LMFDB - Embedded newform 1728.3.q.e.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(449,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("1728.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.q (of order $6$, degree $2$, not 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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.2
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1728.449
Dual form			1728.3.q.e.1601.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.39898 + 3.69445i) q^{5} +(3.39898 + 5.88721i) q^{7} +O(q^{10})$	$q+(6.39898 + 3.69445i) q^{5} +(3.39898 + 5.88721i) q^{7} +(5.29796 - 3.05878i) q^{11} +(8.39898 - 14.5475i) q^{13} +25.1701i q^{17} +17.5959 q^{19} +(-12.3990 - 7.15855i) q^{23} +(14.7980 + 25.6308i) q^{25} +(16.1969 - 9.35131i) q^{29} +(23.3990 - 40.5282i) q^{31} +50.2295i q^{35} +49.5959 q^{37} +(34.5000 + 19.9186i) q^{41} +(-22.0959 - 38.2713i) q^{43} +(-28.8031 + 16.6295i) q^{47} +(1.39388 - 2.41427i) q^{49} -10.1708i q^{53} +45.2020 q^{55} +(-14.2980 - 8.25493i) q^{59} +(10.6010 + 18.3615i) q^{61} +(107.490 - 62.0593i) q^{65} +(-43.4898 + 75.3265i) q^{67} -30.2555i q^{71} -48.7878 q^{73} +(36.0153 + 20.7934i) q^{77} +(-55.7929 - 96.6361i) q^{79} +(-85.0857 + 49.1243i) q^{83} +(-92.9898 + 161.063i) q^{85} +75.5103i q^{89} +114.192 q^{91} +(112.596 + 65.0073i) q^{95} +(70.2980 + 121.760i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{5} - 6 q^{7}+O(q^{10}) $ 4 * q + 6 * q^5 - 6 * q^7	$ 4 q + 6 q^{5} - 6 q^{7} - 18 q^{11} + 14 q^{13} - 8 q^{19} - 30 q^{23} + 20 q^{25} + 6 q^{29} + 74 q^{31} + 120 q^{37} + 138 q^{41} - 10 q^{43} - 174 q^{47} - 112 q^{49} + 220 q^{55} - 18 q^{59} + 62 q^{61} + 234 q^{65} + 22 q^{67} + 40 q^{73} + 438 q^{77} - 86 q^{79} - 66 q^{83} - 176 q^{85} + 300 q^{91} + 372 q^{95} + 242 q^{97}+O(q^{100}) $ 4 * q + 6 * q^5 - 6 * q^7 - 18 * q^11 + 14 * q^13 - 8 * q^19 - 30 * q^23 + 20 * q^25 + 6 * q^29 + 74 * q^31 + 120 * q^37 + 138 * q^41 - 10 * q^43 - 174 * q^47 - 112 * q^49 + 220 * q^55 - 18 * q^59 + 62 * q^61 + 234 * q^65 + 22 * q^67 + 40 * q^73 + 438 * q^77 - 86 * q^79 - 66 * q^83 - 176 * q^85 + 300 * q^91 + 372 * q^95 + 242 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	6.39898	+	3.69445i	1.27980	+	0.738891i	0.976811	−	0.214105i	$-0.0686834\pi$
$5$	6.39898	+	3.69445i	1.27980	+	0.738891i	0.302985	+	0.952995i	$0.402017\pi$
$6$		0			0
$7$	3.39898	+	5.88721i	0.485568	+	0.841029i	0.999862	−	0.0165847i	$-0.00527930\pi$
$7$	3.39898	+	5.88721i	0.485568	+	0.841029i	−0.514294	+	0.857614i	$0.671946\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.29796	−	3.05878i	0.481633	−	0.278071i	−0.239464	−	0.970905i	$-0.576972\pi$
$11$	5.29796	−	3.05878i	0.481633	−	0.278071i	0.721097	+	0.692835i	$0.243638\pi$
$12$		0			0
$13$	8.39898	−	14.5475i	0.646075	−	1.11904i	−0.337977	−	0.941154i	$-0.609743\pi$
$13$	8.39898	−	14.5475i	0.646075	−	1.11904i	0.984052	−	0.177881i	$-0.0569242\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			25.1701i			1.48059i	0.672279	+	0.740297i	$0.265315\pi$
$17$			25.1701i			1.48059i	−0.672279	+	0.740297i	$0.734685\pi$
$18$		0			0
$19$	17.5959			0.926101			0.463050	−	0.886332i	$-0.346755\pi$
$19$	17.5959			0.926101			0.463050	+	0.886332i	$0.346755\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12.3990	−	7.15855i	−0.539086	−	0.311241i	0.205622	−	0.978631i	$-0.434078\pi$
$23$	−12.3990	−	7.15855i	−0.539086	−	0.311241i	−0.744708	+	0.667390i	$0.767411\pi$
$24$		0			0
$25$	14.7980	+	25.6308i	0.591918	+	1.02523i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	16.1969	−	9.35131i	0.558515	−	0.322459i	−0.194034	−	0.980995i	$-0.562157\pi$
$29$	16.1969	−	9.35131i	0.558515	−	0.322459i	0.752549	+	0.658536i	$0.228824\pi$
$30$		0			0
$31$	23.3990	−	40.5282i	0.754806	−	1.30736i	−0.190665	−	0.981655i	$-0.561064\pi$
$31$	23.3990	−	40.5282i	0.754806	−	1.30736i	0.945471	−	0.325707i	$-0.105602\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			50.2295i			1.43513i
$36$		0			0
$37$	49.5959			1.34043			0.670215	−	0.742167i	$-0.266202\pi$
$37$	49.5959			1.34043			0.670215	+	0.742167i	$0.266202\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	34.5000	+	19.9186i	0.841463	+	0.485819i	0.857761	−	0.514048i	$-0.171855\pi$
$41$	34.5000	+	19.9186i	0.841463	+	0.485819i	−0.0162980	+	0.999867i	$0.505188\pi$
$42$		0			0
$43$	−22.0959	−	38.2713i	−0.513859	−	0.890029i	−0.999871	−	0.0160771i	$-0.994882\pi$
$43$	−22.0959	−	38.2713i	−0.513859	−	0.890029i	0.486012	−	0.873952i	$-0.338451\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−28.8031	+	16.6295i	−0.612831	+	0.353818i	−0.774073	−	0.633097i	$-0.781784\pi$
$47$	−28.8031	+	16.6295i	−0.612831	+	0.353818i	0.161242	+	0.986915i	$0.448450\pi$
$48$		0			0
$49$	1.39388	−	2.41427i	0.0284465	−	0.0492707i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	10.1708i		−	0.191902i	−0.995386	−	0.0959509i	$-0.969411\pi$
$53$		−	10.1708i		−	0.191902i	0.995386	−	0.0959509i	$-0.0305892\pi$
$54$		0			0
$55$	45.2020			0.821855
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−14.2980	−	8.25493i	−0.242338	−	0.139914i	0.373913	−	0.927464i	$-0.378016\pi$
$59$	−14.2980	−	8.25493i	−0.242338	−	0.139914i	−0.616251	+	0.787550i	$0.711349\pi$
$60$		0			0
$61$	10.6010	+	18.3615i	0.173787	+	0.301008i	0.939741	−	0.341887i	$-0.111066\pi$
$61$	10.6010	+	18.3615i	0.173787	+	0.301008i	−0.765954	+	0.642896i	$0.777733\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	107.490	−	62.0593i	1.65369	−	0.954758i
$66$		0			0
$67$	−43.4898	+	75.3265i	−0.649101	+	1.12428i	0.334236	+	0.942489i	$0.391522\pi$
$67$	−43.4898	+	75.3265i	−0.649101	+	1.12428i	−0.983338	+	0.181787i	$0.941812\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	30.2555i		−	0.426134i	−0.977038	−	0.213067i	$-0.931655\pi$
$71$		−	30.2555i		−	0.426134i	0.977038	−	0.213067i	$-0.0683453\pi$
$72$		0			0
$73$	−48.7878			−0.668325			−0.334163	−	0.942515i	$-0.608454\pi$
$73$	−48.7878			−0.668325			−0.334163	+	0.942515i	$0.608454\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	36.0153	+	20.7934i	0.467731	+	0.270045i
$78$		0			0
$79$	−55.7929	−	96.6361i	−0.706239	−	1.22324i	−0.966243	−	0.257634i	$-0.917057\pi$
$79$	−55.7929	−	96.6361i	−0.706239	−	1.22324i	0.260004	−	0.965608i	$-0.416276\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−85.0857	+	49.1243i	−1.02513	+	0.591859i	−0.915585	−	0.402123i	$-0.868272\pi$
$83$	−85.0857	+	49.1243i	−1.02513	+	0.591859i	−0.109544	+	0.993982i	$0.534939\pi$
$84$		0			0
$85$	−92.9898	+	161.063i	−1.09400	+	1.89486i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			75.5103i			0.848431i	0.905561	+	0.424215i	$0.139450\pi$
$89$			75.5103i			0.848431i	−0.905561	+	0.424215i	$0.860550\pi$
$90$		0			0
$91$	114.192			1.25486
$92$		0			0
$93$		0			0
$94$		0			0
$95$	112.596	+	65.0073i	1.18522	+	0.684287i
$96$		0			0
$97$	70.2980	+	121.760i	0.724721	+	1.25525i	0.959089	+	0.283106i	$0.0913648\pi$
$97$	70.2980	+	121.760i	0.724721	+	1.25525i	−0.234367	+	0.972148i	$0.575302\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	28.1969	−	16.2795i	0.279178	−	0.161183i	−0.353873	−	0.935293i	$-0.615136\pi$
$101$	28.1969	−	16.2795i	0.279178	−	0.161183i	0.633051	+	0.774110i	$0.281802\pi$
$102$		0			0
$103$	−67.7929	+	117.421i	−0.658183	+	1.14001i	0.322903	+	0.946432i	$0.395341\pi$
$103$	−67.7929	+	117.421i	−0.658183	+	1.14001i	−0.981086	+	0.193574i	$0.937992\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	35.3409i		−	0.330289i	−0.986269	−	0.165144i	$-0.947191\pi$
$107$		−	35.3409i		−	0.330289i	0.986269	−	0.165144i	$-0.0528090\pi$
$108$		0			0
$109$	−53.5959			−0.491706			−0.245853	−	0.969307i	$-0.579068\pi$
$109$	−53.5959			−0.491706			−0.245853	+	0.969307i	$0.579068\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−143.076	−	82.6047i	−1.26615	−	0.731015i	−0.291897	−	0.956450i	$-0.594286\pi$
$113$	−143.076	−	82.6047i	−1.26615	−	0.731015i	−0.974258	+	0.225435i	$0.927620\pi$
$114$		0			0
$115$	−52.8939	−	91.6149i	−0.459947	−	0.796651i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−148.182	+	85.5527i	−1.24522	+	0.718930i
$120$		0			0
$121$	−41.7878	+	72.3785i	−0.345353	+	0.598170i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			33.9588i			0.271670i
$126$		0			0
$127$	11.9796			0.0943275			0.0471637	−	0.998887i	$-0.484982\pi$
$127$	11.9796			0.0943275			0.0471637	+	0.998887i	$0.484982\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−13.3082	−	7.68347i	−0.101589	−	0.0586525i	0.448345	−	0.893861i	$-0.352014\pi$
$131$	−13.3082	−	7.68347i	−0.101589	−	0.0586525i	−0.549934	+	0.835208i	$0.685347\pi$
$132$		0			0
$133$	59.8082	+	103.591i	0.449685	+	0.778878i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	47.7122	−	27.5467i	0.348265	−	0.201071i	−0.315656	−	0.948874i	$-0.602225\pi$
$137$	47.7122	−	27.5467i	0.348265	−	0.201071i	0.663921	+	0.747803i	$0.268891\pi$
$138$		0			0
$139$	50.4898	−	87.4509i	0.363236	−	0.629143i	−0.625255	−	0.780420i	$-0.715005\pi$
$139$	50.4898	−	87.4509i	0.363236	−	0.629143i	0.988491	+	0.151277i	$0.0483386\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	102.762i		−	0.718619i
$144$		0			0
$145$	138.192			0.953047
$146$		0			0
$147$		0			0
$148$		0			0
$149$	187.389	+	108.189i	1.25764	+	0.726100i	0.972616	−	0.232419i	$-0.0746641\pi$
$149$	187.389	+	108.189i	1.25764	+	0.726100i	0.285027	+	0.958519i	$0.407997\pi$
$150$		0			0
$151$	76.7929	+	133.009i	0.508562	+	0.880855i	0.999951	+	0.00991488i	$0.00315606\pi$
$151$	76.7929	+	133.009i	0.508562	+	0.880855i	−0.491389	+	0.870940i	$0.663511\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	299.459	−	172.893i	1.93199	−	1.11544i
$156$		0			0
$157$	40.9847	−	70.9876i	0.261049	−	0.452150i	−0.705472	−	0.708738i	$-0.749265\pi$
$157$	40.9847	−	70.9876i	0.261049	−	0.452150i	0.966521	+	0.256588i	$0.0825983\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	97.3271i		−	0.604516i
$162$		0			0
$163$	−55.2122			−0.338725			−0.169363	−	0.985554i	$-0.554171\pi$
$163$	−55.2122			−0.338725			−0.169363	+	0.985554i	$0.554171\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	133.803	+	77.2512i	0.801216	+	0.462582i	0.843896	−	0.536507i	$-0.180256\pi$
$167$	133.803	+	77.2512i	0.801216	+	0.462582i	−0.0426802	+	0.999089i	$0.513590\pi$
$168$		0			0
$169$	−56.5857	−	98.0093i	−0.334827	−	0.579937i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	22.8031	−	13.1654i	0.131810	−	0.0761003i	−0.432645	−	0.901564i	$-0.642420\pi$
$173$	22.8031	−	13.1654i	0.131810	−	0.0761003i	0.564455	+	0.825464i	$0.309086\pi$
$174$		0			0
$175$	−100.596	+	174.237i	−0.574834	+	0.995641i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	266.700i		−	1.48995i	−0.667094	−	0.744973i	$-0.732462\pi$
$179$		−	266.700i		−	1.48995i	0.667094	−	0.744973i	$-0.267538\pi$
$180$		0			0
$181$	58.4041			0.322674			0.161337	−	0.986899i	$-0.448419\pi$
$181$	58.4041			0.322674			0.161337	+	0.986899i	$0.448419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	317.363	+	183.230i	1.71548	+	0.990431i
$186$		0			0
$187$	76.9898	+	133.350i	0.411710	+	0.713103i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	99.5602	−	57.4811i	0.521258	−	0.300948i	−0.216191	−	0.976351i	$-0.569364\pi$
$191$	99.5602	−	57.4811i	0.521258	−	0.300948i	0.737449	+	0.675403i	$0.236030\pi$
$192$		0			0
$193$	−108.490	+	187.910i	−0.562123	+	0.973626i	0.435188	+	0.900340i	$0.356682\pi$
$193$	−108.490	+	187.910i	−0.562123	+	0.973626i	−0.997311	+	0.0732863i	$0.976651\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	171.105i		−	0.868555i	−0.900779	−	0.434278i	$-0.857004\pi$
$197$		−	171.105i		−	0.868555i	0.900779	−	0.434278i	$-0.142996\pi$
$198$		0			0
$199$	−62.0000			−0.311558			−0.155779	−	0.987792i	$-0.549789\pi$
$199$	−62.0000			−0.311558			−0.155779	+	0.987792i	$0.549789\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	110.106	+	63.5698i	0.542395	+	0.313152i
$204$		0			0
$205$	147.177	+	254.917i	0.717934	+	1.24350i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	93.2225	−	53.8220i	0.446040	−	0.257522i
$210$		0			0
$211$	−64.7020	+	112.067i	−0.306645	+	0.531124i	−0.977626	−	0.210350i	$-0.932540\pi$
$211$	−64.7020	+	112.067i	−0.306645	+	0.531124i	0.670981	+	0.741474i	$0.265873\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	326.529i		−	1.51874i
$216$		0			0
$217$	318.131			1.46604
$218$		0			0
$219$		0			0
$220$		0			0
$221$	366.161	+	211.403i	1.65684	+	0.956576i
$222$		0			0
$223$	−49.1867	−	85.1939i	−0.220568	−	0.382036i	0.734412	−	0.678704i	$-0.237458\pi$
$223$	−49.1867	−	85.1939i	−0.220568	−	0.382036i	−0.954981	+	0.296668i	$0.904125\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−383.651	+	221.501i	−1.69009	+	0.975775i	−0.735659	+	0.677352i	$0.763127\pi$
$227$	−383.651	+	221.501i	−1.69009	+	0.975775i	−0.954434	+	0.298423i	$0.903539\pi$
$228$		0			0
$229$	−56.0051	+	97.0037i	−0.244564	+	0.423597i	−0.962009	−	0.273018i	$-0.911978\pi$
$229$	−56.0051	+	97.0037i	−0.244564	+	0.423597i	0.717445	+	0.696615i	$0.245311\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			80.8526i			0.347007i	0.984833	+	0.173503i	$0.0555088\pi$
$233$			80.8526i			0.347007i	−0.984833	+	0.173503i	$0.944491\pi$
$234$		0			0
$235$	−245.747			−1.04573
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−37.3888	−	21.5864i	−0.156438	−	0.0903197i	0.419737	−	0.907646i	$-0.362122\pi$
$239$	−37.3888	−	21.5864i	−0.156438	−	0.0903197i	−0.576176	+	0.817326i	$0.695456\pi$
$240$		0			0
$241$	140.904	+	244.053i	0.584664	+	1.01267i	0.994917	+	0.100696i	$0.0321071\pi$
$241$	140.904	+	244.053i	0.584664	+	1.01267i	−0.410253	+	0.911972i	$0.634560\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	17.8388	−	10.2992i	0.0728113	−	0.0420376i
$246$		0			0
$247$	147.788	−	255.976i	0.598331	−	1.03634i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	15.5131i		−	0.0618050i	−0.999522	−	0.0309025i	$-0.990162\pi$
$251$		−	15.5131i		−	0.0618050i	0.999522	−	0.0309025i	$-0.00983814\pi$
$252$		0			0
$253$	−87.5857			−0.346189
$254$		0			0
$255$		0			0
$256$		0			0
$257$	312.035	+	180.153i	1.21414	+	0.700986i	0.963659	−	0.267135i	$-0.0860770\pi$
$257$	312.035	+	180.153i	1.21414	+	0.700986i	0.250484	+	0.968121i	$0.419410\pi$
$258$		0			0
$259$	168.576	+	291.981i	0.650871	+	1.12734i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	42.4296	−	24.4967i	0.161329	−	0.0931435i	−0.417162	−	0.908832i	$-0.636975\pi$
$263$	42.4296	−	24.4967i	0.161329	−	0.0931435i	0.578491	+	0.815689i	$0.303642\pi$
$264$		0			0
$265$	37.5755	−	65.0827i	0.141794	−	0.245595i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			281.700i			1.04721i	0.851961	+	0.523606i	$0.175413\pi$
$269$			281.700i			1.04721i	−0.851961	+	0.523606i	$0.824587\pi$
$270$		0			0
$271$	−89.5959			−0.330612			−0.165306	−	0.986242i	$-0.552861\pi$
$271$	−89.5959			−0.330612			−0.165306	+	0.986242i	$0.552861\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	156.798	+	90.5273i	0.570174	+	0.329190i
$276$		0			0
$277$	42.1969	+	73.0872i	0.152336	+	0.263853i	0.932086	−	0.362238i	$-0.117987\pi$
$277$	42.1969	+	73.0872i	0.152336	+	0.263853i	−0.779750	+	0.626091i	$0.784654\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	23.2673	−	13.4334i	0.0828019	−	0.0478057i	−0.458027	−	0.888938i	$-0.651444\pi$
$281$	23.2673	−	13.4334i	0.0828019	−	0.0478057i	0.540829	+	0.841132i	$0.318110\pi$
$282$		0			0
$283$	90.4898	−	156.733i	0.319752	−	0.553827i	−0.660684	−	0.750664i	$-0.729734\pi$
$283$	90.4898	−	156.733i	0.319752	−	0.553827i	0.980436	+	0.196837i	$0.0630671\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			270.811i			0.943594i
$288$		0			0
$289$	−344.535			−1.19216
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−407.985	−	235.550i	−1.39244	−	0.803925i	−0.398854	−	0.917014i	$-0.630592\pi$
$293$	−407.985	−	235.550i	−1.39244	−	0.803925i	−0.993585	+	0.113089i	$0.963925\pi$
$294$		0			0
$295$	−60.9949	−	105.646i	−0.206762	−	0.358123i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−208.278	+	120.249i	−0.696580	+	0.402171i
$300$		0			0
$301$	150.207	−	260.166i	0.499027	−	0.864340i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			156.660i			0.513639i
$306$		0			0
$307$	−464.747			−1.51383			−0.756917	−	0.653511i	$-0.773295\pi$
$307$	−464.747			−1.51383			−0.756917	+	0.653511i	$0.773295\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	218.348	+	126.063i	0.702083	+	0.405348i	0.808123	−	0.589014i	$-0.200484\pi$
$311$	218.348	+	126.063i	0.702083	+	0.405348i	−0.106039	+	0.994362i	$0.533817\pi$
$312$		0			0
$313$	−98.1061	−	169.925i	−0.313438	−	0.542891i	0.665666	−	0.746250i	$-0.268147\pi$
$313$	−98.1061	−	169.925i	−0.313438	−	0.542891i	−0.979104	+	0.203359i	$0.934814\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	98.9847	−	57.1488i	0.312255	−	0.180280i	−0.335680	−	0.941976i	$-0.608966\pi$
$317$	98.9847	−	57.1488i	0.312255	−	0.180280i	0.647935	+	0.761696i	$0.275633\pi$
$318$		0			0
$319$	57.2071	−	99.0857i	0.179333	−	0.310613i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			442.891i			1.37118i
$324$		0			0
$325$	497.151			1.52970
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−195.802	−	113.046i	−0.595143	−	0.343606i
$330$		0			0
$331$	27.2980	+	47.2815i	0.0824712	+	0.142844i	0.904311	−	0.426875i	$-0.140385\pi$
$331$	27.2980	+	47.2815i	0.0824712	+	0.142844i	−0.821840	+	0.569719i	$0.807052\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−556.581	+	321.342i	−1.66143	+	0.959230i
$336$		0			0
$337$	118.884	−	205.913i	0.352771	−	0.611016i	−0.633963	−	0.773363i	$-0.718573\pi$
$337$	118.884	−	205.913i	0.352771	−	0.611016i	0.986734	+	0.162347i	$0.0519063\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	286.289i		−	0.839558i
$342$		0			0
$343$	352.051			1.02639
$344$		0			0
$345$		0			0
$346$		0			0
$347$	108.349	+	62.5553i	0.312245	+	0.180275i	0.647931	−	0.761699i	$-0.275635\pi$
$347$	108.349	+	62.5553i	0.312245	+	0.180275i	−0.335686	+	0.941974i	$0.608968\pi$
$348$		0			0
$349$	−269.985	−	467.627i	−0.773595	−	1.33991i	−0.935581	−	0.353113i	$-0.885123\pi$
$349$	−269.985	−	467.627i	−0.773595	−	1.33991i	0.161986	−	0.986793i	$-0.448210\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−254.490	+	146.930i	−0.720934	+	0.416232i	−0.815096	−	0.579325i	$-0.803316\pi$
$353$	−254.490	+	146.930i	−0.720934	+	0.416232i	0.0941622	+	0.995557i	$0.469983\pi$
$354$		0			0
$355$	111.778	−	193.604i	0.314866	−	0.545364i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	422.550i		−	1.17702i	−0.808490	−	0.588509i	$-0.799715\pi$
$359$		−	422.550i		−	1.17702i	0.808490	−	0.588509i	$-0.200285\pi$
$360$		0			0
$361$	−51.3837			−0.142337
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−312.192	−	180.244i	−0.855320	−	0.493819i
$366$		0			0
$367$	131.358	+	227.519i	0.357924	+	0.619943i	0.987614	−	0.156904i	$-0.0501513\pi$
$367$	131.358	+	227.519i	0.357924	+	0.619943i	−0.629690	+	0.776847i	$0.716818\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	59.8775	−	34.5703i	0.161395	−	0.0931814i
$372$		0			0
$373$	60.9847	−	105.629i	0.163498	−	0.283187i	−0.772623	−	0.634865i	$-0.781056\pi$
$373$	60.9847	−	105.629i	0.163498	−	0.283187i	0.936121	+	0.351679i	$0.114389\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	314.166i		−	0.833331i
$378$		0			0
$379$	641.151			1.69169			0.845846	−	0.533428i	$-0.179096\pi$
$379$	641.151			1.69169			0.845846	+	0.533428i	$0.179096\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−640.681	−	369.897i	−1.67280	−	0.965789i	−0.966063	−	0.258308i	$-0.916835\pi$
$383$	−640.681	−	369.897i	−1.67280	−	0.965789i	−0.706733	−	0.707481i	$-0.749832\pi$
$384$		0			0
$385$	153.641	+	266.114i	0.399067	+	0.691204i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	75.7520	−	43.7355i	0.194735	−	0.112430i	−0.399462	−	0.916750i	$-0.630803\pi$
$389$	75.7520	−	43.7355i	0.194735	−	0.112430i	0.594197	+	0.804319i	$0.297470\pi$
$390$		0			0
$391$	180.182	−	312.084i	0.460823	−	0.798168i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	824.496i		−	2.08733i
$396$		0			0
$397$	−483.090			−1.21685			−0.608425	−	0.793611i	$-0.708199\pi$
$397$	−483.090			−1.21685			−0.608425	+	0.793611i	$0.708199\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−317.682	−	183.414i	−0.792224	−	0.457390i	0.0485212	−	0.998822i	$-0.484549\pi$
$401$	−317.682	−	183.414i	−0.792224	−	0.457390i	−0.840745	+	0.541432i	$0.817882\pi$
$402$		0			0
$403$	−393.055	−	680.791i	−0.975323	−	1.68931i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	262.757	−	151.703i	0.645595	−	0.372734i
$408$		0			0
$409$	−267.641	+	463.567i	−0.654379	+	1.13342i	0.327671	+	0.944792i	$0.393736\pi$
$409$	−267.641	+	463.567i	−0.654379	+	1.13342i	−0.982049	+	0.188625i	$0.939597\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	112.233i		−	0.271751i
$414$		0			0
$415$	−725.949			−1.74927
$416$		0			0
$417$		0			0
$418$		0			0
$419$	605.620	+	349.655i	1.44539	+	0.834499i	0.998202	−	0.0599386i	$-0.0190905\pi$
$419$	605.620	+	349.655i	1.44539	+	0.834499i	0.447193	+	0.894438i	$0.352424\pi$
$420$		0			0
$421$	−180.772	−	313.107i	−0.429388	−	0.743722i	0.567431	−	0.823421i	$-0.307937\pi$
$421$	−180.772	−	313.107i	−0.429388	−	0.743722i	−0.996819	+	0.0796989i	$0.974604\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−645.131	+	372.466i	−1.51795	+	0.876391i
$426$		0			0
$427$	−72.0653	+	124.821i	−0.168771	+	0.292320i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			463.747i			1.07598i	0.842952	+	0.537989i	$0.180816\pi$
$431$			463.747i			1.07598i	−0.842952	+	0.537989i	$0.819184\pi$
$432$		0			0
$433$	−689.514			−1.59241			−0.796206	−	0.605026i	$-0.793163\pi$
$433$	−689.514			−1.59241			−0.796206	+	0.605026i	$0.793163\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−218.171	−	125.961i	−0.499248	−	0.288241i
$438$		0			0
$439$	310.772	+	538.274i	0.707910	+	1.22614i	0.965631	+	0.259917i	$0.0836951\pi$
$439$	310.772	+	538.274i	0.707910	+	1.22614i	−0.257721	+	0.966219i	$0.582972\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−698.843	+	403.477i	−1.57752	+	0.910784i	−0.582320	+	0.812960i	$0.697855\pi$
$443$	−698.843	+	403.477i	−1.57752	+	0.910784i	−0.995204	+	0.0978236i	$0.968812\pi$
$444$		0			0
$445$	−278.969	+	483.189i	−0.626897	+	1.08582i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	317.554i		−	0.707248i	−0.935388	−	0.353624i	$-0.884949\pi$
$449$		−	317.554i		−	0.707248i	0.935388	−	0.353624i	$-0.115051\pi$
$450$		0			0
$451$	243.706			0.540368
$452$		0			0
$453$		0			0
$454$		0			0
$455$	730.711	+	421.876i	1.60596	+	0.927201i
$456$		0			0
$457$	−285.843	−	495.094i	−0.625477	−	1.08336i	−0.988448	−	0.151557i	$-0.951571\pi$
$457$	−285.843	−	495.094i	−0.625477	−	1.08336i	0.362972	−	0.931800i	$-0.381762\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−478.550	+	276.291i	−1.03807	+	0.599330i	−0.919286	−	0.393591i	$-0.871233\pi$
$461$	−478.550	+	276.291i	−1.03807	+	0.599330i	−0.118784	+	0.992920i	$0.537899\pi$
$462$		0			0
$463$	60.1663	−	104.211i	0.129949	−	0.225078i	−0.793708	−	0.608299i	$-0.791852\pi$
$463$	60.1663	−	104.211i	0.129949	−	0.225078i	0.923657	+	0.383221i	$0.125185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	880.440i		−	1.88531i	−0.333767	−	0.942656i	$-0.608320\pi$
$467$		−	880.440i		−	1.88531i	0.333767	−	0.942656i	$-0.391680\pi$
$468$		0			0
$469$	−591.284			−1.26073
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−234.127	−	135.173i	−0.494982	−	0.285778i
$474$		0			0
$475$	260.384	+	450.998i	0.548176	+	0.949469i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−593.793	+	342.826i	−1.23965	+	0.715713i	−0.969023	−	0.246971i	$-0.920565\pi$
$479$	−593.793	+	342.826i	−1.23965	+	0.715713i	−0.270628	+	0.962684i	$0.587231\pi$
$480$		0			0
$481$	416.555	−	721.495i	0.866019	−	1.49999i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			1038.85i			2.14196i
$486$		0			0
$487$	391.131			0.803143			0.401571	−	0.915828i	$-0.368464\pi$
$487$	391.131			0.803143			0.401571	+	0.915828i	$0.368464\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	166.469	+	96.1111i	0.339042	+	0.195746i	0.659848	−	0.751399i	$-0.270621\pi$
$491$	166.469	+	96.1111i	0.339042	+	0.195746i	−0.320807	+	0.947145i	$0.603954\pi$
$492$		0			0
$493$	235.373	+	407.679i	0.477431	+	0.826935i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	178.120	−	102.838i	0.358391	−	0.206917i
$498$		0			0
$499$	304.692	−	527.742i	0.610605	−	1.05760i	−0.380534	−	0.924767i	$-0.624260\pi$
$499$	304.692	−	527.742i	0.610605	−	1.05760i	0.991139	−	0.132832i	$-0.0424070\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	232.130i		−	0.461491i	−0.973014	−	0.230746i	$-0.925883\pi$
$503$		−	232.130i		−	0.461491i	0.973014	−	0.230746i	$-0.0741165\pi$
$504$		0			0
$505$	240.576			0.476387
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−223.136	−	128.827i	−0.438381	−	0.253099i	0.264530	−	0.964377i	$-0.414783\pi$
$509$	−223.136	−	128.827i	−0.438381	−	0.253099i	−0.702910	+	0.711278i	$0.748117\pi$
$510$		0			0
$511$	−165.829	−	287.224i	−0.324518	−	0.562081i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−867.610	+	500.915i	−1.68468	+	0.972650i
$516$		0			0
$517$	−101.732	+	176.204i	−0.196773	+	0.340821i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	484.088i		−	0.929152i	−0.885533	−	0.464576i	$-0.846207\pi$
$521$		−	484.088i		−	0.929152i	0.885533	−	0.464576i	$-0.153793\pi$
$522$		0			0
$523$	644.384			1.23209			0.616046	−	0.787711i	$-0.288734\pi$
$523$	644.384			1.23209			0.616046	+	0.787711i	$0.288734\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1020.10	+	588.955i	1.93567	+	1.11756i
$528$		0			0
$529$	−162.010	−	280.610i	−0.306257	−	0.530454i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	579.530	−	334.592i	1.08730	−	0.627752i
$534$		0			0
$535$	130.565	−	226.146i	0.244047	−	0.422702i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	17.0542i		−	0.0316405i
$540$		0			0
$541$	332.302			0.614237			0.307118	−	0.951671i	$-0.400635\pi$
$541$	332.302			0.614237			0.307118	+	0.951671i	$0.400635\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−342.959	−	198.008i	−0.629283	−	0.363317i
$546$		0			0
$547$	−157.329	−	272.501i	−0.287621	−	0.498174i	0.685621	−	0.727959i	$-0.259531\pi$
$547$	−157.329	−	272.501i	−0.287621	−	0.498174i	−0.973241	+	0.229785i	$0.926198\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	285.000	−	164.545i	0.517241	−	0.298629i
$552$		0			0
$553$	379.278	−	656.928i	0.685855	−	1.18793i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	664.080i		−	1.19224i	−0.802894	−	0.596122i	$-0.796707\pi$
$557$		−	664.080i		−	1.19224i	0.802894	−	0.596122i	$-0.203293\pi$
$558$		0			0
$559$	−742.333			−1.32797
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−211.024	−	121.835i	−0.374821	−	0.216403i	0.300741	−	0.953706i	$-0.402766\pi$
$563$	−211.024	−	121.835i	−0.374821	−	0.216403i	−0.675563	+	0.737302i	$0.736099\pi$
$564$		0			0
$565$	−610.358	−	1057.17i	−1.08028	−	1.87110i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	502.155	−	289.919i	0.882522	−	0.509524i	0.0110330	−	0.999939i	$-0.496488\pi$
$569$	502.155	−	289.919i	0.882522	−	0.509524i	0.871489	+	0.490415i	$0.163155\pi$
$570$		0			0
$571$	−356.843	+	618.070i	−0.624944	+	1.08243i	0.363608	+	0.931552i	$0.381545\pi$
$571$	−356.843	+	618.070i	−0.624944	+	1.08243i	−0.988552	+	0.150882i	$0.951789\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	423.728i		−	0.736918i
$576$		0			0
$577$	829.433			1.43749			0.718746	−	0.695273i	$-0.244717\pi$
$577$	829.433			1.43749			0.718746	+	0.695273i	$0.244717\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−578.409	−	333.945i	−0.995541	−	0.574776i
$582$		0			0
$583$	−31.1102	−	53.8844i	−0.0533623	−	0.0924261i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	777.480	−	448.878i	1.32450	−	0.764699i	0.340054	−	0.940406i	$-0.389555\pi$
$587$	777.480	−	448.878i	1.32450	−	0.764699i	0.984442	+	0.175707i	$0.0562212\pi$
$588$		0			0
$589$	411.727	−	713.131i	0.699026	−	1.21075i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			378.065i			0.637547i	0.947831	+	0.318774i	$0.103271\pi$
$593$			378.065i			0.637547i	−0.947831	+	0.318774i	$0.896729\pi$
$594$		0			0
$595$	−1264.28			−2.12484
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−822.438	−	474.835i	−1.37302	−	0.792712i	−0.381711	−	0.924282i	$-0.624665\pi$
$599$	−822.438	−	474.835i	−1.37302	−	0.792712i	−0.991307	+	0.131569i	$0.957998\pi$
$600$		0			0
$601$	−252.308	−	437.011i	−0.419814	−	0.727139i	0.576107	−	0.817375i	$-0.304571\pi$
$601$	−252.308	−	437.011i	−0.419814	−	0.727139i	−0.995920	+	0.0902356i	$0.971238\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−534.798	+	308.766i	−0.883964	+	0.510357i
$606$		0			0
$607$	−429.954	+	744.702i	−0.708326	+	1.22686i	0.257151	+	0.966371i	$0.417216\pi$
$607$	−429.954	+	744.702i	−0.708326	+	1.22686i	−0.965478	+	0.260486i	$0.916117\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			558.682i			0.914373i
$612$		0			0
$613$	655.253			1.06893			0.534464	−	0.845191i	$-0.320513\pi$
$613$	655.253			1.06893			0.534464	+	0.845191i	$0.320513\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	147.227	+	85.0013i	0.238617	+	0.137765i	0.614541	−	0.788885i	$-0.289341\pi$
$617$	147.227	+	85.0013i	0.238617	+	0.137765i	−0.375924	+	0.926650i	$0.622675\pi$
$618$		0			0
$619$	270.531	+	468.573i	0.437045	+	0.756983i	0.997460	−	0.0712282i	$-0.0226918\pi$
$619$	270.531	+	468.573i	0.437045	+	0.756983i	−0.560415	+	0.828212i	$0.689359\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−444.545	+	256.658i	−0.713555	+	0.411971i
$624$		0			0
$625$	244.490	−	423.469i	0.391184	−	0.677550i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1248.33i			1.98463i
$630$		0			0
$631$	260.788			0.413293			0.206646	−	0.978416i	$-0.433745\pi$
$631$	260.788			0.413293			0.206646	+	0.978416i	$0.433745\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	76.6571	+	44.2580i	0.120720	+	0.0696977i
$636$		0			0
$637$	−23.4143	−	40.5547i	−0.0367571	−	0.0636652i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−585.418	+	337.991i	−0.913289	+	0.527288i	−0.881488	−	0.472206i	$-0.843458\pi$
$641$	−585.418	+	337.991i	−0.913289	+	0.527288i	−0.0318012	+	0.999494i	$0.510124\pi$
$642$		0			0
$643$	−378.318	+	655.267i	−0.588364	+	1.01908i	0.406082	+	0.913837i	$0.366895\pi$
$643$	−378.318	+	655.267i	−0.588364	+	1.01908i	−0.994447	+	0.105241i	$0.966439\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	554.770i		−	0.857450i	−0.903435	−	0.428725i	$-0.858963\pi$
$647$		−	554.770i		−	0.857450i	0.903435	−	0.428725i	$-0.141037\pi$
$648$		0			0
$649$	−101.000			−0.155624
$650$		0			0
$651$		0			0
$652$		0			0
$653$	174.499	+	100.747i	0.267227	+	0.154283i	0.627627	−	0.778514i	$-0.284026\pi$
$653$	174.499	+	100.747i	0.267227	+	0.154283i	−0.360400	+	0.932798i	$0.617360\pi$
$654$		0			0
$655$	−56.7724	−	98.3328i	−0.0866755	−	0.150126i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−89.7122	+	51.7954i	−0.136134	+	0.0785969i	−0.566520	−	0.824048i	$-0.691711\pi$
$659$	−89.7122	+	51.7954i	−0.136134	+	0.0785969i	0.430386	+	0.902645i	$0.358377\pi$
$660$		0			0
$661$	109.207	−	189.152i	0.165215	−	0.286161i	−0.771517	−	0.636209i	$-0.780502\pi$
$661$	109.207	−	189.152i	0.165215	−	0.286161i	0.936732	+	0.350048i	$0.113835\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			883.834i			1.32907i
$666$		0			0
$667$	−267.767			−0.401450
$668$		0			0
$669$		0			0
$670$		0			0
$671$	112.328	+	64.8523i	0.167403	+	0.0966503i
$672$		0			0
$673$	−394.429	−	683.170i	−0.586075	−	1.01511i	−0.994740	−	0.102428i	$-0.967339\pi$
$673$	−394.429	−	683.170i	−0.586075	−	1.01511i	0.408665	−	0.912684i	$-0.365994\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−634.550	+	366.358i	−0.937297	+	0.541149i	−0.889112	−	0.457690i	$-0.848677\pi$
$677$	−634.550	+	366.358i	−0.937297	+	0.541149i	−0.0481850	+	0.998838i	$0.515344\pi$
$678$		0			0
$679$	−477.883	+	827.717i	−0.703804	+	1.21902i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1353.33i		−	1.98145i	−0.135883	−	0.990725i	$-0.543387\pi$
$683$		−	1353.33i		−	1.98145i	0.135883	−	0.990725i	$-0.456613\pi$
$684$		0			0
$685$	407.080			0.594277
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−147.959	−	85.4243i	−0.214745	−	0.123983i
$690$		0			0
$691$	−368.257	−	637.840i	−0.532934	−	0.923068i	−0.999260	−	0.0384555i	$-0.987756\pi$
$691$	−368.257	−	637.840i	−0.532934	−	0.923068i	0.466327	−	0.884613i	$-0.345577\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	646.166	−	373.064i	0.929736	−	0.536783i
$696$		0			0
$697$	−501.353	+	868.369i	−0.719301	+	1.24587i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1068.34i		−	1.52403i	−0.647561	−	0.762014i	$-0.724211\pi$
$701$		−	1068.34i		−	1.52403i	0.647561	−	0.762014i	$-0.275789\pi$
$702$		0			0
$703$	872.686			1.24137
$704$		0			0
$705$		0			0
$706$		0			0
$707$	191.682	+	110.667i	0.271120	+	0.156531i
$708$		0			0
$709$	136.944	+	237.194i	0.193151	+	0.334547i	0.946293	−	0.323311i	$-0.104796\pi$
$709$	136.944	+	237.194i	0.193151	+	0.334547i	−0.753142	+	0.657858i	$0.771463\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−580.247	+	335.006i	−0.813811	+	0.469854i
$714$		0			0
$715$	379.651	−	657.575i	0.530980	−	0.919685i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			654.423i			0.910185i	0.890444	+	0.455092i	$0.150394\pi$
$719$			654.423i			0.910185i	−0.890444	+	0.455092i	$0.849606\pi$
$720$		0			0
$721$	−921.706			−1.27837
$722$		0			0
$723$		0			0
$724$		0			0
$725$	479.363	+	276.761i	0.661191	+	0.381739i
$726$		0			0
$727$	−583.166	−	1010.07i	−0.802155	−	1.38937i	−0.918195	−	0.396128i	$-0.870354\pi$
$727$	−583.166	−	1010.07i	−0.802155	−	1.38937i	0.116041	−	0.993244i	$-0.462980\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	963.292	−	556.157i	1.31777	−	0.760816i
$732$		0			0
$733$	439.146	−	760.623i	0.599108	−	1.03768i	−0.393845	−	0.919177i	$-0.628855\pi$
$733$	439.146	−	760.623i	0.599108	−	1.03768i	0.992953	−	0.118508i	$-0.0378112\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			532.103i			0.721984i
$738$		0			0
$739$	593.151			0.802640			0.401320	−	0.915938i	$-0.368552\pi$
$739$	593.151			0.802640			0.401320	+	0.915938i	$0.368552\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	47.0561	+	27.1679i	0.0633326	+	0.0365651i	0.531332	−	0.847164i	$-0.321692\pi$
$743$	47.0561	+	27.1679i	0.0633326	+	0.0365651i	−0.467999	+	0.883729i	$0.655025\pi$
$744$		0			0
$745$	799.398	+	1384.60i	1.07302	+	1.85852i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	208.059	−	120.123i	0.277783	−	0.160378i
$750$		0			0
$751$	−455.570	+	789.071i	−0.606618	+	1.05069i	0.385175	+	0.922844i	$0.374141\pi$
$751$	−455.570	+	789.071i	−0.606618	+	1.05069i	−0.991793	+	0.127850i	$0.959192\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1134.83i			1.50309i
$756$		0			0
$757$	−1272.22			−1.68061			−0.840304	−	0.542115i	$-0.817624\pi$
$757$	−1272.22			−1.68061			−0.840304	+	0.542115i	$0.817624\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−399.480	−	230.640i	−0.524940	−	0.303074i	0.214013	−	0.976831i	$-0.431346\pi$
$761$	−399.480	−	230.640i	−0.524940	−	0.303074i	−0.738954	+	0.673756i	$0.764680\pi$
$762$		0			0
$763$	−182.171	−	315.530i	−0.238757	−	0.413539i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−240.177	+	138.666i	−0.313138	+	0.180790i
$768$		0			0
$769$	−269.439	+	466.682i	−0.350376	+	0.606868i	−0.986315	−	0.164871i	$-0.947279\pi$
$769$	−269.439	+	466.682i	−0.350376	+	0.606868i	0.635940	+	0.771739i	$0.280613\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1036.29i		−	1.34061i	−0.742087	−	0.670304i	$-0.766164\pi$
$773$		−	1036.29i		−	1.34061i	0.742087	−	0.670304i	$-0.233836\pi$
$774$		0			0
$775$	1385.03			1.78713
$776$		0			0
$777$		0			0
$778$		0			0
$779$	607.059	+	350.486i	0.779280	+	0.449918i
$780$		0			0
$781$	−92.5449	−	160.292i	−0.118495	−	0.205240i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	524.520	−	302.832i	0.668179	−	0.385773i
$786$		0			0
$787$	−706.096	+	1222.99i	−0.897199	+	1.55399i	−0.0661406	+	0.997810i	$0.521069\pi$
$787$	−706.096	+	1222.99i	−0.897199	+	1.55399i	−0.831059	+	0.556185i	$0.812265\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1123.09i		−	1.41983i
$792$		0			0
$793$	356.151			0.449119
$794$		0			0
$795$		0			0
$796$		0			0

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.39898 + 3.69445i) q^{5} +(3.39898 + 5.88721i) q^{7} +O(q^{10})\)	\(q+(6.39898 + 3.69445i) q^{5} +(3.39898 + 5.88721i) q^{7} +(5.29796 - 3.05878i) q^{11} +(8.39898 - 14.5475i) q^{13} +25.1701i q^{17} +17.5959 q^{19} +(-12.3990 - 7.15855i) q^{23} +(14.7980 + 25.6308i) q^{25} +(16.1969 - 9.35131i) q^{29} +(23.3990 - 40.5282i) q^{31} +50.2295i q^{35} +49.5959 q^{37} +(34.5000 + 19.9186i) q^{41} +(-22.0959 - 38.2713i) q^{43} +(-28.8031 + 16.6295i) q^{47} +(1.39388 - 2.41427i) q^{49} -10.1708i q^{53} +45.2020 q^{55} +(-14.2980 - 8.25493i) q^{59} +(10.6010 + 18.3615i) q^{61} +(107.490 - 62.0593i) q^{65} +(-43.4898 + 75.3265i) q^{67} -30.2555i q^{71} -48.7878 q^{73} +(36.0153 + 20.7934i) q^{77} +(-55.7929 - 96.6361i) q^{79} +(-85.0857 + 49.1243i) q^{83} +(-92.9898 + 161.063i) q^{85} +75.5103i q^{89} +114.192 q^{91} +(112.596 + 65.0073i) q^{95} +(70.2980 + 121.760i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{5} - 6 q^{7}+O(q^{10}) \) 4 * q + 6 * q^5 - 6 * q^7	\( 4 q + 6 q^{5} - 6 q^{7} - 18 q^{11} + 14 q^{13} - 8 q^{19} - 30 q^{23} + 20 q^{25} + 6 q^{29} + 74 q^{31} + 120 q^{37} + 138 q^{41} - 10 q^{43} - 174 q^{47} - 112 q^{49} + 220 q^{55} - 18 q^{59} + 62 q^{61} + 234 q^{65} + 22 q^{67} + 40 q^{73} + 438 q^{77} - 86 q^{79} - 66 q^{83} - 176 q^{85} + 300 q^{91} + 372 q^{95} + 242 q^{97}+O(q^{100}) \) 4 * q + 6 * q^5 - 6 * q^7 - 18 * q^11 + 14 * q^13 - 8 * q^19 - 30 * q^23 + 20 * q^25 + 6 * q^29 + 74 * q^31 + 120 * q^37 + 138 * q^41 - 10 * q^43 - 174 * q^47 - 112 * q^49 + 220 * q^55 - 18 * q^59 + 62 * q^61 + 234 * q^65 + 22 * q^67 + 40 * q^73 + 438 * q^77 - 86 * q^79 - 66 * q^83 - 176 * q^85 + 300 * q^91 + 372 * q^95 + 242 * q^97

Introduction

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