LMFDB - Embedded newform 324.4.e.i.217.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,4,Mod(109,324)] 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("324.109"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,16] 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:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.49787136.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.2
Root			$-0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	324.217
Dual form			324.4.e.i.109.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.33918 - 9.24773i) q^{5} +(-11.7477 + 20.3477i) q^{7} +(2.45505 - 4.25227i) q^{11} +(-0.752273 - 1.30297i) q^{13} +99.2990 q^{17} +28.5045 q^{19} +(76.3454 + 132.234i) q^{23} +(5.48636 - 9.50266i) q^{25} +(120.227 - 208.239i) q^{29} +(-128.486 - 222.545i) q^{31} +250.893 q^{35} +359.468 q^{37} +(-54.2499 - 93.9636i) q^{41} +(-205.730 + 356.334i) q^{43} +(155.885 - 270.000i) q^{47} +(-104.518 - 181.031i) q^{49} +702.578 q^{53} -52.4318 q^{55} +(239.595 + 414.991i) q^{59} +(-249.230 + 431.678i) q^{61} +(-8.03304 + 13.9136i) q^{65} +(166.216 + 287.894i) q^{67} +884.729 q^{71} +305.000 q^{73} +(57.6825 + 99.9091i) q^{77} +(351.693 - 609.150i) q^{79} +(210.373 - 364.377i) q^{83} +(-530.175 - 918.290i) q^{85} -1610.00 q^{89} +35.3500 q^{91} +(-152.191 - 263.602i) q^{95} +(-198.009 + 342.962i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} - 116 q^{13} + 448 q^{19} - 616 q^{25} - 368 q^{31} + 1336 q^{37} - 656 q^{43} - 1716 q^{49} + 2880 q^{55} - 1004 q^{61} - 320 q^{67} + 2440 q^{73} + 64 q^{79} - 612 q^{85} - 6976 q^{91}+ \cdots - 2024 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 - 116 * q^13 + 448 * q^19 - 616 * q^25 - 368 * q^31 + 1336 * q^37 - 656 * q^43 - 1716 * q^49 + 2880 * q^55 - 1004 * q^61 - 320 * q^67 + 2440 * q^73 + 64 * q^79 - 612 * q^85 - 6976 * q^91 - 2024 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−5.33918	−	9.24773i	−0.477551	−	0.827142i	0.522118	−	0.852873i	$-0.325142\pi$
$5$	−5.33918	−	9.24773i	−0.477551	−	0.827142i	−0.999669	+	0.0257312i	$0.991809\pi$
$6$		0			0
$7$	−11.7477	+	20.3477i	−0.634318	+	1.09867i	0.352342	+	0.935871i	$0.385386\pi$
$7$	−11.7477	+	20.3477i	−0.634318	+	1.09867i	−0.986659	+	0.162799i	$0.947948\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.45505	−	4.25227i	0.0672932	−	0.116555i	−0.830416	−	0.557144i	$-0.811897\pi$
$11$	2.45505	−	4.25227i	0.0672932	−	0.116555i	0.897709	+	0.440589i	$0.145230\pi$
$12$		0			0
$13$	−0.752273	−	1.30297i	−0.0160495	−	0.0277985i	0.857889	−	0.513835i	$-0.171776\pi$
$13$	−0.752273	−	1.30297i	−0.0160495	−	0.0277985i	−0.873939	+	0.486036i	$0.838442\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	99.2990			1.41668			0.708340	−	0.705872i	$-0.249445\pi$
$17$	99.2990			1.41668			0.708340	+	0.705872i	$0.249445\pi$
$18$		0			0
$19$	28.5045			0.344178			0.172089	−	0.985081i	$-0.444948\pi$
$19$	28.5045			0.344178			0.172089	+	0.985081i	$0.444948\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	76.3454	+	132.234i	0.692135	+	1.19881i	0.971137	+	0.238523i	$0.0766632\pi$
$23$	76.3454	+	132.234i	0.692135	+	1.19881i	−0.279001	+	0.960291i	$0.590003\pi$
$24$		0			0
$25$	5.48636	−	9.50266i	0.0438909	−	0.0760213i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	120.227	−	208.239i	0.769846	−	1.33341i	−0.167801	−	0.985821i	$-0.553666\pi$
$29$	120.227	−	208.239i	0.769846	−	1.33341i	0.937646	−	0.347591i	$-0.113000\pi$
$30$		0			0
$31$	−128.486	−	222.545i	−0.744414	−	1.28936i	−0.950468	−	0.310822i	$-0.899396\pi$
$31$	−128.486	−	222.545i	−0.744414	−	1.28936i	0.206054	−	0.978541i	$-0.433938\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	250.893			1.21168
$36$		0			0
$37$	359.468			1.59719			0.798597	−	0.601866i	$-0.205576\pi$
$37$	359.468			1.59719			0.798597	+	0.601866i	$0.205576\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−54.2499	−	93.9636i	−0.206644	−	0.357918i	0.744011	−	0.668167i	$-0.232921\pi$
$41$	−54.2499	−	93.9636i	−0.206644	−	0.357918i	−0.950655	+	0.310249i	$0.899588\pi$
$42$		0			0
$43$	−205.730	+	356.334i	−0.729615	+	1.26373i	0.227430	+	0.973794i	$0.426968\pi$
$43$	−205.730	+	356.334i	−0.729615	+	1.26373i	−0.957046	+	0.289937i	$0.906366\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	155.885	−	270.000i	0.483789	−	0.837948i	−0.516037	−	0.856566i	$-0.672593\pi$
$47$	155.885	−	270.000i	0.483789	−	0.837948i	0.999827	+	0.0186183i	$0.00592674\pi$
$48$		0			0
$49$	−104.518	−	181.031i	−0.304718	−	0.527787i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	702.578			1.82088			0.910438	−	0.413645i	$-0.135745\pi$
$53$	702.578			1.82088			0.910438	+	0.413645i	$0.135745\pi$
$54$		0			0
$55$	−52.4318			−0.128544
$56$		0			0
$57$		0			0
$58$		0			0
$59$	239.595	+	414.991i	0.528689	+	0.915715i	0.999440	+	0.0334498i	$0.0106494\pi$
$59$	239.595	+	414.991i	0.528689	+	0.915715i	−0.470752	+	0.882266i	$0.656017\pi$
$60$		0			0
$61$	−249.230	+	431.678i	−0.523124	+	0.906078i	0.476514	+	0.879167i	$0.341900\pi$
$61$	−249.230	+	431.678i	−0.523124	+	0.906078i	−0.999638	+	0.0269106i	$0.991433\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.03304	+	13.9136i	−0.0153289	+	0.0265504i
$66$		0			0
$67$	166.216	+	287.894i	0.303082	+	0.524954i	0.976832	−	0.214005i	$-0.0686510\pi$
$67$	166.216	+	287.894i	0.303082	+	0.524954i	−0.673750	+	0.738959i	$0.735318\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	884.729			1.47885			0.739423	−	0.673242i	$-0.235099\pi$
$71$	884.729			1.47885			0.739423	+	0.673242i	$0.235099\pi$
$72$		0			0
$73$	305.000			0.489008			0.244504	−	0.969648i	$-0.421375\pi$
$73$	305.000			0.489008			0.244504	+	0.969648i	$0.421375\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	57.6825	+	99.9091i	0.0853706	+	0.147866i
$78$		0			0
$79$	351.693	−	609.150i	0.500868	−	0.867529i	−0.499131	−	0.866526i	$-0.666348\pi$
$79$	351.693	−	609.150i	0.500868	−	0.867529i	0.999999	−	0.00100260i	$-0.000319137\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	210.373	−	364.377i	0.278210	−	0.481875i	−0.692730	−	0.721197i	$-0.743592\pi$
$83$	210.373	−	364.377i	0.278210	−	0.481875i	0.970940	+	0.239323i	$0.0769254\pi$
$84$		0			0
$85$	−530.175	−	918.290i	−0.676536	−	1.17179i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1610.00			−1.91752			−0.958761	−	0.284212i	$-0.908268\pi$
$89$	−1610.00			−1.91752			−0.958761	+	0.284212i	$0.908268\pi$
$90$		0			0
$91$	35.3500			0.0407218
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−152.191	−	263.602i	−0.164363	−	0.284684i
$96$		0			0
$97$	−198.009	+	342.962i	−0.207266	+	0.358995i	−0.950852	−	0.309645i	$-0.899790\pi$
$97$	−198.009	+	342.962i	−0.207266	+	0.358995i	0.743587	+	0.668640i	$0.233123\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	136.483	−	236.395i	0.134461	−	0.232893i	−0.790930	−	0.611906i	$-0.790403\pi$
$101$	136.483	−	236.395i	0.134461	−	0.232893i	0.925391	+	0.379013i	$0.123736\pi$
$102$		0			0
$103$	198.541	+	343.883i	0.189930	+	0.328969i	0.945227	−	0.326414i	$-0.105840\pi$
$103$	198.541	+	343.883i	0.189930	+	0.328969i	−0.755297	+	0.655383i	$0.772507\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−360.870			−0.326043			−0.163022	−	0.986622i	$-0.552124\pi$
$107$	−360.870			−0.326043			−0.163022	+	0.986622i	$0.552124\pi$
$108$		0			0
$109$	120.568			0.105948			0.0529740	−	0.998596i	$-0.483130\pi$
$109$	120.568			0.105948			0.0529740	+	0.998596i	$0.483130\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−612.311	−	1060.55i	−0.509747	−	0.882908i	−0.999936	−	0.0112917i	$-0.996406\pi$
$113$	−612.311	−	1060.55i	−0.509747	−	0.882908i	0.490189	−	0.871616i	$-0.336928\pi$
$114$		0			0
$115$	815.243	−	1412.04i	0.661059	−	1.14499i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1166.54	+	2020.50i	−0.898625	+	1.55646i
$120$		0			0
$121$	653.445	+	1131.80i	0.490943	+	0.850339i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1451.97			−1.03894
$126$		0			0
$127$	−1991.46			−1.39144			−0.695722	−	0.718311i	$-0.744915\pi$
$127$	−1991.46			−1.39144			−0.695722	+	0.718311i	$0.744915\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	788.743	+	1366.14i	0.526052	+	0.911149i	0.999539	+	0.0303481i	$0.00966160\pi$
$131$	788.743	+	1366.14i	0.526052	+	0.911149i	−0.473487	+	0.880801i	$0.657005\pi$
$132$		0			0
$133$	−334.864	+	580.001i	−0.218318	+	0.378139i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−418.004	+	724.005i	−0.260675	+	0.451503i	−0.966422	−	0.256962i	$-0.917279\pi$
$137$	−418.004	+	724.005i	−0.260675	+	0.451503i	0.705746	+	0.708465i	$0.250612\pi$
$138$		0			0
$139$	1364.30	+	2363.04i	0.832509	+	1.44195i	0.896042	+	0.443969i	$0.146430\pi$
$139$	1364.30	+	2363.04i	0.832509	+	1.44195i	−0.0635329	+	0.997980i	$0.520237\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.38747			−0.00432008
$144$		0			0
$145$	−2567.65			−1.47056
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1388.04	−	2404.16i	−0.763173	−	1.32185i	−0.941207	−	0.337829i	$-0.890307\pi$
$149$	−1388.04	−	2404.16i	−0.763173	−	1.32185i	0.178035	−	0.984024i	$-0.443026\pi$
$150$		0			0
$151$	381.514	−	660.801i	0.205610	−	0.356127i	−0.744717	−	0.667381i	$-0.767415\pi$
$151$	381.514	−	660.801i	0.205610	−	0.356127i	0.950327	+	0.311253i	$0.100749\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1372.02	+	2376.41i	−0.710991	+	1.23147i
$156$		0			0
$157$	−352.743	−	610.969i	−0.179312	−	0.310577i	0.762333	−	0.647185i	$-0.224054\pi$
$157$	−352.743	−	610.969i	−0.179312	−	0.310577i	−0.941645	+	0.336607i	$0.890720\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3587.54			−1.75613
$162$		0			0
$163$	1587.60			0.762886			0.381443	−	0.924392i	$-0.375427\pi$
$163$	1587.60			0.762886			0.381443	+	0.924392i	$0.375427\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	330.432	+	572.325i	0.153111	+	0.265197i	0.932370	−	0.361506i	$-0.117737\pi$
$167$	330.432	+	572.325i	0.153111	+	0.265197i	−0.779258	+	0.626703i	$0.784404\pi$
$168$		0			0
$169$	1097.37	−	1900.70i	0.499485	−	0.865133i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1018.69	−	1764.42i	0.447684	−	0.775411i	−0.550551	−	0.834801i	$-0.685582\pi$
$173$	1018.69	−	1764.42i	0.447684	−	0.775411i	0.998235	+	0.0593907i	$0.0189158\pi$
$174$		0			0
$175$	128.905	+	223.269i	0.0556815	+	0.0964433i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2579.01			−1.07690			−0.538448	−	0.842659i	$-0.680989\pi$
$179$	−2579.01			−1.07690			−0.538448	+	0.842659i	$0.680989\pi$
$180$		0			0
$181$	1851.65			0.760400			0.380200	−	0.924904i	$-0.375855\pi$
$181$	1851.65			0.760400			0.380200	+	0.924904i	$0.375855\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1919.26	−	3324.26i	−0.762741	−	1.32111i
$186$		0			0
$187$	243.784	−	422.246i	0.0953329	−	0.165121i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−28.4831	+	49.3341i	−0.0107904	+	0.0186895i	−0.871370	−	0.490626i	$-0.836768\pi$
$191$	−28.4831	+	49.3341i	−0.0107904	+	0.0186895i	0.860580	+	0.509316i	$0.170101\pi$
$192$		0			0
$193$	−907.014	−	1570.99i	−0.338281	−	0.585920i	0.645828	−	0.763483i	$-0.276512\pi$
$193$	−907.014	−	1570.99i	−0.338281	−	0.585920i	−0.984110	+	0.177562i	$0.943179\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−873.334			−0.315850			−0.157925	−	0.987451i	$-0.550480\pi$
$197$	−873.334			−0.315850			−0.157925	+	0.987451i	$0.550480\pi$
$198$		0			0
$199$	−5063.60			−1.80376			−0.901882	−	0.431982i	$-0.857814\pi$
$199$	−5063.60			−1.80376			−0.901882	+	0.431982i	$0.857814\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2824.78	+	4892.66i	0.976653	+	1.69161i
$204$		0			0
$205$	−579.300	+	1003.38i	−0.197366	+	0.341848i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	69.9801	−	121.209i	0.0231609	−	0.0401158i
$210$		0			0
$211$	−215.389	−	373.064i	−0.0702747	−	0.121719i	0.828747	−	0.559624i	$-0.189054\pi$
$211$	−215.389	−	373.064i	−0.0702747	−	0.121719i	−0.899022	+	0.437904i	$0.855721\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4393.71			1.39371
$216$		0			0
$217$	6037.69			1.88878
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−74.6999	−	129.384i	−0.0227369	−	0.0393815i
$222$		0			0
$223$	1631.84	−	2826.43i	0.490027	−	0.848751i	−0.509907	−	0.860229i	$-0.670320\pi$
$223$	1631.84	−	2826.43i	0.490027	−	0.848751i	0.999934	+	0.0114779i	$0.00365361\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	374.339	−	648.375i	0.109453	−	0.189578i	−0.806096	−	0.591785i	$-0.798423\pi$
$227$	374.339	−	648.375i	0.109453	−	0.189578i	0.915549	+	0.402207i	$0.131757\pi$
$228$		0			0
$229$	1493.50	+	2586.81i	0.430974	+	0.746469i	0.996957	−	0.0779474i	$-0.0248366\pi$
$229$	1493.50	+	2586.81i	0.430974	+	0.746469i	−0.565983	+	0.824417i	$0.691503\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3785.23			1.06429			0.532144	−	0.846654i	$-0.321387\pi$
$233$	3785.23			1.06429			0.532144	+	0.846654i	$0.321387\pi$
$234$		0			0
$235$	−3329.18			−0.924136
$236$		0			0
$237$		0			0
$238$		0			0
$239$	304.188	+	526.868i	0.0823274	+	0.142595i	0.904249	−	0.427005i	$-0.140431\pi$
$239$	304.188	+	526.868i	0.0823274	+	0.142595i	−0.821922	+	0.569600i	$0.807098\pi$
$240$		0			0
$241$	−2671.70	+	4627.53i	−0.714106	+	1.23687i	0.249197	+	0.968453i	$0.419833\pi$
$241$	−2671.70	+	4627.53i	−0.714106	+	1.23687i	−0.963303	+	0.268416i	$0.913500\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1116.08	+	1933.11i	−0.291036	+	0.504090i
$246$		0			0
$247$	−21.4432	−	37.1407i	−0.00552388	−	0.00956764i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6475.89			1.62850			0.814252	−	0.580512i	$-0.197147\pi$
$251$	6475.89			1.62850			0.814252	+	0.580512i	$0.197147\pi$
$252$		0			0
$253$	749.727			0.186304
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.64532	−	4.58182i	−0.000642064	−	0.00111209i	0.865704	−	0.500556i	$-0.166871\pi$
$257$	−2.64532	−	4.58182i	−0.000642064	−	0.00111209i	−0.866346	+	0.499444i	$0.833538\pi$
$258$		0			0
$259$	−4222.93	+	7314.34i	−1.01313	+	1.75479i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3523.47	−	6102.83i	0.826108	−	1.43086i	−0.0749616	−	0.997186i	$-0.523883\pi$
$263$	3523.47	−	6102.83i	0.826108	−	1.43086i	0.901070	−	0.433675i	$-0.142783\pi$
$264$		0			0
$265$	−3751.19	−	6497.25i	−0.869561	−	1.50612i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−787.057			−0.178393			−0.0891965	−	0.996014i	$-0.528430\pi$
$269$	−787.057			−0.178393			−0.0891965	+	0.996014i	$0.528430\pi$
$270$		0			0
$271$	5076.65			1.13795			0.568975	−	0.822355i	$-0.307340\pi$
$271$	5076.65			1.13795			0.568975	+	0.822355i	$0.307340\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−26.9386	−	46.6590i	−0.00590712	−	0.0102314i
$276$		0			0
$277$	−408.891	+	708.220i	−0.0886927	+	0.153620i	−0.906959	−	0.421219i	$-0.861602\pi$
$277$	−408.891	+	708.220i	−0.0886927	+	0.153620i	0.818266	+	0.574840i	$0.194936\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2618.75	−	4535.80i	0.555948	−	0.962929i	−0.441882	−	0.897073i	$-0.645689\pi$
$281$	2618.75	−	4535.80i	0.555948	−	0.962929i	0.997829	−	0.0658560i	$-0.0209778\pi$
$282$		0			0
$283$	1049.03	+	1816.97i	0.220347	+	0.381652i	0.954913	−	0.296885i	$-0.0959477\pi$
$283$	1049.03	+	1816.97i	0.220347	+	0.381652i	−0.734566	+	0.678537i	$0.762614\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2549.25			0.524312
$288$		0			0
$289$	4947.29			1.00698
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1245.03	−	2156.45i	−0.248243	−	0.429970i	0.714795	−	0.699334i	$-0.246520\pi$
$293$	−1245.03	−	2156.45i	−0.248243	−	0.429970i	−0.963039	+	0.269364i	$0.913187\pi$
$294$		0			0
$295$	2558.48	−	4431.42i	0.504951	−	0.874601i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	114.865	−	198.952i	0.0222168	−	0.0384806i
$300$		0			0
$301$	−4833.71	−	8372.23i	−0.925616	−	1.60321i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5322.72			0.999273
$306$		0			0
$307$	2068.21			0.384492			0.192246	−	0.981347i	$-0.438423\pi$
$307$	2068.21			0.384492			0.192246	+	0.981347i	$0.438423\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2455.78	+	4253.54i	0.447764	+	0.775550i	0.998240	−	0.0593006i	$-0.0188871\pi$
$311$	2455.78	+	4253.54i	0.447764	+	0.775550i	−0.550476	+	0.834851i	$0.685554\pi$
$312$		0			0
$313$	−2648.17	+	4586.76i	−0.478221	+	0.828304i	−0.999688	−	0.0249677i	$-0.992052\pi$
$313$	−2648.17	+	4586.76i	−0.478221	+	0.828304i	0.521467	+	0.853272i	$0.325385\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	715.401	−	1239.11i	0.126754	−	0.219544i	−0.795663	−	0.605739i	$-0.792878\pi$
$317$	715.401	−	1239.11i	0.126754	−	0.219544i	0.922417	+	0.386195i	$0.126211\pi$
$318$		0			0
$319$	−590.325	−	1022.47i	−0.103611	−	0.179459i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2830.47			0.487590
$324$		0			0
$325$	−16.5090			−0.00281770
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3662.58	+	6343.77i	0.613752	+	1.06305i
$330$		0			0
$331$	2667.26	−	4619.82i	0.442917	−	0.767156i	−0.554987	−	0.831859i	$-0.687277\pi$
$331$	2667.26	−	4619.82i	0.442917	−	0.767156i	0.997905	+	0.0647034i	$0.0206101\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1774.91	−	3074.24i	0.289474	−	0.501384i
$336$		0			0
$337$	4335.79	+	7509.81i	0.700847	+	1.21390i	0.968169	+	0.250296i	$0.0805280\pi$
$337$	4335.79	+	7509.81i	0.700847	+	1.21390i	−0.267322	+	0.963607i	$0.586139\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1261.76			−0.200376
$342$		0			0
$343$	−3147.54			−0.495484
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2685.86	+	4652.05i	0.415518	+	0.719698i	0.995483	−	0.0949434i	$-0.0302670\pi$
$347$	2685.86	+	4652.05i	0.415518	+	0.719698i	−0.579965	+	0.814642i	$0.696934\pi$
$348$		0			0
$349$	2520.76	−	4366.09i	0.386629	−	0.669660i	−0.605365	−	0.795948i	$-0.706973\pi$
$349$	2520.76	−	4366.09i	0.386629	−	0.669660i	0.991994	+	0.126288i	$0.0403062\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2425.89	−	4201.76i	0.365771	−	0.633533i	−0.623129	−	0.782119i	$-0.714139\pi$
$353$	2425.89	−	4201.76i	0.365771	−	0.633533i	0.988900	+	0.148586i	$0.0474721\pi$
$354$		0			0
$355$	−4723.72	−	8181.73i	−0.706223	−	1.22321i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3143.59			0.462151			0.231075	−	0.972936i	$-0.425776\pi$
$359$	3143.59			0.462151			0.231075	+	0.972936i	$0.425776\pi$
$360$		0			0
$361$	−6046.49			−0.881541
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1628.45	−	2820.56i	−0.233526	−	0.404479i
$366$		0			0
$367$	−4990.91	+	8644.52i	−0.709873	+	1.22954i	0.255030	+	0.966933i	$0.417915\pi$
$367$	−4990.91	+	8644.52i	−0.709873	+	1.22954i	−0.964904	+	0.262604i	$0.915419\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8253.69	+	14295.8i	−1.15501	+	2.00054i
$372$		0			0
$373$	3032.42	+	5252.30i	0.420945	+	0.729099i	0.996032	−	0.0889935i	$-0.0283650\pi$
$373$	3032.42	+	5252.30i	0.420945	+	0.729099i	−0.575087	+	0.818092i	$0.695032\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−361.773			−0.0494224
$378$		0			0
$379$	11238.7			1.52321			0.761603	−	0.648044i	$-0.224413\pi$
$379$	11238.7			1.52321			0.761603	+	0.648044i	$0.224413\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7344.43	−	12720.9i	−0.979851	−	1.69715i	−0.662896	−	0.748711i	$-0.730673\pi$
$383$	−7344.43	−	12720.9i	−0.979851	−	1.69715i	−0.316955	−	0.948441i	$-0.602660\pi$
$384$		0			0
$385$	615.955	−	1066.86i	0.0815375	−	0.141227i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−677.833	+	1174.04i	−0.0883483	+	0.153024i	−0.906813	−	0.421533i	$-0.861492\pi$
$389$	−677.833	+	1174.04i	−0.0883483	+	0.153024i	0.818465	+	0.574557i	$0.194826\pi$
$390$		0			0
$391$	7581.02	+	13130.7i	0.980534	+	1.69833i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7511.01			−0.956759
$396$		0			0
$397$	−9070.50			−1.14669			−0.573345	−	0.819314i	$-0.694354\pi$
$397$	−9070.50			−1.14669			−0.573345	+	0.819314i	$0.694354\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1641.92	−	2843.89i	−0.204473	−	0.354157i	0.745492	−	0.666515i	$-0.232215\pi$
$401$	−1641.92	−	2843.89i	−0.204473	−	0.354157i	−0.949965	+	0.312358i	$0.898881\pi$
$402$		0			0
$403$	−193.314	+	334.829i	−0.0238949	+	0.0413871i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	882.513	−	1528.56i	0.107480	−	0.186162i
$408$		0			0
$409$	−3109.65	−	5386.07i	−0.375947	−	0.651159i	0.614521	−	0.788900i	$-0.289349\pi$
$409$	−3109.65	−	5386.07i	−0.375947	−	0.651159i	−0.990468	+	0.137741i	$0.956016\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−11258.8			−1.34143
$414$		0			0
$415$	−4492.88			−0.531438
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2984.55	−	5169.40i	−0.347983	−	0.602724i	0.637908	−	0.770113i	$-0.279800\pi$
$419$	−2984.55	−	5169.40i	−0.347983	−	0.602724i	−0.985891	+	0.167388i	$0.946467\pi$
$420$		0			0
$421$	−3269.54	+	5663.01i	−0.378498	+	0.655578i	−0.990844	−	0.135012i	$-0.956893\pi$
$421$	−3269.54	+	5663.01i	−0.378498	+	0.655578i	0.612346	+	0.790590i	$0.290226\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	544.790	−	943.604i	0.0621793	−	0.107698i
$426$		0			0
$427$	−5855.76	−	10142.5i	−0.663654	−	1.14948i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11929.1			−1.33319			−0.666596	−	0.745419i	$-0.732250\pi$
$431$	−11929.1			−1.33319			−0.666596	+	0.745419i	$0.732250\pi$
$432$		0			0
$433$	−4047.92			−0.449262			−0.224631	−	0.974444i	$-0.572118\pi$
$433$	−4047.92			−0.449262			−0.224631	+	0.974444i	$0.572118\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2176.19	+	3769.27i	0.238218	+	0.412606i
$438$		0			0
$439$	6062.11	−	10499.9i	0.659063	−	1.14153i	−0.321796	−	0.946809i	$-0.604286\pi$
$439$	6062.11	−	10499.9i	0.659063	−	1.14153i	0.980859	−	0.194722i	$-0.0623803\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6218.80	+	10771.3i	−0.666962	+	1.15521i	0.311787	+	0.950152i	$0.399072\pi$
$443$	−6218.80	+	10771.3i	−0.666962	+	1.15521i	−0.978749	+	0.205060i	$0.934261\pi$
$444$		0			0
$445$	8596.07	+	14888.8i	0.915714	+	1.58606i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2925.50			0.307489			0.153745	−	0.988111i	$-0.450867\pi$
$449$	2925.50			0.307489			0.153745	+	0.988111i	$0.450867\pi$
$450$		0			0
$451$	−532.745			−0.0556231
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−188.740	−	326.907i	−0.0194467	−	0.0336827i
$456$		0			0
$457$	8661.76	−	15002.6i	0.886608	−	1.53565i	0.0427492	−	0.999086i	$-0.486388\pi$
$457$	8661.76	−	15002.6i	0.886608	−	1.53565i	0.843859	−	0.536565i	$-0.180278\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6694.22	+	11594.7i	−0.676314	+	1.17141i	0.299769	+	0.954012i	$0.403090\pi$
$461$	−6694.22	+	11594.7i	−0.676314	+	1.17141i	−0.976083	+	0.217398i	$0.930243\pi$
$462$		0			0
$463$	5741.58	+	9944.71i	0.576315	+	0.998207i	0.995897	+	0.0904898i	$0.0288433\pi$
$463$	5741.58	+	9944.71i	0.576315	+	0.998207i	−0.419582	+	0.907717i	$0.637823\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−18016.7			−1.78526			−0.892629	−	0.450793i	$-0.851141\pi$
$467$	−18016.7			−1.78526			−0.892629	+	0.450793i	$0.851141\pi$
$468$		0			0
$469$	−7810.64			−0.769001
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1010.15	+	1749.64i	0.0981964	+	0.170081i
$474$		0			0
$475$	156.386	−	270.869i	0.0151063	−	0.0261649i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8526.11	−	14767.7i	0.813294	−	1.40867i	−0.0972521	−	0.995260i	$-0.531005\pi$
$479$	8526.11	−	14767.7i	0.813294	−	1.40867i	0.910546	−	0.413407i	$-0.135661\pi$
$480$		0			0
$481$	−270.418	−	468.378i	−0.0256341	−	0.0443996i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4228.82			0.395919
$486$		0			0
$487$	3327.54			0.309620			0.154810	−	0.987944i	$-0.450523\pi$
$487$	3327.54			0.309620			0.154810	+	0.987944i	$0.450523\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2986.10	+	5172.07i	0.274462	+	0.475381i	0.969999	−	0.243108i	$-0.0781671\pi$
$491$	2986.10	+	5172.07i	0.274462	+	0.475381i	−0.695538	+	0.718490i	$0.744834\pi$
$492$		0			0
$493$	11938.4	−	20677.9i	1.09062	−	1.88902i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10393.6	+	18002.2i	−0.938058	+	1.62476i
$498$		0			0
$499$	−3467.45	−	6005.80i	−0.311071	−	0.538791i	0.667523	−	0.744589i	$-0.267354\pi$
$499$	−3467.45	−	6005.80i	−0.311071	−	0.538791i	−0.978595	+	0.205798i	$0.934021\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5751.15			−0.509803			−0.254902	−	0.966967i	$-0.582043\pi$
$503$	−5751.15			−0.509803			−0.254902	+	0.966967i	$0.582043\pi$
$504$		0			0
$505$	−2914.83			−0.256848
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6006.24	+	10403.1i	0.523029	+	0.905913i	0.999641	+	0.0267988i	$0.00853134\pi$
$509$	6006.24	+	10403.1i	0.523029	+	0.905913i	−0.476612	+	0.879114i	$0.658135\pi$
$510$		0			0
$511$	−3583.06	+	6206.04i	−0.310186	+	0.537258i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2120.09	−	3672.10i	0.181403	−	0.314199i
$516$		0			0
$517$	−765.409	−	1325.73i	−0.0651115	−	0.112776i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13667.9			−1.14933			−0.574666	−	0.818388i	$-0.694868\pi$
$521$	−13667.9			−1.14933			−0.574666	+	0.818388i	$0.694868\pi$
$522$		0			0
$523$	−22339.9			−1.86779			−0.933895	−	0.357548i	$-0.883613\pi$
$523$	−22339.9			−1.86779			−0.933895	+	0.357548i	$0.883613\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12758.6	−	22098.5i	−1.05460	−	1.82661i
$528$		0			0
$529$	−5573.74	+	9653.99i	−0.458103	+	0.793457i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−81.6215	+	141.373i	−0.00663306	+	0.0114888i
$534$		0			0
$535$	1926.75	+	3337.23i	0.155702	+	0.269684i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1026.39			−0.0820218
$540$		0			0
$541$	22676.0			1.80207			0.901034	−	0.433749i	$-0.142809\pi$
$541$	22676.0			1.80207			0.901034	+	0.433749i	$0.142809\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−643.735	−	1114.98i	−0.0505956	−	0.0876341i
$546$		0			0
$547$	−5016.26	+	8688.42i	−0.392102	+	0.679141i	−0.992727	−	0.120390i	$-0.961585\pi$
$547$	−5016.26	+	8688.42i	−0.392102	+	0.679141i	0.600624	+	0.799531i	$0.294919\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3427.01	−	5935.75i	0.264964	−	0.458932i
$552$		0			0
$553$	8263.19	+	14312.3i	0.635419	+	1.10058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12557.1			0.955224			0.477612	−	0.878571i	$-0.341502\pi$
$557$	12557.1			0.955224			0.477612	+	0.878571i	$0.341502\pi$
$558$		0			0
$559$	619.059			0.0468397
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10053.8	−	17413.8i	−0.752609	−	1.30356i	−0.946554	−	0.322545i	$-0.895462\pi$
$563$	−10053.8	−	17413.8i	−0.752609	−	1.30356i	0.193945	−	0.981012i	$-0.437872\pi$
$564$		0			0
$565$	−6538.48	+	11325.0i	−0.486860	+	0.843266i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2695.40	+	4668.58i	−0.198589	+	0.343966i	−0.948071	−	0.318058i	$-0.896969\pi$
$569$	−2695.40	+	4668.58i	−0.198589	+	0.343966i	0.749482	+	0.662025i	$0.230303\pi$
$570$		0			0
$571$	−7115.10	−	12323.7i	−0.521467	−	0.903207i	−0.999688	−	0.0249680i	$-0.992052\pi$
$571$	−7115.10	−	12323.7i	−0.521467	−	0.903207i	0.478221	−	0.878239i	$-0.341282\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1675.43			0.121514
$576$		0			0
$577$	−4781.42			−0.344979			−0.172490	−	0.985011i	$-0.555181\pi$
$577$	−4781.42			−0.344979			−0.172490	+	0.985011i	$0.555181\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4942.82	+	8561.21i	0.352948	+	0.611323i
$582$		0			0
$583$	1724.86	−	2987.55i	0.122533	−	0.212233i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6368.41	+	11030.4i	−0.447790	+	0.775594i	−0.998242	−	0.0592724i	$-0.981122\pi$
$587$	−6368.41	+	11030.4i	−0.447790	+	0.775594i	0.550452	+	0.834867i	$0.314455\pi$
$588$		0			0
$589$	−3662.45	−	6343.54i	−0.256211	−	0.443771i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11262.9			0.779954			0.389977	−	0.920825i	$-0.372483\pi$
$593$	11262.9			0.779954			0.389977	+	0.920825i	$0.372483\pi$
$594$		0			0
$595$	24913.4			1.71655
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1841.96	−	3190.37i	−0.125643	−	0.217621i	0.796341	−	0.604848i	$-0.206766\pi$
$599$	−1841.96	−	3190.37i	−0.125643	−	0.217621i	−0.921984	+	0.387227i	$0.873433\pi$
$600$		0			0
$601$	4712.16	−	8161.71i	0.319822	−	0.553948i	−0.660629	−	0.750713i	$-0.729710\pi$
$601$	4712.16	−	8161.71i	0.319822	−	0.553948i	0.980451	+	0.196765i	$0.0630435\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6977.72	−	12085.8i	0.468900	−	0.812159i
$606$		0			0
$607$	−1355.64	−	2348.03i	−0.0906486	−	0.157008i	0.817136	−	0.576445i	$-0.195561\pi$
$607$	−1355.64	−	2348.03i	−0.0906486	−	0.157008i	−0.907784	+	0.419438i	$0.862227\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−469.071			−0.0310582
$612$		0			0
$613$	−15014.7			−0.989298			−0.494649	−	0.869093i	$-0.664703\pi$
$613$	−15014.7			−0.989298			−0.494649	+	0.869093i	$0.664703\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12955.3	+	22439.3i	0.845319	+	1.46414i	0.885344	+	0.464937i	$0.153923\pi$
$617$	12955.3	+	22439.3i	0.845319	+	1.46414i	−0.0400251	+	0.999199i	$0.512744\pi$
$618$		0			0
$619$	−14573.2	+	25241.5i	−0.946278	+	1.63900i	−0.193105	+	0.981178i	$0.561856\pi$
$619$	−14573.2	+	25241.5i	−0.946278	+	1.63900i	−0.753173	+	0.657823i	$0.771478\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18913.8	−	32759.7i	1.21632	−	2.10673i
$624$		0			0
$625$	7066.50	+	12239.5i	0.452256	+	0.783331i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	35694.8			2.26271
$630$		0			0
$631$	−10221.9			−0.644893			−0.322446	−	0.946588i	$-0.604505\pi$
$631$	−10221.9			−0.644893			−0.322446	+	0.946588i	$0.604505\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	10632.8	+	18416.5i	0.664485	+	1.15092i
$636$		0			0
$637$	−157.252	+	272.369i	−0.00978111	+	0.0169414i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5870.53	+	10168.1i	−0.361735	+	0.626543i	−0.988246	−	0.152869i	$-0.951149\pi$
$641$	−5870.53	+	10168.1i	−0.361735	+	0.626543i	0.626512	+	0.779412i	$0.284482\pi$
$642$		0			0
$643$	−7861.89	−	13617.2i	−0.482181	−	0.835162i	0.517610	−	0.855617i	$-0.326822\pi$
$643$	−7861.89	−	13617.2i	−0.482181	−	0.835162i	−0.999791	+	0.0204546i	$0.993489\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6458.34			0.392432			0.196216	−	0.980561i	$-0.437135\pi$
$647$	6458.34			0.392432			0.196216	+	0.980561i	$0.437135\pi$
$648$		0			0
$649$	2352.87			0.142309
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1829.48	+	3168.76i	0.109637	+	0.189898i	0.915623	−	0.402037i	$-0.131698\pi$
$653$	1829.48	+	3168.76i	0.109637	+	0.189898i	−0.805986	+	0.591935i	$0.798364\pi$
$654$		0			0
$655$	8422.48	−	14588.2i	0.502433	−	0.870239i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5239.89	+	9075.76i	−0.309738	+	0.536482i	−0.978305	−	0.207170i	$-0.933575\pi$
$659$	−5239.89	+	9075.76i	−0.309738	+	0.536482i	0.668567	+	0.743652i	$0.266908\pi$
$660$		0			0
$661$	−9909.38	−	17163.5i	−0.583102	−	1.00996i	−0.995109	−	0.0987812i	$-0.968506\pi$
$661$	−9909.38	−	17163.5i	−0.583102	−	1.00996i	0.412008	−	0.911180i	$-0.364828\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7151.59			0.417032
$666$		0			0
$667$	36715.0			2.13135
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1223.74	+	2119.58i	0.0704054	+	0.121946i
$672$		0			0
$673$	13934.0	−	24134.4i	0.798094	−	1.38234i	−0.122762	−	0.992436i	$-0.539175\pi$
$673$	13934.0	−	24134.4i	0.798094	−	1.38234i	0.920856	−	0.389903i	$-0.127491\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7200.21	−	12471.1i	0.408754	−	0.707983i	−0.585996	−	0.810314i	$-0.699297\pi$
$677$	7200.21	−	12471.1i	0.408754	−	0.707983i	0.994750	+	0.102331i	$0.0326300\pi$
$678$		0			0
$679$	−4652.31	−	8058.04i	−0.262945	−	0.455433i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8387.44			0.469892			0.234946	−	0.972008i	$-0.424509\pi$
$683$	8387.44			0.469892			0.234946	+	0.972008i	$0.424509\pi$
$684$		0			0
$685$	8927.20			0.497942
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−528.530	−	915.441i	−0.0292241	−	0.0506176i
$690$		0			0
$691$	−10587.8	+	18338.7i	−0.582895	+	1.00960i	0.412239	+	0.911076i	$0.364747\pi$
$691$	−10587.8	+	18338.7i	−0.582895	+	1.00960i	−0.995134	+	0.0985283i	$0.968587\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14568.5	−	25233.4i	0.795131	−	1.37721i
$696$		0			0
$697$	−5386.96	−	9330.49i	−0.292749	−	0.507055i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7467.68			0.402354			0.201177	−	0.979555i	$-0.435523\pi$
$701$	7467.68			0.402354			0.201177	+	0.979555i	$0.435523\pi$
$702$		0			0
$703$	10246.5			0.549720
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3206.73	+	5554.22i	0.170582	+	0.295457i
$708$		0			0
$709$	−9532.58	+	16510.9i	−0.504942	+	0.874585i	0.495042	+	0.868869i	$0.335153\pi$
$709$	−9532.58	+	16510.9i	−0.504942	+	0.874585i	−0.999984	+	0.00571568i	$0.998181\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	19618.7	−	33980.6i	1.03047	−	1.78483i
$714$		0			0
$715$	39.4430	+	68.3173i	0.00206306	+	0.00357332i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6863.35			0.355994			0.177997	−	0.984031i	$-0.443038\pi$
$719$	6863.35			0.355994			0.177997	+	0.984031i	$0.443038\pi$
$720$		0			0
$721$	−9329.62			−0.481904
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1319.21	−	2284.95i	−0.0675784	−	0.117049i
$726$		0			0
$727$	−7391.78	+	12802.9i	−0.377092	+	0.653143i	−0.990638	−	0.136517i	$-0.956409\pi$
$727$	−7391.78	+	12802.9i	−0.377092	+	0.653143i	0.613546	+	0.789659i	$0.289743\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−20428.7	+	35383.6i	−1.03363	+	1.79030i
$732$		0			0
$733$	−2000.99	−	3465.82i	−0.100830	−	0.174642i	0.811197	−	0.584773i	$-0.198816\pi$
$733$	−2000.99	−	3465.82i	−0.100830	−	0.174642i	−0.912027	+	0.410131i	$0.865483\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1632.27			0.0815815
$738$		0			0
$739$	16889.8			0.840733			0.420367	−	0.907354i	$-0.361901\pi$
$739$	16889.8			0.840733			0.420367	+	0.907354i	$0.361901\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15258.1	−	26427.8i	−0.753386	−	1.30490i	−0.946173	−	0.323662i	$-0.895086\pi$
$743$	−15258.1	−	26427.8i	−0.753386	−	1.30490i	0.192787	−	0.981241i	$-0.438247\pi$
$744$		0			0
$745$	−14822.0	+	25672.4i	−0.728907	+	1.26250i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4239.40	−	7342.86i	0.206815	−	0.358214i
$750$		0			0
$751$	−6931.10	−	12005.0i	−0.336777	−	0.583314i	0.647048	−	0.762449i	$-0.276003\pi$
$751$	−6931.10	−	12005.0i	−0.336777	−	0.583314i	−0.983824	+	0.179135i	$0.942670\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8147.88			−0.392757
$756$		0			0
$757$	−15205.3			−0.730049			−0.365024	−	0.930998i	$-0.618939\pi$
$757$	−15205.3			−0.730049			−0.365024	+	0.930998i	$0.618939\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2420.42	+	4192.30i	0.115296	+	0.199699i	0.917898	−	0.396816i	$-0.129885\pi$
$761$	2420.42	+	4192.30i	0.115296	+	0.199699i	−0.802602	+	0.596515i	$0.796552\pi$
$762$		0			0
$763$	−1416.40	+	2453.28i	−0.0672047	+	0.116402i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	360.482	−	624.373i	0.0169703	−	0.0293935i
$768$		0			0
$769$	8528.08	+	14771.1i	0.399909	+	0.692663i	0.993714	−	0.111946i	$-0.0357083\pi$
$769$	8528.08	+	14771.1i	0.399909	+	0.692663i	−0.593805	+	0.804609i	$0.702375\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10350.7			0.481618			0.240809	−	0.970573i	$-0.422587\pi$
$773$	10350.7			0.481618			0.240809	+	0.970573i	$0.422587\pi$
$774$		0			0
$775$	−2819.69			−0.130692
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1546.37	−	2678.39i	−0.0711225	−	0.123188i
$780$		0			0
$781$	2172.05	−	3762.11i	0.0995163	−	0.172367i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3766.72	+	6524.15i	−0.171261	+	0.296633i
$786$		0			0
$787$	−3200.44	−	5543.32i	−0.144960	−	0.251078i	0.784398	−	0.620258i	$-0.212972\pi$
$787$	−3200.44	−	5543.32i	−0.144960	−	0.251078i	−0.929358	+	0.369180i	$0.879639\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	28773.1			1.29337
$792$		0			0
$793$	749.955			0.0335834
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15790.6	+	27350.1i	0.701795	+	1.21554i	0.967836	+	0.251582i	$0.0809509\pi$
$797$	15790.6	+	27350.1i	0.701795	+	1.21554i	−0.266041	+	0.963962i	$0.585716\pi$
$798$		0			0
$799$	15479.2	−	26810.7i	0.685374	−	1.18710i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	748.791	−	1296.94i	0.0329069	−	0.0569964i
$804$		0			0
$805$	19154.5	+	33176.6i	0.838643	+	1.45257i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3349.34			−0.145558			−0.0727791	−	0.997348i	$-0.523187\pi$
$809$	−3349.34			−0.145558			−0.0727791	+	0.997348i	$0.523187\pi$
$810$		0			0
$811$	5186.82			0.224579			0.112290	−	0.993676i	$-0.464182\pi$
$811$	5186.82			0.224579			0.112290	+	0.993676i	$0.464182\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8476.48	−	14681.7i	−0.364317	−	0.631015i
$816$		0			0
$817$	−5864.23	+	10157.1i	−0.251118	+	0.434949i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13332.5	+	23092.6i	−0.566758	+	0.981654i	0.430126	+	0.902769i	$0.358469\pi$
$821$	−13332.5	+	23092.6i	−0.566758	+	0.981654i	−0.996884	+	0.0788846i	$0.974864\pi$
$822$		0			0
$823$	13152.1	+	22780.2i	0.557053	+	0.964844i	0.997741	+	0.0671832i	$0.0214012\pi$
$823$	13152.1	+	22780.2i	0.557053	+	0.964844i	−0.440688	+	0.897660i	$0.645265\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−10768.0			−0.452769			−0.226385	−	0.974038i	$-0.572691\pi$
$827$	−10768.0			−0.452769			−0.226385	+	0.974038i	$0.572691\pi$
$828$		0			0
$829$	−11960.9			−0.501110			−0.250555	−	0.968102i	$-0.580613\pi$
$829$	−11960.9			−0.501110			−0.250555	+	0.968102i	$0.580613\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10378.6	−	17976.2i	−0.431687	−	0.747704i
$834$		0			0
$835$	3528.47	−	6111.49i	0.146237	−	0.253290i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7177.07	+	12431.1i	−0.295328	+	0.511523i	−0.975061	−	0.221937i	$-0.928762\pi$
$839$	−7177.07	+	12431.1i	−0.295328	+	0.511523i	0.679733	+	0.733459i	$0.262096\pi$
$840$		0			0
$841$	−16714.4	−	28950.2i	−0.685325	−	1.18702i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−23436.2			−0.954117
$846$		0			0
$847$	−30706.0			−1.24566
$848$		0			0
$849$		0			0
$850$		0			0
$851$	27443.7	+	47533.9i	1.10547	+	1.91474i
$852$		0			0
$853$	6065.55	−	10505.8i	0.243470	−	0.421703i	−0.718230	−	0.695806i	$-0.755047\pi$
$853$	6065.55	−	10505.8i	0.243470	−	0.421703i	0.961700	+	0.274103i	$0.0883808\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9376.46	+	16240.5i	−0.373739	+	0.647334i	−0.990137	−	0.140100i	$-0.955258\pi$
$857$	−9376.46	+	16240.5i	−0.373739	+	0.647334i	0.616399	+	0.787434i	$0.288591\pi$
$858$		0			0
$859$	5646.25	+	9779.58i	0.224269	+	0.388446i	0.956100	−	0.293041i	$-0.0946671\pi$
$859$	5646.25	+	9779.58i	0.224269	+	0.388446i	−0.731831	+	0.681487i	$0.761334\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	5959.72			0.235077			0.117538	−	0.993068i	$-0.462500\pi$
$863$	5959.72			0.235077			0.117538	+	0.993068i	$0.462500\pi$
$864$		0			0
$865$	−21755.8			−0.855166
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1726.85	−	2990.99i	−0.0674101	−	0.116758i
$870$		0			0
$871$	250.079	−	433.150i	0.00972861	−	0.0168504i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	17057.3	−	29544.1i	0.659019	−	1.14145i
$876$		0			0
$877$	−8912.47	−	15436.9i	−0.343162	−	0.594374i	0.641856	−	0.766825i	$-0.278165\pi$
$877$	−8912.47	−	15436.9i	−0.343162	−	0.594374i	−0.985018	+	0.172451i	$0.944831\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−44756.5			−1.71156			−0.855781	−	0.517339i	$-0.826923\pi$
$881$	−44756.5			−1.71156			−0.855781	+	0.517339i	$0.826923\pi$
$882$		0			0
$883$	−7241.57			−0.275989			−0.137995	−	0.990433i	$-0.544066\pi$
$883$	−7241.57			−0.275989			−0.137995	+	0.990433i	$0.544066\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16668.9	−	28871.4i	−0.630990	−	1.09291i	−0.987350	−	0.158557i	$-0.949316\pi$
$887$	−16668.9	−	28871.4i	−0.630990	−	1.09291i	0.356360	−	0.934349i	$-0.384018\pi$
$888$		0			0
$889$	23395.1	−	40521.5i	0.882618	−	1.52874i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4443.42	−	7696.23i	0.166510	−	0.288404i
$894$		0			0
$895$	13769.8	+	23850.0i	0.514273	+	0.890746i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−61789.9			−2.29234
$900$		0			0
$901$	69765.2			2.57960
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9886.31	−	17123.6i	−0.363129	−	0.628959i
$906$		0			0
$907$	9254.13	−	16028.6i	0.338785	−	0.586794i	−0.645419	−	0.763829i	$-0.723317\pi$
$907$	9254.13	−	16028.6i	0.338785	−	0.586794i	0.984205	+	0.177035i	$0.0566506\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7352.80	+	12735.4i	−0.267409	+	0.463165i	−0.968192	−	0.250209i	$-0.919501\pi$
$911$	−7352.80	+	12735.4i	−0.267409	+	0.463165i	0.700783	+	0.713374i	$0.252834\pi$
$912$		0			0
$913$	−1032.95	−	1789.13i	−0.0374434	−	0.0648538i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−37063.8			−1.33474
$918$		0			0
$919$	−44333.5			−1.59132			−0.795662	−	0.605741i	$-0.792877\pi$
$919$	−44333.5			−1.59132			−0.795662	+	0.605741i	$0.792877\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−665.558	−	1152.78i	−0.0237347	−	0.0411096i
$924$		0			0
$925$	1972.17	−	3415.90i	0.0701023	−	0.121421i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15114.0	+	26178.3i	−0.533773	+	0.924522i	0.465449	+	0.885075i	$0.345893\pi$
$929$	−15114.0	+	26178.3i	−0.533773	+	0.924522i	−0.999222	+	0.0394470i	$0.987440\pi$
$930$		0			0
$931$	−2979.24	−	5160.20i	−0.104877	−	0.181653i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5206.43			−0.182105
$936$		0			0
$937$	22568.0			0.786834			0.393417	−	0.919360i	$-0.371293\pi$
$937$	22568.0			0.786834			0.393417	+	0.919360i	$0.371293\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−17542.5	−	30384.6i	−0.607727	−	1.05261i	−0.991614	−	0.129233i	$-0.958748\pi$
$941$	−17542.5	−	30384.6i	−0.607727	−	1.05261i	0.383888	−	0.923380i	$-0.374585\pi$
$942$		0			0
$943$	8283.46	−	14347.4i	0.286052	−	0.495456i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17124.1	+	29659.7i	−0.587600	+	1.01775i	0.406946	+	0.913452i	$0.366594\pi$
$947$	−17124.1	+	29659.7i	−0.587600	+	1.01775i	−0.994546	+	0.104301i	$0.966740\pi$
$948$		0			0
$949$	−229.443	−	397.407i	−0.00784831	−	0.0135937i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	38627.8			1.31299			0.656494	−	0.754331i	$-0.272039\pi$
$953$	38627.8			1.31299			0.656494	+	0.754331i	$0.272039\pi$
$954$		0			0
$955$	608.304			0.0206118
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9821.20	−	17010.8i	−0.330702	−	0.572792i
$960$		0			0
$961$	−18122.0	+	31388.2i	−0.608304	+	1.05361i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9685.41	+	16775.6i	−0.323093	+	0.559613i
$966$		0			0
$967$	7814.39	+	13534.9i	0.259869	+	0.450107i	0.966207	−	0.257768i	$-0.0829871\pi$
$967$	7814.39	+	13534.9i	0.259869	+	0.450107i	−0.706337	+	0.707875i	$0.749654\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−26637.8			−0.880379			−0.440190	−	0.897905i	$-0.645089\pi$
$971$	−26637.8			−0.880379			−0.440190	+	0.897905i	$0.645089\pi$
$972$		0			0
$973$	−64109.9			−2.11230
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−888.789	−	1539.43i	−0.0291043	−	0.0504101i	0.851106	−	0.524993i	$-0.175932\pi$
$977$	−888.789	−	1539.43i	−0.0291043	−	0.0504101i	−0.880211	+	0.474583i	$0.842599\pi$
$978$		0			0
$979$	−3952.63	+	6846.15i	−0.129036	+	0.223498i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	23817.2	−	41252.6i	0.772788	−	1.33851i	−0.163241	−	0.986586i	$-0.552195\pi$
$983$	23817.2	−	41252.6i	0.772788	−	1.33851i	0.936029	−	0.351922i	$-0.114472\pi$
$984$		0			0
$985$	4662.89	+	8076.36i	0.150834	+	0.261253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−62826.0			−2.01997
$990$		0			0
$991$	−33719.5			−1.08086			−0.540432	−	0.841388i	$-0.681739\pi$
$991$	−33719.5			−1.08086			−0.540432	+	0.841388i	$0.681739\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	27035.5	+	46826.8i	0.861389	+	1.49197i
$996$		0			0
$997$	200.389	−	347.083i	0.00636547	−	0.0110253i	−0.862825	−	0.505503i	$-0.831307\pi$
$997$	200.389	−	347.083i	0.00636547	−	0.0110253i	0.869191	+	0.494477i	$0.164640\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.e.i.217.2		8
3.2	odd	2	inner	324.4.e.i.217.3		8
9.2	odd	6		324.4.a.e.1.2	✓	4
9.4	even	3	inner	324.4.e.i.109.2		8
9.5	odd	6	inner	324.4.e.i.109.3		8
9.7	even	3		324.4.a.e.1.3	yes	4
36.7	odd	6		1296.4.a.z.1.3		4
36.11	even	6		1296.4.a.z.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.4.a.e.1.2	✓	4	9.2	odd	6
324.4.a.e.1.3	yes	4	9.7	even	3
324.4.e.i.109.2		8	9.4	even	3	inner
324.4.e.i.109.3		8	9.5	odd	6	inner
324.4.e.i.217.2		8	1.1	even	1	trivial
324.4.e.i.217.3		8	3.2	odd	2	inner
1296.4.a.z.1.2		4	36.11	even	6
1296.4.a.z.1.3		4	36.7	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.33918 - 9.24773i) q^{5} +(-11.7477 + 20.3477i) q^{7} +(2.45505 - 4.25227i) q^{11} +(-0.752273 - 1.30297i) q^{13} +99.2990 q^{17} +28.5045 q^{19} +(76.3454 + 132.234i) q^{23} +(5.48636 - 9.50266i) q^{25} +(120.227 - 208.239i) q^{29} +(-128.486 - 222.545i) q^{31} +250.893 q^{35} +359.468 q^{37} +(-54.2499 - 93.9636i) q^{41} +(-205.730 + 356.334i) q^{43} +(155.885 - 270.000i) q^{47} +(-104.518 - 181.031i) q^{49} +702.578 q^{53} -52.4318 q^{55} +(239.595 + 414.991i) q^{59} +(-249.230 + 431.678i) q^{61} +(-8.03304 + 13.9136i) q^{65} +(166.216 + 287.894i) q^{67} +884.729 q^{71} +305.000 q^{73} +(57.6825 + 99.9091i) q^{77} +(351.693 - 609.150i) q^{79} +(210.373 - 364.377i) q^{83} +(-530.175 - 918.290i) q^{85} -1610.00 q^{89} +35.3500 q^{91} +(-152.191 - 263.602i) q^{95} +(-198.009 + 342.962i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} - 116 q^{13} + 448 q^{19} - 616 q^{25} - 368 q^{31} + 1336 q^{37} - 656 q^{43} - 1716 q^{49} + 2880 q^{55} - 1004 q^{61} - 320 q^{67} + 2440 q^{73} + 64 q^{79} - 612 q^{85} - 6976 q^{91}+ \cdots - 2024 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 - 116 * q^13 + 448 * q^19 - 616 * q^25 - 368 * q^31 + 1336 * q^37 - 656 * q^43 - 1716 * q^49 + 2880 * q^55 - 1004 * q^61 - 320 * q^67 + 2440 * q^73 + 64 * q^79 - 612 * q^85 - 6976 * q^91 - 2024 * q^97

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