LMFDB - Embedded newform 648.3.m.a.377.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,3,Mod(377,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.377");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	648.m (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:	$17.6567211305$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			377.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	648.377
Dual form			648.3.m.a.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.12372 - 3.53553i) q^{5} +(-6.00000 - 10.3923i) q^{7} +O(q^{10})$	$q+(-6.12372 - 3.53553i) q^{5} +(-6.00000 - 10.3923i) q^{7} +(-4.89898 + 2.82843i) q^{11} +(4.00000 - 6.92820i) q^{13} +9.89949i q^{17} -16.0000 q^{19} +(34.2929 + 19.7990i) q^{23} +(12.5000 + 21.6506i) q^{25} +(-25.7196 + 14.8492i) q^{29} +(2.00000 - 3.46410i) q^{31} +84.8528i q^{35} +30.0000 q^{37} +(18.3712 + 10.6066i) q^{41} +(4.00000 + 6.92820i) q^{43} +(-14.6969 + 8.48528i) q^{47} +(-47.5000 + 82.2724i) q^{49} +49.4975i q^{53} +40.0000 q^{55} +(-68.5857 - 39.5980i) q^{59} +(7.00000 + 12.1244i) q^{61} +(-48.9898 + 28.2843i) q^{65} +(44.0000 - 76.2102i) q^{67} -28.2843i q^{71} -80.0000 q^{73} +(58.7878 + 33.9411i) q^{77} +(-50.0000 - 86.6025i) q^{79} +(-112.677 + 65.0538i) q^{83} +(35.0000 - 60.6218i) q^{85} +148.492i q^{89} -96.0000 q^{91} +(97.9796 + 56.5685i) q^{95} +(56.0000 + 96.9948i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{7}+O(q^{10}) $ 4 * q - 24 * q^7	$ 4 q - 24 q^{7} + 16 q^{13} - 64 q^{19} + 50 q^{25} + 8 q^{31} + 120 q^{37} + 16 q^{43} - 190 q^{49} + 160 q^{55} + 28 q^{61} + 176 q^{67} - 320 q^{73} - 200 q^{79} + 140 q^{85} - 384 q^{91} + 224 q^{97}+O(q^{100}) $ 4 * q - 24 * q^7 + 16 * q^13 - 64 * q^19 + 50 * q^25 + 8 * q^31 + 120 * q^37 + 16 * q^43 - 190 * q^49 + 160 * q^55 + 28 * q^61 + 176 * q^67 - 320 * q^73 - 200 * q^79 + 140 * q^85 - 384 * q^91 + 224 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−6.12372	−	3.53553i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$5$	−6.12372	−	3.53553i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$6$		0			0
$7$	−6.00000	−	10.3923i	−0.857143	−	1.48461i	−0.874643	−	0.484768i	$-0.838904\pi$
$7$	−6.00000	−	10.3923i	−0.857143	−	1.48461i	0.0174999	−	0.999847i	$-0.494429\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.89898	+	2.82843i	−0.445362	+	0.257130i	−0.705869	−	0.708342i	$-0.749443\pi$
$11$	−4.89898	+	2.82843i	−0.445362	+	0.257130i	0.260508	+	0.965472i	$0.416110\pi$
$12$		0			0
$13$	4.00000	−	6.92820i	0.307692	−	0.532939i	−0.670165	−	0.742212i	$-0.733777\pi$
$13$	4.00000	−	6.92820i	0.307692	−	0.532939i	0.977857	+	0.209274i	$0.0671099\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			9.89949i			0.582323i	0.956674	+	0.291162i	$0.0940417\pi$
$17$			9.89949i			0.582323i	−0.956674	+	0.291162i	$0.905958\pi$
$18$		0			0
$19$	−16.0000			−0.842105			−0.421053	−	0.907036i	$-0.638339\pi$
$19$	−16.0000			−0.842105			−0.421053	+	0.907036i	$0.638339\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	34.2929	+	19.7990i	1.49099	+	0.860826i	0.999947	−	0.0103075i	$-0.00328104\pi$
$23$	34.2929	+	19.7990i	1.49099	+	0.860826i	0.491047	+	0.871133i	$0.336614\pi$
$24$		0			0
$25$	12.5000	+	21.6506i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−25.7196	+	14.8492i	−0.886884	+	0.512043i	−0.872922	−	0.487860i	$-0.837778\pi$
$29$	−25.7196	+	14.8492i	−0.886884	+	0.512043i	−0.0139622	+	0.999903i	$0.504444\pi$
$30$		0			0
$31$	2.00000	−	3.46410i	0.0645161	−	0.111745i	−0.831963	−	0.554831i	$-0.812783\pi$
$31$	2.00000	−	3.46410i	0.0645161	−	0.111745i	0.896479	+	0.443086i	$0.146116\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			84.8528i			2.42437i
$36$		0			0
$37$	30.0000			0.810811			0.405405	−	0.914137i	$-0.367130\pi$
$37$	30.0000			0.810811			0.405405	+	0.914137i	$0.367130\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	18.3712	+	10.6066i	0.448077	+	0.258698i	0.707018	−	0.707196i	$-0.250040\pi$
$41$	18.3712	+	10.6066i	0.448077	+	0.258698i	−0.258940	+	0.965893i	$0.583373\pi$
$42$		0			0
$43$	4.00000	+	6.92820i	0.0930233	+	0.161121i	0.908782	−	0.417271i	$-0.137014\pi$
$43$	4.00000	+	6.92820i	0.0930233	+	0.161121i	−0.815759	+	0.578392i	$0.803680\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−14.6969	+	8.48528i	−0.312701	+	0.180538i	−0.648134	−	0.761526i	$-0.724451\pi$
$47$	−14.6969	+	8.48528i	−0.312701	+	0.180538i	0.335434	+	0.942064i	$0.391117\pi$
$48$		0			0
$49$	−47.5000	+	82.2724i	−0.969388	+	1.67903i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			49.4975i			0.933915i	0.884280	+	0.466957i	$0.154650\pi$
$53$			49.4975i			0.933915i	−0.884280	+	0.466957i	$0.845350\pi$
$54$		0			0
$55$	40.0000			0.727273
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−68.5857	−	39.5980i	−1.16247	−	0.671152i	−0.210575	−	0.977578i	$-0.567534\pi$
$59$	−68.5857	−	39.5980i	−1.16247	−	0.671152i	−0.951895	+	0.306425i	$0.900867\pi$
$60$		0			0
$61$	7.00000	+	12.1244i	0.114754	+	0.198760i	0.917681	−	0.397317i	$-0.130059\pi$
$61$	7.00000	+	12.1244i	0.114754	+	0.198760i	−0.802927	+	0.596077i	$0.796725\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−48.9898	+	28.2843i	−0.753689	+	0.435143i
$66$		0			0
$67$	44.0000	−	76.2102i	0.656716	−	1.13747i	−0.324744	−	0.945802i	$-0.605278\pi$
$67$	44.0000	−	76.2102i	0.656716	−	1.13747i	0.981461	−	0.191664i	$-0.0613885\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	28.2843i		−	0.398370i	−0.979962	−	0.199185i	$-0.936171\pi$
$71$		−	28.2843i		−	0.398370i	0.979962	−	0.199185i	$-0.0638295\pi$
$72$		0			0
$73$	−80.0000			−1.09589			−0.547945	−	0.836514i	$-0.684590\pi$
$73$	−80.0000			−1.09589			−0.547945	+	0.836514i	$0.684590\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	58.7878	+	33.9411i	0.763477	+	0.440794i
$78$		0			0
$79$	−50.0000	−	86.6025i	−0.632911	−	1.09623i	−0.986954	−	0.161005i	$-0.948526\pi$
$79$	−50.0000	−	86.6025i	−0.632911	−	1.09623i	0.354042	−	0.935229i	$-0.384807\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−112.677	+	65.0538i	−1.35755	+	0.783781i	−0.989293	−	0.145944i	$-0.953378\pi$
$83$	−112.677	+	65.0538i	−1.35755	+	0.783781i	−0.368256	+	0.929725i	$0.620045\pi$
$84$		0			0
$85$	35.0000	−	60.6218i	0.411765	−	0.713197i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			148.492i			1.66845i	0.551421	+	0.834227i	$0.314086\pi$
$89$			148.492i			1.66845i	−0.551421	+	0.834227i	$0.685914\pi$
$90$		0			0
$91$	−96.0000			−1.05495
$92$		0			0
$93$		0			0
$94$		0			0
$95$	97.9796	+	56.5685i	1.03136	+	0.595458i
$96$		0			0
$97$	56.0000	+	96.9948i	0.577320	+	0.999947i	0.995785	+	0.0917143i	$0.0292346\pi$
$97$	56.0000	+	96.9948i	0.577320	+	0.999947i	−0.418466	+	0.908233i	$0.637432\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	55.1135	−	31.8198i	0.545678	−	0.315048i	−0.201699	−	0.979448i	$-0.564646\pi$
$101$	55.1135	−	31.8198i	0.545678	−	0.315048i	0.747377	+	0.664400i	$0.231313\pi$
$102$		0			0
$103$	22.0000	−	38.1051i	0.213592	−	0.369953i	−0.739244	−	0.673438i	$-0.764817\pi$
$103$	22.0000	−	38.1051i	0.213592	−	0.369953i	0.952836	+	0.303485i	$0.0981503\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	136.000			1.24771			0.623853	−	0.781542i	$-0.285566\pi$
$109$	136.000			1.24771			0.623853	+	0.781542i	$0.285566\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.3712	+	10.6066i	0.162577	+	0.0938637i	0.579081	−	0.815270i	$-0.303412\pi$
$113$	18.3712	+	10.6066i	0.162577	+	0.0938637i	−0.416504	+	0.909134i	$0.636745\pi$
$114$		0			0
$115$	−140.000	−	242.487i	−1.21739	−	2.10858i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	102.879	−	59.3970i	0.864526	−	0.499134i
$120$		0			0
$121$	−44.5000	+	77.0763i	−0.367769	+	0.636994i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−20.0000			−0.157480			−0.0787402	−	0.996895i	$-0.525090\pi$
$127$	−20.0000			−0.157480			−0.0787402	+	0.996895i	$0.525090\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−48.9898	−	28.2843i	−0.373968	−	0.215910i	0.301223	−	0.953554i	$-0.402605\pi$
$131$	−48.9898	−	28.2843i	−0.373968	−	0.215910i	−0.675191	+	0.737643i	$0.735939\pi$
$132$		0			0
$133$	96.0000	+	166.277i	0.721805	+	1.25020i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	77.1589	−	44.5477i	0.563204	−	0.325166i	−0.191227	−	0.981546i	$-0.561247\pi$
$137$	77.1589	−	44.5477i	0.563204	−	0.325166i	0.754430	+	0.656380i	$0.227913\pi$
$138$		0			0
$139$	−60.0000	+	103.923i	−0.431655	+	0.747648i	−0.997016	−	0.0771956i	$-0.975403\pi$
$139$	−60.0000	+	103.923i	−0.431655	+	0.747648i	0.565361	+	0.824843i	$0.308737\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			45.2548i			0.316467i
$144$		0			0
$145$	210.000			1.44828
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−111.452	−	64.3467i	−0.747999	−	0.431857i	0.0769717	−	0.997033i	$-0.475475\pi$
$149$	−111.452	−	64.3467i	−0.747999	−	0.431857i	−0.824970	+	0.565176i	$0.808808\pi$
$150$		0			0
$151$	98.0000	+	169.741i	0.649007	+	1.12411i	0.983361	+	0.181665i	$0.0581486\pi$
$151$	98.0000	+	169.741i	0.649007	+	1.12411i	−0.334354	+	0.942448i	$0.608518\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−24.4949	+	14.1421i	−0.158032	+	0.0912396i
$156$		0			0
$157$	−119.000	+	206.114i	−0.757962	+	1.31283i	0.185927	+	0.982564i	$0.440471\pi$
$157$	−119.000	+	206.114i	−0.757962	+	1.31283i	−0.943888	+	0.330265i	$0.892862\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	475.176i		−	2.95140i
$162$		0			0
$163$	−168.000			−1.03067			−0.515337	−	0.856987i	$-0.672333\pi$
$163$	−168.000			−1.03067			−0.515337	+	0.856987i	$0.672333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−146.969	−	84.8528i	−0.880056	−	0.508101i	−0.00937926	−	0.999956i	$-0.502986\pi$
$167$	−146.969	−	84.8528i	−0.880056	−	0.508101i	−0.870677	+	0.491855i	$0.836319\pi$
$168$		0			0
$169$	52.5000	+	90.9327i	0.310651	+	0.538063i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−180.037	+	103.945i	−1.04068	+	0.600836i	−0.920026	−	0.391858i	$-0.871832\pi$
$173$	−180.037	+	103.945i	−1.04068	+	0.600836i	−0.120654	+	0.992695i	$0.538499\pi$
$174$		0			0
$175$	150.000	−	259.808i	0.857143	−	1.48461i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	203.647i		−	1.13769i	−0.822444	−	0.568846i	$-0.807390\pi$
$179$		−	203.647i		−	1.13769i	0.822444	−	0.568846i	$-0.192610\pi$
$180$		0			0
$181$	−216.000			−1.19337			−0.596685	−	0.802475i	$-0.703516\pi$
$181$	−216.000			−1.19337			−0.596685	+	0.802475i	$0.703516\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−183.712	−	106.066i	−0.993036	−	0.573330i
$186$		0			0
$187$	−28.0000	−	48.4974i	−0.149733	−	0.259345i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	323.333	−	186.676i	1.69284	−	0.977362i	0.740634	−	0.671909i	$-0.234525\pi$
$191$	323.333	−	186.676i	1.69284	−	0.977362i	0.952207	−	0.305454i	$-0.0988081\pi$
$192$		0			0
$193$	89.0000	−	154.153i	0.461140	−	0.798718i	−0.537878	−	0.843023i	$-0.680774\pi$
$193$	89.0000	−	154.153i	0.461140	−	0.798718i	0.999018	+	0.0443049i	$0.0141073\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	207.889i		−	1.05528i	−0.849469	−	0.527638i	$-0.823078\pi$
$197$		−	207.889i		−	1.05528i	0.849469	−	0.527638i	$-0.176922\pi$
$198$		0			0
$199$	196.000			0.984925			0.492462	−	0.870334i	$-0.336097\pi$
$199$	196.000			0.984925			0.492462	+	0.870334i	$0.336097\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	308.636	+	178.191i	1.52037	+	0.877788i
$204$		0			0
$205$	−75.0000	−	129.904i	−0.365854	−	0.633677i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	78.3837	−	45.2548i	0.375041	−	0.216530i
$210$		0			0
$211$	28.0000	−	48.4974i	0.132701	−	0.229846i	−0.792016	−	0.610501i	$-0.790968\pi$
$211$	28.0000	−	48.4974i	0.132701	−	0.229846i	0.924717	+	0.380655i	$0.124302\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	56.5685i		−	0.263109i
$216$		0			0
$217$	−48.0000			−0.221198
$218$		0			0
$219$		0			0
$220$		0			0
$221$	68.5857	+	39.5980i	0.310343	+	0.179176i
$222$		0			0
$223$	−14.0000	−	24.2487i	−0.0627803	−	0.108739i	0.832927	−	0.553383i	$-0.186663\pi$
$223$	−14.0000	−	24.2487i	−0.0627803	−	0.108739i	−0.895707	+	0.444644i	$0.853330\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	73.4847	−	42.4264i	0.323721	−	0.186900i	−0.329329	−	0.944215i	$-0.606822\pi$
$227$	73.4847	−	42.4264i	0.323721	−	0.186900i	0.653050	+	0.757315i	$0.273489\pi$
$228$		0			0
$229$	−92.0000	+	159.349i	−0.401747	+	0.695846i	−0.993937	−	0.109953i	$-0.964930\pi$
$229$	−92.0000	+	159.349i	−0.401747	+	0.695846i	0.592190	+	0.805798i	$0.298263\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			304.056i			1.30496i	0.757806	+	0.652481i	$0.226272\pi$
$233$			304.056i			1.30496i	−0.757806	+	0.652481i	$0.773728\pi$
$234$		0			0
$235$	120.000			0.510638
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−195.959	−	113.137i	−0.819913	−	0.473377i	0.0304735	−	0.999536i	$-0.490298\pi$
$239$	−195.959	−	113.137i	−0.819913	−	0.473377i	−0.850386	+	0.526159i	$0.823632\pi$
$240$		0			0
$241$	−192.000	−	332.554i	−0.796680	−	1.37989i	−0.921767	−	0.387745i	$-0.873254\pi$
$241$	−192.000	−	332.554i	−0.796680	−	1.37989i	0.125086	−	0.992146i	$-0.460079\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	581.754	−	335.876i	2.37451	−	1.37092i
$246$		0			0
$247$	−64.0000	+	110.851i	−0.259109	+	0.448790i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	197.990i		−	0.788804i	−0.918938	−	0.394402i	$-0.870952\pi$
$251$		−	197.990i		−	0.788804i	0.918938	−	0.394402i	$-0.129048\pi$
$252$		0			0
$253$	−224.000			−0.885375
$254$		0			0
$255$		0			0
$256$		0			0
$257$	79.6084	+	45.9619i	0.309760	+	0.178840i	0.646819	−	0.762643i	$-0.276099\pi$
$257$	79.6084	+	45.9619i	0.309760	+	0.178840i	−0.337059	+	0.941484i	$0.609432\pi$
$258$		0			0
$259$	−180.000	−	311.769i	−0.694981	−	1.20374i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−342.929	+	197.990i	−1.30391	+	0.752813i	−0.981072	−	0.193641i	$-0.937970\pi$
$263$	−342.929	+	197.990i	−1.30391	+	0.752813i	−0.322838	+	0.946454i	$0.604637\pi$
$264$		0			0
$265$	175.000	−	303.109i	0.660377	−	1.14381i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	83.4386i		−	0.310181i	−0.987900	−	0.155090i	$-0.950433\pi$
$269$		−	83.4386i		−	0.310181i	0.987900	−	0.155090i	$-0.0495669\pi$
$270$		0			0
$271$	140.000			0.516605			0.258303	−	0.966064i	$-0.416837\pi$
$271$	140.000			0.516605			0.258303	+	0.966064i	$0.416837\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−122.474	−	70.7107i	−0.445362	−	0.257130i
$276$		0			0
$277$	124.000	+	214.774i	0.447653	+	0.775358i	0.998233	−	0.0594243i	$-0.0189265\pi$
$277$	124.000	+	214.774i	0.447653	+	0.775358i	−0.550579	+	0.834783i	$0.685593\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−145.745	+	84.1457i	−0.518664	+	0.299451i	−0.736388	−	0.676560i	$-0.763470\pi$
$281$	−145.745	+	84.1457i	−0.518664	+	0.299451i	0.217724	+	0.976010i	$0.430137\pi$
$282$		0			0
$283$	−184.000	+	318.697i	−0.650177	+	1.12614i	0.332903	+	0.942961i	$0.391972\pi$
$283$	−184.000	+	318.697i	−0.650177	+	1.12614i	−0.983080	+	0.183178i	$0.941362\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	254.558i		−	0.886963i
$288$		0			0
$289$	191.000			0.660900
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−180.037	−	103.945i	−0.614462	−	0.354760i	0.160247	−	0.987077i	$-0.448771\pi$
$293$	−180.037	−	103.945i	−0.614462	−	0.354760i	−0.774710	+	0.632317i	$0.782104\pi$
$294$		0			0
$295$	280.000	+	484.974i	0.949153	+	1.64398i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	274.343	−	158.392i	0.917535	−	0.529739i
$300$		0			0
$301$	48.0000	−	83.1384i	0.159468	−	0.276207i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	98.9949i		−	0.324574i
$306$		0			0
$307$	−552.000			−1.79805			−0.899023	−	0.437902i	$-0.855722\pi$
$307$	−552.000			−1.79805			−0.899023	+	0.437902i	$0.855722\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−29.3939	−	16.9706i	−0.0945141	−	0.0545677i	0.451998	−	0.892019i	$-0.350711\pi$
$311$	−29.3939	−	16.9706i	−0.0945141	−	0.0545677i	−0.546512	+	0.837451i	$0.684045\pi$
$312$		0			0
$313$	111.000	+	192.258i	0.354633	+	0.614242i	0.987055	−	0.160382i	$-0.0512726\pi$
$313$	111.000	+	192.258i	0.354633	+	0.614242i	−0.632422	+	0.774624i	$0.717939\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−385.795	+	222.739i	−1.21702	+	0.702646i	−0.964279	−	0.264889i	$-0.914665\pi$
$317$	−385.795	+	222.739i	−1.21702	+	0.702646i	−0.252739	+	0.967535i	$0.581331\pi$
$318$		0			0
$319$	84.0000	−	145.492i	0.263323	−	0.456089i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	158.392i		−	0.490377i
$324$		0			0
$325$	200.000			0.615385
$326$		0			0
$327$		0			0
$328$		0			0
$329$	176.363	+	101.823i	0.536059	+	0.309494i
$330$		0			0
$331$	260.000	+	450.333i	0.785498	+	1.36052i	0.928701	+	0.370830i	$0.120927\pi$
$331$	260.000	+	450.333i	0.785498	+	1.36052i	−0.143202	+	0.989693i	$0.545740\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−538.888	+	311.127i	−1.60862	+	0.928737i
$336$		0			0
$337$	−24.0000	+	41.5692i	−0.0712166	+	0.123351i	−0.899435	−	0.437055i	$-0.856021\pi$
$337$	−24.0000	+	41.5692i	−0.0712166	+	0.123351i	0.828218	+	0.560406i	$0.189355\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			22.6274i			0.0663561i
$342$		0			0
$343$	552.000			1.60933
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−308.636	−	178.191i	−0.889440	−	0.513518i	−0.0156808	−	0.999877i	$-0.504992\pi$
$347$	−308.636	−	178.191i	−0.889440	−	0.513518i	−0.873759	+	0.486359i	$0.838325\pi$
$348$		0			0
$349$	−105.000	−	181.865i	−0.300860	−	0.521104i	0.675471	−	0.737386i	$-0.263940\pi$
$349$	−105.000	−	181.865i	−0.300860	−	0.521104i	−0.976331	+	0.216282i	$0.930607\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	236.376	−	136.472i	0.669620	−	0.386605i	−0.126313	−	0.991990i	$-0.540314\pi$
$353$	236.376	−	136.472i	0.669620	−	0.386605i	0.795933	+	0.605385i	$0.206981\pi$
$354$		0			0
$355$	−100.000	+	173.205i	−0.281690	+	0.487902i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			593.970i			1.65451i	0.561825	+	0.827256i	$0.310099\pi$
$359$			593.970i			1.65451i	−0.561825	+	0.827256i	$0.689901\pi$
$360$		0			0
$361$	−105.000			−0.290859
$362$		0			0
$363$		0			0
$364$		0			0
$365$	489.898	+	282.843i	1.34219	+	0.774912i
$366$		0			0
$367$	−86.0000	−	148.956i	−0.234332	−	0.405876i	0.724746	−	0.689016i	$-0.241957\pi$
$367$	−86.0000	−	148.956i	−0.234332	−	0.405876i	−0.959078	+	0.283140i	$0.908624\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	514.393	−	296.985i	1.38650	−	0.800498i
$372$		0			0
$373$	−1.00000	+	1.73205i	−0.00268097	+	0.00464357i	−0.867363	−	0.497676i	$-0.834187\pi$
$373$	−1.00000	+	1.73205i	−0.00268097	+	0.00464357i	0.864682	+	0.502320i	$0.167520\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			237.588i			0.630207i
$378$		0			0
$379$	−736.000			−1.94195			−0.970976	−	0.239176i	$-0.923123\pi$
$379$	−736.000			−1.94195			−0.970976	+	0.239176i	$0.923123\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$	−240.000	−	415.692i	−0.623377	−	1.07972i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−398.042	+	229.810i	−1.02324	+	0.590770i	−0.915042	−	0.403358i	$-0.867843\pi$
$389$	−398.042	+	229.810i	−1.02324	+	0.590770i	−0.108202	+	0.994129i	$0.534509\pi$
$390$		0			0
$391$	−196.000	+	339.482i	−0.501279	+	0.868240i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			707.107i			1.79014i
$396$		0			0
$397$	338.000			0.851385			0.425693	−	0.904868i	$-0.360030\pi$
$397$	338.000			0.851385			0.425693	+	0.904868i	$0.360030\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	128.598	+	74.2462i	0.320694	+	0.185153i	0.651702	−	0.758475i	$-0.274055\pi$
$401$	128.598	+	74.2462i	0.320694	+	0.185153i	−0.331008	+	0.943628i	$0.607389\pi$
$402$		0			0
$403$	−16.0000	−	27.7128i	−0.0397022	−	0.0687663i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−146.969	+	84.8528i	−0.361104	+	0.208484i
$408$		0			0
$409$	−72.0000	+	124.708i	−0.176039	+	0.304909i	−0.940520	−	0.339737i	$-0.889662\pi$
$409$	−72.0000	+	124.708i	−0.176039	+	0.304909i	0.764481	+	0.644646i	$0.222995\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			950.352i			2.30109i
$414$		0			0
$415$	920.000			2.21687
$416$		0			0
$417$		0			0
$418$		0			0
$419$	661.362	+	381.838i	1.57843	+	0.911307i	0.995079	+	0.0990872i	$0.0315923\pi$
$419$	661.362	+	381.838i	1.57843	+	0.911307i	0.583351	+	0.812220i	$0.301741\pi$
$420$		0			0
$421$	−340.000	−	588.897i	−0.807601	−	1.39881i	−0.914521	−	0.404538i	$-0.867432\pi$
$421$	−340.000	−	588.897i	−0.807601	−	1.39881i	0.106920	−	0.994268i	$-0.465901\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−214.330	+	123.744i	−0.504307	+	0.291162i
$426$		0			0
$427$	84.0000	−	145.492i	0.196721	−	0.340731i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			5.65685i			0.0131250i	0.999978	+	0.00656248i	$0.00208892\pi$
$431$			5.65685i			0.0131250i	−0.999978	+	0.00656248i	$0.997911\pi$
$432$		0			0
$433$	−482.000			−1.11316			−0.556582	−	0.830793i	$-0.687887\pi$
$433$	−482.000			−1.11316			−0.556582	+	0.830793i	$0.687887\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−548.686	−	316.784i	−1.25557	−	0.724906i
$438$		0			0
$439$	378.000	+	654.715i	0.861048	+	1.49138i	0.870919	+	0.491427i	$0.163525\pi$
$439$	378.000	+	654.715i	0.861048	+	1.49138i	−0.00987076	+	0.999951i	$0.503142\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−171.464	+	98.9949i	−0.387053	+	0.223465i	−0.680882	−	0.732393i	$-0.738403\pi$
$443$	−171.464	+	98.9949i	−0.387053	+	0.223465i	0.293830	+	0.955858i	$0.405070\pi$
$444$		0			0
$445$	525.000	−	909.327i	1.17978	−	2.04343i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	77.7817i		−	0.173233i	−0.996242	−	0.0866166i	$-0.972394\pi$
$449$		−	77.7817i		−	0.173233i	0.996242	−	0.0866166i	$-0.0276055\pi$
$450$		0			0
$451$	−120.000			−0.266075
$452$		0			0
$453$		0			0
$454$		0			0
$455$	587.878	+	339.411i	1.29204	+	0.745959i
$456$		0			0
$457$	−232.000	−	401.836i	−0.507659	−	0.879291i	−0.999961	−	0.00886613i	$-0.997178\pi$
$457$	−232.000	−	401.836i	−0.507659	−	0.879291i	0.492302	−	0.870424i	$-0.336156\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	42.8661	−	24.7487i	0.0929850	−	0.0536849i	−0.452786	−	0.891619i	$-0.649570\pi$
$461$	42.8661	−	24.7487i	0.0929850	−	0.0536849i	0.545771	+	0.837934i	$0.316237\pi$
$462$		0			0
$463$	−10.0000	+	17.3205i	−0.0215983	+	0.0374093i	−0.876623	−	0.481179i	$-0.840209\pi$
$463$	−10.0000	+	17.3205i	−0.0215983	+	0.0374093i	0.855024	+	0.518588i	$0.173542\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			695.793i			1.48992i	0.667109	+	0.744960i	$0.267532\pi$
$467$			695.793i			1.48992i	−0.667109	+	0.744960i	$0.732468\pi$
$468$		0			0
$469$	−1056.00			−2.25160
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−39.1918	−	22.6274i	−0.0828580	−	0.0478381i
$474$		0			0
$475$	−200.000	−	346.410i	−0.421053	−	0.729285i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−582.979	+	336.583i	−1.21707	+	0.702678i	−0.964291	−	0.264845i	$-0.914679\pi$
$479$	−582.979	+	336.583i	−1.21707	+	0.702678i	−0.252783	+	0.967523i	$0.581346\pi$
$480$		0			0
$481$	120.000	−	207.846i	0.249480	−	0.432112i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	791.960i		−	1.63291i
$486$		0			0
$487$	388.000			0.796715			0.398357	−	0.917230i	$-0.369580\pi$
$487$	388.000			0.796715			0.398357	+	0.917230i	$0.369580\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	−147.000	−	254.611i	−0.298174	−	0.516453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−293.939	+	169.706i	−0.591426	+	0.341460i
$498$		0			0
$499$	−48.0000	+	83.1384i	−0.0961924	+	0.166610i	−0.910106	−	0.414376i	$-0.864000\pi$
$499$	−48.0000	+	83.1384i	−0.0961924	+	0.166610i	0.813913	+	0.580986i	$0.197333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	424.264i		−	0.843467i	−0.906720	−	0.421734i	$-0.861422\pi$
$503$		−	424.264i		−	0.843467i	0.906720	−	0.421734i	$-0.138578\pi$
$504$		0			0
$505$	−450.000			−0.891089
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−268.219	−	154.856i	−0.526953	−	0.304237i	0.212822	−	0.977091i	$-0.431735\pi$
$509$	−268.219	−	154.856i	−0.526953	−	0.304237i	−0.739775	+	0.672855i	$0.765068\pi$
$510$		0			0
$511$	480.000	+	831.384i	0.939335	+	1.62698i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−269.444	+	155.563i	−0.523192	+	0.302065i
$516$		0			0
$517$	48.0000	−	83.1384i	0.0928433	−	0.160809i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			722.663i			1.38707i	0.720423	+	0.693535i	$0.243948\pi$
$521$			722.663i			1.38707i	−0.720423	+	0.693535i	$0.756052\pi$
$522$		0			0
$523$	352.000			0.673040			0.336520	−	0.941676i	$-0.390750\pi$
$523$	352.000			0.673040			0.336520	+	0.941676i	$0.390750\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	34.2929	+	19.7990i	0.0650718	+	0.0375692i
$528$		0			0
$529$	519.500	+	899.800i	0.982042	+	1.70095i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	146.969	−	84.8528i	0.275740	−	0.159199i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	537.401i		−	0.997034i
$540$		0			0
$541$	936.000			1.73013			0.865065	−	0.501660i	$-0.167277\pi$
$541$	936.000			1.73013			0.865065	+	0.501660i	$0.167277\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−832.827	−	480.833i	−1.52812	−	0.882262i
$546$		0			0
$547$	−28.0000	−	48.4974i	−0.0511883	−	0.0886607i	0.839296	−	0.543675i	$-0.182968\pi$
$547$	−28.0000	−	48.4974i	−0.0511883	−	0.0886607i	−0.890484	+	0.455014i	$0.849634\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	411.514	−	237.588i	0.746850	−	0.431194i
$552$		0			0
$553$	−600.000	+	1039.23i	−1.08499	+	1.87926i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			321.026i			0.576349i	0.957578	+	0.288175i	$0.0930483\pi$
$557$			321.026i			0.576349i	−0.957578	+	0.288175i	$0.906952\pi$
$558$		0			0
$559$	64.0000			0.114490
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−406.615	−	234.759i	−0.722230	−	0.416979i	0.0933431	−	0.995634i	$-0.470245\pi$
$563$	−406.615	−	234.759i	−0.722230	−	0.416979i	−0.815573	+	0.578655i	$0.803578\pi$
$564$		0			0
$565$	−75.0000	−	129.904i	−0.132743	−	0.229918i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−353.951	+	204.354i	−0.622058	+	0.359146i	−0.777670	−	0.628673i	$-0.783599\pi$
$569$	−353.951	+	204.354i	−0.622058	+	0.359146i	0.155612	+	0.987818i	$0.450265\pi$
$570$		0			0
$571$	−408.000	+	706.677i	−0.714536	+	1.23761i	0.248602	+	0.968606i	$0.420029\pi$
$571$	−408.000	+	706.677i	−0.714536	+	1.23761i	−0.963138	+	0.269007i	$0.913305\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			989.949i			1.72165i
$576$		0			0
$577$	242.000			0.419411			0.209705	−	0.977765i	$-0.432750\pi$
$577$	242.000			0.419411			0.209705	+	0.977765i	$0.432750\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1352.12	+	780.646i	2.32723	+	1.34362i
$582$		0			0
$583$	−140.000	−	242.487i	−0.240137	−	0.415930i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−342.929	+	197.990i	−0.584205	+	0.337291i	−0.762803	−	0.646631i	$-0.776177\pi$
$587$	−342.929	+	197.990i	−0.584205	+	0.337291i	0.178597	+	0.983922i	$0.442844\pi$
$588$		0			0
$589$	−32.0000	+	55.4256i	−0.0543294	+	0.0941012i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	205.061i		−	0.345803i	−0.984939	−	0.172901i	$-0.944686\pi$
$593$		−	205.061i		−	0.345803i	0.984939	−	0.172901i	$-0.0553142\pi$
$594$		0			0
$595$	−840.000			−1.41176
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−514.393	−	296.985i	−0.858753	−	0.495801i	0.00484173	−	0.999988i	$-0.498459\pi$
$599$	−514.393	−	296.985i	−0.858753	−	0.495801i	−0.863594	+	0.504187i	$0.831792\pi$
$600$		0			0
$601$	−143.000	−	247.683i	−0.237937	−	0.412119i	0.722185	−	0.691700i	$-0.243138\pi$
$601$	−143.000	−	247.683i	−0.237937	−	0.412119i	−0.960122	+	0.279581i	$0.909804\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	545.011	−	314.663i	0.900845	−	0.520103i
$606$		0			0
$607$	50.0000	−	86.6025i	0.0823723	−	0.142673i	−0.821896	−	0.569637i	$-0.807084\pi$
$607$	50.0000	−	86.6025i	0.0823723	−	0.142673i	0.904268	+	0.426964i	$0.140417\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			135.765i			0.222200i
$612$		0			0
$613$	770.000			1.25612			0.628059	−	0.778166i	$-0.283850\pi$
$613$	770.000			1.25612			0.628059	+	0.778166i	$0.283850\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	753.218	+	434.871i	1.22077	+	0.704815i	0.965083	−	0.261944i	$-0.0843635\pi$
$617$	753.218	+	434.871i	1.22077	+	0.704815i	0.255692	+	0.966758i	$0.417697\pi$
$618$		0			0
$619$	−128.000	−	221.703i	−0.206785	−	0.358162i	0.743915	−	0.668274i	$-0.232967\pi$
$619$	−128.000	−	221.703i	−0.206785	−	0.358162i	−0.950700	+	0.310112i	$0.899633\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1543.18	−	890.955i	2.47701	−	1.43010i
$624$		0			0
$625$	312.500	−	541.266i	0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			296.985i			0.472154i
$630$		0			0
$631$	196.000			0.310618			0.155309	−	0.987866i	$-0.450363\pi$
$631$	196.000			0.310618			0.155309	+	0.987866i	$0.450363\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	122.474	+	70.7107i	0.192873	+	0.111355i
$636$		0			0
$637$	380.000	+	658.179i	0.596546	+	1.03325i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	557.259	−	321.734i	0.869359	−	0.501924i	0.00222361	−	0.999998i	$-0.499292\pi$
$641$	557.259	−	321.734i	0.869359	−	0.501924i	0.867135	+	0.498073i	$0.165959\pi$
$642$		0			0
$643$	−60.0000	+	103.923i	−0.0933126	+	0.161622i	−0.908903	−	0.417007i	$-0.863079\pi$
$643$	−60.0000	+	103.923i	−0.0933126	+	0.161622i	0.815591	+	0.578630i	$0.196412\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	707.107i		−	1.09290i	−0.837491	−	0.546450i	$-0.815979\pi$
$647$		−	707.107i		−	1.09290i	0.837491	−	0.546450i	$-0.184021\pi$
$648$		0			0
$649$	448.000			0.690293
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−113.901	−	65.7609i	−0.174428	−	0.100706i	0.410244	−	0.911976i	$-0.365443\pi$
$653$	−113.901	−	65.7609i	−0.174428	−	0.100706i	−0.584672	+	0.811270i	$0.698777\pi$
$654$		0			0
$655$	200.000	+	346.410i	0.305344	+	0.528870i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1028.79	+	593.970i	−1.56113	+	0.901320i	−0.563989	+	0.825783i	$0.690734\pi$
$659$	−1028.79	+	593.970i	−1.56113	+	0.901320i	−0.997143	+	0.0755371i	$0.975933\pi$
$660$		0			0
$661$	65.0000	−	112.583i	0.0983359	−	0.170323i	−0.812660	−	0.582738i	$-0.801981\pi$
$661$	65.0000	−	112.583i	0.0983359	−	0.170323i	0.910996	+	0.412415i	$0.135315\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1357.65i		−	2.04157i
$666$		0			0
$667$	−1176.00			−1.76312
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−68.5857	−	39.5980i	−0.102214	−	0.0590134i
$672$		0			0
$673$	−305.000	−	528.275i	−0.453195	−	0.784956i	0.545388	−	0.838184i	$-0.316382\pi$
$673$	−305.000	−	528.275i	−0.453195	−	0.784956i	−0.998582	+	0.0532277i	$0.983049\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−522.966	+	301.935i	−0.772476	+	0.445989i	−0.833757	−	0.552131i	$-0.813815\pi$
$677$	−522.966	+	301.935i	−0.772476	+	0.445989i	0.0612813	+	0.998121i	$0.480481\pi$
$678$		0			0
$679$	672.000	−	1163.94i	0.989691	−	1.71419i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	989.949i		−	1.44941i	−0.689057	−	0.724707i	$-0.741975\pi$
$683$		−	989.949i		−	1.44941i	0.689057	−	0.724707i	$-0.258025\pi$
$684$		0			0
$685$	−630.000			−0.919708
$686$		0			0
$687$		0			0
$688$		0			0
$689$	342.929	+	197.990i	0.497719	+	0.287358i
$690$		0			0
$691$	−316.000	−	547.328i	−0.457308	−	0.792081i	0.541509	−	0.840695i	$-0.317853\pi$
$691$	−316.000	−	547.328i	−0.457308	−	0.792081i	−0.998818	+	0.0486136i	$0.984520\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	734.847	−	424.264i	1.05733	−	0.610452i
$696$		0			0
$697$	−105.000	+	181.865i	−0.150646	+	0.260926i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	29.6985i		−	0.0423659i	−0.999776	−	0.0211829i	$-0.993257\pi$
$701$		−	29.6985i		−	0.0423659i	0.999776	−	0.0211829i	$-0.00674324\pi$
$702$		0			0
$703$	−480.000			−0.682788
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−661.362	−	381.838i	−0.935449	−	0.540082i
$708$		0			0
$709$	−28.0000	−	48.4974i	−0.0394922	−	0.0684026i	0.845604	−	0.533811i	$-0.179241\pi$
$709$	−28.0000	−	48.4974i	−0.0394922	−	0.0684026i	−0.885096	+	0.465409i	$0.845907\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	137.171	−	79.1960i	0.192386	−	0.111074i
$714$		0			0
$715$	160.000	−	277.128i	0.223776	−	0.387592i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	616.597i		−	0.857576i	−0.903405	−	0.428788i	$-0.858941\pi$
$719$		−	616.597i		−	0.857576i	0.903405	−	0.428788i	$-0.141059\pi$
$720$		0			0
$721$	−528.000			−0.732316
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−642.991	−	371.231i	−0.886884	−	0.512043i
$726$		0			0
$727$	−186.000	−	322.161i	−0.255846	−	0.443138i	0.709279	−	0.704928i	$-0.249021\pi$
$727$	−186.000	−	322.161i	−0.255846	−	0.443138i	−0.965125	+	0.261790i	$0.915687\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−68.5857	+	39.5980i	−0.0938245	+	0.0541696i
$732$		0			0
$733$	260.000	−	450.333i	0.354707	−	0.614370i	−0.632361	−	0.774674i	$-0.717914\pi$
$733$	260.000	−	450.333i	0.354707	−	0.614370i	0.987068	+	0.160304i	$0.0512474\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			497.803i			0.675445i
$738$		0			0
$739$	1040.00			1.40731			0.703654	−	0.710543i	$-0.251551\pi$
$739$	1040.00			1.40731			0.703654	+	0.710543i	$0.251551\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−538.888	−	311.127i	−0.725286	−	0.418744i	0.0914089	−	0.995813i	$-0.470863\pi$
$743$	−538.888	−	311.127i	−0.725286	−	0.418744i	−0.816695	+	0.577069i	$0.804196\pi$
$744$		0			0
$745$	455.000	+	788.083i	0.610738	+	1.05783i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	90.0000	−	155.885i	0.119840	−	0.207569i	−0.799864	−	0.600181i	$-0.795095\pi$
$751$	90.0000	−	155.885i	0.119840	−	0.207569i	0.919704	+	0.392612i	$0.128428\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1385.93i		−	1.83567i
$756$		0			0
$757$	152.000			0.200793			0.100396	−	0.994948i	$-0.467989\pi$
$757$	152.000			0.200793			0.100396	+	0.994948i	$0.467989\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−471.527	−	272.236i	−0.619615	−	0.357735i	0.157104	−	0.987582i	$-0.449784\pi$
$761$	−471.527	−	272.236i	−0.619615	−	0.357735i	−0.776719	+	0.629847i	$0.783117\pi$
$762$		0			0
$763$	−816.000	−	1413.35i	−1.06946	−	1.85236i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−548.686	+	316.784i	−0.715366	+	0.413017i
$768$		0			0
$769$	263.000	−	455.529i	0.342003	−	0.592366i	−0.642802	−	0.766032i	$-0.722228\pi$
$769$	263.000	−	455.529i	0.342003	−	0.592366i	0.984805	+	0.173667i	$0.0555615\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1421.28i		−	1.83866i	−0.393487	−	0.919330i	$-0.628731\pi$
$773$		−	1421.28i		−	1.83866i	0.393487	−	0.919330i	$-0.371269\pi$
$774$		0			0
$775$	100.000			0.129032
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−293.939	−	169.706i	−0.377328	−	0.217851i
$780$		0			0
$781$	80.0000	+	138.564i	0.102433	+	0.177419i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1457.45	−	841.457i	1.85662	−	1.07192i
$786$		0			0
$787$	−536.000	+	928.379i	−0.681067	+	1.17964i	0.293588	+	0.955932i	$0.405151\pi$
$787$	−536.000	+	928.379i	−0.681067	+	1.17964i	−0.974655	+	0.223711i	$0.928183\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	254.558i		−	0.321819i
$792$		0			0
$793$	112.000			0.141236
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1328.85	−	767.211i	−1.66731	−	0.962623i	−0.969078	−	0.246754i	$-0.920636\pi$
$797$	−1328.85	−	767.211i	−1.66731	−	0.962623i	−0.698235	−	0.715869i	$-0.746031\pi$
$798$		0			0
$799$	−84.0000	−	145.492i	−0.105131	−	0.182093i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	391.918	−	226.274i	0.488068	−	0.281786i
$804$		0			0
$805$	−1680.00	+	2909.85i	−2.08696	+	3.61471i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	134.350i		−	0.166070i	−0.996547	−	0.0830348i	$-0.973539\pi$
$809$		−	134.350i		−	0.166070i	0.996547	−	0.0830348i	$-0.0264613\pi$
$810$		0			0
$811$	544.000			0.670777			0.335388	−	0.942080i	$-0.391132\pi$
$811$	544.000			0.670777			0.335388	+	0.942080i	$0.391132\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1028.79	+	593.970i	1.26231	+	0.728797i
$816$		0			0
$817$	−64.0000	−	110.851i	−0.0783354	−	0.135681i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	760.567	−	439.113i	0.926390	−	0.534852i	0.0407223	−	0.999171i	$-0.487034\pi$
$821$	760.567	−	439.113i	0.926390	−	0.534852i	0.885668	+	0.464319i	$0.153701\pi$
$822$		0			0
$823$	434.000	−	751.710i	0.527339	−	0.913378i	−0.472153	−	0.881516i	$-0.656523\pi$
$823$	434.000	−	751.710i	0.527339	−	0.913378i	0.999492	−	0.0318615i	$-0.0101435\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1414.21i			1.71005i	0.518585	+	0.855026i	$0.326459\pi$
$827$			1414.21i			1.71005i	−0.518585	+	0.855026i	$0.673541\pi$
$828$		0			0
$829$	−1208.00			−1.45718			−0.728589	−	0.684952i	$-0.759823\pi$
$829$	−1208.00			−1.45718			−0.728589	+	0.684952i	$0.759823\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−814.455	−	470.226i	−0.977738	−	0.564497i
$834$		0			0
$835$	600.000	+	1039.23i	0.718563	+	1.24459i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	925.907	−	534.573i	1.10358	−	0.637155i	0.166424	−	0.986054i	$-0.446778\pi$
$839$	925.907	−	534.573i	1.10358	−	0.637155i	0.937160	+	0.348900i	$0.113445\pi$
$840$		0			0
$841$	20.5000	−	35.5070i	0.0243757	−	0.0422200i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	742.462i		−	0.878653i
$846$		0			0
$847$	1068.00			1.26092
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1028.79	+	593.970i	1.20891	+	0.697967i
$852$		0			0
$853$	641.000	+	1110.24i	0.751465	+	1.30158i	0.947112	+	0.320902i	$0.103986\pi$
$853$	641.000	+	1110.24i	0.751465	+	1.30158i	−0.195647	+	0.980674i	$0.562681\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−900.187	+	519.723i	−1.05039	+	0.606445i	−0.922760	−	0.385376i	$-0.874072\pi$
$857$	−900.187	+	519.723i	−1.05039	+	0.606445i	−0.127634	+	0.991821i	$0.540738\pi$
$858$		0			0
$859$	516.000	−	893.738i	0.600698	−	1.04044i	−0.392017	−	0.919958i	$-0.628223\pi$
$859$	516.000	−	893.738i	0.600698	−	1.04044i	0.992715	−	0.120482i	$-0.0384441\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	56.5685i		−	0.0655487i	−0.999463	−	0.0327744i	$-0.989566\pi$
$863$		−	56.5685i		−	0.0655487i	0.999463	−	0.0327744i	$-0.0104343\pi$
$864$		0			0
$865$	1470.00			1.69942
$866$		0			0
$867$		0			0
$868$		0			0
$869$	489.898	+	282.843i	0.563749	+	0.325481i
$870$		0			0
$871$	−352.000	−	609.682i	−0.404133	−	0.699979i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	231.000	−	400.104i	0.263398	−	0.456219i	−0.703745	−	0.710453i	$-0.748490\pi$
$877$	231.000	−	400.104i	0.263398	−	0.456219i	0.967143	+	0.254234i	$0.0818234\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			247.487i			0.280916i	0.990087	+	0.140458i	$0.0448576\pi$
$881$			247.487i			0.280916i	−0.990087	+	0.140458i	$0.955142\pi$
$882$		0			0
$883$	8.00000			0.00906002			0.00453001	−	0.999990i	$-0.498558\pi$
$883$	8.00000			0.00906002			0.00453001	+	0.999990i	$0.498558\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	774.039	+	446.891i	0.872648	+	0.503824i	0.868227	−	0.496167i	$-0.165260\pi$
$887$	774.039	+	446.891i	0.872648	+	0.503824i	0.00442069	+	0.999990i	$0.498593\pi$
$888$		0			0
$889$	120.000	+	207.846i	0.134983	+	0.233798i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	235.151	−	135.765i	0.263327	−	0.152032i
$894$		0			0
$895$	−720.000	+	1247.08i	−0.804469	+	1.39338i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			118.794i			0.132140i
$900$		0			0
$901$	−490.000			−0.543840
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1322.72	+	763.675i	1.46157	+	0.843840i
$906$		0			0
$907$	100.000	+	173.205i	0.110254	+	0.190965i	0.915872	−	0.401469i	$-0.131500\pi$
$907$	100.000	+	173.205i	0.110254	+	0.190965i	−0.805619	+	0.592434i	$0.798167\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	68.5857	−	39.5980i	0.0752862	−	0.0434665i	−0.461884	−	0.886940i	$-0.652826\pi$
$911$	68.5857	−	39.5980i	0.0752862	−	0.0434665i	0.537171	+	0.843474i	$0.319493\pi$
$912$		0			0
$913$	368.000	−	637.395i	0.403067	−	0.698132i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			678.823i			0.740264i
$918$		0			0
$919$	−884.000			−0.961915			−0.480958	−	0.876744i	$-0.659711\pi$
$919$	−884.000			−0.961915			−0.480958	+	0.876744i	$0.659711\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−195.959	−	113.137i	−0.212307	−	0.122575i
$924$		0			0
$925$	375.000	+	649.519i	0.405405	+	0.702183i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−932.031	+	538.108i	−1.00326	+	0.579234i	−0.909212	−	0.416334i	$-0.863315\pi$
$929$	−932.031	+	538.108i	−1.00326	+	0.579234i	−0.0940505	+	0.995567i	$0.529982\pi$
$930$		0			0
$931$	760.000	−	1316.36i	0.816327	−	1.41392i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			395.980i			0.423508i
$936$		0			0
$937$	78.0000			0.0832444			0.0416222	−	0.999133i	$-0.486747\pi$
$937$	78.0000			0.0832444			0.0416222	+	0.999133i	$0.486747\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	231.477	+	133.643i	0.245990	+	0.142023i	0.617927	−	0.786236i	$-0.287973\pi$
$941$	231.477	+	133.643i	0.245990	+	0.142023i	−0.371937	+	0.928258i	$0.621306\pi$
$942$		0			0
$943$	420.000	+	727.461i	0.445387	+	0.771433i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	499.696	−	288.500i	0.527662	−	0.304646i	−0.212402	−	0.977182i	$-0.568129\pi$
$947$	499.696	−	288.500i	0.527662	−	0.304646i	0.740064	+	0.672537i	$0.234795\pi$
$948$		0			0
$949$	−320.000	+	554.256i	−0.337197	+	0.584042i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1336.43i		−	1.40234i	−0.712993	−	0.701171i	$-0.752661\pi$
$953$		−	1336.43i		−	1.40234i	0.712993	−	0.701171i	$-0.247339\pi$
$954$		0			0
$955$	−2640.00			−2.76440
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−925.907	−	534.573i	−0.965492	−	0.557427i
$960$		0			0
$961$	472.500	+	818.394i	0.491675	+	0.851607i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1090.02	+	629.325i	−1.12956	+	0.652150i
$966$		0			0
$967$	−10.0000	+	17.3205i	−0.0103413	+	0.0179116i	−0.871150	−	0.491017i	$-0.836625\pi$
$967$	−10.0000	+	17.3205i	−0.0103413	+	0.0179116i	0.860808	+	0.508929i	$0.169958\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			118.794i			0.122342i	0.998127	+	0.0611709i	$0.0194835\pi$
$971$			118.794i			0.122342i	−0.998127	+	0.0611709i	$0.980517\pi$
$972$		0			0
$973$	1440.00			1.47996
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−363.749	−	210.011i	−0.372312	−	0.214955i	0.302156	−	0.953259i	$-0.402294\pi$
$977$	−363.749	−	210.011i	−0.372312	−	0.214955i	−0.674468	+	0.738304i	$0.735627\pi$
$978$		0			0
$979$	−420.000	−	727.461i	−0.429009	−	0.743066i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1028.79	+	593.970i	−1.04658	+	0.604242i	−0.921689	−	0.387929i	$-0.873191\pi$
$983$	−1028.79	+	593.970i	−1.04658	+	0.604242i	−0.124888	+	0.992171i	$0.539857\pi$
$984$		0			0
$985$	−735.000	+	1273.06i	−0.746193	+	1.29244i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			316.784i			0.320307i
$990$		0			0
$991$	−116.000			−0.117053			−0.0585267	−	0.998286i	$-0.518640\pi$
$991$	−116.000			−0.117053			−0.0585267	+	0.998286i	$0.518640\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1200.25	−	692.965i	−1.20628	−	0.696447i
$996$		0			0
$997$	−65.0000	−	112.583i	−0.0651956	−	0.112922i	0.831585	−	0.555397i	$-0.187434\pi$
$997$	−65.0000	−	112.583i	−0.0651956	−	0.112922i	−0.896781	+	0.442475i	$0.854100\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.3.m.a.377.1		4
3.2	odd	2	inner	648.3.m.a.377.2		4
4.3	odd	2		1296.3.q.k.1025.1		4
9.2	odd	6	inner	648.3.m.a.593.1		4
9.4	even	3		72.3.e.a.17.2	yes	2
9.5	odd	6		72.3.e.a.17.1	✓	2
9.7	even	3	inner	648.3.m.a.593.2		4
12.11	even	2		1296.3.q.k.1025.2		4
36.7	odd	6		1296.3.q.k.593.2		4
36.11	even	6		1296.3.q.k.593.1		4
36.23	even	6		144.3.e.a.17.1		2
36.31	odd	6		144.3.e.a.17.2		2
45.4	even	6		1800.3.l.a.1601.1		2
45.13	odd	12		1800.3.c.a.449.1		4
45.14	odd	6		1800.3.l.a.1601.2		2
45.22	odd	12		1800.3.c.a.449.3		4
45.23	even	12		1800.3.c.a.449.2		4
45.32	even	12		1800.3.c.a.449.4		4
63.13	odd	6		3528.3.d.a.1961.1		2
63.41	even	6		3528.3.d.a.1961.2		2
72.5	odd	6		576.3.e.h.449.2		2
72.13	even	6		576.3.e.h.449.1		2
72.59	even	6		576.3.e.a.449.2		2
72.67	odd	6		576.3.e.a.449.1		2
144.5	odd	12		2304.3.h.a.2177.4		4
144.13	even	12		2304.3.h.a.2177.3		4
144.59	even	12		2304.3.h.h.2177.4		4
144.67	odd	12		2304.3.h.h.2177.3		4
144.77	odd	12		2304.3.h.a.2177.1		4
144.85	even	12		2304.3.h.a.2177.2		4
144.131	even	12		2304.3.h.h.2177.1		4
144.139	odd	12		2304.3.h.h.2177.2		4
180.23	odd	12		3600.3.c.c.449.3		4
180.59	even	6		3600.3.l.l.1601.1		2
180.67	even	12		3600.3.c.c.449.2		4
180.103	even	12		3600.3.c.c.449.4		4
180.139	odd	6		3600.3.l.l.1601.2		2
180.167	odd	12		3600.3.c.c.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.e.a.17.1	✓	2	9.5	odd	6
72.3.e.a.17.2	yes	2	9.4	even	3
144.3.e.a.17.1		2	36.23	even	6
144.3.e.a.17.2		2	36.31	odd	6
576.3.e.a.449.1		2	72.67	odd	6
576.3.e.a.449.2		2	72.59	even	6
576.3.e.h.449.1		2	72.13	even	6
576.3.e.h.449.2		2	72.5	odd	6
648.3.m.a.377.1		4	1.1	even	1	trivial
648.3.m.a.377.2		4	3.2	odd	2	inner
648.3.m.a.593.1		4	9.2	odd	6	inner
648.3.m.a.593.2		4	9.7	even	3	inner
1296.3.q.k.593.1		4	36.11	even	6
1296.3.q.k.593.2		4	36.7	odd	6
1296.3.q.k.1025.1		4	4.3	odd	2
1296.3.q.k.1025.2		4	12.11	even	2
1800.3.c.a.449.1		4	45.13	odd	12
1800.3.c.a.449.2		4	45.23	even	12
1800.3.c.a.449.3		4	45.22	odd	12
1800.3.c.a.449.4		4	45.32	even	12
1800.3.l.a.1601.1		2	45.4	even	6
1800.3.l.a.1601.2		2	45.14	odd	6
2304.3.h.a.2177.1		4	144.77	odd	12
2304.3.h.a.2177.2		4	144.85	even	12
2304.3.h.a.2177.3		4	144.13	even	12
2304.3.h.a.2177.4		4	144.5	odd	12
2304.3.h.h.2177.1		4	144.131	even	12
2304.3.h.h.2177.2		4	144.139	odd	12
2304.3.h.h.2177.3		4	144.67	odd	12
2304.3.h.h.2177.4		4	144.59	even	12
3528.3.d.a.1961.1		2	63.13	odd	6
3528.3.d.a.1961.2		2	63.41	even	6
3600.3.c.c.449.1		4	180.167	odd	12
3600.3.c.c.449.2		4	180.67	even	12
3600.3.c.c.449.3		4	180.23	odd	12
3600.3.c.c.449.4		4	180.103	even	12
3600.3.l.l.1601.1		2	180.59	even	6
3600.3.l.l.1601.2		2	180.139	odd	6

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	648.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.12372 - 3.53553i) q^{5} +(-6.00000 - 10.3923i) q^{7} +O(q^{10})\)	\(q+(-6.12372 - 3.53553i) q^{5} +(-6.00000 - 10.3923i) q^{7} +(-4.89898 + 2.82843i) q^{11} +(4.00000 - 6.92820i) q^{13} +9.89949i q^{17} -16.0000 q^{19} +(34.2929 + 19.7990i) q^{23} +(12.5000 + 21.6506i) q^{25} +(-25.7196 + 14.8492i) q^{29} +(2.00000 - 3.46410i) q^{31} +84.8528i q^{35} +30.0000 q^{37} +(18.3712 + 10.6066i) q^{41} +(4.00000 + 6.92820i) q^{43} +(-14.6969 + 8.48528i) q^{47} +(-47.5000 + 82.2724i) q^{49} +49.4975i q^{53} +40.0000 q^{55} +(-68.5857 - 39.5980i) q^{59} +(7.00000 + 12.1244i) q^{61} +(-48.9898 + 28.2843i) q^{65} +(44.0000 - 76.2102i) q^{67} -28.2843i q^{71} -80.0000 q^{73} +(58.7878 + 33.9411i) q^{77} +(-50.0000 - 86.6025i) q^{79} +(-112.677 + 65.0538i) q^{83} +(35.0000 - 60.6218i) q^{85} +148.492i q^{89} -96.0000 q^{91} +(97.9796 + 56.5685i) q^{95} +(56.0000 + 96.9948i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{7}+O(q^{10}) \) 4 * q - 24 * q^7	\( 4 q - 24 q^{7} + 16 q^{13} - 64 q^{19} + 50 q^{25} + 8 q^{31} + 120 q^{37} + 16 q^{43} - 190 q^{49} + 160 q^{55} + 28 q^{61} + 176 q^{67} - 320 q^{73} - 200 q^{79} + 140 q^{85} - 384 q^{91} + 224 q^{97}+O(q^{100}) \) 4 * q - 24 * q^7 + 16 * q^13 - 64 * q^19 + 50 * q^25 + 8 * q^31 + 120 * q^37 + 16 * q^43 - 190 * q^49 + 160 * q^55 + 28 * q^61 + 176 * q^67 - 320 * q^73 - 200 * q^79 + 140 * q^85 - 384 * q^91 + 224 * q^97

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