LMFDB - Embedded newform 648.4.i.p.433.2

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-43})$
Defining polynomial:	$x^{4} - x^{3} - 10 x^{2} - 11 x + 121$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.2
Root			$3.08945 - 1.20635i$ of defining polynomial
Character	$\chi$	$=$	648.433
Dual form			648.4.i.p.217.2

$q$-expansion

$f(q)$	$=$	$q+(4.67891 - 8.10411i) q^{5} +(-4.17891 - 7.23808i) q^{7} +O(q^{10})$	$q+(4.67891 - 8.10411i) q^{5} +(-4.17891 - 7.23808i) q^{7} +(14.5367 + 25.1783i) q^{11} +(-18.3945 + 31.8603i) q^{13} +28.4313 q^{17} +73.7891 q^{19} +(-15.2524 + 26.4179i) q^{23} +(18.7156 + 32.4164i) q^{25} +(92.4680 + 160.159i) q^{29} +(95.4313 - 165.292i) q^{31} -78.2109 q^{35} +160.495 q^{37} +(104.789 - 181.500i) q^{41} +(-58.4727 - 101.278i) q^{43} +(-140.799 - 243.870i) q^{47} +(136.573 - 236.552i) q^{49} -397.450 q^{53} +272.064 q^{55} +(189.073 - 327.485i) q^{59} +(148.762 + 257.663i) q^{61} +(172.133 + 298.143i) q^{65} +(413.968 - 717.014i) q^{67} -729.073 q^{71} +1105.44 q^{73} +(121.495 - 210.436i) q^{77} +(532.134 + 921.683i) q^{79} +(295.239 + 511.369i) q^{83} +(133.027 - 230.410i) q^{85} +227.588 q^{89} +307.476 q^{91} +(345.252 - 597.995i) q^{95} +(-642.827 - 1113.41i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4q - 4q^{5} + 6q^{7} + O(q^{10}) $	$ 4q - 4q^{5} + 6q^{7} - 10q^{11} + 40q^{13} - 68q^{17} + 68q^{19} + 98q^{23} - 16q^{25} + 120q^{29} + 200q^{31} - 540q^{35} + 960q^{37} + 192q^{41} + 334q^{43} + 300q^{47} + 410q^{49} - 136q^{53} + 1588q^{55} + 620q^{59} - 200q^{61} + 1370q^{65} + 1406q^{67} - 2780q^{71} + 3604q^{73} + 804q^{77} + 334q^{79} - 500q^{83} + 1100q^{85} - 180q^{89} + 2820q^{91} + 1222q^{95} + 200q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	4.67891	−	8.10411i	0.418494	−	0.724853i	−0.577294	−	0.816536i	$-0.695891\pi$
$5$	4.67891	−	8.10411i	0.418494	−	0.724853i	0.995788	+	0.0916830i	$0.0292247\pi$
$6$		0			0
$7$	−4.17891	−	7.23808i	−0.225640	−	0.390820i	0.730871	−	0.682515i	$-0.239114\pi$
$7$	−4.17891	−	7.23808i	−0.225640	−	0.390820i	−0.956511	+	0.291696i	$0.905781\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	14.5367	+	25.1783i	0.398453	+	0.690142i	0.993535	−	0.113524i	$-0.0362137\pi$
$11$	14.5367	+	25.1783i	0.398453	+	0.690142i	−0.595082	+	0.803665i	$0.702880\pi$
$12$		0			0
$13$	−18.3945	+	31.8603i	−0.392441	+	0.679727i	−0.992771	−	0.120025i	$-0.961702\pi$
$13$	−18.3945	+	31.8603i	−0.392441	+	0.679727i	0.600330	+	0.799752i	$0.295036\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	28.4313			0.405623			0.202812	−	0.979218i	$-0.434992\pi$
$17$	28.4313			0.405623			0.202812	+	0.979218i	$0.434992\pi$
$18$		0			0
$19$	73.7891			0.890967			0.445484	−	0.895290i	$-0.353032\pi$
$19$	73.7891			0.890967			0.445484	+	0.895290i	$0.353032\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−15.2524	+	26.4179i	−0.138276	+	0.239500i	−0.926844	−	0.375447i	$-0.877489\pi$
$23$	−15.2524	+	26.4179i	−0.138276	+	0.239500i	0.788568	+	0.614947i	$0.210823\pi$
$24$		0			0
$25$	18.7156	+	32.4164i	0.149725	+	0.259331i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	92.4680	+	160.159i	0.592099	+	1.02555i	0.993949	+	0.109840i	$0.0350340\pi$
$29$	92.4680	+	160.159i	0.592099	+	1.02555i	−0.401850	+	0.915705i	$0.631633\pi$
$30$		0			0
$31$	95.4313	−	165.292i	0.552902	−	0.957654i	−0.445161	−	0.895450i	$-0.646854\pi$
$31$	95.4313	−	165.292i	0.552902	−	0.957654i	0.998063	−	0.0622040i	$-0.0198129\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−78.2109			−0.377716
$36$		0			0
$37$	160.495			0.713115			0.356558	−	0.934273i	$-0.383950\pi$
$37$	160.495			0.713115			0.356558	+	0.934273i	$0.383950\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	104.789	−	181.500i	0.399154	−	0.691355i	−0.594468	−	0.804119i	$-0.702637\pi$
$41$	104.789	−	181.500i	0.399154	−	0.691355i	0.993622	+	0.112765i	$0.0359706\pi$
$42$		0			0
$43$	−58.4727	−	101.278i	−0.207372	−	0.359179i	0.743514	−	0.668721i	$-0.233158\pi$
$43$	−58.4727	−	101.278i	−0.207372	−	0.359179i	−0.950886	+	0.309541i	$0.899824\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−140.799	−	243.870i	−0.436970	−	0.756854i	0.560484	−	0.828165i	$-0.310615\pi$
$47$	−140.799	−	243.870i	−0.436970	−	0.756854i	−0.997454	+	0.0713113i	$0.977282\pi$
$48$		0			0
$49$	136.573	−	236.552i	0.398173	−	0.689656i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−397.450			−1.03007			−0.515037	−	0.857168i	$-0.672222\pi$
$53$	−397.450			−1.03007			−0.515037	+	0.857168i	$0.672222\pi$
$54$		0			0
$55$	272.064			0.667002
$56$		0			0
$57$		0			0
$58$		0			0
$59$	189.073	−	327.485i	0.417208	−	0.722625i	−0.578450	−	0.815718i	$-0.696342\pi$
$59$	189.073	−	327.485i	0.417208	−	0.722625i	0.995657	+	0.0930929i	$0.0296754\pi$
$60$		0			0
$61$	148.762	+	257.663i	0.312246	+	0.540826i	0.978848	−	0.204588i	$-0.0655854\pi$
$61$	148.762	+	257.663i	0.312246	+	0.540826i	−0.666602	+	0.745413i	$0.732252\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	172.133	+	298.143i	0.328468	+	0.568924i
$66$		0			0
$67$	413.968	−	717.014i	0.754839	−	1.30742i	−0.190615	−	0.981665i	$-0.561048\pi$
$67$	413.968	−	717.014i	0.754839	−	1.30742i	0.945454	−	0.325755i	$-0.105619\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−729.073			−1.21866			−0.609332	−	0.792915i	$-0.708562\pi$
$71$	−729.073			−1.21866			−0.609332	+	0.792915i	$0.708562\pi$
$72$		0			0
$73$	1105.44			1.77236			0.886178	−	0.463344i	$-0.153351\pi$
$73$	1105.44			1.77236			0.886178	+	0.463344i	$0.153351\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	121.495	−	210.436i	0.179814	−	0.311447i
$78$		0			0
$79$	532.134	+	921.683i	0.757845	+	1.31263i	0.943948	+	0.330095i	$0.107081\pi$
$79$	532.134	+	921.683i	0.757845	+	1.31263i	−0.186103	+	0.982530i	$0.559586\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	295.239	+	511.369i	0.390442	+	0.676266i	0.992508	−	0.122181i	$-0.0389887\pi$
$83$	295.239	+	511.369i	0.390442	+	0.676266i	−0.602066	+	0.798447i	$0.705655\pi$
$84$		0			0
$85$	133.027	−	230.410i	0.169751	−	0.294017i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	227.588			0.271059			0.135529	−	0.990773i	$-0.456726\pi$
$89$	227.588			0.271059			0.135529	+	0.990773i	$0.456726\pi$
$90$		0			0
$91$	307.476			0.354201
$92$		0			0
$93$		0			0
$94$		0			0
$95$	345.252	−	597.995i	0.372865	−	0.645821i
$96$		0			0
$97$	−642.827	−	1113.41i	−0.672878	−	1.16546i	−0.977084	−	0.212853i	$-0.931725\pi$
$97$	−642.827	−	1113.41i	−0.672878	−	1.16546i	0.304206	−	0.952606i	$-0.401609\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	842.367	+	1459.02i	0.829888	+	1.43741i	0.898126	+	0.439739i	$0.144929\pi$
$101$	842.367	+	1459.02i	0.829888	+	1.43741i	−0.0682378	+	0.997669i	$0.521738\pi$
$102$		0			0
$103$	427.201	−	739.935i	0.408674	−	0.707844i	−0.586068	−	0.810262i	$-0.699325\pi$
$103$	427.201	−	739.935i	0.408674	−	0.707844i	0.994741	+	0.102418i	$0.0326580\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1369.73			−1.23754			−0.618768	−	0.785574i	$-0.712368\pi$
$107$	−1369.73			−1.23754			−0.618768	+	0.785574i	$0.712368\pi$
$108$		0			0
$109$	1475.06			1.29620			0.648099	−	0.761556i	$-0.275564\pi$
$109$	1475.06			1.29620			0.648099	+	0.761556i	$0.275564\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.50943	+	7.81057i	−0.00375409	+	0.00650227i	−0.867896	−	0.496745i	$-0.834528\pi$
$113$	−4.50943	+	7.81057i	−0.00375409	+	0.00650227i	0.864142	+	0.503248i	$0.167862\pi$
$114$		0			0
$115$	142.729	+	247.213i	0.115735	+	0.200459i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−118.812	−	205.788i	−0.0915248	−	0.158526i
$120$		0			0
$121$	242.867	−	420.658i	0.182470	−	0.316047i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1520.00			1.08762
$126$		0			0
$127$	2076.05			1.45055			0.725274	−	0.688460i	$-0.241713\pi$
$127$	2076.05			1.45055			0.725274	+	0.688460i	$0.241713\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	887.207	−	1536.69i	0.591723	−	1.02489i	−0.402278	−	0.915518i	$-0.631781\pi$
$131$	887.207	−	1536.69i	0.591723	−	1.02489i	0.994000	−	0.109376i	$-0.0348852\pi$
$132$		0			0
$133$	−308.358	−	534.091i	−0.201038	−	0.348208i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	605.344	+	1048.49i	0.377504	+	0.653855i	0.990698	−	0.136076i	$-0.0434492\pi$
$137$	605.344	+	1048.49i	0.377504	+	0.653855i	−0.613195	+	0.789932i	$0.710116\pi$
$138$		0			0
$139$	366.614	−	634.994i	0.223711	−	0.387478i	−0.732221	−	0.681067i	$-0.761516\pi$
$139$	366.614	−	634.994i	0.223711	−	0.387478i	0.955932	+	0.293589i	$0.0948496\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1069.59			−0.625477
$144$		0			0
$145$	1730.60			0.991160
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−257.781	+	446.489i	−0.141733	+	0.245489i	−0.928149	−	0.372208i	$-0.878601\pi$
$149$	−257.781	+	446.489i	−0.141733	+	0.245489i	0.786416	+	0.617697i	$0.211934\pi$
$150$		0			0
$151$	576.945	+	999.299i	0.310935	+	0.538555i	0.978565	−	0.205938i	$-0.0660246\pi$
$151$	576.945	+	999.299i	0.310935	+	0.538555i	−0.667630	+	0.744493i	$0.732691\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−893.028	−	1546.77i	−0.462773	−	0.801546i
$156$		0			0
$157$	684.754	−	1186.03i	0.348085	−	0.602901i	−0.637824	−	0.770182i	$-0.720165\pi$
$157$	684.754	−	1186.03i	0.348085	−	0.602901i	0.985909	+	0.167281i	$0.0534987\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	254.953			0.124802
$162$		0			0
$163$	−3796.90			−1.82452			−0.912259	−	0.409613i	$-0.865664\pi$
$163$	−3796.90			−1.82452			−0.912259	+	0.409613i	$0.865664\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1024.45	+	1774.39i	−0.474695	+	0.822196i	−0.999580	−	0.0289774i	$-0.990775\pi$
$167$	−1024.45	+	1774.39i	−0.474695	+	0.822196i	0.524885	+	0.851173i	$0.324108\pi$
$168$		0			0
$169$	421.782	+	730.547i	0.191981	+	0.332520i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1012.61	+	1753.90i	0.445016	+	0.770789i	0.998053	−	0.0623667i	$-0.0198648\pi$
$173$	1012.61	+	1753.90i	0.445016	+	0.770789i	−0.553038	+	0.833156i	$0.686531\pi$
$174$		0			0
$175$	156.422	−	270.931i	0.0675679	−	0.117031i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3142.50			−1.31219			−0.656094	−	0.754679i	$-0.727792\pi$
$179$	−3142.50			−1.31219			−0.656094	+	0.754679i	$0.727792\pi$
$180$		0			0
$181$	37.6685			0.0154689			0.00773446	−	0.999970i	$-0.497538\pi$
$181$	37.6685			0.0154689			0.00773446	+	0.999970i	$0.497538\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	750.943	−	1300.67i	0.298435	−	0.516904i
$186$		0			0
$187$	413.298	+	715.852i	0.161622	+	0.279937i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2117.06	−	3666.86i	−0.802016	−	1.38913i	−0.918287	−	0.395915i	$-0.870427\pi$
$191$	−2117.06	−	3666.86i	−0.802016	−	1.38913i	0.116271	−	0.993218i	$-0.462906\pi$
$192$		0			0
$193$	−1914.65	+	3316.27i	−0.714090	+	1.23684i	0.249219	+	0.968447i	$0.419826\pi$
$193$	−1914.65	+	3316.27i	−0.714090	+	1.23684i	−0.963309	+	0.268394i	$0.913507\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1072.38			0.387838			0.193919	−	0.981018i	$-0.437880\pi$
$197$	1072.38			0.387838			0.193919	+	0.981018i	$0.437880\pi$
$198$		0			0
$199$	−1078.53			−0.384196			−0.192098	−	0.981376i	$-0.561529\pi$
$199$	−1078.53			−0.384196			−0.192098	+	0.981376i	$0.561529\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	772.831	−	1338.58i	0.267202	−	0.462808i
$204$		0			0
$205$	−980.597	−	1698.44i	−0.334087	−	0.578656i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1072.65	+	1857.89i	0.355009	+	0.614893i
$210$		0			0
$211$	1808.08	−	3131.69i	0.589922	−	1.02177i	−0.404321	−	0.914617i	$-0.632492\pi$
$211$	1808.08	−	3131.69i	0.589922	−	1.02177i	0.994242	−	0.107157i	$-0.0341747\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1094.35			−0.347136
$216$		0			0
$217$	−1595.19			−0.499027
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−522.980	+	905.828i	−0.159183	+	0.275713i
$222$		0			0
$223$	430.574	+	745.777i	0.129298	+	0.223950i	0.923405	−	0.383828i	$-0.125394\pi$
$223$	430.574	+	745.777i	0.129298	+	0.223950i	−0.794107	+	0.607778i	$0.792061\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1542.98	−	2672.52i	−0.451150	−	0.781414i	0.547308	−	0.836931i	$-0.315653\pi$
$227$	−1542.98	−	2672.52i	−0.451150	−	0.781414i	−0.998458	+	0.0555169i	$0.982319\pi$
$228$		0			0
$229$	−3095.65	+	5361.81i	−0.893301	+	1.54724i	−0.0574083	+	0.998351i	$0.518284\pi$
$229$	−3095.65	+	5361.81i	−0.893301	+	1.54724i	−0.835893	+	0.548892i	$0.815050\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4325.48			1.21619			0.608093	−	0.793866i	$-0.291935\pi$
$233$	4325.48			1.21619			0.608093	+	0.793866i	$0.291935\pi$
$234$		0			0
$235$	−2635.13			−0.731477
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1559.24	+	2700.68i	−0.422003	+	0.730931i	−0.996135	−	0.0878315i	$-0.972006\pi$
$239$	−1559.24	+	2700.68i	−0.422003	+	0.730931i	0.574132	+	0.818763i	$0.305340\pi$
$240$		0			0
$241$	−361.462	−	626.071i	−0.0966134	−	0.167339i	0.813667	−	0.581331i	$-0.197468\pi$
$241$	−361.462	−	626.071i	−0.0966134	−	0.167339i	−0.910281	+	0.413991i	$0.864134\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1278.03	−	2213.61i	−0.333267	−	0.577235i
$246$		0			0
$247$	−1357.32	+	2350.94i	−0.349652	+	0.605615i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6373.89			1.60285			0.801427	−	0.598093i	$-0.204075\pi$
$251$	6373.89			1.60285			0.801427	+	0.598093i	$0.204075\pi$
$252$		0			0
$253$	−886.877			−0.220385
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1902.41	+	3295.07i	−0.461748	+	0.799771i	−0.999048	−	0.0436203i	$-0.986111\pi$
$257$	−1902.41	+	3295.07i	−0.461748	+	0.799771i	0.537300	+	0.843391i	$0.319444\pi$
$258$		0			0
$259$	−670.695	−	1161.68i	−0.160907	−	0.278699i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−729.311	−	1263.20i	−0.170993	−	0.296169i	0.767774	−	0.640721i	$-0.221364\pi$
$263$	−729.311	−	1263.20i	−0.170993	−	0.296169i	−0.938767	+	0.344552i	$0.888031\pi$
$264$		0			0
$265$	−1859.63	+	3220.98i	−0.431080	+	0.746653i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2240.99			0.507940			0.253970	−	0.967212i	$-0.418264\pi$
$269$	2240.99			0.507940			0.253970	+	0.967212i	$0.418264\pi$
$270$		0			0
$271$	−3077.92			−0.689927			−0.344963	−	0.938616i	$-0.612109\pi$
$271$	−3077.92			−0.689927			−0.344963	+	0.938616i	$0.612109\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−544.128	+	942.457i	−0.119317	+	0.206663i
$276$		0			0
$277$	−1545.35	−	2676.63i	−0.335203	−	0.580588i	0.648321	−	0.761367i	$-0.275471\pi$
$277$	−1545.35	−	2676.63i	−0.335203	−	0.580588i	−0.983524	+	0.180779i	$0.942138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2420.90	+	4193.12i	0.513946	+	0.890180i	0.999869	+	0.0161788i	$0.00515008\pi$
$281$	2420.90	+	4193.12i	0.513946	+	0.890180i	−0.485923	+	0.874001i	$0.661517\pi$
$282$		0			0
$283$	−359.725	+	623.062i	−0.0755598	+	0.130873i	−0.901330	−	0.433134i	$-0.857408\pi$
$283$	−359.725	+	623.062i	−0.0755598	+	0.130873i	0.825770	+	0.564007i	$0.190741\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1751.62			−0.360260
$288$		0			0
$289$	−4104.66			−0.835470
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2625.01	+	4546.65i	−0.523395	+	0.906548i	0.476234	+	0.879319i	$0.342002\pi$
$293$	−2625.01	+	4546.65i	−0.523395	+	0.906548i	−0.999629	+	0.0272288i	$0.991332\pi$
$294$		0			0
$295$	−1769.31	−	3064.54i	−0.349198	−	0.604829i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−561.120	−	971.889i	−0.108530	−	0.187979i
$300$		0			0
$301$	−488.704	+	846.460i	−0.0935829	+	0.162090i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2784.17			0.522692
$306$		0			0
$307$	−5823.45			−1.08261			−0.541306	−	0.840826i	$-0.682070\pi$
$307$	−5823.45			−1.08261			−0.541306	+	0.840826i	$0.682070\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	196.407	−	340.187i	0.0358110	−	0.0620265i	−0.847564	−	0.530693i	$-0.821932\pi$
$311$	196.407	−	340.187i	0.0358110	−	0.0620265i	0.883375	+	0.468666i	$0.155265\pi$
$312$		0			0
$313$	−558.235	−	966.891i	−0.100809	−	0.174607i	0.811209	−	0.584756i	$-0.198810\pi$
$313$	−558.235	−	966.891i	−0.100809	−	0.174607i	−0.912018	+	0.410150i	$0.865476\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	654.340	+	1133.35i	0.115935	+	0.200805i	0.918153	−	0.396226i	$-0.129680\pi$
$317$	654.340	+	1133.35i	0.115935	+	0.200805i	−0.802218	+	0.597031i	$0.796347\pi$
$318$		0			0
$319$	−2688.36	+	4656.38i	−0.471848	+	0.817264i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2097.92			0.361397
$324$		0			0
$325$	−1377.06			−0.235033
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1176.77	+	2038.22i	−0.197196	+	0.341553i
$330$		0			0
$331$	3099.55	+	5368.58i	0.514703	+	0.891492i	0.999854	+	0.0170616i	$0.00543113\pi$
$331$	3099.55	+	5368.58i	0.514703	+	0.891492i	−0.485151	+	0.874430i	$0.661236\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3873.84	−	6709.68i	−0.631792	−	1.09430i
$336$		0			0
$337$	−403.851	+	699.491i	−0.0652794	+	0.113067i	−0.896818	−	0.442400i	$-0.854127\pi$
$337$	−403.851	+	699.491i	−0.0652794	+	0.113067i	0.831538	+	0.555467i	$0.187461\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5549.03			0.881223
$342$		0			0
$343$	−5149.64			−0.810655
$344$		0			0
$345$		0			0
$346$		0			0
$347$	746.608	−	1293.16i	0.115504	−	0.200059i	−0.802477	−	0.596683i	$-0.796485\pi$
$347$	746.608	−	1293.16i	0.115504	−	0.200059i	0.917981	+	0.396624i	$0.129818\pi$
$348$		0			0
$349$	249.305	+	431.809i	0.0382378	+	0.0662299i	0.884511	−	0.466519i	$-0.154492\pi$
$349$	249.305	+	431.809i	0.0382378	+	0.0662299i	−0.846273	+	0.532749i	$0.821159\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6002.66	−	10396.9i	−0.905069	−	1.56762i	−0.820825	−	0.571180i	$-0.806486\pi$
$353$	−6002.66	−	10396.9i	−0.905069	−	1.56762i	−0.0842436	−	0.996445i	$-0.526847\pi$
$354$		0			0
$355$	−3411.27	+	5908.49i	−0.510004	+	0.883352i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6842.79			−1.00599			−0.502993	−	0.864291i	$-0.667768\pi$
$359$	−6842.79			−1.00599			−0.502993	+	0.864291i	$0.667768\pi$
$360$		0			0
$361$	−1414.17			−0.206177
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5172.26	−	8958.61i	0.741721	−	1.28470i
$366$		0			0
$367$	56.0606	+	97.0999i	0.00797368	+	0.0138108i	0.869985	−	0.493079i	$-0.164129\pi$
$367$	56.0606	+	97.0999i	0.00797368	+	0.0138108i	−0.862011	+	0.506890i	$0.830795\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1660.91	+	2876.78i	0.232426	+	0.402574i
$372$		0			0
$373$	−988.085	+	1711.41i	−0.137161	+	0.237570i	−0.926421	−	0.376489i	$-0.877131\pi$
$373$	−988.085	+	1711.41i	−0.137161	+	0.237570i	0.789260	+	0.614059i	$0.210464\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6803.63			−0.929455
$378$		0			0
$379$	−669.717			−0.0907679			−0.0453840	−	0.998970i	$-0.514451\pi$
$379$	−669.717			−0.0907679			−0.0453840	+	0.998970i	$0.514451\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5620.25	+	9734.57i	−0.749821	+	1.29873i	0.198087	+	0.980185i	$0.436527\pi$
$383$	−5620.25	+	9734.57i	−0.749821	+	1.29873i	−0.947908	+	0.318544i	$0.896806\pi$
$384$		0			0
$385$	−1136.93	−	1969.22i	−0.150502	−	0.260677i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1882.99	−	3261.43i	−0.245427	−	0.425092i	0.716825	−	0.697254i	$-0.245595\pi$
$389$	−1882.99	−	3261.43i	−0.245427	−	0.425092i	−0.962252	+	0.272161i	$0.912262\pi$
$390$		0			0
$391$	−433.644	+	751.093i	−0.0560878	+	0.0971468i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9959.22			1.26861
$396$		0			0
$397$	−2005.69			−0.253558			−0.126779	−	0.991931i	$-0.540464\pi$
$397$	−2005.69			−0.253558			−0.126779	+	0.991931i	$0.540464\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2777.83	+	4811.34i	−0.345930	+	0.599169i	−0.985522	−	0.169546i	$-0.945770\pi$
$401$	−2777.83	+	4811.34i	−0.345930	+	0.599169i	0.639592	+	0.768715i	$0.279103\pi$
$402$		0			0
$403$	3510.83	+	6080.93i	0.433962	+	0.751645i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2333.08	+	4041.01i	0.284143	+	0.492150i
$408$		0			0
$409$	7438.24	−	12883.4i	0.899260	−	1.55756i	0.0708190	−	0.997489i	$-0.477439\pi$
$409$	7438.24	−	12883.4i	0.899260	−	1.55756i	0.828441	−	0.560076i	$-0.189228\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3160.48			−0.376555
$414$		0			0
$415$	5525.59			0.653592
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4749.82	−	8226.92i	0.553803	−	0.959216i	−0.444192	−	0.895932i	$-0.646509\pi$
$419$	4749.82	−	8226.92i	0.553803	−	0.959216i	0.997996	−	0.0632842i	$-0.0201575\pi$
$420$		0			0
$421$	2998.70	+	5193.90i	0.347144	+	0.601271i	0.985741	−	0.168270i	$-0.0538182\pi$
$421$	2998.70	+	5193.90i	0.347144	+	0.601271i	−0.638597	+	0.769541i	$0.720485\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	532.109	+	921.640i	0.0607320	+	0.105191i
$426$		0			0
$427$	1243.32	−	2153.50i	0.140910	−	0.244064i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15607.1			−1.74424			−0.872121	−	0.489291i	$-0.837256\pi$
$431$	−15607.1			−1.74424			−0.872121	+	0.489291i	$0.837256\pi$
$432$		0			0
$433$	−1204.46			−0.133678			−0.0668390	−	0.997764i	$-0.521291\pi$
$433$	−1204.46			−0.133678			−0.0668390	+	0.997764i	$0.521291\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1125.46	+	1949.35i	−0.123199	+	0.213387i
$438$		0			0
$439$	−8492.29	−	14709.1i	−0.923269	−	1.59915i	−0.794322	−	0.607497i	$-0.792174\pi$
$439$	−8492.29	−	14709.1i	−0.923269	−	1.59915i	−0.128947	−	0.991651i	$-0.541160\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3584.85	+	6209.14i	0.384473	+	0.665926i	0.991696	−	0.128605i	$-0.0410499\pi$
$443$	3584.85	+	6209.14i	0.384473	+	0.665926i	−0.607223	+	0.794531i	$0.707717\pi$
$444$		0			0
$445$	1064.86	−	1844.39i	0.113437	−	0.196478i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	4648.71			0.488611			0.244305	−	0.969698i	$-0.421440\pi$
$449$	4648.71			0.488611			0.244305	+	0.969698i	$0.421440\pi$
$450$		0			0
$451$	6093.16			0.636177
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1438.65	−	2491.82i	0.148231	−	0.256744i
$456$		0			0
$457$	−472.024	−	817.569i	−0.0483158	−	0.0836855i	0.840856	−	0.541259i	$-0.182052\pi$
$457$	−472.024	−	817.569i	−0.0483158	−	0.0836855i	−0.889172	+	0.457573i	$0.848719\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2549.51	−	4415.88i	−0.257576	−	0.446134i	0.708016	−	0.706196i	$-0.249590\pi$
$461$	−2549.51	−	4415.88i	−0.257576	−	0.446134i	−0.965592	+	0.260062i	$0.916257\pi$
$462$		0			0
$463$	458.859	−	794.767i	0.0460583	−	0.0797753i	−0.842077	−	0.539357i	$-0.818667\pi$
$463$	458.859	−	794.767i	0.0460583	−	0.0797753i	0.888135	+	0.459582i	$0.152001\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19073.2			1.88994			0.944969	−	0.327160i	$-0.106092\pi$
$467$	19073.2			1.88994			0.944969	+	0.327160i	$0.106092\pi$
$468$		0			0
$469$	−6919.74			−0.681287
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1700.00	−	2944.49i	0.165256	−	0.286232i
$474$		0			0
$475$	1381.01	+	2391.98i	0.133400	+	0.231056i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9163.06	+	15870.9i	0.874052	+	1.51390i	0.857769	+	0.514036i	$0.171850\pi$
$479$	9163.06	+	15870.9i	0.874052	+	1.51390i	0.0162836	+	0.999867i	$0.494817\pi$
$480$		0			0
$481$	−2952.24	+	5113.42i	−0.279855	+	0.484724i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12030.9			−1.12638
$486$		0			0
$487$	18087.7			1.68302			0.841512	−	0.540238i	$-0.181666\pi$
$487$	18087.7			1.68302			0.841512	+	0.540238i	$0.181666\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1930.14	+	3343.10i	−0.177405	+	0.307275i	−0.940991	−	0.338432i	$-0.890104\pi$
$491$	−1930.14	+	3343.10i	−0.177405	+	0.307275i	0.763586	+	0.645706i	$0.223437\pi$
$492$		0			0
$493$	2628.98	+	4553.53i	0.240169	+	0.415985i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3046.73	+	5277.09i	0.274979	+	0.476278i
$498$		0			0
$499$	7222.33	−	12509.4i	0.647928	−	1.12224i	−0.335689	−	0.941973i	$-0.608969\pi$
$499$	7222.33	−	12509.4i	0.647928	−	1.12224i	0.983617	−	0.180271i	$-0.0576974\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2074.88			0.183925			0.0919625	−	0.995762i	$-0.470686\pi$
$503$	2074.88			0.183925			0.0919625	+	0.995762i	$0.470686\pi$
$504$		0			0
$505$	15765.4			1.38921
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2489.86	−	4312.56i	0.216819	−	0.375542i	−0.737015	−	0.675877i	$-0.763765\pi$
$509$	2489.86	−	4312.56i	0.216819	−	0.375542i	0.953834	+	0.300335i	$0.0970984\pi$
$510$		0			0
$511$	−4619.54	−	8001.27i	−0.399914	−	0.692672i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3997.67	−	6924.17i	−0.342055	−	0.592457i
$516$		0			0
$517$	4093.50	−	7090.15i	0.348224	−	0.603142i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8212.49			−0.690587			−0.345293	−	0.938495i	$-0.612221\pi$
$521$	−8212.49			−0.690587			−0.345293	+	0.938495i	$0.612221\pi$
$522$		0			0
$523$	12630.6			1.05602			0.528009	−	0.849239i	$-0.322939\pi$
$523$	12630.6			1.05602			0.528009	+	0.849239i	$0.322939\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2713.23	−	4699.46i	0.224270	−	0.388447i
$528$		0			0
$529$	5618.23	+	9731.06i	0.461760	+	0.799791i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3855.09	+	6677.22i	0.313288	+	0.542631i
$534$		0			0
$535$	−6408.82	+	11100.4i	−0.517902	+	0.897032i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7941.32			0.634614
$540$		0			0
$541$	22132.0			1.75883			0.879416	−	0.476054i	$-0.157933\pi$
$541$	22132.0			1.75883			0.879416	+	0.476054i	$0.157933\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6901.69	−	11954.1i	0.542451	−	0.939553i
$546$		0			0
$547$	2271.34	+	3934.07i	0.177542	+	0.307511i	0.941038	−	0.338301i	$-0.109852\pi$
$547$	2271.34	+	3934.07i	0.177542	+	0.307511i	−0.763496	+	0.645812i	$0.776519\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6823.13	+	11818.0i	0.527541	+	0.913728i
$552$		0			0
$553$	4447.48	−	7703.26i	0.342000	−	0.592361i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22330.0			−1.69866			−0.849330	−	0.527862i	$-0.822994\pi$
$557$	−22330.0			−1.69866			−0.849330	+	0.527862i	$0.822994\pi$
$558$		0			0
$559$	4302.31			0.325525
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7093.39	−	12286.1i	0.530996	−	0.919712i	−0.468350	−	0.883543i	$-0.655151\pi$
$563$	7093.39	−	12286.1i	0.530996	−	0.919712i	0.999346	−	0.0361685i	$-0.0115153\pi$
$564$		0			0
$565$	42.1985	+	73.0899i	0.00314213	+	0.00544233i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5469.16	−	9472.86i	−0.402951	−	0.697932i	0.591130	−	0.806577i	$-0.298682\pi$
$569$	−5469.16	−	9472.86i	−0.402951	−	0.697932i	−0.994081	+	0.108645i	$0.965349\pi$
$570$		0			0
$571$	6070.81	−	10514.9i	0.444931	−	0.770642i	−0.553117	−	0.833104i	$-0.686562\pi$
$571$	6070.81	−	10514.9i	0.444931	−	0.770642i	0.998047	+	0.0624614i	$0.0198950\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1141.83			−0.0828132
$576$		0			0
$577$	−14080.0			−1.01587			−0.507936	−	0.861395i	$-0.669592\pi$
$577$	−14080.0			−1.01587			−0.507936	+	0.861395i	$0.669592\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2467.56	−	4273.93i	0.176199	−	0.305185i
$582$		0			0
$583$	−5777.62	−	10007.1i	−0.410437	−	0.710898i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12597.5	+	21819.5i	0.885781	+	1.53422i	0.844816	+	0.535057i	$0.179710\pi$
$587$	12597.5	+	21819.5i	0.885781	+	1.53422i	0.0409650	+	0.999161i	$0.486957\pi$
$588$		0			0
$589$	7041.79	−	12196.7i	0.492618	−	0.853239i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24086.1			−1.66796			−0.833978	−	0.551798i	$-0.813942\pi$
$593$	−24086.1			−1.66796			−0.833978	+	0.551798i	$0.813942\pi$
$594$		0			0
$595$	−2223.64			−0.153210
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13917.4	+	24105.7i	−0.949332	+	1.64429i	−0.202495	+	0.979283i	$0.564905\pi$
$599$	−13917.4	+	24105.7i	−0.949332	+	1.64429i	−0.746836	+	0.665008i	$0.768428\pi$
$600$		0			0
$601$	8056.88	+	13954.9i	0.546833	+	0.947143i	0.998489	+	0.0549509i	$0.0175002\pi$
$601$	8056.88	+	13954.9i	0.546833	+	0.947143i	−0.451656	+	0.892192i	$0.649166\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2272.71	−	3936.44i	−0.152725	−	0.264528i
$606$		0			0
$607$	1462.99	−	2533.97i	0.0978268	−	0.169441i	−0.812958	−	0.582322i	$-0.802144\pi$
$607$	1462.99	−	2533.97i	0.0978268	−	0.169441i	0.910785	+	0.412881i	$0.135478\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10359.7			0.685939
$612$		0			0
$613$	−11350.9			−0.747896			−0.373948	−	0.927450i	$-0.621996\pi$
$613$	−11350.9			−0.747896			−0.373948	+	0.927450i	$0.621996\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7907.60	−	13696.4i	0.515961	−	0.893671i	−0.483867	−	0.875142i	$-0.660768\pi$
$617$	7907.60	−	13696.4i	0.515961	−	0.893671i	0.999828	−	0.0185297i	$-0.00589853\pi$
$618$		0			0
$619$	6164.41	+	10677.1i	0.400272	+	0.693292i	0.993759	−	0.111552i	$-0.0355823\pi$
$619$	6164.41	+	10677.1i	0.400272	+	0.693292i	−0.593486	+	0.804844i	$0.702249\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−951.068	−	1647.30i	−0.0611617	−	0.105935i
$624$		0			0
$625$	4772.50	−	8266.21i	0.305440	−	0.529037i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4563.08			0.289256
$630$		0			0
$631$	25545.6			1.61165			0.805826	−	0.592152i	$-0.201722\pi$
$631$	25545.6			1.61165			0.805826	+	0.592152i	$0.201722\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9713.64	−	16824.5i	0.607046	−	1.05143i
$636$		0			0
$637$	5024.41	+	8702.54i	0.312519	+	0.541298i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6361.66	−	11018.7i	−0.391998	−	0.678960i	0.600715	−	0.799463i	$-0.294882\pi$
$641$	−6361.66	−	11018.7i	−0.391998	−	0.678960i	−0.992713	+	0.120503i	$0.961549\pi$
$642$		0			0
$643$	7088.18	−	12277.1i	0.434729	−	0.752972i	−0.562545	−	0.826767i	$-0.690178\pi$
$643$	7088.18	−	12277.1i	0.434729	−	0.752972i	0.997273	+	0.0737946i	$0.0235109\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−25409.1			−1.54395			−0.771975	−	0.635653i	$-0.780731\pi$
$647$	−25409.1			−1.54395			−0.771975	+	0.635653i	$0.780731\pi$
$648$		0			0
$649$	10994.0			0.664952
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6748.22	+	11688.3i	−0.404408	+	0.700455i	−0.994252	−	0.107062i	$-0.965856\pi$
$653$	−6748.22	+	11688.3i	−0.404408	+	0.700455i	0.589845	+	0.807517i	$0.299189\pi$
$654$		0			0
$655$	−8302.32	−	14380.0i	−0.495265	−	0.857824i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6047.54	+	10474.7i	0.357479	+	0.619172i	0.987539	−	0.157374i	$-0.0503030\pi$
$659$	6047.54	+	10474.7i	0.357479	+	0.619172i	−0.630060	+	0.776547i	$0.716970\pi$
$660$		0			0
$661$	5002.37	−	8664.35i	0.294356	−	0.509840i	−0.680479	−	0.732768i	$-0.738228\pi$
$661$	5002.37	−	8664.35i	0.294356	−	0.509840i	0.974835	+	0.222928i	$0.0715614\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5771.11			−0.336533
$666$		0			0
$667$	−5641.42			−0.327491
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4325.02	+	7491.15i	−0.248831	+	0.430988i
$672$		0			0
$673$	−4955.53	−	8583.23i	−0.283836	−	0.491618i	0.688490	−	0.725246i	$-0.258274\pi$
$673$	−4955.53	−	8583.23i	−0.283836	−	0.491618i	−0.972326	+	0.233627i	$0.924940\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9424.73	−	16324.1i	−0.535040	−	0.926716i	−0.999161	−	0.0409446i	$-0.986963\pi$
$677$	−9424.73	−	16324.1i	−0.535040	−	0.926716i	0.464122	−	0.885771i	$-0.346370\pi$
$678$		0			0
$679$	−5372.63	+	9305.67i	−0.303656	+	0.525948i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	17224.5			0.964973			0.482486	−	0.875903i	$-0.339734\pi$
$683$	17224.5			0.964973			0.482486	+	0.875903i	$0.339734\pi$
$684$		0			0
$685$	11329.4			0.631932
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7310.91	−	12662.9i	0.404243	−	0.700170i
$690$		0			0
$691$	3977.32	+	6888.91i	0.218964	+	0.379257i	0.954492	−	0.298238i	$-0.0963989\pi$
$691$	3977.32	+	6888.91i	0.218964	+	0.379257i	−0.735527	+	0.677495i	$0.763066\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3430.71	−	5942.16i	−0.187243	−	0.324315i
$696$		0			0
$697$	2979.29	−	5160.28i	0.161906	−	0.280430i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3771.49			−0.203206			−0.101603	−	0.994825i	$-0.532397\pi$
$701$	−3771.49			−0.203206			−0.101603	+	0.994825i	$0.532397\pi$
$702$		0			0
$703$	11842.8			0.635362
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7040.35	−	12194.2i	0.374512	−	0.648673i
$708$		0			0
$709$	−141.333	−	244.795i	−0.00748640	−	0.0129668i	0.862258	−	0.506469i	$-0.169050\pi$
$709$	−141.333	−	244.795i	−0.00748640	−	0.0129668i	−0.869744	+	0.493503i	$0.835716\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2911.10	+	5042.18i	0.152906	+	0.264840i
$714$		0			0
$715$	−5004.49	+	8668.04i	−0.261759	+	0.453379i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8779.02			−0.455358			−0.227679	−	0.973736i	$-0.573114\pi$
$719$	−8779.02			−0.455358			−0.227679	+	0.973736i	$0.573114\pi$
$720$		0			0
$721$	−7140.94			−0.368852
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3461.19	+	5994.96i	−0.177304	+	0.307100i
$726$		0			0
$727$	1242.74	+	2152.49i	0.0633985	+	0.109809i	0.895982	−	0.444089i	$-0.146473\pi$
$727$	1242.74	+	2152.49i	0.0633985	+	0.109809i	−0.832584	+	0.553899i	$0.813139\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1662.45	−	2879.45i	−0.0841150	−	0.145691i
$732$		0			0
$733$	5089.78	−	8815.75i	0.256474	−	0.444225i	−0.708821	−	0.705388i	$-0.750773\pi$
$733$	5089.78	−	8815.75i	0.256474	−	0.444225i	0.965295	+	0.261163i	$0.0841060\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24071.0			1.20307
$738$		0			0
$739$	3318.26			0.165175			0.0825874	−	0.996584i	$-0.473682\pi$
$739$	3318.26			0.165175			0.0825874	+	0.996584i	$0.473682\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1333.34	+	2309.41i	−0.0658350	+	0.114030i	−0.897064	−	0.441900i	$-0.854304\pi$
$743$	−1333.34	+	2309.41i	−0.0658350	+	0.114030i	0.831229	+	0.555930i	$0.187638\pi$
$744$		0			0
$745$	2412.26	+	4178.16i	0.118629	+	0.205471i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5723.96	+	9914.18i	0.279237	+	0.483653i
$750$		0			0
$751$	−9768.80	+	16920.1i	−0.474659	+	0.822133i	−0.999579	−	0.0290186i	$-0.990762\pi$
$751$	−9768.80	+	16920.1i	−0.474659	+	0.822133i	0.524920	+	0.851151i	$0.324095\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10797.9			0.520498
$756$		0			0
$757$	−30037.2			−1.44217			−0.721084	−	0.692848i	$-0.756356\pi$
$757$	−30037.2			−1.44217			−0.721084	+	0.692848i	$0.756356\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−20577.1	+	35640.6i	−0.980184	+	1.69773i	−0.318544	+	0.947908i	$0.603194\pi$
$761$	−20577.1	+	35640.6i	−0.980184	+	1.69773i	−0.661641	+	0.749821i	$0.730140\pi$
$762$		0			0
$763$	−6164.16	−	10676.6i	−0.292474	−	0.506579i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6955.84	+	12047.9i	0.327459	+	0.567175i
$768$		0			0
$769$	−10939.1	+	18947.0i	−0.512969	+	0.888488i	0.486918	+	0.873448i	$0.338121\pi$
$769$	−10939.1	+	18947.0i	−0.512969	+	0.888488i	−0.999887	+	0.0150403i	$0.995212\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	23321.1			1.08512			0.542561	−	0.840016i	$-0.317455\pi$
$773$	23321.1			1.08512			0.542561	+	0.840016i	$0.317455\pi$
$774$		0			0
$775$	7144.23			0.331133
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7732.29	−	13392.7i	0.355633	−	0.615974i
$780$		0			0
$781$	−10598.3	−	18356.9i	−0.485581	−	0.841050i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6407.81	−	11098.6i	−0.291343	−	0.504621i
$786$		0			0
$787$	−8155.78	+	14126.2i	−0.369406	+	0.639829i	−0.989473	−	0.144719i	$-0.953772\pi$
$787$	−8155.78	+	14126.2i	−0.369406	+	0.639829i	0.620067	+	0.784549i	$0.287105\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	75.3780			0.00338829
$792$		0			0
$793$	−10945.6			−0.490152
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11480.7	−	19885.2i	0.510249	−	0.883778i	−0.489680	−	0.871902i	$-0.662886\pi$
$797$	11480.7	−	19885.2i	0.510249	−	0.883778i	0.999929	−	0.0118755i	$-0.00378019\pi$
$798$		0			0
$799$	−4003.08	−	6933.54i	−0.177245	−	0.306997i
$800$		0			0
$801$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(4.67891 - 8.10411i) q^{5} +(-4.17891 - 7.23808i) q^{7} +O(q^{10})\)	\(q+(4.67891 - 8.10411i) q^{5} +(-4.17891 - 7.23808i) q^{7} +(14.5367 + 25.1783i) q^{11} +(-18.3945 + 31.8603i) q^{13} +28.4313 q^{17} +73.7891 q^{19} +(-15.2524 + 26.4179i) q^{23} +(18.7156 + 32.4164i) q^{25} +(92.4680 + 160.159i) q^{29} +(95.4313 - 165.292i) q^{31} -78.2109 q^{35} +160.495 q^{37} +(104.789 - 181.500i) q^{41} +(-58.4727 - 101.278i) q^{43} +(-140.799 - 243.870i) q^{47} +(136.573 - 236.552i) q^{49} -397.450 q^{53} +272.064 q^{55} +(189.073 - 327.485i) q^{59} +(148.762 + 257.663i) q^{61} +(172.133 + 298.143i) q^{65} +(413.968 - 717.014i) q^{67} -729.073 q^{71} +1105.44 q^{73} +(121.495 - 210.436i) q^{77} +(532.134 + 921.683i) q^{79} +(295.239 + 511.369i) q^{83} +(133.027 - 230.410i) q^{85} +227.588 q^{89} +307.476 q^{91} +(345.252 - 597.995i) q^{95} +(-642.827 - 1113.41i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4q - 4q^{5} + 6q^{7} + O(q^{10}) \)	\( 4q - 4q^{5} + 6q^{7} - 10q^{11} + 40q^{13} - 68q^{17} + 68q^{19} + 98q^{23} - 16q^{25} + 120q^{29} + 200q^{31} - 540q^{35} + 960q^{37} + 192q^{41} + 334q^{43} + 300q^{47} + 410q^{49} - 136q^{53} + 1588q^{55} + 620q^{59} - 200q^{61} + 1370q^{65} + 1406q^{67} - 2780q^{71} + 3604q^{73} + 804q^{77} + 334q^{79} - 500q^{83} + 1100q^{85} - 180q^{89} + 2820q^{91} + 1222q^{95} + 200q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about