LMFDB - Embedded newform 729.3.b.a.728.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,3,Mod(728,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("729.728");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	729.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$19.8638112719$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			728.10
Character	$\chi$	$=$	729.728
Dual form			729.3.b.a.728.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.53349i q^{2} +1.64842 q^{4} -0.0504138i q^{5} +9.85801 q^{7} -8.66177i q^{8} -0.0773090 q^{10} -12.2543i q^{11} +12.2245 q^{13} -15.1171i q^{14} -6.68905 q^{16} +14.7703i q^{17} -12.5317 q^{19} -0.0831031i q^{20} -18.7918 q^{22} -9.23045i q^{23} +24.9975 q^{25} -18.7461i q^{26} +16.2501 q^{28} +44.3623i q^{29} -31.6092 q^{31} -24.3895i q^{32} +22.6500 q^{34} -0.496980i q^{35} +14.3960 q^{37} +19.2172i q^{38} -0.436673 q^{40} -68.9640i q^{41} +40.4256 q^{43} -20.2002i q^{44} -14.1548 q^{46} +1.94368i q^{47} +48.1804 q^{49} -38.3333i q^{50} +20.1511 q^{52} +47.7390i q^{53} -0.617787 q^{55} -85.3879i q^{56} +68.0291 q^{58} -73.7363i q^{59} -5.93231 q^{61} +48.4723i q^{62} -64.1572 q^{64} -0.616285i q^{65} -48.8013 q^{67} +24.3475i q^{68} -0.762113 q^{70} +9.94797i q^{71} +88.4943 q^{73} -22.0761i q^{74} -20.6574 q^{76} -120.803i q^{77} -111.986 q^{79} +0.337221i q^{80} -105.755 q^{82} -19.4647i q^{83} +0.744625 q^{85} -61.9921i q^{86} -106.144 q^{88} -39.1940i q^{89} +120.509 q^{91} -15.2156i q^{92} +2.98061 q^{94} +0.631770i q^{95} -0.386290 q^{97} -73.8841i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67}+ \cdots - 24 q^{94}+O(q^{100}) $ 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 1.53349i	− 0.766744i	−0.923594	−	0.383372i	$-0.874763\pi$
$2$	− 1.53349i	− 0.766744i	0.923594	−	0.383372i	$-0.125237\pi$
$3$	0	0
$3$	0	0
$4$	1.64842	0.412104
$5$	− 0.0504138i	− 0.0100828i	−0.999987	−	0.00504138i	$-0.998395\pi$
$5$	− 0.0504138i	− 0.0100828i	0.999987	−	0.00504138i	$-0.00160473\pi$
$6$	0	0
$7$	9.85801	1.40829	0.704144	−	0.710057i	$-0.251331\pi$
$7$	9.85801	1.40829	0.704144	+	0.710057i	$0.251331\pi$
$8$	− 8.66177i	− 1.08272i
$9$	0	0
$10$	−0.0773090	−0.00773090
$11$	− 12.2543i	− 1.11403i	−0.830503	−	0.557014i	$-0.811947\pi$
$11$	− 12.2543i	− 1.11403i	0.830503	−	0.557014i	$-0.188053\pi$
$12$	0	0
$13$	12.2245	0.940347	0.470174	−	0.882574i	$-0.344191\pi$
$13$	12.2245	0.940347	0.470174	+	0.882574i	$0.344191\pi$
$14$	− 15.1171i	− 1.07980i
$15$	0	0
$16$	−6.68905	−0.418066
$17$	14.7703i	0.868839i	0.900711	+	0.434419i	$0.143046\pi$
$17$	14.7703i	0.868839i	−0.900711	+	0.434419i	$0.856954\pi$
$18$	0	0
$19$	−12.5317	−0.659562	−0.329781	−	0.944058i	$-0.606975\pi$
$19$	−12.5317	−0.659562	−0.329781	+	0.944058i	$0.606975\pi$
$20$	− 0.0831031i	− 0.00415515i
$21$	0	0
$22$	−18.7918	−0.854174
$23$	− 9.23045i	− 0.401324i	−0.979661	−	0.200662i	$-0.935691\pi$
$23$	− 9.23045i	− 0.401324i	0.979661	−	0.200662i	$-0.0643093\pi$
$24$	0	0
$25$	24.9975	0.999898
$26$	− 18.7461i	− 0.721005i
$27$	0	0
$28$	16.2501	0.580362
$29$	44.3623i	1.52974i	0.644187	+	0.764868i	$0.277196\pi$
$29$	44.3623i	1.52974i	−0.644187	+	0.764868i	$0.722804\pi$
$30$	0	0
$31$	−31.6092	−1.01965	−0.509826	−	0.860278i	$-0.670290\pi$
$31$	−31.6092	−1.01965	−0.509826	+	0.860278i	$0.670290\pi$
$32$	− 24.3895i	− 0.762173i
$33$	0	0
$34$	22.6500	0.666176
$35$	− 0.496980i	− 0.0141994i
$36$	0	0
$37$	14.3960	0.389082	0.194541	−	0.980894i	$-0.437678\pi$
$37$	14.3960	0.389082	0.194541	+	0.980894i	$0.437678\pi$
$38$	19.2172i	0.505715i
$39$	0	0
$40$	−0.436673	−0.0109168
$41$	− 68.9640i	− 1.68205i	−0.540997	−	0.841025i	$-0.681953\pi$
$41$	− 68.9640i	− 1.68205i	0.540997	−	0.841025i	$-0.318047\pi$
$42$	0	0
$43$	40.4256	0.940130	0.470065	−	0.882632i	$-0.344230\pi$
$43$	40.4256	0.940130	0.470065	+	0.882632i	$0.344230\pi$
$44$	− 20.2002i	− 0.459096i
$45$	0	0
$46$	−14.1548	−0.307712
$47$	1.94368i	0.0413550i	0.999786	+	0.0206775i	$0.00658232\pi$
$47$	1.94368i	0.0413550i	−0.999786	+	0.0206775i	$0.993418\pi$
$48$	0	0
$49$	48.1804	0.983274
$50$	− 38.3333i	− 0.766666i
$51$	0	0
$52$	20.1511	0.387521
$53$	47.7390i	0.900736i	0.892843	+	0.450368i	$0.148707\pi$
$53$	47.7390i	0.900736i	−0.892843	+	0.450368i	$0.851293\pi$
$54$	0	0
$55$	−0.617787	−0.0112325
$56$	− 85.3879i	− 1.52478i
$57$	0	0
$58$	68.0291	1.17291
$59$	− 73.7363i	− 1.24977i	−0.780717	−	0.624884i	$-0.785146\pi$
$59$	− 73.7363i	− 1.24977i	0.780717	−	0.624884i	$-0.214854\pi$
$60$	0	0
$61$	−5.93231	−0.0972510	−0.0486255	−	0.998817i	$-0.515484\pi$
$61$	−5.93231	−0.0972510	−0.0486255	+	0.998817i	$0.515484\pi$
$62$	48.4723i	0.781811i
$63$	0	0
$64$	−64.1572	−1.00246
$65$	− 0.616285i	− 0.00948130i
$66$	0	0
$67$	−48.8013	−0.728378	−0.364189	−	0.931325i	$-0.618654\pi$
$67$	−48.8013	−0.728378	−0.364189	+	0.931325i	$0.618654\pi$
$68$	24.3475i	0.358052i
$69$	0	0
$70$	−0.762113	−0.0108873
$71$	9.94797i	0.140112i	0.997543	+	0.0700561i	$0.0223178\pi$
$71$	9.94797i	0.140112i	−0.997543	+	0.0700561i	$0.977682\pi$
$72$	0	0
$73$	88.4943	1.21225	0.606126	−	0.795369i	$-0.292723\pi$
$73$	88.4943	1.21225	0.606126	+	0.795369i	$0.292723\pi$
$74$	− 22.0761i	− 0.298326i
$75$	0	0
$76$	−20.6574	−0.271808
$77$	− 120.803i	− 1.56887i
$78$	0	0
$79$	−111.986	−1.41755	−0.708775	−	0.705434i	$-0.750752\pi$
$79$	−111.986	−1.41755	−0.708775	+	0.705434i	$0.750752\pi$
$80$	0.337221i	0.00421526i
$81$	0	0
$82$	−105.755	−1.28970
$83$	− 19.4647i	− 0.234514i	−0.993102	−	0.117257i	$-0.962590\pi$
$83$	− 19.4647i	− 0.234514i	0.993102	−	0.117257i	$-0.0374102\pi$
$84$	0	0
$85$	0.744625	0.00876030
$86$	− 61.9921i	− 0.720839i
$87$	0	0
$88$	−106.144	−1.20618
$89$	− 39.1940i	− 0.440382i	−0.975457	−	0.220191i	$-0.929332\pi$
$89$	− 39.1940i	− 0.440382i	0.975457	−	0.220191i	$-0.0706681\pi$
$90$	0	0
$91$	120.509	1.32428
$92$	− 15.2156i	− 0.165387i
$93$	0	0
$94$	2.98061	0.0317086
$95$	0.631770i	0.00665021i
$96$	0	0
$97$	−0.386290	−0.00398237	−0.00199118	−	0.999998i	$-0.500634\pi$
$97$	−0.386290	−0.00398237	−0.00199118	+	0.999998i	$0.500634\pi$
$98$	− 73.8841i	− 0.753919i
$99$	0	0
$100$	41.2063	0.412063
$101$	47.7119i	0.472395i	0.971705	+	0.236197i	$0.0759012\pi$
$101$	47.7119i	0.472395i	−0.971705	+	0.236197i	$0.924099\pi$
$102$	0	0
$103$	−18.3308	−0.177969	−0.0889843	−	0.996033i	$-0.528362\pi$
$103$	−18.3308	−0.177969	−0.0889843	+	0.996033i	$0.528362\pi$
$104$	− 105.886i	− 1.01813i
$105$	0	0
$106$	73.2072	0.690634
$107$	− 68.8842i	− 0.643778i	−0.946777	−	0.321889i	$-0.895682\pi$
$107$	− 68.8842i	− 0.643778i	0.946777	−	0.321889i	$-0.104318\pi$
$108$	0	0
$109$	9.57392	0.0878341	0.0439171	−	0.999035i	$-0.486016\pi$
$109$	9.57392	0.0878341	0.0439171	+	0.999035i	$0.486016\pi$
$110$	0.947368i	0.00861244i
$111$	0	0
$112$	−65.9407	−0.588757
$113$	0.505174i	0.00447057i	0.999998	+	0.00223528i	$0.000711513\pi$
$113$	0.505174i	0.00447057i	−0.999998	+	0.00223528i	$0.999288\pi$
$114$	0	0
$115$	−0.465342	−0.00404645
$116$	73.1277i	0.630411i
$117$	0	0
$118$	−113.074	−0.958252
$119$	145.605i	1.22357i
$120$	0	0
$121$	−29.1682	−0.241059
$122$	9.09712i	0.0745665i
$123$	0	0
$124$	−52.1051	−0.420203
$125$	− 2.52056i	− 0.0201645i
$126$	0	0
$127$	−104.552	−0.823246	−0.411623	−	0.911354i	$-0.635038\pi$
$127$	−104.552	−0.823246	−0.411623	+	0.911354i	$0.635038\pi$
$128$	0.826147i	0.00645427i
$129$	0	0
$130$	−0.945064	−0.00726973
$131$	201.703i	1.53971i	0.638217	+	0.769857i	$0.279672\pi$
$131$	201.703i	1.53971i	−0.638217	+	0.769857i	$0.720328\pi$
$132$	0	0
$133$	−123.537	−0.928853
$134$	74.8362i	0.558479i
$135$	0	0
$136$	127.937	0.940710
$137$	193.901i	1.41534i	0.706545	+	0.707668i	$0.250253\pi$
$137$	193.901i	1.41534i	−0.706545	+	0.707668i	$0.749747\pi$
$138$	0	0
$139$	28.7336	0.206717	0.103358	−	0.994644i	$-0.467041\pi$
$139$	28.7336	0.206717	0.103358	+	0.994644i	$0.467041\pi$
$140$	− 0.819231i	− 0.00585165i
$141$	0	0
$142$	15.2551	0.107430
$143$	− 149.803i	− 1.04757i
$144$	0	0
$145$	2.23648	0.0154240
$146$	− 135.705i	− 0.929486i
$147$	0	0
$148$	23.7306	0.160342
$149$	− 248.832i	− 1.67001i	−0.550241	−	0.835006i	$-0.685464\pi$
$149$	− 248.832i	− 1.67001i	0.550241	−	0.835006i	$-0.314536\pi$
$150$	0	0
$151$	−146.208	−0.968262	−0.484131	−	0.874996i	$-0.660864\pi$
$151$	−146.208	−0.968262	−0.484131	+	0.874996i	$0.660864\pi$
$152$	108.547i	0.714122i
$153$	0	0
$154$	−185.250	−1.20292
$155$	1.59354i	0.0102809i
$156$	0	0
$157$	104.167	0.663482	0.331741	−	0.943370i	$-0.392364\pi$
$157$	104.167	0.663482	0.331741	+	0.943370i	$0.392364\pi$
$158$	171.730i	1.08690i
$159$	0	0
$160$	−1.22957	−0.00768481
$161$	− 90.9939i	− 0.565179i
$162$	0	0
$163$	126.261	0.774609	0.387304	−	0.921952i	$-0.373406\pi$
$163$	126.261	0.774609	0.387304	+	0.921952i	$0.373406\pi$
$164$	− 113.681i	− 0.693180i
$165$	0	0
$166$	−29.8489	−0.179812
$167$	244.586i	1.46458i	0.680991	+	0.732292i	$0.261549\pi$
$167$	244.586i	1.46458i	−0.680991	+	0.732292i	$0.738451\pi$
$168$	0	0
$169$	−19.5613	−0.115747
$170$	− 1.14187i	− 0.00671690i
$171$	0	0
$172$	66.6383	0.387432
$173$	− 91.9435i	− 0.531465i	−0.964047	−	0.265733i	$-0.914386\pi$
$173$	− 91.9435i	− 0.531465i	0.964047	−	0.265733i	$-0.0856138\pi$
$174$	0	0
$175$	246.425	1.40814
$176$	81.9697i	0.465737i
$177$	0	0
$178$	−60.1035	−0.337660
$179$	63.3253i	0.353773i	0.984231	+	0.176886i	$0.0566025\pi$
$179$	63.3253i	0.353773i	−0.984231	+	0.176886i	$0.943398\pi$
$180$	0	0
$181$	154.991	0.856305	0.428153	−	0.903706i	$-0.359165\pi$
$181$	154.991	0.856305	0.428153	+	0.903706i	$0.359165\pi$
$182$	− 184.800i	− 1.01538i
$183$	0	0
$184$	−79.9521	−0.434522
$185$	− 0.725758i	− 0.00392302i
$186$	0	0
$187$	180.999	0.967911
$188$	3.20400i	0.0170426i
$189$	0	0
$190$	0.968811	0.00509900
$191$	57.3066i	0.300035i	0.988683	+	0.150017i	$0.0479329\pi$
$191$	57.3066i	0.300035i	−0.988683	+	0.150017i	$0.952067\pi$
$192$	0	0
$193$	−9.18758	−0.0476040	−0.0238020	−	0.999717i	$-0.507577\pi$
$193$	−9.18758	−0.0476040	−0.0238020	+	0.999717i	$0.507577\pi$
$194$	0.592370i	0.00305345i
$195$	0	0
$196$	79.4215	0.405212
$197$	82.9506i	0.421069i	0.977586	+	0.210535i	$0.0675204\pi$
$197$	82.9506i	0.421069i	−0.977586	+	0.210535i	$0.932480\pi$
$198$	0	0
$199$	36.6216	0.184028	0.0920141	−	0.995758i	$-0.470670\pi$
$199$	36.6216	0.184028	0.0920141	+	0.995758i	$0.470670\pi$
$200$	− 216.522i	− 1.08261i
$201$	0	0
$202$	73.1655	0.362206
$203$	437.325i	2.15431i
$204$	0	0
$205$	−3.47674	−0.0169597
$206$	28.1100i	0.136456i
$207$	0	0
$208$	−81.7704	−0.393127
$209$	153.567i	0.734771i
$210$	0	0
$211$	−308.822	−1.46361	−0.731806	−	0.681513i	$-0.761322\pi$
$211$	−308.822	−1.46361	−0.731806	+	0.681513i	$0.761322\pi$
$212$	78.6939i	0.371197i
$213$	0	0
$214$	−105.633	−0.493612
$215$	− 2.03801i	− 0.00947911i
$216$	0	0
$217$	−311.604	−1.43596
$218$	− 14.6815i	− 0.0673462i
$219$	0	0
$220$	−1.01837	−0.00462896
$221$	180.559i	0.817010i
$222$	0	0
$223$	309.791	1.38920	0.694598	−	0.719398i	$-0.255582\pi$
$223$	309.791	1.38920	0.694598	+	0.719398i	$0.255582\pi$
$224$	− 240.432i	− 1.07336i
$225$	0	0
$226$	0.774678	0.00342778
$227$	166.873i	0.735125i	0.929999	+	0.367562i	$0.119808\pi$
$227$	166.873i	0.735125i	−0.929999	+	0.367562i	$0.880192\pi$
$228$	0	0
$229$	−232.472	−1.01516	−0.507581	−	0.861604i	$-0.669460\pi$
$229$	−232.472	−1.01516	−0.507581	+	0.861604i	$0.669460\pi$
$230$	0.713596i	0.00310259i
$231$	0	0
$232$	384.257	1.65628
$233$	113.428i	0.486815i	0.969924	+	0.243408i	$0.0782652\pi$
$233$	113.428i	0.486815i	−0.969924	+	0.243408i	$0.921735\pi$
$234$	0	0
$235$	0.0979885	0.000416972 0
$236$	− 121.548i	− 0.515035i
$237$	0	0
$238$	223.284	0.938168
$239$	35.1840i	0.147213i	0.997287	+	0.0736067i	$0.0234510\pi$
$239$	35.1840i	0.147213i	−0.997287	+	0.0736067i	$0.976549\pi$
$240$	0	0
$241$	256.223	1.06317	0.531583	−	0.847006i	$-0.321597\pi$
$241$	256.223	1.06317	0.531583	+	0.847006i	$0.321597\pi$
$242$	44.7290i	0.184831i
$243$	0	0
$244$	−9.77892	−0.0400775
$245$	− 2.42896i	− 0.00991413i
$246$	0	0
$247$	−153.194	−0.620217
$248$	273.792i	1.10400i
$249$	0	0
$250$	−3.86525	−0.0154610
$251$	391.707i	1.56059i	0.625414	+	0.780293i	$0.284930\pi$
$251$	391.707i	1.56059i	−0.625414	+	0.780293i	$0.715070\pi$
$252$	0	0
$253$	−113.113	−0.447086
$254$	160.330i	0.631219i
$255$	0	0
$256$	−255.362	−0.997508
$257$	285.655i	1.11150i	0.831350	+	0.555749i	$0.187568\pi$
$257$	285.655i	1.11150i	−0.831350	+	0.555749i	$0.812432\pi$
$258$	0	0
$259$	141.916	0.547939
$260$	− 1.01589i	− 0.00390729i
$261$	0	0
$262$	309.308	1.18057
$263$	452.311i	1.71981i	0.510451	+	0.859907i	$0.329478\pi$
$263$	452.311i	1.71981i	−0.510451	+	0.859907i	$0.670522\pi$
$264$	0	0
$265$	2.40671	0.00908192
$266$	189.443i	0.712192i
$267$	0	0
$268$	−80.4450	−0.300168
$269$	277.616i	1.03203i	0.856580	+	0.516014i	$0.172585\pi$
$269$	277.616i	1.03203i	−0.856580	+	0.516014i	$0.827415\pi$
$270$	0	0
$271$	310.311	1.14506	0.572530	−	0.819884i	$-0.305962\pi$
$271$	310.311	1.14506	0.572530	+	0.819884i	$0.305962\pi$
$272$	− 98.7990i	− 0.363231i
$273$	0	0
$274$	297.345	1.08520
$275$	− 306.327i	− 1.11392i
$276$	0	0
$277$	−344.135	−1.24236	−0.621182	−	0.783666i	$-0.713347\pi$
$277$	−344.135	−1.24236	−0.621182	+	0.783666i	$0.713347\pi$
$278$	− 44.0626i	− 0.158499i
$279$	0	0
$280$	−4.30473	−0.0153740
$281$	440.366i	1.56714i	0.621305	+	0.783569i	$0.286603\pi$
$281$	440.366i	1.56714i	−0.621305	+	0.783569i	$0.713397\pi$
$282$	0	0
$283$	14.9232	0.0527322	0.0263661	−	0.999652i	$-0.491606\pi$
$283$	14.9232	0.0527322	0.0263661	+	0.999652i	$0.491606\pi$
$284$	16.3984i	0.0577408i
$285$	0	0
$286$	−229.721	−0.803220
$287$	− 679.848i	− 2.36881i
$288$	0	0
$289$	70.8396	0.245120
$290$	− 3.42961i	− 0.0118262i
$291$	0	0
$292$	145.876	0.499574
$293$	162.024i	0.552984i	0.961016	+	0.276492i	$0.0891719\pi$
$293$	162.024i	0.552984i	−0.961016	+	0.276492i	$0.910828\pi$
$294$	0	0
$295$	−3.71733	−0.0126011
$296$	− 124.695i	− 0.421267i
$297$	0	0
$298$	−381.580	−1.28047
$299$	− 112.838i	− 0.377384i
$300$	0	0
$301$	398.516	1.32397
$302$	224.207i	0.742408i
$303$	0	0
$304$	83.8250	0.275740
$305$	0.299070i	0 0.000980559i
$306$	0	0
$307$	−139.049	−0.452929	−0.226465	−	0.974019i	$-0.572717\pi$
$307$	−139.049	−0.452929	−0.226465	+	0.974019i	$0.572717\pi$
$308$	− 199.134i	− 0.646539i
$309$	0	0
$310$	2.44367	0.00788282
$311$	180.481i	0.580326i	0.956977	+	0.290163i	$0.0937096\pi$
$311$	180.481i	0.580326i	−0.956977	+	0.290163i	$0.906290\pi$
$312$	0	0
$313$	579.457	1.85130	0.925650	−	0.378381i	$-0.123519\pi$
$313$	579.457	1.85130	0.925650	+	0.378381i	$0.123519\pi$
$314$	− 159.738i	− 0.508721i
$315$	0	0
$316$	−184.601	−0.584179
$317$	− 505.603i	− 1.59496i	−0.603343	−	0.797482i	$-0.706165\pi$
$317$	− 505.603i	− 1.59496i	0.603343	−	0.797482i	$-0.293835\pi$
$318$	0	0
$319$	543.630	1.70417
$320$	3.23441i	0.0101075i
$321$	0	0
$322$	−139.538	−0.433348
$323$	− 185.096i	− 0.573053i
$324$	0	0
$325$	305.582	0.940251
$326$	− 193.620i	− 0.593926i
$327$	0	0
$328$	−597.351	−1.82119
$329$	19.1609i	0.0582397i
$330$	0	0
$331$	−226.894	−0.685480	−0.342740	−	0.939430i	$-0.611355\pi$
$331$	−226.894	−0.685480	−0.342740	+	0.939430i	$0.611355\pi$
$332$	− 32.0859i	− 0.0966444i
$333$	0	0
$334$	375.069	1.12296
$335$	2.46026i	0.00734407i
$336$	0	0
$337$	−95.9662	−0.284766	−0.142383	−	0.989812i	$-0.545476\pi$
$337$	−95.9662	−0.284766	−0.142383	+	0.989812i	$0.545476\pi$
$338$	29.9970i	0.0887486i
$339$	0	0
$340$	1.22745	0.00361016
$341$	387.349i	1.13592i
$342$	0	0
$343$	−8.07926	−0.0235547
$344$	− 350.157i	− 1.01790i
$345$	0	0
$346$	−140.994	−0.407498
$347$	− 258.794i	− 0.745803i	−0.927871	−	0.372901i	$-0.878363\pi$
$347$	− 258.794i	− 0.745803i	0.927871	−	0.372901i	$-0.121637\pi$
$348$	0	0
$349$	274.885	0.787635	0.393818	−	0.919189i	$-0.371154\pi$
$349$	274.885	0.787635	0.393818	+	0.919189i	$0.371154\pi$
$350$	− 377.890i	− 1.07969i
$351$	0	0
$352$	−298.877	−0.849082
$353$	264.666i	0.749762i	0.927073	+	0.374881i	$0.122316\pi$
$353$	264.666i	0.749762i	−0.927073	+	0.374881i	$0.877684\pi$
$354$	0	0
$355$	0.501515	0.00141272
$356$	− 64.6081i	− 0.181483i
$357$	0	0
$358$	97.1085	0.271253
$359$	211.851i	0.590115i	0.955479	+	0.295058i	$0.0953389\pi$
$359$	211.851i	0.590115i	−0.955479	+	0.295058i	$0.904661\pi$
$360$	0	0
$361$	−203.957	−0.564978
$362$	− 237.677i	− 0.656566i
$363$	0	0
$364$	198.650	0.545741
$365$	− 4.46134i	− 0.0122228i
$366$	0	0
$367$	−645.185	−1.75800	−0.878999	−	0.476823i	$-0.841788\pi$
$367$	−645.185	−1.75800	−0.878999	+	0.476823i	$0.841788\pi$
$368$	61.7429i	0.167780i
$369$	0	0
$370$	−1.11294	−0.00300795
$371$	470.612i	1.26850i
$372$	0	0
$373$	−703.189	−1.88523	−0.942613	−	0.333889i	$-0.891639\pi$
$373$	−703.189	−1.88523	−0.942613	+	0.333889i	$0.891639\pi$
$374$	− 277.560i	− 0.742139i
$375$	0	0
$376$	16.8357	0.0447759
$377$	542.308i	1.43848i
$378$	0	0
$379$	−352.196	−0.929278	−0.464639	−	0.885500i	$-0.653816\pi$
$379$	−352.196	−0.929278	−0.464639	+	0.885500i	$0.653816\pi$
$380$	1.04142i	0.00274058i
$381$	0	0
$382$	87.8789	0.230050
$383$	− 270.255i	− 0.705626i	−0.935694	−	0.352813i	$-0.885225\pi$
$383$	− 270.255i	− 0.705626i	0.935694	−	0.352813i	$-0.114775\pi$
$384$	0	0
$385$	−6.09015	−0.0158186
$386$	14.0890i	0.0365001i
$387$	0	0
$388$	−0.636767	−0.00164115
$389$	− 220.071i	− 0.565735i	−0.959159	−	0.282867i	$-0.908714\pi$
$389$	− 220.071i	− 0.565735i	0.959159	−	0.282867i	$-0.0912856\pi$
$390$	0	0
$391$	136.336	0.348686
$392$	− 417.328i	− 1.06461i
$393$	0	0
$394$	127.204	0.322852
$395$	5.64567i	0.0142928i
$396$	0	0
$397$	−116.035	−0.292278	−0.146139	−	0.989264i	$-0.546685\pi$
$397$	−116.035	−0.292278	−0.146139	+	0.989264i	$0.546685\pi$
$398$	− 56.1588i	− 0.141102i
$399$	0	0
$400$	−167.209	−0.418023
$401$	190.185i	0.474277i	0.971476	+	0.237139i	$0.0762096\pi$
$401$	190.185i	0.474277i	−0.971476	+	0.237139i	$0.923790\pi$
$402$	0	0
$403$	−386.407	−0.958826
$404$	78.6491i	0.194676i
$405$	0	0
$406$	670.631	1.65180
$407$	− 176.413i	− 0.433448i
$408$	0	0
$409$	−23.7884	−0.0581624	−0.0290812	−	0.999577i	$-0.509258\pi$
$409$	−23.7884	−0.0581624	−0.0290812	+	0.999577i	$0.509258\pi$
$410$	5.33154i	0.0130037i
$411$	0	0
$412$	−30.2168	−0.0733416
$413$	− 726.894i	− 1.76003i
$414$	0	0
$415$	−0.981290	−0.00236455
$416$	− 298.150i	− 0.716707i
$417$	0	0
$418$	235.493	0.563381
$419$	− 342.992i	− 0.818598i	−0.912400	−	0.409299i	$-0.865773\pi$
$419$	− 342.992i	− 0.818598i	0.912400	−	0.409299i	$-0.134227\pi$
$420$	0	0
$421$	−494.015	−1.17343	−0.586716	−	0.809793i	$-0.699579\pi$
$421$	−494.015	−1.17343	−0.586716	+	0.809793i	$0.699579\pi$
$422$	473.575i	1.12221i
$423$	0	0
$424$	413.505	0.975247
$425$	369.219i	0.868750i
$426$	0	0
$427$	−58.4808	−0.136957
$428$	− 113.550i	− 0.265304i
$429$	0	0
$430$	−3.12526	−0.00726805
$431$	− 519.074i	− 1.20435i	−0.798365	−	0.602174i	$-0.794301\pi$
$431$	− 519.074i	− 1.20435i	0.798365	−	0.602174i	$-0.205699\pi$
$432$	0	0
$433$	290.297	0.670433	0.335216	−	0.942141i	$-0.391191\pi$
$433$	290.297	0.670433	0.335216	+	0.942141i	$0.391191\pi$
$434$	477.840i	1.10101i
$435$	0	0
$436$	15.7818	0.0361968
$437$	115.673i	0.264698i
$438$	0	0
$439$	334.107	0.761065	0.380532	−	0.924768i	$-0.375741\pi$
$439$	334.107	0.761065	0.380532	+	0.924768i	$0.375741\pi$
$440$	5.35113i	0.0121617i
$441$	0	0
$442$	276.885	0.626437
$443$	332.554i	0.750687i	0.926886	+	0.375343i	$0.122475\pi$
$443$	332.554i	0.750687i	−0.926886	+	0.375343i	$0.877525\pi$
$444$	0	0
$445$	−1.97592	−0.00444027
$446$	− 475.060i	− 1.06516i
$447$	0	0
$448$	−632.463	−1.41175
$449$	336.195i	0.748764i	0.927274	+	0.374382i	$0.122145\pi$
$449$	336.195i	0.748764i	−0.927274	+	0.374382i	$0.877855\pi$
$450$	0	0
$451$	−845.107	−1.87385
$452$	0.832738i	0.00184234i
$453$	0	0
$454$	255.898	0.563652
$455$	− 6.07534i	− 0.0133524i
$456$	0	0
$457$	−794.119	−1.73768	−0.868839	−	0.495094i	$-0.835134\pi$
$457$	−794.119	−1.73768	−0.868839	+	0.495094i	$0.835134\pi$
$458$	356.493i	0.778368i
$459$	0	0
$460$	−0.767078	−0.00166756
$461$	285.585i	0.619491i	0.950819	+	0.309746i	$0.100244\pi$
$461$	285.585i	0.619491i	−0.950819	+	0.309746i	$0.899756\pi$
$462$	0	0
$463$	−134.318	−0.290104	−0.145052	−	0.989424i	$-0.546335\pi$
$463$	−134.318	−0.290104	−0.145052	+	0.989424i	$0.546335\pi$
$464$	− 296.742i	− 0.639530i
$465$	0	0
$466$	173.940	0.373262
$467$	− 268.851i	− 0.575699i	−0.957676	−	0.287849i	$-0.907060\pi$
$467$	− 268.851i	− 0.575699i	0.957676	−	0.287849i	$-0.0929402\pi$
$468$	0	0
$469$	−481.084	−1.02577
$470$	− 0.150264i	0 0.000319711i
$471$	0	0
$472$	−638.687	−1.35315
$473$	− 495.388i	− 1.04733i
$474$	0	0
$475$	−313.260	−0.659495
$476$	240.018i	0.504240i
$477$	0	0
$478$	53.9542	0.112875
$479$	95.3283i	0.199015i	0.995037	+	0.0995076i	$0.0317268\pi$
$479$	95.3283i	0.199015i	−0.995037	+	0.0995076i	$0.968273\pi$
$480$	0	0
$481$	175.984	0.365872
$482$	− 392.915i	− 0.815176i
$483$	0	0
$484$	−48.0814	−0.0993417
$485$	0.0194743i	0 4.01533e-5i
$486$	0	0
$487$	−210.939	−0.433139	−0.216569	−	0.976267i	$-0.569487\pi$
$487$	−210.939	−0.433139	−0.216569	+	0.976267i	$0.569487\pi$
$488$	51.3843i	0.105296i
$489$	0	0
$490$	−3.72478	−0.00760159
$491$	− 597.474i	− 1.21685i	−0.793611	−	0.608426i	$-0.791801\pi$
$491$	− 597.474i	− 1.21685i	0.793611	−	0.608426i	$-0.208199\pi$
$492$	0	0
$493$	−655.243	−1.32909
$494$	234.920i	0.475547i
$495$	0	0
$496$	211.435	0.426281
$497$	98.0672i	0.197318i
$498$	0	0
$499$	921.299	1.84629	0.923145	−	0.384451i	$-0.125609\pi$
$499$	921.299	1.84629	0.923145	+	0.384451i	$0.125609\pi$
$500$	− 4.15494i	− 0.00830988i
$501$	0	0
$502$	600.678	1.19657
$503$	− 921.173i	− 1.83136i	−0.401910	−	0.915679i	$-0.631654\pi$
$503$	− 921.173i	− 1.83136i	0.401910	−	0.915679i	$-0.368346\pi$
$504$	0	0
$505$	2.40534	0.00476305
$506$	173.457i	0.342800i
$507$	0	0
$508$	−172.346	−0.339263
$509$	− 414.391i	− 0.814127i	−0.913400	−	0.407064i	$-0.866553\pi$
$509$	− 414.391i	− 0.814127i	0.913400	−	0.407064i	$-0.133447\pi$
$510$	0	0
$511$	872.378	1.70720
$512$	394.899i	0.771287i
$513$	0	0
$514$	438.048	0.852234
$515$	0.924124i	0.00179442i
$516$	0	0
$517$	23.8185	0.0460706
$518$	− 217.627i	− 0.420128i
$519$	0	0
$520$	−5.33812	−0.0102656
$521$	808.534i	1.55189i	0.630802	+	0.775944i	$0.282726\pi$
$521$	808.534i	1.55189i	−0.630802	+	0.775944i	$0.717274\pi$
$522$	0	0
$523$	150.743	0.288227	0.144114	−	0.989561i	$-0.453967\pi$
$523$	150.743	0.288227	0.144114	+	0.989561i	$0.453967\pi$
$524$	332.490i	0.634523i
$525$	0	0
$526$	693.613	1.31866
$527$	− 466.876i	− 0.885912i
$528$	0	0
$529$	443.799	0.838939
$530$	− 3.69065i	− 0.00696350i
$531$	0	0
$532$	−203.641	−0.382784
$533$	− 843.051i	− 1.58171i
$534$	0	0
$535$	−3.47272	−0.00649106
$536$	422.706i	0.788631i
$537$	0	0
$538$	425.720	0.791301
$539$	− 590.418i	− 1.09540i
$540$	0	0
$541$	−764.505	−1.41313	−0.706567	−	0.707646i	$-0.749757\pi$
$541$	−764.505	−1.41313	−0.706567	+	0.707646i	$0.749757\pi$
$542$	− 475.858i	− 0.877967i
$543$	0	0
$544$	360.240	0.662205
$545$	− 0.482658i	0 0.000885611i
$546$	0	0
$547$	290.503	0.531084	0.265542	−	0.964099i	$-0.414449\pi$
$547$	290.503	0.531084	0.265542	+	0.964099i	$0.414449\pi$
$548$	319.630i	0.583266i
$549$	0	0
$550$	−469.748	−0.854087
$551$	− 555.934i	− 1.00896i
$552$	0	0
$553$	−1103.96	−1.99632
$554$	527.726i	0.952574i
$555$	0	0
$556$	47.3650	0.0851888
$557$	566.868i	1.01772i	0.860850	+	0.508858i	$0.169932\pi$
$557$	566.868i	1.01772i	−0.860850	+	0.508858i	$0.830068\pi$
$558$	0	0
$559$	494.183	0.884049
$560$	3.32433i	0.00593630i
$561$	0	0
$562$	675.295	1.20159
$563$	− 1006.63i	− 1.78798i	−0.448088	−	0.893989i	$-0.647895\pi$
$563$	− 1006.63i	− 1.78798i	0.448088	−	0.893989i	$-0.352105\pi$
$564$	0	0
$565$	0.0254678	4.50757e−5 0
$566$	− 22.8846i	− 0.0404321i
$567$	0	0
$568$	86.1670	0.151703
$569$	138.660i	0.243691i	0.992549	+	0.121846i	$0.0388812\pi$
$569$	138.660i	0.243691i	−0.992549	+	0.121846i	$0.961119\pi$
$570$	0	0
$571$	−523.510	−0.916831	−0.458415	−	0.888738i	$-0.651583\pi$
$571$	−523.510	−0.916831	−0.458415	+	0.888738i	$0.651583\pi$
$572$	− 246.938i	− 0.431710i
$573$	0	0
$574$	−1042.54	−1.81627
$575$	− 230.738i	− 0.401283i
$576$	0	0
$577$	954.956	1.65504	0.827518	−	0.561439i	$-0.189752\pi$
$577$	954.956	1.65504	0.827518	+	0.561439i	$0.189752\pi$
$578$	− 108.632i	− 0.187944i
$579$	0	0
$580$	3.68665	0.00635629
$581$	− 191.883i	− 0.330264i
$582$	0	0
$583$	585.009	1.00345
$584$	− 766.518i	− 1.31253i
$585$	0	0
$586$	248.462	0.423997
$587$	− 748.217i	− 1.27465i	−0.770597	−	0.637323i	$-0.780042\pi$
$587$	− 748.217i	− 1.27465i	0.770597	−	0.637323i	$-0.219958\pi$
$588$	0	0
$589$	396.116	0.672523
$590$	5.70048i	0.00966183i
$591$	0	0
$592$	−96.2957	−0.162662
$593$	− 164.211i	− 0.276916i	−0.990368	−	0.138458i	$-0.955785\pi$
$593$	− 164.211i	− 0.276916i	0.990368	−	0.138458i	$-0.0442145\pi$
$594$	0	0
$595$	7.34053	0.0123370
$596$	− 410.179i	− 0.688219i
$597$	0	0
$598$	−173.035	−0.289356
$599$	155.052i	0.258852i	0.991589	+	0.129426i	$0.0413134\pi$
$599$	155.052i	0.258852i	−0.991589	+	0.129426i	$0.958687\pi$
$600$	0	0
$601$	741.550	1.23386	0.616930	−	0.787018i	$-0.288376\pi$
$601$	741.550	1.23386	0.616930	+	0.787018i	$0.288376\pi$
$602$	− 611.119i	− 1.01515i
$603$	0	0
$604$	−241.011	−0.399025
$605$	1.47048i	0.00243055i
$606$	0	0
$607$	−831.480	−1.36982	−0.684909	−	0.728629i	$-0.740158\pi$
$607$	−831.480	−1.36982	−0.684909	+	0.728629i	$0.740158\pi$
$608$	305.642i	0.502700i
$609$	0	0
$610$	0.458621	0.000751837 0
$611$	23.7606i	0.0388880i
$612$	0	0
$613$	394.616	0.643745	0.321873	−	0.946783i	$-0.395688\pi$
$613$	394.616	0.643745	0.321873	+	0.946783i	$0.395688\pi$
$614$	213.230i	0.347281i
$615$	0	0
$616$	−1046.37	−1.69865
$617$	− 6.70929i	− 0.0108741i	−0.999985	−	0.00543703i	$-0.998269\pi$
$617$	− 6.70929i	− 0.0108741i	0.999985	−	0.00543703i	$-0.00173067\pi$
$618$	0	0
$619$	−360.374	−0.582188	−0.291094	−	0.956694i	$-0.594019\pi$
$619$	−360.374	−0.582188	−0.291094	+	0.956694i	$0.594019\pi$
$620$	2.62682i	0.00423681i
$621$	0	0
$622$	276.766	0.444961
$623$	− 386.375i	− 0.620185i
$624$	0	0
$625$	624.809	0.999695
$626$	− 888.590i	− 1.41947i
$627$	0	0
$628$	171.710	0.273424
$629$	212.633i	0.338049i
$630$	0	0
$631$	−788.173	−1.24909	−0.624543	−	0.780991i	$-0.714715\pi$
$631$	−788.173	−1.24909	−0.624543	+	0.780991i	$0.714715\pi$
$632$	970.002i	1.53481i
$633$	0	0
$634$	−775.336	−1.22293
$635$	5.27088i	0.00830060i
$636$	0	0
$637$	588.982	0.924619
$638$	− 833.649i	− 1.30666i
$639$	0	0
$640$	0.0416492	6.50769e−5 0
$641$	1166.84i	1.82034i	0.414236	+	0.910169i	$0.364049\pi$
$641$	1166.84i	1.82034i	−0.414236	+	0.910169i	$0.635951\pi$
$642$	0	0
$643$	−291.818	−0.453839	−0.226920	−	0.973914i	$-0.572865\pi$
$643$	−291.818	−0.453839	−0.226920	+	0.973914i	$0.572865\pi$
$644$	− 149.996i	− 0.232913i
$645$	0	0
$646$	−283.842	−0.439384
$647$	547.391i	0.846045i	0.906119	+	0.423023i	$0.139031\pi$
$647$	547.391i	0.846045i	−0.906119	+	0.423023i	$0.860969\pi$
$648$	0	0
$649$	−903.588	−1.39228
$650$	− 468.606i	− 0.720932i
$651$	0	0
$652$	208.131	0.319220
$653$	437.312i	0.669697i	0.942272	+	0.334849i	$0.108685\pi$
$653$	437.312i	0.669697i	−0.942272	+	0.334849i	$0.891315\pi$
$654$	0	0
$655$	10.1686	0.0155246
$656$	461.304i	0.703207i
$657$	0	0
$658$	29.3829	0.0446549
$659$	7.22049i	0.0109567i	0.999985	+	0.00547837i	$0.00174383\pi$
$659$	7.22049i	0.0109567i	−0.999985	+	0.00547837i	$0.998256\pi$
$660$	0	0
$661$	883.980	1.33734	0.668669	−	0.743560i	$-0.266864\pi$
$661$	883.980	1.33734	0.668669	+	0.743560i	$0.266864\pi$
$662$	347.939i	0.525587i
$663$	0	0
$664$	−168.599	−0.253914
$665$	6.22800i	0.00936541i
$666$	0	0
$667$	409.484	0.613919
$668$	403.179i	0.603562i
$669$	0	0
$670$	3.77278	0.00563102
$671$	72.6964i	0.108340i
$672$	0	0
$673$	474.384	0.704880	0.352440	−	0.935834i	$-0.385352\pi$
$673$	474.384	0.704880	0.352440	+	0.935834i	$0.385352\pi$
$674$	147.163i	0.218343i
$675$	0	0
$676$	−32.2452	−0.0477000
$677$	− 320.197i	− 0.472965i	−0.971636	−	0.236482i	$-0.924005\pi$
$677$	− 320.197i	− 0.472965i	0.971636	−	0.236482i	$-0.0759945\pi$
$678$	0	0
$679$	−3.80805	−0.00560832
$680$	− 6.44978i	− 0.00948497i
$681$	0	0
$682$	593.995	0.870960
$683$	− 937.935i	− 1.37326i	−0.727008	−	0.686629i	$-0.759090\pi$
$683$	− 937.935i	− 1.37326i	0.727008	−	0.686629i	$-0.240910\pi$
$684$	0	0
$685$	9.77529	0.0142705
$686$	12.3894i	0.0180604i
$687$	0	0
$688$	−270.409	−0.393036
$689$	583.586i	0.847005i
$690$	0	0
$691$	−58.8384	−0.0851496	−0.0425748	−	0.999093i	$-0.513556\pi$
$691$	−58.8384	−0.0851496	−0.0425748	+	0.999093i	$0.513556\pi$
$692$	− 151.561i	− 0.219019i
$693$	0	0
$694$	−396.857	−0.571839
$695$	− 1.44857i	− 0.00208428i
$696$	0	0
$697$	1018.62	1.46143
$698$	− 421.532i	− 0.603914i
$699$	0	0
$700$	406.212	0.580303
$701$	591.380i	0.843623i	0.906683	+	0.421812i	$0.138606\pi$
$701$	591.380i	0.843623i	−0.906683	+	0.421812i	$0.861394\pi$
$702$	0	0
$703$	−180.406	−0.256623
$704$	786.203i	1.11677i
$705$	0	0
$706$	405.862	0.574875
$707$	470.344i	0.665268i
$708$	0	0
$709$	440.059	0.620676	0.310338	−	0.950626i	$-0.399558\pi$
$709$	440.059	0.620676	0.310338	+	0.950626i	$0.399558\pi$
$710$	− 0.769067i	− 0.00108319i
$711$	0	0
$712$	−339.490	−0.476811
$713$	291.767i	0.409210i
$714$	0	0
$715$	−7.55214	−0.0105624
$716$	104.387i	0.145791i
$717$	0	0
$718$	324.871	0.452467
$719$	576.771i	0.802185i	0.916038	+	0.401092i	$0.131369\pi$
$719$	576.771i	0.802185i	−0.916038	+	0.401092i	$0.868631\pi$
$720$	0	0
$721$	−180.705	−0.250631
$722$	312.766i	0.433193i
$723$	0	0
$724$	255.490	0.352887
$725$	1108.95i	1.52958i
$726$	0	0
$727$	−599.131	−0.824114	−0.412057	−	0.911158i	$-0.635189\pi$
$727$	−599.131	−0.824114	−0.412057	+	0.911158i	$0.635189\pi$
$728$	− 1043.83i	− 1.43383i
$729$	0	0
$730$	−6.84140	−0.00937179
$731$	597.096i	0.816821i
$732$	0	0
$733$	−554.170	−0.756030	−0.378015	−	0.925800i	$-0.623393\pi$
$733$	−554.170	−0.756030	−0.378015	+	0.925800i	$0.623393\pi$
$734$	989.383i	1.34793i
$735$	0	0
$736$	−225.126	−0.305878
$737$	598.027i	0.811434i
$738$	0	0
$739$	1416.40	1.91665	0.958324	−	0.285685i	$-0.0922210\pi$
$739$	1416.40	1.91665	0.958324	+	0.285685i	$0.0922210\pi$
$740$	− 1.19635i	− 0.00161669i
$741$	0	0
$742$	721.677	0.972611
$743$	− 301.649i	− 0.405988i	−0.979180	−	0.202994i	$-0.934933\pi$
$743$	− 301.649i	− 0.405988i	0.979180	−	0.202994i	$-0.0650671\pi$
$744$	0	0
$745$	−12.5446	−0.0168383
$746$	1078.33i	1.44548i
$747$	0	0
$748$	298.362	0.398880
$749$	− 679.061i	− 0.906624i
$750$	0	0
$751$	−497.385	−0.662296	−0.331148	−	0.943579i	$-0.607436\pi$
$751$	−497.385	−0.662296	−0.331148	+	0.943579i	$0.607436\pi$
$752$	− 13.0014i	− 0.0172891i
$753$	0	0
$754$	831.622	1.10295
$755$	7.37088i	0.00976276i
$756$	0	0
$757$	1028.94	1.35924	0.679620	−	0.733565i	$-0.262145\pi$
$757$	1028.94	1.35924	0.679620	+	0.733565i	$0.262145\pi$
$758$	540.089i	0.712518i
$759$	0	0
$760$	5.47225	0.00720033
$761$	− 37.6380i	− 0.0494586i	−0.999694	−	0.0247293i	$-0.992128\pi$
$761$	− 37.6380i	− 0.0494586i	0.999694	−	0.0247293i	$-0.00787239\pi$
$762$	0	0
$763$	94.3798	0.123696
$764$	94.4652i	0.123646i
$765$	0	0
$766$	−414.432	−0.541034
$767$	− 901.391i	− 1.17522i
$768$	0	0
$769$	1134.21	1.47491	0.737455	−	0.675397i	$-0.236028\pi$
$769$	1134.21	1.47491	0.737455	+	0.675397i	$0.236028\pi$
$770$	9.33917i	0.0121288i
$771$	0	0
$772$	−15.1450	−0.0196178
$773$	509.525i	0.659153i	0.944129	+	0.329577i	$0.106906\pi$
$773$	509.525i	0.659153i	−0.944129	+	0.329577i	$0.893094\pi$
$774$	0	0
$775$	−790.149	−1.01955
$776$	3.34595i	0.00431180i
$777$	0	0
$778$	−337.476	−0.433774
$779$	864.234i	1.10942i
$780$	0	0
$781$	121.905	0.156089
$782$	− 209.070i	− 0.267352i
$783$	0	0
$784$	−322.281	−0.411073
$785$	− 5.25145i	− 0.00668974i
$786$	0	0
$787$	265.671	0.337574	0.168787	−	0.985653i	$-0.446015\pi$
$787$	265.671	0.337574	0.168787	+	0.985653i	$0.446015\pi$
$788$	136.737i	0.173524i
$789$	0	0
$790$	8.65756	0.0109589
$791$	4.98001i	0.00629584i
$792$	0	0
$793$	−72.5196	−0.0914497
$794$	177.937i	0.224103i
$795$	0	0
$796$	60.3677	0.0758389
$797$	448.390i	0.562597i	0.959620	+	0.281299i	$0.0907651\pi$
$797$	448.390i	0.562597i	−0.959620	+	0.281299i	$0.909235\pi$
$798$	0	0
$799$	−28.7087	−0.0359308
$800$	− 609.676i	− 0.762095i
$801$	0	0
$802$	291.647	0.363649
$803$	− 1084.44i	− 1.35048i
$804$	0	0
$805$	−4.58735	−0.00569857
$806$	592.550i	0.735174i
$807$	0	0
$808$	413.269	0.511472
$809$	357.505i	0.441910i	0.975284	+	0.220955i	$0.0709174\pi$
$809$	357.505i	0.441910i	−0.975284	+	0.220955i	$0.929083\pi$
$810$	0	0
$811$	1269.44	1.56528	0.782638	−	0.622477i	$-0.213874\pi$
$811$	1269.44	1.56528	0.782638	+	0.622477i	$0.213874\pi$
$812$	720.893i	0.887800i
$813$	0	0
$814$	−270.527	−0.332343
$815$	− 6.36532i	− 0.00781020i
$816$	0	0
$817$	−506.600	−0.620074
$818$	36.4793i	0.0445957i
$819$	0	0
$820$	−5.73112	−0.00698917
$821$	− 1160.45i	− 1.41346i	−0.707483	−	0.706730i	$-0.750170\pi$
$821$	− 1160.45i	− 1.41346i	0.707483	−	0.706730i	$-0.249830\pi$
$822$	0	0
$823$	−826.651	−1.00444	−0.502218	−	0.864741i	$-0.667483\pi$
$823$	−826.651	−1.00444	−0.502218	+	0.864741i	$0.667483\pi$
$824$	158.777i	0.192690i
$825$	0	0
$826$	−1114.68	−1.34949
$827$	− 86.0989i	− 0.104110i	−0.998644	−	0.0520550i	$-0.983423\pi$
$827$	− 86.0989i	− 0.104110i	0.998644	−	0.0520550i	$-0.0165771\pi$
$828$	0	0
$829$	−633.473	−0.764141	−0.382071	−	0.924133i	$-0.624789\pi$
$829$	−633.473	−0.764141	−0.382071	+	0.924133i	$0.624789\pi$
$830$	1.50480i	0.00181301i
$831$	0	0
$832$	−784.291	−0.942657
$833$	711.637i	0.854307i
$834$	0	0
$835$	12.3305	0.0147671
$836$	253.143i	0.302802i
$837$	0	0
$838$	−525.974	−0.627654
$839$	− 736.588i	− 0.877936i	−0.898503	−	0.438968i	$-0.855344\pi$
$839$	− 736.588i	− 0.877936i	0.898503	−	0.438968i	$-0.144656\pi$
$840$	0	0
$841$	−1127.02	−1.34009
$842$	757.565i	0.899721i
$843$	0	0
$844$	−509.068	−0.603161
$845$	0.986161i	0.00116705i
$846$	0	0
$847$	−287.540	−0.339481
$848$	− 319.329i	− 0.376567i
$849$	0	0
$850$	566.192	0.666109
$851$	− 132.882i	− 0.156148i
$852$	0	0
$853$	−1640.54	−1.92326	−0.961628	−	0.274357i	$-0.911535\pi$
$853$	−1640.54	−1.92326	−0.961628	+	0.274357i	$0.911535\pi$
$854$	89.6795i	0.105011i
$855$	0	0
$856$	−596.659	−0.697032
$857$	− 341.730i	− 0.398751i	−0.979923	−	0.199376i	$-0.936109\pi$
$857$	− 341.730i	− 0.398751i	0.979923	−	0.199376i	$-0.0638914\pi$
$858$	0	0
$859$	306.808	0.357168	0.178584	−	0.983925i	$-0.442848\pi$
$859$	306.808	0.357168	0.178584	+	0.983925i	$0.442848\pi$
$860$	− 3.35949i	− 0.00390638i
$861$	0	0
$862$	−795.994	−0.923427
$863$	− 1546.03i	− 1.79146i	−0.444596	−	0.895731i	$-0.646653\pi$
$863$	− 1546.03i	− 1.79146i	0.444596	−	0.895731i	$-0.353347\pi$
$864$	0	0
$865$	−4.63522	−0.00535864
$866$	− 445.167i	− 0.514050i
$867$	0	0
$868$	−513.653	−0.591766
$869$	1372.32i	1.57919i
$870$	0	0
$871$	−596.573	−0.684928
$872$	− 82.9271i	− 0.0950999i
$873$	0	0
$874$	177.383	0.202955
$875$	− 24.8478i	− 0.0283974i
$876$	0	0
$877$	−270.460	−0.308392	−0.154196	−	0.988040i	$-0.549279\pi$
$877$	−270.460	−0.308392	−0.154196	+	0.988040i	$0.549279\pi$
$878$	− 512.349i	− 0.583541i
$879$	0	0
$880$	4.13241	0.00469592
$881$	277.534i	0.315022i	0.987517	+	0.157511i	$0.0503469\pi$
$881$	277.534i	0.315022i	−0.987517	+	0.157511i	$0.949653\pi$
$882$	0	0
$883$	−668.927	−0.757562	−0.378781	−	0.925486i	$-0.623657\pi$
$883$	−668.927	−0.757562	−0.378781	+	0.925486i	$0.623657\pi$
$884$	297.637i	0.336693i
$885$	0	0
$886$	509.968	0.575584
$887$	1100.92i	1.24117i	0.784138	+	0.620587i	$0.213106\pi$
$887$	1100.92i	1.24117i	−0.784138	+	0.620587i	$0.786894\pi$
$888$	0	0
$889$	−1030.68	−1.15937
$890$	3.03005i	0.00340455i
$891$	0	0
$892$	510.664	0.572494
$893$	− 24.3576i	− 0.0272762i
$894$	0	0
$895$	3.19247	0.00356701
$896$	8.14416i	0.00908947i
$897$	0	0
$898$	515.551	0.574110
$899$	− 1402.26i	− 1.55980i
$900$	0	0
$901$	−705.118	−0.782594
$902$	1295.96i	1.43676i
$903$	0	0
$904$	4.37570	0.00484038
$905$	− 7.81370i	− 0.00863393i
$906$	0	0
$907$	919.255	1.01351	0.506756	−	0.862090i	$-0.330845\pi$
$907$	919.255	1.01351	0.506756	+	0.862090i	$0.330845\pi$
$908$	275.077i	0.302948i
$909$	0	0
$910$	−9.31646	−0.0102379
$911$	169.748i	0.186332i	0.995651	+	0.0931659i	$0.0296987\pi$
$911$	169.748i	0.186332i	−0.995651	+	0.0931659i	$0.970301\pi$
$912$	0	0
$913$	−238.526	−0.261256
$914$	1217.77i	1.33235i
$915$	0	0
$916$	−383.211	−0.418352
$917$	1988.39i	2.16836i
$918$	0	0
$919$	120.333	0.130939	0.0654697	−	0.997855i	$-0.479145\pi$
$919$	120.333	0.130939	0.0654697	+	0.997855i	$0.479145\pi$
$920$	4.03069i	0.00438118i
$921$	0	0
$922$	437.942	0.474991
$923$	121.609i	0.131754i
$924$	0	0
$925$	359.864	0.389042
$926$	205.975i	0.222435i
$927$	0	0
$928$	1081.98	1.16592
$929$	− 1004.57i	− 1.08134i	−0.841233	−	0.540672i	$-0.818170\pi$
$929$	− 1004.57i	− 1.08134i	0.841233	−	0.540672i	$-0.181830\pi$
$930$	0	0
$931$	−603.782	−0.648530
$932$	186.977i	0.200619i
$933$	0	0
$934$	−412.280	−0.441413
$935$	− 9.12487i	− 0.00975922i
$936$	0	0
$937$	748.423	0.798743	0.399372	−	0.916789i	$-0.369228\pi$
$937$	748.423	0.798743	0.399372	+	0.916789i	$0.369228\pi$
$938$	737.736i	0.786499i
$939$	0	0
$940$	0.161526	0.000171836 0
$941$	− 1481.28i	− 1.57416i	−0.616851	−	0.787080i	$-0.711592\pi$
$941$	− 1481.28i	− 1.57416i	0.616851	−	0.787080i	$-0.288408\pi$
$942$	0	0
$943$	−636.569	−0.675046
$944$	493.226i	0.522485i
$945$	0	0
$946$	−759.671	−0.803035
$947$	470.662i	0.497003i	0.968631	+	0.248502i	$0.0799381\pi$
$947$	470.662i	0.497003i	−0.968631	+	0.248502i	$0.920062\pi$
$948$	0	0
$949$	1081.80	1.13994
$950$	480.380i	0.505663i
$951$	0	0
$952$	1261.20	1.32479
$953$	847.528i	0.889326i	0.895698	+	0.444663i	$0.146677\pi$
$953$	847.528i	0.889326i	−0.895698	+	0.444663i	$0.853323\pi$
$954$	0	0
$955$	2.88905	0.00302518
$956$	57.9980i	0.0606673i
$957$	0	0
$958$	146.185	0.152594
$959$	1911.48i	1.99320i
$960$	0	0
$961$	38.1409	0.0396887
$962$	− 269.870i	− 0.280530i
$963$	0	0
$964$	422.363	0.438136
$965$	0.463181i	0 0.000479980i
$966$	0	0
$967$	−103.479	−0.107011	−0.0535053	−	0.998568i	$-0.517039\pi$
$967$	−103.479	−0.107011	−0.0535053	+	0.998568i	$0.517039\pi$
$968$	252.648i	0.261000i
$969$	0	0
$970$	0.0298637	3.07873e−5 0
$971$	171.424i	0.176543i	0.996096	+	0.0882717i	$0.0281344\pi$
$971$	171.424i	0.176543i	−0.996096	+	0.0882717i	$0.971866\pi$
$972$	0	0
$973$	283.256	0.291116
$974$	323.472i	0.332106i
$975$	0	0
$976$	39.6815	0.0406573
$977$	− 1721.61i	− 1.76213i	−0.472991	−	0.881067i	$-0.656826\pi$
$977$	− 1721.61i	− 1.76213i	0.472991	−	0.881067i	$-0.343174\pi$
$978$	0	0
$979$	−480.296	−0.490598
$980$	− 4.00394i	− 0.00408565i
$981$	0	0
$982$	−916.219	−0.933013
$983$	1312.00i	1.33469i	0.744747	+	0.667346i	$0.232570\pi$
$983$	1312.00i	1.33469i	−0.744747	+	0.667346i	$0.767430\pi$
$984$	0	0
$985$	4.18186	0.00424554
$986$	1004.81i	1.01907i
$987$	0	0
$988$	−252.527	−0.255594
$989$	− 373.146i	− 0.377297i
$990$	0	0
$991$	1928.53	1.94604	0.973022	−	0.230713i	$-0.0741059\pi$
$991$	1928.53	1.94604	0.973022	+	0.230713i	$0.0741059\pi$
$992$	770.933i	0.777150i
$993$	0	0
$994$	150.385	0.151293
$995$	− 1.84624i	− 0.00185551i
$996$	0	0
$997$	1242.23	1.24596	0.622982	−	0.782237i	$-0.285921\pi$
$997$	1242.23	1.24596	0.622982	+	0.782237i	$0.285921\pi$
$998$	− 1412.80i	− 1.41563i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.3.b.a.728.10		30
3.2	odd	2	inner	729.3.b.a.728.21		30
27.2	odd	18		81.3.f.a.17.4		30
27.4	even	9		243.3.f.b.107.4		30
27.5	odd	18		243.3.f.d.26.4		30
27.7	even	9		243.3.f.c.134.2		30
27.11	odd	18		243.3.f.a.215.2		30
27.13	even	9		81.3.f.a.62.4		30
27.14	odd	18		27.3.f.a.20.2	✓	30
27.16	even	9		243.3.f.d.215.4		30
27.20	odd	18		243.3.f.b.134.4		30
27.22	even	9		243.3.f.a.26.2		30
27.23	odd	18		243.3.f.c.107.2		30
27.25	even	9		27.3.f.a.23.2	yes	30
108.79	odd	18		432.3.bc.a.401.4		30
108.95	even	18		432.3.bc.a.209.4		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.20.2	✓	30	27.14	odd	18
27.3.f.a.23.2	yes	30	27.25	even	9
81.3.f.a.17.4		30	27.2	odd	18
81.3.f.a.62.4		30	27.13	even	9
243.3.f.a.26.2		30	27.22	even	9
243.3.f.a.215.2		30	27.11	odd	18
243.3.f.b.107.4		30	27.4	even	9
243.3.f.b.134.4		30	27.20	odd	18
243.3.f.c.107.2		30	27.23	odd	18
243.3.f.c.134.2		30	27.7	even	9
243.3.f.d.26.4		30	27.5	odd	18
243.3.f.d.215.4		30	27.16	even	9
432.3.bc.a.209.4		30	108.95	even	18
432.3.bc.a.401.4		30	108.79	odd	18
729.3.b.a.728.10		30	1.1	even	1	trivial
729.3.b.a.728.21		30	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	729.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.53349i q^{2} +1.64842 q^{4} -0.0504138i q^{5} +9.85801 q^{7} -8.66177i q^{8} -0.0773090 q^{10} -12.2543i q^{11} +12.2245 q^{13} -15.1171i q^{14} -6.68905 q^{16} +14.7703i q^{17} -12.5317 q^{19} -0.0831031i q^{20} -18.7918 q^{22} -9.23045i q^{23} +24.9975 q^{25} -18.7461i q^{26} +16.2501 q^{28} +44.3623i q^{29} -31.6092 q^{31} -24.3895i q^{32} +22.6500 q^{34} -0.496980i q^{35} +14.3960 q^{37} +19.2172i q^{38} -0.436673 q^{40} -68.9640i q^{41} +40.4256 q^{43} -20.2002i q^{44} -14.1548 q^{46} +1.94368i q^{47} +48.1804 q^{49} -38.3333i q^{50} +20.1511 q^{52} +47.7390i q^{53} -0.617787 q^{55} -85.3879i q^{56} +68.0291 q^{58} -73.7363i q^{59} -5.93231 q^{61} +48.4723i q^{62} -64.1572 q^{64} -0.616285i q^{65} -48.8013 q^{67} +24.3475i q^{68} -0.762113 q^{70} +9.94797i q^{71} +88.4943 q^{73} -22.0761i q^{74} -20.6574 q^{76} -120.803i q^{77} -111.986 q^{79} +0.337221i q^{80} -105.755 q^{82} -19.4647i q^{83} +0.744625 q^{85} -61.9921i q^{86} -106.144 q^{88} -39.1940i q^{89} +120.509 q^{91} -15.2156i q^{92} +2.98061 q^{94} +0.631770i q^{95} -0.386290 q^{97} -73.8841i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67}+ \cdots - 24 q^{94}+O(q^{100}) \) 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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