LMFDB - Embedded newform 1296.3.q.i.593.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,3,Mod(593,1296)] 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("1296.593"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,10,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.2
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1296.593
Dual form			1296.3.q.i.1025.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.34847 - 4.24264i) q^{5} +(2.50000 - 4.33013i) q^{7} +(-7.34847 - 4.24264i) q^{11} +(0.500000 + 0.866025i) q^{13} -25.4558i q^{17} -29.0000 q^{19} +(-7.34847 + 4.24264i) q^{23} +(23.5000 - 40.7032i) q^{25} +(-14.6969 - 8.48528i) q^{29} +(-5.00000 - 8.66025i) q^{31} -42.4264i q^{35} -25.0000 q^{37} +(-14.6969 + 8.48528i) q^{41} +(7.00000 - 12.1244i) q^{43} +(-7.34847 - 4.24264i) q^{47} +(12.0000 + 20.7846i) q^{49} +50.9117i q^{53} -72.0000 q^{55} +(80.8332 - 46.6690i) q^{59} +(-11.5000 + 19.9186i) q^{61} +(7.34847 + 4.24264i) q^{65} +(-9.50000 - 16.4545i) q^{67} +101.823i q^{71} -97.0000 q^{73} +(-36.7423 + 21.2132i) q^{77} +(38.5000 - 66.6840i) q^{79} +(102.879 + 59.3970i) q^{83} +(-108.000 - 187.061i) q^{85} +76.3675i q^{89} +5.00000 q^{91} +(-213.106 + 123.037i) q^{95} +(24.5000 - 42.4352i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{7} + 2 q^{13} - 116 q^{19} + 94 q^{25} - 20 q^{31} - 100 q^{37} + 28 q^{43} + 48 q^{49} - 288 q^{55} - 46 q^{61} - 38 q^{67} - 388 q^{73} + 154 q^{79} - 432 q^{85} + 20 q^{91} + 98 q^{97}+O(q^{100}) $ 4 * q + 10 * q^7 + 2 * q^13 - 116 * q^19 + 94 * q^25 - 20 * q^31 - 100 * q^37 + 28 * q^43 + 48 * q^49 - 288 * q^55 - 46 * q^61 - 38 * q^67 - 388 * q^73 + 154 * q^79 - 432 * q^85 + 20 * q^91 + 98 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	7.34847	−	4.24264i	1.46969	−	0.848528i	0.470272	−	0.882522i	$-0.344156\pi$
$5$	7.34847	−	4.24264i	1.46969	−	0.848528i	0.999422	+	0.0339935i	$0.0108226\pi$
$6$		0			0
$7$	2.50000	−	4.33013i	0.357143	−	0.618590i	−0.630339	−	0.776320i	$-0.717084\pi$
$7$	2.50000	−	4.33013i	0.357143	−	0.618590i	0.987482	+	0.157730i	$0.0504176\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.34847	−	4.24264i	−0.668043	−	0.385695i	0.127292	−	0.991865i	$-0.459371\pi$
$11$	−7.34847	−	4.24264i	−0.668043	−	0.385695i	−0.795335	+	0.606171i	$0.792705\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.0384615	+	0.0666173i	0.884615	−	0.466321i	$-0.154421\pi$
$13$	0.500000	+	0.866025i	0.0384615	+	0.0666173i	−0.846154	+	0.532939i	$0.821088\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	25.4558i		−	1.49740i	−0.662908	−	0.748701i	$-0.730678\pi$
$17$		−	25.4558i		−	1.49740i	0.662908	−	0.748701i	$-0.269322\pi$
$18$		0			0
$19$	−29.0000			−1.52632			−0.763158	−	0.646212i	$-0.776352\pi$
$19$	−29.0000			−1.52632			−0.763158	+	0.646212i	$0.776352\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.34847	+	4.24264i	−0.319499	+	0.184463i	−0.651169	−	0.758933i	$-0.725721\pi$
$23$	−7.34847	+	4.24264i	−0.319499	+	0.184463i	0.331670	+	0.943395i	$0.392388\pi$
$24$		0			0
$25$	23.5000	−	40.7032i	0.940000	−	1.62813i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−14.6969	−	8.48528i	−0.506791	−	0.292596i	0.224723	−	0.974423i	$-0.427852\pi$
$29$	−14.6969	−	8.48528i	−0.506791	−	0.292596i	−0.731514	+	0.681827i	$0.761186\pi$
$30$		0			0
$31$	−5.00000	−	8.66025i	−0.161290	−	0.279363i	0.774041	−	0.633135i	$-0.218232\pi$
$31$	−5.00000	−	8.66025i	−0.161290	−	0.279363i	−0.935332	+	0.353772i	$0.884899\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	42.4264i		−	1.21218i
$36$		0			0
$37$	−25.0000			−0.675676			−0.337838	−	0.941204i	$-0.609696\pi$
$37$	−25.0000			−0.675676			−0.337838	+	0.941204i	$0.609696\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−14.6969	+	8.48528i	−0.358462	+	0.206958i	−0.668406	−	0.743797i	$-0.733023\pi$
$41$	−14.6969	+	8.48528i	−0.358462	+	0.206958i	0.309944	+	0.950755i	$0.399690\pi$
$42$		0			0
$43$	7.00000	−	12.1244i	0.162791	−	0.281962i	−0.773078	−	0.634311i	$-0.781284\pi$
$43$	7.00000	−	12.1244i	0.162791	−	0.281962i	0.935869	+	0.352349i	$0.114617\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.34847	−	4.24264i	−0.156350	−	0.0902690i	0.419784	−	0.907624i	$-0.362106\pi$
$47$	−7.34847	−	4.24264i	−0.156350	−	0.0902690i	−0.576134	+	0.817355i	$0.695439\pi$
$48$		0			0
$49$	12.0000	+	20.7846i	0.244898	+	0.424176i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			50.9117i			0.960598i	0.877105	+	0.480299i	$0.159472\pi$
$53$			50.9117i			0.960598i	−0.877105	+	0.480299i	$0.840528\pi$
$54$		0			0
$55$	−72.0000			−1.30909
$56$		0			0
$57$		0			0
$58$		0			0
$59$	80.8332	−	46.6690i	1.37005	−	0.791001i	0.379119	−	0.925348i	$-0.376227\pi$
$59$	80.8332	−	46.6690i	1.37005	−	0.791001i	0.990934	+	0.134347i	$0.0428937\pi$
$60$		0			0
$61$	−11.5000	+	19.9186i	−0.188525	+	0.326534i	−0.944759	−	0.327767i	$-0.893704\pi$
$61$	−11.5000	+	19.9186i	−0.188525	+	0.326534i	0.756234	+	0.654301i	$0.227037\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.34847	+	4.24264i	0.113053	+	0.0652714i
$66$		0			0
$67$	−9.50000	−	16.4545i	−0.141791	−	0.245589i	0.786380	−	0.617743i	$-0.211953\pi$
$67$	−9.50000	−	16.4545i	−0.141791	−	0.245589i	−0.928171	+	0.372154i	$0.878619\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			101.823i			1.43413i	0.697005	+	0.717066i	$0.254515\pi$
$71$			101.823i			1.43413i	−0.697005	+	0.717066i	$0.745485\pi$
$72$		0			0
$73$	−97.0000			−1.32877			−0.664384	−	0.747392i	$-0.731306\pi$
$73$	−97.0000			−1.32877			−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−36.7423	+	21.2132i	−0.477173	+	0.275496i
$78$		0			0
$79$	38.5000	−	66.6840i	0.487342	−	0.844101i	−0.512552	−	0.858656i	$-0.671300\pi$
$79$	38.5000	−	66.6840i	0.487342	−	0.844101i	0.999894	+	0.0145553i	$0.00463326\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	102.879	+	59.3970i	1.23950	+	0.715626i	0.968992	−	0.247091i	$-0.0794748\pi$
$83$	102.879	+	59.3970i	1.23950	+	0.715626i	0.270509	+	0.962718i	$0.412808\pi$
$84$		0			0
$85$	−108.000	−	187.061i	−1.27059	−	2.20072i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			76.3675i			0.858062i	0.903290	+	0.429031i	$0.141145\pi$
$89$			76.3675i			0.858062i	−0.903290	+	0.429031i	$0.858855\pi$
$90$		0			0
$91$	5.00000			0.0549451
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−213.106	+	123.037i	−2.24322	+	1.29512i
$96$		0			0
$97$	24.5000	−	42.4352i	0.252577	−	0.437477i	−0.711657	−	0.702527i	$-0.752055\pi$
$97$	24.5000	−	42.4352i	0.252577	−	0.437477i	0.964235	+	0.265050i	$0.0853885\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	117.576	+	67.8823i	1.16411	+	0.672101i	0.952286	−	0.305206i	$-0.0987252\pi$
$101$	117.576	+	67.8823i	1.16411	+	0.672101i	0.211827	+	0.977307i	$0.432059\pi$
$102$		0			0
$103$	−81.5000	−	141.162i	−0.791262	−	1.37051i	−0.925186	−	0.379515i	$-0.876091\pi$
$103$	−81.5000	−	141.162i	−0.791262	−	1.37051i	0.133924	−	0.990992i	$-0.457242\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			25.4558i			0.237905i	0.992900	+	0.118953i	$0.0379536\pi$
$107$			25.4558i			0.237905i	−0.992900	+	0.118953i	$0.962046\pi$
$108$		0			0
$109$	2.00000			0.0183486			0.00917431	−	0.999958i	$-0.497080\pi$
$109$	2.00000			0.0183486			0.00917431	+	0.999958i	$0.497080\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	95.5301	−	55.1543i	0.845399	−	0.488091i	−0.0136967	−	0.999906i	$-0.504360\pi$
$113$	95.5301	−	55.1543i	0.845399	−	0.488091i	0.859096	+	0.511815i	$0.171027\pi$
$114$		0			0
$115$	−36.0000	+	62.3538i	−0.313043	+	0.542207i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−110.227	−	63.6396i	−0.926278	−	0.534787i
$120$		0			0
$121$	−24.5000	−	42.4352i	−0.202479	−	0.350705i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	186.676i		−	1.49341i
$126$		0			0
$127$	178.000			1.40157			0.700787	−	0.713370i	$-0.252832\pi$
$127$	178.000			1.40157			0.700787	+	0.713370i	$0.252832\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	102.879	−	59.3970i	0.785333	−	0.453412i	−0.0529842	−	0.998595i	$-0.516873\pi$
$131$	102.879	−	59.3970i	0.785333	−	0.453412i	0.838317	+	0.545183i	$0.183540\pi$
$132$		0			0
$133$	−72.5000	+	125.574i	−0.545113	+	0.944163i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−80.8332	−	46.6690i	−0.590023	−	0.340650i	0.175084	−	0.984554i	$-0.443980\pi$
$137$	−80.8332	−	46.6690i	−0.590023	−	0.340650i	−0.765107	+	0.643904i	$0.777314\pi$
$138$		0			0
$139$	−81.5000	−	141.162i	−0.586331	−	1.01555i	−0.994708	−	0.102742i	$-0.967238\pi$
$139$	−81.5000	−	141.162i	−0.586331	−	1.01555i	0.408377	−	0.912813i	$-0.366095\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	8.48528i		−	0.0593376i
$144$		0			0
$145$	−144.000			−0.993103
$146$		0			0
$147$		0			0
$148$		0			0
$149$	29.3939	−	16.9706i	0.197274	−	0.113896i	−0.398109	−	0.917338i	$-0.630333\pi$
$149$	29.3939	−	16.9706i	0.197274	−	0.113896i	0.595383	+	0.803442i	$0.297000\pi$
$150$		0			0
$151$	74.5000	−	129.038i	0.493377	−	0.854555i	−0.506593	−	0.862185i	$-0.669095\pi$
$151$	74.5000	−	129.038i	0.493377	−	0.854555i	0.999971	+	0.00763022i	$0.00242880\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−73.4847	−	42.4264i	−0.474095	−	0.273719i
$156$		0			0
$157$	−121.000	−	209.578i	−0.770701	−	1.33489i	−0.937179	−	0.348848i	$-0.886573\pi$
$157$	−121.000	−	209.578i	−0.770701	−	1.33489i	0.166479	−	0.986045i	$-0.446760\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			42.4264i			0.263518i
$162$		0			0
$163$	−173.000			−1.06135			−0.530675	−	0.847575i	$-0.678061\pi$
$163$	−173.000			−1.06135			−0.530675	+	0.847575i	$0.678061\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	169.015	−	97.5807i	1.01206	−	0.584316i	0.100269	−	0.994960i	$-0.468030\pi$
$167$	169.015	−	97.5807i	1.01206	−	0.584316i	0.911796	+	0.410645i	$0.134696\pi$
$168$		0			0
$169$	84.0000	−	145.492i	0.497041	−	0.860901i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−279.242	−	161.220i	−1.61411	−	0.931910i	−0.988403	−	0.151855i	$-0.951475\pi$
$173$	−279.242	−	161.220i	−1.61411	−	0.931910i	−0.625712	−	0.780054i	$-0.715191\pi$
$174$		0			0
$175$	−117.500	−	203.516i	−0.671429	−	1.16295i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	305.470i		−	1.70654i	−0.521472	−	0.853269i	$-0.674617\pi$
$179$		−	305.470i		−	1.70654i	0.521472	−	0.853269i	$-0.325383\pi$
$180$		0			0
$181$	263.000			1.45304			0.726519	−	0.687146i	$-0.241137\pi$
$181$	263.000			1.45304			0.726519	+	0.687146i	$0.241137\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−183.712	+	106.066i	−0.993036	+	0.573330i
$186$		0			0
$187$	−108.000	+	187.061i	−0.577540	+	1.00033i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	124.924	+	72.1249i	0.654052	+	0.377617i	0.790007	−	0.613098i	$-0.210077\pi$
$191$	124.924	+	72.1249i	0.654052	+	0.377617i	−0.135955	+	0.990715i	$0.543410\pi$
$192$		0			0
$193$	−35.5000	−	61.4878i	−0.183938	−	0.318590i	0.759280	−	0.650764i	$-0.225551\pi$
$193$	−35.5000	−	61.4878i	−0.183938	−	0.318590i	−0.943218	+	0.332174i	$0.892218\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			76.3675i			0.387652i	0.981036	+	0.193826i	$0.0620898\pi$
$197$			76.3675i			0.387652i	−0.981036	+	0.193826i	$0.937910\pi$
$198$		0			0
$199$	−173.000			−0.869347			−0.434673	−	0.900588i	$-0.643136\pi$
$199$	−173.000			−0.869347			−0.434673	+	0.900588i	$0.643136\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−73.4847	+	42.4264i	−0.361994	+	0.208997i
$204$		0			0
$205$	−72.0000	+	124.708i	−0.351220	+	0.608330i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	213.106	+	123.037i	1.01964	+	0.588692i
$210$		0			0
$211$	170.500	+	295.315i	0.808057	+	1.39960i	0.914208	+	0.405246i	$0.132814\pi$
$211$	170.500	+	295.315i	0.808057	+	1.39960i	−0.106151	+	0.994350i	$0.533853\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	118.794i		−	0.552530i
$216$		0			0
$217$	−50.0000			−0.230415
$218$		0			0
$219$		0			0
$220$		0			0
$221$	22.0454	−	12.7279i	0.0997530	−	0.0575924i
$222$		0			0
$223$	−29.0000	+	50.2295i	−0.130045	+	0.225244i	−0.923694	−	0.383132i	$-0.874845\pi$
$223$	−29.0000	+	50.2295i	−0.130045	+	0.225244i	0.793649	+	0.608376i	$0.208179\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−117.576	−	67.8823i	−0.517954	−	0.299041i	0.218143	−	0.975917i	$-0.430000\pi$
$227$	−117.576	−	67.8823i	−0.517954	−	0.299041i	−0.736097	+	0.676876i	$0.763333\pi$
$228$		0			0
$229$	131.000	+	226.899i	0.572052	+	0.990824i	0.996355	+	0.0853034i	$0.0271859\pi$
$229$	131.000	+	226.899i	0.572052	+	0.990824i	−0.424303	+	0.905520i	$0.639481\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	305.470i		−	1.31103i	−0.755182	−	0.655515i	$-0.772451\pi$
$233$		−	305.470i		−	1.31103i	0.755182	−	0.655515i	$-0.227549\pi$
$234$		0			0
$235$	−72.0000			−0.306383
$236$		0			0
$237$		0			0
$238$		0			0
$239$	146.969	−	84.8528i	0.614935	−	0.355033i	−0.159960	−	0.987124i	$-0.551136\pi$
$239$	146.969	−	84.8528i	0.614935	−	0.355033i	0.774894	+	0.632091i	$0.217803\pi$
$240$		0			0
$241$	60.5000	−	104.789i	0.251037	−	0.434809i	−0.712774	−	0.701393i	$-0.752562\pi$
$241$	60.5000	−	104.789i	0.251037	−	0.434809i	0.963812	+	0.266584i	$0.0858950\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	176.363	+	101.823i	0.719850	+	0.415606i
$246$		0			0
$247$	−14.5000	−	25.1147i	−0.0587045	−	0.101679i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			356.382i			1.41985i	0.704278	+	0.709924i	$0.251271\pi$
$251$			356.382i			1.41985i	−0.704278	+	0.709924i	$0.748729\pi$
$252$		0			0
$253$	72.0000			0.284585
$254$		0			0
$255$		0			0
$256$		0			0
$257$	73.4847	−	42.4264i	0.285933	−	0.165083i	−0.350173	−	0.936685i	$-0.613877\pi$
$257$	73.4847	−	42.4264i	0.285933	−	0.165083i	0.636106	+	0.771602i	$0.280544\pi$
$258$		0			0
$259$	−62.5000	+	108.253i	−0.241313	+	0.417966i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	146.969	+	84.8528i	0.558819	+	0.322634i	0.752671	−	0.658396i	$-0.228765\pi$
$263$	146.969	+	84.8528i	0.558819	+	0.322634i	−0.193852	+	0.981031i	$0.562098\pi$
$264$		0			0
$265$	216.000	+	374.123i	0.815094	+	1.41178i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			280.014i			1.04095i	0.853878	+	0.520473i	$0.174244\pi$
$269$			280.014i			1.04095i	−0.853878	+	0.520473i	$0.825756\pi$
$270$		0			0
$271$	−29.0000			−0.107011			−0.0535055	−	0.998568i	$-0.517039\pi$
$271$	−29.0000			−0.107011			−0.0535055	+	0.998568i	$0.517039\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−345.378	+	199.404i	−1.25592	+	0.725106i
$276$		0			0
$277$	191.000	−	330.822i	0.689531	−	1.19430i	−0.282459	−	0.959279i	$-0.591150\pi$
$277$	191.000	−	330.822i	0.689531	−	1.19430i	0.971990	−	0.235023i	$-0.0755165\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−191.060	−	110.309i	−0.679930	−	0.392558i	0.119899	−	0.992786i	$-0.461743\pi$
$281$	−191.060	−	110.309i	−0.679930	−	0.392558i	−0.799829	+	0.600229i	$0.795076\pi$
$282$		0			0
$283$	31.0000	+	53.6936i	0.109541	+	0.189730i	0.915584	−	0.402126i	$-0.131729\pi$
$283$	31.0000	+	53.6936i	0.109541	+	0.189730i	−0.806044	+	0.591856i	$0.798395\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			84.8528i			0.295654i
$288$		0			0
$289$	−359.000			−1.24221
$290$		0			0
$291$		0			0
$292$		0			0
$293$	271.893	−	156.978i	0.927964	−	0.535760i	0.0417967	−	0.999126i	$-0.486692\pi$
$293$	271.893	−	156.978i	0.927964	−	0.535760i	0.886167	+	0.463366i	$0.153358\pi$
$294$		0			0
$295$	396.000	−	685.892i	1.34237	−	2.32506i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.34847	−	4.24264i	−0.0245768	−	0.0141894i
$300$		0			0
$301$	−35.0000	−	60.6218i	−0.116279	−	0.201401i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			195.161i			0.639874i
$306$		0			0
$307$	34.0000			0.110749			0.0553746	−	0.998466i	$-0.482365\pi$
$307$	34.0000			0.110749			0.0553746	+	0.998466i	$0.482365\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−51.4393	+	29.6985i	−0.165400	+	0.0954935i	−0.580415	−	0.814321i	$-0.697110\pi$
$311$	−51.4393	+	29.6985i	−0.165400	+	0.0954935i	0.415015	+	0.909814i	$0.363776\pi$
$312$		0			0
$313$	−119.500	+	206.980i	−0.381789	+	0.661278i	−0.991318	−	0.131486i	$-0.958025\pi$
$313$	−119.500	+	206.980i	−0.381789	+	0.661278i	0.609529	+	0.792764i	$0.291359\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.6969	−	8.48528i	−0.0463626	−	0.0267674i	0.476640	−	0.879099i	$-0.341855\pi$
$317$	−14.6969	−	8.48528i	−0.0463626	−	0.0267674i	−0.523002	+	0.852331i	$0.675188\pi$
$318$		0			0
$319$	72.0000	+	124.708i	0.225705	+	0.390933i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			738.219i			2.28551i
$324$		0			0
$325$	47.0000			0.144615
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−36.7423	+	21.2132i	−0.111679	+	0.0644778i
$330$		0			0
$331$	−249.500	+	432.147i	−0.753776	+	1.30558i	0.192204	+	0.981355i	$0.438437\pi$
$331$	−249.500	+	432.147i	−0.753776	+	1.30558i	−0.945980	+	0.324224i	$0.894897\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−139.621	−	80.6102i	−0.416779	−	0.240627i
$336$		0			0
$337$	72.5000	+	125.574i	0.215134	+	0.372622i	0.953314	−	0.301981i	$-0.0976480\pi$
$337$	72.5000	+	125.574i	0.215134	+	0.372622i	−0.738180	+	0.674603i	$0.764315\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			84.8528i			0.248835i
$342$		0			0
$343$	365.000			1.06414
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−382.120	+	220.617i	−1.10121	+	0.635785i	−0.936539	−	0.350564i	$-0.885990\pi$
$347$	−382.120	+	220.617i	−1.10121	+	0.635785i	−0.164673	+	0.986348i	$0.552657\pi$
$348$		0			0
$349$	60.5000	−	104.789i	0.173352	−	0.300255i	−0.766237	−	0.642558i	$-0.777873\pi$
$349$	60.5000	−	104.789i	0.173352	−	0.300255i	0.939590	+	0.342302i	$0.111207\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	382.120	+	220.617i	1.08249	+	0.624978i	0.931568	−	0.363567i	$-0.118441\pi$
$353$	382.120	+	220.617i	1.08249	+	0.624978i	0.150926	+	0.988545i	$0.451775\pi$
$354$		0			0
$355$	432.000	+	748.246i	1.21690	+	2.10774i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	432.749i		−	1.20543i	−0.797957	−	0.602715i	$-0.794086\pi$
$359$		−	432.749i		−	1.20543i	0.797957	−	0.602715i	$-0.205914\pi$
$360$		0			0
$361$	480.000			1.32964
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−712.802	+	411.536i	−1.95288	+	1.12750i
$366$		0			0
$367$	74.5000	−	129.038i	0.202997	−	0.351602i	−0.746496	−	0.665390i	$-0.768265\pi$
$367$	74.5000	−	129.038i	0.202997	−	0.351602i	0.949493	+	0.313789i	$0.101598\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	220.454	+	127.279i	0.594216	+	0.343071i
$372$		0			0
$373$	180.500	+	312.635i	0.483914	+	0.838164i	0.999829	−	0.0184757i	$-0.00588134\pi$
$373$	180.500	+	312.635i	0.483914	+	0.838164i	−0.515915	+	0.856640i	$0.672548\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	16.9706i		−	0.0450148i
$378$		0			0
$379$	−173.000			−0.456464			−0.228232	−	0.973607i	$-0.573295\pi$
$379$	−173.000			−0.456464			−0.228232	+	0.973607i	$0.573295\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	587.878	−	339.411i	1.53493	−	0.886191i	0.535804	−	0.844342i	$-0.320008\pi$
$383$	587.878	−	339.411i	1.53493	−	0.886191i	0.999124	−	0.0418491i	$-0.0133249\pi$
$384$		0			0
$385$	−180.000	+	311.769i	−0.467532	+	0.809790i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	360.075	+	207.889i	0.925643	+	0.534420i	0.885431	−	0.464771i	$-0.153863\pi$
$389$	360.075	+	207.889i	0.925643	+	0.534420i	0.0402118	+	0.999191i	$0.487197\pi$
$390$		0			0
$391$	108.000	+	187.061i	0.276215	+	0.478418i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	653.367i		−	1.65409i
$396$		0			0
$397$	−286.000			−0.720403			−0.360202	−	0.932875i	$-0.617292\pi$
$397$	−286.000			−0.720403			−0.360202	+	0.932875i	$0.617292\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	293.939	−	169.706i	0.733014	−	0.423206i	−0.0865095	−	0.996251i	$-0.527571\pi$
$401$	293.939	−	169.706i	0.733014	−	0.423206i	0.819524	+	0.573045i	$0.194238\pi$
$402$		0			0
$403$	5.00000	−	8.66025i	0.0124069	−	0.0214895i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	183.712	+	106.066i	0.451380	+	0.260604i
$408$		0			0
$409$	−107.500	−	186.195i	−0.262836	−	0.455246i	0.704158	−	0.710043i	$-0.251325\pi$
$409$	−107.500	−	186.195i	−0.262836	−	0.455246i	−0.966994	+	0.254797i	$0.917991\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	466.690i		−	1.13000i
$414$		0			0
$415$	1008.00			2.42892
$416$		0			0
$417$		0			0
$418$		0			0
$419$	521.741	−	301.227i	1.24521	−	0.718920i	0.275057	−	0.961428i	$-0.411303\pi$
$419$	521.741	−	301.227i	1.24521	−	0.718920i	0.970149	+	0.242508i	$0.0779701\pi$
$420$		0			0
$421$	−227.500	+	394.042i	−0.540380	+	0.935966i	0.458502	+	0.888693i	$0.348386\pi$
$421$	−227.500	+	394.042i	−0.540380	+	0.935966i	−0.998882	+	0.0472723i	$0.984947\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1036.13	−	598.212i	−2.43796	−	1.40756i
$426$		0			0
$427$	57.5000	+	99.5929i	0.134660	+	0.233239i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	330.926i		−	0.767810i	−0.923373	−	0.383905i	$-0.874579\pi$
$431$		−	330.926i		−	0.767810i	0.923373	−	0.383905i	$-0.125421\pi$
$432$		0			0
$433$	218.000			0.503464			0.251732	−	0.967797i	$-0.419000\pi$
$433$	218.000			0.503464			0.251732	+	0.967797i	$0.419000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	213.106	−	123.037i	0.487656	−	0.281548i
$438$		0			0
$439$	187.000	−	323.894i	0.425968	−	0.737798i	−0.570542	−	0.821268i	$-0.693267\pi$
$439$	187.000	−	323.894i	0.425968	−	0.737798i	0.996510	+	0.0834699i	$0.0266002\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.6969	+	8.48528i	0.0331759	+	0.0191541i	0.516496	−	0.856289i	$-0.327236\pi$
$443$	14.6969	+	8.48528i	0.0331759	+	0.0191541i	−0.483320	+	0.875444i	$0.660569\pi$
$444$		0			0
$445$	324.000	+	561.184i	0.728090	+	1.26109i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			483.661i			1.07720i	0.842563	+	0.538598i	$0.181046\pi$
$449$			483.661i			1.07720i	−0.842563	+	0.538598i	$0.818954\pi$
$450$		0			0
$451$	144.000			0.319290
$452$		0			0
$453$		0			0
$454$		0			0
$455$	36.7423	−	21.2132i	0.0807524	−	0.0466224i
$456$		0			0
$457$	11.0000	−	19.0526i	0.0240700	−	0.0416905i	−0.853739	−	0.520700i	$-0.825671\pi$
$457$	11.0000	−	19.0526i	0.0240700	−	0.0416905i	0.877810	+	0.479010i	$0.159004\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−345.378	−	199.404i	−0.749193	−	0.432547i	0.0762091	−	0.997092i	$-0.475718\pi$
$461$	−345.378	−	199.404i	−0.749193	−	0.432547i	−0.825402	+	0.564545i	$0.809052\pi$
$462$		0			0
$463$	−441.500	−	764.700i	−0.953564	−	1.65162i	−0.737621	−	0.675215i	$-0.764051\pi$
$463$	−441.500	−	764.700i	−0.953564	−	1.65162i	−0.215942	−	0.976406i	$-0.569282\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			127.279i			0.272547i	0.990671	+	0.136273i	$0.0435125\pi$
$467$			127.279i			0.272547i	−0.990671	+	0.136273i	$0.956487\pi$
$468$		0			0
$469$	−95.0000			−0.202559
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−102.879	+	59.3970i	−0.217502	+	0.125575i
$474$		0			0
$475$	−681.500	+	1180.39i	−1.43474	+	2.48504i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−338.030	−	195.161i	−0.705699	−	0.407435i	0.103768	−	0.994602i	$-0.466910\pi$
$479$	−338.030	−	195.161i	−0.705699	−	0.407435i	−0.809466	+	0.587166i	$0.800243\pi$
$480$		0			0
$481$	−12.5000	−	21.6506i	−0.0259875	−	0.0450117i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	415.779i		−	0.857276i
$486$		0			0
$487$	−317.000			−0.650924			−0.325462	−	0.945555i	$-0.605520\pi$
$487$	−317.000			−0.650924			−0.325462	+	0.945555i	$0.605520\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−315.984	+	182.434i	−0.643552	+	0.371555i	−0.785982	−	0.618250i	$-0.787842\pi$
$491$	−315.984	+	182.434i	−0.643552	+	0.371555i	0.142429	+	0.989805i	$0.454509\pi$
$492$		0			0
$493$	−216.000	+	374.123i	−0.438134	+	0.758870i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	440.908	+	254.558i	0.887139	+	0.512190i
$498$		0			0
$499$	355.000	+	614.878i	0.711423	+	1.23222i	0.964323	+	0.264728i	$0.0852822\pi$
$499$	355.000	+	614.878i	0.711423	+	1.23222i	−0.252900	+	0.967492i	$0.581384\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			280.014i			0.556688i	0.960481	+	0.278344i	$0.0897856\pi$
$503$			280.014i			0.556688i	−0.960481	+	0.278344i	$0.910214\pi$
$504$		0			0
$505$	1152.00			2.28119
$506$		0			0
$507$		0			0
$508$		0			0
$509$	227.803	−	131.522i	0.447549	−	0.258393i	−0.259245	−	0.965811i	$-0.583474\pi$
$509$	227.803	−	131.522i	0.447549	−	0.258393i	0.706795	+	0.707419i	$0.250141\pi$
$510$		0			0
$511$	−242.500	+	420.022i	−0.474560	+	0.821961i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1197.80	−	691.550i	−2.32583	−	1.34282i
$516$		0			0
$517$	36.0000	+	62.3538i	0.0696325	+	0.120607i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	687.308i		−	1.31921i	−0.751613	−	0.659604i	$-0.770724\pi$
$521$		−	687.308i		−	1.31921i	0.751613	−	0.659604i	$-0.229276\pi$
$522$		0			0
$523$	763.000			1.45889			0.729446	−	0.684039i	$-0.239778\pi$
$523$	763.000			1.45889			0.729446	+	0.684039i	$0.239778\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−220.454	+	127.279i	−0.418319	+	0.241517i
$528$		0			0
$529$	−228.500	+	395.774i	−0.431947	+	0.748154i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−14.6969	−	8.48528i	−0.0275740	−	0.0159199i
$534$		0			0
$535$	108.000	+	187.061i	0.201869	+	0.349648i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	203.647i		−	0.377823i
$540$		0			0
$541$	−313.000			−0.578558			−0.289279	−	0.957245i	$-0.593416\pi$
$541$	−313.000			−0.578558			−0.289279	+	0.957245i	$0.593416\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.6969	−	8.48528i	0.0269669	−	0.0155693i
$546$		0			0
$547$	−69.5000	+	120.378i	−0.127057	+	0.220069i	−0.922535	−	0.385913i	$-0.873886\pi$
$547$	−69.5000	+	120.378i	−0.127057	+	0.220069i	0.795478	+	0.605982i	$0.207220\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	426.211	+	246.073i	0.773523	+	0.446594i
$552$		0			0
$553$	−192.500	−	333.420i	−0.348101	−	0.602929i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			280.014i			0.502719i	0.967894	+	0.251359i	$0.0808776\pi$
$557$			280.014i			0.502719i	−0.967894	+	0.251359i	$0.919122\pi$
$558$		0			0
$559$	14.0000			0.0250447
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−514.393	+	296.985i	−0.913664	+	0.527504i	−0.881608	−	0.471982i	$-0.843539\pi$
$563$	−514.393	+	296.985i	−0.913664	+	0.527504i	−0.0320558	+	0.999486i	$0.510205\pi$
$564$		0			0
$565$	468.000	−	810.600i	0.828319	−	1.43469i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	536.438	+	309.713i	0.942774	+	0.544311i	0.890829	−	0.454339i	$-0.150125\pi$
$569$	536.438	+	309.713i	0.942774	+	0.544311i	0.0519450	+	0.998650i	$0.483458\pi$
$570$		0			0
$571$	−81.5000	−	141.162i	−0.142732	−	0.247219i	0.785792	−	0.618490i	$-0.212255\pi$
$571$	−81.5000	−	141.162i	−0.142732	−	0.247219i	−0.928525	+	0.371271i	$0.878922\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			398.808i			0.693580i
$576$		0			0
$577$	1127.00			1.95321			0.976603	−	0.215050i	$-0.0689913\pi$
$577$	1127.00			1.95321			0.976603	+	0.215050i	$0.0689913\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	514.393	−	296.985i	0.885358	−	0.511162i
$582$		0			0
$583$	216.000	−	374.123i	0.370497	−	0.641720i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	874.468	+	504.874i	1.48972	+	0.860092i	0.999931	−	0.0117465i	$-0.00373911\pi$
$587$	874.468	+	504.874i	1.48972	+	0.860092i	0.489793	+	0.871839i	$0.337072\pi$
$588$		0			0
$589$	145.000	+	251.147i	0.246180	+	0.426396i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	356.382i		−	0.600981i	−0.953785	−	0.300491i	$-0.902850\pi$
$593$		−	356.382i		−	0.600981i	0.953785	−	0.300491i	$-0.0971504\pi$
$594$		0			0
$595$	−1080.00			−1.81513
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−778.938	+	449.720i	−1.30040	+	0.750784i	−0.980472	−	0.196658i	$-0.936991\pi$
$599$	−778.938	+	449.720i	−1.30040	+	0.750784i	−0.319925	+	0.947443i	$0.603658\pi$
$600$		0			0
$601$	83.0000	−	143.760i	0.138103	−	0.239202i	−0.788675	−	0.614810i	$-0.789233\pi$
$601$	83.0000	−	143.760i	0.138103	−	0.239202i	0.926779	+	0.375608i	$0.122566\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−360.075	−	207.889i	−0.595165	−	0.343619i
$606$		0			0
$607$	−261.500	−	452.931i	−0.430807	−	0.746180i	0.566136	−	0.824312i	$-0.308438\pi$
$607$	−261.500	−	452.931i	−0.430807	−	0.746180i	−0.996943	+	0.0781320i	$0.975104\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	8.48528i		−	0.0138875i
$612$		0			0
$613$	335.000			0.546493			0.273246	−	0.961944i	$-0.411903\pi$
$613$	335.000			0.546493			0.273246	+	0.961944i	$0.411903\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−433.560	+	250.316i	−0.702690	+	0.405698i	−0.808349	−	0.588704i	$-0.799638\pi$
$617$	−433.560	+	250.316i	−0.702690	+	0.405698i	0.105659	+	0.994402i	$0.466305\pi$
$618$		0			0
$619$	2.50000	−	4.33013i	0.00403877	−	0.00699536i	−0.863999	−	0.503494i	$-0.832048\pi$
$619$	2.50000	−	4.33013i	0.00403877	−	0.00699536i	0.868038	+	0.496498i	$0.165381\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	330.681	+	190.919i	0.530788	+	0.306451i
$624$		0			0
$625$	−204.500	−	354.204i	−0.327200	−	0.566727i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			636.396i			1.01176i
$630$		0			0
$631$	−245.000			−0.388273			−0.194136	−	0.980975i	$-0.562190\pi$
$631$	−245.000			−0.388273			−0.194136	+	0.980975i	$0.562190\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1308.03	−	755.190i	2.05989	−	1.18928i
$636$		0			0
$637$	−12.0000	+	20.7846i	−0.0188383	+	0.0326289i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	558.484	+	322.441i	0.871269	+	0.503028i	0.867770	−	0.496966i	$-0.165553\pi$
$641$	558.484	+	322.441i	0.871269	+	0.503028i	0.00349951	+	0.999994i	$0.498886\pi$
$642$		0			0
$643$	−41.0000	−	71.0141i	−0.0637636	−	0.110442i	0.832381	−	0.554203i	$-0.186977\pi$
$643$	−41.0000	−	71.0141i	−0.0637636	−	0.110442i	−0.896145	+	0.443762i	$0.853644\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			458.205i			0.708200i	0.935208	+	0.354100i	$0.115213\pi$
$647$			458.205i			0.708200i	−0.935208	+	0.354100i	$0.884787\pi$
$648$		0			0
$649$	−792.000			−1.22034
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−279.242	+	161.220i	−0.427629	+	0.246892i	−0.698336	−	0.715770i	$-0.746076\pi$
$653$	−279.242	+	161.220i	−0.427629	+	0.246892i	0.270707	+	0.962662i	$0.412743\pi$
$654$		0			0
$655$	504.000	−	872.954i	0.769466	−	1.33275i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	896.513	+	517.602i	1.36041	+	0.785436i	0.989679	−	0.143303i	$-0.0457722\pi$
$659$	896.513	+	517.602i	1.36041	+	0.785436i	0.370736	+	0.928738i	$0.379106\pi$
$660$		0			0
$661$	−71.5000	−	123.842i	−0.108169	−	0.187355i	0.806859	−	0.590744i	$-0.201166\pi$
$661$	−71.5000	−	123.842i	−0.108169	−	0.187355i	−0.915029	+	0.403389i	$0.867832\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			1230.37i			1.85017i
$666$		0			0
$667$	144.000			0.215892
$668$		0			0
$669$		0			0
$670$		0			0
$671$	169.015	−	97.5807i	0.251885	−	0.145426i
$672$		0			0
$673$	−335.500	+	581.103i	−0.498514	+	0.863452i	−0.999999	−	0.00171490i	$-0.999454\pi$
$673$	−335.500	+	581.103i	−0.498514	+	0.863452i	0.501484	+	0.865167i	$0.332787\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.34847	+	4.24264i	0.0108545	+	0.00626683i	0.505417	−	0.862875i	$-0.331339\pi$
$677$	7.34847	+	4.24264i	0.0108545	+	0.00626683i	−0.494563	+	0.869142i	$0.664672\pi$
$678$		0			0
$679$	−122.500	−	212.176i	−0.180412	−	0.312483i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			101.823i			0.149083i	0.997218	+	0.0745413i	$0.0237493\pi$
$683$			101.823i			0.149083i	−0.997218	+	0.0745413i	$0.976251\pi$
$684$		0			0
$685$	−792.000			−1.15620
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−44.0908	+	25.4558i	−0.0639925	+	0.0369461i
$690$		0			0
$691$	−425.000	+	736.122i	−0.615051	+	1.06530i	0.375325	+	0.926893i	$0.377531\pi$
$691$	−425.000	+	736.122i	−0.615051	+	1.06530i	−0.990376	+	0.138406i	$0.955802\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1197.80	−	691.550i	−1.72345	−	0.995037i
$696$		0			0
$697$	216.000	+	374.123i	0.309900	+	0.536762i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			280.014i			0.399450i	0.979852	+	0.199725i	$0.0640048\pi$
$701$			280.014i			0.399450i	−0.979852	+	0.199725i	$0.935995\pi$
$702$		0			0
$703$	725.000			1.03129
$704$		0			0
$705$		0			0
$706$		0			0
$707$	587.878	−	339.411i	0.831510	−	0.480072i
$708$		0			0
$709$	600.500	−	1040.10i	0.846968	−	1.46699i	−0.0369339	−	0.999318i	$-0.511759\pi$
$709$	600.500	−	1040.10i	0.846968	−	1.46699i	0.883901	−	0.467673i	$-0.154908\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	73.4847	+	42.4264i	0.103064	+	0.0595041i
$714$		0			0
$715$	−36.0000	−	62.3538i	−0.0503497	−	0.0872082i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			967.322i			1.34537i	0.739928	+	0.672686i	$0.234859\pi$
$719$			967.322i			1.34537i	−0.739928	+	0.672686i	$0.765141\pi$
$720$		0			0
$721$	−815.000			−1.13037
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−690.756	+	398.808i	−0.952767	+	0.550080i
$726$		0			0
$727$	475.000	−	822.724i	0.653370	−	1.13167i	−0.328930	−	0.944354i	$-0.606688\pi$
$727$	475.000	−	822.724i	0.653370	−	1.13167i	0.982300	−	0.187316i	$-0.0599787\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−308.636	−	178.191i	−0.422210	−	0.243763i
$732$		0			0
$733$	−49.0000	−	84.8705i	−0.0668486	−	0.115785i	0.830664	−	0.556774i	$-0.187961\pi$
$733$	−49.0000	−	84.8705i	−0.0668486	−	0.115785i	−0.897513	+	0.440989i	$0.854628\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			161.220i			0.218752i
$738$		0			0
$739$	−1262.00			−1.70771			−0.853857	−	0.520508i	$-0.825742\pi$
$739$	−1262.00			−1.70771			−0.853857	+	0.520508i	$0.825742\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−668.711	+	386.080i	−0.900014	+	0.519624i	−0.877205	−	0.480116i	$-0.840594\pi$
$743$	−668.711	+	386.080i	−0.900014	+	0.519624i	−0.0228095	+	0.999740i	$0.507261\pi$
$744$		0			0
$745$	144.000	−	249.415i	0.193289	−	0.334786i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	110.227	+	63.6396i	0.147166	+	0.0849661i
$750$		0			0
$751$	98.5000	+	170.607i	0.131158	+	0.227173i	0.924123	−	0.382094i	$-0.124797\pi$
$751$	98.5000	+	170.607i	0.131158	+	0.227173i	−0.792965	+	0.609267i	$0.791464\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1264.31i		−	1.67458i
$756$		0			0
$757$	−241.000			−0.318362			−0.159181	−	0.987249i	$-0.550885\pi$
$757$	−241.000			−0.318362			−0.159181	+	0.987249i	$0.550885\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−433.560	+	250.316i	−0.569724	+	0.328930i	−0.757039	−	0.653370i	$-0.773355\pi$
$761$	−433.560	+	250.316i	−0.569724	+	0.328930i	0.187315	+	0.982300i	$0.440021\pi$
$762$		0			0
$763$	5.00000	−	8.66025i	0.00655308	−	0.0113503i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	80.8332	+	46.6690i	0.105389	+	0.0608462i
$768$		0			0
$769$	−215.500	−	373.257i	−0.280234	−	0.485380i	0.691208	−	0.722656i	$-0.257079\pi$
$769$	−215.500	−	373.257i	−0.280234	−	0.485380i	−0.971442	+	0.237276i	$0.923745\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			254.558i			0.329312i	0.986351	+	0.164656i	$0.0526515\pi$
$773$			254.558i			0.329312i	−0.986351	+	0.164656i	$0.947349\pi$
$774$		0			0
$775$	−470.000			−0.606452
$776$		0			0
$777$		0			0
$778$		0			0
$779$	426.211	−	246.073i	0.547126	−	0.315883i
$780$		0			0
$781$	432.000	−	748.246i	0.553137	−	0.958061i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1778.33	−	1026.72i	−2.26539	−	1.30792i
$786$		0			0
$787$	62.5000	+	108.253i	0.0794155	+	0.137552i	0.902998	−	0.429645i	$-0.141361\pi$
$787$	62.5000	+	108.253i	0.0794155	+	0.137552i	−0.823582	+	0.567197i	$0.808028\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	551.543i		−	0.697273i
$792$		0			0
$793$	−23.0000			−0.0290038
$794$		0			0
$795$		0			0
$796$		0			0
$797$	734.847	−	424.264i	0.922016	−	0.532326i	0.0377385	−	0.999288i	$-0.487985\pi$
$797$	734.847	−	424.264i	0.922016	−	0.532326i	0.884278	+	0.466961i	$0.154651\pi$
$798$		0			0
$799$	−108.000	+	187.061i	−0.135169	+	0.234120i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	712.802	+	411.536i	0.887673	+	0.512498i
$804$		0			0
$805$	180.000	+	311.769i	0.223602	+	0.387291i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1120.06i		−	1.38450i	−0.721660	−	0.692248i	$-0.756620\pi$
$809$		−	1120.06i		−	1.38450i	0.721660	−	0.692248i	$-0.243380\pi$
$810$		0			0
$811$	322.000			0.397041			0.198520	−	0.980097i	$-0.436386\pi$
$811$	322.000			0.397041			0.198520	+	0.980097i	$0.436386\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1271.29	+	733.977i	−1.55986	+	0.900585i
$816$		0			0
$817$	−203.000	+	351.606i	−0.248470	+	0.430363i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	756.892	+	436.992i	0.921915	+	0.532268i	0.884246	−	0.467022i	$-0.154673\pi$
$821$	756.892	+	436.992i	0.921915	+	0.532268i	0.0376696	+	0.999290i	$0.488007\pi$
$822$		0			0
$823$	134.500	+	232.961i	0.163426	+	0.283063i	0.936095	−	0.351746i	$-0.114412\pi$
$823$	134.500	+	232.961i	0.163426	+	0.283063i	−0.772669	+	0.634809i	$0.781079\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			25.4558i			0.0307809i	0.999882	+	0.0153905i	$0.00489913\pi$
$827$			25.4558i			0.0307809i	−0.999882	+	0.0153905i	$0.995101\pi$
$828$		0			0
$829$	−1105.00			−1.33293			−0.666466	−	0.745536i	$-0.732194\pi$
$829$	−1105.00			−1.33293			−0.666466	+	0.745536i	$0.732194\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	529.090	−	305.470i	0.635162	−	0.366711i
$834$		0			0
$835$	828.000	−	1434.14i	0.991617	−	1.71753i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	102.879	+	59.3970i	0.122620	+	0.0707950i	0.560056	−	0.828455i	$-0.310780\pi$
$839$	102.879	+	59.3970i	0.122620	+	0.0707950i	−0.437435	+	0.899250i	$0.644113\pi$
$840$		0			0
$841$	−276.500	−	478.912i	−0.328775	−	0.569455i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1425.53i		−	1.68701i
$846$		0			0
$847$	−245.000			−0.289256
$848$		0			0
$849$		0			0
$850$		0			0
$851$	183.712	−	106.066i	0.215877	−	0.124637i
$852$		0			0
$853$	−695.500	+	1204.64i	−0.815358	+	1.41224i	0.0937133	+	0.995599i	$0.470126\pi$
$853$	−695.500	+	1204.64i	−0.815358	+	1.41224i	−0.909071	+	0.416641i	$0.863207\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	470.302	+	271.529i	0.548777	+	0.316837i	0.748629	−	0.662990i	$-0.230713\pi$
$857$	470.302	+	271.529i	0.548777	+	0.316837i	−0.199851	+	0.979826i	$0.564046\pi$
$858$		0			0
$859$	422.500	+	731.791i	0.491851	+	0.851911i	0.999956	−	0.00938424i	$-0.00298714\pi$
$859$	422.500	+	731.791i	0.491851	+	0.851911i	−0.508105	+	0.861295i	$0.669654\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1247.34i		−	1.44535i	−0.691188	−	0.722675i	$-0.742913\pi$
$863$		−	1247.34i		−	1.44535i	0.691188	−	0.722675i	$-0.257087\pi$
$864$		0			0
$865$	−2736.00			−3.16301
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−565.832	+	326.683i	−0.651130	+	0.375930i
$870$		0			0
$871$	9.50000	−	16.4545i	0.0109070	−	0.0188915i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−808.332	−	466.690i	−0.923808	−	0.533361i
$876$		0			0
$877$	−575.500	−	996.795i	−0.656214	−	1.13660i	−0.981588	−	0.191011i	$-0.938823\pi$
$877$	−575.500	−	996.795i	−0.656214	−	1.13660i	0.325374	−	0.945586i	$-0.394510\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1298.25i		−	1.47361i	−0.676107	−	0.736804i	$-0.736334\pi$
$881$		−	1298.25i		−	1.47361i	0.676107	−	0.736804i	$-0.263666\pi$
$882$		0			0
$883$	−677.000			−0.766704			−0.383352	−	0.923602i	$-0.625230\pi$
$883$	−677.000			−0.766704			−0.383352	+	0.923602i	$0.625230\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	896.513	−	517.602i	1.01073	−	0.583542i	0.0993213	−	0.995055i	$-0.468333\pi$
$887$	896.513	−	517.602i	1.01073	−	0.583542i	0.911404	+	0.411513i	$0.135000\pi$
$888$		0			0
$889$	445.000	−	770.763i	0.500562	−	0.867000i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	213.106	+	123.037i	0.238640	+	0.137779i
$894$		0			0
$895$	−1296.00	−	2244.74i	−1.44804	−	2.50809i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			169.706i			0.188772i
$900$		0			0
$901$	1296.00			1.43840
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1932.65	−	1115.81i	2.13552	−	1.23294i
$906$		0			0
$907$	2.50000	−	4.33013i	0.00275634	−	0.00477412i	−0.864644	−	0.502385i	$-0.832456\pi$
$907$	2.50000	−	4.33013i	0.00275634	−	0.00477412i	0.867400	+	0.497611i	$0.165789\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−867.119	−	500.632i	−0.951832	−	0.549541i	−0.0581828	−	0.998306i	$-0.518531\pi$
$911$	−867.119	−	500.632i	−0.951832	−	0.549541i	−0.893650	+	0.448765i	$0.851864\pi$
$912$		0			0
$913$	−504.000	−	872.954i	−0.552026	−	0.956138i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	593.970i		−	0.647731i
$918$		0			0
$919$	−1190.00			−1.29489			−0.647443	−	0.762114i	$-0.724162\pi$
$919$	−1190.00			−1.29489			−0.647443	+	0.762114i	$0.724162\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−88.1816	+	50.9117i	−0.0955381	+	0.0551589i
$924$		0			0
$925$	−587.500	+	1017.58i	−0.635135	+	1.10009i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−411.514	−	237.588i	−0.442965	−	0.255746i	0.261890	−	0.965098i	$-0.415654\pi$
$929$	−411.514	−	237.588i	−0.442965	−	0.255746i	−0.704854	+	0.709352i	$0.748988\pi$
$930$		0			0
$931$	−348.000	−	602.754i	−0.373792	−	0.647426i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			1832.82i			1.96024i
$936$		0			0
$937$	1775.00			1.89434			0.947172	−	0.320727i	$-0.103927\pi$
$937$	1775.00			1.89434			0.947172	+	0.320727i	$0.103927\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	139.621	−	80.6102i	0.148375	−	0.0856644i	−0.423975	−	0.905674i	$-0.639365\pi$
$941$	139.621	−	80.6102i	0.148375	−	0.0856644i	0.572350	+	0.820010i	$0.306032\pi$
$942$		0			0
$943$	72.0000	−	124.708i	0.0763521	−	0.132246i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−492.347	−	284.257i	−0.519902	−	0.300166i	0.216992	−	0.976173i	$-0.430375\pi$
$947$	−492.347	−	284.257i	−0.519902	−	0.300166i	−0.736895	+	0.676008i	$0.763709\pi$
$948$		0			0
$949$	−48.5000	−	84.0045i	−0.0511064	−	0.0885189i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1807.36i			1.89650i	0.317524	+	0.948250i	$0.397149\pi$
$953$			1807.36i			1.89650i	−0.317524	+	0.948250i	$0.602851\pi$
$954$		0			0
$955$	1224.00			1.28168
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−404.166	+	233.345i	−0.421445	+	0.243321i
$960$		0			0
$961$	430.500	−	745.648i	0.447971	−	0.775908i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−521.741	−	301.227i	−0.540665	−	0.312153i
$966$		0			0
$967$	350.500	+	607.084i	0.362461	+	0.627801i	0.988365	−	0.152099i	$-0.0486032\pi$
$967$	350.500	+	607.084i	0.362461	+	0.627801i	−0.625904	+	0.779900i	$0.715270\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			381.838i			0.393242i	0.980480	+	0.196621i	$0.0629968\pi$
$971$			381.838i			0.393242i	−0.980480	+	0.196621i	$0.937003\pi$
$972$		0			0
$973$	−815.000			−0.837616
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−279.242	+	161.220i	−0.285816	+	0.165016i	−0.636053	−	0.771645i	$-0.719434\pi$
$977$	−279.242	+	161.220i	−0.285816	+	0.165016i	0.350238	+	0.936661i	$0.386101\pi$
$978$		0			0
$979$	324.000	−	561.184i	0.330950	−	0.573222i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	36.7423	+	21.2132i	0.0373778	+	0.0215801i	0.518572	−	0.855034i	$-0.326464\pi$
$983$	36.7423	+	21.2132i	0.0373778	+	0.0215801i	−0.481195	+	0.876614i	$0.659797\pi$
$984$		0			0
$985$	324.000	+	561.184i	0.328934	+	0.569730i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			118.794i			0.120115i
$990$		0			0
$991$	475.000			0.479314			0.239657	−	0.970858i	$-0.422965\pi$
$991$	475.000			0.479314			0.239657	+	0.970858i	$0.422965\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1271.29	+	733.977i	−1.27767	+	0.737665i
$996$		0			0
$997$	767.000	−	1328.48i	0.769308	−	1.33248i	−0.168631	−	0.985679i	$-0.553935\pi$
$997$	767.000	−	1328.48i	0.769308	−	1.33248i	0.937939	−	0.346801i	$-0.112732\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.q.i.593.2		4
3.2	odd	2	inner	1296.3.q.i.593.1		4
4.3	odd	2		162.3.d.a.107.1		4
9.2	odd	6		432.3.e.d.161.1		2
9.4	even	3	inner	1296.3.q.i.1025.1		4
9.5	odd	6	inner	1296.3.q.i.1025.2		4
9.7	even	3		432.3.e.d.161.2		2
12.11	even	2		162.3.d.a.107.2		4
36.7	odd	6		54.3.b.a.53.2	yes	2
36.11	even	6		54.3.b.a.53.1	✓	2
36.23	even	6		162.3.d.a.53.1		4
36.31	odd	6		162.3.d.a.53.2		4
72.11	even	6		1728.3.e.l.1025.2		2
72.29	odd	6		1728.3.e.f.1025.2		2
72.43	odd	6		1728.3.e.l.1025.1		2
72.61	even	6		1728.3.e.f.1025.1		2
180.7	even	12		1350.3.b.b.1349.2		4
180.43	even	12		1350.3.b.b.1349.3		4
180.47	odd	12		1350.3.b.b.1349.4		4
180.79	odd	6		1350.3.d.d.701.1		2
180.83	odd	12		1350.3.b.b.1349.1		4
180.119	even	6		1350.3.d.d.701.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.b.a.53.1	✓	2	36.11	even	6
54.3.b.a.53.2	yes	2	36.7	odd	6
162.3.d.a.53.1		4	36.23	even	6
162.3.d.a.53.2		4	36.31	odd	6
162.3.d.a.107.1		4	4.3	odd	2
162.3.d.a.107.2		4	12.11	even	2
432.3.e.d.161.1		2	9.2	odd	6
432.3.e.d.161.2		2	9.7	even	3
1296.3.q.i.593.1		4	3.2	odd	2	inner
1296.3.q.i.593.2		4	1.1	even	1	trivial
1296.3.q.i.1025.1		4	9.4	even	3	inner
1296.3.q.i.1025.2		4	9.5	odd	6	inner
1350.3.b.b.1349.1		4	180.83	odd	12
1350.3.b.b.1349.2		4	180.7	even	12
1350.3.b.b.1349.3		4	180.43	even	12
1350.3.b.b.1349.4		4	180.47	odd	12
1350.3.d.d.701.1		2	180.79	odd	6
1350.3.d.d.701.2		2	180.119	even	6
1728.3.e.f.1025.1		2	72.61	even	6
1728.3.e.f.1025.2		2	72.29	odd	6
1728.3.e.l.1025.1		2	72.43	odd	6
1728.3.e.l.1025.2		2	72.11	even	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(7.34847 - 4.24264i) q^{5} +(2.50000 - 4.33013i) q^{7} +(-7.34847 - 4.24264i) q^{11} +(0.500000 + 0.866025i) q^{13} -25.4558i q^{17} -29.0000 q^{19} +(-7.34847 + 4.24264i) q^{23} +(23.5000 - 40.7032i) q^{25} +(-14.6969 - 8.48528i) q^{29} +(-5.00000 - 8.66025i) q^{31} -42.4264i q^{35} -25.0000 q^{37} +(-14.6969 + 8.48528i) q^{41} +(7.00000 - 12.1244i) q^{43} +(-7.34847 - 4.24264i) q^{47} +(12.0000 + 20.7846i) q^{49} +50.9117i q^{53} -72.0000 q^{55} +(80.8332 - 46.6690i) q^{59} +(-11.5000 + 19.9186i) q^{61} +(7.34847 + 4.24264i) q^{65} +(-9.50000 - 16.4545i) q^{67} +101.823i q^{71} -97.0000 q^{73} +(-36.7423 + 21.2132i) q^{77} +(38.5000 - 66.6840i) q^{79} +(102.879 + 59.3970i) q^{83} +(-108.000 - 187.061i) q^{85} +76.3675i q^{89} +5.00000 q^{91} +(-213.106 + 123.037i) q^{95} +(24.5000 - 42.4352i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{7} + 2 q^{13} - 116 q^{19} + 94 q^{25} - 20 q^{31} - 100 q^{37} + 28 q^{43} + 48 q^{49} - 288 q^{55} - 46 q^{61} - 38 q^{67} - 388 q^{73} + 154 q^{79} - 432 q^{85} + 20 q^{91} + 98 q^{97}+O(q^{100}) \) 4 * q + 10 * q^7 + 2 * q^13 - 116 * q^19 + 94 * q^25 - 20 * q^31 - 100 * q^37 + 28 * q^43 + 48 * q^49 - 288 * q^55 - 46 * q^61 - 38 * q^67 - 388 * q^73 + 154 * q^79 - 432 * q^85 + 20 * q^91 + 98 * q^97

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