LMFDB - Embedded newform 1008.3.cd.i.415.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(415,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.415"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 2])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.cd (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-1,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.1364138928.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 35x^{4} + 364x^{2} + 972 $ x^6 + 35x^4 + 364x^2 + 972
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			415.3
Root			$4.07390i$ of defining polynomial
Character	$\chi$	$=$	1008.415
Dual form			1008.3.cd.i.991.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.29832 - 3.98081i) q^{5} +(4.95955 - 4.93992i) q^{7} +(-0.161228 + 0.0930852i) q^{11} +13.7090 q^{13} +(11.6528 + 20.1833i) q^{17} +(28.5074 + 16.4587i) q^{19} +(-21.4911 - 12.4079i) q^{23} +(1.93542 + 3.35224i) q^{25} -24.0641 q^{29} +(1.08871 - 0.628570i) q^{31} +(-8.26626 - 31.0966i) q^{35} +(17.7736 - 30.7848i) q^{37} +77.2563 q^{41} -41.9125i q^{43} +(-52.3787 - 30.2408i) q^{47} +(0.194309 - 48.9996i) q^{49} +(17.2983 + 29.9616i) q^{53} +0.855760i q^{55} +(1.78257 - 1.02917i) q^{59} +(-15.5484 + 26.9306i) q^{61} +(31.5078 - 54.5731i) q^{65} +(-65.3950 + 37.7558i) q^{67} -30.7190i q^{71} +(22.7736 + 39.4451i) q^{73} +(-0.339786 + 1.25812i) q^{77} +(0.943915 + 0.544970i) q^{79} +39.4358i q^{83} +107.128 q^{85} +(76.1113 - 131.829i) q^{89} +(67.9907 - 67.7216i) q^{91} +(131.038 - 75.6550i) q^{95} -25.9359 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{5} + 11 q^{7} + 3 q^{11} - 44 q^{13} - 8 q^{17} + 30 q^{19} + 24 q^{23} - 14 q^{25} - 34 q^{29} + 39 q^{31} - 90 q^{35} + 6 q^{37} + 136 q^{41} - 258 q^{47} + 157 q^{49} + 89 q^{53} + 63 q^{59}+ \cdots - 266 q^{97}+O(q^{100}) $ 6 * q - q^5 + 11 * q^7 + 3 * q^11 - 44 * q^13 - 8 * q^17 + 30 * q^19 + 24 * q^23 - 14 * q^25 - 34 * q^29 + 39 * q^31 - 90 * q^35 + 6 * q^37 + 136 * q^41 - 258 * q^47 + 157 * q^49 + 89 * q^53 + 63 * q^59 - 50 * q^61 + 184 * q^65 - 348 * q^67 + 36 * q^73 - 221 * q^77 - 3 * q^79 + 536 * q^85 + 202 * q^89 + 62 * q^91 + 432 * q^95 - 266 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.29832	−	3.98081i	0.459665	−	0.796163i	−0.539278	−	0.842128i	$-0.681303\pi$
$5$	2.29832	−	3.98081i	0.459665	−	0.796163i	0.998943	+	0.0459650i	$0.0146363\pi$
$6$		0			0
$7$	4.95955	−	4.93992i	0.708507	−	0.705703i
$7$	4.95955	−	4.93992i	0.708507	−	0.705703i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.161228	+	0.0930852i	−0.0146571	+	0.00846229i	−0.507311	−	0.861763i	$-0.669360\pi$
$11$	−0.161228	+	0.0930852i	−0.0146571	+	0.00846229i	0.492654	+	0.870226i	$0.336027\pi$
$12$		0			0
$13$	13.7090			1.05454			0.527271	−	0.849697i	$-0.323215\pi$
$13$	13.7090			1.05454			0.527271	+	0.849697i	$0.323215\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	11.6528	+	20.1833i	0.685462	+	1.18725i	0.973292	+	0.229573i	$0.0737329\pi$
$17$	11.6528	+	20.1833i	0.685462	+	1.18725i	−0.287830	+	0.957682i	$0.592934\pi$
$18$		0			0
$19$	28.5074	+	16.4587i	1.50039	+	0.866249i	1.00000		0.000447947i	$0.000142586\pi$
$19$	28.5074	+	16.4587i	1.50039	+	0.866249i	0.500388	+	0.865801i	$0.333191\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−21.4911	−	12.4079i	−0.934394	−	0.539472i	−0.0461952	−	0.998932i	$-0.514710\pi$
$23$	−21.4911	−	12.4079i	−0.934394	−	0.539472i	−0.888198	+	0.459460i	$0.848043\pi$
$24$		0			0
$25$	1.93542	+	3.35224i	0.0774167	+	0.134090i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−24.0641			−0.829798			−0.414899	−	0.909868i	$-0.636183\pi$
$29$	−24.0641			−0.829798			−0.414899	+	0.909868i	$0.636183\pi$
$30$		0			0
$31$	1.08871	−	0.628570i	0.0351198	−	0.0202764i	−0.482337	−	0.875986i	$-0.660212\pi$
$31$	1.08871	−	0.628570i	0.0351198	−	0.0202764i	0.517457	+	0.855709i	$0.326879\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−8.26626	−	31.0966i	−0.236179	−	0.888474i
$36$		0			0
$37$	17.7736	−	30.7848i	0.480368	−	0.832022i	−0.519378	−	0.854545i	$-0.673836\pi$
$37$	17.7736	−	30.7848i	0.480368	−	0.832022i	0.999746	+	0.0225223i	$0.00716968\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	77.2563			1.88430			0.942150	−	0.335192i	$-0.108801\pi$
$41$	77.2563			1.88430			0.942150	+	0.335192i	$0.108801\pi$
$42$		0			0
$43$		−	41.9125i		−	0.974710i	−0.873204	−	0.487355i	$-0.837962\pi$
$43$		−	41.9125i		−	0.974710i	0.873204	−	0.487355i	$-0.162038\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−52.3787	−	30.2408i	−1.11444	−	0.643422i	−0.174464	−	0.984664i	$-0.555819\pi$
$47$	−52.3787	−	30.2408i	−1.11444	−	0.643422i	−0.939976	+	0.341242i	$0.889153\pi$
$48$		0			0
$49$	0.194309	−	48.9996i	0.00396548	−	0.999992i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	17.2983	+	29.9616i	0.326383	+	0.565313i	0.981791	−	0.189962i	$-0.0608366\pi$
$53$	17.2983	+	29.9616i	0.326383	+	0.565313i	−0.655408	+	0.755275i	$0.727503\pi$
$54$		0			0
$55$			0.855760i			0.0155593i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.78257	−	1.02917i	0.0302131	−	0.0174435i	−0.484817	−	0.874615i	$-0.661114\pi$
$59$	1.78257	−	1.02917i	0.0302131	−	0.0174435i	0.515030	+	0.857172i	$0.327781\pi$
$60$		0			0
$61$	−15.5484	+	26.9306i	−0.254891	+	0.441485i	−0.964866	−	0.262742i	$-0.915373\pi$
$61$	−15.5484	+	26.9306i	−0.254891	+	0.441485i	0.709975	+	0.704227i	$0.248706\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	31.5078	−	54.5731i	0.484736	−	0.839587i
$66$		0			0
$67$	−65.3950	+	37.7558i	−0.976044	+	0.563519i	−0.901074	−	0.433666i	$-0.857220\pi$
$67$	−65.3950	+	37.7558i	−0.976044	+	0.563519i	−0.0749708	+	0.997186i	$0.523886\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	30.7190i		−	0.432662i	−0.976320	−	0.216331i	$-0.930591\pi$
$71$		−	30.7190i		−	0.432662i	0.976320	−	0.216331i	$-0.0694091\pi$
$72$		0			0
$73$	22.7736	+	39.4451i	0.311967	+	0.540344i	0.978788	−	0.204875i	$-0.0656787\pi$
$73$	22.7736	+	39.4451i	0.311967	+	0.540344i	−0.666821	+	0.745218i	$0.732345\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.339786	+	1.25812i	−0.00441281	+	0.0163392i
$78$		0			0
$79$	0.943915	+	0.544970i	0.0119483	+	0.00689835i	0.505962	−	0.862556i	$-0.331138\pi$
$79$	0.943915	+	0.544970i	0.0119483	+	0.00689835i	−0.494014	+	0.869454i	$0.664471\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			39.4358i			0.475130i	0.971372	+	0.237565i	$0.0763493\pi$
$83$			39.4358i			0.475130i	−0.971372	+	0.237565i	$0.923651\pi$
$84$		0			0
$85$	107.128			1.26033
$86$		0			0
$87$		0			0
$88$		0			0
$89$	76.1113	−	131.829i	0.855183	−	1.48122i	−0.0212925	−	0.999773i	$-0.506778\pi$
$89$	76.1113	−	131.829i	0.855183	−	1.48122i	0.876475	−	0.481447i	$-0.159889\pi$
$90$		0			0
$91$	67.9907	−	67.7216i	0.747151	−	0.744194i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	131.038	−	75.6550i	1.37935	−	0.796368i
$96$		0			0
$97$	−25.9359			−0.267380			−0.133690	−	0.991023i	$-0.542683\pi$
$97$	−25.9359			−0.267380			−0.133690	+	0.991023i	$0.542683\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−32.6134	−	56.4881i	−0.322905	−	0.559288i	0.658181	−	0.752860i	$-0.271326\pi$
$101$	−32.6134	−	56.4881i	−0.322905	−	0.559288i	−0.981086	+	0.193572i	$0.937993\pi$
$102$		0			0
$103$	−120.084	−	69.3306i	−1.16586	−	0.673112i	−0.213162	−	0.977017i	$-0.568376\pi$
$103$	−120.084	−	69.3306i	−1.16586	−	0.673112i	−0.952703	+	0.303904i	$0.901710\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	75.5872	+	43.6403i	0.706423	+	0.407853i	0.809735	−	0.586796i	$-0.199611\pi$
$107$	75.5872	+	43.6403i	0.706423	+	0.407853i	−0.103312	+	0.994649i	$0.532944\pi$
$108$		0			0
$109$	−32.3073	−	55.9578i	−0.296397	−	0.513375i	0.678912	−	0.734220i	$-0.262452\pi$
$109$	−32.3073	−	55.9578i	−0.296397	−	0.513375i	−0.975309	+	0.220845i	$0.929118\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	187.299			1.65752			0.828758	−	0.559607i	$-0.189048\pi$
$113$	187.299			1.65752			0.828758	+	0.559607i	$0.189048\pi$
$114$		0			0
$115$	−98.7868	+	57.0346i	−0.859015	+	0.495953i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	157.497	+	42.5361i	1.32350	+	0.357446i
$120$		0			0
$121$	−60.4827	+	104.759i	−0.499857	+	0.865777i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	132.709			1.06167
$126$		0			0
$127$		−	153.264i		−	1.20680i	−0.797437	−	0.603402i	$-0.793811\pi$
$127$		−	153.264i		−	1.20680i	0.797437	−	0.603402i	$-0.206189\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−167.531	−	96.7241i	−1.27886	−	0.738352i	−0.302224	−	0.953237i	$-0.597729\pi$
$131$	−167.531	−	96.7241i	−1.27886	−	0.738352i	−0.976639	+	0.214885i	$0.931062\pi$
$132$		0			0
$133$	222.689	−	59.1963i	1.67435	−	0.445085i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	108.451	+	187.842i	0.791611	+	1.37111i	0.924969	+	0.380043i	$0.124091\pi$
$137$	108.451	+	187.842i	0.791611	+	1.37111i	−0.133358	+	0.991068i	$0.542576\pi$
$138$		0			0
$139$		−	0.732659i		−	0.00527093i	−0.999997	−	0.00263546i	$-0.999161\pi$
$139$		−	0.732659i		−	0.00527093i	0.999997	−	0.00263546i	$-0.000838895\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.21029	+	1.27611i	−0.0154566	+	0.00892384i
$144$		0			0
$145$	−55.3072	+	95.7948i	−0.381429	+	0.660654i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	74.5137	−	129.062i	0.500092	−	0.866185i	−0.499908	−	0.866079i	$-0.666633\pi$
$149$	74.5137	−	129.062i	0.500092	−	0.866185i	1.00000		0.000106336i	$-3.38477e-5\pi$
$150$		0			0
$151$	−35.8891	+	20.7206i	−0.237676	+	0.137222i	−0.614108	−	0.789222i	$-0.710484\pi$
$151$	−35.8891	+	20.7206i	−0.237676	+	0.137222i	0.376432	+	0.926444i	$0.377151\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	5.77863i		−	0.0372815i
$156$		0			0
$157$	108.224	+	187.449i	0.689324	+	1.19394i	0.972057	+	0.234746i	$0.0754257\pi$
$157$	108.224	+	187.449i	0.689324	+	1.19394i	−0.282733	+	0.959199i	$0.591241\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−167.880	+	44.6267i	−1.04273	+	0.277185i
$162$		0			0
$163$	92.1419	+	53.1981i	0.565288	+	0.326369i	0.755265	−	0.655420i	$-0.227508\pi$
$163$	92.1419	+	53.1981i	0.565288	+	0.326369i	−0.189977	+	0.981788i	$0.560841\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	266.346i		−	1.59489i	−0.603394	−	0.797443i	$-0.706185\pi$
$167$		−	266.346i		−	1.59489i	0.603394	−	0.797443i	$-0.293815\pi$
$168$		0			0
$169$	18.9379			0.112059
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−63.1981	+	109.462i	−0.365307	+	0.632730i	−0.988825	−	0.149079i	$-0.952369\pi$
$173$	−63.1981	+	109.462i	−0.365307	+	0.632730i	0.623519	+	0.781809i	$0.285703\pi$
$174$		0			0
$175$	26.1586	+	7.06480i	0.149478	+	0.0403703i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−138.665	+	80.0585i	−0.774667	+	0.447254i	−0.834537	−	0.550952i	$-0.814265\pi$
$179$	−138.665	+	80.0585i	−0.774667	+	0.447254i	0.0598702	+	0.998206i	$0.480931\pi$
$180$		0			0
$181$	−108.709			−0.600604			−0.300302	−	0.953844i	$-0.597087\pi$
$181$	−108.709			−0.600604			−0.300302	+	0.953844i	$0.597087\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−81.6991	−	141.507i	−0.441617	−	0.764902i
$186$		0			0
$187$	−3.75754	−	2.16942i	−0.0200938	−	0.0116012i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.6656	+	11.3539i	0.102961	+	0.0594446i	0.550596	−	0.834772i	$-0.314400\pi$
$191$	19.6656	+	11.3539i	0.102961	+	0.0594446i	−0.447635	+	0.894216i	$0.647734\pi$
$192$		0			0
$193$	−147.353	−	255.223i	−0.763487	−	1.32240i	−0.941043	−	0.338287i	$-0.890152\pi$
$193$	−147.353	−	255.223i	−0.763487	−	1.32240i	0.177556	−	0.984111i	$-0.443181\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−296.976			−1.50749			−0.753746	−	0.657166i	$-0.771755\pi$
$197$	−296.976			−1.50749			−0.753746	+	0.657166i	$0.771755\pi$
$198$		0			0
$199$	223.142	−	128.831i	1.12131	−	0.647391i	0.179579	−	0.983744i	$-0.442527\pi$
$199$	223.142	−	128.831i	1.12131	−	0.647391i	0.941736	+	0.336352i	$0.109193\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−119.347	+	118.875i	−0.587918	+	0.585591i
$204$		0			0
$205$	177.560	−	307.543i	0.866146	−	1.50021i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.12826			−0.0293218
$210$		0			0
$211$			84.2895i			0.399476i	0.979849	+	0.199738i	$0.0640091\pi$
$211$			84.2895i			0.399476i	−0.979849	+	0.199738i	$0.935991\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−166.846	−	96.3285i	−0.776027	−	0.448040i
$216$		0			0
$217$	2.29445	−	8.49559i	0.0105735	−	0.0391502i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	159.749	+	276.694i	0.722848	+	1.25201i
$222$		0			0
$223$			170.727i			0.765591i	0.923833	+	0.382795i	$0.125039\pi$
$223$			170.727i			0.765591i	−0.923833	+	0.382795i	$0.874961\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−66.4950	+	38.3909i	−0.292930	+	0.169123i	−0.639262	−	0.768989i	$-0.720760\pi$
$227$	−66.4950	+	38.3909i	−0.292930	+	0.169123i	0.346333	+	0.938112i	$0.387427\pi$
$228$		0			0
$229$	−190.772	+	330.426i	−0.833064	+	1.44291i	0.0625335	+	0.998043i	$0.480082\pi$
$229$	−190.772	+	330.426i	−0.833064	+	1.44291i	−0.895597	+	0.444866i	$0.853251\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.4090	+	23.2251i	−0.0575495	+	0.0996786i	−0.893365	−	0.449332i	$-0.851662\pi$
$233$	−13.4090	+	23.2251i	−0.0575495	+	0.0996786i	0.835815	+	0.549011i	$0.184995\pi$
$234$		0			0
$235$	−240.766	+	139.006i	−1.02454	+	0.591517i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			429.135i			1.79554i	0.440460	+	0.897772i	$0.354815\pi$
$239$			429.135i			1.79554i	−0.440460	+	0.897772i	$0.645185\pi$
$240$		0			0
$241$	−143.915	−	249.269i	−0.597159	−	1.03431i	−0.993238	−	0.116094i	$-0.962963\pi$
$241$	−143.915	−	249.269i	−0.597159	−	1.03431i	0.396079	−	0.918216i	$-0.370371\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−194.612	−	113.390i	−0.794334	−	0.462818i
$246$		0			0
$247$	390.809	+	225.634i	1.58222	+	0.913496i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			112.212i			0.447060i	0.974697	+	0.223530i	$0.0717581\pi$
$251$			112.212i			0.447060i	−0.974697	+	0.223530i	$0.928242\pi$
$252$		0			0
$253$	4.61996			0.0182607
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.42828	−	9.40205i	0.0211217	−	0.0365839i	−0.855271	−	0.518180i	$-0.826610\pi$
$257$	5.42828	−	9.40205i	0.0211217	−	0.0365839i	0.876393	+	0.481597i	$0.159943\pi$
$258$		0			0
$259$	−63.9255	−	240.479i	−0.246816	−	0.928491i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−32.3048	+	18.6512i	−0.122832	+	0.0709170i	−0.560157	−	0.828386i	$-0.689259\pi$
$263$	−32.3048	+	18.6512i	−0.122832	+	0.0709170i	0.437325	+	0.899303i	$0.355926\pi$
$264$		0			0
$265$	159.029			0.600108
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.3346	+	26.5603i	0.0570059	+	0.0987372i	0.893120	−	0.449818i	$-0.148511\pi$
$269$	15.3346	+	26.5603i	0.0570059	+	0.0987372i	−0.836114	+	0.548556i	$0.815178\pi$
$270$		0			0
$271$	268.261	+	154.881i	0.989894	+	0.571515i	0.905243	−	0.424895i	$-0.139689\pi$
$271$	268.261	+	154.881i	0.989894	+	0.571515i	0.0846512	+	0.996411i	$0.473022\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.624089	−	0.360318i	−0.00226941	−	0.00131025i
$276$		0			0
$277$	−145.499	−	252.012i	−0.525269	−	0.909792i	−0.999567	−	0.0294279i	$-0.990631\pi$
$277$	−145.499	−	252.012i	−0.525269	−	0.909792i	0.474298	−	0.880364i	$-0.342702\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−433.358			−1.54220			−0.771100	−	0.636714i	$-0.780293\pi$
$281$	−433.358			−1.54220			−0.771100	+	0.636714i	$0.780293\pi$
$282$		0			0
$283$	93.4602	−	53.9593i	0.330248	−	0.190669i	−0.325703	−	0.945472i	$-0.605601\pi$
$283$	93.4602	−	53.9593i	0.330248	−	0.190669i	0.655951	+	0.754803i	$0.272268\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	383.157	−	381.640i	1.33504	−	1.32976i
$288$		0			0
$289$	−127.078	+	220.105i	−0.439715	+	0.761609i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−134.703			−0.459736			−0.229868	−	0.973222i	$-0.573829\pi$
$293$	−134.703			−0.459736			−0.229868	+	0.973222i	$0.573829\pi$
$294$		0			0
$295$		−	9.46145i		−	0.0320727i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−294.622	−	170.100i	−0.985357	−	0.568896i
$300$		0			0
$301$	−207.045	−	207.867i	−0.687856	−	0.690589i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	71.4704	+	123.790i	0.234329	+	0.405870i
$306$		0			0
$307$			532.540i			1.73466i	0.497736	+	0.867329i	$0.334165\pi$
$307$			532.540i			1.73466i	−0.497736	+	0.867329i	$0.665835\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	443.216	−	255.891i	1.42513	−	0.822799i	0.428399	−	0.903590i	$-0.359078\pi$
$311$	443.216	−	255.891i	1.42513	−	0.822799i	0.996731	+	0.0807901i	$0.0257444\pi$
$312$		0			0
$313$	−43.6965	+	75.6845i	−0.139605	+	0.241803i	−0.927347	−	0.374202i	$-0.877917\pi$
$313$	−43.6965	+	75.6845i	−0.139605	+	0.241803i	0.787742	+	0.616005i	$0.211250\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−71.6108	+	124.034i	−0.225902	+	0.391273i	−0.956590	−	0.291439i	$-0.905866\pi$
$317$	−71.6108	+	124.034i	−0.225902	+	0.391273i	0.730688	+	0.682712i	$0.239199\pi$
$318$		0			0
$319$	3.87982	−	2.24002i	0.0121624	−	0.00702199i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			767.164i			2.37512i
$324$		0			0
$325$	26.5327	+	45.9561i	0.0816392	+	0.141403i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−409.162	+	108.766i	−1.24365	+	0.330595i
$330$		0			0
$331$	−528.430	−	305.089i	−1.59647	−	0.921720i	−0.992161	−	0.124966i	$-0.960118\pi$
$331$	−528.430	−	305.089i	−1.59647	−	0.921720i	−0.604304	−	0.796754i	$-0.706549\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			347.100i			1.03612i
$336$		0			0
$337$	243.994			0.724018			0.362009	−	0.932175i	$-0.382091\pi$
$337$	243.994			0.724018			0.362009	+	0.932175i	$0.382091\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.117021	+	0.202687i	−0.000343170	+	0.000594389i
$342$		0			0
$343$	−241.091	−	243.976i	−0.702888	−	0.711300i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−312.604	+	180.482i	−0.900875	+	0.520121i	−0.877484	−	0.479606i	$-0.840780\pi$
$347$	−312.604	+	180.482i	−0.900875	+	0.520121i	−0.0233911	+	0.999726i	$0.507446\pi$
$348$		0			0
$349$	−375.119			−1.07484			−0.537420	−	0.843314i	$-0.680601\pi$
$349$	−375.119			−1.07484			−0.537420	+	0.843314i	$0.680601\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−76.8294	−	133.072i	−0.217647	−	0.376976i	0.736441	−	0.676502i	$-0.236505\pi$
$353$	−76.8294	−	133.072i	−0.217647	−	0.376976i	−0.954088	+	0.299526i	$0.903171\pi$
$354$		0			0
$355$	−122.287	−	70.6022i	−0.344469	−	0.198880i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−46.9502	−	27.1067i	−0.130780	−	0.0755061i	0.433182	−	0.901306i	$-0.357391\pi$
$359$	−46.9502	−	27.1067i	−0.130780	−	0.0755061i	−0.563963	+	0.825800i	$0.690724\pi$
$360$		0			0
$361$	361.280	+	625.755i	1.00078	+	1.73339i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	209.365			0.573602
$366$		0			0
$367$	130.929	−	75.5921i	0.356756	−	0.205973i	−0.310901	−	0.950442i	$-0.600631\pi$
$367$	130.929	−	75.5921i	0.356756	−	0.205973i	0.667657	+	0.744469i	$0.267297\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	233.800	+	63.1436i	0.630188	+	0.170198i
$372$		0			0
$373$	−316.054	+	547.422i	−0.847330	+	1.46762i	0.0362517	+	0.999343i	$0.488458\pi$
$373$	−316.054	+	547.422i	−0.847330	+	1.46762i	−0.883582	+	0.468276i	$0.844875\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−329.896			−0.875056
$378$		0			0
$379$			316.045i			0.833892i	0.908931	+	0.416946i	$0.136900\pi$
$379$			316.045i			0.833892i	−0.908931	+	0.416946i	$0.863100\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	237.606	+	137.182i	0.620380	+	0.358177i	0.777017	−	0.629480i	$-0.216732\pi$
$383$	237.606	+	137.182i	0.620380	+	0.358177i	−0.156637	+	0.987656i	$0.550065\pi$
$384$		0			0
$385$	4.22739	+	4.24419i	0.0109802	+	0.0110239i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−117.536	−	203.579i	−0.302150	−	0.523338i	0.674473	−	0.738299i	$-0.264371\pi$
$389$	−117.536	−	203.579i	−0.302150	−	0.523338i	−0.976623	+	0.214961i	$0.931038\pi$
$390$		0			0
$391$		−	578.348i		−	1.47915i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.33884	−	2.50503i	0.0109844	−	0.00634186i
$396$		0			0
$397$	−284.092	+	492.061i	−0.715596	+	1.23945i	0.247133	+	0.968982i	$0.420512\pi$
$397$	−284.092	+	492.061i	−0.715596	+	1.23945i	−0.962729	+	0.270468i	$0.912822\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−115.696	+	200.391i	−0.288519	+	0.499729i	−0.973456	−	0.228873i	$-0.926496\pi$
$401$	−115.696	+	200.391i	−0.288519	+	0.499729i	0.684938	+	0.728602i	$0.259829\pi$
$402$		0			0
$403$	14.9252	−	8.61709i	0.0370353	−	0.0213824i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			6.61785i			0.0162601i
$408$		0			0
$409$	−33.8249	−	58.5865i	−0.0827016	−	0.143243i	0.821708	−	0.569909i	$-0.193021\pi$
$409$	−33.8249	−	58.5865i	−0.0827016	−	0.143243i	−0.904409	+	0.426666i	$0.859688\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.75675	−	13.9100i	0.00909624	−	0.0336804i
$414$		0			0
$415$	156.987	+	90.6362i	0.378281	+	0.218400i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			111.646i			0.266459i	0.991085	+	0.133229i	$0.0425347\pi$
$419$			111.646i			0.266459i	−0.991085	+	0.133229i	$0.957465\pi$
$420$		0			0
$421$	509.147			1.20937			0.604687	−	0.796463i	$-0.293298\pi$
$421$	509.147			1.20937			0.604687	+	0.796463i	$0.293298\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−45.1063	+	78.1263i	−0.106132	+	0.183827i
$426$		0			0
$427$	55.9220	+	210.371i	0.130965	+	0.492673i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	659.682	−	380.868i	1.53059	−	0.883684i	0.531251	−	0.847215i	$-0.321722\pi$
$431$	659.682	−	380.868i	1.53059	−	0.883684i	0.999335	−	0.0364694i	$-0.0116111\pi$
$432$		0			0
$433$	−218.413			−0.504418			−0.252209	−	0.967673i	$-0.581157\pi$
$433$	−218.413			−0.504418			−0.252209	+	0.967673i	$0.581157\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−408.436	−	707.431i	−0.934635	−	1.61884i
$438$		0			0
$439$	−562.488	−	324.753i	−1.28129	−	0.739756i	−0.304210	−	0.952605i	$-0.598392\pi$
$439$	−562.488	−	324.753i	−1.28129	−	0.739756i	−0.977085	+	0.212849i	$0.931726\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	653.200	+	377.125i	1.47449	+	0.851298i	0.999587	−	0.0287397i	$-0.00914940\pi$
$443$	653.200	+	377.125i	1.47449	+	0.851298i	0.474904	+	0.880038i	$0.342483\pi$
$444$		0			0
$445$	−349.857	−	605.969i	−0.786195	−	1.36173i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	566.657			1.26204			0.631021	−	0.775766i	$-0.282636\pi$
$449$	566.657			1.26204			0.631021	+	0.775766i	$0.282636\pi$
$450$		0			0
$451$	−12.4559	+	7.19142i	−0.0276184	+	0.0159455i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−113.322	−	426.305i	−0.249060	−	0.936933i
$456$		0			0
$457$	350.908	−	607.791i	0.767852	−	1.32996i	−0.170873	−	0.985293i	$-0.554659\pi$
$457$	350.908	−	607.791i	0.767852	−	1.32996i	0.938725	−	0.344666i	$-0.112008\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	475.670			1.03182			0.515911	−	0.856642i	$-0.327454\pi$
$461$	475.670			1.03182			0.515911	+	0.856642i	$0.327454\pi$
$462$		0			0
$463$			564.736i			1.21973i	0.792505	+	0.609866i	$0.208777\pi$
$463$			564.736i			1.21973i	−0.792505	+	0.609866i	$0.791223\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−674.760	−	389.573i	−1.44488	−	0.834203i	−0.446712	−	0.894678i	$-0.647405\pi$
$467$	−674.760	−	389.573i	−1.44488	−	0.834203i	−0.998170	+	0.0604753i	$0.980738\pi$
$468$		0			0
$469$	−137.819	+	510.298i	−0.293857	+	1.08806i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.90144	+	6.75749i	0.00824828	+	0.0142864i
$474$		0			0
$475$			127.418i			0.268249i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	204.157	−	117.870i	0.426214	−	0.246075i	−0.271518	−	0.962433i	$-0.587526\pi$
$479$	204.157	−	117.870i	0.426214	−	0.246075i	0.697732	+	0.716358i	$0.254192\pi$
$480$		0			0
$481$	243.659	−	422.031i	0.506568	−	0.877402i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−59.6090	+	103.246i	−0.122905	+	0.212878i
$486$		0			0
$487$	−304.438	+	175.767i	−0.625129	+	0.360919i	−0.778863	−	0.627194i	$-0.784203\pi$
$487$	−304.438	+	175.767i	−0.625129	+	0.360919i	0.153734	+	0.988112i	$0.450870\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			586.350i			1.19420i	0.802169	+	0.597098i	$0.203680\pi$
$491$			586.350i			1.19420i	−0.802169	+	0.597098i	$0.796320\pi$
$492$		0			0
$493$	−280.416	−	485.694i	−0.568794	−	0.985181i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−151.750	−	152.353i	−0.305331	−	0.306544i
$498$		0			0
$499$	−253.489	−	146.352i	−0.507994	−	0.293291i	0.224014	−	0.974586i	$-0.428084\pi$
$499$	−253.489	−	146.352i	−0.507994	−	0.293291i	−0.732009	+	0.681295i	$0.761417\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			577.054i			1.14722i	0.819127	+	0.573612i	$0.194458\pi$
$503$			577.054i			1.14722i	−0.819127	+	0.573612i	$0.805542\pi$
$504$		0			0
$505$	−299.825			−0.593712
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−410.890	+	711.683i	−0.807250	+	1.39820i	0.107512	+	0.994204i	$0.465712\pi$
$509$	−410.890	+	711.683i	−0.807250	+	1.39820i	−0.914762	+	0.403994i	$0.867622\pi$
$510$		0			0
$511$	307.803	+	83.1299i	0.602354	+	0.162681i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−551.984	+	318.688i	−1.07181	+	0.618812i
$516$		0			0
$517$	11.2599			0.0217793
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−45.0485	−	78.0263i	−0.0864654	−	0.149763i	0.819549	−	0.573009i	$-0.194224\pi$
$521$	−45.0485	−	78.0263i	−0.0864654	−	0.149763i	−0.906015	+	0.423246i	$0.860891\pi$
$522$		0			0
$523$	130.787	+	75.5102i	0.250072	+	0.144379i	0.619797	−	0.784762i	$-0.287215\pi$
$523$	130.787	+	75.5102i	0.250072	+	0.144379i	−0.369725	+	0.929141i	$0.620548\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	25.3732	+	14.6493i	0.0481466	+	0.0277974i
$528$		0			0
$529$	43.4102	+	75.1887i	0.0820609	+	0.142134i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1059.11			1.98707
$534$		0			0
$535$	347.448	−	200.599i	0.649435	−	0.374952i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.52981	+	7.91822i	0.00840411	+	0.0146906i
$540$		0			0
$541$	−271.652	+	470.515i	−0.502129	+	0.869714i	0.497868	+	0.867253i	$0.334117\pi$
$541$	−271.652	+	470.515i	−0.502129	+	0.869714i	−0.999997	+	0.00246052i	$0.999217\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−297.010			−0.544973
$546$		0			0
$547$			908.509i			1.66089i	0.557097	+	0.830447i	$0.311915\pi$
$547$			908.509i			1.66089i	−0.557097	+	0.830447i	$0.688085\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−686.005	−	396.065i	−1.24502	−	0.718812i
$552$		0			0
$553$	7.37350	−	1.96006i	0.0133336	−	0.00354442i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−297.094	−	514.583i	−0.533383	−	0.923847i	−0.999240	−	0.0389864i	$-0.987587\pi$
$557$	−297.094	−	514.583i	−0.533383	−	0.923847i	0.465857	−	0.884860i	$-0.345746\pi$
$558$		0			0
$559$		−	574.581i		−	1.02787i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	606.259	−	350.024i	1.07684	−	0.621712i	0.146796	−	0.989167i	$-0.453104\pi$
$563$	606.259	−	350.024i	1.07684	−	0.621712i	0.930041	+	0.367455i	$0.119771\pi$
$564$		0			0
$565$	430.475	−	745.604i	0.761902	−	1.31965i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	416.677	−	721.705i	0.732297	−	1.26837i	−0.223603	−	0.974680i	$-0.571782\pi$
$569$	416.677	−	721.705i	0.732297	−	1.26837i	0.955899	−	0.293694i	$-0.0948848\pi$
$570$		0			0
$571$	796.086	−	459.621i	1.39420	−	0.804940i	0.400420	−	0.916332i	$-0.368864\pi$
$571$	796.086	−	459.621i	1.39420	−	0.804940i	0.993777	+	0.111392i	$0.0355309\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	96.0577i		−	0.167057i
$576$		0			0
$577$	−5.50000	−	9.52628i	−0.00953206	−	0.0165100i	0.861220	−	0.508232i	$-0.169701\pi$
$577$	−5.50000	−	9.52628i	−0.00953206	−	0.0165100i	−0.870752	+	0.491722i	$0.836368\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	194.810	+	195.584i	0.335301	+	0.336633i
$582$		0			0
$583$	−5.57796	−	3.22044i	−0.00956769	−	0.00552391i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	965.911i		−	1.64550i	−0.568401	−	0.822752i	$-0.692438\pi$
$587$		−	965.911i		−	1.64550i	0.568401	−	0.822752i	$-0.307562\pi$
$588$		0			0
$589$	41.3819			0.0702578
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−513.816	+	889.955i	−0.866468	+	1.50077i	−0.000886063		1.00000i	$0.500282\pi$
$593$	−513.816	+	889.955i	−0.866468	+	1.50077i	−0.865582	+	0.500767i	$0.833051\pi$
$594$		0			0
$595$	531.307	−	529.204i	0.892953	−	0.889419i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−695.499	+	401.547i	−1.16110	+	0.670362i	−0.951567	−	0.307440i	$-0.900528\pi$
$599$	−695.499	+	401.547i	−1.16110	+	0.670362i	−0.209533	+	0.977802i	$0.567194\pi$
$600$		0			0
$601$	702.916			1.16958			0.584789	−	0.811186i	$-0.301177\pi$
$601$	702.916			1.16958			0.584789	+	0.811186i	$0.301177\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	278.017	+	481.540i	0.459533	+	0.795935i
$606$		0			0
$607$	−185.611	−	107.162i	−0.305783	−	0.176544i	0.339255	−	0.940695i	$-0.389825\pi$
$607$	−185.611	−	107.162i	−0.305783	−	0.176544i	−0.645038	+	0.764150i	$0.723159\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−718.061	−	414.573i	−1.17522	−	0.678515i
$612$		0			0
$613$	369.332	+	639.701i	0.602498	+	1.04356i	0.992441	+	0.122719i	$0.0391614\pi$
$613$	369.332	+	639.701i	0.602498	+	1.04356i	−0.389943	+	0.920839i	$0.627505\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−32.2398			−0.0522526			−0.0261263	−	0.999659i	$-0.508317\pi$
$617$	−32.2398			−0.0522526			−0.0261263	+	0.999659i	$0.508317\pi$
$618$		0			0
$619$	354.274	−	204.540i	0.572332	−	0.330436i	−0.185748	−	0.982597i	$-0.559471\pi$
$619$	354.274	−	204.540i	0.572332	−	0.330436i	0.758080	+	0.652161i	$0.226137\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−273.745	−	1029.79i	−0.439399	−	1.65296i
$624$		0			0
$625$	256.623	−	444.484i	0.410597	−	0.711174i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	828.453			1.31710
$630$		0			0
$631$			428.212i			0.678625i	0.940674	+	0.339312i	$0.110194\pi$
$631$			428.212i			0.678625i	−0.940674	+	0.339312i	$0.889806\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−610.116	−	352.251i	−0.960813	−	0.554726i
$636$		0			0
$637$	2.66379	−	671.738i	0.00418177	−	1.05453i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	432.657	+	749.385i	0.674973	+	1.16909i	0.976477	+	0.215622i	$0.0691779\pi$
$641$	432.657	+	749.385i	0.674973	+	1.16909i	−0.301504	+	0.953465i	$0.597489\pi$
$642$		0			0
$643$		−	43.7834i		−	0.0680924i	−0.999420	−	0.0340462i	$-0.989161\pi$
$643$		−	43.7834i		−	0.0680924i	0.999420	−	0.0340462i	$-0.0108393\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	377.136	−	217.740i	0.582900	−	0.336538i	−0.179385	−	0.983779i	$-0.557411\pi$
$647$	377.136	−	217.740i	0.582900	−	0.336538i	0.762285	+	0.647241i	$0.224077\pi$
$648$		0			0
$649$	−0.191601	+	0.331862i	−0.000295225	+	0.000511344i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	335.432	−	580.986i	0.513679	−	0.889718i	−0.486195	−	0.873850i	$-0.661616\pi$
$653$	335.432	−	580.986i	0.513679	−	0.889718i	0.999874	−	0.0158676i	$-0.00505103\pi$
$654$		0			0
$655$	−770.081	+	444.607i	−1.17570	+	0.678789i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	18.6391i		−	0.0282839i	−0.999900	−	0.0141419i	$-0.995498\pi$
$659$		−	18.6391i		−	0.0282839i	0.999900	−	0.0141419i	$-0.00450167\pi$
$660$		0			0
$661$	129.755	+	224.742i	0.196301	+	0.340003i	0.947326	−	0.320270i	$-0.103774\pi$
$661$	129.755	+	224.742i	0.196301	+	0.340003i	−0.751025	+	0.660273i	$0.770440\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	276.161	−	1022.53i	0.415280	−	1.53765i
$666$		0			0
$667$	517.164	+	298.584i	0.775358	+	0.447653i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	5.78930i		−	0.00862787i
$672$		0			0
$673$	−5.29772			−0.00787179			−0.00393590	−	0.999992i	$-0.501253\pi$
$673$	−5.29772			−0.00787179			−0.00393590	+	0.999992i	$0.501253\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−312.667	+	541.556i	−0.461843	+	0.799935i	−0.999053	−	0.0435135i	$-0.986145\pi$
$677$	−312.667	+	541.556i	−0.461843	+	0.799935i	0.537210	+	0.843448i	$0.319478\pi$
$678$		0			0
$679$	−128.630	+	128.121i	−0.189441	+	0.188691i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−597.735	+	345.103i	−0.875162	+	0.505275i	−0.869060	−	0.494707i	$-0.835276\pi$
$683$	−597.735	+	345.103i	−0.875162	+	0.505275i	−0.00610161	+	0.999981i	$0.501942\pi$
$684$		0			0
$685$	997.019			1.45550
$686$		0			0
$687$		0			0
$688$		0			0
$689$	237.143	+	410.745i	0.344185	+	0.596146i
$690$		0			0
$691$	−909.589	−	525.151i	−1.31634	−	0.759987i	−0.333200	−	0.942856i	$-0.608128\pi$
$691$	−909.589	−	525.151i	−1.31634	−	0.759987i	−0.983137	+	0.182869i	$0.941462\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.91658	−	1.68389i	−0.00419651	−	0.00242286i
$696$		0			0
$697$	900.256	+	1559.29i	1.29162	+	2.23714i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	865.251			1.23431			0.617155	−	0.786842i	$-0.288285\pi$
$701$	865.251			1.23431			0.617155	+	0.786842i	$0.288285\pi$
$702$		0			0
$703$	1013.36	−	585.063i	1.44148	−	0.832237i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−440.795	−	119.048i	−0.623472	−	0.168384i
$708$		0			0
$709$	601.050	−	1041.05i	0.847744	−	1.46834i	−0.0354731	−	0.999371i	$-0.511294\pi$
$709$	601.050	−	1041.05i	0.847744	−	1.46834i	0.883217	−	0.468965i	$-0.155373\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−31.1968			−0.0437543
$714$		0			0
$715$			11.7317i			0.0164079i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	268.331	+	154.921i	0.373200	+	0.215467i	0.674855	−	0.737950i	$-0.264206\pi$
$719$	268.331	+	154.921i	0.373200	+	0.215467i	−0.301656	+	0.953417i	$0.597539\pi$
$720$		0			0
$721$	−938.051	+	249.358i	−1.30104	+	0.345850i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−46.5742	−	80.6688i	−0.0642402	−	0.111267i
$726$		0			0
$727$		−	753.887i		−	1.03698i	−0.855083	−	0.518492i	$-0.826494\pi$
$727$		−	753.887i		−	1.03698i	0.855083	−	0.518492i	$-0.173506\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	845.934	−	488.400i	1.15723	−	0.668126i
$732$		0			0
$733$	−194.286	+	336.512i	−0.265055	+	0.459089i	−0.967578	−	0.252572i	$-0.918724\pi$
$733$	−194.286	+	336.512i	−0.265055	+	0.459089i	0.702523	+	0.711661i	$0.252057\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.02902	−	12.1746i	0.00953734	−	0.0165191i
$738$		0			0
$739$	363.390	−	209.803i	0.491732	−	0.283902i	−0.233561	−	0.972342i	$-0.575038\pi$
$739$	363.390	−	209.803i	0.491732	−	0.283902i	0.725293	+	0.688441i	$0.241704\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	86.3362i		−	0.116199i	−0.998311	−	0.0580997i	$-0.981496\pi$
$743$		−	86.3362i		−	0.116199i	0.998311	−	0.0580997i	$-0.0185041\pi$
$744$		0			0
$745$	−342.513	−	593.250i	−0.459749	−	0.796309i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	590.459	−	156.959i	0.788329	−	0.209558i
$750$		0			0
$751$	438.068	+	252.919i	0.583314	+	0.336776i	0.762449	−	0.647048i	$-0.223997\pi$
$751$	438.068	+	252.919i	0.583314	+	0.336776i	−0.179136	+	0.983824i	$0.557330\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			190.490i			0.252305i
$756$		0			0
$757$	−240.329			−0.317475			−0.158737	−	0.987321i	$-0.550742\pi$
$757$	−240.329			−0.317475			−0.158737	+	0.987321i	$0.550742\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	250.415	−	433.732i	0.329061	−	0.569950i	−0.653265	−	0.757129i	$-0.726601\pi$
$761$	250.415	−	433.732i	0.329061	−	0.569950i	0.982326	+	0.187179i	$0.0599346\pi$
$762$		0			0
$763$	−436.657	−	117.930i	−0.572290	−	0.154561i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24.4374	−	14.1089i	0.0318610	−	0.0183949i
$768$		0			0
$769$	−713.336			−0.927615			−0.463807	−	0.885936i	$-0.653517\pi$
$769$	−713.336			−0.927615			−0.463807	+	0.885936i	$0.653517\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−105.072	−	181.990i	−0.135928	−	0.235434i	0.790024	−	0.613076i	$-0.210068\pi$
$773$	−105.072	−	181.990i	−0.135928	−	0.235434i	−0.925951	+	0.377643i	$0.876735\pi$
$774$		0			0
$775$	4.21424	+	2.43309i	0.00543773	+	0.00313947i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2202.37	+	1271.54i	2.82718	+	1.63227i
$780$		0			0
$781$	2.85949	+	4.95278i	0.00366132	+	0.00634158i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	994.934			1.26743
$786$		0			0
$787$	−1284.22	+	741.445i	−1.63179	+	0.942116i	−0.648253	+	0.761425i	$0.724500\pi$
$787$	−1284.22	+	741.445i	−1.63179	+	0.942116i	−0.983540	+	0.180691i	$0.942167\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	928.921	−	925.245i	1.17436	−	1.16972i
$792$		0			0
$793$	−213.153	+	369.193i	−0.268794	+	0.465564i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−445.840			−0.559398			−0.279699	−	0.960088i	$-0.590235\pi$
$797$	−445.840			−0.559398			−0.279699	+	0.960088i	$0.590235\pi$
$798$		0			0
$799$		−	1409.57i		−	1.76416i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−7.34351	−	4.23978i	−0.00914509	−	0.00527992i
$804$		0			0
$805$	−208.192	+	770.865i	−0.258623	+	0.957596i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−84.1885	−	145.819i	−0.104065	−	0.180246i	0.809291	−	0.587408i	$-0.199852\pi$
$809$	−84.1885	−	145.819i	−0.104065	−	0.180246i	−0.913356	+	0.407162i	$0.866518\pi$
$810$		0			0
$811$			1142.96i			1.40932i	0.709545	+	0.704660i	$0.248900\pi$
$811$			1142.96i			1.40932i	−0.709545	+	0.704660i	$0.751100\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	423.544	−	244.533i	0.519685	−	0.300041i
$816$		0			0
$817$	689.827	−	1194.82i	0.844342	−	1.46244i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	72.4494	−	125.486i	0.0882454	−	0.152845i	−0.818524	−	0.574472i	$-0.805207\pi$
$821$	72.4494	−	125.486i	0.0882454	−	0.152845i	0.906770	+	0.421627i	$0.138541\pi$
$822$		0			0
$823$	−7.54995	+	4.35897i	−0.00917370	+	0.00529644i	−0.504580	−	0.863365i	$-0.668353\pi$
$823$	−7.54995	+	4.35897i	−0.00917370	+	0.00529644i	0.495406	+	0.868661i	$0.335019\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	227.873i		−	0.275541i	−0.990464	−	0.137771i	$-0.956006\pi$
$827$		−	227.873i		−	0.275541i	0.990464	−	0.137771i	$-0.0439937\pi$
$828$		0			0
$829$	27.2629	+	47.2207i	0.0328865	+	0.0569611i	0.882000	−	0.471249i	$-0.156197\pi$
$829$	27.2629	+	47.2207i	0.0328865	+	0.0569611i	−0.849114	+	0.528210i	$0.822863\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	991.239	−	567.063i	1.18996	−	0.680748i
$834$		0			0
$835$	−1060.27	−	612.149i	−1.26979	−	0.733113i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1009.63i			1.20337i	0.798732	+	0.601687i	$0.205504\pi$
$839$			1009.63i			1.20337i	−0.798732	+	0.601687i	$0.794496\pi$
$840$		0			0
$841$	−261.918			−0.311436
$842$		0			0
$843$		0			0
$844$		0			0
$845$	43.5254	−	75.3882i	0.0515094	−	0.0892168i
$846$		0			0
$847$	217.535	+	818.338i	0.256830	+	0.966160i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−763.948	+	441.066i	−0.897706	+	0.518291i
$852$		0			0
$853$	1153.22			1.35196			0.675978	−	0.736921i	$-0.263721\pi$
$853$	1153.22			1.35196			0.675978	+	0.736921i	$0.263721\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−255.480	−	442.503i	−0.298109	−	0.516340i	0.677594	−	0.735436i	$-0.263023\pi$
$857$	−255.480	−	442.503i	−0.298109	−	0.516340i	−0.975703	+	0.219096i	$0.929689\pi$
$858$		0			0
$859$	−500.132	−	288.751i	−0.582226	−	0.336148i	0.179792	−	0.983705i	$-0.442458\pi$
$859$	−500.132	−	288.751i	−0.582226	−	0.336148i	−0.762017	+	0.647557i	$0.775791\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−645.308	−	372.569i	−0.747750	−	0.431714i	0.0771303	−	0.997021i	$-0.475424\pi$
$863$	−645.308	−	372.569i	−0.747750	−	0.431714i	−0.824880	+	0.565307i	$0.808758\pi$
$864$		0			0
$865$	290.499	+	503.159i	0.335837	+	0.581687i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.202915			−0.000233503
$870$		0			0
$871$	−896.503	+	517.596i	−1.02928	+	0.594255i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	658.177	−	655.573i	0.752203	−	0.749226i
$876$		0			0
$877$	282.403	−	489.137i	0.322011	−	0.557739i	−0.658892	−	0.752237i	$-0.728975\pi$
$877$	282.403	−	489.137i	0.322011	−	0.557739i	0.980903	+	0.194499i	$0.0623080\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	743.531			0.843963			0.421981	−	0.906605i	$-0.361335\pi$
$881$	743.531			0.843963			0.421981	+	0.906605i	$0.361335\pi$
$882$		0			0
$883$		−	715.022i		−	0.809764i	−0.914369	−	0.404882i	$-0.867313\pi$
$883$		−	715.022i		−	0.809764i	0.914369	−	0.404882i	$-0.132687\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1278.48	−	738.131i	−1.44135	−	0.832166i	−0.443414	−	0.896317i	$-0.646233\pi$
$887$	−1278.48	−	738.131i	−1.44135	−	0.832166i	−0.997940	+	0.0641509i	$0.979566\pi$
$888$		0			0
$889$	−757.113	−	760.122i	−0.851646	−	0.855030i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−995.452	−	1724.17i	−1.11473	−	1.93076i
$894$		0			0
$895$			736.001i			0.822347i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−26.1990	+	15.1260i	−0.0291424	+	0.0168253i
$900$		0			0
$901$	−403.149	+	698.275i	−0.447447	+	0.775000i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−249.849	+	432.751i	−0.276076	+	0.478178i
$906$		0			0
$907$	1.66164	−	0.959350i	0.00183202	−	0.00105772i	−0.499084	−	0.866554i	$-0.666330\pi$
$907$	1.66164	−	0.959350i	0.00183202	−	0.00105772i	0.500916	+	0.865496i	$0.332997\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1568.49i			1.72172i	0.508840	+	0.860861i	$0.330074\pi$
$911$			1568.49i			1.72172i	−0.508840	+	0.860861i	$0.669926\pi$
$912$		0			0
$913$	−3.67089	−	6.35817i	−0.00402069	−	0.00696404i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1308.69	+	347.882i	−1.42714	+	0.379370i
$918$		0			0
$919$	366.523	+	211.612i	0.398828	+	0.230264i	0.685978	−	0.727622i	$-0.259374\pi$
$919$	366.523	+	211.612i	0.398828	+	0.230264i	−0.287150	+	0.957886i	$0.592708\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	421.128i		−	0.456260i
$924$		0			0
$925$	137.598			0.148754
$926$		0			0
$927$		0			0
$928$		0			0
$929$	141.536	−	245.148i	0.152353	−	0.263884i	−0.779739	−	0.626105i	$-0.784648\pi$
$929$	141.536	−	245.148i	0.152353	−	0.263884i	0.932092	+	0.362221i	$0.117982\pi$
$930$		0			0
$931$	812.011	−	1393.65i	0.872192	−	1.49694i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17.2721	+	9.97204i	−0.0184728	+	0.0106653i
$936$		0			0
$937$	−480.799			−0.513126			−0.256563	−	0.966528i	$-0.582590\pi$
$937$	−480.799			−0.513126			−0.256563	+	0.966528i	$0.582590\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	240.048	+	415.776i	0.255099	+	0.441844i	0.964922	−	0.262535i	$-0.0845586\pi$
$941$	240.048	+	415.776i	0.255099	+	0.441844i	−0.709823	+	0.704380i	$0.751225\pi$
$942$		0			0
$943$	−1660.32	−	958.586i	−1.76068	−	1.01653i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−284.108	−	164.030i	−0.300008	−	0.173210i	0.342439	−	0.939540i	$-0.388747\pi$
$947$	−284.108	−	164.030i	−0.300008	−	0.173210i	−0.642446	+	0.766331i	$0.722080\pi$
$948$		0			0
$949$	312.205	+	540.754i	0.328983	+	0.569815i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	882.514			0.926038			0.463019	−	0.886348i	$-0.346766\pi$
$953$	882.514			0.926038			0.463019	+	0.886348i	$0.346766\pi$
$954$		0			0
$955$	90.3956	−	52.1899i	0.0946551	−	0.0546491i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1465.79	+	395.875i	1.52846	+	0.412799i
$960$		0			0
$961$	−479.710	+	830.882i	−0.499178	+	0.864601i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1354.66			−1.40379
$966$		0			0
$967$			1428.30i			1.47704i	0.674231	+	0.738520i	$0.264475\pi$
$967$			1428.30i			1.47704i	−0.674231	+	0.738520i	$0.735525\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	959.579	+	554.013i	0.988238	+	0.570560i	0.904747	−	0.425949i	$-0.140060\pi$
$971$	959.579	+	554.013i	0.988238	+	0.570560i	0.0834910	+	0.996509i	$0.473393\pi$
$972$		0			0
$973$	−3.61928	−	3.63366i	−0.00371971	−	0.00373449i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−181.175	−	313.804i	−0.185440	−	0.321191i	0.758285	−	0.651923i	$-0.226038\pi$
$977$	−181.175	−	313.804i	−0.185440	−	0.321191i	−0.943725	+	0.330732i	$0.892704\pi$
$978$		0			0
$979$			28.3393i			0.0289472i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1139.99	+	658.175i	−1.15971	+	0.669558i	−0.951234	−	0.308470i	$-0.900183\pi$
$983$	−1139.99	+	658.175i	−1.15971	+	0.669558i	−0.208474	+	0.978028i	$0.566850\pi$
$984$		0			0
$985$	−682.546	+	1182.21i	−0.692941	+	1.20021i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−520.045	+	900.744i	−0.525829	+	0.910763i
$990$		0			0
$991$	1231.66	−	711.100i	1.24285	−	0.717558i	0.273174	−	0.961965i	$-0.411927\pi$
$991$	1231.66	−	711.100i	1.24285	−	0.717558i	0.969673	+	0.244407i	$0.0785933\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1184.38i		−	1.19033i
$996$		0			0
$997$	−660.969	−	1144.83i	−0.662958	−	1.14828i	−0.979835	−	0.199811i	$-0.935967\pi$
$997$	−660.969	−	1144.83i	−0.662958	−	1.14828i	0.316876	−	0.948467i	$-0.397366\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.cd.i.415.3		6
3.2	odd	2		336.3.be.f.79.1	yes	6
4.3	odd	2		1008.3.cd.h.415.3		6
7.4	even	3		1008.3.cd.h.991.3		6
12.11	even	2		336.3.be.d.79.1	✓	6
21.2	odd	6		2352.3.m.n.1471.6		6
21.5	even	6		2352.3.m.o.1471.1		6
21.11	odd	6		336.3.be.d.319.1	yes	6
28.11	odd	6	inner	1008.3.cd.i.991.3		6
84.11	even	6		336.3.be.f.319.1	yes	6
84.23	even	6		2352.3.m.n.1471.3		6
84.47	odd	6		2352.3.m.o.1471.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.be.d.79.1	✓	6	12.11	even	2
336.3.be.d.319.1	yes	6	21.11	odd	6
336.3.be.f.79.1	yes	6	3.2	odd	2
336.3.be.f.319.1	yes	6	84.11	even	6
1008.3.cd.h.415.3		6	4.3	odd	2
1008.3.cd.h.991.3		6	7.4	even	3
1008.3.cd.i.415.3		6	1.1	even	1	trivial
1008.3.cd.i.991.3		6	28.11	odd	6	inner
2352.3.m.n.1471.3		6	84.23	even	6
2352.3.m.n.1471.6		6	21.2	odd	6
2352.3.m.o.1471.1		6	21.5	even	6
2352.3.m.o.1471.4		6	84.47	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cd (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.29832 - 3.98081i) q^{5} +(4.95955 - 4.93992i) q^{7} +(-0.161228 + 0.0930852i) q^{11} +13.7090 q^{13} +(11.6528 + 20.1833i) q^{17} +(28.5074 + 16.4587i) q^{19} +(-21.4911 - 12.4079i) q^{23} +(1.93542 + 3.35224i) q^{25} -24.0641 q^{29} +(1.08871 - 0.628570i) q^{31} +(-8.26626 - 31.0966i) q^{35} +(17.7736 - 30.7848i) q^{37} +77.2563 q^{41} -41.9125i q^{43} +(-52.3787 - 30.2408i) q^{47} +(0.194309 - 48.9996i) q^{49} +(17.2983 + 29.9616i) q^{53} +0.855760i q^{55} +(1.78257 - 1.02917i) q^{59} +(-15.5484 + 26.9306i) q^{61} +(31.5078 - 54.5731i) q^{65} +(-65.3950 + 37.7558i) q^{67} -30.7190i q^{71} +(22.7736 + 39.4451i) q^{73} +(-0.339786 + 1.25812i) q^{77} +(0.943915 + 0.544970i) q^{79} +39.4358i q^{83} +107.128 q^{85} +(76.1113 - 131.829i) q^{89} +(67.9907 - 67.7216i) q^{91} +(131.038 - 75.6550i) q^{95} -25.9359 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{5} + 11 q^{7} + 3 q^{11} - 44 q^{13} - 8 q^{17} + 30 q^{19} + 24 q^{23} - 14 q^{25} - 34 q^{29} + 39 q^{31} - 90 q^{35} + 6 q^{37} + 136 q^{41} - 258 q^{47} + 157 q^{49} + 89 q^{53} + 63 q^{59}+ \cdots - 266 q^{97}+O(q^{100}) \) 6 * q - q^5 + 11 * q^7 + 3 * q^11 - 44 * q^13 - 8 * q^17 + 30 * q^19 + 24 * q^23 - 14 * q^25 - 34 * q^29 + 39 * q^31 - 90 * q^35 + 6 * q^37 + 136 * q^41 - 258 * q^47 + 157 * q^49 + 89 * q^53 + 63 * q^59 - 50 * q^61 + 184 * q^65 - 348 * q^67 + 36 * q^73 - 221 * q^77 - 3 * q^79 + 536 * q^85 + 202 * q^89 + 62 * q^91 + 432 * q^95 - 266 * q^97

Introduction

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