LMFDB - Embedded newform 729.3.b.a.728.26

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [729,3,Mod(728,729)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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			728.26
Character	$\chi$	$=$	729.728
Dual form			729.3.b.a.728.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.62787i q^{2} -2.90572 q^{4} +6.25111i q^{5} -8.11762 q^{7} +2.87564i q^{8} -16.4271 q^{10} +8.11386i q^{11} +9.70958 q^{13} -21.3321i q^{14} -19.1797 q^{16} +15.0276i q^{17} -17.8645 q^{19} -18.1640i q^{20} -21.3222 q^{22} -26.3335i q^{23} -14.0764 q^{25} +25.5156i q^{26} +23.5875 q^{28} +15.1718i q^{29} +32.7337 q^{31} -38.8992i q^{32} -39.4906 q^{34} -50.7442i q^{35} +31.6193 q^{37} -46.9457i q^{38} -17.9759 q^{40} -16.5200i q^{41} -83.8240 q^{43} -23.5766i q^{44} +69.2010 q^{46} +13.1702i q^{47} +16.8958 q^{49} -36.9909i q^{50} -28.2133 q^{52} -25.7140i q^{53} -50.7206 q^{55} -23.3433i q^{56} -39.8696 q^{58} -22.7110i q^{59} +34.0883 q^{61} +86.0200i q^{62} +25.5035 q^{64} +60.6957i q^{65} +8.10443 q^{67} -43.6660i q^{68} +133.349 q^{70} -41.2871i q^{71} -80.8091 q^{73} +83.0914i q^{74} +51.9093 q^{76} -65.8652i q^{77} -27.6316 q^{79} -119.894i q^{80} +43.4126 q^{82} -17.5898i q^{83} -93.9392 q^{85} -220.279i q^{86} -23.3325 q^{88} +28.2885i q^{89} -78.8187 q^{91} +76.5176i q^{92} -34.6095 q^{94} -111.673i q^{95} +168.631 q^{97} +44.4001i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.62787i	1.31394i	0.753918	+	0.656968i	$0.228161\pi$
$2$	2.62787i	1.31394i	−0.753918	+	0.656968i	$0.771839\pi$
$3$	0	0
$3$	0	0
$4$	−2.90572	−0.726429
$5$	6.25111i	1.25022i	0.780536	+	0.625111i	$0.214946\pi$
$5$	6.25111i	1.25022i	−0.780536	+	0.625111i	$0.785054\pi$
$6$	0	0
$7$	−8.11762	−1.15966	−0.579830	−	0.814737i	$-0.696881\pi$
$7$	−8.11762	−1.15966	−0.579830	+	0.814737i	$0.696881\pi$
$8$	2.87564i	0.359455i
$9$	0	0
$10$	−16.4271	−1.64271
$11$	8.11386i	0.737623i	0.929504	+	0.368812i	$0.120235\pi$
$11$	8.11386i	0.737623i	−0.929504	+	0.368812i	$0.879765\pi$
$12$	0	0
$13$	9.70958	0.746891	0.373445	−	0.927652i	$-0.378176\pi$
$13$	9.70958	0.746891	0.373445	+	0.927652i	$0.378176\pi$
$14$	− 21.3321i	− 1.52372i
$15$	0	0
$16$	−19.1797	−1.19873
$17$	15.0276i	0.883977i	0.897021	+	0.441988i	$0.145727\pi$
$17$	15.0276i	0.883977i	−0.897021	+	0.441988i	$0.854273\pi$
$18$	0	0
$19$	−17.8645	−0.940238	−0.470119	−	0.882603i	$-0.655789\pi$
$19$	−17.8645	−0.940238	−0.470119	+	0.882603i	$0.655789\pi$
$20$	− 18.1640i	− 0.908198i
$21$	0	0
$22$	−21.3222	−0.969190
$23$	− 26.3335i	− 1.14493i	−0.819928	−	0.572467i	$-0.805987\pi$
$23$	− 26.3335i	− 1.14493i	0.819928	−	0.572467i	$-0.194013\pi$
$24$	0	0
$25$	−14.0764	−0.563055
$26$	25.5156i	0.981367i
$27$	0	0
$28$	23.5875	0.842411
$29$	15.1718i	0.523166i	0.965181	+	0.261583i	$0.0842445\pi$
$29$	15.1718i	0.523166i	−0.965181	+	0.261583i	$0.915756\pi$
$30$	0	0
$31$	32.7337	1.05593	0.527963	−	0.849267i	$-0.322956\pi$
$31$	32.7337	1.05593	0.527963	+	0.849267i	$0.322956\pi$
$32$	− 38.8992i	− 1.21560i
$33$	0	0
$34$	−39.4906	−1.16149
$35$	− 50.7442i	− 1.44983i
$36$	0	0
$37$	31.6193	0.854575	0.427287	−	0.904116i	$-0.359469\pi$
$37$	31.6193	0.854575	0.427287	+	0.904116i	$0.359469\pi$
$38$	− 46.9457i	− 1.23541i
$39$	0	0
$40$	−17.9759	−0.449398
$41$	− 16.5200i	− 0.402928i	−0.979496	−	0.201464i	$-0.935430\pi$
$41$	− 16.5200i	− 0.402928i	0.979496	−	0.201464i	$-0.0645698\pi$
$42$	0	0
$43$	−83.8240	−1.94940	−0.974698	−	0.223526i	$-0.928243\pi$
$43$	−83.8240	−1.94940	−0.974698	+	0.223526i	$0.928243\pi$
$44$	− 23.5766i	− 0.535831i
$45$	0	0
$46$	69.2010	1.50437
$47$	13.1702i	0.280216i	0.990136	+	0.140108i	$0.0447450\pi$
$47$	13.1702i	0.280216i	−0.990136	+	0.140108i	$0.955255\pi$
$48$	0	0
$49$	16.8958	0.344813
$50$	− 36.9909i	− 0.739819i
$51$	0	0
$52$	−28.2133	−0.542564
$53$	− 25.7140i	− 0.485169i	−0.970130	−	0.242585i	$-0.922005\pi$
$53$	− 25.7140i	− 0.485169i	0.970130	−	0.242585i	$-0.0779952\pi$
$54$	0	0
$55$	−50.7206	−0.922193
$56$	− 23.3433i	− 0.416845i
$57$	0	0
$58$	−39.8696	−0.687407
$59$	− 22.7110i	− 0.384933i	−0.981304	−	0.192466i	$-0.938351\pi$
$59$	− 22.7110i	− 0.384933i	0.981304	−	0.192466i	$-0.0616486\pi$
$60$	0	0
$61$	34.0883	0.558824	0.279412	−	0.960171i	$-0.409860\pi$
$61$	34.0883	0.558824	0.279412	+	0.960171i	$0.409860\pi$
$62$	86.0200i	1.38742i
$63$	0	0
$64$	25.5035	0.398492
$65$	60.6957i	0.933780i
$66$	0	0
$67$	8.10443	0.120962	0.0604808	−	0.998169i	$-0.480737\pi$
$67$	8.10443	0.120962	0.0604808	+	0.998169i	$0.480737\pi$
$68$	− 43.6660i	− 0.642147i
$69$	0	0
$70$	133.349	1.90499
$71$	− 41.2871i	− 0.581508i	−0.956798	−	0.290754i	$-0.906094\pi$
$71$	− 41.2871i	− 0.581508i	0.956798	−	0.290754i	$-0.0939062\pi$
$72$	0	0
$73$	−80.8091	−1.10697	−0.553487	−	0.832858i	$-0.686703\pi$
$73$	−80.8091	−1.10697	−0.553487	+	0.832858i	$0.686703\pi$
$74$	83.0914i	1.12286i
$75$	0	0
$76$	51.9093	0.683017
$77$	− 65.8652i	− 0.855393i
$78$	0	0
$79$	−27.6316	−0.349767	−0.174884	−	0.984589i	$-0.555955\pi$
$79$	−27.6316	−0.349767	−0.174884	+	0.984589i	$0.555955\pi$
$80$	− 119.894i	− 1.49868i
$81$	0	0
$82$	43.4126	0.529421
$83$	− 17.5898i	− 0.211925i	−0.994370	−	0.105963i	$-0.966208\pi$
$83$	− 17.5898i	− 0.211925i	0.994370	−	0.105963i	$-0.0337924\pi$
$84$	0	0
$85$	−93.9392	−1.10517
$86$	− 220.279i	− 2.56138i
$87$	0	0
$88$	−23.3325	−0.265142
$89$	28.2885i	0.317849i	0.987291	+	0.158924i	$0.0508026\pi$
$89$	28.2885i	0.317849i	−0.987291	+	0.158924i	$0.949197\pi$
$90$	0	0
$91$	−78.8187	−0.866140
$92$	76.5176i	0.831713i
$93$	0	0
$94$	−34.6095	−0.368187
$95$	− 111.673i	− 1.17551i
$96$	0	0
$97$	168.631	1.73846	0.869232	−	0.494405i	$-0.164614\pi$
$97$	168.631	1.73846	0.869232	+	0.494405i	$0.164614\pi$
$98$	44.4001i	0.453062i
$99$	0	0
$100$	40.9020	0.409020
$101$	− 69.8283i	− 0.691369i	−0.938351	−	0.345684i	$-0.887647\pi$
$101$	− 69.8283i	− 0.691369i	0.938351	−	0.345684i	$-0.112353\pi$
$102$	0	0
$103$	69.8627	0.678279	0.339139	−	0.940736i	$-0.389864\pi$
$103$	69.8627	0.678279	0.339139	+	0.940736i	$0.389864\pi$
$104$	27.9212i	0.268473i
$105$	0	0
$106$	67.5730	0.637481
$107$	121.432i	1.13488i	0.823416	+	0.567438i	$0.192065\pi$
$107$	121.432i	1.13488i	−0.823416	+	0.567438i	$0.807935\pi$
$108$	0	0
$109$	−119.633	−1.09755	−0.548777	−	0.835969i	$-0.684907\pi$
$109$	−119.633	−1.09755	−0.548777	+	0.835969i	$0.684907\pi$
$110$	− 133.287i	− 1.21170i
$111$	0	0
$112$	155.693	1.39012
$113$	124.364i	1.10056i	0.834979	+	0.550282i	$0.185480\pi$
$113$	124.364i	1.10056i	−0.834979	+	0.550282i	$0.814520\pi$
$114$	0	0
$115$	164.613	1.43142
$116$	− 44.0850i	− 0.380043i
$117$	0	0
$118$	59.6817	0.505777
$119$	− 121.988i	− 1.02511i
$120$	0	0
$121$	55.1653	0.455912
$122$	89.5797i	0.734260i
$123$	0	0
$124$	−95.1149	−0.767055
$125$	68.2848i	0.546278i
$126$	0	0
$127$	−117.460	−0.924882	−0.462441	−	0.886650i	$-0.653026\pi$
$127$	−117.460	−0.924882	−0.462441	+	0.886650i	$0.653026\pi$
$128$	− 88.5769i	− 0.692007i
$129$	0	0
$130$	−159.501	−1.22693
$131$	− 4.71281i	− 0.0359756i	−0.999838	−	0.0179878i	$-0.994274\pi$
$131$	− 4.71281i	− 0.0359756i	0.999838	−	0.0179878i	$-0.00572601\pi$
$132$	0	0
$133$	145.018	1.09036
$134$	21.2974i	0.158936i
$135$	0	0
$136$	−43.2139	−0.317749
$137$	220.790i	1.61161i	0.592183	+	0.805804i	$0.298266\pi$
$137$	220.790i	1.61161i	−0.592183	+	0.805804i	$0.701734\pi$
$138$	0	0
$139$	−35.7655	−0.257306	−0.128653	−	0.991690i	$-0.541065\pi$
$139$	−35.7655	−0.257306	−0.128653	+	0.991690i	$0.541065\pi$
$140$	147.448i	1.05320i
$141$	0	0
$142$	108.497	0.764065
$143$	78.7822i	0.550924i
$144$	0	0
$145$	−94.8406	−0.654073
$146$	− 212.356i	− 1.45449i
$147$	0	0
$148$	−91.8766	−0.620788
$149$	167.736i	1.12575i	0.826543	+	0.562873i	$0.190304\pi$
$149$	167.736i	1.12575i	−0.826543	+	0.562873i	$0.809696\pi$
$150$	0	0
$151$	−83.7181	−0.554425	−0.277212	−	0.960809i	$-0.589411\pi$
$151$	−83.7181	−0.554425	−0.277212	+	0.960809i	$0.589411\pi$
$152$	− 51.3719i	− 0.337973i
$153$	0	0
$154$	173.086	1.12393
$155$	204.622i	1.32014i
$156$	0	0
$157$	−74.4442	−0.474167	−0.237084	−	0.971489i	$-0.576192\pi$
$157$	−74.4442	−0.474167	−0.237084	+	0.971489i	$0.576192\pi$
$158$	− 72.6124i	− 0.459572i
$159$	0	0
$160$	243.163	1.51977
$161$	213.765i	1.32773i
$162$	0	0
$163$	254.830	1.56337	0.781687	−	0.623670i	$-0.214359\pi$
$163$	254.830	1.56337	0.781687	+	0.623670i	$0.214359\pi$
$164$	48.0025i	0.292698i
$165$	0	0
$166$	46.2238	0.278456
$167$	− 158.599i	− 0.949697i	−0.880067	−	0.474849i	$-0.842503\pi$
$167$	− 158.599i	− 0.949697i	0.880067	−	0.474849i	$-0.157497\pi$
$168$	0	0
$169$	−74.7240	−0.442154
$170$	− 246.860i	− 1.45212i
$171$	0	0
$172$	243.569	1.41610
$173$	298.081i	1.72301i	0.507747	+	0.861506i	$0.330479\pi$
$173$	298.081i	1.72301i	−0.507747	+	0.861506i	$0.669521\pi$
$174$	0	0
$175$	114.267	0.652953
$176$	− 155.621i	− 0.884211i
$177$	0	0
$178$	−74.3386	−0.417633
$179$	− 29.3291i	− 0.163850i	−0.996639	−	0.0819248i	$-0.973893\pi$
$179$	− 29.3291i	− 0.163850i	0.996639	−	0.0819248i	$-0.0261067\pi$
$180$	0	0
$181$	83.4039	0.460795	0.230397	−	0.973097i	$-0.425997\pi$
$181$	83.4039	0.460795	0.230397	+	0.973097i	$0.425997\pi$
$182$	− 207.126i	− 1.13805i
$183$	0	0
$184$	75.7255	0.411552
$185$	197.655i	1.06841i
$186$	0	0
$187$	−121.932	−0.652042
$188$	− 38.2688i	− 0.203557i
$189$	0	0
$190$	293.463	1.54454
$191$	156.949i	0.821722i	0.911698	+	0.410861i	$0.134772\pi$
$191$	156.949i	0.821722i	−0.911698	+	0.410861i	$0.865228\pi$
$192$	0	0
$193$	−41.4693	−0.214867	−0.107433	−	0.994212i	$-0.534263\pi$
$193$	−41.4693	−0.214867	−0.107433	+	0.994212i	$0.534263\pi$
$194$	443.141i	2.28423i
$195$	0	0
$196$	−49.0945	−0.250482
$197$	− 210.660i	− 1.06934i	−0.845062	−	0.534669i	$-0.820436\pi$
$197$	− 210.660i	− 1.06934i	0.845062	−	0.534669i	$-0.179564\pi$
$198$	0	0
$199$	−175.531	−0.882065	−0.441032	−	0.897491i	$-0.645388\pi$
$199$	−175.531	−0.882065	−0.441032	+	0.897491i	$0.645388\pi$
$200$	− 40.4785i	− 0.202393i
$201$	0	0
$202$	183.500	0.908415
$203$	− 123.159i	− 0.606695i
$204$	0	0
$205$	103.269	0.503749
$206$	183.590i	0.891215i
$207$	0	0
$208$	−186.227	−0.895320
$209$	− 144.950i	− 0.693542i
$210$	0	0
$211$	−260.854	−1.23628	−0.618138	−	0.786070i	$-0.712113\pi$
$211$	−260.854	−1.23628	−0.618138	+	0.786070i	$0.712113\pi$
$212$	74.7175i	0.352441i
$213$	0	0
$214$	−319.107	−1.49115
$215$	− 523.993i	− 2.43718i
$216$	0	0
$217$	−265.720	−1.22452
$218$	− 314.382i	− 1.44212i
$219$	0	0
$220$	147.380	0.669908
$221$	145.912i	0.660234i
$222$	0	0
$223$	−389.217	−1.74537	−0.872684	−	0.488286i	$-0.837622\pi$
$223$	−389.217	−1.74537	−0.872684	+	0.488286i	$0.837622\pi$
$224$	315.769i	1.40968i
$225$	0	0
$226$	−326.812	−1.44607
$227$	348.432i	1.53494i	0.641083	+	0.767471i	$0.278485\pi$
$227$	348.432i	1.53494i	−0.641083	+	0.767471i	$0.721515\pi$
$228$	0	0
$229$	−364.524	−1.59181	−0.795904	−	0.605423i	$-0.793004\pi$
$229$	−364.524	−1.59181	−0.795904	+	0.605423i	$0.793004\pi$
$230$	432.583i	1.88080i
$231$	0	0
$232$	−43.6286	−0.188054
$233$	− 189.671i	− 0.814037i	−0.913420	−	0.407018i	$-0.866569\pi$
$233$	− 189.671i	− 0.814037i	0.913420	−	0.407018i	$-0.133431\pi$
$234$	0	0
$235$	−82.3282	−0.350333
$236$	65.9918i	0.279626i
$237$	0	0
$238$	320.570	1.34693
$239$	306.099i	1.28075i	0.768063	+	0.640374i	$0.221221\pi$
$239$	306.099i	1.28075i	−0.768063	+	0.640374i	$0.778779\pi$
$240$	0	0
$241$	−294.905	−1.22367	−0.611837	−	0.790984i	$-0.709569\pi$
$241$	−294.905	−1.22367	−0.611837	+	0.790984i	$0.709569\pi$
$242$	144.967i	0.599039i
$243$	0	0
$244$	−99.0509	−0.405946
$245$	105.618i	0.431092i
$246$	0	0
$247$	−173.457	−0.702255
$248$	94.1302i	0.379557i
$249$	0	0
$250$	−179.444	−0.717775
$251$	182.909i	0.728723i	0.931258	+	0.364361i	$0.118713\pi$
$251$	182.909i	0.728723i	−0.931258	+	0.364361i	$0.881287\pi$
$252$	0	0
$253$	213.666	0.844530
$254$	− 308.670i	− 1.21524i
$255$	0	0
$256$	334.783	1.30775
$257$	268.364i	1.04422i	0.852879	+	0.522109i	$0.174855\pi$
$257$	268.364i	1.04422i	−0.852879	+	0.522109i	$0.825145\pi$
$258$	0	0
$259$	−256.673	−0.991016
$260$	− 176.364i	− 0.678325i
$261$	0	0
$262$	12.3847	0.0472697
$263$	351.976i	1.33831i	0.743122	+	0.669156i	$0.233344\pi$
$263$	351.976i	1.33831i	−0.743122	+	0.669156i	$0.766656\pi$
$264$	0	0
$265$	160.741	0.606569
$266$	381.088i	1.43266i
$267$	0	0
$268$	−23.5492	−0.0878701
$269$	164.243i	0.610570i	0.952261	+	0.305285i	$0.0987518\pi$
$269$	164.243i	0.610570i	−0.952261	+	0.305285i	$0.901248\pi$
$270$	0	0
$271$	−128.850	−0.475462	−0.237731	−	0.971331i	$-0.576404\pi$
$271$	−128.850	−0.475462	−0.237731	+	0.971331i	$0.576404\pi$
$272$	− 288.225i	− 1.05965i
$273$	0	0
$274$	−580.209	−2.11755
$275$	− 114.214i	− 0.415323i
$276$	0	0
$277$	150.218	0.542303	0.271152	−	0.962537i	$-0.412596\pi$
$277$	150.218	0.542303	0.271152	+	0.962537i	$0.412596\pi$
$278$	− 93.9872i	− 0.338083i
$279$	0	0
$280$	145.922	0.521149
$281$	− 460.999i	− 1.64057i	−0.571958	−	0.820283i	$-0.693816\pi$
$281$	− 460.999i	− 1.64057i	0.571958	−	0.820283i	$-0.306184\pi$
$282$	0	0
$283$	−68.8575	−0.243313	−0.121656	−	0.992572i	$-0.538821\pi$
$283$	−68.8575	−0.243313	−0.121656	+	0.992572i	$0.538821\pi$
$284$	119.969i	0.422425i
$285$	0	0
$286$	−207.030	−0.723880
$287$	134.103i	0.467259i
$288$	0	0
$289$	63.1711	0.218585
$290$	− 249.229i	− 0.859411i
$291$	0	0
$292$	234.808	0.804139
$293$	− 510.766i	− 1.74323i	−0.490191	−	0.871615i	$-0.663073\pi$
$293$	− 510.766i	− 1.74323i	0.490191	−	0.871615i	$-0.336927\pi$
$294$	0	0
$295$	141.969	0.481251
$296$	90.9255i	0.307181i
$297$	0	0
$298$	−440.789	−1.47916
$299$	− 255.687i	− 0.855141i
$300$	0	0
$301$	680.452	2.26064
$302$	− 220.001i	− 0.728479i
$303$	0	0
$304$	342.636	1.12709
$305$	213.090i	0.698655i
$306$	0	0
$307$	52.6035	0.171347	0.0856734	−	0.996323i	$-0.472696\pi$
$307$	52.6035	0.171347	0.0856734	+	0.996323i	$0.472696\pi$
$308$	191.386i	0.621382i
$309$	0	0
$310$	−537.721	−1.73458
$311$	354.825i	1.14092i	0.821326	+	0.570459i	$0.193235\pi$
$311$	354.825i	1.14092i	−0.821326	+	0.570459i	$0.806765\pi$
$312$	0	0
$313$	168.249	0.537536	0.268768	−	0.963205i	$-0.413384\pi$
$313$	168.249	0.537536	0.268768	+	0.963205i	$0.413384\pi$
$314$	− 195.630i	− 0.623026i
$315$	0	0
$316$	80.2897	0.254081
$317$	509.988i	1.60880i	0.594091	+	0.804398i	$0.297512\pi$
$317$	509.988i	1.60880i	−0.594091	+	0.804398i	$0.702488\pi$
$318$	0	0
$319$	−123.102	−0.385899
$320$	159.425i	0.498204i
$321$	0	0
$322$	−561.748	−1.74456
$323$	− 268.461i	− 0.831149i
$324$	0	0
$325$	−136.676	−0.420541
$326$	669.661i	2.05418i
$327$	0	0
$328$	47.5056	0.144834
$329$	− 106.911i	− 0.324956i
$330$	0	0
$331$	−339.342	−1.02520	−0.512601	−	0.858627i	$-0.671318\pi$
$331$	−339.342	−1.02520	−0.512601	+	0.858627i	$0.671318\pi$
$332$	51.1110i	0.153949i
$333$	0	0
$334$	416.779	1.24784
$335$	50.6617i	0.151229i
$336$	0	0
$337$	−68.2380	−0.202487	−0.101243	−	0.994862i	$-0.532282\pi$
$337$	−68.2380	−0.202487	−0.101243	+	0.994862i	$0.532282\pi$
$338$	− 196.365i	− 0.580962i
$339$	0	0
$340$	272.961	0.802826
$341$	265.597i	0.778876i
$342$	0	0
$343$	260.610	0.759795
$344$	− 241.047i	− 0.700719i
$345$	0	0
$346$	−783.320	−2.26393
$347$	173.098i	0.498840i	0.968395	+	0.249420i	$0.0802400\pi$
$347$	173.098i	0.498840i	−0.968395	+	0.249420i	$0.919760\pi$
$348$	0	0
$349$	459.582	1.31685	0.658427	−	0.752645i	$-0.271222\pi$
$349$	459.582	1.31685	0.658427	+	0.752645i	$0.271222\pi$
$350$	300.278i	0.857939i
$351$	0	0
$352$	315.623	0.896655
$353$	− 402.250i	− 1.13952i	−0.821812	−	0.569759i	$-0.807036\pi$
$353$	− 402.250i	− 1.13952i	0.821812	−	0.569759i	$-0.192964\pi$
$354$	0	0
$355$	258.090	0.727015
$356$	− 82.1984i	− 0.230895i
$357$	0	0
$358$	77.0731	0.215288
$359$	− 227.763i	− 0.634437i	−0.948352	−	0.317219i	$-0.897251\pi$
$359$	− 227.763i	− 0.634437i	0.948352	−	0.317219i	$-0.102749\pi$
$360$	0	0
$361$	−41.8587	−0.115952
$362$	219.175i	0.605455i
$363$	0	0
$364$	229.025	0.629190
$365$	− 505.147i	− 1.38396i
$366$	0	0
$367$	−235.467	−0.641599	−0.320800	−	0.947147i	$-0.603952\pi$
$367$	−235.467	−0.641599	−0.320800	+	0.947147i	$0.603952\pi$
$368$	505.067i	1.37247i
$369$	0	0
$370$	−519.413	−1.40382
$371$	208.736i	0.562631i
$372$	0	0
$373$	652.752	1.75000	0.875002	−	0.484119i	$-0.160860\pi$
$373$	652.752	1.75000	0.875002	+	0.484119i	$0.160860\pi$
$374$	− 320.421i	− 0.856742i
$375$	0	0
$376$	−37.8726	−0.100725
$377$	147.312i	0.390748i
$378$	0	0
$379$	625.053	1.64922	0.824608	−	0.565705i	$-0.191396\pi$
$379$	625.053	1.64922	0.824608	+	0.565705i	$0.191396\pi$
$380$	324.490i	0.853922i
$381$	0	0
$382$	−412.442	−1.07969
$383$	106.540i	0.278171i	0.990280	+	0.139086i	$0.0444163\pi$
$383$	106.540i	0.278171i	−0.990280	+	0.139086i	$0.955584\pi$
$384$	0	0
$385$	411.731	1.06943
$386$	− 108.976i	− 0.282321i
$387$	0	0
$388$	−489.994	−1.26287
$389$	593.632i	1.52605i	0.646371	+	0.763023i	$0.276286\pi$
$389$	593.632i	1.52605i	−0.646371	+	0.763023i	$0.723714\pi$
$390$	0	0
$391$	395.729	1.01209
$392$	48.5862i	0.123944i
$393$	0	0
$394$	553.587	1.40504
$395$	− 172.728i	− 0.437287i
$396$	0	0
$397$	374.819	0.944129	0.472064	−	0.881564i	$-0.343509\pi$
$397$	374.819	0.944129	0.472064	+	0.881564i	$0.343509\pi$
$398$	− 461.273i	− 1.15898i
$399$	0	0
$400$	269.980	0.674951
$401$	274.406i	0.684303i	0.939645	+	0.342152i	$0.111156\pi$
$401$	274.406i	0.684303i	−0.939645	+	0.342152i	$0.888844\pi$
$402$	0	0
$403$	317.831	0.788661
$404$	202.901i	0.502231i
$405$	0	0
$406$	323.646	0.797158
$407$	256.554i	0.630354i
$408$	0	0
$409$	−551.474	−1.34835	−0.674174	−	0.738573i	$-0.735500\pi$
$409$	−551.474	−1.34835	−0.674174	+	0.738573i	$0.735500\pi$
$410$	271.377i	0.661894i
$411$	0	0
$412$	−203.001	−0.492722
$413$	184.360i	0.446391i
$414$	0	0
$415$	109.956	0.264954
$416$	− 377.695i	− 0.907921i
$417$	0	0
$418$	380.911	0.911270
$419$	87.5877i	0.209040i	0.994523	+	0.104520i	$0.0333306\pi$
$419$	87.5877i	0.209040i	−0.994523	+	0.104520i	$0.966669\pi$
$420$	0	0
$421$	296.259	0.703704	0.351852	−	0.936056i	$-0.385552\pi$
$421$	296.259	0.703704	0.351852	+	0.936056i	$0.385552\pi$
$422$	− 685.491i	− 1.62439i
$423$	0	0
$424$	73.9440	0.174396
$425$	− 211.534i	− 0.497728i
$426$	0	0
$427$	−276.716	−0.648047
$428$	− 352.846i	− 0.824407i
$429$	0	0
$430$	1376.99	3.20230
$431$	− 586.175i	− 1.36003i	−0.733196	−	0.680017i	$-0.761972\pi$
$431$	− 586.175i	− 1.36003i	0.733196	−	0.680017i	$-0.238028\pi$
$432$	0	0
$433$	415.367	0.959277	0.479639	−	0.877466i	$-0.340768\pi$
$433$	415.367	0.959277	0.479639	+	0.877466i	$0.340768\pi$
$434$	− 698.278i	− 1.60894i
$435$	0	0
$436$	347.621	0.797296
$437$	470.435i	1.07651i
$438$	0	0
$439$	334.358	0.761636	0.380818	−	0.924650i	$-0.375642\pi$
$439$	334.358	0.761636	0.380818	+	0.924650i	$0.375642\pi$
$440$	− 145.854i	− 0.331486i
$441$	0	0
$442$	−383.438	−0.867506
$443$	− 451.642i	− 1.01951i	−0.860320	−	0.509754i	$-0.829737\pi$
$443$	− 451.642i	− 1.01951i	0.860320	−	0.509754i	$-0.170263\pi$
$444$	0	0
$445$	−176.835	−0.397381
$446$	− 1022.81i	− 2.29330i
$447$	0	0
$448$	−207.028	−0.462116
$449$	191.013i	0.425420i	0.977115	+	0.212710i	$0.0682289\pi$
$449$	191.013i	0.425420i	−0.977115	+	0.212710i	$0.931771\pi$
$450$	0	0
$451$	134.041	0.297209
$452$	− 361.366i	− 0.799482i
$453$	0	0
$454$	−915.635	−2.01682
$455$	− 492.705i	− 1.08287i
$456$	0	0
$457$	−385.752	−0.844097	−0.422049	−	0.906573i	$-0.638689\pi$
$457$	−385.752	−0.844097	−0.422049	+	0.906573i	$0.638689\pi$
$458$	− 957.923i	− 2.09153i
$459$	0	0
$460$	−478.320	−1.03983
$461$	433.054i	0.939379i	0.882832	+	0.469689i	$0.155634\pi$
$461$	433.054i	0.939379i	−0.882832	+	0.469689i	$0.844366\pi$
$462$	0	0
$463$	552.856	1.19407	0.597037	−	0.802214i	$-0.296345\pi$
$463$	552.856	1.19407	0.597037	+	0.802214i	$0.296345\pi$
$464$	− 290.990i	− 0.627134i
$465$	0	0
$466$	498.430	1.06959
$467$	− 215.313i	− 0.461056i	−0.973066	−	0.230528i	$-0.925955\pi$
$467$	− 215.313i	− 0.461056i	0.973066	−	0.230528i	$-0.0740454\pi$
$468$	0	0
$469$	−65.7887	−0.140274
$470$	− 216.348i	− 0.460315i
$471$	0	0
$472$	65.3086	0.138366
$473$	− 680.136i	− 1.43792i
$474$	0	0
$475$	251.468	0.529406
$476$	354.464i	0.744672i
$477$	0	0
$478$	−804.389	−1.68282
$479$	− 226.490i	− 0.472839i	−0.971651	−	0.236420i	$-0.924026\pi$
$479$	− 226.490i	− 0.472839i	0.971651	−	0.236420i	$-0.0759740\pi$
$480$	0	0
$481$	307.010	0.638274
$482$	− 774.974i	− 1.60783i
$483$	0	0
$484$	−160.295	−0.331188
$485$	1054.13i	2.17347i
$486$	0	0
$487$	−443.130	−0.909919	−0.454959	−	0.890512i	$-0.650346\pi$
$487$	−443.130	−0.909919	−0.454959	+	0.890512i	$0.650346\pi$
$488$	98.0255i	0.200872i
$489$	0	0
$490$	−277.550	−0.566428
$491$	− 667.456i	− 1.35938i	−0.733499	−	0.679690i	$-0.762114\pi$
$491$	− 667.456i	− 1.35938i	0.733499	−	0.679690i	$-0.237886\pi$
$492$	0	0
$493$	−227.996	−0.462466
$494$	− 455.823i	− 0.922719i
$495$	0	0
$496$	−627.822	−1.26577
$497$	335.153i	0.674352i
$498$	0	0
$499$	−57.4845	−0.115199	−0.0575997	−	0.998340i	$-0.518345\pi$
$499$	−57.4845	−0.115199	−0.0575997	+	0.998340i	$0.518345\pi$
$500$	− 198.416i	− 0.396833i
$501$	0	0
$502$	−480.663	−0.957495
$503$	− 488.672i	− 0.971515i	−0.874094	−	0.485757i	$-0.838544\pi$
$503$	− 488.672i	− 0.971515i	0.874094	−	0.485757i	$-0.161456\pi$
$504$	0	0
$505$	436.504	0.864365
$506$	561.487i	1.10966i
$507$	0	0
$508$	341.305	0.671861
$509$	− 284.252i	− 0.558452i	−0.960225	−	0.279226i	$-0.909922\pi$
$509$	− 284.252i	− 0.558452i	0.960225	−	0.279226i	$-0.0900778\pi$
$510$	0	0
$511$	655.978	1.28371
$512$	525.459i	1.02629i
$513$	0	0
$514$	−705.227	−1.37204
$515$	436.720i	0.847999i
$516$	0	0
$517$	−106.861	−0.206694
$518$	− 674.505i	− 1.30213i
$519$	0	0
$520$	−174.539	−0.335651
$521$	40.6415i	0.0780067i	0.999239	+	0.0390033i	$0.0124183\pi$
$521$	40.6415i	0.0780067i	−0.999239	+	0.0390033i	$0.987582\pi$
$522$	0	0
$523$	349.901	0.669028	0.334514	−	0.942391i	$-0.391428\pi$
$523$	349.901	0.669028	0.334514	+	0.942391i	$0.391428\pi$
$524$	13.6941i	0.0261338i
$525$	0	0
$526$	−924.949	−1.75846
$527$	491.909i	0.933414i
$528$	0	0
$529$	−164.452	−0.310873
$530$	422.406i	0.796993i
$531$	0	0
$532$	−421.380	−0.792067
$533$	− 160.403i	− 0.300943i
$534$	0	0
$535$	−759.083	−1.41885
$536$	23.3054i	0.0434802i
$537$	0	0
$538$	−431.611	−0.802251
$539$	137.090i	0.254342i
$540$	0	0
$541$	145.531	0.269003	0.134501	−	0.990913i	$-0.457057\pi$
$541$	145.531	0.269003	0.134501	+	0.990913i	$0.457057\pi$
$542$	− 338.602i	− 0.624727i
$543$	0	0
$544$	584.562	1.07456
$545$	− 747.842i	− 1.37219i
$546$	0	0
$547$	486.029	0.888535	0.444268	−	0.895894i	$-0.353464\pi$
$547$	486.029	0.888535	0.444268	+	0.895894i	$0.353464\pi$
$548$	− 641.554i	− 1.17072i
$549$	0	0
$550$	300.139	0.545708
$551$	− 271.037i	− 0.491900i
$552$	0	0
$553$	224.303	0.405611
$554$	394.754i	0.712552i
$555$	0	0
$556$	103.924	0.186914
$557$	− 3.26556i	− 0.00586276i	−0.999996	−	0.00293138i	$-0.999067\pi$
$557$	− 3.26556i	− 0.00586276i	0.999996	−	0.00293138i	$-0.000933088\pi$
$558$	0	0
$559$	−813.896	−1.45599
$560$	973.257i	1.73796i
$561$	0	0
$562$	1211.45	2.15560
$563$	664.749i	1.18073i	0.807138	+	0.590363i	$0.201015\pi$
$563$	664.749i	1.18073i	−0.807138	+	0.590363i	$0.798985\pi$
$564$	0	0
$565$	−777.411	−1.37595
$566$	− 180.949i	− 0.319697i
$567$	0	0
$568$	118.727	0.209026
$569$	75.9350i	0.133453i	0.997771	+	0.0667267i	$0.0212556\pi$
$569$	75.9350i	0.133453i	−0.997771	+	0.0667267i	$0.978744\pi$
$570$	0	0
$571$	−294.091	−0.515046	−0.257523	−	0.966272i	$-0.582906\pi$
$571$	−294.091	−0.515046	−0.257523	+	0.966272i	$0.582906\pi$
$572$	− 228.919i	− 0.400208i
$573$	0	0
$574$	−352.407	−0.613949
$575$	370.680i	0.644661i
$576$	0	0
$577$	1058.59	1.83465	0.917325	−	0.398139i	$-0.130344\pi$
$577$	1058.59	1.83465	0.917325	+	0.398139i	$0.130344\pi$
$578$	166.006i	0.287207i
$579$	0	0
$580$	275.580	0.475138
$581$	142.787i	0.245761i
$582$	0	0
$583$	208.639	0.357872
$584$	− 232.378i	− 0.397907i
$585$	0	0
$586$	1342.23	2.29049
$587$	138.902i	0.236630i	0.992976	+	0.118315i	$0.0377493\pi$
$587$	138.902i	0.236630i	−0.992976	+	0.118315i	$0.962251\pi$
$588$	0	0
$589$	−584.772	−0.992822
$590$	373.077i	0.632334i
$591$	0	0
$592$	−606.447	−1.02440
$593$	− 145.309i	− 0.245040i	−0.992466	−	0.122520i	$-0.960902\pi$
$593$	− 145.309i	− 0.245040i	0.992466	−	0.122520i	$-0.0390975\pi$
$594$	0	0
$595$	762.563	1.28162
$596$	− 487.394i	− 0.817775i
$597$	0	0
$598$	671.913	1.12360
$599$	− 22.9725i	− 0.0383515i	−0.999816	−	0.0191757i	$-0.993896\pi$
$599$	− 22.9725i	− 0.0383515i	0.999816	−	0.0191757i	$-0.00610420\pi$
$600$	0	0
$601$	−927.369	−1.54304	−0.771522	−	0.636203i	$-0.780504\pi$
$601$	−927.369	−1.54304	−0.771522	+	0.636203i	$0.780504\pi$
$602$	1788.14i	2.97033i
$603$	0	0
$604$	243.261	0.402750
$605$	344.844i	0.569991i
$606$	0	0
$607$	523.972	0.863216	0.431608	−	0.902061i	$-0.357946\pi$
$607$	523.972	0.863216	0.431608	+	0.902061i	$0.357946\pi$
$608$	694.916i	1.14295i
$609$	0	0
$610$	−559.973	−0.917988
$611$	127.877i	0.209291i
$612$	0	0
$613$	70.1697	0.114469	0.0572346	−	0.998361i	$-0.481772\pi$
$613$	70.1697	0.114469	0.0572346	+	0.998361i	$0.481772\pi$
$614$	138.235i	0.225139i
$615$	0	0
$616$	189.404	0.307475
$617$	− 664.561i	− 1.07708i	−0.842598	−	0.538542i	$-0.818975\pi$
$617$	− 664.561i	− 1.07708i	0.842598	−	0.538542i	$-0.181025\pi$
$618$	0	0
$619$	−970.754	−1.56826	−0.784131	−	0.620596i	$-0.786891\pi$
$619$	−970.754	−1.56826	−0.784131	+	0.620596i	$0.786891\pi$
$620$	− 594.574i	− 0.958990i
$621$	0	0
$622$	−932.436	−1.49909
$623$	− 229.636i	− 0.368596i
$624$	0	0
$625$	−778.765	−1.24602
$626$	442.136i	0.706288i
$627$	0	0
$628$	216.314	0.344449
$629$	475.162i	0.755424i
$630$	0	0
$631$	−189.318	−0.300028	−0.150014	−	0.988684i	$-0.547932\pi$
$631$	−189.318	−0.300028	−0.150014	+	0.988684i	$0.547932\pi$
$632$	− 79.4585i	− 0.125725i
$633$	0	0
$634$	−1340.18	−2.11386
$635$	− 734.255i	− 1.15631i
$636$	0	0
$637$	164.051	0.257538
$638$	− 323.496i	− 0.507047i
$639$	0	0
$640$	553.704	0.865163
$641$	− 692.705i	− 1.08066i	−0.841452	−	0.540332i	$-0.818299\pi$
$641$	− 692.705i	− 1.08066i	0.841452	−	0.540332i	$-0.181701\pi$
$642$	0	0
$643$	1064.60	1.65568	0.827841	−	0.560962i	$-0.189569\pi$
$643$	1064.60	1.65568	0.827841	+	0.560962i	$0.189569\pi$
$644$	− 621.141i	− 0.964505i
$645$	0	0
$646$	705.481	1.09208
$647$	− 352.755i	− 0.545217i	−0.962125	−	0.272608i	$-0.912114\pi$
$647$	− 352.755i	− 0.545217i	0.962125	−	0.272608i	$-0.0878863\pi$
$648$	0	0
$649$	184.274	0.283935
$650$	− 359.167i	− 0.552564i
$651$	0	0
$652$	−740.464	−1.13568
$653$	468.433i	0.717355i	0.933462	+	0.358678i	$0.116772\pi$
$653$	468.433i	0.717355i	−0.933462	+	0.358678i	$0.883228\pi$
$654$	0	0
$655$	29.4603	0.0449775
$656$	316.849i	0.483001i
$657$	0	0
$658$	280.947	0.426972
$659$	85.9405i	0.130410i	0.997872	+	0.0652052i	$0.0207702\pi$
$659$	85.9405i	0.130410i	−0.997872	+	0.0652052i	$0.979230\pi$
$660$	0	0
$661$	123.112	0.186251	0.0931256	−	0.995654i	$-0.470314\pi$
$661$	123.112	0.186251	0.0931256	+	0.995654i	$0.470314\pi$
$662$	− 891.747i	− 1.34705i
$663$	0	0
$664$	50.5819	0.0761775
$665$	906.520i	1.36319i
$666$	0	0
$667$	399.526	0.598990
$668$	460.845i	0.689888i
$669$	0	0
$670$	−133.132	−0.198705
$671$	276.588i	0.412202i
$672$	0	0
$673$	−636.613	−0.945934	−0.472967	−	0.881080i	$-0.656817\pi$
$673$	−636.613	−0.945934	−0.472967	+	0.881080i	$0.656817\pi$
$674$	− 179.321i	− 0.266054i
$675$	0	0
$676$	217.127	0.321194
$677$	165.309i	0.244179i	0.992519	+	0.122089i	$0.0389595\pi$
$677$	165.309i	0.244179i	−0.992519	+	0.122089i	$0.961041\pi$
$678$	0	0
$679$	−1368.88	−2.01603
$680$	− 270.135i	− 0.397257i
$681$	0	0
$682$	−697.954	−1.02339
$683$	− 1304.33i	− 1.90970i	−0.297085	−	0.954851i	$-0.596014\pi$
$683$	− 1304.33i	− 1.90970i	0.297085	−	0.954851i	$-0.403986\pi$
$684$	0	0
$685$	−1380.18	−2.01487
$686$	684.849i	0.998322i
$687$	0	0
$688$	1607.72	2.33680
$689$	− 249.672i	− 0.362368i
$690$	0	0
$691$	−361.817	−0.523613	−0.261807	−	0.965120i	$-0.584318\pi$
$691$	−361.817	−0.523613	−0.261807	+	0.965120i	$0.584318\pi$
$692$	− 866.140i	− 1.25165i
$693$	0	0
$694$	−454.879	−0.655445
$695$	− 223.574i	− 0.321689i
$696$	0	0
$697$	248.257	0.356179
$698$	1207.72i	1.73026i
$699$	0	0
$700$	−332.027	−0.474324
$701$	893.344i	1.27438i	0.770705	+	0.637192i	$0.219904\pi$
$701$	893.344i	1.27438i	−0.770705	+	0.637192i	$0.780096\pi$
$702$	0	0
$703$	−564.863	−0.803504
$704$	206.932i	0.293937i
$705$	0	0
$706$	1057.06	1.49725
$707$	566.840i	0.801753i
$708$	0	0
$709$	−690.613	−0.974066	−0.487033	−	0.873384i	$-0.661921\pi$
$709$	−690.613	−0.974066	−0.487033	+	0.873384i	$0.661921\pi$
$710$	678.228i	0.955251i
$711$	0	0
$712$	−81.3475	−0.114252
$713$	− 861.992i	− 1.20896i
$714$	0	0
$715$	−492.476	−0.688778
$716$	85.2220i	0.119025i
$717$	0	0
$718$	598.532	0.833610
$719$	− 482.097i	− 0.670511i	−0.942127	−	0.335255i	$-0.891177\pi$
$719$	− 482.097i	− 0.670511i	0.942127	−	0.335255i	$-0.108823\pi$
$720$	0	0
$721$	−567.119	−0.786573
$722$	− 109.999i	− 0.152354i
$723$	0	0
$724$	−242.348	−0.334735
$725$	− 213.564i	− 0.294571i
$726$	0	0
$727$	345.033	0.474598	0.237299	−	0.971437i	$-0.423738\pi$
$727$	345.033	0.474598	0.237299	+	0.971437i	$0.423738\pi$
$728$	− 226.654i	− 0.311338i
$729$	0	0
$730$	1327.46	1.81844
$731$	− 1259.67i	− 1.72322i
$732$	0	0
$733$	539.274	0.735708	0.367854	−	0.929884i	$-0.380093\pi$
$733$	539.274	0.735708	0.367854	+	0.929884i	$0.380093\pi$
$734$	− 618.777i	− 0.843021i
$735$	0	0
$736$	−1024.35	−1.39178
$737$	65.7582i	0.0892241i
$738$	0	0
$739$	737.474	0.997935	0.498968	−	0.866621i	$-0.333713\pi$
$739$	737.474	0.997935	0.498968	+	0.866621i	$0.333713\pi$
$740$	− 574.331i	− 0.776123i
$741$	0	0
$742$	−548.532	−0.739262
$743$	1041.25i	1.40141i	0.713450	+	0.700706i	$0.247132\pi$
$743$	1041.25i	1.40141i	−0.713450	+	0.700706i	$0.752868\pi$
$744$	0	0
$745$	−1048.54	−1.40743
$746$	1715.35i	2.29939i
$747$	0	0
$748$	354.299	0.473662
$749$	− 985.736i	− 1.31607i
$750$	0	0
$751$	241.036	0.320953	0.160476	−	0.987040i	$-0.448697\pi$
$751$	241.036	0.320953	0.160476	+	0.987040i	$0.448697\pi$
$752$	− 252.600i	− 0.335904i
$753$	0	0
$754$	−387.117	−0.513418
$755$	− 523.331i	− 0.693154i
$756$	0	0
$757$	32.7615	0.0432781	0.0216391	−	0.999766i	$-0.493112\pi$
$757$	32.7615	0.0432781	0.0216391	+	0.999766i	$0.493112\pi$
$758$	1642.56i	2.16696i
$759$	0	0
$760$	321.131	0.422541
$761$	319.353i	0.419650i	0.977739	+	0.209825i	$0.0672894\pi$
$761$	319.353i	0.419650i	−0.977739	+	0.209825i	$0.932711\pi$
$762$	0	0
$763$	971.140	1.27279
$764$	− 456.049i	− 0.596923i
$765$	0	0
$766$	−279.972	−0.365499
$767$	− 220.515i	− 0.287503i
$768$	0	0
$769$	−476.207	−0.619255	−0.309628	−	0.950858i	$-0.600204\pi$
$769$	−476.207	−0.619255	−0.309628	+	0.950858i	$0.600204\pi$
$770$	1081.98i	1.40516i
$771$	0	0
$772$	120.498	0.156086
$773$	− 1296.29i	− 1.67696i	−0.544929	−	0.838482i	$-0.683444\pi$
$773$	− 1296.29i	− 1.67696i	0.544929	−	0.838482i	$-0.316556\pi$
$774$	0	0
$775$	−460.772	−0.594544
$776$	484.921i	0.624899i
$777$	0	0
$778$	−1559.99	−2.00513
$779$	295.123i	0.378848i
$780$	0	0
$781$	334.998	0.428934
$782$	1039.93i	1.32983i
$783$	0	0
$784$	−324.056	−0.413337
$785$	− 465.359i	− 0.592814i
$786$	0	0
$787$	860.953	1.09397	0.546984	−	0.837143i	$-0.315776\pi$
$787$	860.953	1.09397	0.546984	+	0.837143i	$0.315776\pi$
$788$	612.117i	0.776799i
$789$	0	0
$790$	453.908	0.574567
$791$	− 1009.54i	− 1.27628i
$792$	0	0
$793$	330.983	0.417381
$794$	984.977i	1.24053i
$795$	0	0
$796$	510.043	0.640758
$797$	580.125i	0.727886i	0.931421	+	0.363943i	$0.118570\pi$
$797$	580.125i	0.727886i	−0.931421	+	0.363943i	$0.881430\pi$
$798$	0	0
$799$	−197.916	−0.247705
$800$	547.560i	0.684450i
$801$	0	0
$802$	−721.103	−0.899131
$803$	− 655.674i	− 0.816530i
$804$	0	0
$805$	−1336.27	−1.65996
$806$	835.218i	1.03625i
$807$	0	0
$808$	200.801	0.248516
$809$	− 661.323i	− 0.817457i	−0.912656	−	0.408729i	$-0.865972\pi$
$809$	− 661.323i	− 0.817457i	0.912656	−	0.408729i	$-0.134028\pi$
$810$	0	0
$811$	−168.725	−0.208045	−0.104023	−	0.994575i	$-0.533171\pi$
$811$	−168.725	−0.208045	−0.104023	+	0.994575i	$0.533171\pi$
$812$	357.865i	0.440721i
$813$	0	0
$814$	−674.192	−0.828245
$815$	1592.97i	1.95457i
$816$	0	0
$817$	1497.48	1.83290
$818$	− 1449.20i	− 1.77164i
$819$	0	0
$820$	−300.069	−0.365938
$821$	− 151.578i	− 0.184626i	−0.995730	−	0.0923132i	$-0.970574\pi$
$821$	− 151.578i	− 0.184626i	0.995730	−	0.0923132i	$-0.0294261\pi$
$822$	0	0
$823$	224.822	0.273173	0.136587	−	0.990628i	$-0.456387\pi$
$823$	224.822	0.273173	0.136587	+	0.990628i	$0.456387\pi$
$824$	200.900i	0.243810i
$825$	0	0
$826$	−484.474	−0.586530
$827$	− 338.159i	− 0.408899i	−0.978877	−	0.204449i	$-0.934460\pi$
$827$	− 338.159i	− 0.408899i	0.978877	−	0.204449i	$-0.0655404\pi$
$828$	0	0
$829$	−803.612	−0.969376	−0.484688	−	0.874687i	$-0.661067\pi$
$829$	−803.612	−0.969376	−0.484688	+	0.874687i	$0.661067\pi$
$830$	288.950i	0.348132i
$831$	0	0
$832$	247.628	0.297630
$833$	253.904i	0.304806i
$834$	0	0
$835$	991.423	1.18733
$836$	421.184i	0.503809i
$837$	0	0
$838$	−230.169	−0.274665
$839$	853.942i	1.01781i	0.860823	+	0.508905i	$0.169950\pi$
$839$	853.942i	1.01781i	−0.860823	+	0.508905i	$0.830050\pi$
$840$	0	0
$841$	610.816	0.726298
$842$	778.532i	0.924623i
$843$	0	0
$844$	757.968	0.898067
$845$	− 467.108i	− 0.552791i
$846$	0	0
$847$	−447.811	−0.528703
$848$	493.185i	0.581587i
$849$	0	0
$850$	555.885	0.653982
$851$	− 832.645i	− 0.978431i
$852$	0	0
$853$	−799.115	−0.936829	−0.468414	−	0.883509i	$-0.655175\pi$
$853$	−799.115	−0.936829	−0.468414	+	0.883509i	$0.655175\pi$
$854$	− 727.174i	− 0.851492i
$855$	0	0
$856$	−349.193	−0.407936
$857$	1553.26i	1.81244i	0.422809	+	0.906219i	$0.361044\pi$
$857$	1553.26i	1.81244i	−0.422809	+	0.906219i	$0.638956\pi$
$858$	0	0
$859$	−28.2520	−0.0328895	−0.0164447	−	0.999865i	$-0.505235\pi$
$859$	−28.2520	−0.0328895	−0.0164447	+	0.999865i	$0.505235\pi$
$860$	1522.58i	1.77044i
$861$	0	0
$862$	1540.39	1.78700
$863$	951.550i	1.10261i	0.834305	+	0.551304i	$0.185869\pi$
$863$	951.550i	1.10261i	−0.834305	+	0.551304i	$0.814131\pi$
$864$	0	0
$865$	−1863.34	−2.15415
$866$	1091.53i	1.26043i
$867$	0	0
$868$	772.107	0.889524
$869$	− 224.199i	− 0.257997i
$870$	0	0
$871$	78.6906	0.0903451
$872$	− 344.022i	− 0.394521i
$873$	0	0
$874$	−1236.24	−1.41447
$875$	− 554.310i	− 0.633497i
$876$	0	0
$877$	−228.193	−0.260197	−0.130098	−	0.991501i	$-0.541529\pi$
$877$	−228.193	−0.260197	−0.130098	+	0.991501i	$0.541529\pi$
$878$	878.651i	1.00074i
$879$	0	0
$880$	972.805	1.10546
$881$	1551.18i	1.76071i	0.474318	+	0.880354i	$0.342695\pi$
$881$	1551.18i	1.76071i	−0.474318	+	0.880354i	$0.657305\pi$
$882$	0	0
$883$	1563.59	1.77077	0.885384	−	0.464859i	$-0.153895\pi$
$883$	1563.59	1.77077	0.885384	+	0.464859i	$0.153895\pi$
$884$	− 423.978i	− 0.479614i
$885$	0	0
$886$	1186.86	1.33957
$887$	− 702.161i	− 0.791613i	−0.918334	−	0.395807i	$-0.870465\pi$
$887$	− 702.161i	− 0.791613i	0.918334	−	0.395807i	$-0.129535\pi$
$888$	0	0
$889$	953.496	1.07255
$890$	− 464.699i	− 0.522134i
$891$	0	0
$892$	1130.95	1.26789
$893$	− 235.279i	− 0.263470i
$894$	0	0
$895$	183.339	0.204848
$896$	719.034i	0.802493i
$897$	0	0
$898$	−501.959	−0.558974
$899$	496.629i	0.552424i
$900$	0	0
$901$	386.419	0.428878
$902$	352.243i	0.390514i
$903$	0	0
$904$	−357.625	−0.395603
$905$	521.367i	0.576096i
$906$	0	0
$907$	1614.51	1.78006	0.890030	−	0.455903i	$-0.150683\pi$
$907$	1614.51	1.78006	0.890030	+	0.455903i	$0.150683\pi$
$908$	− 1012.44i	− 1.11503i
$909$	0	0
$910$	1294.77	1.42282
$911$	− 612.962i	− 0.672846i	−0.941711	−	0.336423i	$-0.890783\pi$
$911$	− 612.962i	− 0.672846i	0.941711	−	0.336423i	$-0.109217\pi$
$912$	0	0
$913$	142.721	0.156321
$914$	− 1013.71i	− 1.10909i
$915$	0	0
$916$	1059.20	1.15634
$917$	38.2568i	0.0417195i
$918$	0	0
$919$	628.091	0.683451	0.341725	−	0.939800i	$-0.388989\pi$
$919$	628.091	0.683451	0.341725	+	0.939800i	$0.388989\pi$
$920$	473.368i	0.514531i
$921$	0	0
$922$	−1138.01	−1.23428
$923$	− 400.881i	− 0.434323i
$924$	0	0
$925$	−445.085	−0.481173
$926$	1452.84i	1.56894i
$927$	0	0
$928$	590.171	0.635961
$929$	− 1253.56i	− 1.34936i	−0.738109	−	0.674681i	$-0.764281\pi$
$929$	− 1253.56i	− 1.34936i	0.738109	−	0.674681i	$-0.235719\pi$
$930$	0	0
$931$	−301.836	−0.324206
$932$	551.129i	0.591340i
$933$	0	0
$934$	565.816	0.605799
$935$	− 762.209i	− 0.815197i
$936$	0	0
$937$	−920.264	−0.982139	−0.491070	−	0.871120i	$-0.663394\pi$
$937$	−920.264	−0.982139	−0.491070	+	0.871120i	$0.663394\pi$
$938$	− 172.884i	− 0.184312i
$939$	0	0
$940$	239.222	0.254492
$941$	683.451i	0.726303i	0.931730	+	0.363152i	$0.118299\pi$
$941$	683.451i	0.726303i	−0.931730	+	0.363152i	$0.881701\pi$
$942$	0	0
$943$	−435.030	−0.461325
$944$	435.590i	0.461430i
$945$	0	0
$946$	1787.31	1.88934
$947$	− 841.784i	− 0.888895i	−0.895805	−	0.444448i	$-0.853400\pi$
$947$	− 841.784i	− 0.888895i	0.895805	−	0.444448i	$-0.146600\pi$
$948$	0	0
$949$	−784.623	−0.826789
$950$	660.825i	0.695606i
$951$	0	0
$952$	350.794	0.368481
$953$	1475.15i	1.54790i	0.633244	+	0.773952i	$0.281723\pi$
$953$	1475.15i	1.54790i	−0.633244	+	0.773952i	$0.718277\pi$
$954$	0	0
$955$	−981.105	−1.02734
$956$	− 889.436i	− 0.930373i
$957$	0	0
$958$	595.187	0.621281
$959$	− 1792.29i	− 1.86892i
$960$	0	0
$961$	110.495	0.114979
$962$	806.783i	0.838652i
$963$	0	0
$964$	856.911	0.888912
$965$	− 259.229i	− 0.268631i
$966$	0	0
$967$	324.748	0.335830	0.167915	−	0.985801i	$-0.446297\pi$
$967$	324.748	0.335830	0.167915	+	0.985801i	$0.446297\pi$
$968$	158.635i	0.163879i
$969$	0	0
$970$	−2770.12	−2.85580
$971$	− 1203.37i	− 1.23931i	−0.784873	−	0.619657i	$-0.787272\pi$
$971$	− 1203.37i	− 1.23931i	0.784873	−	0.619657i	$-0.212728\pi$
$972$	0	0
$973$	290.331	0.298387
$974$	− 1164.49i	− 1.19558i
$975$	0	0
$976$	−653.802	−0.669879
$977$	189.328i	0.193785i	0.995295	+	0.0968927i	$0.0308904\pi$
$977$	189.328i	0.193785i	−0.995295	+	0.0968927i	$0.969110\pi$
$978$	0	0
$979$	−229.529	−0.234453
$980$	− 306.895i	− 0.313158i
$981$	0	0
$982$	1753.99	1.78614
$983$	1264.70i	1.28657i	0.765627	+	0.643285i	$0.222429\pi$
$983$	1264.70i	1.28657i	−0.765627	+	0.643285i	$0.777571\pi$
$984$	0	0
$985$	1316.86	1.33691
$986$	− 599.144i	− 0.607651i
$987$	0	0
$988$	504.017	0.510139
$989$	2207.38i	2.23193i
$990$	0	0
$991$	−1394.54	−1.40721	−0.703604	−	0.710593i	$-0.748427\pi$
$991$	−1394.54	−1.40721	−0.703604	+	0.710593i	$0.748427\pi$
$992$	− 1273.32i	− 1.28358i
$993$	0	0
$994$	−880.740	−0.886056
$995$	− 1097.26i	− 1.10278i
$996$	0	0
$997$	1008.71	1.01175	0.505875	−	0.862607i	$-0.331170\pi$
$997$	1008.71	1.01175	0.505875	+	0.862607i	$0.331170\pi$
$998$	− 151.062i	− 0.151365i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.3.b.a.728.26		30
3.2	odd	2	inner	729.3.b.a.728.5		30
27.2	odd	18		81.3.f.a.17.1		30
27.4	even	9		243.3.f.b.107.1		30
27.5	odd	18		243.3.f.d.26.1		30
27.7	even	9		243.3.f.c.134.5		30
27.11	odd	18		243.3.f.a.215.5		30
27.13	even	9		81.3.f.a.62.1		30
27.14	odd	18		27.3.f.a.20.5	✓	30
27.16	even	9		243.3.f.d.215.1		30
27.20	odd	18		243.3.f.b.134.1		30
27.22	even	9		243.3.f.a.26.5		30
27.23	odd	18		243.3.f.c.107.5		30
27.25	even	9		27.3.f.a.23.5	yes	30
108.79	odd	18		432.3.bc.a.401.5		30
108.95	even	18		432.3.bc.a.209.5		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.20.5	✓	30	27.14	odd	18
27.3.f.a.23.5	yes	30	27.25	even	9
81.3.f.a.17.1		30	27.2	odd	18
81.3.f.a.62.1		30	27.13	even	9
243.3.f.a.26.5		30	27.22	even	9
243.3.f.a.215.5		30	27.11	odd	18
243.3.f.b.107.1		30	27.4	even	9
243.3.f.b.134.1		30	27.20	odd	18
243.3.f.c.107.5		30	27.23	odd	18
243.3.f.c.134.5		30	27.7	even	9
243.3.f.d.26.1		30	27.5	odd	18
243.3.f.d.215.1		30	27.16	even	9
432.3.bc.a.209.5		30	108.95	even	18
432.3.bc.a.401.5		30	108.79	odd	18
729.3.b.a.728.5		30	3.2	odd	2	inner
729.3.b.a.728.26		30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.62787i q^{2} -2.90572 q^{4} +6.25111i q^{5} -8.11762 q^{7} +2.87564i q^{8} -16.4271 q^{10} +8.11386i q^{11} +9.70958 q^{13} -21.3321i q^{14} -19.1797 q^{16} +15.0276i q^{17} -17.8645 q^{19} -18.1640i q^{20} -21.3222 q^{22} -26.3335i q^{23} -14.0764 q^{25} +25.5156i q^{26} +23.5875 q^{28} +15.1718i q^{29} +32.7337 q^{31} -38.8992i q^{32} -39.4906 q^{34} -50.7442i q^{35} +31.6193 q^{37} -46.9457i q^{38} -17.9759 q^{40} -16.5200i q^{41} -83.8240 q^{43} -23.5766i q^{44} +69.2010 q^{46} +13.1702i q^{47} +16.8958 q^{49} -36.9909i q^{50} -28.2133 q^{52} -25.7140i q^{53} -50.7206 q^{55} -23.3433i q^{56} -39.8696 q^{58} -22.7110i q^{59} +34.0883 q^{61} +86.0200i q^{62} +25.5035 q^{64} +60.6957i q^{65} +8.10443 q^{67} -43.6660i q^{68} +133.349 q^{70} -41.2871i q^{71} -80.8091 q^{73} +83.0914i q^{74} +51.9093 q^{76} -65.8652i q^{77} -27.6316 q^{79} -119.894i q^{80} +43.4126 q^{82} -17.5898i q^{83} -93.9392 q^{85} -220.279i q^{86} -23.3325 q^{88} +28.2885i q^{89} -78.8187 q^{91} +76.5176i q^{92} -34.6095 q^{94} -111.673i q^{95} +168.631 q^{97} +44.4001i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67}+ \cdots - 24 q^{94}+O(q^{100}) \) 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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