i	$-0.627468\pi$
$173$	−1774.10	−0.779667	−0.389834	+	0.920885i	$0.627468\pi$
$174$	0	0
$175$	8735.09	3.77320
$176$	601.801	0.257741
$177$	0	0
$178$	147.892	0.0622750
$179$	−890.113	−0.371677	−0.185839	−	0.982580i	$-0.559500\pi$
$179$	−890.113	−0.371677	−0.185839	+	0.982580i	$0.559500\pi$
$180$	0	0
$181$	−2883.82	−1.18427	−0.592135	−	0.805838i	$-0.701715\pi$
$181$	−2883.82	−1.18427	−0.592135	+	0.805838i	$0.701715\pi$
$182$	1730.00	0.704594
$183$	0	0
$184$	3259.17	1.30581
$185$	1947.25	0.773863
$186$	0	0
$187$	299.553	0.117142
$188$	−234.980	−0.0911580
$189$	0	0
$190$	143.662	0.0548546
$191$	−1120.48	−0.424478	−0.212239	−	0.977218i	$-0.568076\pi$
$191$	−1120.48	−0.424478	−0.212239	+	0.977218i	$0.568076\pi$
$192$	0	0
$193$	−2777.72	−1.03598	−0.517992	−	0.855385i	$-0.673320\pi$
$193$	−2777.72	−1.03598	−0.517992	+	0.855385i	$0.673320\pi$
$194$	−2143.22	−0.793166
$195$	0	0
$196$	−3497.03	−1.27443
$197$	1266.30	0.457971	0.228985	−	0.973430i	$-0.426459\pi$
$197$	1266.30	0.457971	0.228985	+	0.973430i	$0.426459\pi$
$198$	0	0
$199$	−5113.40	−1.82150	−0.910752	−	0.412955i	$-0.864497\pi$
$199$	−5113.40	−1.82150	−0.910752	+	0.412955i	$0.864497\pi$
$200$	5424.33	1.91779
$201$	0	0
$202$	−214.536	−0.0747262
$203$	−4971.23	−1.71878
$204$	0	0
$205$	3876.73	1.32079
$206$	2164.43	0.732052
$207$	0	0
$208$	−1169.29	−0.389786
$209$	123.858	0.0409924
$210$	0	0
$211$	−1318.35	−0.430138	−0.215069	−	0.976599i	$-0.568998\pi$
$211$	−1318.35	−0.430138	−0.215069	+	0.976599i	$0.568998\pi$
$212$	3533.15	1.14461
$213$	0	0
$214$	−821.003	−0.262255
$215$	877.699	0.278412
$216$	0	0
$217$	−4660.01	−1.45780
$218$	−1090.93	−0.338932
$219$	0	0
$220$	2946.19	0.902874
$221$	−582.026	−0.177155
$222$	0	0
$223$	−812.504	−0.243988	−0.121994	−	0.992531i	$-0.538929\pi$
$223$	−812.504	−0.243988	−0.121994	+	0.992531i	$0.538929\pi$
$224$	−5481.14	−1.63493
$225$	0	0
$226$	−1404.78	−0.413470
$227$	−1890.51	−0.552763	−0.276382	−	0.961048i	$-0.589135\pi$
$227$	−1890.51	−0.552763	−0.276382	+	0.961048i	$0.589135\pi$
$228$	0	0
$229$	1368.43	0.394884	0.197442	−	0.980315i	$-0.436737\pi$
$229$	1368.43	0.394884	0.197442	+	0.980315i	$0.436737\pi$
$230$	4648.60	1.33269
$231$	0	0
$232$	−3087.05	−0.873597
$233$	3535.33	0.994023	0.497012	−	0.867744i	$-0.334431\pi$
$233$	3535.33	0.994023	0.497012	+	0.867744i	$0.334431\pi$
$234$	0	0
$235$	−760.023	−0.210972
$236$	−519.402	−0.143264
$237$	0	0
$238$	−509.854	−0.138861
$239$	−1515.24	−0.410095	−0.205047	−	0.978752i	$-0.565735\pi$
$239$	−1515.24	−0.410095	−0.205047	+	0.978752i	$0.565735\pi$
$240$	0	0
$241$	−6717.46	−1.79548	−0.897738	−	0.440530i	$-0.854791\pi$
$241$	−6717.46	−1.79548	−0.897738	+	0.440530i	$0.854791\pi$
$242$	1050.00	0.278912
$243$	0	0
$244$	4040.80	1.06019
$245$	−11310.8	−2.94948
$246$	0	0
$247$	−240.653	−0.0619934
$248$	−2893.78	−0.740949
$249$	0	0
$250$	4420.65	1.11835
$251$	2291.38	0.576217	0.288109	−	0.957598i	$-0.406974\pi$
$251$	2291.38	0.576217	0.288109	+	0.957598i	$0.406974\pi$
$252$	0	0
$253$	4007.76	0.995912
$254$	−773.114	−0.190982
$255$	0	0
$256$	−2075.31	−0.506668
$257$	2236.18	0.542760	0.271380	−	0.962472i	$-0.412520\pi$
$257$	2236.18	0.542760	0.271380	+	0.962472i	$0.412520\pi$
$258$	0	0
$259$	−2857.42	−0.685526
$260$	−5724.40	−1.36543
$261$	0	0
$262$	2583.69	0.609241
$263$	6393.87	1.49910	0.749550	−	0.661948i	$-0.230270\pi$
$263$	6393.87	1.49910	0.749550	+	0.661948i	$0.230270\pi$
$264$	0	0
$265$	11427.7	2.64905
$266$	−210.812	−0.0485929
$267$	0	0
$268$	3215.92	0.732997
$269$	2688.95	0.609473	0.304736	−	0.952437i	$-0.401432\pi$
$269$	2688.95	0.609473	0.304736	+	0.952437i	$0.401432\pi$
$270$	0	0
$271$	−1057.95	−0.237143	−0.118572	−	0.992945i	$-0.537832\pi$
$271$	−1057.95	−0.237143	−0.118572	+	0.992945i	$0.537832\pi$
$272$	344.605	0.0768189
$273$	0	0
$274$	2231.61	0.492030
$275$	6670.22	1.46265
$276$	0	0
$277$	1904.17	0.413035	0.206518	−	0.978443i	$-0.433787\pi$
$277$	1904.17	0.413035	0.206518	+	0.978443i	$0.433787\pi$
$278$	2255.10	0.486518
$279$	0	0
$280$	−11371.4	−2.42705
$281$	−6520.37	−1.38424	−0.692122	−	0.721781i	$-0.743324\pi$
$281$	−6520.37	−1.38424	−0.692122	+	0.721781i	$0.743324\pi$
$282$	0	0
$283$	130.376	0.0273854	0.0136927	−	0.999906i	$-0.495641\pi$
$283$	130.376	0.0273854	0.0136927	+	0.999906i	$0.495641\pi$
$284$	5004.00	1.04554
$285$	0	0
$286$	1321.05	0.273130
$287$	−5688.76	−1.17002
$288$	0	0
$289$	−4741.47	−0.965086
$290$	−4403.09	−0.891581
$291$	0	0
$292$	−3866.77	−0.774951
$293$	321.150	0.0640333	0.0320166	−	0.999487i	$-0.489807\pi$
$293$	321.150	0.0640333	0.0320166	+	0.999487i	$0.489807\pi$
$294$	0	0
$295$	−1679.96	−0.331563
$296$	−1774.40	−0.348430
$297$	0	0
$298$	593.446	0.115361
$299$	−7787.00	−1.50613
$300$	0	0
$301$	−1287.95	−0.246631
$302$	628.562	0.119767
$303$	0	0
$304$	142.486	0.0268819
$305$	13069.6	2.45366
$306$	0	0
$307$	3944.87	0.733374	0.366687	−	0.930344i	$-0.380492\pi$
$307$	3944.87	0.733374	0.366687	+	0.930344i	$0.380492\pi$
$308$	−4323.28	−0.799811
$309$	0	0
$310$	−4127.44	−0.756202
$311$	−7046.04	−1.28471	−0.642354	−	0.766408i	$-0.722042\pi$
$311$	−7046.04	−1.28471	−0.642354	+	0.766408i	$0.722042\pi$
$312$	0	0
$313$	−230.283	−0.0415858	−0.0207929	−	0.999784i	$-0.506619\pi$
$313$	−230.283	−0.0415858	−0.0207929	+	0.999784i	$0.506619\pi$
$314$	−2990.16	−0.537403
$315$	0	0
$316$	−1495.95	−0.266309
$317$	−6574.02	−1.16477	−0.582387	−	0.812911i	$-0.697881\pi$
$317$	−6574.02	−1.16477	−0.582387	+	0.812911i	$0.697881\pi$
$318$	0	0
$319$	−3796.10	−0.666271
$320$	−558.195	−0.0975126
$321$	0	0
$322$	−6821.41	−1.18057
$323$	70.9237	0.0122177
$324$	0	0
$325$	−12960.1	−2.21199
$326$	−408.579	−0.0694144
$327$	0	0
$328$	−3532.62	−0.594683
$329$	1115.27	0.186890
$330$	0	0
$331$	8326.42	1.38266	0.691331	−	0.722538i	$-0.257025\pi$
$331$	8326.42	1.38266	0.691331	+	0.722538i	$0.257025\pi$
$332$	2642.21	0.436777
$333$	0	0
$334$	−367.091	−0.0601386
$335$	10401.6	1.69642
$336$	0	0
$337$	9021.52	1.45826	0.729130	−	0.684375i	$-0.239925\pi$
$337$	9021.52	1.45826	0.729130	+	0.684375i	$0.239925\pi$
$338$	288.684	0.0464566
$339$	0	0
$340$	1687.06	0.269099
$341$	−3558.44	−0.565104
$342$	0	0
$343$	6324.03	0.995526
$344$	−799.791	−0.125354
$345$	0	0
$346$	2305.81	0.358270
$347$	−133.107	−0.0205925	−0.0102962	−	0.999947i	$-0.503277\pi$
$347$	−133.107	−0.0205925	−0.0102962	+	0.999947i	$0.503277\pi$
$348$	0	0
$349$	−1322.67	−0.202868	−0.101434	−	0.994842i	$-0.532343\pi$
$349$	−1322.67	−0.202868	−0.101434	+	0.994842i	$0.532343\pi$
$350$	−11353.1	−1.73385
$351$	0	0
$352$	−4185.47	−0.633768
$353$	−12515.7	−1.88709	−0.943543	−	0.331251i	$-0.892529\pi$
$353$	−12515.7	−1.88709	−0.943543	+	0.331251i	$0.892529\pi$
$354$	0	0
$355$	16185.0	2.41975
$356$	718.091	0.106907
$357$	0	0
$358$	1156.89	0.170792
$359$	12654.2	1.86035	0.930174	−	0.367118i	$-0.119656\pi$
$359$	12654.2	1.86035	0.930174	+	0.367118i	$0.119656\pi$
$360$	0	0
$361$	−6829.67	−0.995725
$362$	3748.13	0.544191
$363$	0	0
$364$	8400.05	1.20957
$365$	−12506.7	−1.79351
$366$	0	0
$367$	−13307.8	−1.89281	−0.946407	−	0.322976i	$-0.895317\pi$
$367$	−13307.8	−1.89281	−0.946407	+	0.322976i	$0.895317\pi$
$368$	4610.52	0.653098
$369$	0	0
$370$	−2530.86	−0.355602
$371$	−16769.1	−2.34666
$372$	0	0
$373$	−6552.61	−0.909602	−0.454801	−	0.890593i	$-0.650290\pi$
$373$	−6552.61	−0.909602	−0.454801	+	0.890593i	$0.650290\pi$
$374$	−389.331	−0.0538284
$375$	0	0
$376$	692.561	0.0949896
$377$	7375.74	1.00761
$378$	0	0
$379$	11236.0	1.52284	0.761418	−	0.648262i	$-0.224504\pi$
$379$	11236.0	1.52284	0.761418	+	0.648262i	$0.224504\pi$
$380$	697.556	0.0941681
$381$	0	0
$382$	1456.30	0.195054
$383$	−6353.36	−0.847628	−0.423814	−	0.905749i	$-0.639309\pi$
$383$	−6353.36	−0.847628	−0.423814	+	0.905749i	$0.639309\pi$
$384$	0	0
$385$	−13983.3	−1.85105
$386$	3610.23	0.476051
$387$	0	0
$388$	−10406.5	−1.36162
$389$	−3788.91	−0.493844	−0.246922	−	0.969035i	$-0.579419\pi$
$389$	−3788.91	−0.493844	−0.246922	+	0.969035i	$0.579419\pi$
$390$	0	0
$391$	2294.94	0.296828
$392$	10306.8	1.32799
$393$	0	0
$394$	−1645.82	−0.210445
$395$	−4838.52	−0.616335
$396$	0	0
$397$	−2601.66	−0.328901	−0.164451	−	0.986385i	$-0.552585\pi$
$397$	−2601.66	−0.328901	−0.164451	+	0.986385i	$0.552585\pi$
$398$	6645.92	0.837010
$399$	0	0
$400$	7673.41	0.959176
$401$	1698.36	0.211501	0.105750	−	0.994393i	$-0.466276\pi$
$401$	1698.36	0.211501	0.105750	+	0.994393i	$0.466276\pi$
$402$	0	0
$403$	6913.99	0.854616
$404$	−1041.68	−0.128281
$405$	0	0
$406$	6461.15	0.789807
$407$	−2181.96	−0.265739
$408$	0	0
$409$	−5511.51	−0.666324	−0.333162	−	0.942870i	$-0.608116\pi$
$409$	−5511.51	−0.666324	−0.333162	+	0.942870i	$0.608116\pi$
$410$	−5038.61	−0.606925
$411$	0	0
$412$	10509.4	1.25670
$413$	2465.20	0.293715
$414$	0	0
$415$	8545.99	1.01086
$416$	8132.29	0.958457
$417$	0	0
$418$	−160.979	−0.0188367
$419$	6137.26	0.715573	0.357786	−	0.933803i	$-0.383532\pi$
$419$	6137.26	0.715573	0.357786	+	0.933803i	$0.383532\pi$
$420$	0	0
$421$	9168.60	1.06140	0.530701	−	0.847559i	$-0.321929\pi$
$421$	9168.60	1.06140	0.530701	+	0.847559i	$0.321929\pi$
$422$	1713.47	0.197655
$423$	0	0
$424$	−10413.3	−1.19273
$425$	3819.52	0.435939
$426$	0	0
$427$	−19178.5	−2.17357
$428$	−3986.40	−0.450210
$429$	0	0
$430$	−1140.75	−0.127935
$431$	−12535.8	−1.40099	−0.700494	−	0.713658i	$-0.747037\pi$
$431$	−12535.8	−1.40099	−0.700494	+	0.713658i	$0.747037\pi$
$432$	0	0
$433$	7163.93	0.795095	0.397548	−	0.917582i	$-0.369861\pi$
$433$	7163.93	0.795095	0.397548	+	0.917582i	$0.369861\pi$
$434$	6056.65	0.669881
$435$	0	0
$436$	−5297.05	−0.581841
$437$	948.898	0.103872
$438$	0	0
$439$	−2988.31	−0.324884	−0.162442	−	0.986718i	$-0.551937\pi$
$439$	−2988.31	−0.324884	−0.162442	+	0.986718i	$0.551937\pi$
$440$	−8683.36	−0.940825
$441$	0	0
$442$	756.463	0.0814056
$443$	11257.1	1.20732	0.603660	−	0.797242i	$-0.293708\pi$
$443$	11257.1	1.20732	0.603660	+	0.797242i	$0.293708\pi$
$444$	0	0
$445$	2322.60	0.247420
$446$	1056.02	0.112116
$447$	0	0
$448$	819.102	0.0863816
$449$	4960.39	0.521371	0.260685	−	0.965424i	$-0.416052\pi$
$449$	4960.39	0.521371	0.260685	+	0.965424i	$0.416052\pi$
$450$	0	0
$451$	−4344.01	−0.453551
$452$	−6820.92	−0.709799
$453$	0	0
$454$	2457.10	0.254004
$455$	27169.2	2.79937
$456$	0	0
$457$	−4911.00	−0.502685	−0.251343	−	0.967898i	$-0.580872\pi$
$457$	−4911.00	−0.502685	−0.251343	+	0.967898i	$0.580872\pi$
$458$	−1778.56	−0.181455
$459$	0	0
$460$	22571.4	2.28782
$461$	3089.54	0.312135	0.156068	−	0.987746i	$-0.450118\pi$
$461$	3089.54	0.312135	0.156068	+	0.987746i	$0.450118\pi$
$462$	0	0
$463$	2821.14	0.283174	0.141587	−	0.989926i	$-0.454779\pi$
$463$	2821.14	0.283174	0.141587	+	0.989926i	$0.454779\pi$
$464$	−4367.02	−0.436926
$465$	0	0
$466$	−4594.90	−0.456770
$467$	−483.354	−0.0478950	−0.0239475	−	0.999713i	$-0.507623\pi$
$467$	−483.354	−0.0478950	−0.0239475	+	0.999713i	$0.507623\pi$
$468$	0	0
$469$	−15263.4	−1.50277
$470$	987.808	0.0969451
$471$	0	0
$472$	1530.84	0.149285
$473$	−983.492	−0.0956047
$474$	0	0
$475$	1579.28	0.152552
$476$	−2475.61	−0.238381
$477$	0	0
$478$	1969.37	0.188445
$479$	5570.85	0.531396	0.265698	−	0.964056i	$-0.414398\pi$
$479$	5570.85	0.531396	0.265698	+	0.964056i	$0.414398\pi$
$480$	0	0
$481$	4239.51	0.401881
$482$	8730.73	0.825050
$483$	0	0
$484$	5098.31	0.478805
$485$	−33658.8	−3.15127
$486$	0	0
$487$	453.859	0.0422306	0.0211153	−	0.999777i	$-0.493278\pi$
$487$	453.859	0.0422306	0.0211153	+	0.999777i	$0.493278\pi$
$488$	−11909.5	−1.10475
$489$	0	0
$490$	14700.8	1.35533
$491$	−2787.17	−0.256178	−0.128089	−	0.991763i	$-0.540884\pi$
$491$	−2787.17	−0.256178	−0.128089	+	0.991763i	$0.540884\pi$
$492$	0	0
$493$	−2173.73	−0.198580
$494$	312.778	0.0284870
$495$	0	0
$496$	−4093.62	−0.370583
$497$	−23750.1	−2.14354
$498$	0	0
$499$	−325.118	−0.0291669	−0.0145834	−	0.999894i	$-0.504642\pi$
$499$	−325.118	−0.0291669	−0.0145834	+	0.999894i	$0.504642\pi$
$500$	21464.6	1.91985
$501$	0	0
$502$	−2978.12	−0.264781
$503$	11441.8	1.01424	0.507120	−	0.861875i	$-0.330710\pi$
$503$	11441.8	1.01424	0.507120	+	0.861875i	$0.330710\pi$
$504$	0	0
$505$	−3369.24	−0.296889
$506$	−5208.91	−0.457637
$507$	0	0
$508$	−3753.88	−0.327857
$509$	8398.29	0.731332	0.365666	−	0.930746i	$-0.380841\pi$
$509$	8398.29	0.731332	0.365666	+	0.930746i	$0.380841\pi$
$510$	0	0
$511$	18352.5	1.58878
$512$	−8730.11	−0.753554
$513$	0	0
$514$	−2906.39	−0.249407
$515$	33991.8	2.90846
$516$	0	0
$517$	851.632	0.0724463
$518$	3713.81	0.315010
$519$	0	0
$520$	16871.6	1.42282
$521$	−9762.74	−0.820947	−0.410474	−	0.911873i	$-0.634637\pi$
$521$	−9762.74	−0.820947	−0.410474	+	0.911873i	$0.634637\pi$
$522$	0	0
$523$	−14406.0	−1.20445	−0.602227	−	0.798325i	$-0.705720\pi$
$523$	−14406.0	−1.20445	−0.602227	+	0.798325i	$0.705720\pi$
$524$	12545.2	1.04588
$525$	0	0
$526$	−8310.16	−0.688860
$527$	−2037.65	−0.168427
$528$	0	0
$529$	18537.3	1.52357
$530$	−14852.6	−1.21728
$531$	0	0
$532$	−1023.60	−0.0834188
$533$	8440.33	0.685912
$534$	0	0
$535$	−12893.7	−1.04195
$536$	−9478.32	−0.763808
$537$	0	0
$538$	−3494.85	−0.280062
$539$	12674.2	1.01283
$540$	0	0
$541$	22048.2	1.75217	0.876087	−	0.482153i	$-0.160145\pi$
$541$	22048.2	1.75217	0.876087	+	0.482153i	$0.160145\pi$
$542$	1375.02	0.108971
$543$	0	0
$544$	−2396.70	−0.188892
$545$	−17132.8	−1.34659
$546$	0	0
$547$	6165.00	0.481895	0.240947	−	0.970538i	$-0.422542\pi$
$547$	6165.00	0.481895	0.240947	+	0.970538i	$0.422542\pi$
$548$	10835.6	0.844662
$549$	0	0
$550$	−8669.33	−0.672112
$551$	−898.784	−0.0694909
$552$	0	0
$553$	7100.10	0.545980
$554$	−2474.87	−0.189796
$555$	0	0
$556$	10949.7	0.835200
$557$	12519.9	0.952400	0.476200	−	0.879337i	$-0.342014\pi$
$557$	12519.9	0.952400	0.476200	+	0.879337i	$0.342014\pi$
$558$	0	0
$559$	1910.91	0.144584
$560$	−16086.3	−1.21388
$561$	0	0
$562$	8474.57	0.636082
$563$	−21124.7	−1.58135	−0.790674	−	0.612237i	$-0.790270\pi$
$563$	−21124.7	−1.58135	−0.790674	+	0.612237i	$0.790270\pi$
$564$	0	0
$565$	−22061.7	−1.64273
$566$	−169.451	−0.0125840
$567$	0	0
$568$	−14748.4	−1.08949
$569$	22782.7	1.67856	0.839279	−	0.543700i	$-0.182977\pi$
$569$	22782.7	1.67856	0.839279	+	0.543700i	$0.182977\pi$
$570$	0	0
$571$	−12990.7	−0.952092	−0.476046	−	0.879420i	$-0.657930\pi$
$571$	−12990.7	−0.952092	−0.476046	+	0.879420i	$0.657930\pi$
$572$	6414.39	0.468879
$573$	0	0
$574$	7393.72	0.537645
$575$	51101.9	3.70626
$576$	0	0
$577$	−25759.9	−1.85858	−0.929289	−	0.369353i	$-0.879579\pi$
$577$	−25759.9	−1.85858	−0.929289	+	0.369353i	$0.879579\pi$
$578$	6162.52	0.443473
$579$	0	0
$580$	−21379.3	−1.53057
$581$	−12540.5	−0.895468
$582$	0	0
$583$	−12805.1	−0.909663
$584$	11396.6	0.807525
$585$	0	0
$586$	−417.400	−0.0294243
$587$	4369.06	0.307207	0.153604	−	0.988133i	$-0.450912\pi$
$587$	4369.06	0.307207	0.153604	+	0.988133i	$0.450912\pi$
$588$	0	0
$589$	−842.515	−0.0589393
$590$	2183.46	0.152358
$591$	0	0
$592$	−2510.12	−0.174266
$593$	−3241.12	−0.224447	−0.112223	−	0.993683i	$-0.535797\pi$
$593$	−3241.12	−0.224447	−0.112223	+	0.993683i	$0.535797\pi$
$594$	0	0
$595$	−8007.15	−0.551699
$596$	2881.49	0.198038
$597$	0	0
$598$	10120.8	0.692092
$599$	4294.37	0.292927	0.146463	−	0.989216i	$-0.453211\pi$
$599$	4294.37	0.292927	0.146463	+	0.989216i	$0.453211\pi$
$600$	0	0
$601$	1277.55	0.0867093	0.0433546	−	0.999060i	$-0.486195\pi$
$601$	1277.55	0.0867093	0.0433546	+	0.999060i	$0.486195\pi$
$602$	1673.95	0.113331
$603$	0	0
$604$	3052.00	0.205603
$605$	16490.0	1.10813
$606$	0	0
$607$	17305.3	1.15717	0.578584	−	0.815623i	$-0.303606\pi$
$607$	17305.3	1.15717	0.578584	+	0.815623i	$0.303606\pi$
$608$	−990.973	−0.0661008
$609$	0	0
$610$	−16986.7	−1.12749
$611$	−1654.71	−0.109562
$612$	0	0
$613$	−12812.9	−0.844225	−0.422112	−	0.906544i	$-0.638711\pi$
$613$	−12812.9	−0.844225	−0.422112	+	0.906544i	$0.638711\pi$
$614$	−5127.18	−0.336997
$615$	0	0
$616$	12742.1	0.833430
$617$	3963.52	0.258615	0.129307	−	0.991605i	$-0.458725\pi$
$617$	3963.52	0.258615	0.129307	+	0.991605i	$0.458725\pi$
$618$	0	0
$619$	26937.7	1.74914	0.874570	−	0.484899i	$-0.161144\pi$
$619$	26937.7	1.74914	0.874570	+	0.484899i	$0.161144\pi$
$620$	−20040.9	−1.29816
$621$	0	0
$622$	9157.79	0.590344
$623$	−3408.22	−0.219177
$624$	0	0
$625$	32971.0	2.11015
$626$	299.300	0.0191093
$627$	0	0
$628$	−14518.8	−0.922554
$629$	−1249.44	−0.0792026
$630$	0	0
$631$	−2401.37	−0.151501	−0.0757504	−	0.997127i	$-0.524135\pi$
$631$	−2401.37	−0.151501	−0.0757504	+	0.997127i	$0.524135\pi$
$632$	4409.03	0.277503
$633$	0	0
$634$	8544.30	0.535232
$635$	−12141.6	−0.758778
$636$	0	0
$637$	−24625.7	−1.53172
$638$	4933.81	0.306162
$639$	0	0
$640$	−29156.5	−1.80080
$641$	22872.8	1.40940	0.704698	−	0.709508i	$-0.251083\pi$
$641$	22872.8	1.40940	0.704698	+	0.709508i	$0.251083\pi$
$642$	0	0
$643$	27051.9	1.65913	0.829566	−	0.558409i	$-0.188588\pi$
$643$	27051.9	1.65913	0.829566	+	0.558409i	$0.188588\pi$
$644$	−33121.5	−2.02666
$645$	0	0
$646$	−92.1801	−0.00561420
$647$	−12947.9	−0.786760	−0.393380	−	0.919376i	$-0.628694\pi$
$647$	−12947.9	−0.786760	−0.393380	+	0.919376i	$0.628694\pi$
$648$	0	0
$649$	1882.45	0.113856
$650$	16844.4	1.01645
$651$	0	0
$652$	−1983.87	−0.119163
$653$	−16218.4	−0.971934	−0.485967	−	0.873977i	$-0.661533\pi$
$653$	−16218.4	−0.971934	−0.485967	+	0.873977i	$0.661533\pi$
$654$	0	0
$655$	40576.3	2.42053
$656$	−4997.34	−0.297429
$657$	0	0
$658$	−1449.52	−0.0858788
$659$	26252.1	1.55180	0.775901	−	0.630855i	$-0.217296\pi$
$659$	26252.1	1.55180	0.775901	+	0.630855i	$0.217296\pi$
$660$	0	0
$661$	24649.9	1.45049	0.725243	−	0.688493i	$-0.241727\pi$
$661$	24649.9	1.45049	0.725243	+	0.688493i	$0.241727\pi$
$662$	−10821.9	−0.635355
$663$	0	0
$664$	−7787.41	−0.455136
$665$	−3310.75	−0.193061
$666$	0	0
$667$	−29082.7	−1.68828
$668$	−1782.42	−0.103239
$669$	0	0
$670$	−13519.0	−0.779531
$671$	−14645.0	−0.842567
$672$	0	0
$673$	4424.70	0.253432	0.126716	−	0.991939i	$-0.459556\pi$
$673$	4424.70	0.253432	0.126716	+	0.991939i	$0.459556\pi$
$674$	−11725.3	−0.670094
$675$	0	0
$676$	1401.71	0.0797514
$677$	−18153.6	−1.03057	−0.515287	−	0.857018i	$-0.672315\pi$
$677$	−18153.6	−1.03057	−0.515287	+	0.857018i	$0.672315\pi$
$678$	0	0
$679$	49391.3	2.79155
$680$	−4972.29	−0.280410
$681$	0	0
$682$	4624.93	0.259674
$683$	−19887.7	−1.11417	−0.557087	−	0.830454i	$-0.688081\pi$
$683$	−19887.7	−1.11417	−0.557087	+	0.830454i	$0.688081\pi$
$684$	0	0
$685$	35046.9	1.95485
$686$	−8219.39	−0.457460
$687$	0	0
$688$	−1131.41	−0.0626955
$689$	24880.1	1.37570
$690$	0	0
$691$	8550.10	0.470711	0.235355	−	0.971909i	$-0.424375\pi$
$691$	8550.10	0.470711	0.235355	+	0.971909i	$0.424375\pi$
$692$	11195.9	0.615037
$693$	0	0
$694$	173.001	0.00946256
$695$	35415.9	1.93295
$696$	0	0
$697$	−2487.48	−0.135179
$698$	1719.09	0.0932212
$699$	0	0
$700$	−55125.1	−2.97647
$701$	19280.9	1.03884	0.519421	−	0.854518i	$-0.326148\pi$
$701$	19280.9	1.03884	0.519421	+	0.854518i	$0.326148\pi$
$702$	0	0
$703$	−516.612	−0.0277161
$704$	625.477	0.0334851
$705$	0	0
$706$	16266.7	0.867146
$707$	4944.06	0.262999
$708$	0	0
$709$	19384.3	1.02679	0.513394	−	0.858153i	$-0.328388\pi$
$709$	19384.3	1.02679	0.513394	+	0.858153i	$0.328388\pi$
$710$	−21035.8	−1.11191
$711$	0	0
$712$	−2116.44	−0.111400
$713$	−27261.9	−1.43193
$714$	0	0
$715$	20746.8	1.08515
$716$	5617.29	0.293196
$717$	0	0
$718$	−16446.8	−0.854860
$719$	11746.0	0.609253	0.304627	−	0.952472i	$-0.401468\pi$
$719$	11746.0	0.609253	0.304627	+	0.952472i	$0.401468\pi$
$720$	0	0
$721$	−49880.0	−2.57646
$722$	8876.58	0.457551
$723$	0	0
$724$	18199.1	0.934206
$725$	−48403.0	−2.47951
$726$	0	0
$727$	10042.8	0.512333	0.256167	−	0.966633i	$-0.417540\pi$
$727$	10042.8	0.512333	0.256167	+	0.966633i	$0.417540\pi$
$728$	−24757.6	−1.26041
$729$	0	0
$730$	16255.1	0.824148
$731$	−563.170	−0.0284947
$732$	0	0
$733$	−23541.9	−1.18627	−0.593136	−	0.805102i	$-0.702111\pi$
$733$	−23541.9	−1.18627	−0.593136	+	0.805102i	$0.702111\pi$
$734$	17296.3	0.869778
$735$	0	0
$736$	−32065.7	−1.60592
$737$	−11655.3	−0.582538
$738$	0	0
$739$	−39871.6	−1.98471	−0.992354	−	0.123422i	$-0.960613\pi$
$739$	−39871.6	−1.98471	−0.992354	+	0.123422i	$0.960613\pi$
$740$	−12288.6	−0.610458
$741$	0	0
$742$	21795.0	1.07833
$743$	11387.7	0.562281	0.281141	−	0.959667i	$-0.409287\pi$
$743$	11387.7	0.562281	0.281141	+	0.959667i	$0.409287\pi$
$744$	0	0
$745$	9319.94	0.458331
$746$	8516.48	0.417977
$747$	0	0
$748$	−1890.41	−0.0924066
$749$	18920.3	0.923009
$750$	0	0
$751$	−26156.9	−1.27094	−0.635471	−	0.772125i	$-0.719194\pi$
$751$	−26156.9	−1.27094	−0.635471	+	0.772125i	$0.719194\pi$
$752$	979.716	0.0475087
$753$	0	0
$754$	−9586.31	−0.463014
$755$	9871.43	0.475839
$756$	0	0
$757$	9147.11	0.439178	0.219589	−	0.975593i	$-0.429528\pi$
$757$	9147.11	0.439178	0.219589	+	0.975593i	$0.429528\pi$
$758$	−14603.5	−0.699767
$759$	0	0
$760$	−2055.92	−0.0981263
$761$	41329.2	1.96870	0.984350	−	0.176225i	$-0.0563886\pi$
$761$	41329.2	1.96870	0.984350	+	0.176225i	$0.0563886\pi$
$762$	0	0
$763$	25140.9	1.19287
$764$	7071.10	0.334847
$765$	0	0
$766$	8257.51	0.389499
$767$	−3657.57	−0.172187
$768$	0	0
$769$	−3722.69	−0.174569	−0.0872846	−	0.996183i	$-0.527819\pi$
$769$	−3722.69	−0.174569	−0.0872846	+	0.996183i	$0.527819\pi$
$770$	18174.2	0.850586
$771$	0	0
$772$	17529.6	0.817231
$773$	−34945.5	−1.62600	−0.813002	−	0.582261i	$-0.802168\pi$
$773$	−34945.5	−1.62600	−0.813002	+	0.582261i	$0.802168\pi$
$774$	0	0
$775$	−45372.8	−2.10302
$776$	30671.1	1.41885
$777$	0	0
$778$	4924.48	0.226929
$779$	−1028.51	−0.0473045
$780$	0	0
$781$	−18135.9	−0.830925
$782$	−2982.74	−0.136397
$783$	0	0
$784$	14580.3	0.664192
$785$	−46959.8	−2.13512
$786$	0	0
$787$	−26135.3	−1.18377	−0.591883	−	0.806024i	$-0.701615\pi$
$787$	−26135.3	−1.18377	−0.591883	+	0.806024i	$0.701615\pi$
$788$	−7991.33	−0.361268
$789$	0	0
$790$	6288.66	0.283216
$791$	32373.6	1.45521
$792$	0	0
$793$	28454.9	1.27423
$794$	3381.40	0.151135
$795$	0	0
$796$	32269.4	1.43688
$797$	−20345.9	−0.904250	−0.452125	−	0.891955i	$-0.649334\pi$
$797$	−20345.9	−0.904250	−0.452125	+	0.891955i	$0.649334\pi$
$798$	0	0
$799$	487.664	0.0215924
$800$	−53367.8	−2.35855
$801$	0	0
$802$	−2207.37	−0.0971881
$803$	14014.2	0.615880
$804$	0	0
$805$	−107129.	−4.69042
$806$	−8986.16	−0.392710
$807$	0	0
$808$	3070.17	0.133674
$809$	21515.4	0.935034	0.467517	−	0.883984i	$-0.345149\pi$
$809$	21515.4	0.935034	0.467517	+	0.883984i	$0.345149\pi$
$810$	0	0
$811$	−19685.9	−0.852363	−0.426181	−	0.904638i	$-0.640142\pi$
$811$	−19685.9	−0.852363	−0.426181	+	0.904638i	$0.640142\pi$
$812$	31372.3	1.35585
$813$	0	0
$814$	2835.91	0.122111
$815$	−6416.64	−0.275785
$816$	0	0
$817$	−232.857	−0.00997139
$818$	7163.35	0.306187
$819$	0	0
$820$	−24465.1	−1.04190
$821$	−23501.4	−0.999029	−0.499515	−	0.866305i	$-0.666488\pi$
$821$	−23501.4	−0.999029	−0.499515	+	0.866305i	$0.666488\pi$
$822$	0	0
$823$	25153.6	1.06537	0.532684	−	0.846314i	$-0.321183\pi$
$823$	25153.6	1.06537	0.532684	+	0.846314i	$0.321183\pi$
$824$	−30974.6	−1.30953
$825$	0	0
$826$	−3204.03	−0.134967
$827$	−31043.6	−1.30531	−0.652655	−	0.757655i	$-0.726345\pi$
$827$	−31043.6	−1.30531	−0.652655	+	0.757655i	$0.726345\pi$
$828$	0	0
$829$	16479.2	0.690404	0.345202	−	0.938528i	$-0.387810\pi$
$829$	16479.2	0.690404	0.345202	+	0.938528i	$0.387810\pi$
$830$	−11107.3	−0.464505
$831$	0	0
$832$	−1215.29	−0.0506401
$833$	7257.52	0.301871
$834$	0	0
$835$	−5765.08	−0.238933
$836$	−781.636	−0.0323366
$837$	0	0
$838$	−7976.65	−0.328817
$839$	−10409.3	−0.428329	−0.214164	−	0.976798i	$-0.568703\pi$
$839$	−10409.3	−0.428329	−0.214164	+	0.976798i	$0.568703\pi$
$840$	0	0
$841$	3157.71	0.129473
$842$	−11916.5	−0.487731
$843$	0	0
$844$	8319.80	0.339312
$845$	4533.71	0.184573
$846$	0	0
$847$	−24197.7	−0.981633
$848$	−14731.0	−0.596537
$849$	0	0
$850$	−4964.26	−0.200321
$851$	−16716.4	−0.673363
$852$	0	0
$853$	1701.91	0.0683147	0.0341573	−	0.999416i	$-0.489125\pi$
$853$	1701.91	0.0683147	0.0341573	+	0.999416i	$0.489125\pi$
$854$	24926.5	0.998790
$855$	0	0
$856$	11749.2	0.469134
$857$	13987.3	0.557524	0.278762	−	0.960360i	$-0.410076\pi$
$857$	13987.3	0.557524	0.278762	+	0.960360i	$0.410076\pi$
$858$	0	0
$859$	−845.375	−0.0335784	−0.0167892	−	0.999859i	$-0.505344\pi$
$859$	−845.375	−0.0335784	−0.0167892	+	0.999859i	$0.505344\pi$
$860$	−5538.95	−0.219624
$861$	0	0
$862$	16292.8	0.643776
$863$	16970.3	0.669381	0.334691	−	0.942328i	$-0.391368\pi$
$863$	16970.3	0.669381	0.334691	+	0.942328i	$0.391368\pi$
$864$	0	0
$865$	36212.3	1.42342
$866$	−9311.01	−0.365359
$867$	0	0
$868$	29408.2	1.14998
$869$	5421.72	0.211645
$870$	0	0
$871$	22646.1	0.880981
$872$	15612.1	0.606297
$873$	0	0
$874$	−1233.29	−0.0477307
$875$	−101876.	−3.93602
$876$	0	0
$877$	−3402.23	−0.130998	−0.0654989	−	0.997853i	$-0.520864\pi$
$877$	−3402.23	−0.130998	−0.0654989	+	0.997853i	$0.520864\pi$
$878$	3883.92	0.149289
$879$	0	0
$880$	−12283.7	−0.470550
$881$	9753.40	0.372986	0.186493	−	0.982456i	$-0.440288\pi$
$881$	9753.40	0.372986	0.186493	+	0.982456i	$0.440288\pi$
$882$	0	0
$883$	42672.3	1.62632	0.813159	−	0.582042i	$-0.197746\pi$
$883$	42672.3	1.62632	0.813159	+	0.582042i	$0.197746\pi$
$884$	3673.03	0.139748
$885$	0	0
$886$	−14631.0	−0.554783
$887$	−13027.3	−0.493139	−0.246570	−	0.969125i	$-0.579303\pi$
$887$	−13027.3	−0.493139	−0.246570	+	0.969125i	$0.579303\pi$
$888$	0	0
$889$	17816.7	0.672164
$890$	−3018.71	−0.113694
$891$	0	0
$892$	5127.52	0.192469
$893$	201.637	0.00755601
$894$	0	0
$895$	18168.6	0.678560
$896$	42784.5	1.59524
$897$	0	0
$898$	−6447.06	−0.239578
$899$	25822.1	0.957972
$900$	0	0
$901$	−7332.50	−0.271122
$902$	5645.94	0.208414
$903$	0	0
$904$	20103.4	0.739634
$905$	58863.5	2.16209
$906$	0	0
$907$	−2184.13	−0.0799591	−0.0399796	−	0.999200i	$-0.512729\pi$
$907$	−2184.13	−0.0799591	−0.0399796	+	0.999200i	$0.512729\pi$
$908$	11930.5	0.436045
$909$	0	0
$910$	−35312.1	−1.28636
$911$	−15349.5	−0.558236	−0.279118	−	0.960257i	$-0.590042\pi$
$911$	−15349.5	−0.558236	−0.279118	+	0.960257i	$0.590042\pi$
$912$	0	0
$913$	−9576.07	−0.347121
$914$	6382.87	0.230992
$915$	0	0
$916$	−8635.84	−0.311502
$917$	−59542.2	−2.14423
$918$	0	0
$919$	−50672.7	−1.81887	−0.909433	−	0.415850i	$-0.863484\pi$
$919$	−50672.7	−1.81887	−0.909433	+	0.415850i	$0.863484\pi$
$920$	−66525.0	−2.38398
$921$	0	0
$922$	−4015.50	−0.143431
$923$	35237.6	1.25662
$924$	0	0
$925$	−27821.6	−0.988940
$926$	−3666.66	−0.130123
$927$	0	0
$928$	30372.2	1.07437
$929$	40538.7	1.43168	0.715841	−	0.698263i	$-0.246043\pi$
$929$	40538.7	1.43168	0.715841	+	0.698263i	$0.246043\pi$
$930$	0	0
$931$	3000.80	0.105636
$932$	−22310.7	−0.784131
$933$	0	0
$934$	628.219	0.0220085
$935$	−6114.36	−0.213862
$936$	0	0
$937$	13492.4	0.470413	0.235206	−	0.971945i	$-0.424423\pi$
$937$	13492.4	0.470413	0.235206	+	0.971945i	$0.424423\pi$
$938$	19838.0	0.690548
$939$	0	0
$940$	4796.33	0.166424
$941$	−22597.2	−0.782835	−0.391417	−	0.920213i	$-0.628015\pi$
$941$	−22597.2	−0.782835	−0.391417	+	0.920213i	$0.628015\pi$
$942$	0	0
$943$	−33280.3	−1.14926
$944$	2165.57	0.0746645
$945$	0	0
$946$	1278.25	0.0439319
$947$	−21796.6	−0.747933	−0.373967	−	0.927442i	$-0.622003\pi$
$947$	−21796.6	−0.747933	−0.373967	+	0.927442i	$0.622003\pi$
$948$	0	0
$949$	−27229.4	−0.931405
$950$	−2052.60	−0.0701000
$951$	0	0
$952$	7296.40	0.248401
$953$	40073.5	1.36213	0.681064	−	0.732224i	$-0.261518\pi$
$953$	40073.5	1.36213	0.681064	+	0.732224i	$0.261518\pi$
$954$	0	0
$955$	22870.9	0.774957
$956$	9562.31	0.323501
$957$	0	0
$958$	−7240.47	−0.244185
$959$	−51428.2	−1.73170
$960$	0	0
$961$	−5585.45	−0.187488
$962$	−5510.12	−0.184671
$963$	0	0
$964$	42392.3	1.41635
$965$	56697.8	1.89137
$966$	0	0
$967$	−20836.9	−0.692935	−0.346468	−	0.938062i	$-0.612619\pi$
$967$	−20836.9	−0.692935	−0.346468	+	0.938062i	$0.612619\pi$
$968$	−15026.3	−0.498930
$969$	0	0
$970$	43746.6	1.44806
$971$	5902.12	0.195065	0.0975325	−	0.995232i	$-0.468905\pi$
$971$	5902.12	0.195065	0.0975325	+	0.995232i	$0.468905\pi$
$972$	0	0
$973$	−51969.7	−1.71230
$974$	−589.884	−0.0194056
$975$	0	0
$976$	−16847.5	−0.552538
$977$	−17199.6	−0.563218	−0.281609	−	0.959529i	$-0.590868\pi$
$977$	−17199.6	−0.563218	−0.281609	+	0.959529i	$0.590868\pi$
$978$	0	0
$979$	−2602.56	−0.0849623
$980$	71379.9	2.32668
$981$	0	0
$982$	3622.51	0.117718
$983$	−8184.53	−0.265561	−0.132780	−	0.991145i	$-0.542390\pi$
$983$	−8184.53	−0.265561	−0.132780	+	0.991145i	$0.542390\pi$
$984$	0	0
$985$	−25847.3	−0.836104
$986$	2825.22	0.0912507
$987$	0	0
$988$	1518.70	0.0489032
$989$	−7534.73	−0.242255
$990$	0	0
$991$	−43473.7	−1.39353	−0.696765	−	0.717300i	$-0.745378\pi$
$991$	−43473.7	−1.39353	−0.696765	+	0.717300i	$0.745378\pi$
$992$	28470.8	0.911238
$993$	0	0
$994$	30868.2	0.984989
$995$	104373.	3.32546
$996$	0	0
$997$	42253.7	1.34221	0.671107	−	0.741361i	$-0.265819\pi$
$997$	42253.7	1.34221	0.671107	+	0.741361i	$0.265819\pi$
$998$	422.558	0.0134026
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.4.a.h.1.4		6
3.2	odd	2		43.4.a.b.1.3	✓	6
12.11	even	2		688.4.a.i.1.4		6
15.14	odd	2		1075.4.a.b.1.4		6
21.20	even	2		2107.4.a.c.1.3		6
129.128	even	2		1849.4.a.c.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.3	✓	6	3.2	odd	2
387.4.a.h.1.4		6	1.1	even	1	trivial
688.4.a.i.1.4		6	12.11	even	2
1075.4.a.b.1.4		6	15.14	odd	2
1849.4.a.c.1.4		6	129.128	even	2
2107.4.a.c.1.3		6	21.20	even	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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q-1.29971 q^{2} -6.31076 q^{4} -20.4116 q^{5} +29.9522 q^{7} +18.5998 q^{8} +O(q^{10})\)	\(q-1.29971 q^{2} -6.31076 q^{4} -20.4116 q^{5} +29.9522 q^{7} +18.5998 q^{8} +26.5291 q^{10} +22.8719 q^{11} -44.4397 q^{13} -38.9292 q^{14} +26.3118 q^{16} +13.0970 q^{17} +5.41527 q^{19} +128.813 q^{20} -29.7268 q^{22} +175.226 q^{23} +291.634 q^{25} +57.7586 q^{26} -189.021 q^{28} -165.972 q^{29} -155.581 q^{31} -182.996 q^{32} -17.0222 q^{34} -611.374 q^{35} -95.3991 q^{37} -7.03827 q^{38} -379.652 q^{40} -189.928 q^{41} -43.0000 q^{43} -144.339 q^{44} -227.743 q^{46} +37.2349 q^{47} +554.137 q^{49} -379.039 q^{50} +280.448 q^{52} -559.862 q^{53} -466.852 q^{55} +557.106 q^{56} +215.715 q^{58} +82.3042 q^{59} -640.304 q^{61} +202.210 q^{62} +27.3469 q^{64} +907.085 q^{65} -509.592 q^{67} -82.6519 q^{68} +794.607 q^{70} -792.932 q^{71} +612.727 q^{73} +123.991 q^{74} -34.1745 q^{76} +685.065 q^{77} +237.047 q^{79} -537.066 q^{80} +246.850 q^{82} -418.683 q^{83} -267.330 q^{85} +55.8874 q^{86} +425.413 q^{88} -113.788 q^{89} -1331.07 q^{91} -1105.81 q^{92} -48.3944 q^{94} -110.534 q^{95} +1649.00 q^{97} -720.216 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8	\( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8} + 57 q^{10} + 28 q^{11} + 56 q^{13} + 184 q^{14} - 54 q^{16} - 19 q^{17} - 75 q^{19} - 135 q^{20} - 504 q^{22} - 131 q^{23} + 105 q^{25} - 44 q^{26} - 404 q^{28} - 515 q^{29} + 237 q^{31} - 558 q^{32} - 107 q^{34} - 198 q^{35} + 269 q^{37} - 527 q^{38} + 613 q^{40} - 471 q^{41} - 258 q^{43} + 428 q^{44} - 67 q^{46} - 415 q^{47} + 350 q^{49} - 1335 q^{50} - 8 q^{52} - 450 q^{53} - 1732 q^{55} + 780 q^{56} - 1055 q^{58} - 356 q^{59} - 1328 q^{61} - 1603 q^{62} + 466 q^{64} + 62 q^{65} - 632 q^{67} - 571 q^{68} - 1902 q^{70} + 144 q^{71} + 864 q^{73} - 1207 q^{74} + 1005 q^{76} - 2660 q^{77} - 1613 q^{79} - 2399 q^{80} + 1673 q^{82} + 682 q^{83} + 84 q^{85} + 258 q^{86} - 608 q^{88} - 3378 q^{89} - 3900 q^{91} - 3491 q^{92} + 3197 q^{94} + 79 q^{95} - 55 q^{97} - 2398 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8 + 57 * q^10 + 28 * q^11 + 56 * q^13 + 184 * q^14 - 54 * q^16 - 19 * q^17 - 75 * q^19 - 135 * q^20 - 504 * q^22 - 131 * q^23 + 105 * q^25 - 44 * q^26 - 404 * q^28 - 515 * q^29 + 237 * q^31 - 558 * q^32 - 107 * q^34 - 198 * q^35 + 269 * q^37 - 527 * q^38 + 613 * q^40 - 471 * q^41 - 258 * q^43 + 428 * q^44 - 67 * q^46 - 415 * q^47 + 350 * q^49 - 1335 * q^50 - 8 * q^52 - 450 * q^53 - 1732 * q^55 + 780 * q^56 - 1055 * q^58 - 356 * q^59 - 1328 * q^61 - 1603 * q^62 + 466 * q^64 + 62 * q^65 - 632 * q^67 - 571 * q^68 - 1902 * q^70 + 144 * q^71 + 864 * q^73 - 1207 * q^74 + 1005 * q^76 - 2660 * q^77 - 1613 * q^79 - 2399 * q^80 + 1673 * q^82 + 682 * q^83 + 84 * q^85 + 258 * q^86 - 608 * q^88 - 3378 * q^89 - 3900 * q^91 - 3491 * q^92 + 3197 * q^94 + 79 * q^95 - 55 * q^97 - 2398 * q^98

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