LMFDB - Embedded newform 1728.3.q.e.1601.2

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			1601.2
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1728.1601
Dual form			1728.3.q.e.449.2

$q$-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 $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

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

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	0.302985	−	0.952995i	$-0.402017\pi$
$5$	6.39898	−	3.69445i	1.27980	−	0.738891i	0.976811	+	0.214105i	$0.0686834\pi$
$6$		0			0
$7$	3.39898	−	5.88721i	0.485568	−	0.841029i	−0.514294	−	0.857614i	$-0.671946\pi$
$7$	3.39898	−	5.88721i	0.485568	−	0.841029i	0.999862	+	0.0165847i	$0.00527930\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.29796	+	3.05878i	0.481633	+	0.278071i	0.721097	−	0.692835i	$-0.243638\pi$
$11$	5.29796	+	3.05878i	0.481633	+	0.278071i	−0.239464	+	0.970905i	$0.576972\pi$
$12$		0			0
$13$	8.39898	+	14.5475i	0.646075	+	1.11904i	0.984052	+	0.177881i	$0.0569242\pi$
$13$	8.39898	+	14.5475i	0.646075	+	1.11904i	−0.337977	+	0.941154i	$0.609743\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	25.1701i		−	1.48059i	−0.672279	−	0.740297i	$-0.734685\pi$
$17$		−	25.1701i		−	1.48059i	0.672279	−	0.740297i	$-0.265315\pi$
$18$		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	−0.744708	−	0.667390i	$-0.767411\pi$
$23$	−12.3990	+	7.15855i	−0.539086	+	0.311241i	0.205622	+	0.978631i	$0.434078\pi$
$24$		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	0.752549	−	0.658536i	$-0.228824\pi$
$29$	16.1969	+	9.35131i	0.558515	+	0.322459i	−0.194034	+	0.980995i	$0.562157\pi$
$30$		0			0
$31$	23.3990	+	40.5282i	0.754806	+	1.30736i	0.945471	+	0.325707i	$0.105602\pi$
$31$	23.3990	+	40.5282i	0.754806	+	1.30736i	−0.190665	+	0.981655i	$0.561064\pi$
$32$		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	−0.0162980	−	0.999867i	$-0.505188\pi$
$41$	34.5000	−	19.9186i	0.841463	−	0.485819i	0.857761	+	0.514048i	$0.171855\pi$
$42$		0			0
$43$	−22.0959	+	38.2713i	−0.513859	+	0.890029i	0.486012	+	0.873952i	$0.338451\pi$
$43$	−22.0959	+	38.2713i	−0.513859	+	0.890029i	−0.999871	+	0.0160771i	$0.994882\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−28.8031	−	16.6295i	−0.612831	−	0.353818i	0.161242	−	0.986915i	$-0.448450\pi$
$47$	−28.8031	−	16.6295i	−0.612831	−	0.353818i	−0.774073	+	0.633097i	$0.781784\pi$
$48$		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	$0.0305892\pi$
$53$			10.1708i			0.191902i	−0.995386	+	0.0959509i	$0.969411\pi$
$54$		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	−0.616251	−	0.787550i	$-0.711349\pi$
$59$	−14.2980	+	8.25493i	−0.242338	+	0.139914i	0.373913	+	0.927464i	$0.378016\pi$
$60$		0			0
$61$	10.6010	−	18.3615i	0.173787	−	0.301008i	−0.765954	−	0.642896i	$-0.777733\pi$
$61$	10.6010	−	18.3615i	0.173787	−	0.301008i	0.939741	+	0.341887i	$0.111066\pi$
$62$		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	−0.983338	−	0.181787i	$-0.941812\pi$
$67$	−43.4898	−	75.3265i	−0.649101	−	1.12428i	0.334236	−	0.942489i	$-0.391522\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			30.2555i			0.426134i	0.977038	+	0.213067i	$0.0683453\pi$
$71$			30.2555i			0.426134i	−0.977038	+	0.213067i	$0.931655\pi$
$72$		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	0.260004	+	0.965608i	$0.416276\pi$
$79$	−55.7929	+	96.6361i	−0.706239	+	1.22324i	−0.966243	+	0.257634i	$0.917057\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−85.0857	−	49.1243i	−1.02513	−	0.591859i	−0.109544	−	0.993982i	$-0.534939\pi$
$83$	−85.0857	−	49.1243i	−1.02513	−	0.591859i	−0.915585	+	0.402123i	$0.868272\pi$
$84$		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	$-0.860550\pi$
$89$		−	75.5103i		−	0.848431i	0.905561	−	0.424215i	$-0.139450\pi$
$90$		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	−0.234367	−	0.972148i	$-0.575302\pi$
$97$	70.2980	−	121.760i	0.724721	−	1.25525i	0.959089	−	0.283106i	$-0.0913648\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	28.1969	+	16.2795i	0.279178	+	0.161183i	0.633051	−	0.774110i	$-0.281802\pi$
$101$	28.1969	+	16.2795i	0.279178	+	0.161183i	−0.353873	+	0.935293i	$0.615136\pi$
$102$		0			0
$103$	−67.7929	−	117.421i	−0.658183	−	1.14001i	−0.981086	−	0.193574i	$-0.937992\pi$
$103$	−67.7929	−	117.421i	−0.658183	−	1.14001i	0.322903	−	0.946432i	$-0.395341\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			35.3409i			0.330289i	0.986269	+	0.165144i	$0.0528090\pi$
$107$			35.3409i			0.330289i	−0.986269	+	0.165144i	$0.947191\pi$
$108$		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	−0.974258	−	0.225435i	$-0.927620\pi$
$113$	−143.076	+	82.6047i	−1.26615	+	0.731015i	−0.291897	+	0.956450i	$0.594286\pi$
$114$		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	−0.549934	−	0.835208i	$-0.685347\pi$
$131$	−13.3082	+	7.68347i	−0.101589	+	0.0586525i	0.448345	+	0.893861i	$0.352014\pi$
$132$		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	0.663921	−	0.747803i	$-0.268891\pi$
$137$	47.7122	+	27.5467i	0.348265	+	0.201071i	−0.315656	+	0.948874i	$0.602225\pi$
$138$		0			0
$139$	50.4898	+	87.4509i	0.363236	+	0.629143i	0.988491	−	0.151277i	$-0.0483386\pi$
$139$	50.4898	+	87.4509i	0.363236	+	0.629143i	−0.625255	+	0.780420i	$0.715005\pi$
$140$		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	0.285027	−	0.958519i	$-0.407997\pi$
$149$	187.389	−	108.189i	1.25764	−	0.726100i	0.972616	+	0.232419i	$0.0746641\pi$
$150$		0			0
$151$	76.7929	−	133.009i	0.508562	−	0.880855i	−0.491389	−	0.870940i	$-0.663511\pi$
$151$	76.7929	−	133.009i	0.508562	−	0.880855i	0.999951	−	0.00991488i	$-0.00315606\pi$
$152$		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	0.966521	−	0.256588i	$-0.0825983\pi$
$157$	40.9847	+	70.9876i	0.261049	+	0.452150i	−0.705472	+	0.708738i	$0.749265\pi$
$158$		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	−0.0426802	−	0.999089i	$-0.513590\pi$
$167$	133.803	−	77.2512i	0.801216	−	0.462582i	0.843896	+	0.536507i	$0.180256\pi$
$168$		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	0.564455	−	0.825464i	$-0.309086\pi$
$173$	22.8031	+	13.1654i	0.131810	+	0.0761003i	−0.432645	+	0.901564i	$0.642420\pi$
$174$		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	$0.267538\pi$
$179$			266.700i			1.48995i	−0.667094	+	0.744973i	$0.732462\pi$
$180$		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	0.737449	−	0.675403i	$-0.236030\pi$
$191$	99.5602	+	57.4811i	0.521258	+	0.300948i	−0.216191	+	0.976351i	$0.569364\pi$
$192$		0			0
$193$	−108.490	−	187.910i	−0.562123	−	0.973626i	−0.997311	−	0.0732863i	$-0.976651\pi$
$193$	−108.490	−	187.910i	−0.562123	−	0.973626i	0.435188	−	0.900340i	$-0.356682\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			171.105i			0.868555i	0.900779	+	0.434278i	$0.142996\pi$
$197$			171.105i			0.868555i	−0.900779	+	0.434278i	$0.857004\pi$
$198$		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	0.670981	−	0.741474i	$-0.265873\pi$
$211$	−64.7020	−	112.067i	−0.306645	−	0.531124i	−0.977626	+	0.210350i	$0.932540\pi$
$212$		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	−0.954981	−	0.296668i	$-0.904125\pi$
$223$	−49.1867	+	85.1939i	−0.220568	+	0.382036i	0.734412	+	0.678704i	$0.237458\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−383.651	−	221.501i	−1.69009	−	0.975775i	−0.954434	−	0.298423i	$-0.903539\pi$
$227$	−383.651	−	221.501i	−1.69009	−	0.975775i	−0.735659	−	0.677352i	$-0.763127\pi$
$228$		0			0
$229$	−56.0051	−	97.0037i	−0.244564	−	0.423597i	0.717445	−	0.696615i	$-0.245311\pi$
$229$	−56.0051	−	97.0037i	−0.244564	−	0.423597i	−0.962009	+	0.273018i	$0.911978\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	80.8526i		−	0.347007i	−0.984833	−	0.173503i	$-0.944491\pi$
$233$		−	80.8526i		−	0.347007i	0.984833	−	0.173503i	$-0.0555088\pi$
$234$		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	−0.576176	−	0.817326i	$-0.695456\pi$
$239$	−37.3888	+	21.5864i	−0.156438	+	0.0903197i	0.419737	+	0.907646i	$0.362122\pi$
$240$		0			0
$241$	140.904	−	244.053i	0.584664	−	1.01267i	−0.410253	−	0.911972i	$-0.634560\pi$
$241$	140.904	−	244.053i	0.584664	−	1.01267i	0.994917	−	0.100696i	$-0.0321071\pi$
$242$		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	$0.00983814\pi$
$251$			15.5131i			0.0618050i	−0.999522	+	0.0309025i	$0.990162\pi$
$252$		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	0.250484	−	0.968121i	$-0.419410\pi$
$257$	312.035	−	180.153i	1.21414	−	0.700986i	0.963659	+	0.267135i	$0.0860770\pi$
$258$		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	0.578491	−	0.815689i	$-0.303642\pi$
$263$	42.4296	+	24.4967i	0.161329	+	0.0931435i	−0.417162	+	0.908832i	$0.636975\pi$
$264$		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	$-0.824587\pi$
$269$		−	281.700i		−	1.04721i	0.851961	−	0.523606i	$-0.175413\pi$
$270$		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	−0.779750	−	0.626091i	$-0.784654\pi$
$277$	42.1969	−	73.0872i	0.152336	−	0.263853i	0.932086	+	0.362238i	$0.117987\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	23.2673	+	13.4334i	0.0828019	+	0.0478057i	0.540829	−	0.841132i	$-0.318110\pi$
$281$	23.2673	+	13.4334i	0.0828019	+	0.0478057i	−0.458027	+	0.888938i	$0.651444\pi$
$282$		0			0
$283$	90.4898	+	156.733i	0.319752	+	0.553827i	0.980436	−	0.196837i	$-0.0630671\pi$
$283$	90.4898	+	156.733i	0.319752	+	0.553827i	−0.660684	+	0.750664i	$0.729734\pi$
$284$		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	−0.993585	−	0.113089i	$-0.963925\pi$
$293$	−407.985	+	235.550i	−1.39244	+	0.803925i	−0.398854	+	0.917014i	$0.630592\pi$
$294$		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	−0.106039	−	0.994362i	$-0.533817\pi$
$311$	218.348	−	126.063i	0.702083	−	0.405348i	0.808123	+	0.589014i	$0.200484\pi$
$312$		0			0
$313$	−98.1061	+	169.925i	−0.313438	+	0.542891i	−0.979104	−	0.203359i	$-0.934814\pi$
$313$	−98.1061	+	169.925i	−0.313438	+	0.542891i	0.665666	+	0.746250i	$0.268147\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	98.9847	+	57.1488i	0.312255	+	0.180280i	0.647935	−	0.761696i	$-0.275633\pi$
$317$	98.9847	+	57.1488i	0.312255	+	0.180280i	−0.335680	+	0.941976i	$0.608966\pi$
$318$		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	−0.821840	−	0.569719i	$-0.807052\pi$
$331$	27.2980	−	47.2815i	0.0824712	−	0.142844i	0.904311	+	0.426875i	$0.140385\pi$
$332$		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	0.986734	−	0.162347i	$-0.0519063\pi$
$337$	118.884	+	205.913i	0.352771	+	0.611016i	−0.633963	+	0.773363i	$0.718573\pi$
$338$		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	−0.335686	−	0.941974i	$-0.608968\pi$
$347$	108.349	−	62.5553i	0.312245	−	0.180275i	0.647931	+	0.761699i	$0.275635\pi$
$348$		0			0
$349$	−269.985	+	467.627i	−0.773595	+	1.33991i	0.161986	+	0.986793i	$0.448210\pi$
$349$	−269.985	+	467.627i	−0.773595	+	1.33991i	−0.935581	+	0.353113i	$0.885123\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−254.490	−	146.930i	−0.720934	−	0.416232i	0.0941622	−	0.995557i	$-0.469983\pi$
$353$	−254.490	−	146.930i	−0.720934	−	0.416232i	−0.815096	+	0.579325i	$0.803316\pi$
$354$		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	$0.200285\pi$
$359$			422.550i			1.17702i	−0.808490	+	0.588509i	$0.799715\pi$
$360$		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	−0.629690	−	0.776847i	$-0.716818\pi$
$367$	131.358	−	227.519i	0.357924	−	0.619943i	0.987614	+	0.156904i	$0.0501513\pi$
$368$		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	0.936121	−	0.351679i	$-0.114389\pi$
$373$	60.9847	+	105.629i	0.163498	+	0.283187i	−0.772623	+	0.634865i	$0.781056\pi$
$374$		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	−0.706733	+	0.707481i	$0.749832\pi$
$383$	−640.681	+	369.897i	−1.67280	+	0.965789i	−0.966063	+	0.258308i	$0.916835\pi$
$384$		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	0.594197	−	0.804319i	$-0.297470\pi$
$389$	75.7520	+	43.7355i	0.194735	+	0.112430i	−0.399462	+	0.916750i	$0.630803\pi$
$390$		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	−0.840745	−	0.541432i	$-0.817882\pi$
$401$	−317.682	+	183.414i	−0.792224	+	0.457390i	0.0485212	+	0.998822i	$0.484549\pi$
$402$		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	−0.982049	−	0.188625i	$-0.939597\pi$
$409$	−267.641	−	463.567i	−0.654379	−	1.13342i	0.327671	−	0.944792i	$-0.393736\pi$
$410$		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	0.447193	−	0.894438i	$-0.352424\pi$
$419$	605.620	−	349.655i	1.44539	−	0.834499i	0.998202	+	0.0599386i	$0.0190905\pi$
$420$		0			0
$421$	−180.772	+	313.107i	−0.429388	+	0.743722i	−0.996819	−	0.0796989i	$-0.974604\pi$
$421$	−180.772	+	313.107i	−0.429388	+	0.743722i	0.567431	+	0.823421i	$0.307937\pi$
$422$		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	$-0.819184\pi$
$431$		−	463.747i		−	1.07598i	0.842952	−	0.537989i	$-0.180816\pi$
$432$		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	−0.257721	−	0.966219i	$-0.582972\pi$
$439$	310.772	−	538.274i	0.707910	−	1.22614i	0.965631	−	0.259917i	$-0.0836951\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−698.843	−	403.477i	−1.57752	−	0.910784i	−0.995204	−	0.0978236i	$-0.968812\pi$
$443$	−698.843	−	403.477i	−1.57752	−	0.910784i	−0.582320	−	0.812960i	$-0.697855\pi$
$444$		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	$0.115051\pi$
$449$			317.554i			0.707248i	−0.935388	+	0.353624i	$0.884949\pi$
$450$		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	0.362972	+	0.931800i	$0.381762\pi$
$457$	−285.843	+	495.094i	−0.625477	+	1.08336i	−0.988448	+	0.151557i	$0.951571\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−478.550	−	276.291i	−1.03807	−	0.599330i	−0.118784	−	0.992920i	$-0.537899\pi$
$461$	−478.550	−	276.291i	−1.03807	−	0.599330i	−0.919286	+	0.393591i	$0.871233\pi$
$462$		0			0
$463$	60.1663	+	104.211i	0.129949	+	0.225078i	0.923657	−	0.383221i	$-0.125185\pi$
$463$	60.1663	+	104.211i	0.129949	+	0.225078i	−0.793708	+	0.608299i	$0.791852\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			880.440i			1.88531i	0.333767	+	0.942656i	$0.391680\pi$
$467$			880.440i			1.88531i	−0.333767	+	0.942656i	$0.608320\pi$
$468$		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	−0.270628	−	0.962684i	$-0.587231\pi$
$479$	−593.793	−	342.826i	−1.23965	−	0.715713i	−0.969023	+	0.246971i	$0.920565\pi$
$480$		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	−0.320807	−	0.947145i	$-0.603954\pi$
$491$	166.469	−	96.1111i	0.339042	−	0.195746i	0.659848	+	0.751399i	$0.270621\pi$
$492$		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	0.991139	+	0.132832i	$0.0424070\pi$
$499$	304.692	+	527.742i	0.610605	+	1.05760i	−0.380534	+	0.924767i	$0.624260\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			232.130i			0.461491i	0.973014	+	0.230746i	$0.0741165\pi$
$503$			232.130i			0.461491i	−0.973014	+	0.230746i	$0.925883\pi$
$504$		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	−0.702910	−	0.711278i	$-0.748117\pi$
$509$	−223.136	+	128.827i	−0.438381	+	0.253099i	0.264530	+	0.964377i	$0.414783\pi$
$510$		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	$0.153793\pi$
$521$			484.088i			0.929152i	−0.885533	+	0.464576i	$0.846207\pi$
$522$		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	−0.973241	−	0.229785i	$-0.926198\pi$
$547$	−157.329	+	272.501i	−0.287621	+	0.498174i	0.685621	+	0.727959i	$0.259531\pi$
$548$		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	$0.203293\pi$
$557$			664.080i			1.19224i	−0.802894	+	0.596122i	$0.796707\pi$
$558$		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	−0.675563	−	0.737302i	$-0.736099\pi$
$563$	−211.024	+	121.835i	−0.374821	+	0.216403i	0.300741	+	0.953706i	$0.402766\pi$
$564$		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	0.871489	−	0.490415i	$-0.163155\pi$
$569$	502.155	+	289.919i	0.882522	+	0.509524i	0.0110330	+	0.999939i	$0.496488\pi$
$570$		0			0
$571$	−356.843	−	618.070i	−0.624944	−	1.08243i	−0.988552	−	0.150882i	$-0.951789\pi$
$571$	−356.843	−	618.070i	−0.624944	−	1.08243i	0.363608	−	0.931552i	$-0.381545\pi$
$572$		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	0.984442	−	0.175707i	$-0.0562212\pi$
$587$	777.480	+	448.878i	1.32450	+	0.764699i	0.340054	+	0.940406i	$0.389555\pi$
$588$		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	$-0.896729\pi$
$593$		−	378.065i		−	0.637547i	0.947831	−	0.318774i	$-0.103271\pi$
$594$		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	−0.991307	−	0.131569i	$-0.957998\pi$
$599$	−822.438	+	474.835i	−1.37302	+	0.792712i	−0.381711	+	0.924282i	$0.624665\pi$
$600$		0			0
$601$	−252.308	+	437.011i	−0.419814	+	0.727139i	−0.995920	−	0.0902356i	$-0.971238\pi$
$601$	−252.308	+	437.011i	−0.419814	+	0.727139i	0.576107	+	0.817375i	$0.304571\pi$
$602$		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	−0.965478	−	0.260486i	$-0.916117\pi$
$607$	−429.954	−	744.702i	−0.708326	−	1.22686i	0.257151	−	0.966371i	$-0.417216\pi$
$608$		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	−0.375924	−	0.926650i	$-0.622675\pi$
$617$	147.227	−	85.0013i	0.238617	−	0.137765i	0.614541	+	0.788885i	$0.289341\pi$
$618$		0			0
$619$	270.531	−	468.573i	0.437045	−	0.756983i	−0.560415	−	0.828212i	$-0.689359\pi$
$619$	270.531	−	468.573i	0.437045	−	0.756983i	0.997460	+	0.0712282i	$0.0226918\pi$
$620$		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	−0.0318012	−	0.999494i	$-0.510124\pi$
$641$	−585.418	−	337.991i	−0.913289	−	0.527288i	−0.881488	+	0.472206i	$0.843458\pi$
$642$		0			0
$643$	−378.318	−	655.267i	−0.588364	−	1.01908i	−0.994447	−	0.105241i	$-0.966439\pi$
$643$	−378.318	−	655.267i	−0.588364	−	1.01908i	0.406082	−	0.913837i	$-0.366895\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			554.770i			0.857450i	0.903435	+	0.428725i	$0.141037\pi$
$647$			554.770i			0.857450i	−0.903435	+	0.428725i	$0.858963\pi$
$648$		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	−0.360400	−	0.932798i	$-0.617360\pi$
$653$	174.499	−	100.747i	0.267227	−	0.154283i	0.627627	+	0.778514i	$0.284026\pi$
$654$		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	0.430386	−	0.902645i	$-0.358377\pi$
$659$	−89.7122	−	51.7954i	−0.136134	−	0.0785969i	−0.566520	+	0.824048i	$0.691711\pi$
$660$		0			0
$661$	109.207	+	189.152i	0.165215	+	0.286161i	0.936732	−	0.350048i	$-0.113835\pi$
$661$	109.207	+	189.152i	0.165215	+	0.286161i	−0.771517	+	0.636209i	$0.780502\pi$
$662$		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	0.408665	+	0.912684i	$0.365994\pi$
$673$	−394.429	+	683.170i	−0.586075	+	1.01511i	−0.994740	+	0.102428i	$0.967339\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−634.550	−	366.358i	−0.937297	−	0.541149i	−0.0481850	−	0.998838i	$-0.515344\pi$
$677$	−634.550	−	366.358i	−0.937297	−	0.541149i	−0.889112	+	0.457690i	$0.848677\pi$
$678$		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	$0.456613\pi$
$683$			1353.33i			1.98145i	−0.135883	+	0.990725i	$0.543387\pi$
$684$		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	0.466327	+	0.884613i	$0.345577\pi$
$691$	−368.257	+	637.840i	−0.532934	+	0.923068i	−0.999260	+	0.0384555i	$0.987756\pi$
$692$		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	$0.275789\pi$
$701$			1068.34i			1.52403i	−0.647561	+	0.762014i	$0.724211\pi$
$702$		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	−0.753142	−	0.657858i	$-0.771463\pi$
$709$	136.944	−	237.194i	0.193151	−	0.334547i	0.946293	+	0.323311i	$0.104796\pi$
$710$		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	$-0.849606\pi$
$719$		−	654.423i		−	0.910185i	0.890444	−	0.455092i	$-0.150394\pi$
$720$		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	0.116041	+	0.993244i	$0.462980\pi$
$727$	−583.166	+	1010.07i	−0.802155	+	1.38937i	−0.918195	+	0.396128i	$0.870354\pi$
$728$		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	0.992953	+	0.118508i	$0.0378112\pi$
$733$	439.146	+	760.623i	0.599108	+	1.03768i	−0.393845	+	0.919177i	$0.628855\pi$
$734$		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	−0.467999	−	0.883729i	$-0.655025\pi$
$743$	47.0561	−	27.1679i	0.0633326	−	0.0365651i	0.531332	+	0.847164i	$0.321692\pi$
$744$		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	−0.991793	−	0.127850i	$-0.959192\pi$
$751$	−455.570	−	789.071i	−0.606618	−	1.05069i	0.385175	−	0.922844i	$-0.374141\pi$
$752$		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	−0.738954	−	0.673756i	$-0.764680\pi$
$761$	−399.480	+	230.640i	−0.524940	+	0.303074i	0.214013	+	0.976831i	$0.431346\pi$
$762$		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	0.635940	−	0.771739i	$-0.280613\pi$
$769$	−269.439	−	466.682i	−0.350376	−	0.606868i	−0.986315	+	0.164871i	$0.947279\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1036.29i			1.34061i	0.742087	+	0.670304i	$0.233836\pi$
$773$			1036.29i			1.34061i	−0.742087	+	0.670304i	$0.766164\pi$
$774$		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	−0.831059	−	0.556185i	$-0.812265\pi$
$787$	−706.096	−	1222.99i	−0.897199	−	1.55399i	−0.0661406	−	0.997810i	$-0.521069\pi$
$788$		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

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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