LMFDB - Embedded newform 387.6.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,6,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, 6, names="a")

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
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:	$62.0685382676$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 $ x^10 - 2x^9 - 256x^8 + 266x^7 + 21986x^6 - 10450x^5 - 719484x^4 + 384582x^3 + 8437093x^2 - 5752252*x - 22734604
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.48720$ 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)

$f(q)$	$=$	$q-2.48720 q^{2} -25.8138 q^{4} -101.308 q^{5} +15.7005 q^{7} +143.795 q^{8} +O(q^{10})$	\(q-2.48720 q^{2} -25.8138 q^{4} -101.308 q^{5} +15.7005 q^{7} +143.795 q^{8} +251.974 q^{10} -394.739 q^{11} +666.939 q^{13} -39.0503 q^{14} +468.396 q^{16} -172.038 q^{17} -1280.86 q^{19} +2615.15 q^{20} +981.796 q^{22} -569.882 q^{23} +7138.34 q^{25} -1658.81 q^{26} -405.290 q^{28} +6328.41 q^{29} +7795.01 q^{31} -5766.42 q^{32} +427.892 q^{34} -1590.59 q^{35} +16252.3 q^{37} +3185.75 q^{38} -14567.6 q^{40} -7454.95 q^{41} +1849.00 q^{43} +10189.7 q^{44} +1417.41 q^{46} +5628.68 q^{47} -16560.5 q^{49} -17754.5 q^{50} -17216.2 q^{52} -22460.1 q^{53} +39990.3 q^{55} +2257.65 q^{56} -15740.0 q^{58} +9061.48 q^{59} -18280.8 q^{61} -19387.8 q^{62} -646.417 q^{64} -67566.3 q^{65} -27428.5 q^{67} +4440.95 q^{68} +3956.11 q^{70} +12860.9 q^{71} -63446.5 q^{73} -40422.7 q^{74} +33063.8 q^{76} -6197.60 q^{77} -1911.26 q^{79} -47452.4 q^{80} +18542.0 q^{82} +52124.9 q^{83} +17428.8 q^{85} -4598.84 q^{86} -56761.4 q^{88} -66185.2 q^{89} +10471.3 q^{91} +14710.8 q^{92} -13999.7 q^{94} +129761. q^{95} +37497.8 q^{97} +41189.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8}+O(q^{10}) $ 10 * q - 8 * q^2 + 202 * q^4 - 138 * q^5 + 60 * q^7 - 294 * q^8	$ 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8} - 17 q^{10} - 745 q^{11} + 1917 q^{13} - 1936 q^{14} + 5354 q^{16} - 4017 q^{17} - 2404 q^{19} - 1311 q^{20} - 5836 q^{22} - 1733 q^{23} + 7120 q^{25} + 1484 q^{26} - 15028 q^{28} - 6996 q^{29} - 4899 q^{31} + 7554 q^{32} - 27033 q^{34} - 7084 q^{35} + 1466 q^{37} - 13905 q^{38} - 93211 q^{40} - 10297 q^{41} + 18490 q^{43} + 36140 q^{44} + 17991 q^{46} - 48592 q^{47} + 29458 q^{49} - 983 q^{50} + 14232 q^{52} - 127165 q^{53} + 106672 q^{55} + 7780 q^{56} - 10305 q^{58} - 99372 q^{59} + 17408 q^{61} - 28265 q^{62} + 47202 q^{64} - 54484 q^{65} - 2021 q^{67} - 192151 q^{68} - 33194 q^{70} - 11286 q^{71} + 49892 q^{73} + 125431 q^{74} - 249803 q^{76} - 98144 q^{77} - 91524 q^{79} - 12251 q^{80} - 158909 q^{82} + 105203 q^{83} - 87212 q^{85} - 14792 q^{86} - 461824 q^{88} + 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 7259 q^{94} + 305340 q^{95} + 108383 q^{97} - 354656 q^{98}+O(q^{100}) $ 10 * q - 8 * q^2 + 202 * q^4 - 138 * q^5 + 60 * q^7 - 294 * q^8 - 17 * q^10 - 745 * q^11 + 1917 * q^13 - 1936 * q^14 + 5354 * q^16 - 4017 * q^17 - 2404 * q^19 - 1311 * q^20 - 5836 * q^22 - 1733 * q^23 + 7120 * q^25 + 1484 * q^26 - 15028 * q^28 - 6996 * q^29 - 4899 * q^31 + 7554 * q^32 - 27033 * q^34 - 7084 * q^35 + 1466 * q^37 - 13905 * q^38 - 93211 * q^40 - 10297 * q^41 + 18490 * q^43 + 36140 * q^44 + 17991 * q^46 - 48592 * q^47 + 29458 * q^49 - 983 * q^50 + 14232 * q^52 - 127165 * q^53 + 106672 * q^55 + 7780 * q^56 - 10305 * q^58 - 99372 * q^59 + 17408 * q^61 - 28265 * q^62 + 47202 * q^64 - 54484 * q^65 - 2021 * q^67 - 192151 * q^68 - 33194 * q^70 - 11286 * q^71 + 49892 * q^73 + 125431 * q^74 - 249803 * q^76 - 98144 * q^77 - 91524 * q^79 - 12251 * q^80 - 158909 * q^82 + 105203 * q^83 - 87212 * q^85 - 14792 * q^86 - 461824 * q^88 + 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 7259 * q^94 + 305340 * q^95 + 108383 * q^97 - 354656 * 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 $

$2$	−2.48720	−0.439679	−0.219840	−	0.975536i	$-0.570553\pi$
$2$	−2.48720	−0.439679	−0.219840	+	0.975536i	$0.570553\pi$
$3$	0	0
$3$	0	0
$4$	−25.8138	−0.806682
$5$	−101.308	−1.81226	−0.906128	−	0.423004i	$-0.860976\pi$
$5$	−101.308	−1.81226	−0.906128	+	0.423004i	$0.860976\pi$
$6$	0	0
$7$	15.7005	0.121107	0.0605534	−	0.998165i	$-0.480713\pi$
$7$	15.7005	0.121107	0.0605534	+	0.998165i	$0.480713\pi$
$8$	143.795	0.794361
$9$	0	0
$10$	251.974	0.796811
$11$	−394.739	−0.983623	−0.491811	−	0.870702i	$-0.663665\pi$
$11$	−394.739	−0.983623	−0.491811	+	0.870702i	$0.663665\pi$
$12$	0	0
$13$	666.939	1.09453	0.547265	−	0.836959i	$-0.315669\pi$
$13$	666.939	1.09453	0.547265	+	0.836959i	$0.315669\pi$
$14$	−39.0503	−0.0532481
$15$	0	0
$16$	468.396	0.457418
$17$	−172.038	−0.144378	−0.0721890	−	0.997391i	$-0.522998\pi$
$17$	−172.038	−0.144378	−0.0721890	+	0.997391i	$0.522998\pi$
$18$	0	0
$19$	−1280.86	−0.813985	−0.406992	−	0.913432i	$-0.633422\pi$
$19$	−1280.86	−0.813985	−0.406992	+	0.913432i	$0.633422\pi$
$20$	2615.15	1.46191
$21$	0	0
$22$	981.796	0.432479
$23$	−569.882	−0.224629	−0.112314	−	0.993673i	$-0.535826\pi$
$23$	−569.882	−0.224629	−0.112314	+	0.993673i	$0.535826\pi$
$24$	0	0
$25$	7138.34	2.28427
$26$	−1658.81	−0.481242
$27$	0	0
$28$	−405.290	−0.0976946
$29$	6328.41	1.39733	0.698666	−	0.715448i	$-0.253777\pi$
$29$	6328.41	1.39733	0.698666	+	0.715448i	$0.253777\pi$
$30$	0	0
$31$	7795.01	1.45684	0.728421	−	0.685129i	$-0.240254\pi$
$31$	7795.01	1.45684	0.728421	+	0.685129i	$0.240254\pi$
$32$	−5766.42	−0.995478
$33$	0	0
$34$	427.892	0.0634800
$35$	−1590.59	−0.219476
$36$	0	0
$37$	16252.3	1.95169	0.975844	−	0.218469i	$-0.0701061\pi$
$37$	16252.3	1.95169	0.975844	+	0.218469i	$0.0701061\pi$
$38$	3185.75	0.357892
$39$	0	0
$40$	−14567.6	−1.43958
$41$	−7454.95	−0.692604	−0.346302	−	0.938123i	$-0.612563\pi$
$41$	−7454.95	−0.692604	−0.346302	+	0.938123i	$0.612563\pi$
$42$	0	0
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	10189.7	0.793471
$45$	0	0
$46$	1417.41	0.0987646
$47$	5628.68	0.371673	0.185837	−	0.982581i	$-0.440500\pi$
$47$	5628.68	0.371673	0.185837	+	0.982581i	$0.440500\pi$
$48$	0	0
$49$	−16560.5	−0.985333
$50$	−17754.5	−1.00435
$51$	0	0
$52$	−17216.2	−0.882938
$53$	−22460.1	−1.09830	−0.549151	−	0.835723i	$-0.685049\pi$
$53$	−22460.1	−1.09830	−0.549151	+	0.835723i	$0.685049\pi$
$54$	0	0
$55$	39990.3	1.78258
$56$	2257.65	0.0962024
$57$	0	0
$58$	−15740.0	−0.614378
$59$	9061.48	0.338898	0.169449	−	0.985539i	$-0.445801\pi$
$59$	9061.48	0.338898	0.169449	+	0.985539i	$0.445801\pi$
$60$	0	0
$61$	−18280.8	−0.629029	−0.314514	−	0.949253i	$-0.601842\pi$
$61$	−18280.8	−0.629029	−0.314514	+	0.949253i	$0.601842\pi$
$62$	−19387.8	−0.640544
$63$	0	0
$64$	−646.417	−0.0197271
$65$	−67566.3	−1.98357
$66$	0	0
$67$	−27428.5	−0.746474	−0.373237	−	0.927736i	$-0.621752\pi$
$67$	−27428.5	−0.746474	−0.373237	+	0.927736i	$0.621752\pi$
$68$	4440.95	0.116467
$69$	0	0
$70$	3956.11	0.0964992
$71$	12860.9	0.302780	0.151390	−	0.988474i	$-0.451625\pi$
$71$	12860.9	0.302780	0.151390	+	0.988474i	$0.451625\pi$
$72$	0	0
$73$	−63446.5	−1.39348	−0.696740	−	0.717324i	$-0.745367\pi$
$73$	−63446.5	−1.39348	−0.696740	+	0.717324i	$0.745367\pi$
$74$	−40422.7	−0.858117
$75$	0	0
$76$	33063.8	0.656627
$77$	−6197.60	−0.119123
$78$	0	0
$79$	−1911.26	−0.0344549	−0.0172275	−	0.999852i	$-0.505484\pi$
$79$	−1911.26	−0.0344549	−0.0172275	+	0.999852i	$0.505484\pi$
$80$	−47452.4	−0.828959
$81$	0	0
$82$	18542.0	0.304524
$83$	52124.9	0.830520	0.415260	−	0.909703i	$-0.363691\pi$
$83$	52124.9	0.830520	0.415260	+	0.909703i	$0.363691\pi$
$84$	0	0
$85$	17428.8	0.261650
$86$	−4598.84	−0.0670505
$87$	0	0
$88$	−56761.4	−0.781351
$89$	−66185.2	−0.885698	−0.442849	−	0.896596i	$-0.646032\pi$
$89$	−66185.2	−0.885698	−0.442849	+	0.896596i	$0.646032\pi$
$90$	0	0
$91$	10471.3	0.132555
$92$	14710.8	0.181204
$93$	0	0
$94$	−13999.7	−0.163417
$95$	129761.	1.47515
$96$	0	0
$97$	37497.8	0.404648	0.202324	−	0.979319i	$-0.435151\pi$
$97$	37497.8	0.404648	0.202324	+	0.979319i	$0.435151\pi$
$98$	41189.3	0.433231
$99$	0	0
$100$	−184268.	−1.84268
$101$	35537.7	0.346646	0.173323	−	0.984865i	$-0.444550\pi$
$101$	35537.7	0.346646	0.173323	+	0.984865i	$0.444550\pi$
$102$	0	0
$103$	177955.	1.65279	0.826395	−	0.563091i	$-0.190388\pi$
$103$	177955.	1.65279	0.826395	+	0.563091i	$0.190388\pi$
$104$	95902.2	0.869451
$105$	0	0
$106$	55862.8	0.482901
$107$	−43589.8	−0.368065	−0.184033	−	0.982920i	$-0.558915\pi$
$107$	−43589.8	−0.368065	−0.184033	+	0.982920i	$0.558915\pi$
$108$	0	0
$109$	−49084.5	−0.395711	−0.197856	−	0.980231i	$-0.563398\pi$
$109$	−49084.5	−0.395711	−0.197856	+	0.980231i	$0.563398\pi$
$110$	−99464.0	−0.783762
$111$	0	0
$112$	7354.05	0.0553964
$113$	34530.0	0.254390	0.127195	−	0.991878i	$-0.459403\pi$
$113$	34530.0	0.254390	0.127195	+	0.991878i	$0.459403\pi$
$114$	0	0
$115$	57733.7	0.407085
$116$	−163360.	−1.12720
$117$	0	0
$118$	−22537.7	−0.149006
$119$	−2701.08	−0.0174852
$120$	0	0
$121$	−5231.89	−0.0324859
$122$	45468.0	0.276571
$123$	0	0
$124$	−201219.	−1.17521
$125$	−406584.	−2.32742
$126$	0	0
$127$	−10008.6	−0.0550637	−0.0275318	−	0.999621i	$-0.508765\pi$
$127$	−10008.6	−0.0550637	−0.0275318	+	0.999621i	$0.508765\pi$
$128$	186133.	1.00415
$129$	0	0
$130$	168051.	0.872133
$131$	−336036.	−1.71083	−0.855415	−	0.517943i	$-0.826698\pi$
$131$	−336036.	−1.71083	−0.855415	+	0.517943i	$0.826698\pi$
$132$	0	0
$133$	−20110.1	−0.0985791
$134$	68220.2	0.328209
$135$	0	0
$136$	−24738.1	−0.114688
$137$	1217.37	0.00554141	0.00277071	−	0.999996i	$-0.499118\pi$
$137$	1217.37	0.00554141	0.00277071	+	0.999996i	$0.499118\pi$
$138$	0	0
$139$	−157629.	−0.691989	−0.345995	−	0.938237i	$-0.612458\pi$
$139$	−157629.	−0.691989	−0.345995	+	0.938237i	$0.612458\pi$
$140$	41059.2	0.177048
$141$	0	0
$142$	−31987.8	−0.133126
$143$	−263267.	−1.07660
$144$	0	0
$145$	−641119.	−2.53232
$146$	157804.	0.612684
$147$	0	0
$148$	−419534.	−1.57439
$149$	501641.	1.85109	0.925545	−	0.378639i	$-0.123608\pi$
$149$	501641.	1.85109	0.925545	+	0.378639i	$0.123608\pi$
$150$	0	0
$151$	−430865.	−1.53780	−0.768899	−	0.639370i	$-0.779195\pi$
$151$	−430865.	−1.53780	−0.768899	+	0.639370i	$0.779195\pi$
$152$	−184180.	−0.646598
$153$	0	0
$154$	15414.7	0.0523761
$155$	−789698.	−2.64017
$156$	0	0
$157$	157952.	0.511418	0.255709	−	0.966754i	$-0.417691\pi$
$157$	157952.	0.511418	0.255709	+	0.966754i	$0.417691\pi$
$158$	4753.68	0.0151491
$159$	0	0
$160$	584186.	1.80406
$161$	−8947.43	−0.0272041
$162$	0	0
$163$	405846.	1.19645	0.598223	−	0.801330i	$-0.295874\pi$
$163$	405846.	1.19645	0.598223	+	0.801330i	$0.295874\pi$
$164$	192441.	0.558712
$165$	0	0
$166$	−129645.	−0.365162
$167$	−166792.	−0.462789	−0.231394	−	0.972860i	$-0.574329\pi$
$167$	−166792.	−0.462789	−0.231394	+	0.972860i	$0.574329\pi$
$168$	0	0
$169$	73514.3	0.197995
$170$	−43349.0	−0.115042
$171$	0	0
$172$	−47729.8	−0.123018
$173$	287751.	0.730973	0.365487	−	0.930817i	$-0.380903\pi$
$173$	287751.	0.730973	0.365487	+	0.930817i	$0.380903\pi$
$174$	0	0
$175$	112076.	0.276640
$176$	−184894.	−0.449927
$177$	0	0
$178$	164616.	0.389423
$179$	−522579.	−1.21904	−0.609522	−	0.792769i	$-0.708639\pi$
$179$	−522579.	−1.21904	−0.609522	+	0.792769i	$0.708639\pi$
$180$	0	0
$181$	81268.9	0.184386	0.0921930	−	0.995741i	$-0.470612\pi$
$181$	81268.9	0.184386	0.0921930	+	0.995741i	$0.470612\pi$
$182$	−26044.2	−0.0582817
$183$	0	0
$184$	−81946.0	−0.178436
$185$	−1.64649e6	−3.53696
$186$	0	0
$187$	67910.0	0.142014
$188$	−145298.	−0.299822
$189$	0	0
$190$	−322742.	−0.648592
$191$	608799.	1.20751	0.603755	−	0.797170i	$-0.293671\pi$
$191$	608799.	1.20751	0.603755	+	0.797170i	$0.293671\pi$
$192$	0	0
$193$	184959.	0.357423	0.178712	−	0.983902i	$-0.442807\pi$
$193$	184959.	0.357423	0.178712	+	0.983902i	$0.442807\pi$
$194$	−93264.7	−0.177915
$195$	0	0
$196$	427490.	0.794851
$197$	−200949.	−0.368910	−0.184455	−	0.982841i	$-0.559052\pi$
$197$	−200949.	−0.368910	−0.184455	+	0.982841i	$0.559052\pi$
$198$	0	0
$199$	−233653.	−0.418253	−0.209127	−	0.977889i	$-0.567062\pi$
$199$	−233653.	−0.418253	−0.209127	+	0.977889i	$0.567062\pi$
$200$	1.02646e6	1.81453
$201$	0	0
$202$	−88389.5	−0.152413
$203$	99359.2	0.169226
$204$	0	0
$205$	755248.	1.25518
$206$	−442610.	−0.726697
$207$	0	0
$208$	312392.	0.500658
$209$	505604.	0.800654
$210$	0	0
$211$	200513.	0.310054	0.155027	−	0.987910i	$-0.450454\pi$
$211$	200513.	0.310054	0.155027	+	0.987910i	$0.450454\pi$
$212$	579781.	0.885981
$213$	0	0
$214$	108417.	0.161831
$215$	−187319.	−0.276366
$216$	0	0
$217$	122386.	0.176433
$218$	122083.	0.173986
$219$	0	0
$220$	−1.03230e6	−1.43797
$221$	−114739.	−0.158026
$222$	0	0
$223$	347051.	0.467338	0.233669	−	0.972316i	$-0.424927\pi$
$223$	347051.	0.467338	0.233669	+	0.972316i	$0.424927\pi$
$224$	−90535.7	−0.120559
$225$	0	0
$226$	−85883.2	−0.111850
$227$	212111.	0.273212	0.136606	−	0.990625i	$-0.456381\pi$
$227$	212111.	0.273212	0.136606	+	0.990625i	$0.456381\pi$
$228$	0	0
$229$	−470629.	−0.593049	−0.296524	−	0.955025i	$-0.595828\pi$
$229$	−470629.	−0.593049	−0.296524	+	0.955025i	$0.595828\pi$
$230$	−143595.	−0.178987
$231$	0	0
$232$	909991.	1.10999
$233$	952248.	1.14911	0.574553	−	0.818467i	$-0.305176\pi$
$233$	952248.	1.14911	0.574553	+	0.818467i	$0.305176\pi$
$234$	0	0
$235$	−570231.	−0.673567
$236$	−233911.	−0.273383
$237$	0	0
$238$	6718.12	0.00768786
$239$	1.63690e6	1.85365	0.926824	−	0.375497i	$-0.122528\pi$
$239$	1.63690e6	1.85365	0.926824	+	0.375497i	$0.122528\pi$
$240$	0	0
$241$	−695944.	−0.771848	−0.385924	−	0.922531i	$-0.626117\pi$
$241$	−695944.	−0.771848	−0.385924	+	0.922531i	$0.626117\pi$
$242$	13012.8	0.0142834
$243$	0	0
$244$	471897.	0.507426
$245$	1.67771e6	1.78568
$246$	0	0
$247$	−854253.	−0.890931
$248$	1.12088e6	1.15726
$249$	0	0
$250$	1.01126e6	1.02332
$251$	−657886.	−0.659123	−0.329561	−	0.944134i	$-0.606901\pi$
$251$	−657886.	−0.659123	−0.329561	+	0.944134i	$0.606901\pi$
$252$	0	0
$253$	224955.	0.220950
$254$	24893.5	0.0242104
$255$	0	0
$256$	−442266.	−0.421778
$257$	599821.	0.566486	0.283243	−	0.959048i	$-0.408590\pi$
$257$	599821.	0.566486	0.283243	+	0.959048i	$0.408590\pi$
$258$	0	0
$259$	255169.	0.236363
$260$	1.74415e6	1.60011
$261$	0	0
$262$	835788.	0.752217
$263$	1.62868e6	1.45194	0.725968	−	0.687728i	$-0.241392\pi$
$263$	1.62868e6	1.45194	0.725968	+	0.687728i	$0.241392\pi$
$264$	0	0
$265$	2.27539e6	1.99040
$266$	50017.8	0.0433432
$267$	0	0
$268$	708034.	0.602167
$269$	−1.99405e6	−1.68017	−0.840087	−	0.542451i	$-0.817496\pi$
$269$	−1.99405e6	−1.68017	−0.840087	+	0.542451i	$0.817496\pi$
$270$	0	0
$271$	−1.89207e6	−1.56500	−0.782501	−	0.622650i	$-0.786056\pi$
$271$	−1.89207e6	−1.56500	−0.782501	+	0.622650i	$0.786056\pi$
$272$	−80581.8	−0.0660411
$273$	0	0
$274$	−3027.84	−0.00243644
$275$	−2.81778e6	−2.24686
$276$	0	0
$277$	−2.11477e6	−1.65601	−0.828007	−	0.560717i	$-0.810525\pi$
$277$	−2.11477e6	−1.65601	−0.828007	+	0.560717i	$0.810525\pi$
$278$	392055.	0.304253
$279$	0	0
$280$	−228718.	−0.174343
$281$	80948.7	0.0611567	0.0305784	−	0.999532i	$-0.490265\pi$
$281$	80948.7	0.0611567	0.0305784	+	0.999532i	$0.490265\pi$
$282$	0	0
$283$	1.01158e6	0.750814	0.375407	−	0.926860i	$-0.377503\pi$
$283$	1.01158e6	0.750814	0.375407	+	0.926860i	$0.377503\pi$
$284$	−331990.	−0.244247
$285$	0	0
$286$	654798.	0.473361
$287$	−117046.	−0.0838791
$288$	0	0
$289$	−1.39026e6	−0.979155
$290$	1.59459e6	1.11341
$291$	0	0
$292$	1.63780e6	1.12409
$293$	−786392.	−0.535143	−0.267572	−	0.963538i	$-0.586221\pi$
$293$	−786392.	−0.535143	−0.267572	+	0.963538i	$0.586221\pi$
$294$	0	0
$295$	−918001.	−0.614170
$296$	2.33699e6	1.55034
$297$	0	0
$298$	−1.24768e6	−0.813886
$299$	−380076.	−0.245863
$300$	0	0
$301$	29030.2	0.0184686
$302$	1.07165e6	0.676138
$303$	0	0
$304$	−599948.	−0.372332
$305$	1.85199e6	1.13996
$306$	0	0
$307$	106306.	0.0643744	0.0321872	−	0.999482i	$-0.489753\pi$
$307$	106306.	0.0643744	0.0321872	+	0.999482i	$0.489753\pi$
$308$	159984.	0.0960947
$309$	0	0
$310$	1.96414e6	1.16083
$311$	82617.8	0.0484365	0.0242183	−	0.999707i	$-0.492290\pi$
$311$	82617.8	0.0484365	0.0242183	+	0.999707i	$0.492290\pi$
$312$	0	0
$313$	1.38431e6	0.798681	0.399341	−	0.916803i	$-0.369239\pi$
$313$	1.38431e6	0.798681	0.399341	+	0.916803i	$0.369239\pi$
$314$	−392859.	−0.224860
$315$	0	0
$316$	49336.9	0.0277942
$317$	−964944.	−0.539329	−0.269665	−	0.962954i	$-0.586913\pi$
$317$	−964944.	−0.539329	−0.269665	+	0.962954i	$0.586913\pi$
$318$	0	0
$319$	−2.49807e6	−1.37445
$320$	65487.3	0.0357505
$321$	0	0
$322$	22254.1	0.0119611
$323$	220356.	0.117522
$324$	0	0
$325$	4.76084e6	2.50020
$326$	−1.00942e6	−0.526052
$327$	0	0
$328$	−1.07198e6	−0.550178
$329$	88373.0	0.0450122
$330$	0	0
$331$	−3.21355e6	−1.61218	−0.806092	−	0.591790i	$-0.798422\pi$
$331$	−3.21355e6	−1.61218	−0.806092	+	0.591790i	$0.798422\pi$
$332$	−1.34554e6	−0.669966
$333$	0	0
$334$	414844.	0.203479
$335$	2.77873e6	1.35280
$336$	0	0
$337$	835398.	0.400699	0.200350	−	0.979724i	$-0.435792\pi$
$337$	835398.	0.400699	0.200350	+	0.979724i	$0.435792\pi$
$338$	−182845.	−0.0870544
$339$	0	0
$340$	−449904.	−0.211068
$341$	−3.07700e6	−1.43298
$342$	0	0
$343$	−523886.	−0.240437
$344$	265876.	0.121139
$345$	0	0
$346$	−715695.	−0.321394
$347$	−2.19943e6	−0.980589	−0.490295	−	0.871557i	$-0.663111\pi$
$347$	−2.19943e6	−0.980589	−0.490295	+	0.871557i	$0.663111\pi$
$348$	0	0
$349$	2.90995e6	1.27886	0.639428	−	0.768851i	$-0.279171\pi$
$349$	2.90995e6	1.27886	0.639428	+	0.768851i	$0.279171\pi$
$350$	−278754.	−0.121633
$351$	0	0
$352$	2.27623e6	0.979175
$353$	−2.63861e6	−1.12704	−0.563518	−	0.826104i	$-0.690552\pi$
$353$	−2.63861e6	−1.12704	−0.563518	+	0.826104i	$0.690552\pi$
$354$	0	0
$355$	−1.30292e6	−0.548714
$356$	1.70849e6	0.714477
$357$	0	0
$358$	1.29976e6	0.535988
$359$	37197.0	0.0152325	0.00761626	−	0.999971i	$-0.497576\pi$
$359$	37197.0	0.0152325	0.00761626	+	0.999971i	$0.497576\pi$
$360$	0	0
$361$	−835506.	−0.337428
$362$	−202132.	−0.0810707
$363$	0	0
$364$	−270304.	−0.106930
$365$	6.42765e6	2.52534
$366$	0	0
$367$	2.27179e6	0.880447	0.440224	−	0.897888i	$-0.354899\pi$
$367$	2.27179e6	0.880447	0.440224	+	0.897888i	$0.354899\pi$
$368$	−266930.	−0.102749
$369$	0	0
$370$	4.09515e6	1.55513
$371$	−352635.	−0.133012
$372$	0	0
$373$	2.42341e6	0.901892	0.450946	−	0.892551i	$-0.351087\pi$
$373$	2.42341e6	0.901892	0.450946	+	0.892551i	$0.351087\pi$
$374$	−168906.	−0.0624404
$375$	0	0
$376$	809373.	0.295243
$377$	4.22066e6	1.52942
$378$	0	0
$379$	−2.23180e6	−0.798101	−0.399050	−	0.916929i	$-0.630660\pi$
$379$	−2.23180e6	−0.798101	−0.399050	+	0.916929i	$0.630660\pi$
$380$	−3.34963e6	−1.18998
$381$	0	0
$382$	−1.51421e6	−0.530917
$383$	1.13758e6	0.396263	0.198131	−	0.980175i	$-0.436513\pi$
$383$	1.13758e6	0.396263	0.198131	+	0.980175i	$0.436513\pi$
$384$	0	0
$385$	627868.	0.215882
$386$	−460031.	−0.157152
$387$	0	0
$388$	−967963.	−0.326422
$389$	−1.97034e6	−0.660188	−0.330094	−	0.943948i	$-0.607080\pi$
$389$	−1.97034e6	−0.660188	−0.330094	+	0.943948i	$0.607080\pi$
$390$	0	0
$391$	98041.1	0.0324315
$392$	−2.38131e6	−0.782710
$393$	0	0
$394$	499800.	0.162202
$395$	193626.	0.0624412
$396$	0	0
$397$	−5.13128e6	−1.63399	−0.816996	−	0.576644i	$-0.804362\pi$
$397$	−5.13128e6	−1.63399	−0.816996	+	0.576644i	$0.804362\pi$
$398$	581143.	0.183897
$399$	0	0
$400$	3.34357e6	1.04487
$401$	2.92332e6	0.907852	0.453926	−	0.891039i	$-0.350023\pi$
$401$	2.92332e6	0.907852	0.453926	+	0.891039i	$0.350023\pi$
$402$	0	0
$403$	5.19880e6	1.59456
$404$	−917365.	−0.279633
$405$	0	0
$406$	−247126.	−0.0744053
$407$	−6.41542e6	−1.91973
$408$	0	0
$409$	−2.10623e6	−0.622582	−0.311291	−	0.950315i	$-0.600761\pi$
$409$	−2.10623e6	−0.622582	−0.311291	+	0.950315i	$0.600761\pi$
$410$	−1.87845e6	−0.551875
$411$	0	0
$412$	−4.59370e6	−1.33328
$413$	142270.	0.0410428
$414$	0	0
$415$	−5.28068e6	−1.50511
$416$	−3.84585e6	−1.08958
$417$	0	0
$418$	−1.25754e6	−0.352031
$419$	787697.	0.219192	0.109596	−	0.993976i	$-0.465044\pi$
$419$	787697.	0.219192	0.109596	+	0.993976i	$0.465044\pi$
$420$	0	0
$421$	−2.73861e6	−0.753051	−0.376526	−	0.926406i	$-0.622881\pi$
$421$	−2.73861e6	−0.753051	−0.376526	+	0.926406i	$0.622881\pi$
$422$	−498717.	−0.136324
$423$	0	0
$424$	−3.22964e6	−0.872448
$425$	−1.22806e6	−0.329798
$426$	0	0
$427$	−287018.	−0.0761796
$428$	1.12522e6	0.296912
$429$	0	0
$430$	465900.	0.121513
$431$	−2.23642e6	−0.579911	−0.289955	−	0.957040i	$-0.593640\pi$
$431$	−2.23642e6	−0.579911	−0.289955	+	0.957040i	$0.593640\pi$
$432$	0	0
$433$	−7.32309e6	−1.87704	−0.938522	−	0.345219i	$-0.887804\pi$
$433$	−7.32309e6	−1.87704	−0.938522	+	0.345219i	$0.887804\pi$
$434$	−304398.	−0.0775741
$435$	0	0
$436$	1.26706e6	0.319213
$437$	729937.	0.182844
$438$	0	0
$439$	4.05522e6	1.00428	0.502138	−	0.864788i	$-0.332547\pi$
$439$	4.05522e6	1.00428	0.502138	+	0.864788i	$0.332547\pi$
$440$	5.75039e6	1.41601
$441$	0	0
$442$	285378.	0.0694808
$443$	−2.23385e6	−0.540810	−0.270405	−	0.962747i	$-0.587158\pi$
$443$	−2.23385e6	−0.540810	−0.270405	+	0.962747i	$0.587158\pi$
$444$	0	0
$445$	6.70510e6	1.60511
$446$	−863185.	−0.205479
$447$	0	0
$448$	−10149.1	−0.00238908
$449$	3.90186e6	0.913389	0.456695	−	0.889623i	$-0.349033\pi$
$449$	3.90186e6	0.913389	0.456695	+	0.889623i	$0.349033\pi$
$450$	0	0
$451$	2.94276e6	0.681262
$452$	−891352.	−0.205212
$453$	0	0
$454$	−527564.	−0.120126
$455$	−1.06083e6	−0.240223
$456$	0	0
$457$	−184098.	−0.0412343	−0.0206172	−	0.999787i	$-0.506563\pi$
$457$	−184098.	−0.0412343	−0.0206172	+	0.999787i	$0.506563\pi$
$458$	1.17055e6	0.260751
$459$	0	0
$460$	−1.49033e6	−0.328388
$461$	4.89953e6	1.07375	0.536873	−	0.843663i	$-0.319605\pi$
$461$	4.89953e6	1.07375	0.536873	+	0.843663i	$0.319605\pi$
$462$	0	0
$463$	−2.72830e6	−0.591479	−0.295740	−	0.955269i	$-0.595566\pi$
$463$	−2.72830e6	−0.591479	−0.295740	+	0.955269i	$0.595566\pi$
$464$	2.96420e6	0.639165
$465$	0	0
$466$	−2.36843e6	−0.505238
$467$	−3.18682e6	−0.676185	−0.338093	−	0.941113i	$-0.609782\pi$
$467$	−3.18682e6	−0.676185	−0.338093	+	0.941113i	$0.609782\pi$
$468$	0	0
$469$	−430641.	−0.0904031
$470$	1.41828e6	0.296154
$471$	0	0
$472$	1.30299e6	0.269207
$473$	−729873.	−0.150001
$474$	0	0
$475$	−9.14319e6	−1.85936
$476$	69725.1	0.0141050
$477$	0	0
$478$	−4.07130e6	−0.815010
$479$	−4.93825e6	−0.983410	−0.491705	−	0.870762i	$-0.663626\pi$
$479$	−4.93825e6	−0.983410	−0.491705	+	0.870762i	$0.663626\pi$
$480$	0	0
$481$	1.08393e7	2.13618
$482$	1.73095e6	0.339365
$483$	0	0
$484$	135055.	0.0262058
$485$	−3.79884e6	−0.733325
$486$	0	0
$487$	534531.	0.102129	0.0510646	−	0.998695i	$-0.483739\pi$
$487$	534531.	0.102129	0.0510646	+	0.998695i	$0.483739\pi$
$488$	−2.62868e6	−0.499676
$489$	0	0
$490$	−4.17281e6	−0.785124
$491$	−4.65073e6	−0.870597	−0.435299	−	0.900286i	$-0.643357\pi$
$491$	−4.65073e6	−0.870597	−0.435299	+	0.900286i	$0.643357\pi$
$492$	0	0
$493$	−1.08872e6	−0.201744
$494$	2.12470e6	0.391724
$495$	0	0
$496$	3.65115e6	0.666386
$497$	201923.	0.0366687
$498$	0	0
$499$	−215791.	−0.0387955	−0.0193977	−	0.999812i	$-0.506175\pi$
$499$	−215791.	−0.0387955	−0.0193977	+	0.999812i	$0.506175\pi$
$500$	1.04955e7	1.87749
$501$	0	0
$502$	1.63629e6	0.289803
$503$	6.45815e6	1.13812	0.569060	−	0.822296i	$-0.307307\pi$
$503$	6.45815e6	1.13812	0.569060	+	0.822296i	$0.307307\pi$
$504$	0	0
$505$	−3.60026e6	−0.628211
$506$	−559508.	−0.0971471
$507$	0	0
$508$	258361.	0.0444189
$509$	−3.88028e6	−0.663847	−0.331924	−	0.943306i	$-0.607698\pi$
$509$	−3.88028e6	−0.663847	−0.331924	+	0.943306i	$0.607698\pi$
$510$	0	0
$511$	−996142.	−0.168760
$512$	−4.85626e6	−0.818705
$513$	0	0
$514$	−1.49188e6	−0.249072
$515$	−1.80283e7	−2.99528
$516$	0	0
$517$	−2.22186e6	−0.365587
$518$	−634657.	−0.103924
$519$	0	0
$520$	−9.71568e6	−1.57567
$521$	−2.45313e6	−0.395937	−0.197969	−	0.980208i	$-0.563434\pi$
$521$	−2.45313e6	−0.395937	−0.197969	+	0.980208i	$0.563434\pi$
$522$	0	0
$523$	−1.91833e6	−0.306668	−0.153334	−	0.988174i	$-0.549001\pi$
$523$	−1.91833e6	−0.306668	−0.153334	+	0.988174i	$0.549001\pi$
$524$	8.67436e6	1.38010
$525$	0	0
$526$	−4.05087e6	−0.638386
$527$	−1.34104e6	−0.210336
$528$	0	0
$529$	−6.11158e6	−0.949542
$530$	−5.65936e6	−0.875139
$531$	0	0
$532$	519118.	0.0795220
$533$	−4.97200e6	−0.758076
$534$	0	0
$535$	4.41600e6	0.667029
$536$	−3.94407e6	−0.592970
$537$	0	0
$538$	4.95959e6	0.738738
$539$	6.53708e6	0.969196
$540$	0	0
$541$	−6.89363e6	−1.01264	−0.506320	−	0.862346i	$-0.668995\pi$
$541$	−6.89363e6	−1.01264	−0.506320	+	0.862346i	$0.668995\pi$
$542$	4.70597e6	0.688099
$543$	0	0
$544$	992042.	0.143725
$545$	4.97266e6	0.717130
$546$	0	0
$547$	4.78180e6	0.683319	0.341659	−	0.939824i	$-0.389011\pi$
$547$	4.78180e6	0.683319	0.341659	+	0.939824i	$0.389011\pi$
$548$	−31424.9	−0.00447016
$549$	0	0
$550$	7.00840e6	0.987898
$551$	−8.10578e6	−1.13741
$552$	0	0
$553$	−30007.7	−0.00417273
$554$	5.25986e6	0.728115
$555$	0	0
$556$	4.06901e6	0.558215
$557$	1.16813e7	1.59534	0.797672	−	0.603091i	$-0.206065\pi$
$557$	1.16813e7	1.59534	0.797672	+	0.603091i	$0.206065\pi$
$558$	0	0
$559$	1.23317e6	0.166914
$560$	−745026.	−0.100392
$561$	0	0
$562$	−201336.	−0.0268893
$563$	−9.15272e6	−1.21697	−0.608484	−	0.793566i	$-0.708222\pi$
$563$	−9.15272e6	−1.21697	−0.608484	+	0.793566i	$0.708222\pi$
$564$	0	0
$565$	−3.49817e6	−0.461021
$566$	−2.51599e6	−0.330117
$567$	0	0
$568$	1.84933e6	0.240516
$569$	4.46016e6	0.577523	0.288762	−	0.957401i	$-0.406757\pi$
$569$	4.46016e6	0.577523	0.288762	+	0.957401i	$0.406757\pi$
$570$	0	0
$571$	3.82289e6	0.490684	0.245342	−	0.969437i	$-0.421100\pi$
$571$	3.82289e6	0.490684	0.245342	+	0.969437i	$0.421100\pi$
$572$	6.79593e6	0.868478
$573$	0	0
$574$	291118.	0.0368799
$575$	−4.06801e6	−0.513112
$576$	0	0
$577$	1.15777e7	1.44772	0.723858	−	0.689949i	$-0.242367\pi$
$577$	1.15777e7	1.44772	0.723858	+	0.689949i	$0.242367\pi$
$578$	3.45786e6	0.430514
$579$	0	0
$580$	1.65497e7	2.04278
$581$	818387.	0.100582
$582$	0	0
$583$	8.86588e6	1.08032
$584$	−9.12327e6	−1.10693
$585$	0	0
$586$	1.95592e6	0.235291
$587$	−2.16236e6	−0.259020	−0.129510	−	0.991578i	$-0.541340\pi$
$587$	−2.16236e6	−0.259020	−0.129510	+	0.991578i	$0.541340\pi$
$588$	0	0
$589$	−9.98429e6	−1.18585
$590$	2.28325e6	0.270038
$591$	0	0
$592$	7.61251e6	0.892738
$593$	−1.42792e7	−1.66751	−0.833753	−	0.552137i	$-0.813812\pi$
$593$	−1.42792e7	−1.66751	−0.833753	+	0.552137i	$0.813812\pi$
$594$	0	0
$595$	273641.	0.0316876
$596$	−1.29493e7	−1.49324
$597$	0	0
$598$	945326.	0.108101
$599$	−1.13983e7	−1.29799	−0.648997	−	0.760791i	$-0.724811\pi$
$599$	−1.13983e7	−1.29799	−0.648997	+	0.760791i	$0.724811\pi$
$600$	0	0
$601$	−665939.	−0.0752053	−0.0376026	−	0.999293i	$-0.511972\pi$
$601$	−665939.	−0.0752053	−0.0376026	+	0.999293i	$0.511972\pi$
$602$	−72204.0	−0.00812026
$603$	0	0
$604$	1.11223e7	1.24051
$605$	530033.	0.0588728
$606$	0	0
$607$	−1.35927e7	−1.49738	−0.748692	−	0.662918i	$-0.769318\pi$
$607$	−1.35927e7	−1.49738	−0.748692	+	0.662918i	$0.769318\pi$
$608$	7.38596e6	0.810304
$609$	0	0
$610$	−4.60628e6	−0.501217
$611$	3.75398e6	0.406808
$612$	0	0
$613$	−1.08635e7	−1.16767	−0.583833	−	0.811874i	$-0.698448\pi$
$613$	−1.08635e7	−1.16767	−0.583833	+	0.811874i	$0.698448\pi$
$614$	−264405.	−0.0283041
$615$	0	0
$616$	−891182.	−0.0946269
$617$	1.62403e6	0.171744	0.0858720	−	0.996306i	$-0.472632\pi$
$617$	1.62403e6	0.171744	0.0858720	+	0.996306i	$0.472632\pi$
$618$	0	0
$619$	3.82440e6	0.401177	0.200589	−	0.979676i	$-0.435715\pi$
$619$	3.82440e6	0.401177	0.200589	+	0.979676i	$0.435715\pi$
$620$	2.03851e7	2.12978
$621$	0	0
$622$	−205487.	−0.0212965
$623$	−1.03914e6	−0.107264
$624$	0	0
$625$	1.88830e7	1.93362
$626$	−3.44307e6	−0.351164
$627$	0	0
$628$	−4.07735e6	−0.412552
$629$	−2.79601e6	−0.281781
$630$	0	0
$631$	−4.03571e6	−0.403503	−0.201751	−	0.979437i	$-0.564663\pi$
$631$	−4.03571e6	−0.403503	−0.201751	+	0.979437i	$0.564663\pi$
$632$	−274829.	−0.0273697
$633$	0	0
$634$	2.40001e6	0.237132
$635$	1.01396e6	0.0997894
$636$	0	0
$637$	−1.10448e7	−1.07848
$638$	6.21321e6	0.604316
$639$	0	0
$640$	−1.88568e7	−1.81978
$641$	−1.03975e7	−0.999505	−0.499753	−	0.866168i	$-0.666576\pi$
$641$	−1.03975e7	−0.999505	−0.499753	+	0.866168i	$0.666576\pi$
$642$	0	0
$643$	−703463.	−0.0670986	−0.0335493	−	0.999437i	$-0.510681\pi$
$643$	−703463.	−0.0670986	−0.0335493	+	0.999437i	$0.510681\pi$
$644$	230967.	0.0219450
$645$	0	0
$646$	−548069.	−0.0516718
$647$	1.39550e7	1.31060	0.655300	−	0.755369i	$-0.272542\pi$
$647$	1.39550e7	1.31060	0.655300	+	0.755369i	$0.272542\pi$
$648$	0	0
$649$	−3.57692e6	−0.333348
$650$	−1.18412e7	−1.09929
$651$	0	0
$652$	−1.04765e7	−0.965151
$653$	−4.10881e6	−0.377079	−0.188540	−	0.982066i	$-0.560375\pi$
$653$	−4.10881e6	−0.377079	−0.188540	+	0.982066i	$0.560375\pi$
$654$	0	0
$655$	3.40431e7	3.10046
$656$	−3.49187e6	−0.316810
$657$	0	0
$658$	−219802.	−0.0197909
$659$	−4.28147e6	−0.384042	−0.192021	−	0.981391i	$-0.561504\pi$
$659$	−4.28147e6	−0.384042	−0.192021	+	0.981391i	$0.561504\pi$
$660$	0	0
$661$	−6.69686e6	−0.596166	−0.298083	−	0.954540i	$-0.596347\pi$
$661$	−6.69686e6	−0.596166	−0.298083	+	0.954540i	$0.596347\pi$
$662$	7.99274e6	0.708844
$663$	0	0
$664$	7.49528e6	0.659732
$665$	2.03732e6	0.178650
$666$	0	0
$667$	−3.60645e6	−0.313881
$668$	4.30553e6	0.373323
$669$	0	0
$670$	−6.91126e6	−0.594799
$671$	7.21615e6	0.618727
$672$	0	0
$673$	−1.29636e7	−1.10329	−0.551644	−	0.834080i	$-0.685999\pi$
$673$	−1.29636e7	−1.10329	−0.551644	+	0.834080i	$0.685999\pi$
$674$	−2.07780e6	−0.176179
$675$	0	0
$676$	−1.89768e6	−0.159719
$677$	−1.00392e7	−0.841833	−0.420916	−	0.907099i	$-0.638291\pi$
$677$	−1.00392e7	−0.841833	−0.420916	+	0.907099i	$0.638291\pi$
$678$	0	0
$679$	588735.	0.0490055
$680$	2.50617e6	0.207844
$681$	0	0
$682$	7.65312e6	0.630053
$683$	−1.37778e7	−1.13013	−0.565063	−	0.825048i	$-0.691148\pi$
$683$	−1.37778e7	−1.13013	−0.565063	+	0.825048i	$0.691148\pi$
$684$	0	0
$685$	−123329.	−0.0100425
$686$	1.30301e6	0.105715
$687$	0	0
$688$	866065.	0.0697556
$689$	−1.49795e7	−1.20212
$690$	0	0
$691$	−2.12787e7	−1.69532	−0.847658	−	0.530544i	$-0.821988\pi$
$691$	−2.12787e7	−1.69532	−0.847658	+	0.530544i	$0.821988\pi$
$692$	−7.42795e6	−0.589663
$693$	0	0
$694$	5.47044e6	0.431145
$695$	1.59691e7	1.25406
$696$	0	0
$697$	1.28253e6	0.0999969
$698$	−7.23763e6	−0.562287
$699$	0	0
$700$	−2.89310e6	−0.223161
$701$	−2.07790e7	−1.59709	−0.798547	−	0.601933i	$-0.794398\pi$
$701$	−2.07790e7	−1.59709	−0.798547	+	0.601933i	$0.794398\pi$
$702$	0	0
$703$	−2.08169e7	−1.58864
$704$	255166.	0.0194040
$705$	0	0
$706$	6.56274e6	0.495534
$707$	557960.	0.0419812
$708$	0	0
$709$	4.16247e6	0.310982	0.155491	−	0.987837i	$-0.450304\pi$
$709$	4.16247e6	0.310982	0.155491	+	0.987837i	$0.450304\pi$
$710$	3.24062e6	0.241258
$711$	0	0
$712$	−9.51708e6	−0.703564
$713$	−4.44224e6	−0.327249
$714$	0	0
$715$	2.66711e7	1.95108
$716$	1.34898e7	0.983381
$717$	0	0
$718$	−92516.4	−0.00669742
$719$	1.69124e7	1.22007	0.610033	−	0.792376i	$-0.291156\pi$
$719$	1.69124e7	1.22007	0.610033	+	0.792376i	$0.291156\pi$
$720$	0	0
$721$	2.79399e6	0.200164
$722$	2.07807e6	0.148360
$723$	0	0
$724$	−2.09786e6	−0.148741
$725$	4.51743e7	3.19188
$726$	0	0
$727$	1.27084e7	0.891776	0.445888	−	0.895089i	$-0.352888\pi$
$727$	1.27084e7	0.891776	0.445888	+	0.895089i	$0.352888\pi$
$728$	1.50571e6	0.105296
$729$	0	0
$730$	−1.59869e7	−1.11034
$731$	−318098.	−0.0220174
$732$	0	0
$733$	1.32015e6	0.0907535	0.0453767	−	0.998970i	$-0.485551\pi$
$733$	1.32015e6	0.0907535	0.0453767	+	0.998970i	$0.485551\pi$
$734$	−5.65040e6	−0.387114
$735$	0	0
$736$	3.28618e6	0.223613
$737$	1.08271e7	0.734249
$738$	0	0
$739$	2.42511e7	1.63351	0.816753	−	0.576988i	$-0.195772\pi$
$739$	2.42511e7	1.63351	0.816753	+	0.576988i	$0.195772\pi$
$740$	4.25022e7	2.85320
$741$	0	0
$742$	877073.	0.0584825
$743$	1.31970e7	0.877009	0.438504	−	0.898729i	$-0.355508\pi$
$743$	1.31970e7	0.877009	0.438504	+	0.898729i	$0.355508\pi$
$744$	0	0
$745$	−5.08203e7	−3.35465
$746$	−6.02750e6	−0.396543
$747$	0	0
$748$	−1.75302e6	−0.114560
$749$	−684381.	−0.0445752
$750$	0	0
$751$	−1.06321e7	−0.687891	−0.343945	−	0.938990i	$-0.611763\pi$
$751$	−1.06321e7	−0.687891	−0.343945	+	0.938990i	$0.611763\pi$
$752$	2.63645e6	0.170010
$753$	0	0
$754$	−1.04976e7	−0.672455
$755$	4.36502e7	2.78688
$756$	0	0
$757$	2.44667e7	1.55180	0.775900	−	0.630856i	$-0.217296\pi$
$757$	2.44667e7	1.55180	0.775900	+	0.630856i	$0.217296\pi$
$758$	5.55094e6	0.350908
$759$	0	0
$760$	1.86590e7	1.17180
$761$	−2.20641e7	−1.38110	−0.690550	−	0.723285i	$-0.742631\pi$
$761$	−2.20641e7	−1.38110	−0.690550	+	0.723285i	$0.742631\pi$
$762$	0	0
$763$	−770652.	−0.0479233
$764$	−1.57154e7	−0.974076
$765$	0	0
$766$	−2.82938e6	−0.174228
$767$	6.04345e6	0.370934
$768$	0	0
$769$	2.58590e7	1.57687	0.788434	−	0.615119i	$-0.210892\pi$
$769$	2.58590e7	1.57687	0.788434	+	0.615119i	$0.210892\pi$
$770$	−1.56163e6	−0.0949188
$771$	0	0
$772$	−4.77450e6	−0.288327
$773$	1.57927e7	0.950620	0.475310	−	0.879818i	$-0.342336\pi$
$773$	1.57927e7	0.950620	0.475310	+	0.879818i	$0.342336\pi$
$774$	0	0
$775$	5.56435e7	3.32782
$776$	5.39199e6	0.321436
$777$	0	0
$778$	4.90064e6	0.290271
$779$	9.54872e6	0.563770
$780$	0	0
$781$	−5.07672e6	−0.297821
$782$	−243848.	−0.0142594
$783$	0	0
$784$	−7.75687e6	−0.450709
$785$	−1.60018e7	−0.926821
$786$	0	0
$787$	−2.14880e7	−1.23669	−0.618344	−	0.785908i	$-0.712196\pi$
$787$	−2.14880e7	−1.23669	−0.618344	+	0.785908i	$0.712196\pi$
$788$	5.18726e6	0.297593
$789$	0	0
$790$	−481587.	−0.0274541
$791$	542139.	0.0308084
$792$	0	0
$793$	−1.21922e7	−0.688491
$794$	1.27625e7	0.718432
$795$	0	0
$796$	6.03148e6	0.337397
$797$	1.54431e7	0.861170	0.430585	−	0.902550i	$-0.358307\pi$
$797$	1.54431e7	0.861170	0.430585	+	0.902550i	$0.358307\pi$
$798$	0	0
$799$	−968344.	−0.0536615
$800$	−4.11627e7	−2.27394
$801$	0	0
$802$	−7.27088e6	−0.399164
$803$	2.50448e7	1.37066
$804$	0	0
$805$	906448.	0.0493007
$806$	−1.29305e7	−0.701094
$807$	0	0
$808$	5.11014e6	0.275362
$809$	5.05884e6	0.271756	0.135878	−	0.990726i	$-0.456614\pi$
$809$	5.05884e6	0.271756	0.135878	+	0.990726i	$0.456614\pi$
$810$	0	0
$811$	2.95840e7	1.57944	0.789722	−	0.613465i	$-0.210225\pi$
$811$	2.95840e7	1.57944	0.789722	+	0.613465i	$0.210225\pi$
$812$	−2.56484e6	−0.136512
$813$	0	0
$814$	1.59564e7	0.844063
$815$	−4.11156e7	−2.16826
$816$	0	0
$817$	−2.36830e6	−0.124132
$818$	5.23861e6	0.273737
$819$	0	0
$820$	−1.94958e7	−1.01253
$821$	−4.56831e6	−0.236536	−0.118268	−	0.992982i	$-0.537734\pi$
$821$	−4.56831e6	−0.236536	−0.118268	+	0.992982i	$0.537734\pi$
$822$	0	0
$823$	3.99029e6	0.205355	0.102677	−	0.994715i	$-0.467259\pi$
$823$	3.99029e6	0.205355	0.102677	+	0.994715i	$0.467259\pi$
$824$	2.55890e7	1.31291
$825$	0	0
$826$	−353853.	−0.0180457
$827$	−7.46365e6	−0.379479	−0.189739	−	0.981835i	$-0.560764\pi$
$827$	−7.46365e6	−0.379479	−0.189739	+	0.981835i	$0.560764\pi$
$828$	0	0
$829$	−1.22180e7	−0.617467	−0.308734	−	0.951149i	$-0.599905\pi$
$829$	−1.22180e7	−0.617467	−0.308734	+	0.951149i	$0.599905\pi$
$830$	1.31341e7	0.661768
$831$	0	0
$832$	−431121.	−0.0215919
$833$	2.84903e6	0.142260
$834$	0	0
$835$	1.68973e7	0.838692
$836$	−1.30516e7	−0.645874
$837$	0	0
$838$	−1.95916e6	−0.0963740
$839$	1.83694e7	0.900929	0.450464	−	0.892794i	$-0.351258\pi$
$839$	1.83694e7	0.900929	0.450464	+	0.892794i	$0.351258\pi$
$840$	0	0
$841$	1.95376e7	0.952536
$842$	6.81147e6	0.331101
$843$	0	0
$844$	−5.17601e6	−0.250115
$845$	−7.44760e6	−0.358818
$846$	0	0
$847$	−82143.3	−0.00393426
$848$	−1.05202e7	−0.502383
$849$	0	0
$850$	3.05444e6	0.145006
$851$	−9.26189e6	−0.438405
$852$	0	0
$853$	−3.51699e7	−1.65500	−0.827501	−	0.561465i	$-0.810238\pi$
$853$	−3.51699e7	−1.65500	−0.827501	+	0.561465i	$0.810238\pi$
$854$	713871.	0.0334946
$855$	0	0
$856$	−6.26798e6	−0.292377
$857$	1.23215e7	0.573076	0.286538	−	0.958069i	$-0.407496\pi$
$857$	1.23215e7	0.573076	0.286538	+	0.958069i	$0.407496\pi$
$858$	0	0
$859$	2.32476e6	0.107497	0.0537484	−	0.998555i	$-0.482883\pi$
$859$	2.32476e6	0.107497	0.0537484	+	0.998555i	$0.482883\pi$
$860$	4.83541e6	0.222940
$861$	0	0
$862$	5.56244e6	0.254975
$863$	−1.53407e7	−0.701163	−0.350582	−	0.936532i	$-0.614016\pi$
$863$	−1.53407e7	−0.701163	−0.350582	+	0.936532i	$0.614016\pi$
$864$	0	0
$865$	−2.91515e7	−1.32471
$866$	1.82140e7	0.825297
$867$	0	0
$868$	−3.15924e6	−0.142326
$869$	754449.	0.0338907
$870$	0	0
$871$	−1.82931e7	−0.817038
$872$	−7.05809e6	−0.314337
$873$	0	0
$874$	−1.81550e6	−0.0803929
$875$	−6.38358e6	−0.281867
$876$	0	0
$877$	1.21207e7	0.532142	0.266071	−	0.963953i	$-0.414274\pi$
$877$	1.21207e7	0.532142	0.266071	+	0.963953i	$0.414274\pi$
$878$	−1.00861e7	−0.441559
$879$	0	0
$880$	1.87313e7	0.815383
$881$	−208397.	−0.00904589	−0.00452294	−	0.999990i	$-0.501440\pi$
$881$	−208397.	−0.00904589	−0.00452294	+	0.999990i	$0.501440\pi$
$882$	0	0
$883$	−3.57371e7	−1.54247	−0.771237	−	0.636548i	$-0.780362\pi$
$883$	−3.57371e7	−1.54247	−0.771237	+	0.636548i	$0.780362\pi$
$884$	2.96184e6	0.127477
$885$	0	0
$886$	5.55604e6	0.237783
$887$	−4.64203e7	−1.98107	−0.990533	−	0.137276i	$-0.956165\pi$
$887$	−4.64203e7	−1.98107	−0.990533	+	0.137276i	$0.956165\pi$
$888$	0	0
$889$	−157140.	−0.00666858
$890$	−1.66769e7	−0.705734
$891$	0	0
$892$	−8.95871e6	−0.376993
$893$	−7.20952e6	−0.302537
$894$	0	0
$895$	5.29415e7	2.20922
$896$	2.92239e6	0.121610
$897$	0	0
$898$	−9.70472e6	−0.401598
$899$	4.93300e7	2.03569
$900$	0	0
$901$	3.86398e6	0.158571
$902$	−7.31924e6	−0.299537
$903$	0	0
$904$	4.96523e6	0.202078
$905$	−8.23320e6	−0.334155
$906$	0	0
$907$	−2.81640e7	−1.13678	−0.568390	−	0.822759i	$-0.692434\pi$
$907$	−2.81640e7	−1.13678	−0.568390	+	0.822759i	$0.692434\pi$
$908$	−5.47541e6	−0.220395
$909$	0	0
$910$	2.63849e6	0.105621
$911$	−8.25253e6	−0.329451	−0.164726	−	0.986339i	$-0.552674\pi$
$911$	−8.25253e6	−0.329451	−0.164726	+	0.986339i	$0.552674\pi$
$912$	0	0
$913$	−2.05758e7	−0.816919
$914$	457889.	0.0181299
$915$	0	0
$916$	1.21487e7	0.478402
$917$	−5.27593e6	−0.207193
$918$	0	0
$919$	−1.14244e6	−0.0446217	−0.0223108	−	0.999751i	$-0.507102\pi$
$919$	−1.14244e6	−0.0446217	−0.0223108	+	0.999751i	$0.507102\pi$
$920$	8.30179e6	0.323372
$921$	0	0
$922$	−1.21861e7	−0.472104
$923$	8.57746e6	0.331401
$924$	0	0
$925$	1.16014e8	4.45818
$926$	6.78583e6	0.260061
$927$	0	0
$928$	−3.64923e7	−1.39101
$929$	3.37164e7	1.28175	0.640873	−	0.767647i	$-0.278572\pi$
$929$	3.37164e7	1.28175	0.640873	+	0.767647i	$0.278572\pi$
$930$	0	0
$931$	2.12116e7	0.802046
$932$	−2.45812e7	−0.926963
$933$	0	0
$934$	7.92627e6	0.297305
$935$	−6.87984e6	−0.257365
$936$	0	0
$937$	−4.06327e7	−1.51191	−0.755955	−	0.654623i	$-0.772827\pi$
$937$	−4.06327e7	−1.51191	−0.755955	+	0.654623i	$0.772827\pi$
$938$	1.07109e6	0.0397484
$939$	0	0
$940$	1.47198e7	0.543355
$941$	−4.51496e7	−1.66219	−0.831093	−	0.556134i	$-0.812284\pi$
$941$	−4.51496e7	−1.66219	−0.831093	+	0.556134i	$0.812284\pi$
$942$	0	0
$943$	4.24844e6	0.155579
$944$	4.24436e6	0.155018
$945$	0	0
$946$	1.81534e6	0.0659524
$947$	1.85178e7	0.670986	0.335493	−	0.942043i	$-0.391097\pi$
$947$	1.85178e7	0.670986	0.335493	+	0.942043i	$0.391097\pi$
$948$	0	0
$949$	−4.23149e7	−1.52520
$950$	2.27410e7	0.817523
$951$	0	0
$952$	−388400.	−0.0138895
$953$	3.17478e7	1.13235	0.566176	−	0.824284i	$-0.308422\pi$
$953$	3.17478e7	1.13235	0.566176	+	0.824284i	$0.308422\pi$
$954$	0	0
$955$	−6.16763e7	−2.18832
$956$	−4.22546e7	−1.49530
$957$	0	0
$958$	1.22824e7	0.432385
$959$	19113.3	0.000671103 0
$960$	0	0
$961$	3.21331e7	1.12239
$962$	−2.69595e7	−0.939234
$963$	0	0
$964$	1.79650e7	0.622636
$965$	−1.87379e7	−0.647742
$966$	0	0
$967$	4.45707e7	1.53279	0.766396	−	0.642368i	$-0.222048\pi$
$967$	4.45707e7	1.53279	0.766396	+	0.642368i	$0.222048\pi$
$968$	−752318.	−0.0258055
$969$	0	0
$970$	9.44847e6	0.322428
$971$	−2.22017e7	−0.755679	−0.377840	−	0.925871i	$-0.623333\pi$
$971$	−2.22017e7	−0.755679	−0.377840	+	0.925871i	$0.623333\pi$
$972$	0	0
$973$	−2.47485e6	−0.0838045
$974$	−1.32949e6	−0.0449041
$975$	0	0
$976$	−8.56265e6	−0.287729
$977$	3.20865e6	0.107544	0.0537719	−	0.998553i	$-0.482876\pi$
$977$	3.20865e6	0.107544	0.0537719	+	0.998553i	$0.482876\pi$
$978$	0	0
$979$	2.61259e7	0.871193
$980$	−4.33082e7	−1.44047
$981$	0	0
$982$	1.15673e7	0.382783
$983$	4.20453e7	1.38782	0.693910	−	0.720061i	$-0.255886\pi$
$983$	4.20453e7	1.38782	0.693910	+	0.720061i	$0.255886\pi$
$984$	0	0
$985$	2.03578e7	0.668558
$986$	2.70788e6	0.0887027
$987$	0	0
$988$	2.20515e7	0.718698
$989$	−1.05371e6	−0.0342556
$990$	0	0
$991$	1.08440e7	0.350755	0.175377	−	0.984501i	$-0.443885\pi$
$991$	1.08440e7	0.350755	0.175377	+	0.984501i	$0.443885\pi$
$992$	−4.49494e7	−1.45025
$993$	0	0
$994$	−502224.	−0.0161225
$995$	2.36710e7	0.757981
$996$	0	0
$997$	3.29287e7	1.04915	0.524574	−	0.851365i	$-0.324225\pi$
$997$	3.29287e7	1.04915	0.524574	+	0.851365i	$0.324225\pi$
$998$	536715.	0.0170576
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.6.a.e.1.5		10
3.2	odd	2		43.6.a.b.1.6	✓	10
12.11	even	2		688.6.a.h.1.10		10
15.14	odd	2		1075.6.a.b.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.6	✓	10	3.2	odd	2
387.6.a.e.1.5		10	1.1	even	1	trivial
688.6.a.h.1.10		10	12.11	even	2
1075.6.a.b.1.5		10	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q-2.48720 q^{2} -25.8138 q^{4} -101.308 q^{5} +15.7005 q^{7} +143.795 q^{8} +O(q^{10})\)	\(q-2.48720 q^{2} -25.8138 q^{4} -101.308 q^{5} +15.7005 q^{7} +143.795 q^{8} +251.974 q^{10} -394.739 q^{11} +666.939 q^{13} -39.0503 q^{14} +468.396 q^{16} -172.038 q^{17} -1280.86 q^{19} +2615.15 q^{20} +981.796 q^{22} -569.882 q^{23} +7138.34 q^{25} -1658.81 q^{26} -405.290 q^{28} +6328.41 q^{29} +7795.01 q^{31} -5766.42 q^{32} +427.892 q^{34} -1590.59 q^{35} +16252.3 q^{37} +3185.75 q^{38} -14567.6 q^{40} -7454.95 q^{41} +1849.00 q^{43} +10189.7 q^{44} +1417.41 q^{46} +5628.68 q^{47} -16560.5 q^{49} -17754.5 q^{50} -17216.2 q^{52} -22460.1 q^{53} +39990.3 q^{55} +2257.65 q^{56} -15740.0 q^{58} +9061.48 q^{59} -18280.8 q^{61} -19387.8 q^{62} -646.417 q^{64} -67566.3 q^{65} -27428.5 q^{67} +4440.95 q^{68} +3956.11 q^{70} +12860.9 q^{71} -63446.5 q^{73} -40422.7 q^{74} +33063.8 q^{76} -6197.60 q^{77} -1911.26 q^{79} -47452.4 q^{80} +18542.0 q^{82} +52124.9 q^{83} +17428.8 q^{85} -4598.84 q^{86} -56761.4 q^{88} -66185.2 q^{89} +10471.3 q^{91} +14710.8 q^{92} -13999.7 q^{94} +129761. q^{95} +37497.8 q^{97} +41189.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8}+O(q^{10}) \) 10 * q - 8 * q^2 + 202 * q^4 - 138 * q^5 + 60 * q^7 - 294 * q^8	\( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8} - 17 q^{10} - 745 q^{11} + 1917 q^{13} - 1936 q^{14} + 5354 q^{16} - 4017 q^{17} - 2404 q^{19} - 1311 q^{20} - 5836 q^{22} - 1733 q^{23} + 7120 q^{25} + 1484 q^{26} - 15028 q^{28} - 6996 q^{29} - 4899 q^{31} + 7554 q^{32} - 27033 q^{34} - 7084 q^{35} + 1466 q^{37} - 13905 q^{38} - 93211 q^{40} - 10297 q^{41} + 18490 q^{43} + 36140 q^{44} + 17991 q^{46} - 48592 q^{47} + 29458 q^{49} - 983 q^{50} + 14232 q^{52} - 127165 q^{53} + 106672 q^{55} + 7780 q^{56} - 10305 q^{58} - 99372 q^{59} + 17408 q^{61} - 28265 q^{62} + 47202 q^{64} - 54484 q^{65} - 2021 q^{67} - 192151 q^{68} - 33194 q^{70} - 11286 q^{71} + 49892 q^{73} + 125431 q^{74} - 249803 q^{76} - 98144 q^{77} - 91524 q^{79} - 12251 q^{80} - 158909 q^{82} + 105203 q^{83} - 87212 q^{85} - 14792 q^{86} - 461824 q^{88} + 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 7259 q^{94} + 305340 q^{95} + 108383 q^{97} - 354656 q^{98}+O(q^{100}) \) 10 * q - 8 * q^2 + 202 * q^4 - 138 * q^5 + 60 * q^7 - 294 * q^8 - 17 * q^10 - 745 * q^11 + 1917 * q^13 - 1936 * q^14 + 5354 * q^16 - 4017 * q^17 - 2404 * q^19 - 1311 * q^20 - 5836 * q^22 - 1733 * q^23 + 7120 * q^25 + 1484 * q^26 - 15028 * q^28 - 6996 * q^29 - 4899 * q^31 + 7554 * q^32 - 27033 * q^34 - 7084 * q^35 + 1466 * q^37 - 13905 * q^38 - 93211 * q^40 - 10297 * q^41 + 18490 * q^43 + 36140 * q^44 + 17991 * q^46 - 48592 * q^47 + 29458 * q^49 - 983 * q^50 + 14232 * q^52 - 127165 * q^53 + 106672 * q^55 + 7780 * q^56 - 10305 * q^58 - 99372 * q^59 + 17408 * q^61 - 28265 * q^62 + 47202 * q^64 - 54484 * q^65 - 2021 * q^67 - 192151 * q^68 - 33194 * q^70 - 11286 * q^71 + 49892 * q^73 + 125431 * q^74 - 249803 * q^76 - 98144 * q^77 - 91524 * q^79 - 12251 * q^80 - 158909 * q^82 + 105203 * q^83 - 87212 * q^85 - 14792 * q^86 - 461824 * q^88 + 62682 * q^89 - 295304 * q^91 - 183783 * q^92 + 7259 * q^94 + 305340 * q^95 + 108383 * q^97 - 354656 * q^98

Introduction

L-functions

Modular forms

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